Abstract

Using the quasi-particle approach, we studied the problem of the reflection of quadratic spatial solitons from an interface between two χ(2) media with slightly different linear and nonlinear properties. The possibility of soliton capture by an interface associated with nonlinear surface wave excitation is shown. The calculations are carried out for the well-known single as well as a novel type of multihump soliton, for which we obtain a new analytical expression in the nonlocal limit for the first time to our knowledge.

© 2002 Optical Society of America

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References

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  1. A. D. Boardman, K. Xie, and A. Sangarpaul, “Stability of scalar spatial solitons in cascadable nonlinear media,” Phys. Rev. A 52, 4099–4106 (1995).
    [CrossRef] [PubMed]
  2. A. A. Sukhorukov, “Approximate solutions and scaling transformations for quadratic solitons,” Phys. Rev. E 61, 4530–4539 (2000).
    [CrossRef]
  3. I. A. Kol’chugina, V. A. Mironov, and A. M. Sergeev, “On the structure of stationary solitons in systems with nonlocal nonlinearity,” Pis’ma Zh. Eksp. Teor. Fiz. 31, 333–336 (1980) (in Russian).
  4. A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces,” Phys. Rev. A 39, 1809–1827 (1989).
    [CrossRef] [PubMed]
  5. A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, “Theoretical demonstration of beam scanning and switching using spatial solitons in a photorefractive crystal,” J. Mod. Opt. 47, 1941–1957 (2000).
    [CrossRef]
  6. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).
  7. A. D. Capobianco, C. De Angelis, A. Laureti Palma, and G. F. Nalesso, “Beam dynamics at the interface between second-order nonlinear dielectrics,” J. Opt. Soc. Am. B 14, 1956–1960 (1997).
    [CrossRef]
  8. The possibility of the capture of the solitons by the interface between two second-order nonlinear media is more or less obvious, at least in the cascading limit, since this situation corresponds to an effective cubic medium, for which nonlinear surface waves are well known (see, e.g., Ref. 4).
  9. A. D. Boardman, University of Salford, Joule Physics Laboratory, School of Science, The Crescent, Salford M54WT, UK, and W. Ilecki, University of Salford, Physics Department, Maxwell Building, Salford M54WT, UK (personal communication, March, 2001).

2000 (2)

A. A. Sukhorukov, “Approximate solutions and scaling transformations for quadratic solitons,” Phys. Rev. E 61, 4530–4539 (2000).
[CrossRef]

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, “Theoretical demonstration of beam scanning and switching using spatial solitons in a photorefractive crystal,” J. Mod. Opt. 47, 1941–1957 (2000).
[CrossRef]

1997 (1)

1995 (1)

A. D. Boardman, K. Xie, and A. Sangarpaul, “Stability of scalar spatial solitons in cascadable nonlinear media,” Phys. Rev. A 52, 4099–4106 (1995).
[CrossRef] [PubMed]

1989 (1)

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces,” Phys. Rev. A 39, 1809–1827 (1989).
[CrossRef] [PubMed]

1980 (1)

I. A. Kol’chugina, V. A. Mironov, and A. M. Sergeev, “On the structure of stationary solitons in systems with nonlocal nonlinearity,” Pis’ma Zh. Eksp. Teor. Fiz. 31, 333–336 (1980) (in Russian).

Aceves, A. B.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces,” Phys. Rev. A 39, 1809–1827 (1989).
[CrossRef] [PubMed]

Boardman, A. D.

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, “Theoretical demonstration of beam scanning and switching using spatial solitons in a photorefractive crystal,” J. Mod. Opt. 47, 1941–1957 (2000).
[CrossRef]

A. D. Boardman, K. Xie, and A. Sangarpaul, “Stability of scalar spatial solitons in cascadable nonlinear media,” Phys. Rev. A 52, 4099–4106 (1995).
[CrossRef] [PubMed]

Bontemps, P.

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, “Theoretical demonstration of beam scanning and switching using spatial solitons in a photorefractive crystal,” J. Mod. Opt. 47, 1941–1957 (2000).
[CrossRef]

Capobianco, A. D.

De Angelis, C.

Ilecki, W.

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, “Theoretical demonstration of beam scanning and switching using spatial solitons in a photorefractive crystal,” J. Mod. Opt. 47, 1941–1957 (2000).
[CrossRef]

Kol’chugina, I. A.

I. A. Kol’chugina, V. A. Mironov, and A. M. Sergeev, “On the structure of stationary solitons in systems with nonlocal nonlinearity,” Pis’ma Zh. Eksp. Teor. Fiz. 31, 333–336 (1980) (in Russian).

Mironov, V. A.

I. A. Kol’chugina, V. A. Mironov, and A. M. Sergeev, “On the structure of stationary solitons in systems with nonlocal nonlinearity,” Pis’ma Zh. Eksp. Teor. Fiz. 31, 333–336 (1980) (in Russian).

Moloney, J. V.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces,” Phys. Rev. A 39, 1809–1827 (1989).
[CrossRef] [PubMed]

Nalesso, G. F.

Newell, A. C.

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces,” Phys. Rev. A 39, 1809–1827 (1989).
[CrossRef] [PubMed]

Palma, A. Laureti

Sangarpaul, A.

A. D. Boardman, K. Xie, and A. Sangarpaul, “Stability of scalar spatial solitons in cascadable nonlinear media,” Phys. Rev. A 52, 4099–4106 (1995).
[CrossRef] [PubMed]

Sergeev, A. M.

I. A. Kol’chugina, V. A. Mironov, and A. M. Sergeev, “On the structure of stationary solitons in systems with nonlocal nonlinearity,” Pis’ma Zh. Eksp. Teor. Fiz. 31, 333–336 (1980) (in Russian).

Sukhorukov, A. A.

A. A. Sukhorukov, “Approximate solutions and scaling transformations for quadratic solitons,” Phys. Rev. E 61, 4530–4539 (2000).
[CrossRef]

Xie, K.

A. D. Boardman, K. Xie, and A. Sangarpaul, “Stability of scalar spatial solitons in cascadable nonlinear media,” Phys. Rev. A 52, 4099–4106 (1995).
[CrossRef] [PubMed]

Zharov, A. A.

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, “Theoretical demonstration of beam scanning and switching using spatial solitons in a photorefractive crystal,” J. Mod. Opt. 47, 1941–1957 (2000).
[CrossRef]

J. Mod. Opt. (1)

A. D. Boardman, P. Bontemps, W. Ilecki, and A. A. Zharov, “Theoretical demonstration of beam scanning and switching using spatial solitons in a photorefractive crystal,” J. Mod. Opt. 47, 1941–1957 (2000).
[CrossRef]

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

Phys. Rev. A (2)

A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces,” Phys. Rev. A 39, 1809–1827 (1989).
[CrossRef] [PubMed]

A. D. Boardman, K. Xie, and A. Sangarpaul, “Stability of scalar spatial solitons in cascadable nonlinear media,” Phys. Rev. A 52, 4099–4106 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (1)

A. A. Sukhorukov, “Approximate solutions and scaling transformations for quadratic solitons,” Phys. Rev. E 61, 4530–4539 (2000).
[CrossRef]

Pis’ma Zh. Eksp. Teor. Fiz. (1)

I. A. Kol’chugina, V. A. Mironov, and A. M. Sergeev, “On the structure of stationary solitons in systems with nonlocal nonlinearity,” Pis’ma Zh. Eksp. Teor. Fiz. 31, 333–336 (1980) (in Russian).

Other (3)

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

The possibility of the capture of the solitons by the interface between two second-order nonlinear media is more or less obvious, at least in the cascading limit, since this situation corresponds to an effective cubic medium, for which nonlinear surface waves are well known (see, e.g., Ref. 4).

A. D. Boardman, University of Salford, Joule Physics Laboratory, School of Science, The Crescent, Salford M54WT, UK, and W. Ilecki, University of Salford, Physics Department, Maxwell Building, Salford M54WT, UK (personal communication, March, 2001).

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Figures (7)

Fig. 1
Fig. 1

Spectrum of eigenvalues of normalized energy flux P versus ν.

Fig. 2
Fig. 2

Transverse structures of the family of multihump spatial solitons in the nonlocal limit (ν=80). (1) FF field; (2) SH field. Solid curves, analytical solutions [Eqs. (12), (14)]; dashed curves, corresponding numerical solutions. (a) n=1, (b) n=2, (c) n=3.

Fig. 3
Fig. 3

Effective potential versus xs in the cascading limit for various values of μL and μNL: β=1.3, ν=0.3. (a) μL=-0.004, μNL=0.1; (b) μL=0.004, μNL=0.1; (c) μL=0.004, μNL=-0.1.

Fig. 4
Fig. 4

Effective potential versus xs for triple-peak solitons in the nonlocal limit for various values of μL and μNL: β=1.3, ν=80. (a) μL=-0.04, μNL=-0.2; (b) μL=-0.004, μNL=-0.2; (c) μL=0, μNL=-0.2.

Fig. 5
Fig. 5

Trajectories of triple-peak spatial solitons with initial effective kinetic energies (1/2)M(dxs/dz)2 close to a separatrix associated with the saddle point at the maximum effective potential indicated by the arrow in Fig. 4(b). Trajectories 1 and 2 correspond to an initial kinetic energy somewhat below the maximum potential, 3 to a kinetic energy close (slightly above) to the maximum, and 4 to a kinetic energy higher than the plateau of the effective potential.

Fig. 6
Fig. 6

Regions of parameters (shaded) where soliton capture by the interface can take place (Δ2ω/Δω=1).

Fig. 7
Fig. 7

Numerical simulations. A soliton in the cascading limit is launched to a nonlinear interface at various angles. The soliton and interface parameters correspond to the potential illustrated in Fig. 3(b). (a) transmission, (b) capture, (c) reflection.

Equations (52)

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ΔEω+k02ωEω+k02κEω*E2ω=0,
ΔE2ω+4k022ωE2ω+2k02κEω2=0,
Eω,2ω=Ψ1,2(x)exp(-ih1,2z),
d2Ψ1dx2-(h12-k02ω)Ψ1+k02κΨ1Ψ2=0,
d2Ψ2dx2-4(h12-k022ω)Ψ2+2k02κΨ12=0.
U(x)=Ψ1(x)(h12-k02ω)κk022,
V(x)=Ψ2(x)(h12-k02ω)κk02,
x=x(h12-k02ω)1/2
d2Udx2-U+UV=0,
d2Vdx2-δ2V+U2=0,
VU/δ2,
d2Udx2-U+U3/δ2=0.
U=2δ sec h(x),V=2 sec h2(x).
U=V=(3/2)sec h2(x/2).
V=12δ-U2(x)exp(-δ|x-x|)dx.
d2Udx2+12δ-U2(x)e(-δ|x-x|)dx-1U=0.
-U2(x)e(-δ|x-x|)dx
U2(x)-e(-δ|x-x|)dx=2U2(x)/δ.
V(x)=12δ-U2(x)exp(-δ|x-x|)dx12δ exp(-δ|x|)-U2(x)dx=12δ exp(-δ|x|)P0,
U+P02δ exp(-δ|x|)-1U=0.
U(ζ)=AJν(Pν exp[-|ζ|]),
dJν(s)dss=Pν=0.
ω,2ω(x)=ω,2ω+Δω,2ωI(x)
κ(x)=κ0+ΔκI(x),
I(x)=1,x>00,x<0
Δκκ1,Δωω1,Δ2ω2ω1.
ΔEω+k02ωEω+k02κEω*E2ω
=-(k02ΔωEω+k02ΔκEω*E2ω)I(x)=F1,
ΔE2ω+4k022ωE2ω+2k02κEω2
=-(4k02Δ2ωE2ω+2k02ΔκEω2)I(x)=F2,
μ=maxΔκκ,Δωω,Δ2ω2ω
Eω,2ω={Ψ1,2[x-xs(z)]+ζ1,2[x-xs(z)]}×exp-ih1,2z-ih1,2dxsdz[x-xs(z)],
d2ζ1dx2-(h12-k02ω)ζ1+κk02Ψ1ζ2+κk02Ψ2ζ1*=S1,
d2ζ2dx2-4(h12-k022ω)ζ2+2κk02Ψ1ζ1=S2,
S1,2=h1,22Ψ1,2-dΨ1,2dx2dxsdz2+d2xsdz2dΨ1,2dx+2h1,22Ψ1,2+F1,2.
η1=2Ψ1(x),η2=Ψ2(x),
η3=2iΨ1(x),η4=iΨ2(x).
-2dΨ1dx Re(S1)+dΨ2dx Re(S2)dx=0,
-[2Ψ1 Im(S1)+Ψ2 Im(S2)]dx=0.
Md2xsdz2=F(xs),
M=-U24+V2-β2-14β2-δ2dUdx2+dVdx2dx
F=14β2-δ2[-μNL(β2-1)ξκ(xs)+μL(4-δ2)ξ(xs)].
μL=Δω/ω,μNL=Δκ/κ0,β2=2ω/ω,
ξ(xs)=U2(-xs)+8(Δ2ω/Δω)V2(-xs),
ξκ(xs)=V(-xs)U2(-xs)+3-xsV(x)U2(x)dx.
F=-Weffxs.
U(x)=Um sec hp(x/p),V(x)=Vm sec h2(x/p),
Um2=2δ2Vm2Vm-1,p=1Vm-1,
δ2=4(Vm-1)32-Vm.
ξ=Um2 sec h2pxsp+8Δ2ωΔωVm2 sec h4xsp,
ξκ=Um2Vmp-2p+1sec h2p+2xsp.
Θ0=(4-δ2)p+1p-2Um2+8(Δ2ω/Δω)Vm2Um2Vm,

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