Abstract

In this work, the question of the coherency matrix propagation of a light beam is addressed by means of the analysis of interpolation processes between two physical situations. These physical situations are defined according to the second order statistical properties of the underlying process. Two states of a light beam or the path in a medium to go from a physical situation at distance z1 to another one at distance z2 is related to the correlation between both these physical situations. Equivalence classes are derived from the definition of a group action on the set of coherency matrices. The geodesic curves on each equivalence class define the process of interpolation. The general solution is derived as a symbolic equation, and the solution is explicitly developed for the situation of uncorrelated statistical processes. Interpolating coherency matrix in this particular case describes the propagation of a light beam into a uniform nondepolarizing medium characterized by a differential Jones matrix determined by the far points of the interpolation curve up to a unitary matrix.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 38, 671–685 (1948).
    [CrossRef]
  2. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4×4 matrix calculus,” J. Opt. Soc. Am. 68, 1756–1767 (1978).
    [CrossRef]
  3. R. Barakat, “Exponential versions of the Jones and Mueller–Jones polarization matrices,” J. Opt. Soc. Am. A 13, 158–163 (1996).
    [CrossRef]
  4. C. Brosseau, Fundamentals of Polarized Light: a Statistical Optics Approach (Wiley, 1998).
  5. J. W. Goodman, Statistical Optics (John Wiley, 1985).
  6. P. Réfrégier, “Symmetries in coherence theory of partially polarized light,” J. Math. Phys. 48, 033303 (2007).
    [CrossRef]
  7. P. Réfrégier, “Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations,” Opt. Lett. 30, 3117–3119 (2005).
    [CrossRef]
  8. W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. (Academic, 1986).
  9. V. Devlaminck and P. Terrier, “Geodesic distance on non-singular coherency matrix space in polarization optics,” J. Opt. Soc. Am. A 27, 1756–1763 (2010).
    [CrossRef]
  10. S. Helgason, Differential Geometry and Symmetric Spaces (Academic, 1962).
  11. X. Pennec, P. Fillard, and N. Ayache, “A Riemannian framework for tensor computing,” Int. J. Comput. Vis. 66, 41–66 (2006).
    [CrossRef]
  12. J. Burbea and C. Rao, “Entropy differential metric, distance and divergence measures in probability spaces: a unified approach,” J. Multivariate Anal. 12, 575–596 (1982).
    [CrossRef]

2010 (1)

2007 (1)

P. Réfrégier, “Symmetries in coherence theory of partially polarized light,” J. Math. Phys. 48, 033303 (2007).
[CrossRef]

2006 (1)

X. Pennec, P. Fillard, and N. Ayache, “A Riemannian framework for tensor computing,” Int. J. Comput. Vis. 66, 41–66 (2006).
[CrossRef]

2005 (1)

1996 (1)

1982 (1)

J. Burbea and C. Rao, “Entropy differential metric, distance and divergence measures in probability spaces: a unified approach,” J. Multivariate Anal. 12, 575–596 (1982).
[CrossRef]

1978 (1)

1948 (1)

Ayache, N.

X. Pennec, P. Fillard, and N. Ayache, “A Riemannian framework for tensor computing,” Int. J. Comput. Vis. 66, 41–66 (2006).
[CrossRef]

Azzam, R. M. A.

Barakat, R.

Boothby, W. M.

W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. (Academic, 1986).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: a Statistical Optics Approach (Wiley, 1998).

Burbea, J.

J. Burbea and C. Rao, “Entropy differential metric, distance and divergence measures in probability spaces: a unified approach,” J. Multivariate Anal. 12, 575–596 (1982).
[CrossRef]

Devlaminck, V.

Fillard, P.

X. Pennec, P. Fillard, and N. Ayache, “A Riemannian framework for tensor computing,” Int. J. Comput. Vis. 66, 41–66 (2006).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (John Wiley, 1985).

Helgason, S.

S. Helgason, Differential Geometry and Symmetric Spaces (Academic, 1962).

Jones, R. C.

Pennec, X.

X. Pennec, P. Fillard, and N. Ayache, “A Riemannian framework for tensor computing,” Int. J. Comput. Vis. 66, 41–66 (2006).
[CrossRef]

Rao, C.

J. Burbea and C. Rao, “Entropy differential metric, distance and divergence measures in probability spaces: a unified approach,” J. Multivariate Anal. 12, 575–596 (1982).
[CrossRef]

Réfrégier, P.

Terrier, P.

Int. J. Comput. Vis. (1)

X. Pennec, P. Fillard, and N. Ayache, “A Riemannian framework for tensor computing,” Int. J. Comput. Vis. 66, 41–66 (2006).
[CrossRef]

J. Math. Phys. (1)

P. Réfrégier, “Symmetries in coherence theory of partially polarized light,” J. Math. Phys. 48, 033303 (2007).
[CrossRef]

J. Multivariate Anal. (1)

J. Burbea and C. Rao, “Entropy differential metric, distance and divergence measures in probability spaces: a unified approach,” J. Multivariate Anal. 12, 575–596 (1982).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Lett. (1)

Other (4)

W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. (Academic, 1986).

S. Helgason, Differential Geometry and Symmetric Spaces (Academic, 1962).

C. Brosseau, Fundamentals of Polarized Light: a Statistical Optics Approach (Wiley, 1998).

J. W. Goodman, Statistical Optics (John Wiley, 1985).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (38)

Equations on this page are rendered with MathJax. Learn more.

N = d J d z J 1 ,
d S d z = mS ,
m = Λ ( N I + I N ) Λ ,
Λ = 1 2 [ 1 1 0 0 0 0 1 i 0 0 1 i 1 1 0 0 ] .
A B = [ a 11 B a 1 n B a m 1 B a m n B ] .
Φ = E ( k ) E ( k ) k ,
Φ = j = 0 3 S j σ j ,
G ( z 1 , z 2 ) = E ( k , z 2 ) E ( k , z 1 ) k = K E ( k , z 2 ) E ( k , z 1 ) f ( k , z 1 , z 2 ) d k ,
Σ = Ω ( k , z 1 , z 2 ) Ω ( k , z 1 , z 2 ) k = [ H ( z 1 ) G ( z 1 , z 2 ) G ( z 1 , z 2 ) H ( z 2 ) ] .
Σ 1 ( z 1 , z 2 ) Σ 2 ( z 1 , z 2 )  ⇔  Γ = [ Π 1 0 0 Π 2 ] G L ( 2 , C ) G L ( 2 , C ) / Σ 2 = Γ Σ 1 Γ = φ ( Γ , Σ 1 ) ,
x HDP ( 4 ) , φ ( e , x ) = x  ∀ ( g 1 , g 2 ) LG × LG , x HDP ( 4 ) , φ ( g 1 , φ ( g 2 , x ) ) = φ ( g 1 g 2 , x ) ,
O ( Σ 0 ) = { Σ = Γ Σ 0 Γ , Γ LG } .
γ ( p , X ) ( z ) = Exp p [ z Log p ( x ) ] with x = Exp p ( X ) .
x = [ H 1 G G H 2 ]
p = [ H 1 ( H 1 1 / 2 H 2 1 / 2 G ) H 1 1 / 2 H 2 1 / 2 G H 1 ]
g x g = p for g = [ Id 0 0 H 1 1 / 2 H 2 1 / 2 ] .
g = p 1 / 2 = [ H 1 1 / 2 0 0 H 2 1 / 2 ] .
γ ( p , X ) ( z ) = φ [ p 1 / 2 , γ ( Id , Y ) ( z ) ] = p 1 / 2 exp ( z Y ) ( p 1 / 2 ) .
Y = d φ ( p 1 / 2 , X ) = p 1 / 2 X ( p 1 / 2 ) with X = Log p ( x ) .
γ ( p , X ) ( z ) = Exp p ( z Y ) = p 1 / 2 exp [ z p 1 / 2 X ( p 1 / 2 ) ] ( p 1 / 2 ) = p 1 / 2 exp [ z log [ p 1 / 2 x ( p 1 / 2 ) ] ] ( p 1 / 2 ) .
p = [ H 1 0 0 H 1 ] and x = [ H 1 0 0 H 2 ]
γ ( p , X ) ( z ) = [ H 1 1 / 2 0 0 H 1 1 / 2 ] exp [ z log ( [ Id 0 0 H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] ) ] [ ( H 1 1 / 2 ) 0 0 ( H 1 1 / 2 ) ] , γ ( p , X ) ( z ) = [ H 1 0 0 H 1 1 / 2 exp [ z log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] ] ( H 1 1 / 2 ) ] = [ H 1 0 0 γ sm ( p , X ) ( z ) ] .
W = H 1 1 / 2 log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] ( H 1 1 / 2 ) .
H ( H 1 , W ) ( z ) = exp ( z N ) H 1 [ exp ( z N ) ] with W = NH 1 + H 1 N ,
N = 1 2 [ W H 1 1 + H 1 1 / 2 F ( H 1 1 / 2 ) ] ,
N = 1 2 H 1 1 / 2 [ log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] + F ] ( H 1 1 / 2 ) .
[ log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] , F ] = 0
J ( z ) = H 1 1 / 2 exp [ z 2 log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] ] · exp [ z 2 F ] ( H 1 1 / 2 ) .
J ( z ) = exp [ z 2 F ] ,
J ( z ) = exp [ z 2 log ( H 2 ) ] · exp [ z 2 F ] .
d H H 2 = 1 2 tr [ ( H 1 d H ) 2 ] ,
A , B C = 1 2 tr [ C 1 A C 1 B ] .
H ( z ) = exp ( z N ) H 1 [ exp ( z N ) ]
N = 1 2 H 1 1 / 2 [ log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] + F ] ( H 1 1 / 2 ) ,
log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] .
exp ( z N ) = H 1 1 / 2 exp [ z 2 [ log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] + F ] ] ( H 1 1 / 2 ) , exp ( z N ) = H 1 1 / 2 exp [ z 2 log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] ] · exp [ z 2 F ] ( H 1 1 / 2 ) .
H ( z ) = H 1 1 / 2 exp [ z 2 log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] ] exp [ z 2 F ] exp [ z 2 F ] exp [ z 2 log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] ] ( H 1 1 / 2 ) .
H ( z ) = H 1 1 / 2 exp [ z log [ H 1 1 / 2 H 2 ( H 1 1 / 2 ) ] ] ( H 1 1 / 2 ) .

Metrics