Abstract

The properties of a pencil of light as defined approximately in the geometric optics ray tracing method are investigated. The vector Kirchhoff integral is utilized to accurately compute the electromagnetic near field in and around the pencil of light with various beam base sizes, shapes, propagation directions and medium refractive indices. If a pencil of light has geometric mean cross section size of the order p times the wavelength, it can propagate independently to a distance p2 times the wavelength, where most of the beam energy diffuses out of the beam region. This is consistent with a statement that van de Hulst made in a classical text on light scattering. The electromagnetic near fields in the pencil of light are not uniform, have complicated patterns within short distances from the beam base, and the fields tend to converge to Fraunhofer diffraction fields far away from the base.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  11. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
    [Crossref]
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    [Crossref]
  13. L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
    [Crossref]
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    [Crossref]
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    [Crossref]
  18. L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
    [Crossref]
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    [Crossref]
  20. B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044 (2017).
    [Crossref]
  21. A. Konoshonkin, A. Borovoi, N. Kustova, H. Okamoto, H. Ishimoto, Y. Grynko, and J. Förstner, “Light scattering by ice crystals of cirrus clouds: From exact numerical methods to physical-optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 195, 132–140 (2017).
    [Crossref]
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    [Crossref]
  23. M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics (Cambridge University Press, 1999).
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2019 (1)

P. Yang, J. Ding, R. L. Panetta, K.-N. Liou, G. W. Kattawar, and M. I. Mishchenko, “On the convergence of numerical computations for both exact and approximate solutions for electromagnetic scattering by nonspherical dielectric particles (invited review),” Prog. Electromagn. Res. 164, 27–61 (2019).
[Crossref]

2017 (2)

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044 (2017).
[Crossref]

A. Konoshonkin, A. Borovoi, N. Kustova, H. Okamoto, H. Ishimoto, Y. Grynko, and J. Förstner, “Light scattering by ice crystals of cirrus clouds: From exact numerical methods to physical-optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 195, 132–140 (2017).
[Crossref]

2014 (2)

A. Borovoi, A. Konoshonkin, and N. Kustova, “The physical-optics approximation and its application to light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 146, 181–189 (2014).
[Crossref]

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

2012 (1)

C. Liu, R. Lee Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Radiat. Transf. 113(13), 1728–1740 (2012).
[Crossref]

2011 (1)

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

2007 (1)

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

2003 (1)

1997 (1)

1996 (4)

1995 (2)

1993 (1)

1991 (1)

1989 (1)

1982 (1)

1979 (1)

Baum, B. A.

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

Bhatia, A. B.

M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics (Cambridge University Press, 1999).

Bi, L.

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

Born, M.

M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics (Cambridge University Press, 1999).

Borovoi, A.

A. Konoshonkin, A. Borovoi, N. Kustova, H. Okamoto, H. Ishimoto, Y. Grynko, and J. Förstner, “Light scattering by ice crystals of cirrus clouds: From exact numerical methods to physical-optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 195, 132–140 (2017).
[Crossref]

A. Borovoi, A. Konoshonkin, and N. Kustova, “The physical-optics approximation and its application to light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 146, 181–189 (2014).
[Crossref]

Borovoi, A. G.

Cai, Q.

Carlson, B. E.

Clemmow, P. C.

M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics (Cambridge University Press, 1999).

Ding, J.

P. Yang, J. Ding, R. L. Panetta, K.-N. Liou, G. W. Kattawar, and M. I. Mishchenko, “On the convergence of numerical computations for both exact and approximate solutions for electromagnetic scattering by nonspherical dielectric particles (invited review),” Prog. Electromagn. Res. 164, 27–61 (2019).
[Crossref]

Förstner, J.

A. Konoshonkin, A. Borovoi, N. Kustova, H. Okamoto, H. Ishimoto, Y. Grynko, and J. Förstner, “Light scattering by ice crystals of cirrus clouds: From exact numerical methods to physical-optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 195, 132–140 (2017).
[Crossref]

Gabor, D.

M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics (Cambridge University Press, 1999).

Grishin, I. A.

Grynko, Y.

A. Konoshonkin, A. Borovoi, N. Kustova, H. Okamoto, H. Ishimoto, Y. Grynko, and J. Förstner, “Light scattering by ice crystals of cirrus clouds: From exact numerical methods to physical-optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 195, 132–140 (2017).
[Crossref]

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

Hu, Y.

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

Ishimoto, H.

A. Konoshonkin, A. Borovoi, N. Kustova, H. Okamoto, H. Ishimoto, Y. Grynko, and J. Förstner, “Light scattering by ice crystals of cirrus clouds: From exact numerical methods to physical-optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 195, 132–140 (2017).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley & Sons, Inc., 1975).

Kattawar, G. W.

P. Yang, J. Ding, R. L. Panetta, K.-N. Liou, G. W. Kattawar, and M. I. Mishchenko, “On the convergence of numerical computations for both exact and approximate solutions for electromagnetic scattering by nonspherical dielectric particles (invited review),” Prog. Electromagn. Res. 164, 27–61 (2019).
[Crossref]

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044 (2017).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

Konoshonkin, A.

A. Konoshonkin, A. Borovoi, N. Kustova, H. Okamoto, H. Ishimoto, Y. Grynko, and J. Förstner, “Light scattering by ice crystals of cirrus clouds: From exact numerical methods to physical-optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 195, 132–140 (2017).
[Crossref]

A. Borovoi, A. Konoshonkin, and N. Kustova, “The physical-optics approximation and its application to light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 146, 181–189 (2014).
[Crossref]

Kustova, N.

A. Konoshonkin, A. Borovoi, N. Kustova, H. Okamoto, H. Ishimoto, Y. Grynko, and J. Förstner, “Light scattering by ice crystals of cirrus clouds: From exact numerical methods to physical-optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 195, 132–140 (2017).
[Crossref]

A. Borovoi, A. Konoshonkin, and N. Kustova, “The physical-optics approximation and its application to light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 146, 181–189 (2014).
[Crossref]

Lacis, A.

M. Mischenko, L. Travis, and A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002).

Lee Panetta, R.

C. Liu, R. Lee Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Radiat. Transf. 113(13), 1728–1740 (2012).
[Crossref]

Liou, K. N.

Liou, K.-N.

P. Yang, J. Ding, R. L. Panetta, K.-N. Liou, G. W. Kattawar, and M. I. Mishchenko, “On the convergence of numerical computations for both exact and approximate solutions for electromagnetic scattering by nonspherical dielectric particles (invited review),” Prog. Electromagn. Res. 164, 27–61 (2019).
[Crossref]

Liu, C.

C. Liu, R. Lee Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Radiat. Transf. 113(13), 1728–1740 (2012).
[Crossref]

Lock, J. A.

Macke, A.

Mischenko, M.

M. Mischenko, L. Travis, and A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002).

Mishchenko, M. I.

P. Yang, J. Ding, R. L. Panetta, K.-N. Liou, G. W. Kattawar, and M. I. Mishchenko, “On the convergence of numerical computations for both exact and approximate solutions for electromagnetic scattering by nonspherical dielectric particles (invited review),” Prog. Electromagn. Res. 164, 27–61 (2019).
[Crossref]

A. Macke, M. I. Mishchenko, K. Muinonen, and B. E. Carlson, “Scattering of light by large nonspherical particles: ray-tracing approximation versus T-matrix method,” Opt. Lett. 20(19), 1934 (1995).
[Crossref]

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8(6), 871 (1991).
[Crossref]

Muinonen, K.

Okamoto, H.

A. Konoshonkin, A. Borovoi, N. Kustova, H. Okamoto, H. Ishimoto, Y. Grynko, and J. Förstner, “Light scattering by ice crystals of cirrus clouds: From exact numerical methods to physical-optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 195, 132–140 (2017).
[Crossref]

Panetta, R. L.

P. Yang, J. Ding, R. L. Panetta, K.-N. Liou, G. W. Kattawar, and M. I. Mishchenko, “On the convergence of numerical computations for both exact and approximate solutions for electromagnetic scattering by nonspherical dielectric particles (invited review),” Prog. Electromagn. Res. 164, 27–61 (2019).
[Crossref]

Stokes, A. R.

M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics (Cambridge University Press, 1999).

Sun, B.

Taylor, A. M.

M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics (Cambridge University Press, 1999).

Travis, L.

M. Mischenko, L. Travis, and A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley and Sons, 1957).

Wayman, P. A.

M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics (Cambridge University Press, 1999).

Weickmann, H. K.

Wendling, P.

Wendling, R.

Wilcock, W. L.

M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics (Cambridge University Press, 1999).

Wolf, E.

M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics (Cambridge University Press, 1999).

Yang, P.

P. Yang, J. Ding, R. L. Panetta, K.-N. Liou, G. W. Kattawar, and M. I. Mishchenko, “On the convergence of numerical computations for both exact and approximate solutions for electromagnetic scattering by nonspherical dielectric particles (invited review),” Prog. Electromagn. Res. 164, 27–61 (2019).
[Crossref]

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044 (2017).
[Crossref]

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

C. Liu, R. Lee Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Radiat. Transf. 113(13), 1728–1740 (2012).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

P. Yang and K. N. Liou, “Light scattering by hexagonal ice crystals: solutions by a ray-by-ray integration algorithm,” J. Opt. Soc. Am. A 14(9), 2278 (1997).
[Crossref]

P. Yang and K. N. Liou, “Geometric-optics–integral-equation method for light scattering by nonspherical ice crystals,” Appl. Opt. 35(33), 6568 (1996).
[Crossref]

P. Yang and K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13(10), 2072 (1996).
[Crossref]

P. Yang and K. N. Liou, “Light scattering by hexagonal ice crystals: comparison of finite-difference time domain and geometric optics models,” J. Opt. Soc. Am. A 12(1), 162 (1995).
[Crossref]

Yurkin, M. A.

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

Zhang, X.

Appl. Opt. (7)

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

J. Quant. Spectrosc. Radiat. Transf. (6)

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

C. Liu, R. Lee Panetta, and P. Yang, “Application of the pseudo-spectral time domain method to compute particle single-scattering properties for size parameters up to 200,” J. Quant. Spectrosc. Radiat. Transf. 113(13), 1728–1740 (2012).
[Crossref]

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, Y. Hu, and B. A. Baum, “Scattering and absorption of light by ice particles: Solution by a new physical-geometric optics hybrid method,” J. Quant. Spectrosc. Radiat. Transf. 112(9), 1492–1508 (2011).
[Crossref]

A. Borovoi, A. Konoshonkin, and N. Kustova, “The physical-optics approximation and its application to light backscattering by hexagonal ice crystals,” J. Quant. Spectrosc. Radiat. Transf. 146, 181–189 (2014).
[Crossref]

A. Konoshonkin, A. Borovoi, N. Kustova, H. Okamoto, H. Ishimoto, Y. Grynko, and J. Förstner, “Light scattering by ice crystals of cirrus clouds: From exact numerical methods to physical-optics approximation,” J. Quant. Spectrosc. Radiat. Transf. 195, 132–140 (2017).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Prog. Electromagn. Res. (1)

P. Yang, J. Ding, R. L. Panetta, K.-N. Liou, G. W. Kattawar, and M. I. Mishchenko, “On the convergence of numerical computations for both exact and approximate solutions for electromagnetic scattering by nonspherical dielectric particles (invited review),” Prog. Electromagn. Res. 164, 27–61 (2019).
[Crossref]

Other (4)

M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics (Cambridge University Press, 1999).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley & Sons, Inc., 1975).

M. Mischenko, L. Travis, and A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002).

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley and Sons, 1957).

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Figures (10)

Fig. 1.
Fig. 1. Illustration of Huygens–Fresnel principle.
Fig. 2.
Fig. 2. The near field Poynting vectors on the xoz plane for beam base sizes d = 5, 10 and 15 times the wavelength. The medium refractive index is m = 1. The propagation direction of the field on the beam base is along the z axis. The contour plots show the Poynting vector magnitude $|\bar{{{\textbf {S}}}}|$ , and the arrows show the Poynting vector directions. The lengths of the arrows are proportional to the square root of the corresponding Poynting vector magnitude. The red dashed lines indicate the distance from the beam base l = d2 times the wavelength. Upper plot: The beam base shape is a square with side width d; Lower plot: The beam base shape is a circle with diameter d.
Fig. 3.
Fig. 3. The near field Poynting vectors on the xoz plane for beam base sizes d = 10 times the wavelength. The medium refractive index is m = 1. The propagation directions of the field on the beam base have angles θ = 0°, 30°, and 60° relative to the z axis. The contour plots show the Poynting vector magnitude $|\bar{{{\textbf {S}}}}|$ , and the arrows show the Poynting vector directions. The lengths of the arrows are proportional to the square root of the corresponding Poynting vector magnitude. The red solid lines indicate the distance from the beam base l = d2 times the wavelength, and the red dashed lines indicate the distance from the beam base l′ = d2cosθ times the wavelength. Left plot: The beam base shape is a square with side width d; Right plot: The beam base shape is a circle with diameter d.
Fig. 4.
Fig. 4. The near field Poynting vectors on the beam cross section planes at various distances from the beam base. The cross section planes are perpendicular to the propagation directions of the field on the beam base. The beam base is a square, and has side width d = 15 times the wavelength. The medium refractive index is m = 1. The propagation direction of the field on the beam base is along the z axis. The primed x, y and z are the Cartesian coordinates with z′ along the beam propagation direction. The contour plots show the Poynting vector magnitude $|\bar{{{\textbf {S}}}}|$ , and the arrows show the Poynting vector directions. The lengths of the arrows are proportional to the square root of the corresponding Poynting vector magnitude.
Fig. 5.
Fig. 5. Same as Fig. 4, except the beam base shape is a circle with diameter d = 15 times the wavelength.
Fig. 6.
Fig. 6. The ratios between the energy fluxes crossing the beam cross section computed by the vector Kirchhoff integral (PK) and ray tracing (PR). The beam bases are circles, and have diameters d = 5, 10 and 15 times the wavelength. The propagation directions of the field on the beam base have angles θ = 0° and 60° relative to the z axis. The medium refractive index is m = 1. The black dots on the curves indicate the distance from the beam base l = d2cosθ times the wavelength.
Fig. 7.
Fig. 7. The ratios between the power crossing the beam cross section computed by the vector Kirchhoff integral (PK) and ray tracing (PR). The beam bases are square, and have side width d = 10 times the vacuum wavelength. The propagation direction of the field on the beam base is along the z axis. Four medium refractive indices m (=1.0, 1.3, 1.5, and 2.0) are considered. The black dots on the curves indicate the distance from the beam base l = md2 times the wavelength in a vacuum.
Fig. 8.
Fig. 8. The scattering phase functions of a hexagonal column with aspect ratio (2a/H) 1 and size parameter (ka) 100 computed by the II-TM, PGOMS, and PGOMV methods. Left: refractive index m=1.308; Right: refractive index m=1.05.
Fig. 9.
Fig. 9. The Poynting vector component along the propagation direction of the field on the beam base at the distance l=225 times the wavelength from the beam base, and y = 0. The beam base size is 15 times the wavelength. The medium refractive index is m = 1. The numerical results for the square and circle beam bases are fitted by two analytical equations.
Fig. 10.
Fig. 10. The far field Poynting vector magnitude $|\bar{{{\textbf {S}}}}|$ on the beam cross section planes at 1000 times wavelength from beam base. For viewing purpose, the contour plots are the quarter root of $|\bar{{{\textbf {S}}}}|$ . The cross section planes are perpendicular to the propagation directions of the field on the beam base. The medium refractive index is m = 1. The primed x, y and z are the Cartesian coordinates with z′ along the beam propagation direction. The propagation direction of the field on the beam base is along the z axis. Left: The beam base is a square, and has side width d = 5 times the wavelength; Right: The beam base shape is a circle with diameter d = 15 times the wavelength.

Equations (14)

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u Q 1 = P u P i exp ( i k r ) r λ K ( χ ) d S ,
u Q = exp ( i k l ) u P ,
u Q 1 = P u P i exp ( i k l ) l λ d S .
E ( r ) = E 0 exp [ i ( k r ω t ) ] ,
H ( r ) = B ( r ) = × E ( r ) i k 0 ,
H ( r ) = B ( r ) = k × E ( r ) k 0 .
S ¯ = c 8 π Re ( E × H ) ,
S ¯ = c 8 π k 0 Re ( | E | 2 ) k .
E ( r ) = S { i k 0 [ n × B ( r ) ] G + [ n × E ( r ) ] × G + [ n E ( r ) ] G } d r 2 ,
G ( r , r ) = exp ( i k | r r | ) 4 π | r r | .
E ( r ) = 8 π k 0 m c exp [ i ( k r ω t ) ] ( 1 , 0 , 0 ) ,
P = S S ¯ ( r ) k ~ d 2 r ,
S ¯ k ~ = A ( sin κ x κ x ) 4 + B ,
S ¯ k ~ = A [ J 1 ( κ x ) κ x ] 2 + B ,

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