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Dynamics of electrons acceleration in presence of crossed laser field
Accepted Manuscript Dynamics of Electrons Acceleration in Presence of Crossed Laser Field E.N. Frolov, A.V. Dik, S.B. Dabagov PII: DOI: Reference:
S0168-583X(13)00312-1 http://dx.doi.org/10.1016/j.nimb.2013.03.017 NIMB 59362
To appear in:
Nucl. Instr. and Meth. in Phys. Res. B
Received Date: Revised Date:
30 November 2012 14 March 2013
Please cite this article as: E.N. Frolov, A.V. Dik, S.B. Dabagov, Dynamics of Electrons Acceleration in Presence of Crossed Laser Field, Nucl. Instr. and Meth. in Phys. Res. B (2013), doi: http://dx.doi.org/10.1016/j.nimb. 2013.03.017
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
NIM B Nuclear Instruments and Methods B 00 (2013) 1–8
Dynamics of Electrons Acceleration in Presence of Crossed Laser Field Frolov E.N.a,b , Dik A.V.b , and Dabagov S.B.b,c,d a NR
Tomsk Polytechnic University, Tomsk, Russia P.N. Lebedev Physical Institute, Moscow, Russia c NRNU MEPhI, Moscow, Russia d INFN Laboratori Nazionali di Frascati, Frascati, Italy b RAS
Abstract In this work we have analyzed non-relativistic electrons’ dynamics in presence of both standing electromagnetic wave field of crossed laser beams and accelerating electrical field. Special interest has been paid to defining the conditions of charged particles’ bound states formation in such fields. We have demonstrated that a part of electron beam may be trapped by the effective potential of a standing wave. The phenomenology of electron channeling in the field of crossed laser beams is developed. c 2012 Published by Elsevier Ltd. Keywords: Nonrelativistic electron beams, laser-electron interaction, channeling PACS: 52.38.-r, 52.38.Hb, 41.75.Fr
1. Introduction Due to various possible scientific and technological applications [1, 2, 3, 4] production of trains of ultrashort electron beams is nowadays an active field of research. A common way to get such beams of high quality parameters (i. e. with a high peak current and a small both energy spread and emittance) is photoemission. But there are common problems with the length and intensity of the beams generated in such a way. Long laser pulse duration negatively influences on the beam size, and, moreover, can result in cathode sputtering. On the other hand, the number of emitted electrons is directly proportional to the time of irradiation. The main purpose of the SPARC LAB COMB experiment held at the Laboratori Nazionali di Frascati is the modulation of ultrashort electron beams during the process of their generation. Superposed incident and reflected laser beams can be used as a modulating structure in such experiments. Low-emittance beam generation as well as electron dynamics at accelerators are usually the subjects of main attention. But now, drawing a great attention of researchers, studies on space charge effects are becoming more and more urgent and topical. At the same time, the behaviours of electron ensemble in a field of crossed laser beams and accelerating electrical field (Fig. 1) are still to be examined. Such a field configuration might be formed near the cathode by both incident and reflected laser beams. Typically, the accelerating field dominates electron dynamics with respect to the electromagnetic fields present in the system. Hence, the detailed picture of electron distribution just near the cathode surface due to the presence of other fields is not of main interest. On the other hand, potentially useful and insufficiently investigated [5] phenomena, such as trapping of some electrons of the ensemble (bunch) by the potential wells formed by the superposition of optical laser fields with successful electrons’ channeling in the channels of electromagnetic standing waves, become possible under defined conditions [6]. 1
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Figure 1. General scheme of analysed system, where k is the wave vector of a laser beam, the indexes (i, r, s), which correspond to the incident, reflected and superposed waves, respectively, and Ea is the constant accelerating electrical field. The xy-plane coincides with the cathode surface, while the z-axis is orthogonal to that surface; the y-axis is orthogonal to the picture plane. The scheme outlines the triangular region, where standing electromagnetic waves are formed. The period of electromagnetic field for standing waves is defined by λE , while standing wave channels parallel to the cathode surface are characterized by λE /2.
Indeed, the total laser field is characterized by standing waves that form potential wells. The space orientation of channels representing these wells depends on both laser beam parameters and coefficient of reflection from the cathode surface. In the ideal case of total laser beam reflection we obtain the picture of standing waves parallel to the cathode surface; in other words, standing waves are perpendicular to accelerating field. Obviously, at specific conditions a fraction of electrons might be trapped into that channels. The bound motion of such electrons are similar to planar channeling of charged particles in crystals when a particle trajectory becomes limited within the space, a channel, formed by the averaged potential well of main crystallographic planes or axes [7], while in considered case such channel is created by the superposition of two electromagnetic fields. In this paper we present a simplified analysis of such a process and prove the possibility of separation of the original electron beam into accelerated and trapped parts; the latter may channel in the direction transversal (defined by crossed laser field) to the accelerating field. 2. Electron acceleration in presence of standing electromagnetic waves Let us examine a single non-relativistic electron being accelerated by a constant electric field Ea = (0, 0, −E) in presence of two crossed laser beams (Fig. 1). For simplicity, the xy-plane coincides with the cathode surface, while the z-axis is orthogonal to the surface; hence, the electric field is directed towards negative direction of z. Both electromagnetic waves of the laser (incident and reflected) have been chosen to be polarized in xz-plane, and thus, a two-dimensional case is considered (y-axis is orthogonal to the picture plane). Crossed laser field is defined by Ei = E0 (− cos θ, 0, sin θ) sin (ω0 t − ki r) and Er = E0 (cos θ, 0, sin θ) sin (ω0 t − kr r), and let us consider the ideal case of total laser reflection from the cathode (the coefficient of reflection equals to 1). Constant accelerating field is Ea = (0, 0, −E). The bidimentional equation of motion in the combined field can be written as ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎛ ⎞ 0 ⎜⎜⎜ x¨ ⎟⎟⎟ ⎜⎜⎜ cos θ ⎟⎟⎟ ⎜⎜⎜ sin (kz cos θ) cos (ω0 t + kx sin θ) ⎟⎟⎟ ⎜⎜⎜ 0 ⎟⎟⎟ ⎟⎟⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ + 2E0 e ⎜⎜⎝ m ⎜⎜⎝ ⎟⎟⎠ = −eE ⎜⎜⎝ (1) ⎠ cos (kz cos θ) sin (ω0 t + kx sin θ) z¨ 0 − sin θ −1 Due to the presence of high frequency electromagnetic field, the electron trajectory at defined conditions looks like frequent oscillations δxi along a smooth curve x¯i : xi = x¯i + δxi [8]. Neglecting the oscillations in the equation of motion (small by the definition, δxi ∼ eE0 /mω20 ), an effective potential energy of electron interaction in combined field can be written as follows Ue f f = −eEz +
(eE0 )2 sin2 θ cos (2kz cos θ) + U0 , mω20 2
(2)
/ Nuclear Instruments and Methods B 00 (2013) 1–8
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where U0 is the parameter for normalization, hence, can be omitted in optimized cases. For mentioned above parameters of the laser and its orientation the effective energy (2) depends only on z-coordinate. Defining α = ekE02 sin2 θ cos θ/(mω20 E) and φ = 2kz cos θ and neglecting the constant term (since it does not affect the electron movement), we introduce the following function f (φ) = (2Ue f f k cos θ)/(eE) = −φ + α cos φ
(3)
Here α ∼ A0 /Aac is the ratio of the work of a laser field to move the charge through the δx distance to the work of a constant accelerating field to move the charge through the space-period length.
Figure 2. Dimensionless potential energy f (φ) dependence on dimensionless coordinate φ with different values of α-parameter (curves A, B, C, D and E). Here θ = π/6, ω0 = 7 · 1015 c−1 ; the electrical field intensity is defined as E = 3.6 · 10−14 · E02 /α [V/m] (α 0). For E0 = 108 ÷ 1011 V/m and α = 4 we get the potential well depth of U ≈ 10−4 ÷102 eV. The space period of f (φ) is 2π, or, recalculated in units of z, is λU = π/(k cos θ) ≡ λE /2 that corresponds to the width of standing wave interaction potential.
The function f (φ) is the dimensionless effective potential energy of electron interaction in a superposed field (Fig. 2). Obviously, the appearance of electron bound states becomes possible when α > 1, while for α ≤ 1 only the electron density modulation can be observed (no bound states) [9]. It is important to note that the interaction potential formed by a standing wave can be split in two parts: the attracting potential responsible for electrons trapping (seen as a potential well mostly below the inclined rectilinear in the figure) and the reflecting potential. Obviously, the widths of these two sections of the potential depend on the parameter α, but their sum remains a constant equal to the half period of a standing wave (λU = λE /2). Performing substitution of the parameter z(t) by new one φ(t) = 2kz(t) cos θ in z-projection of a particle smooth trajectory m¨z(t) = eE + eEα sin (2kz cos θ) , (4) and defining constants b = (eE0 k sin 2θ)2 /(2m2 ω20 ), a = −2eEk cos θ/m, we get simple equation ¨ = a + b sin φ , φ(t)
(5)
that in new variables transforms the condition α > 1 into b > a. The dependence shown in Fig. 3 proves the condition b > a to be necessary, but not sufficient, condition of electron trapping. Similar curves could be achieved for two particles with equal initial velocities and different initial z-coordinates in a trapping field (b > a). For example, if all projections of initial velocity are equal to zero and the particle is positioned to be in a potential energy peak (maximum), it moves with acceleration in the direction of accelerating field. And, on the contrary, the particle being in a potential energy well (minimum) becomes bound and successfully oscillates in the well. 3
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Figure 3. Time evolution of the dimensionless coordinate φ(t). Initial conditions are the same for both particles shown, while the values of field parameters a and b differ for A and B curves. The binding conditions are fulfilled for the 1 st particle (curve A) and not fulfilled for the 2nd particle (curve B). As a result, φA -coordinate (which corresponds to z-coordinate in a real space) is confined within corresponding channel.
Total electron velocity is also defined by smooth motion velocity Vi and frequent oscillations velocity d(δx)i /dt: vi = Vi + d(δx)i /dt. Initial speed defines smooth trajectory of a charged particle and might be high enough to prevent particle trapping in a bound state. If Vi > 0 and kinetic energy Ek = mVi2 /2 > αeE/(k cos θ), then the electron could not be trapped in the field with a such E-magnitude. In order to reveal these velocities V we solve the following system of equations x0 = X0 +
2eE0 mω20
z0 = Z0 −
2eE0 mω20
cos(θ) sin (kZ0 cos θ) cos (kX0 sin θ) , (6) sin θ cos(kZ0 sin θ) cos (kX0 sin θ) ,
where (x0 , z0 ) are the arbitrary initial coordinates. Finally, we derive the expression for initial value of z-velocity 2eE0 sin θ cos (kZ0 cos θ) cos (kX0 sin θ) , mω0 analysis of which in comparison with simulation results is briefly described below. vz0 = −
(7)
3. Numerical modeling To define vz0 for each electron the system of transcendent algebraic Eqs. (6) for unknown Z0 and X0 is numerically solved for initial coordinates x0 and z0 for every electron. We have used a matrix form of Newton’s method to find a solution of the system (6) because of its applicability and convergence rate [10], obtaining a complete set of initial conditions for the electron ensemble. Solution of the system obtained by means of Newton’s method has shown, that {Z0 , X0 } slightly differ from the proper initial coordinates. Indeed, the difference between vz0 (x0i , zi0 ) and vz0 (X0i , Z0i ) values is three order less then vz0 (X0i , Z0i ) where x0i and zi0 corresponds to X0i and Z0i . The latter proves the feasibility of using the initial coordinates to define vz0 projection. This velocity varies, for instance, harmonically with the amplitude about 106 cm/s for the laser field E0 ∼ 108 V/m. Thus, some electrons might have total energy lower than the height of potential barrier and, consequently, be trapped in a bound motion. This model enables us to solve the equations of motion for all the particles of a system in the fields of described configuration. The electrons, once extracted from the photocathode, are subject to the Lorentz force dp/dt = q (E + 1/c [v × H]), where v is the electron velocity and p = mγv is the momentum. 4
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Obviously, due to nonrelativistic origin of electrons their interaction with magnetic component of electromagnetic waves becomes negligible; hence, for simplicity in routine calculations we can omit the magnetic component of Lorentz force. We have used the unmodified Euler method to solve the 1-st order differential equation for p. For considered configuration of the fields in order to form the bound states the magnitude of accelerating field should be reduced to meet the condition α > 1 (or b > a) as above mentioned. Existence of bound states for known conditions is confirmed by our simulation results. Moreover, our modeling has shown that for the first 200 ps the maximum speed of particles ensemble remains three orders less than the speed of light. This fact justifies neglecting of electrons interaction with magnetic field. We examine bidimensional case (xz-plane) keeping in mind that standing wave channels have a planar structure and are perpendicular to z axis. The particles spatial distribution in xz-plane at t = 200 ps is shown in Fig. 4. As seen, in spite of z-accelerating field effect, some electrons have been trapped by the effective potential wells, and are not accelerated by applied electric field. The result totally correlates with the assumption of possibility for electrons to be trapped into bound states of described fields. The z-coordinate of trapped electrons is confined in the interval of one space half-period of standing electromagnetic wave near their initial positions. This fact well correlates with particle oscillations in a potential well. Besides, all trapped electrons might acquire nonzero x drift velocity (channeling in the field of standing waves).
Figure 4. Trapped and accelerated particles coordinates x(z) at t = 200 ps. For a better electron distribution representation we have scaled the selected region (from the ellipse to the circle) that allows revealing a periodic structure. Some of trapped electrons can move along the wave vector ks (Fig. 1), oscillating between standing wave peaks, which form an effective potential well. This behavior can be considered as electron channeling in the field of standing electromagnetic waves.
These simulation results confirm the possibility of electron binding in the field of crossed electromagnetic waves that is usually responsible for either beam modulation, or beam bunching. At specific conditions (nonzero transverse initial velocity of electron) the phenomenon of electron trapping by the field of standing electromagnetic waves could result in successful electron channeling transverse to the wave vector of standing wave (in considered case, orthogonal to electric accelerating field). Obviously, neglecting interaction between electrons ensemble in our model should be reasonable. This interaction is not only an interesting one, it may change essentially the results discussed above. Since we deal with nonrelativistic case, in evaluating the influence of electrons mutual interaction, we neglect the influence of a beam magnetic field on the electron dynamics. Thus, the electrostatic potential induced by all electrons at the point ri is defined as follows ϕi = −
N ji
e r j − ri
(8)
Assuming that electrons perform small oscillation δri around a smooth trajectory Ri , we can rewrite the potential in the following way (still supposing | δr || r |) 5
/ Nuclear Instruments and Methods B 00 (2013) 1–8
ϕi ≈ −
N ji
N e δr − δr j i R j − Ri − ≈ ϕ¯ i + δϕi , 3 R j − Ri ji R j − Ri e
6
(9)
where ϕ¯ i slowly changes in time, while a small function δϕi (| δϕi || ϕ¯ i |) is quickly oscillating. Then, the condition for bound states can be defined as α(t) =
eE02 k sin2 (θ) cos θ mω20
E+
∂ϕ(t) ¯ ∂z
−1 >1
(10)
As seen, here α depends additionally on z-coordinate, i.e. α ≡ α(z, t). Some conclusions about the condition to trap the electron may be formulated even without an explicit form of ϕ(z, ¯ t). Assuming that at every moment of time we have E > ϕ¯ z , i.e. the interaction force between electrons is less than the force of their interaction with electrostatic field, the binding condition for non-interacting electrons is fulfilled (the laser field E0 and electrostatic field E intensities satisfy the condition α > 1). Moreover, in this case the electron beam is characterized by a plane of symmetry z = z0 (t) providing the function ϕ¯ with opposite signs to the left and right of z0 (t). Hence, at every moment for half of the electrons the condition to be trapped is fulfilled. In more precise way mutual electron interaction was taken into account in numerical modeling of considered experiment (Fig. 1) that was performed by means of the KARAT simulation code [11]. The code is created for solving the problems of electrodynamics and plasma physics of high system complexity aiming at simulation of processes that take place in both electron and ion beam devices. The results acquired (Fig. 5) have shown that including Coulomb interaction between beam electrons in the evaluation still permits the standing wave channels (potentials) to bound some of the electrons.
Figure 5. The results of photo electron beam evolution simulated by means of the KARAT code. As expected, a part of electrons near the cathode surface becomes trapped by the effective potential of standing waves formed due to the interference of incident and reflected laser beams used for electron photoinjection.
Anyhow, due to the space charge effects, trapped electrons could get additional kinetic energy that allows them to easily leave a bound state. Hence, to keep electrons trapped by the effective potential, the channels are required to have higher potential barriers, and this fact qualitatively correlates with the derived expression (10). As seen, in Eq. (10) an additional term ∂ϕ(t)/∂z ¯ appears in denominator leaving possible the phenomenon of electrons trapping (as well as its successful channeling) in crossed electromagnetic waves. 6
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4. Conclusion The presented work is a result of our studies on kinetics of electron photoemission from the cathode under laser irradiation. We have analyzed the problem of electron motion in the combined field of two electromagnetic waves (crossed laser beams) and accelerating field. A condition for electron trapping in such field was derived and numerical experiments for electron beam were conducted. The condition α > 1 proved to be true for electron bound states occurrence: the electron beam can be separated into accelerated and trapped parts. The distance between neighbor maxima of trapped electron’s concentration conforms to the distance between neighbor minima of effective potential energy z = π/(k cos θ). Generally, the number of trapped particles depends on both field parameters and beam geometry in phase-space. The phenomenon has been investigated without consideration of interaction of electrons with each other, as well as with a cathode surface. The feasibility of this approach was confirmed by a simple numerical integration of the equations of motion for a single particle, and also by means of more detailed simulations, which take into account the electron mutual interaction. The influence of trapped beam part on the acceleration of the rest of electrons will take place in case of strong mutual interaction (i. e. high electron density). When the beam is separated into two parts, the first part is trapped in standing wave field, the second one is accelerated and is not influenced by the standing wave, so its emittance remains unchanged. For the first part emittance degradation is possible, but for the second emittance remains unchanged. The condition of bound state occurrence for interacting electrons case is defined in general terms. If electrons interaction force is less than electrostatic field force, that equivalent to the relation E > ϕ¯ z , the condition of bound state forming for non-interacting case α > 1 is fulfilled. Moreover, if there is a plane of symmetry for the beam, then electron would be trapped. As shown, potential well width is defined by the period of electromagnetic standing wave λE and the parameter α, while its depth depends on the intensity of a laser field U =
1 √ 2 + α − 1 − π , 2 arcsin α 2mω20 α
e2 E02 sin2 θ
(11)
where α ≥ 1. As seen, the increase of laser intensity by one order, keeping the parameters θ, ω0 , and α constant, leads to the growth of the well depth by two orders. The accelerating field could be time-dependent, meeting the condition T acc >> T l , where T acc and T l are the periods of accelerating and laser fields, respectively. The described process of electron trapping and channeling in standing wave field structures can be used for beam shaping, focusing and transportation. That can be realized by variation of the form, orientation and other parameters of standing wave region (such as a laser wavelength, its intensity and form). Planar, axial and bent channels of such origin may become important to shape charged beams. Their inner geometry may be formed by means of waveguiding or reflecting structure techniques. For instance, circular standing waves can be formed in a capillary. This is equal to creating axial concentric channels (potential wells). Moreover, bent channels can be realized via multiple reflection of a laser from a solid surface, similar to the phenomenon of x-ray propagation in capillary optical systems [12]. The process of laser wave propagation in capillaries and its interaction with the electron beam as well as the beam deflection/reflection/collimation by means of electromagnetic standing waves will be covered in next papers. Additionally, electromagnetic radiation emitted by channeled electrons in such field structures will also be a subject of our future investigation. Acknowledgements The present work benefited form the input of K.P. Artyomov (IHCE SB RAS) who provided valuable comments and conducted KARAT computer experiment. Also the authors thank M. Ferrario (LNF INFN) for his interest and assistance to the research summarized here. References [1] G. Mesyats, M. Yalandin, High-power picosecond electronics, Physics-Uspekhi 48 (3) (2005) 211–229. [2] V. Kuznetsov, Free-electron lasers, Physics-Uspekhi 22 (11) (1979) 934–938. [3] M. Ferrario, D. Alesini, Direct measurement of the double emittance minimum in the beam dynamics of the sparc high-brightness photoinjector, Phys. Rev. Lett. 99 (23) (2007) 234801.
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[4] T. Shintake, Review of the Worldwide SASE FEL Development, in: Proceedings of the 22nd Particle Accelerator Conference (PAC), Albuquerque, NM, USA, 2007, pp. 89–93. [5] A. Andreev, S. Akhmanov, Interaction of relativistic particles with intense interference optical fields, JETP 99 (1990) 1668 – 1678. [6] S.B. Dabagov, A.V. Dik and E.N. Frolov, Channeling of a free electron in a field of a standing laser wave, Preprint of LPI of RAS (15). [7] D. Gemmell, Channeling and related effects in the motion of charged particles through crystals, Rev. Mod. Phys. 46 (1974) 129–227. [8] L. Landau, E. Lifshitz, Mechanics, Russian Edition, Vol. 1 of Course of Theoretical Physics, Nauka, Moscow, 1988. [9] A. Podlesnaya, A. Dik, S. Dabagov, M. Ferrario, On electron beam motion near the sparc photoinjector cathode, Journal of Physics: Conference Series 236 (2010) 012035. [10] A. Samarsky, A. Gulin, Numerical Methods, Russian Edition, Nauka, Moscow, 1989. [11] V. Tarakanov, Users Manual for Code KARAT, Berkeley Research Associates (1992). [12] S. Dabagov, Channeling of neutral particles in micro- and nanocapillaries, Physics-Uspekhi 46 (10) (2003) 1053–1075.
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S0168-583X(13)00312-1 http://dx.doi.org/10.1016/j.nimb.2013.03.017 NIMB 59362
To appear in:
Nucl. Instr. and Meth. in Phys. Res. B
Received Date: Revised Date:
30 November 2012 14 March 2013
Please cite this article as: E.N. Frolov, A.V. Dik, S.B. Dabagov, Dynamics of Electrons Acceleration in Presence of Crossed Laser Field, Nucl. Instr. and Meth. in Phys. Res. B (2013), doi: http://dx.doi.org/10.1016/j.nimb. 2013.03.017
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
NIM B Nuclear Instruments and Methods B 00 (2013) 1–8
Dynamics of Electrons Acceleration in Presence of Crossed Laser Field Frolov E.N.a,b , Dik A.V.b , and Dabagov S.B.b,c,d a NR
Tomsk Polytechnic University, Tomsk, Russia P.N. Lebedev Physical Institute, Moscow, Russia c NRNU MEPhI, Moscow, Russia d INFN Laboratori Nazionali di Frascati, Frascati, Italy b RAS
Abstract In this work we have analyzed non-relativistic electrons’ dynamics in presence of both standing electromagnetic wave field of crossed laser beams and accelerating electrical field. Special interest has been paid to defining the conditions of charged particles’ bound states formation in such fields. We have demonstrated that a part of electron beam may be trapped by the effective potential of a standing wave. The phenomenology of electron channeling in the field of crossed laser beams is developed. c 2012 Published by Elsevier Ltd. Keywords: Nonrelativistic electron beams, laser-electron interaction, channeling PACS: 52.38.-r, 52.38.Hb, 41.75.Fr
1. Introduction Due to various possible scientific and technological applications [1, 2, 3, 4] production of trains of ultrashort electron beams is nowadays an active field of research. A common way to get such beams of high quality parameters (i. e. with a high peak current and a small both energy spread and emittance) is photoemission. But there are common problems with the length and intensity of the beams generated in such a way. Long laser pulse duration negatively influences on the beam size, and, moreover, can result in cathode sputtering. On the other hand, the number of emitted electrons is directly proportional to the time of irradiation. The main purpose of the SPARC LAB COMB experiment held at the Laboratori Nazionali di Frascati is the modulation of ultrashort electron beams during the process of their generation. Superposed incident and reflected laser beams can be used as a modulating structure in such experiments. Low-emittance beam generation as well as electron dynamics at accelerators are usually the subjects of main attention. But now, drawing a great attention of researchers, studies on space charge effects are becoming more and more urgent and topical. At the same time, the behaviours of electron ensemble in a field of crossed laser beams and accelerating electrical field (Fig. 1) are still to be examined. Such a field configuration might be formed near the cathode by both incident and reflected laser beams. Typically, the accelerating field dominates electron dynamics with respect to the electromagnetic fields present in the system. Hence, the detailed picture of electron distribution just near the cathode surface due to the presence of other fields is not of main interest. On the other hand, potentially useful and insufficiently investigated [5] phenomena, such as trapping of some electrons of the ensemble (bunch) by the potential wells formed by the superposition of optical laser fields with successful electrons’ channeling in the channels of electromagnetic standing waves, become possible under defined conditions [6]. 1
/ Nuclear Instruments and Methods B 00 (2013) 1–8
2
Figure 1. General scheme of analysed system, where k is the wave vector of a laser beam, the indexes (i, r, s), which correspond to the incident, reflected and superposed waves, respectively, and Ea is the constant accelerating electrical field. The xy-plane coincides with the cathode surface, while the z-axis is orthogonal to that surface; the y-axis is orthogonal to the picture plane. The scheme outlines the triangular region, where standing electromagnetic waves are formed. The period of electromagnetic field for standing waves is defined by λE , while standing wave channels parallel to the cathode surface are characterized by λE /2.
Indeed, the total laser field is characterized by standing waves that form potential wells. The space orientation of channels representing these wells depends on both laser beam parameters and coefficient of reflection from the cathode surface. In the ideal case of total laser beam reflection we obtain the picture of standing waves parallel to the cathode surface; in other words, standing waves are perpendicular to accelerating field. Obviously, at specific conditions a fraction of electrons might be trapped into that channels. The bound motion of such electrons are similar to planar channeling of charged particles in crystals when a particle trajectory becomes limited within the space, a channel, formed by the averaged potential well of main crystallographic planes or axes [7], while in considered case such channel is created by the superposition of two electromagnetic fields. In this paper we present a simplified analysis of such a process and prove the possibility of separation of the original electron beam into accelerated and trapped parts; the latter may channel in the direction transversal (defined by crossed laser field) to the accelerating field. 2. Electron acceleration in presence of standing electromagnetic waves Let us examine a single non-relativistic electron being accelerated by a constant electric field Ea = (0, 0, −E) in presence of two crossed laser beams (Fig. 1). For simplicity, the xy-plane coincides with the cathode surface, while the z-axis is orthogonal to the surface; hence, the electric field is directed towards negative direction of z. Both electromagnetic waves of the laser (incident and reflected) have been chosen to be polarized in xz-plane, and thus, a two-dimensional case is considered (y-axis is orthogonal to the picture plane). Crossed laser field is defined by Ei = E0 (− cos θ, 0, sin θ) sin (ω0 t − ki r) and Er = E0 (cos θ, 0, sin θ) sin (ω0 t − kr r), and let us consider the ideal case of total laser reflection from the cathode (the coefficient of reflection equals to 1). Constant accelerating field is Ea = (0, 0, −E). The bidimentional equation of motion in the combined field can be written as ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎛ ⎞ 0 ⎜⎜⎜ x¨ ⎟⎟⎟ ⎜⎜⎜ cos θ ⎟⎟⎟ ⎜⎜⎜ sin (kz cos θ) cos (ω0 t + kx sin θ) ⎟⎟⎟ ⎜⎜⎜ 0 ⎟⎟⎟ ⎟⎟⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟⎟⎠ ⎜⎜⎝ ⎟⎟⎠ + 2E0 e ⎜⎜⎝ m ⎜⎜⎝ ⎟⎟⎠ = −eE ⎜⎜⎝ (1) ⎠ cos (kz cos θ) sin (ω0 t + kx sin θ) z¨ 0 − sin θ −1 Due to the presence of high frequency electromagnetic field, the electron trajectory at defined conditions looks like frequent oscillations δxi along a smooth curve x¯i : xi = x¯i + δxi [8]. Neglecting the oscillations in the equation of motion (small by the definition, δxi ∼ eE0 /mω20 ), an effective potential energy of electron interaction in combined field can be written as follows Ue f f = −eEz +
(eE0 )2 sin2 θ cos (2kz cos θ) + U0 , mω20 2
(2)
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where U0 is the parameter for normalization, hence, can be omitted in optimized cases. For mentioned above parameters of the laser and its orientation the effective energy (2) depends only on z-coordinate. Defining α = ekE02 sin2 θ cos θ/(mω20 E) and φ = 2kz cos θ and neglecting the constant term (since it does not affect the electron movement), we introduce the following function f (φ) = (2Ue f f k cos θ)/(eE) = −φ + α cos φ
(3)
Here α ∼ A0 /Aac is the ratio of the work of a laser field to move the charge through the δx distance to the work of a constant accelerating field to move the charge through the space-period length.
Figure 2. Dimensionless potential energy f (φ) dependence on dimensionless coordinate φ with different values of α-parameter (curves A, B, C, D and E). Here θ = π/6, ω0 = 7 · 1015 c−1 ; the electrical field intensity is defined as E = 3.6 · 10−14 · E02 /α [V/m] (α 0). For E0 = 108 ÷ 1011 V/m and α = 4 we get the potential well depth of U ≈ 10−4 ÷102 eV. The space period of f (φ) is 2π, or, recalculated in units of z, is λU = π/(k cos θ) ≡ λE /2 that corresponds to the width of standing wave interaction potential.
The function f (φ) is the dimensionless effective potential energy of electron interaction in a superposed field (Fig. 2). Obviously, the appearance of electron bound states becomes possible when α > 1, while for α ≤ 1 only the electron density modulation can be observed (no bound states) [9]. It is important to note that the interaction potential formed by a standing wave can be split in two parts: the attracting potential responsible for electrons trapping (seen as a potential well mostly below the inclined rectilinear in the figure) and the reflecting potential. Obviously, the widths of these two sections of the potential depend on the parameter α, but their sum remains a constant equal to the half period of a standing wave (λU = λE /2). Performing substitution of the parameter z(t) by new one φ(t) = 2kz(t) cos θ in z-projection of a particle smooth trajectory m¨z(t) = eE + eEα sin (2kz cos θ) , (4) and defining constants b = (eE0 k sin 2θ)2 /(2m2 ω20 ), a = −2eEk cos θ/m, we get simple equation ¨ = a + b sin φ , φ(t)
(5)
that in new variables transforms the condition α > 1 into b > a. The dependence shown in Fig. 3 proves the condition b > a to be necessary, but not sufficient, condition of electron trapping. Similar curves could be achieved for two particles with equal initial velocities and different initial z-coordinates in a trapping field (b > a). For example, if all projections of initial velocity are equal to zero and the particle is positioned to be in a potential energy peak (maximum), it moves with acceleration in the direction of accelerating field. And, on the contrary, the particle being in a potential energy well (minimum) becomes bound and successfully oscillates in the well. 3
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Figure 3. Time evolution of the dimensionless coordinate φ(t). Initial conditions are the same for both particles shown, while the values of field parameters a and b differ for A and B curves. The binding conditions are fulfilled for the 1 st particle (curve A) and not fulfilled for the 2nd particle (curve B). As a result, φA -coordinate (which corresponds to z-coordinate in a real space) is confined within corresponding channel.
Total electron velocity is also defined by smooth motion velocity Vi and frequent oscillations velocity d(δx)i /dt: vi = Vi + d(δx)i /dt. Initial speed defines smooth trajectory of a charged particle and might be high enough to prevent particle trapping in a bound state. If Vi > 0 and kinetic energy Ek = mVi2 /2 > αeE/(k cos θ), then the electron could not be trapped in the field with a such E-magnitude. In order to reveal these velocities V we solve the following system of equations x0 = X0 +
2eE0 mω20
z0 = Z0 −
2eE0 mω20
cos(θ) sin (kZ0 cos θ) cos (kX0 sin θ) , (6) sin θ cos(kZ0 sin θ) cos (kX0 sin θ) ,
where (x0 , z0 ) are the arbitrary initial coordinates. Finally, we derive the expression for initial value of z-velocity 2eE0 sin θ cos (kZ0 cos θ) cos (kX0 sin θ) , mω0 analysis of which in comparison with simulation results is briefly described below. vz0 = −
(7)
3. Numerical modeling To define vz0 for each electron the system of transcendent algebraic Eqs. (6) for unknown Z0 and X0 is numerically solved for initial coordinates x0 and z0 for every electron. We have used a matrix form of Newton’s method to find a solution of the system (6) because of its applicability and convergence rate [10], obtaining a complete set of initial conditions for the electron ensemble. Solution of the system obtained by means of Newton’s method has shown, that {Z0 , X0 } slightly differ from the proper initial coordinates. Indeed, the difference between vz0 (x0i , zi0 ) and vz0 (X0i , Z0i ) values is three order less then vz0 (X0i , Z0i ) where x0i and zi0 corresponds to X0i and Z0i . The latter proves the feasibility of using the initial coordinates to define vz0 projection. This velocity varies, for instance, harmonically with the amplitude about 106 cm/s for the laser field E0 ∼ 108 V/m. Thus, some electrons might have total energy lower than the height of potential barrier and, consequently, be trapped in a bound motion. This model enables us to solve the equations of motion for all the particles of a system in the fields of described configuration. The electrons, once extracted from the photocathode, are subject to the Lorentz force dp/dt = q (E + 1/c [v × H]), where v is the electron velocity and p = mγv is the momentum. 4
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Obviously, due to nonrelativistic origin of electrons their interaction with magnetic component of electromagnetic waves becomes negligible; hence, for simplicity in routine calculations we can omit the magnetic component of Lorentz force. We have used the unmodified Euler method to solve the 1-st order differential equation for p. For considered configuration of the fields in order to form the bound states the magnitude of accelerating field should be reduced to meet the condition α > 1 (or b > a) as above mentioned. Existence of bound states for known conditions is confirmed by our simulation results. Moreover, our modeling has shown that for the first 200 ps the maximum speed of particles ensemble remains three orders less than the speed of light. This fact justifies neglecting of electrons interaction with magnetic field. We examine bidimensional case (xz-plane) keeping in mind that standing wave channels have a planar structure and are perpendicular to z axis. The particles spatial distribution in xz-plane at t = 200 ps is shown in Fig. 4. As seen, in spite of z-accelerating field effect, some electrons have been trapped by the effective potential wells, and are not accelerated by applied electric field. The result totally correlates with the assumption of possibility for electrons to be trapped into bound states of described fields. The z-coordinate of trapped electrons is confined in the interval of one space half-period of standing electromagnetic wave near their initial positions. This fact well correlates with particle oscillations in a potential well. Besides, all trapped electrons might acquire nonzero x drift velocity (channeling in the field of standing waves).
Figure 4. Trapped and accelerated particles coordinates x(z) at t = 200 ps. For a better electron distribution representation we have scaled the selected region (from the ellipse to the circle) that allows revealing a periodic structure. Some of trapped electrons can move along the wave vector ks (Fig. 1), oscillating between standing wave peaks, which form an effective potential well. This behavior can be considered as electron channeling in the field of standing electromagnetic waves.
These simulation results confirm the possibility of electron binding in the field of crossed electromagnetic waves that is usually responsible for either beam modulation, or beam bunching. At specific conditions (nonzero transverse initial velocity of electron) the phenomenon of electron trapping by the field of standing electromagnetic waves could result in successful electron channeling transverse to the wave vector of standing wave (in considered case, orthogonal to electric accelerating field). Obviously, neglecting interaction between electrons ensemble in our model should be reasonable. This interaction is not only an interesting one, it may change essentially the results discussed above. Since we deal with nonrelativistic case, in evaluating the influence of electrons mutual interaction, we neglect the influence of a beam magnetic field on the electron dynamics. Thus, the electrostatic potential induced by all electrons at the point ri is defined as follows ϕi = −
N ji
e r j − ri
(8)
Assuming that electrons perform small oscillation δri around a smooth trajectory Ri , we can rewrite the potential in the following way (still supposing | δr || r |) 5
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ϕi ≈ −
N ji
N e δr − δr j i R j − Ri − ≈ ϕ¯ i + δϕi , 3 R j − Ri ji R j − Ri e
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(9)
where ϕ¯ i slowly changes in time, while a small function δϕi (| δϕi || ϕ¯ i |) is quickly oscillating. Then, the condition for bound states can be defined as α(t) =
eE02 k sin2 (θ) cos θ mω20
E+
∂ϕ(t) ¯ ∂z
−1 >1
(10)
As seen, here α depends additionally on z-coordinate, i.e. α ≡ α(z, t). Some conclusions about the condition to trap the electron may be formulated even without an explicit form of ϕ(z, ¯ t). Assuming that at every moment of time we have E > ϕ¯ z , i.e. the interaction force between electrons is less than the force of their interaction with electrostatic field, the binding condition for non-interacting electrons is fulfilled (the laser field E0 and electrostatic field E intensities satisfy the condition α > 1). Moreover, in this case the electron beam is characterized by a plane of symmetry z = z0 (t) providing the function ϕ¯ with opposite signs to the left and right of z0 (t). Hence, at every moment for half of the electrons the condition to be trapped is fulfilled. In more precise way mutual electron interaction was taken into account in numerical modeling of considered experiment (Fig. 1) that was performed by means of the KARAT simulation code [11]. The code is created for solving the problems of electrodynamics and plasma physics of high system complexity aiming at simulation of processes that take place in both electron and ion beam devices. The results acquired (Fig. 5) have shown that including Coulomb interaction between beam electrons in the evaluation still permits the standing wave channels (potentials) to bound some of the electrons.
Figure 5. The results of photo electron beam evolution simulated by means of the KARAT code. As expected, a part of electrons near the cathode surface becomes trapped by the effective potential of standing waves formed due to the interference of incident and reflected laser beams used for electron photoinjection.
Anyhow, due to the space charge effects, trapped electrons could get additional kinetic energy that allows them to easily leave a bound state. Hence, to keep electrons trapped by the effective potential, the channels are required to have higher potential barriers, and this fact qualitatively correlates with the derived expression (10). As seen, in Eq. (10) an additional term ∂ϕ(t)/∂z ¯ appears in denominator leaving possible the phenomenon of electrons trapping (as well as its successful channeling) in crossed electromagnetic waves. 6
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4. Conclusion The presented work is a result of our studies on kinetics of electron photoemission from the cathode under laser irradiation. We have analyzed the problem of electron motion in the combined field of two electromagnetic waves (crossed laser beams) and accelerating field. A condition for electron trapping in such field was derived and numerical experiments for electron beam were conducted. The condition α > 1 proved to be true for electron bound states occurrence: the electron beam can be separated into accelerated and trapped parts. The distance between neighbor maxima of trapped electron’s concentration conforms to the distance between neighbor minima of effective potential energy z = π/(k cos θ). Generally, the number of trapped particles depends on both field parameters and beam geometry in phase-space. The phenomenon has been investigated without consideration of interaction of electrons with each other, as well as with a cathode surface. The feasibility of this approach was confirmed by a simple numerical integration of the equations of motion for a single particle, and also by means of more detailed simulations, which take into account the electron mutual interaction. The influence of trapped beam part on the acceleration of the rest of electrons will take place in case of strong mutual interaction (i. e. high electron density). When the beam is separated into two parts, the first part is trapped in standing wave field, the second one is accelerated and is not influenced by the standing wave, so its emittance remains unchanged. For the first part emittance degradation is possible, but for the second emittance remains unchanged. The condition of bound state occurrence for interacting electrons case is defined in general terms. If electrons interaction force is less than electrostatic field force, that equivalent to the relation E > ϕ¯ z , the condition of bound state forming for non-interacting case α > 1 is fulfilled. Moreover, if there is a plane of symmetry for the beam, then electron would be trapped. As shown, potential well width is defined by the period of electromagnetic standing wave λE and the parameter α, while its depth depends on the intensity of a laser field U =
1 √ 2 + α − 1 − π , 2 arcsin α 2mω20 α
e2 E02 sin2 θ
(11)
where α ≥ 1. As seen, the increase of laser intensity by one order, keeping the parameters θ, ω0 , and α constant, leads to the growth of the well depth by two orders. The accelerating field could be time-dependent, meeting the condition T acc >> T l , where T acc and T l are the periods of accelerating and laser fields, respectively. The described process of electron trapping and channeling in standing wave field structures can be used for beam shaping, focusing and transportation. That can be realized by variation of the form, orientation and other parameters of standing wave region (such as a laser wavelength, its intensity and form). Planar, axial and bent channels of such origin may become important to shape charged beams. Their inner geometry may be formed by means of waveguiding or reflecting structure techniques. For instance, circular standing waves can be formed in a capillary. This is equal to creating axial concentric channels (potential wells). Moreover, bent channels can be realized via multiple reflection of a laser from a solid surface, similar to the phenomenon of x-ray propagation in capillary optical systems [12]. The process of laser wave propagation in capillaries and its interaction with the electron beam as well as the beam deflection/reflection/collimation by means of electromagnetic standing waves will be covered in next papers. Additionally, electromagnetic radiation emitted by channeled electrons in such field structures will also be a subject of our future investigation. Acknowledgements The present work benefited form the input of K.P. Artyomov (IHCE SB RAS) who provided valuable comments and conducted KARAT computer experiment. Also the authors thank M. Ferrario (LNF INFN) for his interest and assistance to the research summarized here. References [1] G. Mesyats, M. Yalandin, High-power picosecond electronics, Physics-Uspekhi 48 (3) (2005) 211–229. [2] V. Kuznetsov, Free-electron lasers, Physics-Uspekhi 22 (11) (1979) 934–938. [3] M. Ferrario, D. Alesini, Direct measurement of the double emittance minimum in the beam dynamics of the sparc high-brightness photoinjector, Phys. Rev. Lett. 99 (23) (2007) 234801.
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[4] T. Shintake, Review of the Worldwide SASE FEL Development, in: Proceedings of the 22nd Particle Accelerator Conference (PAC), Albuquerque, NM, USA, 2007, pp. 89–93. [5] A. Andreev, S. Akhmanov, Interaction of relativistic particles with intense interference optical fields, JETP 99 (1990) 1668 – 1678. [6] S.B. Dabagov, A.V. Dik and E.N. Frolov, Channeling of a free electron in a field of a standing laser wave, Preprint of LPI of RAS (15). [7] D. Gemmell, Channeling and related effects in the motion of charged particles through crystals, Rev. Mod. Phys. 46 (1974) 129–227. [8] L. Landau, E. Lifshitz, Mechanics, Russian Edition, Vol. 1 of Course of Theoretical Physics, Nauka, Moscow, 1988. [9] A. Podlesnaya, A. Dik, S. Dabagov, M. Ferrario, On electron beam motion near the sparc photoinjector cathode, Journal of Physics: Conference Series 236 (2010) 012035. [10] A. Samarsky, A. Gulin, Numerical Methods, Russian Edition, Nauka, Moscow, 1989. [11] V. Tarakanov, Users Manual for Code KARAT, Berkeley Research Associates (1992). [12] S. Dabagov, Channeling of neutral particles in micro- and nanocapillaries, Physics-Uspekhi 46 (10) (2003) 1053–1075.
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