The simulation of a long time series of atmosphericspeckles is required for a number of astronomicalapplications in the field of high-spatial-resolution im-aging from the ground. Several methods have beenimplemented for this purpose, the majority of whichfollow two main approaches. The first is based onthe simulation of one or more atmospheric wavefronts, larger than the telescope aperture, that arethen shifted with respect to the pupil. One standardfor the simulation of large random phase screens isthe fast-Fourier-transform- ~FFT-! based method ofMcGlamery.1 The generation of a time series ofspeckles by shifting the phase screen in front of thepupil was discussed, among others, by Jakobsson.2This approach is simple, fast, and computer effectivebut suffers from well-known limitations in the simu-lation of the lower spatial frequencies of atmosphericturbulence. Some methods that were implementedin order to reduce this problem are referenced anddescribed below. The second approach is based onthe simulation of a set of random wave fronts with asize comparable with the pupil and characterized bythe appropriate spatial and temporal statistics for
The author is with the Department of Astronomy, Trieste Uni-versity, Via G. B. Tiepolo 11, 34131 Trieste, Italy.
Received 11 July 1997; revised manuscript received 15 October1997.
Kolmogorov as well as for von Karman turbulencespectra. Each simulated phase screen is then usedto generate the corresponding speckle. One imple-mentation based on the space–time covariance ma-trix of the phases over the pupil was realized byRoggemann et al.3 Recently a different implemen-tation by a set of Fourier-series-transform-basedmodal expansions of the wave front over the pupilwas realized by Welsh.4 This approach could gener-ate accurate simulations that would be useful for finetemporal evolution analysis and particularly for thestudy of anisoplanatic effects. The major drawbackof the method of Roggemann et al.3 is the high de-mand of computational accuracy and memory,whereas the method of Welsh4 could necessitate asignificant amount of computing time for large pu-pils.
Within this framework it might be useful to godeeper into the estimate of the performance of FFT-based simulators inasmuch as little has been pub-lished on the phase-structure function and phasevariance over the pupil of FFT-based simulations oflarge wave fronts, in particular for rectangular for-mats and von Karman statistics. The von Karmancondition is important because at least it approxi-mates the atmospheric turbulence spectrum ob-served during astronomical measurements, as shownby Buscher et al.5 Johansson and Gavel6 report datafor the Kolmogorov and von Karman phase-structurefunctions of FFT-based phase-screen simulators withlow-spatial-frequency boosters for only square for-mats and separations up to 1 m. These data are not
20 July 1998 y Vol. 37, No. 21 y APPLIED OPTICS 4605
sufficient to characterize the performance on pupilsin the 10-m range that is typical of new generationtelescopes. In particular, no information is given inRef. 6 on the possible asymmetries of the phase-structure function in the case of long rectangularFFT-based phase screens. Moreover, no informa-tion has been reported for the phase variance over thepupil. The temporal behavior of the Zernike aber-rations for rectangular FFT-based phase screens hasbeen studied by Jakobsson2 but in this study there isno reference to possible effects of the aspect ratio onthe low-frequency deviations from theory, found inthe simulations.
In Section 2 the formulation of the FFT-basedphase-screen simulator with low-spatial-frequencyboosting is briefly reviewed. Section 3 reports thephase-structure function and phase variance over thepupil of a number of simulations of large single-layerwave fronts with particular attention to the asymme-tries of the phase-structure function on rectangularformats. Section 4 reports a method to reduce thedeviations from theory found in the tests on rectan-gular formats together with sample results that aresummarized in the conclusion.
2. Fast-Fourier-Transform-Based Phase-ScreenSimulators with Low-Spatial-Frequency Boosting
In its simplest approach, the generation of a temporalsequence of speckles under the near-field and Tay-lor’s frozen-flow hypothesis is done by means of sim-ulation of a single static phase screen suitably largerthan the telescope aperture, which is then shifted bysteps in front of the pupil. The short-term temporaldecorrelation observed between speckles can be sim-ulated approximately by means of a Markov processto evolve the wave front being shifted, as shown byGlindemann et al.7 Two or more statistically inde-pendent wave-front simulations can be combined insuch a way as to generate the multilayer model of theturbulent atmosphere. This model is required tomimic more accurately the temporal effects, as dis-cussed by Roddier8 and Glindemann and Rees.9 Allthese methods make use of large phase screens of asize proportional to the time interval to be sampled inthe simulation. With typical speckle lifetimes of,say, 10 ms, a simulation of 10 s will generate 1000speckles. With a typical wind speed of 10 mys thiscorresponds to a phase-screen step of 0.1 m and atotal length of the phase screen of 100 m, a figurecomparable with the outer scale length of the turbu-lent atmosphere. The phase-screen size should inany case be comparable with or larger than the outerscale length of the turbulence in order to include themajor fraction of the turbulent power in the simula-tion.
The numerical generation of such large phasescreens is usually implemented by means of the FFT-based method of McGlamery1 and allows simulationsof square as well as rectangular formats substan-tially limited only by the computer memory availablefor the FFT. The rectangular format is attractivebecause it optimizes the memory allocation for the
4606 APPLIED OPTICS y Vol. 37, No. 21 y 20 July 1998
maximum length of the phase screen at a given ap-erture of the pupil and allows the simulation of alonger time series of speckles at a given memory.One can generate the FFT-based random phasescreen by filtering, in the Fourier domain, a normalnoise with a filter of spectral response equal to thesquare root of the atmospheric turbulence spectrum,usually characterized by Kolmogorov or von Karmanstatistics. The formulation used in this paper, re-ported with minor formal changes from Johanssonand Gavel,6 is described by the following equations:
fmn 5 (m952Nxy2
Nxy221
(n952Nyy2
Nyy221
hm9n9fm9n9
3 exp@i2p~m9myNx 1 n9nyNy!#, (1)
fm9n9 5 0.15132~Gx Gy!21y2r0
25y6
3 @~m9yGx!2 1 ~n9yGy!
2 1 L022#211y12, (2)
where ~fmn, m 5 2Nxy2, Nxy2 2 1, n 5 2Nyy2,Nyy2 2 1! is the simulated random phase screen ofsize ~Gx, Gy! in meters and ~Nx, Ny! in pixels, hm9n9 isa complex Hermitian normal ~0, 1y=2! noise, fm9n9 isthe filter that approximates the square root of the vonKarman spectrum, r0 is the Fried parameter, L0 isthe outer scale length, and f0, 0 5 0 to obtain zero-mean phase.
The sizes of the phase screen and the support arrayset the minimum and the maximum spatial frequen-cies of the atmospheric turbulence spectrum that canbe sampled in the simulation. It is well known thatthe FFT-based method is not critical at higher spatialfrequencies, whereas it is so at lower ones becausethe array size soon exceeds any practicable memorysize for adequate sampling of the outer scale rangeof the turbulence. The undersampling of lower spa-tial frequencies will then deteriorate the simulationof any low-order turbulence effects, such as the tiltof the wave front. Several solutions have beenproposed in order to overcome this problem. Unfor-tunately, the simple approach of increasing thephase-screen size at a fixed array size is not practicalbecause it undersamples the higher spatial frequen-cies of the turbulence as well as the pupil. One run-ning approach consists of adding to the FFT-basedphase screen a statistically independent phase screenthat accurately simulates just the lower spatial fre-quency range of the turbulence not sampled by theFFT-based method.
The low-frequency phase screen can be simulatedby means of the Zernike polynomial expansionmethod of Roddier,10 which shows superior perfor-mance under Kolmogorov statistics. In this methodthe low-frequency wave front is simulated by low-order Zernike modal expansion over the pupil andthen added to the FFT-based phase screen by use ofNoll’s statistics.11 However, as pointed out by Ja-kobsson,2 the method of Roddier10 may suffer fromnumerical problems of the large formats needed bythe simulation of a long time series of speckles and isnot generally used for such applications.
One approach intrinsically fit for large and possiblyrectangular formats consists of adding statisticallyindependent subharmonic random phase screens of asize multiple of the FFT-based phase-screen size tothe FFT-based phase screen. Several techniquesare available for this purpose, such as the Fourier-series-transform-based methods of Herman andStrugala,12 Lane et al.,13 and Johansson and Gavel.6A detailed comparison of the performances of thesemethods is reported in Ref. 6. The method of Her-man and Strugala12 is based on the simulation of arandom subharmonic phase screen that is an exactmultiple of the FFT-based phase screen. The sub-harmonic phase screen is then sampled on the grid ofthe FFT support array. This method is straightfor-ward but yet poor at lower spatial frequencies, evenfor large subharmonic phase screens. The method ofLane et al.13 can be described as an iterative appli-cation of the method of Herman and Strugala12 to asubharmonic phase screen that is three times largerthan the current phase screen, with the first iterationstarting from the FFT-based phase-screen size.This method samples the lower spatial frequencies ofthe turbulence much better than the previous methodand is faster to compute. The method of Johanssonand Gavel6 is a hybrid of the two previous methods.It works like the method of Lane et al.13 but on asubharmonic screen half of the size of the current one.This method works well, particularly in the Kolmog-orov case, at the cost of some extra computing time.However, it should be noted that all these methodsare based on the Fourier series transform and not onthe FFT, and they are intrinsically time-consumingin the simulation of large formats.
The subharmonic phase-screen simulator used inthis paper is that of Lane et al.13 which was selectedfor its favorable performance to computing speed ra-tio. The formulation is that of Johansson and Gav-el,6 with minor formal changes, and is described bythe following equations:
fmnLF 5 (
p51
Np
(m9521
1
(n9521
1
hm9n9LF f m9n9
LF
3 exp@i2p32p~m9myNx 1 n9nyNy!#, (3)
f m9n9LF 5 0.15132~Gx Gy!
21y2r025y632p@~32pm9yGx!
2
1 ~32pn9yGy!2 1 L0
22#211y12, (4)
where LF identifies the variables specific to the low-spatial-frequency phase screen realized with the con-tribution of Np subharmonics, and f00
LF 5 0 for thezero-mean phase. The overall phase screen is thesum of the FFT-based and subharmonic phasescreens.
3. Phase-Structure Function and Phase Variance overthe Pupil
The simulator used for the tests was realized with anIDL 4.0 software procedure that implements Eqs.~1!–~4!. The phase screen is computed with NxyNy 5GxyGy. This sets equal cutoff spatial frequencies
and possibly different spatial-frequency bins on thetwo axes. Any residual mean value is removed fromthe phase after the FFT and after the computation ofeach subharmonic in order to give a zero piston con-tribution and decrease the numerical problems onhigher-order subharmonics. The tests were com-puted for a value of the Fried parameter of r0 5 0.1 m.Other tests done with r0 5 1.0 m have shown quali-tatively similar results.
A. Phase-Structure Function
The phase-structure function can be computed fromthe average of a large set of statistically independentrealizations. Roddier10 reports good estimates fromsets of 1000 realizations. However, it is possible toevaluate analytically the expected phase-structurefunction of the simulated phase screen through theautocorrelation of the filter used for the Fourier-transform-based simulations, as proposed by Johan-sson and Gavel.6 This avoids the long computingtime otherwise needed, with the drawback that thephase-structure function can be computed only up toseparations that are half of the size of the phasescreen, which is the approach that we used. Wedescribe the formulation by the following equationsreported from Johansson and Gavel6:
Df~m, n! 5 2@Bf~0, 0! 2 Bf~m, n!#, (5)
Bf~m, n! 5 (m952Nxy2
Nxy221
(n952Nyy2
Nyy221
f m9n92
3 exp@i2p~m9myNx 1 n9nyNy!#, (6)
where D is the expected phase-structure function ofthe phase screen and B is the autocorrelation of thefilter defined by Eq. ~2!. Once BLF is taken as theautocorrelation of the filter defined by Eq. ~4!, thesame formulation is used for the phase-structurefunction of the subharmonic phase screen when Eq.~6! is replaced with
BmnLF 5 (
p51
Np
(m9521
1
(n9521
1
~ f m9n9LF !2
3 exp@i2p32p~m9myNx 1 n9nyNy!#. (7)
The overall phase-structure function is the sum of thephase-structure functions of the FFT-based and sub-harmonic phase screens. The data obtained fromthe simulations should then be compared with thetheoretical phase-structure function for the vonKarman spectrum. One convenient expression ofthe von Karman phase-structure function was givenby Herman and Strugala12:
D~r! 5 6.16r025y3$~3y5!@L0y~2p!#5y3
2 @rL0y~4p!#5y6K5y6~2pryL0!yG~11y6!%, (8)
where D~r! is the phase-structure function, r is theseparation, K5y6~2! is a modified fractional Besselfunction of the third kind, and G~2! is the gammafunction. At infinite L0 this function approximates
20 July 1998 y Vol. 37, No. 21 y APPLIED OPTICS 4607
the well-known Kolmogorov phase-structure functionD`~r! 5 6.88~ryr0!5y3.
The results of the tests done on the phase-structurefunction are presented in Fig. 1 for square formatsand in Fig. 2 for rectangular formats. The specialfunctions needed for the calculation of the theoreticalphase-structure function were evaluated by means ofMathematica 2.2 software.
The von Karman phase-structure function is rea-sonably well approximated on square formats atscreen sizes comparable with the outer scale lengthwith just a few subharmonics, as shown in Fig. 1~a!.In this case three subharmonics allow a fair approx-imation of the von Karman function with a ratio ofthe phase-screen size to the outer scale length of 2y3with a maximum separation of 10 m and an outerscale length of 30 m. In this case the contribution ofthe subharmonic phase screen is important becauseof the relatively low ratio of the phase-screen size tothe outer scale length. The accuracy of the phase-structure function improves and the relative weightof the subharmonic contribution decreases quickly asthe ratio of the phase-screen size to the outer scalelength increases up to 1 ~and beyond!, as shown in
Fig. 1. Phase-structure functions of FFT-based phase-screen sim-ulations for square formats with additional subharmonic phasescreens: DFFT is the radial section from the center of the phase-structure function of the FFT-based phase screen, DLF is the samefor the subharmonic phase screen, and D Von Karman the theo-retical phase structure functions ~thick curve!. ~a! Phase screenof 20 m 3 20 m and 128 3 128 pixels, r0 5 0.1 m, L0 5 30 m, threesubharmonics; ~b! phase screen of 40 m 3 40 m and 128 3 128pixels, r0 5 0.1 m, L0 5 30 m, no subharmonics.
4608 APPLIED OPTICS y Vol. 37, No. 21 y 20 July 1998
Fig. 1~b!. This implies a minor role of the subhar-monic booster for the large formats typical of simu-lations of a long time series of speckles. Theseresults are consistent with those obtained by otherauthors, e.g., Jakobsson2 and Welsh,4 and extendtheir validity to telescope pupils in the 10-m range.
The results for rectangular formats, shown in Fig.2, are completely different. The most evident effectis the break of the circular symmetry of the phase-structure function, which is evident in the isophotesof the phase-structure function images shown in Fig.2~a! for the FFT-based phase screen and in Fig. 2~b!for the subharmonic phase screen. The X and the Ysections from the center of the phase-structure func-tion are shown in Fig. 2~c!. It should be noted thatthe sections corresponding to the longer side of thephase screen exceed the theoretical values. Theasymmetry is due to the fact that the simulationswere done with NxyNy 5 GxyGy in order to get equal
Fig. 2. Phase-structure functions of FFT-based phase-screen sim-ulations for rectangular formats with additional subharmonicphase screens: Phase-structure function image of ~a! the FFT-based phase screen ~DFFT!; ~b! the subharmonic phase screen~DLF!, rebinned to 128 3 64 pixels in the reproduction. ~c! X andY sections from the center of the phase-structure function.DFFT~x! and DFFT~y! are the X and the Y sections of the phase-structure function of the FFT-based phase screen, respectively,DLF~x! and DLF~y! are the same for the subharmonic phasescreen, D~x! is the X section of the overall phase-structure functionsum of DFFT~x! and DLF~x!, D~y! is the Y-section sum of DFFT~y!and DLF~y!, and D Von Karman is the theoretical phase structurefunction ~thick curve!. The phase screen is 8 m 3 4 m and 512 3256 pixels, r0 5 0.1 m, L0 5 10 m, three subharmonics.
X and Y sampling steps and this forced different fxand fy spatial-frequency bins. The lower spatial-frequency range of the turbulence is then more un-dersampled on the axis that corresponds to thesmaller side of the phase screen. The asymmetry ofthe phase-structure function of one single-layerFourier-transform-based simulation is then propor-tional to the aspect ratio of the phase screen. Longand narrow phase screens could show severe asym-metries and distortions of any parameter dependingon the accuracy of the phase-structure function.This could reduce the advantage of using rectangularFFT-based phase screens because of the need forlower aspect ratios for better accuracy at a givencomputer memory. It should also be noted that themajor axes of the isophotes of the phase-structurefunctions, the FFT-based phase screens, and the sub-harmonic phase screens are mutually perpendicular.This tends to average out the asymmetry of the over-all phase-structure function in the case of substantialsubharmonic contributions or in the case of multi-layer simulations with mutually perpendicular windvectors. However, this does not improve the accu-racy of each individual wave-front simulation. Oneapproximate solution to the distortions of the phase-structure function dependent on the rectangular for-mat of the phase screen is reported in Section 4.
B. Phase Variance over the Pupil
The variance sf2 of the phase over circular pupils open
in the phase screen with apertures d ranging from~dyr0! 5 1 up to the maximum allowed by the screensize can be computed by means of the expressionsgiven by Fried14 and the phase-structure functionD~r!, as follows:
where the term FC~r, d! in Eq. ~9! is replaced by FL~r,d! if the phase variance is computed after removal ofthe tip–tilt contribution.
The results for a 10-m square phase screen withouter scale lengths of 10, 30, and 100 m, and infinityare presented in Fig 3. All the data were obtainedby averaging 100 statistically independent realiza-tions ~200 for the Kolmogorov reference case! andwere normalized by the factor ~dyr0!5y3 from the Kol-mogorov case. The relatively higher dispersion ofdata at lower ~dyr0! values is due to the decreasednumber of pixels in the corresponding pupils. Asshown by Figs. 3~a! and 3~b!, the theoretical distri-bution is well approximated for screen sizes compa-rable with outer scale lengths up to 30 m withtypically up to five subharmonics. The approxima-
tion then deteriorates for screen sizes smaller thanthe outer scale length. As shown by Fig. 3~d! thiseffect cannot be recovered by the addition of moresubharmonics. This is consistent with similar re-sults obtained for the phase-structure function byJohansson and Gavel.6 The effect was further in-vestigated on simulations of phase screens with re-moval of the tip–tilt contribution. The resultsreported in Fig. 4 show a good approximation to the-ory, even if the data dispersion is somewhat largerthan their standard deviation because of the rela-tively low number of tests used. This confirms thatthe deviations from theory without tip–tilt removalare due mostly to the low-frequency region of theturbulence not sampled in any Fourier-transform-based phase-screen simulation.
The rectangular format yields more complex re-sults that depend on the size as well as on the aspectratio of the phase screen, as shown in Fig. 5. In thistest the phase variance was computed for three cir-cular pupils of equal aperture placed at the two endsand the center of the rectangular phase screen.
The test shown in Fig. 5~a! was done with an aspectratio of 1:8 and a value of the smaller side of thephase screen of 20% of the outer scale length, whichshows a large deviation from theory of the data of thecenter pupil, whereas the data of the two end pupilsnearly overlap with the theoretical distribution.This is due to the position-dependent asymmetry ofthe phase-structure function along the phase screenand can imply strongly different levels of accuracy ofthe speckles within the simulated time series. Theeffect is dependent mainly on the ratio of the smallerside of the phase screen to the outer scale length, asshown by the test case of Fig. 5~b!, which was donewith an aspect ratio of 1:8 and with the smaller sideof the phase screen equal to the outer scale length.The data for the three pupils are relatively good, withsome power excess at lower separations. These re-
Fig. 3. Normalized phase variance over circular pupils of 100FFT-based phase-screen simulations for square formats with ad-ditional subharmonic phase screens. There is one sigma errorbars. All phase screens are 10 m 3 10 m and 128 3 128 pixels,with r0 5 0.1 m: ~a! L0 5 10 m, three subharmonics; ~b! L0 5 30m, five subharmonics; ~c! L0 5 100 m, five subharmonics; ~d! infi-nite L0, ten subharmonics. Case ~d! used 200 simulations. Thetheoretical values are plotted by continuous curves.
20 July 1998 y Vol. 37, No. 21 y APPLIED OPTICS 4609
sults show that the phase variance over the pupil issomewhat less critical than the phase-structure func-tion versus the aspect ratio of the phase screen, eventhough it is sensitive to its size in much the same wayas the phase-structure function.
Fig. 4. Normalized phase variance over circular pupils of 200FFT-based phase-screen simulations for square formats with ad-ditional subharmonic phase screens and tip–tilt removal. Thereis one sigma error bar, and the phase screen is 10 m 3 10 m and128 3 128 pixels, with r0 5 0.1 m, infinite L0, and ten subharmon-ics. The theoretical values are plotted by the solid line.
Fig. 5. Normalized phase variance over circular pupils of 100FFT-based phase-screen simulations for rectangular formats withadditional subharmonic phase screens. There is one sigma errorbar. The phase screens have ~1024 3 128! pixels, with r0 5 0.1 m,L0 5 10 m, three subharmonics, and a size of ~a! 16 m 3 2 m, ~b!80 m 3 10 m. The theoretical values are plotted by continuouscurves.
4610 APPLIED OPTICS y Vol. 37, No. 21 y 20 July 1998
4. Compensation for Aspect-Ratio Effects on thePhase-Structure Function
A. Symmetrization of Fast-Fourier-Transform-BasedPhase-Structure Function
One simple, although only approximate, method ofcompensation of the effects of the rectangular formaton the phase-structure function of FFT-based phasescreens is to weight the FFT spectrum properly.The aspect ratio of the spectrum is controlled mostlyby the relative weights of the spectral power at spa-tial frequencies f0,21 and f0,11 for the X axis and f21,0and f11,0 for the Y axis. Setting the weights allowsthe spectrum to be approximately symmetrized in theregion of interest at the center of the FFT-basedphase screen. The weights approximate the valuesof the aspect ratio and its reciprocal at large aspectratios. However, it is convenient to optimize theweights individually against the symmetry of thephase-structure function, particularly at lower aspectratios and smaller outer scale lengths.
This technique was successfully applied to aspectratios from 1:2 to 1:32, with the results shown in Fig.6. The isophotes of the phase-structure function im-ages are approximately circular to ;60% of the cen-tral region of interest. The improvement ofsymmetry of the phase-structure function is clearlyevident in the corresponding X and Y sections. Theresults for the aspect ratio 1:2, shown in Fig. 6~b!,include a subharmonic component that does not addfurther symmetry errors since it was computed bymeans of the method described in subsection 4.Bwithout affecting the FFT compensation. The othercases with aspect ratios of 1:4, 1:16, and 1:32 werecomputed with no subharmonic contribution. Theresults shown were obtained by individual optimiza-tion, for each aspect ratio, of the square-root weightsapplied to the terms of spatial frequency f0, 21, f0, 11,f21, 0, and f11, 0 of the FFT spectrum.
B. Symmetrization of the Subharmonic Phase-StructureFunction
The spectral weighting method used for the FFT-based phase-structure function could also be usedfor the subharmonic phase-structure function.However, it is possible to use directly a large squaresubharmonic phase screen computed on a properlyreduced grid in a relatively shorter time and theninterpolate the result back to the original FFT-based phase-screen format. The phase-structurefunction results are symmetric, which is due to thesymmetry of the phase screen, whereas the comput-ing time remains short since the Fourier seriestransform is computed on a small support array.This is possible because the subharmonic phasescreen contains only spatial periods that are longerthan the FFT-based phase screen, thus allowing asampling step that is much larger than the corre-sponding step for the FFT-based phase screen.The formulation of Johansson and Gavel6 of the
Fig. 6. Symmeterized phase-structure functions ofFFT-based phase-screen simulations for rectangularformats with additional subharmonic phase screensand compensation for the aspect-ratio effects on theFFT-based phase-structure function. Wx and Wy arethe X and the Y square-root spectral weights used forthe compensation. The phase screens are ~a!, ~b! 4 m3 2 m and 256 3 128 pixels, with three subharmon-ics; ~c!, ~d! 64 m 3 16 m and 512 3 128 pixels, with nosubharmonics; ~e!, ~f ! 256 m 3 16 m and 2048 3 128pixels, with no subharmonics; ~g!, ~h! 512 m 3 16 mand 4096 3 128 pixels, with no subharmonics. In allcases r0 5 0.1 m, L0 5 30 m, and NG 5 8. All imagesshow the central region of 256 3 128 pixels rebinnedto 128 3 64 pixels in the reproduction. DFFT marksthe FFT-based phase-structure function image andDFFT~x! and DFFT~y! are its X and Y sections, re-spectively, DLF, DLF~x!, and DLF~y! do the same forthe subharmonic phase screen, D~x! is the X sectionof the overall phase structure function sum ofDFFT~x! and DLF~x!, D~y! is the Y-section sum ofDFFT~y! and DLF~y!, and D Von Karman is the the-oretical phase-structure function ~thick curve!.
subharmonic booster of Lane et al.13 given in Eqs.~3! and ~4! was then modified as follows:
frsLFG 5 (
p51
Np
(m9521
1
(n9521
1
hm9n9LF f m9n9
LFG
3 exp@i2p32p~m9iyNx 1 n9jyNx!#, (12)
where i 5 2Nxy2 1 r~NyyNG!, j 5 2Nyy2 1 s~NyyNG!,and
f m9n9LFG 5 0.15132Gx
21r025y632p@~32pm9yGx!
2
1 ~32pn9yGx!2 1 L0
22#211y12, (13)
where frsLFG, @r 5 0, NG~NxyNy!, s 5 0, NG# is the
reduced-grid subharmonic phase screen and ~NG 11! is the number of samples in the smaller side ofthe rectangular subharmonic phase screen on thereduced grid. The relative gain in computing timeapproximates @Nyy~NG 1 1!#2. A typical value NG5 8 yields large gains, even for moderately large
phase screens. The interpolation to the originalgrid can be done by fast and memory-effective bi-linear algorithms without substantial losses of ac-curacy.
The results obtained with this method are shownin Fig. 7. There is evident improvement comparedwith the corresponding data of Fig. 2. The iso-photes of the subharmonic phase-structure functionimage shown in Fig. 7~a! are circular and the X andthe Y sections shown in Fig. 7~b! overlap. Thetruncation evident at lower values of the separationis due to the use of a bilinear interpolator to restorethe original format. This effect is quite small, asone can see by comparison with the values com-puted with the standard formulation and is of littlepractical importance. If wanted, such a distortioncan be minimized by an increase in the value of NGor by use of a bicubic or two-dimensional splineinterpolator at the cost of some extra computingtime.
20 July 1998 y Vol. 37, No. 21 y APPLIED OPTICS 4611
C. Phase Variance over the Pupil for SymmetrizedPhase-Structure Functions
The symmetrization of the phase-structure functionin the case of rectangular phase screens improves theaccuracy of the phase variance over the pupil, asshown in Fig. 8. The data of Fig. 8~a! can be directlycompared with those of Fig. 5~a!. The improvementis evident, even if not particularly large, because ofthe low ratio of the smaller side of the screen size tothe outer scale length, equal to 0.2. However, thedata for the center pupil now become close to theoryand the dispersion of the data of the three pupils isclearly smaller. The data of Fig. 8~b! are good andfully comparable with the data of Fig. 5~b! in thesame range of separations, but with a ratio of thesmaller side of the screen size to the outer scalelength of only 0.4, against the value of 1.0 of Fig. 5~b!.This gain is due to the positive effects of the symme-trization of the phase-structure function on the phasevariance over a pupil.
5. Conclusions
The accuracy of the phase-structure function and thephase variance over the pupil of FFT-based simula-tions of large atmospheric wave fronts with a vonKarman turbulence spectrum depends on the ratio ofthe phase-screen size to the outer scale length, asalready known, and also for rectangular formats onthe aspect ratio of the phase screen. The undersam-pling of the lower-spatial-frequency range of the at-
Fig. 7. Symmetrized phase-structure function of the subhar-monic phase screen with compensation for the aspect-ratio effects.The phase screen is 8 m 3 4 m and 512 3 256 pixels, with r0 5 0.1m, L0 5 10 m, three subharmonics, and NG 5 8. ~a! Phase-structure function image ~DLF! of the subharmonic phase screenrebinned to 128 3 64 pixels in the reproduction, ~b! X and Ysections form the center of the phase-structure function of thesubharmonic phase screen. DLF~x! and DLF~y! are the X and Ysections of DLF, which overlap in the figure. The radial section ofthe phase-structure function of the subharmonic phase screen atnominal FFT resolution is plotted by the dotted curve.
4612 APPLIED OPTICS y Vol. 37, No. 21 y 20 July 1998
mospheric turbulence intrinsic to the FFT-basedphase-screen simulators plays a minor role for thelarge formats typical of the simulation of a long timeseries of speckles from large telescopes on squareformats, whereas this is true only in the averagesense for rectangular formats. The use of a subhar-monic phase-screen booster might improve the accu-racy of the simulation to acceptable levels for outerscale lengths much larger than the phase screen or itssmaller side for rectangular formats.
The rectangular formats force an asymmetry pro-portional to the aspect ratio on the phase-structurefunction and smaller distortions on the phase vari-ance over a pupil. This effect could limit the use oflong and narrow FFT-based phase screens to onlylower aspect ratios. However, the aspect-ratio-dependent asymmetry of the phase-structure func-tion can be compensated for by weighting the FFTspectrum in the FFT-based phase-screen simulatorand use of a fast symmetrized subharmonic phase-screen booster. This asymmetry compensationmethod achieves a running compromise between per-formance and computer resources for the generationof a long time series of speckles from simulated
Fig. 8. Normalized phase variance over circular pupils of 100FFT-based phase-screen simulations for rectangular formats withadditional subharmonic phase screens and compensation for theaspect-ratio effects on the phase-structure function. There is onesigma error bar. Wx and Wy are the X and the Y square-rootspectral weights used for the compensation. The phase screensare ~a! 16 m 3 2 m and 1024 3 128 pixels, with r0 5 0.1 m, L0 510 m, three subharmonics, and NG 5 8; ~b! 32 m 3 4 m and 1024 3128 pixels, with r0 5 0.1 m, L0 5 10 m, one subharmonic, and NG
5 8. The theoretical values are plotted by the continuous curves.
single-layer atmospheric wave fronts with a largerectangular format.
This study was supported by a research contract ofUniversita di Trieste, Dipartimento di Astronomia.C. Morossi of Osservatorio Astronomico di Triestecontributed extremely useful discussions and criti-cism on the phase-variance analysis. The currentversion of this paper relied heavily on the thoroughanalysis of the original manuscript and the detailedsuggestions of the anonymous referees, gratefully ac-knowledged here for their help.
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