Abstract

The Huygens–Fresnel wave principle is found to extend far beyond its traditional use. Using only elementary mathematical means, it is shown in particular that this principle suffices to provide the exact solution of Sommerfeld’s problem of plane wave diffraction by a perfectly conducting half-plane. A more exact formulation of the wave principle itself is proposed.

© 2007 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1991).
  2. A. Sommerfeld, Optics, Vol. IV of Lectures on Theoretical Physics (Academic, 1954).
  3. F. Gori, Opt. Commun. 48, 67 (1983).
    [CrossRef]
  4. F. Gori, Atti Fond. Giorgio Ronchi 38, 593 (1983).
  5. S. Anokhov, "Plane wave diffraction by perfectly transparent half-plane," J. Opt. Soc. Am. A doc. ID 79704 (posted 11 April 2007, in press).
  6. S. A. Schelkunoff, Electromagnetic Waves, 3rd ed. (Van Nostrand, 1943).
  7. G. T. Markov and D. M. Sazonov, Antennas (Energiya, 1975) (in Russian).
  8. V. V. Nikolskiy and T. I. Nikolskaya, Electrodynamics and Radio Wave Propagation, 3rd ed. (Science Publishing, 1989) (in Russian).
  9. Yu. V. Pimenov, V. I. Volman, and A. D. Muravcov, Technical Electrodynamics (Radio i Svyaz, 2002) (in Russian).
  10. A. S. Marathay and J. F. McCalmont, J. Opt. Soc. Am. A 18, 2585 (2001).
    [CrossRef]
  11. A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, J. Opt. Soc. Am. A 17, 2199 (2000).
    [CrossRef]
  12. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle, 2nd ed. (Clarendon, 1950).

2001 (1)

2000 (1)

1983 (2)

F. Gori, Opt. Commun. 48, 67 (1983).
[CrossRef]

F. Gori, Atti Fond. Giorgio Ronchi 38, 593 (1983).

Atti Fond. Giorgio Ronchi (1)

F. Gori, Atti Fond. Giorgio Ronchi 38, 593 (1983).

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

F. Gori, Opt. Commun. 48, 67 (1983).
[CrossRef]

Other (8)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1991).

A. Sommerfeld, Optics, Vol. IV of Lectures on Theoretical Physics (Academic, 1954).

S. Anokhov, "Plane wave diffraction by perfectly transparent half-plane," J. Opt. Soc. Am. A doc. ID 79704 (posted 11 April 2007, in press).

S. A. Schelkunoff, Electromagnetic Waves, 3rd ed. (Van Nostrand, 1943).

G. T. Markov and D. M. Sazonov, Antennas (Energiya, 1975) (in Russian).

V. V. Nikolskiy and T. I. Nikolskaya, Electrodynamics and Radio Wave Propagation, 3rd ed. (Science Publishing, 1989) (in Russian).

Yu. V. Pimenov, V. I. Volman, and A. D. Muravcov, Technical Electrodynamics (Radio i Svyaz, 2002) (in Russian).

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle, 2nd ed. (Clarendon, 1950).

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Figures (3)

Fig. 1
Fig. 1

Scheme of the plane wave diffraction by a half-plane ( x 0 ) .

Fig. 2
Fig. 2

Directivity diagram of the elementary Huygens source.

Fig. 3
Fig. 3

Variable polar coordinates ρ and β.

Equations (7)

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Е Hu ( r , δ ) = exp ( i k r ) ( 1 + sin δ 2 k r ) 1 2 ,
E Σ forw ( P ) = C 0 exp [ i k ρ ( x ) ] [ 1 + sin β ( x ) 2 k ρ ( x ) ] 1 2 d x ,
E Σ forw ( P ) = A o k ( 1 i ) 2 2 π r exp ( i k ρ ) k ( ρ z j ) d ρ .
E Σ forw ( P ) = A o ( 1 i ) 2 π exp ( i k z j ) U exp ( i μ 2 ) d μ ,
E Σ forw ( P ) = A o exp ( i k z j ) [ 1 2 + 1 i 2 π 0 U exp ( i μ 2 ) d μ ] .
E Σ forw ( P ) = A o exp [ i k R cos ( θ α ) ] [ 1 2 + 1 i 2 π 0 U exp ( i μ 2 ) d μ ] .
E Σ ref ( P ) = A o exp [ i k R cos ( θ + α ) ] [ 1 2 + 1 i 2 π 0 V exp ( i μ 2 ) d μ ] ,

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