Basic Rf Theory, Waveguides and Cavities
BASIC RF THEORY, WAVEGUIDES AND CAVITIES
G. Déme CERN, Geneva, Switzerland
ABSTRACT After a brief mathematical introduction, Maxwell's equations are discussed in their most general form as known today. Plane waves are considered in unbounded media, with the laws of reflection and refraction at a planar interface. A general theory of waveguides and cavities is then developed in detail. Periodically loaded waveguides are given as an example of accelerating structure. The parameters which characterize the interaction of resonant cavities with a particle beam are introduced, for both longitudinal and transverse motions.
1 . MATHEMATIC AL REPRESENTATION OF PHYSICAL VARIABLES AS A FUNCTION OF TIME
1 . 1 Sinusoidal variables (time-harmonic electromagnetic fields)
Phasors
Real sinusoidal variables
a(t) = am cos (cot + tp) are represented as
Rea¤’(°"*°)=l,,,] ReA[u] cf (1.1) where A el"' (with to 2 0) is called a phasor and A = dm ef'? is its complex amplitude.
When the differential equations determining the physical variable a(t) are linear with real coefficients, any complex solution of the type A elm yields two real solutions which represent physical quantities: Re(A ew)l = |A| cos (mz + q>) Irn(A e)lw = |A| sin (mz + rp)
Since these solutions merely differ by the time origin, it is sufficient to keep only the first one.
A (capital letter) is the complex amplitude of the real sinusoidal variable a(z) written as a lower case letter.
IAI is the amplitude am; arg (A) = o is the phase of a(t).
In Eq. (1.1), the phasor A el"' may be replaced by its complex conjugate; in fact both conventions are used in the literature. Here we will adhere to the elw convention because it is always used in RF engineering, in particular in all measuring instruments; the e·i°" convention is often used in theoretical pyromhsics. one conventionIn order to topy p_ytranslate (e usefulness results of phasors f stems from the fact that they theare eigenfunctions other, simlrelaceof the operator [dt, jb-i). with eigenvalue Th jo). The exponential eJ°" can be factorized out of all linear differential equations; f`mally the equations are written using the complex amplitudes of all variables, with 8/32 replaced by jco. This formalism also applies to sinusoidal variables which are exponentially damped with time. In that case to is complex:
jm = jwl — otl where co1,ot] are real positive quantities (1.2) OCR Output
le .=2 lR(lEoj e. =l2` Re(e) 2 2* 4 ·2|2
OCR OutputVector (or cross) product
[5.;;*] = EXH; - EyH; 7 (1.5)
1 -· -• * . .. Time ··averae g[l of e >< h= Re5[] E x H (1-6) C·0l'l’\§l€X vector
1 . 2 General time variation
Fourier vamforms
For a general time variation, we use Fourier transforms:
1 im +°° (1.7) - Jwt -dn —j¤¤¢ *00 f(z)-T jF(m) e dw F(co)- jf(z) e dt
This representation uses positive and negative frequencies, i.e. a two-sided frequency spectrum.
If j(t) is real, F (-0)) = F *(c0) (1) and
f(t)=Re i [ F ( to )e"°‘ dco (1-8)
Complex representation of the variable f(t), as a superposition of phasors with positive frequencies (one-sided Bequency spectrum).
Laplace transforms
Convergence condition: With to real, in (1.7) y‘(t)| must decrease fast enough when z -> :i:
To get around this problem, we use unilateral Fourier transforms:
F+(m) = jf(t) e`J°°'dz F_(co) = _[f(t) e`}°xdt (1-9)
With p = c +j0),
F+#=p °° [)J lf(*) 0 ==""d* F-rN0)p ° K} J -... ¤""d* (*-10) OCR Output
F .,. may be continued analytically for c 2 a where a S 0 F - may be continued analytically for c S b where 0 S b
Let H(t) be Heaviside's unit step ftmction:
H(t)-·=O I/2for t
F +(t0) is the Fourier transform of the causal function j(t) H(z); F +(p/ J) is its Laplace transform.
(1) This relation holds for 0) real; for to complex we would have F (-0)) = F *(0J*). Inversion formula for Laplace tran.sfonn.· f(t)H(t)=-jF+;ep'dp1 c+j¤¤ 5;} t (£j) C··_]°° as c f(¢)H(—r)1 c+j¤¤= ?]+ 2mI C- ,.1.F.<="‘dp 1 C S b f(t)=—jc+j¤=· L 21¤1 C- ,.. 1eP‘dp ascsb (1.11)
Hilbert trartsfomts
The Hilbert transform of a complex f1mctionf(x) defmed on the real axis is another complex function g(y) also defmed on the real axis by
%lf(x)]= 8(>’)=%lt:%%dx (1.12)
where the integral is taken as a Cauchy principal value.
It can be shown that
l"°° gw — l— H[g
Examples
1) Hilbert transform of elw with respect to t . ,. %t¢’°"= [] $21¤>¢$i% rdt =. c"l1:%dt, 1<¤¢· = .iefli:?@dt , ·°' `w =j Q sgn(m) el‘°‘ (1.14)
2) Hilbert transform of eJ°°·’ with respect to 0J
1¤>¢ H,,,e’°°’=lf;-;d0a=j. []lLTC co - co sgn(t) w. u (1.15)
Causal fimction in time: f(t) = 0 for t < O.
Its Fourier transform then reads
F(w) = [ f (t) e"‘°’dt
With (1.15)
¤t[F<<»>1 = iii: —@Tr<¤> elect = To wilt: &¤—1“ = -11d* rw ¤. ,"°"’ TC (D " (D0 0 K (D — (D 3·[[Re F(m)]=rm Fw) . , H,[F(w)] = —] F(0J ) hence Hum F(w)]=_Rc Fm,) (1.16) OCR Output
Causal function in frequency: F (ca) = 0 for on < 0.
Its Fotuier transform reads °j‘°' f(z)= $f;F(w)E edc0
With (1.14)
1 dz 1 ’ 1 °° 1 ,, dt · j °° ... —· it =—;,,—-F,-i2.,, me =—d F -ig-f°"=—dzi,-,21t) F J Wl J »{(wl °°’ g °’ (WJ ·° g°’ "°’° _ . , at[R¤f(i)]=-1mf(»·) 3·£[f(t)] — j f(t ) hence H[Imf(!)]=Rcf(t,) (1.17)
Application: In (1.8) we write
jF(w) e’°"dw = f(z)+ jg(z) = A(¢) em=') s(t) with f,g and A, tp real (1.18)
The complex function s(t) is of the type f(t) in (1.17); therefore
a£[f(:)] = -g(i’) ic. go') = -%l;;%%¤1i“£ (1.19) rpartTo the isreal obtained functiongt) as [f(z)]. we can associate a complex signal s(z) whose real part is f(t), and whose imaginary The modulus and argument of s(z) allow a natural defmition of the instantaneous amplitude, phase and frequency of the real function j(t) as [1, 2, 3, 4] d r ) Am = Ist:)
Moreover +¤¤2 ju): +¤¤-icl [ locA (z)dz= I dz;fF(m)1 ecc dmtjF * [(co) _y! e dm —X —00 0 2°°°° 2°° 2*+3 2+°°2 R · * ·· -F ‘2()d gdmgdw F(u>)F (to) 5(m—w )=ijdu>|F(m)] -;j c¤| (m)| - { f z z
from Parseval's formula applied to (1.7). Therefore, since A2(t) =j2(t) + g2(t),
"2+°2`°2 Fr<»>¤»=fg<¤>¤¢=§fA<¢>¤¤ wv OCR Output
This relation is similar to (1.4b).
1 .3 Useful vector identities
We will make repeated use of the following vector identities: z1·[Bxa]=5·[a¤a]=s·[ax5] [a><[5xa]]=(a-e)5-(5-5)e [z1><5]·[z><&`]=(a-s)(5-J)-(a·J)(5-e) div(U5)=E1’·gradU+ U div& div(U gradV)=gradU~gradV+U AV C11I’l(U§)=[gI3dUX5]+ U curlii curlgradV =0 d1v[axt`5]=5-¤¤i1a- zi-e¤r115 dnvcuina :0
2 . MAXWELL'S EQUATIONS AND SOME APPLICATIONS
2 . 1 Maxwell's equations
J = free electric current density p = free electric charge density 1,,, = free magnetic current density pm = free magnetic charge density
Although magnetic currents and charges do not exist in nature, they are useful mathematical tools as equivalent sources for electromagnetic fields ([5]. P. 46 and 52).
curlFI=]+°%? divD=p (2.1)
em1E=-i d1vB=p,,, m gf Since div curl = O,
dnvhzo (2.2) g anvi,+%??=o,,
Equations (2.2) express the conservation of electric and magnetic charges.
C onsrizuzive relaiions
These relations link the inductiorns D, E to the fields E, I? ; they describe tlne electromagrnetic properties of a medium. In general
D = e0E+° F B=. ucl? + p.0M (2-3)
where
P = electric polarization = electric dipole moment density u0M = magrnetic polarization = magnetic dipole moment density M = magnetisation (equivalent electric current == curl M) The definition (2.3) introduces an asymmetry between I3 and M. This is an historical definition ([6],_p. 12) and the most common one in the literature; but some authors do not write a factor No in front of M.
At a given time, P. and M depend on the previous history of the sample. For linear media, by superposition P(t) can be written as a convolution OCR Output - f .. +¤· .. P(t) = jp(z — t') E (z')dz'= fp(z — z') E (z')d:’ (2.4) where p(t)=0 for t <0 . which is the definition of a causal fxmction lll time.
In frequency domain with 01 real, (2.4) becomes (here the Fourier transforms are represented by the same symbols as the time variables)
P(<·>> = p(w) E (w) with p(—<¤> = p`<<¤) so that
D(0>) = e(w) E (co) where e(m)= so + p(m)
The relation between D and E is simple only in frequency domain. In general, e(c0) is complex:
e(c¤)= e’( Since p(t) is a causal function, from (1,13) Re p(m) = Re [s(m) - :50)] and lm p(w) = lm [s(c0)] are Hilbert transforms of each other: 4-no n • _ 4-oe n 1 ·c"(w) = i { —-—-ag (°{) 80 ot €·(w) - 80 Z l { --8 ,(°°) dw (2.7) rc ___ to -0) 1t___ w -0J Since e' is an even and e" is an odd function of 0), Eq. (2.7) transform in the Kramers·Kronig relations: g~(...> Z 2 ;.....d£< €"(0J) E 0 would imply e'(co) E eo. For any dielectric medium, there are frequencies where e"(c0) == 0, which corresponds to a time lag between the polarization P(c0) and the applied electric field E (co), i.e. hysteresis. We will show later (Eq. 2.38) that to t»;"(oJ) 2 0 in all cases whereas [z-:'(c0)—£0] may take any sign. The loss tangent of the dielectric is defined as lan 5 = L (2.9) OCR Output A typical behaviour of e(c0) near a resonance is shown in Fig.2. 1. (s' — so ), even E", odd anomalous dispersion Fig.2.l Absorption and dispersion in the neighbourhood of a resonance frequency A simplified theory of dielectrics ([7], p. 32-7; [8],p. 19) yields @4% Wn, ,,,,<<0,,, _ e(to)+2.¤;O ,, mn — cn + jotnw Similarly, we write B(m) = u(m)H(m) (2.10) Wim Mw) = u’(¤>) — 1u"(w> (2-11) The variation of Ll with frequency is more complicated than for zz. Nevertheless, the resonant type behaviour shown in Fig. 2.1 also applies for the effective tt., of a right—hand (+) circularly polarized wave in a ferrite ([8], P- 298). Up to now we have implicitly assumed that the medium is isotropic, so that e and tt are scalar quantities. For anisotropic materials, e and tt are tensors of second rank, i.e. matrices (of order 3 in 3 dimensions). If e, it are symmetrical tensors, the material is reciprocal; if ta, tt are asymmetrical tensors, the material is non—reciprocal (like ferrites) ([9], p. 409). Loosely speaking, reciprocity means the possibility of interchanging source and detector (or generator and receiver) without affecting the results of measurements. In particular, in a reciprocal medium the propagation constant of a wave is the same for the two opposite directions of propagation. The condition for reciprocity thus reads e = e it = u (2.12) where a tilde represents the transposed matrix. It can be shown ([9], P. 53) that the condition for a medium to be lossless reads €=€=8],L=]:l=1L* + * + (2.13) where the + superscript represents the hermitian conjugate matrix. In nonlinear materials (like ferromagnetic materials), e and it depend on the field strength. For chiral media, the constitutive relations take the general form D = EE + QI-? (2.14) E = ng + [LI] where Q and T] have the same dimension as ,/eo ug . This is the only form of constitutive relations which is Lorentz covariant ([9], p. 7). In the most general case s, tt, Q, 11 are second-rank tensors and the medium is called bianisotropic (bi because the constitutive relations involve both E and Irl ). It can be shown ([9], p.409) that the conditions (2.12) for reciprocity are supplemented by g = -1] (2.15) whereas the conditions (2.13) for a medium to be lossless must be supplemented by ([9], p. 53) €:‘]:l* : 1]+ These relations show that Q or 11 cannot vanish without the other one also vanishing. Example: Media which rotate the polarization plane of linearly polarized light. This is called natural optical activity; it does not require an extemal magnetic field as the Faraday rotation (which is nonreciprocal) ([8], P. 297). OCR Output In isotropic media with natural optical activity ([10], p. 19) é--f~/SM-E-r ¤=N¤¤ -ii-r 1- k a 1- k a where k2 = 0J2z»:|.t , and a is a length smaller than atomic dimensions; for quartz, a == 0.01 A as deduced from ([13]. P. 6-248). Research is now going on to develop chiral materials for microwave frequencies. Conduction current Ohm's law states that j = GE (2.17) where G is the electric conductivity of the medium. Strictly speaking, this relation is again only valid in frequency domain: Nez 00 2 2 G=—————— with G=s<»t , 0>= (2.18) -00pp——- 1+ 10J*C OSm e where N is the number of free electrons per unit volume, cop is their plasma frequency, and 1: is the mean free time between collisions of the free electrons with the ion lattice ([7], p. 32-11; [11], p. 121; [12], p. 287). For copper at room temperature, 1: = 2.4 x 10-14 s; therefore, the approximation G = Gg is valid up to frequencies such that cm: : 1, which corresponds to f = 6600 GHz or 7t = 45 um. In the microwave range, we will thus consider tl1at G is independent of frequency. Remark Due to the Hall effect ([13], p. 9-27), in the presence of a magnetic field the current density is such that I = e(E + R[f X El) where R is called the Hall coefficient. From this relation it is easy to derive {1-+-(GRB·)2}j = e{E + ¤1z[E' >< B`] + (¤12)2(E - 1§)§} (2.19) For copper at room temperature, G = 5.8 x 107 Q-1 m-1 and R = -5.5 x 10-11 m3 C-1 ([13], p. 9-39) hence GR = -3.2 x 10-3 (Tesla)-1. The relative correction introduced by (2.19) with respect to Ohm's law (2.17) is of the order GRB, that is -0.032 for B = 10 Tesla = 100 kG; this is negligible for all practical magnetic fields. Total current Conduction current + displacement current = GE + jmli = (G + j<0s)E When 0) at 0, the effect of a conduction current can be represented by the complex permittivity e+=e-€+=Ei·j(··9)·j@ ju) to co (2.20) At a given frequency, the dielectric losses (s") cannot be distinguished from electric conductivity; in fact (2.20) shows that the electric conductivity may be considered as part of the complex permtttivity. OCR Output OCR OutputWe see that sw=E-a6+F1,sE (2.24) which is the correct expression for the work done in all cases, in particular when there is hysteresis, and part or all of the work is transformed into heat. For linear, nondispersive and lossless media, where 2 and p are independent of field strength, of frequency, and satisfy (2.13), we have when Q = n = 0: 1 — -· 1 — -· W=-E-D+—H—B (2.25) 2 2 For isotropic media (€.|»l Siialar and real) this reduces to w=lu?2 +lu1?2 (2.26) 2 2 Complex Poynting vector and stored energy for time-harmonic electromagnetic fields In the following, we consider a complex frequency 01 = ml + jul jo) = —ul + jml with ul << ml (2.27) Then, to first order in ul we have ds . e(o>)= c(0>l)+ ;ul—(u)l) +. (2.28) dw and similar expressions for u, §, 1] (which are supposed to be second rank tensors). The relation div [E•> To first order in ul we have dm dco dm dm = uloJc+§6(*) jwle +. (2.30) OCR Output 1 1 (_¤1+j(D1)u(( + jcq = —¢11u + (01 + jmlu — jot? gg- = —oq £(mu)+ jmlu +... (2.31) Therefore div[E ><=H]** E ·f— [Hg(;()]E[E%0()1]E-0t1me+jw1s—*** E-¤1—m§+*** jw§H + IT]idw ·-¤i w) + Mm+]? I·—¤i T ]idco lwu) + iw IF! d- g :5.;-pr [ ] .,;...** {5;g g..Iiv>< ggi ff* 9dw()cr] mt·; *dm() + m§d(D **+ (0m) + ° * %R(E7)e~ .is the power delivered to the current j; R 1 . . . . e[-0t1{ 1= —2otlZRe{ BW }= -5; where W = 7Re{ } is the electromagnetic energy stored in a unit volume; 1 ·· -·=•= -•>•· 5 . -· 1 -·=•= -· -·* -· . . . Re)w1]-—E·D(0]5 (@1)+ H[ ·B(d>1)= -Im —E ·D(o)1)- H ·B(to1)] is the power dissipated as heat by hysteresis. For the density of stored electromagnetic energy we thus obtain ‘* d + ‘“*` ‘* -..+1 WL - 4ReE {dw ws E()dm + H (mu)H + Hi[ co§ ' dm+ r]E()]} (2.33) and for the power lost as heat by hysteresis: Physmsis%{E?) = 1m—+* gz- 11q+5(i* = 1m-Eui $§{°* - if iu? + if ·(>;* - n‘} )E(2.34) Let us remember that in (2.33) 0) must be taken as real. Also, by taking the hermitian conjugate of the first term in (2.33) we see that Re-—-o)c={E* q(+)E} {E* Re-——u>e (l()E} dw dm With Q = 1] = 0, the expression (2.33) has been given by Landau and Lifshitz ([14], p. 255) and by Collin ([8], p. 28; [15], P. 16). When 2, tt are scalar quantities it becomes W:W W =--- ’ E l——— d ' H—- e+ m2 <<¤€>|I+(¤>¤)I l d ··12 4dw 4dm where We, Wm are the electric and magnetic energy densities. This expression loses its validity in regions of anomalous dispersion ([14], p. 256) where it would yield negative values for W. At frequencies where dispersion is negligible, it reduces to the well known formula 2 '2 w = we + wm = §€·|E|+ §tt·|H (2.36) OCR Output which is the time average of (2.26). 12 When e, tt, @,1] are scalar quantities the power loss by hysteresis (2.34) becomes - 2 _ 2 a .. " Ph=ysteresis + f‘L|E| + Im _ This expression must be positive for any field E , Irl; this requires that + - g U we"2 0 mu"2 0 e"u"2 (2.38) In the following we will restrict ourselves to isotropic and homogeneous media where Q = n = O. 2 . 2 Boundary conditions at an interface between two different media Normal components ¤§ = Rus gun E2 **2 °2 E1 **1 °1 Fig. 2.2 Closed surface for div equations. E is a unit vector normal to the interface Electric surface charge div D = p (free electric charge) I52·d.§—D1·dS=psdS_• H (—D2 11: 5)ps(free electric charge surface density) (239) Magnetic surface charge div B = p m ri —— = pm (free magnetic charge surface density) :> O (240) OCR Output Tangential components ds dh ds Fig. 2.3 Closed contour for curl equations. is is the surface density of electric current. 13 8D curl H = f + (2.41) 8: With ls representing a unit vector along s, this yields H2-as-F11as:Fsds·nXs=g-[aXds·]=dr-X[i] [La] (H)!2 - H2= xs(] X ,7or[a X H2()] - H1= is (222) BB mi E = -f (2.43) ,,. Gt (EE)t[Z]2—1=—msXaor ((E%)]aX2-1:-i,,.s :0 (224) Remark. The condition (H2 -H2)' = [Q Xa] entails E-(curlI?2 ——cur1 1:11): E·curl xii] From the definition of curl, this is >< H]-413 line integral of[E] XEalong a closed loop of areaS which is also {xs {rt X dr] = -d1v is the surface divergence of is (by dcfmition of div). Combining with (2.41) we have div? — z-( (f -i )+(D -51)2at S — 2 1 2 With (2.39) this reads div§+E·(j2—]·1)+g%¥=O (245) OCR Output Conservation of iree electric charge at the interface From (2.43) there is a similar equation for the conservation of free magnetic charge at the interface. In particular, if there is no free surface electric current along the interface, (H2 · H1)! = O entails 14 *-.-·-1-=07j i5‘)8t( ¤(2 1+n 2 D1) from Maxwell's equation (2.41). This relation means that the total current (conduction + displacement) is continuous when flowing through the interface. When rewritten as (02 +jO.)€2) E·E2 —(01 +jwe1) a-171:0 (2.46) it also shoyvs that even if there is no surface current, there is a free surface electric charge p S = ( $271 · E2 - elif - E1) at the interface between two media with non zero conductivity, unless °2 = 2 G1 $1 Similarly, since there is no free magnetic charge nor current along the interface, (E2 - E1)! = 0 always entails —O -•`°~ -• Bat — ( B2 1)= O ” from Maxwell's equation (2.43). _ Therefore, when co at 0 it is sufficient to impose the continuity of the tangential components of E and H ; eventually the discontinuity of the normal component of D will yield the free surface electric charge. Case when the medium (1) is infinitely conducting (0 = <>¤) From j = 0E we have E = O inside the conductor. It is just sufficient to impose E2, = O at the outer surface of the conductor. From (2.43) % = O inside the conductor. When to ¢ O this requires E1 = till?] = 0 ; by (2.42) Hzt then determines the electric current is at the surface of the conductor. 2.3 Plane waves in isotropic, homogeneous and linear media In such media 8, tt are scalar quantities independent of the position in space and of the field intensity. .7 = GE fm = O From (2.2) curl H = (0 + jcoe)E curl E = ·j0JuH The other equations div E = O div H = O are automatically satisfied. The Laplacian vector operator ([16], P. 23) A E grad div — curl curl is such that (AE)& = A(E§) in cartesian coordinates, where Q = x, y, z. 15 OCR Output Maxwell's equations are equivalent to AE = j0au(o + jc0e)E with div E = 0 with the same equation for H . Let ke = _jwu(G + jms) (2.47) with Re(k) _ A <·> The previous equation reads 2 AE+kE=0 with rhvrr:0 (2.48) The simplest solution to this equation is a plane wave E : Er, e·f’°·’ (2.49) where E0 is a complex vector independent of the coordinates k is a complex vector with components kx, ky, kl k —F=k,x+k,,y+kZz In cartesian coordinates, (2.48) reads A(E&)+k”E,; = O 2 2 2 2 hence (-jkx)+ (-jk,)+ (—jkz)+ k= 0 Of ki + k§ + ze} = k2 (250) The condition div E = 0 reads —yjkxEx — jkEy — jkZEz = O or { . E = O (2.51) OCR Output With (2.49), the magnetic field H can be obtained as <*> The genera] expression for k2 is k2 = —jw(u'·ju")(o + me"+jc0e') = [c02e'u'—wu"(o + 0>e")l — jw[u’(o + c0e")+ e'(wu")] In the transparency ranges where s' > 0 and u‘> O, from (2.38) we have @ _ , wk) 1r¤(k) S 0 Therefore me condition Re(k)/on > 0 as equivalent to Irhtk) s 0. 16 OCR OutputOCR OutputI ·• 2 I 1• I2 P = -—2¤2¤ J d = —-——— 1 dz|| Z |1‘ For P1 remaining iinitc when dz —> O, we must have ig = 0: from Eq. (2.42) H , is then continuous at the interface. The only possible case for a non—zero surface current is when 0 = ·><> 90 L92 E2 **2 °2 e, u1 o1 0 I\/-T Ov Fig. 2.4 Plane interface between two different media Incident plane wave Referring to Fig. 2.4, following (2.49), (2.51) and (2.63) the incident wave in medium 2 reads /<»7·F - 1 .·· " gl = gc 6-1z<> . —.=(), H==—·"0XE1no] Czi E] I (2.66) Reflected plane wave The reflected wave ir1 medium 2 reads __ .. l ·· ER = E2 c—jk2i2-? .. a2.ER=0, HR=—”2"ER[] C2 (2.67) Transmitted plane wave The transmitted wave in medium 1 reads E F'- - iki it -7 = 3 r_1 ¤ A _ .ETZLIC-/ii ’ ;l•1~%.:0’ (2.68) Boundary conditions: Continuity of tangential components of E and H at z = O EI: + ER: = ET: HI: ‘*' HR: = HT: (2-69) 19 OCR Output Determination of Fi land Fig In order to satisfy the boundary conditions at every point of the interface, the arguments of the exponentials must be identical at z = O: k2(nOxx + nOyy) : k2(n2xx + n2yy) Z k1(n1xx + nlyy) hence nOx Z ”2x I k2"Ox : klnlx nOy I n2y Ik2nOy : klnly (2.70) where, by (2.57): ”gx+”3v+"gz=”2x+”2v+n2z :”fx+n?v+”fz :1 Therefore 2 2 _ "OZ ‘ "ZZ md I l In order to distinguish the reflected wave from the incident wave, we must take ngz = -—nOZ. (2.72) However, the branch of nlz cannot be determined from the boundary conditions, because both determinations of nlz satisfy these conditions. In order to specify that the transmitted wave is receding from the interface, we must take Im(k1n1z) < 0 or Re(k1n1Z)> 0 when 1m(1qn1Z) = 0 (2.73) On physical grounds, such conditions are also satisfied by the incident and reflected waves: I1n(/c2n0Z)< O lm(k2n2z) > O (2.74) or Re(k2nOZ) > O when Irn(k2n0z ) = O ; Re(k2n2Z) < 0 when Im(k2n2Z ) = O The above relations (2.70 to 2.73) giving 711, E2, contain all the laws of reflection and refraction. Determination of the tangential components ofthe reflected and transmitted waves at the interface of two media. Relation between the tangential components of E and H Using (2.58) and (2.63) it is possible to write for any plane wave Ex Hx =EYZ HYor _ at z=O E,=Z H, (2.75) OCR Output where Z is a 2 x 2 matnx which we now determine. From (2.63) and (2.58): - C 2 _ Q 2 2 Ex —§[Hyn, —— Hzny] = T[Hynz + (Hxnx + Hyny)ny] - -I;-[Hxnxny + H),(ny + nz Ey = §[HZnx — Hxnz]1{[ = ;3——(Hxnx + Hyny)nx -] Hxng= §[(; —Hxn+ n——g) Hynxny ] 20 Remembering (2.57) we thus have Z _ ` 2 2 2 A nzny (ny + nz); _nxny (1- nz) (2 76) ‘ 2 2 ` 2 Z —(nz + nz ) —nzny nz ·(l — ny) —nxny It is easy to verify that 1 0 2 — 2 Z -——§{A_ —_ I O hence Z I - —?5Z Reflection and transmission matrices For the tangential electric tield, these 2 x 2 matrices R, T are defined by the relations E2, = R E0, E1, = T E0, (2.77) The boundary conditions (2.69) entail that T = 1 + R (2.78) and that Z0H0r + Z2H2z = Z1H1r with H0: + H2: = H1: Using (2.70) and (2.72) in (2.76) we notice that Z2 = —Zg. Therefore 2Z0H0z =(Z1+ Z0)H1r· For the magnetic field: H1, =(z1+ Z0)_12Z0HO, (2.79) For the electric field: E1: = ZlHlt = 2Zl(Zl + Z0)¢1EOz (2-80) With (2.77) and (2.37) this yields T=2Z1(Z1+Z0)`* R=(Z1—Z0)(Z1+Z0) (281) TA = (Z1 + Z0)(2Z1)_1 RA I (Z1 + Z0 )(Z1 ‘ Z0) Eigenvectors of R and T Let us write (2.76) as §LZ=liO 1}+ nz? —n£ {Q -1 0 ny —nxny Let I hO lUL Ll 0]: - U ence1 O {O A 2.82 ( ) OCR Output 21 Then nz ZU _ {1 0]+ ng nxny C 0 l nxny ng It is straightforward to see that the eigenvectors of the last matrix, which are also eigenvectors of ZU, are ny and nx (2.83) Remembering (2.57) we have aw»»&zu‘) ""] {"C -*1; -*1; =C **y(-t. nifl 8)t **y ”y i.e. zunyi “”x = -inz "’xny zu n" =nx -; ny nz ”y (2.84) The expression (2.81) for R can be rewritten as R = (21 — z0)UU‘* (21 + 29)-.1 = (zla - z0U)(z1U + ZOU) (2.85) From (2.70) it is obvious that both eigenvectors are the same for the three waves O, 1, 2. This means that the eigenvectors for the incident wave are simultaneously eigenvectors of Z()U, ZIU and therefore they are also eigenvectors of (Z1U—ZgU), (Z] U +Z0U), (Z JU +Z0U)-1 and iinally of R, T. From (2.84) and (2.85) we obtain the eigenvalues pl, pg of the reflection matrix R as: 1) for the eigenvector {Emil { nay ] Eoy `nox i.e. when n0xEm»+ n0yE0y = 0 (with similar relations for waves 1, 2): pl = {i - + 2.] nlz ”0z nlz ”0z (2. 86) 2) for the eigenvector Eox not Evy ”¤y i.e. when n,,,E,,y — n,,yE,,x = 0 (with similar relations for waves 1, 2): 02 = (Crm; — C2*¤()z)(Ct*¤iz +C2*1()z) @87) OCR Output Physical interpretation ofthe eigenvectors of R, T They correspond to particular polarizations of the incident plane wave. 22 Case 1 For the three waves O, 1, 2 we have nxEx + nyEy = O (2.88) With (2.58) this entails E Z = 0 for all waves. If the ratio nay/nm, is real, the relation (2.88) means that the electric field is normal to the plane of incidence. Case 2 For the three waves 0, 1, 2 we have nxEy — nyEx = O (2.89) With (2.63) this entails HZ : 0 for all waves. If the ratio nay/nw, is real, the relation (2.89) means that the electric field lies in the plane of incidence. The eigenvalues of the reflection matrix R, i.e. the reflection coefficients p are different for the two cases. They are given by Fresnel formulae (2.86) and (2.87), which Fresnel derived for real Q and ii. Remarks l) Considering the propagation along z, the two polarizations correspond respectively to TE (EZ = O) and TM (H, = 0) waves. 2) For the magnetic field, from (2.79) and (2.78) we obtain the transmission and reflection matrices as TH =(21 +20)-1220 RH = -·(Z1+ZO)—l(Z1—Z0) (2.90) Therefore U"RHU = —U‘*(21 + Z0)_1(Z1 - 20 )u = -(2,u + 20U)`*(21U - 20U) (2.91) The eigenvectors (2.83) of ZU are thus also eigenvectors of U·1RHU, which means that n n ll H Il M Ul Wl Tnx ny ny “"x are eigenvectors of RH , with the same eigenvalues as (2.91). Comparing with (2.85) we see that the eigenvalues of RH are the same as the eigenvalues of R, but opposite in sign: the reflection coefficients for the tangential magnetic field are opposite in sign to the reflection coefficients for the tangential electric field. Surface impedance At the interface z = O the relation between the tangential components of the fields is given by the matrix equation (2.75) for each one of the incident, reflected and transmitted wave. l) TE wave When H, is the first eigenvector (2.92), by (2.84) this relation simplifies to ”y C ”y - ”y E,=ZU =—— with H,=U —nx nz —n,, —nx OCR Output 23 Using (2.82) this can be rewritten as E, {=£U2”’ ]nz —nx =£onz H or in vector form: — C E, = z,1,[Fi] >< zwhere 2, (2.93) Considering the propagation along z, this relation is analogous to (2.63); ZS is called the surface impedance for either the incident, reflected or transmitted wave. For the incident wave, ZOS = 3 ,102 g For the reflected wave Z2; = = —ZO n2z For the transmitted wave, Z1 = Sl S n The reflection coefficient (2.86) then reads Q = (Zn · Zos)(Zis + Zos) (2.94) which is identical to the reflection coefficient in a waveguide, as given later in (4.21). 2) TM wave When H, is the second eigenvector (2.92), all the above still applies with Q/nz simply replaced by Qnz; in particular we have Zs = CM Laws of reflection and refraction When the ratio nay/nm is real, it is possible by rotating the coordinate system about the z——axis, to choose it so that nay = 0; the plane y = O is then the plane of incidence. When the media are lossless and EO is real, from Fig. 2.4 we have n0x=sin 90 n»2x=sin 92 n1z=sin 91 Hoy =O ngy =O nl), =O (2.96) n0z = cos 90 n2z = —cos 92 nlz = cos G1 Equations (2.70) and (2.72) then yield 92 = 90 ·(}€2}.l2SlIl B0 = yltlllllsin 91 (2.97) OCR Output which are the laws of reflection and refraction. Total reflection at a dielectric inzerface We assume that 24 There is total reflection if the transmitted wave in medium l decays exponentially from the interface; that is if nf, < 0 . From (2.70) we have ki = kinéx = kin?. or with (2.57): kg = k§1(3,) - H= kj1—2 (f) n, where n6‘Z > 0 , nfz < 0 Therefore kg < kg < kg (2.98) L €2& With (2.96) this is only possible if gi = —> %—>1 (2.99) This property is used in dielectric waveguides. Light seems to penetrate at some distance from the interface in medium 1: it is the Goos-Hanchen shift ([17]. P. 25). From (2.71) and (2.96) the penetration depth (analogous to the skin depth in good conductors) is the inverse of Re(jk1n1Z) = Rej\//cg -k§(1~ n,= Re t/k§ sr¤002 -1cf but the ejfective penetration depth is the inverse of q·tjk§ San00z -k$ where q is a correction factor ([17], p. 27). For TE waves, q = 1; for TM waves, assuming tt] = ug, E · 2 2 - - () q- 0 srn 90 cos 90 > When does the incident or the reflected wave vanish? The formulae (2.80) presuppose that det (Z; + Z0) if O. If det (Z1 + Z0) = 0, det Trl ; det R-} = 0. From (2.76) it is then possible to have E2, ¢ 0 and E1, ¢ O whilst E0, = O; in this case the incident wave vanishes. If det (Zi — Z0) = 0, det R = 0. It is then possible to have Eg, = O whilst Eg, ¢ 0: in this case the reflected wave vanishes. The equivalent conditions det (ZIU i ZOU) = 0 amount to saying that one of the eigenvalues of (ZIU i ZOU) is zero; from (2.83) this reads Cl g ;l: = 0 for TE waves , Qlnlz :t Qznoz = 0 for TM waves ”1z ”0z With (2.71) this can be rewritten as &— S; = 0 for TE waves , Qlnlz —§2nz = O for TM waves (2.100) nlz nz where nz represents either ng; or ng;. according to which one of the reflected or incident waves exists. Case 1: TE waves With (2.71) and (2.72) the first condition (2.100) transforms into 25 OCR Output 13:% 1%:% 1%% lkinizlki: — 1<%(1—: 13)i <1<2».> ’ where by (2.64) kl; = cou. Therefore 2 2 2 2 2 2 2 111 ki · k2 + (k2”z) ki ‘ k2 M1 *7:*;;:1* OI`72 •·;···=·—·-1 1*2 (kznz) (km) U2 Provided ut ¢ ug, the condition (2.101) determines (k2nz)2. The branch of kgnz is then chosen such as to satisfy the first equation (2.100); from (2.74) Hz is then interpreted as being ngz if Im (kznz) < 0 or Re (kznz) > 0 when Im(k2nz) = 0, and being ngz if Im (kgnz) > 0 or Re (kgnz) < 0 when Im (kznz) = O. Case 2 : TM waves With (2.71) and (2.72) the second condition (2.100) transforms into L <1 Therefore _ ,. (Gl + (kznz) (G2 + Jm€2) Provided (G1 + jms;) ¢ (G2 + jcoeg), the condition (2.102) determines (k2nz)2. The branch of kgnz is then chosen such as to satisfy the second equation (2.100); following the same procedure as above, nz is then interpreted as being either ngz or ngz. When ul = ug, the wave is a TM wave; Eq. (2.47) then converts (2.102) into gGi +1¤>¤i i Z +1 H2 G2 + ](1)€2 If nz is nnz, the wave is incident from z = -¤¤; the particular value of ngz determines an incidence angle (Brewster angle, real or complex) for which the reflected wave vanishes. When both media are lossless and ui = ug, using (2.96) it is easy to see that (2.102) yields tan2Bn = 3 (2.103) It should be noticed that when both media are lossless, by (2.64) Q1, Q2 are real and positive; from (2.100) and (2.73), (2.74) it appears that Hz must be ngz. If nz is nz; , by the previous remark one at least of the two media is lossy; from (2.101) or (2.102) kgngz is complex, as is also Iqniz by (2.71). There is no incident wave; both reflected and transmitted waves propagate along the interface and decay exponentially on both sides along the normal to the interface: they form what is called a surface wave ([6], p. 516; [15], p. 697). Transmitted wave inside a good conductor In Fig. 2.4, we suppose that G2 = 0; 61 is considered to be given by (2.18) in order to include the case of very high frequencies. From (2.47) we then have 26 OCR Output k2 = wl/s2y.t2 kl = l[—jcottlSL1+ ]0)Tl —-- + jtnsl with Im(kl) S 0 (2.104) In (2.104) and in the following, or represents the D.C. conductivity. Orders of magnitude. For copper at room temperature, Gl = 5.8 x 107 Q4 m‘1, *:1 = 2.4 x 10*14 s; we take el = eg because we are neglecting the effects of the bound electrons in the metal ([7], p. 32-10). Using (2.18) we can write + j0J£l = j0J£0n" hence kl = ([:0:-zoltln2z (2.105) 1+jml where (2.106) 2 :4-$@4-@-+11wn(+ iam) and -‘K=(wpr)2 l"· Therefore (mp·;)‘4=]_6.]()5 mp·;=4()() fp=;€’;=2_6·1015Hz (}:p=0.115 lim) 1 We define a critical frequency fc such that (DC'C=l fC=££=—L=6.6·1O120 Hz (}»C=4SlL1'¤) 21: 21:*: We have to consider three frequency ranges for which (2.106) and (2.105) take the following approximate forms: 1) when f < fc , ctr: 2) when fc n2 =1—- tug- , kl = \/(mz —- (,0%)£0l,L0 (2.108) The fields are attenuated exponentially in the metal, with a decay length of the order of P-:0110 c/to = 18 nm. ln copper at room temperature, the mean free path of conduction$2 electron[18] s is 2 nm l, p. 288) which is larger than the skin depth; in that case the local form (2.17) of Ohm's law is no longer valid, and the skin effect becomes anomalous ([12], p. 308; [18], p. 308). This frequency range is also the range where resonances of the bound electrons show up in ci. 3) when f >fp (i.e. 7: < 115 nm), Eq. (2.108) still applies but now to > top and ki is real: the metal becomes transparent. When the wavelength ultimately becomes comparable with the mesh size of the crystal lattice, the metal can no longer be considered as a continuous medium; this is the frequency range of X—rays, which is dominated by diffraction phenomena. 27 OCR Output Coming back to the microwave range, we rewrite (2.107) as ki = J_‘—j<»»1¤`1 = 1sg¤<<»>— 11 ./%-9 or jk. = [1+ jsg¤<¤»>1 I/@@2 <2.w9> From (2.47) we have z kgkl wezu’](DI··L1G1 Jwe E u k91 “ €0H1Iwle is .2.Z.Z;Z.Za;J.k1 91 and I2I Z.AZfHZ.1O- K1, (2110) the last figure being for copper at 20° C. Along z, the propagation constant of the transmitted wave is, with (2.71) and (2.73): nin. =r.1k5—k%1—~.an(5) <2-Mw Whatever iS Moz, the transmitted wave in the conductor varies as -jk1Z ..II+j sgn(w)I Z=€_I;1+j $gI\((D)Ig C = C where 5 Z /.2.. |<¤Iu1¤1 is me skin depth. (2.112) For copper at 20 °C, 5 = 66.09 ttm [fMHZ} ·1/2 50 Hz l kHz l MI-Iz 1 GHz I 100 GHz 9.35 mm I 2.09 mm I 66.1 ttm I 2.09 ttm I 0.21 ttm 21 Z kl O` 6.92-10 ' I 3.09-10 V I 0.979-10 ° I 3.09-10 " I 3.09·10 From (2.58) ;1·1·E1 :0 0I' n1xE1x +I1lyE1y+7l1ZE]z :0 where by (2.70) and (2.73) nl, =%n0x , nl), =%n0y , nu =1, i.e. E1=1z (2.113) Therefore IE 1,1 << IEIXI or lE1yl; the same applies for H1. This means that inside a good conductor, the fields are almost parallel to the conductor surface. Surface impedance of a good conductor Because ni, = 1, the surface impedances (2.93) and (2.95) are practically the same for both TE and TM transmitted waves. Therefore, at the surface of the metal, the boundary condition (2.93): 28 OCR Output E, = zs1,[Fi][F] X z: zs1, X a (2.114) applies with the same value Z, = Q1 for any wave polarization; E = lz is a unit vector on the inward normal to the metal surface. From (2.64) and (2.110), the surface impedance of the metal is thus wu . Zs :;1=y_'J`jlg1 = [1+ 1 S2¤<<¤>lRs (1) where Rs = Mh( 201 (2.115) Since Z=S l1E-, 1[Z]S\EGE << /E Using (2.63), (2.113) and (2.115) we see that inside the metal 1;e=;52.(;1\...<<1 WElliE2 K‘?B (2.116) Wm (i1|]—11| H1 61 Joule power per unit area of the metallic surface It is given by the inward flux of the Poynting vector into the metal surface: ` 15 Zglé, ReX 9;]12} . Using (2.114) this becomes 1 ~ — ~ — l — 2 15 = {5[()RezSH, X 12X:]}{E|[ H·1Z= RezSH, } hence H =:RS|Ht1 - 2 ¤ (2.117) OCR Output where Rs [in Q] is the surface resistance of the metal. From (2.115) and (2.112) it reads R : S aww? 2 _; 26] ($15 The last expression is the resistance of a metal sheet with thickness 5. For copper at 20 °C, Rs = 0.2609 mQ[fMHZ]=1/2 3.690 mQ at 200 MHz 8.250 mQ at 1 GHz 29 2 . S Applications of the complex Poynting vector In isotropic media where Q = n = O and at frequencies where dispersion is negligible, (2.32) reduces to —div{E?] > Remembering (2.6) and (2.11): e’°‘ = e' + je", ti = tt' - ju", this relation can be rearranged as —div;[] -E ><1?1*= g i-f°22§[||2 * + $e··E+ u||] "1?-%[e||u|?|] ’E+2 'I . ..101 ,@1 ,. ·2 + 011— +————*2 wil1).*}** e +— E lK 1..1*11**8 (2.118) When dispersion is negligible, the frequency is far from regions of anomalous dispersion; therefore |u"| << u' |e"| << e' Moreover, we shall always assume that ot; << Icoll, so that the last term in (2.118) is well approximated by a_m , -· an2 , ·-· E 2 |rn · Using (2.36) and (2.37), the integration of (2.118) over a volume V bounded by a closed surface S yields §%[E> In the presence of a current generator, in (2.119) f = ei + fe (2.120) where o‘E is a conduction current and J 8 is an impressed current which, when oppositely directed to E , represents a power generator. Then 1** 1 -*2 1*** ' 1* ** 6ej`:E-J*dV=j;_-GIEI dV+j:E·J dV=P+j:E-J dV (2.121) P is the Joule power dissipated in V; the opposite of the second term is the complex power produced in V by impressed currents. Example 1: Cavity excited by a transmission line, with no impressed currents in the cavity. lf the surface S contains a cross-section of the transmission line and surrounds the cavity walls ata distance of several skirt depths, the inward flux of the complex Poynting vector differs from zero only m the cross—section of the line, and (2.119) becomes VI* = P + 2 j(n(w,,, - we) (2.122) where P includes the power dissipated inside the cavity (by conduction and by hysteresis) and in the cavity walls; we have taken ct] = 0 because the RF generator maintains the field level constant. 30 OCR Output ,?§`° v,1 4 RF generator Fig. 2.5 A transmission iinc coupled to a cavity Thc input impedance seen at S is R + jX = { = t 4fw··WW‘2·7J M whereas the input admittance is 2P _ W —W pe G+B=§= _ 41m in (2.124) Equations (2.123) and (2.124) give an interpretation of the reactance X or the susceptance B in terms of (Wm — We) Example 2: Closed metallic cavity. Imaginary frequency shzft due to power losses If we now consider the cavity to be completely closed, the inward flux of the complex Poynting vector is zero in the metal, at some distance from the cavity walls. Fields can be maintained inside the cavity by impressed volume currents J °; this is the basis for excitation of cavities by electric currents (see Section 6). Without impressed volume currents in the cavity the fields are damped, and (2.119) becomes O=P—2ot1(We+W,,,)+2jm1(W,,,—We) (2.125) Equating real and imaginary parts, we obtain 2ot1= land I wm Z we : %(wm +we) (2.126) where P is the total power dissipated inside the cavity and in the cavity walls; W = We + Wm is the total electromagnetic energy stored inside the cavity and in the cavity walls. This situation occurs when the fields oscillate at well defined eigenfrequencies, which are the resonances of the cavity. The ratio P/W is independent of the field level. It defines the quality factor Q of the cavity at a particular resonance through the relation liil - Z (2.127) so that 31 OCR Output 0,= Lg (2.128) 1 2Q The complex frequency (2.27) reads m=u> +j0t =m 1+--————j Sgnwl) or m2=m21+————j Sgn(m1) 1 1 1 1 2Q -Q (2.129) Let us now apply (2.119) by taking the surface S to be the inside surface of the cavity walls: _ I lcavity st¤rmglE1 * ·•* X H—• ids = Pcaviry · 2¤i(% + Wm)c,,v,,y + 2J¤>i(Wm · We)ca,,w (2-130) where Pcavity is the power dissipated inside the cavity. Applying the boundary condition (2.114) at the surface of the cavity we obtain [EHX1,, :,[1;H] X, ·1,, :z,|H, where Zs is the surface impedance of the metal; from (2.115) ZS = [1+j sgn(<¤1)]RS Therefore the flux of the complex Poynting vector going into the metallic walls is _ l cavity SurfaceI -· 5[E -· X-· H] I'; ` dS = lcavity surface5ZSlH!ll -· 2 dS 2; [1+ J sgn(0)1 lcavity surface;R¤'lHtll -· 2 ds = [1 + j sgn(to1 - Pwau (2.131) because the real part of this flux is the power dissipated in the cavity walls. We may now rewrite (2.130) 0 ~ [1+ j sg¤(t¤1)]1>w,_u + 1>,,,,,y - 2¤1(w, +w,,,)wiw + 2jm1(W,,, —W,,)w, Subtracting from (2.125) yields O = —j sgn(c0l)Pw,,11 — 20t1(We +W,,,)waH + 2jc)1(W,,, —- W€)wau (2.132) The imaginary part yields Pwai = 2l¤>iI(Wm · We)w,u (2333) In this relation the term We is not significant because the derivation of (2.132) is based on the approximate surface impedance (2.115), which implies that in a metal, as shown by (2.116), We is neglected with respect to Wm. The real term in (2.132) can thus as well be written as 2ct1(W,,, — Wgwau. Using (2.133) and (2.128) this reads & P = .L P ml! wan 2Q wall lt is thus very small with respect to the first term in (2.132) and should be neglected. 32 OCR Output Real frequency shit due to the penetration of the fields in the cavity walls By analogy with (2.127), we define quality factors QW, Qd related to the losses in the walls and inside the cavity by A = -—PWa“ and w = —-P°°”“’ (2.134) QW Mem Qa Worn so that i = L + i (2.135) Q Qw Qd When the losses inside the cavity are due to the conductivity 0 and the hysteresis of a uniform dielectric with perrnittivity e, using (2.121), (2.37), (2.36) and (2.126) we simply have [= j-o|E|+—-— { 2 e|Ei)dV &"2 ,, 2 2 G + (DIS ; - = --T Q4 Imll { Eels] av |<°1l¤ From (2.126) we have taken Www] = 2WS. In the expression (2.134) of 1/QW, Pwd; is the real pan of the flux (2.131) of the complex Poynting vector going into the metallic walls; it is proportional to the surface resistance RS. On the other hand, the boundary condition (2.114) involves the surface impedance ZS, which entails an imaginary part (i.e. a reactive part) in the flux (2.131). A complete treatment ([12], p. 360) shows that in (2.135), the term 1/QW is proportional to ZS, which means that it should be multiplied by [1 + j sgn ((.01)]. In (2.129), 1/Q should thus be replaced by 1 -1+j Sg“(°°l)¤ 1 (2.137) Qcomplex Qw Qd yielding tl)2 =wlg i + J Sgn(w1)_1 + ——l Sgn(w1 (2.138) t QW )] Q4 Since the reactive pan of ZS produces a real frequency shift, we had to replace ml in (2.129) by some 010. From (2.138) cog is the resonant frequency of the same cavity with perfectly conducting walls and lossless dielectric; as such it is a real quantity. Comparing (2.138) to (2.129) we see that to? z (1),%[Li 1-=[LI og Qw1+ Qw (2.139) The penetration of the fields into the cavity walls due to the skin effect always reduces the resonant frequency computed for the cavity with perfectly conducting walls. As shown by (2.139) the correct frequency shift is given by Q,.) ; it will not be obtained correctly be assuming that the walls of the cavity recede uniformly in all directions by a distance 8. Relation between the actual complex frequency and the frequency of the cavity with perfectly conducting walls and bssless dielectric It is obtained from (2.138), which we write as 33 OCR Output m1g {l—j+ sgn Re(c0) j sgn)Q4 Re(0J)g=m (2.140) With (2.135) this can also be rewritten as w2+.}..-[1)Qw Q=m§ (2_]4]) Let us remember that wg is the real frequency of the cavity with perfectly conducting walls and lossless dielectric. The correction factor in (2.140) will appear again in the theory of lossy waveguides and lossy resonant cavities (see Eq. (6.67) in section 6). 34 OCR Output 3 . WAVEGUIDES 3 . 1 Monochromatic waves along cylindrical conductors with arbitrary cross section The conductors are parallel to Oz. A_ll me_dia are supposed to be reciprocal, isotropic and uniform in space. Helmholtz's equations read (with J = GE, p = O ): AE+kE=02 with div 1;:0 (31) AII+k1?=0?2 with div 1-1:0 where A E grad div ~ curl curl A = A + i- .L 2 3_2 ( ) 82 k‘ = —jc0u(o + jmc) with Im(k) S 0 (3.3) HG=Q (D2 2 2 k= (08].L = j;€,}.1., (3_4) where 8 : -— , M, : -£— r r 80 No Because of the form (3.2) of the Laplacian operator, Helml1oltz's equations admit of solutions ei’YZ in z. By using Eq. (1.11) with z instead of t, the wave dependence on z can be represented as a linear superposition of complex exponential waves. In the following, we therefore consider only waves of the E(x,y)-ewhere]0)t`yz Y = 0t+j[5 , (12 0 and we factorize the complete exponential (in t and z) out of the equations. Remark; When Y = 0, the two independent solutions e"Y’ and eY’ coalesce into a single one (a constant); a second independent solution si then obtained by taking ;()lim - y—>0e`YZ — ey':=2Y z For EZ, HZ Helmholtz's equations reduce to: A LE, + kgzsz = 0 where kg = k2 + y2 (3.5) OCR Output 3 .2 Equations for the transverse components of the fields With lz representing a unit vector along Oz, tirst observe that 35 - GE 2 - QE Cl1I`lxE_L=—··§é·:'YE>, CUYlyEJ_=·§=—'YEx hence curl_[_L=[E) 7L[EiZ[ >< (3-6) whereas cur1E,Z=gmd[i)[ Ez X 1,:][ gmdlaz X 1, or °¤’liEzz=(l) [grad!. ><1z (3.7) Similar relations apply for H . Therefore, Cl11’i_[_E = [(y1EL +gradLEZ> From Maxwell's equations (2.1) and the constitutive relations (2.5), (2.10) we then obtain: (0+ joae)E_L[Hl +y><,= l[ —gradLH, [ ><1Z This system can be solved for the transverse components of the fields, yielding Ei = -%gr¤diEz -to 8f¤d1Hz%[c ><1Z (3.10) HL =%[ gmdLEZ X 12-gmdlyz]é With Maxwell's curl equations, the conditions div E = 0, div H = 0 are automatically satisfied. They read div El=yEZ , div HL=yH2 (3.11) OCR Output This can be verified at once from (3.10), by using (3.5) and the relations div[gradlEZ> which are direct consequences of (3.7). 3 . 3 Transverse deflecting force The transverse deflecting force for particles with unit charge and velocity V = vlzis -• ...... -• -• _. . FL V=EL+v> Using the first equation of (3.9), this becomes 36 F = -———-v ju) — jmlvE - d) E i Yi Emi z Putting h = C0 / V (3.12) where v is considered to be independent of z, this equation may be written as _ . _ . 8. F·¤"”%;f;(i)ziz = E¢’”’- r¤dE·¤"“ (3.13) in which form it no longer contains Y; therefore it is valid for any variation of fields along z. When v depends on z, just replace eihz by eiwldz/V. This phase factor accounts for the fact that the fields oscillate as eJ‘°’, where I = I0 + v is the time when the particle passes at position z. 3.3.1 Panofsky-Wenzel Theorem . ‘ -·· a —·¢‘· ]c0_[Fi_e]hZ { = [E_Le’hz]0 — j'grad_,_EZ ejhzdzl (3-14) Transverse momentum Vamshcs whcn gained by unit charge Ei (O):EJ.(l}=0 This relation shows that when integrated between two positions where E L is negligible, the transverse momentum transferred to the charge only depends on grad L EZ. In particular, there is no transverse kick to the particle when EZ E O. The relation (3.14) applies at a given frequency. It describes a mathematical equivalence between the total transverse momentum gained by a unit charge and an integral over grad L Ez; but the actual deflecting force is given by (3.13). In particular, if (3.14) vanishes, the total transverse kick given by the electromagnetic field to the particle is zero; but at some z the deflecting force may be different from zero and can induce synchrotron radiation. Example: Consider a relativistic particle passing between two parallel plates of length Z along z and infinitely wide along x. _y__f_I _’X _ __ `f__ iEtfH _ _, ., ` ` : Fig. 3.1 A TEM wave betweenwo ppll OCR Z tarallelOutput lates The plates guide a TEM standing wave along z: 37 E,,=Eqc0s/cz h=9=2=k v c HX = j ig-‘-Lsinkz0 QD is the impedance of free space 1) lf we neglect the fringing fields at both ends of the plates, we are left with the first term of (3.14): _ . L _ . _• . [EJ_e’hZ]0 =1y[Ey(£)-ejhf — Ey(0)] = E01y[cosI<.€-ejhl -1] = Eoiyejulcoskl — {jh!=] 0yjM E1e·j sin hl since h = k 2) If we integrate from —·><> to —w><> we are left with the second term of (3.14): [gradLEz ·eJ"Zdz ¢ 0 because of the fringing fields near z = 0 and z = Z. 3 . 4 Classification of waves Once the longitudinal fields EZ, HZ are known, the transverse fields can be obtained from (3.10). We must distinguish two cases. a) case kc! = 0 With (3.5). Y=jk, i.e. B = k when 0 = 0. Such waves are synchronous with particles travelling along z with the velocity of light. For the transverse fields to stay fmite in (3.10), we must have gradiiz = —CIgr¤diHz ><1Z (3.15) whew (316) V o + Jose , wave impedance of the medium From (3.9), ——— 1 Ei = C[Hi ><1z]·;gr¤diEz;] With (3.15) there are two possibilities: 1) grad_LEz ¢0 with grad_LHz ¢ 0 Since both E, and Hz differ from zero, this is a hybrid HE or EH wave. Such waves are usedto _ deflect ultra-relativistic particles in RF separators [19]. They describe the particular space-harmonic which is synchronous with the beam in the deflecting mode of a periodically loaded (e.g. disk-loaded) waveguide. 2) grad LE, = 0 with grad LH, = 0 If k ¢ 0, from (3.17) ‘ 1li=£1i¤[?i.] (3.18) OCR Output This is the case when Ez = C1 with Hz = C2. 38 Waves with E, = C1 ¢ O, Hz = 0 are used to accelerate ultra-relativistic particles. They describe the particular space-harmonic which is synchronous with the beam in the accelerating mode of a periodically loaded waveguide. Fields with E, = O, H Z = C2 ¢ 0 can exist for k = 0 in metallic pipes. They are the solenoidal tmiform magnetic field produced by azimuthal D.C. currents in the pipe walls. Waves with Ez = 0, HZ = O are TEM (transverse electromagnetic) waves, satisfying (3.18). Since EZ = 0, a pure TEM wave does not accelerate nor deflect charged particles. For TEM waves, curl: 131 Z Curlzg = "jwll Hz = O (3.19a) and, from (3.11): div EL=Y EZ=O Cllflz HL = Curlzl? Z (G "l‘ Ez = 0 div HL = 'Y H Z = O b) case kc! ¢ 0 From (3.10) the transverse fields are obtained by superposition of two kinds of waves: 1. When E, = O: TE (transverse electric) orH (because HZ at O) wave. From (3.9), i EL =$[% 1>< i]z ZOH = &(wave impedance) (3,20) For H waves, curlZEL = —j0Jp Hz ¢ 0 3.21 ( a) div EL=‘Y Ez=0 Similarly CUIIZHL =(G+j0J€) EZ = O 3.21b ( ) div HL = Y Hz ¢ O 2. When H, = 0: TM (transverse magnetic) or E (because EZ ¢ 0) wave. From (3.9), E = —····Y .. H.. l 0+jmli if zl = -·———· OE 6+jmE 3.22 ( ) For E waves, C\1\’lZEL = —j0)|,L Hz = O 3.23 ( 8) div EL=Y EZ¢O Similarly curlZHL =(o+jmc) E, #:0 3.23b ( ) OCR Output div HL = Y HZ = 0 39 Remark: The classification of waves according to the conditions (3.19), (3.21) or (3.23) in two dimensions is similar to the classification of modes as solenoidal or irrotational in a resonant cavity in three dimensions (see Section 6.2). Hybrid waves In a smooth waveguide containing a single homogeneous and isotropic dielectric, where both metal and dielectric are lossless, the most general field may be considered as a superposition of E and H waves; moreover these waves can be excited independently (see Section 3.6). If one of the above conditions is not fulfilled, i.e. if 1) the waveguide is not smooth, but it is periodically loaded (see Section 5); 2) the cross-section of the waveguide contains several isotropic dielectrics; 3) the waveguide contains an anisotropic dielectric (as a ferrite); 4) either the metal or the dielectric is lossy; then all six components of the electromagnetic field are present in the waveguide. This means that a wave is no longer E or H, but it is a mixture of E and H called hybrid wave. The real guide can be reduced to the above ideal guide by continuous variation of one or more parameters (for example, the periodic loading can be made vanishingly small, the dielectric constant of a second dielectric present in the guide can be made very close to co, ...). By such a transformation the original hybrid wave will be reduced to one of the E or H wave of the ideal guide. If, by continuous transformation, tl1e hybrid wave is reduced to an H (or E) wave of the ideal guide, it is called an HE (or EH) wave; but this limiting procedure does not always yield consistent results ([19], p. 211). There is one important exception to the above rule for hybrid waves. If the guide has rotational symmetry about its axis, i.e. if it is a circular guide, and if the fields have also rotational symmetry about the axis, then the waves are still pure E or pure H waves. Since in a circular guide, the fields vary as cos mq: or sin mq), this means that waves with m = O are E orH waves; waves with m ¢ O are EH or HE waves in any real guide. The most important modes for beam dynamics are m = 0: longitudinal or accelerating modes; and m = 1: transverse or deflecting modes. 3 . 5 Surface impedance at the boundary of a medium When the boundary conditions at the surface of a medium are expressed by e, = z,1,[Fa] X (3.24) where EHH, are the field components tangent to the surface whilst E is a unit vector on the normal to the surface pointing towards the medium, then Z, is called the surface impedance at the boundary of the medium. Eggmggz If the medium is a metal with conductivity 6, ) Z, =C».a..t = [1+1 sen (w)],—=[1+jlg;i sen (<¤)]R. (3.25) where R, = 1/(05) is the surface resistance of the metal, 5 being the skin depth. lt should be remembered that the boundary condition (3.24) on a metallic surface is based on the approximation 40 OCR Output W —’?— = EE << l in a metal Wm o which entails Zsl << tl? The boundary condition (3.24) also assumes that the skin depth 5 is small with respect to the thickness of the metallic surface; therefore it carmot be applied down to zero frequency. 3 . 6 Waveguides with ZS : 0 (perfect conductors) Boundary conditions From (3.9) we have (see Fig. 3.3) % YES = ·f¤>L1H,. · (3-26) Bs GH' (0 + j0Je)Es = —·yH,_ (3.27) Bn On the waveguide walls, the tangential electric iield must vanish so that E, = O , EZ = 0 on the waveguide walls (3.28) Therefore, (3.26) requires that mu H,. = O, i.e. H,, = O (3.29) as long as to ¢ O. When to = O, the boundary conditions for H no longer take a simple form. In what follows we restrict our study to cases where (3.29) still applies; this is the case, for example, when the waveguide walls are superconductors. From (3.29) and (3.27) we then obtain the boundary conditions for the magnetic field: GH . O on the waveguide walls. (3.30) Z H,, = O , nT = 3.6.1 Guides with external field The cross section of the guide extends to infinity. It can be shown that the only possible waves are TEM waves with 2 I n = O n=l This condition keeps the power flux of the wave finite. In particular, it prevents the propagation of waves along a single, perfectly conducting wire. 41 OCR Output "` ~`_ Fig. 3.2. N (=3) parallel, _ "·—( 0 infinitely long conductors III OPCII Sp8.CC From (3.19a), curlzEJ_ = 0 div El = 0 Therefore E L = —gradJ_V with A LV = O This reduces to an electrostatic problem in the transverse plane; the magnetic field can then be obtained from (3.18). Examples: Lecher line (N = 2); polyphase lines (N > 2). 3.6.2 Guides with internal field S /f1· Fig. 3.3 Cross section of a waveguide with intemal field ds Z The cross section S of the guide is bounded by metallic walls of finite extent. Dgjferential equations, from (3.5): forHwave: A.LHz +k:Hz :0 (3.31) OCR Output for E wave: A.1.Ez + k:Ez = 0 Boundary conditions From (3.10), 42 _ y BE a?jmp BHrrrC E; · itC ? o+jwe8E, y GH, H _n —k3;EBn 2 -- _ C7 C As long as kf ¢ 0, the boundary conditions GH (3.32) , . E, = O ,n T = O on the perimeter s entail the other conditions E S = 0, H,, = 0. When kg = 0, the full set of boundary conditions (3.28), (3.30) must be imposed. Since the conditions (3.32) are different for Ez and Hz, the corresponding eigenvalues ki are different for E and H waves, which makes these waves independent solutions of Maxwell's equations. But it is important to notice that these waves can exist only in smooth, perfectly conducting pipes filled with a homogeneous isotropic dielectric; in all other cases* the waves are hybrid (EZ ¢ 0 and HZ 1: 0). Multiply the f1rst equation (3.31) by H, and integrate over S : BH 2 2 §SH;$£dS·ISlg1’3dLHziL dS+k3IS|HZl dS=0 hence kj is real ?. 0. A similar equation applies for E,. case k? = 0 From (3.33), kf = 0 implies gradLHz = O and grad_L E, = 0. Therefore Hz = C in case of an H wave; because of the boundary conditions (3.32): Ez E O, which means that kg = 0 is not an eigenvalue for E waves. Since EZ = 0, the first equation of (3.2lb) entails curl, Hl = O hence H_L = —gradJ_\p where the scalar potential ur can be multivalued if the cross section of the waveguide is multiply connected; the boundary condition (3.29) reads LL = O on s (3.34) Bn The second equation of (3.21b) entails Alu; + YH, = O in S Therefore, since H, is a constant in S, OW —-ds H dS = O idn +V JS Hence; YH, = 0 and A_,_\p = 0 in S (3.35) Except for waves with no azimuthal variation in a circular pipe with finite conductivity; such waves are pure E or H waves. 43 OCR Output If Hz ¢ 0, the first condition entails Y = O hence k = 0. By analogy with (3.33), the second equation combined with (3.34) entails, if the domain S is simply connected: grad_L\;!=O or HL=O Let us now suppose that k ¢ 0 (i.e. on ¢ 0). Then, with (3.5), Y at 0 and with (3.35), H 2 E 0 Therefore the wave is TEM. From (3.19a), curlzEJ_ = O and div E_L = O hence E L = —grad_LV with W = constant on si (32,6) and AJ_V=O in S (3.37) The sg represent the disconnected parts of the border of S. Equation (3.37) implies O=_l_gA_LV dS=Z§-C2‘ ds Or Zqi=0 [ S; all where qi represents the charge that the part sg carries per unit length along z. Equation (3.37) also implies =•= 2 O=,l_gV*ALV d.S=Z\Qav §-— ds —j|gradLV| dS i S; an S which entails, if the domain S is simply connected: gradLV=O or EL=O Therefore a TEM wave cannot exist in a domain S which is simply connected. When the domain is multiply connected, TEM waves exist because V can take different values V; on the different conductors. Once EL has been determined by (3.36) and (3.37), H, is obtained from (3.18) as HL:-}[ElXi,] (2.38) C lassqicazion of eigenvalues 1) Hz = C at 0 is only possible if lc? = O and to = 0; this case corresponds to a D.C. uniform magnetic field in the waveguide, which behaves as a solenoid. The fact that co must be zero is also obvious from the boundary condition 0= 2) When S is simply connected (hollow guide), there is a countable infinite set of kf values starting from O (with Hz = C) for H waves and from a positive value for E waves. Moreover ([20], p. 292), smallest positive kf forH waves < smallest kf forE waves The corresponding eigenfunctions H Z or EZ form a complete orthogonal system of scalar functions in S. 44 OCR Output 3) When S is multiply cormcctcd (cablc with N conductors), kf = 0 is also possible with (N — 1) independent TEM waves. The transverse electric field E [ is the same as in the (N — 1) independent electrostatic distributions of charges on the N conductors. Interpretation ofthe eigenvalues kg In all cases, from (3.5), -Y2 = k2 — kg . k If k > kc 'Y = jB propagating wave k = kc Y = O cut- off frequency Density of eigenvalues {Weyl 's theorem) The number N(k§) of modes which have a cut-off frequency smaller than kc can be evaluated by an asymptotic formula valid for large kc. This formula was first derived by H. Weyl ([21], p. 59). Let S be the area, and [ be the perimeter of the waveguide cross-section. It can be shown that the distribution of eigenvalues kf of the scalar Helmholtz equation (3.31) combined with the boundary conditions (3.32) satisfies the following relations: L /5M for H modes: N(k3) =- -2-- + 41: 41: forE modes: A N(k3)=Q-M41: 41: 3 .7 Wavcguides with ZS ¢ 0 They may guide waves having an exponential decay along the normal extemal to the surface of the guide. These waves are called surface waves ([15], p. 697; [20], p. 377). Examples: Sommerfeld wave on a single resistive wire (E wave); wave along a dielectric coating; wave outside an optical fibre. 3 . 8 Dispersion in waveguides Fig. 3.4 Dispersion diagram for a waveguide du . glopezv = —- (group velocity at P) ‘° %‘*~\¤ cn Slgpggyp: Ll- (phase velocity at P) 45 OCR Output For a lossless waveguide, the dispersion relation is given by (3.5): k2 = kg + B2 (3.39) lf the dielectric has zero conductivity, (3.4) applies and up > c(e,u, )` 1/2 "‘ {MB #44546 eu ` -ug < c(e,u,) l/2 When ug < up, the dispersion is called normal. lf the dielectric has 0 at 0, ug > Up and the dispersion is anomalous. The energy velocity UZ is defined by P = ue W1 where P is the power flux of a wave travelling along the guide, and W1 is the energy stored per unit length of the guide in the travelling wave. For a lossless waveguide, we shall prove in (3.65) that upu, = X (3.41) hence from (3.40): ue = ug It should be emphasized that this equality is tied to the absence of losses. For a guide with finite conductivity or with dielectric losses y = cx +jB, P = PO e·2°‘2 for a travelling wave, and P1 is the power lost per unit length of the guide: £ R = —= 20.P = 2(!.U,W{ dz As in (2.127), the quality factor Q(cu) of the guide in the passband is defmed by: Iwl Pt — = — 3.42) ( Q M where we take 0) as real. Therefore 2u = A (3.43) 1>eQ Dispersion relation For a waveguide with perfectly conducting walls and lossless dielectric (G = O, £" = 0), the dispersion relation is simply (3.5): wi eu = kg - y 2 where k; is obtained from Helrnholtz's equations combined with the boundary conditions (3.32). In order to take wall losses and dielectric losses (0 ¢ O or e" ¢ 0) into account, assuming they are small, we simply replace in (3.44) the frequency mc of the lossless case by the expression (2.141); this yields the dispersion relation 46 OCR Output c02eu1+[Qw L — =Q /:30 — y2 where y = or + j (3.45) Equating the imaginary parts we obtain me = a>2eu%‘-Y2 (3.46) Since B = co/up, the comparison of (3.46) with (3.43) yields in the passband: ueup = Sp which is identical to (3.41) obtained for a lossless waveguide. Remark: In the stopband, the dispersion relation (3.45) no longer applies if QW and Q are still defmed by (3.42). Nevertheless, (3.45) still applies if (2.141) is used as a definition for QW and Q. Obviously, such a definition of Q is consistent with the classical definitions (2.127), (2.134) in the passband; it has the advantage of providing an analytic continuation of the classical definitions in the stopband [22], where it yields simpler expressions. For example, from (3.3) and (2.20) a dielectric conductivity 0 has the effect to replace k=2 0Je;,L2 w2e’u1———G(j{—j + = m2e'u1———-—jSgn(w) (D8 p Qd where the last expression follows from (2.140); therefore, with the new definition, 1 ° `“ (3.48) Q4 lwle The definition (3.48) yields the same result as (2.136), which however used the equality Wm = We. The new defmition of Qd extends the validity of (2.136) to cases where Wm ¢ We; such a simple relation (3.48) would not be obtained with the classical definition (2.134) in the stopband, because it would require Wm Wg, which to first order is fulfilled in the passband but not in the stopband, as shown later by Eq. (3.62). Similarly [22], one may consider that imperfectly conducting walls produce a complex factor Qw complex Such that 2 isgn (w) Jsgn (wl _ 2 “° `"'"` `°)" {lbEw complex (3-49) If we compare with (2.140) we obtain 1 _ .+ j sgn(c0) Qw complex Qw which is natural because 1/QW com lex is proportional to the wall impedance ZS = [1 + j sgn(c0)]R,. In fact, the generalized definition of Q leads to ([22], p. 303) uw: Z lR.<<»;Fa where I?1(y) , 1:1 (~Y) represent the magnetic fields of waves travelling in the positive and in the negative z direction along the lossy waveguide. From Eqs. (4.8) and (4.10) it will be seen that 47 OCR Output if F1(7)~]F1y + H,i, |e"" , men H(-y) -]-197 +H,i, |e hence Fl(v)·1?(-7) ~ [-Hi +H? which should be used in (3.50). A first approximation is obtained when in (3.50), H(7) , H(-7) are replaced by the fields in a lossless waveguide; then within the passband -7 = 7* and one may take I?(—7) = C1:I(7)* where C is a constant, so that (3.50) then reduces to the classical definition (2.134). 3 . 9 Longitudinal and transverse energy densities From (3.8) and Maxwell’s equations (2.1) we derived (3.9), which we here rewrite in its complex conjugate form (assuming e, tt, on to be real quantities): 7 * Fi + J¤>u1><[?i L] = —gr¤d.tE£ (0 - jt0e)E1 +7II_*°[;jT] >< ,= —grad_LH;{ >< 1, (351) By taking the altemate form of the equations in (3.8) one would obtain Y.t[Ei]FI•- X z+f¢¤|11 = ‘I gT¤d1.Ez X1: (G+j0)£)_1_Xz+'YI¥_L=—gT3dJ_HZ[Ei] 3 52 ( I ) Let us dot multiply (3.51) respectively by ali, and EL! _ _ ··_ {ele,] L +)o>ett -• 1, -•·*{E,H] > At this point we assume that s, tt do not vary in space. Then, from (3.11), div (eil) = yes, whereas div (li), xi= —i, ·curl El = —curl, E = jcott H, Therefore the previous equations become 7e|EL|-72lE,|=—jmeu*`22 > In a similar manner we dot multiply (3.52) respectively by H; and : 48 7l, {LE7;] x 1+ jmu |FIL|=L -z°[lH;] X-gradJ_EZ = Ez div Z[TF1]([l-{1]) >< 1- div EZ, ><1 (0 + jc¤e)uiz -[E_L >< + 7u[I?IL|=2 —p1—}1—grad_,_HZ = Hz div div(gl) Hzu But div [lz >< = —iZ ·cur1 H1 = —curlZ H' = -(c - jwe)EQ and from (3.11), div ( ) = 7p.H;* so that the previous equations become jmu|1?L|‘ + (G — jt¤s)|1;,|‘[E = yX 17·*] z - i([iHIDdiv i;Z, X _ -•—· -•=· 2, -'Y|.LIH_L\=•¤ 2 -7 tt|HZ| = (o+t—•¤•¤ jc0s)uE xH -lz —d1vHzpH_7_[]() If dispersion, i.e. the variation of s and tt with co is neglected, following (2.36) the time averages of the transverse and longitudinal energy densities per unit length of the waveguide are defmed by: . .. 1WL=¢l€|EJ"·’ — 2 WeI wfihlds— 2 wi ` W =H¤IE.I"¤S W =}l¤|”z|d$2 Then W‘=Wj+wQ' W"‘=Wj‘+vI{'” Wi=Wf+Wi“ Vlé=Vli'+ll£'” WL + VK = W' + W The flux of the complex Poynting vector through the waveguide cross section is separated into real and imaginary parts as(*) i — ~. — . I;)-[ExH ]·dS =P+;Q (356) Integrating (3.53) and (3.54) over the waveguide cross section (with s, tt, 0, assumed to be constant) yields (see Fig. 3.3): 27`Wf - 27%* = -jmsu(P + jQ) -%§E;gE ind; (3.57a) Z—KE jcoWf J + 2jm W2"' = 7`(P + jQ) -%§H;E°· Tsds (3.57b) 2j0J wf + 2-—(g) jcoWf = 7 (P+ jQ) +é§Ez1-7* ·l_,ds (3.57c) * 27 Wi" — 2~f%’” = (6 + i<¤s)u(P + JQ) “%§HzpHTndt (3-57*1) When the waveguide walls are perfect conductors, all the contour integrals in (3.57) vanish because of the boundary conditions (tl) In this section, Q is the imaginary part of the complex power flow along the waveguide. Hopefully, there should be no confusion with the rest of this chapter, where Q always represents a quality factor. 49 OCR Output [E >< E] = O r-Z · I? = O In this case, by separating real and imaginary parts in (3.57a), (3.57d), (3.57b), (3.57c) we obtain 20t(Wj — uy) = mug 2;s(w; + vw) = (DSLIP 2¤(Wf‘— W ) = ¤w°—¤>¤uQ 2B(W£‘ + PK") = ¤uQ + ¤>€t1P (3.5 8) 2;Wf=otP+BQ 2m(Wf—VlQ'”)=—tzQ+[3P 2%WQ‘=otP—BQ 2c0(WL"‘—l4Q‘)=txQ+BP This set of equations is equivalent to — 2¤w-(j W=;) metro 2|sw+(j W;) = cospf 2¤wL() - wz = any 2BW’"() - we= eng (3.59) 9(wj+w;’)=¤1> m(w”*-w*)=nQ ii -·%( Wf·Wze)=BQ ¢°(Wi -%)=BP The eight equations have been linked together in four pairs. The pair which contains (WI + VW) > O implies that 2otB = topo (3.60) With this relation, each pair reduces to a single equation; therefore (3.59) is equivalent to (3.60) and the following four equations: 2on(Wj — Hf) = weuQ 2B W' = u>euP a>(w"-w‘)=¤Q ¤>(wL -vr;)=BP (361) which entail P=% Remembering (3.3) and (3.5), it should be noticed that (3.60) is equivalent to 1mk=1mk2(§)(%) +:0 (3-63) which confirms that is real for a waveguide with perfectly conducting walls. In the particular case where 0 = 0, i.e. when the waveguide dielectric is also lossless, we have aB = 0. There are two possibilities: l) Propagating waves: 0: = O (passband) Then Q = 0 and Wm = we = 1/2 W. The power flow across the waveguide is purely real; it is given by P=·`,§(Wi·Wz)=1>p(Wi -**2) Si¤¤¢ Up <3.64a>OCR Output 50 or by P = = 1>eW hence De (3.64b) lt follows that P ep up WL + WZ ‘ lt thus appears that the difference between ue and Up is tied to the existence of longitudinal fields; in particular ue = up for a TEM wave only. 2) Evanescent waves: B = O (stopband) Then P = O and W _4_ = WZ . The power flow across the waveguide is purely reactive; it is given by Q = E()-W" — W' (3.66a) vt by 2 2 HI hl £7(Wf · ll?) = —§(Wi - WC ) (3.66b) The expression (3.66a) can be obtained directly by applying (2.119), with J = 0 and otl = O, to the volume enclosed between two cross sections of the waveguide at z and (z + dz). The relations (3.58) can be generalized to include the case of waveguides containing lossy and nonreciprocal media [23]. The relation (3.64a) is also valid for dielectric waveguides ([17], p. 43). 3.10 Attenuation of waveguides We suppose that the attenuation is only due to resistive wall losses. In the passband where ot is small, from (3.62) we may take Wm = We. Using (2.117) and (2.36) in (3.42) we obtain 1+R.lH.I -· 2 M 2 = (3.67) Q 1 W iis FLIHI_ 2 ds where the upper integral is taken on the perimeter of the waveguide and the lower integral is taken over the cross-section (see Fig. 3.3). These integrals are computed for a lossles waveguide. Using (3.4) and (3.5), the attenuation is then computed from (3.46) as 2q = M& = l<1L._ W (3_68) QW B QW ,/18 -k§ This formula obviously breaks down when k = kc; in this case one should compute B from (3.45) and not from (3.5), which only applies for a lossless waveguide. H wave By (3.10) we can express both integrals in (3.67) in terms of Hz. 51 OCR Output BH 2 2 » 2 — 2 §\H,\ ds = ;|HL[ ds + f|Hz|2ds = th?] ds + §lHz|ds (3.69) C With (3.33) and (3.5): — 2 7 2 — 2 v 2 k2 [S|H| as = ;S|gmdLH.[ as + ;S|Hz|l2 ]CdS = +1;S|H,| as2 = FjS|Hz| as (3.70) " C Therefore (3.67) becomes k2 M gg‘,-— -1] ———J·1 ds +BH k+k§ as2 +1:+.1 2 QW P2 k ISV-IZ|LdS Ol` %`22&` (,,1§·ds _ Rs kc 8s+{§IHz\2d52 2 igll· frlds 2kc2 Bs k% Q,. u ;S|H,|2ds This can be rewritten as 2 *-=&A+B5L{%i] W Ll (3.71) where 1 BH 2 1 BH ;{‘TsZ‘ ds ¥IHz]2d$ ·i;5`·§ ISIHZI ds ISiHzI ds are coefficients independent of frequency. From (3.68) and (3.71) we have 2u:-l1——9—21+13-£~-1-A+B£22 -1/2 2 1/2 2 -1/2 2 %{];2J {j";i}‘i‘\ {%) {7];] C (3.73) where we have used the Icoll/2 dependence of R , shown in (2.115). It is easy to see that as a function of frequency, ot reaches a minimum when 2 .=-+1++12 E22(§)\!2(§5 £ kc 2 A 4 A A (3.74) OCR Output From (3.72) we always have ” A .1+113. kg Bs 52 Nevertheless, for all waveguide cross-sections where an explicit solution Hz can be found, we have the more stringent inequality B/A > 0. Therefore, since (3.74) is an ever increasing function of B/A, the minimum of attenuation is reached at a frequency $2- > 3 (3.75) In particular, if A = 0 (which means GH,/Bs E 0 on the waveguide periphcfy). the minimum of attenuation recedes to kz = =>¤; in that case the attenuation is an ever decreasing function of frequency. This property makes very attractive the H modes with HZ = constant at a given z on the waveguide periphery; such modes exist as the rotationally symmetrical Hm modes in circular and coaxial waveguides (see later). E wave By (3.10) we can express both integrals in (3.67) in terms of E,. §|— 2l — 2 0+ jcoeBE H dS= H d.9=————~ kg Z t§|· tli} I (3.76) With (3.33): -· 2 — jS|H| ds:2 0{Sly;] + jcoe dS=|—T2—| jS|gmdl1;,|2 o +jjI dS=\7—-kcms C 2 2jS|E,| as (3.77) Therefore (3.67) becomes tele1 BE ds R =-iA where A=—-$7 (3.78) QW u jS|E,| dS A is a coefficient independent of frequency. From (3.68) and (3.78) we have 2ot=4% Q k1-A (3.79) Comparing with (3.73), we immediately see from (3.74) where B = 0 that the attenuation ot reaches a minimum when (3.80) 3.11 Some waveguides of simple shape 3.11.1 Rectangular waveguide When the cross section of the waveguide is a rectangle of sides a, b (with a 2 b), the Helmholtz equation is separable in the transverse coordinates x, y. H waves Hz = Hm,,cos-xcosy(L1Ej(§§) a (3.81) OCR Output 53 2 2 {) a Em,b n=O, l, 2, The mode with the lowest cut-off frequency, apart from H, = C (for m = rz = O), is the H10 mode. From (3.10) we obtain: Ex = HM';*}**%?(?}(‘2'“} €¤°$xS1¤··yHx = é?[’%’£ll?lHm S*¤·*¤°Sy _ By = ··H-·S1¤xc¤Sy”» %%“(?}(?l = H~»·%·=¤S?*]Smé.((?y)(3.82) '£ E waves Ez = E,,msinxsin——(E )(néEy) a (3.83) 2 2 k2=m,[@] {2*5] ,.:1.a b 2 The mode with the lowest cut-off frequency is the E I 1 mode. [ Ex = ——E,,mé?(%)(%c) cosxsin—y Hx = Em -sinxcos-y%(?)(%gi) Ey = —E,,,,,si11xcos é%(?)(££y) · y(3.84) ,,m H= —E%E-cosl-r?x)sin(—rgy] . Remarlc The degeneracy of cutoff frequencies for H and E waves in a rectangular waveguide is removed when wall losses are taken into account ([12], p. 352; [15], p. 350). 3.11.2 Circular waveguide When the cross section of the waveguide is a circle of radius a, the Helmholtz equation is again separable in the transverse coordinates r, tp. In the following, Jm(z) is the Bessel function of the first kind of order m and argument z. H waves HZ = Hm,,Jm(kcr)cosm(p where kca = jbm, aroot of J,,,(kCa)= O ($85) The integer n = 0, 1, .. counts the zeros of J ’,,,(x); n = O corresponds to kca = 0 when m = 0, and to kca > O when m ¢ O. (k,a)“ zJ2 n (+—&2)- n—nz 4-21-; 4 m = 0, 1, 2, n= 0,1,2, For historical reasons these modes are designated as H,,,',,+1 when m at 0, although the notation Hm,1 would be more logical. The mode with the lowest cut-off frequency, apart from HZ = C (for m = 0, n = 0, kc = 0), is the H 1 1 mode. E, = Hm —-LJ,,,(kCr)sinm 54 E-waves Ez = E,,,,,Jm(kCr)cosm 2 2 (kca)L=n+(—@)- - 41:2-2-i 4 m=0, 1, 2, n=1, 2, The mode with the lowest cut-off frequency is the E0] mode. E, = —E,,,,, TL./',,,(kCr)cosmq> H, = ——E,,m t G `Ljmg m Jm(kcr)sinm III Eq, = Emc L—Jm(kCr)sinm Remarks: 1) If m >> n, the fields are concentrated in an annulus near the periphery of the waveguide: these are called whispering gallery modes. 2) We have assumed an azimuthal variation as cos mq) for Hz and EZ. When m ¢ O, we can as well take sin mcp which corresponds to the other polarization of the wave. Since both polarizations have the same cutoff frequency, all modes with m ¢ 0 are doubly degenerate. This degeneracy is due to the rotational symmetry of the waveguide about the z-axis; it is broken as soon as the rotational symmetry disappears (for example, in an elliptical waveguide), 3.11.3 Coaxial transmission line The waveguide cross section is the space between two coaxial circular cylinders of radii a, (with a < b). Since this domain is multiply connected with N = 2 different conductors, the waveguide can support (N — 1 = 1) TEM wave. The Helmholtz equation is separable in the transverse coordinates r, cp. In the following, Y,,,(z) is the Bessel function of the second kind of order m and argument z. TEM wave The electrostatic potential V between the cylinders is V=V0l0g¥j with va :0, vb zvozog (3.89) 1 V From (3.36) and (3.38), . O 0 E, = —— Hq, = —E, = —— (3.90) r Q Qr The total current on the outer conductor is 21: [b=-§H(Dd.s`=TVO and Z Ii=0_ [:1 The characteristic impedance of the line is thus Zc:@Kq.:Ll0g2 (3.91) OCR Output Ib 27E C1 55 H waves Y ' Y V k b Hz =Hm,,£”@——{'—iQ]&—Qcosm(p where N JY"(k°b) ”}( C ) l=O (3.92) |i]‘Im(kCa) Ym(kCa) Jm(kca) YI7l(kCa) kzb-(c( 2: 2) 4 ()()b+a2 sZ2'-T :0,1,2,... :1,2, an1t+m+m n For n = O this relation must be replaced by -1 6 W kb2 -4 21-cl 2M) ml —‘— 4 4 b — a XKLE) i 6 2-1 (W ——- 62 2-11 3 bw + 5.9( m ) 6+21 + 3.5.7.9( m )b+a For historical reasons these modes are designated as H,,,,,,+1 when m ¢ 0, although some authors designate them with the more logical rotation H,,,,, ([20], p. 329). The mode with the lowest cut-off frequency, apart from Hz = C (for m = O, n = 0, kc = O), is the H 1 1 mode. E waves J k Y J k b Y k b Ez =E,,,,J—-"ig£——’i]f£-C-1cosm kb—f()()(b a2 = nrc2 + 4mz+ -a m = 0, 1, 2, n = 1, 2, The mode with the lowest cut-off frequency is the E0] mode. 3.11.4 Radial line 0 l Z Fig. 3.5 E wave in a radial line A radial line is the space between two parallel plates a distance l apart, where an electromagnetic wave propagates radially (outwards or inwards). The basic waves are the same as in a coaxial guide, except that in a coaxial guide we have standing waves in the radial direction and travelling waves in the longitudinal direction, whereas in a radial line we have travelling waves in the radial direction and standing waves m the longitudinal direction. In the radial direction, an inward travelling wave is represented by a Hankel function HQ(krr);) an outward travelling wave is represented by a Hankel function Hg) (k,r). These functions are singular at r = O. Along z, the propagation constant B must take the values 56 OCR Output B=§-?- n=0,1,2. (3.94) from the boundary conditions at z = 0 and z = I. Along r, the propagation constant k, is then given by (3.5) as k? = 18 - ($2 (3.95) H waves (with respect to the z-axis) We shall write the fields for a radially outgoing wave; but the formulae are the same for an ingoing wave. H, = H,,,,H§3(k,r)eos) mq) sin [32 n = 1,2. (3.96) Decomposing sin Bz into two travelling waves, from (3.10) we obtain E, = Hm (k,r)sinm )' n) (3.97) Eq, = H,,,,, %H5,?(k,T)COSI71(§) sinbz Hq, = —H,,m E-i-Hf3(k,r)sinm(prf eos[3z E waves (with respect ta the z—axis) E, = E,,,,,HS,?(k,r)cosm(pcos[3z) n = 0,1,2. (3.98) Decomposing cos Bz into two travelling waves, from (3.10) we obtain )' @) E, = -12,,,,, %Hff(k,r)eesmq) sms; H, = -12,,,,r #5 THfj(k,r)s1nm(p cosBz T) )' (3.99) E., = 1:,,,,,cI' Tr %;—H$3(k,r)simn The most important mode is the one with no variation along ep (m = 0) and no variation along z (rt = 0), i.e. the EDO mode. Since B = 0, from (3.95) k, = k. This mode has only two field components: 0+ jcoe (2)’ _ _ E, - Em(2 ) H0 (kr) Hq, - -EO0 T'--H0 (kr) (3.100) Along r, it is a TEM wave with a wave impedance ” aa- k H£ When kr >> 1, H(kr)(2) ;2)= _j H1( (kr) and 57 Ejk , ——- = ——— = b 2.64 C Y ( ) Hq) G + jm Plots of the first modes in waveguides of simple shape can be found in the literature ([24], p. 59 to 84; [25]). 4 . IMPEDANCE TRANSFORMATIONS 4 . 1 Orthogonality of waveguide modes Because the cross section of a waveguide is two—dimensional, possible modes (and values for kj ) are characterized by two indices. In the following we use a single letter n or m to represent the full set of indices. At a given frequency, the set of all possible (TEM, H and E) modes in a waveguide with a homogeneous isotropic dielectric and perfectly conducting walls form an orthogonal and complete system, which can be used as a basis for expanding the most general fields in the waveguide ([26], p. 121). By using thc identity U · AV + grad U~ grad V = div (U grad V) (4-1) it is straightforward to deduce from the Helmholtz equations (3.31) combined with the boundary conditions (3.32) that, for E waves: (4.2) kcm.|.EznEzmdS + lgf¤diEm ‘gY¤diEzm 45 = fEzn*5—d$z = 0 ham (kg - k§,_)[ Emsmds = 0 lf kg, ¢ kg", , the eigenvalues are said to be non·degenerate, and [EZ,,EZ,,,dS = O for n =¢ m (43) If kg, = kgm, by taking appropriate linear combinations of solutions of the Helmholz equation, one can choose a suitable basis of eigenfunctions Em, Em such that (4.3) still holds. Introducing (4.3) in (4.2) yields { grad [12,,, · grad [1;,,,, as = 0 for n ¢ m (4.4) With (3.10) this entails E,,_·E,_r1s=0, |H,,_·I·fM_dS=0,j[EH] ,X,—as=0, wm (4.5) F0rH waves, Ez is replaced by HZ in (4.3) and (4.4), whereas (4.5) remains unchanged. Since the Helmholz equation and the boundary conditions (3.32) are real, the eigenfunctions EZ, HZ may be taken as real quantities. From (3.10) it then follows that . EL, H L, have a constant phase throughout a cross section S of the waveguide; therefore (4.3) and (4.5) may also be written as 58 OCR Output I EznE;mdS = 0 l HznH;mdS = 0 {EM-EL,,d$=0 jHJJ,·1?],,,ds=0 (4-6) [EH;]j°,,· X· Sdg = 0 for n at m (valid with or without *) By using the identity [grad U >< grad V] = curl (U grad V) which entails _f[grad Uxgrad V]·d§=§U?ds=-§V?ds (4.7) s s it is easy to see that the orthogonality relations (4.6) also remain valid when n is an E wave while m is an H wave. Finally, when one of the waves is TEM (in a multiply connected domain), one uses (3.36) to (3.38) and (3.10) together with the identities (4.1) and (4.7) to prove the validity of (4.6) in this case also. If n and m are both TEM waves, they can be chosen to be orthogonal since they correspond to the same degenerate eigenvalue kf = 0 4 . 2 Reflection at an obstacle Fig. 4.1 An obstacle in the waveguide at z = 0 Wave travelling in the positive z-direction E=+ EAQIEJJ, `Y"Z + Emi, ]e (4.8) H_+ = 2A,,+ _[HU, __ +HZ,,1z]e _ Z Y" El}: = Z0nI.Ln[?i] X z (4-9) where Z0 is given by (3.20) for an H wave and by (3.22) for an E wave; Z0 is simply Q for a TEM wave. Wave travelling in the negative z-direction By changing the sign of Y in (3.10) or by considering the image of E+ in a symmetry plane, we may write -= A,jEJ,, - Emi,w la H`E;[__ = ZA,f—H_L,,” [fl. _ +__ Hz,,1Zle Z Y" Consider a wave E+ impinging on an obstacle at z = 0. In general, the boundary conditions on the obstacle require the presence of all possible modes in its vicinity. However, if the frequency is low enough (lower than the second cut·off frequency). only the mode having the lowest cut—off frequency (dommant 59 OCR Output mode) will propagate away from the obstacle, so that, at some distance, the reflected wave E “ will essentially reduce to the dominant mode. A particular mode is completely determined by the amplitudes of EL or HL in the two travelling waves at some point in the cross section of the waveguide. Having chosen such a point, one may write _ H+ = 2 E] = Ege YZ L 2. (4.1 1) EI = Egeyz H; = J} The wave impedance Zo is defined with respect to a direction of propagation, and changes sign with the latter. The total wave in this particular mode reads EL = E§e+`YZ Ege yz (4.12) H=I ——Ee"YZL 2;[;,;] —EeYZ Analogy with transmission lines: V I ZC or Z0 Transverse fields in waveguides: E L H )_ Z0 Reflection coeficient (ofthe zransverse electric field) Since in the following we only use the transverse fields, we drop the subscript .L. By definition, the reflection coefficient of the transverse electric field is = .. P E, (4.13) In particular, ` pv Z 7% = lpol °]¢0 For the transverse magnetic field, the reflection coefficient is (—p). From (4.11), p = po ::272 = |p0| c2‘”"`j(2B"""°) (4.14) Let us remember that since the reflecting obstacle is located at z = 0 (see Fig. 4.1), in (4.14) z is always negative. Therefore, when ot > O, lpl decreases toward the RF generator; when ot = O, Ipl is independent of 4 .3 Standing waves At every position z in the waveguide, E=E+(1+p) H=H+(1-p) (4.15) If the attenuation is zero, lE+l, lH+| and Ipl do not depend on z. Then, along the waveguide: Elm = \E"I (1+ IPI) |El,,,;,. = \E+I (1-lPD (4.16) OCR Output 60 Voltage standing wave ratio (VSWR) The VSWR is dciincd as S = E? 2 1 along the waveguide (4,17) If thc attenuation is ncgligiblc, (4.16) yiclds 1+P $·1 -n _-l_|D| or |r>I—S+1 (4.18) The voltage standing wave ratio S is thus a measure of lpl. When Ipl = 1, S = ·><>: this is the case of a pure standing wave with total reflection, where the reflected wave has the same amplitude as the incident wave. When p = 0, S = 1: this is the case of a pure travelling wave, and the waveguide is said to be matched. From (4.14), this condition is independent of z. Amplitude variation of E. H along the waveguide For simplicity's sake we assume that the attenuation ot is negligible. From (4.14) and (4.15): QD-E = ‘1 + |p|+2 2|p|eos(2Bz + cpo) = (1 —2 + 4|p|cos|3z 2 {Q;-5 + 2IHIL · 2 TU =1+2 — 2|p\cos(2Bz + Q10) = (1- + 4\p|sm Bz + ¢0 Y E| is maximum when |H| is minimum, and vice versa (see Fig. 4.2). ILI or ULI lvl lrrfl 2 ` F \ ;"` S /’ I _¢’S =3 ` \ \ I 'va ‘\ ` " f` lf \` ]_ ‘ |-—y¢g$—-· 2" —-- —-—-—ra---y-weZ; S `<${ —//— =l \ \\>"; 0 7 \_s_,Q 2n Fig. 4.2 Amplitude variation of E and H with B2. In the figure, for S = ¤<= we have taken $0 = TC (i.e. po = -1) 9/Q 0 Fig. 4.3 Variation of (1 t p) in the complex plane 1¤& / 61 OCR Output 2H ...... · ' or a1'$ afs } ,x2 Eg } ,* / : ,1 // :v/' •!/ / 1} /1/ ,, : / ’/ E 7*J .:Z ...... ¤ ...... ""` .·•" ...... I {.·'€ I l / slope S for E, é for H ` I:x · ’ :,’ / :1,/ Z'} /l1 / jr slope E for E, S for H i/ ,1 t {x' i — Bz 2TT Fig. 4.4 Phase variation of E and H with Bz. The figure is drawn for the case $0 = tr. Phase variation of E, H along the waveguide From (4.14) when ct = 0, p = |p0|m ewhere tp = 2Bz + $0 The variation of (1 :i: p) with ¢ is shown in Fig. 4.3. The phase variation of E and H with [iz follows from (4.15); it is shown in Fig. 4.4. When S = ·><·, the phase of E (or H) is constant except for jumps of 180° at each node of the standing wave. Moreover, E and H are always in quadrature. 4.4 Impedance transformations At a distance z along the waveguide, the impedance Z and the admittance Y are defined in terms of the transverse fields by: I H (4.19) Z E Using (4.15) and (4.11) this becomes ET 1+ p 1 + p Z = ··--—· = Z H+ 1_ p 0 1 __ p 4.20 ( ) Z, like Z0, is defined with respect to a direction of propagation; it changes sign with the latter, as [E >< Fl * 12. The direction of propagation is chosen to be positive from the generator towards the load. As 62 OCR Output is evident from (4.20), the only quantity which appears in the equations is the nonnalized impedance z Z/Z0 or the normalized admittance y = Y/Y0 = 1/z. Relation between z, y, p From (4.20), at every position z along the waveguide, Z ' I 1 p ·— 1 — 0 (4.21) 1 _ p L y y = ·——P = 1+o 1+ These bilinear transformations in the complex plane transform circles into circles (a straight line being considered as a circle of infinite radius). Smith Chart The Smith chan is the disk Ipl S 1 in the complex p-plane, where the circles of constant r = Re(z)or _g = Re(y), and x = Im(z) or b = Irn(y) have been drawn and labelled (see Fig. 4.5). From (4.14) with on = 0, when z is varied, p = po e2JBZ simply rotates around the origin in the p-plane; so one immediately reads the transformation of z or y from A to B. Normally the p-plane is presented in a way which corresponds to z, with the points p = -1, O, +1 appearing from left to right on the real axis. It is seen from (4.21) that adding l80° to the phase of p transforms z into y; therefore the same chart can be used for y, but in that case the points p = -1, 0, +1 appear from right to left on the real axis. ¤=c0nstant qonsya-,y .-r:consfanl* gxgnsyany Z·Pl¤¤€ y-plane x cr b =1 ’° / / A / 2 \ asn "% [for? 9-plane Fig. 4.5 Smith Chart: Transforming from z- or y-plane to p-plane 63 OCR Output Chain matrtlt of a length I of waveguide E,[ EO l E° Fig. 4.6 A waveguide section of length Z H-¤ I ·-“ l**· 0 z to the'ghe e ds chain E0, H0matrix at the is output.the matrix A which relates the tields E.;, H.; at the input of the waveguide section From (4.12) we have E_) = Eje+ll E;e`Yl E0 = EQ + EQ l [ _ .. g 1 Th¢r<=f<>r¢ EZ = £(F0 + ZOHO) E5 = HBO · ZOHO) °$h(Y£) Za$mh(Y£) and E_l ;E0 [-[_l Z —sinh(y!) cosh(y£) HO (422) EO The chain matrix thus reads cosh(y£) ZOsinh(y£) Al') = L Z0 sinh(yl) coshwty (423) It is readily veritied that A`l(£) = A(—l) (4.24) and det A=det A`1 =1 (4.25) The last equation is simply an expression of reciprocity, because the relation between the chain matrix A and the scattering matrix S is such that det A = -A-Eg S21 Impedance tramfonnation by a length I of waveguide From (4.19), at z = —l D_[ D0 z=—— whereas at z = 0, z_=—- ' ZOH-4 ZOHO OCR Output where in ZL the subscript L stands for "load" 64 From (4.22): tanh l anh Z z==i.+ (Y) y:y1.+t (Y) (426) 1+ zLtanh(Y£) 1+ yLtanh(Y£) Particular cases: 1) zL= 1 then z = 1 for any l. This corresponds to p = 0, i.e. to a matched waveguide. 2) ZL = O tl1en z = tanh(Yl). If ot = 0, Y = jB and Z = Z,,j tan([3£) (4.27) Such a short-circuited length of waveguide can provide any reactance according to the value of Z. Impedance matching When a waveguide is matched: 1 . Its input impedance is independent of its length. 2. There is no reflected power towards the generator. 3. VSWR = 1, which means that there is no voltage nor current peak along the waveguide. There are several means for matching a given load YL to a waveguide (at a given frequency). For example: Fig. 4.7 Matching of a load YL with a shunt susceptance at z = —£ By using a properly chosen length l of guide to arrive at C (Fig. 4.5), one crosses the g = 1 circle in the Smith chart, where the admittance appears as Y0 + jB. It is then sufficient to cormect at that point a shunt susceptance —-jB, which can be obtained with a short-circuited length of waveguide (called "stub"). In practice, it is easy to vary the stub length by using a sliding short—circuit in a piece of waveguide, but it is not so easy to vary the distance between the stub and the load. If this distance is fixed, several stubs are generally needed to match the load; in fact it is always possible to match any load by using three equidistant stubs at a fixed distance of 3/8 lg apart. S . PERIODICALLY LOADED WAVEGUIDES 5 . 1 Chain matrix The simplest example of a periodic structure is a waveguide which is periodically loaded with lossless obstacles (infinitely thin diaphragrns, ...). For the dominant mode, these infinitely thin obstacles behave as series or shunt reactances. We assume that the cell length L is long enough for the evanescent modes from one obstacle to be negligible at the position of the next obstacle. In the following we consider the case where the obstacle is equivalent to a shunt susceptance Y. 65 OCR Output 1 E+ _ ZIB Y=Y Y, Fig. 5.1 A waveguide with periodic loading of spacing L Boundary conditions between 2 and 3 (see Fig. 5.1): (continuity of transverse electric field) Fa = E; (5.1) H2 = E5- + Y (defmition of Y) E2 Ez Then VH I °lE2l H3 'Yyo 1 H2 If Y0 is the propagation constant in the unloaded waveguide, from (4.22) and (4.24) we have: [1]22c6s¤(y,,L) H2 ;{O -Y0s1¤h(y0L) -—s1¤1r(y01.) cosh(y0L) E1] H1 The chain matrix T for a full period of the structure is such that lE’l=TlE1l H3 H1 M I hence T I¢<>Sh(v0L) —gsmh(y0L) (5.3) { Y y ° I ‘Yo$i¤h(YoL) ¤<>Sh(v0L) and from (4.25), det T= 1 (5.4) By Fl0quet's theorem, there are particular solutions of Maxwell's equations, called travelling waves, such that for a translation of one period L along the structure: E L E (Z + ) = {YL (Z) H ( z + L) H (2) (5.5) OCR Output 66 Y = ot + j|3 is the propagation constant for a wave travelling in the positive z-direction along the periodic structure. (Let us remember that YO is the propagation constant for a wave travelling in the positive z direction along the unloaded waveguide.) When YL is purely imaginary we put YL = j0 where 0 is the phase-shift per period of the structure. Comparing (5.5) with (5.2) shows that e"YL is an eigenvalue of the chain matrix T. By (5.4) the product of the two eigenvalues of T is l; therefore the other eigenvalue is eYL, which corresponds to a wave travelling in the negative z—direction along the periodic structure (this is reciprocity). The sum of the two eigenvalues of T is given by eYL + {YL = Tr(T) (56) 5 . 2 Dispersion diagram Using (5.3) and (5.6) we obtain the dispersion relation cosh(YL) = cos 9 = cosh(YOL) + %sinh(Y0L) Dependence 0fY on frequency The admittance of a lossless one-port network can be written as an infinite series of terms, each representing the admittance of a series-resonant circuit at a resonant frequency of the network ([27]. P. 87): Y = 2 —;—Jw where 01,, is the nth series-resonant frequency of the network, and Ln > 0 is the equivalent self-inductance of the network at frequency mn. The frequencies w,, are supposed to be numbered in increasing order, starting with wo = 0 (if LO <<><>). We can rewrite (5.8) as l "` l 3 "‘ l . . na.Y = —- am: ————————--—-—. atm: —¤»>¤»: + ME ""” 2 For low frequencies (to << ml), this expression approximates to Y=+—+jmC where C = X-?-—>0 °° (5_9) mL., ,,:1L,,co?, Dependence of y on YO The susceptance Y = yY,, is independent of the direction of propagation; therefore it is an even function of Y0. Since Y0 is an odd function of YO, y must be an odd fimction of YO. In the unloaded, lossless guide: a) for an H wave, from (3.20) we have YO = YO,(jmtt); therefore, using (5.9) and (3.5) we obtain :l’L..-k22= ..]’L-k22 ZQ We C +70 £§LO 8 (LO E E (5.10) Corresponding to LO = ¤<>, a purely capacitive term -k2C0/e may be obtainedwith a dielectric rod across a rectangular waveguide ([28]. P. 122). For metallic inductive obstacles, the tenn tt/LO is the most important one (at low frequencies); for metallic capacitive obstacles, in all examples published in the literature [24, 28], the bracket in (5.10) vanishes and y = YOCO/e. 67 OCR Output (5.11) Y0 kLLO 8 For a purely capacitive obstacle (LO = ·>¤), again y = YOCG/e. We shall here consider only two very simple cases: 1) inductive obstacle (H wave): 1 u 1 =——=——·2K I (5.12) Y0 LO YOL 2) capacitive obstacle (H or E wave): J’=YO¥=Y0L·2K2 (5-13) In (5.12) and (5.13), K1 and K2 are real, positive, dimensionless constants. Inductive obstacles With (5.12) the dispersion relation (5.7) reads cosh(YL) = cosh(Y0L) + K1 where YL = 0tL + JBL (5_]4) Y 0 For the lossless tmloaded waveguide, YOL is either real or purely imaginary; therefore cosh(YL) is always real, which entails sinh(0tL) sin(BL) = 0 hence 0tL = O or BL = nrt (5.15) When 0tL = O, YL = j9, which corresponds to a passband of the loaded waveguide; whereas BL = mt, YL = 0z.L +jn1r corresponds to a stopband of the loaded waveguide. Above the smooth guide cut—off, (5.14) reads Sm cosh(YL) = cos(BL) = cos(B0L) + K1 L gii with K1 > 0 (5.16) 0 This relation is plotted in Fig. 5.2 for K 1 = 4. ln the unloaded waveguide, when to increases from 0 iz the dispersion diagram, one first follows the co-axis up to mc, afterwards one follows the hyperbola k2 = kc + B0. For the loaded waveguide, the cut-off frequency of the first passband is reached when B0K L=0 i.e. @4-tan§g£=§ This happens for O < BOL < 1c: it is the beginning of the first passband; when BOL = rc, BL = it is the end of the first passband. For a slightly higher frequency, cosh(YL) < -1 and YL = 0tL + jx: this is the second stopband. For increasing frequency, YL comes back to jr: when Mm M Z Ji 2 2 2 68 OCR Output cosh (y/L) UU-.-.····.•-`··U.-l--`·`.`°°` `°'` ' ''‘'` "‘°•i'·-···\...... . . · ` I · ' N l • Q O I V l • N ' . ’ N ’· ·· ••-¤•••••••••¤•••••••••\••••••••••§••••••••••¤••••••••••••••••••••••••• 2p .. N I mi QZ ¤ E -3l- E CUZ CU I <¤I NZ ·¤ I .-QZ ..¤Z ¤ wi O Z CUZ U Z ui E -¤F Q¤ ¤- I ¤· I 2n BL lm Fig. 5.2 Variation of cosh(YL) with BOL for inductive obstacles (H wave in unloaded guide). Ir1 the figure K1 = 4. This happens for 1: < BOL < 21:; it is the beginning of the second passband. When BOL = 21:, BL = 21:: this is the end of the second passband. More generally, when BOL = n1:, BL = mt is the end of the nih passband (n = 1, 2, ...). C apaciti ve obstacles With (5.13) the dispersion relation (5.7) reads cosh(YL) = cosh(y0L) + K2(·y0L) sinh(·y0L) where YL = 0tL + JBL (5.17) Again, cosh(‘yL) is always real. Above the smooth guide cut-off, this relation becomes ¤¤sh(y1.) = c¤s(BL) = eos(B0L)— 1<2(|s01.) sin(B0L) with K2 > 0 (5.18) This relation is plotted in Fig. 5.3 for K2 = 0.2. For the loaded waveguide, the first passband starts when BL = O, i.e. when BGL = O; the cut—off frequency of the smooth guide is thus also the cut-off frequency of the loaded guide. The first passband ends when BL = 1:, i.e. when 2 2 2K2 ' 69 OCR Output »c0sh(yL) ...... {...... ?...... l...... E E —| k•·•••••·•••.•·••h•••' ;•••••••¤•••••••¤•••••••••••••••`••••••••••••••,••••••••••••••••••••*•• -2F- 'E -4,}. E:3 u Z 2rr BL an Fig. 5.3 Variation of cosh(yL) with BOL for capacitive obstacles (H or E wave in unloaded guide). In the figure K2 = 0.2. this happens for O < BOL < 1:. For a slightly higher frequency, yL = 0:L + j1:: this is the second stopband. YL comes back to j7C when BOL = 1:, which is the beginning of the second passband. The second passband ends when BL = 21:, i.e. when 2 2 2K2 this happens for 1: < BOL < 21:. Generally, when BOL = n1:, BL = n1: is the beginning of the (n+1)‘ passband (n = 0, 1, 2, ...). Finally, the (k,B) dispersion diagram is obtained by using BO as a parameter and computing k from k2 2 kc-· + B0 while B is deduced from (5.16) or (5.18); the result is shown in Fig. 5.4. Remarks 1) From Fig. 4.2, the distance z between nodes in a standing wave with infinite VSWR is such that Boz = 1:. Therefore, when BOL = nr:. for standing waves with S = ¤¤ in the smooth guide, L is an exact multiple of the distance between nodes. One may then insert infinitely thin diaphragms at the nodes of E _L without perturbing the fields, which yields BL = MTE for the loaded guide at the same frequency. This explains why in the dispersion diagram, the points BOL = n1: of the unloaded guide also belong to the dispersion curves of the loaded guide. There is however one exception to this rule: the BOL = O point of an H wave in the smooth guide (see Fig. 5.2). Indeed, such a wave would have HZ ¢ 0 independent of z (since B = 0), which would violate the boundary conditions on the diaphragms. 70 OCR Output \\ x 2 ki ’/ / , xx .*7 lz / x \, `> 1 / I /3 ,x/ ~ 5 X _ I- Z \ c I s Q \ / /\ \\/ \ / _ · 2· 2: O unloaded waveguide inductive obstacles y = ZX,/(YL) ··· ----·· capacitive obstacles y = 2K2·(YL) Fig. 5.4 Dispersion diagram for a periodically loaded waveguide. BOL and BL are the phase shifts per cell in the unloaded and a loaded waveguide respectively. For the sake of clarity, the reflection of the curves about the vertical lines BL = n1: has not been drawn. The figure corresponds to the case kcL = 7E, K1: 4, K2 -‘= 0.2. 2) The dispersion relation (5.7) only determines YL up to a change of sign, and BL modulo 21:. It follows that the dispersion diagram is symmetrical with respect to the vertical axis BL = O, and is periodic with period 21: in BL; it is therefore also symmetrical with respect to all vertical lines BL = mc, where rz is any integer. As a consequence, it is sufficient to consider the interval O < BL < rt. Because of the 21: periodicity of the dispersion curves for the loaded waveguide, any straight line Up = (D/B = U crosses every passband at one point. Therefore, whatever a particle velocity U may be, in every passband of a loaded guide there is a frequency for which the phase velocity is equal to 0. This is in contrast with the unloaded guide where Up > 1/Vert. 3) When the loading y of the waveguide is increased, for example by decreasing the hole radius of the irrses (which are capacitive obstacles for an E wave), the width of the stopbands increases while the width of the passbands decreases. For maximum loading (i.e. no hole in the irises), the passbands have zero width; the dispersion curves reduce to horizontal lines passing through the points BOL = n1: of the unloaded guide, which is natural because the cells in the loaded guide are then completely decoupled. Resonanz coupling and backward waves Although the above examples are over simplified, they yield a correct qualitative picture of the _ d1spersron diagram for a periodic structure. They become increasingly inaccurate at higher frequencies, when several modes can propagate in the smooth waveguide, and when the approximation (5.9) has to be replaced by the correct expression (5.8) for the shunt susceptance of the obstacles. In particular, (5.8) must be used when the first resonance of the obstacles is close to the cut-off frequency of the smooth guide; the obstacles then produce a resonant coupling between cells, thereby modifying qualitatively the dispersion diagram. An example is a disk-loaded waveguide with slots in the irises, i.e. the slotted iris structure, for which the first passband is of backward wave type ([29], p. 681). The same is true for the travelling wave accelerating structure of the CERN SPS [30], which is a circular waveguide periodically loaded with horizontal bars; the bars behave as 1/2 resonators whose resonant frequency lies below the E0; mode cut off of the cylindrical envelope. 71 OCR Output Most often, in the lower passbands, the frequency is increasing or decreasing monotonously within the interval O < BL < 1:. However, even in lossless periodic structures, dispersion curves may exhibit a local maximum or minimum within the passband, i.e. when BL ¢ nrt. In the stopband which starts at such local extremum, 0tL ¢ 0 while BL ¢ n1:; this means that cosh(yL) is then complex [31]. Such a case occurs for some range of the hole diameter in a disk—loaded waveguide, for the hybrid EH ll wave which is used in RF separators ([19], p. 252). It occurs more often at higher frequencies, when several modes propagate in the unloaded waveguide and their dispersion curves cross when folded into the interval O < 6 < TE . In Fig. 5.5 are shown the dispersion curves for two modes (Eql, E0;) in an unloaded circular waveguide. These curves are the dispersion curves for vanishingly small loading of the guide. When the loading is increased, the curves are progressively deformed, as it appears in Fig. 5.4. It can be shown ([32], Chap. 6,7, 8)(*) that if they correspond to modes which are coupled through the lossless obstacles, the curves do not cross (as the unperturbed curves do at points A, B, C in Fig. 5.5), but they split into two curves, like the two branches of a hyperbola. If the dispersion curves of the unloaded waveguide cross with group velocities of opposite signs, one branch of the dispersion curve of the loaded waveguide has a local maximum and the other has a local minimum. In the stopband between these local extrema, cosh(*yL) takes two different complex conjugate values corresponding to the two coupled modes (see Fig. 5.6). The crossing point is locally a centre of symmetry for the dispersion curves. Loading the waveguide not only provides coupling between modes, but also shifts the frequency of the crossing pomt. If kl and kg (with kl < k2) are the frequencies at the extrema, the dispersion curves are locally represented by . l 2 =j czk+ /c——--kE + b 1 cl/(k—k1)(k—k2) (5.19) where cz, b, c are real coefficients. In the passbands k < kl or k > k2, we have k+ k l 2 ¤t=0, B=a Tk-- we C,/(k-kl)(k—k2) (5.20) In the stopband kl < k < kg we have k+ k l 2 ¤=4=c,/(k—kl)(k2-k) , [sw 7k--— + 1) B of (5.21) is represented by the dotted line in Fig. 5 .6(b). lf 7= ot + jB is a complex propagation constam at a given frequency, so are —Y, Y°` and hence -Y". Indeed, -7 represents a wave travelling in the opposite direction, which (except for the sign) has the same propagation constant when the structure does not contain nonreciprocal media. When losses are neglected, thea solution Helmholtz is alsoequation a solution: and the thereforeyisbound conditions also fora possible the fields propagation are real, constant. so that the complex conjugate of zl Summarizing, if ’y= ot + jB is a propagation constant, so are tu + jB and :l:0t — jB. In the half plane B > 0 they correspond to the two waves (5.21) when there are extrema at kl, kg inside the interval 0 < BL < 1:. In the opposite case, there is only one wave for each direction of propagation; this requires that either ot could not be distinguished from -0t (which means that ot = 0), or that BL could not be distinguished from —BL modulo 21: (which means that BL = n1:): in the first situation we are in a normal passband (without extremum), in the second situation we are in a normal stopband at BL = mc. V`) It should be noted that lc in this reference is our B. 72 OCR Output '/ ki. ¤Q:_,»· T 1 ‘-· '‘'' '"`---•QQ E0]. (21) (b) ii. V Fig. 5.5 Dispersion curves for an Fig. 5.6 Close-up view of crossing points. The unloaded waveguide, folded into the dispersion curves of the unloaded waveguide cross with interval 0 < BL < tz. The figure group velocities of the same sign in (a), of opposite corresponds to a circular waveguide of signs in (b). On the dotted line in (b), cosh(YL) is radius b, with L/b = 4/3. complex. If there is no stopband inside the interval 0 < BL < rc, kl = kg and (5.19) reduces to B=(atc)k-+{@) b (5.22) OCR Output This represents the simple crossing of the dispersion curves of two (uncoupled) modes. 73 curl curl(curl F) + k3(curl F) = 0 or A(curl F) + k3(curl F) = O OCR Outputand A(div F)+k§(a1v F) = 0 (6.7) hence grad div( grad div )F +g kgrad( div F ) = 0 0, A(gmd div F) + k3(grad atv F) = 0 Therefore, if we know a vector solution F of the_ Helmholtz equation (6.6), div F is a solution of the scalar Helmholtz equation (6.7), whereas curl F and grad div F are both solutions of the vector Helmholtz equation (6.6), all with the same eigenvalue kg Suppose now we have an_electric field solution E of the Helmholtz equation subject to the boundary conditions (6.4); then if curl E ¢ 0, it is a solution of the Helmholtz equation which on S satisfies the boundary conditions i°t·curl E = O (6.8) and [E x curl(curl = [F1 >< grad div E] + k3[E >< F] = 0 Indeed, the conditions (6.4) mean that the tangential component of E, and div E vanish along the surface S; this, together with the Helmholtz equation, immediately entails (6.8). Comparing (6.8) with (6.5) we see that curl E satisfies the boundary conditions for H _ If on the other hand div E at O, then grad div E is a solution of the Helmholtz equation which on S satisfies the boundary conditions [E >< grad div E`] = O and div (grad div E) = —k2 div E = 0 (6.9) Both conditions follow from div E = O gn S. Comparing (6.9) with (6.4) we see that grad div E satisfies the same boundary conditions as E. Similarly, if we haye a magnetic iield solution H of the Helmholtz equation subject to the boundary conditions (6.5), curl H is a solution of the Helmholtz equation which on S satisfies the boundary conditions [smut H): 0 and div(curl H) = 0 (6.10) Indeed, the first of these conditions follows from (6.5) while the second one is an identity. Comparing (6.10) with (6.4) we see that curl H satisfies the boundary conditions for E. If on the other hand div H ¢ O, then grad div H is a solution of the Helmholtz equation which on S satisfies the boundary conditions ii-graddiv H=ri-curlcurl H - kg H-H=O (6.11) and [E> The first of these conditions folLows from the Helmholtz equation and from (6.5), which entails that the tangential component of curl H vanishes along_the surface S; the second condition is an identity. Comparing (6.11) with (6.5) we see that grad div H satisfies the same boundary conditions as H . Some properties ofthe eigenvalues kj For two arbitrary vector fields F , G we have 75 OCR Output G-graddiv F = atv(G div F)-atv F·d1v G G·curl curl F = divlcurl F x G]+curl F ~curl G Suppose, to be general, that F is a solution to the inhomogeneous Helmholtz equation AF+k’F=K (612) By taking the dot product of this equation with G and using the above relations we obtain d1v(c`$ atv F+[G> a-G div F+ E·[G> (6.13) _[(div_ F·div_ __G+curl __ F·cur1G)dV+k2jF·G ______dV=_[G·K __ dV If we take K = 0 (then k=2 kg) and G ==t F, either set of boundary conditions (6.4) or (6.5) for F makes the surface integral in (6.13) vanish; so we have 2 ·· 2 . - 2 -· 2 _ kcjlF|dV='[[|d1VFI+|CuI1F| ]dV (6.14) erefore, the eigenvalue kf is a real number 2 O, and the eigenvector F may be taken as real. Moreover, kj = 0 is only possible when div F = 0 and curl F = 0. __ lt; (6.13), F satisfies the inhomogeneous Helrnholz equation (6.12) while G is arbitrary. Let us take G = FB where FB satisfies the homogeneous Helmoltz equation AFB + k§FB = 0 Here kf; represents the eigenvalue which corresponds to FB. Tlhen there is_a_ second relagon similar to (6.13) where K = O, F is replaced by Fg , k2 is replaced by kg = kg, and G is taken as F. Taking the difference with (6.13) yields {pt-Fg div F-a-F atv Fg +n-[Fg mnt ][F-n-Fxcun F§|) as +k-kjF-Fg(§)2 dV=jF§·I? av (6.15) Note in passing that this relation is basic for the computation of the frequency shift due to any wall perturbation in a cavity ([20], p. 414). Now we take F = FA where FA satisfies the homogeneous Helmholtz equation AFA + = O Then in (6.15), F is FA, k2 is ki and K is 0; hence 76 OCR Output {ta-F} div FA -a-FA div“' FB+a-[FBxcur1FA-H-FA][ > +k-k{FA-Fg(§§) at/:0 (6.16) If both FA, FB satisfy the same set of boundary conditions on S, it being either (6.4) or (6.5), the surface integral in (6.16) vanishes; therefore the eigenmodes F A , FB are orthogonal over the cavity volume when A af B. 6 . 2 Classification of modes Starting from a vector solution F of the Helmholtz__equation (6.6) subject to either boundary conditions (6.4) or (6.5), the derived vector fields curl curl F (when curl F ¢ 0) qnd grad div F (when div F ¢ O) are both solutions of the Helmhgltz equation with the same eigenvalue kc ; moreover they both satisfy the same boundary conditions as F. We have to consider four possibilities ([26], p. 121 and p. 1 73 ): 1) curl F¢0, atv F ¢0 From (6.14) kf ¢ 0; then the Helmholtz equation (6.6) may be written as F = 7é—curl curl F —égrad div F (6.17) which shows that F is the gum of two parts Fl and F ;(, each of which satisfies the_Helmholtz equation with the same eigenvalue kc , and also satisfies the same boundary conditions as F. The first part Fl (solenoidal) has zero divergence; the second part F ;_ (irrotational) has zero curl. lg the fields F , G in (6.13) are taken as the solenoidal and the irrotational part of F , since I? = O and k 2= kc 1; O, it follows immediately that these two parts are orthogonal over the cavity volume. Therefore the field F in (6.17) can be considered as the superposition of two orthogonql solutions to the Helmholtz equation, which are degenerate since they correspond to the same eigenvalue kc When the degeneracy is removed (in most cases a slight deformation of the cavity wall is sufficient to achieve such an effect), the solution to the Helmholtz equation splits into a pure solenoidal field and a pure irrotational field having different eigenfrequencies. We are then led to three remaining possibilities. 2) curl F ¢O, div F=O Assume that F satisfies the set of boundary conditions (6.4); we shall then write it as EB (with an ordinary subscript l for solenoidal modes). It hgs been shown in (6.8) that curl E, is a solution of the Helmholtz equation with the same eigenvalue kl , which satisfies the set of boundary conditions (6.5); therefore we can let curl F, = a Ht where a is some scalar constant ¢ 0 (6.18) H, is a solenoidal solution of the Helmholtz equation subject to the boundary conditions (6.5). Again, by (6.10) curl H, is a solenoidal solution of the Helmholtz equation which satisfies the set of boundary conditions (6.4); therefore we anticipate that curl HB = b F! where b is some scalar constant ¢ 0 (6.19) OCR Output From (6.18) and (6.19) we have curl curl Fl = ab Fl 77 With ab = kj ¢ 0 this is precisely the Helmholtz equation (6.6) for E,. The same result is obtained if one fl1'_§[3S§l11'1’lCS that F (= H,) satisfies tqe set of boundary conditions (6.5). Therefore, the solenoidal modes E,, H, have the same eigenvalue kl and are related by (6.18), (6.19) and (6.20). In particular, with a = —jc0!tt and b = -jwle, they correspond to solutions of Maxwell's equations: curl E, = —jt0ltL H, , curl H! = jays E, with togstt = kg >0 . (6.21) Moreover, the volume integrals are linked by (6.14). Taking F = E, we obtain k2] | ..E, 2| at/=|l can.. E,] dvzmipf2 a \2 H, z| dV 4 2 4 2 hence 2} |El { dV= u| |H, I dV (6.22) The same result is obtained by taking F = H,. It simply expresses the equality (2.126) of time averaged electric and magnetic energies in a solenoidal mode. 3) curlF=O, div .F¢O With F satisfying either set of boundary conditions (6.4) or (6.5), Q has been shown in_(6.9) and (6.11) that grad div F satisfies the same set of boundary conditions as F. In fact, for curl F = 0 the Helmholtz equation grad div F + k%F = O where from (6.14) kf ¢ 0 (6.23) shows that F and grad div F are the same vector fields, up to a scalar multiplier (— kf). When F satisfies the set of boundary conditions (6.4), we shall designate it by El (with a Greek subscript X for irrotational modes). Since curl Ek = 0, we may write El = grad ¢)_ Inserting (6.24) into (6.23) yields grad(A¢;_ + k{q>;_ ) = 0 or A Since qa is only determined up to an additive constant and since kf ¢ 0, one can always choose the additive constant such that A¢,_ + k{¢X = 0, k{ ¢ 0. (6.25) The boundary conditions (6.4) require that on S, ¢>;_ = constant and A¢;_ = 0. With (6.25). both conditions are satisfied when ¢;_ = 0 on S (6.26) OCR Output From (6.23) divided by (— k{) and (6.24) it appears that one may take 1 — . ¢;_ = ·E d1V E 78 Equation (6.7) shows that this determination of tp), satisfies (6.25); from (6.4) it is obvious that it also satisfies (6.26). Multiplying (6.25) by ¢,_ and integrating over the cavity volume yields 3 _ , 5 0};-§%ds - j|E,_ |‘4v + ki; [ q),_ [in/:0 (6.27) With (6.26) this confirms that k{ is real and positive. When F satisfies the set of boundary conditions (6.5), we shall designate it by Hx. Again we may write Hx = grad Wx with Aw) + k,fqr,_ = O, k,f ¢ 0 (6.29) The boundary conditions (6.5) require that @:0 on s. (6.30) Bn Again, one may take Wx = -7% div Hx; that it satisfies (6.30) is obvious from (6.5) because (6.23) entails . grad 1 —-div— 7 ki HK = Hx Similarly, from (6.29) we deduce -22 §‘V;~%;?dS “ ll}ix |dV + kiil I \V)t |dV=O (6.31) From (6.30) it again follows that kf is real and positive. Remarks a) Since the boundary conditions (6.26) and (6.30) are d_ifferent__for Ex and FIX, _the eigenvalues kfare in _ general different; therefore there is no relation between Ex and Hx. Physically, El (or H,_) is a static field (co = 0) produced by surface charges (or currents) such that the boundary conditions (6.4) or (6.5) are fulfilled. When ki ¢ 0, div F ¢ 0: the solution of the Helmholtz equation (6.23) also requires the existence of a volume distribution of electric (or magnetic) charges. b) The solutions of the scalar Helmholtz equation (6.25) or (6.29) subject to the boundary conditions (6.26) or (6.30), form a complete orthogonal basis for expanding a scalar function in the cavity volume. In the case of Neumann's boundary condition (6.30), to the eigenvalue k.,_ = 0 there corresponds a non-zero eigenfunction xp), = constant, which must be included in the set of eigenfunctions in order to have it complete. 4) CuIIF=O, divF=O From (6.14) this is the only case where the eigenvalue is zero. Such modes could be considered to be either solenoidal or inotational; but since they show no re ation between E and H type of modes, we shall consider them mainly as irrotational and designate them with a Greek subscript. Because their eigenvalue ki = 0 is the smallest possible, we shall numbt-Er them with X = 0. On the other hand, these modes might equally well be considered as solenoidal with kl = 0. 79 OCR Output From curl F = 0 one can still write (6.24) or (6.28); the condition div F = 0 still takes the fomi of (6.25) or (6.29) where now ki = 0. Types of fields for k,f = 0 When F satisfies the set of botmdary conditions (6.4), one must have $0 = constant $0; on each disconnected part S; of S (6.32) With k{ = 0, (6.25) entails E §@ dg I 0 (6.33) i S; Bn whereas (6.27) yields 2¢5i §?d_g§2 Z I I EO |dV (6.34) i 5, n If the surface S is in one piece, (6.34) together with (6.33) implies E0 = 0. If the surface S is made up of several disconnected parts, $0 can take different $0; values on the different pans, and E0 is the electrostatic field between several unconnected conductors which are raised to different potentials (see Fig. 6.1b). Cut for magnetic scalar potential Bing b) Tubular ’ C) d) conductor Fig. 6.1 The four basic types of cavities: a) simply connected, with a single surface ( E0 = 0, F10 = 0) b) simply connected, with a surface in several parts (FO ¢ 0, 1:10 = 0) c) multiply connected, with a single surface (EO = 0, go ¢ 0) d) multiply connected, with a surface in several parts (E0 ze 0, H0 ¢ 0). When F satisfies the set of boundary cgnditions (6.5),__$0 still has to satisfy (6.30) on S. If $0 is univalued, (6.30) together with (6.31) where kl = 0, implies HO = 0. To have HO 1: 0 requires $0 to be multivalued, which is possible only if a current (and therefore a metallic conductor) makes a loop inside the cavity. By introducing a cut inside the cavity volume, in the form of a surface having the loop and the cavity walls as boundaries, such that it prevents a full turn around the metallic conductor (see Fig. 6.1c), one makes $0 univalued but taking different values $0+ and $0- on both sides of the cut. Since H0 = grad $0 is continuous across the cut, the difference ($0+ — $0.) must be a constant along the cut. With (6.30), the only surface integrals which are left in (6.31) are the integrals on both sides of the cut; therefore -• 2 * * it nh i(v0+—v0-)T ds=(w¤+—¤»0-)i rb as = MHD I dv·<6-35> cuz cut Tn+ flux of the magnetic field through the cut HO is then the D.C. magnetic field produced by D.C. surface currents on the cavity walls, such that ri ·HO = 0 on the walls (which corresponds to superconducting walls). 80 OCR Output 6 . 3 Expansion of the electromagnetic field inside a cavity Once all the solenoidal modes El,}il and the irrotational modes Iil of a cavity have been determined, one may represent the fields in the cavity as E = Zdlgl + Zdxix and H = Zh!]-il + z J. z >. where different E (or I? ) modes are orthogonal over the cavity volume, i.e. {EA-E} at/:0 and yF1A·1?§ dvzo when A it B Here A and B represent any kind of subscripts, Latin or Greek. If A and B modes are both either solenoidal or irrotational, (6.37) follows from (6.16). If one mode is solenoidal and the other is irrotational, (6.37) follows from (6.13) where k2 is taken either as k2 or as kg , provided one of them is different from zero; but (6.13) ensures the orthogonality of a solenoidal and an irrotational mode even when ki = kg :4 O. The fact that irrotational modes are orthogonal to solenoidal modes proves that no one of these sets of modes is complete; both are needed to form a complete basis of vector ftmctions. Since the eigenvalues kj of the Helmholtz equation (6.6) and the boundary conditions (6.4) or (6.5) are real, the eigenvectors E A or H can be taken as real except for some constant complex factor . Therefore the orthogonality relations?6.37) may be written with or without an asterisk; the advantage of writing them with an asterisk is that when A = B, the integrals in (6.37) are real and positive. From (6.37) it follows inunediately that E - F dV F1 -17* dV 48 48 <6-38> EB | dV HB | dV One of the most illuminating ways of deriving the coefficients (6.38) is to use Maxwel1's equations written in a completely symmetrical form with respect to electric and magnetic quantities ([33], p. 191): curl H = i 4 joe E (6.39) curl E = -f,,, · jon 1:1 (6.40) In (6.40) 1,,, is_a fictitious magnetic current density, which ultimately will be made equal to zerg. The electric current J will be expanded by using the same base vectors as E, while the magnetic current J ,,, will be expanded by using the same base vectors as H. In the following, e and it are assumed to be real; then, with (6.21), wl is also real. Expansion ofthe electric current density From (6.39) we have ji-Eidv =j(6ur1 F1- joe E)-Egdv = '|'(div[I:I>< Ef,]+1?-mt E;)dV—jmejE·E;dV ` or yi-E,§dv-§ax-Efnzs=jF1-can[H] Ej§dv—jmejE-Ej§dv (6.41) The left-hand side of (6.41) can be considered as the scalar product of EI, with an electric current volume density im, = i -E[ X ?]1· apt) (6.42) OCR Output 81 where n is the distance from the cavity surface, measured along the normal to the surface; —[E >< H ] would be the electric current surface density if H = O outside the surface (i.e. inside the metal). Because of the boundary conditions (6.4) for E A ,the surface integral in (6.41) vanishes. For the solenoidal modes, (6.41) becomes, using (6.21): i-E}dv=—j6>e|E-Ejdv4jm,p|H-Hjdv. (6.43) For the irrotational modes, (6.41) simply reads f·E,{dv=-jweli-Eid)/. (6.44) Expansion ofthe magnetic current density From (6.40) we have 7,,, -F1f,dv = |(—cur1 E- join)?fj([E?f] J?-1,t1v 4 -d1vX1,- E-curl F1j,dv-)j' jwnH·H§dV or jim-P1f,dv4§aX-1?,§t1s=-jE·6ur1Hjdv-jmnjé-1?f,dv.[ E] (6.45) The left-hand side of (6.45) can be considered as the scalar product of HQ with a magnetic current volume density Tm ,0,,,, = fm +[E] E ><· 5(n) (6.46) where [Ft >< E] would be the magnetic current sgrface density if E = 0 outside the surface (i.e. inside the metal). With the boundary conditions (6.5) for H A ,the surface integral in (6.45) does not vanish. For the solenoidal modes, (6.45) becomes with (6.21): ‘ ° ` jim ·F1§4v 4§a[?] X1:·1?§ds = j(6,e)E - Ejdv - jmpjé -F1}dv. (6.47) For the irrotational modes, (6.45) simply reads ’ ’ )f,,,·F1{dv4§aX-F1,[ds=-jwpm-F1,[dv.[E] (6.48) Solenoidal mode coejficients ag, b; They can be computed from the system of equations (6.43), (6.47); the result is (2k- k)E}) -E_§t1v =j(3)|.1_’·.7‘E;dV 4 jm,t((;i,,,[E] Hjdv Bi)4 ;a X ds(6.49a) (2k- k))F1§ - Hjdv 4 mq} · Ejdv 4 jmqjim izjdv[E]Hf) 4;a X -ds(6.49b) Irrotational mode coejficients ag, b;_ They are given straightforwardly by (6.44) and (6.48): " "'! 1 “‘ "* E- dV=——- J-E dV Ex i at (6.5021) OCR Output jmg 82 " " 1 '•·• lr * -•·• * jH·HdV=—e jJ ·H dV+§ EXE -H dS (6.50b) Using (6.49) and (6.50) yields the ag and bg coefficients through (6.38). When remembering (6.22), the resulting expressions (6.36) for E and H agree with those given by Van Bladel ([33], p. 299). The inptational E,_ modes with X ¢ 0, having div Ex ¢ O, are necessary in order to represent a nop zero S div E = p, which occur_s if there are electric charges in the cavity. Similarly, the irrgtational Hx modes with 7t ¢ 0, having div Hx at 0, are necessary in order to represent a non—zero [J. div H = pm. In fact, although there 5 no volume density pm of magnetic charge inside the cavity, there is a fictitious surface )derisgtyGm p. . = -|.1H · ri associated with the fictitious magnetic current surface density [Fi >< E] ([34], Remark: It is interesting to verify directly that the expressions (6.36) for E` and I·l do satisfy Maxwell‘s equations (6.39) and (6.40). From (6.21): curlH—jc0s-; . E= Z(jc0£ebl —jwea[)E, —jme2aXE;_ With (6.22), (6.38) and (6.43), (6.44) this expression transforms into -• ..* -• -•* curlY II—jtoe E = XE! +ZEk =J insideV. (6.51) I tlaldv >— tletldv This is Maxwell's equation (6.39). Similarly, from (6.21): curl F -1wu F1 = Z(1<¤nwi — J¤>ube)F1e — Jwuibiiii With (6.22), (6.38) and (6.47), (6.48) this expression transforms into * -* * Curl- E_jmuH=ZHl_ - Y ·I?dV+ ”><-fdS , E.7 -l*k#+ZHk-I—ldV+ "><·HdS l m z flE m l;7t f'7. lIHzldV lIHildV = 1,,,[E] +aX·a(n) This is Maxwell's equation (6.40) except for the last temi in (6.52). Since this term is zero inside V, Maxwell`s equation (6.40) is again satisfied inside V; but the 5(n) term is responsible for producing non zero coefficients in its own expansion. Eject of a finite wall impedance In (6.49), the surface integral prevents the coefficients ag, b; from becoming infinite when k2 = kj In both (6.49) and (6.50b), the surface integral can be split as °‘ ;(nX)-HjhzsE = jaX-zas + 5aXE]-1?1§,ds (6.53) OCR Output holes[E]Fjg[walls On the metallic walls of the cavity, we apply the approximate boundary condition (3.24): 1;, = [F]z,1, X nor n [E]X = 2,19,. 83 Since E ·HA = O on S, the last integral in (6.53) becomes, with (6.36): ja><·1,¢1.s = {231:,1-I·f;dS = gb", jzs1?,,,·F1f,ds+ gb, ;zs1?V-f1f,(1s (6.54) wal[E]F§ls walls ’” walls V walls Introducing (6.53), (6.54) in (6.49b) and (6.50b), while remembering (6.38) yields (_ 2•¤ 2_ k —·—k, )blj|Hl|2 dV—)meZb,,, -·jZ,H,,,~H,dS—;ms2bv -·¤ fZ_,H(,-·—·=•= ·H,dS ’” walls V walls °‘ ‘ juni-zji-E}¤1v+j(.)eljf,,,»F1}.1v+ [7]jH> z md mlétldv + $-2% 12.9... —Hi¤S+ #2b. 12.5. aids fw M walls JW v walls =-4ji,,,-H{dv+L[Imp ;n><1.1?1{as[§] holes The set (6.55), (6.56) constitutes an infinite system of linear equations for the bis and the b;,'s. In order to obtain a simple approximate solution to this system, we shall only keep in (6.55) the diagonal tenn m = Z, and in (6.56) the diagonal term v = X., because the diagonal temis are likely to be the most important ones. Then (6.55) reduces to ()_2 k 2-k, -• j|H,| 2 dv- jms-· 2 jzsH,asa,|| iwalls me;}-E§av+j656|jim—H§av + jaX-Fzjds[E] (6.57) holes while (6.56) reduces to · ~ .. j|H,(|l -· dV 2 +e l {zs11las|| Imp wallsb,_ Z -2.Jam ;1,,,-• -•*·H,_av + jn[] X ···•*1;-11,4.5 | . (6.58) holes With (6.22) and the definition (2.127) of Q we have, for a solenoidal mode Z: IR.~(¢¤z) Igzlhds Im I 1 _ wall (659) d l Q¤ it lml dV If we defme the Q factor of an irrotational mode X by the same formula (6.59) with l replaced by 7t, we shall have in both cases, using (3.25): — L . · jz,(¢6) |HA] as = [1 + J sgn(m)], 5R,.(mA) |HA|2 as1+ j sgn(w) — 2 = 7-,/|mA| uj|HA| dV . (6.60) walls to A walls lt should be noticed that even when 0J)( = O, the ratio (/|0J;(| / Q2 which appears in (6.60) is still perfectly defined by (6.59). Therefore (6.57) and (6.58) becomes, with (6.38): 84 OCR Output k2 - k? + k2l?@‘-*2&|F»F‘,/|lQ! °° ;11}w = jmlqf- Efdv + joel 53,,,-F1§4v + 5a><·H§ds[E] | (6.61) holes ‘ and 1+[}l@-@,}l&\ l QW w )F1.F1{av :L{Imp -.,;1,,.n;dv + [E]ja>< -Bids]. (6.62) holes gand ( ignally,. 1): the solenoidal mode coefficients for the electric field are readily obtained by combining (6.43) k2..k;+k2 jgggdv Q1 <¤ = jc0ttl+[,/I, } Q! EC]·E (D ];dV+jcoltt |j]m + _[E><·IY;dS|[E] (6.63) holes whereas the irrotational mode coefficients for the electric field are still given directly by (6.44): j1§-Egdvz-J-ji-1§;dv (6.64) Remark: From the derivation above, it is clear that the expressions (6.61) to (6.63) are only a first approximation; but it is very difficult to go beyond this approximation when taking the wall losses into account. This approximation is essentially valid in the vicinity of each solenoidal mode co;. 6 .4 Excitation of cavities A cavity can be excited: a) by an electric current inside the cavity. The current may be carried by a probe, a loop, or it may be a convection current. b) by a tangential E on the surface of the holes (which is equivalent to a surface magnetic current). __ Using (6.61) to (6.64) in (6.36) and (6.38), we can write the expressions for the phasors E (0)) and H (co): - - -.. jwu l+——— 1_j Sgn(°)),M&| - -. Ev [J·EVdV -· Q,) 0) [J-E,,dV · E(<¤)=·Z·T·T··ZE¤%€1 . () m..r V rIE»l dv ~ ,.2-,.2,/pl) 1+.+ Q7! Sew(D tlal W j ri X-1?,fds+’ {fm ·I?;dV Jmu [E]holes - ZE'! (665) OCR Output » ,,2_ k2,+i—)n{QH sgntwl 0) ' h (tr,) dV 85 ;ax[E] ·I?I;dS+[.7,,, -Iii; dV mm) = _; holes h V -I sgml _ lIHv|‘ jax-1?,jk1s+j7,,,[E] H; dV - ZH, ho! (6.66) _ n k2_k2 1+1-J Sg¤(¤>) j|H,,|_ 2 dv Q,. cc In these expressions we have used the equality (6.22) between electric and magnetic stored energies for solenoidal modes. The rigorous expansion coefficients (6.49), (6.50) would yield expressions similar to (6.65), (6.66), except that all brackets containing Q-factors would be replaced by 1 (there would be no terms contammg Q factors), and that the surface integrals, instead of being restricted to the holes, would be extended over the whole surface S enclosing the cavity. Excitation of cavities by holes has been treated by Kurokawa [26, 34]. In what follows we shall only consider excitation by electric currents. The terms corresponding to current excitation m (6.65) and (6.66) are essentially the same as given by Collin ([8], p. 359) and Van Bladel ([33], p. 299). The need for the irrotational mode terms has been emphasized many times in the literature. Their meaning in the particular case of a rectangular cavity has been discussed by Schelkunoff [35]; it has also been shown by measurements that they are necessary for representing the fields in a rectangular cavity excited by a probe, especially at frequencies below the first resonance of the cavity [36]. Frequency behaviour of the resonant terms in (6.65) The last terms in (6.65) behave like 1-1 sg¤<<¤> ]wu{1+ Q7! (0 N _jSgI1(m)9-J - ,fli`] ...1-..* Qn bC0 jmu+Qn I- J $g¤(€°)(D +]C0€ (6.67) + BE [,6+@,/[6. 1+ j s to) ,,])Qkk k .—-6,k ~ The square bracket in the last expression represents the admittance of a parallel resonant circuit whose two branches are a capacitance 8/k,., and an inductance |.t/k,, having an intemal impedance 1+ ° oo rt \(D mnl ll { kn which is proportional to the surface impedance ZS(c0) of the cavity walls. 86 OCR Output In (6.67), the 1-1 sg¤(<¤) Q,. cn term is a small correction which is only important in the vicinity of the resonant frequency wn; therefore, we shall replace it by [1-j sg11(0J)]/Q,.. The inverse of (6.67) then becomes +]me= +jws=-i'%—;+jme _ _[) jmul +QH 1-1 Sg¤(v>)jmu1+;l_{](] QH 1 sg¤(w)jmu1_QH 1(] Sg¤(¤>) Qn ,2 ·2 kn k,, =————++———+)0>e (6.68) I<¤|uQ,. wu ·2 » 2 k‘ " where kn = ton att = l 1+ — and con is the nm resonant frequency of the cavity, as corrected for the finite skin depth. Again, in (6.68), the real part k,,"/(lt,o|ttQ,,) is only important when to is close to con; therefore we shall replace it by its value for 0) = con. By using (6.69), the inverse of (6.67) is transformed approximately into to e ‘ ~ oe »¤ ,.,Qn .--,,.,mm a m.,., ,, Qn Jo) f l 9211 Qn |(Dn1 (0 Qn where we have put tan ¢ = QnA coIQTI _ E0* (6.70) Finally (6.67) is approximated by ‘“ will +*%§~@J|%l] . " —· Q" (6.71) _ 2 2 1-1 Sg¤(<¤) ¤>,, Hf mn 9 I<¤,.I¤ _ k,, k 1+ —————— Q,. w Now, by definition, W _ P,. Q7! Wil where Pn and Wn represent the power dissipated in the walls, and the energy stored in the cavity for the n solenoidal mode. From (6.22), 1 - 2 Wn Z 2 Welecuic = Z8 dV With (6.71) the nth resonant term in (6.65) may thus be written as 87 OCR Output ·•·•·•* Em) : - . (672) 1+ J tan ¢ 2P,, 6 .5 Beam coupling impedance ZH and Z _L The cavity is excited by the convection current of a charged particle beam; the particles with velocity 1) traverse the cavity on the z-axis. The beam current is then a function of (t- z/1) ) which is supposed to be modulated at frequency 0); therefore the physical current is -J-Z Re \I(0)·ecfil “ b= Re eJ°°‘·Ib(O)e " to which there corresponds a complex amplitude Ib = Ib(0)e""’ where h (6.73) If to is close to a resonant frequency oa"' of the cavity, it is sufficient to keep only a single resonant term of the type (6.72). Longitudinal coupling impedance If a particle passes the plane z = O at time to, it will pass the plane z at time t = 1,, + z/u (neglecting the change in 0 during the traversal of the cavity). The real voltage gain experienced by the particle is RejEZ jco(t,,e dz =Re[e_[Ez +£) . .h ° el]°°’ zdz Therefore the complex amplitude of the ejfective accelerating voltage reads +¤¤ .h iq, = 512, ef zdz (6.74) Since this formula takes into account the finite velocity of the particles, it automatically includes the transit time factor. Strictly speaking, in (6.74) EZ depends on the transverse coordinates of the particle, which in general vary during the cavity traversal. For the sake of simplicity, in the case of ultrarelativistrc particles such a variation is neglected, and the integral (6.74) is taken at fixed transverse coordinates. ln fact, the integral (6.74) is the Fourier transfomi of EZ with respect to z, for a propagation constant Y = jh. From the Helmholtz equation (3.5) for the Fourier transform of Ez, its transverse variation is given by terms .I,,,(k,;r) cos mo similar to (3.87), with kc defined as in (3.5); that is to o kg -"22 2-**2 2 k %7]7o --7---7 c B Y where B, Y here represent the relativistic factors for a particle with velocity *0. It follows that li|(r,cp)§m = (2B7)m m=O ·I[VC", cos mcp + Vsm sin mtp] (6.75) OCR Output 88 where Vcm, Vsm are independent of r, liI(¤) = W10 = 0) · [O kr f-) BY (676) In this case of rotational symmetry, using (6.75), for any m VH (r,q>) can be deduced from its value at a single r > 0. This is extremely useful in numerical computations, when the cavity has a gap m a beam pipe of inner radius as it is then sufficient to compute (6.74) at r = a, where E, vanishes everywhere except in the gap. Using (6.73), it should be noticed that (6.74) can also be written as W = %`“lEz]g dz (677) M0) With (6.73) and (6.74), the driving term due to the beam reads ` I i · E§,dv = 1,,(0)j EQ, e‘f"’dz = Ib(0) - iq.; (6.78) By definition, the longitudinal impedance mn by the beam is Z, [S2] = ..@ ' (6.79) nw) where li", is tl1e effective accelerating voltage induced by the beam itself. Using (6.72), (6.74) and (6.78) it becomes 1 I {En, ejnzdz Z =———— " 1+ j tan ¢ 2P,, Remembering (6.70) this may be rewritten as to 1 + JQ,. — 2"’~(D 4 || (680) where IE ejdz .V "Z2 I hz'2| |In Run[Q]=I =| 2P,, 2P,, Ru is the longitudinal shunt resistance of the cavity at its nih resonant frequency; it is usually taken on the _cavity axis. For the beam, in the vicinity of a resonance, the cavity behaves as a parallel resonant circuit. Remarks: 1) In (6.80), the shunt resistance Ru is defined in terms of the power loss P in the cavity wallsf0r a _given level I VII I of the effective accelerating voltage on the cavity axis. This defmition holds irrespective of 89 OCR Output RF source; in (6.80) the RF source is supposed to be the beam, but the satne definition applies when the RF source is an extemal generator. 2) The shunt resistance Ru is defined in (6.80) by the same formula = Q [q' 2P which is used in the theory of electric circuits. Although this definition was also used for the first proton linac [37], the builders of the early electron linacs in Stanford [38] defmed the shunt resistance as N l' II g IIIHCu R P Since they have been widely followed in the Westem world (but not in the USSR), one should never forget that (linac R") = 2 (circuit R") Transverse coupling impedance In (3.14), if we use B = 0/c instead of tt in the integral giving the transverse momentum gained by a unit charge when traversing a cavity, this integral acquires the dimensions of a voltage, which 1S called the ejfeczi ve deflecting voltage produced by the cavity: —· Vi. =-·· I FJ.dz cjThz The integral is supposed to be taken in the beam pipe, from a region upstream to a region downstream of the cavity, at distances far enough from the cavity for the electromagnetic fields to be negligible. Then from the Panofsky—Wenzel theorem (3.14) we have V] = —-#_[g1·ad_,_Ez·ejhzdz In general grad LE, depends on the transverse coordinates of the particle, which vary during the cavity traversal. As in (6.74), such a variation is again neglected, and the integral (6.81) is taken at fixed transverse coordinates. In this case (6.81) and (6.74) are related by: V] =—%gradJ_(\4|) (6.82) For a mode deflecting in the x-direction, E, is zero on the z—axis and reverses sign with x; moreover it is approximately independent of y. Its Taylor expansion reads BE , . Ez = x + higher order terms (6.83) 6x ,, For a beam close to the z-axis, (6.81) then becomes Vx = —-L ejzdz htaken on z - axis (6.84) jk dx 0 If the beam traverses the cavity at a fixed distance xo off-axis, with (6.73) and (6.83) the driving term reads 90 OCR Output { i -1§,[t1v" = 1,,(0)x0 { %``jh ezdz (6.85) 0 By definition, the transverse coupling impedance seen by the beam is sznrl = ji- [ ] Ib(0)x0 (6.86) where Ib(0)x,, is the dipole moment of the beam current. Using (6.72), (6.84) and (6.85) it becomes , l 1 IaxoGEM` jh; 2 ———-]% e dz .2 1 Zx= _k1+jtanq> 2P,l k 1+]tan¢ 2P,. Remembering (6.70) this may be rewritten as - k ' R elm 11=¢?‘”‘* to 0) . 1 + )Q,, ? 1°~°° 1 i l‘l (6.87) where }$njgradLE,,_,| ejhzdzl I I2 R Q = Ml 1 O I ·· lug 2P,, 2P,, R L is the transverse shunt resistance of the cavity at its nm resonant frequency; it is taken on the cavity axis. At resonance the transverse impedance (6.87) is real and positive; it is in order to achieve this result that the factor j appears in the defmition (6.86). Moreover, at resonance (6.87) yields the important relation Z_]_S2m'=klm`-R_L[Q][1];[1] where kn =:2' (6.88) Some authors use (6.81) without k in the denominator of VL; then, in (6.87) one uses Rl,,om·=[2] ;’[2] k,m·- RM[Q] Remarks 1) In (6.87) the shunt resistance R L is defined ir1 terms of the power loss P in the cavity walls for a given level IV _L| of the deflecting voltage on the cavity axis. Again this definition holds irrespective of the RF source;in (6.87) the RF source is supposed to be the beam, but the same definition applies when the RF _ source is an extemal generator. In fact, the latter definition has been used as a figure of merit for RF separators of high energy particles ([19]. p. 268). The same shunt resistance R L appears in the theory of beam breakup in electron linacs. 91 OCR Output 2) As for the longitudinal shunt resistance, there is a "linac" definition of R _L as V "linac" R. = L! instead of the "circuit" dehnition used in (6.87). Therefore one should not forget that (linac RL) = 2(circuit RL) Finally, it is worthwhile noticing that when used in a detlecting mode, a cavity constitutes an excellent transverse pick·up for the beam; in that case R L should not be too large in order to prevent beam instabilities. Example: Transverse impedance due 20 a gap in a circular vacuum chamber 90 9 - . Fig. 6.2 A gap g in a circular chamber of radius a An off-axis beam at polar coordinates (ro, rpc) excites all modes of (6.75). We now consider only the dipole mode m = 1: W|(r, and MK2?) W|(a) = Vgap u 2 is the amplitude (with respect to cp) of the effective voltage induced by the beam at r = a; the gap induced voltage Vgap can in principle be computed once the surroundings at r > a are known. The transverse deilecting voltage is then obtained by applying (6.82) to (6.89): . V = -.. V · V(a) I (10)+1 (xr) Li. I - Z -L. J.__;.. 1 8 \6|(a) Kd l1(i For ultrarelativistic particles, By >> 1 and wa = ka/(By) << 1; (6.90) then reduces to V a V a . 1%-% ¤<>s( 92 hence —— -· V V_L=V,1,+V(1=—-—%p¢ 1,,, (6.91) From (6.86) we obtain -· - 1 V (cz) -· zl nmka 1 =n.<¤)r. -- A-- 1 el ] ’ 6.92 ( ) where the ratio Vu(a)/[I;,(0)r0] is independent of the beam dipole moment I),(O)?Q,. It appears that the transverse kick experienced by a unit charge is in the direction of the beam offset Q. Multipolar modes The electric field (6.83) corresponds to m = 1 in (6.75). For m > l, from (6.74) and (6.75) the effective voltage Vun in the nm mode has the form lim = [Em ejhzdz = Kn r'” ?:;(mcp) + higher order terms (6.93) where cos mq), sin mq) correspond to the two different polaiizations of the same mode. For a beam passing at (ro, (po) the driving term reads, with (6.78): jj - E`;dV = Ib(O)-| WI; at the beam positionl = Ib(O) · K; rf ?;;(m Summing up the conuibutions of the two mode polarizations we obtain Wm].? · E';dV = Ib(O) |K,,|2 rmrg" [cos(mq>)cos(mq)0)+sin(m Finally, with (6.72) the effective accelerating voltage induced by the beam reads Ei V =———;—-— I (O) rmrm cos m((p—(p0) "" 1+) mq) 2P" " (6.96) ,. By analogy wtih (6.79), the longitudinal coupling impedance is defmed as ~ - Vub “ Q L 2m = [ 1 M0) r"'&C"¢¤Sm(9— q,[0J:2"*]=-f-- @ Ll m = 0, 1, 2, 1+ j tan qa 2P, (6.98) , where Kn is defined by (6.93). This formula generalizes (6.80) to any m. A longitudinal impedance expressed in Q can be defmed as z,,[:2L‘2”‘] Zum} = —T6? (6.99) OCR Output 93 The transverse detlecting voltage is obtained by applying (6.82) to (6.96): .. K Vu-. = g .. .. _ 1b(0) mfmqfgl [1,008 m((p—(p0)· 1¢S1I1)7l((p—(p())I n [lm cos (m - l)( The vector inside the square bracket is simple only when m = 0 (it then reduces to 1, )and when m = 1 (it then reduces to 1,,,. By analogy with (6.86), the transverse coupling impedance is defined as (6.101) Z :11:2***+* = ·-lLL—@mr = 1,2, J ] J (0) ""1rJ” m With (6.100) and (6.98) it reads Z_LQL`"'+1=[2] Z"QL'"‘1,i[2] cos{ rn(q>— A transverse impedance expressed ir1 Q can be defined as ELiE‘° -2m+l §“ Z [g]=;._ -1- |,2m"1 (6103) I With (6.99) the relation (6.102) then becomes @[9] = %Z,,[£2] - [Leos m(q>— (po)- lwsin m(q>— 94 OCR Output[26] K. Kurokawa, An introduction to the theory of microwave circuits, Academic Press, New York, 1969. [27] S.A. Schelkunoff: Representation of impedance functions in terms of resonant frequencies, Proc. Inst. Radio Engrs., Vol. 32 (1944). P. 83-90. [28] L. Lewin, Theory of waveguides, Newnes-Butterworths, London, 1975. [29] G. D6me, Review and survey of accelerating structures, in Linear Accelerators (edited by P.M. Lapostolle and A.L. Septier), North-Holland, Amsterdam, 1970. [30] G. D6me, The SPS acceleration system. Travelling wave drift tube Structure for the CERN SPS, 1976 Proton Linear Accelerator Conference, Chalk River, Canada, AECL-5677 (Nov. 1976), p. 138-147; or CERN-SPS/ARF/77-11 (1977). [31] 1.A. Aleksandrov, V.A. Vagin, and V.I. Kotov, Waves with complex propagation constants in an iris loaded waveguide, Soviet Physics-Technical Physics, Vol. 11 (1967), p. 1486-1490. [32] J.R. Pierce, Almost all about waves, The MIT Press, Cambridge, Mass., 1974; also J.R. Pierce and P.K. Tien, Coupling of modes in helixes, Proc. IRE, Vol. 42 (1954), p. 1389-1396. [33] J. Van Bladel, Electromagnetic fields, Revised printing, Hemisphere Publishing Corporation,Washington, 1985. [34] K. Kurokawa, The expansions of electromagnetic fields in cavities, IRE Trans. Microwave Theory Tech., Vol. MTI`-6 (1958), p. 178-187. [35] S.A. Schelkunoff, On representation of electromagnetic fields in cavities in terms of natural modes of oscillation, J. Appl. Phys., Vol. 26 (1955), p. 1231-1234. [36] H.A. Méndez, On the theory of low frequency excitation of cavity resonators, [EEE Trans. Microwave Theory Tech., Vol. M'I'I`-18 (1970), p. 444-448. [37] L.W. Alvarez, H. Bradner, J.V. Franck, H. Gordon, J.D. Gow, L.C. Marshall, F. Oppenheimer, W.K.H. Panofsky, C. Richman, J.R. Woodyard, Berkeley proton linear accelerator, Rev. Sci. Instr. Vol. 26 (1955), p. 111-133. See equation (6), p. 115. [38] M. Chodorow, E.L. Ginzton, W.W. Hansen, R.L. Kyhl, R.B. Neal, W.K.H. Panofsky, Stanford High-Energy Linear Electron Accelerator (Mark HI), Rev. Sci. Instr., Vol. 26 (1955). P. 134-204. See foomote 5, p. 138. 96—