Photonic Hall effect in absorbing media S. Wiebel, 1,3 A. Sparenberg, 1 G. L. J. A. Rikken, 1, * D. Lacoste, 2 and B. A. van Tiggelen 2 1 Grenoble High Magnetic Field Laboratory, Max Planck Institut fu ¨r Festko ¨rperforschung, CNRS, Boı ˆte Postale 166, 38042 Grenoble Cedex 9, France 2 Laboratoire de Physique et Mode ´lisation des Milieux Condense ´s, CNRS, UMR No. 5493, Maison des Magiste `res, Universite ´ Joseph Fourier, Boı ˆte Postale 166, 38042 Grenoble Cedex 5, France 3 Fakulta ¨t fu ¨r Physik, Universita ¨t Konstanz, Universita ¨tsstrasse 10, 78457 Konstanz, Germany Received 17 May 2000 We describe an experimental and theoretical study of the effect of optical absorption on the photonic Hall effect in a passive matrix containing magnetoactive scatterers. We find that for the case of absorbing scatterers, the magnetotransverse light current changes sign and increases with increasing absorption. Good agreement is obtained with numerical calculations. For the case of an absorbing matrix, no effect was observed. PACS numbers: 42.25.Bs, 78.20.Ls, 94.10.Gb Recently it was shown both theoretically 1,2 and experi- mentally 3,4 that light diffusing in a disordered medium subject to a magnetic field can show behavior that bears a strong phenomenological resemblance to well-known elec- tronic magnetotransport effects. In particular, the electronic Hall effect and magnetoresistance were found to have pho- tonic analogues. This may be surprising at first sight, as pho- tons do not carry electric charge and therefore should not couple to a magnetic field. However, such a coupling is in- directly provided through the induced polarization at optical frequencies. Nevertheless, conceptual differences between the two cases do exist. In the electronic case, the number of diffusing particles—the electrons or holes—is usually con- served. This only holds true for the photonic Hall effect, if optical absorption is negligible. Here we will show that in- troducing optical absorption drastically affects the photonic Hall effect. The effect of a static magnetic field B on the optical prop- erties of an isotropic medium is described by the refractive index tensor n( B), given up to first order in B by 5 n ij  B = n +i    ij +i V k  ijl B l , 1 where k is the vacuum wave vector, n +i  is the complex refractive index of the medium, and V the complex Verdet constant, Re V determining the strength of magnetic circular birefringence the Faraday effect and Im V determining the strength of magnetic circular dichroism. In principle, Maxwell’s equations plus the known magneto-optical material parameters as expressed in Eq. 1, enable us to exactly calculate the effect of a magnetic field on the diffusion of light in strongly disordered media. In practice however, this is an insurmountable task. Theoretical simplifications have to be made, as was first done in Refs. 1 and 2. The relevant transport quantity for light diffusion is the second-rank diffusion tensor relating the diffuse photon flux density to the gradient of the photon density. The off- diagonal elements of this tensor represent the photonic Hall effect PHE. This effect, which is linear in B for VBl * 1, is most conveniently characterized by the normalized magnetotransverse light current , given by   B   1 B I L  B  -I R  B  I L  B  +I R  B  . 2 For the definition of I L and I R , see Fig. 1. Recently, a method has been developed to calculate  numerically for scattering spheres of arbitrary size and arbi- trary refractive index. The magnetic-field dependent scatter- ing cross section from a single dielectric Faraday-active sphere was obtained to first order in the field 6. With this solution for the single-scattering problem, the PHE in mul- tiple light scattering was calculated using transport theory for light in nonabsorbing media 7. Below we will compare the results of an extension of such a calculation with our experi- mental results. Our present experiment deals with scatterers made of magneto-optically active material with refractive index n s +i  s and Verdet constant V s . They are randomly distributed with a volume fraction f in an isotropic matrix with refractive index n m and negligible Verdet constant V m . The case of a nonabsorbing magnetoactive matrix containing inactive scatterers was discussed in Ref. 8. If the transport mean- free path l * of the light is much smaller than the geometrical dimensions of the sample, the propagation of light is diffu- sive. The effect of the magnetic field on the diffusion can be described by the dimensionless parameter VBl * 9. The ex- periments we present here are in the range VBl * 1. The quantity VBl * determines the average number of Faraday rotations of the electric polarization vector of the light be- tween subsequent scattering events. The same parameter has been shown to be important for the suppression of coherent backscattering in magnetic fields, both experimentally 10,11 and numerically 12. A similar situation occurs in diffusive electronic magnetotransport. There, the magnetic field effect is determined by the dimensionless parameter  c  , being the average number of cyclotron orbits an elec- tron completes between subsequent scattering events. Simple free-electron models show that the resulting Hall angle is proportional to  c  and that the longitudinal electronic mag- netoresistance is proportional to (  c  ) 2 13. Also in the *Author to whom correspondence should be sent. FAX: +33 4 76 85 56 10. Email address: rikken@polycnrs-gre.fr PHYSICAL REVIEW E DECEMBER 2000 VOLUME 62, NUMBER 6 PRE 62 1063-651X/2000/626/86364/$15.00 8636 ©2000 The American Physical Society