Leonard Bergstein

A Huygens’ Principle for Uniaxially Anisotropic Media*†

A Huygens’ principle for uniaxially anisotropic media is developed using a plane-wave representation of the electromagnetic field. The field emanating from an illuminated aperture in a uniaxial medium (with an optic axis which is arbitrarily oriented with respect to the axis of the aperture) may be represented as the superposition of a transverse electric (or TE) field and a transverse magnetic (or TM) field with respect to the optic axis of the medium. For the TE field, the field at an observation point and the aperture field distribution are related in the same manner as in isotropic media. An exact, closed-form expression is derived for the TM field, specifying the field at an observation point in terms of the field distribution over the aperture. A Huygens’ principle for anisotropic media emerges as a special case of this expression. The interpretation arrived at is that the field at an observation point is obtained by superposition of elliptical wavelets emanating from the aperture, with due regard to their phase differences when they reach the point in question.

The Frustrated Total Reflection Filter.I. Spectral Analysis

The method of modal expansion is used to develop a theory for the analysis of frustrated total reflection (FTR) filters or similar layered systems of finite lateral extent. The theory is applied to determine the (spectral) transmission properties of the FTR filter of finite lateral extent (for an incident plane wave propagating in direction of maximum transmission). It is found that the finiteness of the lateral dimensions of the filter has a great effect on the transmission properties of the filter. The results are in agreement with available experimental data and clearly show that the unbounded wave theory (i.e., an analysis which assumes that both the incident electromagnetic radiation and the filter are of infinite lateral extent) cannot be applied to FTR filters even when the lateral filter dimensions are several orders of magnitude larger than the wavelength of the incident radiation.

Analysis of speckle noise in bar-code scanning systems

Laser beams used for bar-code scanning exhibit speckle noise generated by the roughness of the surface on which bar codes are printed. Statistical properties of a photodetector signal that integrates a time-varying speckle pattern falling on its aperture are analyzed in detail. We derive simple closed-form expressions for the autocorrelation function and the power spectral density of the detector current for scanning beams with arbitrary field distributions. Theoretical calculations are illustrated by numerical simulations as well.

Angular Spectra of Optic Cavities

The resonant modes of optic interferometer cavities are investigated in the angular spectrum domain. The investigation is based on an integral equation (governing the relation between the normal modes and cavity geometry) which is derived by using the self-consistent Rayleigh formulation for solving diffraction problems. For plane-parallel cavities this integral equation can be solved by means of a series expansion of orthogonal functions characteristic of the cavity geometry without making any assumption about the relative magnitudes of end reflector dimensions and separation. The solution also provides, in addition to a comparison with the solutions of the approximated Huygens’ integral equation as found by other investigators, a direct way for obtaining the angular plane-wave spectrum (or the radiation pattern) of the beam emerging from the cavity. The particular cases of plane-parallel cavities with infinite-strip and circular end reflectors are considered in this paper.

Speckle noise in bar-code scanning systems—power spectral density and SNR

Laser-based flying-spot scanners are strongly affected by speckle that is intrinsic to coherent illumination of diffusing targets. In such systems information is usually extracted by processing the derivative of a photodetector signal that results from collecting over the detectors aperture the scattered light of a laser beam scanning a bar code. Because the scattered light exhibits a time-varying speckle pattern, the signal is corrupted by speckle noise. In this paper we investigate the power spectral density and total noise power of such signals. We also analyze the influence of speckle noise on edge detection and derive estimates for a signal-to-noise ratio when a laser beam scans different sequences of edges. The theory is illustrated by applying the results to Gaussian scanning beams for which we derive closed form expressions.

Diffraction by Plane Apertures in Uniaxially Anisotropic Media

A plane aperture is considered radiating in a uniaxially anisotropic medium that is otherwise assumed to be linear and homogeneous. A given but arbitrary aperture field distribution can be represented as the superposition of a transverse electric (TE) field and a transverse magnetic (TM) field relative to the direction of the optic axis of the medium. The existence of such a representation is used to show that once this decomposition is accomplished, the aperture illumination can be related to its corresponding diffraction field by considerations that involve only the propagation of electromagnetic waves in isotropic media. Considering the TE and TM fields separately, and beginning with existing diffraction formulas for each of these fields, the diffraction field may be obtained by solving two equivalent problems (one associated with the TE field and the other with the TM field), each of which requires the determination of a diffraction field produced by an illuminated aperture radiating in an isotropic medium. The well-known techniques and available results for apertures radiating in isotropic media may henceforth be applied to determine the diffraction fields produced by apertures radiating in uniaxially anisotropic media.

Further Comments on the Theory of the Frustrated Total Reflection Filter

In view of the comments made by Iogansen it would be worthwhile to compare first the two methods of analysis. The method developed by Iogansen is based on an approximate solution of the wave equation by means of orthogonal waves. When the direction of propagation of the incident wave is such that the angle between the wave normal and the normal to the boundary surfaces of the layers is sufficiently large, the orthogonal waves approximate closely the solution to the wave equation. However, in order to relate the transmitted field in each of the layers to the incident field, further approximations are necessaiy. In particular, in the approach used by Iogansen the orthogonal waves are expanded in a Taylor series and only the first linear term of the series is retained. Clearly, such an expansion yields a reasonable approximation to the actual solution if, and only if, the thickness of each of the layers is of the order of a wavelength of the incident light or smaller. When the thickness of the layers is large compared with the wavelength, the approximation becomes poor and the method of analysis breaks down completely. The approximation is therefore limited to the analysis of frustrated total equation filters or similar resonant systems involving a limited number of layers of limited thickness. In contrast, the method of plane wave expansion as proposed in our paper does not basically involve any approximations (assuming of course that the model of blackbody boundaries is accepted). It can, therefore, be used independently of the thickness and the number of layers involved. All results of interest

Speckle revisited: analysis of speckle noise in bar-code scanning systems

Laser beams used for bar-code scanning exhibit speckle noise generated by the roughness of the surface on which bar-codes are printed. Statistical properties of a photodetector signal that integrates a time-varying speckle pattern falling on its aperture are analyzed in detail. We derive simple closed form expressions for the auto-correlation function and power spectral density of the detector current for general form scanning beams with arbitrary field distributions. Theoretical calculations are illustrated by numerical simulations.

Design of extended depth-of-focus laser beams using orthogonal beam expansions

Laser beams with extended depth of focus have many practical applications, such as scanning printed bar codes. Previous work has concentrated on synthesizing such beams by approximating the nondiffracting Bessel beam solution to the wave equation. In this paper, we introduce an alternate novel synthesis method that is based on maintaining a minimum MTF value (contrast) over the largest possible distance. To achieve this, the coefficients of an orthogonal beam expansion are sequentially optimized to this criterion. One of the main advantages of this method is that it can be easily generalized to noncircularly symmetrical beams by the appropriate choice of the beam expansion basis functions. This approach is found to be very useful for applications that involve scanning of the laser beam.