Abstract

We investigated the gain of two-pump fiber-optic parametric amplifiers (OPAs) when the signal is close to one pump. We considered all eight types of OPAs introduced in [J. Opt. Soc. Am. B 20, 2425 (2003) ], and we extended the analysis of that paper from a four- to a six-wave model. We started from six coupled-wave equations and derived exact expressions for the parametric gain coefficients in all cases. The results were verified by simulations based on the split-step Fourier method. We found a variety of possible behaviors: the gain near the pumps may be higher or lower than the gain at the center wavelength, depending on the type of OPA and the operating conditions. In addition, the gain spectrum generally exhibits narrow peaks or dips on either side of the pumps. However, for one type of OPA the gain predicted by the six-wave model is the same as that predicted by the four-wave model, and the gain spectrum is very smooth near the pump.

© 2008 Optical Society of America

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References

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  1. C. J. McKinstrie and S. Radic, "Parametric amplifiers driven by two pump waves with dissimilar frequencies," Opt. Lett. 27, 1138-1140 (2002).
    [CrossRef]
  2. T. Tanemura and K. Kikuchi, "Unified analysis of modulational instability induced by cross-phase modulation in optical fibers," J. Opt. Soc. Am. B 20, 2502-2514 (2003).
    [CrossRef]
  3. J. L. Blows and P. F. Hu, "Cross-talk-induced limitations of two-pump parametric amplifiers," J. Opt. Soc. Am. B 21, 989-995 (2004).
    [CrossRef]
  4. A. Bogris, D. Syvridis, P. Kylemark, and P. A. Andrekson, "Noise characteristics of dual-pump fiber-optic parametric amplifiers," J. Lightwave Technol. 23, 2788-2795 (2005).
    [CrossRef]
  5. F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnitto, "Spurious four-wave mixing in two-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 434-436 (2004).
    [CrossRef]
  6. Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
    [CrossRef]
  7. P. Parolari, L. Marazzi, E. Rognoni, and M. Martinelli, "Influence of pump parameters on two-pump optical parametric amplification," J. Lightwave Technol. 23, 2424-2530 (2005).
    [CrossRef]
  8. S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centani, and A. R. Chraplyvy, "Multiple-band bit-level switching in two-pump fiber parametric devices," IEEE Photon. Technol. Lett. 16, 852-854 (2004).
    [CrossRef]
  9. J. D. Marconi, J. M. Chavez Boggio, and H. L. Fragnito, "Nearly 100 nm bandwidth of flat gain with a double-pumped fiber optic parametric amplifier," in Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference 2007 (OFC/NFOEC) (IEEE, 2007), paper OWB1.
    [PubMed]
  10. M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, "Fiber optical parametric amplifiers with linearly- or circularly-polarized waves," J. Opt. Soc. Am. B 20, 2425-2433 (2003).
    [CrossRef]
  11. M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky "Widely tunable spectrum translation and wavelength exchange by four-wave mixing in optical fibers," Opt. Lett. 21, 1906-1908 (1996).
    [CrossRef] [PubMed]
  12. http://www.photonics.incubadora.fapesp.br/portal/download.
  13. M. E. Marhic, K. K.-Y. Wong, and L. G. Kazovsky, "Fiber optical parametric amplifiers with circularly polarized pumps," Electron. Lett. 39, 350-351 (2003).
    [CrossRef]
  14. F. Yaman, Q. Lin, and G. P. Agrawal, "Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 431-433 (2004).
    [CrossRef]

2005 (3)

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

P. Parolari, L. Marazzi, E. Rognoni, and M. Martinelli, "Influence of pump parameters on two-pump optical parametric amplification," J. Lightwave Technol. 23, 2424-2530 (2005).
[CrossRef]

A. Bogris, D. Syvridis, P. Kylemark, and P. A. Andrekson, "Noise characteristics of dual-pump fiber-optic parametric amplifiers," J. Lightwave Technol. 23, 2788-2795 (2005).
[CrossRef]

2004 (4)

J. L. Blows and P. F. Hu, "Cross-talk-induced limitations of two-pump parametric amplifiers," J. Opt. Soc. Am. B 21, 989-995 (2004).
[CrossRef]

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centani, and A. R. Chraplyvy, "Multiple-band bit-level switching in two-pump fiber parametric devices," IEEE Photon. Technol. Lett. 16, 852-854 (2004).
[CrossRef]

F. Yaman, Q. Lin, and G. P. Agrawal, "Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 431-433 (2004).
[CrossRef]

F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnitto, "Spurious four-wave mixing in two-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 434-436 (2004).
[CrossRef]

2003 (3)

2002 (1)

1996 (1)

Electron. Lett. (1)

M. E. Marhic, K. K.-Y. Wong, and L. G. Kazovsky, "Fiber optical parametric amplifiers with circularly polarized pumps," Electron. Lett. 39, 350-351 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

F. Yaman, Q. Lin, and G. P. Agrawal, "Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 431-433 (2004).
[CrossRef]

F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnitto, "Spurious four-wave mixing in two-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 434-436 (2004).
[CrossRef]

Q. Lin, R. Jiang, C. F. Marki, C. J. McKinstrie, R. Jopson, J. Ford, G. P. Agrawal, and S. Radic, "40-Gb/s optical switching and wavelength multicasting in a two-pump parametric device," IEEE Photon. Technol. Lett. 17, 2376-2378 (2005).
[CrossRef]

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centani, and A. R. Chraplyvy, "Multiple-band bit-level switching in two-pump fiber parametric devices," IEEE Photon. Technol. Lett. 16, 852-854 (2004).
[CrossRef]

J. Lightwave Technol. (2)

P. Parolari, L. Marazzi, E. Rognoni, and M. Martinelli, "Influence of pump parameters on two-pump optical parametric amplification," J. Lightwave Technol. 23, 2424-2530 (2005).
[CrossRef]

A. Bogris, D. Syvridis, P. Kylemark, and P. A. Andrekson, "Noise characteristics of dual-pump fiber-optic parametric amplifiers," J. Lightwave Technol. 23, 2788-2795 (2005).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (2)

Other (2)

J. D. Marconi, J. M. Chavez Boggio, and H. L. Fragnito, "Nearly 100 nm bandwidth of flat gain with a double-pumped fiber optic parametric amplifier," in Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference 2007 (OFC/NFOEC) (IEEE, 2007), paper OWB1.
[PubMed]

http://www.photonics.incubadora.fapesp.br/portal/download.

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Figures (4)

Fig. 1
Fig. 1

Notation scheme for the OPAs with LP or CP SOPs.

Fig. 2
Fig. 2

Gains spectra obtained with the SSFM for: (a) the shortest pump wavelength in the y direction, the other pump in the x direction, and the signal in the x direction; (b) the shortest pump wavelength CP to the left, the other pump CP to the right, and the signal CP to the right; (c) an RRRR OPA.

Fig. 3
Fig. 3

Gains spectra for the same input parameters as in Fig. 2a but using different fiber lengths: (a) 500 m and (b) 1000 m . Solid curve: SSFM; dashed curve: analytical solution of the FWM.

Fig. 4
Fig. 4

Gains spectra for the same input parameters as in Fig. 2 but using (a) YXXY and (b) LRRL configurations. Solid curve: SSFM; dashed curve: analytical solution of the four-wave model.

Tables (1)

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Table 1 Parameters for Two-Pump OPAs

Equations (66)

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i d A k d Z = [ A k , A k * , A k ] + 2 [ A l , A l * , A k ] , k = 1 , 2 , l = 3 k ,
i d A k d Z = ( a k k P k + 2 a k l P l ) A k , k = 1 , 2 , l = 3 k ,
i d A 3 d z = 2 [ A 1 , A 1 * , A 3 ] + 2 [ A 2 , A 2 * , A 3 ] + 2 [ A 1 , A 2 , A 4 * ] e i Δ β 1234 z + [ A 1 , A 1 , A 5 * ] e i Δ β 1135 z + 2 [ A 1 , A 2 * , A 6 ] e i Δ β 1623 z ,
i d A 4 d z = 2 [ A 1 , A 1 * , A 4 ] + 2 [ A 2 , A 2 * , A 4 ] + 2 [ A 1 , A 2 , A 3 * ] e i Δ β 1234 z + [ A 2 , A 2 , A 6 * ] e i Δ β 2246 z + 2 [ A 1 * , A 2 , A 5 ] e i Δ β 2514 z ,
i d A 5 d z = 2 [ A 1 , A 1 * , A 5 ] + 2 [ A 2 , A 2 * , A 5 ] + 2 [ A 1 , A 2 , A 6 * ] e i Δ β 1256 z + [ A 1 , A 1 , A 3 * ] e i Δ β 1135 z + 2 [ A 1 , A 2 * , A 4 ] e i Δ β 1425 z ,
i d A 6 d z = 2 [ A 1 , A 1 * , A 6 ] + 2 [ A 2 , A 2 * , A 6 ] + 2 [ A 1 , A 2 , A 5 * ] e i Δ β 1256 z + [ A 2 , A 2 , A 4 * ] e i Δ β 2246 z + 2 [ A 1 * , A 2 , A 3 ] e i Δ β 2316 z ,
i d A 3 d Z = 2 ( a 13 P 1 + a 23 P 2 ) A 3 + 2 c 1234 ( z ) A 10 A 20 A 4 * exp [ i ( φ 1 + φ 2 ) ] + 2 c 1135 ( z ) A 10 2 A 5 * exp ( 2 i φ 1 ) + 2 c 1623 ( z ) A 10 A 20 * A 6 exp [ i ( φ 1 φ 2 ) ] ,
i d A 4 d Z = 2 ( a 14 P 1 + a 24 P 2 ) A 4 + 2 c 1234 ( z ) A 10 A 20 A 3 * exp [ i ( φ 1 + φ 2 ) ] + c 2246 ( z ) A 20 2 A 6 * exp ( 2 i φ 2 ) + 2 c 2514 ( z ) A 10 * A 20 A 5 exp [ i ( φ 2 φ 1 ) ] ,
i d A 5 d Z = 2 ( a 15 P 1 + a 25 P 2 ) A 5 + 2 c 1256 ( z ) A 10 A 20 A 6 * exp [ i ( φ 1 + φ 2 ) ] + c 1135 ( z ) A 10 2 A 3 * exp ( 2 i φ 1 ) + 2 c 1425 ( z ) A 10 A 20 * A 4 exp [ i ( φ 1 φ 2 ) ] ,
i d A 6 d Z = 2 ( a 16 P 1 + a 26 P 2 ) A 6 + 2 c 1256 ( z ) A 10 A 20 A 5 * exp [ i ( φ 1 + φ 2 ) ] + c 2246 ( z ) A 20 2 A 4 * exp ( 2 i φ 2 ) + 2 c 2316 ( z ) A 10 * A 20 A 3 exp [ i ( φ 2 φ 1 ) ] ,
c k l m n ( z ) = a k l m n exp ( i Δ β k l m n z ) ,
a k l m n = [ e ̂ k , e ̂ l , e ̂ m * , e ̂ n * ] .
i d B 3 d Z = [ ( 2 a 13 a 11 ) P 1 + ( 2 a 23 2 a 21 ) P 2 ] B 3 + 2 a 1234 A 10 A 20 B 4 * + a 1135 A 10 2 B 5 * + 2 a 1623 A 10 A 20 * B 6 ,
i d B 4 * d Z = [ ( 2 a 14 a 22 ) P 1 + ( 2 a 24 2 a 21 ) P 2 ] B 4 * + 2 a 1234 * A 10 * A 20 * B 3 + a 2246 * ( A 20 * ) 2 B 6 + 2 a 2514 * A 10 A 20 * B 5 * ,
i d B 5 * d Z = [ ( 2 a 15 a 11 ) P 1 + ( 2 a 25 2 a 21 ) P 2 ] B 5 * + 2 a 1256 * A 10 * A 20 * B 6 + a 1135 * ( A 10 * ) 2 B 3 + 2 a 1425 * A 10 * A 20 B 4 *
i d B 6 d Z = [ ( 2 a 16 a 22 ) P 1 + ( 2 a 26 2 a 21 ) P 2 ] B 6 + 2 a 1256 A 10 A 20 B 5 * + a 2246 A 20 2 B 4 * + 2 a 2316 A 10 * A 20 B 3 ,
d d z [ B 3 B 4 * B 5 * B 6 ] = i γ [ b 33 2 a 1234 A 10 A 20 a 1135 A 10 2 2 a 1623 A 10 A 20 * 2 a 1234 * A 10 * A 20 * b 44 2 a 2514 * A 10 A 20 * a 2246 * ( A 20 * ) 2 a 1135 * ( A 10 * ) 2 2 a 1425 * A 10 * A 20 b 55 2 a 1256 * A 10 * A 20 * 2 a 2316 A 10 * A 20 a 2246 A 20 2 2 a 1256 A 10 A 20 b 66 ] [ B 3 B 4 * B 5 * B 6 ] ,
b 33 = [ ( 2 a 13 a 11 ) P 1 + ( 2 a 23 2 a 21 ) P 2 ] ,
b 44 = [ ( 2 a 14 a 22 ) P 1 + ( 2 a 24 2 a 21 ) P 2 ] ,
b 55 = [ ( 2 a 15 a 11 ) P 1 + ( 2 a 25 2 a 21 ) P 2 ] ,
b 66 = [ ( 2 a 16 a 22 ) P 1 + ( 2 a 26 2 a 21 ) P 2 ] .
B ̃ ( z ) = exp ( N z ) B ̃ ( 0 ) = m = 0 ( N z ) m m ! B ̃ ( 0 ) .
M = [ c 33 2 a 1234 a 1135 2 a 1623 2 a 1234 * c 44 2 a 2514 * a 2246 * a 1135 * 2 a 1425 * c 55 2 a 1256 * 2 a 2316 a 2246 2 a 1256 c 66 ] ,
c 33 = 2 ( a 13 + a 23 a 12 ) a 11 ,
c 44 = 2 ( a 14 + a 24 a 12 ) a 22 ,
c 55 = 2 ( a 15 + a 25 a 12 ) a 11 ,
c 66 = 2 ( a 16 + a 26 a 12 ) a 22 .
M = [ M a M b M b M a ] ,
M a = [ a b b a ] , M b = [ c d d c ] ,
M = [ a b c d b a d c c d a b d c b a ] .
Det ( M ) = ( a + b + c + d ) ( a + b c d ) ( a b + c d ) ( a b c + d ) = [ ( a + b ) 2 ( c + d ) 2 ] [ ( a b ) 2 ( c d ) 2 ] .
Det ( M λ I ) = λ 4 2 ( a 2 b 2 c 2 + d 2 ) λ 2 + Det ( M ) = 0 .
λ = ± [ ( a ± d ) 2 ( b ± c ) 2 ] 1 2 ,
λ 1 = [ ( a + d ) 2 ( b + c ) 2 ] 1 2 , λ 2 = λ 1 ,
λ 3 = [ ( a d ) 2 ( b c ) 2 ] 1 2 , λ 4 = λ 3 .
V 1 , 2 = [ b + c a d + λ 1 , 2 a d + λ 1 , 2 b + c ] , V 3 , 4 = [ b + c a d λ 3 , 4 a + d + λ 3 , 4 b c ] .
( λ 4 λ 3 ) ( V 1 V 2 ) + ( λ 1 λ 2 ) ( V 3 V 4 ) = 2 ( λ 4 λ 3 ) ( λ 1 λ 2 ) [ 0 1 0 0 ] t ,
C 1 = B 3 + B 4 , C 2 = B 5 + B 6 , C 3 = B 3 B 4 , C 4 = B 5 B 6 ,
d d ξ [ C 1 C 2 ] = M 1 [ C 3 C 4 ] , d d ξ [ C 3 C 4 ] = M 2 [ C 1 C 2 ] ,
M 1 = [ ( a b ) ( c d ) ( c d ) ( a b ) ] , M 2 = [ ( a + b ) ( c + d ) ( c + d ) ( a + b ) ] .
d 2 d ξ 2 [ C 1 C 2 ] = M 1 M 2 [ C 1 C 2 ] , d 2 d ξ 2 [ C 3 C 4 ] = M 2 M 1 [ C 3 C 4 ] .
M 1 M 2 = [ u v v u ] , M 2 M 1 = [ u v v u ] ,
u + v = ( a + d ) 2 ( b + c ) 2 = λ 1 2 , u v = ( a d ) 2 ( b c ) 2 = λ 3 2 .
d 2 C 1 d ξ 2 = u C 1 + v C 2 , d 2 C 2 d ξ 2 = v C 1 + u C 2 ,
d 2 C 3 d ξ 2 = u C 3 v C 4 , d 2 C 4 d ξ 2 = v C 3 + u C 4 .
d 2 ( C 1 + C 2 ) d ξ 2 = ( u + v ) ( C 1 + C 2 ) = λ 1 2 ( C 1 + C 2 ) ,
d 2 ( C 1 C 2 ) d ξ 2 = ( u v ) ( C 1 C 2 ) = λ 3 2 ( C 1 C 2 ) .
C 1 + C 2 = A exp ( λ 1 ξ ) + B exp ( λ 1 ξ ) ,
C 1 C 2 = C exp ( λ 3 ξ ) + D exp ( λ 3 ξ ) ,
C 10 = B 30 + B 40 = B 30 , C 20 = B 50 + B 60 = 0 ,
C 30 = B 30 B 40 = B 30 , C 40 = B 50 B 60 = 0.
A + B = C 10 + C 20 = B 30 , C + D = C 10 C 20 = B 30 .
d C 1 d ξ z = 0 = ( a b ) C 30 + ( c d ) C 40 = ( a b ) B 30 ,
d C 2 d ξ z = 0 = ( c d ) C 30 ( a b ) C 40 = ( c d ) B 30 .
d ( C 1 + C 2 ) d ξ z = 0 = ( a b c + d ) B 30 = λ 1 ( A B ) ,
d ( C 1 C 2 ) d ξ z = 0 = ( a b + c d ) B 30 = λ 3 ( C D ) .
A = B 30 2 ( 1 + a b c + d λ 1 ) , B = B 30 2 ( 1 a b c + d λ 1 ) ,
C = B 30 2 ( 1 + a b + c d λ 3 ) , D = B 30 2 ( 1 a b + c d λ 3 ) .
C 3 + C 4 = E exp ( λ 3 ξ ) + F exp ( λ 3 ξ ) ,
C 3 C 4 = G exp ( λ 1 ξ ) + H exp ( λ 1 ξ ) ,
E = B 30 2 ( 1 + a + b c d λ 3 ) , F = B 30 2 ( 1 a + b c d λ 3 ) ,
G = B 30 2 ( 1 + a + b + c + d λ 1 ) , H = B 30 2 ( 1 a + b + c + d λ 1 ) .
g 3 = 1 4 [ ( 1 + a + d λ 1 ) exp ( λ 1 ξ ) + ( 1 a + d λ 1 ) exp ( λ 1 ξ ) + ( 1 + a d λ 3 ) exp ( λ 3 ξ ) + ( 1 a d λ 3 ) exp ( λ 3 ξ ) ] = 1 2 [ cosh ( λ 1 ξ ) + a + d λ 1 sinh ( λ 1 ξ ) + cosh ( λ 3 ξ ) + a d λ 3 sinh ( λ 3 ξ ) ] .
g 3 = 1 2 [ 1 + ( a + d ) ξ + cosh ( λ 3 ξ ) + a d λ 3 sinh ( λ 3 ξ ) ] , if λ 1 = 0 ,
g 3 = 1 2 [ 1 + ( a d ) ξ + cosh ( λ 1 ξ ) + a + d λ 1 sinh ( λ 1 ξ ) ] , if λ 3 = 0 ,
g 3 = 1 + a ξ , if λ 1 = λ 3 = 0 .

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