Abstract

In this paper, we first review the existing analytical methods used to analyze step discontinuities and tapers in optical dielectric waveguides. We then develop an exact numerical technique for determining the transmitted, reflected, and radiated power that is due to a step discontinuity or a taper in an optical dielectric waveguide. We demonstrate the method by evaluating step discontinuities and tapers in various symmetric and asymmetric slab waveguides. Unlike with previous analytical methods, no physical approximations are made throughout the entire analysis. For this reason, our method can be used to evaluate a step discontinuity or a taper in a dielectric waveguide of arbitrary cross section, regardless of the magnitude of the geometrical mismatch between the input and the output waveguide sections.

© 1986 Optical Society of America

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  1. D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, “EBES: A practical electron beam exposure system,”IEEE Trans. Electron Devices ED-22, 385–392 (1975).
    [Crossref]
  2. V. Ramaswamy, M. D. Divino, “Low-loss bends for integrated optics,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1981), p. 142.
  3. N. Tzoar, R. Pascone, “Radiation loss in tapered waveguides,”J. Opt. Soc. Am. 71, 1107–1114 (1981).
    [Crossref]
  4. A. W. Snyder, “Radiation losses due to variations of radius of dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-18, 608–615 (1970).
    [Crossref]
  5. M. Kuznetsov, H. A. Hauss, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1514 (1983).
    [Crossref]
  6. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  7. F. Sporleder, H. G. Unger, Waveguide Tapers Transitions and Couplers (Institute of Electrical and Electronics Engineers, New York, 1979).
  8. A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [Crossref]
  9. W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
  10. D. Marcuse, “Radiation losses of step-tapered waveguides,” Appl. Opt. 19, 3676–3681 (1981).
    [Crossref]
  11. A. W. Snyder, “Coupled mode theory for optical fibers,”J. Opt. Soc. Am. 62, 1267–1277 (1972).
    [Crossref]
  12. D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).
  13. V. Ramaswamy, P. G. Suchoski, “Power loss at a step discontinuity in an asymmetrical dielectric slab waveguide,” J. Opt. Soc. Am. A 1, 754–759 (1984).
    [Crossref]
  14. J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
    [Crossref]
  15. R. Baets, P. E. Lagasse, “Calculation of radiation loss in integrated-optic tapers and Y-junction,” Appl. Opt. 21, 1972–1978 (1982).
    [Crossref] [PubMed]
  16. P. Danielson, “Two-dimensional propagation beam analysis of an electrooptic waveguide modulator,” IEEE J. Quantum Electron. QE-20, 1093–1097 (1984).
    [Crossref]
  17. J. Van Rooney, J. Van der Donk, P. E. Lagasse, “Beam-propagation method: analysis and assessment,”J. Opt. Soc. Am. 71, 803–810 (1981).
    [Crossref]
  18. W. J. Tomlinson, J. P. Gordon, P. W. Smith, A. E. Kaplan, “Reflection of a Gaussian beam at a nonlinear interface,” Appl. Opt. 21, 2041–2051 (1982).
    [Crossref] [PubMed]
  19. P. G. Suchoski, master’s degree thesis (University of Florida, Gainesville, Fla., 1985).

1984 (2)

P. Danielson, “Two-dimensional propagation beam analysis of an electrooptic waveguide modulator,” IEEE J. Quantum Electron. QE-20, 1093–1097 (1984).
[Crossref]

V. Ramaswamy, P. G. Suchoski, “Power loss at a step discontinuity in an asymmetrical dielectric slab waveguide,” J. Opt. Soc. Am. A 1, 754–759 (1984).
[Crossref]

1983 (1)

M. Kuznetsov, H. A. Hauss, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1514 (1983).
[Crossref]

1982 (2)

1981 (3)

1977 (1)

W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

1975 (1)

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, “EBES: A practical electron beam exposure system,”IEEE Trans. Electron Devices ED-22, 385–392 (1975).
[Crossref]

1973 (1)

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

1972 (1)

1970 (2)

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).

A. W. Snyder, “Radiation losses due to variations of radius of dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-18, 608–615 (1970).
[Crossref]

Baets, R.

Burns, W. K.

W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).

Collier, R. J.

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, “EBES: A practical electron beam exposure system,”IEEE Trans. Electron Devices ED-22, 385–392 (1975).
[Crossref]

Danielson, P.

P. Danielson, “Two-dimensional propagation beam analysis of an electrooptic waveguide modulator,” IEEE J. Quantum Electron. QE-20, 1093–1097 (1984).
[Crossref]

David, S. A.

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, “EBES: A practical electron beam exposure system,”IEEE Trans. Electron Devices ED-22, 385–392 (1975).
[Crossref]

Divino, M. D.

V. Ramaswamy, M. D. Divino, “Low-loss bends for integrated optics,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1981), p. 142.

Feit, M. D.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Fleck, J. A.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Gordon, J. P.

Hauss, H. A.

M. Kuznetsov, H. A. Hauss, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1514 (1983).
[Crossref]

Herriot, D. R.

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, “EBES: A practical electron beam exposure system,”IEEE Trans. Electron Devices ED-22, 385–392 (1975).
[Crossref]

Kaplan, A. E.

Kuznetsov, M.

M. Kuznetsov, H. A. Hauss, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1514 (1983).
[Crossref]

Lagasse, P. E.

Marcuse, D.

D. Marcuse, “Radiation losses of step-tapered waveguides,” Appl. Opt. 19, 3676–3681 (1981).
[Crossref]

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Pascone, R.

Ramaswamy, V.

V. Ramaswamy, P. G. Suchoski, “Power loss at a step discontinuity in an asymmetrical dielectric slab waveguide,” J. Opt. Soc. Am. A 1, 754–759 (1984).
[Crossref]

V. Ramaswamy, M. D. Divino, “Low-loss bends for integrated optics,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1981), p. 142.

Smith, P. W.

Snyder, A. W.

A. W. Snyder, “Coupled mode theory for optical fibers,”J. Opt. Soc. Am. 62, 1267–1277 (1972).
[Crossref]

A. W. Snyder, “Radiation losses due to variations of radius of dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-18, 608–615 (1970).
[Crossref]

Sporleder, F.

F. Sporleder, H. G. Unger, Waveguide Tapers Transitions and Couplers (Institute of Electrical and Electronics Engineers, New York, 1979).

Stafford, J. W.

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, “EBES: A practical electron beam exposure system,”IEEE Trans. Electron Devices ED-22, 385–392 (1975).
[Crossref]

Suchoski, P. G.

Tomlinson, W. J.

Tzoar, N.

Unger, H. G.

F. Sporleder, H. G. Unger, Waveguide Tapers Transitions and Couplers (Institute of Electrical and Electronics Engineers, New York, 1979).

Van der Donk, J.

Van Rooney, J.

Yariv, A.

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

Appl. Opt. (3)

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Bell Syst. Tech. J. (1)

D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).

IEEE J. Quantum Electron. (4)

P. Danielson, “Two-dimensional propagation beam analysis of an electrooptic waveguide modulator,” IEEE J. Quantum Electron. QE-20, 1093–1097 (1984).
[Crossref]

M. Kuznetsov, H. A. Hauss, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. QE-19, 1505–1514 (1983).
[Crossref]

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).

IEEE Trans. Electron Devices (1)

D. R. Herriot, R. J. Collier, S. A. David, J. W. Stafford, “EBES: A practical electron beam exposure system,”IEEE Trans. Electron Devices ED-22, 385–392 (1975).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, “Radiation losses due to variations of radius of dielectric or optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-18, 608–615 (1970).
[Crossref]

J. Opt. Soc. Am. (3)

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

Other (4)

P. G. Suchoski, master’s degree thesis (University of Florida, Gainesville, Fla., 1985).

V. Ramaswamy, M. D. Divino, “Low-loss bends for integrated optics,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1981), p. 142.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

F. Sporleder, H. G. Unger, Waveguide Tapers Transitions and Couplers (Institute of Electrical and Electronics Engineers, New York, 1979).

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Figures (6)

Fig. 1
Fig. 1

Step discontinuity in a symmetric slab waveguide.

Fig. 2
Fig. 2

Step-tapered waveguide transition.

Fig. 3
Fig. 3

Power loss (PLOSS/PIN) versus width 1 for a waveguide with a small value of Δn: TE mode, nf = 1.512, ns = 1.51, λ = 0.6328 μm, width 2 = constant = 0.4 μm. Curve 1, modified step-transition method; curve 2, exact numerical method.

Fig. 4
Fig. 4

Power loss (PLOSS/PIN) versus width 1 for a waveguide with a large value of Δn: TE mode, nf = 1.52, ns = 1.51, λ = 0.6328 μm, width 2 = constant = 0.3 μm. Curve 1, modified step-transition method; curve 2, exact numerical method.

Fig. 5
Fig. 5

Analysis of step discontinuity in a symmetric slab waveguide: TE mode, nf = 1.512, ns = 1.51, λ = 0.6328 μm. Curves 1 and 2 were obtained by using the modified step-transition method, whereas the exact numerical method was used for curves 3 and 4. For curves 1 and 3, width 2 = 0.4 μm, width 1 varies. For curves 2 and 4, width 1 = 0.4 μm, width 2 varies.

Fig. 6
Fig. 6

Power loss versus taper length for linear step-tapered wave- guides composed of M = 2, 4, 6, 10, and 15 steps: TE mode, nf = 1.517, ns = 1.515, nc = 1.00, λ = 0.6328 μm, dIN = 2.5 μm, dOUT = 4.0 μm. (a) Exact numerical method, (b) coupled-mode theory, (c) step-transition method.

Tables (2)

Tables Icon

Table 1 Effect of Panel Size on Numerical Resultsa

Tables Icon

Table 2 Effect of Panel Size on Numerical Resultsa

Equations (25)

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d a 0 ( z ) d z = I g g ( z ) a 0 ( z ) + 0 k 0 n s I g r ( κ s , z ) q T ( κ s , z ) × exp { j 0 z [ β 0 ( z ) - β ( κ s ) ] d z } d κ s ,
d q T ( κ s , z ) d z = I g r ( κ s , z ) a 0 ( z ) exp { j 0 z [ β ( κ s ) - β 0 ( z ) ] d z } .
I g g ( z ) = ω 0 4 [ β 0 ( z - ) - β 0 ( z + ) ] - d r ( x , z ) d z E 0 * ( z - ) E 0 ( z + ) d x .
q T ( κ s , L ) = 0 L I g r ( κ s , z ) exp { j 0 z [ β ( κ s ) - β 0 ( z ) ] d z } d z .
P LOSS = 0 k 0 n s q T ( κ s , L ) 2 d κ s .
( 1 + a R ) e y ( I ) + 0 k 0 n s q R ( κ s ) e y ( R ) ( κ s ) d κ s = c T e y ( T ) + 0 k 0 n s q T ( κ s ) e y ( T ) ( κ s ) d κ s ,
( 1 - a R ) β 1 e y ( I ) - 0 k 0 n s q R ( κ s ) B ( κ s ) e y ( R ) ( κ s ) d κ s = c T β 2 e y ( T ) + 0 k 0 n s q T ( κ s ) β ( κ s ) e y ( T ) ( κ s ) d κ s ,
q T ( κ s ) = [ β 1 + β ( κ s ) ] 4 ω μ 0 I ,
q R ( κ s ) = [ β 1 - β ( κ s ) ] 4 ω μ 0 I ,
P LOSS P in = 0 k 0 n s [ q R ( κ s ) 2 + q T ( κ s ) 2 ] d κ s .
q R ( κ s ) = q R 1 ( κ s ) + i = 2 M c T ( i - 1 ) q R i ( κ s ) × exp { - j [ β T i + ( i - 1 ) β ( κ s ) ] Δ z }
q T ( κ s ) = q T 1 ( κ s ) + i = 2 M c T ( i - 1 ) q T i ( κ s ) × exp { - j [ β T i - ( i - 1 ) β ( κ s ) ] Δ z } ,
E ( x , z + Δ z ) = exp ( - j k 0 n s Δ z ) exp ( - j Δ z 2 k 0 n s T 2 ) × exp { - j k 0 n s Δ z 2 [ n 2 ( x , z + Δ z / 2 ) n s 2 - 1 ] } × E ( x , z ) + 0 ( Δ z ) 3 ,
d 2 E y ( x , z ) d x 2 + d 2 E y ( x , z ) d z 2 + k 0 2 n 2 ( x , z ) E y ( x , z ) = 0.
d 2 E d x 2 | i = E i - 1 - 2 E i + E i + 1 Δ x 2
d E y ( x , z = 0 + ) d z = - j β 1 E y ( x , z = 0 - ) ,
( 1 + a R ) - e y ( I ) e y ( I ) d x + 0 k 0 n s q R ( κ s ) [ - e y ( I ) e y ( R ) ( κ s ) d x ] d κ s = c T - e y ( I ) e y ( T ) d x + 0 k 0 n s q T ( κ s ) [ - e y ( I ) e y ( T ) ( κ s ) d x ] d κ s .
2 ω μ 0 β 1 ( 1 + a R ) = c T - e y ( I ) e y ( T ) d x + 0 k 0 n s q T ( κ s ) [ - e y ( I ) e y ( T ) ( κ s ) d x ] d κ s .
2 ω μ 0 β 1 ( 1 + a R ) = c T - e y ( I ) e y ( T ) d x + 4 Δ κ s 3 i = 1 odd N - 1 q T ( κ s i ) - e y ( I ) e y ( T ) ( κ s i ) d x + 2 Δ κ s 3 i = 2 even N - 2 q T ( κ s i ) - e y ( I ) e y ( T ) ( κ s i ) d x ,
( 1 - a R ) β 1 - e y ( I ) e y ( T ) d x - 4 Δ κ s 3 × i = 1 odd N - 1 β ( κ s i ) q R ( κ s i ) - e y ( T ) e y ( R ) ( κ s i ) d x - 2 Δ κ s 3 i = 2 even N - 2 β ( κ s i ) q R ( κ s i ) × - e y ( T ) e y ( R ) ( κ s i ) d x = 2 ω μ 0 c T ,
q R ( κ s i ) ( 2 ω μ 0 β r i ) = c T - e y ( T ) e y ( R ) ( κ s i ) d x + 4 Δ κ s 3 j = 1 odd N - 1 q T ( κ s j ) - e y ( T ) ( κ s j ) e y ( R ) ( κ s i ) d x + 2 Δ κ s 3 j = 2 even N - 2 q T ( κ s j ) - e y ( T ) ( κ s j ) e y ( R ) ( κ s i ) d x .
β 1 ( 1 - a R ) - e y ( I ) e y ( T ) ( κ s i ) d x - 4 Δ κ s 3 × j = 1 odd N - 1 β ( κ s j ) q R ( κ s j ) - e y ( R ) ( κ s j ) × e y ( T ) ( κ s i ) d x - 2 Δ κ s 3 j = 2 even N - 2 β ( κ s j ) q R ( κ s j ) × - e y ( R ) ( κ s j ) e y ( T ) ( κ s i ) = 2 ω μ 0 q T ( κ s i ) .
[ c T 1 exp ( - j β 2 Δ z ) + a R 2 ] e y 2 ( I ) + 0 k 0 n s { q R 2 ( κ s ) + q T 1 ( κ s ) exp [ - j β ( κ s ) Δ z ] } e y 2 ( R ) ( κ s ) d κ s = c T 2 e y 3 ( T ) + 0 k 0 n s q T 2 ( κ s ) e y 3 ( T ) ( κ s ) d κ s
[ β 2 c T 1 exp ( - j β 2 Δ z ) - a R 2 β 2 ] e y 2 ( I ) - 0 k 0 n s { q T 1 ( κ s ) exp [ - j β ( κ s ) Δ z ] - q R 2 ( κ s ) } β ( κ s ) e y 2 ( R ) ( κ s ) d κ s = c T 2 β 3 e y 3 ( T ) + 0 k 0 n s β ( κ s ) q T 2 ( κ s ) e y 3 ( T ) ( κ s ) d κ s .
[ A ] x = b ,

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