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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 2 — Jan. 16, 2012
  • pp: 1530–1544
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Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms

Fengzhao Dai, Feng Tang, Xiangzhao Wang, Peng Feng, and Osami Sasaki  »View Author Affiliations


Optics Express, Vol. 20, Issue 2, pp. 1530-1544 (2012)
http://dx.doi.org/10.1364/OE.20.001530


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Abstract

A numerical orthogonal transformation method for reconstructing a wavefront by use of Zernike polynomials in lateral shearing interferometry is proposed. The difference fronts data in two perpendicular directions are fitted to numerical orthonormal polynomials instead of Zernike polynomials, and then the orthonormal coefficients are used to evaluate the Zernike coefficients of the original wavefront by use of a numerical shear matrix. Due to the fact that the dimensions of the shear matrix are finite, the high-order terms of the original wavefront above a certain order have to be neglected. One of advantages of the proposed method is that the impact of the neglected high-order terms on the outcomes of the lower-order terms can be decreased, which leads to a more accurate reconstruction result. Another advantage is that the proposed method can be applied to reconstruct a wavefront on an aperture of arbitrary shape from its difference fronts. Theoretical analysis and numerical simulations shows that the proposed method is correct and its reconstruction error is obviously smaller than that of Rimmer-Wyant method.

© 2012 OSA

1. Introduction

In this paper, we propose a numerical orthogonal transformation method to reconstruct the wavefront from difference fronts based on Zernike polynomials. By using this method, the sensitivity of the outcomes of lower-order terms to the remaining high-order terms can be decreased, and a consequent result is that the remaining error is reduced and then the reconstruction accuracy is improved. This method can be implemented easily from Rimmer-Wyant method, and can be applied to reconstruct a wavefront on an aperture of arbitrary shape from its difference fronts. By theoretical analysis and numerical calculations, it is confirmed that the accuracy of the proposed method is superior to that of Rimmer-Wyant method.

2. Numerical orthogonal transformation

It is supposed that a wavefront W(x,y) under test is represented by the first Jfringe Zernike polynomials [20

20. J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI of Applied Optics and Optical Engineering Series (Academic, 1992), 28.

] as
W(x,y)=j=2JajZj(x,y).
(1)
wherexandyare normalized in Cartesian coordinates, Zj(x,y) denotes thejthfringe Zernike polynomials, and ajis the corresponding weighting coefficient. In Eq. (1) the pistonZ1was omitted because it is of no concern. Assuming that the two lateral shearing interfergrams in two perpendicular shear directions have been obtained within the two overlap regionsΣxandΣyon a series of discrete points, as shown in Figs. 1(a)
Fig. 1 Schematic diagrams of lateral shearing interferograms. (a) Lateral shearing interferogaram when the shearing is in the x direction. (b) Lateral shearing interferogram when the shearing is in the y direction.
and 1(b). And the shear ratiosin the two shear directions is assumed identical in this paper, thus the number of measurement pointsNfor the two difference fronts is the same. The two difference fronts on these discrete points can be evaluated from the two lateral shearing interferograms.

The difference fronts are also expanded by Zernike polynomials. For the moment, we analyze the difference front when the shearing is in thexdirection. Then we have
ΔWx(x,y;s)=W(x+s,y)W(xs,y)=j=1JAx,jZj(x,y).
(2)
whereAx,j denotes thejthZernike coefficient ofΔWx(x,y;s), and the summation starts from j=1since the piston of the difference front cannot be omitted because it relates to some terms of the original wavefront, e.g. tilt and coma. Note that some high-orders of Ax,j may be zero, but they still be reserved to facilitate the derivation subsequently. Substituting Eq. (1) and the expression of Zernike polynomials in Cartesian coordinates into Eq. (2), the relationship between the Zernike coefficients of the difference front and the original wavefront can be derived. It is written in matrix form for simplicity as
Ax=Nxa.
(3)
whereAxandaareJ×1and (J1)×1vectors, respectively, they represent the arrays of Zernike coefficients of the difference front and the original wavefront, respectively. Nxis a shear matrix that relates the Zernike coefficients of the original wavefront and the difference front.

However, the Zernike coefficients of the difference fronts are not independent of each other, because Zernike polynomials are orthogonal basis functions over unit circle but not orthogonal over the overlap regionΣxof two circular beams where the difference front lying in. Thus, it can be expected that the use of orthogonal polynomials instead of Zernike polynomials to expand the difference fronts and a new shear matrix to relate the Zernike coefficients of the original wavefront and the orthogonal coefficients of the difference fronts may reduce the reconstruction error. Indeed, as can be seen below, the reconstruction of the original wavefront with orthogonal transformation results in a smaller remaining error and a more accurate reconstruction result.

2.1. Calculation of the numerical orthonormal polynomials

FxTFx=FxTZxMxT.
(6)
ZxTFx=ZxTZxMxT.
(7)

BecauseFxare orthonormal over the data set, so we have(FxTFx)/N=I, whereIis J×J identity matrix. Use this condition to Eq. (6), we get

MxZxTFx=NI.
(8)

Substitute Eq. (7) to Eq. (8) and letMx=(QxT)1, we obtain

QxTQx=(ZxTZx)/N.
(9)

Equation (9) can be solved forQxuniquely with, e.g, Cholesky decomposition due to the fact that ZxTZxis symmetric positive definite matrix [24

24. G. M. Dai and V. N. Mahajan, “Nonrecursive determination of orthonormal polynomials with matrix formulation,” Opt. Lett. 32(1), 74–76 (2007). [CrossRef] [PubMed]

], and then conversion matrixMxcan be determined fromMx=(QxT)1.The expressions of Zernike polynomials in Cartesian coordinate are known, and then the matrix Zxcan be calculated by substitute the coordinate values of the measurement points into the expressions of Zernike polynomials. Therefore, the numerical orthonormal polynomials Fx can be determined from Eq. (5) with the matrixZxand the conversion matrixMx.

2.2 Derivation of the corresponding shear matrix

By expanding the difference front within Σxwith the fistJ numerical orthonormal polynomials{Fx,j(xn,yn)}, the corresponding orthogonal coefficients are obtained as J×1 vectorBx. Since the fact that each term in the set{Fx,j(xn,yn)}is a linear combination of the Zernike polynomials, the difference front expanded with the orthonormal polynomials {Fx,j(xn,yn)}is identical to that expanded with corresponding set of Zernike polynomials. Then, we have a matrix formula,
ΔWx=ZxAx=FxBx.
(10)
whereΔWxisN×1vector representing the difference front values inNpoints within the region Σx.Substituting Eq. (5) to Eq. (10), the relationship between the orthonormal coefficients Ax and the Zernike coefficientsBxcan be obtained as

Ax=MxTBx.
(11)

Substituting Eq. (11) into Eq. (3), we have

Bx=(MxT)1Nxa=Hxa.
(12)

Here, Hx=(MxT)1Nx denotes the shear matrix that relates the Zernike coefficients of the original wavefront and the orthonormal coefficients of difference front.

2.3 Wavefront reconstruction

For the analysis of the difference front within Σywhere the shearing is in theydirection, we find that, similar to Eq. (3) and Eq. (12),
Ay=Nya.
(13)
By=(MyT)1Nya=Hya.
(14)
whereAyandByareJ×1vectors, which represent Zernike coefficients and orthonormal coefficients of the difference front withinΣy, respectively. Myis the conversion matrix which transforms Zernike polynomials to the numerical polynomials{Fy,j(xn,yn)}. The numerical polynomials are orthonormal over the discrete points withinΣy. Ny and Hy=(MyT)1Ny are the shear matrixes corresponding toNxandHx, respectively. All the four shear matrixesNx, Ny,Hxand Hy are J×(J1)upper triangular matrixes whose matrix elements are functions of shear ratiosor zero. Note that, NxandNyare given in analytical form, while HxandHyare given in numerical form.

Combing Axand Ayinto a single vector and matrixNxandNyinto a single matrix, a matrix formula is obtained from Eq. (3) and Eq. (13) as
(AxAy)=(NxNy)a.
(15)
which can be abbreviated to

A=Na.
(16)

Appling the same treatment to Eq. (12) and Eq. (14), a matrix formula is obtained as
(BxBy)=(HxHy)a.
(17)
which can also be simplified as

B=Ha.
(18)

Since Eqs. (16) and (18) are obviously over determined equations, the least-square solutions of the two equations are
a^=NA^.
(19)
a^=HB^.
(20)
whereA^andB^are estimation ofAandBobtained by fitting the two difference fronts ΔWx(xn,yn) and ΔWy(xn,yn)to Zernike polynomials and numerical orthonormal polynomials, respectively, and N=(NTN)1NTandH=(HTH)1HTare generalized inverse of matrixes N andH, respectively. Equations (19) and (20) are the representation of Rimmer-Wyant method and the proposed method, respectively.

We note that a similar method by using elliptical orthogonal transformation has been presented in Ref. 16

16. G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. 35(31), 6162–6172 (1996). [CrossRef] [PubMed]

. In this method, the difference front inxdirection withinΣxwas reduced to an elliptical region. And the elliptical Zernike polynomials which are orthogonal over this elliptical region were generated from Zernike polynomials by coordinate transformation. Then the reduced difference front was expanded into the obtained elliptical Zernike polynomials. The shear matrix that relates the elliptical Zernike coefficients of difference front and the Zernike coefficients of the original wavefront were derived from a double integral. The difference front inydirection withinΣywas analyzed following the same way.

However, the obtained elliptical Zernike polynomials are orthogonal over the full elliptical region, but not orthogonal over the discrete points at which the difference front are measured. Moreover, the applications of this method are limited to the situation that the wavefront under test is circular because the overlap region of the two interference beams cannot be approximated by an ellipse when the wavefront under test is generated from a non-circle aperture such as annular aperture.

For the proposed method, the numerical orthogonal transformation is implemented directly on the discrete points of the difference front data set, and the obtained numerical polynomials are orthogonal over these discrete points rather than the full overlap region. In addition, the orthogonality of the obtained numerical polynomials is not influenced by the shape of the region. Thus, the proposed method can be applied on apertures of arbitrary shape.

The fact that the outcomes of the lower-order terms are influenced by the remaining high-order terms was pointed out and illustrated by an example shown in Ref.16

16. G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. 35(31), 6162–6172 (1996). [CrossRef] [PubMed]

. However, the authors did not explain the effect of orthogonal transformation on the reduction of this influence. This effect will be made clear with a cross-coupling formula derived in the following section.

3. Impact of remaining high-order terms

Here, we assume the number J stands for infinity. Therefore, any practical wavefront can be represented completely by Eq. (1). Now, returning to Eq. (10), under the assumption that J stands for infinity, the column number of the matrixZxis extended to infinity. If the matrixZxand the vectorAxare splited into two blocks, a formula can be obtained from Eq. (10) as

ΔWx=ZxAx=(ZxfZxr)(AxfAxr)=ZxfAxf+ZxrAxr.
(21)

Here, ZxfandZxrare the two blocks containing the firstKand the remaining columns of the matrixZx,respectively. AxfandAxrare the firstKand the remaining elements of the vectorAx.

3.1 Least-square fitting of difference fronts

Since the difference front cannot be fitted with infinite terms of Zernike polynomials, we assume it is fitted by the firstKterms, and then the corresponding coefficients can be obtained by means of least-square fitting,
A^xf=ZxfΔWx.
(22)
whereA^xfis the estimation ofAxf,andZxf=(ZxfTZxf)1ZxfTis the generalized inverse of matrixZxf. Substituting Eq. (21) into Eq. (22), we obtain
A^xf=Zxf(ZxfAxf+ZxrAxr)=Axf+CxAxr.
(23a)
whereCx=ZxfZxr. A similar expression can be obtained for the fitting of the difference front withinΣy where the shearing is in theydirection,
A^yf=Ayf+CyAyr.
(23b)
whereA^yfis the estimation ofAyfwhich is the firstKZernike coefficients of the difference front, Ayris the Zernike coefficients of the remaining high-order terms, and Cy=ZyfZyr, whereZyf=(ZyfTZyf)1ZyfTis the generalized inverse of matrixZyf, and Zyfand Zyrare the firstKand the remaining columns of the matrixZy,respectively, andZyis N×Jmatrix representing each of JZernike polynomials over theNdata points within the regionΣy.

3.2 Splitting of the shear matrixes

To facilitate the derivation, Eq. (3) is written in another form as

(AxfAxr)=(Nxf1Nxf2Nxr1Nxr2)(afar)=(Nxf1Nxf20Nxr2)(afar).
(24a)

Here, the shear matrixNxwith infinite rows and columns is splited into four blocks by a horizontal line under theKthrow and a vertical line behind the(K1)thcolumns. So, the block Nxf1is K×(K1)matrix, and the dimensions of other blocks can also be deduced easily. In addition, Nxr1=0is becauseNxis an upper triangular matrix. Vectorsafandarare arrays of the firstK1and the remaining elements of vectora, respectively.

The shear matrixNycan also be splited into four blocks in the same way, and a similar expression as Eq. (24a) for the analysis of the difference front inyshear direction can be obtained as

(AyfAyr)=(Nyf1Nyf2Nyr1Nyr2)(afar)=(Nyf1Nyf20Nyr2)(afar).
(24b)

From Eq. (24a) and Eq. (24b) the following formula can be obtained easily,

Axf=Nxf1af+Nxf2ar
(25a)
Axr=Nxr2ar.
(25b)
Ayf=Nyf1af+Nyf2ar.
(25c)
Ayr=Nyr2ar.
(25d)

3.3 Reconstruction with finite-dimensional shear matrixes

As discussed in Sec.1, the wavefront cannot be reconstructed completely due to the fact that the dimensions of the shear matrix are finite. As Eq. (19), the estimation of the firstKZernike coefficients of the original wavefront except piston can be obtained by shear matrixNxf1, Nyf1and the estimation coefficientsA^xfand A^yfas
a^f=Nf1(A^xfA^yf)withNf1=(Nxf1Nyf1).
(26)
whereNf1=(Nf1TNf1)1Nf1Tis generalized inverse of matrixNf1.Substituting Eqs. (23a) and (23b) into Eq. (26), we obtain

a^f=Nf1(Axf+CxAxrAyf+CyAyr).
(27)

Substituting Eqs. (25a) - (25d) to Eq. (27), we have
a^f=Nf1(Nxf1af+Nxf2ar+CxNxr2arNyf1af+Nyf2ar+CyNyr2ar)=Nf1{(Nxf1Nyf1)af+(Nxf2Nyf2)ar+(CxNxr2CyNyr2)ar}.=af+TZar
(28)
where

TZ=Nf1Nf2+Nf1Ncr2,Nf2=(Nxf2Nyf2)andNcr2=(CxNxr2CyNyr2).

TZis referred to as cross-coupling matrix, and it represents the impact of the remaining high-order terms on the outcomes of the lower-order terms when the wavefront is reconstructed by use of Eq. (19). Note that this cross-coupling matrix is different from the cross-coupling matrix derived by Herrmann [12

12. J. Herrmann, “Cross coupling and aliasing in modal wavefront estimation,” J. Opt. Soc. Am. 71(8), 989–992 (1981). [CrossRef]

] and the cross-talk matrix derived by Dai [18

18. G.- Dai, “Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 13(6), 1218–1225 (1996). [CrossRef]

].

3.4 Same analysis of numerical orthogonal transformation method

Although Eq. (28) is deduced from the situation that the Zernike polynomials are used as a basis to expand the difference fronts, it can also be used in the case that the numerical orthonormal polynomials are used as a basis to expand the difference fronts. To analyze the proposed method, by imitating Eq. (28) we obtain
a^f=af+THar.
(29)
where

TH=Hf1Hf2+Hf1Hcr2,Hf2=(Hxf2Hyf2)andHcr2=(DxHxr2DyHyr2).

Similar toCx=ZxfZxrandCy=ZyfZyr, we haveDx=FxfFxrandDy=FyfFyr. Note that, due to the orthonormality ofFxandFy, we have

Dx=FxfFxr=(FxfTFxf)1FxfTFxr=0.
(30a)
Dy=FyfFyr=(FyfTFyf)1FyfTFyr=0.
(30b)

Substituting Eqs. (30). (a) and (b) to the expression ofHcr2, the result ofHcr2=0 is obtained, and then substituting this result to the expression ofTH, we obtainTH=Hf1Hf2.

4. Numerical simulation

4.1 Simulation condition

To confirm the proposed method and the theory analysis discussed above, numerical simulations were implemented. A digitized wavefront W(xn,yn) filtered by a circle pupil was generated over a 256×256 square grid by the first 20 fringe Zernike polynomials, that was J=20.The corresponding coefficients were generated randomly, but the last five elements were multiplied by101.This attenuation followed the assumption that all the terms that gave significant contributions to the wavefront were contained in the first 15 terms, and the contributions of the last five terms were very weak compared with those of the first 15 terms. Noise-free difference fronts data in two perpendicular shear directions was calculated with the same shear ratio ofs=0.1. The test wavefront and the difference fronts in two directions are shown in Figs. 2(a)
Fig. 2 Simulation condition, (a) simulated wavefront under testW(xn,yn), (b) difference front ΔWx(xn,yn) when the shearing was in thexdirection, (c) difference front ΔWy(xn,yn) when the shearing was in theydirection and (d) random Zernike coefficients of the wavefront under test.
to 2(c). The Zernike coefficients of the test wavefront are shown in Fig. 2(d). For comparison, the wavefront were reconstructed from the two difference fronts by Rimmer-Wyant method as Eq. (19) and by the proposed method as Eq. (20), respectively.

4.2 Reconstruction without remaining high-order terms

First, we consider the reconstruction without the remaining high-order terms, that is, with K=J=20. In this case, all of the Zernike coefficients of the original wavefront are represented by one or more elements of the shear matrixNandH. The dimensions of both the two shear matrixesNandHwere40×19.The results of the evaluation of the Zernike coefficients of the original wavefront are shown in Table 1

Table 1. Input and evaluated Zernike coefficients of the original wavefront by Rimmer-Wyant method (M1) and the proposed method (M2) when the reconstruction was performed without and with the affections of the remaining high-order terms

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. WhenK=20, the evaluated coefficients of both Rimmer-Wyant method and the proposed method are identical to the input coefficients. These results make it clear that both the two methods are capable to reconstruct the wavefront without error under the condition that all the terms of original wavefront are included in the analysis, that is there is no remaining high-order terms. Unfortunately, this condition can only be met in simulation because the existence of remaining high-order terms is inevitable in the analysis of a practical wavefront in general case, as discussed in Sec. 1.

4.3 Reconstruction under the impact of remaining high-order terms

To examine the impact of the remaining high-order terms on the outcomes of the lower-order terms, the wavefront was reconstructed under the condition thatK=15, that is, the last five terms were the remaining high-order terms. The dimensions of both the two shear matrixes N andHwere shrunk to30×14. The results of the evaluation of the Zernike coefficients of the original wavefront are also shown in Table 1. The value of “Percentage Error” column was calculated by the formula ofrj=(aja^j)/aj×100, where ajis the input coefficient, and a^jis the corresponding evaluated value by the two methods. The calculation results of rjare also diagramed in Fig. 3(d)
Fig. 3 Original wavefront and reconstructed results of the two methods: (a) a part of test wavefront Wf, (b) the reconstruction result of Rimmer-Wyant method, (c) the reconstruction result of the proposed method, (d) the percentage error of the retrieved Zernike coefficients by Rimmer-Wyant method and the proposed method, (e) the difference between the reconstruction result W1 of Rimmer-Wyant method and Wf, and(f) the difference between the reconstruction result W2 of the proposed method and Wf .
for visualization. It can be clearly seen from the “Percentage Error” column of Table.1 that all of the absolute values of the coefficients error that were retrieved by the proposed method are smaller than those by Rimmer-Wyant method. And the proposed method retrieved the input coefficients with small differences of less than 1% in most case, while these differences are higher than30%in some case by Rimmer-Wyant method.

To facilitate the analysis, the wavefront under test is divided into two parts. One part is denoted by Wf which represents the contributions of the first 15 terms, while the other part denoted byWrdescribes the contributions of the remaining 5 terms. In this simulation, we evaluate the reconstruction accuracy ofWf.The wavefront was reconstructed by use of the evaluated Zernike coefficients shown in “K=15”column of Table 1. The notations of W1andW2 were assigned to represent the reconstruction results of Rimmer-Wyant method and the proposed method, respectively. The contour plots of these wavefronts and the reconstruction errors of the two methods are shown in Fig. 3. The reconstruction error of the proposed method is obviously smaller than that of Rimmer-Wyant method, as can be clearly seen from Figs. 3(e) and 3(f).

4.4 Comparison of RMS and PV value

The root mean square (RMS) and peak-to-valley (PV) value were used to characterize the reconstruction accuracy. The RMS and PV value of the original wavefront and the reconstruction error of the two methods are shown in Table 2

Table 2. RMS and PV values of the original wavefront and of the reconstruction error

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. The RMS reconstruction error is6.25%for Rimmer-Wyant method and 0.81%for the proposed method. In other words, the remaining error of the proposed method is about1/8of that of Rimmer-Wyant method. Moreover, we know from Table 2 that the PV reconstruction error of the proposed method is about 1/10of that of Rimmer-Wyant method. In brief, the reconstruction accuracy of the proposed method is superior to that of Rimmer-Wyant method, which can also be confirmed from Figs. 3(a) to 3(c). Note that the reconstruction accuracy will change with some parameters, such as the number of the Zernike polynomialsKused in the reconstruction and the shear ratios, but the fact that the proposed method is more accurate will not change.

4.5 Evaluation of the cross-coupling matrix

To explain the reason why the reconstruction accuracy of the proposed method is superior to that of Rimmer-Wyant method under the impact of remaining high-order terms, the cross-coupling matrixes TZandTHof the two methods were evaluated. The two cross-coupling matrixes TZand TH are shown in Figs. 4(a)
Fig. 4 Cross-coupling matrix of the two methods, (a) the coupling-matrixTZof Rimmer-Wyant method, (b) the cross-coupling matrixTHof the proposed method
and 4(b). As discussed in Sec.1, the cross-coupling matrix manifests the impact of the remaining high-order terms on the outcomes of the lower-order terms. For example, as shown in Fig. 4(a), the high-order coefficient a16 has affections on the estimation of all the lower-order coefficients, especiallya14anda15, when Rimmer-Wyant method is used. When the proposed method is used, the coefficienta16just affects ona4anda9with more slight level, as shown in Fig. 4(b).

On the other hand, the cross-coupling matrix also manifests the sensitivity of the outcomes of the lower-order terms to the remaining high-order terms. For example, as shown in Fig. 4(a), the evaluation of the coefficienta13is sensitive to almost all of the five remaining terms, especially toa16anda19, when Rimmer-Wyant method is used. When the proposed method is used, the evaluation ofa13is not sensitive to any one of the five remaining terms, as can be seen from Fig. 4(b). Thus the calculation error ofa13of the proposed method is far below than that of Rimmer-Wyant method which can also be confirmed from Table 1.

Anyway, it can be clearly seen from Fig. 4 that the level of cross coupling ofTHis far below than that of TZ.Therefore the sensitivity of the outcomes of lower-order terms to the remaining high-order terms can be decreased by the proposed numerical orthogonal transformation method which leads to a smaller remaining error and a more accurate reconstruction result than Rimmer-Wyant method.

4.6. Comparison of the computation time

To compare the computation time of the proposed method and Rimmer-Wyant method, several simulations were implemented in different sample sizes. The computation time of the proposed method was divided into the following two parts: one part is the time to do the numerical orthogonal transformation and the other part is the time to reconstruct the wavefront with the new shear matrixes and numerical orthogonal polynomials. The total reconstruction time is the summation of the two parts. It can be clearly seen from Table 3

Table 3. Comparison of Computation Time of the Two Methods for Different Sample Sizes

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that the proposed method is faster than Rimmer-Wyant method. The reason is that the time to obtain the orthogonal coefficients of difference fronts is shorter than the time to obtain the Zernike coefficients of difference fronts as follows: the orthogonal coefficients of the difference front in x direction can be obtained directly byA^x=FxTΔWx/N, while the Zernike coefficients are obtained by means of least-squares fitting as A^x=(ZxTZx)1ZxTΔWxwhich needs more computation time. The time difference is longer than the time to operate the numerical orthogonal transformation. Note that our calculation was performed on a personal computer equipped with a 2.80GHZ Pentium-4 processor and the software was MATLAB (Version 7.8.0).

4.7. Simulation with a general wavefront

5. Conclusion

Acknowledgments

This work was supported by the Grant from the National Natural Science foundation of China under no. 60938003.

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G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. 35(31), 6162–6172 (1996). [CrossRef] [PubMed]

17.

H. van Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36(13), 2788–2790 (1997). [CrossRef] [PubMed]

18.

G.- Dai, “Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 13(6), 1218–1225 (1996). [CrossRef]

19.

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980). [CrossRef]

20.

J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI of Applied Optics and Optical Engineering Series (Academic, 1992), 28.

21.

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71(1), 75–85 (1981). [CrossRef]

22.

R. Upton and B. Ellerbroek, “Gram-Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape,” Opt. Lett. 29(24), 2840–2842 (2004). [CrossRef] [PubMed]

23.

V. N. Mahajan and G. M. Dai, “Orthonormal polynomials for hexagonal pupils,” Opt. Lett. 31(16), 2462–2464 (2006). [CrossRef] [PubMed]

24.

G. M. Dai and V. N. Mahajan, “Nonrecursive determination of orthonormal polynomials with matrix formulation,” Opt. Lett. 32(1), 74–76 (2007). [CrossRef] [PubMed]

25.

V. N. Mahajan and G. M. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24(9), 2994–3016 (2007). [CrossRef] [PubMed]

26.

G. M. Dai and V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: error analysis,” Appl. Opt. 47(19), 3433–3445 (2008). [CrossRef] [PubMed]

27.

M. Hasegawa, C. Ouchi, T. Hasegawa, S. Kato, A. Ohkubo, A. Suzuki, K. Sugisaki, M. Okada, K. Otaki, K. Murakami, J. Saito, M. Niibe, and M. Takeda, “Recent progress of EUV wavefront metrology in EUVA,” Proc. SPIE 5533, 27–36 (2004). [CrossRef]

28.

Y. Zhu, S. Odate, A. Sugaya, K. Otaki, K. Sugisaki, C. Koike, T. Koike, and K. Uchikawa, “Method for designing phase-calculation algorithms for two-dimensional grating phase-shifting interferometry,” Appl. Opt. 50(18), 2815–2822 (2011). [CrossRef] [PubMed]

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.3940) Instrumentation, measurement, and metrology : Metrology
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: November 11, 2011
Revised Manuscript: December 15, 2011
Manuscript Accepted: December 26, 2011
Published: January 10, 2012

Citation
Fengzhao Dai, Feng Tang, Xiangzhao Wang, Peng Feng, and Osami Sasaki, "Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms," Opt. Express 20, 1530-1544 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1530


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References

  1. D. Malacara, Optical Shop Testing, 3rd ed, (CRC Press, Taylor& Francis, 2007).
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  11. P. Liang, J. Ding, Z. Jin, C. S. Guo, and H. T. Wang, “Two-dimensional wave-front reconstruction from lateral shearing interferograms,” Opt. Express14(2), 625–634 (2006). [CrossRef] [PubMed]
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  13. K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A3(11), 1852–1861 (1986). [CrossRef]
  14. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt.14(1), 142–150 (1975). [PubMed]
  15. W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng.36(3), 905–913 (1997). [CrossRef]
  16. G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt.35(31), 6162–6172 (1996). [CrossRef] [PubMed]
  17. H. van Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt.36(13), 2788–2790 (1997). [CrossRef] [PubMed]
  18. G.- Dai, “Modal wavefront reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A13(6), 1218–1225 (1996). [CrossRef]
  19. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am.70(8), 998–1006 (1980). [CrossRef]
  20. J. C. Wyant and K. Creath, Basic Wavefront Aberration Theory for Optical Metrology, Vol. XI of Applied Optics and Optical Engineering Series (Academic, 1992), 28.
  21. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am.71(1), 75–85 (1981). [CrossRef]
  22. R. Upton and B. Ellerbroek, “Gram-Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape,” Opt. Lett.29(24), 2840–2842 (2004). [CrossRef] [PubMed]
  23. V. N. Mahajan and G. M. Dai, “Orthonormal polynomials for hexagonal pupils,” Opt. Lett.31(16), 2462–2464 (2006). [CrossRef] [PubMed]
  24. G. M. Dai and V. N. Mahajan, “Nonrecursive determination of orthonormal polynomials with matrix formulation,” Opt. Lett.32(1), 74–76 (2007). [CrossRef] [PubMed]
  25. V. N. Mahajan and G. M. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A24(9), 2994–3016 (2007). [CrossRef] [PubMed]
  26. G. M. Dai and V. N. Mahajan, “Orthonormal polynomials in wavefront analysis: error analysis,” Appl. Opt.47(19), 3433–3445 (2008). [CrossRef] [PubMed]
  27. M. Hasegawa, C. Ouchi, T. Hasegawa, S. Kato, A. Ohkubo, A. Suzuki, K. Sugisaki, M. Okada, K. Otaki, K. Murakami, J. Saito, M. Niibe, and M. Takeda, “Recent progress of EUV wavefront metrology in EUVA,” Proc. SPIE5533, 27–36 (2004). [CrossRef]
  28. Y. Zhu, S. Odate, A. Sugaya, K. Otaki, K. Sugisaki, C. Koike, T. Koike, and K. Uchikawa, “Method for designing phase-calculation algorithms for two-dimensional grating phase-shifting interferometry,” Appl. Opt.50(18), 2815–2822 (2011). [CrossRef] [PubMed]

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