Abstract

A general, accurate, and explicit expression has been derived for the angular shift of a Gaussian beam that is partially reflected by a dielectric interface. This result holds for arbitrary incidence angles, including the vicinity of the Brewster angle, and asymptotically approaches the results obtained by others for incidence away from that angle. A series of reflected beam profiles is shown to clarify the angular-shift phenomenon and to illustrate the beam-distortion effect that occurs at, or near, Brewster incidence.

© 1985 Optical Society of America

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References

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  1. B. R. Horowitz, T. Tamir, J. Opt. Soc. Am. 61, 586 (1971), and references therein.
    [CrossRef]
  2. J. W. Ra, H. L. Bertoni, L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
    [CrossRef]
  3. Y. M. Antar, W. M. Boerner, Can. J. Phys. 52, 962 (1974).
  4. I. A. White, A. W. Snyder, C. Pask, J. Opt. Soc. Am. 67, 703 (1977).
    [CrossRef]
  5. M. McGuirk, C. K. Caniglia, J. Opt. Soc. Am. 67, 103 (1977).
    [CrossRef]

1977 (2)

1974 (1)

Y. M. Antar, W. M. Boerner, Can. J. Phys. 52, 962 (1974).

1973 (1)

J. W. Ra, H. L. Bertoni, L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
[CrossRef]

1971 (1)

Antar, Y. M.

Y. M. Antar, W. M. Boerner, Can. J. Phys. 52, 962 (1974).

Bertoni, H. L.

J. W. Ra, H. L. Bertoni, L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
[CrossRef]

Boerner, W. M.

Y. M. Antar, W. M. Boerner, Can. J. Phys. 52, 962 (1974).

Caniglia, C. K.

Felsen, L. B.

J. W. Ra, H. L. Bertoni, L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
[CrossRef]

Horowitz, B. R.

McGuirk, M.

Pask, C.

Ra, J. W.

J. W. Ra, H. L. Bertoni, L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
[CrossRef]

Snyder, A. W.

Tamir, T.

White, I. A.

Can. J. Phys. (1)

Y. M. Antar, W. M. Boerner, Can. J. Phys. 52, 962 (1974).

J. Opt. Soc. Am. (3)

SIAM J. Appl. Math. (1)

J. W. Ra, H. L. Bertoni, L. B. Felsen, SIAM J. Appl. Math. 24, 396 (1973).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Geometry and coordinate systems of the incident and reflected beams. The direction zr indicates the geometrical ray axis of the reflected beam. The actual direction of the reflected beam, as determined by the peak of the beam profile, has an angular shift indicated by Δθm.

Fig. 2
Fig. 2

Reflected beam profile for |Fr|2 in the far field as a function of the angular domain θθB for different values of δ = θiθB, kw0 = 10, z = 5w0, and n = 1.1. The thick vertical arrows show the location of the beam axis predicted by geometrical optics. To exhibit both peaks and their magnitude relation, the solid curve in (e) is repeated on a logarithmic scale, as shown by the dashed lines.

Fig. 3
Fig. 3

Angular shift Δθm as a function of the incidence parameter (θiθB) for kw0 = 10 and n = 1.1. The angular shift predicted by this analysis is shown by the solid lines, while the conventional result is shown by the dashed curve.

Equations (14)

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F r = 1 2 π - r ( θ ) A ( k x ) exp [ i ( k x x r + k z z r ) ] d k x ,
k x = k sin ( θ - θ i ) ,
k z = ( k 2 - k x 2 ) 1 / 2 .
r ( θ ) = n ( 1 - n 2 sin 2 θ ) 1 / 2 - cos θ n ( 1 - n 2 sin 2 θ ) 1 / 2 + cos θ ,
A ( k x ) = π w 0 exp [ - ( k x w 0 / 2 ) 2 ] .
F r = [ r ( θ i ) + 2 i x r k w r 2 r ( θ i ) + ( 1 - 2 x r 2 w r 2 ) r ( θ i ) ( k w r ) 2 ] w 0 w r × exp [ i k z r - ( x r / w r ) 2 ] ,
w r 2 = w 0 2 + i 2 z r k .
F r = r ( θ i ) + r ( θ i ) Δ θ + 1 2 r ( θ i ) ( Δ θ ) 2 ( k w 0 2 / 2 z r ) 1 / 2 × exp [ - ( k w 0 / 2 ) 2 ( Δ θ ) 2 ] .
Δ θ m = r ( θ i ) ( k w 0 ) 2 r ( θ i ) - r ( θ i ) 2 r ( θ i ) ± { [ r ( θ i ) ( k w 0 ) 2 r ( θ i ) - r ( θ i ) 2 r ( θ i ) ] 2 + 2 ( k w 0 ) 2 } 1 / 2 .
Δ θ m = A ( k w 0 ) 2 - δ 2 ± { [ δ 2 - A ( k w 0 ) 2 ] 2 + 2 ( k w 0 ) 2 } 1 / 2 ,
A = 2 n 4 + n 2 + 1 n
δ = θ i - θ B .
Δ θ m = 2 ( k w 0 ) 2 r ( θ i ) r ( θ i ) .
1 ( k w 0 ) 2 r ( θ i ) r ( θ i ) < 2 k w 0 r ( θ i ) 2 r ( θ i ) .

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