Abstract

A thorough search is performed for all cases in which solutions of the quadratically coupled nonlinear equations for χ(2) cascading may be expressed in terms of hyperbolic functions. This reveals two new classes (IV and VI) in addition to classes I–III, which have appeared in many recent papers. These new solutions describe pulses with a nonzero cw background in one or both modes. They include bright–dark, brighter–brighter, twin-hole, and dark–dark solutions, and each contains an adjustable parameter related to the phase shift across the pulse. For a planar waveguide, any of the class VI profiles may arise for any choice of pulse orientation parameters κJ giving pp˜>0, whereas any of the class IV profiles may arise if pp˜<0. In each case, the choice then determines the remaining pulse parameters in the exact representation.

© 1998 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fibre,” Phys. Rev. A 38, 2011–2017 (1988).
    [CrossRef] [PubMed]
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    [CrossRef]
  4. D. F. Parker and G. K. Newboult, “Coupled nonlinear Schrödinger equations arising in fibre optics,” J. Phys. Colloq. 50, 137–146 (1989).
    [CrossRef]
  5. L. Wang and C. C. Yang, “Existence of coupled solitary waves in optical fibers owing to mutual support between bright and dark pulses,” Opt. Lett. 15, 474–476 (1990).
    [CrossRef] [PubMed]
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  7. Yu. S. Kivshar, A. V. Buryak, and D. F. Parker, “Coupling between bright and dark solitons,” Phys. Lett. A 215, 57–62 (1996).
    [CrossRef]
  8. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
    [CrossRef] [PubMed]
  9. K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. M. J. Werner and P. D. Drummond, “Strongly coupled nonlinear parametric solitary waves,” Opt. Lett. 19, 613–615 (1994).
    [CrossRef] [PubMed]
  12. A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).
    [CrossRef]
  13. C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2): χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
    [CrossRef]
  14. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal-processing, mode-locking, pulse-compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
    [CrossRef]
  15. A. V. Buryak, “Solitons due to quadratic nonlinearities,” Ph.D. thesis (Australian National University, Canberra, 1996).
  16. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Optical solitary waves induced by cross-phase modulation,” Opt. Lett. 13, 871–873 (1988).
    [CrossRef] [PubMed]
  17. V. V. Afanasjev, Yu. S. Kivshar, V. V. Konotop, and V. N. Serkin, “Dynamics of coupled dark and bright solitons,” Opt. Lett. 14, 805–807 (1989).
    [CrossRef]
  18. P. Ferro and S. Trillo, “Periodical waves, domain walls, and modulational instability in dispersive quadratic media,” Phys. Rev. E 51, 4944–4951 (1995).
    [CrossRef]
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    [CrossRef] [PubMed]
  20. H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity,” in Proceedings of the Twentieth Australian Conference on Optical Fibre Technology (IREE Society, Milsons Point, Australia, 1995), pp. 259–262.
  21. Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339–342 (1974).
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    [CrossRef] [PubMed]
  23. A. V. Porubov and D. F. Parker, “Some general periodic solutions to coupled nonlinear Schrödinger equations,” submitted to Wave Motion.

1996 (3)

Yu. S. Kivshar, A. V. Buryak, and D. F. Parker, “Coupling between bright and dark solitons,” Phys. Lett. A 215, 57–62 (1996).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal-processing, mode-locking, pulse-compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Yu. S. Kivshar and V. V. Afanasjev, “Drift instability of dark solitons in saturable media,” Opt. Lett. 21, 1135–1137 (1996).
[CrossRef] [PubMed]

1995 (3)

P. Ferro and S. Trillo, “Periodical waves, domain walls, and modulational instability in dispersive quadratic media,” Phys. Rev. E 51, 4944–4951 (1995).
[CrossRef]

A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).
[CrossRef]

A. V. Buryak and Yu. S. Kivshar, “Dark solitons in dispersive quadratic media,” Opt. Lett. 20, 834–836 (1995).
[CrossRef] [PubMed]

1994 (2)

1993 (2)

K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
[CrossRef] [PubMed]

M. J. Werner and P. D. Drummond, “Simulton solutions for the parametric amplifier,” J. Opt. Soc. Am. B 10, 2390–2393 (1993).
[CrossRef]

1992 (2)

1990 (1)

1989 (3)

V. V. Afanasjev, Yu. S. Kivshar, V. V. Konotop, and V. N. Serkin, “Dynamics of coupled dark and bright solitons,” Opt. Lett. 14, 805–807 (1989).
[CrossRef]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. QE-25, 2674–2682 (1989).
[CrossRef]

D. F. Parker and G. K. Newboult, “Coupled nonlinear Schrödinger equations arising in fibre optics,” J. Phys. Colloq. 50, 137–146 (1989).
[CrossRef]

1988 (3)

1974 (1)

Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339–342 (1974).

Afanasjev, V. V.

Buryak, A. V.

Yu. S. Kivshar, A. V. Buryak, and D. F. Parker, “Coupling between bright and dark solitons,” Phys. Lett. A 215, 57–62 (1996).
[CrossRef]

A. V. Buryak and Yu. S. Kivshar, “Dark solitons in dispersive quadratic media,” Opt. Lett. 20, 834–836 (1995).
[CrossRef] [PubMed]

A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).
[CrossRef]

Christodoulides, D. N.

DeSalvo, R.

Drummond, P. D.

Ferro, P.

P. Ferro and S. Trillo, “Periodical waves, domain walls, and modulational instability in dispersive quadratic media,” Phys. Rev. E 51, 4944–4951 (1995).
[CrossRef]

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal-processing, mode-locking, pulse-compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
[CrossRef] [PubMed]

Hayata, K.

K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
[CrossRef] [PubMed]

Joseph, R. I.

Karamzin, Yu. N.

Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339–342 (1974).

Kivshar, Yu. S.

Konotop, V. V.

Koshiba, M.

K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
[CrossRef] [PubMed]

Menyuk, C. R.

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2): χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
[CrossRef]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. QE-25, 2674–2682 (1989).
[CrossRef]

Newboult, G. K.

D. F. Parker and G. K. Newboult, “Coupled nonlinear Schrödinger equations arising in fibre optics,” J. Phys. Colloq. 50, 137–146 (1989).
[CrossRef]

Parker, D. F.

Yu. S. Kivshar, A. V. Buryak, and D. F. Parker, “Coupling between bright and dark solitons,” Phys. Lett. A 215, 57–62 (1996).
[CrossRef]

D. F. Parker and G. K. Newboult, “Coupled nonlinear Schrödinger equations arising in fibre optics,” J. Phys. Colloq. 50, 137–146 (1989).
[CrossRef]

Schiek, R.

Serkin, V. N.

Sheik-Bahae, M.

Sipe, J. E.

M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fibre,” Phys. Rev. A 38, 2011–2017 (1988).
[CrossRef] [PubMed]

Stegeman, G. I.

Sukhorukov, A. P.

Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339–342 (1974).

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal-processing, mode-locking, pulse-compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2): χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
[CrossRef]

Tratnik, M. V.

M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fibre,” Phys. Rev. A 38, 2011–2017 (1988).
[CrossRef] [PubMed]

Trillo, S.

P. Ferro and S. Trillo, “Periodical waves, domain walls, and modulational instability in dispersive quadratic media,” Phys. Rev. E 51, 4944–4951 (1995).
[CrossRef]

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Optical solitary waves induced by cross-phase modulation,” Opt. Lett. 13, 871–873 (1988).
[CrossRef] [PubMed]

Vanherzeele, H.

Wabnitz, S.

Wang, L.

Werner, M. J.

Wright, E. M.

Yang, C. C.

IEEE J. Quantum Electron. (1)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. QE-25, 2674–2682 (1989).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. Colloq. (1)

D. F. Parker and G. K. Newboult, “Coupled nonlinear Schrödinger equations arising in fibre optics,” J. Phys. Colloq. 50, 137–146 (1989).
[CrossRef]

JETP Lett. (1)

Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339–342 (1974).

Opt. Lett. (9)

Opt. Quantum Electron. (1)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal-processing, mode-locking, pulse-compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996).
[CrossRef]

Phys. Lett. A (2)

A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).
[CrossRef]

Yu. S. Kivshar, A. V. Buryak, and D. F. Parker, “Coupling between bright and dark solitons,” Phys. Lett. A 215, 57–62 (1996).
[CrossRef]

Phys. Rev. A (1)

M. V. Tratnik and J. E. Sipe, “Bound solitary waves in a birefringent optical fibre,” Phys. Rev. A 38, 2011–2017 (1988).
[CrossRef] [PubMed]

Phys. Rev. E (1)

P. Ferro and S. Trillo, “Periodical waves, domain walls, and modulational instability in dispersive quadratic media,” Phys. Rev. E 51, 4944–4951 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993).
[CrossRef] [PubMed]

Other (3)

A. V. Buryak, “Solitons due to quadratic nonlinearities,” Ph.D. thesis (Australian National University, Canberra, 1996).

H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity,” in Proceedings of the Twentieth Australian Conference on Optical Fibre Technology (IREE Society, Milsons Point, Australia, 1995), pp. 259–262.

A. V. Porubov and D. F. Parker, “Some general periodic solutions to coupled nonlinear Schrödinger equations,” submitted to Wave Motion.

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Figures (2)

Fig. 1
Fig. 1

Profiles of |u|, Re u, Im u, |v|, Re v, and Im v for class IV with α=0, and for various values of μ and for the stated values of the parameters A6r2-pp˜, B6r2p: (a) =-5π/12 (A=0.25, B=0.1), (b) μ=-sin-1[5+1)/23] (A=0.5, B=0.15), (c) μ=-π/4 (A=0.5, B=0.25), (d) μ=-π/6 (A=0.5, B=0.25), (e) μ=-π/12 (A=1.0, B=0.5), (f) μ=0 (A=1.0, B=0.5), (g) μ=sin-1 [(5-1)/23] (A=0.5, B=0.5), (h) μ=π/3(A=0.25,B=0.15).

Fig. 2
Fig. 2

Profiles as in Fig. 1, but for the dark–dark and brighter–brighter profiles of class VI with α=0, with various values of the parameter λY/X and with A6r2pp˜, B6r2p as specified: (a) λ=8 (A=1.0, B=0.25), (b) λ=3 (A=1.0, B=0.5), (c) λ=0.9 (A=1.0, B=0.5), (d) λ=0.75 (A=0.5, B=0.5), (e) λ=0.25 (A=0.5, B=0.25), (f) λ=-0.5 (A=1.0, B=0.5), (g) λ=-0.6 (A=0.5, B=0.7), (h) λ=-0.125 (A=0.25, B=0.25).

Equations (74)

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iM AZ+PJL 2AyJyL=J*2A*B,
iM˜ BZ+iγ˜J ByJ+Δ˜B+P˜JL 2ByJyL=JA2,
2A=exp(iθ)J-1u(ζ),2B=exp(2iθ)J-1v(ζ),
ζκJyJ-M-1Z,θβJyJ-M-1σZ.
pu(ζ)+iΠu(ζ)+Σu(ζ)=u*(ζ)v(ζ),
p˜v(ζ)+iΠ˜v(ζ)+Σ˜v(ζ)=u2(ζ),
pPJLκJκL,p˜P˜JLκJκL,
Π1+2PJLβJκL,
Π˜M˜M-1+4P˜JLβJκL+γ˜JκJ,
Σ-σ-PJLβJβL,
Σ˜Δ˜-2γ˜JβJ-2σM˜M-1-4P˜JLβJβL.
u=aS2+bST+cS+dT+e,
v=a˜S2+b˜ST+c˜S+d˜T+e˜,
S=-rST,T=rS2,S2+T2=1,
S=r2S(1-2S2),T=-2r2S2T,
S4:-6r2pa=a*a˜-b*b˜,
S3T:-6r2pb=a*b˜+b*a˜,
S3:-2r2pc+i2Πrb=a*c˜+c*a˜-b*d˜-d*b˜,
S2T:-2r2pd-i2Πra=a*d˜+d*a˜+b*c˜+c*b˜,
S2:4r2pa+iΠrd+Σa=a*e˜+e*a˜+b*b˜+c*c˜-d*d˜,
ST:r2pb-iΠrc+Σb=b*e˜+e*b˜+c*d˜+d*c˜,
S:r2pc-iΠrb+Σc=c*e˜+e*c˜+b*d˜+d*b˜,
T:Σd=d*e˜+e*d˜,
1:Σe=d*d˜+e*e˜,
6r2pU±+U˜±U*=0,6r2p˜U˜±+U±2=0.
2r2pX±+U*X˜±+U˜±X*=±2rΠU±,
r2p˜X˜±+U±X±=±rΠ˜U˜±.
X±=ρ±X exp(iα),
X˜±=±(ρ±)2(3r2p˜)-1Y exp(2iα),
a=12(ρ++ρ-)exp(iα),
b=12i(ρ--ρ+)exp(iα),
c=12(ρ--ρ+)X exp(iα),
d=12i(ρ++ρ-)X exp(iα),
a˜=-12q˜-1(ρ+2+ρ-2)exp(2iα),
b˜=12iq˜-1(ρ+2-ρ-2)exp(2iα),
c˜=q˜-1(ρ+2-ρ-2)Y exp(2iα),
d˜=-iq˜-1(ρ+2+ρ-2)Y exp(2iα),
2PJLβJκL+1=0=4P˜JLβJκL+γ˜JκJ+M˜/M,
4r2PJLκJκL+PJLβJβL
=-σ=(2r2P˜JLκJκL+2P˜JLβJβL+γ˜JβJ-12Δ˜)M/M˜.
u=6r2-pp˜ exp(iα)(X+i tanh rζ)sech rζ,
v=6r2p exp(2iα)[16F-(μ)+sech2 rζ+2iY tanh rζ],forpp˜<0.
Π=2tan μ-23 sec μrp,
Π˜=2(2 tan μ+3 sec μ)rp˜,
Σr2p=F+(μ)-1,Σ˜=0,
F±(μ)=X(9Y-2X)±(2+3Y/X).
2PJLβJκL+1
=2tan μ-23 sec μrPJLκJκL,
4P˜JLβJκL+γ˜JκJ+M˜/M
=2(2 tan μ+3 sec μ)rP˜JLκJκL,
[F+(μ)-1]r2PJLκJκL+PJLβJβL
=-σ=(2P˜JLβJβL+γ˜JβJ-12Δ˜)M/M˜,
2 tan-1[12Y/F-(μ)].
u=6r2-pp˜ exp(iα)(1/3+i tanh rζ)sech rζ,
v=6r2p exp(2iα)(sech2 rζ-4/9),
u=exp(iα)pΣ˜Π2p+ir tanh rζ,
r2=Σ2p-Π2p2,
d=12i(ρ++ρ-)X exp(iα)=i(ΣΣ˜)1/2 exp(iα)cos δ,withΓ=α+12π
d˜=-iq˜-1(ρ+2+ρ-2)Y exp(2iα)=-iΣ exp(2iα)sin 2δ.
18(1+L)(1+Q2)X2=LGG˜,
6(1+Q2)Y=QLG,
ρ=6r22pp˜L.
u=6r2pp˜ exp(iα)(sech2 rζ+iX tanh rζ-QX),
v=-6r2p exp(2iα)[sech2 rζ+2iY tanh rζ+(Q-1-Q)Y],forpp˜>0,
G=-4-4X2+6[QX+(Q-Q-1)Y+3XY],
G˜=-4+12XY-4Y2+12QX-6X2.
G=6(Q+Q-1)Y,G˜Y=6QX2,
3QX-6Q-1Y+9XY=2+2X2,
3QX2=6XY(Q+Y)-2Y3-2Y-3X2Y.
(2λ-1)3QX=2λ+λ(2λ2-6λ+3)X2,
3(QX)2+(9λ-2)X2(QX)-2QX=6λX2,
Π=(6Y-4X)rp=2X(3λ-2)rp,
Π˜=(6X-2Y)rp˜=2X(3-λ)rp˜,
=6λ(QX+X2)/QX)r2p,˜=6λ(QX)r2p˜.

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