Abstract

We improve the accuracy of numerical simulations for short fiber optical parametric amplifiers (OPAs). Instead of using the usual coarse-step method, we adopt a model for birefringence and dispersion which uses fine-step variations of the parameters. We also improve the split-step Fourier method by exactly treating the nonlinear ellipse rotation terms. We find that results obtained this way for two-pump OPAs can be significantly different from those obtained by using the usual coarse-step fiber model, and/or neglecting ellipse rotation terms.

© 2008 Optical Society of America

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References

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  1. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, "Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
    [CrossRef]
  2. C. R. Menyuk, "Nonlinear pulse propagation in birefringent optical fibers," IEEE J. Quantum Electron. QE-23, 174-176 (1987).
    [CrossRef]
  3. A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997).
    [CrossRef]
  4. M. E. Marhic, G. M. Williams, L. Goldberg, and J. M. P. Delavaux, "Tunable fiber optical parametric wavelength converter with 900 mW of CW output power at 1665 nm," Proc. SPIE 6103, 165-176 (2006).
    [CrossRef]
  5. P. K. A. Wai and C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technol. 14, 148-157 (1996).
    [CrossRef]
  6. F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, "Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 1292-1294 (2004).
    [CrossRef]
  7. 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]
  8. M. Farahmand and M. de Sterke, "Parametric amplification in presence of dispersion fluctuations," Opt. Express. 12, 136-142 (2004).
    [CrossRef] [PubMed]
  9. www.photonics.incubadora.fapesp.br/portal/download.
  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, A. A. Rieznik, and H. L. Fragnito, "Investigation of the gain spectrum near the pumps of two-pump fiber-optic parametric amplifiers," J. Opt. Soc Am. B 25, 22-30 (2008).
    [CrossRef]
  12. M. Karlsson and J. Brentel, "Autocorrelation function of the polarization-mode dispersion vector," Opt. Lett. 24, 939-941 (1999).
    [CrossRef]

2008 (1)

M. E. Marhic, A. A. Rieznik, and H. L. Fragnito, "Investigation of the gain spectrum near the pumps of two-pump fiber-optic parametric amplifiers," J. Opt. Soc Am. B 25, 22-30 (2008).
[CrossRef]

2004 (3)

F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, "Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 1292-1294 (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]

M. Farahmand and M. de Sterke, "Parametric amplification in presence of dispersion fluctuations," Opt. Express. 12, 136-142 (2004).
[CrossRef] [PubMed]

2003 (1)

1999 (1)

1997 (2)

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, "Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
[CrossRef]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997).
[CrossRef]

1996 (1)

P. K. A. Wai and C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technol. 14, 148-157 (1996).
[CrossRef]

1987 (1)

C. R. Menyuk, "Nonlinear pulse propagation in birefringent optical fibers," IEEE J. Quantum Electron. QE-23, 174-176 (1987).
[CrossRef]

Agrawal, G. P.

F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, "Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 1292-1294 (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]

Benedetto, S.

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997).
[CrossRef]

Brentel, J.

Carena, A.

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997).
[CrossRef]

Curri, V.

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997).
[CrossRef]

de Sterke, M.

M. Farahmand and M. de Sterke, "Parametric amplification in presence of dispersion fluctuations," Opt. Express. 12, 136-142 (2004).
[CrossRef] [PubMed]

Farahmand, M.

M. Farahmand and M. de Sterke, "Parametric amplification in presence of dispersion fluctuations," Opt. Express. 12, 136-142 (2004).
[CrossRef] [PubMed]

Fragnito, H. L.

M. E. Marhic, A. A. Rieznik, and H. L. Fragnito, "Investigation of the gain spectrum near the pumps of two-pump fiber-optic parametric amplifiers," J. Opt. Soc Am. B 25, 22-30 (2008).
[CrossRef]

Gaudino, R.

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997).
[CrossRef]

Karlsson, M.

Kazovsky, L. G.

Lin, Q.

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. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, "Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 1292-1294 (2004).
[CrossRef]

Marcuse, D.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, "Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
[CrossRef]

Marhic, M. E.

M. E. Marhic, A. A. Rieznik, and H. L. Fragnito, "Investigation of the gain spectrum near the pumps of two-pump fiber-optic parametric amplifiers," J. Opt. Soc Am. B 25, 22-30 (2008).
[CrossRef]

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]

Menyuk, C. R.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, "Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
[CrossRef]

P. K. A. Wai and C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technol. 14, 148-157 (1996).
[CrossRef]

C. R. Menyuk, "Nonlinear pulse propagation in birefringent optical fibers," IEEE J. Quantum Electron. QE-23, 174-176 (1987).
[CrossRef]

Poggiolini, P.

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997).
[CrossRef]

Radic, S.

F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, "Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 1292-1294 (2004).
[CrossRef]

Rieznik, A. A.

M. E. Marhic, A. A. Rieznik, and H. L. Fragnito, "Investigation of the gain spectrum near the pumps of two-pump fiber-optic parametric amplifiers," J. Opt. Soc Am. B 25, 22-30 (2008).
[CrossRef]

Wai, P. K. A.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, "Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
[CrossRef]

P. K. A. Wai and C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technol. 14, 148-157 (1996).
[CrossRef]

Wong, K. K. Y.

Yaman, F.

F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, "Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 1292-1294 (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]

IEEE J. Quantum. Electron. (1)

C. R. Menyuk, "Nonlinear pulse propagation in birefringent optical fibers," IEEE J. Quantum Electron. QE-23, 174-176 (1987).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, "A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber," IEEE J. Sel. Areas Commun. 15, 751-765 (1997).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

F. Yaman, Q. Lin, S. Radic, and G. P. Agrawal, "Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers," IEEE Photon. Technol. Lett. 16, 1292-1294 (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]

J. Lightwave Technol. (2)

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, "Application of the Manakov-PMD Equation to Studies of Signal Propagation in Optical Fibers with Randomly Varying Birefringence," J. Lightwave Technol. 15, 1735-1746 (1997).
[CrossRef]

P. K. A. Wai and C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence," J. Lightwave Technol. 14, 148-157 (1996).
[CrossRef]

J. Opt. Soc Am. B (1)

M. E. Marhic, A. A. Rieznik, and H. L. Fragnito, "Investigation of the gain spectrum near the pumps of two-pump fiber-optic parametric amplifiers," J. Opt. Soc Am. B 25, 22-30 (2008).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express. (1)

M. Farahmand and M. de Sterke, "Parametric amplification in presence of dispersion fluctuations," Opt. Express. 12, 136-142 (2004).
[CrossRef] [PubMed]

Opt. Lett. (1)

Other (2)

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

M. E. Marhic, G. M. Williams, L. Goldberg, and J. M. P. Delavaux, "Tunable fiber optical parametric wavelength converter with 900 mW of CW output power at 1665 nm," Proc. SPIE 6103, 165-176 (2006).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Total angle θ(z) along a fiber with L = 200 m and Lb = 50 m, using a coarse-step model (dashed red) or the Wai-Menyuk model (solid blue); both curves are such that their values are the same after each Lb . (b) Variation of Δn(z) along the same fiber, obtained in a similar manner.

Fig. 2.
Fig. 2.

Schematic representation of the structure of the SSFM algorithm in one fiber segment. D is the dispersion; LBR=linear birefringence; NL=nonlinearity; FT=Fourier transform; IFT=inverse FT. L=>C and C=>L indicate transformation from a linear to a circular basis and vice-versa, respectively.

Fig. 3.
Fig. 3.

Gain spectra for an OPA with two orthogonal CP pumps, using the total angle rotations shown in Fig. 1(a). The colors match the corresponding θ(z) and Δn(z) curves in Figs. 1(a) and 1(b).

Fig. 4.
Fig. 4.

(a), (b) and (c): Gain spectra for an OPA with two orthogonal CP pumps, as Fig. 3 but with three other realizations of the random process for the determination of the linear parameters of the fiber. (d): Average gains after 50 realizations of the random process. Solid blue: fine-step method for the linear parameters. dashed red: coarse-step method.

Fig. 5.
Fig. 5.

Gain spectra for an OPA with two orthogonal CP pumps using an isotropic fiber with the same parameters as in Fig.3. We show the analytical result using the four-wave model (dotted gray), and the results obtained using the SSFM with (solid blue) and without (dashed red) the ER terms. The input signal SOP was the same as that of the short-wavelength pump.

Fig. 6.
Fig. 6.

Gain spectra for three different two-pump OPAs with orthogonal CP pumps, with or without the ER terms (solid blue and dashed red curves, respectively). The three OPAs use the same fiber as in Fig. 3, whose birefringence parameters are shown in Fig. 1(a) and 1(b) (solid blue lines). The input signal has the same SOP as the short-wavelength pump. (a): L = 200 m, P 0 = 2 W; (b): L = 20 m, P 0 = 20 W; (c) L = 2 m, P 0 = 200 W. Observe that the solid blue curve in (a) is the same as in Fig. 3. Also, the horizontal scale is different for each graph, since the gain bandwidth increases considerably for high pump powers.

Equations (23)

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E x ( z , t ) = 1 2 { C x ( z , t )   exp [ i ( β c z ω c t ) ] + c . c . }
E y ( z , t ) = 1 2 { C y ( z , t ) exp [ i ( β c z ω c t ) ] + c . c . } ,
β c = β x ( ω c ) + β y ( ω c ) 2 = β ( ω c )
C x z i Δ β xy 2 C x n = 1 i n + 1 β x ( n ) n ! d n C x d t n = i γ [ ( P x + 2 3 P y ) C x + 1 3 ( C y ) 2 C x * ]
C y z + i Δ β xy 2 C y n = 1 i n + 1 β y ( n ) n ! n C y t n = i γ [ ( P y + 2 3 P x ) C y + 1 3 ( C x ) 2 C y * ] ,
D x ( z + dz , Ω ) = D x ( z , Ω ) exp [ i ( β β c + Δ n Ω 2 c ) dz + i Φ 0 ]
D y ( z + dz , Ω ) = D y ( z , Ω ) exp [ i ( β β c Δ n Ω 2 c ) dz i Φ 0 ]
d C x dz = i γ [ ( P x + 2 3 P y ) C x + 1 3 C y 2 C x * ] d C y dz = i γ [ ( P y + 2 3 P x ) C y + 1 3 C x 2 C y * ] .
d C x dz = i γ ( P x + 2 3 P y ) C x d C y dz = i γ ( P y + 2 3 P x ) C y .
C x ( z + dz ) = C x ( z ) exp i γ dz ( P x + 2 P y 3 )
C y ( z + dz ) = C y ( z ) exp i γ dz ( P y + 2 P x 3 )
d C r dz = 2 3 ( P r + 2 P l ) C r d C l dz = 2 3 ( P l + 2 P r ) C l ,
C r ( z + dz ) = C r ( z ) exp [ 2 3 ( P r + 2 P l ) dz ] = C r ( z ) exp ( i θ r )
C l ( z + dz ) = C l ( z ) exp [ 2 3 ( P l + 2 P r ) dz ] = C l ( z ) exp ( i θ l ) .
[ C x ( z + dz ) C y ( z + dz ) ] = exp ( P 0 dz ) [ cos ( δθ ) sin ( δθ ) sin ( δθ ) cos ( δθ ) ] [ C x ( z ) C y ( z ) ]
δ θ = γdz 3 ( P l P r ) = iγdz 3 ( C x C y * C x * C y ) .
β = β x + β y 2 .
β = β ( 0 ) + β ( 1 ) Ω + β ( 2 ) Ω 2 2 + β ( 3 ) Ω 3 6 + β ( 4 ) Ω 4 24
β x = β + b β y = β b ,
b = β x β y 2 .
b = ω ( n x n y 2 c ) = ω Δ n 2 c
b = b ( 0 ) + b ( 1 ) Ω + b ( 2 ) Ω 2 2 + ,
b ( 0 ) = ω c Δ n 2 c , b ( 1 ) = Δ n 2 c , b ( m ) = 0 for m 2

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