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Documentation LabII

1. Input Output pulse in incoming laser pulse pulse reflected reflected laser pulse R1 reflectivity of front surface pulse transmitted transmitted laser pulse R2 reflectivity of back surface material material between front and back side d material thickness alpha angle of incidence The Fabry Perot interferometer consists of some material with two partially reflective coatings on both sides The material may be air or any of the transparent materials available in the data base The coatings are assumed to be ideal and impose no additional phase on the pulse The amplitude and the phase change for the transmitted pulse F and the reflected pulse E introduced by the Fabry Perot interferometer are exp id 2 i A 4 Er H jat 1 r r exp id a r r2 exp id oap 4 4 H H f riro A a where is the incoming laser pulse t and tz are the transmission coefficients for the transitions from air to the medium and the medium to air respectively The amplitude reflection coefficients are r y R and r y Ro 5 2d Z cos0 4 5 Co 1 sind sina 4 6 n where n is the index of refraction of the medium between the two interfaces w is the laser frequency co the speed of light in vacuum and the angle of propagation in the medium 32 M GirRES TOURNOIS INTERFEROMETER pulse in pulse out GT control eu Input Output pulse in incoming laser pulse pulse out outpu
2. MH Input Output pulse 1 the input pulses have to dep SFM cluster that is compatible pulse 2 be connected such that the parameters out to depleted SFM pulse 3 crystal wizard knows the group velocity ratio of cg over the group central wavelengths minimal SFM params everything but angles desired angular required precision precision essential SFM parameters velocities of the pulses information on which angle is the phase matching angle calculated phase matching angle indices out comment phase matching angle The crystal wizard is a tool to determine the optimal angular orientation of a nonlinear crystal for a specific sum frequency mixing situation Therefore it requires a control panel similar to that of the SFM Crystal depleted three wave mixing crystal but without theta or phi angles The output is a control panel compatible to that of SFM Crystal depleted three wave mixing Additional inputs are the three pulses of the three wave mixing and an angular precision value for example 0 05 deg The crystal wizard deter mines whether theta or phi is the phase matching angle finds the optimal phase matching angle and in addition optimizes the other angle for a maximal de 59 E AMPLIFIER Input amplifier contral pulse in r pulse out multipass in a multipass out Output TF pulse in input angle pump duration delay to 1st pulse pump fluence left pum
3. Input Output pulse in input pulse spectrum laser spectrum and min ionization minimum ionization atomic lines stage neutral 0 elements array of strings singly ionized 1 etc length equal to max ionization maximum ionization number of samples stage spectral resolution spectral resolution list of elements list of elements to include The module spectral lines allows to display atomic emission spectra together with the laser spectrum to find out what lines are within the laser bandwidth One needs to specify a list of elements for example Hg and Ar for a mercury lamp with argon buffer gas and the minimum and maximum ionization stage Ionization stage zero refers to the neutral state one to the singly ionized atom etc The spectral resolution determines the width of the atomic lines and is determined by the spectrometer used The module expects the following file structure Rb I Rb I Rb I Rb I Rb I Rb I Rb I Rb I Rb I Rb I 779 7691 300 775 9436 60 780 027 90000 792 526 5 792 554 4 794 760 45000 827 141 40 827 171 30 860 396 2000 886 8512 40 where the first line labels the ionization stage the second the transition wavelength in nm and the third the relative intensity The module can handle files downloaded from the NIST atomic database 92 Chapter 8 Optimizers Here we introduce five optimization algorithms i e a simplex downhill combined with simulated annealing two genetic algorithms an
4. The ladders parallel module calculates the space time evolution of a multi level quan tum system Fig 7 1 and its driving fields The multi level system can be described mathematically by the differential equations in the rotating wave approximation RWA 87 y gt AO Figure 7 1 Parallel connection of three level ladder systems The resonant transitions are driven by the electric fields Ea t 4 Ea t eo c c and E t 4 E t eet c c N amp 4 e 7 1 k 1 Ck t Wa Wkg Ck ike Cg i Gy 72 N Ce Oke gt W iy c i Qk Ck 7 3 k 1 Oke reEa t 2h and yy dpEg t 2h are the Rabi frequencies of the transitions g k and k 7 Since a decay process e g ionization is included in Eq 7 3 the atomic system is not closed for y gt 0 The evolution in space time of the driving fields 4 t and t is given by the simplified wave equation in the RWA Oy ak 10 s Mo WB CO 5 seta py em PE Bayn 1 Ho WA Co t ww where Pa y t and Pg y t are calculated by 88 N Paly t 2N X Lng Ch Ct k 1 N NY dyk Cy cp Y t k 1 Pay t and N is the particle density This means that the system of equations combines MAXWELLs wave equation with BLOCHs quantum mechanical description of the field matter interaction and has to be solved simultaneously In the case of Ex t Ep t E t only one wave equation has to be solved with w N
5. E t E t r for SD FROG All FROG s implemented here assume that the nonlinear crystal you are using has a sufficiently broad spectral acceptance to support the full laser bandwidth If you want to see what happens if this is not the case you may construct a FROG by using more fundamental LABI modules 63 E FWHM temporal FHM me FHM FUHH pulse out pulse in TF Input Output pulse in input pulse pulse out output pulse temporal FWHM FWHM of temporal intensity spectral FWHM FWHM of spectral intensity The FWHM in time module calculates the FWHM Full Width at Half Maximum of the temporal and spectral intensity distribution no matter what kind of slowly varying envelope the pulse has It may lead to funny results whenever the intensity is heavily structured 64 E INTENSITY temporal intensity FHM t y sigma t a pulse out Bae control pulse in Input Output pulse in input pulse pulse out output pulse rel abs relative or absolute intensity temporal intensity time axis FWHM t FWHM of the temporal intensity time step detector time step sigma t 2nd order moment of the samples number of samples temporal intensity The intensity module gives you a number of useful information on the pulse The pulse is going in on the left and coming out on the right so you may use it at any place in your virtual experiment Its outputs are a plot and two numbers The
6. Other crystals may be added upon request One may choose between type I ooe and type II eoe phase matching Choosing the mode delay 0 allows to ignore a possibly present time delay between the two input pulses With the sinc 2 theory mode it is possible to ignore effects on the mixing process caused by phase modulation This is achieved by simply using the absolute value of the phase matching term only 56 nwwa n wr w2 5 7 sinc Ak w we 1 5 8 n w4 wa Since this approximation is often found in textbooks especially when dealing with the mixing of three monochromatic fields this mode of operation allows to compare results of simulations with results given in the literature In addition the influence of a limited spectral bandwidth acceptance may be studied without the influence of phase modulation for example caused by a difference in the group velocities The calculated output waveform has twice the number of sample points as the input pulses which is caused by the convolution One may stick to that or one may reduce the number of samples by using every second sample point or all samples in only half of the spectral window 57 M SFM CRYSTAL DEPLETED THREE WAVE MIXING dep SFM parameters in of z steps pulse 1 e or ony pulse 2 o pulse 3 f e or o deff out y pulse 1 out ta pulse 2 out pulse 3 out evolution MH Input Output puls
7. Pa ly t Pgly t Ply t AN ikg Ch Cly t dak Cop cly t To solve numerically MAXWELLs wave equation and the differential equations for the probability amplitudes in particular for fast decay processes faster than the pulse du ration and a broad temporal window the Predictor Corrector Method slow modus is implemented ee Inth Sf tees Yn v 1 h v Yn 1 Yn Du Aea a 2n Yn 2 where v 0 1 resulting in yo Yn h f tn Yn h yy Yn z pon yo ks Yn h y2 Yn wv 2 UTENA T Pa In the case of the fast modus the first derivative is approximated by Ynt1 Yn 1 2h Toned 7 4 The minimal number of involved states is 3 one three level ladder To create a two level system only one dipole moment should be 0 To examine the field matter interaction without propagation e g the of z steps can be 2 Minimum the propagation length 0 005 mm and the particle density 1 x 10 cm 89 E DIATOMIC MOLECULE Diatomic molecule H pee potential surfaces p diff wavelength a pycited state axis re Coord state dynamics ground state axis excited state population ground state population pulse in TR Input Output pulse in input pulse potential surfaces the two Morse grid size spatial grid size potentials xmin minimum atom separation diff resonance wave dx spatial step size mass 1 mass of atom 1 mass 2 mass o
8. function 44 e Square The amplitude and or phase of the pulse is modulated with a periodic square function of amplitude e Sawtooth The amplitude and or phase of the pulse is modulated with a periodic sawtooth function of amplitude For sinusoidal square and sawtooth the parameters are amplitude A offset A period icity dt and phase The modulations are calculated according to flw Asin dt w wo o A where f stands for either amplitude or phase modulation and sin may be replaced by the square or sawtooth function A negative amplitude is converted to a phase value keeping in mind that 1 e AO fw lt 0 seo S fe f w gt 0 The cluster arbitrary contains three columns the absolute frequency with units the amplitude modulation a modulation of one leaves the original frequency component un changed and the phase modulation in radians 45 M PIXELATED SHAPER amplitude control phase contro arbitrary amplitude phase pulse out shaper window setup eae Input Output pulse in incoming laser pulse pulse out output pulse amplitude control type of amplitude modulation shaper window full spectral phase control type of phase modulation window arbitrary arbitrary amplitude and phase amplitude applied amplitude properties shaper properties modulation setup setup geometry phase applied phase modulation This pulse shaper has been designed to simulate a
9. group delay are zero as they have no influence on the pulse width 93 The feedback signal or the fitness is the FWHM of the signal That is the signal needs to be minimized in order to achieve the desired goal Also note that the actual experiment needs to be located in the false part of the vi The true section is called only once by the optimizer algorithm and this happens at the very beginning Here the boundaries of the search space have to be defined start vector o _ 1 00E 4_ 1 00 5_1 1 00E 6 defined search space out m o 1 00E 5__11 00E 6 Figure 8 2 Definition of the search space Figure 8 2 shows the true case where the search space for the optimization is de fined Allowed values for second order phases are 10 10 fs for third order phases 10 10 fs and for fourth order phases 10 10 fst respectively Figure 8 3 Front panel Figure 8 3 shows the front panel after optimization has been accomplished The view graph shows the original and the optimized pulse shape It is recommended to always open both the experiment vi in order to follow the progress and the optimizer vi Arrange them so that both front panels are visible and follow their progress as the optimization process runs Things are quite similar for the other four optimizer vis Again we will explain an example vi as shown in fig 8 4 The task is also quite similar A Gaussian shaped la
10. output pulse noise control type of noise Add noise adds a certain type of noise to the input laser pulse Presently the options are multiply add amplitude noise and multiply add phase noise Either the spectral amplitude or the spectral phase is multiplied with a white noise of specified amplitude An amplitude of 0 1 for example will add ten percent noise or multiply the existing amplitude by 1 0 1 noise depending on whether add or multiply is selected 22 M READ PULSE FROM FILE path dialog if empty sn pulse out decimalpoint a ee rere Input Output GS path path to file pulse out laser pulse decimalpoint set to false for instead of Read Pulse reads a pulse from a file The format is as follows nop auto wO THz diameter mm tO fsl frq E amp phi w fluence mJ cm number of samples simulation mode center frequency energy per unit area beam diameter first order phase frequency amplitude residual phase 23 M READ PULSE FROM SPREADSHEET auto avausnersentettesseenerenecees path pulse out delimiter Tab fluence or Gs Input Output path path to file pulse out laser pulse auto simulation mode delimiter tab default space etc fluence fluence value 2 n number of samples Read Pulse from Spreadsheet reads a pulse from a spreadsheet file
11. value the pulse has no phase modulation In our analytic Gaussian model this requires A 0 2 2 Short Pulse Propagation in the Linear Regime Linear elements are those which influence the spectral amplitude of the pulse or its spectral phase in a linear fashion This is almost always the case if the pulse passes through optical elements and the intensity is moderate Linear pulse manipulations in the spectral domain may be expressed as follows Eout w Ejn w H w ce 2 30 The spectral amplitude as well as the phase of the input field F w may be changed by H w and w respectively The generated output field is F w Almost all elements in the lab which happen to be in the laser beam are dispersive and modify the spectral phase of the pulse Dispersion in most cases is an unwanted side effect that one would rather avoid in the first place but sometimes it is used on purpose for example in CPA chirped pulse amplification The phase is usually expressed in terms of the coefficients of a Taylor series expansion OO 1 P w kl Oa CA I PR w wo k 0 ek ze w wo 2 31 k 0 with O 6 w w wo Let us inspect the first two terms a little more careful The one corresponding to k 0 represents the absolute phase of the pulse It is this term that determines the relative phase between the slowly varying amplitude and the rapid field oscillations The next term 13 corresponding to k 1 c
12. Electronics John Wiley amp Sons Chichester 1989 Chapter 2 Ultrashort Laser Pulses 2 1 General First of all we will restrict ourselves to the time dependence of the electric field and omit the spatial part This is equivalent to the viewpoint that a detector is located at a fixed position and the field only varies with time Since the electric field of the laser pulse is in principle a measurable quantity it must be real For most calculations however it is much more convenient to use the analytic signal a complex quantity and in a moment we will see why So we start with an electric field described by E t A t cos t A t e 0 1 A t e ee 2 1 where A t is time dependent amplitude and t the time dependent phase The Fourier transform of the electric field is E w F E t i dt E t e A w e 2 2 and the inverse Fourier transform is equivalently OO 1 E t FHE w 7 J dw E w e 2 3 T One might ask why do we consider the Fourier transformation of a time varying laser pulse There are many reasons to do this The spectral analysis yields information on the wavelengths contained in the pulse and also provides a hint whether a pulse may be compressed in time or is as short as it can possibly get The propagation of ultrashort laser pulses through dispersive media depends strongly on the frequency and finally the concept of Fourier transformation is a powerfu
13. File to save zE path to config file to read mm A nmm i error out error in File 110 settings l annealing parameter MH Input Output The simplex downhill is a combination of the deterministic simplex downhill algorithm and simulated annealing Simulated annealing does contribute the stochastic part to the oth erwise deterministic simplex downhill strategy and makes it a very powerful search algorithm for small search spaces Simulated Annealing amp Simplex 4 241 control current fies 2 96 1 o A ture too oF runs of nirs number of optimizations mani average Figure 8 7 Front panel Figure 8 7 shows the first control tab The input parameters control the simplex opti mization These are the annealing parameters k and a which determine the rate with which the annealing temperature decreases To 1 2 j lt k where j is the number of the actual experiment The temperature starts at T and goes all the way down to Tmin The input T is irrelevant The number of runs determines how many subsequent optimizations are performed A boolean button allows to determine whether the feedback signal or the fitness should be minimized or maximized Figure 8 8 shows the second control tab It displays the actual number of the present run the number of individual experiments that have been performed within this run the final fitness of the run and the best fitness
14. Input Output pulse in incoming laser pulse pulse out output laser pulse lines mm grating constant phi2 2nd order spectral phase angle of incidence refers to first grating delta distance between gratings measured at center frequency grating mirror distance from grating 2 to end mirror grating size width of both gratings diffraction order 1 Again let s start defining all relevant parameters as shown in fig 4 5 We split the whole optical path in two parts The first represents the wavelength dependent distance between the two gratings and the second the distance from the second grating to the end mirror The total phase is two times the sum of the two parts plus a relative phase term Figure 4 5 Grating compressor 4nrD d where BoC is the distance between the second grating and the end mirror along the 2w O w l BoC D tan 8 tan 6o sina tan 8 4 16 cos Co 41 center wavelength Again we assume 1st order diffraction We again may calculate the second and third order Taylor coefficients D f XN 1 eee 4 17 mc d cos o one Sie a OU eee Ar sin Bo 14 4 18 gt DAG d cos Bo d cos z e which is just the opposite of the grating stretcher if we set D 2g cos bo 4 19 So in principle a grating compressor should be able to fully compensate for the phase modulation introduced by a grating stretcher Finally we also consider
15. The format is equivalent to that used by the FROG program Femtosoft The first column gives the wavelength in nm the second the spectral intensity in units of J m nm the third column the phase the fourth column the real part and the last column the imaginary part of the spectral field in units of V m nm Obviously there is too much information here but we decided to simply use the format of the FROG software in order to facilitate the import of retrieved pulses Pulses are also correctly loaded if the last two columns are missing If the third column is missing the phase is assumed to be zero for all wavelengths If auto is true the the spectrum is automatically sampled if false then the number of samples must be specified and the loaded spectrum is interpolated accordingly If the fluence is specified then the amplitude of the electric field is recalculated such the the fluence is equal to the value specified 24 Chapter 4 Linear Elements All elements have at least one control panel that you need to connect It allows you to control the important parameters for the simulation and it also allows to change things and watch in real time how the laser pulse intensity or the spectrum or whatever you are looking at changes First there are a number of standard elements such as a delay a telescope etc nothing of great importance but sometimes they come in handy Then there are a number of linear elements mostly complex optical
16. column the phase the fourth column the real part and the last column the imaginary part of the spectral field in units of V m nm Obviously there is too much information here but we decided to simply use the format structure of the FROG software in order to ensure compatibility 70 M TAYLOR PHASE spectral phase order pulse in I temporal phase Taylor coeff temporal phase spectral phase Taylor coeff TF Input Output pulse in input pulse temporal phase Taylor Taylor coefficients of order maximum order temporal phase of series expansion temporal phase plot to check fit spectral phase Taylor Taylor coefficients of spectral phase spectral phase plot to check fit The Taylor phase module expands the spectral and the temporal phase in a Taylor series and outputs the Taylor coefficients of both up to the order you specify You may check the quality of the polynomial fit by inspecting the two plots 71 E TIME BANDWIDTH PRODUCT Awa pulse in u pulse out at bandwidth product Input Output pulse in input pulse pulse out output pulse bandwidth product time frequency bandwidth product TF The time bandwidth product module calculates for a given input pulse the product of the temporal intensity FWHM and the spectral intensity FWHM FWHM FWHMsi If you prefer ArAw simply multiply the result by 27 T2 M INTENSITY AUTOCORRELATION control
17. combines correlation and interferometry The signal is given by I T fe Et Elt r where n is the order of the autocorrelation A linear correlation corresponds to n 1 a second order to n 2 and a third order to n 3 75 2D DISTRIBUTION 20 control pulse in TF Input Output pulse in input pulse 2D distribution intensity plot time step time step t axis x scale information field size sampling grid size y axis y scale information save option specify output format path to file file name type select distribution y axis frequency or wavelength The 2D distribution calculates a time frequency distribution from the electric field of the pulse You may select between e Spectrogram 2 Se Ben f dt E t E t t e 6 2 WFFT windowed fast Fourier transform The function f t is a real Gaussian of some appropriate width 2 S w t wro see 6 3 e Wigner Ville S w t per 1 2 EG27 je 6 4 T e Page t 2 S w t a dt E t eT 6 5 Ot n e Rihaczek i S w t Tz E t E w e 6 6 76 With the time step and the field size you determine the total time window and from this the frequency step and the frequency window are derived The resulting intensity plot may be saved in jpeg format to the specified file name You can also select whether you would like the y axis to be frequency or wavelength TT E FROG FROG control pulse in n Fr
18. determined using Parseval s theorem T 1 T E0Con 2 2 2 0 amp 0 ip dt E t oa deo w n z 2 25 E0Con w wo S w n exp je 2 26 Integrating the spectral intensity S w over w must again yield the total pulse energy W egconAw V 2T There is one thing you should never forget Once the instantaneous intensity I t is known never perform a Fourier transform in order to calculate the spectrum S w Calculate the electric field first Fourier transform the field and then calculate the spectral intensity This is the only way to obtain the correct spectrum Try both ways and you will see that a direct Fourier transformation leads to an erroneous result The last thing in this chapter is the definition of the full width at half maximum FWHM For the Gaussian pulse the intensity FWHM is W 2 27 12 81n 2 1 A Aw FWHM 2 28 and the spectral FWHM FWHM AwvV21n2 2 29 It is easy to see that the product of both is independent of any specific pulse parameters except the linear chirp Note that we only allowed for a linear chirp in the first place Usually the time bandwidth product is different for different pulse shapes Gaussian sech etc and is written in terms of frequency rather than angular frequency Therefore a measurement of the temporal and the spectral FWHM gives already some information on the phase modulation of the pulse If the product of both numbers is equal to the minimum
19. expression in this context is the so called B integral The B integral is a measure of the accumulated nonlinear phase and serves as a criterion whether nonlinear effects play a role or not It takes into account the fact that the intensity might change as the pulse propagates along the z coordinate It is defined by the following relation d d 2 ed Of de nal Nol z J Z f de nal Nol z 2 35 C A 0 0 a As a rule of thumb self phase modulation becomes a serious issue when the B integral exceeds 3 or 4 although effects of self phase modulation may be seen much earlier In order to account for both dispersion and self phase modulation one often uses an algorithm called split step Fourier transform This algorithm splits the material in many slices Each slice is again subdivided into three regions Now the first region takes into account one half of the linear dispersion Then one performs a Fourier transformation to the time domain accounts for the temporal phase due to self phase modulation of the full slice in the second region and Fourier transforms back The third region then takes care of the missing half of the linear dispersion This is done for all slices and eventually if the number of slices is high enough the algorithm will converge and lead to the true output pulse 2 3 2 Three Wave Mixing Processes The equations that describe three wave mixing processes are coupled nonlinear differential equations one differen
20. laser pulse Brewster angle automatic alignment optimal prism Min deviation automatic alignment separation optimized value exact approx prism angle prism base material angle of inc distance tip tip prism mirror in prism 1 in prism 2 exact ray tracing or prism angle approximate solution adjust prism angle angle of inc base size of both prisms prism material adjust angle of incidence distance between the prism tips distance prism 2 and mirror point of incidence prism 1 point of incidence prism 2 calculated prism angle calculated angle of incidence The intelligent prism compressor is nothing else but astandard prism compressor which automatically adjusts the distance between the two prisms such that the second order phase present in the input pulse vanishes and the peak power is maximized 36 M GRATING STRETCHER grating strecher pulse in pulse Out ocs Input Output pulse in incoming laser pulse pulse out output laser pulse lines mm grating constant phi2 2nd order spectral phase angle of incidence refers to first grating focal length of the two mirrors or lenses g displacement see text grating mirror distance from grating 2 to end mirror grating size width of both gratings diffraction order 1 Grating stretchers are found in many different variations Here we only implemented the two most commonly used consisting of a sin
21. of all runs performed so far It shows the temperature 97 Figure 8 8 Front panel and the actual parameters The button run next allows to manually start the next run without waiting for final convergence E IAP Lab2 LV61 examples optimizer experiments simplex_exp vi E IAP Lab2 LV61 examples optimizer experiments simplex bes E IAP Lab2 LV61 examples optimizer experiments simplex log f Figure 8 9 Front panel Figure 8 9 shows the file structure used Most importantly the file information of the experiment vi that is used Then a result file may be generated that saves the final parameters of all runs and finally a log file that saves the results of all experiments of all runs Optionally one may generate a config file that saves the actual optimization parameters for later use If the config file load is not empty then the panel settings are overwritten by the settings of the config file 98 M GENETIC ALGORITHM W annala BAAS ocs Input Output none none The genetic algorithm is an optimization routine that relies on reproduction and mu tation Genetic Algorithm controls fittest fies stop 1 3E 20 j 1 0E 20 ofindividuen efso Cre bitsjgen a7 ofelte fz no loa file prob of crossover 1 00 7 init re prop of mutation Pfo 10 1 0E 19 8 3E 18 1 1 r7 50 100 150 o maw fizz Figure 8 10 Front panel Figure 8 10 shows the first control
22. plot shows the temporal intensity and the two numbers are the temporal FWHM and the second moment As mentioned above the FWHM may lead to funny results whenever the pulse intensity becomes heavily structured In this case the second moment leads to much more reliable results if you need a number that specifies the temporal spread The rel abs selector allows to toggle between a relative time axis that is the pulse is always centered around time zero or an absolute time axis that is the first order phase t is included and the time axis is centered around that time 65 M INSTANTANEOUS FREQUENCY nea instantaneous frequency pulze in y pulse out reljabs m an Input Output pulse in input pulse pulse out output pulse instantaneous frequency instantaneous frequency distribution The instantaneous frequency module plots the relative instantaneous frequency Rel ative here means relative to the laser s center frequency So if the pulse has no chirp you d expect the relative frequency to be zero everywhere 66 M PROBE pulse in GE ns pulse out Input Output TF pulse in input pulse pulse out output pulse The probe module was originally designed as a debugging tool But it seemed to be very useful so we made a detector out of it The output of probe are four plots showing the spectral amplitude and phase and the temporal amplitude and phase probe allows you to access the
23. shows the best individuum of the actual generation Figure 8 18 allows to select whether the optimization stops after a given number of generations or manually by pressing the stop button 102 E IAP Lab2 L 61 examples optimizer experiments ga_shg vi E AP Lab2 L 61 examples optimizer experiments es log E IAP Lab2 L 61 examples optimizer experiments es bes Figure 8 16 Front panel Figure 8 17 Front panel Figure 8 18 Front panel 103 M ADAPTIVE STRATEGY TF Input Output none none The adaptive algorithm is an optimization routine that relies on a large number of mutations The mutation rate itself is adjusted according to the success rate control files mutation fittest stop Adaptive Strateg 1 9643 of individuals 32 1 06 3 fofelte Jz initialize ofall pixel random 116427 1 i 1 i u o so 100 150 20 25 maof b Ji worst Figure 8 19 Front panel Figure 8 19 shows the first control tab The input parameters are the number of individ uals per generation the number of elite and whether all individuals of the first generation are created from random numbers or load from a previously saved individuum Two boolean buttons select whether the feedback signal is minimized or maximized and whether a log file is generated Figure 8 20 shows the second control tab It displays the file structure Most importantly the file information of the exp
24. tab The input parameters are the number of indi viduals per generation the bits gene the number of elite the probability of crossover and the probability of mutation The bits gene determine the resolution of the discrete sampling of the search space and the number of elite determines the number of the best individuals that move on from generation to generation without experiencing any crossover or mutation Three buttons allow to set whether the feedback signal is minimized or maximized whether a log file is generated and whether all individuals of the first generation are created from random numbers or loaded from a previously saved individuum Figure 8 11 shows the second control tab It displays the file structure Most importantly the file information of the experiment vi that is used Then a log file and finally the file which saves the best individuum Figure 8 12 shows the best individuum of the actual generation Figure 8 13 allows to select whether the optimization stops after a given number of generations or manually by pressing the stop button 99 E I4P Lab2 L 61 examples optimizer experiments ga_shq vi E IAP Lab2 L 61 examples optimizer experiments ga2 log E IAP Lab2 L 61 examples optimizer experiments ga2 bes Figure 8 11 Front panel Figure 8 12 Front panel Figure 8 13 Front panel 100 GENETIC ALGORITHM 2 as Input Output none none The genetic algorithm 2
25. the time samples of the pulse the target will be resampled automatically 48 Chapter 5 Nonlinear Elements The nonlinear elements implemented in LABI allow to simulate the most commonly found nonlinear effects There is self phase modulation which appears in almost any material when the intensity gets high enough and there are frequency mixing processes Because the spectral width of the laser pulses is very broad the pulse spectra have to be convoluted which sometimes leads to counterintuitive results For example it is not always so easy to predict what the second harmonic spectrum might look like if the fundamental pulse has some funny shaped spectral phase Peter Blattnig has programmed the optical fiber and most of the demo vi s which illustrate some important effects Fred van Goor from the University of Twente has used the amplifier vi to simulate a number of published multi pass amplifier designs He generously agreed to let us include his results in the present manual 49 M TRANSPARENT NONLINEAR MEDIUM nonlinear material pulse out NL B integral pulse in TF Input Output pulse in incoming laser pulse pulse out output pulse glass material phi2 2nd order spectral phase Z thickness B integral B integral A transparent nonlinear medium has the same linear dispersive properties as the pre viously described linear transparent medium In addition it handles nonlinear effects due to s
26. 1 E w3 WwW nwi w3 w1 2 43 et Ak wi w2 L _ 1 ba n wi w2 Ak w w2 L 2 W3 det 2 45 cks Here EF and Es are the spectral fields of the incoming pulses which are constant and Es is the spectral field of the generated pulse The constant factor is determined by the center frequency w3 the effective nonlinearity der units pm V of the specified nonlinear interaction process which depends on the crystal orientation the vacuum phase velocity Co and the wave vector at the center frequency k3 The phase mismatch is Ak k3 ka k 16 2 3 3 Optical fiber propagation Another important nonlinear element is the optical fiber which is frequently used for a process called super continuum generation To describe pulse propagation along an optical fiber we need to include two more nonlinear effects i e self steepening and the Raman effect and find OeE E 7 gt m am 7 im m 7 h 2 0 EY E 1 dn RIM E T n P 2 46 Wo where we have combined the instantaneous electronic response and the Raman response of the medium to R t 1 f t f RRC 2 47 The relative weight is governed by the constant f which for fused silica is approximately 0 18 17 Chapter 3 Lasers First we introduce the numerical structure of a pulse A virtual laser pulse is a cluster of different data types A cluster in G is equivalent to a structure in C or a record in Pascal It consis
27. A VIRTUAL FEMTOSECOND LASER LABORATORY version 5 0 by Thomas Feurer Copyright 2000 by Thomas Feurer All rights reserved under International Copyright Conventions Published in Bern Switzerland version February 19 2009 http www lab2 de Contents Introduction Ultrashort Laser Pulses Pek GEA no She HR ev we SG ew Ge ae amp RS we BOHR A BEG 2 2 Short Pulse Propagation in the Linear Regime 2 3 Short Pulse Propagation in the Nonlinear Regime 2 3 1 Self phase Modulation 0 0 0 ee ns 2 3 2 Three Wave Mixing Processes 000 52 eee 2 0 0 Optical iber propagation 2 ses sae ea bee De ee we es Lasers Linear Elements Nonlinear Elements Detectors Interactions Optimizers 18 25 49 62 85 93 Chapter 1 Introduction Modern science without lasers is like a lake without water boring and desert like Over the past decades lasers made it possible to observe motions in nature with unprecedented temporal resolution Let us illustrate this by one example Everybody who has taken pictures with dad s analog camera knows that the little son playing hockey may only be captured in a sharp image if the shutter time is very very short A fraction of a millisecond is what state of the art cameras are able to do It is fast enough for most hockey players but it is not fast enough to capture much more tiny hockey players in nature such as single molecul
28. FWHM t pulse jr L sigma t ado a aLtocorrelation TF Input Output pulse in pulse in FWHM t FWHM of the correlation time step detector time step sigma t 2nd moment of the correlation samples number of samples autocorrelation intensity autocorrelation type type of correlation The intensity autocorrelation measures the intensity autocorrelation of the incom ing pulse It can be second or third order and it can be background free or the opposite 73 M INTENSITY CROSS CORRELATION control EETA pulse 1 in fal pulse 2 in cross Beacrosscorrelation orr Input Output pulse 1 in pulse in FWHM t FWHM of the correlation pulse 2 in pulse in sigma t 2nd moment of the correlation time step detector time step cross correlation intensity cross correlation samples number of samples type type of correlation The intensity cross correlation measures the intensity cross correlation of the two incoming pulses opposite 74 It can be second or third order and it can be background free or the M INTERFEROMETRIC AUTOCORRELATION order pulse in i Second order interferometr TF Input Output pulse in input pulse second order inter 2rd order interferometric correlation order order of nonlinearity The interferometric autocorrelation has the great advantage that it allows to get some information on the phase of the pulse In principle such a setup
29. an ideal second harmonic generation process Ideal in a sense that the phase matching condition is always perfectly fulfilled no matter how broad the input spectrum Bgua t E2 t 5 6 Note that the fluence of the second harmonic field is without meaning as the ideal SHG process assumes some fictive conversion efficiency 59 M SFM CRYSTAL NONDEPLETED THREE WAVE MIX ING nondep SFM parameters nou nou i p deff pulse 1 B or Oo i a nou ou pulse 2 0 a PLS 3 e or o FastMode aiaia F GUA Input Output pulse in e or o Ist pulse in pulse out e output pulse pulse in o 2nd pulse in deff effective nonlinear crystal crystal material constant type phase matching type length crystal length theta angle k opt ax phi rotation angle delay 0 ignore delay between input pulses sinc theory no group velocity effects output array full half or every second fast mode no interpolations SFM Crystal nondepleted three wave mixing simulates a three wave mixing pro cess in a nonlinear crystal such as second harmonic generation sum frequency mixing or difference frequency mixing In order to save computation time we assume that both funda mental input beams are not substantially depleted by the conversion process As shown in an earlier chapter this leaves us with only one differential equation to solve Presently we have only implemented BBO LBO KDP AgGaSo and LilO3
30. and most of them have to be thrown away in the retrieval process At the moment there are two save options available namely jpeg format and retrieval format If you chose one of those make sure to specify a path and a filename The color map used for the jpeg format is equal to that used by Femtosoft which makes it easier to compare simulated to measured or retrieved FROG traces If you select retrieval format an output file will be generated that can be read directly into the FROG 3 0 software Check the box use header information and specify order delay and read in as constant frequency 79 M KNIFE EDGE CORRELATOR Page control Page pulse in pam ig axis gee vcd axis correlation MH Input Output pulse in input pulse Page integral over Page function time step detector time step t axis x scale information field size sampling grid size lambda axis y scale information save option specify output format path to file file name rise time rise time of the shutter The knife edge correlator is something we have been using in the lab quite often to temporally overlap two beams of different center frequency One beam needs to be strong enough to create a plasma The other beam is either diffracted or absorbed depending on the electron density in the plasma If you record the intensity of the weak beam as a function of the delay between the two pulses it turns out that the whole setup is more
31. ansmission as a function of the wavelength This file should have the extension bs and must be located in the folder beamsplitters The files are text files and consist of three columns The first is the wavelength in units of meter the second the reflectivity and the third the transmission If you only specify a few values the module will interpolate in order to determine all necessary values The absorption is calculated from the transmission and the reflectivity values The following table shows an example of a low pass filter with a cutoff wavelength at 800 nm Below 800 nm it is fully transparent and above 800 nm completely opaque wavelength nm reflectivity transmission 2 0000E 7 0 1 3 0000E 7 0 1 4 0000E 7 0 1 5 0000E 7 0 1 6 0000E 7 0 1 7 0000E 7 0 1 7 9900E 7 0 1 8 0100E 7 1 0 9 0000E 7 1 0 1 0000E 6 1 0 1 1000E 6 1 0 ZF E BEAMCOMBINER pulse 1 in pulse 2 in pulse out TF Input Output pulse A in first incoming pulse pulse A B out combined pulse pulse B in second incoming pulse The beamcombiner is designed to act as a beamsplitter that is used to combine two laser pulses Of course in a collinear fashion because all calculations are one dimensional The two input pulses may have different center wavelengths spectral bandwidths or even a time delay with respect to each other None of the two pulses looses energy which of course is not true for a real beamsplitter In case the tw
32. aser vis Note LABI assumes a uniform intensity profile across the specified beam diameter That is the fluence is determined through W Fo nr 3 1 with the pulse energy W and the beam radius r d 2 If you assume a Gaussian beam profile F r Foexp 5 3 2 0 with a beam waist of w d 2 measured at 13 5 i e 1 e the fluence at the beam center r 0 is 2W Fo 2 Tr 3 3 which is twice as high as for a uniform beam profile That is to say if you d like to perform the simulations at the peak intensity of a pulse with a Gaussian spatial beam profile having a waist of d 2 then you have to artificially increase the energy by a factor of two 19 M GAUSSIAN PULSE Gauss temporal Gauss spectral sampling TF Input Output energy energy pulse out laser pulse beam diameter beam diameter Xo center wavelength Ad spectral FWHM AT temporal FWHM sampling number of samples resolution sampling mode Gaussian pulse produces a laser pulse with a Gaussian shaped slowly varying amplitude The pulse characteristics are either specified in frequency or in time domain two different clusters Only one input is required In both cases one needs to specify the pulse energy the diameter of the transverse mode profile and the center wavelength Ag In addition in frequency domain the FWHM of the laser spectrum AA and in time domain the FWHM of the temporal intensi
33. axis In addition it may be seen that a quadratic phase modulation in time leads to a quadratic phase modulation in the frequency domain Splitting the frequency distribution in the negative and the positive component leads to E t 2 10 Hex E w exp 2 11 Peep Che a 2 2 12 Ce eee we Dae 2 2 13 1 0 05 To 2 n 00 oO 0 5 1 0 40 20 0 20 40 time Figure 2 1 The slowly varying envelope and the rapid oscillations of a Gaussian pulse E t are shown In the time domain we obtain E t O T exp at iA t iat 2 14 ee Aw Aw is 2 E t NETON exp agt iA t it 2 15 After having defined the fields we have to answer the question what do we measure Here is where the problems start What do we measure First of all this question has to be stated in a different way If we use this specific detector what do we measure Let s start for example with a pyro electric detector The laser pulse is absorbed in a ideally non reflecting layer which subsequently gets warmer and changes its electrical resistivity This change then is monitored on an analog display The whole device has a time constant of a few tens of milliseconds So what do we measure Not surprisingly the pulse energy Or what happens if we use a photodiode with a time constant of ten femtoseconds With this diode we will not resolve the rapid oscillations of the laser field but we will be abl
34. bined the instantaneous electronic response and the Raman response of the medium to R t 1 f t fRr t 5 5 The relative weight is governed by the constant f which for fused silica is approximately 0 18 If you switch off dispersion then the sum first term on the right hand side will be set to zero If you switch of the Raman effect then f is set to zero If you switch off 2i self steepening then is set to zero and if you switch off SPM then y is set to zero 52 M OPTICAL FIBER AMPLIFIER fiber arnplifier control pulse in Seen seat pulse out B gamma gain dB TF Input Output pulse in incoming laser pulse pulse out output pulse fiber amplifier control fiber properties gamma nonlinear constant gain dB overall gain in dB The vi optical fiber amplifier adds to the vi optical fiber the option to account for gain The fiber control is identical to that of the optical fiber In addition you find two more control clusters where one specifies the gain medium and the other details on the inversion profile Besides specifying the gain medium and the dopant concentration in weight percent one has the option to switch on or off gain saturation or gain dispersion effects For each gain medium there must exist a file containing the necessary data If you d like to add a new gain medium the best thing is to copy an existing one and change the numbers appropriately Details on the inversion pr
35. cording to Fourier optics frequency mapping You may select between linear frequency mapping and geometric If you select the first then the relation between pixel number and frequency is assumed to be linear With this you may simulate what happens if you do not calibrate your setup correctly If you select the latter the frequency mapping is calculated from the optical arrangement and everything is as it should be 47 M ITERATIVE FFT SHAPER iterations target annann mm plot delimiter pulse out pulse in phase to shaper Pi Steps ee j e Input Output pulse in incoming laser pulse pulse out output pulse iterations the number of iterations phase to Pi steps use m phase jumps only shaper desired spectral phase target file name plot target and output pulse delimiter tab default space etc The iterative FFT shaper module allows to specify target pulse shape i e the tem poral intensity as a function of time It uses an iterative Fourier transform algorithm to evaluate the phase pattern necessary to transform any input intensity as close as possible to the desired target intensity by phase only shaping The number of iterations must be speci fied The target intensity profile must be available as a file in the folder iffttargets The file contains two columns separated by a delimiter The first column is the time in femtoseconds and the second the intensity If the time samples do not coincide with
36. e 1 e or o 1st pulse in pulse 1 out output pulse 1 pulse 2 o0 2st pulse in pulse 2 out output pulse 2 pulse 3 e or o 3rd pulse in pulse 3 out output pulse 3 crystal crystal material deff effective nonlinear type phase matching type constant length crystal length evolution chart evolution of spectra of z steps integration steps theta angle k opt ax phi rotation angle delay 0 ignore delay between input pulses fast mode no interpolations recommended In principle SFM Crystal depleted three wave mixing is a very powerful tool and is capable of handling any three wave mixing process process Dispersion effects to all orders of Taylor coefficients are included since we use the Sellmeier equations to calculate the index of refraction Quite naturally back conversion and other processes occurring as the fields propagate through the crystal are accurately incorporated As mentioned earlier we assume w3 w1 w2 which in the case of difference frequency generation may be rearranged to wy w3 w1 Note that you need to consider this by connecting the pulses to the right inputs of SFM Crystal depleted three wave mixing 58 Mi CRYSTAL WIZARD desired angular precision minimal SFM params pulse 1 e or o pulse 2 o pulse 3 f e or o t P roug velocity indices out none p Y ee be comment phase matching angle dep SFM parameters out
37. e to measure the temporal evolution of the slowly varying envelope So what we measure is mostly dependent on the type of detector which we are using A detector with a time constant large compared to the rapid oscillations but fast compared to the temporal envelope will measure a quantity called the instantaneous intensity I t t T 2 1 I t cocon T i dt E t 2 16 t T 2 10 electric field 04 02 0 0 0 2 04 frequency Figure 2 2 The electric field of a real Gaussian pulse E w shows two distinct contributions For a Gaussian pulse the instantaneous intensity is 2 Aw 2 nyi a 1 A where o is the dielectric constant of vacuum co the speed of light in vacuum and n the index of refraction of the transparent material the pulse is propagating in Obviously the integration over one period reflects the fact that we have a detector not able to resolve the field oscillations The instantaneous intensity is measured in energy per unit time and per unit area If the finite area A of the detector is considered we obtain io an ap l 2 17 P t fo I t 2 18 A If the response time of the detector is even longer the pulse energy W is measured W 1 dt P t 2 19 which for a Gaussian is We Eon Aw 2 20 V2T Making use of the slowly varying wave approximation it is relatively easy to show that the following equalities hold Oe econ t e t 2egcon Et t E t 2 21 A further useful quantity is the in
38. elf phase modulation SPM The numerical simulation uses a split step Fourier transform algorithm Whereas dispersive effects are incorporated in the spectral domain SPM is treated in the time domain The integration z step width is dynamically adjusted so that the nonlinear phase per integration interval is always 7 20 i e T c Az 5 1 20 WeN2lmax l The output B integral yields _ 2m an The linear material properties come from material files in the folder chii The transparent nonlinear medium requires nonlinear material properties and those are stored in the folder chi3 The material files in this folder provide the nonlinear index of refraction commonly known as nz plus the Raman response Here we only need ng As an example we inspect fused silica SQ1 To simulate the propagation of a short pulse through a piece of silica we need two files B dz nI z 5 2 1 chi1 SQ1 vi for the linear material properties and 2 chi3 X3_SQ1 vi for the nonlinear material properties If you wish to add a new material make sure you know the nonlinear index of refraction Then copy the file X3_SQ1 for example to X3_SF6 edit the material constants and you re done 50 M OPTICAL FIBER pulse in i pulse out gamma ges Input Output pulse in incoming laser pulse pulse out output pulse fiber control fiber properties gamma nonlinear constant Even more sophisticated than the transparent nonli
39. eriment vi that is used Then a log file and finally the file which saves the best individuum Figure 8 21 shows a statistics of the mutations Individuum zero is the best one the second best etc In green are the successful mutations and in red the unsuccessful A mutation is rated successful if the individuum can increase its rating by at least one rank in the next generation Figure 8 22 shows the best individuum of the actual generation Figure 8 23 allows to select whether the optimization stops after a given number of generations or manually by pressing the stop button 104 E IAP Lab2 LV61 examples optimizer experiments ga_shg vi E IAP Lab2 LV61 examples optimizer experiments as log E IAP Lab2 LV61 examples optimizer experiments as bes Figure 8 20 Front panel Figure 8 21 Front panel Figure 8 22 Front panel 105 Figure 8 23 Front panel 106
40. es or atoms Clever people have thought of an alternative way to capture images of fast moving objects Darken the room leave the camera shutter open all the time and illuminate the object you want to image with a short flash of light People who like clubbing know what we mean it s nothing but a strobe light Many images like this in a row make a nice movie The shortest strobe currently available is a sub femtosecond laser A single pulse is about 0 000000000000005 seconds long pulse width ps 2 a 0 01 1E 3 0 1 10 1000 spectral bandwidth nm Figure 1 1 Temporal pulse width full width at half maximum versus spectral bandwidth for a Ti Sapphire laser Because a short burst of light must be supported by a lot of spectral components the 4 spectrum of a femtosecond laser pulse is usually some tens or hundreds of nanometers broad In fig 1 1 we show how broad the spectrum needs to be to support a laser pulse of a given temporal duration The lasers center wavelength is 800 nm Obviously a pulse width of 100 ps requires a spectral bandwidth of 0 01 nm whereas a pulse width of 10 fs already needs about 100 nm spectral bandwidth Now every optical component such as a lens a window a dielectric mirror etc are dispersive meaning that they influence the different spectral components of a short laser in a different way That as one might guess sometimes has dramatic effects on the laser pulse In other cases well cont
41. evolutionary algorithm and an adaptive algorithm All algorithms are designed to optimize a single feedback signal or fitness by continuously varying a set of parameters for example the phase values of a spatial light modulator While the simplex downhill algorithm is strong for a low number of parameters and fails to converge for a large search space the other four are well suited for large search spaces All optimizer vis call an experiment vi Its input are the parameters an array of numbers to optimize and its output is a single feedback a dimensionless number The experiment vi is called by the optimizer vi whenever it needs to evaluate a new set of parameters Its file and path information must be entered in the appropriate field The experiment vi may consist of LABH modules as shown in the following two examples It may also communicate with real devices control their setting through the parameters and read out a real feedback signal for example from a photodiode We first start with a virtual experiment for the simplex downhill algorithm which is shown in fig 8 1 Figure 8 1 The virtual experiment A Gaussian shaped pulse is stretched by a linear dispersive medium and the task is to compress it through a pulse shaping apparatus The parameters to optimize are the second to fourth order phase terms of a Taylor expansion Note that the zeroth absolute phase and first order phase
42. f atom 2 dipole matrix dipole matrix elements ground state D dissociation energy beta width x0 equilibrium separation C asymptotic energy excited state D dissociation energy beta width x0 equilibrium separation C asymptotic energy simulation stop time to stop simulation excited state wavepacket excited state population ground state population ground state wavepacket length vs atomic separation 2D wavepacket motion final population final population 2D wavepacket motion The module diatomic molecule simulates the interaction of a pulse and a diatomic molecule which is characterized by an electronic ground and an excited state The molecule is represented by a pair of Morse potentials the two masses and the transition dipole matrix It is necessary to specify the spatial grid that is used to calculate the space time propagation of the wavepackets The initial ground state wavepacket is calculated from the parameters of the ground state potential You may plot the difference between the two potentials in terms of wavelength as a function of the internuclear separation to determine the resonant wavelengths During the calculations a pop up window shows the evolution of 90 both wavepackets At the end of the calculation a two dimensional plot shows the space time distribution of the wavepackets on both potential surfaces 91 M SPECTRAL LINES info B EES SHectrum TF
43. ficiency The pulse lengths of the stretched pulses are such that the amplifiers operate at a safe peak intensity well below or sometimes close to the damage threshold The simulations have been performed with pulse durations of about 90 ps It was not possible to use the published pulse lengths because the number of samples became too large when increasing the chirp However we believe that this will not influence the simulation results much because we are dealing with energies and not with intensity waveforms Barty Opt Lett 19 1994 p1442 Itani Opt Comm 134 1997 p134 Yamakawa Opt Lett 23 1998 p1468 Barty Opt Lett 21 1996 p668 Zhou Opt Lett 20 1995 p64 Walker Optics Express 5 1999 p196 Wang J Opt Soc Am B 16 1999 p1790 SOO BS WN 61 Chapter 6 Detectors To fully characterize a femtosecond laser pulse it is necessary to measure the amplitude as well as the phase of it either in the spectral or in the time domain All electronic devices however are too slow to come up to this challenge And because no direct methods are appli cable indirect techniques have to be used All of those techniques rely on an instantaneous nonlinear effect In LABI we have implemented a large number of those indirect techniques but we have also implemented rather unrealistic devices which are extremely useful for debugging pur poses All detectors have a number of different outputs Some may require an inpu
44. format is equal to that used by Femtosoft which makes it easier to compare simulated to measured or retrieved FROG traces If you select retrieval format an output file will be generated that can be read directly into the FROG 3 0 software Check the box use header information and specify order delay and read in as constant frequency 78 E XFROG ZFROG control pulse 1 in gds i FROG pulse 2 jn t axis lambda axis TF Input Output pulse 1 in input pulse THG FROG THG spectrogram pulse 2 in input pulse t axis x scale information time step FROG time step lambda axis y scale information field size sampling grid size save option specify output format path to file file name XFROGS are identical to FROGs except the two pulses may be different In addition XFROG offers the option to use difference frequency generation as a nonlinear process which doesn t make sense for FROGs The XFROG module requires a grid size and a time step The larger the grid size the longer it takes to calculate the FROG trace Since Fourier transformation automatically determines the frequency step size once you specify the time step there is no need to input both We decided to use the time step as input because in most experiments a stepper or a dc motor driving a translation stage determines the resolution Most spectrometer usually measure far too many spectral samples typically 512 to 2048 samples
45. frequency mixing Depending on the order of the process there are two three or even more pulses involved So far we have implemented the most commonly encountered nonlinear effects such as three wave frequency mixing and self phase modulation 2 3 1 Self phase Modulation Self phase modulation is in principle a third order x effect and leads to a time varying phase The three waves involved are three replica of the original pulse Just as a spectral phase leaves the spectrum of the pulse unchanged but has dramatic effects on the temporal intensity a temporal phase causes the opposite it leaves the temporal intensity the same but 14 changes the spectrum An easy but somewhat sloppy way to express self phase modulation is by introducing an intensity dependent index of refraction n I no nel 2 34 which in turn causes an extra phase in the time domain b t n I t z no nol t z Pult no I t z so it is essentially the product of intensity J times the propagation distance z that de termines the nonlinear phase contribution In principle one would expect the same result as one changes the material thickness z but keeps the product Iz constant by adjusting the intensity J This is not quite true especially for large z since dispersion effects tend to broaden the pulse as it propagates which leads to a decreasing intensity J and consequently a decrease in the nonlinear phase contribution A frequently used
46. g from reference 5 as the peak intensity is unrealistic high Maybe the guessed beam diameter of the output is not correct in this case Table 5 1 Comparison between published results and simulations Number of passes n pump fluence Fyump seed fluence Fin amplified seed fluence Fout pump beam diameter dyump Seed beam diameter dseea pump energy Wpump seed energy Win simulated amplified energy W im measured amplified energy Wexp length of TiSa crystal Lrisa diameter of TiSa crystal drisa pulse duration T and maximum intensity Imax n Fpump Fin Fout dpump din Wpump Win Wsim Wexp Lisa drTisa Ti T Imax ref rem J cm mJ cm J cm mm mm mJ mJ mJ mJ mm mm ps GW cm 4 2 82 45 84 1 21 6 20 5 00 850 9 237 6 235 13 00 13 00 0 15 300 4 03 1 good 3 3 70 57 69 1 72 8 50 8 00 2100 29 864 6 790 12 00 20 00 0 15 330 5 21 2 ok 4 1 99 28 29 1 23 6 70 6 00 700 8 347 8 340 20 00 0 15 1700 0 72 3 good 4 2 95 15 92 1 25 5 00 4 00 580 2 157 1 160 10 00 12 00 0 15 950 1 32 4 good 4 2 55 19 89 0 99 5 00 4 00 500 2 5 124 4 120 7 00 10 00 0 23 40 24 75 5 good 4 3 16 15 28 1 44 5 50 5 00 750 3 282 7 280 18 00 12 00 0 15 800 1 80 6 good 4 2 48 61 89 1 01 12 00 12 00 2800 70 1142 3 1340 10 00 20 00 0 15 80 7 10 T ok 4 2 63 15 59 1 10 4 00 3 50 330 1 50 105 8 100 15 00 0 25 80 13 75 7 good The simulations agree reasonably good or good with the published results Most sys tems operate above the saturation fluence at the last pass yielding maximum ef
47. gle grating one or two spherical mirrors and a flat end mirror Fig 4 2 shows the definition of the angles and when they are positive and when negative positive _ negative 1 order B Figure 4 2 Definition of the angles O order order In the same manner we define the diffraction orders left of the zero order positive and to the right hand side of zero order negative The grating equation is 2 b arcsin oem sin a wd where M is the diffraction order and d the grating constant 37 4 12 Figure 4 3 a Stretcher geometry b If the setup a is folded with respect to the plane between the two focusing mirrors the most commonly used stretcher geometry emerges c Second Oeffner triplet This setup is aberration free and another commonly used stretcher Rand R 2 are the radii of curvature of the two spherical mirrors Figure 4 3 shows the two most commonly used stretcher geometries The overall stretch ing factor depends on the grating dispersion the angle of incidence and the displacement g Using the more detailed graph shown in fig 4 4 allows to calculate the phase modulation as a function of the frequency B w Z 8f 2r 4g Ap sin B sina Brg cos fo Co cos 3 d tan 8 4 13 where d is the grating constant a the angle of incidence on the grating the diffraction angle which is negative and the diffraction angle also negative at
48. ibes the different laser sources chapter 4 all the available linear elements chapter 5 all the available nonlinear elements chapter 6 all the detectors that have been implemented and chapter 7 a selection of modules simulating laser matter interaction The very last chapter 8 introduces optimization algorithms which are commonly used in combination with pulse shaping to reach specific goals A couple of examples will be given So enjoy reading and more importantly working with LABH Thomas For further reading we recommend the following textbooks gt G P AGRAWAL Nonlinear Fiber Optics Academic Press San Diego 2001 gt S A AKHMANOV V A VYSLOUKH A S CHIRKIN Optics of Femtosecond Laser Pulses AIP New York 1992 R W Boyp Nonlinear Optics Academic Press San Diego 1992 C C Davis Lasers and electro optics Cambridge University Press Cambridge 1996 J C DIELS W RUDOLPH Ultrashort Laser Pulse Phenomena Academic Press San Diego 1996 gt V G DMITRIEV G G GURZADYAN D N NIKOGOSYAN Handbook of Nonlinear Op tical Crystals Springer Berlin 1997 gt S HUARD Polarization of Light John Wiley amp Sons Chichester 1997 gt S MUKAMEL Nonlinear Optical Spectroscopy Oxford University Press New York 1995 gt B E A SALEH M C TEICH Fundamentals of Photonics John Wiley amp Sons New York 1991 gt A YARIV Quantum
49. ing dipole moments the atomic density and the interaction length In addition in manual mode the number of z integration steps has to be specified After the calculations the diagonal and separately the off diagonal elements of the density matrix may be plotted as a function of time With an appropriate choice of the transition wavelengths and the initial population dis tribution one may either simulate a V type a A type or a ladder type system 86 Mi PARALLEL LADDER SYSTEM parameters of Quantum system modus of integration iterations pulse A in pulse E in pulse A out pulse B out ion state intermediates ground state RN Input Output pulse in A pulse in B modus of integration iterations transition wavelengths dipole moments mx dipole moments dx of states Min 3 Tion particle density propagation length initial values of amplitudes input pulse A input pulse B slow fast ion state of z steps intermediates wavelengths of g k ground state and g gt 7 dipole moments of transition ls gt Ik dipole moments of transition k gt l number of involved states lifetime of upper state y y 1 Tion particle density propagation length ce to cx to and c to pulse out A pulse out B output pulse A output pulse B ion state population population of intermediate states ground state population
50. is a identical to the genetic algorithm except that the gens may be binned Binning can be reduced manually as the optimization proceeds For example setting binning equal to two means that always two genes have the same value and the total number of genes reduces by a factor of 2 Figure 8 14 Front panel 101 M EVOLUTIONARY STRATEGY Ca TF Input Output none none The evolutionary strategy is an optimization routine that relies on mutation mainly In contrast to the genetic algorithm the search space parameters are stored as floating point numbers cnt fles test ston Evolutionary Strategy 7 8E 19 of parents 10 of chidren 22 binning tfi Initiale random tiers o 50 100 150 200 255 Bani E worst Figure 8 15 Front panel Figure 8 15 shows the first control tab The input parameters are the number of parents per generation the number of children produced per generation the amount of binning and whether the initial individuals are generated from random numbers or initialized from a file i e a previously stored individuum Two boolean buttons select whether the feedback signal is minimized or maximized and whether a log file is generated Figure 8 16 shows the second control tab It displays the file structure Most importantly the file information of the experiment vi that is used Then a log file and finally the file which saves the best individuum Figure 8 17
51. ism 2 point of incidence prism 2 In the case of more complex optical systems the situation is somewhat more complicated However in most cases the problem turns out to be a geometric one in a sense that one needs to find the optical path length as a function of frequency for a given optical system One may even use ray tracing programs to do that job A first widely used optical system is a prism compressor It consists of four prisms or more frequently of two prisms and a folding mirror Figure 4 1 shows a typical geometry of a prism compressor The beam impinges on the first prism gets dispersed passes the second prism is reflected at the folding mirror and goes all the way back In order to separate the output from the incoming beam the folding mirror is usually tilted in the vertical dimension such that the output beam is on top of the incoming or vice versa Alternatively a roof mirror may be used Both prisms are identical and their apex angle is a The index of refraction is determined by the corresponding Sellmeier equation Usually a prism compressor allows to adjust both positive and negative chirps depending on the distance between the two prisms if the input pulse was bandwidth limited That is 34 Figure 4 1 Prism compressor because the glass material of the prisms itself introduces some positive phase modulation and only if the distance between the prisms exceeds a certain threshold the overall phase modulation bec
52. ition of moments will appear a few times throughout the text The n order moment of x is defined nr _ J do 2 f z fax f z where f x is a distribution function Now we have most of the basic tools and it is time to apply the formalism developed so far to Gaussian pulses A Gaussian pulse has a Gaussian envelope function and we restrict ourselves to phases quadratic in time and we omit the constant phase If we do this many interesting effects can be calculated analytically The reason for choosing a Gaussian shaped envelope is simply that it allows to calculate most things analytically with very little mathematical effort 2 9 Aw Aw 25 AAW gt I AA exp e l Cos nt a AX Figure 2 1 shows the electric field E t for a pulse with A 0 The rapid oscillations increase with time reach a maximum and decrease again This behavior is determined by the slowly varying envelope i e the Gaussian The reason to choose this form is that in most cases we start with a pulse having a fixed spectral width Aw and treat effects that do not change this width The constant factor makes the Fourier transform of this pulse relatively simple w wo 1 7A w wo 1 iA Aw p Aw Clearly the Fourier transform has two distinct frequency distributions around w and wo see fig 2 2 The slowly varying envelope approximation breaks down when the two frequency distribution start to overlap at the origin of the frequency
53. l tool for solving equations Since the electric field is real i e E t E t we see immediately that Peo f dt E t e f dt E t e t E w 2 4 This is a useful identity for a number of the following calculations Inspecting equa tion 2 1 shows that the Fourier transform has two distinct distributions which are symmetric with respect to the origin of the frequency axis Now have you ever seen a spectrometer that can measure at negative frequencies or wavelength Certainly not so it is sometimes more convenient to start in the frequency domain by defining a field having only positive frequencies The consequence is that the electric field in the time domain becomes complex Again we end up with an unrealistic situation since we use complex instead of real Fourier transformation None of these problems would occur would we use a real Fourier trans formation for example the cos Fourier transformation but the mathematical efforts would increase substantially and so nobody uses it ee as wt eae wt E t F fw E w e dw E w e 2 5 0 oo with E w w gt 0 A E w l Oe uci 2 6 The same holds true for the negative frequency components They give rise to a field E t It is easy to show that the original field is reconstructed by E t E t E t and E w E w E w 2 7 Now we want to become somewhat more specific and adapt the formalism to laser pulses we have to deal with in the laborator
54. most fully characterize a femtosecond laser pulse The only missing thing is the absolute phase of the pulse but other than that you get the amplitude and the relative phase from a FROG 62 measurement There is some mathematics involved in that too because you need to process the measured FROG trace run an iterative algorithm and if you have done everything as you should you ll get the field FROG in principle is conceptually easy take any intensity correlation setup and spectrally resolve the correlation signal This is why there are many different types of FROG s out there The most common ones are SHG FROGs because they are easy to build and require very little energy to operate They have one disadvantage however and that is they have no idea in which direction the time axis points For example a SHG FROG never allows you to distinguish between a positive and a negative phase modulation if they only differ by the sign The THG FROG resolves this problem but needs much more optical elements to align and a little more energy so does the PG FROG The FROG signal is a function of both the frequency and the delay time 3 2 F w T fu ftr e 6 1 where the function f t is determined by the nonlinear process you re employing f t 7 E t EX t 7 for DFG FROG f t 7 E t En t 7 for SFG FROG f t 7 E t Fx t 7 for THG FROG f t 7 E t E t 7 for PG FROG f t T E t E t r for TG FROG f t T
55. n on or off all effects related to pixelation This way you may find out which experimentally observed details in the shaped waveform are due to the pixelated nature of the modulator gaps If switched on two neighboring pixels are separated by a gap The width of the gap is in units of pixel width For example 0 04 means that the gap width is 0 04 times the pixel width The phase modulation of all gaps is zero and the amplitude modulation is equal to 1 wraps Most pixelated devices have a maximum accessible range of phase values typically a few times 27 Mathematically we don t need a phase larger than 27 because 37 for example leads to the same result as 37 modulus 27 1r That is whenever the phase modulation exceeds 27 we may subtract 27 this process is known as phase wrapping You may specify the wrap point in terms of 27 that is wraps may occur whenever the phase exceeds 27 1 or 4 2 ete crosstalk pixel If turned on the amplitude and phase pattern are convoluted with a Gaus sian The width is given in units of pixel With this option you may simulate a mis aligned setup where the modulator is not exactly in the focal point of the lens curved mirror diffraction If turned on the phase and amplitude modulation pattern is convoluted with the spatial resolution of the optical system The system resolution is assumed to be Gaussian and the width is determined by the beam diameter and the focal length of the lens curved mirror ac
56. near mediumis the optical fiber Besides linear dispersion and self phase modulation it also accounts for self steepening and the Raman effect The fiber control allows you to specify the following values dispersion Here you need to specify the material which is responsible for the dispersive properties of the fiber This may be a simple material file such as fused silica SQ1 or it may be a file which contains the dispersion properties of a specific complex photonic bandgap fiber nonlinear The nonlinear properties may be specified independent of the linear properties This allows to use a fused silica core SQ1 which specifies the nonlinear properties together with any dispersion file which is dominated for example by the hole structure surrounding the core length The fiber length effects Here you may switch on or off linear or nonlinear effects in order to study their influence for example on super continuum generation The integration step width is dynamically adjusted so that the nonlinear phase never exceeds a specified limit The nonlinear constant is calculated to _ Won2 7 CAcg where w is the center frequency and Aep is the effective area calculated from the beam diameter d simply through Agg 7d 4 The vi numerically integrates the nonlinear differ ential equation for the slowly varying electric field envelope 5 3 MAST 8 i Re IE iy ft 2S een fan RENEE WP We OT 5 4 51 where we have com
57. ntensity FWHM FWHM of spectral intensity spectrum spectral intensity vs wavelength relative spectrum relative frequency spectrum The ideal spectrometer has a number of useful outputs First of all there are two graphics the wavelength spectrum and the relative frequency spectrum In addition you ll find three numbers the center wavelength the spectral FWHM in units of nm and the second moment The second moment is a useful quantity when the spectrum becomes heavily structured for example due to self phase modulation and the FWHM becomes a somewhat meaningless number The spectrometer is ideal in a sense that it has no device limited spectral resolution The resolution is only determined by the number of samples and the full spectral window you are using In addition neither the grating nor the detector have a wavelength dependent sensitivity 81 M SPECTROMETER WITH RESOLUTION pulse in center wavelength vie FHM spectrum wavelength resolution o Input Output pulse in input pulse center wavelength center wavelength wavelength resolution spectral resolution FWHM FWHM of spectral of spectrometer intensity spectrum spectral intensity vs wavelength The spectrometer with resolution is almost the same as the ideal spectrometer except that you need to specify a device resolution in units of nm 82 ENERGY METER laser rep rate pulse Mi ave pover fluence energy TF Input Output p
58. o beams have different diameters the diameter of the second beam is adjusted to match that of the first beam that is when the two beams are physically combined they end up having the same diameter Also note that the combination is non interferometric which in practice means the the two beams are not really collinear but have a small angle 28 DELAY delay time delay stage pulse in ee TE aa pulse out TF Input Output pulse in incoming laser pulse pulse out delayed pulse delay fs temporal delay delay stage mimics a mechanical delay stage The delay simulates a mechanical delay line The delay may be specified in terms of the time delay or in terms of the optical path the beam has to propagate If you enter the optical path the delay corresponds to the time it takes to travel the path back and forth just as it is the case for a real mechanical delay 29 M TELESCOPE scale diameter scale area pulse in f pulse out TF Input Output pulse in incoming laser pulse pulse out output laser pulse scale diameter relative change of diameter scal area relative change of pulse area telescope simulates a telescope It does nothing but to increase or decrease the fluence energy per unit area corresponding to the magnification you choose This you may do in two different ways First by specifying the magnification of the beam diameter or second of the beam area A value la
59. ofile are first of all the longitudinal inversion profile AN z You may select constant In which case the inversion profile is assumed to be constant exponential The gain profile decays exponentially with the decay constant specified by a Fermi Dirac The gain profile resembles a Fermi Dirac distribution in order to account for saturation in the pumping process z marks the position to which the gain is roughly constant and after which it decays exponentially Again a determines the decay constant from file This option allows to load an arbitrary inversion profile from a file which has two columns separated by a tab The first column is the longitudinal position scaled by the fiber length That is the first column contains numbers between zero and one which must be evenly spaced The second column contains the upper state population in arbitrary units No matter what profile you pick the absolute values are automatically adjusted so that the overall small signal gain at the center wavelength of the pulse is as specified by the 53 control gO dB It may happen that the simulated small signal gain is smaller than the specified one One possible reason might be that the dopant concentration and the length of the fiber are not large enough to allow for the desired small signal gain 54 M IDEAL SHG pulse in sn emma pulse out TF Input Output pulse in input pulse pulse output pulse Ideal SHG simulates
60. og axis lambda axis orr Input Output pulse in input pulse FROG spectrogram time step FROG time step t axis x scale information field size sampling grid size lambda axis y scale information save option specify output format path to file file name type nonlinear process The FROG module mixes two replicas of the input pulse in a nonlinear medium and the spectrum of the generated nonlinear signal is detected as a function of the time delay between the two replicas The FROG trace is displayed as a function of delay time x axis and wavelength y axis Connect the scale information to a property node of the intensity plot to get properly calibrated axis The FROG module requires a grid size and a time step The larger the grid size the longer it takes to calculate the FROG trace Since Fourier transformation automatically determines the frequency step size once you specify the time step there is no need to input both We decided to use the time step as input because in most experiments a stepper or a dc motor driving a translation stage determines the resolution Most spectrometer usually measure far too many spectral samples typically 512 to 2048 samples and most of them have to be thrown away in the retrieval process At the moment there are two save options available namely jpeg format and retrieval format If you chose one of those make sure to specify a path and a filename The color map used for the jpeg
61. omes negative Therefore one may fine tune a prism compressor by moving one of the prisms perpendicular to its base in order to introduce slightly more or less glass Spectral clipping occurs whenever the spectrum at the second prism is larger than the prism itself In other words spectral clipping happens when one of the two following equations is true S5 Wmaz lt 0 4 10 for the highest frequency and B 3 sin a 2 for the lowest frequency where B is the length of the prisms base and s is the distance from the tip of the second prism to the point where a specific frequency enters the second prism If you activate the button use Brewster angle the VI automatically adjusts the input angle at the first prism such that the center frequency enters the first prism and also the sec ond prism at Brewsters angle The other choice you have is to specify use minimum deviation which causes the apex angle of the prism to be adjusted such that the center frequency ex periences minimal deviation or in other words it passes the first prism parallel to its base There are two choices to calculate the compressor phase First through exact ray tracing and second through an approximate solution which may be found in any textbook 85 Wmin 4 11 39 M INTELLIGENT PRISM COMPRESSOR prism compressor pulse in a pulse out otma distance tip tip GS Input Output pulse in incoming laser pulse pulse out output
62. onsists of a constant factor multiplied by w wo Recalling the properties of the Fourier transformation shows that a phase term linear in frequency shifts the whole pulse in the time domain by a constant time delay on the time axis Therefore the first derivative with respect to frequency corresponds to the time a pulse needs to travel through the dispersive medium and may be used to determinate the group velocity All terms of higher order generally change the pulse envelope in some way It may be shown that all even order Taylor coefficients change the slowly varying envelope in a symmetric fashion whereas all odd order Taylor coefficients cause an asymmetric change In many textbooks the phase is expressed as follows Ow Blw z Si ACOE A w wo a Bez w wo 2 32 k 0 with Bk A The number per definition is a wave vector or a phase per unit length Suppose the pulse traverses a transparent but dispersive medium with thickness d and assume the medium s dispersion is fully characterized by 3 only then we immediately see that Poult Fo Bin we seo Ein t Bid 2 33 Since the pulse propagates with the group velocity we find that 3 1 v or vg 1 61 d 2 3 Short Pulse Propagation in the Nonlinear Regime Nonlinear effects involve products of electric fields in the time domain and since this is equivalent to a convolution in the frequency domain nonlinear processes usually lead to
63. or less the temporal equivalent to a knife edge experiment What you measure is the integral of the intensity up to the delay time If you spectrally disperse the signal and check out the math it turns out that what you measure is the integral of the Page distribution assuming that the shutter is infinitely fast 2 Seale f dr e E r Q t 7 6 7 The infinitely fast shutter is incorporated by using the Heaviside step function O t 7 Even in the case the shutter rise time is finite equation 6 7 is a pretty good description So all you need to do is to calculate the derivative of the time frequency distribution with respect to time and you obtain the Page distribution of the pulse The Page distribution has the advantage in contrast to the spectrogram FROG that you can invert it in a straight forward way to extract the electric field No need to iterate here The knife edge correlator has two outputs and that is the correlation signal and the time integral over the Page function What you need to specify is the rise time of the shutter If you start with a small rise time and increase it step by step you will also see the influence of a finite rise time on the appearance of the time frequency distribution 80 E SPECTROMETER sigma pulse in center wavelength FHE spectrum relative spectrum TF Input Output pulse in input pulse center wavelength center wavelength sigma second moment of spectral i
64. p fluence right concentration crystal length medium effects input pulse 0 deg or Brewster angle duration of pump pulse delay between pump and first pass pump fluence left pump fluence right active ion concentration crystal length laser medium switch on off pulse out output pulse The amplifier simulates a one or two sided longitudinally pumped laser amplifier In order to avoid reflection losses at the crystal the angle of incidence may be chosen to be equal to Brewster angle instead of normal incidence The amplifier is characterized by its concentration of active ions and the rod length Both pump beams have the same duration but may have different energy densities and the delay between the pump beam and the first pass must be specified The module accounts for saturation in the amplification process for dispersive effects due to the resonant amplification and for dispersive effects of the host material The number of iterations is automatically determined If one wants to simulate a two or other multi pass amplifier the inversion cluster must be connected as shown in the tutorials We also need to specify whether the direction of the seed beam through the amplifier crystal alternates as in a regenerative amplifier or does not alternate as in a ring type multi pass amplifier Fred van Goor from the University of Twente did simulate a number of published Ti sapphire amplifiers We are very happy that he agreed
65. re either specified in frequency or in time domain two different clusters Only one input is required In both cases one needs to specify the pulse energy the diameter of the transverse mode profile and the center wavelength Aj In addition in frequency domain the FWHM of the laser spectrum AA and in time domain the FWHM of the temporal intensity Av are mandatory inputs If nothing else is done all subsequent calculations run in auto mode that is fields are automatically resampled if necessary and iterations are adjusted such that outputs converge to the percent level In case the sampling cluster is connected all simulations run in manual mode and sampling is determined by the values in this cluster The number of samples is always a power of two and there are three options for the type of resolution High spectral The spectral window is such that the laser spectrum just fits in and the spectral resolution is highest High temporal Here sampling is adjusted so that the temporal resolution is high and the full time window is only slightly larger than the temporal FWHM of the pulse Of course spectral resolution is very poor here Balanced In this mode the sampling is such that both temporal and spectral amplitude are sampled with a somewhat balanced resolution Time window Here you can adjust the time window to a fixed value 21 M ADD NOISE noise control pulse in TF Input Output pulse in input pulse pulse out
66. real laboratory setup as realistic as possible The amplitude and phase modulation controls are identical to those of the ideal shaper except the selection arbitrary as type of amplitude or phase modulation is ignored Note new here is two additional control panels i e the properties and the setup control Setup allows to choose between two alternative experimental schemes First a zero dis persion compressor consisting of two gratings and two lenses or two curved mirrors and second a zero dispersion compressor where the gratings are replaced by prisms In both cases you need to specify the focal length of the two lenses or the two curved mirrors When you pick the grating setup you need to specify the number of lines mm the diffrac tion order and the difference between the angle of incidence and the diffraction angle at the center wavelength If you pick the prism setup you need to specify the prism material and the prism angle apex From this information the program calculates the correspondence between pixel and frequency In addition to the zero dispersion compressor geometry you need to provide information on the pixelated modulator itself Here the program needs to know the number of pixels the pixel width and the wavelength of the center pixel Properties allows to switch on or off various effects connected to the pixelated nature of the modulator device In total there are 6 options 46 pixelated This switch allows to tur
67. rger than one increases the beam diameter and vice versa 30 TRANSPARENT DISPERSIVE ELEMENT dispersive element pulse in pulse out TF Input Output pulse in incoming laser pulse pulse out output laser pulse dispersive element specify the glass material phi2 second order phase and thickness The transparent linear medium is usually just a piece of some glass material This may be a vacuum window a lens or by combining two of those elements also an achromatic lens or more complicated arrangements may be simulated The phase change introduced by just a piece of transparent but dispersive material of a given length is w P w n w z 4 1 Co where n w is the index of refraction and z the thickness of the material The coefficients of the Taylor series are easily calculated Z 2 aala e on wo An 4 2 To simulate such a material the index of refraction must be known Usually it is tabulated or given as a Sellmeier equation In LABH many glasses have been implemented through the corresponding Sellmeier equation The most prominent ones are fused silica SQ1 BK7 and SF10 etc You may just copy and paste any of the files under a new name the material you wish to add in the folder chi1 and change the parameters accordingly It is that easy to add a new material to the list 31 FABRY PEROT INTERFEROMETER FP control pulse in reflected pulse out transmitted pulse out wii
68. rolled dispersion is used on purpose for intentional pulse manipulations Another striking feature of ultrashort laser pulses is that the peak intensity can be enormously high although the energy is at a ridicu lously low level All sorts of nonlinear material responses show up and need to be considered when doing experiments with pulses so short A familiar nonlinear effect is breaking the optics whenever the laser intensity is larger than the damage threshold intensity To circum vent damaging of optics especially in laser amplifiers lead to the introduction of what is now known as chirped pulse amplification CPA To do such things as stretching or compressing pulses special optical designs are required In the next few chapters we introduce a modular programming tool we called it LAB I because it was in fact in lab 2 of our institute where we spent so many nights trying to figure out why the heck this or that stupid thing happened that allows to simulate most of the experimental setups in a rather intuitive way It is almost like a virtual laboratory Elements are placed on a white board and connected by laser beams just as one would set up an experiment in the lab We have chosen to embed our package in LabView since the graphic programming language G allows in a unique way to program something that actually also looks like a virtual experiment Chapter 2 summarizes the basic math we have used for all of the programming Chapter 3 descr
69. se quantities at whatever position in the virtual experiment you need to look at them 67 RUN TIME DIFFERENCE pulse 2 in pulse 1 in time delay orr Input Output pulse 1 in 1st pulse in time delay time delay between 1st and 2nd pulse pulse 2 in 2nd pulse in run time difference determines the delay time between two pulses by calculating the difference of the first order phase terms This is sometimes useful if you want to calculate the time difference between two pulses having a different center frequency but which have traveled through the same piece of glass and now you want to know how far are they apart in time 68 M SAVE PULSE To FILE path dialog if empty sn pulse in TF Input Output pulse in input pulse path specify file name dialog if empty The module save to file takes a pulse at any point in the virtual experiment and stores it to a file You may load it later using the corresponding read pulse module 69 M SAVE PULSE TO SPREADSHEET path pulse in TF Input Output pulse in input pulse path specify file name dialog if empty The save pulse to spreadsheet module stores a pulse to a spreadsheet file The format is equivalent to that used by the FROG program Femtosoft The first column gives the wavelength in nm the second the square of the absolute value spectral intensity in units of J m nm the third
70. ser pulse is stretched by a 2 cm thick piece of SF14 and compressed by a pixelated pulse shaper operated 94 in phase only mode The pulse shaper has 128 pixel where each pixel covers 0 75 nm giving a full shaper window of 96 nm which is roughly four times the FWHM of the spectrum The center pixel coincides with the center wavelength of the laser The pulse itself is sampled in manual mode with 1024 samples B search space Figure 8 4 Virtual experiment Figure 8 5 shows the true case structure where the boundaries of the search space are fixed Here all 128 pixels may vary between 0 and a maximum phase of 27 Figure 8 5 Definition of the search space The front panel is shown in fig 8 6 Besides the pulse spectrum we see the shaper setting i e the value of all pixel in a bar graph and the pulse before and after the glass window and after the shaper As the search space is large convergence is usually reached only after many many iterations Use either one of the four genetic type algorithms together with the example experiment to see their performance When you want to design your own experiment use the experiment 95 Figure 8 6 Front panel vi that comes with LABI rename it throw away everything except the input array and the feedback indicator and insert your own code 96 Mm SIMPLEX DOWNHILL Write config File 7 p e EE E ee path to config
71. setups which are designed to stretch or compress short pulses In addition all of them have a little intelligent brother that automatically adjusts some parameters in order to optimize for maximum peak intensity at the output Also you ll find most standard glass materials 25 M IDEAL BEAMSPLITTER absorption transmission d pulse in en pulse out ws reflected pulse out iti Input Output pulse in input laser pulse transmitted pulse transmitted laser pulse absorption absorption 0 1 reflected pulse reflected laser pulse transmission transmission 0 1 The ideal beamsplitter is used to split the incoming beam in two outgoing replica The transmission 7 0 1 as well as the absorption A 0 1 are wavelength independent and the reflectivity is R 1 A T Make sure the sum of A T is always lower than 1 A real beamsplitter where the transmitted beam actually passes through some dispersive material is easily simulated by introducing a piece of dispersive material in the transmitted beam path 26 M BEAMSPLITTER Bearnsplitter pulse in transmitted pulse out reflected pulse out OMH Input Output pulse in input laser pulse transmitted pulse transmitted laser pulse beamsplitter file name reflected pulse reflected laser pulse The beamsplitter is a more realistic version of the module ideal beamsplitter You may specify a file that lists the reflectivity as well as the tr
72. spectral clipping that occurs whenever the second grating is too small We always assume that the center frequency impinges on the center of the second grating The effective transmission therefore is D sin 2o S T 0 cos Bg cos 8 ze 2 4 20 W 1 Dsin 8 fo lt S cos Bocos 2 where S is the lateral size of the grating 42 M INTELLIGENT GRATING COMPRESSOR grating compressor pulse in pulse out optimal delta oss Input Output pulse in incoming laser pulse pulse out output laser pulse lines mm grating constant optimal delta optimal grating separation angle of incidence refers to first grating delta distance between gratings measured at center frequency grating mirror distance from grating 2 to end mirror grating size width of both gratings diffraction order 1 The intelligent grating compressor is nothing else but a standard grating compressor that adjusts automatically the distance between the two gratings such that the second order phase present in the input pulse is minimized and the peak intensity of the out going pulse is maximal 43 M IDEAL SHAPER amplitude control phase control arbitrary e pulse in pulse out TF Input Output pulse in incoming laser pulse pulse out output laser pulse amplitude control specify amplitude modulation phase control specify phase modulation arbitrary arra
73. stantaneous frequency w t Suppose you are able to measure a spectrum of the laser say every femtosecond and you plot the center frequency 11 of each of these spectra as a function of time you have a good idea of what the instanta neous frequency is A more sophisticated definition is given by two dimensional distribution functions Knowing the phase t the instantaneous frequency is derived in the following way w t O t y t wo 2 22 For a Gaussian pulse we find Aw A w t wo 5 aa Obviously a quadratic phase leads to a linear change of the instantaneous frequency This is called a linear chirp A quadratic chirp corresponds to a cubic phase and so on Now suppose an ideal spectrometer is used to measure the spectrum of the pulse Because we operate in the frequency domain there is no such thing as time anymore Each infinitely narrow frequency component corresponds to an infinitely long wave Later we will see what influence the finite resolution of a real spectrometer has on the measurements We have already noted that no spectrometer is able to measure negative frequencies therefore the spectrum is 2 23 S w n w B w 2 24 where 7 w contains all specific information on the spectrometer and the detection unit used i e how the response of the spectrometer mirrors gratings prisms photomultiplier CCD etc depends on the frequency For an ideal spectrometer 7 is a constant and can be
74. t which if not connected is set to a useful default value Correlation measurements are the most useful tools to determine the temporal intensity in femtosecond world They are conceptually easy and may be realized in different variations depending on what exactly it is you want to measure There are background free versions and those which do have background there are second and third order versions depending on what nonlinear process you are using and there are interferometric and intensity correlation versions depending on whether the two beams interact in a collinear or non collinear fashion Background free versions have the great advantage that they may be used to measure deep correlations that is to measure the pulse intensity over many orders of magnitude Second order as compared to third order correlations have intrinsic time symmetry which prevents you from telling your lab mate whether the little bump showing up in the measure ment is a pre or a post pulse Interferometric version in contrast to intensity correlations give you some additional information on the phase of the pulse but they are almost never background free All intensity correlators in LABH have either one or two input pulses so they allow for auto as well as cross correlation measurements FROG is an abbreviation for Frequency Resolved Optical Gating and seems to become one of the standard diagnostic tools in femtosecond laser science It allows to al
75. t laser pulse reflectivity reflectivity of front side material medium between front and back side d thickness of the medium alpha angle of incidence The Gires Tournois interferometer is a special case of the Fabry Perot with the back side being 100 reflective The phase change introduced by the Gires Tournois interferometer is 1 si w arctan G 4 7 2V R R 1 cos where R r is the intensity reflectivity of the front surface The angle 6 is 2d cosd 4 8 Co l 1 sind sina 4 9 n where n is the index of refraction of the medium between the semitransparent and the fully reflective surface w is the laser frequency co the speed of light in vacuum and the angle of propagation in the medium 33 M PRISM COMPRESSOR prism compressor pulse in pulse out prism angle angle of incidence orr Input Output pulse in incoming laser pulse pulse out outgoing laser pulse Brewster angle automatic alignment phi2 2nd order spectral phase Min deviation automatic alignment prism angle calculated prism angle exact approx exact ray tracing or angle of inc calculated angle of approximate solution incidence prism angle adjust prism angle prism base base size of both prisms material prism material angle of inc adjust angle of incidence distance tip tip distance between the prism tips prism mirror distance prism 2 and mirror in prism 1 point of incidence prism 1 in pr
76. tal interaction processes between light and matter The resonant three level system numerically solves the coupled Maxwell Bloch equations and has been programmed by Ralf Netz He is also the author of the ladder system vi The diatomic molecule solves the time dependent Schrodinger equation for a diatomic molecule with a ground and one excited state and has been programmed by Tamas Rozgonyi 85 M RESONANT THREE LEVEL SYSTEM diagonals parameters of atomic system of z steps JETT pulse out SiN Input Output pulse in input pulse pulse out output pulse wavelength of 13 transition 1st resonance diagonals diagonal elements dipole moment of 13 transition wavelength of 12 transition dipole moment of 12 transition cell length z steps density initial values of density matrix Pil p22 p33 p31 p21 p32 of density matrix off diagonal elements of density matrix 1st dipole moment 2nd resonance 2nd dipole moment interaction length number of integration steps particle density off diagonals population state 1 population state 2 population state 3 off diagonal 31 off diagonal 21 off diagonal 32 The resonant three level system simulates the interaction of a short laser pulse with a resonant three level system The simulation is based on solving the coupled Maxwell Bloch equations The three level system is characterized by two resonance wavelengths two correspond
77. the center wave length The diffraction order in this case is 1 The same formula may be used for the Oeffner triplet if the focal length f is replaced by Rj Rz 2 R From the phase we may calculate the second and third order phase term by calculating the second and third derivative with respect to frequency 38 Figure 4 4 Detailed stretcher geometry 2gAL AL 2 1 4 14 i Tc d cos 6o a 3g AL 2 j AL sin Bo 14 4 1 T dj cos po d cos i oy Obviously the second order dispersion is positive and that the third order dispersion is negative 39 M INTELLIGENT GRATING STRETCHER grating strecher pulse in pulse out optimal g oss Input Output pulse in incoming laser pulse pulse out output laser pulse lines mm grating constant optimal g optimal displacement g angle of incidence focal length g grating mirror grating size diffraction order refers to first grating of the two mirrors or lenses displacement see text distance from grating 2 to end mirror width of both gratings 1 The intelligent grating stretcher is nothing else but a standard grating stretcher that adjusts automatically the displacement g such that the second order phase present in the input pulse is minimized and the peak intensity of the out going pulse is maximal 40 M GRATING COMPRESSOR Grating compressor pulse in sit pulse out ors
78. tial equation for each of the three fields These equations need to be integrated along the direction of propagation in order to simulate three wave mixing processes in a nonlinear crystal such as second harmonic generation sum frequency mixing 15 or difference frequency mixing Energy conservation requires that w w2 w3 The interplay between the three field Ej 1 2 3 is described by oe ay f dwz E u Eig w wa Akoro 2 36 0 F u2 ino dwr E w1 E3 w1 wy tAken 2 37 0 E3 ws irs fd E w Elw w1 eTiAkwr wwr 2 38 with 2 Ee oi 2 39 de 5 x 2 40 Aour e wi w2 nglwi w2 wznalwa w n w1 2 41 Co where w is a frequency component corresponding to spectrum F and n w is the wave length dependent index of refraction The coupled set of three differential equations is integrated along the direction of propagation z The final spectral fields are obtained by propagating the solution of the differential equations with the corresponding wave vector kj which incorporates the linear dispersive effects Ej Het 2 42 In case the conversion efficiency is small the above set of differential equations may be substantially simplified This is usually the case for thin crystals and or rather low intensi ties If we assume that both fundamental input beams are not depleted by the conversion process 0 F12 0 only one differential equation is left to solve E3 w3 r f dy E w
79. to publish his results here thank you Fred The TiSa amplifier vi is very helpful for the design of a multi pass Ti sapphire amplifier We compared the simulation results see table 5 1 with a number of amplifier systems published in the literature We selected systems reported after 1994 which were 60 designed to operate at maximum efficiency In table 5 1 six different amplifiers are simulated The fluences where calculated using the published input and output energies and beam diameters Also the published lengths and diameters of the TiSa rods and the number of passes were used in the simulations Some papers did not mention the size of the input or output beam or the length of the rod In these cases we made a guess We believe that these guesses are reasonable because the diameter of the pump beam should be slightly larger than that of the input and output beam The Titanium concentration was not given in most papers we guessed 0 15 This value turns out to be not so critical The diameter of the output beam was not given in the papers We have chosen it equal to the diameter of the input beam We also compared the peak intensities of the six systems calculated from the published output energies pulse lengths of the stretched pulses and the given or guessed output beam diameter It turns out that some of the systems operate close to the damage threshold of Ti sapphire which is about 5GW cm2 We are not sure about the values resultin
80. ts of the following elements Figure 3 1 Laser pulse cluster The first item is the number of sample points and determines how many points are used to represent a pulse in time or frequency space The value is always a power of 2 which allows to use the FF T algorithm fast Fourier transformation The pulse is encoded in the frequency domain and we only consider positive frequencies The frequency vector is stored relative to the center frequency wo of the laser pulse The vector w contains the relative frequency vector and the corresponding spectral amplitudes and phases are stored in the vector E w and phi w Because the energy per unit area i e the fluence is stored separately the spectral amplitude is usually normalized to a value appropriate for numerical 18 calculations The beam diameter is stored in diameter First order phases time delays are accumulated in to Using such an encoding allows to consider large delay times with relatively modest sampling rates which would otherwise violate the Nyquist limit An important flag is auto It determines whether all simulations are done with a variable or a fixed number of samples Variable means that at the input of every element the program tries to estimate the ideal number of sampling points needed to propagate the pulse through that element and resamples the pulse to that ideal sampling grid Auto mode is switched on or off by appropriate choice of parameters in all source or l
81. ty Av are mandatory inputs If nothing else is done all subsequent calculations run in auto mode that is fields are automatically resampled if necessary and iterations are adjusted such that outputs converge to the percent level In case the sampling cluster is connected all simulations run in manual mode and sampling is determined by the values in this cluster The number of samples is always a power of two and there are four options for the type of resolution High spectral The spectral window is such that the laser spectrum just fits in and the spectral resolution is highest High temporal Here sampling is adjusted so that the temporal resolution is high and the full time window is only slightly larger than the temporal FWHM of the pulse Of course spectral resolution is very poor here Balanced In this mode the sampling is such that both temporal and spectral amplitude are sampled with a somewhat balanced resolution Time window Here you can adjust the time window to a fixed value 20 E SECH SQUARE PULSE sech temporal sech spectral sampling TF Input Output energy energy pulse out laser pulse beam diameter beam diameter Ao center wavelength Ad spectral FWHM AT temporal FWHM sampling number of samples resolution sampling mode Sech Square Pulse is similar to Gaussian Pulse The only difference is the slowly varying envelope is a sech rather than a Gaussian The pulse characteristics a
82. ulse in input pulse fluence energy per unit area laser rep rate repetition rate ave power average power energy pulse energy The fluence of the laser pulse may be measured with the energy meter The device integrates the temporal intensity and outputs the pulse energy per unit area If you need you may also use the energy meter whenever you need a photo diode somewhere Just think of it as a calibrated photo diode If you specify a laser repetition rate you may also look at the average power 83 E TPC DIODE pulse 1 T pulse 2 in TPC signal Input Output pulse Ain Ist input pulse TPC signal output signal pulse B in 2nd input pulse TF Conventional photo diodes with a large bandgap may sometimes be used to measure the pulse intensity That is because the generated photo current depends on the square of the intensity since you need two photons to produce one electron hole pair if the bandgap is hwo lt Egap lt 2hwo So the signal you d measure in this case is s fare 6 8 The TPC diode has two inputs Usually both inputs get the same pulse but you may want to use two different pulses to realize something like a correlator 84 Chapter 7 Interactions In addition to the standard elements we have included special elements which have been programmed by users of LABI You will find a resonant three level system a n level ladder system and a diatomic molecule Both allow you to study fundamen
83. y What are the specific features of say a Ti sapphire laser pulse It s central wavelength is around 800 nm the pulse width is typically around 100 fs with some exceptions where pulses do get as short as 5 fs Because the oscillation period of the field at this wavelength is about 2 7 fs the pulses are usually longer than one period If we record a spectrum we see that there is a broad distribution around the center wavelength with a width of some tens of nanometers These experimentally observed characteristics influence the mathematical description in the following way Identifying a central wavelength wo and a width Aw in the spectrum satisfying the condition Aw wo lt 1 is equivalent to At T gt 1 This means that the electric field of the laser pulse can be split into a product of a slowly varying envelope At and a rapid oscillation with a period T 2r w The electric field in the time domain is then written 1 1 E t A e ei etot e t elt 2 8 2 where A t is the slowly varying amplitude y t the slowly varying phase and e t a complex quantity combining the slowly varying parts The definition of wo is not unique and there are various ways to define it A convenient one is to use the central frequency i e the frequency where the spectral amplitude has a global maximum If the spectrum is structured and the maximum is not easily identified it is more convenient to use the first frequency moment The defin
84. y with frequency phase and amplitude modulation The ideal shaper simulates an ideal pulse shaper which modulates the spectral ampli tude and or phase This shaper shows no pixelation effects There are a number of options available for amplitude and phase shaping none No amplitude and or phase modulation is applied interpolate The amplitude and or phase modulation specified in the array arbitrary is used The array contains three columns i e the absolute frequency the amplitude and the phase modulation The frequency axis must not necessarily coincide with the internally used sampling of the pulse The amplitude and phase values are resampled by interpolation to match the internal sampling of the pulse The frequency vector must contain absolute frequencies arbitrary The amplitude and or phase modulation specified in the array arbitrary is used The number of amplitude and or phase values MUST coincide with the number of samples of the incoming pulse The frequency vector is not considered at all as no interpolation is performed Assume the pulse is sampled with 256 individual samples then the array amplitude has to contain 256 values which then are used to modulate the 256 amplitude values of the incoming pulse Taylor Only the phase modulation can be specified in terms of Taylor coefficients The units of the n th order coefficient is fs Sinusoidal The amplitude and or phase of the pulse is modulated with a sinusoidal

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