Walter K. Kahn

Theory of mutual coupling among minimum-scattering antennas

A novel and rigorou formulation for mutual impedance among a class of idealized but realizable antennas is presented. Unlike past treatments of mutual coupling, which have proceeded from structurally specified antennas, the present treatment deals with a class of antennas, all the electromagnetic properties of which are rigorously expressed explicitly in terms of their radiation patterns alone. Employing a network formulation based on a scattering representation of electromagnetic fields in terms of spherical (or cylindrical) modes, the mutual impedances between such antennas are computed exactly. Several alternate representations for the mutual impedance are derived, one of which is an integral representation involving the power pattern function for both real and complex angles. In special cases, these exact results agree with published results obtained by applying well-known approximate techniques to structurally specified antennas. An important and illuminating example is provided by an infinite planar array of antennas radiating into a half-space for which the present formulation yields the well-known grating-lobe series representation of the active impedance without an explicit reference to structural properties of the radiating elements.

Unstable Optical Resonators

A technique, firmly based on a development from ray optics, is presented for calculating the loss due to the finite sizes of curved mirrors when these form an unstable optical resonator. If paraxial rays launched within such a resonator are confined near the resonator axis, the resonator is termed stable; otherwise it is termed unstable, and is known to have high losses. Siegman has recently presented a geometrical method, brilliantly constructed ad hoc, for calculating these losses in unstable resonators, and indicated where these might be advantageous in laser application. The ray optical theory presented here, which employs the concept of ray modes in an equivalent beam waveguide, is shown to yield results equivalent to those of Siegman for all cases considered by him. However, being derived from conventional ray optics, the validity of the formulas is independently established, and these formulas are immediately applicable to re-entrant resonators and resonators containing inhomogeneous media. The fractional loss per resonator pass is equal to 1-|lambda(2)|, where |lambda(2)| < this 1 is an eigenvalue of the transfer matrix T, representing the corresponding ray transformation.

Application of conformal mapping approximation techniques: parallel conductors of finite dimensions

This paper describes a novel approach to the application of conformal mapping to capacitance evaluation. A particular structure composed of an array of identical lines and located below a conductive plate is studied. Results show the application of conformal mapping can reduce computing time when using three-dimensional electrostatic modeling and it can be the basis of algorithms of practical applications, especially in the semiconductor industry.

Element Efficiency: A Unifying Concept for Array Antennas

The antenna-element efficiency parameter quantifies the effect of mutual coupling in array antennas. In general arrays, this efficiency varies from element to element. The average value of these element efficiencies is a significant summary parameter, and plays a prominent role in circumscribing array performance. A useful formula gives the minimum number of antennas necessary to realize a prescribed peak gain envelope. The important special case of an array of identical elements situated in identical array environments is termed an idempotent array. In particular, an infinite idempotent array models a large planar array. There, the unique element efficiency has been directly related to the rms value of the conventional active reflection coefficient as it varies when the beam is (phased) electronically steered. The element-efficiency formulation also provides a framework for the rigorous derivation of important theoretical results in the theory of phased arrays.

Rays and Ray Envelopes within Stable Optical Resonators Containing Focusing Media

In stable resonators any given initially paraxial rays remain close to the axis of the structure and are, in fact, confined within well-defined contours-the envelope of the ray system. Previously, envelopes of rays in empty resonators had been found and their form identified with the variation of the spot size. This geometric optical approach is extended to general resonators, comprising arbitrary arrangements of lenses and convergent or divergent inhomogeneous focusing media. An invariant quadratic form involving parameters descriptive of any of the ray segments that result from a given initial ray segment leads to a differential equation satisfied by the ray segments and their envelope in portions of the resonator. A maximum-minimum problem for the envelope is formulated and solved. In convergent media the envelope function is found to be periodically modulated. The period of the modulation depends only on the properties of the convergent medium; the location of relative maxima and minima, as well as their ratio, depends on both the medium and associated optics. In special cases, results are compared with available solutions of the corresponding electromagnetic problem. A particularly simple resonator is analyzed, and envelope characteristics correlated with the stability limits.

Hamiltonian analysis of beams in an optical slab guide

The formal analogy between Hamiltonian classical mechanics and quantum mechanics on the one hand and geometrical optics and physical optics on the other hand is systematically presented. When the time is replaced by the axial coordinate of a cylindrically uniform optical system and Hamilton’s principle replaced by Fermat’s principle, classical particle trajectories correspond to rays and quantum-mechanical wave functions to physical-optics fields. The operator formalism of quantum mechanics then provides elegant solutions for problems associated with propagation of beams in a gradient-index multimode optical slab guide.

Comparison of various image induction (II) methods with physical optics (PO) for the far-field computation of flat-sectioned segmented reflectors

Kottlers (1923) extension of Kirchhoffs diffraction integral to electromagnetic fields yields the copolarized and cross-polarized fields of segmented reflectors. For flat sections, the Maggi-Rubinowicz (1888, 1917) potential can be used to transform Kottlers surface integral into a line integral resulting in an expression composed entirely of line integrals. Computation is simplified by the use of the Asvestas (1985) potential which eliminates the need to compute a geometrical optics term required by the original Maggi-Rubinowicz potential. In computing the far fields, a further simplification is realized by considering the antenna in reception rather than in transmission as an involved dyadic potential is then replaced by a simple vector potential. This is an exact-analysis method in the context of the image-induction model which, in theory, provides results which are very close to the physical optics (PO) model for the transmitting antenna. An approximate closed-form method is obtained by applying the Gordon (1975) transform to Silvers (1949) vector far field equations.

Dihedral optical resonators and prism beam waveguides.

A simple theory is developed for optical resonators, in which the mirrors are each composed of two planes forming a nearly straight dihedral angle and an equivalent beam-waveguide comprising a sequence of narrow angle prisms. Geometric optical considerations yield the variation of the intensity within the resonator in a manner analogous to (although not entirely as satisfying as in) the case of the spherical mirror resonator. Modes of the continuous medium, identified as the limit of particular sequences of closely spaced weak prisms, characterized by a parameter Omega, are found in terms of Airy functions. An appropriate spot size parameter is given by [Omega/2k0(2)]1/3, where k(0) is the wavenumber.

Element efficiency: a unifying concept for array antennas

The efficiency of the mth elementary (lossless, reciprocal) antenna within the environment of an entire array of N terminated antenna elements may be defined as the ratio of the realized gain, g, relative to power available at the mth element, to the directive gain, g/sub d/ relative to the power radiated from that element. A physical arrangement of such a general array of N elements is indicated. With respect to the N antenna ports the array is characterized by its normalized voltage scattering matrix.

Decomposition of electromagnetic boundary conditions at planar interfaces with applications to TE and TM field solutions

Electromagnetic fields in homogeneous source-free regions can be decomposed into fields that are TE and TM with respect to a particular reference direction (e.g., the z direction). If transverse sources exist, both TE and TM fields may be excited simultaneously. This paper considers the case of two infinite regions having a common planar interface and prescribed sources (surface currents) on the interface. The source currents are decomposed in a manner consistent with the decomposition of the fields. Accordingly, a procedure is established for describing the boundary conditions at the interface in terms of the longitudinal field components E/sub z/, H/sub z/ and the surface currents J~/sub s/. The development is unique in that the continuity of the transverse field components at the boundary are not explicitly considered but interpreted in terms of z-directed fields. This boundary condition approach is shown to give results consistent with those obtained by matching the tangential fields at the interface using vector transforms. A simple example illustrating the procedure using a ring of current in free-space is presented.