Multiple-wave propagation and causality

ANNALS

OF

PHYSICS:

1,

1-45

Multiple-Wave

(1965)

Propagation

and

Causality

E. GERJUOY University

of Pittsburgh,

Pittsburgh,

Pennsylvania

The implications of the principle of causality (loosely stated, that the response to an incident signal cannot begin until the signal arrives) are investigated for the transmission and reflection of electromagnetic waves b!multiple-wave media (i.e., media, such as plasmas, which can propagate waves of several different wave numbers K(w) at the same frequency w). Although many of the results obviously have wider validity, for concreteness the paper takes the medium to be in fact a plasma, while for simplicity the analysis is restricted to the special case of one-dimensional wave propagation along .z, with the plasma confined to a finite slab 0 5 z 5 d. The plasma may be lossy, need not be uniform, and may be stable or unstable. The conclusions drawn include: Causalit,y can be preserved even though the transmission amplitude Z’(o) and reflection amplitude -4(w) have singularities in the upper half o-plane. Poles of a(,), T(w) m . I m w > 0 typically occur with slabs of unstable plasma. In stable plasmas, the existence of a branch point at Im w > 0 for one i?(w) implies there is at least one other K2(u) with the same branch point. In stable plasmas it is impossible to excite one such nonsingly-valued S2(w) wave without exciting at least one other such wave. With isotropic plasmas the singularities of T(w), A(w) in Im w > 0 are symmetrically located about the imaginary axis. Despite some complications associated with the limiting behavior of the dispersion equation at infinite ( w I, under ordinary circumstances, with slabs of stable plasma, A(w) and Z’(O) - 1 each should obey Kramer+Kronig relations. The results are illustrated by reference to two actual plasma dispersion relations-the warm collisionless plasma (stable) and the two-stream plasma (unstable). I. INTRODUCTION

AND SUMMARY

Although in many wave propagation problems it is customary and convenient to assume the waves are traveling through a medium which at each circular frequency o is represented completely by a single index of refraction n(w), there also are many circumstances for which this simplifying assumption is not justifiable. For example, actual plasmas, whether stable or unstable, generally are too complicated to be so represented; in particular, actual plasmas typically propagate multiple waves, i.e., propagate waves of several different wave numbers K(w) = wn(w)/c. The ultimate objective of this paper is to examine the implications of the principle of causality (loosely stated, that the response to an incident signal cannot begin until the signal arrives) for transmission and reflection of electromagnetic waves by such multiple-wave media. This question does not appear

2

GERJUOY

to have been examined previously; indeed the whole class of multiple-wave propagation problems has received little theoretical attention heretofore. Although for concreteness we speak of “plasma” throughout this paper, of course the results pertain to other media as well, e.g., in the propagation of light through isotropic dielectrics a laser-wherein the photon density can grow by stimulated emission-is the analogue of an unstable plasma. The results of this paper also could bear on the interaction of helicons with transverse acoustic waves in metals (1) . As will be seen, the implications of causality for transmission and reflection by multiple-wave systems are somewhat unexpected. For this reason Sections II-VI of this paper are devoted to a thorough discussion of the implications of causality for transmission and reflection by media wherein a single n(w) suffices. This discussion is the foundation for our examination of the multiple-wave case in Sections VII and VIII. It will be observed that even for single-wave media our conclusions concerning the implications of causality are not wholly conventional. Consider now the reflection and transmission of a plane pulse of electromagnetic radiation, e.g., a radar pulse, by an isotropic but not necessarily uniform plasma; for simplicity suppose-as is supposed throughout this paper-the propagation is one-dimensional along z. Let E(z, 2) be the electric field, polarized parallel to the z direction say; for harmonic solutions E(x, t) = E(x, W)e?“. Then when a single n(z, W) suffices, the differential equation obeyed by the transverse components E(z, U) is ‘g

+ ii2n2E = 0

where k = W/C. The plasma may be stable or unstable; since the plasma need not be collisionless, in general n’(z, W) is a complex number. In the absence of plasma n’ = 1, e.g., if the plasma is confined to the slab 0 _I x s d, then n2 = 1 for z < 0 and for x > d. For given real W, if n’(z, w) is reasonably well behaved and approaches unity sufficiently rapidly as 1x ) -P ~0, e.g., if ) n2(x, w) - 1 1 is integrable at sufficiently large ( x /, implying lim (z[n’(x) - l]} + 0, (2) z-*m

Equation (1) has a solution of the form1 E(z, w) = edkr+ A(w)e-“’ E(x, 0) = T(w)eik”

“‘-z

(3)

1 It is difficult to find a simple reference wholly appropriate to the case n’(z) not purely real, but the assertion in the text is fairly obvious, and can be inferred from powerful theorems on the asymptotic behavior of systems of real linear differential equations. Cf., e.g., E. A. CODDINGTON AND N. LEVINSON, “Theory of Ordinary Differential Equations,” problem 29, p. 104. McGraw Hill, New York, 1955.

MULTIPLE

WAVES

AND

CAUSALITY

3

where T(w) and A(w) represent respectively the transmission and reflection amplitudes of the plasma at frequency W. In general the functions T(w) and A (w), defined by (3) for real w only, have analytic continuations into the complex w-plane. One obvious subject for study, which has received considerable attention in the literature, is the connection between the properties of ,n’(z, w) and the behavior of T(w) or A (u) in the complex w-plane. Of special interest are the conditions n’(z, w) must satisfy in order to guarantee that T(W) and A(w) are bounded holomorphic (nonsingular analytic) functions of w in the upper half w-plane Im w > 0. These conditions on n’(z, w) are of interest because the assertion that T(w) and A (w) are bounded and holomorphic for Im w > 0 usually (Z-4) is regarded as an unavoidable consequence of the principle of causality for solutions of the wave equation. The same assertion, supplemented by the additional provision that A(w), T(w) approach constants A, , T, as j w ) + ~0 for Im w 2 0, guarantees (2-d) A(w) - A, and T(o) - To each obey the RramersBronig relations, i.e., guarantees that the real and imaginary parts of A (w) - A, are Hilbert transforms of each other. As it happens, the connection between the properties of n’(z, w) and the behavior of A(w) or T(w) for Im w > 0 seems to have been examined carefully only when: (a) n’(z, w) is (2) not necessarily real but is essentially independent of z as, e.g., when the plasma is a lossy but uniform slab in the interval 0 _S z 5 d; (b) n’(z, w) may depend on z but is necessarily purely real, as e.g., when the plasma is nonuniform but collisionless. Actually, however, case (b) has been treated mainly not in connection with plasmasbut rather (5) in connection with the nonrelativistic Schrodinger equation, where it corresponds to the problem of scattering by a Hermitian (real) potential. To be precise, the one-dimensional time-dependent and time-independent Schrodinger equations are respectively: (4a)

---n2d24 + (V - nw>J,= 0 2m d.9

With the substitution w = fik2/2m, Eq. (4b) becomes $+A?’

1#=O

211&V(Z) l-------ii%2 [

Equation (5) is identical with (1) if one puts 1 - 2mV(z) = n2(z , W) fi21c2 where of course now for given k in (6) the value of w in (6) and in (1) is ck, not fik2/27n.

4

GEHJUOY

cONI~ITIONS ox n?(w) To fill the obvious gap in the literature, Sections II-IV carefully examine the connection between the properties of n2(z, W) and the analytic behavior of A(w), T(U) when: (c) N~(x, W) is neither purely real nor independent of x as, e.g., when the plasma is nonuniform and lossy. This case corresponds as well to a complex potential V in Eq. (4b) as, e.g., in the so-called optical model of the nucleus (6). To avoid some mathematical complications associated with the fact that Eqs. (3) are only asymptotically valid in the limits x ---f f 00, we do not permit the x-dependence of n2(z, w) to be arbitrary, but confine our attention to plasmas having finite thickness only, i.e., we assume n* = 1 except in the slab 0 2 z 5 d. Also we must assume of course that n’(z, W) is a reasonably well-behaved function of x in 0 5 z 5 d, e.g., that n’(z, W) at fixed w is continuous and has a continuous derivative dn2/dx except at a finite number of points. With these restrictions we are able to prove that (even for lossy plasmas) the following three conditions on n(x, w) suffice to guarantee T(w), A(w) are bounded holomorphic functions for Im w > 0: (i) In the upper half plane (CQ = Im w 2 0) [n2(wl

+

and, on the positive imaginary

iwz)]*

=

n?( -

iwz)

(7)

axis (w 1 = 0) where (7) implies n2 is purely real, d(iw2)

(ii) In the first quadrant

WI +

( w1 1 0,

w2

2 0 2 0)

Im w"n"(w)

L 0

(iii) In the upper half plane (Im w > 0) n"(w) is holomorphic, erty that n’(w) approaches unity sufficiently rapidly as 1w 1 ---f lim ( n’(w) w+-=

(8)

- 1 1 5 -CCI w I2

(9) with the prop00, e.g.,

(10)

where p is a constant independent of w and of x. Moreover on the entire real w-axis n’(w), though not necessarily free from singularities (e.g., branch points) remains bounded except possibly near w = 0, where however w”n”( w) is bounded. Since the above conditions (i)-(iii) are independent of z, they apply equally well to uniform and nonuniform plasmas. As a matter of fact, except that we have stated them somewhat more carefully than usual, (i)-(iii) in essence are the conditions customarily (2) cited as sufficient for causality to hold in unbounded uniform plasmas, where there is no return reflection from the surface z = d. These same conditions suffice to guarantee n(w) - 1 obeys the KramersKronig relations. Actually the inequality (lo), which we use to prove that A(w)

MULTIPLE

WAVES

AND

c n

CSUSSLITY

and T(w) approach 0 and 1 respectively as j w j -+ CQin the upper half w-planeand therefore that A (0) and T(w) - 1 obey the Kramer+Kronig relations-does not seem to have been stated previously. The inequality (9), which is an assertion about w2n2 only, as one expects from the form of Eq. (l), is slightly more general than the typically quoted corresponding condition on ML(W), namely that Im MZ(W) _2 0 when Im w 2 0 (along with the implicit assumption that Re ML(U) 2 0 in the first quadrant); Ey. (9) is consistent with either choice of the sign of n = &Z/G. The result A (ti), T(w) have no singularities in the upper half o-plane is principally a consequence of (9), although (8) must be used. The condition (7) is imposed (4) primarily to guarantee Eq. (1) preserves reality, i.e., to guarantee a real incident eiectromagnetic wave on the plasma produces real transmitted and reflected waves; Eq. (7) implies that for Im o > 0 [T(w,

+ z&)J*

= T( -0-q + icoP)

(lla)

[A(w,

+ iw2)]*

= A( -Lt1 + i&)

(lib)

so that the singularities of T(w), A(w) in the upper half w-plane are symmetrically located about the imaginary axis, and need be investigated only in the first quadrant Re (cr 2 0. Moreover, it is sufficient to investigate only the singularities of T(w) because, as Section II proves, the singularities of A(w) and T(w) coincide and are of the same order whenever they do occur (,of course only when the aforementioned conditions on n’(w) are violated). PRESERVATION

OF

C;~USALITY

For the plasma slabs of thickness d on which we are concentrating, the principle of causality postulates that physically allowed solutions to Eq. (1) cannot be transmitted through the slab in a time less than d/c. As remarked following Eq. (3), this postulate customarily is thought to imply T(w) is bounded and holomorphic for Im w > 0. Thus the significance of our proof that conditions (i)-( iii) suffice to keep T(w) bounded and holomorphic for x-dependent refractive indices might be paraphrased as, “if the plasma is everywhere locally causal, then it is necessarily globally causal.” The dispersion relations encountered in plasmas (7-Q), however, frequently do not satisfy the above conditions (i)(iii). In particular n’(w) typically has branch points in the upper half w-plane, in violation of condition (iii) (see Section VIII); condition (ii) also is often violated. Correspondingly, it is possible to construct actual examples wherein T(w) has singularities in the upper half w-plane. Since we are not disposed to abandon our belief that information cannot be transmitted through the plasma at a speed exceeding 3 X 10sm/sec, we can only conclude that solutions consistent with causality can be constructed even when T(w) is not holomorphic for Im LJ > 0.

6

GERJUOT

The explicit demonstration that causality can be preserved even though T(w) and A(w) have singularities in Im w > 0 is given in Section V; however preserving causality when there are singularities in Im w > 0 necessarily has the consequence that the transmitted and reflected signals grow exponentially with time at long times t > 0. Therefore singularities of I’(w) and A(w) in 11nw > 0 are not physically unreasonable with unstable plasmas, which characteristically can propagate growing waves. With stable plasmas, on the other hand, one may hypothesize that T(w) and L4(0) must be nonsingular in the upper half w-plane, in which event the transmitted and reflected signals preserve causality without unbounded growth at times t > 0. Section VI shows that poles of T(w), A(w), in Im o > 0 indeed will occur with unstable plasmas; in fact with sufficiently thick (d -+ m ) slabs, poles will occur even for uniform (n independent of Z) unstable plasmas capable of propagating single waves only. With sufficiently thick uniform single-wave plasma slabs, a simple criterion for T(w), A (w), to have poles in Im w > 0 is that there be a line in the upper half o-plane on which arg C%?(U) E arg W%“(W) = 211aa, 11% an integer; it is pointed out that the usual definition of an unstable plasma amounts to the assertion that such a line exists. No actual single-wave plasma dispersion equations are examined in Section VI, but an actual unstable multiplewave plasma, whose dispersion equation does satisfy the criterion for T(w), A(w) to have poles, is discussedin Section VIII. RESULTS

FOR MULTIPLE-WAVE

PLSSMAS

The results quoted thus far hold strictly only for isotropic single-wave plasmas, becausethey follow from Sections II-VI whose starting point is Eq. (1) . However, Section VII makes it clear that-except for (hopefully unlikely) special circumstances-Sections II-VI and our inferences therefrom should apply to multiple-wave plasmas as well. As an illustration of this remark, under ordinary circumstances poles of A(U) and T ( CO)coincide and are of the same order for nonuniform multiple-wave plasma slabs (recall the statements following Eq. ( 11) ) ; still there are special complicating circumstances when this assertion is not true. The complications arise becausewith multiple-wave plasmas there are extra degrees of freedom requiring the introduction of supplementary boundary conditions-in addition to the usual continuity conditions on the tangential components of the electromagnetic fields-to make T(w) and A ( W) determinate. Consequently, in multiple wave plasmas there are circumstances when two waves of different wave number at the same w can cancel identically at the plasma boundary z = 0, meaning one can find nonvanishing sets of plasma waves which connect to an identically zero vacuum electromagnetic field; it is under such special circumstances that the poles of A(w) and T(w) need not coincide. Sections II and VII show that under ordinary circumstances with multiple-

MULTIPLE

WAVES

AND

CAUSALITY

7

wave slabs (as with single-wave slabs) poles of T(U) or A(w) occur at those special frequencies where equivalently: ( CY) the boundary conditions, including the supplemental conditions, do not uniquely specify the solution; (p) there exist purely outgoing harmonic solutions, i.e., solutions which in the regions to left and right of the slab have the form (17) below. However, in the special circumstances that at w = w0 nonvanishing sets of multiple-plasma waves can connect to identically zero vacuum electromagnetic fields at both z = 0 and 2 = d, the amplitudes A ((J), T(w) need not have poles at w = w. even though the boundary conditions do not uniquely determine the solution at such w = w. , Similarly, A(w) and T(w) - 1 should obey the Kramers-Kronig relations with stable multiple-wave plasma slabs (recall the statements preceding Eq. (II)), but there could be complications from the fact that in multiple-wave plasmas, as Section VIII illustrates, the limit of n”(w) as ) w 1-+ 00 need not be unity. In other words, some plasmas seemingly can propagate waves whose associated electromagnetic fields are purely transverse, but which nevertheless do not propagate with the free space velocity c at infinite w. For A(w) and T(w) - 1 to obey the Kramers-Kronig relations in their usual form, A (w) and T(w) must approach 0 and 1 respectively as j w 1--) COin Im w 2 0; the condition n”(w) -+ 1, though perhaps stronger than absolutely necessary, is the foundation for the proof in Section IV that A(w), T(w) have these limits. Section VIII further shows that the dispersion equations for multiple-wave plasmas-whether stable or unstable-typically have roots K’(w) which have branch points in Im w > 0. If such branch points were branch points of T(W), A ( W) as well, solutions preserving causality would grow exponentially with time at t > 0, as explained previously. Thus, certainly for any stable multiple-wave plasmas, the boundary conditions and dispersion equation F(K, U) = 0 must be consistent with the requirement that the electromagnetic waves within the plasma are single-valued functions of w everywhere in Im w > 0, even though individual roots K”(w) may be multiple-valued. From this requirement of single-valuedness one can infer that in stable plasmas the existence of a branch point at Im w > 0 for one n”(w) implies there is at least one other n”(w) with the same branch point. There is the further remarkable conclusion (which should be capable of experimental confirmation) that in stable plasmas it must be impossible to excite one of these nonsingly-valued waves without exciting at least one other such wave. Both illustrative multiple-wave plasmas discussed in Section VIII-the warm collisionless plasma (stable) and the two-stream plasma (unstable)-obey the single-valuedness requirement although in both cases the dispersion equation has two roots K,‘(W), IQ’(w) which are multiple-valued in Im w > 0. In the case of the warm plasma it has been argued that one of the two roots K’(O) of the dispersion equation (Kp2 say) is inconsistent with the approximations used to

8

CEHJCOY

derive the dispersion equation and therefore should he discarded. The considerations of the preceding paragraph imply either that this argument is incorrect, or else that the other root K,’ really is a singly-valued function of w. The two-stream plasma discussed in Section VIII illustrates the fact that with nonisotropic plasmas the singularities of T(w), A(w) are less simply mirrored in the imaginary axis than was asserted following Eq. (11). In particular, with slabs of this two-stream plasma, the poles of T(w) for incident right circularly polarized waves are the mirror images, in the imaginary axis, of the poles for incident left circularly polarized waves. With isotropic plasmas, however, whether single-wave or multiple-wave, the reality requirement does imply the singularities of T(o), A(w) in Im w > 0 are symmetrically located about the imaginary axis. PLASMA-SCHRODINGER

CORRESPONDENCE

It is noteworthy that even when n2(x, w) in the time-independent differential equation ( 1) obeys conditions (i )-( iii), the identical solutions (3) to the identical time-independent differential equation (4b) correspond in quantum mechanics to time-dependent solutions of (4a) representing instantaneous’ transmission through the potential layer of thickness d, in other words, to timedependent solutions having no immediately obvious connection with causality. This formal correspondence between Eqs. (1) and (4b) implies an occasionally useful correspondence between the singularities of A(W), T(W) for single-wave plasma slabs and the singularities of the scattering matrix for solutions to the time-independent single-particle Schrodinger equation. For example, if n’(z, W) corresponds to a Hermitian interaction, any poles of A(w), T(w) in Im w > 0 lie on the imaginary axis; if this interaction has no bound states, A(W) and T(W) have no poles in Im w > 0. We add that an actual n’(z, w) which satisfies (i)(iii), and yet through (6) corresponds in (4b) to a physically sensible purely real V(z) independent of w is the very commonly employed

representing a cold collisionlessplasma (lo), and corresponding to (13b) 2 L. A. MACCOLL, Phys. Rev. 40, 621 (1932). This result stems from the fact that infinite wave velocities are possible in the nonrelativistic Schrodinger case. The same fact is the source of the difficulties in developing-from strict causality-the theory of the scattering matrix for the nonrelativistic Schrodinger equation. See ref. 6.

MULTIPLE

WAVES

AND

9

CAUSALITY

where wp , defined by (13b), is the (circular) plasma frequency. To make I’ a function of x, it is necessary merely that the electron density N per cc be z-dependent. It also is noteworthy that-despite the physical reasonableness of the aforementioned hypothesis that with stable plasmas T(w), A(w) must be nonsingular in the upper half w-plane-with single wave slabs one can construct mathematical n*(w) for which A(w), T(w) h ave poles in Irn w > 0, although these n”(w) have no singularities in Im w > 0 and satisfy the usual criteria for stability (see Section VI). Physical examples of such n”(w) are not cited, and probably do not occur for actual single-wave plasmas. However, it undoubtedly is equally possible to hypothesize dispersion equations F(K, w) = 0 which normally would be characterized as corresponding to multiple-wave stable plasmas, but which nevertheless would yield poles of T(w), A(w) with some slabs. Thus, because actual multiple-wave plasmas can have very complicated and not always obvious properties, perhaps one should not overlook the possibility that the usual criterion for instability-which examines solely the time evolution in unbounded plasmas of initial fields proportional to ezXr, K real-is insufficiently comprehensive. We conclude this section with the observation that it would be delusory to pretend our analysis is entirely unexceptionable from a strict mathematical standpoint, but we feel certain that any result we term “proved” is basically correct and could be established rigorously. II.

CONNECTION

WITH

WRONSKIAN

We now proceed to prove the assertion that conditions (i)-(iii) guarantee T(U), A (0) for solutions of (1) are bounded holomorphic functions in Im w > 0. To do so, we first must review the connection between T(w) and the Wronskian (11) W( U- , U+ ) of certain solutions U- , U+ to (1) . Because we are confining ourselves to plasmas lying only in the finite interval 0 5 x 5 d, the form (3), instead of being merely asymptotically valid, now is an exact solution of (1) in the regions x < 0 and z > d. in other words, we have E(x, w) = eikz + A(w)eeikz

Xl
E’(z, w) = T(w)c?”

d
(14)

where x1 and XTare arbitrary chosen points, to the left and right of the origin respectively. Limiting the range of z to ~1 < x < x2 instead of - ~0 < z < 00 is unnecessary for real k, but convenient for complex k, as will be seen. On the real w-axis this solution E(z, w) to (l), and therefore the amplitudes T(w), A(w) as well, are specified uniquely by the form (14) together with the requirements

10

GERJUOY

that E and dE/dz be everywhere continuous. Otherwise, there would be another solution to (1) El(z, W) = eikE + Al(w)eeik” 2&=;;; (15) El(z, co) = T&)eik’ 2 implying in turn there is a solution E, = E - I& of form Ez(x, u) = A2(u)e-ik” = (A - Al)eCik” zl < z < 0 (16) E2(z, w) = Tz(w)eik” 3 (T - Tl)eik’ d d into a function which is identically zero for all z < 0. Thus if (14) were not unique, (1) would have a solution of form E(z, u) = e-ikz E(z, w) = T’(w)eik’

;;;;;

2

(17)

We shall prove below, however, that when conditions (i)-(iii) hold solutions to (1) of form (17) cannot exist for real k. For complex k = W/C in the upper half w-plane the solution (14) remains acceptable as z --$ 00 ; is not defined as z --+ - 00 where eikz diverges; but does remain valid for all z > z1 . Thus, just as when k is real, for complex k in the upper half w-plane (14) is a unique solution to (1) unless there exist solutions to (1) of the form (17). The advantage of dealing with slabs of finite thickness, rather than with plasmas of infinite extent, is that for complex k the above connection between the existence of (17) and the uniqueness of (14) cannot be immediately inferred from a merely asymptotically valid form (3) (which though presumably valid at z = - co for k real, diverges there when k has a positive imaginary part). When Im w > 0, Eq. (1) has a solution of the form u-(2, w) = eCikr

2 < 0

(18)

at z = - 00. The quantities u- and du-/dz which may be termed “outgoing” obviously are bounded for all z 5 0. Moreover, because of condition (iii), which means n”(w) is bounded, Eq. (1) cannot carry a bounded u- and du4d.z into an unbounded u- or du-/dx in the finite interval 0 5 x 5 d where n’(.z, w) # 1. Thus the solution U-(X, W) to (1)) defined by (18) for z < 0, is a bounded and otherwise well-behaved function of z for all - 00 < z < z2 . Furthermore, because n”(w) is holomorphic, u-(2, w) also must3 be a holomorphic function of w 3 For z >= 0 the function u-(2, du-/dz = -G/c. Were it not for clude u- is a holomorphic function (see footnote l), pp. 36-37 and merely need repIace Eq. (1) for

w) is specified by the conditions that at t = 0 u- = 1 and the fact that du-/dz at t = 0 depends on w, one could conof o from theorems proved by Coddington and Levinson problem 7, p. 40. To make these theorems applicable, one u by the set v = o-‘du/dz, dv/dz = - (wn*/c’)u.

MULTIPLE

WAVES

AND

11

CAUSALITY

in the z-interval - ~4 < x < zz . Similarly, Eq. (1) has an “outgoing” solution u+(x, w) at x = co which is holomorphic in the upper half w-plane for x1 < z < ~0, and which for z > d is defined by u+(x, cd) = eikz The Wronskian

x > d

(19)

of U- , U+ is (II)

For arbitrary functions U(Z), v(x), W(u, v) is a function of x. However, from (1) it follows that for any two solutions EI(z, w), &.(z, U) belonging to the same w

g W(E1 ,Ez) = 0 In particular, therefore, is a function of w only. WCs-,

u+)

W(u-

, u+) is independent

of z, i.e., W(u-

, u+) in (20)

EVA~LuATED

The solution U+ of (19) is unique by definition, the values ikz u+ = e du+ -= dz

since one simply

starts

with

ikeikt

at x = x2 say, after which the differential equation determines u+(z) at all values z < z2 . Because of the peculiar way we have defined E(x, w) in (14), namely E(z, w) - eikz is outgoing at x = f ~0, E(x, w) is not necessarily uniquely defined. But it is clear from (14) and (19) that, unless T(w) is zero or infinite, E(z, a> = T(w)u+(~ w>, w h ere for complex w the quantity T(w) now is defined by its analytic continuation from the real w-axis; for real w, as previously asserted, (14) does uniquely specify T(w). Hence E(x, U) is unique except at values of w where T(w) is zero or singular. Moreover, to the left of the origin u+(x, 6~) = I_

1

T(w)

Thus, evaluating

W(u-

ikr

e

,

A(u)

e--ikz

Zl
(22)

T(w)

, u+) at any point .zl < 2 < 0 2ik = __ T(w)

(23)

Since u- , ZL+ (and their derivatives) are holomorphic, Eq. (23) shows that the only singularities of T( u ) in the upper half w-plane are poles, at those values of w where W(U_) U-t) = 0 (24)

12

GERJUOY

Zeros of T(w) do not occur in the upper half w-plane, because W(u- , u+) is bounded. Consequently, R(x, U) is unique except where (24) is satisfied. In fact, if the Wronskian of L , u+ is identically zero, then U- and U+ must be linearly dependent (11) . Therefore, recalling (21)) where (24) holds u- = cu+

Xl < z < 52 (25)

and conversely, C some constant. From (25), after taking note of ( 18) and (19), it follows4 that the poles of T( w) in the upper half w-plane occur at equivalently: the roots of (24); the values of w at which (14) is not unique; the values of w at which ( 1) has a solution of form ( 17). The solution (14) would have the form (17), with T’ = T/A, if T(w) and A (u) were to become infinite together, so that in (14) the term eikr becomes negligible at z1 < z < 0. Thus the fact that the poles of T(w) coincide with the values of w where (17) holds can be understood on the basis of the following argument that T(w) and A(w) must become infinite together. In Eq. (14), if T(W) has a pole at some point w = w0 , then E(x, w,) is infinite at all points 2 > d, and in particular is infinite at x = d. Then according to the boundary conditions E(z, wO)is infinite at z = d on the plasma side of the boundary. ?jow supposeA (~0) in (14) is finite at 0 = w0. Then E’(z, w,) and dE/& will be finite On the plasma side of the boundary z = 0. But, as remarked following Eq. (18)) Eq. (1) cannot carry a bounded E and dE/dx at x = 0 into an infinite E and dE/dz at z = d. Hence if T(w) is infinite at w = w, , so is A(w), implying eikO* really is negligible in (14), as presumed in the preceding paragraph. Further, the poles of T(w) and A (u) must be of the same order at w = w0, since otherwise one could divide (14) by an appropriate power of (k - k,) and, by the same argument as above, infer that in the neighborhood of Ic = X-,the bounded differential equation ( 1) carries a bounded E at z = 0 into an unbounded E at x = d. This argument amounts to a somewhat less rigorous proof (than via introduction of the Wronskian, as above) that the poles of T(w) are associated with solutions of form (17) to (1). This argument does not show, as we have proved above, that when conditions (i)-(iii) hold the only possible singularities of T(W) for Im w > 0 are poles. In particular, because U- , U+ are unique (i.e., can have but a single value at each w) , we have proved T(W) is a single-valued function in Im w > 0. 4 This conclusion, and much of the remaining eralization, to the complex 7~’ case, of previously tentials. (B. Friedman and E. Gerjuoy, New York matics Research Group, Research Report CX-4 how to generalize the very elegant and powerful Rev. 89, 1072 (1953), because he makes strong use ity

material in this section, is simply a genestablished results for Hermitian poUniversity Washington Square Mathe(1952), unpublished.) It is less obvious methods of Van Kampen (6) and Phys. of energy conservation, i.e., of Hermitic-

MULTIPLE

III.

WAVES

PROOF

T(o)

AND

13

CAUSALITY

IS HOLOMORPHIC

We now shall prove that in fact (1) has no solutions of form ( 17) in the upper half w-plane when conditions (i)-(iii) are obeyed. Let w’ be the mirror of w in the imaginary axis, i.e., w’ = -wl + iw2 . Similarly the mirror of k = W/C is Ic’. Then the solution E(x, w’) to (1) having the form (14) is E(~, w’) = e--iklz-kzr + ~(~~)~~k.‘zfkz2 Zl
(27)

it is seen that E(z, w’) * satisfies the same differential equation (1) as does E (x, W) . Taking the complex conjugate of (26), E(x, a’)* = eik2+ &w’)*e-“k” T(x,

w’)*

=

Zl
d < z < 2.2

T(w’)*2””

(‘28)

which shows E(z, w’)* satisfies the same boundary conditions, i.e., has the same asymptotic behavior at x = =t ~0, as does E(z, CO).We conclude E(x, w’)* = E(x, w)

In1 w 2 0

(29)

which in turn implies Eq. (11). According to Eq. (II), if T(W) has a pole at w = w. = w1 f iws in the first + iwsin the second quadrant. quadrant, then T(w) also has a pole at w,’ = -wl ’ according to the previous section, there are solutions E(x, w,) and At w = wo, E(x, wo’) to (lb) of form (17). Moreover, by the same reasoning as led to (29) E(z, wo’) = E(z, a,,)* = eik’z’kzz

Xl
E(z, w,‘) = E(z, wo)* = T’(wo)*e-iklr-kZ”

d < x < x?

(30)

where in (30) we now explicitly are choosing Ic, = w,/c in the first quadrant, i.e., kl > 0, kp > 0. From Eq. (1) for these solutions E(z, w,) and E(z, wo’) d’E( w,) + ko’n’(w,)E(w,) Hz, wo’) dx2 [ E(z, wo) 9’ [

+ ?&“(w,‘) E( w,‘)

$ W [E(w,‘), E(w,)] + L%%‘(w,) - k:2n2(w,l)l Eb,)E(wo’)

1 1

= 0

(31a)

= 0

(31b)

= 0

(32)

14

GERJUOY

so that, integrating Eq. (32) over the range x1 < x < z?where E(x, w,), E(x, u,‘) certainly are defined, and recalling (7 ) , (27), and (30)) W[E(w,‘),E(~,)],,

- W[E(U,‘),E(~,)I~,

+ 2iIm ~i2c~zk~~y~z,w,!j~(~,~,)/2 = 0 (33)

Because each of E(z, wO), E( x, a,‘) is proportional to exp( - knx) at z > d, the first Wronskian in (33) is proportional to exp( -2/&x2). The point x2 has been quite arbitrary so far, and can be chosen as far to the right as we like. Thus the first Wronskian in (33) can be made arbitrarily small. Similarly, because each of E(x, w,), E(z, a,‘) is proportional to exp(&) at x < 0, the second Wronskian in (33) can be arbitrarily small. Thus Eq. (33) cannot hold unless 22 Im k,‘d(x, w,)l E(x, @,)I2 = 0 (34) sz1 But off the real or imaginary axis in the first quadrant Im k: = 2klkz > 0; also for z > d or for x < 0, n’(w,) has been postulated equal to unity. Hence, for & off the real or imaginary axis in the first quadrant, Im k02n2(x,wO) > 0 when x < 0 or z > d. Equation (34) cannot be satisfied, therefore, unless Im ko2n2(z,w,) < 0 somewherein the interval 0 5 z 5 d. Recalling our postulated relation (9), we conclude Eq. (33) cannot hold. Correspondingly there are no points w = w0 in the upper half w-plane at which (1) can have a solution of form (17) except possibly for w0lying on the real or imaginary axis. Therewith we have proved the only singularities of T( w) in the upper half w-plane lie on the real or imaginary axes. CASE ko2 PURE REAL

Supposenext Ic, lies on the imaginary axis, so that kt is pure real and negative. According to (8)) k,‘n’(z, w,) now is pure real and 5 0 at all z, so that (34) holds and (33) is satisfied. However, integrating (31a) from x1 to 22 and then integrating by parts, we have, using (30),

The first term in (35)) like the Wronskian in (33)) can be made negligibly small by choosing z1and z2far enough to the left and right respectively. Therefore Eq. (35) cannot hold if iLo2n2(z,w,) 5 0 everywhere, since this will make the left side of (35) intrinsically negative. We conclude that there can be no singularities of T(w) on the positive imaginary axis.

MULTIPLE

WAVES

AND

15

CAUSALITY

This completes the demonstration that T(w) has no singularities in the upper half w-plane, Im w > 0. As a matter of fact we also can prove that T(w) has no poles on the real w-axis. For real k, the Wronskians in (33) do not vanish. Instead we have, using (17) and (30)

Substituting

W[,%‘(U,‘),

E(w,)],,

= WIT’*e-ikoz,

T’eik“z]zz = 2ik, I T’(w,)

W[E(W,‘),

E(w,)],,

= W[eikor, e-ikoE]rl = -2iko

1’

(36)

Eq. (36) in (33) yields 1T’(w,) i2 + 1 + $ IrnLl 0

dzk~n2(z, w,) 1E(z, w,) 1’ = 0

(37)

where, as is perfectly permissible for real k, , we now have replaced z1 and x2 by - 00 and 00 respectively. By virtue of (9), Eq. (37) cannot hold for real k, 2 0. Thus everywhere outgoing solutions of (1) cannot be found for real w, i.e., in accordance with our previous discussion T(w) has no poles on the real axis. If in (31a) (instead of E(w,), E(w,‘) of f orms (17) and (30) respectively) we introduce E(z, w), E(x, w’) of forms (14) and (26) respectively, then for real w Eq. (37) is replaced by 1T(w)

1’ + 1A(w)

I2 = 1 - k Im 1,

dzk2n2(z, u) 1E(z, w) I2

(38)

For real k the integrand in (38) vanishes except over the region 0 s x 5 d occupied by the plasma. Moreover, because of (9), for real k 1 0 the value of 1 T(O) I2 cannot exceed unity, i.e., again we see T(w) has no poles on the real axis. The right side of (38) is rl, with the integral representing the losses in the plasma. If n2(z, U) is pure real, the plasma has no losses, and I T(u) I ’ + I A(w) I ’ = 1

(39)

as one expects. IV.

PROOF

T(o)

BOUNDED

AT

INFINITY

We shall show, using (10) in the form

that for Im u 2 0 lim T(w) = 1 o+m

(41)

Actually occurring indices of refraction in plasmas often, but not always, obey (10). In fact (see Section VIII below) n”(w) may not even approach unity as

16

GERJUOY

LL,3 M. However, since (10) guarantees (41), probably a much weaker inequality than ( 10) sutlices to guarantee T(w) ( or B (w) ) is bounded at infinity in the upper half w-plane. The reason a separate argument is needed to prove T(w) bounded at infinity (i.e., the reason one cannot apply the proof from the previous section that T(w) is bounded in the upper half w-plane), is that when j w ) is infinite: eikEvanishes for x infinitesimally to the right of z = d; ezk*is infinite for x infinitesimally to the left of x = 0; in a distance d the differential equation now may be able to carry an unbounded function into a vanishing function. Thus the arguments in Section II that U- , U+ are well-defined, and that singularities of T(U) coincide with solutions ( 17)) break down. Hence we cannot claim immediately that T(w) is regular at infinity, but instead must actually evaluate the limit of T(w) as w + m in the upper half w-plane. Introduce the free space outgoing Green’s function (12) G/(z, x’, w) satisfying

$$ + k2Gf

= 6(z - z’)

(42)

which: behaves like (is proportional to) eCkz as x + - co; behaves like eikz as z ---f + 03; is continuous and has a continuous derivative everywhere except at x = z’, where ClG dG 1 (43) dz z=t’+ - &Y z=zr- = As is well known (12) --ikzeikz’ GAx, x’, w> = ‘=Gr(z,

e z’,

w)

=

ikz

___

x < 2’ (44)

--ikz’

e

2ik

2 > 2’

Moreover it is obvious from (44) that Gf is well-defined for all z, x’ provided In1 co1 0. In terms of Gf , the solution (14) obeys the integral equation (IS) E(x, cd) = eikr + ld dz’Gb,

z’, co)[k2 - k2n2(d, w)]E(z’, w)

(4.5)

From (44) and (45) for x > d E(z, w) = eikz + &

eikz Id &‘[k2 - k2n2(z’, co)]ewikt’E(~‘, W) d < z

(46)

0

where, for 0 $ z 5 d E(z, w) = eikz + $k eikr 1’ dz’[k’ - k2n2(z’, ~)]e-~~*‘E(z’, W) 0 I

1

2tk

e--ikz

’

sz

dz’[k’ - k2n2(z’, ~)]e~~“E(z’, w)

(47) 0 6 x <= cl

MULTIPLE

Equation

WAVES

(47) can be rewritten

e-ik2E(~, 0) = 1 + $% j’

AND

17

CAUSALITY

as

dz’[P -

k2n2(z’, ~)]e+“E(z’,

w)

0

+ &

(48)

sd dz’[2G2- k2n2(z’, ~)]e?~~‘E(z’, I

O~x~d

w)eZik(z’-z)

In the upper half plane Im I% 2 0, the exponential factor exp [2ik(.z’ - z)] I in the second integral on the right side of (48), where x’ 2 z. Thus, calling u(z, w) = PE(x, for sufficiently

Equation

w)

1

(49)

large 1 w j Eqs. (40) and (48) imply

(50) means that for sufficiently

(z&w) But, comparing

large ( k /

Imax5 (1 - k2>’

0 6 .z 5 d (51)

Eqs. (46) and (14) T(w)

-

1 = kk Sd dz’[k2 - lcZn2(x’, o)]v(x’,

cd)

(52)

0

so that as 1 k I+ I T(w)

00, using (40) and (51) in (52)

- 1 I s 2+j

$ d / 4.2, w) lmax 5 2+

f

(

1 - &

>

-l

(53)

Since q and d are finite constants independent of k, as 1 w 1-+ w the right side of (53) becomes arbitrarily small, i.e., T(w) -+ 1, Q.E.D. Similarly, one can prove 9 (w) -+ 0 as w -+ 00 in the upper half w-plane. Sections II-IV have proved A(w) and T(w) - 1 obey the Kramers-Kronig relations when conditions (i)(iii) hold. V.

TIME-DEPENDENT

SOLUTIONS

We now shall examine the construction of time-dependent solutions in terms of the time-independent solutions to (1) we have been discussing. Let EJz, t) be the incident pulse, in the region z > 0. Then we may write

&(qt) = Eo(t- z/e>= & s,dd’(~)ei”““”

(54)

18

GERJUOY

Choosing the origin of time so that the pulse reaches z = 0 at the instant t = 0, i.e., Ei(O, t) = E,(t) = 0 for t < 0, F(o)

=

s0

o dtEo( t)eiwt

(55)

Assuming E,(t) is a bounded reasonably behaved function, F(w) evidently is a bounded holomorphic function in the upper half w-plane Im w > 0. Hence the contour C can be supposed to run from - m to 00 infinitesimally above the real o-axis. We shall denote this particular contour by C,, . Customarily (3)) one assumes the pulse transmitted through the plasma must be Et(z, t) = &S,

*

dcdF(u)T(o)ei(kP-Wf)

d
(56)

where T(o) is the transmission amplitude defined by (14). If T(o) is bounded at infinity, the contour C, in (56) can be closed on the infinite semicircle in the upper half w-plane whenever t < z/c. Thus, if T(w) is holomorphic for Im w > 0, one concludes Et(x, t) = 0

t < z/c,

z > d

(57)

consistent with causality. Conversely, if one postulates causality, then in general (56) is not consistent with (57) unless T(w) is holomorphic and bounded for Imw > 0. The result (56) for z > d presumes (3) the complete time-dependent solution corresponding to the input pulse (54), is E(z, t) = & l

0

dwF(w)E(z,

w)eeiwt

where E(z, U) is the solution (14) to (l), which for complex w remains defined for x1 < z < z2, and which in the regions z1 < z < 0 and d < z < z2 has the explicit forms shown in (14). Equation (58) is not the most general timedependent solution corresponding to (54) however. For finite t > 0 and z > z1 the contour C in (54) can be deformed arbitrarily into the upper half w-plane, except that to guarantee convergence the contour should remain on the real o-axis at o = f co ; the resultant integral of form (58)) now running over C rather than C0 , also will be a time-dependent solution corresponding to the same input pulse Ei(.z, t). Moreover the solution integrated over C need be identical with the solution integrated over C0 only if no singularities of E(z, CA) are crossed in deforming the contour from C, to C. Of course there is only one physically acceptable time-dependent solution corresponding to given Ei(z, t). When singularities of E(z, w) occur in the upper

MULTIPLE

WAVES

AND

19

CAUSALITY

half w-plane, therefore, the ambiguity in the contour presumably could be resolved by returning to the original time-dependent equations (e.g., Maxwell’s .equations and the constitutive relations for the plasma) from which Eq. (1) ultimately was inferred. This elaborate procedure for resolving the ambiguity ,can be avoided simply by postulating that the correct choice of contour must be consistent with causality. In particular, if T(w) remains bounded at infinity but has a finite number of poles wj and branch points Wb in the upper half w-plane, and C’ is a contour lying above all these singularities then E’(z,

t)

=

&

I,,

dwF(w)E(z,

w)eP

is a time-dependent solution corresponding to the input pulse (54) which is zero for all t < z/c, z > d; as previously, to guarantee convergence, C’ should remain on the real w-axis at w = =t 00. Similarly (59) ensures the reflected wave, on the x < 0 side of the plasma, is zero at times t < -x/c. If all In? Wb 5 0, (58) and (59) imply E’(z, t) = E(x, t) - & 2ai c F(Wi).wZ, j

wi)e--iwlt

where E,(x, wj) is the residue of E(z, w) at w = wj . If some Im Wb > 0, the solutions E’(x, 1) and E(z, t) may differ by integrals along branch cuts through wb , whether or not T(w) has poles in the upper half w-plane. We add that choosing the contour C’ symmetric about the imaginary axisso that if w lies on C’ so does the mirrored point w’ = - w* defined in Section IIIit is easily seen that (59) keeps E’(z, t) real if the initial E,(t) is real.5 One notes that if dw is a differential element on C’ in the first quadrant, then the mirrored element in the second quadrant, drawn in the positive direction along the contour, is + (cZw)*. Thus, using (29) and qw’)*

= F(w)

(61)

(which follows directly from (55) when E,(I) is real), the contribution to the integral (59) from C’ in the second quadrant is the complex conjugate of the contribution from C’ in the first quadrant. Of course use of (29) to prove E’(z, t) real is not unexceptionable, because according to Sections II-IV the mere existence of singularities of T(w) in the upper half w-plane implies some or all of (7)-(10) do not hold. With actual n2(w), however, (29) surely will hold on C’ if C’ is drawn sufficiently far above all singularities of T(w). It is necessary to 6 It as (11) reason reality

is worth remarking that time reversibility, which often is used to infer relations such and (29)-as, e.g., Hilgevoord (8), p. G3-does not hold in lossy media. It is for this that the present paper lays such heavy stress on (and makes so much use of) the requirement that incident initially real waves must remain real for all time.

20

GERJUOY

grant that (7) holds at infinity in the upper half w-plane, as it will if (10) continues to hold at infinity for example; then assuming only that ,n’(w) has isolated singularities, so that it is analytically continuable, Eq. (7) will hold everywhere in the upper half w-plane. GROWING

WAVES

It is proved in Section VI that T(w) for a uniform slab typically contains poles in the upper half w-plane when the slab is composed of an “unstable” plasma, i.e., of a plasma (9) in which (when unbounded, - ~0 < x < 00 ) an initial electric field (of form EOeiKa, K real) can grow exponentially with time. This result is consistent with Eq. (60). Because Im wj > 0, the exp( -iwjt) terms in (60) increase with increasing d at fixed z, and approach infinity exponentially as t + 00; with branch points Wb in the upper half w-pIane, growth as t + co also will be manifested by the branch cut integrals which may have to be included on the right side of (60). Presumably an incident radar pulse cannot excite such growing waves in a slab composed of “stable” plasma, which contains no reservoirs of energy capable of conversions into electromagnetic radiation. Since growing waves are absent from E(x, t) defined by (-58)) one can hypothesize that conditions (i)-(iii) of Section I characteristically are obeyed by stable plasmas, but not by unstable plasmas. To put it differently, the customary belief that causality implies conditions (i)-(iii) probably is correct for stable plasmas obeying Eq. (1). However it must be remembered that actual plasmas often are anisotropic and/or propagate more than one type of wave at given W; in either of these circumstances Eq. (1) is not immediately applicable, and the theory presented thus far must be generalized (see Sections VII and VIII). In an unstable plasma an infinitesimal disturbance can grow to finite amplitude in an infinite time. With an unstable plasma, therefore, even if at times t --$ 00 (when the incident pulse is emitted a long distance to the left of the plasma) there seem to be no other waves present, nevertheless (57) should not be required to hold unless at large negative times the amplitudes of the waves present in the plasma are precisely zero. Otherwise, these plasma waves can build up and produce signals outside of the plasma at times t < 0 when the incident pulse has not yet reached the plasma. To put it differently, an even more general solution of (1) than (59) replaces F’(wj) in (60) by F’(wi) where F’(wj) are quite independent of the incident Ei(x, t). Only the special choice F’(wj) = F(wj) satisfies (57), and this choice corresponds to the solution (59), i.e., to using the contour C’ instead of C, in (58), where C’ runs above all singularities of T(W). The above considerations are reinforced by the observation that (as is well known) by using the Fcdtung property of the Fourier transform Eq. (58) can be put in the form (14) E(z,

t)

=

s’

cZt’Eil(t -cc

-

t’)&(Z,

t’)

(62)

MULTIPLE

WAVES

AND

CAUSALITY

21

In Eq. (62) (63) obtained putting F(w) = 1 in (58), is the response of the system to an infinitesimally short unit pulse E,(t) = 6(t) incident on the pIasma at time t = 0. The interpretation of (62) is that the wave field at time t is compounded solely from the responses the incident wave induces at all times t’ < t. If the plasma contains a wave capable of growing to finite amplitude at finite times t though starting from an infinitesimal amplitude at t = - cc,, then the form (62) would not necessarily be expected to completely represent the wave field present at finite times. In summary, if T(U) is bounded and holomorphic for Im w > 0, the timedependent solution (58) is consistent with causality. If T(w) has singularities in the upper half w-plane, but a contour C’ can be drawn lying above all of them, then (59) is consistent with causality; in this circumstance, however, there are growing waves within and outside the plasma at times t > 0. Such growing waves are not understandable with slabs composed of stable plasma, but are believable with slabs of unstable plasma. With unstable plasmas, moreover, requiring (57) need not be consistent with the actual physical situation in the plasma at the large negative times when the incident pulse starts out toward the plasma. We remark that when growing waves exp( - hjt) occur in the plasma then, as in (60), outside the plasma (even though the free space wave equation (1) with n’ = 1 is satisfied there) the field cannot be expressed as a Fourier integral over real frequencies only. The reason, of course, is that these growing waves in the plasma produce a field outside the plasma which for any z becomes unbounded as t 3 m ; unbounded functions generally are not expressible as Fourier integrals over real frequencies only. On the other hand, for fixed t the reflected and transmitted fields external to the plasma always are bounded as 1x / ---f cc (because Im Icj = WJC is >O), so that at any given instant tl the spatial dependence of E(z, &) outside the plasma is expressible as a Fourier integral over real wave numbers only. Correspondingly, the time-dependent wave generated by E( z, tl) will propagate as if composed of real frequencies only, as in (56)) but this wave will not represent the actual total field external to the plasma because of continued radiation (into this external region) by the growing plasma wave. Failure to recognize that even in the free space region the actual field need not be expressible as a Fourier integral (56) over real frequencies only may be the reason for the common but mistaken conclusion that (57) requires holomorphic T(U) for Im w >O. Certainly there is nothing very novel about our replacement of C, by a contour C’ running over complex frequencies above all singularities of T(w).

22

GERJUOY

The results of this section also can be interpreted in the following way.6 With a pole of T(w), A(w) in Im w > 0 is connected an exponentially increasing solution (at times t > 0) whose physical necessity is understandable for unstable plasmas. If one chooses to exclude this solution, as (58) does, and as is done automatically by imposing square integrability, then the remaining solutions are unphysical and in fact turn out to be acausal. From this viewpoint the fact that (58) violates causality when poles of T(w), A(w) in Im w > 0 exist comes about because one has illegitimately imposed a kind of boundary condition at t -F 00, namely that the solution has to remain finite. In an unstable plasma an arbitrary incident field starting at t = 0 in general will trigger the unstable solutions; if one wants to exclude this solution one must perform special operations on the medium ahead of time, which means one must then expect the system will respond at times t < 0. VI.

GEOMETRIES

PRODUCING

POLES

We shall show that the inequalities (8b) and (9), which were the foundation for our proof in Section III that T(U) has no poles for Im w > 0, are necessary in the sense that plasma geometries producing poles of T(w) in the upper half w-plane can be found under circumstances violating these inequalities. In fact, poles are produced with uniform plasma slabs, to which we henceforth confine our attention. The necessity for (8) is trivially obvious from the known theory (15) of the scattering matrix associated with the Schrodinger equation (5) ; to every bound state of energy - 1E 1there corresponds a pole of the scattering matrix (i.e., of T(w) in the present one-dimensional case) at a value of Ic = ilc2 = ( -202 / E 1fizy on the positive imaginary axis. Now for a one-dimensional square well on the full open interval - 00 5 x s co (specifically, V(z) = 0 except in 0 5 x 2 d where V(z) = -V, with V, > 0) there always is (16) at least one bound state 0 < ) E 1 < V, . By Eq. (6)) this V(x) and lc correspond to a uniform slab in which n2 < 0 on the positive imaginary axis. Of course the quadratically integrable bound state is a solution to (1) of form (17). We next examine the possibility of finding poles off the imaginary axis when (9) is violated. On the imaginary axis, when n’(w) is real and negative, K2(u) = k2n2 is real and positive, i.e., arg K2 = 2mn, m an integer. It will be shown that poles can be produced in the vicinity of any line arg K’(u) = 2m?r lying above the real w-axis. Note that no such line can exist if conditions (i)-(iii) hold, 6 This interpretation was suggested by Professor Van Kampen, who also pointed out that in Dirac’s classical theory of the electron one has to put up with a “small” acausality to eliminate the self-accelerating solution. A similar result holds for the quantum mechanical electron. P. A. M. DIRAC, Proc. Roy. Sot. 167A, 148 (1938); N. G. VAN KAMPEN, K. Danske Videnskab. Selskab. Mat-fys. Medd. 26, No. 15 (1951).

MULTIPLE

WAVES

AND

23

CAUSALITY

because on one side or the other of the line Im K2 will have to be <0, in violation of (9) ; when K2(w) is analytic, the Cauchy conditions (17) forbid Im K2 to be positive on both sides of a line Im K2 = 0. When conditions (i)-( iii) do not hold, the behavior of n*(o) and K2(w) may be very complicated. To restrict the behavior somewhat, let us retain the inequality (10) on the circle at infinity in the upper half w-plane. Then on this circle we may define n (w ) = n2(o) = 1. Further let K(w) = d@(u) be defined by the relation K = kn so that on the circle at infinity in the first quadrant of the w plane 0 S argK 5 r/2,0 5 arg K2 5 ?r. Consider the analytic continuations of n, n2, K, K2 as w in the first quadrant moves from infinity to finite values, assuming for simplicity that n(w) has no poles for Im w > 0, but recognizing that branch cuts may be required to keep n(w)-and perhaps even n”(w)single-valued. The inequality (9) fails as soon as K2(w) continues into the third or fourth quadrants of the K2-plane, i.e., as soon as w crosses a line Im K2(w) = 0. Of course, there may be many lines Im K2 = 0 in the first quadrant, and on any such line arg K2 may be an odd rather than an even multiple of a. We postulate, however, that there is at least one line arg K2 = 2m?r. AS w crosses this line, K2(w) moves from the first to the fourth quadrant. Correspondingly K(W), whose argument on this line is mr, moves either from the first to the fourth quadrant (when m is even) or from the third to the second quadrant (m odd). Moreover, because w lies in the first quadrant, in the vicinity of the line K(W) = wn/c real and positive (m even) n(w) must lie in the fourth quadrant; if m is odd, n(w) lies in the second quadrant in the vicinity of arg K = mn. LOCI

OF

POLES

Suppose we have the specific situation arg K(w) = 0 on a line in the first quadrant of the w-plane. For a uniform plasma slab of thickness d, the transmission coefficient obtained from (14) together with E(z, w) = BleiKZ + B2eeiKt

0 % x 5 d

(64)

is -ikd

e T(w)

Zeros of the denominator

= cos Kd - (i/a)[(l/n)

of (65)) corresponding

+ n] sin Kd

(65)

to poles of T(w), occur when

n(w) - 1 iKd ~ K(w) k eiKd = fl (66) n(w) + 1 e K(w) + k There are no poles in the vicinity of n(w) = 0, however, where both n and K vanish in the denominator of (65), In the present situation, as explained above, n(w) lies in the fourth quadrant P(w) =

24

GEKJUOT

in the vicinity of argK(w) = 0. Thus / (n - I)(TL + I)-‘/ < 1 in (66). On the other hand, if K lies in the fourth quadrant, the magnitude of P(W) can be made arbitrarily large by choosing d sufficiently large. Also arg eiKd increases monotonically with d. It follows that with arbitrarily large d Eq. (66) has roots lying arbitrarily close to the line arg K(U) = 0. At these roots Im K is negative but arbitrarily small, while Re K has the value required to make the phase of P(W) in (66) exactly equal to jr, j some integer. Now, having established the existence of solutions to (66) in the vicinity of arg K = 0 for large d, we can permit d to decrease, thereby generating a series of branches K = Kj(d) in the first quadrant of the w-plane on which the roots of (66)-and therefore the poles of (65)-move continuously as d is continuously varied. The subscript j indexes the phase jr of P(w) at the root. All branches converge asymptotically to the line arg K = 0 at large d, and may end on the real or imaginary axes, or on a branch cut, for some minimum dj ; with decreasing d, Kj(d) penetrates increasingly into the fourth quadrant of the K-plane, but never crosses lines Re n(w) = 0, where n(w) is leaving the fourth quadrant. Obviously the above argument also demonstrates the existence of poles of T(w) associated with any line (in the first quadrant of the w-plane) on which arg K is an even multiple of 7r. Essentially the same argument demonstrates poles when arg K = ‘ms, with in odd. The sole difference is that n(w) lies in the second quadrant, so that J (n - 1) (n + 1)-r 1 > 1 in (66) ; consequently for large d the roots of (66) now lie on the Im K > 0 side of the line arg K = m?r, and penetrate increasingly into the second quadrant of the K-plane as d decreases. Because Re w > 0 in the first quadrant, the time-dependent solution ei(9z--wt) in the interior of a uniform plasma represents a plane wave traveling to the right when Re K > 0, i.e., when K lies in the first or fourth quadrant. When K = wn/c is in the first quadrant, the inequality (9) is obeyed; when K is in the fourth quadrant, Im K2 < 0. Correspondingly, when K is in the first quadrant, the plane wave is spatially damped in its direction of propagation, i.e., decreases with increasing x at fixed t; when K is in the fourth quadrant the wave is spatially growing. Now in (64) let us agree that eiKz represents the wave traveling to the right; in other words we are choosing the sign of K = *\I?? so that Re K > 0, ignoring the fact that this choice may make K = -h rather than K = kn as heretofore. The wave eiKz reflects at z = d into a wave BzeviK” of amplitude Bz = (K - k) (K + k)--le2iKd A wave eei”” reflects at z = 0 into a wave BleiKr of amplitude B, = (K - k)(K

+ lc)-’

MULTIPLE

WAVES AND CAUSALITY

25

Thus (66) is just the condition for the wave ezKz,starting out in the plasma at x = 0 and reflected at x = d, to have exactly unit amplitude at x = 0 after a second reflection there. Moreover, since some fraction of the wave energy is transmitted out of the plasma at each reflection, it is understandable that 1B, ) from (67b) is < 1 when K lies in the first quadrant or in the fourth quadrant near arg K = 0. Correspondingly, it is understandable that with w in the first quadrant roots of (66) near arg K = 0 occur only when K lies in the fourth quadrant, so that a wave propagating from .z = 0 to x = d and back can grow spatially. Of course (67b), not (67a), also is the ratio of the actual reflected wave amplitude at z = d (namely B,e-““) to th e actual incident wave amplitude at z = d (namely eiKd). The foregoing interpretation of (66) is especially useful for purposes of the next section. CONNECTION

WITH PLASMA

INSTABILITY

Plasma dispersion equations relate w to K. The analysis of this paper has been concerned with examining the roots K = K(w) of these dispersion equations, for real and complex w in the upper half w-plane. It is more usual (7-9) to analyse plasma dispersion equations by examining the roots w = o(K) as functions of real K. This latter analysis is suited primarily to situations in which one seeks E(,z, t) within the plasma at times t > 0, starting from given E,(z) = E(x, 0) and &6(z) = (&!Z/dt)l=~. Specifically, to find E(z, t) one expands E,(z) and J&(z) in plane waves ezKz,noting that in unbounded plasmas K must be real in order that these expansions remain convergent at z = f 00. Thus, since the time dependence of any plane wave still is exp [--iw(K)t], one concludes that in stable plasmas Im w(K) always is 50 for real K, whereas unstable plasmas are characterized by the fact that somew(K) lie above the real w-axis for someranges of real K. In other words, for unstable plasmasthere are lines o = w(K) (above the real w-axis) on which K is pure real, i.e., on which arg K” = 2nm, na an integer; no such lines can be found at Im w > 0 for stable plasmas. As argued following Eq. (61), if n’(w) is assumedanalytic except for isolated singularities, any lines arg K’ = 2rn?rin the upper half w-plane must be symmetrically located about the imaginary axis. Thus with actual n”(w) and Im w > 0, if lines arg K* = 2m?roccur at all off the real and imaginary w-axes, they occur within the interior of the first quadrant of the w-plane. The above paragraph shows a sufficiently thick uniform slab composed of a plasma customarily characterized as unstable assuredly has poles of T(w) in the upper half w-plane. In this connection it is noteworthy that poles of T(w) are not associated with all lines Im K’(W) = 0. In particular, (66) does not have roots in the neighborhood of lines where arg K’(W) equals an odd multiple of ?r, so that K(U) lies on the positive or negative imaginary axes. When K = hx lies on the positive imaginary axis for w in the first quadrant, e.g., arg K(W) = r/2,

26

GERJUOY

then n(u) lies in the first quadrant; thus for values of w in the vicinity of argK(w) = r/2, both factors of P(U) in (66) are < 1 in absolute value. Similarly, when K(w) lies on the negative imaginary axis, e.g., arg K(W) = --a/2 then n(w) lies in the third quadrant, and both factors of (66) have absolute values > 1. The results just quoted confirm a theorem that can be proved from Eq. (35), namely, that in the first quadrant of the w-plane Eq. (1) has no solutions of form (17) if Re K’(x, w) I 0 at all points in the slab 0 Z z 5 d. This theorem does not mean that in nonuniform plasmas poles are forbidden unless K’(z, W) lies in the fourth quadrant for some value of z; for instance, it may be possible to have poles in the first quadrant of the w-plane if (9) is violated by having K2( x, w ) move from the first to the third quadrant as z is varied. With uniform plasma slabs, however, poles can occur only at those values of w in the first quadrant where K’(w) lies in the fourth quadrant. This implication of the cited theorem is not an obvious consequence of (66), but can be deduced from that equation. The ranges of n(w) and K(w) wherein first quadrant poles of T(w) can occur with uniform slabs are illustrated in Fig. 1. The discussion of Eq. (66) has made it apparent that roots of (66) occur only when n(w) and K(w) lie in the same quadrant, either the second or the fourth. Suppose that as o moves from infinity to finite values (along a line of constant arg w for instance) n(w) rotates in the clockwise direction from B to P in Fig. 1; then the line arg K2(u) = 0 is crossed at A. Suppose on the other hand n(w) rotates in the counterclockwise direction, from B to Q, but there are no points w in the first quadrant for which arg K reaches ?r; now K2 has reached the fourth quadrant from the second and third quadrants, not by crossing a line arg K2 = 2nm. Then although the plasma normally would be termed stable, there are poles of T(w) in the first quadrant of the w-plane for some finite range of d. As the slab thickness d -+ 0~) these poles move out of the first quadrant, and their associated growing waves exp ( - iwjt) disappear from (60). An illustrative (not actual) K2(co) for which poles of this latter type-not associated with plasma instability as it is usually defined, and having an unphysical dependence on slab thickness-would occur is K2(W)

= g (w + ib)* c2 (w + ia)”

with 6 > a > 0. This K2(co) is everywhere holomorphic for Im w 2 0; approaches w”/c” as w + ~0; obeys (27), i.e., is consistent with the reality conditions (7) and (8) ; and for all finite w in the first quadrant satisfies (see Fig. 2) 0 < arg K2 = 27 + 4p - 4a = 27 + 49 < 2~ - 2y < 21r because (y + (o) < 1r/2. However, letting b -+ a and a + 0, arg K2 can be made arbitrarily close to 21r for a range of w on and immediately above the posi-

MULTIPLE

WAVES

AND

27

CAUSALITY

(a 1

(b)

1. Ranges of K(w) and n(w) permitting a uniform slab to have a pole of T(w) in the first quadrant of the w-plane. The positions of k = w/c and -k are shown in (a), and are held fixed; B = arg k. Then as n(w) traces the circle shown in (b), K(W) = k(w) traces the circle shown in (a); corresponding points are labeled by the same letter in the two diagrams. Only in the shaded areas DOE and AOH are both K(w) and n(w) located in the same second or fourth quadrant. Of the shaded areas, only the cross hatched J’OE and AOJ are consistent with the requirement that K*(W) lies in the fourth quadrant. FIG.

tive real w-axis. Thus this K’(o) violates (9) and in fact, recalling (38), for a range of real w will cause ( T 1’ + ( A 1’ t o exceed unity with any slab thickness d. Yet the only values of w consistent with real K lie in the lower half w-plane. As discussedin Section V, poles of T(o) should not occur with slabs of actual stable plasmas containing no energy reservoirs. In particular, unphysical poles of the above type will not occur with stable plasmas obeying Eq. (1) if one postulates conditions (i)-(iii) are obeyed. Most actual plasmas do not obey Eq. (1) however (see Section VII). Moreover, as the warm collisionless plasma discussedin Section VIII evidences, it would be incorrect to maintain that in actual stable plasmas capable of propagating waves, each individual n’(w) must obey conditions (i)-( iii). VII.

MULTIPLE-WAVE

PROPAGATION

Actual plasmas generally (7-9) are too complicated to be described by Eq. (I), which assumes the plasma is completely represented by a single given

28

GERJUOY

w - PLANE

FIG.

2.

Angles in the w-plane for K* = P(w + ib)d/(w + ia)4.

n2(w). The complications stem primarily from two sources: the plasma may not be isotropic and/or the electromagnetic fields alone may not sufficiently specify the state of the plasma. In either event, it becomes possible to have several waves of different K(w) associated with the same given circular frequency W. Thus, excepting some very trivial cases, such as the so-called cold plasma described by (13a) when there are no collisions, it has not been made apparent that the results of the preceding sections are applicable to actual plasmas. In this section it will be argued that indeed the preceding sections largely are applicable; in particular it will be argued that transmission coefficients T(w) for sufficiently thick uniform slabs of many (if not all) actual unstable plasmas typically will have poles in the upper half w-plane. In the simplest one-dimensional (independent of 2, y) plasmas to which Sections I-VI directly apply, a vacuum electromagnetic wave moving along +x and polarized along 2, impinging on a semi-infinite plasma at z = 0 say, excites in the plasma only transverse waves polarized along 2. Under somewhat less simple circumstances (e.g., with an external magnetic field parallel to x) E,

MULTIPLE

WAVES

AND

29

CAUSALITY

and E, may be coupled in the plasma, but it still may be possible to find two uncoupled independently polarized transverse waves propagating along +Z (e.g., left and right circularly polarized waves). In this event, after projecting the incident vacuum wave onto corresponding polarization states, each independently polarized component transmitted into the semi-infinite x > 0 region may obey Eq. (l), although of course n’(w) need not be the same for both polarizations. However, n”( ti) now also need not be the same for waves of identical polarization moving along +z and --z (e.g., left, i.e., clockwise circularly polarized waves as seen by an observer looking toward negative z move with different speeds to the right and left in the presence of an external magnetic field). Under the relatively uncomplicated circumstances described, therefore, Eq. (1) no longer need be valid when the plasma has a second boundary at x = cl, from which waves moving to the right can be reflected without change of polarization (e.g., looking toward negative x left circularly polarized waves incident on x = d are reflected as left circularly polarized waves). With an external magnetic field oblique to x, the situation is still more complicated. Kow the plasma waves may be neither purely transverse nor purely longitudinal, and an incident arbitrarily polarized vacuum wave at z = 0 can excite three (not two as heretofore) independent outgoing plasma waves moving along +x. Hence the four boundary conditions at x = O-namely, continuity dE.Jdx and E,/dz, or equivalently continuity of the tangential of E,, Eu, components of E and H-no longer suffice to determine the amplitudes of the outgoing waves; since there now are five waves going out from z = O-namely the three in the plasma plus the two independently polarized reflected vacuum waves-a fifth boundary condition, presumably continuity of the z-component of the electric displacement, is required. Even more complicated and more numerous types of waves become possible when the electromagnetic fields within the plasma can couple to other quantities, e.g., to the velocities or densities of plasma streams; in fact, there may be (18) an infinite set of different wave numbers K(W) associated with the same given circular frequency w. For each additional outgoing plasma wave which can be excited by an incident vacuum wave at a plasma boundary, an additional homogeneous boundary condition, presumably on the velocities or densities of the streaming speciesand hopefully consistent with the actual physical situation, must be introduced7 to make the problem of calculating plasma reflection and transmission coefficients determinate. In any given physical situation, once sensible supplementary boundary conditions making the plasma reflection and transmission coefficients determinate 7 Only problems conditions.

very recently does there appear to have been any involving multiple wave propagation and requiring J. D6, Phys. Fluids 6, 1772 (1963).

serious attention supplementary

to actual boundary

30

GERJUOY

have been found, the set of coupled equations generalizing Eq. (1) and yielding the complete set of plasma waves probably can be treated along the lines of the preceding sections. Whatever these equat,ions and boundary conditions may be, however, Eq. (14) continues to hold in the regions xl < z < 0 and d < x < zz exterior to the plasma. Then, as in Section II, ordinarily there are uniquely specified solutions u- , u+ , defined by Eqs. (18) and (19) respectively. Let T(w)-’ now be defined as the coefficient of eikein the expression for U+ to the left of the plasma, as in Eq. (22). It follows,8 essentially as previously, that: T(w) is given by (23) ; where T(w) has a pole, (24) and (25) hold, and the equations have an everywhere outgoing solution of form (17), thereby implying (14) does not uniquely define E(x, w); where T(w) is defined and $0, the function E(.z, w) defined by (14) equals T( w) u+ and is unique. Kate that because Eq. (1) no longer holds, Eq. (21) need not be valid within the plasma, so that W(U- , u+) to the left of the plasma now need not equal W(u- , u+) at x > d; nevertheless the knowledge that (24) holds over an extended range x > d guarantees (25), granted that U- , U+ are indeed unique. Of course it also is necessary to grant that the additional boundary conditions and equations generalizing (1) are reasonably well behaved functions of Z, and in the upper half w-plane have only isolated singularities as functions of o. BEHAVIOR

UNDER

SPECIAL

CIRCUMSTANCES

The proviso “ordinarily” in the above paragraph takes into account the fact that with multiple-wave plasmas (many K(w) possible) there can be special circumstances, of the sort discussedfollowing Eq. (73) below, wherein the assertions of the preceding paragraph are not wholly true. Essentially in these special circumstances Eqs. (18) and (19) do not properly define U+ and/or u- , an eventuality not possible with the simple Eq. (1). With U+ and/or U- not well-defined, the arguments of Section II and of the above paragraph break down. To illustrate such special circumstances and the complications they produce, let us examine transmission through a unifoim plasma slab in which propagate multiple waves corresponding to the warm collisionless plasma discussed in Section VIII below. In this case: transverse and longitudinal oscillations are uncoupled; plane waves polarized along x are uncoupled from waves polarized along y; there are four independent transverse waves polarized along x and propagating parallel to Z, with wave numbers &K,(w), &K@(w) ; and of course there are four independent waves polarized along y. Instead of (64), therefore, the fields along 2 in the uniform plasma slab 0 6 z s d can be and must be expressed in the form E(z, w) = Blpz * Some of the assertions results on nonself-adjoint note l), chap. 12.

+ BZpQS + &-in the remainder boundary-vaIue

+ &-‘Q”

0 5 z 5 d

(68)

of this section are illustrations of very general problems. Coddington and Levinson (see foot-

MULTIPLE

The boundary

conditions

WAVES

AND

31

CAUSALITY

at x = 0 yield, using (14),

1 + A = BI + Bz + BB + B, k - kA = K,BI 0 = a&

(69a>

+ KBBp - K,Bs + a&

+ a&

- KoB,

+ a&

(69b) (69c)

Equations (69a) and (69b) correspond respectively to continuity of E and dE/dz. Equation (69c) expresses the required supplementary boundary condition on the fields within the plasma. The numbers al, a2, a3, a4 are independent of B1 , B, , Ba , B4 but are otherwise arbitrary and may depend on W. However, except possibly at isolated values of W, Eqs. (69) must be a set of three simultaneous equations sufficing to determine the three outgoing amplitudes A, B1 , Bz in terms of the incoming amplitudes (in this case 1, B3 , B4 of which at the moment B3 , B4 happen still to be unknown). Likewise, Eqs. (69) must suffice to determine the three incoming amplitudes in terms of the outgoing amplitudes. Thus we must have @=S+J

(70)

where So is a 3 X 3 matrix depending only on W, whose determinant does not vanish except possibly at isolated values of w. Similarly the boundary conditions at z = d yield ikd Te = BleiKad+ BzeiQd + ~~~-iJhd + B4e-iKbd (7la) kTeikd = K,&eiK”d + KgB2ei”“d - KaB3e-iK”d - &B4e-iKBd Wb) 0 = blBleiKed+ bzB2eiKBd + b3B3eeiKad + b4B4ewiKpd

(71c)

where (71~) expressesthe required supplementary boundary condition at x = d, and the b’s need not coincide with the a’s since the physical conditions at the z = d boundary need not coincide with the physical conditions at z = 0. Then Eqs. (71) imply, corresponding to (70),

In (72) as in (70), the column matrices on the left and right are respectively the amplitudes of the outgoing and incoming waves at the plasma boundary. When the physical conditions at z = 0 and z = d are identical, b, = a3 , b, = a4 , b3 = al , b4 = a2, i.e., the supplementary conditions on incoming and outgoing plasma waves are the same at each boundary, and indeed in this event it can be seen from (69) and (71) that Sd = So. Furthermore, as explained earlier in

32

GERJUOT

this section, under “ordinary” circumstances poles of T(w), A (w) should occur when and only when 1 on the right side of (70) can be replaced by zero, i.e., when the six homogeneous simultaneous equations (72) and

(I)=

SpJ

(73)

can be solved for nonvanishing A, T, Bl , & , & , Ba . Such solutions exist, of course, only when the coefficients of A, T, & , Bz , & , B, in (69) and (71) have a vanishing determinant; the values of A, T, etc., then are proportional to a set of minors of this determinant. By properly choosing the ratio of B1 to B, , it is possible to make T = 0 in (72), i.e., with (68) one can find many sets of plasma waves which connect to an identically zero vacuum electromagnetic field. From the fact that this possibility does not exist for the simpler nonmultiple-wave case described by Eq. (1)) we inferred in Section II that A2 and T, in (16) must vanish together, or equivalently that T(w) and A(w) have poles of the same order at the same w = w. . Of course, usually this choice of BI/& is not consistent with (70) ; however T = 0 will be consistent with (70) for special choices of the coefficients in (69) and (71). (In this special circumstance the solution U+ defined by (19) will be infinite to the left of z = d.) Thus, there are special choices of the coefficients in (69) and (71) for which the aforementioned determinantal solution of (72) and (73) yields T = 0 and A finite, corresponding to a pole of A and not of T, or to A having a higher order pole than T. Likewise, because the ratio of B, to B, can be chosenso as to make A = 0 in (73), there may be poles w = w0 of T(W) but not of A(U), or T(U) may have a higher order pole than A(U) at w = oo. These are illustrations of special circumstances wherein the singularities of T(w), A(U) for multiple-wave plasmas have a more complicated behavior than would be inferred from Section II. Moreover, although generally the wave set (68) cannot simultaneously connect to identically zero vacuum fields at both z = 0 and z = d, there are extraordinary circumstances for which even such sets exist. For example, suppose for simplicity Sd is identical with So and suppose further that each of K, , KA is real while K,/KB is rational. Then d can be so chosen that exp (iK,d) = exp (iK&) = 1 in (72), in which event the simultaneous solution of (72) and (73) makes B1 = B3 , BB = B, ; in addition the first row of Sd = So, whose elements play no role in determining B1,/B2 (see Eq. (78) below), can be chosen so as to yield T = A = 0. When for w = w0Eq. (68) can connect to identically zero vacuum fields at z = 0 and x = d, Eqs. (69) and (71) do not have a unique solution, but nevertheless A (0) and T(w) need not have poles at w = w. . Other special circumstances arise when Eq. (71~) happens to be identical with (69c). There may be physical

MULTIPLE

WriVES

AND CAUSALITY

33

reasonswhy someor all of the special circumstances mentioned in this paragraph cannot occur, but these reasons are not apparent to the writer. DEMONSTRATION

POLES OCCUR

Having noted the kinds of complications which can occur in multiple-wave plasmas, let us confine our attention to the more frequent “ordinary” circumstances wherein the poles of T(w) surely coincide with the values of w at which purely outgoing solutions exist. It now will be argued that with sufficiently thick uniform slabs of actual plasma obeying the specific Eqs. (68) we have been considering, such purely outgoing solutions exist in the first quadrant of the w-plane for a wide class of unstable plasmas. To be precise, we shall demonstrate Eqs. (72) and (73) have nonvanishing solutions whenever there is a line in the first quadrant of the w-plane on which either arg Ka2(w) = 2n1a and Kg’(u) obeys (9), or else arg Kfl’(w) = 2)~ and K,“(W) obeys (9). As has been explained in Section VI: if any root &Km(w), &KS(w) of the plasma dispersion equation can be pure real on a line in the first quadrant of the w-plane, the plasma is characterized as unstable; in a region where (9) is obeyed, arg K’(U) # 2nz7r,since Im K’(U) < 0 on one side or the other of a line arg K’(U) = 21~; with u in the first quadrant, ei(Kz--wt)represents a plane wave traveling to the right when Re K > 0; with Re K > 0, if (9) is to be obeyed Im K > 0, so that ei(Kr--wt)is spatially damped. In other words, we shall prove Eqs. (72) and (73) have nonvanishing solutions in domains of the first quadrant where one of K,‘(U), Kg’(a) satisfies the usual criterion for instability, whereas the other wave number propagates only the spatially damped waves characteristically associated with stability. Assume the plasma dispersion equation makes each of K:(w), K;(W) consistent with (9) 011 at least part of the circle at infinity in the first quadrant; in other words assume the domain of instability does not include the entire range of infinite complex w. This very weak assumption ahnost certainly is obeyed in actual plasmas. For instance, this assumption is obeyed if each na2 = K,‘/k’, nB2 = K~‘/IC2 approaches a constant, not necessarily unity as w + Q); the stable warm plasma and the unstable two-stream plasma discussed in Section VIII satisfy this particular criterion for stability at infinite complex W. Then, if the plasma is unstable, as w moves along any path from infinite to finite values one or the other of Km2(w), Kb2(u) eventually will violate (9). If on at least one such path this first violation of (9) is a crossing of K,’ or Kp2 into the fourth quadrant rather than into the third quadrant, then the plasma is of the type postulated at the end of the preceding paragraph. In particular the two-stream plasma is of this type. Doubtless poles of T(W) also occur in domains where K,‘(w) , KB’( W) are less restricted, e.g., where both Ka2, KB2 lie in the fourth quadrant. For a domain

34

GERJUOY

restricted as described, however, the demonstration that poles exist is greatly simplified, as will be seen, and is very similar to the previous demonstration in Section VI that poles exist for plasmas obeying Eq. (1). Restricting the domain of Km2, Kp2 as described also eliminates nonphysical behavior of the poles, analogous to the behavior discussed at the end of Section VI. Now consider an arbitrary pair of waves BleiKoz, B2eiKBZstarting out in the plasma at x = 0. We suppose w lies in the first quadrant and choose the signs of K, = f(K.2)l”, KB = f(K:)1’2 consistent with Re K,(w) > 0, Re K@(U) > 0, so that each of these waves propagates in the direction of increasing z. When these waves are incident at the boundary z = d they will produce transmitted waves Teikr and reflected waves B3 exp ( - iK,z), B, exp ( - iKpz) satisfying (72). According to (73)) these reflected waves returning to x = 0 will produce a transmitted wave Aeikz and rereflected waves B1’exp (iK,z) , Bz’ exp (iKp) satisfying (since there is no incident wave from outside the plasma)

(2)=so(~iJ

(74)

In general the rereflected amplitudes Bl, B2’ need not be identical with the original amplitudes B1 , B2 . However, because BI , B4 satisfy (72)) it is obvious that the values of 01 for which BI = B1 , B2/ = Bz are precisely those w for which (72) and (73) h ave nonvanishing solutions, i.e., are the poles of T(o), A(u). The above result is the generalization, to multiple-wave plasmas, of our previous interpretation of Eq. (66). The generalization of Eq. (66) itself is obtained as follows. Equation (72) can be written in the form

(ii =FSdF@

(75)

where the diagonal matrix --ikd

0 iKop

F=

e

eiKp!

0

(76)

From (75) and (76)

B3 ,$e%d s” 0 (

~(K,+K$

23 e

=

i(K,+Kg)d

B4

s;2

e

sf3

e2iK@d

(77)

MULTIPLE

which,

when substituted

s;,

R=

s;,

e2%d

+

CAUSALITY

35

(5::)=R(E:>

(7%)

WAVES

AND

in (74) yields

SC’,

s;,

ei(K,+K,@

s;,

st, xi, e2%d + s”,, si, ei(K,+Kg)d

c&

ei(K,+K$d

+

cJ’,

s;,

e2iK+d

s;, s;, ei(Ka+Kg)d + s;, s;, e2iK,d

The condition for B1 = B, , B; = B2, i.e., the condition A (w), is that the determinant

for poles of T(U) and

IR-ll=O Equation M(w,

d)

~

(79) can be written r22e2i(Kcz+Kdd

_

(78b)

(79)

in the form

s;2,$2e2iK-d

-

&3s;2e2iK~d

+

rllei(K”+R@)d

=

- 1

(80)

where rz2 , rll are independent of d but depend on &f, ~!5$j, which in turn are related to the reflection amplitudes of incident waves exp (iK,z), exp (iKg) at z = d, and of exp (- iK,x), exp ( -iKax) at x = 0. Now suppose, as explained above, that Ka2 say lies in the vicinity of a line arg Ka2 = 2mn, while Kb’ obeys (9). Then, because So and Sd have been defined so as to be independent of d, for sufficiently large d the term f$2L5$2exp(2iK,d) dominates the left side of (SO). Thus, just as in the case of Eq. (66), at arbitrarily large d Eq. (80) has roots lying arbitrarily close to the line arg K,’ = 2m?r, with the value of Re K, adjusted so that the phase of &2S&exp(2iK,d) exactly equals Qr, j some integer. Presumably, if we could examine explicit expressions for A’!& , S& as we were able to examine P(w) in Eq. (66), we would see that / Sz2S& j is < 1, so that the roots surely lie on the Im K, < 0 side of arg Ka2 = 2ma, and penetrate increasingly into the fourth quadrant of the K, plane as d decreases. In the absence of such explicit expressions we observe from (70) that B1 = S!? is the amplitude at z = 0 of the reflected wave B, exp (iK,x) when a unit amplitude e-iK@, unaccompanied by any B4 exp (- iK~2), is incident from the right onto the boundary 2 = 0. Hence, just as in our interpretation of Eq. (66), ( Xi, / should be < 1, granting that energy is conserved on reflection. Similarly one can argue that 1 Sf2 ( should be
30

GEKTUOT

outgoing waves; the inference from this interpretation that (granting energy conservation on reflection) purely outgoing solutions cannot occur if the plasma propagates only spatially damped waves; the determinantal Eq. (SO) expressing this interpretation; the argument that this determinantal equation has roots in the neighborhood of arg K,’ = 27r11rif all other wave numbers Kg2, K,“, . . . , obey (9) in this neighborhood; the argument that the unstable plasma will be of this restricted type if the first violation of (9) along any path from infinity to finite w is a crossing (by one of Km2, Kp’, Ky2, +. . ,) into the fourth rather than the third quadrant. Strictly, Eq. (80) and the discussion thus far pertain only to plasmas in which wave numbers along +z and --x are the same, i.e., in which K+ = K- = K. It seemsclear, however, that the arguments of this section will remain relevant to a wide variety of anisotropic plasmas,except that for large d the poles now are found in the neighborhood of lines where (K,+ + K,- ) is pure real and >O, assuming Re Km+ and Re K,- are each >O. Of course once poles or other singularities of T(w), A(w) have been demonstrated to exist in the upper half o-plane, the main content of Section V-namely the possibility and implications of satisfying causality despite the presence of these singularities-is completely applicable to multiple-wave plasmas. VIII.

ILLUSTRATIVE

PLASMA

DISPERSION

FORMULAS

We proceed to examine-in the light of our analysis-dispersion formulas for two illustrative multiple-wave plasmas, one stable and the other unstable. Because we have been concerned primarily with the reflection and transmission of electromagnetic waves, and (as (10) makes explicit) normally expect the plasma to be essentially indistinguishable from vacuum in the limit w -+ w (see the discussion below, however) we shall consider only transverse waves in these illustrative plasmas. It will be seenthat the various K*(W) implied by these dispersion formulas have branch points in the upper half w-plane. The possible existence of such branch points, which are characteristic of multiple-wave dispersion formulas, was ignored in the previous section. Branch points of K*(W) cannot be wholly disregarded because they can be branch points of T(W), A (w) as well; in this event (as discussedin Section V) the reflected and transmitted time-dependent signals satisfying causality may grow exponentially with time for t > 0, whether or not T(w), A(W) have poles at Im w > 0. WARM

COLLISIONLEHS

PLASMA

The dispersion equation for transverse waves in a warm collisionlessplasma is (9) &,” - c2K2)(w” - xcaKZ) - w*c+,~= 0

(81)

MULTIPLE

WAVES

AND

37

CAUSALITY

where x << 1 is the ratio of the thermal energy ?cT to the electron rest energy no? and wP is defined by (13b). Equation (81) is a quadratic in w2, whose roots are

L?(K) = Mb,” + (1 + xm21

(82b)

It can be seen that for real K either sign of the radical in (82a) makes m2 real and >O, i.e., the warm plasma satisfies the usual criterion for stability. Letting K = wn/c, Eq. (81) becomes xn4 which is a quadratic

(1 + x)n2 + 1 - $

equation in n’, with

1 + x =F [(I n 2 =------

= 0

(83)

roots

+ X)” - 4x(1 - 0JPZ/W2)l”’ 2x

(84)

In the limit as w -+ 00, the upper and lower signs in (84) yield respectively 2 2 12, = l - (1 wpx)> ng2

=

;

+

(1

?:)u2

Equation (85a) is consistent with (10) ; Eq. (85b) is not. However each of Eqs. (85) satisfies (9) on the circle at infinity in the first quadrant, as discussed in section 7. Equation (&Fib) corresponds to a wave whose phase velocity approaches 1c/no 1 = (kT/m) l/Z as w ---f w. At first sight it is difficult to see how in the high frequency limit an actual collection of charges can propagate electromagnetic waves at a velocity different from the free space velocity c. On further reflection the result (85b) appears understandable. The existence of four (rather than two, as in vacuum) independent transverse waves polarized along x and propagating along fz results from coupling of the electromagnetic oscillations to pressure oscillations of the plasma particles, as can be seen from the derivation (9) of (81). Indeed the quantity (kT/n~)~‘~ obviously is related to the velocity of electron sound waves; thus the w + m limits of n,’ and nB2 suggest the true electromagnetic oscillations and the pressure oscillations are becoming uncoupled at high frequencies, as one anticipates. It then follows that the existence of the P-type waves will have no unphysical implications, provided incident vacuum electromagnetic waves excite these P-type waves less and less readily

35

GEItJUOY

as (J --j m ; if this is the case, at w = m a transverse electromagnetic wave incident on a warm plasma slab generates only waves with nz = 1. Note that unless actual calculations-starting from Eq. (68) and introducing the correct supplementary boundary conditions (69c) and (7lc)-supprt this conjecture that the waves (85b) are suppressed at high frequencies, the boundedness of T(w) as w + 00 will have to be re-examined. The proof in Section IV is not valid for multiple waves, and surely is not obviously generalizable when waves violating (10) can be excited at infinite w. It might be maintained that the limit n* = x-l as w ---f 00 means the lower sign root in (84) is an artificial result of the approximations made in deriving (81), and should be discarded. Discarding the P-type waves (85b) also has been suggested (9) on the grounds that Eq. (81) is valid only in the approximation w2 >> xc2K2; it can be seen that use of the lower sign in (84) corresponds to use of the lower sign in (82a), and does not lead to w* >> xc2K2, whereas use of the upper sign does. However, discarding the lower sign roots in (82a) and (84) leads to other difficulties, as will be shown. These difficulties stem from the fact that neither na2 nor ng2 satisfies conditions (i)-( iii) ; correspondingly, neither ne2 nor nf12satisfies the Kramer+Kronig relations. The wave numbers K, , Kp corresponding respectively to use of the upper and lower signs in (84) are: c2Ka2(a) = co2(1 ___+ 2x c2K&o)

xl

-

(1 - xl

= W2(12X+‘) + v

2x

w(w - is)““(w + is)‘;’ w(o - is)“‘(w

+ is)“‘”

(86a) (86b)

where s =

2w 1-x

2

x1’2 >

0

Suppose Kb2 is discarded, so that K, becomesthe only relevant wave number7 and the plasma becomes describable by Eq. (1). With Eq. (l), any simple zeros of K*(O) in the upper half w-plane also are branch points of K(w). But Eq. (1) is a function of K2(w), not of K(w), so that computed T(w), A(w) using Eq. (1) will be single-valued as long as K’(W) is single-valued, whether or not K(w) has branch points. In the case of a uniform plasma the correctness of this assertion is obvious from the form of (64), which merely postulates E(z, w) is a linear combination of eigz and edigZwith coefficients B1 , B2 to be determined from the boundary conditions; changing K to -K may change B1 to BP , but does not alter the actual field within the plasma. Correspondingly, one can verify directly from (65) that T ( w) is an even function of n(w) at fixed w, with of course K = cm/c.

MULTIPLE

WAVES

AND

39

CAUSALITY

To keep K2(w) from (86a) single-valued, and thereby to uniquely specify T(w) in the upper half w-plane, a cut or cuts through w = &ti must be introduced as, e.g., in Fig. 3(a) or Fig. 3(b). For either is consistent with (85a) provided one defines

arg (w - is) = arg (w k2 (WI I

of these two figures,

+is> = 0

K,‘(W)

(87)

W - PLANE

(a) k'(w)

/ / J /

\

/

\

/ \

/ CO

Go -

0”

-w

(b) FIG. 3. Cuts and contours in the w-plane for a pulse transmitted through a warm plasma. In (a) the contour C’ can be closed at infinity in the upper half o-plane, consistent with causality. The contour Co in (b) cannot be so closed. The contour C’ can be deformed to C”.

40

GERJUOY

when w = m on the positive real w-axis. Sow consider the transmitted pulse through a uniform plasma slab, where l’(w) is determined in terms of K,*(W) by Eq. (65). With the cuts of Fig. 3(b) it is not obvious that, the causality requirement (57) can be satisfied. For example, using (.56) for the transmitted pulse and closing the contour C0 at infinity in the upper half w-plane via the dashed contour shown in Pig. 3(b), necessarily involves integration along the cut. Using Fig. 3(a), causality obviously is satisfied if, as was done in (39), C’, in (.56) is replaced by the contour C’ shown. For times t > x/c, however, the transmitted pulse is evaluated by closing the contour in the lower half w-plane, and Et(x, t) will grow exponentially with increasing t unless C’ can be replaced by a contour on which Im w 5 0. Granting T(w) has no poles for Im w 2 0, the contour C’ can be deformed to the dashed contour 6” shown in Fig. 3(a), but must run into the upper half w-plane along each side of the cut. Thus the Et(x, t) consistent with causality will grow exponentially at large t unless, despite the cut, T(w) has the same value on both sides of the positive imaginary axis, i.e., unless T(w) is single-valued despite the cut, in which event the integrals on the left and right sides of the cut cancel. The transmission coefficient T(w) from (6.5) is not single-valued across the cut when K” is computed from (86a). On the other hand, T(w) computed from Eq. (68) using both (86a) and (86b) will be single-valued. As explained below Eq. (86) in connection with Eq. (64), the field within the plasma computed from (68) will be single-valued provided the set of wave numbers fK, , &Kp to the left of the cut coincides with the set fK, , ~k’o to the right of the cut. These sets do coincide if Fig. 3 (a) and (87) are used for (86b) as well as (86a), because the square root in (86) merely changes sign as the cut is crossed, so that the quantities Ka2, K8’ on the right of the cut go over respectively into the quantities Ko2, K,” on the left of the cut. Of course, the supplementary boundary conditions (69~) and (71~) also must be consistent with the requirement that the fields within the plasma are single valued; in other words the requirement that causality be preserved without growing waves at long times t > 0 imposes constraints on the forms of the physically allowable supplementary boundary conditions. Note also that Ka2, K8’ from (86) are consist’ent with the requirement that a real incident wave produces real reflected and transmitted waves. In the present multiple-wave case, to ensure satisfying this requirement, evidently (7) or (27) must be replaced by the condition that the set of quantities K,“(o)*, Ko2(u)* are identical with the set Ka2( o’), Kg*( w’), where as always w’ is the mirror of W. That these sets are identical can be seen directly from the complex conjugate of Eq. (81). With the identifications (86), Km2(w)* = Km2(w’) and Ko2(u)* = Ko2(ti’). However neither Km2 nor KB’ are pure real on the imaginary axis in the neighborhood of the cut. On the imaginary axis above the cut Km2 and Kb2 each are pure real and obey (8). As with the causality requirement discussed in the

MULTIPLE

WAVES

AND

41

CAUSALITY

preceding paragraph, the reality requirement imposes constraints on the forms of the physically allowable supplementary boundary conditions. Because a warm plasma in thermal equilibrium should have no internal reservoirs of energy capable of sustaining external exponentially growing electromagnetic waves, the foregoing paragraphs provide strong reasons for retaining the lower sign roots in (82a) and (84). Dropping (86b) could be made consistent with the requirement of no exponentially growing waves at t > 0, provided (86a) is replaced by a rational well behaved function consistent with conditions (i)-(iii); for instance one might try to choose this function so that it agrees closely with (86a) only in the domain W* >> xc%* where (81) is assuredly valid. Similar considerations pertain whenever one attempts to estimate transmission of a pulse through a (isotropic) stable multiple-wave plasma slab by means of Eq. (l), on the basis that for one reason or the other only one type of wave is strongly excited in the plasma at the frequencies of interest. As Section VI has shown in connection with poles of T(w), and as this section has demonstrated in connection with branch points, if Eq. (1) is supposed to apply then unless the actual n”(w) obtained from the multiple-wave dispersion equation is approximated by a function obeying conditions (i)-(iii), incorrect unphysical results may be inferred at large t. Assuming, as discussed above, that section 4 generalizes, we have shown A(U) and T(w) - 1 for the multiple-wave warm collisionless plasma obey the Kramers-Kronig relations. These relations would not be obeyed if (86b) were dropped. The foregoing discussion leads to the inference that in stable plasmas the existence of a branch point at Im w > 0 for one n’(u) implies there is at least one other n”(w) with the same branch point, so that cancellation of the troublesome branch line integrals can occur. All boundary conditions and other couplings to the plasma’s external environment must be consistent with this assertion to ensure causality can be preserved without exponentially growing waves at t > 0. TWO-STREAM

TRANSVERSE

INSTABILITY

Probably the simplest illustration of unstable transverse electromagnetic waves in a plasma is the so-called two-stream transverse instability. This instability occurs when an external magnetic field B, is imposed on a plasma which, though carrying no current, is composed of two streams of relativistic electrons moving in opposite directions along the magnetic field. The dispersion equation determining the index of refraction n(w) for left circularly polarized waves (clockwise as seen by an observer looking toward negative z) is (9)

1

w* -

.f2 7Zw,2

Ylh w 1

-

w

+

+ ,Fw,"-

~1

we

+ u2K]

71

UI K

=

'

(88)

42

GERJUOY

where fi , fi represent respectively the fraction of electrons moving with speeds respectively to B, ; the plasma ions Ul , % ; n2 , u1 are parallel and antiparallel have mass Al; fiul = f2uz ; oe = eB,/mc > 0 is the electron cyclotron (circular) frequency;

m Wi = -We; M

and

y = (1 - u*/c~)~‘~.

Equation (88), though a fifth degree equation for w as a function of K, is a fourth degree equation for K as a function of w. Thus Eq. (88) describes five independent types of waves of which only four can propagate at any given circular frequency w. As K approaches f 00, one sees that w = w(K) from (88) must have the five roots &cK, wi , -ulK, uzK. Hence there are five real roots w(K) at infinite real K. Moreover one sees that on the circle at infinity in the upper half w-plane the four roots K = K(w) are infinite and correspond to indices of refraction n = cK/w = f 1, -c/u1 , c/u2 . The latter two values of n correspond to phase velocities -ul , up at infinite w, and are understandable much as in the case of the warm plasma. The four independent K(w) result from coupling of electromagnetic oscillations in the bulk of the plasma to density and velocity oscillations in the streams. Presumably these oscillations become uncoupled at high frequencies, where therefore it is reasonable that a laboratory observer should find some types of electric field perturbations appear to move with precisely the stream velocities. Also presumably these low-velocity waves cannot be excited within a plasma by infinite-frequency electromagnetic waves incident on the plasma from the external vacuum. As real K2 > 0 is decreased from infinity, two of the aforementioned real roots w = w(K) became imaginary in a rather narrow range of K2-but sufficient to make the plasma unstable for transverse oscillations-provided the electron streams are sufficiently relativistic. To conveniently illustrate the behavior of the roots K = K(w) in the upper half w-plane, set u1 = uz = u in (88), therewith simplifying (88) to a tractable quadratic function of K’. One finds

2&?K2

= (w + yw,)’ + ,B2 w2 - * w(

where fi = U/C, e = m/M, leading terms are

and Y(w)

Y(w) = (1 - p”>“(w” + w5w,(4y-’

* Dwll”? EW,- yw, > w - ewe (sga)

is a sixth degree polynomial

in

w

whose

- 2e)

+ W4[W,2(E2+ 2 + 4r-2 - 8y%)

+ 2P2w,2Y-2(~ +

7111+ . **

(8% >

The two possible values Kn2, Kp2 at each value of w correspond respectively to the upper and lower signs of the radical in (89a). In general each of Ku2, Ko2 has six branch points in the w-plane. To ensure T(w) remains single-valued, the set Km’, Kb2 to the right of any cut through these branch points must coincide

MULTIPLE

WAVES

AND

CAUSALITY

43

with the set Ka2, Ko2 to the left of the cut. That these sets do coincide can be seen directly from (89), just as was seen previously from (86). However it is simpler and more general to argue that these sets must coincide because-no matter how the cuts are drawn-Eq. (88) is a single-valued function of w and so determines a single-valued set of roots K(w), although any one root K(w) need not be single-valued. Obviously the same argument could have been used previously in connection with Eq. (81). Probably the supplementary boundary conditions (69c) and (71~) are consistent with the requirement that the fields within the plasma are single-valued, so that T(w) indeed remains single-valued in this two-stream case. However, because the main physical consequence of multiple-valued T(w) is growing waves at t > 0 in solutions consistent with causality, perhaps one should not insist T(W) must be single-valued with unstable plasmas. Correspondingly, although doubtless in this two-stream plasma it is not possible to excite one of the Ka2, Kb2 waves without exciting the other, perhaps no general conclusion of this nature is legitimate for unstable plasmas. Suppose the cuts through each branch point of Wbof (89) are drawn consistent with arg (w - Wb) = 0 at w = CD on the positive real axis, as in (87). Then one finds: the positive sign in (89a) yields the roots K,’ = u*/u’ at o = m corresponding to the value n,” = $1~~ found above; the negative sign in (89a) yields the values Kb2 = w”/c” at w = ~4, corresponding to nb2 = 1; the behavior of na’(w) at w = ~0 is consistent with (10). At all w W2c2(K~ + K;)

= (w + vd

p2eJgcd + P”(w” - YW,~) - O e

(90)

which is single-valued. The imaginary part of each term in (90) is 20 for u in the first quadrant of the w-plane. Consequently, as assumed in Section VII to simplify locating the roots of (80), where arg Km2 = 0 in the first quadrant, Im Kl 2 0 and vice versa. Granting the plasma is indeed unstable, i.e, that the streams are sufficiently relativistic, we conclude that with sufficiently thick two-stream plasma slabs, T(w) should have poles at Im w > 0. With such slabs, solutions consistent with causality grow exponentially at t > 0 whether or not T(w) is single-valued at Im w > 0. Taking the complex conjugate of (88), one seesthe set of four roots K(w)* are not identical with the set K(w’). Nevertheless (88) is consistent with the reality requirement. A transverse vacuum pulse incident on the plasma from the left can be written in the form

(91)

44

GERJUOP

where

e* = -& (i f d-lj) (92)

F*(w)

= +p.(“)

7= +lF,(o)l

and F, , F, are defined as in (55) in terms of E, , E, . The quantities F+(U) and F-(w) represent amplitudes of right and left circularly polarized waves respectively, as is readily verified by examining the time dependence of either the real or imaginary parts of c*e?. Moreover E*,* = ET and from Eqs. (55) and (61), since E, is real,

F*(w)* for Im w > 0. Conversely,

= F,(d)

as in Section V, Eq. (93b) together with

E*(z, w)* = E&x, w’)

@a)

ensure the integral (59) is real when C’ is a contour in the upper half w-plane symmetric about the imaginary axis. Evidently E+ (z, W) in (94a) are defined as the projections of the total E(x, a) on E& respectively. Equation (94a) is the generalization of (29) to circularly polarized waves; the corresponding generalization of (lla) obviously is T*(w)*

= T,(d)

(94b) Note Eq. (94b) means the singularities of the individual T+(w) or T-(w) need not be symmetrically located in the upper half w-plane in this two-stream case. If we had chosen incident circularly polarized waves in the previously considered warm plasma case, the singularities of the individual T+(w) and T-(w) would have been symmetrically located, however, because the warm plasma is isotropic. For the two-stream plasma under present consideration the solution in a uniform slab corresponding to an incident unit amplitude left circularly polarized plane wave of complex circular frequency w has the form

E-(2, W) = Z B-, exp (iK”x) a=1

0 5 25 d

(95a)

where K-*(w) are the wave numbers of the independent left circularly polarized waves. Equation (95a) is the obvious generalization of (68) ; because (88) is not an even function of K we now do not assume the K-” occur in pairs AK, , &Kg . For an incident unit amplitude right circularly polarized wave, the solution in the slab is

E+(x, W) = i B+B exp (iK+“z) B=l

0 5 z 5 d

(95b)

MULTIPLE

WAVES AND CAUSALITY

4.5

As in the warm plasma case discussed above, it now is clear that the reality requirements will be satisfied if the set K+‘(u’) is identical with the set - [K-*(w)]*; of course the amplitudes B+‘(u’), B- a(w’) also must be consistent with (94a), so that (again as in the warm plasma case) there are constraints on the forms of the physically allowable supplementary boundary conditions (69~) and (71~). The dispersion equation (9) for right circularly polarized waves is obtained from (88) simply by changing the signs of the cyclotron frequencies wj and (Jo. Thus one seesthat the equation for K+‘(w’) is identical with the equation for [-K-*(w)]*, consistent with the reality requirement. ACKNOWLEDGMENTS I am indebted to Dr. Bruce Robinson and toDr. Werner of preliminary versions of this paper. Helpful discussions Bernstein, Carl Oberman and Sidney Borowita also have

RECEIVED

APRIL

Teutsch for very careful readings with Drs. Clifford Gardner, Ira been much appreciated.

8, 1964 REFERENCES

1. J. J. QUINN AND S. RODRIGUEZ, Bull. Am. Phys. Sm. 9, No. 1, p. 57 (1964). 2. N. G. VAN KAMPEN, J. phys. rad. 22, 179 (1961). 3. J. HILGEVOORD, “Dispersion Relations and Causal Description,” esp. chap. 3. North Holland, Amsterdam, 1960. 4. L. D. La~o.4~ AND E. M. LIFSHITZ, “Electrodynamics of Continuous Media,” pp. 247-260. Pergamon, Oxford, 1960. 5. N. G. VAN KAMPEN, Phys. Rev. 91, 1267 (1953). 6. H. FESHBACH AND C. E. PORTER, Phys. Rev. 96,448 (1954); F. L. FRIEDMAN AND V. F. WEISSKOPF, in W. PAULI, “Niels Bohr and the Development of Physics,” p. 134. Pergamon, Oxford, 1955. 7. V. L. GINZBURG, “Propagation of Electromagnetic Waves in Plasma.” Gordon and Breach, New York, 1961. 8. T. H. STIX, “The Theory of Plasma Waves.” McGraw Hill, New York, 1962. 9. I. B. BERNSTEIN AND S. K. TREHSN, Nucl. Fusion 1, 3 (1960). 10. W. B. THOMPSON, “An Introduction to Plasma Physics,” pp. 9-14. Pergamon, Oxford, 1962. 11. R. COURINT, “Differential and Integral Calculus,” Vol. 2, p. 440. Blackie, Glasgow, 1936. II. P. M. MORSE AND H. FESHBACH, “Methods of Theoretical Physics,” pp. 810-811. McGraw Hill, New York, 1953. 13. P. M. MORSE AND H. FESHBACH, ibid, p. 1671. “Introduction to the Theory of Fourier Integrals,” p. 51. Oxford 14. E. C. TITCHM.~RSH, Univ. Press, 1948. 16. W. HEISENBERG, 2. Natwforsch. 1, 608 (1946); C. MOLLER, Kgl. DansAe Videnskab. Selskab., Mat-fys Medd. 22, No. 19 (1946); ibid. 23 No. 1 (1947); B. FRIEDMaN AND E. GERJUOY, see footnote 4. pp. 34-38. McGraw Hill, New York, 1955. 16. L. I. SCHIFF, “Quantum Mechanics,” 17. P. M. MORSE .~ND H. FESHB~CH, ibid., pp. 357, 369. 18. N. G. VAN KAMPEN, Physica 21, 949 (1955).