Abstract

Closed-form formulae have been analytically derived to estimate crosstalk due to stimulated Raman scattering (SRS) in a multipumped, broad, and flattened gain distributed Raman amplifier (DRA). The derived formulae are applied to study the crosstalk due to SRS in a multipumped DRA having different pumping configurations. System bounds for a typical wavelength division multiplexing sys tem employing DRA have been evaluated theoretically. Performance of DRA for other parameters such as wavelength separation, input signal power, and bit rate of the system has also been investigated.

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References

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  1. R. H. Stolen, “Nonlinearity in fiber transmission,” Proc. IEEE 68, 1232–1236 (1980).
    [CrossRef]
  2. A. R. Chrapylvy and P. S. Henry, “Performance degradation due to stimulated Raman scattering in wavelength-division-multiplexed optical-fiber systems,” Electron. Lett. 19, 641–643 (1983).
    [CrossRef]
  3. A. R. Chrapylvy, “Optical power limits in multichannel wavelength-division-multiplexed systems due to stimulated Raman scattering,” Electron. Lett. 20, 58–59 (1984).
    [CrossRef]
  4. X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, “Input power limits and maximum capacity in long-haul WDM lightwave systems due to stimulated Raman scattering,” Opt. Commun. 107, 358–360 (1994).
    [CrossRef]
  5. J. Wang, X. Sun, and M. Zhang, “Effect of group velocity dispersion on stimulated Raman crosstalk in multichannel transmission systems,” IEEE Photon. Technol. Lett. 10, 540–542 (1998).
    [CrossRef]
  6. D. Cotter and A. M. Hill, “Stimulated Raman crosstalk in optical transmission system: Effects of group velocity dispersion,” Electron. Lett. 20, 185–187 (1984).
    [CrossRef]
  7. G. P. Aggarwal, Nonlinear Fiber Optics (Academic, 1995), Chap. 8.
  8. S. Norimatsu and T. Yamamoto, “Waveform distortion due to stimulated Raman scattering in wideband WDM transmission systems,” J. Lightwave Technol. 19, 159–171 (2001).
    [CrossRef]
  9. M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8, 548–558 (2002).
    [CrossRef]
  10. S. Namiki and Y. Emori, “Ultra broadband Raman amplifiers pumped and gain equalized by wavelength-division multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7, 3–16 (2001).
    [CrossRef]
  11. F. Forghieri, R. W. Tkach, and A. R. Chrapylvy, “Fiber nonlinearities and their impact on transmission systems,” in Optical Fiber Telecommunications III A, I.P.Kaminov and T.L.Koch, eds. (Academic, 1997), pp. 196–264.
    [CrossRef]
  12. D. N. Christodoulides and R. I. Joseph, “Theory of stimulated Raman scattering in optical fibers in pulse walk off regime,” IEEE J. Quantum Electron. 25, 273–279 (1989).
    [CrossRef]
  13. D. N. Christodoulides and R. B. Jander, “Evolution of stimulated Raman crosstalk in wavelength division multiplexed systems,” IEEE Photon. Technol. Lett. 8, 1722–1724 (1996).
    [CrossRef]
  14. F. Forgheiri, R. W. Tkach, and A. R. Chrapylvy, “Effect of modulation statistics on Raman crosstalk in WDM systems,” IEEE Photon. Technol. Lett. 7, 998–1000 (2000).
  15. K.-P. Ho, “Statistical properties of stimulated Raman crosstalk in WDM systems,” J. Lightwave Technol. 18, 915–921(2000).
    [CrossRef]
  16. T. Yamamoto and S. Norimatsu, “Statistical analysis on stimulated Raman crosstalk in dispersion managed fiber links,” J. Lightwave Technol. 21, 2229–2239 (2003).
    [CrossRef]
  17. Anamika and V. Priye, “Power penalty in WDM system due to stimulated Raman crosstalk,” presented at Photonics 2010, Guwahati, India, 11–15 December 2010.
  18. M. S. Kao and J. Wu, “Signal light amplification by stimulated Raman scattering in an N-channel WDM optical fiber communication system,” J. Lightwave Technol. 7, 1290–1299, (1989).
    [CrossRef]
  19. I. S. Gradshetyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed. (Academic, 2000).

2003 (1)

2002 (1)

M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8, 548–558 (2002).
[CrossRef]

2001 (2)

S. Namiki and Y. Emori, “Ultra broadband Raman amplifiers pumped and gain equalized by wavelength-division multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7, 3–16 (2001).
[CrossRef]

S. Norimatsu and T. Yamamoto, “Waveform distortion due to stimulated Raman scattering in wideband WDM transmission systems,” J. Lightwave Technol. 19, 159–171 (2001).
[CrossRef]

2000 (2)

F. Forgheiri, R. W. Tkach, and A. R. Chrapylvy, “Effect of modulation statistics on Raman crosstalk in WDM systems,” IEEE Photon. Technol. Lett. 7, 998–1000 (2000).

K.-P. Ho, “Statistical properties of stimulated Raman crosstalk in WDM systems,” J. Lightwave Technol. 18, 915–921(2000).
[CrossRef]

1998 (1)

J. Wang, X. Sun, and M. Zhang, “Effect of group velocity dispersion on stimulated Raman crosstalk in multichannel transmission systems,” IEEE Photon. Technol. Lett. 10, 540–542 (1998).
[CrossRef]

1996 (1)

D. N. Christodoulides and R. B. Jander, “Evolution of stimulated Raman crosstalk in wavelength division multiplexed systems,” IEEE Photon. Technol. Lett. 8, 1722–1724 (1996).
[CrossRef]

1994 (1)

X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, “Input power limits and maximum capacity in long-haul WDM lightwave systems due to stimulated Raman scattering,” Opt. Commun. 107, 358–360 (1994).
[CrossRef]

1989 (2)

D. N. Christodoulides and R. I. Joseph, “Theory of stimulated Raman scattering in optical fibers in pulse walk off regime,” IEEE J. Quantum Electron. 25, 273–279 (1989).
[CrossRef]

M. S. Kao and J. Wu, “Signal light amplification by stimulated Raman scattering in an N-channel WDM optical fiber communication system,” J. Lightwave Technol. 7, 1290–1299, (1989).
[CrossRef]

1984 (2)

D. Cotter and A. M. Hill, “Stimulated Raman crosstalk in optical transmission system: Effects of group velocity dispersion,” Electron. Lett. 20, 185–187 (1984).
[CrossRef]

A. R. Chrapylvy, “Optical power limits in multichannel wavelength-division-multiplexed systems due to stimulated Raman scattering,” Electron. Lett. 20, 58–59 (1984).
[CrossRef]

1983 (1)

A. R. Chrapylvy and P. S. Henry, “Performance degradation due to stimulated Raman scattering in wavelength-division-multiplexed optical-fiber systems,” Electron. Lett. 19, 641–643 (1983).
[CrossRef]

1980 (1)

R. H. Stolen, “Nonlinearity in fiber transmission,” Proc. IEEE 68, 1232–1236 (1980).
[CrossRef]

Aggarwal, G. P.

G. P. Aggarwal, Nonlinear Fiber Optics (Academic, 1995), Chap. 8.

Anamika,

Anamika and V. Priye, “Power penalty in WDM system due to stimulated Raman crosstalk,” presented at Photonics 2010, Guwahati, India, 11–15 December 2010.

Chrapylvy, A. R.

F. Forgheiri, R. W. Tkach, and A. R. Chrapylvy, “Effect of modulation statistics on Raman crosstalk in WDM systems,” IEEE Photon. Technol. Lett. 7, 998–1000 (2000).

A. R. Chrapylvy, “Optical power limits in multichannel wavelength-division-multiplexed systems due to stimulated Raman scattering,” Electron. Lett. 20, 58–59 (1984).
[CrossRef]

A. R. Chrapylvy and P. S. Henry, “Performance degradation due to stimulated Raman scattering in wavelength-division-multiplexed optical-fiber systems,” Electron. Lett. 19, 641–643 (1983).
[CrossRef]

F. Forghieri, R. W. Tkach, and A. R. Chrapylvy, “Fiber nonlinearities and their impact on transmission systems,” in Optical Fiber Telecommunications III A, I.P.Kaminov and T.L.Koch, eds. (Academic, 1997), pp. 196–264.
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides and R. B. Jander, “Evolution of stimulated Raman crosstalk in wavelength division multiplexed systems,” IEEE Photon. Technol. Lett. 8, 1722–1724 (1996).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, “Theory of stimulated Raman scattering in optical fibers in pulse walk off regime,” IEEE J. Quantum Electron. 25, 273–279 (1989).
[CrossRef]

Cotter, D.

D. Cotter and A. M. Hill, “Stimulated Raman crosstalk in optical transmission system: Effects of group velocity dispersion,” Electron. Lett. 20, 185–187 (1984).
[CrossRef]

Ebskamp, F.

X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, “Input power limits and maximum capacity in long-haul WDM lightwave systems due to stimulated Raman scattering,” Opt. Commun. 107, 358–360 (1994).
[CrossRef]

Emori, Y.

S. Namiki and Y. Emori, “Ultra broadband Raman amplifiers pumped and gain equalized by wavelength-division multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7, 3–16 (2001).
[CrossRef]

Forgheiri, F.

F. Forgheiri, R. W. Tkach, and A. R. Chrapylvy, “Effect of modulation statistics on Raman crosstalk in WDM systems,” IEEE Photon. Technol. Lett. 7, 998–1000 (2000).

Forghieri, F.

F. Forghieri, R. W. Tkach, and A. R. Chrapylvy, “Fiber nonlinearities and their impact on transmission systems,” in Optical Fiber Telecommunications III A, I.P.Kaminov and T.L.Koch, eds. (Academic, 1997), pp. 196–264.
[CrossRef]

Gradshetyn, I. S.

I. S. Gradshetyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed. (Academic, 2000).

Henry, P. S.

A. R. Chrapylvy and P. S. Henry, “Performance degradation due to stimulated Raman scattering in wavelength-division-multiplexed optical-fiber systems,” Electron. Lett. 19, 641–643 (1983).
[CrossRef]

Hill, A. M.

D. Cotter and A. M. Hill, “Stimulated Raman crosstalk in optical transmission system: Effects of group velocity dispersion,” Electron. Lett. 20, 185–187 (1984).
[CrossRef]

Ho, K.-P.

Islam, M. N.

M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8, 548–558 (2002).
[CrossRef]

Jander, R. B.

D. N. Christodoulides and R. B. Jander, “Evolution of stimulated Raman crosstalk in wavelength division multiplexed systems,” IEEE Photon. Technol. Lett. 8, 1722–1724 (1996).
[CrossRef]

Jorgensen, B. F.

X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, “Input power limits and maximum capacity in long-haul WDM lightwave systems due to stimulated Raman scattering,” Opt. Commun. 107, 358–360 (1994).
[CrossRef]

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Theory of stimulated Raman scattering in optical fibers in pulse walk off regime,” IEEE J. Quantum Electron. 25, 273–279 (1989).
[CrossRef]

Kao, M. S.

M. S. Kao and J. Wu, “Signal light amplification by stimulated Raman scattering in an N-channel WDM optical fiber communication system,” J. Lightwave Technol. 7, 1290–1299, (1989).
[CrossRef]

Namiki, S.

S. Namiki and Y. Emori, “Ultra broadband Raman amplifiers pumped and gain equalized by wavelength-division multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7, 3–16 (2001).
[CrossRef]

Norimatsu, S.

Pedersen, R. J.

X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, “Input power limits and maximum capacity in long-haul WDM lightwave systems due to stimulated Raman scattering,” Opt. Commun. 107, 358–360 (1994).
[CrossRef]

Priye, V.

Anamika and V. Priye, “Power penalty in WDM system due to stimulated Raman crosstalk,” presented at Photonics 2010, Guwahati, India, 11–15 December 2010.

Ryzhik, I. M.

I. S. Gradshetyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed. (Academic, 2000).

Stolen, R. H.

R. H. Stolen, “Nonlinearity in fiber transmission,” Proc. IEEE 68, 1232–1236 (1980).
[CrossRef]

Sun, X.

J. Wang, X. Sun, and M. Zhang, “Effect of group velocity dispersion on stimulated Raman crosstalk in multichannel transmission systems,” IEEE Photon. Technol. Lett. 10, 540–542 (1998).
[CrossRef]

Tkach, R. W.

F. Forgheiri, R. W. Tkach, and A. R. Chrapylvy, “Effect of modulation statistics on Raman crosstalk in WDM systems,” IEEE Photon. Technol. Lett. 7, 998–1000 (2000).

F. Forghieri, R. W. Tkach, and A. R. Chrapylvy, “Fiber nonlinearities and their impact on transmission systems,” in Optical Fiber Telecommunications III A, I.P.Kaminov and T.L.Koch, eds. (Academic, 1997), pp. 196–264.
[CrossRef]

Wang, J.

J. Wang, X. Sun, and M. Zhang, “Effect of group velocity dispersion on stimulated Raman crosstalk in multichannel transmission systems,” IEEE Photon. Technol. Lett. 10, 540–542 (1998).
[CrossRef]

Wu, J.

M. S. Kao and J. Wu, “Signal light amplification by stimulated Raman scattering in an N-channel WDM optical fiber communication system,” J. Lightwave Technol. 7, 1290–1299, (1989).
[CrossRef]

Yamamoto, T.

Zhang, M.

J. Wang, X. Sun, and M. Zhang, “Effect of group velocity dispersion on stimulated Raman crosstalk in multichannel transmission systems,” IEEE Photon. Technol. Lett. 10, 540–542 (1998).
[CrossRef]

Zhang, X.

X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, “Input power limits and maximum capacity in long-haul WDM lightwave systems due to stimulated Raman scattering,” Opt. Commun. 107, 358–360 (1994).
[CrossRef]

Electron. Lett. (3)

A. R. Chrapylvy and P. S. Henry, “Performance degradation due to stimulated Raman scattering in wavelength-division-multiplexed optical-fiber systems,” Electron. Lett. 19, 641–643 (1983).
[CrossRef]

A. R. Chrapylvy, “Optical power limits in multichannel wavelength-division-multiplexed systems due to stimulated Raman scattering,” Electron. Lett. 20, 58–59 (1984).
[CrossRef]

D. Cotter and A. M. Hill, “Stimulated Raman crosstalk in optical transmission system: Effects of group velocity dispersion,” Electron. Lett. 20, 185–187 (1984).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. N. Christodoulides and R. I. Joseph, “Theory of stimulated Raman scattering in optical fibers in pulse walk off regime,” IEEE J. Quantum Electron. 25, 273–279 (1989).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8, 548–558 (2002).
[CrossRef]

S. Namiki and Y. Emori, “Ultra broadband Raman amplifiers pumped and gain equalized by wavelength-division multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7, 3–16 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

J. Wang, X. Sun, and M. Zhang, “Effect of group velocity dispersion on stimulated Raman crosstalk in multichannel transmission systems,” IEEE Photon. Technol. Lett. 10, 540–542 (1998).
[CrossRef]

D. N. Christodoulides and R. B. Jander, “Evolution of stimulated Raman crosstalk in wavelength division multiplexed systems,” IEEE Photon. Technol. Lett. 8, 1722–1724 (1996).
[CrossRef]

F. Forgheiri, R. W. Tkach, and A. R. Chrapylvy, “Effect of modulation statistics on Raman crosstalk in WDM systems,” IEEE Photon. Technol. Lett. 7, 998–1000 (2000).

J. Lightwave Technol. (4)

Opt. Commun. (1)

X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, “Input power limits and maximum capacity in long-haul WDM lightwave systems due to stimulated Raman scattering,” Opt. Commun. 107, 358–360 (1994).
[CrossRef]

Proc. IEEE (1)

R. H. Stolen, “Nonlinearity in fiber transmission,” Proc. IEEE 68, 1232–1236 (1980).
[CrossRef]

Other (4)

F. Forghieri, R. W. Tkach, and A. R. Chrapylvy, “Fiber nonlinearities and their impact on transmission systems,” in Optical Fiber Telecommunications III A, I.P.Kaminov and T.L.Koch, eds. (Academic, 1997), pp. 196–264.
[CrossRef]

G. P. Aggarwal, Nonlinear Fiber Optics (Academic, 1995), Chap. 8.

I. S. Gradshetyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 6th ed. (Academic, 2000).

Anamika and V. Priye, “Power penalty in WDM system due to stimulated Raman crosstalk,” presented at Photonics 2010, Guwahati, India, 11–15 December 2010.

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of forward and backward pumped DRA.

Fig. 2
Fig. 2

Variation of SRS crosstalk standard deviation (dB) with signal wavelength for forward pumped DRA and SMF.

Fig. 3
Fig. 3

Variation of SRS crosstalk standard deviation (dB) with signal wavelength for backward pumped DRA and SMF.

Fig. 4
Fig. 4

Variation of SRS crosstalk standard deviation (dB) with signal wavelength for Δ λ = 1 , 2.5, and 5 nm wavelength separation for forward pumped DRAs.

Fig. 5
Fig. 5

Variation of SRS crosstalk standard deviation (dB) with signal wavelength for Δ λ = 1 , 2.5 and 5 nm wavelength separation for backward pumped DRAs.

Fig. 6
Fig. 6

Variation of SRS crosstalk standard deviation (dB) with signal wavelength for power input = 0 , 5, and 10 dBm for forward pumped DRAs.

Fig. 7
Fig. 7

Variation of SRS crosstalk standard deviation (dB) with signal wavelength for power input = 0 , 5, and 10 dBm for backward pumped DRAs.

Fig. 8
Fig. 8

Variation of SRS crosstalk standard deviation (dB) with signal wavelength for bit rate = 2.5 Gbps , 10 Gbps , and 40 Gbps for forward pumped DRAs.

Fig. 9
Fig. 9

Variation of SRS crosstalk standard deviation (dB) with signal wavelength for bit rate = 2.5 Gbps , 10 Gbps , and 40 Gbps for backward pumped DRAs.

Fig. 10
Fig. 10

Variation of SRS crosstalk standard deviation (dB) with power input for bit rate = 2.5 Gbps , 10 Gbps , and 40 Gbps for forward pumped DRAs.

Fig. 11
Fig. 11

Variation of SRS crosstalk standard deviation (dB) with power input for bit rate = 2.5 Gbps , 10 Gbps , and 40 Gbps for backward pumped DRAs.

Tables (1)

Tables Icon

Table 1 Parameters for Evaluating Crosstalk Variance

Equations (46)

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P i ( z , t ) = P i ( 0 , τ ) exp { α z x i ( z , t ) } ,
x i ( z , t ) = j = 1 N x i j ( z , t )
x i j ( z , t ) = k = + b k q i j ( t z v i k T )
q i j ( t ) = K 0 L p ( t d i j z ) e α z g ( z ) d z ,
K = g R ψ A eff f i f j ( f i f j ) ( | f i f j | 15 THz ) = 0 ( | f i f j | > 15 THz ) .
g ( z ) = exp ( γ ( λ s λ p P so + P po ) A eff α ( 1 e α z ) ) ,
g ( z ) = exp ( γ ( λ s λ p P so + P P L ) A eff α ( e α ( z L ) e α L ) ) ,
Q i j ( ω ) = K 0 L P ( ω ) e j d i j z ω e α z exp ( K ( 1 e α z ) ) d z = K P ( ω ) 0 L e ( α + d i j ω ) z e K ( 1 e α z ) d z ,
Q i j ( ω ) = K P ( ω ) e K 1 e L t ( α + j d i j ω ) t e K t α d t .
Q i j ( ω ) = A P ( ω ) 1 e L 1 K t α t ( α + 1 + j d i j ω ) d t ,
Q i j ( ω ) = A P ( ω ) [ 1 e ( α + j d i j ω ) L α + j d i j ω ] A K P ( ω ) [ 1 e ( 2 α + j d i j ω ) L 2 α + j d i j ω ] .
P ( ω ) = 2 P 0 T sin ω T 2 ω T 2 .
μ x = Q ( 0 ) 2 T = K P ( 0 ) e K L e ( α ) K P ( 0 ) e K K L e ( 2 α ) .
α x 2 = 1 8 π T | Q ( ω ) | 2 d ω .
Q i j ( ω ) = K 0 L P ( ω ) e j d i j z ω e α z exp ( K ( e α ( z L ) e α L ) ) d z .
Q i j ( ω ) = K P ( ω ) L 0 e ( α + d i j ω ) ( L m ) e K ( e α m e α L ) d m = K P ( ω ) 0 L e ( α + d i j ω ) ( L m ) e K ( e α m e α L ) d m ,
Q i j ( ω ) = K P ( ω ) e K e α L K 1 e L t ( α + j d i j ω ) t e K t α d t ,
Q i j ( ω ) = A P ( ω ) e ( α + j d i j ω ) L 1 e L t ( α 1 + j d i j ω ) ( 1 + K t α ) d t ,
Q i j ( ω ) = A P ( ω ) e ( α + j d i j ω ) L [ e ( α + j d i j ω ) L 1 α + j d i j ω ] + A P ( ω ) K e ( α + j d i j ω ) L [ e ( j d i j ω ) L 1 j d i j ω ] = A P ( ω ) [ 1 e ( α + j d i j ω ) L α + j d i j ω ] + A P ( ω ) K e α L [ 1 e ( j d i j ω ) L j d i j ω ] .
μ x = Q ( 0 ) 2 T = K P ( 0 ) e K e α L L e ( α ) .
σ x 2 = 1 8 π T | Q ( ω ) | 2 d ω .
| Q ( ω ) | 2 = | M + N | 2 = ( M + N ) × ( M + N ) * = | M | 2 + | N | 2 + M N * + M * N = | M | 2 + | N | 2 + 2 × Re ( M N * ) .
| M | 2 = A 2 α 2 + ( d i j ω ) 2 [ ( 1 e α L ) 2 + 4 e α L sin 2 ( d i j ω L 2 ) ] ,
| N | 2 = A 2 K 2 4 α 2 + ( d i j α ) 2 [ ( 1 e 2 α L ) 2 + 4 e 2 α L sin 2 ( d i j ω L 2 ) ] ,
2 Re ( M N * ) = 2 A 2 K ( 2 α 2 + d i j 2 ω 2 ) ( α 2 + d i j 2 ω 2 ) ( 4 α 2 + d i j 2 ω 2 ) [ 1 + e 3 α L ( e 2 α L + e α L ) ( 1 2 sin 2 ( d i j ω L 2 ) ) + α d i j ω sin d i j ω L ( α 2 + d i j 2 ω 2 ) ( 4 α 2 + d i j 2 ω 2 ) [ e α L e 2 α L ] ] .
2 α 2 + d i j 2 ω 2 ( α 2 + d i j 2 ω 2 ) ( 4 α 2 + d i j 2 ω 2 ) = ( 1 3 ) ( α 2 + d i j 2 ω 2 ) + ( 2 3 ) ( 4 α 2 + d i j 2 ω 2 ) .
sin 2 ω T 2 ω × sin 2 ( d i j ω L 2 ) = 1 4 [ 2 sin 2 ω T 2 + 2 sin 2 ( d i j ω L 2 ) sin 2 ( T + d i j L ) ω 2 sin 2 ( T d i j L ) ω 2 ]
sin 2 ω T 2 ω 2 ( ω 2 + ( α d i j ) 2 ) d ω = 2 π 4 | α d i j | 3 ( e α L w + α L w 1 ) .
4 P 0 2 T 2 8 π T sin 2 ω T 2 ω 2 T 2 4 | M | 2 = A 2 P 0 2 α 3 L w [ ( 1 e α L ) 2 ( e α L w + α L w 1 ) + e α L { 2 ( e α L w + e α L 1 ) e α | L + L w | + e α | L L w | + α ( | L + L w | | L L w | ) } ] ,
4 P 0 2 T 2 8 π T sin 2 ω T 2 ω 2 T 2 4 | N | 2 = A 2 K 2 P 0 2 8 α 3 L w [ ( 1 e 2 α L ) 2 ( e 2 α L w + 2 α L w 1 ) + e 2 α L { 2 ( e 2 α L w + e 2 α L 1 ) ( e 2 α | L + L w | + e 2 α | L L w | + 2 α ( | L + L w | | L L w | ) } ] ,
4 P 0 2 T 2 8 π T sin 2 ω T 2 sin 2 ω 2 T 2 4 2 Re ( M N * ) = 2 A 2 K P 0 2 3 α 3 L w [ ( 1 + e 3 α L e 2 α L e α L ) ( e α L w + α L w 1 ) + 1 2 ( e α L + e 2 α L ) { 2 ( e α L w + e α L 1 ) ( e α | L + L w | + e α | L L w | + α ( | L + L w | | L L w | ) } ] A 2 K P 0 2 6 α 3 L w [ ( 1 + e 3 α L e 2 α L e α L ) ( e 2 α L w + 2 α L w 1 ) + 1 2 ( e α L + e 2 α L ) { 2 ( e 2 α L w + e 2 α L 1 ) ( e 2 α | L + L w | + e 2 α | L L w | + 2 α ( | L + L w | | L L w | ) } ] A 2 K P 0 2 3 α 3 L w [ sign ( d i j L ) ( 2 2 e α L ) + sign ( T d i j L ) ( 1 e α | L L w | ) sign ( T + d i j L ) ( 1 e α | L + L w | ) ] + A 2 K P 0 2 12 α 3 L w [ sign ( d i j L ) ( 2 2 e 2 α L ) + sign ( T d i j L ) ( 1 e 2 α | L L w | ) sign ( T + d i j L ) ( 1 e 2 α | L + L w | ) ] .
σ x 2 = K 1 { ( 1 e α L ) 2 P 1 ( α ) + e α L P 2 ( α ) } + K 8 { ( 1 e 2 α L ) 2 P 1 ( 2 α ) + e 2 α L P 2 ( 2 α ) } 2 K 3 { M α P 1 ( α ) + N α P 2 ( α ) } K 6 { M α P 1 ( 2 α ) + N α P 2 ( 2 α ) } K 3 { sign ( d i j L ) ( 2 2 e α L ) + sign ( γ ) ( 1 e | γ | | α d i j | ) + sign ( β ) ( 1 e | β | | α d i j | ) } + K 12 { sign ( d i j L ) ( 2 2 e 2 α L ) + sign ( γ ) ( 1 e | γ | | 2 α d i j | ) + sign ( β ) ( 1 e | β | | 2 α d i j | ) } .
| M | 2 = A 2 α 2 + ( d i j ω ) 2 [ ( 1 e α L ) 2 + 4 e α L sin 2 ( d i j ω L 2 ) ] ,
| N | 2 = A 2 K 2 e 2 α L ( d i j ω ) 2 [ 4 sin 2 ( d i j ω L 2 ) ] ,
2 Re ( M N * ) = [ 2 A 2 K ( α 2 + d i j 2 ω 2 ) ( d i j 2 ω 2 ) { ( 1 + e α L ) d i j 2 ω 2 ( cos d i j ω L 1 ) α d i j ω sin d i j ω L ( 1 e α L ) } ] .
4 P 0 2 T 2 8 π T sin 2 ω T 2 ω 2 T 2 4 | M | 2 = A 2 P 0 2 α 3 L w [ ( 1 e α L ) 2 ( e α L w + α L w 1 ) + e α L { 2 ( e α L w + e α L 1 ) ( e α | L + L w | + e α | L L w | + α ( | L + L w | | L L w | ) } ] ,
4 P 0 2 T 2 8 π T sin 2 ω T 2 ω 2 T 2 4 | N | 2 = 8 A 2 K 2 P 0 2 e 2 α L 3 d i j 2 T min ( T 2 4 , d i j 2 L 2 4 ) { 3 max ( T 2 , d i j L 2 ) min ( T 2 , d i j L 2 ) } ,
4 P 0 2 T 2 8 π T sin 2 ω T 2 ω 2 T 2 4 2 Re ( M N * ) = A 2 K e α L P 0 2 α 3 L w [ ( 1 + e α L ) { 2 α L + | L w L | | α | + | L w + L | | α | 2 e 2 α L + e | L w L | | α | + e | L w + L | | α | } ] + 2 A 2 K e α L P 0 2 α 3 L w [ ( 1 + e α L ) ( e 2 α L w + 2 α L w 1 ) ] + A 2 K e α L P 0 2 T α d i j [ ( 1 e α L ) F ( T , d i j L ) ] A 2 K e α L P 0 2 α 3 L w [ ( 1 e α L ) { sign ( d i j L ) ( 2 2 e α L ) + sign ( T d i j L ) ( 1 e | L w L | | α | ) sign ( T + d i j L ) ( 1 e | L w + L | | α | ) } ] ,
F ( T , d i j L ) = sign ( d i j L + 2 T ) ( d i j L 2 8 + ( d i j L ) T 2 + T 2 ) sign ( d i j L ) d i j L 2 4 + sign ( d i j L + 2 T ) ( d i j L 2 8 + ( d i j L ) T 2 T 2 2 ) .
σ x 2 = A 2 P s 2 α 3 L w K 2 + 8 A 2 K 2 P s 2 e 2 α L 3 d i j 2 T K 3 A 2 K e α L P s 2 α 3 L w ( K 4 + K 5 ) + 2 A 2 K e α L P s 2 α 3 L w K 6 .
K 2 = [ ( 1 e α L ) 2 ( e α L w + α L w 1 ) + e α L { 2 ( e α L w + e α L 1 ) ( e α | L + L w | + e α | L L w | + α ( | L + L w | | L L w | ) } ] ,
K 3 = min ( T 2 4 , d i j 2 L 2 4 ) × { 3 max ( T 2 , d i j L 2 ) min ( T 2 , d i j L 2 ) } ,
K 4 = ( 1 + e α L ) { 2 α L + | L w L | | α | + | L w + L | | α | 2 e 2 α L + e | L w L | | α | + e | L w + L | | α | } ,
K 5 = ( 1 e α L ) { sign ( d i j L ) ( 2 2 e α L ) + sign ( T d i j L ) ( 1 e | L w L | | α | ) + sign ( T + d i j L ) ( 1 e | L w + L | | α | ) } ,
K 6 = [ ( 1 + e α L ) ( e 2 α L w + 2 α L w 1 ) ] ,
g R

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