Abstract

Power loss at an abrupt but small step discontinuity in an asymmetrical single-mode waveguide is analyzed. An integral expression is given for the power loss suffered by the transverse-electric and transverse-magnetic modes as a function of wavelength, film index, cover index, substrate index, and input and output waveguide widths so that any slab-waveguide discontinuity can be analyzed. Normalized power-loss plots are presented for two specific, commonly used waveguides with different degrees of asymmetry. It is shown that the magnitude of the power loss at a discontinuity depends more on the mismatch in the modal effective width than on the geometrical mismatch in waveguide width. In particular, close to cutoff, guides exhibit severe power loss even for small step size, whereas waveguides that are sufficiently far away from cutoff can tolerate a relatively large step discontinuity with surprisingly low power loss.

© 1984 Optical Society of America

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References

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  1. A. F. Milton, W. K. Burns, “Mode coupling in optical waveguide horns,” IEEE J. Quantum Electron. QE-13, 828–835 (1977).
    [CrossRef]
  2. A. R. Nelson, “Coupling optical waveguides by tapers,” Appl. Opt. 14, 3012–3015 (1975).
    [CrossRef] [PubMed]
  3. R. K. Winn, J. H. Harris, “Coupling from multimode to single-mode linear waveguides using horn-shaped structures,” IEEE Trans. Microwave Theory Appl. MTT-23, 92–97 (1975).
    [CrossRef]
  4. D. Marcuse, “Radiation losses of tapered dielectric slab waveguides,” Bell Syst. Tech. J. 49, 273–290 (1970).
  5. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  6. V. Shevchenko, Continuous Transitions in Open Waveguides (Golem, Boulder, Colo., 1971).
  7. D. Marcuse, Theory Of Dielectric Optical Waveguides (Academic, New York, 1974).
  8. G. H. Owyang, Foundations of Optical Waveguides (Elsevier, New York, 1951).
  9. For a description of Romberg integration, see A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).
  10. H. Kogelnik, V. Ramaswamy, “Scaling rules for thin film optical waveguides,” Appl. Opt. 13, 1857–1862 (1974).
    [CrossRef] [PubMed]

1977 (1)

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

1975 (2)

R. K. Winn, J. H. Harris, “Coupling from multimode to single-mode linear waveguides using horn-shaped structures,” IEEE Trans. Microwave Theory Appl. MTT-23, 92–97 (1975).
[CrossRef]

A. R. Nelson, “Coupling optical waveguides by tapers,” Appl. Opt. 14, 3012–3015 (1975).
[CrossRef] [PubMed]

1974 (1)

1970 (1)

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

Burns, W. K.

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

Harris, J. H.

R. K. Winn, J. H. Harris, “Coupling from multimode to single-mode linear waveguides using horn-shaped structures,” IEEE Trans. Microwave Theory Appl. MTT-23, 92–97 (1975).
[CrossRef]

Kogelnik, H.

Marcuse, D.

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

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

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

Milton, A. F.

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

Nelson, A. R.

Owyang, G. H.

G. H. Owyang, Foundations of Optical Waveguides (Elsevier, New York, 1951).

Ralston, A.

For a description of Romberg integration, see A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).

Ramaswamy, V.

Shevchenko, V.

V. Shevchenko, Continuous Transitions in Open Waveguides (Golem, Boulder, Colo., 1971).

Winn, R. K.

R. K. Winn, J. H. Harris, “Coupling from multimode to single-mode linear waveguides using horn-shaped structures,” IEEE Trans. Microwave Theory Appl. MTT-23, 92–97 (1975).
[CrossRef]

Appl. Opt. (2)

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. (1)

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

IEEE Trans. Microwave Theory Appl. (1)

R. K. Winn, J. H. Harris, “Coupling from multimode to single-mode linear waveguides using horn-shaped structures,” IEEE Trans. Microwave Theory Appl. MTT-23, 92–97 (1975).
[CrossRef]

Other (5)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

V. Shevchenko, Continuous Transitions in Open Waveguides (Golem, Boulder, Colo., 1971).

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

G. H. Owyang, Foundations of Optical Waveguides (Elsevier, New York, 1951).

For a description of Romberg integration, see A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).

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Figures (5)

Fig. 1
Fig. 1

Asymmetrical slab waveguide.

Fig. 2
Fig. 2

Step discontinuity of different waveguide widths.

Fig. 3
Fig. 3

(a) Power loss (Prad/Pin) versus (width 2/width 1) for TE modes: nf = 1.61, ns = 1.515, nc = 1.00, λ = 0.6328 μm, width 1 (2d1) = 0.3 μm. Case 1, small guide to large guide; case 2, large guide to small guide. (b) Relative difference in power radiated [(Case 1 − Case 2)/Case 2] at the discontinuity, from the results of (a).

Fig. 4
Fig. 4

(a) Power loss (Prad/Pin) versus (width 2/width 1) for TE and TM modes: Sputtered Corning 7059 glass film on Corning 7059 substrate. (b) Power loss (Prad/Pin) versus (width 2/width 1) for TE and TM modes: Boriosilicate film on Corning 7059 substrate.

Fig. 5
Fig. 5

Effective modal width versus waveguide width for nf = 1.61, ns = 1.515, nc = 1.00.

Equations (31)

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2 e y x 2 + ( k 0 2 n 2 - β 2 ) e y = 0
e y = { A cos ( κ f d + ϕ ) exp [ - α c ( x - d ) ] for x d A cos ( κ f x + ϕ ) for - d x d A cos ( κ f d - ϕ ) exp [ α s ( x + d ) ] for x - d ,
κ f = ( k 0 2 n f 2 - β 2 ) 1 / 2 , α s = ( β 2 - k 0 2 n s 2 ) 1 / 2 , α c = ( β 2 - k 0 2 n c 2 ) 1 / 2 ,
ϕ = tan - 1 ( α c / κ f ) - κ f d .
tan ( 2 κ f d ) = κ f ( α c + α s ) κ f 2 - α c α s ,
P δ i j = β i 2 ω μ 0 - e y i e y j d x ,
A = [ 4 ω μ 0 P β ( 2 d + 1 α c + 1 α s ) ] 1 / 2 .
e y = { [ B cos ( κ f d ) + C sin ( κ f d ) ] exp [ - α c ( x - d ) ] for x d B cos ( κ f x ) + C sin ( κ f x ) for - d x d [ B cos ( κ f d ) - C sin ( κ f d ) ] cos [ κ s ( x + d ) ] + ( κ f / κ s ) [ B cos ( κ f d ) + C sin ( κ f d ) ] sin [ κ s ( x + d ) ] for x - d ,
B = [ κ f cos ( κ f d ) + α c sin ( κ f d ) ] C [ κ f sin ( κ f d ) - α c cos ( κ f d ) ] , κ s = ( κ 0 2 n s 2 - β 2 ) 1 / 2 ,
- e y ( β i ) e y ( β j ) d x = 2 ω μ 0 P β i δ ( β i - β j ) .
C 2 = 4 ω μ 0 P π β { [ ( D cos ( κ f d ) - sin ( κ f d ) ] 2 + κ f 2 κ s 2 [ D sin ( κ f d ) + cos ( κ f d ) ] 2 } - 1 ,
e y = { F i [ cos ( κ f d ) + G i sin ( κ f d ) ] cos [ κ c ( x - d ) ] + F i ( κ f / κ c ) [ G i cos ( κ f d ) - sin ( κ f d ) ] sin [ κ c ( x - d ) ] for x d F i [ cos ( κ f x ) + G i sin ( κ f x ) ] for - d x d F i [ cos ( κ f d ) - G i sin ( κ f d ) ] cos [ κ s ( x + d ) ] + F i ( κ f / κ s ) [ sin ( κ f d ) + G i cos ( κ f d ) ] sin [ κ s ( x + d ) ] for x - d ,
2 cos 2 ( κ f d ) + κ f 2 sin 2 ( κ f d ) ( 1 κ c 2 + 1 κ s 2 ) + G m G n × [ 2 sin 2 ( κ f d ) + κ f 2 cos 2 ( κ f d ) ( 1 κ c 2 + 1 κ s 2 ) ] + ( G m + G n ) × [ κ f 2 sin ( κ f d ) cos ( κ f d ) ( 1 κ s 2 - 1 k c 2 ) ] = 0
G m = - G n = [ 2 cos 2 ( κ f d ) + κ f 2 sin 2 ( κ f d ) ( 1 n s 2 + 1 n c 2 ) 2 sin 2 ( κ f d ) + κ f 2 cos 2 ( κ f d ) ( 1 n s 2 + 1 n c 2 ) ] 1 / 2 .
F n , m 2 = 4 ω μ 0 P n , m π β { [ cos ( κ f d ) + G n , m sin ( κ f d ) ] 2 + [ cos ( κ f d ) - G n , m sin ( κ f d ) ] 2 + κ f 2 κ c 2 × [ G n , m cos ( κ f d ) - sin ( κ f d ) ] 2 + κ f 2 κ s 2 [ sin ( κ f d ) + G n , m cos ( κ f d ) ] 2 } - 1 .
e y ( I ) + a R e y ( R ) + 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 ,
h x ( I ) + a R h x ( R ) + 0 k 0 n s q R ( κ s ) h x ( R ) ( κ s ) d κ s = C T h x ( T ) + 0 k 0 n s q T ( κ s ) h x ( T ) ( κ s ) d κ s ,
h x = - β ω μ 0 e y for waves traveling in the - z direction , h x = β ω μ 0 e y for waves traveling in the + z direction .
β 1 e y ( I ) - a R β 1 e y ( R ) - 0 k 0 n s β r ( κ s ) q R ( κ s ) e y ( R ) ( κ s ) d κ s = C T β 2 e y ( T ) + 0 k 0 n s β r ( κ s ) q T ( κ s ) e y ( T ) ( κ s ) d κ s ,
- e y ( I ) e y ( T ) ( κ ^ s ) d x + 0 k 0 n s q R ( κ s ) × [ - e y ( R ) ( κ s ) e y ( T ) ( κ ^ s ) d x ] d κ s = C T - e y ( T ) e y ( T ) ( κ ^ s ) d x + 0 k 0 n s q T ( κ s ) [ - e y ( T ) ( κ s ) e y ( T ) ( κ ^ s ) d x ] d κ s .
- e y ( I ) e y ( T ) ( κ ^ s ) d x + q R ( κ ^ s ) 2 ω μ 0 β r ( κ ^ s ) = 2 ω μ 0 β r ( κ ^ s ) q T ( κ ^ s ) .
β 1 - e y ( I ) e y ( T ) ( κ ^ s ) d x - 2 ω μ 0 q R ( κ ^ s ) = 2 ω μ 0 q T ( κ ^ s ) .
q T ( κ s ) = β 1 + β r ( κ s ) 4 ω μ 0 I ,
q R ( κ s ) = β 1 - β r ( κ s ) 4 ω μ 0 I .
I = - e y ( I ) e y ( T ) ( κ s ) d x ,
P rad P in = 0 k 0 n s q R ( κ s ) 2 d κ s + 0 k 0 n s q T ( κ s ) 2 d κ s .
P rad P in = 0 k 0 n s q R ( β r ) 2 β r κ s d β r + 0 k 0 n s q T ( β r ) 2 β r κ s d β r .
P rad P in = - k 0 n s k 0 n s [ ( β r + β 1 ) I 4 ω μ 0 ] 2 β r κ s d β r ,
I = - e y ( I ) e y ( T ) ( κ s ) d x .
P rad P in = - k 0 n s k 0 n s [ ( β r + β 1 ) I 4 ω ɛ 0 ] 2 β r κ s d β r ,
I = - 1 n 2 ( x ) h y ( I ) h y ( T ) ( κ s ) d x .

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