J. Allen Cox

Mathematical studies in rigorous grating theory

We consider the diffraction of a time-harmonic wave incident upon a grating (or periodic) structure. We study mathematical issues that arise in the direct modeling, inverse, and optimal design problems. Particular attention is paid to the variational approach and to finite-element methods. For the direct problem various results on existence, uniqueness, and numerical approximations of solutions are presented. Convergence properties of the variational method and sensitivity to TM polarization are examined. Our recent research on inverse diffraction problems and optimal design problems is also discussed.

Guided-mode grating resonant filters for VCSEL applications

A brief summary of both VCSEL technology and guided-mode grating resonant filters (GMGRFs) is presented. We then discuss benefits and issues of integrating the two technologies, emphasizing control of wavelength, polarization, and laser cavity modes. We present a GMGRF design suitable for a 980 nm InGaAs VCSEL and show that a significant loss (-4%) in reflectivity results from the slight loss associated with the minimum mirror conductivity required to inject current through the mirror. Experimental data are presented at 850 mm for gratings designed for and fabricated on fused silica substrates and illustrate that GMGRFs are also very sensitive to other forms of loss such as scatter caused by roughness in the grating lines. We suggest a hybrid approach of a GMGRF on a reduced distributed Bragg reflector stack as a means to circumvent the high sensitivity to loss in the GMGRF.

Modal analysis of homogeneous optical waveguides by the boundary integral formulation and the Nyström method

The optical field in a weakly guiding homogeneous waveguide satisfies scalar Helmholtz equations in both the core and the cladding and the transmission conditions on the boundary. Two different systems of boundary integral equations are derived for numerical solutions of the discrete propagation constants of the optical field; one of them is in the form of Fredholm integral equations of the second kind, and the other is a mixed first and second kind. The Nystrom method is used to solve the boundary integral equations numerically. The numerical results show that the two boundary integral formulations are both very efficient in the numerical simulations of homogeneous waveguides but that the second kind is superior because it controls spurious modes better.

Integral equation method for biperiodic diffraction structures

We describe an integral approach to the rigorous solution of Maxwells equations for diffraction from a biperiodic grating having arbitrary profile and periodicity. Emphasis is placed on the implementation of solutions and is addressed to the engineering community. The method described is amenable to mathematical convergence analysis. A numerical example is given which indicates convergence of solutions.

Point-Source Location Using Hexagonal Detector Arrays

For staring sensors, improved performance in the location of point sources can be achieved by use of an array of hexagonal detectors instead of the usual array of square or rectangular detectors. This improvement is demonstrated by calculating the accuracy of the centroid algorithm as a function of signal-to-noise ratio and blur spot size for both types of detector arrays. The probability density function for the centroid random variable is derived and is used to perform all noise analysis. The analysis indicates that the algorithm error is reduced by as much as a factor of 3, the sensitivity to noise is reduced by 17%, the computational load is decreased by 23%, and the data storage requirement is reduced by 22%. The clutter-induced noise, as measured by the clutter equivalent target, is essentially identical for square and hexagonal detectors of the same area.

OPTICAL PERFORMANCE OF HIGH-ASPECT LIGA GRATINGS II

The analysis of an IR tunable filter that uses a transmission grating with variable spacing is extended to include incident spherical waves instead of plane waves and to illustrate specifically the effects of finite conductivity in the metal. The model is based on a rigorous vector solution of Maxwell’s equations implemented in a finite elements solver. The modeling results based on tabulated optical properties of nickel are compared with experimental measurements made with filters fabricated in permalloy by the lithographie, galvanoformung, abformung (LIGA) process. The theoretical and experimental results compare quite well and suggest that the resolution of the filter is severely reduced at the shorter wavelengths (less than 4 ?m). At longer wavelengths, the resolution is limited by the aspect ratio of the LIGA structures. A spectral resolution of 0.1 ?m appears to be practical.

Advantages Of Hexagonal Detectors And Variable Focus For Point-Source Sensors

Analytical results are presented showing improved detection and location performance of point sources in a staring sensor using a hexagonal detector array or having the capability to adjust the size of its blur spot (variable focus). Square and hexagonal detector arrays are compared for detection probability versus focus, for location accuracy versus focus and signal-to-noise ratio, and for location accuracy versus blur size and detector array fill factor. We find that hexagonal detectors provide better detection probability when the point source lies near a corner of the detector, but otherwise the two detector types give comparable performance. Hexagonal detectors generally give better location accuracy than square detectors. Variable focus is shown to be advantageous for maximizing performance for both detection and location of point sources.

Application of diffractive optics to infrared imagers

An overview is presented of diffractive optics and micro-optics technologies and their application to infrared imaging systems. The technology overview describes several fabrication methods, emphasizing the importance of diamond machining for infrared systems and materials, and discusses various optical functions commonly implemented with diffractive elements. Applications are discussed both as a general survey by citing an extensive list of references and as a more detailed presentation of three examples.

Microlens array for staring infrared imager

This paper describes the fabrication and testing of 64 X 96 arrays of microlenses used for fill-factor enhancement of uncooled infrared detector arrays. Each lenslet represents a f/0.9 Fresnel phase lens at 10 micrometers wavelength. All arrays were etched into silicon wafers as either 8-level or 16-level staircase kinoforms. When integrated with a detector array having 20 fill factor, these microlens arrays were capable of increasing the magnitude of the measured signal with f/2.2 fore-optics by 2.5-fold.

Point Source Location Sensitivity Analysis

This paper presents the results of an analysis of point source location accuracy and sensitivity as a function of focal plane geometry, optical blur spot, and location algorithm. Five specific blur spots are treated: gaussian, diffraction-limited circular aperture with and without central obscuration (obscured and clear bessinc, respectively), diffraction-limited rectangular aperture, and a pill box distribution. For each blur spot, location accuracies are calculated for square, rectangular, and hexagonal detector shapes of equal area. The rectangular detectors are arranged on a hexagonal lattice. The two location algorithms consist of standard and generalized centroid techniques. Hexagonal detector arrays are shown to give the best performance under a wide range of conditions.