Abstract

An asymmetry measure is introduced to characterize thin-film optical waveguides that are asymmetric in refractive index. Together with the usual normalized frequency this allows the plotting of universal charts from which the guide cutoff, the effective guide index, and the effective guide thickness can be determined by the use of simple scaling rules. The minimum value of the effective guide thickness is found to be a simple function of wavelength and the film and substrate indices.

© 1974 Optical Society of America

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References

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  1. J. E. Goell, R. D. Standley, Proc. IEEE 58, 1504 (1970).
    [CrossRef]
  2. S. E. Miller, IEEE J. Quantum Electron. QE-8, 199 (1972).
    [CrossRef]
  3. P. K. Tien, Appl. Opt. 10, 2395 (1971).
    [CrossRef] [PubMed]
  4. N. S. Kapany, J. J. Burke, J. Opt. Soc. Am. 51, 1067 (1961).
    [CrossRef]
  5. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  6. W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
    [CrossRef]
  7. E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).
  8. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  9. H. Kressel, “Semiconductor Lasers,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-DuBois, Eds. (North Holland, Amsterdam, 1972).
  10. I. P. Kaminow, J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
    [CrossRef]
  11. V. Ramaswamy, Appl. Opt. 13, 1363 (1974).
    [CrossRef] [PubMed]
  12. N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 82.
  13. H. Kogelnik, H. P. Weber, J. Opt. Soc. Am. 64, 174 (1974).
    [CrossRef]

1974 (2)

1973 (1)

I. P. Kaminow, J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

1972 (1)

S. E. Miller, IEEE J. Quantum Electron. QE-8, 199 (1972).
[CrossRef]

1971 (2)

1970 (1)

J. E. Goell, R. D. Standley, Proc. IEEE 58, 1504 (1970).
[CrossRef]

1969 (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

1965 (1)

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

1961 (1)

Anderson, W. W.

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

Burke, J. J.

N. S. Kapany, J. J. Burke, J. Opt. Soc. Am. 51, 1067 (1961).
[CrossRef]

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 82.

Carruthers, J. R.

I. P. Kaminow, J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

Gloge, D.

Goell, J. E.

J. E. Goell, R. D. Standley, Proc. IEEE 58, 1504 (1970).
[CrossRef]

Kaminow, I. P.

I. P. Kaminow, J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

Kapany, N. S.

N. S. Kapany, J. J. Burke, J. Opt. Soc. Am. 51, 1067 (1961).
[CrossRef]

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 82.

Kogelnik, H.

Kressel, H.

H. Kressel, “Semiconductor Lasers,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-DuBois, Eds. (North Holland, Amsterdam, 1972).

Marcatili, E. A. J.

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

Marcuse, D.

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

Miller, S. E.

S. E. Miller, IEEE J. Quantum Electron. QE-8, 199 (1972).
[CrossRef]

Ramaswamy, V.

Standley, R. D.

J. E. Goell, R. D. Standley, Proc. IEEE 58, 1504 (1970).
[CrossRef]

Tien, P. K.

Weber, H. P.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

I. P. Kaminow, J. R. Carruthers, Appl. Phys. Lett. 22, 326 (1973).
[CrossRef]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, Bell Syst. Tech. J. 48, 2071 (1969).

IEEE J. Quantum Electron. (2)

W. W. Anderson, IEEE J. Quantum Electron. QE-1, 228 (1965).
[CrossRef]

S. E. Miller, IEEE J. Quantum Electron. QE-8, 199 (1972).
[CrossRef]

J. Opt. Soc. Am. (2)

Proc. IEEE (1)

J. E. Goell, R. D. Standley, Proc. IEEE 58, 1504 (1970).
[CrossRef]

Other (3)

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

H. Kressel, “Semiconductor Lasers,” in Laser Handbook, F. T. Arecchi, E. O. Schultz-DuBois, Eds. (North Holland, Amsterdam, 1972).

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 82.

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Figures (6)

Fig. 1
Fig. 1

Asymmetric slab waveguide configuration.

Fig. 2
Fig. 2

Guide index b as a function of normalized frequency V for the lowest three TE-mode orders for various degrees of asymmetry.

Fig. 3
Fig. 3

Guide index b for the single-TE-mode regime. Cutoff values are indicated by circles.

Fig. 4
Fig. 4

Normalized effective guide thickness W as a function of normalized frequency V for the fundamental TE mode.

Fig. 5
Fig. 5

Guide index b as a function of normalized frequency V for the fundamental TM mode with asymmetry a = aM = 1 and ns/nf = 0.7 − 1.

Fig. 6
Fig. 6

Normalized effective guide thickness W as a function of normalized frequency V for the fundamental TM mode for various degrees of asymmetry a and ns/nf = 0.7 − 1.

Tables (2)

Tables Icon

Table I Asymmetry Measures for TE Modes (aE) and TM Modes (aM)

Tables Icon

Table II Waveguides Parameters

Equations (41)

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V = k f ( n f 2 n s 2 ) 1 / 2 .
N = β / k
κ f = m π + ϕ s + ϕ c m = 0,1,2 , ,
κ 2 = k 2 ( n f 2 N 2 ) .
γ s 2 = k 2 ( N 2 n s 2 ) , γ c 2 = k 2 ( N 2 n c 2 ) .
b = ( N 2 n s 2 ) / ( n f 2 n s 2 ) N 2 = n s 2 + b ( n f 2 n s 2 ) .
N n s + b ( n f n s ) ,
tan ϕ s = γ s / κ , tan ϕ c = γ c / κ .
V ( 1 b ) 1 / 2 = m π + tan 1 [ b / ( 1 b ) ] 1 / 2 + tan 1 [ ( b + a ) / ( 1 b ) ] 1 / 2 .
a = ( n s 2 n c 2 ) / ( n f 2 n s 2 ) .
V 0 = tan 1 ( a ) 1 / 2 ,
( f / λ ) 0 = [ 1 / 2 π ( n f 2 n s 2 ) 1 / 2 ] tan 1 ( a ) 1 / 2 .
( f / λ ) 0 = ( 1 4 ) ( n f 2 n s 2 ) 1 / 2 .
V m = V 0 + m π .
m = ( 2 f / λ ) ( n f 2 n s 2 ) 1 / 2 .
b V = 2 ( 1 b ) / [ V + 1 ( b ) 1 / 2 + 1 ( b + a ) 1 / 2 ] = 2 ( 1 b ) / W .
b V | b = 0 = 0.
tan 1 x = ( π / 2 ) ( 1 / x ) + for x 1
b = 1 ( m + 1 ) 2 π 2 / { V + 1 + [ 1 / ( 1 + a ) ] 1 / 2 } 2 ,
[ There is no Eq . ( 20 ) ]
w = f + ( 1 / γ s ) + ( 1 / γ c ) .
W = k w ( n f 2 n s 2 ) 1 / 2 .
W = V + [ 1 / ( b ) 1 / 2 ] + [ 1 / ( b + a ) ] 1 / 2 .
W = V + 1 + 1 / ( 1 + a ) 1 / 2 , for V 1.
W min = 4.40
w min / λ = 0.7 / ( n f 2 n s 2 ) 1 / 2 ,
tan ϕ s = ( n f 2 / n s 2 ) ( γ s / κ ) tan ϕ c = ( n f 2 / n c 2 ) ( γ c / κ ) .
b = [ ( N 2 n s 2 ) / ( n f 2 n s 2 ) ] [ n f 2 / ( n s 2 q s ) ] ,
N 2 = [ n f 2 ( 1 b ) + n s 2 b ] q s
q s = N 2 n f 2 + N 2 n s 2 1 = n s 2 / n f 2 ( 1 b ) + b n s 4 / n f 4 .
V [ ( q s ) 1 / 2 n f / n s ] ( 1 b ) 1 / 2 = m π + tan 1 ( b 1 b ) 1 / 2 + tan 1 [ b + a ( 1 b d ) 1 b ] 1 / 2 .
a = ( n f 4 / n c 4 ) · [ ( n s 2 n c 2 ) / ( n f 2 n s 2 ) ]
d = ( 1 n s 2 / n f 2 ) ( 1 n c 2 / n f 2 ) .
( n f 2 / n s 2 ) q s 1 + 2 b ( 1 n s 2 / n f 2 ) .
V E ( b , a ) = V M ( b , a ) [ ( q s ) 1 / 2 n f / n s ]
b = 1 ( m + 1 ) 2 π 2 / [ V ( n f 2 / n s 2 ) + 1 + 1 / ( 1 + a a d ) 1 / 2 ] 2
1 + a a d = ( n s 4 / n c 4 ) [ ( n f 2 n c 2 ) / ( n f 2 n s 2 ) ] .
w = f + [ 1 / ( q s γ s ) ] + [ 1 / ( q c γ c ) ] ,
q c = N 2 n f 2 + N 2 n c 2 1 = q s × [ ( 1 b ) ( 1 + n f 2 n c 2 n f 2 n s 2 ) + b n s 2 n c 2 ] .
W [ ( q s ) 1 / 2 n f / n s ] = V [ ( q s ) 1 / 2 n f / n s ] + 1 / [ q s ( n s / n f ) 2 ( b ) 1 / 2 ] + 1 / { q c ( n c / n f ) 2 [ b + a ( 1 b d ) ] 1 / 2 } .
q c n c 2 / n f 2 = 1 ( 1 b ) ( 1 n s 2 / n f 2 ) ( 1 + n c 2 / n f 2 ) .

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