The reflectivity of a relativistic flying plasma mirror has been studied by Bulanov [60]
and H.-C Wu [63]. Their procedures are different but construct the same relation between density profile of electrons and the reflectivity. Here, we first introduce Bulanov’s method and then the one proposed by H.-C Wu.
Consider the electromagnetic wave polarized in z direction and propagates along x axis. The vector potential Az(x, t) satisfies the wave equation
∂2Az
∂t2 − c2∂2Az
∂x2 + ω2pAz = 0, (3.1)
where ωpis the plasma frequency. Consider Az is in the form
Az = Az(x)e−iω0t, (3.2)
Eq.(3.1) becomes
∂2Az
∂x2 +ω02− ωp2
c2 Az = 0. (3.3)
After normalizing by defining a(x) = eAz(x)/mec2and performing the Lorentz transfor-mation to a frame moving with phase velocity of the plasma wave, which is defined by ζ = γph(x− vpht). Eq.(3.3) then becomes
d2
dζ2 + (ω′0)2− ωp′2(ζ) c2
a(ζ) = 0, (3.4)
where “′ ” denotes quantities in mirror’s proper frame. We can represent the solution to Eq.(3.4) as
a(ζ) = b+exp(isζ) + b−exp(−isζ), (3.5)
where s ≡ ω0′/c and b+(ζ) and b−(ζ) are the amplitudes of reflected and transmitted waves. In the limit ζ → −∞, b+(ζ) is the amplitude of the incident wave and b−(−∞) = ρ is the amplitude of the reflected wave. For ζ → +∞, b+(∞) is equal to the amplitude of transmitted wave and b−(∞) = 0. Therefore, with definition of the reflective and transmit coefficient R and T, we have|b+(−∞)|2 = 1,|b−(−∞)|2 = R,|b+(∞)|2 = T and b−(∞) = 0.
Because two unknown functions b+(ζ) and b−(ζ) are introduced, instead of one a(ζ), we need to impose extra conditions on the solution. The derivative of vector potential is required to satisfy
da
dζ = is[b+(ζ) exp(isζ)− b−(ζ) exp(−isζ)], (3.6)
that is
db+
dζ exp(isζ) = −db−
dζ exp(−isζ). (3.7)
After substituting Eq.(3.5) into Eq.(3.4) and taking Eq.(3.6) into account, the system of solution can be expressed in the form [64]
d solution of system (3.8) corresponds to a known approximation in quantum mechanics with the potential considered as a perturbation [65]. By integrating the equation of db−/dζ from ζ =−∞ to ∞, we have
Due to the small reflectivity, b+and b−in the right hand side can be replaced by the zeroth order solution (a plane wave): a(0)(ζ) = exp(isζ), that is b(0)+ = 1 and b(0)− = 0. This leads
After redefining the variable ζ =−ζ , the result is
ρ = i 2s
Z ∞
−∞
ν(ζ) exp(−2isζ)dζ. (3.11)
Eq.(3.11) constructs the relation between the electron density profile (ν(ζ)) and the ratio between incident wave and reflected wave ρ.
On the other hand, the method utilized in [63] is shown as follows. Starting from the wave equation of the vector potential,
∂2
where A is the vector potential of the radiation, S is the source term and “′” is used to indicate quantities in the mirror’s proper frame. The solution of A′ can be written in the form
A′(x, t) = A′(0)+ Z Z
dx′dt′G(x− x′, t− t′)S′(x′, t′), (3.13)
where A′(0) is the zeroth order solution which corresponds to the solution of Eq.(3.12) without source, that is a planar wave. G(x− x′, t− t′) =−(c/2)H[(t − t′)− |x − x′|/c]
is the Green’s function of Eq.(3.12) that satisfies
∂2
H is the Heaviside step function. The vector potential can be expressed in the iterative style
If the contribution of the source term is small compared to the zeroth order term, that is when reflectivity is small, Eq.(3.15) is a perturbative equation to describe the vector po-tential. Here, we only keep terms up to the first order. Consider the zeroth order solution, a right-moving plane wave, to be A(0) = A0exp[i(ωst− ksx)] where A0, ωsand ksare the amplitude, the angular frequency, and the wave number of incident wave, respectively.
The transmitted and reflected part of the incident wave can then be written as
A′t≈ A′(0)+ A′(1)(x→ +∞, t), (3.16)
A′r≈ A′(1)(x→ −∞, t), (3.17)
respectively. After substituting the Green’s function into the first order term of Eq.(3.15), we have
where the property of the step function is used to arrive at the second equation. After carrying out the integration over t′
A′(1)(x, t) = iA0
It can be found that the reflected wave is left-moving as expected. The ratio of amplitude between reflected wave and incident wave is
A′r0
It is noteworthy that the result is the same no matter using the method proposed by Bulanov et al. (Eq.(3.11)) or the one by H.-C, Wu (3.21) for calculating the reflection coefficient.
After substituting the definition of ωpand considering the ratio between amplitude of
elec-tric field, Eq.(3.21) can be expressed as
It is clear that the reflected electric field depends on the electron density distribution of the flying plasma mirror, n(x). Here we discuss three different density distributions: Slab [63], Cusp [66], and Square-Root Lorentzian Distribution (SRLD), defined as
Slab : nslab(x) = npeak[H(x + 2D)− H(x)], (3.23)
where n0 and npeak are the unperturbed background plasma density and peak density of the distribution, separately. γ and β are the Lorentz factor and the normalized velocity, respectively, calculated from the phase velocity of the flying mirror. To compare the results from different density distributions, we unify the definition of mirror density in these three distributions. For Slab and SRLD, the peak density and thickness of the mirror can be associated with wave-breaking limit of the background plasma, under which the flying mirror contains half of the total electrons within the volume encompassed by the nonlinear plasma wavelength. Therefore, we have
Z λN P/4 [39] (see also Sec.(2.3)), λpis the linear plasma wavelength and a0the normalized vector potential of a linearly polarized driver pulse. Then, we can link npeak with background
density n0, such as in SRLD case
npeak = λN Pn0
4L sinh−1(λN P/4L). (3.28)
Besides, to make the comparison on an equal basis, we consider the same peak density of both Slab and SRLD distributions. The cusp distribution is with infinite peak density therefore can not be normalized in this way. With these considerations, Eq.(3.23)-(3.25) can be written as
where the normalization constants are defined as
Cslab ≡q
1 + a20/2, (3.32)
Csrld ≡
p1 + a20/2
sinh−1(λN P/4L). (3.33)
The respective parameters used and suitable scene of these different distributions are explained as below. The Slab Distribution is a simplified model to describe the flying mirror with the thickness of the slab is defined as 2D. This may be an approximated model in the interaction between intense laser and a solid target when all the electrons in the thin film are pushed away and formed a slab flying mirror. The Cusp Distribution is derived from the 1D cold, collisionless plasma theory and the nonlinear coupled wave equation at the wave breaking situation [60]. The SRLD is a fitting function that we deduced from the PIC simulation. From 1D PIC simulations, the peak density of the flying mirror was found to be not as spiky as the Cusp Distribution but more rounded instead. Actually, the singularity in the Cusp Distribution at the wave-breaking point may suggest the breakdown of the cold plasma description. For a typical Laser Wakefield Accelerator scheme, warm
plasma theory should take place as the flying mirror approaches the wave-breaking point, which in turn renders the maximum density finite [67]. Without solving the complex equations based on the warm plasma theory, we deduced the SRLD distribution as a good approximation to the flying mirror density near the wave-breaking limit (see the Inset of Fig.(3.1)). Here, L is the characteristic thickness of the flying mirror.
The reflectivity in terms of the photon number can be calculated from Eq.(3.22) by R ≡ |Er′/E0′|2. To transform the density function (Eq.(3.31)) into mirror’s frame, it can be noted that the total electrons number in the distribution, N , is a Lorentz invariant. Thus, we have
Here, we take the SRLD case as an example Z ∞
To arrive at the right hand side, the transformation x′ = γx and dx′ = γdx between mirror’s frame and lab frame are used. Accordingly, the SRLD in mirror’s frame is
n′(x′) = npeak
γ s
(γL)2
(x′)2+ (γL)2. (3.36)
The integration in Eq.(3.22) can then be carried out Z ∞
−∞
dx′n′e−2iks′x′ = 2npeakLK0(2γLk′s), (3.37)
where K0 is modified Bessel function of the second kind [68]. This gives the amplitude of the reflected wave for a SQLD mirror
|Er′(x, t)|
The definition ωp = n0e2/meϵ0 is used to simplified the equation. This leads to the
Note that this expression is calculated in the mirror’s frame. In the lab frame, this repre-sents the reflectivity in terms of number of photons (from here simply referred as “reflec-tivity”).
The reflectivity for three different distributions are summarized as follows, parameters are all expressed with quantities in the lab frame [60, 63],
Slab: Rslab(ωs) =
where the sinc function is defined as sinc(x)≡ sin(x)/x, Γ is the gamma function. From Eq.(3.40)-(3.42), it is clear that the reflectivity quickly decays as the frequency of incident wave ωsincreases. In addition, the reflectivity decreases as γ increases. This means that there exists a trade-off between high reflectivity and high frequency in the reflected wave.
The tendencies of such decrease in reflectivity are different among the three different density distributions of the sinc, the exponential (ωs−8/3) and the K0 functions, respec-tively. The decaying and oscillating behavior of the sinc function has been explained as the result of the modulations due to the constructive and destructive interferences [63]. It should be noted that the argument in the sinc function and K0 are of the same form, de-fined as s≡ 4γ2Lωs/c = 2πL/λr, where λr ≡ 2πc/(4γ2ωs) is the reflected wavelength in the lab frame. As s ≫ 1, both sinc and K0 functions decay quickly, which in turn highly suppress the reflectivity. Therefore, s can serve as a parameter to define the qual-ity of the flying mirror. A good mirror is one whose thickness is roughly the same order of magnitude as the doubly Doppler shifted wavelength, i.e., L ≤ O(λr). This explains
why in an experiment one usually tunes the collision point at the wave-breaking limit so as to minimize the thickness of the flying mirror [49, 50], which is an optimum point for trade-off between the reflectivity and the frequency of the reflected wave.
To examine the validity of Eq.(3.40)-Eq.(3.42), we numerically study the property of relativistic flying mirror traversing a uniform plasma in the underdense regime with PIC simulations in 1D Cartesian geometry. The 1D configuration is a good approximation to the case of a driver pulse with a large focal spot in a higher dimension. This corresponds to the condition that r ≫ λp where r is the spot radius of the driver pulse and λp is the wavelength of the background plasma. The simulations are performed with the fully relativistic electromagnetic PIC code EPOCH [69].
In our simulation, the relativistic flying mirror is generated by a highly intense driver pulse (referred to as the ”driver” from here on), which enters from the left boundary and propagates in the +x direction. Along its way, the driver induces a flying mirror (wake-field) that follows behind it. The incident wave (referred to as the ”source”) enters, on the other hand, from the right boundary and propagates in the−x direction. The collision point between the flying mirror and the source is tuned in such a way that the wave-breaking condition is reached with the flying mirror thickness minimized. Below we use subscripts
“m”,“d”,“s” “r” to denote quantities that are associated with the flying mirror, the driver, the source and the reflected pulse, respectively.
The driver is characterized by the wavelength λd= 800nm and the normalized vector potential ad = 5.0. The temporal profile is Gaussian with full-width-at-half-maximum (FWHM) duration of τd ≈ λp/2, which is chosen to excite the wakefield resonantly. The driver is linearly polarized with the electric field pointing in the y-direction.
To study the dependence of reflectivity on the source frequency, ωs, several source wavelengths are chosen: 266nm, 400nm, 800nm, 1600nm, 2400nm and 4000nm. The normalized vector potential as = 0.004 is set to be small enough to prevent the recoil effect [55, 70]. The temporal profile is Gaussian with FWHM duration τs = 2Ts, where Ts = λs/c is the source cycle period. To distinguish the reflected pulse from the driver, we set the source linearly polarized in z-direction.
The background plasma density is uniform with a density np = 0.025nc, where nc≡ meϵ0ω2d/e2 is the critical plasma density with respect to the driver. The simulation box size is 80µm in the x direction with 160, 000 cells. For shorter λs, the finer the grid size so as to guarantee the spatial resolution is sufficient for tracking the blue-shifted reflected pulse. In our strictest case, the resolution of the Cartesian grid size is roughly 8.3 cells per reflected wavelength λr, which is estimated by λr ≈ λs/4γm2. Outflow conditions are applied to each simulation boundary for both electromagnetic waves and quasi-particles.
1 2 3 4 5 6 7 8
Figure 3.1: Reflectivity of the relativistic flying mirror as a function of the source fre-quency ωs. Solid lines are calculated from different reflectivity models (Eq.(3.40)-(3.42)).
Distinct symbols are PIC simulation results with different λs. SRLD model agrees well with PIC results and the cusp model approaches SRLD when a longer wavelength source is applied. Inset: Comparison between three density distribution (Eq.(3.29)-(3.31)) mod-els and the density of flying mirror from PIC simulations. Note that the PIC data (circles) is almost overlapped by SRLD (blue line).
The comparison between the analytic formula and the simulation result on reflectivity is shown in Fig.(3.1). Parameters used in the analytic formulas (Eq.(3.40)-(3.42)) are ωp = 3.72×1014sec−1, ad= 5, n0 = 4.35×1025m−3, γ = 4.08 and L = 1.12nm. The first three parameters are fixed in the simulation setup while the last two are the values of the flying mirror at the collision point.The rightmost PIC data point is the one with the source wavelength λs = 266nm which corresponds to the frequency tripling of the frequency of the conventional 800nm Ti:Sapphire laser. In this setup, the double-Doppler-shifted wavelength λr ≈ 4nm corresponds to the water-window X-ray wavelength, which can
be a useful tool for life science research. The reflectivity in terms of the photon number is R ≈ 5 × 10−8, which is deduced from the ratio of the electric fields in the frequency spectrum between the reflected and the incident waves.
The reflectivity formula with the Cusp Distribution (Eq.(3.41)) may be a good esti-mation for long wavelength sources (λs ≥ λd), such as the leftmost point in Fig.(3.1) or the case in [70] with λs = 5λd. However, as shown in Fig.(3.1), the discrepancy be-tween Eq.(3.41) and the PIC result is found to grow as ωs increases. With λs = 266nm (λs = λd/3), we found that Eq.(3.41) tends to over-estimate the reflectivity by roughly two orders of magnitude. Within these three different models, the SRLD reflectivity for-mula gives the best agreement with the sifor-mulation results. This may not be surprising because one additional parameter, the thickness, was introduced in SRLD.
In the simulation, the resolution of the cell depends on the reflected wavelength λr. For a higher source frequency, a higher resolution is needed. However, with the help of Eq.(3.42), one can estimate the reflectivity directly from the property of the flying mir-ror. This helps to greatly accelerate the process to search for an appropriate experimental parameter space.