Abstract

Some recent developments in the treatment of evanescent waves are reviewed. Such new methods as evanescent wave tracking, complex ray, and complex-source-point techniques are discussed and illustrated by examples, which include Gaussian beams, shadow formation due to evanescent fields, propagation on curved dielectric layers, and modal fields in graded-index films and fibers.

© 1976 Optical Society of America

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References

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  1. J. B. Keller, Proceedings of the Symposium on Applied Mathematics (McGraw-Hill, New York, 1958), p. 27.
    [Crossref]
  2. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, Englewood Cliffs, N. J., 1973), Secs. 1.6 and 1.7.
  3. S. Choudhary and L. B. Felsen, IEEE Trans. Antennas Propag. AP-21, 730, 1973.
  4. G. Otis, J. Opt. Soc. Am. 64, 1545, 1974.
    [Crossref]
  5. S. Choudhary and L. B. Felsen, “Propagation in inhomogeneous slab wave guides” paper presented at the IEEE-MTT-S International Microwave Symposium, Cherry Hill, N. J., June 14–16, 1976.
  6. S. Choudhary and L. B. Felsen, Proc. IEEE 62, 1530, 1974.
    [Crossref]
  7. J. B. Keller and W. Streifer, J. Opt. Soc. Am. 61, 40, 1971.
    [Crossref]
  8. Yu. A. Kravtsov, Proceedings of the 4th All Union Diffraction of Waves (Nauka, Moscow, 1967).
  9. L. B. Felsen, Philips Res. Rep. 30, 187, 1975.
  10. G. A. Deschamps, Electron. Lett. 7, No. 23 (1971).
    [Crossref]
  11. J. A. Arnaud, Appl. Opt. 8, 1909, 1969.
    [Crossref] [PubMed]
  12. L. B. Felsen, “Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams,” Symp. Mat. (to be published).
  13. S. Y. Shin and L. B. Felsen, Appl. Phys. 5, 239, 1974.
    [Crossref]
  14. L. B. Felsen and S. Y. Shin, IEEE Trans. Microwave Theory Tech. MTT-23, 150, 1975.
    [Crossref]
  15. S. Y. Shin and L. B. Felsen, “Gaussian Beam Tracking in an Optical Fiber,” in preparation.
  16. W-Y. D. Wang and G. A. Deschamps, Proc. IEEE 62, 1541, 1974.
    [Crossref]
  17. E. Gowan and G. A. Deschamps, Antenna Lab., Univ of Illinois, Urbana, Ill., , 1970.
  18. Ref. 2, Sec. 6.5.
  19. A. Green, H. Bertoni, and L. B. Felsen, Progress Report to Joint Services Technical Advisory Committee, Polytechnic Institute of New York, 1974.
  20. A. Green, Doctoral dissertation (Polytechnic Institute, New York, 1976).
  21. H. Jeffreys, Asymptotic Approximations (Oxford U. P., 1962), Sec. 2.5.
  22. L. B. Felsen and S. Choudhary, invited paper presented at the Colloquium on the Optics of Guided Waves, Paris, April 1975; Rev. Opt. 6, No. 5, 297, 1975.
  23. A. W. Snyder and J. D. Love, IEEE Trans. Microwave Theory Tech. MTT-23, 134, 1975.
    [Crossref]
  24. E. T. Kornhauser and A. D. Yaghjian, Radio Sci. 2, 229, 1967.
  25. S. Choudhary and L. B. Felsen, “Propagation in graded-index fibers,” (unpublished).
  26. C. Imbert, invited paper presented at the Colloquium on the Optics of Guided Waves, Paris, April, 1975; Rev. Opt. 6, No. 5, 1975.
  27. J. W. Ra, H. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. (Soc. Ind. Appl. Math) 24, 396, 1973.
    [Crossref]
  28. S. Y. Shin and L. B. Felsen, “Lateral ray and beam shifts associated with total reflection” (unpublished).
  29. H. Kogelnik and H. P. Weber, J. Opt. Soc. Am. 64, 174, 1974.
    [Crossref]

1975 (3)

L. B. Felsen, Philips Res. Rep. 30, 187, 1975.

L. B. Felsen and S. Y. Shin, IEEE Trans. Microwave Theory Tech. MTT-23, 150, 1975.
[Crossref]

A. W. Snyder and J. D. Love, IEEE Trans. Microwave Theory Tech. MTT-23, 134, 1975.
[Crossref]

1974 (5)

H. Kogelnik and H. P. Weber, J. Opt. Soc. Am. 64, 174, 1974.
[Crossref]

W-Y. D. Wang and G. A. Deschamps, Proc. IEEE 62, 1541, 1974.
[Crossref]

S. Y. Shin and L. B. Felsen, Appl. Phys. 5, 239, 1974.
[Crossref]

G. Otis, J. Opt. Soc. Am. 64, 1545, 1974.
[Crossref]

S. Choudhary and L. B. Felsen, Proc. IEEE 62, 1530, 1974.
[Crossref]

1973 (2)

S. Choudhary and L. B. Felsen, IEEE Trans. Antennas Propag. AP-21, 730, 1973.

J. W. Ra, H. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. (Soc. Ind. Appl. Math) 24, 396, 1973.
[Crossref]

1971 (2)

1969 (1)

1967 (1)

E. T. Kornhauser and A. D. Yaghjian, Radio Sci. 2, 229, 1967.

Arnaud, J. A.

Bertoni, H.

J. W. Ra, H. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. (Soc. Ind. Appl. Math) 24, 396, 1973.
[Crossref]

A. Green, H. Bertoni, and L. B. Felsen, Progress Report to Joint Services Technical Advisory Committee, Polytechnic Institute of New York, 1974.

Choudhary, S.

S. Choudhary and L. B. Felsen, Proc. IEEE 62, 1530, 1974.
[Crossref]

S. Choudhary and L. B. Felsen, IEEE Trans. Antennas Propag. AP-21, 730, 1973.

L. B. Felsen and S. Choudhary, invited paper presented at the Colloquium on the Optics of Guided Waves, Paris, April 1975; Rev. Opt. 6, No. 5, 297, 1975.

S. Choudhary and L. B. Felsen, “Propagation in graded-index fibers,” (unpublished).

S. Choudhary and L. B. Felsen, “Propagation in inhomogeneous slab wave guides” paper presented at the IEEE-MTT-S International Microwave Symposium, Cherry Hill, N. J., June 14–16, 1976.

Deschamps, G. A.

W-Y. D. Wang and G. A. Deschamps, Proc. IEEE 62, 1541, 1974.
[Crossref]

G. A. Deschamps, Electron. Lett. 7, No. 23 (1971).
[Crossref]

E. Gowan and G. A. Deschamps, Antenna Lab., Univ of Illinois, Urbana, Ill., , 1970.

Felsen, L. B.

L. B. Felsen and S. Y. Shin, IEEE Trans. Microwave Theory Tech. MTT-23, 150, 1975.
[Crossref]

L. B. Felsen, Philips Res. Rep. 30, 187, 1975.

S. Choudhary and L. B. Felsen, Proc. IEEE 62, 1530, 1974.
[Crossref]

S. Y. Shin and L. B. Felsen, Appl. Phys. 5, 239, 1974.
[Crossref]

S. Choudhary and L. B. Felsen, IEEE Trans. Antennas Propag. AP-21, 730, 1973.

J. W. Ra, H. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. (Soc. Ind. Appl. Math) 24, 396, 1973.
[Crossref]

S. Y. Shin and L. B. Felsen, “Lateral ray and beam shifts associated with total reflection” (unpublished).

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, Englewood Cliffs, N. J., 1973), Secs. 1.6 and 1.7.

S. Choudhary and L. B. Felsen, “Propagation in graded-index fibers,” (unpublished).

L. B. Felsen and S. Choudhary, invited paper presented at the Colloquium on the Optics of Guided Waves, Paris, April 1975; Rev. Opt. 6, No. 5, 297, 1975.

A. Green, H. Bertoni, and L. B. Felsen, Progress Report to Joint Services Technical Advisory Committee, Polytechnic Institute of New York, 1974.

L. B. Felsen, “Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams,” Symp. Mat. (to be published).

S. Y. Shin and L. B. Felsen, “Gaussian Beam Tracking in an Optical Fiber,” in preparation.

S. Choudhary and L. B. Felsen, “Propagation in inhomogeneous slab wave guides” paper presented at the IEEE-MTT-S International Microwave Symposium, Cherry Hill, N. J., June 14–16, 1976.

Gowan, E.

E. Gowan and G. A. Deschamps, Antenna Lab., Univ of Illinois, Urbana, Ill., , 1970.

Green, A.

A. Green, Doctoral dissertation (Polytechnic Institute, New York, 1976).

A. Green, H. Bertoni, and L. B. Felsen, Progress Report to Joint Services Technical Advisory Committee, Polytechnic Institute of New York, 1974.

Imbert, C.

C. Imbert, invited paper presented at the Colloquium on the Optics of Guided Waves, Paris, April, 1975; Rev. Opt. 6, No. 5, 1975.

Jeffreys, H.

H. Jeffreys, Asymptotic Approximations (Oxford U. P., 1962), Sec. 2.5.

Keller, J. B.

J. B. Keller and W. Streifer, J. Opt. Soc. Am. 61, 40, 1971.
[Crossref]

J. B. Keller, Proceedings of the Symposium on Applied Mathematics (McGraw-Hill, New York, 1958), p. 27.
[Crossref]

Kogelnik, H.

Kornhauser, E. T.

E. T. Kornhauser and A. D. Yaghjian, Radio Sci. 2, 229, 1967.

Kravtsov, Yu. A.

Yu. A. Kravtsov, Proceedings of the 4th All Union Diffraction of Waves (Nauka, Moscow, 1967).

Love, J. D.

A. W. Snyder and J. D. Love, IEEE Trans. Microwave Theory Tech. MTT-23, 134, 1975.
[Crossref]

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, Englewood Cliffs, N. J., 1973), Secs. 1.6 and 1.7.

Otis, G.

Ra, J. W.

J. W. Ra, H. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. (Soc. Ind. Appl. Math) 24, 396, 1973.
[Crossref]

Shin, S. Y.

L. B. Felsen and S. Y. Shin, IEEE Trans. Microwave Theory Tech. MTT-23, 150, 1975.
[Crossref]

S. Y. Shin and L. B. Felsen, Appl. Phys. 5, 239, 1974.
[Crossref]

S. Y. Shin and L. B. Felsen, “Gaussian Beam Tracking in an Optical Fiber,” in preparation.

S. Y. Shin and L. B. Felsen, “Lateral ray and beam shifts associated with total reflection” (unpublished).

Snyder, A. W.

A. W. Snyder and J. D. Love, IEEE Trans. Microwave Theory Tech. MTT-23, 134, 1975.
[Crossref]

Streifer, W.

Wang, W-Y. D.

W-Y. D. Wang and G. A. Deschamps, Proc. IEEE 62, 1541, 1974.
[Crossref]

Weber, H. P.

Yaghjian, A. D.

E. T. Kornhauser and A. D. Yaghjian, Radio Sci. 2, 229, 1967.

Appl. Opt. (1)

Appl. Phys. (1)

S. Y. Shin and L. B. Felsen, Appl. Phys. 5, 239, 1974.
[Crossref]

Electron. Lett. (1)

G. A. Deschamps, Electron. Lett. 7, No. 23 (1971).
[Crossref]

IEEE Trans. Antennas Propag. (1)

S. Choudhary and L. B. Felsen, IEEE Trans. Antennas Propag. AP-21, 730, 1973.

IEEE Trans. Microwave Theory Tech. (2)

L. B. Felsen and S. Y. Shin, IEEE Trans. Microwave Theory Tech. MTT-23, 150, 1975.
[Crossref]

A. W. Snyder and J. D. Love, IEEE Trans. Microwave Theory Tech. MTT-23, 134, 1975.
[Crossref]

J. Opt. Soc. Am. (3)

Philips Res. Rep. (1)

L. B. Felsen, Philips Res. Rep. 30, 187, 1975.

Proc. IEEE (2)

S. Choudhary and L. B. Felsen, Proc. IEEE 62, 1530, 1974.
[Crossref]

W-Y. D. Wang and G. A. Deschamps, Proc. IEEE 62, 1541, 1974.
[Crossref]

Radio Sci. (1)

E. T. Kornhauser and A. D. Yaghjian, Radio Sci. 2, 229, 1967.

SIAM J. Appl. Math. (Soc. Ind. Appl. Math) (1)

J. W. Ra, H. Bertoni, and L. B. Felsen, SIAM J. Appl. Math. (Soc. Ind. Appl. Math) 24, 396, 1973.
[Crossref]

Other (15)

S. Y. Shin and L. B. Felsen, “Lateral ray and beam shifts associated with total reflection” (unpublished).

S. Choudhary and L. B. Felsen, “Propagation in graded-index fibers,” (unpublished).

C. Imbert, invited paper presented at the Colloquium on the Optics of Guided Waves, Paris, April, 1975; Rev. Opt. 6, No. 5, 1975.

E. Gowan and G. A. Deschamps, Antenna Lab., Univ of Illinois, Urbana, Ill., , 1970.

Ref. 2, Sec. 6.5.

A. Green, H. Bertoni, and L. B. Felsen, Progress Report to Joint Services Technical Advisory Committee, Polytechnic Institute of New York, 1974.

A. Green, Doctoral dissertation (Polytechnic Institute, New York, 1976).

H. Jeffreys, Asymptotic Approximations (Oxford U. P., 1962), Sec. 2.5.

L. B. Felsen and S. Choudhary, invited paper presented at the Colloquium on the Optics of Guided Waves, Paris, April 1975; Rev. Opt. 6, No. 5, 297, 1975.

S. Y. Shin and L. B. Felsen, “Gaussian Beam Tracking in an Optical Fiber,” in preparation.

L. B. Felsen, “Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams,” Symp. Mat. (to be published).

Yu. A. Kravtsov, Proceedings of the 4th All Union Diffraction of Waves (Nauka, Moscow, 1967).

S. Choudhary and L. B. Felsen, “Propagation in inhomogeneous slab wave guides” paper presented at the IEEE-MTT-S International Microwave Symposium, Cherry Hill, N. J., June 14–16, 1976.

J. B. Keller, Proceedings of the Symposium on Applied Mathematics (McGraw-Hill, New York, 1958), p. 27.
[Crossref]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice Hall, Englewood Cliffs, N. J., 1973), Secs. 1.6 and 1.7.

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Figures (5)

FIG. 1
FIG. 1

Phase paths and paths of constant phase.

FIG. 2
FIG. 2

Various evanescent fields in a homogeneous medium with n = 1. The phase paths are shown solid and the equiphase contours dashed. The incremental lengths ds and dt denote the spacing between adjacent equiphase contours and phase paths, respectively.

FIG. 3
FIG. 3

Phase paths and complex rays for Gaussian beam. The complex wave numbers and initial points are related by η i = i ŷ 0 ( i ) / b.

FIG. 4
FIG. 4

Geometric optical domains for plane-wave diffraction by a half-plane. The formulation in (52) applies outside the shaded regions.

FIG. 5
FIG. 5

Evanescent field exterior to a curved slab waveguide. The schematization in terms of real rays and a real caustic applies approximately when the energy leakage per unit length is small.

Equations (67)

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[ 2 + k 2 n 2 ( r ) ] u ( r ) = 0 ,
u ( r ) ~ Ā ( r ) exp [ i k S ( r ) ] + exp [ i k S ( r ) ] n = 1 Ā n ( r ) ( i k ) n ,             k large ,
( S ) 2 = n 2 ,
2 S + 2 ( S · ) ln Ā = 0 ,
Ā n 2 S + 2 ( S · ) Ā n = - 2 Ā n - 1 ,             n = 1 , 2 , 3 ,
S ( r ) = R ( r ) + i I ( r ) ,             R and I real
Ā ( r ) = exp [ w ( r ) + i v ( r ) ] ,             w and v real
( R ) 2 - ( I 2 ) = n 2
1 2 2 R + R · w - I · v = 0 ,
1 2 2 I + R · v + I · w = 0 ,
Ā n 2 R - Ā n 2 I + 2 ( R · Ā n - I · Ā n ) = - 2 Ā n - 1 ,
Ā n 2 I + Ā n 2 R + 2 ( I · Ā n + R · Ā n ) = - 2 Ā n - 1 ,
s 0 = R / β ,             t 0 = I / α ,             with β = R ,             α = I .
β 2 - α 2 = n 2 ,
d d s ( β s 0 ) = β ,             d d t ( α t 0 ) = α .
α 2 = α 1 d t 1 / d t 2
β 2 = β 1 d s 1 / d s 2 .
1 2 · ( β s 0 ) + β d w d s - α d v d t = 0
1 2 · ( α t 0 ) + β d v d s + α d w d t = 0 ,
t 0 K = d s 0 / d s ,             s 0 K ˆ = d t 0 / d t ,
K = d β / ( β d t ) ,             K ˆ = - d α / ( α d s ) .
u ( r ) ~ exp [ i k ψ ( r ) ] ,
ψ ( r ) = S ( r ) + n = 0 A n ( r ) ( i k ) n + 1 ,             A n = w n + i v n ,
2 S · A n + j = 0 n = 1 A j · A n - j - 1 + 2 A n - 1 = 0 ,             n = 1 , 2 , .
2 ( β d w n d s - α d v n d t ) + j = 0 n - 1 ( w j · w n - j - 1 - v j · v n - j - 1 ) + 2 w n - 1 = 0 ,
2 ( α d w n d t + β d v n d s ) + j = 0 n - 1 ( v j · w n - j - 1 + w j · v n - j - 1 ) + 2 v n - 1 = 0.
u ( y , 0 ) = exp [ - k I ( y , 0 ) ] ,             I ( y , 0 ) = y 2 / 2 b ,             b > 0 ,
α ( y , z ) = α ( y 0 , 0 ) D ,             D ( 1 + z 2 / b 2 ) - 1 / 2 ,
R = ( 1 + 1 2 y 0 2 / b 2 ) z , I = y 0 2 / 2 b , w = ln D 1 / 2 , v = - 1 2 tan - 1 ( z / b ) ,
u ( y , z ) ~ D - 1 / 2 exp [ i k ( 1 + 1 2 y 2 D 2 / b 2 ) z - k y 2 D 2 / 2 b - 1 2 i tan - 1 ( z / b ) ] ,
u ( y , z ) = C - exp [ i k P ( η ) ] d η , P ( η ) = η y + κ z + i b η 2 / 2 ,
η s = y [ z / κ ( η s ) - i b ] - 1 ,
η s y ( z - i b ) - 1 ,             with             κ ( η s ) 1 - 1 2 y 2 / ( z - i b ) 2 .
u ( y , z ) ~ C ( 2 π - i k d 2 P / d η s 2 ) 1 / 2 e i k P ( η s ) ,
u ( y , z ) ~ C ( 2 π - i k ( i b - z ) ) 1 / 2 × exp [ i k z + i k ( y 2 / 2 ) ( z - i b ) - 1 ] .
y - y 0 = τ z ,             τ = η / κ .
ŷ = ŷ 0 = τ z ,
Re ŷ - ŷ 0 = τ z .
ŷ - ŷ = η κ ( z ˆ - z ˆ ) ,             η = Q ¯ ŷ ,             κ = Q ¯ z ˆ ,
ŷ = 0 ,             z ˆ = i b .
G f ( r , r ) = 1 4 i H 0 ( 1 ) ( k D ) , D = [ ( y - y ) 2 + ( z - z ) 2 ] 1 / 2 ,             r = ( y , z ) ,
G f ( r , r ) ~ ( - i 8 π k ) - 1 / 2 D - 1 / 2 exp ( i k D ) .
D ˆ ~ Re z ˆ - i b + ( Re ŷ ) 2 / 2 ( Re z ˆ - i b ) ,
G f ( r , r ˆ ) = ( 4 π D ˆ ) - 1 exp ( i k D ˆ ) , D ˆ = [ ρ 2 + ( z - i b ) 2 ] 1 / 2 ,             ρ 2 = x 2 + y 2 ,
D ˆ ~ z - i b + ρ 2 / 2 ( z - i b ) ,             ρ 2 z 2 + b 2 ,
H x f = - k m ( μ ) 1 / 2 x y D ˆ 2 G f , H y f = k m ( μ ) 1 / 2 x 2 + ( z - i b ) 2 D ˆ 2 G f , H z f = - k m ( μ ) 1 / 2 y ( z - i b ) D ˆ 2 G f ,
E x f = - k m z - i b D ˆ G f ,             E y f = 0 ,             E z f = k m x D ˆ G f ,
u ( r ) = 1 2 exp [ i k ρ cos ( θ - θ 0 ) ] Q ( - i p k ρ ) - 1 2 exp [ - i k ρ cos ( θ + θ 0 ) ] Q ( - i q k ρ ) ,
p = 2 e i π / 4 sin 1 2 ( θ 0 - θ ) ,             q = 2 e i π / 4 cos 1 2 ( θ 0 + θ ) ,
Q ( τ ) = 2 π - 1 / 2 τ e - ν 2 d ν .
Q ( τ ) ~ 2 U ( - Re τ ) + e - τ 2 τ π 1 / 2 × [ 1 + n = 1 ( - 1 ) n 1 × 3 × 5 × × ( 2 n - 1 ) 2 n τ 2 n ] ,
u ( r ) ~ u g + u d ,
u g = exp [ i k ρ cos ( θ - θ 0 ) ] U ( θ - θ 0 ) - exp [ - i k ρ cos ( θ + θ 0 ) ] U ( θ + θ 0 - π )
u d = [ exp ( i π / 4 ) 2 ( 2 π k ρ ) 1 / 2 ( csc θ 0 - θ 2 - sec θ 0 + θ 2 ) ] exp ( i k ρ ) .
ρ 1 cos ( θ θ 0 ) M / 2 k ,
θ 0 = φ + i φ ,             φ and φ real ,
θ ¯ = φ + tan - 1 ( sinh φ ) .
β 2 = β 1 ρ 1 / ρ 2 ,             α 2 = ( β 1 2 ρ 1 2 / ρ 2 2 - 1 ) 1 / 2 ;
I 2 = I 1 + β 1 ρ 1 ( tanh τ 2 - tanh τ 1 - τ 2 + τ 1 ) , τ i = cosh - 1 ( β 1 ρ 1 / ρ i ) ;
R 2 = R 1 = β 1 s 1 ;
v 2 = v 1 ,             w 2 = - 1 2 ln [ ( ρ 2 sinh τ 2 ) / ( ρ 1 sinh τ 1 ) ] .
u 2 = u 1 H ν ( 1 ) ( κ ρ 2 ) / H ν ( 1 ) ( k ρ 1 ) ,             ν = k β 1 ρ 1 ,
β = n 0 ,             n 2 ( x ) = n 0 2 - α 2 ( x ) ,
w x = - 1 2 α α x + n 0 p α ,
w = ln α - 1 / 2 + n 0 p d x α ( x ) .
u ~ [ α ( x ) ] - 1 / 2 exp ( i k n 0 z - k α ( x ) d x - i p z + n 0 p [ α ( x ) ] - 1 d x ) ,
n 2 ( x ) = n 0 2 sech 2 b x ,             b = const ,