Surface Wave Echo in a Semi-Bounded Plasma

Plasma echo theory is revisited to apply it to a semi-bounded plasma. Spatial echoes associated with plasma surface wave propagating in a semi-bounded plasma are investigated by calculating the second order electric field produced by external charges and satisfying the boundary conditions at the interface. The boundary conditions are two-fold: the specular reflection condition and the electric boundary condition. The echo spots are determined in terms of the perpendicular coordinate to the interface and the parallel coordinate along which the wave propagates. This improves the earlier works in which only the perpendicular coordinate is determined. In contrast with the echo in an infinite medium, echoes in a bounded plasma can occur at various spots. The diversity of echo occurrence spots is due to the discontinuity of the electric field at the interface that satisfies the specular reflection boundary condition. Physically, the diversity appears to be owing to the reflections of the waves from the interface.


Introduction
Plasma echoes in an infinite plasma have long been known theoretically [1] [2] as well as experimentally [3].Spatial echoes were theoretically investigated in a static situation where the non-propagating electric field is directed perpendicular to the interface of a semi-bounded plasma [4] [5].If the perpendicular direction is designated as the x direction, the electric field E as well as the distribution function f is spatially onedimensional: ( ) , where 0 x > ( 0 x < ) is the plasma (vacuum) region.In this case, the corresponding Vlasov equation takes the form of a first order differential equation, and can be solved by satisfying the specular reflection boundary condition at the interface 0 . This differential equation approach with the specular reflection boundary condition for a semi-bounded plasma has been shown to be entirely equivalent with the Fourier transform (with respect to x) under the recipe that the ( ) E x is extended into the region 0 x < in an odd function manner, ( ) ( ) E x E x = − − [5].This odd function extension of ( ) E x gives rise to a surface term in the Fourier transform of the Poisson equation, which plays a significant role in the determination of the echo spots.It appears that this surface term, which the earlier authors entirely neglected, gives rise to diversity of echo spots [5].Physically, the surface term manifests the reflection of the electric field at the boundary.
The echo phenomena is the result of a quadratic interaction of the two primary waves launched by two external charges at different locations (spatial echoes) or different times (temporal echoes).In response to the external charges, the plasma distribution function ( ) , , f x v t is modulated with the exponential phase of the linear response function.This term is called the free streaming term since x vt = is the characteristic line of the Vlasov equation for a free particle.This rapidly modulating exponential phase makes the ( ) , , f x v t more and more oscillatory as t or x increases, and consequently, d f v ∫ will become vanishingly small due to almost complete phase mixing.Therefore, in the first order, the phase mixing obliterates any appreciable effect on the macroscopic variable such as density perturbation.However, the second order distribution function which is a product of two first order distribution functions is not phase-mixed when or where the condition for a constructive interference is met, thereby the second order electric field does not vanish, resulting in an echo.It is evident from the expression for the product of two free-streaming exponentials .In this work, we investigate spatial echoes in a semi-bounded plasma, taking a full account of the boundary terms which originate from the oddly continuation of the electric field.This work is an extension of the earlier paper by Lee and Lee [5]; the distribution function and the electric field are now spatially two-dimensional, allowing for the z-dependance.Therefore, the echoes are associated with the surface wave which is propagating in the z-direction.The second order electric field endowed with the additional z-dependance can be Fourierinverted by contour integration with unstraightforward analytic exercise, and delineating the echo condition requires extra complexity.The important boundary term is the discontinuity of the perpendicular electric field at the interface that is necessary to have the specular reflection boundary condition satisfied [5].The diversity of echo occurrence spots has been experimentally reported [7] and can be explained by this boundary term.The identification of the echo spot associated with surface wave appears to be useful in experimental point of view [7].

Formulation of the Problem
We consider a plasma consisting of electrons and stationary ions, the latter forming the uniform background.The plasma is assumed to occupy the half-space 0 x ≥ .The region 0 x < is assumed to be a vacuum.The perturbed electron distribution function ( ) , , f t r v and the electric field ( ) , E t r will depend on x and zcoordinates with the y coordinate ignored since y direction has a translational invariance.We have the nonlinear Vlasov equation and the Poisson equation to describe the electrostatic perturbation: with ˆˆˆ, , where f is a two-dimensional distribution function, and 0 ρ represents the external charges: 0 k is introduced to make the argument of the δ-function dimensionless, and 1 2 → means the replica of the preceding term with the subscript 1 replaced by subscript 2. We solve the simultaneous Equations ( 1) and (2) for a given ( ) 0 , x t ρ as prescribed by Equation (3).In mathematical terms, we have an inhomogeneous system, driven by the source term in Equation (3).The responses f and E should be determined by 0 ρ .
The kinetic equation is supplemented by the kinematic boundary condition which we assume to be the specular reflection condition

( ) ( )
, , 0, , , 0, This specular reflection boundary condition is automatically satisfied by extending the electric field component ( ) x E x in odd function manner into the region 0 x < , i.e., ( ) ( ) . Assuming that the external perturbation is small, we solve Equations ( 1) and ( 2) by successive approximation.First, the linear solution of Equation (1) will be obtained for f with the boundary condition (4).Substituting this solution in Equation ( 2) yields an integral equation for the electric field which is solved by Fourier transform.Then the linear solution will be used to obtain the higher order solutions.We work only up to the second order.The higher order distribution function should also satisfy the boundary condition (4).The electric field should satisfy the electric boundary conditions: the normal component of the electric displacement ( ) x D x and the tangential electric field z E are continuous across the interface.In this work, the Fourier transform is defined by Let us Fourier transform Equations ( 1)-( 3) with respect to t and z to write is derived from the discontinuity of x E at 0 x = .This N-term is characteristic of a semi-bounded plasma and responsible for the diversity of surface wave echoes, as compared with an infinite plasma.The external charges are Fourier transformed to Equations ( 5) and ( 6) constitute a set of nonlinear simultaneous equations.We solve the set of equations by successive approximations in terms of perturbation series: 5) and ( 6) order by order, we have The quantity ( ) 9) should be determined in terms of the vacuum field from the electric field boundary condition: electric displacement ( ) x D x is continuous across the interface 0 x = , ( ) ( ) ( ) ( ) where ( ) ( ) equals to the vacuum electric field 0 E .

Linear Solution
Equations ( 8) and ( 9), and 0 is the dielectric function ( p ω is the plasma frequency).N is determined from the electric boundary condition as shown in the following.We need the normal component of electric displacement, where J is the current: where we used Equation (8).The above quantity equals to ( ) . This statement can be most easily proved by assuming 0 f a Maxwellian.Use . Then, (−1)-term vanishes upon integration, and we have ( ) Using the above result, we obtain To invert Equation (15), we write e e e 1 2 2π In the above integral, we take the limit 0 x + → .Evaluating the integral by residue theorem gives Note that we set up the contour encircling the upper half plane since 0 In either case, the integral is found to be π i .Then, Equation ( 16) takes the form where ( ) where + (−) sign corresponds to 0 The above equality can be easily proven by using the contour winding the lower half plane.Taking the limit 0 Using the above equation in Equation ( 13) gives For an infinite plasma without boundary, we have 0 N = in Equation ( 13), and the plasma electric field is given by Note that in Equation (20), the 0 E -term and ( ) 1 , z I L k -term are the boundary terms which are non-existent in an infinite plasma.
In the static situation where the electric field is nonpropagating, we put 0 z k = in Equation (20), and the electric field reduces to Equation (23) in Lee and Lee [5]: )

Second Order Solution and Echo Occurrence
Next, we deal with the second order equations, Equations ( 10) and (11).Using Equation (10) in Equation ( 11) yields, owing to the electrostatic nature of where Q stands for Substituting the first order solutions [Equations ( 8) and (20)], into the above equations, we can write where I stands for the exponential function as given by Equation ( 18).Since we don't know yet which sign should be chosen, we keep on using the symbol I. Equation ( 23) is to be used for investigation of echo occurrence.The various cross terms in the product (AB) are the candidates of echo resonances to see if the condition for vanishing phase can be met.
We choose to investigate a cross term which is 1-term in A multiplied by 2-term in B. With this term, the t-inversion of Equation ( 23) can be easily carried out by simply putting where In the above equation, we can assume that the poles associated with the dielectric functions contribute negligibly in the dk ∫ -or dk′ ∫ -integral. [The dominant contribution comes from the free-streaming poles.]Also we assume 0 f to be a Maxwellian.Then we have where 1 can be assumed to contribute nothing to the inversion integral in the following, due to phase mixing.Thus, Equation (26) can be further simplified as Let us write explicitly the inversion integral of Equation ( 28) with respect to k: This equation will be examined in view of the possibility of the vanishing phase.
(1) First, we shall consider the interference of two exponential terms in Equation ( 29): The important singularities are: the double pole at ( ) + and the simple poles associated respectively with 2 0 k = and 2 0 k′ = and 1 0 We shall consider only these four poles.Singularities at 0 ε = are not important.Therefore we can put and all the ε's can be taken out of the integral.The residue at the double pole is obtained by taking and substituting for ( ) , suppressing the unessential factor.

Integral d x k′
∫ can be easily done by picking up the pole at For definiteness we assume 2 1 L L > .Then the contour in x k′ -plane should encircle the upper half x k′ -plane, and in order for the pole to lie in the upper x k′ -plane, the imaginary part of ( ) k′ is only a function of z k′ per Equation (31), and we can write the second part of Equation (30) as where ( ) ( ) ∫ -integral depends on the sign of θ ′ : when 0 θ ′ > ( 0 θ ′ < ), the contour must wind the upper (lower) z k′ -plane.The location of the poles depends upon the sign of x v .Sorting out the relevant cases, we carry out the integral for J ′ : Next, taking on the first part of the integral in Equation (30) ( ) ( ) ∫  , we have two cases: 1) 2 x L < In this case, the x k contour must encircle the lower half plane and the x k -integral does not vanish under the provision Then, the integral can be written as where ( ) Analogously to the foregoing calculation in J ′ , the above integral depends on the sign of θ : x L > Repeating a similar analysis, we obtain where ( ) Now, we have to multiply J and J ′ .In doing it, note that ( ) ( ) ( ) ( ) ( ) 0 H x H x − = .Nonzero results surviving the velocity integral are obtained in the following four cases: a) Let us first consider case a).Using Equations ( 35) and (41), we obtain where ( ) ( ) Using Equations ( 34), (39), and (46), we can obtain the exponential phases: Thus Equation ( 45) can be written in the form Therefore the velocity integrals in Equation ( 29) survive the phase mixing when 0 where an echo is given rise to.The electric field can be obtained by velocity integral in the form (see Equation ( 29))  denotes the obvious integrand.Next, let us calculate case (b).Using Equations ( 36) and ( This equation is identical with Equation (45) if

( )
x H v and ( ) x H v − are interchanged in the latter.Thus, this case can give rise to an echo at the same spot as predicted by Equation (53).The corresponding electric field is obtained by a similar velocity integral to Equation (54) but over different range of x v .The cases (a) and (b) predict the same echo spot because they yield the same imaginary phase i ϕ .One more task: the various inequality conditions set forth to specify the contour in the contour integrations need to be checked against the echo coordinate found in Equation (54).Let us consider the inequalities 0 θ < and 0 θ ′ > postulated in the case (a).Using Equation (52), the inequality 0 θ < can be written in the form ( ) which is the condition 0 θ ′ > .Therefore the conditions 0 θ < and 0 θ ′ > imply each other.Also we can ascertain that the echo x-coordinate is in accord with the condition (2) Next, we consider the product of two boundary terms, ( , , where C is a nonessential constant factor.For definiteness, we assume where ( ) The contour of d z k′ ∫ -integral should encircle the upper z k′ -plane.Since the relevant singularity should be located in the upper z k′ -plane, the residue is calculated from ∫ ) in Equation ( 57) can be carried out in the form , π exp exp To carry out ( ) ∫  in Equation (59), let us assume that Using Equation (60) in Equation (62) yields where we have . Therefore, we obtain , π exp exp , π exp exp The above two equations and Equation (61) yield H v H v ± ± .Since we are interested in the echo spots, we pay attention only to the exponential phases: e , e , ( ) e , e Straightly we can identify: From above, the imaginary phases are obtained as ( ) In Equation (75), echo x corresponding to upper signs and echo x corresponding to lower signs are mutually exclusive because if one of them is inside the plasma the other is necessarily is outside the plasma.We add that the condition 0 ϑ < amounts to ( ) , which poses no problem in as much as we have ample liberty in choosing the sign of z x v v .

Discussion
In Section 3, the plasma electric field was determined in terms of the vacuum electric field.Judicious application of the boundary conditions at the interface enables one to determine the plasma electric field entirely in terms of the external charges without introducing the vacuum electric field 0 E .Inverting Equation (13), we can write ( ) ( ) ( ) is the electrostatic dispersion relation of the surface wave in a semi-bounded plasma [10].
In the investigation of echo occurrence, 0 E in Equation ( 20) can be discarded because echoes are given rise to by interference of influences of the external charges.This amounts to saying that the denominator , L S .In the static situation, the z-coordinate is irrelevant.The echo spot given by Equation (53) corresponds to echo x in Equation (45) in Lee and Lee [5].The echo spot given by Equations ( 74) and ( 75) is surface wave-proper.Our search for the echo spots are not exhaustive; we put aside many other product terms in (AB) in Equations ( 24) and (25).It appears that we have diversity of echoes in a bounded plasma, which was also experimentally reported [7].The diversity seems to be due to reflections of the wave at the interface.
In reality, bounded plasmas are usual rather than exceptional.Important literatures to get acquainted with this field are References [8] and [9], among others.Surface wave dispersion relation in a plasma slab is derived in Ref. [10].An exact nonlinear solution of a surface wave excited by external charges is obtained in Ref. [11].
enforce the boundary condition (12).By definition, Clearly, this integral manifests the nature of a step function.By equating the quantity on the right hand side of Equation (17) to the x-component of the vacuum electric field ( 0 E ≡ ), we obtain in cases (a) and (b), the premise and the result are consistent.For the cases of (c) and (d), we state without repeating a similar algebra that the imaginary part of the phase is still obtained by Equation (50) [the real part of the phase is different].Although the echo spot is predicted by the same equation as Equation (53), these cases of (c) and (d) are not acceptable because are contradictory to each other.We have the conclusion: an echo occurs where 2 x L < and the echo coordinates are predicted by Equation (53).
are defined in Equation (33).Then the last integral ( d z k′ ready to evaluate d z k ∫ -integral in Equation (59):

E
turn to the vacuum solution.is the vacuum electric field, the quantity designated by the same symbol 0 E in Equation (19). ) should be compared with Equation (20).Eliminating the vacuum field introduces the denominator play any role in the determination of echo locations.Equations (53) and (74) and (75) are the main results of this work in locating the echo spots associated with the surface wave in a semi-bounded plasma launched by the oscillating external charges at ( ) ( ) 67)Now we are ready to carry out the velocity integral in Equation (59) by substituting Equation (63) into it.Because of the step functions