Contains research objectives and reports on two research projects.Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E)National Science Foundation (Grant GK-3370

Chu, L. J.

Glaser, J. I.

Granek, H.

Haus, Hermann A.

Kong, Jin Au

Penfield, Paul L., Jr.

DSpace@MIT

XVII. ELECTRODYNAMICS OF MEDIA*Academic and Research StaffProf. L. J. Chu Prof. J. I. Glaser L. FrenkelProf. H. A. Haus Prof. J. A. Kong T. E. SullivanProf. P. Penfield, Jr.Graduate StudentsB. L. Diamond E. R. Kellet, Jr. E. E. Stark, Jr.H. Granek E. F. McCann L-h. WangP. W. Hoff C. M. WatsonRESEARCH OBJECTIVES1. Study of the relativistic theory of quadrupolar media has led to an asymmetricenergy momentum tensor. Asymmetry of the energy momentum tensor of a closed sys-tem poses questions concerning the appropriateness of fundamental postulates under-lying some formulations of electrodynamics of media. Further study will concentrateon the symmetry problem.Also under study are problems of interaction of both electromagnetic and acousticwaves with moving media, and quantization of electromagnetic fields in material mediain the presence of a charged particle. These problems are relevant to the phenomenaof Brillouin scattering, Cerenkov radiation, and recent cascade shower experiments.2. Studies of the nonlinear interactions in CO2 lasers are concerned with cross-relaxation phenomena and amplification of cavity-dumped pulses shorter than the inverselinewidth of the CO 2 amplifier. Cross relaxation among different rotational levels willbe studied in its effect of preventing hole burning and the Lamb dip. The study of short-pulse amplification will be refined through elimination of the cw background from thecavity-dumped pulse and increased power output through pulsing of the discharge.3. Interest in quantum noise of lasers has been renewed by the expectation that neweffects should be observable in a CO 2 laser near threshold. Quantum analyses are inprogress to treat systems in which the noise spectrum is comparable in width with therelaxation times of the medium.L. J. Chu, H. A. Haus, J. A. Kong, P. Penfield, Jr.A. AN EXPERIMENTAL DETERMINATION OF CROSSRELAXATION IN CO 2Experiments are in progress to study the relative strength of the various relaxationprocesses that tend to reduce the saturation of a given rotational level in the CO 2 laser,namely, the relaxation of the velocity groups across the linewidth, the relaxation of thepopulation into the other rotational levels, and the net relaxation out of the vibrationallevel, dominated by vibrational-vibrational coupling. In this experiment (see Fig. XVII- 1)*This work was supported by the Joint Services Electronics Programs (U. S. Army,U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E).QPR No. 96(XVII. ELECTRODYNAMICSMECHANICAL OSCILLATOR TO SYNCHRONOUS OSCILLATORTURNER AMPLIFIERTO X - YRECORDERSATURATING SPECTROMETERABSORBER PROBEDETECTORX INPUT FROM FREOUENCY REFERENCERAMP VOLTAGE FROM CHOPPERFig. XVII-1. Schematic of experiment probing the rotational lines of CO 2 .a saturating CO 2 laser tuned to a 10. 6-i R line is chopped (chopper in position 1) at alow frequency and modulates the population of a laser amplifier. A low-intensity beamcolinear with the saturating beam is used to probe the effects of the modulation on thegain or absorption of other lasing lines of the amplifier. The probe beam is the outputof a piezoelectrically tuned oscillator that can be swept across the linewidth of manyP transitions. (The sweep of the piezoelectric oscillator has been calibrated in fre-quency by beating this laser against a fixed laser on the same line.) The probe beam isseparated beyond the amplifier by a grating, and after detection is synchronously narrow-band amplified at the chopper frequency. Thus only the effect on the gain of the probelines through modulation of the amplifier population is observed. A sample output isshown in Fig. XVII-2b for gain (discharged gas) and XVII-2c (cold gas). With the chopperin position 2, the saturating beam blocked, and the discharge on and off we can producegain curves for the probe line, as shown in Fig. XVII-2a.For a saturating line of R (20) used in the present experiments, all of the P transi-tions can be affected by the radiation field only through the rotational cross coupling, withthe exception of P (22), which shares a common upper level with R (20), and P (20),which shares a common lower level. With the discharge turned on in the amplifier, theradiation field attempts to produce a Lamb dip in the upper lasing level; however,because of velocity cross relaxation, the entire level will tend to be pulled down to someextent and, because of rotational relaxation, other lines can show either the Lamb dip,or the over-all pull-down, or both. Likewise, in this case the lower levels areQPR No. 96OF MEDA)(XVII. ELECTRODYNAMICS OF MEDIA)4x102SCALE COMPARED WITH (Ib) AND (Ic)I MHzI MHzFig. XVII-2. (a)(b)(c)Gain measurement of probe line P(12),0. 6 Torr N2, 0. 6 Torr CO .Change in absorption on probe line P(12),saturating line R(20).Gain measurement of probe line P(12),0. 6 Torr N2, 0. 6 Torr CO 2 .correspondingly filled because of the radiation field. Without the discharge, the gas inthe amplifier provides absorption, and the saturating laser pumps population into theupper level, thereby reducing absorption on all levels. The process is controlled bythese same parameters that depend upon the relaxation times which now, with a dis-charge, assume different values.In a previous report, the cross relaxation applicable to a multilevel upperQPR No. 96---- ---------~--- -- ---- ~ IELa~ I I __---_ _ ___ ___(XVII. ELECTRODYNAMICS OF MEDIA)vibrational state with an entirely depleted lower vibrational state is treated. A simpleextension of this approach to include the lower level includes the cases for both gain andabsorption:N (e) N(e)2 ) N f(v) f W(v) f(v) dvAn g = - - + W(v) (1)2 f 1 2  2 2N(e) N(e) N (e)J2 2 f(v) f W(v) f(v) dv A2/z 2An- ,, + W(v) (2)J 2 (e) g f 9 g Y? Y2 T22 N y2 + 1/T2 + A2/z22where1 TS- (3)' (22y + 2yz+- y+ +1 T z z' (4)S(,+A) y + +T Z1 (T A (5)with similar solutions for the lower level with a change in sign. From initial measure-ments we have determined that the relative pull-downs are J-independent for 0. 6 TorrCO 2 and 0. 6 Torr N 2.At this pressure there was no Lamb dip; hence, we may assume that the second termin brackets in Eqs. 1 and 2 is negligible compared with the first term.If we look at the ratio of change in gain to gain on the lasing line to that on thenonlasing lines, we getT2 y2 T + 1 = 4 (experimentally).2 A 1 +z 2+-+AZTQPR No. 96(XVII. ELECTRODYNAMICS OF MEDIA)We must have - > A in order to see this effect where y might be smaller than eitherTfrom physical arguments, and z can be calculated from equilibrium population to beapproximately 14. If we take the equality between I/T and y, we find_ _ 1=3 or y A.2A/zFurthermore, we know that y cannot be too small compared with A because the systemwould not have been able to reach a new equilibrium in which the changes in the lasinglevel are appreciably different from changes in the other lines. An appropriate value-1of A has been given as ~107 sec- for -1 Torr from the work by Cheo.3 Also, no Lambdip was observed, even at 0. 3 Torr CO 2 and 0. 3 Torr N2, in this amplifier experiment.H. GranekReferences1. H. A. Haus and P. W. Hoff, "Cross Relaxation in a Multilevel System," QuarterlyProgress Report No. 94, Research Laboratory of Electronics, M. I. T. , July 15,1969, pp. 80-85.2. See graphs (Fig. V-i) in H. A. Haus and P. W. Hoff, "Effect of Cross Relaxationon Lamb Dip," Quarterly Progress Report No. 93, Research Laboratory of Elec-tronics, M. I. T. , April 15, 1969, pp. 43-48.3. P. K. Cheo and R. L. Abrams, "Rotational Relaxation Rate of CO 2 Laser Levels,"Appl. Phys. Letters 14, 47 (1969).B. SOLUTION OF WAVEGUIDE SCATTERING PROBLEMS BYFINITE-DIFFERENCE GREEN'S DYADICS1. IntroductionThis report deals with a finite-difference Green's dyadic method for the numericalcalculation of the induced current distribution and equivalent circuit for time-harmonicTM modes in a two-dimensional parallel-plate waveguide impinging on a class of metallicobstacles. The class of obstacles is that with obstacle sides either parallel or perpen-dicular to the waveguide axis. In a previous report the finite-difference method wasused to solve TE mode scattering problems in parallel-plate waveguide. The TM prob-lem involves a Green's dyadic rather than a scalar Green's function because inducedcurrents flow in two dimensions rather than in one.The equivalent susceptance of a transverse semidiaphragm that was computed byThis work was supported in part by the National Science Foundation (GrantGK-3370).QPR No. 96(XVII. ELECTRODYNAMICS OF MEDIA)using a 50-element approximation for the induced current distribution is 2. 06% greaterthan the exact value; this computation takes 1. 98 min on the IBM 360/65 computer, andcosts $6. 62. This report gives a formulation of the finite-difference Green's dyadic forTM modes and the finite-difference boundary-value problem. The round-off error,sampling error, and computation costs vs number of current elements and the inducedcurrent distribution for the semidiaphragm are presented.2. Finite-Difference Green's DyadicAs described in a previous report, the time-harmonic Maxwell's equations can beapproximated by the method of finite differences, so that each field and current com-ponent is evaluated at a different point of a cubic lattice. 2 The finite-difference Green'sdyadic for TM modes in a parallel-plate waveguide is defined as the electric field solu-tion to the finite-difference Maxwell's equations, subject to boundary conditions at theparallel plates for a discrete electric current distribution that is nonzero only at onepoint. The geometry of the parallel-plate waveguide and the lattice points of interestI E AND J POINTO- E AND J POINTz z* H POINTYa KFig. XVII-3. Geometry of lattice and parallel-plate waveguide.QPR No. 96(XVII. ELECTRODYNAMICS OF MEDIA)is two-dimensional, and is shown in Fig. XVII-3. The coordinates x, y, and z areCartesian, and the field and current components are independent of y. The complexmagnetic field component of interest is H (x, z) and the associated complex electricfield components are E x(x, z) and E (x, z); the complex electric current densities arex zjtgiven by Jx (x, z) and Jz(x, z). The time factor e , where co is the angular frequency,will be omitted. Arrows indicate the direction of the electric field associated with eachlattice point, and the numbering scheme is such that each lattice point is labeled by twointegers, p and r.Ex(p, r) = Ex(x, Z) x= (p+ 1 / 2 )az z=raEz(p, r) = Ez(x, z)I =paz= (p+1/2)aH y(p, r) = H (x, z) x=(p+1/2)a (1)z=(p+l/2)aThe electric current density components are numbered exactly as the electric field com-ponents. The finite-difference form of Maxwell's equations is given byE (p, r) - E (p+l, r) + Ex (p, r+l) - Ex (p, r) = -j aH y(p, r) (2)-H (p, r) + H (p, r-1) = joE aE (p, r) + aJ (p, r) (3)-H (p-1, r) + H y(p, r) = jw aE (p, r) + aJ (p, r), (4)where a is the length of the side of one cell, Eo is the permittivity of free space, and[o is the permeability of free space. The boundary conditions at the parallel plates areTable XVII-1. Comparison of finite-difference TM modes and exact TM modes.Finite-Difference TM Mode Exact TM ModeHy cos s p+ -/M e s cos (mwTx/L) e-NZEx (e S-1)/jwEo cos sw p+ /M) e (y/j.o) cos (mwrx/L) e-YZ-wrEz -(2 sin(sT/2M)/jE 0o) sin (slTp/M) e s -((mT/L)/jo ) sin(mx/L) e - y zw = j2 sin-1 ((ka/2)2 sin 2 (sT/2M)) = j k 2 - (m/L) 2s = 0, 1,2,...M-1 m = 0, 1,2,...QPR No. 96(XVII. ELECTRODYNAMICS OF MEDIA)obtained by choosing the origin of the lattice so that the Ez points fall along the plateboundaries.E (p, r) = 0 for p = 0 and p = M.Z ' (5)The finite-difference TM modes are solutions to Eqs. 2-5 with J = 0 and J = 0. Thesex zmodes are compared with the exact TM modes in Table XVII-1. Here, k is the propa-gation constant of free space and L = Ma. The properties of these modes have beendiscussed elsewhere. 3Green's dyadic has 4 terms, Gxx Gxz, Gzx G defined byEx(p, r) - Gxx(P, r, px, rx) Jx + Gxz(p, r, Pz, rz) JzE (p, r) Gzx(p, r, Px rx )x + Gzz(p, r, Pz9 r )Jzz '  x'z  Z z(6)(7)where (Px, rx) denotes the location of an x-directed current density element of value jx'and (Pz, rz) denotes the location of a z-directed current density element of value j . Thedyadic is obtained by solving Eqs. 2-5, with J and J given byx zJ = 6 6x x ppx rrxJ = j 6 6z zPP rrzz z(8)(9)where 6 denotes a Kronecker delta function. The solutions arefollowing completeness relations for the modess=M-16 = (2/M) E cos sp+ /Ms=0s=M-16 PP =(2/M)s=Oobtained by using thecos (sTr(px+ 1 /Msin (sTrp/M) sin (sTrpz/M),1where E = for s = 0 and 1 for s # 0.s 2G = (-1/jwE )6 6 + ( 'xx o pp rr wE MIx 0/s=M-1s=Ocos s )) coss 2 sinh w(2us(r-rx)-us(r-rx-1)-u s(r-rx+1))QPR No. 96(10)(11)(12)(S Tr ( M ))s= -, 2 / 1 o= sin(ST') Cos(XVII. ELECTRODYNAMICS OF MEDIA)(s (P+ )) sin (sipzM MG =G (. 1 (2xz jwE MG . 1 2G = 0 (Ozx jeoLs=ls=M-ls=lsinh wssin sin ) Cos (s( 12M nMsinh ws(13)(u (r-r x)-u s(r-rx +1))s x s x(14)G = (-i/jwco)6pp 6 + (2~ o)()s=M- 2 sin 2s=1sin sin zu s(r-r z ),sinh w szswhere us is given byUs(r) = exp(-ws r ). (16)Green's dyadic has been calculated for the x-directed current element shown inFig. XVII-4. Here, G xx/Z and G x/Z , where Zo is the characteristic impedance offree space, are given for 10 different points. The frequency and dimensions have beenchosen so that 2 modes are above cutoff.3. Finite-Difference Boundary-Value Problem SolutionThe finite-difference solution to the problem of determining the current distributioninduced on a metallic obstacle in a parallel-plate waveguide by a TM field is obtainedin three steps.1. The current distribution on the surface of the obstacle is approximated by a setof x- and z-directed current elements having unknown amplitudes.2. The Green's dyadic is used to compute the electric field produced at each currentelement location, under the assumption that each current element is excited one at atime.3. The boundary condition that the total electric field, incident plus scattered, van-ishes at each current-element location yields a set of simultaneous linear equations forthe current amplitudes.The x-directed current-element locations will be labeled by the coordinates (Pi, ri)'the current amplitudes by ji, and the value of the incident x-directed electric field atthat element by E(i), where i is an index running from 1 to Nx9 the number of x-directedQPR No. 96(15)k U s 1-1 z )-U s I-I Z_ -1/1(XVII. ELECTRODYNAMICS OF MEDIA)p(x)M = 10054 L =0.9 INCH3 X/L= 1.121 EXCITED ELEMENT6 78910CENTER OFWAVEGUIDE0 5 1 r (z)0 5 101 2 3 4 52.0x1.0x4.0x3.0x2.0x1.0x 1-1.0x 104.0x 103.0x 102.0x 10-1.0x 10-6 7 8 9 10ELEMENT NUMBERFig. XVII-4.6 7 8 9 10ELEMENT NUMBERGreen' s dyadic for x-directed current element.current elements. The same labels will be used for the corresponding z-directed vari-ables with i running from N + 1 to N + N , where N is the number of z-directed cur-rent elements. The scattered field produced at each current element location, e(i), isthen related to the current amplitudes byN +Nx ze(i) = Y gikjk' (17)k=lwhereGxx(Pi, ri' Pk' rk)G xz(Pi'., ri' Pk rk)Gzxz (Pi, ri' k'rk)Gzz(Pi' ri' Pk' rk)1< i, k < N1<i< Nx; N + 1 k< N +NSx zN + 1 < i, k, <N + NX X ZQPR No. 96(18)(XVII. ELECTRODYNAMICS OF MEDIA)The boundary condition at the current elements is given byE(i) + e(i) = 0 1 i< N +Nxand yields the following set of Nx + N simultaneous equations:N +Nx zI gikk + e(i) = 0.k=lSolution to Eq. 20 for jk canelectric field of the scatteredexpressions.(TEM) E x =xI-rAe -w r+w rewhereA = __-1 oi1A A jwE M=( i-) ( -)j Eo)(I- -Ibe accomplished by Gaussian elimination. The x-directedTEM (s = 0) mode can then be computed from the followingr > r.1r < r.1Nx rkw°tanh (wo/2) Z jk ek=lNx -rkw °tanh (w /2) jk ek=14. Practical Limitations of the MethodThe practical limitations on the use of the method described here are sampling ordiscretization errors inherent in the finite-difference approximation, computation round-off error, and computation cost or time. The scattering of a TEM mode from a trans-verse semidiaphragm (capacitive iris) was solved in order to evaluate these limitations.The induced current distribution was obtained from Eq. 20 with N = 50 and N = 0 byx zGaussian elimination, and is shown in Fig. XVII-5. The incident electric field strengthwas +377 V/m at r = 0, M = 100, and X/L = 2. 1, where k is the free-space wavelength.The accuracy of the computation was tested by recomputing the scattered field at eachcurrent element location; these residual-field errors are also plotted in Fig. XVII-5The equivalent susceptance of the transverse semidiaphragm was computed and comparedwith a value computed to 6-place accuracy by means of an exact series given by Collin. 4These values are plotted against wavelength in Fig. XVII-6a. The discretizationerror, which is defined as the percentage difference between the computed and exactQPR No. 96(19)(20)(21)(22)TRANSVERSE SEMIDIAPHRAGMREAL PART OF JIMAG. PART OF J0LYZ0.03S0.02 r,0.010.10 0.20 0.30 0.40 0.50POSITION x/LFig. XVII-5.Ncaomuu 3.0zLU2.0z> 1.0aInduced current distribution and residual fielderror of transverse semidiaphragm.FINITE-DIFFERENCE VALUE FOR50 CURRENT ELEMENTSEXACT VALUE-I I I I I I2.0 2.2 2.4 2.6 2.8 3.0X/L(a)z0.0300.02 cu.O0.01 oZ010 20 30 40Nx NUMBER OF CURRENT ELEMENTSFig. XVII-6. (a) Computed and exact equivalent susceptance oftransverse semidiaphragm vs wavelength.(b) Discretization error, round-off error and com-putation costs vs number of current elements.QPR No. 962.0x1041.0x104(XVII. ELECTRODYNAMICS OF MEDIA)susceptance, is plotted in Fig. XVII-6b against number of current elements. The com-putation costs and the round-off error, which is defined as the maximum residual-fielderror, are also plotted against number of current elements in Fig. XVII-6b. The costof computation on the IBM 360/65 computer was $3. 34 per 450K byte-min for central-processing unit time.J. I. GlaserReferences1. J. I. Glaser, "Solution of Waveguide Obstacle Problems by a Finite-DifferenceMethod," Quarterly Progress Report No. 93, Research Laboratory of Electronics,M. I. T. , April 15, 1969, pp. 49-65.2. J. I. Glaser, "Maxwell's Equations for Discrete Space," Quarterly Progress ReportNo. 92, Research Laboratory of Electronics, M. I. T., January 15, 1969, pp. 211-219.3. Ibid. , pp. 218-219.4. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill Book Company, New Yorkand London, 1960), p. 452, Prob. 10. 6.QPR No. 96

Electrodynamics of Media

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Electrodynamics of Media

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