Abstract

Phase shifts suffered by evanescent electromagnetic waves traversing an air gap between two glass prisms have been measured. Because the evanescent waves propagate parallel to the glass-to-air interface, they should be able to penetrate in a direction normal to the interface, without change of phase, although the amplitude is expected to decrease with depth of penetration. This has been confirmed by the experiment, which showed that the phase shifts are substantially independent of the width of the air gap.

© 1971 Optical Society of America

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References

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  1. G. Quincke, Ann. Phys. Chem. 127, 1 (1866).
  2. E. E. Hall, Phys. Rev. 15, 73 (1902).
  3. D. D. Coon, Am. J. Phys. 34, 240 (1966).
    [Crossref]
  4. R. N. Smartt, Appl. Opt. 9, 970 (1970).
    [Crossref]
  5. For example, see K. H. Drexhage, Sci. Am. 222, 108 (March1970).
    [Crossref]
  6. H. Nassenstein, Phys. Letters 28A, 249 (1968).
  7. O. Bryngdahl, J. Opt. Soc. Am. 59, 1645 (1969).
    [Crossref]
  8. C. Schaefer and G. Gross, Ann. Phys. (Leipzig) 32, 648 (1910).
  9. W. Culshaw and D. S. Jones, Proc. Phys. Soc. (London) B66, 859 (1953).
  10. J. J. Brady, R. O. Brick, and M. D. Pearson, J. Opt. Soc. Am. 50, 1080 (1960).
    [Crossref]
  11. H. D. Raker and G. R. Valenzuela, Trans. IRE MTT-10, 392 (1962).
    [Crossref]
  12. J. J. Taub, H. J. Hindin, O. F. Hinckelmann, and M. L. Wright, IEEE MTT-11, 338 (1963).
  13. R. G. Fellers, Proc. IEEE 55, 1003 (1967).
    [Crossref]
  14. C. J. Bouwkamp, Repts. Progr. Phys. 17, 39 (1954).
    [Crossref]
  15. E. Wolf, Proc. Phys. Soc. (London) 74, 269 (1959).
    [Crossref]
  16. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, 1st ed. (Pergamon, New York, 1966).
  17. G. C. Sherman, J. Opt. Soc. Am. 57, 1160 (1967); J. Opt. Soc. Am. 57, 1490 (1967).
    [Crossref]
  18. J. R. Shewell and E. Wolf, J. Opt. Soc. Am. 58, 1596 (1968).
    [Crossref]
  19. E. Lalor, J. Opt. Soc. Am. 58, 1235 (1968).
    [Crossref]
  20. A. Walther, J. Opt. Soc. Am. 58, 1256 (1968); J. Opt. Soc. Am. 59, 1325 (1969).
    [Crossref]
  21. G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).
  22. R. Asby and E. Wolf, J. Opt. Soc. Am. 61, 52 (1971).
    [Crossref]
  23. C. Carniglia and L. Mandel, Phys. Rev. D 3, 280 (1971).
    [Crossref]
  24. A first account of this experiment was presented at the 1970 Spring Meeting of the Optical Society of America [, J. Opt. Soc. Am. 60, 738 A (1970)].
  25. See, for example, M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 38.
  26. The notation used here is that of Ref. 23.
  27. For the purpose of setting up the basic equations, we take the dielectric constant of air to be unity, and choose our units so that the vacuum velocity of light c= 1.
  28. It should be emphasized that, although the phase shift of the evanescent wave was measured by a holographic technique, the holograms were not produced by the evanescent waves, as in Refs. 6 and 7.
  29. For a discussion of surface layers on polished glass surfaces, see, for example, L. Holland, The Properties of Glass Surfaces, 1st ed. (Chapman and Hall, London, 1966).
  30. A. Vašiček, J. Opt. Soc. Am. 37, 145, 979 (1947).
    [Crossref]

1971 (2)

R. Asby and E. Wolf, J. Opt. Soc. Am. 61, 52 (1971).
[Crossref]

C. Carniglia and L. Mandel, Phys. Rev. D 3, 280 (1971).
[Crossref]

1970 (3)

A first account of this experiment was presented at the 1970 Spring Meeting of the Optical Society of America [, J. Opt. Soc. Am. 60, 738 A (1970)].

R. N. Smartt, Appl. Opt. 9, 970 (1970).
[Crossref]

For example, see K. H. Drexhage, Sci. Am. 222, 108 (March1970).
[Crossref]

1969 (1)

1968 (4)

1967 (2)

1966 (1)

D. D. Coon, Am. J. Phys. 34, 240 (1966).
[Crossref]

1963 (1)

J. J. Taub, H. J. Hindin, O. F. Hinckelmann, and M. L. Wright, IEEE MTT-11, 338 (1963).

1962 (1)

H. D. Raker and G. R. Valenzuela, Trans. IRE MTT-10, 392 (1962).
[Crossref]

1960 (2)

1959 (1)

E. Wolf, Proc. Phys. Soc. (London) 74, 269 (1959).
[Crossref]

1954 (1)

C. J. Bouwkamp, Repts. Progr. Phys. 17, 39 (1954).
[Crossref]

1953 (1)

W. Culshaw and D. S. Jones, Proc. Phys. Soc. (London) B66, 859 (1953).

1947 (1)

1910 (1)

C. Schaefer and G. Gross, Ann. Phys. (Leipzig) 32, 648 (1910).

1902 (1)

E. E. Hall, Phys. Rev. 15, 73 (1902).

1866 (1)

G. Quincke, Ann. Phys. Chem. 127, 1 (1866).

Asby, R.

Born, M.

See, for example, M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 38.

Bouwkamp, C. J.

C. J. Bouwkamp, Repts. Progr. Phys. 17, 39 (1954).
[Crossref]

Brady, J. J.

Brick, R. O.

Bryngdahl, O.

Carniglia, C.

C. Carniglia and L. Mandel, Phys. Rev. D 3, 280 (1971).
[Crossref]

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, 1st ed. (Pergamon, New York, 1966).

Coon, D. D.

D. D. Coon, Am. J. Phys. 34, 240 (1966).
[Crossref]

Culshaw, W.

W. Culshaw and D. S. Jones, Proc. Phys. Soc. (London) B66, 859 (1953).

Drexhage, K. H.

For example, see K. H. Drexhage, Sci. Am. 222, 108 (March1970).
[Crossref]

Fellers, R. G.

R. G. Fellers, Proc. IEEE 55, 1003 (1967).
[Crossref]

Gross, G.

C. Schaefer and G. Gross, Ann. Phys. (Leipzig) 32, 648 (1910).

Hall, E. E.

E. E. Hall, Phys. Rev. 15, 73 (1902).

Hinckelmann, O. F.

J. J. Taub, H. J. Hindin, O. F. Hinckelmann, and M. L. Wright, IEEE MTT-11, 338 (1963).

Hindin, H. J.

J. J. Taub, H. J. Hindin, O. F. Hinckelmann, and M. L. Wright, IEEE MTT-11, 338 (1963).

Holland, L.

For a discussion of surface layers on polished glass surfaces, see, for example, L. Holland, The Properties of Glass Surfaces, 1st ed. (Chapman and Hall, London, 1966).

Jones, D. S.

W. Culshaw and D. S. Jones, Proc. Phys. Soc. (London) B66, 859 (1953).

Lalor, E.

Mandel, L.

C. Carniglia and L. Mandel, Phys. Rev. D 3, 280 (1971).
[Crossref]

Nassenstein, H.

H. Nassenstein, Phys. Letters 28A, 249 (1968).

Pearson, M. D.

Quincke, G.

G. Quincke, Ann. Phys. Chem. 127, 1 (1866).

Raker, H. D.

H. D. Raker and G. R. Valenzuela, Trans. IRE MTT-10, 392 (1962).
[Crossref]

Schaefer, C.

C. Schaefer and G. Gross, Ann. Phys. (Leipzig) 32, 648 (1910).

Sherman, G. C.

Shewell, J. R.

Smartt, R. N.

Taub, J. J.

J. J. Taub, H. J. Hindin, O. F. Hinckelmann, and M. L. Wright, IEEE MTT-11, 338 (1963).

Toraldo di Francia, G.

G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).

Valenzuela, G. R.

H. D. Raker and G. R. Valenzuela, Trans. IRE MTT-10, 392 (1962).
[Crossref]

Vašicek, A.

Walther, A.

Wolf, E.

R. Asby and E. Wolf, J. Opt. Soc. Am. 61, 52 (1971).
[Crossref]

J. R. Shewell and E. Wolf, J. Opt. Soc. Am. 58, 1596 (1968).
[Crossref]

E. Wolf, Proc. Phys. Soc. (London) 74, 269 (1959).
[Crossref]

See, for example, M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 38.

Wright, M. L.

J. J. Taub, H. J. Hindin, O. F. Hinckelmann, and M. L. Wright, IEEE MTT-11, 338 (1963).

Am. J. Phys. (1)

D. D. Coon, Am. J. Phys. 34, 240 (1966).
[Crossref]

Ann. Phys. (Leipzig) (1)

C. Schaefer and G. Gross, Ann. Phys. (Leipzig) 32, 648 (1910).

Ann. Phys. Chem. (1)

G. Quincke, Ann. Phys. Chem. 127, 1 (1866).

Appl. Opt. (1)

IEEE (1)

J. J. Taub, H. J. Hindin, O. F. Hinckelmann, and M. L. Wright, IEEE MTT-11, 338 (1963).

J. Opt. Soc. Am. (9)

Nuovo Cimento (1)

G. Toraldo di Francia, Nuovo Cimento 16, 61 (1960).

Phys. Letters (1)

H. Nassenstein, Phys. Letters 28A, 249 (1968).

Phys. Rev. (1)

E. E. Hall, Phys. Rev. 15, 73 (1902).

Phys. Rev. D (1)

C. Carniglia and L. Mandel, Phys. Rev. D 3, 280 (1971).
[Crossref]

Proc. IEEE (1)

R. G. Fellers, Proc. IEEE 55, 1003 (1967).
[Crossref]

Proc. Phys. Soc. (London) (2)

W. Culshaw and D. S. Jones, Proc. Phys. Soc. (London) B66, 859 (1953).

E. Wolf, Proc. Phys. Soc. (London) 74, 269 (1959).
[Crossref]

Repts. Progr. Phys. (1)

C. J. Bouwkamp, Repts. Progr. Phys. 17, 39 (1954).
[Crossref]

Sci. Am. (1)

For example, see K. H. Drexhage, Sci. Am. 222, 108 (March1970).
[Crossref]

Trans. IRE (1)

H. D. Raker and G. R. Valenzuela, Trans. IRE MTT-10, 392 (1962).
[Crossref]

Other (6)

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, 1st ed. (Pergamon, New York, 1966).

See, for example, M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), p. 38.

The notation used here is that of Ref. 23.

For the purpose of setting up the basic equations, we take the dielectric constant of air to be unity, and choose our units so that the vacuum velocity of light c= 1.

It should be emphasized that, although the phase shift of the evanescent wave was measured by a holographic technique, the holograms were not produced by the evanescent waves, as in Refs. 6 and 7.

For a discussion of surface layers on polished glass surfaces, see, for example, L. Holland, The Properties of Glass Surfaces, 1st ed. (Chapman and Hall, London, 1966).

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Figures (10)

Fig. 1
Fig. 1

Multiple reflections and propagation across the air gap.

Fig. 2
Fig. 2

The phase difference between the optical fields at directly opposite points on two sides of the air gap, as a function of separation a, for several different angles of incidence. θc is the critical angle. n0 is taken to be 1.5159.

Fig. 3
Fig. 3

Illustrating the extra path difference introduced in the displacement of one prism (a) normal to the glass-to-air interface and (b) parallel to the glass-to-air interface.

Fig. 4
Fig. 4

Illustrating the method of producing a small displacement Δy by a much larger displacement Δx of a stressed leaf spring.

Fig. 5
Fig. 5

Illustrating the use of Fizeau interference fringes to measure the separation of the prisms.

Fig. 6
Fig. 6

The optical arrangement for interference between the reference beam and the main beam traversing the air gap.

Fig. 7
Fig. 7

The method of scanning interference fringes by longitudinal displacement of a photographically produced grating.

Fig. 8
Fig. 8

Experimental results obtained from a scan of two sets of interference fringes, corresponding to a phase shift of 9° between them. The full curves were obtained by a least-squares fit from Eq. (29). x is the displacement of the photographic grating.

Fig. 9
Fig. 9

Results of phase-shift measurements, for various separations a, with homogeneous waves, i.e., for incidence below the critical angle. The theoretical curve is based on Eq. (14). Only changes of phase shift, not absolute phase shifts, were measured in the experiment. The origin of the experimental points has been adjusted to make the discrepancy zero when extrapolated to zero separation.

Fig. 10
Fig. 10

Results of phase-shift measurements, for various separations a, with evanescent waves, after correction of the data for systematic errors. The full curve is based on Eq. (15).

Equations (35)

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k 1 = K 1 ,
k 2 = K 2 ,
K 1 2 + K 2 2 + K 3 2 = K 2 ,
k 1 2 + k 2 2 + k 3 2 = n 0 2 K 2 .
k 3 = n 0 K cos θ ,
K 3 = ± [ k 3 2 - ( n 0 2 - 1 ) K 2 ] 1 2 = ± K ( 1 - n 0 2 sin 2 θ ) 1 2 .
( TM ) B incident = A ɛ exp i ( k · r - K t ) + c . c .
penetration wave = A ɛ [ 2 k 3 / ( k 3 + n 0 2 K 3 ) ] × exp i ( K · r - K t ) + c . c . ,
1st transmitted wave = A ɛ ( 2 k 3 k 3 + n 0 2 K 3 ) ( 2 K 3 n 0 2 k 3 + n 0 2 K 3 ) × e i K · a exp i [ k · ( r - a ) - K t ] + c . c . ,
penetration wave = A ɛ [ 2 k 3 / ( k 3 + n 0 2 K 3 ) ] e - K 3 z × exp i ( K 1 x + K 2 y - K t ) + c . c .
( TM ) B transmitted = A ɛ ( 2 k 3 k 3 + n 0 2 K 3 ) ( 2 K 3 n 0 2 k 3 + n 0 2 K 3 ) e i K · a exp i [ k · ( r - a ) - K t ] × n = 0 ( n 0 2 K 3 - k 3 n 0 2 K 3 + k 3 ) 2 n exp [ i n ( K - K ( R ) ) · a ] + c . c . = A ɛ 4 k 3 n 0 2 K 3 e i K · a exp i [ k · ( r - a ) - K t ] ( n 0 4 K 3 2 + k 3 2 ) [ 1 - exp i ( K - K ( R ) ) · a ] + 2 n 0 2 K 3 k 3 [ 1 + exp i ( K - K ( R ) ) · a ) ] + c . c . ,
( TM ) transmittance = 4 k 3 K 3 e i K 3 a / [ ( n 0 4 K 3 2 + k 3 2 ) ( 1 - e i 2 K 3 a ) + 2 n 0 2 K 3 k 3 ( 1 + e i 2 K 3 a ) ] = { cos K 3 a - i [ ( n 0 4 K 3 2 + k 3 2 ) / 2 n 0 2 K 3 k 3 ] × sin K 3 a } - 1 ,
K · a = K 3 a , ( K - K ( R ) ) · a = 2 K 3 a .
tan Φ 1 = [ ( n 0 4 K 3 2 + k 3 2 ) / 2 n 0 2 K 3 k 3 ] tan K 3 a = { [ cos 2 θ + n 0 2 ( 1 - n 0 2 sin 2 θ ) ] / 2 n 0 cos θ × ( 1 - n 0 2 sin 2 θ ) 1 2 } tan [ K a ( 1 - n 0 2 sin 2 θ ) 1 2 ] ,
tan Φ 1 = [ ( k 3 2 - K 3 2 n 0 4 ) / 2 n 0 2 K 3 k 3 ] tanh ( K 3 a ) = { [ cos 2 θ + n 0 2 ( n 0 2 sin 2 θ - 1 ) ] / 2 n 0 cos θ × ( n 0 2 sin 2 θ - 1 ) 1 2 } × tanh [ K a ( n 0 2 sin 2 θ - 1 ) 1 2 ] .
tan - 1 [ ( k 3 2 - n 0 4 K 3 2 ) / 2 n 0 2 K 3 k 3 ] ,
Φ 1 ( K 3 = 0 ) = tan - 1 ( 1 2 k 3 a / n 0 2 ) = tan - 1 [ 1 2 a K ( n 0 2 - 1 ) 1 2 / n 0 2 ] .
( TM ) attenuation = cosh 2 K 3 a + [ ( k 3 2 - n 0 4 K 3 2 ) / 2 k 3 n 0 2 K 3 ] 2 sinh 2 K 3 a ,
1 + k 3 2 a 2 / 4 n 0 4 ,
( TE ) transmittance = { cos K 3 a - i [ ( K 3 2 + k 3 2 ) / 2 k 3 K 3 ] sin K 3 a } - 1 .
tan Φ 2 = [ ( k 3 2 - K 3 2 ) / 2 k 3 K 3 ] tanh ( K 3 a ) = [ ( n 0 2 cos 2 θ + 1 ) / 2 n 0 cos θ ( n 0 2 sin 2 θ - 1 ) 1 2 ] × tanh [ a K ( n 0 2 sin 2 θ - 1 ) 1 2 ] ,
( TE ) attenuation = cosh 2 K 3 a + [ ( k 3 2 - K 3 2 ) / 2 k 3 K 3 ] 2 sinh 2 K 3 a ,
n glass sin ( α - θ ) = n air sin ψ .
( OPD ) 1 = - n air h 1 cos ( α - ψ ) .
q = [ h 2 sin α cos ( α - θ - ψ ) ] / cos ( α - θ ) .
( OPD ) 2 = h 2 [ sin α / cos ( α - θ ) ] × [ n glass - n air cos ( α - θ - ψ ) ] .
tan γ = h 1 h 2 = sin α [ n glass - n air cos ( α - θ - ψ ) ] n air cos ( α - θ ) cos ( α - ψ ) .
Δ y 2 ( Δ x ) 2 / l ,
δ 2 / 2 ( n 2 glass - 1 ) 1 2 .
Δ ( α + 1 2 α 2 ) / ( n 2 glass - 1 ) 1 2 .
Ω = [ A cos ( ω x + ϕ ) + B ] - 1 .
A = 1 2 ( 1 / Ω min - 1 / Ω max ) , B = 1 2 ( 1 / Ω min + 1 / Ω max ) ,
ω x i + ϕ = cos - 1 [ ( 1 / Ω i - B ) / A ] ,
( OPD ) ( h 1 / 2 ) [ ( n glass - 1 ) cot γ - 1 ] ,
Δ ( OPD ) / λ - 2.0 ( h 1 / λ ) Δ γ .