Alain L. Fymat

High-resolution interferometric spectrophotopolarimetry

Spectrophotopolarimetric capability can be added to a laboratory interferometer-spectrometer by use of a specially designed module described herein. With the instrument so augmented, high-resolution spectra can be obtained of the Stokes parameters of the reference beam and the beams diffusely reflected or transmitted by a sample medium of interest. For any such beam, the exponential Fourier transforms of the two interferograms obtained with a polarizer-analyzer oriented along the 0 ° and the 90 ° directions provide the spectra of I and Q, separately. Within experimental (and numerical) noise, this I spectrum should be the same as the one obtained with the polarizer removed. The remaining Stokes parameters U and V are obtained with a third interferogram recorded with the polarizer along the 45° direction. The complete theory of this instrument is described including the detailed analysis of the polarization-interferograms it provides.

Jones's matrix representation of optical instruments. I - Beam splitters

A general method is provided for constructing Joness reflection and transmission matrices of any beam splitter. Derivations are presented for the various known configurations. The method uses Abelèss matrices and pays special consideration to the different expressions of Joness matrices relative to the various beams in an interferometric arrangement. The reversibility of the beam splitter in its action on the amplitude or phase, or both, of an incident light is studied. It is finally suggested that, even for an asymmetric beam splitter configuration, the symmetry of the interferogram can still be preserved by adjusting the thickness of the beam splitter in a prescribed manner.

Mie forward scattering: improved semiempirical approximation with application to particle size distribution inversion

The approximations of Penndorf and Shifrin-Punina to the Mie solution at forward scattering angles are extended to smaller size parameter values. The present approximation, Eq. (7), is found to represent accurately the Mie result down to x ~ 0.5-1.0 for refractive index m = 1.33, and to x ~ 2.0 for much larger index values. The implications of this result are discussed relative to the reconstruction of particle size distributions utilizing the Shifrin-Fymat analytical inversion formula of forward scattered intensities.

Remote monitoring of environmental particulate pollution: a problem in inversion of first-kind integral equations

The possibility of remotely sensing the optical properties of scattering particulates from the variations in either the angular or the spectral characteristics, or both, of the radiation they transmit or scatter, is a problem of fundamental importance in the monitoring of environmental particulate pollution. It is shown that the corresponding problem is (or can be brought to) one of inverting first kind Fredholm integral equations. The solution to this problem would also enable one to follow the dynamical evolution of the polluted environment if it can be obtained in a time scale that is comparable to, or shorter than, the time constant of the physical measurements. The direct problem of how given physical parameters of such particles affect the transmission and scattering of incident radiation is first analyzed on the basis of the corresponding radiative transfer problem, including single and multiple scattering, and polarization induced on scattering. The various available methods for reconstructing the size distribution from the observed directly transmitted or scattered light are reviewed, particularly with regard to their main advantages and shortcomings. For direct light transmission, these include: library, iterative, and least-squares methods; the trial-and-error method; the matrix inversion method with smoothing constraint (MIM); the resolution accuracy trade-off method; and the analytical method. A minimization search method with smoothing constraint, an essential modification to the MIM, is also proposed. The corresponding methods for singly and multiply scattered light are likewise reviewed. The proposed forward-scattering method is shown to provide excellent reconstructions from either angular or spectral light measurements under proper experimental conditions. It can also be coupled with a minimization search in order to provide simultaneously the complex refractive index of the particles. The potentialities of other methods for the complete multiple scattering problem-the minimization search method, quasilinearization method, and small-angle Gaussian approximation method-are also studied.

Analytical inversions in remote sensing of particle size distributions. 4:Comparison of Fymat and Box-McKellar solutions in the anomalous diffraction approximation

It is shown that the inverse analytical solutions, provided separately by Fymat and Box-McKellar, for reconstructing particle size distributions from remote spectral transmission measurements under the anomalous diffraction approximation can be derived using a cosine and a sine transform, respectively. Sufficient conditions of validity of the two formulas are established. Their comparison shows that the former solution is preferable to the latter in that it requires less a priori information (knowledge of the particle number density is not needed) and has wider applicability. For gamma-type distributions, and either a real or a complex refractive index, explicit expressions are provided for retrieving the distribution parameters; such expressions are, interestingly, proportional to the geometric area of the polydispersion.

Analytical inversions in remote sensing of particle size distributions. 3: Angular and spectral scattering in the Rayleigh-Gans-Born approximation for particles of various geometrical shapes

Analytical inverse formulas are provided for reconstructing the size distribution of particulates whose scattering patterns can be adequately described by the Rayleigh-Gans-Born (or Shifrin) approximation. The formulas hold for arbitrary polarization states at incidence and scattering of light and for angular or spectral measurements. The particle shapes considered are spheres, spherical shells, thin disks (which may be randomly oriented), and thin rods. Circular cylinders and ellipsoids can also be encompassed by our formulas if these particles are described in terms of equivalent spheres having the same volume.

Inverse atmospheric radiative transfer problems - A nonlinear minimization search method of solution

Abstract The mathematical inversion of the equations of radiative transfer is an important tool for the remote sensing and probing of the atmosphere. These equations take the form of first-kind Fredholm integral equations (extinction and primary scattering of solar radiation, thermal emission from the atmosphere-surface system), ratios of such equations, and nonlinear integrodifferential equations (multiple scattering). A nonlinear minimization search method for solving these various equations is presented and discussed relatively to uniqueness, stability and accuracy of the solution it yields. The method is a direct search in parameter space aimed at minimizing the objective function of the problem considered without resorting to a gradient approach. The proposed algorithm is basically a nonlinear least-squares minimization which relies on a random number generator to find an improved iterate at each step. It makes constant use of the measurements in order to arrive at the solution. It does not depend on the initial guess for well-behaved problems, and does not require any a priori information on the solution. It is quite general and can be applied to any inverse problem which can be reduced to the determination of a certain number, however large, of unknown parameters. The computation times involved tend to increase in proportion to the first power of the number of variables in opposition to most classical optimization methods where the proportion is to the cube of the number of variables. Illustrations are provided in the field of environmental particulate pollution where aerosol physical parameters are sought from measurements of solar extinction ratios.

Polarization Effects in Fourier Spectroscopy. I: Coherency Matrix Representation

A general analytical method using the formalisms of polarization coherency and Joness matrices is provided for the evaluation of all polarization effects in fourier spectroscopy. The method applies to any incident state of arbitrary (complete, random, or partial) polarization. Inversely, it may also be used for determining the intensity and state of polarization of the source of light. TE- and TM-mode reflectivity and transmissivity for beam splitters and the dependence of these quantities on the incident polarization are obtained. It is demonstrated that three different efficiencies for these optical components must be introduced. Interferometer efficiency expressions for the source beam and the detector beam are also derived and shown to be essentially different from the previous efficiencies. Polarization effects of beam splitters, reflectors, and their composite combinations (interferometers) are investigated in detail. General conditions for complete or restricted polarization compensation are derived. Theoretical SNR expressions for both the source beam and the detector beam are also obtained; these formulas specifically account for the incident state of polarization, the polarization effects of the interferometer, and make use of the exact expressions for the appropriate interferometer efficiency. In an Appendix, a brief comparison is made between some usual representations of the state of wave polarization.

Jones’s Matrix Representation of Optical Instruments. 2: Fourier Interferometers (Spectrometers and Spectropolarimeters)

Our method of matrix synthesis of optical components and instruments is applied to the derivation of Joness matrices appropriate for fourier interferometers (spectrometers and spectropolarimeters). These matrices are obtained for both the source beam and the detector beam. In the course of synthesis, Joness matrices of the various reflectors (plane mirrors; retroreflectors: roofed mirror, trihedral and prism cube corner, cats eye) used by these interferometers are also obtained.

An interferometric approach to the measurement of optical polarization

After discussing the desirability of determining the variation of polarization with frequency in planetary spectra, the possibility of measuring the intensity and state of polarization of optical radiation by means of the high resolution Fourier spectroscopic method is discussed. In the proposed experimental arrangement a two-beam interferometer is used with a polarizer in each beam. After recombination the emergent radiation is analyzed with a linear polarizer. It is shown that the interferograms obtained in this way contain information about the four Stokes parameters of the incident radiation. The polarizers introduce an asymmetry in the interferograms requiring full (exponential) transforms for retrieval of the desired data. The effects of the finite range of path difference and the variation of its zero point with frequency are considered, and evaluation of the corresponding phase error with a proper choice of the polarizer settings is discussed. The formalism also takes into account the residual polarization introduced by the beam splitter, and the differential transmission of the two beams. Generally, three independent interferograms are needed for determining the phase error and the four Stokes parameters. Some simple arrangements are described in which the two beams are either both linearly or both circularly polarized. It is hoped that instruments based on the principle described here will be built by workers in the field.