Abstract

We propose and demonstrate multiple shearing interferometry for measuring two-dimensional phase object. Multi-shear interference can effectively eliminate the problem of spectral leakage that results from the single-shear interference. The Fourier coefficients of a two-dimensional wavefront are computed from phase differences obtained from multiple shearing interferograms, which are acquired by a shearing interferometer, and the desired phase is then reconstructed. Numerical and optical tests have confirmed that the multiple shearing interferometry has a higher recovery accuracy than single-shear interferometry and the reconstruction precision increases as the number of shear steps increases.

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    [CrossRef]
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2011 (2)

2009 (2)

2007 (1)

2006 (2)

2005 (1)

2004 (1)

2000 (3)

1999 (1)

1998 (1)

T. Nomura, K. Kamiya, H. Miyashiro, S. Okuda, H. Tashiro, and K. Yoshikawa, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng.22(4), 185–189 (1998).
[CrossRef]

1997 (1)

1996 (3)

1995 (1)

1987 (1)

1986 (1)

1983 (1)

1982 (1)

1980 (1)

1977 (1)

1975 (1)

1947 (1)

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc.59(6), 940–950-2 (1947).
[CrossRef]

Bahk, S. W.

Bates, W. J.

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc.59(6), 940–950-2 (1947).
[CrossRef]

Bon, P.

Chen, J.

Cohen, M.

da Silva, P.

Dainty, C.

Ding, J.

Druart, G.

Dubra, A.

Eiju, T.

Elster, C.

Falldorf, C.

Fan, Y. X.

Flynn, T. J.

Fornaro, G.

Franceschetti, G.

Freischlad, K. R.

Guérineau, N.

Guo, C. S.

Harbers, G.

Hariharan, P.

Heimbach, Y.

Hudgin, R. H.

Idir, M.

Ina, H.

Itoh, M.

Jin, Z.

Jüptner, W.

Kamiya, K.

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]

T. Nomura, K. Kamiya, H. Miyashiro, S. Okuda, H. Tashiro, and K. Yoshikawa, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng.22(4), 185–189 (1998).
[CrossRef]

Kasana, R. S.

Kobayashi, S.

Koliopoulos, C. L.

Kunst, P. J.

Lanari, R.

Legarda-Sáenz, R.

Leibbrandt, G. W. R.

Liang, P.

Malacara, D.

Marroquin, J. L.

Maucort, G.

Mercère, P.

Miyashiro, H.

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]

T. Nomura, K. Kamiya, H. Miyashiro, S. Okuda, H. Tashiro, and K. Yoshikawa, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng.22(4), 185–189 (1998).
[CrossRef]

Monneret, S.

Nomura, T.

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]

T. Nomura, K. Kamiya, H. Miyashiro, S. Okuda, H. Tashiro, and K. Yoshikawa, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng.22(4), 185–189 (1998).
[CrossRef]

Okuda, S.

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]

T. Nomura, K. Kamiya, H. Miyashiro, S. Okuda, H. Tashiro, and K. Yoshikawa, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng.22(4), 185–189 (1998).
[CrossRef]

Oreb, B. F.

Paterson, C.

Primot, J.

Ribak, E. N.

Rimmer, M. P.

Rivera, M.

Rizzi, J.

Rodríguez-Vera, R.

Rosenbruch, K.-J.

Sansosti, E.

Servin, M.

Southwell, W. H.

Takeda, M. H.

Talmi, A.

Tashiro, H.

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]

T. Nomura, K. Kamiya, H. Miyashiro, S. Okuda, H. Tashiro, and K. Yoshikawa, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng.22(4), 185–189 (1998).
[CrossRef]

Tian, X.

Trujillo-Schiaffino, G.

Velghe, S.

Vincent, G.

von Kopylow, C.

Wang, H. T.

Wattellier, B.

Weingärtner, I.

Weitkamp, T.

Wyant, J. C.

Yatagai, T.

Yoshikawa, K.

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]

T. Nomura, K. Kamiya, H. Miyashiro, S. Okuda, H. Tashiro, and K. Yoshikawa, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng.22(4), 185–189 (1998).
[CrossRef]

Zhai, S. H.

Appl. Opt. (10)

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]

R. S. Kasana and K.-J. Rosenbruch, “Determination of the refractive index of a lens using the Murty shearing interferometer,” Appl. Opt.22(22), 3526–3531 (1983).
[CrossRef] [PubMed]

A. Dubra, C. Paterson, and C. Dainty, “Wave-front reconstruction from shear phase maps by use of the discrete Fourier transform,” Appl. Opt.43(5), 1108–1113 (2004).
[CrossRef] [PubMed]

X. Tian, M. Itoh, and T. Yatagai, “Simple algorithm for large-grid phase reconstruction of lateral-shearing interferometry,” Appl. Opt.34(31), 7213–7220 (1995).
[CrossRef] [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]

C. Elster, “Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears,” Appl. Opt.39(29), 5353–5359 (2000).
[CrossRef] [PubMed]

C. Falldorf, Y. Heimbach, C. von Kopylow, and W. Jüptner, “Efficient reconstruction of spatially limited phase distributions from their sheared representation,” Appl. Opt.46(22), 5038–5043 (2007).
[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. Servin, D. Malacara, and J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt.35(22), 4343–4348 (1996).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt.26(13), 2504–2506 (1987).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Opt. Express (4)

Opt. Lett. (3)

Precis. Eng. (1)

T. Nomura, K. Kamiya, H. Miyashiro, S. Okuda, H. Tashiro, and K. Yoshikawa, “Shape measurements of mirror surfaces with a lateral-shearing interferometer during machine running,” Precis. Eng.22(4), 185–189 (1998).
[CrossRef]

Proc. Phys. Soc. (1)

W. J. Bates, “A wavefront shearing interferometer,” Proc. Phys. Soc.59(6), 940–950-2 (1947).
[CrossRef]

Other (1)

M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), 123–172.

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Figures (5)

Fig. 1
Fig. 1

Plot of the noise coefficient for different shear amounts versus the number of sample points.

Fig. 2
Fig. 2

Computer simulation of wavefront reconstruction from multiple phase differences in orthogonal directions: (a) original phase function, (b) x-directional phase difference (4-pixel shear), (c) y-directional phase difference (4-pixel shear), and (d) reconstructed distribution using multi shears of 4, 5, and 6 pixels.

Fig. 3
Fig. 3

Comparison between reconstruction accuracy of single-shear and multi-shear methods: (a) original phase map, (b) difference between original and reconstructed phases obtained using single-shear algorithm, and (c) difference between original and reconstructed phases obtained using multi-shear algorithm.

Fig. 4
Fig. 4

Three-wave lateral shearing interferometer based on SLM with four lenses (L1, L2, L3 and L4) and a rotating ground glass that can lower the spatial coherence of light in order to reduce speckle noise.

Fig. 5
Fig. 5

Optic surface testing with single- and multiple-shearing interferometer based on SLM: reconstructed 2-D phase map (top) and 1-D relief profile (bottom) using (a) single-shear of 4 pixels and (b) multi-shear of 1, 2, 3, 4, 5, 6, and 7 pixels.

Tables (5)

Tables Icon

Table 1 Noise coefficient versus the shear amount s

Tables Icon

Table 2 Deviation of reconstructed phase from original phase using singe-shear versus shear amount s (Units of relative RMS and PV are in percent).

Tables Icon

Table 3 Deviation of the reconstructed phase from the original phase using multi-shear versus shear amount s (Units of relative RMS and PV are in percent)..

Tables Icon

Table 4 Deviation of the recovered phase from the original phase under different spectral leaking(case 1- the phase φ(x, y) in Eq. (18), case 2- the phase “B” in Fig. 4(a))

Tables Icon

Table 5 RMS of the reconstructed phase using multi-shear versus shear amount s ( σ n = 0.100)

Equations (20)

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φ(m,n)= p=0 N1 q=0 N1 α pq Z pq (m,n), m,n=0,1,2,...,N1,
Z pq (m,n)= 1 N exp[ i2π N (pm+qn) ], p,q=0,1,2...,N1,
D x (m,n)=φ(m,n)φ(ms,n),
D y (m,n)=φ(m,n)φ(m,ns),
D x (m,n)= p=0 N1 q=0 N1 α pq Z pq (m,n)[ 1exp( i2πps N ) ],
D y (m,n)= p=0 N1 q=0 N1 α pq Z pq (m,n)[ 1exp( i2πqs N ) ].
F= m=0 N1 n=0 N1 {[ D x (m,n) D x (m,n) ] 2 + [ D y (m,n) D y (m,n)] 2 }.
F α pq =0 .
α pq = 1 4[ sin 2 (πps/N)+ sin 2 (πqs/N)] { [ 1exp( i2πps N ) ]F T pq { D x (m,n)} } +{ [ 1exp( i2πqs N ) ]F T pq { D y (m,n)} },
ε= m=0 N1 n=0 N1 { j=1 K [ D x ' j (m,n) D x j (m,n)] 2 + j=1 K [ D y ' j (m,n) D y j (m,n) ] 2 } , j=1,2...k.
α(p,q)= j=1 K (1 e i2πp s j /N )F T pq { D x ' j (m,n)}+ j=1 K (1 e i2πq s j /N )F T pq { D y ' j (m,n)} j=1 K [ 4 sin 2 ( πp s j N )+4 sin 2 ( πq s j N ) ] ,
α(p,q)= j=1 K [ b pq xj F T pq { D x ' j (m,n)}+ b pq yj F T pq { D y ' j (m,n)} ]
( b pq xj )= 1exp( i2πp s j N ) j=1 K { 4[ sin 2 (πp s j /N)+ sin 2 (πq s j /N)] } ,
( b pq yj )= 1exp( i2πq s j N ) j=1 K { 4[ sin 2 (πp s j /N)+ sin 2 (πq s j /N)] } .
n u i (k,l) n v j (m,n) = σ n 2 δ( u,v )δ( i,j )δ( k,m )δ( l,n ),
σ a 2 (p,q)= j=1 K ( | b pq xj | 2 + | b pq yj | 2 ) σ n 2 .
σ φ 2 = 1 N 2 m,n | n φ ( m,n ) | 2 = 1 N 2 m,n σ a 2 ( p,q ) .
C= σ φ 2 σ n 2 = 1 N 2 p,q j=1 K ( | b pq xj | 2 + | b pq yj | 2 ) ,
C= 1 N 2 p,q 1 j=1 K [4 sin 2 ( πp s j N ) +4 sin 2 ( πq s j N )] .
φ(x,y)=1.014×[( x 2 3× y 2 )x+y+ x 2 y 2 ].

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