II! l ”H I MI W M STUDY OF THE ANKSOTROPY OF WLYCRYSTALLINE FERROMAGNETIC MATERSALS BY MEANS OF THE MJCROWAVE RESONANCE METHOD Thesis for The Degm a! M. 5. MICHIGAN STATE COLLEGE Wiliiam Frankiin Hacker? 1950 n ’ TY UBHMlEs @El/I/I/I/Ia/I/m I W 53‘; "W we 1/3 Ill/lllllIl/ll ,"r + « l ‘7‘" 3 93017 4832 . ‘r‘ 1 ‘fi 7‘“ fi ' 'fi 73 I7 if. ital . i u t ( n. l” . f p This is to certifg that the : ‘ ’i 1.; thesis entitled I The Study of the AnisotrOpy of Poly-crystalline ; Ierromgentic Materials by "can: of the Micro- ' wave Resonance Method 1 ‘ 5; presented by r: uni-m r. Heckort has been accepted towards fulfillment of the requirements for ’ gin “ 3.5. degree mm 4 3 u . ' ,V“ \. Major professor ¢ '_ Date 23 "‘y 1950 . .. -- «n.3,. .a..._'-.t . . H s”.,_'hbgl- .. “- 7 '—u - 3'.“ ' “*1.” “6‘o-‘ v¥¢-.' . t ’ .4“"' 0“ 41.15 $4.; ,. ’A’Q‘\I r ’ . . PLACE IN RETURN BOX to remove this meckout from your record. TO AVOID FINE retum on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE THE STUDY OF THE ANISOTROPY OF POLYCRYSTALLINE FERROMAGNETIC MATERIALS BY'MEANS OF THE MICROWAVE RESONANCE METHOD by Wfllliam.Frank11n Heckert A THESIS Submitted to the School of Greduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1950 ACKNOWEEDGMENT I wish to thank Dr. Robert D. Spence, under ihose direction this work was done, for his many helpful suggestions and patient guidance through- out the course of the problem. I would also like to thank Mr. Charles Kingston for his work on.the construction of apparatus. m Km”, V. \WT _D ' ‘.\ 5' 3’ r. pl M ’1 H! V I. II. III. IV. V. TABLE OF CONTENTS IntrOdUOtion e e e e e e Theory......... Arrangement of apparatus and calibration procedures Experimental Procedure . Data and Interpretation Page 17 35 39 I. Introduction Ferromagnetic Resonance Absorption was originally ob- served by J. H. E. Griffiths1 in 1946 and since then consi- derable work has been reported by other investigators. Among these are Yager and Bozorthz, Kip and Arnold3, and Kittel4. A ferromagnetic sample is used as one wall of a rectangu- lar resonant cavity which terminates a waveguide section. The other end of the waveguide is connected to a microwave oscillator which delivers microwave power to the resonant cavity. The ferromagnetic material is so placed in the cavity that the magnetic vector of the microwave field is constant in direction in the sample. An electromagnet is used to fur- nish a static magnetic field which is applied in the plane of the sample of the resonant cavity and at 90° to the r.f. mag- netic field. Directional couplers inserted between the cavity and oscillator provide means of coupling reflected power to suitable detection apparatus. When the strength of the magnetic field is increased, one observes that the r.f. energy loss in the cavity goes through a maximum. When.the pgggg.lgg§ is plotted as a func- tion.of magnetic field strength a resonance curve is obtained 1. J.H.E.Griffiths, Nature gap, 670 (1946) 2. W.A.Yager and R.M.Bozorth, Phys. Rev. 1g, 80 (1947) s. A.F.Kip and R.D.Arnold, Phys. Rev. '_7_§_. 1556 (1949) 4. C.Kittel, Phys. Rev., 19, 155 (1948) 1 Whose general appearance is shown in Figure l. A \u t D c M o O I n I m t a I a l a! a 9 '. o a— '. Hues. hi2 Graces Figure l The field at which maximum.power loss is observed is called the resonance field. In Figure 1, this value is desig- nated by Hrss‘ Ferromagnetic materials are anisotropic and, therefore, different magnetizing forces are required to magnetize them to saturation in different directions with respect to the crystal axes. .Kip and Arnold have found that the resonance fields are dependent on the orientation of the crystal axes with.respect to the magnetic field. Their experiments were conducted with.a single crystal of iron. It is the purpose of this experiment to study the aniso- tropy of polycrystalline ferromagnetic materials by the reso- nance absorption method. we shall investigate the resonance fields for different orientations of the 100 direction in Silectron with.respect to the static magnetic field. II. Theory we shall first consider resonance absorption from.a simple spectroscOpic point of view. Ferromagnetic materials have been found to have essen- tially zero electronic orbital angular momentum.and the only angular momentum of significance is that due to electron spin. According to weiss, an unmagnetized ferromagnetic material is divided into small domains which are magnetized to saturation. The sum.of the individual spins is equal to the total domain angular momentum as given by this relation "‘ - Zea <1) M-3R .- where E = domain angular momentum and ‘55." electron spin. If a magnetic field H of sufficient strength is applied, the domain spins will all line up in the field direction and the specimen will be magnetized as a single domain. we also note that the large angular momentum due to spin is quantized with respect to an axis parallel to the static magnetic field. An occasional spin will be oriented anti-parallel to the field and will decrease the angular momentum.vector fit Figure 2 shows the quantization of "M? with H. \l 4 9 2 3-1 Figure 2 While this lone spin is so oriented we will have two energy levels looking something like this: Hi>0 ¥S=+L 7.. +3 “-2-; Figure 5 The difference in energy which gives the two energy levels follows from.a simple classical argument: When a spin is ori- ented as shown in Figure 4 H A z [is 9 Figure 4 with a magnetic momenth,its energy is given by the relation (2) E= —f‘.- H=-)‘SH‘°‘*9 If FL,is oriented parallel to the field direction, we have the most stable energy state and therefore a lower energy than if lie were oriented anti-parallel to H in which case we would have a higher and, consequently, less stable energy state. The resonance condition occurs when we have a transition from.an energy level having a lower energy to one having higher energy; in this case When the spin vector parallel to the field is "flipped" over to position.Where the vector is anti-parallel to the field. The reversal of the spin is accomplished by the microwave magnetic field and the energy required to "flip” these vectors accounts for the energy loss in the resonant cavity as the static magnetic field is varied. The relaxation time is very short so the relaxation mechanism.establishes a surplus in the lower state and absorption occurs rather than emission. The energy Which the radio frequency field must supply to reverse the spin is given by the equation (3) AE Z )(Vmut. This energy is constant in value and transitions between energy levels occur When the difference in energy of the two levels is equal to fiKWhLi. The expression for energy difference between the two levels represented by oppositely oriented spine is equal to (4) WflB Ht: J’Cv‘fl-‘l‘ and is represented diagrammatically'by Figure 5. “2:0 ,"I l l \ OXPB Hz \‘ g _l_ Figure 5 If our line widths were of infinitely narrow width in Figure 5, the transitions, theoretically, would occur for only one value of Hz and the absorption curve would be infinitely sharp. However, the lines are of finite widths and consequently 5 transitions can occur for more than one value of Hz. Thus we get a resonance curve similar to the one shown on page 2. That is, we get transitions other than at the point given by equation (4). The finite line width can be explained in the following way: The field in the interior of the specimen is not quite that one would expect from classical predictions, obtained by taking the shape of the specimen into account. This happens because the local fields in the neighborhood of individual spins vary as the spins are "flipped" from one position to another. Thus, at one instant of time, the actual field near a particular spin may be slightly greater than the internal field and a neighboring spin may have an actual field of slightly lower value than the internal field predicted in.the previously-mentioned manner. It is now obvious that in a given specimen we will have spins of varying energy i.e. many of the higher-energy spins will vary a little above and a little below the average energy for that level. The same will hold for the lower energy levels and it will be possible for a variety of transitions to occur instead of a single type from.ane energy level to another. However, most of the transi- tions will occur at the predicted field values. Thus we see that the resonance curve will be of a finite width rather than a single vertical line. Returning to equation (4) which is (4) Jae/at: Oyfie Hg 6 one can easily see that Q (5) Wax-2.: CQ—qmg “E where g is the Lande' g factor and is equal to 2 for a free spin. Setting g = 2, we obtain MC This equation is Just equal to the Larmor frequency condition (n m=¥Ha where Y 3 e/mc and is commonly called the magneto-mechanical ratio. The Larmor precession.may be also used to interpret the resonance phenomenon. By Larmor precession we mean the pre- cession of the electron spin vector about the applied static field direction. The resonance phenomenon.can'be thought of as occurring when.the frequency of the microwave field is equal to the Larmor frequency. While this method of deriving the angular frequency relation is not the one we have used, the method gives identical results. Therefore, when.thinking of ferromagnetic resonance absorption, one can think of the method we have outlined or the Larmor precessionuabout the field direction. we shall now consider the effect of the shape of the specimen on the resonance condition. The field inside the specimen is modified by a demagnetization field which materi- ally alters the field present in the interior. Thus the 7 internal field will not be H; but H1. The two are related by (8) EL=T\E——l{l‘p\ where If is a demagnetization dyadic and II is a magnetization vector. We shall now calculate the resonant frequency for the field given by equation (8). Since the angular momentum of the entire sample is large, we may use the correspondence prin- ciple to reduce the problem to a classical one: 3 will be designated as angular momentum density / unit volume. The magnetizationfi and angular momentum density 3 are related by the equation (9) /V\= Y T The equation of motion / unit volume of material is given by 43.. {T— (10) i where (11) :1: = qu;.+—rp and Tie the torque acting on a unit volume. Ta is the aniso- tropy torque. Equation (11) becomes ‘15:— =Y (M‘fii .771) (12) From this relation we can calculate the susceptibility of the radio frequency field. Let us consider the resonance condition for a general ellipsoid having principal axes parallel to the x, y, and z axes of the Cartesian coordinate system. The demagnetizing factors along these three directions are Nx, Ny, and Nz‘ Thev static applied magnetic field is Hz and the microwave field is Hi. The values of H; in.these three directions are (13) Hi: Hs-NaMx (14) “i: 4“». M‘t‘r (15) \4; : \Az- NaMt and the equation for II; is (16) Hi: : (Ht-NINMJ + I {-N“; M‘s} +1 (“2- NtMi) If we substitute these values in the equation of motion we get the three component equations a.) a. a Y mew—Nam— Mime-m5) (18) M}: Y [M%(ua~N1MD —N\$(\'\%~Nth\+fi‘3—1 (19) ME =- Y [M‘( “N‘t Mtg —M‘t(ut—N1M~O +111] By factoring, we see that (20) a. = 1K“s-N*M1+NaMa\Ma+Taal (21) Mar: "Y (Nth *’ (“‘“NEMQMQ‘F YMade-Tw (22) Ma “‘50 We shall call Hz equal to zero since no Mz terms appear here and the quantity is therefore very small. (MJ: and My are small while Mz is large) gw'i' Now, we shall assume a time dependence €— (23) (“sin e Mx emf We may call (20) equal to (24) as M. = [Y ( (a + (Na-N2) Ma] Ma and (21) equal to (25) Lu) M‘h— .2 EV (Ht +(NV.--N1'i-ll\(\=z—l[\l\‘L + YMa Ht Solving (24) for My gives (26) Mar 2 -—“” Mt (ELL—e aria—Newt] Substituting this My value into (25) we obtain " L M“ ._ _ +§Ns~N2~J a 2 (2'7) YLHE +£~$~N33ng — (Kr—Ht M TMX +Y M Hat From (27) manipulation gives (28) fl "”1- I‘ll)!“ : YMS H“ __ fasten-NamesHum—News] “ ((19.4).-.)th Before continuing, we must point out that the components T and T9. of the anisotropy torque have been temporarily dis- ax Y carded for convenience of manipulation. Later in this discus- sion they will be calculated and added to the H2 expressions. The susceptibility of the r.f. field is given by (29) X: MK ’- M’L HA4. H V‘ 10 and Phi; ____ I: Hz+(N -N \M (30) X" x i i] _ ___.——————“'"_‘ '2...— DJ ’- XLEHVH' (NX’N1)MQ( ui+kNx 441:] (WE) At resonance 3(-—*- c%’ and therefore, (31) us“: (LU “fiWrNfiM*)(“‘*)N“N*)M‘fl where\fl is the resonance frequency i.e. the frequency of the r.f. field. we now consider special cases of equation (51), showing how (51) varies depending on.the shape of the ferromagnetic specimen. For a plane, which is the type of sample we have used here, the demagnetizing factors are Nx I Nz I 0; Ny I ifi‘. Then (32) w: x“ Eur 4+1- Meme] For a sphere, Nx I N = N2 I 4“73 and the resonance equation 9‘ reduces to ‘L 7.. (33) szy H} M w: VH2 Which.is the simple Larmor expression. It is of interest to note also that the magnetization vector M2 = 1700 gauss for iron. Having shown that the resonance condition.changes with the shape of the specimen, we may now consider the effect of 11 crystalline anisotropy on the resonance condition. We may start this by discussing crystalline anisotropy briefly. In a ferromagnetic crystal there is a variation of mag- netic properties with direction. If we consider iron, we see that iron crystals are of a body-centered cubic structure \\0 \\\ Figure 6 in which 100, 110, and 111 are directions with respect to the crystal axes. The 100 direction is known as the direction of easy magnetization, and the 110 as the direction of medium magnetization and the 111 the direction of hard magnetization. In a face-centered cubic crystal such as nickel, the roles of the 100 and 111 directions are reversed from their properties in iron. These variations in magnetization with direction can be shown in a series of magnetization curves drawn for these three directions Figure '7 12 The internal energy density of ferromagnetic crystals depends on the direction of magnetization relative to the crystal axes and is called the anisotropy energy. The direction of magnetization is defined by the d irection cosines 81, $2 and $3 referred to the cubic axes of the crystal. We can expand the energy in a power series of direction cosines and get (34) E: Ko+ \<\ ( STSr-r 9:S:+9: 51:) + Kn ( 5' $52) The major term in the anisotropy energy is the portion ‘L ‘l— 1— 'L 2—. L (35) EM:K\ The anisotropy energy will cause a change in the resonance condition. In a single ferromagnetic crystal the magnetic field required to produce resonance will vary, depending on the direction of the c rystal axes with respect to the magneti- zation of the sample. In a polycrystalline specimen, such as the Silectron used in this experiment, one would expect a resonance condition which is much broader than that of the single crystal. This broadness is introduced by the distribution in direction of the crystal axes which in turn introduces a dis- tribution in field strengths required for the resonance absorp- tion. On page 10 we temporarily abandoned the torques Tax and T ay' We shall now calculate Tax and Tay from mechanics by using the relations 15 ._A (56) I :JKXF :\:\ F€7E =W*(‘AE) and 7.. L 7- 1.. 1.— 1. (57) E:\<\ (S.s,_+ SL33+S.S33 where T is the torque due to anisotropy and E is the major part of the anisotropy energy. Equation (37) is identical with equation (35). The metal used in this experiment was a highly anisotropic metal known as Silectron which has its rolling direction along the 110 direction and we shall use this fact to derive the aniso- tropy torques in the x and y directions. Figure 8 shows the coordinate system used to describe the Silectron and its position, in our experimental arrangement, with respect to Hz and Ex. a. I I "HO l W Suuv we. n» Figure 8 This drawing represents two Cartesian coordinate systems ing‘n)s and X, y, and Z. The direction cosines of the energy given in equation 14 (37) are s3: 1!); (38) S\=_P_/\_§ Sizzivl: \ M M M \ So (37) now becomes (59) E=—‘;—‘&; (M;M~:+ MEM: + WM?) From Figure 8, we can find the values of Mg , MK' and MS components. Mg: 2&9 M‘s-Hag M1 ‘4“ MM is Mw-‘c: Meme-S Ma— (42) MS : —A.~;e Mx ' + Cue M2 Now, we must square these three 'values and place the results in equation (59)). We may simplify the energy expression some- what by discarding all terms not containing Mz i.e. all terms in Mx and My alone. The energy is given by L L 3 E: K‘ [(“3K19939Mx 1);)“: + (1M3Q29-M39Q‘6)MM: (45) tM“ M": IV\ .. M39 M7“; 4mg +mn‘edte\—————M+] Setting Mz = M, (43) becomes E:K\ S:(-3RQV:'BQB)&EL+ Klmgd9-m£9g¢9) Mk (44) M M -m‘e $3 + (“t—$.9— +A-Lte Codi-93F] To conserve space, we shall indicate what the next few steps 15 1) The gradient of E is computed 2) 3.7 is crossed into fit”: giving us an equation for torque. This torque equation will be in the f orm —* (45) T‘L 1: T009. 1:25: Tum“!- Elf—rot? Since we are interested in only Tax and Tay9 we see that these components are equal to (46) T1” 3K (male Egg: (47) “R = _.(.K. m‘eQL‘eEM’i-nmmecfe -K‘Agg’e (04.9 we now return to where we dropped the Tax and Tay compo- nents of the equation of motion. Equations (46) and (4'7) are placed in equations (20) and 21). The solution of the reson- ance formula, equation (51), will then equal (48) waffiuuflNa-NQM +‘L‘EFA1::§_ \(Ha +(Nx-Nt)M1-§_A_'X_[ mite (“f-9)] I The demagnetizing factors Nx, Ny and Nz for a plane surface are substituted into (48) giving (49) N1=YtB HE +4~TM1+3_‘_<_;TA“:_':§ \(Hz - 3:; MLBQGH We may now use (49 to calculate a theoretical curve which we can compare with our experimental curve. In calculating this theoretical curve, we shall use a value of K1 I 2.94 x 105 ergs / cm.3, which was. determined by Kropschots. —___ 5. R.H.Kropschot, Master's Thesis, Michigan State College, (1950) 16 III. Arrangement of Apparatus and Calibration Procedures Before we discuss the apparatus of this experiment in detail we shall consider an overall description, accompanied by a block diagram which is shown on the next page. The system consists of a Shepherd-Pierce 725A/B Klystron oscillator, wave guide sections, a cavity wavemeter, a squeeze section, directional couplers, crystal rectifiers, an audio amplifier, an oscilloscope, and power supplies for the micro- wave oscillator. The r.f. power source was the Shepherd-Pierce Klystron at a frequency of about 9375 mc./sec. The voltage to the resonant cavity is supplied by a 300 volt regulated power sup- ply. The repelling electrode of the Klystron is modulated by an audio frequency square wave which is superposed upon a ‘variable negative D.C. voltage which attains a negative maxi- lnum of about -2OO volts. The Klystron filament voltage is sup- :plied by'a 6.3, v. secondary from.a 60 cycle, 110 volt trans- former. The power generated in the oscillator is coupled, by Ineans of a coaxial probe, into a 3 cm. rectangular waveguide. The power then flows through a cavity wavemeter which is in ‘turn coupled through the waveguide to a variable "flap" attenu- ‘ator, whose function is to decouple the load from the oscillator and.prevent load changes from.varying the oscillator frequency. The next component in the microwave circuit is a squeeze 17 w x_ (swan CAVITY VARIABLE so UEEZE DIRECTIONAL 33%?!“ ) .oscmuon- - «avenue» «ATTENUATOR» < SECTION .. COUPLERS- - e V ‘ - ~ .. .. H(ELECTRO-l 1 MAGNEY J ‘ . . . l h r I i A ‘ w - i . ‘ , . . - I l A eoouuuo O I REPELLER REGULATED . ‘AUDI O CRYSTAL “Lu“ 33:12:: OSCILLO- : :ANPLIFIER RECT'F'ER -ueov. scowe . I 300V L l L noun-us BLOCK DIAGRAM son RESONANCE assonpnou EXPERIMENT . _,....T. _ mw’a Figure 9 Apparatus Block Diagram 18 section. Its function is to change the electrical length of the waveguide. This process serves to give the load a better impedance match with the oscillator. Going from the squeeze section, away from the oscillator, we have two directional couplers, represented by a single block in the diagram. The directional couplers are oriented in such a manner that they provide us with a means of viewing the trans- 'mitted and reflected waves. The coupler oriented to couple out the reflected wave is the one used most since we measure power reflected from the waveguide termination. From the directional couplers, we have two branches. One is the main waveguide which terminates in a resonant cavity. The other branch goes to a crystal rectifier, which sends the detected waves, via coaxial cables, to the audio amplifier. The audio amplifier receives a rectified square wave minus the r.f. component. From the audio amplifier, the ampli- fied square wave is coupled, again via coaxial cable, to an oscilloscope where the square wave can be viewed visually. Having discussed the coupling of the transmitted and reflected waves from the directional couplers, let us follow the progress of the power from the couplers down the waveguide to our resonant cavity. The field components travel through the main waveguide to the resonant cavity. The resonant cavity consists of an iris in the waveguide, plus, of course, the waveguide termination. The iris is mounted perpendicular to the direction of propagation and is about 1/3 of the narrow dimension of the waveguide in diameter. 19 The length of the resonant cavity is one guide wavelength (or ‘multiples of Tug/2) and can be varied depending on whether we have a variable or a fixed cavity. A variable cavity would be achieved by means of a movable shorting plunger as the wave- guide termination. The ferromagnetic sample is situated as follows: In the case where we have a variable cavity, the sample forms part of the wall of the guide in the narrow dimension, and when.the cavity is fixed, the sample is usually placed as a.termina- tion on the waveguide. The last section of our block diagram is shown as the electromagnet. The polepieces are adjacent to the long dimen- sions of the guide i.e. they are parallel to the long dimension. The sample is always directly between the polepieces. The strength of the magnetic field between the polepieces is varied by varying the D.C. current through the coils. This current is measured with a D.C. milliameter in the coil circuit. The field is calibrated as a function of oersteds vs. current through the magnet's coils. This process will be described later. The cavitwaavemeter was calibrated to give a curve of wavemeter setting ( C.W.S. ) versus wavelength. This process .sill also be described later in the report. Having given a general description of the entire apparatus used in the problem, we will now discuss, in more detail, specific pieces of apparatus. 20 POEER SUPPLIES AND VOLTAGE REGULATION Since the stability of a reflex Klystron oscillator is dependent on very well-regulated voltages, it was necessary to devote a considerable amount of time on building a well- regulated power supply for the system. Before we discuss the power supplies used in this experi- ment, let us outline briefly the requirements of a Shepherd- Pierce tube: . A positive, well-regulated, D.C. Voltage, which can be varied from two hundred to three hundred and fifty volts, is necessary.for the resonant cavity of the tube. Also, a steady, negative D.C. voltage, upon which issuperposed a wells regulated square wave form, is required for the repeller electrode of the 723 A/B. Before we enumerate technical details of the power system, let us give a brief resume of the action taken: we had at our disposal a square wave generator which.gave an unregulated waveform. To eliminate the effects of the non- regulated square wave, the experimenter coupled this waveform into a two stage audio amplifier consisting of a simple resis- tance, coupled stage and a cathode follower stage. The resis- tance coupled stage was driven to saturation and cutoff and clipped the top and bottom.of the waveform off. Since the two stage amplifier was driven by regulated voltages, the operation of the resistance coupled stage eliminated the effect of the lack of regulation in the original square wave generator. The cathode follower stage was an attempt to affect an 21 impedance match between the repeller electrode and the square wave generator. The square wave output from.the cathode follower was superposed on a negative battery voltage which went to a potential divider and thence to the repeller electrode of the:ref1ex Klystron. The battery voltage was obtained from a bank of storage batteries. The power supply and voltage regulator supplying voltage to the two-stage amplifier and to the resonant cavitynwere of this nature: The power supply consists of a 5U4-G full-wave rectifier connected to the secondary of a step-up transformer. The filter is a conventional condenser-input pi-type filter. The output of the power supply goes to an electronic voltage regulator. This regulator consists of a regulator and a control tube, one being in parallel with the D.C. output voltage and the other in series. The effect is that the control tube (the one in parallel) furnishes bias voltage to the grid of the series tube which acts as an automatic variable resistor. If the voltage output increases, current through the series tube increases and causes a more positive voltage to be placed on the control grid of the parallel tube. This tube conducts more and therefore its plate voltage draps which makes the grid of the series tube more negative and tends to decrease the<3urrent flow through it, thus lowering the output voltage. Aschematic diagram of the power supply and voltage regu- lator is shown on page 25. The schematic of the two-stage 22 '0 U LC E .f Ec' .' f 300: , ; POWER SUPPLY I _ — - u- _ , _ ' . T t e f vanes: steuuroa ‘ L F“ v i "— x't'“¥ ’fi a ’F V “——\‘ ‘— — — — * Figure 10 Schematic Diagram of Power Supply and veltage Regulator 25 amplifier is omitted because of its simplicity and conven- tional design. This amplifier is on the same chassis as the power supply. THE DIRECTIONAL COUPLER A directional coupler is a device which couples power from a waveguide and is able, at the same time, to differen- tiate between a transmitted and reflected wave. A description of this function will follow after an analysis of the construc- tion of the instrument. A single hole ("Bethe-Hole") coupler was used and is constructed as follows: Two sections of waveguide are fastened together with their long dimensions adjacent. There is a single common hole connecting the waveguides as this drawing illustrates: l———_+i _ —v~—-w—f‘——— ~——-—fi-— —— -—.— i7-—- Figure 11 .we assume an electromagnetic wave incident from.an oscillator on the left side of the diagram. Power will couple into the auxiliary waveguide in a backward direction as shown. Some power will be coupled in the forward direction and will be absorbed by the absorbing material placed there. The 24 reflected wave, incident from the right, will not couple into the auxiliary waveguide in any appreciable amount. we shall now define incident power as P1, gamer coupled in a useful direction as Pr and power absorbed by the absorb- ing material as Pb. Using these quantities we will mathemati- cally define some properties of the "Bethe-Hole" coupler. Directivity is an indication of the quality of a coupler and is given, in decibels, as (50) D: (0 Lynx/m (E: where Pb approaches zero in the ideal case. Coupling gives an indication of the ratio of power inci- dent from.the oscillator to useful power coupled into the auxiliary guide (Pf). Coupling is given by (51) C: ‘0 Mm %" e and is also in decibels. we now consider how coupling takes place through the hole. The power coupled into our auxiliary waveguide is a superposition of two electric fields outside the hole, one produced by the electric field in the guide and the other produced by the magnetic field in the waveguide. Consider the drawing I (s ‘J F'\ | (tulhuat " w ? g, 4 . k I um I . .....-...... '1 (en ! $1 ' _— ~——__ Figure 12 25 Figure 12 represents electric field coupling through the hole, where the:fie1d coupled through the hole is exactly analogous to an oscillating electric dipole. The magnetic field coupling is illustrated thusly: J L _§E : fl 55;; - I‘. wag-I! Iaae" alw- \ee ‘\ Figure 13 The dots and crosses, of course, mean lines of force going into and coming out of the plane of the paper. In the magnetic field coupling, the electric lines of force coupled into the auxiliary guide are set up by a flow of current inside and outside the hole. This current flow induces charges on the outside of the hole which set up the electric field lines as shown. We shall now illustrate the superposition of the two fields. 0: IF : ‘eF? / ' *. M ' in" fill 1‘ / Figure 14 One can easily see'that there is partial reinforcement to the left and partial cancellation to the right as shown by the vectors R and C. This discussion is valid for the TE mode only, which 10 26 is used predominately in microwave work. It is obvious that one has simply to orient the