Abstract

The diffraction effects produced by light incident upon an interface at the critical angle of total reflection are examined for the purpose of explaining the weak illumination recently detected by Acloque and Guillemet, and by Osterberg and Smith. This illumination accompanies the rays reflected from the interface, but it occurs in a manner that cannot be interpreted in conventional geometric-optical terms. We show that a rigorous but simple approach that accounts for first-order-diffraction effects is capable of explaining the experimental results in terms of a lateral wave which in fact may provide the dominant field under certain circumstances. In particular, we examine the special case of a collimated light beam and show that the lateral wave is strongest, and accounts for the observed illumination, when the beam is incident at the critical angle of total reflection. The quasi-optical properties of this wave are emphasized and they are shown to provide straightforward physical interpretations for the various features of the weak-illumination field.

© 1969 Optical Society of America

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References

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  1. H. K. V. Lotsch, J. Opt. Soc. Am. 58, 551 (1968).
    [Crossref]
  2. F. Goos and H. Hänchen, Ann. Physik (6) 1, 333 (1947).
    [Crossref]
  3. H. Maecker, Ann. Physik (6) 4, 409 (1949).
    [Crossref]
  4. H. Ott, Ann. Physik (6) 4, 432 (1949).
    [Crossref]
  5. P. Acloque and C. Guillemet, Compt. Rend. 250, 4328 (1960).
  6. H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1073 (1964).
    [Crossref]
  7. See, for instance: C. B. Officer, Introduction to the Theory of Sound Transmission (McGraw–Hill Book Co., New York, 1958), 192–201; W. N. Ewing, W. S. Jardetzky, and F. Press, Elastic Waves in Layered Media (McGraw–Hill Book Co., New York, 1957), pp. 90–115.
  8. L. B. Felsen, in Electromagnetic Theory, J. Brown, Ed. (Pergamon Press, Oxford, 1967), p. 11.
  9. D. Staiman and T. Tamir, Proc. IEE (London) 113, 1299 (1966).
  10. L. M. Brekhovskikh, Waves in Layered Media (Academic Press Inc., New York, 1960), Ch. IV, p. 234.
  11. T. Tamir, IEEE Trans. Antennas Propagation AP-15, 806 (1967).
    [Crossref]
  12. T. Tamir and L. B. Felsen, IEEE Trans. Antennas Propagation AP-13, 410 (1965).
    [Crossref]
  13. Because L appears as a factor in the denominator of Eq. (5), Elat becomes unbounded as L→ 0. However, Eq. (5) is not valid for small L and must be replaced10 by a different expression that yields a finite value of Elat as L→ 0. Such a case is excluded in the present discussion, since we assume that L is a large quantity.
  14. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill Book Co., New York, 1960), Ch. 11, p. 453.
  15. T. Tamir and A. A. Oliner, Proc. IEE (London) 110, 310 (1963).
  16. H. M. Barlow and A. L. Cullen, Proc. IEE (London) 100 (Part III), 329 (1953).
  17. B. J. Woodward and H. C. Bryant, J. Opt. Soc. Am. 57, 430 (1967).
    [Crossref]
  18. H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1078 (1967).
    [Crossref]

1968 (1)

1967 (3)

1966 (1)

D. Staiman and T. Tamir, Proc. IEE (London) 113, 1299 (1966).

1965 (1)

T. Tamir and L. B. Felsen, IEEE Trans. Antennas Propagation AP-13, 410 (1965).
[Crossref]

1964 (1)

1963 (1)

T. Tamir and A. A. Oliner, Proc. IEE (London) 110, 310 (1963).

1960 (1)

P. Acloque and C. Guillemet, Compt. Rend. 250, 4328 (1960).

1953 (1)

H. M. Barlow and A. L. Cullen, Proc. IEE (London) 100 (Part III), 329 (1953).

1949 (2)

H. Maecker, Ann. Physik (6) 4, 409 (1949).
[Crossref]

H. Ott, Ann. Physik (6) 4, 432 (1949).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, Ann. Physik (6) 1, 333 (1947).
[Crossref]

Acloque, P.

P. Acloque and C. Guillemet, Compt. Rend. 250, 4328 (1960).

Barlow, H. M.

H. M. Barlow and A. L. Cullen, Proc. IEE (London) 100 (Part III), 329 (1953).

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic Press Inc., New York, 1960), Ch. IV, p. 234.

Bryant, H. C.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill Book Co., New York, 1960), Ch. 11, p. 453.

Cullen, A. L.

H. M. Barlow and A. L. Cullen, Proc. IEE (London) 100 (Part III), 329 (1953).

Felsen, L. B.

T. Tamir and L. B. Felsen, IEEE Trans. Antennas Propagation AP-13, 410 (1965).
[Crossref]

L. B. Felsen, in Electromagnetic Theory, J. Brown, Ed. (Pergamon Press, Oxford, 1967), p. 11.

Goos, F.

F. Goos and H. Hänchen, Ann. Physik (6) 1, 333 (1947).
[Crossref]

Guillemet, C.

P. Acloque and C. Guillemet, Compt. Rend. 250, 4328 (1960).

Hänchen, H.

F. Goos and H. Hänchen, Ann. Physik (6) 1, 333 (1947).
[Crossref]

Lotsch, H. K. V.

Maecker, H.

H. Maecker, Ann. Physik (6) 4, 409 (1949).
[Crossref]

Officer, C. B.

See, for instance: C. B. Officer, Introduction to the Theory of Sound Transmission (McGraw–Hill Book Co., New York, 1958), 192–201; W. N. Ewing, W. S. Jardetzky, and F. Press, Elastic Waves in Layered Media (McGraw–Hill Book Co., New York, 1957), pp. 90–115.

Oliner, A. A.

T. Tamir and A. A. Oliner, Proc. IEE (London) 110, 310 (1963).

Osterberg, H.

Ott, H.

H. Ott, Ann. Physik (6) 4, 432 (1949).
[Crossref]

Smith, L. W.

Staiman, D.

D. Staiman and T. Tamir, Proc. IEE (London) 113, 1299 (1966).

Tamir, T.

T. Tamir, IEEE Trans. Antennas Propagation AP-15, 806 (1967).
[Crossref]

D. Staiman and T. Tamir, Proc. IEE (London) 113, 1299 (1966).

T. Tamir and L. B. Felsen, IEEE Trans. Antennas Propagation AP-13, 410 (1965).
[Crossref]

T. Tamir and A. A. Oliner, Proc. IEE (London) 110, 310 (1963).

Woodward, B. J.

Ann. Physik (6) (3)

F. Goos and H. Hänchen, Ann. Physik (6) 1, 333 (1947).
[Crossref]

H. Maecker, Ann. Physik (6) 4, 409 (1949).
[Crossref]

H. Ott, Ann. Physik (6) 4, 432 (1949).
[Crossref]

Compt. Rend. (1)

P. Acloque and C. Guillemet, Compt. Rend. 250, 4328 (1960).

IEEE Trans. Antennas Propagation (2)

T. Tamir, IEEE Trans. Antennas Propagation AP-15, 806 (1967).
[Crossref]

T. Tamir and L. B. Felsen, IEEE Trans. Antennas Propagation AP-13, 410 (1965).
[Crossref]

J. Opt. Soc. Am. (4)

Proc. IEE (London) (3)

D. Staiman and T. Tamir, Proc. IEE (London) 113, 1299 (1966).

T. Tamir and A. A. Oliner, Proc. IEE (London) 110, 310 (1963).

H. M. Barlow and A. L. Cullen, Proc. IEE (London) 100 (Part III), 329 (1953).

Other (5)

L. M. Brekhovskikh, Waves in Layered Media (Academic Press Inc., New York, 1960), Ch. IV, p. 234.

See, for instance: C. B. Officer, Introduction to the Theory of Sound Transmission (McGraw–Hill Book Co., New York, 1958), 192–201; W. N. Ewing, W. S. Jardetzky, and F. Press, Elastic Waves in Layered Media (McGraw–Hill Book Co., New York, 1957), pp. 90–115.

L. B. Felsen, in Electromagnetic Theory, J. Brown, Ed. (Pergamon Press, Oxford, 1967), p. 11.

Because L appears as a factor in the denominator of Eq. (5), Elat becomes unbounded as L→ 0. However, Eq. (5) is not valid for small L and must be replaced10 by a different expression that yields a finite value of Elat as L→ 0. Such a case is excluded in the present discussion, since we assume that L is a large quantity.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill Book Co., New York, 1960), Ch. 11, p. 453.

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Figures (5)

Fig. 1
Fig. 1

Basic experimental configuration for detecting weak illumination (shown dashed) accompanying the reflected portion of a beam incident at the critical angle of total reflection.

Fig. 2
Fig. 2

The field of a line source located above an interface between two semi-infinite dielectric regions (n1 > n2). The source is at S and produces a source image at I. The direct field Edir is given by the ray R, the reflected field Erefl is accounted for by the rays R1 and R2, and the lateral wave is represented by the ray path L1, L, and L2.

Fig. 3
Fig. 3

Field of the lateral wave. Solid and dashed lines denote equi-amplitude and equi-phase contours, respectively.

Fig. 4
Fig. 4

The field of a beam incident and reflected at the critical angle θc. The dashed lines show the lateral-wave domain and their density indicates the relative field irradiance.

Fig. 5
Fig. 5

Geometry for collimated-beam calculations.

Equations (33)

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E = E dir + E refl + E corr + E lat u ( θ 2 - θ c ) ,
E dir = E 0 [ exp ( i k 1 R ) / ( k 1 R ) 1 2 ]
E refl = E 0 A ( θ 2 ) exp [ i k 1 ( R 1 + R 2 ) ] [ k 1 ( R 1 + R 2 ) ] 1 2
E corr = E 0 8 i { exp ( i k 1 R ) ( k 1 R ) 3 2 + D ( θ 2 ) exp [ i k 1 ( R 1 + R 2 ) ] k 1 ( R 1 + R 2 ) ] 3 2 }
E lat = E 0 2 i m n 2 - 1 exp [ i k 1 ( L 1 + L 2 ) + i k 2 L ] ( k 2 L ) 3 2 .
n = n 1 / n 2 = k 1 / k 2 .
m = { m = n 2 for polarization parallel to the plane x z ; m = 1 for polarization normal to the plane x z .
A ( θ 2 ) = n cos θ 2 - m ( 1 - n 2 sin 2 θ 2 ) 1 2 n cos θ 2 + m ( 1 - n 2 sin 2 θ 2 ) 1 2 ,
sin θ c = n - 1 .
E dir + E refl - 2 i E 0 exp ( i k 1 R ) ( k 1 R ) 1 2 z R × [ k 1 z + n ( 1 + z / z ) m ( n 2 - 1 ) 1 2 ] .
E corr - 2 i E 0 exp ( i k 1 R ) ( k 1 R ) 3 2 n 2 m 2 ( n 2 - 1 ) ,
Ē = E ( ξ ) d ξ ,
E 0 ( ξ ) = exp [ - ( ξ / 2 w ) 2 ] 2 π w ,             ( in V / m 2 ) ;
Ē dir = 1 2 π w - exp ( i k 1 R ) ( k 1 R ) 1 2 exp [ - ( ξ / 2 w ) 2 ] d ξ ,
x = ξ cos θ i z = h + ξ sin θ i ,
R = [ ( x - x ) 2 + ( z - z ) 2 ] 1 2 = { x 2 + ( z - h ) 2 - 2 ξ [ x cos θ i + ( z - h ) sin θ i ] + ξ 2 } 1 2 .
x = r 0 sin θ 0 z - h = r 0 cos θ 0 ,
R [ r 0 2 - 2 r 0 ξ ( sin θ 0 cos θ i + cos θ 0 sin θ i ) ] 1 2 r 0 - ξ sin ( θ 0 + θ i ) .
Ē dir = exp ( i k 1 r 0 ) 2 w ( π k 1 r 0 ) 1 / 2 × - exp [ - ( ξ / 2 w ) 2 - i k 1 ξ sin ( θ 0 + θ i ) ] d ξ .
Ē dir = exp ( i k 1 r 0 ) ( k 1 r 0 ) 1 / 2 exp [ - ( k 1 w ) 2 sin 2 ( θ 0 + θ i ) ] .
ϕ = k 1 ( L 1 + L 2 ) + k 2 L = k 2 [ x - x + ( n 2 - 1 ) 1 2 ( z - z ) ] = ϕ ( ξ ) .
Ē lat = 2 i m n 2 - 1 1 2 π w × - e i ϕ ( ξ ) ( k 2 L ) 3 / 2 exp [ - ( ξ / 2 w ) 2 ] d ξ .
Ē lat = 2 i m n 2 - 1 exp { i k 2 [ x + ( n 2 - 1 ) 1 2 ( z + h ) ] } 2 w π ( k 2 L 0 ) 3 2 × - exp { - ( ξ / 2 w ) 2 - i k 2 ξ × [ cos θ i - ( n 2 - 1 ) 1 2 sin θ i ] } d ξ .
Ē lat = 2 i m n 2 - 1 exp { i k 2 [ x + ( n 2 - 1 ) 1 2 ( z + h ) ] } ( k 2 L 0 ) 3 2 × exp { - ( k 2 w ) 2 [ cos θ i - ( n 2 - 1 ) 1 2 sin θ i ] 2 } .
tan θ i ( n 2 - 1 ) - 1 / 2 = tan θ c .
Z ( z ) exp [ i ( κ x - ω t ) ] .
e i τ z exp [ i ( κ x - ω t ) ] ,
κ 2 + τ 2 = k 2 = ( 2 π / λ ) 2
E 0 ( ξ ) = ( 2 b ) - 1 .
Ē lat = 2 i m n 2 - 1 - b b exp { i k 2 [ x - x + ( n 2 - 1 ) 1 2 ( z - z ) ] } 2 b ( k 2 L ) 3 2 d ξ .
Ē lat = 2 i m n 2 - 1 exp { i k 2 [ x + ( n 2 - 1 ) 1 2 ( z + h ) ] } 2 b ( k 2 L 0 ) 3 2 × - b b exp { - i k 2 ξ [ cos θ i - ( n 2 - 1 ) 1 2 sin θ i ] } d ξ .
Ē lat = 2 i m n 2 - 1 exp { i k 2 [ x + ( n 2 - 1 ) 1 2 ( z + h ) ] } ( k 2 L 0 ) 3 2 sin u u ,
u = k 2 b [ cos θ i - ( n 2 - 1 ) 1 2 sin θ i ] .