Small signal gain of a Smith-Purcell free electron laser

In this paper the operability of a Smith-Purcell free-electron laser for an arbitrary grating is considered. An arbitrary grating is described in terms of impedance matrix. The consideration is based on the self-consistent set of Maxwell-ﬂuid equations. The dispersion relation is established for the TM mode of electromagnetic waves. The latter is studied considering ampliﬁcation of the evanescent as well as radiation modes. The small signal gain in various limits is calculated and the dependence of the gain on the electron beam current is analyzed.


I. INTRODUCTION
A traveling charged particle induces radiation when it passes over the metallic grating. This radiation, called Smith-Purcell (SP) radiation [1], has several remarkable properties. Since its first observation, SP radiation has been studied both experimentally and theoretically by many authors [2 -4] (for more recent discussions and other references on the subject see [5]). Owing to the unique properties of SP radiation, it has long been recognized that SP radiation can be a basic mechanism for a compact freeelectron laser (FEL) [6,7].
Recently, there has been a renewed interest in the SP system related to both a possibility of using it to generate intense radiation from the millimeter to optical region [8] and to implement it in nondestructive beam diagnostics [9]. The experiment at Dartmouth College [10,11], where superradiance in the SP system was observed, has stimulated new investigations concerning the SP FEL as an open slow wave structure [12 -16].
In the paper [7], the authors considered the initial incident propagating wave, which is amplified during the multiple reflection upon grating. They predicted a small signal gain of SP FEL to be proportional to I 1=3 b , where I b denotes the electron beam current. The authors of [7] considered magnetized electron beam and described grating in terms of reflection matrix. It an additional requirement was invoked that the reflection matrix is singular at the frequency of the incident propagating wave. But this is actually not the case. As will be shown below, the singularities of the reflection matrix correspond to eigenmodes of grating which are the evanescent waves.
In [12], the authors considered an infinitely thin electron beam passing close to a grating. They assumed that both an evanescent and a propagating mode have the same frequency. The electron beam amplifies the evanescent mode, which scatters into the propagating radiation mode. The gain predicted by this model is / I 1=2 b . A different result has been obtained in Ref. [13]. In this paper it is considered a rectangular grating, assuming that the entire space above the grating is filled by a uniform electron beam. The authors have established the dispersion relation and found the dispersion law !k. It turns out that the dispersion equation allows only evanescent solutions and the operating point of a SP FEL is fixed by the intersection of the dispersion curve with the beam line. The corresponding small signal gain follows the I 1=3 b law. However, the coefficient differs from that of Ref. [7]. In Refs. [14,15] it has also been shown that for a low-energy electron beam the group velocity of the resonant evanescent wave may be negative. This means that the device operates on an absolute instability regime in contrast to the conventional FEL, where a convective instability takes place. At the absolute instability, the SP FEL operates like a backward wave oscillator [17] and without external feedback if the current is above a threshold value called the start current.
In [16], the authors have revised the work [12] with proper evaluation of singularities of reflection matrix for rectangular grating. The authors considered an infinitely thin electron beam extending analysis also to the nonlinear regime by performing one-dimensional numerical simulations. Particularly, they found that the growth rate equation becomes cubic near the singularity of the reflection matrix as obtained in Ref. [7]. Otherwise the growth rate equation is quadratic as obtained in Ref. [12]. The latter regime corresponds to the spectral region, where some of the spectral orders are radiative.
Recently, SP FEL dynamics has also been investigated using the 2D particle-in-cell (PIC) code [18,19]. The numerical results of [18] in general are in agreement with the analysis of [13,14], but the gain calculated in [18] is within a factor of 2 of that predicted in the work [13].
Hence, there is no general consent on the mechanisms of the SP FEL and the small signal gain obtained by different approaches. Taking also into account that there is currently substantial interest in the implementation of a SP FEL, it is of interest to reexamine the theory of SP FEL.
In the present work, the operability of the SP FEL for the arbitrary grating is considered. The method described allows any complex geometry one may wish to consider so long as the geometry repeats itself periodically and has one axis of translational invariance. The consideration is based on the self-consistent set of Maxwell-fluid equations. An arbitrary grating is described in terms of impedance matrix. The electron beam is not assumed to be infinitely thin as it was considered in Refs. [12,16] and it does not occupy the entire space above the grating as was assumed in Ref. [13]. We consider an electron beam of finite thickness to be at a finite height above the grating and in contrast to Ref. [7] there is no external incident wave, and the electron beam is not magnetized. For comparison, we also analyze the case of a magnetized electron beam. We derive a general expression for the dispersion relation and examine various limits of small signal gain depending on the particle beam and grating parameters. Then, obtained results are applied to rectangular grating making comparisons with the results of the works [13,18].
The paper is organized as follows. In Sec. II we solve the self-consistent set of equations which describe FEL dynamics in general and as the most effective case the hydrodynamic instability of a cold electron beam is considered. Rectangular grating is considered in Sec. III. A concluding section summarizes the paper.

II. HIGH-GAIN REGIME OF A SP FEL
In this section, we develop a theory of a SP FEL for an arbitrary grating considering the linear stage of instability and calculate the small signal gain of a SP FEL in the regime of hydrodynamic instability. We take a Cartesian coordinate system: the electrons initially move to the z direction in the vacuum over the grating along the trajectories d x d as it is shown in Fig. 1 and coupled with the TM mode of electromagnetic wave. The grating is ruled parallel to the y axis and its ruled area is assumed to be large enough to ignore any boundary effect. We denote the grating period as g.
Our goal is to obtain the dispersion relation for the TM mode of electromagnetic waves. The roots of the dispersion equation give us the functional dependence !k. The presence of the electron beam leads to complex shift of the characteristic frequencies (or wave numbers). The roots with imaginary part indicate the collective instability, i.e., exponential gain of the corresponding wave.
Following the ansatz developed in the paper [20] for surface Cherenkov FEL, we proceed to calculation of the gain in the regime of hydrodynamic instability. Thus, we will arise from the self-consistent set of the Maxwell and fluid equations. For the TM mode (E y H x H z 0) the Maxwell equations can be written as where j x; z; t j ! x; ze ÿi!t c:c: is the current density. The fluid equations read (with v denoting the local fluid velocity) @ is the Lorenz factor, m is the electron mass, e is the electron charge, and nx; z; t is the electron beam density. Before the interaction for nx; z; t we assume nx; z; tj t0 n 0 x 8 > < > : The set of Eqs.
n n 0 x n ! e ÿi!t c:c: gives Taking into account the periodicity of the grating, the solutions of Eqs. (1) above the grating are expanded in infinite series With the help of Eqs. (11), from Eqs. (7) and (8) where v zL ie Here is the resonance width. Taking into account Eqs. (13) and (14), from Eqs. (9) and (10) for the amplitudes of the current density components we obtain Lÿ1 j x;zL e ik L z ; where is the plasma frequency of the electron beam and Returning to the Maxwell equations (1) with the help of Eqs. (16) and (17), for the tangential component of the field amplitude we obtain the following equation: In region (I) for the tangential component of the field from this equation we have where E 1L is a constant and When q L is imaginary, the solution (20) represents propagating waves. In this case we assume and Eq. (20) corresponds to an outgoing wave to x 1. When q L is real, the solution (20) corresponds to an evanescent wave. To satisfy the boundary condition, that the wave vanish in the limit x ! 1, we will choose the root with positive real part.
In region (II) d x d , where the electron beam exists, we have and the solution can be written in the form In region (III) the solution of Eq. (19) can be written in the form Hence the solutions of the Maxwell equations (1) in the gap between the electron beam and grating can be written in the form Next we consider the boundary conditions. For this purpose we integrate Eq. (19) across the interface between the electron beam and vacuum where x 1 d , x 2 d, and we obtain the boundary condition for the normal component of the wave field Taking also into account the matching conditions for the tangential fields we obtain the coupling equation for E 3L , E ÿ 3L , E 2L , E ÿ 2L , and E 1L . From these equations one can express the amplitude of the reflected wave, as where is the coupling constant and To guide electron beams near the grating surface, usually the strong static magnetic field is applied along the direction of the electron beam motion. At that, the electron beam dynamics is assumed to be one dimensional. The electron beam can be assumed to be magnetized and electron motion -one dimensional, if Here H 0 is the static magnetic field strength, v ? is the electron transversal velocity, and ! c is the cyclotron frequency. For magnetized electron beam, one can omit the density variation effect and the normal component of the current density in Eqs. (1)- (3). In this case Eq. (19) reads In region (II), where the electron beam exists, the solution can be written in the form where With the same analysis performed for an unmagnetized electron beam, one can obtain coupling constant (32) for the magnetized electron beam: Note that the latter is exactly the same as that obtained in Ref. [7]. For the boundary conditions on the grating surface, at x 0, one needs the solution of the Maxwell equations in the grooves (x < 0). The latter is expressed as series of the cavity modes in the grooves. At this stage we do not concretize this solution and represent the boundary conditions imposed by the grating, in terms of an impedance matrix. As the most general relation between the ampli- Here the elements of the impedance matrix are determined by the coupling of each incident harmonic (E ÿ 3J ) with the entire manifold of reflected ones (E 3J ). Taking also into account the coupling of reflected and incident waves due to the presence of the electron beam [see, Eq. (32)], we obtain For a solution of Eq. (41) to exist, the determinant of the coefficients must be zero: This is the most general dispersion relation, and its roots give us the functional dependence !k. Now, we should take into account that the electrons resonantly interact with an evanescent mode traveling along the surface of a grating. Indeed, for the resonant interaction, the phase velocity of the wave should be synchronous with the mean velocity of the electron beam. The wave-electron synchronous implies that and from Eq. (21) we see that for this mode Without loss of generality, one can assume N 0. Hence, near the particle-wave synchronism J can be omitted, except the one for a mode J 0, then J 0 0J ; and the dispersion relation (42) can be rewritten as Here we have introduced the notations and is the corresponding minor. The dispersion relation (45) among other effects includes the space charge effects, the gain dependence on the beam thickness, and the beam height above the grating. Note that Eq. (45) can be written in terms of the reflection matrix: is the element of reflection matrix: The dispersion relation in the form (48) with the coupling constant (38) corresponding to magnetized electron beam is exactly the same as that obtained in Refs. [7,16].
In the absence of the electron beam ( 0 0), the dispersion relation is  In the case of evanescent wave amplification, we expand D!; k about the solution for the no-beam case: D!k 0 ; k 0 0 and write where D k @D=@k. The dispersion relation (55) near the particle-wave synchronism !k 0 ck 0 ; 0 ck 0 ÿ k gives the cubic equation for small signal gain in the cold beam limit: Of the three roots, the root with the negative imaginary part gives rise to exponential gain of the corresponding wave, and we find that the amplitude growth rate is As is seen from Eq. (56), the small signal gain decays exponentially with the beam height above the grating as expÿ2k 0 d=3 0 . The gain dependence on the beam thickness is given by the factor 1 ÿ expÿ2k 0 = 0 1=3 . For the thick electron beam k 0 = 0 1, and when passing very close to a grating k 0 d= 0 1, from Eq. (56) we get This formula is analogous to a small signal gain obtained in [13] for a rectangular grating (with 1). For the thin electron beam k 0 = 0 1, which for magnetized electron beam ( 1=2) corresponds to the setup of Ref. [16], from Eq. (56) we get Here the plasma frequency associated with the electron beam is expressed by the beam current as follows: where I A mc 3 =e 17 kA is the Alfvén current and y is the beam width in the y direction. If one assumes the existence of a propagating mode, then for those ! and k some of q L (L ÿ1; ÿ2; . . . :) are imaginary. In this case D!; k Þ 0 and Eq. (54) becomes G 2 R 00 ck; k ! 2 p 2 3 0 c 2 2 1 ÿ e ÿ2k 0 = 0 e ÿ2k 0 d= 0 : Assuming that ImR 00 ck; k 0, and ReR 00 ck; k > 0 for the small signal gain, one obtains Far away from the grating, the propagating modes correspond to SP radiation. In this regime, the electron beam amplifies the evanescent mode which scatters into the propagating radiation mode. The wavelength of the radiation, observed at the angle # from the direction of the electron beam, is g jLj As follows from Eqs. (51) and (60), the formula for the gain (60) is valid when jR 00 ck; kj 1. The latter is difficult to realize and presumes severe restriction on the grating parameters.
For the thin electron beam k 0 = 0 1, the small signal gain (60) becomes Note that the latter with 1=2 is exactly the same as that obtained in Ref. [12]. This can be understood as follows. In [12] the electron beam dynamics is assumed to be one dimensional which is applicable for magnetized electron beam.
For the high-gain regime it is also necessary to take into account the conditions G max 1 where L int is the interaction length in the z direction. Here by i we denote the set of quantities characterizing the electron beam and by i their spreads. The condition (63) can be written as The first and second terms in the curly brackets of Eq. are the resonance widths due to energetic and angular spreads, and the last term expresses the resonance width due to limited interaction length.

III. RECTANGULAR GRATING
In this section, as an important example of the application of the obtained results, we present the small signal gain of a SP FEL in the case of a rectangular grating. The grating is assumed to have ideal conductivity. We denote width and depth of a groove as w and a, respectively. The solution of the Maxwell equations in the grooves ( ÿ a x 0) is expressed as series of the cavity modes in the grooves. With the help of these solutions it is easy to calculate the impedance matrix Z LJ . The latter is presented in the Appendix of this work. On inserting the impedance matrix Z LJ Eq. (A11) into Eq. (50), we obtain the free dispersion relation for a rectangular grating: Note that the coefficients of Eq. (65) show the existence of the similarity relation in the functional dependence !k with respect to the grating parameters. In other words, the dependence of the normalized angular frequency !g=2c on the normalized wave number kg=2 for one grating coincides with that of another grating if the ratios w=g and a=g are the same for the two gratings.
To find out the functional dependence !k for rectangular grating we solve Eq. (65) using a 5 5 approximation to infinite determinant (we have kept elements with J, L ÿ2, ÿ1, 0, 1, 2). The result of the numerical solution of Eq. (65) is shown in Fig. 2 for a rectangular grating with w=g 0:5 and a=g 0:5. This corresponds to the setup of [18]. The abscissa is the normalized wave number and the ordinate is the normalized angular frequency. As is seen from this figure, the beam line ! ck for 0:548 (at the kinetic energy 100 keV) intersects the dispersion curve at the wave number 0.5684 and frequency 0.3115 of the resonant evanescent wave. The resonant frequency and wave number are close to those of the paper [18]. As we see the group velocity (d!=dk) of the resonant evanescent wave is negative and for this energy of an electron beam the SP FEL operates like a backward wave oscillator.
Next we solve Eq. (45) to find out small signal gain. In Fig. 3 the small signal gain versus the beam current (I) per meter in the y direction is plotted for d=g 0:1 and =g 0:25. We have also plotted gain calculated by the approximate solution (56) (dashed line). As we see, the numerical solution is in agreement with the analytical one, except the region for high current values because of the space charge effects contribution in the amplification process.
For the comparison with the results of PIC simulations [18] where the gain is estimated to be Gg=2 ' 0:0045 I 1=3 cm ÿ1 , we solve Eq. (45) with a coupling constant corresponding to the magnetized electron beam [see Eq. (38)]. The latter comes from the fact that in [18] a superimposed constant magnetic field of 2 T (in the z direction) was assumed. That is, in fact they have magnetized electron beam. In Fig. 4 the small signal gain versus the beam current is plotted for the magnetized electron beam. The solid line corresponds to the numerical solution of Eq. (45), while the dashed line to PIC simulations [18]. As we see, the agreement is quite good.
We have also made calculations for the parameters close to those of the experiment at Dartmouth [10]. We assume that the electron beam fills a region of width 24 m, equal to the diameter of the beam used at experiment. At that, beam height above the grating is assumed to be d 0 and beam current per meter is I ' 53 A=m (this corresponds to beam current I b 1 mA). In Fig. 5 the small signal gain versus the beam energy is plotted for g 173 m, w 62 m, and a 100 m. Within factor 1.2 this result is close to that of [13], where it is assumed that the entire space above the grating is filled by an electron beam. It is easy to see that difference comes from the ''filling factor'' 1 ÿ e ÿ2k 0 = 0 1=3 for a beam of finite height.

IV. CONCLUSION
We have presented a theoretical treatment of the highgain regime of a SP FEL. Considering an electron beam of finite thickness being at a finite height above the grating, we have established the dispersion relation for the TM mode of electromagnetic waves. An arbitrary grating is described in terms of impedance matrix. We have also taken into account electron motion in the normal to the grating surface direction. The dispersion relation has been studied considering amplification of the evanescent as well as radiation modes. An analytical expression for the small signal gain has been derived. The gain dependence on the beam thickness and the beam height above the grating has also been analyzed. We have examined various limits of small signal gain depending on the electron beam and grating parameters. For comparison, we have also analyzed the case of a magnetized electron beam. Then, obtained results have been applied to rectangular grating, making comparisons with the works [13,18]. The numerical results in general are in agreement with analytical ones. The analysis of obtained results shows that the obtained small signal gain is in good agreement with the results of numerical investigations [18].