Mathematical frameworks for representing fields and waves and expressing Maxwell's equations of electromagnetism include vector calculus, differential forms, dyadics, bivectors, tensors, quaternions, and Clifford algebras. Vector notation is by far the most widely used, particularly in applications. Of the more sophisticated notations, differential forms stand out as being close enough to vectors that most practitioners can readily understand the notation, yet at the same time offering unique visualization tools and graphical insight into the behavior of fields and waves. We survey recent papers and book on differential forms and review the basic concepts, notation, graphical representations, and key applications of the differential forms notation to Maxwell's equations and electromagnetic field theory.

An L-band low-noise phased array feed antenna was designed, fabricated, and characterized experimentally. The element design was optimized for an active impedance matching condition with a given set of nonuniform amplitude beamformer coefficients associated with a formed PAF beam. Using measured array S-parameters and modeled array element patterns, the predicted boresight beam equivalent system noise temperature and aperture efficiency at prime focus with uncooled low-noise amplifiers were 42 K and 61% at 1600 MHz. Experimental measurements of receiver noise using an array Y-factor technique matched modeled predictions. The array feed was mounted on the Arecibo radio telescope for on-reflector characterization. The measured boresight beam sensitivity figure of merit (T_sys/η_ap) was 88 K and the 1 dB sensitivity bandwidth was 450 MHz.

Embedded element patterns depend on the loads attached to elements in an antenna array. When modeling antenna arrays, the loading condition used in numerically simulations may be different from the loads used in the operating environment. The ability to transform readily from one set to another is of great utility for both computed and measured data. We derive new expressions for the loading condition transformations that combine practical usefulness with a pleasing mathematical symmetry, compare array patterns synthesized from differently-loaded set of EEPs, and give numerical examples for arrays with different element port loading conditions.

For phased array receivers, the accuracy with which the polarization state of a received signal can be measured depends on the antenna configuration, array calibration process, and beamforming algorithms. A signal and noise model for a dual-polarized array is developed and related to standard polarimetric antenna figures of merit, and the ideal polarimetrically calibrated, maximum-sensitivity beamforming solution for a dual-polarized phased array feed is derived. A practical polarimetric beamformer solution that does not require exact knowledge of the array polarimetric response is shown to be equivalent to the optimal solution in the sense that when the practical beamformers are calibrated, the optimal solution is obtained. To provide a rough initial polarimetric calibration for the practical beamformer solution, an approximate single-source polarimetric calibration method is developed. The modeled instrumental polarization error for a dipole phased array feed with the practical beamformer solution and single-source polarimetric calibration was -10 dB or lower over the array field of view for elements with alignments perturbed by random rotations with 5 degree standard deviation.

Since the conception of radio astronomy [1], researchers have been striving to develop a "zero-noise" broadband receiver. Although this goal has remained elusive, the pursuit of it has led to numerous low-noise circuit innovations [2], [3]. Perhaps the most notable of these innovations are cryogenic low-noise amplifiers (cryo-LNAs) [3]-[6] as this technology has been able to achieve the lowest noise levels to date. While references to cryogenic amplifiers in the literature date back to the early 1950s [7], they are still a subject of much interest among researchers. Cryo-LNAs were initially conceived of to address challenges in the fields of space exploration [8], [9] and radio astronomy [10]. More recently, researchers in other fields have become interested in "zero-noise" receivers, making this a topic of widespread interest among RF and microwave engineers.