# Boosted high-harmonics pulse from a double-sided relativistic mirror.

**ABSTRACT** An ultrabright high-power x- and gamma-radiation source is proposed. A high-density thin plasma slab, accelerating in the radiation pressure dominant regime by an ultraintense electromagnetic wave, reflects a counterpropagating relativistically strong electromagnetic wave, producing extremely time-compressed and intensified radiation. The reflected light contains relativistic harmonics generated at the plasma slab, all upshifted with the same factor as the fundamental mode of the incident light. The theory of an arbitrarily moving thin plasma slab reflectivity is presented.

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**ABSTRACT:**We report on the realistic scheme of intense X-rays and gamma-radiation generation in a laser interaction with thin foils. It is based on the relativistic mirror concept, i.e., a flying thin plasma slab interacts with a counterpropagating laser pulse, reflecting part of it in the form of an intense ultra-short electromagnetic pulse having an up-shifted frequency. A series of relativistic mirrors is generated in the interaction of the intense laser with a thin foil target as the pulse tears off and accelerates thin electron layers. A counterpropagating pulse is reflected by these flying layers in the form of a swarm of ultra-short pulses resulting in a significant energy gain of the reflected radiation due to the momentum transfer from flying layers.Journal of Physics Conference Series 01/2010; 244(2). - SourceAvailable from: Alexei Zhidkov[Show abstract] [Hide abstract]

**ABSTRACT:**An optically dense ionization wave (IW) produced by two femtosecond (approximately 10/30 fs) laser pulses focused cylindrically and crossing each other may become an efficient coherent x-ray converter in accordance with the Semenova-Lampe theory. The resulting velocity of a quasiplane IW in the vicinity of pulse intersection changes with the angle between the pulses from the group velocity of ionizing pulses to infinity allowing a tuning of the wavelength of x rays and their bunching. The x-ray spectra after scattering of a lower frequency and long coherent light pulse change from the monochromatic to high order harmoniclike with the duration of the ionizing pulses.Physical Review Letters 11/2009; 103(21):215003. · 7.73 Impact Factor - SourceAvailable from: Alexei Zhidkov
##### Article: Relativistic spherical plasma waves

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**ABSTRACT:**Tightly focused laser pulses as they diverge or converge in underdense plasma can generate wake waves, having local structures that are spherical waves. Here we report on theoretical study of relativistic spherical wake waves and their properties, including wave breaking. These waves may be suitable as particle injectors or as flying mirrors that both reflect and focus radiation, enabling unique X-ray sources and nonlinear QED phenomena.Physics of Plasmas 01/2011; · 2.38 Impact Factor

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Boosted high harmonics pulse from a double-sided relativistic mirror

T. Zh. Esirkepov,1S. V. Bulanov,1, ∗M. Kando,1A. S. Pirozhkov,1and A. G. Zhidkov2

1Kansai Photon Science Institute, JAEA, Kizugawa, Kyoto 619-0215, Japan

2Central Research Institute of Electric Power Industry, Yokosuka, Kanagawa 240-0196, Japan

(Dated: January, 2009; rev. March, 2009.)

An ultra-bright high-intensity X- and gamma-radiation source is proposed. A high-density thin

plasma slab, accelerating in the radiation pressure dominant regime by a co-propagating ultra-

intense electromagnetic wave, reflects a counter-propagating relativistically strong electromagnetic

wave, producing strongly time-compressed and intensified radiation due to the double Doppler effect.

The reflected light contains relativistic harmonics generated at the plasma slab, all upshifted with

the same factor as the fundamental mode of the incident light.

PACS numbers:52.38.Ph, 52.59.Ye, 52.38.-r, 52.35.Mw, 52.27.Ny

Interaction of electromagnetic (EM) wave with the relativistic mirror has been used by A. Einstein to illustrate basic

effects of special relativity [1]. In modern theoretical physics the consept of relativistic mirror is used for solving a

wide range of problems, such as the dynamical Casimir effect [2], the Unruh radiation [3] and other nonlinear vacuum

phenomena. Relativistic mirrors made by wake waves may lead to an electromagnetic wave intensification [4] resulting

in an increase of pulse power up to the level when the electric field of the wave reaches the Schwinger limit when

electron-positron pairs are created from the vacuum and the vacuum refractive index becomes nonlinearly dependent

on the electromagnetic field strength [5]. In classical electrodynamics EM wave reflected off a moving mirror undergoes

the frequency and electric field magnitude multiplication, a phenomenon called the double Doppler effect. If the EM

wave is co-propagating with respect to the mirror, its frequency and energy decreases upon reflection. If it is counter-

propagating, the reflected light gains energy and becomes frequency-upshifted. For the latter case the multiplication

factor is approximately 4γ2, where γ ? 1 is the Lorentz factor of the mirror, making this effect an attractive basis

for a source of powerful high-frequency EM radiation. Relativistic plasma provides numerous examples of moving

mirrors which can acquire energy from co-propagating EM waves or transfer energy to reflected counter-propagating

EM waves (see [6] and references therein). Manifestation of the Doppler effect in plasma governed by strong collective

fields is seen in the concepts of the sliding mirror [7], oscillating mirror [8], flying mirror [4] and other schemes [9]

which can produce ultra-short pulses of XUV- and X-radiation.

In this paper we discuss the concept of the accelerating double-sided mirror, Fig. 1, which can efficiently reflect

the counter-propagating relativistically strong electromagnetic radiation. The role of the mirror is played by a high-

density plasma slab accelerated by an ultra-intense laser pulse (the driver) in the Radiation Pressure Dominant

(RPD) regime (synonymous to the Laser Piston regime), [10]. Such an acceleration can be viewed as the double

Doppler effect: it is the reflection that allows the energy transfer from the driver pulse to the co-propagating plasma

slab, which acquires the fraction ≈ 1 − (4γ)−2of the driver pulse energy, [10]. The plasma slab also acts as a

mirror for a counter-propagating relativistically strong electromagnetic radiation (the source). As such it exhibits

the properties of the sliding and oscillating mirrors, producing relativistic harmonics. The source pulse should be

sufficiently weaker than the driver, nevertheless it can be relativistically strong. In the spectrum of the reflected

radiation, the fundamental frequency of the incident radiation and the relativistic harmonics and other high-frequency

radiation like bremsstrahlung generated at the plasma slab are multiplied by the same factor, ≈ 4γ2, Fig. 1.

2

4

dd

?

?

? ??

2

4

?

?

???

d ?

?

V

2

(4)

nn

?

?

????

Source

Driver

reflected

reflected

...

Double-sided mirror

FIG. 1: (color) The accelerating double-sided mirror concept.

∗Also at A. M. Prokhorov Institute of General Physics of RAS, Moscow, Russia.

arXiv:0902.0860v3 [physics.plasm-ph] 1 Apr 2009

Page 2

2

Compared with previously discussed schemes, the double-sided mirror concept benefits from a high number of

reflecting electrons (since the accelerating plasma slab initially has solid density and can be further compressed

during the interaction) and from the multiplication of the frequency of all the harmonics (since the interaction is

strongly nonlinear and the mirror is relativistic). This concept opens the way towards extremely bright sources of

ultrashort energetic bursts of X- and gamma-ray, which become realizable with present-day technology enabling new

horizons of laboratory astrophysics, laser-driven nuclear physics, and studying the fundamental sciences, e.g. the

nonlinear quantum electrodynamics effects.

In order to investigate the feasibility of this effect we performed two-dimensional (2D) particle-in-cell (PIC) simu-

lations using the Relativistic ElectroMagnetic Particle-mesh code REMP based on the density decomposition scheme

[12]. The driver laser pulse with the wavelength λd= λ = 2πc/ω, the intensity Id= 1.2 × 1023W/cm2× (1µm/λ)2,

corresponding to the dimensionless amplitude ad = 300, and the duration τd = 20π/ω is focused with the spot

size of Dd = 10λ onto a hydrogen plasma slab with the thickness l = 0.25λ and the initial electron density

ne = 480ncr = 5.4 × 1023cm−3× (1µm/λ)2placed at x = 10λ. The plasma slab transverse size is 28 λ. The

driver pulse shape is Gaussian but without the leading part, starting 5λ from the pulse center along the x-axis. At

the time t = 0, when the driver pulse hits the plasma slab from the left (x < 10λ), the source pulse arrives at another

side of the slab from the right (x > 10.25λ). The driver is p-polarized, i. e., its electric field is directed along the

y-axis. The source pulse is s-polarized (its electric field is along the z-axis). It has the same wavelength as the driver

pulse. Its intensity is Is= 1.2 × 1019W/cm2× (1µm/λ)2, corresponding to the dimensionless amplitude as= 3, its

duration is τs= 120π/ω and its waist size is Ds= 20λ. The source pulse has rectangular profile along the x- and

y-axes; such the profile is not necessary for the desired effect but helps to analyse the results. The simulation box

FIG. 2: (color) (a) The driver and source pulses represented by the y- and z-components of the electric field, respectively, and

the ion density (black). (b) The ion energy (curve) and anglular (grayscale) distributions. Both the frames for t = 37 × 2π/ω.

(c) The electric field z-component, showing the source pulse overlapped with the first two cycles of the reflected radiation at

t = 4 × 2π/ω.

FIG. 3: (color) (a) The electric field component Ez representing the reflected radiation along x-axis for t?= 32 × 2π/ω.

(b) Colorscale: the modulus of the spectrum of Ez(τ) seen along x-axis, taken for each t with the Gaussian filter, Iω(t) =

R+infty

width of the filter and a fast change of the frequency multiplication factor.

−∞

calculated from the fast ion spectrum maximum at the time of reflection. Modes aliasing occurs at later times due to the fixed

Ez(τ)e−iτω−c2(τ−t)2/λ2dτ. Dashed curves: the odd harmonics frequency multiplied by the factor (1 + β)/(1 − β)

Page 3

3

has size of 50 λ with the resolution of 128 steps per λ along the x-axis and 32 λ with the resolution 16 steps per λ

along the y-axis. The number of quasi-particles is 106. We note that the use of a circularly polarized driver pulse

may provide a smoother start of the slab acceleration in the radiation pressure dominant regime [11], nevertheless it

was chosen to be p-polarized in order to easily distiguish between the driver and the source pulses. In addition, our

choice demonstrates the robustness of the double-sided mirror effect. The results are shown in Figs. 2-3, where the

spatial coordinates and time units are in the laser wavelengths and wave periods, respectively.

The driver laser makes a cocoon where it stays confined, Fig. 2(a). At t = 37 × 2π/ω, the ions are accelerated

up to 2.4 GeV while the majority of accelerated ions carry the energy about 1.5 GeV, Fig. 2(b). The accelerating

plasma reflects the source pulse, which becomes chirped and compressed about 10 times, Fig. 2(a). As the mirror

velocity, cβ, increases, the reflected light frequency grows as (1 + β)/(1 − β), thus the electric field profile along the

x-axis becomes more and more jagged, Fig. 3(a). A portion of the source pulse reflected from the curved edges of the

expanding cocoon acquires an inhomogeneous frequency upshift determined by the angle of the reflecting region. At

the begining, the magnitude of the reflected radiation is higher than that of the incident source (∼ 3 times), due to the

double Doppler effect and an enhancement of reflectivity owing to the plasma slab compression under the radiation

pressure exerted by the driver pulse. In an instantaneous proper frame of the accelerating mirror, the frequency of

the source pulse increases with time, thus the mirror becomes more transparent. Correspondingly, the source starts

to be transmitted through the plasma more efficiently, as seen in Fig. 2(a) where the transmitted radiation is focused

owing to a cocoon-like spatial distribution of the plasma.

The reflected radiation has a complex structure of the spectrum, Fig. 3(b). It contains not only the frequency-

multiplied fundamental mode of the source pulse, but also high harmonics due to the nonlinear interaction of the source

with the plasma slab. This is also seen in Fig. 2(c), where the first two consecutive cycles of the reflected radiation

exhibit presence of high harmonics, while the later cycle is compressed together with its harmonics in comparison with

the earlier cycle. The spectrum is also enriched by a continuous component since the mirror moves with acceleration.

The high harmonics generation efficiency is optimal for a certain areal density of the foil, according to the condition

as≈ πnelreλs[7], where re= e2/mec2is the classical electron radius. Initially far from this condition, the accelerated

plasma slab satisfies it at certain time, when harmonics are generated most efficiently.

In order to analytically describe the reflected EM wave we use the approximation of an infinitely thin foil (see

also Refs.[4, 7]), representing a mirror moving along the x-axis with the coordinate XM(t). We consider the one-

dimensional (1D) Maxwell equation

∂2A

∂t2− c2∂2A

∂x2+4πe2nelδ[x − XM(t)]

meγM

A = 0,

(1)

where γM =?1 − (dXM/dt)2c−2?−1/2is the Lorentz factor of the mirror, A is the EM wave vectror-potential, δ is

variables, ξ,η, which are the characteristics of the Maxwell equation,

the Dirac delta function. Let k is the incident wave number. Transfromations to dimensionless variables and to new

¯ x = kx, ¯t = kct,

ξ = (¯ x −¯t)/2, η = (¯ x +¯t)/2,

XM(t) = k−1XM(η − ξ) , A (x,t) =mec2

(2)

(3)

(4)

e

A(ξ,η),

and the property δ(kz) = k−1δ(z) yield the equation

∂2A

∂ξ∂η= χδ[ψ(ξ,η)]

γ(ξ,η)

A,

(5)

where χ = 2nelreλ, λ = 2π/k, and

ψ(ξ,η) = ξ + η − XM(η − ξ) ,

γ(ξ,η) =?1 − X?2

(6)

M(η − ξ)?−1/2.

(7)

We seek the solution to Eq. (5) in the form of the incident, transmitted and reflected waves:

?a1(ξ) + a0e2iη,

Here the factor e2iη= eik(x+ct)represents the incident wave. The solution should satisfy the boundary conditions

at the position of the mirror, ψ(ξ,η) = 0. We introduce new functions ξ0(η) and η0(ξ), which satisfy the following

A(ξ,η) =

ψ(ξ,η) > 0;

a2(η),ψ(ξ,η) ≤ 0.

(8)

Page 4

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expressions

ψ(ξ0(η),η) = 0 for ∀η,

ψ(ξ,η0(ξ)) = 0 for ∀ξ.

(9)

(10)

The requirement that the solution is continuous, A(ξ,η0(ξ) − 0) = A(ξ,η0(ξ) + 0), leads to the following condition:

a1(ξ) + a0e2iη0(ξ)= a2(η0(ξ)).

(11)

The remaining conditions can be obtained from Eqs. (5) and (8). Integrating Eq. (5) over η in the vicinity of η0(ξ)

for fixed ξ and some small ? > 0, we obtain:

∂A

∂ξ

????

η=η0(ξ)+?

η=η0(ξ)−?

= χ

?η0(ξ)+?

η0(ξ)−?

δ[ψ(ξ,η)]A(ξ,η)

γ(ξ,η)dη .

(12)

If X?

the function ψ has a simple zero at (ξ,η0(ξ)), therefore the equation

M(η0(ξ) − ξ) ?= 1 for given ξ, which means that the mirror velocity does not reach the speed of light in vacuum,

?η0+?

η0−?

δ[ψ(ξ,η)]f(ξ,η)dη =f(ξ,η0)

∂ψ/∂η,

(13)

where the derivative ∂ψ/∂η is taken at the point (ξ,η0(ξ)), holds for any integrable function f. Using Eq. (13) we

find that at the limit ? → 0 Eq. (12) gives the magnitude of the jump discontinuity of the derivative Aξ= ∂A/∂ξ at

η = η0(ξ) for fixed ξ:

∂A

∂ξ

????

η=η0(ξ)+0

η=η0(ξ)−0

= χ?(ξ,η0(ξ))A(ξ,η0(ξ)),

(14)

where we introduce the digamma factor, ?,

?(ξ,η) =

?1 + X?

M(η − ξ)

M(η − ξ)1 − X?

?1/2

.

(15)

Similar expression is obtained for the magnitude of the jump discontinuity of the derivative Aη= ∂A/∂η at ξ = ξ0(η)

for fixed η. For the ansatz (8) these expressions give the following two ordinary differential equations (ODE):

?

2ia0e2iη− a?

a?

1(ξ) = χa1(ξ) + a0e2iη0(ξ)?

2(η) =

?(ξ0(η),η)a2(η).

?(ξ,η0(ξ)),

(16)

χ

(17)

The reflected, a1(ξ), and the transmitted, a2(η), waves are determined by Eqs. (11), (16), and (17) which can be

easily reduced to quadratures.

In the simplest case of uniform motion,

X?

M(¯t) = β = const,

(18)

we have

?(ξ,η) = ?0=

?1 + β

1 − β

?1/2

η0(ξ) = −?2

≈ 2γM,

(19)

0ξ.

(20)

The solution to Eqs. (11), (16), and (17) reads

a1= −a0

χ

χ + 2i?0

exp(−2i?2

2i?0

χ + 2i?0

0ξ),

(21)

a2= a0

exp(2iη),

(22)

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so that the reflection coefficient in terms of the number of photons is

R =

????

a1

a0

????

2

≈

(nelreλ)2

(nelreλ)2+ 4γ2

M

,

(23)

thus we recover the result of Ref. [4].

In the case of a mirror moving with a uniform acceleration gkc2, for simplicity we consider the particular trajectory

XM(¯t) = g−1[1 + (g¯t)2]1/2.

(24)

Then we obtain

η0(ξ) = (4g2ξ)−1, ξ0(η) = (4g2η)−1

?(ξ,η0(ξ)) = (2gξ)−1, ?(ξ0(η),η) = 2gη,

(25)

(26)

and the solution to Eqs. (11), (16), (17):

a1(ξ) =χa0

2g

?2ig2ξ?χ

2gΓ

?

?

χ

2g,(2ig2ξ)−1,0

?

?

,

(27)

a2(η) =χa0

2g(−2iη)−χ

2gΓ

χ

2g,−2iη,0+ a0e2iη,

(28)

where Γ(a,z1,z2) =?z2

z1ta−1e−tdt is the generalized incomplete gamma function [13]. At ξ → 0,

?2ig2ξ?χ

where Γ(z) is the Euler gamma function [13]. The frequency of the reflected radiation increases as ξ−1, as in the

case of a perfect mirror of Ref. [14]. However, in our case the mirror reflectivity decreases with time. An observer at

infinity (which corresponds to ξ = 0) see the radiation with the frequency spectrum of the following intensity

a1(ξ) = −χa0

2g

2gΓ

?

χ

2g

?

+ iχa0g exp

?

i

2g2ξ

??ξ + O(ξ2)?,

(29)

Iν≈

πa2

0

2g4(1 + 2g/χ)2

?

1F2

?

1 +

χ

2g;2,2 +

χ

2g;−ν

2g2

??2

,

(30)

where ν is the observed frequency,1F2(a1;b1,b2;z) is the generalized hypergeometric function [13]. Here Iνis defined

as the square of the modulus of the Fourier transform of the function a1(ξ), Iν=

cast out the essential singularity at ν = 0 representing the finite limit a1(ξ → +∞) = −a0. For large ν, the spectral

intensity decreases with frequency, ν, as

?√2πΓ(1 +

ν

???

1

√2π

?+∞

−∞eiνξa1(ξ)dξ

???

2

, where we

Iν∼

a2

4g4

0

χ

2g)

Γ(−χ

2g)

?2g2

?1+χ

2g

+χ

2g

?

cos

?√2ν

g

?

+ sin

?√2ν

g

???2g2

ν

?5/4?2

, ν → ∞.

(31)

Now we consider the case of a mirror oscillating with frequency Ω (normalized on the incident wave frequency)

?

d

d¯t

β(¯t)

?1 − β2(¯t)

?

= g cos(Ω¯t),

(32)

choosing the following trajectory of the mirror

XM(¯t) =1

Ωarctan

?

−

cos(Ω¯t)

?h2− cos2(Ω¯t)

?

.

(33)

where h2= 1 + Ω2/g2. Solving the equation φ(ξ,η0(ξ)) = 0, we obtain

η0(ξ) = −1

ΩarctanH −π

Ω

?Ωξ + arctanh

H =htan(Ωξ) + 1

tan(Ωξ) + h,

π

−1

2

?

,

(34)

(35)

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6

where the function ?z? gives the integer closest to z. The digamma factor, Eq. (15), for η = η0(ξ) reads

?

For Eq. (33) the only bounded solution to Eq. (16) is

?(ξ,η0(ξ)) =

h2− 1

h2+ 1 + 2hsin(2Ωξ)

?1/2

.

(36)

a1(ξ) =χa0

g

+∞

?

Ωξ

E(Ωξ)

E(τ)

e−2iτ

(h2+ 1 + 2hsin(2τ))

?

Ω (h − ie2iτ)

2

Ω dτ

2+Ω

2Ω

,

(37)

E(τ) = exp

?

χ

g(h+1)F

τ −π

4

??

4h

(h+1)2

??

.

(38)

where F(z|m) is the elliptic integral of the first kind with an asymptotic ∝ z for z → ∞ [13].

In conclusion, a solid density plasma slab, accelerated in the radiation pressure dominant regime, can efficiently

reflect a counter-propagating relativistically strong laser pulse (source).

consists of the reflected fundamental mode and high harmonics, all multiplied by the factor ≈ 4γ2, where γ increases

with time. In general, the reflected radiation is chirped due to the mirror acceleration. With a sufficiently short source

pulse being sent with an appropriate delay to the accelerating mirror, one can obtain a high-intense ultra-short pulse

of X-rays.

For the mirror velocities greater than some threshold, the distance between electrons in the plasma slab in the

proper reference frame becomes longer than the incident wavelength. Thus the plasma slab will not be able to afford

the reflection in a coherent manner, where the reflected radiation power is proportional to the square of the number

of particles in the mirror. In this case the reflected radiation becomes linearly proportional to the number of particles.

Even with this scaling one can build an ultra-high power source of short gamma-ray pulses, when the interaction of

the source pulse with a solid-density plasma is in the regime of the backward Thomson scattering.

We can estimate the reflected radiation brightness in two limiting cases. For 2γ < (neλ3

and for the reflected photon energy ?ω the brightness is BM≈ Es(?ω)3λs/4π5?4c3, where Esis the source pulse energy.

For larger γ, the interaction may become incoherent and should be described as Thomson scattering, which gives BT≈

adEs(?ω)2reλ2

which is orders of magnitude greater than any existing or proposed source, [15]. For the same parameters and

λd= 0.8µm, ad= 300, ?ω = 10keV (γ = 40), BT= 3 × 1032photons/mm2mrad2s.

Employing the concept of the accelerating double-sided mirror, one can develop a relatively compact and tunable

ultra-bright high-power X-ray or gamma-ray source, which will considerably expand the range of applications of the

present-day powerful sources and will create new applications and research fields. Implementation in the “water

window” will allow performing a single shot high contrast imaging of biological objects [16]. In atomic physics and

spectroscopy, it will allow performing the multi-photon ionization and producing high-Z hollow atoms. In material

sciences, it will reveal novel properties of matter exposed to the high power X-rays and gamma-rays. In nuclear physics

it will allow studying states of high-Z nucleus. The sources of high-power coherent X-ray and ultra-bright gamma-ray

radiation also pave the way towards inducing and probing the nonlinear quantum electrodynamics processes.

This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for

Scientific Research (A), 29244065, 2008.

The reflected electromagnetic radiation

s)1/6, the reflection is coherent

s/8π4?3c2λ3

d. For Es= 10J, λs= 0.8µm, ?ω = 1keV (γ = 13), BM = 0.8 × 1040photons/mm2mrad2s,

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[15] F. V. Hartemann, et al. Phys. Rev. Lett. 100, 125001 (2008).

[16] R. Neutze, et al., Nature 406, 752 (2000).

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