Abstract

The longitudinal displacement of an elelctromagnetic beam upon reflection from a dielectric slab has been investigated. The experimental results, using 3 cm microwaves, are compared to those predicted by a theory based on a plane-wave approach. The experiment has been completed for both perpendicular and parallel polarization using a slab several wavelengths thick for angles of incidence around 45°. The agreement between experimental values and theoretical predictions is good.

© 1978 Optical Society of America

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References

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  1. F. Goos and H. Hänchen, “Uber das Eindrigen des totalreflektierten Lichtes in das dünnere Medium,” Ann. der Phys. 43, 383–392 (1943).
    [Crossref]
  2. H. C. Bezner, “A microwave study of the Goos-Hänchen shift,” Ph.D. Thesis, McMaster University, Canada (1968) (unpublished).
  3. L.A.A. Read, I. R. Dagg, and G. E. Reesor, “Microwave phase measurements associated with the Goos-Hänchen shift,” Can. J. Phys. 50, 52–56 (1972).
    [Crossref]
  4. V. Akylas, J. Kaur, and T. M. Knasel, “Measurement of the longitudinal shift of radiation at total internal reflection by microwave techniques,” Amer. J. Phys. 44, 77–80 (1976).
    [Crossref]
  5. M. Wong, G. E. Reesor, and L. A. A. Read, “Displacement of an electromagnetic beam upon external dielectric reflection,” Can. J. Phys. 55, 1061–1065 (1977).
    [Crossref]
  6. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  7. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 1.6.4.
  8. W. M. Antar and W. M. Boerner, “Gaussian Beam Interaction with a Planar Dielectric Interface,” Can. J. Phys. 52, 962–972 (1974).
  9. L. A. A. Read, “Microwave Phase Measurements Associated with the Goos-Hänchen Shift,” Ph.D. Thesis, University of Waterloo, Canada (1973) (unpublished).
  10. W. A. Coles, “Near-Field Measurement of a Focussed Microwave Lens,” M. Eng. Thesis, McGill University, Canada (1965) (unpublished).
  11. L. A. A. Read, I. R. Dagg, and G. E. Reesor, “Further Microwave Phase Measurements Associated with the Goos-Hänchen Shift,” Can. J. Phys. 52, 2088–2092 (1974).

1977 (1)

M. Wong, G. E. Reesor, and L. A. A. Read, “Displacement of an electromagnetic beam upon external dielectric reflection,” Can. J. Phys. 55, 1061–1065 (1977).
[Crossref]

1976 (1)

V. Akylas, J. Kaur, and T. M. Knasel, “Measurement of the longitudinal shift of radiation at total internal reflection by microwave techniques,” Amer. J. Phys. 44, 77–80 (1976).
[Crossref]

1974 (2)

W. M. Antar and W. M. Boerner, “Gaussian Beam Interaction with a Planar Dielectric Interface,” Can. J. Phys. 52, 962–972 (1974).

L. A. A. Read, I. R. Dagg, and G. E. Reesor, “Further Microwave Phase Measurements Associated with the Goos-Hänchen Shift,” Can. J. Phys. 52, 2088–2092 (1974).

1972 (1)

L.A.A. Read, I. R. Dagg, and G. E. Reesor, “Microwave phase measurements associated with the Goos-Hänchen shift,” Can. J. Phys. 50, 52–56 (1972).
[Crossref]

1943 (1)

F. Goos and H. Hänchen, “Uber das Eindrigen des totalreflektierten Lichtes in das dünnere Medium,” Ann. der Phys. 43, 383–392 (1943).
[Crossref]

Akylas, V.

V. Akylas, J. Kaur, and T. M. Knasel, “Measurement of the longitudinal shift of radiation at total internal reflection by microwave techniques,” Amer. J. Phys. 44, 77–80 (1976).
[Crossref]

Antar, W. M.

W. M. Antar and W. M. Boerner, “Gaussian Beam Interaction with a Planar Dielectric Interface,” Can. J. Phys. 52, 962–972 (1974).

Bezner, H. C.

H. C. Bezner, “A microwave study of the Goos-Hänchen shift,” Ph.D. Thesis, McMaster University, Canada (1968) (unpublished).

Boerner, W. M.

W. M. Antar and W. M. Boerner, “Gaussian Beam Interaction with a Planar Dielectric Interface,” Can. J. Phys. 52, 962–972 (1974).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 1.6.4.

Coles, W. A.

W. A. Coles, “Near-Field Measurement of a Focussed Microwave Lens,” M. Eng. Thesis, McGill University, Canada (1965) (unpublished).

Dagg, I. R.

L. A. A. Read, I. R. Dagg, and G. E. Reesor, “Further Microwave Phase Measurements Associated with the Goos-Hänchen Shift,” Can. J. Phys. 52, 2088–2092 (1974).

L.A.A. Read, I. R. Dagg, and G. E. Reesor, “Microwave phase measurements associated with the Goos-Hänchen shift,” Can. J. Phys. 50, 52–56 (1972).
[Crossref]

Goos, F.

F. Goos and H. Hänchen, “Uber das Eindrigen des totalreflektierten Lichtes in das dünnere Medium,” Ann. der Phys. 43, 383–392 (1943).
[Crossref]

Hänchen, H.

F. Goos and H. Hänchen, “Uber das Eindrigen des totalreflektierten Lichtes in das dünnere Medium,” Ann. der Phys. 43, 383–392 (1943).
[Crossref]

Kaur, J.

V. Akylas, J. Kaur, and T. M. Knasel, “Measurement of the longitudinal shift of radiation at total internal reflection by microwave techniques,” Amer. J. Phys. 44, 77–80 (1976).
[Crossref]

Knasel, T. M.

V. Akylas, J. Kaur, and T. M. Knasel, “Measurement of the longitudinal shift of radiation at total internal reflection by microwave techniques,” Amer. J. Phys. 44, 77–80 (1976).
[Crossref]

Read, L. A. A.

M. Wong, G. E. Reesor, and L. A. A. Read, “Displacement of an electromagnetic beam upon external dielectric reflection,” Can. J. Phys. 55, 1061–1065 (1977).
[Crossref]

L. A. A. Read, I. R. Dagg, and G. E. Reesor, “Further Microwave Phase Measurements Associated with the Goos-Hänchen Shift,” Can. J. Phys. 52, 2088–2092 (1974).

L. A. A. Read, “Microwave Phase Measurements Associated with the Goos-Hänchen Shift,” Ph.D. Thesis, University of Waterloo, Canada (1973) (unpublished).

Read, L.A.A.

L.A.A. Read, I. R. Dagg, and G. E. Reesor, “Microwave phase measurements associated with the Goos-Hänchen shift,” Can. J. Phys. 50, 52–56 (1972).
[Crossref]

Reesor, G. E.

M. Wong, G. E. Reesor, and L. A. A. Read, “Displacement of an electromagnetic beam upon external dielectric reflection,” Can. J. Phys. 55, 1061–1065 (1977).
[Crossref]

L. A. A. Read, I. R. Dagg, and G. E. Reesor, “Further Microwave Phase Measurements Associated with the Goos-Hänchen Shift,” Can. J. Phys. 52, 2088–2092 (1974).

L.A.A. Read, I. R. Dagg, and G. E. Reesor, “Microwave phase measurements associated with the Goos-Hänchen shift,” Can. J. Phys. 50, 52–56 (1972).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 1.6.4.

Wong, M.

M. Wong, G. E. Reesor, and L. A. A. Read, “Displacement of an electromagnetic beam upon external dielectric reflection,” Can. J. Phys. 55, 1061–1065 (1977).
[Crossref]

Amer. J. Phys. (1)

V. Akylas, J. Kaur, and T. M. Knasel, “Measurement of the longitudinal shift of radiation at total internal reflection by microwave techniques,” Amer. J. Phys. 44, 77–80 (1976).
[Crossref]

Ann. der Phys. (1)

F. Goos and H. Hänchen, “Uber das Eindrigen des totalreflektierten Lichtes in das dünnere Medium,” Ann. der Phys. 43, 383–392 (1943).
[Crossref]

Can. J. Phys. (4)

W. M. Antar and W. M. Boerner, “Gaussian Beam Interaction with a Planar Dielectric Interface,” Can. J. Phys. 52, 962–972 (1974).

M. Wong, G. E. Reesor, and L. A. A. Read, “Displacement of an electromagnetic beam upon external dielectric reflection,” Can. J. Phys. 55, 1061–1065 (1977).
[Crossref]

L.A.A. Read, I. R. Dagg, and G. E. Reesor, “Microwave phase measurements associated with the Goos-Hänchen shift,” Can. J. Phys. 50, 52–56 (1972).
[Crossref]

L. A. A. Read, I. R. Dagg, and G. E. Reesor, “Further Microwave Phase Measurements Associated with the Goos-Hänchen Shift,” Can. J. Phys. 52, 2088–2092 (1974).

Other (5)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Sec. 1.6.4.

L. A. A. Read, “Microwave Phase Measurements Associated with the Goos-Hänchen Shift,” Ph.D. Thesis, University of Waterloo, Canada (1973) (unpublished).

W. A. Coles, “Near-Field Measurement of a Focussed Microwave Lens,” M. Eng. Thesis, McGill University, Canada (1965) (unpublished).

H. C. Bezner, “A microwave study of the Goos-Hänchen shift,” Ph.D. Thesis, McMaster University, Canada (1968) (unpublished).

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Figures (4)

FIG. 1
FIG. 1

The coordinate systems used in the theoretical analysis. The microwave horn is shown on the incident beam axis. For both systems, the y axis is perpendicular to the incident plane.

FIG. 2
FIG. 2

Block diagram of the basic microwave circuit.

FIG. 3
FIG. 3

Results for perpendicular polarization and various angles of incidence. The solid circles are associated with the middle curve.

FIG. 4
FIG. 4

Results for parallel polarization and various angles of incidence. The solid circles are associated with the middle curve.

Equations (15)

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E x ( x , 0 , t ) = E z ( x , 0 , t ] = 0 E y ( x , 0 , t ) = { cos ( π / W ) x exp ( j w t ) x W / 2 0 otherwise ,
E y ( x , z , t ) = ( 1 / 2 π ) 1 / 2 × - E ( k x ) exp [ j ( k x x + k z z - w t ) ] d k x .
E ( k x ) = ( 2 W π ) ( 1 2 π ) 1 / 2 cos ( k x W / 2 ) 1 - ( k x W / π ) 2 .
A ( θ ) = cos ( k W sin θ / 2 ) 1 - ( k W sin θ / π ) 2 ,
E y ( x , z ) = ( W / π 2 ) - π / 2 π / 2 A ( θ ) × exp [ j ( k x sin θ + k z cos θ ) ] k cos θ d θ .
E ( x r , z r ) = ( W / π 2 ) - π / 2 π / 2 Γ ( θ + θ i ) A ( θ ) × exp [ j ( k x r sin θ + k z r cos θ ) ] k cos θ d θ ,
Γ ( θ + θ i ) = R ( θ + θ i ) 1 - exp ( 2 j β ) 1 - [ R 2 ( θ + θ i ) exp ( 2 j β ) ]
R ( θ + θ i ) = cos ( θ + θ i ) - [ ( n 2 / n 1 ) 2 - sin 2 ( θ + θ i ) ] 1 / 2 cos ( θ + θ i ) + [ ( n 2 / n 1 ) 2 - sin 2 ( θ + θ i ) ] 1 / 2 ,
β = k τ [ ( n 2 / n 1 ) 2 - sin 2 ( θ + θ i ) ] 1 / 2 ,
E y ( x , 0 , t ) = E z ( x , 0 , t ) = 0 , E x ( x , 0 , t ) = { exp ( j ω t ) x W / 2 0 otherwise .
E x ( x , z ) = ( W / 2 π ) - π / 2 π / 2 B ( θ ) × exp ( k x sin θ + k z cos θ ) k cos θ d θ ,
B ( θ ) = sin [ ( k W / 2 ) sin θ ] ( k W sin θ ) / 2 .
E ( x r , z r ) = ( W / 2 π ) - π / 2 π / 2 Γ ( θ + θ i ) B ( r ) × exp [ j ( k x r sin θ + k z r cos θ ) ] k cos θ d θ ,
Γ ( θ + θ i ) = { - 1 geometric reflection R ( θ + θ i ) 1 - exp ( 2 j β ) 1 - R 2 ( θ + θ i ) 2 exp ( 2 j β ) reflection from dielectric slab
R ( θ + θ i ) = ( n 2 / n 1 ) 2 cos ( θ + θ i ) - [ ( n 2 / n 1 ) 2 - sin 2 ( θ + θ i ) ] 1 / 2 ( n 2 / n 1 ) 2 cos ( θ + θ i ) + [ ( n 2 / n 1 ) 2 - sin 2 ( θ + θ i ) ] 1 / 2 .