Conceptual designs of a laser plasma accelerator-based EUV-FEL and an all-optical Gamma-beam source

Kazuhisa Nakajima

doi:10.1017/hpl.2014.37

Abstract

Recently, intense research into laser plasma accelerators has achieved great progress in the production of high-energy, high-quality electron beams with GeV-level energies in a cm-scale plasma. These electron beams open the door for broad applications in fundamental, medical, and industrial sciences. Here we present conceptual designs of an extreme ultraviolet radiation source for next-generation lithography and a laser Compton Gamma-beam source for nuclear physics research on a table-top scale.

Keywords

high peak high average power lasers laser wakefield accelerators

1. Introduction

To date, intense research has been carried out on laser plasma acceleration concepts^[1] to achieve high-energy, high-quality electron beams with GeV energies in a cm-scale plasma^[2–6], a 1%-level energy spread^[7], a 1 mm mrad level transverse emittance^[8], and a 1 fs level bunch duration^[9], ensuring that the stability of reproduction is as high as that of present high-power ultra-short-pulse lasers^[10]. Recently, staged laser plasma acceleration^[11–13] has been successfully demonstrated in conjunction with ionization-induced injection^[14–16] and phase-locking acceleration^[17]. Relativistic electron beams from ultraintense laser plasma interactions can be conceived to be compact particle accelerators, inspiring a wide range of applications of unique particle beam and radiation sources, such as THz^{[18, 19]} and X-ray/Gamma-ray radiation^[20–25].

Here we present an extreme ultraviolet (EUV) radiation source for next-generation lithography and a laser Compton Gamma-beam source for nuclear physics research. EUV lithography with wavelengths below 13.5 nm is capable of providing resolution below 30 nm in semiconductor manufacturing. We propose a self-amplified spontaneous emission (SASE) free electron laser (FEL) driven by relativistic electron beams from laser plasma accelerators. For example, this FEL system, capable of generating an average EUV power of 1 kW at 13.5 nm, comprises a fiber-based chirped pulse amplification (CPA) laser delivering a 1 MW average laser power, a 5 cm gas cell-type plasma accelerator producing a 660 MeV electron beam with a 1.6% relative energy spread and a 0.5 nC charge, and a 1 m long undulator with a 15 mm period and a 1.4 T peak magnetic field.

High-quality Gamma beams generated from inverse Compton scattering off relativistic electron beams interacting with an intense laser pulse have aroused interest in photonuclear physics and nuclear astrophysics research, the characterization of nuclear materials or radioactive waste and so on. We present a table-top all-optical laser plasma accelerator-based Gamma-beam source comprising a high-power laser system with synchronous dual outputs, a laser plasma accelerator producing 300–900 MeV electron beams, and scatter optics whereby the laser pulse is focused onto the electron beam to generate a Gamma beam via Compton scattering with photon energies of 2–20 MeV.

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2. Design of laser plasma accelerators for driving electron beams

2.1. Accelerator stage

Most of the laser plasma acceleration experiments that have successfully demonstrated the production of quasi-monoenergetic electron beams with a narrow energy spread have been elucidated in terms of self-injection and acceleration mechanisms in the bubble regime^{[26, 27]}, where a drive laser pulse with wavelength , peak power , intensity , and focused spot radius is characterized by a normalized vector potential with respect to the electron rest energy , given for the linear polarization as

(1)

In these experiments, electrons are self-injected into a nonlinear wake, often referred to as a bubble, i.e., a cavity void of plasma electrons consisting of a spherical ion column surrounded by a narrow electron sheath, formed behind the laser pulse instead of a periodic plasma wave in the linear regime. The phenomenological theory of nonlinear wakefields in the bubble (blowout) regime^[26] describes the accelerating wakefield

in the bubble frame moving in a plasma with velocity

, i.e.,

, where

is the plasma wavenumber evaluated with a plasma frequency

, an unperturbed on-axis electron density

and the classical electron radius

, and

is the non-relativistic wave-breaking field, approximately given by

. In the bubble regime for

, since an electron-evacuated cavity shape is determined by balancing the Lorentz force of the ion sphere exerted on the electron sheath with the ponderomotive force of the laser pulse, the bubble radius

is approximately given as

^[27]. Thus, the maximum accelerating field is given by

, where

represents a factor taking into account the difference between the theoretical estimation and the accelerating field reduction due to the beam loading effects.

Here we consider the self-guided case, where a drive laser pulse propagates in a homogeneous density plasma. The equations of longitudinal motion of an electron with normalized energy and longitudinal velocity are written approximately as^[28]

(2)

where

(

) is the longitudinal coordinate of the bubble frame moving at a velocity of

, taking into account diffraction at the laser pulse front that etches back at a velocity

^[27] with laser wavenumber

, and

is assumed. Integrating Equations (2), the energy and phase of the electron can be calculated as^[28]

(3)

where

is the injection energy. Hence, the maximum energy gain is obtained at

(4)

where

is the correction factor of the relativistic factor for the group velocity in a uniform plasma for a self-guided pulse, i.e.,

, obtained from^[28]

(5)

and

is the critical plasma density. The dephasing length

for the self-guided bubble regime is given by

(6)

For a given energy gain

, the operating plasma density is determined from Equation (4) as

(7)

The accelerator length equal to the dephasing length becomes

(8)

while the pump depletion length due to pulse-front erosion is given by

(9)

The dephasing length should be less than the pump depletion length, i.e.,

. Thus, the required pulse duration for self-guiding of the drive laser pulse is given by

(10)

The matched spot radius becomes

(11)

where

is the dimensionless matched spot radius given by^[28]

(12)

The corresponding matched power is calculated as

(13)

The required pulse energy becomes

(14)

2.2. Beam loading effects

In laser wakefield acceleration, an accelerated electron beam induces its own wakefield and cancels the laser-driven wakefield. Assuming the beam loading efficiency defined by the fraction of plasma wave energy absorbed by particles of the bunch with a root mean square (r.m.s.) radius , the beam-loaded field is given by , where is the accelerating field without beam loading, given by for the bubble regime . Thus, a loaded charge is calculated as^[29]

(15)

Using the plasma density Equation (7), the loaded charge is given by

(16)

Therefore, the field reduction factor

for accelerating charge

up to energy

is obtained by solving the equation

(17)

where the coefficient

is defined as

(18)

2.3. Injector stage

Electron beams can be produced and accelerated in the injector stage driven by the same laser pulse as that in the accelerator stage, relying on a self-injection mechanism such as the expanding bubble self-injection mechanism^[30] or an ionization-induced injection scheme with a short mixed gas cell^{[14–16, 31]}, where tunnel ionization leads to electron trapping near the centre of the laser wakefield. Here we consider the ionization-induced injection scheme. According to theoretical considerations in ionization-induced injection^[31], for trapping electrons ionized at the peak of the laser electric field, the minimum laser intensity is given by . At a plasma density in the injector, the required minimum laser field is . The maximum number of trapped electrons saturates at approximately at a gas length for a plasma density with a nitrogen concentration of and laser parameters of and due to the beam loading effects and initially trapped particle loss from the separatrix in the phase space. From the particle-in-cell (PIC)-simulation results^[31], the trapped electron density scales as

(19)

The energy spread is also proportional to both the mixed gas length and the nitrogen concentration. In a injector with gas length

, the electron charge

trapped inside a bunch with radius

is estimated as

(20)

An electron charge of 500 pC will be trapped via the ionization-induced injection mechanism in an injector gas cell with a 2 mm length and a nitrogen concentration of

2.4. Design of a SASE FEL

In the SASE FEL process, coupling the electron bunch with a co-propagating undulator radiation field induces an energy modulation of electrons that yields current modulation of the bunch due to the dispersion of the undulator dipole fields, known as microbunching. It means that the electrons are grouped into small bunches separated by a fixed distance that resonantly coincides with the wavelength of the radiation field. Consequently, the radiation field can be amplified coherently. In the absence of an initial resonant radiation field, a seed may build up from spontaneous incoherent emission in the SASE process.

The design of the FEL-based EUV light source is carried out using one-dimensional FEL theory as follows^[32]. The FEL amplification takes place in an undulator with undulator period and peak magnetic field at a resonant wavelength given by

(21)

where

is the relativistic factor of the electron beam energy

, and

is the undulator parameter, which is related to the maximum electron deflection angle

. In the high-gain regime required for the operation of a SASE FEL, an important parameter is the Pierce parameter

, given by

(22)

where

is the beam current,

is the Alfven current,

is the r.m.s transverse size of the electron bunch, and the coupling factor is

for a helical undulator and

for a planar undulator, where

and

are Bessel functions of the first kind. Another important dimensionless parameter is the longitudinal velocity spread

of the beam normalized by the Pierce parameter:

(23)

where

is the relativistic r.m.s. energy spread,

is the r.m.s. transverse emittance,

is the beta function provided by the guiding field (undulator plus external focusing) and

is the normalized emittance, defined as

, assuming that the beta function is constant along the length of the undulator. The

-folding gain length

over which the power grows exponentially according to

is given by

(24)

In order to minimize the gain length, one needs a large Pierce parameter

and a normalized longitudinal velocity spread

sufficiently low compared to unity, which means a sufficiently small energy spread

and

. This expression applies to a moderately small beam size

such that the diffraction parameter

, where

is defined as

(25)

The saturation length

required to saturate the amplification can be expressed as

(26)

where

and

are the input power and the saturated power, which are related to the electron beam power

according to

(27)

where

is the number of electrons per wavelength, given by

Case	A	B	C	D	E
Laser
Laser wavelength ()	1	1	1	1	1
Average laser power (MW)	1.63	1.24	1.19	1.22	1.27
Repetition rate (MHz)	1.22	0.515	0.315	0.223	0.168
Laser energy per pulse (J)	1.34	2.40	3.79	5.52	7.57
Peak power (TW)	29	43	59	75	93
Pulse duration (fs)	46	56	65	73	82
Matched spot radius ()	19	23	27	30	34
Laser plasma accelerator
Electron beam energy (MeV)	243	427	659	937	1257
Plasma density ()	8.3	5.6	4.2	3.2	2.6
Accelerator length (mm)	18	32	51	74	102
Charge per bunch (nC)	0.5	0.5	0.5	0.5	0.5
Field reduction factor	0.223	0.267	0.302	0.325	0.364
Bunch duration (fs)	10	10	10	10	10
Energy spread (%)	1.1	1.5	1.6	1.6	1.6
Transverse beam size ()	25	25	25	25	25
Peak current (kA)	50	50	50	50	50
Average beam power (kW)	148	110	104	104	105
Efficiency of laser to beam (%)	9.1	8.9	8.7	8.5	8.3
Free electron laser
Undulator period (mm)	5	10	15	20	25
Radiation wavelength (nm)	13.5	13.5	13.5	13.5	13.5
Gap (mm)	1	2	3	4	5
Peak magnetic field (T)	1.425	1.425	1.425	1.425	1.425
Undulator parameter	0.666	1.33	2.00	2.66	3.33
Pierce parameter (%)	1.12	1.51	1.60	1.60	1.57
Gain length (mm)	41	61	86	115	146
Saturation length (mm)	499	721	1016	1355	1723
Number of periods	100	72	68	68	69
Spectral bandwidth (%)	1.0	1.4	1.5	1.5	1.5
R.m.s. radiation cone angle ()	116	97	82	71	63
Input power (MW)	0.94	3.03	5.26	7.48	9.72
Saturated power (GW)	82	194	317	451	596
Duration of EUV pulse (fs)	10	10	10	10	10
Average EUV power (kW)	1	1	1	1	1
Efficiency of EUV generation (%)	0.061	0.081	0.084	0.082	0.079

Table 1. Parameters for laser plasma accelerator-based EUV FEL light sources.

View all Tables

For an EUV light source based on a FEL, a planar undulator comprising alternating dipole magnets is used, e.g., a pure permanent magnet (PPM) undulator with (Nd–Fe–B) blocks or a hybrid undulator comprising PPMs and ferromagnetic poles, e.g., a high saturation cobalt steel such as Vanadium Permendur or a simple iron. For a hybrid undulator, the thickness of the pole and magnet is optimized in order to maximize the peak field. The peak field of the gap is estimated in terms of the gap and period according to for a gap range , where , and for the hybrid undulator with Vanadium Permendur. Table 1 summarizes design examples for a fiber laser-driven laser plasma accelerator-based FEL-produced EUV radiation source at 13.5 nm wavelength using undulators with periods 5 mm (Case A), 10 mm (Case B), 15 mm (Case C), 20 mm (Case D), and 25 mm (Case E), all cases of which have the same gap:period ratio 0.2, e.g., mm (Case A), 2 mm (Case B), 3 mm (Case C), 4 mm (Case D), and 5 mm (Case E), respectively. The bunch duration of the electron beam in the injector stage at a plasma density of is assumed to be 10 fs full-width at half-maximum (FWHM), based on a measurement of the electron bunch duration in a recent laser wakefield acceleration experiment^[33]. The relative energy spread of the accelerated electron beam with an injection energy of , where is the final beam energy in the accelerator stage, is assumed to be of the order of 10% in the injector stage. After acceleration up to 10 times higher energy in the accelerator stage, the relative energy spread at the final beam energy is reduced to due to adiabatic damping in the longitudinal beam dynamics. The transverse beam size is tuned by employing a beam focusing system. Figure 1 shows a schematic of the EUV light source based on a compact FEL driven by a fiber laser-based plasma accelerator.

2.5. Design of all-optical Gamma-beam source

The design of a Gamma-beam source based on inverse Compton scattering is carried out by using a result of quantum electrodynamics on photon–electron interactions, namely, the Klein–Nishina formula, which gives the differential cross section of photons scattered from a single electron in the lowest order of quantum electrodynamics. In Compton scattering of a laser photon with energy ( for laser wavelength off a beam electron, the maximum energy of the scattered photon is given by , where is the relativistic factor for an electron beam energy with electron rest mass and the factor . In the laboratory frame, the differential cross section of Compton scattering^[34] is given by

(28)

where

is the energy of a scattered photon normalized by the maximum photon energy and

(

with the classical electron radius

. In the laboratory frame, the scattering angle

of the photon is given by

. Integrating the differential cross section over

, the total cross section of Compton scattering becomes

(29)

This total cross section leads to a cross section of Thomson scattering

for an electron beam energy

. The fractional cross section for the photon energy range

is given by

(30)

with

. All photons in this energy range are scattered in the forward direction within a half-cone angle

. For an electron beam interacting with a laser pulse at an angle of

in the horizontal plane (

-plane), the luminosity representing the probability of collisions between electron and laser beams per unit cross section per unit time is obtained by

, where

is the number of electrons contained in the electron bunch,

is the number of photons per laser pulse,

is the repetition rate of laser pulses, and

is the area where the two beams overlap, given by

(31)

where

and

are the r.m.s. horizontal and vertical sizes of the electron beam,

is the r.m.s. bunch length of the electron beam,

and

are the r.m.s. horizontal and vertical spot sizes of the laser beam, and

is the r.m.s. pulse length of the laser beam. For a head-on collision providing efficient Gamma-beam production, the crossing angle between the electron and laser beams is chosen to be

. Tuning the beam focusing system and the interaction optics so as to give

, the luminosity turns out to be

, where

is the laser spot radius at the interaction point. Using

and

, where

is the charge of the electron bunch and

is the energy of a scatter pulse with peak power

and duration

, the luminosity is calculated as

(32)

where

is the focused intensity of the scatter pulse at the interaction point. Thus the Gamma-beam flux is given by

(33)

The fractional Gamma-beam flux with photon energy spread

is estimated as

(34)

Table 2 summarizes design examples for an all-optical laser plasma accelerator-based Gamma-beam source at photon energies 2.5 MeV (Case A), 5 MeV (Case B), 10 MeV (Case C), 15 MeV (Case D), and 20 MeV (Case E), respectively. Figure 2 is a schematic illustration of the Gamma-beam source based on inverse Compton scattering off relativistic electron beams driven by a laser plasma accelerator.

Case	A	B	C	D	E
Laser plasma accelerator
Laser wavelength ()	0.8	0.8	0.8	0.8	0.8
Repetition rate (Hz)	10	10	10	10	10
Laser energy per pulse (J)	1.78	2.56	3.68	4.55	5.31
Peak power (TW)	41	52	66	77	85
Pulse duration (fs)	43	49	55	59	62
Matched spot radius ()	18	20	23	24	26
Electron beam energy (MeV)	326	461	654	802	928
Plasma density ()	9.2	7.3	5.7	4.9	4.5
Accelerator length (mm)	24	34	50	61	72
Charge per bunch (nC)	0.5	0.5	0.5	0.5	0.5
Bunch duration (fs)	10	10	10	10	10
Transverse beam size ()	25	25	25	25	25
Compton scatter
Photon energy (MeV)	2.5	5	10	15	20
Laser peak power (TW)	10	10	10	10	10
Pulse duration (fs)	250	250	250	250	250
Pulse energy (J)	2.5	2.5	2.5	2.5	2.5
Laser spot radius ()	25	25	25	25	25
Focused intensity ()	1	1	1	1	1
Repetition rate (Hz)	10	10	10	10	10
Luminosity ()	10	10	10	10	10
Total cross section (mb)	660	658	655	653	651
Total photon flux ()	6.60	6.58	6.55	6.53	6.51
Spectral bandwidth (%)	1.0	1.0	1.0	1.0	1.0
Scattering angle within 1% BW ()	313	222	157	128	111
Cross section within 1% BW (mb)	9.80	9.77	9.73	9.69	9.66
Photon flux within 1% BW ()	0.980	0.977	0.973	0.969	0.966

Table 2. Parameters for all-optical laser plasma accelerator-based Gamma-beam sources.

View all Tables

3. Conclusion

We present methods for producing EUV light at a wavelength of 13.5 nm from a SASE FEL generated by electron beams from a laser plasma accelerator driven by a fiber-based CPA laser and also for producing a Gamma beam with photon energies of 1–20 MeV via inverse Compton scattering off relativistic electron beams from a laser plasma accelerator. For these practical applications of laser plasma accelerators, it is essential to employ high average power, high efficiency drive lasers operating at high repetition pulse rates (of the order of 300 kHz); the corresponding average power of 1 MW means that the EUV FEL is capable of producing an average radiation power of 1 kW at a wavelength of 13.5 nm and the all-optical Gamma beam source can produce a high-quality photon flux of at 10 MeV energy within a 1% bandwidth. One such high average power laser is a coherent combining fiber laser system^[35], comprising a plurality of amplifying fibers wherein an initial laser pulse is distributed and amplified to a 1 mJ level, intended for grouping together the elementary pulses amplified in the fiber in order to form a single amplified global laser pulse with a 1 J level energy.

In both radiation sources, beam transport and imaging from the laser plasma accelerator to the undulator or a focal point of the scatter laser pulse is provided by a beam focusing system that comprises Halbach-type permanent quadrupole magnets made of NdFeB-type rare-earth magnets with a high remanent field^{[36, 37]}. According to simulation results on ionization-induced injection at a plasma density ^[31], the normalized emittance is assumed to be inside the wakefield. The transverse beam size in the beam transport optics is given by , where is the beta function of the beam optics at the undulator or the scattering point. For Case C in Table 1, the beta function should be set to inside the undulator. The electron beam, after passing through the undulator or being scattered by the scatter laser pulse, is bent by the dipole field of a permanent magnet (a beam separator) made of NdFeB material and dumped to a copper beam dump with a water cooling element, while the EUV radiation or the Gamma beam is extracted from a beam separator and directed to an EUV lithography scanner or a photon beam irradiation system.

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