Abstract

Absolute instabilities of counterpropagating pump beams in a dispersive Kerr medium, placed inside a Fabry–Perot cavity, are analytically studied by use of the analysis and the results of part I [J. Opt. Soc. B 14, 607 (1998)]. Our approach allows characterization of such a complicated nonlinear system in terms of a doubly resonant optical parametric oscillator. We consider the growth of modulation-instability sidebands associated with each pump beam when weak probe signals are injected through one of the mirrors of the Fabry–Perot cavity. The results are used to obtain the threshold condition for the onset of the absolute instability and the growth rate for the unstable sidebands in the above-threshold regime. As expected, the well-known Ikeda instability is recovered at low modulation frequencies. The effects of the group-velocity dispersion are found to become quite important at high modulation frequencies. Although the absolute instability dominates in the anomalous-dispersion regime, it exists even in the normal-dispersion regime of the nonlinear medium. Below the instability threshold, our analysis provides analytic expressions for the probe transmittivity and the reflectivity of the phase-conjugated signal that is generated through a four-wave-mixing process.

© 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. M. Yu, C. J. McKinstrie, and G. P. Agrawal, J. Opt. Soc. Am. B 14, 607 (1998).
    [CrossRef]
  7. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, New York, 1993), Chap. 7.
  8. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 9.
  9. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  10. M. Yu, G. P. Agrawal, and C. J. McKinstrie, J. Opt. Soc. Am. B 12, 1126 (1995), and references therein.
    [CrossRef]
  11. R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992).
  12. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, Orlando, Fla., 1985).

1998

M. Yu, C. J. McKinstrie, and G. P. Agrawal, J. Opt. Soc. Am. B 14, 607 (1998).
[CrossRef]

1995

1992

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

1984

1981

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

Agrawal, G. P.

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]

McKinstrie, C. J.

Penman, C.

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

Silberberg, Y.

Yu, M.

J. Opt. Soc. Am. B

Opt. Commun.

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

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

Other

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

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, Orlando, Fla., 1985).

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]

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).

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Figures (3)

Fig. 1
Fig. 1

Gain spectra of modulation instability in the case of anomalous dispersion obtained by plotting ln|Geff(Ωr)| as a function of Ωr. (a) Pump–power ratio S=|A20|2/|A10|2=1, and ψrf+ψrb, ϕ20-ϕ10, and ψrf-ψrb are multiples of 2π. L=lγ|A10|2=1 and L=1.7 for the solid and dashed curves, respectively. (b) Same as (a) except that S=1/3, and the solid and the dashed curves are for L=1.7 and 2.6, respectively. (c) Same as (a) except that L=1, and the solid and the dashed curves are for ϕ20-ϕ10=π/4 and π/2, respectively. (d) Same as (a) except that L=1, and the solid and the dashed curves are for ψrf+ψrb=π/2 and π, respectively. In all cases, the three horizontal loss lines represent the mirror loss -ln(rfrb) for rfrb=4% (upper line), 30% (middle line), and 50% (lower line).  

Fig. 2
Fig. 2

Gain spectra of modulation instability in the case of normal dispersion. (a) Pump–power ratio S=|A20|2/|A10|2=1, the phases ψrf+ψrb and ψrf-ψrb are multiples of 2π, and ϕ20-ϕ10=π/2. The dashed and the solid curves are for L=1 and 2.6, respectively. (b) Same as (a) except that S=1/3, and the dashed and the solid curves are for L=2.6 and 5, respectively. (c) Same as (a) except that L=1, ψrf+ψrb=π/2, and the dashed and the solid curves are for ϕ20-ϕ10=0 and π/2 respectively. In all cases, the three horizontal loss lines are the same as in Fig. 1.

Fig. 3
Fig. 3

The transmissivity and reflectivity of an external probe plotted as a function of the pump–probe detuning. Only the upper and lower envelopes, associated with fast oscillations on the scale of mode spacing, are shown for a FP cavity with rfrb=4%. The solid and dashed curves are for L=1 and L=1.5, respectively. Other parameters are the same as in Fig. 1(a). (a) Probe transmissivity; (b) transmissivity of the four-wave-mixing frequency component; (c) probe reflectivity; and (d) reflectivity of the four-wave-mixing frequency component.

Equations (42)

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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)].
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,
=ωW/ωD=|β2|γP/β12,
δA1(ω, l)=exp(iβ1ωl)Mf(ω, l)δA1(ω, 0),
δA2(ω, 0)=exp(iβ1ωl)Mb(ω, l)δA2(ω, l).
Y1(ω)=(β2ω2/2+γ|A10|2)2-(γ|A10|2)2,
r1(ω)=(Y1-β2ω2/2)/(γ|A10|2)-1.
δA1(ω, 0)=TfδAi(ω)+RfδA2(ω, 0),
δA2(ω, l)=RbδA1(ω, l),
Rj=rj exp(iψrj)00rj exp(-iψrj)(j=f, b),
Tf=tf exp(iϕtf)00tf exp(-iϕtf).
δA1(ω, 0)=[1-exp(2iβ1ωl)RfMbRbMf]-1TfδAi(ω),
[1-exp(2iβ1ωl)RfMbRbMf]δA1(ω, 0)=0
D(ω)=|1-exp(2iβ1ωl)RfMbRbMf|=0.
δAt(ω)=TbδA1(ω, l),δAr(ω)=TfδA2(ω, 0),
Tj=tj exp(iϕtj)00tj exp(-iϕtj),(j=f, b),
δAt(ω)TδAi(ω),=exp(iβ1ωl)TbMf×(1-exp(i2β1ωl)RfMbRbMf)-1TfδAi(ω),
δAr(ω)RδAi(ω),=exp(i2β1ωl)TfMbRbMf(1-exp(i2β1ωl)×RfMbRbFMf)-1TfδAi(ω),
D(ω)=1-rfrb exp(i2β1ωl)G(ω)+[rfrb exp(i2β1ωl)]2=0,
G(ω)=Mf11Mb11 exp[i(ψrf+ψrb)]+Mf21Mb12 exp[i(ψrf-ψrb)]+c.c.
rfrbG±eff(ω)exp(2iβ1ωl)=1,
G±eff(ω)=(G±G2-4)/2.
|G±eff(ω)|>1/(rfrb).
G=(exp[i(2Y1l+ψrf+ψrb)]×{1-r12exp[-i2(ψrf+ψrb)]}+exp[-i(2Y1l+ψrf+ψrb)]×{1-r12 exp[i2(ψrf+ψrb)]})/(1-r12)+2[cos(θf-θb)-cos(ψrf+ψrb)]×(γ|A10|2l)2 sinc2(Y1l),
G±eff=exp(i2Y1l).
exp{iL[2Ω/s(Ω2/2+1)2-1]}=1/(rfrb),
Ωi=[ln|exp[±iLs(Ωr2/2+1)2-1]|+ln(rfrb)]/(2L).
ωi=[ln|G+eff(ωr)|+ln(rfrb)]/(2lβ1).
G+effG={exp(-iY2l)[r1r2 exp(iθf)-exp(-iθf)][r1r2 exp(iθb)-exp(-iθb)]-exp(iY2l)[r1 exp(iθf)-r2 exp(-iθf)]×[r1 exp(iθb)-r2 exp(-iθb)]}×exp(-iY1l)/[(1-r12)(1-r22)],
T11(ω)=exp[i(β1ωl+ϕtb+ϕtf)]tbtf{Mf11-rfrb×exp[i(2β1ωl-ψrf-ψrb)]Mb22}/D(ω),
T12(ω)=exp[i(β1ωl+ϕtb-ϕtf)]tbtf{Mf12+rfrb×exp[i(2β1ωl+ψrf-ψrb)]Mb12}/D(ω),
R11(ω)=exp[i(2β1ωl+ϕtf+ϕtf)]tftfrb×{exp(-iψrb)Mf21Mb12+exp(iψrb)Mf11Mb11-exp[i(2β1ωl-ψrf)]rfrb}/D(ω),
R12(ω)=exp[i(2β1ωl+ϕtf-ϕtf)]tftfrb×[exp(-iψrb)Mf22Mb12+exp(iψrb)Mf12Mb11]/D(ω),
T11(ω)=exp[i(β1ωl+ϕtb+ϕtf)]tbtfMf11/×[1-rfrbexp(i2β1ωl)G],
T12(ω)=exp[i(β1ωl+ϕtb-ϕtf)]tbtfMf12/×[1-rfrbexp(i2β1ωl)G],
R11(ω)=exp[i(2β1ωl+ϕtf+ϕtf)]tftfrb×[exp(-iψrb)Mf21Mb12+exp(iψrb)Mf11Mb11]/[1-rfrb×exp(i2β1ωl)G],
R12(ω)=exp[i(2β1ωl+ϕtf-ϕtf)]tftfrb×[exp(-iωrb)Mf22Mb12+exp(iψrb)Mf12Mb11]/[1-rfrb×exp(i2β1ωl)G].
A10=A1itf exp(iϕtf)+A20 exp(ilΔ2)rf exp(iϕrf),
A20=A2itb exp(iϕtb)+A10 exp(ilΔ1)rb exp(iϕrb).

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