Abstract

A previously developed theory for the transmission of radiation in over moded hollow circular waveguides is extended to cover walls of arbitrary material. The extension relies on approximations to the Fresnel reflection coefficients near grazing angles of incidence for a material characterized by its complex refractive index.

© 1987 Optical Society of America

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References

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  1. B. S. Frost, P. M. Gourlay, N. R. Heckenberg, S. T. Shanahan “Geometrical Optics Treatment of Circular Lightguides,” Appl. Opt. 24, 4414 (1985).
    [Crossref] [PubMed]
  2. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  3. Expressions for δ∥ given at the beginning of Sec. II.B of Ref. 1 contain an error in sign. There is also a typographical error (/) in δ∥ for a metal. Fortunately these errors do not affect any final expressions given in Ref. 1.
  4. H. E. Bennett, J. M. Bennett “Validity of the Drude Theory for Silver, Gold and Aluminum in the Infrared,” in Proceedings, International Colloqium on Optical Properties and Electronic Structure of Metals and Alloys, Paris, 1965 (North-Holland, Amsterdam, 1966), pp. 175–88.
  5. The correction to FGHS Eq. (73) consists of replacing 4 by 2 in the fourth line and replacing ¼ by ½ in the definition of H3 which followed.
  6. J. P. Crenn, “Optical Theory of Gaussian Beam Transmission Through a Hollow Circular Waveguide,” Appl. Opt. 21, 4533 (1982).
    [Crossref] [PubMed]
  7. J. P. Crenn, “Gaussian Beam Transmission through Circular Waveguide with Conducting Wall Material,” Appl. Opt. 24, 3648 (1985).
    [Crossref] [PubMed]

1985 (2)

1982 (1)

Bennett, H. E.

H. E. Bennett, J. M. Bennett “Validity of the Drude Theory for Silver, Gold and Aluminum in the Infrared,” in Proceedings, International Colloqium on Optical Properties and Electronic Structure of Metals and Alloys, Paris, 1965 (North-Holland, Amsterdam, 1966), pp. 175–88.

Bennett, J. M.

H. E. Bennett, J. M. Bennett “Validity of the Drude Theory for Silver, Gold and Aluminum in the Infrared,” in Proceedings, International Colloqium on Optical Properties and Electronic Structure of Metals and Alloys, Paris, 1965 (North-Holland, Amsterdam, 1966), pp. 175–88.

Crenn, J. P.

Frost, B. S.

Gourlay, P. M.

Heckenberg, N. R.

Shanahan, S. T.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Appl. Opt. (3)

Other (4)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Expressions for δ∥ given at the beginning of Sec. II.B of Ref. 1 contain an error in sign. There is also a typographical error (/) in δ∥ for a metal. Fortunately these errors do not affect any final expressions given in Ref. 1.

H. E. Bennett, J. M. Bennett “Validity of the Drude Theory for Silver, Gold and Aluminum in the Infrared,” in Proceedings, International Colloqium on Optical Properties and Electronic Structure of Metals and Alloys, Paris, 1965 (North-Holland, Amsterdam, 1966), pp. 175–88.

The correction to FGHS Eq. (73) consists of replacing 4 by 2 in the fourth line and replacing ¼ by ½ in the definition of H3 which followed.

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Equations (57)

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α = ν 2 cos 2 ϕ ν 2 sin ϕ ν 2 cos 2 ϕ + ν 2 sin ϕ , α = ν 2 cos 2 ϕ sin ϕ ν 2 cos 2 ϕ + sin ϕ .
ν 2 cos 2 ϕ = ν 2 1 ( 1 + sin 2 ϕ ν 2 1 ) 1 / 2 ν 2 1 + sin 2 ϕ 2 ν 2 1 + ,
| ν 2 1 | ½ sin 2 ϕ .
α = 1 σ sin ϕ 1 + σ sin ϕ where σ = ν 2 ν 2 1 , α = σ sin ϕ σ + sin ϕ where σ = ν 2 1 .
| α | 1 2 Re σ sin ϕ + | σ | 2 sin 2 ϕ 1 + 2 Re σ sin ϕ + | σ | 2 sin 2 ϕ ,
| α | min = 1 cos ( arg σ ) 1 + cos ( arg σ )
sin ϕ m = 1 | σ | .
tan δ 2 Im σ sin ϕ | σ | 2 sin 2 ϕ 1
| α | 1 2 Re σ | σ | 2 sin ϕ , tan σ 2 Im σ | σ | 2 sin ϕ ,
| α | = exp ( β ξ ) for ξ < ξ m , = exp ( β ξ m 2 / ξ ) for ξ > ξ m , }
δ = π + ξ for ξ < ξ m , = ξ m 2 ( π 2 ξ m 2 ) ξ m / ξ for ξ > ξ m , }
ξ m = 1 / | σ | , β = 2 Re σ , = 2 Im σ .
| α | = exp ( β ξ ) ,
δ = π ξ ,
β = 2 Re σ | σ | 2 , = 2 Im σ | σ | 2 .
σ σ ν = 2 q exp ( i π / 4 ) ,
I n ± = 1 r z exp ( z a β ξ m 2 ) P ( θ , ξ n ) f p ( θ ) ξ n ,
J = 0 ξ m P ¯ ( θ , ξ ) exp ( z a β ξ 2 ) ξ d ξ + exp ( z a β ξ m 2 ) ξ m P ¯ ( θ , ξ ) ξ d ξ ,
J = 0 P ¯ ( θ , ξ ) exp ( z a β ξ 2 ) ξ d ξ .
z a > 2 β t 0 2 ,
T ( z ) = a z t 0 2 ( 1 β + 1 β ) .
T ( z ) = T ( z ) + T ( z ) .
I ( z , r , θ ) W ( z ) = 1 π ar f ( θ ) T + f ( θ ) T T ,
T = 1 2 l [ 1 + ( l 1 ) exp ( 2 l ξ m 2 / t 0 2 ) ] ,
T = 1 2 l ,
l p = 1 + ½ t 0 2 β p z a ,
max { 1 β ξ m 2 , 2 t 0 2 β } < z a < 2 t 0 2 β ,
T = 1 2 l ( 1 exp ( l ) ) if ξ u < ξ m ,
= 1 2 l [ 1 + ( l 1 z a β ξ m 2 ) exp ( z a β ξ m 2 ) ] if ξ u > ξ m ,
T = 1 2 l ( 1 exp ( l ) ] ,
l p = ξ u 2 β p z a .
I ( z , r , θ ) = I + I ,
I p = f p ( θ ) J p ( θ , z ) ar .
J = ( 2 π ) 1 W ( 0 ) exp ( 2 t 1 2 / t 0 2 ) × ( l 1 { 1 ½ exp ( K 2 ) ( exp ( 2 K 0 F 0 ) + exp ( 2 K 0 F 0 ) ) + ½ π F exp ( F 2 ) [ erf ( K F ) erf ( K + F ) ] + π F exp ( F 2 erf ( F ) } + exp ( K 2 ) { ½ [ exp ( 2 K 0 F 0 ) + exp ( 2 K 0 F 0 ) ] + ½ π F 0 exp ( K 0 2 + F 0 2 ) [ erfc ( K 0 F 0 ) erfc ( K 0 + F 0 ) ] } ) ,
J = ( 2 π ) 1 W ( 0 ) exp ( 2 t 1 2 / t 0 2 ) × l 1 [ 1 + π F exp ( F 2 ) erf ( F ) ] .
K p ( z ) = 2 l p ξ m t 0 , F p ( z , θ ) = 2 l p t 1 t 0 cos ( θ θ 1 ) ,
W ( z ) = 0 a r d r 0 2 π d θ I ,
T p ( z ) = exp ( G p 2 t 1 2 / t 0 2 ) 2 l p { 1 + H p ( θ 1 ) [ 1 1 exp ( G p ) G p ] } ,
G p ( z ) = 2 t 1 2 t 0 2 l p , H ( θ 1 ) = H ( θ 1 ) = f ( θ 1 ) f ( θ 1 ) = 2 h ( θ 1 ) .
h ( θ ) = ½ [ | A x | 2 | A y | 2 ] cos 2 θ + | A x | | A y | cos ( δ x δ y ) sin 2 θ .
A = A x cos θ + A y sin θ , A = A x sin θ + A y cos θ .
| E χ | 2 | A | 2 | α | 2 n cos 2 ( θ χ ) + | A | 2 | α | 2 n sin 2 ( θ χ ) 2 Re [ A A * α n α * n ] sin ( θ χ ) cos ( θ χ ) .
| A | 2 f ( θ ) = | A x | 2 cos 2 θ + | A y | 2 sin 2 θ + 2 | A x | | A y | cos ( δ x δ y ) sin θ cos θ ,
| A | 2 f ( θ ) = | A x | 2 sin 2 θ + | A y | 2 cos 2 θ 2 | A x | | A y | cos ( δ x δ y ) sin θ cos θ ,
A A * f a ( θ ) = | A x | | A y | cos ( δ x δ y ) ( cos 2 θ sin 2 θ ) ( | A x | 2 | A y | 2 ) sin θ cos θ + i | A x | | A y | sin ( δ x δ y ) .
I χ ( z , r , θ ) = 1 ar { J ( θ , z ) f ( θ ) cos 2 ( θ χ ) + J ( θ , z ) f ( θ ) sin 2 ( θ χ ) 2 Re ( J a ( θ , z ) f a ( θ ) ) sin ( θ χ ) cos ( θ χ ) } .
J a = 0 ξ m ξ d ξ P ¯ ( θ , ξ ) exp [ z a ( β + β 2 i + 2 ) ξ 2 ] + exp { ½ z a ( β ξ m 2 + i ( π 2 ξ m 2 ) ξ m ) } × ξ m ξ d ξ P ¯ ( θ , ξ ) exp [ ½ i z a ( π + ξ m 2 ) ξ ] × exp [ ½ z a ( β i ) ξ 2 ] .
T χ ( z ) = ½ T [ 1 + ( 1 + 2 T a T ) h ( χ ) ] ,
T a ( z ) = Re { 1 2 l a [ 1 exp ( 2 l a ξ m 2 / t 0 2 ) ] + 1 2 l a exp ( L + Q 2 ) [ exp ( ( K a + Q ) 2 ) π Q erfc ( K a + Q ) ] } ,
l a = 1 + ½ t 0 2 z a ( β + β 2 i + 2 ) l a = 1 + 1 4 t 0 2 z a ( β i ) , L = ½ z a [ β ξ m 2 + i ( π 2 ξ m 2 ) ξ m ] , K a = 2 l a ξ m t 0 , Q = i z t 0 ( π + ξ m 2 ) 4 a 2 l a .
T a ( z ) = Re { 1 2 l a [ 1 exp ( l a ) ] }
Re { 1 2 l a [ 1 exp ( l a ξ m 2 / ξ u 2 ) ] + 1 2 l a exp ( L + Q 2 ) [ exp ( ( K + Q ) 2 ) exp ( ( l a + Q ) 2 ) + π Q erf ( K + Q ) π Q erf ( l a + Q ) ] } ,
l a = ξ u 2 z a ( β + β 2 i + 2 ) , l a = ½ ξ u 2 z a ( β i ) , L = ½ z a [ β ξ m 2 + i ( π 2 ξ m 2 ) ξ m ] , K = l a ξ m ξ u , Q = i z ξ u ( π + ξ m 2 ) 4 a l a .
J a = ( 2 π ) 1 W ( 0 ) exp ( 2 t 1 2 / t 0 2 ) { l a 1 [ 1 ½ exp ( K a 2 ) [ exp ( 2 K 0 F 0 ) + exp ( 2 K 0 F 0 ) ] + ½ π F a exp ( F a 2 ) [ erf ( K a F a ) erf ( K a + F a ) ] + π F a exp ( F a 2 ) erf ( F a ) ] + l a 1 exp ( L ) × ( ½ exp [ ( F a Q ) 2 ] { exp [ ( K a F a + Q ) 2 ] + π ( F a Q ) erfc ( K a F a + Q ) } + ½ exp [ ( F a + Q ) 2 ] { exp [ ( K a + F a + Q ) 2 ] π ( F a + Q ) erfc ( K a + F a + Q ) } ) } ,
K a = 2 l a ξ m t 0 F a = 2 l a t 1 t 0 cos ( θ θ 1 ) ,
T χ ( z ) = 1 4 exp ( 2 t 1 2 / t 0 2 ) ( l 1 exp ( G ) [ H 1 ( χ ) + H 2 ( θ 1 , χ ) F 2 ( G ) + H 3 ( θ 1 , χ ) F 3 ( G ) ] + l 1 exp ( G ) [ H 1 ( χ ) H 2 ( θ 1 , χ ) F 2 ( G ) + H 3 ( θ 1 , χ ) F 3 ( G ) ] 2 Re { l a 1 exp ( G a ) [ h ( χ ) + 2 i | A x A y | sin ( δ x δ y ) × sin ( 2 θ 1 2 χ ) F 2 ( G a ) + H 3 ( θ 1 , χ ) F 3 ( G a ) ] } ) ,
H 1 = 1 + h ( χ ) , H 2 = cos ( 2 θ 1 2 χ ) + 2 h ( θ 1 ) , H 3 = h ( 2 θ 1 χ ) , F 2 ( G ) = 1 1 exp ( G ) G , F 3 ( G ) = 1 2 [ 2 + exp ( G ) ] G + 6 [ 1 exp ( G ) ] G 2 , G p ( z ) = 2 t 1 2 t 0 2 l p ( p = , , a ) .

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