Abstract

Crossed grating 3×3 beam lateral shear interferometry for extended range wave front sensing is presented. A Fresnel diffraction pattern of two multiplicatively superimposed linear diffraction gratings each generating three diffraction orders is recorded. A simple solution employs a common crossed binary amplitude Ronchi grating with spatial filtering. Digital processing of a single-shot pattern includes separating multidirectional pairs of orthogonal lateral shear interferograms, retrieving second harmonics of their intensity distribution, and calculating shearing phases. Single-frame automatic fringe pattern processing based on the Hilbert–Huang transform is used for this purpose. Using second harmonics extends the aberration measurement range since they encode self-imaging free two-beam interferograms without contrast modulations. Experimental works corroborate the principle and capabilities of the proposed approach.

© 2016 Optical Society of America

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References

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2016 (1)

P. Han and J. Weng, J. Opt. 18, 055606 (2016).

2015 (3)

2014 (4)

J. Azana and H. Guillet de Chatellus, Phys. Rev. Lett. 112, 213902 (2014).
[Crossref]

M. Wielgus, K. Patorski, P. Etchepareborda, and A. Federico, Opt. Express 22, 10775 (2014).
[Crossref]

Z. Sunderland, K. Patorski, M. Wielgus, and K. Pokorski, Proc. SPIE 9441, 944111 (2014).
[Crossref]

M. Trusiak, M. Wielgus, and K. Patorski, Opt. Lasers Eng. 52, 230 (2014).
[Crossref]

2013 (1)

2011 (1)

2009 (1)

2000 (1)

1997 (1)

1996 (1)

1995 (1)

1986 (2)

1984 (1)

J. Jahns, A. W. Lohmann, and J. Ojeda Castaneda, Opt. Acta 31, 313 (1984).
[Crossref]

1980 (2)

K. Patorski, Opt. Laser Technol. 12, 267 (1980).
[Crossref]

R. Sudol, Proc. SPIE 240, 155 (1980).
[Crossref]

1978 (1)

1975 (1)

1974 (2)

P. Hariharan, W. H. Steel, and J. C. Wyant, Opt. Commun. 11, 317 (1974).
[Crossref]

J. C. Wyant, Appl. Opt. 13, 200 (1974).
[Crossref]

1973 (1)

1971 (2)

S. Yokozeki and T. Suzuki, Appl. Opt. 10, 1575 (1971).
[Crossref]

A. W. Lohmann and D. Silva, Opt. Commun. 2, 413 (1971).
[Crossref]

1964 (1)

1881 (1)

Rayleigh, Philos. Mag. 11(67), 196 (1881).
[Crossref]

1836 (1)

F. Talbot, Philos. Mag. 9, 401 (1836).

Azana, J.

J. Azana and H. Guillet de Chatellus, Phys. Rev. Lett. 112, 213902 (2014).
[Crossref]

Chen, J.

Cornejo-Rodriguez, A.

A. Cornejo-Rodriguez, in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007).

Dai, F.

J. Li, F. Tang, X. Wang, F. Dai, and H. Zhang, Appl. Opt. 54, 8070 (2015).
[Crossref]

J. Li, F. Tang, X. Wang, F. Dai, P. Feng, and S. Li, J. Opt. 17, 065401 (2015).
[Crossref]

Ding, J.

Etchepareborda, P.

Fan, Y. X.

Farrant, D. I.

Federico, A.

Feng, P.

J. Li, F. Tang, X. Wang, F. Dai, P. Feng, and S. Li, J. Opt. 17, 065401 (2015).
[Crossref]

Fried, D. L.

D. L. Fried, in Adaptive Optics for Astronomy, C243 of NATO Advanced Study Institute Series (Kluwer Academic, 1994), pp. 25–57.

Guerineau, N.

Guillet de Chatellus, H.

J. Azana and H. Guillet de Chatellus, Phys. Rev. Lett. 112, 213902 (2014).
[Crossref]

Han, P.

P. Han and J. Weng, J. Opt. 18, 055606 (2016).

Harbers, G.

Hariharan, P.

P. Hariharan, W. H. Steel, and J. C. Wyant, Opt. Commun. 11, 317 (1974).
[Crossref]

Hibino, K.

Jahns, J.

J. Jahns, A. W. Lohmann, and J. Ojeda Castaneda, Opt. Acta 31, 313 (1984).
[Crossref]

Koike, C.

Koike, T.

Kunst, P. J.

Leibbrandt, G. W. R.

Li, J.

J. Li, F. Tang, X. Wang, F. Dai, P. Feng, and S. Li, J. Opt. 17, 065401 (2015).
[Crossref]

J. Li, F. Tang, X. Wang, F. Dai, and H. Zhang, Appl. Opt. 54, 8070 (2015).
[Crossref]

Li, S.

J. Li, F. Tang, X. Wang, F. Dai, P. Feng, and S. Li, J. Opt. 17, 065401 (2015).
[Crossref]

Lohmann, A. W.

J. Jahns, A. W. Lohmann, and J. Ojeda Castaneda, Opt. Acta 31, 313 (1984).
[Crossref]

A. W. Lohmann and D. Silva, Opt. Commun. 2, 413 (1971).
[Crossref]

Marston, P. L.

Odate, A.

Ojeda Castaneda, J.

J. Jahns, A. W. Lohmann, and J. Ojeda Castaneda, Opt. Acta 31, 313 (1984).
[Crossref]

Oreb, B. F.

Otaki, K.

Patorski, K.

K. Patorski, M. Trusiak, and K. Pokorski, Opt. Lett. 40, 1089 (2015).
[Crossref]

M. Wielgus, K. Patorski, P. Etchepareborda, and A. Federico, Opt. Express 22, 10775 (2014).
[Crossref]

Z. Sunderland, K. Patorski, M. Wielgus, and K. Pokorski, Proc. SPIE 9441, 944111 (2014).
[Crossref]

M. Trusiak, M. Wielgus, and K. Patorski, Opt. Lasers Eng. 52, 230 (2014).
[Crossref]

M. Trusiak, K. Patorski, and K. Pokorski, Opt. Express 21, 28359 (2013).
[Crossref]

K. Patorski, J. Opt. Soc. Am. A 3, 1862 (1986).
[Crossref]

K. Patorski, Appl. Opt. 25, 4192 (1986).
[Crossref]

K. Patorski, Opt. Laser Technol. 12, 267 (1980).
[Crossref]

K. Patorski, in Progress in Optics, E. Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 1–108.

Pokorski, K.

Primot, J.

Rayleigh,

Rayleigh, Philos. Mag. 11(67), 196 (1881).
[Crossref]

Rimmer, M. P.

Ronchi, V.

Silva, D.

A. W. Lohmann and D. Silva, Opt. Commun. 2, 413 (1971).
[Crossref]

Sogno, L.

Steel, W. H.

P. Hariharan, W. H. Steel, and J. C. Wyant, Opt. Commun. 11, 317 (1974).
[Crossref]

Sudol, R.

R. Sudol, Proc. SPIE 240, 155 (1980).
[Crossref]

Sugaya, A.

Sugisaki, K.

Sunderland, Z.

Z. Sunderland, K. Patorski, M. Wielgus, and K. Pokorski, Proc. SPIE 9441, 944111 (2014).
[Crossref]

Suzuki, T.

Talbot, F.

F. Talbot, Philos. Mag. 9, 401 (1836).

Tang, F.

J. Li, F. Tang, X. Wang, F. Dai, and H. Zhang, Appl. Opt. 54, 8070 (2015).
[Crossref]

J. Li, F. Tang, X. Wang, F. Dai, P. Feng, and S. Li, J. Opt. 17, 065401 (2015).
[Crossref]

Trusiak, M.

Uchikawa, K.

Wang, H. T.

Wang, X.

J. Li, F. Tang, X. Wang, F. Dai, P. Feng, and S. Li, J. Opt. 17, 065401 (2015).
[Crossref]

J. Li, F. Tang, X. Wang, F. Dai, and H. Zhang, Appl. Opt. 54, 8070 (2015).
[Crossref]

Ward, B. K.

Weng, J.

P. Han and J. Weng, J. Opt. 18, 055606 (2016).

Wielgus, M.

M. Trusiak, M. Wielgus, and K. Patorski, Opt. Lasers Eng. 52, 230 (2014).
[Crossref]

M. Wielgus, K. Patorski, P. Etchepareborda, and A. Federico, Opt. Express 22, 10775 (2014).
[Crossref]

Z. Sunderland, K. Patorski, M. Wielgus, and K. Pokorski, Proc. SPIE 9441, 944111 (2014).
[Crossref]

Wyant, J. C.

Yokozeki, S.

Zhai, S. H.

Zhang, H.

Zhu, Y.

Appl. Opt. (11)

J. Opt. (2)

J. Li, F. Tang, X. Wang, F. Dai, P. Feng, and S. Li, J. Opt. 17, 065401 (2015).
[Crossref]

P. Han and J. Weng, J. Opt. 18, 055606 (2016).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

J. Jahns, A. W. Lohmann, and J. Ojeda Castaneda, Opt. Acta 31, 313 (1984).
[Crossref]

Opt. Commun. (2)

P. Hariharan, W. H. Steel, and J. C. Wyant, Opt. Commun. 11, 317 (1974).
[Crossref]

A. W. Lohmann and D. Silva, Opt. Commun. 2, 413 (1971).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

K. Patorski, Opt. Laser Technol. 12, 267 (1980).
[Crossref]

Opt. Lasers Eng. (1)

M. Trusiak, M. Wielgus, and K. Patorski, Opt. Lasers Eng. 52, 230 (2014).
[Crossref]

Opt. Lett. (1)

Philos. Mag. (2)

F. Talbot, Philos. Mag. 9, 401 (1836).

Rayleigh, Philos. Mag. 11(67), 196 (1881).
[Crossref]

Phys. Rev. Lett. (1)

J. Azana and H. Guillet de Chatellus, Phys. Rev. Lett. 112, 213902 (2014).
[Crossref]

Proc. SPIE (2)

Z. Sunderland, K. Patorski, M. Wielgus, and K. Pokorski, Proc. SPIE 9441, 944111 (2014).
[Crossref]

R. Sudol, Proc. SPIE 240, 155 (1980).
[Crossref]

Other (3)

K. Patorski, in Progress in Optics, E. Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 1–108.

A. Cornejo-Rodriguez, in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007).

D. L. Fried, in Adaptive Optics for Astronomy, C243 of NATO Advanced Study Institute Series (Kluwer Academic, 1994), pp. 25–57.

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Figures (4)

Fig. 1.
Fig. 1. Schematic geometry of the three-beam Ronchi test. G, diffraction grating; OL1, objective under test; OL2, collimating objective. For presentation clarity, only two diffraction orders are shown, the 0 (solid line) and the + 1 (dashed line) ones. The axial distance between the grating G and the tested beam focus plane (with spatial filter included when using binary gratings) is denoted as z . Focal lengths of OL1 and OL2 are f 1 and f 2 , respectively.
Fig. 2.
Fig. 2. Interference pattern of the lens under test obtained in the optical system of Fig. 1 for z M d 2 / λ ( M = 4 ) 63    mm . A square mask inserted in the frequency plane passed 3 × 3 lowest diffraction orders of the crossed grating G to form the fringe pattern.
Fig. 3.
Fig. 3. Modulus of the Fourier spectrum of the grating image shown in Fig. 2. The comatic shape of the diffraction spots stems from the testing of the lens’ spherical aberration by lateral shear interferometry.
Fig. 4.
Fig. 4. Shearing phase distributions (wrapped, cosine, and continuous) calculated from second harmonics of lateral shear patterns along the x and y directions extracted from the image of Fig. 2. Linear terms related to the grating axial location z , shown in Fig. 1, were subtracted. Note the obtained continuous phase distributions in spite of the presence of zero contrast bands and parasitic fringes in Fig. 2.

Equations (3)

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E ( x , y , OP ) = a 0 exp [ i k ψ ( x , y ) ] + a 1 exp { i k [ λ z x f 2 d + λ 2 z 2 d 2 + ψ ( x λ f 2 d , y ) ] } + a 1 exp { i k [ λ z x f 2 d + λ 2 z 2 d 2 + ψ ( x + λ f 2 d , y ) ] } ,
I ( x , y , OP ) = a 0 2 + 2 a 1 2 + 4 a 0 a 1 cos { π [ λ z d 2 + Δ 2 λ 2 ψ ( x , y ) x 2 ] } cos { π [ 2 z x f 2 d + 2 Δ λ ψ ( x , y ) x + Δ 3 3 λ 3 ψ ( x , y ) x 3 ] } + 2 a 1 2 cos { 2 π [ 2 z x f 2 d + 2 Δ λ ψ ( x , y ) x + Δ 3 3 λ 3 ψ ( x , y ) x 3 ] } .
Ψ ( x , y ) ψ ( x Δ , y ) + Δ ψ ( x , y ) x Δ 2 2 2 ψ ( x , y ) x 2 + Δ 3 6 3 ψ ( x , y ) x 3 ,

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