Abstract

The exact solution is found for plane wave diffraction by an arbitrary phase step. The analysis is performed by using the Huygens–Fresnel principle and the superposition integral, where every secondary wave was identified with the surface element field of the actual electromagnetic wave. The dependence of the total field structure on the height of the phase step is analyzed. The formation algorithm is demonstrated for the primary wave component of the edge diffraction, which has a singular nature and determines nearly all physical properties of this phenomenon.

© 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, "Mathematische Theorie der Diffraction," Math. Ann. 47, 317-374 (1896).
    [CrossRef]
  3. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  4. S. A. Schelkunoff, Electromagnetic Waves, 3rd ed. (D. Van Nostrand, 1943).
  5. V. V. Nikolskiy and T. I. Nikolskaya, Electrodynamics and Radio Wave Propagation, 3rd ed. (Science Publishing, 1989) (in Russian).
  6. A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, "Structure of an edge-dislocation wave originating in plane-wave diffraction by a half-plane," J. Opt. Soc. Am. A 17, 2199-2207 (2000).
    [CrossRef]
  7. S. Anokhov, "Physical approach to analytic simulation of Fresnel integrals," J. Opt. Soc. Am. A 24, 197-203 (2007).
    [CrossRef]
  8. J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
    [CrossRef]
  9. P. Ya. Ufimtsev, "Elementary edge waves and the physical theory of diffraction," Electromagnetics 11, 125-160 (1991).
    [CrossRef]
  10. L. B. Felsen, "Evanescent waves," J. Opt. Soc. Am. 66, 751-760 (1976).
    [CrossRef]
  11. R.C.Hansen, ed., Microwave Scanning Antennas (Academic, 1964).
  12. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, "Optical wavefront dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
    [CrossRef]

2007 (1)

2000 (1)

1995 (1)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, "Optical wavefront dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
[CrossRef]

1991 (1)

P. Ya. Ufimtsev, "Elementary edge waves and the physical theory of diffraction," Electromagnetics 11, 125-160 (1991).
[CrossRef]

1976 (1)

1974 (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

1896 (1)

A. Sommerfeld, "Mathematische Theorie der Diffraction," Math. Ann. 47, 317-374 (1896).
[CrossRef]

Anokhov, S.

Anokhov, S. P.

Basistiy, I. V.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, "Optical wavefront dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Born, M.

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

Felsen, L. B.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Khizhnyak, A. I.

Lymarenko, R. A.

Nikolskaya, T. I.

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

Nikolskiy, V. V.

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

Nye, J. F.

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Schelkunoff, S. A.

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

Sommerfeld, A.

A. Sommerfeld, "Mathematische Theorie der Diffraction," Math. Ann. 47, 317-374 (1896).
[CrossRef]

Soskin, M. S.

Ufimtsev, P. Ya.

P. Ya. Ufimtsev, "Elementary edge waves and the physical theory of diffraction," Electromagnetics 11, 125-160 (1991).
[CrossRef]

Vasnetsov, M. V.

Wolf, E.

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

Electromagnetics (1)

P. Ya. Ufimtsev, "Elementary edge waves and the physical theory of diffraction," Electromagnetics 11, 125-160 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Math. Ann. (1)

A. Sommerfeld, "Mathematische Theorie der Diffraction," Math. Ann. 47, 317-374 (1896).
[CrossRef]

Opt. Commun. (1)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, "Optical wavefront dislocations and their properties," Opt. Commun. 119, 604-612 (1995).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336, 165-190 (1974).
[CrossRef]

Other (5)

R.C.Hansen, ed., Microwave Scanning Antennas (Academic, 1964).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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

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

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

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Figures (9)

Fig. 1
Fig. 1

Scheme of plane wave diffraction by a perfectly transparent 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.

Fig. 4
Fig. 4

Three families of elementary sources.

Fig. 5
Fig. 5

Two variants of oblique incidence of a plane wave. Open symbols represent the directivity diagrams of elementary waves without the phase shift. Filled symbols are the same with a π phase shift.

Fig. 6
Fig. 6

Relative (a) amplitude and (b) phase of the d-wave in the vicinity of the edge dislocation. (c), (d) Same characteristics for another means of notion.

Fig. 7
Fig. 7

Topological structure of the amplitude distribution of the d-wave near the edge of the transparency at a different incidence for plane wave geometry.

Fig. 8
Fig. 8

Field distribution of the backward wave near the edge of the transparency under normal incidence of the plane wave. (a) Spatial shape of an amplitude in a 3 λ × 3 λ area. (b) Field profile in the sections of z = 0 (solid curve) and x = 0 (dotted curve).

Fig. 9
Fig. 9

Relative amplitude of the total field in the region z < 0 at a phase step height of π 2 .

Equations (19)

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E pl ( t , z ) = A o exp ( i ω t + i k z ) ,
E el ( r , δ ) = exp ( i k r ) ( 1 + sin δ 2 k r ) 1 2 ,
E Σ ( x j , z j ) = C { 0 exp ( i [ k ρ ( x ) + φ ] ) ( 1 + sin β ( x ) 2 k ρ ( x ) ) 1 2 d x + 0 + exp [ i k ρ ( x ) ] ( 1 + sin β ( x ) 2 k ρ ( x ) ) 1 2 d x } ,
E Σ ( x j , 0 ) = C + exp ( i k ( ξ x j ) ) k ( ξ x j ) d ξ .
E Σ ( x j , 0 ) = 2 C k + exp ( i η 2 ) d η = 2 C π k exp ( i π 4 ) .
C = A o k 2 π exp ( i π 4 ) .
E A + B ( P ) = [ exp ( i φ ) + 1 ] exp ( i k ρ B ) ( 1 + sin β B 2 k ρ B ) 1 2 .
E Σ ( P ) = A o k 2 π exp ( i π 4 ) { [ 1 + exp ( i φ ) ] 0 exp [ i k ρ ( x ) ] ( 1 + sin β ( x ) 2 k ρ ( x ) ) 1 2 d x + 2 x j 0 exp [ i k ρ ( x ) + i φ ] ( 1 + sin β ( x ) 2 k ρ ( x ) ) 1 2 d x } ,
E Σ ( P ) = A o k 2 π exp ( i π 4 ) { [ 1 + exp ( i φ ) ] r exp ( i k ρ ) k ( ρ z j ) d ρ + 2 z j r exp [ i k ρ + i φ ] k ( ρ z j ) d ρ } .
E Σ ( P ) = A o π exp ( i k z j i π 4 ) { [ 1 + exp ( i ϕ ) ] 0 exp ( i μ 2 ) d μ + [ 1 exp ( i φ ) ] 0 U exp ( i μ 2 ) d μ } ,
U = ± k ( x j 2 + z j 2 z j ) .
E Σ ( P ) = A o exp ( i k z j ) { 1 + exp ( i φ ) 2 + [ 1 exp ( i φ ) ] 1 i 2 π 0 U exp ( i μ 2 ) d μ } .
E pl ( R , θ ) = A o exp [ i k R cos ( θ α ) ] ,
E Σ ( P ) = A o exp [ i k R cos ( θ α ) ] { 1 + exp ( i φ ) 2 + [ 1 exp ( i φ ) ] 1 i 2 π 0 U exp ( i μ 2 ) d μ } ,
U = 2 k R sin [ ( θ α ) 2 ]
E d ( P ) = 2 A o exp [ i k R cos ( θ α ) ] 1 i 2 π 0 U exp ( i μ 2 ) d μ .
E Som ( P ) = A o exp [ i k R cos ( θ α ) ] ( 1 2 + 1 i 2 π 0 U exp ( i μ 2 ) d μ ) ,
0 U exp ( i μ 2 ) d μ = 0 U cos ( μ 2 ) d μ + i 0 U sin ( i μ 2 ) d μ = C ( U ) + i S ( U ) ,
E back ( P ) = A o exp [ i k R cos ( θ α ) ] { 2 ( 1 i ) 2 π 0 U exp ( i μ 2 ) d μ 1 } .

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