Lecture 3. Dispersion and waves in cold plasma. Review and extension of the previous lecture. Basic ideas. Kramers-Kronig relations

Size: px

Start display at page:

Download "Lecture 3. Dispersion and waves in cold plasma. Review and extension of the previous lecture. Basic ideas. Kramers-Kronig relations"

Willis Woods
5 years ago
Views:

1 Lecture 3 Dierion and wave in cold lama Review and extenion of the reviou lecture Baic idea At the reviou lecture, we dicued how to roerly earch for eigenmode (or quai-eigenmode) of a dierive medium. In ummary, thi involve the following te: find the time-ace rereentation of the conductivity, Σt, x; take the Fourier tranform in x and the Lalace tranform in t to get Σw, k for w with large enough Im w; continue the reult analytically into the whole comlex-w lane (a rocedure we are yet to dicu); calculate Εw, k 1 w Σw, k; calculate i j c k i k j k i j w Ε i j w, k; find the eigenfrequencie q k from det q k, k ; find the correonding olarization vector R q k from q k, kr q k. In contrat to the naïve alication of the Fourier tranform to the field equation, here, all function are well defined, and Σw, k i not jut the Fourier image of Σt, x but intead the analytic continuation of the temoral Lalace tranform combined with the atial Fourier tranform. Thi how that, a a hortcut, one can derive the dierion relation by relacing t, k, but one mut kee in mind the true definition of Σw, k that enter the final reult, and one mut alo realize that the mode obtained thi way are generally quaimode rather than the true eigenmode of the ytem. Kramer-Kronig relation Since we tarted dicuing comlex frequencie, let u alo briefly dicu ome roertie of reone function uch a Σt, x and Σw, k that we will need further on. We will be intereted in the temoral/frequency deendence, o the coordinate/wavevector deendence will be omitted in thi ection for brevity. Alo note that the following dicuion alie not jut to Σ but alo to Χ and, even more generally, to any reone function of any table ytem. Hence we adot the following general terminology. Conider an arbitrary ytem whoe reone j to an external force E i linear, i.e., jt t t ' Σt t ' Et ', where Σ i ome reone function, or Green function. (For examle, j can be a current denity and E can be an electric field, but the argument below doe not require thi interretation.) Let u aume that the ytem i table, which mean, by definition, that it reone eventually fade away; i.e., Σt. Hence, the integral that determine it Lalace image,

2 lecture-3.nb Σw w t Σt t, converge at all w with Im w. Auming that Σt fade away fater than any ower of t, the ame argument alo alie to integral J n w w t t n Σt t, which can be recognized a derivative of Σw [namely, J n w Σ n w]. We conclude then that all derivative of Σw are well defined at Im w. Therefore, Σw i a mooth function in that region. We ummarize thi in the form of the following theorem: Theorem: Σw i analytic in the uer half of the comlex-w lane. Comment: In contrat, the analytic continuation of Σw to the lower half of the comlex-w lane i not guaranteed to be mooth everywhere in the region Im w ; i.e., there can be oint to which continuing the function analytically i imoible in rincile. For examle, a we dicued at the reviou lecture, the conductivity of colliional cold nonmagnetized lama cale a Σw w Ν 1 with Ν, o it ha a firt-order ole at w Ν. Alo, note that Σw JΣ w t Σt t w t t w Σt w t Σ' t t w w t w Σt w w t Σ' t t Σ w w JΣ'. (Here, we aumed that the function Σt i alo nice enough, uch that Σ and JΣ' are finite.) Since Σ t i determined by microcoic hyical rocee, which are never intantaneou, the time cale of Σ i never zero. Thi mean that there alway exit a art of w axi where w 1. There, JΣ'w JΣ, o Σw Σw, o Σw at w. Comment: The above argument (and thu the dicuion below too) doe not aly to reone function that involve immediate (i.e., delta-haed) reone. An examle i the dielectric function, Εt, defined in the firt lecture. Unlike the conductivity and the ucetibility, Ε ha a finite aymtotic ectral amlitude, Εw 1. Likewie, alying the above argument to the ucetibility require a modification. Thi i becaue the ucetibility characterize the medium olarization, i.e., charge dilacement from equilibrium (ee elementary E&M). If a zero-frequency, i.e., tatic field i alied to the medium, the olarization at t doe not fade away but rather aroache a finite limit. (Think of a harmonic ocillator erturbed by a tatic force a an examle.) Hence, Χ i ingular at w, o it i analytic only at Im w, a ooed to Σ, which i analytic at Im w. In combination with the fact that Σ, the analyticity of Σw in the uer half of the comlex-w lane lead

3 lecture-3.nb 3 to ome intereting roertie of Σ r w Re Σw and Σ i w Im Σw, which can be derived a follow. Conider real and the integral Σw w, C w where C i a contour that goe along the real axi yet encircle the ole at w from above. On one hand, clearly, one can rewrite a Σw PV w Π Σ, w where PV denote the Cauchy rincial value. On the other hand, one can cloe the contour C through a emicircle C with radiu R at Im w, ince the integral over C i identically zero. [Thi i becaue, at w, the integrand decreae fater than 1w due to Σw.] But Σww i analytic everywhere within the cloed contour, o any integral over any cloed contour in thi region i zero. Hence, we get, or, equivalently, Σ Π PV Σw w. w Taking the real and imaginary art of thi give that Σ r and Σ i are connected via the Hilbert tranform; namely, Σ r 1 Π PV Σ i w w, w Σ i 1 Π PV Σ r w w. w Thee are known a Kramer-Kronig relation (KKR). In articular, they how that having nontrivial Σ i at ome frequencie imlie alo having nontrivial Σ r at ome (oibly, different) frequencie, and vice vera. In thi ene, dierive media are alway diiative media and vice vera. Comment: In alication to ecific ytem, the aumtion of tability that underlie the KKR can be ubtle. In articular, in homogeneou lama, the reone of a current to an external field i table, o the lama conductivity atifie the KKR indeed. However, the reone of the elf-conitent field in a lama to external current i not necearily table. (Later in thi coure, we will how that lama can uort variou intabilitie.) Hence, the reone function of the field doe not necearily atify the KKR. For examle, for electrotatic ocillation in nonmagnetized lama, the correonding reone function i Ε, k 1. [Thi i een from Gau law, Ε E Ρ ext, where Ρ ext i the external charge denity.] Since Ε, k can have zero at Im (which correond to untable mode; we will tudy them later), the reone function Ε, k 1 i not necearily analytic in the uer half of the comlex- lane. Further te Now that we have dicued the general rocedure for obtaining the dierion relation and baic roertie of reone function uch a the conductivity, we will aly thi knowledge to tudy ecific wave in ecific lama. We tart by reviiting the imle examle of cold magnetized lama that we dicued at the reviou lecture. From now on, we adot the aumtion of colliionle lama for clarity. However, thi aumtion doe not exclude diiation, a we will dicover later within a kinetic aroach.

4 4 lecture-3.nb Cold nonmagnetized lama reviited Previouly, we obtained the following equation for the linearized induced current in cold nonmagnetized lama: t j E. Intead of imly relacing t, let u work in the acetime rereentation and imly integrate the equation for j with reect to time. Thi give j t, x x j t E t ', x t '. Here, the firt term, j x, i ome integration contant (more reciely, a function of x), which can be undertood a an external current, and the econd term rereent the induced current, j ind. One can infer Σ a the current roduced by a delta-haed field (ee Lecture 1), namely, Σt, x t t ' x t ' x. Alternatively, one may imly notice that the exreion for j t, x can be rewritten a j t, x x j t t ' 3 x' x x' E t ', x', o one obtain the ame reult: Σt t ', x x' jt, x Et ', x' x x'. Comment: The reence of the delta function in x ignifie that the reone function Σ i local in ace, o there i no atial dierion. (The abence of the atial dierion i due to the fact that we neglected the lama temerature. Thermal effect will be reintroduced later during the coure.) In contrat, temoral dierion i reent in the ene that Σt, x doe not decreae in time, i.e., the current in cold colliionle lama retain information about external field forever. Now we can calculate the Lalace-Fourier image of thi (from now on, we will ue and w interchangeably and omit the bar in ymbol denoting Lalace image), Σ, k t t 3 x kx Σt, x t t kx x 3 x t t. Here, we aumed Im for the latter integral to converge, but the reult i analytic at all (excet ), o it analytic continuation to the whole comlex lane i given by the ame formula. In other word, we can aly thi exreion for Σ at all comlex (excet, where Σ i inherently not analytic, o no analytic continuation i oible by definition). Finally, the dielectric function i found to be Ε 1 Σ 1. Note that, reviouly, the naïve rocedure of relacing t with in the differential equation for the current led

5 lecture-3.nb 5 u to the ame reult. Wave General dierion relation A we already dicued, the correonding dierion tenor can be exreed a follow:, k Ε Ε N, Ε N t where N k c refraction index vector, and axe are choen uch that k i ointed along x axi. The correonding dierion relation, det, can be written a follow: Ε N Ε. Hence, it i een that two tye of wave are oible, which we will now conider in detail. Longitudinal electrotatic lama ocillation One olution of the dierion relation correond to Ε, i.e.,. Thi exlain the term lama frequency ; namely, haen to be a natural frequency of lama ocillation. (Conequently, it i alo common to refer to them a the lama ocillation, a ooed to other wave and ocillation that can exit in lama.) Thee wave are nondiiative and can have any hae velocity v kk k, becaue, while the frequency i fixed, k i not. In contrat, their grou velocity v g k k i identically zero. The wave olarization i found from the field equation, ke. At, thi equation become N N E x E y E z, o E y E z. In other word, the field i arallel to k, o the wave i longitudinal and therefore electrotatic. Take-home quetion: Derive a PDE for the electron-denity erturbation in thi wave uing the linearized continuity equation for the electron denity, the linearized momentum equation for the electron velocity, and the Poion equation for the electrotatic otential. (Since we already know that thee wave are electrotatic, uing the comlete et of Maxwell equation i not neceary; the Poion equation i a good enough ubtitute.) In electron-ion lama, one ha e (due to m i m e ), in which cae the lama ocillation can be interreted a ocillation of electron relative to the tationary neutralizing ion background. Thee ocillation were dicovered by Langmuir and Tonk in the 19 [Phy. Rev. 33, 195 (199)], o they are alo known a (electron) Langmuir ocillation. Tranvere electromagnetic wave The other olution of the dierion relation correond to Ε N, which lead to k c. The frequencie are real, o thee wave do not diiate either. However, unlike Langmuir ocillation, thee are tranvere electromagnetic wave (E x ); i.e., the field i erendicular to the wave vector. In fact, at,

6 6 lecture-3.nb they are the ame a vacuum light wave. The lama effect become imortant at lower frequencie. In articular, thee wave cannot roagate at ; e.g., a wave launched from vacuum toward a dene lama will be reflected. Finally, we notice that the tranvere electromagnetic wave have a uerluminal hae velocity, Υ k k c k c (o they are called fat ), and their grou velocity i given by Υ g c Υ c. There are no other wave in cold nonmagnetized lama. However, the ituation change dratically when a tatic magnetic field i imoed, a we will dicu next. Cold magnetized lama Plama dierive roertie Baic equation Under the ame aumtion a above (linear dynamic in cold colliionle lama), yet with nonzero background magnetic field B that we chooe arallel to z axi, we get t v e E m v e z for any given ecie. [We introduced e B m c and temorarily omitted the index for brevity. Note that i generally a igned quantity.] Thi give the following equation for the velocity comonent: t Υ x e E x m Υ y, t Υ y e E y m Υ x, t Υ z e E z m. We already know how to olve the latter equation, o let u focu on the former two. Secifically, let u multily the econd equation by and add it to the firt equation. Uing the notation Υ Υ x Υ y and E E x E y, one can write the reult a follow: t Υ Υ e E m. For the correonding current, thi give t j j E. One may recognize thi a a driven Schrödinger equation. We will alo encounter later in other context, o we olve it in the aendix in detail. Conductivity and ucetibility A een from the aendix, the equation for the field-induced current (for any given ecie) equal t, j x t tt' E t ', x t '. Correondingly, the comonent of thi current are

7 lecture-3.nb 7 xt, j x Re t tt' E t ', x t ' t co t t ' E xt ', x in t t ' E yt ', x t ', jyt, x Im t tt' E t ', x t ' t in t t ' E xt ', x co t t ' E yt ', x t ', The conductivity i obtained from here jut like for nonmagnetized lama and i found to be Σ xx t, x Σ yy t, x x co t, Σ xy t, x Σ yx t, x x in t, Σ zz t, x x, Σ xz Σ zx Σ yz Σ zy. The correonding ectral rereentation, at Im, are Σ xx, k Σ yy, k Σ xy, k Σ yx, k Σ zz, k.,, Like in the cae of nonmagnetized lama, the conductivity i indeendent of k (i.e., there i no atial dierion), o below we omit the argument k. Accordingly, the linear element of the ucetibility tenor are Χ xx Χ yy Σ xx Χ xy Χ yx Σ xy Χ zz.,, Take-home roblem: Show that the ame reult are obtained if one imly relace t with in the equation for j. Dielectric tenor [memorize thi reult] We now um u over ecie and obtain the dielectric tenor in the form Ε S D D S P where we introduced, a in [Stix],,

8 8 lecture-3.nb S 1 1 R L, D P 1 and alo 1 R L,, R 1, L 1. Although thi reult wa derived for Im, it i analytic at Im too. Therefore, thi reult alie at all. Note that thi dielectric tenor i alo Hermitian, which we will later how to be ignificative of nondiiative medium. The only excetion are the oint and ±. We will learn how to deal with thee excetion later. Aendix: Solution of a driven Schrödinger equation Conider a driven Schrödinger equation of the form t Ψ H Ψ F, auming that H i contant, and F i a recribed function of time. Let u earch for a olution in the form Ψt H t t. Then, t Ψ H Ψ H t t. Subtituting thi into the original equation lead to H t t F, o, for the driven art of, we obtain t t H t' Ft ' t ', where i an integration contant. Hence, for Ψ we get Ψt H t t H tt' Ft ' t '. The former term here, which we will denote a Ψ t, i a olution of a homogeneou equation, t Ψ H Ψ, while the econd term i a driven olution. Hence it i convenient to rereent the reult alo a Ψt Ψ t tgt t ' Ft ' t ', where G i the Green function (or the roagator of the homogeneou equation), Gt H t.

9 lecture-3.nb 9 Comment: More generally, H can be a matrix. Then, the ame olution alie, excet G mut be undertood a a matrix exonential. If, in addition, H i time-deendent, G mut be undertood a ordered (a.k.a. timeordered ) matrix exonential, Gt ex tht ' t '.

Figure 1 Siemens PSSE Web Site

Figure 1 Siemens PSSE Web Site Stability Analyi of Dynamic Sytem. In the lat few lecture we have een how mall ignal Lalace domain model may be contructed of the dynamic erformance of ower ytem. The tability of uch ytem i a matter of