Abstract

A comparison is made between approximations using the geometrical theory of diffraction (GTD) and Kirchhoff theory and the exact solution using Maxwell’s equations for the case of Fraunhofer diffraction of electromagnetic radiation incident upon a long, thin, perfectly conducting strip with the electric field vector polarized perpendicular to the strip axis. Strip widths from approximately 0.3λ to 3λ are considered. Glancing angles of incidence are taken from 4° to 90°. Irradiances are compared as a function of diffraction angle in the region on the same side of the strip as the incident radiation.

© 1976 Optical Society of America

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References

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  1. L. A. DeAcetis, I. Lazar, Appl. Opt. 9, 1691 (1970). The authors point out several errors in this publication: (a) the captions under Figs. 3 and 10 should be interchanged, likewise those under Figs. 4 and 7, 5 and 8, and 6 and 9; (b) the strip width indicated in the caption for Fig. 10 should read 1.424λ, not 1.24λ; (c) Eq. (6) should read the same as Eq. (4) of this paper; (d) in the Results section, the fourth sentence should read “In Figs. 2 through 6, … for the case where u = 84°.”
  2. B. Sieger, Ann. Phys. 27, 626 (1908).
  3. P. Morse, P. Rubenstein, Phys. Rev. 54, 895 (1938).
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 8.
  5. J. B. Keller, J. Opt. Soc. Am. 52, 116 (1962).
  6. J. Bowman, T. Senior, P. Uslenghi, Electromagnetic and Acoustical Scattering by Simple Shapes (Elsevier, New York, 1969), Chap. 4.
  7. I. Lazar, Ph.D. thesis, New York University (1962) (Univ. Microfilms, Ann Arbor, Mich., 63-7193).
  8. Computation Laboratory of the National Applied Mathematics Laboratories, National Bureau of Standards, Tables Relating to Mathieu Functions (Columbia U.P., New York, 1951).
  9. J. Yu, R. Rudduck, IEEE Trans. Antennas Propag. AP-15, 662 (1967). The sine terms in their Eqs. (2a) should have for their arguments (θ + θ0)/2.

1970 (1)

1967 (1)

J. Yu, R. Rudduck, IEEE Trans. Antennas Propag. AP-15, 662 (1967). The sine terms in their Eqs. (2a) should have for their arguments (θ + θ0)/2.

1962 (1)

1938 (1)

P. Morse, P. Rubenstein, Phys. Rev. 54, 895 (1938).

1908 (1)

B. Sieger, Ann. Phys. 27, 626 (1908).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 8.

Bowman, J.

J. Bowman, T. Senior, P. Uslenghi, Electromagnetic and Acoustical Scattering by Simple Shapes (Elsevier, New York, 1969), Chap. 4.

DeAcetis, L. A.

Keller, J. B.

Lazar, I.

Morse, P.

P. Morse, P. Rubenstein, Phys. Rev. 54, 895 (1938).

Rubenstein, P.

P. Morse, P. Rubenstein, Phys. Rev. 54, 895 (1938).

Rudduck, R.

J. Yu, R. Rudduck, IEEE Trans. Antennas Propag. AP-15, 662 (1967). The sine terms in their Eqs. (2a) should have for their arguments (θ + θ0)/2.

Senior, T.

J. Bowman, T. Senior, P. Uslenghi, Electromagnetic and Acoustical Scattering by Simple Shapes (Elsevier, New York, 1969), Chap. 4.

Sieger, B.

B. Sieger, Ann. Phys. 27, 626 (1908).

Uslenghi, P.

J. Bowman, T. Senior, P. Uslenghi, Electromagnetic and Acoustical Scattering by Simple Shapes (Elsevier, New York, 1969), Chap. 4.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 8.

Yu, J.

J. Yu, R. Rudduck, IEEE Trans. Antennas Propag. AP-15, 662 (1967). The sine terms in their Eqs. (2a) should have for their arguments (θ + θ0)/2.

Ann. Phys. (1)

B. Sieger, Ann. Phys. 27, 626 (1908).

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

J. Yu, R. Rudduck, IEEE Trans. Antennas Propag. AP-15, 662 (1967). The sine terms in their Eqs. (2a) should have for their arguments (θ + θ0)/2.

J. Opt. Soc. Am. (1)

Phys. Rev. (1)

P. Morse, P. Rubenstein, Phys. Rev. 54, 895 (1938).

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), Chap. 8.

J. Bowman, T. Senior, P. Uslenghi, Electromagnetic and Acoustical Scattering by Simple Shapes (Elsevier, New York, 1969), Chap. 4.

I. Lazar, Ph.D. thesis, New York University (1962) (Univ. Microfilms, Ann Arbor, Mich., 63-7193).

Computation Laboratory of the National Applied Mathematics Laboratories, National Bureau of Standards, Tables Relating to Mathieu Functions (Columbia U.P., New York, 1951).

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Figures (12)

Fig. 1
Fig. 1

Geometry for strip diffraction. Plane radiation is incident upon the strip at glancing angle u and diffracted into angle θ.

Fig. 2
Fig. 2

Comparison of the irradiances predicted by exact, Kirchhoff, and first-order geometrical theories for Fraunhofer diffraction by a strip. The radiation is incident at a glancing angle u of 90°, and the strip width w equals 3.183λ. All curves are normalized so that the irradiance at pattern maximum is unity: ×, exact; ○, Kirchhoff; - - -, first-order GTD.

Fig. 3
Fig. 3

Single strip Fraunhofer diffraction with u = 44°, w = 2.847λ: ×, exact; ○, Kirchhoff; ——, higher-order GTD.

Fig. 4
Fig. 4

Single strip Fraunhofer diffraction with u = 44°, w = 2.251λ: ×, exact; ○, Kirchhoff; ——, higher-order GTD.

Fig. 5
Fig. 5

Single strip Fraunhofer diffraction with u = 44°, w = 1.424λ: ×, exact; ○, Kirchhoff; - - -, first-order GTD; ——, higher-order GTD.

Fig. 6
Fig. 6

Single strip Fraunhofer diffraction with u = 44°, w = 1.007λ: ×, exact; ○, Kirchhoff; - - -, first-order GTD; ——, higher-order GTD.

Fig. 7
Fig. 7

Single strip Fraunhofer diffraction with u = 34°, w = 2.847λ: ×, exact; ○, Kirchhoff; ——, higher-order GTD.

Fig. 8
Fig. 8

Single strip Fraunhofer diffraction with u = 24°, w = 2.847λ: ×, exact; ○, Kirchhoff; ——, higher-order GTD.

Fig. 9
Fig. 9

Single strip Fraunhofer diffraction with u = 14°, w = 2.847λ: ×, exact; ○, Kirchhoff; ——, higher-order GTD.

Fig. 10
Fig. 10

Single strip Fraunhofer diffraction with u = 4°, w = 2.847λ: ×, exact; ○, Kirchhoff; - - -, first-order GTD; ——, higher-order GTD.

Fig. 11
Fig. 11

Single strip Fraunhofer diffraction with u = 14°, w = 3.183λ: ×, exact; ○, Kirchhoff; ——, higher-order GTD.

Fig. 12
Fig. 12

Single strip Fraunhofer diffraction with u = 4°, w = 3.183λ: ×, exact; ○, Kirchhoff; ——, higher-order GTD.

Tables (2)

Tables Icon

Table I Relative Irradiance of the Principal Maximum as a Function of Glancing Angle for Strip Widths of 2.251λ and 2.847λ

Tables Icon

Table II Relative Irradiance of the Principal Maximum as a Function of Strip Width for a Glancing Angle of Incidence of 44°

Equations (5)

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I E = B ( r ) [ ( m Q o m 1 + f 0 , m 2 ) 2 + ( m Q o m f 0 , m 1 + f 0 , m 2 ) 2 ] ,
Q o m ( u , θ ) = S o m ( u ) S o m ( θ ) M m 0 ,
f 0 , m = N o m ( μ ) J o m ( μ ) | μ = 0 .
I KI = A ( k w ) 2 ( sin u + sin θ ) 2 sin 2 ( γ w / 2 ) / ( γ w / 2 ) 2 ,
I GTD = B { cos 2 ( γ w / 2 ) sin 2 [ ( u + θ ) / 2 ] + sin 2 ( γ w / 2 ) sin 2 [ ( u - θ ) / 2 ] } ,

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