Abstract

The calculation of a propagation constant is an interesting problem in the study of planar waveguides. We present a new method for solving the problem of a planar dielectric waveguide with arbitrary refractive-index variation for either TE or TM modes, and this method is simpler than other methods. Taking a parabolic refractive-index profile as an example, we show that high accuracy can be attained with this new method, provided that the waveguide is divided into layers of proper number and thickness.

© 1993 Optical Society of America

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References

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  1. J. N. Polky, J. Opt. Soc. Am. 64, 274 (1974).
    [CrossRef]
  2. T. Tamir, Integrated Optics, 2nd ed. (Springer-Verlag, Berlin, 1979), Chap. 2.
  3. Z. H. Wang, S. R. Seshadri, J. Opt. Soc. Am. A 6, 142 (1989).
    [CrossRef]
  4. J. Chilwell, I. Hodgekindon, J. Opt. Soc. Am. A 1, 742 (1984).
    [CrossRef]
  5. L. M. Walpita, J. Opt. Soc. Am. A 2, 595 (1985).
    [CrossRef]
  6. H.-C. Huang, Z.-H. Wang, Electron. Lett. 17, 202 (1981).
    [CrossRef]

1989 (1)

1985 (1)

1984 (1)

1981 (1)

H.-C. Huang, Z.-H. Wang, Electron. Lett. 17, 202 (1981).
[CrossRef]

1974 (1)

Chilwell, J.

Hodgekindon, I.

Huang, H.-C.

H.-C. Huang, Z.-H. Wang, Electron. Lett. 17, 202 (1981).
[CrossRef]

Polky, J. N.

Seshadri, S. R.

Tamir, T.

T. Tamir, Integrated Optics, 2nd ed. (Springer-Verlag, Berlin, 1979), Chap. 2.

Walpita, L. M.

Wang, Z. H.

Wang, Z.-H.

H.-C. Huang, Z.-H. Wang, Electron. Lett. 17, 202 (1981).
[CrossRef]

Electron. Lett. (1)

H.-C. Huang, Z.-H. Wang, Electron. Lett. 17, 202 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Other (1)

T. Tamir, Integrated Optics, 2nd ed. (Springer-Verlag, Berlin, 1979), Chap. 2.

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Figures (2)

Fig. 1
Fig. 1

Geometry of a multilayer planar waveguide.

Fig. 2
Fig. 2

Parabolic index profile (solid curve) and the practical guide profile (dashed line) that is approximated by it.

Tables (2)

Tables Icon

Table 1 Comparison of the Calculated β of the TE0 Mode with the Exact Value for the Parabolic Profile

Tables Icon

Table 2 Comparison of the Calculated β of the TE1 Mode with the Exact Value for the Parabolic Profile

Equations (29)

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i ω H x ( x , z ) = - z E y ( x , z ) ,
i ω H z ( x , z ) = x E y ( x , z ) ,
- i ω E v E y ( x , z ) = z H x ( x , z ) - x H z ( x , z ) ,
E v { E c , n , x > a n E f , j , a j + 1 > x > a j             ( j = 1 , 2 , 3 , , n - 1 ) E f , 0 , a 1 > x > - b 1 E f , - j , - b j > x > - b j + 1             ( j = 1 , 2 , 3 , , m - 1 ) E s , - m , - b m > x .
( 2 x 2 + 2 z 2 + ω 2 E v ) E y ( x , z ) = 0.
E y ( x ) = { D n exp [ - α c , n ( x - a n ) ] ,             x > a n D j exp [ i k f , j ( x - a j ) ] + E j exp [ - i k f , j ( x - a j + 1 ) ] ,             a j + 1 > x > a j ,             j = 1 , 2 , 3 , , ( n - 1 ) G 0 exp [ i k f , 0 ( x + b 1 ) ] + E 0 exp [ - i k f , 0 ( x - a 1 ) ] ,             a 1 > x > - b 1 F j exp [ - i k f , - j ( x + b j ) ] + G j exp [ i k f , - j ( x + b j + 1 ) ] ,             - b j > x > - b j + 1 ,             j = 1 , 2 , 3 , , ( m - 1 ) F m exp [ α s , - m ( x + b m ) ] ,             x < - b m
α v = ( β 2 - ω 2 E v ) 1 / 2 ,             v = c , n ; s , - m ,
k f , l = ( ω 2 E f , l - β 2 ) 1 / 2 , l = ( n - 1 ) , ( n - 2 ) , , 1 , 0 , - 1 , , - ( m - 1 ) .
exp [ i k f , ( j - 1 ) ( a j - a j - 1 ) ] + E j - 1 / D j - 1 exp [ i k f , ( j - 1 ) ( a j - a j - 1 ) ] - E j - 1 / D j - 1 = 1 i tan Φ c , j ,
exp [ i k f , 0 ( a 1 + b 1 ) ] + E 0 / G 0 exp [ i k f , 0 ( a 1 + b 1 ) ] - E 0 / G 0 = 1 i tan Φ c , 1 ,
tan Φ c , n = α c , n / k f , ( n - 1 ) ,
tan Φ c , j = tan α c , j tan [ - k f , j ( a j + 1 - a j ) + Φ c , ( j + 1 ) ] ,             j = 1 , 2 , 3 , , ( n - 1 ) ,
tan α c , j = k f , j / k f , ( j - 1 ) ,             j = 1 , 2 , 3 , , ( n - 1 ) .
E 0 / G 0 = exp { i [ k f , 0 ( a 1 + b 1 ) - 2 Φ c , 1 ] } .
G 0 / E 0 = exp { i [ k f , 0 ( a 1 + b 1 ) - 2 Φ s , - 1 ] } ,
tan Φ s , - j = tan α s , - j tan [ - k f , - j ( b j + 1 - b j ) + Φ s , - ( j + 1 ) ] ,             j = 1 , 2 , 3 , , ( m - 1 ) ,
tan Φ s , - m = α s , - m / k f , - ( m - 1 ) ,
tan α s , - j = k f , - j / k f , - ( j - 1 ) ,             j = 1 , 2 , , ( m - 1 ) .
k f , 0 ( a 1 + b 1 ) - Φ c , 1 - Φ s , 1 - p π = 0             ( p = 0 , 1 , 2 , ) ,
β i = β i - 1 - D ( β i - 1 ) / D ( β i - 1 ) ,             i = 1 , 2 , ,
D ( β ) = k f , 0 ( a 1 + b 1 ) - Φ c , 1 - Φ s , - 1 - p π ,
D ( β ) = - [ β k f , 0 ( a 1 + b 1 ) + Φ c , 1 Φ s , - 1 ] ,
( Φ c , j ) = cos Φ c , j { β ( 1 k f , j - 1 2 - 1 k f , j 2 ) tan Φ c , j + ( tan 2 α c , j + tan 2 Φ c , j ) 1 tan α c , j × [ - β k f , j ( a j + 1 - a j ) + ( Φ c , j + 1 ) ] } ,             j = 1 , 2 , , ( n - 1 ) .
tan Φ c , n = E f , ( n - 1 ) α c , n / E c , n k f , ( n - 1 ) ,
tan α c , j = E f , ( j - 1 ) k f , j / E f , j k f , ( j - 1 ) ,             j = 1 , 2 , , ( n - 1 ) ,
tan Φ s , - m = E f , - ( m - 1 ) α s , - m / E s , - m k f , - ( m - 1 ) ,
tan α s , - j = E f , - ( j - 1 ) k f , - j / E f , - j k f , - ( j - 1 ) ,             j = 1 , 2 , , ( m - 1 ) .
n 2 ( x ) = n f 2 ( 1 - x 2 / x 0 2 )
n ( x ) n f ( 1 - ½ x 2 / x 0 2 ) ,

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