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# Adv Electromag Waves ECE 6341

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ECE 6345 Fall 2006 Prof David R Jackson ECE Dept Notes 20 Overview In this set of notes we apply the SDI method to find the field of a finite current sheet Start with the field produced by an infinite phased current sheet derived in Notes 19 Apply superposition to find the field of the finite current sheet Fourier integral Identify a closedform expression for the Fourier transform of the field Identify a TEN model where voltage and current now represent the Fourier transform of the field quantities As an example calculate the field due to a rectangular patch on top of a substrate Finite Source For the infinite current sheet we have 1 x y 10 This produces a field E x y Note the i subscript on the voltage Where functions denotes the p A A voltage due to a one EIO ZEuo XEvo Amp parallel current source 2 L VTM Z VTE 2 whiz 12 iKTEltZgtl150 399 Finite Source cont Recall that 1 N 150 lt2ngt21sltkwkyngtnkxnky Hence p A M 1 N A Em m ltzgtlt27gt2 1skxmkwu AkxAky 2VFE 2 27102 is Wyn2 AkxAky where 22ltkmykyngt WMltZgtKWltWgt f fltkxmkyngt ViTE Z ViTE ktm zgt ktm ltij kin Finite Source cont Adding the contributions from all the phased current sheets we have ejkxmxkyy Akx Aky Zs km kw Finite Source cont From this we can identify gt kw92 22743 VTE2JS 19 or A From this we can make the following TEN identifications TM A N TM A V ZZEtkx9kyaz Is kxaky We Etkxkyz 13132 9110 k x9 y TEN The TEN models are shown below VTM VTE O 0 0 Z TM TE TM A 1 TE A 1 7 1M Zl Patch Fields Patch Fields 39cont Patch Fields cont kx TM ky TEZ FJV z E V U ijzTM2 KTEZ L39Q k Zn Mm lt WEltZgtamp Jm kx 2 TM E V ZJsx inn1wJsxltkxkygte JWdkxdky Ex 3 3 2 WW2 Jsx w l 00 00 t 10 Patch Fields cont Define Also we can write Patch Fields cont The spectraldomain Green s function is the Fourier transform ofthe spatialdomain Green s function Ex j ijx x x y39 y zz Jsx ltx y gt dx dy 700 700 Gxx JSX From the convolution property of Fourier transforms Patch Fields cont Note more generally xx ny Jsx yx ny sz o 1Q Patch Fields cont From the TEN we have VH0 1 Z 0 1 2 K 0 YOT leT cotkzlh 1 TM 080 Y0 k 20 TM 081 Y1 k 21 k TE 0 IO Z 010 k TE 1 fl Z 011 Patch Fields cont Define the denominator term as DT k K5 0 so that Wag YJM mm cotkzlh DTEUc Y0 1Y1 cotkzlh We then have Patch Fields cont Polar Coordinates Use the following change of variables ky dkx dky kt dkt d k 00 00 27roo a jjdkxdkyjjktdktdlt3 I kquot 4 ktdktd 005525 k sin y Polar Coordinates cont Using 2 2 1 00 00 k 1 k 1 39 xx y Exltxy0gtw J z W H WWW CD we have Poles Poles occur when the either of the following conditions are satisfied DTM kt 0 kt kt DTEUQ o kt 2 kg Poles cont This coincides with the wellknown Transverse Resonance Equation TRE for determining the characteristic equation of a guided mode 1 V 4 V V V 7 Y T Kirchhoff s laws I I 1 Hence YTM YTM so that C Imw jKWcmfh 0 0 20 C Poles cont Comparison poles TRE surfacewave mode YBTM Z 0 7OTM COtkZIh 0 YOTM YTM kZO kzo YTM YTM 1 kZI 1 kzl 12 12 kzo kg kf kzo 2 435 2 2 12 k21 kf kf 1 2 kzl k1 TMO 21 A similar comparison holds for the TE case Poles cont Hence we have the conclusion that That is the poles are located at the wavenumbers of the guided modes the surfacewave modes 22 Poles cont The complex plane thus has poles on the real axis at the wavenumbers of the surface waves Imk t t X IXIR k 23 Path of Integration The path avoids the poles by going above them 1m kt lossy case LR hR I L I C k0 x k1 Re kt 1m kt lossy case This path may be used for numerical computation LR gt C hR i I k0 x k1 Re kt 24 Path of Integration cont The path avoids the poles by going above them 1m kt lossless case LR C hR x I k0 k1 Re kt hR 005 k0 L k 1 1 Practical note If hR is too small we are too close R 1 to the pole If hR is too large there is too much typical Choices roundoff error due to exponential growth in the 25 snn and cos functions Branch Points To explain why we have branch points consider the TM function DTM kt W ijM comm 080 081 t k h kw Jk210021 Branch Points cont 12 kw k k2gt ko kt12k0 12 kt 12 jkt k0 kt Note the representation of the square root of 1 as j is arbitrary here jkt k0 2 k k0 2 1ng kt lt kogt kt 4c k0 k0 r 7 k0 27 Branch Points cont kzo Jkz k0 12 kt k0 12 Jgk k0 k k0 em em Branch cuts are necessary to prevent the angles from changing by 27 Imk t Note the shape of the k branch cuts is arbitrary but vertical cuts are shown here 2 1 Rek 28 Branch Points cont kzo kt k0 12 kt k0 12 We obtain the correct signs for kzo if we choose the following principal branches 29 Branch Points cont The wavenumber kzo is then uniquely defined everywhere in the complex plane kzo Jkr k0 12 kt k0 12 Jk k0 k k0 em eW Imk t 2 1 Rek 30 Riemann Surface The Riemann surface is a pair of complex planes connected by ramps where the branch cuts used to be The angles change continuously over the surface All possible values of the wavenumber are found on the surface Imk t M 2 Rck Example top sheet 17r4 27r6 bottom sheet 1 7r 4 27 2 7r6 31 Riemann Surface cont Riemann surface forz 2 y z requ top sheet r c 7r lt lt 7 lt x bottom sheet top view 7 lt lt 37 Note a horizontal branch out has been arbitrarily Chosen 32 Riemann Surface cont Riemann surface forz 2 x 3D view top bottom DC x D B side view 1 top View 33 Sommerfeld Branch Cuts Sommerfeld branch cuts are a convenient choice for theoretical purposes discussed more in ECE 6341 ImUcZo 0 on branch out Imk t k02 k2 12 1m kzo lt 0 top sheet k0 Z Im kzo gt 0 bottom sheet Rek t k0 k0 34 Complex Plane Ex ff218 d 1aFkta dict top sheet 35 POLYTECHNIC INSTITUTE OF BROOKLYN GRADUATE CENTER Electrophysics Department 1963 Short Course on quotMicrowave Field and Network Techniquesquot Tuesday June 4 1963 Radiating periodic strucutures Analysis in terms of 1 vs diagrams A A Oliner Contents 1 Introduction 1 2 Propagation in closed periodically loaded waveguides 2 21 Space harmonics 2 3 k vs 3 Diagrams and Mode Coupling77 3 4 Some Properties of the Dispersion Curves and the Space Harmonics 5 5 Propagation Along Open Periodic Structures 8 51 The k vs 3 Diagram Applied to Open Periodic Structures 8 6 Application to radiating structures 13 61 Periodically Modulated Slow Wave Antennas 13 62 Periodic Fast Wave Structures 17 7 Frequency Scanning Antennas 20 8 Diffraction Grating 23 References 29 Radiating periodic structures analysis in terms of k vs 6 diagrams 1 Introduction Dispersion plots in the form of k vs 3 diagrams have long been found useful or understanding the behavior of electromagnetic periodic structures in closed waveguides Such structures have found application in linear accelerators traveling wave tubes exible corrugated waveguide certain types of lters and delay lines etc Applications such as arti cial dielectrics and corru gated surface wave lines involve periodic structures which a r e essentially open in character since a portion of the guide crosssection is unbounded but in general only the bound wave prop erties have been considered In recent years it has been recognized that these k vs 3 diagrams may also be used to advantage in explaining the behavior of radiating periodic structures The purpose of these notes is to present an understanding of the new features which arise when these diagrams are used in connection with radiating periodic structures Section B which is devoted to closed periodic structures only establishes the point of view adopted here and clari es the basic notation and concepts The terms space harmonic and mode coupling77 are reviewed and the k vs 3 diagram is described The characteristic differences in the features of the k vs 3 diagrams when they are descriptive of open structures are discussed in Sec C The concepts of radiation region surface wave and leaky wave are brought into context and the changes occur ring when mode coupling effects arise in radiating structures are touched upon The concepts and procedures discussed in Secs B and C are then applied in Sec D to a variety of radiating periodic structures The properties of several types of antennas are examined by the use of the k vs 3 diagrams a procedure is employed which permits one to obtain a considerable amount of information in a rapid fashion without actually solving any eld problerns Application is made to periodically modlilated slowwave antennas fast wave arrays and frequency scanning serpen tine structures Finally the case 0 scattering by a diffraction grating is treated by these methods as an illustration of the fact that k vs 3 diagrams are useful not only for an understanding of guiding and radiation characteristics but also for the description of scattering phenomena 80563 2 2 Propagation in closed periodicallyloaded waveguides In order to establish the point 0 View adopted here and to clarify some 0 the concepts we shail rst restrict our attention to the propagation characteristics in closed periodically loaded waveguides The modi cations and additional features introduced by the open nature of the structure are considered in Sec C 21 Space harmonics Let us consider a typical periodically loaded waveguide of the form shown in Fig 1 period 1 represents the length of a unit cell and z is the transmission direction If the time dependence is chosen as 57 and if the cross sectional dependence is suppressed Floquet s theorem may be stated as W2 e jkOZPWL 1 where Pz Pz d 2 and where represents some eld quantity These relations state that the eld behavior is describable in terms of a fundamental traveling wave with propagation wavenumber kg and a standing wave Pz which is the same in each period 1 and which represents the local variations due to the periodicity which is present This standing wave may alternatively be represented as a superposition of traveling z the representation is accomplished by expanding the periodic function Pz in a Fourier series as 132 Z Pne sz 3 it is readily seen that expansion 3 also satis es condition When 3 is inserted into 1 the latter becomes 132 Z P e iWMTW 4 00 132 Z Pne jkM 5 n7oo 80563 3 with 27139 knk0j n0i1i2 6 Relation 5 for the eld quantity indicates that the eld is expressible in terms of an in nite nurnber of traveling wave components the separate wave components are called space harmonics in analogy to the familiar time harmonic expansion for an arbitrary function in time The values of kn are thus seen to represent physically the propagation wavenumbers of these space harmonic contributions to the total eld It should be noted that the space harmonics do not exist independently and are not modal solutions by themselves they are portions of a total solution A customary analytic procedure utilizes the standing wave formulation for the losai periodic effects implicit in Floquet s theorem 1 and 2 and explicitly considers the fundamental n 0 wave only Such considerations are suf cient to permit the determination of all the macroscopic properties of the periodic structure The alternative representation in ternis of space harmonics permits the exploration of certain microscopic properties and adds insight into many features of the behavior of the periodic structures Space harmonic considerations have found wide use in treatments 0 microwave tubes and have recently been introduced into antenna analyses 3 k vs Diagrams and Mode Coupling77 In order to examine the way in which the space harmonics in uente the dispersion curves let us rst consider a guiding structure before the introduction of periodicity This guiding structure possesses a propagation waveiiumber which Sy itself is characterized by a dispersion curve Figure 7a exhibits this dispersion curve for the case of an air lled TEM line in the form of a k vs 3 diagram Such a diagram differs only slightly from the more familiar w vs 3 diagram but is preferable for our needs as will be shown below Quantity k is the free space wavenumber with 27139 k 7 7 A lt gt where is the free space wavenumber and is related to w by w k 7 Auxp060 8 C where c is the velocity of light in vacuum The propagation wavenumber k may be complex and is then written as k 3 7 ja 9 so that the phase constant is simply the real part of k From 8 one sees that a k vs 3 diagram differs from an an vs 3 diagram only by a constant in the ordinate value Since the phase and group velocities are de ned respectively as 11 g lt10 80563 4 do 7 11 M lt gt these quantities are given by the location of a point on a dispersion curve in an to vs 3 plot and its slope at that point In a k vs 3 diagram however the points location and slope represent respectively the phase and group velocities normalized to free space velocity as U9 U 7 E 7 lt12 1 dk 9 a 13 The latter formulation is more useful or radiation problems as will be seen later Referring to the dispersion curve itself in Fig 2a it is seen to consist of two straight lines at i45o since 3 ik for an air lled TEM line The line with the negative slope refers to the wave traveling in the negative 2 direction Note that for this example which was chosen for 45 45 0 41rd 21rd 0 21rd 41rld 6 b Fig 2 k VS 5 diagram and dispersion curves for an unloaded TEM line a Dispersion curve for an air lled TEM line b Same as a but including all the space harmonics of the unloaded line when in nitesimal periodic loading is introduced simplicity the relative phase and group velocities are equal to each other and to i1 Now let us introduce periodic elements in an in nitesimal fashion so that the original guiding structure is really unperturbed The effect of these negligible periodic perturbations is thus to introduce the in nite number of space harmonics but to leave undisturbed the shape of the initial dispersion curve Each of these space harmonics will also possess a dispersion curve of its own and in the unperturbed state de ned above each space harmonic does not interact with any other space harmonic In view of relation 6 the curves for the space harmonics are all displaced by 2n7rd from each other as shown in Fig 2b The solid lines in Fig 2b represent the original n 0 dispersion curves of the unloaded line the dashed lines represent the space harmonic dispersion curves introduced by the periodicity but still representative of the unloaded line We now follow the point of view introduced by Pierce1 rather generally and by Pierce and Tien2 to explain qualitatively the dispersion curve behavior for a tape helix This point of view known as mode coupling asserts that the dispersion curves of the loaded structure may be obtained by taking into account the coupling occurring between the modes of the unloaded structure 80563 5 Pierce1 points out that for passive structures the coupling may be of either of the two types shown in Figs 3a and 3b The coupling in Fig 3a is of the directional coupler type and occurs when the two interacting modes both possess the same sign for the group velocity The coupling shown in Fig 3b which is pertinent for modes with oppositely directed group veloc ities results in a stop band Within such a stop band k is complex and therefore unattenuated propagation is not possible The dashed lines refer to the modes of the unloaded line which are being coupled together by the introduction 0 the periodic elements the solid lines represent the resulting dispersion curve for the loaded line We now employ these ideas of mode coupling to determine qualitatively the shape of the dispersion curves or a TEM line periodically loaded with small perturbing elements Reference to Fig 2b and comparison with Fig 3 indicates that the only type of coupling which occurs here is that of Fig 3b We therefore expect that stop bands will be present and that the resulting dispersion curves for the loaded structure will be roughly like those shown by the solid curves in Fig 4a The precise shape will depend on the nature of the perturbing periodic elements Although B has a constant value of mrd in the stop bands it is customary for closed structures to indicate the values of B only when k is purely real Fig 3 Modecoupling e ects a Directional coupler behavior b Stop band behavior 4 Some Properties of the Dispersion Curves and the Space Harmonics It is seen that the various branches of the complete dispersion curve in Fig 4a are assigned numbers according to which space harmonic it corresponds The n 0 branches evidently correspond to the original basic wave while the others follow directly from relation 6 with those of negative slope being the mirror image of those of positive slope This identi cation according to space harmonic number is of particular value in the analysis of radiating structures The dispersion curve is repetitive every Bd 27139 along the abscissa axis but in general not along the ordinate kd axis The vertical strip from 3d in to Bd 71 corresponds to the rst Brillouin zone for these structures which are periodic in one dimension only Suppose now that the periodically loaded structure is excited at one end only and that the structure is either in nitely long or is terminated in a matched non reflecting load It is known 80563 6 that all waves excited by a source must possess a group velocity corresponding to the propagation of energy away from the source Hence for uni directional excitation in the positive 2 direction all of the space harmonics must possess positive group velocities so that only those branches of the dispersion curve which have positive slope are applicable This situation is illustrated by the solid curves in Fig 4b The dashed portions of the complete dispersion curve are not appropriate for such unidirectional excitation and become valid only if a termination or discontinuity is present which sets up a backward traveling wave 0 land I a 5 q I 9 K I x x P 3 1 A 5quot Fig 4 Cd VS pd diagrams for a periodicallyloaded closed TEM line a Complete dispersion curve With space harmonic labeling indicated b Unidirectional excitation Inspection of Fig 4b also indicates that all of the space harmonics corresponding to a given frequency kd possess the same slope and therefore the same velocity This result is in agreement with physical expectations since the enecgy in the total eld must travel as a unit The phase velocities of the various space harmonics are seen to be very different from each other however Some phase velocities are fast compared to the freespace velocity While others are slow compared to it This fact may be observed either directly from Fig 4b or by noting that Ui2 n012 14 0 30 71 80563 7 In addition it is seen that for all space harmonics for which 3 is negative the phase and group velocities are of opposite sign These waves are called backward waves It is noted however that they are actually forward traveling and that backward waves should not be confused with backward traveling waves In a stop band the propagation wavenumbers km of all the space harmonics are complex Since the kn values of the various harmonics differ from each other by 27md which is a real number the attenuation constant a must be the same for all of the space harmonics and k is then written as kn lt30 7 ja 15 While the complete eld solution is seen to consist of an in nite number of space harmonics and the complete dispersion curve of an in nite number of branches in practice only a nite number of these is signi cant The fact is that in general the amplitude of a space harmonic drops off sharply as the number n of the harmonic moves away from n 0 Except near the stop bands the amplitude of the n 0 space harmonic is usually dominant At the stop bands the amplitudes of the two space harmonics which cross to produce the stop band are large and equal so as to produce the standing wave associated with the stop band Near the stop bands the amplitudes of the other space harmonics generally increase somewhat The actual distribution of amplitudes among the space harmonics obviously depends on the structure under consideration Although the periodicity exists only along one dimension the guiding structure has cross section dimensions and the eld possesses a transverse as well as a longitudinal variation Then unless the structure can at every cross section plane be characterized by the same single trans verse wavenumber as in a uniform waveguide loaded periodically by a series of dielectric slabs each of which completely lls the guide cross section each space harmohic will possess a different transverse wavenumber The transverse wavenumber km of the nth space harmonic is given by 2 2 kt k2 7 k0 71 16 using If n is such that k0 2372 gt k2 as will be the case for most of the space d harmonics at any given frequency then the space harmonic is a slow wave and km is imaginary The transverse variation for this space harmonic thus resemblesthat present in a waveguide below cut off For example suppose that the structure under consideration is a parallel plate waveguide periodically loqded by shunt capacitive irises with their openings along the guide mid plane The space harmonics which are slow waves would then decay transversely toward this mid plane since the space harmonics with higher values of 71 would decay more rapidly the eld at the mid plane due to the slow space harmonics would consist essentially of contributions from the lowest few harmonics only If some space harmonics are present for which k0 2n2 lt k2 then these space harmonics are fast waves and the corresponding km values are real Thus in the example above the transverse eld distribution will be trigonometric in nature and the contribution to the eld at the mid plane may be high The signi cance ofa particular space harmonic in a given application 80563 8 may thus depend not only on its amplitude but also on its transverse dependence This feature will be strikingly evident when we consider radiation effects associated with open structures Space harmonics are not to be confused with higher modes Higher modes with respect to the longitudinal direction are set up at each of the periodically located discontinuities in a periodic structure These higher modes are all below cut off longitudinally however in contrast to the space harmonics which are all propagating above cut off in the longitudinal direction These higher modes are essentialiy taken into account by treating the discontinuities as lumped insofar as the dominant mode is the unloaded waveguide is concerned The higher modes are ignored thereafter We conclude this section by pointing out that some signi cant effects can be produced on k vs 3 diagrams by certain symmetries in the periodic structure The typical diagram of Fig 4a shows the presence of stop bands due to coupling between the originally non coupling space harmonics of the unloaded waveguide The introduction of the actual periodic elements produced the coupling between these space harmonics and thereby created the stop bands Now it can be shown3 that if certain symmetries are present in the periodic structure not all of these space harmonics of the unloaded guide couple and simple crossings rather than stop bands are produced for particular pairs of harmonics Two classes of symmetry that fall into this category are the screw and the glide symmetries The screw is a translation rotation symmetry while the glide corresponds to a translation re ection symmetry Further information on the effects of these symmetries is available3 but this material will not be included here 5 Propagation Along Open Periodic Structures We designate as open structures those whose cross sections are unbounded in some transverse direction Such structures usually possess a well de ned interface which separates the guiding region from the external open region We shall be concerned with the eld behavior in this open region and shall regard the region as being bounded on one side by the interface The 2 direction is then taken as the direction along the interface on which the guiding occurs k and kt again represent the propagation wavenumbers along the guiding direction and perpendicular to it Examples of open periodic structures are corrugated surfaces modulated surfacewave antennas slot arrays helices etc 51 The k vs 6 Diagram Applied to Open Periodic Structures The k vs 3 diagram is also applicable to open periodic structures but certain signi cant and characteristic differences arise here as compared to its use with closed guides Consider the kd vs Bd diagram shown in Fig 5 If a point on the dispersion curve for the open structure lies anywhere within the lined region then that space harmonic corresponds to a fast wave The related transverse wavenumber k is then real according to 16 In closed waveguides a real value of k signi ed a trigonometric rather than a hyperbolic type of transverse eld distribution for the space harmonic in question and the longitudinal wavenumber k remained purely real ln open structures however there is no opposite wall to re ect the wave which is transversely above 80563 9 cut off for real values of km and as a result radiation of energy occurs away from the interface and kn becomes complex The lined region is therefore properly labeled as the radiation region Now consider a point such a s the one marked a on a dispersion curve located inside one of the triangles Since the other space harmonic solutions differ from this one by 3d 2n7r we see that all other space harmonic solutions also lie within the sequence of triangles and that no space harmonic solution exists in the radiation region None of the space harmonics is radiating therefore under these conditions Hence if a space harmonic solution occurs within one of these triangles the total eld is purely bound to the interface and the guiding structure below it If the solution appears at point b however then at least one space harmonic solution appears within the radiation region and the total wave is no longer completely bound Sensiper4 was the rst to point out the signi cance of these triangles He in fact was interested in completely bound solutions only for the case of the tape helix which of course is an open structure and he termed the region outside the triangles as the forbidden region since solutions in that region did not yield purely real values of kn The discussion above neglects certain ne structure considerations wherein the solution may lie within a triangle but near to one of its sides under such conditions a space harmoiuc may be radiating even though its phase velocity is slow This point will be touched on in Sec 4 For most purposes however the above discussion is valid If the space harmonic solutions lie within the triangles therefore the total eld is completely bound the propagation wavenumbers kn of all of the space harmonics are purely real and all values of km are purely imaginary Such a solution is called a surface waue and its properties are considered further in Sec 2 below If at least one space harmonic solution lies in the radiation region and all space harmonic solutions are therefore outside the abovementioned triangles the total eld is no longer com pletely bound and radiation away from the interface is produced by those solutions which lie in the radiation region The values of km for all of the space harmonics are then complex with the same value of attenuation constant 04 for all of them The values of km are correspondingly also complex the space harmonics which appear in the radiation region of the k vs 3 diagram and which therefore contribute to the radiation possess values of km which are essentially real but which contain a small attenuation 04m contribution since kn is complex Such space harmonics are said to be essentially radiating Those space harmonic solutions which are outside the radia tion region have km values which are essentially imaginary but which contain a small phase 3m contribution since k is complex such space harmonics are called essentially bound Now that we have made clear that in a radiating periodic structure the space harmonics are not either completely bound or completely radiating we shall for convenience drop the qualifying word essentially These radiating space harmonic solutions are called leaky waues and are discussed further in Sec 3 From a knowledge of only the periodicity and the dispersion curve for the unloaded structure one can very quickly obtain information regarding the number of radiating beams the angles in which these beams point the transverse dependences of each of these beams etc The kd vs 3d diagrams furnish a graphical and visual mechanism for nding these results which may be regarded as the kinematic features of the problem Not determinable in this fashion is the ampli tude distribution among the various space harmonics or therefore the actual power radiated in SC563 10 each beam and the total eld distribution This amplitude information can be found only after solving the complete problem for the particular structure in question The beamwidth of the radiating beams is also not determinable unless one knows the length of the radiating structure and the equivalent aperture distribution The number of radiating beams at any given frequency is directly equal to the number of space harmonic solutions lying in the radiation region at that value of kd The beam angle 9 from broadside is given by sin 0 17 as seen in Fig 6a The relation to angle 1 shown in Fig 6b is seen to be sin 0 tan 1 18 A location on the kd axis of Fig 6b is therefore seen to correspond to broadside radiation while points lying on the 450 and 745 lines correspond to forward end re and backward end re respectively The full value of these kd vs 3d diagrams with respect to the kinematic features of the radiation will be evident when we apply them later to speci c structures Surface Waves A surface wave on an interface possesses a eld which is completely bound to the interface The propagation wavenumber along the interface corresponds to that for a slow wave and the eld decays exponentially away from the interface in a direction transverse to it This familiar wave type occurs on a variety of uniform open structures the most common of these being a dielectric slab For an open periodic structure a completely bound total eld does not occur unless every space harmonic possesses a slow phase velocity This is not always possible for a given structure but when it is it occurs for suf ciently low frequencies or suf ciently small periods In terms of the kd vs 3d diagrams a surface wave is possible when the solutions appear inside the sequence of triangles shown in Fig 5 The best known example of an open periodic structure which supports a surface wave is probably the corrugated surface in the range for which the frequency and the corrugation spacings are appropriately low A dielectric slab which is periodically grooved or otherwise loaded furnishes another example A helix is a third example In all of these cases the surface wave is no longer maintained when the frequency or the period ie the product kd is increased suf ciently Leaky Waves When the value of kd is made suf ciently high a surface wave is no longer possible and one or more space harmonics will become radiating In terms of the kd vs Bd diagrams radiation will occur when the solutions lie outside the set of triangles and the space harmonics which radiate are those which lie in the radiation region These radiating harmonics are leaky waves and possess properties identical to leaky waves on uniform structures SC563 11 Leaky waves on uniform structures were rst recognized in connection with so called leaky wave antennas the best known of these being a slitted rectangular waveguide The eld distri bution associated with this leaky wave is shown in Fig 7a in which the density of the rays is a measure of the radiated power density The leaky wave is characterized by a positive attenu ation constant 04 and phase constant 3 along the interface so that more power is radiated per unit length near the beginning of the cut It is also seen that the eld increases as one moves perpendicularly from the waveguide wall up to a maximum value related to the source location and then is zero above that height The eld therefore exists only in the wedgeshaped region shown Since this wave type would diverge at in nity in the transverse plane if it were de ned everywhere it cannot be spectral ie a proper mode The leaky wave of Fig 7a is thus seen to be a forward wave and to be non spectral A backward leaky wave is also a possible wave type and its eld distribution is shown in Fig 7b The eld lines show that in the forward direction the wave is again characterized by a positive 04 along the interface however since the wave radiates in the backward direction the eld is seen to decrease transversely This transverse decay rather than increase permits this wave type to be spectral Fig 7 b also shows the region over which the wave is de ned We have thus observed that two types of leaky wave are possible a forward leaky wave which is non spectral and a backward leaky wave which is spectral these two types being shown respectively in Figs 7 a and 7 The spectral or non spectral character of these waves should not be of concern here however as they manifest themselves only in the regions shown in Figs 7 a and b within which they are physically well de ned We shall next identify the variaus beam directions of the radiating space harmonics with these leaky waves Recall rst that k2 193 k3 19 so that k2 33 7 a2 7 52 7 04 7 23 043 7 amp 20 where n 30 2 7L Since the imaginary part of k2 is zero we have that 04 27139 at 7 7 lt30 21 Since the longitudinal attenuation constant 04 which is the same for all space harmonics and the transverse phase constant 3 of the outgoing nth space harmonic are both positive the sign of the transverse attenuation constant 04m depends on the sign of 30 If 30 277 gt 0 the wave is a forward wave and from 21 am lt 0 and the wave is non spectral If 30 2 rm lt O we have a backward wave and from 21 am gt 0 and the wave is spectral These conclusions agree exactly with the leaky waves described in Figs 7a and 7 respectively With reference to the kd vs Bd diagram these considerations are summarized in Fig 8 More sophisticated considerations which are beyond the scope of this discussion indicate that the leaky waves are actually exponentially decaying in any radial direction from the origin of the excitation and that they therefore cannot contribute directly to the radiation eld They do however strongly in uence the shape of the radiation pattern which exhibits peaks that are SC563 12 shifted only slightly in angle from the leaky wave directions Further information on leaky waves is available in the literature references ModeCoupling Effect in the Radiation Region In Sec B it was pointed out that in closed periodic structures stop bands are produced in the dispersion curve k vs 3 diagram by coupling between pairs of space harmonics of the unloaded vaveguide This coupling is created by the introduction of the periodic loading and is loosely termed mode coupling When two such space harmonics of an unloaded open structure cross and couple inside one of the triangles indicating that for this solution the total eld is purely bound a stop band is produced in the same manner as for a completely closed waveguide since no radiation effects are present Such7 a stop band is produced for example at Bd mr 71 odd for periodically modulated slow wave structures as mentioned in Sec D1 When two such space harmonics cross outside the sequence 0 triangles the effects produced on the dispersion curve are different because of the attendant radiation The usual stop band is characterized by a range of frequencies or which k becomes complex and by zero slope for the dispersion curve at the edges of the stop band When radiation is present the wavenumber k is already complex and the abovementioned mode coupling does not result in a band effect There is a signi cant effect produced on the eld behavior however The amplitude of the radiating space harmonic drops drastically as a result of this coupling and the amount of radiated power is similarly reduced In addition the amplitudes of the pertinent pair of oppositely directed space harmonics become eqtial resulting in a standing wave effect As a result the input impedance to the radiating structure is altered signi cantly and the amount of power is markedly reduced at these mode coupling points Such modecoupling intersections in the radiation region are known to occur at Bd mr 71 even It is shown in Sec D that radiation at broadside falls into this category so that the behavior corresponding to that angle should be suspect It is well known however that arrays customarily cannot be designed to point precisely at broadside but are made to point a few degrees off broadside As an array i s scaraed through broadside the radiated power drops and the input VSWR rises sharply These well known experimental observations are in agreement with the modecoupling effects described above At the end of Sec B it is pointed out that these mode coupling effects for certain pairs of space harmonics can be avoided by the use of particular symmetries in the periodic structure The same symmetry considerations are applicable to open periodic structures The deleterious behavior at broadside referred to above can therefore be eliminated by an appropriate choice of symmetry in the radiating elements These considerations will not be pursued further here however Another type of mode coupling occurs at the crossing of the dispersion curve with one of the sides of a triangle The side of a triangle can be shown to correspond to a plane wave skimming along the surface The mode coupling here is thus due to the interaction between the particular space harmonic in question and a plane wave traveling along the surface This coupling produces a characteristic ne structure just inside the triangle the details of this structure differ SC563 13 depending on whether the slopes of the space harmonic and the triangle side have the same or opposite signs These details are somewhat involved and are beyond the scope of this discussion We will simply comrnent that complex solutions do penetrate somewhat into the triangle in the vicinity of this coupling region Although the effects associated with mode coupling77 as discussed above can be 0 considerable signi cance at times for example the broadside radiation case they are touched upon here only in passing 2nd as a caution The kd vs Bd diagrams are useful in revealing information on many other aspects of the behavior of radiating periodic structures and we wish rather to stress some of these 6 Application to radiating structures The space harmonic concept and the properties of the kd vs Bd diagrams are used in this section to display the radiation characteristics 0 several types of open periodic structures The antenna types to be considered are periodically modulated slow wave antennas fast wave antennas such as slotted waveguide arrays and frequency scanning antennas The radiating waves in all of these cases are leaky waves of the type described in Sec C3 The kd vs Bd diagrams are useful not only for an understanding 0 guiding and radiation characteristics but also for the description of scattering phenomena To illustrate the treatment 0 this class of applications in Sec 4 below we consider the case of scattering by a diffraction grating cod 45 45 Radiati 11 region x x x B x I O A I 74 4n 0 27r 47r id Fig 5 Cd VS 5d diagram for an open structure indicating the radiation region and the triangles or purely bound solutions 61 PeriodicallyModulated SlowWave Antennas The class of structures whose behavior is to be described next includes any slowwave structure which is periodically modulated Familiar surfacewave structures which fall into this category SC563 14 cod 45 kod 45 0 and id b Fig 6 Radiating space harmonic a Angle of radiation b Location on kd VS 5d diagram may be exempli ed by dielectric slabs dielectric rods Goubau lines or disk on rod structures when these are modulated periodically by grooving or surrounding by an array of disks or varying the cross section sinusoidally etc Other examples arc helices with suf ciently large spacing helices loaded periodically by probes rectangular waveguide serpentine structures with slot loading and even slot arrays in dielectric loaded waveguides with the value of dielectric constant suf ciently high to produce a phase velocity slow compared to the freespace velocity It is clear then that the discussion below applies to a rather large class of microwave structures Figure 9 illustrates several of these structures We wish to obtain information regarding the principal radiation features of such structures without explicitly solving any eld problems If we know only the periodicity and the dispersion curve of the structure before the periodic loading the space harmonic concept and its utilization in the kd vs 3d diagram permits the ready abstraction of many important radiation character istics This procedure is useful not only for qualitative understanding but for small loadings for semi quantitative results as well The unloaded dispersion curves for the various slow wave structures shown in Fig 9 differ somewhat from each other but a simple approximation to all of them is a straight line of appropriate slope emanating from the origin of the kd vs Bd diagram This simpli ed curve corresponds to assuming that the equivalent surface reactance associated with the unloaded structure is frequency independent The discussion below assumes the use of such a dispersion curve it is a simple matter to replace this curve by the actual unloaded dispersion curve if desired and follow the same procedure Figure 10 shows this straight line dispersion curve for a slow wave guide together with several space harmonic solutions superimposed on a kd vs Bd diagram for the case of uni directional excitation The slope of the line corresponds to a relative phase velocity U C equal to 05 Since the equivalent surface reactance normalized to free space impedance is related to the phase velocity by vi 22 CW7 the above value of phase velocity is equivalent to specifying a nornialized surface reactance 7 SC563 15 Fig 7 Leaky ane eld distributions a Forward nonspectral ane b Backward spectral ane When the space harmonic solutions lie within the sequence of triangles the total eld i s completely bound to the periodically modulated slow wave structure Stop bands located at 3d 7139 37139 are also present within the triangles but are not shown in Fig 10 because they are of lesser interest at this time These stop bands are in the same manner as those of Fig 4 ie due to coupling between the forward and backward directed space harmonics of the unloaded structure of which only the forward directed set are shown in Fig 10 k d 45quot 45 Leaky wave is Leaky wave is backward and forward and non spectml spectral 0 M Fig 8 Spectral character of leaky anes in thekd VS 5d diagram 1 When the n 0 dispersion curve crosses the side of its triangle the n 71 curve is seen to just enter the radiation region At this value of kd the n 71 space harmonic is the only one in the radiation region The angle of radiation is seen from Fig 6 to correspond to backward end re ie 01 790 measured from broadside As kd increases ie as either the frequency or the spacing increases the radiated beam lifts up from the backward end re position and approaches broadside 01 0 If we follow the n 71 space harmonic as kd increases further we note that it passes through broadside and continues swinging through a sequence of angles until it reaches forward end re For stiii higher values of kd the space harmonic becomes essentially bound and no longer contributes to the radiation SC563 16 a b e9eth wwmwm 0 d Fig 9 Ebramples of periodially modulated slowwave sstructures a Striploaded dielectric slab b Grooved dielectric slab C Gaubou line with periodic rings d Sinusoidally modulated diskonrod structure For some value of kd during the excursion of the n 71 space harmonic between broadside and forward end re we see from Fig 10 that the n 72 space harmonic just enters the radiation region Again upon entry it points in the backward end re direction and then lifts up and approaches broadside as the value of kd increases eventuaiiy terminating as a bound space harmonic after passing through forward end re For a range of kd it is clear that both then 71 and the n 72 space harmonics are radiating simultaneously but at different angles of course For the particular choice of U gtc made in Fig 10 namely 05 the n 73 space harmonic just enters the radiation region and points in the backward end re direction as the n 71 space harmonic reaches the other side of the radiation region and radiates in the forward end re direction For a larger value of phase velocity and therefore a smaller value of surface reactance and a steeper slope for the straight line dispersion curve in Fig 10 three or more beams radiating space harmonics may be present simultaneously Such a situation is illustrated in Fig 11 in which the n 71 72 and 73 space harmonics are seen to be radiating at the value of kd indicated by the dashed horizontal line The angles of radiation are seen to be somewhere near forward end ire broadside and backward end re respectively The number of radiating beams at any speci ed value of kd may be determined as indicated above the radiation angles are found via Eqs 17 or 18 and the transverse eld dependence if desired is obtainable from 16 For some applications such asfrequency scanning we desire that ohly a single radiating beam be present while this beam is scanned from 790 to 90 This situation is readily achieved by making the phase velocity suf ciently slow ie increasing the surface reactance suf ciently thereby producing a small enough slope for the straight line dispersion curve Figure 12 demon strates a typical case of this type it is seen that while the n 71 space harmonic is in the radiation region no other space harmonic is radiating In fact for a short range of frequencies just above that for which forward end re occurs the total eld is completely bound The condi tion on the phase velocity such that only one beam is present while the n 71 space harmonic scans is seen by inspection to be SC563 17 it lt l c 7 3 The value of the longitudinal attenuation constant a associated with the radiation or the relative power radiated by each beam cannot be found from these simple considerations but requires the inspection of the actual amount of perturbation introduced by the periodic loadings An example of a periodic slow wave structure for which a rigorous analysis is available is a sinusoidally modulated reactance surface dispersion curves and relative amplitudes have been evaluated for various parameter values These periodically modulated slow wave structures are useful as leaky wave antennas They have the virtue that they can be designed easily to radiate at any desired angle with or without other beams present and that no phase reversals are required as with fast wave structure for angles near broadside or in the backward direction By applying a small loading per unit length a small value of a can be obtained with an attendant large aperture and narrow far eld beam Such antennas however are inherently narrow band The end re case requires special attention and will not be elaborated on here The actual leaky wave feature vanishes at this limiting angle and the structure becomes a modi ed surface wave antenna 23 62 Periodic FastWave Structures We next consider the behavior of a class of structures possessing a fast rather than a slow phase veloclty along the longitudinal direction Such structures also have a non zero cut off wavenumber The category of fast wave arrays includes all the familiar slotted and probe loaded waveguides with both reversed and unreversed loadings We again proceed to gain insight into the behavior of this class of structures from the knowledge oi the periodicity and the dispersion curve for the unloaded structure Again we neglect at this stage the various modecoupling effects since we wish at rst to determine only the principal radiation features Let us employ the dispersion curve for an ordinary air lled waveguide with a non zero cut off wavenumber kc kt and rst consider the case of period 1 small compared to kc say an array of closely spaced small holes Under these conditions ktd ltlt 7139 for dominant mode in rectangular guide for example kt kc 7ra so that ktd 7rda and is small when 1 lt a The kd vs 3d diagram for these circumstances is shown in Fig 13 which includes only the n 0 and n 71 space harmonics for uni directional excitation We see that for the lower frequencies the n 0 space harmonic radiates and that is the only beam present until kd becomes slight greater than 7139 Only one beam is therefore present for a considerable range in frequency An important distinction form the case of slow wave structures is evident there the n 0 space harmonic was always bound The above case for which the spacing d between periodic elements is small is actually the so called leaky wave antenna case in which the periodicity plays an important role Essentially the same behavior would be obtained if the array of holes say were replaced by a continuous long slit Two well known leaky wave antennas with closely spaced periodic elements are shown in Fig 14 We note from Fig 13 that the range of scan angles 1s limited for the radiating n 0 space harmonic One cannot approach broadside too closely because the structure is at cut off there SC563 18 led 45 45 41r 21r 0 21139 41r ll Fig 10 Cd VS 5d diagram for a periodicallyloaded slowwave structure possessing a frequencyindependent surface reactance 4m 47r 21r 0 21r 41r d Fig 11 Cd VS 5d diagram for a slowwave structure with a basic n 0 wave which is only slightly slow and unless the forward end re region is avoided the n 71 space harmonic will also become radiating In addition7 the n 0 space harmonic cannot radiate in the backward direction for forward traveling excitation If the spacing d between successive identical periodic elements is now increased considerably so that d is now of the order of a7 the relevant kd vs Bd diagram becomes that presented in Fig 15 Note that the value of k relative to ld determines the location of the dispersion curve relative to the triangles It is now easy to achieve many beams simultaneously Fig 15 illustrates how three simultaneous beams can be obtained at a given frequency denoted by the horizontal dashed line We also see that the forward radiating n 0 space harmonic is always present SC563 19 Fig 12 Cd VS 5d diagram for a slowwave structure with a Very slow basic n 0 wave Thus if we wish to radiate in the backward direction with the n 71 space harmonic we must accept the simultaneous presence of the forward beam The standard technique for eliminating this unwanted n 0 beam is to employ a has se reversal between successive periodic elements Several familiar examples of arrays which employ such phase reversal techniques are illustrated in Fig 16 The insertion of a phase reversal between successive elements produces an added phase shift per period of 7139 so that the effective value of B is altered In fact Be ectived 1 7r7 so that if we assume that the mutual impedance effects between neighboring elements are affected negligibly by the phase reversals the new dispersion curves are the same as the old ones except for a shift to the right of 7139 in the kd vs Bd diagram Alternatively one can retain the dispersion curves as before and instead shift the radiation region by 7139 to the left The kd vs Bd diagram of Fig 17 adopts the latter choice retaining the same dispersion curves as those shown in Fig 15 It is seen from Fig 17 that the phase reversal has removed the n 0 space harmonic from the radiation region and that the n 71 beam can now be scanned from backward end re past broadside and up to some angle in the forward direction before the n 72 space harmonic enters the radiation region One also sees that in order to suppress the n 0 space harmonic completely we must have ktd lt 7139 For the non phase reversed case of Fig 15 we see that broadside for the n 71 beam occurs when the n 0 space harmonic has the value Bd 27139 i e the periodic elements are Ag apart When phase reversal between elements is introduced this periodic spacing need now be only Ag2 since the extra contribution of 7139 is supplied by the phase reversal This familiar observation is in accord with Fig 17 in which the n 0 space harmonic is now at 3d 7139 when the n 71 beam radiates at broadside We should also recall the remarks of Sec C which point out that broadside radiation corresponds to a modecoupling point and that at precisely broadside the power radiated drops and the input SC563 20 VSWR rises These comments also agree with antenna experience Simple analytic expressions for the angle of the radiating beam with and without phase reversal are readily obtainable from expression 17 Without phase reversal one has i 27139 sin 0 an 25 or A A sin0nTggn n07172 26 With phase reversal 17 becomes sin 0 e eZivem r usinx 24 i 7 30 27139 7r s1n0ni k kdnkd 28 or 1 sin0nTggltn gt n07172 29 It can be seen from Figs 13 and 17 or Eqs 25 and 26 that if one desires a beam near forward endiire with a periodic fast wave structure it is best to use the n 0 space harmonic with an unreversed array with kd ltlt n For a beam near broadside or pointing in the backward direction one should use a reversed structure and employ the n 71 space harmonic Two differences may also be noted between the fast and slow wave structures First the n 0 space harmonic is always bound for slow wave structures so that phase reversal is unnecessary and second in the phase reversed fast wave structures the n 71 beam does not eventually pass through forward end re and become a bound space harmonic for suf ciently high frequency as is the case for the slow wave structures Procedures for obtaining narrow beams by making the radiation per unit length small thereby achieving a wide aperture are similar to those for the periodic slow wave structures and are very well hown and widely employed Priodically loaded open TEM lines such as a two wire line periodically loaded with identical elements are at the traiisition between the fast and the slow wave structures If the effect of the periodic loading is such as to slow down the wave even slightly the considerations of Sec D1 apply If the loading speeds up the wave radiation will occur near the forward end re direction via the n 0 space harmonic lf radiation in other directions is desired it is necessary to phase reverse as discussed above for non TEM lines 7 FrequencyScanning Antennas One desires in a frequency scan antenna a structure which permits a radiating beam to scan from 790 to 90 or perhaps a smaller range rapidly with only a very small change in frequency during which process only a single radiating beam is present From the discussion in Secs SC563 21 27r O 21r Fig 13 Cd VS 5d diagram for a fastwave structure with Closely spaced periodic elements kd ltlt 7r D1 and 2 a periodic slow wave structure is obviously needed in fact the basic wave must be suf ciently slow to insure that only a single beam will be present during the entire scan process Figure 12 and relation 23 have already considered the conditions under which the n 71 beam satis es these requirements Several types of periodically modulated surface wave structures would be suitable for this purpose It is customary however to employ a serpentine arrangement involving either a TEM line or a waveguide possessing a non zero value of k Such a serpentine arrangement loaded by a series of slots is shown in Fig 18 The gross aspects of the serpentine structure are dictated by a few simple considerations If the line is a TEM line then the phase velocity is given simply by i 7 30 neglecting the small corrections due to the discontinuities appearing at the bends If the structure is composed of non TEM waveguide then the relation becomes 11 o lg ii 31 c L or vi 32 C L417 kidM In addition ktd must be somewhat lass than 7139 or smaller than that as will be eviddent from the dispersion diagram The slots shown in Fig 18 must be centrally placed in the serpentine arrangement and can be seen to be automatically phase reversed due to the reversal of the direcbion of the transverse electric eld in the guide The guide dimension shown is b the narrow guide dimension if rectangular waveguide is employed the slots are thus strongly excited For a TEM feed line SC563 22 b Fig 14 Leaky wave antennas with Closely spaced periodic elements a Holey waveguide b Periodically slitted at waveguide kd 45 45 41r 27r 0 21r 47r d Fig 15 led VS 5d diagram for a fastane structure with kd near to 7r such as a coaxial line the slot orientation shown is inappropriate one can use either slots oriented perpendicularly to the ones shown or other radiating elements such as circular holes If the slots were placed instead at line A at the bends the slots would no longer be automatically phase reversed and the period d would be doubled The phase velocity would be unchanged of course but 1 in 27 and 28 would be replaced by d2 The kd vs Bd diagram suitable for the TEM line is presented in Fig 19 for the case of the slots shown in Fig 18 The radiation region is shifted by 7139 because of the phase reversals of the slots As seen the n 71 beam scans completely for a small variation in frequency When the slots at line A i e at the bends are used instead the radiation region is not shifted but the period is doubled The resulting kd vs Bd diagram appears in Fig 20 Comparison with Fig 19 shows that the frequency range for scan is identica1 for the two cases but that one must be SC563 23 l ESE a b Fig 16 Arrays employing phase reversal between successive periodic elements a Rotated series slot waveguide array7 b Longitudinal shunt slot waveguide array 45 7r 0 id Fig 17 Cd vs 5d diagram for a fastwave structure employing phase reversal between successive periodic elements same as that in Fig 15 except for the phase reversal careful in the case corresponding to Fig 20 in that the n 72 space harmonic should not enter the radiation region during the scan process When non TEM waveguide is employed in the serpentine structure7 it is necessary to use the phase reversed slot arrangement to remove the n 0 space harmonic from the radiation region since that harmonic does not permit scanning in the backward direction The appropriate kd vs 3d diagram is shown in Fig 217 where the dispersion curves a r e asymptotic to those for the TEM case It is seen that the curve in the radiation region is atter than that for the TEM case7 so that the frequency variation required for sianning is generally smaller even though the center frequency is higher It is evident that thekd vs Bd diagranm lends insight into the behavior of these frequency scanning antennas7 and can be useful in their design 8 Diffraction Grating It is known that if a plane wave is incident at an angle on a periodic plane re ecting surface a re ection grating7 with period small compared to wavelength7 only a re ected wave is produced When the periodic spacing becomes suf ciently large7 however7 higher order diffracted waves are SC563 24 I I b4 m iii ib Fig 18 Serpentine Waveguide frequencyscanning array With slot loading kd 45 45 27r 1r 0 21r d Fig 19 Cd VS 5d diagram for a serpentine arrangement employing a TEM line and phasereversed slots also created A plane wave incident on such a grating is shown in Fig 22 The plane wave7 of propagation wavenumber k has wavenumber components kmm along the periodic surface and kmm perpendicular to it In addition7 the angle of incidence 0inc with respect to the normal is related to kmm by k i sin0im 33 The periodicity along the surface introduce an in nite number of space harmonics7 all related to kzinc by SC563 25 Fig 20 Cd VS 5d diagram for a serpentine arrangement employing a TEM line and unreVersed slots With twice the slot spasing as that for Fig 19 led 45 45 27r 7 0 2n Fig 21 led VS 5d diagram for a serpentine arrangement employing a rectangular Waveguide With phasereversed slots 27139 kzm kzinc l The corresponding wavenumbers km in the z direction are given by km 4 k2 7 35 2 2 kmm k2 7 kzinc 7 or 0r7 in View of 297 SC563 26 H H z Fig 22 Plane Wave incident on a plane periodic re ecting surface di raction grating 415quot 45 n 0 n 2 6 4 27r 0 Zr 439quot kd Fig 23 Cd VS kzd diagram for plane Wave incidence on a di raction grating k i 2 2n 17 SID 01m The angles of radiated beams are obtained from k sin0n Zn or sin 0 sin 0incgn 37 38 39 One should note the similarity between expressions 307 32 and 347 and 67 16 and 177 respectively7 when kz and k are replaced by B and kt for the guided wave case Whether a particular space harmonic will or will not be a propagating higher order diffracted beam is determined by whether km in 32 or 33 is real or imaginary If it is indeed real7 the angle of the beam is given by 34 or 35 SC563 27 The kd vs kzd diagram of Fig 23 analogous to the kd vs Bd diagram graphically portrays these relations in a simple manner and permits one to visually determine the number of diffracted beams The line corresponding to the incident plane wave is determined by the angle of incidence 01m and can be located via the relation tan 11an sin 01m 40 The re ected beam 71 0 has the same value of kz as the incident wave7 so that the same line applies to both the incident wave and the directly rr ected wave The space harmonics7 obeying 27 are parallel to the line for the incident wave7 and all have the same slope since the wave is incident from one side only Then7 depending on the value of kd7 various numbers of diffracted beams are obtainable It is seen from Fig 23 that if the value of kd is too low only the incident and re ected waves are present As kd is raised7 the n 71 space harmonic enters the radiation region and becomes the rst higher order diffracted wave7 which appears at backward end re as in thr case of guided waves The dashed line represents some higher value of kd frequency or grating spacing for which four real outgoing beams are present the re ected wave and three diffracted waves The amplitudes of the various beams depend on the precise characteristics of the grating surface7 and are not obtainable by the above considerations References EOE P 01 F 1 00 3 Pierce J R 1954 Coupling of modes of propagation7 J Appl Phys 25 179183 Pierce J R e P K Tien 1954 Coupling of modes in helices Proc lRE7 42 13841396 Crepeau P J e P R Mclsaac 1964 Consequencies of symmetry in periodic structures7 Proceedings of the IEEE 52 3343 Sensiper S 1951 Electromagnetic Wave Propagation on Helical Conductors7 Thesis Ml T also 1955 Electromagnetic Wave Propagation on Helical Structures A Review and Survey of Recent Progress Proc IRE 43 149161 Goldstone L 0 e A A Oliner 1959 Leaky Wave antennas l Rectangular Waveguides Trans IRE 7 307319 Oliner A A 1962 Leaky Waves in Electromagnetic Phenomena7 Proceedings of the Symposium on Elece tromagnetic Theory and Antennas Copenhagen Denmark Tamir T e A A Oliner 1963 Guided complex Waves Part 1 Fields at an interface Proc IEE 110 310324 Tamir T e A A Oliner 1963 Guided complex Waves Part II Relation to radiation patterns Proc IEE 110 325334 Oliner A A e A Hessel 1959 Guided Waves on sinusoidallymodulated reactance surfaces IRE Trans Antennas Propagat7 7 201 208

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