Abstract

The analysis of planar strip-loaded optical waveguides with the vectorial method of lines is described. The main feature of this method is its semianalytical approach, which yields accurate results with less computational effort than other techniques. Hybrid-mode dispersion curves and field and intensity distributions of a sample waveguide are presented.

© 1991 Optical Society of America

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References

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  1. R. Pregla and W. Pascher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), Chap. 6, pp. 381–446.
  2. R. Pregla, “About the nature of the method of lines,” AEU Arch. Electron. Uebertragungstech. Electron. Commun. 41(6), 368–370 (1987).
  3. H. Diestel, “A method for calculating the guided modes of strip-loaded optical waveguides with arbitrary index profile,” IEEE J. Quantum Electron. QE-20, 1288–1293 (1984).
    [Crossref]
  4. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 6, pp. 224–257.
  5. K. Yasuura, K. Shimohara, and T. Miyamoto, “Numerical analysis of a thin-film waveguide by mode-matching method,” J. Opt. Soc. Am. 70, 183–191 (1980).
    [Crossref]
  6. Working Group I, European Cooperation in the Field of Scientific and Technical Research Project 216, “Comparison of different modelling techniques for longitudinally invariant integrated optical waveguides,” IEE Proc. 136J (5), 273–280 (1989).
  7. H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), Chap. 3, pp. 189–289.

1989 (1)

Working Group I, European Cooperation in the Field of Scientific and Technical Research Project 216, “Comparison of different modelling techniques for longitudinally invariant integrated optical waveguides,” IEE Proc. 136J (5), 273–280 (1989).

1987 (1)

R. Pregla, “About the nature of the method of lines,” AEU Arch. Electron. Uebertragungstech. Electron. Commun. 41(6), 368–370 (1987).

1984 (1)

H. Diestel, “A method for calculating the guided modes of strip-loaded optical waveguides with arbitrary index profile,” IEEE J. Quantum Electron. QE-20, 1288–1293 (1984).
[Crossref]

1980 (1)

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 6, pp. 224–257.

Diestel, H.

H. Diestel, “A method for calculating the guided modes of strip-loaded optical waveguides with arbitrary index profile,” IEEE J. Quantum Electron. QE-20, 1288–1293 (1984).
[Crossref]

Miyamoto, T.

Pascher, W.

R. Pregla and W. Pascher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), Chap. 6, pp. 381–446.

Pregla, R.

R. Pregla, “About the nature of the method of lines,” AEU Arch. Electron. Uebertragungstech. Electron. Commun. 41(6), 368–370 (1987).

R. Pregla and W. Pascher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), Chap. 6, pp. 381–446.

Shimohara, K.

Unger, H.-G.

H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), Chap. 3, pp. 189–289.

Yasuura, K.

AEU Arch. Electron. Uebertragungstech. Electron. Commun. (1)

R. Pregla, “About the nature of the method of lines,” AEU Arch. Electron. Uebertragungstech. Electron. Commun. 41(6), 368–370 (1987).

IEE Proc. (1)

Working Group I, European Cooperation in the Field of Scientific and Technical Research Project 216, “Comparison of different modelling techniques for longitudinally invariant integrated optical waveguides,” IEE Proc. 136J (5), 273–280 (1989).

IEEE J. Quantum Electron. (1)

H. Diestel, “A method for calculating the guided modes of strip-loaded optical waveguides with arbitrary index profile,” IEEE J. Quantum Electron. QE-20, 1288–1293 (1984).
[Crossref]

J. Opt. Soc. Am. (1)

Other (3)

R. Pregla and W. Pascher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), Chap. 6, pp. 381–446.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960), Chap. 6, pp. 224–257.

H.-G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977), Chap. 3, pp. 189–289.

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Figures (15)

Fig. 1
Fig. 1

Examples for planar strip-loaded optical waveguides: (a) Strip-loaded film guide, (b) rib waveguide.

Fig. 2
Fig. 2

Discretization lines.

Fig. 3
Fig. 3

Layer with arbitrary thickness d and interfaces A and B.

Fig. 4
Fig. 4

Dispersion characteristics of first three modes of a rib waveguide: ●, mode-matching method for HE00 and EH00; ×, mode-matching method5 for HE01.

Fig. 5
Fig. 5

Integrated optical waveguide on InP, according to Ref. 6, operating at λ = 1.55 μm: w = 2.4 μm, d = 0.2 μm, t is 0.0, 0.2, 0.4 μm.

Fig. 6
Fig. 6

Convergence behavior of normalized propagation constant B for HE00 mode for the waveguide of Fig. 5: N is the number of lines; h is the discretization distance, h = a/N; B0 is the extrapolated value of B.

Fig. 7
Fig. 7

Normalized x component of the electric field: nc = 1.0, nl = 3.81/2, ns = 1.51/2.

Fig. 8
Fig. 8

Calculated strip-loaded film guide: a = 9.943 μm, d = 0.3 μm, nl = 3.17 (InP), w = 2.4 μm, t = 2.4 μm, nf = 3.38 (InGaAsP), h1 = 1.0 μm, h3 = 3.4 μm, ns = 3.17 (InP), h2 = 33.0 μm, nc = 1.0, discretization distance is h = w/21.

Fig. 9
Fig. 9

Dispersion characteristics of first five modes of strip-loaded film guide depicted in Fig. 8: B = [(kz/k0)2ns2]/(nf2ns2); v = k0d(nf2ns2)1/2; points, calculated values.

Fig. 10
Fig. 10

Field distribution of HE00 mode, normalized x component of electric field at y = 0: wavelength λ = 1.55 μm.

Fig. 11
Fig. 11

Intensity distribution of HE00 mode, normalized z component of Poynting vector at y = 0: wavelength λ = 1.55 μm.

Fig. 12
Fig. 12

Intensity distribution of HE00 mode in the half-structure, contour lines of the normalized z component of Poynting vector: wavelength λ = 1.55 μm.

Fig. 13
Fig. 13

Field distribution of HE01 mode, normalized x component of electric field at y = 0: wavelength λ = 1.55 μm.

Fig. 14
Fig. 14

Intensity distribution of HE01 mode, normalized z component of Poynting vector at y = 0: wavelength λ = 1.55 μm.

Fig. 15
Fig. 15

Intensity distribution of HE01 mode in the half structure, contour lines of normalized z component of Poynting vector: wavelength λ = 1.55 μm.

Tables (1)

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Table 1 Normalized Propagation Constant B for the Fundamental Quasi-TE Mode for the Waveguide of Figure 5a

Equations (15)

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2 x 2 ψ h + 2 y 2 ψ h + [ n 2 ( x ) k 0 2 - k z 2 ] ψ h = 0 ,
n 2 ( x ) x ( 1 n 2 ( x ) x ψ e ) + 2 y 2 ψ e + [ n 2 ( x ) k 0 2 - k z 2 ] ψ e = 0 ,
k 0 2 = ω 2 μ 0 0 .
n 2 ( x ) diag [ n 2 ( x i e , h ) ] = n e , h , h x ψ h D ψ h , h x ψ e - D t ψ e , h 2 2 x 2 ψ h - D t D P h ψ h , h 2 n 2 ( x ) x ( 1 n 2 ( x ) x ψ e ) - n e D ( n h ) - 1 D t P e ψ e .
D = [ 1 - 1 1 - 1 1 - 1 ] , P h = [ 2 - 1 - 1 2 - 1 - 1 2 - 1 - 1 2 ]
[ - h - 2 P + k 0 2 n + I ( d 2 d y 2 - k z 2 ) ] ψ = 0 ,
ψ = S ψ ¯ ,
( - h - 2 P + k 0 2 n ) S = S diag ( λ i ) ,
[ I d 2 d y 2 - diag ( k y i 2 ) ] ψ ¯ = 0
k y i 2 = k z 2 - λ i ,
ψ ¯ i ( y ) = A i cosh ( k y i y ) + B i sinh ( k y i y )
[ d d y ψ ¯ A d d y ψ ¯ B ] = [ γ α α γ ] [ - ψ ¯ A ψ ¯ B ] ,
γ = diag [ k y i / tanh ( k y i d ) ]
α = diag [ k y i / sinh ( k y i d ) ] .
[ Y ¯ ( k z ) ] E ¯ = 0 ,

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