Abstract

We propose an approach based on geometric phase for performing several types of shearing interferometry through a robust, compact, common-path setup. The key elements are two identical parallel plates with spatially varying birefringence distributions, which perform the shearing by writing opposite geometric phases on the two circular polarization components of the linearly polarized incident wavefront. This setup allows the independent control of the shearing magnitude and relative phase of the two wavefront replicas. The approach is first illustrated for the simplest case of lateral shearing, and then extended to other geometries where the magnitude and direction of the shear vary smoothly over the wavefront.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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2018 (1)

G. C. T. Malhorta, Phys. Rev. Lett. 120, 233602 (2018).
[Crossref]

2017 (2)

B. Piccirillo, M. F. Picardi, L. Marrucci, and E. Santamato, Eur. J. Phys. 38, 034007 (2017).
[Crossref]

L. Gu, L. Liu, S. Hu, A. Zeng, and H. Huang, Opt. Rev. 24, 600 (2017).
[Crossref]

2015 (2)

T. Ling, Y. Yang, D. Liu, X. Yue, J. Jiang, J. Bai, and Y. Shen, Appl. Opt. 54, 8913 (2015).
[Crossref]

B. Piccirillo, S. Slussarenko, L. Marrucci, and E. Santamato, Nat. Commun. 6, 8606 (2015).
[Crossref]

2014 (1)

2013 (1)

L. Wang, L. Liu, Z. Luan, J. Sun, and Y. Zhou, Optik 124, 1215 (2013).
[Crossref]

2010 (1)

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 97, 241104 (2010).
[Crossref]

2009 (3)

2007 (1)

2006 (2)

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

S. Velghe, J. Primot, N. Guérineau, R. Haidar, S. Demoustier, M. Cohen, and B. Wattellier, Proc. SPIE 6292, 62920E (2006).
[Crossref]

2003 (1)

2001 (1)

1987 (1)

M. Berry, J. Mod. Opt. 34, 1401 (1987).
[Crossref]

1985 (1)

1980 (1)

M. V. R. K. Murty and R. P. Shukla, Opt. Eng. 19, 19 (1980).
[Crossref]

1977 (1)

S. Debrus, Opt. Commun. 20, 257 (1977).
[Crossref]

1974 (1)

Y. Y. Hung, Opt. Commun. 11, 132 (1974).
[Crossref]

1973 (1)

1970 (1)

1961 (1)

P. Hariharan and D. Sen, J. Sci. Instrum. 38, 428 (1961).
[Crossref]

1956 (1)

S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).
[Crossref]

1947 (1)

W. J. Bates, Proc. Phys. Soc. 59, 940 (1947).
[Crossref]

1942 (2)

F. Zernike, Physica 9, 686 (1942).
[Crossref]

F. Zernike, Physica 9, 974 (1942).
[Crossref]

Bai, J.

Bates, W. J.

W. J. Bates, Proc. Phys. Soc. 59, 940 (1947).
[Crossref]

Berry, M.

M. Berry, J. Mod. Opt. 34, 1401 (1987).
[Crossref]

Bon, P.

Cohen, M.

S. Velghe, J. Primot, N. Guérineau, R. Haidar, S. Demoustier, M. Cohen, and B. Wattellier, Proc. SPIE 6292, 62920E (2006).
[Crossref]

D’Ambrosio, V.

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 97, 241104 (2010).
[Crossref]

Debrus, S.

S. Debrus, Opt. Commun. 20, 257 (1977).
[Crossref]

Delisle, C.

Demoustier, S.

S. Velghe, J. Primot, N. Guérineau, R. Haidar, S. Demoustier, M. Cohen, and B. Wattellier, Proc. SPIE 6292, 62920E (2006).
[Crossref]

Dixon, P. B.

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
[Crossref]

Falldorf, C.

Griffin, D. W.

Gu, L.

L. Gu, L. Liu, S. Hu, A. Zeng, and H. Huang, Opt. Rev. 24, 600 (2017).
[Crossref]

Guérineau, N.

S. Velghe, J. Primot, N. Guérineau, R. Haidar, S. Demoustier, M. Cohen, and B. Wattellier, Proc. SPIE 6292, 62920E (2006).
[Crossref]

Haidar, R.

S. Velghe, J. Primot, N. Guérineau, R. Haidar, S. Demoustier, M. Cohen, and B. Wattellier, Proc. SPIE 6292, 62920E (2006).
[Crossref]

Hariharan, P.

P. Hariharan and D. Sen, J. Sci. Instrum. 38, 428 (1961).
[Crossref]

Howell, J. C.

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
[Crossref]

Hu, S.

L. Gu, L. Liu, S. Hu, A. Zeng, and H. Huang, Opt. Rev. 24, 600 (2017).
[Crossref]

Huang, H.

L. Gu, L. Liu, S. Hu, A. Zeng, and H. Huang, Opt. Rev. 24, 600 (2017).
[Crossref]

Hung, Y. Y.

Y. Y. Hung, Opt. Commun. 11, 132 (1974).
[Crossref]

Jiang, J.

Jordan, A. N.

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
[Crossref]

Jüptner, W.

Karimi, E.

Kopylow, C. v.

Kothiyal, M. P.

Lee, H.-H.

Ling, T.

Liu, D.

Liu, L.

L. Gu, L. Liu, S. Hu, A. Zeng, and H. Huang, Opt. Rev. 24, 600 (2017).
[Crossref]

L. Wang, L. Liu, Z. Luan, J. Sun, and Y. Zhou, Optik 124, 1215 (2013).
[Crossref]

Luan, Z.

L. Wang, L. Liu, Z. Luan, J. Sun, and Y. Zhou, Optik 124, 1215 (2013).
[Crossref]

Ma, Y.

Malacara, D.

D. Malacara, Optical Shop Testing, Wiley Series in Pure and Applied Optics (Wiley, 2007).

Malhorta, G. C. T.

G. C. T. Malhorta, Phys. Rev. Lett. 120, 233602 (2018).
[Crossref]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

Marrucci, L.

B. Piccirillo, M. F. Picardi, L. Marrucci, and E. Santamato, Eur. J. Phys. 38, 034007 (2017).
[Crossref]

B. Piccirillo, S. Slussarenko, L. Marrucci, and E. Santamato, Nat. Commun. 6, 8606 (2015).
[Crossref]

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 97, 241104 (2010).
[Crossref]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[Crossref]

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

Maucort, G.

Monneret, S.

Murty, M. V. R. K.

M. V. R. K. Murty and R. P. Shukla, Opt. Eng. 19, 19 (1980).
[Crossref]

M. V. R. K. Murty, Appl. Opt. 9, 1146 (1970).
[Crossref]

Osten, S.

Pancharatnam, S.

S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).
[Crossref]

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

Park, S.-H.

Picardi, M. F.

B. Piccirillo, M. F. Picardi, L. Marrucci, and E. Santamato, Eur. J. Phys. 38, 034007 (2017).
[Crossref]

Piccirillo, B.

B. Piccirillo, M. F. Picardi, L. Marrucci, and E. Santamato, Eur. J. Phys. 38, 034007 (2017).
[Crossref]

B. Piccirillo, S. Slussarenko, L. Marrucci, and E. Santamato, Nat. Commun. 6, 8606 (2015).
[Crossref]

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 97, 241104 (2010).
[Crossref]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[Crossref]

Primot, J.

S. Velghe, J. Primot, N. Guérineau, R. Haidar, S. Demoustier, M. Cohen, and B. Wattellier, Proc. SPIE 6292, 62920E (2006).
[Crossref]

Santamato, E.

B. Piccirillo, M. F. Picardi, L. Marrucci, and E. Santamato, Eur. J. Phys. 38, 034007 (2017).
[Crossref]

B. Piccirillo, S. Slussarenko, L. Marrucci, and E. Santamato, Nat. Commun. 6, 8606 (2015).
[Crossref]

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 97, 241104 (2010).
[Crossref]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[Crossref]

Sen, D.

P. Hariharan and D. Sen, J. Sci. Instrum. 38, 428 (1961).
[Crossref]

Shen, Y.

Shukla, R. P.

M. V. R. K. Murty and R. P. Shukla, Opt. Eng. 19, 19 (1980).
[Crossref]

Slussarenko, S.

B. Piccirillo, S. Slussarenko, L. Marrucci, and E. Santamato, Nat. Commun. 6, 8606 (2015).
[Crossref]

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 97, 241104 (2010).
[Crossref]

Starling, D. J.

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
[Crossref]

Sun, J.

L. Wang, L. Liu, Z. Luan, J. Sun, and Y. Zhou, Optik 124, 1215 (2013).
[Crossref]

Velghe, S.

S. Velghe, J. Primot, N. Guérineau, R. Haidar, S. Demoustier, M. Cohen, and B. Wattellier, Proc. SPIE 6292, 62920E (2006).
[Crossref]

Wang, K.

Wang, L.

L. Wang, L. Liu, Z. Luan, J. Sun, and Y. Zhou, Optik 124, 1215 (2013).
[Crossref]

Wattellier, B.

P. Bon, G. Maucort, B. Wattellier, and S. Monneret, Opt. Express 17, 13080 (2009).
[Crossref]

S. Velghe, J. Primot, N. Guérineau, R. Haidar, S. Demoustier, M. Cohen, and B. Wattellier, Proc. SPIE 6292, 62920E (2006).
[Crossref]

Wyant, J. C.

Yang, Y.

You, J.-H.

Yue, X.

Zeng, A.

L. Gu, L. Liu, S. Hu, A. Zeng, and H. Huang, Opt. Rev. 24, 600 (2017).
[Crossref]

Zernike, F.

F. Zernike, Physica 9, 686 (1942).
[Crossref]

F. Zernike, Physica 9, 974 (1942).
[Crossref]

Zhou, Y.

L. Wang, L. Liu, Z. Luan, J. Sun, and Y. Zhou, Optik 124, 1215 (2013).
[Crossref]

Zito, G.

Appl. Opt. (5)

Appl. Phys. Lett. (1)

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, Appl. Phys. Lett. 97, 241104 (2010).
[Crossref]

Eur. J. Phys. (1)

B. Piccirillo, M. F. Picardi, L. Marrucci, and E. Santamato, Eur. J. Phys. 38, 034007 (2017).
[Crossref]

J. Mod. Opt. (1)

M. Berry, J. Mod. Opt. 34, 1401 (1987).
[Crossref]

J. Sci. Instrum. (1)

P. Hariharan and D. Sen, J. Sci. Instrum. 38, 428 (1961).
[Crossref]

Nat. Commun. (1)

B. Piccirillo, S. Slussarenko, L. Marrucci, and E. Santamato, Nat. Commun. 6, 8606 (2015).
[Crossref]

Opt. Commun. (2)

Y. Y. Hung, Opt. Commun. 11, 132 (1974).
[Crossref]

S. Debrus, Opt. Commun. 20, 257 (1977).
[Crossref]

Opt. Eng. (1)

M. V. R. K. Murty and R. P. Shukla, Opt. Eng. 19, 19 (1980).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Opt. Rev. (1)

L. Gu, L. Liu, S. Hu, A. Zeng, and H. Huang, Opt. Rev. 24, 600 (2017).
[Crossref]

Optik (1)

L. Wang, L. Liu, Z. Luan, J. Sun, and Y. Zhou, Optik 124, 1215 (2013).
[Crossref]

Phys. Rev. Lett. (3)

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

G. C. T. Malhorta, Phys. Rev. Lett. 120, 233602 (2018).
[Crossref]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009).
[Crossref]

Physica (2)

F. Zernike, Physica 9, 686 (1942).
[Crossref]

F. Zernike, Physica 9, 974 (1942).
[Crossref]

Proc. Indian Acad. Sci. A (1)

S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).
[Crossref]

Proc. Phys. Soc. (1)

W. J. Bates, Proc. Phys. Soc. 59, 940 (1947).
[Crossref]

Proc. SPIE (1)

S. Velghe, J. Primot, N. Guérineau, R. Haidar, S. Demoustier, M. Cohen, and B. Wattellier, Proc. SPIE 6292, 62920E (2006).
[Crossref]

Other (1)

D. Malacara, Optical Shop Testing, Wiley Series in Pure and Applied Optics (Wiley, 2007).

Supplementary Material (1)

NameDescription
» Supplement 1       The document contains details about generalization of the method in the main text.

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Figures (7)

Fig. 1.
Fig. 1. Sketch of a GPSI. The linearly polarized light leaving the polarizer P can be described as the sum of two identical wavefronts with opposite circular polarizations. Two identical Λ -plates with separation ζ introduce a controlled shear between wavefront replicas, and an analyzer A selects the polarization component used for studying their interference. The bottom right inset shows the fast-axis distribution of the Λ -plates we fabricated, superposed to their experimental fringe pattern. The spatial period Λ = ( 1.32 ± 0.01 ) mm corresponds to a fast-axis rotation of π .
Fig. 2.
Fig. 2. Sketch of the experimental apparatus. (a) A He–Ne laser beam passes through a variable attenuator (VA) and is expanded with a telescope (BE, 3 × Mag). Two metallic mirrors ( M 1 and M 2 ), placed on flip mounts, enable including/excluding a spatial light modulator (SLM) for generating test wavefronts holographically. When M 1 and M 2 are down, optical components can be inserted into the apparatus and analyzed. The half-wave plate (HWP) is used to rotate the input linear polarization with respect to the axis of the analyzer A. A telescope is used to image onto the complementary metal-oxide semiconductor (CMOS) camera the test beam’s transverse cross section. The lenses L 1 and L 2 have focal lengths f 1 = 500 mm and f 2 = 100 mm , respectively.
Fig. 3.
Fig. 3. Linear shearing interferograms of light focused by lenses implemented holographically via a SLM. In each row, from the top down, f S = 400 mm , 500 mm, and 600 mm. The first and third columns contain, respectively, the experimental and theoretical profiles for ψ = 0 , while the second and fourth columns show the same for ψ = 90 ° . The plates’ separation is ζ = 12.0 mm .
Fig. 4.
Fig. 4. Linear shearing interferograms for light focused by a cylindrical lens of focal length f C = 400 mm with different orientations, implemented on a SLM. The rows correspond to angles γ = 0 ° , 30 ° , and 60° between the lens axis and the shearing direction x . The first and third rows show the experimental and theoretical profiles for ψ = 0 ° , while the second and fourth rows show the same for ψ = 90 ° . The plates’ separation is ζ = 12.0 mm .
Fig. 5.
Fig. 5. Linear shearing interferograms (with ζ = 20 mm ) for a beam emerging from a q -plate ( q = 1 ). The top row shows the measured profiles, and the bottom row shows the theoretical predictions. From left to right, the columns correspond to ψ = 0 ° , 90 ° , 86 ° , and 94°, respectively. Note that the sign of the x derivative can be inferred from either of the latter two.
Fig. 6.
Fig. 6. Sketch of a radial GPSI. This device is identical to the GPSI in Fig. 1, but with the Λ -plates replaced by GPLs with focal length f = ( 130 ± 1 ) mm . The bottom right inset shows the fast-axis distribution of the GPLs we fabricated, superimposed onto the experimental polarization fringe pattern of the GPL.
Fig. 7.
Fig. 7. Radial shearing interferogram of a cylindrical lens phase profile with focal length f = ( 400 ± 1 ) mm . The derivative pattern can be obtained as the difference between the interference pattern at ψ = + 45 ° and ψ = 45 ° , ψ being the mismatch angle between the input polarization direction and the axis of the analyzer A. On the upper line, we show a numerical simulation of the derivative pattern; on the lower line, we show the experimentally observed pattern.

Equations (3)

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E ( x , y , 0 ) = E ( x , y , 0 ) x = 1 2 E ( x , y , 0 ) ( C + + C ) .
E ± ( x , y , z ) = 1 2 E ( x λ z Λ , y , z ) exp ( ± i 2 π Λ x ) C ± ,
I ( ψ ) | E + e i ψ + E e i ψ | 2 = | S | 2 cos 2 ψ + | D | 2 sin 2 ψ + Im ( S D * ) sin 2 ψ ,

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