Abstract

Radiative transfer in scattering media with spatially varying refractive indices, such as plasma with density fluctuations, is considered. It has been shown that singularities of diffuse radiation intensity can appear in the scattered field if the gradient of the refractive index is strong enough. To do that, we solve the scalar radiative transfer equation approximately and then analyze the solution qualitatively. Examples of the analytic singular solutions of the scalar radiative transfer equation in flat layered and spherically symmetric media, typically occurring in remote sensing applications, are provided. Conditions of formation of these singularities are discussed. Monte Carlo simulation results exhibiting singularities are compared with the derived analytical solutions.

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References

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  1. C. E. Siewert, “On the singular components of the solution to the searchlight problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 33, 551–554 (1985).
    [Crossref]
  2. T. A. Germogenova, Local Properties of Solutions to the Transport Equation (Nauka, 1986) [in Russian].
  3. R. Sanchez, “On the singular structure of the uncollided and first-collided components of the green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
    [Crossref]
  4. Y. A. Ilyushin and V. P. Budak, “Narrow beams in scattering media: the advanced small-angle approximation,” J. Opt. Soc. Am. A 28, 1358–1363 (2011).
    [Crossref]
  5. Y. A. Ilyushin and V. P. Budak, “Narrow-beam propagation in a two-dimensional scattering medium,” J. Opt. Soc. Am. A 28, 76–81 (2011).
    [Crossref]
  6. W. S. Helliwell, “Finite-difference solution to the radiative-transfer equation for in-water radiance,” J. Opt. Soc. Am. A 2, 1325–1330 (1985).
    [Crossref]
  7. V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering,” in Light Scattering Reviews (Springer, 2010), pp. 147–203.
  8. D. L. Feinstein, F. E. Butler, K. R. Piech, and A. Leonard, “Radiative transport analysis of electromagnetic propagation in isotropic plasma turbulence,” Phys. Fluids 15, 1641–1651 (1972).
    [Crossref]
  9. L. S. Dolin and E. A. Sergeeva, “A model of irradiance distribution for a directed point source in an infinite weakly absorbing turbid medium,” Radiophys. Quantum Electron. 44, 858–865 (2001).
    [Crossref]
  10. Y. A. Ilyushin, “Backscattering halo from the beam in the scattering medium with highly forward peaked phase function: is it feasible?” J. Opt. Soc. Am. A 29, 1986–1991 (2012).
    [Crossref]
  11. S. F. Smerd and K. C. Westfold, “The characteristics of radio-frequency radiation in an ionized gas, with applications to the transfer of radiation in the solar atmosphere,” Philos. Mag. 40(307), 831–848 (1949).
    [Crossref]
  12. G. Bekefi, Radiation Processes in Plasmas, Wiley Series in Plasma Physics (Wiley, 1966).
  13. M. L. Shendeleva, “Radiative transfer in a turbid medium with a varying refractive index: comment,” J. Opt. Soc. Am. A 21, 2464–2467 (2004).
    [Crossref]
  14. V. L. Granatstein and S. J. Buchsbaum, “Limits of validity of born approximation in microwave scattering from turbulent plasma,” Phys. Fluids (1994) 10, 1851–1853 (1967).
    [Crossref]
  15. L. Prikhodko, A. Vologdin, and I. Shirokov, “Dispersion and spatial autocorrelation of the phase and group signal paths in a randomly inhomogeneous medium with regular refraction,” Proc. SPIE 10035, 100351G (2016).
    [Crossref]
  16. L. Prikhodko, I. Shirokov, and A. Padokhin, “Fluctuations of the eikonal of an extraordinary wave reflected from the inhomogeneous ionospheric plasma,” Proc. SPIE 10833, 108331C (2018).
    [Crossref]
  17. A. Vologdin and L. Prikhod’ko, “The autocorrelation function of the plane wave phase in the case of oblique sounding of a randomly inhomogeneous planar stratified medium,” J. Commun. Technol. Electron. 49, 1141–1144 (2004).
    [Crossref]
  18. D. L. Feinstein and V. L. Granatstein, “Scalar radiative transport model for microwave scattering from a turbulent plasma,” Phys. Fluids (1994) 12, 2658–2668 (1969).
    [Crossref]
  19. K. M. Watson, “Multiple scattering of electromagnetic waves in an underdense plasma,” J. Math. Phys. 10, 688–702 (1969).
    [Crossref]
  20. R. W. Ziolkowski and N. Engheta, Introduction, History, and Selected Topics in Fundamental Theories of Metamaterials (Wiley, 2006), Chap. 1, pp. 1–41.
  21. D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
    [Crossref]
  22. P. Ben-Abdallaha and B. Ni, “Radiative transfer in strongly lossy inhomogeneous thin films,” J. Quant. Spectrosc. Radiat. Transfer 78, 481–488 (2003).
    [Crossref]
  23. V. Y. Soloviev, “Light transport in refractive turbid media,” J. Opt. Soc. Am. A 33, 383–390 (2016).
    [Crossref]
  24. V. Y. Soloviev, “Polarized light transport in refractive weak scattering media,” J. Opt. Soc. Am. A 33, 1323–1330 (2016).
    [Crossref]
  25. E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
    [Crossref]
  26. Y. A. Ilyushin, “Coherent backscattering enhancement in refracting media: diffusion approximation,” J. Opt. Soc. Am. A 30, 1305–1309 (2013).
    [Crossref]
  27. Y. A. Ilyushin, “Weak localization in media with refractive-index gradient: the diffusion approximation,” Radiophys. Quantum Electron. 57, 730–736 (2015).
    [Crossref]
  28. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  29. F. Olver and W. Rheinbolt, Asymptotics and Special Functions (Elsevier Science, 2014).
  30. C. J. Coleman, “On the generalization of Snell’s law,” Radio Sci. 39, 1–4 (2004).
    [Crossref]
  31. V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 109, 220–234 (2008).
    [Crossref]
  32. M. A. Janssen, ed., Atmospheric Remote Sensing by Microwave Radiometry, Vol. 6 of Wiley Series in Remote Sensing and Image Processing, 1st ed. (Wiley, 1993).
  33. V. Sadovnichy, A. Tikhonravov, V. Voevodin, and V. Opanasenko, “‘Lomonosov’: supercomputing at Moscow State University,” in Contemporary High Performance Computing: From Petascale toward Exascale (Chapman & Hall/CRC Computational Science, 2013), pp. 283–307.
  34. Y. Ilyushin, “Numerical simulations of singular radiative fields in scattering media with refraction,” Zenodo: Version 1, 18 November 2021, https://doi.org/10.5281/zenodo.5709883.

2018 (1)

L. Prikhodko, I. Shirokov, and A. Padokhin, “Fluctuations of the eikonal of an extraordinary wave reflected from the inhomogeneous ionospheric plasma,” Proc. SPIE 10833, 108331C (2018).
[Crossref]

2016 (3)

L. Prikhodko, A. Vologdin, and I. Shirokov, “Dispersion and spatial autocorrelation of the phase and group signal paths in a randomly inhomogeneous medium with regular refraction,” Proc. SPIE 10035, 100351G (2016).
[Crossref]

V. Y. Soloviev, “Light transport in refractive turbid media,” J. Opt. Soc. Am. A 33, 383–390 (2016).
[Crossref]

V. Y. Soloviev, “Polarized light transport in refractive weak scattering media,” J. Opt. Soc. Am. A 33, 1323–1330 (2016).
[Crossref]

2015 (1)

Y. A. Ilyushin, “Weak localization in media with refractive-index gradient: the diffusion approximation,” Radiophys. Quantum Electron. 57, 730–736 (2015).
[Crossref]

2013 (1)

2012 (1)

2011 (2)

2008 (1)

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 109, 220–234 (2008).
[Crossref]

2004 (3)

C. J. Coleman, “On the generalization of Snell’s law,” Radio Sci. 39, 1–4 (2004).
[Crossref]

M. L. Shendeleva, “Radiative transfer in a turbid medium with a varying refractive index: comment,” J. Opt. Soc. Am. A 21, 2464–2467 (2004).
[Crossref]

A. Vologdin and L. Prikhod’ko, “The autocorrelation function of the plane wave phase in the case of oblique sounding of a randomly inhomogeneous planar stratified medium,” J. Commun. Technol. Electron. 49, 1141–1144 (2004).
[Crossref]

2003 (1)

P. Ben-Abdallaha and B. Ni, “Radiative transfer in strongly lossy inhomogeneous thin films,” J. Quant. Spectrosc. Radiat. Transfer 78, 481–488 (2003).
[Crossref]

2002 (1)

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
[Crossref]

2001 (1)

L. S. Dolin and E. A. Sergeeva, “A model of irradiance distribution for a directed point source in an infinite weakly absorbing turbid medium,” Radiophys. Quantum Electron. 44, 858–865 (2001).
[Crossref]

2000 (1)

R. Sanchez, “On the singular structure of the uncollided and first-collided components of the green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
[Crossref]

1986 (1)

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref]

1985 (2)

C. E. Siewert, “On the singular components of the solution to the searchlight problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 33, 551–554 (1985).
[Crossref]

W. S. Helliwell, “Finite-difference solution to the radiative-transfer equation for in-water radiance,” J. Opt. Soc. Am. A 2, 1325–1330 (1985).
[Crossref]

1972 (1)

D. L. Feinstein, F. E. Butler, K. R. Piech, and A. Leonard, “Radiative transport analysis of electromagnetic propagation in isotropic plasma turbulence,” Phys. Fluids 15, 1641–1651 (1972).
[Crossref]

1969 (2)

D. L. Feinstein and V. L. Granatstein, “Scalar radiative transport model for microwave scattering from a turbulent plasma,” Phys. Fluids (1994) 12, 2658–2668 (1969).
[Crossref]

K. M. Watson, “Multiple scattering of electromagnetic waves in an underdense plasma,” J. Math. Phys. 10, 688–702 (1969).
[Crossref]

1967 (1)

V. L. Granatstein and S. J. Buchsbaum, “Limits of validity of born approximation in microwave scattering from turbulent plasma,” Phys. Fluids (1994) 10, 1851–1853 (1967).
[Crossref]

1949 (1)

S. F. Smerd and K. C. Westfold, “The characteristics of radio-frequency radiation in an ionized gas, with applications to the transfer of radiation in the solar atmosphere,” Philos. Mag. 40(307), 831–848 (1949).
[Crossref]

Akkermans, E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref]

Bekefi, G.

G. Bekefi, Radiation Processes in Plasmas, Wiley Series in Plasma Physics (Wiley, 1966).

Ben-Abdallaha, P.

P. Ben-Abdallaha and B. Ni, “Radiative transfer in strongly lossy inhomogeneous thin films,” J. Quant. Spectrosc. Radiat. Transfer 78, 481–488 (2003).
[Crossref]

Buchsbaum, S. J.

V. L. Granatstein and S. J. Buchsbaum, “Limits of validity of born approximation in microwave scattering from turbulent plasma,” Phys. Fluids (1994) 10, 1851–1853 (1967).
[Crossref]

Budak, V. P.

Y. A. Ilyushin and V. P. Budak, “Narrow beams in scattering media: the advanced small-angle approximation,” J. Opt. Soc. Am. A 28, 1358–1363 (2011).
[Crossref]

Y. A. Ilyushin and V. P. Budak, “Narrow-beam propagation in a two-dimensional scattering medium,” J. Opt. Soc. Am. A 28, 76–81 (2011).
[Crossref]

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 109, 220–234 (2008).
[Crossref]

V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering,” in Light Scattering Reviews (Springer, 2010), pp. 147–203.

Butler, F. E.

D. L. Feinstein, F. E. Butler, K. R. Piech, and A. Leonard, “Radiative transport analysis of electromagnetic propagation in isotropic plasma turbulence,” Phys. Fluids 15, 1641–1651 (1972).
[Crossref]

Coleman, C. J.

C. J. Coleman, “On the generalization of Snell’s law,” Radio Sci. 39, 1–4 (2004).
[Crossref]

Dez, V. L.

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
[Crossref]

Dolin, L. S.

L. S. Dolin and E. A. Sergeeva, “A model of irradiance distribution for a directed point source in an infinite weakly absorbing turbid medium,” Radiophys. Quantum Electron. 44, 858–865 (2001).
[Crossref]

Engheta, N.

R. W. Ziolkowski and N. Engheta, Introduction, History, and Selected Topics in Fundamental Theories of Metamaterials (Wiley, 2006), Chap. 1, pp. 1–41.

Feinstein, D. L.

D. L. Feinstein, F. E. Butler, K. R. Piech, and A. Leonard, “Radiative transport analysis of electromagnetic propagation in isotropic plasma turbulence,” Phys. Fluids 15, 1641–1651 (1972).
[Crossref]

D. L. Feinstein and V. L. Granatstein, “Scalar radiative transport model for microwave scattering from a turbulent plasma,” Phys. Fluids (1994) 12, 2658–2668 (1969).
[Crossref]

Germogenova, T. A.

T. A. Germogenova, Local Properties of Solutions to the Transport Equation (Nauka, 1986) [in Russian].

Granatstein, V. L.

D. L. Feinstein and V. L. Granatstein, “Scalar radiative transport model for microwave scattering from a turbulent plasma,” Phys. Fluids (1994) 12, 2658–2668 (1969).
[Crossref]

V. L. Granatstein and S. J. Buchsbaum, “Limits of validity of born approximation in microwave scattering from turbulent plasma,” Phys. Fluids (1994) 10, 1851–1853 (1967).
[Crossref]

Helliwell, W. S.

Ilyushin, Y. A.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Klyuykov, D. A.

V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering,” in Light Scattering Reviews (Springer, 2010), pp. 147–203.

Korkin, S. V.

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 109, 220–234 (2008).
[Crossref]

V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering,” in Light Scattering Reviews (Springer, 2010), pp. 147–203.

Lemonnier, D.

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
[Crossref]

Leonard, A.

D. L. Feinstein, F. E. Butler, K. R. Piech, and A. Leonard, “Radiative transport analysis of electromagnetic propagation in isotropic plasma turbulence,” Phys. Fluids 15, 1641–1651 (1972).
[Crossref]

Maynard, R.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref]

Ni, B.

P. Ben-Abdallaha and B. Ni, “Radiative transfer in strongly lossy inhomogeneous thin films,” J. Quant. Spectrosc. Radiat. Transfer 78, 481–488 (2003).
[Crossref]

Olver, F.

F. Olver and W. Rheinbolt, Asymptotics and Special Functions (Elsevier Science, 2014).

Opanasenko, V.

V. Sadovnichy, A. Tikhonravov, V. Voevodin, and V. Opanasenko, “‘Lomonosov’: supercomputing at Moscow State University,” in Contemporary High Performance Computing: From Petascale toward Exascale (Chapman & Hall/CRC Computational Science, 2013), pp. 283–307.

Padokhin, A.

L. Prikhodko, I. Shirokov, and A. Padokhin, “Fluctuations of the eikonal of an extraordinary wave reflected from the inhomogeneous ionospheric plasma,” Proc. SPIE 10833, 108331C (2018).
[Crossref]

Piech, K. R.

D. L. Feinstein, F. E. Butler, K. R. Piech, and A. Leonard, “Radiative transport analysis of electromagnetic propagation in isotropic plasma turbulence,” Phys. Fluids 15, 1641–1651 (1972).
[Crossref]

Prikhod’ko, L.

A. Vologdin and L. Prikhod’ko, “The autocorrelation function of the plane wave phase in the case of oblique sounding of a randomly inhomogeneous planar stratified medium,” J. Commun. Technol. Electron. 49, 1141–1144 (2004).
[Crossref]

Prikhodko, L.

L. Prikhodko, I. Shirokov, and A. Padokhin, “Fluctuations of the eikonal of an extraordinary wave reflected from the inhomogeneous ionospheric plasma,” Proc. SPIE 10833, 108331C (2018).
[Crossref]

L. Prikhodko, A. Vologdin, and I. Shirokov, “Dispersion and spatial autocorrelation of the phase and group signal paths in a randomly inhomogeneous medium with regular refraction,” Proc. SPIE 10035, 100351G (2016).
[Crossref]

Rheinbolt, W.

F. Olver and W. Rheinbolt, Asymptotics and Special Functions (Elsevier Science, 2014).

Sadovnichy, V.

V. Sadovnichy, A. Tikhonravov, V. Voevodin, and V. Opanasenko, “‘Lomonosov’: supercomputing at Moscow State University,” in Contemporary High Performance Computing: From Petascale toward Exascale (Chapman & Hall/CRC Computational Science, 2013), pp. 283–307.

Sanchez, R.

R. Sanchez, “On the singular structure of the uncollided and first-collided components of the green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
[Crossref]

Sergeeva, E. A.

L. S. Dolin and E. A. Sergeeva, “A model of irradiance distribution for a directed point source in an infinite weakly absorbing turbid medium,” Radiophys. Quantum Electron. 44, 858–865 (2001).
[Crossref]

Shendeleva, M. L.

Shirokov, I.

L. Prikhodko, I. Shirokov, and A. Padokhin, “Fluctuations of the eikonal of an extraordinary wave reflected from the inhomogeneous ionospheric plasma,” Proc. SPIE 10833, 108331C (2018).
[Crossref]

L. Prikhodko, A. Vologdin, and I. Shirokov, “Dispersion and spatial autocorrelation of the phase and group signal paths in a randomly inhomogeneous medium with regular refraction,” Proc. SPIE 10035, 100351G (2016).
[Crossref]

Siewert, C. E.

C. E. Siewert, “On the singular components of the solution to the searchlight problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 33, 551–554 (1985).
[Crossref]

Smerd, S. F.

S. F. Smerd and K. C. Westfold, “The characteristics of radio-frequency radiation in an ionized gas, with applications to the transfer of radiation in the solar atmosphere,” Philos. Mag. 40(307), 831–848 (1949).
[Crossref]

Soloviev, V. Y.

Tikhonravov, A.

V. Sadovnichy, A. Tikhonravov, V. Voevodin, and V. Opanasenko, “‘Lomonosov’: supercomputing at Moscow State University,” in Contemporary High Performance Computing: From Petascale toward Exascale (Chapman & Hall/CRC Computational Science, 2013), pp. 283–307.

Voevodin, V.

V. Sadovnichy, A. Tikhonravov, V. Voevodin, and V. Opanasenko, “‘Lomonosov’: supercomputing at Moscow State University,” in Contemporary High Performance Computing: From Petascale toward Exascale (Chapman & Hall/CRC Computational Science, 2013), pp. 283–307.

Vologdin, A.

L. Prikhodko, A. Vologdin, and I. Shirokov, “Dispersion and spatial autocorrelation of the phase and group signal paths in a randomly inhomogeneous medium with regular refraction,” Proc. SPIE 10035, 100351G (2016).
[Crossref]

A. Vologdin and L. Prikhod’ko, “The autocorrelation function of the plane wave phase in the case of oblique sounding of a randomly inhomogeneous planar stratified medium,” J. Commun. Technol. Electron. 49, 1141–1144 (2004).
[Crossref]

Watson, K. M.

K. M. Watson, “Multiple scattering of electromagnetic waves in an underdense plasma,” J. Math. Phys. 10, 688–702 (1969).
[Crossref]

Westfold, K. C.

S. F. Smerd and K. C. Westfold, “The characteristics of radio-frequency radiation in an ionized gas, with applications to the transfer of radiation in the solar atmosphere,” Philos. Mag. 40(307), 831–848 (1949).
[Crossref]

Wolf, P. E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref]

Ziolkowski, R. W.

R. W. Ziolkowski and N. Engheta, Introduction, History, and Selected Topics in Fundamental Theories of Metamaterials (Wiley, 2006), Chap. 1, pp. 1–41.

Ann. Nucl. Energy (1)

R. Sanchez, “On the singular structure of the uncollided and first-collided components of the green’s function,” Ann. Nucl. Energy 27, 1167–1186 (2000).
[Crossref]

J. Commun. Technol. Electron. (1)

A. Vologdin and L. Prikhod’ko, “The autocorrelation function of the plane wave phase in the case of oblique sounding of a randomly inhomogeneous planar stratified medium,” J. Commun. Technol. Electron. 49, 1141–1144 (2004).
[Crossref]

J. Math. Phys. (1)

K. M. Watson, “Multiple scattering of electromagnetic waves in an underdense plasma,” J. Math. Phys. 10, 688–702 (1969).
[Crossref]

J. Opt. Soc. Am. A (8)

J. Quant. Spectrosc. Radiat. Transfer (4)

C. E. Siewert, “On the singular components of the solution to the searchlight problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 33, 551–554 (1985).
[Crossref]

V. P. Budak and S. V. Korkin, “On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering,” J. Quant. Spectrosc. Radiat. Transfer 109, 220–234 (2008).
[Crossref]

D. Lemonnier and V. L. Dez, “Discrete ordinates solution of radiative transfer across a slab with variable refractive index,” J. Quant. Spectrosc. Radiat. Transfer 73, 195–204 (2002).
[Crossref]

P. Ben-Abdallaha and B. Ni, “Radiative transfer in strongly lossy inhomogeneous thin films,” J. Quant. Spectrosc. Radiat. Transfer 78, 481–488 (2003).
[Crossref]

Philos. Mag. (1)

S. F. Smerd and K. C. Westfold, “The characteristics of radio-frequency radiation in an ionized gas, with applications to the transfer of radiation in the solar atmosphere,” Philos. Mag. 40(307), 831–848 (1949).
[Crossref]

Phys. Fluids (1)

D. L. Feinstein, F. E. Butler, K. R. Piech, and A. Leonard, “Radiative transport analysis of electromagnetic propagation in isotropic plasma turbulence,” Phys. Fluids 15, 1641–1651 (1972).
[Crossref]

Phys. Fluids (1994) (2)

D. L. Feinstein and V. L. Granatstein, “Scalar radiative transport model for microwave scattering from a turbulent plasma,” Phys. Fluids (1994) 12, 2658–2668 (1969).
[Crossref]

V. L. Granatstein and S. J. Buchsbaum, “Limits of validity of born approximation in microwave scattering from turbulent plasma,” Phys. Fluids (1994) 10, 1851–1853 (1967).
[Crossref]

Phys. Rev. Lett. (1)

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref]

Proc. SPIE (2)

L. Prikhodko, A. Vologdin, and I. Shirokov, “Dispersion and spatial autocorrelation of the phase and group signal paths in a randomly inhomogeneous medium with regular refraction,” Proc. SPIE 10035, 100351G (2016).
[Crossref]

L. Prikhodko, I. Shirokov, and A. Padokhin, “Fluctuations of the eikonal of an extraordinary wave reflected from the inhomogeneous ionospheric plasma,” Proc. SPIE 10833, 108331C (2018).
[Crossref]

Radio Sci. (1)

C. J. Coleman, “On the generalization of Snell’s law,” Radio Sci. 39, 1–4 (2004).
[Crossref]

Radiophys. Quantum Electron. (2)

Y. A. Ilyushin, “Weak localization in media with refractive-index gradient: the diffusion approximation,” Radiophys. Quantum Electron. 57, 730–736 (2015).
[Crossref]

L. S. Dolin and E. A. Sergeeva, “A model of irradiance distribution for a directed point source in an infinite weakly absorbing turbid medium,” Radiophys. Quantum Electron. 44, 858–865 (2001).
[Crossref]

Other (9)

T. A. Germogenova, Local Properties of Solutions to the Transport Equation (Nauka, 1986) [in Russian].

V. P. Budak, D. A. Klyuykov, and S. V. Korkin, “Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering,” in Light Scattering Reviews (Springer, 2010), pp. 147–203.

G. Bekefi, Radiation Processes in Plasmas, Wiley Series in Plasma Physics (Wiley, 1966).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

F. Olver and W. Rheinbolt, Asymptotics and Special Functions (Elsevier Science, 2014).

R. W. Ziolkowski and N. Engheta, Introduction, History, and Selected Topics in Fundamental Theories of Metamaterials (Wiley, 2006), Chap. 1, pp. 1–41.

M. A. Janssen, ed., Atmospheric Remote Sensing by Microwave Radiometry, Vol. 6 of Wiley Series in Remote Sensing and Image Processing, 1st ed. (Wiley, 1993).

V. Sadovnichy, A. Tikhonravov, V. Voevodin, and V. Opanasenko, “‘Lomonosov’: supercomputing at Moscow State University,” in Contemporary High Performance Computing: From Petascale toward Exascale (Chapman & Hall/CRC Computational Science, 2013), pp. 283–307.

Y. Ilyushin, “Numerical simulations of singular radiative fields in scattering media with refraction,” Zenodo: Version 1, 18 November 2021, https://doi.org/10.5281/zenodo.5709883.

Data Availability

Data underlying the results presented in this paper are available in [34].

34. Y. Ilyushin, “Numerical simulations of singular radiative fields in scattering media with refraction,” Zenodo: Version 1, 18 November 2021, https://doi.org/10.5281/zenodo.5709883.

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Figures (3)

Fig. 1.
Fig. 1. Ray trajectories in the flat layered medium with the exponential gradient of the refractive index. Values of the direction cosine ${\mu _0}$ at the origin are shown by numerical labels.
Fig. 2.
Fig. 2. Integrated photon number $N(\mu)$. To show all curves in the same plot, each is normalized by the arbitrary constant. Values of the logarithmic derivative of the refractive index $-\gamma /\varepsilon$ and estimated powers of the singularity $\hat \alpha$ are shown by the numbers at the left and right end of each curve, respectively.
Fig. 3.
Fig. 3. Powers of the singularity $\alpha$ for different values of $\gamma /\varepsilon$ and $b/\varepsilon$. Numerical estimates from Monte Carlo simulations $\hat \alpha$ (45) are shown by the points along with the theoretical curves $\alpha$ (32). Values of $b/\varepsilon$ are shown by the numerical labels near each curve.

Equations (45)

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d d s ( L n 2 ) + ε L n 2 = S n 2 + e n 2 ,
S ( r , Ω ) = Λ ε 4 π L ( r , Ω ) x ( Ω , Ω ) d Ω
( L n 2 ) ( s ) = e ε s s e + ε s S ( r ( s ) , Ω ( s ) ) + e ( r ( s ) , Ω ( s ) ) n 2 ( s ) d s ,
L n 2 = s 0 e ε s S ( r ( s ) , Ω ( s ) ) + e ( r ( s ) , Ω ( s ) ) n 2 ( s ) d s ,
( D n 2 ϕ ( r ) n 2 ) + μ a ϕ ( r ) = E ( r ) ,
E ( r ) = 4 π e d Ω
ϕ ( r ) = 4 π L ( r , Ω ) d Ω
D = 1 3 ( μ a + ( 1 g ) μ s ) ,
n ( z ) = n ( 0 ) e γ z
e ( r, Ω ) = e 0 ( Ω ) e b z ,
E ( z ) = e 0 ( Ω ) e b z x ( Ω , Ω ) d Ω d Ω = e 0 ( Ω ) e b z d Ω = E 0 exp ( b z ) .
a = 2 3 μ s ( 1 g ) .
ϕ ( z ) = E 0 D b ( 2 γ + b ) μ a e b z + E 0 D b ( 2 γ + b ) μ a e ( γ + b γ 2 + μ a / D ) a + ( γ γ 2 + μ a / D ) z .
L ( r , Ω ) = 1 4 π ϕ ( r ) + 3 4 π ( J Ω ) ,
J ( r ) = D ϕ ( r ) + 2 D n ( r ) ϕ ( r ) n ( r ) .
J ( z ) = D z ϕ ( z ) + 2 D n ( z ) ϕ ( z ) z n ( z ) ,
L ( z , Ω ) = 1 4 π ϕ ( z ) + 3 4 π J μ z .
x ( Ω Ω ) = 4 π n = 0 m = j j x j Y j m ( Ω ) Y j m ( Ω ) ,
S ( z , Ω ) + e ( z , Ω ) = S 1 exp ( b z ) f 1 ( μ z ) + S 2 exp ( γ z γ 2 + μ a / D z ) f 2 ( μ z ) ,
S + e = exp ( b z ) f ( μ z ) ,
n ( z ) sin θ = n 0 sin θ 0 = c o n s t
d z = μ z d s
γ ( z z 0 ) = ln ( 1 + μ 0 2 e γ ( s s 0 ) + 1 μ 0 2 e γ ( s s 0 ) ) ,
γ ( s s 0 ) = ln ( e γ z ± μ 0 2 + e 2 γ ( z z 0 ) 1 μ 0 + 1 ) .
( z t u r n z 0 ) = ln ( 1 μ 0 2 ) 2 γ .
L = e Φ ( s ) f ( s ) d s ,
Φ ( s ) = log ( e s ( 1 2 e s γ ( 1 μ 0 ) + 1 2 e s γ ( μ 0 + 1 ) ) b γ 2 ) ,
γ < 1 + b 2 ,
exp ( 2 γ s ) = ( μ 0 2 1 ) b 2 4 γ ( μ 0 2 1 ) b + 4 γ 2 4 γ 2 μ 0 2 + μ 0 2 1 ( b + 2 γ + 1 ) 2 ( μ 0 + 1 ) 2 .
Φ ( s ) = γ ( b + 2 γ 1 ) ( b + 2 γ + 1 ) b + 2 γ .
L 1 2 π Φ ( s ) exp ( Φ ( s ) ) f ( s ) .
L ( μ 0 ) ( 1 μ 0 ) 1 1 + b 2 γ ( 1 μ 0 ) α .
n ( r ) = exp ( γ r ) / r
n ( r ) r sin θ = c o n s t .
1 r 2 d d r ( r 2 D n 2 ( r ) d d r ϕ ( r ) n 2 ( r ) ) + μ a ϕ ( r ) = E ( r ) .
E ( r ) e ε r r 2 ,
S ( z , Ω ) = i exp ( β i r ) f i ( μ r ) r m i ,
ϕ ( z ) = 4 π B ( T ) 4 π B ( T ) e ( γ γ 2 + μ a / D ) ( z + a ) .
L ( z , Ω ) = L 0 δ ( Ω ) exp ( ε z ) .
e ( r , Ω ) = Λ ε 4 π L 0 δ ( Ω ) exp ( ε z ) x ( Ω , Ω ) d Ω = Λ ε 4 π L 0 exp ( ε z ) x ( Ω , 0 ) .
E = Λ ε 4 π L 0 δ ( Ω ) exp ( ε z ) x ( Ω , Ω ) d Ω d Ω = Λ ε L 0 exp ( ε z ) δ ( Ω ) d Ω = L 0 Λ ε exp ( ε z ) .
ϕ ( z ) = 0 G ( z , z ) E ( z ) d z ,
d d z ( D n ( z ) 2 d d z G ( z , z ) n ( z ) 2 ) + μ a G ( z , z ) = δ ( z z ) ,
N ( μ ) μ 0 1 L ( 0 , μ ) μ d μ μ 0 1 ( 1 μ ) α μ d μ ( 1 μ ) ( 1 + α ) .
d ln N ( μ ) d ln ( 1 μ ) 1 = α ^ .

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