Abstract

A generalized analysis of the wave equation in a lens-like medium by the perturbation method shows that the usual analysis of the TEM approximation is not correct for the case in which the electric permittivity distribution is almost symmetric with the center axis of the lens-like medium. The correct electromagnetic wave modes are presented.

© 1973 Optical Society of America

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References

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  1. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and M. Matsumura, IEEE J. Quantum Electron. 6, 606 (1970).
    [Crossref]
  2. A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).
    [Crossref]
  3. J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), p. 343
  4. J. C. Slater, Quantum Theory of Matter (McGraw–Hill, New York, 1951), Ch. 4.

1970 (1)

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and M. Matsumura, IEEE J. Quantum Electron. 6, 606 (1970).
[Crossref]

1964 (1)

A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).
[Crossref]

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and M. Matsumura, IEEE J. Quantum Electron. 6, 606 (1970).
[Crossref]

Kitano, I.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and M. Matsumura, IEEE J. Quantum Electron. 6, 606 (1970).
[Crossref]

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and M. Matsumura, IEEE J. Quantum Electron. 6, 606 (1970).
[Crossref]

Marcatili, A. J.

A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).
[Crossref]

Matsumura, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and M. Matsumura, IEEE J. Quantum Electron. 6, 606 (1970).
[Crossref]

Slater, J. C.

J. C. Slater, Quantum Theory of Matter (McGraw–Hill, New York, 1951), Ch. 4.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), p. 343

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and M. Matsumura, IEEE J. Quantum Electron. 6, 606 (1970).
[Crossref]

Bell Syst. Tech. J. (1)

A. J. Marcatili, Bell Syst. Tech. J. 43, 2887 (1964).
[Crossref]

IEEE J. Quantum Electron. (1)

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and M. Matsumura, IEEE J. Quantum Electron. 6, 606 (1970).
[Crossref]

Other (2)

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), p. 343

J. C. Slater, Quantum Theory of Matter (McGraw–Hill, New York, 1951), Ch. 4.

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Figures (1)

F. 1
F. 1

Polarization configuration of the fields, (a) field given by Eq. (33), (b) field given by Eq. (34), (c) field given by Eq. (35), (d) field given by Eq. (36).

Equations (40)

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2 E + ( E ) + ω 2 μ E = 0 ,
t 2 E x + x ( 1 x E x + 1 y E y ) + ( ω 2 μ β 2 ) E x = 0 ,
t 2 E y + y ( 1 x E x + 1 y E y ) + ( ω 2 μ β 2 ) E y = 0 ,
t 2 E z j β ( 1 x E x + 1 y E y ) + ( ω 2 μ β 2 ) E z = 0 ,
E = ( i E x + j E y + k E z ) e j β z ,
t 2 = 2 x 2 + 2 y 2 .
= ( 0 ) { 1 k ( 0 ) 2 ( g x 2 x 2 + g y 2 y 2 ) } ,
E x = p q e xpq U p q ,
E y = p q e ypq U p q ,
E z = p q e z p q U p q ,
U p q = k ( 0 ) ( g x g y ) 1 4 ( π p ! q ! ) 1 4 H p { ( 2 g x ) 1 2 k ( 0 ) x } H q { ( 2 g y ) 1 2 k ( 0 ) y } × exp { k ( 0 ) 2 2 ( g x x 2 g y y 2 ) } ,
( 2 g x ) 1 2 k ( 0 ) x U p q = ( p + 1 ) 1 2 U ( p + 1 ) q + p 1 2 U ( p 1 ) q ,
2 ( 2 g x ) 1 2 k ( 0 ) U p q x = ( p + 1 ) 1 2 U ( p + 1 ) q + p 1 2 U ( p 1 ) q .
{ β ( 0 ) p q 2 β 2 } e xpq + p q { a ( g x , g y ; p , q ; p , q ) e x p q + b ( g x , g y ; p , q ; p , q ) e y p q } = 0 ,
a ( g u g υ ; i , j , i , j ) = { k ( 0 ) 2 g u 2 { 1 + 3 2 ( 2 i + 1 ) g u + 1 2 ( 2 j + 1 ) g υ } ( i = i , j = j ) , k ( 0 ) 2 g u 2 { ( i + 1 ) ( i + 2 ) } 1 2 ( i = i + 2 , j = j ) , k ( 0 ) 2 g u 2 { ( i 1 ) i } 1 2 ( i = i 2 , j = j ) , 0 ( for all other i , j ) ;
b ( g u , g υ ; i , j , i , j ) = { k ( 0 ) 2 g u 1 2 g υ 3 2 { ( i + 1 ) ( j + 1 ) } 1 2 ( i = i + 1 , j = j + 1 ) , k ( 0 ) 2 g u 1 2 g υ 3 2 { ( i + 1 ) j } 1 2 ( i = i + 1 , j = j 1 ) , k ( 0 ) 2 g u 1 2 g υ 3 2 { ( j + 1 ) } 1 2 ( i = i 1 , j = j + 1 ) , k ( 0 ) 2 g u 1 2 g υ 3 2 { ( i j ) } 1 2 ( i = i 1 , j = j 1 ) , 0 ( for other i , j ) ;
β ( 0 ) p q 2 = k ( 0 ) 2 [ 1 { ( 2 p + 1 ) g x + ( 2 q + 1 ) g y } ] ,
{ β ( 0 ) p q 2 β 2 } e ypq + p q { a ( g y , g x ; q , p ; q , p ) e y p q + b ( g y , g x ; q , p ; q , p ) e x p q } = 0 .
β ( 0 ) 2 | a | , | b | ,
| β ( 0 ) τ s 2 β ( 0 ) m n 2 | or < | a | , | b | .
β ( 0 ) p q 2 β ( 0 ) m n 2 = 2 k ( 0 ) 2 [ { ( m + n ) ( p + q ) } g + { ( m n ) ( p q ) } g δ ] ,
g = ( g x + g y ) / 2 , δ = ( g x g y ) / ( g x + g y ) .
r + s = m + n .
[ { β ( 0 ) r s 2 k ( 0 ) 2 g x 2 } β 2 ] e x r s k ( 0 ) 2 g x 1 2 g y 3 2 × [ { ( r + 1 ) s } 1 2 e y ( r + 1 ) ( s 1 ) { r ( s + 1 ) } 1 2 e y ( r 1 ) ( s + 1 ) ] = 0 ,
[ { β ( 0 ) r s 2 k ( 0 ) 2 g y 2 } β 2 ] e yrs k ( 0 ) 2 g x 1 2 g y 3 2 × [ { r ( s + 1 ) } 1 2 e x ( 1 ) ( s + 1 ) { ( r + 1 ) s } 1 2 e x ( r + 1 ) ( s 1 ) ] = 0 .
| β ( 0 ) r s 2 β ( 0 ) m n 2 | | a | , | b | ,
{ ( β ( 0 ) m n 2 k ( 0 ) 2 g x 2 ) β 2 } e xmn = 0 ,
{ ( β ( 0 ) m n 2 k ( 0 ) 2 g y 2 ) β 2 } e ymn = 0 .
β 2 = β ( 0 ) m n 2 k ( 0 ) 2 g x 2 , e xmn 0 , all other coefficients are zero ;
β 2 = β ( 0 ) m n 2 k ( 0 ) 2 g y 2 , e ymn 0 , all other coefficients are zero .
β 2 = β ( 0 ) m n 2 ( m + n + 1 2 t ) k ( 0 ) 2 g 2 ,
t = 0 , 1 , 2 , ( m + n 1 ) , ( m + n ) .
β 2 = β ( 0 ) m n 2 , e x 01 = e y 10 , all other coefficients are zero ;
β 2 = β ( 0 ) m n 2 , e x 10 = e y 01 , all other coefficients are zero ;
β 2 = β ( 0 ) m n 2 2 k ( 0 ) 2 g 2 , e x 01 = e y 10 , all other coefficients are zero ;
β 2 = β ( 0 ) m n 2 2 k ( 0 ) 2 g 2 , e x 10 = e y 10 , all other coefficients are zero .
β ( 0 ) p q 2 β 2 β ( 0 ) p q 2 β ( 0 ) m n 2 = 2 k ( 0 ) 2 { ( m p ) g x + ( n q ) g y } .
e x ( m + 2 ) n = 1 4 { ( m + 1 ) ( m + 2 ) } 1 2 g x e xmn , e x ( m 2 ) n = 1 4 { ( m 1 ) m ) } 1 2 g x e xmn , e y ( m + 1 ) ( n + 1 ) = 1 4 { ( m + 1 ) ( m + 1 ) } 1 2 2 ( g y / g x ) 1 2 1 + g y / g x g x e xmn , e y ( m + 1 ) ( n 1 ) = 1 4 { ( m + 1 ) n } 1 2 2 ( g y / g x ) 1 2 1 g y / g x g x e xmn , e y ( m 1 ) ( n + 1 ) = 1 4 { m ( n + 1 ) } 1 2 2 ( g y / g x ) 1 2 1 g y / g x g x e xmn , e y ( m 1 ) ( n 1 ) = 1 4 { m n ) } 1 2 2 ( g y / g x ) 1 2 1 + g y / g x g x e xmn ,
β 2 = β ( 0 ) m n 2 k ( 0 ) 2 g x 2 { 1 + 2 ( 2 m + 1 ) g x + 1 2 ( 2 n + 1 ) g y + ( m + n + 1 ) g y 2 ( g x / g y + 1 ) + ( m n ) g y 2 ( g x / g y 1 ) } .
e z ( m + 1 ) n = j ( g x 2 ) 1 2 ( m + 1 ) 1 2 e xmn , e z ( m 1 ) n = j ( g x 2 ) 1 2 m 1 2 e xmn ,