Abstract

Nonspecular phenomena involving Gaussian beams reflected at a dielectric interface include the well-known lateral displacement, the more recently investigated focal and angular shifts, and a fourth newly recognized effect termed beam-waist modification. In the past, expressions for the magnitudes of these effects have been obtained for incidence angles θi that are not too close to the critical angle of total reflection θc, but results for θi near θc have been reported only for lateral displacement. We have therefore examined all the four effects and derived explicit expressions that hold for arbitrary values of θi in the vicinity of θc. These expressions are consistent with the results obtained by others for θiθc as well as with those for the lateral displacement around θi= θc. We find that the nonspecular effects are finite at θc, and we include a set of universal curves to clarify the behavior and interrelations of these phenomena.

© 1987 Optical Society of America

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References

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  1. H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).
  2. B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,”J. Opt. Soc. Am. 61, 586–594 (1971).
    [CrossRef]
  3. J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
    [CrossRef]
  4. Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).
  5. M. McGuirk, C. K. Carniglia, “An angular spectrum representation approach to the Goos-Hänchen shift,”J. Opt. Soc. Am. 67, 103–107 (1977).
    [CrossRef]
  6. C. K. Carniglia, K. R. Brownstein, “Focal shift and ray model for total internal reflection,”J. Opt. Soc. Am. 67, 121–123 (1977).
    [CrossRef]
  7. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
    [CrossRef]
  8. H. M. Lai, F. C. Cheng, W. K. Tang, “Goos–Hänchen effect around and off the critical angle,” J. Opt. Soc. Am. A 3, 550–557 (1986).
    [CrossRef]
  9. Note that ui here plays the same role as (−δ)1/2in Ref. 2.
  10. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 337, Eq. (3.462).
  11. W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), Chap. 8, p. 331.
  12. See Ref. 11, p. 324.
  13. C. C. Chan, T. Tamir, “Angular shift of a Gaussian beam reflected near the Brewster angle,” Opt. Lett. 10, 378–380 (1985).
    [CrossRef] [PubMed]
  14. I. A. White, A. W. Snyder, C. Pask, “Directional change of beams undergoing partial reflection,”J. Opt. Soc. Am. 67, 703–705 (1977).
    [CrossRef]
  15. W. Nasalski, T. Tamir, “Composite beam-shifting effects under critical reflection conditions,” J. Opt. Soc. Am. A 3(13), P124 (1986).
  16. J. J. Cowan, B. Aničin, “Longitudinal and transverse displacements of a bounded microwave beam at total internal reflection,”J. Opt. Soc. Am. 67, 1307–1314 (1977).
    [CrossRef]
  17. H. L. Bertoni, C. W. Hsue, T. Tamir, “Non-specular reflection of convergent beams from liquid-solid interface,” Trait. Signal, 2, 201–205 (1985).
  18. P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,” J. Acoust. Soc. Am. (to be published).

1986 (3)

1985 (2)

C. C. Chan, T. Tamir, “Angular shift of a Gaussian beam reflected near the Brewster angle,” Opt. Lett. 10, 378–380 (1985).
[CrossRef] [PubMed]

H. L. Bertoni, C. W. Hsue, T. Tamir, “Non-specular reflection of convergent beams from liquid-solid interface,” Trait. Signal, 2, 201–205 (1985).

1977 (4)

1974 (1)

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

1973 (1)

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

1971 (1)

1970 (1)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).

Adler, L.

P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,” J. Acoust. Soc. Am. (to be published).

Anicin, B.

Antar, Y. M.

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

Bertoni, H. L.

H. L. Bertoni, C. W. Hsue, T. Tamir, “Non-specular reflection of convergent beams from liquid-solid interface,” Trait. Signal, 2, 201–205 (1985).

J. W. Ra, H. L. Bertoni, 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. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

Brownstein, K. R.

Carniglia, C. K.

Chan, C. C.

Cheng, F. C.

Chimenti, D. E.

P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,” J. Acoust. Soc. Am. (to be published).

Cho, K.

P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,” J. Acoust. Soc. Am. (to be published).

Cowan, J. J.

Felsen, L. B.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 337, Eq. (3.462).

Horowitz, B. R.

Hsue, C. W.

H. L. Bertoni, C. W. Hsue, T. Tamir, “Non-specular reflection of convergent beams from liquid-solid interface,” Trait. Signal, 2, 201–205 (1985).

Lai, H. M.

Lotsch, H. K. V.

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).

Magnus, W.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), Chap. 8, p. 331.

McGuirk, M.

Nagy, P. B.

P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,” J. Acoust. Soc. Am. (to be published).

Nasalski, W.

W. Nasalski, T. Tamir, “Composite beam-shifting effects under critical reflection conditions,” J. Opt. Soc. Am. A 3(13), P124 (1986).

Oberhettinger, F.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), Chap. 8, p. 331.

Pask, C.

Ra, J. W.

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

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 337, Eq. (3.462).

Snyder, A. W.

Soni, R. P.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), Chap. 8, p. 331.

Tamir, T.

T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
[CrossRef]

W. Nasalski, T. Tamir, “Composite beam-shifting effects under critical reflection conditions,” J. Opt. Soc. Am. A 3(13), P124 (1986).

C. C. Chan, T. Tamir, “Angular shift of a Gaussian beam reflected near the Brewster angle,” Opt. Lett. 10, 378–380 (1985).
[CrossRef] [PubMed]

H. L. Bertoni, C. W. Hsue, T. Tamir, “Non-specular reflection of convergent beams from liquid-solid interface,” Trait. Signal, 2, 201–205 (1985).

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

Tang, W. K.

White, I. A.

Can. J. Phys. (1)

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

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

Opt. Lett. (1)

Optik (1)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).

SIAM J. Appl. Math. (1)

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

Trait. Signal (1)

H. L. Bertoni, C. W. Hsue, T. Tamir, “Non-specular reflection of convergent beams from liquid-solid interface,” Trait. Signal, 2, 201–205 (1985).

Other (5)

P. B. Nagy, K. Cho, L. Adler, D. E. Chimenti, “Focal shift of convergent ultrasonic beams reflected from a liquid–solid interface,” J. Acoust. Soc. Am. (to be published).

Note that ui here plays the same role as (−δ)1/2in Ref. 2.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 337, Eq. (3.462).

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), Chap. 8, p. 331.

See Ref. 11, p. 324.

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Figures (5)

Fig. 1
Fig. 1

Geometry of the reflecting interface. The coordinates of the incident beam and the geometrical-optics reflected beam are given by (xi, zi) and (xr, zr), respectively.

Fig. 2
Fig. 2

Universal curve for L′/Lc′ versus v as given by Eq. (31), and for α/αc versus −v as given by Eq. (57), with wr = w.

Fig. 3
Fig. 3

Universal curve for F′/|Fc′| versus v as given by Eq. (44), and for −μ/μc versus −v as given by Eq. (50), with wr = w.

Fig. 4
Fig. 4

Magnitude (a) and phase (b) of the approximated reflectance function r(θ) versus the spatial angle θ for parallel polarization and n = 1.491. The expansion is taken at θi = θc + 5° and includes up to the second-order term. The exact values of r(θ) are shown by dashed lines for comparison.

Fig. 5
Fig. 5

Variation of L′/Lc′ versus θi in the vicinity of θc, as calculated from Eq. (30), together with Eqs. (24)(28), for parallel polarization and n = 1.491.

Equations (74)

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G i ( x i , z i ) = ( w / w i ) exp [ - ( x i / w i ) 2 + i k z i ] ,
w i 2 = w 2 + i ( 2 z i / k ) ,
G r ( x r , z r ) = k w 2 π - r ( θ ) exp [ - ( k w s / 2 ) 2 + i k ( s x r + c z r ) ] d s ,
s = s ( θ ) = sin ( θ - θ i ) ,
c = c ( θ ) = cos ( θ - θ i ) ,
r ( θ ) = cos θ - m ( sin 2 θ c - sin 2 θ ) 1 / 2 cos θ + m ( sin 2 θ c - sin 2 θ 1 / 2 ,
m = { 1 for perpendicular polarization n 2 for parallel polarization .
u 2 = u i 2 - s ,
u i = [ sin ( θ c - θ i ) ] 1 / 2 .
r ( θ ) r ( θ i ) [ 1 - ( u - u i ) R 1 ( θ i ) + ( u - u i ) 2 R 2 ( θ i ) 2 ] ,
R 1 ( θ i ) = 4 m sin θ i [ sin ( θ c + θ i ) ] 1 / 2 ( cos θ c cos θ i ) 2 1 1 - ( m U u i ) 2 ,
R 2 ( θ i ) = R 1 ( θ i ) { 4 sin θ i [ sin ( θ c + θ i ) ] 1 / 2 m + U u i 1 + m U u i + [ 2 cot θ i - tan ( θ c - θ i 2 ) - cot ( θ c + θ i ) ] u i } ,
U = [ sin ( θ c + θ i ] 1 / 2 sec θ i .
c 1 - s 2 2 = 1 - 1 2 ( u 2 - u i 2 ) 2 .
G r ( x r , z r ) = ( w / w r ) r ( θ i ) ( 1 + g ) exp [ - ( x r / w r ) 2 + i k z r ] ,
w r 2 = w 2 + i ( 2 z r / k )
g = R s ( θ i ) [ u i - ( k w r ) - 1 / 2 C 1 / 2 ( β ) ] - i ( x r / k w r 2 ) R 2 ( θ i ) ,
R s ( θ i ) = R 1 ( θ i ) + u i R 2 ( θ i ) ,
C ν ( β ) = [ 2 exp ( β 2 + i π ) ] 1 / 4 D ν ( β ) ,
β = - 2 ( x r w r + i k w r u i 2 2 ) .
G r ( x r , z r ) = [ ( w / w r ) r ( θ i ) exp ( i k z r ) ] P ( x r , z r ) ,
P ( x r , z r ) = ( 1 + g ) exp [ - ( x r / w r ) 2 ] .
1 + g = exp [ ln ( 1 + g ) ] exp [ b 0 + b 1 ( x r / w r ) + b 2 ( x r / w r ) 2 ] ,
b 0 = ln ( 1 + g 0 ) ,
b 1 = 1 1 + g 0 [ R s ( θ i ) C - 1 / 2 ( i v r ) ( 2 k w r ) 1 / 2 - i k w r R 2 ( θ i ) ] ,
b 2 = 1 4 [ R s ( θ i ) C - 3 / 2 ( i v r ) ( 1 + g 0 ) ( k w r ) 1 / 2 - 2 b 1 2 ] ,
g 0 = R s ( θ i ) [ u i - ( k w r ) - 1 / 2 C 1 / 2 ( i v r ) ] ,
v r = - ( k w r / 2 ) u i 2 = ( k w r / 2 ) sin ( θ i - θ c ) .
P ( x r , z r ) exp [ b 0 + ( 1 - b 2 ) ( L w r ) 2 ] × exp [ - ( 1 - b 2 ) ( x r - L w r ) 2 ] ,
L = L + i L = b 1 w r 2 ( 1 - b 2 ) ,
L / w r = 2 - 3 / 2 R 1 ( θ i ) ( k w r ) - 1 / 2 C - 1 / 2 ( i v r ) ,
C ν ( β ) ~ [ 2 exp ( i π ) ] 1 / 4 β ν [ 1 - ν ( ν - 1 ) 2 β 2 + ] ,
L = L { R 1 ( θ i ) 2 k sin 1 / 2 ( θ i - θ c ) for θ i > θ c 0 for θ i < θ c .
C ν ( β ) C ν ( 0 ) = [ 2 exp ( i π ) ] 1 / 4 2 ν / 2 Γ ( 1 / 2 ) Γ ( 1 / 2 - ν / 2 ) ,
L c = c l m ( w λ tan θ c ) 1 / 2 ,
v = ( k w / 2 ) sin ( θ i - θ c ) ,
g | R s ( θ i ) ( 2 k w ) 2 u i 3 | 1 ( 2 k w ) 2 [ n ( n 2 + 1 ) 1 / 2 n - ( n 2 - 1 ) 1 / 2 + 2 ( 3 n 4 + 2 n 2 + 2 ) ) n ] 1 θ i - θ b + 1
w f 2 = w r 2 1 - b 2 = w 2 + 2 i k ( z r - F ) ,
F = F + i F = i k w r 2 b 2 2 ( 1 - b 2 ) .
F { Re [ R s ( θ i ) ] 4 k sin 3 / 2 ( θ i - θ c ) for θ i > θ c 0 for θ i < θ c ,
F s n λ cos θ i π ( n 2 sin 2 θ i - 1 ) 3 / 2 .
F p ( 1 + p ) F s ( n 2 + 1 ) sin 2 θ i - 1 ,
p = 2 ( n 2 sin 2 θ i - 1 ) ( n 2 + 1 ) sin 2 θ i ( n 2 + 1 ) sin 2 θ i - 1 .
F / w r = i 8 R 1 ( θ i ) ( k w r ) 1 / 2 C - 3 / 2 ( i v r ) .
F c = - c f m [ ( w 3 / λ ) tan θ c ] 1 / 2 ,
F = - d L / d θ i ,
w m 2 = w f 2 z r = F = w 2 ( 1 + μ ) ,
μ = 2 F k w 2 .
μ = Re [ ( w r w ) 2 ( b 2 1 - b 2 ) ] .
μ = R 1 ( θ i ) 4 ( k w ) 1 / 2 Re [ ( w r w ) 3 / 2 C - 3 / 2 ( i v r ) ] .
μ { 0 for θ i > θ c - R 1 ( θ i ) 2 ( k w ) 2 sin 3 / 2 ( θ c - θ i ) for θ i < θ c .
μ c = c μ m ( λ w tan θ c ) 1 / 2 ,
- μ ( - v ) μ c = F ( v ) F c .
k w 2 ( 1 + μ ) ( x r - L ) = 2 L ( z r - F ) .
tan α = ( x r = L ) / ( z r - F ) .
α tan α = 2 L k w 2 ( 1 + μ ) .
α = R 1 ( θ i ) 2 ( k w ) 3 / 2 Im [ ( w r w ) 1 / 2 C - 1 / 2 ( i v r ) ] ,
μ = - d α / d θ i .
α { 0 for θ i > θ c R 1 ( θ i ) ( k w ) 2 sin 1 / 2 ( θ c - θ i ) for θ i < θ c .
R 1 ( θ i ) = 2 sin 1 / 2 ( θ c - θ i ) r ( θ i ) d r d θ | θ = θ i .
α = 2 ( k w ) 2 r ( θ i ) d r d θ | θ = θ i .
α c = c α m ( λ w ) 3 / 2 ( tan θ c ) 1 / 2 ,
α ( - v ) α c = L ( v ) L c .
x r = L + ( z r - F ) α = S ,
S = w Re ( b 1 ) 2 [ 1 - Re ( b 2 ) ] ,
R 1 ( θ i ) = R c = m ( 8 tan θ c ) 1 / 2 ,
R 2 ( θ i ) = R c 2 ( 1 + A u i ) ,
R s ( θ i ) = R c ( 1 + R c u i ) ,
A = [ 4 ( 1 - m 2 ) tan θ c + 2 cot θ c - cot 2 θ c ] / R c .
b 1 1 - g 0 ( k w r ) 1 / 2 [ 2 - 1 / 2 R s ( θ i ) C - 1 / 2 ( 0 ) - i ( k w r ) - 1 / 2 R 2 ( θ i ) ] .
b 1 = R c ( k w r ) 1 / 2 [ B ( 1 + i ) + i 1 + 2 R c - A ( k w r ) 1 / 2 R c u i ] ,
G j = k w r ( θ i ) 2 π - R j ( θ i ) j ! ( u i - u ) j × exp [ - ( k w s 2 ) 2 + i k ( s x r + c z r ) ] d s ,
G j = ( - ) j w w r R j j ! 2 ( j - 1 ) / 4 ( i k w r ) j / 2 C j / 2 ( β ) exp [ - ( x r / w r ) 2 + i k z r ] .
= | G 3 G 1 | = 1 3 k w Γ ( 1 / 4 ) Γ ( - 1 / 4 ) R 3 ( θ c ) R 1 ( θ c ) = 0.371 2 + n 2 - 8 m 2 k w ( n 2 - 1 ) 1 / 2 ,

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