Abstract

The method for designing the incoherent pump spectrum for the distributed fiber Raman amplifiers (DFRAs) of low gain ripple is studied, in which the wavelength of the maximum power spectral density can be assigned. The assigned wavelength is called the main pump wavelength. Incoherent pump spectrum is described with the power spectral density function (PSDF) that comprises a set of piece-wise continuous functions. PSDF is optimized for the minimum gain ripple with the least-square minimization method. An extremum pump wavelength condition is applied to PSDF. A proper initial trial PSDF is given so that the optimized PSDF converges to the desired result and the power spectral density at the extremum pump wavelength is the maximum. With this design method, we show the optimized PSDFs for the DFRAs using backward pumping and bidirectional pumping. The gain ripples of considered DFRAs are less than 0.1 dB for 20-dB ON-OFF Raman gain over 70-nm bandwidth. The reduction of average effective noise figure with shorter main pump wavelength is shown and investigated.

© 2007 Optical Society of America

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  1. D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, L. Knopp, C. Lu, Y. Shen, G. Rodes, S. Vote, P. Wang, and X. Zhu, "Raman amplification using high-power incoherent semiconductor pump sources," OFC 2003, Paper PD47.
  2. T. Zhang, X. Zhang, and G. Zhang, "Distributed fiber Raman amplifiers with incoherent pumping," IEEE Photon. Technol. Lett. 17, 1175-1177 (2005).
    [CrossRef]
  3. B. Han, X. Zhang, G. Zhang, Z. Lu, and G. Yang, "Composite broad-band fiber Raman amplifiers using incoherent pumping," Opt. Express 14, 3752-2762 (2006).
  4. S. Wen, "Design of the pump power spectrum for the distributed fiber Raman amplifiers using incoherent pumping, " Opt. Express 13, 6023-6032 (2005).
  5. E. Lichtman, R. Waarts, and A. Friesem, "Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fiber," J. Lightwave. Technol. 7, 171-1174 (1989).
    [CrossRef]
  6. X. Zhou, M. Birk, and S. Woodward, "Pump-noise induced FWM effect and its reduction in a distributed Raman fiber amplifiers," IEEE Photon. Technol. Lett. 14, 1686-1688 (2002).
    [CrossRef]
  7. T. Kung, C. Chang, J. Dung, and S. Chi, "Four-wave mixing between pump and signal in a distributed Raman amplifier," J. Lightwave. Technol. 21, 1164-1170 (2003).
    [CrossRef]
  8. J. Bouteiller, L. Leng, and C. Headley, "Pump-pump four-wave mixing in distributed Raman amplified systems," J. Lightwave. Technol. 22, 723-732 (2004).
    [CrossRef]
  9. S. Sugliani, G. Sacchi, G. Bolognini, S. Faralli, and F. Pasquale, "Effective suppression of penalties induced by parametric nonlinear interaction in distributed Raman amplifiers based on NZ-DS fibers," IEEE Photon. Technol. Lett. 16, 81-83 (2004).
    [CrossRef]
  10. H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, "1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology," IEEE Photon. Technol. Lett. 12, 903-905 (2000).
    [CrossRef]
  11. M. Islam, "Raman amplifiers for telecommunications," IEEE J. Sel. Tops. Quantum Electron. 8, 548-559 (2002).
    [CrossRef]
  12. V. Perlin and G. Winful, "On distributed Raman amplification for ultrabroad-band long-haul WDM systems," J. Lightwave. Technol. 20, 409-416 (2002).
    [CrossRef]
  13. J. Bromage, "Raman amplification for fiber communication systems," J. Lightwave. Technol. 22, 79-93 (2004).
    [CrossRef]
  14. I. Mandelbaum and M. Bolshtyansky, "Raman amplifier model in singlemode optical fiber," IEEE Photon. Technol. Lett. 15, 1704-1706 (2003).
    [CrossRef]
  15. S. Wen, T.-Y. Wang, and S. Chi, "Self-consistent pump depletion method to design optical transmission systems amplified by bidirectional Raman pumps," Int. J. Nonlinear Opt. Phys. 1, 595-608 (1992).
    [CrossRef]
  16. J. Moré, B. Garbow, and K. Hillstrom, User Guide for MINPACK-1, Argonne National Laboratory Report ANL-80-74, (Argonne National Laboratory, Argonne, Illinois, 1980).

2006 (1)

2005 (2)

T. Zhang, X. Zhang, and G. Zhang, "Distributed fiber Raman amplifiers with incoherent pumping," IEEE Photon. Technol. Lett. 17, 1175-1177 (2005).
[CrossRef]

S. Wen, "Design of the pump power spectrum for the distributed fiber Raman amplifiers using incoherent pumping, " Opt. Express 13, 6023-6032 (2005).

2004 (3)

J. Bouteiller, L. Leng, and C. Headley, "Pump-pump four-wave mixing in distributed Raman amplified systems," J. Lightwave. Technol. 22, 723-732 (2004).
[CrossRef]

S. Sugliani, G. Sacchi, G. Bolognini, S. Faralli, and F. Pasquale, "Effective suppression of penalties induced by parametric nonlinear interaction in distributed Raman amplifiers based on NZ-DS fibers," IEEE Photon. Technol. Lett. 16, 81-83 (2004).
[CrossRef]

J. Bromage, "Raman amplification for fiber communication systems," J. Lightwave. Technol. 22, 79-93 (2004).
[CrossRef]

2003 (2)

I. Mandelbaum and M. Bolshtyansky, "Raman amplifier model in singlemode optical fiber," IEEE Photon. Technol. Lett. 15, 1704-1706 (2003).
[CrossRef]

T. Kung, C. Chang, J. Dung, and S. Chi, "Four-wave mixing between pump and signal in a distributed Raman amplifier," J. Lightwave. Technol. 21, 1164-1170 (2003).
[CrossRef]

2002 (3)

M. Islam, "Raman amplifiers for telecommunications," IEEE J. Sel. Tops. Quantum Electron. 8, 548-559 (2002).
[CrossRef]

V. Perlin and G. Winful, "On distributed Raman amplification for ultrabroad-band long-haul WDM systems," J. Lightwave. Technol. 20, 409-416 (2002).
[CrossRef]

X. Zhou, M. Birk, and S. Woodward, "Pump-noise induced FWM effect and its reduction in a distributed Raman fiber amplifiers," IEEE Photon. Technol. Lett. 14, 1686-1688 (2002).
[CrossRef]

2000 (1)

H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, "1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology," IEEE Photon. Technol. Lett. 12, 903-905 (2000).
[CrossRef]

1992 (1)

S. Wen, T.-Y. Wang, and S. Chi, "Self-consistent pump depletion method to design optical transmission systems amplified by bidirectional Raman pumps," Int. J. Nonlinear Opt. Phys. 1, 595-608 (1992).
[CrossRef]

1989 (1)

E. Lichtman, R. Waarts, and A. Friesem, "Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fiber," J. Lightwave. Technol. 7, 171-1174 (1989).
[CrossRef]

Birk, M.

X. Zhou, M. Birk, and S. Woodward, "Pump-noise induced FWM effect and its reduction in a distributed Raman fiber amplifiers," IEEE Photon. Technol. Lett. 14, 1686-1688 (2002).
[CrossRef]

Bolognini, G.

S. Sugliani, G. Sacchi, G. Bolognini, S. Faralli, and F. Pasquale, "Effective suppression of penalties induced by parametric nonlinear interaction in distributed Raman amplifiers based on NZ-DS fibers," IEEE Photon. Technol. Lett. 16, 81-83 (2004).
[CrossRef]

Bolshtyansky, M.

I. Mandelbaum and M. Bolshtyansky, "Raman amplifier model in singlemode optical fiber," IEEE Photon. Technol. Lett. 15, 1704-1706 (2003).
[CrossRef]

Bouteiller, J.

J. Bouteiller, L. Leng, and C. Headley, "Pump-pump four-wave mixing in distributed Raman amplified systems," J. Lightwave. Technol. 22, 723-732 (2004).
[CrossRef]

Bromage, J.

J. Bromage, "Raman amplification for fiber communication systems," J. Lightwave. Technol. 22, 79-93 (2004).
[CrossRef]

Chang, C.

T. Kung, C. Chang, J. Dung, and S. Chi, "Four-wave mixing between pump and signal in a distributed Raman amplifier," J. Lightwave. Technol. 21, 1164-1170 (2003).
[CrossRef]

Chi, S.

T. Kung, C. Chang, J. Dung, and S. Chi, "Four-wave mixing between pump and signal in a distributed Raman amplifier," J. Lightwave. Technol. 21, 1164-1170 (2003).
[CrossRef]

S. Wen, T.-Y. Wang, and S. Chi, "Self-consistent pump depletion method to design optical transmission systems amplified by bidirectional Raman pumps," Int. J. Nonlinear Opt. Phys. 1, 595-608 (1992).
[CrossRef]

Dung, J.

T. Kung, C. Chang, J. Dung, and S. Chi, "Four-wave mixing between pump and signal in a distributed Raman amplifier," J. Lightwave. Technol. 21, 1164-1170 (2003).
[CrossRef]

Faralli, S.

S. Sugliani, G. Sacchi, G. Bolognini, S. Faralli, and F. Pasquale, "Effective suppression of penalties induced by parametric nonlinear interaction in distributed Raman amplifiers based on NZ-DS fibers," IEEE Photon. Technol. Lett. 16, 81-83 (2004).
[CrossRef]

Friesem, A.

E. Lichtman, R. Waarts, and A. Friesem, "Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fiber," J. Lightwave. Technol. 7, 171-1174 (1989).
[CrossRef]

Han, B.

Headley, C.

J. Bouteiller, L. Leng, and C. Headley, "Pump-pump four-wave mixing in distributed Raman amplified systems," J. Lightwave. Technol. 22, 723-732 (2004).
[CrossRef]

Islam, M.

M. Islam, "Raman amplifiers for telecommunications," IEEE J. Sel. Tops. Quantum Electron. 8, 548-559 (2002).
[CrossRef]

Iwatsuki, K.

H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, "1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology," IEEE Photon. Technol. Lett. 12, 903-905 (2000).
[CrossRef]

Kani, J.

H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, "1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology," IEEE Photon. Technol. Lett. 12, 903-905 (2000).
[CrossRef]

Kung, T.

T. Kung, C. Chang, J. Dung, and S. Chi, "Four-wave mixing between pump and signal in a distributed Raman amplifier," J. Lightwave. Technol. 21, 1164-1170 (2003).
[CrossRef]

Leng, L.

J. Bouteiller, L. Leng, and C. Headley, "Pump-pump four-wave mixing in distributed Raman amplified systems," J. Lightwave. Technol. 22, 723-732 (2004).
[CrossRef]

Lichtman, E.

E. Lichtman, R. Waarts, and A. Friesem, "Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fiber," J. Lightwave. Technol. 7, 171-1174 (1989).
[CrossRef]

Lu, Z.

Mandelbaum, I.

I. Mandelbaum and M. Bolshtyansky, "Raman amplifier model in singlemode optical fiber," IEEE Photon. Technol. Lett. 15, 1704-1706 (2003).
[CrossRef]

Masuda, H.

H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, "1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology," IEEE Photon. Technol. Lett. 12, 903-905 (2000).
[CrossRef]

Pasquale, F.

S. Sugliani, G. Sacchi, G. Bolognini, S. Faralli, and F. Pasquale, "Effective suppression of penalties induced by parametric nonlinear interaction in distributed Raman amplifiers based on NZ-DS fibers," IEEE Photon. Technol. Lett. 16, 81-83 (2004).
[CrossRef]

Perlin, V.

V. Perlin and G. Winful, "On distributed Raman amplification for ultrabroad-band long-haul WDM systems," J. Lightwave. Technol. 20, 409-416 (2002).
[CrossRef]

Sacchi, G.

S. Sugliani, G. Sacchi, G. Bolognini, S. Faralli, and F. Pasquale, "Effective suppression of penalties induced by parametric nonlinear interaction in distributed Raman amplifiers based on NZ-DS fibers," IEEE Photon. Technol. Lett. 16, 81-83 (2004).
[CrossRef]

Sugliani, S.

S. Sugliani, G. Sacchi, G. Bolognini, S. Faralli, and F. Pasquale, "Effective suppression of penalties induced by parametric nonlinear interaction in distributed Raman amplifiers based on NZ-DS fibers," IEEE Photon. Technol. Lett. 16, 81-83 (2004).
[CrossRef]

Sumida, M.

H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, "1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology," IEEE Photon. Technol. Lett. 12, 903-905 (2000).
[CrossRef]

Suzuki, H.

H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, "1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology," IEEE Photon. Technol. Lett. 12, 903-905 (2000).
[CrossRef]

Tada, Y.

H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, "1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology," IEEE Photon. Technol. Lett. 12, 903-905 (2000).
[CrossRef]

Takachio, N.

H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, "1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology," IEEE Photon. Technol. Lett. 12, 903-905 (2000).
[CrossRef]

Waarts, R.

E. Lichtman, R. Waarts, and A. Friesem, "Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fiber," J. Lightwave. Technol. 7, 171-1174 (1989).
[CrossRef]

Wang, T.-Y.

S. Wen, T.-Y. Wang, and S. Chi, "Self-consistent pump depletion method to design optical transmission systems amplified by bidirectional Raman pumps," Int. J. Nonlinear Opt. Phys. 1, 595-608 (1992).
[CrossRef]

Wen, S.

S. Wen, "Design of the pump power spectrum for the distributed fiber Raman amplifiers using incoherent pumping, " Opt. Express 13, 6023-6032 (2005).

S. Wen, T.-Y. Wang, and S. Chi, "Self-consistent pump depletion method to design optical transmission systems amplified by bidirectional Raman pumps," Int. J. Nonlinear Opt. Phys. 1, 595-608 (1992).
[CrossRef]

Winful, G.

V. Perlin and G. Winful, "On distributed Raman amplification for ultrabroad-band long-haul WDM systems," J. Lightwave. Technol. 20, 409-416 (2002).
[CrossRef]

Woodward, S.

X. Zhou, M. Birk, and S. Woodward, "Pump-noise induced FWM effect and its reduction in a distributed Raman fiber amplifiers," IEEE Photon. Technol. Lett. 14, 1686-1688 (2002).
[CrossRef]

Yang, G.

Zhang, G.

B. Han, X. Zhang, G. Zhang, Z. Lu, and G. Yang, "Composite broad-band fiber Raman amplifiers using incoherent pumping," Opt. Express 14, 3752-2762 (2006).

T. Zhang, X. Zhang, and G. Zhang, "Distributed fiber Raman amplifiers with incoherent pumping," IEEE Photon. Technol. Lett. 17, 1175-1177 (2005).
[CrossRef]

Zhang, T.

T. Zhang, X. Zhang, and G. Zhang, "Distributed fiber Raman amplifiers with incoherent pumping," IEEE Photon. Technol. Lett. 17, 1175-1177 (2005).
[CrossRef]

Zhang, X.

B. Han, X. Zhang, G. Zhang, Z. Lu, and G. Yang, "Composite broad-band fiber Raman amplifiers using incoherent pumping," Opt. Express 14, 3752-2762 (2006).

T. Zhang, X. Zhang, and G. Zhang, "Distributed fiber Raman amplifiers with incoherent pumping," IEEE Photon. Technol. Lett. 17, 1175-1177 (2005).
[CrossRef]

Zhou, X.

X. Zhou, M. Birk, and S. Woodward, "Pump-noise induced FWM effect and its reduction in a distributed Raman fiber amplifiers," IEEE Photon. Technol. Lett. 14, 1686-1688 (2002).
[CrossRef]

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

M. Islam, "Raman amplifiers for telecommunications," IEEE J. Sel. Tops. Quantum Electron. 8, 548-559 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (5)

T. Zhang, X. Zhang, and G. Zhang, "Distributed fiber Raman amplifiers with incoherent pumping," IEEE Photon. Technol. Lett. 17, 1175-1177 (2005).
[CrossRef]

X. Zhou, M. Birk, and S. Woodward, "Pump-noise induced FWM effect and its reduction in a distributed Raman fiber amplifiers," IEEE Photon. Technol. Lett. 14, 1686-1688 (2002).
[CrossRef]

S. Sugliani, G. Sacchi, G. Bolognini, S. Faralli, and F. Pasquale, "Effective suppression of penalties induced by parametric nonlinear interaction in distributed Raman amplifiers based on NZ-DS fibers," IEEE Photon. Technol. Lett. 16, 81-83 (2004).
[CrossRef]

H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, and M. Sumida, "1-Tb/s (100 × 10 Gb/s) super-dense WDM Transmission with 25-GHz channel spacing in the zero-dispersion region employing distributed Raman amplification technology," IEEE Photon. Technol. Lett. 12, 903-905 (2000).
[CrossRef]

I. Mandelbaum and M. Bolshtyansky, "Raman amplifier model in singlemode optical fiber," IEEE Photon. Technol. Lett. 15, 1704-1706 (2003).
[CrossRef]

Int. J. Nonlinear Opt. Phys. (1)

S. Wen, T.-Y. Wang, and S. Chi, "Self-consistent pump depletion method to design optical transmission systems amplified by bidirectional Raman pumps," Int. J. Nonlinear Opt. Phys. 1, 595-608 (1992).
[CrossRef]

J. Lightwave. Technol. (5)

T. Kung, C. Chang, J. Dung, and S. Chi, "Four-wave mixing between pump and signal in a distributed Raman amplifier," J. Lightwave. Technol. 21, 1164-1170 (2003).
[CrossRef]

J. Bouteiller, L. Leng, and C. Headley, "Pump-pump four-wave mixing in distributed Raman amplified systems," J. Lightwave. Technol. 22, 723-732 (2004).
[CrossRef]

V. Perlin and G. Winful, "On distributed Raman amplification for ultrabroad-band long-haul WDM systems," J. Lightwave. Technol. 20, 409-416 (2002).
[CrossRef]

J. Bromage, "Raman amplification for fiber communication systems," J. Lightwave. Technol. 22, 79-93 (2004).
[CrossRef]

E. Lichtman, R. Waarts, and A. Friesem, "Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fiber," J. Lightwave. Technol. 7, 171-1174 (1989).
[CrossRef]

Opt. Express (2)

Other (2)

J. Moré, B. Garbow, and K. Hillstrom, User Guide for MINPACK-1, Argonne National Laboratory Report ANL-80-74, (Argonne National Laboratory, Argonne, Illinois, 1980).

D. Vakhshoori, M. Azimi, P. Chen, B. Han, M. Jiang, L. Knopp, C. Lu, Y. Shen, G. Rodes, S. Vote, P. Wang, and X. Zhu, "Raman amplification using high-power incoherent semiconductor pump sources," OFC 2003, Paper PD47.

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Figures (10)

Fig. 1.
Fig. 1.

(a). Pump power spectral density functions (PSDFs) and (b) effective noise figures for the DFRAs using the backward pumping with λ e = 1420 nm, where the cases with null and non-null IT-PSDFs are shown. The bandwidth Δλ w and the maximum PSD p 0 are indicated in the figures.

Fig. 2.
Fig. 2.

(a). Pump power spectral density functions (PSDFs) and (b) effective noise figures for the DFRAs using the backward pumping with λ e = 1380 nm, where the cases with null and non-null IT-PSDFs are shown. The bandwidth Δλ w = 40 nm. The maximum PSD p 0 is indicated in the figures.

Fig. 3.
Fig. 3.

Pump power spectral density functions (PSDFs) and (b) effective noise figures for the DFRAs using the backward pumping with λ e = 1380 nm, where the cases with non-null ITPSDFs are shown. The bandwidth Δλ w = 30 nm. The maximum PSD p 0 is indicated in the figures.

Fig. 4.
Fig. 4.

(a). Pump power spectral density functions (PSDFs) and (b) effective noise figures for the DFRAs using the bidirectional pumping with λ e = 1410 nm, where the cases with null and non-null IT-PSDFs are shown. The bandwidth Δλ w and the maximum PSD p 0 are indicated in the figures.

Fig. 5.
Fig. 5.

Pump power spectral density functions (PSDFs) and (b) effective noise figures for the DFRAs using the bidirectional pumping with λ e = 1370 nm, where the cases with null and non-null IT-PSDFs are shown. The bandwidth Δλ w = 25 nm. The maximum PSD p 0 is indicated in the figures.

Fig. 6.
Fig. 6.

Output forward ASEN PSDs for the DFRAs using the bidirectional pumping with λ e = 1370 nm and 1410 nm. The case with λ e = 1370 nm takes the IT-PSDF with Δλ w = 25 nm and p 0 = 20 mW/100GHz. The case with λ e = 1370 nm takes the null IT-PSDF.

Fig. 7.
Fig. 7.

Evolutions of the forward ASEN PSDs at 1530 nm, 1570 nm, and 1600 nm for the DFRA using the bidirectional pumping with λ e = 1370 nm, and 1410 nm shown in Fig. 6.

Fig. 8.
Fig. 8.

Evolutions of the PSDs at several pump wavelengths for the DFRA using the bidirectional pumping with λ e = 1370 nm shown in Fig. 6, in which the evolutions of the co-pump PSDs at 1370 nm and 1430 nm and the evolutions of the counter-pump PSDs at 1456 nm and 1500 nm are shown.

Fig. 9.
Fig. 9.

Pump power spectral density functions (PSDFs) for the DFRAs using bidirectional pumping, in which the cases with λ e ranging from 1350 nm to 1390 nm are shown. The PSDFs of co-pump and counter-pump are shown in (a) and (b), respectively. The bandwidth Δλ w and the maximum PSD p 0 of initial trial solutions are shown in text.

Fig. 10.
Fig. 10.

Effective noise figures of the DFRAs using bidirectional pumping with the pump power spectral density functions given in Fig. 9.

Equations (12)

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

f i ( ν ) = j = 0 M a ij ( ν ν i 1 ) j , ν i 1 ν ν i ,
f N ( ν ) = j = 0 M a Nj ( ν ν N ) j , ν N 1 ν ν N .
P i ( ν ) = f i ( ν ) , ν i 1 ν ν i .
f i ( ν i ) = f i + 1 ( ν i ) ,
f′ i ( ν i ) = f′ i + 1 ( ν i ) ,
f o ( ν 0 ) = 0 ,
f N ( ν N ) = 0 ,
f′ o ( ν 0 ) = 0 ,
f′ N ( ν N ) = 0 .
f′ o ( ν e ) = 0 ,
f t ( ν ) = { p 0 ( ν ν 0 ) ν e ν 0 exp [ ( ν ν e Δv w ) 2 ] , 0 ν ν e , p 0 ν ν N ν e ν N , ν e ν ν N
ENF = 1 G on off ( 1 + P ASE + hv Δ v ) ,

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