Abstract

We have measured the diffraction pattern of the edge of a transparent sheet for plane-polarized microwaves (λ = 3.0 cm) for angles of incidence from 0° to 45°. The material was expanded polyurethane (Styrofoam) with index of refraction 1.0150 (that of air being 1.0003). The thickness of the sheet, and thus of the rectangular edge, varied from 1.7 to 13.5 wavelengths. Our initial purpose was to study the diffraction of microwaves by phase objects, as an analog that might indicate general ways of improving the contrast in images of phase objects obtained with an electron microscope and, in particular, ways of explaining electron edge diffraction. The high contrast of as much as 2 to 1 of irradiance of the edge pattern for material of such high transparency (99.5% for thickness of 3.4 wavelengths) was surprising. We have explained the major features of our results on the basis of Huygens’s principle and Fresnel’s equations for reflection and refraction at the surface of the edge of finite thickness.

© 1974 Optical Society of America

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References

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  1. C. L. Andrews, Optics of the Electromagnetic Spectrum (Prentice–Hall, Englewood Cliffs, N. J., 1964).
  2. R. D. Kodis, J. Appl. Phys. 23, 249 (1952).
    [Crossref]
  3. L. Barkley, B. A. Horvath, and T. J. F. Pavlasek, J. Opt. Soc. Am. 63, 673 (1973).
    [Crossref]
  4. J. D. Barrett and F. S. Harris, J. Opt. Soc. Am. 52, 637 (1962).
    [Crossref]
  5. B. N. Harden, Proc. IEE (Lond.), Pt. III. 99, 229 (1952).
  6. C. R. Carpenter, Am. J. Phys. 39, 120 (1971).
    [Crossref]
  7. A. Golab and C. L. Andrews, Am. J. Phys. 39, 121 (1971).
    [Crossref]
  8. C. L. Andrews, J. Appl. Phys. 21, 761 (1950).
    [Crossref]
  9. C. R. Carpenter, Ph.D. thesis (State University of New York at Albany, 1972) (University Microfilms, Ann Arbor, Mich., Order No. 7 325 701).
  10. R. O. Dell, C. R. Carpenter, and C. L. Andrews, J. Opt. Soc. Am. 62, 902 (1972).
    [Crossref]
  11. J. Komrska, in Advances in Electronics and Electron Physics, Vol. 30, edited by L. Marton (Academic, New York, 1971), p. 139.
    [Crossref]
  12. J. N. Turner, Ph.D. thesis (State University of New York at Buffalo, 1973) (University Microfilms, Ann. Arbor, Mich., Order No. 7 319 245).
  13. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 41–51.
  14. C. Ferreira-Lina, A. Howie, and P. F. Linington, in Proceedings of the Fifth European Congress on Electron Microscopy (University of Manchester, Institute of Physics, London, 1972), p. 418.

1973 (1)

1972 (1)

1971 (2)

C. R. Carpenter, Am. J. Phys. 39, 120 (1971).
[Crossref]

A. Golab and C. L. Andrews, Am. J. Phys. 39, 121 (1971).
[Crossref]

1962 (1)

1952 (2)

B. N. Harden, Proc. IEE (Lond.), Pt. III. 99, 229 (1952).

R. D. Kodis, J. Appl. Phys. 23, 249 (1952).
[Crossref]

1950 (1)

C. L. Andrews, J. Appl. Phys. 21, 761 (1950).
[Crossref]

Andrews, C. L.

R. O. Dell, C. R. Carpenter, and C. L. Andrews, J. Opt. Soc. Am. 62, 902 (1972).
[Crossref]

A. Golab and C. L. Andrews, Am. J. Phys. 39, 121 (1971).
[Crossref]

C. L. Andrews, J. Appl. Phys. 21, 761 (1950).
[Crossref]

C. L. Andrews, Optics of the Electromagnetic Spectrum (Prentice–Hall, Englewood Cliffs, N. J., 1964).

Barkley, L.

Barrett, J. D.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 41–51.

Carpenter, C. R.

R. O. Dell, C. R. Carpenter, and C. L. Andrews, J. Opt. Soc. Am. 62, 902 (1972).
[Crossref]

C. R. Carpenter, Am. J. Phys. 39, 120 (1971).
[Crossref]

C. R. Carpenter, Ph.D. thesis (State University of New York at Albany, 1972) (University Microfilms, Ann Arbor, Mich., Order No. 7 325 701).

Dell, R. O.

Ferreira-Lina, C.

C. Ferreira-Lina, A. Howie, and P. F. Linington, in Proceedings of the Fifth European Congress on Electron Microscopy (University of Manchester, Institute of Physics, London, 1972), p. 418.

Golab, A.

A. Golab and C. L. Andrews, Am. J. Phys. 39, 121 (1971).
[Crossref]

Harden, B. N.

B. N. Harden, Proc. IEE (Lond.), Pt. III. 99, 229 (1952).

Harris, F. S.

Horvath, B. A.

Howie, A.

C. Ferreira-Lina, A. Howie, and P. F. Linington, in Proceedings of the Fifth European Congress on Electron Microscopy (University of Manchester, Institute of Physics, London, 1972), p. 418.

Kodis, R. D.

R. D. Kodis, J. Appl. Phys. 23, 249 (1952).
[Crossref]

Komrska, J.

J. Komrska, in Advances in Electronics and Electron Physics, Vol. 30, edited by L. Marton (Academic, New York, 1971), p. 139.
[Crossref]

Linington, P. F.

C. Ferreira-Lina, A. Howie, and P. F. Linington, in Proceedings of the Fifth European Congress on Electron Microscopy (University of Manchester, Institute of Physics, London, 1972), p. 418.

Pavlasek, T. J. F.

Turner, J. N.

J. N. Turner, Ph.D. thesis (State University of New York at Buffalo, 1973) (University Microfilms, Ann. Arbor, Mich., Order No. 7 319 245).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 41–51.

Am. J. Phys. (2)

C. R. Carpenter, Am. J. Phys. 39, 120 (1971).
[Crossref]

A. Golab and C. L. Andrews, Am. J. Phys. 39, 121 (1971).
[Crossref]

J. Appl. Phys. (2)

C. L. Andrews, J. Appl. Phys. 21, 761 (1950).
[Crossref]

R. D. Kodis, J. Appl. Phys. 23, 249 (1952).
[Crossref]

J. Opt. Soc. Am. (3)

Proc. IEE (Lond.), Pt. III. (1)

B. N. Harden, Proc. IEE (Lond.), Pt. III. 99, 229 (1952).

Other (6)

C. R. Carpenter, Ph.D. thesis (State University of New York at Albany, 1972) (University Microfilms, Ann Arbor, Mich., Order No. 7 325 701).

J. Komrska, in Advances in Electronics and Electron Physics, Vol. 30, edited by L. Marton (Academic, New York, 1971), p. 139.
[Crossref]

J. N. Turner, Ph.D. thesis (State University of New York at Buffalo, 1973) (University Microfilms, Ann. Arbor, Mich., Order No. 7 319 245).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), pp. 41–51.

C. Ferreira-Lina, A. Howie, and P. F. Linington, in Proceedings of the Fifth European Congress on Electron Microscopy (University of Manchester, Institute of Physics, London, 1972), p. 418.

C. L. Andrews, Optics of the Electromagnetic Spectrum (Prentice–Hall, Englewood Cliffs, N. J., 1964).

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Figures (11)

Fig. 1
Fig. 1

Microwave optical systems. (a) Anechoic chamber, source, detection system, and normally oriented Stryofoam sheet. (b) Orientation relative to detection plane of Styrofoam edge cut at an angle that defines the angle X. (c) Third orientation of Styrofoam edge that defines the angle Θ. The surface labeled e was shown to be the important interface, and is referred to in the text as the surface e.

Fig. 2
Fig. 2

Edge patterns plotted as irradiance vs distance from optic axis. Each curve represents a different value of b, as defined in Fig. 1; ●, b = 2λ; ——, b = 8λ – – –, b = 16λ; +, b = 32λ. The electric vector is polarized parallel to the plane of incidence.

Fig. 3
Fig. 3

Edge patterns plotted as irradiance vs distance from the optic axis. Same parameters as Fig. 2, except that the electric vector is polarized normal to the plane of incidence.

Fig. 4
Fig. 4

Contours of constant irradiance for square-edged Styrofoam blocks as a function of b. The electric vector is polarized parallel to the plane of incidence.

Fig. 5
Fig. 5

Contours of constant irradiance for square-edged Styrofoam blocks as a function of b. The electric vector is polarized normal to the plane of incidence.

Fig. 6
Fig. 6

Thickness dependence of the irradiance of the first maximum on the Styrofoam side of the edge (top half of the plot), and of the first minimum on the free-space side of the edge (bottom half of the plot). The electric vector is polarized parallel to the plane of incidence. ×, t = 40.64 cm; △, t = 20.32 cm; ●, t = 10.16 cm.

Fig. 7
Fig. 7

Thickness dependence of the irradiance of the first maximum on the Styrofoam side of the edge (top half of the plot), and of the first minimum on the free-space side of the edge (bottom half of the plot). The electric vector is polarized normal to the plane of incidence. ×, t = 40.64 cm; △, t = 20.32 cm; ●, t = 10.16 cm; ○, t = 5.08 cm.

Fig. 8
Fig. 8

Irradiance of fringe pattern for b = 8λ, t = 20.32 cm, as a function of the angle X. ——, X = 0°; – – –, X = 10°; +, X = 20°; ○, X = 30°; — · —, X = 45°.

Fig. 9
Fig. 9

Geometric reconstruction of the irradiance distribution in the plane of observation.

Fig. 10
Fig. 10

Plots of Fresnel’s equations of reflection and refraction for a Styrofoam–air interface, with the microwaves incident from the air side of the interface.

Fig. 11
Fig. 11

Plots of Fresnel’s equations of reflection and refraction for a Styrofoam–air interface, with the microwaves incident from the Styrofoam side of the interface.

Equations (1)

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T P = 2 { ( sin β t cos β i ) / sin ( β i + β t ) cos ( β i - β t ) } A P , T N = 2 { ( sin β t cos β i ) / sin ( β j + β t ) } A N , R P = { tan ( β i - β t ) / tan ( β i + β t ) } A P , R N = - { sin ( β i - β t ) / sin ( β i + β t ) } A N ,