Jones Matrix Formalism Research Articles

Polarization eigenstates for twisted-nematic liquid-crystal displays

We derive theoretical expressions for the eigenvalues and the eigenvectors for a twisted-nematic liquid-crystal display (LCD) as a function of the twist angle and the birefringence by use of the Jones-matrix formalism. These polarization eigenvectors are of particular interest for phase-only transmission because they propagate unchanged through the display. We find that the eigenvectors are elliptically polarized and that the ellipticity changes as a function of the birefringence of the LCD (which is proportional to the external voltage applied to the display). We can define an average eigenvector over a desired range for the applied voltage. We show, using Jones matrices, how this average eigenvector can be generated using a quarter-wave plate and a linear polarizer having appropriate orientation angles. Using this average eigenvector, we show that superior phase-only operation can be obtained over a given operating range for the LCD compared with other approaches.

Polarization states of a single-mode (microchip) Nd/sup 3+/:YAG laser. I. Theory

We present a microscopic mathematical model for the polarization states of a single-frequency Nd/sup 3+/:YAG laser. It is a plane wave, mean field, vector model carried to all orders in the laser field. The crystal is assumed to be optically pumped longitudinally with a laser of specified polarization. For D/sub 2/ site symmetry and an odd number of electrons, we establish the phase relationships between the components of the electric dipole matrix elements between the Kramers states. These relationships are central in determining the site-specific coupling between both, the pump and laser fields to the Nd/sup 3+/ ions. The laser cavity is assumed to be linear and quasi-isotropic. The residual optical anisotropies are included using a round-trip Jones matrix formalism.

Stokes parameters as a Minkowskian four-vector

It is noted that the Jones-matrix formalism for polarization optics is a six-parameter two-by-two representation of the Lorentz group. It is shown that the four independent Stokes parameters form a Minkowskian four-vector, just like the energy-momentum four-vector in special relativity. The optical filters are represented by four-by-four Lorentz-transformation matrices. This four-by-four formalism can deal with partial coherence described by the Stokes parameters. A four-by-four matrix formulation is given for decoherence effects on the Stokes parameters, and a possible experiment is proposed. It is shown also that this Lorentz-group formalism leads to optical filters with a symmetry property corresponding to that of two-dimensional Euclidean transformations.

Jones-matrix formalism as a representation of the Lorentz group

It is shown that the two-by-two Jones-matrix formalism for polarization optics is a six-parameter two-by-two representation of the Lorentz group. The attenuation and phase-shift filters are represented respectively by the three-parameter rotation subgroup and the three-parameter Lorentz group for two spatial and one time dimensions. It is noted that the Lorentz group has another three-parameter subgroup which is like the two-dimensional Euclidean group. Possible optical filters having this Euclidean symmetry are discussed in detail. It is shown also that the Jones-matrix formalism can be extended to some of the non-orthogonal polarization coordinate systems within the framework of the Lorentz-group representation.

Generalized Abelès relations for an anisotropic thin film of an arbitrary dielectric tensor

The extended Jones matrix method is applied to one dielectrically anisotropic, homogeneous thin film at oblique incidence. Standard boundary conditions are imposed on resultant electric- and magnetic-field vectors at interfaces. Thus simple matricial relations are obtained for transmitted and reflected electric-field amplitudes at the two interfaces. In the limits of isotropy, they reduce to four well-known Abelès relations, and thus they may be considered as generalized Abelès relations for dielectrically anisotropic thin films. These matricial relations include multiple reflections while dealing with total fields. Thus they provide new insights into the 2 x 2 extended Jones matrix formalism.

Investigation of Mueller matrices of anisotropic nonhomogeneous layers in application to an optical model of the cornea

We consider a system of anisotropic layers as an optical model of the eye cornea. Effective refraction indices for normally incident light are calculated with the assumption that each layer consists of closely packed uniform cylinders (fibrils). Jones matrix formalism is used to describe light propagation through the cornea. We calculate the Jones matrices from the experimentally measured Mueller matrices. Two algorithms are used for this purpose. The experiments have shown that ~20% of cornea area studied had the structure well described by the helical model proposed.

Numerical study of electromagnetic wave propagation in twisted birefringent layers by the spectral moments method

This paper presents a first attempt at using the spectral moments method (SMM) to solve Maxwell's equations in twisted anisotropic media in the presence of defects. This numerical method, previously developed in condensed matter physics, allows computation of Green functions for very large systems. The dynamic matrix of the discretized system is built from the medium parameters. Green functions, calculated for a given source, representing a point source at infinity and given receiver, are developed as a continued fraction whose coefficients are related to the moments and directly computed from the dynamic matrix. In this study we compute the light transmitted through thin surface-stabilized ferroelectric liquid crystal cells with a chevron structure and a twisted director distribution. The efficiency and accuracy of the method are analysed by comparing the results obtained by SMM with the analytical solution obtained using the Jones matrix formalism. Finally, we apply SMM to compute the transmitted light with different director configurations. We show, by comparisons with experimental data, that the simplest director configuration is certainly the most probable.

Multiaxis laser eigenstates

We derive a theoretical model based on a generalized Jones matrix formalism to calculate the eigenstates of lasers with N propagation axes. Spatial separation of the beams is realized inside the cavity by a collection of double-refraction crystals. This method allows one to adapt the mode volume to the geometry of the active medium. N-forked eigenstates are isolated, and the possibility of scaling up the TEM00 output power of a monomode laser is shown. All these features are tested experimentally in two-, four-, and eight-axis lasers and show good agreement with the theoretical predictions.

Measurement of the Carbon Isotopic Composition of Methane Using Helicoidal Laser Eigenstates

The spatially generalized Jones matrix formalism is used to design a laser cavity to make intracavity measurements of the carbon isotopic composition of methane. The method is based on a double optical lever effect for helicoidally polarized eigenstates, permitting to measure successively the ~~CH4 and ~~CH4 concentrations. To choose the probed isotope, one simply tunes the frequency of the laser by Zeeman effect. The experiment exhibits a good agreement with the predictions and permits to measure the ~~CH4/~~CH4 composition ratio of methane with an uncertainty of the order of +0.07% for a sample containing only 6 X 10~~ mole of methane.

Numerical modeling of the cornea's lamellar structure and birefringence properties.

A model of the cornea's lamellar structure is proposed that is capable of explaining experimental results obtained for the transmission of normal-incidence polarized light through rabbit and bovine cornea. The model consists of a large number of planar lamellae, each approximated as a uniaxial birefringent layer, stacked one upon another with various angular orientations. Polarized light transmission through the composite system is modeled theoretically by use of the Jones matrix formalism. The light transmission is calculated numerically for a large number of model lamellae arrangements, each generated from a statistical description, and histograms are constructed of various properties of the light transmission, including the minimum and maximum cross-polarized output intensities. It is demonstrated that various structural and optical parameters of the lamellae arrangements of actual corneas may be estimated by comparison of the calculations with detailed experimental data. Certain characteristics of the histograms are identified that permit a clear distinction between random and partially ordered systems. Comparisons with previously published experimental data provide strong evidence that the lamellae orientations are not entirely random, but rather a significant fraction are oriented in a fixed, preferred direction.

Spatially resolved eigenstates for traveling and standing waves in ring lasers

The spatially generalized Jones matrix formalism is applied to ring lasers with several propagation axes. Local spatial separation is theoretically and experimentally shown to be a convenient means for controlling the coupling, frequencies, and oscillation regime of the eigenstates of a ring laser. It is applied successively to four locally circularly polarized eigenstates, four linearly polarized eigenstates, and a combination of two traveling-wave and one standing-wave linearly polarized eigenstates. In all cases, novel types of optical diodes based on either the Zeeman or the Faraday effect permit the selection of the desired eigenstates. In particular, three kinds of ring laser gyroscope designs that incorporate these optical diodes are shown to circumvent the problem of frequency locking. The first two sustain two counterpropagating frequency-biased traveling waves with orthogonal polarizations. The third one exhibits the Sagnac effect on a single traveling wave, the standing-wave eigenstate being used as a local oscillator. In all cases, fluctuations in the bias can be eliminated, and good agreement is observed between experiments and the predictions obtained from a spatially generalized Jones matrix formalism.

Double-helicoidal eigenstates in lasers

Longitudinal segmented helicoidal eigenstates in laser resonators, with different field distributions of either identical or opposite handedness, are studied theoretically and experimentally. The eigenmodes and frequencies are derived in the framework of the Jones-matrix formalism. Double helices can be applied to birefringence compensation for helicoidal waves. A novel Fabry–Perot laser cavity is realized in which both spatial hole-burning suppression and birefringence compensation are simultaneously achieved.

Polarization analysis of LP/sub 01/ and LP/sub 11/ intermodal interference in highly birefringent bow-tie optical fibers

A polarization analysis of bimodal highly birefringent fibers in which LP/sub 01/, LP/sub 11/, interference takes place is performed. The polarization state of the interference pattern is theoretically described in terms of recently proposed spatial Stokes parameters on the basis of coupled-mode equations and Jones matrix formalism. The general quasimonochromatic case is considered and depolarization as well as interference fading are accounted for. Special attention is paid to application of the results to fiber-optic sensors, and the basic schemes for the detection of the interference beat signal are discussed. The theoretical conclusions are supported by experiments in which the modal polarization and intermodal interferometric response to elongation of a bow-tie bimodal fiber is investigated. Measurements are carried out on both a separate fiber and on a multifiber assembly at various input conditions. >

Theoretical and experimental study of eigenmodes and eigenstates in ring lasers. Applications to gyrometry and to the detection of small effects

A spatial and vectorial description of the eigenmodes and eigenstates of ring lasers is provided. After having studied the tangential and sagittal mode size dynamics due to resonant lenslike effects in one- and two isotope- ring lasers, the non-reciprocity of these mechanisms is shown to explain the biases observed in ring laser gyroscopes. The role of diffraction in the lock-in region is also isolated, allowing us to observe the so-called reverse Sagnac effect. The spatial separation of laser eigenstates is investigated and permits then to build cavities with several propagation axes whose eigenstates must be calculated in the framework of a spatially generalized Jones matrix formalism. This allows, in the case of Fabry-Perot cavity lasers, to control the coupling between eigenstates, to create forked eigenstates, and to build a two-tunable-frequency laser, and, in the case of ring lasers, to cancel the coupling between two counterpropagating eigenstates. The study of the stability and dynamics of eigenstates allows, first, to control the stability of the two eigenstates of various lasers, second, to measure small effects like the Goos-Hanchen shift or to build a mean-field magnetometer in the case where no eigenstate exists, and, third, to investigate the behavior of the four circularly polarized eigenstates of a ring laser in order to create an oscillation regime with two biased counterpropagating eigenstates, whose application as a ring laser gyroscope is discussed.

Laser eigenstates in the framework of a spatially generalized Jones matrix formalism: errata

Laser eigenstates in the framework of a spatially generalized Jones matrix formalism: errata

Resultant mode pattern and polarization in a LP01, LP02 two-mode linearly birefringent optical fibre

Modal interference between the LP01 and LP02 polarization modes propagating along a linearly birefringent bimodal optical fibre is considered. On the basis of Jones matrix formalism, radial Stokes parameters containing information about the resultant polarization and mode pattern are derived in the monochromatic and the quasimonochromatic cases. Particular cases with different polarization inputs are considered. The application of the obtained results to fibre-optic sensors and wavelength filter structures is discussed.

Optical studies of thin layers of smectic-C materials

The optical behaviour of surface-stabilized ferroelectric liquid crystal samples cannot be described using a uniform director profile. The simplest model based on the chevron layer structure and the surface alignment is a triangular director profile. Analytical expressions for the resulting transmitted light were calculated using the Jones matrix formalism and fitted to experimental data results of a smectic-C host made using an optical multichannel analyser (OMA). This simple model gives an excellent fit, with results that are consistent with X-ray measurements of the chevron structure.

Laser eigenstates in the framework of a spatially generalized Jones matrix formalism

A theoretical model is developed to predict the longitudinal and transversal distributions of polarization and intensity and the eigenfrequencies of the laser eigenstates in the case where the transversal degeneracy of the eigenstates is broken up by a birefringent element inducing a polarization walk-off. This formalism, which generalizes the Jones matrix formalism, is necessary for interpreting the existence of ordinary and extraordinary eigenstates that have different paths in the cavity. Moreover, it leads to the prediction of so-called forked eigenstates, in which a single laser eigenstate itself is split into two spatially separated orthogonally polarized branches in one part of the cavity containing an extra normal birefringent plate. This formalism also allowed us to design and realize a two-frequency laser in which two eigenstates can oscillate simultaneously at easily adjustable frequencies. Experimental evidence of these phenomena was gathered and showed excellent agreement with the theoretical predictions.

Picosecond circular dichroism spectroscopy: a Jones matrix analysis

Measurement of time-resolved circular dichroism signals can be complicated by transient linear dichroism and birefringence effects. These signals arise from using a polarized light beam for excitation and from imperfect modulation of the circularly polarized probe beam. In a polarization-modulation experiment designed to measure the time dependence of circular dichroism, a small amount of pump-induced linear dichroism coupled with birefringence of optics would dominate the observed signals, rendering an accurate measurement virtually impossible. We examine these effects within the Jones matrix formalism. These calculations demonstrate how one can experimentally eliminate transient linear polarization signals, permitting accurate measurement of the transient circular dichoism of a sample with picosecond resolution. Theory and experiment are compared for time-dependent data on myoglobin following the photoelimination of CO from carbonmonoxy myoglobin.

Jones matrix treatment of electromagnetic wave propagation in anisotropic stratified media

Light propagation in anisotropic stratified media may be conveniently described, in the quasi-adiabatic limit, by the so-called Jones matrix formalism. In systems like liquid crystals, the quasi adiabatic limit occurs when the director rotates very slightly over distances of the order of the light wavelength. This limit is of relevant interest in many optical devices. As known, the general solution of the Maxwell's equations in anisotropic stratified media may be obtained by using the 4 × 4 Berreman's matrix method. However, as long as the polarization properties of the light transmitted through the medium are concerned, a much simpler approach is possible, which consists in completely neglecting the solutions of the Berreman's equations corresponding to the waves propagating in the backward direction. In this case, the number of differential equations describing light propagation is reduced from 4 to 2. The 2 × 2 matrix introduced in this approximated fromalism is termed the Jones matrix of the problem. In this paper, we show that in general case of oblique incidence of light is possible to properly transform the original Berreman's matrix into a form suitable for reduction to a 2 × 2 Jones matrix.