Electromagnetic scattering from anisotropic materials, part II: Computer code and numerical results in two dimensions

The two-dimensional problem of oblique scattering by penetrable cylinders of arbitrary cross section made of materials which are linear, lossy, anisotropic and possibly inhomogeneous is considered. The materials are characterized by arbitrary tensor susceptibilitiesbar{x}_{ec}andbar{x}_{m}. The frequency-domain volume integrodifferential equations satisfied by the electric and magnetic fields and obtained in a previous paper (Part 1) are analyzed numerically. Optimal ordering of the unknowns and transverse electric-transverse magnetic (TE-TM) decomposition in the matrix formulation of the problem are discussed. The cross section of the scatterer is broken down into a triangular mesh. The field components at the vertices of the triangles are the unknowns; within each triangle, each field component is a linear combination of its values at the vertices. Computed field distributions inside the scatterer are found to be in excellent agreement with results obtained by other methods.     

Fast evaluation of logarithms in fields of characteristic two

A method for determining logarithms in GF(2^{n})is presented. Its asymptotic running time isO(exp (cn^{1/3} log^{2/3} n))for a small constantc, while, by comparison, Adleman's scheme runs in timeO(exp (c^{'}n^{1/2} log^{1/2} n )). The ideas give a dramatic improvement even for moderate-sized fields such as GF(2^{127}), and make (barely) possible computations in fields of size around2^{400}. The method is not applicable to GF(q)for a large primeq.     

Threshold of the optical backward-wave parametric oscillator

We present a discussion of the threshold condition for the optical backward-wave parametric oscillator, taking into account diffraction due to the finite transverse extent of the fields, and using three transverse modes of both the forward- and backward-wave fields. The coupled differential equations are solved numerically, and the threshold is obtained by minimizing the pump field with respect to the parameters of the forward- and backward-wave fields. Denoting the confocal parameters byb_{1}, b_{2}, and b3for the backward, forward, and pump waves, respectively, and if the length of the crystal is 1, we find that for1/b_{3} geq 2, we may setb_{1} = b_{2} =b_{3}. For most purposes, the phase-matching condition may be taken ask_{2} = k_{1} + k_{3}. Also, when calculating the threshold, it is adequate to consider only the two lowest order transverse modes of the forward-and backward-wave fields.     

Rayleigh scattering and power conservation

In the usual formulation of Rayleigh scattering in terms of the quasi-static dipole moments of the scatterer, the scattered field is evaluated exactly to the lowest order(k_{0}a)^{3}whereais a characteristic dimension of the scatterer. Although the scattered power may be found correctly to order(k_{0}a)^{6}in terms of the scattered field, power conservation does not hold. To achieve power conservation the scattered field must be determined exactly to order(k_{0}a)^{6}. It is shown that the term of order(k_{0}a)^{6}may be found by including radiation reaction as part of the polarizing field. The scattered power can then be found by using the forward scatter theorem (optical extinction theorem) and leads to power conservation correct to lowest order.     

On the eigenfunction expansion of electromagnetic dyadic Green's functions

A relatively simple approach is described for developing the complete eigenfunction expansion of time-harmonic electric (bar{E}) and magnetic (bar{H}) fields within exterior or interior regions containing an arbitrarily oriented electric current point source. In particular, these results yield directly the complete eigenfunction expansion of the electric and magnetic dyadic Green's functionsbarbar{G}_{e}andbarbar{G}_{m}that are associated withbar{E}andbar{H}, respectively. This expansion ofbarbar{G}_{e}andbarbar{G}_{m}contains only the solenoidal type eigenfunctions. In addition, the expansion ofbarbar{G}_{e}also contains an explicit dyadic delta function term which is required for making that expansion complete at the source point. The explicit dyadic delta function term inbarbar{G}_{e}is found readily from a simple condition governing the behavior of the eigenfunction expansion at the source point, provided one views that condition in the light of distribution theory. These general expressions for the eigenfunction expansion ofbarbar{G}_{e}andbarbar{G}_{m}reduce properly to those obtained previously for special geometries by Tai.     

The solutions of nonaxisymmetric fields in SELFOC fibers with longitudinal imperfections

The solutions of nonaxisymmetric fields in SELFOC fibers with longitudinal imperfections are investigated. The refractive index is assumed to be of the formK = varepsilon/varepsilon_{0} = K_{0} - K_{2}(z) r^{2}(K_{2}(z)is a gradually varying function ofz). The analytic solution is obtained. When K2is a constant, the solution is reduced to that of perfect SEL-FOC fibers, which is well known. Further the relations of modes conversion in imperfect fibers are obtained.     

On the capacity of certain additive non-Gaussian channels

A model of an additive non-Gaussian noise channel with generalized average input energy constraint is considered. The asymptotic channel capacityC_{zeta}(S), for large signal-to-noise ratioS, is found under certain conditions on the entropyH_{ tilde{ zeta}}( zeta)of the measure induced in function space by the noise processzeta, relative to the measure induced bytilde{zeta}, where is a Gaussian process with the same covariance as that ofzeta. IfH_{ tilde{zeta}}( zeta) < inftyand the channel input signal is of dimensionM< infty, thenC_{ zeta}(S)= frac{1}{2}M ln(1 + S/M) + Q_{zeta}( M ) + {o}(1), where0 leq Q_{ zeta}( M ) leq H_{ tilde{ zeta}}( zeta). If the channel input signal is of infinite dimension andH_{ tilde{ zeta}}( zeta) rightarrow 0forS rightarrow infty, thenC_{ zeta}(S) = frac{1}{2}S+{o}(1).     

Normal basis of finite fieldGF(2^m)(Corresp.)

Massey and Omura recently developed a new multiplication algorithm for Galois fields based on the normal basis representation. This algorithm shows a much simpler way to perform multiplication in finite field than the conventional method. The necessary and sufficient conditions are presented for an element to generate a normal basis in the field GF(2^{m}), wherem = 2^{k}p^{n}andp^{n}has two as a primitive root. This result provides a way to find a normal basis in the field.     

Mutual admittance characteristics for two-element parallel prolate spheroidal antenna systems

Spheroidal dipole antenna systems consisting of two thin center-fed parallel prolate spheroids in various configurations (side-by-side and collinear) are considered, The resultant EM fields are represented as modal expansions in terms of spheroidal vector eigenfunctions. The translational addition theorems for spheroidal functions developed by the authors in a previous paper play the central role in the formulation of a system matrix [G] which transforms the primary. EM excitations of the fed dipoles into their scattered responses. From the knowledge of the [G] matrix, the mutual admittances of the dipole system are obtained and plotted against center-to-center separation of the dipoles for side-by-side and collinear dipole configurations. The major to minor axial ratios of 10:1 and 100:1 of the prolate spheroidal dipoles are considered for presenting various curves. However, for the side-by-side configuration, due to a radius of convergence of the translational relation for outgoing wave to incoming wave transformation, the separationdof the spheroids is restricted tod > {a_{1}, a_{2}}_{max}, whosea_{1}anda_{2}are the semimajor axial lengths of the two spheroids.     

Alias-free randomly timed sampling of stochastic processes

The notion of alias-free sampling is generalized to apply to random processesx(t)sampled at random timest_n; sampling is said to be alias free relative to a family of spectra if any spectrum of the family can be recovered by a linear operation on the correlation sequence{r(n)}, wherer(n) = E[x(l_{m+n}) overline{x(t_m)}]. The actual sampling timest_nneed not be known to effect recovery of the spectrum ofx(t). Various alternative criteria for verifying alias-free sampling are developed. It is then shown that any spectrum whatsoever can be recovered if{t_n}is a Poisson point process on the positive (or negative) half-axis. A second example of alias-free sampling is provided for spectra on a finite interval by periodic sampling (fort leq t_oort geq t_o) in which samples are randomly independently skipped (expunged), such that the average sampling rate is an arbitrarily small fraction of the Nyquist rate. A third example shows that randomly jittered sampling at the Nyquist rate is alias free. Certain related open questions are discussed. These concern the practical problems involved in estimating a spectrum from imperfectly known{ r(n) }.     

Multiple detection of a slowly fluctuating target (Corresp.)

A "slowly" fluctuating target is assumed to keep its radar cross section constant for the duration of several(M)dwells on target. To resolve multiple range and/or Doppler ambiguities, the received signal, which is presumably coherently processed (i.e., predetection integrated or matched filtered) over each dwell, must often be tested against a threshold, {em independently} of those on other dwells. Such a procedure is referred to as {em multiple detection}. A technique for the evaluation of a tight lower bound on the multiple-detection probabilityP_{M}, under Swerling case I statistics for the cross section, is presented in term of an infinite series and worked out in detail forP_{2}andP_{3}. Estimates on the computation error due to the truncation of the series are derived. Numerical results indicate thatP_{3}comes much closer toP_{1}than top_{1}^{3}or even toP_{1}P_{2}; at an expected signal-to-noise ratio of13dB and atP_{1} = 0.51, it obtains thatP_{3} geq 0.40, whereasP_{1}P_{2} = 0.23andp_{1}^{3} = 0.17.     

Electromagnetic reflection from an extended turbulent medium: Cumulative forward-scatter single-backscatter approximation

The backscatter cross sectionQfor high-frequency irradiated turbulent dielectric media, many mean free pathsL_{1}wide, is computed. The lengthL_{1}is the distance into the medium over which the mean electric field decreases in amplitude by a factore^{-1}. Previous calculations have always been restricted toL ll L_{1}. It is found thatQincreases from the Born approximationQ = Q_{1}for medium widthL ll L_{1}toQ = 2Q_{1}forL gg L_{1}, and the theory is valid as long asL ll (kL_{0})^{5/3} L_{1}, a significant improvement over the Born approximation, when the macroscaleL_{0}is much larger than the wavelength2_{pi}k^{-1}. The improvement is due to incorporation of the dominant effects of cumulative forward scattering in the local electric field in the medium. A rigorous and a heuristic derivation are given. The transitional behavior is discussed and a simple physical interpretation is given.     

Flush mounted rectangular cavity slot antennas--Theory and design

The loaded rectangular cavity slot antenna is analyzed using variational methods in conjunction with simplified equivalent circuit techniques to derive accurate design guides for efficiency, bandwidth, and resonant frequency. The aperture admittance is computed and the effects of a compound aperture plane iris and of material loading are analyzed. The aperture admittance of all such loaded cavity antennas is proportional tomu_{r},sqrt{mu_{r}/epsilon_{r}}, or1/epsilon_{r}, which characteristic lends to a common method of optimization of|T|^{2}(transmission cofficient). Experimental results include: 1) measurements of aperture field; 2) a comparison of theoretical and experimental value of bandwidth, efficiency, resonant frequency, and beam pattern for several experimental models; and 3) the measurement of the effect of applied dc magnetic field.     

Capacity theorems for the relay channel

A relay channel consists of an inputx_{l}, a relay outputy_{1}, a channel outputy, and a relay senderx_{2}(whose transmission is allowed to depend on the past symbolsy_{1}. The dependence of the received symbols upon the inputs is given byp(y,y_{1}|x_{1},x_{2}). The channel is assumed to be memoryless. In this paper the following capacity theorems are proved. 1)Ifyis a degraded form ofy_{1}, thenC : = : max !_{p(x_{1},x_{2})} min ,{I(X_{1},X_{2};Y), I(X_{1}; Y_{1}|X_{2})}. 2)Ify_{1}is a degraded form ofy, thenC : = : max !_{p(x_{1})} max_{x_{2}} I(X_{1};Y|x_{2}). 3)Ifp(y,y_{1}|x_{1},x_{2})is an arbitrary relay channel with feedback from(y,y_{1})to bothx_{1} and x_{2}, thenC: = : max_{p(x_{1},x_{2})} min ,{I(X_{1},X_{2};Y),I ,(X_{1};Y,Y_{1}|X_{2})}. 4)For a general relay channel,C : leq : max_{p(x_{1},x_{2})} min ,{I ,(X_{1}, X_{2};Y),I(X_{1};Y,Y_{1}|X_{2}). Superposition block Markov encoding is used to show achievability ofC, and converses are established. The capacities of the Gaussian relay channel and certain discrete relay channels are evaluated. Finally, an achievable lower bound to the capacity of the general relay channel is established.     

Repeated use of codes which detect deception (Corresp.)

A senderAwants to sendNmessagesx_{i} = 1 , ldots ,N, chosen from a set containingMdifferent possible messages,M > N, to a receiverB. Everyx_{i}has to pass through the hands of a dishonest messengerC. ThereforeAandBagree on a mathematical transformationfand a secret parameter, or keyk, that will be used to produce the authenticatory_{i} = f(x_{i} , k ), which is sent together withx. The key is chosen at random from a set ofLelements.Cknowsfand can find all elements in the setG(x_{i},y_{i}) = {k|f(x_{i}, k) = y_{i}}given enough time and computer resources.Cwants to changex_{i} into x^{prime}withoutBsuspecting. This means thatCmust find the new anthenticatory^{prime} = f(x^{prime} , k). SinceG(x_{i},y_{i})can be found for any(x_{i},y_{i}), it is obvious thatCwill always succeed unlessG(x_{i},y_{i})contains more than one element. Here it is proved that the average probability of success forCis minimized if (a)G(x_{i}, y_{i})containsL^{(N-1)/N}elements and (b) each new known pair(x_{j}, y_{j})will diminish this set of solutions by a factor ofL^{-l/N}. The minimum average probability will then beL^{-l/N}.     

Nonlinear interaction of electromagnetic waves in bounded plasmas

A method is presented for the solution of second-order electromagnetic fields in a bounded plasma which is excited by two high-frequency electromagnetic waves. Due to nonlinearities inherent in any plasma, incoming waves are mixed, creating sum and difference frequencies. The signal from the difference frequencyomega_{1}-omega_{2}, whereomega_{1}andomega_{2}are source frequencies, can be of considerable consequence whenomega_{1}-omega_{2}is sufficiently low to cause a significant reaction in a plasma as resonant conditions develop. In particular, the problem considered in this paper is the optimization of second-order fields in a uniform plasma confined to a circular waveguide including reference to diagnostic applications.     

Higher dimensional orthogonal designs and applications

The concept of orthogonal design is extended to higher dimensions. A properg-dimensional design[d_{ijk cdots upsilon}]is defined as one in which all parallel(g-1)-dimensional layers, in any orientation parallel to a hyper plane, are uncorrelated. This is equivalent to the requirement thatd_{ijk cdots upsilon} in {0, pm x_{1}, cdots , pm x_{t} }, wherex_{1}, cdots , x_{t}are commuting variables, and thatsum_{p} sum_{q} sum_{r} cdots sum_{y} d_{pqr cdots ya} d_{pqr cdots yb} = left( sum_{t} s_{i}x_{i}^{2} right)^{g-1} delta ab,where(s{1}, cdots , s{t})are integers giving the occurrences ofpm x_{1}, cdots , pm x_{t}in each row and column (this is called the type(s_{1}, cdot ,s_{t})^{g-1})and(pqr cdots yz)represents all permutations of(ijk cdots upsilon). This extends an idea of Paul J. Shlichta, whose higher dimensional Hadamard matrices are special cases withx_{1}, cdots , x_{t} in {1,- 1}, (s_{1}, cdots, s_{t})=(g), and(sum_{t}s_{i}x_{i}^{2})=g. Another special case is higher dimensional weighing matrices of type(k)^{g}, which havex_{1}, cdots , x_{t} in {0,1,- 1}, (s_{1}, cdots, s_{t})=(k), and(sum_{t}s_{i}x_{i}^{2})=k. Shlichta found properg-dimensional Hadamard matrices of size(2^{t})^{g}. Proper orthogonal designs of type     

Electromagnetic reflection from a curved dielectric interface

The reflection of a high-frequency electromagnetic field from an arbitrarily curved dielectric interface is considered. The fields are expanded in asymptotic series ofk^{-1}, known as Luneburg-Kline expansions. Based on a ray method the zero- and first-order terms ofk^{-1}of the reflected and transmitted field are evaluated at the interface. Associated with the fields at the interface, effective surface current densities can be used to determine the reflected and transmitted field at points away from the interface, which is done analytically for the reflected far field in the case of plane wave incidence. The result consists of a frequency-independent term, which is related to geometrical optics solution, and a term ofk^{-1}, which is a useful extension of geometrical optics solution in some cases.     

On the determination of the terminal impedances of a nonuniform transmission line

The problem of determining the terminal impedance values of a lossless nonuniform transmission line is addressed. The transmission line is assumed to have discontinuities at input and output with local reflection coefficientsGamma_{1}andGamma_{2}, respectively. A set of formulas is given which allowsGamma_{1}andGamma_{2}to be recovered once high-frequency data forS_{11}(omega) is specified; the formulas extend the generality of earlier theory to the case whereGamma_{1}orGamma_{2}may be either positive or negative. It is also shown thatS_{11} (omega)) magnitude and phase information is necessary to uniquely determineGamma_{1}andGamma_{2}.     

Backscattering from a Gaussian-distributed perfectly conducting rough surface

An analytical approach to the problem of scattering by composite random surfaces is presented. The surface is assumed to be Gaussian so that the surface height can be split (in the mean-square sense) into large (zeta_{l}) and small (zeta_{s}) scale components relative to the electromagnetic wavelength. A first-order perturbation approach developed by Burrows is used wherein the scattering solution for the large-scale structure is perturbed by the small-scale diffraction effects. The scattering from the large-scale structure (the zeroth-order perturbation solution) is treated via geometrical optics since4k_{0}^{2}bar{zeta_{l}^{2}} gg 1. The first-order perturbation result comprises a convolution in wavenumber space of the height spectrum, the shadowing function, a polarization dependent factor, the joint density function for the large-scale slopes, and a truncation function which restricts the convolution to the domain corresponding to the small-scale height spectrum. The only "free" parameter is the surface wavenumber separating the large and small height contributions. For a given surface height spectrum, this wavenumber can be determined by a combination of mathematical and physical arguments.