Electromagnetic Wave Propagation and Source Radiation in Space-Time Periodic Media Citation Elachi, Charles (1971) Electromagnetic Wave Propagation and Source Radiation in Space-Time Periodic Media. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/WYFR-GR84. https://resolver.caltech.edu/CaltechETD:etd-10302003-143115 Abstract The electromagnetic wave equations for the fields, potentials and Hertz vectors are derived and a Lorentz gauge is given for space-time dependent media. Electromagnetic wave propagation, electric and magnetic dipole radiation, and Cerenkov and transition radiation in sinusoidally space-time periodic dielectric, plasma and uniaxial plasma are studied and numerous radiation patterns are given. A special radiation effect in the uniaxial plasma is investigated. Finally the study is extended to general space-time periodic media (i.e., ε = ε 0 ε r [1 + ε 1 f(Kz - Ωt)] where f(ξ) is a periodic function). Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Electromagnetic wave propagation ; Source radiation ; Space-time periodic media Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Minor Option: Business Economics and Management Awards: Caltech Distinguished Alumni Award, 2021. Allan D. Emil Memorial Award, 2016. William Procter Prize for Scientific Achievement, 2008. NASA Distinguished Service Medal, 1999. Nevada Medal. IEEE Fellow. Thesis Availability: Public (worldwide access) Research Advisor(s): Papas, Charles Herach Group: GALCIT, Caltech Distinguished Alumni Award Thesis Committee: Unknown, Unknown Defense Date: 17 May 1971 Record Number: CaltechETD:etd-10302003-143115 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-10302003-143115 DOI: 10.7907/WYFR-GR84 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 4318 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 06 Nov 2003 Last Modified: 20 Jun 2024 22:00 Thesis Files Preview PDF (Elachi_c_1971.pdf) - Final Version See Usage Policy. 3MB Repository Staff Only: item control page

ELECTROMAGNETIC WAVE PROPAGATION AND SOURCE RADIATION IN SPACE-TIME PERIODIC MEDIA Thesis by Charles Elachi In Partial Fulfillment of the Requirements For the Degree of ~ Doctor of Philosophy California Institute of Technology Pasadena, California 1971 (Submitted: 17 May 1971) -iti- ACKNOWLEDGMENTS I would like to express my deep appreciation to Professor Charles H. Papas for his guidance, inspiration and encouragement during the course of this work and throughout my studies at the Califormia Institute of Technology. I acknowledge with thanks the efforts of Mrs. Ruth Stratton in typing the manuscript. This work is dedicated to my dear parents. — ABSTRACT The electromagnetic wave equations for the fields, potentials and Hertz vectors are derived and a Lorentz gauge is given for space-time dependent media. Electromagnetic wave propagation, electric and magnetic dipole radiation, and Cerenkov and transition radiation in sinusoidally space-time periodic dielectric, plasma and uniaxial plasma are studied and numerous radiation patterns are given. A special radiation effect in the uniaxial plasma is inves- tigated. Finally the study is extended to general space-time periodic media (i.e., € = eéo¢ 1 + €, f (Kz - @t)] where £(€) is a periodic function). Il. TIt. Iv. VI. TABLE OF CONTENTS Introduction Space-Time Periodic Dielectric A. Wave equations B. General solution C. Reflection and transmission: half space D. Guided waves E, Reflection anc transmission: slab F. Dipole radiation G. Different approach Space-Time Periodic Isotropic Plasma A. Infinite and half space B. Dipole radiation Space-Time Periodic Uniaxial Media A. Wave in infinite and half space fi B. Dipole radiation Cerenkov Radiation in Space~Time Periodic Media A. Fields B. Radiation and Spectrum C. Transition radiation in plasma Waves in General Space-Time Periodic Media A. Dielectric B. Plasma 21 31 32 40 61 68 68 75 80 81 92 101 101 108 115 118 118 122 ~_~Vo VIL. Conclusion Appendix A: Appendix 5: Appendix C: Appendix D: References Potentials, Lorentz Gauge and Hertz Vectors in Space-Time Dependent Media Amplitude Ratios Steepest Descent Method Caustics 124 125 130 132 133 135 -j-+ I. INTRODUCTION Considerable interest has recently been stimulated by phenomena that involve the interaction between two waves in the presence of liquids, solids or plasma, including such applications as diffraction of light by ultrasonic waves [1], parametric processes in nonlinear media [2], optical visualization of sonic waves [3,4], Bragg scattering in plasma [5], and others. These processes usually involve a stronger wave (of the elastic, electromagnetic, or other type) that produces a space-time periodic modulation of the properties of a medium which in turn causes a diffraction of the weaker wave (usually of the electromagnetic type). The configuration which is customarily selected to represent the two-wave interactions expresses the periodic modulation caused by the strong wave as a traveling wave variation of the permittivity along one particular direction in the medium (i.e., €(z,t) = e,e, {1 + €,f (Kz - 2t)] where £(§) is a periodic fwmction); this allows linearization of the problem. Two limiting cases are also of great interest: periodically time-dependent media (i.e., K = 0) and periodically stratified media (i.e., 2 = 0). Plane waves in periodically time-dependent media were studied by Holberg and Kunz [6]. Plane wave propagation, dipole radiation, Cerenkov and transition radiation in sinusoidally stratified media were extensively investigated in the last decade by numerous authors [7-14]: For sinusoidally space-time periodic media, only the plane wave and guided wave propagation problems in dielectric and -2- isotropic plasma were studied [15-20]. In our present work, the problem of electromagnetic waves in sinusoidally space-time periodic media is formulated in a compact form such that the basic results and diagrams apply, with minor changes, to different problems (plane wave propagation, electric and magnetic dipole radiation, and Cerenkov and transition radiation), and to dif- ferent media (dielectric, plasma, anisotropic and uniaxial media). Our method is then extended to the generally space-time periodic media and all the analytic results obtained in the sinusoidal case are found to be valid,with slight modifications, in the general periodic case. A method using the Hertz vectors is also investigated, the Hertz vectors wave equations are derived, and a new Lorentz gauge is given for space-time dependent media. In most of our study no limit on the modulation amplitude oat is imposed (except that ¢€ > 0 for the dielectric and N > O for the plasma density), and the analytic solutions are exact (approximations are used only in the numerical computations). _ ~3~ it. SPAC tJ ~TIME PERIODIC DIELECTRIC The medium constants are taken as BF BS = free space magnetic permeability © u E(z,t) = eje fi + €,cos(Kz - 2Qt)] 1 where € free space electric permittivity € = relative permittivity of the undisturbed dielectric aT = amplitude of the relative permittivity change due to the disturbance The disturbance is a plane wave propagating in the z direction with a velocity Va = Q/K A. Wave Equations 1. Field equations: Maxwell's equations for the current-free medium are: VxEs “Ky 3E (24.1) VxXH= S [e(z,t)E] (24.2) V+H= 0 (24.3) V+ [e(z,c)E] = 0 (2.4) This set of equations gives the wave equation in a space~time dependent media: n2 VE = uy lel2, cE] + VE (2A. 5) at 2 “9 on 3 - ‘ \ VH = Ue BE [e(z,t) re seth x V e(z,t)] (2A. 6) -4- The electromagnetic field can be subdivided into a T.E. field (i.e., transverse electric where E= Ee and H= Het+He), _ ~y = xx 2-z en and a T.M. field (i.e., transverse magnetic where H = H £, and E= Ee+Ee) relative to the z-direction (Fig. 1). For the T.E. = xx ZZ field the vector E is normal to Ve(z,t), then equation 2A.4 becomes it Ve [e(z,t)E] e(z,t)V° E+ E* Ve(z,t) = e(z,t)VeE= 0 and the wave equation reduces to n2 VE - u — = (cE) = 0 (24.7) fe) 2 ot where ESeE £, . For the T.M. field, no such simplification is pos- sible. Cael ore nang mm wwe eacen ae eorsts ew ome enome po $e eres enue ne ewunes. guns! | | | | | SH a ae i TE wave | pees oie ecre sorseau deg aver saaetoeewty eore exnvarnce Fig. 1. TE end TM Fields The electromagnetic field can de described by the electric Hertz vector Il , and magnetic Hertz vector M which are defined by (see Appendix A): aM Ee -20K 0K ESV KS (24.8) l on He-=—VxVxM-V x= (24.9) _ Ho _ dt ‘The Hertz vectors have the properties that, in our problem, their wave equations are scalar equations. In fact, as the vector currents considered in this study are parallel to the z-axis, the e.m. field can be determined completely from the z components of the two Hertz vectors. Outside the source volume, we get: T = Ile M = Me (24.10) - “Zz _ —z 2 3 1 al 971 Vou +e Bn S ee) ~ Ue ae = 0 (24.11) 2 3 OM VM - Us SE (e ave 0 (24.12) 2.2 a where Vo =V" - 7° See Appendix A for the detailed computation. OZ B. General Solution l. Transverse electric waves: a. Dispersion equation and wave vector diagram. The electric field E= E(x,2,t)e, is a solution of the equation VE - 5} (cE) = 0 (2B.1) Using Floquet's theorem in conjunction with the principle of superposition, the solution for the electric field can be written idx +i(kz - wt) “y ain(Kz - Mt) E(x,z,t) = EA =O or oo E(x,z,t) = } Ee neo iéx + i(k + nK)z - idm + nQ)t (2B.2) where: EN = amplitude of the harmonics which are generated by the interaction with the disturbance 6 = x-component of the wave vector «K = z-component of the wave vector for the n= Q harmonic: $6 is fixed by the incident wave in the half-space problem or the wave- guide geometry for the waveguide problem. K is determined from the dispersion equation. s The dispersion equation is found by replacing, in the wave equa- tion 2B-1, E by its expression from 23-2. We have Fi ike - 9%) é(z,t) = eject + €, cos (Kz - “2t)j] = eoe fh + € + “= et (Kz - Qt), so idx + i(K + nkK)z - i(w t+ nQ)t cEs ee.) Be PS y ze +itkK + nk + Kz - iWwe 124+ Q)t + n +ee £1 or 2 ~7- ey p 20x + i(k + nk -K)z - i(wtnQ- De €¢-> } & @ or 2 n a £1 £1 idx + i(k + nKk)z - i(wt¢nMt = l ££, 7 TF Fa-1 * > =n? 9 eR --J B2(z 4 Ff. . + ie 5 yetOx ilk #nK)z2 - i(w+nk)t uy 5 2 n 2 “nel 2 “ntl t n where B/ = pe € (wt n2)* n oor and v*E - . ) (624 (ck + nk)*Jz_et* + 26 + nKk)z -— i(wtnQ)t n Putting these expressions into the wave equation, we get € € ir a; > * tz 2 2 n+l - 6 E (kK + nK) EW 2 Y (8) (E+ aeD n n x eiox + i(k + nK)z - i(wenQ)e = 0 To satisfy this equation for all values of x, z and t , the variables EO must be the solution of the infinite set of homogeneous 4 equations € € 2 2 2 1 2 1 2 _ (8 - 6° = («K + nK) JE, + 5 Beet 5 BS n-1 7 0 (2B. 3) or DEL + Eel + E-l = 0 | (2B.4) where 2 6* + (Kk + nx)? 2, er h- 2 n This system of equations will nave a nontrivial solution if ~g- its determinant A(K,é,w) vanishes, so if Oo ACK,6,) (2B. 5) anc this is the dispersion equation. Solving 23.5 we can plot the corresponding wave vector diagram. In order to have an insight on the wave vector diagram without any numerical computation, let us study the limit when > e) Oo. If the modulation (i.e., disturbance) amplitude goes to zero, then from 2B.3 we get Bo - 6° ~ (« + nk)? = 0 or 2 2 2 6” + (K + nk)” = Woe, &,(w + nf) which can be written ve cxen? eh Ge ny? R2 Q where y = 6/K X = K/K B = WYNEE ° oo Tf K = 2h € ¢€ [e} oo fF R = K/K re) This is the equation of a family of circles centered at X= =n with radius =@ +n) . For e&, 5 QO only the n= 0 harmonic has physi- cal meaning. ~9- Figure 2 gives the wave vector diagram for €, > 0, R= 3.5 1 and w/fMm 2.5 . On this diagram we see that there are intersection points between two modes, so when ey # 0, strong interactions will occur around these points. Solving the dispersion equation and plotting the wave vector diagram for €, = 0.25, R= 3.5 and w/ = 2.5, we get stop bands near the intersection points (Fig. 3). In these bands the solution for K is complex and «Kk = K + ik, . A very important result which comes from this diagram is that, for a fixed value of 6 , there is an infinite number of solutions for « which satisfy the dispersion equation, so we have K = KG where q Yvepresents all integers e¢]-~,+°[ and the Ka are real, imaginary or complex. The expression of the field, using the principle of superposition has now to be written +o “oO v E(x,z,t) = ) , £ eidxt i(k tnK)z -i(wtnQ)t (23.6) where EL is the amplitude of the nth harmonic of the qt mode. The nq properties of Ko will be studied later. To be able to see some of the effects due to the motion of the disturbance, let us compare the wave vector diagram of Fig. 3 to the diagram on Fig. 4 which corresponds to a stationary disturbance (i.e., &= 0 or v, = 0). Let A = wavelength of the disturbance X = wavelength of the e.m. wave in the undisturbed nedium v_ = e@.m. wave velocity 1/ Vue EL Sorz = g/m :WeIGeTp 102D3A BAEK ‘Z °3ta eo -~10- ~Ll- the diagram is symmeciric relative to this axis Fig. 3. Dielectric wave vector diagram: ¢€, = 0.25, R= 3.5, w/R= 2.5. Dashed line corresponds to é, + 0 Pig. 4. Dielectric wave vector diagram: €) = 0.25, va ™ 0, A/A = 1.4, B= © , Dashed line corresponds to Ey *~O., = disturbance velocity = K/K, = vo/v Vv [e} = — Va d E | ul >>| >- In both Figs. 3 and 4 we have e€, = 0.25 and A/A = 1.4. Three in- 1 teresting effects caused by the motion of the disturbance appear: 1) 2) In Fig. 3 the diagram envelope is an oblique straight line with an inclination angle a = sin” (v,/v,) in the stationary disturbance case Kg = tK Y + qK where Ko is the value of «k for the fundamental mode, and the field expression is + - Es eidx + i(zk + nK + qK)z dwt pr q “4 . + C4: _ 4 = y E etdx + i (tc + mK) z iwt m where m=n+q and E = ) E . mo n,men As soon as there is a motion of the disturbance, the relation between the different Ky is lost, degeneracy and shifting of the stop-bands occur. This implies that for a fixed value of 6 some modes will propagate, some will be attenuated and some will be cut off. This is a major change relative to the stationary case where all modes behave in the same way, they are all propagating, all attenuated or all cut off, depending on the value of é. ~13- 3) For the fundamental mode the two solutions for kK are not any more equal in absolute value, as in the stationary case, because of the dissymmetry introduced by the motion of the disturbance in one direction. Also, the two solu- tions for each of the other modes are not any more symmetric relative to the center of the corresponding circle in the wave vector diagram. In both moving and stationary cases, for e, # 0 there are inflection points in the wave vector diagram. These points will play a major role when we study the dipole radiation. b. Harmonic amplitudes The infinite system of linear equations 1B.4 has now to be writ- ten Dag Eq + Etlia + Eo-1,q = Q (2B.7) where 2 6o+ (K + nk)* g D «—f{l- 5 ] nq &) 8 n se As we take K = Kg » this system has a nontrivial solution, but since it is homogeneous we only can determine the values of C = E /E * nq nq oq It is straightforward to find that (see Appendix B) RA Ea z a == for n> 0 n-1l,q DO - 3 | fa nq. i 7 for n< 0 n+l ,q Deg - _ n-1,q Then we have E E E E aw 2G Nog x Boog isd for n> QO nq —E E E E O44 n-1,q n-2 ,q Ord E E E E we od, 29, otha... ed for n<0O (28.8) nq Eo Eotl,q E4244 034 and the values of Eoq will be determined by: -~ the boundary conditions for the half-space; -- the source condition for the dipole. c. Convergence A last factor to complete the solution is to study the conver- gence of the system 23.7. The convergence condition for a system of difference equations Dn + Viel + Vi-l = 0 is given by |1im D | > 2 (Poincare theorem [16,21]) nwo © This gives 2 |i - | >2 “3h > or — R> vite for R 1 Ro< Ll-eé for R<1 So the solution will converge for all values of R or Va outside of the region io- 1 or Vv Vv fe) < va < Oo This divergence region is called the sonic region, in which there is a blow up of all the harmonics. This condition of conver=- gence excludes from our study the case of a disturbance created by an e.m. wave in the dielectric. (This last remark is valid only for an isotropic dielectric; for plasma and anisotropic media, see the cor- responding chapters). 2. Transverse magnetic field a. Dispersion equation We have E=s= Fe +Ee - ROK 2"Z H= He ~y The field can be found from the electric Hertz vector wave equation 2A.11, or directly from Maxwell's equations. Here, the second method is used. The field equations are: 3H 8E OE — x z oH VM ER He oe G2 Gx 7 7 Mo Be (28-9) ~16- ou 3 = ; "Sy * SE (€ E..) (23.10) VxH»s = (cE) = ~ 9b 3E 3 = 7 SC (e E.) (23.11) . oH Ve H#®O > — = 0 (2B.12) a by oE.. r) Ve C€e€E) = 0 > Ee 73% 7 3p (cE) = 0 (28.13) Using Floquet's theorem and the superposition principle, as we did in the T.E. case, the field: expressions can be written: Hs y wee TACK + nk)z - i@ +nQ)t E= J pet + i(k + nk)z - idw t+ nQ)t n (Be +E e )eroX i(k tnk)z ~ iQ #tnQ)t naxx “nz-z a aed Putting these expressions in 2B.9, 2B.10 and 2B.11, and follow- ing the same procedure as in the T.E. case (i.e., equating the elements corresponding to the same harmonic), we get (K + nK)E - cE = HW, (@ + nv) Ho € E + a i i “(CK + nK)H €,e.(w +n) (EL t+ 5 Entlxt 5) En-1,x ey f1 OH, = = €5€ (an) (E+ > Eutl,2* 2 Ey-1,z? Eliminating qo the above system reduces to ~l7- a nx * n+1,x ul ea-1,x" CFaz = 0 (28.14) ' = Dez ul Eotl,2 * Ea-1,2° Cex 0 (28.15) where 2 K + nK,2 1 th 2 6 2 ple [1- G97] 1 n 2 6(K + nk) G == n e, ge n From 2B.15 we have T po ow-zg -Lz Ls nx G. n,z G. nt+1,z Go n-1,z D! _ atl , 1 E 1 E n+1,x Ciel ntl ,z Gael n+2 ,z C41 n,z t ; esl _1t, ._1 n-1,x GG, n-l,z Co-1 n,z Goel n-2,z Putting these expressions in 2B.14 gives 2 DD'-G ' r 1 1 n ntl —— es + JE + + ]E GC. Goel Goi nz GC. G+ nt+1,z aes 1 l + [+=] E ——E —_ = 0 (2B.16) GC. Ci n-1,z Corl n+2,z G -1 2 2,Z or in a matrix form M|] > jE] = 0 (2B.17) -18- where [E.| is a colum vector with elements a {||| is a matrix with elements: DD. - G? 1 l M a 5 a G * G ? n n+l n-1 1 M _ vay n+l n,ntl G G n n+1 D D! M, n-1 a+ 3 ? n n-1 1 M — n,nmt2 Cel 1 M = n,n-2 G2 and all the other elements are equal to zero. To get a nontrivial solution for |B the determinant of ||M|| must vanish A'(6,k,w) = Det ||M|| = 0 ° (2B.18) This equation is the dispersion relation and it gives the wave vector diagram for the T.M. case. In the limit ¢€, + O , taking the highest order of Il/e equation 2B.16 reduces to 1 > B.D. > 6 ( GC. ) Enz = 0 and the corresponding dispersion equation becomes DD = ce , and for B.#0 -19- 2 2 2. 3 (K + nk)” + 6 B which is the same as in the T.E. case. This is to be expected because the dispersion equation for T.E. and T.M. waves in homogene- ous isotropic media (i.e., when E, = 0) is unique. The wave vector diagram in Fig. 2 applies for the T.M. wave when 1 >0Q . For ey # 0 the T.M. wave vector diagram is similar to the T.E. diagram in Fig. 3 (but not exactly the same), and K is multivalued. Then the field expression is H lox + i(k + nK)z - i@ + nQ)t nq a1 u ol ae) Oo )etd* Fikt nK)z - L(w+n%)t fos (E e +E e nqx-x TGZ-z - 2-7 The values of Kg are determined from the dispersion equation or the corresponding wave vector diagram. The same remarks made about the effects of the disturbance or its motion on the T.E. field apply to the T.M. field too. b. Harmonic amplitudes E Equation 2B.17 gives the ratios Coq = de ; then using 2B.9 Eoqz and 28.15 we get. E 1 1 1 BI = 9X _ ~- {+ c! +— cc! +—— ¢! ] nq E a2 Gq ng Cog ntl,q Cag n-1,q Fag Ew, e e, (2B.19) ' = = = 1 — t —_ ' And Eoqz 6 (Ce 73 Cotl.q ar) C3-1,4! where wo = w + nQ and the set Eoaz is determined by the -~20- boundary conditions or the source conditions. For 6 = 0 (i.e., normal incidence), equations 2B.19 are not valid and we must use 23.14 and 23.15. c. Convergence When n~* © equation 2B.16 becomes AE + BE +1,z + BE 1,2 * Eat2,z + En-2,2 = 0 where Ae2+¢2 (1 - Ry al B= = (2 -R*) L This is a fourth-order difference equation, and from Poincare theorem [18,21] the divergence or sonic region is given by -2(B + 1) A > A 2(B = 1) for B>4 ~2(B + 1) A > A B/G + 2 for B<4 or replacing A and B by their expressions in terms of Ri: lee < R° <x lt ey for x <2(1 - e,) o< RF < lte, for R” > 2(1 - ,) These inequalities can be written in one relation as l-e, < RK <«< l te for all R 1 | eel So the sonic region for the T.E. and T.M. fields is the same. ~21- C. Reflection and Transmission: Half-Space We assume a wave is incident upon a semi-infinite medium where = Kz - & e(z,t) ee [2 + €, cos (Xz it) J with an incidence angle 9, - The reflected and transmitted waves will contain harmonics with frequencies w= +nQ , because of the inter- action with the disturbance in the dielectric (see Fig. 5). €® €o G | Ite, cos(Kz-Qt) ee ee ~iniQ w+2 w wrinid. yw+inia G vf i i bo 1 _ 1 | Fig. 5. Half space case ~22— 1. T.E. incident wave (Fig. 5): The normalized incident wave is i(B,sin 9.x + B cos 0,2 - wt) Ete e i(8 sin O.xt 8 ,cos Oz - wt) H, =-cos 0, ve Jue * (2.1) ix i (one) i(8 sin 8.x + B cos 9,2 - wt) Hee = sin 8, Ye Th, e 2 2 where By ye, The x component of the wave vector is the same for the inci-~ dent wave and all the components of the transmitted and reflected waves. The reflected wave can be written: _ i(B,sin 9.x - B, {cos 02 - wit) Ez. =e ) RL e “J a (2€.2) i(B,sin @,x-8, cos 9 -w _t) 1 i in rm n He veh, l Ricos 0 e . where Bin * Labatt We must have (B,sin 9 2 + (B. cos @ )* = g? therefore i 1 in rn in ’ cos“0 + (Y Pointe = jl rn w L n or sin 8, sin a =? (2C. 3) l+-en oD The transmitted wave is ~2 3 ft fur uF . - ist nKk)z + i8 sin 0x wit ze = 2, ») to Eq e na (2C.4) K > K at 4 i ~ . ; 5. Ss n _ (kh 2 + iB sin 65x wit tx f q UoY, dong The values of Ke are determined from the wave vector diagram with 6 = 8, sin 6, . At the boundary (z = 0) the continuity of E and q gives ewtw t a t -iuwt n e + ) R, e = )} ) te Faq n nq and So -t _ [Fo ~iw, t ; po Kink +nK ~iwit — Coos 8, RL cos 9 m® a t Ee Mo nq o%n 494 These equations must be satisfied for all values of t , therefore the terms with the same frequency must be equal. This gives the set of equations: §(n,0) +R = - to a (2¢.5) - = an: SOU 6(n,0) cos e, R cos oon : B t £E (26.6) where 6(1,4) is the Kroenicker delta i a) 6 (4,4) i Lo) é(i,3) for i# j Eliminating R, we get -24- K nk - s ) Guo 6(n,o) [cos 8+ cos e.) cos an )) tz ad ) 7 tend q q in but 6(n,o) cos On = 6(n,o) cos Oo = O(n,o) cos 8, so K t+nK 26(n,0) cos 6. = ) (cos On + 3+—) c, Ena q in This equation can be written )) Baa T = 26(n,o) cos o q where K aK sin’®, K tak B = (cos 9 + sc = l- + Cc nq mo Bin 7 84 Gen 4? Bin } Ra E a 2d ng x Oq T = teE ° q q oq or in a matrix form Bl] + [tls |v (2.7) where [[B|] = matrix with elements BO [T| = column vector with elements T . V. = 2 cos @, for n = 0 n L |v| = column vector with cienncel = 0 for n #0 ~25- Equation 2C-7 gives IT] = {[Bi]~~ iy and the transmitted field is completely determined by i(k,+ nK)z + iB sin Ox - iwit EL = } ) Cc 8 6T e Writing 2C.5 in a matrix form, we have iv'| + [rR] = |[cl) + men in| = -[v'] + [le v| (2C.8) where [|c|| = matrix with elements C. ; Viel forn=0 [v'| = column vector with elements V7 0 for n # 0 [R| = column vector with elements RA This equation gives all the reflection coefficients. a. Reflection angle The reflection angles are given by 2C.3 sin 6 L sin O = 5 ltea- W The high frequency harmonics (n > 0) will be reflected near the normal, the low frequency harmonics (n < 0) will be reflected near the boundary, and the harmonics with n < iors + sin 8,) will not be reflected at all. So in reality, the reflected field will contain only -26- the harmonics with frequencies W > W = w sin @ co i (see Fig. 5). b. Transmission angle The transmitted wave is L(k + nK)z + i8 sin Ox - iw t E. =e )} ») C. T e Yag m4 and the wave vector components are ke = 6 = 8, sin 9, = YUE, W sin 9, x = Ky) + nk = Kk, + nk The group angle y (di.e., angle between the group velocity vector v_ and the vector e) is different from the phase angle 9 “2g ~Z C (i.e., angle between the wave vector k and 2): These two angles are given by: k vue. w sin 9 " tgQ@, ss 29 8 Ming k, Ko + nK Vox oK Tava ” Vos *“ 36 For a certain mode q all the harmonics n will have the same group angle, which can be found from the slope at the correspond- ing point on the wave vector diagram. ~27- ec. Transmitted wave vector The incident wave fixes the value of 6&6 6 = B. sin o, or 6 By ri Ys = x7 x sin eo. = ve_ x sin 9 where A = disturbance wavelength \ = e.m. wavelength = 27 / we € In Fig. 6 the mode q= 0 is represented by the curve ABCDEFGH, and the dotted line corresponds to a complex K. Let us examine the characteristics of the transmitted wave for different values of the incident angle (see Fig. 6): “Y= Y, | This corresponds to ve” sin 8, > 1. The modes q* 0,-1,-2 are cut off and the major contribution comes from the mode q@=il1. “Y, = Yo : The mode q= 0 gives a backward propagating wave (negative slope on the diagram) and an exponentially attenuated for- ward wave (intersection with the dotted line in the stop band). This possibility of having only a propagating backward wave for a certain mode is due to the motion of the disturbance. The modes q = -1,-2 are still cut off. “Y, = Y3 : The mode q=0 gives a propagating forward wave but an exponentially attenuated backward wave. *xaTdwod ST 10209A BARA BY VIBYUA spueq dojs 03 puodseii10) sqacd paqjop ay, ‘werzeyTp 10}00A aaeMy ~28- -29— 2. %T.M. incident wave (Fig. 5) The normalized incident wave is i(B,sin 9,x + B,cos 0.z) -iuwt - : i 1 L i E = (cos @,e - sin 8.e)e i(8 sin Ox + B, cos Q,2) - iwt H=ve /t ee * =_ 0 "0 ~y the reflected wave is i(B,sin 0,x+8, cos 9 z~wWt) E = ) (R.cos 8 e +R sin @ e De 4 A in mm n —r n mx Tn rn—-z Ey i(8 sin 05% + B, cos 4% - wt) H.. mu -@e ) — Re Va Ho where sin 8, w ; sin 9 = os 6h sin 8 mn Lead wo i and the transmitted wave is i8,sin Ox tik + nK)z-iwt E, = } ) (Ee +E ede 4 n “tf q nqx-x nqz—z if, sin 95x + i(k + nK)z - iw it i. =e ) ) Ke e “Vu q q (a slightly different notation than in the T.E. case is used.) At z= 0 -the boundary conditions give, after equating the terms with the same frequency, [Ho 6(o,n) - Rg, = JE ) Hog (20.9) o gq §(0,n) cos 8. + R cos o = L En ox (2C.10) Eliminating R, » we get 26 (o,n) cos G, = ) Nag T q where T = &£ q oqz N= B' -A' cos 9 = Bl =A! Jr- & sin 0,)? nq nq rn ng ng wo 1 Ea B' os aude (see 23.19) nq E oqz H Atos =a (see 2B.19) ng E oqz 6(o,n) = Kroenecker delta Using a matrix form, we can write iIsil+ [zl = Iv (26.11) and tty wd | [tT] = [inj[~~ + |v! (20.12) where ° [IN|] = matrix with elements No |t| = column vector with elements qT, V_= 2 cos @, for n=0 |v| = column vector with elements Viz 0 for n # 0 The reflection column vector [RI ,» which has the different reflection coefficients RQ, as elements, is given by [Rl = - VeTu |fat|| + fyi? + [vj = [ve (20.13) of Ho ~3]- and the transmitted fleld is completely determined by i(6,sin @,)x+i(K +mK)z-iuw t BE =)) @'e +cle)Te * * q n —t of G nq-x new~z’ “gq 48 sin Ox titki+ nKk)z - wit D. Guided Waves Let us consider a cylindrical waveguide with axis parallel to the z-direction, filled with a dielectric which is submitted to a traveling disturbance such that —€ = €(z,t) = éje.(1 + €,cos(Kz - Nt) ] 1 The value of 6 is fixed by the geometrical characteristics of the waveguide. For a rectangular waveguide of dimensions L and & , mT, 2 TT, 2 Spt | + oD where m and p are integers representing the mode excited in the waveguide by the source, not the modes caused by the disturbance. For a circular waveguide of radius R , we have = ' Ld Joy Sp ®? 0 ox J, Sup) 0 depending on the excited mode. Having the values of 6 , we follow the same methods used in the half space problem to determine the values of Ko » the cut-off ~32— frequencies, the behavior of the forward and backward waves, and the amplitudes of different harmonics. E. Transmission and Reflection: Slab (e) << 1) << 1 (see Fig. 7). As €, << 1, In this section we take 1 “1 the electric field expression is i(Kz-wt) + idx ai (Kz = 2t) + ent (Kz Mt), E= +E, El (2E.1) where we neglect all the other harmonics. Let us consider the T.E. case where E = E £, . The systen of equations 2B.4 becomes £1 ey dF, + > EA + > Ei = Q (2E.2) © 1 d, 5, + = E, = 0 (2E. 3) at where ©) _ 574 (Kk + ax)? d =D =ii1- n 21 Pea n From 2E.3 and 2E.4 we see that usually E, and E_4.<< Ey and to satisfy 2E.2 we must have n Oo d f) which gives the dispersion equation in the homogeneous medium Parallel interaction w~ Q i Antiperatel interaction ~33- Le > fb oF | bo | br ; ft in fb | ae Pood bei | | | | ear ex eee mom meres Fig. 7. Transmission and reflection from a slab =34- 2 2 2 6 + KT BS But if qo = 0 , and at the same time dy or day = 0 , then there is energy conversion. 1. Up-conversion Let us suppose the parameters in the problem are such that and ==> 674 (ec +x)? = gC + 2)? Qu R oO This occurs if K? + 2K /s2 - 6° = x + 2K 8 (2E.5) where K_ = vy € €.Q oor ° In this case we still have Bi << an but we cannot say any more that Ey << Ey » and the system of equations becomes fy GF + > EL = Q £1 djE, + > Ey = Q This system solution is not trivial if 2 a dod = = (2E.6) Solving equation 2E.6, we get -35- K =k ce PaPo ‘20 (2E.7) zo "1%k VkO+E . ZO zo a) ko Bo 7 8 1 Ey or Q “70 E W k +K ° zO 2 = (2 + Ry 2 W where and this gives k ZO k +K zo "| fe be (2E.8) FL E Oo ° We see that E, is of the order of Ey and this up-conversion effect occurs only when 2E.5 is satisfied. For normal incidence (6 = 0), the up-conversion condition becomes but as Bo is positive, then K must be negative (i.e., the disturb- ance must propagate opposite to the e.m. wave) and it must be such that [K| > K, » then iK| - Ky B, * 2 Three interesting remarks must be mentioned: 1) The condition of up-conversion, 2E.5, is exactly the first- order Bragg condition with moving disturbance. This can be shown from Fig. 8. angles of incidence 6, (8 ~36- 2) For oblique incidence, up-conve = angle between it c'd the disturbance wave vector) such that o<6, < T ~ rsion can occur for all the e.m. wave vector and where cos } = KU/k = Val 3) The power: conversion ratio is PL - a + Qy2 | ko W jk +K fe) zo Py and Py are power densities. To get th e total power we must multiply by the cross sections of the incident and reflected waves. From Fig. 9 we have but So i} L cos 9 a ee a Sy Ly cos 9, [k + Ki ZO cos 0. = ST iL ZO cos 98, = i BY PS. PS fe) WtQL 2 Ko uy 8 = Kot z| i and this is the Manley~Rowe relation. -37- @ Fig. 8. Successive position of a disturbance front at different times. Bragg effect. AB = A een coon eeese b exe V Czy e eeu em —a eae he, _ i Fig. 9. Oblique incidence and reflection ~38- For a slab of thickness L (see Fig. 7), if we write the field expressions in the two media and satisfy the bowdary conditions, we get for the normal incidence bd ul reflection coefficient ii - 5 [PE earn GE [OE = W . 4 2 T = transmission coefficient 1 e,L [sol - Ko?) 4 2 cosh ( The reflected wave frequency is w+ and the transmitted wave fre- quency is Ww. When L*>© , then T = O R= i wt 7 wW +2 and in|? . & : 2. Down-conversion Let us suppose the parameters of the problem are such that 6+ x = £6 | a. z oO | for) th + ron ras i “a we bt u WD This occurs if -39~ (2E.9) 0 “2 2 2 x2 - ox yp -s°s x - 2x8 fo fe) rome) and following the same method Then Ely will be of the order used in the up-conversion case, Kak te BP Ko zo “l 4k k -K zo zo we get “lL , Wa? Keo | P W k= K| ce) fe] zO Psy w - 2 PS. Ww oO 2 and for the slab at normal incidence (Fig. 7) K+ k e.L fe) 1 2 2 Rei [x Er, tanh ( 5 K'~ KE } T= i €,L 2 2 cosh + K- Ko } For K < Ky the reflection and transmission coefficients become K+kK e.L _ / ° 1” /,2 2 R=-i ZL. kK tang = Ko K fo) 1 T= e_L la re ,2 cos 37 Ko - K } and if the slab thickness is such that -40- 1 ape ,2 1 3 KK 7 (p +>) where p is an integer, then R and T*®. This means that there is auto-oscillation and the slab will radiate, on one side, a wave of frequency W and on the other, a wave of frequency of w-Q. F. Dipole Radiation We assume a disturbance propagating parallel to the dipole moment in the z-direction (Fig. 10). 1. Magnetic dipole The magnetic dipole will radiate a wave with an electric field parallel to the disturbance wave front z = E(p,z,t) =9 therefore the wave equation is identical to the T.E. equation 24.7 2 97¢ E # outside of the source volume. 1 Lo) Using the separation of variables method, we have and 2 2 d 1 dpx QZ CoA Ze [(S —] 4+ X ee Ke = OO do bs do: 32° La 547 which can be separated in the two equations -~41- jZ AK —i — —~ Pos TaGius > Se” lo Magnetic dipole EE eg Fig. 10. Electric dipole Dipole in a space-time periodic medium d 1 cox 2 —— fo + 6 = . dp GS dp: x 0 (2F.2) ad nz : 3 ~y 284. 5¢2 = 0 (2F. 3) fe) 2 oz r) where 6 is the separation variable. The solution for X which satisfies the radiation condition at infinity and the finiteness condition at the origin is X = B, (5p) where B (Sp) = J 659) for p<a B (6p) = H) (6p) for p>a Using Floquet's theorem the solution for Z is L(K #nK)z - i(wenQ)t Z(z,t) =) Ee n with K = K(é) . So = i(k+nk)z - idwenQ)t XZ = ) EB, (6p) e n Here 6 is not fixed as it was in the half space problem, and to get the field we have to integrate over all the values of 6 using a weighting function (or, we can integrate over K , because K and 6 are related by the dispersion equation). So the electric field expression is _ i(k+ nk)z - iw,t dk (2F.4) b $F Wk) J EB, (6p) e n -43- Putting the expression of E into the wave equation 2F.1 we get the same dispersion equation 2B.5, and wave vector diagram (Fig. 2.3) as for the T.E. wave studied previously. So for each value of K , there are many values for 46 and the field is finally given by * i(<tnK)z~ iw t | W«) Eg B, (Spe dk (2F.5) E=e,)} =p qn - with 5, = § 6%) . The magnetic field can be determined from oH and we find -iwt > i (KK) z a= - |] de n | =, W_ CK) E gS B, (6 ,pe dk oqgn n Lys ~iwt [2 i (Kink) z —_ e W_(K) E. («) B (6 p)e dK H ~ UL, qn J we q nq oq (2F.7) All the 5,66) are known functions-of K and can be found from the dispersion equation or the wave vector diagram. The weighting functions WK) are determined from Ampere's law at the source (i.e., dipole). At the source we have [H,(p> a) - H(p<a)] = - 1,6(z) et (2F.8) pra where I, = current in the loop of radius a , and w is its frequency. Also E. | | is continuous d'pra ~44- The field expressions are ~dlwit fs) zZ Ko qn 2a q nq o q -iw t ° 6 Yo gn 1%, a ndo'a -iwt * E,(p > a) phe Wo Bagh (6 pre dic qn te oF qn q “ng “1g” and the weighting functions are not the same for 9 > a and QO <a. The continuity equation gives (1) : W' HL 6 a = W J,(6o a and from Ampere's law (equation 2F.8) we get # " (1) $ i(<+nK)z- twit lim ) } | [WoHo* (60) - W'S o)] St EL. e dk prya qn 3 4 q a 4 nm 4 = ~Luwt = ip 1 § (ede Replacing Wy by its expression function of Wo ,» and taking the limit p77 a=, we obtain (1) ~ wil) es - He (6 add, 8) Hy (6 a5, (608) ; Eno fi nk) z duit W q. Jy (6,4) qo, Oo~d pme4 = iy,1 6(z)e** ~45- This reduces to ow a .“ +nK) - iwit ) I | apessey G (a) ° "dk = - $m, al jb(ze™ ga Qt where we used the following properties of the Bessel functions: (1) (1), . qe Jy HY Jy iY Jy YI, so = 1 Jy Jo = ' %y XG Bessel equation Wronskian = 2 Finally, taking a-+0O we get i(KktnK)z - iwit rr E e qn 1 Sy 39 where m, = ma‘, = magnetic moment. To solve this equality for Wo we write 6(z) form §(z) = | erk2 dK Then equation 2F.9 becomes tut (2F.9) in an integral Em i(K«z - wt) * W | [ ) y = Ea ein (Kz Qt) + 2 Qe 00 qn q and this is satisfied if dk = 0 =46- W fe in(Kz - Qe) my ) ) fs) Eq © ar, qn q This relation must be true for all values of z and t , so we must have ; Cag a, = 0 for n #0 um ) c a= - 28 q q q where we took W. at = 5, “04 E na” = (defined by equation 2B.8) q oq or in a matrix form [Icll+ Ja] = jal and (2F.10) -l Ja} = |jc]]™ + [ml where ||C|| = matrix with elements C nq | o.| = column vector with elements 0a n= 0 for n #0 | m| = column vector with elements og m= - Z for n= 0 Equation 2F.10 determines a| and the field expression is completely determined by ~47— ~ilwit E(p>a)=JSJe 3 | ow (kK) 6 (K) C_(K) HAP (5 pyettMK)2 ay q q nq i q qn 00 (2F.11) The integral can be evaluated in the far field by the steepest descent method (see later). 2. Electric dipole (Fig. 10) The field has the components H = He mm) E = Ee +Ee _ e-— zz Maxwell's equations outside of the source volume give oN Je E oz dt VxHe a oO (2F.12) de E 19 Zz 2 3p (pH) TE dE. oz JE oH VN EC Ho Se > Fp 7 Ge 7 Mo be (2.23) Using Floquet's theorem, and following the same method used for the magnetic dipole problem, we can write (1) ; L(KinK)z - iw it Ho = WH H S51 | Whe AY Cee ak ~438- —_ _4 (1) (1) E Li cx 1B acto (69s, + Bo Hy (6,92) a i(Kink)z - iwit x e dk Putting these expressions in 2F.12 and 2F.13, equating the terms with the same frequency, and eliminating Haq we get DCE +E +E +G _éE nq nqp ntl,qo n-l,qp nq nqz 1 D'_ £E + E +E +G _éE nq nqz ntl1,qz n-1,qz nq nqp where D_, D! and G are defined by 2B.14 and 2B.15. These nq’ “ng nq equations are identical to the equation obtained for the T.M. field in Section II-B.2. The weighting functions are determined by the source condition lim 279 Hy = I p> 0 wt where I = ~iwp 6(z) et » and p = dipole moment. This condition gives *t (1) i(kimK)z - iwit lim 2no ) ) | WH HS~’(6 p)e dk +0 qaqi *'s p qn} = -iwp 6(z) eg Lwt Taking the limit p+ OQ we get ” OW i(kinK)z - iwt z- iw q n ei ~Lwe i) | & Bag e dk = [ up 6(z) e qn am 4 a Writing 6§6(z) = f ettz d< , this equation becomes Tina ans -49- W . . . is won ~ 8) wp] eb (KZ — 8 a Sg This relation is satisfied for all z and t if or using a matrix form ' = OP : M4 Abe i 1 = 2 G, Ana 0 for n#0 W o* = Ey Zz G q q Enq 1 ~— bal : And =F (defined by 23.19) oqz I[A'||+]a] = [P| rari of = ANTE + [P| P= 0 for n#0 is a column vector with elements P= wp/4 Having lal, the field is completely determined by i(KktnK)z - iw t =e i | ALK) a (kK) 85 (0) Hy) (6 pe dk -50- 3. Asymptotic evaluation of the integral Only the magnetic dipole field is studied in this section. The same method, using the corresponding dispersion equation, can be used for the electric dipole field. The electric field is given by 2F.11 ao ~iwit E=P De % | aso He pret* ax qn qqnql ‘4 When p +e, T ume) i> (1) _s 2 4 Hy (6,9) > <i 798, e therefore, in the far field we have TT foe) -i- irF (K) -iwet = / 2 4 1/2 q n E i Treingd ° L l | So OC? dK cee] where Fo) = (KinK)cos 0 + 6 sin 8 Le = 0° + 2" and = £ tgo z Using the steepest descent method when r+ ™ , we find (see Appendix C) a [6" | qs x Cag Ss® where Aus = aK.) ; Fos = Bok.) -5]- the index s means the value at the saddle point, and the summation over j means summation over all the saddle points (it is possible to get more than one.) The saddle points are determined by de, Gk = 0 KK s or dé = § =6 = ~cotg@d q qs K= K From the wave vector diagrams in Figs. 3 and 4 which correspond to a moving and a stationary disturbance, we computed ds /dk and plotted 9 function of K/K in Figs. ll and 12. These plots give, for each value of 9 , the corresponding values of K = Ke » and then we get the 8 = Sus from the wave vector diagrams, From these plots we see that: 1) Stationary disturbance ; -~For a given 9, there is one common 85 for all the modes and the values of Ks differ by a multiple of K --There is symmetry relative to 96 = 90° --The number of saddle points per mode is the same for all modes. For instance, in Fig. 12, there is one saddle point per mode wnen 9 < 62°, and three saddle points per mode for 62° < 9 < 90°. 2) Moving disturbance -~For a given 6, each mode has a set So Ks unrelated to the other mode sets -52- peysep ay Q + ts 0} spuods32z109 aUuTT © org = Ia fee ‘ sz'o = '3 ‘672 = y/m ‘Ge =X OFTIDeTaTp e 203 (A/"M)F =O PAINQ “TT *3TA O81 -- O21 -- -53- QO +« th 03 spuodsazi09 osUuT{ P T A OFAILOTP B AOZ CTAE)Y: =@ 2AIND ‘"7T °3ta poysep vy, *° ¢Z7°O = 3 *¥°T = V/Y ‘0 = N —_ sooihep ug V -54- ~~The symmetry, relative to @ = 90°, ts lost --The number of saddle points per mode is not the same for all modes. 4, Caustics A very interesting effect appears wnen ey # 0 and this is the existence of inflection points, near the interaction region between two modes (i.e., the stop band), in the wave vector diagram. This corresponds to extremas on the curve dé/dk« . At an inflection point 6" = 0, and the field given by equation 2F.14 is very large. This is a focalization and radiation enhancement effect, due to the inhomogeneities of the medium, on a certain surface called "Caustic". In our problem the caustics are conical surfaces. For a detailed physical explanation, see Appendix C. The slope of §S(k) at the point where 6" = 0 gives the caustic angle 9. dé s tang 8. = at inflection point From Figs. 4 and 12, which correspond to a stationary distur- bance we find a. = 62° and G. = 118° for all the modes If A/A or Ey changes we may get more or less caustics. For .a moving disturbance (Figs. 3 and 11) we find eo. = 120° for the mode q = -1 ~55=- @ = 37° and a. 93° for the mode q = 0 @ = 56°, @ = 82° and o.= 120° for q=1 >. Field amplitude and radiation pattern The magnetic dipole field expression is -24 8 z° = r sin 0 : d l SC Oo 8 "ng “se? 7 ty ~iwt + irk, 1 Z (1 - sign 6.) Let us take -1 gt licll then from 2F.10, we get ei ' % 4 Yo Coq So, for the nth harmonic we get [E \* = ( Boo 7 y Y: Sas - (c' C¢ y2 + n 2x sin 0/ fc [6N | oq “nq q 3 qs Wom, 6 8 1/2 ir(F -F_) qs ms qs ms p ) ( Coq "om ng Snm® (vote 9” ) ) q j 2 The second term in the expression of [EI is oscillatory and : th we can drop it if we average over r . So the average of the n” har- monic field is Mon 2,1/2 Iz, “ Ox sin 0 [ ; its om 5! (Co Cue or -~56- Lm 2 2 2 oie Sos 2 ele! = Gai) PT RD oq Sag? qj Sas If Ey > 0 then Cad = OQ for n #0 Cc = J] og c'c =] oq og 6 2 4 TST BoE SEL sin @ and there is only one mode and one saddle point. So 2 hom 2\ = SS Lege sin“e E > If we want to normalize the pattern such that for e, =O we 2 2 2 th get xr [EI = sin 0 , then the n° harmonic pattern for oat # O must be multiplied by a normalization factor equal to Kono, 2 2,1 Qo 0 {¢ 5) ) Hoe Ge ] Then we get the pattern 2. 2 2 c= © Ld ry ) 0 e Tonks sin 25 qj a “oq “ng Figure 12' gives the dipole pattern for a stationary disturbance with = 0.25 and Ae 0.8 -57- In Figs. 13 and 14 we present the patterns of the fundamental wave and the w+ harmonic for uw Vv €,2 0.25 , 52.5 and Ra —2= 3.5 2 Vv d The pattern for the w- harmonic is very small and was not plotted. Let us examine the features of these patterns: 1) Stationary disturbance For A/A = 1.25 (Fig. 12), there are four caustics at 9. = 43°, 147°, 82° and 98°, and the pattern is symmetric relative to @ = 90°, For A/A = 0.71 (wave-vector diagram on Fig. 4), there are only two caustics at G. = 62° and 118°. For larger values of A/A we get more caustics, and for smaller values of A/A we might get no caustics. In fact, if A >> A , the e.m. wave will not see the disturbance, and no focusing effect occurs. 2) Moving disturbance e In Fig. 13 we present the fundamental pattern for e, = 0.25, A/A = 0.71 and R= 2.5 The caustic angles are 8. = 94° and Q. = 37°, Comparing these values to the corresponding stationary disturbance caustic angles (i.e., 8. = 118° and 6. = 62°), we see that the moving disturbance pushed both caustics upward in the +z direction. The fundamental mode contributes mostly to the fundamental wave (i.e., frequency WwW) pattern. The other modes contribute less -58~ i> G) TSE EY oe Yer ae aa penepeto=conar area iG 130 150 170. @ Fig. 12'. Radiation pattern of a dipole in a disturbed dielectric w/@= 1000, R= @, v= 0, A/A = 1.25, €, = 0.25 The dashed line corresponds to €, = 0 -59= / O.2-- - o wot Coal O10 SO SO 7O SO HO 130 150 I70 8 Pig. 13.° Fundamental radiation pattern of a dipole in a disturbed dielectric: w/M= 2.5, R= 3.5, € 1” 0.25 . The dashed line corresponds to ¢, = 0 ~60- ssemuermemnucmere warns ET ae mostly from mode q=i -+ 1.4 mosily from mode q=O <2 — ita arg RAT ie é ree EES O10 30 50 760 SO HO ISO 150 170 =- | p | | from a} : mOCe | 7 from mode q=O i a J Fig. 14, Harmonic (w+) radiation pattern of a dipole in a disturbed dielectric: w/Q = 2.5, R= 3.5, &,* 0.25 -~6l- except near a caustic, and at this angle the fundamental pattern has a peak. In Fig. 14 we give the pattern for the harmonic w+. The mode q*= 1 gives the peaks at 0 = 56° and 82°, and the mode q = 0 gives the peaks at 6 = 94° and 37°. G. Different Approach The problem of e.m. waves in a sinusoidally space-time periodic medium.can‘be.solved by a different method which deserves mentioning. This method will be studied in this section and compared to the one used up to now. We use the Hertz vectorswave equations 2A.12 and 2A.13 2 2 3d. ,1 Ol oI Voll +e 55 eon 7 hoe Tz = QO ot 2 P) oM VM - HS oe (© 55 = Q Making the following changes in the independent and dependent variables Q Kz K I. Bez-7t 1a E Dae 0 2 k 1/2 1/2 Te ay i a ae K- k K-k where 2 K £9) = = -~62= Kg) = ue(g) 9” Qt €(>) = e.e [1 + E, cos Ko] (o stands here for z- =) we get 2 4 2 2 2 oP aS a Ss VoP + F(E) P= 0 (26.1) d& Q™ (K"= k°(§))~ an KT=- k™(&) 2 4 2 2 2 an _K Ae 5 oe > VGN + G(E) N= 0 (26.2) 0& N° (Ko= ko (E))~ on K= k™(&) where kk! + kt? k k! 2 G(—) = 7 tS yy K*- k K*- k 2 3,e',2 . lie" k kt et ki 4+ k kM k k' (2 FQ) = -F-—)° + 5D - eo + + ( ) 4*e 2°€ Kee 2 € Ke. «2 Kr. Ke and a prime means derivation relative to € The equations for P and N are similar, and we will study only the N equation. ° The separation of variables method can be used to solve the N equation (this is the objective of all the above transformations). Writing NCEn,x,y) = ¥,(n) Yo (x,y) ¥4(E) and using y and 6 as separation variables, we get ay, dn ~63- 2 2 Vo XY, + 5 % * 0 (26.4) 2 a 5 5 + J(g) 5 = 0 (26.5) as 4 2 I 2.2 K R where JE) = GE? y? - 2 6? + te) Q" XxX - k Ko= k and J(&) = periodic function of & KE _ Ke - &t Taking y= 5 7 » equation 2G.5 becomes d7y oo — 2+ (a+2 J acos 2ny) Y,= 0 (26.6) 2 fe n 3 dy n=] where we took “o 4 JI@) = a+ 2 )} a_cos 2nw 2 fe) n K n=] This is Hill's differential equation and its solution is Hill's func- tion u(w;y,6) . So ¥g = ulsy,d) where is the independent variable and y and 6 are parameters. The solutions for Yy and Yo are straightforward, and N is given by N = u(v;y,6) et* + tn in Cartesian coordinates with no y dependence, or N= udysy,6) en" B (6p) ~64- in polar coordinates with axial symmetry (3, = Bessel function). Hill's function can be written under a Floquet form [22] ° Qy I u(sd,y) = J (8) ete¥ * Zany n=0 where C. are parameters and 8 is the Hill exponent which is given by tV¥a_ sin? = = A(0) sin’ > where A(O) = Det||Al| =] 1f mn mn an [A] | = matrix with elements A = —_ if mn mn a - 4m ° a = coefficients in Hill's equation (2G.6) 1. Half Space N is given by N(E,n,x) = u(wsy,6) ete tL . ) c_ (Bye 2nt8)Y + tn + 16x or , 8 N(x,z,t) = ) c_ (Bye INCRAnRE) LF Keke) + idx + Lyn n Developing et!" in a Fourler series, and as the time depen- er (w+hQ)t dence must be of the form » we get Ge t2K)z - L(wt+ 2Q)t + idx N(x,z,t) = ) ) Ay e q qk “4 where -65- Y +6 an: and Yq is determined from the system of equations y R? +o. w+ GQ + S. J (p2- 1)2- ef I (2G.7) Iva ein” TB = (0) sin’ — From these two equations we can eliminate 8 and get ¥, 69) or K_ 6): 2. Dipole Radiation For the magnetic dipole we have N= usy,6) e*Y7 B (50) = u(y38)e"¥P" B60) This is an elementary solution and we must integrate over all values of 8 to get the total field, so ° N= | w(B) u(y3B) er Pn B (6p) 48 where w(8) is a weighting function and 6 = 6(§) . The magnetic field can be found from The magnetic field must be such that: -~ Ho is continuous at p = a (see Fig. 10) ~66= ~~ [H,(0> a) - Hp <a)],. =~ 1, 6(z - ze WF -- H finite at the origin -~ H satisfies the radiation condition at infinity where z 5 is the dipole position on the z-axis (2, = 0 if the dipole is at the origin). From these conditions, after long computation, we can determine w(B) and we get finally 2.2 co Xv- k(E) 8 ee eee Ve ME | v8) wcos8) K7~ k2(E) where y= = gE E Kz K n= | q 0%) do 0 m © dipole moment and we used the Hill's functions orthogonality relation = 2n $e u" (x, 38) u(x;8) dB = 6 (x -x,) lf v qr 9 M+ 0) » we get k(q) = 0 -67- ke | 2 A we Y “ Th <0 ie M = = omy | u" (28) us8) a? (8p) dp —0o which is the result found by Casey [11], with a difference of a fac~ tor iW because of the Hertz vector definition. The Hertz vector method gives the field in terms of Hill's function, but it is rather a long method and it does not give a clear picture of the modes and the harmonics. This led us to use the first method in our study. ~68- Til. SPACE-TIME PERIODIC ISOTROPIC PLASMA In this chapter we consider an isotropic plasma submitted to a propagating sinusoidal disturbance, which leads to a plasma density N(z,t) = Nj + Nj cos (Kz - 2Qt)] A. Wave in Infinite and Half-Space 1. TE waves (Fig. 1) Maxwell, Newton and the continuity equations are oH OE VX E@-U5 oP VxXH= €) a5 + Nay 1 VeH= 0 Ve E=—o0 — _ € ° dv . 32 . — = V (Nqv) + ae 0 evs gE and for the TE wave E= E(x,z,t) £, Taking E™= 0A/dt it is straightforward to get the wave equation for AG@=Ae) 2 2. oA 2 VA - Woe 52 - Hye QW, (zt) A 0 (34.1) where . w"(z,t) = we [1 + Nicos(Kz - Qt)] (3A.2) Pp.” po 1 and 2 2 Noa G@ = -~69- Using Floquet's theorem and the superposition principle as we did in Chapter II, we can write ns-}<o i(KinK)z + idx - dwt As )} Ale Teo where WO = WwW + 2Q n Putting the expression of A in equation 3A.1 and equating the elements of the same frequency, we get the system of equations N 2 2 92, 22 1 22 (6+ (ktmK)” - Bo + Biol Ant Bro Ane t Any) 79 (3A3) or DA, +t Aner Age * 2 (3A.4) where w 2 Lg 2 a2. 22 Dd. Ny 92 [Oo (Kink)" - Bo + Boa] (3A.5) po B50 * YOK obo 2 2 2 BS = Ue w= He, w + nQ) To have a nontrivial solution, the determinant A(6,k,w) of this system must vanish A(S,K,w) = Det Oe) = 0 (3A.6) This is the dispersion equation. Solving the dispersion equation we get the wave vector diagram 6(K) , or the Brillouin diagram K(w) . -70- In Fig. 15, we plotted the wave vector diagram correspond- ing to Ny = 0,25 ; w/R = 2.5 , W : Pe 0.75, re Sa fe 355 K Vv fe) d where c ™ e.m. wave velocity in vacuum V5 = disturbance velocity The modes with index n , such that w+ nQ < w, » are cut off. In Fig. 15, this corresponds to wW-~ Ww n< #o— = -0.51 or The wave vector diagram envelope is a hyperbola. Its analytic expression can be found by taking the limit Ny -~O. If Ny +0 , the dispersion equation becomes (from equation 3A. 3) -7l- 0 = Ww 03 spuodsa1109 sUTT{ Od poysep ayy * ¢67°O = Ln *cL°0 = M/ ™ S6°Z = B/M *G'€ = Y twesr8eypp 10qDd|aA aAeM “CT *BTd xX SO O I- c- <x} a \ edojcaud\\ gsb AV ~72- 2 2 2 2 6°+ (K+nK) ~ 8+ 857 0 or 2 2 2 2 wT 850 Y“+ (X¢n)* = a i (3A.7) R 2 where Y= 8 and X=< K K This represents a family of circles centered at X= =n with radius . 1/2 1; 2 2 Toil io “Bd . The envelope of this family can easily be found to be W, 2 2 y* _ (x - iy = Ooo 1 R29? which is the equation of a hyperbola centered at with an asymptote yo —L_ (x = >) 2 R= 1 2 For each value of 6 (or K) there is an infinite number of values for Kk (or 6) which are not related by any simple relation because of the disturbance motion. So the expression of A must be i(k,+ nK)z + iéx - dwit nq JA and as E= 7, ? then i(k + nK)z + idx - int q n with -73- EO ss - iwa nn aq q and the relative amplitudes of the different harmonics are E W A W E WwW oA W ng oq oO oq where Cag is defined by 28.8 with 2 1 2 2 2 2 Dag = Ny a [6+ (K+nK)~ - Be + B50! po Finally, the convergence condition lim |p | > 2 (Poincare's theorem) n nro is always satisfied and there is no sonic region of divergence. This is a major difference from the dielectric case. Therefore, for the plasma, our study is valid even if the disturbance is due to an elec- tromagnetic wave. 2. TM waves (Fig. 1) Here we start directly from Maxwell, Newton and the continu- ity equations. Following the same method used in Chapter II for the dielectric, we can write n=-}co i({kinK)z + idx - iwt n H=e } He n=-}o i(K+nK)z + idx - twit A } (A. e +A ede = nx —x nz —z a=0o and we get the infinite system of equations -74- 2 DD'-G DoD 1 1 n n+l fy + JA + (P+ ja GC. Ctl Gon nz Go G+ n+1,z D D! n n-1 1 1 + [—+——J A +——A +——A = 0 (3A. 8) 6 Gy Oo nel,z Go) at2,z Gy 722 This system is identical to 2B.16 except here we have 2 2 2 2 Dd. ma ae [(K+mK)~ =~ om + Boo! 1" po a 2 2 22 2 D, N 3¢ [6 Bn + Boo! Ll" po G * 26 (K+nK) n N 3° Ll’ po The dispersion equation is A'(S,K,w) = Det (rere) = 0 The characteristic equation of the system 3A:8 when n+ © is 0” +AD +1 = 0 where 2 are g Ll” po fe) 2 2 2 (K"~ KD n and the convergence condition lim |A| > 2 n.+o is always satisfied. (Even if K= Ky it can be shown that the ~75- convergence condition is satisfied.) Therefore, for the TM wave too, there is no sonic region. 3. Reflection and transmission: half-space The problem can be treated as in the case of the dielectric. The only differences are: 1) We have to use the corresponding Dag in all the expressions. 2) The solution of the system 3A.8 gives Ang and we have E a - WA . nq n ng B. Dipole Radiation The method of solution is identical to the method used in Chapter II for the dielectric, and only the final results and patterns for the magnetic dipole are given here. The magnetic dipole electric field is found to be ° ; (1) £¢knK}z = iw t En 2 d L | 0,00 g 4H (Soe an and the normalized average power pattern is 9 G (®) = (2 “n 1 ) ) as_ (c' ¢ 2 n 0 we w «2 sin“@ qj \s" | oq md Pp Oo qs where the notations are the same as in Section IIF-3, except that Ja = =|[cl] > + [a The factor i/w. appears because the source condition applies to E , not A. ~76- In Fig. 16 we plotted the curve 9 = — arctan & function of K/K which corresponds to the wave vector diagram of Fig. 15. This curve gives, for each radiation angle @ , the corresponding modes and saddle points. In Figs. 17 and 18 we give the radiation patterns for the fundamental wave (frequency WwW) and the first harmonic (w+). As in the dielectric case, there are caustics and no symmetry relative - to @= 90°. The harmonic (w-2) is cut off because w-Q < Wo . -J7J- 0 « Ty 03 spuodseiio> auy{ peysep au SZ°O = Ln “cl1°0 = fon ‘o°Z = 9/m Sg’e = y :eusetTd @ 1oj OTA Y: = @ ®AIND ‘OT *3Ta i 3? ot OL1- OSI- Of! - Oll- en je -78- moce qei | 0.2-> a i f 8 3 4 i 010 30 50 70 €0 HO 180 150 170 Fig. 17. Fundamental radiation pattern in a disturbed isotropic plasma w/Q2= 2.5, R= 3.5, 0, sft 0.75, N, = 0.25 The dashed line correapones to N, +0. o> 8 ova G mostly from q=O | -- mosiy from ql 6.2 or a i Lo} _——j a ee i 10 SO4050 70 Sd ti0 130 170 Fig. 18. Harmonic (w+) radiation pattern in a disturbed plasma w/2 = 2.5, R= 3.5, w/w = 0.75, N. = 0.25, Po 1 ~80= iV. SPACE-TIME PERIODIC UNIAXIAL MEDIA The electric permittivity tensor of a uniaxial medium can be written e! 0 0 | e = lle) =|;0 e 0 | 0 ] ei in a Cartesian coordinate system, where the z axis is parallel to the optical axis. For the uniaxial plasma, the static magnetic field is taken parallel to the z-axis (Fig. 19). We assume the medium to be perturbed by a sinusoidal distur- bance propagating in the z-direction, such that e' = e'(z,t) = ee, [2 + €,cos (Kz -2t)] 1 ce" a e' (zt) = eee {1 + €,cos(Kz -Qt)] 3 for the dielectric, and a N(z,t) = Nol + N, cos (Kz = 2t)] for the plasma. ~81- Ce teed ed ore eter ere evens woo emery F won cree A Optical axis or —w/ static Bo ees x y 2 Fig. 19. TE and TM waves in unaxial media A. Wave in Infinite and Half-Space 1. TE wave (Fig. 19) We find for the electric field wave equation 2 VE ~ Hy —. (e'®) = 0 (for the dielectric) dt 2 375 VE-wue = 0 (for the plasma) oo 4,2 ot with E#E = . These equations are to be expected because, as E is in a plane normal to the anisotropy axis, it does not see the anisotropy, and we get the same equation as for the isotropic medium. For the uniaxial plasma, as Bo is infinite, only the electric field z-component, if it exists, can interact with the plasma (see next sec- tion). -82- 2. TM wave (Fig. 19) Here E= Ee +E£e and H = He _ xx Zz _ —y and the electric field sees the anisotropy of the medium. This leads to drastic changes relative to the isotropic case. a. Uniaxial dielectric Maxwell's equations are oH EOE, yu VX BR hy oe 2 7 Mo Oe (4A-1) = ale) Se" E » From Floquet's theorem and the superposition principle we can write H = ) A idx + i(KinK)z - i(wing)t , E= J (Ee +E e yet (KtnK) z + idx - i(wint)t a ne2-z Putting these expressions in 4A.1 and 4A.2, equating the components with the same frequency, and eliminating La and Ex as we did in the isotropic dielectric problem, we get DoD! - GC. Gr D Dp! ye 1 1 n nt1 fo el Ht | + Go Gi+l Gel nz Go Go+l ntl ,z ~83- es : l tt op] +——E +-o> E = 0 (4A. 3) Go Ci-l n-1,z Ciel nt2 ,z Gel n-2,z where «27 te 2 S_y2 dD = E _ en, d= E [1 -G a) ] 12 . 1 BS 3 Bo G - 2 6 (K+nK) Gg! = 2. S(KinK) n € 12 n e. -m2 1 8 3 Bo n oo 2 wy = ve 2 Bo HEEL oy Bn u ofor an or in a matrix form | EI = 0 (44.4) | which is similar to 23.17 but with different elements, and Det ||m|| = 0 (44.5) is the dispersion equation. From equation 4A.4 we can compute the relative amplitudes E/E and then from Maxwell's equations we get E/E and nz’ “oz nx’ “o H/Ea, . To get an idea about the wave vector diagram, let us take the limits €&,* QO and é, > 0. At these limits, equation 44.3 becomes & 2 WwW e it Q r or e) a) (X + n)* +T —s y? = on 2 e R'2 r where X= K/K y= 6/K Ria See — Lb _ Va vivu ee! door This is a family of ellipses centered at X= ~n with axes ratio Ce" /e) . The envelope of this family of ellipses is a straight line myt/2 c 1 W Y= & ——__ &- 5) xr [a 12_ l So the wave vector diagram for a uniaxial dielectric looks like the isotropic dielectric diagram (Fig. 3), with the circles replaced by ellipses. The characteristic equation of the system 44.3 is ot Bo? + Ap? +Botl= QO where A= [1-R")} +2 . “173 Bet a-r") +h 1 3 Rt a Vv d c' = speed of light in the * direction From Poincare’ convergence Theorem [18,21], the sonic region is found equal to € LSB <lte for R19 <1+—+- 2 Lt-€ _ L _ Ee 1 € 0 < RY < l#e for Ri >14—-+- 2 - _ 1 € 1 3 Remarking that € L-2¢.,+—~2e1-¢c,+e¢(4+-1) > 1-e 1 E, i l'e, _ 1 then the sonic region is given by l-e < RY 2 l < 1+ e, for ail R' 1 and the value of E4 has no effect on the sonic region. b. Uniaxial plasma Maxwell, Newton and the continuity equations are aH BE VxHee apts (4A. 7) VeH=0 ot dv Ror = gE + qv x B, (4A. 8) J = Nav (44.9) and B = Be. = static magnetic field. cane) omz From 4A.8 and 44.9 we have J. 22,2 cn — x dew’ meaty? “SS (44.10) where w,=4B . “BO m-o 4 Let J, j*e, , then |Z x w,| = Jo We ; but, as the medium is uniaxial (Bo oo), then we must have Jn =O and oA Taking E = rd » equation 44.10 reduces to 2 J Ew (2st) A, & and equations 4A.6 and 4A.7 become XA = om Vxa UH (44.11) 378 ; VxH = € —=+ew Ae (44.12) - Oo 4 2 op 22 Ct Using Floquet's theorem we can write i(kinK)z + idx - iw it He ) He e “Ya i(kinK)z + idx - iw t A= ) (A:e +A ede _ i Axx nz —z with E = -iy A . Putting these expressions in 44.11 and nx/z n “nx/z 44.12, then equating the elements with the same frequency, we find “6A, + (kink) AL = UH ~ (KinK) H = ie a A n on nx ~ oH a ie wr A - te w [A +—A n on uz [e) po" nz 2 nel,z + “2. Solving this system we get bon Ek 2 Anz BF - (kK) te 65 w H =- A n 3? _ (xenk) “ nz n DiAng + Antl,z + An-1,z = 0 where 2 2 2 n 6 Dy Ny ae (1 - 3 cme po n or in a matrix form (mil + [AL] = 0 and A(6,K,w) = Det ||M|| = 0 is the dispersion equation. > 7 N Ny An-1,2 (44.13) (44.14) (44.15) In the limit Ny +0 the dispersion equation reduces to (from 44.13) re 87 + (1 - —B2) (eenk)? = go - 8? n po wo or 2 2 2 aye “50 2 97 o fe) ve = 3) en)" = we RQ where (4A.16) 6 K YX x? X KR? Ww» This is a family of conics, and the wave vector diagram will consist of -~ hyperbolas for n such that w+ nQ < 0 -~ ellipses for n such that w+nQ > w All the special and interesting properties encountered in a uniaxial plasma are studied in the following three cases: Case (1): N, = 0.25, 2=0, w 24 WoW > wW OO 3 “po 1 po The wave vector diagram shown in Fig. 20 is a family of ellipses which interact strongly near their intersection points and lead to the apparition of stop-bands. As the group propagation angle is given by %y = -are cotg = , then there is propagation for all values of © , which is to be expected because Ww > Woo" Case (2): N, = 0.25, 250, w= 0.8 Woe Wy) i po For this case we get the diagram of Fig. 21 which consists of a family of hyperbolas with stop-bands near the interaction points. For the limit Ny + 0 we know that there is group propagation only inside a cone with half angle c= are sin — . in fact, we see on Po Fig. 21 that, at this limit, the slope of the curves is jasymptote slope| = < |slope| < But for Ny # 0 the slope near the stop-bands goes to zero, and this -39- allows propagation for all angles ©. This is a special effect due to the presence of the disturbance and it will be explained physically in a later section. Case (3): N, = 0.25 , R= 3.5, w® 2.52, w ~4 Ww 1 3 Po Here there are harmonics at frequencies un , and the plasma frequency Wo will be in the middle of the spectrum. Therefore, we expect to have a mixed wave vector diagram which contains ellipses and hyperbolas. Also, there is the disturbance motion effect. In Fig. 22 we plotted the corresponding wave vector diagram, and we remark the following factors: -- The modes (q > 0) are ellipses because wW+q > woo -~ The modes (q < 0) are hyperbolas because w+ qf < ro ~~ Because of the disturbance motion, the ellipses are not equal and the hyperbolas' asymptotic slopes are not the same. In this case too, all the modes propagate for all @, even the modes which, in the limit Ny + Q , were limited to a certain cone of propagation. Finally, the relative amplitudes of the harmonics can be com~ puted from equation 44.14, and from Poincare convergence theorem we find that there is no sonic region. For the reflection coefficients from a half-space, we can use the results of Section II.C with Dag given by equation 44.13. ~90- * go « Igy 03 spuodsaii0d auyT peysep ayy 'T = V/Y ‘Sz°0 = Ty *si'O = often ‘Qo = Pa reused [PEXepUN ev 10s WeaseTp JoJIeA VAeM “OZ “3TH X | GO 0 GO- I- oy oy = =<. ~=D ~sb sb b =b > 9 « 'y 03 spuodsasi09 autT paysep auL ° 9° = V/V *¢z°0 = th ‘cz°t = a7 dn ‘9 = Pa reusetd [eExeyzun e z0J werZeTp 10390A BAeM “TZ *3Td x Sl | GO 0 GO- I- GI- O< TN 03 spuodsaii0d SULT paysep ayy 6Z°0 = ty *“cl°O = oP *¢°Z = y/M SG°e€ = y teusetd [erxezun e 10jy weiseTp A0OJVAaA |AeM ‘*7z °3TA 0) I- aN T -92- B. Dipole Radiation The dipole moment, the optical axis (or 5, for the uniaxial medium), and the disturbance propagation vector are supposed to be parallel to the z-axis. For a magnetic dipole, the electric field is in a plane normal to the z-direction, and it does not see the anisotropy as we showed for the TE wave, thus: ~- In a uniaxial dielectric the radiation pattern is iden- tical to the pattern in an isotropic dielectric with e = e'(z,t). ; ~- In a uniaxial plasma, the radiation pattern is identical to the pattern in vacuum (i.e., € = E43 therefore, only the electric dipole is studied in detail. l. Electric dipole in a uniaxial dielectric The wave-vector diagram for a uniaxial dielectric is the same as the isotropic dielectric diagram (Fig. 3) with the circles replaced by ellipses, because eo # en therefore, all the results % found in Section IIF.2 are valid here if we use the corresponding Dog and diagram. 2. Electric dipole in a uniaxial plasma The wave vector diagram is given in Fig. 22, and the field expression and radiation pattern are found in the same way as for the isotropic plasma. We get caustics, but also there are the cones of radiation on which surface the field is very large (see Appendix D). The cones of radiation appear only for some modes (with q such that arql < Win) but because of the inhomogeneity created by the -93~ disturbance, the radiation is not limited any more to the inside of- the cone. Let us consider two phase rays, 1 and 2, inside the phase cone angle (Fig. 23); then if at a point A we take the vector sum of the rays 2 and 1' (which is the reflected part of 1 from an inhomo~ geneity surface), we find the resulting ray outside the phase cone, in the previously forbidden region. — we wa cone — =e eh | «e —_—— oo oe a —_ em ame (oe eeplUleeePerlll eee ee mee Oe eUll lll Fig. 23. Radiation in the cut-off cone -94- This was verified by computing and plotting the radiation pattern for a case where we expect to have cones. In Figs. 24 and 25 we plotted 0 = £8) and the radiation pattern of an electric dipole in a uniaxial plasma with stationary disturbance (i.e., N(z) = NG + Nicos Kz)) and wW > Wo’ No major 1 difference with the isotropic case occurs. K In Figs. 26, 27, 28, 29 we plotted @-= £ (=) and the pat- terns for the harmonics (Ww, wif, we) of an electric dipole in a perturbed uniaxial plasma with R= <= 3.5, w/= 2.5, wety , N, = 0.25 d The harmonic (w-2) pattern contains caustics and cones because the mode q = ~1 in Fig. 22 has a hyperbolic wave vector diagram (i.e., w-R < wa: The disturbance motion has an effect only on the caustic angles, but not on the cone angle (the cone angles are 53° and 127° = 180°-53°). The fundamental (Ww) pattern contains caustics but no clear cones. But for 36° < @< 53° there is a slight increase in the field due to the up-conversion from the mode q2-1l which has a cone (i.e., large field) in that region. ~95- "O< th 0} spuodsal10d sUuTT peysep syy Pp "¢Z°0 aly *cl°0 = nn ‘0 = 8 ‘0 = A ‘mo =¥Y ‘:eusetd [Terxetun e& 10j (x/ says =@ oeAInja ‘47 °3Tq 08i - scouSep ulg 9 ~96- i ; ‘N / ond ! i ! , ~~ 60 10 30 SO 70 SO HO Fig. 25. Radiation pattern in a disturbed uniaxial plasma va * 0, 2= 0, w 62 = 0.75, Ny = 0.25. The dashed line corresponds to the pattern in vacuun. R = 0, I30 150 170180 @ > ~97- T 0 <« N 03 spuodsei109 auT{T paysep ayuz d 6Z°O = th *cl°O = m9 m §¢°z = U/m *o°€ = yY seusetd [erxezun e Io; (x/*a)3 =@ eAIND “97 °3Tq - O81: |-=b epow --OS] “7 , Gl | o7 HL oo a = 7 “ 7 OL 7 4 “7 “7 cof O=b spow 0 sooi$ep ul 9 -98- \ caustic SS caustic ve 7 So caustic of mode q=O | one caustic of mode q=-i . region of the cone of moce q=-| = ! ! l ! ! { 2 010 30 50 70 930 HO 130 150 170180 Fig. 27. Fundamental radiation pattern of a dipole in a disturbed uniaxial plasma: R= 3.5, w/Q = 2.5, wou = 0.75, Ny = 0.25 ~99- caustic cone =e COUSTIC YN cone &G_, cone qe | | coustic | 4 | | -; caustic Aq=O | | | | ; | i} LL a: I i O10 30 50 SO II0 150 180 Fig. 28. Harmonic (w-) radiation pattern in a disturbed uniaxial plasma: R= 3.5, w/®= 2.5,W, fw= 0.75, N, = 0.25 ~-100- ccustic of mode zl lar from mode q=0 ot fh ea } i Lod Me. O10 3sO 50 70 90 Ii0 130 150 170180 Fig. 29: Harmonic (w+) radiation pattern in a disturbed uniaxial plasma: R= 3.5, w/Q = 2.5, 06% = 0.75, No = 0.25 Vm -101- V.__ CERENKOV RADIATION IN SPACE-TIME PERIODIC MEDIA In this chapter we study the radiation from a moving electric charge, with constant velocity, in sinusoidally space-time periodic media. In a dielectric (Sections A and B), we get transition radiation and, under some conditions, Cerenkov radiation. In a plasma (Section C), there is only transition radiation. The charge is assumed to move parallel to the disturbance wave vector (i.e., z-direction). A. Fields The current density due to the charged particle in cylindrical coordinates is ev = —P - Ee 8, Tap SP) 6 vo) where e = charge of the particle v_ = velocity of the particle The radiated electromagnetic field is transverse magnetic relative to the z-axis, and most of the basic results found in the study of the electric dipole are valid here. Two factors must be taken into account ~~ the current expression is different ~~ the radiatec wave spectrum is continuous not discrete, and we have to integrate over all the frequencies. 1. Field expression The magnetic field can be written -102- oO 3 He» ) ?) | | Wy (Kw) a (5 per (imi 2 = £ (wand) dk dw n q OD we (5A.1) with H= Hey » and the source condition is 2T0H = ev, 6(z - v%) (54.2) +0 Putting the expression of H in 5A.2, we get 7) | | BEd g gh(eink)e-s(win)e a gy Md, a 4 = —P 6(z - vo) or if f ~ian V. ”- | )} ete ) a At ei (kink) z dk - if e Pie tue dw = 0 m0 OD q where qo Atos 0 (determined in Section II.F.2) ng E oqz . WE g Og2 a = q Sy S(ax) = = 8 (x) 6(x) = | etn dw This equation must be satisfied for all t , therefore 7») | a at eitkinK)z - int oC ie eltw/v,)z nq qaq OO and to have this relation too satisfied for all z and t , we must have P ' a )} aati = 0 for n #0 (5A.4) q 1 . ie ) oar 3 (5A.5) q or in a matrix form \ja*|| + Jol = |r| (5A. 6) where P=0O forn #0 [P| = column vector with elements n Pot 1e/4 As K is related to w , then the, integration over K can be deleted, and the H field is completely determined by . ' (1) H l L | a, @) 6, () Ag Hy (6,0) i(—+nk)z - Lwin) t xe Yp du) (5A. 7) and in the far field oT co) 2a , gl/2 axe H Vip e y ) | OAigg fH (5A. 8) qo where X= 6, sin O+ < + nK)cos 8 = (win) = . (54.9) P o = r sin 6 -104- From the above expressions we see that a band of the spectrum near the frequency un will propagate with a phase angle = arctg wut = arctg > (54.10) The integral in 54.8 can be computed by the steepest descent method. For r+ with t/r staying finite, we get (see Appendix C) 6 ixrt+iZ-sign 6") 772i qs 8 4 qs r sin @ l La “Gs Ana 8 \o. | © qs where the index s means the value at w= oo saddle point, and W is given by x = 0 (5A.11) WW s or ds cos 8 t sin @ =i += -== 0 (54.12) URW P : 2. Radiation cones From equation 5A.12 we have rcos 0+v6'(@ ) rsin9 = vt Pq s or zt Vo q 645?” = vt (see Fig. 30) This is the equation of a cone with axis z , vertex at vt (which is the position of the particle) and half angle % Gs given by ~105- l tg % as = iG, Hence, for a given mode of radiation, the band of the spectrum around We (and w+ nf) will be concentrated in space on the sur- face of a cone attached to the particle. ei lootg $s)P = Vol . Seummee aal o_o exe mie eee 0 ee | | 9 osI\ : ae | “ | Fig. 30. Cerenkov and transition radiation geometry -106- 3. Brillouin dtiagran From Section II.F.2, where we studied the electric dipole radia- tion, we have (the subscript q is omitted for clarity of notation) DE. + E44+1,0 + E-1,0 7 Gea = 0 DE ae + Entl,z + En-1,z * Fn = 0 where 2 Kink, 2 De fl- CO) nf) B, 2 6,2 Dias [l- @°) 1 B. g = 2. Skink) nn € 3? n or in a matrix form IIpil+ le] + liell- ig] = 0 "ll + fel + Hell + [e,] = 0 where Ile]| = diagonal matrix with elements Go {| D|| or [|p' || = tridiagonal matrix with diagonal elements Dd, Cor Dd.) and off-diagonal elements equal to l , Solving this system for [E_ | we get g{|~* + | C{iel] - |p] D'||)- lz] = 0 -107- and the dispersion equation is given by or pet(||cl| - |{pl| + ile{|7* + |[p'{[) = 0 At the limit €) + 0 the dispersion equation becomes D Di «a ¢? nn n 67 + (kank)* = B As K ® w/v, , then or where 32 7 Cita Cy? Vv Vv c P ve - ve (vi- Vv )? y2 _ PB c x2 2 d 2 2 2 Vv vioe-v c c Pp 2 vo VAY vioe-Vv c Pp yY = 6/K , X = w/ Kv, v._= e.m. wave velocity v, = disturbance velocity v_ = particle velocity (5A.13) (5A.14) (54.15) (54.16) -108- B. Radiation and Spectrum In this section we study the angular distribution and the spectrum of the radiated field using the Brillouin diagram, for dif- ferent relative values of Vos %, and Ya: 1. Stationary disturbance (vg = 0) We assume a sinusoidally stratified media with —€& =e € (1 + €,cos Kz) or 1 then the limit dispersion equation 5A.16 becomes 2 vin-ve 2 2 v, Vv Voe- Vv c c with 2 Yo x = X+tn 5} 5} Vionwr-Vv cp In Fig. 31 we give the Brillouin diagram corresponding to dif- ferent values of vac and é, 7 0. For v Iv. <1, the diagram consists of a family of ellipses (only 2 ellipses are shown), and only the modes q< 0 appear. For v5/%Q > 1 the diagram consists of a family of hyperbolas (for gq # 0) with asymptotes parallel to the fundamental mode branch (q = 0), which is a straight line. For at # 0 strong interactions occur near the intersection points (Fig. 32). In Fig. 33 we plotted the radiation angle 1 v_é' Pq ) o = arete ( -109- function of X= w/Kv,, , and in Fig. 34 we give the Brillouin diagrams and the radiation angle curves for specific modes. In Fig. 34a va/%e <i and ¢ = -1l. The spectrum is limited to 1 < xX < 1 Vv Vv 1+ y -—2 Vv Vv c c or Kv_v Kv_v Pf 6 y <« —P& vVitv vr-v P c Pp and there is radiation in all directions. This is a transition type radiation and it is caused by the inhomogeneity of the mediun, In Fig. 34b v,/%, >1l and q#=0Q0. Most of the radiation is concentrated on the Cerenkov cone surface, with cone angle Vv ¢C ¢ = arcsin — c Vv P but some parts of the spectrum are radiated inside the cone. In Fig. 34c v,/% >1 and q = -1 .* The radiation spectrum is Kv_v Ww > Po _ Vet Vo The lower frequencies are radiated backward (i.e., ¢ = 0) and the higher frequencies are radiated in the Cerenkov cone. In all the diagrams an inflection point leads to a caustic in the radiation pattern. The above results, which are obtained by an elegant use of the Brillouin diagram without involved computations, match perfectly with -110- Fig. 31. SBrillouin diagram for e, + 0 -lll- <V Fig. 32. Brillouin diagram for ey 7 0 Vv -112~ Vp S Ver Vg tO SO0° - 30° Fig. 33. Radiation angle $ fmetion of X = w/Kv, -113- AY 1Is0A b(degrees) a)rnode q=-!| I20;- a)mode q=-| Vp < Ve Vp <Vo 80Pr 40 = A L o Ve 3/2 x vo-Vp b) mode q=O 60 - Vp >Ve P| ANA 20 > > X /2 | X A c)mode q==| Vp >Ve 60f-- Pol 40- 20 - it i > ! l — “a Fi/2 3/2 y 1/2 3/2 \ Vpt Ve Fig. 34. Brillouin diagram and radiation angle for different modes and different cases €, # 0 -114- the results of other authors [13] who used long numerical computations of Hill's functions. 2. Moving disturbance (vg # 0) We assume v,<v (a) vio<v The corresponding Brillouin diagram is a family of ellipses for the modes q < 0 which are the only ones to radiate. Each mode has a series of harmonic spectrum bands given by (vo /vQ) 1 (v/v) +1 rats wy) L + nQ2 < w nq [ale (v/v)? -ait nit JA for v <v.<v P c d (vi/v,) +1 (v_/v,) - 1 ec 6d ' cud lala Gy et * nd <u, < ial® Gy) 17 ni for v,<vi<v d P c For a certain mode and harmonic the Higher part of the spec- trum is radiated forward (¢ = 180°) and the lower part backward ° (b) vo>vi>dV The corresponding Brillouin diagram is a family of hyperbolas for q #0 anda straight line for q=0. The fundamental mode (q = 0) corresponds to the Cerenkov radiation, and the other modes will radiate inside the Cerenkov cone over a spectrum -115- vor Vg @® > qKv ——£ for q> 0 P v-v c Var Va “aK + < W > akv,, ve v, for q< 0 For the special case v.v_ = ve we get dp Vv w > |q| + Q for all q c C. Transition Radiation in Plasma We follow the same procedure as in the previous section and we use the relations obtained in Chapter III for the isotropic plasma. Only the final results are given here. In the limit N, +0 the dispersion equation is 1 2 Wwev)? w 2 p. 2 2 ‘Yq ‘> po,2 Yo + (1 2 ® qq 5 Com) vVvoe~- Vv Cc c c Pp where 2 Vo 7 YnYg x = X¥+q—j—FS VvVoe- V c P x #= w/Kv,, > Y= O6/K As always Vy < Yo in a plasma, then the Brillouin diagram is a family of ellipses and only the modes uo, (ee q £7 Kv v= Vv dp will radiate. The radiation is of the transition type. -116- Each mode spectrum will consist of harmonic bands given by —yY tai¢tal < w < at + nQ Yq q 7 q Yq where = ae 2 _ _ 2 2 _ 2 Yq 2 5 est Kov Va Vv Wo%c v» ¢c ve Two special cases deserve more study: Va* % and va * 0 1) vi =v The radiating modes correspond to w Vo + Vv < ~ Po = 4 2 Vo Vo and the radiation bands are Vv Vv —P_. - ——_P__. nQ + vt v (lq ]a a) 4% < n+ vat y ([q|@ + a) P with vetv 1/2 as (9207 . we c By / po vo ‘> An interesting fact in this case is that the modes do not intersect each other. The radiating modes correspond to ~117- Wo a) qs ~~ fi- P c and the spectrum bands are viv viv lalk 4*s-b sw < |qlk—PS 5+ vioew-Vv q VvVoee- Vv c P c P with 2 qKv_v W b- “| eee es - Vv Viow-V c P c These results (for Va = 0) which were obtained with a minimum amount of computation compared very well with the results obtained by other authors [14] for Ny = 0.4 and different values of Vy? Wor" at the expense of long numerical computations. ~118- VI. WAVES IN GENERAL SPACE-TIME PERIODIC MEDIA Most of the results found in our study of the sinusoidally space-time periodic media are extended, in this chapter, to general space-time periodic medta. We assume a medium submitted to a propagating periodic dis- turbance, such that € = €(2,t) = ee (2 + €,<(Kz = Qt)] for a dielectric N= N(z,t) # Nj + N, f (Kz - Qt)] for a plasma where f(n) is a normalized periodic function which can be developed in a Fourier series oo £(n) = ) a em m=—0co A. Dielectric 1. TE waves (and magnetic dipole) The electric field wave equation is 2 v*E _ 7" 3 = = 0 dt with EE & . Using Floquet's theorem we can write sd 2 : 7 ) E elox + i(K+nK)z - i(win?)t ~119- (for the dipole eidx is replaced by a) (6p)). Putting the expres-~ sion of E in the wave equation with = ° im(Kz-"t) exec (lte, } aie ] n= we get idx + i(KinK)z - iwt VE = - J [624 (Kenx) 2] ze n n idx +i(K+nK)z - iw t cE = ec. l (E+ €, L a Ey-m® 2 idx +i(K+nK)z - iw t oe £ 2 o ..2 ~ ) BET f » an E-m® ot n m and, equating the terms with the same frequency DE, + } ap Ey = 0 (6A. 1) RAn where 1 oe (xtnk)? Ds» {1 - Jta (6A. 2) n E 2 fe) 1 B n or in a matrix form [Mi] + |E] = 0 (6A. 3) where M., = as, for i# j {|m|| = matrix with elements J J Mi = Dy The dispersion equation is given by Det ||M{] = 0 -120- the ratios C. = E/E, are found from equation 64.3, and all the results of Chapter II can be used here. 2. TM waves We take L(K+nK)z + idx - iw it H = He idx +i(ktnK)z - fut E=)}) (Ee +E ee Putting the expressions of E and H in Maxwell's equations, equat- ing the terms with the same frequency, and eliminating Ho we get GE = 0 DEax tT ) au-2 Box t n nz n Ldn ' = DEL + ) ap EB +GeE 0 n nz Qin nx where 1 (K4nk)? D =#=— {1 - ~~ —Jt+a n fy g* fo) poet - G71 +a 1 n G = 6 (xtnk) n 2 & BA or in a matrix form IIoli + lef + Ici = lz,] = 0 (64.4) mr] lz] + Usll + lel = 0 -121- where [|pl| , [}>" || and Ilo] | are matrices with elements Dis = Ds = ayy for i # j Dir OE Dig = DE Gia = G, and Gis = 0 for i # j Solving 6A.4 for ED we find [Ml] + E.| = 0 (6A.5) where Im] = [lel] - [pl] = [Jel|[7*-{Jor]] Having the expression for [Im], we can apply the results found in Chapter II and Chapter V. 3. Sonic region s For the TE wave we have the infinite order difference equation DE, +.) a. B= 0 (64.6): . Rén If we suppose a> Q for | m| > xr , the characteristic equa- tion becomes r a_P tosses # DOU tees + a1? + a. 0 From Poincare theorem, the sonic or divergence -122- region is equal to the regions where two roots have the same amplitude. B. Plasma 1. TE wave The A wave equation is 2 2 OA 2 _ Vv A- HOE, 5 77 HAE w, (z 0) A = Q t This leads to DA + ) a. A,= 0 nn adn n-2 “2 or j[m|l - [Al = 0 where , Bo 67 (eenk)? Dewy 2 } +a, 1 8 po 2. TM wave From Maxwell's and Newton's equations wé get D Aax * wie 45-2 Apy + cy nz = 0 (6A. 7) TA = Dan ) Fak Ag, + a nx 0 Ltn where , Be - (K+nk)? 1 B po 1 po - 8 DPeyi- 5 } +a, 1 B -123- ‘i N B30 i(KtnK)z + Idx - iw t Je aQ t> ir E= The system 6A.7 can be written in a matrix form and the general results of Chapters III, IV, V are valid here. No sonic region exists in the plasma. -124- VII. CONCLUSION In this work we investigated the problem of electromagnetic plane wave propagation and source radiation in different media (diel- ectric, plasma, uniaxial plasma) submitted to a traveling sinusoidal or general periodic disturbance. The sources considered were: electric dipole, magnetic dipole and uniformly moving charged particle. The basic method of solution consists in using the Floquet theorem in conjunction with the principle of superposition to solve the wave equation or Maxwell's equations. The wave vector diagram and the Brillouin diagram were used extensively to study and explain some special effects without involved mathematical computations. Many interesting effects were studied in detail: parametric conversion and interaction for TE and TM waves, Manley~Rowe relation for oblique incidence, caustics in the dipole patterns, disturbance motion effect on the radiation angles, radiation in the cut-off cone of a uniaxial plasma, Cerenkov and transition radiation and the generalization to any space-time periodic disturbance. ° Many dipole radiation patterns are given for different media and different cases. Finally the potentials and Hertz vectors in space-time depen- dent media were studied and a Lorentz gauge for these media was given. ~125- Appendix A POTENTIALS, LORENTZ GAUGE AND HERTZ VECTORS IN A SPACE-TIME DEPENDENT MEDIA Maxwell's equations in a space-time dependent media are aB VXER- a (1) aD V-D= p (3) V:Be 0 (4) and D=e(r,t) E > Be HoH (5) The field can be expressed in terms of a scalar, and vector potential ¢$ and A , equations 1 and 4 being satisfied by aA Be VxA E = -Vo - 3 (6) These potentials are not unique, in fact if (> A.) is a set of poten- tials representing a certain field, then ee = ' - ~ 2Y " As A+ W g= 9 (7) where yw is any scalar function, will represent the same field also. Equations 2, 3, 5, 6 lead to the coupled equations Q? [> VX VX At S Ce | d+u Se Vo) = ugk (8) oP) et OA Ve(e Vo) +¥(e =) = - 0 (9) -126- By an appropriate choice of Ww , we can choose a gauge which uncouples one of the above equations. We take 9 VeAt xe (use o> = 0 (10) (This gauge reduces to Lorentz gauge for € = constant) then equation 8 becomes 9 A 3 1 VxVXAtU SF ESD - 55 eve | y+ a) = uot (11) t So the electromagnetic field can be determined by finding the scalar and vector potentials (i.e., four scalar functions); but since they are related by equation 10, only three functions are necessary and this is where the Hertz vector comes in. The electric Hertz vector is defined by oll l AT Ho Te eereved Equation 10 is automatically satisfied and, from equation 11, we get 2 a" Ve ° VI- pem-Vel—e=- | I (12) _ fe) ate — € = t with the field outside the source volume given by er 4 1 Eup 7 VEU Ds - EU xvHr ot _ oll H=-Vx5> If e(r;t) = e(z,t) and I= Te, » then equation 12. becomes -127- ; 37m ven , Va- ue —z = [-z ve - | te, t This implies II = I &, and the problem reduces to the determination of one scalar function Ii from the wave equation 2 2 ° 1 oll o° TI Voll re dz ‘€ a2 Hoe 2 1 ot t 2 2 2 where V, = VO - on 2 2 Oz For the magnetic potentials we start with Maxwell's equations with magnetic sources [23, p.12] aD aB VxEe-J -3F (14) Ve Be op. (16) From 13 and 15 we can write A H = Vo +> = x D V Ae — m t where A. is an’electric vector potential and e, is a magnetic.sca~ lar potential. These potentials are not unique and two sets of potentials (¢° AD) or (oA) will represent the same field if they are related by AST AR + WU — 7) o=-¢-2 (18) ~128- where W dis any scalar function. Taking the gauge A. 2% VG + ae = 8 the A, wave equation is found as 2 1 ov AS A. VEG VE AD tH Ta VV GD = Sy We define a magnetic Hertz vector by ou A, = &9F a) = Equation 19 is automatically satisfied and equation 20 gives aM ve OM VM - Ul ae (Ee ap > | Vx (= * 3P? = | J t t (19) (20) (21) (22) Let us suppose e(r,t) = €(z,t). Writing equation 22 for the three components of M we get ‘OM A) a 2 3 x 3 Ve x V M7 Ho Oe Ce otf ba (le ot { Jax t OM oM a, ey Se te XY) - & VE} YX) f V My Mo Ot te at dz (| e | 5t | aay t oM OM OM 2 Bp 2% VE (x4 yy. VM Ho ae Ge 7 es oe xt | 3, t If the source is such that J =J_e J =0,J3 #0 =~n m—z mx ny -129- then M= Me, because there is no source term to induce the two other components, and the M wave equation becomes aM E s Vx 3 oM weeLyv eet hiceeae-Lvxoxm _ Hy = ot ot u _ -130- Appendix B AMPLITUDE RATIOS We have the difference equation DEL t+ Eu. t E17 0 (1) which can be written too Do-1 Eel + mm iu E-2 = 0 (2) These two equations give Burl a Ei-1 ~ D, BO EA =~E._o Do-l n-l and ntl a Bi-l + Dd, n-2 dD, Di-l n-1 So E E n no *a-2 7 “n-1 “n-1 n-2 n-1 EnHl and EA =e 1 Earl D- 1 n E-2 Do-1 + E n-3 Making the same computation for Eo -2/En-3 » and so on, we get -131- -132- Appendix C STEEPEST DESCENT METHOD Let us suppose we have to compute the integral I= | g(t) eth) gy for large x , where h(t) and g(t) are analytic, and h(t) is real on the real axis. Let ty be defined by A'(t)) = 0 , then we have two cases: 1) h"(t) #0 ; then we have (t-¢t > h(t) = h(t) + 7 h(t ») + eee and we get 1/2 ixh(t_)+4 2 sign(h") I= [27 __) g(t_)e ° 4 x{h'"(t | ° ° tS is called the saddle point. 2) hM(t 5) = 0 ; then (t-t > = ttt h(t). h(t.) + 31 h (t,) and 1/3 ixh(t,) l= V5) rap a(ta)e -133- Appendix D CAUSTICS The caustic effect can be explained from the wave vector diagram without any mathematical computations. This diagram shows directly the group and phase velocity directions. For instance, in Fig. a, the radiated wave which corresponds to the point A has a phase velocity vector x such that 9 = (v2) and a group velocity vector ¥, such that @ = (vw, 92) Inversely, if we want to find the points on the wave vector diagram which radiate in the direction 9 , we have to find the points on the diagram where the normal to the tangent makes an angle @ with the k, axis. 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