Optics Communications 237 (2004) 257–266

www.elsevier.com/locate/optcom

Significance of thresholding processing in centroidbased gradient wavefront sensors: effective modulation of the

wavefront derivative

J. Arines *, J. Ares

Departamento of Applied Physics, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Galicia, Spain

Received 6 October 2003; received in revised form 12 April 2004; accepted 13 April 2004

Abstract

The use of thresholding processing is a very extended task in centroid based gradient wavefront sensors. In this paper

we develop the expressions that include the thresholding process in the relation between the wavefront derivative and

the centroid of a thresholded intensity distribution. We also analyze through numerical simulations the effective in-

fluence of thresholding on the wavefront slope. The results of the developed simulations remark: the relevancy of the

centroid shift induced by the interaction between thresholding processing and comatic aberrations; and the nonlinear

relation between the centroid and the phase gradient in presence of thresholding processing. Particularization of these

results to Shack–Hartmann sensors has also been done.

� 2004 Elsevier B.V. All rights reserved.

PACS: 95.75; 42.87; 42.15.D; 42.79.P; 42.15.F; 42.30

Keywords: Shack–Hartmann; Centroid; Thresholding; Wavefront sensing; Optical testing

1. Introduction

Broadly speaking centroid based gradient

wavefront sensors obtain phase information frommeasurements of the mean slope of the field cal-

culated over a certain spatial region. They have

been extensively used in a very wide range of ap-

plications; range sensing [1], ocular aberrometry

* Corresponding author. Tel.: +34981563100x13530; fax:

+34981590485.

E-mail address: fajap@usc.es (J. Arines).

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2004.04.019

[2,3], lens testing [4,5] and adaptive optics [6]. In

particular, the Shack–Hartmann sensor [7,8] gets

the phase information from the local measure-

ments of the mean slope of the field, calculatedover each one of the elements of the microlenses

array. Moreover, they use to compute the local

gradient from the displacement of the centroid of

an image distribution, from another centroid po-

sition of reference.

Many works have been devoted to study, more

or less straightforwardly, the different error sour-

ces which affect centroid based gradient wavefrontsensors. On the one hand, some practical aspects

ed.

Fig. 1. Schematic of the relationship between the pupil plane

and the image plane.

258 J. Arines, J. Ares / Optics Communications 237 (2004) 257–266

have been extensively studied, as for example: In-

fluence of read noise [9,10], missalignements [1,11],

photon flux [6,12], wavefront statistics [13], and so

on. On the other hand, analysis of the significance

of the phase estimation algorithms, least-squares

[14] or minimum variance [15], local [16] or modal[17] estimation, and significance of the measure-

ment model has also been performed [18].

However, one important step that has been

normally neglected for further analysis is the

thresholding processing. The thresholding pro-

cessing is outstanding into the data reduction task

of all these sensors. This kind of image processing

is extensively used with the aim of discarding thebackground and the noisy pixels of the objects

(image focal regions) that can respectively decrease

the accuracy and precision of the centroiding task

[19]. However, into the vast amount of studies

related with centroid based wavefront sensors, no

more than some advises or mentions regarding the

use of thresholding processing can be found. In

particular, the most specific studies, that we haveknown, only warns of the centroid shift error in-

duced when changing the threshold level [20,21]

without developing further analytical explana-

tions. Additionally the clever work developed by

Alexander et al. [22] put a small warning about the

shifting effect that thresholding induces on the

centroiding operation. Their contribution is lim-

ited to some particular simulated cases for a po-sitioning triangulation sensor.

In a precedent paper we have shown the effects

that thresholding induces on the centroid variance

calculated over a noisy set of detected intensity

distributions [23]. With this idea we have purposed

an algorithm to estimate the optimum threshold

value in minimum variance centroid sense. Taking

a different way, in this paper we show a new pointof view for understanding the effect that thres-

holding the intensity distribution performs over

the centroid, and so, over the gradient phase

measurement. By separating this effect from the

random noise influence, we quantify the bias in the

centroid due to thresholding in an ideal case

without noise. This new sight put attention to the

way that thresholding processing modifies, in aneffective way, the impinging field estimated from

centroid measurements.

2. Theoretical description

In analysing the situation exposed in Fig. 1

where an optical field impinges on a lens, Primot et

al. [7] shown that, by neglecting scintillation, thecentroid in the u-direction (XC) of an irradiance

distribution Iðu; vÞ detected at the focal plane of an

optical system is related with the spatial phase

/ðx; yÞ of the impinging field according to:

XC ¼R R

uIðu; vÞdudvR RIðu; vÞdudv

¼ kf2pAS

Z ZSrx/ðx; yÞdxdy; ð1Þ

where ðu; vÞ are the transversal spatial coordinatesin the focal plane, AS denotes the area of the pupilaperture,

R RS dxdy denotes the integration over

the pupil area, f denotes the focal length of the

lens and rx/ is the x-derivative of the phase.

Broadly speaking, it is clear from Eq. (1) that

the centroid provides information about the spa-

tial average of the phase x-gradient. It is also evi-

dent from Eq. (1), that the centroid has additive

property in phases, it means, the centroid thatcorresponds to a total phase /ð1þ2Þ ¼ /1 þ /2, is

equal to the sum of the centroids related with each

one of its components /1 and /2, ðXCð/ð1þ2ÞÞ ¼XCð/1Þ þ XCð/2ÞÞ.

However as it was appointed in the introduc-

tion, in practices the centroid is not directly cal-

culated over the original intensity distribution but

a thresholded one. Now, for answering how andwhen (or when not) the Eq. (1) still stems valid, we

have to obtain the expression that relates the op-

J. Arines, J. Ares / Optics Communications 237 (2004) 257–266 259

tical field at the pupil plane with the centroid of the

corresponding thresholded intensity distribution.

In this way, the general definition of the cen-

troid of an irradiance distribution in the u-direc-tion can be expressed as:

XC ¼

R Ruv

uIðu; vÞdudvR Ruv

Iðu; vÞdudv : ð2Þ

It is straightforward to insert the case of a thres-

holded irradiance distribution as:

XC ¼Ruv uIðu; vÞHðu; vÞdudvRuv Hðu; vÞIðu; vÞdudv ð3Þ

by defining a Heaviside function Hðu; vÞ as:

Hðu; vÞ ¼ 1 Iðu; vÞP T0 Iðu; vÞ < T ;

�ð4Þ

where T is the threshold level.

On the other hand, by considering a space-in-

variant system under incoherent illumination, the

image distribution Iðu; vÞ may be described

through the convolution integral [24]:

Iðu; vÞ ¼Z Z

jKðu� n; v� gÞj2jOgðu; vÞj2 dndg;

ð5Þ

where the kernel Kðu; vÞ is the Fourier Transform

of the pupil function P ðx; yÞ (see (A.2)), and

Ogðu; vÞ is the geometrical image of the object.Consequently, moving on, we can introduce Eq.

(5) in the numerator of Eq. (3) for obtaining:ZuvuIðu; vÞHðu; vÞdudv

¼Zuv

ZngjOgðn; gÞj2jKðu

�� n; v� gÞj2 dndg

�� uHðu; vÞdudv

¼ZngjOgðn; gÞj2 dndg

ZuvuHðu; vÞ

� jKðu� n; v� gÞj2 dudv: ð6Þ

Then we replace K�ðu� nv� gÞ by its FourierTransform, and change the order of integration

(see (A.2)) to obtain:

ZngjOgðn;gÞj2 dndg

ZuvuKðu� n;v� gÞHðu;vÞdudv

�ZxyP �ðx;yÞexp i

2pkd

ððu�

� nÞxþðv� gÞyÞ�dxdy

¼ZngjOgðn;gÞj2dndg

�ZxyP �ðx;yÞexp

�� i

2pkd

ðnxþ gyÞ�dxdy

�ZuvuHðu;vÞKðu� n;v� gÞs

� exp i2pkd

ðux�

þ vyÞ�dudv; ð7Þ

where k is the wavelength and d is the distance

from the pupil to the image plane.

Now, using the shifting and derivative proper-

ties of the Fourier Transform (see (A.3) and (A.4)),

we obtain the final expression for the numerator of

Eq. (3):ZuvuIðu; vÞHðu; vÞdudv

¼ �ikd2p

ZngjOgðn; gÞj2 dndg

�ZxyP �ðx; yÞrx P ðx; yÞ

�� ~Hðx; yÞ

�dxdy;

ð8Þwhere ~Hðx; yÞ is the Fourier Transform of thethreshold mask Hðu; vÞ.

Following a similar procedure for the denomi-

nator of Eq. (3), we arrive at:ZuvIðu; vÞHðu; vÞdudv

¼ZngjOgðn; gÞj2 dndgZ

xyP �ðx; yÞ P ðx; yÞ

�� ~Hðx; yÞ

�dxdy: ð9Þ

Finally we get the desired expression for the

centroid of the thresholded intensity distributionin function of the pupil function and the Fourier

transform of the thresholding mask:

XC¼Real

8<:� i

kd2p

Rxy P

�ðx;yÞrx P ðx;yÞ� ~Hðx;yÞ� �

dxdyRxy P

�ðx;yÞ P ðx;yÞ� ~Hðx;yÞ� �

dxdy

9=;:

ð10Þ

260 J. Arines, J. Ares / Optics Communications 237 (2004) 257–266

The obtained expression ismore general thanEq.

(1), mainly because we have introduced a new pa-

rameter in the description of the centroid, the

Fourier Transform of the thresholding function~Hðx; yÞ, but also, as other authors did before us [25],because we have not limited the study to the focalplane or considered constant amplitude over the

pupil plane.

The introduction of ~Hðx; yÞ in the expression of

the centroid (Eq. (10)) does not complicate its use.

In fact it is very easy to arrive at expression (1)

from Eq. (10). The first step consists of assuming

that no thresholding processing is applied. Math-

ematically this assumption can be expressed asHðu; vÞ ¼ 1 8u; v, and so ~Hðx; yÞ ¼ dðx; yÞ. Intro-ducing this equality on Eq. (10) and performing the

convolution of P ðx; yÞ with a delta function we get:

XC ¼ Real

(� i

kd2p

Rxy P

�ðx; yÞrxPðx; yÞdxdyRxy jP ðx; yÞj

2dxdy

):

ð11ÞThen, assuming that we are imaging at the focal

plane, using the derivative of a complex variable

(see (A.7)), and considering that the centroid must

be a real variable we get:

XC ¼ kf2p

Rxy jCðx; yÞj

2rx/ðx; yÞdxdyRxy jCðx; yÞj

2dxdy

: ð12Þ

Finally, considering constant amplitude of the

pupil function, we arrive again at Eq. (1):

XC ¼ kf2pAS

Zxyrx/ðx; yÞdxdy: ð13Þ

This expression shows that the centroid provides

information of the mean phase slope over the pupil

function, or equivalently the mean direction of

propagation of the rays that passes through thepupil. Comparing Eq. (12) with Eq. (13) we observe

that when the amplitude of the pupil is not con-

stant, as for example when using an apodization

amplitude filter, we get that the centroid can be

shifted depending on the filter, whichmodulates the

contribution of the different parts of the wavefront

to the final centroid position. However, in its actual

form, it is not so easy to make a similar analysisfrom Eq. (10). Nevertheless we can rearrange it to

make easier its interpretation. We obtain two ex-

pressions by using some properties of the convo-

lution that can be found in the appendix (A.6).ZxyP �ðx; yÞrx Pðx; yÞ

�� ~Hðx; yÞ

�dxdy

¼ZxyrxPðx; yÞ P �ðx; yÞ

�� ~Hðx; yÞ

�dxdy

ð14:aÞ

¼Zxy

~Hðx; yÞrx P �ðx; yÞð � Pðx; yÞÞdxdy

¼ QZxy

~Hðx; yÞrxOTFðx; yÞdxdy; ð14:bÞ

where the OTF is the Optical Transfer Function

and Q is the normalization factor of the OTF [24].

Consider again that Cðx; yÞ ¼ c, constant over

the pupil. Using Eq. (A.6) we get:ZxyP �ðx; yÞrx P ðx; yÞ

�� ~Hðx; yÞ

�dxdy

¼ ijcj2Zxyrx/ðx; yÞ ~Hðx; yÞ

�h� exp f � i/ðx; yÞg

�� exp i/ðx; yÞf g

idxdy: ð15Þ

By comparing this expression with the numera-

tor of Eq. (12), we can notice that the term into

brackets behaves as an apodization filter able to

modify the contribution of the different parts of the

wavefront to the final centroid depending mainly

on the form of ~Hðx; yÞ. Again, if ~Hðx; yÞ ¼ dðx; yÞ,we obtain the numerator of Eq. (13). In order to see

the effect that thresholding the intensity distribu-tion performs over the pupil plane, we show in

Fig. 2 the evaluation of P �ðx; yÞrxðP ðx; yÞ�~Hðx; yÞÞ for three different threshold levels, and the

corresponding shapes of the Heaviside functions.

We simulated a comatic phase, in terms of Zernike

polynomials [26], with normalized coefficient value

a8 ¼ 0:5 rad. In this figure same grey levels repre-

sent same value of the phase gradient. It is easy tonote from this figure that the presence of thres-

holding induces the effective modulation of the

phase gradient over the pupil plane, changing the

amount of rays with the same direction of propa-

gation, and so the average direction of propagation

of the bound of rays.

In the same way, expression (14.b) allows to

relate the centroid with the mean of the gradient of

Fig. 2. Evaluation of P �ðx; yÞrxðPðx; yÞ � ~Hðx; yÞÞ for three

different threshold levels: (a) no thresholding processing; (b)

T ¼ 2; (d) T ¼ 10, and the corresponding Heaviside functions (c)

and (e).

J. Arines, J. Ares / Optics Communications 237 (2004) 257–266 261

the optical transfer function (computed over the

pupil plane) weighted by ~Hðx; yÞ:

XC ¼ Real

(� i

kd2p

Rxy~Hðx; yÞrxOTFðx; yÞdxdyR

xy~Hðx; yÞOTFðx; yÞdxdy

):

ð16Þ

Again we can analyze the case when nothresholding processing is considered (it means~Hðx; yÞ ¼dðx; yÞ). In this situation the integral

corresponds to the value of the function that goes

with the delta function at the origin of coordinates

XC ¼ Real�� i

kd2p

rxOTFð0; 0ÞOTFð0; 0Þ

�: ð17Þ

Finally, by evaluating the x-gradient of the OTF at

the origin we obtain:

XC ¼ kd2p

rxUOTFð0; 0Þ; ð18Þ

where rx/OTFð0; 0Þ is the x-derivative of the phaseof the OTF, taken at the origin.

At this time, as it was also pointed out in

[22,27], by comparing Eq. (18) with Eq. (1) we

observe that, theoretically, we only need to knowrx/OTFð0; 0Þ to recover the mean direction of

propagation of the impinging field. This result re-

marks two important concepts:

• It is not essential to fulfil the Whittaker–Sha-

noon sampling theorem in a possible spatially

discretizated image detection to achieve a pos-

terior accurate centroid determination. (Big ad-

vantage of the centroid regarding other subpixellocation techniques.)

• The centroid computation accuracy is very sen-

sitive to background influence, due to its strong

contribution to the lowest spatial frequencies.

(Main drawback of the centroid regarding other

subpixel location techniques.)

Regarding all of this, it is curious to observe at

Eq. (16) that the thresholding processing, which isjust employed to overcome this last drawback,

may lead to a spurious contribution of higher

spatial frequency components through the ~Hðx; yÞmodulation of the centroid integral.

Now, with this theoretical description, we can

easily realize how thresholding may distort the real

value of the centroid in the trial of eliminating the

background influence. So it is clearly shown thewell-known practical awkward situation: ‘‘Thres-

holding? yes, but not too much, we can alter the

measurements’’. To show, how and when is ‘‘too

much thresholding’’ it is devoted the next section

by means of numerical simulation in a significant

set of cases.

3. Computer simulation

The simulation arrangement (see Fig. 1) con-

sists of a collimated optical field (k ¼ 0:633 lm)impinging on a lens with focal length f ¼ 20 cm

and aperture radius R ¼ 2 mm. The observation

plane was placed at the focal plane of the lens.

For doing this, we simulated a circular pupil

centred into a 1024� 1024 sampled rectangular

matrix. The zero padding was chosen so that the

pupil diameter was 193 pixels. For simplicity, we

262 J. Arines, J. Ares / Optics Communications 237 (2004) 257–266

always assumed constant amplitude for the pupil

field, moreover the extension to include the effects

of amplitude fluctuations are straightforward.

The image was computed through the Fast

Fourier transform of the pupil function, and was

supposed to be detected by a noiseless CCDcamera with 6 lm pixel size. The intensity maxi-

mum of each one of the simulated images was set

to 256 to simulate an optimal use of the detection

dynamical range of the camera.

The phase profiles were represented in terms of

Zernike polynomials Zi. We simulated four differ-

ent pupil fields with phases: /T ¼ a2Z2; /D ¼ a4Z4;

/A ¼ a5Z5; /C ¼ a8Z8, (also known as tilt, defocus,astigmatism and 3rd order coma, respectively).

Their corresponding coefficient values were: a2 ¼ 2

rad, a4 ¼ 1 rad, a5 ¼ 0:5 rad and a8 ¼ 1 rad. Ad-

ditionally we also simulated another phase that

was obtained from the sum of the phases men-

tioned above (/Total ¼ /T þ /D þ /A þ /C).

In practical terms for the simulated lens, the

selected value for the tilt coefficient correspondswith a point-like source situated 0.012� off-axis.

Similarly, the selected defocus coefficient corre-

sponds to an axial shift of the image plane equal to

6.7 mm in relation with the lens focal plane (see

Appendix B).

Once we have simulated the focal image, we

obtained the Heaviside function Hðu; vÞ for each of

the tested threshold levels by simply evaluating Eq.(4). Following we inserted the pupil function and

the Fourier Transform of the Heaviside function

in Eq. (10) in order to obtain the centroid value.

The centroid value obtained through this way was

compared with the one obtained by computing the

centroid of the simulated thresholded image

through Eq. (3), in order to test the validity of the

simulations. The fitting achieved by comparingthese centroid values was nearly perfect, finding

differences of the order of 10�11 rad, what shows

the accuracy of the performed simulations.

Fig. 3. Comparison of the evolution of ax versus threshold level

for different phases, /T , /D, and /A.

4. Results

In this section we are going to show the evolu-tion of the centroid value in function of the

threshold level for the different phases mentioned

above. To make easier the interpretation of the

results we represented ax ¼ Xc � R=ð2p=f � kÞ(the average of the phase x-gradient, normalized to

the pupil radius), instead of the centroid value XC.

This option let us to directly compare the value of

the Zernike coefficients with the mean slope of thewavefront avoiding scaling terms (as f , k, As).

Fig. 3 shows the evolution of ax in function of

the threshold level for /T (open triangle), /D

(cross) and /A (open square). It can be observed

that the possible centroid shift induced by thres-

holding is almost negligible (0.3� the pixel detec-

tor size at maximum) in comparison with the non-

thresholded case, drawn as grey solid line in thisfigure. The reason for this centroid independence

on thresholding is that for these kinds of phases,

the generated focal distributions are symmetric in

relation to its centroid without thresholding. Thus,

the Heaviside function and its Fourier Transform

would be also symmetric for any value of the

threshold level. Consequently the centroid value

would be not shifted due to thresholding process-ing. The small oscillating difference between the

thresholded and non-thresholded cases is caused

by the spatial discretization of the focal spot which

can slightly break the theoretical symmetry of the

focal image.

In Fig. 4 we present some ax curves corre-

sponding to comatic phases (/C) generated with

Fig. 4. Evolution of ax for different thresholding levels. Several

magnitudes of Zernike comatic component Z8, are shown.

Fig. 5. Comparison between the aTotal and the total sum

(aTþaAþaDþaC) for different thresholding levels.

J. Arines, J. Ares / Optics Communications 237 (2004) 257–266 263

three different values of the normalized coefficient

a8 ¼ ð1; 0:7; 0:3Þ rad. It can be observed in thisfigure: first the dependence of the centroid shift on

the threshold value, and second that this depen-

dence is the more important the bigger the mag-

nitude of the comatic coefficient is. In the case of

a8 ¼ 1 rad (drawn as open circles) the figure shows

a maximum equivalent shift of 7 detector pixels

from the case without thresholding. For a8 ¼ 0:3(open triangles) the maximum equivalent dis-placement is nearly 1.7 pixels.

In a comatic spot the energy is nonsymetrically

spread out into a comet shaped flare. Indeed al-

most the 55 percent of the energy is concentrated

near the nominal paraxial position since the rest of

the light is smoothly scattered into the last tail of

the comatic spot that extends up to three times

farther than the width of the main lobe. This as-pect is reflected in the curves of Fig. 4 as a steep

fall when the threshold overcomes the energy of

the widest tail. To understand this effect it is im-

portant to note that, for the centroid computation,

there are a lot of tail pixels significantly weighted

by its off-centre position (see Eq. (3)). Conse-

quently, the lack of the tail induces an important

change in the centroid position.In terms of a pupil plane point of view, thres-

holding the tail means that we are avoiding the

phase contribution of most of the off-centre pixels

of the thresholding function H to Eq. (16). By

means of the Fourier displacement theorem, it is

easy to realize that these pixels have significant

phase information in the Fourier space. Following

this logic it is possible to understand how thres-

holding the first tail induces the fast decrease ob-

served in the curves.

Finally as a third result, in Fig. 5 we representaTotal, i.e. the average of the normalized x-gradientof a field with phase /Total, as open circles, and the

sum (aTþaAþaDþaC) as crosses. Conversely to the

expected from Eq. (1) which states that there is a

linear relation between centroid and phase gradi-

ent, we can observe that the centroid generated by

a uniform amplitude field with phase /Total is not

equal to the sum of the centroids related with eachone of its phase components.

5. Discussion

We have presented in this work the analytical

expressions that relates the centroid of a thres-

holded image with the phase of the impinging fieldover the pupil and with the Fourier Transform of

the thresholding function Hðu; vÞ. Besides, we haveshown in this work two main results: the relevancy

of the centroid shift induced by thresholding pro-

cessing for comatic aberrations; and the nonlinear

264 J. Arines, J. Ares / Optics Communications 237 (2004) 257–266

relation between the centroid and the phase gra-

dient in presence of thresholding processing and

comatic aberrations.

The lack of literature related with these results

make us think that most of the centroid

based gradient wavefront sensor users did not payattention to the possible effects induced by thres-

holding processing, considering that this pre-pro-

cessing step do not alter any interesting

magnitude, removing only the undesirable back-

ground or excessively noisy pixels. In this sense,

the ignorance of the effects induced by threshold-

ing makes the user to erroneously estimate the

amount of certain optical aberration; whileknowing the possible effects of thresholding, make

the user be prevented of the possible alteration

of the centroid and thus allowing the avoidance of

the propagation of this instrumental error in the

wavefront estimation. The comprehension of the

effects of thresholding is the first step to develop

tools to correct the inaccuracies accompanied by

its use.In relation to this, we show now the implica-

tions of the results presented in this paper for the

particular case of Shack–Hartmann wavefront

sensors (extension to other centroid based gradient

wavefront sensors can be easily done).

The Shack–Hartmann wavefront sensor obtains

the phase gradient over each microlens by sub-

traction of two centroids, one obtained from theunknown phase and the other from the reference

phase.

This differential way of computing the phase

gradient brought the belief that the static aberra-

tions of the microlenses are not important, in-

ducing no error in the wavefront estimation due to

the cancellation of their effects during the sub-

traction process. This is nearly true when nothresholding is applied, but when thresholding is

used, the presence of comatic aberrations in the

microlenses can shift the centroids if different

threshold levels were selected for processing the

reference and the unknown phase. In the particu-

lar case that the unknown incoming phase could

be considered to be a plane wave over each co-

matic microlens, the use of the same threshold le-vel (or a very close one) could be an acceptable

solution (when computing the centroids of the

unknown and reference phases) in order to mini-

mize this gradient shift during the subtraction

process.

However, it is very noticeable to remark that the

total elimination of the gradient shift would be un-

achievable due to the nonlinearity induced bythresholding processing, when for instance, the

unknown wavefront can not be considered plane

over each microlens. Traditionally the phase of the

field that impinges over each microlens is assumed

to beplane. This assumption can be nearly correct in

certain situations where the impinging field present

smooth aberrations. Nevertheless centroid based

wavefront sensors are extensively used nowadays insituations were the wavefront aberrations are so

high that cannot be considered plane over each

microlens. This would be the case of a Shack–

Hartmann sensor used for ocular aberrometry.

On the other hand, a spherical microlens (ap-

erture radius¼ 400 lm and focal length ¼ 7 mm)

will suffer a comatic aberration equal to a8 � 1 rad

when imaging a point-like object situated 10� off-axis (see Appendix C). This can be the case for

some of the microlenses of a SH positioning sensor

[28] measuring the position of a non cooperative

point-like luminous object.

Furthermore, the linear phase estimation (so

frequent in a SH) is also affected by this lack of

linearity between phase and centroids. Now we

know that this assumption can fail in presence ofthresholding processing (as was shown in Fig. 5).

This result, joined to the one described above,

propose the advisability, if possible, of avoiding

the comatic aberrations of each of the microlenses

that compound the SH, or alternately the necessity

of developing more suitable optimal wavefront

estimators in the sense that they have information

of the threshold level that was used for the cen-troid computation. So thus, the linear estimators

would take into account the possible centroid shift

and the lost of linearity in the relation between the

centroid and the phase gradient.

Acknowledgements

This work has been supported by the Ministerio

de Ciencia y Tecnolog�ia, Plan Nacional de Inves-

J. Arines, J. Ares / Optics Communications 237 (2004) 257–266 265

tigaci�on Cient�ıfica, Desarrollo e Innovaci�on Tec-

nol�ogica, DPI2002-04370-C02 and FEDER.

Appendix A. Some Fourier Transform useful math-

ematical relations

In this section we present the definition of some

variables and some properties that were used

through the text.

Definition of the pupil function:

P ðx; yÞ ¼ Cðx; yÞ expfi/ðx; yÞg: ðA:1ÞDefinition of the Fourier Transform

Kðu; vÞ ¼ F fP ðx; yÞg

¼ZuvPðx; yÞ exp

�� i

2pkd

ðuxþ vyÞ�dxdy:

ðA:2Þ

Shifting property of the Fourier Transform

F �1fKðu� n; v� gÞg

¼ P ðx; yÞ exp i2pkd

ðnx�

þ gyÞ�: ðA:3Þ

Property of the derivative of the Fourier Trans-

form

rxðP ðx; yÞ � ~Hðx; yÞÞ

¼ F �1

�� i

2pkd

uKðu; vÞHðu; vÞ�: ðA:4Þ

Derivative of a convolution

rxðf ðx; yÞ � gðx; yÞÞ¼ rxðf ðx; yÞÞ � gðx; yÞ ¼ f ðx; yÞ � rxðgðx; yÞÞ:

ðA:5Þ

Property of convolution

f ðx; yÞðgðx; yÞ � hðx; yÞÞ¼ gðx; yÞðf ðx; yÞ � hðx; yÞÞ¼ hðx; yÞðf ðx; yÞ � gðx; yÞÞ: ðA:6Þ

Derivative of a complex function

rxP ðx; yÞ ¼ rxðCðx; yÞÞ expfi/ðx; yÞgþ iCðx; yÞ expfi/ðx; yÞgrxð/ðx; yÞÞ:

ðA:7Þ

Appendix B. Zernike coefficients versus image shifts

The transversal shift w of the focal spot (in

decimal degrees) can be calculated in function of

the Zernike tilt-coefficient (a2) through the nextexpression [28]:

w ¼ 2a2R

k2p

180

p

� �; ðB:1Þ

where R is the pupil radius, and k is the wave-

length.

Likewise, considering that the axial image po-

sition (z0) can be obtained in the paraxial domain

by [28]

z0 ¼ 2pk

R2

4ffiffiffi3

pa04

: ðB:2Þ

We can relate the axial image shift (Dz) with the

well-known defocus aberration coefficient Da4 ¼a04 � a4:

Dz ¼ z0 � z ¼ R2z

4ffiffiffi3

pDa4z k

2p

� þ R2

� z; ðB:3Þ

where, a04 and a4 are respectively, the Zernike co-

efficients of the image and reference wavefronts

and z is the reference axial image position. For

instance, the condition z ¼ f provides the fo-

cal shift induced by certain amount of defocusaberration Da4.

Appendix C. From coma seidel to zernike

Off-axis illumination induces comatic aberra-

tion in an ideally thin spherical lens. An approach

to the magnitude of this aberration is provided bythe primary Seidel Coma coefficient SII:

SII ¼R3U2 �U

4

nþ 1

nðn� 1ÞB

þ 2nþ 1

nC�; ðC:1Þ

where R is the radius of the pupil, n the refractive

index of the lens, U the power of the lens equal to

U ¼ ðn� noÞðC1 � C2Þ, no the refractive index of

the material surrounding the lens, C1 and C2 arerespectively the curvatures of the first and second

surfaces of the lens, B the shape factor B ¼ðC1 þ C2Þ=ðC1 � C2Þ, C ¼ ðU1 þ U 0

2Þ=ðU1 � U 02Þ,

266 J. Arines, J. Ares / Optics Communications 237 (2004) 257–266

where U1 and U 02 are respectively the angles of the

object and image marginal rays, and �U is the angle

of the object field. For more details on the ex-

pression see [29].

For small field angles, the primary Seidel co-matic coefficient is approximately related with the

Zernike coefficient by SII � 3a8 [29,30]. The valid-

ity of this relation was successfully tested by sim-

ulation for the situations considered in the text by

means of the commercial software OSLO� LT

EDITION Rev. 6.1.

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