Abstract

We describe and analyze a method by which an optical polarization state is mapped to an image sensor. When placed in a Bayesian framework, the analysis allows a priori information about the polarization state to be introduced into the measurement. We show that when such a measurement is applied to a single photon, it eliminates exactly one fully polarized state, offering an important insight about the information gained from a single photon polarization measurement.

© 2013 OSA

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References

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  1. C.A. Skinner, “The polarimeter and its practical applications,” J. Franklin Inst.196, 721–750 (1923).
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  8. W. Sparks, T. Germer, J. MacKenty, and F. Snik, “Compact and robust method for full Stokes spectropolarimetry,” Appl. Opt.51, 5495–5511 (2012).
    [CrossRef] [PubMed]
  9. A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express15, 8411–8421 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8411 .
    [CrossRef] [PubMed]
  10. A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007), http://ao.osa.org/abstract.cfm?id=119864 .
    [CrossRef]
  11. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express18, 10777–10785 (2010).
    [CrossRef] [PubMed]
  12. R. D. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” Proc. Spie8227, 82270Q–82270Q-8 (2012).
    [CrossRef]
  13. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A.64, 052312–27 (2001).
    [CrossRef]

2012

R. D. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” Proc. Spie8227, 82270Q–82270Q-8 (2012).
[CrossRef]

W. Sparks, T. Germer, J. MacKenty, and F. Snik, “Compact and robust method for full Stokes spectropolarimetry,” Appl. Opt.51, 5495–5511 (2012).
[CrossRef] [PubMed]

2010

2008

2007

2006

2001

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A.64, 052312–27 (2001).
[CrossRef]

2000

1999

1985

1923

C.A. Skinner, “The polarimeter and its practical applications,” J. Franklin Inst.196, 721–750 (1923).
[CrossRef]

Alonso, M. A.

Azzam, R.

Beckley, A. M.

R. D. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” Proc. Spie8227, 82270Q–82270Q-8 (2012).
[CrossRef]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express18, 10777–10785 (2010).
[CrossRef] [PubMed]

Brady, D.

Brown, T. G.

R. D. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” Proc. Spie8227, 82270Q–82270Q-8 (2012).
[CrossRef]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express18, 10777–10785 (2010).
[CrossRef] [PubMed]

A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express15, 8411–8421 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8411 .
[CrossRef] [PubMed]

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007), http://ao.osa.org/abstract.cfm?id=119864 .
[CrossRef]

Chenault, D.

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (CRC Press, 1992).

DeHoog, E.

Dereniak, E. L.

Germer, T.

Goldstein, D.

Gori, F.

Guo, J.

James, D. F. V.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A.64, 052312–27 (2001).
[CrossRef]

Kudenov, M.

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A.64, 052312–27 (2001).
[CrossRef]

Luo, H.

MacKenty, J.

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A.64, 052312–27 (2001).
[CrossRef]

Oka, K.

Ramkhalawon, R. D.

R. D. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” Proc. Spie8227, 82270Q–82270Q-8 (2012).
[CrossRef]

Schwegerling, J.

Shaw, J.

Skinner, C.A.

C.A. Skinner, “The polarimeter and its practical applications,” J. Franklin Inst.196, 721–750 (1923).
[CrossRef]

Snik, F.

Sparks, W.

Spilman, A. K.

A. K. Spilman and T. G. Brown, “Stress-induced focal splitting,” Opt. Express15, 8411–8421 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8411 .
[CrossRef] [PubMed]

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007), http://ao.osa.org/abstract.cfm?id=119864 .
[CrossRef]

Tyo, J.

White, A. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A.64, 052312–27 (2001).
[CrossRef]

Appl. Opt.

J. Franklin Inst.

C.A. Skinner, “The polarimeter and its practical applications,” J. Franklin Inst.196, 721–750 (1923).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A.64, 052312–27 (2001).
[CrossRef]

Proc. Spie

R. D. Ramkhalawon, A. M. Beckley, and T. G. Brown, “Star test polarimetry using stress-engineered optical elements,” Proc. Spie8227, 82270Q–82270Q-8 (2012).
[CrossRef]

Other

E. Collett, Polarized Light: Fundamentals and Applications (CRC Press, 1992).

A. K. Spilman and T. G. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.26, 61–66 (2007), http://ao.osa.org/abstract.cfm?id=119864 .
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Experimental setup, in which a partially-polarized beam is prepared by combining two orthogonally polarized laser beams, spatially filtering and recollimating them. This beam then passes through the center of the SEO element shown in the inset, and through a left-circular analyzer, and is focused by a lens onto a CCD. (a) Experimental image of contours of equal (half-wave) birefringence of the SEO. (b) Theoretical model of birefringence at the central region of the SEO. The aperture size used in the experiments, corresponding to cR = 0.8π, is illustrated with the small blue circle in the inset.

Fig. 2
Fig. 2

Comparison of simulated (a–f) and measured (g–l) PSFs, for fully polarized incident light whose polarization is right-circular (a,g), left-circular (b,h), horizontal (c,i), vertical (d,j), and linear at +45° (e,k) and −45° (f,l).

Fig. 3
Fig. 3

(a,b) Plots of P(xn|s) (with black corresponding to zero) for given xn as a function of s, over a cross-section of the Poincaré sphere that includes the origin as well as the points of zero and maximum probability, for (a) ū = 0 and (b) ū = (0, 0, 0.5). (c) Plot of q(s) over the surface of the sphere for the case of two detected photons.

Fig. 4
Fig. 4

Widths of the standard deviations of the polarization measurements corresponding to several values of s0 within the s1s3 slice of the Poincaré sphere, for (a) the proposed SEO-based polarimetric system, (b) for the optimal polarimeter; and (c) for a standard polarimeter. The radius of the green circles represents the extent of the standard deviation in the direction normal to the plane.

Fig. 5
Fig. 5

Measured PSFs for incident light in different states of polarization: (a–b) fully polarized elliptical states; (c–f) partially polarized states with degree of polarization as listed in Table 1.

Tables (1)

Tables Icon

Table 1 Measured polarization states for the PSFs in Fig. 5. The right hand columns show: root sum square (RSS) difference when compared to a commercial polarimeter; equivalent volume uncertainty; difference in the degree of polarization (ΔDoP); measured DoP.

Equations (25)

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S 0 = | β | 2 + | α | 2 , S 1 = 2 Re α * β , S 2 = 2 Im α * β , S 3 = | β | 2 | α | 2 .
𝕁 ( r , φ ) = cos ( c 2 r ) 𝕀 + i sin ( c 2 r ) ( φ ) ,
I U ( x ; s ) = 1 2 [ I R + I L + ( I R I L ) s 3 ] + I R I L ( s 1 cos ϕ s 2 sin ϕ ) ,
I R , L ( x ) = A ( ρ ) sin 2 cos 2 ( c ρ 2 ) ,
I R , L ( x ) = | k f 0 R A ( r ) sin cos ( c r 2 ) J 1 , 0 ( k r ρ / f ) r d r | 2 ,
I U ( x ; s ) = I R ( x ) + I L x 2 [ 1 + u ( x ) s ] ,
u ( x ) = ( 2 I R I L I R + I L cos ϕ , 2 I R I L I R + I L sin ϕ , I R I L I R + I L ) .
P ( x | s ) = I U ( x ; s ) I U ( x ; s ) d 2 x = w ( x ) 1 + u ( x ) s 1 + u ¯ s ,
w ( x ) = I R ( x ) + I L ( x ) Φ R + Φ L , Φ R , L = I R , L ( x ) d 2 x ,
f ¯ = f ( x ) w ( x ) d 2 x .
P ( x 1 x 2 x N | s ) = n = 1 N P ( x n | s ) .
P ( x ) = P ( s ) P ( x | s ) d 3 s ,
P ( x 1 x 2 x N ) = n = 1 N P ( x n ) ,
P ( s | x 1 x 2 x N ) = P ( s ) P ( x 1 x 2 x N | s ) P ( x 1 x 2 x N ) = P ( s ) n = 1 N P ( x n | s ) P ( x n ) .
P ( s | I ˜ ) = P ( s ) i [ P ( x i | s ) P ( x i ) ] I ˜ i = P ( s ) exp [ q ( s ) q 0 ] ,
q ( s ) = i I ˜ i ln P ( x i | s ) , q 0 = i I ˜ i ln P ( x i ) .
d d s exp [ q ( s ) ] | s 0 = exp [ q ( s 0 ) ] i I ˜ i P ( x i | s 0 ) d d s P ( x i | s ) | s 0 N a exp [ q ( s 0 ) ] [ d d s i P ( x i | s ) ] s 0 0 ,
exp [ q ( s ) ] exp [ q ( s 0 ) ] exp [ ( s s 0 ) ( s s 0 ) ] .
H m , n = 2 q s m s n | s 0 N 1 + u ¯ s 0 [ ( u m u n 1 + u s 0 ) ¯ u ¯ m u ¯ n 1 + u ¯ s 0 ] ,
w ρ d ρ sin θ d θ = d u z = d u z d ρ d ρ ,
( I R + I L ) ρ d d ρ ( I R I L I R + I L ) .
I R = I 0 ρ 2 R 2 , I L = I 0 ( 1 ρ 2 R 2 ) ,
𝕁 ( ρ , ϕ ) = 1 ρ 2 R 2 𝕀 + i ρ R ( ϕ ) ,
P St ( s | I ˜ St ) = P 0 i = 1 6 ( 1 + u i St s ) I ˜ i St ,
P St ( s | I ˜ St ) = P 0 n = 1 3 ( 1 + s n ) N 6 ( 1 + s 0 n ) ( 1 s n ) N 6 ( 1 s 0 n ) , H m , n St = 2 q St s m s n | s 0 N δ m , n 3 ( 1 s 0 n 2 ) ,

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