On the refractive properties of media with poles or zeros in the index of refraction

The refractive properties of inhomogeneous fibers are examined with emphasis being placed on the limiting situation where the index of refraction possesses poles or zeros. If, e.g., the index of refraction isn(xi) = xi^{m}, xi = r/a, whereais the radius of the fiber andman arbitrary constant, it is found that energy integrability is satisfied ifm > -1. Whenm leq -1energy infinities occur. The ray behavior of such media is examined in terms of geometrical optics, and corrections to geometrical optics are obtained by an asymptotic analysis of the exact solution. Form > 0, the lens is of the diverging type, and when the angle of incidence isalpha = 0, geometrical optics predicts that rays "reflect" at various angles from the origin (depending on the value ofm). Whenm < 0, rays "wrap" around the origin several times with a zero radius of curvature before they leave the lens (alpha = 0). For-1 < m < -frac{1}{2}, it is found that whenalpha= [(2m + 1)/2m] picaustics occur (alpha = 0, pi/2excluded). Pictorial diagrams show the behavior of these caustics and the correction coefficients to geometrical optics are obtained.     

A uniform asymptotic theory of electromagnetic diffraction by a curved wedge

Diffraction of an arbitrary electromagnetic optical field by a conducting curved wedge is considered. The diffracted field according to Keller's geometrical theory of diffraction (GTD) can be expressed in a particularly simple form by making use of rotations of the incident and reflected fields about the edge. In this manner only a single scalar diffraction coefficient is involved. Near to shadow boundaries where the GTD solution is not valid, a uniform theory based on the Ansatz of Lewis, Boersma, and Ahluwalia is described. The dominant terms, to the order ofk^{-1/2}included, are used to compute the field exactly on the shadow boundaries. In contrast with the uniform theory of Kouyoumjian and Pathak, some extra terms occur: one depends on the edge curvature and wedge angle; another on the angular rate of change of the incident or reflected field at the point of observation.     

On the accuracy of physical optics

The first-order correction to the physical optics (PO) solution for the axial radiation field from parabolic reflector antennas is found. The correction is of orderk^{-1}and is in quadrature with the PO solution. By means of the correction term the accuracy of the PO surface integral is round to be better than0.22(lambda/D)^{2}dB on axis, whereDis the diameter of the reflector. Thus, the PO solution for the directivity is extremely accurate for commun reflector sizes of several wavelengthslambda.     

Diffraction by a circular aperture in a unidirectionally conducting screen

The diffraction of a normally incident plane electromagnetic wave with wave numberkby a circular aperture of radiusain a unidirectionally conducting plane screen of zero thickness and infinite extent is considered. In the limit of largeka, the ratio of the transmission cross section to the geometrical optics valuepi a^{2}, is found up to the order(ka)^{-3/2}.     

Electromagnetic reflection from a conducting surface: Geometrical optics solution

The reflection of electromagnetic waves from a smooth conducting surface at high frequencies is studied in this paper. Both the incident and reflected fields are expanded in asymptotic series of the formE = exp [iks(r)]Sigmamin{m=0}max{infty}(ik)^{-m}e_{m}(r), wheresis the phase function, and {e_{m}} are amplitude vectors. Explicit formulas based on a ray technique are given for calculating the first two orders of the electric field, magnetic field, and surface current. When the conducting surface is a paraboloid (or a sphere) and the incident field is a plane wave in its axial direction, our solution recovers the exact solution (or the first two orders of the exact asymptotic solution), As a special case, our result is compared with the work of Schensted. It appears that the latter is only partially correct.     

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.     

Diffraction of an arbitrary plane electromagnetic wave by a half-plane

The asymptotic form of the diffracted field produced by a plane electromagnetic wave of arbitrary polarization incident obliquely on a conducting half-plane is expressed very simply in terms of the incident and reflected wavesF_{1}andF_{2}, respectively, rotated to bring their wave vectorsk_{1}andk_{2}onkin the direction of observation. The diffraction coefficients becomepm(sin psi_{i}/2)^{-1}wherepsi_{i}is the angle betweenk_{i} (i = 1,2)and the direction of observationk.     

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.     

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.     

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.     

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.     

4C1--Nonlinear effects in an NMR experiment

We have studied multiple-photon transition in a low-field nuclear magnetic resonance (NMR) experiment. It is well known that a saturation of the NMR occurs when using a sufficiently high alternating magnetic field which induces multiple-photons transitions. We show theoretically that these multiphoton transitions are accompanied by an harmonic generation. For an-photon transition, this generation occurs mostly on thenth and (n pm 1)th harmonic. For an irradiating field2H_{1} cos omegat of pulsation ω such thatnomega= gamma H_{0}=omega_{0}where H0is the steady field and θ is the angle between H0and H1the amplitudesSmin{n}max{n}andSmin{n}max{n+1}of these components are given bySmin{n}max{n} = f_{n}(theta) (H_{1})^{n},Smin{n}max{n+1} = cos theta f_{n}(theta) (H_{1})^{n+1}. We have verified these equations in a low-field NMR experiment (H_{0} = 0.7Gs/s), polarizing first a flowing liquid in a high magnetic field. The liquid then flows in a modified Bloch spectrometer. The receiving coil is still perpendicular to the steady field but it is possible to adjust the angle θ between the axis of the emission coils (H1field) and the steady-field H0. Using a synchronous detection at the output of the receiving coil on the pulsation(n - 1) omega, nomega, or(n + 1) omega, we have directly detected multiphoton transistions. The above equations have been verified forn = 2ton = 5. There are slight discrepancies at high excitation which may be explained taking into account a large saturation.     

On optimal codes for binary asymmetric channels (Corresp.)

Let{ v_{1}, cdots , v_{N} }be a code ofNvectors whose components consist ofmones and(n - m)zeros. The vectors are transmitted with equal probability over the asymmetric channel whose error probabilities for transmitted zeros and ones equal respectively0andp, 0 < p < 1(such channels arise in optical communication). The problem of determining for fixed values of the parametersN, n, m, pthose codes which achieve minimum error probability is studied. The problem is solved forN leq 3. Partial results are obtained for allN.     

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.     

Threshold-voltage instability of n-channel MOSFET's under bias-temperature aging

New experimental evidence of positive threshold-voltage shift caused by interface state generation under positive bias-temperature (BT) aging is presented. Interface states were estimated for MOSFET's using low-frequency (8-Hz)C-Vmeasurement, which was carried out by a lock-in technique. Generated acceptor-type interface states are distributed between the midgap and the conduction-band edge in the forbidden gap. Time(t) and temperature(T) dependence for threshold-voltage shift (deltaV_{T}) is represented experimentally asdeltaV_{T}infin log (t/t_{0}), wheret_{0}^{-1} infin exp (-1.0 eV/kT). The positive VTshift appears faster for MOSFET's fabricated with dry O2oxides as gate insulator than for those with HCI oxides. It is also shown that the VTshift is always larger than the flat-band voltage shift caused by interface state generation under negative BT aging. Generated interface states are distributed in the entire forbidden gap, differing from the case of positive BT aging.     

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.     

The panchromatic principle in optical filtering

The panchromatic optical correlator is a coherent correlator which uses a slightly broadened light spectrum to correlate simultaneously a number of receiver channels against a continuous range of Doppler distortions of the transmitted signal. The relation between spatial frequencyomega_{x}and aperture distancexin a coherent optical correlator isomega_{x} = - 2 pi x/ lambda f, whereinlambdais the light wavelength andfis the lens focal length. Since signals reflected from a target moving with uniform radial velocity undergo a scale factor adjustment of the signal spectrum, the Doppler distortions resulting from target motion can be simulated by a range oflambda. The panchromatic correlator is actually a superposition of many monochromatic correlators with eachlambdacorresponding to a particular Doppler distortion. The correlator output can be color separated by a prism to resolve Doppler information, or left unresolved to provide a Doppler invariant correlation regardless of the invariance properties of the chosen signal. Inserting a diffraction grating in the proper aperture changes the relation above toomega_{x} = -2 pi x/ lambda f - omega_{h}, whereomega_{h}is a frequency shift. This form permits correlation against a Doppler-distorted signal, heterodyned an amountomega_{h}to use the time-bandwidth product of the correlator more efficiently.     

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.     

On certain projective geometry codes (Corresp.)

LetVbe an(n, k, d)binary projective geometry code withn = (q^{m}-1)/(q - 1), q = 2^{s}, andd geq [(q^{m-r}-1)/(q - 1)] + 1. This code isr-step majority-logic decodable. With reference to the GF(q^{m}) = {0, 1, alpha , alpha^{2} , cdots , alpha^{n(q-1)-1} }, the generator polynomialg(X), ofV, hasalpha^{nu}as a root if and only ifnuhas the formnu = i(q - 1)andmax_{0 leq l < s} W_{q}(2^{l} nu) leq (m - r - 1)(q - 1), whereW_{q}(x)indicates the weight of the radix-qrepresentation of the numberx. LetSbe the set of nonzero numbersnu, such thatalpha^{nu}is a root ofg(X). LetC_{1}, C_{2}, cdots, C_{nu}be the cyclotomic cosets such thatSis the union of these cosets. It is clear that the process of findingg(X)becomes simpler if we can find a representative from eachC_{i}, since we can then refer to a table, of irreducible factors, as given by, say, Peterson and Weldon. In this correspondence it was determined that the coset representatives for the cases ofm-r = 2, withs = 2, 3, andm-r=3, withs=2.