The gyrotropic characteristics of hexaferrite ceramics.

Transcription

1 Te gyrotropic caracteristics of exaferrite ceramics. D H Martin, Bin Yang and R S Donnan. July Introduction. Hig-performance non-reciprocal devices aving quasi-optical structures and operating at millimetre-wave/submillimetre-wave frequencies above 9 GHz ave recently been reported. In suc a device a signal beam, propagating in free-space, passes normally troug a plane plate of a magnetic material tat is uniformly magnetised perpendicularly to its surfaces. Te diameter of te plate is muc greater tan te signal-wavelengt (te beam passing troug it must ave a widt tat assures ig directivity), typically 1 mm or so. Te plate s tickness is determined by te requirement tat te plane of polarisation of a linearly-polarised signal beam passing troug it will suffer a 45-degree Faraday rotation, wic typically requires a tickness of one or two millimetres. Te diameter of te plate is tus muc greater tan its tickness, it is a tin plate, and wen it is near-uniformly magnetised to near-saturation perpendicularly to its surfaces (magnetisation M), a large uniform demagnetising field, -M, acts inside it. Provision of a surrounding magnet to compensate te large demagnetising field over te plate would be impractical - a magnet tat would produce suc a field over te full widt of te plate, witout obtruding on te passage of te signal beam troug te plate, would be extremely large and massive. In order tat a plate retain uniform magnetisation at nearsaturation value on removal from te large external magnet in wic it as been poled, in spite of te large demagnetising field tat will ten be acting inside it, te material of te plate must ave ig uniaxial anisotropy (a near-rectangular ysteresis loop) and a coerecive field, H c, at least as large as te demagnetising field, ie. Hc / M 1. Only a few of te commercially-available non-metallic permanent-magnet materials (certain finegrained, grain-oriented, exaferrite ceramics) satisfy tis requirement. [Tese criteria do not apply for te ferrite materials used in waveguide-mounted non-reciprocal devices operating at frequencies below 9 GHz.; an in-waveguide magnetic element in suc a case would ave a rod-like form, and terefore a small demagnetising factor, and would be of small volume, so an external magnet of modest scale could compensate te demagnetising field over te volume of te element - a magnetically-soft material would ten be acceptable.] A candidate material must of course ave appropriate magneto-optical properties at frequencies as well as te static permanent-magnet properties referred to above. Te large magneto-crystalline anisotropy in te grains of exaferrite ceramics gives tese materials teir permanent-magnet properties and also teir magnetic-resonance beaviour at mmwave frequencies wic promises well-developed gyrotropic properties at iger frequencies into te submm-wave range. Te suppliers data seets for tese materials contain extensive information pertinent to te use of tese materials in permanent magnets but little tat is relevant to te nice applications as mm-wave/submm-wave gyrotropic materials. Intending users of tese materials for te latter purposes must screen samples from many suppliers to find any tat as te required properties. Only a

2 few of tese materials ave been found to ave sufficiently low mm-wave/submm-wave attenuations. Te purpose of tis Tecnical Note is to present a formal caracterisation of te uniaxial gyrotropic beaviour exibited by fine-grained, grain-oriented, exaferrite ceramics. Te grains in suc a material are of single-domain size (a few tens of microns) and teir exagonal axes are well, but not perfectly, aligned (a spread of a few degrees about te mean direction wic defines te axis of te material), bot of tese structural aspects being essential for te ig uniaxial anisotropy and ig coercivity of te material. Te spontaneous magnetization as te same magnitude in eac grain and, wen te material is poled, it varies sligtly in direction from grain to grain. Tis random inomogeneity would result in some break-up of te spatial coerence of a mm/submm-wave signal beam propagating in te direction of te axis of te material but tis break-up will be weak because te grains are muc smaller tan te wavelengt of suc a beam. Te outcome will be a main component of te beam wic will ave te spatially-coerent form caracteristic of a beam in an omogeneous uniaxial gyrotropic material (albeit wit some attenuation), wit a weak spatially-incoerent component, generated at te inomogeneities, in te form of diffuse scattering of signal-power out of te spatiallycoerent beam, over a wide angular range. Tis diffusely-scattered power will be registered as a weak attenuation of te spatially-coerent component as it propagates troug te material. (Tis is analogous to te well-known (among microwave exponents) division of a signal-beam into spatially-coerent and spatially-incoerent components on reflection at a metallic surface wic as sort-range random deviations from a smoot profile; te form of te coerent component is determined by te underlying smoot profile and te spatially-incoerent component is diffusely scattered at te random deviations. It is analogous also to te diffraction of X-rays from a crystal lattice tat is disturbed by termal vibrations; sarp diffraction spots caracteristic of te undisturbed lattice are obtained superimposed on a weak diffuse background). Tis is te model on wic te formal magneto-optical caracterization of a fine-grained, grainoriented exaferrite ceramic presented in tis Tecnical Note is based, ie. te material will be modeled as a uniformly spontaneously magnetized omogeneous uniaxial gyrotropic material wit enancement of te imaginary parts of its permittivity and permeability tensors to represent te loss of power from te spatially-coerent signalbeam due to diffuse-scattering at inomogeneities in te real material.. Te RF permeability and permittivity tensors of a uniaxial gyrotropic material. An RF magnetic field, r (, ω), of angular frequencyω, acting at a point, r, in a uniaxial magnetic material aving a uniform spontaneous magnetization, Mr (), and a static magnetic field H(r), bot in te direction of te axis, induces a local RF deviation of Mr () from te axis, ie. an induced RF magnetic induction, br (, ω), related linearly to te RF magnetic field acting at r tus

3 [ ] br (, ω) μ. μ(, rω). r (, ω) were μ is te scalar permeability of te vacuum and [ μ(, r ω) ] is te local relative permeability tensor of te material. [ μ(, r ω) ] as te same form and value at all points witin te omogeneous material and we can denote it [ μ( ω )]. Te symmetries of te material, including in particular te axial-vector symmetries of M and of H, determine te following form for te relative permeability tensor: μω ( ) iκω ( ) μω ( ) iκω ( ) μω ( ) + 1 [ ] were te 3-axis is parallel to te axis of te material. Te placing of te zero elements in tis form registers te uniaxial caracter of te material and te equal magnitudes, but opposite signs, of te non-zero off-diagonal elements express te gyrotropic caracter. Taking te 33 element to be unity implies tat realizable RF magnetic fields will induce negligible canges in te magnitude of te spontaneous magnetization, Mr (). Similarly, a local RF electric field in te material, er (, ω), will induce a local RF (, ω) ε. ε.(, ω) ε is te scalar permittivity of dielectric displacement, dr [ ] er, were te vacuum and [ ε ( ω )] is te relative permittivity tensor of te material. As for te RF permeability, te symmetries of te material (in particular tose of te axial vectors ε ( ω ) : M and of H ) determine te following gyrotropic form for [ ] εω ( ) iηω ( ) εω ( iηω ( ) εω ( ). εl [ ] Te intrinsic symmetries of a uniaxial gyrotropic material determine te above forms for its RF permeability and permittivity tensors but not te magnitudes of te non-zero elements nor teir variations wit frequency wic are determined by details of te local dynamical responses of atoms and electrons in te material to applied rf electromagnetic fields. It is necessary to know tose values if te material is to be assessed for application in a non-reciprocal device. Tis requires measurement of te complex reflectances and transmittances of sample plates of te material over a range of frequency followed by analysis of te results to determine te complex values of tese elements. It can be elpful, but not essential, to seek an interpretation of te results of suc determinations in terms of local dynamical responses of atoms and electrons in te material to applied rf electromagnetic fields. In te case of a exaferrite, a simple model of precessional resonance of te spontaneous magnetization in te large anisotropy field gives a good

4 account of te measured frequency-dependencies of te elements in te permeability tensor. 3. Te caracteristic plane-wave modes tat propagate along te axis of a uniaxial gyrotropic material; te refractive indices and wave-impedances of te material. A material caracterised by a permeability tensor and a permittivity tensor aving te gyrotropic forms above will support self-sustaining, independent, circularly-polarised plane-wave fields propagating along te axis; ie. tese plane-waves are caracteristic, or normal, modes of te system. Eac of te circularly-polarised transverse RF fields making up suc a wave,, b, e and d, as te following form (as a Jones vector): fx ( ω, z, t) fy ( ω, z, t) 1 f.exp iω( t k. z). i were te z-axis is in te direction of te material s axis, were f ( ω, zt, ) denotes any of tese constituent fields and f is its complex amplitude. Waves of bot senses of circularpolarization are included ere using a notation we sall use trougout. Were a subscript is concerned, te upper alternative (+) denotes te clockwise circularpolarization looking in te direction of M, and te lower ( ) denotes te anti-clockwise circular-polarization. Were an in-line sign-alternative (eiter or m ) is concerned, te upper sign is to be taken for te + polarization, and te lower for te polarization. Te magnitudes of te complex propagation constants, k, for suc waves, and te relative magnitudes of te amplitudes of te four constituent fields in tese waves, are determined as follows by te requirements tat te fields satisfy (a) te Maxwell equations for time-armonic electromagnetic fields and (b) te local relationsips represented by te permeability and permittivity tensors. Te Maxwell equations for time-armonic electromagnetic fields are e iωb iωd. d b If e and are independent of x and y (a plane-wave), te z-components of e and must be zero and terefore (from te curl Maxwell equations) b and d are transverse to te z-axis at all z, t; tence, via te constitutive relationsips, e and are also transverse. To begin wit we sall take te tensor permittivity to be diagonal (ie. as no gyrotropic caracter). Te dependencies of te field components on z (as exp ikz (. ωt) ) in tat case lead, via te curl Maxwell equations, to te following relationsips between d, e, b and at all z, t:

5 d ( k/ ω) d ( k/ ω) x y y x e ( k/ εε ω) e ( k/ εε ω) x y y x b ( k / εε ω ) b ( k / εε ω ) x x y y Te latter can be written bx x b y ( k / ωεε ) y. Tere is a second relationsip between b and at eac point and time as expressed by te permeability tensor: bx μ iκ x b y μ iκ μ y 1 x μ iκ Tus te field at eac z, t is to be an eigen-vector of te operator y + iκ μ, wit eigen-value ( k / ω εμε ). Te x and y components of te constituent fields must tus be related at eac z, t tus: μ iκ x k x iκ μ y ωμεε + y and tis determines te polarization states of te caracteristic waves. Tere are two solutions, labeled ere +/-, eigen-vectors: x (,) z t y (,) z t 1 exp i( ωt k z) i eigen-values: k /( ω μεε ) μ as can be confirmed straigtforwardly by substitution. is an arbitrary wave-amplitude, and te permeabilities μ ( μ κ). Tese caracteristic plane-wave fields can be recognized as circularly-polarised, tat labeled wit subscript + aving clockwise circular polarization, and tat wit subscript aving anticlockwise polarization, wen looking in te positive-z direction, tat is to say in te direction of M.

6 It is straigtforward to follow a similar procedure to tat above but wit te permittivity, as well as te permeability, aving gyrotropic form (as is allowed by te intrinsic symmetries of te material); te result for te (square of) te wave-number is ten k ω μμ ω εε ω ( ( ). ( )) were μ μ κ and ε ε η. Eac element of te permeability and permittivity tensors, μ, κ, ε, η( ω ), will be complex wit a positive real part and a small imaginary part to allow for dissipative processes in te material s electromagnetic response. Writing k α + iβ wit α and β real, te form of te plane-wave factor is exp i( ωt k z) exp i( ωt α z).exp β z and it can be seen tat a positive/negative value for α refers to a wave propagating in te positive/negative z-direction and tat β must ave positive value for a wave propagating in te positive z-direction (positive α ), and negative value for a wave propagating in te negative z-direction (negative α ) since a wave must decline in amplitude in te direction of its propagation. It is important to recall tat te sense of circular polarization is labeled +/ for clockwise/anticlockwise rotation as seen at eac point in te wave wen looking in te direction of M irrespective of te direction of propagation of te wave, parallel or antiparallel to M. Given te form above for te -field in a caracteristic plane-wave, te oter fields in te wave, b,d,e, can be directly determined using te relations obtained at te start of tis Section. Eac of tese constituent fields is everywere transverse and is circularlypolarised in te same sense as te -field and wit te same wave-number. Te amplitude of te b-field is simply related to tat of te -field by te scalar permeability μ μ κ : b (,) z t (,) z t b z t z t x x μμ y(,) y(,) and similarly te amplitude of te d-field is simplyε ε ε ( ε η) times tat of te e- field. Te amplitude of te e-field is related to tat of te -field tus: e (,) z t μ μ (,) z t x x s. i. ey(,) z t ε (,) ε y z t

7 were s denotes if te wave is propagating in te positive z direction (parallel to M), and m if it is propagating in te negative z direction (anti-parallel to M). Te i ere implies tat e and differ in pase by nearly π /, ie. tey are nearly perpendicular to eac oter (not exactly so because μ andε will ave small imaginary parts). We now identify te complex refractive index and te wave-impedance for suc a caracteristic wave propagating along te axis of a uniaxial gyrotropic material. Te complex refractive index, n. Te complex refractive index, n, for a circularly-polarised plane-wave of frequency ω, propagating along te axis of a gyrotropic material, is te wave s complex wavenumber, k, divided by te wave-number of a plane-wave of te same frequency propagating in te same direction in vacuo, k( ω ) : k is n k / k + ω / c for propagation in te positive z-direction and ω / c for propagation in te negative z-direction. n terefore takes te same value for waves propagating parallel and antiparallel to M. It follows from te result k ω. μμ. εε tat te refractive index is related to te elements of te permeability and permittivity tensors tus: n μ. ε (positive root) were μ ( μ κ) and ε ( ε η). Te reduced wave-impedance, z ( ω ). Te wave-impedance, Z, is a measure of te ratio of te electric and magnetic field amplitudes in te circularly-polarised caracteristic plane-wave, specifically siz.. e / were s denotes if te wave is propagating in te positive z direction (parallel to M), and m if it is propagating in te negative z direction (anti-parallel to M). By tis definition, waves aving te same sense of polarisation but opposite directions of propagation, parallel and anti-parallel to M, ave identical wave-impedances. From te results above, tis means tat te reduced wave-impedance, z Z / Z, were

8 Z μ / ε is te impedance of free-space, oms, is related to te elements of te permeability and permittivity tensors tus z μ / ε (positive root). were μ ( μ κ) and ε ( ε η). Te complex refractive indices and te reduced wave-impedances of te caracteristic waves propagating along te axis of a uniaxial gyrotropic material can be regarded as magneto-optical constants of te material. Tey are uniquely related to te local rf permeability and permittivity tensors of te material, as sown above, so tat, given te values of te former, te values of te latter can be calculated, and vice-versa: n μ. ε ; z μ / ε μ n. z ; ε n / z were μ ( μ κ) and ε ( ε η). Tus n, Z are alternative to μ, ε as specifications of te gyrotropic properties of te material. n, Z are more directly measurable tan μ, ε and are also more directly invoked wen analyzing te operation of a magneto-optical device; μ, ε on te oter and are invoked wen te microscopic origins of gyrotropic beaviour are under investigation. Faraday rotation of a plane of polarisation. Te circularly polarised caracteristic plane-waves described in te previous Section propagate along te axis of a uniaxial gyrotropic material witout cange in form. An arbitrary wave propagating along te axis can be represented as a superposition of caracteristic plane waves. For example two suc waves aving te same frequency and direction of propagation, but opposite senses of circular polarization and arbitrary complex amplitudes, propagate independently but te resultant field is not a true planewave because te pattern of polarization of te field is not te same in all transverse cross-sections, ie. te pattern canges wit z. A special case of interest is tat of two suc waves aving equal amplitudes. Taking, first, te case wen te two waves propagate in te positive z-direction, ie. parallel to M, wit wave-numbers k α + iβ. If losses in te material are negligible ( β β ), te resultant -field is exp i( α+ z ωt) exp i( α z ωt) + i i cos( δ z) exp i( αz ωt) sin( δ z)

9 were α ( α + α ) / and δ ( α α ) /. + + Inspection of tis field sows tat witin any given transverse (constant-z) plane te field is linearly polarized (uniform in direction and in magnitude) but te direction of te polarization varies from one transverse-plane to te next, rotating continuously wit increasing z, at rate δ ( α+ α ) / radians per unit increment of pat; tis wave-field is terefore not plane-polarised. Tis rotation of te direction of a linear polarization is known as Faraday rotation. If α > α+ te rotation is clockwise wen looking in te positive z direction, ie. parallel to M. Taking now te case of two superposed equal-amplitude circularly-polarised plane-waves propagating antiparallel to M, ie. in te negative z-direction; te resultant magnetic field is exp i( α+ z+ ωt) exp i( α z ωt) + + i i cosδ z exp i( αz+ ωt) sinδ z were α ( α + α ) / and δ ( α α ) /. + + Tis sows a Faraday rotation tat is identical in bot rate and sense (as seen looking along M) to tat found above for te two plane-waves propagating in superposition in te positive z-direction, parallel to M. Tis identity of te Faraday rotation for te two directions of propagation, parallel and anti-parallel to M, establises non-reciprocal beaviour. Loss processes were neglected above in order to demonstrate te nature of Faraday rotation as simply as possible. Loss processes in real materials will give rise to differing attenuations for two superposed circularly-polarised plane-waves so te initial linear polarization of te resultant wave becomes progressively more elliptical as it propagates wit a Faraday rotation of te axes of te elliptical polarization. Furtermore, toug a linearly-polarised free-space plane-wave migt be incident normally on a plate of uniaxial gyromagnetic material, te caracteristic waves of opposite circular polarization witin te plate will not usually be of precisely equal amplitude, even wen tere are impedance-matcing dielectric layers at te surfaces of te plate. Te ABCD transfer matrices of a plate (following Section) are used to analyse te non-reciprocal properties of suc a plate. 4. Te ABCD transfer matrices of a plate of uniaxial gyrotropic material. Te optical transfer properties of a tin plate of a uniaxial gyrotropic material, isolated in vacuo or in air, wit a static and uniform permanent magnetisation, M, and a static and uniform demagnetising field H D M inside te plate, bot normal to te surfaces of

10 te plate, are expressed as circular-polarisation transfer matrices, te elements of wic depend on te material s refractive indices and wave-impedances and on te tickness of te plate. Te caracteristic plane-wave fields tat propagate along te axis inside suc a plate, bot parallel and anti-parallel to M, are circularly-polarised wit te forms establised in te preceding Section. Te optical ABCD transfer matrix of suc a plate for eac sense of circular polarisation expresses te relationsips tat exist between te resultant field amplitudes just inside one of te two surfaces of te plate to tose just inside te oter, as follows. Te x transfer matrices Τ of suc a plate are defined by e 1 1 e Τ were e 1, 1 are te complex amplitudes of te transverse, circularly-polarised, electric and magnetic fields just inside one face of te plate (face 1) and e, are tose just inside te oter face of te plate (face ). Eac of tese fields is a sum of two components, one contributed by a plane-wave propagating in te plate parallel to M, and te oter contributed by a plane-wave propagating antiparallel to M wit te same sense of circular polarization.using sub-symbol arrows to indicate te directions of propagation of te component wave-fields (and omitting te time-dependence exp iωtof eac field component): at face 1: at face : e e e 1 r 1 + s 1 e er + es er 1exp i( ω / c) n d + es 1exp i( ω / c) n d 1 r 1 + s 1 r + s r 1exp i( ω / c) n d + s 1exp i( ω / c) n d were d is te tickness of te plate and n is te refractive index of te material of te plate. Hence at face 1: at face : e 1 iz r 1miZ s 1 e iz r 1exp i( ω/ c) n d miz s 1exp i( ω/ c) n d 1 r 1+ s 1 r 1exp i( ω/ c) n d + s 1exp i( ω/ c) n d were Z is te wave-impedance for te material of te plate. Te contributory fields r 1, s 1 can now be eliminated from tese simultaneous equations to leave just te total fields e 1, and 1 e,, and te resulting equations are e 1 1 e Τ

11 wen Τ as te form: cosφ Z sinφ T 1 m sin φ cos φ Z were Z are te wave-impedances for te material of te plate and φ are te complex angles φ ( ω / cnd ). Te fields at face in terms of te fields at face 1 are: e Τ 1 e 1 1 were Τ 1 is te inverse of Τ wic, for a matrix of tis form, is given simply by canging te signs of te off-diagonal elements. Tese ABCD transfer matrices are used in determining te optical properties of te gyrotropic plate in isolation or in combination wit oter dielectric plates, as illustrated in Sections 5,6 and 7 below. 5. Te circular-polarisation normal-incidence transmittances and reflectances of an isolated uniaxial gyrotropic plate. Wen a circularly-polarised wave is incident normally on an isolated plate of a uniaxial gyrotropic material wit magnetisation, M, perpendicular to te surfaces of te plate, in order to satisfy te boundary conditions it is necessary to ave te following plane-wave fields, all of te same frequency and all wit te same sense of circular polarisation: in te space before te plate: in te plate: in te space after te plate: a plane-wave incident normally on te plate, and a plane-wave normally reflected from te plate; a plane-wave propagating parallel to M and a plane-wave propagating antiparallel to M; a plane-wave propagating normally away from te plate. Te relative amplitudes of tese five plane-waves will be determined by te boundary conditions at bot faces. We ave, from te preceding Section,

12 e 1 1 e Τ were e 1, 1 are te complex amplitudes of te transverse circularly-polarised electric and magnetic fields just inside te first face of te plate, e, are tose of te fields just inside te second face, and T is te normal-incidence transfer matrix of te plate. Te general interfacial electromagnetic boundary condition is tat te tangential components of te electric and magnetic fields be continuous troug eac face of te plate.tis means tat te transverse fields just outside te plate, in vacuum/air, are equal to tose just inside, ie. er 1 es i r s 1r e 1 1 and e r r t e were te subscripts irt,, indicate te incident, reflected and transmitted component fields just outside te plate, respectively, and te sub-symbol arrow indicates te sense of propagation ( being te sense of M). Tus er 1 es i r s 1r er T r t Dividing troug by e r 1 and writing r es r / er i (ie. te normal-incidence complex reflectance of te plate) and t er t / er i (a normal-incidence complex transmittance of te plate), and noting tat ( r 1/ er 1) i ( r / er ) t mi/ Z and ( s 1/ es 1) r i/ Z were Z is te vacuum wave-impedance (te waves outside an isolated plate propagate in vacuum/air), we ave 1 1 r i/ Z + i/ Z m t. T 1 i/ Z m Tis matrix equation represents two simultaneous equations in r and t (te upper and lower rows) wic can be solved to give r t ( AZ m ib) ( icz + DZ ) ( AZ ib) ( icz DZ ) m + + Z ( AZ ib) ( icz DZ ) m + +

13 were A, B, C, D denote te elements in T, ie. T A C B D Substitution of te explicit forms for A, B, C, D from te preceding Section ten gives { } { } { } { } r r exp i( ω/ c). n. d (1 r ) exp i( ω/ c). n. d ; t 1 1 r 1 r exp i( ω/ c). n. d 1 r exp i( ω/ c). n. d z 1 were r 1 z + 1, were z are te reduced wave-impedances of te material of te plate (tese can be identified as single-surface reflectances). Tese expressions can serve as bases for te determination of z and n for a candidate material from measurements of te reflectances and transmittances of a plate of te material. In te analysis above, te direction of propagation of te incident circularly-polarised plane-wave was taken to be parallel to te magnetisation in te plate, M. A corresponding analysis for a plane-wave incident on te plate in te direction anti-parallel to M follows a similar course in wic te inverse transfer matrix is used; te resulting expressions for te reflectance and transmittance prove to be te same as tose above. 6. Te transmittance and reflectance of an isolated uniaxial gyrotropic plate for an arbitrarily-polarized normally-incident plane-wave. A plane-wave in vacuo (or air) wave can be represented as a superposition of ortogonal linearly-polarised plane-waves aving arbitrary complex amplitudes. If tis plane-wave is incident normally on an isolated uniaxial gyrotropic plate, eac of te constituent linearly-polarised components will generate bot co-polar and cross-polar linearlypolarised transmitted and reflected plane-waves (in vacuo/air). Te properties of te plate can tus be stated in te form of reflectance and transmittance matrices as follows. Taking te z-axis to be te direction of M, of te axis of te plate, and of incidence, te incident plane-wave can be specified as a two-component Jones vector, te upper element being te complex amplitude of te component tat is linearly-polarised in te x direction, and te lower element tat of te component polarized in te y direction. Te reflectance and transmittance matrices ten operate on tis vector to generate vectors for te reflected and transmitted plane-waves.

14 Te reflectance matrix takes te form r r co cr rcr rco since tis form meets te basic requirement tat an incident circularly-polarised planewave will be reflected as a circularly-polarised plane-wave of te same sense: rco rcr 1 1 ( rco ircr ) rcr rco i m i Tis establises te relationsip between te elements of tis reflectance matrix, rco and rcr ( ω ), and te reflectances of te plate for circularly-polarised planewaves, r ( ω ), as treated in te previous Section: r rco m ircr. or r ( r + r ) / ; r i( r r ) / co + cr + Te reason for using te symbols rco and rcr ( ω ) for te elements of te reflectance matrix is now clear since letting te matrix operate on linearly-polarised plane-waves gives: rco rcr 1 rco 1 rco rcr rcr 1 rco rcr and rco rcr rcr r co r cr + 1 rcr r co 1 r co 1 from wic rco and rcr ( ω ) are clearly identified as co-polar and cross-polar reflectances respectively. Tere is a similar argument for transmittances to tat above for reflectances, leading to a transmittance matrix of te form t t co cr tcr tco were t and t ( ω ) are co-polar and cross-polar transmittances respectively and co cr t ( t + t ) / ; t i( t t ) / co + cr + ( ) t ω being te transmittances of te plate for circular polarizations as treated in te preceding Section.

15 Te significance of te results above is tat values for te circular-polarisation reflectances of a plate can be determined from measured values of te plate s co- and cross-polar reflectances; te former are required in determining values for a material s refractive indices and wave-impedances but te latter are more readily measured over a wide frequency range at ig resolution. 7. Te transmittance and reflectance of a uniaxial gyrotropic plate wit dielectric matcing layers at its surfaces. Dielectric layers will usually be applied to te surfaces of a gyrotropic plate used in a quasi-optical non-reciprocal device to matc as nearly as possible an incident signal beam into (and out of) te plate. Te circular-polarisation ABCD transfer matrices of a layer of an isotropic dielectric material ave te same form as te transfer matrices of te gyrotropic plate as given above, but in tis case te refractive indices and waveimpedances, and terefore also te transfer matrices temselves, are te same for te two circular-polarizations. Te ABCD transfer matrix of te tree-layer sandwic is ten given straigtforwardly by series multiplication of te ABCD transfer matrices of te separate layers (te simplicity of tis procedure is te basic reason for introducing ABCD transfer matrices). Once te ABCD transfer matrix of te sandwic as been determined in tis way, it is a straigtforward matter to determine te transmittances and reflectances of te sandwic since te formulae for te circular-polarisation transmittance and reflectance of te sandwic is given by standard forms r t ( AZ m ib) ( icz + DZ ) ( AZ ib) ( icz DZ ) m + + Z ( AZ ib) ( icz DZ ) m + + were A,B,C, and D are te elements in te transfer matrix of te sandwic. Te ticknesses of te dielectric layers will be cosen, of course, to acieve optimal matcing to te gyrotropic material. 8. Te frequency-dependencies of te elements of te permeability tensor of an exaferrite. Measured variations wit frequency of te values of te elements in te gyrotropic permeability tensors of a exaferrite are broadly accounted for by te following simple dynamical model. In suc a material te energy density of te orbital and spin motions of te atomic electrons tat give rise to te local spontaneous magnetisation, M, increases sarply from an absolute minimum value wen M deviates, wit negligible cange in its magnitude, from a particular crystallograpic direction in te local crystalline environment te axis

16 of te material (ie. te material exibits very strong uniaxial magnetic anisotropy). Tis implies tat, if M is caused to deviate from te axis by an applied transverse magnetic field, it will experience a restoring couple wic can be expressed as M ( H + H ) were H A is a large effective field representing te anisotropy and H is a (true) static magnetic field, bot of wic act along te axis. Now M is intrinsically associated wit te electrons angular momentum density, J M /γ, were γ is te ratio of te magnetic moment of an electron to its associated angular momentum, wic as a value close to tat of ( μ e/ m) were e/ m is te carge-to-mass ratio for an electron. Te unforced classical equation of motion for te electron system is terefore A dj dt dm M HE ; tat is γ M H dt E were HE HA + H (neglecting any damping effect). Tis is a gyroscopic equation of motion and its solution is precession of M around te direction of H E, at frequencyω γ H E, in te clockwise sense looking along H E sinceγ μ e/ m as negative value. If tere is an applied local rf magnetic field transverse to te axis,.exp iωt, te equation of motion is forced : d M γ M ( H E +.exp iωt) dt and substituting M Mkˆ + m exp iωt, were m exp iωt is te induced rf magnetization transverse to te axis, leads directly to te following relations between m and : mx my y HE M m γ γ y i m x i x ω + ω Solving tese simultaneous equations for mx, my, m z in terms of x, y, z gives directly mx ωω m iωω m x 1 m y iωω m ωω m + y ω ω were te caracteristic frequencies are ωm γ M and ω γ H E. Te rf magnetic induction, b μ ( + ), m is tus

17 bx μω ( ) iκω ( ) x b y μ iκ μ + y 1 ωmω ωmω in wic μω ( ) 1 + and κω ( ) ω ω ω ω and ω γ ( H + H) and ωm γ M A Te values of te elements in te permeability tensor in tis model tus exibit resonance beaviours; te two caracteristic frequencies, ω and, ωm being measures of te frequency, and of te strengt, of te resonance, respectively. If te static magnetic field H acting in te material is M, as in a tin plate wit te axis of te material normal to te plate s surfaces, ω ω r ω m were ω r γ H A. Measurements on exaferrites confirm suc resonant beaviours wit values of ω, ω m typically 45 GHz, 1 GHz respectively, in good accord wit measured values of te magnetocrystalline anisotropy and spontaneous magnetization.