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Progress of Theoretical Physics

Electromagnetic Gyration: -- Hamiltonian Dynamics of the Stokes Parameters --

Hiroshi Kuratsuji, Robert S. Botet, Ryohei Seto

Published Feb 1, 2007• 15 citations
Publisher

Abstract

Starting with a Schrödinger-type equation derived from the Maxwell theory of electromagnetism, a general formalism is presented for the change of light polarization in nonlinear birefringent media, which we call “optical gyration” or more generally, “electromagnetic gyration” and applications are considered. The theory is formulated in terms of the Hamiltonian dynamics of the Stokes parameters which turns out to be the equation for a classical spin. As applications, we consider two models: (a) As the first application, we analyze solvable models admitting analytic solutions, which exhibit the structure change in the behavior of the Stokes parameter caused by the mutual interplay between linear and nonlinear birefringence. (b) The second application is the optical analogue of the so-called nuclear magnetic resonance (NMR) or Rabi resonance. This is investigated in the case of purely linear birefringence and coexisting of linear and nonlinear birefringence.

DOI
10.1143/ptp.117.195

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Dec 11, 1992•0 citations
Starting with a Schrödinger-type equation derived from the Maxwell theory of electromagnetism, a general formalism is presented for the change of light polarization in nonlinear birefringent media, which we call “optical gyration” or more generally, “electromagnetic gyration” and applications are considered. The theory is formulated in terms of the Hamiltonian dynamics of the Stokes parameters which turns out to be the equation for a classical spin. As applications, we consider two models: (a) As the first application, we analyze solvable models admitting analytic solutions, which exhibit the structure change in the behavior of the Stokes parameter caused by the mutual interplay between linear and nonlinear birefringence. (b) The second application is the optical analogue of the so-called nuclear magnetic resonance (NMR) or Rabi resonance. This is investigated in the case of purely linear birefringence and coexisting of linear and nonlinear birefringence.
The equations of propagation for the polarisation components of an electromagnetic wave in non-linear birefringent media are often termed “non-linear Schrödinger equations”. In this letter, we pursue the analogy between the quantum dynamics of a 2-states oscillator and the time evolution of the wave polarisation. More specifically, we define exactly the optical analogue of the conditions for nuclear magnetic resonance, leading in the optical medium to resonant oscillations of the polarisation state when birefringence is periodically modulated along the wave propagation direction in addition to a static longitudinal magnetic field. Circular polarisation then exhibits Rabi oscillations. The non-linear influence of light intensity on the shape of the resonant polarisation line is discussed precisely.
The theory of optical birefringence is discussed from the point of view of the scattering of polarized photons by molecules using the framework of formal scattering theory and quantum electrodynamics. In Part One, those features from noncovariant quantum electrodynamics relevant to the molecular scattering of photons are reviewed. Chapter One contains an outline of classical electrodynamics in the Coulomb gauge, including the Stokes parameter and density matrix formulations of polarization. Chapter Two deals with the quantization of the radiation field, and the occupation number representation is defined. The quantum mechanical description of polarized light is outlined in Chapter Three: operator equivalents of the density matrix and Stokes parameters are invoked, the parameters themselves being obtained from the expectation values of the Stokes operators with the state vector of the photon. The Stokes parameter and density matrix are then applied to the polarization of a general beam containing an arbitrary number of photons, and some new methods are developed. The interaction of the electromagnetic field with bound electrons is formulated in Chapter Four in terms of a canonically transformed interaction Hamiltonian in which the multipole terms are explicit. Since each term is then gauge-invariant, this Hamiltonian is particularly suited to the application of diagrammatic perturbation theory which follows from the iteration of the scattering matrix in the interaction representation (Chapter Five). Also the extension of the theory to multiphoton processes is facilitated "because the field strength is explicit in the transformed Hamiltonian. The S-matrix connects the initial and final asymptotic states of the scattering system, and in Chapter Six the state vector and density operator of the scattered photon beam are obtained from those of the incident beam. Part Two contains the application of the above methods to a variety of linear and nonlinear optical birefringence effects in the absence or presence of external fields. In Chapter Seven, general equations are obtained for the polarization changes imposed on a forward scattered photon in terms of matrix elements of the reaction operator between plane polarized photon base states; whereas Chapter Eight extends this treatment to an n-photon beam, thereby encompassing the hyperpolarizability contributions to the polarization changes of the forward scattered beam. It is then demonstrated that the fundamental process in optical birefringence can be interpreted as an infinitesimal unitary transformation of the polarization spinor. Also the S-matrix approach to birefringence is compared with the methods used in crystal optics, and with the classical treatments of the scattering of polarized light by molecules. The general equations obtained for the polarization changes of the forward scattered beam are subsequently interpreted in the light of the physical circumstances appropriate to the bire- fringence effect of interest. In Chapters Nine, Ten, and Eleven, diagrammatic perturbation theory is used to obtain explicit expressions, in terms of molecular polarizability and rotation tensors, for the linear and nonlinear contributions to the polarization changes involved in natural optical activity, and in birefringence phenomena induced by static and optical electric and magnetic fields. The nonforward scattering of polarized photons by molecules is discussed in Part Three. General equations are calculated in Chapter Twelve for the azimuth, ellipticity, and depolarization ratio of the Rayleigh beam at any scattering angle and for any polarization of the incident beam. Explicit expressions in terms of molecular quantities are then obtained for the polarization at scattering angles of 90° and almost zero from both optically inactive and optically active molecules. It is demonstrated that a determination of the precise polarization state of the scattered beam can provide useful molecular information additional to that obtained from depolarization ratios alone. This is because the depolarization ratio depends on only the diagonal elements of the polarization density matrix and so does not convey all the information about the system. Raman scattering processes are contained in the formulation and the polarization of the Raman light at any scattering angle can be obtained. This is shown to be particularly useful for predicting the polarization of stimulated Raman light for any Raman-active liquid placed within a laser cavity. By extending the nonforward scattering formulation to incorporate multiphoton processes, equations are obtained in Chapter Thirteen for the ellipticity, azimuth, and depolarization ratio of the second harmonic wave at any scattering angle as a function of the incident polarization. Several specific situations are subsequently investigated. Finally, it is shown in Chapter Fourteen that the near-forw- ard Rayleigh and Raman scattering formulations provide an explanation for the observed polarization, as a function of the initial polarization, of the self-trapped filaments that arise from the self-focusing of an intense laser beam.
The analogy between polarization optics effects and the dynamics of a spin in a magnetic field is analyzed based on the fact that the Bloch equation for a spin in a magnetic field is formally identical to the evolution equation for the quasi-spin vector of a light propagating in an optically anisotropic medium. Among the effects discussed within this framework are light polarization beats in a birefringent medium; optical activity suppression in the media with linear birefringence; suppression of linear birefringence in the media with spatially modulated gyration; 'adiabatic' transformation of light polarization in media of smoothly varying anisotropy, and two- and three-pulse polarization echoes.
Abstract A general approach to the discussion of optical birefringence phenomena is presented. It is based on the S-matrix approach to photon scattering in a nonrelativistic quantum field theoretical formalism and the Stokes operator description of polarization. The use of a canonically transformed interaction Hamiltonian in combination with diagrammatic perturbation theory enables the response of a system to electromagnetic radiation to be calculated in a straightforward and consistent manner to any order in the perturbation, and the effect of static fields is included easily. As an illustration, the angle of rotation of a beam of plane-polarized light by a solution of optically active molecules is calculated and the intensity dependence of the angle determined. The latter phenomenon should become apparent at high photon flux densities particularly when ω, the frequency of the incident light, or 2ω are similar to electronic excitation frequencies in the optically active molecules.
We present a model calculation of optical bistability and polarisation evolution in optically birefringent media based on the interaction of three waves.Second-order optical nonlinearity (described by the third rank susceptibility tensor) is assumed to be dominant.A bistable light transmission is analysed in terms of standard Fabry-Perot model of feedback, and is a direct consequence of the results obtained for a semi-infinite medium.A second effect considered in the paper is evolution of polarisation state of the waves involved in these processes.We derived a closed formula for intensity-dependent refractive index, which corresponds to that used in the case of third-order nonlinear media.As a result, we calculated both intensity • and Stokes parameters of transmitted light.
The usefulness of Brown's polarization formalism as a paradigm for the dynamics of analogous linear systems is demonstrated. By connecting the rotations of the Lorentz group with birefringence and the Lorentz group boosts with dichroism, this formalism has offered closed-form solutions, expressed in the language of the Lorentz group, for the 4 X 4 Mueller matrices of polarization optics. First, we discuss the dynamics of a generic two-state system satisfying a Schrodinger-like differential equation. This equation is cast into a four dimensional form. Next, we show the solution of the latter to be a Lorentz transformation. Then, we analyze several specific examples. The examples include: (1) precession, (2) a magnetic dipole in a magnetic field, (3) a charged particle in an electric field, and (4) time- independent perturbation theory with phenomenological damping. We argue the approach of Brown's formalism provides a convenient conceptual framework to describe the physics and to interpret the measurement results of such systems. In addition, this approach broadens the role of the Lorentz group to areas other than special relativity and electromagnetic scattering.