# American Institute of Mathematical Sciences

February  2021, 26(2): 1171-1195. doi: 10.3934/dcdsb.2020158

## Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium

 1 CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France 2 Department of Mathematics, University of California, Irvine CA 92697, USA

* Corresponding author: Knut Sølna

Received  September 2019 Revised  February 2020 Published  February 2021 Early access  May 2020

Fund Project: This research is supported by AFOSR grant FA9550-18-1-0217, NSF grant 1616954

The weak localization or enhanced backscattering phenomenon has received a lot of attention in the literature. The enhanced backscattering cone refers to the situation that the wave backscattered by a random medium exhibits an enhanced intensity in a narrow cone around the incoming wave direction. This phenomenon can be analyzed by a formal path integral approach. Here a mathematical derivation of this result is given based on a system of equations that describes the second-order moments of the reflected wave. This system derives from a multiscale stochastic analysis of the wave field in the situation with high-frequency waves and propagation through a lossy medium with fine scale random microstructure. The theory identifies a duality relation between the spreading of the wave and the enhanced backscattering cone. It shows how the cone, its regularity and width relate to the statistical structure of the random medium. We discuss how this information in particular can be used to estimate the internal structure of the random medium based on observations of the reflected wave.

Citation: Josselin Garnier, Knut Sølna. Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1171-1195. doi: 10.3934/dcdsb.2020158
##### References:
 [1] V. A. Banakh and I. N. Smalikho, Determination of optical turbulence intensity by atmospheric backscattering of laser radiation, Atmospheric and Oceanic Optics, 24 (2011), 457. [2] Y. N. Barabanenkov, Wave corrections for the transfer equation for backward scattering, Izv. Vyssh. Uchebn. Zaved. Radiofiz., 16 (1973), 88-96. [3] R. Bi, J. Dong and K. Lee, Coherent backscattering cone shape depends on the beam size, Appl. Optics, 51 (2012), 6301-6306. [4] I. R. Capoglu, J. D. Rogers, A. Taflove and V. Backman, Accuracy of the Born approximation in calculating the scattering coefficient of biological continuous random media, Opt. Lett., 34 (2009), 2679-2681. [5] J. Chrzanowski, J. Kirkiewicz and Yu. A. Kravtsov, Influence of enhanced backscattering phenomenon on laser measurements of dust and aerosols content in a turbulent atmosphere, Phys. Lett. A, 300 (2002), 298-302. [6] J. H. Churnside and J. J. Wilson, Enhanced backscatter of a reflected beam in atmospheric turbulence, Appl Opt., 32 (1993), 2651-2655. [7] M. V. de Hoop, J. Garnier and K. Sølna, Enhanced and specular backscattering in random media, Waves in Random and Complex Media, 22 (2012), 505-530.  doi: 10.1080/17455030.2012.728299. [8] D. de Wolf, Electromagnetic reflection from an extended turbulent medium: Cumulative forward-scatter single-backscatter approximation, IEEE Trans. Antennas Propagat, 19 (1971), 254-262. [9] D. de Wolf, Discussion of radiatlve transfer methods applied to electromagnetic reflection from turbulent plasma, IEEE Trans. Antennas Propagat, 20 (1972), 805-807. [10] J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer, New York, 2007. [11] J. Garnier and K. Sølna, Random backscattering in the parabolic scaling, J. Stat. Phys., 131 (2008), 445-486.  doi: 10.1007/s10955-008-9488-0. [12] J. Garnier and K. Sølna, Coupled paraxial wave equations in random media in the white-noise regime, Ann. Appl. Probab., 19 (2009), 318-346.  doi: 10.1214/08-AAP543. [13] J. Garnier and K. Sølna, Wave backscattering by point scatterers in the random paraxial regime, SIAM J. Multiscale Model. Simul., 3 (2014), 1309-1334.  doi: 10.1137/140953757. [14] J. Garnier and K. Sølna, White-noise paraxial approximation for a general random hyperbolic system, SIAM J. Multiscale Model. Simul., 13 (2015), 1022-1060.  doi: 10.1137/15M101556X. [15] J. Garnier and K. Sølna, Fourth-moment Analysis for wave propagation in the white-noise paraxial regime, Arch. Rat. Mech. Anal., 220 (2016), 37-81.  doi: 10.1007/s00205-015-0926-2. [16] J. Garnier and K. Sølna, Imaging through a scattering medium by speckle intensity correlations over incident angle, Inverse Problems, 34 (2018), 094003, 22pp. doi: 10.1088/1361-6420/aacfb0. [17] J. Garnier and K. Sølna, Non-invasive imaging through random media, SIAM J. Appl. Math., 78 (2018), 3296-3315.  doi: 10.1137/18M1171977. [18] A. K. Glaser, Y. Chen and J. T. C. Liu, Fractal propagation method enables realistic optical microscopy simulations in biological tissues, Optica, 3 (2016), 861-869. [19] K. S. W. Gong and C. J. R. Shappard, Model for light scattering in biological tissue and cells based on random rough nonspherical particles, Appl. Optics, 48 (2009), 1153-1157. [20] Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali and V. Backman, Coherent backscattering spectroscopy, Opt. Lett., 29 (2004), 1906-1908. [21] Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali, P. P. Subramanian, P. Pradhan and V. Backman, Low-coherence enhanced backscattering: Review of principles and applications for colon cancer screening, J. Biomed. Opt., 11 (2006), 041125. [22] Y. L. Kim, Y. Liu, V. M. Turzhitsky, R. K. Wali, H. K. Roy and V. Backman, Depth-resolved low-coherence enhanced backscattering, Opt. Lett., 30 (2007), 741-743. [23] G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura and R. Kaiser, Coherent backscattering of light by atoms, Phys. Rev. Lett., 83 (1999), 5266-5269. [24] J. Liu, Z. Xu, Q. Song, R. L. Konger and Y. L. Kim, Enhancement factor in low-coherence enhanced backscattering and its applications for characterizing experimental skin carcinogenesis, J. Biomed. Opt., 15 (2010), 037011. [25] N. Mutyal, A. Radosevich, B. Gould, J. D. Rodgers, A. Gomes, V. Turzhitsky and V. Backman, A fiber optic probe design to measure depth- limited optical properties in-vivo with with Low-coherence Enhanced Backscattering (LEBS) Spectroscopy, Opt. Express, 20 (2012), 19643-19657. [26] J. P. Nolan, Multivariate elliptically contoured stable distributions: Theory and estimation, Computational Statistics, 28 (2013), 2067-2089.  doi: 10.1007/s00180-013-0396-7. [27] A. J. Radosevisch, N. M. Nikhil, J. D. Rogers, B. Gould, T. A. Hensing, D. Ray, V. Backman and H. K. Roy, Buccal spectral markers for lung cancer risk stratification, Plos One, 9 (2014), e10157. [28] J. D. Rogers, I. R. Capoglu and V. Backman, Nonscalar elastic light scattering from continuous random media in the Born approximation, Opt. Lett., 34 (2009), 1891-1893. [29] J. D. Rogers, A. J. Radosevich, J. Yi and V. Backman, Modeling light scattering in tissue as continuous random media using a versatile refractive index correlation function, IEEE J. Sel. Top. Quant., 20 (2014), 7000514. [30] Y. M. Sebrebrennikova and L. H. Garcia-Rubio, Modeling and interpretation of extinction spectra of oriented nonspherical composite particles: application to biological cells, Appl. Optics, 49 (2010), 4460-4471. [31] C. J. R. Sheppard, Fractal model of light scattering in biological tissue and cells, Opt. Lett., 32 (2007), 142-144. [32] A. Tourin, A. Derode, P. Roux, B. A. van Tiggelen and M. Fink, Time-dependent coherent backscattering of acoustic waves, Phys. Rev. Lett., 79 (1997), 3637-3639. [33] V. Turzhitsky, A. J. Radosevich, J. D. Rogers, N. N. Mutyal and V. Backman, Measurement of optical scattering properties with low-coherence enhanced backscattering spectroscopy, J. Biomed. Opt., 16 (2011), 067007. [34] V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy and V. Backman, Characterization of light transport in scattering media at subdiffusion length scales with Low-coherence Enhanced Backscattering, IEEE J. Sel. Top. Quant., 16 (2010), 619-626. [35] M. P. van Albada and A. Lagendijk, Observation of weak localization of light in a random medium, Phys. Rev. Lett., 55 (1985), 2692–2695. doi: 10.1103/PhysRevLett.55.2692. [36] M. C. W. van Rossum and Th. M. Nieuwenhuizen, Multiple scattering of classical waves: Microscopy, mesoscopy, and diffusion, Rev. Mod. Phys., 71 (1999), 313-371.  doi: 10.1103/RevModPhys.71.313. [37] P. E. Wolf and G. Maret, Weak localization and coherent backscattering of photons in disordered media, Phys. Rev. Lett., 55 (1985), 2696–2699. [38] P. E. Wolf, G. Maret, E. Akkermans and R. Maynard, Optical coherent backscattering by random media: An experimental study, Journal de Physique, 49 (1988), 63-75. [39] M. Xu and R. R. Alfano, Fractal mechanisms of light scattering in biological tissue and cells, Opt. Lett., 30 (2005), 3051-3053. [40] K. M. Yoo, G. C. Tang and R. R. Alfano, Coherent backscattering of light from biological tissues, Appl. Opt., 29 (1990), 3237-3239.

show all references

##### References:
 [1] V. A. Banakh and I. N. Smalikho, Determination of optical turbulence intensity by atmospheric backscattering of laser radiation, Atmospheric and Oceanic Optics, 24 (2011), 457. [2] Y. N. Barabanenkov, Wave corrections for the transfer equation for backward scattering, Izv. Vyssh. Uchebn. Zaved. Radiofiz., 16 (1973), 88-96. [3] R. Bi, J. Dong and K. Lee, Coherent backscattering cone shape depends on the beam size, Appl. Optics, 51 (2012), 6301-6306. [4] I. R. Capoglu, J. D. Rogers, A. Taflove and V. Backman, Accuracy of the Born approximation in calculating the scattering coefficient of biological continuous random media, Opt. Lett., 34 (2009), 2679-2681. [5] J. Chrzanowski, J. Kirkiewicz and Yu. A. Kravtsov, Influence of enhanced backscattering phenomenon on laser measurements of dust and aerosols content in a turbulent atmosphere, Phys. Lett. A, 300 (2002), 298-302. [6] J. H. Churnside and J. J. Wilson, Enhanced backscatter of a reflected beam in atmospheric turbulence, Appl Opt., 32 (1993), 2651-2655. [7] M. V. de Hoop, J. Garnier and K. Sølna, Enhanced and specular backscattering in random media, Waves in Random and Complex Media, 22 (2012), 505-530.  doi: 10.1080/17455030.2012.728299. [8] D. de Wolf, Electromagnetic reflection from an extended turbulent medium: Cumulative forward-scatter single-backscatter approximation, IEEE Trans. Antennas Propagat, 19 (1971), 254-262. [9] D. de Wolf, Discussion of radiatlve transfer methods applied to electromagnetic reflection from turbulent plasma, IEEE Trans. Antennas Propagat, 20 (1972), 805-807. [10] J.-P. Fouque, J. Garnier, G. Papanicolaou and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer, New York, 2007. [11] J. Garnier and K. Sølna, Random backscattering in the parabolic scaling, J. Stat. Phys., 131 (2008), 445-486.  doi: 10.1007/s10955-008-9488-0. [12] J. Garnier and K. Sølna, Coupled paraxial wave equations in random media in the white-noise regime, Ann. Appl. Probab., 19 (2009), 318-346.  doi: 10.1214/08-AAP543. [13] J. Garnier and K. Sølna, Wave backscattering by point scatterers in the random paraxial regime, SIAM J. Multiscale Model. Simul., 3 (2014), 1309-1334.  doi: 10.1137/140953757. [14] J. Garnier and K. Sølna, White-noise paraxial approximation for a general random hyperbolic system, SIAM J. Multiscale Model. Simul., 13 (2015), 1022-1060.  doi: 10.1137/15M101556X. [15] J. Garnier and K. Sølna, Fourth-moment Analysis for wave propagation in the white-noise paraxial regime, Arch. Rat. Mech. Anal., 220 (2016), 37-81.  doi: 10.1007/s00205-015-0926-2. [16] J. Garnier and K. Sølna, Imaging through a scattering medium by speckle intensity correlations over incident angle, Inverse Problems, 34 (2018), 094003, 22pp. doi: 10.1088/1361-6420/aacfb0. [17] J. Garnier and K. Sølna, Non-invasive imaging through random media, SIAM J. Appl. Math., 78 (2018), 3296-3315.  doi: 10.1137/18M1171977. [18] A. K. Glaser, Y. Chen and J. T. C. Liu, Fractal propagation method enables realistic optical microscopy simulations in biological tissues, Optica, 3 (2016), 861-869. [19] K. S. W. Gong and C. J. R. Shappard, Model for light scattering in biological tissue and cells based on random rough nonspherical particles, Appl. Optics, 48 (2009), 1153-1157. [20] Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali and V. Backman, Coherent backscattering spectroscopy, Opt. Lett., 29 (2004), 1906-1908. [21] Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali, P. P. Subramanian, P. Pradhan and V. Backman, Low-coherence enhanced backscattering: Review of principles and applications for colon cancer screening, J. Biomed. Opt., 11 (2006), 041125. [22] Y. L. Kim, Y. Liu, V. M. Turzhitsky, R. K. Wali, H. K. Roy and V. Backman, Depth-resolved low-coherence enhanced backscattering, Opt. Lett., 30 (2007), 741-743. [23] G. Labeyrie, F. de Tomasi, J.-C. Bernard, C. A. Müller, C. Miniatura and R. Kaiser, Coherent backscattering of light by atoms, Phys. Rev. Lett., 83 (1999), 5266-5269. [24] J. Liu, Z. Xu, Q. Song, R. L. Konger and Y. L. Kim, Enhancement factor in low-coherence enhanced backscattering and its applications for characterizing experimental skin carcinogenesis, J. Biomed. Opt., 15 (2010), 037011. [25] N. Mutyal, A. Radosevich, B. Gould, J. D. Rodgers, A. Gomes, V. Turzhitsky and V. Backman, A fiber optic probe design to measure depth- limited optical properties in-vivo with with Low-coherence Enhanced Backscattering (LEBS) Spectroscopy, Opt. Express, 20 (2012), 19643-19657. [26] J. P. Nolan, Multivariate elliptically contoured stable distributions: Theory and estimation, Computational Statistics, 28 (2013), 2067-2089.  doi: 10.1007/s00180-013-0396-7. [27] A. J. Radosevisch, N. M. Nikhil, J. D. Rogers, B. Gould, T. A. Hensing, D. Ray, V. Backman and H. K. Roy, Buccal spectral markers for lung cancer risk stratification, Plos One, 9 (2014), e10157. [28] J. D. Rogers, I. R. Capoglu and V. Backman, Nonscalar elastic light scattering from continuous random media in the Born approximation, Opt. Lett., 34 (2009), 1891-1893. [29] J. D. Rogers, A. J. Radosevich, J. Yi and V. Backman, Modeling light scattering in tissue as continuous random media using a versatile refractive index correlation function, IEEE J. Sel. Top. Quant., 20 (2014), 7000514. [30] Y. M. Sebrebrennikova and L. H. Garcia-Rubio, Modeling and interpretation of extinction spectra of oriented nonspherical composite particles: application to biological cells, Appl. Optics, 49 (2010), 4460-4471. [31] C. J. R. Sheppard, Fractal model of light scattering in biological tissue and cells, Opt. Lett., 32 (2007), 142-144. [32] A. Tourin, A. Derode, P. Roux, B. A. van Tiggelen and M. Fink, Time-dependent coherent backscattering of acoustic waves, Phys. Rev. Lett., 79 (1997), 3637-3639. [33] V. Turzhitsky, A. J. Radosevich, J. D. Rogers, N. N. Mutyal and V. Backman, Measurement of optical scattering properties with low-coherence enhanced backscattering spectroscopy, J. Biomed. Opt., 16 (2011), 067007. [34] V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy and V. Backman, Characterization of light transport in scattering media at subdiffusion length scales with Low-coherence Enhanced Backscattering, IEEE J. Sel. Top. Quant., 16 (2010), 619-626. [35] M. P. van Albada and A. Lagendijk, Observation of weak localization of light in a random medium, Phys. Rev. Lett., 55 (1985), 2692–2695. doi: 10.1103/PhysRevLett.55.2692. [36] M. C. W. van Rossum and Th. M. Nieuwenhuizen, Multiple scattering of classical waves: Microscopy, mesoscopy, and diffusion, Rev. Mod. Phys., 71 (1999), 313-371.  doi: 10.1103/RevModPhys.71.313. [37] P. E. Wolf and G. Maret, Weak localization and coherent backscattering of photons in disordered media, Phys. Rev. Lett., 55 (1985), 2696–2699. [38] P. E. Wolf, G. Maret, E. Akkermans and R. Maynard, Optical coherent backscattering by random media: An experimental study, Journal de Physique, 49 (1988), 63-75. [39] M. Xu and R. R. Alfano, Fractal mechanisms of light scattering in biological tissue and cells, Opt. Lett., 30 (2005), 3051-3053. [40] K. M. Yoo, G. C. Tang and R. R. Alfano, Coherent backscattering of light from biological tissues, Appl. Opt., 29 (1990), 3237-3239.
Physical interpretation of the scattering of a plane wave by a random medium. The output wave in direction $A$ is the superposition of many different scattering paths. One of these paths is plotted as well as the reversed path. The phase difference between the two outgoing waves is $k e = k d \sin A$
The backscattering enhancement cone in Eq. (81) (normalized by $\pi^2 P_{\rm tot}$). Here we use the Matérn covariance function (55). In the left plot $p = .6$, while in the right plot $p = .9$ so that the medium fluctuations are smoother in the right plot. In the plots the narrowest cones with largest peak values correspond to the largest $\beta$ values
Notations used in the paper
 $c_o$ background speed of propagation of the medium $\sigma_o$ background attenuation of the medium $\ell_z$ longitudinal correlation radius of the random medium $\ell_x$ transverse correlation radius of the random medium $\sigma$ standard deviation of the random medium $\omega$ (angular) frequency of the source $r_0$ radius of the source $\rho_0$ correlation radius of the source ${{{\boldsymbol k}}}_0$ transverse wavevector of the source $\lambda_o = \frac{2\pi c_o}{\omega}$ wavelength $L_{\rm att} = \frac{c_o}{2\sigma_o}$ attenuation length $\zeta_L= \frac{L}{L_{\rm att}}$ relative propagation distance $K_z= \frac{2\omega \ell_z}{c_o}$ relative wavenumber $\alpha= \frac{c_o^2}{2\sigma_o \omega \ell_x^2}$ strength of diffraction $\beta= \frac{ \omega^2 \sigma^2 \ell_z}{8 c_o \sigma_o}$ strength of forward scattering $\overline{D}_o$ cross spectral density central value (see Eq. (26)) $P_{\rm tot}$ mean reflected power (see Eq. (31))
 $c_o$ background speed of propagation of the medium $\sigma_o$ background attenuation of the medium $\ell_z$ longitudinal correlation radius of the random medium $\ell_x$ transverse correlation radius of the random medium $\sigma$ standard deviation of the random medium $\omega$ (angular) frequency of the source $r_0$ radius of the source $\rho_0$ correlation radius of the source ${{{\boldsymbol k}}}_0$ transverse wavevector of the source $\lambda_o = \frac{2\pi c_o}{\omega}$ wavelength $L_{\rm att} = \frac{c_o}{2\sigma_o}$ attenuation length $\zeta_L= \frac{L}{L_{\rm att}}$ relative propagation distance $K_z= \frac{2\omega \ell_z}{c_o}$ relative wavenumber $\alpha= \frac{c_o^2}{2\sigma_o \omega \ell_x^2}$ strength of diffraction $\beta= \frac{ \omega^2 \sigma^2 \ell_z}{8 c_o \sigma_o}$ strength of forward scattering $\overline{D}_o$ cross spectral density central value (see Eq. (26)) $P_{\rm tot}$ mean reflected power (see Eq. (31))
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