Characteristic functions of Hartmann-Shack wavefront sensors and laser-ray-tracing aberrometers

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3700 J. Opt. Soc. Am. A/Vol. 24, No. 12 /December 2007 Salvador Bará

Characteristic functions of Hartmann–Shackwavefront sensors and laser-ray-tracing

aberrometers

Salvador Bará

Área de Óptica, Departamento de Física Aplicada, Universidade de Santiago de Compostela,15782 Santiago de Compostela, Galiza, Spain

*Corresponding author: [email protected]

Received March 5, 2007; revised August 10, 2007; accepted September 16, 2007;posted October 10, 2007 (Doc. ID 80637); published November 14, 2007

It is shown that the aberration estimated at any point of the pupil using wavefront slope aberrometers such asHartmann–Shack wavefront sensors or laser ray tracers is a spatial average of the actual aberration weightedby a characteristic function that depends on the aberrometer design and on the estimation procedure. Thischaracteristic function, whose explicit form is given here for wavefront slope aberrometers using either modalor zonal estimators, may be useful in analyzing some basic aspects of the aberrometer performance. It is alsoinstrumental in establishing the links between the statistical properties of the actual and the estimated aber-rations. Explicit formulas are given to show in terms of this function how the bias arises in the first- andsecond-order statistics of the retrieved aberrations. This approach is mathematically equivalent to the analysisof the effects of modal coupling (cross-coupling and aliasing). It may provide, however, some complementaryinsight. © 2007 Optical Society of America

OCIS codes: 330.4300, 330.5370, 010.7350, 330.7310, 000.5490.

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. INTRODUCTIONavefront slope sensors have become widespread tools in

ifferent fields of science and technology, such as high-esolution imaging with ground-based telescopes [1–3],tmospheric optics [4–7], optical metrology [8–12], andosition and displacement sensing [13]. They have beenuccesfully applied also to physiological optics, vision sci-nce, and clinical practice in optometry and ophthalmol-gy, where devices such as Hartmann–Shack wavefrontensors (HS) [14–16], laser ray tracers (LRT) [17–19], andpatially resolved refractometers (SRR) [20–22] have suc-essfully demonstrated their ability to measure eye aber-ations. Different designs are available and most systemsan be built using relatively inexpensive off-the-shelfptoelectronic components. Comparisons have been mademong the outcomes of several kinds of eye aberrometers23–27], with the overall result of a reasonable agreementnd compatibility of the data obtained with them.There is, complementarily, a growing interest in assess-

ng the absolute performance of these devices to deter-ine the best sampling configurations—number, size,

eometry, and spatial distribution of the samplingubpupils—and data reduction procedures. In eye aber-ometry it is usual to retrieve the aberrations using linearodal estimations [28,29], the zonal versions [29,30] hav-

ng received somewhat less attention probably due in parto the widespread use of the Zernike polynomials [31–33].ithin the scope of the modal description of wavefront

ensors, performance is primarily assessed in terms of theccuracy and uncertainty of the estimated aberration co-fficients, from which other useful metrics are derivedrms phase error, etc.). It is well known that the measured

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berration coefficients are affected, to a greater or lesserxtent, by an estimation bias arising from modal cou-ling, i.e., cross-coupling and aliasing [34–37]. Theecond-order statistics of the aberration coefficients hasn additional bias arising from noise propagation.In this paper it is shown that the estimated aberration

ˆ �r� can be expressed as a spatial average of the actualberration W�r�, weighted over the pupil by an aberrom-ter characteristic function H�r ,r�� that depends on theberrometer design and the chosen reconstruction matrix.his weighted averaging takes place irrespective ofhether W�r� is obtained by a modal or by a zonal recon-

truction method. In terms of this function the statisticsf the estimated aberrations can be easily related to thetatistics of the actual ones. The function also providesome visual insight into the way aberrometers work, andt may be a potentially useful complement to the tools al-eady available to optimize aberrometer design.

The relationship linking W�r� to W�r� cannot in generale inverted, and this implies that there is some amount ofnformation unavoidably lost in the estimation process, aact well known from previous studies about modal cou-ling. There is in principle an infinity of different actualberrations that would give rise to exactly the same esti-ated one W�r�. As usual in any physical measurement,

his uncertainty may be sensibly solved from a decision-aking standpoint if some a priori statistical information

bout the magnitudes being measured is available. Thisas been successfully done in atmospheric optics, whereeasonable models for the statistics of the aberrations in-uced by atmospheric refractive-index fluctuations were

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Salvador Bará Vol. 24, No. 12 /December 2007 /J. Opt. Soc. Am. A 3701

stablished long ago [38] starting from sound first physi-al principles.

However in eye aberrometry we still lack these models,nd this gives rise to an interesting methodological prob-em: The main sources of data available on eye aberrationtatistics are the estimated aberration datasets obtainedith different kinds of aberrometers, but they are affectedy a finite (however small) amount of bias. This bias maybscure some of the actual statistical features of the eyeamples. Assessing the soundness of these data, i.e.,valuating the accuracy and precision of the aberrometerssed to obtain them, requires in turn the knowledge ofhe actual (as opposed to “estimated”) statistics of the ab-rrations they intend to measure. A few comments on thisssue are made in Section 5 of this paper.

. CENTROID DISPLACEMENTS ASEIGHTED AVERAGES OF THE EYEBERRATIONost wavefront slope sensors sample the aberrations at

ifferent locations on the eye pupil, either measuring theeviations of the centroids of the sampling beams fromome reference positions (as in the HS and LRT devices)r introducing in the sampling beams suitable amounts ofilt to compensate for these deviations, performing in thisay a null test as is the case of the SRR. All these proce-ures are based on the well-known result that in a homo-eneous medium the irradiance centroid of a light beamropapagates along a straight line whose slope is given byhe spatial average of the transverse wavefront gradientW�r� weigthed by the normalized local irradiance i�r�. Ifc�0� is the position vector giving the �x ,y� coordinates ofhe centroid at any initial plane XY�z=0�, then afterropagating a distance z it becomes [39–41]

�c�z� = �c�0� + z� i�r� � W�r�d2r, �1�

here d2r=dxdy, the integral symbol is used as a conve-ient shorthand notation indicating two-dimensional in-egration in the XY plane with infinite limits, the �� /�x ,� /�y� operator acts on the transverse �x ,y� coordi-ates, and i�r�=I�r��I�r�d2r, where I�r� is the absolute ir-adiance of the beam.

This result, which is a particular case of the optical ver-ion of the Ehrenfest theorem [42,43], can be straightfor-ardly deduced within the scalar parabolic (Fresnel) ap-roximation using some analytical properties of the Diracelta distributions and their first-order derivatives [41].The centroid �c�z� appearing in Eq. (1) is the first mo-ent of the normalized irradiance i�r� and in principle itust be computed by integrating that irradiance over thehole detection plane. In practice, due to the constraints

mposed by the finite size and the noise statistics of thevailable detectors, this centroid is computed from irradi-nce measurements taken in a finite subdomain of thislane in a region which should be wide enough to avoidndesired truncation errors while keeping the noiseropagation within acceptable limits. There is a huge

mount of literature devoted to optimal centroid estima-ion, an issue which will not be dealt with here (see, e.g.,44–48]).

Note that there are some practically relevant differ-nces when computing experimentally the centroid withequential aberrometers (those that sample one region ofhe pupil at a time, such as the LRT, SRR, or the scanningartmann sensor [49,50]) and with “one-step” aberrom-

ters (those that subdivide simultaneoulsly the input pu-il in a set of sampling subapertures and detect the cen-roid positions of the irradiance distributions produced byach one at the detection plane, such as the HS. While inhe first group, the whole detector area is in principlevailable to compute each individual centroid, in the sec-nd group this area has to be segmented in order to applyq. (1) separately to the centroids of the irradiance distri-utions produced by each individual subpupil [41]. Someare must be exercised in the latter case to avoid crossalk between the irradiances produced by adjacent sam-ling subpupils (e.g., between the focal regions of neigh-oring microlenses in a HS). A detailed discussion of theonditions under which Eq. (1) can be applied indepen-ently to each individual microlens in a HS ensuring neg-igible cross talk can be found in [51].

The raw measurements mS, given by the aberrometert the S=1, . . . ,N sampling subpupils, i.e., the angularisplacements of the centroids, are the spatial averages ofhe aberration slopes weighted by the normalized irradi-nce distributions and corrupted by noise. Thus, at theth sampling subpupil, the two components of the angu-

ar displacement of the centroid are given by the vector

mS =� �S�r� � W�r�d2r + �S, �2�

here the integral is extended to the whole pupil planeXY ,z=0�, �S�r� is the generalized subpupil function ac-ounting for the geometrical limits of the Sth subpupil,ncluding its irradiance distribution normalized such that�S�r�d2r=1, and vS is the measurement noise.

For HS or SRR under uniform illumination �S�r� isimply the aperture function of the Sth sampling subpu-il (square, circular, etc.), while for LRT �S�r� is the nor-alized irradiance of the Gaussian sampling beamlet. If,

s usual, all sampling subpupils are displaced versions ofbasic form ��r� centered at different locations rS but

therwise identical, we have �S�r�=��r−rS�.The noiseless term in Eq. (2), which is the spatial aver-

ge of the local aberration slope weighted by �S�r�, can betraightforwardly transformed to express it as a spatialverage of the aberration itself. Integrating Eq. (2) byarts and taking into account that the generalized subpu-il function �S�r� for every S vanishes at infinity gives52]

mS = −� ��S�r��W�r�d2r + �S. �3�

Equation (3) tells us that the raw data provided by theavefront slope aberrometer at any subpupil S are in fact

he spatial averages of the actual aberrations weighted byinus the gradient of the corresponding subpupil func-

ion. This form of expressing the measurements, which

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as been extensively used by the astronomy and atmo-pheric optics community, is particularly useful in estab-ishing the links between the actual and the estimated ab-rrations.

Several particular cases of generalized subpupil func-ions ��r� and their directional derivatives −��r� /�x arehown in Fig. 1 [the −��r� /�y derivatives are obtained byoating these −� /�x images 90° counterclockwise). The up-er row of this figure displays the shapes of the square,ircular, and Gaussian subpupils defined, respectively, as

�sq�r� =1

d2 rect� r

d� = d−2, if x � d/2 and y � d/2

0, otherwise,

�4�

�cir�r� =1

�R2 circ� r

R� = 1/�R2, if r � R

0, otherwise, �5�

�Ga�r� =1

��2 exp�−r2

�2� , �6�

here r= r, and d, R, and � are the subpupil size, radius,nd rms Gaussian spot size, respectively. The lower row ofig. 1 displays the shapes of their corresponding gradi-nts, which are given by

��sq�r� =1

d2 rect� y

d��x +d

2� − ��x −d

2��x

+1

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d��y +d

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ig. 1. Upper row: shape of the generalized aperture functions�r� for (left to right): square subpupils of size d, circular subpu-ils of radius R=d /2, and Gaussian sampling beams of rmsidth �=d /121/2. Lower row, left to right: shape of the directionalerivatives −��r� /�x of the upper row functions. The brightnessf each image has been normalized to 1. The scale bar on theight indicates the normalized value associated with each grayevel.

��Ga�r� =− 2

��4 exp�−r2

�2�r, �9�

here x, y, are unitary vectors along the directions X and, respectively, and r=xx+yy. The brightness of eachubimage of Fig. 1 has been normalized to 1.

Note that the aberrometers with uniformly illuminatedubpupils, e.g., the conventional HS or the SRR, averagenly the aberration W�r� along the subpupil rims, gettingo information about the phase in the inner pupil regions.or instance, substituting Eq. (7) into Eq. (3) and inte-rating the variables of the Dirac delta distributions, weet for the uniformly illuminated square-subpupil HS

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d� 1

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−1

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yS+d/2

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xS+d/2

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hat is, the components of the angular displacements ofhe centroids (noise aside) are given by the differences be-ween the line averages of the aberration along the micro-ens borders orthogonal to the displacement direction, allivided by d. Aberrometers with uniformly illuminatedircular sampling pupils also perform line averages of theberration along the subpupil rim, weighted in this casey cos � (the x component) or sin � (the y component), �eing arctan�y /x�, the azimuthal angle. LRTs, in turn,erform a wider averaging, showing two regions of oppo-ite weighting sign [the different signs of the two regionsrise from the sign of the x or y coordinate in r as indi-ated by Eq. (9)] and displayed in the lower-right image ofig. 1.The spatial averaging indicated by Eq. (3) implies the

ell-known fact, mentioned above, that there are an infi-ite number of different aberrations W�r� that will giveise to the same value of the measurement at any givenubpupil.

. ACTUAL AND ESTIMATEDBERRATIONS: THE ABERROMETERHARACTERISTIC FUNCTIONn estimate W�r� of the actual aberration W�r� can be ob-

ained in several ways from the measurements providedy the sensor. The approach most frequently used in eyeberrometry is modal and linear. The estimated aberra-ion is expanded as a linear combination of a finite subsetf basis functions defined over the system pupil, and thexpansion coefficients are estimated as linear combina-ions of the sensor measurements (angular displacementsf centroids). Zonal estimation procedures, in turn, com-ute the estimated aberration at each point of the pupils a one-step linear combination of the measurements.

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ince the measurements are spatial averages of the ac-ual aberration [Eq. (3)], it follows that in either approachhe estimated aberration W�r�, is itself a weighted spatialverage of the actual aberration W�r�. The differences be-ween modal and zonal approaches lie in the differentorm of the weighting functions used in each case, butoth methods share the fundamental property of retriev-ng the estimated aberration as a weighted spatial aver-ge of the actual one.In what follows we will assume for concreteness that aodal estimation procedure is used (the zonal version is

asily obtained following essentially the same steps). Theensor measurements are arranged as a 2N�1 columnector m, whose Sth element mS is given by

mS = −� ��S�r�/��W�r�d2r + �S, �11�

here � /�� stands for � /�x if S=1, . . . ,N, and for � /�y if=N+1, . . . ,2N. The measurement noise is arraged like-ise to form the noise vector �. Defining a 2N�1 columnector ��r� such that their elements �S=1, . . . ,2N� areiven by

��r��S = ��S�r�/��, �12�

nd adopting the convention that the integral symbol op-rates on a vector as the integral of each of its elements,he 2N�1 measurement vector can be written as

m = −��r�W�r�d2r + �. �13�

The modal expansion of the actual aberration in termsf any definite set of basis functions (e.g., Zernike polyno-ials) is given in its most general form by

W�r� = i=1

aiZi�r� = ZT�r�a, �14�

here in the right side of the equality the aberration isxpressed formally using the �1 vectors Z�r� and a,hose elements are the basis functions (evaluated at r)nd the actual coefficients of the aberration, respectively.he superscript T stands for transpose. The estimated ab-

ig. 2. Gray level plot of H�r ,r�� for a HS with 69 unvignetted, eormalized to the pupil radius) with a 100% fill factor distributifferent estimation points r is represented, one at the geometricant (right). Their positions are indicated in each case by the bri

rration is similarly expressed, but now as a truncated se-ies of the form

W�r� = i=1

M

aiZi�r� = ZMT �r�a, �15�

here M is the number of modes included in the expan-ion, ZM is an M�1 column vector formed by the basisunctions corresponding to these M modes (evaluated at) and a is the M�1 column vector of estimated aberra-ion coefficients whose elements are the �ai�, i=1, . . . ,M.hese coefficients are estimated from the sensor measure-ents using a reconstruction matrix R (e.g., the conven-

ional least-squares one) of size M�2N, such that aRm. Substituting Eq. (13) for m, and substituting theesult for a in Eq. (15) we get

W�r� =� H�r,r��W�r��d2r� + ZMT �r�R�, �16�

here the characteristic function of the aberrometer�r ,r�� is given by

H�r,r�� = − ZMT �r�R��r��. �17�

Equation (16) shows that the estimated aberrationˆ �r� is the spatial average of the actual one, W�r��,eighted by the characteristic function H�r ,r�� and cor-

upted by the noise propagation term ZMT �r�R�. Note that

�r ,r�� is a scalar function, since the dimensions of itshree constituting factors are 1�M, M�2N, and 2N�1,espectively. This function gives us a direct measure ofow much the actual aberration at every point r� of theye pupil contributes to our estimated phase at r. It acts,roadly speaking, as an “aberration spread function”ending to “blur” the actual eye aberration pattern, as es-imated by the sensor. It also has some relevant conse-uences with respect to the relationship between the ac-ual and estimated statistics of the aberration (which areealt with in Section 4 below). This function expresses theay the aberrometer works, meaning in this context by

aberrometer” not only the physical device [which deter-ines the form of the elements of the vector ��r��] but

lso the observer’s choice of basis functions �ZMT �r�� and

illuminated square subpupils of width d=0.1818 (in length unitssquare array. The value of H�r ,r�� as a function of r� for two

er of the pupil (left) and the other in the upper-right pupil quad-spot within the pupil. See other parameters in the text.

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stimation matrix R (least-squares, Bayesian, etc.). In or-er to get an unbiased reconstruction of any actual aber-ation, leaving aside noise, H�r ,r�� should be equal to airac delta distribution ��r−r��. This situation cannot,owever, be attained in practice. There is hence an un-voidable bias given by

W�r� − W�r� =� �H�r,r�� − ��r − r��W�r��d2r�, �18�

hich—unlike zero-mean random noise—cannot be elimi-ated by averaging sets of measurements.Several examples of H�r ,r�� are displayed as graylevel

mages in Figs. 2–4. They were calculated for two differ-nt points r, one at the geometrical center of the pupilleft image) and the other in the upper-right pupil quad-ant (right image). Their positions are indicated in eachase by the brightest spot within the pupil. Figure 2hows the values of H�r ,r�� for a HS with 69 unvignetted,venly illuminated square subpupils of width d=0.1818in length units normalized to the pupil radius) with a00% fill factor. The least-squares reconstruction matrix

included the first 35 Zernike modes (excluding piston)nd was calculated with a small modeling error (pur-osely introduced) using the derivatives of the Zernikeodes at the centers of the subpupils instead of their

rradiance-weighted averages. As expected from Eq. (7),nly the actual phase along the pupil rims of the squareicrolenses contributes to the estimated aberration.

Fig. 3. Gray level plot of H�r ,r�� for the same HS wa

ig. 4. Gray level plot of H�r ,r�� for a LRT aberrometer with thean rms spot size �=d /121/2.

Figure 3 displays H�r ,r�� for a HS similar to that de-cribed above, but with microlenses of smaller size (d re-uced to the 70% of the former value).Figure 4 shows the characteristic function of a LRT de-

ice with the same square sampling pattern as the previ-us ones and with a Gaussian rms spot size �=d /121/2.

The aberrometer characteristic function can easily beeduced for a zonal estimation procedure. In the zonal ap-roach the aberration at a set of pupil points is estimatedirectly as a linear combination of the sensor measure-ents, whose weights LS�r� �S=1, . . . ,2N� depend on the

hosen reconstruction model. Arranging these weights ascolumn vector L�r�, the zonal estimation is obtained as

ˆ �r�=LT�r�m. Following the same steps as above, allquations may be rewritten for the zonal case substitut-ng LT�r� for ZM

T �r�R. This leads to the explicit form of theonal characteristic function H�r ,r��=−LT�r��r��.

. ACTUAL AND ESTIMATED STATISTICSF THE EYE ABERRATION

he aberrometer characteristic function helps to establishhe link between the statistical properties of the actualnd of the estimated aberrations. Denoting by anglerackets the operation of averaging over a set of aberra-ions (e.g., the ensemble of aberrations of a given popula-ion, the fluctuations of the aberration of a given subject,epeated measurements of a fixed target, or any mixed

t sensor as in Fig. 2, but with subpupils of 70% size.

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�W�r�� =� H�r,r��W�r��d2r�. �19�

Substituting Eq. (16) for W�r� it is easy to see that thepatial correlation of the estimated aberration,

W�r1 ,r2�= �W�r1�W�r2�� is related to the spatial correla-ion of the actual aberration BW�r1 ,r2�= �W�r1�W�r2�� by

BW�r1,r2� =�r��

r�

H�r1,r��H�r2,r��BW�r�,r��d2r�d2r�

+ ZMT �r1�RC�RTZM�r2�, �20�

here C�= ��T� is the measurement noise correlationatrix (size 2N�2N), and the last term accounts for the

ontribution of the noise propagation to the estimated cor-elations.

Similarly, the relationship between the estimated

W�r1 ,r2��W�r1�−W�r2��2� and actual DW�r1 ,r2��W�r1�−W�r2��2� aberration structure functions is

DW�r1,r2� =− 1

2 �r��

r�

�H�r1,r�� − H�r2,r��H�r1,r��

− H�r2,r��DW�r�,r��d2r�d2r� + �ZMT �r1�

− ZMT �r2��RC�RT�ZM�r1� − ZM�r2��, �21�

here once again the last term is due to the propagationf noise.

The spatial spectrum of the estimated eye aberrationan be defined as

�� =� A�r�W�r�exp�i2�� · r�d2r, �22�

being the two-dimensional spatial frequency and A�r�he unit-radius circle aperture function. Substituting Eq.16) for W�r� and expressing the result as a function of thepatial spectrum of the actual aberration ��, one gets

�� =� G��,− ��d2�� + QMT ��R�, �23�

here in the last term, which once again accounts for theropagation of noise, QM�� is a M�1 column vectorhose elements are the Fourier transforms of the basis

unctions �Zi�r�� evaluated at the spatial frequency �. Ifhe basis functions are the Zernike polynomials their Fou-ier transforms are essentially radial Bessel functionsith azimuthal factors, whose explicit form may be found

n the classic paper by Noll [53]. G is the Fourier trans-orm of the characteristic function H�r ,r�� with respect toll its spatial variables and defined as

G��,�� =�r�

r�

A�r�A�r��H�r,r��

�exp�i2�� · r + �� · r��d2r�d2r. �24�

rom Eq. (23), the power spectral density of the estimated

berrations, ��= ��*��, is related to the correla-ion of the power spectrum of the actual ones, �� ,��*��, by

�� =��

��

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+ QMT ��RC�RTQM

* ��. �25�

. ADDITIONAL REMARKS ANDONCLUSIONSquation (16) shows that the estimated aberration W�r�t any point r of the pupil is an average of the actual ab-rration W�r�� weighted by the characteristic function�r ,r��, plus a noise propagation term. This spatial aver-

ging gives rise to a well-known fact: the measurementrocess causes an unavoidable loss of information sincehere are in principle infinitely many different actual ab-rrations W�r�� which will give rise to exactly the sameˆ �r� for a given aberrometer and estimation procedure.he magnitude of the estimation error as indicated by Eq.

18) is determined not only by the aberrometer designthrough the characteristic function) but also by the mag-itude of the actual aberration.This situation is common to most physical measure-ent processes and is deeply rooted in the fact that actual

berrations have, in principle, more degrees of freedomhan the number of modes the aberrometer may measure36].

As a side note, this does not mean that the aberrationatasets obtained by current eye aberrometers in labora-ory or clinical settings are unacceptably inaccurate: Onhe contrary, there is growing experimental evidence sup-orting the accuracy of the measurements made withtate-of-the-art aberrometers. This evidence comes inart from absolute calibrations (i.e., comparisons of thestimated aberrations with those measured by indepen-ent means, e.g., by interferometry using optical elementsroducing fixed patterns of aberrations [27]) and partlyrom comparisons of the outcomes of different kinds ofavefront slope aberrometers that give compatible re-

ults [23–26]. The main reason behind this accuracy isrobably tied to the favorable matching between the cur-ent aberrometer designs (including under this concepthe way in which the reconstruction matrix is computed)nd the aberration structure of nonpathological eyes. Theodal coupling between the estimated modes (cross-

oupling) can be efficiently canceled using a properly com-uted least-squares reconstruction matrix [34–36]. Onhe other hand, the modal coupling arising from the non-stimated higher-order modes of the actual aberrationaliasing), as well as the truncation error [36], are keptmall due in part to the fact that the modal coefficients ofhe eye show a general trend of decreasing magnitude forncreasing aberration order. The existence of this trend,hich on general grounds may be anticipated from thenatomy and physiology of the eye, is a valuable piece of ariori knowledge for eye aberrometer design.

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Although for many practical applications the estima-ion error can be kept within perfectly acceptable limits, iteems advisable to take into account the detailed way theberrometers work in order to get more definite answersor at least two of the open questions of present day aber-ometry: What is the basic statistical structure of the eyeberrations? What is the most efficient aberrometer de-ign? Both questions are strongly tied together becausehe performance of any given aberrometer is always rela-ive to the population it measures, since it depends notnly on the aberrometer sampling geometry, signal-to-oise ratio, and reconstruction method, but also on theopulation statistics [35,36]. Knowing the actual statis-ics is a key step for the optimal design of aberrometers.ince the statistical properties of the estimated aberra-ions are in fact filtered versions of the actual ones (as in-icated in Section 4), a sensible approach to deal with thisssue seems to be (i) to elaborate reasonable models forhe actual statistics based on educated guesses arisingrom independent measurements and physiological opticsasic principles, (ii) to predict—using for example thequations of Section 4—which would be the observed sta-istics were those models true, and finally (iii) to checkhether these predictions are compatible or not with thexperimental datasets. This kind of mild Popperian ap-roach may be of some help to fine tune the already de-ailed knowledge about the spatial and temporal struc-ure of the eye aberrations that the research communityas accumulated in the past decades.If the aberrations are estimated in a modal way, the de-

cription of the aberrometer performance in terms of theharacteristic function is mathematically equivalent tohe description based on the traditional analysis of modaloupling (cross-coupling and aliasing). The characteristicunction H�r ,r�� plays, in the pupil position space, a roleimilar to the one played by the coupling matrix (vari-usly denoted by Cs,f, RA, or H in [34–36], respectively) inhe Zernike coefficient space. The modal coupling matrixndicates how much any actual aberration mode contrib-tes to the value of the estimated one. Likewise, the char-cteristic function H�r ,r�� indicates how much the actualhase at points r� contributes to the value of the esti-ated phase at r. It accounts for what could be called by

nalogy the “spatial coupling” of the aberration.In conclusion, the aberrometer characteristic function

�r ,r�� may be a potentially useful tool for analyzing theerformance of different kinds of wavefront sensors, irre-pective of whether they rely on modal or zonal estima-ors. The amount of information it provides is, with re-pect to its content, equivalent to that provided by theoupling matrix already used in different studies, but itpproaches the aberrometric landscape from a differentiewpoint. Both approaches underscore the unavoidableoss of information inherent to the measurement process.ven if in many practical situations this loss of informa-

ion may successfully be kept within acceptable limits, itmphasizes the need of elaborating statistical models forhe actual eye aberrations that take into account the ef-ects of aberrometer filtering and give rise to predicitionsompatible with the observed (estimated) datasets.

CKNOWLEDGMENTShis work has been supported by the Spanish Ministerioe Educación y Ciencia, Plan Nacional de Investigaciónientífica, Desarrollo e Innovación Tecnológica �I+D+i�,rant FIS2005-05020-C03-02 and FEDER.

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Documents

Characteristic functions of Hartmann-Shack wavefront sensors and laser-ray-tracing aberrometers