Abstract

We develop a new method to analytically obtain the beam spread function (BSF) for light radiative transfer in oceanic environments. The BSF, which is defined as the lateral distribution of the (scalar) irradiance with increasing depth in response to a uni-directional beam emanating from a point source in an infinite ocean, must in general be obtained by solving the three-dimensional (3D) radiative transfer equation (RTE). By taking advantage of the highly forward-peaked scattering property of the ocean particles, we assume, for a narrow beam source, the dependence of radiance on polar angle and azimuthal angle is deliberately separated; only single scattering takes place in the azimuthal direction while multiple scattering still occurs in the polar direction. This assumption enables us to reduce the five-variable 3D RTE to a three-variable two-dimensional (2D) RTE. With this simplification, we apply Fourier spectral method to both spatial and angular variables so that we are able to analytically solve the 2D RTE and obtain the 2D BSF accordingly. Using the relations between 2D and 3D solutions acquired during the process of simplification, we are able to obtain the 3D BSF in explicit form. The 2D and 3D analytical solutions are validated by comparing with Monte Carlo radiative transfer simulations. The 2D analytical BSF agrees excellently with the Monte Carlo result. Despite assumptions of axial symmetry and spike-like azimuthal profile of the radiance in deriving the 3D BSF, the comparisons to numerical simulations are very satisfactory especially for limited optical depths (< O(5)) for single scattering albedo values typical in the ocean. The explicit form of the analytical BSF and the significant gain in computational efficiency (several orders higher) relative to RTE simulations make many forward and inverse problems in ocean optics practical for routine applications.

© 2015 Optical Society of America

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References

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  1. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  2. S. Q. Duntley, “Underwater lighting by submerged lasers and incandescent sources,” Tech. Rep., Scripps Institution of Oceanography, San Diego, Calif. (1971).
  3. K. J. Voss and A. L. Chapin, “Measurement of the point spread function in the ocean,” Appl. Opt. 29, 3638–3642 (1990).
    [Crossref] [PubMed]
  4. K. J. Voss, “Simple empirical model of oceanic point spread function,” Appl. Opt. 30, 2647–2651 (1991).
    [Crossref] [PubMed]
  5. W. Hou, D. J. Gray, A. D. Weidemann, and R. A. Arnone, “Comparison and validation of point spread models for imaging in natural waters,” Opt. Express. 16(13), 9958–9965 (2008).
    [Crossref] [PubMed]
  6. C. D. Mobley, Light and Water: Radiative Transfer In Natural Waters (Academic, 1994).
  7. G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of radiative transfer in the earth atmosphere-ocean system: I. flux in the atmosphere and ocean,” J. Phys. Oceanog 2(13), 139–145 (1972).
    [Crossref]
  8. R. Sanchez and N. J. McCormick, “Analytic beam spread function for ocean optics applications,” Appl. Opt. 41(30), 6276–6288 (2002).
    [Crossref] [PubMed]
  9. E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).
    [Crossref]
  10. G. C. Pomraning, A. K. Prinja, and J. W. VanDenburg, “An asymptotic model for the spreding of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).
  11. C. Börgers and E. W. Larsen, “Asymptotic derivation of the Fermi pencil-beam approximation,” Nucl. Sci. Eng. 123, 343–357 (1996).
  12. W. H. Wells, “Theory of small angle scattering,” in Optics of the Sea (Interface and In-Water Transmission and Imaging), P. Halley, ed., Lecture Series no. 61 (North Atlantic Treaty Organization, Advisory Group for Aerospace Research and Development, Neuilly-sur-Seine, France, 1973).
  13. R. E. Walker, Marine Light Field Statistics (Wiley, 1994).
  14. J. W. McLean, J. D. Freeman, and R. E. Walker, “Beam spread function with time dispersion,” Appl. Opt. 37(21), 4701–4711 (1998).
    [Crossref]
  15. W. Wei, X. Zhang, Y. Chao, and X. Zhou, “An analytical model of the power spatial distribution for underwater optical wireless communication,” Optica Applicata XLII(1), 157–166 (2002).
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    [Crossref]
  18. V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14(1), L13–L19 (2004).
    [Crossref]
  19. A. Liemert and A. Kienle, “Green’s functions for the two-dimensional radiative transfer equation in bounded media,” J. Phys. A: Math. Theor. 45, 175201 (2012).
    [Crossref]
  20. Z. Xu and D. K. P. Yue, “Monte Carlo radiative transfer simulation for the near-ocean-surface high-resolution downwelling irradiance statistics,” Opt. Eng. 53, 051408 (2014). doi:
    [Crossref]
  21. T. J. Petzold, “Volume scattering functions for selected ocean water,” Tech. Rep., Scripps Institution of Oceanography, San Diego, Calif. (1972).
  22. L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [Crossref]
  23. L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phy. Rev. E 79, 036607 (2009).
    [Crossref]
  24. J. P. Boyd, Chebyshev and Fourier Spectral Methods (DOVER Publications, 2000).
  25. Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011). doi:
    [Crossref]
  26. Z. Xu, X. Guo, L. Shen, and D. K. P. Yue, “Radiative transfer in ocean turbulence and its effect on underwater light field,” J. Geophys. Res. 117, C00H18 (2012)., doi:
    [Crossref]

2014 (1)

Z. Xu and D. K. P. Yue, “Monte Carlo radiative transfer simulation for the near-ocean-surface high-resolution downwelling irradiance statistics,” Opt. Eng. 53, 051408 (2014). doi:
[Crossref]

2012 (2)

A. Liemert and A. Kienle, “Green’s functions for the two-dimensional radiative transfer equation in bounded media,” J. Phys. A: Math. Theor. 45, 175201 (2012).
[Crossref]

Z. Xu, X. Guo, L. Shen, and D. K. P. Yue, “Radiative transfer in ocean turbulence and its effect on underwater light field,” J. Geophys. Res. 117, C00H18 (2012)., doi:
[Crossref]

2011 (1)

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011). doi:
[Crossref]

2009 (1)

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phy. Rev. E 79, 036607 (2009).
[Crossref]

2008 (1)

W. Hou, D. J. Gray, A. D. Weidemann, and R. A. Arnone, “Comparison and validation of point spread models for imaging in natural waters,” Opt. Express. 16(13), 9958–9965 (2008).
[Crossref] [PubMed]

2004 (1)

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14(1), L13–L19 (2004).
[Crossref]

2002 (2)

W. Wei, X. Zhang, Y. Chao, and X. Zhou, “An analytical model of the power spatial distribution for underwater optical wireless communication,” Optica Applicata XLII(1), 157–166 (2002).

R. Sanchez and N. J. McCormick, “Analytic beam spread function for ocean optics applications,” Appl. Opt. 41(30), 6276–6288 (2002).
[Crossref] [PubMed]

2001 (2)

1998 (1)

1996 (1)

C. Börgers and E. W. Larsen, “Asymptotic derivation of the Fermi pencil-beam approximation,” Nucl. Sci. Eng. 123, 343–357 (1996).

1992 (1)

G. C. Pomraning, A. K. Prinja, and J. W. VanDenburg, “An asymptotic model for the spreding of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).

1991 (1)

1990 (1)

1972 (1)

G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of radiative transfer in the earth atmosphere-ocean system: I. flux in the atmosphere and ocean,” J. Phys. Oceanog 2(13), 139–145 (1972).
[Crossref]

1941 (1)

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Arnone, R. A.

W. Hou, D. J. Gray, A. D. Weidemann, and R. A. Arnone, “Comparison and validation of point spread models for imaging in natural waters,” Opt. Express. 16(13), 9958–9965 (2008).
[Crossref] [PubMed]

Billard, B. D.

Börgers, C.

C. Börgers and E. W. Larsen, “Asymptotic derivation of the Fermi pencil-beam approximation,” Nucl. Sci. Eng. 123, 343–357 (1996).

Boyd, J. P.

J. P. Boyd, Chebyshev and Fourier Spectral Methods (DOVER Publications, 2000).

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Chao, Y.

W. Wei, X. Zhang, Y. Chao, and X. Zhou, “An analytical model of the power spatial distribution for underwater optical wireless communication,” Optica Applicata XLII(1), 157–166 (2002).

Chapin, A. L.

Duntley, S. Q.

S. Q. Duntley, “Underwater lighting by submerged lasers and incandescent sources,” Tech. Rep., Scripps Institution of Oceanography, San Diego, Calif. (1971).

Florescu, L.

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phy. Rev. E 79, 036607 (2009).
[Crossref]

Freeman, J. D.

Gehman, V. M.

Gennaro, T. L.

Gray, D. J.

W. Hou, D. J. Gray, A. D. Weidemann, and R. A. Arnone, “Comparison and validation of point spread models for imaging in natural waters,” Opt. Express. 16(13), 9958–9965 (2008).
[Crossref] [PubMed]

Greenstein, J. L.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Guo, X.

Z. Xu, X. Guo, L. Shen, and D. K. P. Yue, “Radiative transfer in ocean turbulence and its effect on underwater light field,” J. Geophys. Res. 117, C00H18 (2012)., doi:
[Crossref]

Henyey, L. C.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

Hou, W.

W. Hou, D. J. Gray, A. D. Weidemann, and R. A. Arnone, “Comparison and validation of point spread models for imaging in natural waters,” Opt. Express. 16(13), 9958–9965 (2008).
[Crossref] [PubMed]

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).
[Crossref]

Katsev, I. L.

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).
[Crossref]

Kattawar, G. W.

G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of radiative transfer in the earth atmosphere-ocean system: I. flux in the atmosphere and ocean,” J. Phys. Oceanog 2(13), 139–145 (1972).
[Crossref]

Kienle, A.

A. Liemert and A. Kienle, “Green’s functions for the two-dimensional radiative transfer equation in bounded media,” J. Phys. A: Math. Theor. 45, 175201 (2012).
[Crossref]

Larsen, E. W.

C. Börgers and E. W. Larsen, “Asymptotic derivation of the Fermi pencil-beam approximation,” Nucl. Sci. Eng. 123, 343–357 (1996).

Liemert, A.

A. Liemert and A. Kienle, “Green’s functions for the two-dimensional radiative transfer equation in bounded media,” J. Phys. A: Math. Theor. 45, 175201 (2012).
[Crossref]

Markel, V. A.

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phy. Rev. E 79, 036607 (2009).
[Crossref]

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14(1), L13–L19 (2004).
[Crossref]

McCormick, N. J.

McLean, J. W.

Mobley, C. D.

C. D. Mobley, Light and Water: Radiative Transfer In Natural Waters (Academic, 1994).

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean water,” Tech. Rep., Scripps Institution of Oceanography, San Diego, Calif. (1972).

Plass, G. N.

G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of radiative transfer in the earth atmosphere-ocean system: I. flux in the atmosphere and ocean,” J. Phys. Oceanog 2(13), 139–145 (1972).
[Crossref]

Pomraning, G. C.

G. C. Pomraning, A. K. Prinja, and J. W. VanDenburg, “An asymptotic model for the spreding of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).

Prinja, A. K.

G. C. Pomraning, A. K. Prinja, and J. W. VanDenburg, “An asymptotic model for the spreding of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).

Sanchez, R.

Schotland, J. C.

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phy. Rev. E 79, 036607 (2009).
[Crossref]

Shen, L.

Z. Xu, X. Guo, L. Shen, and D. K. P. Yue, “Radiative transfer in ocean turbulence and its effect on underwater light field,” J. Geophys. Res. 117, C00H18 (2012)., doi:
[Crossref]

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011). doi:
[Crossref]

Swanson, N. L.

VanDenburg, J. W.

G. C. Pomraning, A. K. Prinja, and J. W. VanDenburg, “An asymptotic model for the spreding of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).

Voss, K. J.

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011). doi:
[Crossref]

K. J. Voss, “Simple empirical model of oceanic point spread function,” Appl. Opt. 30, 2647–2651 (1991).
[Crossref] [PubMed]

K. J. Voss and A. L. Chapin, “Measurement of the point spread function in the ocean,” Appl. Opt. 29, 3638–3642 (1990).
[Crossref] [PubMed]

Walker, R. E.

Wei, W.

W. Wei, X. Zhang, Y. Chao, and X. Zhou, “An analytical model of the power spatial distribution for underwater optical wireless communication,” Optica Applicata XLII(1), 157–166 (2002).

Weidemann, A. D.

W. Hou, D. J. Gray, A. D. Weidemann, and R. A. Arnone, “Comparison and validation of point spread models for imaging in natural waters,” Opt. Express. 16(13), 9958–9965 (2008).
[Crossref] [PubMed]

Wells, W. H.

W. H. Wells, “Theory of small angle scattering,” in Optics of the Sea (Interface and In-Water Transmission and Imaging), P. Halley, ed., Lecture Series no. 61 (North Atlantic Treaty Organization, Advisory Group for Aerospace Research and Development, Neuilly-sur-Seine, France, 1973).

Xu, Z.

Z. Xu and D. K. P. Yue, “Monte Carlo radiative transfer simulation for the near-ocean-surface high-resolution downwelling irradiance statistics,” Opt. Eng. 53, 051408 (2014). doi:
[Crossref]

Z. Xu, X. Guo, L. Shen, and D. K. P. Yue, “Radiative transfer in ocean turbulence and its effect on underwater light field,” J. Geophys. Res. 117, C00H18 (2012)., doi:
[Crossref]

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011). doi:
[Crossref]

Yue, D. K. P.

Z. Xu and D. K. P. Yue, “Monte Carlo radiative transfer simulation for the near-ocean-surface high-resolution downwelling irradiance statistics,” Opt. Eng. 53, 051408 (2014). doi:
[Crossref]

Z. Xu, X. Guo, L. Shen, and D. K. P. Yue, “Radiative transfer in ocean turbulence and its effect on underwater light field,” J. Geophys. Res. 117, C00H18 (2012)., doi:
[Crossref]

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011). doi:
[Crossref]

Zege, E. P.

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).
[Crossref]

Zhang, X.

W. Wei, X. Zhang, Y. Chao, and X. Zhou, “An analytical model of the power spatial distribution for underwater optical wireless communication,” Optica Applicata XLII(1), 157–166 (2002).

Zhou, X.

W. Wei, X. Zhang, Y. Chao, and X. Zhou, “An analytical model of the power spatial distribution for underwater optical wireless communication,” Optica Applicata XLII(1), 157–166 (2002).

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Appl. Opt. (5)

Astrophys. J. (1)

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[Crossref]

J. Geophys. Res. (2)

Z. Xu, D. K. P. Yue, L. Shen, and K. J. Voss, “Patterns and statistics of in-water polarization under conditions of linear and nonlinear ocean surface waves,” J. Geophys. Res. 116, C00H12 (2011). doi:
[Crossref]

Z. Xu, X. Guo, L. Shen, and D. K. P. Yue, “Radiative transfer in ocean turbulence and its effect on underwater light field,” J. Geophys. Res. 117, C00H18 (2012)., doi:
[Crossref]

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

J. Phys. A: Math. Theor. (1)

A. Liemert and A. Kienle, “Green’s functions for the two-dimensional radiative transfer equation in bounded media,” J. Phys. A: Math. Theor. 45, 175201 (2012).
[Crossref]

J. Phys. Oceanog (1)

G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of radiative transfer in the earth atmosphere-ocean system: I. flux in the atmosphere and ocean,” J. Phys. Oceanog 2(13), 139–145 (1972).
[Crossref]

Nucl. Sci. Eng. (2)

G. C. Pomraning, A. K. Prinja, and J. W. VanDenburg, “An asymptotic model for the spreding of a collimated beam,” Nucl. Sci. Eng. 112, 347–360 (1992).

C. Börgers and E. W. Larsen, “Asymptotic derivation of the Fermi pencil-beam approximation,” Nucl. Sci. Eng. 123, 343–357 (1996).

Opt. Eng. (1)

Z. Xu and D. K. P. Yue, “Monte Carlo radiative transfer simulation for the near-ocean-surface high-resolution downwelling irradiance statistics,” Opt. Eng. 53, 051408 (2014). doi:
[Crossref]

Opt. Express. (1)

W. Hou, D. J. Gray, A. D. Weidemann, and R. A. Arnone, “Comparison and validation of point spread models for imaging in natural waters,” Opt. Express. 16(13), 9958–9965 (2008).
[Crossref] [PubMed]

Optica Applicata (1)

W. Wei, X. Zhang, Y. Chao, and X. Zhou, “An analytical model of the power spatial distribution for underwater optical wireless communication,” Optica Applicata XLII(1), 157–166 (2002).

Phy. Rev. E (1)

L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phy. Rev. E 79, 036607 (2009).
[Crossref]

Waves Random Media (1)

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Media 14(1), L13–L19 (2004).
[Crossref]

Other (8)

T. J. Petzold, “Volume scattering functions for selected ocean water,” Tech. Rep., Scripps Institution of Oceanography, San Diego, Calif. (1972).

J. P. Boyd, Chebyshev and Fourier Spectral Methods (DOVER Publications, 2000).

W. H. Wells, “Theory of small angle scattering,” in Optics of the Sea (Interface and In-Water Transmission and Imaging), P. Halley, ed., Lecture Series no. 61 (North Atlantic Treaty Organization, Advisory Group for Aerospace Research and Development, Neuilly-sur-Seine, France, 1973).

R. E. Walker, Marine Light Field Statistics (Wiley, 1994).

C. D. Mobley, Light and Water: Radiative Transfer In Natural Waters (Academic, 1994).

E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, 1991).
[Crossref]

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

S. Q. Duntley, “Underwater lighting by submerged lasers and incandescent sources,” Tech. Rep., Scripps Institution of Oceanography, San Diego, Calif. (1971).

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Figures (5)

Fig. 1
Fig. 1 Axisymmetric geometry of the problem: uni-directional point source illuminating in positive z-axis under the azimuthal single scattering approximation.
Fig. 2
Fig. 2 Evaluation of the convergence of the 2D analytical solution. Relative error (RE) as the function of N for ω0=0.1, 0.5 and 0.9 at lateral position (a) x = 0.1, (b) x = 0.2, and (c) x = 0.3.
Fig. 3
Fig. 3 2D scalar irradiance solution at optical depth z = 2 under generalize delta function source with ε=0.06, 0.03, 0.015 and 0.
Fig. 4
Fig. 4 Comparisons of the lateral 2D scalar irradiance E s ( 2 ) between Monte Carlo simulations and analytical solutions for (a) optical depth z=2, for different single scattering albedos ω0=0.1, 0.5 and 0.9; (b) ω0=0.5 for z=1, 2, and 3. All analytical BSFs are obtained using N=320 modes.
Fig. 5
Fig. 5 Comparisons of the lateral 3D scalar irradiance E s ( 3 ) between Monte Carlo simulations and analytical solutions for single scattering albedos ω0=0.1, 0.5 and 0.9 at (a) z=1.05, (b) z=2.89, and (c) z=5.

Equations (48)

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

E ( 3 ) ( r , z ) = 0 2 π 0 π L ( r , z , θ , ψ ) sin θ d θ d ψ .
cos θ L z + sin θ ( cos ψ L r sin ψ r L ψ ) + L = ω 0 0 2 π 0 2 π L ( θ , ψ ) Φ ( θ , ψ ; θ , ψ ) sin θ d θ d ψ + 1 π r δ ( r ) δ ( z ) δ ( θ ) δ ( ψ ) ,
L ( r , z , θ , ψ ) = L ^ ( r , z , θ ) δ ( ψ ) .
cos θ L ^ z + sin θ ( L ^ r + L ^ r ) + L = 2 π ω 0 0 π L ^ ( θ ) p ( θ θ ) sin θ d θ + 1 π r δ ( r ) δ ( z ) δ ( θ ) .
p ( θ θ ) = 2 π | sin ( θ θ ) | n = 0 g n P n [ cos ( θ θ ) ] ,
g n = 2 n + 1 2 1 1 Φ ( x ) P n ( x ) d x .
I ( r , z , θ ) = π r L ^ ( r , z , θ ) ,
cos θ I ( r , z , θ ) z + sin θ I ( r , z , θ ) r + I ( r , z , θ ) = ω 0 0 π p ( θ θ ) I ( r , z , θ ) d θ + δ ( r ) δ ( z ) δ ( θ ) .
cos θ I ( x , z , θ ) z + sin θ I ( x , z , θ ) x + I ( x , z , θ ) = ω 0 0 2 π p ( θ θ ) I ( x , z , θ ) d θ + δ ( x ) δ ( z ) δ ( θ ) .
E ( 2 ) ( x , z ) = 0 2 π I ( x , z , θ ) d θ .
E ( 3 ) ( r , z ) = 1 π r E ( 2 ) ( | x | , z ) .
I ( x , z , θ ) = 1 ( 2 π ) 2 I ^ ( k x , k z , θ ) e i ( x k x + z k z ) d k x d k z ,
I ^ ( k x , k z , θ ) = I ( x , z , θ ) e i ( x k x + z k z ) d x d z ,
( i k cos ( θ θ k ) + ) I ^ ( k , θ k , θ ) = ω 0 0 2 π p ( θ θ ) I ^ ( k , θ k , θ ) d θ + S ( k , θ k , θ ) .
I ^ ( k , θ k , θ ) = I ^ ( k , θ k , α ) = m = I m ( k , θ k ) e i m α , and
I m ( k , θ k ) = 1 2 π 0 2 π I ^ ( k , θ k , α ) e i m α d α .
p ( θ θ ) = 1 2 π m = p m e i m ( θ θ ) , and
p m = 0 2 π p ( θ θ ) e i m ( θ θ ) d θ .
S m ( k , θ k ) = 1 2 π 0 2 π δ ( θ ) e i m α d α = 1 2 π e i m θ k .
i k 2 ( I m 1 + I m + 1 ) + ( 1 ω 0 p m ) I m = S m .
i k 2 A X ¯ + B X ¯ = S ¯ ,
X ¯ = [ I N I 0 I N ] T = [ x 1 x N x 2 N + 1 ] T ,
S ¯ = [ S N S 0 S N ] T = [ s 1 s N s 2 N + 1 ] T ,
s = S N 1 = 1 2 π e i ( N 1 ) θ k .
A n , m = δ n , m + 1 + δ n + 1 , m , and B n , m = β n δ n , m ,
C = B 1 / 2 , and D = CAC .
X ¯ = C ( i k 2 D + I ) 1 C S ¯ ,
D n , m = 1 β n β m ( δ n , m + 1 + δ n + 1 , m ) .
D = V Σ V 1 ,
x n = = 1 2 N + 1 m = 1 2 N + 1 V n m V m β n β 1 i k λ m / 2 + 1 s .
I ^ ( k , θ k , θ ) = = 1 2 N + 1 m = 1 2 N + 1 V N + 1 , m V m β N + 1 β 1 i k λ m / 2 + 1 e i ( N 1 ) θ k .
E ( 2 ) ( k , θ k ) = 0 2 π I ^ ( k , θ k , θ ) d θ .
E ( 2 ) ( x , z ) = E ( 2 ) ( ρ , θ ρ ) = 1 ( 2 π ) 2 0 2 π 0 k E ( 2 ) ( k , θ k ) e i k ρ cos ( θ k θ r ) d k d θ k .
e i k ρ cos ( θ k θ ρ ) = n = i n J n ( k ρ ) e i n ( θ k θ ρ ) ,
E ( 2 ) ( ρ , θ ρ ) = 1 2 π = 1 2 N + 1 m = 1 2 N + 1 i N 1 V n m V m β N + 1 β e i ( N 1 ) θ ρ 0 k i k / 2 λ m + 1 J N 1 ( k ρ ) d k .
V 2 N + 2 n , m = ( 1 ) m 1 V n m , V n , 2 N + 2 m = ( 1 ) n 1 V n m , V 2 N + 2 n , 2 N + 2 m = ( 1 ) m + n V n m , β m = β 2 N + 2 m , λ m = λ 2 N + 2 m , λ N + 1 = 0 .
E ( 2 ) ( ρ , θ ρ ) = 4 π m = 1 N V N + 1 , m 2 β N + 1 λ m 2 K 0 ( 2 ρ λ m ) + 1 2 π n = 1 N m = 1 N V N + 1 , m V n + N + 1 , m β N + 1 β n + N + 1 { [ 1 + ( 1 ) n ] [ 1 + ( 1 ) n + m 1 ] cos [ n ( θ ρ + π / 2 ) ] H n R ( ρ , λ 2 m + 1 ) + 2 λ m [ 1 + ( 1 ) n + 1 ] [ 1 + ( 1 ) n + m ] sin [ n ( θ ρ + π / 2 ) ] [ 1 ρ H n I ( ρ , λ 2 m + 1 ) ] } ,
H n R ( ρ , λ ) = 0 k J n ( k ρ ) 1 + k 2 λ 2 / 4 d k = 4 λ 2 n F 1 2 ( 1 ; 1 + n 2 , 1 n 2 ; ρ 2 λ 2 ) 2 π λ 2 I n ( 2 ρ λ ) csc ( n π 2 ) ,
H n I ( ρ , λ ) = 0 J n ( k ρ ) 1 + k 2 λ 2 / 4 d k = 4 ρ λ 2 ( n 2 1 ) F 1 2 ( 1 ; 3 2 n 2 , 3 2 + n 2 ; ρ 2 λ 2 ) π λ I n ( 2 ρ λ ) sec ( n π 2 ) .
λ m b = 2 cos ( m π 2 N + 1 ) ,
V n m b = 1 N + 1 sin ( n m π 2 N + 2 ) .
E s ( 2 ) ( ρ , θ ρ ) = E ( 2 ) ( ρ , θ ρ ) E b ( 2 ) ( ρ , θ ρ ) .
E s ( 2 ) ( x , z ) = E ( 2 ) ( x 2 + z 2 , arctan x z ) E b ( 2 ) ( x 2 + z 2 , arctan x z ) .
E s ( 3 ) ( r , z ) = 1 π r [ E ( 2 ) ( r 2 + z 2 , arctan r z ) E b ( 2 ) ( r 2 + z 2 , arctan r z ) ] .
p ( Φ ) = p ( θ θ ) = 1 2 π 1 g 2 1 + g 2 2 g cos ( θ θ ) , g = 0.924 ,
RE = | E s , N ( 2 ) E s , ( 2 ) | | E s , ( 2 ) | ,
S ( x , z , θ ) = δ ( x ) δ ( z ) 1 π ε exp ( θ 2 2 ε 2 ) .
Φ ( Θ ) = 1 4 π 1 g 2 ( 1 + g 2 2 g cos Θ ) 3 / 2 , g = 0.924 ,

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