Grating Electromagnetic Theory User Guide
Contents
1. AIR Figure 12 Bragg Fresnel linear zone plate The abscissae xn are e and the wavelength Figure 13 Crossed grating which depends on the groove shape slit mirror etc While such an approach can predict that 100 of the inci dent energy can be concentrated inside one order it is not capable of fully taking into account the polarization effects neither can it account for the existence of Wood anomalies Although improved by several authors such an approach fails in the resonance domain where A d ra tio is close to unity typically lying between 0 2 and 5 The approach can be used only in the short wavelength regime and even there can be erroneous under high incident angles However the scalar approach is capable of bring ing some physical insight and making good predictions for near normal incidence Because the approach is not reliable in general we concentrate here on electromag netic theories starting with the simplest The Rayleigh Theory Let us consider the grating illustrated in Fig 1 with an incident plane wave having an electric field vector parallel to Oz axis TE P or s po larization The z component of the electric field can be written as El x y A exp ik x sin 0 y cos 6 assuming exp iwf time dependence Such function obvi ously satisfies a pseudo periodicity property qual to x Vna where a 4 FA cos Op F is the focal length E d y E x yex2. JOURNAL OF IMAGING SCIENCE AND TECHNOLOGY Volume 41 Number 4 July August 1997 amp amp Grating Electromagnetic Theory User Guide M Nevi re and E Popovt Laboratoire d Optique Electromagn tique case 262 Facult des Sciences et Techniques de Saint J r me Av Escadrille Normandie Niemen 13397 Marseille Cedex 20 France Recent state of art applications of diffraction gratings and stratified materials with one or several modulated interfaces impose spe cific requirements on electromagnetic grating theory A review of such applications is presented here with an emphasis on the aspects of the theoretical methods required for their efficiency prediction and optimization A brief review of the basic ideas of the various electromagnetic theories used is given The specific domain of validity for each theory is discussed together with advantages and shortcomings The aim is to serve as a guide in selecting the most appropriate theoretical method for handling specific grating problems Journal of Imaging Science and Technology 41 315 323 1997 Introduction Two centuries after the discovery of gratings by Ritten house gratings and more complicated periodic structures have become common not only in spectroscopy but also in numerous domains of physics such as acoustics solid state physics nonlinear optics x ray instrumentation optical communications and information processing Moreover gratings began to appea
3. Basic Principles of Rigorous Electromagnetic Theories Until recently grating properties were taught in uni versities in the frame of scalar optics Grating were as sumed to be a periodic collection of slits or small mirrors and their diffraction phenomenon was analyzed with the Kirchhoff diffraction theory in the Fraunhofer approxi mation The Fraunhofer equation which determines the direction of diffracted orders was established and the dis persive properties of gratings were taught But little was said about the distribution of energy among the various propagating orders The normalized intensity was found to be a product of the normalized interference function and the normalized intensity function I of a single slit The intensity function falls off its maximum slowly in com parison with the interference function Thus the intensity diffracted by the grating was found to consist of sharp peaks due to the interference function modulated by Io Vol 41 No 4 July August 1997 317 Va Figure 9 Bragg Fresnel grating v and v are the refractive indices of the low and high index layers which have thicknesses e and e respectively and D is the period of the multilayer D e e3 Figure 10 Multilevel grating Figure 11 Dammanan grating burnal of Imaging Science and Technology Nevi re and Popov caama a l Grating Electromagnetic Theory User Guide
4. 10 1996 fon leave from the Institute of Solid State Physics Bulgarian Academy of Sciences 72 Tzarigradsko Chaussee Blvd 1784 Sofia Bulgaria 1997 IS amp T The Society for Imaging Science and Technology be used further Such gratings are usually produced by diamond ruling Holographic recording beam etching and lithographic methods produce in general symmetrical pro files with sinusoidal lamellar rectangular or trapezoi dal grooves Electron microscopy reveals that for very high groove frequencies e g 6000 groove mm as well as very low ones e g echelles widely used in astronomy the real profiles have much more complicated form different from the classical four types mentioned below In far infrared and millimeter range of wavelengths the metal can be assumed to have infinite conductivity In the visible region and for shorter wavelengths the finite con ductivity complicates the grating response and requires different theoretical methods Transmission gratings are frequently used as beam split ters and as grisms Fig 2 to transform a telescope or a camera into a spectroscope Under special conditions the diffraction compensates the refraction and a dispersive diffraction order can propagate in the initial direction Fig 2 Transmission gratings require a transparent lossless grating material in contrast to reflection gratings Grating couplers combine a dielectric grating and an optical waveguide Fi
5. Rigorous coupled wave analysis of dielectric surface reliet grat ings J Opt Soc Am 72 1385 1392 1982 C Botten M S Craig R C McPhedran J L Adams and J R Andrewartha a The dielectric lamellar diffraction grating Opt Acta 28 413 428 1981 b The finitely conducting lamellar diffraction grating Opt Acta 28 1087 1102 1984 a J R Andrewartha G H Derrick and R C McPhedran A general modal theory for reflection gratings Opt Acta 28 11 1501 1516 1981 b J Y Suratteau M Cadilhac and R Petit Sur la d termination num rique des efficacit s de certain r seaux di lectriques profonds J Optics Paris 14 273 282 1983 L Li Multilayer modal method for diffraction gratings of arbitrary pro file depth and permittivity J Opt Soc Am A 10 2581 2591 1993 D Maystre a Sur la diffraction d une onde plane par un r seau m tallique de conductivit finie Opt Commun 6 50 54 1972 b A new general integral theory for dielectric coated gratings J Opt Soc Am 68 490 495 1978 J Chandezon D Maystre G Raoult a A new theoretical method for diffraction gratings and its numerical application J Optics Paris 14 235 241 1980 b J Chandezon M T Dupuis G Cornet and D Maystre Multicoated gratings A differential formalism applicable in the entire optical region J Opt Soc Am 72 839 846 1982 G Derrick R McPhedran D Maystre and M Nevi re Crossed
6. a sone vey er eses sne Modal eey me oy sees 0 sosse integral verse t wasee tae tee aiii ians jis Coordinate transform wess s N arees ter sesse 0 tease Conformal mapping TABLE IL Ability shown by Asterisks of Grating Theories to Deal with Exotic Gratings Problems Nonlinear optics Crossed SHG in SHGin Kerr Methods Bare Bare Multilayer Phase Multilevel Dammann Multilayer Anisotropic Bragg X ray metalic dielectric x ray gratings gratings gratings coated chiral etc Fresnel gratings dielectrics metals effect domain echelle echelle echelle holograms gratings media gratings Rayleigh nee ee Differential sesse arnee sasoe tee nase tees T ne s eses s wrens tener Moharam seste see aen anew amp Gaylord Modal u res see se ae Integral e wers asses sess s wee sesse nm on Coordinate teeee tentel aeiee whee ete ee transform SHG second harmonic generation the coordinate transformation suitable for these problems is consideredin Ref 20 method ete but we limit ourselves here to the methods that have proved themselves in tackling many grating problems and that have the corresponding computer codes in regular use Grating Theory User Guide The diversity of grating problems and existing theories today is so great that it is not obvious for an engineer or a scientist which method is best suited to resolve the prob lem he encounters in his work To try to hel
7. lamellar one only if represented as a few rectan gular steps Recently an interesting generalization to arbi trary profiles has been proposed The main advantage is that each of the modes represents a solution of Maxwell s equation and the boundary conditions inside the different media of the corrugated region so evaluation of the elec tromagnetic field characteristics inside the grooves becomes Nevi re and Popov possible with great precision The main difference in com parison to the classical differential method and the method of Moharam and Gaylord is that the classical model method does not require a Fourier representation of permittivity and the field components at both sides of the corrugated inter face and so can easily deal with highly reflecting surfaces Integral Theory The integral method was the first rigorous grating theory The method was developed by sev eral authors in circa 1966 for perfectly conducting gratings and generalized in 1972 for finite conductivity The ba sic principle of the integral method can be understood more easily for perfectly conducting substrates When an inci dent plane wave falls on the grating surface the wave in duces a surface current js M at each point M of the surface When propagating along the grating the surface current radiates a diffracted field E P at a given point P above the surface Provided jg is known E P can be found through the Kirchhoff Huygens formula us
8. of the permittivity increases infinitely and Eq 2 cannot be used Conformal mapping is used then to transform the grating surface into a plane making the corrugated surface equivalent to a phase grating on a plane surface which latter problem can be easily resolved by the differ ential theory Method of Moharam and Gaylord For lamellar rectangular laminar profiles the function e x y does not depend on the vertical coordinate y and a solution of Eq 5 or its equivalent for TM polarization can be found with out numerical integration using the eigenvalue eigenvec tor technique The field in the modulated region is represented as a superposition of modes in the form Ca exp i ce y where are the eigenvalues of matrix V The unknown coefficients c are determined from the boundary condi tions at the limits of the modulated region y 0 and y a This method was originally called the modal method and nowadays is known as rigorous coupled wave theory The theory uses the same differential equations and basic functions as the differential theory The difference is in the numerical method specific to the lamellar profile Because each profile can be more or less precisely repre sented in a staircase approximation the method of Moharam and Gaylord has been generalized to arbitrary profiles It then appears to be quite similar to the classi cal differential theory While in differential theory the discretization of x
9. shape are used in integrated optics and in the x ray domain for focusing and beam shaping If only the groove period is varied we ob tain a linear zone plate When etched inside a planar mul tilayer structure it is called a Bragg Fresnel linear zone plate Fig 12 used in x ray microscopy and spectroscopy Some gratings have discontinuous profiles e g grid or rod gratings made of dielectric or metallic rods The latter are used as frequency filters in infrared selective absorb ers of solar energy or radar furtivity In photolithography chromium masks are made of rectangular rods strips sometimes called Ronchi gratings and are used to cast their shadow onto a photoresist layer which represents a combination of a rod grating and several plane layers Phase or volume gratings do not contain corrugated in terfaces but rather a plane layer with a periodic modula tion of the refractive index The result is that lamellar and rectangular rod gratings belong to both relief and phase grating types The next complications appears when going to two di mensional geometry with a modulation of the index or cor rugation of the surface made in two directions The result is called a crossed grating as illustrated in Fig 13 Such devices can be useful in solar absorption beam splitting and memory storage Their other name bi grating is often confused with a one dimensional grating having two differ ent periods of modulation Nevi r
10. 62 1973 a M Nevi re G Cerutti Maori and M Cadilhac Sur une nouvelle m thode de r solution du probl me de la diffraction d une onde plane par un r seau infiniment conducteur Opt Commun 3 48 52 1971 b M Nevi re P Vincent and R Petit Sur la th orie du r seau conducteur et ses applications a l optique Nouv Rev Opt 5 65 77 1974 c P Vincent Differential methods Chap 4 in Electromag Grating Electromagnetic Theory User Guide B te 10 14 12 14 15 17 18 20 netic Theory of Gratings R Petit Ed Springer Verlag Berlin 1980 L Li Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings J Opt Soc Am A 13 1024 1035 1956 M Nevi re and F Montiel Electromagnetic theory of multilayer grat ings and zone plates Chap 6 in Multilayer and Grazing incidence X Ray EUV Optics lll R B Hoover and A B C Walker Eds Proc SPIE 4996 A K Cousins and S Gottschalk Applications of the impedance for matism to diffraction gratings with multiple coating layers Appi Opt 29 4268 4271 1990 M Nevi re M Cadilhac and R Petit Applications of conformat map pings to the diffraction of electromagnetic waves by a grating IEEE Trans Ant Propag AP 21 37 46 1973 M G Moharam and T K Gaylord a Rigorous coupled wave analy sis of planar grating diffraction J Opt Soc Am 71 811 818 1977 b
11. be a continuous function so edges are excluded but this is true of all the electromagnetic meth ods and fortunately nature does not allow edges In prac tice the slope of the steepest part of the profile is much more important when g x becomes large not only does its Fourier representation slowly converge but the trans formation of the coordinate system degenerates The de rivative with respect to the first and second coordinates tend to each other 3 3 a X However numerically this limitation is not so severe as appears sinusoidal gratings with h d gt 10 have a very steep slope but only along a limited part of the profile and can be successfully treated while lamellar gratings are excluded by definition and triangular gratings with limited asymmetry of the profile can be treated without difficulties Other theoretical methods for grating diffraction do ex ist like the conformal mapping technique for finite con ductivity the finite element method the fictitious sources Vol 41 No 4 July August 1997 321 TABLE I Ability shown by Asterisks of Grating Theorles to Deal with Simple Gratings Problems Methods Perfectly Dielectric Real metals Deep Deep Binary Dielectric Multilayer Low conducting gratings dielectric metallic gratings coated dielectric modulated gratings gratings gratings gratings gratings gratings Rayleigh gt erer Differential eens ae tases e e wesse ansie Moharam amp Gaylord sers
12. cterized by volume distributed sources which are difficult to include in the integral equation In addi tion working in real space instead of the transformed one requires numerical derivatives if all the field components are searched near the surface As a result of its generality the integral theory is able to deal with practically any kind of grating including some limiting cases where it is the only available method Ex amples here are echelle gratings used in 50 100 or even higher orders and at high angles of incidence and highly conducting very deep gratings with arbitrary profiles etc This advantage is obtained at the cost of more complex mathematics larger codes and longer computation times as well as larger memory storage requirements The com plexity of the theory also makes more difficult its adapta tion to phase gratings anisotropic media profiles with interpenetration etc Method of Coordinate Transformation An effort to combine the relative simplicity of the differential theory with the advantage of the integral theory of not crossing Grating Electromagnetic Theory User Guide the profile results in another approach based on a coor dinate transformation Ysy ag x 8 that transforms the grating surface y g x into a plane Y constant Maxwell equations can be integrated at each side of this plane separately and the solutions then matched along the surface Although the transformation is curvi
13. e and Popoy a oe oe ne _ __ Figure 5 Two examples of multi layer dielectric diffraction gratings a a i coe a A Figure 6 Grating coupler with double surface corrugation n NAAA le N E N z Si AAA pp Ne ee 1 gee Lee ee Figure 7 Stack of gratings for the XUV region Another problem that can be treated numerically using grating theories comes from the necessity to interpret the images of the photon scanning tunelling microscope When a grating surface is studied it is necessary to analyze the near field picture as diffracted by a periodic surface in Grating Electromagnetic Theory User Guide 0 d Figure 8 Multilayer coated grating presence of a single object namely the tip of the scanning device optical fiber This structure can be modelled by substituting the single tip with a collection of periodic tips with a distance much larger than the grating period to avoid parasitic coupling The geometry is then reduced to two gratings with different periods one being a multiple of the other In addition to geometrical complexity the gratings can be made of anistropic biaxial or chiral material Also they can be used in nonlinear optics for second harmonic gen eration Kerr effect and optical bistability etc This points out how complex grating problems may become The next section will explain how to tackle them
14. g 3 They are used in integrated optics to couple an incident beam into the slab or vice versa Dielectric coated gratings are mainly used in vacuum UV with a dielectric layer typically MgF to prevent y Figure 1 Schematic representation of a bare metallic grating k is the incident wave vector 315 Figure 3 A grating coupler aluminum from oxidation Fig 4 A similar technique is used in the visible region to protect silver gratings from tarnishing Multilayer dielectric gratings consist of a dielectric grat ing replicated on the top or at the bottom of a stack of plane dielectric layers with alternating high and low re fractive indices and a quarter of wavelength thickness Fig 5 They are used in high power laser optics to achieve peak absolute efficiency typically more than 96 at a single wavelength Another kind of multilayer grating consisting of a stack of modulated interfaces is used in the XUV and x ray do mains Their theoretical treatment requires us to distin guish between the two main cases profiles without and with an interpenetration No interpenetration occurs when the bottom of the upper profile lies higher than the top of the lower profile A grating coupler with double corruga 316 Journal of Imaging Science and Technology MgF gt Al Figure 4 Dielectric coated grating for VUV region tion Fig 6 is an example of a thin stack of identical pro files without interpenet
15. grat ings A theory and its applications Appl Phys 18 39 52 1972 Vol 41 No 4 July August 1997 323
16. ing the Green function technique so that E P GP M g M ds one grating 7 period where 9 M is proportional toj M G P M is the Green function which is known and the integration is carried over the curvilinear coordinate s along the grating profile However the problem is more complex than just a single integration because the surface current is not produced only by the incident wave but results also from the dif fracted field radiated from the other points of the grating Thus j 41 depends both on E and on the values of the current js M at each other point M of the profile differ ent from M This is why Eq 7 is transformed into an inte gral equation for M The situation becomes even more complicated for finitely conducting gratings The key point of the theory is the numerical resolution of an inte gral equation with a singular kernel Fortunately the singularities can be eliminated analytically and then the integral equation is transformed into a linear set of alge braic equations by discretization along the grating profile The greatest advantage of the integral formalism when compared to the differential method is that it follows the profile without crossing it As a result it is not neces sary to develop into Fourier series some quantities that exhibit a jump over the surface Sometimes however this can be a disadvantage as in the case of nonlinear dielec trics chara
17. linear and nonorthogonal fortunately it is possible to reduce Maxwell equations to a set of ordinary differen tial equations with constant coefficients which can be written in a matrix form dF ae TF 9 Because matrix T is independent of Y the solution of Eq 9 can be expressed in terms of eigenvalues Pm and eigenvec tors C of T Fm F Cmne by 10 5 After that the expansions Eq 10 above and below the corrugation are matched on the flat boundary Y const taking into account the appropriate outgoing wave condi tions The diffraction order amplitudes can be obtained by backward transformation of the basis Eq 10 after the amplitudes of the field expansion are determined Because this method does not require crossing the profile during a search for the solution but only when matching the dif ferent expansions it is able to deal with deep gratings independent of polarization and refractive index and in cluding multilayer gratings The method has relatively simple computer codes and short computation times com parable to those of the differential method The ease of incorporating several overcoating layers that follow the initial profile may be stressed The main limitation con cerns the types of profile with which the method can deal This limitation comes from the nature of the coordinate transformation 8 that does not allow noncontinuous func tions g x Strictly speaking the method requires the de rivative of g x
18. n ear optics Nevi re and Popov Conclusion The limitations advantages and potential of the most important grating theories have been discussed No theory is actually universal in the sense that no computer code that should be able to analyze any grating problem still exist But for most classical gratings problems several methods are available The aim of this paper is to guide scientists and engineers in their choice a References 4 2 3 D Rittenhouse An optical problem proposed by F Hopkinson and solved J Am Phil Soc 201 202 206 1786 M Born and E Wolf Principles of Optics Pergamon Press New York 1959 A Mar chal and G W Stroke Sur l origine des effets de polarization et de diffraction dans les r seaux optiques C A Acad Sci Paris B 249 2042 2044 1959 P Beckmann and A Spizzicerio The Scattering of Electromagnetic Waves from Rough Surfaces Pergamon Press New York 1 963 A Madden and J Strong Concepts of Classical Optics Freeman San Francisco 1958 O M Lord Rayleigh On the dynamical theory of gratings Proc Royal Soe London Ser A 79 399 416 1907 M Cadilhac Some mathematical aspects of the grating theory Chap 2 in Electromagnetic Theory of Gratings R Petit Ed Springer Verlag Bertin 1980 H Ikuno and K Yasuura improved point matching method with appli cation to scattering from a periodic surface IEEE Trans Ant Propag AP 21 657 6
19. on may appear when dealing with deep gratings or stacks of interpenetrating gratings Re cent developments gathered under the names of R ma trix and S matrix propagation algorithms eliminate this difficulty The S matrix algorithm is easier for use Similar difficulties can arise when a low modulation grat ing is combined with a thick stack of plane layers The problems are solved using similar methods the S matrix algorithm or the impedance method The only numerical problem that remains is linked with the truncation of the set in Eq 5 required for numerical implementation of the theory It assumes that the field is correctly described by a limited 2N 1 number of Fourier components in Eq 3 Obviously the number N will be different if the field and its normal derivative are continous across the grating sur face as it is in the TE case of polarization or strongly discontinuous as happens for the normal derivative for metallic gratings in TM polarization Numerical experi ence shows that the truncation parameter N can vary be tween 4 and 100 depending on the polarization groove depth and grating material Because computation time is roughly proportional to N it can vary by a factor of 15 000 depending on the problem Small N leads to very short computation times less than a second on most of the small workstations 320 Journal of Imaging Science and Technology Note that for perfectly conducting metals the modulus
20. oretically for profiles with edges the method can be used succesfully at low blaze angles for triangular grooves and for low groove depths for lamellar gratings In addition the variational formu lation of the Rayleigh hypothesis known as the Yasuura method is rigorous whatever the groove shape may be Differential Theory Instead of using two analytical expressions of E x y and matching them on the grating profile the differential theory distinguishes between the homogeneous regions y 0 a where Rayleigh expansion such as Eq 3 are valid and a modulated region 0 lt y lt a where a numerical integration is carried out In the modu lated region the Helmholtz equation is changed into AE x y kee x y E x y 0 4 Introducing Eq 1 into Eq 4 leads to a system of ordi nary differential equations with nonconstant coefficients of the form E y VOEO 5 where E y is a column vector with components E y and V y is a square matrix whose elements depend on the Fourier components y of the permittivity Vian Y Em n Y Yamn A numerical integration of Eq 5 is performed using a standard algorithm Matching the numerical solution with the Rayleigh expansions at the boundaries of the modu lated region y 0 and y a enables us to find the Rayleigh coefficients B in Eq 3 and thus the field everywhere Numerical instabilities from growing exponential func tions during the integrati
21. p him in this aim we first distinguish between two kinds of grating problems The first kind are simple grating problems and concern relief gratings made with homogeneous isotropic materials that are used in linear optics in the resonance domain The modulated region will be limited to one or a few less than 3 nonseparable interfaces plus a stack of plane interfaces and other separable modulated interfaces if any The second kind are exotic situations and include bare echelles multilevel and Dammann gratings A d lt lt 1 phase gratings multilayer coated gratings and echelles gratings ruled on anisotropic materials gratings used in nonlinear optics crossed gratings Bragg Fresnel gratings and any kind of x ray gratings Table I lists the relative merit of the theories used to confront the various types of simple grating problems The criteria of simplicity short computation time and univer sality groove shape TE and TM polarization etc are added to the performance of computation in view of at tributing the number of asterisks for each method The general impression is that if we except the low modulated case a grating with high but not infinite conductivity 822 Journal of Imaging Science and Technology will be analyzed through the integral theory or through the coordinate transformation method For dielectric grat ings the differential theory is preferred The same crite ria are used to construct Table II which gi
22. plijd with y k sin The Floquet theorem requires that the total field E x y is quasi periodical thus E x y exp iyd can be expanded into Fourier series i e E x y SE y expliy x a n with Yn Yy nK 27 K d In each homogeneous region defined by y g 0 a Max well equations lead to a Helmholtz equation AE x y kB v7 E x y 0 2 where v is the refractive index of the medium ky w e and c is the light velocity Substitution of Eq 1 into Eq 2 leads to an analytical expression for E y E x y A exp i yox Boy 5 B expli y Bry 3 n with Br kov Ya In metals the corresponding expression includes the metal complex refractive index unless the metal is perfectly conducting Expression 3 is called Rayleigh expansion The Rayleigh method assumes that Eq 3 is valid not only out side the modulated region but also inside the grooves This Vol 41 No 4 July August 1997 319 enables one to explicitely write the boundary conditions but this hypothesis has been shown to be not valid in gen eral A detailed discussion can be found in Ref 7 In par ticular the Rayleigh method is valid for sinusoidal perfectly conducting gratings when the groove depth to period ratio does not exceed 0 142 Outside this domain however some times the method gives acceptable results for the far field while the errors in the near field are greater Although never valid the
23. r in common use in CD players as safety features on credit cards and bank notes as well as in variety of display and advertising applications The term grating is no longer restricted to periodically modulated surfaces and is also used for modulated devices with varying periods curved grooves or varying groove shapes such as Fresnel planar lenses and diffractive op tic elements Such structures are used in integrated op tics to achieve beam focusing and beam shaping Their diffraction properties can still be related to the correspond ing gratings as far as an adiabatic variation of the groove geometry can be assumed i e the groove change is car ried out over a large number of grating periods The following section reviews the various types of grat ings quasi gratings and periodically modulated stratified media in use in various domains of science The next sec tion presents a brief description of the most commonly used theoretical diffraction methods The final section before the conclusion provides a user guide to grating theories devoted to helping scientists select the method most suit able for a given particular problem Review of the Grating Problems The classical grating problem consists of a bare metal lic grating used in reflection for spectroscopic purposes Figure 1 represents a triangular groove grating usually called an echelette grating and defines some notations to KEEFER Original manuscript received November
24. ration It can be optimized to couple as high as 80 of the incident beam into the waveguide In a more complicated case Fig 7 the high diffraction efficiency 45 in the XUV region wavelength A 15 nm is obtained by combining profiles with interpenetration e g 2 and 3 and without e g 1 and 2 The case of iden tical profiles is called a multilayer coated grating and the layers are usually thinner than the profile depth i e in terpenetration occurs Fig 8 Such gratings are used in the two extremities of the spectrum Another alternative high efficient grating for x ray is the Bragg Fresnel mul tilayer grating as shown in Fig 9 Echelles are echelette gratings used in high orders typi cally 10 to 500 They can be bare Fig 1 or multilayer coated Fig 8 depending on the spectral domain The low wavelength to period A d ratio requires special theoreti cal methods The multilevel gratings used in diffractive optics to pro duce multiple beams from a single incident ray are illus trated in Fig 10 Usually they also work with extremely low Md ratios In comparison Dammann gratings as illustrated in Fig 11 consist of several bumps of different width within a single period Not only do they also work in a low A d region but in addition their profile is more complicated to introduce in grating theories Dammann gratings have also become a common element in diffractive optics Gratings with varying period or groove
25. ves an idea of the versatility and universality of the theories when con fronted with new problems To illustrate how these tables can be used let us go back to the grating problems described in the above section Review of Grating Problems and give our preferred theory for each problem Keep in mind that in most cases several theories could work The metallic grating in Fig 1 would be studied with the integral method the transmis sion gratings in Figs 2 and 3 with the differential theory as well as the dielectric coated grating for VUV region except if the groove depth is high in which case we would go back to the integral method The multilayer dielectric grating in Fig 5 would be studied with the differential method the lamellar grating coupler in Fig 6 with the Moharam and Gaylord method The stack of gratings in Figs 7 and 8 as well as the Bragg Fresnel gratings and zone plate in Figs 9 and 12 would be studied with the differential method the multilevel grating in Fig 10 with the integral method the Dammann grating in Fig 11 with the differential theory the crossed grating in Fig 12 with coordinate transformation In addition phase gratings and holograms would be studied with the differential tech nique and grid gratings with the differential or integral theory depending on whether the rods are dielectric or metallic These examples give an idea of the choice that should be made for various problems encountered in li
26. y is made during the numerical inte gration the Moharam and Gaylord method makes the discretization by substituting the real profile with a staircase function Thus problems due to numerical in stabilities are quite similar in both methods as are the approaches for their resolution Classical Modal Method For a steplike lamellar rectangular profile use of the Fourier series expansion 1 is not even necessary with respect to the x axis A solution of the Maxwell s equations can be found in closed form in each of the grooves and lamellae in Cartesian coordinates Fala Y Up xe 6 so that the function u x has a different form inside the grooves and inside the lamellae determined by the opti cal index of the media The boundary conditions are then applied on the vertical groove walls Owing to periodicity a discrete set of values exist for p called modal constants The total field is then represented as a sum over all the modes of the corrugated system and the coefficients in the modal expansion are determined from the boundary conditions on the interfaces between the corrugated and the homogeneous media Unfortunately for highly conduct ing materials again the modal constants are spread over the complex plane and cannot be easily located Several different techniques have been proposed The main disadvantage of the modal method is that it is too narrowly specialized and can be applied to profiles other than the
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