Abstract

We develop a modal decomposition technique to describe wave propagation in dynamically varying waveguides such as those predicted to occur in photosensitive materials. In particular, we use this method to examine the self-writing of a channel waveguide in a slab of photosensitive glass. This method is up to two orders of magnitude faster than the numerical technique used previously, which is based on a conventional beam-propagation treatment. Hence it is a powerful tool for exploring the waveguides that form under a wide range of conditions.

© 1997 Optical Society of America

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References

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  1. T. M. Monro, C. M. de Sterke, L. Poladian, “Self-writing a waveguide in glass using photosensitivity,” Opt. Commun. 119, 523–526 (1995).
    [CrossRef]
  2. T. M. Monro, C. M. de Sterke, L. Poladian, “Investigation of waveguide growth in photosensitive germano-silicate glass,” J. Opt. Soc. Am. B 13, 2824–2832 (1996).
    [CrossRef]
  3. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, San Diego, Calif., 1991), Chap. 3.
  4. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Part 3.
  5. R. März, Integrated Optics: Design and Modeling (Artec House, London, 1995), Chap. 5.
  6. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.
  7. I. W. Busbridge, “Some integrals involving Hermite polynomials,” J. London Math. Soc. 23, 135–141 (1948).
  8. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 19.
  9. E. Caglioti, B. Crosignani, P. Di Porto, “Hamiltonian description of nonlinear propagation in optical fibers,” Phys. Rev. A 38, 4036–4042 (1988).
    [CrossRef] [PubMed]
  10. T. Ueda, W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
    [CrossRef] [PubMed]
  11. D. J. Mitchell, A. W. Snyder, L. Poladian, “Selfguided beam interaction: method of invariants,” Electron. Lett. 27, 848–849 (1991).
    [CrossRef]
  12. A. S. Kewitsch, A. Yariv, “Self-focusing and self-trapping of optical beams upon photopolymerization,” Opt. Lett. 21, 24–26 (1996).
    [CrossRef] [PubMed]
  13. A. S. Kewitsch, A. Yariv, “Nonlinear optical properties of photoresists for projection lithography,” Appl. Phys. Lett. 68, 455–457 (1996).
    [CrossRef]
  14. C. M. de Sterke, J. E. Sipe, “Ideal mode expansion for planar optical waveguides: application to the TM–TM coupling coefficient for grating structure,” J. Opt. Soc. Am. A 7, 636–645 (1990).
    [CrossRef]
  15. V. Mizrahi, S. LaRochelle, G. I. Stegeman, J. E. Sipe, “Physics of photosensitive-grating formation in optical fibers,” Phys. Rev. A 43, 433–438 (1991).
    [CrossRef] [PubMed]
  16. C. M. de Sterke, S. An, J. E. Sipe, “Growth dynamics of phase gratings in optical fibres,” Opt. Commun. 83, 315–321 (1991).
    [CrossRef]
  17. S. J. Frisken, “Light-induced optical waveguide uptapers,” Opt. Lett. 18, 1035–1037 (1993).
    [CrossRef] [PubMed]

1996 (3)

1995 (1)

T. M. Monro, C. M. de Sterke, L. Poladian, “Self-writing a waveguide in glass using photosensitivity,” Opt. Commun. 119, 523–526 (1995).
[CrossRef]

1993 (1)

1991 (3)

V. Mizrahi, S. LaRochelle, G. I. Stegeman, J. E. Sipe, “Physics of photosensitive-grating formation in optical fibers,” Phys. Rev. A 43, 433–438 (1991).
[CrossRef] [PubMed]

C. M. de Sterke, S. An, J. E. Sipe, “Growth dynamics of phase gratings in optical fibres,” Opt. Commun. 83, 315–321 (1991).
[CrossRef]

D. J. Mitchell, A. W. Snyder, L. Poladian, “Selfguided beam interaction: method of invariants,” Electron. Lett. 27, 848–849 (1991).
[CrossRef]

1990 (2)

1988 (1)

E. Caglioti, B. Crosignani, P. Di Porto, “Hamiltonian description of nonlinear propagation in optical fibers,” Phys. Rev. A 38, 4036–4042 (1988).
[CrossRef] [PubMed]

1948 (1)

I. W. Busbridge, “Some integrals involving Hermite polynomials,” J. London Math. Soc. 23, 135–141 (1948).

An, S.

C. M. de Sterke, S. An, J. E. Sipe, “Growth dynamics of phase gratings in optical fibres,” Opt. Commun. 83, 315–321 (1991).
[CrossRef]

Busbridge, I. W.

I. W. Busbridge, “Some integrals involving Hermite polynomials,” J. London Math. Soc. 23, 135–141 (1948).

Caglioti, E.

E. Caglioti, B. Crosignani, P. Di Porto, “Hamiltonian description of nonlinear propagation in optical fibers,” Phys. Rev. A 38, 4036–4042 (1988).
[CrossRef] [PubMed]

Crosignani, B.

E. Caglioti, B. Crosignani, P. Di Porto, “Hamiltonian description of nonlinear propagation in optical fibers,” Phys. Rev. A 38, 4036–4042 (1988).
[CrossRef] [PubMed]

de Sterke, C. M.

T. M. Monro, C. M. de Sterke, L. Poladian, “Investigation of waveguide growth in photosensitive germano-silicate glass,” J. Opt. Soc. Am. B 13, 2824–2832 (1996).
[CrossRef]

T. M. Monro, C. M. de Sterke, L. Poladian, “Self-writing a waveguide in glass using photosensitivity,” Opt. Commun. 119, 523–526 (1995).
[CrossRef]

C. M. de Sterke, S. An, J. E. Sipe, “Growth dynamics of phase gratings in optical fibres,” Opt. Commun. 83, 315–321 (1991).
[CrossRef]

C. M. de Sterke, J. E. Sipe, “Ideal mode expansion for planar optical waveguides: application to the TM–TM coupling coefficient for grating structure,” J. Opt. Soc. Am. A 7, 636–645 (1990).
[CrossRef]

Di Porto, P.

E. Caglioti, B. Crosignani, P. Di Porto, “Hamiltonian description of nonlinear propagation in optical fibers,” Phys. Rev. A 38, 4036–4042 (1988).
[CrossRef] [PubMed]

Frisken, S. J.

Kath, W. L.

T. Ueda, W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

Kewitsch, A. S.

A. S. Kewitsch, A. Yariv, “Self-focusing and self-trapping of optical beams upon photopolymerization,” Opt. Lett. 21, 24–26 (1996).
[CrossRef] [PubMed]

A. S. Kewitsch, A. Yariv, “Nonlinear optical properties of photoresists for projection lithography,” Appl. Phys. Lett. 68, 455–457 (1996).
[CrossRef]

LaRochelle, S.

V. Mizrahi, S. LaRochelle, G. I. Stegeman, J. E. Sipe, “Physics of photosensitive-grating formation in optical fibers,” Phys. Rev. A 43, 433–438 (1991).
[CrossRef] [PubMed]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 19.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Part 3.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, San Diego, Calif., 1991), Chap. 3.

März, R.

R. März, Integrated Optics: Design and Modeling (Artec House, London, 1995), Chap. 5.

Mitchell, D. J.

D. J. Mitchell, A. W. Snyder, L. Poladian, “Selfguided beam interaction: method of invariants,” Electron. Lett. 27, 848–849 (1991).
[CrossRef]

Mizrahi, V.

V. Mizrahi, S. LaRochelle, G. I. Stegeman, J. E. Sipe, “Physics of photosensitive-grating formation in optical fibers,” Phys. Rev. A 43, 433–438 (1991).
[CrossRef] [PubMed]

Monro, T. M.

T. M. Monro, C. M. de Sterke, L. Poladian, “Investigation of waveguide growth in photosensitive germano-silicate glass,” J. Opt. Soc. Am. B 13, 2824–2832 (1996).
[CrossRef]

T. M. Monro, C. M. de Sterke, L. Poladian, “Self-writing a waveguide in glass using photosensitivity,” Opt. Commun. 119, 523–526 (1995).
[CrossRef]

Poladian, L.

T. M. Monro, C. M. de Sterke, L. Poladian, “Investigation of waveguide growth in photosensitive germano-silicate glass,” J. Opt. Soc. Am. B 13, 2824–2832 (1996).
[CrossRef]

T. M. Monro, C. M. de Sterke, L. Poladian, “Self-writing a waveguide in glass using photosensitivity,” Opt. Commun. 119, 523–526 (1995).
[CrossRef]

D. J. Mitchell, A. W. Snyder, L. Poladian, “Selfguided beam interaction: method of invariants,” Electron. Lett. 27, 848–849 (1991).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

Sipe, J. E.

V. Mizrahi, S. LaRochelle, G. I. Stegeman, J. E. Sipe, “Physics of photosensitive-grating formation in optical fibers,” Phys. Rev. A 43, 433–438 (1991).
[CrossRef] [PubMed]

C. M. de Sterke, S. An, J. E. Sipe, “Growth dynamics of phase gratings in optical fibres,” Opt. Commun. 83, 315–321 (1991).
[CrossRef]

C. M. de Sterke, J. E. Sipe, “Ideal mode expansion for planar optical waveguides: application to the TM–TM coupling coefficient for grating structure,” J. Opt. Soc. Am. A 7, 636–645 (1990).
[CrossRef]

Snyder, A. W.

D. J. Mitchell, A. W. Snyder, L. Poladian, “Selfguided beam interaction: method of invariants,” Electron. Lett. 27, 848–849 (1991).
[CrossRef]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 19.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Part 3.

Stegeman, G. I.

V. Mizrahi, S. LaRochelle, G. I. Stegeman, J. E. Sipe, “Physics of photosensitive-grating formation in optical fibers,” Phys. Rev. A 43, 433–438 (1991).
[CrossRef] [PubMed]

Ueda, T.

T. Ueda, W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

Yariv, A.

A. S. Kewitsch, A. Yariv, “Nonlinear optical properties of photoresists for projection lithography,” Appl. Phys. Lett. 68, 455–457 (1996).
[CrossRef]

A. S. Kewitsch, A. Yariv, “Self-focusing and self-trapping of optical beams upon photopolymerization,” Opt. Lett. 21, 24–26 (1996).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

A. S. Kewitsch, A. Yariv, “Nonlinear optical properties of photoresists for projection lithography,” Appl. Phys. Lett. 68, 455–457 (1996).
[CrossRef]

Electron. Lett. (1)

D. J. Mitchell, A. W. Snyder, L. Poladian, “Selfguided beam interaction: method of invariants,” Electron. Lett. 27, 848–849 (1991).
[CrossRef]

J. London Math. Soc. (1)

I. W. Busbridge, “Some integrals involving Hermite polynomials,” J. London Math. Soc. 23, 135–141 (1948).

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

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

Opt. Commun. (2)

C. M. de Sterke, S. An, J. E. Sipe, “Growth dynamics of phase gratings in optical fibres,” Opt. Commun. 83, 315–321 (1991).
[CrossRef]

T. M. Monro, C. M. de Sterke, L. Poladian, “Self-writing a waveguide in glass using photosensitivity,” Opt. Commun. 119, 523–526 (1995).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (3)

V. Mizrahi, S. LaRochelle, G. I. Stegeman, J. E. Sipe, “Physics of photosensitive-grating formation in optical fibers,” Phys. Rev. A 43, 433–438 (1991).
[CrossRef] [PubMed]

E. Caglioti, B. Crosignani, P. Di Porto, “Hamiltonian description of nonlinear propagation in optical fibers,” Phys. Rev. A 38, 4036–4042 (1988).
[CrossRef] [PubMed]

T. Ueda, W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

Other (5)

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 19.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, San Diego, Calif., 1991), Chap. 3.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Part 3.

R. März, Integrated Optics: Design and Modeling (Artec House, London, 1995), Chap. 5.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

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Figures (9)

Fig. 1
Fig. 1

View of a glass slab showing the initial diffraction of the Gaussian beam.

Fig. 2
Fig. 2

Axial intensity at T=0. M=10,p=1. Solid curve, exact result; dashed curve, adaptive modal method result; dotted curve, nonadaptive result.

Fig. 3
Fig. 3

(a) Field intensity (b) and refractive index at T=0.052, M=N=10 for p=1. Solid curves, BPM simulation results; dashed curves, adaptive modal method results; dotted curves, nonadaptive results.

Fig. 4
Fig. 4

As for Figs. 3(a) and 3(b) except that T=0.221.

Fig. 5
Fig. 5

(a) Field intensity and (b) refractive index at T=2.45 for p=2, M=N=10 for p=2. Solid curves, BPM simulation results; dashed curves, adaptive modal method results; dotted curves, nonadaptive results.

Fig. 6
Fig. 6

As for Figs. 5(a) and 5(b) except that T=5.95.

Fig. 7
Fig. 7

View of the slab of glass from the top. Dashed curves, the form of the incident beam in air; dotted curves, the form of the refracted beam in the glass. As discussed in the text, the beam waist in air is d from the input face.

Fig. 8
Fig. 8

Position of the intensity primary eye as predicted by the adaptive modal method for p=1 as a function of the curvature of the incident beam for different times. The solid, dotted, short-dashed and long-dashed curves correspond to T=0, 0.01, 0.02, and 0.1, respectively. The large dots are BPM simulation results at these times.

Fig. 9
Fig. 9

Waveguide nonuniformity as predicted by the adaptive modal method for p=1 as a function of the curvature of the incident beam for different times. Our definition of nonuniformity is described in the text. The solid, dotted, and dashed curves correspond to T=0.01, 0.02, and 0.1, respectively. The solid dots, open dots, and crosses are the BPM results at these times.

Tables (2)

Tables Icon

Table 1 Speed and Accuracy of the Adaptive Modal Method for p=1 a

Tables Icon

Table 2 Speed and Accuracy of the Adaptive Modal Method for p=2 a

Equations (28)

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E(x, y, z, t)=F(x)E(y, z, t)exp[i(kz-ωt)]yˆ+c.c.,
i Eζ+12 2Eη2+Δnn E=0,
T Δnn=(EE*)p,
E(η, 0, t)=1w exp-η22w2.
zR=kW22.
E(η, ζ, T)=j=0M-1aj(ζ, T)Fjβ(η),
Δn(η, ζ, T)n=j=0N-1cj(ζ, T)Fjβ(η),
ajζ=ik=0M-1akJjk2+akl=0N-1clKjkl-βζ k=0M-1akIjk.
cjT=k,l=0M-1akal*Kjkl-βT k=0N-1ckIjk,
cjT=k,l,m,n=0M-1akal*aman*Ljklmn-βT k=0N-1ckIjk.
Ijk=-Fjβ(η) Fkβ(η)β dη,
Jjk=-Fjβ(η) 2Fkβ(η)η2 dη,
Kjkl=-Fjβ(η)Fkβ(η)Flβ(η)dη,
Ljklmn=-Fjβ(η)Fkβ(η)Flβ(η)Fmβ(η)Fnβ(η)dη.
Fjβ(η)=12j(2j)! βπ1/4 exp(-βη2/2)H2j(βη),
Δnn=Δn(0)n-βη22.
Ijk=14β {-[2k(2k-1)]1/2δj-1,k+[(2k+2)(2k+1)]1/2δj,k-1},
Jjk=β2 {-(1+4k)δj,k+[2k(2k-1)]1/2δj,k-1+[(2k+2)(2k+1)]1/2δj-1,k},
Kjkl=β1/4π-3/4(-23)qΓ(q+12)[32(2j)!(2k)!(2l)!]1/2×r,s,t(-2j)s+t(-2k)t+r(-2l)r+sr!s!t!(12-q)r+s+t( 23)r+s+t,
Ljklmn=β3/4Ljklmn,
βW(ζ, T)=-Δn(η, ζ, T)dη-η2Δn(η, ζ, T)dη.
βW(ζ, 0)=-(EE*)pdη-η2(EE*)pdη,
βWT=4β j=0N-1k=0N-1c˙jckrjrk(k-j)l=0N-1cl(1+4l)rl2,
rl=(2l)!2ll!,
β(ζ, T)=CβW(ζ, T).
βT=4Cβ j=0N-1k=0N-1c˙jckrjrk(k-j)l=0N-1cl(1+4l)rl2,
=1qmax q=0qmax-1[Δnmod(qΔζ)-Δnsim(qΔζ)]2,
E(η, 0, T)=w-idkawk-1/2 exp-η22w2-ik2dka,

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