Abstract

The exact solution of the scattering of obliquely incident plane waves by an elliptical dielectric cylinder is obtained. Each expansion coefficient of the scattered or transmitted wave is coupled to all coefficients of the series expansion for the incident wave except when the elliptical cylinder degenerates to a circular one. Both polarizations of the incident wave are considered: one with the incident electric vector in the axial direction, and the other with the incident magnetic vector in the axial direction. In the general case of oblique incidence, the scattered field contains a significant cross-polarized component which vanishes at normal incidence.

© 1964 Optical Society of America

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References

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  1. P. Debye, Physik Z. 9, 775 (1908).
  2. J. R. Wait, Can. J. Phys. 33, 189 (1955).
    [Crossref]
  3. H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).
  4. C. Yeh, J. Math. Phys. 4, 65 (1963).
    [Crossref]
  5. N. McLachlan, Theory and Application of Mathieu Function (Oxford University Press, London1951).
  6. C. Yeh, J. Appl. Phys. 33, 3235 (1962).
    [Crossref]

1963 (1)

C. Yeh, J. Math. Phys. 4, 65 (1963).
[Crossref]

1962 (1)

C. Yeh, J. Appl. Phys. 33, 3235 (1962).
[Crossref]

1955 (1)

J. R. Wait, Can. J. Phys. 33, 189 (1955).
[Crossref]

1908 (1)

P. Debye, Physik Z. 9, 775 (1908).

Debye, P.

P. Debye, Physik Z. 9, 775 (1908).

McLachlan, N.

N. McLachlan, Theory and Application of Mathieu Function (Oxford University Press, London1951).

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

Wait, J. R.

J. R. Wait, Can. J. Phys. 33, 189 (1955).
[Crossref]

Yeh, C.

C. Yeh, J. Math. Phys. 4, 65 (1963).
[Crossref]

C. Yeh, J. Appl. Phys. 33, 3235 (1962).
[Crossref]

Can. J. Phys. (1)

J. R. Wait, Can. J. Phys. 33, 189 (1955).
[Crossref]

J. Appl. Phys. (1)

C. Yeh, J. Appl. Phys. 33, 3235 (1962).
[Crossref]

J. Math. Phys. (1)

C. Yeh, J. Math. Phys. 4, 65 (1963).
[Crossref]

Physik Z. (1)

P. Debye, Physik Z. 9, 775 (1908).

Other (2)

N. McLachlan, Theory and Application of Mathieu Function (Oxford University Press, London1951).

H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

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Figures (1)

F. 1
F. 1

Plane wave incident on an elliptical dielectric cylinder. Arrow indicates the direction of incident wave.

Equations (31)

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exp [ i k 0 ( x cos ϕ sin θ + y sin ϕ sin θ + z cos θ ) ] = exp { i k 0 [ q sin θ ( cosh ξ cos η cos ϕ + sinh ξ sin η sin ϕ ) + z cos θ ] } = 2 n = 0 [ 1 p 2 n C e 2 n ( ξ , γ 0 2 ) c e 2 n ( η , γ 0 2 ) c e 2 n ( ϕ , γ 0 2 ) + 1 s 2 n + 2 S e 2 n + 2 ( ξ , γ 0 2 ) s e 2 n + 2 ( η , γ 0 2 ) s e 2 n + 2 ( ϕ , γ 0 2 ) + i p 2 n + 1 C e 2 n + 1 ( ξ , γ 0 2 ) c e 2 n + 1 ( η , γ 0 2 ) c e 2 n + 1 ( ϕ , γ 0 2 ) + i s 2 n + 1 S e 2 n + 1 ( ξ , γ 0 2 ) s e 2 n + 1 ( η , γ 0 2 ) s e 2 n + 1 ( ϕ , γ 0 2 ) ] × e i k 0 z cos θ ,
C e n ( ξ , γ 0 2 ) = C e n ( ξ ) , c e n ( η , γ 0 2 ) = c e n ( η ) , S e n ( ξ , γ 0 2 ) = S e n ( ξ ) , s e n ( η , γ 0 2 ) = s e n ( η ) , C e n ( ξ , γ 1 2 ) = C e n * ( ξ ) , c e n ( ξ , γ 1 2 ) = c e n * ( η ) , S e n ( ξ , γ 1 2 ) = S e n * ( ξ ) , s e n ( η , γ 1 2 ) = s e n * ( η ) , M e n ( 1 ) , ( 2 ) ( ξ , γ 0 2 ) = M e n ( 1 ) , ( 2 ) ( ξ ) , N e n ( 1 ) , ( 2 ) ( ξ , γ 0 2 ) = N e n ( 1 ) , ( 2 ) ( ξ ) ,
E z i = E 0 [ the right - hand side of Eq . ( 1 ) ]
H z i = 0 .
E z s = 2 E 0 n = 0 [ A 2 n p 2 n M e 2 n ( 1 ) ( ξ ) c e 2 n ( η ) c e 2 n ( ϕ ) + B 2 n + 2 s 2 n + 2 N e 2 n + 2 ( 1 ) ( ξ ) s e 2 n + 2 ( η ) s e 2 n + 2 ( ϕ ) + i A 2 n + 1 p 2 n + 1 M e 2 n + 1 ( 1 ) ( ξ ) c e 2 n + 1 ( η ) c e 2 n + 1 ( ϕ ) + i B 2 n + 1 s 2 n + 1 N e 2 n + 1 ( 1 ) ( ξ ) s e 2 n + 1 ( η ) s e 2 n + 1 ( ϕ ) ] × e i k 0 z cos θ ,
H z s = 2 E 0 n = 0 [ C 2 n p 2 n M e 2 n ( 1 ) ( ξ ) c e 2 n ( η ) c e 2 n ( ϕ ) + D 2 n + 2 s 2 n + 1 N e 2 n + 2 ( 1 ) ( ξ ) s e 2 n + 2 ( η ) s e 2 n + 2 ( ϕ ) + i C 2 n + 1 p 2 n + 1 M e 2 n + 1 ( 1 ) ( ξ ) c e 2 n + 1 ( η ) c e 2 n + 1 ( ϕ ) + i D 2 n + 1 s 2 n + 1 N e 2 n + 1 ( 1 ) ( ξ ) s e 2 n + 1 ( η ) s e 2 n + 1 ( ϕ ) ] × e i k 0 z cos θ ,
E z t = 2 E 0 n = 0 [ F 2 n p 2 n * C e 2 n * ( ξ ) c e 2 n * ( η ) c e 2 n ( ϕ ) + G 2 n + 2 s 2 n + 2 * S e 2 n + 2 * ( ξ ) s e 2 n + 2 * ( η ) s e 2 n + 2 ( ϕ ) + i F 2 n + 1 p 2 n + 1 * C e 2 n + 1 * ( ξ ) c e 2 n + 1 * ( η ) c e 2 n + 1 ( ϕ ) + i G 2 n + 1 s 2 n + 1 * S e 2 n + 1 * ( ξ ) s e 2 n + 1 * ( η ) s e 2 n + 1 ( ϕ ) ] × e i k 0 z cos θ ,
H z t = 2 E 0 n = 0 [ P 2 n p 2 n * C e 2 n * ( ξ ) c e 2 n * ( η ) c e 2 n ( ϕ ) + Q 2 n + 2 s 2 n + 2 * S e 2 n + 2 * ( ξ ) s e 2 n + 2 * ( η ) s e 2 n + 2 ( ϕ ) + i P 2 n + 1 p 2 n + 1 * C e 2 n + 1 * ( ξ ) c e 2 n + 1 * ( η ) c e 2 n + 1 ( ϕ ) + i Q 2 n + 1 s 2 n + 1 * S e 2 n + 1 * ( ξ ) s e 2 n + 1 * ( η ) s e 2 n + 1 ( ϕ ) ] × e i k 0 z cos θ ,
[ r . h . s . of ( 3 ) with ξ = ξ 0 ] + [ r . h . s . of ( 5 ) with ξ = ξ 0 ] = [ r . h . s . of ( 7 ) with ξ = ξ 0 ] ,
[ r . h . s . of ( 6 ) with ξ = ξ 0 ] = [ r . h . s . of ( 8 ) with ξ = ξ 0 ] ,
1 k 0 2 sin 2 θ { i k 0 cos θ η [ r . h . s . of ( 3 ) with ξ = ξ 0 ] + i k 0 cos θ η [ r . h . s . of ( 5 ) with ξ = ξ 0 ] i ω μ 0 ξ 0 [ r . h . s . of ( 6 ) with ξ = ξ 0 ] } = 1 ( k 1 2 k 0 2 cos 2 θ ) { i k 0 cos θ η × [ r . h . s . of ( 7 ) with ξ = ξ 0 ] i ω μ 1 ξ 0 [ r . h . s . of ( 8 ) with ξ = ξ 0 ] } ,
1 k 0 2 sin 2 θ { i ω 0 ξ 0 [ r . h . s . of ( 3 ) with ξ = ξ 0 ] i ω 0 ξ 0 [ r . h . s . of ( 5 ) with ξ = ξ 0 ] i k 0 cos θ η [ r . h . s . of ( 6 ) with ξ = ξ 0 ] } = 1 ( k 1 2 k 0 2 cos 2 θ ) { i ω 1 ξ 0 × [ r . h . s . of ( 7 ) with ξ = ξ 0 ] i k 0 cos θ η [ r . h . s . of ( 8 ) with ξ = ξ 0 ] } ,
c e m * ( η ) = n = 0 α m , n c e n ( η ) ,
s e m * ( η ) = n = 0 β m , n s e n ( η ) ,
d d η c e m ( η ) = n = 0 γ m , n s e n ( η ) ,
d d η s e m ( η ) = n = 0 χ m , n c e n ( η ) ,
1 p n [ C e n ( ξ 0 ) + A n M e n ( 1 ) ( ξ 0 ) ] c e n ( ϕ ) = m = 0 F m p m * C e m * ( ξ 0 ) c e m ( ϕ ) α m , n ,
1 s n [ S e n ( ξ 0 ) + B n N e n ( 1 ) ( ξ 0 ) ] s e n ( ϕ ) = m = 0 G m s m * S e m * ( ξ 0 ) s e m ( ϕ ) β m , n ,
C n p n M e n ( 1 ) ( ξ 0 ) c e n ( ϕ ) = m = 0 P m p m * C e m * ( ξ 0 ) c e m ( ϕ ) α m , n ,
D n s n N e n ( 1 ) ( ξ 0 ) s e n ( ϕ ) = m = 0 Q m s m * S e m * ( ξ 0 ) s e m ( ϕ ) β m , n ,
( 0 μ 0 ) 1 2 cos θ ( 1 k 0 2 sin 2 θ k 1 2 k 0 2 cos 2 θ ) { m = 0 1 s m [ S e m ( ξ 0 ) + B m N e m ( 1 ) ( ξ 0 ) ] s e m ( ϕ ) χ m , n } C n p n M e n ( 1 ) ( ξ 0 ) c e n ( ϕ ) = ( μ 1 μ 0 ) ( k 0 2 sin 2 θ k 1 2 k 0 2 cos 2 θ ) m = 0 p m p m * C e m * ( ξ 0 ) c e m ( ϕ ) α m , n ,
( 0 μ 0 ) 1 2 cos θ ( 1 k 0 2 sin 2 θ k 1 2 k 0 2 cos 2 θ ) { m = 0 1 p m [ C e m ( ξ 0 ) + A m M e m ( 1 ) ( ξ 0 ) ] c e m ( ϕ ) γ m , n } D n s n N e n ( 1 ) ( ξ 0 ) s e n ( ϕ ) = ( μ 1 μ 0 ) k 0 2 sin 2 θ k 1 2 k 0 2 cos 2 θ m = 0 Q m s m * S e m * ( ξ 0 ) s e m ( ϕ ) β m , n ,
1 s n [ S e n ( ξ 0 ) + B n N e n ( 1 ) ( ξ 0 ) ] s e n ( ϕ ) + ( μ 0 0 ) 1 2 cos θ ( 1 k 0 2 sin 2 θ k 1 2 k 0 2 cos 2 θ ) m = 0 C m p m M e m ( 1 ) ( ξ 0 ) c e m ( ϕ ) χ m , n = k 0 2 sin 2 θ k 1 2 k 0 2 cos 2 θ 1 0 m = 0 G m s m * S e m * ( ξ 0 ) s e m ( ϕ ) β m , n ,
1 p n [ C e n ( ξ 0 ) + A n M e n ( 1 ) ( ξ 0 ) ] c e n ( ϕ ) + ( μ 0 0 ) 1 2 cos θ ( 1 k 0 2 sin 2 θ k 1 2 k 0 2 cos 2 θ ) m = 0 D m s m N e m ( 1 ) ( ξ 0 ) s e m ( ϕ ) χ m , n = k 0 2 sin 2 θ k 1 2 k 0 2 cos 2 θ 1 0 m = 0 F m p m * C e m * ( ξ 0 ) c e m ( ϕ ) α m , n , ( n = 0 , 1 , 2 , 3 , 4 ) .
m = 0 F m s m n e + Q m t m n e = 0 ,
m = 0 F m u m n e + Q m υ m n e = w n e , ( n = 0 , 1 , 2 , 3 ) ,
m = 0 G m s m n 0 + P m t m n 0 = 0 ,
m = 0 G m u m n 0 + P m υ m n 0 = w n 0 , ( n = 0 , 1 , 2 , 3 ) ,
s m n e = ( 0 μ 0 ) 1 2 cos θ ( 1 x 2 ) f m r = 0 α m r γ r n , s m n 0 = ( 0 μ 0 ) 1 2 cos θ ( 1 x 2 ) g m r = 0 β m r χ r n , t m n e = [ μ 1 μ 0 x 2 g m b n b n g m ] β m n , t m n 0 = [ μ 1 μ 0 x 2 h m a n a n h m ] α m n , u m n e = [ f m a n a n x 2 1 0 f m ] α m n , u m n 0 = [ g m b n b n x 2 1 0 g m ] β m n , υ m n e = ( μ 0 0 ) 1 2 cos θ ( 1 x 2 ) g m r = 0 β m r χ r n , υ m n 0 = ( μ 0 0 ) 1 2 cos θ ( 1 x 2 ) f m r = 0 α m r γ r n , w n 0 = c n a n a n c n , w n e = d n b n b n d n .
c n = ( 1 / p n ) C e n ( ξ 0 ) c e n ( ϕ ) , c n = ( 1 / p n ) C e n ( ξ 0 ) c e n ( ϕ ) , d n = ( 1 / s n ) S e n ( ξ 0 ) s e n ( ϕ ) , d n = ( 1 / s n ) S e n ( ξ 0 ) s e n ( ϕ ) , a n = ( 1 / p n ) M e n ( 1 ) ( ξ 0 ) c e n ( ϕ ) , a n = ( 1 / p n ) M e n ( 1 ) ( ξ 0 ) c e n ( ϕ ) , b n = ( 1 / s n ) N e n ( 1 ) ( ξ 0 ) s e n ( ϕ ) , b n = ( 1 / s n ) N e n ( 1 ) ( ξ 0 ) s e n ( ϕ ) , f n = ( 1 / p n * ) C e n * ( ξ 0 ) c e n ( ϕ ) , f n = ( 1 / p n * ) C e n * ( ξ 0 ) c e n ( ϕ ) , g n = ( 1 / s n * ) S e n * ( ξ 0 ) s e n ( ϕ ) , g n = ( 1 / s n * ) S e n * ( ξ 0 ) s e n ( ϕ ) , x 2 = k 0 2 sin 2 θ / ( k 1 2 k 0 2 cos 2 θ ) .
α m , n = 0 2 π c e m * ( η ) c e n ( η ) d η / 0 2 π c e n 2 ( η ) d η , β m , n = 0 2 π c e m * ( η ) c e n ( η ) d η / 0 2 π c e n 2 ( η ) d η , γ m , n = 0 2 π c e m ( η ) s e n ( η ) d η / 0 2 π s e n 2 ( η ) d η , χ m , n = 0 2 π s e m ( η ) c e n ( η ) d η / 0 2 π c e n 2 ( η ) d η ,