Abstract

Optical components manipulating both polarization and phase of wave fields find more and more applications in today’s optical systems. In particular, the polarization orientation may vary across the aperture. New measurement techniques and evaluation algorithms are needed to simultaneously characterize the properties of such elements. In this Letter, a general measurement algorithm for locally linear polarization distributions is presented, extending the methods of phase shifting interferometry to the simultaneous determination of polarization and phase. A class of evaluation algorithms is derived, and some example algorithms are described and tested for their resilience against systematic and stochastic stepping errors.

© 2012 Optical Society of America

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References

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  1. E. Hasman, G. Biener, A. Niv, and V. Kleiner, Prog. Opt. 47, 215 (2005).
    [CrossRef]
  2. Q. Zhan, Adv. Opt. Photon. 1, 1 (2009).
    [CrossRef]
  3. D. Goldstein, Polarized Light (Marcel Dekker, 2003).
  4. H. Schreiber and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), p. 547.
  5. Q. Zhan and J. R. Leger, Opt. Commun. 213, 241 (2002).
    [CrossRef]
  6. G. M. Lerman and U. Levy, Opt. Express 17, 23234 (2009).
    [CrossRef]
  7. V. Nercissian, F. Geier, A. Berger, S. Dmitriev, Z. Ghadyani, I. Harder, K. Mantel, and N. Lindlein, in Proceedings of 113th Annual Meeting of the DGaO (Deutsche Gesellschaft für angewandte Optik e.V., 2012).

2009

2005

E. Hasman, G. Biener, A. Niv, and V. Kleiner, Prog. Opt. 47, 215 (2005).
[CrossRef]

2002

Q. Zhan and J. R. Leger, Opt. Commun. 213, 241 (2002).
[CrossRef]

Berger, A.

V. Nercissian, F. Geier, A. Berger, S. Dmitriev, Z. Ghadyani, I. Harder, K. Mantel, and N. Lindlein, in Proceedings of 113th Annual Meeting of the DGaO (Deutsche Gesellschaft für angewandte Optik e.V., 2012).

Biener, G.

E. Hasman, G. Biener, A. Niv, and V. Kleiner, Prog. Opt. 47, 215 (2005).
[CrossRef]

Bruning, J. H.

H. Schreiber and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), p. 547.

Dmitriev, S.

V. Nercissian, F. Geier, A. Berger, S. Dmitriev, Z. Ghadyani, I. Harder, K. Mantel, and N. Lindlein, in Proceedings of 113th Annual Meeting of the DGaO (Deutsche Gesellschaft für angewandte Optik e.V., 2012).

Geier, F.

V. Nercissian, F. Geier, A. Berger, S. Dmitriev, Z. Ghadyani, I. Harder, K. Mantel, and N. Lindlein, in Proceedings of 113th Annual Meeting of the DGaO (Deutsche Gesellschaft für angewandte Optik e.V., 2012).

Ghadyani, Z.

V. Nercissian, F. Geier, A. Berger, S. Dmitriev, Z. Ghadyani, I. Harder, K. Mantel, and N. Lindlein, in Proceedings of 113th Annual Meeting of the DGaO (Deutsche Gesellschaft für angewandte Optik e.V., 2012).

Goldstein, D.

D. Goldstein, Polarized Light (Marcel Dekker, 2003).

Harder, I.

V. Nercissian, F. Geier, A. Berger, S. Dmitriev, Z. Ghadyani, I. Harder, K. Mantel, and N. Lindlein, in Proceedings of 113th Annual Meeting of the DGaO (Deutsche Gesellschaft für angewandte Optik e.V., 2012).

Hasman, E.

E. Hasman, G. Biener, A. Niv, and V. Kleiner, Prog. Opt. 47, 215 (2005).
[CrossRef]

Kleiner, V.

E. Hasman, G. Biener, A. Niv, and V. Kleiner, Prog. Opt. 47, 215 (2005).
[CrossRef]

Leger, J. R.

Q. Zhan and J. R. Leger, Opt. Commun. 213, 241 (2002).
[CrossRef]

Lerman, G. M.

Levy, U.

Lindlein, N.

V. Nercissian, F. Geier, A. Berger, S. Dmitriev, Z. Ghadyani, I. Harder, K. Mantel, and N. Lindlein, in Proceedings of 113th Annual Meeting of the DGaO (Deutsche Gesellschaft für angewandte Optik e.V., 2012).

Mantel, K.

V. Nercissian, F. Geier, A. Berger, S. Dmitriev, Z. Ghadyani, I. Harder, K. Mantel, and N. Lindlein, in Proceedings of 113th Annual Meeting of the DGaO (Deutsche Gesellschaft für angewandte Optik e.V., 2012).

Nercissian, V.

V. Nercissian, F. Geier, A. Berger, S. Dmitriev, Z. Ghadyani, I. Harder, K. Mantel, and N. Lindlein, in Proceedings of 113th Annual Meeting of the DGaO (Deutsche Gesellschaft für angewandte Optik e.V., 2012).

Niv, A.

E. Hasman, G. Biener, A. Niv, and V. Kleiner, Prog. Opt. 47, 215 (2005).
[CrossRef]

Schreiber, H.

H. Schreiber and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), p. 547.

Zhan, Q.

Q. Zhan, Adv. Opt. Photon. 1, 1 (2009).
[CrossRef]

Q. Zhan and J. R. Leger, Opt. Commun. 213, 241 (2002).
[CrossRef]

Adv. Opt. Photon.

Opt. Commun.

Q. Zhan and J. R. Leger, Opt. Commun. 213, 241 (2002).
[CrossRef]

Opt. Express

Prog. Opt.

E. Hasman, G. Biener, A. Niv, and V. Kleiner, Prog. Opt. 47, 215 (2005).
[CrossRef]

Other

V. Nercissian, F. Geier, A. Berger, S. Dmitriev, Z. Ghadyani, I. Harder, K. Mantel, and N. Lindlein, in Proceedings of 113th Annual Meeting of the DGaO (Deutsche Gesellschaft für angewandte Optik e.V., 2012).

D. Goldstein, Polarized Light (Marcel Dekker, 2003).

H. Schreiber and J. H. Bruning, in Optical Shop Testing, D. Malacara, ed. (Wiley, 2007), p. 547.

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Figures (3)

Fig. 1.
Fig. 1.

Principle of a polarization and phase measuring two-beam interferometer with LS, light source; DET, detection and imaging; BS, beam splitter; and BC, beam combiner.

Fig. 2.
Fig. 2.

Reconstructed phase Φ ϵ and difference to the original phase Φ Φ ϵ .

Fig. 3.
Fig. 3.

Remaining phase error PV ( Φ Φ ϵ ) over PV of a linear ϵ 1 (u.l.), quadratic ϵ 2 (l.r.), cubic ϵ 3 (l.l.) and stochastic Rand ( ϵ 1 , 2 , 3 ) (l.r.) introduced stepping error with PV ( φ i φ i ϵ ) for different algorithms.

Equations (18)

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E Obj = E x + E y e i Φ ( e x cos Ω + e y sin Ω ) .
H i = I 0 [ 1 + V cos ( Φ φ i ) cos ( Ω ω i ) ] ,
c⃗ i = A ^ i · b⃗ ,
c⃗ i a⃗ i · H i , A ^ i a⃗ i · a⃗ i T , a⃗ i ( 1 cos a p i sin a p i cos a n i sin a n i ) , b⃗ I 0 V 2 ( 2 V cos b p sin b p cos b n sin b n ) ,
b p Φ + Ω , a p i φ i + ω i , b n Φ Ω , a n i φ i ω i .
[ 1 cos a p i sin a p i cos a n i sin a n i cos a p i ( cos a p i ) 2 cos a p i sin a p i cos a n i cos a p i cos a p i sin a n i sin a p i cos a p i sin a p i ( sin a p i ) 2 cos a n i sin a n i sin a n i sin a p i cos a n i cos a n i cos a p i cos a n i sin a p i ( cos a n i ) 2 cos a n i sin a n i sin a n i cos a p i sin a n i sin a n i sin a p i cos a n i sin a n i ( sin a n i ) 2 ] .
i = 1 N c⃗ i = A ^ · b⃗ , A ^ i = 1 N A ^ i .
b⃗ = A ^ 1 · i = 1 N c⃗ i .
b p = arctan b ( 3 ) b ( 2 ) ( Φ + Ω ) mod 2 π , b n = arctan b ( 5 ) b ( 4 ) ( Φ Ω ) mod 2 π , I 0 = b ( 1 ) , V = b ( 3 ) 2 + b ( 2 ) 2 + b ( 5 ) 2 + b ( 4 ) 2 b ( 1 ) ,
φ i k i · δ φ , k i { 0 ; ; N φ 1 } , ω i l i · δ ω , l i { 0 ; ; N ω 1 } ,
δ φ , ω = z φ , ω · 2 π N φ , ω 1 or δ φ , ω = z φ , ω · 2 π N φ , ω .
δ φ = 2 π N 1 , δ ω = 2 π N ,
b⃗ = ( 2.61 H 1 4.24 H 3 + 2.62 H 5 0.30 H 1 0.69 H 2 + 0.32 H 3 + 0.26 H 4 0.19 H 5 0.30 H 1 0.50 H 2 0.23 H 3 + 0.81 H 4 0.38 H 5 0.69 H 2 1.92 H 1 + 3.92 H 3 0.26 H 4 2.42 H 5 2.85 H 3 0.50 H 2 1.92 H 1 + 0.81 H 4 1.23 H 5 )
δ φ = 2 π N , δ ω = 4 π N .
b⃗ = ( 1 N i = 1 N H i 2 N i = 1 N cos ( 6 π k i N ) H i 2 N i = 1 N sin ( 6 π k i N ) H i 2 N i = 1 N cos ( 2 π k i N ) H i 2 N i = 1 N sin ( 2 π k i N ) H i ) .
δ φ , ω = 2 π N φ , ω and k i = [ 0 N φ , 0 N φ , 0 N φ , , 0 N φ ] , l i = [ 0 0 , 1 1 , , N ω 1 N ω 1 ] ,
b⃗ = ( 1 N ω N φ i = 1 N ω N φ H i 2 N ω N φ i = 1 N ω N φ cos ( 2 π k i N φ + 2 π l i N ω ) H i 2 N ω N φ i = 1 N ω N φ sin ( 2 π k i N φ + 2 π l i N ω ) H i 2 N ω N φ i = 1 N ω N φ cos ( 2 π k i N φ 2 π l i N ω ) H i 2 N ω N φ i = 1 N ω N φ sin ( 2 π k i N φ 2 π l i N ω ) H i ) .
φ i ϵ = ( φ i ) + ( ϵ 1 φ i ) + ( ϵ 2 φ i ) 2 + ( ϵ 3 φ i ) 3 , ω i ϵ = ( ω i ) + ( ϵ 1 ω i ) + ( ϵ 2 ω i ) 2 + ( ϵ 3 ω i ) 3 .

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