Transport Mean Free Path for Magneto-Transverse Light Diffusion

– We derive an expression for the transport mean free path l∗⊥ associated with magneto-transverse light diffusion for a random collection of Faraday-active Mie scatterers. This expression relates the magneto-transverse diffusion in multiple scattering directly to the magneto-transverse scattering of a single scatterer. Magneto-transverse light diffusion more popularly known as the “Photonic Hall Effect” (PHE) has been predicted theoretically some three years ago [1], and has been confirmed experimentally one year later [2]. Phenomenologically, the effect has many similarities to the well-known electronic Hall effect: Given a diffusion current J, the presence of an external and constant magnetic field creates a flow in the “magneto-transverse” direction qB×J, with q the charge of the current carriers. By the non-existence of photon charge, the physics of the PHE seems different, and compares perhaps better to the so-called Beenakker-Senftleben effect in dilute gazes [3]. The evident driver behind the electronic Hall effect is the Lorentz force acting on a charged particle while colliding with the impurities. According to Ref. [1] the PHE finds its origin in the Faraday effect for dielectric scatterers, that slightly changes their scattering amplitude. The charge q is replaced by a material parameter V with the symmetry of charge, quantifying the Faraday effect of the particle’s material. In a homogeneous medium the Faraday effect implies a rotation V B per unit length of the polarization vector of linearly polarized light. Two other magneto-optical effects in multiple scattering, such as the suppression of coherent backscattering in a magnetic field [4, 5], and “Photonic MagnetoResistance” [6] are known to exist, and basically originate from the same Faraday effect. In an isotropic medium, Fick’s phenomenological law relates the diffusion current to the energy-density gradient ∇ρ according to J = −D0∇ρ [7]. D0 is the conventional diffusion constant for radiative transfer and is usually related to the transport mean free path l and the transport velocity vE according to D = 1 3 vEl . The velocity is relevant only for dynamical experiments. In order to describe stationary phenomena like the PHE, we can put Typeset using EURO-LTEX 2 EUROPHYSICS LETTERS vE = 1. Fick’s law applies to media much bigger than l ∗ and, when supplied by boundary conditions involving the incident flux, can be solved for the emerging current. In a magnetic field, the diffusion constant must be replaced by a second-rank tensor. By Onsager’s relation Dij(B) = Dji(−B), the part linear in the external magnetic field must be an antisymmetric tensor, and Fick’s law becomes, J = −D(B)·∇ρ = −D0∇ρ−D⊥B×∇ρ . (1) The term containingD⊥ describes a magneto-transverse diffusion current. In analogy toD0, we shall define the transport mean free path l ⊥ for magneto-transverse diffusion as D⊥ = 1 3 vEl ∗ ⊥ . For the electronic Hall effect, l ⊥ would be proportional to the Hall conductivity σxy, whose sign is determined by the charge of the current carriers. Similarly, l ⊥ of the PHE can have both signs depending on the scattering. Anisotropy in the scattering cross-section quantified by the familiar anisotropy factor 〈cos θ〉 [8] is well known to make the transport mean free path l different from the extinction length l [9]. The latter is the average distance between two subsequent scattering events. In this Letter we will show that the PHE can be understood as a generalization of this anisotropy factor to magneto light diffusion, which establishes a difference in scattering between “upward” and “downward” directions (with respect to the plane of incident light and magnetic field). To this end we use our solution for the Faraday-active dielectric sphere [10] to relate the PHE in multiple scattering, quantified by l ⊥ , directly to the PHE of one single particle. Although such a link may be physically clear, it is not evident from previous work [1]. A microscopic approach provides both the exact sign as well as the role of anisotropy, which will enable us to conclude this Letter with a realistic comparison to reported experiments [2]. The magneto-active dielectric sphere has been discussed by Ford etal. [11]. Experiments and symmetry arguments show the PHE to be linear in B. Therefore we have developed a linear perturbation formula for the scattering amplitude Tkk′(B) of one magneto-active dielectric sphere [10]. The differential cross-section, proportional to the modulus squared of the scattering amplitude must satisfy the reciprocity relation dσ/dΩ(p → p,B) = dσ/dΩ(−p → −p,−B). A magneto-cross-section proportional to (p̂ · B̂) or (p̂ · B̂) is parity-forbidden since B is a pseudo-vector and p a vector. Together with the rotational symmetry of the sphere it must have the form, 1 σtot dσ dΩ (p → p,B) = F0(θ) + det(p̂, p̂ , B̂) F1(θ) , (2) where cos θ = p̂ · p̂, σtot is the total cross-section and det(A,B,C) = A · (B×C) the scalar determinant that can be constructed from three vectors. The second term in Eq. (2) will be called the magneto cross-section. For a small Rayleigh scatterer one finds and F1(θ) ∼ V B cos θ/k. The antisymmetry of this magneto cross-section between forward scattering and backscattering causes the PHE to vanish. The Mie solution breaks this symmetry and a PHE was seen to emerge [10]. We will use field techniques developed in Refs. [9, 12] to calculate the magneto-transverse transport mean free path for a random collection of identical Faraday-active dielectric spheres. The four-rank tensor Lijkl,pp′(Ω,q) linearly connects field correlations 〈El(ω +Ω/2,p+ q/2) Ēj(ω − Ω/2,p− q/2) 〉 of incident and outgoing fields in space-time; Ω is the Laplace of time, and q is the Fourier variable of space. On long time and length scales (Ω,q → 0) and without absorption it takes the diffuse form, Lijkl,pp′ (Ω,q,B) = lik(p,q,B)llj(−p ,−q,−B) −iΩ + q ·D(B)·q . (3) Transport Mean Free Path for Magneto-Transverse Light Diffusion 3 The symmetric form of the nominator is imposed by the reciprocity principle. Rigorous transport theory yields [12], l(p,q,B) = i [G(p,B)−G(p,B)]− iG(p,B) · Γ(p,q,B) ·G(p,B) . (4) G(p,B) denotes the Dyson Green’s tensor of the ensemble-averaged electric field to be specified later; the asterisk denotes hermitean conjugation in polarization space. We left out explicit reference to the optical frequency ω = pc0. The tensor Γ(p,q,B) is linear in q. In real space, the wavenumber q corresponds to the gradient in Fick’s law (1). The exact relation between Γ and the diffusion tensor is [1, 12],