Diversity selection for phase-diverse phase retrieval

1490 J. Opt. Soc. Am. A/Vol. 20, No. 8 /August 2003 B. H. Dean and C. W. Bowers

Diversity selection for phase-diverse phaseretrieval

Bruce H. Dean

Optics Branch: Design & Analysis Group/Code 551, NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771

Charles W. Bowers

UV/Optical Astronomy Branch/Code 681, NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771

Received September 19, 2002; revised manuscript received January 21, 2003; accepted March 18, 2003

Wavefront-sensing performance is assessed for focus-diverse phase retrieval as the aberration spatial fre-quency and the diversity defocus are varied. The analysis includes analytical predictions for optimal diversityvalues corresponding to the recovery of a dominant spatial-frequency component in the pupil. The calculationis shown to be consistent with the Cramer–Rao lower bound by considering a sensitivity analysis of the point-spread function to the spatial frequency being estimated. A maximum value of diversity defocus is also cal-culated, beyond which wavefront-sensing performance decreases as diversity defocus is increased. The re-sults are shown to be consistent with the Talbot imaging phenomena, explaining multiple periodic regions ofmaximum and minimum contrast as a function of aberration spatial frequency and defocus. Wavefront-sensing performance for an iterative-transform phase-retrieval algorithm is also considered as diversity defo-cus and aberration spatial frequency are varied.

OCIS codes: 100.5070, 120.5050, 110.6760.

1. INTRODUCTIONPhase-diverse phase retrieval is an image-basedwavefront-sensing method that utilizes point-source im-ages modulated in a controlled way (i.e., with a diversityfunction1) to recover optical phase information. In prin-ciple, any known aberration can serve as a diversity func-tion, but defocus is often the simplest to implement andexhibits no angular dependence as a function of the pupilcoordinates.2 In earlier analysis,3–5 certain defocus set-tings have been shown to enhance phase-retrieval estima-tion performance based on numerical calculations of theCramer–Rao lower bound (for general references on theCramer–Rao bound, see Refs. 6–9; additional applica-tions to wavefront sensing are addressed in Refs. 10–12).But no general predictions have been derived to showwhich diversity values are optimal for a given wavefront-sensing application or even to motivate a general under-standing of why certain defocus values are preferable toothers. In this paper, a theoretical basis for the phenom-ena is discussed, as well as how optimal diversity func-tions can be utilized to enhance phase-retrieval estima-tion performance. An important corollary result is thatproper diversity selection really defines how wavefrontsensing and control should be implemented, i.e., systemsthat are designed for lower-spatial-frequency recoveryshould be distinguished from those designed to recoverhigher-spatial-frequency aberrations.13,14 This view-point provides a general guide for understanding how di-versity functions should be implemented and is discussedfurther in Section 10.

To motivate the approach discussed in the paper fromthe perspective of wavefront-sensing algorithms, Fig. 1depicts the mechanism of classical iterative-transform

phase retrieval (termed ‘‘error reduction’’ in the earlierliterature15), which operates by iteratively enforcing con-straints between conjugate Fourier domains. A phasemap is thus produced that is consistent with the mea-sured data in both domains to within some convergencecriterion. This basic mechanism gives a simple picture ofhow good contrast in the aberration ‘‘signals’’ can lead toimproved wavefront-sensing performance. Conversely, ifthe data have low signal-to-noise ratio or if the aberrationcontent is only weakly coupled to the intensity data (orboth), convergence can be slow or yield inaccurate results.This conceptual framework applies to other estimationmethods as well; e.g., parametric phase retrieval (e.g.,Ref. 16) attempts to minimize an objective function Owith respect to the unknown aberration parameters. Ifthe objective function is weakly coupled to the data, then,similarly, the gradient of the objective function will ex-hibit a weak functional dependence with respect to theparameters being estimated. In such cases, a numericalsolution minimizing O may converge slowly or yield inac-curate results. A quantitative numerical description ofthe viewpoint described above follows from the Cramer–Rao lower bound, which is discussed further in Section 7.

The paper is organized as follows: In Section 2, thespectral power corresponding to a defocused phase grat-ing is calculated. In Section 3, these results are appliedto derive optimal diversity defocus values that yield maxi-mum contrast in the spectral power for image-basedwavefront sensing. A maximum value of diversity defo-cus is also calculated in Section 4, beyond whichwavefront-sensing performance decreases as diversity de-focus is increased. In Section 5, the fundamental roleplayed by diffraction in the recovery of aberration infor-

B. H. Dean and C. W. Bowers Vol. 20, No. 8 /August 2003 /J. Opt. Soc. Am. A 1491

Fig. 1. Iterative-transform phase retrieval [F(•) denotes Fourier transform].

Fig. 2. Defocus 1 sinusoidal ‘‘aberration.’’

mation is emphasized, and the alternating contrastmaxima and minima that occur in the spectral power as afunction of defocus are shown to be consistent with theTalbot imaging phenomena. In Section 6, a transferfunction analysis is presented to locate ‘‘windows’’ of peakresponse as a function of spatial frequency and defocus.In Section 7, consistency with the Cramer–Rao lowerbound is discussed. Wavefront-sensing performance isdiscussed in Section 8 to demonstrate that proper diver-sity selection can lead to improved wavefront-sensing con-vergence and accuracy when utilizing iterative-transformphase retrieval. Finally, in Section 9, a dominant spatialfrequency for the Zernike basis functions of a given orderis calculated. These results are then applied to deriveoptimal diversity defocus values as a function of Zernikeorder.

2. SPECTRAL POWER OF A DEFOCUSEDPHASE GRATINGConsider the calculation of image irradiance in a defo-cused image plane when the exit pupil of the system isconjugate to a phase grating (phase gratings are dis-cussed, for instance, in Ref. 17; earlier analysis on the

propagation of spectral power utilizing a phase grating inthe context of wavefront sensing has been presented inRefs. 18–20). This model is particularly relevant whenconsidering the wavefront spatial-frequency content ofsystems with deformable mirrors,21 in addition to beingconvenient for analytical work.

Assuming a monochromatic point source, the calcula-tion is modeled as a Fresnel propagation through an ar-bitrary paraxial optical system of focal length f. In thisinstance, a cancellation of quadratic phase terms occurs,and therefore the complex amplitude of the point-spreadfunction, PSFca (or amplitude-spread function), is theFourier transform of the complex pupil A. For now, con-sider only one-dimensional (1-D) coordinate dependencein the pupil, and let A(x) denote the aperture function.The following notation is used: k 5 2p/l, i2 5 21, m isthe ‘‘aberration’’ amplitude, a is the defocus parameter,and x is the pupil coordinate as shown in Fig. 2. ThePSFca is expressed as

PSFca 5 F~A! 5 F(A~x !exp$ik@ax2 1 m sin~2pn0x !#%).(1)

Assuming that km ! 1, the pupil function is then ap-proximated by


Fig. 3. Plot of sin(pn02/8a) versus defocus for n0 5 10 cycles/aperture.

A ' @1 1 imk sin~2pn0x !#A~x !exp~ikax2!, (2)

so that Eq. (1) becomes

PSFca~n! 5 F $@1 1 imk sin~2pn0x !#%

^ F @exp~ikax2!# ^ F@A~x !#, (3)

where ^ denotes convolution and F (•) is the Fouriertransform operator. Note from Eq. (3) that the Fouriertransform of the aperture function acts simply as an over-all ‘‘modulator’’ of the result, i.e., the multiplier termF @A(x)# exhibits no functional dependence on n0 , typi-cally given by the Airy function for a circular aperture.Therefore, only the first two terms in Eq. (3) determinethe condition for maximum and minimum contrast as afunction of spatial frequency and defocus. With this ob-servation, we leave the aperture geometry arbitrary inEq. (3) and drop the F @A(x)# term for now. The conse-quences for neglecting the aperture will be discussedfurther in Section 5. Equation (3) then becomes( n [ x/lf# is a dimensionless image plane spatial-frequency coordinate and f# denotes f-number)

PSFca 5 F~A!

5 H d ~n ! 1mk

2@d ~ n 1 n0! 2 d ~n 2 n0!#J

^ exp~2ipln2/2a !, (4)

which further simplifies to

PSFca 51

2 XexpS 2 ipl

2an2D

1mk

2 H expF 2 ipl

2a~n 1 n0!2G

2 expF 2 ipl

2a~n 2 n0!2G J C. (5)

Equation (5) takes a more convenient form when ex-pressed in terms of the defocus parameter a (waves/radian), defined by first noting that

~a/l!x2 → ax2 5ax2

~D/2!2⇒a 5

4l

D2a, (6)

where D/2 is the aperture radius and the normalized pu-pil coordinate is x [ 2x/D. In terms of the dimension-less variable n0 5 Dn0 (cycles/aperture), the complex am-plitude of the PSF is expressed finally as

PSFca 51

2 XexpS 2 ip

8an2D 1

mk

2 H expF 2 ip

8a~ n 1 n0!2G

2 expF 2 ip

8a~ n 2 n0!2G J C. (7)

The intensity is the complex-conjugate square of Eq. (7).Given that mk ! 1, the result simplifies to

I~ n ! 5 PSFca 3 PSFca

5 1 2 2mk sin~pn02/8a !sin@2p~n0/8a !n#, (8)

again noting that diffraction due to the aperture stop hasbeen ignored in this portion of the calculation. The ef-fects due to the aperture stop are discussed further inSection 5 and illustrated numerically in Section 6.

3. OPTIMAL DIVERSITY DEFOCUSConsidering the sin(pn0

2/8a) term of Eq. (8), the conditionfor maximum contrast in n0 as a function of defocus fol-lows from

pn02

8a5

p

2~2n 6 1 ! ⇒ amax 5 6

n02

4~2n 7 1 !,

n 5 0, 1, 2,..., (9)

since

sinFp

2~2n 6 1 !G 5 7cos~np!, n 5 0, 1, 2,... .

(10)

That is, for a particular phase grating frequency n0 , thereexists a discrete set of defocus locations amax at whichfringe contrast will be maximum. For example, the val-


ues of sin(pn02/8a) are plotted in Fig. 3 as a function of

defocus for the numerical example of n0 5 10 cycles/aperture. Heuristically, wavefront-sensing accuracy forn0 is expected to be greatest at these diversity values.

A couple of comments should be made with regard toEq. (9). For n @ 1, the predicted optimal defocus valuestend toward zero. However, in practice, zero defocus re-sults in zero contrast for n0 . This behavior is indicatedin Fig. 3, where we note that as a → 0 the frequency ofoscillation of the sin(pn0

2/8a) term approaches infinity,and therefore the contrast differences between variousmaxima and minima are not resolved, i.e., the 61 diffrac-tion orders will fall within the Airy disk, yielding a zerofringe contrast. Second, the analysis that we have pre-sented is based on the assumption that the PSF is treatedas a continuous function, i.e., measured with a perfect de-tector with continuous sampling. This assumption ne-glects the fact that in practice the PSF is sampled over afinite grid (a detector array). As a result, nonzero defo-cus values are preferred when the PSF data are under-sampled over a finite grid, despite the prediction of Eq.(9). That is the primary reason for including a nonzerodefocus value is to ‘‘spread out’’ the PSF response to sucha degree as to indicate how the aberrations are modulat-ing the intensity data. A detailed study of sampling ef-fects is beyond the scope of this paper and will be consid-ered in a future analysis.

The 1 sign in the denominator of Eq. (9) represents acontrast reversal in the spectral power with respect to thesignal contrast predicted by the negative-sign case. Bothcases yield a maximum in the signal n0 and originatefrom the 7 sign in Eq. (10). Therefore we make no dis-tinction between these two cases insofar as wavefrontsensing is concerned. The 6 sign preceding Eq. (9) [andEq. (11) below] is independent of the 7 sign in Eq. (10)and shows that maximum contrast is obtained for equallyspaced defocus values placed on either side of the imageplane by inspection of Eq. (8). The implication for wave-front sensing is that a diversity image can be placed oneither side of focus, while still maintaining maximum con-trast in n0 according to Eq. (9). This result has been ob-tained independently from an earlier Cramer–Raoanalysis.22 General consistency between the Cramer–

Rao analysis and the results above are considered furtherin Section 7.

A basic observation from Eq. (9) is that amax increasesquadratically with n0 . Therefore, to the extent that es-timating n0 is aided by stronger spectral power in thedata, the recovery of higher-spatial-frequency aberrationsignals will generally benefit from larger defocus values.This interpretation is supported in Section 7 [see relation(21)] by an estimation-theoretic analysis. Conversely, itfollows that lower-spatial-frequency aberrations will gen-erally benefit from smaller, diversity defocus settings.The wavefront-sensing implications are discussed furtherin Section 8. Referring again to Eq. (8), minima in thespectral power occur when

pn02

8a5 np ⇒ amin 5 6

n02

8n, n 5 0, 1, 2,... . (11)

The predictions for maximum or minimum contrastwere derived assuming a 1-D pupil function but are alsovalid for the two-dimensional (2-D) case, since spatial fre-quencies in orthogonal directions separate as convolutionproducts by inspection of Eqs. (2) and (3). As a consis-tency check, several defocus values are calculated fromthe predictions for the maxima and the minima and thencompared with their 2-D numerical counterparts bymeans of fast Fourier transforms (FFTs). These compari-sons are made in Section 6.

4. MAXIMUM DIVERSITY DEFOCUSIn addition to the contrast maxima and minima that oc-cur (e.g., Fig. 3), a theoretical value of maximum diversitydefocus exists, beyond which there is a penalty in imagingcontrast for further increasing focus. This value corre-sponds to the last contrast reversal/maxima, n 5 0, in Eq.(9),

arev,max 5 6n0

2

4, (12)

and in the limit as a → `,

Fig. 4. Position of maximum diversity defocus for n0 5 10 cycles/aperture.


lima → `

$sin~pn02/8a !% 5 0, (13)

corresponding to a contrast minimum. Therefore, arbi-trarily increasing the magnitude of diversity defocus be-yond the value predicted by Eq. (12) can only hinderwavefront-sensing performance. Alternatively, the limitin Eq. (13) is consistent with a contrast minimum at thedefocus value of infinity, corresponding to n 5 0 in Eq.(11). A specific numerical example is illustrated in Fig. 4.

The existence of a maximum value for diversity defocusis consistent with earlier Cramer–Rao studies of qua-dratic curvature wavefront sensing,4 but for a differentreason. In Ref. 4, this effect is attributed to the opticaltransfer function (OTF) developing zeros as defocus is in-creased, leading to a reduction in contrast at the focalplane. But the present analysis shows that for aberra-tion recovery through phase retrieval, the penalty in im-aging contrast is due to the fact that information on n0 isshifted to ‘‘dc’’ in the image as a → `. Furthermore,‘‘windows’’ of spectral power corresponding to n0 areshown to exist at the defocus values predicted by Eq. (9).The locations of these windows in the spectral power arecalculated from the OTF in Section 6.

5. ROLE OF APERTURE GEOMETRY ANDTALBOT IMAGINGBy inspection of Eqs. (8) and (11), the minimum-contrastcase corresponds to the intensity value

Imin 5 1, (14)

implying that the minimum-contrast case contains no in-formation on the signal n0 when aperture geometry is ne-glected. As expected, the corresponding Cramer–Raobound is infinite, as shown in Section 7 [see Eq. (22)].This result not only identifies the estimation-theoreticconsequence of neglecting aperture geometry but also em-phasizes the fundamental role played by diffraction in therecovery of aberration information from intensity data.In Section 6, comparisons are made between the analyti-cal predictions above [Eqs. (9) and (11)] and their 2-D nu-merical counterparts by means of FFTs. In particular, avisual inspection of the minimum-contrast result in Fig. 6below indicates that spatial-frequency information corre-sponding to n0 is ‘‘dispersed’’ at the aperture boundary be-cause of diffraction by the aperture stop. Indeed, it isthis diffracted ‘‘information’’ due to the aperture responsethat allows the recovery of n0 from the minimum-contrastintensity data, as illustrated in Fig. 8 below.

The Talbot imaging phenomena23 consist of periodic‘‘self-images’’ formed by an amplitude grating under co-herent illumination in the Fresnel zone. The grating be-haves as an image-forming system or ‘‘lens’’ as a result ofinterference that results from the diffractive nature of thegrating. An important corollary result is that theFresnel zone propagation of spectral power is mathemati-cally equivalent to the Fraunhofer propagation of a defo-cused pupil plane. This is due to the quadratic phase fac-tors carried in each case. In Section 3, we have shownthat a defocused phase grating yields an imaging phe-nomenon completely analogous to Talbot imaging, but for

a phase grating rather than an amplitude grating. Thelocations of the ‘‘Talbot’’ image planes are thus marked byregions of maximum and minimum contrast as a functionof spatial frequency as well as contrast reversal in thespectral power. The locations of the maxima and the con-trast reversals give predictions for the optimal diversitydefocus values for wavefront sensing, as demonstrated inSection 8.

6. OPTICAL TRANSFER FUNCTIONThe (incoherent) optical transfer function (OTF) is de-fined as the Fourier transform of the PSF (or, equiva-lently, the autocorrelation of the pupil). Therefore, usingEq. (8), we calculate the OTF as a function of spatial fre-quency and defocus:

OTF~v ! 5 F @I~ n !#

5 d ~v! 1 imk sin~pn02/8a !

3 @d ~v 2 v0! 2 d ~v 1 v0!#, (15)

where d is the Kronecker delta function and vc is the in-coherent cutoff frequency, given in

v 5 lf#v 5 v/vc , (16)

with v0 defined as

v0 [ n0/8a. (17)

Equation (17) is the main result, locating ‘‘windows’’ ofthe OTF peak response as a function of spatial frequencyand defocus. The result in Eq. (15) is idealized in thesense that the locations of peak response are marked bydelta functions, but when aperture geometry is included,the delta functions are replaced by a ‘‘finite-spread’’ func-tion because of the aperture response. This effect is il-lustrated in Fig. 5, which gives a graphical comparison ofthe analytical prediction [Eq. (17)] with the corresponding2-D numerical OTF calculation. The calculation as-sumes an unobscured circular aperture and propagationwith FFTs. The same comparison is repeated in Fig. 6for the minimum-contrast case.

Typical numerical values for the various contrast casesare listed in Table 1 for n0 5 10 cycles/aperture. Thevalues follow from Eqs. (9) and (11), using n 5 0, 1, 2.The 1 and 2 subscripts in the maximum-contrast casesdistinguish between the 1 and 2 signs in the denomina-tor of Eq. (9) (i.e., the 1 case is a contrast reversal): Aspointed out above, Figs. 5 and 6 give a graphical compari-son of the analytical predictions (for n 5 1), as listed inTable 1, and their 2-D numerical counterparts. The 1case (a contrast reversal) is omitted, since it is nearlyidentical to the contrast maxima, aside from a p/2 phaseshift in the ‘‘band’’ structure of the defocused intensitydata. For simplicity, an unobscured circular aperture isassumed as well as 512 3 512 Nyquist-sampled PSFs.As noted earlier in Section 3, information on the signal n0is shifted to dc in the image as defocus is increased be-yond the theoretical value predicted in Eq. (12). This re-sult is corroborated by the OTF calculation above [and bythe Cramer–Rao bound below—see the comments after


Fig. 5. Comparison of transfer functions and analytical prediction for contrast maxima for n 5 1 and n0 5 10 cycles/aperture; defocus5 8.33 waves.

Fig. 6. Comparison of transfer functions and analytical prediction for contrast minima for n 5 1 and n0 5 10 cycles/aperture; defocus5 12.5 waves.

relation (21)], since v0 → 0 as a → ` in Eq. (17), i.e., v0decreases toward the dc component of the transfer func-tion.

7. CONSISTENCY WITH THECRAMER–RAO LOWER BOUNDThe Cramer–Rao lower bound is calculated from a statis-tical model relating the parameters being estimated tonoise in the measured data. As a result, the calculationprovides the basis for a physical understanding of how es-timation errors are affected by signal-to-noise ratio, diver-sity functions, sampling, and so on. In this section, theCramer–Rao theorem is briefly reviewed, and its applica-tion to focus-diverse wavefront sensing is discussed. Wethen demonstrate consistency between the Cramer–Raobound and the analytical results of Section 3.

The statistical model is given by the Poisson probabil-ity density function for photon-limited images. TheFisher information array for phase-diverse imaging is de-fined as the negative expectation of the log-likelihoodHessian matrix (Refs. 5 and 24):

Fab [ 2E~]ab2 L ! 5 Np (

m51

M

(i, j

~]aPSFm!~]bPSFm!/PSF,

(18)

Table 1. Summary of Defocus ValuesCorresponding to Contrast Maxima and Minimafor n Ä 0, 1, 2, Assuming n0Ä10 Cycles/Aperture

nuamax(2)u(waves)

uamax(1)u(waves) uaminu (waves)

0 25.0 25.0 `

1 8.33 5 12.52 3.57 2.78 6.25


where ]a [ ]/]aa , ]ab2 [ ]2/]aa]ab , the ab are the aber-

ration coefficients, the (i, j) refer to pixel samples, L isthe log-likelihood function, and E( • ) denotes ensembleaverage over a statistically meaningful distribution of ab-errations. In Eq. (18), the Fisher array has been normal-ized to the total photon count Np 5 ( (i, j)PSF, and M isthe total number of diversity channels.

The Cramer–Rao theorem6 states that

Var~ ab! [ sab

2 > Fbb21~a!, (19)

where Var(•) is the variance, F21 is the inverse of theFisher array, and ab denotes an unbiased estimator forthe bth aberration parameter. An intuitive understand-ing of the Cramer–Rao theorem is obtained by inspectionof Eq. (18) [and Eq. (19)]: The error bounds in estimationperformance are inversely proportional to the expected‘‘curvature’’ of the log-likelihood function L, that is, to thesensitivity of L on the parameters being estimated.Therefore it will be easier to produce more accurate esti-mates from data sets in which L is strongly influenced bythe parameters that we wish to estimate. Conversely, fordata corrupted by noise and other degrading factors, thefunctional dependence on the parameters will be reduced,and therefore the bounds will be correspondingly higher.This theoretical framework indicates that a study of thepropagation of spectral power associated with aberration‘‘signals’’ can be used to identify optimal recovery configu-rations as a function of the aberration spatial frequencyand provides a quantitative basis in estimation theory forthe results in this paper.

Substituting Eq. (8) into Eq. (18) gives a simple consis-tency check on the above interpretation. Note, however,that this calculation is not really the Cramer–Rao bound,since the statistical interpretation has been ignored [byassuming a single aberration realization, i.e., a sin(•)phase term]. Nevertheless, given that the bound is in-versely proportional to the gradient of the PSF with re-spect to the aberration parameter being estimated, thecalculation gives an accurate indicator of how theCramer–Rao bound will behave analytically as a functionof spatial frequency and defocus. The partial derivativeof Eq. (18) is thus calculated, and the Fisher informationhas the following proportionality relation:

F } ~]nI !2u n → n05 @kmpn0 cos~pn0

2/4a !sin~pn02/8a !/2a#2.

(20)

The presence of the sin(pn02/8a) term in relation (20)

shows that the contrast maxima and minima are identicalto the earlier predictions derived in Section 3 but are de-rived independently based on the propagation of spectralpower for a defocused phase grating. Numerical corre-spondence is illustrated in Fig. 7 as a function of defocus,assuming, as above, that n0 5 10 cycles/aperture for asingle diversity channel. Hence the predictions for opti-mal contrast based on the propagation of spectral powerare consistent with the estimation-theoretic lower bound(compare Figs. 3 and 4 with Fig. 7). In Fig. 7, a variety ofhigher-order maxima begin to dominate as a → 0 becauseof the 1/a2 dependence of the gradient and the fact thatsampling effects have been ignored, as pointed out earlierin the discussion following Eq. (10). A proper math-

ematical analysis of sampling effects will introduce appro-priate deweighting factors for smaller defocus, reducingthe relative magnitudes predicted by relation (20), theanalysis of which is beyond the scope of this paper.

The Fisher information corresponding to themaximum-contrast case is calculated by substituting amaxinto relation (20) to get

Fmax } F2km~1 2 4n !

n0G2

. (21)

For a fixed numerator in relation (21), the 1/n0 depen-dence implies that the estimation problem becomes moredifficult as n0 increases. By inspection of the result in re-lation (20), this result is simply an alternative statementof the fact that the spectral content of n0 is shifted to dc inthe image as a → `. Finally, the minimum-contrastcase yields

Fmin 5 0, (22)

and therefore the Cramer–Rao bound is infinite, consis-tent with the earlier result in Eq. (14), since no informa-tion on n0 is present when aperture geometry is ne-glected.

8. WAVEFRONT-SENSING IMPLICATIONSThe presence (or the absence) of n0 in the PSF intensitydata is striking upon comparison of the maximum andminimum-contrast cases of Figs. 5 and 6 and indicates thepotential benefits of properly matching diversity defocusvalues to the desired spatial frequency of recovery. Asexpected, there are wavefront-sensing implications aswell. Here we consider several simulations to investi-gate the wavefront recovery performance for a focus-diverse iterative-transform phase-retrieval algorithm asdefocus and spatial frequency are varied. Although wehave not attempted to advocate any one algorithmic ap-proach in this paper, the iterative-transform (or point-by-point phase-recovery) method is in fact especially suitedfor our discussion of mid-spatial-frequency aberrations.Indeed, the recovery of mid to high spatial frequenciesutilizing the parametric approach is impractical giventhat such parameterized formulations require solving forthousands of aberration basis coefficients, as indicated inthe fitting results of Fig. 17 below.

The core algorithm is based on the Gerchberg–Saxton25

iterative-transform method, as discussed by Misell.26

Roddier and Roddier27 utilized this method for analyzingthe Hubble PSFs and have discussed additional details ofit in Ref. 28. The algorithm has also been utilized byFienup et al.,29 and Lyon has conducted a detailed simu-lation study of the algorithm.30 This approach has beenutilized by Redding et al.,31,32 as the baseline wavefront-sensing method for NASA’s Next Generation Space Tele-scope (subsequently named the James Webb Space Tele-scope). A bandpass sensitivity analysis has beendiscussed in Refs. 13 and 33. Other phase-retrievalmethods have also been discussed in the literature basedon the transport of intensity (e.g., Ref. 34).

The first simulation addresses the recovery of a singlepupil spatial-frequency component n0 , i.e., a sinusoid ab-


Fig. 7. Fisher information versus defocus for a phase grating assuming n0 5 10 cycles/aperture. Values predicted by Eqs. (9) and (11)are also identified.

Fig. 8. Phase-retrieval result for minimum-contrast case; n 5 1, n0 5 10 cycles/aperture; defocus 5 12.5 waves.

Fig. 9. Phase-retrieval result for maximum-contrast case; n 5 1, n0 5 10 cycles/aperture; defocus 5 8.33 waves.


Fig. 10. Phase-retrieval convergence comparison for maximum- and minimum-contrast cases.

erration as a function of defocus. Two additional simula-tions are then considered to examine wavefront-sensingperformance for aberrations having more complicatedspatial-frequency dependence. In the first instance, alow-order optical aberration is introduced to demonstratethe advantage of utilizing a smaller diversity defocusvalue. In the second, the higher-spatial-frequency‘‘print-through’’ associated with a deformable mirror is ex-amined to address an instance where larger diversity de-focus values can be advantageous to wavefront-sensingaccuracy. As an example of this type of mid-spatial-frequency recovery, we routinely perform such phase re-trievals on the NASA Goddard wave-front-control testbedto register actuators and to flatten the wavefront to ap-proximately l/40 rms. In this instance, m of relation (2)is approximately 3/40 (and km , 1), consistent with theinterferometer data discussed in Ref. 35.

For simplicity, monochromatic Nyquist-sampled PSFsare generated for the simulations. Additive noise, detec-tor quantization effects, and blurring functions such asjitter and charge diffusion are neglected, since it is ex-pected that some bias (estimation bias for the Misell–Gerchberg–Saxton algorithm has been discussed in Ref.36) will be introduced into the aberration estimate due tothese external effects alone. The analysis of estimationbias in image-based wavefront sensing as a function ofPSF blurring functions is beyond the scope of this paper,since the sensitivity to these factors is expected to be al-gorithm and perhaps implementation dependent. Hencewe consider only a single algorithm implementation (theauthors’), as discussed above, to demonstrate the mainpoints of the paper.

The first simulation begins with the calculation of twosymmetrically placed defocused PSFs for input to a modi-fied Misell–Gerchberg–Saxton estimation procedure.For convenience, the numerical value n0 5 10 cycles/aperture is selected, and then the diversity defocus valuesare derived according to Eqs. (9) and (11), respectively.The minimum-contrast wavefront-sensing result is shown

in Fig. 8, while the maximum-contrast case is given forcomparison in Fig. 9. In each case, the algorithm is ex-ecuted for an identical (fixed) number of iterations andthen terminated. After only three iterations, themaximum-contrast phase-retrieval estimate has nearly

Fig. 11. Trefoil power spectral density (PSD).


converged to the correct value with an rms error of ap-proximately 4%, while the minimal-contrast case con-verges at a much slower rate, with an rms error of ap-proximately 28%. The convergence is plotted in Fig. 10,

showing the rms difference between individual iterates asa function of the number of iterations. After 25 itera-tions; the algorithm converges (in each case) to an rmsdifference between iterates of approximately l/250. The

Fig. 12. Diversity point-spread function (PSF) data and phase-retrieval (PR) results for the trefoil aberration.

Fig. 13. Phase-retrieval convergence comparison for phase-retrieval results shown in Fig. 12.


final estimate for the maximum-contrast phase-retrievalresult differs by only approximately 3% from the truephase value, while the minimum-contrast estimate differsby 10%. Additional discussion is given in Section 10.

The second simulation compares the estimation perfor-mance between the maximum- and minimum-contrast di-versity cases for a phase error of lower spatial frequency.The trefoil aberration is arbitrarily selected (Zernike or-der 4, term 7 out of 15 terms in the Malacara ordering37),which is approximated by a sinusoidal spatial-frequencycomponent of approximately n0 5 1.5 cycles/aperture, asillustrated by the power spectral density (PSD) calcula-tion shown in Fig. 11 (diversity defocus values impliedby a given spatial frequency are derived from the Zernikebasis order in Section 9). In this instance, Eq. (9) pre-dicts a maximum-contrast defocus value of approximatelyamax ' 61 wave for n 5 0. The first image of the inputdata (a symmetrically placed diversity pair) to the phase-retrieval algorithm is shown in Fig. 12, where we notethat the larger-defocus pair was generated for a some-what arbitrarily selected 612 waves, since amin → ` forn 5 0 (i.e., spectral power shifts to dc in the image, asnoted earlier in Section 6).

The phase-retrieval algorithm discussed earlier recov-ers a point-by-point phase map, which in turn is subse-quently decomposed into a Zernike basis set. By inspec-tion of Fig. 12, one can easily see the trefoil contentstrongly modulating the smaller-defocus PSF. But inthe larger-defocus case, the trefoil is barely visible. The

Fig. 14. Deformable mirror ‘‘print-through’’ PSD.

rms difference between the input and recovered phasesis also shown in Fig. 12, where we note that for a fixednumber of iterations (50 in this instance), the smaller de-focus data yield a better result when measured in termsof algorithm convergence. The wavefront-sensing con-vergence is plotted in Fig. 13, showing the rms differencebetween individual phase-retrieval iterates. Comparedwith the purely sinusoidal result of Fig. 10, the differencein phase-retrieval convergence between the two casesis slightly slower. This is likely due to the morecomplicated spatial-frequency dependence of the trefoilaberration compared with that of a pure sinusoid. Nev-ertheless, the smaller-defocus case converges after ap-proximately five iterations, while the larger-defocus casehas not converged after 50 iterations, as indicated by thepresence of trefoil in the phase residual difference of Fig.12.

Finally, consider the case of a mid-spatial-frequencyphase error (ripple) such as that produced by a deform-able mirror. The phase residual used for generating thePSFs was obtained from a Zygo interferometer data setafter flattening a Xinetics 349 channel deformable mirror,as discussed in Ref. 35. The Xinetics deformable mirrorhas 21 actuators along the maximum x and y pupil direc-tions, and therefore n0x 5 n0y is approximately 10 cycles/aperture, as illustrated by the PSD calculation shown inFig. 14. The optimal diversity defocus value for themaximum-contrast case is calculated by using Eq. (9) andlisted in Table 1 for n 5 1, giving amax 5 68.33. Thesmaller defocus value is chosen based on the minimum-contrast prediction using Eq. (11), giving amin 5 62.5waves of defocus for n 5 5. In this instance, one can eas-ily see the actuator residuals in the defocused PSFs, butin the smaller-defocus case, the residual is not as clearlydefined. Two sets of defocused PSFs (each consisting of apositive/negative defocus pair) were simulated and areshown in Fig. 15. Although phase-retrieval convergenceis relatively quick in the smaller-defocus case, the finalresult is less accurate (biased) compared with that of thelarger-defocus case, where we note that actuator residu-als are still present in the difference between the startingand recovered phase aberrations for the smaller defocusdata. Therefore, in this instance, we see that larger de-focus can be advantageous for the recovery of higher-spatial-frequency phase aberrations based on accuracy.Additional discussion is given in Section 10.

9. DIVERSITY DEFOCUS IMPLIED BY THEZERNIKE BASIS FUNCTIONSIn this section, a statistical calculation of the frequencycontent implied by the Zernike basis functions of a givenorder is considered (for convenience, we adopt theMalacara-ordered Zernike functions37). The calculationsuggests a ‘‘target’’ aberration spatial-frequency contentthat one may match to a given Zernike order in units ofcycles per aperture. Therefore, using Eq. (9), one canthen calculate optimal diversity defocus values as a func-tion of the desired spatial frequency of wavefront-sensingrecovery, expressed in terms of a given Zernike polyno-mial order.


Fig. 15. Diversity PSF data and phase-retrieval results for a deformable mirror ‘‘print-through’’ phase residual.

For convenience, we adopt an unobscured circular ap-erture and normalize the coefficients to peak–valleyunits. The mean PSD is calculated by first generating anumber of aberration realizations for a given Zernike or-der N and then averaging the peak PSDs obtained in eachcase. This mean PSD value for a given Zernike order isintended to represent a target spatial frequency that canbe represented in the Zernike basis of a given order. Forsimplicity, the (N 1 1)(N 1 2)/2 Zernike aberration coef-ficients for a given realization are taken from a Gaussiandistribution. Similar aberration realizations have beendiscussed in Ref. 5 for calculation of the Cramer–Raobounds. Analytical PSD calculations from the Zernikebasis set have been discussed in Ref. 38.

The numerical results are summarized in Fig. 16,which plots the peak PSD (dominant spatial frequency)versus the Zernike order, from order 4 (15 terms) to order12 (91 terms). Assuming a linear relationship betweenthe peak PSD and the Zernike order allows one to ex-trapolate these results to higher orders by using a linearfit to the mean PSD values, as shown in Fig. 16:

PSDmean 5 bZorder 1 c,

b 5 0.1787, c 5 1.64 cycles/aperture. (23)

Fig. 16. Dominant PSD versus Zernike order.


Applying these results to the mid-spatial-frequency de-formable mirror actuator residual shows that Zernike or-der 50 (1326 terms) is sufficient to represent approxi-mately 10 cycles/aperture according to Eq. (23).However, it should be noted that these results dependsolely on the assumed linearity of the PSD means of Fig.16 through higher orders than 12. The standard devia-tion about these mean values is significant, and, thereforeorder-of-magnitude estimates are all that should beclaimed from Eq. (23).

Of course, higher-order decompositions result in betterapproximations, but only up to a point. This limitation isdue to the fact that higher-frequency noise or ‘‘spiky’’phase data cannot be accurately represented by using aZernike basis (or any other basis) for a fixed number ofdata values across the pupil. Once this limitation hasbeen reached for a given phase map, the fit saturates, andthen no significant improvement in the fitting is realizedby increasing the order of the fit. This behavior is illus-trated in Fig. 17, which plots the rms difference betweenthe deformable mirror actuator residual and the Zernikefit to these data as a function of the Zernike fit order.The plot also gives a consistency check on the relationshipimplied by Eq. (23); e.g., at approximately Zernike order45–50 (1081–1326 terms), the curve begins to level out toapproximately l/100 (rms fit error for 10 cycles/aperture)as the noise floor in the data is reached. A single datapoint is also illustrated in Fig. 17, showing the deform-able mirror data, the Zernike fit to these data (e.g., order50), and the difference between the fit and the data.

Finally, setting n0 [ PSDmean [i.e., substituting Eq. (23)into Eq. (9)] gives a prediction for the maximum-contrastdiversity defocus values as a function of the desired spa-tial frequency of wavefront-sensing recovery, expressed interms of a given Zernike polynomial order:

amax 5 6~bZorder 1 c !2/~16n 7 4 !, n 5 0, 1, 2,...,

(24)

assuming the numerical values for b and c listed in Eq.(23). A similar expression is obtained upon substitutionfor the contrast minima, plotted in Fig. 18 for several val-ues of n 5 0 and 1.

10. SUMMARY AND DISCUSSIONThe analysis that we have considered is based on the ob-servation that pupil phase aberrations may be decom-posed into different spatial-frequency regimes (low1 mid 1 high) and that specific defocus values tend toemphasize (or minimize) spectral power corresponding todifferent spatial frequencies in the pupil. Therefore, tothe extent that one can assume that an aberrated wave-front is composed of a dominant spatial frequency, one canuse Eq. (9) to predict the defocus values corresponding tomaximum contrast in the aberration signal. Conversely,Eq. (11) predicts diversity defocus values that should beavoided. As discussed in Section 5, this behavior is com-pletely analogous to the Talbot imaging phenomena for aphase grating, explaining multiple periodic regions ofmaximum and minimum contrast as a function of spatialfrequency and defocus.

Since image-based wavefront-sensing methods operatefrom intensity data, wavefront-sensing performance istherefore expected to vary as the spectral power corre-sponding to aberration spatial frequency varies with de-focus. In Section 8, several simulations were conductedto indicate some details of this sensitivity. The threemain pupil phase aberrations considered were (a) a puresinusoidal frequency component, (b) a low-order opticalphase error, and (c) a mid-spatial-frequency optical phaseerror. In (a), accelerated convergence is the dominantimprovement obtained by properly matching diversity de-focus. Although some deviation in accuracy exists forphase-retrieval results between the optimal and subopti-mal diversity values, improved convergence is really the

Fig. 17. Rms (Zernike fit minus data) versus Zernike order for a deformable mirror ‘‘print-through’’ phase residual.


main advantage when the phase aberration consists of asingle spatial-frequency component. In (b), convergenceis still the greatest performance increase, although some-what slower than in (a) as a result of the more compli-cated spatial-frequency dependence of a low-order opticalphase error compared with that of a pure sinusoid. Fi-nally, in (c), wavefront-sensing accuracy is more impor-tant than convergence when choosing between optimaland suboptimal diversity values. The fact that larger-defocus values are generally better suited for higher-spatial-frequency recovery is simple to understand whenone considers that larger-defocus intensity data (at leastthose avoiding minimum-contrast values) support higherspatial frequencies in the pupil. Therefore it should notbe too surprising that defocus settings tending to empha-size certain spatial frequencies will be optimal comparedwith those that do not.

An important conclusion is that no one diversity valueis best for all wavefront-sensing applications. Intelligentalgorithm implementation is just as important as algo-rithm selection; i.e., by properly matching diversityvalues to the desired spatial frequency of recovery,one can potentially realize significant performance gainsin both accuracy and convergence. A corollary result isthat proper diversity selection helps to emphasize a fun-damental property for wavefront-sensing and -controlsystems, i.e., systems that are designed for lower-spatial-frequency recovery should be distinguished from those de-signed to recover higher-spatial-frequency aberrations.13

An additional characteristic is that since lower-spatial-frequency phase-retrieval systems require relativelysmall diversity defocus values, smaller PSF sizes can beexploited to minimize FFT array sizes when utilizing theiterative-transform approach. Conversely, larger defocusvalues require additional array size. As a result, thecomputational burden can generally be reduced for lower-spatial-frequency phase-retrieval systems. Thereforebuilding a ‘‘real-time’’ image-based wavefront sensor ismore feasible based on the recovery of lower-spatial-frequency aberrations.

Finally, in an earlier analysis,13 we have shown thatthe wavefront-sensing estimation burden increases as the

Fig. 18. Diversity defocus values implied by Zernike basis order.

aberration spatial frequency increases. This is simply aresult of the fact that many more aberration coefficientsmust be estimated for higher-spatial-frequency recovery.In such cases, it is straightforward to show that if one uti-lizes a greater number of diversity images, the Cramer–Rao bound can be lowered,5,13 leading to an improvementin wavefront-sensing performance when compared withthat of the lower-spatial-frequency wavefront-sensingproblem. However, even in the lower-spatial-frequencycase, the placement of two images on either side of focusis favorable for estimation, since an important cancella-tion occurs that can help to resolve confusion between amisplaced central obscuration and coma.39

ACKNOWLEDGMENTSWe thank Joe Green and Dave Redding for stimulatingdiscussion. We also thank anonymous reviewers forhelpful comments.

Corresponding author Bruce H. Dean may be reachedby e-mail, [email protected]; phone, 301-286-8238;and fax, 301-286-0204.

REFERENCES AND NOTES1. R. A. Gonsalves, ‘‘Phase retrieval and diversity in adaptive

optics,’’ Opt. Eng. 21, 829–832 (1982).2. The purpose of a diversity function in wavefront sensing is

to ‘‘spread out’’ the optical response to expose aberration in-formation. Since defocus has only angular dependence, noone part of the point-spread function is emphasized morethan any other.

3. J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin,‘‘Hubble Space Telescope characterized by using phase-retrieval algorithms,’’ Appl. Opt. 32, 1747–1767 (1993).

4. J. R. Fienup, B. J. Thelen, R. G. Paxman, and D. A. Car-rara, ‘‘Comparison of phase diversity and curvature wave-front sensing,’’ in Adaptive Optical System Technologies, D.Bonaccini and R. Tyson, eds., Proc. SPIE 3353, 930–940(1998).

5. D. J. Lee, M. C. Roggemann, and B. M. Welsh, ‘‘Cramer–Rao analysis of phase-diverse wave-front sensing,’’ J. Opt.Soc. Am. A 16, 1005–1015 (1999).

6. H. L. Van Trees, Detection, Estimation, and ModulationTheory (Wiley, New York, 1968).

7. A. D. Whalen, Detection of Signals in Noise (Academic, NewYork, 1971).

8. S. M. Kay, Fundamentals of Statistical Signal Processing(Prentice-Hall, Englewood Cliffs, N.J., 1993).

9. R. V. Hogg and A. T. Craig, Introduction to MathematicalStatistics, 4th ed. (Macmillan, New York, 1978).

10. J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R.Fienup, and C. C. Wackerman, ‘‘Cramer–Rao lower boundon wavefront sensor error,’’ Opt. Eng. 25, 586–592 (1986).

11. J. N. Cederquist and C. C. Wackerman, ‘‘Phase-retrieval er-ror: a lower bound,’’ J. Opt. Soc. Am. A 4, 1788–1792(1987).

12. J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Rob-inson, and D. Kryskowski, ‘‘Wave-front estimation fromFourier intensity measurements,’’ J. Opt. Soc. Am. A 6,1020–1026 (1989).

13. B. H. Dean and R. Lyon, ‘‘Cramer–Rao bounds for focus di-verse wavefront sensing with deformable mirror ‘print-through’,’’ in Computational Optics and Imaging for SpaceApplications, R. Lyon, ed. (NASA/Goddard Space FlightCenter, Greenbelt, Md., 2000), pp. 127–142.

14. B. H. Dean, ‘‘Cramer–Rao analysis for phase-diverse-phase-retrieval: diversity functions and broadband phase-


retrieval,’’ presented at the Wavefront Sensing & ControlsConference, sponsored by Kamuela Optical Associates, Ko-hala Coast, Hawaii, November 13–16, 2000.

15. J. R. Fienup, ‘‘Phase retrieval algorithms: a comparison,’’Appl. Opt. 21, 2758–2769 (1982).

16. W. H. Southwell, ‘‘Wave-front analyzer using a maximumlikelihood algorithm,’’ J. Opt. Soc. Am. A 3, 396–399 (1977).

17. W. B. Wetherell, ‘‘The calculation of image quality,’’ in Ap-plied Optics and Optical Engineering, R. Shannon and J.Wyant, eds. (Academic, New York, 1980), Vol. VIII, p. 172.

18. B. H. Dean, ‘‘Phase-retrieval performance as a function ofdefocus and aberration frequency,’’ presented for the NASANext Generation Space Telescope Technical Memoranda,NASA/Goddard Space Flight Center, Greenbelt, Md., Au-gust 20, 2000.

19. C. Bowers, ‘‘Some considerations regarding the propagationof spectral power from pupil to defocused image planes,’’presented for the NASA Next Generation Space TelescopeTechnical Memoranda, NASA/Goddard Space Flight Cen-ter, Greenbelt, Md., April 10, 2001.

20. B. H. Dean, ‘‘Fresnel zone propagation of spectral powerwith applications to image-based wavefront sensing,’’ pre-sented for the NASA Next Generation Space TelescopeTechnical Memoranda, NASA/Goddard Space Flight Cen-ter, Greenbelt, Md., July 8, 2001.

21. The frequency content in this instance is associated withthe finite spacing of actuators placed in a periodic array be-hind a thin mirror face sheet. Ironically, the ‘‘print-through’’ associated with such systems is a result of thehigh-frequency correction capability of these many-actuatorcontrol systems. For example, for a Xinetics 349 channeldeformable mirror, there are 21 actuators spanning theclear aperture, yielding approximately 10 cycles/aperturealong a single spatial-frequency direction.

22. R. Paxman and B. Thelen, ‘‘Wavefront sensing fordeployable-optic systems,’’ presented at the WavefrontSensing and Controls Conference, sponsored by KamuelaOptical Associates, Kohala Coast, Hawaii, November 13–16, 2000.

23. J. W. Goodman, Introduction to Fourier Optics, 2nd ed.(McGraw-Hill, New York, 1996), pp. 87–89.

24. W. Sun, ‘‘Cramer–Rao lower bound analysis on estimationaccuracy for phase retrieval,’’ M. S. thesis, directed by T.Schulz (Michigan Technological University, Houghton,Mich., 1998).

25. R. W. Gerchberg and W. O. Saxton, ‘‘A practical algorithmfor the determination of phase from image and diffractionplane pictures,’’ Optik (Stuttgart) 35, 237–246 (1972).

26. D. L. Misell, ‘‘A method for the solution of the phase prob-lem in electron microscopy,’’ J. Phys. D 6, L6–L9 (1973).

27. C. Roddier and F. Roddier, ‘‘Wavefront reconstruction fromdefocused images and the testing of ground-based opticaltelescopes,’’ J. Opt. Soc. Am. A 10, 2277–2287 (1993).

28. C. Roddier and F. Roddier, ‘‘Combined approach to theHubble Space Telescope wavefront distortion analysis,’’Appl. Opt. 32, 2992–3008 (1993).

29. J. R. Fienup, J. C. Marron, R. G. Paxman, T. J. Schulz, J. H.Seldin, and B. Thelen, ‘‘Image inversion analysis of the

Hubble Space Telescope,’’ in Final Report: Jet PropulsionLaboratory Contract 958892 on the Hubble Space TelescopeOptical Telescope Assembly Analysis, August 1991 (Jet Pro-pulsion Laboratory, Pasadena, Calif., 1991).

30. R. G. Lyon, ‘‘DCATT wavefront sensing and optical controlstudy,’’ Developmental Comparative Active Telescope Test-bed Rep. WFSC-0001, February 22, 1999, http://jansky.gsfc.nasa.gov/OSCAR/.

31. D. Redding, S. Basinger, A. Lowman, F. Shi, P. Bely, R.Burg, and G. Mosier, ‘‘Wavefront sensing and control for aNext Generation Space Telescope,’’ in Space Telescopes andInstruments V, P. Bely and J. Breckinridge, eds., Proc. SPIE3356, 758–772 (1998).

32. D. Redding, S. Basinger, A. Lowman, F. Shi, C. Bowers, L.Burns, P. Davila, B. Dean, M. Fitzmaurice, G. Mosier, B.Perkins, P. Petrone, T. Norton, M. Wilson, and L. Wheeler,‘‘Wavefront control for a segmented deployable space tele-scope,’’ in UV, Optical, and IR Space Telescopes and Instru-ments, J. B. Breckinridge and P. Jakobsen, eds., Proc. SPIE4013, 546–558 (2000).

33. B. H. Dean, ‘‘White-light phase-retrieval analysis,’’ pre-sented for the NASA Next Generation Space TelescopeTechnical Memoranda, NASA/Goddard Space Flight Cen-ter, Greenbelt, Md., October 11, 2000.

34. T. E. Gureyev, A. Roberts, and K. A. Nugent, ‘‘Phase re-trieval with the transport-of-intensity equation: matrixsolution with the use of Zernike polynomials,’’ J. Opt. Soc.Am. A 12, 1932–1941 (1995).

35. B. H. Dean and R. Boucarut, ‘‘Deformable mirror opticalcalibration and test results,’’ presented at the Next Genera-tion Space Telescope Science and Technology Exposition,Woods Hole, Mass., September 13–16, 1999.

36. C. Roddier and F. Roddier, ‘‘Reconstruction of Hubble SpaceTelescope wavefront distortion from stellar images taken atvarious focal positions,’’ in Final Report: Jet PropulsionLaboratory Contract 958893 on Hubble Space Telescope Op-tical Telescope Assembly Analysis, May 1991 (Jet Propul-sion Laboratory, Pasadena, Calif., 1991), pp. 2–3.

37. D. Malacara and S. DeVore, ‘‘Interferogram evaluation andwavefront fitting,’’ in Optical Shop Testing, D. Malacara, ed.(Wiley-Interscience, New York, 1997).

38. B. M. Levine, E. A. Martinsen, A. Wirth, A. Jankevics, M.Toledo-Quinones, F. Landers, and T. Bruno, ‘‘Horizontalline-of-sight turbulence over near-ground paths and impli-cations for adaptive optics corrections in laser communica-tions,’’ Appl. Opt. 37, 4553–4560 (1998).

39. If inaccurate pupil data are applied in the iterative-transform sense (Fig. 1), the pupil amplitude and the phaseinformation become mixed, resulting in artificial aberra-tions in the phase estimate. For example, a pupil ampli-tude function consisting of a shifted central obscuration in atwo-mirror system can lead to phase-retrieval estimateswith artificial coma. But this ambiguity is resolved whenutilizing diversity images on both sides of focus since thedefocused PSF is asymmetric about focus (for a shifted cen-tral obscuration), while true coma produces a symmetricPSF about focus.

Documents

Diversity selection for phase-diverse phase retrieval