Abstract

We propose a method of effectively extending the stimulated Brillouin scattering (SBS) gain bandwidth in a single-mode optical fiber to reduce group-velocity-dispersion (GVD)-dependent pulse spread of SBS slow light. This can be done by overlapping doublet SBS gain spectra synthesized from a single pump laser. Numerical calculations are performed to verify our proposed method. We find that there exists the optimum spectral separation between two center frequencies of the doublet SBS gain spectrum with respect to the inherent spectral width of the pump laser, which makes it possible to effectively reduce the signal pulse broadening due to GVD. We show that the maximum time delay of the amplified signal pulse can be approximately two times longer than that by a previously reported method using a single broadband pump laser.

© 2008 Optical Society of America

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References

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  1. D. J. Gauthier, “Slow light brings faster communications,” Phys. World 18, 30-32 (2005).
  2. D. Dahan and G. Eisenstien, “Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering,” Opt. Express 13, 6234-6248 (2005).
    [CrossRef] [PubMed]
  3. Y. A. Vlasov, M. O. Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
    [CrossRef] [PubMed]
  4. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
    [CrossRef] [PubMed]
  5. M. G. Herráez, K. Y. Song, and L. Thévenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express 14, 1395-1400 (2006).
    [CrossRef]
  6. K. Y. Song, M. G. Herráez, and L. Thévenaz, “Gain-assisted pulse advancement using single and double Brillouin gain peaks in optical fibers,” Opt. Express 13, 9758-9765 (2005).
    [CrossRef] [PubMed]
  7. M. G. Herráez, K. Y. Song, and L. Thévenaz, “Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering,” Appl. Phys. Lett. 87, 081113 (2005).
    [CrossRef]
  8. M. D. Stenner, M. A. Neifeld, Z. M. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express 13, 9995-10002 (2005).
    [CrossRef] [PubMed]
  9. T. Schneider, M. Junker, K. U. Lauterbach, and R. Henker, “Distortion reduction in cascaded slow light delays,” Electron. Lett. 42, 1110-1111 (2006).
    [CrossRef]
  10. Z. W. Lu, Y. K. Dong, and Q. Li, “Slow light in multi-line Brillouin gain spectrum,” Opt. Express 15, 1871-1877 (2007).
    [CrossRef] [PubMed]
  11. R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73, 063812 (2006).
    [CrossRef]
  12. Z. M. Zhu and D. J. Gauthier, “Nearly transparent SBS slow light in an optical fiber,” Opt. Express 14, 7238-7245 (2006).
    [CrossRef] [PubMed]
  13. A. M. C. Dawes, Z. M. Zhu, and D. J. Gauthier, “Improving the bandwidth of SBS-based slow light delay,” in Proceedings of the Conference on Lasers and Electro-Optics (CLEO'06) (2006), paper CThW1.
  14. Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “12-GHz-bandwidth SBS slow light in optical fibers,” in Proceedings of the Optical Fiber Communication Conference (OFC) (2006), paper PDP1.
  15. Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201-206 (2007).
    [CrossRef]
  16. K. Y. Song and K. Hotate, “25 GHz bandwidth Brillouin slow light in optical fibers,” Opt. Lett. 32, 217-219 (2007).
    [CrossRef] [PubMed]
  17. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
    [CrossRef]
  18. A. Minardo, R. Bernini, and L. Zeni, “Low distortion Brillouin slow light in optical fibers using AM modulation,” Opt. Express 14, 5866-5876 (2006).
    [CrossRef] [PubMed]
  19. B. Macke and B. Ségard, “Pulse normalization in slow-light media,” Phys. Rev. A 73, 043802 (2006).
    [CrossRef]

2007 (3)

2006 (6)

T. Schneider, M. Junker, K. U. Lauterbach, and R. Henker, “Distortion reduction in cascaded slow light delays,” Electron. Lett. 42, 1110-1111 (2006).
[CrossRef]

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73, 063812 (2006).
[CrossRef]

M. G. Herráez, K. Y. Song, and L. Thévenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express 14, 1395-1400 (2006).
[CrossRef]

A. Minardo, R. Bernini, and L. Zeni, “Low distortion Brillouin slow light in optical fibers using AM modulation,” Opt. Express 14, 5866-5876 (2006).
[CrossRef] [PubMed]

Z. M. Zhu and D. J. Gauthier, “Nearly transparent SBS slow light in an optical fiber,” Opt. Express 14, 7238-7245 (2006).
[CrossRef] [PubMed]

B. Macke and B. Ségard, “Pulse normalization in slow-light media,” Phys. Rev. A 73, 043802 (2006).
[CrossRef]

2005 (8)

D. Dahan and G. Eisenstien, “Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering,” Opt. Express 13, 6234-6248 (2005).
[CrossRef] [PubMed]

K. Y. Song, M. G. Herráez, and L. Thévenaz, “Gain-assisted pulse advancement using single and double Brillouin gain peaks in optical fibers,” Opt. Express 13, 9758-9765 (2005).
[CrossRef] [PubMed]

M. D. Stenner, M. A. Neifeld, Z. M. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express 13, 9995-10002 (2005).
[CrossRef] [PubMed]

D. J. Gauthier, “Slow light brings faster communications,” Phys. World 18, 30-32 (2005).

Y. A. Vlasov, M. O. Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

M. G. Herráez, K. Y. Song, and L. Thévenaz, “Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering,” Appl. Phys. Lett. 87, 081113 (2005).
[CrossRef]

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Bernini, R.

Bigelow, M. S.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Boyd, R. W.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Boyle, M. O.

Y. A. Vlasov, M. O. Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Camacho, R. M.

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73, 063812 (2006).
[CrossRef]

Dahan, D.

Dawes, A. M. C.

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201-206 (2007).
[CrossRef]

M. D. Stenner, M. A. Neifeld, Z. M. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express 13, 9995-10002 (2005).
[CrossRef] [PubMed]

A. M. C. Dawes, Z. M. Zhu, and D. J. Gauthier, “Improving the bandwidth of SBS-based slow light delay,” in Proceedings of the Conference on Lasers and Electro-Optics (CLEO'06) (2006), paper CThW1.

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “12-GHz-bandwidth SBS slow light in optical fibers,” in Proceedings of the Optical Fiber Communication Conference (OFC) (2006), paper PDP1.

Dong, Y. K.

Eisenstien, G.

Gaeta, A. L.

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Gauthier, D. J.

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201-206 (2007).
[CrossRef]

Z. M. Zhu and D. J. Gauthier, “Nearly transparent SBS slow light in an optical fiber,” Opt. Express 14, 7238-7245 (2006).
[CrossRef] [PubMed]

M. D. Stenner, M. A. Neifeld, Z. M. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express 13, 9995-10002 (2005).
[CrossRef] [PubMed]

D. J. Gauthier, “Slow light brings faster communications,” Phys. World 18, 30-32 (2005).

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “12-GHz-bandwidth SBS slow light in optical fibers,” in Proceedings of the Optical Fiber Communication Conference (OFC) (2006), paper PDP1.

A. M. C. Dawes, Z. M. Zhu, and D. J. Gauthier, “Improving the bandwidth of SBS-based slow light delay,” in Proceedings of the Conference on Lasers and Electro-Optics (CLEO'06) (2006), paper CThW1.

Hamann, H. F.

Y. A. Vlasov, M. O. Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Henker, R.

T. Schneider, M. Junker, K. U. Lauterbach, and R. Henker, “Distortion reduction in cascaded slow light delays,” Electron. Lett. 42, 1110-1111 (2006).
[CrossRef]

Herráez, M. G.

Hotate, K.

Howell, J. C.

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73, 063812 (2006).
[CrossRef]

Junker, M.

T. Schneider, M. Junker, K. U. Lauterbach, and R. Henker, “Distortion reduction in cascaded slow light delays,” Electron. Lett. 42, 1110-1111 (2006).
[CrossRef]

Lauterbach, K. U.

T. Schneider, M. Junker, K. U. Lauterbach, and R. Henker, “Distortion reduction in cascaded slow light delays,” Electron. Lett. 42, 1110-1111 (2006).
[CrossRef]

Li, Q.

Lu, Z. W.

Macke, B.

B. Macke and B. Ségard, “Pulse normalization in slow-light media,” Phys. Rev. A 73, 043802 (2006).
[CrossRef]

McNab, S. J.

Y. A. Vlasov, M. O. Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Minardo, A.

Neifeld, M. A.

Okawachi, Y.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Pack, M. V.

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73, 063812 (2006).
[CrossRef]

Schneider, T.

T. Schneider, M. Junker, K. U. Lauterbach, and R. Henker, “Distortion reduction in cascaded slow light delays,” Electron. Lett. 42, 1110-1111 (2006).
[CrossRef]

Schweinsberg, A.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Ségard, B.

B. Macke and B. Ségard, “Pulse normalization in slow-light media,” Phys. Rev. A 73, 043802 (2006).
[CrossRef]

Sharping, J. E.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Song, K. Y.

Stenner, M. D.

Thévenaz, L.

Vlasov, Y. A.

Y. A. Vlasov, M. O. Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Willner, A. E.

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201-206 (2007).
[CrossRef]

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “12-GHz-bandwidth SBS slow light in optical fibers,” in Proceedings of the Optical Fiber Communication Conference (OFC) (2006), paper PDP1.

Zeni, L.

Zhang, L.

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201-206 (2007).
[CrossRef]

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “12-GHz-bandwidth SBS slow light in optical fibers,” in Proceedings of the Optical Fiber Communication Conference (OFC) (2006), paper PDP1.

Zhu, Z. M.

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201-206 (2007).
[CrossRef]

Z. M. Zhu and D. J. Gauthier, “Nearly transparent SBS slow light in an optical fiber,” Opt. Express 14, 7238-7245 (2006).
[CrossRef] [PubMed]

M. D. Stenner, M. A. Neifeld, Z. M. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express 13, 9995-10002 (2005).
[CrossRef] [PubMed]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “12-GHz-bandwidth SBS slow light in optical fibers,” in Proceedings of the Optical Fiber Communication Conference (OFC) (2006), paper PDP1.

A. M. C. Dawes, Z. M. Zhu, and D. J. Gauthier, “Improving the bandwidth of SBS-based slow light delay,” in Proceedings of the Conference on Lasers and Electro-Optics (CLEO'06) (2006), paper CThW1.

Appl. Phys. Lett. (1)

M. G. Herráez, K. Y. Song, and L. Thévenaz, “Optically controlled slow and fast light in optical fibers using stimulated Brillouin scattering,” Appl. Phys. Lett. 87, 081113 (2005).
[CrossRef]

Electron. Lett. (1)

T. Schneider, M. Junker, K. U. Lauterbach, and R. Henker, “Distortion reduction in cascaded slow light delays,” Electron. Lett. 42, 1110-1111 (2006).
[CrossRef]

J. Lightwave Technol. (1)

Nature (1)

Y. A. Vlasov, M. O. Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Opt. Express (7)

Opt. Lett. (1)

Phys. Rev. A (3)

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A 71, 023801 (2005).
[CrossRef]

B. Macke and B. Ségard, “Pulse normalization in slow-light media,” Phys. Rev. A 73, 043802 (2006).
[CrossRef]

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73, 063812 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Phys. World (1)

D. J. Gauthier, “Slow light brings faster communications,” Phys. World 18, 30-32 (2005).

Other (2)

A. M. C. Dawes, Z. M. Zhu, and D. J. Gauthier, “Improving the bandwidth of SBS-based slow light delay,” in Proceedings of the Conference on Lasers and Electro-Optics (CLEO'06) (2006), paper CThW1.

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “12-GHz-bandwidth SBS slow light in optical fibers,” in Proceedings of the Optical Fiber Communication Conference (OFC) (2006), paper PDP1.

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Figures (6)

Fig. 1
Fig. 1

Brillouin spectrum (solid curve) constructed by two Stokes gain profiles (gains 1 and 2, dashed curves) and two anti-Stokes loss profiles (losses 1 and 2, dashed-dotted curves) with a doublet broadband pump, where Δ Δ ω p = 0.596 and Δ ω p = 0.477 Ω B .

Fig. 2
Fig. 2

Time delay of the signal pulse at the central frequency ω 0 as a function of Δ Δ ω p for different values of Δ ω p .

Fig. 3
Fig. 3

Dependences of (a) the Brillouin gain, (b) the Brillouin phase shift, and (c) the relative delay on the frequency detuning ( ω ω 0 ) Ω B and on the relative spectral separation between two gain peaks Δ Δ ω p , where Δ ω p = 0.477 Ω B .

Fig. 4
Fig. 4

Dependence of the normalized k 2 and k 3 on Δ Δ ω p .

Fig. 5
Fig. 5

Spectral dependences of (a) the Brillouin gain when Δ Δ ω p = 0.707 under Δ ω p = 0.453 Ω B with k 2 = 0 and (b) the relative delay when Δ Δ ω p = 0.596 under Δ ω p = 0.477 Ω B with k 3 = 0 . Results for a single broadband pump case (dotted curve) are also plotted as compared with that for a double broadband pump case (solid curve). Also shown in Fig. 5a are the two constituent gain profiles (dashed curve) that construct the doublet.

Fig. 6
Fig. 6

(a) Time delay dependence of the signal pulse on gain. (b) Dependence of the pulse broadening factor of the signal pulse on relative delay for the single broadband pump case (dashed curves) and the double broadband pump case (solid curves) with Δ ω p = 0.477 Ω B and T in = 37 ps . Note that Δ Δ ω p and k 3 are set at 0.596 and 0, respectively, for the double pump case.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

E ( z , ω ) = E ( 0 , ω ) H ( ω ) ,
H ( ω ) = exp [ i k ( ω ) z ] = exp ( i z n 0 ω c ) exp [ g ( ω ) 2 ] .
g ( ω ) = g 0 ( ω ) I p ( ω p ) = + g 0 ( ω ω p ) I p ( ω p ) d ω p ,
g 0 ( ω ω p ) = g 0 z { i γ [ ω ( ω p Ω B ) ] + i γ } ,
I p ( ω p ) = I p 0 π Δ ω p { exp [ ( ω p ( ω p 0 Δ ) Δ ω p ) 2 ] + exp [ ( ω p ( ω p 0 + Δ ) Δ ω p ) 2 ] } .
g ( ω ) = G [ e ξ 1 2 erfc ( i ξ 1 ) + e ξ 2 2 erfc ( i ξ 2 ) ] ,
H ( ω ) 2 = exp [ Re [ g ( ω ) ] ] = exp [ G ( e ξ 1 2 + e ξ 2 2 ) ] ,
H ( ω ) = Im [ g ( ω ) 2 ] = G [ e ξ 1 2 0 ξ 1 e t 2 d t + e ξ 2 2 0 ξ 2 e t 2 d t ] π .
Δ T d ( ω c ) = d Im [ g ( ω ) 2 ] d ω ω c = 2 G π Δ ω p [ 1 ξ 1 e ξ 1 2 0 ξ 1 e t 2 d t ξ 2 e ξ 2 2 0 ξ 2 e t 2 d t ] ,
Δ T d ( ω 0 ) = 2 G π Δ ω p [ 1 2 ξ 1 e ξ 1 2 0 ξ 1 e t 2 d t ] .
k 1 = n 0 c + d d ω [ g ( ω ) 2 i z ] ω 0 = n 0 c + Δ T d ( ω 0 ) z ,
k 2 = d 2 d ω 2 [ g ( ω ) 2 i z ] ω 0 = 2 i G z ( Δ ω p ) 2 ( 1 2 ξ 1 2 ) e ξ 1 2 ,
k 3 = d 3 d ω 3 [ g ( ω ) 2 i z ] ω 0 = 8 G z π ( Δ ω p ) 3 [ ( 3 ξ 1 2 ξ 1 3 ) e ξ 1 2 0 ξ 1 e t 2 d t + ξ 1 2 1 ] ,

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