Abstract

This paper presents a comprehensive analytical study of temporal modulation instabilities in a finite, nonlinear, dispersive medium in which two counterpropagating pump beams interact through a Kerr-type nonlinearity. The analysis includes self- and cross-phase modulations, group-velocity dispersion, four-wave mixing, and reflections occurring at the two facets of the dispersive Kerr medium. The use of a new method based on a small-parameter analysis has resulted in a physically transparent model in terms of a doubly resonant optical parametric oscillator that allows characterization of the complicated nonlinear system in a familiar language. The effects of boundary reflections are shown to be very important. In the low-frequency limit, in which dispersive effects are negligible, our results reduce to those obtained previously. At high frequencies, dispersive effects lead to new instabilities both in the normal- and anomalous-dispersion regions of the dispersive Kerr medium. The anomalous-dispersion case is discussed in detail after including weak boundary reflections. The growth rate and the threshold for the absolute instability are obtained in an analytical form for arbitrary pump–power ratios. Our analytic results are in agreement with previous numerical work done by neglecting boundary reflections and assuming equal powers for the counterpropagating pump beams.

© 1998 Optical Society of America

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References

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  1. W. J. Firth, Opt. Commun. 39, 343 (1981).
    [CrossRef]
  2. Y. Silberberg and I. Bar-Joseph, J. Opt. Soc. Am. B 1, 662 (1984).
    [CrossRef]
  3. W. J. Firth and C. Paré, Opt. Lett. 13, 1096 (1989); W. J. Firth, C. Paré, and A. FitzGerald, J. Opt. Soc. Am. B 7, 1087 (1990).
    [CrossRef]
  4. C. T. Law and A. E. Kaplan, Opt. Lett. 14, 734 (1989); J. Opt. Soc. Am. B 8, 58 (1991).
    [CrossRef] [PubMed]
  5. W. J. Firth and C. Penman, Opt. Commun. 94, 183 (1992).
    [CrossRef]
  6. R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).
  7. K. Ikeda, Opt. Commun. 30, 257 (1979).
    [CrossRef]
  8. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  9. R. Vallée, Opt. Commun. 81, 419 (1991); Opt. Commun. 93, 389 (1992).
    [CrossRef]
  10. M. B. van der Mark, J. M. Schins, and A. Lagendijk, Opt. Commun. 98, 120 (1993).
    [CrossRef]
  11. G. Steinmeyer, D. Jaspert, and F. Mitschke, Opt. Commun. 104, 379 (1994).
    [CrossRef]
  12. M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989); M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1989).
    [CrossRef]
  13. M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992); Opt. Lett. 17, 745 (1992).
    [CrossRef]
  14. D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
    [CrossRef] [PubMed]
  15. M. Haelterman, Opt. Lett. 17, 792 (1992).
    [CrossRef]
  16. A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981).
  17. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, New York, 1993), Chap. 7.
  18. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 9.
  19. Y. Yu, C. McKinstrie, and G. P. Agrawal, J. Opt. Soc. Am. B 14, 617 (1998).
    [CrossRef]

1998

Y. Yu, C. McKinstrie, and G. P. Agrawal, J. Opt. Soc. Am. B 14, 617 (1998).
[CrossRef]

1994

G. Steinmeyer, D. Jaspert, and F. Mitschke, Opt. Commun. 104, 379 (1994).
[CrossRef]

1993

M. B. van der Mark, J. M. Schins, and A. Lagendijk, Opt. Commun. 98, 120 (1993).
[CrossRef]

1992

M. Haelterman, Opt. Lett. 17, 792 (1992).
[CrossRef]

W. J. Firth and C. Penman, Opt. Commun. 94, 183 (1992).
[CrossRef]

1985

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

1984

1981

W. J. Firth, Opt. Commun. 39, 343 (1981).
[CrossRef]

1979

K. Ikeda, Opt. Commun. 30, 257 (1979).
[CrossRef]

Agrawal, G. P.

Y. Yu, C. McKinstrie, and G. P. Agrawal, J. Opt. Soc. Am. B 14, 617 (1998).
[CrossRef]

Bar-Joseph, I.

Firth, W. J.

W. J. Firth and C. Penman, Opt. Commun. 94, 183 (1992).
[CrossRef]

W. J. Firth, Opt. Commun. 39, 343 (1981).
[CrossRef]

Haelterman, M.

Ikeda, K.

K. Ikeda, Opt. Commun. 30, 257 (1979).
[CrossRef]

Jaspert, D.

G. Steinmeyer, D. Jaspert, and F. Mitschke, Opt. Commun. 104, 379 (1994).
[CrossRef]

Lagendijk, A.

M. B. van der Mark, J. M. Schins, and A. Lagendijk, Opt. Commun. 98, 120 (1993).
[CrossRef]

McKinstrie, C.

Y. Yu, C. McKinstrie, and G. P. Agrawal, J. Opt. Soc. Am. B 14, 617 (1998).
[CrossRef]

McLaughlin, D. W.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

Mitschke, F.

G. Steinmeyer, D. Jaspert, and F. Mitschke, Opt. Commun. 104, 379 (1994).
[CrossRef]

Moloney, J. V.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

Newell, A. C.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

Penman, C.

W. J. Firth and C. Penman, Opt. Commun. 94, 183 (1992).
[CrossRef]

Schins, J. M.

M. B. van der Mark, J. M. Schins, and A. Lagendijk, Opt. Commun. 98, 120 (1993).
[CrossRef]

Silberberg, Y.

Steinmeyer, G.

G. Steinmeyer, D. Jaspert, and F. Mitschke, Opt. Commun. 104, 379 (1994).
[CrossRef]

van der Mark, M. B.

M. B. van der Mark, J. M. Schins, and A. Lagendijk, Opt. Commun. 98, 120 (1993).
[CrossRef]

Yu, Y.

Y. Yu, C. McKinstrie, and G. P. Agrawal, J. Opt. Soc. Am. B 14, 617 (1998).
[CrossRef]

J. Opt. Soc. Am. B

Y. Yu, C. McKinstrie, and G. P. Agrawal, J. Opt. Soc. Am. B 14, 617 (1998).
[CrossRef]

Y. Silberberg and I. Bar-Joseph, J. Opt. Soc. Am. B 1, 662 (1984).
[CrossRef]

Opt. Commun.

W. J. Firth, Opt. Commun. 39, 343 (1981).
[CrossRef]

K. Ikeda, Opt. Commun. 30, 257 (1979).
[CrossRef]

W. J. Firth and C. Penman, Opt. Commun. 94, 183 (1992).
[CrossRef]

M. B. van der Mark, J. M. Schins, and A. Lagendijk, Opt. Commun. 98, 120 (1993).
[CrossRef]

G. Steinmeyer, D. Jaspert, and F. Mitschke, Opt. Commun. 104, 379 (1994).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

D. W. McLaughlin, J. V. Moloney, and A. C. Newell, Phys. Rev. Lett. 54, 681 (1985).
[CrossRef] [PubMed]

Other

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981).

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, New York, 1993), Chap. 7.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 9.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

R. Vallée, Opt. Commun. 81, 419 (1991); Opt. Commun. 93, 389 (1992).
[CrossRef]

M. Nakazawa, K. Suzuki, and H. A. Haus, IEEE J. Quantum Electron. 25, 2036 (1989); M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1989).
[CrossRef]

M. Haelterman, S. Trillo, and S. Wabnitz, Opt. Commun. 91, 401 (1992); Opt. Lett. 17, 745 (1992).
[CrossRef]

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).

W. J. Firth and C. Paré, Opt. Lett. 13, 1096 (1989); W. J. Firth, C. Paré, and A. FitzGerald, J. Opt. Soc. Am. B 7, 1087 (1990).
[CrossRef]

C. T. Law and A. E. Kaplan, Opt. Lett. 14, 734 (1989); J. Opt. Soc. Am. B 8, 58 (1991).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Schematic illustration of a finite dispersive Kerr medium of length l in which two counterpropagating pump waves interact nonlinearly with each other. The front and back surface are labeled f and b, respectively.

Fig. 2
Fig. 2

Normalized growth rate Ωi/ of the absolute instability plotted as a function of normalized frequency Ωr for =1 ×10-4. (a) S=1 (equal pump powers) with L=9 (dashed curve) and L=12 (solid curve). (b) S=0.5 (unequal pump powers) with L=12 (dashed curve) and L=20 (solid curve).

Fig. 3
Fig. 3

Threshold condition for absolute instability to occur plotted in the ΩrL plane by use of =1×10-4 for (a) S=1 and (b) S=0.5.

Fig. 4
Fig. 4

Effects of weak boundary reflections on the absolute instability shown in Figs. 2 and 3 for =1×10-4, rf=rb=5 ×10-4, and S=1. (a) Ωi/ versus frequency Ωr for L=9, θf=θb=π/2 (dashed curve), and θf=θb=0 (solid curve). (b) Threshold curves in the ΩrL plane, for ϕrf=ϕrb=k0l=0 and ϕ20=ϕ10.

Equations (87)

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i A1z+iβ1 A1t=12 β2 2A1t2-γ(|A1|2+2|A2|2)A1,
-i A2z+iβ1 A2t=12 β2 2A2t2-γ(|A2|2+2|A1|2)A2,
E1(t, z)=Re{A1(t, z)exp(ik0z-iω0t)},
E2(t, z)=Re{A2(t, z)exp[ik0(l-z)-iω0t]},
A1s(t, z)=A10 exp[iγ(|A10|2+2|A20|2)z],
A2s(t, z)=A20 exp[iγ(|A20|2+2|A10|2)(l-z)],
δA1(t, z)=δA¯1(t, z)exp[iγ(|A10|2+ 2|A20|2)z],
δA2(t, z)=δA¯2(t, z)exp[iγ(|A20|2+2|A10|2)(l-z)].
(i/z+β1ω+β2ω2/2+γ|A10|2)δA1(ω, z)
+γ[A102δA1*(-ω, z)+2A10A20*δA2(ω, z)
+2A10A20δA2*(-ω, z)]=0,
(i/z-β1ω+β2ω2/2+γ|A10|2)δA1*(-ω, z)
+γ[A10*2δA1(ω, z)+2A10*A20δA2*(-ω, z)
+2A10*A20*δA2(ω, z)]=0,
(-i/z+β1ω+β2ω2/2+γ|A20|2)δA2(ω, z)
+γ[A202δA2*(-ω, z)+2A20A10*δA1(ω, z)
+2A20A10δA1*(-ω, z)]=0,
(-i/z-β1ω+β2ω2/2+γ|A20|2)δA2*(-ω, z)
+γ[A20*2δA2(ω, z)+2A20*A10δA1*(-ω, z)
+2A20*A10*δA1(ω, z)]=0,
δA1(ω, z)δA1*(-ω, z)δA2(ω, z)δA2*(-ω, z)=exp(ik1+)1r1+e1++e1+-c1+exp(ik1-z)r1-1e1-+e1--c2+exp[ik2+(l-z)]e2++e2+-1r2+c3+exp[ik2-(l-z)]e2-+e2--r2-1c4,
k1±(ω)β1ω±Y1(ω),
k2±(ω)β1ω±Y2(ω),
Y1(ω)=(β2ω2/2+γ|A10|2)2-(γ|A10|2)2,
Y2(ω)=(β2ω2/2+γ|A20|2)2-(γ|A20|2)2.
r1+(ω)(Y1-β2ω2/2-γ|A10|2)/(γA102),
r1-(ω)(Y1-β2ω2/2-γ|A10|2)/(γA10*2),
r2+(ω)(Y2-β2ω2/2-γ|A20|2)/(γA202),
r2-(ω)(Y2-β2ω2/2-γ|A20|2)/(γA20*2),
e1++(ω)-(Y1-β2ω2/2)A20/(β1ωA10),
e1+-(ω)(Y1-β2ω2/2)A20*/(β1ωA10),
e1-+(ω)-(Y1-β2ω2/2)A20/(β1ωA10*),
e1--(ω)(Y1-β2ω2/2)A20*/(β1ωA10*),
e2++(ω)-(Y2-β2ω2/2)A10/(β1ωA20),
e2+-(ω)(Y2-β2ω2/2)A10*/(β1ωA20),
e2-+(ω)-(Y2-β2ω2/2)A10/(β1ωA20*),
e2--(ω)(Y2-β2ω2/2)A10*(β1ωA20*).
cf+=c1+r1-c2,cb+=c3+r2-c4,
cf-*=r1+c1+c2,cb-*=r2+c3+c4.
δA1(ω, z)=exp(iβ1ωz)Mf(ω, z)cf+exp[iβ1ω(l-z)]Mbf(ω, l-z)cb,
δA2(ω, z)=exp[iβ1ω(l-z)]Mb(ω, l-z)cb+exp(iβ1ωz)Mfb(ω, z)cf,
Mf(ω, z)=11-r1+r1- 1r1-r1+1×exp(iY1z)00exp(-iY1z)×1-r1--r1+1,
Mfb(ω, z)=11-r2+r2- e1++e1-+e1+-e1--×exp(iY1z)00exp(-iY1z)×1-r1--r1+1.
Mf11(ω, z)=[exp(iY1z)-r1+r1-×exp(-iY1z)]/(1-r1+r1-),
Mf12(ω, z)=r1-[-exp(iY1z)+exp(-iY1z)]/(1-r1+r1-),
Mf21(ω, z)=r1+[exp(iY1z)-exp(-iY1z)]/(1-r1+r1-),
Mf22(ω, z)=(-r1+r1- exp(iY1z)+exp(-iY1z))/(1-r1+r1-),
Mfb11(ω, z)=[e1++ exp(iY1z)-e1-+r1+×exp(-iY1z)]/(1-r1+r1-),
Mfb12(ω, z)=[-e1++r1- exp(iY1z)+e1-+×exp(-iY1z)]/(1-r1+r1-),
Mfb21(ω, z)=[e1+- exp(iY1z)-e1--r1+×exp(-iY1z)]/(1-r1+r1-),
Mfb22(ω, z)=[-e1+-r1- exp(iY1z)+e1--×exp(-iY1z)]/(1-r1+r1-).
Mf11=1+iγ|A10|2z,
Mf12=iγA102z,
Mf21=-iγA10*2z,
Mf22=1-iγ|A10|2z,
Mfb11=-γA10*A20/(β1ω),
Mfb12=-γA10A20/(β1ω),
Mfb21=γA10*A20*/(β1ω),
Mfb22=γA10A20*/(β1ω).
δA1(ω, z)δA2(ω, z)=exp(iβ1ωz)Mf(ω, z)Mfb(ω, z)cf+exp[iβ1ω(l-z)]×Mbf(ω, l-z)Mb(ω, l-z)cb.
δA1(ω, 0)=RfδA2(ω, 0),
δA2(ω, l)=RbδA1(ω, l),
Rf=rf exp(iψrf)00rf exp(-iψrf),
[1-RfMfb(ω, 0)]cf=exp(iβ1ωl)[RfMb(ω, l)-Mbf(ω, l)]cb,
[1-RbMbf(ω, 0)]cb=exp(iβ1ωl)[RbMf(ω, l)-Mfb(ω, l)]cf.
D(ω)= |1-exp(i2β1ωl)[RfMb(ω, l)-Mbf(ω, l)]
×[1-RbMbf(ω, 0)]-1[RbMf(ω, l)
-Mfb(ω, l)]×[1-RfMfb(ω, 0)]-1| =0.
D(ω)= |1-exp(i2β1ωl)[RfMb(ω, l)-Mbf(ω, l)]×[RbMf(ω, l)-Mfb(ω, l)]| =0.
|1-exp(i2β1ωl)U| =1-exp(i2β1ωl)TrU+exp(i4β1ωl)|U|,
U=[RfMb(ω, l)-Mbf(ω, l)][RbMf(ω, l)-Mfb(ω, l)].
D(ω)=1-exp(i2β1ωl)g(ω)=0,
g(ω)=(Rf11Mb11-Mbf11)(Rb11Mf11-Mfb11)+(Rf11Mb12-Mbf12)(Rb11*Mf12*+Mfb12*)+(Rf11*Mb12*+Mbf12*)(Rb11Mf12-Mfb12)+(Rf11*Mb11*+Mbf11*)(Rb11*Mf11*+Mfb11*).
g(ω)={exp(-iY2l)[r1e2+e2+r1r2rf exp(iθf)-rf exp(-iθf)][r2e1+e1+r2r1rb exp(iθb)-rb exp(-iθb)]-exp(iY2l)[r1e2+e2+r1rf exp(iθf)-r2rf exp(-iθf)][r2e1+e1+r1rb exp(iθb)-r2rb exp(-iθb)]}×exp(-iY1l)/[(1-r12)(1-r22)],
r1(ω)=(Y1-β2ω2/2)/(γ|A10|2)-1,
r2(ω)=(Y2-β2ω2/2)/(γ|A20|2)-1,
e1(ω)=(r1+1)γ|A10A20|/(β1ω),
e2(ω)=(r2+1)γ|A10A20|/(β1ω),
θf=ϕrf+k0l+γ(|A20|2+2|A10|2)l+ϕ20-ϕ10,
θb=ϕrb+k0l+γ(|A10|2+2|A20|2)l+ϕ10-ϕ20,
g(ω)=-i γ2|A10|2|A20|2β22ω22β12Y1Y2 exp(-iY1l)sin(Y2l).
g(ω)=2g¯(Ω, L),
g¯(Ω, L)=-SΩ2 exp[L1-(Ω2/2-1)2] sin[L(Ω2/2-S)2-S2]21-(Ω2/2-1)2(Ω2/2-S)2-S2.
1-2 exp(i2ΩL/)g¯(Ω, L)=0.
Ωi= ln[2|g¯(Ωr, L)|]/(2L).
ωi=ln|g(ωr)|/(2lβ1).
L=ln(4-Ω2)-2 ln Ω(4-Ω2)1/2,

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