R914 Philips Res. Repts 30, 357*-375*, 1975Issue in honour of C. J. Bouwkamp

EIGENMODES, QUASIMODES ANDQUASIPARTICLES *)

by N. MARCUVITZ

Polytechnic Institute of New YorkFarmingdale, N. Y., U.S.A.

(Received February 18, 1975)

~bstract

A generallinear field supports both a discrete and continuous spectrumof modes or source-free field solutions. Although only eigenmodes arenecessary for a complete representation of fields excited by sources,noneigen discrete modes (quasimodes) provide a useful and rapidlyconvergent alternative to the continuum eigenmode part of a fieldrepresentation. Quasimode field representations, and their interpreta-tion in terms of quasiparticles, are discussed both generally and for thespecial case of a vector electron plasma field, for which the completeset of eigenmodes is found via a resolvent or characteristic Green's func-tion method.

1. Introduetion

As is known, many wave types or modes are capable of excitation by sourcesin general linear systems. Some are discrete or continuous eigenmodes that,taken together, provide complete representations of excited fields. Other typesmay be noneigenmodes, not members of a complete orthogonal set, but theynevertheless play a useful role in field representations. If the system admits acontinuous eigenspectrum, both eigen and noneigen modes are usually presentand one must determine which of the possible discrete modes are eigen andwhich are not. Discrete modes may be chosen to be either oscillatory, guidedwave, etc., each being distinguished by a specific wave structure or polarization.In the following we shall restrict the discussion to oscillatory modes withprescribed spatial periodicity. Although the analysis will be applicable to generallinear systems, a specific composite (electromagnetic-charged particle) systemwill be treated in some detail to illustrate how one distinguishes between eigenand noneigen discrete modes, howone determines complete sets of eigenmodesby Green's function techniques, and how one utilizes the noneigen modes.Since discrete modes are source-free wave solutions, we shall first review thegeneral features of their determination.

*) It is with great pleasure that I offer this contribution as part of a well deserved testimonialissue for Chris Bouwkamp. These comments on modes recall for me Chris' elegant useof eigenmode techniques in his classic treatment of diffraction by a circular aperture.

358* N. MARCUVITZ

Source-free, homogeneous, and stationary linear systems can be generallydescribed in terms of homogeneous first-order field equations of the form

L(\l , i~)'1P(r, t) = 0i bt

(1)

where L is usually an n X n matrix operator and 'Ijl an n-component wavevector.Such equations admit plane-wave solutions of the form exp [i(k. r - wt)]where the spatial and temporal periodicities, k and os, are related by the de-terminental equation

detL(k, w) = O. (2)

Oscillatory solutions of (1) have the form

'Ijler, t) = P..(k) exp [i (k • r - w..(k) t)] (3)

where for k given w..(k) defines the ath root of the determinental equation (2)and distinguishes the dispersion relation for the ath oscillatory mode; PaCk)is determined from eq. (1) on use of (2) and (3). As is generally known, theoscillatory modes (3) give rise to a complete set of orthogonal eigenmodes inthe polarization (spin) space spanned by the eigenveetors Pik), a = 1, 2,... , n, for fixed k.The relevant eigenvalue problem is readily obtained by decomposition of

the matrix operator L into spatial and temporal components, viz. 1)

L=M(\l)+ W~,i i bt

(4a)

where M and Ware n Xn matrices. Thus for oscillatory solutions (3), eq. (1)assumes the form of an eigenvalue problem:

MPa = Wa W1Pa

where M, Pa, Wa are k-dependent; the adjoint eigenvalue problem is

(4b)

(4c)

where M+, W+ are transposed conjugate (Hermitean adjoint) matrices andPa +, Wa* are the adjoint eigenveetors and eigenvalues. From eqs (4) one con-ventionally obtains the biorthogonality property of the eigenveetors as

(Sa)

where ( , ) defines the Hermitean inner product in the space spanned by thevectors Pa, Na is a k-dependent normalization constant, and 6a,/) is the Kro-necker delta which is unity if Wa = WIJ and zero otherwise. It is desirable torephrase both the orthogonality property (Sa) and the completeness of the

EIGENMODES, QUASI MODES AND QUASIPARTICLES

IJl" in terms of a "completeness relation", which provides a representation ofthe identity operator as

from which one can infer (Sa). In general the "summation" index spans adiscrete oeand a continuous oe' spectrum of eigenvalues cv,,; in the case ofcontinuous indices oe',fJ' the Kronecker delta in (5a) is to be replaced by adelta function <5(oe'- fJ').For n finite and k given, the zeros cv" of the determinental equation define

the n discrete eigenvalues of the operator M (provided M is a "complete" ornormal operator). For infinite 11, the picture may be quite different. In thislatter case, some of the zeros of eq. (2) distinguish discrete eigenvalues of Mbut others do not. Since the zeros of the determinental equation yield dis-persion relations for the discrete oscillatory modes which the system can sup-port, the question arises as to how to distinguish the zeros characteristic ofeigenmodes and those characteristic of noneigenmodes. This problem arises ina number of different fields; its importance stems from the need to ascertainthose zeros of (2) which correspond to source-free solutions that are membersof a complete eigenset and those that correspond to noneigen solutions andhence are not to be included in the complete set. Depending on the field, thenoneigen solutions are termed leaky-wave, complex-resonance, radioactive-state,Landau, etc., solutions. In the following we shall term them quasimode solu-tions. They arise only when the operator M possesses a continuous spectrumof eigenvalues, or equivalently if the operator L(k, cv) is a nonanalytic func-tion of cv.Quasimode solutions generally correspond to complex roots of the deter-

minental equation (2).When M =M+ is an Hermitean operator and hence itseigenvalues cv" = cv" * are real, the presence of complex roots clearly under-scores the noneigen nature of the quasimode solutions. Despite their lack ofmembership in a complete eigenset, quasimodes frequently play an importantrole in field representations. The solution of an initial-value problem for thefield defined by eq. (1), or of a source-excited inhomogeneous version of eq. (1),can be represented as a superposition of eigenmode contributions via a wellknown procedure. Such field representations in general comprise both discreteand continuous eigenmode contributions. Quasimodes provide an alternative,and usually rapidly convergent representation of the continuous eigenmodecontribution. In fact in problems wherein only a continuous eigenspectrumexists, one or two quasimode types may provide an adequate representationof the field response to arbitrary excitation; this feature represents one of theimportant applications of quasimodes.

359*

(Sb)

360* N. MARCUVITZ

In sec. 2 we shall introduce the characteristic Green's function, or resolventoperator, for the linear field described by eq. (1). This operator, whose singular-ities are determined by eq. (2), permits a ready distinction between the eigenand non eigen 100ts of eq. (2); also alternative representations of such Green'sfunctions clarify the utility of quasimodes in the solution of field problemswhen a continuous eigenvalue spectrum exists. In sec. 3 we consider, as aspecial example, the linearized electron plasma field and summarize the charac-teristic Green's function technique for determination of the complete set ofeigenrnodes and as well the quasimodes of this field for given k, In sec. 4quasipartic1e concepts are introduced to illustrate how k-dependent quasi-mode wave packets evolve in space and time.

2. Characteristic Green's function operator

The deeper significanee of the determinental equation (2) becomes evidenton introduetion of an appropriate source-excited or Green's "function"problem. For fields of spatial form exp (i k •r) one can define a unique timedependent Green's function (matrix operator) G(t, t') for eq. (1) by

L(k, i~) G(t, t') = [M + W ~JG(t, t') = <5(t- t') (6a)()t i bt

and the requirement that

G(t, t') = 0 for t < t'. (6b)

The Laplace representation

1 00+ la

G(t, t') = - J G(w) exp [-i w (t- t')] dw,2n

- 00+ la

(7)

with a chosen to be a suitably large positive number so as to satisfy (6b),introduces the spectral operator G(w), which on transformation of eq. (6) evi-dently satisfies the operator equation

L(k, w) G(w) = [M - wW] G(w) = 1. (8)

The matrix operator G(w) = I/L(k, co), the so-called characteristic Green'sfunction or resolvent operator, is a singular function of complex w. It mani-festly has pole singularities at the zeros of det L(k, w) and, if L(k, w) is non-analytic in ca, branch line singularities. For kinetically described many-particlesystems, such as the electron-plasma example treated in sec. 3, G(w) possessesa branch-line singularity on the real co-axis. In consequence, a two-sheetedRiemann surface, with the two sheets connected via a branch line cut alongthe real co-axis, must be introduced to depict the dependence of G(w) on corn-

EIGENMODES. QUASIMODES AND QUASIPARTICLES

plex w. One sheet of the Riemann surface, the so-called "physical branch", isso defined that it contains only the eigenvalue type of pole singularities ; theother "nonphysical branch" will contain the noneigen singularities.The singularities of G(w) depict, among others, the natural resonances or

eigenfrequencies of a system, and thus there should be a completeness rela-tion associated with G(w). This well known completeness relation can be simplyinferred by division of eq. (8) by 2niw and counter-clockwise contourintegration of the result over an infinitely large contour C centered at w = 0,whence there is obtained (cf. ref. 2)

1 =__1_ ,c W G(w) dw, (9)2ni j

c

where the physical branch, on which the contour C is taken, is defined by therequirement that

f_M_:_(W_) dw-+O.

c

The knowledge of all singularities of G(w) on this physical branch leads, onevaluation ofthe residue and branch-line contributions from the contour integralin (9), to a result of the form (5b) - i.e. to the identity operator which revealsthe eigenvectors, and their normalizations, for both the discrete and continuousspectrum.The above distinction between the physical and nonphysical branches of the

Riemann surface for G(w) = I/L(k, w) provides the basis for distinguishingbetween roots of detL(k, w) that correspond to eigenrnodes and to quasi-modes. The poles of G(w), or roots of det L(k, w), that lie on the physicalbranch distinguish the eigenmodes, whereas the poles, or roots, on the non-physical branch define the quasimodes. Figure 1 is illustrative of a singularitypicture for a typical component of the matrix operator G(w). The presenceof a single pole (root) at w = 0 indicates the system in question supports only

"'-plane "'-plane

branch line branch line®

® ®®

® ® ®®

(a) physical sheet (b) na"physlcal sheet

Fig. 1. Singularities of G(w); circles: poles, heavy lines: branch lines.

361*

(10)

362* N. MARCUVITZ

a single eigenrnode, whereas the multiple complex poles (roots) on the non-physical branch are indicative of multiple quasimodes.

The preceding observations imply the identity

1 f ~ v.r:1=-- WG(co)dco= W---2ni £........J Na

C a,a'

(11)

which follows from the explicit evaluation of the contour integral in (9)., Forthe example of interest (cf. sec. 3 and fig. I), G(co) has both pole and branch-line singularities on the physical sheet. The residue contributions to the integralin (11) from the poles of G(co) will be shown to yield discrete (ex) terms in theeigen representation in (11) while the branch-line 'contribution will yield thecontinuous spectrum (ex') terms.It is of interest to compare the implications of the representations in (7) and

in (11). From (7) one infers, since -i W G(t'+, t') = 1 is implied by eqs (6),that

1 oo+la

1=- J WG(co)dco2ni-oo+la

(12)

where an exponential convergence factor exp (-icoLl) (with Ll = 0+) is im-plied for Im co < O. On the other hand, on contour deformation eq. (11)implies

1 oo+~ oo-~

1 = - J WG(co) dco- _1_ J WG(co) dw2ni 2ni

-oo+~ -oo-~

(13)

where the real positive number a is chosen sufficiently large so that deformationof the contour Cin (11) to that given in (13) is such as to retain all G(co) singular-ities within the segment of the co-plane bounded by the lines co= +ia andco = =ia. If the integrals in (13), taken over the indicated upper and lowercontours on the physical sheet, are deformed into the upper and lower shoresofthe branch line (cut) along the real co-axis, one obtains, on taking cognizanceof the G(co) poles in the physical sheet, a residue sum and a branch-line integralthat reproduce the completeness relation in (11). On the other hand if in theintegral representation (12) the contour along Im co= +a in the physicalsheet is deformed through the cut along the real axis onto the lower half ofthe unphysical sheet and into a contour approaching co-- -iet:), then becauseof the implied convergence factor in (12) the integral over this deformed con-tour vanishes. The residue contributions at the G(co) poles encountered in thedeformation arise from poles coa+ which lie in the upper half of the physicalsheet and poles cop in the lower half of the unphysical sheet; this residue sum

provides an alternative representation of (12) that may be written in the form

I w Pa. Pa.+ I W Pp Pp +1= +

Na. Np(14)

EIGEN MODES. QUASIMODES AND QUASIPARTICLES 363*

,,+ p

In (14) the discrete quasimode contributions at W = wp(k) are phrased in thesame form as the discrete eigenmode contributions at w = wcx+(k). Complexpoles at w,,+ are indicative of unstable modes (if they exist) whereas the com-plex poles wp imply damped modes; furthermore, it should be observed thatthe lJ'" possess orthogonality properties whereas the lJ'p do not (in the usualsense).The identity operator representations in (11) and (14) appear in clearer per-

spective if alternatively phrased as Green's function (operator) representations.Thus, paralleling the familiar Green's function representation in (7), one hasinstead of (11) for t > t' the eigen representation

I Pa.lJ'cx+G(t, t') = i exp [-i w" (t- t')]+

N""

I lJ'",P",++ i exp [-i W", (t- t')].N".

(15)

ex'

Correspondingly, instead of (14), one has for t > t' the mixed representation

I P" P,,+G(t, t') = i exp [-i W" (t- t')] +

N",,+

I PplJ'p++ i exp [-i wp (t- t')].Np

(16)

p

Although, because of the notation, (15) and (16) appear very similar, it shouldbe noted that (16) is usually far more convenient to use in applications becausethe f3 sum is generally rapidly convergent whereas the IX' "sum" in (15) reallyis an integral. Multiplication of (15) or (16) by a relatively arbitrary excitationvector CPk(t') and integration over t' yields a representation of the transformedresponse lJ'k'(t) to the source CPk(t). Further multiplication by exp (ik. r) andintegration over all k then leads to a representation of the space-time dependentresponse 1p(r, t) to the source vector cp(r,t), as defined by the field equationL 1jJ = cp.3. Electron plasma field

As an explicit illustration of the above we shall evaluate the characteristic

364* N. MARCUVITZ

Green's function operator and complete set of eigenveetors for a collisionless,isotropic, electron plasma field 3). At the linear kinetic level a sourceless non-relativistic electron plasma is described by the normalized Maxwell- Vlasovequations:

bE f--c\lxH- vfdvbt

= 0,

bHe \lxE+-

bt

- \l .J«. E + Gt + v. \l)f = 0,

(17)=0,

where the vector electric field E = E(r, t) is normalized to the critical fieldEe = m mpa/e, the magnetic field H = H(r, t) to (80/#0)1/2 Ee, and the veloc-ity integrals of the electron distribution functionf=f(v, r, t) and its homo-geneous background component fo = fo(v) are both normalized to the back-ground plasma density no; e, m are electron charge and mass; 80' #0 are thepermittivity and permeability of vacuum; c = 1/(Jlo 80)1/2 the speed of lightand v the electron velocity variable are both normalized to the electron thermalspeed a = (2kb T/m)1/2; mp = (no e2/m 80)112 is the electron plasma frequency;time t is normalized to I/mp and distances r to the Debye length a/mp; all inMKS units.Equations (17) are manifestly expressible in the matrix form (1) if one defines

1 b \l- -1 -e-x1 iv Ei bt

\l 1 <>L- e-xl - -1 0 ,"P- H (18a)

i <>t

i \l .r; 0 -i (bbt+ v , \l) r, f

with the matrix product L"P reproducing the left-hand members of eq. (17).In conformity with general matrix notation, the elements of the matrix L arepartitioned into vector or dyadic elements in 3-vector or co-vector space;1_ (c5IJ) is the unit dyadic in 3-vector space representative of E or H,J, - (c5(v- v') is the unit "dyadic" in an co-vector space representative ofthe velocity variable v, and v and \l .J« are co-vector elements. On trans-formation to the k, m basis one finds

[

-m1L(k, m) - k c x I

i \l .i:

iv ]

-(m _Ok. v) r,(18b)

-kex1-m1

o

EIGENMODES, QUASIMODES AND QUASIPARTICLES

from which the component operators M and Ware readily identified; one notesthe weighting operator W = 1. The characteristic Green's function G(co), i.e.,the inverse of L(k, co) as defined in (8), is evidently representable by the matrix

whose dyadic, vector, and scalar elements can be inferred from (8). Thus forexample the first column elements are defined for Im wik =1= 0 by

whence[

kx(kxl) c2 1 f v \lvJo J-co 1+ - - dv . Gu = 1

CO2 to k u- colk

kcxlG21 = --- . Gu,

co

For isotropic Jo(v) it is convenient to introduce vector decompositions, longi-tudinal and transverse to the propagation vector k = k ko, by

v = uko + vT,lL = ko ko = 1- IT, IT = -ko x(ko x I),

whence on decomposition of Gll into (diagonal) scalar longitudinal Gll andtransverse G11 components via

Gll = Gll IL+ Gu IT,

one finds from (20) (on integration by parts using Jo -)- 0 as v ---+ (0)

1 [ 1 f 'bJol'bu J-1Gu=-- 1-- dv,W k2 U - wik

coIm- =1= 0,

k

~ 1 [ k2c2

1 f Jo dv J-1 coGu =-- 1- -- + - , Im- =1= O.w co2 co k u - osjk k

Since the integrands in (22) have a singularity at u = wik, the integrals in(22), and hence Gll, are undefined on the real co-axis for real k. To defineunambiguously Gll(w) the co-plane may be viewed as a two-sheeted Riemannsurface with a branch line on the real eo-axis. To distinguish the physical sheet,on which Gu(co) vanishes at co ---+ 00, requires the ability to decompose theintegrals in (22) into parts regular in the upper and lower halves of this sheetand vanishing at 00.

With the notation

365*

(19)

(20)

(21)

(22)

366* N. MARCUVITZ

PIemelj's theorem provides "regular decompositions" of 'Y}(u) into parts'Y}±(v) regular, respectively, in the upper/lower half planes of 'V and vanishingat 00 in their half planes of regularity, viz.

eo1 f 'Y}(u)

'Y}±('JI)= ± - -- dw,2ni u- 'V-<Xl

Im'JI ~O, (23a)

provided 'Y}(u) is integrable from - 00 < u < +00. It is to be noted thatalthough the decompositions 'Y}±('V) in (23a) are regular for Im 'V ~ 0, respec-tively, in the physical sheet thereby defined, they may be "analytically con-tinued" into the unphysical sheet where they may have singularities. Onallowing 'V in the physical sheet to approach the real axis, one obtains from(23a) in the limit Im'V = 0:

P f<Xl 'Y}(u) 'Y}('V)'Y}±('V)= ±-. --du+-,

2m U-'V 2-<XlIm'V= 0, (23b)

where P denotes the Cauchy principal part. As a special case of (23), one hasthe delta function decompositions

1 1t5±('V- u) = ± - --,

2ni u- 'VIm'V ~ 0 (24a)

P 1 t5(u-'V)=±---+---

2ni u- 'V 2'Im'V= O. (24b)

With the regular decompositions (23) one can now unambiguously extendthe definition of Gll in (22) to all values of 'V. The desired extension for thelongitudinal components Gal (a = 1, 2, 3) becomes for all 'V in the physicalsheet

ko koG111L = -k--±-C-)'- 'Ve 'V

G21 = 0, (25)

wheree(v) = 'Y}(u) Fo(vT),

e±('V) = 1 ~ 2ni'Y}±('V).

In a similar manner one generalizes the transverse Gal components in (22) to

where

~ 1GIl= - ---IT

k ê±(v)

~ 1G21 = - ckoxlT,

kv ê±(v)

~ fj(u) 2:n; c5±(u - v)G31 = + \J vT Fo(vT),

ê±(v)

(26)

EIGENMODES. QUASIMODES AND QUASIPARTICLES 367*

all v in the physical sheet, via:

1'f)(u) = - go(u),

k2

c2ê±(v) = v- - ± 2:n;i n±(v).

v

With the knowledge of Gll one can employ the completeness relation (11)to evaluate explicitly the eigenveetors Pa. and their adjoints Pa.+. If we denote

(27)

then from (11) and (25) one finds that the longitudinal Ea components aregiven by

I Ea Ea + * ko ko f ko ko f dv---=--- Gll(w)dw=-- ---.

Na 2:n;i 2:n;i v 8±(V)c c

For a "passive" plasma with io(v) MaxweIlian (for example), v 8+(V) has azero at v = 0, whence evaluating the residue at v = 0 and the branch-linecontribution along Im v = 0 within the contour C, one has

(28a)

where2:n;i'YJ(v)

-----=18(v)12

1 1 2:n;i 'YJ(v)8+(V) c(v)'

1 J 1 (Jio8(0) = 8±(0) = 1- - - - dv.k2 u (Ju

Similarly from the longitudinal component of G31 in (25) one obtains

368* N. MARCUVITZ

whence on evaluation of the residue and branch-line contribution within thecontour C

(28b)

IX

If for simplicity one adopts the normalization Ea+ = ko for the longitudinalcomponent of Ea+, then via eqs (27) and (28) one identifies a discrete longi-tudinal oscillatory eigenmode with eigenvalue v = 0, and eigenvector andnormalization given by

Na = e(O). (29a)

Similarly the branch-line contribution in (28) yields a continuum of eigen-modes with real eigenvalues - 00 < v < +00, and eigenveetors and normal-izations

(29b)

where1')(u)le(v)12 [c5+(U- v) c5_(u- V)]

qJv(u) = + ---1')('11) e+(v) e-(v)

= e+(u) c5+(v- u) + e-(u) c5_(v- u).

It is evident that, for the electron plasma being considered, det L(k, co)possesses only one zero on the physical sheet. However, as noted in sec. 2,det L(k, co)may have zeros on the unphysical sheet. Examination of e+(v) inthe lower half plane of the unphysical sheet reveals complex zeros at v = VI>

i.e. detL(k, co) does indeed have complex zeros at co= kv" Via analyticcontinuation into the unphysical sheet as sketched in connection with relations(12) and (14), one identifies from the expressions for G in (25) and fromthe v = v, residue contributions to the integral in (12) the longitudinal quasi-mode vector and normalization

-------~-----------~~~~.,. - - --., ~...,....---~\

(30)

EIGENMODES. QUASIMODES AND QUASIPARTICLES 369*

with 8+(VI) = O. It should be observed that if the electron plasma were un-stable (by appropriate choice of fo), there would be eigenveetors of the sameform as the quasimode vector in (30) but VI would lie in the upper half planeof the physical sheet.The integration procedure employed in eqs (28) and (29) to infer the longi-

tudinal waveveetors from the longitudinal Green's functions in (25) can berepeated for the transverse Green's functions in (26). Thus from (26) and (11)one infers from the transverse 611 the transverse completeness relation:

which by contour integration yields the branch line contribution:

IT JOO

[ 1 1 ]= 2ni ê-(v) - ê+(v) dv.

-00

(31a)

Similarly from the 621 component

I n,E<z+ * 1 f ~ c ko X IT f dv---=-- G21dw=---N<z 2ni 2ni V ê±(v)C C

00 •c ko X IT J [1 1 ] dv

= 2ni ê-(v) - ê+(v) -;.-00

(3Ib)

From the transverse G31 component in (26) one infers

(31c)

Equations (31) imply a continuous spectrum of eigenvalues -00 < v < +00

with eigenveetors and normalizations given by (if one adopts the adjoint nor-malization Ea+ = To):

[

Ck~:To 1P.'~ ik 'i,(u)T,~ ",T F,(y') ,

(32)

370* N. MARCUVITZ

where the transverse (to ko) unit dyadic IT = To' To' + To" To", and in (32)To is the unit vector To' or To" which defines the polarization of the two dis-tinct types of transverse eigenvectors, and where

For a stable plasma the form (26) of ê±(,,) implies that there are no discreteeigenveetors nor significant quasimodes ; i.e. no significant zeros of det L(k, co)arise from the transverse structure of the field.

Adjoint waveveetors P" +

The determination of the adjoint waveveetors lP"+, corresponding to P"and with components defined in (27), is based on the matrix Green's functionelements G33 and G22• From the inverse of the matrix operator L(k, co) in(I8b), on~ finds that G33 = G33(V, Vi) is defined by

(u-::!_) G33- _1_ \lvlo. (IL + h ).fv G33 dv = t5(v- Vi) (33)k CO k 1- P c2 / co2 k

whose form implies that G33 can be decomposed into parts even and odd inT .

V , VIZ., gv(u, Ui) T g.(u, Ui) T 'T

G33(V, v) = Fo(V ) - \lyT Fo(v ). v . (34)k k

The even part defines a longitudinal contribution gv(u, Ui) which may be de-termined by substitution of (34) into (33) and integration over vT as

(u- v) s, + 'YJ(u)[f e.du +~] = t5(u- u'), Im v =1= 0; (35a)

while the odd part defines a transverse contribution gv(u, Ui) determined onsubstitution of (34) into (33), niultiplication by v", and integration over v", by

(u- v) i.+ ij(u) f i- du = t5(u- u'),v (1- C2/V2)

Im v =1= O. (35b)

EIGENMODES. QUASIMODES AND QUASIPARTICLES

Notation is the same as in eqs (23) et seq. Integration of eqs (35) over u andelimination of the g. and ë. integrals from (35) then leads to the explicit ex-pressions:

1 [ 1J(u)u'[v (u' - v) ]g.(u, u') = -- r5(u- u')- ,

u- v 1+ J [1J(u)/(u- v)] duIm v i= 0, (36a)

1 [ fJ(u)/(u' - v) ]g.(u, u') = -- r5(u- u')- ,Imv i= O. (36b)u- v . v- eZ/v + f [fJ(u)/(u- v)] du

Use of the regular decompositions (23) permits one to extend eqs (36) to all vin the physical sheet, viz.

[1J(u) u' r5±(u' - V)]ss». u') = ± Zni r5±(u- v) 6(u- u') =F 2ni , (37a)

ve±(v)

[n(u) r5±(u' - V)]

i.(u, u') = ± 2ni r5(u- v) 6(u- u') =F 2'Jti .ê±(v)

From the completeness relation (11) and from (27) and (37a) one infers bymeans of the longitudinal part G3l of G33 the following longitudinal relationfor la:

I lala+* 1 f Fo(vT) f-- = - -. G33L dco =- --.- g.(u, u') dv

Nrt 2nz 2nzc crt

which, on evaluation of the residue at the pole of g. and the branch-line con-tribution along Im v = 0, becomes

= 1J(u)Fo(vT) + Fo(vT) Xu e(O)

Comparison of (38) and (29a) then permits the identification of the .discretelongitudinal v = 0 adjoint eigenvector as

and, on comparison with (29b), the continuous adjoint eigenveetors as

371*

(37b)

(39a)

(39b)

372* N. MARCUVITZ

where

[c5+(U - v) c5_(u - v)J 113(1')12 cpv(U)

àV(u)= + --=--.13+(1') 13-(1') '1](1') 'I](u)

One can now verify that the eigenveetors and their adjoints satisfy the bi-orthogonality property (5a)..The analytic continuation procedure used for finding (30) can be employed

with the representation (12) involving G3l to obtain, via contour deformationand evaluation of the residue at the complex zero Vi of 13+(1') in the unphysicalsheet, the longitudinal adjoint quasimode wavevector:

(40)

as used in the representation (I4).A procedure similar to the above, but based on the transverse part of G33

and as well on G22, leads to the determination of the transverse adjoint eigen-vectors as

Tocko xTo

(41)

vT • To CPv+(u)

ik

where To = To' or To" and

6+(u- v) 6_(u- v)cp v +(u) = ê+(v) + ê-(v) .

From the magnetic field G22 representation and (11) one also identifies astatic magnetic field eigenvector with eigenvalue v = 0 as

(42)

4. Quasiparticles

The discrete oscillatory quasimodes considered above are distinguished by acomplex frequency W{J= ro{J - iy *). For each mode type (J, knowledge of thek-dependent w{J permits the determination of the phase velocity, wp/k, of a

*) Note that wp is the real part of the complex frequency wp.

EIGENMODES. QUASIMODES AND QUASIPARTICLES 373*

single k wave, or the group velocity \lk wp of a wavepacket centered at k.Fields excited by sources or evolving from a prescribed initial state usuallygive rise to wavepackets, each distinguished by a group velocity and wave struc-ture characteristic of the mode type. Finitely extended wavepackets can beregarded as composed of "point" wavepackets, or quasiparticles, each with aposition rtCt) and momentum kl(t). To elucidate this view we shall first employthe completeness relation (14) to decompose fields into their constituent quasi-mode types. Such quasimode representations are particularly useful for systemsadmitting a continuous spectrum of oscillatory eigenmodes.

For example, in linear systems wherein the eigenspectrum is continuous, let'1fJ(r,O)represent an initially prescribed field at t = O. From (14), on multi-plication by exp (i k • r) and integration over k, one obtains 4)

L PoCk) dk'1fJ(r,O)= J -- ap(k, 0) exp (i k , r)--

NP(k) (2:n:)3(43)

wherep

ap(k,O) = J (Po +, W'1fJ(r, 0)) exp (-i k. r) dr.

The 13th quasimode contribution to (43) at time t then follows from eqs (1)and (4) as

whereJ Pp(k) dk

'1fJp(r,t) = -- ap(k, t) exp (i k . r) --NP(k) (2:n:)3

(~~ + WP(k)) ap(k, t) = O.i "Dt

(44a)

(44b)

Equation (44a) is descriptive of a wavepacket if the latter has formed by time t.A coarse but useful description of the evolution of this wavepacket as a func-tion of r, t is provided by the "energetic" measure of '1fJp(r,t) given by theHermitean inner product ('1fJp, W'1fJp). To introduce this measure one definesthe adjoint to the representation (44) of '1fJp,obtained from (14), as

dk''1fJp(r',t) = J Pp +(k') ap"(k', t) exp (i k' • r') (2:n:)3 (45a)

where

\,(45b)

/

Using eqs (44), (45), and the normality property (5a), one finds with ril = r - r'and kil = k- k':

374* N. MARCUVITZ

()dk dk"

"Pp(r',t), W"Pp(r, t) = f f ap(k, t) ap+*(k', t) exp [i (k. r" + k" . r')]--(2:7t)6(46a)

(46b)f dk= Fik, r', t) exp (i k . r")--

(2:7t)3

where the spectral measure Fp is defined bydk"

Fik, r, t) = f ap(k, t) ap+*(k', t) exp (ik" • r) --. (47)(2:7t)3

The spectral function Fp(k, r, t) will be shown to be interpretable as a time-dependent density in a k, r phase space used to depict the dynamics of quasi-particles of momentum ket), position r(t), and "energy" WO. To deduce thisinterpretation we shall assume a weakly inhomogeneous background so thatwp = wp(k, r) becomes weakly dependent on r; for simplicity of notation, thethereby implied dependence of ao and ap+ on r will be left implicit. Fromeqs (44b) and (45b), one infers

[~~ + wik, r)- wp*(k', r')] ao(k, t) ap+*(k', t) = O.i ()t

In successive steps one then obtains

f f [~:t + woCk,r' + r")- wp*(k- k", r')] ao(k, t) ap+*(k', t) X

dk dk"X exp [i(k" . r' + k , r")] -- = 0

(2:7t)6

and, on replacing k" by V'ji when acting on exp (i k" . r'), and r" by V kjiwhen operating on exp (i k . r") only or by - V kji when acting only on ao ap+,and using (47):

feXP(ik.r")[~~+ WP(k,i'- ~k)_ WO*(k- ~' ,r')]FP(k,r',t)~ ...t ()t l t . (2:7t)3

=0,and by uniqueness of this Fourier transform, the defining equation for Fp be-comes

[~:t +wp(k,r- ~k)_Wp*(k_ ~ ,r)]Fp(k,r,t)=o. (48)

Setting wp(k, r) = woCk,r) - i roCk,r), one can expand (48) for systemswherein Wp is weakly dependent on k, r as:

[()~+ V k wp, V- V wp, V k + ... ] Fp = -2 r« Fp. (49)

EIGENMODES. QUASIMODES AND QUASIPARTICLES

I

Equation (49) is evidently a "kinetic equation for waves" (or quasiparticles),with higher-order diffusion terms in rand k-space omitted for simplicity. Inits indicated form, (49) defines Fp(k, r,t) as the k, r phase space density attime t of quasiparticles of position rCt) and momentum ket). Along the trajee-tory ("characteristic") defined by

dr _- = V' k wp,dt

dk- =-\1 Wpdt

the indicated kinetic equation (49) becomes simply

d-Fp =-2ypFpdt

which implies that, as one moves with a quasipartiele along the trajectory (50),quasiparticles are being annihilated at the rate 2yp per second. If Fp(k, r, 0)is chosen to be consistent with the prescribed initial condition 'IjJ(r,0), one canreadily deduce therefrom Fp(k, r, t) through solution of (49) by means of (50)and (51). Knowledge of how the quasipartiele phase space density evolves inr, t then permits calculation of the evolution of the "envelope" of the wave-packet (44a) from (46b) by

f dk('IjJp,W"pp) = Fp(k, r, t) -- ,

(2:n)3

whose interpretations as a quasipartiele "fluid density" is apparent.It should be remarked that the analysis in eqs (46)-(52) is patterned on a

procedure in turbulence theory wherein 'IjJpand ap are weakly correlated in rand k, respectively; in this stochastic analysis many of the approximationsimplied above are more evidently justified.

Acknowledgement

This work was supported in part by the Office of Naval Research underContract No. NOOOI4-67-A-0438-0016.

REFERENCES1) L. Felsen and N. Marcuvitz, Radiation and scattering of waves, Prentice Hall, Inc.,

1973, Ch. 1.2) B. Friedman, Principles and techniques of applied mathematics, John Wiley and Sons,

Inc., 1956, p, 214 et seq.R. Newton, Scattering theory of waves and particles, McGraw-Hill Book Co., 1966,sec. 7.3.N. Marcuvitz, Comm. on pure and applied Math. 4, 263-315, 1951; see secs 3a and 3b.

3) N. G. van Kampen, Physica 21, 949, 1955.K. M. Case, Annals of Physics 7, 349, 1959.N. Marcuvitz, Symposia mathematica, VII, Academic Press (to be published 1975) orPolytechnic Inst. of N.Y., EP Report No. 74-137.

4) N. Marcuvitz, IEEE Trans. on Electron Devices ED-I7, 252-257, 1970.

375*

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