Abstract

The effect of the polarization-mode dispersion (PMD) of an acousto-optic tunable filter (AOTF) is studied both theoretically and experimentally. A coupled-mode method derived from Maxwell’s equations is proposed to study the evolution of the polarization of light and hence the deterministic dynamics of the PMD inside an AOTF. It is found that the PMD value of a typical AOTF is around several picoseconds and is adjustable by tuning the strength of the applied sound signal. A set of experimental data acquired by the wavelength scanning method is employed to check the validity of the proposed method. A high degree of similarity is observed between the experimental and the theoretical data.

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References

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  1. C. D. Poole and R. E. Wagner, "Phenomenological approach to polarization dispersion in long single-mode fibers," Electron. Lett. 22, 1029-1030 (1986).
    [CrossRef]
  2. A. O. Dal Forno, A. Paradisi, R. Passy, and J. P. von der Weid, "Experimental and theoretical modeling of polarization-mode dispersion in single-mode fibers," IEEE Photon. Technol. Lett. 12, 296-298 (2000).
    [CrossRef]
  3. N. Gisin, R. Passy, J. C. Bishoff, and B. Perny, "Experimental investigations of the statistical properties of polarization mode dispersion in single mode fibers," IEEE Photon. Technol. Lett. 5, 819-821 (1993).
    [CrossRef]
  4. G. J. Foschini and C. D. Poole, "Statistical theory of polarization dispersion in single-mode fibers," J. Lightwave Technol. 9, 1439-1456 (1991).
    [CrossRef]
  5. B. W. Hakki, "Polarization mode dispersion in a single mode fiber," J. Lightwave Technol. 14, 2202-2208 (1996).
    [CrossRef]
  6. R. Caponi, M. Potenza, M. Schiano, M. Artiglia, I. Joindot, C. Geiser, B. Huttner, and N. Gisin, "Deterministic nature of polarisation mode dispersion in fibre amplifiers," in 24th European Conference on Optical Communication (IEEE, 1998), Vol. 1, pp. 543-544.
  7. J.-J. Kao, H.-T. Wu, and C.-W. Tarn, "Theoretical and experimental studies of polarization mode dispersion of an electro-optic Mach-Zehnder modulation," Appl. Opt. 44, 5422-5428 (2005).
    [CrossRef] [PubMed]
  8. C.-W. Tarn, "Spatial coherence property of a laser beam during acousto-optic interaction," J. Opt. Soc. Am. A 16, 1395-1401 (1999).
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  9. C.-W. Tarn, "Spatial Fourier transform approach to the study of polarization changing and beam profile deformation of light during Bragg acousto-optic interaction with longitudinal and shear ultrasonic waves in isotropic media," J. Opt. Soc. Am. A 14, 2231-2242 (1997).
    [CrossRef]
  10. C.-W. Tarn and R.-S. Huang, "Polarization changing and beam profile deformation of light during the isotropic Raman-Nath acousto-optic interaction," Appl. Opt. 37, 7496-7503 (1998).
    [CrossRef]
  11. S. E. Harris and R. W. Wallace, "Acousto-optic tunable filter," J. Opt. Soc. Am. 59, 744-747 (1969).
    [CrossRef]
  12. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984) Chap. 10, pp. 366-404.
  13. A. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge U. Press, 1989), Chaps. 16-19.
  14. J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, 1990), Chap. 2.
  15. J. Xu and R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, 1992).
  16. D.Derickson, ed., Fiber Optic Test and Measurement (Prentice Hall, 1998), Chap. 12.
  17. L. Moller and L. Buhl, "Method for PMD vector monitoring in picosecond pulse transmission systems," J. Lightwave Technol. 19, 1125-1129 (2001)
    [CrossRef]

2005 (1)

2001 (1)

2000 (1)

A. O. Dal Forno, A. Paradisi, R. Passy, and J. P. von der Weid, "Experimental and theoretical modeling of polarization-mode dispersion in single-mode fibers," IEEE Photon. Technol. Lett. 12, 296-298 (2000).
[CrossRef]

1999 (1)

1998 (1)

1997 (1)

1996 (1)

B. W. Hakki, "Polarization mode dispersion in a single mode fiber," J. Lightwave Technol. 14, 2202-2208 (1996).
[CrossRef]

1993 (1)

N. Gisin, R. Passy, J. C. Bishoff, and B. Perny, "Experimental investigations of the statistical properties of polarization mode dispersion in single mode fibers," IEEE Photon. Technol. Lett. 5, 819-821 (1993).
[CrossRef]

1991 (1)

G. J. Foschini and C. D. Poole, "Statistical theory of polarization dispersion in single-mode fibers," J. Lightwave Technol. 9, 1439-1456 (1991).
[CrossRef]

1986 (1)

C. D. Poole and R. E. Wagner, "Phenomenological approach to polarization dispersion in long single-mode fibers," Electron. Lett. 22, 1029-1030 (1986).
[CrossRef]

1969 (1)

Appl. Opt. (2)

Electron. Lett. (1)

C. D. Poole and R. E. Wagner, "Phenomenological approach to polarization dispersion in long single-mode fibers," Electron. Lett. 22, 1029-1030 (1986).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

A. O. Dal Forno, A. Paradisi, R. Passy, and J. P. von der Weid, "Experimental and theoretical modeling of polarization-mode dispersion in single-mode fibers," IEEE Photon. Technol. Lett. 12, 296-298 (2000).
[CrossRef]

N. Gisin, R. Passy, J. C. Bishoff, and B. Perny, "Experimental investigations of the statistical properties of polarization mode dispersion in single mode fibers," IEEE Photon. Technol. Lett. 5, 819-821 (1993).
[CrossRef]

J. Lightwave Technol. (3)

G. J. Foschini and C. D. Poole, "Statistical theory of polarization dispersion in single-mode fibers," J. Lightwave Technol. 9, 1439-1456 (1991).
[CrossRef]

B. W. Hakki, "Polarization mode dispersion in a single mode fiber," J. Lightwave Technol. 14, 2202-2208 (1996).
[CrossRef]

L. Moller and L. Buhl, "Method for PMD vector monitoring in picosecond pulse transmission systems," J. Lightwave Technol. 19, 1125-1129 (2001)
[CrossRef]

J. Opt. Soc. Am. (1)

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

Other (6)

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984) Chap. 10, pp. 366-404.

A. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge U. Press, 1989), Chaps. 16-19.

J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, 1990), Chap. 2.

J. Xu and R. Stroud, Acousto-Optic Devices: Principles, Design, and Applications (Wiley, 1992).

D.Derickson, ed., Fiber Optic Test and Measurement (Prentice Hall, 1998), Chap. 12.

R. Caponi, M. Potenza, M. Schiano, M. Artiglia, I. Joindot, C. Geiser, B. Huttner, and N. Gisin, "Deterministic nature of polarisation mode dispersion in fibre amplifiers," in 24th European Conference on Optical Communication (IEEE, 1998), Vol. 1, pp. 543-544.

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Figures (12)

Fig. 1
Fig. 1

Schematic representation of the change and interaction of light with different polarizations inside an AOTF.

Fig. 2
Fig. 2

Schematic representation of the cause of PMD inside an AOTF.

Fig. 3
Fig. 3

Phase-matching condition for an anisotropic AOTF. (a) Zero- and 1 -order light, (b) the zero and + 1 -order light.

Fig. 4
Fig. 4

Relationship between the x y z and k DB ( ζ ξ η ) coordinate systems.

Fig. 5
Fig. 5

Experimental setup of the DGD measurement of an AOTF.

Fig. 6
Fig. 6

Power exchange between two coupled light waves, ψ 0 o and ψ ( + 1 ) e , as functions of the sound intensity. The shear sound frequency equals 43.760 MHz , and the wavelength of the incident light equals 1540 nm .

Fig. 7
Fig. 7

Power exchange between two coupled light waves, ψ 0 e and ψ ( 1 ) o , as functions of the sound intensity. The shear sound frequency equals 44.168 MHz , and the wavelength of the incident light equals 1540 nm .

Fig. 8
Fig. 8

Comparison of experimental and simulation results of Stokes parameters as a function of wavelength of the zero-order light with the applied sound signal intensity equal 1.1 × 10 5 W mm 2 .

Fig. 9
Fig. 9

Experiential and theoretical results for the DGD as a function of the sound intensity for the zero-order light.

Fig. 10
Fig. 10

Simulation results of Stokes parameters of the first-order light with the applied sound signal equal to 1.1 × 10 5 W mm 2 .

Fig. 11
Fig. 11

Simulation of the DGD of the first-order light as a function of the sound intensity.

Fig. 12
Fig. 12

Simulation results of the Stokes parameters of the 1 -order light with the applied sound signal equal to 1.1 × 10 5 W mm 2 .

Tables (1)

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Table 1 AOTF Parameters

Equations (72)

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2 D ¯ ( r , t ) = μ 0 ε ¯ ¯ 2 t 2 D ¯ ( r , t ) + μ o Δ ε ¯ ¯ 2 t 2 D ¯ ( r , t ) .
Δ ε ¯ ¯ = ε 0 n ¯ ¯ 2 [ [ 1 2 ( p 11 + p 12 ) + p 66 ] S x x 1 2 ( p 11 p 12 ) S x y p 44 S x z + [ 1 2 ( p 11 + p 12 ) p 66 ] S y y + p 13 S z z 1 2 ( p 11 p 12 ) S x y [ 1 2 ( p 11 + p 12 ) p 66 ] S x x p 44 S y z + [ 1 2 ( p 11 + p 12 ) + p 66 ] S y y + p 13 S z z p 44 S x z p 44 S y z p 31 S x x + p 31 S y y + p 33 S z z ] n ¯ ¯ 2 ,
S x y = 1 2 A 0 exp ( j Ω t j K x ) ,
S x z = 0 ,
S y z = 0 ,
Δ ε ¯ ¯ ( r , t ) = 1 2 ε 0 n ¯ ¯ 2 A 0 cos ( Ω t K x ) [ 0 p 11 p 12 0 p 11 p 12 0 0 0 0 0 ] n ¯ ¯ 2 .
D ¯ ( x , z , t ) = m = D ¯ m ( x , z ) exp ( j ω 0 t + j m Ω t ) = 1 2 m = { ψ m o ( x , z ) exp [ j ( ω 0 t + m Ω t k m o x sin θ m o k m o z cos θ m o ) ] ψ m e ( x , z ) exp [ j ( ω 0 t + m Ω t k m e x sin θ m e k m e z cos θ m e ) ] ψ m k ( x , z ) exp [ j ( ω 0 t + m Ω t k m k x sin θ m k k m k z cos θ m k ) ] } + c.c. ,
2 D ¯ m ( r ) + ω 0 2 μ 0 T ¯ ¯ m ε ¯ ¯ 0 T ̿ m 1 D ¯ m ( r ) = 1 2 ω 0 2 μ 0 ε 0 T ̿ m n ̿ 2 p ̿ n ̿ 2 S T ̿ m 1 1 D ¯ m 1 ( r ) + 1 2 ω 0 2 μ 0 ε 0 T ̿ m n ̿ 2 p ̿ n ̿ 2 S T ̿ m + 1 1 D ¯ m + 1 ( r ) ,
T ̿ m o , e = [ sin ϕ m cos ϕ m 0 cos θ m o , e cos ϕ m cos θ m o , e sin ϕ m sin θ m o , e sin θ m o , e cos ϕ m sin θ m o , e sin ϕ m cos θ m o , e ] .
ψ 0 o ( η ) η = j k 0 ( p 11 p 12 ) A 2 n 0 o cos θ 0 o exp [ 2 j Δ β 0 ( + 1 ) o e η ] ψ ( + 1 ) e ( η ) ,
ψ ( + 1 ) e ( η ) η = j k 0 ( p 11 p 12 ) A 2 n ( + 1 ) e cos θ ( + 1 ) e exp [ 2 j Δ β ( + 1 ) 0 e o η ] ψ 0 o ( η ) ,
ψ 0 e ( η ) η = j k 0 ( p 11 p 12 ) A 2 n 0 e cos θ 0 e exp [ 2 j Δ β 0 ( 1 ) e o η ] ψ ( 1 ) o ( η ) ,
ψ ( 1 ) o ( ( η ) ) η = j k 0 ( p 11 p 12 ) A 2 n ( 1 ) o cos θ ( 1 ) o exp [ 2 j Δ β ( 1 ) 0 o e η ] ψ 0 e ( η ) ,
Δ β 0 ( + 1 ) o e = 1 2 ( k 0 o cos θ 0 o k ( + 1 ) e cos θ 1 e ) ,
Δ β ( + 1 ) 0 e o = 1 2 ( k ( + 1 ) e cos θ 1 e k 0 o cos θ 0 o ) ,
Δ β 0 ( 1 ) e o = 1 2 ( k 0 e cos θ 0 e k ( 1 ) o cos θ ( 1 ) o ) ,
Δ β ( 1 ) 0 o e = 1 2 ( k ( 1 ) o cos θ ( 1 ) o k 0 e cos θ 0 e ) .
ψ ¯ 0 ( η = 0 ) = ψ 0 o ( η = 0 ) a ̂ 0 + ψ 0 e ( η = 0 ) a ̂ e ,
= ψ i n c o ( η = 0 ) a ̂ o + ψ i n c e ( η = 0 ) a ̂ e ,
ψ ¯ ( 1 ) ( η = 0 ) = ψ ( 1 ) o ( η = 0 ) a ̂ 0 = 0 ,
ψ ¯ ( + 1 ) ( η = 0 ) = ψ ( + 1 ) e ( η = 0 ) a ̂ e = 0 ,
ψ 0 o ( η ) = [ cos ( g 1 η ) j Δ β 01 o e sin ( g 1 η ) 2 g 1 ] exp ( j 2 Δ β 0 ( + 1 ) o e η ) ψ i n c o ( η ) ,
ψ ( + 1 ) e ( η ) = [ j k 0 ( p 11 p 12 ) A sin ( g 1 η ) 4 n ( + 1 ) e cos θ ( + 1 ) e g 1 ] exp ( j 2 Δ β ( + 1 ) 0 e o η ) ψ i n c o ( η ) ,
ψ 0 e ( η ) = [ cos ( g 2 η ) j Δ β 0 ( 1 ) e o sin ( g 2 η ) 2 g 2 ] exp ( j 2 Δ β 0 ( 1 ) e o η ) ψ i n c e ( η ) ,
ψ ( 1 ) o ( η ) = [ j k 0 ( p 11 p 12 ) A sin ( g 2 η ) 4 n ( 1 ) o cos θ ( 1 ) o g 2 ] exp ( j 2 Δ β ( 1 ) 0 o e η ) ψ i n c e ( η ) ,
g 1 = ( 1 4 Δ β 0 ( + 1 ) o e 2 + k 0 2 ( p 11 p 12 ) 2 A 2 16 n 0 o n ( + 1 ) e cos θ 0 o cos θ ( + 1 ) e ) ,
g 2 = ( 1 4 Δ β 0 ( 1 ) e o 2 + k 0 2 ( p 11 p 12 ) 2 A 2 16 n 0 e n ( 1 ) o cos θ 0 e cos θ ( 1 ) o ) .
E x = 1 n o 2 ε 0 ψ 0 o ( η ) exp [ j ( w t k 0 η ) ] ,
E y = 1 n e 2 ( θ e ) ε 0 ψ 0 e ( η ) exp [ j ( w t k e η ) ] ,
S 1 = E x 2 E y 2 E x 2 + E y 2 ,
= [ cos ( g 1 L o ) n o 2 ε 0 ψ i n c o ] 2 [ cos ( g 2 L e ) n e 2 ( θ e ) ε 0 ψ i n c e ] 2 [ cos ( g 1 L o ) n o 2 ε 0 ψ i n c o ] 2 + [ cos ( g 2 L e ) n e 2 ( θ e ) ε 0 ψ i n c e ] 2 ,
S 2 = 2 Re [ E x E y * ] E x 2 + E y 2 ,
= 2 cos ( g 1 L o ) n o 2 ε 0 cos ( g 2 L e ) n e 2 ( θ e ) ε 0 cos ( k e L e k o L o ) ψ i n c o ψ i n c e [ cos ( g 1 L o ) n o 2 ε 0 ψ i n c o ] 2 + [ cos ( g 2 L e ) n e 2 ( θ e ) ε 0 ψ i n c e ] 2 ,
S 3 = 2 Im [ E x E y * ] E x 2 + E y 2 ,
= 2 cos ( g 1 L o ) n o 2 ε 0 cos ( g 2 L e ) n e 2 ( θ e ) ε 0 sin ( k e L e k o L o ) ψ i n c o ψ i n c e [ cos ( g 1 L o ) n o 2 ε 0 ψ i n c o ] 2 + [ cos ( g 2 L e ) n e 2 ( θ e ) ε 0 ψ i n c e ] 2 ,
d s ¯ o u t ( l , ω ) d ω = Ω ¯ ( ω ) × s ¯ o u t ( l , ω ) ,
Ω ¯ ( ω ) = 1 sin θ d s ¯ o u t ( ω ) d ω 1 s ¯ o u t ( ω ) ,
Δ τ = Ω ¯ ( ω ) = Ω 1 2 + Ω 2 2 + Ω 3 2 = [ d s 1 d ω ] 2 + [ d s 2 d ω ] 2 + [ d s 3 d ω ] 2 ,
d s 1 d ω = { [ 2 n o 4 cos ( g 1 L o ) sin ( g 1 L o ) g 1 L 0 ω ψ i n c o ψ i n c e + 2 n e 4 ( θ e ) cos ( g 2 L e ) sin ( g 2 L e ) g 2 L e ω ψ i n c o ψ i n c e ] × [ 1 n o 4 cos 2 ( g 1 L o ) ψ i n c o 2 + 1 n e 4 ( θ e ) cos 2 ( g 2 L e ) ψ i n c e 2 ] 1 } ( [ cos 2 ( g 1 L o ) n o 4 ψ i n c o 2 cos 2 ( g 2 L e ) n e 4 ( θ e ) ψ i n c e 2 ] [ 2 n o 4 cos ( g 1 L o ) sin ( g 1 L o ) g 1 L o ω ψ i n c o ψ i n c e + 2 n e 4 ( θ e ) cos ( g 2 L e ) sin ( g 2 L e ) g 2 L e ω ψ i n c o ψ i n c e ] × { [ 1 n o 4 cos 2 ( g 1 L o ) ψ i n c o 2 + 1 n e 4 ( θ e ) cos 2 ( g 2 L e ) ψ i n c e 2 ] 2 } 1 ) ,
d s 2 d ω = { 2 cos ( k e L e k o L o ) 1 n o 2 n e 2 ( θ e ) [ sin ( g 1 L o ) cos ( g 2 L e ) g 1 L o ω ψ i n c o ψ i n c e + cos ( g 1 L o ) sin ( g 2 L e ) g 2 L e ω ψ i n c o ψ i n c e ] × [ 1 n o 4 cos 2 ( g 1 L o ) ψ i n c o 2 + 1 n e 4 ( θ e ) cos 2 ( g 2 L e ) ψ i n c e 2 ] 1 } ( 2 cos ( k e L e k o L o ) cos ( g 1 L o ) cos ( g 2 L e ) n o 2 n e 2 ( θ e ) [ 2 n o 4 cos ( g 1 L o ) sin ( g 1 L o ) g 1 L o ω ψ i n c o ψ i n c e + 2 n e 4 ( θ e ) cos ( g 2 L e ) sin ( g 2 L e ) g 2 L e ω ψ i n c o ψ i n c e ] × { [ 1 n o 4 cos 2 ( g 1 L o ) ψ i n c o 2 + 1 n e 4 ( θ e ) cos 2 ( g 2 L e ) ψ i n c e 2 ] 2 } 1 ) ,
d s 3 d ω = { 2 sin ( k e L e k o L o ) 1 n o 2 n e 2 ( θ e ) [ sin ( g 1 L o ) cos ( g 2 L e ) g 1 L o ω ψ i n c o ψ i n c e + cos ( g 1 L o ) sin ( g 2 L e ) g 2 L e ω ψ i n c o ψ i n c e ] × [ 1 n o 4 cos 2 ( g 1 L o ) ψ i n c o 2 + 1 n e 4 ( θ e ) cos 2 ( g 2 L e ) ψ i n c e 2 ] 1 } ( 2 sin ( k e L e k o L o ) cos ( g 1 L o ) cos ( g 2 L e ) n o 2 n e 2 ( θ e ) [ 2 n o 4 cos ( g 1 L o ) sin ( g 1 L o ) g 1 L o ω ψ i n c o ψ i n c e + 2 n e 4 ( θ e ) cos ( g 2 L e ) sin ( g 2 L e ) g 2 L e ω ψ i n c o ψ i n c e ] × { [ 1 n o 4 cos 2 ( g 1 L o ) ψ i n c o 2 + 1 n e 4 ( θ e ) cos 2 ( g 2 L e ) ψ i n c e 2 ] 2 } 1 ) ,
E x = 1 n o 2 ε 0 ψ 0 o ( η ) exp [ j ( w t k 0 η ) ] ,
E y = 1 n e 2 ( θ e ) ε 0 ψ ( + 1 ) e ( η ) exp [ j ( w t + Ω t k e η ) ] .
S 1 = E x 2 E y 2 E x 2 + E y 2
= [ cos ( g 1 L o ) n o 2 ε 0 ψ i n c o ] 2 [ sin ( g 1 L e ) n e 2 ( θ e ) ε 0 ψ i n c o ] 2 [ cos ( g 1 L o ) n o 2 ε 0 ψ i n c o ] 2 + [ sin ( g 1 L e ) n e 2 ( θ e ) ε 0 ψ i n c o ] 2
= [ cos ( g 1 L o ) n o 2 ε 0 ] 2 [ sin ( g 1 L e ) n e 2 ( θ e ) ε 0 ] 2 ,
S 2 = 2 Re [ E x E y * ] E x 2 + E y 2
= 2 cos ( g 1 L o ) n o 2 ε 0 sin ( g 1 L e ) n e 2 ( θ e ) ε 0 cos ( k e L e k o L o + Ω t ) ψ i n c o ψ i n c o [ cos ( g 1 L o ) n o 2 ε 0 ψ i n c o ] 2 + [ sin ( g 1 L e ) n e 2 ( θ e ) ε 0 ψ i n c o ] 2
= 2 cos ( g 1 L o ) n o 2 ε 0 sin ( g 1 L e ) n e 2 ( θ e ) ε 0 cos ( k e L e k o L o + Ω t ) ,
S 3 = 2 Im [ E x E y * ] E x 2 + E y 2
= 2 cos ( g 1 L o ) n o 2 ε 0 sin ( g 1 L e ) n e 2 ( θ e ) ε 0 sin ( k e L e k o L o + Ω t ) ψ i n c o ψ i n c o [ cos ( g 1 L o ) n o 2 ε 0 ψ i n c o ] 2 + [ sin ( g 1 L e ) n e 2 ( θ e ) ε 0 ψ i n c o ] 2
= 2 cos ( g 1 L o ) n o 2 ε 0 sin ( g 1 L e ) n e 2 ( θ e ) ε 0 sin ( k e L e k o L o + Ω t ) .
Δ τ = Ω ¯ ( ω ) = Ω 1 2 + Ω 2 2 + Ω 3 2 = [ d s 1 d ω ] 2 + [ d s 2 d ω ] 2 + [ d s 3 d ω ] 2 ,
d s 1 d ω = [ 2 n o 4 ε 0 2 cos ( g 1 L o ) sin ( g 1 L o ) g 1 L 0 ω 2 n e 4 ( θ e ) ε 0 2 cos ( g 1 L e ) sin ( g 1 L e ) g 1 L e ω ] ,
d s 2 d ω = 2 cos ( k e L e k o L o + Ω t ) n o 2 n e 2 ( θ e ) ε 0 2 [ sin ( g 1 L o ) sin ( g 1 L e ) g 1 L o ω + cos ( g 1 L o ) cos ( g 1 L e ) g 1 L e ω ] ,
d s 3 d ω = 2 sin ( k e L e k o L o + Ω t ) n o 2 n e 2 ( θ e ) ε 0 2 [ sin ( g 1 L o ) sin ( g 1 L e ) g 1 L o ω + cos ( g 1 L o ) cos ( g 1 L e ) g 1 L e ω ] .
E x = 1 n o 2 ε 0 ψ ( 1 ) o ( η ) exp [ j ( w t Ω t k 0 η ) ] ,
E y = 1 n e 2 ( θ e ) ε 0 ψ 0 e ( η ) exp [ j ( w t k e η ) ] .
S 1 = E x 2 E y 2 E x 2 + E y 2
= [ sin ( g 2 L o ) n o 2 ε 0 ψ i n c e ] 2 [ cos ( g 2 L e ) n e 2 ( θ e ) ε 0 ψ i n c e ] 2 [ sin ( g 2 L o ) n o 2 ε 0 ψ i n c e ] 2 + [ cos ( g 2 L e ) n e 2 ( θ e ) ε 0 ψ i n c e ] 2
= [ sin ( g 2 L o ) n o 2 ε 0 ] 2 [ cos ( g 2 L e ) n e 2 ( θ e ) ε 0 ] 2 ,
S 2 = 2 Re [ E x E y * ] E x 2 + E y 2
= 2 sin ( g 2 L o ) n o 2 ε 0 cos ( g 2 L e ) n e 2 ( θ e ) ε 0 cos ( k e L e k o L o Ω t ) ψ i n c e ψ i n c e [ sin ( g 2 L o ) n o 2 ε 0 ψ i n c e ] 2 + [ cos ( g 2 L e ) n e 2 ( θ e ) ε 0 ψ i n c e ] 2
= 2 sin ( g 2 L o ) n o 2 ε 0 cos ( g 2 L e ) n e 2 ( θ e ) ε 0 cos ( k e L e k o L o Ω t ) ,
S 3 = 2 Im [ E x E y * ] E x 2 + E y 2
= 2 sin ( g 2 L o ) n o 2 ε 0 cos ( g 2 L e ) n e 2 ( θ e ) ε 0 sin ( k e L e k o L o Ω t ) ψ i n c e ψ i n c e [ sin ( g 2 L o ) n o 2 ε 0 ψ i n c e ] 2 + [ cos ( g 2 L e ) n e 2 ( θ e ) ε 0 ψ i n c e ] 2
= 2 sin ( g 2 L o ) n o 2 ε 0 cos ( g 2 L e ) n e 2 ( θ e ) ε 0 sin ( k e L e k o L o Ω t ) ,
Δ τ = Ω ¯ ( ω ) = Ω 1 2 + Ω 2 2 + Ω 3 2 = [ d s 1 d ω ] 2 + [ d s 2 d ω ] 2 + [ d s 3 d ω ] 2 ,
d s 1 d ω = [ 2 n o 4 ε 0 2 cos ( g 2 L o ) sin ( g 2 L o ) g 2 L 0 ω 2 n e 4 ( θ e ) cos ( g 2 L e ) sin ( g 2 L e ) g 2 L e ω ] ,
d s 2 d ω = 2 cos ( k e L e k o L o Ω t ) n o 2 n e 2 ( θ e ) ε 0 2 [ cos ( g 2 L o ) cos ( g 2 L e ) g 2 L o ω ψ i n c o ψ i n c e + sin ( g 2 L o ) sin ( g 2 L e ) g 2 L e ω ] ,
d s 3 d ω = 2 sin ( k e L e k o L o Ω t ) n o 2 n e 2 ( θ e ) ε 0 2 [ cos ( g 2 L o ) cos ( g 2 L e ) g 2 L o ω ψ i n c o ψ i n c e + sin ( g 2 L o ) sin ( g 2 L e ) g 2 L e ω ] .
Δ τ = N e λ s t a r t λ s t o p 2 ( λ s t a r t λ s t o p ) c ,

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