Abstract

The problem of reflection of a Gaussian beam from a parallel-sided dielectric slab is studied from the angular-spectrum point of view. It is shown that the path of the peak of the reflected profile undergoes three types of effects with reference to its expected geometrical path: a lateral displacement independent of the propagation distance, a focal shift, and an angular shift that results in a lateral shift proportional to the total propagation distance between the beam waist and the point of observation and independent of the location in between, where reflection takes place. These effects are illustrated by computer simulating some typical cases. The generality of the analysis and its applicability to many different situations are noted. The results of several authors on related problems are shown to be special cases of the general results. The relevance of the analysis and the results to an interesting series of recent experiments in the microwave region is noted.

© 1985 Optical Society of America

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References

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  1. J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” Soc. Ind. Appl. Math. J. Appl. Math. 24, 396–413 (1973).
    [CrossRef]
  2. Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a plane dielectric interface,” Can. J. Phys. 52, 962–972 (1974).
  3. I. A. White, A. W. Snyder, C. Pask, “Directional change of beams undergoing partial reflection,” J. Opt. Soc. Am. 67, 703–705 (1977).
    [CrossRef]
  4. S. Kosaki, H. Sakurai, “Characteristics of a Gaussian beam at a dielectric interface,” J. Opt. Soc. Am. 68, 508–514 (1978).
    [CrossRef]
  5. L. A. A. Read, M. Wong, G. E. Reesor, “Displacement of an electromagnetic beam upon reflection from a dielectric slab,” J. Opt. Soc. Am. 68, 319–322 (1978).
    [CrossRef]
  6. L. A. A. Read, G. E. Reesor, “Displacement of a microwave beam upon transmission through a dielectric slab.”Can. J. Phys. 57, 1409–1413 (1979).
    [CrossRef]
  7. H. K. V. Lotsch, “Reflection and refraction of a beam of light at plane interface,” J. Opt. Soc. Am. 58, 551–561 (1968).This reference contains an extensive list of references on Goos–Hänchen effect.
    [CrossRef]
  8. B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971).
    [CrossRef]
  9. E. Collett, E. Wolf, “Beams generated by Gaussian quasihomogeneous sources,” Opt. Commun. 32, 27–31 (1980).
    [CrossRef]
  10. This terminology is appropriate, for Ẽ0(f) represents the angular distribution of field in the far-zone (Fraunhofer) region.See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 61.
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 54.
  12. A. E. Siegman, An Introduction of Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8;H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  13. V. Shah, T. Tamir, “Absorption and lateral shift of beams incident upon lossy multilayered media,” J. Opt. Soc. Am. 73, 37–44 (1983).
    [CrossRef]
  14. S.-Y. Lee, N. Marcuvitz, “Beam reflection from lossy dielectric layers,” J. Opt. Soc. Am. 73, 1714–1718 (1983).
    [CrossRef]
  15. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).
  16. C. K. Carniglia, K. R. Brownstein, “Focal shift and ray model for total internal reflection,” J. Opt. Soc. Am. 67, 121–122 (1977).
    [CrossRef]
  17. M. McGuirk, C. K. Carniglia, “An angular spectrum approach to the Goos–Hänchen shift,” J. Opt. Soc. Am. 67, 103–107 (1977).
    [CrossRef]
  18. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 359.
  19. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 40.
  20. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), App. III.
  21. W. J. Tomlinson, J. P. Gordon, P. W. Smith, A. E. Kaplan, “Reflection of a Gaussian beam at a nonlinear interface,” Appl. Opt. 21, 2041–2051 (1982).
    [CrossRef] [PubMed]
  22. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 60.
  23. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 62.

1983

1982

1980

E. Collett, E. Wolf, “Beams generated by Gaussian quasihomogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

1979

L. A. A. Read, G. E. Reesor, “Displacement of a microwave beam upon transmission through a dielectric slab.”Can. J. Phys. 57, 1409–1413 (1979).
[CrossRef]

1978

1977

1974

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

1973

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

1971

1968

Antar, Y. M.

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

Bertoni, H. L.

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

Boerner, W. M.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), App. III.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 359.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 40.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 60.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 62.

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

Brownstein, K. R.

Carniglia, C. K.

Collett, E.

E. Collett, E. Wolf, “Beams generated by Gaussian quasihomogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

Felsen, L. B.

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

Goodman, J. W.

This terminology is appropriate, for Ẽ0(f) represents the angular distribution of field in the far-zone (Fraunhofer) region.See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 61.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 54.

Gordon, J. P.

Horowitz, B. R.

Kaplan, A. E.

Kosaki, S.

Lee, S.-Y.

Lotsch, H. K. V.

Marcuvitz, N.

McGuirk, M.

Pask, C.

Ra, J. W.

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

Read, L. A. A.

L. A. A. Read, G. E. Reesor, “Displacement of a microwave beam upon transmission through a dielectric slab.”Can. J. Phys. 57, 1409–1413 (1979).
[CrossRef]

L. A. A. Read, M. Wong, G. E. Reesor, “Displacement of an electromagnetic beam upon reflection from a dielectric slab,” J. Opt. Soc. Am. 68, 319–322 (1978).
[CrossRef]

Reesor, G. E.

L. A. A. Read, G. E. Reesor, “Displacement of a microwave beam upon transmission through a dielectric slab.”Can. J. Phys. 57, 1409–1413 (1979).
[CrossRef]

L. A. A. Read, M. Wong, G. E. Reesor, “Displacement of an electromagnetic beam upon reflection from a dielectric slab,” J. Opt. Soc. Am. 68, 319–322 (1978).
[CrossRef]

Sakurai, H.

Shah, V.

Siegman, A. E.

A. E. Siegman, An Introduction of Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8;H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Smith, P. W.

Snyder, A. W.

Tamir, T.

Tomlinson, W. J.

White, I. A.

Wolf, E.

E. Collett, E. Wolf, “Beams generated by Gaussian quasihomogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), App. III.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 60.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 40.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 359.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 62.

Wong, M.

Appl. Opt.

Can. J. Phys.

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

L. A. A. Read, G. E. Reesor, “Displacement of a microwave beam upon transmission through a dielectric slab.”Can. J. Phys. 57, 1409–1413 (1979).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

E. Collett, E. Wolf, “Beams generated by Gaussian quasihomogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

Soc. Ind. Appl. Math. J. Appl. Math.

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

Other

This terminology is appropriate, for Ẽ0(f) represents the angular distribution of field in the far-zone (Fraunhofer) region.See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 61.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 54.

A. E. Siegman, An Introduction of Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8;H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 359.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 40.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), App. III.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 60.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 62.

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Figures (6)

Fig. 1
Fig. 1

The geometry used in the analysis. The beam profile is measured with respect to the axis of the geometric propagation (i.e., plane-wave propagation at the main profile angle). The total distance of propagation from the beam waist (z = 0) to the point of observation is z0 + z1. Note that the dielectric slab of index N > 1 is assumed immersed in vacuum.

Fig. 2
Fig. 2

Amplitude of the reflection function, r(F), where F = sin θ and θ is the angle at which a plane wave travels with respect to the surface normal. Normal incidence corresponds to F = 0 and grazing incidence to F = 1. The optical thickness of the film is N × D, where N is the index of refraction and D is the thickness of the film.

Fig. 3
Fig. 3

Profiles of a beam in f space before reflection and in real space before and after reflection. (A) The amplitude of the reflection function and the antenna function. (B) The product r(f)0(f), which is the effective reflected antenna function. (C) The launched beam. (D) Reflective profiles after various distances of propagation. The angle was chosen so that two peaks appear in the far field. The amplitudes shown are normalized so that peak has a value of unity. All graphs are shown with respect to the geometric path. On the f axis, negative values are toward normal incidence, whereas positive values are toward grazing incidence. On the transverse axis, negative values are toward the normal to the surface, whereas positive values are away from the normal.

Fig. 4
Fig. 4

Phase [δ(f)] of the reflection function r(f) for the parameters of Fig. 3. A phase shift of 2π has been suppressed at the angle for zero reflection (f = 0.1).

Fig. 5
Fig. 5

Same as Fig. 3, except that the angle of incidence is 39°.

Fig. 6
Fig. 6

Same parameters as Fig. 3, but the beam width is increased by factors of 2 and 4, respectively, for (A) and (B). The amplitudes shown are normalized so that peak has a value of unity.

Equations (29)

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E z ( x ) = z ( f ) exp ( i 2 π f x ) d f .
f = λ 1 sin α ,
E 0 ( x ) = A exp [ ( x 2 / 2 σ 2 ) ] ,
0 ( f ) = A π ( 1 / 2 ) exp [ ( 2 π σ ) 2 f 2 / 2 ] .
P ( f ) = exp [ i ( π z λ f 2 ) ] .
r ( f ) = r 12 [ 1 exp ( 2 i β ) ] 1 r 12 2 exp ( 2 i β ) .
r ( f ) = r ( f ) exp [ i δ ( f ) ] ,
δ ( f ) = δ ( 0 ) + δ ( 0 ) f + 1 2 ! δ ( 0 ) f 2 + .
L = δ ( 0 ) / 2 π .
D f = δ ( 0 ) / ( 2 π λ ) .
β ( f ) = integral multiple of π ,
2 D ( N 2 sin 2 θ ) 1 / 2 / λ = integer .
d d f [ r ( f ) 0 ( f ) ] f m = 0 .
f m = ( 2 π σ ) 2 r ( f m ) r ( f m ) ( 2 π σ ) 2 r ( 0 ) r ( 0 ) ,
Δ α = sin 1 ( λ f m ) λ f m = λ ( 2 π σ ) 2 r ( 0 ) r ( 0 ) .
E z ( x ) = 0 ( f ) r ( f ) exp [ i ψ ( f ) ] d f ,
ψ ( f ) = δ ( f ) π λ z f 2 + 2 π f x .
δ ( f s ) + f s δ ( f s ) 2 π λ z f s + 2 π x = 0 .
f s = x L ( Z + D f ) .
| E z ( x ) | = [ π 2 ψ ( f s ) ] 1 / 2 0 ( f s ) r ( f s ) = { 2 [ λ ( z + D f ) ] 1 / 2 } 1 E 0 [ x L λ ( z + D f ) ] r [ x L λ ( z + D f ) ] .
x m L λ ( z + D f ) ( 2 π σ ) 2 r ( 0 ) r ( 0 )
x m = L + r ( 0 ) r ( 0 ) [ λ z ( 2 π σ ) 2 + λ D f ( 2 π σ ) 2 ] .
r = r 12 1 exp ( 2 i β ) 1 r 12 2 exp ( 2 i β ) ,
r 12 = ( cos θ N cos θ ) / ( cos θ + N cos θ ) ,
β = ( 2 π D / λ ) N cos θ ,
sin θ = F / N , cos θ = [ 1 ( F / N 2 ] 1 / 2 , N cos θ = ( N 2 F 2 ) 1 / 2 , cos θ = ( 1 F 2 ) 1 / 2 .
β = ( 2 π D / λ ) ( N 2 F 2 ) 1 / 2 ,
r 12 = ( 1 F 2 ) 1 / 2 ( N 2 F 2 ) 1 / 2 ( 1 F 2 ) 1 / 2 + ( N 2 F 2 ) 1 / 2 .
F = ( 1 λ 2 f 2 ) 1 / 2 sin θ i + λ f cos θ i .

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