Estimation of the wavefront of a spatially coherentbeam is important in a number of tasks, such as mea-suring flatness in the semiconductor and computerindustries, measuring the amount of aberrations in-troduced by optical elements and by the human eye,analyzing the quality of a laser beam, and measuringthe atmospheric turbulence in astronomy. One of themost extensively used wavefront sensor in such tasksis the Shack–Hartmann wavefront sensor [1]. Herethe beam under investigation is incident on a two-dimensional (2D) array of tiny lenses known as lens-lets. Each lenslet focuses a small portion of the beampassing through it onto the plane of the photodetec-tor array. In the case of a plane wavefront parallel tothe plane of the lenslet array, the focal spots corre-sponding to all the lenslets describe a 2D arrayhaving the same dimensions as the lenslet array.The area of the photodetector array can be equally
divided into a number of detector subapertures, suchthat each focal spot lies within the respective detec-tor subaperture. When the test wavefront is incidenton the lenslet array, the focal spots in the detectorplane get displaced depending on the amount oflocal slope through the corresponding lenslet. Bymeasuring the shifts of the focal spot centroids rela-tive to the original positions, the test wavefront canbe estimated.
Accuracy and speed are two important parametersof the Shack–Hartmann-type wavefront sensor, andalready a number of efforts have been put to improvethese two parameters. Accuracy of the measuredwavefront depends on the amount of cross talk be-tween two adjacent detector subapertures. Suchcross talk can be minimized by using a sparse arrayof lenslets in conjunction with a movable mask [2] orby sequentially projecting small portions of the wave-front [3]. The accuracy and dynamic range of theestimated wavefront can also be improved by incor-porating a phase precorrecting element prior to thelenslet array [4]. However, such methods are notsuitable for real-time operations, or they need prior
information of the test wavefront. Most of the work toincrease the speed of a zonal wavefront sensor uti-lizes the advantage of the complementary metal-oxide semiconductor (CMOS)-based photodetectortechnology [5,6].
It is noticed that in the case of the Shack–Hartmann wavefront sensing scheme, the requirednumber rows of the detector array is a multiple ofthe number of rows of the lenslet array. This putsa limit on the achievable frame rate of the photode-tector device, because for both CCD and CMOSimage sensors, a smaller number of rows in the de-tector array can facilitate a higher number of imageframes per second [7–9]. The amount of cross talk be-tween adjacent subapertures can be decreased by in-creasing the separation between two nearby focalspots. However, this will lead to an increase in therequired number of rows of the photodetector array,resulting in a decrease in the frame rate. Fortu-nately, there is another way to increase the framerate and to reduce the cross talk between two nearbysubapertures, that is, by reducing the number ofrows in the focal spot array.
It has been shown that an arrangement of diffrac-tion gratings can be used to measure aberrations in amicroscope setup [10]. Further, a 2D array of grat-ings arranged in a geometry similar to the lensletarray, followed by a focusing lens, can measure thewavefront profile of an incident beam [11]. In this pa-per, we describe how the array of gratings can be con-figured to generate an array of focal spots in thedetector plane with a number of rows that can beas small as one. We have implemented the proposedwavefront sensing scheme using a liquid-crystal spa-tial light modulator (LCSLM) assembly. We presenthere proof-of-concept experimental results that showthat the test wavefront can be measured with areduced number rows of the focal spot array in a sim-ilar way as the 2D focal spot array in the case of theShack–Hartmann wavefront sensor.
2. Theoretical Considerations
We consider a sinusoidal amplitude grating describedusing theCartesian coordinates ðξ; ηÞ. The gratinghasa circular aperture of radiusw. Assuming the gratingrulings to be along the ξ axis, the transmittance func-tion of the grating can be written as
tgðξ; ηÞ ¼�12þ 12sinð2πνξÞ
�circ
�ρw
�; ð1Þ
where ν is the frequency of the grating, ρ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiξ2 þ η2
p,
and
circ�ρw
�¼
�1 if ρ
w ≤ 10 otherwise : ð2Þ
If the amplitude grating is kept at a distance f behindadiffraction-limited lens,where f is the focal length ofthe lens and the grating is illuminated by a plane
monochromatic wave of amplitude A, then the fieldin the focal plane in front of the lens is given by [12]
Uðu; vÞ ¼ eikf
iλf
Z∞
−∞
Z∞
−∞
tgðξ; ηÞe−i2πξuþηvλf dξdη: ð3Þ
It can be seen that the double integration above isnothing but a 2D Fourier transform of tgðξ; ηÞ ex-pressed using spatial frequency coordinates f x ¼ u
λfand f y ¼ v
λf . For two real-valued 2D functions g andh, one can write using the correlation theorem,F½gh� ¼ F½g� ⊗ F½h�, where F½� � �� is the 2D Fouriertransform operation and ⊗ denotes correlationoperation defined as
fg ⊗ hgðx; yÞ ¼Z
∞
−∞
Z∞
−∞
hðx − ξ; y − ηÞgðξ; ηÞdξdη: ð4Þ
Thus, Eq. (3) can be written as
Uðu; vÞ ¼ eikf
iλf F�12þ 12sinð2πνξÞ
�⊗ F
�circ
�ρw
��:
ð5Þ
Writing sin x ¼ 12i ðeix − e−ixÞ and δðf x; f yÞ ¼R
∞−∞
R∞−∞ e−2πiðξf xþηf yÞdξdη, where δð� � �Þ is the Dirac-
delta function:
F
�12þ 12sinð2πνξÞ
�¼ 1
2
�δðf x; f yÞ þ
12i
fδðf x − ν; f yÞ
− δðf x þ ν; f yÞg�: ð6Þ
Also
F
�circ
�ρw
��¼ SArðf x; f yÞ; ð7Þ
where
Arðf x; f yÞ ¼J1ð2πw
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2x þ f 2y
qÞ
πwffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2x þ f 2y
q ;
J1ð� � �Þ being the Bessel function of the first kind, or-der 1 and S is the area of the circular aperture of thelens. FromEqs. (6) and (7) and using the properties ofDirac-delta function
Uðf x; f yÞ ¼Seikf
i2λf
�Arðf x; f yÞ
þ 12i
fArðf x − ν; f yÞ − Arðf x þ ν; f yÞg�: ð8Þ
The first term on the right of Eq. (8) corresponds tothe undeviated or 0 order beam, while the second andthird terms correspond to the þ1 and −1 orderbeams. It is seen that the locations of the �1 orderfocal spots with respect to the 0 order focal spot de-pend on the value of the frequency ν. Choosing ν such
that there is minimum overlap between the threeterms, one may write the expression of intensity inthe plane ðu; vÞ as
Iðf x; f yÞ ¼S2
4λ2f 2�A2r ðf x; f yÞ
þ 14fA2
r ðf x − ν; f yÞ þ A2r ðf x þ ν; f yÞg
�: ð9Þ
Considering the general case when the grating rul-ings make an angle with the ξ axis, such thatν2 ¼ ν2x þ ν2y , where νx;y are the spatial frequenciesof the grating along the ξ and η axes, using the aboveresults, one can write
I ¼ S2
4λ2f 2�A2r ðf x; f yÞ þ
14fA2
r ðf x − νx; f y − νyÞ
þ A2r ðf x þ νx; f y þ νyÞg
�: ð10Þ
If it is now assumed that the incident wavefront hasa slope relative to the plane of the grating such thatthe incident field can be written as Aei2πðmxξþmyηÞ,Eq. (10) becomes
I ¼ S2
4λ2f 2�A2r ðf x −mx; f y −myÞ
þ 14fA2
r ðf x − νx −mx; f y − νy −myÞ
þ A2r ðf x þ νx −mx; f y þ νy −myÞg
�: ð11Þ
It is evident from Eqs. (10) and (11) that the focalspot shift of one of the orders, say the þ1 order, inthe case of a tilted incident wave, relative to the focalspot position in the case of zero slope plane incidentwave (referred to as the reference beam), is a functionof the horizontal and vertical slopes of the tiltedwave. However, realizing a perfect sinusoidal ampli-tude grating is more difficult than a perfect squarewave amplitude grating with the same periodicity.Such a square wave amplitude grating can easilybe implemented with an LCSLM. We now considera square wave amplitude grating of frequency ν withthe rulings along the ξ axis. If the amplitude justafter the gratings varies from 0 to 1, then the trans-mittance function of the grating can be written usingFourier series analysis as
tgðξ; ηÞ ¼12þXn
2nπ sinðn2πνξÞ; ð12Þ
where the summation is over all the odd integers n.Thus, if the transmittance function of the sinusoidalamplitude grating in Eq. (1) is replaced by the ex-pression in Eq. (12), then one can obtain an expres-sion similar to Eq. (9) as
Iðf x; f yÞ ¼S2
4λ2f 2�A2r ðf x; f yÞ
þXn
4
π2n2 fA2r ðf x � nν; f yÞg
�: ð13Þ
Therefore, in addition to the 0 order there exists�1; � 3; � 5; � � � orders at locations �ν; � 3ν; �5ν; � � � along the f x axis. It is to be noted that inten-sity of the higher order focal spots decreases as 1=n2,where n is the order of a given focal spot, such thatthe intensity of the �1 orders is much larger relativeto the higher orders. Considering the square wavegrating to be arbitrarily oriented and the incidentwavefront to have an arbitrary slope, one may obtainthe expression of intensity in the (u; v) plane as
I ¼ S2
4λ2f 2�A2
r ðf x −mx; f y −myÞ
þXn
4
π2n2 fA2r ðf x � nνx −mx; f y � nνy −myÞg
�:
ð14Þ
Thus, like the sinusoidal amplitude grating, the focalspot shift corresponding to theþ1 order, in the case ofthe square wave amplitude grating, is also a functionof the slope of the incident wavefront. We now consid-er a 2D array of such square wave gratings, each witha given spatial frequency. For various gratings, νxincreases uniformly along a row and νy increases uni-formly along a column so that Δνx and Δνy are theincrements of νx and νy between adjacent gratings.With appropriate choices of Δνx;y and initial valuesof νx;y, the þ1 order focal spots due to all the gratingscan be made to describe a 2D array similar to the fo-cal spot array in the case of the Shack–Hartmannwavefront sensor. Thus measuring the shifts of thefocal spot centroids corresponding to the þ1 orderspots, the incident wavefront profile can be esti-mated using known methods [13]. Furthermore, itis noticed that so long as the square wave gratingis correctly described and the lens aperture is largeenough to transmit the diffraction order, the focalspot shift for a given grating is independent of thespatial frequency (νx; νy) of the of grating. Therefore,it is possible to specify the spatial frequencies of thegratings such that þ1 order focal spots describe anarray of smaller number of rows compared to thenumber of rows in the grating array. The incidentwavefront can be estimated by measuring the focalspot shifts in the focal spot array in the same manneras the focal spot array with same number of rows asthe grating array. We note here that the light effi-ciency in the þ1 order for a binary amplitude gratingis given by 1=π2 ð¼ 10:1%Þ. However, the proposedscheme can also be applied to an array of binaryphase gratings (the incident beam is phase modu-lated with a phase delay of 0 or π), in which case
the diffraction efficiency in the þ1 order increasesto 4=π2 ð¼ 40:5%Þ.3. Design of the Array of Diffraction Gratings
We consider a 2D array of square wave amplitudegratings having the dimensions N ×N. Let ν0x andν0y be the spatial frequencies of the grating element(1; 1). We assume that Ny is the number of rowsand Nxi is the number of columns of the ith row, cor-responding to the þ1 order focal spot array to be pro-duced, such that
PNyi¼1 Nxi ¼ N2. The required spatial
frequencies of the gratings element ði; jÞ is given by
νx ¼ ν0x þ ðRemfðjþ ði − 1ÞN − 1Þ;Sijg − S0ijÞΔνx
νy ¼ ν0y þ ðim − 1ÞΔνy; ð15Þ
where Sij ¼Pim
i0¼1 Nxi0 and S0ij ¼
Pim−1i0¼1 Nxi0 such that
im is the smallest integer satisfying the conditionSij ≥
ðjþ ði − 1ÞNÞ and Remfx; yg denotes the integer re-mainder of x after division by y. The values of ν0x,ν0y,Δνx, andΔνy are chosen such that for the gratingelement (N;N), ðνx; νyÞ ≤ ð0:5 lines per pixelÞ. How-ever, these maximum values of (νx; νy) are achievableonly if the lens diameter is large enough to transmitthe corresponding þ1 diffraction order.
Figures 1(a)–1(c) show patterns representing thetransmittance functions of the array of binary dif-fraction gratings of dimension 4 × 4 to produce þ1 or-der focal spot array of dimensions 4 × 4, 2 × 8, and1 × 16, respectively. Each grating is defined over acircular area, with the dark portion indicatingtransmittance ¼ 0 and the white portion represent-ing transmittance ¼ 1. The þ1 order focal spotsobtained by performing the Fourier transform opera-tion numerically over the transmittance functionsof the array of gratings, similar to those inFigs. 1(a)–1(c), are seen in Figs. 1(d)–1(f).
4. Estimation of the Wavefront
The wavefront of the beam under observation can beestimated using both zonal as well as modal estima-
tion methods [13]. However, in this work we estimatethe wavefront in terms of a number of Zernikemodes.Considering a grating array of dimension N ×N, theeffective area of the photodetector array can be di-vided into N2 detector subapertures of equal areas.Because the position of each focal spot is a functionof the spatial frequencies of the respective gratingrulings, hence, irrespective of the number of rowsin the focal spot array, each focal spot can be mappedto its respective grating. Let us assume that ϕoðξ; ηÞis the phase profile of the incident beam. The coordi-nates (ξ; η) are normalized in such a way that at theperiphery of the largest circle described inside thegrating array,
file to be a linear combination of M Zernike modes,
ϕðξ; ηÞ ¼XMk¼1
akZkðξ; ηÞ; ð16Þ
where ak is the coefficient of the single-index Zernikecircular polynomial Zkðξ; ηÞ [14]. For each grating(i; j) the corresponding subaperture is described bya function gijðξ; ηÞ. Let A be a matrix of dimension2N2 ×M, where the first N2 rows are ∂Zk=∂ξ andthe next N2 rows are ∂Zk=∂η, at the geometricalcenters of gijðξ; ηÞ. The M columns correspond to thederivatives of the M different Zernike modes. The lo-cal slope of the wavefront portion through each grat-ing subaperture can be obtained from the shift of thefocal spot centroid relative to the ideal centroid posi-tion. If T is a columnmatrix of 2N2 elements contain-ing the horizontal slopes in the first N2 and verticalslopes in the next N2 elements, then we may write
a ¼ ðA†AÞ−1A†T: ð17Þ
Here, a is a column matrix of M elements containingthe coefficients of the M Zernike modes, † representsthe matrix transpose, and −1 represents the matrixinversion.
Fig. 1. (a) Patterns representing the transmittance functions of the grating arrays of the dimensions 4 × 4 to produce aþ1 order focal spotarray comprising four rows, (b) two rows, and (c) one row, while (d)–(f) are the corresponding þ1 order spots obtained numerically.
We have implemented the array of square wave grat-ings using an LCSLM assembly. The LCSLM panel isa 2D array of liquid-crystal (LC) cells. The lighttransmittance properties of each cell can be alteredon application of an electric field. By using a compu-ter interface, it is possible to send an image or a pat-tern to the LCSLM panel such that different pixelvalues in the image correspond to LC cells withdifferent transmittance values and reconfigure thepanel at least at the video rate. Thus, with the appro-priate pattern displayed onto the panel, the LCSLMacts as an array of binary holograms to the incidentlaser beam. We employ a computer-generated holo-graphy technique [15,16] to compute the binary am-plitude holograms. Each binary hologram can becomputed in such a way that the array of þ1 orderfocal spots corresponds to an incident wavefront withan additional user defined phase profile ϕ. Whenϕ ¼ 0, no phase profile is added to the incident wa-vefront and the array of binary holograms becomesan array of square wave gratings as seen in Fig. 1.
A schematic of the basic experimental arrange-ment is shown in Fig. 2. A 2D array of binary ampli-tude holograms of the dimensions 4 × 4 is generatedby a computer program and displayed on the LCSLMpanel over an area of 384 × 384 LC cells. Therefore,each subaperture in the array has a diameter of 96LC cells. With the availability of a greater number ofLCSLM pixels, one can write such an array of aneven larger dimension. The laser beam whose wave-front is to be measured is incident on the LCSLMpanel kept in plane G at a distance f behind lensL, where f is the focal length of the lens. One thusgets, in plane D, the Fourier transform of the ampli-tude profile in G. In Fig. 2 we show the focusing ofonly the 0 and þ1 orders. The CCD camera to mea-sure the centroid positions of the þ1 order spots iskept in plane D, and the image grabbed is sent tothe computer, controlling the LCSLM. The computerprogram also can send only a single large hologram(described over 384 × 384 pixels) instead of an arrayof 4 × 4 holograms, so that the detector plane con-tains only one þ1 order. For both cases (i.e., singlehologram and 4 × 4 holograms in the panel), the com-puter program can write holograms on the LCSLM toincorporate a linear combination of eight single-
index Zernike modes, ϕ ¼ a4Z4 þ a5Z5 þ a6Z6 þa7Z7 þ a8Z8 þ a9Z9 þ a10Z10 þ a11Z11, into the inci-dent beam such that the �1 orders have the phaseprofile �ϕ, the �3 orders have the phase profile�3ϕ, and so on. Using the data sent by the CCD,the computer program measures the incident wave-front in terms of the coefficients of these same eightZernike modes.
6. Results and Discussion
In order to demonstrate that the setup can correctlymeasure the incident wavefront, we have holographi-cally added the phase profile ϕaðξ; ηÞ ¼ −1 × Z4ðξ; ηÞþ1 × Z6ðξ; ηÞ þ 0:5 × Z7ðξ; ηÞ − 0:4 × Z9ðξ; ηÞ to the refer-ence beam. The computer program is used to producethe þ1 order focal spot array of dimensions 4 × 4,2 × 8, and 1 × 16. For each dimension of the focal spotarray, the þ1 order focal spots are recorded, firstwhen the þ1 order carries the phase informationof the reference beam, and then when the referencebeam is aberrated with ϕa. From the shift of the þ1order focal spot centroids in the second case relativeto the first case, the computer program estimates thecoefficients (i.e., ajm) of the eight Zernike modes.
The measured coefficients of the Zernike modes forall three dimensions of the focal spot array are shownin the Table 1. Figures 3 (i) (a)–(c) show the initialþ1order spots (corresponding to the reference beam) forthe three dimensions of the focal spot array, whileFigs. 3 (ii) (a)–(c) show the same after the aberrationϕa has been incorporated. Using the measured coeffi-cients, the program then computes the correctionphase, ϕc ¼
Pj − ajmZj. The correction phase is then
incorporated to the aberrated reference beambywrit-ing holograms using ϕ ¼ ϕa þ ϕc. Figures 3 (iii)(a)–(c) show the focal spot arrays for the three dimen-sions after the þ1 order has been corrected. For allthree states of aberrations (i.e. unaberrated, aber-rated, and corrected) the program also switches to asingle hologram to record the focal spot of the singleþ1 order, shown in the insets near the respectivefigures of the þ1 order arrays. Figure 4(a) showsthe three-dimensional plots of the phase profile ϕa,and Figs. 4(b)–4(d) show the phase profilesϕm ¼ −ϕc, measured using the focal spot array of di-mensions 4 × 4, 2 × 8, and 1 × 16, respectively. Thevariances in the measured phase profiles relative tothe actual phase profile, defined as hjϕmðξ; ηÞ
Table 1. Coefficients of the Zernike Modes Applied aja andMeasured ajm Using Focal Spot Arrays Comprising Four
−ϕaðξ; ηÞj2i, are found to be 0.014, 0.035, and 0.011 forthe focal spot array dimensions 4 × 4, 2 × 8, and1 × 16, respectively. One of the possible causes forthe minimum error in the case of the single focal spotrow, can be the fact that in the case of a single row theprobability of cross talks between nearby detectorsubapertures is minimum. The number of effectiveCCD rows has decreased from 64 × 4, in the case of fo-cal spot array dimension 4 × 4, to 64 × 1, in the case offocal spot array dimension 1 × 16. As mentioned ear-lier, the reduced number of rows in the photodetectorarray can lead to a faster frame rate of the wavefrontsensor. Moreover, as the number of effective rows inthe CCD array decreases, the maximum duration oftime, after exposure and before the charge in aCCD pixel is read, also decreases. This will lead toa reduced amount of thermal noise associated witha pixel (especially in the row to be read at the end),thus, contributing to more precise determination ofthe focal spot centroids.
7. Conclusion
We have introduced a zonal wavefront sensingscheme that requires a reduced number of rows ofthe detector array. The wavefront sensor comprisesa 2D array of square wave amplitude gratings in con-junction with a focusing lens. The wavefront to be
measured is incident on the grating array, and theresulting þ1 orders are focused onto the photodetec-tor array. The wavefront is estimated by measuringthe shifts of the þ1 order spots. We have shown howthe spatial frequencies of the square wave gratingscan be specified so that theþ1 order focal spot arrayscan have a smaller number of rows relative to thenumber of rows of the grating array. Experimentalresults show that measurement precision by thewavefront sensor with only one focal spot row is com-parable to the same by the wavefront sensor with agreater number of focal spot rows. The effective num-ber of rows of the detector array can be decreased upto a factor of 1=N in the case of a single row of thefocal spot array compared to an array of dimensionN ×N. This will primarily lead to sensing of thewavefront at a faster rate using readily availableCCD or CMOS image sensors. The proposed scheme,without the wavefront correcting part, can also beimplemented with a fixed phase plate without theneed of the spatial light modulator.
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Fig. 3. (Color online) (a) þ1 order focal spot arrays of dimensions4 × 4, (b) 2 × 8 and (c) 1 × 16, (i) corresponding to the referencebeam, (ii) after adding aberrations to the reference beam, and(iii) after aberration correction. The inset adjacent to each figureis the corresponding single hologram þ1 order focal spot.
Fig. 4. (Color online) (a) Three-dimensional plots of the wave-front added and (b) measured wavefronts usingþ1 order focal spotarrays of dimensions 4 × 4, (c) 2 × 8, and (d) 1 × 16. The color barrepresents the phase values in radians.
