Signatures of the Carrier Envelope Phase in Nonlinear Thomson Scattering

Abstract

High-energy radiation can be generated by colliding a relativistic electron bunch with a high-intensity laser pulse—a process known as Thomson scattering. In the nonlinear regime the emitted radiation contains harmonics. For a laser pulse whose length is comparable to its wavelength, the carrier envelope phase changes the behavior of the motion of the electron and therefore the radiation spectrum. Here we show theoretically and numerically the dependency of the spectrum on the intensity of the laser and the carrier envelope phase. Additionally, we also discuss what experimental parameters are required to measure the effects for a beamed pulse.

1. Introduction

Ultra-short laser pulses, for which the temporal length is of the order of the wavelength, are effective tools to measure ultra-fast events or can be used for nonlinear light–matter interactions [1,2,3]. For the latter, the exact shape of the field is important, i.e., the phase of the oscillations with respect to the envelope (carrier envelope phase; ${\varphi }_{cep}$) and is therefore important to control in experiments. For intensities up to ${10}^{14}$–${10}^{15}$ W/cm${}^2$, methods exist to measure ${\varphi }_{cep}$ by utilizing ionization processes [4,5,6,7]. In relativistic laser pulses, the carrier envelope phase cannot be distinguished through ionization; however, other light–matter interactions are susceptible to ${\varphi }_{cep}$ and could be utilized for measuring it. One such interaction is Thomson or Compton scattering, the conversion of low- to high-energy photons through the collision with charged particles. The former can be described using classical electrodynamics when the collision is elastic, i.e., the recoil of the charge is negligible ($\chi =\frac{\gamma \hslash {\omega }_l}{m_e{c}^2}\ll 1$ where $\gamma$ is the electron’s Lorentz factor, ℏ is the reduced Planck constant, ${\omega }_l$ is the frequency of the laser pulse, $m_e$ is the mass of the electron and c is the speed of light). In this regime the process has been studied with regard to the laser strength parameter $a_0$, the shape of the laser pulse both longitudinally and transversely [8,9,10,11,12], the bandwidth of the emitted radiation [13,14] and methods to reduce it [15]. In [16], it was shown that the phase of the field of the emitted radiation is linked to that of the laser pulse. Moreover, [17] found numerically that Thomson or Compton scattering can be used as a method to measure the laser pulse parameters in the plane wave limit by comparing the difference in emission angles. Here we show on an analytical basis how ${\varphi }_{cep}$ is visible in the spectrum and what the experimental requirements are for its measurement.

2. Thomson Scattering

In the classical regime an accelerating charge emits radiation, or alternatively the radiation is fully described by the motion of the charge. This description is called the Lienard–Wiechert potential, for which the intensity per unit frequency and steradian is given by [18].

$$\frac{d^{2}I}{d\omega d\Omega }=\frac{e^2}{4{\pi }^{2}c}{\left|\omega \sum _{i=0}^{N_e}{\int }_{-\infty }^{\infty }dt\widehat{n}\times \widehat{n}\times {\overrightarrow{\beta }}_{i}exp\left[i\frac{\omega }{c}\left(ct-\widehat{n}·{\overrightarrow{r}}_i\right)\right]\right|}^2,$$

(1)

where e is the electric charge, $\widehat{n}$ is the unit vector from the source to the detector and in spherical coordinates it is given by $\widehat{n}=(cos\left(\phi \right)sin\left(\vartheta \right),sin\left(\phi \right)sin\left(\vartheta \right),cos\left(\vartheta \right))$, $N_e$ is the total number of electrons, and ${\overrightarrow{\beta }}_i$ and ${\overrightarrow{r}}_i$ are the velocity and trajectory of the $i^{\mathrm{th}}$ charge. The motion of the electron is then calculated according to the Lorentz force

$$\frac{d{U}^{\mu }}{ds}=-\frac{e}{m_e{c}^2}{F}^{\mu \nu }{U}_{\nu }=-({\partial }^{\mu }{a}^{\nu }-{\partial }^{\nu }{a}^{\mu })U_{\nu },$$

(2)

where $U^{\mu }\equiv \frac{d{X}^{\mu }}{ds}=\gamma \left(\begin{array}{c}1\\ \overrightarrow{\beta }\end{array}\right)$, $F^{\mu \nu }$ is the electromagnetic field tensor and $a^{\mu }=\frac{e{A}^{\mu }}{m_e{c}^2}$ is the normalized vector potential depending on the four position of the particle. We define a linearly polarized laser pulse as

$$\overrightarrow{a}=a_0\mathcal{E}\left(\zeta \right)\Psi \left(\overrightarrow{r}\right)cos\left(\zeta +{\varphi }_{cep}\right)\widehat{x},$$

(3)

where $\mathcal{E}$ and $\Psi$ are the temporal and spatial envelope functions, respectively, and $\zeta =k_l(ct-z)$. For the envelope functions we use the following normalization: $\mathcal{E}\left(0\right)=1$ and $\Psi \left(0\right)=1$. A well-suited temporal profile for a pulse with temporal length of the order of $N_c$ is $\mathcal{E}\left(\zeta \right)=\sec \mathrm{h}\left(\frac{\zeta }{{\sigma }_l}\right)$[19] where ${\sigma }_l=\frac{\pi N_c}{log(2+\sqrt{3})}$ is the length of the pulse. With this definition the FWHM of the laser pulse is an integer number times its wavelength.

Given that with our definition the laser pulse travels in $+\widehat{z}$, the angle for backscattered radiation is $\vartheta =\pi$. For numerical simulations we use a particle tracker based on the Vay method [20] to obtain the trajectory of the electron, which is subsequently used to calculate the spectrum according to Equation (1).

2.1. Single Electron—Plane Wave Dynamics

In the plane wave approximation (Equation (3) with $\Psi =1$) the motion of the electron has an exact analytical solution

$$U^{\mu }=\left(\begin{array}{c}\gamma +\frac{{\left(a\right)}^2}{2}\gamma (1-\beta )\\ a^1\\ a^2\\ -\gamma \beta +\frac{{\left(a\right)}^2}{2}\gamma (1-\beta )\end{array}\right).$$

(4)

Figure 1 shows the effect of ${\varphi }_{cep}$ on the motion of the electron. The electric field of the emitted radiation will have the imprint of ${\varphi }_{cep}$[16]; however, this does not lead directly to a visible effect on the spectrum. We must then examine Equation (1) where the electron motion is affected by ${\varphi }_{cep}$. Substitution of Equation (4) into Equation (1) and setting $\vartheta =\pi$ leads to [10,11,21].

$$\frac{d^{2}I}{d\omega d\Omega }=\frac{e^2}{4{\pi }^{2}c}{\left|\nu {\sum }_{m=-\infty }^{\infty }\frac{e^{i(2m+1){\varphi }_{cep}}}{2}{\int }_{-\infty }^{\infty }d\zeta a_0\mathcal{E}\left(\zeta \right)\left[{\mathcal{J}}_m\left(B\right)+{\mathcal{J}}_{m+1}\left(B\right)\right]exp\left[i{\int }_{-\infty }^{\zeta }d{\zeta }^{\prime }(2m+1)+\nu \left(1+\frac{a_0^2{\mathcal{E}}^2\left({\zeta }^{\prime }\right)}{2}\right)\right]\right|}^2,$$

(5)

where ${\mathcal{J}}_m$ is the Bessel function of the first kind and $B=\nu {\left(\frac{a_0\mathcal{E}\left(\zeta \right)}{2}\right)}^2$ and $\nu =\frac{\omega }{{(1+\beta )}^2{\gamma }^2{\omega }_l}$ is the normalized emitted frequency. Using the stationary phase approximation, the frequencies where the peak intensities of the harmonics are located are given by [10,11,21]

$${\nu }_{\mathcal{H}}\left(\zeta \right)=\frac{\mathcal{H}}{1+\frac{a_0^2{\mathcal{E}}^2\left(\zeta \right)}{2}},$$

(6)

where $\mathcal{H}=2m+1$ is the harmonic order. The frequency is red-shifted due to the drift velocity ($\langle U^3\rangle$) of the nonlinear interaction as shown in Figure 1, right panel.

Quantitatively, Equation (5) reads as $|{\sum }_m{f}_m{|}^2={\left|{\sum }_{\mathcal{H}}{f}_{{\nu }_{\mathcal{H}}}\right|}^2$ where $f_{{\nu }_{\mathcal{H}}}$ represents the contribution to the spectrum of a single harmonic order. For low values of $a_0$ the harmonics are separated, as is shown in the left panels of Figure 2, and therefore the solution of Equation (5) can be read as $|f_{{\nu }_1}{|}^2+{\left|f_{{\nu }_3}\right|}^2+\cdots$. In this case, ${\varphi }_{cep}$ appears as a global phase and thus does not influence the spectrum. If we increase $a_0$, the lower bound of the emitted frequency decreases (${\nu }_{\mathcal{H}}\left(0\right)$), yet the upper bound remains the same (${\nu }_{\mathcal{H}}(\pm \infty )$). Therefore, for a given value of $a_0$, frequencies from different harmonic orders, emitted at different times of the interaction, will start to overlap and can be calculated using

$$\begin{array}{cc}\hfill 0=& {\nu }_{{\mathcal{H}}_i}(\zeta =\pm \infty )-{\nu }_{{\mathcal{H}}_j}(\zeta =0),\hfill \\ \hfill a_{0\Delta \mathcal{H}}=& \sqrt{2\frac{{\mathcal{H}}_j-{\mathcal{H}}_i}{{\mathcal{H}}_i}},\hfill \end{array}$$

(7)

where ${\mathcal{H}}_i$ is a lower harmonic than ${\mathcal{H}}_j$. For example, the third harmonic starts to overlap with the first when $a_0=2$, as can be seen in Figure 2. In this case, the solution of the spectrum can be read as $|f_{{\nu }_1}{|}^2+2\mathcal{R}e\left(f_{{\nu }_1}{f}_{{\nu }_3}^{*}\right)+{\left|f_{{\nu }_3}\right|}^2+\cdots$, where $f^{*}$ is the complex conjugate. The cross term is the interference of the two different harmonic orders and depends on ${\varphi }_{cep}$. Due to the interference, the peak intensity of a harmonic shifts and scales with

$${\nu }_{\mathcal{H},cep}\propto \frac{\mathcal{H}}{1+\frac{a_0^2}{2}}\left(1\pm \frac{{sin}^2\left({\varphi }_{cep}\right)}{\pi N_c}\right),$$

(8)

as is shown in Figure 3. We can see that the spectrum is symmetric around ${\varphi }_{cep}=\frac{\pi }{2}$, thus there is not a unique solution to the frequency shift due to ${\varphi }_{cep}$ (e.g., $\Delta {\nu }_{\mathcal{H},cep}({\varphi }_{cep}=\frac{\pi }{4})$ = $\Delta {\nu }_{\mathcal{H},cep}({\varphi }_{cep}=\frac{3\pi }{4})$). Therefore, the interval for which ${\varphi }_{cep}$ can be determined from a given spectrum is $[0,\frac{\pi }{2}]+k\frac{\pi }{2}$, where k is an integer.

The cone of emission depends on the energy of the electron and for the fundamental Thomson line scales with $\vartheta \sim \pi -\frac{1}{\gamma }$. To include the higher harmonics, the range of the acceptance angle needs to be larger, namely $\pi \left(1-\frac{3}{4\gamma }\right)\le \vartheta \le \pi$. The resolution of ${\varphi }_{cep}$ can be greatly improved by choosing low electron energy. Consequently, the emitted frequency is also low energy, an order of magnitude larger than the laser frequency, improving diagnostics. The effect of the on-axis radiation can be generalized for emissions in all directions, i.e., ${\varphi }_{cep}$ has a distinct effect on the spectrum when harmonics overlap, as is shown in Figure 4.

2.2. Electron Bunch—Plane Wave Laser Pulse

The choices of electron parameters are $N_e={10}^3$, (normalized emittance) $ϵ={10}^{-6}$ mm mrad and $\frac{{\sigma }_{\gamma }}{\gamma }={10}^{-3}$ (rms). The radiation emitted is collected in a fraction of the emission cone in order (${\vartheta }_{max}=\frac{1}{10\gamma }$). This is required due to the angular dependency of the emitted radiation, as Figure 4 indicates. In Figure 5 the ${\varphi }_{cep}$ dependency is clearly visible. For higher values of $a_0$ the peaks of the harmonics are more downshifted and are spaced closer together (Equation (6)). However, the effect remains visible for $a_0=5$ when $\mathcal{H}⩾20$.

2.3. Electron Bunch—Beamed Laser Pulse

Electrons traversing the beamed laser pulse will experience different intensity values depending on their transverse position ($a_0\Psi \left(\overrightarrow{r}\right)$). The result is that the nonlinear broadening will be different for each electron and could obscure the dependency of ${\varphi }_{cep}$. The broadening of the spectrum due to the spatial profile, when (transverse) ponderomotive scattering is omitted, scales as

$$\Delta {\nu }_{\mathcal{E}=1,\Psi }=\frac{\frac{a_0^2}{2}}{1+\frac{a_0^2}{2}}\frac{1-{\Psi }^2}{1+\frac{a_0^2{\Psi }^2}{2}}$$

(9)

In order to see the effect of ${\varphi }_{cep}$ this contribution to the spectral broadening has to be much smaller than Equation (8). The choice of the spatial profile is a Gaussian beam. We use laser beam width $W_0=45\mu$m, by which the zeroth order Gaussian beam can be used and is given by

$$\Psi \left(\overrightarrow{r}\right)=\frac{q\left(0\right)}{q\left(z\right)}exp\left[-i\frac{{\omega }_l}{c}\frac{x^2+y^2}{2q\left(z\right)}\right],$$

(10)

where $q\left(z\right)=z+i\frac{W_0^2{\omega }_l}{2c}$. Figure 6 shows the the broadening due to a spatial profile. For the ${\varphi }_{cep}$ dependency to be visible, the width of an electron bunch needs to be a fraction of the width of the laser pulse.

3. Discussion

Here we showed that for a short laser pulse the effect of ${\varphi }_{cep}$ can be measured in the Thomson spectrum, where the recoil of the electron is negligible and classical electron dynamics can be used. The recoil of the electron is determined through $\chi =\frac{\gamma \hslash {\omega }_l}{m_e{c}^2}$. This parameter, however, does not include the stochastic effects for the emission of higher harmonics. In [22] it was shown that the classical description does not describe higher harmonics sufficiently for an electron energy of $\gamma =80$ and instead the scattering process should be treated quantum mechanically (Compton scattering). The electrons of $\gamma =2$ and 10 present in this work yield a value of the quantum parameter $\chi$ 8–40 times less than that which is described in [22], reducing the deviation of the classical from the quantum description. Moreover, for an electron energy of $\gamma =10$ and a laser photon energy of the order of 1 eV, the emitted radiation remains sub-keV, photon energies that can be measured with high enough precision to observe the CEP dependency, i.e., the change in position of the peak intensity of a harmonic (Equation (8)). Another advantage of using low-energy electrons is the (relatively) small size of the beamline and components necessary to obtain them.

In this work we used a pulse with length $N_c=5$ (${\tau }_{FWHM}=2$ fs). For shorter pulses the effect will be stronger, as the frequency shift due to ${\varphi }_{cep}$ scales with the length of the laser pulse. However, our description of the pulse (Equation (3)) no longer satisfies Maxwell’s equations to a good approximation. For longer pulses the effect is reduced, but since the shift scales with the harmonic one could look at higher harmonics for a better resolution. This can already be seen in Figure 6, bottom right panel.

For a realistic laser pulse, i.e., with a spatial profile, there is a stringent requirement to measure ${\varphi }_{cep}$: the gradient must be sufficiently small in the range of the width of the electron bunch. Here we showed this for a relatively large laser spot size ($W_0=45$ $\mathsf{\mu }$m) for which an electron bunch with $W_e=15$ $\mathsf{\mu }$m or smaller is required. Electron bunches with such characteristics have experimentally been obtained using laser wakefield acceleration (e.g., [23]). Thus, for laser pulses used in experiments, e.g., [24] $W_0=20$ $\mathsf{\mu }$m, [25] $W_0=30$ $\mathsf{\mu }$m, our proposed method is viable.

4. Conclusions

We have shown analytically and numerically how the spectrum of (inverse) Thomson scattering is influenced by the carrier envelope phase, making it a potential candidate as a diagnostic tool to measure ${\varphi }_{cep}$ at the interaction point. The interval for which ${\varphi }_{cep}$ can be determined through harmonic interference in the spectrum is $[0,\frac{\pi }{2}]+k\frac{\pi }{2}$ due to the symmetry in the spectrum. The application of such a diagnostic would require $a_0>1$ ($I\sim {10}^{18}$ W/cm${}^2$) in order for the harmonics of the emitted radiation to overlap. The physical reason can be understood from the classical model where the time of emission of the same frequency occurs at different times during the interaction. Due to the interference, the peak of a harmonic shifts proportionally to the harmonic number and ${\varphi }_{cep}$ and inversely with the length of the laser pulse. For larger harmonic numbers (consequently higher values of $a_0$) the effect is more visible. Further, we showed that for a realistic laser pulse, the transverse size of the electron beam needs to be at least three times smaller than that of the laser pulse for ${\varphi }_{cep}$ to be measurable.