Abstract

We introduce a formalism that provides a unified solution for the gain spectrum of fiber optical parametric amplifiers (OPAs) using either linearly or circularly polarized waves. There are 12 basic types, including five that have not been previously investigated to our knowledge. We provide a simple method for calculating the maximum gain and nonlinear phase mismatch for such OPAs. All these OPAs have similar elliptical graphs for parametric gain versus propagation constant mismatch, with a width equal to four times the height. This implies that, when fiber dispersion is taken into account, any two-pump OPA can be used to obtain the same gain spectrum as any other OPA in the same fiber, provided that the pump power is adjusted for equal maximum gain and that real solutions exist for the resulting equation for the required pump spacing. In that sense, all these fiber OPAs form an equivalence class.

© 2003 Optical Society of America

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References

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  1. R. H. Stolen, M. A. Bosch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. 6, 213–215 (1981).
    [CrossRef] [PubMed]
  2. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
    [CrossRef]
  3. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1991).
  4. K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
    [CrossRef]
  5. M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Fiber optical parametric amplifier with circularly-polarized pumps,” IEE Electron. Lett. 39, 350–351 (2003).
    [CrossRef]
  6. R. W. Hellwarth, A. Owyoung, and N. George, “Origin of the nonlinear refractive index of liquid CCl4,” Phys. Rev. A 4, 2342–2347 (1971).
    [CrossRef]
  7. M.-C. Ho, M. E. Marhic, Y. Akasaka, and L. G. Kazovsky, “200-nm bandwidth fiber optical amplifier combining parametric and Raman gain,” J. Lightwave Technol. 19, 977–981 (2001).
    [CrossRef]
  8. M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Broadband fiber-optical parametric amplifiers,” Opt. Lett. 21, 573–575 (1996).
    [CrossRef] [PubMed]
  9. M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky, “Broadband fiber-optical parametric amplifiers and wavelength converters with low-ripple Chebyshev gain spectra,” Opt. Lett. 21, 1354–1356 (1996).
    [CrossRef] [PubMed]
  10. M.-C. Ho, M. E. Marhic, K. Y. Wong, and L. G. Kazovsky, “Narrow-linewidth idler generation in fiber four-wave mixing and parametric amplification by dithering two pumps in opposition of phase,” J. Lightwave Technol. 20, 469–476 (2002).
    [CrossRef]
  11. S. Radic, C. Mckinstrie, and R. Jopson, “Polarization dependent parametric gain in amplifiers with orthogonally multiplexed optical pumps,” in Optical Fiber Communication Conference, Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003), pp. 508–510.
  12. C. R. Menyuk, “Application of multiple-length-scale methods to the study of optical fiber transmission,” J. Eng. Math. 36, 113–136 (1999).
    [CrossRef]
  13. S. V. Chernikov and J. R. Taylor, “Measurement of normalization factor of n2 for random polarization in optical fibers,” Opt. Lett. 21, 1559–1561 (1996).
    [CrossRef] [PubMed]

2003 (1)

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

2002 (2)

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
[CrossRef]

M.-C. Ho, M. E. Marhic, K. Y. Wong, and L. G. Kazovsky, “Narrow-linewidth idler generation in fiber four-wave mixing and parametric amplification by dithering two pumps in opposition of phase,” J. Lightwave Technol. 20, 469–476 (2002).
[CrossRef]

2001 (1)

1999 (1)

C. R. Menyuk, “Application of multiple-length-scale methods to the study of optical fiber transmission,” J. Eng. Math. 36, 113–136 (1999).
[CrossRef]

1996 (3)

1982 (1)

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

1981 (1)

1971 (1)

R. W. Hellwarth, A. Owyoung, and N. George, “Origin of the nonlinear refractive index of liquid CCl4,” Phys. Rev. A 4, 2342–2347 (1971).
[CrossRef]

Akasaka, Y.

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

Bosch, M. A.

Chernikov, S. V.

Chiang, T.-K.

George, N.

R. W. Hellwarth, A. Owyoung, and N. George, “Origin of the nonlinear refractive index of liquid CCl4,” Phys. Rev. A 4, 2342–2347 (1971).
[CrossRef]

Hellwarth, R. W.

R. W. Hellwarth, A. Owyoung, and N. George, “Origin of the nonlinear refractive index of liquid CCl4,” Phys. Rev. A 4, 2342–2347 (1971).
[CrossRef]

Ho, M.-C.

Kagi, N.

Kazovsky, L. G.

Lin, C.

Marhic, M. E.

Menyuk, C. R.

C. R. Menyuk, “Application of multiple-length-scale methods to the study of optical fiber transmission,” J. Eng. Math. 36, 113–136 (1999).
[CrossRef]

Owyoung, A.

R. W. Hellwarth, A. Owyoung, and N. George, “Origin of the nonlinear refractive index of liquid CCl4,” Phys. Rev. A 4, 2342–2347 (1971).
[CrossRef]

Park, Y.

Stolen, R. H.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

R. H. Stolen, M. A. Bosch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. 6, 213–215 (1981).
[CrossRef] [PubMed]

Taylor, J. R.

Uesaka, K.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
[CrossRef]

Wong, K. K. Y.

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

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
[CrossRef]

Wong, K. Y.

Yang, F. S.

IEE Electron. Lett. (1)

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

IEEE J. Quantum Electron. (1)

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
[CrossRef]

J. Eng. Math. (1)

C. R. Menyuk, “Application of multiple-length-scale methods to the study of optical fiber transmission,” J. Eng. Math. 36, 113–136 (1999).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Lett. (4)

Phys. Rev. A (1)

R. W. Hellwarth, A. Owyoung, and N. George, “Origin of the nonlinear refractive index of liquid CCl4,” Phys. Rev. A 4, 2342–2347 (1971).
[CrossRef]

Other (2)

S. Radic, C. Mckinstrie, and R. Jopson, “Polarization dependent parametric gain in amplifiers with orthogonally multiplexed optical pumps,” in Optical Fiber Communication Conference, Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003), pp. 508–510.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1991).

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Figures (2)

Fig. 1
Fig. 1

Signal gain versus signal wavelength for an XXX-type one-pump fiber OPA. λ0=λp=1550 nm, β2=0, β4=-8×10-56 s4 m-1, γ=18 W-1 km-1, L=0.333 km, P0=0.5 W.

Fig. 2
Fig. 2

Graphs showing the equivalence of several fiber OPAs. The curves above the Δβ axis correspond to g versus Δβ, whereas the curves below correspond to Δβ versus (Δωs)4. They can be used to graphically find g for any Δωa for the various OPA types. Solid lines, XXX; dotted–dashed lines, XXXX or RLRL. The pump powers are P0 for XXX, P0 for XXXX, and 1.5P0 for RLRL.

Tables (2)

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Table 1 Parameters for One-Pump OPAs

Tables Icon

Table 2 Parameters for Two-Pump OPAs

Equations (50)

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dA3dz=ip3A3+ir exp[i(p1+p2-Δβ)z]A4*,
dA4dz=ip4A4+ir exp[i(p1+p2-Δβ)z]A3*,
Ck=Akexpiκ2-pkz,k=3, 4,l=7-k.
d2Ckdz2-g2Ck=0,k=3, 4,
g2=|r|2-κ22.
Ck(z)=Ak(0)cosh(gz)+igκ2 Ak(0)+rAl*(0)sinh(gz),
k=3, 4,l=7-k.
h3(z)=A3(z)A3(0)=cosh(gz)+i κ2gsinh(gz)×expip3-κ2z,
h4(z)=A4(z)A3*(0)=i rgsinh(gz)expip4-κ2z.
Gi(z)=rgsinh(gz)2,
Gs(z)=Gi(z)+1.
Gi,max(z)=|sinh(rz)|2,Gs,max(z)=|cosh(rz)|2.
two-pumpOPA:u=2(a13+a14-a12)-a11,
one-pumpOPA:u=2(2a13-a11).
g2=-14 (κ-2r)(κ+2r)=-14 (Δβ+ΔβNL-2r)(Δβ+ΔβNL+2r).
δβgmax=4.
Δβ=2m=1β2m(2m)! [(Δωs)2m-(Δωp)2m]=2[βe(Δωs)-βe(Δωp)],
βe(Δωs)=[β(ωc+Δωs)+β(ωc-Δωs)]/2=m=0β2m(2m)! (Δωs)2m.
g2(Δωs)=r2-14 [2βe(Δωs)-2βe(Δωp)+ΔβNL]2,
g2(Δωs)=r2-14 [2βe(Δωs)-2βe(Δωp)+ΔβNL]2.
2βe(Δωp)+ΔβNL=2βe(Δωp)-ΔβNL.
2Ez2-n2c22Et2=-μ02PNLt2,
PNL(z, t)=ε0χ(3)EEE,
PNL(z, t)=ε0χxxxx(3)(E  E)E.
E(z, t)=ρ2k=14{Ak(z)exp[i(βkz-ωkt)]+Ak*(z)exp[-i(βkz-ωkt)]},
ρ22Alz2+2iβlAlziρβlAlz=-χxxxx(3)ωl2c2 {(E  E)E}l×exp[-i(βlz-ωlt)],
Alz=i χxxxx(3)ωlρnc {(E  E)E}lexp[-i(βlz-ωlt)].
[a, b, c]=[(a  b)c+(b  c)a+(c  a)b]/3,
[a, b, c, d]=[a, b, c]d=[(a  b)(c  d)+(b  c)(a  d)+(c  a)(b  d)]/3.
Rl=[Al, Al*, Al]+2jl=14[Aj, Aj*, Al]+2[Am, An, Ak*]exp(iΔβklmnz).
R1=[A1, A1*, A1]+2jl=14[Aj, Aj*, A1]+2[A3, A4, A1*]exp(iΔβz),
Rl=[Al, Al*, Al]+2jl=14[Aj, Aj*, Al]+[A1, A1, Ak*]exp(-iΔβz),
l=3, 4,k=7-l,
Alz=iγRl,
dAkdz=iγ([Ak, Ak, Ak*]+2[Ak, Al, Al*]),
k=1, 2,l=3-k.
dAkdz=iγ([e^k, e^k*, e^k, e^k*]Pk+2[e^k, e^k*, e^l, e^l*]Pl)Ak=iγ(akkPk+2aklPl)Ak=ipkAk,
k=1, 2,l=3-k,
Ak(z)=Ak(0)exp(ipkz),k=1, 2.
dAkdz=iγ(2ak1P1+2ak2P2)Ak+2iγ[A1, A2, A1*]e^k*exp(-iΔβz)
=2iγ(ak1P1+ak2P2)Ak+2iγA1A2[e^1, e^2, e^k*, e^l*]Al*exp(-iΔβz),
k=3, 4,l=7-k
=2iγ(ak1P1+ak2P2)Ak+irkAl*×exp[i(p1+p2-Δβ)z]=ipkAk+irkAl*exp[i(p1+p2-Δβ)z],
pk=2γ(ak1P1+ak2P2),
rk=2γA1(0)A2(0)[e^1, e^2, e^k*, e^l*],
k=3, 4,l=7-k,
dAkdz=ipkAk+irkAl*exp[i(2p1-Δβ)z],
k=3, 4,l=7-k,
pk=2γak1P1,rk=γ[A1(0)]2[e^1, e^1, e^k*, e^l*],
k=3, 4,l=7-k,

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