Proceedings of the Royal Society of London A Mathematical and Physical Sciences
A quantum electrodynamical theory of differential scattering based on a model with two chromophores II. Differential Raman scattering of circularly polarized light
Abstract The theory of differential scattering described in part I (preceding paper) is applied to the Raman process. Here, a distinction between inequivalent and equivalent chromophores is required. For systems with inequivalent chromophores, the leading contribution to the differential intensity of scattering involves the interference of second- and fourth-order probability amplitudes; in near- and far-zone limits, it depends on the inverse square of the group separation. For systems with equivalent chromophores, the spectrum should, in general, feature a doublet. The dominant contribution to the differential intensity comes from the second-order-second-order interference term and has a different sign for each doublet component. In many cases these contributions cancel and the leading term becomes the second-order-fourth-order interference term as in the case of inequivalent chromophores.
Abstract The theory of differential scattering described in part I (preceding paper) is applied to the Raman process. Here, a distinction between inequivalent and equivalent chromophores is required. For systems with inequivalent chromophores, the leading contribution to the differential intensity of scattering involves the interference of second- and fourth-order probability amplitudes; in near- and far-zone limits, it depends on the inverse square of the group separation. For systems with equivalent chromophores, the spectrum should, in general, feature a doublet. The dominant contribution to the differential intensity comes from the second-order-second-order interference term and has a different sign for each doublet component. In many cases these contributions cancel and the leading term becomes the second-order-fourth-order interference term as in the case of inequivalent chromophores.
In this paper a quantum electrodynamical theory of cooperative Raman scattering in fluids is developed. The process is one in which pairs of molecules undergo concerted Raman transitions via an intermolecular energy transfer mechanism. The formalism is also extended to chromophore pairs in which intramolecular energy transfer is involved. Using the Power-Zienau-Woolley multipolar Hamiltonian, intensity equations of the Kramers-Heisenberg type are derived which are valid over all regions of molecular separation. In the near zone it is illustrated how the intensity of cooperative scattering is simply that obtained by consideration of the instantaneous response of one molecule to the scalar field generated by another. Within this regime it is shown that the rate of scattering has the familiar ${R}^{\mathrm{\ensuremath{-}}6}$ dependence on intermolecular separation R associated with interacting induced dipoles. In the far zone it is demonstrated that the fully retarded and causal interaction between molecules is mediated through virtual photon coupling, and that this leads to a result for the intensity that dies off with ${R}^{\mathrm{\ensuremath{-}}2}$. This long-range-interaction contribution to the overall scattering intensity is generally ignored in collisional treatments of intermolecular interactions, yet is likely to be of more significance than higher-order multipole terms in near-zone calculations. Finally it is shown how bands in a Raman spectrum due to cooperative scattering effects may be identified by their unique pressure characteristics, or by observation of light scattering that occurs outside the interaction volume of the laser beam. In the case where the scattering occurs between molecules or chromophores held in a fixed orientation with respect to one another, it is demonstrated how detection may be facilitated through the manifestation of a differential scattering effect.
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 review deals with various variants of classification of the types of secondary emission of light by molecules; cases are distinguished in which the secondary emission occurs without additional processes that find their way between photon emission and absorption; the advisability of differentiating between the cases of resonance fluorescence and resonance scattering is noted. Reference is made to papers that have made contributions to the development of quantum-mechanical and classical theory of Raman scattering and to the development of the concepts of the relations between electron-vibrational absorption spectra, fluorescence spectra, and scattering excitation near and near and far resonance. Conditions are considered of excitation of the spectrum at different relations between the electronic-oscillator damping constant, the atom vibration frequency, the rotational frequencies, and the difference between the frequency of the incident light and the frequency of the electronic transition. The conditions for the excitation of typical light scattering, typical resonance fluorescence, and certain intermediate types, in which individual features of scattering are combined with certain features of fluorescence (phenomena of the transition type) are noted. The results of investigations of different models of the scattering centers, their advantages, shortcomings, and limits of applicability are compared. The most interesting results of the experimental investigations of resonant scattering of light by molecules and crystals are noted.
A theoretical description of the resonant coherent hyper-Raman scattering for an arbitrary system–bath interaction is presented. Particularly the relationship between the coherent hyper-Raman scattering (CHRS) and coherent Raman scattering (CRS) signals is established without alluding to any specific model of optical broadening. The permanent dipole moment difference between the excited and ground states of a given chromophore is shown to be proportional to the ratio of the CHRS to CRS signals, and it is proved that this result is valid regardless of the model for the chromophore–bath interaction.
We present a theory of Raman Scattering in the Hubbard model. The scattering of light has resonant and non-resonant contributions. The resonant term gives rise to scattering by spin degrees of freedom in the insulating case, for which we derive a general form of the effective scattering Hamiltonian. The later involves four terms with distinct symmetry properties, one of them being the “chiral” spin operator ∑ Si · (Sj×Sk). The corresponding four distinct correlation function can be measured directly in the experiments with different scattering geometries. The nonresonant term contributes to the scattering in the doped case and is shown to probe the fluctuations of the “stress tensor”. This quantity is not conserved and hence its fluctuations at small q inherent in optical experiments, need not be small, in striking contrast with the density fluctuation in usual metals.
Abstract An achiral system bathed in an intense beam of circularly polarized light shows chiral behaviour which may be detected with an auxiliary beam of polarized light. In this paper, expressions are derived for differential absorption rates, optical rotation of a plane polarized probe beam and circular differential scattering intensities. The theory of these two, three and four photon processes is developed by using quantum electrodynamics in electric dipole approximation. Two types of circular dichroism are distinguished: (a) the transition exhibiting induced dichroism is one-photon allowed, with chirality induced by scattering of the laser beam, or (b) the transition is two-photon allowed with absorption of a photon from each beam. In both, the induced circular dichroism is linearly dependent on the intensity of the inducing beam. Circular dichroism of type (a) is also discussed from the dressed-molecule viewpoint, and it is shown how a canonical transformation may be used to recover the results more simply. The possibility of observing Doppler-free circular dichroism is indicated. The induced optical rotation is analysed as a two-state process with transitions involving change of polarization only. The angle of rotation is linearly dependent on the intensity of the intense beam; its sign depends on the helicity. It is shown that, for non-forward Rayleigh scattering of the probe beam in the presence of an intense beam of light, the scattering intensity depends on the relative helicities of the two beams, and the differential intensity is proportional to the incident intensities.