Hall Effect of Light

Abstract

We derive the semiclassical equation of motion for the wave-packet of light taking into account the Berry curvature in momentum space. This equation naturally describes the interplay between orbital and spin angular-momenta, i.e., the conservation of total angular-momentum of light. This leads to the shift of wave-packet motion perpendicular to the gradient of dielectric constant, i.e., the polarization-dependent Hall effect of light. An enhancement of this effect in photonic crystals is also proposed.PACS numbers: 42.15. Eq, 03.65.Sq, 03.65.Vf, The similarity between geometrical optics and particle dynamics has been the guiding principle to develop the quantum mechanics in its early stage. One can consider a trajectory or ray of light at the length scale much larger than the wavelength λ. By setting = c = 1, the equation for the eikonal ψ in the geometrical optics reads [∂ µ ψ(x)] 2 = [∇φ(r)] 2 − n(r) 2 = 0, where x µ = (r, t) is the four-dimensional coordinates, ψ(x) = φ(r) − n(r)t, and n(r) is the refractive index slowly varying within the wavelength [1]. This equation is identical to the Hamilton-Jacobi equation for the particle motion by replacing ψ by the action S. The equation for the ray of light can be derived from the eikonal equation in parallel to the Hamiltonian equation of motion. Deviation from this geometrical optics is usually treated in terms of the diffraction theory. However, most of the analyses have been done in terms of the scalar diffraction theory, neglecting its vector nature. However, the light has the degree of freedom of the polarization, which is represented by the spin S = 1 parallel or anti-parallel to the wavevector k. It is known that this spin produces the Berry phase when the light is guided by the optical fiber with torsion [2]. Also the adiabatic change of the polarization even without the change of k produces the phase change called Pancharatnam phase [3]. In this Letter, we will show that the trajectory or ray of light itself is affected by the Berry phase [4], which leads to various nontrivial effects including the Hall effect.For simplicity, we focus on an isotropic, nonmagnetic medium; the refractive index n(r) is real and scalar, but is not spatially uniform. Let us consider the wavefunction for the wave-packet with the wavevector centered at k c and position centered at r c . Because of the conjugate relation of the position and wavevector, both of them inevitably have finite width of distribution. Therefore, the relative phase of the wavefunctions at k and k + dk matters, which is the Berry connection or gauge field Λ k defined by [Λ k , where e λk is the polarization vector of λ-polarized photon, and λ = ± correspond to the right-and left-circular polarization. Here, it is noted that the SU(2) gauge field Λ k is 2 × 2 matrix. The corresponding Berry curvature (BC) or field strength is given byDue to the masslessness of a photon, it is diagonal in the basis of the right and left circular polarization as given by Ω k = σ 3 k/k 3 [a massive case will be discussed late...

Emergent functions of quantum materials

“…With a relativistic spin-orbit interaction, this leads to the intrinsic spin Hall effect 66 , the quantized version of which has led to the discovery of topological insulators, described in the next section. Another extension is to consider the Berry phase for neutral particles, such as photons 77,78 and magnons [79][80][81] .…”

Imaging mesoscopic spin Hall flow: Spatial distribution of local spin currents and spin densities in and out of multiterminal spin-orbit coupled semiconductor nanostructures

“…27 suggest that nonequilibrium spin Hall accumulation S z m = 0 will appear only in the four corners at the lead/2DEG interfaces (due to J z y = 0 existing within a spin relaxation length L SO wide region around such interfaces), in contrast to the Keldysh formalism applied to finite-size 2DEG in the Landauer two-terminal setup where non-zero spin accumulation (with opposite sign on the two lateral edges 1,2 ) is found along the whole lateral edge. 9,10 Within the formalism of bond spin currents of Sec. I (which represent the lattice version of the same quantity J z y studied in Ref.…”

“…When such current hits the sample boundary, it will deposit nonequilibrium spin accumulation 6,7,8,9,10 at the lateral edges of the sample attached to two longitudinal electrodes, 9,10 as detected optically in recent breakthrough spin Hall experiments. 1,2 However, there are several apparently disconnected mechanisms capable of inducing the spin Hall currents.…”

Relaxor ferroelectricity induced by electron correlations in a molecular dimer Mott insulator

“…4(a). However, ferroelectricity or charge disproportionation in a dimer is driven intrinsically by the hybridization of two orbitals with different parity 24 . Such partial hybridization produces electrically polar excited states, considered to result from collective phenomena induced by the inter dimer Coulomb correlation, V (Fig.…”