A general description of linear optical birefringence effects is presented in terms of the effect of forward scattering on the photon polarization density matrix. The approach enables explicit statements to be given for the polarization changes involved in any linear birefringence phenomenon in terms of R matrix elements between plane-polarized base states. Expressions in terms of molecular quantities are listed for the R matrix elements involved in some typical natural and induced birefringence effects. This description of birefringence is related to the infinitestimal unitary transformations of the density matrix and rotations in Poincaré space.
A general description of linear optical birefringence effects is presented in terms of the effect of forward scattering on the photon polarization density matrix. The approach enables explicit statements to be given for the polarization changes involved in any linear birefringence phenomenon in terms of R matrix elements between plane-polarized base states. Expressions in terms of molecular quantities are listed for the R matrix elements involved in some typical natural and induced birefringence effects. This description of birefringence is related to the infinitestimal unitary transformations of the density matrix and rotations in Poincaré space.
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.
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.
The theoretical dependence of optical birefringence phenomena on the statistical nature of the light is investigated. Photon S-matrix calculations lead to expressions relating the change induced in the statistical nature of the beam to the molecular properties of the target sample. A hypothetical generalized optical birefringence experiment is analysed: in the experiment the coincidences of photons arriving at spatially separated detectors sensitive to different polarizations are observed with and without the sample in the beam, and the difference in the photon correlations depend on the initial statistical composition and polarization of the beam and the molecular properties of the sample. When the beam is generated by a thermal source the difference of the signals (with and without the sample) depends on the product of the polarizability and optical rotation tensors. When the light is generated by a single-mode coherent source the statistical properties of the light are such as to eliminate the second-order interaction terms, and so the experiment yields the hyperpolarizability and hyper-rotation tensors automatically and with the explicit exclusion of the polarization and rotation tensor contributions.
Polarization effects in stimulated Raman and stimulated Rayleigh scattering and self focusing are investigated from the viewpoint of the scattering of polarized photons by molecules. Both stimulated vibrational Raman scattering in liquids and stimulated rotational Raman scattering in gases are considered, and it is demonstrated that the polarization changes and relative power thresholds for different laser polarizations should be somewhat different to those predicted in other theories. For example, the ellipticity eta 0 of the stimulated rotational Raman photons in the forward direction from Delta J=+or-2 transitions in gaseous symmetric top molecules is given in terms of the initial ellipticity eta 1 by sin 2 eta 0 = - 5 / 7 sin 2 eta 1 , whereas conventional theories predict that eta 0 and eta 1 are equal in magnitude and opposite in sign. Furthermore it is shown that deviations from this relation could provide a measure of the degree of optical orientation of the molecules.
In this chapter we build up the basic concepts of quantum mechanics using two simple examples, following a heuristic approach which is more inductive than deductive. We start with a familiar phenomenon, that of the polarization of light, which will allow us to introduce the necessary mathematical formalism. We show that the description of polarization leads naturally to the need for a two-dimensional complex vector space, and we establish the correspondence between a polarization state and a vector in this space, referred to as the space of polarization states. We then move on to the quantum description of photon polarization and illustrate the construction of probability amplitudes as scalar products in this space. The second example will be that of spin 1/2, where the space of states is again two-dimensional. We construct the most general states of spin 1/2 using rotational invariance. Finally, we introduce dynamics, which allows us to follow the time evolution of a state vector. The analogy with the polarization of light will serve as a guide to constructing the quantum theory of photon polarization, but no such classical analog is available for constructing the quantum theory for spin 1/2. In this case the quantum theory will be constructed without reference to any classical theory, using an assumption about the dimension of the space of states and symmetry principles. The polarization of light and photon polarization The polarization of an electromagnetic wave The polarization of light or, more generally, of an electromagnetic wave, is a familiar phenomenon related to the vector nature of the electromagnetic field. Let us consider a plane wave of monochromatic light of frequency ω propagating in the positive z direction.
A theoretical study of the polarization dependence of stimulated Rayleigh-wing scattering and the optical-frequency Kerr effect has been undertaken. When the incident light is circularly polarized, a striking difference in gain is found for the co- and counter-rotating senses of polarization of the scattered light---the former being suppressed and the latter being exceptionally favored. The analysis begins with a model for the Kerr effect, which involves the alignment of anisotropic molecules in an electric field, but the results can be immediately generalized to any light scattering process of tensor symmetry. The nonlinear problem of the propagation of an intense elliptically polarized light wave in a Kerr-active medium is shown to have a solution in which the vibrational ellipse undergoes self-precession and self-retardation. The stimulated scatterings or instabilities of such a self-precessing and self-retarding light wave are obtained for the backward and forward directions. Birefringence, optical activity, and linear and circular dichroisms are some of the phenomena which result, but the forward direction yields substantially different results from the backward direction because of Stokes-anti-Stokes coupling.