Peter R. Conwell

Resonant spectra of dielectric spheres

The natural resonant frequencies and poles associated with the electromagnetic modes of a dielectric sphere with a relative index of refraction of 1.4 have been calculated for size parameters ranging from 1 to 50. Determining pole locations in the complex plane entailed the computation of spherical Bessel functions for large complex arguments. The symbolic programming language reduce was used to provide independent verifications of the convergence and accuracy of the numerical Bessel function routines required in these computations. To determine pole locations, we used a standard zero-finding routine to find the zeros of the scattering coefficient denominators. In addition, we used a separate zero-counting routine in conjunction with the search routine to ensure that all poles within a given region of the complex plane were found. The real parts of the calculated poles agree with the location of peaks in the resonance spectrum (calculated for real frequency excitation), whereas the imaginary parts are related to the widths of these peaks. The intensity inside the sphere, averaged over all spherical angles, was computed as a function of radius. When the particle is excited at resonance, the internal intensity exhibits a sharp peak near, but not on, the surface. The intensity was found to be the strongest when the particle is driven at resonant frequencies whose poles have small imaginary components in the complex plane.

Sizing dielectric spheres and cylinders by aligning measured and computed resonance locations: algorithm for multiple orders

An algorithm for determining the size of dielectric spheres and cylinders by aligning measured and computed resonance locations is presented. The orders of the resonance locations need not be known a priori. The algorithm is applicable to several types of scattering and emission spectra of spheres and cylinders if the index of refraction including dispersion is known and uniform, or nearly uniform, throughout the sphere or cylinder. The algorithm performs well when tested with groups of computed resonance locations of spheres (synthetic data) and with measured fluorescence emission spectra of spheres exhibiting as many as 5 orders of resonance.

Fixed-weight networks can learn

A theorem describing how fixed-weight recurrent neural networks can approximate adaptive-weight learning algorithms is proved. The theorem applies to most networks and learning algorithms currently in use. It is concluded from the theorem that a system which exhibits learning behavior may exhibit no synaptic weight modifications. This idea is demonstrated by transforming a backward error propagation network into a fixed-weight system

Efficient automated algorithm for the sizing of dielectric microspheres using the resonance spectrum

We describe an efficient algorithm for the sizing of single microspheres having a known index of refraction as a function of wavelength. The algorithm employs a peak-detection routine that determines several resonant frequencies in the radiation scattered from the particle. These measured resonances are then compared with entries from a library of stored resonance locations in order to determine a few neighborhoods within which the size of the particle is likely to lie. A final step finds a local minimum of the cost function within each neighborhood, and the size estimate is determined by selecting the smallest of these local minima. The algorithm has modest computational and memory requirements, and it requires no analysis of complicated features of the resonance spectrum that would call for human intervention. Hence it could be automated for nearly real-time operation using a microprocessor. When applied to the measured resonance spectrum of a fluorescent polystyrene sphere, the algorithm finds the radius with an accuracy limited only by such factors as surface roughness, asphericity, and imperfect knowledge of the refractive index. The algorithm is currently limited to use with first-order resonances.

Learning algorithms and fixed dynamics

The authors discuss the equivalence of learning algorithms and nonlinear dynamic systems whose differential equations have fixed coefficients. They show how backpropagation transforms into a fixed-weight recursive neural network suitable for VLSI or optical implementations. The transformation is quite general and implies that understanding physiological networks may require one to determine the values of fixed parameters distributed throughout a network. Equivalently, a particular synaptic weight update mechanism such as Hebbian learning could likely be used to implement many known learning algorithms. The authors use the transformation process to illustrate why a network whose only variable weights are hidden-layer thresholds is capable of universal approximation.<<ETX>>

Universal approximation by phase series and fixed-weight networks

In this note we show that weak (specified energy bound) universal approximation by neural networks is possible if variable synaptic weights are brought in as network inputs rather than being embedded in a network. We illustrate this idea with a Fourier series network that we transform into what we call a phase series network. The transformation only increases the number of neurons by a factor of two.

Methuselah networks and optimal control

The authors describe a conceptual framework for optimal neural control called the Methuselah network. A cost surface constructed by the Methuselah network allows it to assess the performance of control schemes over long intervals. The network balances the cost of control with the need for exploratory control schemes. The environment or plant is controlled by a fixed-weight subnetwork whose weights are chosen to minimize long-term cost. The authors describe how an appropriate fixed-weight network can mimic adaptive-weight learning algorithms. This implies that the Methuselah network could be capable of developing learning algorithms