Abstract

Unconventional imaging and focusing properties are obtained when odd powers of the radial coordinate are present in a radial gradient-index profile. We calculate the transmittance function for this kind of medium under a paraxial approximation by a quasi-geometrical approach. Likewise, we analyze the pupil effect; that is, we evaluate the intensity distribution diffracted by a gradient-index axicon at the image plane.

© 1994 Optical Society of America

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References

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  1. E. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps.3 and 4.
  2. Ch. Frére, O. Bryngdahl, “Computer-generated holograms: reconstruction of curves in 3-D,” Opt. Commun. 60, 369–372 (1986).
    [CrossRef]
  3. J. Linares, C. Gómez-Reino, “Focal curves generated by one-dimensional GRIN tapers,” Opt. Lett. 15, 1258–1260 (1990).
    [CrossRef]
  4. J. Linares, R. M. González, C. Gómez-Reino, “Three-dimensional focal curves generated by one-dimensional GRIN tapers,” in Gradient-Index Optical Systems, Vol. 9 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 130–133.
  5. R. M. González, J. Linares, C. Gómez-Reino, “Three-dimensional focal curves generated by one-dimensional gradient-index tapers,” Appl. Opt. 31, 5171–5177 (1992).
    [CrossRef] [PubMed]
  6. L. M. Soroko, “Axicon and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, pp. 109–160.
    [CrossRef]
  7. Ch. Frére, D. Leseberg, O. Bryngdahl, “Computer generated holograms of three-dimensional objects composed of line segments,” J. Opt. Soc. Am. A 3, 726–730 (1986).
    [CrossRef]
  8. R. Blankebecler, J. J. Hagerty, G. E. Rindone, M. A. Wickson, “Prescribed profile gradient-index glasses of macrodimensions and large AN,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1354, 130–133 (1990).
  9. M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Apps. A and B.
  10. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 2.
  11. C. Gómez-Reino, E. Larrea, M. V. Pérez, “Complex amplitude propagation through a quadratic index medium: characteristic point function,” Appl. Opt. 22, 2927–2929 (1983).
    [CrossRef] [PubMed]
  12. C. Gómez-Reino, J. Linares, “Optical path integrals in gradient-index media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), Chaps. 3 and 4.
  14. M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).s
    [CrossRef]
  15. W. L. Emkey, C. A. Jack, “Analysis and evaluation of graded-index fiber lenses,” J. Lightwave Technol. LT-5, 1156–1164 (1987).
    [CrossRef]
  16. J. Linares, C. Gómez-Reino, “Diffraction-limited efficiency between SMF connected by GRIN lens,” J. Mod. Opt. 38, 597–604 (1991).
    [CrossRef]
  17. E. Marchand, “Axicon gradient lenses,” Appl. Opt. 29, 4001–4002 (1990).
    [CrossRef] [PubMed]
  18. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Chap. 4.
  19. C. Gómez-Reino, E. Acosta, M. V. Pérez, “Limitation in the cone of light through GRIN lenses: stops, pupils and vignetting,” Jpn. J. Appl. Phys. 31, 1582–1585 (1992).
    [CrossRef]
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 3 and 4.
  21. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 355–389.
  22. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 5.

1992 (2)

C. Gómez-Reino, E. Acosta, M. V. Pérez, “Limitation in the cone of light through GRIN lenses: stops, pupils and vignetting,” Jpn. J. Appl. Phys. 31, 1582–1585 (1992).
[CrossRef]

R. M. González, J. Linares, C. Gómez-Reino, “Three-dimensional focal curves generated by one-dimensional gradient-index tapers,” Appl. Opt. 31, 5171–5177 (1992).
[CrossRef] [PubMed]

1991 (1)

J. Linares, C. Gómez-Reino, “Diffraction-limited efficiency between SMF connected by GRIN lens,” J. Mod. Opt. 38, 597–604 (1991).
[CrossRef]

1990 (2)

1987 (2)

C. Gómez-Reino, J. Linares, “Optical path integrals in gradient-index media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
[CrossRef]

W. L. Emkey, C. A. Jack, “Analysis and evaluation of graded-index fiber lenses,” J. Lightwave Technol. LT-5, 1156–1164 (1987).
[CrossRef]

1986 (3)

Ch. Frére, O. Bryngdahl, “Computer-generated holograms: reconstruction of curves in 3-D,” Opt. Commun. 60, 369–372 (1986).
[CrossRef]

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).s
[CrossRef]

Ch. Frére, D. Leseberg, O. Bryngdahl, “Computer generated holograms of three-dimensional objects composed of line segments,” J. Opt. Soc. Am. A 3, 726–730 (1986).
[CrossRef]

1983 (1)

Acosta, E.

C. Gómez-Reino, E. Acosta, M. V. Pérez, “Limitation in the cone of light through GRIN lenses: stops, pupils and vignetting,” Jpn. J. Appl. Phys. 31, 1582–1585 (1992).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 2.

Blankebecler, R.

R. Blankebecler, J. J. Hagerty, G. E. Rindone, M. A. Wickson, “Prescribed profile gradient-index glasses of macrodimensions and large AN,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1354, 130–133 (1990).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), Chaps. 3 and 4.

Bryngdahl, O.

Ch. Frére, O. Bryngdahl, “Computer-generated holograms: reconstruction of curves in 3-D,” Opt. Commun. 60, 369–372 (1986).
[CrossRef]

Ch. Frére, D. Leseberg, O. Bryngdahl, “Computer generated holograms of three-dimensional objects composed of line segments,” J. Opt. Soc. Am. A 3, 726–730 (1986).
[CrossRef]

Cuadrado, J. M.

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).s
[CrossRef]

Emkey, W. L.

W. L. Emkey, C. A. Jack, “Analysis and evaluation of graded-index fiber lenses,” J. Lightwave Technol. LT-5, 1156–1164 (1987).
[CrossRef]

Frére, Ch.

Ch. Frére, D. Leseberg, O. Bryngdahl, “Computer generated holograms of three-dimensional objects composed of line segments,” J. Opt. Soc. Am. A 3, 726–730 (1986).
[CrossRef]

Ch. Frére, O. Bryngdahl, “Computer-generated holograms: reconstruction of curves in 3-D,” Opt. Commun. 60, 369–372 (1986).
[CrossRef]

Ghatak, A. K.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Apps. A and B.

Gómez-Reino, C.

C. Gómez-Reino, E. Acosta, M. V. Pérez, “Limitation in the cone of light through GRIN lenses: stops, pupils and vignetting,” Jpn. J. Appl. Phys. 31, 1582–1585 (1992).
[CrossRef]

R. M. González, J. Linares, C. Gómez-Reino, “Three-dimensional focal curves generated by one-dimensional gradient-index tapers,” Appl. Opt. 31, 5171–5177 (1992).
[CrossRef] [PubMed]

J. Linares, C. Gómez-Reino, “Diffraction-limited efficiency between SMF connected by GRIN lens,” J. Mod. Opt. 38, 597–604 (1991).
[CrossRef]

J. Linares, C. Gómez-Reino, “Focal curves generated by one-dimensional GRIN tapers,” Opt. Lett. 15, 1258–1260 (1990).
[CrossRef]

C. Gómez-Reino, J. Linares, “Optical path integrals in gradient-index media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
[CrossRef]

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).s
[CrossRef]

C. Gómez-Reino, E. Larrea, M. V. Pérez, “Complex amplitude propagation through a quadratic index medium: characteristic point function,” Appl. Opt. 22, 2927–2929 (1983).
[CrossRef] [PubMed]

J. Linares, R. M. González, C. Gómez-Reino, “Three-dimensional focal curves generated by one-dimensional GRIN tapers,” in Gradient-Index Optical Systems, Vol. 9 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 130–133.

González, R. M.

R. M. González, J. Linares, C. Gómez-Reino, “Three-dimensional focal curves generated by one-dimensional gradient-index tapers,” Appl. Opt. 31, 5171–5177 (1992).
[CrossRef] [PubMed]

J. Linares, R. M. González, C. Gómez-Reino, “Three-dimensional focal curves generated by one-dimensional GRIN tapers,” in Gradient-Index Optical Systems, Vol. 9 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 130–133.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 3 and 4.

Hagerty, J. J.

R. Blankebecler, J. J. Hagerty, G. E. Rindone, M. A. Wickson, “Prescribed profile gradient-index glasses of macrodimensions and large AN,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1354, 130–133 (1990).

Jack, C. A.

W. L. Emkey, C. A. Jack, “Analysis and evaluation of graded-index fiber lenses,” J. Lightwave Technol. LT-5, 1156–1164 (1987).
[CrossRef]

Larrea, E.

Leseberg, D.

Linares, J.

R. M. González, J. Linares, C. Gómez-Reino, “Three-dimensional focal curves generated by one-dimensional gradient-index tapers,” Appl. Opt. 31, 5171–5177 (1992).
[CrossRef] [PubMed]

J. Linares, C. Gómez-Reino, “Diffraction-limited efficiency between SMF connected by GRIN lens,” J. Mod. Opt. 38, 597–604 (1991).
[CrossRef]

J. Linares, C. Gómez-Reino, “Focal curves generated by one-dimensional GRIN tapers,” Opt. Lett. 15, 1258–1260 (1990).
[CrossRef]

C. Gómez-Reino, J. Linares, “Optical path integrals in gradient-index media,” J. Opt. Soc. Am. A 4, 1337–1341 (1987).
[CrossRef]

J. Linares, R. M. González, C. Gómez-Reino, “Three-dimensional focal curves generated by one-dimensional GRIN tapers,” in Gradient-Index Optical Systems, Vol. 9 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 130–133.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Chap. 4.

Marchand, E.

E. Marchand, “Axicon gradient lenses,” Appl. Opt. 29, 4001–4002 (1990).
[CrossRef] [PubMed]

E. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps.3 and 4.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 5.

Pérez, M. V.

C. Gómez-Reino, E. Acosta, M. V. Pérez, “Limitation in the cone of light through GRIN lenses: stops, pupils and vignetting,” Jpn. J. Appl. Phys. 31, 1582–1585 (1992).
[CrossRef]

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).s
[CrossRef]

C. Gómez-Reino, E. Larrea, M. V. Pérez, “Complex amplitude propagation through a quadratic index medium: characteristic point function,” Appl. Opt. 22, 2927–2929 (1983).
[CrossRef] [PubMed]

Rindone, G. E.

R. Blankebecler, J. J. Hagerty, G. E. Rindone, M. A. Wickson, “Prescribed profile gradient-index glasses of macrodimensions and large AN,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1354, 130–133 (1990).

Sodha, M. S.

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Apps. A and B.

Soroko, L. M.

L. M. Soroko, “Axicon and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, pp. 109–160.
[CrossRef]

Wickson, M. A.

R. Blankebecler, J. J. Hagerty, G. E. Rindone, M. A. Wickson, “Prescribed profile gradient-index glasses of macrodimensions and large AN,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1354, 130–133 (1990).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), Chaps. 3 and 4.

Appl. Opt. (3)

J. Lightwave Technol. (1)

W. L. Emkey, C. A. Jack, “Analysis and evaluation of graded-index fiber lenses,” J. Lightwave Technol. LT-5, 1156–1164 (1987).
[CrossRef]

J. Mod. Opt. (1)

J. Linares, C. Gómez-Reino, “Diffraction-limited efficiency between SMF connected by GRIN lens,” J. Mod. Opt. 38, 597–604 (1991).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

C. Gómez-Reino, E. Acosta, M. V. Pérez, “Limitation in the cone of light through GRIN lenses: stops, pupils and vignetting,” Jpn. J. Appl. Phys. 31, 1582–1585 (1992).
[CrossRef]

Opt. Acta (1)

M. V. Pérez, C. Gómez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986).s
[CrossRef]

Opt. Commun. (1)

Ch. Frére, O. Bryngdahl, “Computer-generated holograms: reconstruction of curves in 3-D,” Opt. Commun. 60, 369–372 (1986).
[CrossRef]

Opt. Lett. (1)

Other (11)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 3 and 4.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 355–389.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 5.

L. M. Soroko, “Axicon and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. XXVII, pp. 109–160.
[CrossRef]

J. Linares, R. M. González, C. Gómez-Reino, “Three-dimensional focal curves generated by one-dimensional GRIN tapers,” in Gradient-Index Optical Systems, Vol. 9 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 130–133.

R. Blankebecler, J. J. Hagerty, G. E. Rindone, M. A. Wickson, “Prescribed profile gradient-index glasses of macrodimensions and large AN,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1354, 130–133 (1990).

M. S. Sodha, A. K. Ghatak, Inhomogeneous Optical Waveguides (Plenum, New York, 1977), Apps. A and B.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 2.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1984), Chaps. 3 and 4.

E. Marchand, Gradient Index Optics (Academic, New York, 1978), Chaps.3 and 4.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Chap. 4.

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Figures (6)

Fig. 1
Fig. 1

Variation of the axicon gradient profile with the transverse coordinate r.

Fig. 2
Fig. 2

Ray trajectories within the medium when it is illuminated by a spherical wave. z′ represents the normalized thickness, z′ = bz: (a) a = 0.0125 mm−1, b = 0.1 mm−1, c = 0.625 mm, bd 1 = 2π, R = 1 mm, d 0 = 10 mm; (b) a = 0.01 mm−1, b = 0.1 mm−1, c = 0.5 mm, bd 1 = 2π, R = 1 mm, d 0 = 10 mm.

Fig. 3
Fig. 3

Ray trajectories inside the medium when it is illuminated by a plane wave. z′ represents the normalized thickness, z′ = bz: (a) a = 0.0125 mm−1, b = 0.1 mm−1, c = 0.625 mm, bd 1 = 2π, R = 1 mm; (b) a = 0.01 mm−1, b = 0.1 mm−1, c = 0.5 mm, bd 1 = 2π, R = 1 mm.

Fig. 4
Fig. 4

Effective stop position as function of thickness d 1. In this particular case, a = 0.0125 mm−1, b = 0.1 mm−1, R = 1 mm.

Fig. 5
Fig. 5

Intensity distribution in the image plane for spherical illumination. The solid curve represents the intensity when the exponential amplitude of the TF is considered, and the dashed curve is the intensity for the unit amplitude of the TF. a = 0.0125 mm−1, b = 0.1 mm−1, bd 1 = π/4, R = 1 mm, d 0 = 20 mm, n 0 = 1.56, λ = 1.56 × 10−3 mm.

Fig. 6
Fig. 6

Intensity distribution obtained when c* = 0, a = 0.0125 mm−1, b = 0.1 mm−1, R = 1 mm, d 0 = 20 mm, n 0 = 1.56, λ = 1.56 × 10−3 mm.

Equations (55)

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n 2 = n 0 2 [ 1 + a ( θ ) r - b 2 ( θ ) r 2 ] ,
d a d θ , d b d θ 1.
L ( r , r ˙ , θ ˙ , z ) = n ( 1 + r ˙ 2 + r 2 θ ˙ 2 ) 1 / 2 ,
l = n d z d s = n ( 1 + r ˙ 2 + r 2 θ ˙ 2 ) 1 / 2 .
L = ( n 2 / l 0 ) .
l 0 r 2 θ ˙ = k ,
r ¨ = 1 2 l 0 d d r ( n 2 - k 2 r 2 ) ,
r ˙ 2 = 1 l 0 2 ( n 2 - k 2 r 2 ) - 1.
θ ( z ) = θ 0 ,
r ¨ = 1 2 l 0 2 d n 2 d r .
r ¨ = - n 0 2 b 2 l 0 2 ( r - c ) ,
c = a 2 b 2 .
r ( z ) = c + ( r 0 - c ) H 2 ( z ) + r ˙ 0 H 1 ( z ) ,
H 1 ( z ) = l 0 n 0 b sin ( n 0 b l 0 z ) ,
H 2 ( z ) = cos ( n 0 b l 0 z ) ,
r ˙ 0 = p 0 2 + q 0 2 l 0 ,
p 0 2 = q 0 2 = sin 2 ( α )
l 0 2 = [ n 2 ( r 0 ) - sin 2 ( α ) ] 1 / 2 ,
H 1 ( z ) = 1 b sin ( b z ) ,
H 2 ( z ) = cos ( b z ) .
θ ( z ) = θ 0 ,
r ( z ) = c [ 1 - H 2 ( z ) ] + r 0 [ H 2 ( z ) + H 1 ( z ) n 0 d 0 ] .
z im n = 1 b tan - 1 ( - n 0 d 0 b ) + n π ,
r im n = c [ 1 + ( - 1 ) n ( 1 + n 0 2 d 0 2 b 2 ) 1 / 2 ] .
b z im n = ( n + 1 2 ) π ,
r im = c .
r 0 u = c + [ ( R - c ) 2 ( 1 + D 0 2 ) - c 2 D 0 2 ] 1 / 2 ( 1 + D 0 2 ) ,
z u = 1 b tan - 1 ( r 0 u D 0 r 0 u - c ) ,
D 0 = 1 n 0 d 0 b ,
R eff ( d 1 ) = { R if d 1 < z u c [ 1 - H 2 ( d 1 ) + r 0 u [ H 2 ( d 1 ) + H 1 ( d 1 ) n 0 d 0 ] if z u < d 1 < z im 1 .
z 0 = 1 b cos - 1 ( c - R R ) .
r 0 l = c - [ ( R - c ) 2 ( 1 + D 0 2 ) - c 2 D 0 2 ] 1 / 2 ( 1 + D 0 2 ) ,
z l = 1 b tan - 1 ( r 0 l D 0 r 0 l - c ) .
R eff ( d 1 ) = { c [ 1 - H 2 ( d 1 ) ] if z im 1 < d 1 < z 0 R if z 0 < d 1 < z l c [ 1 - H 2 ( d 1 ) ] + r 0 l [ H 2 ( d 1 ) + H 1 ( d 1 ) n 0 d 0 ] if z l < d 1 < z im 2 .
R eff ( d 1 ) = c [ 1 - H 2 ( d 1 ) ]             if z im 1 < d 1 < z im 2 .
R ef ( d 1 ) = c [ 1 - H 2 ( d 1 ) ] + r 0 u [ H 2 ( d 1 ) + H 1 ( d 1 ) / n 0 d 0 ] .
t p ( r 0 , r 1 ; d 1 ) = exp ( - 1 2 n 0 0 d 1 2 S p d z ) exp ( i k S p ) ,
S p ( r 0 , r 1 ; d 1 ) = 0 d 1 L p d z ,
S p ( r 0 , r 1 ; d 1 ) = n 0 2 [ ( r 1 - c ) r ˙ 1 - ( r 0 - c ) r ˙ 0 ] + n 0 d ˜ ,
d ˜ = d 1 [ 1 + 1 2 ( a 2 b ) 2 ] .
t p ( r 0 , r 1 ; d 1 ) = { A exp [ i k n 0 ( H ˙ 1 + n 0 d 0 H ˙ 2 ) 2 ( H 1 + n 0 d 0 H 2 ) ( r 1 - c * ) 2 ] exp ( - i k 2 d 0 r 0 2 ) for r 1 < R eff 0 for r 1 > R eff ,
A = n 0 d 0 H 1 + n 0 d 0 H 2 exp [ c * 2 r 1 ln ( H 2 + H 1 n 0 d 0 ) ] × exp [ i k n 0 c 2 H ˙ 2 2 ( H ˙ 1 + n 0 d 0 H ˙ 2 ) ] exp ( i k n 0 d ˜ ) ,
c * ( d 1 ) = c [ 1 - 1 H ˙ 1 ( d 1 ) + n 0 d 0 H ˙ 2 ( d 1 ) ] .
r 2 = c * ( d 1 ) ,
d 2 = H 1 ( d 1 ) + n 0 d 0 H 2 ( d 1 ) n 0 [ H ˙ 1 ( d 1 ) + n 0 d 0 H ˙ 2 ( d 1 ) ] .
φ ( r 1 , d 1 ) = t p ( r 0 , r 1 ; d 1 ) exp ( i k n 0 2 d 0 r 0 2 ) .
φ ( r 1 , d 1 ) = 1 H 2 ( d 1 ) exp ( i k n 0 d ˜ ) exp { c 2 r 1 ln [ H 2 ( d 1 ) ] } × exp [ i k n 0 H ˙ 2 ( d 1 ) 2 H 2 ( d 1 ) ( r 1 - c ) 2 ] .
r 2 = a 2 b 2 ,
f ( d 1 ) = - H 2 ( d 1 ) n 0 H ˙ 2 ( d 1 ) ,
φ ( r 2 , d 2 ) = 1 i λ d 2 exp ( i k d 2 ) R 2 exp [ i k 2 d 2 ( r 2 - r 1 ) 2 ] × φ ( r 1 , d 1 ) r 1 d r 1 d θ 1 ,
φ ( r 2 , d 2 ) n 0 d 0 d 2 ( H 1 + n 0 d 0 H 2 ) exp [ i k 2 d 2 ( r 2 - c * 2 ) ] × 0 R ef exp [ c * 2 r 1 ln ( H 2 + H 1 n 0 d 0 ) ] × exp ( i k d 2 c * r 1 ) J 0 ( k d 2 r 2 r 1 ) r 1 d r 1 ,
I ( r 2 , d 2 ) = φ ( r 2 , d 2 ) φ * ( r 2 , d 2 ) .
φ ( r 2 , d 2 ) = α ( r 2 , d 2 ) c * ( r 2 2 - c * 2 ) 3 / 2 ,
α ( r 2 , d 2 ) = n 0 d 0 d 2 2 π ( H 1 + n 0 d 0 H 2 ) exp [ i k ( d 2 + n 0 d ˜ ) ] × exp [ i k 2 d 2 ( r 2 2 - c * 2 ) ] ,
I ( r 2 , d 2 ) c * 2 ( r 2 2 - c * 2 ) .

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