SYMMETRICAL WAVEGUIDE
JUNCTIONS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT. OP GEZAG VAN DE RECTOR MAG-NIFICUS. Dr O. BOTTEMA. HOOGLERAAR IN DE AFDELING DER ALGEMENE WETEN-SCHAPPEN, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP W O E N S D A G 27FEBRUAR1 1952, DES NAMIDDAGS TE 4 UUR
DOOR
ANTON EDUARD PANNENBORG
NATUURKUNDIG INGENIEURGEBOREN TE s-GRAVENHAGE
Dit proefschrifl is fjoedgekeurd door de promotor Prof. Dr R. Kronig
page
INTRODUCTION 7 Chap. I. FUNDAMENTAL THEOREMS 9
I. 1. General remarks 9 I. 2. Fields in waveguides 10 I. 3. Scattering matrix 11 I. 4. Shift in position of terminal reference planes 12
I. 5. Symmetry of the scattering matrix 13
I. 6. Lo.ssless junctions 14 I. 7. Frequency dependence of a unitary scattering matrix . 15
Chap. 11. RESONATORS 18 I I . 1. Quality factor 18 I I . 2. Fundamental considerations 10
11. 3. Resonators with one output lead 21 II. 4. Resonators with two output leads 26 II. 5. Losses in the output circuit 30 Chap. I I I . SYMMETRY ANALYSIS OF WAVEGUIDE
JUNCTIONS 32 I I I . 1. Representation of a symmetry group 32
I I I . 2. Relation between the symmetry of a junction and its
scattering matrix 34 I I I . 3. Field distribution in a symmetry plane 38
Chap. IV. JUNCTIONS OF FOUR WAVEGUIDES W I T H
TWO-FOLD PLANAR SYMMETRY 40
IV. 1. The general case 40 IV. 2. The nature of the solution 44
IV. 3. The degenerate case 44 IV. 4. The Mave matrix 47 IV. 5. Shift of the reference planes 50
Chap. V. DIRECTIONAL COUPLERS 51 V. 1. Condition for perfect directivity 51
V. 2. Bethe-hole coupler 52 V. 3. Systems of identical junctions connected in cascade . 53
V. 4. Frequency dependence of directional couplers . . . . 60
V. 5. Variable attenuator 62 V. 6. Standard matching transformer 65
Chap. VI. RESONANT SLOTS 71
VI 1. Single slots 71 VI. 2. Measurements on single slots 76
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INTRODUCTION
The physical configuration of microwave circuits is in all but the simplest cases too intricate to enable one to solve Maxwell's equations subject to the boundary conditions imposed. For practical purposes, however, it is not necessary to know explicitly the components of the electromagnetic field at every point of the interior of the structure. It often suffices to know how the structure affects the field in the leads connecting it to the outside. This approach to the behaviour of microwave circuits, sometimes called transducer theory, has been made up till now in the majority of cases with the aid of equivalent circuits. It has proved itself to be a powerful tool for the analysis of waveguide junctions. The physical insight so obtained is very illuminating, especially for the radio engineer accustomed to low-frequency lumped-circuit concepts.
On the other hand for structures with more than two or three output leads the equivalent circuit merely serves to provide a well-defined framework in which numerical values obtained from experiments can be inserted. In these more intricate cases the value of the representation as a guide to better understanding seems at least doubtful; anyway much of the charm of the method is lost. The way out of this difficulty is found in an alter-native method, where the description of the field in the output leads of a junction is made by specifying not the ratios of electric and magnetic components, but rather the ratios of amplitudes of travelUng waves incident on and emergent from the junction. In analogy with other branch-es of physics the matrix exprbranch-essing the relation between thbranch-ese wave am-plitudes is called the scattering matrix of the junction. Up till now the scat-tering matrix has been used to a limited extent only and — for the reasons stated above — especially in the case of junctions with more than two output leads.
Apart from the general theorems supplied to transducer theory by Maxwell's equations, a source of information is often available in the form of the symmetries pertaining to the structure under consideration. In very simple cases the consequences of symmetry can be found by in-spection. For more intricate situations a formalism has been developed by Dicke ^); it introduces symmetry operators, subject to which the solu-tion of Maxwell's equasolu-tions must remain invariant. In this way it sometimes becomes possible to construct a number of special solutions, the most general solution then being a linear combination of these. The number of parameters needed to specify the behaviour of a junction can thus be con-siderably reduced. For the symmetrical structures with many output leads, where the formalism is most useful, the scattering matrix is preferable
to impedance or admittance representations. Therefore the symmetry formalism usually is applied to the former method.
As stated above, the scattering matrix has found little application for simple circuits. Especially in the case of resonators the impedance or ad-mittance description has been used almost exclusively. This is quite logical from a historical point of view as work in the microwave range originates from a gradual extension of techniques well known for longer wavelengths. In this way it was natural to transfer the famihar concept of an LCR circuit to microwave resonators. That this procedure is quite correct was proved subsequently by several authors, e.g. by Slater ^).
On the other hand Tomonaga ^) has shown how the scattering matrix can be used equally well for the description of the behaviour of resonators. The results are, of course, identical with those from calculations on an impedance basis and whilst to the radio engineer the simple LCR repre-sentation for a resonator is very attractive, the scattering description should appeal for analogous reasons to the physicist. Furthermore the latter method seems to involve slightly less specific assumptions about the nature of the resonant structure.
In Chap. 1 a short recapitulation is given of the fundamental properties of the scattering matrix. This seems justified since elsewhere in literature'),*) the theorems are derived, using almost invariably the impedance matrix as an intermediate station. Chapter II contains a slight extension of Tomo-naga's theory together with some examples demonstrating the practical usefulness of this theory. Chapter 111 gives an outline of the symtaetry formalism. In Chap. IV the general properties of structures with four output leads having a high degree of symmetry are derived. In Chap. V these results are applied to waveguide systems with special reference to directional couplers, while finally Chap. VI deals with analogous systems incorporating resonant elements.
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CHAPTER I. FUNDAMENTAL THEOREMS
I. 1. General remarksAs in the following pages we shall be concerned with certain applications of transducer theory to microwave structures and as these investigations will be carried out with the aid of the scattering matrix, it seems proper to start with the derivation of some fundamental properties of this matrix. Because extensive demonstrations constituting the rigorous foundation of transducer theory can be found elsewhere'),*), we will confine ourselves to a recapitulation of those proofs that are of direct importance for the contents of the following chapters. In this connection it is supposed that the reader is familiar with waveguide theory.
Before embarking on the subject we will state some conventions and restrictions that will be adhered to throughout.
(i) The time factor is understood to be e""'. Unless specifically stated otherwise m is real and the fields vary harmonically with time. (ii) All structures considered are linear and passive. Anisotropic media
are allowed with the limitation that the tensors describing their permittivity, permeability and conductivity must be symmetrical. (iii) It is possible to surround a structure by a closed surface of finite dimensions on which all field quantities are zero except where this surface cuts the output leads.
(iv) The leads, in which the terminal fields relating to a structure are defined, consist of ideal (i.e. lossless and cylindrical) waveguides, not containing anisotropic materials.
(v) In each output lead power is transported by one mode only.
(vi) Terminal surfaces are drawn at sufficient distance from any discon-tinuity so that the effect of non-propagating modes can be ignored. (vii) Rationalized MKS(Giorgi) units are used.
Some points need some comment. Condition (iii) merely requires that the structure and its output leads are effectively shielded. Point (v) con-stitutes no important restriction. In fact, because of the orthogonality properties of the characteristic solutions (called modes) of Maxwell's equa-tions for an ideal waveguide, a system of two modes in one waveguide is within the formalism of transducer theory fully equivalent to a system of two modes each in a different waveguide. Point (vi), finally, is only of significance when the interconnection between two transducers is consider-ed. It states that the connecting leads should be sufficiently long to prevent non-propagating modes excited in the interior of one transducer from reaching any discontinuity in the interior of the other transducer. If this condition is not fulfilled, the behaviour of the combination of two structures cannot be accurately predicted from the transducer data of the separate
structures. Indeed, it should always be remembered that transducer theory combined with the necessary experiments gives for most applications suf-ficient, but never complete information on the repartition of the electro-magnetic field.
As a consequence of points (i), (ii), and (vii) Maxwell's equations have the form
curl H = (— iojE + (T) E , i
. [ (1) curl E =: ifo/yH , N
the permittivity e, the permeability /( and the conductivity a being inde-pendent of time and of the field vectors.
I. 2. Fields in waveguides
We next recall without proof some results from waveguide theory as far as they are needed in later sections.
Let the z-axis be parallel to the axis of the waveguide. The general ex-pressions for the electric and magnetic fields of a waveguide mode can be written as follows ^):
H = H,(a e-'f' - b e'"') + kH.{a e"'"' + b e'"'). S ^ ' Here E^ and Hz are the longitudinal components; k represents the unit vector in the positive z-direction. The transverse components Ej and Hj are vector functions (derivable from Ez and Hz) lying in a plane perpendi-cular to the z-axis; a and b are amplitude coefficients for the waves in the negative and the positive z-direction respectively. The propagation con-stant jH is real for propagating modes. As the z-dependenee has been written explicitly in eqs (2), the quantities Ez, Hz, E( and H( are functions of the transverse coordinates only.
Equations (2) are valid for all three classes of modes: transverse electro-magnetic or TEM(i?2 = 0, Hz = 0), transverse electric or TE(£2 = 0) and transverse magnetic or TM(Hj = 0).
The transverse electric and magnetic fields are proportional to each other; the relation is
E, = Z„ (k X H,) , (3) where ZQ is a constant which is real for propagating modes.
If a section of waveguide for which the field components are given by eqs (2) is fed by a generator sending power in the negative z-direction, where the guide is terminated by a passive load, it is natural to define a reflection coefficient as the ratio of the amplitudes of the waves incident on and reflected by the load. As the transverse electric field is most easuy
— 11 —
accessible to measurement, we base the definition on this component; the reflection coefficient / " t h u s derived from eqs (2) is
ae '^ a
To determine the expression for power flow, first the z-component of the time average Poynting vector S is formed, viz.
S, = i k(E X H* + E* X H ) . (5)
The asterisk denotes the complex conjugate. Substituting eqs (2) and (3) in eq. (5) it is clear that only the transverse field components contribute
S. = ' ƒ £ \\l>\' - |«I1 = i Z, |if,p )|6p - |a|^(. (6)
The power flow P in the positive z-direction is obtained by integrating
Sz over the cross section Q of the guide. But first we introduce the
norma-lization condition. The expression for power flow is brought into its most simple form by putting
^ /" | £ , p dü = Z„ / \H,\^ di3 == 1. (7) With this condition P becomes
p = iH-H"s- (8)
When applying eq. (8) to the setup with generator and load as used for deriving eq. (4) it is to be noted that it represents the power flow from the load towards the generator.
I. 3. Scattering matrix
Consider a waveguide junction subject to the conditions stated in Sec. 1. In each of the N output leads a coordinate system is introduced, the z-axis being parallel to the axis of the waveguide and pointing outward from the junction. The electromagnetic field in each lead can be written in the form of eqs (2), indicating the presence in output lead (n) of an incident wave of amplitude a„ and an outgoing wave of amplitude b,i. The linearity of Maxwell's equations together with condition (ii). Sec. 1, requires that the amplitudes 6„ of the outgoing waves are linear functions of the amplitudes a„ of the incident waves. Grouping the quantities a,i and b„ together in column matrices A and B respectivelv, this linear relation can be formally represented by
the elements of the scattering matrix S being independent of the amplitudes, a„ and t„ and of time.
The physical meaning of the elements of S is closely related to the re-flection coefficient F introduced in eq. (4). In fact, applying eq. (9) to the case of a structure with only one lead, i.e. a waveguide terminated by a load, the single remaining element of S is seen to be identical with r{z = 0). In the general case with N output leads the diagonal elements hf the square matrix S of order N relate outgoing waves to waves incident in the same lead; they are of the same nature as a reflection coefficient. The off-diagonal elements, relating the outgoing wave in one lead to waves incident in other leads, may be called transmission coefficients.
If we put all a„ equal to zero except a„,, which situation is in practice achieved by connecting a generator to lead (m), all other leads being ter-minated by a matched load, and choose a^ equal to 1, the elements of the m'th column of S give the amplitudes of the outgoing waves.
I. 4. Shift in position of terminal reference planes
Just as the phase angle of the reflection coefficient 7' defined by eq. (4) depends on the specific position along the lead, the phase angles of the elements of the scattering matrix S also vary with the choice of the origins of the respective z-axes. It is in this connection useful to study the trans-formation of S caused by a change of the origin of any z„-axis. Referring to fig. 1, let the z^-axis be shifted over a distance /„ with respect to the z„-axis away from the junction.
The appropriate phase shift is
<Pn = f^n In • (10) Denoting with a prime the quantities relating to the new origin, we have
«;, = a „ e - " " , bn = b„e""', (11) or in matrix notation
A' = 0-^A, B = 0 B . (12) 0 is a diagonal matrix with elements
^un = e'"", *„m = o(n+m). (13)
\
1
1 - • — 3 n — • 6 / > 1 1 ' « ' " » ' 2n , I ^n , 69649— 13 —
Solving eqs (12) for A a n d B a n d inserting t h e r e s u l t in eq. (9) we o b t a i n
0-'B' = S0A', (14) '*'• JB' = (PS0A'. (15) H e n c e it is seen t h a t t h e s c a t t e r i n g m a t r i x S', r e l a t e d t o t h e new origins,
is c o n n e c t e d t o S by t h e simple r e l a t i o n
S'=0S0. (16)
I. 5 . S y m m e t r y of the scattering m a t r i x
T h e well-known principle of reciprocity in e l e c t r o d y n a m i c s has a direct b e a r i n g o n t h e s c a t t e r i n g m a t r i x . I n fact i t c a n b e d e m o n s t r a t e d t h a t if reciprocity holds, as is g u a r a n t e e d in our case b y t h e conditions set o u t in Sec. 1, t h e scattering m a t r i x is s y m m e t r i c a l .
T h e proof r u n s as follows. Let us consider t h e v e c t o r i d e n t i t y
d i v ( E i X Ha — E^ X Hi) = H2 c u r l E i - E ^ curl H,^—H^ c u r l E g + E g curl H j . (17) H e r e t h e subscripts 1 a n d 2 refer to t w o solutions of Maxwell's e q u a t i o n s for t h e same j u n c t i o n a n d t h e same frequency b u t for different t e r m i n a l conditions. E l i m i n a t i n g t h e curls w i t h t h e aid of eqs (1) we find t h e right-h a n d side of eq. (17) t o be identically zero. H e n c e
d i v ( E i X H 2 - E 2 X H i ) = 0 . (18) E q u a t i o n (18) is now i n t e g r a t e d over t h e v o l u m e V of t h e j u n c t i o n a n d
t h i s i n t e g r a l is t r a n s f o r m e d i n t o a n i n t e g r a l over t h e surface F enclosing t h e j u n c t i o n a n d c u t t i n g t h e o u t p u t leads p e r p e n d i c u l a r l y to their respec-t i v e z-axes. This gives
ƒ div (Ej X Ha - Ea X H , ) d U = ƒ (E^ X H., - E^ X H^) d F = 0 . (19)
V F
d F is n o r m a l t o t h e surface F. As t h e j u n c t i o n h a s b e e n assumed t o be perfectly shielded, t h e only c o n t r i b u t i o n s different from zero in eq. (19) come from those p o r t i o n s i3„ of t h e surface F, where i t c u t s t h e o u t p u t l e a d s . Therefore
f (Ej X Ha - Eg X H i ) d F = S f (Ej X H^ - E^ X H^) k„ d.Q„ = 0 . (20) F " dn
W e n o w identify solution 1 w i t h t h e case, t h a t t h e r e is an incident w a v e in lead (1) only.
Solution 1: o„ = 0 n =1= 1 , ; K == S„i a^ . ^ I n t h e same w a y
S u b s t i t u t i n g eqs (21) a n d (22) t o g e t h e r w i t h eqs (2), (3) a n d (7) into eq. (20) we find t h a t t h e only c o n t r i b u t i o n s come from leads (1) a n d (2). E v a l u a t i n g these we o b t a i n from lead (1)
(oi e-''*''' + Sii aie'"'^') S^^ a^e'^^'^ +
+ Sia a,e'f^'^ (a,e~'f"^ - S,, a,e'^^'^) = 2 S,^ a,a„ (23)
a n d from lead (2)
^ S a i a,e'^'"{a^e-'^"' - S^^ a^e'^"') ~
- {a^e"'"-'- + S^.a.e'f'"') S,^ a,e'^^'' = - 2 S,, a,a, . (24) According t o eq. (20) t h e q u a n t i t i e s (23) a n d (24) a d d u p t o zero. Hence
•->12 = •->21 • (25) N o w i t is clear t h a t only for t h e sake of c l a r i t y h a v e we c o n d u c t e d t h e
a b o v e a r g u m e n t specifically for leads (1) a n d (2). N o t h i n g special has been a s s u m e d a b o u t t h e m . Therefore eq. (25) can be generalized to
or, in m a t r i x n o t a t i o n ,
S = S, (27) where .S represents t h e t r a n s p o s e of S. This completes t h e proof t h a t t h e
s c a t t e r i n g m a t r i x is s y m m e t r i c a l . I t is t o be n o t e d t h a t t h e v a l i d i t y of t h i s result is closely linked w i t h t h e p a r t i c u l a r choice of t h e n o r m a l i z a t i o n condition, eq. (7).
I. 6. Lossless j u n c t i o n s
If a j u n c t i o n is lossless, i t follows from physical considerations t h a t , once a s t a t i o n a r y s t a t e h a s b e e n a t t a i n e d (a> r e a l ) , t h e n e t power flow into t h e j u n c t i o n m u s t be zero. I t will be p r o v e d t h a t as a consequence t h e scattering m a t r i x of t h e j u n c t i o n m u s t be u n i t a r y . W e recall t h a t a m a t r i x is u n i t a r y if t h e p r o d u c t of its t r a n s p o s e w i t h its complex conjugate yields t h e unit m a t r i x . P r e c e d i n g this proof a general t h e o r e m will be derived t h a t will be needed l a t e r on. L e t us consider t h e i d e n t i t y
div (E X H* + E* X H) = H* curl E—E c u r l H * + H c u r l E * — E * curl H . (28) T h e curls in eq. (28) can be e l i m i n a t e d w i t h t h e aid of Maxwell's e q u a t i o n s
(1). I n t h i s connection complex values of o) a r e allowed; on t h e o t h e r h a n d e a n d //- are restricted t o real values * ) . T h e result is
d i v ( E X H * + E * X H ) = i ( f t > - « * ) ) / i | H | 2 + £ ( E | 2 ; - _ 2 o - | £ | 2 . ( 2 9 ) *) This is only a formal restriction. Dielectric losses are already accounted for by a. Losses
of a magnetic nature could be easily included by adding a real term proportional to IH)" on the right hand side of eq. (29).
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This equation is integrated over the volume V; the left hand side can be converted to a surface integral. Thus
f ( E x H * + E * x H ) d F = i ( o j - w * ) ƒ )/*|Hp + e | E | 2 ( d F - 2 j 0 | E p d K (30) F V V
It is to be noted that
l\\jx\H\^ + E\E\^\AV=W (31)
V
is the energy stored in the volume V. Further
\\a\EfAV=I> (32)
V
is the power dissipated in the volume V and finally
1 ƒ (E X H* + E* X H) dF = P (33) F
represents the net power flowing through F out of V. Introducing the symbols W, D and P in eq. (30) this takes the form
P= i{o, — a,*) W—D. (34)
If eq. (30) or eq. (34) is applied to a lossless junction for stationary states, the right hand side vanishes so that
P = 1 ƒ (E X H* + E* X H) dF = 0 . (35) The contributions of the various output leads to eq. (35) can be written
explicitly. Thus
P = * 2 ( 6 A * - « « « « * ) = 0 . (36)
n
Introducing matrix notation we find that
BB*-AA* = (i. (37)
Now B can be eliminated by applying cq. (9), whence
SAS*A*-AA* = 0, (38) A{S S* - I)A* = (). (39)
Since eq. (30) is valid for all values of ^ , it can be concluded that
SS* = I. (40)
Thus the scattering matrix of a lossless junction is unitary. I. 7. Frequency dependence of a unitary scattering matrix
From a reasoning similar to t h a t of the two previous sections a general' expression for the frequency dependence of the scattering matrix of a
lossless junction can be derived .The result is equivalent to Foster's reac-tance theorem in network theory.
Consider the identity div (El X H* + E* X Hi) =
= H | curl El — El curl H* + Hi curl EJ - E* curl H i . (41) Here the subscripts 1 and 2 refer to two solutions of Maxwell's equations for the same lossless structure but for different ,(real) frequencies. With the aid of eqs (1) the curls in the right hand side of eq. (41) can be elimi-nated:
div (El X H* + E* X Hi) = (wi - CO,) IfiHJl* + e EiE*(. (42) As before eq. (42) is integrated over the volume V, yielding
ƒ (El X H | + E | X Hi) dF = i(oj, - wa) ƒ )^i H i . H.* + eEi. Ea*( d V. (43)
F V
For ft)i = «a eq. (43) is seen to become identical with eq. (35) as it should be. Next we assume «a to differ from M^ by an infinitesimal amount dio. The field quantities with index 2 then differ from E^ and Hi by small variations only, viz.
ÓE =^ Eo — E, , ;
(44) Ö H = H 2 - H i . ^ ^ ' Substituting eqs (44) in eq. (43) we obtain to the first order
ƒ (E X èn* + ÖE* X H)dF = - ióo) ƒ )!Ji\HY + e|Ep( d F = - 4 iWdm, (45)
F V
where the subscript 1 has been dropped as it is no longer needed.
On the left hand side of eq. (45) the contributions from the various ter-minal surfaces can be introduced explicitly. With the aid of eqs (2), (3) and (7) we find
f (E X ÓH* + ÓE* X H) dF = S [(a„e''*"=" + 6„e''*'"") {hal c'""'"
-F
— db* e-'""'") + {da* e'i''"" + dbZ e"'""'") (a„ e"'^"'" - b„ e'^'""')] =
= 2^{bnöbl^a„dal). (46)
n
Equation (46) can be written in matrix notation and combined with cq. (45) resulting in
BOB* - Ad A * = - 2 i Wdm. (47)
Now B can be eliminated according to
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S u b s t i t u t i o n of eq. (48) in eq. (47) gives
AS Ö S*A * =-^2iW dm. (49)
T a k i n g t h e complex conjugate of eq. (49) a n d p u t t i n g OS dS
T ^ A ' v^O) oco do)
we finally o b t a i n t h e e q u a t i o n for t h e d e r i v a t i v e of t h e s c a t t e r i n g m a t r i x of a lossless j u n c t i o n w i t h respect t o frequency, viz.
A* 5*^^^-A = 2iW. (51) da>
W e shall n o w a p p l y t h i s general r e s u l t t o t h e special case of a single waveguide t e r m i n a t e d b y a lossless s t r u c t u r e . T h e m a t r i c e s in eq. (51) t h e n reduce t o scalars; S degenerates t o t h e reflection coefficient r{z =^ 0 ) . H e n c e d / ' a*r* - a = 2iW. (52) dcu As t h e t e r m i n a t i o n is lossless, t h e " m a t r i x " F is u n i t a r y or r r* = 1. (53) This e q u a t i o n is satisfied b y p u t t i n g r = e"r, (54) where (p is r e s t r i c t e d t o real values. W e further notice t h a t t h e power
in-cident on t h e j u n c t i o n Pi is given b y
P i = ^ a*a . (55) If eqs (54) a n d (55) a r e s u b s t i t u t e d i n t o eq. (52), t h e final result
dw W
/ = p (56) dm Pi
is o b t a i n e d . This e q u a t i o n is i m p o r t a n t n o t so m u c h for t h e q u a n t i t a t i v e i n f o r m a t i o n which it gives as for t h e fact t h a t t h e r i g h t h a n d side is essen-tially positive. T h u s
dw
, - > 0 , (57) dm
or in words: for a lossless t e r m i n a t i o n t h e p h a s e angle of t h e reflection coefficient always increases with frequency.
CHAPTER I I . RESONATORS
In this chapter we shall be concerned with transducer theory for struc-tures exhibiting resonance phenomena. As stated in the introduction, in writing the next pages we have been inspired by a little-known paper of Tomonaga^). This paper contains much valuable material, but it has one serious shortcoming in t h a t it deals with lossless structures only. Although this idealization is justified for many practical configurations, it forms es-pecially for resonators a prohibitive restriction. We shall, accordingly, extend Tomonaga's theory so as to include the effect of losses.
II. I. Quality factor
We start by reviewing the essential property in which a resonator differs from other junctions. Let us assume t h a t a certain amount of energy has been fed into a junction. If now the junction is left to itself and if it con-tains a resonating element, outgoing exponentially damped waves will appear in the output leads, which are for the moment assumed to be ter-minated by matched loads. In other words the amount of electromagnetic energy, stored inside the junction, is gradually decreasing because of the power loss through the various output leads and the dissipation inside the junction proper. I t is this gradual exponential decrease of the stored energy (and at the same time of the amplitudes of the corresponding out-going waves) which forms the characteristic feature of a resonator.
It is well known that microwave resonators have an infinite number of resonance frequencies. In the following we will develop the theory under the following restrictions. The angular frequency o) shall lie close to one resonance frequency and be far removed from all others. This particular resonance frequency shall moreover be simple, i.e. there shall be only one mode of oscillation associated with it. Both conditions are fulfilled in many practically important cases. The theory, however, may readily be extended so as to include more general possibilities.
The theorem derived in Sec. 1.6, eq. (34),furnishes quantitative informa-tion on the decrement of the oscillainforma-tion in a resonator. We apply this equa-tion
P=i{me~ojt)W~D, (1)
to the case when no power is incident on the junction. Then P stands for the power leaving the junction through the various output leads, D is the power dissipated within the junction and W denotes the stored energy. The damped oscillation is described as an oscillation with a complex angular frequency Wc.
— 19 —
I t will be r e m e m b e r e d t h a t t h e q u a l i t y factor ^ of a r e s o n a t o r can be defined as
Q = ^ - ^ ' (2 D + P ^ '
where CUQ is t h e real p a r t of co^.
E q u a t i o n (2) can be r e w r i t t e n as
^ =
.;V'^
+
?^"^ (^)
or
A p p a r e n t l y Qi is t h e s u m of t h e c o n t r i b u t i o n from i n t e r n a l losses Djm^W = QIJ plus t h e c o n t r i b u t i o n s from each o u t p u t lead individually Pn/'^o^ = QEI- H e r e a n o t a t i o n is employed which is c u s t o m a r y in m i c r o w a v e work. Q^^, t o b e identified w i t h Q in eq. (2), is called t h e loaded Q a n d r e p r e s e n t s t h e q u a l i t y factor of t h e r e s o n a t o r loaded b y all o u t p u t leads. T h e p a r t i a l ^ ' s i n t r o d u c e d in eq. (4) are divided i n t o t h e u n l o a d e d Q, d e n o t e d b y Qu, a n d various e x t e r n a l ^ ' s , d e n o t e d b y ^g^, each repre-s e n t i n g t h e loading effect of one o u t p u t lead.
A t t e n t i o n should be d r a w n t o t h e fact t h a t in eq. (3) t h e t e r m s D a n d P „ a p p e a r in a similar w a y . W h i l s t in eq, (1.30) their different origins are clearly i n d i c a t e d , in eq. (3) t h e r e is left only a formal difference i n n o t a t i o n . T h e s i t u a t i o n is, indeed, n o t altered if t h e t e r m involving D is supposed t o be due t o a n e x t r a o u t p u t lead, t h e j u n c t i o n itself being lossless. T h e a d v a n t a g e gained b y t h i s p o i n t of view is t h a t we can s t a r t calculations w i t h a u n i t a r y *) s c a t t e r i n g m a t r i x for t h e idealized j u n c t i o n , i n t r o d u c i n g t h e i n t e r n a l losses only a t a l a t e r s t a g e . I t should n e v e r be forgotten, how-ever, t h a t this is a purely formal w a y of a p p r o a c h a n d t h a t , whilst N o u t p u t leads of t h e j u n c t i o n a r e accessible for direct m e a s u r e m e n t , t h e fictitious lead n u m b e r e d (A'^ -|- 1) is n o t .
This p r o c e d u r e , viz. t h e c o n c e n t r a t i o n of t h e d i s t r i b u t e d losses w i t h i n t h e j u n c t i o n i n t o a fictitious e x t r a o u t p u t lead, is identical w i t h t h e re-p r e s e n t a t i o n of t h e i n t e r n a l losses b y a l u m re-p e d resistance i n e q u i v a l e n t circuit t h e o r y . C o m p a r e d w i t h t h e l a t t e r our p o i n t of view seems t o be slightly superior in t h a t no further a s s u m p t i o n a b o u t t h e n a t u r e of t h e fic-t i fic-t i o u s o u fic-t p u fic-t lead is necessary whereas we m u s fic-t always assign a definific-te location t o a l u m p e d resistance.
*) The scattering matrix of a lossless junction is unitary. This property is a direct result of the conservation of energy within the junction. The lemma therefore is true only for real frequencies as is also evident from the proof in Sec. I. 6.
T h e r e l a t i o n b e t w e e n t h e d e c r e m e n t of d a m p e d oscillations in a resona-tor a n d its ^ - f a c t o r is o b t a i n e d b y combining eqs (1) a n d (2). T h u s
i{o)c ~ m*) = -"• (5)
I I . 2. F u n d a m e n t a l considerations
R e v e r t i n g n o w o u r a t t e n t i o n t o t h e s c a t t e r i n g m a t r i x we recall t h a t its elements represent t h e r a t i o b e t w e e n a m p l i t u d e s of outgoing a n d incident waves. I n t h e preceding section we h a v e discussed t h e s t a t e of a resonator w i t h only outgoing waves i n t h e o u t p u t leads. A complex frequency Wc was used t o describe these d a m p e d waves. Considering t h e elements of t h e s c a t t e r i n g m a t r i x as functions in t h e complex oj-plane, it is clear t h a t some, if n o t all, elements b e c o m e infinite if t h e frequency a p p r o a c h e s t h e v a l u e m,:- T h e most obvious possibility is t h a t t h i s singularity is a simple pole. Consequently we write t e n t a t i v e l y
^nm = T i^nm, (6)
m — Wc
where pnm a n d Rnm ^re i n d e p e n d e n t of m. B y t h e l a t t e r a s s u m p t i o n we r e s t r i c t t h e validity of eq. (6) a priori t o t h a t frequency r a n g e where t h e n o n - r e s o n a n t frequency-dependence of t h e j u n c t i o n is completely obscured b y t h e properties of t h e r e s o n a t i n g e l e m e n t . E q u a t i o n (6) therefore is a t best valid in t h e n e i g h b o u r h o o d of r e s o n a n c e .
W i t h t h e aid of t h e following a r g u m e n t *) it can be shown t h a t eq. (6) is a plausible a p p r o x i m a t i o n . Suppose t h a t along lead (m) a pulse of in-finitesimal d u r a t i o n is fed into t h e r e s o n a t o r . This incoming signal can be r e p r e s e n t e d b y ") + C0 1 ,• s i n (foT/2) , . a„=lim - -^^ J ' e-""'dm. (7) r->ii 27r . ' O J T / 2 —cc T h e outgoing w a v e s 6„, caused b y a ^ , a r e o b t a i n e d b y m u l t i p l y i n g t h e i n t e g r a n d in eq. (7) b y t h e a p p r o p r i a t e e l e m e n t 5 , , ^ . I n s e r t i n g eq. (6) we h a v e 1 ' " s i n (COT/2) e-'-' On = Pnm h m - ~ ,o " dm + T^i) ITI J on 11 m — w^ - c o 1 ,• sin (mxl2\ . , + i?„,„lim - ~\^-' e-""'dm. (8)
*) This proof was suggested to the author by Prof. Dr H . B. G. Casimir, Director Philips Research Laboratories.
— 21 —
The first term on the right-hand side of eq. (8) can be evaluated by closing the path of integration by a semicircle of infinite radius in the negative imaginary half plane. Taking the limit T —> 0 and applying the theorem of residues we find
+ C0
1 /• sin (wr/2) , ,
b„ = - i Pnme-"'' + Rnm Um ' - - ^ e'"" dm . (9)
— CO
The last term in eq. (9) represents an outgoing pulse of which the amplitude is determined by the non-resonant properties of the junction. The first term on the right in eq. (9) indicates, that part of the energy contained in the incident pulse is temporarily stored within the junction and leaks away only gradually through the output leads. This behaviour agrees exactly with the views developed in Sec. 1 on resonators. The assumption that S„„, has a simple pole for m = mc is thus seen to be well-founded.
From the preceding argument another important conclusion can be drawn. It follows from eq. (9) that the amplitudes of the outgoing damped waves, caused by the pulse incident in lead {m), are proportional to the coefficients pnm- Nothing special has been assumed about lead (m); it is clearly immaterial through which lead the energy, giving rise to the outgoing damped waves, is brought into the resonator. Therefore the ratios of the coefficients p„m ("i fixed) should be independent of m. This fact together with the symmetry condition for the scattering matrix, which property is not restricted to real values of w (cf. Sec. 1, 5), requires, that all Pnm have the form
/'raifi= ^n ^m • ( 1 0 )
II. 3. Resonators with one output lead
Instead of deriving the general theory for resonant junctions with any number of output leads we shall conduct the discussion along simpler lines by working out some practically important examples. Occasionally the obvious extensions to the more general case will be pointed out.
As a first example we shall analyse the behaviour of a resonator with one output lead; the coupling between resonator and output lead is supposed to be lossless. According to the argument developed at the end of Sec. 1 the situation can be schematically represented by fig. 2.1. The actual internal losses are accounted for by the matched termination in lead (2), while the resonant junction between the leads is lossless.
The scattering matrix for this structure is, according to eqs (6) and (10),
S =
/'-'il '^12\ ^^12 ^22 ± - + «11 - J - ? ^ + JRi m — Wc •2 ft) — mc (11) + iÏ2T h e s y m m e t r y condition for S h a s b e e n used in eq. (11). T h e u n i t a r y condition requires | S n | ^ + | S i 2 | ^ = | S i 2 | ^ + | S 2 2 P = l , / ^ll'^12 ~r '^12'^22 = 0 . ) (12) fl) J2)_ ^L '^ I I 69651
Fig. 2.1. Schematic representation of a resonator with one output lead. The matched termination in the (fictitious) lead (2) accounts for the internal losses.
These conditions m u s t be fulfilled for all real values of ft). If eq. (11) is s u b s t i t u t e d in eqs (12), t e r m s a p p e a r in {m — o.)c)^^, (ft) — m*)~^ a n d {m — mc)^\
{m — co?)~^. T h e l a t t e r c a n be e x p a n d e d in p a r t i a l fractions, so t h a t t h e resulting equations t a k e t h e form
+
B- + C=0, (13) where A, B and C are n o t d e p e n d e n t on m. If an e q u a t i o n like eq. (13) is t o hold for all m, t h e n ^ = 0, B = 0 a n d C = 0 (note t h a t ojc is n o t r e a l ) . T h e knowledge t h a t t h e t e r m s C are zero i m m e d i a t e l y leads t o t h e r e s u l t t h a t t h e m a t r i x
R
i?ii i?i^ 1 2 ^ 2 2
(14)
is u n i t a r y . A c t u a l l y t h i s fact could h a v e been deduced directly from eq. (11), a s B r e p r e s e n t s t h e b e h a v i o u r of t h e lossless j u n c t i o n far off resonance.
T h e remaining e q u a t i o n s are o b t a i n e d from t h e coefficients of (ft) — Wc)^ a n d (ft) — ftJ*)"^ in eqs (12) a n d are ^1* l l ^ l l ' + l^2|'( + («1*1^1 + «1*2^2) («C - OJ*) = 0 , n* )|7ri|2 + \n,\^\ + {R*,7z^ + R*,7i,) {0
0
0 . E q u a t i o n s (15) a n d (16) are identical w i t h t h e m a t r i x e q u a t i o n{nn*)n* + {mc-m*c)R*n=o,
(15) (16) (17)— 23 -where the column matrix / 7 is given by
n^r^. (18)
VTta'
From eq. (17) R can be eliminated. As R is unitary, multiplication of eq. (17) by R and a subsequent change to the complex conjugate yields
{nn*)R*n-{ojc-mt)n* = o. (i9)
By combination of eqs (17) and (19) we obtain
{nn*)^ = -{mc-m*c)K (20)
As ft)c is the frequency of a damped oscillation, it has a negative imaginary part; hence the positive root of eq. (20) is the correct one, so that
nn* = \nyf + |7Ta|' = i{o^c - «*). (21)
Equation (5) enables us to introduce the experimentally important pa-rameters ft)Q and Qj-.
WY + W?-"^- (22)
In Sec. 1 we have seen, that the internal losses and the loading due tothe output lead contribute separately to Q~i}. Equation (4) adapted to the present problem can be written as
1 1 1 1 1
— = \ = - - H (23) QL QE QU QI Q2 ' Comparison with eq. (22), where Q2^ also appears as a sum of contributions
from each output lead individually, compels us to conclude that
ft)0 Wo I 12 «^0 0^0 ,c^^
The relation between 77 and B can be derived by insertion of eq. (21) into eq. (17). The complex conjugate of the result is
R77* = in. (25) As was pointed out before R describes the off-resonance behaviour of
the resonator. The condition that the coupling between resonator and output lead is lossless implies total reflection far off resonance. Hence the off-diagonal elements of B must vanish and eq. (25) results in separate equa-tions for TTi and n^. Anticipating the needs for the next section, however, we shall derive the consequences of eq. (25) without imposing any restric-tion on B .
B u t for t h e asterisk eq. (25) would h a v e been an eigenvalue e q u a t i o n for B . I t can be b r o u g h t i n t o this form b y splitting up 77 i n t o m o d u l u s a n d a r g u m e n t , viz.
S u b s t i t u t i o n of eq. (26) in eq. (25) yields
B ^ * | 7 7 | = i 0 | 7 7 | . (27) P r e m u l t i p l i c a t i o n of b o t h sides of eq. (27) b y 0~^ = 0 * gives
0 - i B 0 - i | 7 7 | = B ' | 7 7 | = il77| . (28)
T h e m a t r i x B ' , i n t r o d u c e d in eq. (28), is derived from B b y a simple shift of t h e reference planes in t h e o u t p u t leads, as follows from Sec. 1.4. T h e shift is chosen so as t o m a k e t h e q u a n t i t i e s Ti'n =--- |7r„| real.
E q u a t i o n (28) is a n eigenvalue e q u a t i o n . T h e l i m i t a t i o n it imposes on B ' consists in t h e p r e s c r i p t i o n of a n eigenvalue i with a corresponding eigenvector p r o p o r t i o n a l t o (77|. T h e second eigenvalue of B ' is a priori u n k n o w n b u t its m a g n i t u d e should equal 1, because B ' , like B , is u n i t a r y . T h e second eigenvector is fixed b y t h e condition t h a t it should be real a n d o r t h o g o n a l t o |77|. W e are now able t o form B ' from its eigenvalues a n d eigenvectors. T h e l a t t e r are n o r m a l i z e d t o u n i t y . L e t t h e second eigen-value be i e^"', // being real b u t otherwise u n k n o w n . I n this w a y we o b t a i n
w i t h li 0 1' ^ 1
\
IQ^.
^ Q^)M,
^ <?1 (29) (30)Reference to eq. (24) shows t h a t t h e first c o l u m n of t h e orthogonal m a t r i x M. is, indeed, p r o p o r t i o n a l t o |77|.
E v a l u a t i n g eq. (29) we find for B ' t h e general expression
B ' = ^ ^ ' ' ^ ^ ! ^ . (31) Ql i — 2 ^'- sin»? e'" V2 QL 2 ^''- sin« e"" )'(?1<?2 QL 2 - _=sin»7e"' 1'<?1<?2 QL i - 2 ^ sin?/ e'"
25
I n t h e special case t h a t t h e off-diagonal elements of B ' v a n i s h , eq. (31) reduces t o t h e almost t r i v i a l form
B'
'o
0 (32)T h e reflection coefficient 7^ seen in o u t p u t lead (1), if o u t p u t lead (2) is t e r m i n a t e d b y a m a t c h e d load, becomes identical w i t h S n . I n s e r t i o n of eqs (24) a n d (32) i n t o eq. (11) yields
r' = s[,
(OnQE{(^ — ftJe)
+ i.
(33)
E q u a t i o n (33) represents t h e reflection coefficient in t h e o u t p u t lead of t h e r e s o n a t o r a t t h e p a r t i c u l a r reference plane fixed b y eq, (28).
I t is useful t o eliminate mc from eq. (33) a n d t o replace i t b y q u a n t i t i e s b e t t e r suited t o m e a s u r e m e n t . To this p u r p o s e p u t , as is c u s t o m a r y ,
w Am. (34)
Then
m — (Oc = Am — \{mc — m*) • (35)
W i t h t h e aid of eqs (5) a n d (35) eq. (33) can be b r o u g h t i n t o its final form
F' 2QL
Qj,{l-2iQ^ Am/mo)
+ i.
(36) T h e locus of 7^' in t h e c o m p l e x p l a n e as a function of frequency is a circle of r a d i u s QijQji t a n g e n t t o t h e u n i t circle a t t h e p o i n t i, as is illu-s t r a t e d in fig. 2.2. I t illu-should be r e m e m b e r e d t h a t eq. (36) iillu-s valid for one p a r t i c u l a r reference p l a n e . A shift of t h e reference plane simply corresponds t o a r o t a t i o n of t h e locus a r o u n d t h e origin.II. 4. Resonators with two output leads
As a second e x a m p l e of t h e use of a s c a t t e r i n g m a t r i x , t h e elements of w h i c h can be r e p r e s e n t e d b y eq. (6), we shall discuss t h e case of a r e s o n a t o r w i t h t w o o u t p u t leads a t t a c h e d t o it. I n t h e d i a g r a m of fig. 2.3 t h e o u t p u t leads are designated (1) a n d (2). O u t p u t lead (3) accounts for t h e i n t e r n a l losses of t h e resonator, so t h a t t h e s c a t t e r i n g m a t r i x p e r t i n e n t to t h e rec-t a n g l e c o n rec-t a i n i n g rec-t h e r e s o n a rec-t o r is u n i rec-t a r y . B o rec-t h rec-t h e coupling b e rec-t w e e n t h e r e s o n a t o r and its o u t p u t leads a n d t h e direct (i.e. off-resonance) coupling b e t w e e n leads (1) a n d (2) are a s s u m e d t o be lossless. As t h e m a t -ched load in lead (3) r e p r e s e n t s formally t h e losses w i t h i n t h e r e s o n a t o r , no direct coupling will occur b e t w e e n lead (3) a n d t h e other t w o o u t p u t leads. T h e off-resonance m a t r i x B will therefore be
/ R j l i?i2 0 X B = « l a 7?a2 0 . (37)
^ 0
RJm
(3) (2) 69652Fig. 2.3. Schematical representation of a resonator with two output leads. The matched termination in the (fictitious) lead (3) accounts for the internal losses.
F o r t h e present p r o b l e m we require an obvious extension of some results of t h e preceding section. T h e proof, if needed, can be given in e x a c t ana-logy t o t h e t r e a t m e n t in Sec. 3. I n fact, all m a t r i x e q u a t i o n s derived t h e r e are generally v a h d , irrespective of t h e order of t h e m a t r i c e s , i.e. t h e n u m b e r of o u t p u t leads. T h e m a t r i x B is u n i t a r y . As a consequence t h e s u b m a t r i x is also u n i t a r y . T h e e q u a t i o n s e q u i v a l e n t t o eqs (23) a n d (24) are 1 1 1 1 QL^ Qi^ Q^'^ Q,' ( 3 9 ) "
— 27 —
Vl V 2 V 3 To analyse t h e relation b e t w e e n B a n d 77 i t is useful t o define t h e re-ference p l a n e s again i n such a w a y t h a t t h e elements of 77 b e c o m e real a n d eq. (28) applies. Because of eq. (37), eq. (28) falls a p a r t in s e p a r a t e e q u a t i o n s for Ü33 a n d t h e s u b m a t r i x (38); t h e l a t t e r e q u a t i o n is
R[, R'J \\n,\l \\n,\) ^ ^
T h e conclusions t o be d r a w n from this e q u a t i o n are almost literally t h e s a m e as eqs (29) and (30) of t h e previous section; t h e only difference lies in t h e n o r m a l i z a t i o n of t h e m o d a l m a t r i x Jtf. T h u s ^ 1 ] ^ l 2 ^ - - /^ 0 '^^12 -^^22' ^0 ^ ( n .^.,)M, (42) where1/;
Q E M = M - i = I ^^ ^ | . (43)-1/
Ql H e r e we h a v e i n t r o d u c e d t h e symbol QE t o r e p r e s e n t t h e t o t a l e x t e r n a l loading of t h e r e s o n a t o r , which is given b y1 1 1 — = - H (44) QE Ql Q2 E v a l u a t i n g eq. (42) we find •'^U ^ 1 2 \ 12 22 QE /i-2 -smr^e'" / <?2 \c QE . i„ \ 2 - = ^ - smri e ' 1'<?1<?2 ^ QE . Z sinjy e ^QiQ^ . ^QE . I — 2 — sm»7 e Ql (45)
O u t p u t lead (3) is n o w assumed t o be t e r m i n a t e d b y a m a t c h e d load. T h e s c a t t e r i n g m a t r i x for t h e r e s o n a n t j u n c t i o n b e t w e e n leads (1) a n d (2) is o b t a i n e d b y o m i t t i n g t h e t h i r d row a n d c o l u m n from t h e m a t r i x p e r t a i n -ing t o t h e lossless j u n c t i o n discussed up till now. B y eqs (6), (10) a n d (40) we h a v e
5' =
^0 , ^ , « o Ql (ft) — ojc) " 1 ' ^ 2 («J — «<•) y(?i(?a(ft^-ft>o) ^ 1 2 Q,{oj — ftif) j?;2 -'^22 (46)I t is possible t o eliminate o)c again b y t h e i n t r o d u c t i o n of Am = m —• o)(,. If further eq. (45) is s u b s t i t u t e d in eq. (46), t h e final expression becomes
S'
2QE ^ . ^QE . ,-, h i — 2 — sin?5 e ' • <?i(l-2i(?^zlft>K)^ (?2 ^QL y^i(?a(i-2i(?^/ift)/ft)o) r(?i(?2 2<?,, 2 - .^ .Sim/e"'— QE 2 —_^^= sin??e"'-l(?i^a(l-2ig,.^«K) i<?i^:
2<?. , . ^QE . ,, h i — 2 - sinM e '(?2(l-2ta^zlft)K)^ ^1 '
(47)A n i n s t r u m e n t t o which eq. (47) is applicable is t h e iris-coupled r e a c t i o n w a v e m e t e r shown in fig. 2.4. I t is n o t a n e x a m p l e of eq. (47) in its m o s t general form, as t h e s y m m e t r i c a l location of t h e two o u t p u t leads obviously requires
< ? 1 = ( ? 2 (48)
29 —
After insertion of t h i s condition i n t o eq. (44), eq. (47) can be simplified t o
S' =
/ ^ i —
+1—sin«e"'-QE{l~2iQ^Amlojo) QE{l~2iQj^ Am/m^) QL -j- sinrj e" (49) sinwc"'' — I —— - | - i — sinwe"'; Q,{l^2iQ^Am!<o,)^ QE{l-2iQ, Am/mo)
T h e t r a n s m i s s i o n coefficient from lead (1) t o lead (2) or vice versa is given b y t h e off-diagonal elements in eq. (49). As a function of frequency i t is r e p r e s e n t e d in t h e complex plane b y a circle of r a d i u s QiJQx ^= QLI^QE-T h e position of this circle is d e t e r m i n e d b y t h e v a l u e of )/; its c e n t r e i n p a r t i c u l a r lies on a circle, which we shall call ?/-circle, of r a d i u s ^ c e n t r e d on t h e i m a g i n a r y axis a t t h e p o i n t {QiJ2Q.^)i = {Qil2Qu)i, as is i l l u s t r a t e d i n fig. 2.5.
.+i
Fig. 2.5. Transmission-coefficient diagram for a resonator with two symmetrical outputleads.
F o r t h e ideal r e a c t i o n w a v e m e t e r , in which t h e off-resonance reflection caused b y t h e coupling window is s u i t a b l y c o m p e n s a t e d , t h e a p p r o p r i a t e v a l u e of i] is ^Ji. I t s t r a n s m i s s i o n coefficient is r e p r e s e n t e d in fig. 2.5 b y t h e circle (a).
T h e b e h a v i o u r of a n o n - c o m p e n s a t e d r e a c t i o n w a v e m e t e r is i l l u s t r a t e d b y t h e circle (b) t h e c e n t r e of which lies on t h e ^/-circle slightly off t h e i m a g i n a r y axis.
T h e circle (c), for which rj = 0, shows t h a t eq. (49) also applies t o w a v e -m e t e r s of t h e t r a n s -m i s s i o n t y p e . I t r e p r e s e n t s t h e t r a n s -m i s s i o n coefficient
through a resonator with two symmetrically located output leads between which no direct coupling exists.
Dropping the restriction imposed by eq. (48) for symmetrical output leads we can deduce from eq. (47), for the case rj ^ 0, that the condition for a match in output lead (1) at the resonance frequency is
or 2QL ~ =1, Ql Qu^Qu, Ql (?2 (50) (51)
As a final remark it should be pointed out that the rather artificial difference between series and parallel resonant circuits, usual in equivalent-circuit theory, is completely avoided here.
II. 5. Losses in the output circuit
The theory developed in the preceding section can be made, by a slight modification, to describe the behaviour of a resonator coupled to an ideal waveguide through a circuit of which the losses cannot be neglected. To this purpose the schematical representation of fig. 2.3. is redrawn in fig. 2.6. The output circuit is idealized to a frequency-independent T-junction. Its losses are lumped in the matched load of lead (2), while lead (3) is again included to account for the internal losses of the resonator. The configuration depicted in fig. 2.6 is a useful approximation for the analysis of a "cold" oscillator tube ^), ' ) . Formally figs 2.3 and 2.6 are identical, so that eq. (47) is equally valid in the two cases.
If both leads (2) and (3) are terminated by a matched load, the re-flection coefficient 7^ in lead (1) is given by Sn. Hence, from eq. (47)
F' = - i ~ h i - 2 — sin?7 e"
Qi{l^2iQ,^Amlmy Q,
(52)
The locus of F' in the complex plane is commonly called the ^-circle.
(f) (3)
(2)
— 31 —
Once this circle has been obtained by experiment, all parameters of the resonator can be derived from it as follows:
From the relation between m and F' on the circle Q, can be determined. From the radius of the circle, which equals Q/JQi, Qi is then calculated. It is seen from eq. (52) that the position of the centre of the ^-circle is determined by i] and Q,. For rj = 0 the ^-circle is tangent with its off-resonance point to the unit circle at the point i; its centre lies for other values of rj on the ^/-circle, centred on the imaginary axis, with radius QEIQ^ = QiKQi + Qi)- 1^ fig- 2.7 it is shown how the rj-circle can be derived from the ^-circle by a geometrical construction. As a first step the off-resonance point P must be located "), which can be done by extra-polation only, as our fundamental assumptions in Sec. 2 restrict the validity of eq. (52) to the neighbourhood of resonance. The diameter of the ()-circle through P is drawn and parallel to it the diameter of the unit circle. The latter is to be identified with the imaginary axis; it should be realized t h a t the reference plane for which eq. (52) applies is not known beforehand. The construction of the )/-circle is now a simple matter, because it is known that (i) its centre lies on the imaginary axis, (ii) the centre of the ^-circle falls on it and (iii) it cuts the imaginary axis at a distance (^1 — QL)IQI from the origin. The geometrical consequences are illustrated in fig. 2.7. After the »/-circle has been constructed, Q.^ can be calculated from its radius. Finally ^3 ^ Qy can be found with the aid of eq. (39) from the known values of Qi_, Q^ and
^a-CHAPTER III. SYMMETRY ANALYSIS OF WAVEGUIDE JUNCTIONS
If a waveguide junction exhibits a certain degree of symmetry, the num-ber of constants needed to describe its behaviour as a transducer is less than for the general non-symmetrical junction with the same number of output leads.
A method to perform this reduction of constants has first been given, for lossless structures, by Dicke ^). The scattering matrix of a lossless junction is unitary and can therefore always be brought into diagonal form by a suitable unitary similarity transformation. Dicke's method, according-ly, is one suitable to eigenvalue problems; he developed it largely by treat-ing a number of specific examples.
In order to include dissipative structures. Kerns ^) has lately put the theory on a more rigorous mathematical basis by extensive use of concepts derived from group theory. It is hardly necessary to point out that the methods used are for a large part identical with those developed for the analysis of symmetry problems in molecular physics ^) and quantum me-chanics ^^), 1^).
In this chapter we shall give an outline of Kerns' theory in conjunction with some aspects elaborated by Dicke. It is supposed that the reader is acquainted with the fundamental results of the theory of finite groups. III. 1. Representation of a symmetry group
From the vector notation of Maxwell's equations (1. 1) it is clear that their validity is not restricted to a particular choice of coordinate system. If a new coordinate system is introduced by a real linear transformation from the original system and the field components are transformed accord-ingly, the latter will again satisfy Maxwell's equations. Mathematically we can say that Maxwell's equations are invariant with respect to the real linear group. A particular subgroup of this linear group is the three-dimen-sional rotation reflection group, which in its turn comprises the point-symmetry groups (also called the crystallographic groups).
The invariance of Maxwell's equations, though important, is in itself not very helpful. A new situation arises, however, if in a specific problem there can be found a transformation subject to which the boundary con-ditions also are invariant. Such a transformation is termed a covering opera-tion for the structure under consideraopera-tion. Because we deal only with structures of finite size, a translation cannot be a covering operation. The only possibUities left are reflections in a plane or a point and rotations about an axis. All possible covering operations of a structure, including the iden-tity operator, constitute the symmetry group of the structure.
— 33 —
Let us consider now the electromagnetic field inside a waveguide junction. A covering operation will change the spatial position of the junction and the field, without altering their relative position. But as the junction remains by definition invariant under the operation, we may just as well assume the junction to be fixed in space, whilst the operation affects only the field within it.
In the previous chapters we have discussed, how the electromagnetic behaviour of a junction can be described by the amplitudes of waves in the output leads. Once the amplitudes of the incoming waves, represented by the column matrix yi, are prescribed, the amplitudes B of the emergent waves are determined by the properties of the junction as expressed in terms of the scattering matrix. In Chap. I it has been explained, that the elements a„ and b,i of A and B respectively are to be regarded as coefficients for the transverse electric field E(„ in the various output leads of the junc-tion. In other words we may state, that the fields E(„ constitute the coor-dinate system or basis for the description of the electromagnetic field;
A comprises the coordinates of a possible incident electromagnetic field
relative to this coordinate system and can as such be regarded as a vector. If a covering operation P of the symmetry group of the structure is appUed to A, we obtain another possible field PA. The coefficient a„, des-cribing before the operation the field in lead (n), will after the operation per-tain to the field in lead {m). The normalizing condition, eq. (I. 7), requires, t h a t the basis fields Em and Etm at corresponding points for all leads (ra) and (m) respectively which interchange under the covering operation, are connected by
l^ml^ = \Etm\'. ' (1) It should be noted that the one-mode assumption, condition (v). Sec. 1.1,
is essential here. The origin of the time scale can always be chosen so that
Etn is real for all n. Then there are only two possible solutions of eq. (1),
viz.
Etn = ± Elm • (2)
It follows from eq. (2), that the matrix D{P), which expresses the vector
PA in terms of A, consists only of elements 0, + 1 and — 1 . In each row
and column there is only one non-zero element. The matrix D{P) is there-fore real and orthogonal.
For all operators P of the symmetry group the corresponding matrices
D{P) can be formed. These matrices then constitute a representation of
the group. The basis for this representation consists of the transverse electric fields in the output leads. It is evident that the dimension of the representation equals the number of output leads of the junction.
III. 2. Relation between the symmetry of a junction and its scattering matrix As our aim is the simplification of the scattering matrix of a junction, it is well to consider the connection between this quantity and the repre-sentation of the symmetry group.
Let A and B specify a possible solution of Maxwell's equations for thf junction; then
B = SA. (3) By definition the fields
A'==D{P)A, B = D{P)B, (4) obtained by a covering operation from A and B, constitute also a possible
solution of Maxwell's equations. Hence
B = SA'. (5) Inserting eq. (4) in eq. (3) we have
D{P)B = SD{P)A, (6) or by virtue of eq. (3)
D{P)SA = SD{P)A. (7) Equation (7) is valid for any A, so that for any P
D{P)S = SD{P). (8) Equation (8) tells us that the scattering matrix commutes with all matrices
D{P) of the representation of the symmetry group. This relation is the key to the solution of our problem.
The choice of the transverse electric fields in the output leads as a basis follows in a natural way from the physical properties of the structure and is in this respect the simplest possible. However, this does not imply that it is, mathematically, the most logical choice for the analysis of eq. (8) and in general it will not be so. If then we want to analyse the influence of the symmetry properties of a junction on its electromagnetic behaviour, it will be wise to introduce a new basis, in which the symmetry operators appear in their most clear-cut form.
A change of basis can be achieved by the linear transformation
A = TA, B = TB, (9) where the prime now denotes the quantities with respect to the new
coor-dinate system. The matrices D'{P) representing the symmetry operators in the new basis are given by the similarity transformation
D'{P) = V D{P)T. (10) Likewise the scattering matrix in the new basis becomes
— 35 —
The quantity S' is a scattering matrix in so far that it relates outgoing waves to incident waves. Its physical meaning cannot always be visualized easily. The specific transformation matrix T t h a t is most useful here is the one -for which D'{P) is a completely reduced representation. The matrices £)'(P)
will then consist of a set of square submatrices on the principal diagonal, all other submatrices being zero. Thus
D'{P) =
I>i(P) 0 0 0 D,{P) 0
0 0 D^{P) (12)
The dimension of each submatrix in eq. (12) is independent of P .
I t is clear t h a t in this specific basis, called symmetry basis, the coor-dinates A' fall apart in a number of subsets, each of which is invariant with respect to all operations P . The num^)e^ of elements of any subset equals the dimension of the corresponding irreducible representation.
As D{P) and S are subject to the same similarity transformation, eq. (8) is equally valid in the new coordinate system as in the original basis, so that
D'{P)S' = S'D'{P), (13)
where D'{P) is given by eq. (12). To evaluate the matrix products in eq. (13) ^ve write
""«' s'
S'
^ 1 1 C5i2 O i 3 « ' « ' C' '-'21 '-'22 '323 S'= S',, SL S33 (14)Here S' has been spht up into submatrices by identical dividing lines as have been used in eq. (12). The submatrices on the diagonal will therefore be square; the off-diagonal submatrices are in general rectangular, as the various irreducible representations may have different dimensions. Com-bining eqs (12)-(14) we obtain
" l - ^ l l ^ l ' ^ 1 2 " l ' ^ 1 3
IJ2S21 tJiSii A'2'^23
" 3 ' ^ 3 1 ^ 3 ' ^ 3 2 ^ 3 ' ^ 3 3
SllLfy ^12,1-^2. ^liLfz
'^21*-'l ' ^ 2 2 ' ' 2 ' ^ 2 3 ^ 3
Though for simplicity it has not been indicated explicitly, eq. (15) is valid for all P . The separate equations contained in eq. (15) all have the form
DkSki = SjtiDi. (16)
Because the matrices D'{P) are completely reduced, Schur's Lemma applies to eq. (16). This fundamental theorem states, that a matrix Ski satisfying eq. (16) for all P is either (i) a multiple of the unit matrix, if
Dk and Dl are the same irreducible representations, or (ii) identically zero,
if the irreducible representations Dfc and Dl are non-equivalent. To bring out most clearly the consequences of this theorem we rearrange the irre-ducible representations in eq. (12) so t h a t
D'{P) Dl Dl 0 0 0 Ö 1
D,
0 0 0D,
D,
(17)where the subscript now denotes the species of the irreducible representa-tion. Symbolically we can write D'{P) as the direct sum of its irreducible components, viz.
D'{P) = ^iDi + /laDa + + niDi + ... (18) The coefficient n; denotes the number of times the irreducible representa-tion Di occurs in D'{P).
If the submatrices in S' are ordered in the same fashion as in D'{P), eq. (17), S' becomes, by Schur's Lemma,
S' Sii{l) S,,{1) Sn{l) ^22(1) ' J n i n i ( l ) 0 0 Sii{2) '^n2'i2(^/
te(3) J
(19)— 37 —
I t is seen that a considerable number of submatrices in S' necessarily vanish-es, while all non-zero submatrices Skl{i) are multiples of the unit matrix. Thus
Skl{i) = Skl{i) I, (20)
where Skl{i) is a number. Hence, the number of constants needed to speci-fy S' is reduced, by symmetry reasons alone, from A'^'^ {N is the number of output leads of the junction) to
nl
+ n;+ + ni^+ (21)
No use has been made so far of any of the properties of S' previously derived, such as the requirements for reciprocity {S = S) or for the absence of losses {S S* = ƒ). Fulfilment of additional conditions like these will in general still further reduce the above number.
Summarizing we can state that the division of the coordinates A' of the symmetry basis into invariant subsets is less far-reaching with respect to the scattering matrix S' than with respect to the reduced-out represen-tation of the symmetry group. In fact the number of invariant subsets equals in the former case the number of different irreducible representations, whilst in the latter case it is equal to the total number of irreducible repre-sentations contained in the representation of the group.
The construction of the reduced-out representation, eq. (17), and of the matrix T needed in the similarity transformations, eqs (10) and (11), will now be outlined very briefly.
Character tables for the crystallographic groups can be found in many texts ^), '^). The number of times each irreducible representation appears in the specific representation of the group for the problem on hand can be calculated by a well-known theorem of group theory, involving the charac-ters. The irreducible representations themselves have not been tabulated, but they can in most cases easily be constructed as they are implicitly determined by their known characters. In this way the completely reduced representation D'{P) can be found.
Once both D'{P) and D{P) are known, eq. (10) may be used to obtain a set of linear equations for the individual elements of T. In all cases where one irreducible representation occurs more than once, this set of equations will be found to be insufficient for the determination of all elements of T. In fact only the division of the symmetry basis into subsets that are in-variant with respect to S', eq. (19), is fixed, whereas the choice of the spe-cific coordinates within each subset is arbitrary. This indeterminacy is brought out very elegantly in the method given by Eyring, Walter and Kimball '^) for the calculation of T, which method furthermore is much less tedious than the solution of eq. (10).
The limited freedom in the choice of T may be used with advantage in those cases where it is known a priori that the scattering matrix can be transformed to a diagonal matrix. This is e.g. always possible for a lossless junction the scattering matrix of which has been proved to be unitary. The symmetry basis can then be chosen in such a way that the elements of S', eq. (19), obey the equations
|sfcfc(i)| = l ; ski{i) = 0, k ^ l . (21) for all i.
III. 3. Field distribution in a symmetry plane
By the adoption of the scattering matrix for the description of waveguide junctions we have deliberately relinquished the aim to obtain any in-formation on the field distribution inside the junction. One is forced to accept this for most practical junctions because of the usually unsurmount-able mathematical difficulties encountered in the process of solving Max-well's equations subject to the boundary conditions imposed by the struc-ture.
The symmetry of a junction can, however, provide us with some know-ledge about the field in the interior; in particular some information can be obtained about the distribution in a symmetry plane, point or axis of those fields that correspond to the individual coordinates of the symmetry basis. The insight gained in this respect, though scanty, sometimes enables us to predict qualitatively the effect of alterations in the structure that change the boundary conditions in the symmetry plane, point or axis.
We shall now work out in detail the properties of the field in a symmetry plane, referring the reader for the results in a symmetry point or axis to Dickei).
As reflection in a plane, applied twice in succession, leaves everything unchanged, it constitutes together with the identity operator a group. This group (elements I and F) has two one-dimensional irreducible repre-sentation, which obviously are the trivial and the alternating representa-tions. Thus we have
i)i(J) = l , D,{F) = l; (22) Da(/) = 1 , Da(F) = - l . (23) The representations eqs (22) and (23) correspond to the familiar concept
of solutions even and odd respectively with respect to the symmetry plane. In fig. 3.1 these two different types of solution are illustrated for the electric field vector. Let the field vector on the left-hand side of the symmetry plane be resolved into a component E„ normal to the plane and a component Ej parallel to it. The corresponding components of the field
