Abstract

An improved evaluation of nonspecular effects associated with the reflection of Gaussian beams from stratified dielectric media is described. These effects are derived by examining the angular spectrum of the reflected beam rather than its spatial form and by using expansions that are more accurate than those in previous derivations. The results show that, for beams having waist-to-wavelength ratios w/λ > 10, the four previously known nonspecular effects (lateral, focal, and angular beam shifts and beam-waist modification) are changed only slightly, but substantial differences appear in the lateral and angular beam shifts for ratios w/λ < 10. In addition, we find that the magnitude of the reflected-beam peak can be quite different from that given by geometric-optical considerations.

© 1990 Optical Society of America

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References

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  1. C. K. Carniglia, K. R. Brownstein, “Focal shift and ray mode for total internal reflection,” J. Opt. Soc. Am. 67, 121–122 (1977).
    [CrossRef]
  2. I. A. White, A. W. Snyder, C. Pask, “Directional change of beams undergoing partial reflection,” J. Opt. Soc. Am. 67, 703–705 (1977).
    [CrossRef]
  3. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
    [CrossRef]
  4. W. Nasalski, T. Tamir, L. Lin, “Displacement of the intensity peak in narrow beams reflected at a dielectric interface,” J. Opt. Soc. Am. A 5, 132–140 (1988).
    [CrossRef]
  5. S. Zhang, C. Fan, “Nonspecular phenomena on Gaussian beam reflection at dielectric interfaces,” J. Opt. Soc. Am. A 5, 1407–1409 (1988).
    [CrossRef]
  6. W. Nasalski, “Modified reflectance and geometrical deformations of Gaussian beams reflected at a dielectric interface,” J. Opt. Soc. Am. A 6, 1447–1454 (1989).
    [CrossRef]
  7. R. Simon, T. Tamir, “Nonspecular phenomena in partly coherent beams reflected by multilayered structures,” J. Opt. Soc. Am. A 6, 18–22 (1989).
    [CrossRef]

1989 (2)

1988 (2)

1986 (1)

1977 (2)

Brownstein, K. R.

Carniglia, C. K.

Fan, C.

Lin, L.

Nasalski, W.

Pask, C.

Simon, R.

Snyder, A. W.

Tamir, T.

White, I. A.

Zhang, S.

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Figures (6)

Fig. 1
Fig. 1

Geometry of a multilayered reflecting structure, showing incident beam (xi, zi) coordinates, geometric-optical coordinates (xr, zr), and actual reflected-beam coordinates (xa, za).

Fig. 2
Fig. 2

Normalized differences Δl = |(δxL′)/δx| and Δf = |(δZF′)/δZ| the incidence parameter δ′ = ni sinθi, − κa, for a beam incident from the lower semi-infinite medium. Here the value δ′ = 0 corresponds to θi = 45.25°.

Fig. 3
Fig. 3

Comparison of the lateral beam shifts δx and L′ with δ′ for a beam incident as in Fig. 2 with w/λ = 10.

Fig. 4
Fig. 4

Comparison of maximum values of the angular beam shifts δθ and α with w/λ for beam incidence from both above and below the dielectric slab.

Fig. 5
Fig. 5

Variation of the reflectance modification factor ρ versus δ′ for a beam with w/λ = 10 incident from the lower medium.

Fig. 6
Fig. 6

Maximum value ρmax of the reflectance modification factor ρ versus w/λ for beam incidence from both above and below the dielectric slab.

Equations (36)

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E i ( x i , z i ) = ( w / w i ) exp [ ( x i / w i ) 2 + i k i z i ] ,
w i 2 = w 2 + i ( 2 z i / k i )
E i ( x i , z i ) = k i w 2 π exp [ ( k i w s / 2 ) 2 i k i ( x i s z i c ) ] d s ,
s = sin ( θ θ i ) ,
c = cos ( θ θ i ) ,
c = 1 ( s 2 / 2 ) ,
E i ( x i , z i ) = k i w 2 π exp ( i k i z i ) exp [ ( k i w i s / 2 ) 2 i k i x i s ] d s .
E r ( x r , z r ) = k i w 2 π r ( θ ) exp [ ( k i w s / 2 ) 2 + i k i ( x r s + z r c ) ] d s ,
E r ( x a , z a ) = k i w a 2 π r a ( θ a ) exp ( i k i z a ) × exp [ ( k i w f a σ / 2 ) 2 + i k i x a σ ] d σ ,
w f a 2 = w a 2 + i ( 2 z a / k i ) ,
σ = sin ( θ θ a ) .
x a = ( x r δ x ) cos δ θ ( z r δ z ) sin δ θ ,
z a = ( x r δ x ) sin δ θ + ( z r δ z ) cos δ θ .
r ( θ ) d s exp [ ln r ( θ ) ] d s = R ( σ ) d σ R ( 0 ) exp [ i k i L a σ + i k i F a ( σ 2 / 2 ) ] d σ ,
R ( 0 ) = r ( θ a ) cos ( θ a θ i ) , L a = L a + i L a = i k i R ( σ ) d R d σ | σ = 0 = i k i r ( θ ) d r ( θ ) d θ | θ = θ a , F a = F a + i F a = i k i d d σ [ 1 R ( σ ) d R ( σ ) d σ ] | σ = 0 = d L a d θ | θ = θ a .
s = s 0 + c 0 σ s 0 ( σ 2 / 2 ) ,
c = c 0 s 0 σ c 0 ( σ 2 / 2 ) ,
s 2 = s 0 2 + 2 s 0 c 0 σ + ( c 0 2 s 0 2 ) σ 2 ,
s 0 = sin ( θ a θ i ) = sin δ θ ,
c 0 = cos ( θ a θ i ) = cos δ θ .
E r ( x a , z a ) = k i w 2 π R ( 0 ) exp [ ψ ( σ ) ] d σ ,
ψ ( σ ) = [ ( k i w / 2 ) 2 ( c 0 2 s 0 2 ) + i ( k i / 2 ) ( x r s 0 + z r c 0 F a ) ] σ 2 + i k i [ x r c 0 z r s 0 L a + i ( k i w 2 / 2 ) s 0 c 0 ] σ + i k i [ x r s 0 + z r c 0 + i k i w 2 ( s 0 / 2 ) 2 ] .
ψ ( σ ) = [ ( k i w / 2 ) 2 ( c 0 2 s 0 2 ) + i ( k i / 2 ) ( z a + ξ 1 i F a ) ] σ 2 + i k i [ x a + ξ 2 i ξ 3 ] σ + i k i [ z a + δ x s 0 + δ z c 0 + i k w 2 ( s 0 / 2 ) 2 ] ,
ξ 1 = δ x s 0 + δ z c 0 F a ,
ξ 2 = δ x c 0 δ z s 0 L a ,
ξ 3 = L a ( k i w 2 / 2 ) s 0 c 0 .
δ x = F a sin δ θ + L a cos δ θ ,
δ z = F a cos δ θ L a sin δ θ ,
δ θ ( 1 / 2 ) sin 2 δ θ = 2 L a / k i w 2 .
ψ = [ ( k i w / 2 ) 2 cos 2 δ θ + ( k i F a / 2 ) + i ( k i z a / 2 ) ] σ 2 + i k i x a σ + i k i ( z a + F a ) ( k i w sin δ θ / 2 ) 2 .
w a 2 = w 2 ( cos 2 δ θ + 2 F / k i w 2 ) = ( M w ) 2 ,
E r ( x a , z a ) = k i w 2 π R ( 0 ) exp [ ( k i w sin δ θ / 2 ) 2 + i k i ( z a + F a ) ] × exp [ ( k i w f a σ / 2 ) 2 + i k i x a σ ] d σ .
r a ( θ a ) = r ( θ a ) cos δ θ M exp [ ( k i w sin δ θ / 2 ) 2 + i k i F a ]
ρ = cos δ θ M exp [ ( k i w sin δ θ / 2 ) 2 ]
r ( θ ) = R n i sin θ i κ n n i sin θ i κ p ,
w a w m cos δ θ ,

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