In the previous section, the incident wave was assumed to be a plane wave. In an actual experiment, however, the incident laser pulses are expected to have finite bandwidth. Here we consider such a pulse by a Gaussian wave packet with central frequency ωsand pulse duration τs:
Ein(x, t) = E0e−t2/τs2ei(ksx−ωst). (3.43)
To calculate the reflection of a finite-bandwidth pulse, we can use Fourier transform into the frequency space and discuss different frequency components separately. The electric
field amplitude in frequency space is
The transformation of Ein(ω) into mirror’s proper frame can be realized by noting that the phase of electric field is a Lorentz invariant, that is ksx− ωst = ks′x′ − ωs′t′. The frequency in the co-moving frame is ω′ = [(1 + βm)/(1− βm)]1/2ω. For simplifying the equation form, we consider the ultra-relativistic limit (β → 1) , this gives ωs′ = 2γmωs. Therefore we have the electric field amplitude in mirror’s frame
Ein′ (ω′) = E0τs
It is clear that in this frame the central frequency of the wave packet is blue-shifted to 2γmωs and the pulse duration shortened by the same factor 2γm. The amplitude of the reflected wave can be calculated frequency by frequency
Er′(ω′) = Y′(ω′)· Ein′ (ω′), (3.46)
where Y′(ω′) is the ratio between the amplitude of incident wave and reflected wave for a specific frequency ω′(cf. Eq.(3.22)). Accordingly, the electric field of the reflected wave from a flying mirror with a square-root Lorentzian distribution (Eq.(3.31)) in the mirror’s proper frame can be calculated. After transforming back to the lab frame, the reflected electric field is
It should be noted that, when the background plasma density is sufficiently low ( np/nc ≪ 1), the parametric Doppler effect [60] due to the frequency dispersion in the background medium can be ignored. The exponential term describes a pulse with the central frequency at 4γm2ωs and the pulse duration that is compressed by a factor 4γm2. However, the ω-dependent and decaying term, i.e., K0(Lω/c)/ω, will distort the reflected
spectrum. Fig.(3.2) shows the normalized reflected spectrum with γm = 5, L = 15nm, and λs = 800nm. The red curve is the normalized exponential term. The blue curve is calculated from Eq.(3.47) and the black curve is the value of the decaying term. The distor-tion of the spectral shape, shown in the blue curve, is not evident, while both the frequency and the amplitude at the peak of the spectrum clearly deviate from the red curve.
0.5 1 1.5 2
Figure 3.2: Normalized reflected electric field amplitude calculated by Eq.(3.47) (blue curve) and the naive estimation with ω = 4γm2ω0 (red curve). The black curve shows the decaying term in Eq.(3.47) and is also normalized to the value calculated with ω = 4γm2ω0. The deviation of both the frequency and amplitude at the peak of spectrum is demonstrated.
The deviation ratio between the frequency associated with the maximum amplitude, ωpeak, and the naively estimated frequency, ωest ≡ 4γm2ωs, is defined as
δ≡ ωpeak− ωest
ωest
(3.48)
From Eq.(3.47), δ depends mainly on three parameters: the pulse duration of source τs, the Lorentz factor of the flying mirror γm, and the characteristic thickness of the mirror L.
Fig.(3.3) shows the dependence of δ on τsand γm, which are accessible in an experiment.
τs can be measured with an auto-correlator and γm can be estimated by the background plasma density, γm ≈ ω0/ωp [26], or the energy of the accelerated electrons [50, 71].
The frequency associated with the maximum amplitude, ωpeak, can be calculated through dEr(ω)/dω = 0 at ω = ωpeak. However, due to the existence of the modified Bessel function of the second kind, ωpeak of Eq.(3.47) can not be found analytically. To have a
1 2 3 4 5 6 7 8 9 10
(a) Frequency Deviation Ratio and pulse du-ration
(b) Frequency Deviation Ratio and γm
Figure 3.3: Dependence of δ on the pulse duration T and the Lorentz factor γ with other parameters fixed. The deviation is evident for few cycle source pulse or flying mirror with higher lorentz factor.
sense about the amount of deviation, we numerically solve the deviation with Eq.(3.47) based on the typical parameters of intense lasers: 30fs pulse duration and γm = 4. δ is found to be roughly−1%, which may be hard to detect. However, from Fig.(3.3), it can be seen that the deviation is more significant for fewer-cycle sources and flying mirrors with higher Lorentz factors. This implies that the correction cannot be neglected when few-cycle pulses are employed or high blue-shift reflections are demanded, such as the situation for generating attosecond pulses with relativistic flying mirrors [56].
To study the validity of Eq.(3.47), the code EPOCH [69] was used. Instead of fly-ing mirrors induced by a driver laser, we imposed the mirror as an initial condition. The flying mirror was constructed as an electron sheet with a given longitudinal density distri-bution and propagating in the +x direction with an initially assigned velocity. To prevent electrons from expelling each other during propagation, positive charge (proton) was in-troduced to co-move with the relativistic electron sheet. The interaction between protons and the source is negligible because of their large mass.
We used the simplified Square-Root Lorentzian Distribution, nm(x) = nm,0
pL2m/(x2+ L2m),
to characterize the density distribution of the flying mirror. There are three parameters to be determined: the peak density nm,0, the characteristic thickness Lm, and the Lorentz factor of flying mirror γm. The peak density only affects the reflectivity. We therefore chose nm,0 = 3nc, where nc is the critical density for a 800nm electromagnetic wave, to guarantee that the reflected pulse is intense enough for observation. γm = 4 and Lmranges
from 1nm to 20nm according to the PIC results from the laser-driven flying mirror. The source is a linearly polarized pulse with a wavelength λs= 1.6µm, which is long enough to increase the reflectivity. The normalized vector potential is as = 0.004. The temporal profile is Gaussian with FWHM duration τs = 1.5Tswhere Ts = λs/c is the source cycle period. The source enters from right boundary and propagates in−x direction.The sim-ulation box size is 50um in x direction with 25000 cells. Therefore, the resolution of the Cartesian grid is 12.5 cells per reflected wavelength, λr ≈ 4γm2λs. Boundary conditions remained the same as the setup in the previous section.
0 0.5 1 1.5 2
Figure 3.4: Comparison among the estimated double-Doppler shift frequency (yellow line), the theoretical prediction of ωpeakfrom Eq.(3.47) (blue line), and the PIC simulation result (red dots). γm = 4, nm,0 = 3ncand τs = 1.5Tsare used as the initial condition. The 1D PIC result agrees reasonably with theoretical prediction and the linear dependence of the deviation on mirror thickness is also illustrated.
The comparison between the theoretical prediction of the frequency at the peak ampli-tude from Eq.(3.47) and the PIC simulation results is shown in Fig.(3.4). The horizontal yellow line is the estimated naive frequency 4γm2ωsand the blue one is the maximum value of Eq.(3.47) solved numerically. Red circles are the PIC simulation results with differ-ent characteristic thicknesses Lm. We see that the PIC results are in reasonable agreement with the theoretical prediction from Eq.(3.47). The discrepancy is resulted from the statis-tical fluctuations in the initialization of the SRLD distribution due to the limited number of macro-particles in our PIC simulations. From Fig.(3.4), the magnitude of the deviation, which is always negative, increases linearly as the characteristic thickness of the flying mirror Lm increases, where the slope depends on the Lorentz factor, γm, and the source pulse duration, τs.
In an actual experiment, γmcan be deduced through the measurement of the reflected wave spectrum. Usually, this is estimated from the peak frequency of the reflected spec-trum and the naive double-Doppler shifted relation, ωpeak = 4γmωs. The deviation from this idealized value, as we have shown, can serve as its correction that can further improve the precision of this method.
We have shown in Section II that the plasma mirror thickness is an important parameter that determines the reflectivity and the reflected spectrum. In actual experiments, multiple tools can be employed to diagnose the dynamics of the wakefield, i.e., the mirror, such as the relativistic electron bunch probe [72] and the optical probe [73]. However, the spatial and the temporal resolutions of these methods are still not precise enough to measure the thickness of a flying mirror near the wave-breaking condition, which is typically of tens of nanometer scale. Our investigation shows that the frequency deviation can serve as a diagnosis on the thickness. As Eq.(3.47) shows, the peak frequency of the reflected wave depends on ωs, τs, γm, and Lm. Among them ωsand τs are laser parameters that can be measured accurately. In principle, γm can be determined by conventional methods such as that based on the background plasma density [26] or the accelerated electron energy [50, 71], from which the mirror thickness can be deduced. However, the diagnostic scheme suggested here may require highly stable condition of lasers and plasmas.
3.4 Conclusion
In this section, we extended previous studies on the reflectivity of relativistic flying mir-rors with incident plane waves. We showed that the Square Root Lorentzian Distribution can accurately describe the flying mirror density distribution, and can provide a better estimation about the reflectivity. We defined a dimensionless parameter, s = 2πLm/λr, to characterize the quality of the flying mirror. To attain a high enough reflectivity, the condition, s ≤ 2π, must be satisfied, which means that the mirror must be thinner than the wavelength of the reflected pulse. In our simulations, we demonstrated the feasibil-ity of the generation of the water-window X-ray through plasma mirror reflection based on state-of-the-art laser parameters, which would provide great utility to life science
re-searches. We found that the reflectivity in this case is∼ 5 × 10−8 in photon numbers, which is encouraging. We also found that, for an incident wave with a Gaussian temporal profile, the peak frequency of the reflected spectrum is red-shifted from its expected value, 4γm2ωs. The magnitude of the deviation is positively correlated to the thickness of the mir-ror and its Lorentz factor, but negatively correlated with the duration of the source pulse.
This deviation helps to provide a better description of the reflected spectrum, which can serve as a diagnostic tool about the dynamics of the wakefield. These studies about the reflectivity and the reflected spectrum may benefit future experiments such as AnaBHEL.
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