Abstract

The geometrical interpretation of Gaussian-beam deformations under reflection at a linear dielectric interface is revised. It is proved that, besides the four known geometrical effects, namely, the lateral, focal, and angular shifts and the beam-waist modification, an independent nonspecular effect termed the complex amplitude modification exists and that all these effects are necessary for a complete description of the deformed beam. The new effect is described as a product of reflectance and propagation nonspecular modifications. The amplitude-based and intensity-based definitions of the modified reflection coefficient are given, and substantial differences between them and the Fresnel reflectance are shown. The significance of the propagation modification in the evaluation of the modified reflectance is also explained. Analytical expressions for all the nonspecular effects and for the interrelations among them are derived, and an accurate numerical procedure for their evaluation is discussed.

© 1989 Optical Society of America

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References

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  1. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
    [CrossRef]
  2. F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflection,” Ann. Phys. 1, 333–345 (1947).
    [CrossRef]
  3. M. McGuirk, C. K. Carniglia, “An angular spectrum representation approach to the Goos–Hänchen shift,”J. Opt. Soc. Am. 67, 103–107 (1977).
    [CrossRef]
  4. 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]
  5. 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]
  6. W. Nasalski, “Geometrical, amplitude and phase distortions of beam fields reflected by multilayered media,” in Microwave Physics and Technique, A. Spasov, ed. (World Scientific, Singapore, 1988), pp. 371–375.
  7. P. W. Smith, W. J. Tomlinson, “Nonlinear optical interfaces: switching behavior,” IEEE J. Quantum Electron. QE-20, 30–36 (1984).
    [CrossRef]
  8. C. C. Chan, T. Tamir, “Beam phenomena at and near critical incidence upon a dielectric interface,” J. Opt. Soc. Am. A 4, 655–663 (1987).
    [CrossRef]
  9. J. J. Cowan, B. Aničin, “Longitudinal and transverse displacement of a bounded microwave beam at total internal reflection,”J. Opt. Soc. Am. 67, 1307–1314 (1977).
    [CrossRef]
  10. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 50–161.
  11. W. Nasalski, “Nonspecular reflection by an electro-optically driven interface,” in Proceedings of URSI International Symposium on Electromagnetic Theory (International Union of Radio Science, Stockholm, 1989).

1988 (1)

1987 (1)

1986 (1)

1984 (1)

P. W. Smith, W. J. Tomlinson, “Nonlinear optical interfaces: switching behavior,” IEEE J. Quantum Electron. QE-20, 30–36 (1984).
[CrossRef]

1977 (2)

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]

1947 (1)

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflection,” Ann. Phys. 1, 333–345 (1947).
[CrossRef]

Anicin, B.

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 50–161.

Bertoni, H. L.

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]

Carniglia, C. K.

Chan, C. C.

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]

Goos, F.

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflection,” Ann. Phys. 1, 333–345 (1947).
[CrossRef]

Hänchen, H.

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflection,” Ann. Phys. 1, 333–345 (1947).
[CrossRef]

Lin, L.

McGuirk, M.

Nasalski, W.

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]

W. Nasalski, “Nonspecular reflection by an electro-optically driven interface,” in Proceedings of URSI International Symposium on Electromagnetic Theory (International Union of Radio Science, Stockholm, 1989).

W. Nasalski, “Geometrical, amplitude and phase distortions of beam fields reflected by multilayered media,” in Microwave Physics and Technique, A. Spasov, ed. (World Scientific, Singapore, 1988), pp. 371–375.

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]

Smith, P. W.

P. W. Smith, W. J. Tomlinson, “Nonlinear optical interfaces: switching behavior,” IEEE J. Quantum Electron. QE-20, 30–36 (1984).
[CrossRef]

Tamir, T.

Tomlinson, W. J.

P. W. Smith, W. J. Tomlinson, “Nonlinear optical interfaces: switching behavior,” IEEE J. Quantum Electron. QE-20, 30–36 (1984).
[CrossRef]

Ann. Phys. (1)

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflection,” Ann. Phys. 1, 333–345 (1947).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. W. Smith, W. J. Tomlinson, “Nonlinear optical interfaces: switching behavior,” IEEE J. Quantum Electron. QE-20, 30–36 (1984).
[CrossRef]

J. Opt. Soc. Am. (2)

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

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]

Other (3)

W. Nasalski, “Geometrical, amplitude and phase distortions of beam fields reflected by multilayered media,” in Microwave Physics and Technique, A. Spasov, ed. (World Scientific, Singapore, 1988), pp. 371–375.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 50–161.

W. Nasalski, “Nonspecular reflection by an electro-optically driven interface,” in Proceedings of URSI International Symposium on Electromagnetic Theory (International Union of Radio Science, Stockholm, 1989).

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Figures (4)

Fig. 1
Fig. 1

Interface configuration and coordinate systems; beam-waist centers are placed at origins of the relevant coordinate frames.

Fig. 2
Fig. 2

The normalized focal shift δz and the beam-waist modification increment μ, plotted versus θiθc; kw = 30, n = 1.5 and z0 = 5, evaluation at the interface. Dotted curves, one iteration step; solid curves, four steps.

Fig. 3
Fig. 3

Magnitude of the modified reflectance Rm (four iteration steps, solid curve) and R ^ m (one step, dotted curve) versus θiθc. The Fresnel reflectance R (long-dashed curve) and the intensity-based modified reflectance rm (short-dashed curve) are shown for comparison. Parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Magnitude of the amplitude-based (solid curve) and intensity-based (short-dashed curve) modified reflection coefficients Rm and rm, respectively, and the propagation modification factor Pm (long-dashed curve), versus z0/zF; kw = 30, n = 1.5, θi = θc, and the evaluation is at the interface.

Tables (1)

Tables Icon

Table 1 Magnitude of the Reflected Gaussian Field Evaluated at the Beam Axis xm = 0a

Equations (45)

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G i ( x i , z i ) = ( 2 / π ) 1 / 4 v i - 1 exp [ - ( x i / v i ) 2 + i k w z i ] ,
v i 2 = 1 + i z i / z F ,
z F = k w / 2 ,
G r ( x r , z r ) R ( θ i ) G i ( x r , z r ) = R ( θ i ) ( 2 / π ) 1 / 4 v r - 1 exp [ - ( x r / v r ) 2 + i k w z r ] ,
v r 2 = 1 + i z r / z F ,
L = L + i L = δ ^ x + i δ ^ θ ν ^ 2 k w / 2 ,
F = F + i F = δ ^ z + i ( 1 - ν ^ 2 ) k w / 2 ,
ν ^ = ( 1 + μ ^ ) 1 / 2
G m ( x m , z m ) = A r R ( θ m ) ( 2 / π ) 1 / 4 v r - 1 × exp { - [ ( x r - L ) / v f ] 2 + i k w z r } ,
v f 2 = 1 + i ( z r - F ) / z F ,
k w 1.
x m = ( x r - δ x ) cos δ θ - ( z r - δ z ) sin δ θ ,
z m = ( x r - δ x ) sin δ θ + ( z r - δ z ) cos δ θ .
w m / w = ν = ( 1 + μ ) 1 / 2 ,
θ m = θ i + δ θ .
G m ( x m , z m ) = R m P m ( 2 / π ) 1 / 4 ν 1 / 2 v m - 1 × exp [ - ( x m / v m ) 2 + i k w z m ] ,
v m 2 = ν 2 [ 1 + i z m / ( z F ν 2 ) ] ,
R m G m ( 0 , z m ) / G i ( 0 , z r ) = R ( θ i ) G m ( 0 , z m ) / G r ( 0 , z r ) ,
P m = v m v r - 1 ν - 1 / 2 exp ( - i δ p ) ,
δ p = k w [ z m ( 1 - cos δ θ ) - δ z ]
z m z F ν 2 ;
v m - 2 ν - 2 .
δ θ δ ^ θ 1 ,
z m z F ν ^ 2
v f - 2 ν ^ - 2 [ 1 - i ( z r - δ ^ z ) / ( z F ν ^ 2 ) ] .
δ x = δ ^ x ,
δ z = δ ^ z ,
δ θ = tan - 1 ( δ ^ θ / ( 1 - δ ^ θ 2 / 2 ) ) ,
ν 2 = ν ^ 2 / ( cos 2 δ θ + δ ^ θ sin 2 δ θ ) .
R m = R ( θ m ) A r exp [ ( δ ^ θ k w ν ^ / 2 ) 2 ] ,
P m = v m v r - 1 ν - 1 / 2 exp { - i k w [ z m ( 1 - cos δ θ ) - δ z ] } ,
δ z / z F ~ ( k w ) - 1 / 2 v r 3 / 2 .
P m ν 1 / 2 exp { - i δ p [ 1 - ( k w ) 2 ] } ,
z m ( 1 - cos δ θ ) / δ z ~ ( z m / z F ) ( k w ) - 5 / 2 v r - 1 / 2 ,
δ p - k w δ z ,
S b = 1 2 R e [ ( i k w ) - 1 Z - G b * z G b d x b ] ,
r m S m / S i = [ 2 - ( k w ν ) - 2 ] ( 2 - ( k w ) - 2 ] - 1 P m R m 2
r m P m R m 2 .
R m R ( θ i ) [ 1 + c 1 ( k w ) - 1 / 2 ( 1 + i z m / z F ) - 1 / 4 + c 2 ( k w ) - 1 ( 1 + i z m / z F ) 1 / 2 ] ,
r m = ν [ 1 + z m 2 / ( z F ν 2 ) 2 ] 1 / 2 v r - 2 R m 2 ,
v r 2 = [ 1 + ( δ z + z m cos δ θ ) 2 / z F 2 ] 1 / 2
δ x ~ ( k w ) - 1 / 2 v r 1 / 2 ,
δ z ~ ( k w ) 1 / 2 v r 3 / 2 ,
δ θ ~ ( k w ) - 3 / 2 v r 1 / 2 ,
ν 2 ~ 1 + c 3 ( k w ) - 1 / 2 v r 3 / 2 ,

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