Abstract

Two equivalent forms of a refined discontinuity-free edge-diffraction model describing the structure of a stationary focused wave field are presented that are valid in the framework of the scalar Debye integral representation for a diffracted rotationally symmetric converging spherical wave of a limited yet not-too-low angular opening. The first form describes the field as the sum of a direct quasi-spherical wave and a plurality of edge quasi-conical waves of different orders, the optimum discontinuity-free angular spectrum functions of all the waves being dependent on the polar angle only. According to the second form, the focused field is fully characterized by only three components—the same quasi-spherical wave and two edge quasi-conical waves of the zero and first order, of which the optimum discontinuity-free angular spectrum functions are dependent on both the polar angle and the polar radius counted from the geometrical focus.

© 2010 Optical Society of America

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  1. G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Math. 16, 21-48 (1888).
  2. A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der beugungserscheinungen,” Ann. Phys. 53, 257-278 (1917).
    [CrossRef]
  3. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116-130 (1962).
    [CrossRef] [PubMed]
  4. A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47, 317-374 (1896).
    [CrossRef]
  5. P. Ya. Ufimtsev, Method of Edge Waves in the Physical Theory of Diffraction (English translation, U. S. Air Force Foreign Technology Division, Wright-Patterson AFB, Ohio, Sept. 7, 1971).
  6. A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, “Manifestation of a hidden dislocation wave originated in a plane wave diffraction on a half-plane screen,” Proc. SPIE 3904, 19-26 (1999).
    [CrossRef]
  7. A. I. Khizhnyak, S. P. Anokhov, R. A. Lyramenko, 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]
  8. A. G. Sedukhin, “Discontinuity-free edge-diffraction model for characterization of focused wave fields,” J. Opt. Soc. Am. A 27, 622-631 (2010).
    [CrossRef]
  9. P. V. Polyanskii and G. V. Bogatiryova, “EDW - Edge diffraction wave, edge dislocation wave or whether tertio est datur?” Proc. SPIE 4607, 109-124 (2002).
    [CrossRef]
  10. Y. Z. Umul, “Alternative interpretation of the edge-diffraction phenomenon,” J. Opt. Soc. Am. A 25, 582-587 (2008).
    [CrossRef]
  11. S. Ganci, “An experiment on the physical reality of edge-diffracted waves,” Am. J. Phys. 57, 370-373 (1989).
    [CrossRef]
  12. S. Ganci, “Analytical and experimental remarks on the Sommerfeld's half plane solution,” Optik (in press, doi:10.1016/j.ijleo.2008.11.010).
  13. R. Kumar, “Structure of the boundary diffraction wave revisited,” Appl. Phys. B 90, 379-382 (2009).
    [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics, 7th ed., (Cambridge Univ. Press, 1999), Chap. VIII.
  15. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), Chap. 3.
  16. M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (Dover, 1972), Chap. 9.
  17. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., (Cambridge Univ. Press, 1944), Chap. IX.

2010

2009

R. Kumar, “Structure of the boundary diffraction wave revisited,” Appl. Phys. B 90, 379-382 (2009).
[CrossRef]

2008

2002

P. V. Polyanskii and G. V. Bogatiryova, “EDW - Edge diffraction wave, edge dislocation wave or whether tertio est datur?” Proc. SPIE 4607, 109-124 (2002).
[CrossRef]

2000

1999

A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, “Manifestation of a hidden dislocation wave originated in a plane wave diffraction on a half-plane screen,” Proc. SPIE 3904, 19-26 (1999).
[CrossRef]

1989

S. Ganci, “An experiment on the physical reality of edge-diffracted waves,” Am. J. Phys. 57, 370-373 (1989).
[CrossRef]

1962

1917

A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der beugungserscheinungen,” Ann. Phys. 53, 257-278 (1917).
[CrossRef]

1896

A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47, 317-374 (1896).
[CrossRef]

1888

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Math. 16, 21-48 (1888).

Anokhov, S. P.

A. I. Khizhnyak, S. P. Anokhov, R. A. Lyramenko, 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]

A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, “Manifestation of a hidden dislocation wave originated in a plane wave diffraction on a half-plane screen,” Proc. SPIE 3904, 19-26 (1999).
[CrossRef]

Bogatiryova, G. V.

P. V. Polyanskii and G. V. Bogatiryova, “EDW - Edge diffraction wave, edge dislocation wave or whether tertio est datur?” Proc. SPIE 4607, 109-124 (2002).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed., (Cambridge Univ. Press, 1999), Chap. VIII.

Ganci, S.

S. Ganci, “An experiment on the physical reality of edge-diffracted waves,” Am. J. Phys. 57, 370-373 (1989).
[CrossRef]

S. Ganci, “Analytical and experimental remarks on the Sommerfeld's half plane solution,” Optik (in press, doi:10.1016/j.ijleo.2008.11.010).

Keller, J. B.

Khizhnyak, A. I.

A. I. Khizhnyak, S. P. Anokhov, R. A. Lyramenko, 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]

A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, “Manifestation of a hidden dislocation wave originated in a plane wave diffraction on a half-plane screen,” Proc. SPIE 3904, 19-26 (1999).
[CrossRef]

Kumar, R.

R. Kumar, “Structure of the boundary diffraction wave revisited,” Appl. Phys. B 90, 379-382 (2009).
[CrossRef]

Lymarenko, R. A.

A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, “Manifestation of a hidden dislocation wave originated in a plane wave diffraction on a half-plane screen,” Proc. SPIE 3904, 19-26 (1999).
[CrossRef]

Lyramenko, R. A.

Maggi, G. A.

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Math. 16, 21-48 (1888).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), Chap. 3.

Polyanskii, P. V.

P. V. Polyanskii and G. V. Bogatiryova, “EDW - Edge diffraction wave, edge dislocation wave or whether tertio est datur?” Proc. SPIE 4607, 109-124 (2002).
[CrossRef]

Rubinowicz, A.

A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der beugungserscheinungen,” Ann. Phys. 53, 257-278 (1917).
[CrossRef]

Sedukhin, A. G.

Sommerfeld, A.

A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47, 317-374 (1896).
[CrossRef]

Soskin, M. S.

A. I. Khizhnyak, S. P. Anokhov, R. A. Lyramenko, 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]

A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, “Manifestation of a hidden dislocation wave originated in a plane wave diffraction on a half-plane screen,” Proc. SPIE 3904, 19-26 (1999).
[CrossRef]

Ufimtsev, P. Ya.

P. Ya. Ufimtsev, Method of Edge Waves in the Physical Theory of Diffraction (English translation, U. S. Air Force Foreign Technology Division, Wright-Patterson AFB, Ohio, Sept. 7, 1971).

Umul, Y. Z.

Vasnetsov, M. V.

A. I. Khizhnyak, S. P. Anokhov, R. A. Lyramenko, 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]

A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, “Manifestation of a hidden dislocation wave originated in a plane wave diffraction on a half-plane screen,” Proc. SPIE 3904, 19-26 (1999).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., (Cambridge Univ. Press, 1944), Chap. IX.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed., (Cambridge Univ. Press, 1999), Chap. VIII.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), Chap. 3.

Am. J. Phys.

S. Ganci, “An experiment on the physical reality of edge-diffracted waves,” Am. J. Phys. 57, 370-373 (1989).
[CrossRef]

Ann. Math.

G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Math. 16, 21-48 (1888).

Ann. Phys.

A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der beugungserscheinungen,” Ann. Phys. 53, 257-278 (1917).
[CrossRef]

Appl. Phys. B

R. Kumar, “Structure of the boundary diffraction wave revisited,” Appl. Phys. B 90, 379-382 (2009).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Math. Ann.

A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47, 317-374 (1896).
[CrossRef]

Proc. SPIE

P. V. Polyanskii and G. V. Bogatiryova, “EDW - Edge diffraction wave, edge dislocation wave or whether tertio est datur?” Proc. SPIE 4607, 109-124 (2002).
[CrossRef]

A. I. Khizhnyak, S. P. Anokhov, R. A. Lymarenko, M. S. Soskin, and M. V. Vasnetsov, “Manifestation of a hidden dislocation wave originated in a plane wave diffraction on a half-plane screen,” Proc. SPIE 3904, 19-26 (1999).
[CrossRef]

Other

S. Ganci, “Analytical and experimental remarks on the Sommerfeld's half plane solution,” Optik (in press, doi:10.1016/j.ijleo.2008.11.010).

P. Ya. Ufimtsev, Method of Edge Waves in the Physical Theory of Diffraction (English translation, U. S. Air Force Foreign Technology Division, Wright-Patterson AFB, Ohio, Sept. 7, 1971).

M. Born and E. Wolf, Principles of Optics, 7th ed., (Cambridge Univ. Press, 1999), Chap. VIII.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995), Chap. 3.

M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (Dover, 1972), Chap. 9.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., (Cambridge Univ. Press, 1944), Chap. IX.

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Figures (1)

Fig. 1
Fig. 1

Polar-angle variations of the real and imaginary parts of the angular spectrum functions for different components of a converging spherical wave with a half angular opening Θ = π 4 : (a) the optimum spectrum functions Re { g 0 , 5 ( ϑ ) } and Im { g 0 , 5 ( ϑ ) } of a direct quasi-spherical wave as compared with the spectrum function g ( G , SP ) ( ϑ ) of a geometrical stationary phase wave; (b) the optimum composite functions Re { g 0 , 5 ( C ) ( r ̃ , ϑ ) } and Im { g 0 , 5 ( C ) ( r ̃ , ϑ ) } of an edge quasi-conical wave of the zeroth order as compared with the spectrum function g 0 ( E , SP ) ( ϑ ) of an edge stationary phase wave of the zeroth order; (c) the optimum composite functions Re { g 1 , 5 ( C ) ( r ̃ , ϑ ) } and Im { g 1 , 5 ( C ) ( r ̃ , ϑ ) } of an edge quasi-conical wave of the first order as compared with the spectrum function g 1 ( E , SP ) ( ϑ ) of an edge stationary phase wave of the first order. The families of the optimum composite functions are plotted for the relative radii of the 1st, 2nd, 4th, and 16th minimum of the axial intensity.

Equations (15)

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U D ( x , y , z ) = U ( HS ) ( x , y , z ) U ( C ) ( x , y , z ) ,
U D ( x , y , z ) = i k 2 π 0 s x 2 + s y 2 sin 2 Θ exp ( s x x + s y x + s z z ) s z d s x d s y ,
U ( HS ) ( x , y , z ) = sgn ( z ) exp [ sgn ( z ) i k r ] r ,
U ( C ) ( x , y , z ) = i k 2 π sin 2 Θ < s x 2 + s y 2 < exp ( s x x + s y x + s z z ) s z d s x d s y ,
s z = { 1 s x 2 s y 2 , s x 2 + s y 2 1 sgn ( z ) s x 2 + s y 2 1 , s x 2 + s y 2 > 1 } ,
U D ( r ̃ , ϑ ) = i 2 π Ω 0 Θ A ( ζ ) J 0 ( 2 π r ̃ sin ϑ sin ζ ) exp [ sgn ( z ̃ ) i 2 π r ̃ cos ϑ cos ζ ] sin ζ d ζ .
U DA ( r ̃ , ϑ ) = U ( D ) ( r ̃ , ϑ ) + U ( E ) ( r ̃ , ϑ ) ,
U ( D ) ( r ̃ , ϑ ) = sgn ( z ̃ ) A ( ϑ ) Ω ( Θ ) g 0 , P ( ϑ ) exp [ sgn ( z ̃ ) i 2 π r ̃ ] r ̃ ,
U ( E ) ( r ̃ , ϑ ) = A ( Θ ) Ω ( Θ ) p = 0 P i p g p , P ( ϑ ) J p ( 2 π r ̃ sin ϑ sin Θ ) exp ( i 2 π r ̃ cos ϑ cos Θ ) r ̃ ,
J p + ν ( z ) = h p , ν ( z ) J ν ( z ) h p 1 , ν + 1 ( z ) J ν 1 ( z ) ,
h p , ν ( z ) = n = 0 Int { p 2 } ( 1 ) n ( p n ) ! Γ ( ν + p n ) n ! ( p 2 n ) ! Γ ( ν + n ) ( 2 z ) p 2 n ,
h p + 1 , ν ( z ) = 2 ( p + ν ) z h p , ν ( z ) h p 1 , ν ( z ) , h 1 , ν ( z ) = 0 , h 0 , ν ( z ) = 1.
U ( E ) ( r ̃ , ϑ ) = A ( Θ ) Ω ( Θ ) [ g 0 , P ( C ) ( r ̃ , ϑ ) J 0 ( 2 π r ̃ sin ϑ sin Θ ) + g 1 , P ( C ) ( r ̃ , ϑ ) J 1 ( 2 π r ̃ sin ϑ sin Θ ) ] r ̃ exp ( i 2 π r ̃ cos ϑ cos Θ ) ,
g 0 , P ( C ) ( r ̃ , ϑ ) = { g 0 , P ( ϑ ) , P = 1 g 0 , P ( ϑ ) p = 2 P i p g p , P ( ϑ ) h p 2 , 2 ( 2 π r ̃ sin ϑ sin Θ ) , P 2 }
g 1 , P ( C ) ( r ̃ , ϑ ) = p = 1 P i p g p , P ( ϑ ) h p 1 , 1 ( 2 π r ̃ sin ϑ sin Θ )

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