Abstract

Stimulated Brillouin scattering (SBS) has been recently shown to offer a mechanism for generating tunable all-optical delays in room-temperature single-mode optical fibers at telecommunication wavelengths. This technique makes use of the rapid variation of the refractive index that occurs in the vicinity of the Brillouin gain resonance. When the slow light pulse delay is subject to a constraint on the allowable pulse distortion, it has been shown that the use of a pair of closely-spaced Brillouin gain lines can increase the distortion-constrained delay, with respect to the single-line configuration. In this paper, we numerically and experimentally demonstrate that the same experimental apparatus usually employed for generating a Brillouin gain doublet, can also be used for achieving three equally-spaced Brillouin gain resonances, further increasing the distortion-constrained pulse delay.

© 2006 Optical Society of America

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References

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  1. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594 (1999).
    [CrossRef]
  2. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of Ultraslow Light Propagation in a Ruby Crystal at Room Temperature," Phys. Rev. Lett. 90, 113903 (2003).
    [CrossRef] [PubMed]
  3. 3. J. Sharping, Y. Okawachi, and A. Gaeta, "Wide bandwidth slow light using a Raman fiber amplifier," Opt. Express 13, 6092-6098 (2005).
    [CrossRef] [PubMed]
  4. K. Y. Song, M. G. Herráez, and L. Thévenaz, "Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering," Opt. Express 13, 82-88 (2005).
    [CrossRef] [PubMed]
  5. 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]
  6. K. Song, M. Herráez, and L. Thévenaz, "Long optically controlled delays in optical fibers," Opt. Lett. 30, 1782-1784 (2005).
    [CrossRef] [PubMed]
  7. 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]
  8. H. Cao, A. Dogariu, and L. J. Wang, "Negative group delay and pulse compression in superluminal pulse propagation," IEEE J. Sel. Top. Quantum Electron. 9, 52-58 (2003).
    [CrossRef]
  9. B. Macke and B. Ségard, "Propagation of light-pulses at a negative group-velocity," European Phys. J. D 23, 125-141 (2003).
    [CrossRef]
  10. M. Bashkansky, G. Beadie, Z. Dutton, F. K. Fatemi, J. Reintjes, and M. Steiner, "Slow-light dynamics of large bandwidth pulses in warm rubidium vapor," Phys. Rev. A 72, 033819 (2005).
    [CrossRef]
  11. Z. Dutton, M. Bashkansky, M. Steiner, and J. Reintjes, "Channelization architecture for wide-band slow light in atomic vapors," SPIE 5735, 115-129 (2005).
    [CrossRef]
  12. Q. Sun, Y. V. Rostovtsev, J. P. Dowling, M. O. Scully, and M. S. Zhubairy, "Optically controlled delays for broadband pulses," Phys. Rev. A 72 031802(R) (2005).
    [CrossRef]
  13. M. Stenner, M. Neifeld, Z. Zhu, A. Dawes, and D. Gauthier, "Distortion management in slow-light pulse delay," Opt. Express 13, 9995-10002 (2005).
    [CrossRef] [PubMed]
  14. K. Song, M. González 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]
  15. M. González Herráez, K. Song, and L. Thévenaz, "Arbitrary-bandwidth Brillouin slow light in optical fibers," Opt. Express 14, 1395-1400 (2006).
    [CrossRef] [PubMed]
  16. Z. Zhu, A.M.C. Dawes, D.J. Gauthier, L. Zhang, and A.E. Willner, "12-GHz-Bandwidth SBS Slow Light in Optical Fibers," postdeadline paper PDP1, OFC 2006, Anaheim, CA, Mar. 5-10, 2006.
  17. D. Dahan and G. Eisenstein, "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, 6234-6249 (2005).
    [CrossRef] [PubMed]
  18. A. V. Oppenheim and A. S. Willsky, Signals and Systems, 2nd Ed. (Prentice Hall, Upper Saddle River, 1997).
  19. G. P. Agrawal, Nonlinear fiber optics, 3th Ed. (Academic Press, Boston, 2001).
  20. M. Nikles, L. Thévenaz, and P. A. Robert, "Brillouin gain spectrum characterization in single-mode optical fibers," J. Lightwave Technol. 15, 1842-1851 (1997).
    [CrossRef]

2006 (1)

2005 (10)

D. Dahan and G. Eisenstein, "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, 6234-6249 (2005).
[CrossRef] [PubMed]

3. J. Sharping, Y. Okawachi, and A. Gaeta, "Wide bandwidth slow light using a Raman fiber amplifier," Opt. Express 13, 6092-6098 (2005).
[CrossRef] [PubMed]

K. Y. Song, M. G. Herráez, and L. Thévenaz, "Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering," Opt. Express 13, 82-88 (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]

K. Song, M. Herráez, and L. Thévenaz, "Long optically controlled delays in optical fibers," Opt. Lett. 30, 1782-1784 (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]

M. Bashkansky, G. Beadie, Z. Dutton, F. K. Fatemi, J. Reintjes, and M. Steiner, "Slow-light dynamics of large bandwidth pulses in warm rubidium vapor," Phys. Rev. A 72, 033819 (2005).
[CrossRef]

Z. Dutton, M. Bashkansky, M. Steiner, and J. Reintjes, "Channelization architecture for wide-band slow light in atomic vapors," SPIE 5735, 115-129 (2005).
[CrossRef]

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

K. Song, M. González 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]

2003 (3)

H. Cao, A. Dogariu, and L. J. Wang, "Negative group delay and pulse compression in superluminal pulse propagation," IEEE J. Sel. Top. Quantum Electron. 9, 52-58 (2003).
[CrossRef]

B. Macke and B. Ségard, "Propagation of light-pulses at a negative group-velocity," European Phys. J. D 23, 125-141 (2003).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of Ultraslow Light Propagation in a Ruby Crystal at Room Temperature," Phys. Rev. Lett. 90, 113903 (2003).
[CrossRef] [PubMed]

1999 (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594 (1999).
[CrossRef]

1997 (1)

M. Nikles, L. Thévenaz, and P. A. Robert, "Brillouin gain spectrum characterization in single-mode optical fibers," J. Lightwave Technol. 15, 1842-1851 (1997).
[CrossRef]

Bashkansky, M.

Z. Dutton, M. Bashkansky, M. Steiner, and J. Reintjes, "Channelization architecture for wide-band slow light in atomic vapors," SPIE 5735, 115-129 (2005).
[CrossRef]

M. Bashkansky, G. Beadie, Z. Dutton, F. K. Fatemi, J. Reintjes, and M. Steiner, "Slow-light dynamics of large bandwidth pulses in warm rubidium vapor," Phys. Rev. A 72, 033819 (2005).
[CrossRef]

Beadie, G.

M. Bashkansky, G. Beadie, Z. Dutton, F. K. Fatemi, J. Reintjes, and M. Steiner, "Slow-light dynamics of large bandwidth pulses in warm rubidium vapor," Phys. Rev. A 72, 033819 (2005).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594 (1999).
[CrossRef]

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]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of Ultraslow Light Propagation in a Ruby Crystal at Room Temperature," Phys. Rev. Lett. 90, 113903 (2003).
[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]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of Ultraslow Light Propagation in a Ruby Crystal at Room Temperature," Phys. Rev. Lett. 90, 113903 (2003).
[CrossRef] [PubMed]

Cao, H.

H. Cao, A. Dogariu, and L. J. Wang, "Negative group delay and pulse compression in superluminal pulse propagation," IEEE J. Sel. Top. Quantum Electron. 9, 52-58 (2003).
[CrossRef]

Dahan, D.

Dawes, A.

Dogariu, A.

H. Cao, A. Dogariu, and L. J. Wang, "Negative group delay and pulse compression in superluminal pulse propagation," IEEE J. Sel. Top. Quantum Electron. 9, 52-58 (2003).
[CrossRef]

Dutton, Z.

M. Bashkansky, G. Beadie, Z. Dutton, F. K. Fatemi, J. Reintjes, and M. Steiner, "Slow-light dynamics of large bandwidth pulses in warm rubidium vapor," Phys. Rev. A 72, 033819 (2005).
[CrossRef]

Z. Dutton, M. Bashkansky, M. Steiner, and J. Reintjes, "Channelization architecture for wide-band slow light in atomic vapors," SPIE 5735, 115-129 (2005).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594 (1999).
[CrossRef]

Eisenstein, G.

Fatemi, F. K.

M. Bashkansky, G. Beadie, Z. Dutton, F. K. Fatemi, J. Reintjes, and M. Steiner, "Slow-light dynamics of large bandwidth pulses in warm rubidium vapor," Phys. Rev. A 72, 033819 (2005).
[CrossRef]

Gaeta, A. L.

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]

Gauthier, D.

Gauthier, D. J.

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]

González Herráez, M.

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594 (1999).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594 (1999).
[CrossRef]

Herráez, M.

Herráez, M. G.

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of Ultraslow Light Propagation in a Ruby Crystal at Room Temperature," Phys. Rev. Lett. 90, 113903 (2003).
[CrossRef] [PubMed]

Macke, B.

B. Macke and B. Ségard, "Propagation of light-pulses at a negative group-velocity," European Phys. J. D 23, 125-141 (2003).
[CrossRef]

Neifeld, M.

Nikles, M.

M. Nikles, L. Thévenaz, and P. A. Robert, "Brillouin gain spectrum characterization in single-mode optical fibers," J. Lightwave Technol. 15, 1842-1851 (1997).
[CrossRef]

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]

Reintjes, J.

Z. Dutton, M. Bashkansky, M. Steiner, and J. Reintjes, "Channelization architecture for wide-band slow light in atomic vapors," SPIE 5735, 115-129 (2005).
[CrossRef]

M. Bashkansky, G. Beadie, Z. Dutton, F. K. Fatemi, J. Reintjes, and M. Steiner, "Slow-light dynamics of large bandwidth pulses in warm rubidium vapor," Phys. Rev. A 72, 033819 (2005).
[CrossRef]

Robert, P. A.

M. Nikles, L. Thévenaz, and P. A. Robert, "Brillouin gain spectrum characterization in single-mode optical fibers," J. Lightwave Technol. 15, 1842-1851 (1997).
[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, "Propagation of light-pulses at a negative group-velocity," European Phys. J. D 23, 125-141 (2003).
[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.

Song, K. Y.

Steiner, M.

M. Bashkansky, G. Beadie, Z. Dutton, F. K. Fatemi, J. Reintjes, and M. Steiner, "Slow-light dynamics of large bandwidth pulses in warm rubidium vapor," Phys. Rev. A 72, 033819 (2005).
[CrossRef]

Z. Dutton, M. Bashkansky, M. Steiner, and J. Reintjes, "Channelization architecture for wide-band slow light in atomic vapors," SPIE 5735, 115-129 (2005).
[CrossRef]

Stenner, M.

Thévenaz, L.

Wang, L. J.

H. Cao, A. Dogariu, and L. J. Wang, "Negative group delay and pulse compression in superluminal pulse propagation," IEEE J. Sel. Top. Quantum Electron. 9, 52-58 (2003).
[CrossRef]

Willner, A. E.

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]

Zhu, Z.

Zhu, Z. M.

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]

European Phys. J. D (1)

B. Macke and B. Ségard, "Propagation of light-pulses at a negative group-velocity," European Phys. J. D 23, 125-141 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

H. Cao, A. Dogariu, and L. J. Wang, "Negative group delay and pulse compression in superluminal pulse propagation," IEEE J. Sel. Top. Quantum Electron. 9, 52-58 (2003).
[CrossRef]

J. Lightwave Technol. (1)

M. Nikles, L. Thévenaz, and P. A. Robert, "Brillouin gain spectrum characterization in single-mode optical fibers," J. Lightwave Technol. 15, 1842-1851 (1997).
[CrossRef]

Nature (London) (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594 (1999).
[CrossRef]

Opt. Express (6)

Opt. Lett. (1)

Phys. Rev. A (2)

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]

M. Bashkansky, G. Beadie, Z. Dutton, F. K. Fatemi, J. Reintjes, and M. Steiner, "Slow-light dynamics of large bandwidth pulses in warm rubidium vapor," Phys. Rev. A 72, 033819 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of Ultraslow Light Propagation in a Ruby Crystal at Room Temperature," Phys. Rev. Lett. 90, 113903 (2003).
[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]

SPIE (1)

Z. Dutton, M. Bashkansky, M. Steiner, and J. Reintjes, "Channelization architecture for wide-band slow light in atomic vapors," SPIE 5735, 115-129 (2005).
[CrossRef]

Other (4)

Q. Sun, Y. V. Rostovtsev, J. P. Dowling, M. O. Scully, and M. S. Zhubairy, "Optically controlled delays for broadband pulses," Phys. Rev. A 72 031802(R) (2005).
[CrossRef]

Z. Zhu, A.M.C. Dawes, D.J. Gauthier, L. Zhang, and A.E. Willner, "12-GHz-Bandwidth SBS Slow Light in Optical Fibers," postdeadline paper PDP1, OFC 2006, Anaheim, CA, Mar. 5-10, 2006.

A. V. Oppenheim and A. S. Willsky, Signals and Systems, 2nd Ed. (Prentice Hall, Upper Saddle River, 1997).

G. P. Agrawal, Nonlinear fiber optics, 3th Ed. (Academic Press, Boston, 2001).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Relative delay for the single Lorentzian line (solid blue line), the double Lorentzian line (dashed red line for simulation, circles for experimental results), and the triple Lorentzian line (dotted magenta line for simulation, squares for experimental results) (b) Lorentzian line-center amplitude gain coefficients g01 (solid blue line), g02 (dashed red line), gt01 (dashed-dotted magenta line) and gt02 (dotted magenta line). The fitting gain coefficients extracted from the measured Brillouin gain spectra are represented by circles for the double Lorentzian line, and squares for the triple Lorentzian line (c) Normalized line separation for the double Lorentzian line (δ2/γ, solid blue line) and the triple Lorentzian line (δ3/γ, dashed red line).

Fig. 2.
Fig. 2.

Experimental set-up based on stimulated Brillouin scattering in optical fiber. OI: optical isolator; EOM1, EOM2, EOM3: Mach-Zehnder modulators; FPC: fiber polarization controller; OC: optical circulator; PD1, PD2: photodetectors; OF: optical filter; EDFA: Erbium-doped fiber amplifier; SMF-28: 8-km-long SMF fiber.

Fig. 3.
Fig. 3.

Measurement (blue solid line) and numerical fit (red dashed line) of the Brillouin gain spectrum, related to a triple Lorentzian line configuration optimized for Δb/γ=1.33. The line separation δ3 is 27 MHz, whereas the gain coefficients retrieved from fitting procedure are gt01 =0.59 and gt02 =0.98. The green line represents the phase of the transfer function H3, calculated from Eq. (8) by using the above gain coefficients, and evaluated for z=0.

Fig. 4.
Fig. 4.

Slow light delay induced by SBS in a single-mode optical fiber. The different curves represent the temporal evolution of the Stokes pulses emitted from the fiber in absence (blue line) of the pump beam, and in presence (red line and green line) of the pump beam. The green line refers to the case of a gain doublet configuration, with δ2=2π×21.65 MHz, whereas the red line refers to the case of a gain triplet configuration, with δ3=2π×27 MHz.

Equations (10)

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

A ( ω , z ) = A ( ω , 0 ) exp ( j k ( ω ) z ) ,
k ( ω ) = k 0 + k 1 ( ω ω c ) ,
D a = H max H min H max + H min ,
D p = 1 2 π max [ H ( ω ) ( t p ω + ϕ 0 ) ] ω 0 Δ b ω 0 + Δ b .
H 1 ( ω ) = exp ( i z n 0 ω c ) × exp ( g 1 ( ω ) )
= exp ( z n 0 ω c + g 01 i γ ( ω ω 0 ) + i γ ) ,
H 2 ( ω ) = exp ( i z n 0 ω c ) × exp ( g 2 ( ω ) )
= exp ( z n 0 ω c + g 02 i γ ( ω ω 0 δ 2 ) + i γ + g 02 i γ ( ω ω 0 + δ 2 ) + i γ ) ,
H 3 ( ω ) = exp ( i z n 0 ω c ) × exp ( g 3 ( ω ) )
= exp ( z n 0 ω c + g t 01 i γ ( ω ω 0 ) + i γ + g t 02 i γ ( ω ω 0 δ 3 ) + i γ + g t 02 i γ ( ω ω 0 + δ 3 ) + i γ )

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