Smith-Purcell Radiation from Rough Surfaces

Radiation of a charged particle moving parallel to a inhomogeneous surface is considered. Within a single formalism periodic and random gratings are examined. For the periodically inhomogeneous surface we derive new expressions for the dispersion relation and the spectral-angular intensity. In particular, for a given observation direction two wavelengths are emitted instead of one wavelength of the standard Smith-Purcell effect. For a rough surface we show that the main contribution to the radiation intensity is given by surface polaritons induced on the interface between two media. These polaritons are multiply scattered on the roughness of surface and convert into real photons. The spectral-angular intensity is calculated and its dependence on different parameters is revealed.

Introduction. Smith-Purcell radiation(SP) [1] is originated when a charged particle travels parallel to a plane with diffraction grating. Recent renewed interest in this problem is caused by different applications. Among these applications are length determination for short electron bunches [2], creation of monochromatic light source in the far infrared region [3][4][5][6][7][8], etc.. Various theoretical models were proposed for describing the SP; [9][10][11][12][13], for a brief review of recent theoretical works see [14,15]. Most of these models deal with the periodical grating in the strong scattering regime (see below). However in many situations the interface over which the charge travels is rough . As an example one can mention chamber walls in storage rings. Even the best treated surfaces contain roughness. Radiation appearing when a charged particle moves near a rough surface could be useful for beam diagnostics [16]. The influence of the surface roughness on the transition radiation (originating when particle crosses the interface between two media) was discussed in [17,18]. Roughness-induced radiation for a charged particle sliding over a surface was experimentally observed in [19].
In the present paper we study radiation emitted due to electromagnetic field scattering on inhomogeneities of dielectric constant . We will see below that in the weak scattering regime it is possible to develop a rigorous theory describing both periodical and random grating within a single formalism.
General Relations The geometry of the problem is shown in Fig.1.
A charged particle moves uniformly in the vacuum at the distance d from the plane z = 0 separating vacuum and isotropic medium. We are interested in the radiation field far away from the charge and the interface. The

Maxwell equation for the electric field reads
where j is the current density related to the charge Here v is the velocity of the particle moving on 0x direction and ε( r, ω) is the inhomogeneous dielectric permittivity of the system which for a rough surface can be chosen in the form where Θ(z) is Heaviside ′ s unit step function, h(x, y) is the amplitude of surface roughness. As it follows from Eq.(3), the space z > h(x, y) is vacuum while the space z < h(x, y) is occupied by a medium with isotropic dielectric constant ε(ω). Assuming h(x, y) small and expanding Eq.(3), one gets [20] ε( r, ω) = ε 0 (z, ω) + ε r ( r, ω) where and Thus the total ε is presented as a sum of a regular part ε 0 and an irregular part ε r . To separate the radiation field we decompose electric field E = E 0 + E r analogous to Eq.(4). Here E 0 and E r are the background and radiation fields, respectively. They obey the following equations Note that although the term ε r E r in Eq.(8) is small one should keep it because it causes multiple scattering of electromagnetic field. We will see that multiple scattering effects are very important in radiation from rough surface. Multiple scattering effects in SP radiation for a cluster of dielectric particles were discussed in [21]. At large distances from the system the electromagnetic field can be treated as a plane wave in which electric and magnetic fields equal to each other. Therefore the intensity of radiation at the frequencies [ω,ω + dω] and at solid angles [Ω,Ω + dΩ] can be determined as follows where n is unit vector on the direction of observation point R, Ω is the corresponding solid angle, see Fig.1 and also [14]. As usual at large distances | E r ( R)| 2 behaves as 1/R 2 therefore intensity does not depend on R. The expression Eq.(9) should be averaged over the realizations of random roughness h(x, y). For this reason it is convenient to introduce the Green's functions of Eqs. (7,8) ε 0 (z, ω) ω 2 c 2 δ λµ − ∂ 2 ∂r λ ∂r µ + δ λµ ∇ 2 + +ε r ( r, ω) ω 2 c 2 δ λµ Gµν( r, r ′ , ω) = δ λν δ( r − r ′ ). (11) In Eqs.(10,11) a summation over the repeated indexes is supposed. Solutions of inhomogeneous Eqs. (7,8) can be expressed through the Green's functions Eqs. (10,11). Using Eqs. (7,8) and (10,11) one can represent the averaged radiation intensity tensor < I ij ( R) >=< E ri ( R)E * rj ( R) > in the form where the background electric field E 0µ ( r) is expressed through the bare Green's function Here < ... > means averaging over the surface random profile h(x, y). Note that in the original Smith-Purcell experiment [1], as well as in subsequent works on SP a periodical grating in one direction is used. In this case h(x, y) ≡ h(x) is some periodical function of one coordinate. In the present paper within a single approach we consider both periodical and random gratings. In the random case we suppose that h is a gaussian stochastic process characterized by two parameters where ρ is the two dimensional vector in the xy plane, δ 2 =< h 2 ( ρ) > is the average deviation of surface from the plane z = 0. Correlation function W is characterized by a correlation length σ at which it is essentially decreased.
The Maxwell equations for electric fields Eq.(7,8) and Green's functions Eq.(10,11)should be amended by boundary conditions. As usual, it is required that tangential components of electric field be continuous across the plane z = 0. The exact field, of course, will satisfy the boundary conditions across the surface z = h(x, y) rather than the plane. However this approximation seems reasonable for small roughness λ ≫ δ and is widely used in the literature. The Green's function G µν ( r, r ′ , ω), when considered a function of z for fixed z ′ satisfies the same boundary condition as the µth Cartesian component of electric field.
Green's Functions. The equation for bare Green's function Eq.(10) with correct boundary conditions for arbitrary ε(ω) was solved in [20]. To obtain radiation intensity in vacuum we will need Green's functions in the half space z > 0. In order to simplify the problem we will consider the case when isotropic medium is a metal with very large negative dielectric constant |ε(ω)| ≫ 1. Using expressions for Green's functions from [20] we find the following basic components where G 0 ij ( p|z, z ′ ) is the two-dimensional Fourier transform of G 0 ij ( r, r ′ ) and z > 0. In the coordinate representation Here p and ρ are two-dimensional vectors with Cartesian components p x , p y , 0 and x, y, 0. Also k = ω/c, k 1 and q are determined as follows: In Eq.(18) a branch cut for the square root along the negative real axis is assumed [20]. Other components of Green's function are small over the parameter 1/|ε|. To determine radiation intensity we will need asymptotics of Green's functions at large distances. Substituting Eq.(15) into Eq. (16), one finds where n is the unit vector on the direction of the observation point R = nR, n x,z and n ρ = n 2 x + n 2 y are it's corresponding components. When obtaining Eq. (19) we use asymptotics of Bessel functions for large argument [22]. Eqs. (19) are correct provided that kR ≫ 1, R ρ ≫ ρ and we use approximate equation | R − r| ≈ R − n r.
Radiation Intensity. Spectral-angular radiation intensity Eq.(12)can be represented as a sum of two contributions, I( R, ω) = I 0 ( R, ω)+I D ( R, ω), where I 0 and I D are single scattering and diffusive contributions, respectively [23]. First consider the single scattering contribution to the radiation intensity. Substituting the Green's functions in Eq.(12) by the bare ones, we obtain where (ij) ≡ (xz). The background electric field in the limit |ε(ω)| ≫ 1 can be found from Eqs. (13), (2) and (15) where k 0 = ω/v, γ = (1 − v 2 /c 2 ) −1/2 is the Lorentz factor of the particle and K 1 is the first order Macdonald function. As it follows from Eq.(21) the background electric field and correspondingly radiation intensity is exponentially small when ωd/vγ ≫ 1, see also [25]. One can expect essential intensity provided that ωd/vγ ≪ 1. Far away from the system at the observation point one can use asymptotic expressions for Green's functions Eq. (19). Substituting Eqs. (19) and (21) into Eq. (20), for the spectral-angular radiation intensity I(ω, Ω) = cR 2 I ii ( R)/2, one obtains where L x is the system size in the x direction, g = (ε − 1) 2 δ 2 σ 2 k 4 and β = v/c. When obtaining Eq. (22) we neglect strongly oscillating terms in the limit kR ≫ 1 and suppose that W ( ρ − ρ ′ ) ≡ σ 2 δ( ρ − ρ ′ ). Beside that we substitute the Macdonald function by it's asymptotics for small argument assuming that k 0 d/γ ≪ 1. In the opposite limit as was mentioned above radiation intensity is negligible. The components of unit vector n are determined through the polar θ and azimuthal φ angles of observation direction: n z = cos θ, n ρ = sin θ, n x = sin θ sin φ. We consider radiation into the half-space z > 0 (vacuum) which means θ < π/2. Note that the coupling constant g = k 4 (ε − 1) 2 δ 2 σ 2 in Eq. (22)is a dimensionless parameter. From the condition R ρ ≫ ρ, one obtains a restriction on angles sin θ ≫ L/R, where L is a characteristic size of the system. To avoid misunderstanding note that 1/β 2 dependence of radiation intensity Eq.(22) is correct in an intermediate regime for not very low velocities ωd/vγ ≪ 1. When β → 0,as was mentioned above, radiation disappears. Note that the background field E 0 can originate radiation without any roughness provided that Cherenkov condition v 2 ε > c 2 is fulfilled. Cherenkov radiation is possible for dielectric surfaces with positive large ε. For metallic surfaces in the optical region we are interested in the present paper dielectric constant is negative and Cherenkov radiation is absent.
Periodical case. Analogously one can consider the case when surface grating is a periodical function. For simplicity we will assume that h( ρ) ≡ δ sin 2πx/b, where b is the period of grating. Substituting W (| ρ − ρ ′ |) by sin 2πx/b sin 2πx ′ /b in Eq. (20) and using Eq. (19), after integration , for spectral-angular radiation intensity, one has I SP ( n, ω) = e 2 cβ 2 (23) where g 1 = k 4 (ε − 1) 2 δ 2 and F is determined as follows When obtaining Eq.(23) we keep only the terms proportional to L x . In the most interesting case n y ∼ 0 and dk 0 /γ ≪ 1, substituting K 1 by its asymptotic expression, one finds from Eq.(24),F (0) ∼ π 2 /4. As follows from Eq.(23), because of the δ functions, for a given observation direction, only two discrete wavelengths are emitted This is a generalization of well known Smith-Purcell dispersion relation [1] to the weak scattering (see below) case. For an arbitrary periodical grating one can expand the surface profile h(x) into Fourier series and for each term one can obtain analogous dispersion relation with b substituted by b/m, where m is the diffraction order. Note that the dispersion relation Eq. (25) and the spectral-angular radiation intensity Eq.(23) differ from reported earlier. The reason of those differences are following. First we are considering weak scattering regime instead of strong one considered in above mentioned papers. Our theory is applicable provided that (ε(ω) − 1) 2 δ 2 /λ 2 ≪ 1 although |ε(ω)| ≫ 1. Probably this regime was realized in the experiment on SP radiation in the optical region for shallow gratings [24]. As mentioned in [24] traditional formula of SP radiation is failed to explain the results of experiment in the shallow grating case. Second reason is the boundary conditions. As follows from Eqs. (7,8) Maxwell equations for E 0 and E r contain the same disruptive function ε 0 (z, ω). Therefore both of them should satisfy the same boundary conditions at z = 0. In our consideration this goal is achieved automatically because the Green's functions Eqs. (15,19) satisfy the correct boundary conditions [20]. In contrary, in traditional consideration [11], only the total field E 0 + E r and not they separately, satisfy the boundary conditions at z = 0. Probably this difference leads to different dispersion relation Eq. (25).
where G ij ( ρ 2 , ρ 1 ) ≡ G ij ( ρ 2 , 0, ρ 1 , 0),and where diffusive propagator P is determined by the sum of ladder diagrams see Fig.2. and [26]. All integrations over z coordinates make them equal to 0 because of δ(z) in fluctuation part of dielectric constant Eq.(6). Averaged two-dimensional surface polariton Green's function [27] satisfies the Dyson equation Remind that Gµν( p) ≡ G µν ( p|0, 0),see Eq. (15). Bare Green's functions are determined by Eq. (15). In further we will interested in the behavior of Green's function close to the pole. These values give the main contribution in the limit g → 0. As it follows from Eq.(15) twodimensional Green's functions of surface polariton has a pole at p 2 = k 2 ε/(ε + 1), see [27]. The corresponding velocity of a surface polariton is equal to c (ε + 1)/ε < c. Remind that we consider the case when ε ≪ −1. When electron velocity becomes equal to this velocity a superradiant emission is possible provided that the grating is periodical [6,8]. Close to the pole and for large negative |ε(ω)| ≫ 1 the Green's functions of surface polariton can be represented in the form where α = k 2 ε 2 /ε 2 1 , ε = ε 1 +iε 2 and ε 2 ≪ |ε 1 |. In Eq.(28) α describes the damping of the surface polariton on the flat surface due to the inelastic processes in the medium, i.e. ε 2 (ω). It follows from Eq.(28) that d p pG 0 zx ( p) ≡ 0. Therefore only G 0 zz gives contribution to the integral in Eq. (27). Solving Dyson equation Eq.(27) one can represent the averaged Green's functions in the form where Λ = d p (2π) 2 ImG 0 zz (p) = gk 2 /4ε 1 (ω). Real part of the integral leads to renormalization of the parameters and does not play any role. Integral is calculated in the limit ε 2 → 0. Λ describes the damping of the surface polariton by its roughness-induced conversion into radiative modes [27]. It is convenient also to introduce the polariton mean free path on the rough surface l = k/Λ. Note that the neglected terms in diagram expansion are small on parameter λ/l ≪ 1 [26].
Using Eqs. (29) one sees that the main contribution to the diffusive radiation intensity Eq. (26) give the term proportional to P zzzz which contains a diffusive pole at small momentums. Summing the ladder diagrams in Fig.2 one finds a Bethe-Salpeter equation for diffusive propagator P (K) ≡ P zzzz (K) Here G(p) ≡ G zz (p). Using Eq. (29) and calculating the integral in Eq.(31) in the limit g → 0, one finds P (K) at small Kl ≪ 1 Substituting Eqs. (19) into Eq. (26), for the diffusive contribution to the radiation intensity, one has As follows from Eqs.(32) and (33) radiation intensity diverges. This divergence is caused by the infinite size of the system, see also [23,28]. If one takes into account the finite size of the system the minimal momentum will be of order ∼ 1/L. As was mentioned above the radiation intensity is exponentially small provided that d ≫ γ/k 0 . In the opposite limit d ≪ γ/k 0 substituting K 1 by it's asymptotic expression and integrating Eq.(33), we finally obtain In this consideration weak l ≪ l in ,where l in = k/α = ε 2 1 /kε 2 is the inelastic mean free path of surface polariton, absorption can be taken into account as follows [29]. When L > (ll in ) 1/2 , L in Eq.(34) should be substituted by (ll in ) 1/2 . Comparing single scattering Eq.(22) and diffusive Eq.(34) contributions, one has I D /I 0 ∼ L 2 /l 2 ≫ 1. Therefore the diffusion of surface polaritons is the main mechanism of radiation. Let us make some numerical estimates for the optical region. For Ag at λ ∼ 4500A o ε 1 ∼ −7.5 and ε 2 ∼ 0.24. Taking δ ∼ 50A o and σ ∼ 1000A o [27], one has g ∼ 0.68, l ∼ 7.7λ and l in ∼ 47.94λ. Thus the conditions for diffusive mechanism λ ≪ l ≪ l in , L are fulfilled. Evidently, depending on grating parameters δ, σ emission in other wavelength regions is possible too.
We have considered multiple scattering effects in radiation for uncorrelated roughness. However they are very important for the periodical as well as correlated grating cases too. These cases are more complicate and will be discussed elsewhere later. Our result Eq.(23)for SP radiation intensity with only single scattering contribution is correct in the cases when multiple scattering contribution is negligible. Such a situation can occur for the metals with relatively large absorption when the condition of multiple scattering of polaritons l in ≫ l is not fulfilled.