Hartmann-Shack test with random masks for modal wavefront reconstruction

Hartmann-Shack test with randommasks for modal wavefront

reconstruction

Oleg Soloviev and Gleb VdovinEI Lab., EWI, TU Delft, Mekelweg 4, 2628CD Delft, the Netherlands

[email protected]

Abstract: The paper discusses the influence of the geometry of aHartmann-(Shack) wavefront sensor on the total error of modal wavefrontreconstruction. A mathematical model is proposed, which describes themodal wavefront reconstruction in terms of linear operators. The modelcovers the most general case and is not limited by the orthogonality ofdecomposition basis or by the method chosen for decomposition. The totalreconstruction error is calculated for any given statistics of the wavefrontto be measured. Based on this estimate, the total reconstruction error iscalculated for regular and randomised Hartmann masks. The calculationsdemonstrate that random masks with non-regular Fourier spectra provideabsolute minimum error and allow to double the number of decompositionmodes.

© 2005 Optical Society of America

OCIS codes: (010.1080) Adaptive optics; (010.7350) Wave-front sensing

References and links1. I. Ghozeil, Optical Shop Testing, chap. Hartmann and Other Screen Tests, 2nd ed. (John Wiley & Sons, Inc., New

York, 1992), pp. 367 – 396.2. R. K. Tyson, Principles of adaptive optics, 2nd ed. (Academic Press, Boston, 1998).3. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements,”

J. Opt. Soc. Am. 67, 370 – 375 (1977).4. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am.

69, 393 –399 (1979).5. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998 –

1006 (1980).6. K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using

the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852 – 1861 (1986).7. M. C. Roggemann, “Optical perfomance of fully and partially compensated adaptive optics systems using least-

squares and minimum variance phase reconstructors,” Computers Elect. Engng 18, 451–466 (1992).8. R. G. Lane and M. Tallon, “Wavefront reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31, 6902 –

6908 (1992).9. G.-m. Dai, “Modified Hartmann-Shack Wavefront Sensing and Iterative Wavefront Reconstruction,” in Adaptive

Optics in Astronomy, vol. 2201 of Proceedings of SPIE, (SPIE, 1994), pp. 562 – 573.10. G.-m. Dai, “Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen-

Loeve functions,” J. Opt. Soc. Am. A 12, 2182 – 2193 (1995).11. G.-m. Dai, “Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loeve functions,” J. Opt.

Soc. Am. A 13, 1218–1225 (1996).12. M. C. Roggemann and T. J. Schulz, “Algorithm to increase the largest aberration that can be reconstructed from

Hartmann sensor measurements,” Appl. Opt. 37, 4321–4329 (1998).13. W. Zou and Z. Zhang, “Generalized wave-front reconstruction algorithm applied in a Shack-Hartmann test,”

Appl. Opt. 39, 250 – 268 (2000).

(C) 2005 OSA 14 November 2005 / Vol. 13, No. 23 / OPTICS EXPRESS 9570#8902 - $15.00 USD Received 26 September 2005; revised 7 November 2005; accepted 7 November 2005

mailto:[email protected]

14. Y. Carmon and E. N. Ribak, “Phase retrieval by demodulation of a Hartmann-Shack sensor,” Opt. Commun.215, 285–288 (2003).

15. L. A. Poyneer and J.-P. Veran, “Optimal modal Fourier-transform wavefront control,” J. Opt. Soc. Am. A 22,1515 – 1526 (2005).

16. A. Talmi and E. N. Ribak, “Direct demodulation of Hartmann-Shack patterns,” J. Opt. Soc. Am. A 21, 632 –639 (2004).

17. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. A 71, 989–992(1981).

18. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207 – 211 (1976).19. B. Patterson, “Circular and Annular Zernike Polynomials, Mathematica� Package,”

http://library.wolfram.com/infocenter/MathSource/4483/ (2002). UK Astronomy Technology Centre.20. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Optical Engineering 29, 1174 –

1180 (1990).21. D. W. de Lima Monteiro, O. Akhzar-Mehr, P. M. Sarro, and G. Vdovin, “Single-mask microfabrica-

tion of aspherical optics using KOH anisotropic etching of Si,” Opt. Express 11, 2244 – 2252 (2003),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2244

1. Introduction

The Hartmann or Hartmann-Shack (HS) wavefront sensor [1, 2] is one of the most widelyused sensors in adaptive optics (AO), where a monochromatic light source and/or referencebeam are often not available. The HS sensor measures the averaged local tilts over an array ofsub-apertures (s/a) – holes in a Hartmann screen, and lenslets in a HS test. From these localmeasurements, the wavefront can be reconstructed.

The problem of wavefront reconstruction with a HS test has been addressed in many pub-lications [3–16]. Two general approaches can be distinguished: modal and zonal. Zonal al-gorithms [3–6, 13] build a discrete phase field with finite differences closest to the measuredslopes in the least-squares sense. The phase data in between the measured points are interpo-lated. Modal algorithms [5, 7, 17] approximate the wavefront’s phase by a linear combinationof a number of aperture functions, or modes. According to Southwell [5], the modal approachis superior to zonal estimation in terms of error propagation and computationally easier andfaster.

In adaptive optics, commonly, modal algorithms are used, with different choices of de-composition modes and methods. Zernike polynomials are often chosen as the decomposi-tion basis due to their connection to classical aberrations; Karhunen-Loeve (K-L) functionsor their pseudo-analytical analogues [10] present an optimal basis for compensation of atmo-spheric turbulence; exponential functions are suitable for direct demodulations of Hartman-ngrams [14, 15]; influence functions of the adaptive element are used in all cases when aberra-tions should be corrected.

The error of modal wavefront reconstruction has been analysed by a number of authors [7,9, 11, 12, 17]. Herrmann [17] introduced the basis gradient matrix and used it to describe cross-coupling and aliasing effects; Roggemann [7] proposed using the minimum-variance recon-struction algorithm instead of standard least-squares minimisation, which makes no use ofwavefront’s aberration statistics; Dai [9, 11] developed a clear theory to analyse the total modalreconstruction error and applied it to define the optimal number of decomposition modes forvarious sensor configurations.

In this paper, we address the modal reconstruction error using Dai’s approach of total re-construction error minimisation. Dai [9, 11] has shown that this error is difficult to eliminateand that therefore one should either define the optimal number of decomposition modes for agiven sensor geometry or use iterative methods to increase this number while keeping the errorat the same level. However, iterative methods increase the computation time, a crucial factorin real-time AO systems. We propose to use randomised sensor geometry, which doubles thenumber of correctly reconstructed modes, requires the same computation time and has a smaller


http://library.wolfram.com/infocenter/MathSource/4483/

http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2244

total reconstruction error. We also show that although the randomisation of the sensor geometryvia small movements of s/a centres eliminates cross-talk of low-order aberrations, it does notdecrease the total error and should not be used.

The next section presents the most general description of the modal wavefront reconstructionin matrix, or linear operators notation. A formula for the total reconstruction error is derived ina more general context than that of Dai [11]; it is also valid for not least-squares algorithms. Inthe following sections, the theory is applied to build and compare Hartmann reconstruction ma-trices for five sensor geometries and least-squares minimisation. Then, the total reconstructionerror is calculated for Kolmogorov’s atmosphere turbulence.

2. Mathematical description of HS wavefront sensing

2.1. HS sensor as a linear operator

In modal wavefront sensing with the Hartmann or the Hartmann-Shack (HS) sensor, the incom-ing phase distribution f = f (x,y) is represented by the sensor as some function f , which isa linear combination of some given decomposition modes f j, j = 1 . . . ,N. Because this repre-sentation is based on linear operations – measurements of the wavefront slopes averaged overthe subapertures and least-squares approximation of these slopes, mathematically Hartmannsensing can be represented as a linear operator H , such that

f = H f =N

∑j=1

λ j f j = λ j f j,

where λ j are the coefficients of the modal decomposition1. Due to linearity of H , if the in-coming wavefront can be decomposed over some “native” basis gi, i = 1, . . . ,∞, (eg. Zernikepolynomials for optical shop tests, K-L functions for AO in turbulent media), so one can write

f = cigi + c0,

where the constant (piston) term c0 is not sensed by the HS sensor, then the Hartmann operatorH is fully defined by its matrix H = (h j

i ), i = 1, . . . ,∞, j = 1, . . . ,N. The elements h ji of the

matrix H are given by the Hartmann images of the basis function:

H gi =N

∑j=1

h ji f j,

and thus one has

H f = ( f1, f2, . . . , fN) ·H ·c1

c2

. . .

= h j

i ci f j,

or, in terms of coefficientsλ j = h j

i ci. (1)

The properties of the HS wavefront sensor are thus defined by the properties of the matrix H,which, in turn, depends on the sets of basis and decompositions functions f j and gi, on the HSmask or lenslets geometry, and on the demodulation algorithm. These properties are discussedin the following subsections; first we explore common properties of linear operators mappinginfinite-dimensional space in a finite-dimensional one.

1For brevity, we have adopted tensor notation, namely the upper index denotes the rows and the lower, the columns;thus λ j is a column vector, and f j , a row vector. The same index repeated both as superscript and subscript denotes thesummation by this index. Moreover, to further simplify the formulae, we will use indices always spanning the samerange for a given letter, for instance, j, j1, j2 always run values 1, . . . ,N; i, i1, i2 = 1, . . . ,∞, l = N +1, . . . ,∞, and so on.We will also use this for splitting sums and vectors; for instance, we will write ci = c j +cl and 〈cici〉= 〈c jc j〉+ 〈clcl〉.


2.2. The error of the operator H

For given basis and decomposition functions, one can identify the functions with their decom-position coefficients, and operator H may be regarded as the mapping of an infinitely longvector ci in an N-dimensional vector λ j. Then, according to the well-known fact from linearalgebra, that the dimension of the subspace of all vectors mapped to zero, KerH , is infinity.Hence, for any HS sensor there exists an unlimited choice of aberrations sensed as zero by thesensor, and so for the natural definition for the error ε of the HS sensor as

ε = ‖H f − f‖

the error is unlimited, unless we impose some restrictions on the possible forms of f . Usu-ally, these restrictions are expressed as statical properties of the expansion coefficients ci ofthe sensed function. This can be either Kolmogorov statistics of the Zernike coefficients of theplane wavefront passed through the turbulent media, or some expected values of optical aberra-tions for a given polishing process, etc. These will be transformed by the operator H into theprobability properties of ε . Namely, one has:

〈ε〉 =⟨‖ f − f‖⟩ = ‖〈 f 〉−〈 f 〉‖ = ‖〈ci〉gi −〈λ j〉 f j‖ = ‖〈ci〉gi −〈ci〉h j

i f j‖, (2)

which in often case of⟨ci

⟩= 0 leads to 〈ε〉 = 0. In applications with non-zero expected values

of the decomposition coefficients, as for instance in optical shop tests, Eq. (2) provides a for-mula for calculating the systematic error of the HS sensor.

Using identity (1), the second moment of the error ε can be calculated as

〈ε2〉 =⟨( f − f , f − f )

⟩= 〈( f , f )〉−2

⟨( f , f )

⟩+

⟨( f , f )

⟩= 〈ci1 ci2〉(gi1 ,g

i2)−2〈ciλ j〉(gi, f j)+ 〈λ j1λ j2〉( f j1 , f j2)

= 〈ci1 ci2〉(gi1 ,gi2)−2〈ci1ci2〉h∗i′

j (gi1 , f j)+h j1i1〈ci1ci2〉h∗i′

j ( f j1 , f j2),(3)

where h∗ij denotes element of the transposed matrix H∗. Hence the variation σ2

ε = 〈ε2〉−ε2 canbe calculated, provided that the matrix of cross-correlation 〈ci1ci2〉 and matrix H are known.Note that formula (3) includes infinite summations and in practice should be approximatedwith a finite sum, or explicit estimates should be used for hj

i and 〈ci1 ci2〉.

2.3. Reconstruction in the native basis

Let us now consider the special case where decomposition modes are just the first N basisfunctions

f j = g j, (4)

and let the basis be orthonormal for some dot product ( f ,g),so

(gi,gi′) = δi′

i ,

where δi′i is the Kronecker symbol. The calculations of the second moment of ε are then sim-

plified, as now one can write Eq. (3) as

〈ε2〉 = 〈ci1ci2〉(gi1 ,gi2)−2〈ciλ j〉(gi,g

j)+ 〈λ j1λ j2〉(g j1 ,gj2) = 〈cici〉−2〈c jλ j〉+ 〈λ jλ j〉, (5)

and thus〈ε2〉 = 〈cici〉−2〈c jci〉h∗i

j +h ji1〈ci1ci2〉h∗i2

j . (6)


An “ideal” Hartmann matrix H, given by

h ji = δ j

i , (7)

can be considered for the special case of reconstruction in the same basis. The HS sensorwith such a matrix is an ideal low-order filter in the space of the basis function, giving exactcoefficients for the first N modes and being insensitive to higher-order modes:

H( ∞

∑i=1

cigi)

=N

∑j=1

c jg j.

In this case, λ j = c j, and from Eq. (2) we have

〈ε〉 =∥∥∥ ∞

∑l=N+1

clgl

∥∥∥for the mean error; and from Eq. (5)

〈ε2〉 = 〈cici〉−〈c jλ j〉 = 〈clcl〉 def= σ2tr. (8)

The last term represents a mean-square error due to infinite-series truncation, a minimum pos-sible error for the orthonormal basis. This explains the name “ideal” for such a matrix.

In practice, however, the matrix H is often only an approximation to the ideal one, and theerror increases. If the first N columns of H form an identity matrix and the other columns arenot zero, then Eq. (7) is valid only for i ≤ N

h j′j = δ j′

j . (9)

Using this, after splitting the sums in index i into two sums with indices j and l, we obtain fromEq. (6)

〈ε2〉 = 〈c jc j〉+ 〈clcl〉−2〈c jc j′ 〉h∗ j′j −2〈c jcl〉h∗l

j

+h jj1〈c j1c j2〉h∗ j2

j +h jl1〈cl1 c j2〉h∗ j2

j +h jj1〈c j1cl2〉h∗l2

j +h jl1〈cl1cl2〉h∗l2

j

= 〈c jc j〉+ 〈clcl〉−2〈c jc j〉−2〈c jcl〉h∗lj + 〈c jc j〉+h j

l 〈clc j〉+ 〈c jcl〉h∗lj +h j

l1〈cl1 cl2〉h∗l2

j

= 〈clcl〉+h jl1〈cl1 cl2〉h∗l2

j .

(10)

Now the error is given by the infinite-series truncation σ2tr plus a new term due to aliasing of

the higher-order modes,

σ2al

def= h jl1〈cl1cl2〉h∗l2

j , (11)

an aliasing error. The expression obtained coincides with that of Dai [11].If the first N columns of H do not form the identity matrix, an additional error is accumulates

due to cross-coupling of the decomposition modes. In this case, a general formula (3) should beused for the error, combining truncation, cross-coupling and aliasing errors. Figure 1 illustratesthe relation between different error types and elements of the matrix H.

The importance of H for the error estimation is now obvious. In the next sections we willdescribe the practical way to calculate the matrix.


��

1 0�

0 10

0 1 00 �

��

Infinite series truncation Alliasing Cross-coupling

��

1 0�

0 1

h1N�1 . . .. . .

hNN�1 . . .

��

��

1 0�

0 10

��

��

1 h1N

�

hN1 1

h1N�1 . . .. . .

hNN�1 . . .

��

Fig. 1. Differences between ideal reconstruction matrix H (∞×∞ identity matrix) and non-ideal Hartmann matrix H and corresponding modal reconstruction errors. The series trun-cation error appears due to the discarding in reconstruction high-order modes (marked withblue colour in the diagram). These modes, however, can be sensed as low-order ones (non-zero elements in the “tail” of the matrix, marked with orange), the effect known as aliasingin sampling theory. If the first N columns do not form a unity matrix, low-order modesare also reconstructed with aliasing error, often called cross-coupling or cross-talk in theliterature.

2.4. Dependence of the error on the sensor geometry

To build the matrix H, let us first try to account for the geometry of the mask. Note that themeasuring of the averaged wavefront slopes in HS sensing also represents a linear operator.Indeed, if the slopes are measured over m subapertures ak,k = 1, . . . ,m, then the Hartmanngramcontains encoded information about m pairs of numbers

1Sak

(∫ak

∂ f∂x

dr,∫

ak

∂ f∂y

dr)

,

where Sak is the subaperture’s area. If all subapertures have the same symmetrical form, andhence all characteristic functions of the ak, χak , are obtained from the same even function χa(r):

χak(r) = χa(r− rk),

where ri are the subapertures’ centres, these m pairs can be presented as sampling of the “fil-tered” function, written as

1Sa

∫R2

χa(r− rk)∇ f (r)dr =1Sa

χa ∗ ∇ f

in points rk. This is equivalent to multiplying the filtered wavefront by the sampling function

m

∑i=1

δ(r− rk).

Thus the result of HS-sampling can be presented as a linear operator G such that

G fdef=

1Sa

m

∑k=1

δ(r− rk)(χa ∗ ∇ f ). (12)

Although G f consists of m pairs of numbers, we will represent it as a 2m-vector for simplic-ity. Then in basis gi, operator G is described by its matrix G, formed by G gi, as the i-th column.This “filtered basis’ gradient” matrix contains important information about the “compatibility”of the mask geometry and the basis functions. For instance, for a 91-hole hexagonal mask withZernike polynomials as the basis, only the first 87 columns of G are linearly independent. Thismeans that for any further linear transformation, the dimension of the space of all reconstructedwavefronts will be less than 87, and in the general case, the following inequality holds:

dimImH ≤ rankG .


� flow� fhigh

�� Χa

��k ∆�x � xk�

Fig. 2. A schematic sketch of the sensing of high- and low-order modes for a regular and anirregular sensor geometry in the Fourier domain. Averaging of the slopes over subaperturesacts as a low-pass filter (multiplication with F χa). During this operation high-order modesare transformed into low-order ones (aliasing). Even in the case where filtered functions arequite dissimilar, sampling with a periodic centre distribution (convolution with a periodicalspectrum) could result in significant spectrum distortion and similar spectra (cross-talk).Irregular geometry, with the spectrum of the distribution of centres imitating a δ-function,should not have the cross-talk effect.

Hence the number of possible reconstruction modes is limited by the rank of G. Thus thegeometries which result in matrices G with higher rank are desirable.

It seems extremely difficult to get an analytical dependence of rankG on the mask geometry.However, some insight can be obtained from Eq. (12) considered in the frequency domain:

F(G f

)=

1Sa

F( m

∑k=1

δ(r− rk))∗(

F χa ·F ∇ f).

In this formula, the influences of the subaperture size and the centres are separated. If, as usual,the higher-order modes contain higher frequencies, the term F χa ·F ∇ f leads to sensing ofhigh-frequency modes as low-frequency ones, because F χa is a low-pass filter2. On the other

hand, the term F(

∑mk=1 δ(r−rk)

), given by the position of the centres, broadens the spectrum

of the low-frequency functions. For regular masks, this broadening is performed in a regularway, and the resulting spectrum is similar to one of a high-order mode that is also regular dueto the symmetry. As a result of this operation, some different modes can be seen by the sensoras very similar (see Fig. 2 for illustration). This, in turn, produces ill-defined G. An irregularcentre distribution does not have a periodical spectrum and should increase the rank of thematrix G.

2.5. Dependence on the decomposition functions

The matrix H is defined not only by the gradients matrix G but also by the choice of thedecomposition functions f j, or, more accurately, by the choice of the decomposition vectors f j

2Due to the shape of F χa, located near origin, high frequencies contained in ∇ f will be attenuated after multipli-cation of F ∇ f by F χa


generated by f j. The possible choices for f j are:

• the same basis function g j, e.g. Zernike polynomials;

• the response functions r j of the adaptive optical element;

• an arbitrary set of functions f j satisfying the desired properties, e.g. trigonometrical basisappearing in the Fourier analysis of the Hartmanngrams.

The 2m-vectors f j are obtained from the chosen f j, either by applying the same gradient oper-ator G ,

f j = G f j,

or, in some cases, by sampling unfiltered (one-point gradients) [17] (or filtered with anotherfilter) gradients of the decomposition functions

f j =m

∑k=1

δ(r− rk)∇ f j. (13)

In any case, the coefficients λ j are found as the coefficients of the closest by some norm ‖•‖vector in space generated by f j:

‖G −λ j f j‖ = minx1,...,xN

‖G − x j f j‖. (14)

Independently from the choice of f j, f j,‖•‖, the result of the minimisation can be presented inthe form of some linear operator L :

λ j = L (G f ) = (L ◦G ) f = H f ,

and thusH = L ·G.

From this equation, it is obvious that to minimise the HS sensor error, it is important to choosethe proper matrix L; not the f j or ‖•‖, which can be chosen a posteriori. For instance, if thedecomposition modes coincide with the basis function, one usually uses a pseudoinverse forthe first N columns of G (denoted as GN) as matrix L

L = G+N , (15)

to obtain the first N columns of H closer to the identity matrix and to eliminate the systematicerror. If the systematic error is expected to be zero, the choice of L given by Eq. (15) may notnecessarily provide the minimum variance of the error. A detailed discussion of this topic isbeyond the scope of the present paper, however, and we will restrict ourselves to the case ofEq. (15).

2.6. Gradient measurement and jittering error

There exist two additional sources of error in HS wavefront sensing. The first is the measure-ment error of the spot centres in the Hartmanngram, ek, so we have

λ j = L (G f + ek) = H f +L ek.

If the measurement errors are zero-mean independent variables, uncorrelated with coefficientsci, with a standard deviation σg rad:

〈ek〉 = 0, 〈ek,ek′ 〉 = σ2g δk

k′ ,〈ciek〉 = 0,


(a) Hex91 (b) Hex61 (c) Hex61s

(d) move61 (e) MC61

Fig. 3. 5 masks used for calculation of H, shown in the unit circles. The subaperturesare circles with radius 1

11 , except (b) with radius 19 . Each mask contains 61 subapertures,

except (a) with 91 holes.

then, instead of Eq. (6), from Eq. (5) it follows that

〈ε2〉 = 〈cici〉−2〈c jci〉hij +h j

i1〈ci1 ci2〉hi2

j + l jk1

lk2j σ2

g ,

and we callσ2

meas = l jk1

lk2j σ2

g (16)

the measurement error term.The jitter error appears because the subapertures centres rk are known only with some finite

accuracy. According to Eq. (12), this leads to non-linear changes in matrix G. Usually, theuncertainty is much smaller than the size of the subapertures, and the error is negligible or canbe considered as a part of the measurement error. Moreover, in this article, we compare maskswith regular structures with ones with an irregular pattern, for which the jitter error is irrelevant.

3. Irregular HS masks

To investigate the dependence of the error on the mask regularity, we calculated the matricesH for 5 different HS masks shown in Fig. 3. Masks Hex91 and Hex61 are regular hexagonalmasks with 91 and 61 subapertures (s/a) of maximum size (1/11 and 1/9, resp.). Because thehexagonal mask is the densest one, the s/a size should be decreased to allow some randomisa-tion. We have chosen to use the size of Hex91 for masks with 61 holes. Hex61s is a regularhexagonal grid with smaller s/a; move61 and MC61 represent two irregular masks.Move61 represents a “small-movement” randomisation of the Hex61s. Each of the centres

of s/a in Hex61s has been moved by a random vector uniformly distributed in the circle ofradius 1/9− 1/11 = 2/99 to avoid s/a overlapping. This kind of randomisation seems to be acommon approach for randomising regular structures. For instance, the s/a centres of modifiedHS sensors considered by Dai [9] can be regarded as small-movement modification.


-20 -10 0 10 20-20

-10

0

10

20

(a) Hex61

-20 -10 0 10 20-20

-10

0

10

20

(b) move61

-20 -10 0 10 20-20

-10

0

10

20

(c) MC61

Fig. 4. Fourier transforms of the s/a centres for Hex61, move61, and MC61 masks.

MC61 is a random mask. We obtained it with the Monte-Carlo method by adding a randomvector uniformly distributed in a circle with radius 1− 1/11 as a new s/a centre if it was notcloser than 2/11 to any of the existing centres (to avoid overlapping) and not further away fromany of these than at least 3/10 (to avoid too sparse pattern containing less than 61 points).

For 61 s/a, the maximum possible rank of G is 122. Consider a determinant of the first 122columns of G as a continuous function of s/a centres rk. Then, due to continuous dependence,this determinant is likely to be ill-defined for move61, and a similar matrix H. Again, whenlooking at the Fourier transforms of the s/a centres of different types of masks, we see thatsmall-movement randomisation still has a regular spectrum, while the Monte-Carlo centres donot have any regular pattern in its transform (see Fig. 4). Thus one can expect a bigger rank ofG for a truly random mask geometry.

4. Calculation results

We have used Mathematica� for our calculations. First, we calculated the gradients of the first1000 Zernike polynomials gi in Noll’s [18] form using a free package written by Dr. BrettPatterson [19]. To calculate the filtered basis 1

Saχa ∗ ∇ gi in analytical form for any s/a radius a

and s/a centre rk, we converted gi into polynomials in x and y with the origin translated in rk,and replaced every monomial xiy j by its integral over s/a:

xiy j �→ 1πa2

∫x2+y2≤a2

xiy jdxdy =

ai+ jΓ( i+12 )Γ

(j+12

)

πΓ(

i+ j2 +2

) , i and j are even,

0, otherwise.

Then we calculated numerically matrices G for all masks shown in Fig. 3, and obtained themaximum number N′ of the first linearly independent columns of G for each of the sensorconfigurations. The results are shown in Table 1.

From G we can obtain H for any number N of decomposition modes using the least-squaresapproach given by Eq. (15). Figure 5 presents matrices obtained for N = 40 and N = 80. Noticethat for N = 40, matrix H for MC61 mask contains smaller elements, while regular masks con-tain less non-zero elements. For cross-correlation coefficients 〈ci1 ci2〉 also often form a sparsematrix, one can expect less aliasing error from a matrix with smaller coefficients produced bythe Monte-Carlo Hartmann mask.

For a twice larger number of decomposition modes N = 80 (see Fig. 5(f) – (j)), for regularmasks Hex61 and Hex61s H does not satisfy condition (9), and even low-order aberrationsare sensed as some combination with other modes (cross-talk of modes). This could potentiallyresult in a larger aliasing error, as correlation coefficients usually are greater for low-orderfunctions. Randomized masks and regular Hex91 all have N′ > 80, and thus their first 80


columns form identity matrices (no cross-talk). However, obviously the aliasing error, given bythe “tails” of the matrices, will be larger for Hex91 and move61 masks than for MC61.

Table 1. Properties of the HS sensing for various mask types

Hex91 Hex61 Hex61s move61 MC61

Geometry regular regular regular small movements Monte-CarloNumber of s/a 91 61 61 61 61Radius of s/a 1

1119

111

111

111

N′ 86 62 62 122 122NOpt 29 29 23 26 42σ2

tr +σ2al, rad2,

for D = 2 m, r0 = 20 cm 0.92723 1.01019 1.15096 1.3821 0.880957

To calculate the residual error, we used the turbulence statistics from Roddier’s article [20].For a number of decomposition modes less than N′, formula (10) can be used3. This expressioninvolves infinite sums, which should be approximated for practical purposes. We used Dai’s [9]asymptotical approximation for the infinite-series truncation term 〈clcl〉,

〈clcl〉 0.274N−0.8428(

Dr0

)5/3

,

and approximated the infinite sum in the aliasing term hjl1〈cl1cl2〉hl2

j by a finite one, letting l1, l2change from N + 1 to some L. For matrices H under consideration, the aliasing term can beconsidered to be constant for L > 700 (see Fig. 6 for an illustration), and thus we used L = 700in our calculations.

The results for the contribution of aliasing and series truncation terms in the residual errorare presented in Fig. 7a. The plot presents the variance of truncation and aliasing errors (in(D/r0)5/3rad2) against the number of decomposition modes N. As was expected, the Monte-Carlo randomisation of the s/a centres results in a smaller error and at the same time can be usedfor a bigger number of decomposition modes. Table 1 shows the optimal number of decompo-sition modes NOpt for all the masks and minimum aliasing plus the truncation error calculatedfor (D/r0) = 10.

For each mask we calculated the measurement error with Eq. (16) in terms of the slope-measurement variance σ2

g , which depends strongly on the application parameters, such as de-tector sensitivity and pixel size, spectrum of the light, etc. Added to the truncation and aliasingerror, the measurement error can affect the optimal number of decomposition modes and, ofcourse, the total restoration error. Nevertheless, from Fig. 7b it follows that the total error formask MC61 for N > 40 remains the smallest.

We also calculated H and the reconstruction error for the case of decomposition of averagedgradients by point gradients, given by Eq. (13). In this situation, the least-squares solution ofEq. (14) is given by the pseudoinverse of the point gradients matrix GN :

L = G+N . (17)

3for N > N′, Eq. (10) is not valid, and the variance of the error is given by Eq. (6). The plot of σ2al according to

Eq. (11) for N covering the whole range from 1 to 2m has a big discontinuity in almost monotonic behaviour in pointN = N′. This allows implicit calculation of N′ for a given mask.


(a) Hex91

(b) Hex61

(c) Hex61s

(d) move61

(e) MC61

(f) Hex91

(g) Hex61

(h) Hex61s

(i) move61

(j) MC61

0 1��3

2��3

1 2 4 8 16 32 64 128

(k) Colour scale coding

Fig. 5. Matrices H, obtained for various masks for N = 40 ((a) – (e)) and N = 80 ((f) –(j)) decomposition modes. Absolute values of non-zero elements are indicated by a lineargray-scale level for values in [0,1] and a logarithmic hue scale for values greater than 1 (seeFig. (k)). This figure is advised to be viewed on-screen and with increased zoom level.


0 200 400 600 800 10000

0.001

0.002

0.003

0.004

L

hjl1〈cl1cl2 〉hl2

j , l1, l2 ≤ L

Fig. 6. An infinite sum h jl1〈cl1 cl2〉hl2

j for Hmove61 for N = 15 decomposition modes isapproximated by a finite one with l1, l2 = N +1, . . . ,L and plotted against L.

0 20 40 60 80 100 120

0.05

0.1

0.15

0.2

Hex61s

Hex91

Hex61

move61

MC61

(a) σ2tr +σ2

al, in (D/r0)53

0 20 40 60 80 100 120

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Hex61s

Hex61

Hex91

move61

MC61

(b) σ2meas, in σ2

g

Fig. 7. Sum of aliasing and truncation errors and measurement error for the 5 masks shownin Fig. 3 and the Kolmogorov turbulence statistical model.

(a) N = 40

(b) N = 60

(c) N = 100

Fig. 8. H for decomposition by averaged (top picture in each pair) and point gradients forvarious N.


0 20 40 60 80 100 120

0.02

0.04

0.06

0.08

0.1

Hex61

Hex61Point

MC61

MC61Point

(a) σ2tr +σ2

al, in (D/r0)53

0 20 40 60 80 100 120

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Hex61

Hex61Point

MC61

MC61Point

(b) σ2meas, in σ2

g

Fig. 9. Aliasing and measurement error, in (D/r0)53 and σ2

g resp., for reconstruction withaveraged and point gradients.

The obtained matrices look similar for a small number of decomposition modes. When N in-creases, the “tail” of the matrix of decomposition by point gradients grows faster than that fordecomposition by averaged gradients; also cross-talk (non-zero elements in the first N columns)becomes more apparent (see Fig. 8). The calculated error is shown in the Fig. 9. The truncationand aliasing error are approximately the same for small N, but have larger minimal values andgrow more rapidly when N increases. The measurement error is slightly less for point gradientsdecomposition for small N, but the difference is negligible.

5. Discussion and concluding remarks

In this paper we have developed the reasoning of Dai [11] to the general case of modal wave-front reconstruction. We have shown that independently of the choice of native wavefront basisand reconstruction function set, the modal wavefront reconstruction with the HS wavefront sen-sor can be presented as a linear operator in a space of continuous functions. As this operatormaps infinite-dimensional space onto a finite-dimensional one, there is an infinite-dimensionalspace of function reconstructed as zero by the sensor. As a consequence, no statement about theaccuracy of HS-sensor-based measurements should be made in the lack of a statistical modelof the incoming wavefront.

We have also illustrated the developed model by calculating the total wavefront reconstruc-tion error with Zernike polynomials for the case of Kolmogorov’s turbulence. The error wascalculated for various geometries of the sensor’s array of subapertures. According to the calcu-lations, the use of the 61-hole mask with randomly distributed centres gives better results thanthe regular hexagonal mask with 91 subapertures of the same size.

The results can be different for other wavefront statistics, but based on a comparison of thematrices for Zernike basis, for instance for 40 decomposition modes, we expect better perfor-mance of the random masks. Moreover, measurements with regular masks experience catas-trophical growth of the aliasing error when number of reconstruction modes increases. Thisencourages one to use randomized masks in practice.

Some problems expected for the location of the spot centroids due to the irregular natureof the mask are related to the need for calibration, which, in principle, increases the slopemeasurement error. But in practice, any Hartmann sensor is calibrated first, and as a resultthe slope measurement error increases equally for regular and irregular masks. The associatedmeasurement error of the Hartmann sensor was shown to be smaller for the irregular maskstarting from some number of decomposition modes.


The proposed model allows to estimate the error of the correction with active element’sinfluence functions, which are usually not orthogonal. In this case, the use of irregular masksis not yet justified. Another interesting problem for this case is the construction of the optimaldecomposition operator L , either for a given statistics or for a least-squares approximation.This can be a topic for future research.

An interesting and unexpected fact is the sudden drop of the aliasing error for N = N′ + 1.Compare, for instance, matrices for Hex61 and Hex 91 for N = 80 (Fig. 5(f) and (g)). Althoughthe first N columns do not build an identity matrix anymore for HHex61 and produce somecross-coupling error, the tail of the matrix contains smaller coefficients than that of HHex91,resulting in a smaller aliasing error. This is another possible research topic – minimising thenumber of s/a while keeping approximately the same error.

Some notes on the s/a sizes, fill factor, and signal-to-noise ratio. Due to reduced fill-factorof randomised masks, one can expect also noticeably reduced signal-to-noise ratio in a light-starved environment for HMC61. Again, we should consider separately aliasing and measure-ment terms.

From the signal analysis point of view, the signal-to-noise ratio is defined by signal and noiseparts of power spectrum. If we are interested only in the first N Zernike components, or the low-frequency part, the high-frequency part is related to the noise. Averaging of wavefront slopesover s/a can be considered as prefiltering, limiting the signal bandwidth. For s/a smaller in size,the filtering function is broader, so the high-frequency parts are not lost but replicated as low-frequency ones for regular masks, increasing the aliasing error (see Fig. 2 and also compare theplots of σal for Hex61 and Hex61s in Fig. 7a). Thus for regular geometries signal-to-noiseratio decreases with reduced fill factor. Our conjecture is that random masks should be lessprone to this effect.

From the image analysis point of view, the size of subaperture is related to the signal-to-noiseratio in the recorded Hartmanngram, and affects the gradient measurement error ek. However,in equal light conditions, mask with equal s/a sizes have equal ek. Thus, although the fill factorof HMC61 is approximately 2/3 of that of HHex91, the signal-to-noise ratio and correspondentterm σ2

g remain at the same level. Moreover, it is possible to create microlens array with MC61centre distribution and 100% fill factor, using technology described in [21]. An average lens insuch an array will be 3/2 times bigger, than lens in a regular array with Hex91 geometry, andthe measurement error will be even reduced.

In this article, we performed calculations only for constant s/a size. For trigonometric basis,Fourier transform of circular aperture is a low-pass filter. A hypothetical averaging throughBessel function subaperture would result in an ideal low-pass filter in the Fourier basis, andthus an ideal, projective H. This is not possible for the Zernike basis, as its high-order termscontain a low-frequency part (in the central region). Thus, it would be interesting to investigateH for random masks with changing subaperture size, bigger in the centre and smaller towardsthe edges. This mask could generate H closer to the ideal projection.

Acknowledgments

The authors are very grateful to Dr. Alexey Simonov for fruitful discussions that contributed alot in understanding of theoretical and practical aspects of the problem and to Mirjam J. Niemanfor checking the English.


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Hartmann-Shack test with random masks for modal wavefront reconstruction

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