Abstract

The change of coherence and polarization of an electromagnetic beam modulated by a random anisotropic phase screen passing through any optical system is found within the framework of complex ABCD-matrix theory This means that the formalism can treat imaging and Fourier transform and free-space optical systems, as well as fractional Fourier transform systems, with finite-size limiting apertures of Gaussian transmission shape. Thus, the current paper shall be considered as a continuation, extension, and generalization of a previous work by Shirai and Wolf [J. Opt. Soc. Am. A 21, 1907 (2004) ]. It will be shown that the inclusion of apertures in the optical system strongly influences not only the propagation of spatial coherence but also the degree of polarization of a propagating field. Analytical expressions of coherence and polarization propagation will be given in terms of the matrix elements for any complex optical system.

© 2008 Optical Society of America

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References

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  1. P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051-6060 (2005).
    [CrossRef] [PubMed]
  2. R. Martinez-Herrero and P. M. Mejías, “Relation between degrees of coherence for electromagnetic fields,” Opt. Lett. 32, 1504-1506 (2007).
    [CrossRef] [PubMed]
  3. E. Wolf, “Relation between degrees of coherence for electromagnetic fields,” J. Opt. Soc. Am. A 72, 343-351 (1982).
    [CrossRef]
  4. J. Tervo, T. Setälä, and A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328-330 (2004).
    [CrossRef] [PubMed]
  5. 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]
  6. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263-267 (2003).
    [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. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
    [CrossRef]
  10. 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]
  11. A. E. Siegman, Lasers (University Science, 1986), Chap. 20.
  12. H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931-1948 (1987).
    [CrossRef]
  13. F. Scudieri, M. Bertolotti, and R. Bartolino, “Light scattered by a liquid-crystal--new quasi-thermal source,” Appl. Opt. 13, 181-185 (1974).
    [CrossRef] [PubMed]
  14. J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984), Chap. 2, pp. 42-46.
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  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. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205-2215 (2004).
    [CrossRef]

2008

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

2007

2005

P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13, 6051-6060 (2005).
[CrossRef] [PubMed]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

2004

2003

1994

1987

1982

E. Wolf, “Relation between degrees of coherence for electromagnetic fields,” J. Opt. Soc. Am. A 72, 343-351 (1982).
[CrossRef]

1974

Bartolino, R.

Bertolotti, M.

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984), Chap. 2, pp. 42-46.

Goudail, F.

Hanson, S. G.

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]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

Mandel, L.

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

Martinez-Herrero, R.

Mejías, P. M.

Réfrégier, P.

Scudieri, F.

Setälä, T.

Shirai, T.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7, 232-237 (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]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986), Chap. 20.

Tervo, J.

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]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7, 232-237 (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]

E. Wolf, “Relation between degrees of coherence for electromagnetic fields,” J. Opt. Soc. Am. A 72, 343-351 (1982).
[CrossRef]

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

Yura, H. T.

Appl. Opt.

J. Opt. A, Pure Appl. Opt.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7, 232-237 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

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

Opt. Express

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

A. E. Siegman, Lasers (University Science, 1986), Chap. 20.

J. W. Goodman, “Statistical properties of laser speckle patterns” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1984), Chap. 2, pp. 42-46.

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

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Figures (7)

Fig. 1
Fig. 1

Basic setup for examining the propagation of the degree of polarization and degree of coherence traveling through a complex A B C D optical system.

Fig. 2
Fig. 2

Degree of polarization for free-space propagation as a function of the radial distance R measured in spot size units and the distance measured in units of the Rayleigh length for N = 200 . The angle of incidence of the polarized light and number of scattering elements are (a) θ = 30 ° , N = 200 ; (b) θ = 600 , N = 200 ; (c) θ = 30 ° , N = 20 ; (d) θ = 60 ° , N = 20 .

Fig. 3
Fig. 3

Absolute value of the on-axis spectral degree of coherence given for angles of polarization of the incident beam of 60° (soid curve), 45° (dashed curve), and 30° (dot-dashed curve) for (a) N = 200 ; (b) N = 20 .

Fig. 4
Fig. 4

Degree of polarization for a Fourier transform system as a function of the radial distance R measured in speckle size units for angles of polarization of the incident field of 60° (solid curve), 45° (dashed curve), and 30° (dot-dashed curve) for (a) N = 200 and (b) N = 20 .

Fig. 5
Fig. 5

Degree of polarization for an imaging system as a function of the radial distance R for 60° (solid curve), 45° (dashed curve), and 30° (dot-dashed curve), ρ = 10 μ m .

Fig. 6
Fig. 6

Absolute value of the on-axis spectral degree of coherence given for the imaging system for the following values of the angle of polarization of the incident beam: 60° (solid curve), 45° (dashed curve), and 30° (dot-dashed) for ρ = 40 μ m .

Fig. 7
Fig. 7

Degree of polarization in the vicinity of the focusing position for two values of the polarization angle θ of the incident field: (a) 30°, (b) 60°.

Equations (36)

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W ( r 1 , r 2 , ω ) = [ E x * ( r 1 , ω ) E x ( r 2 , ω ) E x * ( r 1 , ω ) E y ( r 2 , ω ) E y * ( r 1 , ω ) E x ( r 2 , ω ) E y * ( r 1 , ω ) E y ( r 2 , ω ) ] ,
S ( r , ω ) = tr W ( r , r , ω ) ,
η ( r 1 , r 2 , ω ) = tr W ( r 1 , r 2 , ω ) tr W ( r 1 , r 1 , ω ) tr W ( r 2 , r 2 , ω ) ,
P ( r , ω ) = { 1 4 det W ( r , r , ω ) [ tr W ( r , r , ω ) ] 2 } 1 2 .
W out ( r 1 , r 2 , ω ) = T ̂ ( r 1 , ω ) W in ( r 1 , r 2 , ω ) T ̂ ( r 2 , ω ) .
E i ( r , ω ) = d r E i ( r , ω ) G ( r , r ) ,
G ( r , r ) = i k 2 π B exp [ i k 2 B ( A r 2 2 r r + D r 2 ) ] .
W out ( r 1 , r 2 , ω ) = W in ( r 1 , r 2 , ω ) G * ( r 1 , r 1 ) G ( r 2 , r 2 ) d r 1 d r 2 ,
T ̂ ( r , ω ) = [ 1 0 0 exp [ i ϕ ( r , ω ) ] ] .
ϕ ( r 1 , ω ) ϕ ( r 2 , ω ) = ϕ 0 2 exp [ ( r 1 r 2 ) 2 2 σ ϕ 2 ] ,
exp [ i ( ϕ ( r 2 , ω ) ϕ ( r 1 , ω ) ) ] = exp [ 1 2 ( ( ϕ ( r 2 , ω ) ϕ ( r 1 , ω ) ) 2 ) ] ,
exp [ i ( ϕ ( r 2 , ω ) ϕ ( r 1 , ω ) ) ] = exp [ ϕ 0 2 ( 1 exp [ ( r 1 r 2 ) 2 2 σ ϕ 2 ] ) ] .
exp [ ϕ 0 2 ( 1 exp [ ( r 1 r 2 ) 2 2 σ ϕ 2 ] ) ] exp [ ( r 1 r 2 ) 2 2 ρ 2 ] ,
W ( r 1 , r 2 , ω ) = S 0 exp [ r 1 2 + r 2 2 4 σ s 2 ] [ cos 2 [ θ ] 0 0 exp [ ( r 1 r 2 ) 2 2 ρ 2 ] sin 2 [ θ ] ] .
W A B C D ( r 1 , r 2 , ω ) = [ cos 2 [ θ ] W 1 A B C D ( r 1 , r 2 , ω ) 0 0 sin 2 [ θ ] W 2 A B C D ( r 1 , r 2 , ω ) ] ,
W 1 A B C D ( r 1 , r 2 , ω ) = exp [ r 1 2 + r 2 2 4 σ s 2 ] G * ( r 1 , r 1 ) G ( r 2 , r 2 ) d r 1 d r 2 ,
W 2 A B C D ( r 1 , r 2 , ω ) = exp [ r 1 2 + r 2 2 4 σ s 2 ] exp [ ( r 1 r 2 ) 2 2 ρ 2 ] G * ( r 1 , r 1 ) G ( r 2 , r 2 ) d r 1 d r 2 .
W 1 A B C D ( Δ r , R , ω ) = 4 π 2 B 2 k 2 A 2 exp [ k A 2 ( Im [ A C * ] R 2 i Re [ A C * ] R Δ r + Im [ A C * ] Δ r 2 4 ) ] ,
W 2 A B C D ( Δ r , R , ω ) = 4 π 2 B 2 ρ 2 k κ exp { k κ [ ( 1 + Re [ B C * ] Re [ A D * ] Im [ A C * ] k ρ 2 ) R 2 + i ( Im [ B C * ] Im [ A D * ] + Re [ A C * ] k ρ 2 ) R Δ r ( 1 Re [ B C * ] + Re [ A D * ] + Im [ A C * ] k ρ 2 ) Δ r 2 4 ] } .
η lin. pol ( Δ r , R , ω ) = W 1 ( Δ r , R , ω ) + W 2 ( Δ r , R , ω ) tan 2 [ θ ] W 1 ( 0 , R , ω ) + W 2 ( 0 , R , ω ) tan 2 [ θ ]
P lin. pol ( R , ω ) = ( 1 4 W 1 ( 0 , R , ω ) W 2 ( 0 , R , ω ) tan 2 [ θ ] ( W 1 ( 0 , R , ω ) + W 2 ( 0 , R , ω ) tan 2 [ θ ] ) 2 ) 1 2 .
S lin. pol ( R ) W 1 ( 0 , R , ω ) cos 2 [ θ ] + W 2 ( 0 , R , ω ) sin 2 [ θ ] .
η unpol. ( Δ r , R , ω ) = W 1 ( Δ r , R , ω ) + W 2 ( Δ r , R , ω ) W 1 ( 0 , R , ω ) + W 2 ( 0 , R , ω )
P unpol. ( R ) = ( 1 4 W 1 ( 0 , R ) W 2 ( 0 , R ) ( W 1 ( 0 , R ) + W 2 ( 0 , R ) ) 2 ) 1 2 .
S unpol ( R , ω ) W 1 ( 0 , R , ω ) + W 2 ( 0 , R , ω ) .
M ̂ = [ 1 i z z R z i z R 1 ] ,
W 1 , Free Space A B C D ( Δ r , R , ω ) = 4 π 2 z 2 k 2 ( 1 + z 2 z R 2 ) exp [ k z R ( 1 1 + z 2 z R 2 R 2 i z z R 1 + z 2 z R 2 R Δ r + 1 4 ( 1 + z 2 z R 2 ) Δ r 2 ) ] ,
W 2 , Free Space A B C D ( Δ r , R , ω ) = 4 π 2 z 2 k 2 ( 1 + ( 2 N + 1 ) z 2 z R 2 ) exp { k z R 1 + ( 2 N + 1 ) z 2 z R 2 [ R 2 i z ( 2 N + 1 ) z R R Δ r + ( 2 N + 1 ) Δ r 2 4 ] } .
M ̂ Fourier = [ 4 f 2 z R z F 2 i f ( z R + z F ) z R z F f 2 i f 2 z F 1 f 2 i z F 4 f z R z F 2 i f z F ] .
W 1 , Fourier A B C D ( Δ r , R , ω ) = ( π z R k ) 2 exp [ ( ( R 2 + Δ r 2 4 ) k z R 2 f 2 2 i k z R f z F Δ r R ) ] ,
W 2 , Fourier A B C D ( Δ r , R , ω ) = π 2 z R 2 k 2 ( 1 + N ) × exp [ ( k z R 2 f 2 ( 1 + N ) R 2 + k z R 8 f 2 Δ r 2 i k ( 2 + N ) z R f z F ( 1 + N ) Δ r R ) ] .
η lin . pol ( Δ r , 0 , ω ) = exp [ k z R 8 f 2 Δ r 2 ] ,
P lin . pol ( R , ω ) = ( 1 4 ( 1 + N ) exp [ N R 2 2 ( N + 1 ) ] tan 2 θ ( ( 1 + N ) exp [ N R 2 2 ( N + 1 ) ] + tan 2 θ ) 2 ) 1 2 ,
M ̂ imaging = [ 1 + i ( 2 Δ z k s 0 2 4 f k s f 2 ) Δ z 8 i f 2 k s f 2 1 f + 2 i ( 1 k s 0 2 1 k s f 2 ) 1 4 i f k s f 2 ] .
W 1 , imag A B C D ( Δ r , R , ω ) = 4 π 2 ( 64 f 4 + k 2 σ f 4 Δ z 2 ) k 4 σ f 4 exp { [ ( R 2 + Δ r 2 4 ) ( 2 σ 0 2 + 2 σ f 2 ) i k f Δ r R ] } ,
W 2 , imag A B C D ( Δ r , R , ω ) = 4 π 2 ρ 0 2 ( 64 f 4 + k 2 σ f 4 Δ z 2 ) k 2 σ f 2 ( 16 f 2 + k 2 ρ 2 σ f 2 ) exp { [ ( 2 σ 0 2 + 2 σ f 2 Δ z f ρ 2 ) R 2 + ( 2 σ 0 2 + 2 σ f 2 + 2 ρ 2 ) Δ r 2 i k f Δ r R ] } .

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