Abstract

Here we studied the light transport initiated by oblique illumination of a turbid cylinder on its top or barrel with a pencil beam. The pencil beam was approximated by an isotropic point source according to the incident point and angle. Solutions were derived for the steady-state, frequency, and time domains. Time- and spatially resolved reflectance results with different incident angles agreed well with those of Monte Carlo simulations. A method to determine optical properties using the time- and spatially resolved reflectance results along the z direction was proposed for the light incident on the cylinder barrel. Analytical solutions can improve analysis of turbid cylindrical media with incident illumination and effective optical properties reconstruction.

© 2018 Optical Society of America

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References

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2014 (1)

I. Steinberg, O. Harbater, and I. Gannot, “Robust estimation of cerebral hemodynamics in neonates using multilayered diffusion model for normal and oblique incidences,” J. Biomed. Opt. 19, 071406 (2014).
[Crossref]

2013 (3)

A. R. Gardner, A. D. Kim, and V. Venugopalan, “Radiative transport produced by oblique illumination of turbid media with collimated beams,” Phys. Rev. E 87, 2513–2514 (2013).
[Crossref]

A. Liemert and A. Kienle, “Exact and efficient solution of the radiative transport equation for the semi-infinite medium,” Sci. Rep. 3, 2018 (2013).
[Crossref]

R. J. Zemp, “Phase-function corrected diffusion model for diffuse reflectance of a pencil beam obliquely incident on a semi-infinite turbid medium,” J. Biomed. Opt. 18, 067005 (2013).
[Crossref]

2012 (1)

2011 (2)

A. Liemert and A. Kienle, “Light diffusion in a radially N-layered cylinder,” Phys. Rev. E 84, 041911 (2011).
[Crossref]

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[Crossref]

2010 (6)

A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt. 15, 025002 (2010).
[Crossref]

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73, 076701 (2010).
[Crossref]

T. Li, H. U. I. Gong, and Q. Luo, “MCVM: Monte Carlo modeling of photon migration in voxelized media,” J. Innov. Opt. Health Sci. 03, 91–102 (2010).
[Crossref]

A. Zhang, D. Piao, C. F. Bunting, and B. W. Pogue, “Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. I. Steady-state theory,” J. Opt. Soc. Am. A 27, 648–662 (2010).
[Crossref]

A. Liemert and A. Kienle, “Light diffusion in a turbid cylinder. II. Layered case,” Opt. Express 18, 9266–9279 (2010).
[Crossref]

A. Kienle and A. Liemert, “Light diffusion in a turbid cylinder. I. Homogeneous case,” Opt. Express 18, 9456–9473 (2010).
[Crossref]

2008 (2)

A. Bassi, D. J. Cuccia, A. J. Durkin, and B. J. Tromberg, “Spatial shift of spatially modulated light projected on turbid media,” J. Opt. Soc. Am. A 25, 2833–2839 (2008).
[Crossref]

J. Q. Lu, C. Chen, D. W. Pravica, R. S. Brock, and X. H. Hu, “Validity of a closed-form diffusion solution in P1 approximation for reflectance imaging with an oblique beam of arbitrary profile,” Med. Phys. 35, 3979–3987 (2008).
[Crossref]

2005 (1)

2004 (1)

2001 (1)

2000 (1)

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

1998 (1)

1997 (1)

1995 (2)

L. Wang and S. L. Jacques, “Use of a laser beam with an oblique angle of incidence to measure the reduced scattering coefficient of a turbid medium,” Appl. Opt. 34, 2362–2366 (1995).
[Crossref]

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[Crossref]

1994 (1)

1989 (1)

Arendt, J. T.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

Backman, V.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

Baker, W. B.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73, 076701 (2010).
[Crossref]

Bassi, A.

Bays, R.

Brock, R. S.

J. Q. Lu, C. Chen, D. W. Pravica, R. S. Brock, and X. H. Hu, “Validity of a closed-form diffusion solution in P1 approximation for reflectance imaging with an oblique beam of arbitrary profile,” Med. Phys. 35, 3979–3987 (2008).
[Crossref]

Bunting, C. F.

Chance, B.

Chen, C.

J. Q. Lu, C. Chen, D. W. Pravica, R. S. Brock, and X. H. Hu, “Validity of a closed-form diffusion solution in P1 approximation for reflectance imaging with an oblique beam of arbitrary profile,” Med. Phys. 35, 3979–3987 (2008).
[Crossref]

Choe, R.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73, 076701 (2010).
[Crossref]

Cuccia, D. J.

Dögnitz, N.

Durduran, T.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73, 076701 (2010).
[Crossref]

Durkin, A. J.

Duvic, M.

Feng, T. C.

Gannot, I.

I. Steinberg, O. Harbater, and I. Gannot, “Robust estimation of cerebral hemodynamics in neonates using multilayered diffusion model for normal and oblique incidences,” J. Biomed. Opt. 19, 071406 (2014).
[Crossref]

Garcia-Uribe, A.

Gardner, A. R.

A. R. Gardner, A. D. Kim, and V. Venugopalan, “Radiative transport produced by oblique illumination of turbid media with collimated beams,” Phys. Rev. E 87, 2513–2514 (2013).
[Crossref]

Gong, H. U. I.

T. Li, H. U. I. Gong, and Q. Luo, “MCVM: Monte Carlo modeling of photon migration in voxelized media,” J. Innov. Opt. Health Sci. 03, 91–102 (2010).
[Crossref]

Guo, L.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[Crossref]

Gurjar, R.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

Hanlon, E. B.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[Crossref]

Harbater, O.

I. Steinberg, O. Harbater, and I. Gannot, “Robust estimation of cerebral hemodynamics in neonates using multilayered diffusion model for normal and oblique incidences,” J. Biomed. Opt. 19, 071406 (2014).
[Crossref]

Haskell, R. C.

Hu, X. H.

J. Q. Lu, C. Chen, D. W. Pravica, R. S. Brock, and X. H. Hu, “Validity of a closed-form diffusion solution in P1 approximation for reflectance imaging with an oblique beam of arbitrary profile,” Med. Phys. 35, 3979–3987 (2008).
[Crossref]

Itzkan, I.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[Crossref]

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[Crossref]

L. Wang and S. L. Jacques, “Use of a laser beam with an oblique angle of incidence to measure the reduced scattering coefficient of a turbid medium,” Appl. Opt. 34, 2362–2366 (1995).
[Crossref]

Kehtarnavaz, N.

Kienle, A.

Kim, A. D.

A. R. Gardner, A. D. Kim, and V. Venugopalan, “Radiative transport produced by oblique illumination of turbid media with collimated beams,” Phys. Rev. E 87, 2513–2514 (2013).
[Crossref]

Kline, E.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

Li, T.

T. Li, H. U. I. Gong, and Q. Luo, “MCVM: Monte Carlo modeling of photon migration in voxelized media,” J. Innov. Opt. Health Sci. 03, 91–102 (2010).
[Crossref]

Liemert, A.

A. Liemert and A. Kienle, “Exact and efficient solution of the radiative transport equation for the semi-infinite medium,” Sci. Rep. 3, 2018 (2013).
[Crossref]

A. Liemert and A. Kienle, “Spatially modulated light source obliquely incident on a semi-infinite scattering medium,” Opt. Lett. 37, 4158–4160 (2012).
[Crossref]

A. Liemert and A. Kienle, “Light diffusion in a radially N-layered cylinder,” Phys. Rev. E 84, 041911 (2011).
[Crossref]

A. Kienle and A. Liemert, “Light diffusion in a turbid cylinder. I. Homogeneous case,” Opt. Express 18, 9456–9473 (2010).
[Crossref]

A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt. 15, 025002 (2010).
[Crossref]

A. Liemert and A. Kienle, “Light diffusion in a turbid cylinder. II. Layered case,” Opt. Express 18, 9266–9279 (2010).
[Crossref]

Lu, J. Q.

J. Q. Lu, C. Chen, D. W. Pravica, R. S. Brock, and X. H. Hu, “Validity of a closed-form diffusion solution in P1 approximation for reflectance imaging with an oblique beam of arbitrary profile,” Med. Phys. 35, 3979–3987 (2008).
[Crossref]

Luo, Q.

T. Li, H. U. I. Gong, and Q. Luo, “MCVM: Monte Carlo modeling of photon migration in voxelized media,” J. Innov. Opt. Health Sci. 03, 91–102 (2010).
[Crossref]

Marquez, G.

Martelli, F.

Mcadams, M. S.

Mcgillican, T.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

Müller, M. G.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

Patterson, M. S.

Perelman, L. T.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[Crossref]

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

Piao, D.

Pogue, B. W.

Pravica, D. W.

J. Q. Lu, C. Chen, D. W. Pravica, R. S. Brock, and X. H. Hu, “Validity of a closed-form diffusion solution in P1 approximation for reflectance imaging with an oblique beam of arbitrary profile,” Med. Phys. 35, 3979–3987 (2008).
[Crossref]

Prieto, V.

Qiu, L.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[Crossref]

Sassaroli, A.

Steinberg, I.

I. Steinberg, O. Harbater, and I. Gannot, “Robust estimation of cerebral hemodynamics in neonates using multilayered diffusion model for normal and oblique incidences,” J. Biomed. Opt. 19, 071406 (2014).
[Crossref]

Svaasand, L. O.

Tromberg, B. J.

Tsay, T. T.

Turzhitsky, V.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[Crossref]

van den Bergh, H.

Venugopalan, V.

A. R. Gardner, A. D. Kim, and V. Venugopalan, “Radiative transport produced by oblique illumination of turbid media with collimated beams,” Phys. Rev. E 87, 2513–2514 (2013).
[Crossref]

Vitkin, E.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[Crossref]

Wagnières, G.

Wallace, M. B.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[Crossref]

L. Wang and S. L. Jacques, “Use of a laser beam with an oblique angle of incidence to measure the reduced scattering coefficient of a turbid medium,” Appl. Opt. 34, 2362–2366 (1995).
[Crossref]

Wang, L. V.

Wilson, B. C.

Yamada, Y.

Yodh, A. G.

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73, 076701 (2010).
[Crossref]

Zaccanti, G.

Zemp, R. J.

R. J. Zemp, “Phase-function corrected diffusion model for diffuse reflectance of a pencil beam obliquely incident on a semi-infinite turbid medium,” J. Biomed. Opt. 18, 067005 (2013).
[Crossref]

Zhang, A.

Zhang, Q.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[Crossref]

Zonios, G.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

Appl. Opt. (5)

Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[Crossref]

J. Biomed. Opt. (3)

A. Liemert and A. Kienle, “Light diffusion in N-layered turbid media: frequency and time domains,” J. Biomed. Opt. 15, 025002 (2010).
[Crossref]

R. J. Zemp, “Phase-function corrected diffusion model for diffuse reflectance of a pencil beam obliquely incident on a semi-infinite turbid medium,” J. Biomed. Opt. 18, 067005 (2013).
[Crossref]

I. Steinberg, O. Harbater, and I. Gannot, “Robust estimation of cerebral hemodynamics in neonates using multilayered diffusion model for normal and oblique incidences,” J. Biomed. Opt. 19, 071406 (2014).
[Crossref]

J. Innov. Opt. Health Sci. (1)

T. Li, H. U. I. Gong, and Q. Luo, “MCVM: Monte Carlo modeling of photon migration in voxelized media,” J. Innov. Opt. Health Sci. 03, 91–102 (2010).
[Crossref]

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

Med. Phys. (1)

J. Q. Lu, C. Chen, D. W. Pravica, R. S. Brock, and X. H. Hu, “Validity of a closed-form diffusion solution in P1 approximation for reflectance imaging with an oblique beam of arbitrary profile,” Med. Phys. 35, 3979–3987 (2008).
[Crossref]

Nat. Commun. (1)

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[Crossref]

Nature (1)

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Müller, Q. Zhang, G. Zonios, E. Kline, and T. Mcgillican, “Detection of preinvasive cancer cells,” Nature 406, 35–36 (2000).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. E (2)

A. Liemert and A. Kienle, “Light diffusion in a radially N-layered cylinder,” Phys. Rev. E 84, 041911 (2011).
[Crossref]

A. R. Gardner, A. D. Kim, and V. Venugopalan, “Radiative transport produced by oblique illumination of turbid media with collimated beams,” Phys. Rev. E 87, 2513–2514 (2013).
[Crossref]

Rep. Prog. Phys. (1)

T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, “Diffuse optics for tissue monitoring and tomography,” Rep. Prog. Phys. 73, 076701 (2010).
[Crossref]

Sci. Rep. (1)

A. Liemert and A. Kienle, “Exact and efficient solution of the radiative transport equation for the semi-infinite medium,” Sci. Rep. 3, 2018 (2013).
[Crossref]

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Figures (11)

Fig. 1.
Fig. 1. Schematic of a cylinder. The beam is normally incident on the cylinder top at r = ( ρ 0 , θ 0 , z 0 ) . The equivalent isotropic point source is located at r 0 = ( ρ 0 , θ 0 , z 0 ) , where z 0 = z 0 + 1 / μ t .
Fig. 2.
Fig. 2. Scheme of the three situations for light obliquely incident on the cylinder: (a) incident on the top, (b) incident on the barrel in plane θ 0 = 0 , and (c) incident on the barrel in plane z = z 0 .
Fig. 3.
Fig. 3. Schematic of the equivalent image source structure of the boundary conditions of the z -axis of the cylinder for an isotropic source at z = z 0 .
Fig. 4.
Fig. 4. Diagram of the cylinder geometry model composed with voxels.
Fig. 5.
Fig. 5. (a) Time-resolved reflectance and (b) error between analytical solutions and Monte Carlo simulations when light is incident on the top of the cylinder with different incident angles; (c) time-resolved reflectance and (d) error between analytical solutions and Monte Carlo simulations with different incident angles when light is incident on the barrel of the cylinder, as shown in Fig. 2(b); and (e) time-resolved reflectance and (f) error between analytical solutions and Monte Carlo simulations with different incident angles when light is incident on the barrel of the cylinder, as shown in Fig. 2(c).
Fig. 6.
Fig. 6. Time-resolved light distribution (photon flux) of the incidence plane in the cylinder for (a) light obliquely incident on the barrel of the cylinder, as shown in Fig. 2(b). The incidence plane shown here is the plane θ 0 = 0 . (b) Light obliquely incident on the barrel of the cylinder, as depicted in Fig. 2(c). The incidence plane shown here is the plane z = z 0 . The white number at right corner of each figure is the time (ns) after light incident. The color bar of each figure represents the log of intensity of light flux. It is clearly demonstrated that the light is reflected at the boundary of the cylinder and the reflected light is transported in the cylinder.
Fig. 7.
Fig. 7. (a) Comparison of the inverse absorption coefficient from MC results with the real absorption coefficient ( μ a = 0.05 , 0.5, 5    mm 1 ). (b) Percentage error of the inverse absorption coefficient with different oblique incident angles. The squares connected by dashed lines mean the incident light is in the θ 0 = 0 plane, as shown in Fig. 2(b); the circles connected by solid lines mean the incident light is in the z = z 0 plane, as shown in Fig. 2(c). For all cases in this figure, the radius of the cylinders is a = 15    mm , the height of the cylinders is l z = 12    mm , and μ a / μ s = 0.0625 .
Fig. 8.
Fig. 8. (a) Spatially resolved reflectance and (b) error between analytical solutions and Monte Carlo simulations of different incident angles along the θ direction. The light is incident on the top of the cylinder.
Fig. 9.
Fig. 9. Spatially resolved reflectance of different incident angles along the θ direction. (a) Light is incident on the barrel of the cylinder, as shown in Fig. 2(b). (b) MC results for (a). (c) Light is incident on the barrel of the cylinder, as shown in Fig. 2(c). (d) MC results for (c).
Fig. 10.
Fig. 10. Amplitude and the phase of the frequency domain with light incident on the top of the cylinder. (a) Amplitude and (b) error of amplitude between analytical solutions and Monte Carlo simulations; (c) phase and (d) error of phase between analytical solutions and Monte Carlo simulations.
Fig. 11.
Fig. 11. Spatially resolved reflectance of different incident angles along the z direction. (a) Spatially resolved reflectance and (b) the error between analytical solutions and Monte Carlo simulations when light is incident on the barrel of the cylinder, as shown in Fig. 2(b). (c) Spatially resolved reflectance and (d) the error between analytical solutions and Monte Carlo simulations when light is incident on the barrel of the cylinder, as shown in Fig. 2(c).

Tables (1)

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Table 1. Comparison of the Value of μ t for Different Incident Angles

Equations (31)

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r 0 = ( a 2 2 a cos θ t / μ t + 1 / μ t 2 , arctan ( sin θ t / ( a μ t cos θ t ) ) , z 0 ) ,
2 Ψ ( μ a D + i ω D c ) Ψ = 1 D δ ( r r 0 ) ,
( 1 ρ ρ ( ρ ρ ) + 1 ρ 2 2 θ 2 + 2 z 2 ) Ψ ( μ a D + i ω D c ) Ψ = 1 D δ ( r r 0 ) .
( 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 θ 2 + 2 z 2 ) Ψ ( μ a D + i ω D c ) Ψ = 1 D ρ δ ( ρ ρ 0 ) δ ( θ θ 0 ) δ ( z z 0 ) ,
Ψ ( θ , m ) = 0 2 π Ψ ( θ ) cos ( m ( θ θ ) ) d θ , m = 0 , 1 , 2
( 2 ρ 2 + 1 ρ ρ m 2 ρ 2 + 2 z 2 ) Ψ ( μ a D + i ω D c ) Ψ = δ ( ρ ρ 0 ) cos ( m ( θ θ 0 ) ) δ ( z z 0 ) D ρ .
2 z 2 Ψ ( s n 2 + μ a D + i ω D c ) Ψ = 1 D J m ( s n ρ 0 ) cos ( m ( θ θ 0 ) ) δ ( z z 0 ) ,
z b = 1 + R eff 1 R eff 2 D .
R eff = 1.440 n rel 2 + 0.710 n rel 1 + 0.668 + 0.0636 n rel ,
Ψ ( s n , θ , m , z , ω ) = G ( s n , z , ω ) J m ( s n ρ 0 ) cos ( m ( θ θ 0 ) ) ,
2 z 2 G ( s n , z , ω ) ( s n 2 + μ a D + i ω D c ) G ( s n , z , ω ) = 1 D δ ( z z 0 ) .
Ψ ( ρ , θ , z = z b ) = 0 , Ψ ( ρ , θ , z = l z + z b ) = 0 ,
G ( s n , z , ω ) = 1 2 D α k = exp ( α | z z 0 2 k l z 4 k z b | ) k = exp ( α | z + z 0 + 2 z b 2 k l z 4 k z b | ) ,
Ψ ( s n , ϕ , m ) = 0 2 π 0 a ρ Ψ ( ρ , ϕ ) J m ( s n ρ ) cos ( m ( ϕ ϕ ) ) d ρ d ϕ
n = 1 ρ J m ( s n ρ ) J m ( s n ρ 0 ) J m + 1 2 ( a s n ) = a 2 2 δ ( ρ ρ 0 ) ,
Ψ ( r , ω ) = 1 π a 2 m = 0 ( 2 δ m , 0 ) cos ( m ϕ ) m = 1 G ( s n , z , ω ) J m ( s n ρ 0 ) J m ( s n ρ ) J m + 1 2 ( a s n ) ,
J m + 1 ( a s n ) = 2 m a s n J m ( a s n ) J m 1 ( a s n ) = J m 1 ( a s n ) ,
Ψ ( r , ω ) = 1 π a 2 m = cos ( m ϕ ) n = 1 G ( s n , z , ω ) J m ( s n ρ 0 ) J m ( s n ρ ) J m + 1 2 ( a s n ) ,
R ( θ , z , ω ) = D π a m = cos ( m ϕ ) n = 1 G ( s n , z , ω ) J m ( s n ρ 0 ) ( s n J m + 1 ( a s n ) m a J m ( a s n ) ) J m + 1 2 ( a s n ) ,
1 π t exp ( a t ) exp ( b 2 4 t ) ϵ ( t ) exp ( i ω t ) d t = exp ( | b | a + i ω ) a + i ω , a > 0 .
Ψ ( r , t ) = 1 2 π a 2 c D π t exp ( μ a c t ) × m = cos ( m ϕ ) n = 1 exp ( D c s n 2 t ) J m ( s n ρ 0 ) J m ( s n ρ ) J m + 1 2 ( a s n ) × ( k = exp [ ( z z 0 2 k l z 4 k z b ) 2 / 4 D c t ] k = exp [ ( z + z 0 + 2 z b 2 k l z 4 k z b ) 2 / 4 D c t ] )
R ( r , t ) = 1 2 π a 2 c D π t exp ( μ a c t ) × m = cos ( m ϕ ) n = 1 exp ( D c s n 2 t ) J m ( s n ρ 0 ) ( s n J m + 1 ( a s n ) m a J m ( a s n ) ) J m + 1 2 ( a s n ) × k = ( exp [ ( z z 0 2 k l z 4 k z b ) 2 / 4 D c t ] k = exp [ ( z + z 0 + 2 z b 2 k l z 4 k z b ) 2 / 4 D c t ] ) .
G = n = 0 N 1 g ( n ) exp ( in ω Δ t ) ,
A = G R 2 + G I 2 , ϕ = arctan ( G I G R ) ,
R fresnel = 1 2 [ sin 2 ( θ i θ t ) sin 2 ( θ i + θ i ) + tan 2 ( θ i θ t ) tan 2 ( θ i + θ i ) ] ,
lim t d ln R d t = lim t d ln R d R · d R d t = lim t 1 R · d R d t ,
lim t 1 R · d R d t = μ a c x ,
x = m = cos ( m ϕ ) n = 1 ( D c s n 2 ) exp ( D c s n 2 t ) J m ( s n ρ 0 ) ( s n J m + 1 ( a s n ) m a J m ( a s n ) ) J m + 1 2 ( a s n ) × k = exp [ ( z z 0 2 k l z 4 k z b ) 2 / 4 D c t ] k = exp [ ( z + z 0 + 2 z b 2 k l z 4 k z b ) 2 / 4 D c t ] m = cos ( m ϕ ) n = 1 exp ( D c s n 2 t ) J m ( s n ρ 0 ) ( s n J m + 1 ( a s n ) m a J m ( a s n ) ) J m + 1 2 ( a s n ) × k = exp [ ( z z 0 2 k l z 4 k z b ) 2 / 4 D c t ] k = exp [ ( z + z 0 + 2 z b 2 k l z 4 k z b ) 2 / 4 D c t ] .
lim t d ln R d t = μ a c 1 .
Δ x = 3 D sin θ i n 0 / n 1 = sin θ i n 0 μ t n 1 .
Δ x = 3 D sin θ i n 0 / 2 n 1 = sin θ i n 0 2 μ t n 1 .