Abstract

The spatial and spectral properties of three-dimensional photonic jets are studied in a framework employing rigorous Lorentz-Mie theory. The contributions to the field from each spectral component are studied quantitatively and highlight the distinctive features of photonic jets. In particular, the role of evanescent field in photonic jets generated by small spheres is investigated. Secondary lobes in the propagative frequency distribution are also singled out as a distinctive property of photonic jets. It is shown that these differences lead to angular openings of photonic jets at least twice as small as those in comparable ‘Gaussian’ beams.

© 2008 Optical Society of America

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References

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  1. Z. Chen, A. Taflove, and V. Backman, "Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique," Opt. Express 12, 1214-1220 (2004).
    [CrossRef] [PubMed]
  2. X. Li, Z. Chen, A. Taflove, and V. Backman, "Optical analysis of nanoparticles via enhanced backscattering facilitated by 3-D photonic nanojets," Opt. Express 13, 526-533 (2005).
    [CrossRef] [PubMed]
  3. Z. Chen, A. Taflove, X. Li, and V. Backman, "Superenhanced backscattering of light by nanoparticles," Opt. Lett. 31, 196-198 (2006).
    [CrossRef] [PubMed]
  4. A. Heifetz, K. Huang, A. V. Sahkian, X. li, A. Taflove, and V. Backman, "Experimental confirmation of backscattering enhancement induced by a photonic jet," Appl. Phys. Lett. 89, 221118 (2006).
    [CrossRef]
  5. P. Ferrand, J. Wenger, M. Pianta, H. Rigneault, A. Devilez, B. Stout, N. Bonod, and E. Popov, "Direct imaging of photonic nanojet," Opt. Express 16, 6930 - 6940 (2008).
    [CrossRef] [PubMed]
  6. D. Grojo, P. Delaporte, and A. Cros, "Removal of particles by impulsional laser," Journal de Physique IV 127, 145-149 (2005).
    [CrossRef]
  7. S. M. Huang, M. H. Hong, B. Luk�??yanchuk, and T. C. Chong, "Nanostructures fabricated on metal surfaces assisted by laser with optical near-field effects," Appl. Phys. A: Mater. Sci. Process. 77, 293-296 (2003).
  8. W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk�??yanchuk, and Z. Liu, "Near-field laser parallel nanofabrication of arbitrary-shaped patterns," Appl. Phys. Lett. 90, 243101 (2007).
    [CrossRef]
  9. S. Lecler, S. Haacke, N. Lecong, O. Crégut, J. L. Rehspringer, and C. Hirlimann, "Photonic jet driven non-linear optics: example of two-photon fluorescence enhancement by dielectric microspheres," Opt. Express 15, 4935-4942 (2007).
    [CrossRef] [PubMed]
  10. J. Kofler and N. Arnold, "Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie," Phys. Rev. B 73, 235401 (2006).
    [CrossRef]
  11. A. V. Itagi and W. A. Challener, "Optics of photonic nanojets," J. Opt. Soc. Am A 22, 2847-2858 (2005).
    [CrossRef]
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  13. O. Moine and B. Stout, "Optical force calculations in arbitrary beams by use of the vector addition theorem," J. Opt. Soc. Am. B 22, 1620-1631 (2005).
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  16. L. Mandel and E. Wolf, "Some useful mathematical techniques," in Optical coherence and quantum optics, L. Mandel and E. Wolf (Cambridge University press, 1995), pp. 92-146.
  17. H. C. Van de Hulst, "Very large spheres," in Light scattering by small particles, H. C. Van de Hulst (Dover publication, 1981), pp. 200-227.
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    [CrossRef] [PubMed]
  19. M. Born and E. Wolf, "Elements of the theory of diffraction," in Principles of optics, M. Born and E. Wolf (Pergamon press, 1986), pp. 370-458.

2008 (1)

2007 (2)

S. Lecler, S. Haacke, N. Lecong, O. Crégut, J. L. Rehspringer, and C. Hirlimann, "Photonic jet driven non-linear optics: example of two-photon fluorescence enhancement by dielectric microspheres," Opt. Express 15, 4935-4942 (2007).
[CrossRef] [PubMed]

W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk�??yanchuk, and Z. Liu, "Near-field laser parallel nanofabrication of arbitrary-shaped patterns," Appl. Phys. Lett. 90, 243101 (2007).
[CrossRef]

2006 (3)

J. Kofler and N. Arnold, "Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie," Phys. Rev. B 73, 235401 (2006).
[CrossRef]

A. Heifetz, K. Huang, A. V. Sahkian, X. li, A. Taflove, and V. Backman, "Experimental confirmation of backscattering enhancement induced by a photonic jet," Appl. Phys. Lett. 89, 221118 (2006).
[CrossRef]

Z. Chen, A. Taflove, X. Li, and V. Backman, "Superenhanced backscattering of light by nanoparticles," Opt. Lett. 31, 196-198 (2006).
[CrossRef] [PubMed]

2005 (5)

2004 (1)

2003 (1)

S. M. Huang, M. H. Hong, B. Luk�??yanchuk, and T. C. Chong, "Nanostructures fabricated on metal surfaces assisted by laser with optical near-field effects," Appl. Phys. A: Mater. Sci. Process. 77, 293-296 (2003).

1980 (1)

Arnold, N.

J. Kofler and N. Arnold, "Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie," Phys. Rev. B 73, 235401 (2006).
[CrossRef]

Backman, V.

Bonod, N.

Challener, W. A.

A. V. Itagi and W. A. Challener, "Optics of photonic nanojets," J. Opt. Soc. Am A 22, 2847-2858 (2005).
[CrossRef]

Chen, Z.

Chong, T. C.

S. M. Huang, M. H. Hong, B. Luk�??yanchuk, and T. C. Chong, "Nanostructures fabricated on metal surfaces assisted by laser with optical near-field effects," Appl. Phys. A: Mater. Sci. Process. 77, 293-296 (2003).

Crégut, O.

Cros, A.

D. Grojo, P. Delaporte, and A. Cros, "Removal of particles by impulsional laser," Journal de Physique IV 127, 145-149 (2005).
[CrossRef]

Delaporte, P.

D. Grojo, P. Delaporte, and A. Cros, "Removal of particles by impulsional laser," Journal de Physique IV 127, 145-149 (2005).
[CrossRef]

Devilez, A.

Ferrand, P.

Grojo, D.

D. Grojo, P. Delaporte, and A. Cros, "Removal of particles by impulsional laser," Journal de Physique IV 127, 145-149 (2005).
[CrossRef]

Guo, W.

W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk�??yanchuk, and Z. Liu, "Near-field laser parallel nanofabrication of arbitrary-shaped patterns," Appl. Phys. Lett. 90, 243101 (2007).
[CrossRef]

Haacke, S.

Heifetz, A.

A. Heifetz, K. Huang, A. V. Sahkian, X. li, A. Taflove, and V. Backman, "Experimental confirmation of backscattering enhancement induced by a photonic jet," Appl. Phys. Lett. 89, 221118 (2006).
[CrossRef]

Hirlimann, C.

Hong, M. H.

S. M. Huang, M. H. Hong, B. Luk�??yanchuk, and T. C. Chong, "Nanostructures fabricated on metal surfaces assisted by laser with optical near-field effects," Appl. Phys. A: Mater. Sci. Process. 77, 293-296 (2003).

Huang, K.

A. Heifetz, K. Huang, A. V. Sahkian, X. li, A. Taflove, and V. Backman, "Experimental confirmation of backscattering enhancement induced by a photonic jet," Appl. Phys. Lett. 89, 221118 (2006).
[CrossRef]

Huang, S. M.

S. M. Huang, M. H. Hong, B. Luk�??yanchuk, and T. C. Chong, "Nanostructures fabricated on metal surfaces assisted by laser with optical near-field effects," Appl. Phys. A: Mater. Sci. Process. 77, 293-296 (2003).

Itagi, A. V.

A. V. Itagi and W. A. Challener, "Optics of photonic nanojets," J. Opt. Soc. Am A 22, 2847-2858 (2005).
[CrossRef]

Kofler, J.

J. Kofler and N. Arnold, "Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie," Phys. Rev. B 73, 235401 (2006).
[CrossRef]

Lecler, S.

Lecong, N.

Li, L.

W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk�??yanchuk, and Z. Liu, "Near-field laser parallel nanofabrication of arbitrary-shaped patterns," Appl. Phys. Lett. 90, 243101 (2007).
[CrossRef]

Li, X.

Liu, Z.

W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk�??yanchuk, and Z. Liu, "Near-field laser parallel nanofabrication of arbitrary-shaped patterns," Appl. Phys. Lett. 90, 243101 (2007).
[CrossRef]

Luk???yanchuk, B.

S. M. Huang, M. H. Hong, B. Luk�??yanchuk, and T. C. Chong, "Nanostructures fabricated on metal surfaces assisted by laser with optical near-field effects," Appl. Phys. A: Mater. Sci. Process. 77, 293-296 (2003).

Luk???yanchuk, B. S.

W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk�??yanchuk, and Z. Liu, "Near-field laser parallel nanofabrication of arbitrary-shaped patterns," Appl. Phys. Lett. 90, 243101 (2007).
[CrossRef]

Moine, O.

Nevière, M.

Pianta, M.

Popov, E.

Rehspringer, J. L.

Rigneault, H.

Sahkian, A. V.

A. Heifetz, K. Huang, A. V. Sahkian, X. li, A. Taflove, and V. Backman, "Experimental confirmation of backscattering enhancement induced by a photonic jet," Appl. Phys. Lett. 89, 221118 (2006).
[CrossRef]

Stout, B.

Taflove, A.

Wenger, J.

Whitehead, D. J.

W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk�??yanchuk, and Z. Liu, "Near-field laser parallel nanofabrication of arbitrary-shaped patterns," Appl. Phys. Lett. 90, 243101 (2007).
[CrossRef]

Wiscombe, W. J.

Wuang, Z. B.

W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk�??yanchuk, and Z. Liu, "Near-field laser parallel nanofabrication of arbitrary-shaped patterns," Appl. Phys. Lett. 90, 243101 (2007).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. A: Mater. Sci. Process. (1)

S. M. Huang, M. H. Hong, B. Luk�??yanchuk, and T. C. Chong, "Nanostructures fabricated on metal surfaces assisted by laser with optical near-field effects," Appl. Phys. A: Mater. Sci. Process. 77, 293-296 (2003).

Appl. Phys. Lett. (2)

W. Guo, Z. B. Wuang, L. Li, D. J. Whitehead, B. S. Luk�??yanchuk, and Z. Liu, "Near-field laser parallel nanofabrication of arbitrary-shaped patterns," Appl. Phys. Lett. 90, 243101 (2007).
[CrossRef]

A. Heifetz, K. Huang, A. V. Sahkian, X. li, A. Taflove, and V. Backman, "Experimental confirmation of backscattering enhancement induced by a photonic jet," Appl. Phys. Lett. 89, 221118 (2006).
[CrossRef]

J. Opt. Soc. Am A (1)

A. V. Itagi and W. A. Challener, "Optics of photonic nanojets," J. Opt. Soc. Am A 22, 2847-2858 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Journal de Physique IV (1)

D. Grojo, P. Delaporte, and A. Cros, "Removal of particles by impulsional laser," Journal de Physique IV 127, 145-149 (2005).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. B (1)

J. Kofler and N. Arnold, "Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie," Phys. Rev. B 73, 235401 (2006).
[CrossRef]

Other (5)

M. Born and E. Wolf, "Elements of the theory of diffraction," in Principles of optics, M. Born and E. Wolf (Pergamon press, 1986), pp. 370-458.

M. Born and E. Wolf, "Rigorous diffraction theory," in Principles of optics, M. Born and E. Wolf (Pergamon press, 1986), pp. 556-592.

M. Born and E. Wolf, "Optics of metals," in Principles of optics, M. Born and E. Wolf (Pergamon press, 1986), pp. 611-664.

L. Mandel and E. Wolf, "Some useful mathematical techniques," in Optical coherence and quantum optics, L. Mandel and E. Wolf (Cambridge University press, 1995), pp. 92-146.

H. C. Van de Hulst, "Very large spheres," in Light scattering by small particles, H. C. Van de Hulst (Dover publication, 1981), pp. 200-227.

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Figures (9)

Fig. 1.
Fig. 1.

Photonic jet produced when a dielectric sphere of radius R=1 µm, and refractive index Ns is illuminated by a plane wave propagating along the z axis in a homogeneous embedding medium of refractive index N ο(ρ=Ns /N ο=1.2, λ v=525 nm, λ ο=λ v/N ο, k ο R=16). (a) A hot scale map of a photonic jet is displayed on a logarithmic scale of the electric field intensity. (b) A schema for photonic jet parameters is provided: ‘focal’ distance f, maximum enhancement I max, width at f, w ο and diffraction length z r. The amplitude contours of the photonic jet at I(z)/e 2 and I max/e 2 are also displayed.

Fig. 2.
Fig. 2.

Intensity enhancement distribution for a 1 µm radius sphere illuminated by a plane wave at λ v=525 nm with an index contrast of ρ=1.2, λ ο=λ v/N ο=394 nm (a) along the propagation axis (z) (in black), and its Lorentzian fit (in red), z r=800 nm (b) along a transverse axis (x) for z=f=1.57 µm (in black), and its Gaussian fit (in red), w ο=241 nm=0.6λ ο.

Fig. 3.
Fig. 3.

Total spectral amplitude S=(AA*+BB*+CC*)1/2 as a function of normalized spatial frequencies K/k 0 at z=1.05 µm, for a sphere of radius R=1µm, and index contrasts (a) ρ=1.2 and (b) ρ=1.6, illuminated by a plane wave at λ v=525 nm. The low frequencies are respectively fitted by a Gaussian frequency distribution of respective widths K/k ο=0,190 in (a) and K/k ο=0,270 in (b) (red line).

Fig. 4.
Fig. 4.

S as a function of z and K/kο in logarithmic scale with the same parameters as used in Fig. 3(a) and Fig. 3(b) respectively.

Fig. 5.
Fig. 5.

Intensity distribution of the electric field: (a) along the propagation axis (z), and (b) at z=f, for sphere of radius R=1µm illuminated at λ v=525 nm for an index contrast of ρ=1.6. The red curve corresponds to the full intensity while the green curve is the intensity once the evanescent field has been removed.

Fig. 6.
Fig. 6.

Scattered intensity of a photonic jet produced by a sphere of radius 1 µm, illuminated at λ v=525 nm with an index contrast of ρ=1.6. The angles corresponding to the maxima and minima in the spatial frequency expansion of Fig. 3(b) are displayed in direct space by red lines and black lines respectively.

Fig. 7.
Fig. 7.

Amplitude of scattered coefficients of photonic jets in the VSWF expansion; (a) for the magnetic |f (h) n,1| coefficients, (b) for the electric |f (e) n,1| coefficients. R=1µm, ρ=1.6, λ v=525 nm.

Fig. 8.
Fig. 8.

Scattered intensity of the terms of order n=11, (a) |f (h) 11 M 11,1|2, (b) |f (e) 11 N 11,1 |2. R=1µm, ρ=1.6, λ v=525 nm.

Fig. 9.
Fig. 9.

Scattered intensity in log10 hot scale of a “photonic jet” generated with the principal order in the VSWF expansion: n=11, i.e. |f (h) 11 M 11,1+f (e) 11 N 11,1|2. R=1 µm, ρ=1.6, λ v=525 nm

Equations (40)

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E scat ( r , θ , ϕ ) = E 0 n = 1 m = n n f n , m ( h ) M n , m ( k 0 r , θ , ϕ ) + f n , m ( e ) N n , m ( k 0 r , θ , ϕ )
K = K cos ϕ k x + K sin ϕ k y ,
M n , m ( k 0 r , ϕ , z ) = K = 0 ϕ = 0 2 π X n , m ( θ k , ϕ k ) e i ( K · r + k z z ) k z K d K d ϕ k
N n , m ( k 0 r , ϕ , z ) = K = 0 ϕ = 0 2 π Z n , m ( θ k , ϕ k ) e i ( K · r + k z z ) k z K d K d ϕ k
E scat ( r , ϕ , z ) = E 0 0 K k 0 d K k 0 [ A ( K k 0 , ϕ , z ) J 0 ( K r ) { x ̂ + i y ̂ } 2 B ( K k 0 , ϕ , z ) J 1 ( K r ) z ̂ C ( K k 0 , ϕ , z ) J 2 ( K r ) { x ̂ i y ̂ } 2 ]   with A = n = 0 f n ( h ) A n ( h ) + f n ( e ) A n ( e ) B = n = 0 f n ( h ) B n ( h ) + f n ( e ) B n ( e ) C = n = 0 f n ( h ) C n ( h ) + f n ( e ) C n ( e )
E scat ( r , θ , ϕ ) = E 0 n = 1 m = n n f n , m ( h ) M n , m ( k 0 r , θ , ϕ ) + f n , m ( e ) N n , m ( k 0 r , θ , ϕ )
M n, m ( k 0 r , θ , ϕ ) = h n ( + ) ( k 0 r ) X n , m ( θ , ϕ )
N n , m ( k 0 r , θ , ϕ ) = 1 k 0 r [ h n ( + ) ( k 0 r ) Y n , m ( θ , ϕ ) + ( k 0 r h n ( + ) ( k 0 r ) ) Z n , m ( θ , ϕ ) ]
Y n , m ( θ , ϕ ) r ̂ Y n , m ( θ , ϕ ) Z n , m ( θ , ϕ ) r Y n , m ( θ , ϕ ) n ( n + 1 ) X n , m ( θ , ϕ ) Z n , m ( θ , ϕ ) × r ̂
R g { M n , m ( k 0 r , θ , ϕ ) } = j n ( k 0 r ) X n , m ( θ , ϕ )
R g { N n , m ( k 0 r , θ , ϕ ) } = 1 k 0 r [ j n ( k 0 r ) Y n , m ( θ , ϕ ) + ( k 0 r j n ( k 0 r ) ) Z n , m ( θ , ϕ ) ]
E inc ( r , θ , ϕ ) = E 0 n = 1 m = n n a n , m ( h ) R g { M n , m ( k s r , θ , ϕ ) } + a n , m ( e ) R g { N n , m ( k s r , θ , ϕ ) }
E int ( r , θ , ϕ ) = E 0 n = 1 m = n n s n , m ( h ) R g { M n , m ( k s r , θ , ϕ ) } + s n , m ( e ) R g { N n , m ( k s r , θ , ϕ ) }
a n , m ( h ) = 4 π i n X n , m * ( θ , ϕ ) · e i ̂
a n , m ( e ) = 4 π i n 1 Z n , m * ( θ , ϕ ) · e i ̂
e i ̂ = x ̂ + i y ̂ 2
E scat ( r , θ , ϕ ) = E 0 n = 1 m = n n f n , m ( h ) M n , m ( k 0 r , θ , ϕ ) + f n , m ( e ) N n , m ( k 0 r , θ , ϕ )
f n , m ( h ) = ψ n ( k 0 R ) ψ n ( k s R ) ρ ψ n ( k s R ) ψ n ( k 0 R ) ρ ξ n ( k 0 R ) ψ n ( k s R ) ψ n ( k s R ) ξ n ( k 0 R ) a n , m ( h )
f n , m ( e ) = ψ n ( k s R ) ψ n ( k 0 R ) ρ ψ n ( k 0 R ) ψ n ( k s R ) ρ ξ n ( k 0 R ) ψ n ( k s R ) ψ n ( k s R ) ξ n ( k 0 R ) a n , m ( e )
E scat ( r , θ , ϕ ) = E 0 n = 1 m = n n f n , m ( h ) M n , m ( k 0 r , θ , ϕ ) + f n , m ( e ) N n , m ( k 0 r , θ , ϕ )
M n, m ( k 0 r , θ , ϕ ) = h n ( + ) ( k 0 r ) X n , m ( θ , ϕ )
N n , m ( k 0 r , θ , ϕ ) = 1 k 0 r [ h n ( + ) ( k 0 r ) Y n , m ( θ , ϕ ) + ( k 0 r h n ( + ) ( k 0 r ) ) Z n , m ( θ , ϕ ) ]
h n ( k 0 r ) Y n , m ( r ̂ ) = 1 2 π i n k 0 K = 0 ϕ k = 0 2 π K d K d ϕ k Y n , m ( k ̂ ) e ± i k . r k z   sign ( z ) = ±
Y n , m ( θ , ϕ ) = c n , m P n m ( cos θ ) e i m ϕ
c n , m = [ 2 n + 1 4 π ( n m ) ! ( n + m ) ! ] 1 2
χ ̂ 1 = x ̂ i y ̂ 2 , χ ̂ 0 = z ̂ ,   χ ̂ 1 = x ̂ + i y ̂ 2
Y n , l m = Σ μ = 1 1 ( l , m μ ; 1 , μ n , m ) Y l , m μ χ ̂ μ
X n , m = Y n , n m i
Y n , m = n 2 n + 1 Y n , n 1 m + n + 1 2 n + 1 Y n , n + 1 m
Z n , m = n + 1 2 n + 1 Y n , n 1 m n 2 n + 1 Y n , n + 1 m
M n , m ( k 0 r , ϕ , z ) = K = 0 ϕ = 0 2 π X n , m ( θ k , ϕ k ) e i ( K . r + k z z ) k z K d K d ϕ k
N n , m ( k 0 r , ϕ , z ) = K = 0 ϕ = 0 2 π Z n , m ( θ k , ϕ k ) e i ( K . r + k z z ) k z K d K d ϕ k
e i K . r = e i K r cos ( ϕ k ϕ )
0 2 π e i x cos ( ϕ k ϕ ) e i n ϕ k d ϕ k = 2 π i n J n ( x ) e i n ϕ
E scat ( r , ϕ , z ) = E 0 Σ n = 1 0 K k 0 d K k 0 f n ( h ) [ A n ( h ) ( K k 0 , ϕ , z ) J 0 ( K r ) { x ̂ + i y ̂ } 2 B n ( h ) ( K k 0 , ϕ , z ) J 1 ( K r ) z ̂ C n ( h ) ( K k 0 , ϕ , z ) J 2 ( K r ) { x ̂ i y ̂ } 2 ] + f n ( e ) [ A n ( e ) ( K k 0 , ϕ , z ) J 0 ( K r ) { x ̂ + i y ̂ } 2 B n ( e ) ( K k 0 , ϕ , z ) J 1 ( K r ) z ̂ C n ( e ) ( K k 0 , ϕ , z ) J 2 ( K r ) { x ̂ i y ̂ } 2 ]
A n ( h ) = 1 i n + 1 ( n , 0 ; 1 , 1 n , 1 ) c n , 0 P n 0 ( 1 K k 0 ) e i k 0 z 1 K k 0 1 k k 0 B n ( h ) = 1 i n ( n , 1 ; 1 , 0 n , 1 ) c n , 1 P n 1 ( 1 K k 0 ) e i ϕ e i k 0 z 1 K k 0 1 K k 0 C n ( h ) = 1 i n + 1 ( n , 2 ; 1 , 1 n , 1 ) c n , 2 P n 2 ( 1 K k 0 ) e 2 i ϕ e i k 0 z 1 K k 0 1 K k 0
A n ( e ) = 1 i n 1 [ n + 1 2 n + 1 ( n 1 , 0 ; 1 , 1 n , 1 ) c n 1 , 0 P n 1 0 ( 1 K k 0 ) + n 2 n + 1 ( n + 1 , 0 ; 1 , 1 n , 1 ) c n + 1 , 0 P n + 1 0 ( 1 K k 0 ) ] e i k 0 z 1 K k 0 1 K k 0
B n ( e ) = i i n 1 [ n + 1 2 n + 1 ( n 1 , 1 ; 1 , 0 n , 1 ) c n 1 , 1 P n 1 1 ( 1 K k 0 ) + n 2 n + 1 ( n + 1 , 1 ; 1 , 0 n , 1 ) c n + 1 , 1 P n + 1 1 ( 1 K k 0 ) ] e i ϕ e i k 0 z 1 K k 0 1 K k 0
C n ( e ) = 1 i n 1 [ n + 1 2 n + 1 ( n 1 , 2 ; 1 , 1 n , 1 ) c n 1 , 2 P n 1 2 ( 1 K k 0 ) + n 2 n + 1 ( n + 1 , 2 ; 1 , 1 n , 1 ) c n + 1 , 2 P n + 1 2 ( 1 K k 0 ) ] e 2 i ϕ e i k 0 z 1 K k 0 1 K k 0
E scat ( r , ϕ , z ) = E 0 0 K k 0 d K k 0 [ A ( K k 0 , ϕ , z ) J 0 ( K r ) { x ̂ + i y ̂ } 2 B ( K k 0 , ϕ , z ) J 1 ( K r ) z ̂ C ( K k 0 , ϕ , z ) J 2 ( K r ) { x ̂ i y ̂ } 2 ]   with A = n = 0 f n ( h ) A n ( h ) + f n ( e ) A n ( e ) B = n = 0 f n ( h ) B n ( h ) + f n ( e ) B n ( e ) C = n = 0 f n ( h ) C n ( h ) + f n ( e ) C n ( e )

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