Abstract

A model of discontinuity-free edge diffraction is proposed that is valid in the framework of the scalar Debye approximation and describes the formation process and approximate structure of the stationary diffracted field of a monochromatic converging spherical wave of limited angular opening throughout the whole space about the focus. The field is represented semianalytically in terms of the sum of a direct quasi-spherical wave and two edge quasi-conical waves of the zeroth and first order. The angular spectrum amplitudes of all these waves have smooth continuous variations of the real and imaginary parts in polar angle and radius, the separable nonanalytic functions defining the polar-angle variations of the amplitudes being found by optimization techniques.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed.(Cambridge Univ. Press, 1999), Sect. 8.
  2. T. Young, “On the theory of light and colours,” Philos. Trans. R. Soc. London 92, 12-48 (1802).
    [CrossRef]
  3. P. Debye, “Das verhalten von lichtwellen in der nahe eines brennpuktes oder iener brennlinie,” Ann. Phys. 30, 755-776 (1909).
    [CrossRef]
  4. W. H. Carter, “Band-limited angular-spectrum approximation to a spherical scalar wave field,” J. Opt. Soc. Am. 65, 1054-1058 (1975).
    [CrossRef]
  5. H. Weyl, “Ausbreitung elektromagnetischer wellen über einem ebenen leiter,” Ann. Phys. 60, 481-500 (1919).
    [CrossRef]
  6. G. A. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Matem. 16, 21-48 (1888).
    [CrossRef]
  7. A. Rubinowicz, “Die beugungswelle in der Kirchhoffschen theorie der beugungserscheinungen,” Ann. Phys. 53, 257-278 (1917).
    [CrossRef]
  8. A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47, 317-374 (1896).
    [CrossRef]
  9. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116-130 (1962).
    [CrossRef] [PubMed]
  10. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—Part I,” J. Opt. Soc. Am. 52, 615-625 (1962).
    [CrossRef]
  11. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—Part lI,” J. Opt. Soc. Am. 52, 626-637 (1962).
    [CrossRef]
  12. S. Ganci, “An experiment on the physical reality of edge-diffracted waves,” Am. J. Phys. 57, 370-373 (1989).
    [CrossRef]
  13. R. Kumar, “Structure of boundary diffraction wave revisited,” Appl. Phys. B 90, 379-382 (2008).
    [CrossRef]
  14. 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]
  15. Y. Z. Umul, “Alternative interpretation of the edge-diffraction phenomenon,” J. Opt. Soc. Am. A 25, 582-586 (2008).
    [CrossRef]
  16. G. C. Sherman and W. C. Chew, “Aperture and far-field distributions expressed by the Debye integral representation of focused fields,” J. Opt. Soc. Am. 72, 1076-1083 (1982).
    [CrossRef]
  17. J. J. Stamnes, “Waves, rays, and the method of stationary phase,” Opt. Express 10, 740-751 (2002).
    [PubMed]
  18. W. Wang, A. T. Friberg, and E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A 12, 1947-1953 (1995).
    [CrossRef]
  19. E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205-209 (1981).
    [CrossRef]
  20. A. Papoulis, Systems and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, 1968), Chap. 7.
  21. A. G. Sedukhin, “Marginal phase correction of truncated Bessel beams,” J. Opt. Soc. Am. A 17, 1059-1066 (2000).
    [CrossRef]
  22. A. G. Sedukhin, “Refinement of a discontinuity-free edge-diffraction model describing focused wave fields,” J. Opt. Soc. Am. A 26, 632-636 (2010).
    [CrossRef]
  23. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
    [CrossRef]

2010 (1)

A. G. Sedukhin, “Refinement of a discontinuity-free edge-diffraction model describing focused wave fields,” J. Opt. Soc. Am. A 26, 632-636 (2010).
[CrossRef]

2008 (2)

R. Kumar, “Structure of boundary diffraction wave revisited,” Appl. Phys. B 90, 379-382 (2008).
[CrossRef]

Y. Z. Umul, “Alternative interpretation of the edge-diffraction phenomenon,” J. Opt. Soc. Am. A 25, 582-586 (2008).
[CrossRef]

2002 (1)

2000 (2)

1995 (1)

1989 (1)

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

1982 (1)

1981 (1)

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205-209 (1981).
[CrossRef]

1975 (1)

1962 (3)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

1919 (1)

H. Weyl, “Ausbreitung elektromagnetischer wellen über einem ebenen leiter,” Ann. Phys. 60, 481-500 (1919).
[CrossRef]

1917 (1)

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

1909 (1)

P. Debye, “Das verhalten von lichtwellen in der nahe eines brennpuktes oder iener brennlinie,” Ann. Phys. 30, 755-776 (1909).
[CrossRef]

1896 (1)

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

1888 (1)

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

1802 (1)

T. Young, “On the theory of light and colours,” Philos. Trans. R. Soc. London 92, 12-48 (1802).
[CrossRef]

Anokhov, S. P.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed.(Cambridge Univ. Press, 1999), Sect. 8.

Carter, W. H.

Chew, W. C.

Debye, P.

P. Debye, “Das verhalten von lichtwellen in der nahe eines brennpuktes oder iener brennlinie,” Ann. Phys. 30, 755-776 (1909).
[CrossRef]

Friberg, A. T.

Ganci, S.

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

Keller, J. B.

Khizhnyak, A. I.

Kumar, R.

R. Kumar, “Structure of boundary diffraction wave revisited,” Appl. Phys. B 90, 379-382 (2008).
[CrossRef]

Li, Y.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205-209 (1981).
[CrossRef]

Lyramenko, R. A.

Maggi, G. A.

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

Miyamoto, K.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, 1968), Chap. 7.

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Rubinowicz, A.

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

Sedukhin, A. G.

A. G. Sedukhin, “Refinement of a discontinuity-free edge-diffraction model describing focused wave fields,” J. Opt. Soc. Am. A 26, 632-636 (2010).
[CrossRef]

A. G. Sedukhin, “Marginal phase correction of truncated Bessel beams,” J. Opt. Soc. Am. A 17, 1059-1066 (2000).
[CrossRef]

Sherman, G. C.

Sommerfeld, A.

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

Soskin, M. S.

Stamnes, J. J.

Umul, Y. Z.

Vasnetsov, M. V.

Wang, W.

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetischer wellen über einem ebenen leiter,” Ann. Phys. 60, 481-500 (1919).
[CrossRef]

Wolf, E.

W. Wang, A. T. Friberg, and E. Wolf, “Structure of focused fields in systems with large Fresnel numbers,” J. Opt. Soc. Am. A 12, 1947-1953 (1995).
[CrossRef]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205-209 (1981).
[CrossRef]

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—Part I,” J. Opt. Soc. Am. 52, 615-625 (1962).
[CrossRef]

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—Part lI,” J. Opt. Soc. Am. 52, 626-637 (1962).
[CrossRef]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed.(Cambridge Univ. Press, 1999), Sect. 8.

Young, T.

T. Young, “On the theory of light and colours,” Philos. Trans. R. Soc. London 92, 12-48 (1802).
[CrossRef]

Am. J. Phys. (1)

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

Ann. Matem. (1)

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

Ann. Phys. (3)

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

P. Debye, “Das verhalten von lichtwellen in der nahe eines brennpuktes oder iener brennlinie,” Ann. Phys. 30, 755-776 (1909).
[CrossRef]

H. Weyl, “Ausbreitung elektromagnetischer wellen über einem ebenen leiter,” Ann. Phys. 60, 481-500 (1919).
[CrossRef]

Appl. Phys. B (1)

R. Kumar, “Structure of boundary diffraction wave revisited,” Appl. Phys. B 90, 379-382 (2008).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (5)

Math. Ann. (1)

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

Opt. Commun. (1)

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205-209 (1981).
[CrossRef]

Opt. Express (1)

Philos. Trans. R. Soc. London (1)

T. Young, “On the theory of light and colours,” Philos. Trans. R. Soc. London 92, 12-48 (1802).
[CrossRef]

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

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed.(Cambridge Univ. Press, 1999), Sect. 8.

A. Papoulis, Systems and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, 1968), Chap. 7.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Illustrating the notation.

Fig. 2
Fig. 2

Model of discontinuity-free edge diffraction of a focused wave: (a) Huygens waves induced in the central part of the aperture wave front are gradually matched in phase in region II and then generate in region I a discontinuity-free quasi-spherical wave with near-spherical fronts shown by arcs; (b) edge Huygens waves generate in region I Bessel-like waves whose amplitudes are smoothly modulated in radius and polar angle, while the wave fronts retain near-conical shapes shown by oblique straight-line segments. The gradual transition of the intensity of the quasi-spherical wave from the illuminated light region to the dark one occurring near the geometrical shadow boundary is shown by shades of gray, with decrease of the radial factor 1 r 2 .

Fig. 3
Fig. 3

Variations of the real and imaginary parts of the optimum angular spectrum functions of a diffracted converging spherical wave for different half angular openings: (a) Re { g 0 ( ϑ ) } ; (b) Im { g 0 ( ϑ ) } ; (c) Re { g 1 ( ϑ ) } ; (d) Im { g 1 ( ϑ ) } ; (e) Re { g 1 ( ϑ ) } sin ϑ ; (f) Re { g 1 ( ϑ ) } tan Θ sin ϑ .

Fig. 4
Fig. 4

Normalized exact ( I D , solid curves) and approximate ( I D A , dashed curves) intensity distributions of a diffracted uniform converging spherical wave calculated in the framework of Debye’s model and plotted versus the polar radius at the half angular opening Θ equal to (a) π 6 , (b) π 4 , (c) π 3 . The details of these distributions in the close proximity of the focus are shown in insets.

Fig. 5
Fig. 5

Normalized exact ( I D ) and simplified approximate ( I D A ) intensity distributions of a diffracted uniform converging spherical wave along the geometrical shadow boundary in comparison with the intensity distributions ( I L S ) obtained from the Lommel–Struve theory for the cases when ϑ = Θ = π 12 , π 6 , π 4 , π 3 .

Fig. 6
Fig. 6

Reduced forms of the exact ( I D , solid curves) and simplified approximate ( I D A , dashed curves) intensity distributions of a diffracted uniform converging spherical wave obtained by dropping the factor of a common inverse proportion of the wave magnitudes to the radial distance and plotted versus the polar angle at a half angular opening Θ = π 4 and the relative radii of the (a) 1st, (b) 3rd, (c) 9th, (d) 27nd maximum of the axial intensity.

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

f ̃ 1 sin Θ ,
r f .
U D ( r ) = i Ω Ω A ( s x 2 + s y 2 ) exp ( i k n s r ) d Ω ,
U D ( r , ϑ ) = i f exp ( i k n f ) Ω ( Θ ) 0 2 π 0 Θ A ( ζ ) exp ( i k n d ) d sin ζ d ζ d ϕ ,
d = lim r f 0 ( x f x ) 2 + ( y f y ) 2 + ( z f z ) 2 = f + sgn ( z ) r ( sin ζ sin ϑ cos ϕ + cos ζ cos ϑ )
U D ( r , ϑ ) = i Ω ( Θ ) 0 2 π 0 Θ A ( ζ ) exp [ sgn ( z ) i k n r ( sin ζ sin ϑ cos ϕ + cos ζ cos ϑ ) ] sin ζ d ζ d ϕ .
U D ( r ̃ , ϑ ) = i 2 π Ω ( Θ ) 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 .
f ̃ 1 ( 1 cos Θ ) .
J 0 ( 2 π r ̃ sin ϑ sin ζ ) { exp [ i ( 2 π r ̃ sin ϑ sin ζ π 4 ) ] + exp [ i ( 2 π r ̃ sin ϑ sin ζ π 4 ) ] } / 2 π r ̃ sin ϑ sin ζ .
U DA ± ( r ̃ , ϑ ) = i Ω ( Θ ) r ̃ sin ϑ 0 Θ A ( ζ ) sin ζ exp [ i K μ ± ( ζ , ϑ ) ] d ζ ,
μ ± ( ζ , ϑ ) = sgn ( z ̃ ) cos [ ζ ± sgn ( z ̃ ) ϑ ] ± 1 ( 8 r ̃ ) .
U DA ( r ̃ , ϑ ) = U ( E ) ( r ̃ , ϑ ) + U ( G ) ( r ̃ , ϑ ) ,
U ( E ) ( r ̃ , ϑ ) = U + ( E ) ( r ̃ , ϑ ) + U ( E ) ( r ̃ , ϑ ) ,
U ± ( E ) ( r ̃ , ϑ ) = sgn ( z ̃ ) A ( Θ ) sin Θ 2 π Ω ( Θ ) r ̃ r ̃ sin ϑ sin [ Θ ± sgn ( z ̃ ) ϑ ] × exp { sgn ( z ̃ ) i 2 π r ̃ cos [ Θ ± sgn ( z ̃ ) ϑ ] ± i π 4 } ,
U ( G ) ( r ̃ , ϑ ) = sgn ( z ̃ ) A ( ϑ ) g ( G , S P ) ( ϑ ) exp [ sgn ( z ̃ ) i 2 π r ̃ ] Ω ( Θ ) r ̃ ,
g ( G , S P ) ( ϑ ) = { 1 , 0 ϑ < Θ 0 , Θ ϑ < π 2 }
U ( E ) ( r ̃ , ϑ ) = A ( Θ ) sin Θ exp [ sgn ( z ̃ ) i 2 π r ̃ cos ϑ cos Θ ] Ω ( Θ ) r ̃ ( cos 2 ϑ cos 2 Θ ) × [ sgn ( z ̃ ) cos ϑ sin Θ cos ( 2 π r ̃ sin ϑ sin Θ π 4 ) π r ̃ sin ϑ sin Θ + i sin ϑ cos Θ sin ( 2 π r ̃ sin ϑ sin Θ π 4 ) π r ̃ sin ϑ sin Θ ] .
U ( E ) ( r ̃ , ϑ ) = U 0 ( E ) ( r ̃ , ϑ ) + U 1 ( E ) ( r ̃ , ϑ ) ,
U 0 ( E ) ( r ̃ , ϑ ) = sgn ( z ̃ ) A ( Θ ) Ω ( Θ ) g 0 ( E , S P ) ( ϑ ) r ̃ J 0 ( 2 π r ̃ sin ϑ sin Θ ) exp [ sgn ( z ̃ ) i 2 π r ̃ cos ϑ cos Θ ] ,
U 1 ( E ) ( r ̃ , ϑ ) = i A ( Θ ) Ω ( Θ ) g 1 ( E , S P ) ( ϑ ) r ̃ J 1 ( 2 π r ̃ sin ϑ sin Θ ) exp [ sgn ( z ̃ ) i 2 π r ̃ cos ϑ cos Θ ] ,
g 0 ( E , S P ) ( ϑ ) = sin 2 Θ cos ϑ ( cos 2 ϑ cos 2 Θ ) ,
g 1 ( E , S P ) ( ϑ ) = sin Θ cos Θ sin ϑ ( cos 2 ϑ cos 2 Θ ) ,
min g ( ϑ n ) η [ g ( ϑ n ) ] = m | U DA [ g ( ϑ n ) , r ̃ m , ϑ n ] U D ( r ̃ m , ϑ n ) | 2 ,
| Re { g 0 ( ϑ ) } | ϑ Θ 1 and | Re { g 1 ( ϑ ) } | ϑ Θ sin ϑ tan Θ .
| Re { g 0 ( Θ ) } | Θ π 4 1 2 and | Re { g 1 ( Θ ) } | Θ π 4 tan 2 ( Θ 2 ) .
| Re { g 0 ( π 2 ) } | Θ π 4 0 and | Re { g 1 ( π 2 ) } | Θ π 4 2 tan ( Θ 2 ) .
U DA ( r ̃ , ϑ ) = sgn ( z ̃ ) g 0 ( ϑ ) Ω ( Θ ) r ̃ { A ( ϑ ) exp [ sgn ( z ̃ ) i 2 π r ̃ ] A ( Θ ) J 0 ( 2 π r ̃ sin ϑ sin Θ ) exp [ sgn ( z ̃ ) i 2 π r ̃ cos ϑ cos Θ ] } + i g 1 ( ϑ ) Ω ( Θ ) r ̃ A ( Θ ) J 1 ( 2 π r ̃ sin ϑ sin Θ ) exp [ sgn ( z ̃ ) i 2 π r ̃ cos ϑ cos Θ ] .
U DA ( r ̃ , ϑ ) = A ( r ̃ , ϑ ) exp { i sgn ( z ̃ ) arctan [ Y ( r ̃ , ϑ ) X ( r ̃ , ϑ ) ] } ,
A ( r ̃ , ϑ ) = 1 Ω ( Θ ) r ̃ [ { g 0 ( ϑ ) A ( ϑ ) cos [ 2 π r ̃ ( 1 cos ϑ cos Θ ) ] g 0 ( ϑ ) A ( Θ ) J 0 ( 2 π r ̃ sin ϑ sin Θ ) } 2 + { g 0 ( ϑ ) A ( ϑ ) sin [ 2 π r ̃ ( 1 cos ϑ cos Θ ) ] g 1 ( ϑ ) A ( Θ ) J 1 ( 2 π r ̃ sin ϑ sin Θ ) } 2 ] 1 2 ,
X ( r ̃ , ϑ ) = g 0 ( ϑ ) A ( ϑ ) cos ( 2 π r ̃ ) g 0 ( ϑ ) A ( Θ ) J 0 ( 2 π r ̃ sin ϑ sin Θ ) cos ( 2 π r ̃ cos ϑ cos Θ ) + g 1 ( ϑ ) A ( Θ ) J 1 ( 2 π r ̃ sin ϑ sin Θ ) sin ( 2 π r ̃ cos ϑ cos Θ ) ,
Y ( r ̃ , ϑ ) = g 0 ( ϑ ) A ( ϑ ) sin ( 2 π r ̃ ) g 0 ( ϑ ) A ( Θ ) J 0 ( 2 π r ̃ sin ϑ sin Θ ) sin ( 2 π r ̃ cos ϑ cos Θ ) g 1 ( ϑ ) A ( Θ ) J 1 ( 2 π r ̃ sin ϑ sin Θ ) cos ( 2 π r ̃ cos ϑ cos Θ ) .
I DA ( Z ) = | U DA ( r ̃ = z ̃ , ϑ = 0 ) | 2 = ( sin Z Z ) 2 ,
I DA ( R ) = | | U DA ( r ̃ = ρ ̃ , ϑ = π 2 ) | 2 | Θ π 4 [ 2 J 1 ( R ) R ] 2 ,
I DA ( V ) | Θ π 4 = = | | U DA ( r ̃ , Θ ) | 2 | Θ π 4 cos 4 ( Θ 2 )
× [ cos V J 0 ( V ) ] 2 + [ sin V + 2 tan 2 ( Θ 2 ) J 1 ( V ) ] 2 V 2 ,
| I L S ( V ) | sin Θ Θ 1 2 J 0 ( V ) cos V + J 0 2 ( V ) V 2 .
| I DA ( r ̃ , ϑ ) | ϑ Θ | Ω ( Θ ) r ̃ U DA ( r ̃ , ϑ ) | 2 = { cos [ 2 π r ̃ ( 1 cos ϑ cos Θ ) ] J 0 ( 2 π r ̃ sin ϑ sin Θ ) } 2 + { sin [ 2 π r ̃ ( 1 cos ϑ cos Θ ) ] ( sin ϑ tan Θ ) J 1 ( 2 π r ̃ sin ϑ sin Θ ) } 2 .
r ̃ n = ( n + 1 2 ) ( 1 cos Θ ) .

Metrics