Abstract

An algorithm is presented for simulation of guided modes in a multilayer uniaxial structure with each layer characterized by its own ellipsoid of refractive indices and direction of optical axis. The proposed approach is based on presenting an electromagnetic field in each layer as a linear combination of ordinary and extraordinary waves coupled through the boundary conditions. The problem is reduced to two dimensions by considering the waves with a given projection of the wave vector on the plane of the waveguide. No a priori assumption about the guided-mode polarization is required in this method. Hybrid polarized modes appear naturally as solutions of a system of linear equations with respect to the amplitudes of the ordinary and extraordinary waves. The proposed approach covers a wide variety of important practical cases including isotropic waveguides, surface waves at the boundary between positive uniaxial crystal and isotropic medium, surface plasmons at metallic interfaces, uniaxial multilayers in a very general form, and leaky modes in such structures.

© 2003 Optical Society of America

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References

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  1. See, e.g., A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  2. P. J. Linchung, S. Teitler, “4×4 matrix formalism for optics in stratified anisotropic media,” J. Opt. Soc. Am. A 1, 703–705 (1984).
    [CrossRef]
  3. A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
    [CrossRef]
  4. I. Hodgkinson, D. Endelma, “Bound modes in anisotropic multilayer thin-film wave-guides,” Appl. Opt. 29, 4424–4426 (1990).
    [CrossRef] [PubMed]
  5. C. R. Paiva, A. M. Barbosa, “Spectral representation of self-adjoint problems for layered anisotropic wave-guides,” IEEE Trans. Microwave Theory Tech. 39, 330–338 (1991).
    [CrossRef]
  6. G. Tartarini, P. Bassi, S. F. Chen, M. P. Demicheli, D. B. Ostrovsky, “Calculation of hybrid modes in uniaxial planar optical wave-guides—application to proton-exchanged lithium-niobate wave-guides,” Opt. Commun. 101, 424–431 (1992).
    [CrossRef]
  7. J. F. Offersgaard, “Wave-guides formed by multiple layers of dielectric, semiconductor, or metal media with optical loss and anisotropy,” J. Opt. Soc. Am. A 12, 2122–2128 (1995).
    [CrossRef]
  8. J. F. Offersgaard, T. Veng, T. Skettrup, “Accurate method for determining the refractive-index profiles of planar waveguides in uniaxial media with the optical axis normal to the surface,” Appl. Opt. 35, 2602–2609 (1996).
    [CrossRef] [PubMed]
  9. W. H. Press, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1997).
  10. M. I. D’Yakonov, “Novel type of boundary electromagnetic waves,” Zh. Eksp. Teor. Fiz. 94, 119–123 (1988); Sov. Phys. JETP 67, 714 (1988).
  11. D. B. Walker, E. N. Glysis, T. K. Gaylord, “Surface mode at isotropic–uniaxial and isotropic–biaxial interfaces,” J. Opt. Soc. Am. A 15, 248–260 (1998).
    [CrossRef]
  12. A. Boudrioua, P. Moretti, J. C. Loulergue, K. Polgar, “Helium ion-implanted planar waveguide in Y-cut and Z-cut β-BBO (BaB2O4),” Opt. Mater. 14, 31–39 (2000).
    [CrossRef]
  13. L. Torner, F. Canal, J. Hernandezmarco, “Leaky modes in multilayer uniaxial waveguides,” Appl. Opt. 29, 2805–2814 (1990).
    [CrossRef] [PubMed]
  14. L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab wave-guides,” J. Lightwave Technol. 11, 1592–1600 (1993).
    [CrossRef]

2000

A. Boudrioua, P. Moretti, J. C. Loulergue, K. Polgar, “Helium ion-implanted planar waveguide in Y-cut and Z-cut β-BBO (BaB2O4),” Opt. Mater. 14, 31–39 (2000).
[CrossRef]

1998

1996

1995

1993

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab wave-guides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

1992

G. Tartarini, P. Bassi, S. F. Chen, M. P. Demicheli, D. B. Ostrovsky, “Calculation of hybrid modes in uniaxial planar optical wave-guides—application to proton-exchanged lithium-niobate wave-guides,” Opt. Commun. 101, 424–431 (1992).
[CrossRef]

1991

C. R. Paiva, A. M. Barbosa, “Spectral representation of self-adjoint problems for layered anisotropic wave-guides,” IEEE Trans. Microwave Theory Tech. 39, 330–338 (1991).
[CrossRef]

1990

1988

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

M. I. D’Yakonov, “Novel type of boundary electromagnetic waves,” Zh. Eksp. Teor. Fiz. 94, 119–123 (1988); Sov. Phys. JETP 67, 714 (1988).

1984

Barbosa, A. M.

C. R. Paiva, A. M. Barbosa, “Spectral representation of self-adjoint problems for layered anisotropic wave-guides,” IEEE Trans. Microwave Theory Tech. 39, 330–338 (1991).
[CrossRef]

Bassi, P.

G. Tartarini, P. Bassi, S. F. Chen, M. P. Demicheli, D. B. Ostrovsky, “Calculation of hybrid modes in uniaxial planar optical wave-guides—application to proton-exchanged lithium-niobate wave-guides,” Opt. Commun. 101, 424–431 (1992).
[CrossRef]

Boudrioua, A.

A. Boudrioua, P. Moretti, J. C. Loulergue, K. Polgar, “Helium ion-implanted planar waveguide in Y-cut and Z-cut β-BBO (BaB2O4),” Opt. Mater. 14, 31–39 (2000).
[CrossRef]

Canal, F.

Chen, S. F.

G. Tartarini, P. Bassi, S. F. Chen, M. P. Demicheli, D. B. Ostrovsky, “Calculation of hybrid modes in uniaxial planar optical wave-guides—application to proton-exchanged lithium-niobate wave-guides,” Opt. Commun. 101, 424–431 (1992).
[CrossRef]

D’Yakonov, M. I.

M. I. D’Yakonov, “Novel type of boundary electromagnetic waves,” Zh. Eksp. Teor. Fiz. 94, 119–123 (1988); Sov. Phys. JETP 67, 714 (1988).

Demicheli, M. P.

G. Tartarini, P. Bassi, S. F. Chen, M. P. Demicheli, D. B. Ostrovsky, “Calculation of hybrid modes in uniaxial planar optical wave-guides—application to proton-exchanged lithium-niobate wave-guides,” Opt. Commun. 101, 424–431 (1992).
[CrossRef]

Endelma, D.

Gaylord, T. K.

D. B. Walker, E. N. Glysis, T. K. Gaylord, “Surface mode at isotropic–uniaxial and isotropic–biaxial interfaces,” J. Opt. Soc. Am. A 15, 248–260 (1998).
[CrossRef]

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Glysis, E. N.

Hernandezmarco, J.

Hodgkinson, I.

Knoesen, A.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Linchung, P. J.

Loulergue, J. C.

A. Boudrioua, P. Moretti, J. C. Loulergue, K. Polgar, “Helium ion-implanted planar waveguide in Y-cut and Z-cut β-BBO (BaB2O4),” Opt. Mater. 14, 31–39 (2000).
[CrossRef]

Moharam, M. G.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Moretti, P.

A. Boudrioua, P. Moretti, J. C. Loulergue, K. Polgar, “Helium ion-implanted planar waveguide in Y-cut and Z-cut β-BBO (BaB2O4),” Opt. Mater. 14, 31–39 (2000).
[CrossRef]

Offersgaard, J. F.

Ostrovsky, D. B.

G. Tartarini, P. Bassi, S. F. Chen, M. P. Demicheli, D. B. Ostrovsky, “Calculation of hybrid modes in uniaxial planar optical wave-guides—application to proton-exchanged lithium-niobate wave-guides,” Opt. Commun. 101, 424–431 (1992).
[CrossRef]

Paiva, C. R.

C. R. Paiva, A. M. Barbosa, “Spectral representation of self-adjoint problems for layered anisotropic wave-guides,” IEEE Trans. Microwave Theory Tech. 39, 330–338 (1991).
[CrossRef]

Polgar, K.

A. Boudrioua, P. Moretti, J. C. Loulergue, K. Polgar, “Helium ion-implanted planar waveguide in Y-cut and Z-cut β-BBO (BaB2O4),” Opt. Mater. 14, 31–39 (2000).
[CrossRef]

Press, W. H.

W. H. Press, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1997).

Recolons, J.

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab wave-guides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

Skettrup, T.

Tartarini, G.

G. Tartarini, P. Bassi, S. F. Chen, M. P. Demicheli, D. B. Ostrovsky, “Calculation of hybrid modes in uniaxial planar optical wave-guides—application to proton-exchanged lithium-niobate wave-guides,” Opt. Commun. 101, 424–431 (1992).
[CrossRef]

Teitler, S.

Torner, L.

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab wave-guides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

L. Torner, F. Canal, J. Hernandezmarco, “Leaky modes in multilayer uniaxial waveguides,” Appl. Opt. 29, 2805–2814 (1990).
[CrossRef] [PubMed]

Torres, J. P.

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab wave-guides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

Veng, T.

Walker, D. B.

Yariv, A.

See, e.g., A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

See, e.g., A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Appl. Opt.

IEEE Trans. Microwave Theory Tech.

C. R. Paiva, A. M. Barbosa, “Spectral representation of self-adjoint problems for layered anisotropic wave-guides,” IEEE Trans. Microwave Theory Tech. 39, 330–338 (1991).
[CrossRef]

J. Lightwave Technol.

A. Knoesen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar wave-guides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab wave-guides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

G. Tartarini, P. Bassi, S. F. Chen, M. P. Demicheli, D. B. Ostrovsky, “Calculation of hybrid modes in uniaxial planar optical wave-guides—application to proton-exchanged lithium-niobate wave-guides,” Opt. Commun. 101, 424–431 (1992).
[CrossRef]

Opt. Mater.

A. Boudrioua, P. Moretti, J. C. Loulergue, K. Polgar, “Helium ion-implanted planar waveguide in Y-cut and Z-cut β-BBO (BaB2O4),” Opt. Mater. 14, 31–39 (2000).
[CrossRef]

Zh. Eksp. Teor. Fiz.

M. I. D’Yakonov, “Novel type of boundary electromagnetic waves,” Zh. Eksp. Teor. Fiz. 94, 119–123 (1988); Sov. Phys. JETP 67, 714 (1988).

Other

See, e.g., A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

W. H. Press, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1997).

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Figures (7)

Fig. 1
Fig. 1

Multilayer uniaxial structure.

Fig. 2
Fig. 2

Real and imaginary parts of the characteristic determinant versus the trial value of the modal index for an isotropic planar waveguide.

Fig. 3
Fig. 3

(a) Real and imaginary parts of the characteristic determinant versus the trial value of the modal index for a surface wave at the interface between positive uniaxial crystal and isotropic medium, (b) modal index versus guided-mode propagation direction.

Fig. 4
Fig. 4

Real and imaginary parts of the plasmon modal index.

Fig. 5
Fig. 5

Modal indices of an anisotropic waveguide versus guided-mode propagation direction.

Fig. 6
Fig. 6

Modal indices of a multilayer anisotropic waveguide versus guided-mode propagation direction.

Fig. 7
Fig. 7

Light tunneling losses in a multilayer anisotropic waveguide versus optical barrier layer thickness.

Equations (31)

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ci=sin(θi)cos(ϕi)sin(θi)sin(θi)cos(θi).
k×(k×E)+κ2ˆ·E=0,
do(ko)=ko×c|ko×c|,
de(ke)=do(ke)×ke|do(ke)×ke|.
Do=Dodo(ko),
De=Dede(ke).
Eo=Doeo(no, ne, ko, c),eo(no, ne, ko, c)=1no2·do(ko),
Ee=Deee(no, ne, ke, c),ee(no, ne, ke, c)=1ne2[de(ke)·c]c+1no2{de(ke)-[de(ke)·c]c},
Ho=Doho(no, ne, ko, c),ho(no, ne, ko, c)=1κ[ko×eo(no, ne, ko, c)],
He=Deho(no, ne, ke, c),ho(no, ne, ke, c)=1κ[ke×ee(no, ne, ke, c)].
kox2+koy2+koz2=κ2no2,
kex2+key2+kez2=κ211no2(ke·c)2ke2+1ne21-(ke·c)2ke2.
kox=kex=0,
koy=key=κn*.
koz+=κ(no2-n*2)1/2,
koz-=-κ(no2-n*2)1/2,
kez+=χ+(n*, no, ne, θ, ϕ),
kez-=χ-(n*, no, ne, θ, ϕ),
kez2cos2(θ)no2+sin2(θ)ne2+kezκn*1no2-1ne2×sin(2θ)sin(ϕ)+(κn*)2sin2(θ)sin2(ϕ)no2+1-sin2(θ)sin2(ϕ)ne2-1n*2=0.
Di(y, z)=exp(in*κy)[Doi+exp(ikozi+z)+Doi-exp(ikozi-z)+Dei+exp(ikezi+z)+Dei-exp(ikezi-z)].
D0(y, z)=exp(in*κy)[Do0+exp(ikoz0+z)+De0+exp(ikez0+z)],
DN+1(y, z)=exp(in*κy)[Do(N+1)-exp(ikoz(N+1)-z)+De(N+1)-exp(ikez(N+1)-z)].
fpis=f(noi, nei, kpis, c(θi, ϕi)),
Do0+(fo0+)τ+De0+(fe0+)τ=Do1+(fo1+)τ+De1+(fe1+)τ+Do1-(fo1-)τ+De1-(fe1-)τ.
Doi+(foi+)τψoi++Dei+(fei+)τψei++Doi-(foi-)τψoi-+Dei-(fei-)ψei-=Do(i+1)+(fo(i+1)+)τ+De(i+1)+(fe(i+1)+)τ+Do(i+1)-(fo(i+1)-)τ+De(i+1)-(fe(i+1)-)τ,
ψpis=exp(-ikpzisti).
DoN+(foN+)τψoN++DeN+(feN+)τψeN++DoN-(foN-)τψoN-+DeN-(feN-)τψeN-=Do(N+1)-(fo(N+1)-)τ+De(N+1)-(fe(N+1)-)τ.
MˆD=0,
det[Mˆ(n*)]=0.
Do0+(fo0+)τ+De0+(fe0+)τ=Do1-(fo1-)τ+De1-(fe1-)τ,
n*=mdm+d1/2,

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