We propose and analyze a mechanism to accelerate protons in a low-phase-velocity wakefield. The wakefield is shock-excited by the interaction of two counter-propagating laser pulses in a plasma density gradient. The laser pulses consist of a forward-propagating short pulse (less than a plasma period) and a counter-propagating long pulse. The beating of these pulses generates a slow forward-propagating wakefield that can trap and accelerate protons. The trapping and acceleration is accomplished by appropriately tapering both the plasma density and the amplitude of the backward-propagating pulse. An example is presented in which the trapping and accelerating wakefield has a phase velocity varying from $Vph\u22480\u2009$ to $\u22480.15\u2009c\u2009\u2009\u2009(\u223c10\u2009MeV\u2009\u2009\u2009proton\u2009\u2009energy)$ over a distance of ∼1 cm. The required laser intensities, pulse durations, pulse energies, and plasma densities are relatively modest. Instabilities such as the Raman instability are mitigated because of the large plasma density gradients. Numerical solutions of the wakefield equation and simulations using turboWAVE are carried out to support our model.

Laser wakefield acceleration of electrons has proven to be a promising avenue of investigation for the production of high-energy electrons over short distances,^{1–3} with maximum energies up to several GeV having been demonstrated experimentally.^{4} A single-pulse laser wakefield in a plasma has a phase velocity of ∼*c*, and is not suitable for the acceleration of ions, because the initial ion velocity is much less than *c* and must increase by several orders of magnitude.

Current mechanisms for acceleration of ions, in particular, protons, require either large acceleration distances of multiple meters in the case of conventional RF linacs or synchrotrons,^{5} or multi-TW to PW laser systems.^{6,7} Some proposed alternatives include vacuum acceleration in a laser beat wave^{8} (a method also proposed for acceleration of electrons^{1,9}) or in a plasma wave produced by backward Raman scattering;^{10} however, these require laser pulse characteristics which are currently challenging.

The current paper proposes a mechanism for proton acceleration from close to rest up to and beyond ∼10 MeV (∼0.15 *c*) in a slow wakefield (Fig. 1). The slow wakefield is shock-excited by the beat wave generated in a plasma by two counter-propagating laser pulses. The phase velocity and amplitude of the wakefield can be appropriately controlled to permit proton trapping and acceleration. The slow wakefield has been analyzed theoretically, and modeled numerically. We present and discuss an example of proton acceleration to 10 MeV in a distance of ∼1 cm. In addition, a fluid simulation is performed (turboWAVE) showing the excitation of the slow wakefield which would allow acceleration up to 1 MeV (proof of concept). Full simulation of the acceleration process was deemed to be computationally unfeasible at this time.

It can be shown that the equation for the wakefield's electric field $EPW$ in a variable-density plasma with frequency $\omega p(z)$ driven by a ponderomotive force is^{11}

where $m$ and $q$ are the electron mass and charge, respectively. The effects of electron collisions have not been included in Eq. (1), but will be discussed later. The total normalized vector potential of the laser field is $a=qA/mec2=a0+a1,E=\u2212\u2202A/c\u2202t$, where the forward-propagating laser pulse is

and the backward-propagating laser pulse is

The wavenumbers of the pulses are $k0$ and $k1$, the frequencies are $\omega 0$ and $\omega 1$, and $\theta 0$ and $\theta 1$ are the initial phases at $z=0$. The pulse amplitude envelopes are $a\u03020$ and $a\u03021$. Written as a function of the laser pulse parameters, the normalized vector potential is $a=8.6\xd710\u221210\lambda [\mu m]I1/2[W/cm2]$.

The cross term in the product $a\u22c5a$ in Eq. (1) contains the slow-phase-velocity beat wave and excites the slow wakefield, i.e., $a\u22c5a=2\u2009a0\u2009\u22c5a1+\cdots =a\u03020(t\u2212z/c)\u2009a\u03021(t+z/c)\u2009\u2009sin\u2009\Psi (z,t)+\cdots $, where the phase of the beat wave is

where $K=k0+k1$, $\Delta \omega 01=\omega 0\u2212\omega 1$, and $\theta 01=\theta 0+\theta 1$. The forward going pulse, $|a0|<1$, is short compared to a plasma period; hence, the amplitude of the excited fast wakefield is small.

The forward-propagating short pulse with duration $\Delta \tau $ can be represented by $a\u03020=a0\u2009\Delta \tau \u2009\delta (t\u2212z/c)$. Equation (1) has the forward-propagating solution $EWF(z,t)=(\omega p(z)/\omega 0)E\u0302\u20090(z)\u2009sin\u2009\Phi (z,t)$, where

and

The delta-function representation for $a\u03020$ is valid as long as $|\omega p(z)\u2212\Delta \omega 01|\Delta \tau \u226a1$.

The phase velocity of the slow wakefield is

where $Kp(z,t)=K\u2212\Delta \omega 01/c+\omega p(z)/c\u2212(\u2202\omega p/\u2202z)(t\u2212z/c)$ and $K\u2212\Delta \omega 01/c=2k1$. The phase velocity depends only on the characteristics of the backward-propagating laser pulse and the plasma density gradient. In a positive density gradient, the phase velocity increases as a function of time.

The equation of motion for a proton at position $z(t)$ is

where the first term on the right-hand side is the force due to the wakefield, the second term is the vacuum ponderomotive force of the laser fields, and *M* is the proton mass. The vacuum term provides no net gain of energy in the absence of the short pulse.

The phase $\Phi (z(t),t)$ (Eq. (4b)) of a resonant proton will oscillate about a resonant phase $\Phi R$. If the wave's phase velocity is changing, the acceleration $dVz/dt$ of the proton must be equal to the acceleration of the wave, evaluated at the position of the proton: $dVz/dt=dVph/dt=\u2202Vph/\u2202t+Vz\u2202Vph/\u2202z$. The resonant phase is given by

Since $|\u2009sin\u2009\Phi R|\u22641$, Eq. (7) places requirements on the plasma density gradient and wakefield amplitude.

When the proton is close to resonance, its phase can be expressed as a small deviation about the resonant phase, $\Phi (t)=\Phi R+\delta \Phi (t)$. The pendulum-like equation for this phase deviation is

where $\Omega 02(z,t)=\u2212Kp(z,t)(|q|/M)(\omega p(z)/\omega 0)E\u03020(z)\u2009cos\u2009\Phi R$ and $\Omega 1(z,t)=(2/Kp(z,t))(\u2202\omega p(z)/\u2202z)$. For the proton to remain trapped (resonant) and accelerated, the amplitude of the oscillations $\delta \Phi (t)$ should remain small. From Eq. (8), this implies $\Omega 02(z(t),t)>\Omega 12(z(t),t)/4$, $E\u03020<0$, $sin\u2009\Phi R<0$ and $cos\u2009\Phi R>0$, assuming the parameters (frequencies) $\Omega 0$ and $\Omega 1$ vary slowly in time.

The stability conditions implied by Eq. (8) introduce a lower bound on the amplitude of the wakefield. A conservative upper bound is the wave-breaking field^{12}

which is the wave-breaking field of a fast, single-pulse laser wakefield^{2} reduced by the factor $Vph/c$. This upper bound in the wakefield amplitude translates into a limit on the acceleration of the proton, and therefore places a constraint on the plasma density profile. The limit is represented by the inequality $\Omega 1(z(t),t)/\omega p(z(t))\u2272me/M$.

The electron-ion collision frequency is $\nu ei\u2009\u2009[\u2009s\u22121]\u224810\u22125n0[cm\u22123]/Te3/2[eV]$, where $Te$ is the effective temperature of the electrons.^{13} For electrons in a laser field with normalized amplitude *a*, the effective electron temperature is due to the electron quiver velocity. The effective temperature is $Te\u2009\u2009[eV]\u2009\u2009\u223ca2mec2$, and the characteristic collision time, which can be treated as a characteristic time for the damping of the wakefield, is $\tau ei\u2009[s]=1/\nu ei\u22483.64\xd71013a3/n0\u2009[cm\u22123]$. If collisions are included in Eq. (1), a term $\u22124\pi |q|n0\nu ei\delta \upsilon z$ appears on the right-hand-side, where $\delta \upsilon z$ is the axial electron fluid velocity. This term can be neglected if $\nu ei\u226a\omega p(z)$, a condition that is satisfied in our example. In the absence of an external laser field, once the long backward-going pulse has propagated out of the interaction region, the wakefield will be damped. In this case, it may be necessary to include a third laser pulse to extend the damping time.

Instabilities such as the Raman instability can be driven in the acceleration region. This can amplify the forward-propagating short pulse (Backward-Raman-Amplification^{14}). The three-wave instability can also grow from noise via the interaction of the long backward-propagating pulse with the plasma. The growth rate and the condition for suppression for both of these is the same. The stimulated Raman scattering instability will be suppressed in an inhomogeneous plasma if the e-folding length of the instability is longer than the characteristic gradient of the plasma density. The approximate condition for suppression of the instability is^{15}

where the growth rate is $\Gamma 0=a\u03021(t+z/c)\u2009\u2009(\u2009\omega p(z)\u2009\u2009\omega 1/4)1/2$.^{16} In the examples presented in the below paragraph, this condition is easily satisfied in regions where the wakefield has already been excited, but is only marginally satisfied elsewhere. It may be necessary to include a chirp on the long pulse to further suppress growth of the instability from noise, and/or to modify the ratio of pulse amplitudes. Similarly, amplification of the short pulse was observed in simulation in the region where the frequency-matching condition was approximately satisfied, but may be suppressed or amplified depending on the chirp on the backward-propagating pulse. The chirp may also provide an additional control on the phase velocity of the slow wakefield.

To illustrate the acceleration mechanism, a proton test charge can be placed in the analytically derived forward-going slow wakefield. Figs. 2(a) and 2(b) show the proton energy and distance for an accelerated test charge with an initial energy of 10 keV. This energy would require injection of protons rather than trapping from resonant background protons. Injection might be accomplished by direct ponderomotive acceleration by a laser beat wave, for example.^{1,8,9}

The laser parameters for this example correspond to a 5 GW long pulse (828 nm for ∼80 ps) and 1 TW short pulse (800 nm for 20 fs). Both beams are assumed to have a 50 *μ*m spot size, with a Rayleigh length longer than the acceleration distance. For these parameters, the long pulse energy is ∼400 mJ and the short pulse is ∼30 mJ. The laser and plasma density profiles in this example were chosen so that the accelerating field does not significantly exceed wavebreaking, but has an amplitude sufficient to trap protons. The plasma density was chosen to increase quadratically from ∼10^{17 }cm^{−3} to ∼5 × 10^{18 }cm^{−3} over a distance of ∼1 cm. Trapping and acceleration of resonant protons occurs over many hundreds of wavelengths of the slow wakefield and in a large fraction of each wavelength. The final energy of the protons depends on the particular wavelength in which they are trapped.

A full-scale particle in cell (PIC) simulation of the acceleration of protons to ∼MeV energies in a slow wakefield was deemed computationally unfeasible. As a preliminary proof-of-concept, simulation in a fluid model of the excitation of a slow wakefield in a density gradient has been performed, with parameters shown by analysis to be suitable for proton acceleration from 50 keV at $z=0.33\u2009cm$ to ∼1 MeV at $z=0.75\u2009cm$. The electric field of the wakefield is shown in Fig. 3. PIC protons were not placed in the wakefield, as acceleration would take much longer than excitation of the accelerating field.

A mechanism for the acceleration of protons in a laser wakefield has been proposed, which could allow energies up to 10 MeV in a short distance ∼1 cm. The analysis presented here is performed in a one-dimensional limit. This mechanism has the potential to produce high-quality quasi-monoenergetic proton bunches of low emittance. This is because in the linear regime protons can undergo transverse focusing.^{2} The controlled production of high-density gas jets has also been demonstrated experimentally.^{17} One potential application for protons at this energy is the generation of short-lived radioisotopes for use in Positron Emission Tomography (PET). Current production of these radio-pharmaceuticals is limited and expensive. The approximate requirement for this is an average flux 10^{12} protons/s, with energy greater than 5 MeV.^{18,19} For the example given in this paper of acceleration up to 10 MeV, an estimation assuming a $50\u2009\mu m$ laser spot size gives a required repetition rate in the kHz range.

There are several issues and challenges which might impact the proposed mechanism. One of these is the collisional damping of the slow wakefield. Damping can be mitigated by the introduction of a long third laser pulse such that the characteristic collision time is on the ns time scale. For a plasma density $\u2272\u20095\xd71018\u2009cm\u22123$, this requires a normalized pulse amplitude $a>0.05$, equivalent to, for example, a 1 J pulse of a transversely excited atmospheric (TEA) CO_{2} laser.

Another issue to consider is the possible presence of Raman instabilities in the laser plasma. One such instability can be excited by the long backward-going pulse. Raman instabilities can also amplify the forward-going pulse, via Backward Raman Amplification. This will increase the amplitude of the wakefield, which could result in loss of trapping and/or extreme wavebreaking. Although this effect is difficult to analyze, it can be corrected for in experiments by the appropriate choice of pulse amplitude profile on the backward-going wave. The increase in pulse amplitude may also prove useful for exciting the slow wakefield with a lower-intensity pulse than might otherwise be required.

The authors acknowledge useful discussions with A. Ting, H. Milchberg, D. Papadopoulos, and L. Johnson. This work was supported by a DOE Grant, Award No. DE-SC0015516. The authors also thank D. Gordon and NRL for use of the turboWAVE code.