ISSN 1054-660X, Laser Physics, 2006, Vol. 16, No. 9, pp. 1311–1314.
FREE-ELECTRON LASERS AND LASER ACCELERATORS
© MAIK “Nauka /Interperiodica” (Russia), 2006. Original Text © Astro, Ltd., 2006.
Acceleration of Free Electrons in a Symmetric Evanescent Wave B. R. Frandsen, S. A. Glasgow*, and J. B. Peatross Department of Physics and Astronomy, Brigham Young University, Provo, UT, 84602 USA e-mail:
[email protected] Received December 31, 2005
Abstract—The possibility of accelerating free electrons in a vacuum gap between closely spaced dielectric materials is explored. Plane waves impinging symmetrically on the gap from either side at oblique incidence produce an evanescent wave with net electric field along the direction of propagation. Near the critical angle, the evanescent wave propagates at the vacuum speed of light. A theoretical development and numerical simulations show that free electrons in the gap can be accelerated and accumulate energy indefinitely. This approach lies outside the purview of the Lawson–Woodward theorem, which does not apply in the vicinity of a medium. Damage thresholds of materials restrict the light intensity to far below that achievable by current high-power lasers. This limits the particle energy that might be achieved from an accelerator based on this approach. PACS numbers: 41.75.Jv,42.25.–p, 42.68.Ay DOI: 10.1134/S1054660X06090040
*
1. INTRODUCTION Significant effort has gone into using intense laser fields to accelerate charged particles. Advances in highpower laser technology offer a potentially convenient alternative to conventional particle accelerators. Tabletop lasers can reach extraordinary electric fields. For example, an intensity of 1021 W/cm2, achieved in a tight laser focus [1], produces an electric field of a teravolt per centimeter. This would seem to be a very attractive candidate for accelerating bare charged particles except that the field is oscillatory, making a sustained buildup of kinetic energy for an accelerated particle unattainable. The limitations to using laser fields for accelerating charged particles have been well documented both theoretically and experimentally. The Lawson–Woodward Theorem [2, 3] enumerates general conditions under which electromagnetic radiation cannot be used for prolonged acceleration of charged particles. They are the following. (1) Nonlinear effects such as the power radiated by the accelerating charge and the ponderomotive force are negligible. (2) The particle is in vacuum and does not interact with other particles. (3) No static electric or magnetic fields are employed. (4) No refractive or reflective medium is present. Taken together, these conditions imply that the charge to be accelerated is far (compared to the wavelength) from any source of radiation. In this far-field regime, the field may be expressed as a superposition of plane-wave solutions to Maxwell’s equations. In this case, the Lawson–Woodward theorem states that a charge interacting with an electromagnetic field over an * Department of Mathematics.
infinite distance will experience no net acceleration. For a finite pulse, if the interaction between the charge and the field is restricted to a fraction of the pulse, the charge may end up with energy on the order of that gained in one half cycle. The disappointing limitations imposed by the Lawson–Woodward theorem arise from the transverse nature of electromagnetic plane waves and the fact that waves propagate at the maximum speed of any particle, namely, the speed of light in vacuum. One scheme that has been proposed [4] to overcome the transverse nature of electromagnetic waves is the use of a pair of crossed beams. While this setup has the advantage of a longitudinal field component and a vanishing on-axis transverse Field component, the phase velocity of the field at the point of intersection is greater than c. The accelerating particle inevitably falls behind and over an entire pulse cannot gain net energy. The crossed-beam setup lies within the purview of the Lawson–Woodward theorem and therefore is not a viable acceleration scheme [5]. Current methods for using laser radiation to accelerate electrons depend on one or more of the conditions of the Lawson–Woodward theorem being circumvented. For example, plasma wake field accelerators exploit the presence of free charges [6]. Recent experiments have produced energy gains up to the gigaelectronvolts range [7–11], which is the leading technology among laser-driven accelerating techniques. The nonlinear ponderomotive force is exploited in the so-called vacuum beat wave acceleration scheme [5]. By relaxing the assumption of no static fields present, acceleration by means of the inverse free electron laser is possible [12]. This is based on an external oscillating magnetic “wiggler” field that changes direction as the particle slips with respect to the phase of the electromagnetic field. Experimental results for the
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θ
This symmetric arrangement eliminates (near the center) the component of the evanescent field perpendicular to the surfaces, which would be present in the case of only one surface. P-polarized laser radiation is incident on the surface separating the vacuum gap from the refractive medium from both the left and the right at angle θ. If θ exceeds the critical angle, θcrit = sin–1(1/n), then total internal reflection occurs at each surface, and evanescent waves are set up within the vacuum gap. We take the positive z axis to be to the right in Fig. 1 and the positive y axis to be upwards. The electric field inside the gap set up by the incident symmetric plane waves is found to be E0 t k ( z – d )ξ – kzξ - { yˆ iξ [ e +e ] E gap = --------------------– kdξ 1 – re
Fig. 1. Diagram of vacuum-gap evanescent wave accelerator that uses P-polarized radiation.
– zˆ n sin θ [ e inverse free electron laser have shown final energies of 30–50 MeV, although injection energies of about 15 MeV were needed [13]. Finally, a scheme known as the grating accelerator makes acceleration possible by introducing the presence of a refractive index [14]. In this paper, we investigate laser acceleration of charged particles also in the vicinity of a refractive medium. As discussed above, the Lawson–Woodward theorem docs not apply. We examine the field in a narrow gap between two semi-infinite dielectric regions, which arises from identical plane waves impinging from either side at oblique incidence. Beyond the critical angle, the field in the gap is evanescent but is influenced by the dual boundaries. These evanescent waves are exploited in simulations of particle acceleration. In the next section, we present a theoretical development of the fields present in the gap. In Section 3, we provide results of numerical simulations of electron trajectories in the gap. Simulations show that the field produced by this arrangement can accelerate an electron indefinitely. However, damage thresholds of materials limit the maximum intensity of the incident radiation to below 1012 W/cm2. This means that the maximum energy gained per distance is bounded by 10 MeV/cm. 2. FORMULATION OF VACUUM-GAP EVANESCENT WAVE In the vicinity of a refractive medium, evanescent waves can exist, which die off exponentially away from the surface of the medium. Evanescent waves differ from electromagnetic plane waves in that they travel slower than c, so that particles have the chance to “keep up” or stay in phase with the wave. They also have a longitudinal component ideal for accelerating charged particles. The vacuum-gap evanescent wave scheme consists of a region of vacuum sandwiched by regions of refractive medium with real index n on either side (see Fig. 1).
– kzξ
–e
k ( z – d )ξ
(1)
] } exp { i ( nky sin θ – ωt ) },
where 2
2
ξ ≡ n sin θ – 1 .
(2)
This expression is particularly useful for super-critical angles where the parameter ξ is real. The single-boundary Fresnel coefficients are given by [15] 2n t = -------------- , 1 + iγ
–1
r = e
– 2i tan γ
and
(3)
where nξ γ ≡ ------------ . cos θ
(4)
3. PROPERTIES OF VACUUM-GAP ELECTRIC FIELD An electromagnetic field is an attractive candidate for accelerating charged particles if (1) the force has a longitudinal component, (2) it has the possibility of propagating at speeds less than c, and (3) the field does not tend to push the charged particle laterally out of the accelerating region. The wave described by Eq. (1) propagates in the y direction and has a component in that direction. The speed of the wave is v = c/(n sin θ), which at supercritical angles is below c. Its speed approaches c as θ approaches the critical angle. Thus, the first two criteria are satisfied by the electric field in the gap. We next turn our attention to the z component of the electric field to assess whether electrons tend to be pushed into the sides of the gap while being accelerated longitudinally. Evaluating (1) at z = d/2 reveals that, in the center of the gap, the z component of the electric field vanishes while the y component does not. Thus, for an electron on axis, there is no lateral field. Analysis of Eq. (1) shows that, left of the center of the gap (i.e. z < d/2), the z component of the field lags the y component by 90°. To the right of the center (i.e., z > d/2), the z component of the field leads the y comLASER PHYSICS
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ACCELERATION OF FREE ELECTRONS IN A SYMMETRIC EVANESCENT WAVE
ponent by 90°. Owing to this phase difference, the z component of the field is identically zero when the y component is maximum, and this is independent of the value of z. Thus, at this optimal phase, charge particles are not pushed laterally even if they are not centered. Ideally, one would like charged particles to surf slightly in front of where the y component of the field maintains its maximum. In this case, if a charged particle moves too slowly, the y component of the field increases. On the other hand, if the particle moves too fast, it experiences a weakened field. This scenario would facilitate the formation of particle bunches. Unfortunately, the phase of the z component of the field is such that, if the charged particle is ahead of where the y component of the field is maximum, the particle is pushed towards the closer wall. This is true regardless of the sign of the particle’s charge; electrons are appropriately accelerated in the negative phase of the cycle. Although this result is not favorable for the acceleration scheme explored in this paper, we will see in the next section that the magnetic field in large part compensates for this problem of instability in the z component. On the other hand, if a charged particle moves just behind where the y component of the field is maximum, the z component of the field is such that it pushes the particle away from the closer wall. Finally, we examine the field when the incident light is exactly at the critical angle. In this case, Eq. (1) reduces to 4nE 0 - [ yˆ i – zˆ k ( d/2 – z ) ] E gap = -----------------------------------------2 2 kd + 2in / n – 1 × exp { i ( ky – ωt ) }.
(5)
As was mentioned above, the wave is seen to propagate in the y direction at the speed of light ω/k = c. As before, the z component of the electric field vanishes in the center of the gap while the y component does not. An interesting feature is that the strength of y component of the field decreases with the gap thickness kd. To maintain the strongest possible accelerating field, one should keep the gap width small, on the scale of a wavelength. 4. PROPERTIES OF VACUUM-GAP MAGNETIC FIELD As the speed of the electron approaches c, the force on the electron due to the magnetic field becomes nonnegligible. In this section, we describe the magnetic field and examine how the magnetic force affects the equilibrium and stability of acceleration in the gap. The magnetic field associated with (1), as dictated by Maxwell’s equation — × E = –∂B/∂t, is 1 B = xˆ ----------------- E z . cn sin θ LASER PHYSICS
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At the critical angle, the magnetic field becomes Bcrit = xˆ E z /c. If the electron moves at a velocity close to c in the y direction, the force exerted on it due to the magnetic field is – ecyˆ × B = zˆ eE z /(nsinθ), which is equal and opposite to the z component of electric force. (The charge of the electron is written as –e.) The magnetic field tends to remove the instability in the z component of the field for particles ahead of the peak y component of the field. For charged particles behind the peak y component of the field, the magnetic field tends to work against stability. However, since particles will always have a speed somewhat less than c, the magnetic force will not be able to completely counteract the electric force. In summary, from an analytical perspective, the electromagnetic field in a vacuum gap between two regions of a refractive medium due to incident radiation at and beyond the critical angle exhibits favorable characteristics for electron acceleration. First, the speed of the wave can be less than c and can therefore be tuned to the electron’s speed. Second, the field has a component in the direction of propagation. Third, the total lateral force on a charged particle traveling near c tends to vanish. 5. VACUUM-GAP EVANESCENT WAVE ACCELERATOR: NUMERICAL SIMULATIONS In the previous section, we saw that the electromagnetic field in the vacuum gap due to incident radiation at the critical angle and beyond seems promising as a means of accelerating electrons to high energies. In this section, we numerically simulate an electron’s trajectory in the field. The equation of motion for a relativistic charged particle in an electromagnetic field can be written as [16] 2 du' ------- = – 1 – u' [ E' + u' × B' – u' ( u' ⋅ E' ) ], dt'
(7)
where we have employed the following dimensionless variables: t' ≡ ωt, r' ≡ kr, u' ≡ u/c, E' ≡ eE/ ( cωm e ),
(8)
B' ≡ eB/ ( ωm e ). Here, u is the ordinary velocity of the electron, ω is the angular frequency of the incident monochromatic field, and k = ω/c is the magnitude of the associated wave vector in the absence of any medium. We recast (7) as a system of first-order differential equations and used a Runge–Kutta method to solve them.
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55 MeV obtained in approximately 10 cm. Therefore, in order for an electron to reach the gigaelectronvolts range, several meters of total acceleration distance would be necessary.
Velocity, c 1.0000 0.9998
REFERENCES
0.9996 0.9994 0.9992 0.9990 0.9988 0.9986
0
2
4
6
8 10 Time, ωt × 105
Fig. 2. y component of velocity.
The limiting factor for an accelerating field is the damage threshold of the refractive medium, which we set (optimistically) to ~1012 W/cm2. The results of the simulation of this scenario are shown in Fig. 2. The final energy of the electron after being accelerated through a distance of about 10 cm was 55 MeV. The simulation shows very little if any lateral motion, so that the electron never nears the side of the gap. The plot of the y component of the velocity confirms that the electron stays in phase with the wave, since the speed never decreases. The initial velocity was set at 0.85 s. 6. CONCLUSIONS The Lawson–Woodward theorem indicates that charged particles cannot be directly accelerated in a free laser field to energies above the quiver energy obtained in a half cycle of the field. However, if one or more conditions of the Lawson–Woodward theorem is relaxed, accumulated acceleration is possible. In particular, bringing a refractive medium into the vicinity of the charged particle introduces evanescent-style waves into the accelerating field. Evanescent waves are attractive candidates for acceleration, since they can propagate slower than c and they have longitudinal field components. The field structure allows for stable-equilibrium trajectories in the center of the gap. However, the materials’ damage threshold limits the intensities that can be used. With an incident intensity near damage thresholds, a simulation showed electron energies up to
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