Abstract

Fraunhofer diffraction by two long, thin, parallel conducting strips in a plane is investigated experimentally using microwaves whose wavelength is of the order of the strip width. The radiation is polarized with the electric field parallel to the axes of the strips. The results are compared with scalar Kirchhoff theory and first and second order Keller theory. It is found that for this polarization each theory yields good agreement with the results obtained in the laboratory.

© 1968 Optical Society of America

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References

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  1. R. Tremblay, A. Boivin, Appl. Opt. 5, 249 (1966).
    [Crossref] [PubMed]
  2. See, for example, S. T. Moseley, “On-Axis Defocus Characteristics of the Paraboloidal Reflector,” Final Report AF 30 (602)–925, Syracuse University, 1954.
  3. See the 1965 October issueAppl. Opt. [Vol. 4, 1213ff (1965)].
  4. J. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), p. 460.
  5. J. Keller, J. Opt. Soc. Amer., 52, 116 (1962).
    [Crossref]
  6. Ref. 5, p. 120.

1966 (1)

1965 (1)

1962 (1)

J. Keller, J. Opt. Soc. Amer., 52, 116 (1962).
[Crossref]

Boivin, A.

Keller, J.

J. Keller, J. Opt. Soc. Amer., 52, 116 (1962).
[Crossref]

Moseley, S. T.

See, for example, S. T. Moseley, “On-Axis Defocus Characteristics of the Paraboloidal Reflector,” Final Report AF 30 (602)–925, Syracuse University, 1954.

Stratton, J.

J. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), p. 460.

Tremblay, R.

Appl. Opt. (2)

J. Opt. Soc. Amer. (1)

J. Keller, J. Opt. Soc. Amer., 52, 116 (1962).
[Crossref]

Other (3)

Ref. 5, p. 120.

J. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), p. 460.

See, for example, S. T. Moseley, “On-Axis Defocus Characteristics of the Paraboloidal Reflector,” Final Report AF 30 (602)–925, Syracuse University, 1954.

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Figures (9)

Fig. 1
Fig. 1

Kirchhoff theory geometry. Only one strip is shown. Radiation from a point source is incident on a surface element ds at an angle θi and is diffracted into angle θ to the observation point. The strips are part of a closed surface s, which consists of the plane in which the strips lie and a hemisphere at infinity.

Fig. 2
Fig. 2

Geometry for Kirchhoff diffraction by two infinitely long, planar strips (S1,S2) in the Fraunhofer limit. Plane radiation, incident at angle θi, is diffracted into angle θ. The strips are of width w with their axes perpendicular to the page. The distance between strip centers is D. The diffracted angle measured with respect to the direction of the main beam is Ⓗ.

Fig. 3
Fig. 3

Geometry for Keller theory of edge diffraction. A thin screen, perpendicular to the page, has radiation incident upon it at angle α. This gives rise to diffracted rays, one of which emerges at angle β and passes through observation point P. Both α and β are measured relative to a plane that is perpendicular to the edge.

Fig. 4
Fig. 4

Schematic sketch of the spectrometer: AE—angle encoder to measure diffraction angles, CD—crystal detector in a tunable mount, CP—parabolic dish to collect the diffracted radiation, F—supporting frame for the diffracting strips FM—frequency meter to determine the radiation wavelength, and S1,S2—diffracting strips

Fig. 5
Fig. 5

Comparison of the intensities given by first order Keller theory (KE–1), second order Keller theory (KE–2), and experiment (EXPT) for Fraunhofer diffraction by two parallel strips in a plane. Plane radiation of wavelength λ = 0.525 cm is incident on the strips at angle θi = 36° and diffracted into angle Ⓗ. The width w of each strip is 1.220 cm and the distance between strip centers, D, is 1.625 cm (approximately 3λ).

Fig. 6
Fig. 6

Comparison of the intensities given by first order Keller theory (KE–1), second order Keller theory (KE–2), and experiment (EXPT) for Fraunhofer diffraction by two strips where w = 1.220 cm, θi = 36°, λ = 0.525 cm, and D = 2.719 cm (approximately 5λ).

Fig. 7
Fig. 7

Comparison of the intensities given by Kirchhoff theory (KI), second order Keller theory (KE–2), and experiment (EXPT) for Fraunhofer diffraction by two strips where w = 1.475 cm, θi = 46°, λ = 0.533 cm, and D = 2.105 cm (approximately 4λ).

Fig. 8
Fig. 8

Comparison of the intensities given by Kirchhoff theory (KI), second order Keller theory (KE–2) and experiment (EXPT) for Fraunhofer diffraction by two strips where w = 1.220 cm, θi = 46°, λ = 0.525 cm, and D = 3.805 cm (approximately 7λ).

Fig. 9
Fig. 9

Comparison of the intensities given by Kirchhoff theory (KI), second order Keller theory (KE–2) and experiment (EXPT) for Fraunhofer diffraction by two strips where w = 1.475 cm, θi = 46°, λ = 0.533 cm, and D = 4.278 cm (approximately 8λ).

Equations (9)

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ψ = 1 4 π s [ ψ n e i k r r - ψ n ( e i k r r ) ] d s ,
ψ = i k 4 π S 1 , S 2 exp [ i k ( r + r 0 ) ] ( cos θ i + cos θ ) r 0 r d s ,
ψ = A ( cos θ i + cos θ ) [ S 1 e i γ y 1 d y 1 + S 2 e - i γ y 2 d y 2 ] ,
I = ψ 2 = A 2 ( cos θ i + cos θ ) 2 { sin 2 γ ( w / 2 ) [ γ ( w / 2 ) ] 2 cos 2 γ ( D / 2 ) } ,
u = v exp [ i ( π / 4 + k r ) ] 2 ( 2 π k r ) 1 2 { sec [ ( β - α ) / 2 ± csc [ ( β + α ) / 2 ] } ,
u ( 1 ) = B cos ( γ D 2 ) [ sec ( θ i + θ 2 ) cos γ w 2 - i csc ( θ i - θ 2 ) sin γ w 2 ] ,
I ( 1 ) = u ( 1 ) 2 = B 2 [ cos 2 γ ( w / 2 ) cos 2 [ ( θ i + θ ) / 2 ] + sin 2 γ ( w / 2 ) sin 2 [ ( θ i - θ ) / 2 ] ] cos 2 γ D 2 .
u ( 2 ) = B exp { i [ k ( D - w ) - π / 4 ] } { π k [ ( D - w ) / 2 ] } 1 2 × { sin [ ( θ i + θ ) / 2 ] sin [ ( D - w ) γ / 2 ] + i cos [ ( θ i - θ ) / 2 ] × cos [ ( D - w ) γ / 2 ] cos 2 [ ( θ i - θ ) / 2 ] - sin 2 [ ( θ i + θ ) / 2 ] } .
I ( 2 ) = u ( 1 ) + u ( 2 ) 2 .

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