Time Dependence of the Intensity of Diffracted Radiation Produced by a Relativistic Particle Passing through a Natural or Photonic Crystal
aa r X i v : . [ phy s i c s . acc - ph ] J a n Time Dependence of the Intensity ofDiffracted Radiation Produced by aRelativistic Particle Passing through aNatural or Photonic Crystal
V.G. Baryshevsky, A.A. Gurinovich
Abstract
The formulas which describe the time evolution of radiation produced by a rela-tivistic particle moving in a crystal are derived. It is shown that the conditions arerealizable under which parametric (quasi-Cherenkov) radiation, transition radiation,diffracted radiation of the oscillator, surface quasi-Cherenkov and Smith-Purcell ra-diation last considerably longer than the time τ p of the particle flight through thecrystal. The results of carried out experiments demonstrate the presence of addi-tional radiation peak appearing after the electron beam has left the photonic crystal. Introduction
At present, the processes of diffracted radiation of photons by relativistic parti-cles passing through crystals (natural or artificial spatially periodic structures)are intensively studied both theoretically and experimentally. Worthy of men-tion are such types of diffracted radiation as parametric (quasi-Cherenkov)radiation and diffracted radiation of a relativistic oscillator [1–3]. It should benoted, however, that until now, theoretical and experimental analysis of radi-ation produced by a relativistic particle passing through a crystal has focusedon spectral-angular characteristics of radiation. Nevertheless, it was shownin [5, 6] that because of diffraction, photons produced through radiation incrystals have group velocity v pgr , which is appreciably smaller than the veloc-ity v of a relativistic particle. As a result, the situation is possible in whichradiation from the crystal still continues after the particle has passed throughit [5, 6]. This enables studying time evolution of the process of photon radi-ation produced during the particle transmission through the crystal (naturalor photonic), or during the particle flight along the surface of such crystals. Inthe present paper the formulas are derived, which describe the time evolutionof radiation produced by a relativistic particle moving in a crystal. It is shown Preprint submitted to Elsevier August 15, 2018 hat the conditions are realizable under which parametric (quasi-Cherenkov)radiation, transition radiation, diffracted radiation of the oscillator, surfacequasi-Cherenkov and Smith-Purcell radiation last considerably longer thanthe time τ p of the particle flight through the crystal, i.e., much longer than τ p ≤ − s. Let us first recall the conventional consideration of the radiation process incrystals [1, 8].Both the spectral-angular density of radiation energy per unit solid angle W ~nω and the differential number of emitted photons dN ~nω ω = 1 / ~ ω · W ~nω can beeasily obtained if the field ~E ( ~r, ω ) produced by a particle at a large distance ~r from the crystal is known [3] W ~nω = er π (cid:12)(cid:12)(cid:12) ~E ( ~r, ω ) (cid:12)(cid:12)(cid:12) , (1)The vinculum here means averaging over all possible states of the radiatingsystem. In order to obtain ~E ( ~r, ω ), Maxwell’s equation describing the interac-tion of particles with the medium should be solved. The transverse solutioncan be found with the help of Green’s function of this equation, which satisfiesthe expression: G = G + G ω πc (ˆ ε − G, (2) G is the transverse Green’s function of Maxwell’s equation at ˆ ε = 1. It isgiven, for example, in [19].Using G , we can find the field we are concerned with E n ( ~r, ω ) = Z G nl ( ~r, ~r ′ , ω ) iωc j l ( ~r, ω ) d r ′ , (3)where n, l = x, y, z , j l ( ~r, ω ) is the Fourier transformation of the e-th compo-nent of the current produced by a moving beam of charged particles (in thelinear field approximation, the current is determined by the velocity and thetrajectory of a particle, which are obtained from the equation of particle mo-tion in the external field, by neglecting the influence of the radiation field on2he particle motion). Under the quantum-mechanical consideration the cur-rent j should be considered as the current of transition of the particle-mediumsystem from one state to another.According to [3, 8], Green’s function is expressed at r → ∞ through thesolution of homogeneous Maxwell’s equations E ( − ) n ( ~r, ω ) containing incomingspherical waves:lim G nl ( ~r, ~r ′ , ω ) = e ikr r X S e sn E ( − ) s ∗ ~kl ( ~r ′ , ω ) , (4) r → ∞ where ~e s is the unit polarization vector, s − , ~e ⊥ ~e ⊥ ~k .If the electromagnetic wave is incident on a crystal of finite size, then at r → ∞ ~E ( − ) sk ( ~r, ω ) = ~e s e i~k~r + const e ikr r , and one can show that the relation between the solution ~E ( − ) sk and the solutionof Maxwell’s equation ~E (+) ( ~k, ω ) describing scattering of a plane wave by thetarget (crystal), is given by: ~E ( − ) s ∗ ~k = ~E (+) s − ~k (5)Using (3), we obtain E n ( ~r, ω ) = e ikr r iωc X S e sn Z E ( − ) s ∗ ~k ( ~r, ω ) ~j ( ~r ′ , ω ) d r ′ . (6)As a result, the spectral energy density of photons with polarization s can bewritten in the form: W s~n,ω = ω π c (cid:12)(cid:12)(cid:12)(cid:12)Z ~E ( − ) s ∗ ~k ( ~r, ω ) ~j ( ~r, ω ) d r (cid:12)(cid:12)(cid:12)(cid:12) , (7) ~j ( ~r, ω ) = Z e iωt ~j ( ~r, ω ) dt = eQ Z e iωt ~v ( t ) δ ( ~r − ~r ( t )) dt, (8)where eQ is the charge of the particle, ~v ( t ) and ~r ( t ) are the velocity and thetrajectory of the particle at moment t . By introducing (8) into (7) we get dN s~n,ω = e Q ω π ~ c (cid:12)(cid:12)(cid:12)(cid:12)Z ~E ( − ) s ∗ ~k ( ~r ( t ) , ω ) ~v ( t ) e iωt d (cid:12)(cid:12)(cid:12)(cid:12) t. (9)3ntegration in (9) is carried out over the whole interval of the particle motion.It should be noted that the application of the solution of a homogeneousMaxwell’s equation instead of the inhomogeneous one essentially simplifies theanalysis of the radiation problem and enables one to consider various cases ofradiation emission taking into account multiple scattering.Using equations (7)–(9), one can easily obtain the explicit expression for theradiation intensity and that for the effect of multiple scattering on the processunder study [3, 8, 9].Consider, for example, the PXR radiation. Let a particle moving with a uni-form velocity be incident on a crystal plate with the thickness L being L ≪ L c ,where L c = ( ωq ) − / is the coherent length of bremsstrahlung q = θ / θ is the mean square angle of multiple scattering. The latter requirement allowsneglecting the multiple scattering of particles by atoms. A theoretical methoddescribing multiple scattering effect on the radiation process is given in [10].According to (9), in order to determine the number of quanta emitted by aparticle passing through the crystal plate, one should first find the explicitexpressions for the solutions ~E ( − ) s~k . As was mentioned above, the field ~E ( − ) s~k can be found from the relation ~E ( − ) s~k = ( ~E (+) s − ~k ) ∗ if one knows the solution ~E (+) s~k describing the photon scattering by the crystal.In the case of two strong waves excited under diffraction (the so-called two-beam diffraction case [11]), one can obtain the following set of equations fordetermining the wave amplitudes (see [12]): k ω − − χ ∗ ! ~E ( − ) s~k c s χ ∗− ~τ ~E ( − ) s~k τ = 0 k ω − − χ ∗ ! ~E ( − ) s~k τ c s χ ∗ ~τ ~E ( − ) s~k = 0 . (10)Here ~k ~τ = ~k + ~τ , ~τ is the reciprocal lattice vector, χ , χ ~τ are the Fouriercomponents of the crystal susceptibility. It is well known that the crystal isdescribed by a periodic susceptibility (see, for example, [11]: χ ( ~r ) = X ~τ χ ~τ exp( i~τ ~r ) . (11) c s = ~e s ~e s~τ , where ~e s ( ~e s~τ ) are the unit polarization vectors of the incident anddiffracted waves, respectively.The condition for the linear system (10) to be solvable leads to a dispersionequation that determines the possible wave vectors ~k in a crystal. These wave4ectors are convenient to present in the form: ~k µs = ~k + ~κ ∗ µs ~N , κ ∗ µs = ωcγ ε ∗ µs , where µ = 1 , ~N is the unit vector of a normal to the entrance crystal surfacewhich is directed into the crystal, ε s = 14 [(1 + β ) χ − β α B ] ± n [(1 − β ) χ + β α B ] +4 β C s χ ~τ χ ~ − τ o − / . (12) α B = (2 ~k~τ + τ ) k − is the off-Bragg parameter ( α B = 0 if the exact Braggcondition of diffraction is fulfilled), γ = ~n γ · ~N , ~n γ = ~kk , β = γ γ , γ = ~n γτ · ~N , ~n γτ = ~k + ~τ | ~k + ~τ | . The general solution of (10) inside a crystal is: ~E ( − ) s~k ( ~r ) = X µ =1 h ~e s A µ exp( i~k µs ~r ) + ~e sτ A τµ exp( i~k µsτ ~r ) i . (13)Associating these solutions with the solutions of Maxwell’s equations for thevacuum area, one can find the explicit form of ~E ( − ) s~k ( ~r ) throughout the space.It is possible to discriminate several types of diffraction geometries, namely,the Laue (a) and the Bragg (b) schemes are most well known. (a) Let us consider the PXR in the Laue case.In this case, the electromagnetic waves emitted by a particle in both theforward and the diffracted directions leave the crystal through the same surface( k z > , k z + τ z > z -axis is parallel to the normal N (where N is thenormal to the crystal surface being directed inside a crystal). By matching thesolutions of Maxwell’s equations on the crystal surfaces with the help of (10),(12), (13), one can obtain the following expressions for the Laue case: ~E ( − ) s~k = ~e s − X µ =1 ξ ∗ µs e − i ωγ ε ∗ µs L e i~k~r + e s~τ β X µ =1 ξ τ ∗ µs e − i ωγ ε ∗ µs L e i~k τ ~r θ ( − z )+ ~e s − X µ =1 ξ ∗ µs e − i ωγ ε ∗ µs ( L − z ) e i~k~r + e s~τ β X µ =1 ξ τ ∗ µs e − i ωγ ε ∗ µs ( L − z ) e i~k τ ~r θ ( L − z ) θ ( z ) + ~e s e i~k~r θ ( z − L ) , (14)where ξ , s = ∓ ε , s − χ ε s − ε s ) ; ξ τ , s = ∓ c s χ − τ ε s − ε s ) ; θ ( z ) = , if z ≥ , if z < . Substitution of (14) into (9) gives for the Laue case the differential number ofquanta of the forward directed parametric X-rays with the polarization vector ~e s : d N L s dωd Ω = e Q ω π ~ c ( ~e s ~v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X µ =1 , ξ µs e i ωcγ ε µs L ω − ~k~v − ω − ~k ∗ µs ~v × [ e i ( ω − ~k ∗ µs ~v ) T − (cid:12)(cid:12)(cid:12) , (15)where T = L/cγ is the particle time of flight; ~e k [ ~k~τ ]; ~e k [ ~k~e ].One can see that formula (15) looks like the formula which describes thespectral and angular distribution of the Cherenkov and transition radiationsin the matter with the index of refraction n µs = k zµs /k z = 1 + κ µs /k z .The spectral angular distribution for photons in the diffraction direction ~k τ = ~k + ~τ can be obtained from (15) by a simple substitution ~e s → ~e sτ , ξ µs → β ξ τµs ,ξ τ s = ± χ τ c s ε s − ε s ) ~k → ~k τ , ~k µs → ~k τµs = ~k µs + τ. (b) Now let us consider PXR in the Bragg case. In this case, side by side withthe electromagnetic wave emitted in the forward direction, the electromagneticwave emitted by a charged particle in the diffracted direction and leavingthe crystal through the surface of the particle entrance can be observed. Bymatching the solutions of Maxwell’s equations on the crystal surface with thehelp of (10), (12), (13), one can get the formulas for the Bragg diffractionschemes. 6t is interesting that the spectral angular distribution for photons emitted inthe forward direction can be obtained from (15) by the following substitution, ξ µs → γ µs , γ s = 2 ε s − χ (2 ε s − χ ) − (2 ε s − χ ) e i ωγ ( ε s − ε s ) L (16)The spectral angular distribution of photons emitted in the diffracted directioncan be obtained from (15) by substitution ~e s → ~e sτ , ~k → ~k τ , k µs → ~k µτs , ξ µs e i ωγ ε µs L → γ τµs , where γ τ s = − β c s χ τ (2 ε s − χ ) − (2 ε s − χ ) e i ωγ ( ε s − ε s ) L . (17)Let us note that the above formulas fully describe parametric (quasi-Cherenkov)radiation in natural and photonic crystals and they certainly include that con-tribution to radiation, which goes over to ordinary transition radiation, if theradiation is considered outside the region of diffraction reflection. A descrip-tion of diffracted radiation of a relativistic oscillator is given in [1, 2] and thereference therein.Let us take notice of the fact that in photonic crystals built from metal threadswith the diameter smaller than or comparable with λ , the value of χ ( τ ) ispractically independent on τ . As a result, it is possible to effectively exciteradiation in, e.g., the terahetrz range in a lattice with a period of severalmillimeters.When a particle travels in a vacuum near the surface of a spatially periodicmedium, new kinds of radiation arise [13, 14] – surface parametric (quasi-Cherenkov) X-ray radiation (SPXR) and surface DRO (see Figure 1). Thisphenomenon takes place under the condition of uncoplanar surface diffraction,first considered in [15].The solution of Maxwell’s equation ~E (+) ~k ( ~r ) in this case of uncoplanar surfacediffraction was obtained in [15]. It was shown that the surface diffraction inthe two-wave case is characterized by two angles of total reflection (severalangles in the case of multi-wave diffraction [16]). The solution obtained in [16]contains the component, which describes the state that damps with growingdistance from the surface of the medium, both within the material and in thevacuum, and which describes a surface wave, i.e., a wave in which the energyflux is directed along the boundary of the surface of a spatially periodic target7 k ττ t k τ t k τ k k k τ1 k photonic crystal Figure 1. Surface diffraction of a radiated photon (see review [17]). According to [15], this solution, which describes scatteringof a plane wave by the target under the surface diffraction geometry, can bewritten in the form: ~E (+) s~k = e s e i~k~r + A s ( ~k, ω ) e i~k ~r + B s ( ~k, ω ) e i~k ~r , (18)where the wave vector in a vacuum ~k = ( ~k t , ~k ⊥ ), ~k = ( ~k t , − ~k ⊥ ), ~k =( ~k t , − ~k ⊥ ), | ~k ⊥ | = q k − k t , ~k t = ~k t + ~τ , ~k t is the component of the wavevector that is parallel to the surface, ~τ is the reciprocal lattice vector, ω isthe photon frequency. The amplitudes A s and B s are given in [9, 14]. Substi-tuting the solution ~E ( − ) s~k = ( ~E (+) s − ~k ) ∗ into (3), we can find the spectral-angulardistribution of SPXR and DRO. The intensity I ( t ) of radiation produced by a particle which has passed througha crystal can be found with known intensity of the electric field ~E ( ~r, t )) (mag-netic field ~H ( ~r, t )) of the electromagnetic wave, which is produced by thisparticle [18], I ( t ) = c π | ~E ( ~r, t ) | r d Ω , (19)where r is the distance from the crystal, which is assumed to be larger thanthe crystal size.The field ~E ( ~r, t ) can be presented as an expansion in a Fourier series ~E ( ~r, t ) = 12 π Z ~E ( ~r, ω ) e − iωt dω. (20)8ccording to the results obtained in [3,7,9], at a long distance from the crystal,the Fourier component can be written as follows: ~E ( ~r, t ) = e ikr r iωc X s e si Z ~E ( − ) s ∗ ~k ( ~r ′ ω ) ~j ( ~r ′ , ω ) d r ′ . (21)where i = 1 , , x , y , z ), e si is the i -component of the wave polarization vector ~e s ; s = 1 , ~E ( − ) s~k is the solution ofMaxwell’s equations describing scattering of a plane wave with a wave vector ~k = k ~rr and the asymptotic of a converging spherical wave, ~j ( ~r, ω ) = Z ~j ( ~r, t ) e iωt dt (22) ~j ( ~r, ω ) = Q~v ( t ) δ ( ~r − ~r ( t )) is the is the current density of the particle withcharge Q , ~r ( t ) is the particle coordinate at time t .The explicit form of the expressions ~E ( − ) s describing diffraction of the elec-tromagnetic wave in a crystal in the Laue and Bragg cases is given in [3, 8, 12](See Section 1).Now let us take a closer look at the expression for the amplitude A ( ω ) of theemitted wave: A s~k ( ω ) = iωc Z ~E ( − ) s ∗ ~k ( ~r ′ , ω ) ~j ( ~r ′ , ω ) d r ′ . (23)Using (22), (23) can be recast as follows A s~k ( ω ) = iωc Z ~E ( − ) s ∗ ~k ( ~r ′ , ω ) Q~v ( t ) δ ( ~r ′ − ~r ( t )) e iωt dtd r ′ = iωQc Z ~E ( − ) s ∗ ~k ( ~r ( t ) , ω ) ~v ( t ) e iωt dt (24)Recall that ~E ( − ) s ∗ ~k = ~E (+) s − ~k , where the field ~E (+) s − ~k is the solution of Maxwell’sequations describing scattering by a crystal of a plane wave with wave vector( − ~k ) and the asymptotics of a diverging wave at infinity. According to (24),the radiation amplitude is determined by the field ~E ( − ) s~k taken at point ~r ( t ) ofparticle location at time t and integrated over the time of particle motion.Let us consider in more detail the constant motion of a particle in passingthrough the crystal. In this case, parametric quasi-Cherenkov radiation canappear [1, 8], which includes, as a particular case, diffracted transition radia-tion. The explicit formulas for the radiation amplitude in the case of two-wave9iffraction of photons in crystals for the Laue and Bragg geometries are givenin [3, 8, 12] (Section 1).From (20), (21), and (23) follows that the expression for the electromagneticwave emitted by the particle passing through the crystal (natural or photonic)can be presented in a form: ~E i ( ~r, t ) = 12 πr X s e si Z A s~k ( ω ) e − iω ( t − rc ) dω, (25)i.e., ~E i ( ~r, t ) = r P s e si A s~k ( t − rc ).From (25) follows that the time dependence of the form of the pulse I ( ~r, t )( ~E ( ~r, t ))of radiation generated by a particle passing through the crystal is determinedby the dependence of the radiation amplitude A s~k ( ω ) on frequency. Accord-ing to the explicit expression for the radiation amplitudes given in [3, 8, 12],the radiation amplitudes A s~k ( ω ) can be presented as sums proportional to theamplitudes of diffraction reflection from the crystal and to the amplitude ofwave transmission through the crystal. For example, for the case of forwardparametric radiation in the Laue geometry A s~k ( ω ) = Qc ( ~e s ~v ) X µ =1 , ξ µs e i ωγ ε µs L × " ω − ~k~v − ω − ( ~k + κ µs ~N ) ~v e i ( ω − ( ~k + κ µs ~N ) ~u ) Lcγ − (cid:21) (26)Thus, the time dependence of the from of the radiation pulse is determinedby the time dependence of the radiation amplitude A s~k ( t − rc ).By way of example, let us consider the characteristics of the time dependenceof radiation produced by a particle passing through the crystal for a wavepacket passing through the crystal [2, 5, 6]Let us consider the pulse of electromagnetic radiation passing through themedium with the index of refraction n ( ω ). The group velocity of the wavepacket is as follows: v gr = ∂ωn ( ω ) c∂ω ! − = cn ( ω ) + ω ∂n ( ω ) ∂ω , (27)where c is the speed of light, ω is the quantum frequency.10n the X-ray range ( ∼ tens of keV) the index of refraction has the universalform n ( ω ) = 1 − ω L ω , ω L is the Langmuir frequency. Additionally, n − ≃ − ≪
1. Substituting n ( ω ) into (27), one can obtain that v gr ≃ c (cid:18) − ω L ω (cid:19) .It is clear that the group velocity is close to the speed of light. Therefore thetime delay of the wave packet in a medium is much shorter than the timeneeded for passing the path equal to the target thickness in a vacuum.∆ T = lv gr − lc ≃ lc ω L ω ≪ lc . (28)To consider the pulse diffraction in a crystal, one should solve Maxwell’s equa-tions that describe a pulse passing through a crystal. Maxwell’s equations arelinear, therefore it is convenient to use the Fourier transform in time and torewrite these equations as functions of frequency: " − curl curl ~E ~k ( ~r, ω ) + ω c ~E ~k ( ~r, ω ) i + χ ij ( ~r, ω ) E ~k,j ( ~r, ω ) = 0 , (29)where χ ij ( ~r, ω ) is the spatially periodic tensor of susceptibility; i, j = 1 , , (cid:16) k ω − − χ (cid:17) ~E s~k − c s χ − ~τ ~E s~k τ = 0 (cid:16) k τ ω − − χ (cid:17) ~E s~k τ − c s χ ~τ ~E s~k = 0 (30)Here ~k is the wave vector of the incident wave, ~k ~τ = ~k + ~τ , ~τ is the reciprocallattice vector; χ , χ ~τ are the Fourier components of the crystal susceptibility: χ ( ~r ) = X ~τ χ ~τ exp( i~τ ~r ) (31) C s = ~e s ~e s~τ , ~e s ( ~e s~τ ) are the unit polarization vectors of the incident anddiffracted waves, respectively.The solvability condition for the linear system (30) leads to a dispersion equa-tion that determines the possible wave vectors ~k in a crystal. It is convenient11o present these wave vectors as: ~k µs = ~k + æ µs ~N , æ µs = ωcγ ε µs , where µ = 1 , ~N is the unit vector of a normal to the entrance surface of thecrystal, which is directed into the crystal, ε (1 , s = 14 [(1 + β ) χ − βα B ] ± n [(1 + β ) χ − βα B − χ ] + 4 βC S χ ~τ χ − ~τ o / , (32) α B = (2 ~k~τ + τ ) k − is the off-Bragg parameter ( α B = 0 when the Braggcondition of diffraction is exactly fulfilled), γ = ~n γ · ~N , ~n γ = ~kk , β = γ γ , γ = ~n γτ · ~N , ~n γτ = ~k + ~τ | ~k + ~τ | The general solution of equations (29), (30) inside a crystal is: ~E s~k ( ~r ) = X µ =1 h ~e s A µ exp( i~k µs ~r ) + ~e sτ A τµ exp( i~k µsτ ~r ) i (33)Associating these solutions with the solutions of Maxwell’s equation for thevacuum area one can find the explicit expression for ~E s~k ( ~r ) throughout thespace. It is possible to discriminate several types of diffraction geometries,namely, the Laue and the Bragg schemes, which are most well-known [22].In the case of two-wave dynamical diffraction, the crystal can be described bytwo effective indices of refraction n (1 , s = 1 + ε (1 , s ,ε (1 , s = 14 (cid:26) χ (1 + β ) − βα ± q ( χ (1 − β ) + βα ) + 4 βC s χ τ χ − τ (cid:27) . (34)The diffraction is significant in the narrow range near the Bragg frequency,therefore χ and χ τ can be considered as constants and the dependence on ω should be taken into account for α = π −→ τ (2 π −→ τ +2 −→ k ) k = − (2 πτ ) k B c ( ω − ω B ),where k = ωc ; 2 π −→ τ is the reciprocal lattice vector which characterizes the setof planes where the diffraction occurs; Bragg frequency is determined by thecondition α = 0. 12rom (27), (34) one can obtain v (1 , sgr = cn (1 , ( ω ) ± β (2 πτ ) k B ( χ (1 − β )+ βα ) √ ( χ (1 − β )+ βα ) +4 βC s χ τ χ − τ . (35)In the general case ( χ (1 − β ) + βα ) ≃ √ βχ , therefore the term that isadded to n (1 , s ( ω ) in the denominator (35) is of the order of 1. Moreover, v gr significantly differs from c for the antisymmetric diffraction ( | β | ≫ . Itshould be noted that because of the complicated character of the wave fieldin a crystal, one of v ( i ) sgr can appear to be much higher than c and negative.When β is negative the radicand in (35) can become zero (Bragg reflectionthreshold) and v gr → v gr appears to be the function of Ω . This can be easily observed in the conditionsof X-ray-acoustic resonance. The performed analysis allows one to concludethat the center of the X-ray pulse in a crystal can undergo a significant delay∆ T ≫ lc available for experimental investigation. Thus, when β = 10 , l = 0 . l/c ≃ · − , the delay time can be estimated as ∆ T ≃ · − sec.Let us study now the time dependence of the delay law of radiation afterpassing through a crystal. Assuming that B ( ω ) is the reflection or transmissionamplitude coefficients of a crystal, one can obtain the following expression forthe pulse form E ( t ) = 12 π Z B ( ω ) E ( ω ) e − iωt dω = Z B ( t − t ′ ) E ( t ′ ) dt ′ . (36)where E ( ω ) is the amplitude of the electromagnetic wave incident on a crys-talIn accordance with the general theory, for the Bragg geometry, the amplitudeof the diffraction-reflected wave for the crystal width much greater than theabsorbtion length can be written as [22]: B s ( ω ) = (37) − χ τ (cid:26) χ (1 + | β | ) − | β | α − q ( χ (1 − | β | ) − | β | α ) − | β | C s χ τ χ − τ (cid:27) In the absence of resonance scattering, the parameters χ and χ ± τ can beconsidered as constants and frequency dependence is defined by the term13 = − (2 πτ ) k B c ( ω − ω B ). So, B s ( t ) can be found from B s ( t ) = − πχ τ (38) × Z (cid:26) χ (1 + | β | ) − | β | α − q ( χ (1 − | β | ) − | β | α ) − | β | C s χ τ χ − τ (cid:27) e − iωt dω. The Fourier transform of the first term results in δ ( t ) and we can neglect itbecause the delay is described by the second term. The second term can becalculated by the methods of the theory of function of complex argument: B s ( t ) = − i χ τ | β | (2 πτ ) k B ω B J ( a s t ) t e − i ( ω B +∆ ω B ) t θ ( t ) , (39)or B s ( t ) = − i q | β | J ( a s t ) a s t e − i ( ω B +∆ ω B ) t θ ( t ) , (40)where a s = 2 √ C s χ τ χ − τ ω B q | β | (2 πτ ) k B , ∆ ω B = − χ (1 + | β | ) ω B k B | β | (2 πτ ) . Since χ and χ τ are complex, both a s and ∆ ω B have real and imaginaryparts. According to (39)–(40), in the case of Bragg reflection of a short pulse(the pulse frequency bandwidth ≫ frequency bandwidth of the total reflectionrange) both the instantly reflected pulse and the pulse with amplitude under-going damped beatings appear. Beatings period increases with | β | grows and χ τ decrease. Pulse intensity can be written as I s ( t ) ∼ | B s ( t ) | = | β | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J ( a s t ) at (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − Im ∆ ω B t θ ( t ) . (41)It is evident that the reflected pulse intensity depends on the orientationof photon polarization vector ~e s and undergoes the damping oscillations ontime.Let us evaluate the effect. Characteristic values are Im ∆ ω B ∼ Im χ ω B and Im a ∼ Im χ τ ω B √ β . For 10 keV for the crystal of Si Im χ = 1 , · − , for LiH14 m χ = 7 , · − , Im χ τ = 7 · − , for LiF Im χ ∼ − . Consequently,the characteristic time τ for the exponent decay in (41) can be estimated asfollows ( ω B = 10 ):for Si the characteristic time τ ∼ − sec, for LiF the characteristic time τ ∼ − sec, for LiH the characteristic time τ ∼ − sec!!The reflected pulse also undergoes oscillations, the period of which increaseswith growing | β | and decreasing Re χ τ . This period can be estimated for β =10 and Re χ τ ∼ − as T ∼ − sec (for Si, LiH, LiF).When the resolving time of the detecting equipment is greater than the os-cillation period, the expression (41) should be averaged over the period ofoscillations. Then, for the time intervals when Re a s t ≫ , Im ∆ ω B t ≪ I s ( t ) ∼ t − . In the case of multi-wave diffraction, the time delay for the photon exit fromthe crystal will be even more appreciable.For an artificial spatially periodic medium (diffraction grating, photonic crys-tal), the parameter g can vary over a wide range. For example, accordingto [23], for a photonic crystal built from tungsten threads of 100 µm in diame-ter, the parameter g ∼ ω has the value of g ∼ − in a 10 GHz range. Asa result, in this range we have T (10 GHz) ∼ √ β | g | ω B ∼ √ β · − . At the sametime, in the terahertz range, T (1 THz) due to the drop of g ( T increasesproportionally to ω , the parameter a decreases: a ∼ ω B ), we have the period T (1 THz) ∼ √ β · − . As is seen, the oscillations of radiation from photoniccrystals are quite observable.So the time τ ph = Lv gr that the photon spends in the crystal can be longer thanthe flight time τ p = Lv of a relativistic particle in a crystal. Hence, the emis-sion of diffraction-related radiation (quasi-Cherenkov, transition, diffractedradiation of an oscillator, surface parametric radiation and others) producedby a relativistic particle will continue after the particle has left the crystal(see Fig.2) Under diffraction conditions, the crystal acts as a high-qualityresonator [1, 24].It should be noted, of course, that in observation of oscillations, one shouldeither register the moment of particle entrance into the crystal or use a shortbunch of particles with duration much shorter than the oscillation period.In the X-ray range, such situation is typical of electron buches, which areapplied for creating X-ray FELs (DESY). (The bunch duration in such FELs is15ens-hundreds of femptoseconds). In the terahertz range, much longer bunchesare required, so there are not serious experimental problems in this case. Ifthe bunch duration is large in comparison with the duration of the radiationpulse or the time of the electron entrance into the crystal is not registered,which occurs in a conventional experimental arrangement, then the intensity I ( t ) should be integrated over longer observation time intervals. As a result,we, in fact, obtain the expression (1) integrated over all frequencies, i.e., anordinary stationary angular distribution of radiation. If the response time ofthe devices detecting τ D (or the flight time of the particle in a crystal, or thebunch duration) is comparable with the oscillation period, then I ( t ) shouldbe integrated over the interval τ D . In this case oscillations will disappear, butwe will observe the power-law decrease in the intensity of radiation from thecrystal.In according with the above analysis some experiments are carried to observedelay of radiation pulse in a photonic crystal used for VFEL lasing [25–28].In these experiments the additional radiation peak (see Fig.2) is observed atstudies of lasing of VFEL with ”grid” photonic crystals in backward waveoscillator regime. This peak appears when the electron beam has left theresonator. -100 0 100 200 300 400 500Time, nanoseconds0-0.25-0.50-0.75-1 N o r m a li z ed s i gna l voltagebeam currentmicrowave power Figure 2. Detected microwave signal (black curve) synchronized with the beamcurrent and electron gun voltage
It should be mentioned here that backward wave oscillator regime impliesgeneration in presence of Bragg diffraction, therefore, under some conditionsthe group velocity could appear even to be close to 0 (see equation (35)). Theobserved delay (Fig.2) corresponds to v gr ∼ cm/s, i.e. v gr c ∼ − .In travelling wave regime, which corresponds to case of Laue diffraction, suchlong delay can not be obtained (according to (35) for β > v gr changes insignificantly). Particularly, in our experiments with Cherenkovgenerator without diffraction grating no additional peaks are detected, because16he group velocity in this case changes insignificantly due to the same reasonsas in the Laue case.And after all note that diffraction of a pulse of radiation produced by anexternal radiation source in a periodic structure could be accompanied byappearance of several transmitted or reflected radiation pulses (pulses of pho-tons) (see [29]). Conclusion
The formulas which describe the time evolution of radiation produced by a rel-ativistic particle moving in a crystal are derived. It is shown that the conditionsare realizable under which parametric (quasi-Cherenkov) radiation, transitionradiation, diffracted radiation of the oscillator, surface quasi-Cherenkov andSmith-Purcell radiation last considerably longer than the time τ p of the particleflight through the crystal. The results of carried out experiments demonstratethe presence of additional radiation peak appearing after the electron beamhas left the photonic crystal. References [1] V.G. Baryshevsky, Spontaneous and Induced Radiation by RelativisticParticles in Natural and Photonic Crystals. Crystal X-ray Lasers and VolumeFree Electron Lasers (VFEL), LANL e-print arXiv:1101.0783 (physics.acc-ph);(physics.optics)[2] V.G. Baryshevsky, I.Ya. Dubovskaya,
Diffraction phenomena in spontaneousand stimulated radiation by relativistic particles in crystals (Review)(1991) Technical Report Lawrence Berkeley Lab., CA (United States) DOI10.2172/5808050.[3] V.G. Baryshevsky
Channeling, Radiation and Reactions in Crystals at HighEnergy (Bel. State Univers., Minsk, 1982)[4] V.G.Baryshevsky, Diffraction of X-ray pulse in crystals, Izvestia AN BSSRser.phys.-mat. N5 (1989) 109-112.[5] V.G.Baryshevsky Izvestia AN BSSR ser.phys.-mat. N5 (1989) 109-112.[6] V.G.Baryshevsky Diffraction of X-ray pulse in crystals, LANL e-printarXive:physics/9906022v1.[7] V.G. Baryshevsky, Diffraction of X-ray pulse in crystals LANL e-print arxiv:physics/9906022v1.
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Parametric X-RayRadiation in Crystals: Theory, Experiment and Applications (Series: SpringerTracts in Modern Physics, Vol. 213 2005).[9] V.G. Baryshevsky
Nuclear Optics of Polarized Media (Energoatomizdat,Moscow, 1995) [in Russian].[10] V.G. Baryshevsky, A.O. Grubich, Le Tien Hai, Zh. Eksp. Teor. Fiz. (1988)51 [Sov. Phys. JETP (1988) 895].[11] Chang Shih-Lin, Multiple Diffraction of X-Rays in Crystals (Springer-VerlagBerlin Heidelberg New-York Tokyo, 1984).[12] V. G. Baryshevsky, Parametric X-ray radiation at a small angle near thevelocity direction of the relativistic particle Nucl. Instr. Methods
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Some Problems of Modern Physics to 80-th Anniversaryof I.M. Frank (Nauka, Moscow, 1989) 156.[15] V.G. Baryshevsky, Pis’ma Zh. Tekh. Fiz. (1976) 112-114; V.G. Baryshevsky,Zh.Exp.Teor Fiz. (1976) 430-434[Sov. Phys. JETP].[16] V.G. Baryshevsky and I.Ya. Dubovskaya, Phys. Status Solidi [in Russian] (1977) 597.[17] A.V. Andreev, Sov. Phys. Usp. (1985) 70–84.[18] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields in: L.D. Landau,E.M. Lifshitz
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Methods of Theoretical Physics (Mc Graw Hill, NewYork, 1953),[20] V.G.Baryshevsky, K.G.Batrakov, I.Ya.Dubovskaya J.Phys. D: Appl. Phys.24(1991) 1250-1257.[21] CERN COURIER 39, N4 (1999) 11-12[22] Z.G.Pinsker Dynamical scattering of X-rays in crystals (Springer, Berlin, 1988)[23] V.G. Baryshevsky, A.A. Gurinovich, Spontaneous and in- duced parametricand SmithPurcell radiation from electrons moving in a photonic crystal builtfrom the metallic threads NIM B 252 (2006) 92. ,132,132[24] Baryshevsky V.G., Batrakov K.G., Dubovskaya I.Ya., Karpovich V.A.,Rodionova V.M.,
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