Abstract

A lateral shearing interferometer has an advantage over previous wavefront measuring interferometers since it requires no reference. Therefore the lateral shearing interferometer can be a powerful solution to measure a freeform surface. It, however, has some issues to be resolved before it can be implemented. One of them is the orthogonality problem between two shearing directions in LSI. Previous wavefront reconstruction algorithms assume that the shearing directions are perfectly orthogonal to each other and lateral shear is obtained simultaneously in the sagittal and tangential directions. For practical LSI, however, there is no way to guarantee perfect orthogonality between two shearing directions. Motivated by this, we propose a new algorithm that is able to compensate the rotational inaccuracy. The mathematical model is derived in this paper. Computer simulations and experiments are also displayed to verify our algorithm.

© 2013 Optical Society of America

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References

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  1. M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.1.
  2. A. G. Poleshchuk, E. G. Churin, V. P. Koronkevich, V. P. Korolkov, A. A. Kharissov, V. V. Cherkashin, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, and A. G. Verhoglyad, “Polar coordinate laser pattern generator for fabrication of diffractive optical elements with arbitrary structure,” Appl. Opt.38(8), 1295–1301 (1999).
    [CrossRef] [PubMed]
  3. P. Zhou and J. H. Burge, “Coupling of surface roughness to the performance of computer-generated holograms,” Appl. Opt.46(26), 6572–6576 (2007).
    [CrossRef] [PubMed]
  4. H.-G. Rhee and Y.-W. Lee, “Improvement of linewidth in laser beam lithographed computer generated hologram,” Opt. Express18(2), 1734–1740 (2010).
    [CrossRef] [PubMed]
  5. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A19(3), 590–595 (2002).
    [CrossRef] [PubMed]
  6. R. Hu, X. Luo, H. Zheng, Z. Qin, Z. Gan, B. Wu, and S. Liu, “Design of a novel freeform lens for LED uniform illumination and conformal phosphor coating,” Opt. Express20(13), 13727–13737 (2012).
    [CrossRef] [PubMed]
  7. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am.69(7), 972–977 (1978).
    [CrossRef]
  8. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am.70(8), 998–1006 (1980).
    [CrossRef]
  9. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am.71(8), 989–992 (1981).
    [CrossRef]
  10. 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]
  11. G. W. R. Leibbrandt, G. Harbers, and P. J. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt.35(31), 6151–6161 (1996).
  12. 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]
  13. W. Shen, M. Chang, and D. Wan, “Zernike polynomial fitting of lateral shearing interferometry,” Opt. Eng.36(3), 905–913 (1997).
    [CrossRef]
  14. 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]
  15. 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]
  16. F. Dai, F. Tang, X. Wang, P. Feng, and O. Sasaki, “Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms,” Opt. Express20(2), 1530–1544 (2012).
    [CrossRef] [PubMed]
  17. F. Dai, F. Tang, X. Wang, O. Sasaki, and P. Feng, “Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms,” Appl. Opt.51(21), 5028–5037 (2012).
    [CrossRef] [PubMed]
  18. 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).
    [CrossRef] [PubMed]
  19. D. L. Fried, “Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am.67(3), 370–374 (1977).
    [CrossRef]
  20. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am.67(3), 375–378 (1977).
    [CrossRef]
  21. R. H. Hudgin, “Optimal wave-front estimation,” J. Opt. Soc. Am.67(3), 378–382 (1977).
    [CrossRef]
  22. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am.69(3), 393–399 (1979).
    [CrossRef]
  23. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am.70(1), 28–35 (1980).
    [CrossRef]
  24. 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]
  25. F. Dai, F. Tang, X. Wang, and O. Sasaki, “Generalized zonal wavefront reconstruction for high spatial resolution in lateral shearing interferometry,” J. Opt. Soc. Am. A29(9), 2038–2047 (2012).
    [CrossRef]
  26. H.-G. Rhee, Y.-W. Lee, and S.-W. Kim, “Azimuthal position error correction algorithm for absolute test of large optical surfaces,” Opt. Express14(20), 9169–9177 (2006).
    [CrossRef] [PubMed]
  27. D. Malacara and S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.13.
  28. J. W. Goodman, “Analog optical information processing,” in Introduction to Fourier optics, 2nd ed., (McGraw-Hill, 1996), Chap. 8.

2012 (4)

2010 (1)

2007 (1)

2006 (2)

2002 (1)

2000 (1)

1999 (1)

1997 (2)

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]

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

1996 (2)

1986 (1)

1981 (1)

1980 (2)

1979 (1)

1978 (1)

1977 (3)

1975 (1)

Burge, J. H.

Chang, M.

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

Cherkashin, V. V.

Churin, E. G.

Cubalchini, R.

Dai, F.

Ding, J.

Feng, P.

Freischlad, K. R.

Fried, D. L.

Gan, Z.

Guo, C. S.

Harbers, G.

Herrmann, J.

Hu, R.

Hudgin, R. H.

Hunt, B. R.

Jin, Z.

Kamiya, K.

Kharissov, A. A.

Kim, S.-W.

Kiryanov, A. V.

Kiryanov, V. P.

Kokarev, S. A.

Koliopoulos, C. L.

Korolkov, V. P.

Koronkevich, V. P.

Kunst, P. J.

Lee, Y.-W.

Leibbrandt, G. W. R.

Liang, P.

Liu, S.

Luo, X.

Miyashiro, H.

Muschaweck, J.

Nomura, T.

Okuda, S.

Poleshchuk, A. G.

Qin, Z.

Rhee, H.-G.

Ries, H.

Rimmer, M. P.

Sasaki, O.

Shen, W.

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

Southwell, W. H.

Tang, F.

Tashiro, H.

van Brug, H.

Verhoglyad, A. G.

Wan, D.

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

Wang, H. T.

Wang, X.

Wu, B.

Wyant, J. C.

Yoshikawa, K.

Zheng, H.

Zhou, P.

Appl. Opt. (8)

A. G. Poleshchuk, E. G. Churin, V. P. Koronkevich, V. P. Korolkov, A. A. Kharissov, V. V. Cherkashin, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, and A. G. Verhoglyad, “Polar coordinate laser pattern generator for fabrication of diffractive optical elements with arbitrary structure,” Appl. Opt.38(8), 1295–1301 (1999).
[CrossRef] [PubMed]

P. Zhou and J. H. Burge, “Coupling of surface roughness to the performance of computer-generated holograms,” Appl. Opt.46(26), 6572–6576 (2007).
[CrossRef] [PubMed]

F. Dai, F. Tang, X. Wang, O. Sasaki, and P. Feng, “Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms,” Appl. Opt.51(21), 5028–5037 (2012).
[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).
[CrossRef] [PubMed]

G. W. R. Leibbrandt, G. Harbers, and P. J. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt.35(31), 6151–6161 (1996).

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]

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]

J. Opt. Soc. Am. (8)

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

Opt. Eng. (1)

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

Opt. Express (5)

Other (3)

D. Malacara and S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.13.

J. W. Goodman, “Analog optical information processing,” in Introduction to Fourier optics, 2nd ed., (McGraw-Hill, 1996), Chap. 8.

M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., (Wiley, 1992), Chap.1.

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Figures (7)

Fig. 1
Fig. 1

(a) Type 1: Wavefront slopes in x- and y-directions are obtained in order by part rotation. (b) Type 2: Wavefront slopes in x- and y-directions are obtained at the same time. (c) Orthogonality problem between x- and y-shearing-directions.

Fig. 2
Fig. 2

Four measurements scheme to estimate exact αj.

Fig. 3
Fig. 3

(a) Original wavefront. Its peak-to-valley (PV) and root-mean-square (rms) values were about 727.8 nm and 86.8 nm, respectively. (b) Wavefront error with the rotational inaccuracy, in which the direction of α1 was defined in Fig. 2 (maximum error in PV: 20.59 nm at α1 = 88°, maximum error in rms: 3.14 nm at α1 = 88°).

Fig. 4
Fig. 4

(a) Residual wavefront error by using the proposed algorithm (maximum error in PV value: 0.9 nm at α1 = 91.5 o, maximum error in rms: 0.19 nm at α1 = 91 o), and a (b) Wavefront error comparison, with without application of the algorithm. In this graph, the blue and red lines correspond to Fig. 4(a) and Fig. 3(b), respectively.

Fig. 5
Fig. 5

(a) Photographic view of a DVD pickup lens. (b) Measured wavefront of the lens in LSI without (PV: 209.6 nm, rms: 30.4 nm), and (c) with application of the proposed algorithm (PV: 172.8 nm, rms: 30.1 nm). α1, α2, and α3 were estimated to be 90.33°, 179.74° and 270.40 o, respectively. (d) Wavefront error due to the rotational inaccuracy (PV: 8.1 nm, rms: 1.27 nm). (e) Measured Zernike coefficients obtained by the LSI and a commercial Fizeau interferometer.

Fig. 6
Fig. 6

(a) Photographic view of a cast of an f-theta lens and its target area (6 mm x 6 mm). (b) Whole surface shape of the cast (designed shape).

Fig. 7
Fig. 7

Form error of the cast due to the rotational inaccuracy (a) without and (b) with the compensation. The target area was 6 mm x 6 mm, as shown in Fig. 6(a). (c) Form error according to α1.

Equations (15)

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W(r,θ)= l,k R l k (r)[ c lk cos(kθ)+ d lk sin(kθ)]
W(r,θα)= l,k R l k (r)[ c lk cos(kθkα)+ d lk sin(kθkα)] = l,k R l k (r)[( c lk cos(kα) d lk sin(kα))cos(kθ)+( d lk cos(kα)+ c lk sin(kα))sin(kθ)] = l,k R l k (r)[ c lk ' cos(kθ)+ d lk ' sin(kθ)]
[ c lk ' d lk ' ]= [ cosα sinα sinα cosα ][ c lk d lk ]=A[ c lk d lk ]fork=1,and [ c lk ' d lk ' ]= [ cos(kα) sin(kα) sin(kα) cos(kα) ][ c lk d lk ]= A k [ c lk d lk ]forarbitraryk
W(r,θα)= l,k R l k (r)[ c lk cos(kθkα)+ d lk sin(kθkα)] = l,k R l k (r)[ c lk cos(kθ)+ d lk sin(kθ)] cos(kα) + l,k R l k (r)[ d lk cos(kθ)+ c lk sin(kθ)] sin(kα) = k [ W k (r,θ)cos(kα)+ W k (r,θ 90 0 )sin(kα)] = k [ W k (r,θ)cos(kα)+ W ˜ k (r,θ)sin(kα)]
[ c lk ' d lk ' ]= [ cos 90 o sin 90 o sin 90 o cos 90 o ][ c lk d lk ]= [ 0 1 1 0 ][ c lk d lk ]=[ d lk c lk ]
W k (r,θ)= l L R l k (r)[ c lk cos(kθ)+ d lk sin(kθ)] = l L β l k Z l k (x,y) , where Z l k (x,y) R l k (r){ cos sin }(kθ)
W ^ j k x | j=0 = W ^ k (r,θ) x = l L β l k Z l k (x,y) x cos(k α 0 )+ l L β ˜ l k Z l k (x,y) x sin(k α 0 ), W ^ j k x | j=1 = W ^ k (r,θ α 1 ) x = l L β l k Z l k (x,y) x cos(k α 1 )+ l L β ˜ l k Z l k (x,y) x sin(k α 1 ), W ^ j k x | j=2 = W ^ k (r,θ α 2 ) x = l L β l k Z l k (x,y) x cos(k α 2 )+ l L β ˜ l k Z l k (x,y) x sin(k α 2 ), W ^ j k x | j=3 = W ^ k (r,θ α 3 ) x = l L β l k Z l k (x,y) x cos(k α 3 )+ l L β ˜ l k Z l k (x,y) x sin(k α 3 )
D ^ j k = W ^ j k (r,θ) x W ^ 0 k (r,θ) x = l K β l k Z l k (x,y) x [cos(k α j )1]+ l K β ˜ l k Z l k (x,y) x sin(k α j ).
F l k = j=0 N1 { D lj k D ^ lj k } 2 = j=0 N1 { β l k Z l k (x,y) x [cos(k α j )1]+ β ˜ l k Z l k (x,y) x sin(k α j ) D ^ lj k } 2 ,and F j k = l L { D lj k D ^ lj k } 2 = l L { β l k Z l k (x,y) x [cos(k α j )1]+ β ˜ l k Z l k (x,y) x sin(k α j ) D ^ lj k } 2 .
F l k β l k =0, F l k β ˜ l k =0, F j k cos(k α j ) =0,and F j k sin(k α j ) =0.
M[ β l k β ˜ l k ]=N, whereM=[ j=0 N1 [cos(k α j )1] 2 Z l k (x,y) x j=0 N1 sin(k α j )[cos(k α j )1] Z l k (x,y) x j=0 N1 sin(k α j )[cos(k α j )1] Z l k (x,y) x j=0 N1 sin 2 (k α j ) Z l k (x,y) x ], N=[ j=0 N1 D ^ lj k [cos(k α j )1] j=0 N1 D ^ lj k sin(k α j ) ],and
[ β l k β ˜ l k ]= M 1 N.
P[ cos(k α j ) sin(k α j ) ]=Q, whereP=[ l L [ β l k Z l k (x,y) x ] 2 l L β l k β ˜ l k [ Z l k (x,y) x ] 2 l L β l k β ˜ l k [ Z l k (x,y) x ] 2 l L [ β ˜ l k Z l k (x,y) x ] 2 ], Q=[ l L { D ^ lj k β l k Z l k (x,y) x + [ β l k Z l k (x,y) x ] 2 } l L { D ^ lj k β ˜ l k Z l k (x,y) x + β l k β ˜ l k [ Z l k (x,y) x ] 2 } ],and
[ cos(k α j ) sin(k α j ) ]= P 1 Q.
z= C 1 x 2 1+ 1(K+1) C 1 2 x 2 + m=3 6 A m x m + C 2 y 2 (1+ m=3 6 B m x m ) 1+ 1 C 2 y( m=3 6 B m x m ) ) 2 ,

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