Complex-plane Representation Research Articles

Complex plane representation of a Fabry-Perot spectrometer output with multiple apertures

The output function of a Fabry-Perot spectrometer with a finite circular aperture is represented on the complex plane. This technique is illustrated by applying it to the calculation of size and diameter of multiple apertures.

Asymmetric Realizations for Dual-Mode Bandpass Filters

Two analytic synthesis techniques are presented for even-degree dual-mode in-line prototype networks up to degree 14. Commencing with the coupling matrix for the double cross-coupled array, rotational transformations are applied to transform the matrix into the form required for the dual-mode in-line asymmetric structure. Asymmetric here means that the coupling elements (irises, screws) are unequal in value about the physical center of the filter. The necessity for these asymmetric solutions arose when it was discovered that it was impossible to realize certain useful transmission characteristics with the symmetric in-line structure, on account of their transmission zero pattern in the complex-plane representation of the transfer function. Furthermore, because the full coupling matrix is used instead of the even-mode matrix as with the symmetric solution, the asymmetric in-line realization process may be applied to electrically asymmetric matrices, such as those for single ended filters for multiplexer applications. To demonstrate the validity of the theory, a practical model of each type of realization has been constructed.

Modified O'Bryan ellipsometer (MOE) for film-substrate systems

We present a modification of O'Bryan's return-path ellipsometer (J. Opt. Soc. Am. 26 (1936) 122) that extends its applicability to arbitrary surface retardations (not to be restricted to ±90°) at angles of incidence other than the principal angles. This modified version uses a phase-retarding corner-reflector to return the light beam for a second reflection at the sample surface, with a single optical component acting as both polarizer and analyzer. The operation of the ellipsometer is visualized using the complex-plane representation of polarized light.

New properties of the complex-plane representation of polarization

Properties of the complex-plane representation of polarization are derived which include first the resolution of an arbitrary polarization into two orthogonal states, and secondly the definition of nearness functions that express the closeness of the polarization states represented by two points in the complex plane. Applications where these results are significant are discussed.

The effect of an optical system in the complex-plane and the poincaré-sphere representations

General polarization-mapping properties of optical systems are discussed in the Poincaré-sphere representation by use of its relationship to the complex-plane representation. The various complex-plane representations that arise from a change of the bases of the polarization states are interrelated and are related to the Poincaré-sphere

Spacing of the multiple nulls in generalized ellipsometry

For accurate determination of the polarization transfer function of an optical system or the reflection matrix of an anomalous surface in the presence of imperfections in the ellipsometer a large number (e.g., six to twelve) of null measurements might be needed. Better insight is achieved using the circular complex-plane representation to define the conditions under which the nulls tend to be crowded. Instead of setting the compensator at different azimuths in the range of compensation (where a null exists) and adjusting the polarizer and analyzer for null, more uniform and hence improved spacing of the multiple nulls is achieved by setting the analyzer (in the case of the PCS arrangement) at equispaced azimuths in the range from zero to π and nulling by the polarizer and compensator.

A complex plane representation of dielectric and mechanical relaxation processes in some polymers

In a previous paper it was shown that the complex dielectric constant data of polymers can be represented by an empirical dispersion function. In the present work it is shown that the complex polarization of the same data can be represented by a function of the same form but with different values for the constants. This means that a quantitative evaluation of Scaife's remarks can be made. The dispersion parameters for eighteen polymers were determined for the ϵ∗ (ω) or the ϱ∗ (ω) data. In general, as the ratio ϵ 0 ≥ ∞ increases, the ϱ∗ (ω) data become broader and faster than the ϵ∗ (ω) data, thus 1 — α, β and τ 0 decrease. The normalized loss maximum was found to be temperature dependent [ϵ∗ (ω) data] for eleven polymers where ϵ 0 ≥ ∞≈2.0 . However, the normalized loss maximum [ϵ∗ (ω) data] for the two acetates was found to be independent of temperature while the normalized loss maximum for the ϱ∗ (ω) behaved in a way similar to the other polymers. This observation can be traced to a fortuitous compensation of effects encountered with large dispersions. For those cases where the method of reduced variables is applicable ϵ∗ (ω) calculated from the dispersion function is in good agreement with the shifted values of ϵ∗ (ω). The two parameters α and β are shown to be uniquely related to distribution of relaxation times. The agreement between the distribution function calculated from the dispersion function is in good agreement with the approximate methods used to calculate the function from the shifted data. A complex plane plot of the complex compliance is not at all similar to the dielectric dispersions. An empirical transformation procedure is constructed by analogy with the one used to calculate the complex polarization in order to normalize the mechanical data. The locus of this complex deformation resembles the dielectric dispersion with nearly the same values of α, β and τ 0. At very low frequencies, deviations from the assumed behaviour were observed. With polyisobutylene and poly( n-octyl methacrylate) the deviations were in terms of another dispersion. With poly(vinyl acetate) and poly(methyl acrylate) the deviations could be attributed to another low frequency dispersion. The similarity between the dielectric and mechanical dispersions suggests that the following mechanical model can be considered: a spherical inclusion containing the specimen of interest is perfectly bonded to an otherwise continuous homogeneous elastic continuum. Under these conditions the complex distortion of the sphere subjected to a periodic tensile field at infinity is very nearly the empirical complex deformation with Poisson's ratio of 1 2 for both media. It can also be shown for this model that if the distortion of the sphere is time dependent, then there will be in-phase and out-of-phase components to the distortion in a periodic field. In other words it is not necessary to postulate an internal viscosity to account for a macroscopic viscosity. The equilibrium distortion of the sphere is shown to be related to the square of the asymmetry of the orienting segments. The decay of the distortion with time of the removal of stress field is interpreted in terms of transition probabilities.