Abstract

The characteristics of a Gaussian beam that is reflected or transmitted by an interface separating a denser medium from a rarer medium are mainly described by means of numerical integration. An approximate result is shown to compare with the exact value obtained by numerical integration. The cases where the incident angle θ is arbitrary, that is, θ = θc (critical angle), θθc, θθc are treated. Transmitted power in the rarer medium is obtained, from which properties of a lateral wave are found.

© 1978 Optical Society of America

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References

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  1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).
  2. T. Tamir and A. A. Oliner, “Role of the lateral wave in total reflection of light,” J. Opt. Soc. Am. 59, 942–949 (1969).
  3. B. R. Horowitz and T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971).
    [Crossref]
  4. B. R. Horowitz and T. Tamir, “Unified theory of total reflection phenomena at a dielectric interface,” Appl. Phys. 1, 31–38 (1973).
    [Crossref]
  5. S. Nemoto and T. Makimoto, “Reflection and transmission of two-dimensional Gaussian beam at the plane interface of dielectric,” Electron. Commun. (Japan) 54-B, 715–721 (1971).
  6. J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
    [Crossref]
  7. Y. M. Anter and W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).
  8. Y. M. Anter and W. M. Boerner, “A generalized approach to beam wave interaction with a dielectric interface,” Appl. Phys. 7, 295–301 (1975).
    [Crossref]
  9. H. K. V. Lotsch, “Reflection and refraction of a beam of light at a plane interface,” J. Opt. Soc. Am. 58, 551–561 (1968).
    [Crossref]
  10. H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect I,” Optik (Stuttgart) 32, 116–137 (1970).
  11. H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect II,” Optik (Stuttgart) 32, 190–204 (1970).
  12. H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect III,” Optik (Stuttgart) 32, 299–319 (1970).
  13. H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect IV,” Optik (Stuttgart) 32, 553–569 (1971).
  14. McGuirk and C. K. Carniglia; “An angular spectrum representation approach to the Goos-Hänchen shift,” J. Opt. Soc. Am. 67, 103–107 (1977).
    [Crossref]
  15. S. Kozaki and K. Kimura, “Wave path of a Gaussian beam in an inhomogeneous medium,” J. Opt. Soc. Am. 66, 63–64 (1976).
    [Crossref]

1977 (1)

1976 (1)

1975 (1)

Y. M. Anter and W. M. Boerner, “A generalized approach to beam wave interaction with a dielectric interface,” Appl. Phys. 7, 295–301 (1975).
[Crossref]

1974 (1)

Y. M. Anter and W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

1973 (2)

J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[Crossref]

B. R. Horowitz and T. Tamir, “Unified theory of total reflection phenomena at a dielectric interface,” Appl. Phys. 1, 31–38 (1973).
[Crossref]

1971 (3)

S. Nemoto and T. Makimoto, “Reflection and transmission of two-dimensional Gaussian beam at the plane interface of dielectric,” Electron. Commun. (Japan) 54-B, 715–721 (1971).

B. R. Horowitz and T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971).
[Crossref]

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect IV,” Optik (Stuttgart) 32, 553–569 (1971).

1970 (3)

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect I,” Optik (Stuttgart) 32, 116–137 (1970).

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect II,” Optik (Stuttgart) 32, 190–204 (1970).

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect III,” Optik (Stuttgart) 32, 299–319 (1970).

1969 (1)

1968 (1)

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).

Anter, Y. M.

Y. M. Anter and W. M. Boerner, “A generalized approach to beam wave interaction with a dielectric interface,” Appl. Phys. 7, 295–301 (1975).
[Crossref]

Y. M. Anter and W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

Bertoni, H. L.

J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[Crossref]

Boerner, W. M.

Y. M. Anter and W. M. Boerner, “A generalized approach to beam wave interaction with a dielectric interface,” Appl. Phys. 7, 295–301 (1975).
[Crossref]

Y. M. Anter and W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

Carniglia, C. K.

Felsen, L. B.

J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[Crossref]

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).

Horowitz, B. R.

B. R. Horowitz and T. Tamir, “Unified theory of total reflection phenomena at a dielectric interface,” Appl. Phys. 1, 31–38 (1973).
[Crossref]

B. R. Horowitz and T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971).
[Crossref]

Kimura, K.

Kozaki, S.

Lotsch, H. K. V.

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect IV,” Optik (Stuttgart) 32, 553–569 (1971).

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect I,” Optik (Stuttgart) 32, 116–137 (1970).

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect II,” Optik (Stuttgart) 32, 190–204 (1970).

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect III,” Optik (Stuttgart) 32, 299–319 (1970).

H. K. V. Lotsch, “Reflection and refraction of a beam of light at a plane interface,” J. Opt. Soc. Am. 58, 551–561 (1968).
[Crossref]

Makimoto, T.

S. Nemoto and T. Makimoto, “Reflection and transmission of two-dimensional Gaussian beam at the plane interface of dielectric,” Electron. Commun. (Japan) 54-B, 715–721 (1971).

McGuirk,

Nemoto, S.

S. Nemoto and T. Makimoto, “Reflection and transmission of two-dimensional Gaussian beam at the plane interface of dielectric,” Electron. Commun. (Japan) 54-B, 715–721 (1971).

Oliner, A. A.

Ra, J. W.

J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[Crossref]

Tamir, T.

Ann. Phys. (Leipzig) (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).

Appl. Phys. (2)

B. R. Horowitz and T. Tamir, “Unified theory of total reflection phenomena at a dielectric interface,” Appl. Phys. 1, 31–38 (1973).
[Crossref]

Y. M. Anter and W. M. Boerner, “A generalized approach to beam wave interaction with a dielectric interface,” Appl. Phys. 7, 295–301 (1975).
[Crossref]

Can. J. Phys. (1)

Y. M. Anter and W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

Electron. Commun. (Japan) (1)

S. Nemoto and T. Makimoto, “Reflection and transmission of two-dimensional Gaussian beam at the plane interface of dielectric,” Electron. Commun. (Japan) 54-B, 715–721 (1971).

J. Opt. Soc. Am. (5)

Optik (Stuttgart) (4)

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect I,” Optik (Stuttgart) 32, 116–137 (1970).

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect II,” Optik (Stuttgart) 32, 190–204 (1970).

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect III,” Optik (Stuttgart) 32, 299–319 (1970).

H. K. V. Lotsch, “Beam displacement at total reflection: The Goos-Hänchen effect IV,” Optik (Stuttgart) 32, 553–569 (1971).

SIAM J. Appl. Math. (1)

J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[Crossref]

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Figures (6)

FIG. 1
FIG. 1

Ray trajectory for a Gaussian beam in a denser medium and rarer medium.

FIG. 2
FIG. 2

Variations in Z vs θθc.

FIG. 3
FIG. 3

Plots of reflected beam amplitude and phase for aλ = 0.05 θθc = 1°.

FIG. 4
FIG. 4

Equiamplitude and equiphase of the transmitted beam for aλ = 0.1, (a) θ = 60°, (b) θ = θc + 1°, (c) θ = θc.

FIG. 5
FIG. 5

Distribution in the transmitted power in the direction perpendicular to the interface.

FIG. 6
FIG. 6

The transmitted power amplitude and angle in the space (y,z) for aλ = 0.1. (a) θ = 60° > θc, (b) θ = θc.

Equations (50)

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2 E x i y i 2 + 2 E x i z i 2 + k 1 2 E x i = 0 ,
E x i ( y i , z i ) = 1 2 π - Φ ( α , z i ) exp ( - j α y i ) d α ,
d 2 Φ d z i 2 + ( k 1 2 - α 2 ) Φ = 0 ,
Φ = E i ( α ) exp [ - j z i ( k 1 2 - α 2 ) 1 / 2 ] .
E x i ( y i , z i ) = 1 2 π - E i ( α ) × exp [ - j z i ( k 1 2 - α 2 ) 1 / 2 - j α y i ] d α ,
( y i z i ) = ( cos θ - sin θ sin θ cos θ ) ( y z + z 0 ) ,
E x i ( y , z ) = 1 2 π - E i ( α ) × exp [ - j y p - j ( z + z 0 ) ( k 1 2 - p 2 ) 1 / 2 ] d α ,
2 E x i y 2 + 2 E x i z 2 + k 1 2 E x i = 0             for z < 0.
E x r ( y , z ) = 1 2 π - E i ( α ) R ( α ) × exp [ - j y p + j ( z - z 0 ) ( k 1 2 - p 2 ) 1 / 2 ] d α ,
2 E x t y 2 + 2 E x t z 2 + k 2 2 E x t = 0             for z > 0 ,
E x t ( y , z ) = 1 2 π - E i ( α ) T ( α ) × exp [ - j y p - j z ( k 2 2 - p 2 ) 1 / 2 ] d α ,
R ( α ) = ( k 1 2 - p 2 ) 1 / 2 - ( k 2 2 - p 2 ) 1 / 2 ( k 1 2 - p 2 ) 1 / 2 + ( k 2 2 - p 2 ) 1 / 2 ,
T ( α ) = 2 ( k 1 2 - p 2 ) 1 / 2 ( k 1 2 - p 2 ) 1 / 2 + ( k 2 2 - p 2 ) 1 / 2 × exp [ - j z 0 ( k 1 2 - p 2 ) 1 / 2 ] .
( y r z r ) = ( cos θ sin θ - sin θ cos θ ) ( y z - z 0 ) ,
( y t z t ) = ( cos θ t - sin θ t sin θ t cos θ t ) ( y - y 0 z ) ,
E x i ( y i , 0 ) = exp ( - a 2 y i 2 ) ,
E ( α ) = - exp ( - a 2 y i 2 + j α y i ) d y i = π a exp ( - α 2 4 a 2 ) .
( k 2 2 - p 2 ) { > 0 : case 1 , < 0 : case 2 , = 0 : case 3 ,
y p + ( z + z 0 ) ( k 1 2 - p 2 ) 1 / 2 k 1 z i + y i α - ( z i / k 1 ) ( α 2 / 2 ) ,
y p - ( z - z 0 ) ( k 1 2 - p 2 ) 1 / 2 - k 1 z r + y r α + ( z r / k 1 ) ( α 2 / 2 ) ,
x p + y ( k 2 2 - p 2 ) 1 / 2 + y 0 ( k 1 2 - p 2 ) 1 / 2 k 2 z t + k 1 ( z 0 / cos θ ) + u y t α - ( v / k 1 ) ( α 2 / 2 ) ,
R ( α ) R ( 0 ) + R ( 0 ) α + R ( 0 ) ( α 2 / 2 ) ,
T ( α ) T ( 0 ) + T ( 0 ) α + T ( 0 ) ( α 2 / 2 ) ,
R ( 0 ) = n cos θ - ( 1 - n 2 sin 2 θ ) 1 / 2 n cos θ + ( 1 - n 2 sin 2 θ ) 1 / 2 , R ( 0 ) = 2 sin θ k 2 ( 1 - n 2 sin 2 θ ) 1 / 2 R ( 0 ) , R ( 0 ) = 2 [ cos θ + 2 n sin 2 θ ( 1 - n 2 sin 2 θ ) 1 / 2 ] k 2 2 n [ ( 1 - n 2 sin 2 θ ) 1 / 2 ] 3 R ( 0 ) , T ( 0 ) = 2 n cos θ n cos θ + ( 1 - n 2 sin 2 θ ) 1 / 2 , T ( 0 ) = R ( 0 ) , T ( 0 ) = R ( 0 ) , u = cos θ cos θ t ,             v = n cos 2 θ cos 3 θ t z + z 0 cos θ + sin θ cos θ t y t .
E x i ( y i , z i ) 1 ( 1 - j 2 a 2 z i / k 1 ) 1 / 2 × exp ( - ( a y i ) 2 ( 1 - j 2 a 2 z i / k 1 ) - j k 1 z i ) ,
E x r ( y r , z t ) [ R ( 0 ) ( 1 + j 2 a 2 z r / k 1 ) 1 / 2 - j 2 a 2 R ( 0 ) y r ( 1 + j 2 a 2 z r / k 1 ) 3 / 2 + a 2 R ( 0 ) ( 1 + j 2 a 2 z r / k 1 ) 3 / 2 ( 1 - 2 ( a y r ) 2 1 + j 2 a 2 z r / k 1 ) ] × exp ( - ( a y r ) 2 ( 1 + j 2 a 2 z r / k 1 ) + j k 1 z r ) ,
E x t ( y t , z t ) [ T ( 0 ) ( 1 - j 2 a 2 v / k 1 ) 1 / 2 - j 2 a 2 T ( 0 ) y t ( 1 - j 2 a 2 v / k 1 ) 3 / 2 + a 2 T ( 0 ) ( 1 - j 2 a 2 v / k 1 ) 3 / 2 ( 1 - 2 ( a u y t ) 2 1 - j 2 a 2 v / k 1 ) ] × exp [ - ( a u y t ) 2 ( 1 - j 2 a 2 v / k 1 ) - j ( k 2 z t + k 1 z 0 cos θ ) ] ,
R ( α ) = exp [ j 2 g ( α ) ] ,
g ( α ) = tan - 1 [ ( p 2 - k 2 2 ) 1 / 2 / ( k 1 2 - p 2 ) 1 / 2 ] ,
g ( α ) g ( 0 ) + g ( 0 ) + g ( 0 ) ( α 2 / 2 ) ,
g ( 0 ) = tan - 1 ( n 2 sin 2 θ - 1 ) 1 / 2 n cos θ , g ( 0 ) = sin θ k 2 ( n 2 sin 2 θ - 1 ) 1 / 2 , g ( 0 ) = - cos θ k 2 2 n ( n 2 sin 2 θ - 1 ) 3 / 2 .
E x r ( y r , z r ) 1 { 1 + j 2 a 2 [ z r - 2 k 1 g ( 0 ) ] / k 1 } 1 / 2 × exp ( - { a [ y r - 2 g ( 0 ) ] } 2 1 + j 2 a 2 [ z r - 2 k 1 g ( 0 ) ] / k 1 + j [ k 1 z r + 2 g ( 0 ) ] ) .
E x r ( y r , z r ) = 1 ( 1 + { [ ( a λ ) 2 / n π ] [ ( z r / λ ) + Z ] } 2 ) 1 / 4 × exp ( - ( a λ ) 2 [ ( y r / λ ) - Y ] 2 1 + { [ ( a / λ ) 2 / n π ] [ ( z r / λ ) + Z ] } 2 ) ,
λ = 2 π / k 2 ,             n = { 1 / 2 } 1 / 2 , Y = sin θ π ( n 2 sin 2 θ - 1 ) 1 / 2 ,
Z = cos θ π ( n 2 sin 2 θ - 1 ) 3 / 2 .
( z r / λ ) + Z = 0 ,
E x r ( y r , - λ Z ) = exp { - ( a λ ) 2 [ ( y r / λ ) - Y ] 2 } .
E x r ( λ Y , z r ) = 1 ( 1 + { [ ( a λ ) 2 / n π ] [ ( z r / λ ) + Z ] } 2 ) 1 / 4 ,
Z = b + c ( = constant ) ,
P z t ( y , 0 ) = ½ Re ( E x t H y * t ) - 4 n 3 a 2 sin 2 θ cos 2 θ ω μ 0 ( ( n 2 - 1 ) 1 / 2 ( n 2 sin 2 θ - 1 ) 1 / 2 ) × [ y - ( z 0 tan θ + λ tan θ 2 π ( n 2 sin 2 θ - 1 ) 1 / 2 ) ] × exp { - 2 a 2 cos 2 θ [ y - ( z 0 tan θ + λ tan θ 2 π ( n 2 sin 2 θ - 1 ) 1 / 2 ) ] } ,
λ tan θ 2 π ( n 2 sin 2 θ - 1 ) 1 / 2 = Y cos θ
P = { [ p y ( y , z ) ] 2 + [ p z ( y , z ) ] 2 } 1 / 2 ,
Θ = tan - 1 [ p z ( y , z ) / p y ( y , z ) ] ,
T ( α ) = H ( α ) exp [ j g ( α ) ] ,
H ( α ) = 2 ( k 1 2 - p 2 ) 1 / 2 / ( k 1 2 - k 2 2 ) 1 / 2 .
H ( α ) H ( 0 ) + H ( 0 ) α + H ( 0 ) ( α 2 / 2 ) ,
f t ( α ) = y p - j z ( p 2 - k 2 2 ) 1 / 2 + z 0 ( k 1 2 - p 2 ) 1 / 2 , f t ( 0 ) + f t ( 0 ) α + f t ( 0 ) ( α 2 / 2 ) ,
H ( 0 ) = 2 n cos θ ( n 2 - 1 ) 1 / 2 ,             H ( 0 ) = - λ sin θ π ( n 2 - 1 ) 1 / 2 , H ( 0 ) = - λ 2 cos θ 2 π 2 n ( n 2 - 1 ) 1 / 2 , f t ( 0 ) = k 1 [ y sin θ - j ( z / n ) ( n 2 sin 2 θ - 1 ) 1 / 2 + z 0 cos θ ] , f t ( 0 ) = y cos θ - j z n sin 2 θ 2 ( n 2 sin 2 θ - 1 ) 1 / 2 - z 0 sin θ , f t ( 0 ) = - 1 k 1 [ y sin θ - j n z ( sin 2 θ ( n 2 sin 2 θ - 1 ) 1 / 2 + cos 2 θ ( n 2 sin 2 θ - 1 ) 3 / 2 ) + z 0 cos θ ] ,
E x t ( y , z ) = [ H ( 0 ) u ¯ - j 2 a 2 H ( 0 ) m ( u ¯ ) 3 / 2 + a 2 H ( 0 ) ( u ¯ ) 3 / 2 ( 1 - 2 a 2 m 2 u ¯ ) ] × exp [ - ( a m ) 2 u ¯ - k 2 z ( n 2 sin 2 θ - 1 ) 1 / 2 - j k 1 ( y sin θ + z 0 cos θ - g ( 0 ) k 1 ) ] ,
u ¯ = 1 - 2 a 2 n z k 1 ( sin 2 θ ( n 2 sin 2 θ - 1 ) 1 / 2 + cos 2 θ ( n 2 sin 2 θ - 1 ) 3 / 2 ) - j 2 a 2 k 1 [ y sin θ + z 0 cos θ - k 1 g ( 0 ) ] , m = g ( 0 ) - ( y cos θ - z 0 sin θ ) + j z n sin 2 θ ( n 2 sin 2 θ - 1 ) 1 / 2 .