Abstract

The propagation properties of optical planar waveguides with multilayer index profiles are analyzed by the transfer matrix of transmitted and reflected beam amplitudes in multilayers. The propagation wave number for guided-wave modes is obtained from the condition that certain elements in the transfer matrix must be zero. This numerical technique requires much shorter computer times compared with the usual method of solving the eigenvalue equations, obtained by setting the characteristic determinant to zero. The analysis is also applicable either to waveguides that have losses or to certain cases of uniaxial dielectric anisotropy. All waveguides are assumed to be magnetically isotropic. Some examples of the analysis of graded-index profiles and calculations of the effect of metal claddings and prism perturbations on guided modes are given.

© 1985 Optical Society of America

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References

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  1. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 2395–2413 (1971).
    [Crossref] [PubMed]
  2. D. P. Gia Russo, J. H. Harris, “Wave propagation in anisotropic thin film optical waveguides,”J. Opt. Soc. Am. 63, 138–145 (1973).
    [Crossref]
  3. M. O. Vassell, “Structure of optical guided modes in planar multilayers of optically anisotropic materials,”J. Opt. Soc. Am. 64, 166–173 (1974).
    [Crossref]
  4. Y. Yamato, T. Kamiya, H. Yanai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with low index dielectric buffer layer,” IEEE J. Quantum Electron. QE-11, 729–736 (1975).
    [Crossref]
  5. G. B. Hocker, W. K. Burns, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
    [Crossref]
  6. T. Miyamoto, M. Momoda, “Propagation characteristics of a multilayered thin film optical waveguide with buffer layer,”J. Opt. Soc. Am. 72, 1163–1166 (1982).
    [Crossref]
  7. J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, London, 1970), pp. 8–21.
  8. E. F. Kuester, D. C. Chang, “Propagation, attenuation, and dispersion characteristics of inhomogeneous dielectric slab waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 98–106 (1975).
    [Crossref]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 665–718.
  10. V. Ramaswamy, “Ray model of energy and power flow in anisotropic film waveguides,”J. Opt. Soc. Am. 64, 1313–1320 (1974).
    [Crossref]
  11. E. C. Jordon, K. G. Balman, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), pp. 100–111.
  12. The computer-time differential depends on the size of the problem under investigation. As an example, in obtaining the normalized propagation constant of a 10-layer structure, the zero-element method takes less than 5% of the computation time that Vassell’s method requires.
  13. T. Findakly, C. L. Chen, “Diffused optical waveguides with exponential profiles: effect of metal-clad and dielectric overlay,” Appl. Opt. 17, 469–474 (1978).
    [Crossref] [PubMed]

1982 (1)

1978 (1)

1975 (3)

E. F. Kuester, D. C. Chang, “Propagation, attenuation, and dispersion characteristics of inhomogeneous dielectric slab waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 98–106 (1975).
[Crossref]

Y. Yamato, T. Kamiya, H. Yanai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with low index dielectric buffer layer,” IEEE J. Quantum Electron. QE-11, 729–736 (1975).
[Crossref]

G. B. Hocker, W. K. Burns, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
[Crossref]

1974 (2)

1973 (1)

1971 (1)

Balman, K. G.

E. C. Jordon, K. G. Balman, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), pp. 100–111.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 665–718.

Burns, W. K.

G. B. Hocker, W. K. Burns, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
[Crossref]

Chang, D. C.

E. F. Kuester, D. C. Chang, “Propagation, attenuation, and dispersion characteristics of inhomogeneous dielectric slab waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 98–106 (1975).
[Crossref]

Chen, C. L.

Findakly, T.

Gia Russo, D. P.

Harris, J. H.

Hocker, G. B.

G. B. Hocker, W. K. Burns, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
[Crossref]

Jordon, E. C.

E. C. Jordon, K. G. Balman, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), pp. 100–111.

Kamiya, T.

Y. Yamato, T. Kamiya, H. Yanai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with low index dielectric buffer layer,” IEEE J. Quantum Electron. QE-11, 729–736 (1975).
[Crossref]

Kuester, E. F.

E. F. Kuester, D. C. Chang, “Propagation, attenuation, and dispersion characteristics of inhomogeneous dielectric slab waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 98–106 (1975).
[Crossref]

Miyamoto, T.

Momoda, M.

Ramaswamy, V.

Tien, P. K.

Vassell, M. O.

Wait, J. R.

J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, London, 1970), pp. 8–21.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 665–718.

Yamato, Y.

Y. Yamato, T. Kamiya, H. Yanai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with low index dielectric buffer layer,” IEEE J. Quantum Electron. QE-11, 729–736 (1975).
[Crossref]

Yanai, H.

Y. Yamato, T. Kamiya, H. Yanai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with low index dielectric buffer layer,” IEEE J. Quantum Electron. QE-11, 729–736 (1975).
[Crossref]

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

Y. Yamato, T. Kamiya, H. Yanai, “Characteristics of optical guided modes in multilayer metal-clad planar optical guide with low index dielectric buffer layer,” IEEE J. Quantum Electron. QE-11, 729–736 (1975).
[Crossref]

G. B. Hocker, W. K. Burns, “Modes in diffused optical waveguides of arbitrary index profile,” IEEE J. Quantum Electron. QE-11, 270–276 (1975).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

E. F. Kuester, D. C. Chang, “Propagation, attenuation, and dispersion characteristics of inhomogeneous dielectric slab waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 98–106 (1975).
[Crossref]

J. Opt. Soc. Am. (4)

Other (4)

E. C. Jordon, K. G. Balman, Electromagnetic Waves and Radiating Systems (Prentice-Hall, Englewood Cliffs, N.J., 1968), pp. 100–111.

The computer-time differential depends on the size of the problem under investigation. As an example, in obtaining the normalized propagation constant of a 10-layer structure, the zero-element method takes less than 5% of the computation time that Vassell’s method requires.

J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, London, 1970), pp. 8–21.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 665–718.

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Figures (10)

Fig. 1
Fig. 1

Coordinate system on which the theory is based. The wave propagation is in the x direction, and the guide-thickness variation is in the z direction. The waveguide structure is a planar slab, and therefore the y coordinate has no influence on the wave propagation, i.e., the wave propagation is two dimensional.

Fig. 2
Fig. 2

Wave propagation in a uniaxial anistropic medium. Two cases of wave propagation are considered: (top) the electric field in the y direction is influenced only by the ordinary refractive index (no), and (bottom) the electric field in the plane xz is influenced by both the ordinary (no) and the extraordinary (ne) refractive indices.

Fig. 3
Fig. 3

General form of an anistropic multilayer structure. Each layer (j) is of uniform index nj1m. Pjzγ and Pjxγ are propagation constants in the z and x directions, respectively. A and B are forward- and backward-propagation wave amplitudes. Wj is the layer thickness.

Fig. 4
Fig. 4

Characteristic TE0 curve for a multilayer waveguide. The isotropic multilayer structure has a substrate index 1.5124 and a superstrate index 1.000. Each layer in the structure consists of a high-index body (refractive index, 1.80000) 20 Å thick and a low-index termination (refractive index, 1.400) 7 Å thick. Number of layers indicated on the abscissa.

Fig. 5
Fig. 5

Two-layer waveguide. In this case both layers are guiding, i.e., the electric fields are sinusoidal in both the layers. Sometimes it could also be the case that the field in one layer is evanescent.

Fig. 6
Fig. 6

Step-index waveguide. The simplest form of the optical waveguide and also considered a special case of the two-layer or multilayer waveguide.

Fig. 7
Fig. 7

Characteristics curves of an anisotropic step-index asymmetrical waveguide. The normalized propagation constant versus thickness curves for an anisotropic layer (no = 1.525, ne = 1.570) on a substrate (isotropic index, 1.457). The superstrate is air (isotropic index, 1.000).

Fig. 8
Fig. 8

Typical index profiles for a diffused guide. Two types of profiles are considered, Gaussian and exponential. The following parameters are assumed for the evaluation of the propagation constant for the TE0 mode: substrate index, 2.20; maximum index, 2.22; diffusion depth, 1.0 μm.

Fig. 9
Fig. 9

Effect of metal claddings on buffered guided waves. The normalized propagation constant and the attenuation constant are plotted for the TE0 mode as a function of the buffer-layer thickness. The structure consists of substrate index 2.20, guide index 2.210, guide thickness 1.5 μm, buffer-layer index 1.47, and metal-cladding complex index 1.44 + i3.70.

Fig. 10
Fig. 10

Prism perturbation of the guided modes. The curves are for a TE0 mode in a guide (index, 1.7) on a substrate (index, 1.45). The guide thickness is 0.2 μm, and the prism index is 1.778.

Equations (35)

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x 2 n o 2 + y 2 n o 2 + z 2 n e 2 = 1.
[ D x D y D z ] = [ n o 2 0 0 0 n o 2 0 0 0 n e 2 ] [ E x E y E z ] .
n = n e n o / ( n e 2 sin 2 θ + n o 2 cos 2 θ ) 1 / 2 .
p x γ = i β = i k n cos θ , p z γ = k n sin θ = ( n o n e ) 2 ( β 2 - k 2 n e 2 ) 1 / 2             for TM ( γ = 1 ) ,
p x γ = i β = i k n o cos θ , p z γ = k n o sin θ = ( β 2 - k 2 n o 2 ) 1 / 2             for TE ( γ = 0 ) ,
× E j m = l - i ω μ j m l μ 0 H j l , × H j m = l i ω j m l 0 E j 1 .
μ j m l = [ μ j 0 0 0 μ j 0 0 0 μ j ] ,
j m l = [ j 11 0 0 0 j 22 0 0 0 j 33 ] ,
E j y = A j 0 exp [ - p j z 0 ( z - z j - 2 ) ] + B j 0 exp [ p j z 0 ( z - z j - 2 ) ] ,
i ω μ μ 0 H j x = - p j z 0 A j 0 exp [ - p j z 0 ( z - z j - 2 ) ] + p j z 0 B j 0 exp [ p j z 0 ( z - z j - 2 ) ] ,
i ω μ μ 0 H j z = p j x 0 A j 0 exp [ - p j z 0 ( z - z j - 2 ) ] + p j x 0 B j 0 exp [ p j z 0 ( z - z j - 2 ) ] ;
H j y = A j 1 exp [ - p j z 1 ( z - z j - 2 ) ] + B j 1 exp [ p j z 1 ( z - z j - 2 ) ] ,
i ω n j o 2 0 E j x = - p j z 1 A j 1 exp [ - p j z 1 ( z - z j - 2 ) ] + p j z 1 B j 1 exp [ p j z 1 ( z - z j - 2 ) ] ,
i ω n j e 2 0 E j z = p j x 1 A j 1 exp [ - p j z 1 ( z - z j - 2 ) ] + p j x 1 B j 1 exp [ p j z 1 ( z - z j - 2 ) ] ,
[ E j y i ω μ μ 0 H j x ] = [ exp [ - p j z γ ( z - z j - 2 ) ] exp [ p j z γ ( z - z j - 2 ) ] - p j z γ exp [ - p j z γ ( z - z j - 2 ) ] p j z γ exp [ p j z γ ( z - z j - 2 ) ] ] [ A j γ B j γ ] ,             TE ( γ = 0 ) ,
[ H j y i ω 0 E j x ] = [ exp [ - p j z γ ( z - z j - 2 ) ] exp [ p j z γ ( z - z j - 2 ) ] - p j z γ n j o 2 exp [ - p j z γ ( z - z j - 2 ) ] p j z γ n j o 2 exp [ p j z γ ( z - z j - 2 ) ] ] [ A j γ B j γ ] ,             TM ( γ = 1 ) ,
[ A n γ 0 ] = [ α 1 α 2 α 3 α 4 ] [ 0 B 1 γ ] .
α 4 = 1 / ( Γ 2 z γ Γ 4 z γ ) { Γ 4 z γ + Γ 3 z γ × tanh [ p 3 z γ ( z 2 - z 1 ) ] } [ Γ 2 z γ + Γ 1 z γ tanh ( p 2 z γ z 1 ) ] + 1 / ( Γ 3 z γ Γ 4 z γ ) { Γ 3 z γ + Γ 4 z γ × tanh [ p 3 z γ ( z 2 - z 1 ) ] } [ Γ 1 z γ + Γ 2 z γ tanh ( p 2 z γ z 1 ) ] ,
Γ j z γ = p j z γ / n j o 2 γ .
tanh ( p 2 z γ z 1 ) = - Γ 2 z γ ( Γ 4 z γ + Γ 1 z γ ) / ( Γ 2 z γ 2 + Γ 1 z γ Γ 4 z γ ) .
2 p 2 z γ z 1 - 2 ϕ 21 - 2 ϕ 24 = 2 M π ,
Gaussian : n ( x ) = n s + Δ n exp ( - x 2 / D 2 ) , Exponential : n ( x ) = n s + Δ n exp ( - x / D ) ,
β / k Gaussian = 2.2057 , β / k Exponential = 2.2047.
α 4 [ N 1 , N 2 , W 2 , N 3 , W 3 , ( N 4 + i k 4 ) , ( β / k + i Λ ) ] = 0 ,
[ E j y i ω μ o H j x ] = [ exp [ - p j z γ ( z - z j - 2 ) ] e x p [ p j z γ ( z - z j - 2 ) ] - Γ j z γ exp [ - p j z γ ( z - z j - 2 ) ] Γ j z γ exp [ p j z γ ( z - z j - 2 ) ] ] × [ A ¯ j γ B ¯ j γ ] .
[ E 1 y i ω μ μ 0 H 1 x ] = [ exp [ - p 1 z γ ( - z ) ] exp [ p 1 z γ ( - z ) ] - Γ 1 z γ exp [ - p 1 z γ ( - z ) ] Γ 1 z γ exp [ 1 z γ ( - z ) ] ] [ A ¯ 1 γ B ¯ 1 γ ] ,
[ E 2 y i ω μ μ 0 H 2 x ] = [ exp ( - p 2 z γ z ) e x p ( p 2 z γ z ) - Γ 2 z γ exp ( - p 2 z γ z ) Γ 2 z γ exp ( p 2 z γ z ) ] [ A ¯ 2 γ B ¯ 2 γ ] ,
[ E 3 y i ω μ μ 0 H 3 x ] = [ exp [ - p 3 z γ ( z - z 1 ) ] exp [ p 3 z γ ( z - z 1 ) ] - Γ 3 z γ exp [ - p 3 z γ ( z - z 1 ) ] Γ 3 z γ exp [ p 3 z γ ( z - z 1 ) ] ] × [ A ¯ 3 γ B ¯ 3 γ ] ,
[ E 4 y i ω μ μ 0 H 4 x ] = [ exp [ - p 4 z γ ( z - z 2 ) ] exp [ p 4 z γ ( z - z 2 ) ] - Γ 4 z γ exp [ - p 4 z γ ( z - z 2 ) ] Γ 4 z γ exp [ p 4 z γ ( z - z 2 ) ] ] × [ A ¯ 4 γ B ¯ 4 γ ] ,
z = 0 [ E 1 y ( 0 ) H 1 x ( 0 ) ] = [ E 2 y ( 0 ) H 2 x ( 0 ) ] , z = z 1 [ E 2 y ( z 1 ) H 2 x ( z 1 ) ] = [ E 3 y ( z 1 ) H 3 x ( z 1 ) ] , z = z 2 [ E 3 y ( z 2 ) H 3 x ( z 2 ) ] = [ E 4 y ( z 2 ) H 4 x ( z 2 ) ] .
[ A ¯ 4 γ B ¯ 4 γ ]
[ A ¯ 1 γ B 1 γ ] .
[ A ¯ 4 γ B ¯ 4 γ ] = [ 1 1 - Γ 4 z γ Γ 4 z γ ] - 1 M f 4 [ exp [ - p 3 z γ ( z 2 - z 1 ) ] exp [ p 3 z γ ( z 2 - z 1 ) ] - Γ 3 z γ exp [ - p 3 z γ ( z 2 - z 1 ) ] Γ 3 z γ exp [ p 3 z γ ( z 2 - z 1 ) ] M f 3 [ 1 1 - Γ 3 z γ Γ 3 z γ ] - 1 × [ exp ( - p 2 z γ z 1 ) exp ( p 2 z γ z 1 ) - Γ 2 z exp ( - p 2 z γ z 1 ) Γ 2 z exp ( p 2 z γ z 1 ) ] [ 1 1 - Γ 2 z γ Γ 2 z γ ] - 1 M f 2 [ 1 1 - Γ 2 z γ Γ 2 z γ ] M f 1 [ A 1 γ B 1 γ ] [ A 4 γ B 4 γ ] = [ M f 4 M f 3 M f 2 M f 1 ] [ A 1 γ B 1 γ ] .
[ A ¯ n γ B ¯ n γ ] = [ M f n M f n - 1 M f 4 M f 3 M f 2 M f 1 ] [ A ¯ 1 γ B ¯ 1 γ ] .
[ α 1 α 2 α 3 α 4 ] = [ M f n M f n - 1 M f 4 M f 3 M f 2 M f 1 ] .

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