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.

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2011 (1)

2009 (1)

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A, Pure Appl. Opt. 11(4), 045702 (2009).
[CrossRef]

2008 (1)

2007 (3)

2006 (2)

2004 (3)

2000 (1)

1997 (2)

W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36(3), 905–913 (1997).
[CrossRef]

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]

1996 (2)

1986 (1)

1981 (2)

1980 (2)

1979 (1)

1977 (2)

1975 (1)

1974 (1)

Chang, M.

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A, Pure Appl. Opt. 11(4), 045702 (2009).
[CrossRef]

Chang, M. W.

W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36(3), 905–913 (1997).
[CrossRef]

Dai, G.-

Dai, G. M.

Dainty, C.

Ding, J.

Dubra, A.

Ellerbroek, B.

Freischlad, K. R.

Fried, D. L.

Gao, Y.

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A, Pure Appl. Opt. 11(4), 045702 (2009).
[CrossRef]

Guo, C. S.

Harbers, G.

Hasegawa, M.

Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt. 46(27), 6783–6792 (2007).
[CrossRef] [PubMed]

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]

Hasegawa, T.

Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt. 46(27), 6783–6792 (2007).
[CrossRef] [PubMed]

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]

Herrmann, J.

Hudgin, R. H.

Hunt, B. R.

Ishii, M.

Jin, Z.

Kamiya, K.

Kato, S.

Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt. 46(27), 6783–6792 (2007).
[CrossRef] [PubMed]

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]

Kawakami, J.

Koike, C.

Koike, T.

Koliopoulos, C. L.

Kunst, P. J.

Leibbrandt, G. W. R.

Liang, P.

Liu, X.

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A, Pure Appl. Opt. 11(4), 045702 (2009).
[CrossRef]

Liu, Z.

Mahajan, V. N.

Miyashiro, H.

Murakami, K.

Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt. 46(27), 6783–6792 (2007).
[CrossRef] [PubMed]

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]

Niibe, M.

Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt. 46(27), 6783–6792 (2007).
[CrossRef] [PubMed]

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]

Nomura, T.

Odate, S.

Ohkubo, A.

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]

Okada, M.

Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt. 46(27), 6783–6792 (2007).
[CrossRef] [PubMed]

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]

Okuda, S.

Otaki, K.

Ouchi, C.

Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt. 46(27), 6783–6792 (2007).
[CrossRef] [PubMed]

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]

Paterson, C.

Rimmer, M. P.

Saito, J.

Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt. 46(27), 6783–6792 (2007).
[CrossRef] [PubMed]

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]

Shen, W.

W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36(3), 905–913 (1997).
[CrossRef]

Southwell, W. H.

Sugaya, A.

Sugisaki, K.

Suzuki, A.

Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt. 46(27), 6783–6792 (2007).
[CrossRef] [PubMed]

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]

Takeda, M.

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]

Tashiro, H.

Uchikawa, K.

Upton, R.

van Brug, H.

Wan, D. S.

W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36(3), 905–913 (1997).
[CrossRef]

Wang, H. T.

Wyant, J. C.

Yokota, H.

Yoshikawa, K.

Zhu, Y.

Appl. Opt. (8)

Y. Zhu, K. Sugisaki, M. Okada, K. Otaki, Z. Liu, J. Kawakami, M. Ishii, J. Saito, K. Murakami, M. Hasegawa, C. Ouchi, S. Kato, T. Hasegawa, A. Suzuki, H. Yokota, and M. Niibe, “Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography,” Appl. Opt. 46(27), 6783–6792 (2007).
[CrossRef] [PubMed]

M. P. Rimmer, “Method for evaluating lateral shearing interferograms,” Appl. Opt. 13(3), 623–629 (1974).
[CrossRef] [PubMed]

S. Okuda, T. Nomura, K. Kamiya, H. Miyashiro, K. Yoshikawa, and H. Tashiro, “High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials,” Appl. Opt. 39(28), 5179–5186 (2000).
[CrossRef] [PubMed]

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]

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]

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]

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

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]

J. Opt. A, Pure Appl. Opt. (1)

X. Liu, Y. Gao, and M. Chang, “A partial differential equation algorithm for wavefront reconstruction in lateral shearing interferometry,” J. Opt. A, Pure Appl. Opt. 11(4), 045702 (2009).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

W. Shen, M. W. Chang, and D. S. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng. 36(3), 905–913 (1997).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Proc. SPIE (1)

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]

Other (2)

D. Malacara, Optical Shop Testing, 3rd ed, (CRC Press, Taylor& Francis, 2007).

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.

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Figures (4)

Fig. 1
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.

Fig. 2
Fig. 2

Simulation condition, (a) simulated wavefront under test W( x n , y n ) , (b) difference front Δ W x ( x n , y n ) when the shearing was in thexdirection, (c) difference front Δ W y ( x n , y n ) when the shearing was in theydirection and (d) random Zernike coefficients of the wavefront under test.

Fig. 3
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 .

Fig. 4
Fig. 4

Cross-coupling matrix of the two methods, (a) the coupling-matrix T Z of Rimmer-Wyant method, (b) the cross-coupling matrix T H of the proposed method

Tables (3)

Tables Icon

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

Tables Icon

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

Tables Icon

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

Equations (38)

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W(x,y)= j=2 J a j Z j (x,y) .
Δ W x (x,y;s)=W(x+s,y)W(xs,y)= j=1 J A x,j Z j (x,y) .
A x = N x a .
F x,i ( x n , y n )= j=1 J M x,ij Z j ( x n , y n ) .
F x = Z x M x T .
F x T F x = F x T Z x M x T .
Z x T F x = Z x T Z x M x T .
M x Z x T F x =NI .
Q x T Q x = ( Z x T Z x ) /N .
Δ W x = Z x A x = F x B x .
A x = M x T B x .
B x = ( M x T ) 1 N x a= H x a .
A y = N y a .
B y = ( M y T ) 1 N y a= H y a .
( A x A y )=( N x N y )a .
A=Na .
( B x B y )=( H x H y )a .
B=Ha .
a ^ = N A ^ .
a ^ = H B ^ .
Δ W x = Z x A x =( Z xf Z xr )( A xf A xr )= Z xf A xf + Z xr A xr .
A ^ xf = Z xf Δ W x .
A ^ xf = Z xf ( Z xf A xf + Z xr A xr )= A xf + C x A xr .
A ^ yf = A yf + C y A yr .
( A xf A xr )=( N xf1 N xf2 N xr1 N xr2 )( a f a r )=( N xf1 N xf2 0 N xr2 )( a f a r ) .
( A yf A yr )=( N yf1 N yf2 N yr1 N yr2 )( a f a r )=( N yf1 N yf2 0 N yr2 )( a f a r ) .
A xf = N xf1 a f + N xf2 a r
A xr = N xr2 a r .
A yf = N yf1 a f + N yf2 a r .
A yr = N yr2 a r .
a ^ f = N f1 ( A ^ xf A ^ yf )with N f1 =( N xf1 N yf1 ) .
a ^ f = N f1 ( A xf + C x A xr A yf + C y A yr ) .
a ^ f = N f1 ( N xf1 a f + N xf2 a r + C x N xr2 a r N yf1 a f + N yf2 a r + C y N yr2 a r ) = N f1 { ( N xf1 N yf1 ) a f +( N xf2 N yf2 ) a r +( C x N xr2 C y N yr2 ) a r } . = a f + T Z a r
T Z = N f1 N f2 + N f1 N cr2 , N f2 =( N xf2 N yf2 )and N cr2 =( C x N xr2 C y N yr2 ) .
a ^ f = a f + T H a r .
T H = H f1 H f2 + H f1 H cr2 , H f2 =( H xf2 H yf2 )and H cr2 =( D x H xr2 D y H yr2 ) .
D x = F xf F xr = ( F xf T F xf ) 1 F xf T F xr =0 .
D y = F yf F yr = ( F yf T F yf ) 1 F yf T F yr =0 .

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