Abstract

A general method for analyzing a multilayer optical waveguide with nonlinear cladding and a nonlinear substrate is presented. This method can also be used to analyze a waveguide with a multiple-quantum-well structure and a multibranch waveguide with nonlinear cladding and a nonlinear substrate. Numerical simulation results show that the analysis is correct.

© 2002 Optical Society of America

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References

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  1. M. Haruna and J. Koyama, “Electrooptic branching waveguide switches and their application to 1×4 optical switching networks,” J. Lightwave Technol. 1, 223–227 (1983).
    [CrossRef]
  2. H. Sasaki and I. Anderson, “Theoretical and experimental studies on active Y junctions in optical waveguides,” IEEE J. Quantum Electron. 14, 883–892 (1978).
    [CrossRef]
  3. W. E. Martin, “A new waveguide switch/modulator for integrated optics,” Appl. Phys. Lett. 26, 562–564 (1975).
    [CrossRef]
  4. T. J. Cullen and C. D. Wilkinson, “Radiation losses from single-mode optical Y junctions formed by silver-ion exchange in glass,” Opt. Lett. 10, 134–136 (1984).
    [CrossRef]
  5. Y. Murakami and M. Ikeda, “Single-mode optical Y-branching circuit using deposited silica guides,” Electron. Lett. 17, 411–413 (1981).
    [CrossRef]
  6. Z. Weissman, E. Marom, and A. Hardy, “Very low loss Y junction power divider,” Opt. Lett. 14, 293–295 (1989).
    [CrossRef] [PubMed]
  7. M. Izutsu, H. Haga, and T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulator,” J. Lightwave Technol. 1, 285–289 (1983).
    [CrossRef]
  8. S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
    [CrossRef]
  9. J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471–473 (1990).
    [CrossRef] [PubMed]
  10. D. R. Anderson, D. E. Hooton, G. A. Swartzlander, and A. E. Kaplan, “Direct measurement of transverse velocity of dark spatial solitons,” Opt. Lett. 15, 783–785 (1990).
    [CrossRef]
  11. W. K. Burns and A. F. Milton, “Mode conversion in planar dielectric separating waveguides,” IEEE J. Quantum Electron. 11, 32–39 (1975).
    [CrossRef]
  12. M. I. Zutsu, Y. Nakai, and T. Sutena, “Operation mechanism of the single-mode optical-waveguide Y junction,” Opt. Lett. 7, 136–138 (1982).
    [CrossRef]
  13. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
    [CrossRef]
  14. Y. Chung and N. Dagli, “Assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1337 (1990).
    [CrossRef]

1990 (3)

1989 (1)

1988 (1)

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

1985 (1)

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
[CrossRef]

1984 (1)

1983 (2)

M. Haruna and J. Koyama, “Electrooptic branching waveguide switches and their application to 1×4 optical switching networks,” J. Lightwave Technol. 1, 223–227 (1983).
[CrossRef]

M. Izutsu, H. Haga, and T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulator,” J. Lightwave Technol. 1, 285–289 (1983).
[CrossRef]

1982 (1)

1981 (1)

Y. Murakami and M. Ikeda, “Single-mode optical Y-branching circuit using deposited silica guides,” Electron. Lett. 17, 411–413 (1981).
[CrossRef]

1978 (1)

H. Sasaki and I. Anderson, “Theoretical and experimental studies on active Y junctions in optical waveguides,” IEEE J. Quantum Electron. 14, 883–892 (1978).
[CrossRef]

1975 (2)

W. E. Martin, “A new waveguide switch/modulator for integrated optics,” Appl. Phys. Lett. 26, 562–564 (1975).
[CrossRef]

W. K. Burns and A. F. Milton, “Mode conversion in planar dielectric separating waveguides,” IEEE J. Quantum Electron. 11, 32–39 (1975).
[CrossRef]

Aitchison, J. S.

Anderson, D. R.

Anderson, I.

H. Sasaki and I. Anderson, “Theoretical and experimental studies on active Y junctions in optical waveguides,” IEEE J. Quantum Electron. 14, 883–892 (1978).
[CrossRef]

Burns, W. K.

W. K. Burns and A. F. Milton, “Mode conversion in planar dielectric separating waveguides,” IEEE J. Quantum Electron. 11, 32–39 (1975).
[CrossRef]

Chilwell, J. T.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
[CrossRef]

Chung, Y.

Y. Chung and N. Dagli, “Assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1337 (1990).
[CrossRef]

Cullen, T. J.

Dagli, N.

Y. Chung and N. Dagli, “Assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1337 (1990).
[CrossRef]

Desailly, R.

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

Froehly, C.

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

Haga, H.

M. Izutsu, H. Haga, and T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulator,” J. Lightwave Technol. 1, 285–289 (1983).
[CrossRef]

Hardy, A.

Haruna, M.

M. Haruna and J. Koyama, “Electrooptic branching waveguide switches and their application to 1×4 optical switching networks,” J. Lightwave Technol. 1, 223–227 (1983).
[CrossRef]

Hooton, D. E.

Ikeda, M.

Y. Murakami and M. Ikeda, “Single-mode optical Y-branching circuit using deposited silica guides,” Electron. Lett. 17, 411–413 (1981).
[CrossRef]

Izutsu, M.

M. Izutsu, H. Haga, and T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulator,” J. Lightwave Technol. 1, 285–289 (1983).
[CrossRef]

Jackel, J. L.

Kaplan, A. E.

Koyama, J.

M. Haruna and J. Koyama, “Electrooptic branching waveguide switches and their application to 1×4 optical switching networks,” J. Lightwave Technol. 1, 223–227 (1983).
[CrossRef]

Leaird, D. E.

Maneuf, S.

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

Marom, E.

Martin, W. E.

W. E. Martin, “A new waveguide switch/modulator for integrated optics,” Appl. Phys. Lett. 26, 562–564 (1975).
[CrossRef]

Milton, A. F.

W. K. Burns and A. F. Milton, “Mode conversion in planar dielectric separating waveguides,” IEEE J. Quantum Electron. 11, 32–39 (1975).
[CrossRef]

Murakami, Y.

Y. Murakami and M. Ikeda, “Single-mode optical Y-branching circuit using deposited silica guides,” Electron. Lett. 17, 411–413 (1981).
[CrossRef]

Nakai, Y.

Oliver, M. K.

Sasaki, H.

H. Sasaki and I. Anderson, “Theoretical and experimental studies on active Y junctions in optical waveguides,” IEEE J. Quantum Electron. 14, 883–892 (1978).
[CrossRef]

Seaton, C. T.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
[CrossRef]

Shoemaker, R. L.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
[CrossRef]

Silberberg, Y.

Smith, P. W. E.

Smith, S. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
[CrossRef]

Stegeman, G. I.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
[CrossRef]

Sueta, T.

M. Izutsu, H. Haga, and T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulator,” J. Lightwave Technol. 1, 285–289 (1983).
[CrossRef]

Sutena, T.

Swartzlander, G. A.

Valera, J. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
[CrossRef]

Vogel, E. M.

Weiner, A. M.

Weissman, Z.

Wilkinson, C. D.

Zutsu, M. I.

Appl. Phys. Lett. (1)

W. E. Martin, “A new waveguide switch/modulator for integrated optics,” Appl. Phys. Lett. 26, 562–564 (1975).
[CrossRef]

Electron. Lett. (1)

Y. Murakami and M. Ikeda, “Single-mode optical Y-branching circuit using deposited silica guides,” Electron. Lett. 17, 411–413 (1981).
[CrossRef]

IEEE J. Quantum Electron. (4)

W. K. Burns and A. F. Milton, “Mode conversion in planar dielectric separating waveguides,” IEEE J. Quantum Electron. 11, 32–39 (1975).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774–783 (1985).
[CrossRef]

Y. Chung and N. Dagli, “Assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335–1337 (1990).
[CrossRef]

H. Sasaki and I. Anderson, “Theoretical and experimental studies on active Y junctions in optical waveguides,” IEEE J. Quantum Electron. 14, 883–892 (1978).
[CrossRef]

J. Lightwave Technol. (2)

M. Haruna and J. Koyama, “Electrooptic branching waveguide switches and their application to 1×4 optical switching networks,” J. Lightwave Technol. 1, 223–227 (1983).
[CrossRef]

M. Izutsu, H. Haga, and T. Sueta, “Picosecond signal sampling and multiplication by using integrated tandem light modulator,” J. Lightwave Technol. 1, 285–289 (1983).
[CrossRef]

Opt. Commun. (1)

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

Opt. Lett. (5)

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Figures (11)

Fig. 1
Fig. 1

Nonlinear multilayer waveguide structure: (a) n even, (b) n odd.

Fig. 2
Fig. 2

Dispersion relation in a five-layer nonlinear waveguide.

Fig. 3
Fig. 3

Field distribution at the five points A–E in Fig. 2.

Fig. 4
Fig. 4

Dispersion relation in a seven-layer nonlinear waveguide.

Fig. 5
Fig. 5

Field distribution at the five points A–E in Fig. 4.

Fig. 6
Fig. 6

Schematic diagram of a MQW waveguide structure.

Fig. 7
Fig. 7

Field distribution of an optical waveguide in a MQW structure.

Fig. 8
Fig. 8

Multibranched waveguide with nonlinear cladding and substrate.

Fig. 9
Fig. 9

1×3 branch waveguide with nonlinear cladding and substrate.

Fig. 10
Fig. 10

Electric-field distributions of the three-branch waveguide at positions (a) Z1, (b) Z2, (c) Z3, (d) Z4, (e) Z5.

Fig. 11
Fig. 11

Evolution of a wave propagating along a three-branch waveguide.

Equations (55)

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2y=nj2c2 2yt2,j=i, f, c, s,
y(x, z, t)=E(x)exp[i(wt-βk0z)].
nj2=nj02+α|E|2,j=c, s,
Ec(x)=2/α qc/cosh{k0qc[x-xc(n)]}inthecladding,
Ef(x)=Af(n)cos{k0qf[x-xf (n)]}inthefilm,β<nf,
Ef(x)=Af(n)sinh{k0Qf[x-xf (n)]}inthefilm,β>nf,
Ei(x)=Ai(n)cosh{k0qi[x-xi(n)]}intheinteractionlayer,
Es(x)=As2/α qs/cosh[k0qs(x-xs)]inthesubstrate.
qi=β2-nj2,j=c, i, s,
qf=nf2-β2,β<nf,
Qf=β2-nf2,β>nf.
tan(k0qfdf)=qf(qi tanh Ψi+qs tan Ψs)(qf2-qiqs tan Ψi tanh Ψs),β<nf,
tanh(k0Qfdf)=-Qf(qi tanh Ψi+qs tanh Ψs)(Qf2+qiqs tanh Ψi tanh Ψs),β>nf,
Ψi=k0qin-12df+n2di-xi(1),
Ψs=k0qsn+12df+n2di+xs(n+1).
xf(n+1)=-n+12df-n2di+1k0qf tan-1×qsqf tanhk0qsn+12df+n2di+xs(n+1),
xf(n+1)=-n+12df-n2di+1k0Qf tanh-1×-Qfqs tanhk0qsn+12df+n2di+xs(n+1) ,
xc(n+1)=n+12df+n2di-1k0qc tanh-1qfqc tanhk0qf n+12df+n2di-xf(1),
xc(n+1)=n+12df+n2di+1k0qc tanh-1Qfqc tanhk0Qf n+12df+n2di-xf(1),
xs(n+1)=-n+12df-n2di+1k0qs tanh-1qfqs tank0qf n+12df+n2di+xf(n+1),
xs(n+1)=-n+12df-n2di-1k0qs tanh-1Qfqs tanhk0Qf n+12df+n2di+xf(n+1),
Af(1)=2α qc seck0qf n2di+n+12df-xf(1)coshk0qcn2di+n+12df-xc(n+1),
Af(1)=2α qc cschk0Qf n2df+n+12df-xf(1)coshk0qcn2di+n+12df-xc(n+1),
Ai(1)=Af(1) cosk0qf n2di+n-12df-xf(1)coshk0qin2di+n-12df-xi(1),
Ai(1)=Af(1) sinhk0Qf n2di+n-12df-xf(1)coshk0qin2di+n+12df-xi(1).
 Af(n-p)=Ai(n-p-1) coshk0qin-22-pdi+n-32-pdf+xi(n-p-1)cosk0qf n-22-pdi+n-32-pdf+xf(n-f),
Af(n-p)=Ai(n-p-1) coshk0qin-22-pdi+n-32-pdf-xi(n-p-1)sinhk0Qf n-22-pdi+n-32-pdf-xf(n-f),
Ai(n-p)=Af(n-p) cosk0qf n-22-pdi+n-12-pdf+xf(n-p)coshk0qin-22-pdi+n-12-pdf+xi(n-p),
Ai(n-p)=-Af(n-p) sinhk0Qf n-22-pdi+n-12-pdf+xf(n-p)coshk0qin-22-pdi+n-12-pdf+xi(n-p),
Af(n+1)
=Ai(n) coshk0qin2di+n-12df+xi(n)cosk0qf n2di+n-12df+xf (n+1),
Af(n+1)
=-Ai(n) coshk0qin2di+n-12df+xi(n)sinhk0Qf n2di+n-12df+xf (n+1),
As=Af(n+1)α2qs×cosk0qf n2di+n+12df+xf (n+1)sech k0qsn2di+n+12df+xs(n+1),
As=-Af(n+1)α2qs×sinhk0Qf n2di+n+12df+xf (n+1)coshk0qsn2di+n+12df+xs(n+1),
xf(n-p)=-n-12+pdf+-n-22+pdi+1k0qf tan-1-qiqf tanhk0qin-12-pdf+n-22-pdi+xi(n-p),
xf(n-p)=-n-12+pdf+-n-22+pdi+1k0Qf tanh-1Qfqi tanhk0qin-12-pdf+n-22-pdi+xi(n-p),
xi(n-p)=-n-12+pdf+-n2+pdi+1k0qi tanh-1-qfqi tank0qf n-12-pdf+n2-pdi+xf(n-p+1),
xi(n-p)=-n-12+pdf+-n2+pdi+1k0qi tanh-1Qfqi tanhk0Qf n-12-pdf+n2-pdi+xf(n-p+1),
xf(n-p)=-n-12+pdf+-n-22+pdi-1k0qf tan-1-qiqf tanhk0qi-n-12-pdf-n-22-pdi-xi(n-p),
xf(n-p)=-n-12+pdf+-n-22+pdi-1k0Qf tanh-1Qfqi tanhk0qi-n-12-pdf-n-22-pdi-xi(n-p),
xi(n-p)=-n-12+pdf+-n2+pdi-1k0qi tanh-1-qfqi tank0qf -n-12-pdf-n2-pdi-xf(n-p+1),
xi(n-p)=-n-12+pdf+-n2+pdi-1k0qi tanh-1Qfqi tanhk0Qf -n-12-pdf-n2-pdi-xf(n-p+1),
xf(n-p)=-n-12+pdf+-n-22+pdi+1k0qf tan-1-qiqf tanhk0qin-12-pdf+n-22-pdi+xi(n-p),
xf(n-p)=-n-12+pdf+-n-22+pdi+1k0Qf tanh-1Qfqi tanhk0qin-12-pdf+n-22-pdi+xi(n-p),
xi(n-p)=-n-12+pdf+-n2+pdi+1k0qi tanh-1-qfqi tank0qf n-12-pdf+n2-pdi+xf(n-p+1),
xi(n-p)=-n-12+pdf+-n2+pdi+1k0qi tanh-1Qfqi tanhk0Qf n-12-pdf+n2-pdi+xf(n-p+1),
xf(n-p)=-n-12+pdf+-n-22+pdi-1k0qf tan-1-qiqf tanhk0qi-n-12-pdf-n-22-pdi-xi(n-p),
xf(n-p)=-n-12+pdf+-n-22+pdi-1k0Qf tanh-1Qfqi tanhk0qi-n-12-pdf-n-22-pdi-xi(n-p),
xi(n-p)=-n-12+pdf+-n2+pdi-1k0qi tanh-1-qfqi tank0qf -n-12-pdf-n2-pdi-xf(n-p+1),
xi(n-p)=-n-12+pdf+-n2+pdi-1k0qi tanh-1Qfqi tanhk0Qf -n-12-pdf-n2-pdi-xf(n-p+1),
xf n2+12=12di-1k0qf tan-1-qiqf tanhk0qi12di-xin2+12,
xf n2+12=12di-1k0Qf tanh-1Qfqi tanhk0qi12di-xin2+12,
xin2+12=-12di+1k0qi tanh-1-qfqi tank0qf12di+xf n2+32 ,
xin2+12=-12di+1k0qi tanh-1Qfqi tanhk0Qf12di+xfn2+32.

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