Inverse Smith-Purcell effect near rough surfaces
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Inverse Smith-Purcell effect near rough surfaces
Zh.S. Gevorkian , , ∗ and V. Gasparian Yerevan Physics Institute, Alikhanian Brothers St. 2,0036 Yerevan, Armenia. Institute of Radiophysics and Electronics, Ashtarak-2, 0203, Armenia. California State University, Bakersfield, USA ∗ [email protected]: [email protected] Abstract.
Absorption of a photon by an electron moving parallel to a rough surfaceis studied. In the weak scattering regime we have evaluated the absorption probabilityof absorption of a single photon of energy ω . It is shown the absorption probabilitywith diffusional contribution becomes large by a l in /l ≫ nverse Smith-Purcell effect near rough surfaces
1. Introduction
It is well known that a charged particle moving in the vacuum can not emit or absorbphotons due to the energy-momentum conservation laws. On the other hand, emissionor absorption is possible when particle moves in a medium or close to an interface.Cherenkov, Transition, Smith-Purcell radiations are the examples of above mentionedemission (see, for example, [1]). During the recent years the inverse counterparts ofthe mentioned radiations have been observed [2]-[5]. Earlier the inverse Smith-Purcelleffect for periodical metallic gratings was theoretically analyzed in the sub-millimeterwavelength region [6]. The interest to these effects is largely motivated by a possibilityof laser driven acceleration of charged particles. Besides, the inverse Smith-Purcell effectcan be used in electron energy gain spectroscopy [7].In the present paper we investigate the inverse Smith-Purcell effect, namely theabsorption of a photon by an electron which moves parallel to a rough surface. Tothe best of our knowledge, no such calculations have been previously reported. Themain difficulties with rough surfaces arise because it is more difficult to performanalytical calculations when we deal with an arbitrary shaped profile of the gradingfor understandable reasons: there is no general algorithm to calculate the radiatingpart in the reflected waves. At present, most numerical simulations are available asone of the effective tools to analyze and to observe a variety of physical quantitiessuch as electromagnetic fields as functions of time and space, power outflow, radiatedintensity as a function of the radiating angle, etc. (see, e.g., [8] and references therein).Therefore any study of the inverse Smith-Purcell radiation from rough surfaces shouldbe quite important and analytical results are highly desirable. The purpose of thepresent work goes in this direction, in the sense that we provide analytical expressionfor the absorption probability of a photon by an electron moving parallel to a roughsurface. In the diffusion regime we were able to obtain a closed analytic expression forthe absorption probability, taking into account the diffusion contribution. We show thatthe diffusion contribution is dominant compared to the single scattering probability.It has been considered recently, by one of the authors (Gevorkian), the radiationfrom a charged particle moving parallel to rough surfaces [10, 11]. The averagedradiation intensity for a quite general surface random profile was directly calculatedand it is shown that the main contribution to the radiation intensity is determined bythe multiple scattering of polaritons induced by a charge on the surface. We will developan approach, following closely Refs.[10, 11], which allows us to investigate periodical aswell as random surface profiles, different materials from sub-millimeters to optics. Weindicate necessary conditions for absorption to take place.The plan of the work is as follows. In Sec. II we briefly formulate the problemand introduce the basic equation for the absorption probability of a single photon ofenergy ω . In Sec. III we carry out an exhaustive description of our two-dimensionalrough surface and the analytical approach used in absorption probability calculations.In Sec. IV we calculate the absorption probabilities with single and multiple scattering nverse Smith-Purcell effect near rough surfaces Figure 1.
Geometry of the problem. Electron moves parallel to the rough surface atthe plane xy which is illuminated by an external laser field. contributions. It was shown that the diffusion contribution to absorption probabilityis the dominant one. In Sec. V we discuss the utilization of the inverse Smith-Purcelleffect for particle acceleration. Finally, we summarize our results in Sec. VI.
2. Initial Relations
Suppose that a fast electron moves on the positive x direction parallel to a rough surface xy at the distance Z from it. Simultaneously a laser field of frequency ω falls down onthe surface, see Fig.1.Electron wave function can be described asΦ( ~r ) = 1 √ L x ϕ ( Z, Y ) e ipix ¯ h , (1)where L x is the system size in the 0 x direction, ϕ ( Z, Y ) is the wave function in the zy plane, p i is the electron momentum along the direction of motion. After absorbinga photon electron momentum and energy become p f = p i + ¯ hq and E f = E i + ¯ hqv ,respectively ( v is the velocity of the electron and a non-recoil approximation ¯ hq ≪ p i isassumed). For a fast electron one can assume that the wave function ϕ ( Z, Y ) remainsunchanged during the interaction with the photon. We will discuss conditions ofapplicability of this assumption below. Treating electron and photon as a quantummechanical subjects, the absorption probability of absorption of a single photon ofenergy ω can be represented in the form [7] P ( ω ) = (cid:18) e ¯ hω (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)Z dxE x ( x, Y, Z ) e − i ωxv (cid:12)(cid:12)(cid:12)(cid:12) . (2) E x ( x, Y, Z ) is the electric field component along the electron motion direction thatincludes incident as well as scattered from surface fields and Y, Z are the electronconstant coordinates in the perpendicular to motion plane. If the incident field is a nverse Smith-Purcell effect near rough surfaces E x will be substituted by the scattered one. This will be done in the next sections,separately for the situations when single and multiple scattering contributions are takeninto account, while calculating the appropriate absorption probability.
3. Scattered Field
Dielectric constant of the system is described as ε ( ~r ) = θ ( z − h ( x, y ))+ ε ( ω ) θ ( h ( x, y ) − z ),where θ is the step function and ε ( ω ) is the dielectric constant of the isotropic medium, h ( x, y ) is the random profile of the surface. Assuming that h is small and expandingthe ε ( ~r ) in powers of h and keeping linear terms wee get ε ( ~r ) = ε ( z ) + ε r ( ~r ), where ε r ( ~r ) = ( ε − h ( x, y ) δ ( z ). The function ε ( z ) = 1 at z > ε ( z ) = ε ( ω ) at z < E µs ( ~r ) = − ω c ( ε − Z d~r ′ G µν ( ~r, ~r ′ , ω ) h ( ~ρ ′ ) δ ( z ′ ) E ν ( ~r ′ , ω ) . (3) E ν ( ~r ′ , ω ) is the solution of Maxwell equation with the flat interface and because ofthe translational symmetry in the xy plane it can be represented as follows: E ν ( ~r ) = e ~k || ~ρ E ν ( z ), where ~k || and ~ρ are two dimensional vectors in the xy plane. Green’s functionin Eq.(3) obeys the inhomogeneous Maxwell equation " ε ( z ) ω c δ λµ − ∂ ∂r λ ∂r µ + δ λµ ∇ + ε r ( ~r ) ω c δ λµ G µν ( ~r, ~r ′ , ω ) = δ λν δ ( ~r − ~r ′ ) . (4)It is worth noticing that the presence of the δ -function in the expression of ε r will lead tothe different values of any physical quantity at z = 0, while evaluating the integral over z . To avoid the problem with discontinuous physical quantities at z = 0 in our furthercalculations we will take their value at z = 0 + , see also, [12]. Such determination ofintegrals over δ functions give correct answers in the limit | ε | → ∞ . Hence, substitutingEqs.(4) and (3) into Eqs.(2), one has P ( ω ) = (cid:18) e ¯ hω (cid:19) ( ε − ω c Z dxdx ′ d~ρ d~ρ G xν ( x, Y, Z, ~ρ , + ) G ∗ µx ( ~ρ , + , x ′ , Y, Z ) h ( ~ρ ) h ( ~ρ ) E ν ( ~ρ , + ) E ∗ µ ( ~ρ , + ) e − i ωv ( x − x ′ ) . (5)This is a general expression, independent of the model considered and can be appliedfor both, periodical and random grating cases. Below, we will analyze these casesseparately (hereafter the sign + is omitted). In the periodical grating case (photoniccrystal) surface profile is a periodical function h ( ~ρ ) = δ cos ~K ~ρ , where ~K = (2 π/b, π/d )is a two-dimensional vector and b, d are the grating periods in the x and y directions,respectively. In the rough surface case h ( ~ρ ) is a Gaussian distributed random function.First let us consider the photonic crystal case. It is convenient first to present Green’sfunction in the form G µν ( ~r, ~r ′ ) = Z G µν ( ~p | z, z ′ ) e i~p ( ~ρ − ~ρ ′ ) d~p (2 π ) , (6) nverse Smith-Purcell effect near rough surfaces G µν ( ~p | z, z ′ ) is the Fourier transform in the xy plane. In the second step, letus assume that the plane of incidence of external light is xz . Then the backgroundelectric field in Eq.(5) that includes incident and reflected parts, takes the form: E ν ( ~ρ, e ik x x E ν , where k x = ω cos θ/c and θ is the angle between the externalphoton momentum and electron velocity directions. Substituting expressions for Green’sfunction, electric field and h ( ~ρ ) into Eq.(5), one finds P ( ω ) = g πL x G xν (cid:18) ωv , πd | Z, (cid:19) G ∗ µx (cid:18) − ωv , − πd | , Z (cid:19) δ (cid:18) k x + 2 πb − ωv (cid:19) E ν E ∗ µ , (7)where g = ( e/ ¯ hω ) ω c ( ε − δ , L x is the system size in the x direction and δ ( k x = 0)was substituted by L x / π . In the weak scattering regime ( ε − δ /λ ≪ ε r ≡ δ function in Eq.(7) sets the relation between the external light wavelength,incident angle, electron velocity and the gratings period. Interestingly, they are relatedto each other in the same way as in the direct Smith-Purcell effect λ = b (cid:18) β − cosθ (cid:19) , (8)with β = v/c . Note that the dispersion relation depends only on the grating periodin the electron velocity direction. As it was is shown in [12] G xy = G yx ≡ | ε | ≫
1. In this case the main contribution to the Eq.(7) comes from the termcontaining G xz . The explicit expression of the bare Green’s function is (see [12]) G xz ( ~p | z,
0) = − G zx ( ~p | , z ) = − ip x k ε ( ω ) qe iqz k − ε ( ω ) q , (9)where q = √ k − p if k > p and i √ p − k if k < p and k = − ( ε ( ω ) k − p ) / .Substituting Eq.(9) into Eq.(7), one comes after some algebraic manipulations to thefollowing expression for P ( ω ) P ( ω ) = πg L x c | ε ( ω ) | ( γ − + λ β d ) | E z | δ ( ωv − πb − k x ) β ω (cid:20) ( ε ( ω ) β − − λ β d ) / + iε ( ω ) q γ − + λ β d (cid:21) × (10) × exp (cid:16) − πZ q d + γ β λ (cid:17)(cid:20) ( ε ∗ ( ω ) β − − λ β d ) / − iε ∗ ( ω ) q γ − + λ β d (cid:21) , with γ = (1 − β ) − / . The amplitude of electric field includes both incident and reflectedparts, E z = (1 + r ( ω )) E z , where E z is the amplitude of incident field and r ( ω ) is thereflection amplitude that goes to unity in the limit | ε | → ∞ . Because of the exponential nverse Smith-Purcell effect near rough surfaces π d + 1 λ β γ ! / Z ≪ ϕ ( Z, Y ) remains unchangedduring the interaction with the photon. We need this assumption for derivation ofabsorption probability Eq.(2). Essential absorption probability is achieved in the casewhen the imaginary part of ε ( ω ) is small compared to the real part. Such a situationoccurs, for example, for noble metals Au,Ag, Cu and etc at the infrared wavelengths[9]. As an example for gold at photon energy ¯ hω = 1 ev , Reε = −
70 and
Imε = 6 . ε ( ω ) real, for the distances Eq.(11),one obtains P ( ω ) = πg L x c ε ( ω )( γ − + λ β d ) | E z | δ ( ωv − πb − k x ) β ω h ε ( ω ) β − − λ β d + ε ( ω )( γ − + λ β d ) i . (12)Since the absorption probability is positive one gets a condition on the particle velocity v c ≤ ( ε + 1) ε − λ ( ε +1) d (13)Maximum of P ( ω ) is achieved provided that the equality in Eq.(13) holds. It followsfrom Eq.(13) that the wavelength λ ≫ d is suppressed. Taking the limit d → ∞ onereturns, as it should be expected, to the Smith-Purcell geometry with periodicity onlyin the direction of particle motion. In this geometry and for metallic gratings ( ε < − c q ( ε + 1) /ε represents the velocity of induced polariton on the surface.Hence, the condition Eq.(13), means that the maximum of absorption is reached if theparticle velocity equals the polariton velocity.
4. Rough Surface
The above consideration can be generalized to a case when a charged particle travels overa rough surface, by applying a recently developed approach to study the spectrum ofradiation from a surface roughness [10, 11]. It can be shown that in this case absorptionprobability consists of two parts, each of them has different origin and must be evaluatedseparately. One is caused by the single scattering of polaritons another is caused by theirdiffusion, i.e., by the multiple scattering effect. Using Eq.(5), for the single scatteringcontribution to the absorption probability, one obtains P r ( ω ) = g Z dxdx ′ d~ρ ~ρ G xν ( x, Y, Z, ~ρ , + ) G ∗ µx ( ~ρ , + , x ′ , Y, Z ) (14) W ( | ~ρ − ~ρ | ) E ν ( ~ρ ) E ∗ µ ( ~ρ ) e − i ω ( x − x ′ ) v where δ W ( | ~ρ − ~ρ | ) = < h ( ~ρ ) h ( ~ρ ) > is the correlation function of the rough surfaceprofile. We assume that the surface profile fluctuations are uncorrelated. This allows usto substitute the correlation function W by δ function except the cases when finitecorrelation length is needed for divergence reasons, i.e. to avoid the divergences nverse Smith-Purcell effect near rough surfaces λ ≫ σ ,where σ is the correlation length of random surface profile fluctuations. Next, weevaluate the integral Eq.(15), by assuming W ( p ) = πσ e − p σ / . Then, taking theFourier transforms, substituting Eq.(9) into Eq.(15) in the limit Z → Z ≪ βλγ/ π ), we obtain the desired result for the absorption probability with singlescattering contribution P r ( ω ) = πL x g (1 + r )(1 + r ∗ ) | E z | cβ ω F ( ε, β ) (15)where F ( ε, β ) = ε ( ε − ωπσεβ c q (1 + ε )(1 + εγ − ) . (16)Note that in Eqs.(15,16), like before, we assume that imaginary part of ε is small andneglect it compared to real part. For the positive ε (dielectrics), as it follows from F ( ε, β ), the expression under the square root is always positive. As for negative ε (metals), it can be easily checked, that it leads to serious restriction on the energy ofparticle, i.e. γ ≤ − ε . Maximum absorption is achieved when the equality holds,i.e., γ = − ε ( ω ), see Fig.2. For the negative ε a plasmon-polariton is formed onthe surface, see for example Ref. [13]. The pole at p = εk / ( ε + 1) in the Green’sfunction Eq.(9) is manifestation of the plasmon-polariton. It is scattered on theinhomogeneities and gives contribution to the scattered electric field in Eq.(2) and henceto the absorption probability P ( ω ). The expression (15) is the plasmon-polariton singlescattering contribution. To make a further analytical progress in the study of P ( ω ) wewill assume that the following inequality is met: λ << l << l in , L , where l, l in are elasticand inelastic mean free paths of polariton on the surface. In other words we will assumethat the condition of multiple or diffusional scattering of polariton are realized in thesurface. In our calculations of the diffusional contribution to the absorption probabilitywe follow closely Refs. [10, 11]. Further manipulations are completely analogous tothose outlined in Refs.[10, 11] for the case of the radiation problem. Hence, here wepresent the final result without derivation by noting, that the diffusion contribution toabsorption probability is the dominant one P D ( ω ) = 323 l in l P r ( ω ) . (17)Indeed, as seen from Eq.(17), the quantity P D ( ω ) is proportional to P r ( ω ), withprefactor l in /l , which is the average number of polariton scatterings in the system.In the diffusion regime the ratio is large number, i.e l in /l ≫
1, see also [14], justifyingthat the diffusion contribution is dominant. It is important to notice, that one of theadvantages of the random surface profile is that the external light incident angle can bearbitrary instead the certain one in the periodical case.For completeness , we also compare the absorption probability with the probabilityof emission of a photon by a charge particle moving under the same conditions, see nverse Smith-Purcell effect near rough surfaces Figure 2.
Absorption probability dependence on the particle energy.
Fig.1. The probability of emission of a photon of energy ω by an electron moving overa rough surface can be estimated as (following Refs.[10, 11]) P e ( ω ) ≈ e hcβ g ( ω ) L x Z l in ( ω ) l ( ω ) , (18)where g = ( ε − k δ σ and Z ≪ λβγ/ π is the distance from the plane z = 0. UsingEqs.(15), (16), (17) and (18) the ratio of probabilities can be estimated as R = P D ( ω ) P e ( ω ) ≈ Zc | E z | (1 + r )(1 + r ∗ )¯ hω F ( ε, β ) . (19)Now let us estimate numerically R . Before doing so, first we verify numerically theapplicability of the diffusion approximation. Note, that in the weak scattering regimeaverage mean free paths are described by the following expressions: l = 4 | Reε | /kg and l in = ( Reε ) /kImε [10, 11]. For Au at the photon energy ¯ hω = 1 eV Reε = − Imε = 6 .
27 , r = r ∗ ∼
1. Taking for the roughness parameters δ = 10 nm and σ = 100 nm one gets g ∼ .
13 and l ∼ λ and l in ∼ λ . This means that theconditions λ ≪ l ≪ l in of diffusion of polaritons are realized in the system. Now takingelectron energy E = 3 . M ev , Z ∼ λβγ/ π , laser power | E z | c ∼ W m − one findsfrom Eq.(19) that R ∼ nverse Smith-Purcell effect near rough surfaces ε ( ω ) in the optical region.Concluding this section let us note that the ratio R can be made essentially largervia increasing the laser power. This point is topical for the laser driven accelerationapplication of the inverse Smith-Purcell effect.
5. Laser driven acceleration
We now want to discuss utilization of the inverse Smith-Purcell effect for particleacceleration. Metal surfaces with rough or one-dimensional periodic gratings can not beused for acceleration purposes because of the restriction on the energy of particle (seeEq.(13). However, there is an important exception, when the strength of the electricfield, that determines the absorption probability and were scattered from a metal surface,can be resonantly large. To illustrate this, we consider two-dimensional periodicalgrating case. The absorption probability, Eq.(12), is straightforwardly applicable inthis case. Rewriting the restriction condition on the energy, Eq.(13), in the form γ ≤ λ d ( ε + 1) − ε λ d ( ε + 1) , (20)it is easy to see, that the most favorable situation happens when the photon energysatisfies the resonant condition, i.e. the denominator of Eq.(20) becomes zero1 + (cid:18) πcωd (cid:19) ( ε ( ω ) + 1) = 0 (21)Note that we have in mind optical frequencies for which ε ( ω ) is a large negative number.In this case the energy of accelerated particle can be very large. The absorptionprobability Eq.(12) at the resonance photon energy will be large too.The largeness of absorption probability is caused by the resonance enhancementof the scattered field due to the surface plasmon-polaritons. Growth of probability islimited only by the losses in the optical region. In dielectrics restriction on energy isabsent and they can be used in acceleration purposes, see for example, Ref.[15].
6. Summary
In conclusion, we have investigated the absorption of a photon by an electron movingover a rough surface. Optimal conditions that include polarization of incident light,electron energy, material and grating types are indicated. In particular, it is shownthat only p-polarized photon can be absorbed. For metallic surfaces and for relativisticparticles two-dimensional periodical grating is preferable because of the restriction onthe energy of particle. For dielectrics restriction on the energy is absent. nverse Smith-Purcell effect near rough surfaces We are grateful to A.Akopian and F.J.de Abajo for helpful comments. V.G.acknowledges partial support by FEDER and the Spanish DGI under projectno.FSI2010-16430.
References [1] Rullhusen R, Artru X and Dhez P 1998
Novel Radiation Sources Using Relativistic Electrons ,(Singapore:World Scientific)[2] Edighoffer J A, Kimura W D, Pantell R H, Piestrup M A and Wang D Y 1981
Phys.Rev.A Phys.Rev.Lett. Nature
Phys.Rev.Lett. JapaneseJournal of Appl.Physics New Journal of Physics Phys.Rev.B ,4370[10] Gevorkian Zh S 2010 Phys.Rev.ST Accel.Beams , 070705[11] Gevorkian Zh S 2011 EPL , 64004[12] Maradudin A A and Mills D L 1975 Phys.Rev.B , 1392[13] Raether Heinz 1988 Surface Plasmons on Smooth and Rough Surfaces and on Gratings
SpringerTracts in Modern Physics (Berlin:Springer)[14] Gasparian V M and Gevorkian Zh S 2013
Phys.Rev.A ,053807[15] Cowan B W 2003 Phys.Rev. ST-AB6