Abstract

The combination of an angular spectrum representation (in the space–frequency domain) and the second-order Rytov perturbation theory is applied for description of the second-order statistical properties of arbitrary (coherent and partially coherent) stochastic electromagnetic beamlike fields that propagate in a turbulent atmosphere. In particular, we derive the expressions for the elements of the cross-spectral density matrix of the beam, from which its spectral, coherence, and polarization properties can be found. We illustrate the method by applying it to the propagation of several electromagnetic model beams through the atmosphere.

© 2007 Optical Society of America

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References

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  1. G. Gbur and O. Korotkova, "Angular spectrum representation for propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence," J. Opt. Soc. Am. A 24, 745-752 (2007).
    [CrossRef]
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  4. D. F. V. James, "Change of polarization of light beams on propagation in free space," J. Opt. Soc. Am. A 11, 1641-1649 (1994).
    [CrossRef]
  5. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 3, 1-9 (2001).
    [CrossRef]
  6. 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]
  7. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
    [CrossRef]
  8. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
    [CrossRef]
  9. W. Gao, "Changes of polarization of light beams propagating through tissue," Opt. Commun. 260, 749-755 (2006).
    [CrossRef]
  10. W. Gao and O. Korotkova, "Changes in the state of polarization of a random electromagnetic beam propagating through tissue," Opt. Commun. 270, 474-478 (2007).
    [CrossRef]
  11. E. Wolf, "Unified theory of coherence and polarization of statistical electromagnetic beams," Phys. Lett. A 312, 263-265 (2003).
    [CrossRef]
  12. J. Tervo and J. Turunen, "Angular spectrum representation of partially coherent electromagnetic fields," Opt. Commun. 209, 7-16 (2002).
    [CrossRef]
  13. V. I. Tatarskii, "The estimation of light depolarization by turbulent inhomogeneities of atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 10, 1762-1765 (1967).
  14. S. F. Clifford, "The classical theory of wave propagation in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J.Strohbehn, ed. (Springer, 1978).
  15. A. D. Wheelon, Electromagnetic Scintillation, II Week Scattering (Cambridge U. Press, 2003), Chap. 11.

2007 (2)

G. Gbur and O. Korotkova, "Angular spectrum representation for propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence," J. Opt. Soc. Am. A 24, 745-752 (2007).
[CrossRef]

W. Gao and O. Korotkova, "Changes in the state of polarization of a random electromagnetic beam propagating through tissue," Opt. Commun. 270, 474-478 (2007).
[CrossRef]

2006 (1)

W. Gao, "Changes of polarization of light beams propagating through tissue," Opt. Commun. 260, 749-755 (2006).
[CrossRef]

2005 (2)

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (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 (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

2003 (1)

E. Wolf, "Unified theory of coherence and polarization of statistical electromagnetic beams," Phys. Lett. A 312, 263-265 (2003).
[CrossRef]

2002 (1)

J. Tervo and J. Turunen, "Angular spectrum representation of partially coherent electromagnetic fields," Opt. Commun. 209, 7-16 (2002).
[CrossRef]

2001 (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

1994 (1)

1967 (1)

V. I. Tatarskii, "The estimation of light depolarization by turbulent inhomogeneities of atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 10, 1762-1765 (1967).

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

Clifford, S. F.

S. F. Clifford, "The classical theory of wave propagation in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J.Strohbehn, ed. (Springer, 1978).

Dogariu, A.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

Gao, W.

W. Gao and O. Korotkova, "Changes in the state of polarization of a random electromagnetic beam propagating through tissue," Opt. Commun. 270, 474-478 (2007).
[CrossRef]

W. Gao, "Changes of polarization of light beams propagating through tissue," Opt. Commun. 260, 749-755 (2006).
[CrossRef]

Gbur, G.

Gori, F.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

James, D. F. V.

Korotkova, O.

G. Gbur and O. Korotkova, "Angular spectrum representation for propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence," J. Opt. Soc. Am. A 24, 745-752 (2007).
[CrossRef]

W. Gao and O. Korotkova, "Changes in the state of polarization of a random electromagnetic beam propagating through tissue," Opt. Commun. 270, 474-478 (2007).
[CrossRef]

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (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]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

Mandel, L.

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

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

Salem, M.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, "The estimation of light depolarization by turbulent inhomogeneities of atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 10, 1762-1765 (1967).

Tervo, J.

J. Tervo and J. Turunen, "Angular spectrum representation of partially coherent electromagnetic fields," Opt. Commun. 209, 7-16 (2002).
[CrossRef]

Turunen, J.

J. Tervo and J. Turunen, "Angular spectrum representation of partially coherent electromagnetic fields," Opt. Commun. 209, 7-16 (2002).
[CrossRef]

Wheelon, A. D.

A. D. Wheelon, Electromagnetic Scintillation, II Week Scattering (Cambridge U. Press, 2003), Chap. 11.

Wolf, E.

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (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]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

E. Wolf, "Unified theory of coherence and polarization of statistical electromagnetic beams," Phys. Lett. A 312, 263-265 (2003).
[CrossRef]

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

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

V. I. Tatarskii, "The estimation of light depolarization by turbulent inhomogeneities of atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 10, 1762-1765 (1967).

J. Opt. A, Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello and R. Simon, "Partially polarized Gaussian Schell-model beams," J. Opt. A, Pure Appl. Opt. 3, 1-9 (2001).
[CrossRef]

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

Opt. Commun. (4)

W. Gao, "Changes of polarization of light beams propagating through tissue," Opt. Commun. 260, 749-755 (2006).
[CrossRef]

W. Gao and O. Korotkova, "Changes in the state of polarization of a random electromagnetic beam propagating through tissue," Opt. Commun. 270, 474-478 (2007).
[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]

J. Tervo and J. Turunen, "Angular spectrum representation of partially coherent electromagnetic fields," Opt. Commun. 209, 7-16 (2002).
[CrossRef]

Phys. Lett. A (1)

E. Wolf, "Unified theory of coherence and polarization of statistical electromagnetic beams," Phys. Lett. A 312, 263-265 (2003).
[CrossRef]

Waves Random Complex Media (1)

O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, "Changes in the polarization ellipse of random electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 15, 353-364 (2005).
[CrossRef]

Waves Random Media (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent EM beams propagating through turbulent atmosphere," Waves Random Media 14, 513-523 (2004).
[CrossRef]

Other (4)

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

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

S. F. Clifford, "The classical theory of wave propagation in a turbulent medium," in Laser Beam Propagation in the Atmosphere, J.Strohbehn, ed. (Springer, 1978).

A. D. Wheelon, Electromagnetic Scintillation, II Week Scattering (Cambridge U. Press, 2003), Chap. 11.

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Figures (6)

Fig. 1
Fig. 1

Illustraton of the notation relating to the propagation of beams.

Fig. 2
Fig. 2

Evolution of the degree of polarization for a pair of tilted, partially correlated plane waves of equal intensity. (a) μ = 1 ; (b) μ = 0.5 . Here λ = 1 μ m , C n 2 = 10 14 m 2 3 , and Δ k = k u 2 u 1 .

Fig. 3
Fig. 3

Evolution of the degree of polarization for a pair of tilted, partially correlated plane waves with I 1 = 1 , I 2 = 2 , and μ = 0.5 exp [ i π 4 ] . Here (i) represents Δ k = 0 , (ii) represents Δ k = 0.001 k , (iii) represents Δ k = 0.002 k , and (iv) represents Δ k = 0.1 k . All other parameters are as in Fig. 2.

Fig. 4
Fig. 4

On-axis degree of polarization of a Gaussian Schell-model beam as a function of propagation distance L for different values of spatial coherence and source degree of polarization. In all cases, I y = 1 , δ x x = 0.1 mm , σ I = 5 cm , and C n 2 = 10 14 m 2 3 .

Fig. 5
Fig. 5

On-axis degree of polarization of an exponentially correlated beam as a function of propagation distance L for different values of spatial coherence and source degree of polarization. In all cases, I y = 1 , δ x x = 0.1 mm , σ I = 5 cm , and C n 2 = 10 14 m 2 3 . The dashed curve indicates the results for a Gaussian Schell-model beam with the same parameters.

Fig. 6
Fig. 6

On-axis degree of polarization of a beam of mixed-correlation type as a function of propagation distance L for different values of spatial coherence and source degree of polarization. In all cases, I y = 1 , δ x x = 0.1 mm , σ I = 5 cm , and C n 2 = 10 14 m 2 3 .

Equations (51)

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E i ( r , ω ) = a i ( u , ω ) P u ( r , ω ) d 2 u , ( i = x , y ) ,
P u ( r , ω ) = exp [ i k ( u r ) ]
u z = + 1 u 2 .
a i ( u , ω ) = 1 ( 2 π ) 2 E i ( 0 ) ( r , ω ) P u * ( r , ω ) d 2 r , ( i = x , y ) ,
E i ( r , ω ) = a i ( u , ω ) P u T ( r , ω ) d 2 u , ( i = x , y ) ,
W i j ( r 1 , r 2 , ω ) = E i * ( r 1 , ω ) E j ( r 2 , ω ) T , ( i , j = x , y ) ,
W i j ( r 1 , r 2 , ω ) = a i * ( u 1 , ω ) a j ( u 2 , ω ) P u 1 T * ( r 1 , ω ) P u 2 T ( r 2 , ω ) T d 2 u 1 d 2 u 2 ,
( i , j = x , y ) .
W i j ( r 1 , r 2 , ω ) = E i * ( r 1 , ω ) E j ( r 2 , ω ) , ( i , j = x , y ) ,
W i j ( r 1 , r 2 , ω ) = A i j ( u 1 , u 2 , ω ) P u 1 * ( r 1 , ω ) × P u 2 ( r 2 , ω ) d 2 u 1 d 2 u 2 , ( i , j = x , y ) ,
A i j ( u 1 , u 2 , ω ) = a i * ( u 1 , ω ) a j ( u 2 , ω )
A i j ( u 1 , u 2 , ω ) = 1 ( 2 π ) 4 W i j ( 0 ) ( r 1 , r 2 , ω ) P u 1 * ( r 1 , ω ) × P u 2 ( r 2 , ω ) d 2 r 1 d 2 r 2 , ( i , j = x , y ) .
W i j ( r 1 , r 2 , ω ) = A i j ( u 1 , u 2 , ω ) P u 1 T * ( r 1 , ω ) × P u 2 T ( r 2 , ω ) T d 2 u 1 d 2 u 2 , ( i , j = x , y ) .
P u 1 T * ( r 1 , ω ) P u 2 T ( r 2 , ω ) = P u 1 * ( r 1 , ω ) P u 2 ( r 2 , ω ) × exp [ 2 H u 1 , u 2 ( 1 ) ( ω ) + H u 1 , u 2 ( 2 ) ( ω ) ] ,
H u 1 , u 2 ( 1 ) ( ω ) = 2 π 2 k 2 L 0 L κ Φ n ( κ , z ) d κ ,
H u 1 , u 2 ( 2 ) ( ω ) = 4 π 2 k 2 0 L d z 0 κ d κ Φ n ( κ , z ) × J 0 [ κ ( r 2 r 1 ) ( L z ) ( u 2 u 1 ) ] .
E ( 0 ) ( r ; ω ) = [ E x ( 0 ) ( r ; ω ) , E y ( 0 ) ( r ; ω ) ] .
S ( 0 ) ( r ; ω ) = E x ( 0 ) * ( r ; ω ) E x ( 0 ) ( r ; ω ) + E y ( 0 ) * ( r ; ω ) E y ( 0 ) ( r ; ω ) .
( E x ( 0 ) ( r , ω ) a 1 ) 2 + ( E y ( 0 ) ( r , ω ) a 2 ) 2 2 E x ( 0 ) ( r , ω ) a 1 E y ( 0 ) ( r , ω ) a 2 cos δ
= sin 2 δ ,
S ( r ; ω ) = Tr [ W ( r , r ; ω ) ] ,
P ( r ; ω ) = 1 4 Det [ W ( r , r ; ω ) ] Tr 2 [ W ( r , r ; ω ) ] ,
C ( r ; ω ) ϵ x ( r ) 2 2 Re D ( r ; ω ) ϵ x ( r ) ϵ y ( r ) + B ( r ; ω ) ϵ y ( r ) 2 = [ Im D ( r ; ω ) ] 2 ,
B ( r ; ω ) = 1 2 ( W x x W y y + ( W x x W y y ) 2 + 4 W x y 2 ) ,
C ( r ; ω ) = 1 2 ( W x x W y y ( W x x W y y ) 2 + 4 Re [ W x y ] 2 ) ,
D ( r ; ω ) = W x y ,
θ ( r , ω ) = 1 2 arctan [ 2 Re [ W x y ( r , ω ) ] W x x ( r , ω ) W y y ( r , ω ) ] .
ϵ ( r , ω ) = A minor A major ,
A major minor 2 = 1 2 [ ( W x x W y y ) 2 + 4 W x y 2 ± ( W x x W y y ) 2 + 4 Re [ W x y ] 2 ] .
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 ) ] .
Φ n ( κ ) = 0.033 C n 2 exp ( κ 2 κ m 2 ) ( κ 2 + κ 0 2 ) 11 6 ,
E ( ω ) = E 1 ( ω ) x ¯ + E 2 ( ω ) y ¯ ,
W ( 0 ) ( r 1 , r 2 ; ω ) = [ I 1 ( ω ) e i k 1 ( r 2 r 1 ) μ ( ω ) I 1 ( ω ) I 2 ( ω ) e i ( k 2 r 2 k 1 r 1 ) μ * ( ω ) I 1 ( ω ) I 2 ( ω ) e i ( k 1 r 2 k 2 r 1 ) I 2 ( ω ) e i k 2 ( r 2 r 1 ) ] .
P ( 0 ) ( ω ) = ( I 1 ( ω ) I 2 ( ω ) ) 2 + 4 μ ( ω ) 2 I 1 ( ω ) I 2 ( ω ) I 1 ( ω ) + I 2 ( ω ) ,
θ ( 0 ) ( ω ) = 1 2 arctan [ 2 Re [ μ ( ω ) I 1 ( ω ) I 2 ( ω ) ] I 1 ( ω ) I 2 ( ω ) ] ,
ϵ ( 0 ) ( ω ) = ( I 1 I 2 ) 2 + 4 μ ( ω ) 2 I 1 ( ω ) I 2 ( ω ) ( I 1 I 2 ) 2 + 4 Re [ μ ( ω ) ] 2 I 1 ( ω ) I 2 ( ω ) ( I 1 I 2 ) 2 + 4 μ ( ω ) 2 I 1 ( ω ) I 2 ( ω ) + ( I 1 I 2 ) 2 + 4 Re [ μ ( ω ) ] 2 I 1 ( ω ) I 2 ( ω ) .
W ( r 1 , r 2 , ω ) = [ I 1 e i k 1 ( r 2 r 1 ) P 1 T ( r 2 ) P 1 T * ( r 1 ) μ I 1 I 2 e i ( k 2 r 2 k 1 r 1 ) P 2 T ( r 2 ) P 1 T * ( r 1 ) μ * I 1 I 2 e i ( k 1 r 2 k 2 r 1 ) P 1 T ( r 2 ) P 2 T * ( r 1 ) I 2 e i k 2 ( r 2 r 1 ) P 2 T ( r 2 ) P 2 T * ( r 1 ) ] ,
P 1 T ( r 2 ) P 1 T * ( r 1 ) = P 1 * ( r 1 ) P 1 ( r 2 ) exp [ 2 H 11 ( 1 ) + H 11 ( 2 ) ] ,
P 1 T * ( r 1 ) P 2 T * ( r 2 ) = P 1 * ( r 1 ) P 2 ( r 2 ) exp [ 2 H 12 ( 1 ) + H 12 ( 2 ) ] ,
P 2 T * ( r 1 ) P 1 T * ( r 2 ) = P 2 * ( r 1 ) P 1 ( r 2 ) exp [ 2 H 21 ( 1 ) + H 21 ( 2 ) ] ,
P 2 T * ( r 1 ) P 2 T ( r 2 ) = P 2 * ( r 1 ) P 2 ( r 2 ) exp [ 2 H 22 ( 1 ) + H 22 ( 2 ) ] .
W ( r , r , ω ) = [ I 1 μ I 1 I 2 e i ( k 2 k 1 ) r exp [ 2 H 12 ( 1 ) + H 12 ( 2 ) ] μ * I 1 I 2 e i ( k 1 k 2 ) r exp [ 2 H 12 ( 1 ) + H 12 ( 2 ) ] I 2 ] ,
P ( ω ) = ( I 1 I 2 ) 2 + 4 μ 2 I 1 I 2 exp [ 4 H 12 ( 1 ) + 2 H 12 ( 2 ) ] I 1 + I 2 ,
θ ( ω ) = 1 2 arctan [ 2 Re [ μ I 1 I 2 exp [ 4 H 12 ( 1 ) + 2 H 12 ( 2 ) ] ] I 1 I 2 ] ,
ϵ ( ω ) = ( I 1 I 2 ) 2 + 4 μ exp [ 4 H 12 ( 1 ) + 2 H 12 ( 2 ) ] 2 I 1 I 2 ( I 1 I 2 ) 2 + 4 Re [ μ exp [ 4 H 12 ( 1 ) + 2 H 12 ( 2 ) ] ] 2 I 1 I 2 ( I 1 I 2 ) 2 + 4 μ exp [ 4 H 12 ( 1 ) + 2 H 12 ( 2 ) ] 2 I 1 I 2 + ( I 1 I 2 ) 2 + 4 Re [ μ exp [ 4 H 12 ( 1 ) + 2 H 12 ( 2 ) ] ] 2 I 1 I 2 .
P ( ω ) = 2 μ ( ω ) exp [ 2 H 12 ( 1 ) + H 12 ( 2 ) ] .
lim L W ( r , r , ω ) = [ I 1 0 0 I 2 ] .
W i j ( 0 ) ( r 1 , r 2 , ω ) = { I i exp [ r 1 2 + r 2 2 2 σ I 2 ] exp [ ( r 2 r 1 ) 2 2 δ i i 2 ] , i = j 0 , i j } ,
A i i ( u 1 , u 2 , ω ) = 1 8 π 2 I i σ I 2 σ i i 2 exp [ k 2 ( u 2 u 1 ) 2 σ I 2 4 ] exp [ k 2 ( u 2 2 + u 1 2 ) σ i i 2 8 ] ,
σ i i 2 = 1 2 ( 1 4 σ I 2 + 1 2 δ i i 2 ) 1 .
W i j ( 0 ) ( r 1 , r 2 , ω ) = { I i exp [ r 1 2 + r 2 2 2 σ I 2 ] exp [ r 2 r 1 2 δ i i ] , i = j 0 , i j } .

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