Abstract

A theoretical analysis of the Raman amplification in optical fibers and the pump-to-signal relative intensity noise (RIN) transfer has been performed in the spectral domain. An efficient Raman amplification of a monochromatic signal beam by a large-bandwidth pump beam has been demonstrated for a pump bandwidth much smaller than the Raman linewidth. Under the same approximation the pump-to-signal RIN transfer has been calculated in both cases of copropagating and counterpropagating beams in the two limiting cases of modulated monochromatic and smooth-profile large-bandwidth pump beams. At low frequencies the excess of noise evidenced in the case of a modulated monochromatic pump beam did not exist in the case of large-bandwidth pseudoincoherent sources. As this noise reduction can be as large as 13dB for a 40dB net gain of the amplifier, such incoherent pumping sources must be considered for the purpose of low-noise Raman amplifiers.

© 2006 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  2. G. P. Agrawal, Fiber Optics Communication Systems (Wiley-Interscience, 2002).
    [CrossRef]
  3. K. Keita, R. Frey, P. Delaye, D. Dolfi, J.-P. Huignard, and G. Roosen, 'Stimulated Raman scattering for variable gain amplification of small optically carried microwave signals,' Opt. Commun. 263, 300-303 (2006).
    [CrossRef]
  4. M. L. Dakss and P. Melman, 'Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,' J. Lightwave Technol. 3, 806-813 (1985).
    [CrossRef]
  5. C. R. S. Fludger, V. Handerek, and R. J. Mears, 'Pump to signal RIN transfer in Raman fiber amplifiers,' J. Lightwave Technol. 19, 1140-1148 (2001).
    [CrossRef]
  6. M. D. Mermelstein, C. Headley, and J.-C. Bouteiller, 'RIN transfer analysis in pump depletion regime for Raman fibre amplifiers,' Electron. Lett. 38, 403-405 (2002).
    [CrossRef]
  7. C. Martinelli, L. Lorcy, A. Durecu-Legrand, D. Mongardien, S. Borne, and D. Bayart, 'RIN transfer in copumped Raman amplifiers using polarization-combined diodes,' IEEE Photon. Technol. Lett. 17, 1836-1838 (2005).
    [CrossRef]
  8. D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.
  9. K. Petermann, Laser Diode Modulation and Noise, Advances in Optoelectronics (Kluwer Academic, KTP Scientific, 1988).
    [CrossRef]
  10. R. Frey and F. Pradere, 'High-efficiency narrow-linewidth Raman amplification and spectral compression,' Opt. Lett. 5, 374-376 (1980).
    [CrossRef] [PubMed]
  11. A. A. Fotiadi, S. A. Babin, D. V. Churkin, S. I. Kablukov, and E. V. Podivilov, 'Origin of uniform pump-to-Stokes relative intensity noise (RIN) transfer in Raman fiber lasers,' in Proceedings Symposium/LEOS Benelux Chapter (IEEE, 2005).
  12. Y. R. Shen, Nonlinear Optics (Wiley, 1984).

2006 (1)

K. Keita, R. Frey, P. Delaye, D. Dolfi, J.-P. Huignard, and G. Roosen, 'Stimulated Raman scattering for variable gain amplification of small optically carried microwave signals,' Opt. Commun. 263, 300-303 (2006).
[CrossRef]

2005 (1)

C. Martinelli, L. Lorcy, A. Durecu-Legrand, D. Mongardien, S. Borne, and D. Bayart, 'RIN transfer in copumped Raman amplifiers using polarization-combined diodes,' IEEE Photon. Technol. Lett. 17, 1836-1838 (2005).
[CrossRef]

2002 (1)

M. D. Mermelstein, C. Headley, and J.-C. Bouteiller, 'RIN transfer analysis in pump depletion regime for Raman fibre amplifiers,' Electron. Lett. 38, 403-405 (2002).
[CrossRef]

2001 (1)

1985 (1)

M. L. Dakss and P. Melman, 'Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,' J. Lightwave Technol. 3, 806-813 (1985).
[CrossRef]

1980 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

G. P. Agrawal, Fiber Optics Communication Systems (Wiley-Interscience, 2002).
[CrossRef]

Azimi, M.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Babin, S. A.

A. A. Fotiadi, S. A. Babin, D. V. Churkin, S. I. Kablukov, and E. V. Podivilov, 'Origin of uniform pump-to-Stokes relative intensity noise (RIN) transfer in Raman fiber lasers,' in Proceedings Symposium/LEOS Benelux Chapter (IEEE, 2005).

Bayart, D.

C. Martinelli, L. Lorcy, A. Durecu-Legrand, D. Mongardien, S. Borne, and D. Bayart, 'RIN transfer in copumped Raman amplifiers using polarization-combined diodes,' IEEE Photon. Technol. Lett. 17, 1836-1838 (2005).
[CrossRef]

Borne, S.

C. Martinelli, L. Lorcy, A. Durecu-Legrand, D. Mongardien, S. Borne, and D. Bayart, 'RIN transfer in copumped Raman amplifiers using polarization-combined diodes,' IEEE Photon. Technol. Lett. 17, 1836-1838 (2005).
[CrossRef]

Bouteiller, J.-C.

M. D. Mermelstein, C. Headley, and J.-C. Bouteiller, 'RIN transfer analysis in pump depletion regime for Raman fibre amplifiers,' Electron. Lett. 38, 403-405 (2002).
[CrossRef]

Chen, P.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Churkin, D. V.

A. A. Fotiadi, S. A. Babin, D. V. Churkin, S. I. Kablukov, and E. V. Podivilov, 'Origin of uniform pump-to-Stokes relative intensity noise (RIN) transfer in Raman fiber lasers,' in Proceedings Symposium/LEOS Benelux Chapter (IEEE, 2005).

Dakss, M. L.

M. L. Dakss and P. Melman, 'Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,' J. Lightwave Technol. 3, 806-813 (1985).
[CrossRef]

Delaye, P.

K. Keita, R. Frey, P. Delaye, D. Dolfi, J.-P. Huignard, and G. Roosen, 'Stimulated Raman scattering for variable gain amplification of small optically carried microwave signals,' Opt. Commun. 263, 300-303 (2006).
[CrossRef]

Dolfi, D.

K. Keita, R. Frey, P. Delaye, D. Dolfi, J.-P. Huignard, and G. Roosen, 'Stimulated Raman scattering for variable gain amplification of small optically carried microwave signals,' Opt. Commun. 263, 300-303 (2006).
[CrossRef]

Durecu-Legrand, A.

C. Martinelli, L. Lorcy, A. Durecu-Legrand, D. Mongardien, S. Borne, and D. Bayart, 'RIN transfer in copumped Raman amplifiers using polarization-combined diodes,' IEEE Photon. Technol. Lett. 17, 1836-1838 (2005).
[CrossRef]

Fludger, C. R. S.

Fotiadi, A. A.

A. A. Fotiadi, S. A. Babin, D. V. Churkin, S. I. Kablukov, and E. V. Podivilov, 'Origin of uniform pump-to-Stokes relative intensity noise (RIN) transfer in Raman fiber lasers,' in Proceedings Symposium/LEOS Benelux Chapter (IEEE, 2005).

Frey, R.

K. Keita, R. Frey, P. Delaye, D. Dolfi, J.-P. Huignard, and G. Roosen, 'Stimulated Raman scattering for variable gain amplification of small optically carried microwave signals,' Opt. Commun. 263, 300-303 (2006).
[CrossRef]

R. Frey and F. Pradere, 'High-efficiency narrow-linewidth Raman amplification and spectral compression,' Opt. Lett. 5, 374-376 (1980).
[CrossRef] [PubMed]

Han, B.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Handerek, V.

Headley, C.

M. D. Mermelstein, C. Headley, and J.-C. Bouteiller, 'RIN transfer analysis in pump depletion regime for Raman fibre amplifiers,' Electron. Lett. 38, 403-405 (2002).
[CrossRef]

Huignard, J.-P.

K. Keita, R. Frey, P. Delaye, D. Dolfi, J.-P. Huignard, and G. Roosen, 'Stimulated Raman scattering for variable gain amplification of small optically carried microwave signals,' Opt. Commun. 263, 300-303 (2006).
[CrossRef]

Jiang, M.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Kablukov, S. I.

A. A. Fotiadi, S. A. Babin, D. V. Churkin, S. I. Kablukov, and E. V. Podivilov, 'Origin of uniform pump-to-Stokes relative intensity noise (RIN) transfer in Raman fiber lasers,' in Proceedings Symposium/LEOS Benelux Chapter (IEEE, 2005).

Keita, K.

K. Keita, R. Frey, P. Delaye, D. Dolfi, J.-P. Huignard, and G. Roosen, 'Stimulated Raman scattering for variable gain amplification of small optically carried microwave signals,' Opt. Commun. 263, 300-303 (2006).
[CrossRef]

Knopp, K. J.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Lorcy, L.

C. Martinelli, L. Lorcy, A. Durecu-Legrand, D. Mongardien, S. Borne, and D. Bayart, 'RIN transfer in copumped Raman amplifiers using polarization-combined diodes,' IEEE Photon. Technol. Lett. 17, 1836-1838 (2005).
[CrossRef]

Lu, C. C.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Martinelli, C.

C. Martinelli, L. Lorcy, A. Durecu-Legrand, D. Mongardien, S. Borne, and D. Bayart, 'RIN transfer in copumped Raman amplifiers using polarization-combined diodes,' IEEE Photon. Technol. Lett. 17, 1836-1838 (2005).
[CrossRef]

Mears, R. J.

Melman, P.

M. L. Dakss and P. Melman, 'Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,' J. Lightwave Technol. 3, 806-813 (1985).
[CrossRef]

Mermelstein, M. D.

M. D. Mermelstein, C. Headley, and J.-C. Bouteiller, 'RIN transfer analysis in pump depletion regime for Raman fibre amplifiers,' Electron. Lett. 38, 403-405 (2002).
[CrossRef]

Mongardien, D.

C. Martinelli, L. Lorcy, A. Durecu-Legrand, D. Mongardien, S. Borne, and D. Bayart, 'RIN transfer in copumped Raman amplifiers using polarization-combined diodes,' IEEE Photon. Technol. Lett. 17, 1836-1838 (2005).
[CrossRef]

Petermann, K.

K. Petermann, Laser Diode Modulation and Noise, Advances in Optoelectronics (Kluwer Academic, KTP Scientific, 1988).
[CrossRef]

Pinzone, C. J.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Podivilov, E. V.

A. A. Fotiadi, S. A. Babin, D. V. Churkin, S. I. Kablukov, and E. V. Podivilov, 'Origin of uniform pump-to-Stokes relative intensity noise (RIN) transfer in Raman fiber lasers,' in Proceedings Symposium/LEOS Benelux Chapter (IEEE, 2005).

Pradere, F.

Roosen, G.

K. Keita, R. Frey, P. Delaye, D. Dolfi, J.-P. Huignard, and G. Roosen, 'Stimulated Raman scattering for variable gain amplification of small optically carried microwave signals,' Opt. Commun. 263, 300-303 (2006).
[CrossRef]

Shen, Y.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Shen, Y. R.

Y. R. Shen, Nonlinear Optics (Wiley, 1984).

Vakhshoori, D.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Vander Rhodes, G.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Vote, S.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Wang, P. D.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Zhu, X.

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

Electron. Lett. (1)

M. D. Mermelstein, C. Headley, and J.-C. Bouteiller, 'RIN transfer analysis in pump depletion regime for Raman fibre amplifiers,' Electron. Lett. 38, 403-405 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

C. Martinelli, L. Lorcy, A. Durecu-Legrand, D. Mongardien, S. Borne, and D. Bayart, 'RIN transfer in copumped Raman amplifiers using polarization-combined diodes,' IEEE Photon. Technol. Lett. 17, 1836-1838 (2005).
[CrossRef]

J. Lightwave Technol. (2)

M. L. Dakss and P. Melman, 'Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers,' J. Lightwave Technol. 3, 806-813 (1985).
[CrossRef]

C. R. S. Fludger, V. Handerek, and R. J. Mears, 'Pump to signal RIN transfer in Raman fiber amplifiers,' J. Lightwave Technol. 19, 1140-1148 (2001).
[CrossRef]

Opt. Commun. (1)

K. Keita, R. Frey, P. Delaye, D. Dolfi, J.-P. Huignard, and G. Roosen, 'Stimulated Raman scattering for variable gain amplification of small optically carried microwave signals,' Opt. Commun. 263, 300-303 (2006).
[CrossRef]

Opt. Lett. (1)

Other (6)

A. A. Fotiadi, S. A. Babin, D. V. Churkin, S. I. Kablukov, and E. V. Podivilov, 'Origin of uniform pump-to-Stokes relative intensity noise (RIN) transfer in Raman fiber lasers,' in Proceedings Symposium/LEOS Benelux Chapter (IEEE, 2005).

Y. R. Shen, Nonlinear Optics (Wiley, 1984).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

G. P. Agrawal, Fiber Optics Communication Systems (Wiley-Interscience, 2002).
[CrossRef]

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, K. J. Knopp, C. C. Lu, C. J. Pinzone, Y. Shen, G. Vander Rhodes, S. Vote, P. D. Wang, and X. Zhu, 'Raman amplification using high-power incoherent semiconductor pump sources,' in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2003), pp. PD47-P1-3.

K. Petermann, Laser Diode Modulation and Noise, Advances in Optoelectronics (Kluwer Academic, KTP Scientific, 1988).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

RIN transfer versus the frequency detuning for a modulated monochromatic pump beam.

Fig. 2
Fig. 2

Low-frequency RIN transfer versus the net gain for a modulated monochromatic pump beam.

Fig. 3
Fig. 3

Comparison of the RIN transfer for modulated monochromatic (thin curves) and smooth large-bandwidth (bold curves) pump beams. Solid and dotted curves correspond to copropagating and counterpropagating pumping schemes, respectively.

Fig. 4
Fig. 4

Low-frequency RIN transfer versus the net gain for a smooth large-bandwidth pump beam.

Equations (41)

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E ̱ p ̱ ( z , t ) = e ̂ p E p ( z , t ) exp ( i ω p t ) + C.C. ,
E p ( z , t ) = 1 2 π + a p ( z , ν p ) exp [ i k ( ω p + ν p ) z ] exp ( i ν p t ) d ν p ,
j p ( z , ν ) = 1 2 π + E p ( z , t ) 2 exp ( i ν t ) d t ,
j p ( z , ν ) = 1 2 π exp [ i ( δ k δ ω ) ω p ν z ] + a p ( z , ν p ) a p * ( z , ν p ν ) d ν p
j p ( z , ν ) = exp [ i ( k ω ) ω p ν z ] j p ( z , ν s ) = 1 2 π + a p ( z , ν p ) a p * ( z , ν p ν ) d ν p .
I p ( z ) = c n 2 π T j p ( z , 0 ) .
E ̱ x ̱ ( z , t ) = e ̂ x A x ( z , t ) exp ( i ω s t ) + C.C. , x = s , n ,
A x ( z , t ) = 1 2 π + a x ( z , ν s ) exp [ i k ( ω s + ν s ) z ] exp ( i ν s t ) d ν s ,
a s ( z , ν s ) = 2 π A s ( z ) sin ( ν s T 2 ) ν s .
RIN ( z ) = Δ ν W p ( z , ν ) δ P ( z , t ) 2 ,
W p ( z , ν ) = 1 2 π T + T 2 + T 2 P ( z , t ) P ( z , t τ ) exp ( i ν τ ) d t d τ .
P ( z , t ) = c n S 2 π E ( z , t ) 2 = c n S ( 2 π ) 3 2 + j ( z , ν ) exp ( i ν t ) d ν ,
RIN p ( z , ν p ) = 2 Δ ν p T ( 2 π ) 3 2 j p ( z , ν p ) 2 j p ( z , 0 ) 2 .
j s ( z , ν s ) = 1 2 π + a s ( z , ν s ) a n * ( z , ν s ν s ) exp { ± i [ k ( ω s + ν s ) k ( ω s + ν s ν s ) ] z } d ν s ,
RIN s ( z , ν s ) = 2 π ω Δ s T a n ( z , ν s ) 2 A s ( z ) 2 .
ρ = RIN s ( z , ν ) RIN p ( 0 , ν ) = 1 2 π T 2 a n ( z , ν ) 2 A s ( z ) 2 j p ( 0 , 0 ) 2 j p ( 0 , ν ) 2 .
a p ( z , ν p ) = a p ( 0 , ν p ) exp ( α p z 2 ) ,
j p ( z , ν s ) = j p ( 0 , ν s ) exp ( α p z ) ,
a s ( z , ν s ) z = ± i 2 π n s λ s + d ν p + d ν s χ eff ( 3 ) ( ω p + ν p , ω p + ν s ν p ν s , ω s + ν s ) × a p ( z , ν p ) a p * ( z , ν p + ν s ν s ) a s ( z , ν s ) exp [ i Δ k ± ( ν p , ν s , ν s ) z ] α s 2 a s ( z , ν s )
d A s ( z ) d z = ± ( G α s 2 ) A s ( z ) ,
A s ( z ) = A s ( 0 ) exp { G α p [ 1 exp ( α p z ) ] α s 2 z } ,
A s ( z ) = A s ( L ) exp { G α p [ exp ( α p z ) exp ( α p L ) ] α s 2 ( L z ) }
g net ( dB ) = 20 Ln ( 10 ) { G α p [ 1 exp ( α p L ) ] α s L 2 } .
a n ( z , ν s ) z = ± i 2 π n s λ s + d ν p + d ν s χ eff ( 3 ) ( ω p + ν p , ω p + ν s ν p ν s , ω s + ν s ) a p ( z , ν p ) a p * ( z , ν p + ν s ν s ) [ a s ( z , ν s ) + a n ( z , ν s ) ] exp [ i Δ k ± ( ν p , ν s , ν s ) z ] α s 2 a n ( z , ν s ) .
a n ( z , ν s ) z = ± i 4 π 2 n s λ s χ eff ( 3 ) ( ω p , ω p , ω s ) A s ( z ) j p ( z , ν s ) exp [ i Δ k ± ( 1 ) z ] ± i ( 2 π ) 3 2 n s λ s χ eff ( 3 ) ( ω p , ω p , ω s ) { + d ν s j p ( z , ν s ν s ) a n ( z , ν s ) exp [ i Δ k ± ( 2 ) z ] } α s 2 a n ( z , ν s ) ,
E p ( 0 , t ) = A p ( 0 ) [ 1 + m 2 sin ( Ω t + Φ ) ] ,
a p ( 0 , ν p ) = 2 π A p ( 0 ) { δ ( ν p ) i m 4 [ exp ( i Φ ) δ ( ν p + Ω ) + exp ( i Φ ) δ ( ν p Ω ) ] } .
d a n ( z , ν s ) d z = ± [ G exp ( α p z ) α s 2 ] a n ( z , ν s ) ± G j p ( 0 , ν s ) A p ( 0 ) 2 exp ( α p z ) A s ( z ) exp [ i Δ k ± ( 1 ) z ] α s 2 a n ( z , ν s ) .
a n ( z , ν s ) = ± G j p ( 0 , ν s ) A p ( 0 ) 2 A s ( z ) exp { [ i Δ k ± ( 1 ) α p ] z } i Δ k ± ( 1 ) α p .
ρ = G 2 Δ k ± ( 1 ) 2 + α p 2 { 1 2 cos [ Δ k ± ( 1 ) L ] exp ( α p L ) + exp ( 2 α p L ) } .
ρ = ( G α p ) 2 [ 1 exp ( α p L ) ] 2 ,
ρ dB ( dB ) = 20 Ln ( 10 ) Ln [ Ln ( 10 ) 20 g net ( dB ) + α s L 2 ] ,
z 2 π ( 1 v p 1 v s ) Δ ω p
ρ co = G 2 ( G α p ) 2 + Δ k + ( 1 ) 2 { exp ( 2 α p L ) 2 cos [ Δ k + ( 1 ) L ] exp [ ( G + α p ) L ] + exp ( 2 G L ) } ,
ρ count = G 2 ( G + α p ) 2 + Δ k ( 1 ) 2 { exp [ 2 ( G + α p ) L ] 2 cos [ Δ k ( 1 ) L ] exp [ ( G + α p ) L ] + 1 }
E ̱ x ̱ ( z , t ) = e ̂ A x ( z , t ) exp ( i ω x t ) + C.C. , x = 1 , 4 ,
A x ( z , t ) = 1 2 π + a x ( z , ν x ) exp [ i k ( ω x + ν x ) z ] exp ( i ν x t ) d ν x ,
P ̱ 4 ̱ ( z , t ) = 1 ( 2 π ) 3 2 + + + χ eff ( 3 ) ( ω 1 + ν 1 , ω 2 + ν 2 , ω 3 + ν 3 ) a 1 ( z , ν 1 ) a 2 ( z , ν 2 ) a 3 ( z , ν 3 ) exp { i [ k ( ω 1 + ν 1 ) + k ( ω 2 + ν 2 ) + k ( ω 3 + ν 3 ) ] } d ν 1 d ν 2 d ν 3 ,
Δ E 4 ̱ ( z , t ) 1 c 2 2 D 4 ̱ ( z , t ) t 2 = 4 π c 2 2 P 4 ̱ ( z , t ) t 2 ,
[ 2 a 4 ( z , ν 4 ) z 2 2 k ( ω 4 + ν 4 ) a 4 z ( z , ν 4 ) ] ,
d a 4 ( z , ν 4 ) d z = i 2 π n 4 λ 4 + + χ eff ( 3 ) ( ω 1 + ν 1 , ω 4 ω 1 ω 3 + ν 4 ν 1 ν 3 , ω 3 + ν 3 ) a 1 ( z , ν 1 ) a 2 ( z , ν 4 ν 1 ν 3 ) a 3 ( z , ν 3 ) exp ( i Δ k 4 z ) d ν 1 d ν 3 a 4 2 a 4 ( z , ν 4 ) ,

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