Abstract

Longitudinal and transverse shifts of an 8 cm parallel bounded beam of 34.2 GHz (8.77 mm) microwaves totally reflected from a paraffin prism have been investigated. The 45°–45°–90° prism is 18 cm high by 25 cm on the sides and the index of refraction is 1.491. Longitudinal shifts as large as 3 cm have been measured in a single reflection near the critical angle for a beam linearly polarized in the plane of incidence. The shift for perpendicular polarization is approximately half this value. The results are in general agreement with the classical theory for the Goos-Hänchen effect. An incident beam polarized at 45° to the incidence plane produces both parallel and perpendicular polarization shifts with values similar to the above. The shifts for both polarizations are reduced but are still distinctly separate if either a second prism or a metallic reflector is brought into the evanescent wave at millimeter distances from the interface. These results are in accordance with stationary phase calculations for two interfaces. It is found that a small (6 mm) transverse shift results if the prism is illuminated with circularly polarized microwaves.

© 1977 Optical Society of America

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References

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  1. F. Goos and H. Hänchen, Ann. Physik (Leipz.) 1, 333 (1947).
    [Crossref]
  2. See, for example, the review by H. K. V. Lotsch, Optik 32, 116 (1970);Optik 32, 189 (1970);Optik 32, 299 (1971);Optik 32, 553 (1970), and references therein.
  3. O. Bryngdahl, Prog. Opt. XI, 169 (1973).
  4. T. Tamir, Optik 36, 209 (1972);Optik 37, 204 (1973);Optik 38, 269 (1973).Both this reference and Refs. 2 and 3 above contain excellent descriptions of the physical phenomena involved.See also H. Kogelnik, in Integrated Optics, edited by T. Tamir, Topics in Applied Physics,Vol. 7 (Springer, New York, Heidelberg, Berlin, 1975), pp. 15–81.
    [Crossref]
  5. C. von Fragstein, Ann. Phys. (Leipz.) 4, 271 (1949);Remi H. Renard, J. Opt. Soc. Am. 54, 1190 (1964);J. Ricard, Nouv. Rev. Opt. Appl. 1, 5 (1970);the possibility of an energy flow into the evanescent wave was postulated several years before the experiment was performed.See J. Picht, Ann. Phys. (Leipz.) 3, 433 (1929).
    [Crossref]
  6. K. Artmann, Ann. Phys. (Leipz.) 2, 87 (1948).
    [Crossref]
  7. L. Agudin, Phys. Rev. 171, 1385 (1968).
    [Crossref]
  8. F. I. Fedorov, Dokl. Akad. Nauk. SSSR 105, 465 (1955).
  9. C. Imbert, Phys. Rev. D 5, 787 (1972), and references therein;O. Costa de Beauregard, Found. Phys. 2, 111 (1972).
    [Crossref]
  10. O. Costa de Beauregard and C. Imbert, Phys. Rev. D 7, 3555 (1973);A. Mozet, C. Imbert, and S. Huard, C. R. Acad. Sci. Paris B 273, 592 (1971).Goos and Hänchen also observed this effect with a beam linearly polarized at arbitrary incident azimuthal angles.See Ann. Phys. (Leipz.) 5, 251 (1949).
    [Crossref]
  11. Costa de Beauregard and Imbert (Ref. 10) claim that both the transverse and longitudinal shifts are quantized, each having characteristic eigenvalues.The longitudinal shift should then be characterized by the principal linear polarization states, and the transverse shift, by the circular polarization states: K. W. Chiu and J. J. Quinn, [Am. J. Phys. 40, 1847 (1972)] analyze the longitudinal shift by classical electrodynamics in terms of a time delay scattering process, and also arrive at the conclusion that an incident beam of arbitrary polarization splits exactly into p and s polarization states.A similar conclusion was also reached by G. J. Troup and et al., Phys. Rev. Lett. 28, 1540 (1972).
    [Crossref]
  12. L. de Broglie and J. P. Vigier, Phys. Rev. Lett. 28, 1001 (1972).
    [Crossref]
  13. At the completion of our work we became aware of a concurrent study of the longitudinal shift due to microwaves that was made by V. Akylas, J. Kaur, and T. M. Knassel, Appl. Opt. 13, 742 (1974);Am. J. Phys. 44, 77 (1976).In a remarkably similar experiment, they used a paraffin prism but a larger wavelength (3.2 cm). Their measurements were restricted to a single angle of incidence and to illumination with only parallel and perpendicular linear polarization.It also came to our attention that the Goos–Hänchen shift was measured in a general study of the total reflection of microwaves by P. Baumler, Ann. Phys. (Leipz.) 10, 409 (1963).He also used a single angle of incidence with p and s polarizations but a shorter wavelength of 1.46 cm. The results of the authors above were found to be in good agreement with classical theory.
    [Crossref] [PubMed]
  14. Portions of the present work were presented at the Spring Meeting of theOptical Society of America, Washington, D. C., 21–25 April 1974.See J. J. Cowan and B. Aničin, J. Opt. Soc. Am. 64, 525 (1974).
    [Crossref]
  15. H. Wolter, Handbuch der Physik (Springer-Verlag, Berlin, 1956), Vol. 24, p. 472.
  16. B. R. Horowitz and T. Tamir, J. Opt. Soc. Am. 61, 586 (1971).
    [Crossref]
  17. C. K. Carniglia (personal correspondence).
  18. M. McGuirk, C. K. Carniglia, and J. J. Cowan (unpublished).
  19. In the acoustical case this distance is significant [M. A. Breazeale (personal correspondence)].See alsoM. A. Breazeale, L. Adler, and G. W. Scott, J. Acoust. Soc. Am. Suppl. 57, 5–38 (1975).

1974 (1)

1973 (2)

O. Costa de Beauregard and C. Imbert, Phys. Rev. D 7, 3555 (1973);A. Mozet, C. Imbert, and S. Huard, C. R. Acad. Sci. Paris B 273, 592 (1971).Goos and Hänchen also observed this effect with a beam linearly polarized at arbitrary incident azimuthal angles.See Ann. Phys. (Leipz.) 5, 251 (1949).
[Crossref]

O. Bryngdahl, Prog. Opt. XI, 169 (1973).

1972 (4)

T. Tamir, Optik 36, 209 (1972);Optik 37, 204 (1973);Optik 38, 269 (1973).Both this reference and Refs. 2 and 3 above contain excellent descriptions of the physical phenomena involved.See also H. Kogelnik, in Integrated Optics, edited by T. Tamir, Topics in Applied Physics,Vol. 7 (Springer, New York, Heidelberg, Berlin, 1975), pp. 15–81.
[Crossref]

Costa de Beauregard and Imbert (Ref. 10) claim that both the transverse and longitudinal shifts are quantized, each having characteristic eigenvalues.The longitudinal shift should then be characterized by the principal linear polarization states, and the transverse shift, by the circular polarization states: K. W. Chiu and J. J. Quinn, [Am. J. Phys. 40, 1847 (1972)] analyze the longitudinal shift by classical electrodynamics in terms of a time delay scattering process, and also arrive at the conclusion that an incident beam of arbitrary polarization splits exactly into p and s polarization states.A similar conclusion was also reached by G. J. Troup and et al., Phys. Rev. Lett. 28, 1540 (1972).
[Crossref]

L. de Broglie and J. P. Vigier, Phys. Rev. Lett. 28, 1001 (1972).
[Crossref]

C. Imbert, Phys. Rev. D 5, 787 (1972), and references therein;O. Costa de Beauregard, Found. Phys. 2, 111 (1972).
[Crossref]

1971 (1)

1970 (1)

See, for example, the review by H. K. V. Lotsch, Optik 32, 116 (1970);Optik 32, 189 (1970);Optik 32, 299 (1971);Optik 32, 553 (1970), and references therein.

1968 (1)

L. Agudin, Phys. Rev. 171, 1385 (1968).
[Crossref]

1955 (1)

F. I. Fedorov, Dokl. Akad. Nauk. SSSR 105, 465 (1955).

1949 (1)

C. von Fragstein, Ann. Phys. (Leipz.) 4, 271 (1949);Remi H. Renard, J. Opt. Soc. Am. 54, 1190 (1964);J. Ricard, Nouv. Rev. Opt. Appl. 1, 5 (1970);the possibility of an energy flow into the evanescent wave was postulated several years before the experiment was performed.See J. Picht, Ann. Phys. (Leipz.) 3, 433 (1929).
[Crossref]

1948 (1)

K. Artmann, Ann. Phys. (Leipz.) 2, 87 (1948).
[Crossref]

1947 (1)

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

Agudin, L.

L. Agudin, Phys. Rev. 171, 1385 (1968).
[Crossref]

Akylas, V.

Artmann, K.

K. Artmann, Ann. Phys. (Leipz.) 2, 87 (1948).
[Crossref]

Beauregard, O. Costa de

O. Costa de Beauregard and C. Imbert, Phys. Rev. D 7, 3555 (1973);A. Mozet, C. Imbert, and S. Huard, C. R. Acad. Sci. Paris B 273, 592 (1971).Goos and Hänchen also observed this effect with a beam linearly polarized at arbitrary incident azimuthal angles.See Ann. Phys. (Leipz.) 5, 251 (1949).
[Crossref]

Breazeale, M. A.

In the acoustical case this distance is significant [M. A. Breazeale (personal correspondence)].See alsoM. A. Breazeale, L. Adler, and G. W. Scott, J. Acoust. Soc. Am. Suppl. 57, 5–38 (1975).

Bryngdahl, O.

O. Bryngdahl, Prog. Opt. XI, 169 (1973).

Carniglia, C. K.

C. K. Carniglia (personal correspondence).

M. McGuirk, C. K. Carniglia, and J. J. Cowan (unpublished).

Chiu, K. W.

Costa de Beauregard and Imbert (Ref. 10) claim that both the transverse and longitudinal shifts are quantized, each having characteristic eigenvalues.The longitudinal shift should then be characterized by the principal linear polarization states, and the transverse shift, by the circular polarization states: K. W. Chiu and J. J. Quinn, [Am. J. Phys. 40, 1847 (1972)] analyze the longitudinal shift by classical electrodynamics in terms of a time delay scattering process, and also arrive at the conclusion that an incident beam of arbitrary polarization splits exactly into p and s polarization states.A similar conclusion was also reached by G. J. Troup and et al., Phys. Rev. Lett. 28, 1540 (1972).
[Crossref]

Cowan, J. J.

M. McGuirk, C. K. Carniglia, and J. J. Cowan (unpublished).

de Broglie, L.

L. de Broglie and J. P. Vigier, Phys. Rev. Lett. 28, 1001 (1972).
[Crossref]

Fedorov, F. I.

F. I. Fedorov, Dokl. Akad. Nauk. SSSR 105, 465 (1955).

Goos, F.

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

Hänchen, H.

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

Horowitz, B. R.

Imbert, C.

O. Costa de Beauregard and C. Imbert, Phys. Rev. D 7, 3555 (1973);A. Mozet, C. Imbert, and S. Huard, C. R. Acad. Sci. Paris B 273, 592 (1971).Goos and Hänchen also observed this effect with a beam linearly polarized at arbitrary incident azimuthal angles.See Ann. Phys. (Leipz.) 5, 251 (1949).
[Crossref]

C. Imbert, Phys. Rev. D 5, 787 (1972), and references therein;O. Costa de Beauregard, Found. Phys. 2, 111 (1972).
[Crossref]

Kaur, J.

Knassel, T. M.

Lotsch, H. K. V.

See, for example, the review by H. K. V. Lotsch, Optik 32, 116 (1970);Optik 32, 189 (1970);Optik 32, 299 (1971);Optik 32, 553 (1970), and references therein.

McGuirk, M.

M. McGuirk, C. K. Carniglia, and J. J. Cowan (unpublished).

Quinn, J. J.

Costa de Beauregard and Imbert (Ref. 10) claim that both the transverse and longitudinal shifts are quantized, each having characteristic eigenvalues.The longitudinal shift should then be characterized by the principal linear polarization states, and the transverse shift, by the circular polarization states: K. W. Chiu and J. J. Quinn, [Am. J. Phys. 40, 1847 (1972)] analyze the longitudinal shift by classical electrodynamics in terms of a time delay scattering process, and also arrive at the conclusion that an incident beam of arbitrary polarization splits exactly into p and s polarization states.A similar conclusion was also reached by G. J. Troup and et al., Phys. Rev. Lett. 28, 1540 (1972).
[Crossref]

Tamir, T.

T. Tamir, Optik 36, 209 (1972);Optik 37, 204 (1973);Optik 38, 269 (1973).Both this reference and Refs. 2 and 3 above contain excellent descriptions of the physical phenomena involved.See also H. Kogelnik, in Integrated Optics, edited by T. Tamir, Topics in Applied Physics,Vol. 7 (Springer, New York, Heidelberg, Berlin, 1975), pp. 15–81.
[Crossref]

B. R. Horowitz and T. Tamir, J. Opt. Soc. Am. 61, 586 (1971).
[Crossref]

Vigier, J. P.

L. de Broglie and J. P. Vigier, Phys. Rev. Lett. 28, 1001 (1972).
[Crossref]

von Fragstein, C.

C. von Fragstein, Ann. Phys. (Leipz.) 4, 271 (1949);Remi H. Renard, J. Opt. Soc. Am. 54, 1190 (1964);J. Ricard, Nouv. Rev. Opt. Appl. 1, 5 (1970);the possibility of an energy flow into the evanescent wave was postulated several years before the experiment was performed.See J. Picht, Ann. Phys. (Leipz.) 3, 433 (1929).
[Crossref]

Wolter, H.

H. Wolter, Handbuch der Physik (Springer-Verlag, Berlin, 1956), Vol. 24, p. 472.

Am. J. Phys. (1)

Costa de Beauregard and Imbert (Ref. 10) claim that both the transverse and longitudinal shifts are quantized, each having characteristic eigenvalues.The longitudinal shift should then be characterized by the principal linear polarization states, and the transverse shift, by the circular polarization states: K. W. Chiu and J. J. Quinn, [Am. J. Phys. 40, 1847 (1972)] analyze the longitudinal shift by classical electrodynamics in terms of a time delay scattering process, and also arrive at the conclusion that an incident beam of arbitrary polarization splits exactly into p and s polarization states.A similar conclusion was also reached by G. J. Troup and et al., Phys. Rev. Lett. 28, 1540 (1972).
[Crossref]

Ann. Phys. (Leipz.) (2)

C. von Fragstein, Ann. Phys. (Leipz.) 4, 271 (1949);Remi H. Renard, J. Opt. Soc. Am. 54, 1190 (1964);J. Ricard, Nouv. Rev. Opt. Appl. 1, 5 (1970);the possibility of an energy flow into the evanescent wave was postulated several years before the experiment was performed.See J. Picht, Ann. Phys. (Leipz.) 3, 433 (1929).
[Crossref]

K. Artmann, Ann. Phys. (Leipz.) 2, 87 (1948).
[Crossref]

Ann. Physik (Leipz.) (1)

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

Appl. Opt. (1)

Dokl. Akad. Nauk. SSSR (1)

F. I. Fedorov, Dokl. Akad. Nauk. SSSR 105, 465 (1955).

J. Opt. Soc. Am. (1)

Optik (2)

T. Tamir, Optik 36, 209 (1972);Optik 37, 204 (1973);Optik 38, 269 (1973).Both this reference and Refs. 2 and 3 above contain excellent descriptions of the physical phenomena involved.See also H. Kogelnik, in Integrated Optics, edited by T. Tamir, Topics in Applied Physics,Vol. 7 (Springer, New York, Heidelberg, Berlin, 1975), pp. 15–81.
[Crossref]

See, for example, the review by H. K. V. Lotsch, Optik 32, 116 (1970);Optik 32, 189 (1970);Optik 32, 299 (1971);Optik 32, 553 (1970), and references therein.

Phys. Rev. (1)

L. Agudin, Phys. Rev. 171, 1385 (1968).
[Crossref]

Phys. Rev. D (2)

C. Imbert, Phys. Rev. D 5, 787 (1972), and references therein;O. Costa de Beauregard, Found. Phys. 2, 111 (1972).
[Crossref]

O. Costa de Beauregard and C. Imbert, Phys. Rev. D 7, 3555 (1973);A. Mozet, C. Imbert, and S. Huard, C. R. Acad. Sci. Paris B 273, 592 (1971).Goos and Hänchen also observed this effect with a beam linearly polarized at arbitrary incident azimuthal angles.See Ann. Phys. (Leipz.) 5, 251 (1949).
[Crossref]

Phys. Rev. Lett. (1)

L. de Broglie and J. P. Vigier, Phys. Rev. Lett. 28, 1001 (1972).
[Crossref]

Prog. Opt. (1)

O. Bryngdahl, Prog. Opt. XI, 169 (1973).

Other (5)

Portions of the present work were presented at the Spring Meeting of theOptical Society of America, Washington, D. C., 21–25 April 1974.See J. J. Cowan and B. Aničin, J. Opt. Soc. Am. 64, 525 (1974).
[Crossref]

H. Wolter, Handbuch der Physik (Springer-Verlag, Berlin, 1956), Vol. 24, p. 472.

C. K. Carniglia (personal correspondence).

M. McGuirk, C. K. Carniglia, and J. J. Cowan (unpublished).

In the acoustical case this distance is significant [M. A. Breazeale (personal correspondence)].See alsoM. A. Breazeale, L. Adler, and G. W. Scott, J. Acoust. Soc. Am. Suppl. 57, 5–38 (1975).

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Figures (9)

FIG. 1
FIG. 1

Schematic diagram of microwave beam-shift apparatus. X is the beam shift along the hypotenuse, D is the parallel beam shift, d is the measured beam shift, and θ is the angle of incidence.

FIG. 2
FIG. 2

Variation of shift with incidence angle showing comparison of measured shift with the classical stationary phase and bounded beam theories.

FIG. 3
FIG. 3

Variation of shift with incident beam polarization, where ψi is the azimuth angle of the incident electric field.

FIG. 4
FIG. 4

(a) Variation of shift with detector polarization for a fixed incident azimuth angle (ψi = 45°) and where ψr is the azimuth angle of the receiver. (b) Variation of shift at ψi = 45° for detector on p, s, and at ψr = 45°.

FIG. 5
FIG. 5

Effect of metallic reflector on shift with s polarization, where a is the distance the reflector is positioned parallel to the interface.

FIG. 6
FIG. 6

Effect on shift of placing second paraffin prism at distances a parallel to the interface for s and p polarizations.

FIG. 7
FIG. 7

Comparison of experimental results with stationary phase theory, taking into account the presence of a second paraffin prism near the interface.

FIG. 8
FIG. 8

Variation of shifts for varying distances a from interface of the second paraffin prism, when the incident beam is polarized at an azimuth angle of ψi = 45°. The maximum scale readings in decibels are indicated.

FIG. 9
FIG. 9

Transverse beam shift with circular polarization. Curves A and B result from microwaves circularly polarized in opposite senses.

Equations (23)

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Δ ϕ = 2 π ( n 1 ) t / λ ,
sin θ c = 1 / n
X = 1 2 d ( 2 ) 1 / 2 ( 1 + tan θ ) ,
E = A ( S ) e i K ( S x + C y ) r e i α d S ,
ψ S = 0 .
ψ = K ( S x + C y ) + α ,
S ( S x + C y ) + 1 K α S = 0
S ( S x + C y ) = x y tan θ
= x x 0
= X ,
X = 1 K α S .
X p = λ tan θ π ( n 2 S 2 C 2 ) ( n 2 S 2 1 ) 1 / 2 ,
X s = λ tan θ / π ( n 2 S 2 1 ) 1 / 2 .
X = A ( θ ) 2 5 / 4 cos θ Re [ ( w K ) 1 / 2 e i π / 4 exp ( 1 4 γ 0 2 ) D 1 / 2 ( γ 0 ) 1 + A ( θ ) { ( δ ) 1 / 2 [ 2 1 / 4 e i π / 4 / ( K w ) 1 / 2 ] exp ( 1 4 γ 0 2 ) D 1 / 2 ( γ 0 ) } ] ,
A ( θ ) = 4 m cos 2 θ c sin θ cos 1 / 2 θ ( sin θ + sin θ c ) 1 / 2 [ cos 2 θ + m 2 ( sin 2 θ sin 2 θ c ) ]
m = { n 2 p polarization , 1 s polarization .
L = λ / π n sin θ c cos θ c .
R a = R ( 1 γ ) / ( 1 R 2 γ )
= r a e i β ,
γ = e 2 ρ a ,
tan β = [ ( 1 + γ ) / ( 1 γ ) ] tan α .
X X max = ( 1 γ 2 ) ( 1 + A ) 2 4 γ Q k a F ( 1 A ) ( 1 γ ) 2 ( 1 A ) 2 + 4 ( 1 + γ ) 2 A ,
Q = ( n 2 S 2 1 ) 1 / 2 , k = 2 π / λ ; A = { ( n Q / C ) 2 p polarization , ( Q / n C ) 2 s polarization ; F = { n 2 ( n 2 S 2 C 2 ) p polarization , 1 s polarization .