# Correlation of Light Polarization in the Magnetic Media with Non-Spherical Point-Like Inclusions

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Effective Dielectric Tensor

## 3. Correlation Matrix

- 1.
- Anisotropy is along the observation axis ($\overline{a}\Vert z$). Correlations ${W}_{xx}$ and ${W}_{yy}$ are equal. Correlation ${W}_{zz}$ is higher than ${W}_{xx}$ and ${W}_{yy}$ if $a>0$ and vice versa.$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{xx}={\tilde{W}}_{yy}={\tilde{W}}_{xx}^{0}+a({\tilde{W}}_{xx}^{a}+{\tilde{W}}_{yy}^{a})\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{zz}={\tilde{W}}_{zz}^{0}+2a{\tilde{W}}_{zz}^{a}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{xx}^{0}={\textstyle \frac{({\tilde{X}}^{2}-1)sin\tilde{X}+\tilde{X}cos\tilde{X}}{2{\tilde{X}}^{3}}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}{\tilde{W}}_{zz}^{0}={\textstyle \frac{sin\tilde{X}-\tilde{X}cos\tilde{X}}{{\tilde{X}}^{3}}}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{xx}^{a}=-{\textstyle \frac{\tilde{X}(8{\tilde{X}}^{2}+27)cos\tilde{X}+(5{\tilde{X}}^{4}+{\tilde{X}}^{2}-27)sin\tilde{X}}{6{\tilde{X}}^{5}}}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{yy}^{a}={\textstyle \frac{\tilde{X}(16{\tilde{X}}^{2}-81)cos\tilde{X}+(7{\tilde{X}}^{4}-43{\tilde{X}}^{2}+81)sin\tilde{X}}{6{\tilde{X}}^{5}}}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{zz}^{a}={\textstyle \frac{(54-19{\tilde{X}}^{2})sin\tilde{X}+\tilde{X}({\tilde{X}}^{2}-54)cos\tilde{X}}{3{\tilde{X}}^{5}}},\hfill \end{array}$$
- 2.
- Anisotropy is perpendicular to the observation axis ($\overline{a}\perp z$). Correlations ${W}_{xx}$ and ${W}_{yy}$ are different. Correlation ${W}_{yy}$ is higher than ${W}_{xx}$ and ${W}_{zz}$ if the anisotropy is along the y-axis and $a>0$ and vice versa. The same is true if the anisotropy is along x as it corresponds to changing ${W}_{xx}\leftrightarrow {W}_{yy}$.$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{xx}={\tilde{W}}_{xx}^{0}+a{\tilde{W}}_{xx}^{a}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{0.166667em}{0ex}}{\tilde{W}}_{yy}={\tilde{W}}_{xx}^{0}+a{\tilde{W}}_{yy}^{a}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{zz}={\tilde{W}}_{zz}^{0}+a{\tilde{W}}_{zz}^{a}.\hfill \end{array}$$

**Figure 3.**Symmetric ${\tilde{W}}_{\perp}^{S}$ and antisymmetric ${\tilde{W}}_{\perp}^{A}$ normalized contributions of the electric field correlations of the orthogonal polarizations for different directions of the gyration, $\mathbf{g}$, and the anisotropy, $\overline{a}$. Dependence on the gyration value $\left|\mathbf{g}\right|$ was included in the normalization factor. The anisotropy value is $a=0.07$. Five different cases are possible (for $\mathbf{X}\Vert z$): (

**a**) and (

**b**) The anisotropy is along the observation axis, while the gyration is perpendicular to both of them; (

**c**) All three vectors are along the same axis; (

**d**) All three vectors are perpendicular to each other; (

**e**) The gyration is directed along the anisotropy but perpendicular to the observation axis; (

**f**) The gyration is along the observation axis, while the anisotropy is perpendicular to it.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- The anisotropy is along the observation axis, while the gyration is perpendicular to both of them ($\mathbf{g}\perp \overline{a}\Vert z$) (Figure 3a,b of the main text).$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{\perp}^{A}={\tilde{W}}_{\perp}^{A(0,\perp )}+{\textstyle \frac{a((36-13{\tilde{X}}^{2})sin\tilde{X}+\tilde{X}({\tilde{X}}^{2}-36)cos\tilde{X})}{12{\tilde{X}}^{5}}},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{\perp}^{S}={\textstyle \frac{3a((24-13{\tilde{X}}^{2})sin\tilde{X}+\tilde{X}(5{\tilde{X}}^{2}-24)cos\tilde{X})}{4{\tilde{X}}^{5}}},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{\perp}^{A(0,\perp )}={\textstyle \frac{(\tilde{X}cos\tilde{X}-sin\tilde{X})}{4{\tilde{X}}^{3}}}\hfill \end{array}$$
- All of the three vectors are along the same axis ($\mathbf{g}\Vert \overline{a}\Vert z$) (Figure 3c of the main text).$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{\perp}^{A}={\tilde{W}}_{\perp}^{A(0,\Vert )}+{\textstyle \frac{a(({\tilde{X}}^{4}+13{\tilde{X}}^{2}-36)sin\tilde{X}-\tilde{X}({\tilde{X}}^{2}-36)cos\tilde{X})}{6{\tilde{X}}^{5}}},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{\perp}^{S}=0,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{\perp}^{A(0,\Vert )}={\textstyle \frac{((2-{\tilde{X}}^{2})sin\tilde{X}-2\tilde{X}cos\tilde{X})}{4{\tilde{X}}^{3}}}.\hfill \end{array}$$
- All of the three vectors are perpendicular to each other ($\mathbf{g}\perp \overline{a}\perp z$) (Figure 3d of the main text).$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{\perp}^{A}={\tilde{W}}_{\perp}^{A(0,\perp )}-{\textstyle \frac{a((9-5{\tilde{X}}^{2})sin\tilde{X}+\tilde{X}(2{\tilde{X}}^{2}-9)cos\tilde{X})}{12{\tilde{X}}^{5}}},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{\perp}^{S}=-{\textstyle \frac{a(3(44-21{\tilde{X}}^{2})sin\tilde{X}+\tilde{X}(19{\tilde{X}}^{2}-132)cos\tilde{X})}{4{\tilde{X}}^{5}}}.\hfill \end{array}$$
- The gyration is directed along the anisotropy but perpendicular to the observation axis ($\mathbf{g}\Vert \overline{a}\perp z$) (Figure 3e of the main text).$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{\perp}^{A}={\tilde{W}}_{\perp}^{A(0,\perp )}-{\textstyle \frac{a((27-14{\tilde{X}}^{2})sin\tilde{X}+\tilde{X}(5{\tilde{X}}^{2}-27)cos\tilde{X})}{12{\tilde{X}}^{5}}},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\tilde{W}}_{\perp}^{S}=-{\textstyle \frac{a(3(5-2{\tilde{X}}^{2})sin\tilde{X}+\tilde{X}({\tilde{X}}^{2}-15)cos\tilde{X})}{{\tilde{X}}^{5}}}.\hfill \end{array}$$
- The gyration is along the observation axis, while the anisotropy is perpendicular to it ($\overline{a}\perp \mathbf{g}\Vert z$) (Figure 3f of the main text).$$\begin{array}{cc}\hfill {\tilde{W}}_{\perp}^{A}& ={\tilde{W}}_{\perp}^{A(0,\Vert )}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +{\textstyle \frac{a(\tilde{X}(7{\tilde{X}}^{2}-36)cos\tilde{X}+(2{\tilde{X}}^{4}-19{\tilde{X}}^{2}+36)sin\tilde{X})}{12{\tilde{X}}^{5}}},\hfill \\ \hfill {\tilde{W}}_{\perp}^{S}& =-{\textstyle \frac{a(2\tilde{X}(11{\tilde{X}}^{2}-39)cos\tilde{X}+3(3{\tilde{X}}^{4}-16{\tilde{X}}^{2}+26)sin\tilde{X})}{4{\tilde{X}}^{5}}}.\hfill \end{array}$$

## References

- Berne, B.J.; Pecora, R. Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics; Courier Corporation: Chelmsford, MA, USA, 2000. [Google Scholar]
- Brongersma, M.L.; Kik, P.G. Surface Plasmon Nanophotonics; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Skipetrov, S.E. Optical devices: Localizing light with electrons. Nat. Nanotechnol.
**2014**, 9, 335. [Google Scholar] [CrossRef] [PubMed] - Réfrégier, P.; Wasik, V.; Vynck, K.; Carminati, R. Analysis of coherence properties of partially polarized light in 3D and application to disordered media. Opt. Lett.
**2014**, 39, 2362. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wang, Y.; Yan, S.; Kuebel, D.; Visser, T.D. Dynamic control of light scattering using spatial coherence. Phys. Rev. A
**2015**, 92, 013806. [Google Scholar] [CrossRef][Green Version] - Gorodnichev, E.E.; Kuzovlev, A.I.; Rogozkin, D.B. Impact of wave polarization on long-range intensity correlations in a disordered medium. J. Opt. Soc. Am. A
**2016**, 33, 95–106. [Google Scholar] [CrossRef] [PubMed] - Hoskins, J.G.; Schotland, J.C. Acousto-optic effect in random media. Phys. Rev. E
**2017**, 95, 033002. [Google Scholar] [CrossRef][Green Version] - Uchida, H.; Masuda, Y.; Fujikawa, R.; Baryshev, A.; Inoue, M. Large enhancement of Faraday rotation by localized surface plasmon resonance in Au nanoparticles embedded in Bi: YIG film. J. Magn. Magn. Mater.
**2009**, 321, 843–845. [Google Scholar] [CrossRef] - Strudley, T.; Akbulut, D.; Vos, W.L.; Lagendijk, A.; Mosk, A.P.; Muskens, O.L. Observation of intensity statistics of light transmitted through 3D random media. Opt. Lett.
**2014**, 39, 6347–6350. [Google Scholar] [CrossRef][Green Version] - de Aguiar, H.B.; Gigan, S.; Brasselet, S. Polarization recovery through scattering media. Sci. Adv.
**2017**, 3, e1600743. [Google Scholar] [CrossRef][Green Version] - Dogariu, A.; Carminati, R. Electromagnetic field correlations in three-dimensional speckles. Phys. Rep.
**2015**, 559, 1–29. [Google Scholar] [CrossRef][Green Version] - Wolf, E. Unified theory of coherence and polarization of random electromagnetic beams. Phys. Lett. A
**2003**, 312, 263–267. [Google Scholar] [CrossRef] - Wolf, E. Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation. Opt. Lett.
**2003**, 28, 1078. [Google Scholar] [CrossRef] [PubMed] - Jonckheere, T.; Müller, C.A.; Kaiser, R.; Miniatura, C.; Delande, D. Multiple scattering of light by atoms in the weak localization regime. Phys. Rev. Lett.
**2000**, 85, 4269. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wiersma, D.S.; Bartolini, P.; Lagendijk, A.; Righini, R. Localization of light in a disordered medium. Nature
**1997**, 390, 671–673. [Google Scholar] [CrossRef] - Lagendijk, A.; van Tiggelen, B.; Wiersma, D.S. Fifty years of Anderson localization. Phys. Today
**2009**, 62, 24–29. [Google Scholar] [CrossRef][Green Version] - Segev, M.; Silberberg, Y.; Christodoulides, D.N. Anderson localization of light. Nat. Photonics
**2013**, 7, 197–204. [Google Scholar] [CrossRef] - Sperling, T.; Schertel, L.; Ackermann, M.; Aubry, G.J.; Aegerter, C.M.; Maret, G. Can 3D light localization be reached in ‘white paint’? New J. Phys.
**2016**, 18, 013039. [Google Scholar] [CrossRef] - Vynck, K.; Pierrat, R.; Carminati, R. Polarization and spatial coherence of electromagnetic waves in uncorrelated disordered media. Phys. Rev. A
**2014**, 89, 013842. [Google Scholar] [CrossRef][Green Version] - Vynck, K.; Pierrat, R.; Carminati, R. Multiple scattering of polarized light in disordered media exhibiting short-range structural correlations. Phys. Rev. A
**2016**, 94, 033851. [Google Scholar] [CrossRef][Green Version] - Rikken, G.; Van Tiggelen, B. Observation of magnetically induced transverse diffusion of light. Nature
**1996**, 381, 54–55. [Google Scholar] [CrossRef] - Erbacher, F.; Lenke, R.; Maret, G. Multiple light scattering in magneto-optically active media. Europhys. Lett.
**1993**, 21, 551. [Google Scholar] [CrossRef] - Skipetrov, S.; Sokolov, I. Magnetic-field-driven localization of light in a cold-atom gas. Phys. Rev. Lett.
**2015**, 114, 053902. [Google Scholar] [CrossRef] [PubMed][Green Version] - Golubentsev, A.A. Interference correction to the albedo of a strongly gyrotropic medium with random inhomogeneities. Radiophys. Quantum Electron.
**1984**, 27, 506–516. [Google Scholar] [CrossRef] - MacKintosh, F.C.; John, S. Coherent backscattering of light in the presence of time-reversal-noninvariant and parity-nonconserving media. Phys. Rev. B
**1988**, 37, 1884–1897. [Google Scholar] [CrossRef] [PubMed] - van Tiggelen, B.A.; Maynard, R.; Nieuwenhuizen, T.M. Theory for multiple light scattering from Rayleigh scatterers in magnetic fields. Phys. Rev. E
**1996**, 53, 2881–2908. [Google Scholar] [CrossRef][Green Version] - Niyazov, R.A.; Kozhaev, M.A.; Achanta, V.G.; Belotelov, V.I. Polarization Eigenchannels in a Magnetic Uncorrelated Disordered Medium. Phys. Met. Metallogr.
**2022**, 123, 447–450. [Google Scholar] [CrossRef] - Kozhaev, M.A.; Niyazov, R.A.; Belotelov, V.I. Correlation of light polarization in uncorrelated disordered magnetic media. Phys. Rev. A
**2017**, 95, 023819. [Google Scholar] [CrossRef][Green Version] - Mittal, M.; Furst, E.M. Electric Field-Directed Convective Assembly of Ellipsoidal Colloidal Particles to Create Optically and Mechanically Anisotropic Thin Films. Adv. Funct. Mater.
**2009**, 19, 3271–3278. [Google Scholar] [CrossRef] - Li, Z.Y.; Wang, J.; Gu, B.Y. Creation of partial band gaps in anisotropic photonic-band-gap structures. Phys. Rev. B
**1998**, 58, 3721–3729. [Google Scholar] [CrossRef][Green Version] - Sadecka, K.; Gajc, M.; Orlinski, K.; Surma, H.B.; Klos, A.; Jozwik-Biala, I.; Sobczak, K.; Dluzewski, P.; Toudert, J.; Pawlak, D.A. When Eutectics Meet Plasmonics: Nanoplasmonic, Volumetric, Self-Organized, Silver-Based Eutectic. Adv. Opt. Mater.
**2014**, 3, 381–389. [Google Scholar] [CrossRef][Green Version] - Grzelczak, M.; Vermant, J.; Furst, E.M.; Liz-Marzán, L.M. Directed Self-Assembly of Nanoparticles. ACS Nano
**2010**, 4, 3591–3605. [Google Scholar] [CrossRef] - Ding, T.; Song, K.; Clays, K.; Tung, C.H. Fabrication of 3D Photonic Crystals of Ellipsoids: Convective Self-Assembly in Magnetic Field. Adv. Mater.
**2009**, 21, 1936–1940. [Google Scholar] [CrossRef] - Khramova, A.E.; Ignatyeva, D.O.; Kozhaev, M.A.; Dagesyan, S.A.; Berzhansky, V.N.; Shaposhnikov, A.N.; Tomilin, S.V.; Belotelov, V.I. Resonances of the magneto-optical intensity effect mediated by interaction of different modes in a hybrid magnetoplasmonic heterostructure with gold nanoparticles. Opt. Express
**2019**, 27, 33170. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wolf, E. Optics in terms of observable quantities. Il Nuovo C. (1943-1954)
**1954**, 12, 884–888. [Google Scholar] [CrossRef] - Roychowdhury, H.; Wolf, E. Determination of the electric cross-spectral density matrix of a random electromagnetic beam. Opt. Commun.
**2003**, 226, 57–60. [Google Scholar] [CrossRef] - Born, M.; Wolf, E. Principles of Optics: 60th Anniversary Edition, 7 ed.; Cambridge University Press: Cambridge, UK, 2019. [Google Scholar] [CrossRef]
- Sheng, P. Introduction to Wave Scattering, Localization and Mesoscopic Phenomena; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2006; Volume 88. [Google Scholar]
- Akkermans, E.; Montambaux, G. Mesoscopic Physics of Electrons and Photons; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Bharucha-Reid, A. Probabilistic Methods in Applied Mathematics; Elsevier Science: Amsterdam, The Netherlands, 2014; Volume 3. [Google Scholar]
- Brekhovskikh, V.L. Effective dielectric constant in the calculation of the second field moments in a randomly inhomogeneous medium. Zh. Eksp. Teor. Fiz
**1985**, 89, 2013–2020. [Google Scholar] - van Rossum, M.C.W.; Nieuwenhuizen, T.M. Multiple scattering of classical waves: Microscopy, mesoscopy, and diffusion. Rev. Mod. Phys.
**1999**, 71, 313–371. [Google Scholar] [CrossRef]

**Figure 1.**Schematic representation of the light propagation from a source at a point ${\mathbf{r}}_{0}$ inside an infinite magnetic nanocomposite medium with a gyration $\mathbf{g}$ containing scatterers with an anisotropy axis $\overline{a}$. Electromagnetic field detectors are placed at points $\mathbf{r}$ and ${\mathbf{r}}^{\prime}$, $\mathbf{X}\equiv \mathbf{r}-{\mathbf{r}}^{\prime}$.

**Figure 2.**(

**a**) Polarization eigenchannels for 3 types of disordered media are shown on the Poincaré sphere. They are depicted by blue dots for the non-magnetic case, $\mathbf{g}=0$, with possible anisotropy, $a\ne 0$, corresponding to the linear polarization. For the magnetic case, $\mathbf{g}\ne 0$, without anisotropy, $a=0$, they are depicted by red dots, corresponding to the circular polarization. For the general case, $\mathbf{g}\ne 0$, and for $a\ne 0$, they are depicted by purple dots, corresponding to the elliptical polarization. The modification of the polarization in the general case by manipulating the gyration is shown by purple lines. (

**b**) Normalized Stokes parameters are computed for the electric field correlation matrix, $\tilde{W}\left(\mathbf{X}\right)$, for components in the plane that is orthogonal to the gyration direction. Red line shows the oscillation of $\tilde{W}$ as a function of $\mathbf{X}$ for the case $\mathbf{g}\Vert \overline{a}\Vert z$. It is directed along the v-axis, so polarization oscillates between the right and left circularity, being limited to zero for a large value of $\mathbf{X}$ (the black dot, full depolarization). For cases $\mathbf{g}\Vert \overline{a}\Vert y$ (purple curve) and $\mathbf{g}\Vert x,\phantom{\rule{0.166667em}{0ex}}\overline{a}\Vert y$ (blue curve), symmetric contributions appear in the correlation. The polarizations acquire ellipticity, i.e., all Stokes parameters obtain non-zero values. (We assume that the direction of light propagation is parallel to the magnetization. $\left|\mathbf{g}\right|=0.2$, $a=0.1$).

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Niyazov, R.A.; Achanta, V.G.; Belotelov, V.I.
Correlation of Light Polarization in the Magnetic Media with Non-Spherical Point-Like Inclusions. *Magnetism* **2023**, *3*, 1-10.
https://doi.org/10.3390/magnetism3010001

**AMA Style**

Niyazov RA, Achanta VG, Belotelov VI.
Correlation of Light Polarization in the Magnetic Media with Non-Spherical Point-Like Inclusions. *Magnetism*. 2023; 3(1):1-10.
https://doi.org/10.3390/magnetism3010001

**Chicago/Turabian Style**

Niyazov, Ramil A., Venu Gopal Achanta, and Vladimir I. Belotelov.
2023. "Correlation of Light Polarization in the Magnetic Media with Non-Spherical Point-Like Inclusions" *Magnetism* 3, no. 1: 1-10.
https://doi.org/10.3390/magnetism3010001