Contains research objectives, summary of research and reports on two research projects.Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DAAB07-71-C-030

Kyhl, R. L.

Raines, J. K.

Strandberg, Malcom W. P.

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II. MICROWAVE SPECTROSCOPYAcademic and Research StaffProf. M. W. P. StrandbergProf. R. L. KyhlJ. D. KiersteadGraduate StudentsJ. U. Free, Jr.J. K. RainesA. TachagumpuchRESEARCH OBJECTIVES AND SUMMARY OF RESEARCH1. Second-Harmonic SpectroscopyThis research is directed to the discovery and demonstration of useful nonlinearitiesin quantum-mechanical systems. The work includes both theoretical and experimentalinvestigation in the microwave region.It is fairly evident that one should be able to observe nonlinear effects in the inter-action of lasers with matter. A 1-MW laser can create an electric field a few volts perangstrom in matter. Since this is on the order of atomic fields, it is not surprising thatnonlinear interactions are observable in matter not possessing a center-of-inversionsymmetry. In contrast, it is certainly not evident that nonlinear effects can be observedfrom the interaction between matter and microwave sources of a few watts. Resonancebetween the field frequency and the energy levels of the matter, however, enhances thenonlinear effects by many orders of magnitude. Furthermore, if a static magnetic orelectric field is applied, centro-symmetric matter such as a gas can give rise toobservable nonlinear interaction. Since microwave wavelengths are of the order of thesample size, the problem of phase matching, which can be a limitation in the observablelaser nonlinear effect, is easily dealt with.That the nonlinear effect is not trivial compared with the linear effect may be seenfrom the following argument: In normal, linear spectroscopy at microwave frequencies,sample saturation ultimately limits the observable signal size. A small sample in acavity will be saturated with a power on the order of a microwatt. Since we can operateat a frequency of one-half the matter resonance frequency when studying resonant non-linear interactions, saturation can be avoided and a power level of watts may be usedto produce a signal. Under the proper conditions this signal can equal or surpass themaximum available absorption signal. Finally, since the signal is observed at a fre-quency twice that of the source field, the source noise does not introduce an additionallimitation to the sensitivity of the observation.A preliminary experiment has been done to evaluate empirically the sensitivity of asecond-harmonic EPR spectrometer.M. W. P. StrandbergReferences1. M. W. P. Strandberg, Quarterly Progress Report No. 107, Research Laboratoryof Electronics, M.I.T. , October 15, 1972, p. 3.This work is supported by the Joint Services Electronics Programs (U. S. Army,U. S. Navy, and U. S. Air Force) under Contract DAAB07 -71 -C -0300.QPR No. 108(II. MICROWAVE SPECTROSCOPY)2. Dielectric Waveguide CalculationsDielectric waveguides and resonators without metal walls can be used to advantagein microwave networks, The use of materials with very high dielectric constant (rutile,titanates, and tantalates, for example) reduces the size needed for elements in integratedcircuits. The addition of metal walls would in many cases increase losses and compli-cate manufacture.We are continuing to develop programs for calculating field patterns of rectilineardielectric waveguides with c E 0 . The case we are studying first is that of large aspectratio b >> a (thin slab in cross section). The first computer results for the lowest modesare now being checked. These results can be compared directly with those for the vanish-ingly thin dielectric ribbon which we have obtained previously. The work will be extendedto guides with more nearly square cross section and then to rectangular resonators.R. L. KyhlA. WORK COMPLETED1. LOW-TEMPERATURE BEHAVIOR OF RADIO-FREQUENCYSURFACE IMPEDANCE IN METALSJoint Services Electronics Programs (Contract DAAB07 -71 -C -0300)M. W. P. StrandbergThis research has been completed by R. M. Langdon, Jr., and submitted to the Depart-ment of Physics on August 1, 1972 in partial fulfillment of the requirements for the degreeof Doctor of Philosophy. A summary of the thesis follows.The surface impedance of a boundary between two media is defined and is shown toprovide a measure of the energy absorbed or stored in a medium. The behavior of thesurface impedance of a metal in the absence of a magnetic field as a function of frequencyand electron mean-free path is reviewed; "classical" and "anomalous" skin-effect condi-tions are contrasted. The addition of a magnetic field bends the conduction electrons intoorbits about the field direction, changing the "nonlocal conductivity" (current responseto a delta-function electric field) and affecting the surface impedance in a manner thatdepends on electron reflection at the surface. Some previous measurements and calcu-lations of magneto-surface impedance are briefly reviewed.The experimental apparatus was based around a Pound-Knight-Watkins oscillatorwith the sample included in its tank circuit. If the surface impedance of the sample ischanged by a magnetic field, the resulting changes in the oscillator RF amplitude areproportional to the real part of the impedance change. The samples were monocrystal-line wafers of gallium in a dewar of liquid He whose temperature was adjustable, bymeans of a vacuum pump, from -2' to above 12'K. A modulated de magnetic field wasapplied to the sample. Changes in the oscillator RF amplitude were either displayeddirectly in real time as a synchronized oscilloscope trace, or, for small modulationQPR No. 108(II. MICROWAVE SPEC TROSCOPY)excursions, were phase-detected and their amplitude recorded as a measure ofRe [dZ/dH].Two semianalytical mathematical methods are developed for calculating approxi-mately the magneto-surface impedance of a semi-infinite metal. We assume, in the firstcase, completely diffuse surface reflection, and in the second, reflection that is spec-ular for co-angles of incidence less than a critical 0c and diffuse for greater co-angles.For use in the first case a way is found to expand the nonlocal conductivity in theunbounded metal, and its Fourier transform, in powers of H 2 . The result, from a for-mula which assumes the conductivity to be independent of distance from the surface, isa magneto-surface resistance behaving as Z (1 + f 4/3I2) for large f. In the second casethe conductivity is approximated by a pure logarithm of the difference of the argumentswith an amplitude depending on the distance of one argument from the surface. Thisform is used in a variational expression that was solved numerically by computer. Theresults are qualitatively similar to those of a more exact, but completely numerical,published calculation, but depart quantitatively from them in the limit of the longmean-free path.We used the model for specular reflection to interpret the experimental results. Theposition of the minimum in R(H) observed at wRF = 5 MHz, H = several tens of oersteds,and T 2 - 5 K gives a value of 09 ~ 8' for gallium crystals grown in a mold of Lucitecoated with carbon black and -1 for crystals grown in an uncoated mold. The behaviorof the curvature of the Re [Z(H1)] signal vs temperature, however, is not in agreement withthe combined results of the approximate mathematical methods. Further exact numeri-cal calculations must be made to resolve the question of whether this discrepancy arisesout of errors inherent in the approximations or out of differences of gallium from thephysical model assumed. The frequency dependence of the minimum positions Hmmobserved in measurements from 850 kHz to 17 MHz tentatively confirms the predictionof the completely numerical calculation that H co_ c /3mmB. WORK IN PROGRESS1. PROPAGATING WAVES ON A HIGH DIELECTRIC SLABJoint Services Electronics Programs (Contract DAAB07-71-C-0300)J. K. RainesField solutions have been obtained for propagating waves in a slab of high dielectric(E/co - c) material, surrounded by free space. These are solutions of a more generalequation that describes propagating waves in a high dielectric waveguide of any crosssection.QPR No. 108(II. MICROWAVE SPECTROSCOPY)2W 27T z T~iL ; ; E;]K~bKkzET - VT(VT' ET)2+ 227whereET ET(rT)E= rET(rK = modified Bessel function.oThe boundary conditionn-E =0Tis also required for bound waves.E0 0Y/II////////////////E,p-,// 0a°!I/IIII///////////III//- I'T /rT/I//~// / /////////LLLL/Fig. II-1. Coordinate system for slab waveguide. Aspect ratio = a/b.Equations 1 and 2 determine the transverse electric field, and the remaining fieldsfollow uniquely from Maxwell's equationsQPR No. 108I1111 1 111111 11 1 II rl - - - ------ ~~~~~~~~~~Id2r ' V [Ko(kzR)V • P1 E ],T oz TTFig. II-2. Lowest mode symmetric in E .X Fig. II-3. Second mode symmetric in ExxHFig. II-4. Third mode symmetric in E .xQPR No. 108A n[×EvI vS-- MAGNETIC FIELD LINES-- - - ELECTRIC FIELD LINESFig. 1-5. Second symmetric mode of thin guide.HWIDTH OF SHEETFig. II-6. Second symmetric mode of ribbon.QPR No. 108(II. MICROWAVE SPECTROSCOPY)V EE = 0H=- VXE.For a rectangular cross section, the second term on the right side of Eq. 1 describessingularities in the magnetic fringing fields. As the cross section degenerates to a rib-bon, the singularities become logarithmically unbounded. 1 For the slab, the singulari-ties are less severe, and decrease as the wave number kz increases (Fig. II-1).For a slab with aspect ratio 10, Eq. 1 has been solved by using the method ofmoments. 2 The trial functions are Legendre polynomials multiplied by (1 -x 2 ) or (1 -y2)so that Eq. 2 is satisfied identically. These functions proved a fortunate choice. Thefields for the first mode are adequately represented by only three terms, and for thethird mode (symmetric in Ex) only six terms are required.Equation 1 describes an eigenvalue problem. That is, for a given propagation num-ber, solutions exist only for discrete c. Thus in solving for the fields, a dispersioncurve is also obtained. Work continues to obtain solutions for any aspect ratio.Some sample results for 3 modes are shown in Figs. II-2 through 1I-4. In all casesthe aspect ratio is 10:1, and kzab = 1. Figure II-5 is a 3-dimensional sketch ofFig. 11-3. Figure II-6 shows H for the ribbon mode (k = 0) corresponding to the secondy zmode in Fig. 11-3.References1. R. L. Kyhl, "Electromagnetic Waves on High Dielectric Constant Ribbon," QuarterlyProgress Report No. 99, Research Laboratory of Electronics, M. I. T. , October 15,1970, pp. 3-5.2. R. F. Harrington, Field Computation by Method of Moments (The Macmillan Co.,New York, 1968).QPR No. 108

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