Abstract

The attenuation of rays within lossless dielectric structures is determined by two in-principle-exact methods: (a) solution of the appropriate eigenvalue equation, and (b) Poynting’s vector theorem. The methods are applied to cylinders and spheres. There are no trapped rays within the circle or sphere, or any finite structure; however, many rays are only very weakly attenuated.

© 1974 Optical Society of America

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References

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  1. A. W. Snyder and D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
    [Crossref]
  2. Rayleigh, Phil. Mag. 27, 100 (1914).
  3. F. G. Reich, Appl. Opt. 4, 1395 (1965).
    [Crossref]
  4. J. R. Wait, Radio Sci. 2 (New Series), 10005 (1967).
  5. D. S. Jones, The Theory of Electromagnetism (Pergamon, New York, 1964), pp. 562–564.
  6. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
    [Crossref]
  7. A. W. Snyder, D. J. Mitchell, and C. Pask, J. Opt. Soc. Am. 64, 608 (1974).
    [Crossref]
  8. A. W. Snyder and D. J. Mitchell, Electron. Lett. 10(2), 16 (1974).
    [Crossref]
  9. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 293.
  10. J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), pp. 556–557.
  11. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, Natl. Bur. Stand. (U. S.) Appl. Math. Ser. 55 (U. S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965).

1974 (3)

1969 (1)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
[Crossref]

1967 (1)

J. R. Wait, Radio Sci. 2 (New Series), 10005 (1967).

1965 (1)

1914 (1)

Rayleigh, Phil. Mag. 27, 100 (1914).

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 293.

Jones, D. S.

D. S. Jones, The Theory of Electromagnetism (Pergamon, New York, 1964), pp. 562–564.

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 293.

Mitchell, D. J.

Pask, C.

Rayleigh,

Rayleigh, Phil. Mag. 27, 100 (1914).

Reich, F. G.

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
[Crossref]

A. W. Snyder, D. J. Mitchell, and C. Pask, J. Opt. Soc. Am. 64, 608 (1974).
[Crossref]

A. W. Snyder and D. J. Mitchell, Electron. Lett. 10(2), 16 (1974).
[Crossref]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), pp. 556–557.

Wait, J. R.

J. R. Wait, Radio Sci. 2 (New Series), 10005 (1967).

Appl. Opt. (1)

Electron. Lett. (1)

A. W. Snyder and D. J. Mitchell, Electron. Lett. 10(2), 16 (1974).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
[Crossref]

J. Opt. Soc. Am. (2)

Phil. Mag. (1)

Rayleigh, Phil. Mag. 27, 100 (1914).

Radio Sci. (1)

J. R. Wait, Radio Sci. 2 (New Series), 10005 (1967).

Other (4)

D. S. Jones, The Theory of Electromagnetism (Pergamon, New York, 1964), pp. 562–564.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 293.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), pp. 556–557.

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, Natl. Bur. Stand. (U. S.) Appl. Math. Ser. 55 (U. S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965).

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Figures (3)

Fig. 1
Fig. 1

Contours for determining α by the Poynting’s vector theorem; (a) is for the cylinder, i.e., αz; (b) is for the circle or sphere, i.e., αϕ = 2 ImU.

Fig. 2
Fig. 2

Contours for determining Imω by Poynting’s vector theorem; (a) is for the cylinder; (b) is for the circle or sphere.

Fig. 3
Fig. 3

Contour used to determine αz by Poynting’s vector theorem using the radiation field, i.e., far-field expressions. Q = . We take z2 = 0.

Equations (73)

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U J l ( U ) / J l - 1 ( U ) = Q H l ( Q ) / H l - 1 ( Q ) ,
i Q - U tan U = 0.
V 2 = U 2 - Q 2 = ( k 1 ρ ) 2 θ c 2 ,
k 1 = ω ( μ 1 ) 1 2 = 2 π n 1 / λ ,
θ c = { 1 - 2 / 1 } 1 2 = { 1 - ( n 2 / n 1 ) 2 } 1 2 ;
l - 1 = V ( sin θ z / sin θ c ) cos θ ϕ ,
U = V ( sin θ z / sin θ c ) ,
Q = V { ( sin θ z / sin θ c ) 2 - 1 } 1 2 ,
Im U = ( 2 U / π V 2 ) H l ( Q ) H l - 2 ( Q ) - 1 ,
Im U = U Q / V 2 .
P ( z ) = P ( 0 ) e - α z z / ρ ,
α z = 2 Re U ρ Re β Im U = 2 ( Im U ) tan θ z ,
2 t Im ω = α z / ρ .
α = 2 ρ ( μ 1 ) 1 2 Im ω = 2 ρ Im k 1 .
α ϕ = 2 Im l = 2 Im U = 2 ρ Im k 1 ,
α z = 2 ( Im U ) tan θ z = 2 ρ Im k 1 .
- α ρ = 1 P ( z ) { d P ( z ) d z } ,
- α ( d z ρ ) = Re A SA e × h * · r ^ d A / Re A CS e × h * · z ^ d A ,
- Im ω = Re A SA e × h * · r ^ d A / v { e 2 + μ h 2 } d v ,
( Im ω ) v { e 2 + μ h 2 } d v = α ( d z ρ ) Re A CS e × h * · z ^ d A ,
h = ( / μ ) 1 2 z ^ × e .
e × h * · z ^ = ( 1 μ ) 1 2 | J l ( U ( r / ρ ) ) J l ( U ) | 2 ,             r < ρ
= ( μ ) 1 2 | H l ( Q ( r / ρ ) ) H l ( Q ) | 2 ,             r > ρ
e × h * · r ^ = i ( 1 ρ k 1 ) ( 1 μ ) 1 2 [ Q * H l * ( Q ( r / ρ ) ) H l - 1 ( Q ( r / ρ ) ) cos 2 l ϕ - Q H l ( Q ( r / ρ ) ) H l - 1 * ( Q ( r / ρ ) ) sin 2 l ϕ H l - 1 ( Q ) 2 ] .
A CA e × h * · z ^ d A = ρ 2 π ( μ ) 1 2 [ 1 - J l - 2 ( U ) J l ( U ) J l - 1 2 ( U ) ]
= ρ 2 π ( μ ) 1 2 [ 1 - ( Q U ) 2 H l - 2 ( Q ) H l ( Q ) { H l - 1 ( Q ) } 2 ] ,
A SA e × h * · r ^ d A = π i ( Q k 1 ) ( 1 μ ) 1 2 [ H l * ( Q ( r / ρ ) ) H l - 1 ( Q ( r / ρ ) ) - H l ( Q ( r / ρ ) ) H l - 1 * ( Q ( r / ρ ) ) H l - 1 ( Q ) 2 ] d z
= ( 4 / k 1 ) ( 1 / μ ) 1 2 d z H l - 1 ( Q ) - 2 ,
α z = ( 4 π ) ( 1 ρ k 1 ) ( 1 H l - 1 ( Q ) 2 ) [ 1 - ( Q U ) 2 H l - 2 ( Q ) H l ( Q ) { H l - 1 ( Q ) } 2 ] - 1
( 4 / π ) ( 1 / ρ k 1 ) ( U / V ) 2 H l - 1 ( Q ) - 2 ,
d P ( z ) = e - α z / ρ 2 Re A Δ d e ( r 0 ) × h * ( r 0 ) · k ^ d A
= e - α z / ρ 2 d z sin θ Re 0 2 π e ( r 0 ) × h * ( r 0 ) · k ^ r 0 d ϕ ,
P ( z ) = e - α z / ρ Re A r tp e × h * · z ^ d A ,
Im k · ξ r 0 Im Q + z ρ Im β = 0 ,
tan θ = ( Re Q / ρ Re β ) = - ( ρ Im β / Im Q )
θ c 2 α ϕ = ( 4 / π ) ( 1 / k 1 ρ ) H l ( k 1 ρ ) - 2 ,
{ U J l + 3 2 ( U ) / J l + 1 2 ( U ) } = { Q H l + 3 2 ( 2 ) ( Q ) / H l + 1 2 ( 2 ) ( Q ) } ,
P l ( cos θ ) Γ ( l + 1 ) Γ ( l + 3 2 ) ( 1 2 π sin θ ) - 1 2 cos [ ( l + 1 2 ) θ - π / 4 ] ,
- · ( E × H ) = 1 2 t [ E 2 + μ H 2 ] ,
E = ( e e - i ω t + e * e + i ω * t ) / 2 ,
- Re · e × h * - Im ω [ E 2 + μ H 2 ] ,
S ϕ = e × h · ϕ ^ - ( l / k 1 r ) ( 1 / μ ) 1 2 J l 2 ( k 1 r ) e - 2 ϕ Im l ,
S r = e × h · r ^ - i | J l ( U ) H l ( Q ) | 2 ( 1 μ ) 1 2 × H l ( k 2 r ) H l * ( k 2 r ) e - 2 ϕ Im l ,
α ϕ = 2 Im l = - ρ ( Re S r ) r = ρ + / Re 0 ρ S ϕ d r
= ( 2 π l ) 1 H l ( Q ) 2 [ 1 J l 2 ( U ) 0 U J l 2 ( X ) X d x ] - 1 ,
I ( U ) = 0 U 1 J l 2 ( X ) X d x
= I ( ) - U 1 J l 2 ( X ) X d x .
U 1 J l 2 ( X ) X d x 2 π U d x cos 2 γ ( X ) X ( X 2 - l 2 ) 1 2
1 π U d x X ( X 2 - l 2 ) 1 2 = 1 π [ 1 l cos - 1 ( l X ) ] U
( l π l ) sin - 1 ( l U ) .
I ( U ) θ ϕ π l - 1 2 l ( l + 1 )
θ ϕ π l .
I ( U ) J l 2 ( U ) θ ϕ 2 2 cos 2 γ ( U ) .
tan 2 γ ( u ) = l 2 - Q 2 U 2 θ ϕ 2 θ c 2 - θ ϕ 2 θ ϕ 2 ,
I ( U ) J l 2 ( U ) = θ c 2 2 .
α ϕ = 4 π 1 V θ c 1 H l ( Q ) 2 .
Im ω = - 2 ρ ( S r ) r = ρ + / 0 ρ { μ H 2 + 1 E 2 } r d r .
0 ρ { μ H 2 + 1 E 2 } r d r = 1 k 1 2 0 U x d x { J l 2 ( X ) + J l - 1 2 ( X ) - 2 l X J l ( X ) J l ( X ) }
= 1 k 1 2 J l 2 ( U ) [ U 2 { J l - 1 ( U ) J l ( J l ( U ) J l ( U ) ) - ( l - 1 ) U J l - 1 ( U ) J l ( U ) + 1 } - l ]
= 1 k 1 2 J l 2 ( U ) [ ( Q H l - 1 ( Q ) H l ( Q ) ) 2 - l Q H l - 1 ( Q ) H l ( Q ) - ( l - 1 ) Q H l - 1 ( Q ) H l ( Q ) + U 2 - l ]
1 k 1 2 V 2 J l 2 ( U ) ,             V 1
J l ( z ) = ( 2 / π ) 1 2 ( z 2 - l 2 ) 1 4 cos γ ( z ) ,
H l ( 1 ) ( z ) = ( 2 / π ) 1 2 ( z 2 - l 2 ) 1 4 e i γ ( z ) ,
H l ( 2 ) ( z ) = ( 2 / π ) 1 2 ( z 2 - l 2 ) 1 4 e - i γ ( z ) ,
H l ( 2 ) ( z ) = - i ( 2 π ) 1 2 exp [ l cosh - 1 ( l / z ) - ( l 2 - z 2 ) ] ( l 2 - z 2 ) 1 4 ,
H l ( 2 ) ( z ) = - H l ( 1 ) ( z ) .
H l ( 1 ) ( z ) - i ( 2 π ) 1 2 exp [ - ( l / 3 ) { 1 - ( z / l ) 2 } 3 2 ] ( l 2 - z 2 ) 1 4 = - H l ( 2 ) ( z ) .
H l ( l ) 2 = 4 a 2 / l 2 3 ,
z H l ( z ) H l - 1 ( z ) = l + ( l 2 - z 2 ) 1 2 ,
z H l ( 1 ) ( z ) H l - 1 ( 1 ) ( z ) = l + i ( z 2 - l 2 ) 1 2 ,
z H l ( 2 ) ( z ) H l - 1 ( 2 ) ( z ) = l - i ( z 2 - l 2 ) 1 2 .
H l ( 1 ) ( X ) H l + 1 ( 2 ) ( X ) - H l ( 2 ) ( X ) H l + 1 ( 1 ) ( X ) = H l ( 2 ) ( X ) H l ( 1 ) ( X ) - H l ( 1 ) ( X ) H l ( 2 ) ( X ) = 4 i π X ,
H l ( 2 ) ( X ) J l ( X ) - H l ( 2 ) ( X ) J l ( X ) = 2 i π X .