The mathematical representation of each of the four interference laws of Fresnel and Arago is presented. Each of the laws is characterized by a unique Stokes vector. These vectors are arrived at by a matrix analysis of the, interference experiments of Fresnel and Arago in which only polarizers are used. The historical motivation that led Stokes to introduce the parameters that now bear his name is noted. The similarity in the approach taken by Stokes to represent the fundamental properties of the classical optical field in terms of observables is compared with Heisenberg's approach to the formulation of quantum mechanics. The mathematical statements that characterize unpolarized light are developed. These statements are then used to determine the Stokes vectors that describe each of the four interference laws of Fresnel and Arago. In a final section of the paper the mathematical apparatus developed here is used to describe Young's interference experiment with unpolarized light.
The mathematical representation of each of the four interference laws of Fresnel and Arago is presented. Each of the laws is characterized by a unique Stokes vector. These vectors are arrived at by a matrix analysis of the, interference experiments of Fresnel and Arago in which only polarizers are used. The historical motivation that led Stokes to introduce the parameters that now bear his name is noted. The similarity in the approach taken by Stokes to represent the fundamental properties of the classical optical field in terms of observables is compared with Heisenberg's approach to the formulation of quantum mechanics. The mathematical statements that characterize unpolarized light are developed. These statements are then used to determine the Stokes vectors that describe each of the four interference laws of Fresnel and Arago. In a final section of the paper the mathematical apparatus developed here is used to describe Young's interference experiment with unpolarized light.
In this work, we formulate a quantum-mechanical description of interference of electromagnetic fields in Young's interference experiment, thereby taking into account the polarization properties of the field and describing them in terms of quantum analogs of classical Stokes parameters.Commencing with the classical theory of interference of scalar fields, we proceed to a relatively advanced approach to electromagnetic interference, bringing into the equation cross-spectral density tensor, polarization matrix and Stokes parameters to analyze the polarization properties.Subsequently, the same phenomenon is analyzed in the domain of quantum optics, thereby expressing the fields as operators and observables as the expectation values.Firstly, an outline of the scalar approach of the operators in the interference experiment is presented to establish the foundation to base the electromagnetic approach on, followed by a full description of quantum analog of electromagnetic interference in Young's experiment.In particular, Stokes parameters are adopted to calculate the polarization effects in quantum theory. PrefaceThis report attempts to condense and present the theories related to the interference in the canonical Young's double pinhole experiment, prior to heading to investigate the phenomenon with the quantum analysis of electromagnetic fields.My interest in the realm of quantum mechanics has encouraged me to work on this thesis, wherefrom I have learnt a lot during these times.I am really grateful to my supervisors, Prof. Ari T. Friberg and Prof. Jani Tervo for guiding me and believing in me.I will always be indebted for their constant support and confidence on me.I would also like to offer my special thanks to the Faculty of Forestry and Sciences for providing the financial
We present a trajectory-based interpretation of Young's experiment, the Arago-Fresnel laws and the Poisson-Arago spot.This approach is based on the equation of the trajectory associated with the quantum probability current density in the case of massive particles, and the Poynting vector for the electromagnetic field in the case of photons.Both the form and properties of the evaluated photon trajectories are in good agreement with the averaged trajectories of single photons observed recently in Young's experiment by Steinberg's group at the University of Toronto.In the case of the Arago-Fresnel laws for polarized light, the trajectory interpretation presented here differs from interpretations based on the concept of 'which-way' (or 'which-slit') information and quantum erasure.More specifically, the observer's information about the slit that the photons went through is not relevant to the existence of interference; what is relevant is the form of the electromagnetic energy density and its evolution, which will model consequently the distribution of trajectories and their topology.Finally, we also show that the distributions of end points of a large number of evaluated photon trajectories are in agreement with the distributions measured at the screen behind a circular disc, clearly giving rise to the Poisson-Arago spot.
The laws of Arago and Fresnel relate the polarization of light to its capacity for interference. We investigate their quantum counterpart at the level of the fundamental helicity states of a single photon. To this end, we introduce displaced photon helicity states (DPHS), constructed by coherently displacing single-photon helicity eigenstates. The displacement bridges the microscopic spin degree of freedom and macroscopic polarization observables, enabling interference effects that would otherwise be inaccessible to isolated photons. We derive analytical expressions for the Stokes parameters associated with the interference between DPHS and coherent optical fields and show that opposite helicities generate distinct parameter dependences in the resulting fringe structure. The interference constitutes a quantum realization of Arago-Fresnel interference and permits the helicity of an unknown photon to be inferred from the Stokes response of the combined field. Beyond this application, DPHS generalize the framework of displaced nonclassical states from photon-number excitations to the helicity degree of freedom of light.
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.
Any polarization state of light occupies intrinsic quantum uncertainties due to the noncommutative behavior of the Stokes operators, which are manifested as unavoidable quantum fluctuations in the corresponding classical Stokes parameters.In this thesis, we explore such quantum polarization uncertainties exhibited by vectorial light fields in Young's seminal double-slit interference experiment.As a main result, by adopting a proper Stokes operator representation of double-slit interference, we establish general quantum polarization uncertainty relations for the Stokes parameters in the observation plane.The relations show how the lower bounds of the polarization uncertainties are periodically varying across the observation screen and how they are affected by the polarization and coherence properties of the light at the two slits.We also present various noncommutative, interlinked connections between coherence and polarization in the slit plane which have not been considered earlier in the literature.Our work sets the fundamental precision limits imposed by quantum physics for characterizing any polarization state in double-slit interference, which could find use in photonic applications exploiting polarimetric and interferometric techniques for high-resolution quantum-enhanced sensing.
We show, both analytically and experimentally, that under common experimental conditions the interference pattern produced in a classic Young's double-slit experiment is indistinguishable from that generated by means of a doubly refracting uniaxial crystal whose optic axis makes a skew angle with the light propagation direction. The equivalence between diffraction and crystal optics interference experiments, taken for granted by Arago and Fresnel in their pioneering research on the interference of polarized light beams, is thus rigorously proven.