aa r X i v : . [ phy s i c s . acc - ph ] O c t Smith-Purcell radiation on a surface wave
A. A. Saharian ∗ Institute of Applied Problems of Physics25 Nersessian Street, 0014 Yerevan, Armenia
October 17, 2018
Abstract
We consider the radiation from an electron in flight over a surface wave of an arbitraryprofile excited in a plane interface. For an electron bunch the conditions are specified un-der which the overall radiation essentially exceeds the incoherent part. It is shown thatthe radiation from the bunch with asymmetric density distribution of electrons in the lon-gitudinal direction is partially coherent for waves with wavelengths much shorter than thecharacteristic longitudinal size of the bunch.Talk presented at the International Conference”Electrons, Positrons, Neutrons and X-rays Scattering Under External Influences”Yerevan-Meghri, Armenia, October 26-30, 2009
Surface waves have wide applications in various fields of science and technology. In the presenttalk, based on [1]-[4], we discuss the radiation from a charged particle flying over surface acousticwave. The physics of this phenomenon is similar to that for the radiation of particle flying overa diffraction grating (Smith-Purcell radiation). The latter is used for the generation of theradiation in the range of millimeter and submillimeter waves.
Let a charge q move with constant velocity v parallel to a surface wave excited in the planeinterface between homogeneous media with permittivities ε and ε (see figure 1). If the axis z is aligned with the particle trajectory, then the equation of the interface has the form x = x ( z, t ) = − d + f ( k z ∓ ω t ) , (1)where k and ω are the wave number and cyclic frequency, d is the distance from the non-excitedinterface, and f ( u ) is the function describing the wave profile, f ( u + 2 π ) = f ( u ). We assumethat the particle moves in the medium with permittivity ε and will consider the radiation inthis medium. ∗ E-mail: [email protected] x z y q j v (cid:1) k (cid:0) Figure 1: Electron flying over surface wave excited on the plane interface between homogeneousmedia.From the symmetry properties of the problem it follows that the emission angle θ (withrespect to the particle velocity v ) and the frequency ω of the emitted photon are related by theformula ω = m ( k v ∓ ω )1 − β √ ε cos θ , β = vc , (2)where m is an integer. The dependence of the radiation intensity on the distance of the electrontrajectory from the non-excited interface, d , is determined by the factor exp( − ωd Re σ/v ), where σ = (cid:2) (1 − β ε ) ω /ω + β ε sin θ sin ϕ (cid:3) / , ω = ω ± mω , (3)with ϕ being the polar angle with respect to the x axis in the plane perpendicular to the particletrajectory. This factor does not depend on the specific form of the wave profile in (1) and isdetermined by the dependence of the field spectral components for a uniformly moving particleon distance d . For relativistic particles with β ε .
1, one has ω ≈ ω and the radiation atlarge azimuthal angles is exponentially suppressed and the radiation distribution is stronglyanisotropic. For the radiation in the vacuum at ϕ ≪ − ω dv/γ ),with γ = 1 / p − β been the Lorentz factor. In the case β √ ε <
1, the quantity σ is realand the intensity exponentially decreases with increasing distance. The same is the case for thedirections of radiation satisfying the condition sin θ sin ϕ > − / ( β ε ), when β √ ε >
1. For β √ ε > θ sin ϕ < − / ( β ε ), the radiation intensity does not depend on particledistance from a surface of the periodic structure in the absence of absorption. This correspondsto the reflection of Cherenkov radiation emitted in the first medium.In order to determine the radiation intensity we have used two independent approximatemethods. In the first one it is assumed that | ε − ε | ≪ ε . The second method, which is moreappropriate for the problem under consideration, assumes that the amplitude of the surfacewave is small. Assuming that the charge moves in the vacuum ε = 1, and under the condition β √ ε − >
1, the spectral-angular distribution of the radiation intensity (per unit path length)2n the region x > dWdωd
Ω = 2 q ( ε − πc v X m =0 ω sin θ cos ϕ | f m | e − ωσd/v × A σ + A sin θ sin ϕ + A β sin θ sin ϕ (cid:0) β sin θ sin ϕ (cid:1) ( σ + ε sin θ cos ϕ ) δ (cid:18) cos θ − β + mk cω (cid:19) , (4)where d Ω = sin θdθdϕ , and we have assumed that v ≫ ω /k . In (4), f m = π R + π − π du f ( u ) e − imu is the Fourier transform of the profile function, σ = q β ( ε − sin θ sin ϕ ) − , σ = q ε − θ sin ϕ, (5)and A = [ σ σ cos θ − ε (1 − sin θ cos ϕ )] + sin θ [ σ ( σ cos ϕ + sin θ sin ϕ ) − ε cos θ cos ϕ ] ,A = σ ( σ cos θ + ε sin θ cos ϕ ) + β ( σ sin θ cos ϕ + β cos θ sin θ sin ϕ + cos θ ) ,A = σ (1 − β cos θ ) + [cos θ + β sin θ ( σ cos ϕ + sin θ sin ϕ )] . (6)For a sinusoidal surface wave, f ( u ) = a sin u , one has f m = ± ( a/ i ) δ m, ± . H Ω (cid:144) k c L Λ a d W Ñ d Ω Figure 2: Spectral density of radiation intensity as a function of the frequency of the radiatedphoton. The numbers near the curves correspond to the values of the ratio d/λ .We have numerically evaluated the spectral-angular distribution of the radiation intensityfor various values of the parameters in the case of sinusoidal profile. In this case the onlycontribution to the radiation intensity comes from the harmonic m = 1. The results of thesecalculations show that the parameters of the radiation may be effectively controlled by tuningthe characteristics of the surface wave. For an illustration we plot in figure 2 the spectraldistribution of the radiation intensity dWdω = Z π dθ sin θ Z π/ − π/ dϕ dWdωd Ω , (7)3n the case of a sinusoidal surface wave as a function of the frequency of the radiated photon forvarious values of the ratio d/λ (numbers near the curves), where λ = 2 π/k is the wavelengthof the surface wave. The full (dashed) curves correspond to the electron energy E e = 100 MeV( E e = 500 MeV). Note that ω/k c = λ /λ with λ being the wavelength of the radiated photon.The corresponding radiation angle is related to the frequency by the relation cos θ = 1 /β − k c/ω .In accordance with this relation, for relativistic electrons, large values of ω/ ( k c ) correspond tosmall angles θ . For θ ≫ γ − the radiation intensity is relatively insensitive to the particle energy,whereas for θ . γ − the intensity strongly increases with increasing energy. From figure 2 thesuppression of the radiation intensity with increasing d is well seen. In figure 3 we present theradiation intensity dWdωdϕ = Z π dθ sin θ dWdωd Ω , (8)as a function of the azimuthal angle ϕ for various values of θ (numbers near the curves) andfor d/λ = 0 .
15. As in the case of figure 1, the full (dashed) curves correspond to the electronenergy E e = 100 MeV ( E e = 500 MeV). As we have mentioned, for θ ≫ γ − the dependence ofthe radiation intensity on the particle energy is weak and for θ = 30 ◦ , ◦ the full and dashedcurves coincide. For small angles θ the radiation is emitted mainly along ϕ . γ − . j Λ a d W Ñ d Ω d j Figure 3: Spectral-angular density of radiation intensity as a function of the azimuthal angle ofthe radiated photon for various values of θ . In this section, as a source of the radiation we shall consider a cold bunch consisting of N electrons and moving with constant velocity v along the z axis (for coherence effects in theSmith-Purcell radiation see also [5]). The density of the current in the bunch can be written inthe form j = q v N X j =1 δ ( r − R j − v t ) , (9)with R j = ( X j , Y j , Z j ) being the position of the j -th particle at the initial moment t = 0. Thespectral density of the radiation energy flux for a given m , defined by the relation c π Z + ∞−∞ dt [ EH ] = Z ∞ dω X m P ( N ) m ( ω ) , (10)4ith E and H being the electric and magnetic fields, can be written in the form P ( N ) m ( ω ) = P (1) m ( ω ) S N . (11)In (11), P (1) m ( ω ) is the corresponding function for the radiation from a single electron withcoordinates x = y = z = 0 at the initial moment t = 0, and S N = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j =1 exp [ − ω σX j /v − ik y Y j − iω Z j /v ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (12)Assuming that the coordinates of the j -th particle are independent random variables and aver-aging the quantity (12) over the positions of a particle in the bunch, we obtain D P ( N ) m ( ω ) E = P (1) m ( ω ) h S N i , h S N i = N h + N ( N − | h x h y h z | , (13)where we have introduced the notations h = h exp(2 ωX Re σ/v ) i , h l = h exp( iK l l ) i , l = x, y, z,K x = − iωσ/v, K y = k y = ( ω √ ε /c ) sin θ sin ϕ, K z = ω /v. (14)The functions | h l | determine the bunch form factors in the corresponding directions. In (13),the term proportional to N determines the contribution of coherent effects. Conventionallyit is assumed that the coherent radiation is produced at wavelengths equal and longer thanthe electron bunch length. However, as we shall see below, this conclusion depends on thedistribution of electrons in the bunch.As it is seen from (14), the form factors in y and z directions are determined by the Fouriertransforms of the corresponding bunch distributions. First let us consider a Gaussian distribu-tion: f l = 1 √ πb l exp (cid:18) − l b l (cid:19) , l = x, y, z, (15)with b x , b y and b z being the corresponding characteristic sizes of the bunch. Assuming that allparticles of the bunch are in the medium with permittivity ε and, therefore, b x < d − a , with a being the surface wave amplitude, one finds h S N i = N exp (cid:20) ω v (Re σ ) b x (cid:21) (cid:20) N −
1) exp (cid:18) − ω v | σ | b x − k y b y − ω b z v (cid:19)(cid:21) , (16)where the second term in the square brackets determines the relative contribution of the coherenteffects. For a non-relativistic bunch with b y & b x,z , the corresponding exponent is equal toexp[ − (2 πm/λ ) ( b x + b z ) − (2 π/λ ) b y sin θ sin ϕ ](here we consider the case ε = 1). Note that in this case for the wavelength of the radiationone has λ ∼ λ /βm and the coherent effects are exponentially suppressed when λ . b l /β . Fora relativistic bunch the relative contribution of coherent effects is given by N exp {− (2 π/λ ) [( b x + b y ) sin θ sin ϕ + b z ] } , (17)for sin θ sin ϕ > γ − , and by N exp[ − (2 πb x /λγ ) − (2 π/λ ) ( b z + b y sin θ sin ϕ )] , (18)5or the radiation with sin θ sin ϕ . γ − . As we see, in this case the transverse form factoris strongly anisotropic. It follows from (16) that for a real σ , fixed electron number and fixeddistance of the bunch center from the surface wave the radiation intensity exponentially increaseswith increasing b x . This is because the number of electrons passing close to the surface waveincreases. The number of electrons with long distances will also increase. But the contributionof close electrons surpasses the decrease of the intensity due to distant ones.As we have seen, for a Gaussian distribution the relative contribution of coherent effects isexponentially suppressed in the case b i > λ . This result is a consequence of the mathematicalfact that for a function f ( x ) ∈ C ∞ ( R ) one has the estimate F ( u ) ≡ Z + ∞−∞ f ( l ) e iul dl = O ( u −∞ ) , u → + ∞ , (19)where u = 2 πb z /λ for the longitudinal form factor of the relativistic bunch.We have considered the case of Gaussian distribution in the bunch. However, it should benoted that due to various beam manipulations the bunch shape can be highly non-Gaussian. Inthe estimate (19) the continuity condition for the function f ( l ) and for infinite number of itsderivatives is essential. It can be seen that when f ( l ) ∈ C n − ( R ), and the derivative f ( n ) ( l ) isdiscontinuous at point l , we have the asymptotic estimate F ( u ) = ( − iu ) − n − h f ( n ) ( l +) − f ( n ) ( l − ) i , u → + ∞ . (20)Unlike the case of (19), now the form factor for the short wavelengths decreases more slowly, aspower-law, (2 πb l /λ ) − n − . In the coherent part of the radiation this form factor is multiplied bya large number, N , of particles per bunch and the coherent effects dominate under the condition2 πb l /λ < N / n +1) . (21)The radiation intensity is enhanced by the factor N ( λ/ πb l ) n +1) . Since conventionally there are10 − electrons per bunch, the condition (21) can be easily met even in the case 2 πb l /λ > n = 1, N ∼ , the coherent radiation dominates for 100 λ > b l .As an example, let us discuss the case when the electrons are normally distributed in the x and y directions and the distribution function in the z direction has an asymmetric Gaussianform f ( z ) = 2 √ π (1 + p ) b l (cid:20) exp (cid:18) − l p b l (cid:19) θ ( − l ) + exp (cid:18) − l b l (cid:19) θ ( l ) (cid:21) , (22)where θ ( l ) is the unit step function, l = (1 + p ) b l is the characteristic bunch length, parameter p determines the degree of bunch asymmetry. Now the expression for h S N i can be written as h S N i = N exp (cid:18) ω v σ b x (cid:19) (cid:20) N −
1) exp (cid:18) − ω v σ b x − k y b y (cid:19) | F ( ω /v ) | (cid:21) , (23)where F ( u ) = 1 p + 1 (cid:26) e − t + pe − p t − i √ π [ W ( t ) − pW ( pt )] (cid:27) , (24)with the notation W ( t ) = Z t exp( l − t ) dl, t = ub l √ . (25)The expression | F ( u ) | is invariant with respect to the replacement p → /p , b l → b l p thatcorresponds to the mirror reversal of the bunch. When the electron distribution is symmetric6 p = 1), the second summand in the square brackets of (24) vanishes and, as was mentionedearlier, the form factor exponentially decreases for short wavelengths λ < πb l /β . For pt ≫ F ( u ) is found from (24) and has the form F ( u ) ∼ i r π − pu b l p . (26)In the case of a relativistic bunch and for ϕ > γ − , the exponent of the second summand inthe square brackets of (23) is of the order (2 πb i /λ ) , i = x, y . When the transverse size of thebunch is shorter than the radiation wavelength, then the relative contribution of the coherenteffects is ∼ N | F ( ω /v ) | . For an asymmetrical bunch this contribution can be dominant evenin the case when the bunch length is greater than the radiation wavelength. Indeed, accordingto (26) even for weakly asymmetrical bunch we have N | F | ∼ N ( v/ω b z ) , and the radiationis coherent for b z . λN / / (2 π ), where we have taken into account that for an relativisticbunch ω ≫ mω and therefore ω ≈ ω , as was mentioned above. In the case of the bunchwith N ∼ and for 2 πb z /λ .
10 the radiation is coherent. For the radiation ϕ . γ − ,the exponent of the second summand in square brackets of Eq. (23) for ε = 1 is of the order(2 πb x /λγ ) . Hence, for a relativistic bunch the radiation in directions ϕ . γ − can be coherenteven in the case when the transverse size of the bunch is greater than the wavelength. For thisit is sufficient to have the conditions b x . γλ π , b z . λN / π . (27)The second of these conditions is written for weakly asymmetrical bunches. In the case of strongasymmetry the corresponding conditions are less restrictive: b z . λN / / π for pb z ≪ λ .Let f ( l, a ) be a continuous distribution function depending on the parameter a , and lim a → f ( l, a ) = f ( l ). The integral F ( u, a ) for f ( l, a ) uniformly converges and hence lim a → F ( u, a ) = F ( u ). Itfollows from here that the estimate presented above is valid for continuous functions as wellif they are sufficiently close to the corresponding discontinuous function (the correspondingderivative is sufficiently large, see below). Aiming to illustrate this, we consider the asymmetricdistribution f ( z, a l , b l , l ) = 14 l (cid:20) th (cid:18) l + l a l (cid:19) − th (cid:18) l − l b l (cid:19)(cid:21) . (28)For l > a l , b l this function describes a rectangular bunch having exponentially decreasing asym-metric tails with characteristic sizes a l and b l . In the limit a l , b l → l . The explicit evaluation of the expression (19) with thefunction (28) leads to F ( u, a l , b l , l ) = i ul (cid:18) a l e − iul sinh a l − b l e iul sinh b l (cid:19) , (29)with the notations a l ≡ πua l / b l ≡ πub l /
2. From here it follows that if a l ∼ b l ∼ F ∼ ( ul ) − for ul ≫
1. In the limit a l , b l →
0, from (29) one obtains the well known formfactor for the rectangular distribution: F rect ( u, l ) = sin( ul ) / ( ul ). As it is seen, the rectangulardistribution is a good approximation for (28) if a l , b l ≪ λ . The main contribution to (29) comesfrom the bunch tails, i.e. from the parts of bunch with large derivatives of the distributionfunction. This is the case for the general case of distribution function as well: if ul ≫ df /d ( l/l ) & u and in this case F ( u ) ∼ /u , u → ∞ . This can be generalized for higher derivatives as well: if d i f /d ( l/l ) i ≪ u , i = 1 , ..., n −
1, and d n f /d ( l/l ) n & u then F ( u ) ∼ ( ul ) − n .7or a relativistic bunch one has u ∼ π/λ for the form factors in y and z directions and theconclusion can be formulated as follows. If for the distribution function one has λ d i fd ( l/l ) i ≪ π, i = 1 , ..., n − λ d n fd ( l/l ) n & π, (30)then the relative contribution of coherent effects into the radiation intensity is proportional to N ( λ/ πl ) n and the radiation is partially coherent in the case λ < l but λ > πl N − / n ,with l being the characteristic bunch size in the corresponding direction. In this case the maincontribution into the radiation intensity comes from the parts of the bunch with large derivativesof the distribution function in the sense of the second condition in (30). For example, in thecase of asymmetric distribution (22), when ub l ≫ pub l <
1, the main contribution comesfrom the left Gaussian tail with l <
0. For this tail df /d ( l/b l ) ∼ u/ ( pub l ) & u , at l ∼ pb l andtherefore F ( u ) ∼ / ( ub l ). This can be seen directly from the exact relation (24) as well by usingthe asymptotic formula for the function W ( t ). In the present talk we have considered the radiation from a single electron and from an electronbunch of arbitrary structure flying over the surface wave excited in a plane interface. For smallamplitudes of the surface wave the spectral-angular distribution of the radiation intensity froma single electron is given by expression (4). We have shown that the at large angles with respectto the electron trajectory the dependence of the radiation intensity on the electron energy isrelatively week, whereas at small angles the intensity strongly increases with increasing energy.It is demonstrated that the radiation from a bunch can be partially coherent in the range ofwavelengths much shorter than the characteristic longitudinal size of the bunch and the maincontribution to the radiation intensity comes from the parts of the bunch with large derivativesof the distribution function. In this case for short wavelengths the relative contribution ofcoherent effects decreases as a power-law instead of exponentially decreasing. The correspondingconditions for the distribution function are specified. The coherent effects lead to an essentialincrease in the intensity of the emitted radiation.The author gratefully acknowledges the organizers of the International Conference ”Elec-trons, Positrons, Neutrons and X-rays Scattering Under External Influences”, Yerevan-Meghri,October 26-30, 2009, for the financial support to attend the conference.
References [1] A. R. Mkrtchyan, L. Sh. Grigorian, A. A. Saharian, A. H. Mkrtchyan, A. N. Didenko, IzvestiaAN Arm. SSR. Fizika , 62 (1989).[2] A. R. Mkrtchyan, L. Sh. Grigorian, A. A. Saharian, A. N. Didenko, Zhurnal Tekh. Fiz. ,21 (1991).[3] A. R. Mkrtchyan, L. Sh. Grigorian, A. A. Saharian, A. N. Didenko, Acustica , 184 (1991).[4] A. A. Saharian, A. R. Mkrtchyan, L. A. Gevorgian, L. Sh. Grigoryan, B. V. Khachatryan,Nucl. Instr. and Meth. B , 211 (2001).[5] A. R. Mkrtchyan, L. A. Gevorgian, L. Sh. Grigorian, B. V. Khachatryan, A. A. Saharian,Nucl. Instr. and Meth. B145