Evaluation of the Performance of Hartmann Sensors in Strong Scintillation

Evaluation of the performance of Hartmannsensors in strong scintillation

Jeffrey D. Barchers, David L. Fried, and Donald J. Link

A simulation study is presented that evaluates the performance of Hartmann wave-front sensors withmeasurements obtained with the Fried geometry and the Hutchin geometry. Performance is defined interms of the Strehl ratio achieved when the estimate of the complex field obtained from reconstructionis used to correct the distorted wave front presented to the wave-front sensor. A series of evaluations isperformed to identify the strengths and the weaknesses of Hartmann sensors used in each of the twogeometries in the two-dimensional space of the Fried parameter r0 and the Rytov parameter. We foundthat the performance of Hartmann sensors degrades severely when the Rytov number exceeds 0.2 and theratio l�r0 exceeds 1�4 �where l is the subaperture side length� because of the presence of branch pointsin the phase function and the effect of amplitude scintillation on the measurement values produced by theHartmann sensor. © 2002 Optical Society of America

OCIS codes: 010.1080, 010.1300, 010.1330.

1. Introduction

The Shack–Hartmann wave-front sensor in the Friedgeometry is today the most widely used sensor foradaptive optical systems. Ease of fabrication andalignment, robustness, and light efficiency make theHartmann sensor an excellent choice for a wide rangeof applications. Here we evaluate the accuracy ofthe reconstructed wave function obtained throughthe use of measurements obtained from Hartmannwave-front sensors. This accuracy is directly re-lated to the ability of an adaptive optical system tocorrect for the perturbations in the complex field. Inweak scintillation, the measurements obtained by aHartmann sensor are �nearly� equal to the averagegradient across a subaperture �the gradient tilt orG-tilt�. The average gradient is proportional to theaverage phase difference between opposite sides ofthe subaperture. Conventional least-squares recon-struction1,2 uses an approximation based on this factto reconstruct an estimate of the phase function ob-

J. D. Barchers is with the Starfire Optical Range, Directed En-ergy Directorate, U.S. Air Force Research Laboratory, Kirtland AirForce Base, New Mexico 87117. D. L. Fried can be reached atP.O. Box 680, Moss Landing, California 95039. D. J. Link is withthe Science Applications International Corporation, 140 Intra-coastal Pointe Drive, Suite 312, Jupiter, Florida 33477.

Received 22 March 2001; revised manuscript received 16 August2001.

0003-6935�02�061012-10$15.00�0© 2002 Optical Society of America

1012 APPLIED OPTICS � Vol. 41, No. 6 � 20 February 2002

served by the sensor. In the weak scintillation re-gime, where the geometrical-optics approximation isreasonable, studies have been performed to analyzethe performance of Hartmann wave-front sensors.3–5

In strong scintillation, the situation is quite differ-ent. The measurements obtained from a Hartmannsensor are in fact the average of the intensityweighted gradient across a subaperture. This canbe shown to be proportional to the displacement ofthe centroid of the image formed from the wave func-tion6 and, accordingly, is known as the centroid tilt orthe so-called C-tilt. C-tilt�G-tilt anisoplanatism,the difference between the gradient and the centroidtilts, can significantly affect the measurement valuesproduced by the Hartmann sensor and thus the re-constructed wave function. Unfortunately, with theexception of the Rytov theory, which is valid only inweak scintillation, there is no means to evaluate theC-tilt�G-tilt anisoplanatism analytically. Anotherimportant effect that occurs in strong scintillation isdue to the presence of branch points in the phasefunction.7–9 The presence of branch points in thephase function complicates the reconstruction pro-cess and a nonlinear estimation of the phase functionis necessary.10,11 It is not clear in any sense whateffect the presence of branch points in the phase func-tion has on the measurement values produced by theHartmann sensor. Nor is it clear, when we considersubapertures of nontrivial extent, if any reconstruc-tion algorithm can effectively estimate the phasefunction. Further, there is only limited theory to

evaluate any and�or all these effects. Finally, thereare only limited simulation results in the literaturethat evaluate these effects.12

These observations led to a simulation study tocharacterize the performance of Hartmann wave-front sensors in strong scintillation. Lacking astrong theoretical foundation, wave-optics simulationis the only option available to analyze the problem.The performance of the wave-front sensor and recon-structor is characterized by the Strehl ratio achievedwhen the estimated wave function is used to correctthe true wave function. The term wave-front sens-ing is defined in this paper to include three steps:�1� the physics of the sensor that obtains measure-ments related to the complex field to be sensed, �2�the mathematics of the reconstruction process used toestimate the complex field at an array of points in theaperture �a set of points corresponding to the cornersof the subapertures for a Hartmann sensor in theFried geometry�, and �3� the interpolation processused to estimate the complex field within this array ofpoints. In this paper, only the first two steps in thewave-front sensing process are considered and eval-uated. Hartmann wave-front sensors configured toobtain measurements in the Hutchin geometry andin the Fried geometry were considered. Two types ofreconstruction algorithm that estimate the phasefunction were evaluated: a least-squares recon-struction �that by definition cannot detect branchpoints in the phase function� and a complex exponen-tial reconstructor10,11 �that is designed to estimateboth the least-squares and the hidden phases�.

The various sensors and reconstruction algorithmswere evaluated for a range of values of the Rytovnumber and the ratio l�r0, where l is the length of theside of a subaperture and r0 is the Fried coherencelength. In this paper, the Rytov number is the esti-mate of the variance of the logarithm of the ampli-tude of a complex field after propagation through aturbulent medium that is obtained by use of theRytov approximation, given by 0.5631k0

7�6 �0Z

dzCn2�z�z5�6, where k0 � 2�� is the wave number

and Cn2�z� describes the strength of turbulence along

the propagation path.13 The Rytov number hasbeen shown to be a fundamental scaling parameterfor propagation through turbulence.14 The Friedcoherence length, r0, approximately describes thelargest telescope diameter for which nearlydiffraction-limited performance can be expected with-out the use of an adaptive optical system to correct forthe effects of atmospheric turbulence.15 It has beenestablished that, for a given distribution of thestrength of turbulence, the Rytov number and theratio l�r0 nearly completely characterize the perfor-mance of a noise-free adaptive optical system with aninfinite control bandwidth.14

The key results of this study are contained in thispaper. The primary conclusion drawn from thisstudy is that, if l�r0 is greater than 1�4, the perfor-mance of any and all of the above permutations on thewave-front sensing process begins to drop off rapidlywith increasing Rytov number when the Rytov num-

ber exceeds 0.2. Simulation results are shown thatindicate that this effect is due to both the presence ofbranch points in the phase function and to the effectsof amplitude scintillation on the Hartmann sensormeasurements.

In Section 2 we detail our approach for the evalu-ation of wave-front sensors. In Section 3 we presentsimulation results that characterize the estimationaccuracy of Hartmann sensors in the Fried geometryand in the Hutchin geometry. In Section 3 we alsopresent simulation results that indicate that bothbranch points in the phase function and amplitudescintillation degrade the estimation accuracy associ-ated with Hartmann sensors. Section 4 is a sum-mary of the results we obtained.

2. Approach for Evaluation of the Performance ofWave-Front Sensors

We now detail our approach for evaluation of theperformance of wave-front sensors. In Subsection2.A we describe the notation used in this paper. InSubsection 2.B we describe the wave-front sensor ge-ometries and reconstruction methods considered inthis paper. The parameter space of turbulence con-ditions considered are described in Subsection 2.C.

A. Notation for Evaluation of the Performance ofWave-Front Sensors

The accuracy of the wave-front sensing process can beevaluated at the completion of any of three steps:the process of obtaining physical measurements re-lated to the complex field to be measured, the math-ematics of converting those measurements into anestimate of the complex field at a sparse array ofpoints, and finally the process of estimating the com-plex field at a finer resolution. In this paper, per-formance is evaluated only at the completion of thesecond step. A notation to represent the wave-frontsensing process is selected that combines the first twosteps. The first two steps �measurement and recon-struction of the complex field� are represented by

U�r�� wfsrec�U�r��. (1)

The variable U�r�� is used to denote the complex fieldthat is being measured. The complex field is com-posed of its amplitude, A�r��, and phase, ��r��, i.e., U�r�� A�r�� expi��r��. The circumflex is used to identifyan estimate of the complex field. The prime is usedto indicate that the quantity of interest is known orestimated at an array of points. For example, in theFried geometry, the array of points corresponds to thecorners of the Hartmann sensor subapertures.These points are part of the much denser set of pointson which the propagation results were developed.The set of points that correspond to the prime coor-dinates is denoted by �� whereas the set of all pointsin the simulation is denoted by �. The function�wfs

rec� �� is the wave-front sensing operator. Thevariable wfs is used to describe the sensing process.The variable rec is used to describe the reconstruction

20 February 2002 � Vol. 41, No. 6 � APPLIED OPTICS 1013

1

process to estimate the complex field at the points in��.

A performance metric is defined to evaluate theability of a wave-front sensing and reconstructionprocess to estimate the complex field at the points in��. The performance metric is the estimation Strehlratio, S�U1�r��, U2�r�� , for the two complex fields ofinterest, U1�r�� and U2�r��, where U1�r�� is the field to bemeasured, and U2�r�� is an estimate of that field.This Strehl ratio value is formed on the set of pointsin ��, the value being given by the equation

S�U1�r��, U2�r��

�

� �k��

U1�r�k�U*2�r�k��2

� �k��

U1�r�k�U1*�r�k�� k��

U2�r�k�U2*�r�k�� . (2)

B. Wave-Front Sensor Geometries and ReconstructionAlgorithms

We considered two Hartmann wave-front sensor ge-ometries to correspond to the two reconstructor ge-ometries: Fried and Hutchin. Reconstructoralgorithms were formulated with and without noisevariance weighting. The identifiers, wfs and rec,that correspond to each type of sensor and each typeof reconstructor are defined at the end of this subsec-tion.

The two wave-front sensor geometries consideredare the Hartmann sensor in the Fried geometry andin the Hutchin geometry. The measurements ob-tained from a Hartmann sensor are taken to be theaverage of the intensity weighted gradient across theextent of the subaperture,

sC �� dr�W�r��I�r��r��

� dr�W�r��I�r��

, (3)

where sC is the gradient measurement, W�r�� is a win-dow function that describes the extent of the subap-erture, I�r�� is the intensity or modulus squared of thecomplex field, and ��r�� is the gradient of the phasefunction. It is possible to show that the quantityabove, computed in the pupil plane, is exactly pro-portional to the centroid measured by a detector withinfinitesimal angular resolution.6 The quantityabove is known as the C-tilt. If the illuminationpattern is uniform, the C-tilt is exactly equivalent tothe G-tilt given by

sG �� dr�W�r��r��

� dr�W�r��

. (4)


The effect of C-tilt�G-tilt anisoplanatism is not di-rectly addressed in this paper. However, it is inher-ently an important part of the study presented hereand some indirect attempts are made to quantify theeffect of the C-tilt�G-tilt anisoplanatism.

Although the measurements obtained from a Hart-mann sensor in the Fried or the Hutchin geometryare defined by Eq. �3�, the set of points �� on whichthe field is to be estimated differs for each geometry.The relationship between the points of �� and theHartmann sensor’s subapertures is illustrated in Fig.1. In the Fried geometry, the points in �� are thecorners of the Hartmann sensor subapertures. Inthe Hutchin geometry, the points in �� correspond tothe midpoints of opposite edges of the subapertures,and �through use of a beam splitter� there are twodistinct sets of Hartmann sensor subapertures, eachentirely space filling.

Two reconstruction algorithms were used to recon-struct phase measurements from C-tilt derived esti-mates of phase differences. The first was thestandard least-squares reconstruction.1,2 Thismethod is well established but suffers from a defi-ciency in the presence of branch points.8,9 To over-come this deficiency, a complex exponential

Fig. 1. Measurements and points at which the field is to be re-constructed in the �a� Fried geometry and �b� Hutchin geometry.The arrows indicate the measurements produced by the subaper-ture in which it is located or centered. The squares define thesubapertures. For the Hutchin geometry there are two sets of�partially� overlapping subapertures: one set for the x measure-ments and the other set for the y measurements.

reconstructor was developed.10,11 The complex ex-ponential reconstructor is an algorithm that workswith what are called differential phasors, exp�i��,rather than with the phase differences ��. Whereasthe least-squares reconstructor adds phase differ-ences to form an estimate of the phase function, thecomplex exponential reconstructor multiplies differ-ential phasors to form an estimate of the wave func-tion. The complex exponential reconstructoralgorithm is based on the process of reducing gridresolution to a trivially small level �for which thewave function is easily estimated� and then buildingthe wave function back up to the resolution of thesubapertures. We accomplished the reduction ingrid resolution by developing differential phasors forthe reduced resolution grid by combining differentialphasors �to some extent use of multiplication of dif-ferential phasors and to some extent use of averagingof differential phasors�. The reduction is accom-plished in a series of steps, each step reducing thegrid size by a factor of 2 � 2, from �2n � 1� � �2n � 1�to �2n�1 � 1 � 2n�1 � 1�, until the grid size is reducedto a 2 � 2 array, which is solved in a least-squaresmanner. Then the process is reversed, in each stepwe combined the wave function values at the gridpoints with the differential phasors �in a process ofmultiplications and averaging� to develop wave func-tion values on a factor of 2 finer grid, and we repeatedthis process as many times as necessary to recoverresults on a grid of the starting size. The complexexponential reconstructor can be used in the Hutchinor the Fried geometry. In the averaging process, theaveraging can be noise variance weighted.

It should be noted that no additive noise �shotnoise, read noise, amplifier noise, etc.� was includedin the simulations that we performed for this study.The only noise in these simulations is that associatedwith averaging the intensity weighted gradient overthe extent of a subaperture and defining the result-ant C-tilt value as an approximation of a phase dif-ference between two or four points �in the Hutchingeometry and the Fried geometry, respectively�.The error induced by this approximation has beenlabeled the reconstruction formulation approxima-tion error, or formulation error, for short.5 One canmake heuristic arguments that noise induced by av-eraging over a subaperture can increase in the pres-ence of branch points. Branch points correspond toregions of low intensity. Thus, by assigning noisevariances proportional to the inverse of the intensityin a subaperture, it is presumed that some estimationaccuracy can be obtained by penalizing noisy mea-surements. Here we apply variance weighting inexactly this fashion.

Reconstruction of the amplitude was accomplishedby linear interpolation from the amplitude defined atthe centers of subapertures to the amplitude definedat the coordinates in ��, the values at the centers ofthe subapertures being inferred from the total signalintensity collected by the subaperture.

The notation for the variable wfs is as follows:FH, Hartmann sensor in the Fried geometry; HH,

Hartmann sensor in the Hutchin geometry. The no-tation for the variable rec is LS, least-squares recon-struction; CER, complex exponential reconstruction;and CNW, noise variance weighted complex exponen-tial reconstruction. �Note again that the noise vari-ance for each measurement is taken to be the inverseof the total intensity in the subaperture associatedwith each measurement.�

C. Turbulence Conditions

Here we detail the turbulence conditions that wereevaluated. A square aperture was used with 33 �33 equally spaced field points in ��. For the Friedgeometry this corresponds to an array of 32 � 32pairs of x measurements with y measurements. Inthe Hutchin geometry this corresponds to an array of33 � 32 x measurements and 32 � 33 y measure-ments. Five screens with equal spacing betweenscreens were used to approximate a uniform Cn

2 pro-file. Although this is generally not a sufficient num-ber of screens to approximate a uniform turbulenceprofile, it is sufficient to evaluate the performance ofwave-front sensors. The turbulence strength wasvaried so that l�r0 was set to 1�4, 1�2, and 1. Thetotal length of the profile and corresponding spacingbetween screens was defined to set Rytov numbers of0.05, 0.1, 0.2, 0.4, 0.6, and 0.8. By using 4096 �4096 phase screens and propagation array, weavoided phase tears in the propagation kernel �cor-responding to 4096 � 2�z��x

2, where � is the wave-length of propagation, set to 0.85 �m, z is the totallength of propagation, and �x is the grid point spac-ing�. For l�r0 equal to 1�2 and 1, we used eight gridpoint spacings per subaperture. For l�r0 equal to1�4, we used only four grid point spacings per sub-aperture. For l�r0 equal to 1�4 and 1�2, there were16 grid point spacings per r0 and for l�r0 equal to 1,there were eight grid point spacings per r0. We eval-uated 16 realizations for each case.

We confirmed the validity of the resultant fieldsafter propagation by plotting the measured log-amplitude variance as a function of Rytov number foreach value of l�r0, which is shown in Fig. 2. Asexpected, there is good agreement for small values ofthe Rytov number, slight amplification in the inter-mediate regime, and saturation in the large Rytovnumber regime.16 Note that the same screens wereused for l�r0 equal to 1�4 and 1�2; hence only twocurves are shown.

3. Results

Having described the wave-front sensor geometriesand turbulence conditions of interest, here wepresent simulation results that begin to provide anunderstanding of the performance of Hartmann sen-sors in strong scintillation. In Subsection 3.A wecharacterize the ability of a Hartmann sensor in theFried geometry and the Hutchin geometry to esti-mate the complex field at the array of points in ��.In Subsection 3.B, we present an investigation thatattempts to describe the magnitude of two errorsources in the wave-front sensing process: branch


1

points in the phase function and amplitude scintilla-tion.

A. Characterization of the Performance of HartmannSensors

The fundamental quantity of interest is the ability ofa Hartmann sensor to estimate the complex field atan array of points in the aperture as a function ofturbulence conditions. The results for the Hart-mann sensor in the Fried geometry by use of a least-squares and a complex exponential reconstructor areshown in Fig. 3�a�. The results for the Hartmannsensor in the Hutchin geometry by use of a least-squares and a complex exponential reconstructor areshown in Fig. 3�b�. The complex exponential recon-structor is used with noise variance weighting. Thiswas found to yield slightly better performance thanthe complex exponential reconstructor with noiseweights all set to unity, i.e., without noise varianceweighting, the results of which are not shown.

Several observations can be noted from these plots.First, in Fig. 3�a�, when the coherence length of theturbulence is large relative to a subaperture �l�r0 �1�4�, the complex exponential reconstructor withnoise variance weighting forms a better estimate ofthe complex field than a least-squares reconstructorwhen the Rytov number is large. The two curvesbegin to separate at a Rytov number of 0.2, which isconsistent with other results in the literature.9 Asthe coherence length shrinks relative to the subaper-ture side length, l�r0 � 1�2 and l�r0 � 1, there is noperformance improvement that is due to a complexexponential reconstructor. In fact, for l�r0 � 1,there is actually a degradation in the estimation ac-curacy. It is believed that this occurs because thecomplex exponential reconstructor is a nonlinearfunction and has a large noise gain beyond some

Fig. 2. Measured log-amplitude variance of the fields used in thispaper shown as a function of the Rytov number. As expected, theagreement is excellent for small values of the Rytov number, aslight amplification is observed in the intermediate regime, andsaturation occurs for large values of the Rytov number.


threshold.10,11 The authors suggest that the errorthat is induced by averaging the intensity weightedgradient over the extent of a subaperture leads tolarge measurement noise in the gradients obtainedfrom the Hartmann sensor, which leads to this ratherpoor estimation accuracy.

The results for the Hartmann sensor in theHutchin geometry, shown in Fig. 3�b�, are similar.However, the performance degradation for the com-plex exponential reconstructor is not as severe forl�r0 � 1 as it is for the Fried geometry. In addition,the performance improvement when the coherencelength is large relative to a subaperture is more sig-nificant than in the Fried geometry.

The results in Fig. 3 indicate that in both theHutchin geometry and the Fried geometry, even with

Fig. 3. Estimation accuracy of the Hartmann sensor in the �a�Fried geometry and �b� Hutchin geometry by use of a least-squares�solid curve� or a complex exponential reconstructor �dashed curve�for l�r0 � 1�4 �circle�, l�r0 � 1�2 �square�, and l�r0 � 1 �lefttriangle�. When l�r0 � 1�4, a significant improvement in estima-tion accuracy is noted by use of the complex exponential recon-structor when the Rytov number is large. However, when l�r0 �1, there is actually a degradation in performance when the complexexponential reconstructor is applied.

the use of a complex exponential reconstructor, theestimation accuracy of the Hartmann sensor is poorwhen the Rytov number is large, unless the coherencelength of the turbulence is large relative to the sub-aperture side length.

It is interesting to note, as pointed out by the re-viewers, that the estimation accuracy of the Hart-mann sensor in the Hutchin geometry is better thanthat of the Hartmann sensor in the Fried geometry.The authors attribute this difference to two contribu-tions. The first is that the reconstructor noise gainfor a Fried geometry reconstructor is a factor of ap-proximately 2 times that for a Hutchin geometry re-constructor.1,2 The second is that the reconstructorformulation error in the Fried geometry is approxi-mately a factor of 1.44 times that for measurementsobtained with the Hutchin geometry.5,17 Taking themeasurement noise variance associated with themeasurements to be the reconstructor formulationerror, the noise variance associated with the recon-structed phase measurements is expected to benearly a factor of 3 greater for the Fried geometrythan that for the Hutchin geometry. This nearly afactor of 3 increase in the noise variance explains thefact that the estimation accuracy of a Hartmann sen-sor in the Hutchin geometry is better than that of aHartmann sensor in the Fried geometry.

B. Evaluation of Two Error Sources

Having established that the estimation accuracy ofthe Hartmann sensor is poor in strong scintillation,we are interested in examining two possible reasonsfor the poor estimation performance. The first ofthese is branch points in the phase function and thesecond is the random apodization of the amplitudeacross the subapertures. To examine and to isolatethese phenomena, several versions of the complexfield that results from propagation are considered.After propagation, the complex field, U�r��, is charac-terized by both its amplitude, A�r��, and phase, ��r��,i.e., U�r�� A�r�� expi��r��. With the freedom of asimulation environment, there is nothing that pre-vents us from breaking up the wave front into differ-ent versions to evaluate the ability of a Hartmannsensor to estimate different versions of the complexfield. We considered five versions of the complexfield other than the nominal complex field U�r��.These are as follows:

�1� expi��r��, the complex field after propagationwith a uniform amplitude profile;

�2� expi��r��, the complex field after propagationwith the hidden phase removed and a uniform am-plitude profile;

�3� A�r�� expi��r��, the complex field after propa-gation with the hidden phase removed;

�4� expi��r��, the complex field after propagationwith the least-squares phase removed and a uniformamplitude profile;

�5� A�r�� expi��r��, the complex field after propa-gation with the least-squares phase removed.

Each version of the complex field was computed forall points in �, i.e., the simulation grid level. Theleast-squares and hidden phases were isolated by useof the least-squares reconstruction operator, definedas

�� GTG��1GT��G��, (5)

where �� is the principal value operator, and Gis the discrete representation of the phase differ-ence operator in the Hutchin geometry. The ma-trix G is composed of the values 0, 1, and �1 inappropriate locations so that the quantity G� is avector of phase difference measurements. Becausethe operator GT is the discrete representation of thedivergence operator, and the phase difference mea-surements associated with the hidden phase are thecurl of a vector potential, and the divergence of thecurl of a vector potential is zero, the hidden phasecan be removed from the phase function by appli-cation of the least-squares operator to find the least-squares component of the phase function, i.e., �� . Given the least-squares and total phasefunctions, the hidden phase can now be found triv-ially with

�� argexp�i��exp��i�� . (6)

Calculation of the least-squares reconstruction at thesimulation grid level was made practical by use ofsparse matrix techniques and the Cholesky factoriza-tion.18

The comparison of the ability of a wave-front sen-sor to estimate each of the five different versions ofthe complex field provides indicators of the effect ofvarious error sources in the wave-front sensing pro-cess. In Subsection 3.B.1, we examine the effect ofbranch points on estimation accuracy, and in Subsec-tion 3.B.2, we examine the effect of scintillation onestimation accuracy.

1. Effect of Branch Points on Wave-FrontMeasurement and Reconstruction of theLeast-Squares and Hidden PhasesTo obtain an understanding of the fundamental ca-pabilities of the Hartmann wave-front sensor, weevaluated the ability to estimate the least-squaresand hidden phases in the absence of scintillation.We evaluated the quantities

�{expi��r��, exp[i arg(�wfsrec��expi��r�� )]}, (7)

�{expi��r��, exp[i arg(�wfsrec��expi��r�� )]} (8)

for the least-squares and complex exponential recon-structors as a function of Rytov number and the ratiol�r0 for the Hartmann sensor in the Fried geometryand in the Hutchin geometry. Results are shownhere only for the Fried geometry because of the sim-ilarities between the two sets of results. The abovequantities represent the ability of a given reconstruc-tor to form an estimate of either the least-squares orthe hidden phase at the points in �� from measure-ments obtained from a Hartmann sensor in the ab-



sence of scintillation. Evaluation of these quantitieseliminates any corruption that is due to scintillationeffects and eliminates any cross talk in the measure-ment process between the least-squares and the hid-den phases.

The main result is shown in Figs. 4�a�–4�c�. Es-timation accuracy is shown as a function of Rytovnumber for l�r0 � 1�4, 1�2, and 1 in Figs. 4�a�, 4�b�,and 4�c�, respectively. When the coherence length islarge relative to a subaperture, as in Fig. 4�a�, theresults are consistent with expectation. The least-squares reconstructor provides a good estimate of theleast-squares phase ��. The only degradation in theability of the least-squares reconstructor to estimatethe least-squares phase is due to averaging the in-tensity weighted gradient of the phase function overthe extent of a subaperture. However, for large val-ues of the Rytov number, when branch points arepresent and there is significant hidden phase to beestimated �as indicated by the quantity S�, which isdefined as the Strehl loss, computed by use of thevalues of the field at the points in ��, that is due onlyto uncompensated hidden phase ��, the least-squares reconstructor fails to estimate the hiddenphase accurately. In fact, as expected, the Strehlloss that is due to not correcting the hidden phase isalmost exactly equivalent to the estimation accuracywhen the least-squares reconstructor is used. Thisresult is expected because the phase differences as-sociated with the hidden phase function are the curlof the vector potential of the phase function. Theleast-squares reconstructor contains the divergenceoperator, and the divergence of the curl of a vectorpotential is zero. The fact that the Strehl ratio doesnot go to zero when one uses the least-squares recon-structor to estimate the hidden phase is simply be-cause, as indicated by the nonzero value of S�, theStrehl loss that is due to not correcting the hiddenphase is not zero. As can be seen from Fig. 4�a�,when the coherence length is large relative to a sub-aperture side length, the noise variance weightedcomplex exponential reconstructor forms a good esti-mate of the least-squares phase. Furthermore, thenoise variance weighted complex exponential recon-structor also forms a good estimate of the hiddenphase, as suggested by its development.10,11

As the coherence length shrinks relative to a sub-aperture, the story changes significantly. Again,note that the complex exponential reconstructor ex-hibits a severe noise gain when the noise exceedssome threshold.10 Below this threshold, the noisegain is comparable with that of a least-squares re-constructor. However, above this threshold, any

are present�. The least-squares reconstructor ignores the hiddenphase in the same conditions. However, as the coherence lengthshrinks relative to a subaperture �b� and �c�, the nonlinearities inthe CNW lead to significantly reduced performance. In fact, byignoring the hidden phase, the least-squares reconstructor actu-ally forms a better estimate of the hidden phase when l�r

Fig. 4. Estimation accuracy of the Hartmann sensor in the Friedgeometry in the absence of intensity fluctuations shown as a func-tion of Rytov number for l�r0 � 1�4, 1�2, and 1 in �a�, �b�, and �c�,respectively. Each line corresponds to the ability of either theleast-squares �solid curve� or the noise weighted complex exponen-tial reconstructor �dashed curve� to form an estimate of the least-squares �no symbol� or hidden �diamonds� phase at the primecoordinates. Also shown is the Strehl loss that is due to hiddenphase S� �dash–dot curve�. In �a�, when coherence length r0 ismuch larger than a subaperture, it is clear that the CNW �definedin Subsection 2.B� successfully forms a good estimate of the hiddenphase when the Rytov number is large �i.e., many branch points
0 � 1 �c�.

noise is greatly amplified. As the coherence lengthshrinks, the noise associated with averaging over theextent of a subaperture increases, causing the per-formance of the complex exponential reconstructor todegrade. In Figs. 4�b� and 4�c�, it is clear that thecomplex exponential reconstructor does not form asgood of an estimate of the least-squares phase as theleast-squares reconstructor. Furthermore, whenl�r0 � 1, the least-squares reconstructor, by essen-tially ignoring the hidden phase, actually forms abetter estimate of the hidden phase than the complexexponential reconstructor. It is believed that thisresult is due to the fact that, as the coherence lengthapproaches a subaperture side length, the noise vari-ance associated with the estimates of the phase dif-ferences obtained from the sensor, or the formulationerror, has reached a level such that not only will theHartmann sensor measurement values not supportestimation of the hidden phase but they are so noisythat they drive the noise in the output of the complexreconstructor into the severe noise gain regime.10

2. Effect of Scintillation on Wave-FrontMeasurement and Reconstruction of theLeast-Squares and Hidden PhasesThe results presented thus far in Subsection 3.B in-dicate that the use of a Hartmann sensor leads todifficulty in estimation of the hidden phase. Here,the effect of intensity fluctuations on estimation ofthe phase function is evaluated. The followingquantities are taken to be indicators of the effect ofscintillation on estimation accuracy:

Qscint,� �

�{exp[i��r��, exp[i arg(�wfsrec��A�r��expi��r�� )]}

�{expi��r��, exp[i arg(�wfsrec��expi��r�� )]}

,

(9)

Qscint,� �

�{expi��r��, exp[i arg(�wfsrec��A�r��expi��r�� )]}

�{expi��r��, exp[i arg(�wfsrec��expi��r�� )]}

.

(10)

These two quantities are simply the estimation accu-racy with scintillation divided by the estimation ac-curacy without scintillation. Although the abovequantities are useful indicators of the effect of ampli-tude scintillation on estimation accuracy, they arenot a direct measurement of the Strehl loss that isdue to amplitude scintillation.

Results are shown by use of both the least-squaresand the complex exponential reconstructors. Fig-ures 5�a� and 5�b� present the effect of scintillation asa function of Rytov number for l�r0 � 1�4 and l�r0 �1 for the Hartmann sensor in the Fried geometry.As before, the results for the Hutchin geometry aresimilar.

As expected, estimation accuracy of the least-squares phase is reduced by scintillation. However,estimation of the hidden phase actually improves

with scintillation. The authors note immediatelythat improve is a relative term. Although a perfor-mance improvement is observed for some cases, inthese cases the estimation accuracy, despite the im-provement, is still very poor �i.e., 2 � 0.01 is still only0.02�.

The authors hypothesize that the slight improve-ment in estimation of the hidden phase could possiblybe associated with the fact that the measurementsobtained are the integral over the extent of the sub-aperture of the intensity weighted gradient of thephase function. The results in Subsection 3.B.1 in-dicate that the presence of branch points in the phasefunction can lead to a significant reduction in estima-tion accuracy unless the coherence length is largerelative to a subaperture. It is possible that thepresence of branch points in the phase function cor-

Fig. 5. Strehl loss �or gain� that is due to scintillation shown as afunction of the Rytov number for l�r0 � 1�4 �circle� and l�r0 � 1�left triangle� for the Hartmann sensor in the Fried geometry byuse of the �a� least-squares and �b� complex exponential reconstruc-tors. Independent of the reconstructor, while estimation of theleast-squares phase �solid curve� is degraded by scintillation, theestimation accuracy of the hidden phase �dashed curve� actuallyimproves with scintillation.


1

rupts the gradient measurements in a manner simi-lar to that of noise. Thus one might hypothesizethat the noisiest contribution to the gradient mea-surement is concentrated near the branch points inthe phase function. However, the intensity is al-ways small in the vicinity of branch points. Notingagain that the gradient measurements obtained areproportional to the integral over the extent of a sub-aperture of the intensity weighted gradient of thephase function, one can hypothesize that the physicsof the measurement process leads to a natural noisevariance weighting of the contributions to the gradi-ent measurement.

One final point to note is that, when the coherencelength is four times the subaperture side length, scin-tillation has little effect on the estimation accuracy ofthe Hartmann sensor.

4. Discussion

The performance of Hartmann sensors in theHutchin geometry and in the Fried geometry wasevaluated by a wave-optics simulation study. Thewave-front sensing process is viewed in three parts:the physics of obtaining measurements related tothe complex field of interest, the mathematics ofusing those measurements to form an estimate ofthe complex field at a sparse array of points, and theinterpolation process used to form an estimate ofthe complex field at a dense array of points. Onlythe first two steps in the wave-front sensing processwere considered in this paper. Variations on re-construction algorithms were considered. Severalobservations were noted from the simulation study.

The most important observation is that the Hart-mann sensor, for a uniform distribution of thestrength of turbulence, in either the Fried geometryor the Hutchin geometry, has poor estimation accu-racy when the Rytov number is greater than 0.2 andthe ratio l�r0 is greater than 1�4. However, if theratio l�r0 is less than 1�4, a Hartmann sensor, cou-pled with a reconstructor that can be used to estimatethe phase associated with the branch points in thephase function, achieves good estimation accuracy.The criterion l�r0 � 1�4 for accurate estimation of thehidden phase contribution is not general, but ratheris accurate for a uniform distribution of the strengthof turbulence. Obviously, for a different distributionof the strength of turbulence the criterion could bedifferent.

Based on the simulation results presented in thispaper, the authors attribute the poor performanceof the Hartmann sensor primarily to the hiddenphase contribution to the phase function. The sim-ulation results indicate that, for relatively smoothturbulence �l�r0 � 1�4�, the performance of Hart-mann wave-front sensors, in either the Fried geom-etry or the Hutchin geometry, coupled with acomplex exponential reconstructor with noise vari-ance weighting, is quite good, even for large valuesof the Rytov number. When the turbulence issmooth �relative to the size of a subaperture�, theamplitude fluctuations and the hidden phase have a


smooth structure that is resolvable by the smallsubapertures. However, as the coherence lengthshrinks relative to the size of a subaperture, theamplitude fluctuations and hidden phase havehigher frequency structure that causes measure-ment errors that severely reduce estimation accu-racy, regardless of the geometry. This highfrequency structure leads to an increase in the for-mulation error, which acts as a noise source for thecomplex exponential reconstructor. The nonlinearnature of the complex exponential reconstructorleads to a severe noise gain when the noise exceedsapproximately 0.25 rad2.10 It is apparent that thisthreshold has been crossed for values of l�r0 greaterthan 1�4. These results are somewhat inconsis-tent with the conventional wisdom that suggeststhat a Hartmann sensor provides good performancewhen l�r0 � 1. It is clear that, while this assump-tion is valid for weak scintillation, in the strongscintillation regime �Rytov numbers greater than0.2�, a more stringent requirement must be imposedif one is using a Hartmann sensor. The results inthis simulation study indicate that for a uniformdistribution of the strength of turbulence, this re-quirement is somewhere near l�r0 � 1�4.

This research was partially funded by the U.S.Air Force Office of Scientific Research. The au-thors gratefully acknowledge their support. Theauthors also acknowledge and thank several indi-viduals for useful discussions and interactions, spe-cifically David Lee, Troy Rhoadarmer and EarlSpillar of the U.S. Air Force Research Laboratory�Directed Energy Directorate/Starfire OpticalRange. The authors also acknowledge the contri-butions of the following individuals, who not onlyprovided useful discussions but also provided anindependent qualitative confirmation of the resultspresented in this paper that used a least-squaresreconstructor: Glenn Tyler, William Moretti, andTerry Brennan, of the Optical Sciences Company.Finally, the authors sincerely thank the reviewersfor their efforts in assisting with the improvementof this paper.

References and Notes1. D. L. Fried, “Least-square fitting a wave-front distortion esti-

mate to an array of phase-difference measurements,” J. Opt.Soc. Am. 67, 370–374 �1977�.

2. R. H. Hudgin, “Wave-front reconstruction for compensated im-aging,” J. Opt. Soc. Am. 67, 375–379 �1977�.

3. B. M. Welsh, B. L. Ellerbroek, and M. C. Roggemann, “Fun-damental performance comparison of a Hartmann and ashearing interferometer wave-front sensor,” Appl. Opt. 34,4186–4195 �1995�.

4. B. L. Ellerbroek, “First-order performance evaluation ofadaptive-optics systems for atmospheric-turbulence compen-sation in extended-field-of-view astronomical telescopes,” J.Opt. Soc. Am. A 11, 783–805 �1994�.

5. D. L. Fried, “Reconstructor formulation error,” in Optical Pulseand Beam Propagation III, Y. B. Band, ed., Proc. SPIE 4271,1–21 �2001�.

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Report TT-68-50464 �National Technical Information Service,Springfield, Va., 1968�.

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8. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt.Soc. Am. A 15, 2759–2768 �1998�.

9. G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J.Opt. Soc. Am. A 17, 1828–1839 �2000�.

10. D. L. Fried, “Adaptive optics wave function reconstruction andphase unwrapping when branch points are present,” Opt.Commun. 200, 43–72 �2001�; available �with associated soft-ware� from D. L. Fried or J. D. Barchers.

11. D. L. Fried, “Adaptive optics wave-function�wave-front recon-struction: problems and solutions,” in Proceedings of OSAAnnual Meeting �Optical Society of America, Washington,D.C., 2000�, invited talk WR-1.

12. M. C. Roggemann and A. C. Koivunen, “Wave-front sensingand deformable-mirror control in strong scintillation,” J. Opt.Soc. Am. A 17, 911–919 �2000�.

13. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence�Springer-Verlag, New York, 1994�.

14. D. L. Fried, “Scaling laws for propagation through turbulence,”Atmos. Oceanic Opt. 11, 982–990 �1998�.

15. D. L. Fried, “Statistics of a geometric representation of wave-front distortion,” J. Opt. Soc. Am. 55, 1427–1435 �1965�.

16. S. F. Clifford, G. R. Ochs, and R. S. Lawrence, “Saturation ofoptical scintillation by strong turbulence,” J. Opt. Soc. Am. 64,148–154 �1974�.

17. We computed the formulation error in the Hutchin geometryby following the same development as previously published forthe formulation error in the Fried geometry.5

18. G. H. Golub and C. F. Van Loan, Matrix Computations �JohnHopkins U. Press, Baltimore, Md., 1996�.


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Evaluation of the Performance of Hartmann Sensors in Strong Scintillation