PROCEEDINGS OF THE IRE

The Excitation of Circular Polarizationin Microwave Cavities *

M. TINKHAMt AND M. W. P. STRANDBERGt, SENIOR MEMBER, IRE

Summary-The usefulness of exciting a single circularly-polar-ized mode in a microwave cavity is indicated. A matrix method is pre-sented then for the analysis of the operation of various components,such as transition pieces and differential phase shifters on wavespropagating in waveguides with two degenerate orthogonal modes.This method is applied to describe three distinct systems for gener-ating the desired circular polarization in a cavity that is coupleddirectly to the side wall of a waveguide. The final system, which in-volves a minmum of critical adjustments, is described in more de-tail, and its performance is indicated. It is shown that purely circularradiation can be set up along the axis of the cavity. However, thelargest preponderance of one rotating mode over the other which canbe set up, if one averages over the entire cavity volume, is 11.5 to 1.This value is obtained with the TE,,,, modes.

INTRODUCTION

A NUMBER OF microwave experiments and appli-cations require, or would at least be facilitatedby, the excitation of a single circularly-polar-

ized mode in a resonant cavity. Examples from existingwork are (1) determination of AMT selection rules inmicrowave paramagnetic resonance transitions,' and(2) measurement of the complex permeability tensor offerrite materials as a function of the static magneticfield.2 A possible future application would be in anoscillator magnetically tuned with a ferrite-loadedcavity. In all such applications, it is necessary to applya magnetic field of the order of kilogauss along the axisof symmetry of the cavity. Since this requires that thecavity be placed between the polepieces of a magnet, asimple end-on feed to the cavity, from a waveguide inwhich a circularly-polarized wave is propagating, is im-possible unless a hole is bored along the axis of thepolepiece, or unless the gap is made wide enough forbends which do not destroy the circularity of incidentradiation. These considerations indicate the desirabilityof a system for exciting a single circularly-polarizedmode by coupling directly from the side of a piece ofwaveguide which is inserted in the gap. Several suchsystems are described in this paper.

* Original manuscript received by the IRE, October 28, 1954; re-vised manuscript received, January 21, 1955. This work was sup-ported in part by the Signal Corps, the Office of Scientific Research,Air Research and Development Command, and the Office of NavalResearch.

f Dept. of Physics and Research Lab. Electronics, Mass. Inst.Tech., Cambridge, Mass.

' M. Tinkham and M. W. P. Strandberg, "Theory of the finestructure of the molecular oxygen ground state and the measurementof its paramagnetic spectrum," Phys. Rev. (in press). More details aregiven in the Ph.D. Dissertation of M. Tinkham, Dept. of Physics,MIT; 1954.

2 J. 0. Artman and P. E. Tannenwald, "Measurement of per-meability tensor in ferrites," Phys. Rev., vol. 91, p. 1014; 1953. H. G.Beljers and J. L. Snoek, "Gyromagnetic phenomena occurring withferrites," Phillips Tech. Rev., vol. 11, pp. 313-322; 1950.

The essential problem is that of establishing a purelyrotating microwave H field at a point on the wall of thewaveguide. If this is done, a single rotating mode maybe excited in a cylindrical cavity with two degeneratemodes in the following manner. The cavity is attachedto the wall of the waveguide in such a position that thecenter of an end wall lies at the point where the micro-wave field is rotating in the guide, and coupling isaccomplished by means of a circular hole in the wall atthat point. Our theoretical treatment is to first order,that is, we neglect effect of aperture on waves in theguide. In this approximation electric field must benormal to the wall, and hence it could not be used toprovide a rotating electric field in the plane of the hole.To demonstrate the operation of the systems to be

described and to show how other such systems may bestudied, we first present a matrix algebraic techniquefor concisely analyzing the behavior of microwaves in a(square or round) waveguide with two degenerate andorthogonal modes of propagation. It is well-known thatwaves that are circularly polarized about the axis ofthe guide may be set up in such a waveguide by simplyretarding the phase of one component of a wave byir/2 with respect to an equal orthogonal component.3Our problem of establishing at the wall a field circularlypolarized about an axis normal to the wall is sufficientlycomplex to justify setting up this formal technique forquantitative analysis of various configurations sug-gested by more qualitative considerations.

MATRIX METHOD FOR BIMODAL WAVEGUIDEIn this method we follow the scattering matrix ap-

proach.4 The fields are expanded in terms of wavestraveling along the guide (z direction) and polarizedalong x and y, respectively. Suppressing the universaltransverse spatial dependence and the eiwt time de-pendence, we have

4 = e-t z->

EII = e- Ou X, (1)

where the ii's are unit vectors, ,B is the propagation con-stant 2ir/X,, and z is the distance along the guide fromany convenient reference plane. The waves propagating

2 J. R. Eshbach and M. W. P. Strandberg, 'Apparatus for Zee-man effect measurements on microwave spectra," Rev. Sci. Instr.,vol. 23, p. 623; 1952.

4 C. G. Montgomery, R. H. Dicke, and E. M. Purcell, "Principlesof Microwave Circuits," MIT Radiation Lab. Ser., McGraw-HillBook Co., Inc., New York, N. Y., especially ch. 5 and 10; 1948. Asdiscussed further, our conventions in setting up the matrixes differfrom those used in this reference.

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Tinkham and Strandberg: The Excitation of Circular Polarization in Microwave Cavities

in the reverse (left) direction would then be given by

E=1=EII and E' = EVI. Thus the field in any regionof undistorted guide may be expressed by four complexnumbers giving the amplitude and phase of these fourwaves. These numbers are conveniently collected in acolumn vector, with their order defined as follows:

CIICii

Ciii

Civ

Vectors in various regions of guide will. be related bymatrix operators that represent various types of ele-ments introduced. These matrixes are defined to have theproperty they yield the coefficients of the four outgoingwaves when matrix is multiplied into a column vectorcomposed of coefficients of incoming waves from bothsides. To illustrate, if the situation is

-El--

( a

bc

d ) (I

ef

9h )

the matrix of A must satisfy the matrix equation

A

The matrixes of various devices are readily tabulatedby considering simple cases. To handle a complicatedsystem, one then sets up a group of these equations,eliminates the undesired unknowns which describe thewaves inside the unit, and one is left with a new matrixwhich describes the more complex system.

Let us now list a few typical examples. If we introducean element Ef which produces a phase shift 0 and atransmission factor a (depending on the mode), but noreflection or mode conversion, the matrix is simply

conductive, or lossy vanes into the guide in one or theother of the planes of polarization. The quarter-wavepipe described4 for generating a circularly polarizedtraveling wave is an example of such an element withki-4II =wr/2.A short 0J converts waves propagating to the right

to ones propagating to the left with a phase dependenton the location of the short. Since the two polarizationsmay be shorted independently by orthogonal vanes atz, and Zii, respectively, the general matrix is

o 0 ei2OzI 0

o 0 0 -ei Mzj1

_e-i2#zl 0 0 0

O -e-i2tzI 0 0

The factor 3 is here the same for both modes, since weare considering the case with degenerate modes.

If we introduce a "Babinet compensator" [ (in theform of a section of guide squeezed along the diagonal,6we may consider the wave expanded in terms of t and7 components along the diagonals at 45 degrees to thex, y waves. These waves now propagate at differentspeeds because of the distortion that lifts the de-generacy. At the end of the distorted section having alength 1, we re-expand in the original basic waves.Neglecting reflection and the mean phase shift, exp[i2(13+i3)l], the matrix is

cos isin a 0 0 1

isin a cos5 0 0B =

O 0 cosS isin 6o 0 isin 6 cos5

where a is half the differential phase shift 2(i,-3i)l;e.g., 6 = r/4 for a quarter-wave plate.

If we consider an ideal transition piece M whichcouples between two pieces of guide with axes rotated byan angle 0, the matrix is easily seen to be

cos 0

-sin 0T=

0

l O

sinO 0 0

cosO 0 0

0 cos 0 -sin0 sin 0 cos

'00

axi]

0 0 0

le'~iOi 0 0

0 aIetoI 0

0 0 alleii II

We note that this matrix reduces properly to the unitmatrix in the limit of no attenuation or phase shift.5Such a device M may be made by inserting dielectric,

This desirable reduction does not hold.with the conventins .usedby Montgomery, et al. (reference 4), since their matrixes differ fromours by an exchange of the upper and lower two rows. A more impor-tant advantage of our convention is that, by using wave coefficientsrather than terminal voltages as the basis for the matrixes, we obtaina cleaner representation in the guide of the fields which are actuallycoupling into the cavity.

where again we neglect reflection. The turnstile couplerdiscussed in a later section is a special form of such adevice, with 0= +45 degrees and -45 degrees foi thetwo side arms. This matrix also represents the effect ofrotating. the reference axes in circular waveguidethrough an angle 0.

APPLICATION OF THE METHOD TO CIRCULARPOLARIZATION

In this section we shall give three arrangements inwhich the components described above may be used toprovide the proper phase and amplitude relations to

6 For example, Eshbach and Strandberg, loc. cit.

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excite circularly-polarized radiation in a cavity coupledthrough a hole in the sidewall of the waveguide.The first arrangement one might try in analogy to the

problem of generating circular polarization in wave-guides3 would be a Babinet compensator set as a quar-ter-wave plate with a movable short at the end of theline to aid in matching into the cavity. Assuming thatpurely vertically-polarized (I) radiation is incident, andthat the two polarizations are shorted at the same point,ZI=ziI =z0, our matrix method yields the fields indicatedby the column vectors:

cavity4R F;

One would expect to obtain less critical performancewith the addition of a differential phase shifter J toachieve independent phase and amplitude control. Thisarrangement is

covity

=~~~00 f bXa(x (I I)0

-i2zoz~~ ~ ~ ~ ~ ~ ~ -

-ie - ie

We note that purely horizontally-polarized (IV) radi-ation is reflected. From the vector at the right, we may

then construct the following z-dependence of the electricfield in the region where the cavity is coupled:

E = \/2ei#to sin f(z- zo)(ux - iu5).From this expression, the z-dependence ofH at the wallis easily shown to be proportional to

H = e-zo[¢ sin l(z - z)u - i cos 3(z - zo)u7T], (2)

where P= (H.)max/ (Hx;)max. For this to be purely circular,we then must choose the position of the short (zo) withrespect to the coupling hole (z) so that cot j3(z zo) = P.

Reference to the forms of the waveguide modes7 showsthat =X0/k for square waveguide, and 1.841 X,/k, forcircular waveguide (operating in the lowest mode inboth cases). It is also useful to note that

Xg/Xc = [(XC/X)2_ 1-1/2

where the cut-off wavelength X, is given by 2a for rec-

tangular waveguide (a is the width) and by 3.41R forcircular waveguide (R is the radius). Pertinent exam-

ples are that r=1 for 0.90X0.90 inch (I.D.) squareguide at 9,300 mc and =2.38 for 0.94 inch (I.D.) circu-lar guide at the same frequency. Thus for these typicaldimensions, the short should be distant by an oddmultiple of (X,/8) from the hole with the square guide,and displaced approximately (X,/16) from these pointsfor the circular guide. Although this method was suc-

cessfully used in an early model, it suffers from the dis-advantage that the coupling fields at the hole have only1/V.2 their maximum values and are therefore changingrapidly with position. This makes the positioning forexact circularity critical.

7 H. R. L. Lamont, "Wave Guides," Methuen and Co., Ltd., Lon-don, Eng.; 1949.

For a quantitative investigation, we apply the matrixmethod, assuming vertically polarized incident radi-ation but leaving all other parameters general. Afteralgebraic elimination of the unwanted variables efgh, wefind that the field in the region of the cavity is given by

a

b

c

d

e;+

- ei(4r-2#z1)- iei(4ir-2izni)

cos 6

sin a

cos a

sin a

From these coefficients we may construct the E and Hfields as in the previous example. The result for H is

H = r sin a sin ,B(z - zII)e (Oir Oz)U,- i cos a cosf(z - ()3ei(z)X.

The strength of the matrix method is the ease withwhich it yields these general expressions which allow thefull capabilities of an arrangement to be seen at once.Inspecting (3) we see that the conditions for circularityare

XiI - = 4'r = /3ZI

and

r sin a sin ,B(z - ZII) = coS cos O(z - ZI).

These may be satisfied by choosing i- -=+ 7r/2,ZI-Z += ±X/4, and a= ± cot-''. This choice allows thecoupling window to be at the maxima of both couplingfields. The positioning is noncritical and a smallercoupling hole may be used. To carry out this choice ofparameters, the two modes I and I I are shorted aquarter-wave apart by orthogonal vanes, and the twovaried together to the point of strongest coupling. Thedegree of phase shift required in the squeeze section Bdepends on the ratio t. For our example of the squareguide with =1, 5= 7r/4, and the device acts as aquarter-wave plate. For the circular guide example,however, 5= ± 22.8 degrees. In either case, the majorfraction of the reflected power, y2 = (2¢/1 +¢2)2, is in theIV mode. An apparatus of this type also was successfullytested, but the problem of coupling signal out of IV modeand adjustment of jm and [(P are troublesome.

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Tinkham and Strandberg: The Excitation of Circular Polarization in Microwave Cavities

With the experience now obtained it will be realizedthat, essentially, one desires independent amplitude andphase control of the two modes. Simple amplitude con-trol is obtained from a rotation section Fl. Theorthogonal shorting vanes used in the second case(above) provide a simple method of phase control. Theseconsiderations lead to the final model, which may besymbolized as shown*

cavity1)~~~~~~~~~~( x

y/

(ab

d/

Following the same procedure as above, we find

H = r sin 0 sin #(z -zI)e-u.- cos 0 cos ,B(z - zl)e-0Iuz. (4)

The conditions for circularity are zI-zI = ±X,/4, ando = cot-' . The shorts are located so that z -z= z -zi±X,/4=nX5/2 at the hole, giving maximum and non-critical coupling. For these settings, the values of |x|and jyI are (1-¢2)/(1+r2) and 2¢/(1+%2), respec-tively. Thus for r near unity, the greater share of thereflected signal is again in the IV mode. This is noproblem, however, with the turnstile transition couplerof the sort to be described. Also, the settings of 0 andzI-zII are simple mechanical properties, and the moresubtle and troublesome squeeze and phase shift settingof the previously described method are eliminated.

Fig. 1-Schematic diagram of transition coupler forcircular polarization.

THE FINAL DEVICE

In the previous section we have used a matrix formal-ism to indicate several methods for providing circularexcitation to a cavity. We now describe the final device,shown in Figs. 1 and 2, in a more concrete manner andindicate its operation in nonmathematical terms for thesimple case of a square guide with r = 1.The incident wave propagates down the square guide,

polarized along a diagonal. At the far end, the verticalcomponent is shorted by a conducting vane a quarter-wave in front of the end plate, which shorts the hori-zontal component. The reflected components combineto give a wave polarized along the other diagonal

which leaves through the other arm. This output arm isoriented at exactly 90 degrees to the input arm to avoiddirect cross-coupling. (The choke plunger is adjusted foroptimum matching of the input and output arms. Bysymmetry, the same setting is best for both.) The inci-dent and reflected waves combine to set up a standing-wave pattern in which the vertically- and horizontally-polarized waves are 90 degrees out-of-phase (in spaceand time) because of the X,/4 difference in path lengthsto the effective shorting position. The round couplinghole to the cavity is located on the center line of the wallof the square guide at such a distance from the end thatit is at the maximum of both the longitudinal H of thehorizontally-polarized standing wave and the transverseH of the vertically-polarized standing wave. Since r isassumed equal to 1, the two components are of equalintensity and 90 degrees out-of-phase. Thus we suc-ceeded in producing a circularly-polarized radiationfield at the window that couples into the cavity.The circular excitation will not give circular radiation

in the cavity, though, unless the cavity has two de-generate orthogonal modes differing only by a 90-degreerotation about the axis of the cavity. This will be thecase with circular or square TEi,.. or TM lmn modes. Thetwo modes then may be considered to be excited inde-pendently and 90-degrees out-of-phase, with a circu-larly-polarized radiation field as the result. Along theaxis, the field will rotate purely in one direction. How-ever, averaged over the cavity, this is not true. In factif we use TM modes, the energy, on the average, isshared equally between the two senses of rotation. Thatthis is so may be seen qualitatively by noting that thelines of H are closed. In a plane (TM) field, this implies

Fig. 2-Photograph of transition coupler forcircular polarization.

that there is as much rotation in one sense as in theother. If we go to a TE mode, however, longitudinal His allowed, and the loops of H can close in the axialdirection. This enables us to get a net circular polariza-tion over the cavity at the expense of acquiring someaxial fields. If one is dealing with a magnetic materialconfined near the axis of the cavity the fields are purelycircular over the sample, and these averages are of littlerelevance. However, if the material is a gas which fillsthe cavity, they are important.To render this discussion more quantitative, we may

expand field in cavity in terms of rotating unit vectors,

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H = H ( ) + H_ (- ) +Hzuz

and then, for example, define f+, the fraction of theenergy stores in the H+ rotating fields, by

Hr d,f H+2dr

f+ =...

fH2dr

For TM modes, f± =f_=f, ==0. For TE mn modes, thefractions depend on m, only m =1 giving a large pre-ponderance of one sense of circular polarization over theother. For the TElln modes, evaluation of the requiredintegrals shows that

f+ = 11.5f = 0.92(1 - f,)(6)\2 1

fz = 0.086=a 1 + 2.91(an/l)2

where c is the velocity of light, a is the radius of thecavity, I is its length, and v is the resonant frequency.With the dimensions of the cavity actually used,fz=0.26, f+=0.68, and f-=0.06. The ratio of 11.5between f+ and f_ is quite adequate to allow unambigu-ous results, even in this case of a cavity filled with thematerial under study.A photograph of the actual apparatus is given in Fig.

2. Note the tuning screws in the cavity. They are re-quired to balance out the effect of coupling into thewaveguide, which is not equivalent in the two orthogo-nal directions, and to compensate for any other imper-fections which destroy the exact degeneracy of the twomodes. The coupling hole is precisely centered in theend of the cavity and is on the center line of the guidewall. A hole diameter of 0.25 inch gives sufficient cou-pling into the cavity at resonance to reduce the powerat the exit arm to one-third of its value away from thecavity resonance frequency. In mounting the X,/4shorting vane, one must remember that the effectiveshorting plane lies several millimeters behind the edge

of the vane.8 An 0-ring mica seal is used to isolate thegaseous samples studied with the apparatus. To reversethe sense of rotation, the assembly is unscrewed andturned through 180 degrees at the square-flange jointvisible in Fig. 2.

In the measurement arrangement for which it wasactually designed, the device is used as follows. A chartrecording is made of the damping of the cavity resonatorby resonant absorption of the sample gas as a functionof the static magnetic field. The dominant sense of rota-tion is then reversed and another chart is made. Bycontrasting the two charts one can readily distinguishwhich absorptions are caused by H+, IL, and Hz, sincethe strengths of the H+ and H_ absorptions change byfactors of 11.5 in opposite directions, while the H2 ab-sorptions are unchanged. It should probably be repeatedhere that, for a sample which is localized on the axis ofthe cavity, perfect selection of H+ or H_ is theoreticallypossible, and is nearly realizable in practice.As an indication of the degree to which our cavity

circular polarizer approaches ideal behavior, we quotethe following results. The vswr of the input or outputwith optimum adjustment of the choke plunger is 1.2.The cross-coupled power is down by 20 db. The fulltheoretical ratio of f+/f- 12 was observed with gaseoussamples. Thus this simple apparatus gives quite usableperformance. Reduction of the cross-coupling couldprobably be obtained by tapering the input arms intonarrower apertures in the cylindrical section, since thiswould reduce the distortion of the modes of the cylinder,leaving more complete orthogonality.

If one wishes to operate at other frequencies, or if oneuses circular rather than square guide, one must copewith ¢ 1. In that case, the permanent rigid construc-tion shown in Fig. 2 would have to be modified by theinsertion of a rotatable joint in the cylindrical section toallow the angle 0 in (4) to be adjusted properly. As thisis a simple mechanical adjustment, it should introduceno uncertainty in operation.

8 N. Marcuvitz, "Wave Guide Handbook," MIT Radiation Lab.Ser., McGraw-Hill Book Co., Inc., New York, N. Y., ch. 4, p. 172;1951.

CORRECTIONA. H. Zemanian, author of the paper, "Bounds Existing in the Time and Frequency Re-

sponses of Various Types of Networks," which appeared on pages 835-839 of the May, 1954issue of the PROCEEDINGS OF THE IRE, has brought the following correction to the attentionof the editors.The definition of settling time appearing on page 839 should be changed to read: "The settling

time to E, r.8, is the least time beyond which the step response rema.ins greater than r(1 -E)and the impulse response remains within the bounds +E/C, where e is a positive quantity lessthan unity, r is the final value of the step response and 1/C is the initial value of the impulseresponse. This assumes that the input functions are impressed at t = 0."

738 June