1. The search for optical analogues of quantum effects observed in electron transport in solid structures, has attracted attention for more than thirty years [1–3]. The results obtained in this line of research open up new possibilities for manipulating coherent light fields [3] and are important for many applications, e.g., for the creation of elements of optical devices (photoconverters [4], sensors, spectrometers [5, 6], random lasers [7], etc.). As examples illustrating the analogy between optical and quantum electronic phenomena, we can mention Anderson localization of light in random layered structures [8, 9], optical Tamm states in photonic crystals [10], universal fluctuations of optical conductance [11, 12], the photonic analog of the Aharonov–Bohm effect [13, 14].

Although the optical conductance has been discussed in a number of theoretical and experimental works (see [11, 12, 15]) in the context of studying its fluctuations in quasi-one-dimensional (Q1D) systems (waveguides), the contribution of weak localization of electromagnetic waves to the conductance has not yet been studied. Unlike electrons, for which various ways of manipulating the interference contribution to the conductance are known [16–18], this question has not been considered in relation to the optical conductance. The effect of weak localization of light was discussed only in application to the angle-dependent reflectance, where it manifests itself as a coherent enhancement of backscattering (see, e.g., [2, 19, 20] and references therein).

As well known [2, 16–18], the effect of weak localization in metals with impurities is a consequence of the interference of electron waves passing along time-reversed trajectories. It manifests itself, in particular, as a negative magnetoresistance [2, 16–18], i.e., decrease in the negative interference correction to the conductance with increasing magnetic field. In this case, random phase shift between interfering electron waves is due to the Aharonov–Bohm effect [2, 16]. For light waves, the situation is different. The magnetic field influences the interference due to the Faraday effect. Unlike electrons, the magnetic field does not affect the interference of waves if their helicity does not change [21–23]. The mechanism of the interference destruction is triggered only by depolarizing helicity-flip scattering. Phase shifts caused by the Faraday effect add up for the interference of waves with the same helicity, but cancel out for waves with the opposite one. As a result of multiple scattering, random phase shifts occur between waves with the same helicity, which leads to the interference suppression.

In the present work, the interference correction to the optical conductance of a disordered sample of the Faraday medium is calculated. The calculations are based on a system of diffusion equations for two Cooperon modes describing the interference of time-reversed waves with a given helicity. As in the case of negative magnetoresistance of metals with impurities, the interference contribution to the optical conductance is shown to decrease with increasing the magnetic field. In the strong field limit, the effect of “saturation” occurs. The interference contribution to the optical conductance reaches its limiting value, which does not depend on the field strength and is determined by the ratio of the transport mean free path to the helicity-flip scattering mean free path. In the case of a waveguide (Q1D geometry), the interference contribution turns out to be inversely proportional to the field strength and the sample length L. This allows us to conclude that there is a critical value of the field that prevents the transition to the strong Anderson localization regime with increasing L.

2. Consider the transmission of electromagnetic waves through a sample of a magnetoactive medium with scattering centers. It is assumed that the linear sizes of the sample, \({{L}_{x}},{{L}_{y}},L\), significantly exceed the transport mean free path, \({{L}_{x}},{{L}_{y}},L \gg {{l}_{{{\text{tr}}}}}\), and the weak localization condition, \({{k}_{0}}{{l}_{{{\text{tr}}}}} \gg 1\) (\({{k}_{0}}\) is the wavenumber), is fulfilled. The number of transverse modes N that characterize the propagation of electromagnetic waves through the sample is large, \(N = k_{0}^{2}A{\text{/}}(4\pi ) \gg 1\) (\(A = {{L}_{x}}{{L}_{y}}\) is the area of the sample cross-section) [12].

According to [2, 3, 15], the conductance (total transmittance) of the sample is the sum of the transmission coefficients \({{T}_{{ab}}}\) connecting the input and output modes a and b, respectively,

$$T = \sum\limits_{a,b} \;{{T}_{{ba}}}.$$
(1)

In the first approximation in \(1{\text{/}}({{k}_{0}}{{l}_{{{\text{tr}}}}})\), the conductance \(\langle T\rangle \) averaged over random positions of scattering centers is determined by the ladder diagrams and is equal to [2, 3, 11]

$$\langle T\rangle = 2N{\kern 1pt} {{l}_{{{\text{tr}}}}}{\text{/}}L.$$
(2)

The factor 2 appearing in \(\langle T\rangle \) takes into account two independent wave polarizations, either orthogonal linear polarizations or two circular ones, clockwise and counterclockwise. The interference contribution \(\langle \delta T\rangle \) to the conductance is described by the diagram containing one Hikami vertex (see Fig. 1 and [2, 16–18]). The average Green’s function \(\langle {{G}_{{ik}}}\left( {{\mathbf{r}},{\mathbf{r}}'{\kern 1pt} {\text{|}}{\mathbf{h}}} \right)\rangle \) appearing in the diagram describes the propagation of electromagnetic waves in the magnetoactive disordered medium between scattering events. It is determined by the expression [22–26]

$$\begin{gathered} \langle {{G}_{{ik}}}\left( {{\mathbf{r}},{\mathbf{r}}'{\kern 1pt} {\text{|}}{\mathbf{h}}} \right)\rangle = \langle {{G}_{{{\text{scal}}}}}\left( {{\text{|}}{\mathbf{r}} - {\mathbf{r}}'{\text{|}}} \right)\rangle \\ \times \;\left[ {P_{{ik}}^{{( + )}}\left( {\mathbf{n}} \right){{{\text{e}}}^{{ - {\text{i}}{\mathbf{h}}({\mathbf{r}} - {\mathbf{r}}')/2}}} + P_{{ik}}^{{( - )}}\left( {\mathbf{n}} \right){{{\text{e}}}^{{{\text{i}}{\mathbf{h}}({\mathbf{r}} - {\mathbf{r}}')/2}}}} \right], \\ \end{gathered} $$
(3)

where \(\langle {{G}_{{{\text{scal}}}}}\left( {{\text{|}}{\mathbf{r}}{\text{|}}} \right)\rangle \) is the scalar Green’s function [2], \({\mathbf{n}} = ({\mathbf{r}} - {\mathbf{r}}'){\text{/|}}{\mathbf{r}} - {\mathbf{r}}'{\kern 1pt} {\text{|}}\) and

$$P_{{ik}}^{{( \pm )}}\left( {\mathbf{n}} \right) = \frac{1}{2}({{\delta }_{{ik}}} - {{n}_{i}}{{n}_{k}} \pm {\text{i}}{{e}_{{ikj}}}{{n}_{j}})$$
(4)

are the projection operators on the states of light with a given helicity, \({{e}_{{ikj}}}\) is the antisymmetric tensor. The vector h appearing in Eq. (3) is proportional to the strength of the applied magnetic field H: \({\mathbf{h}} = 2V{\kern 1pt} {\mathbf{H}}\) (V is the Verdet constant [27]).

Fig. 1.
figure 1

(a) Interference contribution to the conductance. (b) Hikami vertex [2]. Solid lines denote the disorder-average Green’s functions. Dashed lines denote scattering by centers. The incoming \(i\) and outgoing f ladder propagators contain the summation over modes.

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When calculating the diagram shown in Fig. 1, we assume that the magnetic field is sufficiently weak, \(hl \ll 1\) (l is the mean free path, \(l{\text{/}}{{l}_{{{\text{tr}}}}} = 1 - \langle \cos \vartheta \rangle \), \(\langle \cos \vartheta \rangle \) is the mean cosine of the single-scattering angle), and the phase shift due to the field results from many scattering events. In this approximation, we can neglect the influence of the magnetic field on the single-scattering amplitude and ignore the field when calculating the Hikami vertex (see Fig. 1b). The diagram in Fig. 1 can be calculated by “closing” the incoming propagators in the formula obtained in [28, 29] for the correlation function of intensity fluctuation of the polarized light. The resulting “internal” propagator (see the “loop” in Fig. 1a) describes the interference of waves propagating along time-reversed trajectories. It is determined by the sum of the most-crossed diagrams, or by the Cooperon \(\hat {C} = {{\left\langle {{{G}_{{ik}}}\left( {{\mathbf{r}},{\mathbf{r}}'{\kern 1pt} {\text{|}}{\mathbf{h}}} \right)G_{{jl}}^{*}\left( {{{{\mathbf{r}}}_{1}},{\mathbf{r}}_{1}^{'}{\text{|}}{\mathbf{h}}} \right)} \right\rangle }_{{\text{C}}}}\). Using the equality \(G_{{jl}}^{*}\left( {{{{\mathbf{r}}}_{1}},{\mathbf{r}}_{1}^{'}{\kern 1pt} {\text{|}}{\mathbf{h}}} \right) = G_{{lj}}^{*}\left( {{\mathbf{r}}_{1}^{'},{{{\mathbf{r}}}_{1}}{\text{|}} - {\mathbf{h}}} \right)\) we can relate the cooperon to the sum of ladder diagrams where, in one of the Green’s functions, the polarization indices and coordinates are rearranged, and the direction of the magnetic field is changed to the opposite one (see Fig. 2 and [25, 26]),

$$\begin{gathered} {{\left\langle {{{G}_{{ik}}}\left( {{\mathbf{r}},{\mathbf{r}}'{\text{|}}{\mathbf{h}}} \right)G_{{jl}}^{*}\left( {{{{\mathbf{r}}}_{1}},{\mathbf{r}}_{1}^{'}{\text{|}}{\mathbf{h}}} \right)} \right\rangle }_{{\text{C}}}} \\ = {{\left\langle {{{G}_{{ik}}}\left( {{\mathbf{r}},{\mathbf{r}}'{\text{|}}{\mathbf{h}}} \right)G_{{lj}}^{*}\left( {{\mathbf{r}}_{1}^{'},{{{\mathbf{r}}}_{1}}{\text{|}} - {\mathbf{h}}} \right)} \right\rangle }_{L}}. \\ \end{gathered} $$
(5)
Fig. 2.
figure 2

Converting most-crossed diagrams to ladder ones.

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In the case \(L \gg {{l}_{{{\text{tr}}}}}\), when calculating the incoming and outgoing ladder propagators in the diagram shown in Fig. 1a, we can use the standard diffusion approximation (see, e.g., [28, 29]). Then for the interference contribution to the conductance we arrive at the following expression:

$$\begin{gathered} \langle \delta T({\mathbf{h}})\rangle = - \frac{4}{3}\frac{{{{l}_{{{\text{tr}}}}}}}{{{{L}^{2}}}} \\ \times \;\int d{\mathbf{r}}{{\left\langle {{{G}_{{ik}}}\left( {{\mathbf{r}},{\mathbf{r}}{\text{|}}{\mathbf{h}}} \right)G_{{ki}}^{*}\left( {{\mathbf{r}},{\mathbf{r}}{\text{|}} - {\mathbf{h}}} \right)} \right\rangle }_{L}}. \\ \end{gathered} $$
(6)

If we neglect the vector nature of the electromagnetic field in Eq. (6) (i.e., assume that the Green’s function \({{G}_{{ik}}}\) is proportional to \({{\delta }_{{ik}}}\)), then from Eq. (6) we obtain the well-known result [2, 16–18] for the correction to the electronic conductance for weak localization in magnetic field.

3. When light propagates through a scattering medium, only the intensity of waves with a given helicity can decay at distances exceeding the transport mean free path \({{l}_{{{\text{tr}}}}}\) [30–33]. Linearly polarized waves always decay at distances of the order of \({{l}_{{{\text{tr}}}}}\) [30–33]. Therefore, when calculating the Cooperon in the diffusion approximation, we take into account only slowly decaying circularly polarized modes.

For \({{k}_{0}}l \gg 1\), successive scattering events occur in the wave zone, and the correlator \({{\left\langle {{{G}_{{ik}}}\left( {\mathbf{h}} \right)G_{{lj}}^{*}\left( { - {\mathbf{h}}} \right)} \right\rangle }_{L}}\) appearing in Eq. (6) can be written in the Wigner representation as follows:

$${{\left\langle {{{G}_{{ik}}}\left( {{\mathbf{r}},{\mathbf{r}}'{\text{|}}{\mathbf{h}}} \right)G_{{lj}}^{*}\left( {{{{\mathbf{r}}}_{1}},{\mathbf{r}}_{1}^{'}{\text{|}} - {\mathbf{h}}} \right)} \right\rangle }_{L}}$$
$$ = \int \frac{{d{\mathbf{k}}}}{{{{{(2\pi )}}^{3}}}}\frac{{d{\mathbf{k}}'}}{{{{{(2\pi )}}^{3}}}}{{{\text{e}}}^{{{\text{i}}{\mathbf{k}}\left( {{\mathbf{r}} - {{{\mathbf{r}}}_{1}}} \right) - {\text{i}}{\mathbf{k}}'\left( {{\mathbf{r}}' - {\mathbf{r}}{{'}_{1}}} \right)}}}\frac{{\delta (k - {{k}_{0}})}}{{{{k}^{2}}}}\frac{{\delta (k{\kern 1pt} '\; - {{k}_{0}})}}{{k{\kern 1pt} {{'}^{2}}}}$$
(7)
$$ \times \;{{\Gamma }_{{il,kj}}}\left( {\left. {\frac{{{\mathbf{r}} + {{{\mathbf{r}}}_{1}}}}{2},{\mathbf{n}} = \frac{{\mathbf{k}}}{{{{k}_{0}}}}} \right|\frac{{{\mathbf{r}}' + {\mathbf{r}}_{1}^{'}}}{2},{\mathbf{n}}' = \frac{{{\mathbf{k}}'}}{{{{k}_{0}}}}} \right),$$

where the propagator \({{\Gamma }_{{il,kj}}}\left( {{\mathbf{r}},{\mathbf{n}}{\text{|}}{\mathbf{r}}',{\mathbf{n}}'} \right)\) obeys the transport equation (see Fig. 3 and [23, 24]), which is similar to the radiative transfer equation for electromagnetic waves. When going from the laboratory reference frame to the concomitant one the matrix \({{\Gamma }_{{il,kj}}}\) transforms as

$$\begin{gathered} {{\Gamma }_{{il,kj}}}\left( {\left. {{\mathbf{r}},{\mathbf{n}}} \right|{\mathbf{r}}',{\mathbf{n}}'} \right) = \sum\limits_{\alpha ,\beta ,\gamma ,\delta } e_{i}^{{(\alpha )}}\left( {\mathbf{n}} \right)\left( {e_{l}^{{(\beta )}}\left( {\mathbf{n}} \right)} \right){\text{*}}{\text{,}} \\ {{\Gamma }_{{\alpha \beta ,\gamma \delta }}}\left( {\left. {{\mathbf{r}},{\mathbf{n}}} \right|{\mathbf{r}}',{\mathbf{n}}'} \right)e_{k}^{{(\gamma )}}\left( {{\mathbf{n}}'} \right)\left( {e_{j}^{{(\delta )}}\left( {{\mathbf{n}}'} \right)} \right){\text{*}}, \\ \end{gathered} $$
(8)

where indices \(\alpha ,\beta ,\gamma ,\delta \) take the values \( \pm 1\). Three ve-ctors, n = \((\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )\) and \({{{\mathbf{e}}}^{{( \pm )}}}\left( {\mathbf{n}} \right) = (\partial {\mathbf{n}}{\text{/}}\partial \theta \mp {\text{i}}[{\mathbf{n}} \times \partial {\mathbf{n}}{\text{/}}\partial \theta ]){\text{/}}\sqrt 2 \), form a circular basis (see, e.g., [34]).

Fig. 3.
figure 3

Integral transport equation for the propagator \({{\Gamma }_{{il,kj}}}\).

Full size image

To go to an approximation in which only circularly polarized modes remain, we should keep the terms with \(\alpha = \beta \) and \(\gamma = \delta \) in the sum (8). Then instead of Eq. (8) we obtain [23]

$$\begin{gathered} {{\Gamma }_{{il,kj}}}\left( {\left. {{\mathbf{r}},{\mathbf{n}}} \right|{\mathbf{r}}',{\mathbf{n}}'} \right) \\ = \sum\limits_{\alpha ,\beta } P_{{il}}^{{(\alpha )}}\left( {\mathbf{n}} \right){{\Gamma }_{{\alpha \beta }}}\left( {\left. {{\mathbf{r}},{\mathbf{n}}} \right|{\mathbf{r}}',{\mathbf{n}}'} \right)P_{{jk}}^{{(\beta )}}\left( {{\mathbf{n}}'} \right), \\ \end{gathered} $$
(9)

where \(P_{{il}}^{{(\alpha )}}\left( {\mathbf{n}} \right) = e_{i}^{{(\alpha )}}\left( {\mathbf{n}} \right)\left( {e_{l}^{{(\alpha )}}\left( {\mathbf{n}} \right)} \right){\text{*}}\) are the projection operators (4). Combining Eqs. (6) and (9), we arrive at the following expression for the quantity \(\langle \delta T\rangle \):

$$\langle \delta T({\mathbf{h}})\rangle = - \frac{4}{3}\frac{{{{l}_{{{\text{tr}}}}}}}{{{{L}^{2}}}}\sum\limits_\alpha \left[ {\int d{\mathbf{r}}{{\Gamma }_{{\alpha \alpha }}}\left( {{\mathbf{r}},{\mathbf{r}}} \right)} \right],$$
(10)

where \({{\Gamma }_{{\alpha \alpha }}}\left( {{\mathbf{r}},{\mathbf{r}}'} \right) = \int d{\mathbf{n}}\int d{\mathbf{n}}'{{\Gamma }_{{\alpha \alpha }}}\left( {\left. {{\mathbf{r}},{\mathbf{n}}} \right|{\mathbf{r}}',{\mathbf{n}}'} \right)\) is the density propagator.

If in the original integral transport equation for the propagator \({{\Gamma }_{{il,kj}}}\left( {\left. {{\mathbf{r}},{\mathbf{n}}} \right|bfr',{\mathbf{n}}'} \right)\) (see Fig. 3) we go to the representation (9) and take advantage of the diffusion approximation, then we obtain a system of diffusion equations for the density propagator \({{\Gamma }_{{\alpha \beta }}}\left( {{\mathbf{r}},{\mathbf{r}}'} \right)\) [23]

$$\left( {\begin{array}{*{20}{c}} {{\mathbf{\hat {D}}}_{ - }^{2} + {{\zeta }^{2}}}&{ - {{\zeta }^{2}} - \chi {{{{\mathbf{\hat {D}}}}}_{ + }}{{{{\mathbf{\hat {D}}}}}_{ - }}} \\ { - {{\zeta }^{2}} - \chi {{{{\mathbf{\hat {D}}}}}_{ + }}{{{{\mathbf{\hat {D}}}}}_{ - }}}&{{\mathbf{\hat {D}}}_{ + }^{2} + {{\zeta }^{2}}} \end{array}} \right)$$
$$ \times \;\left( {\begin{array}{*{20}{c}} {{{\Gamma }_{{ + + }}}\left( {{\mathbf{r}},{\mathbf{r}}'} \right)}&{{{\Gamma }_{{ + - }}}\left( {{\mathbf{r}},{\mathbf{r}}'} \right)} \\ {{{\Gamma }_{{ - + }}}\left( {{\mathbf{r}},{\mathbf{r}}'} \right)}&{{{\Gamma }_{{ - - }}}\left( {{\mathbf{r}},{\mathbf{r}}'} \right)} \end{array}} \right)$$
(11)
$$ = 3{{n}_{0}}{{\sigma }_{{{\text{tr}}}}}(1 - \chi )\delta \left( {{\mathbf{r}} - {\mathbf{r}}'} \right)\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right),$$

where \({{{\mathbf{\hat {D}}}}_{ \pm }} = - {\text{i}}\nabla \pm {\mathbf{h}}\),

$${{\zeta }^{2}} = 3n_{0}^{2}{{\sigma }_{{{\text{dep}}}}}(\sigma _{{{\text{tr}}}}^{{{\text{(2)}}}} + {{\sigma }_{{{\text{dep}}}}})\frac{{{{\sigma }_{{{\text{tr}}}}}}}{{{{\sigma }_{{{\text{tr}}}}} + \sigma _{{{\text{tr}}}}^{{(2)}} + {{\sigma }_{{{\text{dep}}}}}}},$$
(12)
$$\chi = \frac{{{{\sigma }_{{{\text{tr}}}}} - \sigma _{{{\text{tr}}}}^{{{\text{(2)}}}} - {{\sigma }_{{{\text{dep}}}}}}}{{{{\sigma }_{{{\text{tr}}}}} + \sigma _{{{\text{tr}}}}^{{{\text{(2)}}}} + {{\sigma }_{{{\text{dep}}}}}}},$$
(13)

\({{n}_{0}}\) is the number of particles per unit volume, σtr = \(\int d{\mathbf{n}}'(1 - ({\mathbf{nn}}')){{a}_{1}}({\mathbf{nn}}')\) and \(\sigma _{{tr}}^{{(2)}} = \int d{\mathbf{n}}'(1 - ({\mathbf{nn}}')){{a}_{2}}({\mathbf{nn}}')\) are the transport scattering cross sections for the intensity and the fourth Stokes parameter, \({{a}_{1}}({\mathbf{nn}}')\) and \({{a}_{2}}({\mathbf{nn}}')\) are the diagonal elements of the single-scattering matrix [34], \({{\sigma }_{{{\text{dep}}}}} = \int d{\mathbf{n}}'({{a}_{1}}({\mathbf{nn}}') - {{a}_{2}}({\mathbf{nn}}'))\) is the helicity-flip scattering cross section [30, 32].

In the absence of the magnetic field, the quantities \({{\Gamma }_{{ \pm \pm }}}\) and \({{\Gamma }_{{ \pm \mp }}}\) can be expressed in terms of the intensity I and the fourth Stokes parameter V as Γ++ = \({{\Gamma }_{{ - - }}} = (I + V){\text{/}}2\) and \({{\Gamma }_{{ + - }}} = {{\Gamma }_{{ - + }}} = (I - V){\text{/}}2\). In this case, Eq. (11) reduces to two independent diffusion equations for I and V, respectively [35].

Under conditions of slow decay of circular polarization [30–32, 36] the difference between \({{a}_{1}}({\mathbf{nn}}')\) and \({{a}_{2}}({\mathbf{nn}}')\) is small and, as a consequence, \({{\sigma }_{{{\text{dep}}}}} \ll {{\sigma }_{{{\text{tr}}}}}\) and \({{\sigma }_{{{\text{tr}}}}} - \sigma _{{{\text{tr}}}}^{{{\text{(2)}}}} \ll {{\sigma }_{{{\text{tr}}}}}\). The parameter χ appearing in Eq. (13) is also small, \(\chi \ll 1\). Under conditions of rapid depolarization, \({{\sigma }_{{{\text{tr}}}}} \sim {{\sigma }_{{{\text{dep}}}}}\), the value of χ increases. For example, in the case of Rayleigh scattering \(\chi = 1{\text{/}}3\) [23], for scattering by Mie particles in the vicinity of the second Kerker point [36] \(\chi = 1{\text{/}}2\). According to the calculations of the quantities entering into Eqs. (12) and (13) [37], the parameter χ reaches 0.6−0.7 for strongly depolarizing scattering from correlated Mie particles.

4. To determine the interference correction \(\langle \delta T\rangle \) to the conductance of a plane slab, we assume that \({{L}_{x}},{{L}_{y}} \gg L \gg {{l}_{{{\text{tr}}}}}\), \({{l}_{{{\text{tr}}}}} = ({{n}_{0}}{{\sigma }_{{{\text{tr}}}}}{{)}^{{ - 1}}}\). When calculating the density propagator \({{\Gamma }_{{\alpha \alpha }}}\) in the case of a relatively strong magnetic field, but under conditions of applicability of the diffusion approximation, \(1{\text{/}}L \ll h \ll 1{\text{/}}{{l}_{{{\text{tr}}}}}\), we can consider the medium to be unbounded. In this sit-uation

$$\langle \delta T({\mathbf{h}})\rangle - \langle \delta T({\mathbf{h}} = 0)\rangle = - \frac{{4\langle T\rangle }}{{\pi k_{0}^{2}{{l}_{{{\text{tr}}}}}}}$$
$$ \times \;\int\limits_0^\infty {{Q}^{2}}dQ\left[ {\frac{{(1 - \chi )\left( {{{Q}^{2}} + {{h}^{2}} + {{\zeta }^{2}}} \right)}}{{2Qh\Phi (Q)}}\ln \frac{{\Phi (Q) + 2Qh}}{{\Phi (Q) - 2Qh}}} \right.$$
(14)
$$\left. { - \left( {\frac{1}{{{{Q}^{2}}}} + \frac{{(1 - \chi )}}{{(1 + \chi ){{Q}^{2}} + 2{{\zeta }^{2}}}}} \right)} \right],$$

where \(\langle T\rangle \) is the optical conductance (2), and

$$\begin{gathered} \Phi (Q) = \left[ {\left( {{{Q}^{2}} + {{h}^{2}}} \right)\left( {{{Q}^{2}} + {{h}^{2}} + 2{{\zeta }^{2}}} \right)} \right. \\ {{\left. { - \;\chi \left( {{{Q}^{2}} - {{h}^{2}}} \right)\left( {\chi \left( {{{Q}^{2}} - {{h}^{2}}} \right) + 2{{\zeta }^{2}}} \right)} \right]}^{{1/2}}}. \\ \end{gathered} $$
(15)

Equation (14) determines the dependence of the interference correction to the conductance on the magnetic field. As is known [2, 16, 18], the interference correction itself diverges in the 3D case (each of the contributions to Eq. (14) is divergent). The measurable quantity is the change in the interference contribution depending on the factor that destroys the interference.

Under conditions of circular polarization memory, \({{\sigma }_{{{\text{dep}}}}} \ll {{\sigma }_{{{\text{tr}}}}}\) (in this case the parameter χ is also small, \(\chi \ll 1\)), two regions can be distinguished in the h‑dependence of \(\langle \delta T\rangle \). For \(h \ll {{n}_{0}}\sqrt {{{\sigma }_{{{\text{tr}}}}}{{\sigma }_{{{\text{dep}}}}}} \) the quantity \(\langle \delta T\rangle \) changes linearly with the magnetic field h,

$$\langle \delta T({\mathbf{h}})\rangle - \langle \delta T({\mathbf{h}} = 0)\rangle = \frac{{2\left\langle T \right\rangle }}{{{{{({{k}_{0}}{{l}_{{{\text{tr}}}}})}}^{2}}}}\left( {h{{l}_{{{\text{tr}}}}}} \right).$$
(16)

According to Eqs. (14)–(16), in the case of \(h \gg 1{\text{/}}L\) the interference contribution \(\langle \delta T\rangle \) does not depend on the direction of the field. The orientation dependence of \(\langle \delta T\rangle \) manifests itself only at \(h \leqslant 1{\text{/}}L\), where the effects due to the finite sizes of the sample become important. However, in this case, the influence of the field on the change in the interference correction is insignificant.

For \(h \geqslant {{n}_{0}}\sqrt {{{\sigma }_{{{\text{tr}}}}}{{\sigma }_{{{\text{dep}}}}}} \) the interference contribution \(\langle \delta T\rangle \) reaches a plateau (see Fig. 4) and tends to

$$\langle \delta T({\mathbf{h}})\rangle - \langle \delta T({\mathbf{h}} = 0)\rangle = \frac{2}{{\sqrt 2 + 1}}\frac{{\langle T\rangle }}{{{{{({{k}_{0}}{{l}_{{{\text{tr}}}}})}}^{2}}}}\sqrt {\frac{{3{{\sigma }_{{{\text{dep}}}}}}}{{{{\sigma }_{{{\text{tr}}}}}}}} .$$
(17)
Fig. 4.
figure 4

(Color online) Change \(\langle \delta T(h)\rangle - \langle \delta T(h = 0)\rangle \) in the interference contribution to the optical conductance versus the magnetic field under conditions of circular polarization memory (\(\chi = 0\), \({{\sigma }_{{{\text{tr}}}}}{\text{/}}{{\sigma }_{{{\text{dep}}}}} = 10\), lower curve) and in the absence of it (Rayleigh scatterers, \(\chi = 1{\text{/}}3\), \({{\sigma }_{{{\text{tr}}}}}{\text{/}}{{\sigma }_{{{\text{dep}}}}} = 1\), upper curve).

Full size image

In the absence of circular polarization memory, \({{\sigma }_{{{\text{dep}}}}} \sim {{\sigma }_{{{\text{tr}}}}}\) (e.g., for Rayleigh scattering \(\chi = 1{\text{/}}3\) and \({{\sigma }_{{{\text{dep}}}}} = {{\sigma }_{{{\text{tr}}}}}\)), the slope in the linear dependence of \(\langle \delta T({\mathbf{h}})\rangle - \langle \delta T({\mathbf{h}} = 0)\rangle \) on h at \(h \ll 1{\text{/}}{{l}_{{{\text{tr}}}}}\) increases, and the transition to the plateau shifts to the region of large \(h \sim 1{\text{/}}{{l}_{{{\text{tr}}}}}\), where the diffusion approach loses its applicability (see Fig. 4).

The appearance of a plateau in the h-dependence of \(\langle \delta T({\mathbf{h}})\rangle - \langle \delta T({\mathbf{h}} = 0)\rangle \) can be explained by decrease in the probability of depolarizing collisions over the length \({{h}^{{ - 1}}}\) with increasing the magnetic field. In a strong field, \(h \gg \zeta \), under conditions of slow decay of circular polarization, \(\chi \ll 1\), the off-diagonal elements in the system of Eqs. (11) can be neglected, and Eq. (11) reduces to two independent equations

$$\begin{gathered} \left\{ {{{{\left( {\frac{1}{{\text{i}}}\nabla \pm {\mathbf{h}}} \right)}}^{2}} + {{\zeta }^{2}}} \right\}{{\Gamma }_{{ \pm \pm }}}\left( {{\mathbf{r}},{\mathbf{r}}'} \right) \\ = 3{{n}_{0}}{{\sigma }_{{{\text{tr}}}}}(1 - \chi )\delta \left( {{\mathbf{r}} - {\mathbf{r}}'} \right). \\ \end{gathered} $$
(18)

Using the transformation

$${{\Gamma }_{{ \pm \pm }}}\left( {{\mathbf{r}} - {\mathbf{r}}'} \right) = {{\Gamma }_{{{\text{scal}}}}}\left( {{\mathbf{r}} - {\mathbf{r}}'} \right){{{\text{e}}}^{{ \mp {\text{i}}{\mathbf{h}}\left( {{\mathbf{r}} - {\mathbf{r}}'} \right)}}}$$
(19)

we can eliminate the magnetic field from Eq. (18). From Eq. (19) it follows that \({{\Gamma }_{{ \pm \pm }}}\left( {{\mathbf{r}} = {\mathbf{r}}'} \right) = \) \({{\Gamma }_{{{\text{scal}}}}}\left( {{\mathbf{r}} = {\mathbf{r}}'} \right)\), and the interference contribution (10) to the conductance turns out to be independent of the magnetic field. The scalar Cooperon \({{\Gamma }_{{{\text{scal}}}}}\left( {{\mathbf{r}} - {\mathbf{r}}'} \right)\) appearing to Eq. (19) satisfies the diffusion equation with the attenuation coefficient ζ, which takes into account helicity-flip scattering.

5. When light propagates alone a waveguide, \({{L}_{x}},{{L}_{y}} \ll L\) (Q1D geometry), the situation changes. In this case, if we do not consider a waveguide with a specially adjusted transverse profile of refractive index, the effect of circular polarization memory is suppressed. With each reflection from the side boundaries of the waveguide, the sign of circular polarization changes to the opposite [38]. In this situation, the main contribution to the interference correction (10) is made by the linear combination of elements \({{\Gamma }_{{ \pm \pm }}} + {{\Gamma }_{{ \pm \mp }}}\), which survives at large lengths L. In the absence of the magnetic field, this combination corresponds to the scalar mode, i.e., the intensity. The difference \({{\Gamma }_{{ \pm \pm }}} - {{\Gamma }_{{ \pm \mp }}}\) decays on the scales of the order of \({{L}_{x}},{{L}_{y}}\) and turns out to be small. Therefore, in the first approximation, we can assume that all elements of \({{\Gamma }_{{\alpha \beta }}}\) are equal to each other, \({{\Gamma }_{{\alpha \beta }}} = \Gamma \), and obey the equation

$$\left[ {{{\nabla }^{2}} - \frac{{1 + \chi }}{{1 - \chi }}{{h}^{2}}} \right]\Gamma \left( {{\mathbf{r}},{\mathbf{r}}'} \right) = - \frac{3}{2}{{n}_{0}}{{\sigma }_{{{\text{tr}}}}}\delta \left( {{\mathbf{r}} - {\mathbf{r}}'} \right)$$
(20)

with the boundary conditions

$${{\left. {\Gamma \left( {{\mathbf{r}},{\mathbf{r}}'} \right)} \right|}_{{z = 0,L}}} = 0,$$
(21)

at the input and output ends of the waveguide and

$${{\left. {\frac{\partial }{{\partial x}}\Gamma \left( {{\mathbf{r}},{\mathbf{r}}'} \right)} \right|}_{{x = \pm {{L}_{x}}/2}}} = 0,\quad {{\left. {\frac{\partial }{{\partial y}}\Gamma \left( {{\mathbf{r}},{\mathbf{r}}'} \right)} \right|}_{{y = \pm {{L}_{y}}/2}}} = 0$$
(22)

on its lateral surface.

The solution of Eq. (20) can be sought as an expansion in eigenmodes \(\cos {{q}_{{xn}}}x\cos {{q}_{{ym}}}y\), where qxn = \((2\pi n{\text{/}}{{L}_{x}})\), \({{q}_{{ym}}} = (2\pi m{\text{/}}{{L}_{y}})\) (\(n,m = 0, \pm 1, \pm 2, \ldots \)). Then Eq. (10) is transformed as follows:

$$\langle \delta T({\mathbf{h}})\rangle = - \frac{8}{3}\frac{{{{l}_{{{\text{tr}}}}}}}{{{{L}^{2}}}}\sum\limits_{\mathbf{q}} \int\limits_0^L dz\Gamma \left( {z,z{\text{|}}{\mathbf{q}}} \right),$$
(23)

where \(\Gamma \left( {z,z{\text{|}}{\mathbf{q}}} \right)\) are the coefficients of the \(\Gamma \left( {{\mathbf{r}},{\mathbf{r}}'} \right)\) expansion in terms of the waveguide transverse eigenmodes. According to Eqs. (20) and (21), these coefficients are determined by the expression

$$\Gamma \left( {z,z|{\mathbf{q}}} \right) = \frac{3}{{4\gamma {{l}_{{{\text{tr}}}}}}}\frac{{\sinh (\gamma z)\sinh \left( {\gamma \left( {L - z} \right)} \right)}}{{\sinh (\gamma L)}},$$
(24)

where

$${{\gamma }^{2}} = {{{\mathbf{q}}}^{2}} + {{\tilde {h}}^{2}},\quad \tilde {h} = \sqrt {\frac{{1 + \chi }}{{1 - \chi }}} {\kern 1pt} {\kern 1pt} {\kern 1pt} h.$$
(25)

In Q1D geometry (\(L \gg {{L}_{x}},{{L}_{y}}\)), the mode with \({\mathbf{q}} = 0\) is characterized by the least attenuation. Keeping only the term with \({\mathbf{q}} = 0\) in Eq. (23), we find the interference contribution to the conductance,

$$\langle \delta T({\mathbf{h}})\rangle = - \frac{1}{{(\tilde {h}L)}}\left[ {{\text{coth}}(\tilde {h}L) - \frac{1}{{\tilde {h}L}}} \right].$$
(26)

The dependence of \(\langle \delta T\rangle \) on the magnetic field is illustrated in Fig. 5. In the limit \(h = 0\), Eq. (26) transforms to the well-known result [39] for the interference contribution to the conductance of scalar waves, \(\langle \delta T\rangle = - 1{\text{/}}3\). For large h, the value of \(\langle \delta T\rangle \) decreases as \( - 1{\text{/}}\tilde {h}L\). Qualitatively, the change in \(\langle \delta T\rangle \) with increasing the field resembles the behavior of the interference correction to the electron conductance [16, 17], but the form of the corresponding functional dependence on the field turns out to be different.

Fig. 5.
figure 5

Interference contribution \(\langle \delta T(h)\rangle \) to the optical conductance of the quasi-one-dimensional sample versus the magnetic field.

Full size image

6. Consider how the interference contribution to the optical conductance changes depending on the ratio \({{\sigma }_{{{\text{dep}}}}}{\text{/}}{{\sigma }_{{{\text{tr}}}}}\) and the magnetic field strength.

In the absence of the magnetic field (\(h = 0\)), if there were no depolarization at all (i.e., there were no mixing of polarizations due to scattering, and in Eq. (6)\(\langle {{G}_{{ik}}}G_{{ki}}^{*}\rangle = 2I\) [35], where I is the intensity), the interference correction \(\langle \delta T\rangle \) to the conductance would be equal to \(2\langle \delta {{T}^{{({\text{sc}})}}}\rangle \), where \(\langle \delta {{T}^{{({\text{sc}})}}}\rangle < 0\) is the corresponding result in the scalar wave approximation. Under conditions of rare helicity-flip scattering events (\({{\sigma }_{{{\text{dep}}}}} \ll {{\sigma }_{{{\text{tr}}}}}\)), the interference contribution acquires an additional term

$$\langle \delta T\rangle = 2\langle \delta {{T}^{{({\text{sc}})}}}\rangle + \frac{{2\langle T\rangle }}{{{{{({{k}_{0}}{{l}_{{{\text{tr}}}}})}}^{2}}}}\sqrt {\frac{{3{{\sigma }_{{{\text{dep}}}}}}}{{{{\sigma }_{{{\text{tr}}}}}}}} .$$
(27)

Relation (27) is determined by the second term appearing in Eq. (14). In the case of rapid depolarization of waves, when the helicity changes in each scattering, \(\langle \delta T\rangle = \langle \delta {{T}^{{({\text{sc}})}}}\rangle \) (this follows directly from Eq. (6) in the case of \(\langle {{G}_{{ik}}}G_{{ki}}^{*}\rangle = I\) [35]).

If there were no helicity-flip scattering of waves, then the interference contribution to the optical conductance would remain unchanged when the magnetic field is applied, \(\langle \delta T\rangle = 2\langle \delta {{T}^{{({\text{sc}})}}}\rangle \) (see Eq. (14)). Depolarization turns on the influence of the magnetic field on \(\langle \delta T\rangle \). Under conditions of slow depolarization (\({{\sigma }_{{{\text{dep}}}}} \ll {{\sigma }_{{{\text{tr}}}}}\)), the change in the interference contribution with the magnetic field is described by Eqs. (16) and (17), and at \(h \gg {{n}_{0}}\sqrt {{{\sigma }_{{{\text{dep}}}}}{{\sigma }_{{{\text{tr}}}}}} \) the quantity \(\langle \delta T\rangle \) tends to the value

$$\langle \delta T\rangle = 2\langle \delta {{T}^{{({\text{sc}})}}}\rangle + \frac{{2\langle T\rangle }}{{{{{({{k}_{0}}{{l}_{{{\text{tr}}}}})}}^{2}}}}\sqrt {\frac{{6{{\sigma }_{{{\text{dep}}}}}}}{{{{\sigma }_{{{\text{tr}}}}}}}} ;$$
(28)

i.e., the difference between \(\langle \delta T\rangle \) and \(2\langle \delta {{T}^{{({\text{sc}})}}}\rangle \) changes by a factor of about 1.5 as the field h changes from zero to high values.

In the case of rapid depolarization, the interference contribution to the conductance tends to zero with increasing the magnetic field. Under conditions of wave diffusion through the Q1D sample (waveguide), it is precisely this case that is realized.

It should be noted that in the Q1D geometry, the magnetic field makes it possible to control the transition to the regime of Anderson localization of light in long waveguides, \(L \sim {{l}_{{{\text{loc}}}}} = N{{l}_{{{\text{tr}}}}}\) [39]. As \(\langle T\rangle \) and \(\langle \delta T\rangle \) decrease as \(1{\text{/}}L\) with increasing L (see Eqs. (2) and (26)), the transition to the Anderson localization regime is violated at \(\tilde {h} > {{h}_{c}} \sim 1{\text{/}}N{{l}_{{{\text{tr}}}}}\). For a multimode optical fiber (\(N \gg 1\)), the critical value of the magnetic field \({{h}_{{\text{c}}}}\) turns out to be much less than the one that causes a noticeable effect in the coherent backscattering from the Faraday medium [21, 23]. In electron transport through Q1D systems (wires), such an effect is difficult to observe, as with increasing in L, the coherence of electron waves is destroyed due to inelastic interactions, temperature, and other factors [17, 18].