Modal wavefront reconstruction for radial shearinginterferometer with lateral shear

Naiting Gu,1,2,3,* Linhai Huang,1,2 Zeping Yang,1,2 Qun Luo,1,2,4 and Changhui Rao1,2

1Institute of Optics and Electronics, Chinese Academy of Sciences, P.O. Box 350, Chengdu 610209, China2The Key Laboratory on Adaptive Optics, Chinese Academy of Sciences, Chengdu 610209, China

3Graduate University of Chinese Academy of Sciences, Beijing 100039, China4College of Opto-Electronics Science and Engineering, National University of Defense Technology, Changsha 410073, China

*Corresponding author: gnt7328@163.com

Received July 18, 2011; revised August 22, 2011; accepted August 24, 2011;posted August 24, 2011 (Doc. ID 151022); published September 15, 2011

In a radial shearing interferometer, a portion of the test beam is magnified and used as the reference for the testedwavefront. However, the reference portion is always off center (lateral shear), which complicates the wavefrontreconstruction. A modal method for solving this problem is presented here. This method uses orthogonal Zernikepolynomials and its matrix formalism to calculate the Zernike coefficient of the wavefront under test. This approachhas easier implementation, is better filtering, and has a more adaptive practical situation. The corresponding math-ematical formula is deduced, and a computer simulation is also made to verify operation of the algorithm. The resultof simulation analysis shows that the proposedmethod is correct and accurate. © 2011 Optical Society of AmericaOCIS codes: 010.1080, 040.1880, 110.1650.

In radial shearing interferometry (RSI), no extra refer-ence wavefront is needed, and the radial shearing inter-ference pattern is directly related to the wavefrontdifference between the two versions of the wavefront un-der test rather than the wavefront under test itself [1].Meanwhile, a common-path configuration can be de-signed in RSI, which gives RSI immunity to environmen-tal disturbances and mechanical vibrations [2]. Becauseof these advantages, such as self-referencing interferenceand antivibration, RSI has been commonly used in opticaltesting [3], corneal topographic inspection [4], wavefrontsensing for adaptive optics [5], and high-power laserbeam characterization [6]. According to the RSI princi-ple, the measured interferometric phases (wavefront dif-ference, hereafter) by pattern analysis algorithm onlyrepresent the difference between these two versions ofthe wavefront under test. So the relationship between thewavefront under test and its radial shearing interfero-gram is indirectly embedded. A variety of approacheshave been proposed to retrieve the wavefront from thewavefront difference in a radial shearing interferometer[7–9]. But they do not consider the influence of lateralshear (i.e., the misalignment between the two versionsof wavefront under test) in RSI. Unfortunately, lateralshear is inevitable in a practical RSI system, and the in-fluence of lateral shear cannot be neglected. Someauthors [10,11] have noticed this problem, and two itera-tive reconstruction methods have been proposed to cor-rect it. They give proper mathematical justification andperform fairly well in the sense of reconstruction preci-sion. In this Letter, we describe a different method, novelto our knowledge, to reconstruct the wavefront undertest in a radial shearing interferometer, which has a cer-tain amount of lateral shear in two orthogonal directions.This method, according our investigation, is the onlymodal reconstruction method to solve the problemcaused by lateral shear in RSI. Compared with thepredecessors’method, the proposed modal wavefront re-construction method reduces effectively the noise accu-mulation and has good error propagation properties. The

proposed reconstruction method is based on Zernikepolynomials and its matrix formalism, and it should leadto an easier implementation in some practical situations.

In this Letter, we use φ1ðx; yÞ and φ2ðx; yÞ to denote thecontracted and the expanded version of the wavefrontunder test, and s is the radial shear ratio for a certainradial shearing interferometer. In the common areabetween φ1ðx; yÞ and φ2ðx; yÞ, i.e., interference area asshown in Fig. 1(a), the wavefront difference can be de-scribed as

Δφðx; yÞ ¼ φ1ðx; yÞ − φ2ðx; yÞ: ð1Þ

The wavefront aberration φ1ðx; yÞ is a contractedversion of the wavefront under test φ0ðx; yÞ, and thereno information is lost. So the wavefront aberrationφ1ðx; yÞ is equivalent with the wavefront under testφ0ðx; yÞ if we normalize the diameter of the origin pupiland the contracted pupil. In other words, we can consid-er φ1ðx; yÞ to be as the wavefront under test φ0ðx; yÞ.Similarly, in the magnified pupil the expanded wavefrontaberration φ2ðx; yÞ can also be considered to be as thewavefront under test φ0ðx; yÞ. For simplicity, we definethe interference area as circleðdÞ, and in this area two

x

y

O

O’ x

y

O(O’)

(a) (b)

Fig. 1. Schematic diagram of the radial shearing interferogramwithout and with lateral shear. (a) Shearogram without lateralshear. (b) Shearogram with lateral shear.

September 15, 2011 / Vol. 36, No. 18 / OPTICS LETTERS 3693

0146-9592/11/183693-03$15.00/0 © 2011 Optical Society of America

versions of wavefront under test can be expressed as�φ1ðx; yÞ ¼ φ0ðx; yÞ;

φ2ðx; yÞ ¼ φ0ðx=s2; y=s2Þ ; ðx; yÞ ∈ circleðdÞ: ð2Þ

Equation (2) is under an ideal alignment condition asshown in Fig. 1(a), and two versions of the wavefront un-der test are aligned ideally. However, in a practical RSIsystem, lateral shear is inevitable. As shown in Fig. 1(b),the smaller circle denotes the contracted beam, and italso is the interference area, i.e., circleðdÞ. For conveni-ence, we define the center of the contracted beam as theorigin O of the Cartesian coordinate system, and the cen-ter position O0 of the expanded beam expresses theamount of lateral shear. Two variables, x0 and y0, are de-fined as the lateral shear at x direction and y direction,respectively, and they can be determined by a special ca-libration process (when a flat glass plate with a þ signcan be inserted at the entrance pupil, the offset betweentwo þ signs at the exit pupil along two directions can bemeasured accurately, and they are lateral shear x0 andy0, respectively). Under this condition, two versions ofthe wavefront under test can be rewritten as

�φ1ðx; yÞ ¼ φ0ðx; yÞ;

φ2ðx; yÞ ¼ φ0ðx=s2 − x0; y=s2 − y0Þ : ð3Þ

Here we suppose that the φ0ðx; yÞ can be described byan N -limited number of Zernike polynomials, and it canbe expressed as

φ0ðx; yÞ ¼XNk¼1

ak · Zkðx; yÞ; ð4Þ

where Zkðx; yÞ is the kth order Zernike polynomial and akis the corresponding weighting coefficient.Correspondingly, the expanded wavefront φ2ðx; yÞ in

circleðdÞ is given by

φ2ðx; yÞ ¼XNk¼1

ak · Pkðx; yÞ; ð5Þ

where Pkðx; yÞ is a portion of Zernike polynomialsZDk ðx; yÞ in the expanded beam area [denoted as

circleðDÞ], i.e., Pkðx; yÞ ¼ ZDk ðx; yÞ; ðx; yÞ ∈ circleðdÞ.

According to the [12], provided an N -limited Zernikedescription of a wavefront on a large pupil, any circularportion inside it can be described by another Zernike en-semble, limited to the same N . So the function Pkðx; yÞcan be written as

Pkðx; yÞ ¼Xkj¼1

bkj · Zjðx; yÞ: ð6Þ

Considering Eqs. (4)–(6), Eq. (1) is rewritten as

Δφðx; yÞ ¼XNk¼1

akZkðx; yÞ −XNk¼1

ak

�Xkj¼1

bkj Zjðx; yÞ�: ð7Þ

From Eq. (7), the relationship between the wavefrontdifference and the Zernike polynomials is expressed, butit is indirect and not clear. Actually, Eq. (7) can be rewrit-ten in its matrix form, and it is expressed as

Δφ¼½a1;a2;���;ak; ���aN �

2666666664

1 0 ��� 0 ��� 00 1 ��� 0 ��� 0... ... . ..

0 . ..

00 0 ��� 1 ��� 0... ... . .. ..

. . ..

00 0 ��� 0 ��� 1

3777777775

2666666664

Z1

Z2

..

.

Zk

..

.

ZN

3777777775

− ½a1;a2;���;ak;���aN �

26666666664

b11 0 ��� 0 ��� 0b21 b22 ��� 0 ��� 0

..

. ... . .

.0 . .

.0

bk1 bk2 ��� bkk ��� 0

..

. ... . .

. ... . .

.0

bN1 bN2 ��� bNk ��� bNN

3777777775

26666666664

Z1

Z2

..

.

Zk

..

.

ZN

37777777775;

ð8Þwhere Δφ and Zk are the compact form of matrixformalism of Δφðx; yÞ and Zkðx; yÞ, respectively.

Finally, Eq. (8) is merged as

Δφ ¼ ABZ; ð9Þ

where A ¼ fa1; a2;…; aNg is a coefficient vector of thewavefront under test; Z ¼ fZ1; Z2;…; ZNgT is a columnvector, which is composed of a set of Zernike polyno-mials; B is a coefficient matrix, and it is

B ¼

26666666664

1 − b11 0 � � � 0 � � � 0−b21 1 − b22 � � � 0 � � � 0

..

. ... . .

. ... . .

. ...

−bk1 −bk2 � � � 1 − bkk � � � 0

..

. ... . .

. ... . .

. ...

−bN1 −bN2 � � � −bNk � � � 1 − bNN

37777777775: ð10Þ

On the other hand, the wavefront difference Δφðx; yÞcan be decomposed into a linear combination of theN -limited Zernike polynomials, and it is shown by

Δφ ¼ CZ; ð11Þwhere C ¼ fc1; c2;…; cNg is the coefficient vector.

Because of the orthogonality of Zernike polynomials,the decomposed result is unique, so we can get

AB ¼ C; ð12Þ

and finally the coefficient vector A can be solved by

A ¼ CBþ; ð13Þwhere Bþ is the generalized inverse of the coefficientmatrix B.

For a certain RSI system, the radial shear ratio s is aconstant, and the lateral shear x0 and y0 can be deter-mined by a calibration process. So the coefficient matrixB can be calculated numerically beforehand. So we justneed to express the wavefront differenceΔφðx; yÞ with aset of Zernike polynomials, and the corresponding coef-ficient vectorC is obtained. Finally, the coefficient vectorA of the wavefront under test φ0ðx; yÞ can be calculated

3694 OPTICS LETTERS / Vol. 36, No. 18 / September 15, 2011

by a one-time matrix operation, and the wavefront undertest is reconstructed.To validate the proposed algorithm, we calculated

Zernike coefficients for the wavefront under test havinga random shape and compared themwith original Zernikecoefficients of the wavefront under test. In our calcula-tion, the maximal order of Zernike polynomials of thewavefront under test is limited to 45, and the samplingnumber of the numeric array is set to 256 × 256. The radialshear ratio s ¼ 1:2, and the relative lateral shear ηx ¼ 10%in the x direction and ηy ¼ −18% in the y direction.A random aberration, which is the combination of the

45 limited Zernike polynomials, is generated as the ran-dom wavefront under test, i.e., φ0ðx; yÞ. Two variables,φrms and φpv, are used to express the RMS and peak tovalley (PTV) of the wavefront, respectively. The three-dimensional (3D) plot of the simulated randomwavefrontφ0ðx; yÞ is shown in Fig. 2(a), and the corresponding coef-ficient vector A0 is plotted in Fig. 2(b). The RMS and PTVof the simulated wavefront under test are 2:000λ and12:285λ, respectively, where λ is the wavelength of the la-ser. According to the radial shear ratio s and the relativelateral shear ηx and ηy, the wavefront differenceΔφðx; yÞcan be calculated easily, and it is shown in Fig. 2(c). Wedecompose the calculated wavefront differenceΔφðx; yÞinto a linear combination of the 45 limited Zernike poly-nomials, and the corresponding coefficients vector, i.e.,the vector C, is obtained and shown in Fig. 2(d).From Eqs. (6)–(10), the coefficient matrix B can be

calculated and is shown in Fig. 3(a). According to the de-scription of Eq. (13), the coefficient vector A is obtained,and it is plotted in Fig. 3(b). Finally, the wavefront undertest is reconstructed from Eq. (4), and its 3D plot isdrawn in Fig. 3(c). The residual error between the wave-front under test and the reconstructed result is shown inFig. 3(d), and its RMS and PTV are 7:003 × 10−14λ and2:896 × 10−13λ, respectively.

In conclusion, an analytic and novel reconstructionmethod is presented for a radial shearing interferometerwith a certain amount of lateral shear. The proposedmethod is based on standard Zernike polynomials andits matrix formalism. The resulting formulation is verysimple. Comparing with the iterative method, the pro-posed method leads to an easier implementation, a morerobust rejection of noise, and a more adaptive practicalsituation. In theory, there are no special limitations to theradial shearing ratio s and the amount of lateral shearalong two orthogonal directions. In this Letter, we makea simple simulation to validate the proposed method, anda detailed analysis of the technique requires at least an in-depth simulation and is beyond the limits of this Letter.

Theauthorsgratefullyacknowledgesupport fromthe in-novation fundsof theChineseAcademyofSciences (CAS).

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Fig. 3. (Color online) Wavefront reconstruction and the cor-responding residual error. (a) Two-dimensional plot of coeffi-cient matrix B. (b) Coefficient vector A calculated fromEq. (13). (c) Reconstructed wavefront under test by substitutingthe vector A into Eq. (4). (d) Residual error between the originwavefront and the reconstructed wavefront.

Fig. 2. (Color online) Simulated wavefront under test and thecorresponding wavefront difference. (a) Randomwavefront un-der test φ0ðx; yÞ. (b) Weighting coefficient vector A0 for eachorder of Zernike polynomials. (c) Wavefront differenceΔφðx; yÞ calculated from φ0ðx; yÞ; the RMS and PTV ofΔφðx; yÞare 2:091λ and 18:057λ, respectively. (d) Coefficient vector C ofZernike polynomials for Δφðx; yÞ.

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