Abstract

Based on the theory of coherence and polarization of stochastic electromagnetic pulses, the analytical expression for the spectral degree of polarization of stochastic spatially and spectrally partially coherent electromagnetic pulses propagating in dispersive media is derived. Numerical calculations are performed to show the dependence of spectral degree of polarization on the spatial correlation length, temporal coherence length, pulse duration, pulse frequency, and medium dispersion. The results are interpreted physically.

© 2009 Optical Society of America

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References

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  1. D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641-1643 (1994).
    [CrossRef]
  2. G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019-2023 (2000).
    [CrossRef]
  3. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21, 1907-1916 (2004).
    [CrossRef]
  4. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35-43 (2005).
    [CrossRef]
  5. H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23, 940-948 (2006).
    [CrossRef]
  6. Z. Y. Chen and J. X. Pu, “Degree of polarization in Young's double-slit interference experiment formed by stochastic electromagnetic beams,” J. Opt. Soc. Am. A 24, 2043-2048 (2007).
    [CrossRef]
  7. E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32, 3400-3401 (2007).
    [CrossRef] [PubMed]
  8. O. Korotkova, T. D. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun. 281, 515-520 (2008).
    [CrossRef]
  9. X. Y. Du and D. M. Zhao, “Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system,” J. Opt. Soc. Am. A 25, 773-779 (2008).
    [CrossRef]
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
  11. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
  12. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53-58 (2002).
    [CrossRef]
  13. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536-1545 (2005).
    [CrossRef]
  14. W. H. Huang, S. A. Ponomarenko, M. Cada, and G. P. Agrawal, “Polarization changes of partially coherent pulses propagating in optical fibers,” J. Opt. Soc. Am. A 24, 3063-3068 (2007).
    [CrossRef]
  15. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  16. E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078-1080 (2003).
    [CrossRef] [PubMed]
  17. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).
  18. G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15-22 (1999).
    [CrossRef]

2008

2007

2006

2005

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536-1545 (2005).
[CrossRef]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35-43 (2005).
[CrossRef]

2004

2003

2002

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53-58 (2002).
[CrossRef]

2000

1999

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15-22 (1999).
[CrossRef]

1994

Agrawal, G. P.

Cada, M.

Chen, Z. Y.

Du, X. Y.

Friberg, A. T.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Huang, W. H.

James, D. F. V.

Korotkova, O.

O. Korotkova, T. D. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun. 281, 515-520 (2008).
[CrossRef]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35-43 (2005).
[CrossRef]

Lajunen, H.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Ponomarenko, S. A.

Pu, J. X.

Roychowdhury, H.

Shirai, T.

Tervo, J.

Turunen, J.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Vahimaa, P.

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536-1545 (2005).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Visser, T. D.

O. Korotkova, T. D. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun. 281, 515-520 (2008).
[CrossRef]

Wolf, E.

O. Korotkova, T. D. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun. 281, 515-520 (2008).
[CrossRef]

E. Wolf, “Polarization invariance in beam propagation,” Opt. Lett. 32, 3400-3401 (2007).
[CrossRef] [PubMed]

H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23, 940-948 (2006).
[CrossRef]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35-43 (2005).
[CrossRef]

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21, 1907-1916 (2004).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078-1080 (2003).
[CrossRef] [PubMed]

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019-2023 (2000).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Wyrowski, F.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53-58 (2002).
[CrossRef]

Zhao, D. M.

J. Opt. Soc. Am. A

D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641-1643 (1994).
[CrossRef]

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019-2023 (2000).
[CrossRef]

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21, 1907-1916 (2004).
[CrossRef]

H. Roychowdhury, G. P. Agrawal, and E. Wolf, “Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber,” J. Opt. Soc. Am. A 23, 940-948 (2006).
[CrossRef]

Z. Y. Chen and J. X. Pu, “Degree of polarization in Young's double-slit interference experiment formed by stochastic electromagnetic beams,” J. Opt. Soc. Am. A 24, 2043-2048 (2007).
[CrossRef]

X. Y. Du and D. M. Zhao, “Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system,” J. Opt. Soc. Am. A 25, 773-779 (2008).
[CrossRef]

H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536-1545 (2005).
[CrossRef]

W. H. Huang, S. A. Ponomarenko, M. Cada, and G. P. Agrawal, “Polarization changes of partially coherent pulses propagating in optical fibers,” J. Opt. Soc. Am. A 24, 3063-3068 (2007).
[CrossRef]

Opt. Commun.

O. Korotkova, T. D. Visser, and E. Wolf, “Polarization properties of stochastic electromagnetic beams,” Opt. Commun. 281, 515-520 (2008).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53-58 (2002).
[CrossRef]

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15-22 (1999).
[CrossRef]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35-43 (2005).
[CrossRef]

Opt. Lett.

Phys. Lett. A

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Other

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

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Figures (6)

Fig. 1
Fig. 1

Contours of the on-axis spectral degree of polarization P ( r , z , ω ) of a stochastic spectrally fully coherent electromagnetic pulse as a function of the propagation distance z and frequency ω ω 0 (a) in fused silica and (b) in free space.

Fig. 2
Fig. 2

Contours of the on-axis spectral degree of polarization P ( r , z , ω ) of stochastic spatially and spectrally partially coherent electromagnetic pulses as a function of the propagation distance z and spatial correlation length δ x x in fused silica for different values of the temporal coherence length (a) T c x = 5 fs and (b) T c x = 7 fs .

Fig. 3
Fig. 3

Contours of the on-axis spectral degree of polarization P ( r , z , ω ) of stochastic spatially and spectrally partially coherent electromagnetic pulses as a function of the propagation distance z and temporal coherence length T c x in fused silica for different values of the frequency (a) ω ω 0 = 1 and (b) 1.1.

Fig. 4
Fig. 4

Contours of the on-axis spectral degree of polarization P ( r , z , ω ) of stochastic spatially and spectrally partially coherent electromagnetic pulses as a function of the propagation distance z and pulse duration T 0 in fused silica for different values of the frequency (a) ω ω 0 = 1 and (b) 1.1.

Fig. 5
Fig. 5

Contours of the on-axis spectral degree of polarization P ( r , z , ω ) of stochastic spatially and spectrally partially coherent electromagnetic pulses as a function of the propagation distance z and frequency ω ω 0 (a) in fused silica and (b) in free space.

Fig. 6
Fig. 6

Distribution of the spectral degree of polarization P ( r , z , ω ) of stochastic spatially and spectrally partially coherent electromagnetic pulses in fused silica for different values of the carrier frequency (a) ω 0 = 1.02 rad fs , (b) 2.36 rad fs and (c) 4.72 rad fs .

Equations (38)

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Γ 0 ( ρ 1 , ρ 2 , t 1 , t 2 ) = [ Γ i j 0 ( ρ 1 , ρ 2 , t 1 , t 2 ) ] = [ E i * ( ρ 1 , t 1 ) E j ( ρ 2 , t 2 ) ] ,
( i = x , y ; j = x , y unless otherwise stated ) ,
Γ i i 0 ( ρ 1 , ρ 2 , t 1 , t 2 ) = I i 0 ( ρ 1 , t 1 ) I i 0 ( ρ 2 , t 2 ) μ i i 0 ( ρ 1 , ρ 2 , t 1 , t 2 ) ,
Γ x y 0 ( ρ 1 , ρ 2 , t 1 , t 2 ) = Γ y x 0 ( ρ 1 , ρ 2 , t 1 , t 2 ) = 0 ,
I i 0 ( ρ , t ) = A i exp [ t 2 T 0 2 ] exp [ ρ 2 2 σ 2 ] ,
μ i i 0 ( ρ 1 , ρ 2 , t 1 , t 2 ) = exp [ ( ρ 1 ρ 2 ) 2 2 δ i i 2 ] exp [ ( t 1 t 2 ) 2 2 T c i 2 ] exp [ i ω 0 ( t 1 t 2 ) ] ,
W i j 0 ( ρ 1 , ρ 2 , ω 1 , ω 2 ) = 1 ( 2 π ) 2 + + Γ i j 0 ( ρ 1 , ρ 2 , t 1 , t 2 ) exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d t 1 d t 2 ,
W 0 ( ρ 1 , ρ 2 , ω 1 , ω 2 ) = [ W i j 0 ( ρ 1 , ρ 2 , ω 1 , ω 2 ) ] ,
W i i 0 ( ρ 1 , ρ 2 , ω 1 , ω 2 ) = T 0 A i 2 π Ω 0 i exp ( ρ 1 2 + ρ 2 2 4 σ 2 ) exp [ ( ρ 1 ρ 2 ) 2 2 δ i i 2 ] exp [ ( ω 1 ω 0 ) 2 + ( ω 2 ω 0 ) 2 2 Ω 0 i 2 ] exp [ ( ω 1 ω 2 ) 2 2 Ω c i 2 ] ,
W x y 0 ( ρ 1 , ρ 2 , ω 1 , ω 2 ) = W y x 0 ( ρ 1 , ρ 2 , ω 1 , ω 2 ) = 0 ,
Ω 0 i = 1 T 0 2 + 2 T c i 2 , ( Spectral width of the i component of the electric vector ) ,
Ω c i = T c i T 0 Ω 0 i . ( Spectral coherence width of the i component of the electric vector ) .
P 0 ( ρ , ω ) = 1 4 Det [ W 0 ( ρ , ρ , ω , ω ) ] { Tr [ W 0 ( ρ , ρ , ω , ω ) ] } 2 = | A x Ω 0 x exp [ ( ω ω 0 ) 2 Ω 0 x 2 ] A y Ω 0 y exp [ ( ω ω 0 ) 2 Ω 0 y 2 ] A x Ω 0 x exp [ ( ω ω 0 ) 2 Ω 0 x 2 ] + A y Ω 0 y exp [ ( ω ω 0 ) 2 Ω 0 y 2 ] | .
W i j ( r 1 , r 2 , z , ω 1 , ω 2 ) = k 1 ( ω 1 ) k 2 ( ω 1 ) 4 π 2 z 2 exp { i [ k 2 ( ω 2 ) k 1 ( ω 1 ) ] z } W i j 0 ( ρ 1 , ρ 2 , ω 1 , ω 2 ) exp { i 2 z [ k 2 ( ω 2 ) ( r 2 ρ 2 ) 2 k 1 ( ω 1 ) ( r 1 ρ 1 ) 2 ] } d 2 ρ 1 d 2 ρ 2 ,
W x x ( r , r , z , ω , ω ) = A x Q x x exp [ r 2 2 σ 2 Q x x ] T 0 2 π Ω 0 x exp [ ( ω ω 0 ) 2 Q 0 x 2 ] ,
W y y ( r , r , z , ω , ω ) = A y Q y y exp [ r 2 2 σ 2 Q y y ] T 0 2 π Ω 0 y exp [ ( ω ω 0 ) 2 Q 0 y 2 ] ,
W x y ( r , r , z , ω , ω ) = W y x ( r , r , z , ω , ω ) = 0 ,
Q x x = 1 + c 2 z 2 σ 2 n 2 ( ω ) ω 2 ( 1 4 σ 2 + 1 δ x x 2 ) ,
Q y y = 1 + c 2 z 2 σ 2 n 2 ( ω ) ω 2 ( 1 4 σ 2 + 1 δ y y 2 ) .
P ( r , z , ω ) = 1 4 D e t [ W ( r , r , z , ω , ω ) ] { T r [ W ( r , r , z , ω , ω ) ] } 2 = | A x Q x x exp [ r 2 2 σ 2 Q x x ] 1 Ω 0 x exp [ ( ω ω 0 ) 2 Ω 0 x 2 ] A y Q y y exp [ r 2 2 σ 2 Q y y ] 1 Ω 0 y exp [ ( ω ω 0 ) 2 Ω 0 y 2 ] A x Q x x exp [ r 2 2 σ 2 Q x x ] 1 Ω 0 x exp [ ( ω ω 0 ) 2 Ω 0 x 2 ] + A y Q y y exp [ r 2 2 σ 2 Q y y ] 1 Ω 0 y exp [ ( ω ω 0 ) 2 Ω 0 y 2 ] | .
P ( r , z , ω ) = | A x Q x x exp [ r 2 2 σ 2 Q x x ] A y Q y y exp [ r 2 2 σ 2 Q y y ] A x Q x x exp [ r 2 2 σ 2 Q x x ] + A y Q y y exp [ r 2 2 σ 2 Q y y ] | .
P ( r , z , ω ) = | A x Ω 0 x exp [ ( ω ω 0 ) 2 Ω 0 x 2 ] A y Ω 0 y exp [ ( ω ω 0 ) 2 Ω 0 y 2 ] A x Ω 0 x exp [ ( ω ω 0 ) 2 Ω 0 x 2 ] + A y Ω 0 y exp [ ( ω ω 0 ) 2 Ω 0 y 2 ] | .
n 2 ( λ ) = 1 + l = 1 3 B l 1 λ l 2 λ 2 ,
z 1 = n ( ω ) ω 2 ( P + P 0 ) F x x ( P 1 ) ( P 0 1 ) F y y ( P + 1 ) ( P 0 + 1 ) ,
z 2 = n ( ω ) ω 2 ( P P 0 ) F x x ( P + 1 ) ( P 0 1 ) F y y ( P 1 ) ( P 0 + 1 ) .
z 1 = z 2 = n ( ω ) ω 2 P 0 F x x ( 1 P 0 ) F y y ( 1 + P 0 ) ,
F i i = c 2 σ 2 ( 1 4 σ 2 + 1 δ i i 2 ) .
z 1 = B 1 P ( B + 1 ) G x x [ 4 B + P ( 4 + B ) ] H ( 1 + P ) ,
z 1 = B 1 + P ( B + 1 ) G x x [ 4 B P ( 4 + B ) ] H ( 1 P ) .
z 1 = z 2 = B 1 G x x ( 4 B ) H ,
B = A x A y Ω 0 y Ω 0 x ,
G x x = c 2 4 σ 2 n 2 ( ω ) ω 2 ( 1 σ 2 + 1 δ x x 2 ) ,
H = 3 c 2 4 σ 4 n 2 ( ω ) ω 2 .
z 1 = n ( ω ) ω α β ( P + 1 ) ( P 1 ) α β F y y ( P + 1 ) + F x x ( P 1 ) ,
z 2 = n ( ω ) ω α β ( P 1 ) ( P + 1 ) α β F y y ( P 1 ) + F x x ( P + 1 ) ,
α = A x A y ,
β = Ω 0 y Ω 0 x exp [ ( ω ω 0 ) 2 Ω 0 y 2 ( ω ω 0 ) 2 Ω 0 x 2 ] .
z = n ( ω ) ω 1 α β α β F y y F x x .

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