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PSI Graphic Transport Framework User Manual

1. PHYSICAL TYPE 2 3 4 57 6 7 8 9 ELEMENT CODE ENTRY ENTRY ENTRY ENTRY ENTRY ENTRY ENTR ENTRY Y BEAN 1 VVVVV x cm 9 mr Y cm mr L cm Po ve percent r m s 1 VVVVV AO mr Ay cm mr AL Ad AP e ADDITION TO vee percent GeV BEAN ENVELOPE c POLE FACE 2 V ANGLE OF ROTATION ROTA TION degrees DRIFT LENGTH 3 V metres BENDING LENGTH FIELD FIELD MAGNET 4 VVWV metres kG GRADIENT n value QUADRIPOLE 5 VV0O LENGTH FIELD HALF metres kG APERTURE cm TRANSFORM 1 6 0 0 0 1 0 UPDATE TRANFORM 2 6 0 0 0 2 0 UPDATE BEAM CENTROID 7 0 SHIFT SHIFT SHIFT y SHIFT SHIFT SHIFT 6 SHIFT x 0 mr cm mr L CM percent ALIGNMENT 8 VVVVV DISPLACEMENT ROTATION DISPLACEMENT DISPLACEMENT ROTATION CODE TOLERANCE ve x mr cm 2 mr NUMB ER REPEAT 9 0 NUMBER OF CONTROL REPEATS FITTING 10 0 I J DESIRED ACCURACY CONSTRAINTS VALUE OF I OF FIT J MATRIX ELEMENTS Note I is used for fitting a beam c matrix element I is used for fitting an RI matrix element I 20 is used for fitting an R2 matrix element ACCELERATOR 11 0 LENGTH E phase WAVELEN metres energy lag GTH gain degrees cm GeV BEAM 12 0 THE FIFTEEN CORRELATIONS AMONG THE SIX ELEMENTS This entry must be proceded by a type Rotated code 1 0 entry Ellipse INPUT OUTPUT 13 0 CONTROL CODE OPTIONS NUMBER A
2. Note The v s following the type codes indicate the parameters which may be varied See section under type code 10 0 for a detailed explanation of vary da controllare perch ho faticato molto nella lettura e interpretazione segni OUTPUT FORMAT General appearance Here we give a brief description of the general appearance of the output and its meaning The user may refer to the sample output shown on pages 22 through 27 It is the printed output resulting from the sample data shown in the section on input format In a simple example it is not possible to show each of the different type codes Several of the type codes produce output which is not characteristic of all other type codes We therefore refer the user to the sections on the various type codes for an explanation of any features peculiar to a given type code The output for each step of a given problem is printed separately The printing for one step is completed before that for the next step is begun Therefore we will describe the output for a single problem step The output shown below is from a problem with two steps Initial listing For each problem step the program begins by printing out the user s input records Listing during the calculation pag 20 The program now begins the calculation If there is no fitting one listing of the beam line will be made If there is fitting there will normally be two listings The first
3. 22 From which it follows that BG y ee By x y t S 00 1 Kaman 2 For terms containing only K the basic differential equations assume the form 139 B 24 Substituting the Taylor s expansion of equation 15 and solving for the nth order terms using a conventional Green s function solution see Ref 1 yields equation 19 above IV Interpretation and Use of Equation 19 For most practical cases of interest K will be a constant over the interval of integration In this event we may define the coupling coefficient of an nth order multipole to an nth order aberration as the partial derivative of equation 19 with respect to the K in question as follows 6 xilxEydxr 6 X Cy 5570 25 Kn KIA where now the interval of integration is over the multipole length L represented by Ka For a distributed multipole component such as in a non uniform field bending magnet equation 25 is used In many cases where a curved entrance or exit pole contour is used or a short multipole magnet is used such that the characteristic first order functions Cy 5 Sy dy are essentially constants over the interval of integration the length of one multipole then the coupling coefficient is best defined as the partial derivative of equation 19 with respect to S as follows S xilxk
4. plane Figure 7 For a quadrupole pair the principal planes are displaced toward and usually beyond the focusing element of the pair as shown in Figures 6 and 7 For any lens system no matter how many elements are involved object p Py P2 q Figure 8 image 1 p 1 q 1 f if p and q are distances measured to the principal planes Then the magnification between object and image planes is M q p Since the quadrupole pair is different in the two planes x and y both situations must be examined The interesting result turns out to be that in the x plane the principal planes are to the right Figure 6 and 88 in the y plane they are to the left Figure 7 Therefore in the y plane the magnification is greater than 1 and in the x plane the magnification is less than 1 Typically for a quadrupole pair the ratio of may be as high as 20 1 and such cases be disastrous if not recognized beforehand This is a first order image distortion For example if the source is a circular spot at A the image at B will appear as a long thin line The situation is different for the Quadrugole triplet 2 1 plane Figure 9 A 1 plane Figure 10 In the symmetric triplet as shown in Figures 9 and 10 the principal planes are located symmetrically about the center of the system although Zx gt Zy This is perhaps the dominant reason why quadrupole triplets are used The magnif
5. 70 0 where x r Rik Tike Xy Xp is the image of the original centroid We may now again use equations 9 and 24 to relate this matrix of second moments to the final beam half widths and correlations Reference 1 Karl L Brown Sam K Howry SLAC Report No 91 1970 2 F R Gantmachcr The Theory of Matrices Chelsea Publishing Co New York 1959 132 A Systematic Procedure for Designing High Resolving Power Beam Transport Systems or Charged Particle Spectrometers The following is a report submitted to the Third International Magnet Symposium held in Hamburg Germany May 197 It is a general description of a suggested procedure for designing systems to any order and includes the derivation of the coupling coefficient of an nth order multipole to any nth order aberration coefficient The report also derives the multipole strengths for the three techniques for introducing multipole components into a system namely 1 pure multipole fields 2 non uniform fields and 3 contoured entrance or exit boundaries of magnets The notation used in this report is identical to TRANSPORT notation except for the following Replace x and y in the report by o and respectively to convert to TRANSPORT notation REPORT SUBMITTED TO THE THIRD INTERNATIONAL mam sYMPosIUM HELD IN HAMBURG GERMANY May 1970 by Karl L Brown Stanford Linear Accelerator Center Stanford California Summary By extrapolating t
6. R KL R Solenoid KS c 1 1 COORDINATE ROTATION Type code 20 0 The transverse coordinates x and y may be rotated through an angle a about the z axis the axis tangent to the central trajectory at the point in question Thus a rotated bending magnet quadrupole or sextupole may be inserted into a beam transport system by preceding and following the element with the appropriate coordinate rotation See examples below The positive sense of rotation is clockwise about the positive z axis There are two parameters to be specified for a coordinate rotation 1 Type 20 0 signifying a beam coordinate rotation 2 The angle of rotation a degrees The angle of rotation may be varied in a first order fitting see type code 10 0 Note This transformation assumes that the units of x and y and and 9 are the same This is always true unless a 15 0 3 0 or a 15 0 4 0 type code has been used See 75 page 45 and Fig 4 page 12 for definitions of x y and z coordinates ExampLes For a bending magnet the beam rotation matrix may be used to specify a rotated magnet Examples No 1 A bend up is represented by rotating the x y coordinates by 90 0 degrees as follows not to exceed 4 spaces if desired 20 90 2 p 1 5 4 B 2 Pe ig 20 90 lt lt return coordinates to their initial orientation
7. 4 used in the derivation of the equation of motion First Order Dispersion pag 163 The spatial dispersion d t Rig of a system at position t is derived 1 using the Green s function integral and the driving term f t o t the dispersion see Table I of SLAC 75 The result is 44 Rus Sx t fy Cx T h COdt e t fos COh Odc 5 where t is the variable of integration Note that h t dt da is the differential angle of bend of the central trajectory at any point in the system Thus first order dispersion is generated only in regions where the central trajectory is deflected i e in dipole elements The angular dispersion is obtained by direct differentiation of d t with respect to t the curvilinear distance along the central trajectory di t Ras s t h t dt c x t 6 where Roi S E gt gt First Order Path Length The first order path length difference is obtained by expanding and integrating Eq 4 and retaining only the first order term i e T t f higher order terms 6 from which Xe f ex COR CO is h t dt 2 6 fa h t dt amp lt 1 Rs5209 Rsg 7 107 Inspection of Eqs 5 6 and 7 yields the following useful theorems Achromaticity A system is defined as being achromatic if Rig R 0 i e if d t d t 0 Therefore
8. 0 Se X De 3 1 51 1 Di 4 The symbols S and S represent six by six matrices whose form will be derived below The two six vectors D and D are translations in the six dimensional space of particle coordinates The three vectors and D formed from the displacement coordinates x y z of Do and D give the displacement due to the misalignment of the origins of the reference coordinate systems These two three vectors are shown in figure 1 III Magnet Misalignment Coordinates The alignment of a rigid magnet has six degrees of freedom three translational and three rotational These are conveniently represented by the six quantities 145 where 6x 6z are the displacements in the x and z directions and 0 and are the rotations about the and z axes respectively The origin of the xyz coordinate system called the pivot is the point about which the misalignments are measured If the pivot point is located at some point on the reference trajectory the x y and z axes of the alignment coordinate system are taken to coincide with the x y and z axes of the beam line reference coordinate system The misalignments form a mathematical group which is the Euclidean group in three dimensions This group is non commutative and the order in which the misalignments are imposed is important if terms of higher order than linear are included In practice however m
9. 5 5 uorjonijsut 2 114 00000 8 00096 000962 100000 0 100000 0 00000 0 100076 1000 S 2 100000 0 100000 0 00050 000 O8vOt t 06 60 091 997 092166 86699781 8669981 86699781 0166476 09000 v 6181 6 2618176 00296 0000 09006 2 olde n 100000 ED 00000 t096 2 682 00000701 vOw 10000072 0000001 t00005 G 100000 v 100000 100000 081 10006678 00000 02 n Se 1000 6 00000 081 0000671 100000 081 10006678 00000 02 bau 0006678 100000 081 10000671 100000 081 1000 S 8 00000 02 Coe 000 S 8 100000 081 0000671 100000 929 00000 100000 9 00000701 COn 100009 8 00000 Von 100000 00001 L 00000 01 10 166689 LC 100000 081 00062701 w Cw 1000060 081 100064 100000 081 00062701 dr 100000 081 1866 71 08 000 SO 000 20 00000 61 C0000 8 0000071 WV38 011070 38 HOIH T3NIIN3S El l o mmommun mium 8 8 8 gis 9 N ce N N 8 eee eee MIN ONT NGM ONE N S ONS 8 8 N 000000 cee 8SSa0S00c500 5 8 O win 3Q0W Q3NISWOO3Uw 48 000 0 00
10. Some typical o matrix constraints are as follows Desired optical condition Typical fitting constraint Horizontal waist o 21 10 2 1 001 FP Vertical waist o 43 0 10 4 3 001 F2 Fit beam size to x max 1 cm 10 1 1 1 001 F3 Fit beam size to y max 2 cm 10 3 3 2 001 FP Limit max beam size to x 2 cm 10 2 1 1 2 01 FS Limit min beam size to y 1 cm 10 1 3 3 1 01 F In general it will be found that achieving a satisfactory beam fit with TRANSPORT is more difficult than achieving an R matrix fit When difficulties are encountered it is suggested that the user help the program by employing sequential step by step fitting procedures when setting up the data for his problem More often than not a failure to fit is caused by the user requesting the program to find a physically unrealizable solution An often encountered example is a violation of Liouville s theorem Beam correlation matrix r fitting constraints Five parameters are needed for a constraint on the i j element of the beam correlation matrix 1 Type code 10 n 2 Code digit 10 1 3 Code digit 1 4 Desired value of the i j matrix element 5 Desired accuracy of fit standard deviation TRANSPORT does not print the beam c matrix directly Instead it prints the beam half widths and represents the off diagonal elements by the correlation matrix
11. plane the distance to a waist is or the distance to the waist is 118 Relationship between a Waist and a Parallel to Point Image parallel to point image in the x 0 plane occurs when R O The R matrix corresponding to this is 0 f TECTED T Sx 5 suddivisione centrale nelle 3 matrici Since 1 for this situation If we assume an erect ellipse o as the beginning of the system the final beam matrix 0 1 is given by substitution of Eq 48 into Eq 37 as follows 2 o 1 12022 0 2022 0 2022 0 51011 0 2 gt 0 gt 2 0 suddivisione centrale 49 for parallel to point imaging Several conclusions may be extracted from this result the first observation is that a waist and a parallel to point image will coincide if Ri R 0 This is equivalent to requiring that the object and image distances measured to the principal planes are both equal to the focal length f of the system The distance to a waist in this example is 7210 _ 12 22022 0 2 5 5 22 0 50 022 1 821011 0 2 gt 022 0 2011 0 5 2022 0 If sys 0 a waist and parallel to point image coincide If sys gt 0 the waist precedes the image and if If s s lt 0 the waist follows the image unless 06 0 0 zero phase space area in which case a waist and an image always coincide
12. 30 0 HLON3ST JNVIdGIN DILSNOVW eat A uojdpiw s i 1VHlIN39 1 7 AYOLOSIVYL AYVYLIGYV L 30 2 0 7 j HI9N31 ONV 447 y X 02 p20 H VA OL LOWNdSXH NI SAILISOd SINZWSIS PIOdLINAW ALVOIGNE SXLLIHVIOd OLLANOW SINSAGIS WIOdQLX4S SIXV X OIIENOVA THL 40 NOILVHIS TII 2 014 019822 310401 35 1 210410 103 104 CENTRAL TRAJECTORY 748A8 dt 0_ FIG 3 COSINE LIKE FUNCTION c t IN MAGNETIC MIDPLANE Rae CENTRAL OBJECT 5 TRAJECTORY imate o Sy 748A7 dS 5 0 2 2 dt 1 0 FIG 4 SINE LIKE FUNCTION s t IN MAGNETIC MIDPLANE Roos 12 105 pt Ap CENTRAL TRAJECTORY R 4p d O 0 748A9 d dx O pop e FIG 5 DISPERSION FUNCTION 4 t IN MAGNETIO MIDPLANE a t Rog 0 _ dCy pie dt i 748A6 FIG 6 COSINE LIKE FUNCTION ey R33 IN THE NON BEND y PLANE Ryg 106 Y plane Sy t OBJECT IMAGE Sy O O _ dSy m Sy O dt 2 5 FIG 7 SINE LIKE FUNCTION s t IN NON BEND PLANE s t dx dy 1 hx dt
13. O m x P 14 The orthogonal transformation indicated by the matrix 0 gives the misalignment parameters in the magnet face coordinate system Finally the transformation of rotational misalignment parameters is again given by equation 11 so that my 0 m 15 V Evaluation of the Relevant Matrices We choose the pivot to be the origin of the aligned magnet entrance face coordinate system Therefore we have 16 For the exit face the matrix 0 transforms from the aligned entrance face coordinate system to the aligned exit face coordinate system The vector P gives the position of the origin of the aligned exit face coordinate system in the aligned entrance face coordinate system In figure 1 it is the vector which reaches from A to B For the exit face of a bending magnet we therefore have cosa 0 sina 0 0 1 0 17 sina 0 cosa 1 cosa P 0 18 psina where p is the radius of curvature of the central trajectory and a is the total bend angle We then derive for the matrices A and cosa 0 0 sina sina 0 0 0 0 1 0 0 0 sina 1 0 0 p 1 cosa A 0 cosa 0 0 0 sina 19 sina 0 0 1 cosa 0 0 0 0 0 0 0 1 121 B 1 341 sina 1 124 1 44 p 1 cosa 1 125 B 1 3as cosa 148 1 1 2 1 242 1 12 1 422 sina 1 136 B 1 246 B 1 316 B 1 426 COSQ 20 All other elements of 1 are zero To ca
14. e y xeye Xeye 9eye 15 66 8 Po ylye 6 yo 5 yl oo The absence of certain terms which might otherwise be expected in Eqs 14 and 15 is due to the fact that horizontal mid plane symmetry has been assumed in the derivation That is the field on the horizontal mid plane is normal to the plane Thus there can be no or 0 term Similarly there can only be even powers of y and such as and in the 2 and in the x equation Also note that there is no y 6 or y 6 term if there is mid plane symmetry TRANSPORT uses a numerical notation to signify the six basic coordinates x 0 oq 12 34 5 6 16 The term has not been introduced here before Its significance is the preservation of the bunch length of a beam such as the SLAC electron beam The first order output from TRANSPORT is a 6 x 6 matrix printout of the R matrix where the labels are implied by row and column position of the elements For example the element appearing at the intersection of row 3 and column 4 Is the coefficient etc O c6 lt 1 1 1 1 1 1 The second order terms are labeled by the convention indicated in Eqs 14 15 and 16 For example becomes 2 and becomes 1 11 9 becomes 1 26 5 Second Order Matrix Normally the matrix method is expected only to apply to the solution of linear i e first
15. over the entire median plane hence the magnetic field B over the whole space The components of the field are expressed in terms of explicitly by B V or 2 1 x oni dum 36 2 1 1 n 2m41 x ym 296 yo a By by 2 2 1 n Qm _ 1 6 1 o5 o5 x ym t x 0 2 0 1 n 2 1 3 The expression for the magnetic field on the midplane is x By xot 4 57 6 At this point deviate from the notation and formalism of Ref 1 and introduce Kn t the multipole strength per unit length and s the total multipole strength of a static magnetic field We rewrite equation 6 as By x 0 t Bp 7 Where Bp xt P is the magnetic rigidity of a particle of momentum P and charge e along the central trajectory from which Kult 55 5269 GF payne i We define S as Sn f Kaldt 9 135 S so defined is the strength of the nth order multipole component of a field over the interval of integration Multipole Strengths for Pure Multipole Fields Consider the scalar potential of an nth order 2 n 1 pole pure multipole element Bor tt 1 sin n 1 10 where x rcos andy 5110 is the field at the pole and is the radial distance to the pole from the central trajectory Expanding as a
16. 12022 0 0 0 0 960 5 B zl 65 For symmetric magnetic systems Rj R Using this property the fact that 1 it follows that 8158 R2 1 So for symmetric magnetic systems Eq 65 reduces to 1 2 ee 0 R21 66 0 1 The above equations may be used to calculate the optimum design parameters for periodic beam transport systems Example No 1 Consider a unit cell of a periodic focusing array consisting of focusing elements only as indicated below Beam Envelope f 1510428 122 The R matrix for the unit cell i e from the principal planes of the first lens to the principal planes of the second lens is R 2 67 ug If now we require that the beam envelope possess symmetry coincident with the lens symmetry i e that erect ellipses occur at me principal planes of each lens and a waist midway in between and furthermore that the beam size at the second lens be kept to a minimum and equal to the beam size at the first lens then substituting Eq 67 into Eq 66 and setting o4 1 to be a minimum yields 011 1 yo1 0 1 611 0 11 if 68 24 022 0 gt 550 where 04 0 0 are measured at the principal planes of the first lens and finally _ 91100 2 x min 011 W 69 Note that the ratio of the maximum to the minimum beam size Eq 69 is independent of th
17. K Halbach Program for Inversion of System Analysis and its Application to the Design of Magnets Second International Conference on Magnet Technology Oxford 1967 The best estimates or optimized values of the varied parameters A are precisely those that minimize the quantity 2 y pem 3 where M number of constraints fy a function selected by the code digits i j on the constraint definition Type Code 10 For example I 1 j 1 means that the transform matrix element R4 is to be constrained EK the desired value of the selected function Sx the desired accuracy of fit i e the standard deviation In our notation this minimum is expressed by 2 ss Xmin 77222 4 is printed at the successful conclusion of run involving parameter fitting 153 Whether or not the optimization Ca 29 is acceptable depends on each application and must be evaluated by the user Values of M N are sometimes but not necessarily regarded as good In particular if M N then an exact solution Ax 0 should be found by TRANSPORT If the resulting fit is acceptable then the following interpretation may be put on the covariance matrix C Let the parameters be changed to values near the optimum A such that they stay within the ellipsoid defined by Then the resulting deviation of the specified constrained quant
18. O 0 3519 9 221 0 3G6cC cl 1 WHOdSNYHI QNC 0000071 00000 0 00000 0 00000 0 00000 0 0000070 Lv98S 11 0000071 0000070 000000 92 1 889 1 0000070 00000 0 00000 1 09 8 1 0000070 00000 0 00000 0 00000 0 00000 00000 1 0000070 00000 0 188 2721 00000 0 00000 0 00000 0 00000 l 409 871 942 676 00000 00000 0 00000 0 00000 00000 1 WHOdSNYHI TITLE CARD The title card is the first card in every problem step of a TRANSPORT data set The title card is always required and must be followed by a O a 1 or 2 card see next section to indicate whether the data to follow is new card or a continuation of a previous data set a 1 card or a 2 card The title must be enclosed within either quotation marks slashes or equal signs on a single card The string may begin and end in any column free field format for example SLAC 20 GEV C or SLAC 20 GEV SPECTROMETER Note that whichever character is used to enclose the title must not be used again within the title itself Example of a DATA SET for a single problem step SENTINEL need not be on separate card Elements Title card INDICATOR CARD pag 29 The second card of the input for each step of a problem is the indicator card If the data which follow describe a new problem a zero 0 is punched in any column
19. 0 53 0 54 0 55 0 61 0 62 0 63 0 64 0 65 o 66 lt Dx The matrix is symmetric so that only a triangle of elements is needed In the printed output this matrix has a somewhat different format for ease of interpretation y CM 21 CM 31 r 32 MR r 41 r 42 r 43 CM 51 r 52 53 54 w 35 N N 6 lt x gt d a 66 PC 61 62 63 64 65 SYL o ij ij Where r ij Te dp edi 34 As a result of the fact that the matrix is position definite the r ij s satisfy the relation Ir ij ls 1 35 The physical meaning of the 5 5 as follows Jo 11 Xmax The maximum half width of the beam envelope in the x plane at the point of the printout Jo 22 Onax The maximum half angular divergence of the beam envelope in the x plane 49 33 Ymax The maximum half height of the beam envelope Jo 44 Pmax The maximum half angular divergence of the beam envelope in the y plane 40 55 amp nax 1 2 the longitudinal extent of the bunch of particles 6 66 6 The half width 1 2 AP P of the momentum interval being transmitted by the system The units appearing next to the o 11 s in the TRANSPORT printout sheet are the units chosen for coordinates x y 2 and 6 AP P respectively To the immediate left of the listing
20. 15 SFNTI EL 16 2 r e COVARIANCE FIT 0 8 0 079 Covariance Matrix 0 883 0 038 USE TRIPLET TO FO M POINT TO POINT IMAGE 1 000000 1 05 nFV 1 3 0 12 0000 QUAD 5 00 1 50000 FT 7 4096 o DRIFT 3 0 0 5000 FT 5 00 3 00000 FT 6 1577 KG DRIFT 3 0 0 5000 FT M a 5 00 1 50000 7 4096 K DRIFT 3 0 7 0000 FT 1 10 0 1 2 0 0 0 005 0 000 LAREL FITL 62 FIT 10 0 3 4 0 0 0 005 0 000 LABEL FIT2 LFHRT I 26 0000 FT IMAGE AT TAPRFET Data Deck for TRANSPORT Run AT TARGET 4 000 IN 3 208 FT 4 000 IM 2 673 FT 4 000 IN 3 208 Constraint on constrain on 155 Interpretation So long as B and B fall inside the shaded area this is the tolerance requirement then the ellipsoid representing the corresponding deviations of the matrix elements and is using Eq 5 107 56 or R2 R2 2 6 0 005 Note that it is not enough to prescribe tolerances AB lt 079 AB lt 038 since there is unshaved area inside the rectangle defined by these values The strongly tilted covariance ellipse i e 1 suggests that the triplet power supplies should be designed so that any drift in magnetic field B causes a compensating drift in the magnetic field B so as to stay inside the shaded area shown in Figure 1
21. Desired value of the I j matrix element Desired accuracy of fit standard deviation UBBWN RP Some typical R2 matrix constraints are as follows The symbol n is normally zero or blank If n 1 then entry 4 is taken to be a lower limit on the matrix element If n 2 entry 4 is taken to be an upper limit Desired optical condition Typical fitting constraint Point to point imaging Horizontal plane 12 0 10 21 2 0 001 FP Vertical plane R 34 0 10 23 4 0 001 F Parallel to point focus Horizontal plane 11 0 10 21 1 0 001 F2 Vertical plane R 33 0 10 23 3 0 001 F2 Achromatic beam Horizontal plane 10 21 6 0 001 F3 R 16 R 26 e 10 22 6 0 001 See type code 6 0 for a complete list of elements which update the R2 matrix o BEAM matrix fitting constraints There are five parameters to be specified when imposing a constraint upon the i j element of a o BEAM matrix 1 Type code 10 n 2 Code digit i i 2 j 3 Code digit j 4 Desired value of the i j matrix element 55 5 Desired accuracy of fit standard deviation The symbol n is normally zero or blank If n 1 then entry 4 is taken to be a lower limit on the matrix element If n 2 entry 4 is taken to be an upper limit If i j then the value inserted in entry 4 is the desired bean size o ii e g x max o 11 etc
22. If one wishes to fit an element of this matrix to a non zero value it is convenient to be able to constrain the matrix element directly Some typical matrix constraints are as follows Desired optical condition Typical fitting constraint Horizontal waist r 21 10 12 1 0 001 FP yy correlation r 34 0 2 10 13 4 0 2 001 F2 First moment constraint In first order known misalignments and centroid shifts cause the centre centroid of the phase ellipsoid to be shifted from the reference trajectory i e they cause the beam to have a non zero first moment The first moments appear in a vertical array to the left of the vertical array giving the Jo ii The units of the corresponding quantities the same It is perhaps helpful to emphasize that the origin always lies on the reference trajectory First moments refer to this origin However the ellipsoid is defined with respect to its centre so the covariance matrix as printed defines the second moment about the mean First moments may be fitted The code digits i 0 andj where j is the index of the quantity being fit Thus 10 1 1 01 constrains the horizontal 1 displacement of the ellipsoid to be 0 1 0 01 cm This constraint is useful in deriving the alignment tolerances of a system or in warning the system designer to offset the element in order to accommodate a centroid shift System length constraint
23. derived in Ref 1 equation 5 is x efle h 1 hx 1 9 b m x x y y 14 hx hx n x 1 hx hx 1291 T T T xB tr B 1 h9B h2B x B E x B y B 18 If this equation is solved to nth order for the Taylor s coefficients of equation 15 it will be observed that the result has the remarkably simple form t n Ecol amp Onde Oe 0 terms containing 19 where the variable of integration is Kk p v X The x have the following meaning x t X t X y t X y t Cy Cy Sx Sy are defined by equation 17 and in general are function of the variable of integration t over the interval of integration Kn is defined by equation 8 and in general is also a function of t G s are Green s fimctions where 138 Gilt T x Olx 2 5 Galt T 5 05 0 G3 t t YOYE 5 0 Gilt t 5 29 Note that the Gi s are Just first order Taylor s coefficients measured from the location of the multipole component to the end of the system t Thus we see that the coupling coefficient to an nth order multipole i
24. sigma matrix and the R2 matrix before each bending magnet with fringe fields and after each misaligned magnet of any type The tolerances may be varied Thus type vary code 8 111111 permits any of the six parameters 2 through 7 above to be adjusted to satisfy whatever BEAM constraints may follow For fitting a misalignment must pertain to a single magnet or single section of the beam line and the results must be displayed in the beam sigma matrix See the section under type code 10 for a discussion of the use of vary codes The meaning of the options for each digit of the three digit code number is given in the following table A The units position specifies the magnet s or section of the beam line to be misaligned CODE INTERPRETATION pag 57 NUMBER XXO The single magnet type code element immediately preceding the align card it to be misaligned A bending magnet with fringe fields should be misaligned using one of the options described below XX1 The last R1 matrix update the start of the beam line or a 6 1 type code entry marks the beginning of the section to be misaligned The misalignment element itself marks the end The section is treated as a unit and misaligned as a whole The misalignments of quadrupole triplets and other combinations involving more than two quadru poles may be studied using this code digit XX2 The last R2 matrix update see type code 6 for a list of elements which update R2 marks
25. 00000 1 wlLlde 401 00000 121 w 150 100000 0 0000S 00000701 006186 00000 O 91721 law 100000 NANN NO t widdSe 00000 1 0000071 0000 00000 1 0000 0000071 000065 5 5 ONOO23S t 1114 01 t 14 t 12356 Lt 1 H30HO 940335 21 166 000 0 00070 66 666 24 00071 00070 000 00070 216 886 WD 069711 lt 20 919 000 0 689671 00070 00070 00070 Wo 009 00070 966 26721 LEO WO SE El 810 W 426751 W SivcL c l n 13180 1667 00070 000 0 766 666 0007 000 0 000 0 00070 216 266 WD 06911 200 980 00070 00070 86671 000 0 000 O 000 0 WO OLE 00070 66 HW 26721 LEO W2 0 01 890 609721 00000 0 C IVIOMg 166 000 O 00070 766 666 94 00071 00070 00070 00070 216 266 W2 069711 200 980 00070 865671 00070 00070 000 0 WO OLE 000 0 66 uw 267021 160 WD 0 01 890 O9 cI 930 069 691 9EE E 00004 00000701 0061876 v N38 000 0 00070 00070 000 0 94 00071 00070 00070 0000 009070 000 WO 0064 000 8Lv 000 O uw 00071 000 0 00070 0000 WD 699 00070 814 dw 00071 000 0 WD 696 000 veL c 930 00000 0 2 41VLOU 00070 00070 00070 00070 00070 23 00071 000 0 00070 00070 000 O 00070 006 000
26. 3 2 B 2 D 20 90 3 returns coordinates to their initial orientation A bend down is accomplished via 20 90 s 2 4 2 20 9 A bend to the left looking in the direction of beam travel is accomplished by rotating the x y coordinates by 180 degrees e g 20 180 2 4 2 20 180 TRAJECTORY _ lt 2 O didascalia del disegno FIELD BOUNDARIES FOR BENDING MAGNETS The TRANSPORT sign conventions for X B R and h are all positive as shown in the figure The positive y direction is out of the paper Positive B s imply transverse focusing Positive R s convex curvatures represent negative sextupole components of strength s h ZR sec3 5 See SLAC 75 page 71 35 x x Tx urs 79 m x e 1933 soo 3 UTS 10329 e413 Terquao jo yABZueT uied 1 pueq e 8ue ey 9 Ay u 5 2 A D I 0 1 5 0 0 ja x 0 UTS A TX UIS J 0 02 0 x x 0 505 UIS J 0 urs T 505 7 4 qeubou 18 414 The field expansion for the midplane of a bending magnet is taken from Eq 18 page 31 of SLAC 75 thereby defining the dimensionless quantities and as follows x 0 t 0 0 4 1 nhx Th
27. A running total of the lengths of the various elements encountered is kept by the program and may be fit The code digits i 0 1 0 Thus the element 10 0 150 5 would make the length of the system prior to this element equal to 150 5 metres Presumably there would be a variable drift length somewhere in the system By redefining the cumulative length via the 16 6 L element partial system lengths may be accumulated and fit AGS machine constraint Provision has been made in the program for fitting the betatron phase shift angle p associated with the usual AGS treatment of magnet systems In the horizontal plane use code digits i 11 j 2 and specify A zi cos 0 5 Ri R3 vert freq No of periods In the vertical plane i 13 j 4 and 1 cos 1 0 5 R33 R44 vert For example if there are 16 identical sectors to a proposed AGS machine and the betatron frequencies per revolution are to be 3 04 and 2 14 for the horizontal and vertical planes respectively then the last element of the sector should be followed by the constraints 10 11 2 190 001 10 13 4 134 001 3 04 2 14 i e 0 190 and 0 134 16 16 For example A typical data listing might be 5 01 3 2 5 01 2 10 11 2 0 190 001 10 13 4 0 134 001 See Courant and Snyder Also note that this constraint is valid only when th
28. Batavia IL 60510 USA 3 IBM or CDC Vers ions Program Library Division DD CERN CH 1211 Gen ve 23 Switzerland The present authors assume responsibility for the contents of this manual but in no way imply that they are solely responsible for the entire evolution of the program In order to make this report available without delay the Appendix has been reproduced directly as published by FNAL MATHEMATICAL FORMULATION OF TRANSPORT General conventions A beam line is comprised of a set of magnetic elements placed sequentially at intervals along an assumed reference trajectory The reference trajectory is here taken to be a path of a charged particle passing through idealized magnets no fringing fields and having the central design momentum of the beam line In TRANSPORT a beam line is described as a sequence of elements Such elements may consist not only of magnets and the intervals between them but also of specifications of the input beam calculations to be done or special configurations of the magnets A certain relation described below of the magnets and their fields to the assumed reference trajectory is considered normal Alternative configurations can be described by means of elements provided for such purposes The two coordinates transverse to the initial reference trajectory are labeled as horizontal and vertical A bending magnet will normally bend in the horizontal plane To allow for other possibilities a c
29. LOYU 2 14 OSL 76 13 0006471 13 000716 930 00000 081 oZ Z 13 000716 00000 0 2618176 13 00052701 n 19 v N38 14 061 08 940 000007081 1 7 13 06 08 43 86671 08 E LAJIS8 000 0 000 0 000 0 000 0 000 0 000 0 l4 000 0 39 00000 Wv38 14 000 O uw 0000 1 WO 00020 MW 0000 1 WD 00020 8 NOITYSIWA 10 3000S2 61 91 OYI3HI wv38 NOlLIn10S34 HOIH 3Q0W Q3NISWOO3H 49 000 0 00070 000 0 000 0 00070 000 0007 610 00071 700 000 000 000 000 910 66010 aw 66071 HW so 00070 WD 602 uw LIO 99S 2 uw 100 WO 21876 13 cee 2 11 cee eLtt 14 eee 72111 00000 SSscco 00000 00000 ctLvGS 0861 000 0 000 0 000 0 000 0 000 0 000 0 00000 0 00000 0 00000 0 00000 1 60000 00000 00000 1SSEO t 89 7 13 14 13 14 13 id 13 0000070 00000 0 00000 0 0000070 00000 0 0000071 00000 00000 11 99 10 000000 00000 PfI t 00000 00000 000000 98826 2 1622 00000 00000 0000070 00000 00000 10000 1 0000070 00000 090000 2908179 6 871 1 930 00000 081 oc 0006678 e 93 86699 81 14 00000 02 Gaw 7 uw 0006578 2 930 000007081 02 14 0000671 930 000007081 Oc 0006578 20 86699 81 14 00000 02 tar
30. a shift in the centroid of the beam and a change in the beam focusing characteristics Two varieties of misalignment are commonly encountered 1 the magnet is displaced and or rotated by a known amount or 2 the actual position of the magnet is uncertain within a given tolerance TRANSPORT has the capability of simulating the misalignment of either single magnets or entire sections of a beam line Any combination of the above alternatives may be simulated through the use of the align element The results may be displayed in either the printed output of the beam sigma matrix or tabulated in a special misalignment table described below There are eight parameters to be specified Type code 8 0 specifying a misalignment The magnet displacement in the horizontal direction cm A rotation about the horizontal axis mt A displacement in the vertical direction cm A rotation about the vertical axis mr 42 6 A displacement in the beam direction cm 7 A rotation about the beam direction mr 8 A three digit code number defined below specifying the type of misalignment The three displacements and three rotations comprise the six degrees of freedom of a rigid body and are used as the six misalignment coordinates The coordinate system employed is that to which the beam is referred at the point it enters the magnet For example a rotation of a bending magnet about the beam direction parameter 7 abov
31. at the end of a problem step Once changed they remain the same for all succeeding problem steps in an input deck until a 0 indicator card is encountered at which time they set to standard TRANSPORT units The units may be reset to standard units by inserting a 15 type code entry The 15 0 elements are the first cards in a deck immediately following the title card and the or 1 indicator card and should not be inserted in any other location They produce no printed output during the calculation their effect being visible only in the output from other elements Exemple To change length to feet width to inches and momentum to MeV c add to the front of the deck the elements 15 8 0 3048 15 1 IN 2 54 15 11 MEV 0 001 The scale factor 0 3048 multiplies a length expressed in the new unit feet to convert it to the reference unit metres etc For the conventional units listed below it is sufficient to stop with the unit name the conversion factor is automatically inserted by the program If units other than those listed below are desired then the unit name and the appropriate conversion factor must be included If the automatic feature is used with older versions of the program there must be no blank spaces between the quotes and the unit name Input output units Type code 15 0 Conversion fact rs for dimension changes versus code digit and label 69 quadied 001 sueeu N que2
32. by element calculating the properties of the beam or other quantities described below where requested Therefore one of the first elements is a specification of the phase space region occupied by the beam entering the system Magnets and intervening spaces and other elements then follow in the sequence in which they occur in the beam line Specifications of calculations to be done or of configurations other than normal are placed in the same sequence at the point where their effect is to be made The transfer matrix R The following of a charged particle through a system of magnetic lenses may be reduced to a process of matrix multiplication At any specified position in the system an arbitrary charged particle is represented by a vector single column matrix X whose components are the positions angles and momentum of the particle with respect to the reference trajectory i e 1 For a more complete description of the mathematical basis of TRANSPORT refer to 51 75 1 and to other references listed at the end of this manual gt 11 9 M Definitions x the horizontal displacement of the arbitrary ray with respect to the assumed central trajectory the angle this ray makes in the horizontal plane with respect to the assumed central trajectory y the vertical displacement of the ray with respect to the assumed central trajectory the vertical angle of the ray with respect to the assumed central tra
33. either first or second order fitting Typical input format for a DRIFT pcc if desired not to exceed 4 spaces between quotes 3 6 DP DRIFT space matrix The first order R matrix for a drift space is as follows oooorr 0 where L the length of the drift space The dimensions of L are those chosen for longitudinal length via a units symbol factor if needed 15 8 5 type code entry if used preceding the BEAM type code 1 0 card If no 15 8 entry is made the units of L will automatically be in metres standard TRANSPORT units WEDGE BENDING MAGNET Type Code 4 0 A wedge bending magnet implies that the central trajectory of the beam enters and exits perpendicularly to the pole face boundaries to include fringing field effects and 1 1 entrance or exit boundaries see type codes 2 0 and 16 0 There are four first order parameters to be specified for the wedge magnet via type code 4 0 1 Type code 4 0 specifying a wedge bending magnet 32 2 The effective length L of the central trajectory in metres 3 The central field strength B O in kG B 0 33 356 p p where p is the momentum in GeV c and pg is the bending radius of the central trajectory in metres 4 The field gradient n value dimensionless where n is defined by the equation B GG0 t B 0 0
34. labeled In later steps indicator card 1 of a problem only those elements to be changed are specified The elements to be changed are identified by their labels If the type code number of an element is negative in a given step of a problem that element will be ignored when the calculation is performed However storage space in the computer will be allocated for the element for possible activation in later steps of the problem In the later step only those parameters to be changed need to be specified The storage space allocated for the parameters of a given element is determined only by the type code The sole exceptions are the continuation codes for type codes 1 0 and 14 0 For example if a fitting constraint is to be ignored in the first step of a problem but activated in a later step it should be indicated in both steps In the first step such an element might appear as 10 FIT In the later step one would then insert 10 T 25 0 0 001 FIT causing a waist constraint to be imposed on the beam Alternatively one can specify the physical parameters in the first step and then in the later step merely indicate that the element is now to be activated The above procedure is therefore equivalent to placing the element 10 1 2 0 0 001 FIT in the first problem step and the element 10 FIT in the later step Vary codes may also be inserted or removed in passing from one problem step to the next F
35. more than one row has variable components If it is desired to read in the second order matrix coefficients for row then the followlng 22 additional numbers may be read in This feature frees the user from making repetitive expensive second order runs through a fixed portion of his system while experimenting with other magnets This is done by reading the full matrix of this portion obtained from a previous run back into the machine as a single arbitrary matrix 9 continuation code 9 10 to 30 the 21 coefficients T i11 T i12 T i13 T ii4 T i15 T i16 T i22 T i23 T i24 T i25 T i26 T i33 T ia4 T ia5 T i36 T i44 T8i45 146 T i55 T i56 T i66 in that order where i is the row number It is necessary to read in the first order matrix row which corresponds to the second order matrix row being read in As in the first order case full rows not different from the 67 identity matrix i e R ii 1 all other R ij 0 and all T ijk 0 need not be read in Input output UNITS Type code 15 0 TRANSPORT is designed with a standard set of units that have been used throughout this manual However to accommodate other units conveniently provision has been made for redefining the units to be employed This is accomplished by insertion of one or more of the following elements There are four parameters to be specified 1 Type code 15 0 Code digit 3 The abbreviation of the unit
36. necessary Since this element is solely an extension of the beam input a 12 0 element must immediately be preceded by a 1 0 BEAM element entry The effect of this element in the printed output is shown only in the beam matrix If the beam matrix is printed automatically it is not printed directly after the BEAM element but only after the correlation matrix has been inserted Output PRINT CONTROL instructions Type code 13 0 A number of control codes which transmit output print instructions to the program have been consolidated into a single type code There are two parameters 1 Type code 13 0 2 Code number The effects of the various code numbers will be described below not in numerical order Several codes are available to control various aspects of the printed output Most type codes produce a line of output that advertises their existence Those that do not usually have an obvious effect upon the remainder of the output and thus make their presence clear Beam matrix print controls 1 2 3 13 1 3 The current beam matrix is printed by this code 13 3 3 The beam c matrix will be printed after every physical element which follows this code 13 2 3 The effect of a previous 13 3 code is cancelled and the beam matrix is printed only when a 13 1 code is encountered or when another 13 3 code is inserted The suppression of the beam matrix is the normal default Transfo
37. of cm mr and the T 236 0 y6 matrix element will have the dimensions mr cm percent Ap p and so forth Misalignment table print control 8 The misalignment summary table is printed wherever a 13 8 element is inserted Its contents are the effects of all previously specified misalignments whose results were to be stored in a table A full description of the table and its contents is to be found in the section on the align element type code 8 Coordinate Layout control 12 One can produce a layout of a beam line in any Cartesian coordinate system one chooses The coordinates printed represent the x y and z position and the angles theta phi and psi respectively of the reference trajectory at the interface between two elements Theta is the angle which the floor projection of the reference trajectory makes with the floor 2 axis Phi is the vertical pitch Psi is a rotation about the reference trajectory In the printed output the values given are those at the exit of the element listed above and at the entrance of the element listed immediately below A request for a layout is specified by placing 13 12 card before the beam card If no additional cards are inserted the reference trajectory of the beam line will be assumed to start at the origin and proceed along the positive z axis The y axis will point up and the x axis to the left One can also specify other starting coordinates and orientations by placing cert
38. of the beam envelope size in a 113 TRANSPORT printout there appears a column of numbers whose values will normally be zero These numbers are the coordinates of the centroid of the beam phase ellipse with respect to the initially assumed central trajectory of the system They may become non zero under one of three circumstances 1 When the misalignment Type Code 8 is used 2 When a Beam Centroid shift Type Code 7 is used or 3 When a 2nd order calculation Type Code 17 is used Physical Interpretation of Various Projections of the 2 dimensional BEAM Ellipse Consider again Eq 30 o 1 o 0 R and expand it in it s most general form for the 2 dimensional x 0 plane case Ri sat oat the result is 0 1 E240 281 R4502 0 12022 0 Ri R2 0 0 R11R22R12R21 021 0 Ry2R22522 0 821 011 0 2R21R22621 0 2 gt 022 0 In the special case when the initial ellipse is erect i e 0 1 0 0 o 1 reduces to R41011 0 Rj2022 0 1 21011 0 512820207 21011 0 R32622 0 Similar results are of course obtained for the y q plane If an arbitrary beam transport system is reduced to the most elementary first order form of representing it as an initial drift distance followed by a lens action between two principal planes and a final drift distance then we observe that for the 2 dimensional representation there are only two basic phase ellip
39. of the fitting procedure The transfer and beam matrices and layout coordinates appear as requested in the listing of the beam line The results of the fitting procedure appear between the two listings 11 these quantities are explained in greater detail below The transfer and beam matrices appear only where requested A re quest for printing of layout coordinates should be made at the beginning of the beam line The coordinates will then be printed after each physi cal element In all cases the quantities printed are the values at the 16 interface between two elements They are evaluated at a point after the element listed above them and before the element listed below For further explanation of calculated quantities the user should read the section on the mathematical formulation of TRANSPORT the appendix to the manual and the section on the appropriate type code For the transfer matrix the appropriate type code is thirteen for the beam matrix it is one and for the coordinate layout it is again thirteen Quantities relevant to the fitting appear between the two listings of the beam line At each iteration of the fitting procedure a line is printed containing the value of the relaxation factor used the value of chi squared before the iteration was made and the corrections made to each of the varied parameters Once the fitting is complete the final chi squared and the covariance matrix are printed For further details the user
40. on the card If the data which follow describe changes to be made in the previous step of a given problem a one 1 or two 2 is punched in any column on the card If a given problem step involves fitting the program will normally list the beam line twice printing each time the sequence of elements along with transfer or beam matrices where specified The first listing uses the parameters of each element before any fitting has taken place The second shows the results of the fitting If a problem involving fitting has several steps the second run of a given step often differs little from the first run of the following step If the second or subsequent step of a problem involves fitting and one wishes to print both runs through the beam line a one 1 is punched on the indicator card If the first listing is to be suppressed a two 2 is punched If no fitting is involved the program will ignore the two and will do one single run through the system If the initial listing is to be deleted 10 is added to the indicator to give 10 11 or 12 In order to be consistent with earlier versions of TRANSPORT an indicator of minus one 1 is interpreted as a two 2 but nine 9 is not interpreted as twelve 12 x DA E The sample problem input shown on page 14 causes TRANSPORT to do a first order calculation with fitting 0 indicator card and then to do a second order calculation 1 indicator card with the data that is the result of the fi
41. order only Provision is made in the program to vary some of the physical parameters of the elements comprising the system and to impose various constraints on the beam design In a first order run one may fit either the TRANSFORM R matrix representing the transformation of an arbitrary ray through the system and or the phase ellipse sigma matrix representing a bundle of rays by the system as transformed In a second order run one may fit either the second order TRANSFORM T matrix or minimize the net contribution of second order terms to the beam sigma matrix The program will normally make a run through the beam line using values for the physical parameters as specified by the user and printing the results If constraints and parameters to be varied are indicated it will attempt to fit To do this it will make an additional series of runs through the beam line Each time it will calculate corrections to be made from the previous step to the varied parameters to try to satisfy the indicated constraints When the constraints are satisfied or the fitting procedure has failed the program will make a final run through the beam line again printing the results In this final run the values of the physical parameters used are those which are the result of the fitting procedure Thus in principle the program is capable of searching for and finding the first or second order solution to any physically realizable problem In pract
42. percent Ap p 8 The momentum change in the central trajectory in GeV c 9 The code digit 0 indicating an r m s addition to the BEAM is being made The units for the r m s addition are the same as those selected for a regular BEAM type code 1 0 entry Thus a typical r m s addition to the BEAM would appear as follows 1 1 2 15 43 0 13 0 1 where the last entry 0 preceding the semicolon signifies r m s addition to the BEAM is being made and the next to the last entry indi cates a central momentum change of 1 GeV c FRINGING FIELDS and POLE FACE ROTATIONS for bending magnets Type code 2 0 pag 37 To provide for fringing fields and or pole face rotations on bending magnets the type code 2 0 element is used There are two parameters 1 Type code 2 0 2 Angle of pole face rotation degrees The type code 2 0 element must either immediately precede a bending magnet type code 4 0 element in which case it indicates an entrance fringing field and pole face rotation or immediately follow a type code 4 0 element exit fringing field and pole face rotation with other data entries between A positive sign of the angle on either 28 entrance or exit pole faces corresponds to a non bend plane focusing action and bend plane defocusing action For example a symmetrically oriented rectangular bending magnet whose total bend is 10 degrees would be represented by
43. point the two are related by the equation 2 _ 1 Y Rix 12 If we continue to assume a distribution centered at the origin the first moments at both initial and final point will be zero The second moments will now be given by 2 2 2 1 ar Dik xf 13 1 or more concisely o Ro pt 14 To first order an ellipsoid at the initial point will transform into an ellipsoid at the final point so that the equation T oOx 1 15 will give the envelope of the particle distribution at the later point B Second Order In second order the transformation on the coordinates effected by the beam line is given by 2 _ 1 1 Y Rix xj Xk 16 We employ here a symmetric T matrix whose off diagonal elements are half those of the T matrix used by Brown The first and second moments of the distribution at the final point are now given by x j Tijk x 17 2 2 1 1 xt Me Rix Rye x m 0 0 0 xP Tike Do aa x For a symmetric on axis initial distribution the first and third moments vanish The problem now reduces to determining the fourth moments of the initial distribution As an extension of previous notation we now denote the fourth moments of the distribution about the initial point by B We consider the coordinate system in which the matrix of second moment
44. should read the section on type code 10 0 and the section on fitting in the appendix Mancano tabelle pagg 22 23 24 25 26 27 2 37 01000 0000070 00000 2 00000 1 eel LIde 00000 100SvL C 13 00000 O 10000S 00000 01 00618 6 100000 0 009 2 w t00000 wlDaSe 00000 0000071 00006 00000 0000S 0000071 00006 000 m 114 VI38 NO 18 1667 166 000 0 000 0 000 0 00070 iE oc oo 00070 566 00070 216 189 00070 000 0 00070 66 00070 6 980 0 000 0 000 O v 000 0 2 6 980 00070 00070 06970691 00070 00070 00070 18 000 0 000 0 000 0 00070 00070 18 00070 000 0 00070 000 O 000 0 00070 00070 000 O 666 886 000 0 000 0 966 666 66 000 0 000 0 66 666 266 000 0 000 0 66 000 0 00070 000 0 000 0 185 000 0 000 0 2 Wo uw WO WO 0000 8020 81v00 01000 0000070 00000 000000 0000070 000000 Lv98S 11 0000071 0000070 0000070 000000 000000 8 00 1 909 8 1 0000070 000000 81700 0071 18892721 0000070 0000070 00000 0 1189 0000070 000000 90000070 00071 000 0 069711 000 0 09671 000 0 000 0 00070 00070 OO0GvL C 00071 000 0 069711 000 0 09671 000 0 696 000 0 0070 0 01 0007
45. t R21 14 in the x plane and if R4 0 15 in the y plane Point to parallel imaging occurs in the x plane if 57 16 and in the y plane if s t Raa 0 17 Focal Lengths A simple ray diagram of a thick lens demonstrates that R and have the following physical interpretations 108 c x t Rai and c y t Ra 2 18 fe 5 where f and f are the system focal lengths in the x and y planes respectively Zero Dispersion For point to point imaging using Eq s 5 and 10 the necessary and sufficient condition for zero dispersion at an image is t dx t Rie f Sx t h t dt 0 19 For parallel to point imaging i e c t O the condition for zero dispersion at the image is 44 Ris frc x e 20 Magnification For monoenergetic point to point imaging in the x plane the magnification is given by t 2 c t 0 And the plane by My R33 Cy t 21 where a negative number means an inverted image First Order Momentum Resolution For point to point imaging the first order momentum resolving power not to be confused with the matrix R is the ratio of the momentum dispersion to the total image size Thus if 2 is the total source size then dx t Nc E 0 2 t 2xoR41 Ri For point to point imaging s t 0 Using Eq 5 the dispersion at an image is t
46. the beginning of the misaligned section The misalignment element marks the end This option makes use of the fact that R2 matrix updates do not affect the R1 matrix A bending magnet with fringing fields or pole face rotations 44 type code 2 should be misaligned using this option See examples 1 and 2 below for an illustration of this An array of quadrupoles provides another example of the use of this option By successive application of align elements the elements of a quadrupole triplet could be misaligned relative to each other and then the triplet as a whole could be misaligned See example 3 below for an illustration of this XX3 All subsequent bending magnets and quadrupoles are independently misaligned by the amount specified This option is useful in conjunction with the tabular display of the misalignment results see below A bending magnet with fring ing fields included is treated as a single unit and misaligned accordingly XX4 All subsequent bending magnets including fringing fields are independently misaligned by the amount specified See XX3 above for further comments XX5 All subsequent quadrupoles are independently misaligned by the amount specified See example 4 below for an illustration of this See XX3 above for further comments B The tens position defines the mode of display of the results of the misalignment XOX beam matrix contains the results of the misalignment The beam matrix
47. the principal planes of the lens system is the source size and is the magnification of the first order image Again we observe that the ratio of the beam size at the source and the beam size at the lens is the criterion determining the proximity of these two quantities Orientation of the Major Axes of a Phase Space Ellipse The matrix equation for a coordinate rotation as shown in Fig 11 is Xo Fig 11 5 dez t is or X Xe The equation of an ellipse in either set of coordinates is X oc X 1where X 595 X x and the transformation from c 0 to o 1 is 1 M o 0 M provided 1 which it does If we assume a general ellipse for o 0 and an erect ellipse for o 1 125 a _ 011 10 61 0 9 9 It follows from Eq 72 that 021 1 0 04 0 MoiMq2 O12 0 022 0 from which 2021 0 Tan 22 80 2 0 11 0 or using the definition _ 021 r zn y 011022 an alternate form of expressing the ellipse orientation is 2721011022 _ 2721 022 011 922 _ 611 011 022 Clearly is dependent upon the units chosen for 011 and except in the obvious case of a i e an erect ellipse Tan 2a 81 SECOND ORDER ABERRATIONS Section III TRANSPORT has the capability of calculating the second order matrix elements aberration coefficients of any static magnetic beam transport syste
48. the three entries 2 5 5 4 j 2 The angle of rotation be varied For example the element 2 1 5 3 would allow the angle to vary from initial guess of 5 degrees to a final value which would say satisfy a vertical focus constraint imposed upon the system See the type code 10 0 section for a complete discussion of vary codes Even if the pole face rotation angle is zero 2 0 3 entries must be included in the data set before and after a type code 4 0 entry if fringing field effects are to be calculated A single type code 2 0 entry that follows one bending magnet and precedes another will be associated with the latter It is extremely important that no data entries be made between a type code 2 0 and a type code 4 0 entry If this occurs it may result in an incorrect matrix multiplication in the program and hence an incorrect physical answer If this rule is violated an error message will be printed Should it be desired to misalign such a magnet an update element must be inserted immediately before the first type 2 0 code entry and the convention appropriate to misalignment of a set of elements applied since indeed three separate transformations are involved See section under type code 8 0 for a discussion of misalignment calculations and the section under type code 6 0 for a discussion of updates The type code signifying a rotated pole face is 2 0 The input format is Label if desir
49. to the matrix elements we have 011 2 011 0 2 d 2 X2 max 0 max 44 022 2 m 022 or 93 7 6 max 85 45 Note the change in sign of the elements for the Drift and the Lens actions indicating the different sense of orientation of the resulting ellipses as illustrated in Fig 8 The Upright Ellipse A case of particular interest in any 2 dimensional phase ellipse projection e g the x 0 plane or the plane is when the off diagonal correlation matrix elements are equal to zero i e an erect ellipse In a field free region this corresponds to a so called waist in the BEAM as illustrated in Fig 9 It is important to understand correctly the meaning of a waist for an existing beam it is the location of the minimum beam size in a given region of the system Although the waist is the minimum beam size in any given beam the minimum beam size achievable at fixed target position by varying the focal strength of the preceding lens system is not the same as the above defined wait See Fig 10 In a field free region the minimum beam spot size achievable at a fixed target position will occur when the preceding lens system is adjusted such that a waist precedes the target position Only in the limit of zero phase space area do these quantities occur at the same location A useful criterion that determines the physical proximity of these quantities is the following if th
50. transfer matrix It too is accumulated by the program as one traverses a series of elements At each point the series is again truncated to second order Normally the program will calculate only the first order terms and their effect If it is desired to include second order effects in a beam line an element is provided which specifies that a second order calculation is to be done For more information the T matrix see the references at the end of the manual The following of a charged particle via TRANSPORT through a system of magnets is thus analogous to tracing rays through a system of optical lenses The difference is that TRANSPORT is a matrix calculation which truncates the problem to either first or second order in a Taylor s expansion about a central trajectory For studying beam optics to greater precision than a second order TRANSPORT calculation permits ray tracing programs which directly integrate the basic differential equation of motion are recommended The beam matrix o In accelerator and beam transport systems the behavior of an individual particle is often of less concern than is the behavior of a bundle of particles the beam of which an individual particle is a member An extension of the matrix algebra of Eq 1 provides a convenient means for defining and manipulating this beam TRANSPORT assumes that the beam may be correctly represented in phase space by an ellipsoid in the six dimensional coordinate sy
51. transformation of a single trajectory in addition to the central trajectory through a static magnetic system We wish now to extend the discussion to include bundles of rays To accomplish this we take advantage of Liouville s Theorem which states that the phase space is conserved through the system While the Theorem is strictly true to all orders a convenient mathematical transformation has only been developed to first order A manifestation of Liouville s Theorem is the fact that 1 Now so long as there is no coupling mechanism between the x plane and y plane of a magnetic optical system which is the case if the midplane symmetry prevails throughout the system then the phase space area in a given plane is also conserved Consider a bundle of rays represented by the parallelogram shown in Figure 15 a representing the phase space distribution of the rays at some initial position If we now look at the phase space distribution of the same bundle after it has drifted down stream we observe the Omax boundary and the x 93 intercept remain unchanged In other words the area of the parallelogram is the same or phase space area has been conserved a Figure 15 For mathematical convenience the parallelogram is rather difficult to work with and hence a phase ellipse is usually used i Figure 16 The phase ellipse transformation for a drift distance is illustrated in Figure 16 Figure 16 a co
52. 0 930 00000 0 000 000 0 069711 070 09671 00070 69 0000 2 7021 000 0 10701 00070 0000 S 5 00000 01 00618 6 09071 000 0 009 000 0 00071 90070 016 000 0 000 1 000 0 046 00070 930 000000 071 000 0 004 0000 09071 000 0 046 00070 000 1 000 0 014 09070 006 72 00071 000 0 00s 000 0 000 1 000 0 005 000 0 00071 0000 oos 000 0 39 0000071 000 1 379698 114 3ONVIHVAOCO 20 3680269 10 300001 70 379711 104300001 SNOLLO3UM4O2 006967561 HIONAT 1113 0701 113 00000 0 00000 0 18955 1 88PCC l 00000 0 00000 0 00000 0 00000 0 8 00 1 4096871 81700 1 WHOJSNYUI 696741 3002 180 E 13180 29721 e IVIONM W vc9 ci v x N38 SvL e e 1 108 69 72 3009 180 E 00070 Wv38 114 Vl38 NO 323HO NYUIlSOJ AO 2 88 T oo W 62126751 HIONGTI 00000 01000 00000 0 1113 070 113 00000 I 00000 000000 000000 000000 00000 0 17985 11 0000071 000000 0000070 929 1 88 7 l 00000 0 0000070 00000 1 S09 8 000000 0000070 0000070 000000 00000 00000 00000 0 00000 0 18892721 000000 000000 0000070 00000 1 5609 8 1 9Geve el 0000070 0000070 0000070 00000 0000071 WHOISNVUL 0000 766 666 00071 00070 0
53. 0 0 24 00070 000 000 0 00071 WD vio 000 000 LIO 000 000 WO 29472 020 100 WO E18 13 96767 9 00070 00070 0070 00070 000 0 24 00070 QUO 000 00071 00071 WO vIO 00071 0007 000 uw 66971 000 000 WD 18672 00071 uw 46272 SOC Ld 66996 0067 I4 ISL E8EZ HW OLE Jd 16 7 862 00070 000 0 000 0 000 0 24 000 0 00070 00070 00070 00070 00070 00070 00070 9187 00070 00070 050 000 0 299 040 10 3009 0000071 W3 00001 00070 00070 00070 000 0 00070 000 0 13 066 991 13 00000 S E 13180 14 0667191 WO 000 672 OISSv E 13 00000701 G x qvnos 13 0667161 13 00009 t E 13180 00000 1 00000 0 00000 0 90000 0 00000 0 00000 0 60000 00000 00000 00000 80720 09800 00000 00000 0 S65018 1 6S5 69 00000 00000 00000 00000 0 LeC50v vS6cL 00000 00000 e SILV 00000 0 00006 00000 6vt80 v L6SvL 73081 00000 0 00000 00000 PCCOL GS 1698271 n WHOASNVUL 0 0 09070 00070 00070 00070 000 0 14 068 871 13 0000171 13 I4 069 071 WD O00vS c 09000 7 13 00000701 G 14 069 051 13 66689 12 IJIUO 14 000 601 000007081 LOM Z l4 0007601 00000 0 2618176 14 00042701 wo y N38 13 061 26 930 000007081 0c
54. 0070 216 686 WO 069711 00070 919 00070 0000 uw 16671 00070 0000 009 000 0 966 00070 WD 2SE E 00070 W 126751 3002 6192772 180 00070 66 666 24 00071 00070 00070 216 266 2 069711 00070 980 00070 7 000 0 uw 165671 000 0 000 000 0 WO OLE 000 0 66 26721 00070 0701 00070 09721 00000 0 00070 66 666 24 00071 0000 0000 16 266 WD 069711 00070 980 00070 00070 uw 16671 00070 00070 00070 WD OLE 000 0 66 uw wect 7 00070 W3 0 01 000 0 609721 930 0697691 9EE E 0000 93 00000701 OO6L8 6 y N38 0000 00070 000 0 00071 000 0 000 0 0000 000 0 WO 000 0 8L 00070 000 0 uw 00071 00070 000 0 WO 69S 000 0 8 7 aw 00071 000 0 2 69S 000 veL 00000 0 1108 000 0 000 7 000 0 24 00071 000 0 0000 006 00070 81 00070 00070 00071 000 0 000 0 000 0 2 69S 000 0 81 00071 000 0 WD 696 000 0 N 21 2 3002 AHVA 6192772 w Ue 000 0 00070 2d 000 000 0 000 0 0000 2 006 000 0 00070 000 0 00071 000 0 00070 000 0 2 009 000 0 000 0 uw 00071 000 0 004 000 0 000 0 A39 0000071 WY38 VI38 NO X23HO 20 I3NIIN3S 101000 00000 0 00000 Z
55. 1ed jo u3uo3 euo 01 4 ue ST 87 0 S C 001 8 L 9 6 19 4009 pue 3161 gt 015 40 5 0129 UOLSA AUOJ 0 61 gt adAy 531 3nd3no 3ndu SPECIAL INPUT PARAMETERS Type code 16 0 A number of constants are used by the program which do not appear as parameters in elements of any other type code A special element has been provided to allow the designer to set their values These special parameter entries must always precede the physical element s to which they apply Once introduced they apply to all succeeding elements in the beam line unless reset to zero or to new values There are three parameters 1 Type code 16 0 2 Code digit 3 Value of the constant A number of such constants have been defined in this manner All have a normal value that is initialized at the beginning of each run Code digits for special parameters 1 1 a second order measure of magnetic field inhomogeneity in bending magnets If x x 2 B x B 0 1 2 gt is the field expansion in the median 9 plane then e 1 is defined as 1 2 gt where pg is measured in units of horizontal beam width normally cm This parameter affects second order calculations only Normally the value if It may be varied in second order fitting
56. 3 M m Mass of the particles comprising the beam in units of the electron mass normally A non zero mass introduces the dependence of pulse length on velocity an important effect in low energy pulsed beams 4 W 2 Horizontal half aperture of bending magnet in the same units as horizontal beam width normally 0 i e effect of horizontal half aperture is ignored 5 5 2 Vertical half aperture of bending magnet in the same units as vertical beam height this parameter must be inserted if the effect of the spatial extent of the fringing fields upon transverse focusing is to be taken into account See type codes 2 0 and 4 0 as a cross reference normally 0 6 L Cumulative length of system in the same units as system length It is set to zero initially then increased by the length of each element and finally printed at the end of the system Ki K 71 This element allows the cumulative length to be reset as desired An integral related to the extent of the fringing field of a bending magnet See section under type code 2 0 and SLAC 75 page 74 for further explanation If the 16 5 g 2 element has been inserted the program inserts a default value of K1 gt unless 16 7 3 element is introduced in which case the program uses the value selected by the user The table below shows typical values for various types of magnet designs A second integral related to the extent of the fringing f
57. 4 0 2to7 The six numbers comprising the row The units must be those used to print the transfer matrix in other words consistent with the BEAM input output 8 Row number 1 to 6 A complete matrix must be read and applied one row at a time Rows that do not differ from the unit transformation need not be read For example 14 1 9 0 0 O 2 3 introduces a transformation matrix whose second row is given but which is otherwise a unit matrix Note that this transformation does not conserve phase space because R 22 0 9 i e the determinant of 1 Any of the components of a row may be varied however there are several restrictions Type code 14 0 elements that immediately follow one another will all be used to form a single transformation matrix If distinct matrices are desired another element must be inserted to separate the type code 14 0 cards Several do nothing elements are available for example a zero length drift 3 0 is a convenient one When the last of a sequence of type code 14 0 cards is read the assembled transformation matrix will be printed in the output Note that 1 0 411 042 41 0 1 f 2 Hence a matrix formed by successive 14 3 0 14 elements is not always equal to the one formed by leaving out the 3 O element If components of a 14 0 card are to be varied it must be the last 14 0 card in its matrix This will force a matrix to be split into factors if
58. 5 f By x 0 t dt Spx x tan B x 14 where h 2 and B The quadrupole strength The radius of curvature of the boundary is related to the sextupole strength as follows 1 2 252 R 1 2 2 3 72 hsec or 3 S 8 The sextupole strength From equation 13 we note that a positive multipole component of the field increases the for a positive x thus a positive sextupole is represented by a concave surface of the entrance or exit boundary The Descrigtion of the Trajectories as a Taylor s Expansion The deviation of an arbitrary trajectory from the central trajectory is described by expressing x and y as functions of t The expressions will also contain yo and where the subscript o indicates that the quantity is evaluated at t o The prime denotes the derivative with respect to t and is the fractional momentum deviation of the ray from that of the central trajectory These five initial boundary values will have the value zero for the central trajectory itself an and y are expressed as a five fold Taylor expansion using these initial boundary values The expansions are written x t y t A A X xIx yox 6 6 xi yox 664 15 Here the parentheses are symbols for the Taylor coefficients the first part of the symbol identifies the coordinate represented by the expansion and the second indicates the term in que
59. 65 TRANSPORT APPENDIX K L Brown and F Rothacker Stanford Linear Accelerator Center Stanford California D C Carey E Fermi National Accelerator Laboratory Batavia Illinois and Ch Iselin CERN Geneva Switzerland December 1977 APPENDIX Table of Contents Introduction 1 Beam Transport Optics a set of lectures given to the SLAC technical staff Part I Introduction Geometric Light Optics vs Magnetic Optics Introduction of Momentum Dispersion into the Matrix Formalism Second Order Matrix Formalism Transformations Involving Many Trajectories Part II Introduction First Order Transformation Matrix Beam Switchyard TRANSPORT Notation Second Order Matrix 2 First Order Matrix Formalism for TRANSPORT First Order R Transfer Matrix Formalism First Order Dispersion First Order Path Length Achromaticity Isochronicity First Order Imaging Focal Lengths Zero Dispersion Magnification First Order Momentum Resolution First Order Phase Ellipse Formalism for TRANSPORT Description of the Sigma BEAM Matrix The Phase Ellipse Matrix used by TRANSPORT Physical Interpretation of Various Projections of the 2 dimensional BEAM Ellipse The Upright Ellipse Relationship between a Waist and a Parallel to Point Image Relationship between a Waist and Point to Point Image Relationship between a Waist and the Smallest Spot Size Achievable at a Fixed Target Position Imaging from an Erect Ellipse to an Erect Elli
60. 81 000 0 00071 000 0 000 O 00070 2 695 000 0 817 uw 00071 00070 WD 6945 000 0 W 21 2 W Sivert c n E 13180 01118141519 NVISSNVO 1235 LI 53080 00070 00070 00070 000 0 00070 0007 000 0 000 0 000 0 000 0 00070 WD 004 000 0 000 0 000 0 uw 00071 00070 0000 WD 0065 000 0 000 0 00071 0000 WD 009 000 0 0000 39 0000071 8 43040 0235 22 62126751 HLONA T 0 99 v O0 9S v tO aL9t v 9v 0 196 748 9 v 0 92 91 40 cS 40 Sv v 40 Gt 41 O wep 0 12 072 v 0 3666 C vl v et v tO 3rv66 G EZ tO dtOG S6 40 eC 21 0 WE 0 986 0 39717 97 90 i 9E 70 926 0 91 0 SS 0 8 0 EE 0 Sc 0 0 we 0 ve 0 3 2270 vC 0 32 0 02 PI 0 t E 60 32 02 EC 0 30087 ec O 99 0 962 92 0 20 399876 922 20 316076 912 64264 0 GE C 70 SIZ 0 3120 vv 2 t0 d 66 2 C 0 vec 0 3 05 6 2 0 2 Elz 0 3902179 90 LO 3611 9 c0 3806 L 991 40 961 40 OF 0 co 3990 921 20 3690 2 91 70 GG 0 Sb 0 SE 0 Sl 0 321171 0 32 072 40 70 vl 8
61. A bend down is accomplished via at 90 degree rotation 20 90 2 2 20 90 A bend to the left looking in the direction of beam travel is accomplished by rotating the x y coordinates by 180 degrees e g 20 180 2 4 2 20 180 Example No 2 A quadrupole rotated clockwise by 60 degrees about the positive z axis would be specified as follows 20 60 Be 7153 20 60 j Beam rotation matrix C S C S S C R S where cos S sina a angle of coordinate rotation about the beam axis blank spaces are zeros e g for a 90 degrees this matrix interchanges rows 1 and 2 with 3 and 4 of the accumulated R matrix as follows 0 0 1 0 ROAD R2 R 14 7 R 31 R 32 R 33 R 34 0 1 R21 R22 R 24 R 41 R 44 1 0 0 1 R 33 R i1 R 12 R 13 R 14 1 0 01 LR 41 R 42 R 44 l R 21 R 22 R 23 R 24 The rest of the matrix is unchanged STRAY MAGNETIC FIELD Type code 21 0 1 Element No 21 0 2 Code No n n 4 horizontal deflection n 2 vertical deflection 3 BL mean value of f Bdz B a E Gaussian random number generator affects beam first moment uncertainty in affects beam second moment Uses the misalignment element 8 to calculate an angular deflection Bdz equal to Th
62. A is a source point and if Ry 0 i e x is independent of 05 then B is image point for monoenergetic particles Under these circumstances 90 is the x plane magnification Ro 1 M because det R 1 And R21 1 1f fact Ra 1 f for the system between and even if A and B are not foci It is now convenient to develop a more general definition of the matrix elements Ri and at the same time introduce the first order matrix transformation for the y non bend plane Consider again a general system where the projection of the central trajectory is allowed to bend in the X plane but is a straight line in the y plane The x plane and y plane matrix transformations may be written as follows For the x plane x Rx Xe 50 c fp 100 d manca quadrettatura nella parte centrale Similarly for the Y plane Ry yo t t 0 law cw e drettat 11 t tral r 5 quadrettatura nella parte centrale The c and s functions may be defined in terms of their initial conditions Let t be the distance measured along the Figure 13 central trajectory Then s 0 0 s 1 where s t zo c 1 Within ideal magnet where the bending radius pg is constant and c are sine and cosine or else sinh and cosh functions Because of this the terminology s a sine like function and c a cosine lik
63. ANSFORM 2 updates take place immediately before and immediately after any bending magnet which has either the entrance or exit fringe fields specified via a type code 2 entry SHIFT IN THE BEAM CENTROID Type code 7 0 Sometimes it is convenient to redefine the beam centroid such that it does not coincide with the TRANSPORT reference trajectory Provision has been made for this possibility via type code 7 9 Seven parameters are required 1 Type code 7 0 2 to 7 the coordinates x y A and 6 defining the shift in the location of the beam centroid with respect to its previous position The units for x y are the same as those chosen for the BEAM type code 1 0 entry normally cm mr cm mr cm and percent Any or all of the six beam centroid shift parameters may be varied in first order fitting The centroid position may then be constrained at any later point in the beam line by this procedure The transformation matrix R2 is updated by this element In order for this code to function properly the initial BEAM entry type code 1 0 must have a non zero phase space volume for example a 1 0 0 0 0 0 BEAM entry is not permissible when calculating a shift in the beam centroid whereas a 1 1 1 1 1 1 1 pt8 entry non zero phase volume is acceptable MAGNET ALIGNMENT TOLERANCES Type code 8 0 The first order effects of the misalignment of a magnet or group of magnets are
64. B is given by the following matrix equation 92 Xx Xo le Ria Ria Ri le 5 6 parte centrale scritta in modo impreciso As in all matrix calculations the order of writing down the elements comprising the system is from right to left The individual matrix elements must be derived from the solution of the equation of motion within each element If this has been done then the calculation for the total system is carried out in the fashion shown by the above equation 4 Second Order Matrix Formalism 1 pag 139 It is possible to extend the 3 x 3 matrix formalism to solve simultaneous sets of power series by generating a second order matrix equation as follows x 3 x 3 second order x 2 first order terms terms e 6 95 6 6 2 x 3 all zero 2 1 6 all other x 6 order terms The R term is obtained by squaring the upper left corner 3 X 3 matrix so as to obtain second order equations for xc X16 6 etc as functions of products of the initial first order variables xe and This is then a convenient mathematical formalism for keeping all the terms desired and dropping those undesired In the above example all first and second order terms are retained and all higher order terms are automatically dropped by the matrix multiplication 5 Transformations Involving Many Trajectories All of the discussion to this point relates to the
65. Figure 16 Beam Transport Optics Part II K L Brown 1 Introduction In Part I the basic concepts of been transport optics were established Starting from the essentials of geometric optics the methods of matrix algebra were introduced with the example of calculating the principal planes of a thick lens The 3 x 3 matrix for the first order bean transport calculations were introduced to take into account the particle moments 2 First Order Transformation Matrix Figure 1 shows a general region containing a magnetic field Arbitra Ray cy Central Ray Figure 1 General Magnetic Field Configuration 95 The matrix presents a convenient way of writing the family of equations which describe the transformation from surface A to surface B If and represent the conditions of a ray entering the system at A then the conditions of the ray at B X and 6 Here is the distance from the central ray to the ray C is the angle between and the parallel to the central ray and is the ratio Ap p where Ap is the difference between the momentum of C and the momentum of the central ray The linear transformation equations are CyXe 5 00 4 6 C Xe 5 0 0 76 1 6 0 0 6 Expressed as a matrix Eq 1 are X1 Sy dy xo 2 9 Cy d 5 P 0 0 1 The equation 6 6 expresses the fact that the magnetic field cannot change the scalar momentum of the pa
66. LIM SAILISOd JAV LVHL SINSATIS AIOdILINN ALVOIGNI SAILINVIOd LAINOVN AHL SLN3ATI3 JIOdNLXAS ANY YOA SIXV X SNVIddIM JILAN VW AHL 40 NOILVALSATII 1 e TRANSFORM 1 update Type code 6 0 1 To re initialize the matrix TRANSFORM 1 the product of the R matrices Rl use type code 6 0 A 6 O 1 card effects an update of the R1 matrix and initiates the accumulation of a new product matrix at the point of the update This facility is often useful for misaligning a set of magnets or fitting only a portion of a system The matrix R1 is updated by no other element It is not used in the calculation of the beam matrix The beam matrix is calculated from the auxiliary transfer matrix R2 described on the next page A TRANSFORM 1 matrix will be printed at any position in the data set where a 13 4 entry is inserted See the following section for the introduction of an auxiliary trans formation matrix R2 TRANSFORM 2 to avoid the need for TRANSFORM 1 up dates The 6 1 5 card also causes an update of the R2 matrix By updating we mean initiating a new starting point for the accumulation multiplication of the R matrix At the point of update the previous accumulation is discontinued When the next element possessing a transfer matrix is encountered the accumulated transfer matrix R1 is set equal to the individual tra
67. PSI Graphic Transport Framework User Manual http aea web psi ch Urs Rohrer MyWeb trans htm Draft rewritten by S Sinigardi and M Camatti based on the original manual by Urs Rohrer v0 8 July 2012 If you would like to contribute correction to this manual please mail me Sinigardi bo infn it INTRODUCTION TRANSPORT 15 a first and second order matrix multiplication computer program intended for the design of static magnetic beam transport systems It has been in existence in various evolutionary versions since 1963 The present version described in this manual includes both first and second order fitting capabilities Many people from various laboratories around the world have contributed either directly or indirectly to the development of TRANSPORT The first order matrix methods were developed by the AGS machine theorists followed by a paper by Penner The extension of the first order matrix methods to include second and higher orders was conceived and developed by Brown Belbeoch and Bounin in Orsay France in 1958 59 The original first order TRANSPORT computer program was written in BALGOL by C H Moore at SLAC in collaboration with H S Butler and S K Howry in 1963 The second order portion of the program was developed and debugged by Howry and Brown also in BALGOL The resulting BALGOL version was translated into FORTRAN by S Kowalski at MIT and later debugged and improved by Kear Howry and Brown
68. RBITRARY R 14 RQ I R 3 2 2 3 R 2 4 2 5 2 6 2 UNITS CONTROL 15 0 CODE UNIT SCALE FACTOR Transport SYMBOL if Dimensions required QUADRATIC 16 0v 1 0 1 in units of tranverse length cm TERM OF Bo BENDING FIELD Po MASS OF 16 0 3 0 M m M mass of x J PARTICLES IN dimensi electron BEAM onless HALF APERTURE 16 0 4 0 w 2 cm OF BENDING MAGNET IN x PLANE HALF APERTURE 16 0 5 0 g 2 cm OF BENDING MAGNET IN y PLANE gsp LENGTH OF 16 0 6 0 SYSTEM metres FRINGE FIELD 16 0 7 0 dimensi COEFFICIENT onless FRINGE FIELD 16 0 8 0 CORRECTION dimensi COEFFICIENT onless CURVATURE OF 16 0v 12 0 1 R1 ENTRANCE FACE 1 metre OF BENDING s MAGNET CURVATURE OF 16 0V 13 0 1 R2 EXIT FACE OF 1 metre BENDING 5 FOCAL PLANE 16 0 15 0 Angle of focal plane ROTATION rotation See type code 16 0 for details INITIAL BEAM 16 0 16 0 LIME X COORDINATE INITIAL BEAM 16 0 17 0 Yo LIME y COORDINATE INITIAL BEAM 16 0 18 0 Zo LIME z COORDINATE INITIAL BEAM 16 0 19 0 05 LIME HORIZONTAL ANGLE INITIAL BEAM 16 0 20 0 Vo LIME VERTICAL ANGLE SECOND ORDER 17 0 CALCULATIONS SEXTUPOLE 18 0V LENGTH FIELD metres kG SOLENOID 19 VV LENGTH FIELD metres kG BEAM ROTATION 20 V ANGLE OF ROTATION deg rees STRAY FIELD 21 0 See later section of report
69. RESPECT TO EACH OTHER SOLENOID Type code 19 0 The solenoid is most often used as a focusing element in systems passing low energy particles Particles in a solenoidal field travel along helical trajectories The solenoid fringing field effects necessary to produce the focusing are included There are three parameters 1 Type code 19 0 2 Effective length of the solenoid metres 3 The field kG A positive field by convention points in the direction of positive z for positively charged particles The length and the field may be varied in first order fitting Both first and second order matrix calculations are available for the solenoid A typical input format is Label if desired not to exceed 4 spaces 19 1 B lt lt First order solenoid matrix Solenoid R matrix Definitions L effective length of solenoid B 0 2Bpe where B O is the field inside the solenoid and is the momentum of the central trajectory C cos KL S sin KL For a derivation of this transformation see report SLAC 4 by R Helm Alternate forms of matrix representation of the solenoid isc sc 4 e k k sc c 5 sc e e 5 is fs Le e k k R Solenoid ks sc 5 c 0 0 0 0 1 0 0 0 1 Rotating the transverse coordinates about the z axis by an angle KL decouples the x and y first order terms i e k KS C 0 0 C is
70. TS The TRANSPORT sign conventions for x and h are all positive as shown in the figure The positive y direction is out of the paper Positive B s imply transverse focusing Positive R s convex curvatures represent negative sextupole components of strength h 2R sec B See SLAC 75 page 71 Tilt to focal plane 16 15 a element pag 106 Very often it is desired to have a listing of the second order aberrations along the focal plane of a system rather than perpendicular to the optic axis i e along the x coordinate If the focal plane makes an angle a with respect to the x axis measured clockwise then provision has been made to rotate to this focal plane and print out the second order aberrations This is achieved by the following procedures Alpha is the focal plane tilt angle in degrees measured from the perpendicular to the optic axis a is normally zero The programming procedure for a tilt in the x bend plane rotation about y axis is 16 15 3 0 a necessary do nothing element 13 4 16 15 a rotate back to zero 3 0 a necessary do nothing element 16 15 O to turn off rotation element The programming procedure for a tilt in the y plane rotation about x axis is 16 15 a 20 90 3 0 20 90 13 4 16 15 rotate back to zero 305 5 16 15 O to turn off rotation element Initial beam Line coordinates and direction Wh
71. The size of the beam at the image is 011 1 Rip 022 0 f 05 max 51 independent of the source size and of the object distance The size of the beam at the waist is o 0 011 0 022 0 022 1 821 011 0 2 gt 022 0 size at waits 52 If S x the two sizes are equal as expected otherwise the size at the waist is always smaller Relationship between a Waist and a Point to Point Image A point to point first order image in the x 0 plane occurs when Riz Sx 0 The matrix representing this case is R 0 e 0 M 0 TE ja Je 1 53 Ra Rz Cx Sx fz M manca suddivisione centrale nelle 3 matrici where R 1 and M is the magnification 119 If we again assume an erect ellipse o as the beginning the final beam matrix o 1 is given by Eq 37 as 811011 0 R11R21011 0 R11R21011 0 R341011 0 2 gt 022 0 manca suddivisione centrale 0 1 54 for point to point imaging Our first observation is that except for a zero source size an image and a waist will coincide only if Ra O Clearly this is not possible with a single lens at least two lenses are needed Such an optical situation is as follows 1510426 The distance to a waist is 29210 R11R21011 0 022 1 821011 0 2 gt 022 0 55 So if 0 a waist and point to point image coincide If gt 0 the waist precedes the
72. a print instruction Calculation specifications such as misalignments and constraints are placed in sequence with the other beam line elements where their effect is to take place The input format of the cards is free field which is described below The data for a given problem step are terminated by the word SENTINEL which need not be punched on a separate card Each element in turn is given by a sequence of items mostly numbers separated by spaces and terminated by a semicolon The items in order are a type code number a vary field the physical parameters and an optional label The type code number identifies the element indicating what sort of entity such as a magnet drift space constraint etc is represented It is an integer number followed by a decimal point The interpretation of the physical parameters which follow is therefore dependent on the type code number The type code numbers and their meanings are summarized in Table 1 If the type code number is negative the element will be ignored in the given problem step However storage for that element will be allocated by the program so that the element may be introduced in a later step of the same problem Storage space for any element in any problem step must be allocated in the first step of the problem The vary field indicates which physical parameters of the element are to be adjusted if there is to be any fitting It is punched immediately no interve
73. ain other cards before the beam card For a description of such cards see type code 16 0 special parameters The calculation of the coordinates is done from the parameters of the physical elements as given in the data Therefore if effective lengths are given for magnetic elements the coordinates printed will be those at the effective field boundary The effects of fringing fields in bending magnets are not taken into account General output format controls 17 18 19 13 17 The subsequent printing of the physical parameters of all physical elements will be suppressed Only the type code and the label will remain This element is useful in conjunction with the 13 19 element which restricts the beam matrix and the transformation matrix each to a single row The elements of these matrices then appear in uninterrupted columns in the output similar to the TRAMP computer code used at the Rutherford Lab CERN and elsewhere 13 18 Only varied elements and constraints will be printed This element in conjunction with the various options on the indicator card can produce a very abbreviated output The entire output of a multistep problem can now easily be printed on a teletype or other terminal 13 19 The beam c and transformation R1 or R2 matrices when printed will occupy a single line Only those elements are printed which will be non zero if horizontal midplane symmetry is maintained The second orde
74. ality but for other applications such as charged particle spectrometers caution is in order in its use and interpretation The equation of an n dimensional ellipsoid may be written in matrix form as follows 0 o 09 x 09 1 27 Where X 0 is the transpose of the coordinate vector 0 and is a real positive definite symmetric matrix The volume of the n dimensional ellipsoid defined by sigma is n TU det the area of the projection one plane is n det 0o vt This is the phase space occupied by the beam As a particle passes through a system of magnets it undergoes the matrix transformation of Eq 3 Combining this transformation with the equation of the initial ellipsoid and using the identity RR I the unity matrix it follows 110 that X 0 R RT 0 X 0 1 from which RX 0 Ro 0 R RX 0 1 28 The equation of the ellipsoid representing the BEAM at the end of the system is thus XC o 1 XO 1 29 where the equation for the sigma matrix at the end may be related to that at the beginning by 1 Ro 0 R 30 In addition to calculating the product matrix R TRANSPORT also computes the signs BEAM matrix at the end of each physical element via Eq 39 All of the important physical parameters of the BEAM ellipsoid may be expressed as functions of the matrix elements of the sigma matrix at the location in question I
75. alue of the i j k matrix element Desired accuracy of the fit standard deviation Note that upper and lower limit constraints are not available for second order fitting Some typical T2 matrix constraints are as follows Desired optical condition Typical fitting constraint Geometric aberration 7122 0 10 21 22 0 001 F1 Chromatic aberration 5 10 23 46 5 001 F2 By using a T2 constraint the user may fit an element of the second order transfer matrix which pertains to any section of the beam One causes an R2 update at the beginning of the section with 6 0 2 element One then places the T2 constraint at the end of the section Any number of such constraints may be imposed This is the only second order constraint that may be used in conjunction with an R2 update If a printing of the T1 matrix is requested via 13 4 element it will be the second order transfer matrix from the last R1 update The comments about phase space weighting made in connection with the T1 constraint are equally valid for the T2 constraint provided the phase space factors are obtained from the beam matrix at the position of the R2 update Second order U BEAM matrix fitting constraint Five parameters must be specified for a constraint on the second order contributions to a beam matrix diagonal element o i Type code 10 0 Code digit i Code digit i The number 0 Desire
76. ance matrix are printed The covariance matrix is symmetric so only a triangular matrix is shown The diagonal elements give the change in each varied parameter needed to produce a unit increase in the chi squared The off diagonal elements give the correlations between the varied parameters The appearance of the chi squared and covariance matrix is COVARIANCE FIT 4 C22 Pin 6 For more details the mathematics of the fitting the user should consult the Appendix For an example of the output of the program he or she should refer to the section on output format ACCELERATION Type code 11 0 An energy gain is reflected in both the divergence and the width of the beam This element provides a simulation of a travelling wave linear accelerator energy gain over a field free drift length i e no externally applied magnetic field There are five parameters Type code 11 0 Accelerator length metres Energy gain GeV phase lag in degrees A wavelength in cm The new beam energy is printed as output The energy of the reference trajectory is assumed to increase linearly over the entire accelerator length If this is not the case an appropriate model may be constructed by combining separate 11 0 elements 11 0 element with a zero energy gain is identical to a drift length None of the parameters may be varied Second order matrix elements have not been incorporated
77. and prints out the second moments of the phase space distribution function in the oj columns In addition it also calculates and prints out the new coordinates of the centroid first moment of the phase space distribution function and tabulates this result to the left of the cj columns in the same manner as it does for a magnet misalignment run Caution should be used in the use and interpretation of the second order phase ellipsoid results especially if it is known or suspected that the phase space distribution resulting from a second order run is not symmetrical about the beam centroid To be certain of the situation in any given design it would be wise to calculate the actual distribution function by using the Monte Carlo computer program TURTLE The actual method used in TRANSPORT by which the second order terms are included in the beam ellipse is described in the following report The reader should bear in mind that the derivation is based on a gaussian initial beam distribution For any other initial distribution the second order effects on the beam ellipsoid should be regarded only as an approximation 1 D C Carey TURTLE A Computer Program for Simulating Charged Particle Beam Transport Systems N A L Report No 64 Fermi National Accelerator Laboratory Batavia Illinois 1971 127 Second Order Contributions to Beam Dimensions David C Carey May 1972 I Introduction The phase space region occupied by an aggregate of
78. ard which must not have a decimal point in any version of TRANSPORT The use of labels pag 15 The use of labels is available for identification of individual elements When inserted for the user s convenience the association of a label with a given element is optional I f the parameters of an element are to be changed between steps of a given problem a label is required The label identifies the element in the earlier step to which the changes specified in the later step are to apply The label may be placed anywhere among the parameters of a given element It should be enclosed in quotes slashes or equal signs Blanks within a label are ignored The maximum length of a label is four non blank characters As an example the following all denote the same drift space DRF 3 1 5 3 3 DRF 15 1 3 15 1 12 On a 15 0 type code element the label may not be the third item This is to avoid ambiguities with the unit name Thus the following are not equivalent This entry is used as the units symbol This entry is used as a label 15 1 FT 15 l FI i If the parameters describing an element are to differ in succeeding steps of a given problem the element must be included in both steps having the same label each time All elements which appear in a problem must be included in the first step indicator card 0 of that problem Only those to be changed in later steps need to be
79. aried 1 82 16 0 13 pole face curvature of bending magnet exit may be varied SEXTUPOLE 18 Ov The field strength may be varied The special parameter cards type code 16 0 once introduced apply to all subsequent magnets in a beam line until another type code 16 O specifying the same parameter is encountered Thus if such a parameter is varied the variation will apply simultaneously to all subsequent magnets to which it pertains The variation will persist until the parameter or vary code attached to the parameter is changed by the introduction of another type code 16 0 card specifying the same parameter Coupled vary codes It is possible to apply the same correction to each of several variables This may be done by replacing the digit 1 in the vary code with one of the digits 2 through 9 or a letter A through Z All such variables whose vary digits are the same regardless of position will receive the same correction For example the three type vary codes 5 0A 5 01 5 0A might represent a symmetric triplet The same correction will be made to the first and third quadrupoles guaranteeing that the triplet will remain symmetric If a vary digit is immediately preceded by a minus sign the computed correction will be subtracted from rather than added to this variable Thus parameters with the same vary digit one of them being preceded by a minus sign will be inversely coupled For example the type vary code
80. at SLAC In 1971 72 D Carey at FNAL completely rewrote the program and developed an efficient second order fitting routine using the coupling coefficient partial derivatives of multipole components to the optics as derived by Browns This version was implemented at SLAC by F Rothacker in the early spring of 1972 and subsequently carried to CERN in April 1972 by K L Brown C Iselin of CERN made further contributions to the program structure and improved the convergence capabilities of the first order fitting routines A standard version of the resulting program has now been adopted at SLAC FNAL and CERN This manual describes the use of this standard version and is not necessarily applicable to other versions of TRANSPORT Copies of this manual may be obtained from 1 Scientific Information Service CERN 1211 Geneva 23 Switzerland Ref CERN 80 04 2 The Reports Office Stanford Linear Accelerator Center P O Box 4349 Stanford CA 94305 USA Ref SLAC 91 without the Appendix 3 The Reports Office Fermi National Accelerator Laboratory P O Box 500 Batavia IL 60510 USA Ref NAL 91 TRANSPORT Appendix available under separate cover The program may be obtained from 1 IBM Version Frank Rothacker TRANSPORT Program Librarian Mail Bin 88 Stanford Linear Accelerator Center P O Box 4349 Stanford CA 94305 USA 2 IBM CDC or PDPIO Versions David C Carey Fermi National Accelerator Laboratory P O Box 500
81. atrices then equation 6 holds using the barred quantities Using the results derived above the matrices and B are now given by 100000 0 0 010 0 00 1000 10000 29 000010 0 0 0000 125 B345 1 1 5 B246 B316 1 11 mancano segni dei vettori su The first two indices for B correspond to the ray coordinates and the third corresponds to the misalignment parameters All other elements of B are zero In order to express the quantities m in terms of m the misalignment parameters at the pivot point we need two items First is the orthogonal matrix 0 giving the three translational coordinates at the magnet face in terms of those at the pivot point 12 Also needed is the three vector which gives the position of the origin of the aligned magnet face coordinate system in the coordinate system of the pivot We now define two three vectors which give the translational and rotational mg parts of the misalignment vector m We also do the same for m Then the contribution of m to m is given by equation 11 so that my 0 m 13 The contribution of mx to m is zero since parallel translations do not affect angles The displacement of a point due to a rotation about the pivot is given by the vector product of the rotation vector and the position vector of the point Therefore the displacements of the magnet face m due to a rotation at the pivot are given by
82. atrix 1 2 An element of the second order auxiliary transfer matrix T2 3 The net contributions of aberrations to a given coordinate of the beam matrix o 4 The strength of sextupoles used in the system The second order matrices are actually computed using the auxiliary matrix T2 Therefore when activating second order fitting one must not include any element which causes an update of the R2 matrix For a complete list of such elements see type code 6 0 The present value of the constrained quantity as well as the desired value is printed in the output In the case of transfer matrix elements this value may be checked by printing the transfer matrix itself Certain other constrained quantities may be checked similarly Exceptions are noted in the explanations following R1 matrix fitting constraints There are five parameters to be specified when imposing a constraint upon the i j element of an R1 matrix 1 Type code 10 n specifying that a fitting constraint follows Code digit i Code digit j Desired value of the i j matrix element Desired accuracy of fit standard deviation Note that any fitting constraint on an R1 matrix element is from the preceding update of the R1 matrix An R1 matrix is updated only by 6 0 1 35 entry The symbol n is normally zero or blank If n 1 then entry 4 is taken to be a lower limit on the matrix element If n 2 entry 4 is taken to be an upper limi
83. below contains two problem steps each beginning with title and indicator cards and terminating with a SENTINEL The first step causes TRANSPORT to do a first order calculation with fitting The second initiates a second order calculation with the data that is a result of that fitting Corresponding elements between the two steps are identified by having the same label type ten element which specifies the fitting condition is labeled 1 It is active for the first order calculation but is turned off for the second order calculation The vary codes for elements DR1 are set to zero for the second order problem The second order element 5 is ineffective during the fitting but causes the program to compute the second order matrices in the second calculation Example of a TRANSPORT Input Deck FORTRAN CHECK ON BETA FIT Title 0 Indicator card 1 5 1 5 1 5 11 17 SEC1 13 3 3 3 2 745 DpRI First problem 2 O 4 9 879 10 5 2 0 Elements arep 3 3 2 745 4 13 4 10 1 2 0 0001 SENTINEL SECOND ORDER Title card 1 Indicator 17 SECI Second problem 3 Elements to be changed 10 FITl SENTINEL SENTINEL Second sentinel signifies end of run As many problems and problem steps as one wishes may be stacked in one job Note that in previous versions of TRANSPORT a decimal point was required with every numerical entry except the indicator c
84. bout the new centroid we now have 49 XOX OM 543 1 Ray 2 ouo n Timna om IV Off Axis Initial Distribution Now consider a gaussian distribution whose center does not coincide with the beam axis Letting the coordinates of the centroid by we have for the coordinates of 200 We let the matrix represent the moments of the distribution about its centroid so that 25 131 COC Lj Oy 26 _ 0 Ep Equation 17 continues to hold for the moments of the distribution about the beam axis vhile equation 22 holds for the moments about the centroid We must therefore express one set of moments in terms of the other Using equations 22 25 and 26 and applying the first part of equation 24 to the initial distribution the initial third and fourth moments are given in terms of the initial first and second moments as follows A UT x 0400 00 X NE 1 CD Dr Xy Xi xj 2 Xk Lu D 0 00 CD ea DO Dx xDD 27 Substituting into equation 17 and rearranging terms we arrive at the following expressions for the first and second moments of the distribution at the final point 2 zx Y Rix ijk x0 28 2 2 H A 2 2 2 2 22 5 ORM 2 2 Dem R 0 Tike x 0 4 Rim n Da 0 Xk 09
85. cations and strengths of the sextupole components required may be selected via the coupling coefficients for the aberrations to be minimized 3 Calculate and make the necessary corrections via ray tracing to the third order aberrations by introducing octupole components into the system Note that an nth order multipole couples with terms of order n or higher but not with terms of order lower than n Thus an octupole component will not disturb the first and second order solutions already found from steps 1 and 2 4 Repeat the above procedure up to the multipole order desired or needed to achieve the design objectives If the design requires a solution to nth order and m multipoles at each order are necessary to minimize the aberrations the number of computer runs previously needed to complete a design was at least Having a knowledge of the coupling coefficients after the first order design has been selected now in principle reduces the number of computer runs required to n Since ray tracing is very time consuming this is indeed a significant saving The following results are applicable to static magnetic charged particle optical systems possessing median plane symmetry As in Ref 1 we shall use a right handed curvilinear coordinate system x y t where x and y are the trans verse coordinates x is the outward normal distance in the median plane away from the central trajectory y is the perpendicular distance from the m
86. charged particles in a beam line is often represented by a higher dimensional ellipsoid Given no further information one might interpret such an ellipsoid as an envelope inside of which particles are distributed uniformly or as giving the scale dimensions of a gaussian distribution The latter case has the advantage that is easily adapted to include higher order effects of the beam line In either case the parameters of the ellipsoid are simply related to the first and second moments and therefore the width of the distribution in any coordinate In first order an ellipsoid at any point in a beam line is transformed into another ellipsoid at any other location in a beam line In second and higher orders a transformation from one location in a beam line to another hill cause the ellipsoid to become distorted One can still however calculate the first and second moments of the distribution and thereby obtain a measure of its dimensions in any coordinate Below we elaborate on the methods for calculating the ellipsoid parameters at any point in the beam line Much of the first order theory can be found in the work of Brown and Howry It is included here for completeness II The Ellipsoid Formalism The position and motion of a particle in a beam line may be represented via a six dimensiona vector 1 DR coordinates x represent respectively the horizontal and vertical displacements at the position of the
87. d the field may be the aperture may not be AXIS SHIFT 7 vvvvvv Any of the axis shift parameters may be varied ALIGN 8 vvvvvvO Any of the alignment parameters may be varied INITIAL 16 0v Any of the three initial position floor coordi COORDINATES nates or two angle coordinates may be varied MATRIX 14 vvvvvvO Any of the first order matrix elements may be varied SOLENOID 19 vv The length and or field may be varied BEAM ROTATION 20 v The angle of rotation may be varied The use of the permissive may rather than the imperative will in discussing variables is meaningful The program will choose the parameters it will vary from among those that it may vary In general it chooses to vary those parameters that have the greatest influence upon the conditions to be fit Second 2 de1 vary codes In a second order run the following parameters may be varied DRIFT 3 v The drift length may be varied Variation of a drift length should be done with caution as it may affect the 52 first order properties of the beam line But inversely coupled drift spaces straddling a sextupole will for example show only second order effects cg te a os 16 0v 1 The normalized quadratic term sextupole component in the midplane expansion for the field of a bending magnet may be varied 95 23 16 0v 12 The pole face curvature of a bending magnet entrance may be v
88. d accuracy of the fit standard deviation If for example one wished to minimize the net contributions of second order aberrations to the horizontal divergence one would insert the following card 10 2 2 0 01 The quantity that is minimized is the net increase due to second order terms in the second moment of the beam about the origin This 59 quantity is treated as the chi squared of the problem so the only meaningful desired value for the fit is zero The square root of this quantity is printed in the output It is computed using the R matrix Therefore once again one must not include any element which updates the R matrix Centroid shifts must not be inserted when doing second order fitting even immediately following the beam card The second order image of the initial beam centroid at some later point in the beam is not necessarily the beam centroid at the later point The parameters printed by TRANSPORT are the new centroid position and the beam matrix about the new centroid One must therefore look at both of these to observe the effects of the fitting procedure It may even happen that an improvement in one parameter will be accompanied by a slight deterioration in the other The beam profile at any point is a function of the initial beam parameters One may therefore impose weights on the effect of the various aberrations by the choice of parameters on the BEAM card One might for example a
89. d particle beam transport systems SLAC Report No 91 1970 The present manual supersedes the above reference 6 K L Brown A systematic procedure for designing high resolving 81 power beam transport systems or charged particle spectrometers Proc 3rd Int Conf on Magnet Technology Hamburg Germany May 1970 348 or SLAC PUB 762 June 1970 7 Suggested Ray Tracing Programs to supplement TRANSPORT David C Carey TURTLE Trace Unlimited Rays Through Lumped Elements Fermilab Report No NAL 64 1971 This is a computer program using TRANSPORT notation and designed to be run using the same data cards as for a previous TRANSPORT run K L Brown and Ch Iselin DECAY TURTLE Trace Unlimited Rays Through Lumped Elements CERN Report 74 2 1974 This is an extension of TURTLE to include particle decay calculations H Enge and S Kowalski have developed a Ray Tracing program using essentially the same terminology as TRANSPORT Any experienced user of TRANSPORT should find it easy to adapt to the M I T program 8 K G Steffen High energy beam optics Interscience Monographs and Texts in Physics and Astronomy Vol 17 John Wiley and Sons New York 1965 SUGGESTED BIBLIOGRAPHY A P Banford The transport of charged particle beams E and F N Spon Ltd London 1966 K G Steffen High energy beam optics Interscience Monographs and Texts in Physics and Astronomy Vol 17 John Wiley and Sons NY 19
90. di t cx t fo se t h t dt 22 from which the first order momentum resolving power becomes dx t cx t 2XeR1 185 23 Equation 23 for the first order resolving power of a system may be expressed in a number of useful forms If we consider a ray particle originating at the source with O and 0 and lying the midplane i e mono energetic point source the first order equation representing the midplane displacement x of this trajectory is x t sx t 24 109 We may then rewrite Equation 23 as follows t 1 rt t x t Rs 25 or we may also write it in the form 2221 _ 1 box nt Ra 2X000 Jy B Bpo J BAA 28 2X080 where f BdA is the magnetic flux inclosed between the central trajectory and the ray represented by Eq 24 and Bp is the magnetic rigidity momentum of the central trajectory Please note however that if the ray crosses the central trajectory or the sign of B changes this changes the sign of the integration Some important observations may be made from Eq s 25 and 26 1 Resolving particles of different momentum requires that a path length difference must exist between the central trajectory and the trajectory defined by Eq 22 The greater the path length difference the greater the resolving power 2 From Eq 24 we may define resolving po
91. dinal extent is useful for pulsed beams It indicates the spread in length of particles in a pulse It does not interact with any other component and may be set to zero if the pulse length is not important The phase ellipse sigma matrix beam parameters may be printed as output after every physical element if activated by a 13 3 element Alternatively individual printouts may be activated by a 13 1 element The projection of the semi axes of the ellipsoid upon each of its six coordinates axes is printed in a vertical array and the correlations among these components indicating the phase ellipse orientations are printed in a triangular array see the following pages The phase ellipse beam matrix The beam matrix carried in the computer has the following construction x A 6 x o 11 21 22 31 32 33 41 42 43 44 o 51 52 53 54 o 55 6 61 62 63 64 o 65 66 The matrix is symmetric so that only a triangle of elements is needed In the printed output this matrix has a somewhat different format for ease of interpretation x A x vo 11 Vo 22 r 21 0 33 CM 31 32 44 MR 41 42 43 55 51 52 53 54 66 PC r 61 r 62 r 63 r 64 r 65 P o ij where r ij Gi T 26 As a result of the
92. dividual matrices of the system elements TRANSPORT calculates and tabulates the product matrix R representing the system The zero elements Ru Roa Rar Ray Rag Rag in the R matrix are a direct consequence of midplane symmetry If midplane symmetry is destroyed these elements will in general become non zero The zero elements in column five occur because the variables x y and 6 are independent of the path length difference 2 The zero s in row six result from the fact that we have restricted the problem to static magnetic fields i e the scalar momentum is a constant of the motion In SLAC report 75 Ref 1 a physical significance has been attached to the non zero matrix elements in the first four rows in terms of their identification with characteristic first order trajectories We include figures showing these characteristic functions as a convenient reference We now wish to relate the elements appearing in column six and those in row five in terms of simple integrals of the characteristic first order matrix elements c t R4 and s t Ry In order to do this we make use of the Green s function integral Eq 43 Section II of SLAC 75 and of the expression for the differential path length in curvilinear coordinates 40 5 NI CASN WSISAS ALVNIGHOOO 014 v 3 u TWHLNSO
93. djust the strength of the correction of the chromatic aberrations by the choice of the Ap p parameter In particular when using a BEAM constraint one should not attempt to minimize or eliminate chromatic aberrations if Ap p is set equal to zero on the beam card type code 1 0 Correlations the 12 card may also be included in the initial beam specification Sextupole strength constraints Five parameters must be specified for a constraint on sextupole strength 1 Type code 10 0 Code digit 18 Code digit 0 The number Desired maximum sextupole field strength single sextupole constraint card applies to all sextupoles which follow The maximum field strength is treated as a standard deviation and may be exceeded on an optimal fit One can employ this constraint to find the optimal locations for sextupoles By placing inversely coupled drift lengths before and after the sextupole its longitudinal position may be varied By constraining the field strength the sextupole can be slid to a position where the coupling coefficients to the aberrations will be largest One will need to experiment with adjusting the maximum field strength to achieve the best configuration Internal constraints A set of upper and lower bounds on the value of each type of parameter is in the memory of the program If a correction is computed for a parameter which would take its value outside this range it is reset to the limit of the range Th
94. e 91 function has been adopted for describing the general case where pe pe t is function of By analogy with previous discussions we observe that whenever s t we are at an image of point A Also at the position where s t O is the magnification of point A at that image c t 1 f where f is the focal length of the system between A and B The dispersion d may be derived from the general differential equations of motion of a charged particle in a static magnetic field 1 The results may always be expressed as a function of s and c as follows dx t st cx t f sy x da and t t d X t s x t c t f s t da where at e is the differential angle of bend of the central trajectory At an image point s t note that d t c t f s x da This approach to the problem may be generalized to include all of the second order aberrations of a system When this is done it is always possible to express these aberrations as functions of the first order matrix elements C Sx d and s Having developed the above physical concepts and mathematical tools we are now in a position to study more complicated systems As an example we consider the general system shown in Figure 14 Li L3 La A B M Figure 14 L drift elements magnetic elements _ The matrix formalism states that in the x plane the transformation from A to
95. e APPENDIX Introduction This appendix has been included as an addition to the manual in an attempt to better acquaint the user with what TRANSPORT does and with the notation and mathematical formalism used in a TRANSPORT calculation The first section Beam Transport Optics Part I and Part II is a rewrite of two lectures given to members of the SLAC technical staff on the elementary matrix algebra of optics We include them here for the benefit of the new user who may need a brief refresher course on charged particle optics and or has a need to become familiar with TRANSPORT notation The new user should also acquaint himself with the contents of the books and other publications listed under references at the end of the manual References 1 and 2 essential if the user is to obtain the maximum value from TRANSPORT The second section of this appendix was written to introduce the mathematical formalism of the first order R matrix and Sigma matrix phase ellipsoid beam optics used in a TRANSPORT calculation and to correlate this with the printed output Section three discusses second order calculations and in particular a procedure for calculating the Sextupole strengths required to minimize and or eliminate second order aberrations in a beam transport system Section four is a brief derivation of the mathematical formalism used by TRANSPORT for calculating magnet alignment tolerances Section five deals with the first ord
96. e is referred to the direction of the beam where it enters the magnet The units employed are the standard TRANSPORT units shown above unless redefined by type code 15 entries If the units are changed the units of the misalignment displacements are those determined by the 15 1 type code entry the units for the misalignment rotations are those determined by the 15 2 type code entry The misalignment of any physical element or section of a beam line may be simulated Misaligned sections of a beam line may be nested A beam line rotation type code 20 may be included in a misaligned section Thus for example one can simulate the misalignment of magnets that bend vertically The arbitrary matrix type code 14 may not be included in a misaligned section A misalignment must never be included in a second order run type code 17 A misalignment element may indicate that a single magnet or section of the beam line is to be misaligned or it may indicate that all subsequent magnets of a given type quadrupoles and or bending magnets are to be misaligned The type of misalignment is specified in the three digit code number and the location of the type code 8 align element depends on the type of misalignment If a misalignment pertains to a single magnet or a single section of the beam line then the misalignment element type code 8 must directly follow that magnet or section of the beam line If a misalignment element indicat
97. e system has been adjusted for the smallest spot size at a fixed position and if the size of the beam at the principal planes of the optical system is large compared to its size at the waist or at the minimum spot size then the location of these quantities the waist and the minimum will closely coincide if on the other hand the size of the beam does not change substantially throughout the system then the locations of a waist and the minimum beam size may and usually 116 will differ substantially The numerical proximity of these two quantities will be discussed in greater detail later in the report In a field free region i e a Drift the distance to a waist from any location may be readily calculated if the o matrix at the location is known Using Eq 36 and the matrix for a Drift Eq 38 we have for the x 0 plane 021 1 04 0 05 0 0 specifying that o 1 shall be at a waist FIG 9 BEAM WAIST DISTANCE ALONG BEAM OL 614 27268 117 139 1 1 1095 31815509 LSATIVWS 1V LSIVM WHOS OL HI9N3HIS 3O0dnuavno O WV 38 Tz SER 55556 suaj 139391 LSIVM 01 Q3LSNCAV HLON3YLS 3 0dnugvno 46 47 011 022 033 044 021 0 022 0 0 044 0 L Similarly for the
98. e current limits are 60 code Limits 1 0 9 lt input beam 2 0 60 lt pole face rotation 60 deg 3 0 0 drift 4 0 0 magnet length 5 0 0 quad length 20 0 360 beam rotation 360 deg These limits apply only when a parameter is being varied Fixed values that exceed this range may be used as desired These constraints were included to avoid physically meaningless solutions Corrections and covariance matrix When the program is fitting it makes a series of runs through the beam line From each run it calculates the chi squared and the corrections to be made to the varied parameters For each iteration a single line is printed containing these quantities The program calculates the corrections to be made using a matrix inversion procedure However because some problems are difficult it proceeds with caution The corrections actually made are sometimes a fixed fraction of those calculated This fraction used as a scaling factor is the first item appearing on the line of printed output The second factor is the chi squared before the calculated corrections are made Following are the corrections to be made to the varied parameters They are in the order in which they appear in the beam line If several parameters are coupled they are considered as one and their position is determined by the first to appear When convergence has occurred the final value of the chi squared and the covari
99. e phase space area and of the lens spacing Example No 2 If the unit cell is a FODO array as follows Beom Envelope f f f 1510429 matrix for the unit cell from the principal planes of the first lens to the principal planes of the third lens is 78 If we now impose the symmetry requirements that erect ellipses occur at the principal planes of each lens and that the beam size o 1 at lens 3 be a minimum and equal to the beam size o 0 at lens 1 then it follows that y 011 1 y 011 0 4 1 236 71 011 0 _ _ 3 5 Gun gt 4 23 123 or _ 041 0 25 TAW 2 055 72 and finaly 0 3003 4140 73 022 0 where 01 and 0 0 are measured at the principal planes of the first lens in the FODO array For a FODO quadrupole array where the field strength is held constant for all elements rather than the focal lengths the results are somewhat different than those above This case may be readily calculated via TRANSPORT using the above results as initial guesses in the calculation Relationship between a First Order Point to Point Image and the Minimum Spot Size Achievable at_a Fixed Target Position This problem is not as easy to explore as were the preceding ones because the question arises the first order image of what If however we restrict the discussion to a thick or thin lens system that does not have intermediate ima
100. e type code signifying a BEND is 4 The input format for a TRANSPORT calculation is to exceed 4 spaces 4 L B E H If n is not included in the data entry the program assumes it to be zero entry for a second order calculation is made via the 16 0 1 0 element Do not confuse this with a pole face rotation The standard units for L and B are metres and kG If desired these units may be changed by 15 0 8 0 and 15 0 9 0 type code entries preceding the BEAM Card 37 MSHLO HOVE OL lO3dS3H HIIM 111504 WAV LVHL 61 ATOdILINW ALVOIGNI SHILIYV10d LANOVW FHL SIN34313 3TOdNLXAS ANY SIXV X 3NVIddIW OILANOWW AHL 40 NOTIVYLSATTI 3 Oodnix3s 31Odnugvno 31049 QUADRUPOLE Type code 5 48 A quadrupole provides focusing in one transverse plane and defocusing in the other There are four parameters to be specified for a TRANSPORT calculation 1 Type code 5 0 specifying a quadrupole 2 Effective magnet length L in metres 3 Field at pole tip B in kG A positive field implies horizontal focusing a negative field vertical focusing 4 Half aperture a in cm Radius of the circle tangent to the pole tips The length and field of a quadrupole may be varied in first order fitting The aperture may not be The strength of the quadrupole is computed from its field aperture and length The horizontal focal length is print
101. e unit cell structure and the corresponding beta functions are both periodic Floor coordinate fitting constraint Five parameters are needed to specify a floor coordinate constraint Type code 10 Code digit 8 Code digit 1 Desired value of floor coordinate Desired accuracy of fit standard deviation The code digit j indicates the floor coordinate to be constrained Its possible values are 1 to 6 indicating the floor x y z theta phi and psi respectively Theta is the angle which the floor projection of the reference trajectory makes with the floor z axis Phi is the vertical pitch Psi is a rotation about the reference trajectory This is also the order in which coordinates are printed in the floor layout activated by the 13 12 element Initial coordinates are given on type codes 16 16 through 16 2 and type code 20 The floor coordinates are actually zero th rather than first order properties of a beam line However in TRANSPORT they may be constrained in a first order fitting run and therefore are included here T1 matrix fitting constraints Five parameters are needed for a constraint on the i j k element of the second order transfer matrix T1 1 Type code 10 0 2 Code digit i 3 Code digit 10j 4 Desired value of the i j k matrix element 5 Desired accuracy of the fit standard deviation Note that upper and lower limit constraints are not available for second o
102. ector single column matrix X whose components are the positions angles and momentum of the particle with respect to a specified reference trajectory X y i e x n 6 where the radial displacement of the arbitrary ray with respect to the assumed central trajectory the angle this ray makes in the radial plane with respect to the assumed central trajectory y the transverse displacement of the ray with respect to the assumed central trajectory the transverse angle of the ray with respect to the assumed central trajectory the path length difference between the arbitrary ray and the central trajectory 6 AP P is the fractional momentum deviation of the ray from the assumed central trajectory The magnetic lens is represented by the square matrix R which describes the action of the magnet on the particle coordinates Thus the passage of a charged particle through the system may be represented by the matrix equation X 1 3 where X O is the initial coordinate vector and X 1 is the final coordinate vector of the particle under consideration R is the transformation matrix for all 101 such particles traversing the system one particle differing from another only by its initial coordinate vector X 0 The traversing of several magnets and interspersing drift spaces is described by the same basic equation but with R now being the product matrix R R n R 3 R 2 R 1 of the in
103. ed 2 p units for are degrees Pole face rotation matrix The first order R matrix for a pole face rotation used in a TRANSPORT calculation is as follows 1 0 0 0 0 0 tank i 0 000 Po T ME 1 o o 1 0 Po 0 0 0 0 00 1 Definitions 29 B angle of rotation of pole face see figure on following page for sign convention of B bending radius of central trajectory g total gap of magnet correction term resulting from spatial extent of fringing fields where Ki 4 1 KiK 2 tan cosp See type code 16 0 for input formats for Ki and TRANSPORT entries See SLAC 75 page 74 for a discussion of 30 gt G 49 TO zs x O E x N f 7 E C didascalia FIELD BOUNDARIES FOR BENDING MAGNETS 31 The TRANSPORT sign conventions for X B R and h are all positive as shown in the figure The positive y direction is out of the paper Positive B s imply transverse focusing Positive R s convex curvatures represent negative sextupole components of strength h 2R sec See SLAC 75 page 71 DRIFT Type code 3 0 A drift space is a field free region through which the beam passes There are two parameters 1 Type code 3 0 specifying a drift length 2 Effective drift length metres The length of a drift space may be varied in
104. ed 2 when a beam centroid shift type code 7 0 is used or 3 when a second order calculation type code 17 0 is used To aid in the interpretation of the phase ellipse parameters listed above an example of an x 0 plane ellipse is illustrated below For further details the reader should refer to the Appendix of this report 1 O22 Y slope 2 2 slope 722 wim 922 Sing x CENTROID TWO DIMENSIONAL BEAM PHASE ELLIPSE The area of the ellipse is given by A Tl det TUXmax 0 Xint TUXint Onax The equation of the ellipse is 2 9 pe Where 011 021 022 y 2 21 By 12 Tage By r m s addition to the BEAM To allow for physical phenomena such as multiple scattering provision has been made in the program to permit an r m s addition to the beam envelope There are nine entries to be included 1 Type code 1 0 specifying a BEAM entry follows 2 The r m s addition to the horizontal beam extent Ax cm 3 The r m S addition to the horizontal beam divergence mr 4 The r m s addition to the vertical beam extent Ay cm 5 The r m s addition to the vertical beam divergence mr 6 The r m s longitudinal beam extent AA cm 7 The r m s momentum spread 5 in
105. ed in parentheses as output A positive focal length indicates horizontal focusing and a negative focal length indicates horizontal defocusing The quantity actually printed is the reciprocal of the 0 x transfer matrix element 1 1 for the quadrupoles Thus two identical quadrupoles of opposite polarity will have different horizontal focal lengths due to the difference between the sine and the hyperbolic sine The type code for a QUAD is 5 0 The input format for a typical data set is Label if desired not to exceed 4 spaces between quotes 5 L B a x H The standard TRANSPORT units for L B and a are metres kG and cm respectively If other units are desired they must be chosen via the appropriate 15 0 type code entries preceding the BEAM type code 1 0 card First order quadrupole matrix cos KaL 0 0 0 0 kgsinkgL cos 0 0 0 0 0 0 cosh y Sinh 0 0 0 0 kgsinhk L 0 0 0 0 0 0 1 0 0 0 0 0 0 1 These elements are for a quadrupole which focuses in the horizontal x plane B positive A vertically y plane focusing quadrupole B negative has the first two diagonal submatrices interchanged Definitions L the effective length of the quadrupole a the radius of the aperture the field at radius 1 where the magnetic rigidity momentum of the central trajectory 39 1 HOVE OL 10 4 H
106. ed it are shown below When the user specifies that the actual position of the magnet s is uncertain within a given tolerance the printout will show a change in the beam sigma matrix resulting from the effects of the misalignment s CEN Thus if one wishes to determine the uncertainty in the beam centroid resulting from uncertainties in the positioning of the magnets the initial beam dimensions should be set to zero i e the beam card entry at the beginning of the system should appear as follows 1 0 0 0 O p 0 If it is desired to know the effect of an uncertainty in position on the beam focusing characteristics then a non zero initial phase space must be specified The printout will then show the envelope of all pos sible rays including both the original beam and the effects of the misalignment If the misalignment is a known amount it may affect the beam centroid as well as the beam dimensions Therefore one should place on the BEAM card the actual dimensions of the beam entering the system For a known misalignment the program requires that the initial beam specified by type code 1 must be given a non zero phase volume to insure a correct printout An align element pertaining to a single magnet or section of the beam line updates the BEAM sigma matrix and the R2 matrix but not the R1 matrix A misalignment element which indicates misalignment of all subsequent magnets of a given type will update the BEAM
107. edian plane t is the distance along the central trajectory and is the curvature of the central trajectory The existence of the median plane requires that the scalar potential be an odd function of y i e 9 The most general form of may therefore be expressed as follows x 2 1 t 0 A2m 1 n 1 n 2m 1 where the coefficients A2m 1n are functions of t In this coordinate system the differential line element dT is given by dT dy 14hx dt The Laplace equation has the form x Oto gi Lan 2 2 V E mn m x B 134 Substitution of 1 into 2 gives the following recursion formula for the where prime means 3 and where it is understood that all coefficients 2m43 n nhAjnsin 1 nh A 2m 1 n 1 2m 1n4 2 3 3n 1 hAgm 1n41 n 3n 1 h Am etn n n E 1 5 A m1n 1 3nhAgm43n 1 3n n 1 h Agms3n 2 n n 1 n 2 Az ia n 3 where prime means and where it is understood that all coefficients A with one or more negative subscripts are zero This recursion formula expresses all the coefficients in terms of the midplane field B x 0 where Ain functions of t 4 0 x Since is odd function of y on the median plane we have B O normal in x direction derivatives on the reference curve defines
108. ement Only one parameter should be specified 1 Type coder 17 0 signifying a second order calculation is to be made To print out the second order T1 matrix terms at a given location in the system the 13 4 5 print control card is used For T2 the 13 24 print control card is used The update rules are the same as those for the corresponding first order matrix See SLAC 75 for definitions of subscripts in the second order T ijk matrix elements The values of the BEAM sigma matrix components may be perturbed from their first order value by the second order aberrations In a second order TRANSPORT calculation the initial beam is assumed to have a Gaussian distribution For exact details the reader should consult the Appendix For the beam matrix to be calculated correctly there should be no elements which update the R2 matrix If a centroid shift is present it must immediately follow the beam type code 1 0 or beam rotated ellipse type code 12 0 card Only second order fitting may be done in a second order run See the section on type code 10 0 for a list of quantities that may be constrained in a second order run If a beam constraint is to be imposed in second order there must be no centroid shifts present anywhere Second order matrices are included in the program for quadrupoles bending magnets including fringing fields the arbitrary matrix sextupoles and solenoids They have not been calculated for
109. en requesting a beam line coordinate layout via a 13 12 element one can employ any coordinate system one desires The position and direction of the beginning of the reference trajectory in this coordinate system are given on elements 16 16 through 16 20 Such cards should be placed before the beam card but after any units changes Their meanings are as follows 16 16 Xe ye and ze respectively the coordinates of 16 17 the initial point of the reference trajectory 16 18 in the units chosen for longitudinal length 16 19 and the initial horizontal and vertical angles 16 20 of the reference trajectory in degrees When specifying the initial orientation of the reference trajectory via the two angles one must give the horizontal angle first The meaning of the two angles is given in the following figure Any of the above five parameters not explicitly specified will be taken to equal zero The initial coordinates may be varied in first order fitting Their values will affect only the beam line floor coordinates and not any beam or transfer matrix element REFERENCE TRAJECTORY X SPECIFICATION OF INITIAL ANGLES AND FOR BEAM LINE LAYOUT SECOND ORDER CALCULATION Type Code 17 0 A second order calculation may be obtained provided no alignments are employed A special element instructs the program to calculate the second order matrix elements It must be inserted immediately following the beam 1 el
110. ence All three methods have been used with pure multipoles dominating the situation for higher energy physics and the other two methods dominating medium and low energy physics applications All three techniques should be considered in any given design situation to be certain that an important economic or practical advantage has not been ignored REFERENCES 1 A First and Second Order Matrix Theory for the Design of Beam Transport Systems and Charged Particle Spectrometers SLAC Report No 75 by K L Brown or Advances in Particle Physics Interscience 1 71 134 1967 2 A General First and Second Order Theory of Beam Transport Optics Proc of the International Symposium on Magnet Technology 1965 p 141 3 Some First and Second Order Magnetic Optics Theorems Applicable to the Design of Beam Transport Systems and Charged Particle Spectrometers Proc of the International Symposium on Magnet Technology 1967 p 40 4 E D Courant and H S Snyder Theory of the Alternating Gradient Synchrotron Annals of Physics 3 pp 1 48 1958 5 K L Brown with Belbeoch and P Bounin First and Second Order Magnetic Optics Matrix Equations for the Midplane of Unifom Field Wedge Magnets Rev Sci Instr 35 481 1964 6 L Brown B Kear S Howry TRANSPORT 360 A Computer Program for Designing Beam Transport Systems SLAC Report No 91 Stanford Linear Accelerator Center Stanford University Stanford Califo
111. er parameter optimization code of TRANSPORT and includes a brief explanation of the covariance matrix that is printed after each first order fit routine BEAM TRANSPORT OPTICS Section I Beam Transport Optics Part I K L Brown 1 Introduction A convenient starting point for this lecture is the equation relating the magnetic rigidity of a particle Bp to the particle momentum P 102 2 99793 33 356 where B is in kilogauss is the bending radius in meters is the particle s momentum in BeV c A note of caution When using this equation for a TRANSPORT calculation it is necessary to use at least 5 significant figures for the constant to avoid round off errors in the readout 2 Geometric Light Optics vs Magnetic Optics To relate geometrical light optics to charged particle optics we begin with the thin lens Figure 1 shows a thin lens with a ray leaving a focal point at an angle 0 impinging on the lens at x As the ray leaves the lens it is at x and going toward a focal point B at an angle of 01 Figure 1 Thin lens optics says that 1 p 1 q 1 f Using this equation it is readily verified that the matrix transformation for the lens action between principal planes is x x ET nella parentesi centrale manca croce divisoria The transformation for a drift distance L is lo I al fel 85 Note that the determinant of the matrix in both example
112. es of the various misalignments as expressed in the coordinate system of the aligned magnet face Then we will express the misalignment of the magnet face in terms of the misalignment parameters about the pivot point A rigid translation of the magnet face will change the X y and z coordinates of a ray by the amount of the displacement The z translation will also introduce a short drift distance 146 positive or negative length at the magnet face and will contribute to B via the transformation matrix of that drift space To determine the effect of a rotational misalignment we form from the ray angles x dx dz and y dy dz and the number 1 dz dz a three vector x y 1 giving the ray direction We let 0 be the three rotational components of the misalignment vector Then including only first order effects this three vector is transformed as 16 0 ry 6 1 6 8 1 0 06 1 1 y mancano f in fondo prima parentesi e segni dei vettori su In the misaligned coordinate system the ray angles become 0 9 mancano segni dei vettori su Thus coordinate rotations about the aligned magnet face x and y axes only shift the ray angles A rotation about the z axis mixes x and y If we let m represent the misalignment parameters relative to the aligned magnet face coordinate system and A and B be the corresponding m
113. es that all subsequent magnets of a given type are to be misaligned it must precede the first of such magnets Further description of the available types of misalignment is given in the table below The results of the misalignment may be displayed in either the beam sigma matrix or in a misalignment table If the results are displayed in the beam sigma matrix then that matrix is altered by the effects of the misalignment The effects of additional misalignments cause further alterations so that at any point along the beam line the beam sigma matrix will contain the combined effects of all previous misalignments The misalignment table can be used to show independently the effects on the beam matrix of a misalignment in each degree of freedom of each misaligned magnet Each new misalignment to be entered in the table creates a new set of six duplicates of the beam matrix Printed for each duplicate beam matrix are the centroid displacement and the beam half width in each of the six beam coordinates Each of the six matrices shows the combined result of the undisturbed beam matrix and the effect of the misalignment in a single coordinate of a single magnet or section of the beam line In a single TRANSPORT run the results of misaligning up to ten magnets or sections of the beam line may be included in the misalign ment table Further requests for entry in the misalignment table will be ignored Examples of such a table and the input which generat
114. fact that the o matrix is positive definite the r ij satisfy the relation lt 1 The full significance of the o ij and the r ij are discussed in detail in the Appendix Description of Beam Matrix The units are always printed with the matrix In brief the meaning of the 11 is as follows 11 Xmax the maximum half width of the beam envelope in the X bend plane at the point of the print out Vo 22 Omax the maximum half angular divergence of the beam envelope in the x bend plane V0 33 Ymax the maximum half height of the beam envelope V0 44 the maximum half angular divergence of the beam envelope in the y non bend plane 55 max one half the longitudinal extent of the bunch of particles 66 6 the half width 1 2 Ap p of the momentum interval being transmitted by the system The units appearing next to the 11 in the TRANSPORT print out are the units chosen for coordinates x A and Ap p respectively To the immediate left of the listing of the beam envelope size in a TRANSPORT print out there appears a column of numbers whose values will normally be zero These numbers are the coordinates of the centroid of the beam phase ellipse with respect to the initially assumed central trajectory of the system They may become non zero under one of three circumstances 1 when the misalignment type code 8 0 is us
115. function of x and y and differentiating we have OF _ Bo pun _ From which Bo kn an Where L is the length of the multipole element For a dipole and the dipole strength is and So For 254 angle of bend of the central trajectory a quadrupole and zz 51 2 For sextupole and 5 2 5 etc for higher order multipoles Multipole Strengths for a Non Uniform Field Expansion From the midplane field expansion of a non uniform magnetic field and evaluated over the length L of the central trajectory is By x 0 t By o 0 t 1 B hx y hxP 12 Ko h nh Bh etc and S evaluated over the length L of the central trajectory is Se a as before 5 1 and S BhL etc 136 Multipole Strengths for Contoured Entrance or Exit Boundary of a Magnet A third method of introducing multipole components is via a curved entrance or exit boundary of a magnet To calculate the multipole strengths in this case we integrate equation 7 holding x constant as follows p B x 0 t dt t dt Bp Spx 13 To relate this to the field boundary we assume B to be a constant inside the effective field boundary and zero outside i e we ignore the finite extent of the fringing field In this sharps cutoff approximation the field boundary Z Z x is _ 1 pl at _ 52 2 7
116. g the bending radius equal to unity To restore ordinary units it is only necessary to insert the bending radius wherever a length is needed dimensionally In this case the matrix then becomes c ps 1 s p S 4 0 0 1 manca quadrettatura If the distance from the entrance plane to the first principal plane is 2 and the distance from the second principal plane to the exit plane is z we can find the values 2 2 by solving the following matrix equation 1 0 s 1 c 1 0 1 0 d 0 1 0 5 c 0 1 O 1 f 1 5 0 0 1110 0 1 0 0 1 0 0 1 The matrix multiplication need only be done for the 2 X 2 matrices as outlined To illustrate matrix multiplication the indicated operations will be given below in natural stages as follows 7 manca quadrettatura CT 725 CZA 1 0 5247 CZ 6 57172 2 ED 1 6 s 571 c manca quadrettatura Note that these transformations do not change the focal length expression 1 f s In order for two matrices to be equal each individual element must be equal to its counterpart in the other matrix Thus we have 255 1 215 1 7 which when solved for 2 and 2 yeld 22 1 c s and z 1 c s 97 If we substitute the trigonometric equivalents and apply standard identities we have 7 z tan a 2 which can be seen from Figure 3 to indicate that the two principal planes are coincident with the symmetr
117. ges between the source and the image under consideration then the following comments are applicable The ratio of the minimum beam size to the size of a first order image at a fixed target position may be calculated using Eq s 56 and 59 From Eq 59 we have 2 min eyes and from Eq 56 the size of a first order image at hue position is image M o obiect 2 obiect where is the magnification of the first order image is the object distance measured to the principal planes and L is the distance to the target measured from the principal planes The ratio of sizes is 11 74 011 lst order image 011 object o lens Using Eq 36 we may write 034 lens oq object 2 02 object p o5 object 75 and since object o22 object a object it follows that the first order image will coincide with the smallest spot size only if the orientation of the initial beam ellipse at the object is such that oa object o4 object 76 or if oq object O 1 e for a point source For an erect ellipse at the source and the lens adjusted to provide a minimum spot size at the target it can be shown that the first order image will 124 always follow the target position the minimum spot size by a distance gii object IMI 7 1 77 011 1 15 x Lens where L is the distance to the target position from
118. gn of B changes this changes the sign of the integration From equation 32 we may define resolving power as the magnetic flux enclosed per unit phase space area 2 per unit momentum Bp of the central ray In any given design one or more of the ebove equations may be used as a guide toward achieving the required resolving power One of the design decisions that must be made is the appropriate choice of the dipole magnet parameters width and length to achieve the required f BdA From first order considerations this choice is dominated primarily by practical and economic factors However a study of the nature of the origin of aberrations see for example Ref 1 suggests that it is advisable to keep the amplitude of s small In order to simultaneously satisfy this requirement and meet the required resolving power R we see from equation 29 that the total angle of bend a of the central trajectory should be chosen as large as is practical Also in general the focal plane angle tends to be more normal to the optic axis for larger a a property usually desired in most designs 2 Dispersion From Reference 1 2 or 3 for point to point imaging 5 the dispersion at the image plane is d t c t s 33 where c t is the magnification at the image plane The dispersion and hence the magnification in the design of a spectrometer is dominated almost entirely by a compromise between the spatial resolution
119. he 6 0 2 entry The last 8 card refers to the misalignment of the whole triplet as a single unit via the R1 matrix update the 6 0 1 entry The comments about the BEAM card type code 1 entry in example 2 above are applicable here also Example N 4 Misaligned quadrupoles in a triplet pag 62 Individual uncertainties in the positions of the quadrupoles in the triplet in example no 3 above may be induced by a single misalignment as follows 8 015 5 1 8 10 5 2 7 10 5 1 8 10 The effect of each misalignment coordinate on each quadrupole will be stored separately in a table This table is printed wherever a 13 8 type code is inserted Mancano tabelle pag 63 64 65 66 del manuale 47 24nd1j 43 Ut UMOYS ST 40132104351 juiid paqestput Aq poonposd ndino 0 013404 oup ur uo 03 so odnipgnb uoi dn si nsoi pue 5 Aq iuounr esru uaar 03 3uonbasqns 1241 sorgidods juouo o pestu oup out jo 3164 Ayava juouudrt estu 51 uwous aq 03 so odnipenb 06000 00000 1 00000 0 0000 00918 0000 100000 00000 91 9 qei 8
120. he effect of all misalignments may then be added into the sigma matrix and thereby be traced through the system Alternatively the effects of separate components of the misalignment vector on individual magnets may be stored in a table This table is traced through the beam line and may be compared with the unperturbed sigma matrix at any later point Details of implementation are described in the TRANSPORT manual Figure Captions Figure 1 Perfectly aligned and misaligned bending magnets With the misaligned magnet the beam line reference trajectory is no longer continuous with that inside the magnet The displacements of the origins of the entrance and exit face reference coordinate systems are shown as De and D respectively Figure 2 Magnet entrance and exit face coordinate systems The misalignment causes both a translation and a rotation of the reference coordinate system eaznbtz NI 100 93 1 30N3U333U 30N3H343M g Q3N9l 1VSIN un NI 1 0 AHOLO3 VH l AHO LO3PhVH L 30N38343H 30N34343H Q3N9I 1V A112343H3d 151 1 39N38343H W3lSAS 3JVNIQHOOO 3993 LIXS LANSVW GANSIIVSIN Q3N9l1V eanbry W3lSAS 3l1VNIQHOOO 3993 SONVYLNS 1 NI J39N3Y343Y Q3N9l 1V Q3N9l IVSIN 152 FIRST ORDER PARAMETER OPTIMIZATION AND COVARIANCE Section V To optimize the selected parameters TRANSPORT u
121. he optical mode and dispersion of the system are determined to a great extent by the choice of the quadrupole components chosen to achieve the first order imaging although it is clear that the dipole elements also influence the first order imaging to a greater or lesser extent depending upon the total angle of bend of the system 4 Aberrations and their Correction A study of the source of second and higher order aberrations see for example Ref 1 suggests that it is advisable to maintain the characteristic first order functions Sx d and cy sy and their derivatives as small as is feasible through the magnetic elements of a system when choosing the first order design This procedure will tend to reduce the initial size of the aberrations and hence simplify the problem of minimizing them by the addition of multipole components to the system design The procedure for minimizing aberrations has already been outlined in the Introduction and as such will not be repeated here The key to the minimization procedure is the coupling coefficient given by the integral expression in equation 19 The best location for the correcting multipole is where the coupling coefficient has its maximum value The preferred method of introducing the multipole components i e via pure multipoles contoured entrance or exit boundaries or non uniform fields is a combination of practical and economical considerations and of course personal taste and experi
122. he systematics of the general first and second order theory of beam transport optics 1 2 3 to include higher order multipole terms it has been possible to evolve a simple step by step procedure for the design of high resolving power static magnetic beam transport systems The choice of the appropriate dipole and quadrupole elements for a given system may be determined once the resolving power solid angle momentum range and detector system of the instrument have been specified The partial derivative of any nth order aberration coefficient with respect to an nth order multipole component located anywhere in the system has been derived From this coupling coefficient the strength and the optimum location of multipole element s to correct or modify a given aberration or group of aberrations is uniquely determined I Introduction Within the last two decades significant advances have been made in the understanding of charged particle optics Perhaps the first major contribution was the development of the theory of the Alternating Gradient Synchrotron A G S by Courant Livingston and Snyder 4 which led to the first order matrix algebra formulation of beam transport optics Subsequent to this a second order matrix algebra was developed by Brown Belbeoch and Bounin 5 followed by the development at SLAC of the digital computer program called TRANSPORT 6 that is widely used today in many laboratories for solving first and second order s
123. ibution it is now an easy matter to calculate the second moments in the new frame since the coordipates are decoupled We arrive at 6 The second moments in the old frame are now Vee Vix Une Vin Ojpexexp Une 0 00 ke Gij 7 Therefore in this case the elements of the matrix o give the second moments of the distribution in the original coordinate system The density function properly normalized now becomes No P Guy whore is the total number of particles Since the matrix 0 is orthogonal the determinants of o and o are equal exp 1 8 The elements of the matrix be put in more convenient form for interpretation The square roots of the diagonal elements may be taken as giving the half widths x of the distribution in a given coordinate while the off diagonal elements may be related to the correlations rij so Xoi Tij 9 0 jj 9 Since for any positive definite symmetric matrix o we have Z2 ll O iiO jj gt 0 10 the correlations must all obey the inequality 1 11 If the ellipsoid is interpreted as describing the envelope of a uniform distribution then the x represent the maximum extents of the beam in the given coordinates III The Effect of a Beam Line A First Order 129 If we now let a be the coordinates of a ray at the initial point ina beam line and gr the coordinates at some later
124. ication is approximately equal in both planes consequently a circular spot can be imaged through the system with much less first order image distortion than is the case for the doublet 3 Introduction of Momentum Dispersion into the Matrix Formalism The foregoing discussion and examples dealt only with monoenergetic first order effects First order dispersion may be taken into account by introducing a 3 X 3 matrix as follows Consider two particles of momentum and Ap passing through the midplane of a static magnetic field as illustrated in Figure 11 Figure 11 Figure 11 Since the scalar momentum of a particle is constant in a static magnetic field the transport equation from A to B may be expressed as 41 le Ro le manca quadrettatura nella parte centrale 0 0 1 Where Ap pe d the spatial momentum dispersion d the derivative of the dispersion the angular momentum dispersion and 1 a carrying tern to generate a square matrix and denote a constant momentum The determinant of the matrix R is equal to 1 as for the 2 x 2 matrix However because of the zeros in the bottom row the fact a aa Rit R22 R12 R21 1 only checks the 2 x 2 matrix and not the terms containing d Consider now a general system from an object point A to an image point i for all 8 The above matrix equation is still valid for midplane trajectories If
125. ice life is not quite so simple The user will find that an adequate knowledge of geometric magnetic optics principles is a necessary prerequisite to the successful use of TRANSPORT He or she should possess a thorough understanding of the first order matrix algebra of beam transport optics and of the physical interpretation of the various matrix elements In other words the program is superb at doing the numerical calculations for the problem but not the physics The user must provide a reasonable physical input if he or she expects complete satisfaction from the program For this reason a list of pertinent reprints and references are included at the end of this manual They should provide assistance to the inexperienced as well as the experienced user INPUT FORMAT FOR TRANSPORT Pag 10 By the TRANSPORT input DATA SET is meant the totality of data read by the program in a single job A DATA SET may consist of one or more problems placed sequentially A problem specifies a calculation or set of calculations to be done on a given beam line A problem in turn may consist of more problem steps The data in the first step of a problem specify the beam line and the calculations to be made The data in succeeding steps of the same problem specify only changes to the data given in the first step A common example of a problem with several steps is sequential fitting In the first step one may specify that certain parameters a
126. ield Default value of 9 unless specified by a 16 8 entry Typical values of and are given below for four types of fringing field boundaries a a linear drop off of the field b a clamped Rogowski fringing field an unclamped Rogowski fringing field d a square edged non saturating magnet 12 13 zero i e 1 Ri 1 R Model K 1 6 3 8 Clamped Rogowski 0 4 4 4 Unclamped Rogowski 0 7 4 4 Square edged magnet 0 45 2 8 R is the radius of curvature in units of longitudinal length normally metres of the entrance face of bending magnets See figure on p Where R is the radius of curvature in units of longitudinal length normally metres of the exit face of bending magnets See figure on p The pole face curvatures 1 R and 1 R affect the system only in second order creating an effective sextupole component in the neighbourhood of the magnet If the parameters are not specified they are assumed to be no curvature and hence no sextupole component Either parameter or both may be varied in second order fitting For most applications is unimportant If you find it is important to your result you should probably be making a more accurate calculation with a ray tracing program see References at the end of the manual TRAJECTORY E lt gt FIELD BOUNDARIES FOR BENDING MAGNE
127. image and if c c lt 0 the waist follows the image The size of the bean at the image x max 011 1 11 011 max 56 and the size of the bean at the waist is o 0 011 0 022 0 57 022 1 821 011 0 2 gt 022 0 size at waits Thus if R4 0 the two sizes are equal since R 1 Otherwise the size at the waist is smaller than the image size Relationship between a Waist and the Smallest Spot Size Achievable at a Fixed Target Position Consider the following general situation Principal Planes of Lens System Being Adjusted mn min Target R fixed lens Waist TARGET f Variable 1 10427 120 Assume that the size of the beam Jo lens at the principal planes of the lens system being adjusted is held constant i e that no other preceding optical elements of the system are being varied and that the remainder of the system may be represented by a general matrix R which is also held constant The focal length is then varied until a minimum spot size Vo min is achieved at the target location The sigma beam matrix at the target position then has the following unique form independent of the orientation of the initial beam ellipse at the lens c at target for a minimum spot size iens R lo x 22 at target 5 s Rip 9 1 ens thus min 11 011 lens or Riz o Xm
128. in 59 min x lens 59 If the position of the waist and the minimum bean size both fall within the the same field free region then the distance to the waist from the target is 021 min Ri2R22 041 waist Z 68 022 ED 12722 gens 90 So if s s 0 the waist and the minimum spot size coincide If gt 0 the waist precedes the target and if RR lt 0 the waist occurs after the target position If the waist and the target positions fall within the same field free region the following simple relationship exists between the beam size at the lens o lens at the waist J oi waist and at the target 1 1 R2 _ 22__ 61 041 waist 041 min lens If now the lens system is readjusted to form a waist at the target position as shown by the dotted lines in Fig 10 the relative size of this waist and the minimum spot size achieved by the previous lens setting is Oy1 minattarget _ 822011 waist at target 62 011 waist at target 044 lens Again we observe that the two quantities approach each other if the size of the beam at the lens is large compared to the beam size at the target There are several cases of special interest that may be derived from the above equations 1 If Rz at the target position then a minimum spot size at the target is also a waist This corresponds to point to parallel imaging from the principal planes of the variable lens sys
129. in the program for the accelerator section The units of parameters 2 3 and 5 are changed by 15 8 15 11 and 15 5 type code entries respectively Accelerator section matrix 03 0 soo 04 urs 0 soo 3 03 ay 03 0 soo gy 04 oq soo I 03 soo 04 1 ur didascalia Definitions L effective length of accelerator sector Eg particle energy at start of sector AE energy gain over sector length phase lag of the reference particle behind the crest of the accelerating wave if is positive then for some 2 gt 0 the particles having this value are riding the crest of the wave the units of are degrees wavelength of accelerating wave the units of A are those of L normally cm This matrix element assumes that gt gt fully relativistic BEAM rotated ellipse Type code 12 0 To allow the output beam from some point in a system to become the input beam of some succeeding system provision has been made for reentering the correlation matrix which appears as a triangular matrix in the beam output See section under type code 1 0 and or the Appendix for definitions There are 16 parameters 1 Type code 12 0 2 to 16 The 15 correlations r ij among the 6 beam components in the order printed by rows Several cards may be used to insert the 15 correlations if
130. ing High Resolution Systems First Order Considerations In many respects the determination of a satisfactory first order magnetic optical design is more difficult to achieve than is the subsequent higher order design This is true not only because the basic equipment configuration is dominated by first order optical considerations but also because the choice of the first order optics affects the magnitude of all higher order aberrations and the ease with which these aberrations may be minimized by introducing multipole components into the design The dominating design parameters that must be clearly specified in order to evolve a first order design are the momentum resolving power the spatial resolution of the particle detector system to be used this determines the momentum dispersion required the required phase space acceptance the solid angle the source size and the momentum range of the instrument and the first order imaging requirements in both the x and y coordinates Given the above specifications assuming they are self consistent the optical mode and physical configuration of the instrument may be determined Often more than one theoretical solution exists in which case the choice is usually resolved by practical or economic considerations In other cases no solution is evident and the basic specifications must be modified accordingly In any event the following equations and discussion are applicable to the solution of the prob
131. ion in the aligned coordinate system of the beam centroid and the sigma matrix For a known misalignment m the centroid is transformed as in equation 25 The sigma matrix is transformed by 149 0 1 Ro 0 R Go mR Ro m G Go 0 mm c 28 where the superscript T indicates a transpose For an uncertainty in position we define a covariance matrix the distribution of possible magnet positions The sigma matrix which represents the beam envelope entering the magnet may contain contributions from both the original beam and from the uncertainty in positions of previous magnets We assume there is no correlation of errors of positioning between any two magnets The beam centroid is unaffected by an uncertainty in position The transformed sigma matrix becomes c 0 Rp R mm F Go 0 mm 6 29 If the original sigma matrix is zero then the resultant sigma matrix represents the uncertainty in the beam centroid upon leaving the magnet If the original sigma matrix encloses a region of _phase space then the resultant sigma matrix represents the envelope of possible particle trajectories including both the undisturbed sigma matrix and the effects of the misalignment VII This model for misalignments has been implemented in the computer program TRANSPORT An arbitrary misalignment m may be imposed on any magnet or section of the beam line Misalignments may also be nested T
132. is printed wherever a 13 1 card is encountered The beam matrix will then contain contributions from all previous misalignments X1X A table is used to store the results of misalignments The effect of up to ten independently misaligned magnets may be shown in the table in a single run The table is printed via a 13 8 card and may be compared with the undisturbed beam matrix printed by a 13 1 card at any point An example of such a table is shown below C The hundreds position distinguishes between an uncertainty in position OXX or a known displacement 1XX Any combination of digits may be used to define the exact circumstances intended Thus code 111 indicates the deliberate displacement of a set of magnets referred to the point where the beam enters the set with the results to be stored in a table Example N 1 A bending magnet with a known misalignment A bending magnet including fringe fields misaligned by a known amount might be represented as follows 3 L 1 6 2 5 2 0 4 1 B n 2 e 3 8 2 102 3 L 2 This represents known rotation of the bending magnet about the incoming beam direction z axis by 2 0 mr The result of this misalignment will be 45 definite shift in the beam centroid and a mixing of the horizontal and vertical coordinates The use of the 6 O 2 Transform 2 update and the misalignment code number XX2 is necessary beca
133. is to be imposed Vary codes Associated with each physical element in a system is a vary code which specifies which physical parameters of the element may be varied This code occupies the fraction portion of the type code specifying the element It has one digit for each parameter the digits having the same order in the code as the physical parameters have on the card A 0 indicates the parameter may not be varied 1 that it may be For instance 3 0 is the combined type 3 and vary code 0 for a drift length which is to remain fixed 3 1 indicates a drift length that may be varied by the virtue of the 1 The type code 4 010 indicates a bending magnet with a variable magnetic field In punching the code 3 O the zero need not be punched In punching the 4 010 code the first zero must be punched but the second zero need not be First order vary codes In a first order run the following parameters marked v may be varied those marked 0 may not be varied BEAM l vvvvvvO All components of the input beam may be varied except the momentum R M S ADDITION 1 00 All components of an r m s addition be varied except the momentum change Ap ROTAT 2 v The pole face angle of a bending magnet may be varied DRIFT 3 v The drift length may be varied BEND 4 vvv The length the field and or the n value may be varied QUAD 5 vvO The length may be varie
134. is type code is not functioning in the present version of the program SENTINEL Each step of every problem in a TRAWSPORT data set must be terminated with the word SENTINEL The word SENTINEL need not be on a separate card For a description of the form of a TRANSPORT data set see the section on input format An entire run consisting of one or several problems is indicated by an additional card containing the word SENTINEL Thus at the end of the entire data set the word SENTINEL will appear twice Acknowledgements R Helm s suggestions and criticisms at SLAC have been invaluable throughout the development of the program and the underlying theory R Pordes has ably assisted D Carey at FNAL during the more recent developments of the program REFERENCES 1 Courant and 11 8 Snyder Theory of the Alternating Gradient Synchrotron Ann Phys NY 2 1 48 1958 2 S Penner Calculations of properties of magnetic deflection systems Rev Sci Instrum 150 160 1961 3 K L Brown Belbeoch and P Bounin First and second order magnetic optics matrix equations for the midplane of uniformfield wedge magnets Rev Sci Instrum 1964 4 K L Brown A first and second order matrix theory for the design of beam transport systems and charged particle spectrometers SLAC Report No 75 or Advances Particle Phys L 71 134 1967 5 K L Brown and S K Howry TRANSPORT 360 a computer program for designing charge
135. isalignment values are sufficiently small so that a first order approach is justified For these reasons we consider only those terms which are of first order in the misalignment parameters IV Transformation of Particle Trajectory Coordinates We now temporarily delete the indices 0 and 1 indicating the entrance and exit magnet faces respectively and consider the effect of a misalignment at a single magnet face Later we will combine the results from the two faces to obtain the net effect of a misalignment When the components of the misalignment vector m are small we may expand the matrix S and the centroid displacement D in the misalignment parameters Retaining only first order terms we have D S Am I Bm 6 The six by six matrix A represents a transformation from the misalignment parameters to the particle coordinates I is the identity matrix and B represents a set of six matrices one for each of the misalignment parameters A single six by six matrix Bm is obtained by multiplying each of the six matrices by its corresponding misalignment parameter and summing the results In terms of the misalignment parameters the particle coordinates in the misaligned reference coordinate system now take the form X Am BXm 7 To derive the forms of the matrices A and B we consider separately the effect of each of the various misalignment parameters on each of the ray coordinates First we derive the effect on the ray coordinat
136. it follows from 5 and 6 that the necessary and sufficient conditions for achromaticity are that fo s t h t dt 0 8 By comparing Eq 7 with Eq 8 we note that if a system is achromatic all particles of the same momentum will have equal first order path lengths through the system Isochronicity It is somewhat unfortunate that this word has been used in the literature to mean equal path lengths since equal path lengths only imply equal transit times for highly relativistic particles Nevertheless from Eq 7 the necessary and sufficient conditions that the first order path length of all particles in dependent of their initial momenta will be the same through a system Ro Rss 0 i e if focx x fs x 0 9 First Order Imaging pag 164 First order point to point imaging in the x plane occurs when x t is independent of the initial angle This can only be so when Sx t 10 Similarly first order point to point imaging occurs in the y plane when Sy t R34 11 First order parallel to point imaging occurs in the x t plane when is independent of the initial particle position x This will occur only if cx t Ru 12 and correspondingly in the y plane parallel to pointimaging occurs when cy t R33 13 A parallel ray entering a system exits parallel to the central trajectory if C
137. ities is bounded by fy Ay1 Xi E259 lt uei 5 This interpretation is strictly true if the constrained functions f are linear in the parameters A In the non linear case it is an approximation valid only in some neighborhood of are linear in Example On the following page is an example of a TRANSPORT data deck and the resulting covariance fit of a first order run We have ask for a point to point image in both the x and y planes by varying the fields of the quadrupole triplet The following definitions and solution are applicable B M Be Bs Ay fa R2 fo amp Oy amp 51 0 005 0 005 2 From the data deck and the TRANSPORT printout shown on the following page we learn that nin 08 N 7 4096 Kg 6 1577 Kg VC 0 079 0 883 0 038 Coviance matrix information The ellipsoid B B B lt 1 then be constructed as shown Note the printout format is COVARIANCE FIT x2 in N 154 C 1 12 ow Tin 1 Example 5 USF TRIPLET TO FORM POINT TO POINT 0 3 15 2 4 15 1 ti 15 8 5 0 0 0 0 0 0 1 05 3 6 5 0 12 7 5 02 1 5 8 0 4 0 1 8 5 0 0 5 3 5 01 5 0 7 0 4 0 12 10 3 0 0 5 11 5 02 1 5 8 4 0 N1 12 3 0 7 0 13 10 1 2 0 005 FITI 15 10 5 0 005 FIT2
138. itudinal separation of the ray from a ray which enters the beam line at the same time as the given ray and travels along the central trajectory 6 is the fractional momentum deviation of the particle from the design momentum of the beam line When a charged particle passes through a perfectly aligned magnet the transformatiqn may be described to first order by the matrix equation X 1 R X 0 2 The sets of six coordinates X and X 1 give the particle position and direction at the entrance and exit faces of the magnet respectively When a magnet is misaligned the central trajectory of the magnet is no longer continuous with the central trajectory of the beam line see figure 1 below In particular at both the entrance and exit faces the reference coordinate system external to the magnet no longer coincides with the reference coordinate system internal to the magnet see figure 2 below The misaligned and aligned reference coordinate systems are related by a translation of origin plus a rotation of axes We continue to use and X 1 to denote respectively the entrance and exit face ray coordinates in the aligned coordinate systems We use a subscript f to denote the ray coordinates 0 1 expressed in the misaligned reference coordinate systems To first order the ray coordinates in the misaligned coordinate systems may be expressed in terms of those in the aligned coordinate systems by an affine transformation
139. jectory the path length difference between the arbitrary ray and the central trajectory 6 Ap p is the fractional momentum deviation of the ray from the assumed central trajectory This vector for a given particle will henceforth be referred to as a ray The magnetic lens is represented to first order by a square matrix R which describes the action of the magnet on the particle coordinates Thus the passage of a charged particle through the system may be represented by the equation X 1 RX 0 1 where X O is the initial coordinate vector and X 1 is the final coordinate vector of the particle under consideration The same transformation matrix R is used for all such particles traversing a given magnet one particle differing from another only by its initial coordinate vector X 0 The traversing of several magnets and interspersing drift spaces is described by the same basic equation but with R now being replaced by the product matrix R t R n R 3 R 2 R 1 of the individual matrices of the system elements This cumulative transfer matrix is automatically calculated by the program and is called TRANSFORM 1 It may be printed where desired as described in later sections This formalism may be extended to second order by the addition of another term The components of the final coordinate vector in terms of the original are now given as 0 T6 0 0 2 jk where T is the second order
140. k This is accomplished by enclosing the comments made on each card within single parentheses Each element must be terminated by a semicolon Optionally a semicolon may be replaced by an asterisk or a dollar sign Spaces before and after the semicolon are allowed but not required If the program encounters a semicolon dollar sign or asterisk before the expected number of parameters has been read in and if the indicator card was a zero 0 the remaining parameters are set to zero If the indicator card was a one 1 or two 2 then the numbers indicated on the card are substituted for the numbers remaining from the previous solution all other numbers are unchanged The free field input format of the data cards makes it considerably easier to prepare input than the standard fixed field formats of FORTRAN Numbers may be punched anywhere on the card and must simply be in the proper order They must be separated by one or more blanks Several elements may be included on the same card and a single element may continue from one card to the next A single number must be all on one card it may not continue from one card to the next The program storage is limited to a total of 2000 locations including type codes and those parameters not punched but implied equal to zero and 500 elements A decimal number e g 2 47 may be represented in any of the following ways 2 47 0024743 0247 02 247 2 247000 5 sample problem
141. lculate A and B for a quadrupole we take the limit 0 with ap the length of the magnet being held fixed Then we have 1 0 01 0 0 0 0 01 0 0 0 L 10 0 0 20000 Vet 0 0 001 0 0 0 0000 and B B as given above VI Effect on the Beam Envelope To first order the coordinates at the misaligned magnet exit face are related to those at the misaligned entrance face by a transfer matrix so that 1 R X 0 22 or X 1 ByX 1 m R X 0 Agm BeX 0 m 23 If we solve for X 1 and discard all terms in m of order higher than first we then derive X 1 R X 0 A RAe m RBe B R X 0 m 24 For later use we define two new matrices F and G given by Ai RAg G 25 so that X 1 RX 0 Fm 4GX 0 m 26 An ensemble of particles in a beam line is often represented as a six dimensional ellipsoid The equation of this ellipsoid may be written in matrix form as follows 1 27 where X is the transpose of the coordinate vector X and is a real positive definite symmetric matrix The square roots of the diagonal terms of the sigma matrix are a measure of the beam size in each coordinate If the centroid of this ellipsoid does not fall on the central trajectory then one needs to specify this centroid position also The sigma matrix then gives the beam dimensions as measured about the centroid The beam envelope entering a misaligned magnet may be described in terms of the posit
142. lem 1 First Order Resolving Power A general equation for the first order resolving power has been derived in References 1 2 and 3 For point to point imaging the first order momentum resolving power R is defined as the ratio of the momentum dispersion at the image plane to the total image size Thus if 2x is the total source size then from Reference 1 we have P R 17 AP 400 _ x fo COh Odr 29 2xocy t Note that h t dt da is the differential angle of bend of the central trajectory of the optical system Equation 29 may be expressed in a number of useful forms If we consider a particle originating at the source with x 0 and 5 and x and lying 0 1n the midplane i e a monoenergetic point source the first order equation of its trajectory is X T 5 30 We may then rewrite equation 29 as follows 141 1 2X9X o Ry xCohCodc 9 31 2 where is the path length difference between the trajectory described by equation 30 and the central trajectory Or we may also write equatien 31 in the form Erg db uri e en where f BdA is the magnetic flux enclosed between the central trajectory and the trajectory described by equation 30 and Bp is the magnetic rigidity of the central trajectory Please note however that if the trajectory of equation 30 crosses the central trajectory or the si
143. line coordinate layout is desired the card specifying that a layout is to be made a 13 0 12 0 element and any initial coordinates see type code 16 0 all precede the BEAM card There are eight entries all positive to be made on the BEAM card 1 The type code 1 0 specifies a BEAM entry follows One half the horizontal beam extent x cmv in standard units One half the horizontal beam divergence 0 mr One half the vertical beam extent y cm One half the vertical beam divergence mr One half the longitudinal beam extent A cm One half the momentum spread 6 in units of percent Ap p The momentum of the central trajectory 0 GeV c 11 eight entries must be made even if they are zero As for all other type codes the last entry must be followed by a semicolon dollar sign or asterisk Thus a typical BEAM entry might be 00 ou AWN Label if desired s meaning x 0 5 cm 2 0 mr y 1 3 cm 2 5 mr A 0 0 cm 11 5 percent Ap p and the central momentum p O 10 0 GeV c The units of the tabulated matrix elements in either the first order or sigma matrix or second order T matrix of a TRANSPORT print out will correspond to the units chosen for the BEAM card For the above example the R 12 matrix element will have the dimensions of cm mr and the T 236 0 matrix element will have the dimensions mr cm percent and so forth The longitu
144. m composed of combinations of bending magnets quadrupoles solenoids sextupoles and interspersed drift spaces It is assumed that mid plane symmetry prevails for any given magnetic element in a system except for solenoids but not necessarily for the system as a whole The notation used in a TRANSPORT printout is described in reference 1 SLAC 75 beginning on page 46 The subscript notation is the same as that used for first order where the subscript 1 means x 2 means 3 means y 4 means o 5 means and 6 means 6 The symbol has been used to signify a first order matrix element and the symbol Tijk will be used to signify a second order matrix element Thus we may write the second order Taylor expansion representing the deviation of an arbitrary trajectory from the central trajectory as Xi t Xj Rgx 0 Dhar 0 0 where x X x and x 6 denotes the subscript notation In an actual computer printout the Tix s are abbreviated as i jk for example 0 6 would appear in a printout as 1 26 followed by the computed value of the aberration coefficient for the system being designed In order to modify the magnitude of any given aberration coefficient it is necessary to introduce multipole component s of the magnetic field of order equal to or less than the order of the aberration Thus sextupole quadrupole and dipole components of the field may all be used to modif
145. n particular the square roots of the diagonal elements aji s are the projection of the ellipse upon the coordinate axes and thus represent the maximum extent of the BEAM in the various coordinate directions The correlation between components the orientation of the ellipse is determined by the off diagonal terms the oij s An illustration of this is given below for 2 dimensional ellipse Description of the Sigma BEAM Matrix Consider a 2 dimensional x 0 plane projection of the general 6 dimensional ellipsoid Let A 021 922 be a real positive definite symmetric matrix the inverse of which is 1 1 022 021 where is the determinant of c The 2 dimensional coordinate vector column matrix and its transpose are x X 6 and X x The expansion of the matrix equation X o X 1 is the equation of the ellipse 2 2 2 X 204 det o 31 The x 0 plane BEAM ellipse represented by Eq 31 is shown in the following figure along with the physical meaning of the sigma matrix elements 111 One 222 11 CENTROID 1358A The area of the ellipse is given by 1 2 A n det o TUXint Omax 32 A Two Dimensional BEAM Phase Ellipse The correlation between x and the orientation of the ellipse depends upon the off diagonal term o3 This correlation is defined as Pa P127 011022 So defined always falls i
146. n the range 1 lt lt 1 The correlation r measures the tilt of the ellipse and the intersection of the ellipse with the coordinate axes Since the det R 1 for all static magnetic beam transport elements it follows that the determinant of o 1 and o 0 are identical under the transforma tion of Eq 30 Hence the phase space area is an invariant under the transformation of Eq 30 This is a statement of Liouville s Theorem for the magnetostatic fields employed and results from the fact that the det 1 It is perhaps worthwhile noting that this 2 dimensional representation of the BEAM matrix has a one to one correspondence with the Courant Snyder treatment of the theory of the Alternating Gradient Synchrotron as follows 011 021 022 en E D Courant and H S Snyder Theory of the Alternating Gradient Synchrotron Annals of Physics 3 pp 1 118 1958 112 The Phase Ellipse Beam Matrix used by TRANSPORT For static magnetic systems possessing midplane symmetry the 0 plane and y plane trajectories are decoupled in first order i e there is no mixing of phase space between the two planes However for mathematical simplicity and to allow for the possibility of more general systems the sigma BEAM matrix used in a TRANSPORT calculation has the following general 6 dimensional construction 9 e 6 0 11 0 21 0 22 0 31 0 32 0 33 0 41 0 42 0 43 o 44 0 51 0 52
147. ning blanks to the right of the decimal point of the type code number See the section under type code 10 0 for an explanation of the use of vary codes The physical parameters are the quantities which describe the physical element represented Such parameters may be lengths magnetic fields apertures rotation angles beam dimensions or other quantities depending on the type code number The meanings for the physical parameters for each type code are described thoroughly in the section for that type code A summary indicating the order in which the physical parameters should be punched is given in Table 1 For any element the first physical parameter is the second 10 entry in Table 1 or the second parameter in the section describing a given element In some cases the parameters of an element do not really refer to physical quantities but will nevertheless be referred to as such in this manual The label if present contains one to four characters and is enclosed by single quotes slashes or equal signs During the calculation the elements will be printed in sequence and the label for a given element will be printed with that element Labels are useful in problems with many elements and or when sequential fitting is used They must be used to identify any element to be changed in succeeding steps of a given problem Provision has been made in the program to allow the user to introduce comments before any type code entry in the data dec
148. nsfer matrix R for that element Accumulation is resumed thereafter AUXILIARY TRANSFORMATION MATRIX R2 Type code 6 O 2 The R1 matrix represents the accumulated transfer matrix from either the beginning of the beam line or the last explicit R1 update 6 O 1 However several elements in TRANSPORT which affect the beam matrix cannot be represented in any transfer matrix To avoid update complications with R1 an auxiliary transfer matrix R2 exists The beam matrix is then calculated from the R2 matrix and the beam matrix at the last R2 update Both the R1 and R2 matrices are normally available for printing However there is no redundancy in computer use since internally to the program only R2 is calculated at each element The matrix R1 is calculated from R2 only as needed The R2 matrix is updated explicitly via a 6 0 2 3 entry It may be printed by 13 24 3 entry Constraints on R2 are imposed similarly to those on R1 For details see the section describing type code 10 0 The complete list of elements which update TRANSFORM 2 is 1 a beam type code 1 0 entry 2 the 6 0 1 entry 3 the 6 2 entry 4 a centroid shift type code 7 0 entry 5 a misalignment type code 8 0 entry 6 a stray field type code 21 0 entry Please note that automatic updates of TRANSFORM 2 occur when an 41 align element type code 8 is inserted specifying the misalignment of all subsequent bending magnets These TR
149. of the particle detectors used at the image plane and the momentum range to be covered by the instrument or in the case of a momentum defining analyzing system by the acceptable momentum defining slit spacings 3 The Selection of the Optical Mode By optical mode we mean the type of imaging e g point to point or parallel to point etc required at the image plane in both the x and y coordinates and the number of intermediate images imposed between the source and image planes The imaging requirements at the image plane are usually dominated by the physics to be performed by the instrument and the nature of the particle detectors used However often especially at low energies the imaging in the y plane may be unimportant as far as the physics requirements are concerned which in turn provides some additional flexibility in the optics design A study of the coupling coefficients to the aberration coefficients equation 19 shows the not surprising result that multipoles located at intermediate images in a system do not couple to aberrations in the plane in which the intermediate image occurs Hence it often proves beneficial to intentionally create an intermediate image in the y plane of an optical system 142 so as to achieve Some degree of orthogonality in the minimizing of x and y aberrations The considerations of 1 2 and 3 above are the determining factors in the selection of the first order solution of a system design T
150. oordinate rotation element is provided Because of such other possibilities when describing bending magnets we shall often speak of the bend and non bend planes The transverse coordinates will also often be labeled x and y while the longitudinal coordinate will be labeled z All magnets are normally considered aligned on the central trajectory A particle following the central trajectory through a magnet experiences a uniform field which begins and ends abruptly at the entrance and exit faces of the idealized magnet Therefore through a bending magnet the reference trajectory is the arc of a circle while through all other magnetic elements it is a straight line To accommodate a more gradual variation of field at the ends of a bending magnet a fringing field element is provided In order to represent an orientation with respect to the reference trajectory other than normal of a magnet or section of a beam line a misalignment element also exists The magnetic field of any magnet except a solenoid is assumed to have midplane symmetry This means that the scalar potential expanded in transverse coordinates about the reference trajectory is taken to be an odd function of the vertical coordinate If a coordinate rotation is included then the potential is odd in the coordinate to which the vertical has been rotated For a bending magnet this will always be in the non bend plane The program TRANSPORT will step through the beam line element
151. or instance one might wish to vary the field of a quadrupole in one step of a problem and then use the fitted value as data in the following step The first step might then contain the element 5 01 5 0 10 0 5 0 QUAD and the following step would contain the element 5 QUAD Since in the second step the first item on the card contains no vary code the vary code is deleted All other parameters not being re specified are left unchanged Several elements may have the same label If as in the above example one wished to vary the field of several quadrupoles in one step then pass the final values to the next step one could give all such elements the same label There might be four quadrupoles all labeled QUAD being varied simultaneously If the data for the next step contain the single element 5 QUAD 13 the vary code on all elements labeled QUAD will be deleted The physical parameters of an element may be changed between steps of of 5 meters 4 5 0 10 0 In 0 0 setting the element BEND All parameters up to and including the one to be changed must be specified The remaining if omitted will be left unchanged from the previous step 4 10 0 END Table 1 Summary of TRANSPORT type codes a problem In the first step a bending magnet may be given a length a succeeding step its length could be increased to 10 meters by in
152. order equations However the method has been extended to include second order terms as discussed in Part I For a more extensive discussion of the second order matrix formalism the reader is referred to SLAC report number 75 by K L Brown FIRST ORDER R MATRIX FORMALISM FOR TRANSPORT Section II Beam transport optics may be reduced to a process of matrix multiplication 1 2 To first order this is represented by the matrix equation using the notation of SLAc 75 x t Eja x 0 1 where 5 x The determinant R 1 This is a direct consequence of the basic equation of 100 motion for a charged particle in a static magnetic field and is a manifestation of Liouville s theorem of conservation of phase space volume See SLAC 75 for a proof that R 1 For static magnetic systems possessing midplane symmetry the six simultaneous linear equations represented by Eq 1 may be expanded in matrix form as follows x t Ry 0 0 0 e t Roy 0 0 0 Rog yO 0 0 0 0 0 Rag 0 0 t Ra R 0 0 1 Reel 5 t 0 0 0 0 0 1116 2 mancano righe verticali parte centrale where the transformation is from an initial position t 0 to a final position t t measured along the assumed central reference trajectory Thus at any specified position in a system an arbitrary charged particle is represented by a v
153. parte Solving for 21 2 we find Z R21 R21 where 21 and 2 are the location of the principal planes as shown Fig 3 The principal planes of any system may be determined by this method Note that 1 f is not affected by the transformation and that the upper right hand matrix element is zero if det 1 The principal planes may coincide may be close together be far apart or in many systems may be located external to all of the elements comprising the system An example of the latter case is a quadrupole pair Some examples of principal plane locations for simple systems follow A quadrupole singlet Figure 4 The principal planes in a single quadrupole are located very close to each other and very near the center of the lens As such a quadrupole singlet may be considered as a thin lens if the object and image distances are measured to the center of the lens A simple uniform field wedge magnet orm field si P 5 87 If the optic axis enters and exits perpendicularly to the pole boundary the principal planes are at the center of the magnet as shown in Figure 5 From this we conclude that a simple wedge bending magnet may be considered as thin lens if the object and image distances are measured to the lens center 0 quadrupole pair defocusing focusing P a A B lt 1 gt plane Figure 6 focusing defocusing 4 gt 1 ES
154. particle and the angles with the axis of the beam line in the same planes The quantity 2 represents the longitudinal position of the particle relative to a particle traveling on the magnetic axis of the system with the central momentum designed for the system The remaining quantity 6 2 gives the fractional deviation of the momentum of the particle from the central design momentum of the system An ellipsoidal hypersurface in this six dimensional space may be represented by the equation xot 1 2 where o is symmetric positive definite matrix We represent this matrix as an inverse for reasons which will become apparent later At this stage the center of the ellipsoid is assumed to lie at the origin of the coordinate system The ellipsoid may be taken to be the envelope of a uniform distribution or the scale in a gaussian distribution giving a particle density 128 exp x o x 3 For any symmetric matrix there exists a coordinate system in which that matrix is diagonal and an orthogonal transformation to that coordinate system Let us represent the orthogonal transformation by the matrix 0 so that Xi 7 0 4 Xj are the coordinates in the frame where the transform of 01 and therefore that of o are diagonal Calling the matrix c transformed to the new frame we now have Vine e0je 5 and equation 1 becomes 1 Specializing to the gaussian distr
155. pse Relationship between a First Order Point to Point Image and the Minimum Spot Size Achievable at a Fixed Target Position Orientation of the Major Axes of a Phase Space Ellipse Table of Contents continued 3 Second Order Aberrations Second Order Contributions to Beam Dimensions I Introduction II The Ellipsoid Formalism III The Effect of a Beam Line IV Off Axis Initial Distribution References A Systematic Procedure for Designing High Resolving Power Beam Transport Systems or Charged Particle Spectrometers I II III IV Introduction Theory Multipole Strengths for Pure Multipole FieldsMultipole Strengths for a Non Uniform Field Expansion Multipole Strengths for a Contoured Entrance or Exit Boundary of a Magnet The Description of the Trajectories as a Taylor s Expansion Solution of the Equation of Motion Interpretation and use of the Coupling Coefficient 83 V A Systematic Procedure for Designing High Resolution Systems First Order Considerations First Order Resolving Power Dispersion The Selection of the Optical Mode Aberrations and their Correction References 4 The Effect of Beam Line Magnet Misalignments I Introduction II Particle Trajectory Coordinates III Magnet Misalignment Coordinates IV Transformation of Particle Trajectory Coordinates V Evaluation of the Relevant Matrices VI Effect on the Beam Envelope VII Implementation 5 First Order Parameter Optimization and Covarianc
156. r transformation matrix will obviously occupy several lines This element in conjunction with the 13 17 element and either the 13 3 element or the 13 6 element will produce output in which the printed matrix elements will occupy single uninterrupted columns For visual appearances it is recommended that if both beam and transformation matrices are desired they be printed in separate steps of a given problem Punched output controls 29 30 31 32 33 34 35 36 If the control is equal to 29 all of the terms in the first order matrix and the x and y terms of the second order matrix are punched If the control is equal to 30 all of the terms of the first order matrix and all second order matrix elements are punched out If the control n is greater than 30 all of the first order terms are punched and the second order matrix elements which correspond to n 30 i e if n 32 the second order theta matrix elements are punched out If n 31 the second order x matrix elements are punched and so forth ARBITRARY TRANSFORMATION input Type code 14 0 To allow for the use of empirically determined fringing fields and other specific perhaps non phase space conserving transformations provision has been 66 made for reading in an arbitrary transformation matrix The first order 6 X 6 matrix is read in row by row There are eight parameters for each row of a first order matrix entry 1 Type code 1
157. rays from the center of the first bending magnet to the center of the last one For the serious student it is a worthwhile exercise to do the BSY problem without the simplification which resulted from introducing the principal planes 4 TRANSPORT pag 150 As an aid to solving beam transport problems a computer program TRANSPORT has been developed at SLAC which takes the greatest amount of labor out of this work The program operates in about the way as the BSY example above was calculated but with some important exceptions Most importantly 1 TRANSPORT has the ability to find the best first order solution given a certain set of constraints 2 TRANSPORT also calculates the transformation of a whole family of rays as found in a beam by means of the concept of phase space which was introduced in Part I 3 TRANSPORT can as an option calculate the second order effects on the beam By second order is meant for example terms which depend net linearly on the displacement x but on x2 or 9 etc To aid in the discussion of TRANSPORT and of the second order terms we now introduce an abbreviated notation By writing out the complete equations for x and y to second order we will adequately have displayed the new notation Xi x xe xe x 6 6 x xG x x xo 6 xo 02 02 6 6 0 6 gt 67 67 x y lye 2 14 ylyo
158. rder fitting Some typical T1 matrix constraints are as follows Desired optical condition Typical fitting constraint Geometric aberration 10 1 22 0 001 1 Chromatic aberration 5 10 3 46 5 001 F2 There must be no updates of the R2 matrix when constraining an element of the 1 matrix There is no limit on the number of constraints which may be imposed If no drift lengths are varied the problem will be linear and the absolute size of the tolerances will be unimportant Only their relative magnitude will be significant Sometimes only a subset of the elements of the matrix Tijk which give significant contributions to beam dimensions need be eliminated In such cases one may wish to minimize the effect of this subset by weighing each matrix element according to its importance 58 One does this by including a constraint for each such matrix element and setting its tolerance equal to the inverse of the phase space factor which the matrix element multiplies For a matrix element acting on an uncorrelated initial phase space the tolerance factor would be 1 XejXe where and are the initial beam half widths specified by the type code 1 0 card T2 matrix fitting constraints Five parameters are needed for a constraint on the i j k element of the second order auxiliary transfer matrix T2 Type code 10 0 Code digit 20 1 Code digit 10j k Desired v
159. re to be varied to satisfy certain constraints Once the desired fit has been achieved the program will then proceed to the next step The data in this step now need specify only which new parameters to vary or old ones no longer to vary or which constraints to add or delete The values of the varied parameters that are passed from one step to the next one are those that are the result of the fitting procedure A problem step contains three kinds of DATA cards the TITLE card the INDICATOR card and the ELEMENT cards The TITLE card contains a string of characters and blanks enclosed by single quotes Whatever is between the quotes will be used as a heading in the output of a TRANSPORT run The second card of the input is the INDICATOR card If the data which follow describe a new problem a zero is punched in any column on the card If the data which follow describe changes to be made in the previous problem step a one 1 or two 2 is punched in any column on the card For further explanation read the Indicator Card section of this manual The remaining cards in the deck for a given problem step contain the DATA describing the beam line and the calculations to be done The DATA consist of a sequence of elements whose order is the same as encountered as one proceeds down the beam line Each element specifies a magnet or portion thereof or other piece of equipment a drift space the initial beam phase space a calculation to be done or
160. rmation matrix print controls 4 5 6 24 13 4 3 The current transformation matrix R1 TRANSFORM 1 is printed by this code If the program is computing a second order matrix this second order transformation matrix will be included in the print out This matrix is cumulative from the last R1 TRANSFORM 1 update The units of the elements of the printed matrix are consistent with the input units associated with the type code 1 0 BEAM entry 13 6 3 The transformation matrix R1 will be printed after every physical element which follows this code The second order matrix will be printed automatically only if the one line form code 13 19 of the transformation is selected The second order matrix will however be printed at each location of a 13 4 element The first order matrix will not be repeated 13 5 5 The automatic printing of R1 will be suppressed and R1 will be printed only when subsequently requested 13 24 5 The TRANSFORM 2 matrix R2 will be printed by this code The format and units of R2 are identical with those of R1 which is printed by the 13 4 code For a list of elements which update the R2 matrix see type code 6 The units of the tabulated matrix elements in either the first order or sigma matrix or second order T matrix of a TRANSPORT print out will correspond to the units chosen for the BEAM card For example the R 12 x 0 matrix element will normally have the dimensions
161. rnia 1970 143 THE EFFECT OF BEAM LINE MAGNET MISALIGNMENTS Karl L Brown and Frank Rothacker Stanford Linear Accelerator Center Stanford California David C Carey Fermi National Accelerator Laboratory Batavia Illinois and Christoph Iselin CERN Geneva Switzerland Operated by Universities Research Association Inc under contract with the Energy Research and Development Administration The misalignment of a magnet in a beam line will cause an alteration of the beam envelope at any later point in that beam line The position of a misaligned magnet may be described in terms of six coordinates three translational and three rotational The effect of a misalignment on a single particle trajectory is derived to first order including bilinear terms A bilinear effect is one which affects the beam line focusing characteristics but not the central ray such as the effect of rotating a quadrupole about its axis The effect on the beam envelope is calculated both for a known magnet displacement and for an uncertain magnet position The formalism has been included in the computer program IRANSPORT I Introduction The effects of magnet misalignments are an important consideration at every stage of beam line design installation and operation The selection of the optical mode determination of surveying accuracy requirements and the choice of correcting elements are all dependent on misalignment information Two types of mi
162. rresponds to a beam which is at its minimum width a waist and Figure 16 b shows the same beam after it has drifted downstream from the waist position The physical meaning of this is that particles entering at 9 O are parallel to the optic axis and therefore cannot change their relative positions with respect to the optic axis that is all particles on the axis act in this manner Those that enter at a given angle continue at the same angle The phase ellipse transformation for a thin lens is illustrated in Figure 17 In passing through a thin lens changes and the x dimension remains constant for a given trajectory after lens action before lens action Figure 17 Stated in other terms Xmax remains constant as does the intercept It is apparent in the given example that the spot size now becomes or can become smaller at the new image because Omax is larger This can be related to the physics of the system by saying that the x magnification is less than unity This fact is observed directly by comparison of the x intercept of the ellipse before and after the lens action It is interesting to observe that a particle initially at c is transformed to c and that particles entering at x do not change their direction 0 is constant If the particles are now allowed to drift the ellipse rotates clockwise when the ellipse if vertical the spot size is at a minimum namely Xmax x as was illustrated in
163. rticle The 6 terms in the x and equations express the momentum dispersion of the system If it happens that A is an object point and B is an image point of the system then is independent of thus sy O In this case is given by cx Xe M the magnification in x plane for 0 if 6 0 then xe f 1 f It must always be true that the determinant of the matrix R is unity Thus for this special case of sy 0 it follows that s 1 3 Beam Switchyard As an example of a system which can be calculated with the matrix method we next consider the beam switchyard of the two mile accelerator Figure 2 shows the three essential elements two bending magnets and a quadrupole lens In common with many beam transport systems this one is designed to be achromatic Mathematically this means that the matrix elements d and d should be zero so that there is no x or 0 dependence on the momentum of the particles Figure 2 The Essentials of the Beam Switchyard 96 As a preliminary step we will find the matrix expression for a bending magnet when measured from the principal planes The matrix for a bending magnet when measured from the ends of the poles is given by S 1 Rbend 5 5 3 0 0 1 quadrettatura where cos ands sina is the deflection angle of the central ray This expression has been normalized by settin
164. s a function only of the first order matrix elements Sx sy d and their derivatives with respect to t From median plane symmetry considerations the allowed aberrations are those with and or y appearing an even number of times in the Taylor coefficient For example 2 and 2 are allowed aberrations whereas x yo y o or y y amp are not allowed and are therefore equal to zero The minus sign is used when y and or y appear 4 8 12 times and the plus sign is used when y and or y appear 2 6 10 times For example for the coefficients 0 the minus sign is applicable whereas for the coefficients 2 and y ygy 2 the plus sign is applicable Equation 19 is derived by observing in the pattern of the solution of the differential equation that an nth order aberration term containing the nth order multipole strength factor K cannot include multipole strength factors of lower order than n or stated physically an nth order multipole cannot couple to aberrations terms of order lower than n This fact allows the recursion formula equation 3 to be reduced to the simple form 21 1 2 21 so far as it applies to the derivation of nth order terms containing only K As a consequence the scalar potential for deriving these terms assumes the Simplied form 55 55 m x 2 1 P x y 0 0 1
165. s is equal to unity This is always the case as will be proved formally later That this is so is a manifestation of Liouville s theorem of conservation of phase spa ce area Consider now a thick lens as illustrated in Figure 3 Figure 3 If is the object distance to the face of the lens L is the corresponding image distance then in general 1 L 1 L 1 f If however we introduce two planes P and located at a distance 2 z from the entrance and exit faces of the lens it is always possible to find 2 and a 7 such that the equation 1 p 1 1q 1 f is valid Where P L 2 q Z When this is so P and are called the principal planes of the lens Now relating the above statement to matrix formalism the matrix transformation for a thick lens between the input and output faces of the lens has the general form 16 1 where as before the det 1 For a general transformation is not necessarily equal to O and R4 not necessarily equal to 1 The principal planes may be located by the transformation R44 ue 1 7 E il 1 74 Raa d 1 f 1 lo 1 2 Using the relation 17z npn z 1 OF _ o il o zi 11 1 the unit matrix the previous equation may be manipulated into the form 12 71 11 R R 7 Ru cz Ra 72 gt 2 74 21 2 D elle xL 1 f 21 22 R21 R22 Z1R21 Manca ripartizione nella prima
166. s o is diagonalized denoting the moments in this frame by 6 Then from equation 7 we have oij Oi Oj OKe 18 Xx We continue to specialize to a gaussian distribution so that the fourth moments will be directly derivable from the second 130 moments In the diagonal frame the coordinates separate and the fourth moments are easily calculated The only ones which are non zero are Ojj for i j and with 01 0 19 646 0107 3 aij ci so that in general Oi ke 0j Opr Six 0110 29 Now if under the transformation the fourth moments transform as 0 0 0 21 then from equation la we finally arrive at Oj Oke 22 Substituting into equation 17 we determine that 0 oi ijk 23 Dee 2 2 Xo 2 5 Note that because of the symmetry properties of both that the two expressions in parentheses in the last term of the second equation represent the same array From a practical standpoint this means that it needs to be calculated only once We see from equation 23 that the centroid of the distribution at the final point no longer coincides with the beam axis Letting represent the matrix of second moments a
167. salignment information are typically needed To assess the general effect of misalignments in the design stage one needs to know the change in beam position and beam line transmission characteristics due to uncertainties in the position of each magnetic element in each separate coordinate Secondly to provide for correcting elements one needs to know the effect on the beam of specific misalignments I In the following we derive a method of determining the effect of magnet misalignments on a particle beam We first define a reference system in which to express misalignments Then we determine the effect of a misalignment on individual particle trajectories Finally we express the effect on the beam envelope which describes the ensemble of particles comprising the beam II Particle Trajectory Coordinates To specify the position and direction of a particle at any instant in time we employ a coordinate system defined with respect to the beam line reference trajectory The z axis is taken to point along the reference trajectory the x axis points to the left and the y axis points up The position and direction of the particle trajectory can then be given by a vector with six components 144 Mo gt II lt lt 1 lt anes The quantities x x y and y are respectively the horizontal displacement and slope and vertical displacement and slope of the ray with respect to the central reference trajectory 2 is the long
168. se transformations of interest 1 An arbitrary DRIFT distance and 2 A LENS action Each of these elementary cases are illustrated on Fig 8 for both a parallelogram as well as ellipse phase space transformations Note that a DRIFT followed by a LENS action is not necessarily equal to a LENS action followed by a DRIFT i e the matrices do not necessarily commute The phase ellipse transformations for a DRIFT and for a LENS action between principal planes as shown in Fig 8 may be readily calculated using the results of Eq 37 The 2 dimensional R matrix representing a drift of distance L is ub R Drift m 4 38 Substituting into Eq 37 we find 2 1 E 0 L 022 0 1 0 2201 39 Lo22 0 022 0 1 1 022 1 Attaching the physical meaning to the matrix elements yields the following interpretations 011 1 61 0 L or x7 x6 max L 06 40 114 DRIFT LENS ACTION CENTRAL TRAJECTORY 4 PRINCIPAL PLANES 3 OF LENS 2 PRINCIPAL PLANES OF LENS 3 135882 Fig 8 115 similary 022 1 022 05 max 41 Note that this transformation assumes that the initial phase ellipse is erect 1 6 5 031 0 0 The 2 dimensional R matrix for a lens actions between principal planes is 1 0 R Lens 2 i 42 f Substitution into Eq 37 yields 011 2 021 2 022 2 43 Again attaching physical meaning
169. see examples below This will be printed on the output listing It must be enclosed in single quotes and is a maximum of three characters long four for energy The format for insertion is the same as for labels 4 The scale factor if needed The scale factor is the size of the new unit relative to the standard TRANSPORT unit For example if the new unit is inches and the standard TRANSPORT unit cm the scale factor is 2 54 N The various units that may be changed are Code Quantit Standard Symbols used in Digit y TRANSPORT Unit SLAC 75 1 0 horizontal and vertical cm transverse dimensions magnet apertures and misalignment displacements 2 0 horizontal and vertical mr 0 0 angles and misalignment rotation angles 3 0 vertical beam extent cm y cm y 1 and bending magnet gap height 4 0 vertical beam divergence only mr 5 0 pulsed beam length cm wave length in accelerator 6 0 momentum spread percent PC 6 7 0 bend pole face rotation and degrees DEG coordinate layout angles t 8 0 length longitudinal metres M of elements layout coordinates and bending magnet pole face curvatures 9 0 magnetic fields kG B 10 0 mass electron mass m 11 0 momentum and GeV c energy gain in accelerator section GeV AE These codes should not be used if the coordinate rotation 20 0 type code is used anywhere in the system Units are not normally restored
170. sequence 3 B 5 01 3 B will allow the quadrupole to move without altering the total system length Vary digits may also be immediately preceded by a plus sign without changing their meaning Thus 5 0A is the same as 5 0 For historical reasons the vary digits 9 and 4 8 and 3 and 7 and 2 are also inversely coupled Inverse coupling may not be used with type codes 1 0 or 8 0 The total number of independent variables in a first order run is limited to 20 by reasons of the mathematical method of fitting and to 10 for a second order run So far as this limit is concerned variables that are tied together count as one Variables within repeat elements type code 9 0 also count only one Possible fitting constraints A variety of possible constraints is available Fitting may be done in either first or second order but not in both simultaneously The order of the constraint must be appropriate to the order of the run A list of constraints available is given below They are explained more fully on later pages First order constraints 1 An element of the first order transfer matrix R1 2 An element of the auxiliary first order matrix R2 3 A o BEAM matrix element 4 The correlations r in the beam coordinates 5 The first moments of the beam 6 The total system length 7 An AGS machine constraint 8 The reference trajectory floor coordinates Second order constraints 1 An element of the second order transfer m
171. ses the method of non linear least squares differential correction as good description of which is found in Ref 1 below pages 390 393 A useful by product of this method is the covariance matrix C printed by the program at the successful conclusion of any run involving parameter fitting In many applications C may be used to estimate tolerances on the fitted parameters subject to the specified tolerances i e the standard deviations of the constraints The covariance matrix C is symmetric This admits a geometrical interpretation as an ellipsoid and is printed in the same suggestive format as is the beam ellipsoid o except that in this case the dimension is equal to N the number of parameters varied The center of the ellipsoid is at a A9 A8 29 1 the N values found by TRANSPORT to be the best estimate of the varied parameters The equation of the covariance ellipsoid is 2 where C4 the first diagonal element printed measures the maximum extent of the ellipsoid along the A axis the first varied parameter in the same sense that 0u measures the maximum extent of the beam ellipsoid along the x axis The off diagonal elements are normalized so that they are lt 1 in magnitude in analogy with the ri of the beam matrix and can be interpreted as measures of the orientation of the covariance ellipsoid 1 2 7 SOLMITZ Analysis of Experiments in Particle Physics Ann Rev Nuc Sci Vol 14 1964
172. soid after the transformation becomes where It can readily be shown that the square roots of the diagonal terms of the sigma matrix are a measure of the beam size in each coordinate The off diagonal terms determine the orientation of the ellipsoid in n dimensional space for TRANSPORT n 6 Thus we may specify the beam at any point in the system via Eq 5 given the initial phase space represented by the matrix elements of o 0 2 See the Appendix of this report or the Appendix of Ref 5 for derivation of these statements The initial beam is specified by the user as one of the first elements of the beam line Normally it is taken to be an upright ellipse centered on the reference trajectory that is there are no correlations between coordinates Both correlations and centroid displacements may be introduced via additional elements The phase ellipse may be printed wherever desired For an interpretation of the parameters printed see the section under type code 1 0 When a second order calculation is specified the second order matrix elements are included in the beam matrix For details on how this is done see the Appendix to this manual Fitting Several types of physical elements have been incorporated in the program to facilitate the design of very general beam transport systems Included are an arbitrary drift distance bending magnets quadrupoles sextupoles solenoids and an accelerator section to first
173. stem described above Particles in a beam are assumed to occupy the volume enclosed by the ellipsoid each point re presenting a possible ray The sum total of all phase points the phase space volume is commonly referred to as the phase space occupied by the beam The validity and interpretation of this phase ellipse formalism must be ascertained for each system being designed However in general for charged particle beams in or emanating from accelerators the firs t order phase ellipse formalism of TRANSPORT is a reasonable representation of physical reality For other applications such as charged particle spectrometers caution is in order in its use and interpretation The equation of an n dimensional ellipsoid may be written in matrix form as follows 0 76 0 1 0 21 2 where X O T is the transpose of the coordinate vector and o is a real positive definite symmetric matrix The volume of the n dimensional ellipsoid defined by sigma is debo The area of the projection in one plane is 1 A m detoj 2 where o is the submatrix corresponding to the given plane This is the phase space occupied by the beam As a particle passes through a system of magnets it undergoes the matrix transformation of Eq 1 Combining this transformation with the equation of the initial ellipsoid and using the identity RR I the unity matrix it follows that from which we derive The equation of the new ellip
174. stion These coefficients are functions of t to be determined The symbol indicates summation over zero and all positive integer values of the exponents A p v X The constant term is zero and the terms that would indicate a coupling between the coordinates x and y are also zero this results from the midplane symmetry Thus we have 1 1 e xlye 0 137 xly e heo e 16 Here the first line is a consequence of choosing the central trajectory as the reference axis while the second and third lines follow directly from considerations of median plane symmetry Since they will appear often in the formalism it is convenient to introduce the following abbreviations for the first order Taylor coefficients x xe e t Sx t 6 d t cy t yly e sx t 17 yl ye When the transverse position of an arbitrary trajectory at position t is written as a first order Taylor s expansion as a function of the initial boundary conditions the above five quantities are just the coefficients appearing in the expansion for the transverse coordinates x and y as follows x t s t x e d4 t 6 higher order terms and y t c t ye S t y higher order terms III Solution of the Equations of Motion The general differential equation of motion of a charged particle ina static magnetic field valid to all orders in x and y and their derivatives as
175. t Some typical R1 matrix constraints are as follows Desired optical condition Typical fitting constraint Point to point imaging Horizontal plane R 12 0 10 1 2 0 0001 F1 Vertical plane R 34 0 10 3 4 0 0001 F2 Parallel to point focus Horizontal plane R 11 0 10 1 1 0 0001 F3 Vertical plane R 33 0 10 3 3 0 0001 F4 Point to parallel transformation Horizontal plane R 22 0 10 2 2 0 0001 F5 Vertical plane R 44 9 10 4 4 0 0001 Achromatic beam Horizontal plane 10 1 6 0 0001 F7 R 16 R 26 0 10 2 6 0 0001 F8 Zero dispersion beam Horizontal plane R 16 0 10 1 6 0 0001 F9 Simultaneous point to point and waist to waist imaging Horizontal plane 10 1 2 0001 F10 R 12 R 21 9 10 2 1 0 0001 F11 Vertical plane 10 3 4 0 0001 F12 R 34 R 43 0 10 4 3 0 0001 F13 Simultaneous parallel to point and to waist transformation Horizontal plane 10 1 1 0 0001 F14 R 11 R 22 0 10 2 2 0 0001 F15 Vertical plane 10 3 3 0 0001 F16 R 33 R 44 0 10 4 4 0001 F17 R2 matrix fitting constraints pag 76 There are five parameters to be specified when imposing a constraint upon the I j element of an R2 matrix Type code 10 n Code digit 20 i Code digit j
176. t 1 e nhx 4 H 1 pe See SLAC 75 page 31 The quantities L B O and n may be varied for first order fitting see type code 10 0 for a discussion of vary codes where The bend angle in degrees and the bend radius in metres are printed in the output A typical first order TRANSPORT input for a wedge magnet is not to exceed 4 spaces 4 1 B e o H If fringing field effects are to be included a type code 2 0 entry must immediately precede and follow the pertinent type code 4 0 entry even if there are no pole face rotations Thus a typical TRANSPORT input for a bending magnet including fringing fields might be o em not to exceed I 4 spaces if desired 2 e 4 L B 0 n 2 2 0 e 3 For non zero pole face rotations a typical data input might be 25 10 4 L B 0 P 2 20 Note that the use of labels is optional and that all data entries may be made on one line if desired The sign conventions for bending magnet entries are illustrated in the following figure For TRANSPORT a positive bend is to the right looking in the direction of particle travel To represent a bend in another sense the coordinate rotation matrix type code 20 0 should be used as follows A bend up is represented by rotating the x y coordinates by 90 0 degrees about the z beam axis as follows Labels not to exceed 4 spaces 1 if desired 20 90 3 2 1 s 4 L B n
177. tatic magnetic beam transport problems In principle the second order matrix formalism my be extended to any order but in practice this approach has proved to be too cumbersome Thus beyond second order it has been more efficient to use computer ray tracing programs which integrate the basic differential equation of motion of the charged particles through the known or assumed magnetic fields The fundamental difficulty with ray tracing has been the required computational time to complete a design involving the minimization of many higher order aberrations 133 In this report we will outline a systematic procedure for the design of high resolution systems based upon the extrapolation of the first and second order theory 1 2 3 to include higher order multipole components A general equation has been derived for the coupling coefficient of an nth order multipole to any given nth order aberration coefficient As will be shown later these coupling coefficients are a function only of the characteristic first order trajectories matrix elements introduced and defined in References 1 and 2 Given this information a systematic procedure for designing high resolution beam transport systems is as follows 1 Find a satisfactory first order solution to the problem using TRANSPORT or its equivalent 2 Calculate and make the necessary corrections to the second order aberrations by introducing sextupole components into the system The best lo
178. tem to the target position Beyond the last lens in the field free region preceding the target Ri a constant if 0 thus we conclude from Eq 59 that in this field free region the minimum spot size achievable at a target is a waist and is independent of the target position Such a system is a Zoom lens 121 2 If there are no lenses beyond the variable lens system i e R is an entirely field free region a drift then is of the form 1 L Raz i 3 In this situation RR L is a positive quantity consequently the waist always precedes a minimum spot size at a target A case of particular interest is when the minimum spot size achievable is equal to the initial beam size at the lens It then follows from Eq 60 that Z L 2 i e a waist occurs midway between the lens and the target From Eq 61 the ratio of the size of the beam at the lens and at the waist is x lens O4 lens __ x waist qlo waist v2 63 Combining this result with Eq 59 L R x lens x min _ lens 011 2 x waist waist or _ 2x waist L O0 waist 64 where L is the longest distance a beam can drift without exceeding its initial size at the lens Imaging from an Erect Ellipse to an Erect Ellipse The general sigma matrix for imaging from an erect ellipse to an erect ellipse may be derived by inspection from Eq 36 by setting 02 1 6 0 0 The result is _ Ri10 11 0
179. the acceleration type code 11 0 element SEXTUPOLE Type code 18 0 Sextupole hexapole magnets are used to modify second order aberrations in beam transport systems The action of a sextupole on beam particles is a second and higher order effect so in first order runs absence of the 17 0 card this element will act as a drift space There are four parameters 1 Type code 18 0 2 Effective length metres 3 Field at pole tips kG Both positive and negative fields are possible see figures below 4 Half aperture cm Radius of circle tangent to pole tips Other orientations of the sextupole may be obtained using the beam rotation element type code 20 0 The pole tip field may be varied in second order fitting It may also be constrained not to exceed a certain specified maximum field See the explanation of vary codes in the section on type code 10 0 Such a constraint allows one to take into account the physical realities of limitations on pole tip fields See SLAC 75 for a tabulation of sextupole matrix elements TRANSPORT input format for a typical data set is Label if desired not to exceed 4 spaces 18 din o 76 310dnix3as 310dnuavno DIPOLE QUADRUPOLE SEXTUPOLE ILLUSTRATION OF THE MAGNETIC MIDPLANE X AXIS FOR DIPOLE QUADRUPOLE AND SEXTUPOLE ELEMENTS THE MAGNET POLARITIES INDICATE MULTIPOLE ELEMENTS THAT ARE POSITIVE WITH
180. the printed TRANSPORT output when calculating except for the repeated listing of the elements they control Vary codes may be used within a repeating unit in the usual fashion However all repetitions of a given varied element will be coupled Repeat cards may be nested four deep By nesting we mean a repeat within a repeat An example is given below Example of Nesting 9 2 3 10 9 3 3 20 9 4 3 50 Outer 9 block 9 3 9 The total length of this sequence is 2 10 3 29 4 50 1 5 1343 VARY CODES and FITTING CONSTRAINTS Type code 10 0 Some not all of the physical parameters of the elements comprising a beam line may be varied in order to fit selected matrix elements In a first order calculation one might fit elements of the R1 or R2 transformation matrices or the beam sigma matrix In second order one might constrain an element of the second order matrix T1 or minimize the net contribution of aberrations to a given beam coordinate Special constraints are also available 51 One may not mix orders in fitting First order vary codes and constraints must be inserted only in a firs calculation and similarly for second order The physical parameters to be varied are selected via Vary Codes attached to the type codes of the elements comprising the system The fitting constraints on matrix elements are selected via type code 10 0 entries placed in the system where the constraint
181. tting COMMENT CARDS Comment cards may be introduced anywhere in the deck where an element would be allowed by enclosing the comments made on each card within single parentheses No parentheses are allowed within the parentheses of any comment card The comments are not stored but appear only in the initial listing of the given problem step Example of the use of comment cards in a data set Title Card 0 THIS IS TEST PROBLEM TO ILLUSTRATE THE USE OF COMMENT CARDS elements COMMENTS MAY ALSO BE MADE BETWEEN TYPE CODE ENTRIES elements SENTINEL LISTING OF AVAILABLE TRANSPORT TYPE CODE ENTRIES INPUT BEAM Type code 1 0 The phase space and the average momentum of the input beam for a TRANSPORT calculation are specified by this element The input is given in terms of the semi axes of a six dimensional erect beam ellipsoid representing the phase space variables x 0 and 6 Each of these six parameters is entered as a positive quantity but should be thought of as x 0 etc hence the total beam width is 2x the total horizontal beam divergence is 2 and so forth Usually the BEAM card is the third card in the deck If other than standard TRANSPORT units are to be used the units specification cards type code 15 0 should precede the BEAM card Standard TRANSPORT units for x y and are cm mr cm mr cm and percent The standard unit for the momentum p O is GeV c Also if a beam
182. use the magnetic array bending magnet fringing fields consists of three type code elements instead of one Example 2 A bending magnet with an uncertain position A bending magnet having an uncertainty of 2 mr in its angular positioning about the incoming beam z axis would be represented as follows 3 L 1 6 2 s 2 e 4 L B n 2 e 8 2 0 002 5 3 L 2 F To observe the uncertainty in the location of the outcoming beam centroid the input BEAM card should have zero phase space dimensions as follows 1 0 0 0 9 p 0 If the beam dimensions on the input BEAM card are non zero the resultant beam sigma matrix will show the envelope of possible rays including both the input beam and the effect of the misalignment Example No 3 A misaligned quadrupole triplet One typical use of both the R1 and R2 matrices is to permit the misalignment of a triplet For example an uncertainty in the positions within the following triplet 5 1 8 10 5 2 7 10 5 1 8 10 may be induced by appropriate 8 elements as noted 6 0 1 5 1 8 10 6 0 2 5 2 7 18 3 5 1 8 10 a es 000 a 002 O01 The first 8 card in the list refers to the misalignment of the third magnet only The second 8 card refers to the misalignment of the second and third magnets as a single unit via the R2 matrix update t
183. v 00064678 e 930 00000 081 02 13 0000871 930 00000 081 oc uw 0006578 ui 86699 81 14 0000002 Eir v 0006678 930 000007081 oc 14 0000S 000000 000000 0000070 000000 00000 0 00000 1 00000 00000 80 20 09800 000000 6 00 82195 00000 00000 000000 t 9vc c 19 00000 00000 0000040 00000 00000 10000 veLvtl 0000070 00000 00000 9281 9 v98tv n 826 WHOdSNYUIS O6r IEZ 103 Z 06 IEZ LVLOU O6 IEZ N38 06v IVIOUx 112 104 Z O6r 112 IJIUI 066602 7 066602 LVLOUF 066 602 ON38 066 681 IVIOU IVIOUS 06v 881 N38 06 891 IVIOUx 0657 891 Z 069891 IdIU0 WHOISNVUL REPETITION Type code 9 pag 67 Many systems include a set of elements that are repeated several times To minimize the chore of input preparation a repeat facility is provided There are two parameters 1 Type code 9 0 2 Code digit If non zero it states the number of repetitions desired from the point it appears If zero it marks the end of a repeating unit For example a total bend of 12 degrees composed of four 3 degree bending magnets each separated by 0 5 metres could be represented by 9 4 4 5 3 5 3 9 0 3 Those elements in this case a bend and drift between the 9 4 and 9 O would be employed four times There is no indication of the 9 0 cards in
184. wer as the magnetic flux inclosed per unit phase space area per unit momentum Bp of the central ray First Order o Matrix Phase Ellipse Formalism for TRANSPORT In accelerators and beam transport systems the behavior of an individual particle is often of less concern than is the behavior of a bundle of particles the BEAM of which an individual particle is a member An extension of the matrix algebra of Eq 3 provides a convenient means for defining and manipulating this BEAM TRANSPORT assumes that the bundle of rays constituting a BEAM may correctly be represented in coordinate phase space by an ellipsoid whose coordinates are the position angle and momentum coordinates of the arbitrary rays in the beam about an assumed central trajectory TRANSPORT is a matrix calculation that truncates the problem to either first or second order in a Taylors expansion about the central trajectory Particles in a BEAM are assumed to lie within the boundaries of the ellipsoid with each point within the ellipsoid representing a possible ray The sum total of all phase points the phase space volume is commonly referred to as the phase space occupied by the BEAM The validity and interpretation of this phase ellipse formalism must be ascertained for each system being designed However in general for charged particle beams in or emanating from accelerators the first order phase ellipse formalism of TRANSPORT is a reasonable representation of physical re
185. will represent the beam line before any fitting has occurred The second will be based on the new values of the physical parameters which were altered by the fitting process If sequential fitting is employed and an indicator card of two 2 is used the first run will be omitted The user should read the section describing the indicator card for further explanation In any listing the elements are printed in order with their labels and physical parameters Elements with negative type code numbers are ignored Each listed element is preceded by the name of that type of element enclosed in asterisks All physical elements are listed in this way Some of the other elements are not explicitly listed but produce their effect in either the calculated quantities or the listing of the beam line For descriptions of individual cases the reader should consult the sections on the type codes Calculated quantities appear in the listing as requested in the input data Important cases will be described in greater detail below The physical parameters for each element are printed with the appropriate units For some elements a calculated quantity not in the input data will appear enclosed in parenthesis Such quantities are explained in the sections under the individual type codes Calculated quantities The important cases of calculated quantities which appear in the output are the transfer matrices the beam matrix the layout coordinates and the results
186. y any given second order aberration But in practice the second order aberrations are usually minimized by only introducing sextupole components so as not to disturb the first order optics of the system It should always be kept in mind however that it may be beneficial to go back and change the first order solution optical mode so as to provide a more favorable situation for correcting aberrations a wise selection 126 of the first order optical mode may in many instances be the deciding factor between the success or failure of a design For a fixed location of a sextupole component the partial derivative of any second order aberration coefficient Tijk with respect to the strength 5 of a given sextupole component is a constant i e Tijk 8 S T hus minimizing a selected group of aberrations is a straight forward problem of solving a set of simultaneous linear equations once the coupling coefficients are known a constant the coupling coefficient of S to The strengths of the sextupole components may be determined directly by TRANSPORT The user may either constrain certain second order matrix elements to certain values or may minimize the net second order contributions to a given component of the beam ellipsoid Second Order Phase Ellipsoid Formalism It will be noted by the user that a second order TRANSPORT calculation modifies the phase ellipsoid printout In a second order run TRANSPORT calculates
187. y plane in the middle of the magnet Figure 3 The Principal Planes of a Simple Bending Magnet are Coincident with the Center Plane The simplified matrix for a bending magnet measured to the principal planes is then 100 s 1 s 8 0 0 1 manca quadrettatura To calculate the transformation matrix for the entire Beam Switchyard system as shown in Fig 2 we write the matrices in opposite order from that in which the beam passes through the elements That this must be true can be seen from the way in which one element alone is calculated by Xi le Ri le 9 6 6 Then for a second element we have 1 10 and so forth If we allow the system to be symmetrical i e 5 5 and L the complete series of matrices for Fig 2 are Resv s 1 1010 01 0 5 1 s 11 0 0 1110 0 1 0 0 1110 0 1 manca quadrettatura We will show the step by step multiplication of the matrices to get The d and d terms 1 0 Ofl L 0 1 L 0 1 do 98 1 L 0 1 0 0 1 L Ls 5 1 s 1 f 1 0 1 s 0 0 1 0 0 1 0 0 1 17 04 14 La 1 2s La 1 2 s 1 La 27 2 1a 2 0 0 1 manca quadrettatura Rasv manca quadrettatura To obtain the required condition that d d 0 we set 2 2 f Then 1 Resy 2 1 1 1 0 13 0 0 1 manca quadrettatura Thus the quadrupole acts as a lens to refocus
188. ydarby sx _ t zl Genet sr sya 26 Examples Assume a situation where the end of the system is a point to point image or the origin i e 5 0 then using equation 26 the coupling coefficients of a sextupole of strength 5 to various second order aberration coefficients are 6 6 _ 24 itl 094 Slx lt _ 3 y t Slx y o _ 2c t cys Sy 27 etc Where the Green s function used in these examples is Gi Sx t cx c t s 5 since Sx t for point to point imaging The aberration and c t are evaluated at the end of the system c t is equal to the magnification M in the examples given The remaining coefficients Sx Sy and d are evaluated at the location of the sextupole 5 The above results in agreement with Table VII of Ref 1 To illustrate a more complex example consider the fourth order aberration coefficient y Iy y o 8 and assume parallel to point imaging in the y coordinate i e cy t 0 The appropriate Green s function is 140 Sy t cy cy t sy sy t cy and the coupling coefficient to a fourth order multipole of strength S is Iy Y o 41 7 85 Sy t cySydx 28 where again the aberration coefficient y Iy y o and s t are evaluated at the end of the optical system and sy and d are evaluated at the location of the fourth order multipole S V A Systematic Procedure for Design

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