Free-electron laser driven by fiber laser-based laser plasma accelerator

Abstract

A Free Electron Laser source includes: a fiber-based laser having a plurality of amplifying fibers wherein an initial laser pulse is distributed and amplified, and element for grouping together the elementary pulses amplified in the fiber in order to form an a single amplified global laser pulse; a laser plasma accelerator wherein the global laser pulse generates relativistic electron beams, a beam focusing system transporting electron beams from the laser plasma accelerator, an undulator wherein relativistic electron beams generate an electromagnetic beam, and a beam separator system, wherein the electron beam and the electromagnetic beam are separated.

Claims

1. A Free Electron Laser source generating an electromagnetic beam presenting a wavelength, called Free Electron Laser wavelength λ.sub.X, belonging to the range from 5 nm to 15 nm, said Free Electron Laser comprising: a fiber-based laser, comprising a plurality of amplifying fibres wherein an initial laser pulse is distributed and amplified, and means for grouping together elementary pulses amplified in said fibre in order to form a single amplified global laser pulse, and comprising a stretching device able to stretch out in time said initial laser pulse, according to a chirped pulse amplification technique and a grating pulse compressor able to compress in time said single amplified global laser pulse, according to the chirped pulse amplification technique, at least one parameter of said fiber-based laser, called a fiber-based laser parameter, being obtained as a function of said Free Electron Laser wavelength λ.sub.X; a laser plasma accelerator wherein, in a bubble regime of said plasma accelerator said global laser pulse generates electron beams, at least one parameter of said laser plasma accelerator, called a laser plasma accelerator parameter, being obtained as a function of said Free Electron Laser wavelength λ.sub.X; a beam focusing system transporting said electron beams from the laser plasma accelerator to an undulator, said undulator, wherein said electron beams generate said electromagnetic beam, a peak magnetic field and a period of said undulator being previously set, at least one parameter of said undulator, called an undulator parameter, being obtained as a function of said Free Electron Laser wavelength λ.sub.X; and a beam separator system, wherein said electron beams and said electromagnetic beam are separated.

2. A Free Electron Laser source according to claim 1 wherein said laser plasma accelerator comprises: a first gas cell filled with mixed gas, and a second gas cell filled with pure helium gas, a gas feeding system.

3. A Free Electron Laser source according to claim 2 wherein said laser plasma accelerator comprise means for modifying a length of said second gas cell.

4. A Free Electron Laser source according to claim 1 wherein said beam separator system comprises a dipole magnet for bending electron beams and a beam dump.

5. A Free Electron Laser source according to claim 1 wherein said electromagnetic beam is a Extreme UltraViolet beam.

6. A Free Electron Laser source according to claim 5 wherein said Extreme UltraViolet beam wavelength is 13.5 nm.

7. A Free Electron Laser source according to claim 1 wherein said beam wavelength is 6.7 nm.

Description

5. BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention will become more fully understood from the following description of preferred but non-limiting embodiments thereof, described in connection with accompanying drawings, wherein:

(2) FIG. 1 shows a schematic diagram of a Free Electron Laser according to an embodiment of the invention, usable as an Extreme UltraViolet light source.

(3) FIG. 2 illustrates a schematic diagram of a coherent combining fiber-based drive laser used in the Free Electron Laser of FIG. 1.

(4) FIG. 3 shows the two-stage gas cell plasma accelerator of the Free Electron Laser of FIG. 1.

(5) FIG. 4 shows the laser plasma electron accelerator system of the Free Electron Laser of FIG. 1, comprising the gas cell plasma accelerator of FIG. 3.

(6) FIG. 5 illustrates schematically electron acceleration mechanism due to laser wakefield.

(7) FIG. 6 shows a schematic view of a miniature permanent magnet quadrupole (PMQ).

(8) FIG. 7 shows an example of a beam focusing system comprising four permanent magnet quadrupoles (PMQ) of the type represented by FIG. 6.

(9) FIG. 8 shows schematically a planar undulator made of pure permanent magnets producing a vertical field.

(10) FIG. 9 shows schematically a hybrid planar undulator made of permanent magnets and ferromagnetic material, e.g., iron or cobalt steel.

(11) FIGS. 10A and 10B show a schematic assembly of the undulator used in the Free Electron Laser of FIG. 1.

(12) FIGS. 11A, 11B and 11C shows a monolithic beam dump system comprising a permanent magnet dipole separating electron beams from Extreme UltraViolet radiation and a electron beam dump.

(13) FIG. 12 illustrates a setup of a compact self-amplified spontaneous emission (SASE) Free-Electron Laser system comprising the beam focusing system of FIG. 4, the compact undulator of FIG. 5 and the beam dump system of FIG. 6.

(14) FIG. 13 illustrates schematically the Extreme UltraViolet light source of FIG. 1, based on a compact Free-Electron Laser driven by a fiber laser-based plasma accelerator according to aspects of an embodiment of the present invention,

(15) FIG. 14 illustrates how the parameters of each component constituting the Free-Electron Laser of FIG. 1, are obtained as a function of the Free-Electron Laser wavelength according to aspects of an embodiment of the present invention

6. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS OF THE INVENTION

6.1 General Description of a Free-Electron Laser According to an Embodiment of the Invention

(16) With initial reference to FIG. 1, there is shown a schematic diagram of a proposed Free-Electron Laser according to one aspect of an embodiment. As shown in FIG. 1, and described in further detailed below, the proposed Free-Electron Laser may include a drive laser system 1 for generating high energy laser pulses at a high repetition rate of the order of 1 MHz and delivering intense laser pulses 3 compressed through a pulse compressor chamber 2 into a laser plasma accelerator chamber 4. Such a laser-driven Free-Electron Laser can be used as an Extreme UltraViolet light source.

(17) Fibre-Type Laser

(18) FIG. 2 shows an item of drive laser system 1 for producing high-energy laser pulses.

(19) In this drive laser system 1, a low-energy laser pulse 810 is produced by an oscillator 81. This pulse 810 is then stretched out in time, according to the chirped pulse amplification (CPA) technique, by a stretching device 82, comprising a pair of diffraction gratings 821 and 822, having the effect of offsetting in time the various spectral components of the original low-energy pulse 810. The stretched pulse 820 then has a lower peak power and a longer duration of the pulse 810.

(20) This laser pulse 820 is then distributed in a plurality of amplifying fibres 831 forming a first amplifying stage 83 of the fibre-type laser amplifier. The various fibres are separated from one another so as to make it possible to cool them effectively. Each of the amplifying fibres 831 comprises a core made from doped material, and is optically pumped, so as to optically amplify the laser pulse flowing in the fibre. The pulse passing through each of these amplifying fibres 831 is then amplified, and is then itself distributed in a plurality of amplifying fibres 841 forming the second amplification stage 84 of the fibre-type laser amplifier. Once again, the pulse passing through each of these amplifying fibres 841 is amplified and is then itself distributed in a plurality of amplifying fibres 851 forming the third amplification stage 85 of the fibre-type laser amplifier. Thus, in each amplification stage, the pulse is amplified in a plurality of fibres independent of one another and then divided so that pulses of lower power are transmitted to each of the higher-level amplification fibres.

(21) The third and last amplification stage 85 then comprises a very large number of amplifying fibres, for example around 10.sup.6. Each of the amplifying fibres of this third stage is extended by a transmission fibre having a very low loss level. The transmission fibres are collected together in a cluster 86 so that the pulses emerging from each of the ends of these transmission fibres are emitted in parallel and juxtaposed. These pulses then form a single amplified global pulse 860.

(22) This amplified global pulse 860 is compressed timewise by a compressor 87, located in the pulse compressor chamber 2. This compressor 87 comprises a pair of diffraction gratings 871 and 872, grouping together in time the various spectral components of the pulse. The pulse 3 emerging from this temporal compressor 17 then has a very high energy and very short duration.

(23) Two Stages Gas Cell Plasma Accelerator with Ionization-Induced Injection

(24) As shown in FIG. 1, the ultrashort intense laser pulse 3 is focused by an off-axis parabolic mirror 5 on the entrance of a two-stage gas cell 6, of which the first cell referred to an injector is filled with a mixed gas, e.g., helium gas mixed with nitrogen, and the second cell referred to as an accelerator is filled with a pure gas, e.g., hydrogen or helium. The gases are fed through a gas flow control system 7 to the two-stage gas cell separately at the different pressures.

(25) As described below, in the injector of the gas cell 6, the said laser pulse 3 excites large-amplitude plasma wakefields, of which an accelerating electric field can trap plasma electrons exclusively out of the inner shell electrons and accelerate them owing to ionization-induced injection. A pre-accelerated electron beam 10 from the injector is further accelerated to the relativistic energy of the order of 1 GeV in the accelerator stage of the gas cell 6, where the laser pulse generates plasma wakefields of the order of 1 GV/cm. A transmitted laser light is directed through a mirror with a beam hole 8 to a recovery box 9 that includes diagnostics and absorbers of the transmitted laser pulses.

(26) This plasma accelerator is particularly advantageous when it is combined with a fiber-based laser, a beam focusing system, an undulator and a beam separator system according to the invention. However, such a laser plasma accelerator comprising a first gas cell filled with mixed gas, and a second gas cell filled with pure helium gas can also be combined with an other type of laser, for producing relativistic electron beams.

(27) Beam Focusing System, Undulator and Separation Chamber

(28) The output electron beam 10 from the laser plasma accelerator chamber 4 is transported into an undulator 13 through a beam focusing system 12, installed in a radiation chamber 11. As described below, the electron beam 10 focused by quadrupole magnetic field of the beam focusing system 12 generates the resonantly amplified Extreme UltraViolet radiation 14 due to self-amplified spontaneous emission (SASE) mechanism when passing through the alternating dipole magnetic fields of the undulator 13 that force the electron bunch on a sinusoidal trajectory.

(29) After passing through the undulator 13, the electron bunch is decelerated so strongly that it becomes non-resonant and could not contribute to the amplification of the Extreme UltraViolet radiation, i.e., the onset of saturation. The decelerated electron beam 10 is separated from the Extreme UltraViolet radiation 14 in the dipole magnetic field of the deflection magnet 16 and dumped to a beam dump 17, while the saturated Extreme UltraViolet radiation 14 is extracted from a beam separation chamber 15 and directed to a Extreme UltraViolet lithography scanner/stepper.

6.2. Detailed Description of the Laser Plasma Accelerator Chamber

(30) Two-Stage Gas Cell

(31) FIG. 3 shows schematically a two-stage gas cell 6 comprising an injector stage 21 and an accelerator stage 24 for efficient electron trapping and acceleration in laser wakefields. As shown in FIG. 4, the two-stage gas cell is set up in a laser plasma accelerator chamber 4 and a compressed laser pulse 3 from the pulse compressor chamber 2 is focused on the entrance of the injector gas cell by an off-axis parabolic mirror 5 with a F-number, e.g., 20.

(32) The injector cell 21 is filled with a mixed gas, e.g., 98% He and 2% N.sub.2, fed through a gas feedthrough 20 from the gas flow control system 7. The accelerator cell 24 is filled with a pure gas, e.g. H.sub.2 or He, fed through a gas feedthrough 24 from the gas flow control system 7. A length of the accelerator stage is variably adjusted with a bellows structure 25 driven by a motorized actuator 26. Such an adjustment can permit to reuse easily the accelerator stage, using said motorized actuator 26, for different Free-Electron laser wavelengths ranging from 5 nm to 15 nm. The laser plasma accelerator chamber 4 is pump out by a vacuum pump system 27 to keep an inside pressure of 10.sup.−3-10.sup.−4 Pa.

(33) Description of the Physical Process

(34) FIG. 5 illustrates schematically a physical process 100 for the wakefield excitation and electron trapping and acceleration in wakefields, which are generated when an intense laser pulse propagates a neutral mixed gas in the injector 21. In FIG. 5, the evolution of plasma electron density is shown in the upper plot 101 and the excited longitudinal wakefield is shown in the lower plot 102.

(35) As shown in the central part 100 of FIG. 5, He and the outer shell electrons up to N.sup.5+ are fully ionized in the leading front of the laser pulse with intensity 1.5×10.sup.16 W/cm.sup.2 to produce plasma electrons in the outer region of the laser pulse, of which the boundary is indicated by a thin dotted line 103. Since two inner shell (K-shell) electrons of N.sup.6+ and N.sup.7+ are ionized at the laser intensity higher than 1×10.sup.19 W/cm.sup.2, the inner shell electrons are produced only near the peak intensity of the laser pulse 3, of which the intensity profile is indicated by a thick dotted line 108 for the normalized laser field a.sub.0≃0.855×10.sup.−9I.sup.1/2 [W/cm.sup.2]λ.sub.L [μm]=2 where I [W/cm.sup.2] is the intensity and λ.sub.L [μm] is the laser wavelength. In the upper plot 101, a thick solid curve 109 indicates the evolution of ionization level of nitrogen (the electron number of ionized nitrogen atom) along the propagation axis. A boundary of the plasma region containing the inner shell electrons from ionized N.sup.6+ and N.sup.7+ is indicated by a thin dashed line 104.

(36) Plasma electrons contained in the boundary 103 are blown out by radiation pressure (ponderomotive force) of the laser pulse 3 with the relativistic intensity a.sub.0 1 and form a narrow dense electron sheath surrounding a spherical ion column behind the laser pulse, often referred to as a bubble 105. Such charge separation generates a strong longitudinal electric field 110 of the order of 100 GV/m at a plasma electron density of 10.sup.18 cm.sup.−3, which is three orders of magnitude higher than an accelerating field of conventional RF accelerators. In the bubble 105, an electron undergoes a strong focusing force simultaneously. Hence, once electrons 10 are trapped into a bubble, they are efficiently accelerated up to high energy of the order of 1 GeV over a dephasing length of the order of 1 cm, where accelerated electrons outrun a proper accelerating phase.

(37) The said inner shell electrons from ionized N.sup.6+ and N.sup.7+ are located near the bubble center on the propagation axis, where the wake potential is a maximum and the expelling ponderomotive force of the laser pulse is a minimum. Contrary to pre-ionized free electrons, whose trajectories move along a narrow sheath outside the bubble, the ionized electrons emitted from the inner shell move close to the bubble axis toward the back of the bubble where the wake potential is a minimum, and eventually trapped into the wakefield in condition that electrons gain a sufficient kinetic energy required for trapping, as shown in the electron trajectory 106, while the electron shown in the trajectory 107, ionized earlier and off-axis, slips over the potential well and is not trapped. This mechanism called as ionization-induced injection occurs at the intensity as low as the optical field ionization threshold for the inner shell electrons of impurity gas and significantly increases the trapped charge. As trapping occurs close to the bubble axis, amplitudes of the betatron oscillation after trapping decrease compared to the self-injection from the electron sheath. According to theoretical considerations on the ionization-induced injection, for trapping electrons ionized at the peak of the laser electric field, the minimum laser intensity is given by 1−γ.sub.p.sup.−1≦0.64a.sub.0.sup.2, where γ.sub.p is the Lorentz factor defined as γ.sub.p=(1−β.sub.p.sup.2).sup.−1/2 and β.sub.p is the phase velocity of the plasma wave. For electrons to be trapped at or in front of the laser envelope, the intensity must be a.sub.0≧21.7 for γ.sub.p=33. The 1D PIC simulations show that the maximum number of trapped electrons is saturated to be approximately N.sub.e max˜5×10.sup.6 μm.sup.−2 at the mixed gas length L.sub.mix=1000λ.sub.0 for the plasma density n.sub.e=0.001n.sub.c (1.7×10.sup.18 cm.sup.−3) with the nitrogen concentration of α.sub.N=1%, and the laser parameters a.sub.0=2 and cτ.sub.0≈15λ.sub.0 due to the beam loading effects and initially trapped particle loss from the separatrix in the phase space, where λ.sub.0 is the laser wavelength and n.sub.c is the critical plasma density defined as n.sub.c=m.sub.eω.sub.L.sup.2/4πe.sup.2=π/(r.sub.eλ.sub.L.sup.2)≃1.115×10.sup.21 [cm.sup.−3]/(λ.sub.L [μm]).sup.2. The number of trapped electrons scales as N.sub.e [μm.sup.−2] 8×10.sup.7α.sub.Nk.sub.pL.sub.mix(n.sub.e/n.sub.c).sup.1/2 for α.sub.Nk.sub.pL.sub.mix≦2. The energy spread is also proportional to both the mixed gas length and the nitrogen concentration. According to the 2D-PIC simulation for a.sub.0=2, the energy spread of a trapped electron beam may scale as δE/E=0.02[%](L.sub.mix/λ.sub.L)(n.sub.e/10.sup.17 cm.sup.−3).sup.−1/2, while the transverse normalized emittance is estimated to be ε.sub.n0≈0.5 [μm]a.sub.0.sup.1/2(n.sub.e/10.sup.17 [cm.sup.−3]).sup.−1/2.

(38) In the bubble (blowout) regime for a.sub.0≧2, 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 R.sub.B is approximately given as k.sub.pR.sub.B≈2√{square root over (a.sub.0)}, where k.sub.p=(4πr.sub.en.sub.e).sup.1/2 is the plasma wavenumber evaluated with the unperturbed on-axis density n.sub.e, and the classical electron radius r.sub.e=e.sup.2/m.sub.ec.sup.2=2.818×10.sup.−13 cm with electron charge e, mass m.sub.e and vacuum light velocity c. The accelerating field E.sub.z is given by E.sub.z/E.sub.0=(½)αk.sub.pR.sub.B, where E.sub.0=mcω.sub.p/e≈96 [GV/m](n.sub.e/10.sup.18 [cm.sup.−3]).sup.1/2 and α represents a factor taking into account the beam loading and the difference between the simulation and theoretical estimation. The maximum energy gain limited due to dephasing is given by
Δγ.sub.max=W.sub.max/m.sub.ec.sup.2≈(⅔)ακ.sub.selfa.sub.0(n.sub.c/n.sub.e),
where κ.sub.self=(a.sub.0.sup.2/8){(1+a.sub.0.sup.2/2).sup.1/2−1−ln([(1+a.sub.0.sup.2/2).sup.1/2+1]/2)}.sup.−1 is a correction factor of the group velocity for a self-guided relativistic laser pulse, of which the relativistic factor related to the group velocity β.sub.g=v.sub.g/c is given by γ.sub.g.sup.2=1/(1−β.sub.g.sup.2)≈κ.sub.self(ω.sub.L.sup.2/ω.sub.p.sup.2)=κ.sub.self(n.sub.c/n.sub.e)=κ.sub.chγ.sub.g0.sup.2, where γ.sub.g0=ω.sub.L/ω.sub.p is the relativistic factor for the linear group velocity for a.sub.0.sup.2 1. The dephasing length L.sub.dp for self-guided bubble regime is given by k.sub.pL.sub.d≈(⅔)k.sub.pR.sub.Bγ.sub.g.sup.2=(4/3)√{square root over (a.sub.0)}κ.sub.self(n.sub.c/n.sub.e). The important parameters of a laser plasma accelerator for reaching a given energy E.sub.b are summarized as follows:

(39) The operating plasma density is determined by

(40) $n_e=\frac{2}{3}\alpha {\kappa }_{\mathrm{self}}{a}_0\frac{n_c}{\Delta {\gamma }_{\mathrm{ma}x}}\approx 1.9\times {10}^{18}\left[{\mathrm{cm}}^{-3}\right]{\kappa }_{\mathrm{self}}{a_0\left(\frac{1\mathrm{\mu m}}{{\lambda }_L}\right)}^2\left(\frac{200\mathrm{MeV}}{E_b/\alpha }\right).$

(41) The accelerator length is set to be equal to the dephasing length as

(42) $L_{\mathrm{acc}}=L_{\mathrm{dp}}\approx \sqrt{\frac{3}{2}}\frac{{\left(\Delta {\gamma }_{\mathrm{ma}x}/\alpha \right)}^{3/2}}{\pi {\kappa }_{\mathrm{self}}^{1/2}{a}_0}{\lambda }_L\approx \frac{3.1\left[\mathrm{mm}\right]}{{\kappa }_{\mathrm{self}}^{1/2}{a}_0}\left(\frac{{\lambda }_L}{1\mathrm{\mu m}}\right){\left(\frac{E_b/\alpha }{200\mathrm{MeV}}\right)}^{3/2}$

(43) The pump depletion length due to pulse front erosion becomes

(44) $L_{\mathrm{pd}}\approx c{\tau }_L\frac{n_c}{n_e}=\frac{3}{2}\frac{c{\tau }_L{\mathrm{\Delta \gamma }}_{\max }\text{/}\alpha }{{\kappa }_{\mathrm{self}}{a}_0}\approx \frac{5\left[\mathrm{mm}\right]}{{\kappa }_{\mathrm{self}}{a}_0}\left(\frac{{\tau }_L}{30\mathrm{fs}}\right)\left(\frac{E_b\text{/}\alpha }{200\mathrm{MeV}}\right)$

(45) The pulse duration required for satisfying a dephasing length longer than a pump depletion length is

(46) ${\tau }_L\ge 18\left[\mathrm{fs}\right]{\kappa }_{\mathrm{self}}^{1\text{/}2}\left(\frac{{\lambda }_L}{1\mu m}\right){\left(\frac{E_b\text{/}\alpha }{200\mathrm{MeV}}\right)}^{1\text{/}2}$

(47) The matched spot radius is given by

(48) $r_m\approx 3.9\left[\mu m\right]\frac{R_m}{\sqrt{{\kappa }_{\mathrm{self}}{a}_0}}\left(\frac{{\lambda }_L}{1\mu m}\right){\left(\frac{E_b\text{/}\alpha }{200\mathrm{MeV}}\right)}^{1\text{/}2}$$\mathrm{where}$$R_m=k_p{r}_m={\left\{\frac{\ln \left(1+a_0^2\text{/}2\right)}{\sqrt{1+a_o^2\text{/}2}-1-2\ln \left[\left(\sqrt{1+a_0^2\text{/}2}+1\right)\text{/}2\right]}\right\}}^{1\text{/}2}$

(49) The corresponding matched power is

(50) $P_L=\frac{k_p^2{r}_L^2{a}_0^2}{32}{P}_c\approx 0.312\left[\mathrm{TW}\right]\frac{a_0{R}_m^2}{{\kappa }_{\mathrm{self}}}\left(\frac{E_b\text{/}\alpha }{200\mathrm{MeV}}\right)$

(51) The required laser pulse energy is given by U.sub.L=P.sub.Lτ.sub.L.

(52) Assuming the beam loading efficiency η.sub.b ≡1−E.sub.z.sup.2/E.sub.M.sup.2 defined by the fraction of the plasma wave energy absorbed by particles of the bunch with the root mean square (r.m.s) radius τ.sub.b, the beam-loaded field is given by E.sub.z=√{square root over (1−η.sub.b)}E.sub.M=αE.sub.M, where E.sub.M is an accelerating field without beam loading. Thus a loaded charge is calculated as

(53) $Q_b\approx \frac{e}{4k_L{r}_e}\frac{{\eta }_b{k}_p^2{\sigma }_b^2}{\left(1-{\eta }_b\right)}\frac{E_z}{E_0}{\left(\frac{n_c}{n_e}\right)}^{1\text{/}2}\approx 76\left[\mathrm{pC}\right]\frac{{\eta }_b{k}_p^2{\sigma }_b^2}{\left(1-{\eta }_b\right)}\frac{E_z}{E_0}{\left(\frac{n_e}{{10}^{18}\left[{\mathrm{cm}}^{-3}\right]}\right)}^{-1\text{/}2}.$

(54) Using the plasma density n.sub.e, the loaded charge is given by

(55) $Q_b\approx 55\left[\mathrm{pC}\right]\frac{{\eta }_b{k}_p^2{\sigma }_b^2}{{\kappa }_{\mathrm{self}}^{1\text{/}2}\sqrt{1-{\eta }_b}}\left(\frac{{\lambda }_L}{1\mu m}\right){\left(\frac{E_b\text{/}\alpha }{200\mathrm{MeV}}\right)}^{1\text{/}2}\approx 55\left[\mathrm{pC}\right]\frac{1-{\alpha }^2}{a^{3\text{/}2}}\frac{k_p^2{\sigma }_b^2}{{\kappa }_{\mathrm{self}}^{1\text{/}2}}\left(\frac{{\lambda }_L}{1\mu m}\right){\left(\frac{E_b}{200\mathrm{MeV}}\right)}^{1\text{/}2}.$

(56) A field reduction factor α for accelerating a charge of electrons Q.sub.b up to an energy E.sub.b is obtained from α.sup.2+Cα.sup.3/2−1=0, where C≡(Q.sub.b/55 pC)κ.sub.self.sup.1/2(k.sub.p.sup.2τ.sub.b.sup.2).sup.−1(1 μm/λ.sub.L).sup.−1(E.sub.b/200 MeV).sup.−1/2.

6.3. Detailed Description of the Beam Focusing System, Undulator and Beam Separator

(57) Beam Focusing System

(58) Beam transport and imaging from the laser plasma accelerator 6 to the undulator 13 is provided by a beam focusing system 12 with short focal length. The field gradient of the two dimensional Halbach-type permanent quadrupole magnet (PMQ) as shown in FIG. 6 is given by B′=2B.sub.r(r.sub.i.sup.−1−r.sub.o.sup.−1), where B.sub.r is the tip field strength, r.sub.i is the bore radius and r.sub.o is the outer radius of PMQ. With B.sub.r=1.45 T for NdFeB material and r.sub.i=2.5 mm, one can obtain the field gradient B′=1160 [T/m](1−2.5 [mm]/r.sub.o). FIG. 6 illustrates schematically a twelve segments Halbach-type PMQ 31, 32 of a set of the quadrupole magnet 36, including a housing 33, 34 and a bracket 35 for supporting and positioning the PMQ. The quadrupole field is composed of four radially wedges of permanent magnet material 31, e.g., Nd.sub.2Fe.sub.14B or SmCO, with a high remanent field, of which the direction of magnetization is indicated by arrows. The outer field closure is formed by eight wedges of permanent magnet material 32. Since the four main wedges are strongly attracted toward of the center of the quadrupole, their mechanical precision and field accuracy can be achieved by the insertion of a nonmagnetic precision cylinder 33 into the center of the PMQ and by the housing case 34 outside the permanent quadrupole magnet (PMQ).

(59) As shown in FIG. 7, the beam focusing system 12 comprises two to four permanent quadrupole magnets (PMQ), e.g. a doublet (FD), a triplet (FDF), or a quadruplet (FFDD) 36-39 installed into a housing chamber 46 with water cooling tubes 44. The longitudinal position of each permanent quadrupole magnet (PMQ) along the electron beam axis is optimized with a computer controlled mover system 40-43, comprising a vacuum linear motion manipulator driven by a stepping motor. Alignment of permanent quadrupole magnets (PMQ) is precisely constrained by a rail system 45.

(60) Undulator

(61) For a Extreme UltraViolet light source based on Free-Electron Laser, a planar undulator comprising alternating dipole magnets 52 is used, e.g., a pure permanent magnet (PPM) undulator with Nd.sub.2Fe.sub.14B blocks 50 as shown in FIG. 8 or a hybrid undulator comprising pure permanent magnets 50 and ferromagnetic poles 51 as shown in FIG. 9, 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. In FIGS. 8 and 9, the arrows represent the direction of magnetization in the magnet blocks, of which a period is λ.sub.u. The minimum distance between the magnet jaws is a gap g. The peak field B.sub.u of the gap is estimated in terms of the gap g and period λ.sub.u according to B.sub.u=a [T]exp[b(g/λ.sub.u)+c(g/λ.sub.u).sup.2] for gap range 0.1<g/λ.sub.u<1, where a=2.076 T, b=−3.24, c=0 for the pure permanent magnets planar undulator, a=3.694 T, b=−5.068, c=1.520 for the hybrid undulator with vanadium permendur, and a=3.381 T, b=−4.730, c=1.198 for the hybrid undulator with iron.

(62) As shown in FIGS. 10A and 10B, the undulator 13 comprises a rectangular box frame 53, a gap adjusting mechanism 54 and cooling elements 55. The permanent magnet blocks 52 is attached to a thick base plate of the box frame 53 made of aluminium material. The alignment and gap of the undulator are adjusted by controlling the distance between two base plates of the box frame 53 with 4 or 6 adjusting mechanisms 54. Two monolithic water cooling elements 55 fabricated from tubing are connected to each magnet block 52.

(63) Beam Separator

(64) As shown in FIGS. 11A, 11B and 11C, a decelerated electron beam 10 after saturation is bent by a dipole field of permanent magnet (a beam separator 16) made of NdFeB material and dumped to a beam dump 17 made of copper with a water cooling element 56. The permanent magnet dipole (PMD) 16, e.g., Halbach-type permanent magnet dipole, comprises 8 wedges of the NdFeB material, of which the magnetization direction is shown by the arrows in FIG. 11B. The dipole field B.sub.D of the Halbach-type permanent magnet dipole is given by B.sub.D=B.sub.r ln(r.sub.o/r.sub.i), where B.sub.r is the tip field strength, r.sub.i is the bore radius and r.sub.0 is the outer radius of the PMD. With B.sub.r=1.45 T for NdFeB material, r.sub.i=5 mm and r.sub.o=100 mm, one can obtain the dipole field B.sub.D=4.34 T. The mechanical precision and field accuracy of the permanent magnet dipole 16 can be achieved by the insertion of a nonmagnetic precision cylinder 57 into the center of the permanent magnet dipole and by the housing case 58 outside the permanent magnet dipole. The electron beam 10 bent by the permanent magnet dipole field 16 is dumped to the copper beam dump 17, while the Extreme UltraViolet radiation 14 is extracted through a narrow Extreme UltraViolet output hole 59 machined in the beam dump 17 at the edge of the permanent magnet dipole bore. The permanent magnet dipole length required for deflecting d≈2r.sub.i [mm] is given by L.sub.PMD [cm]=10[(r.sub.i/3.26 mm)(E.sub.b/1 GeV)].sup.1/2 for B.sub.D=4.34 T (r.sub.o/r.sub.i=20). Since electrons lose energy by a factor of 1/e≃0.37 over the radiation length X.sub.0=1.44 cm via electromagnetic cascades in the copper beam dump, almost all electrons with energy 1 GeV lose their energy inside the copper block with length 10X.sub.0˜15 cm and diameter 7X.sub.0˜10 cm. Both permanent magnet dipole and beam dump are cooled down by the water cooling elements 56.

(65) The Free-Electron Laser Device

(66) As shown in FIG. 12, the Free-Electron Laser device comprising the beam focusing system 12, the undulator 13 and the beam separator 16 and beam dump 17 is installed into vacuum chambers 11, 15 with vacuum pumping systems 28, 29 that maintain a pressure of the order of 10.sup.−4 Pa inside the chamber. The beam dump 17 connected to the beam separator 16 forms a monolithic device.

(67) In self-amplified spontaneous emission (SASE) Free-Electron Laser process, coupling the electron bunch with a copropagating undulator radiation field induces the energy modulation of electrons that yields a current modulation of the bunch due to the dispersion of the undulator dipole fields, called 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. When lacking an initial resonant radiation field, a seed may build up from spontaneous incoherent emission in the self-amplified spontaneous emission (SASE) process.

(68) Design of Free-Electron Laser Based Extreme UltraViolet Light Source

(69) A design of Free-Electron Laser based Extreme UltraViolet light source is made by the one-dimensional Free-Electron Laser theory as follows. The Free-Electron Laser amplication takes place in the undulator with the undulator period λ.sub.u at the resonant wavelength given by

(70) ${\lambda }_X=\frac{{\lambda }_u}{2{\gamma }^2}\left(1+\frac{K^2}{2}\right),$

(71) where γ=E.sub.b/m.sub.ec.sup.2 is the relativistic factor of the electron beam energy E.sub.b, and K.sub.u=0.934B.sub.u [T]λ.sub.u [cm]=γθ.sub.e is the undulator parameter, which is related to the maximum electron deflection angle θ.sub.e.

(72) In the high-gain regime required for the operation of a self-amplified spontaneous emission (SASE) Free-Electron Laser, an important parameter is the Pierce parameter τ.sub.FEL given by

(73) 0${\rho }_{\mathrm{FEL}}={\frac{1}{2\gamma }\left[\frac{I_b}{I_A}{\left(\frac{{\lambda }_u{K}_u{A}_u}{2{\mathrm{\pi \sigma }}_b}\right)}^2\right]}^{1\text{/}3}$

(74) where I.sub.b is the beam current, I.sub.A=17 kA is the Alfven current, σ.sub.b is the root mean square (r.m.s) transverse size of the electron bunch, and the coupling factor is A.sub.u=1 for a helical undulator and A.sub.u=J.sub.0(ξ)−J.sub.1(ξ) for a planar undulator, where ξ=K.sub.u.sup.2/[4(1+K.sub.u.sup.2/2)] and J.sub.0 and J.sub.1 are the Bessel functions of the first kind.

(75) Another important dimensionless parameter is the longitudinal velocity spread Λ of the beam normalized by the Pierce parameter:

(76) ${\Lambda }^2=\frac{1}{{\rho }_{\mathrm{FEL}}^2}\left[{\left(\frac{{\sigma }_{\gamma }}{\gamma }\right)}^2+{\left(\frac{{\mathrm{.Math.\lambda }}_u}{4{\lambda }_X\beta }\right)}^2\right]=\frac{1}{{\rho }_{\mathrm{FEL}}^2}\left[{\left(\frac{{\sigma }_{\gamma }}{\gamma }\right)}^2+{\left(\frac{{\mathrm{.Math.}}_n^2}{2{\sigma }_b^2\left(1+K_u^2\text{/}2\right)}\right)}^2\right],$

(77) where σ.sub.γ/γ is the relative root mean square (r.m.s.) energy spread, ε is the r.m.s. transverse emittance, β=σ.sub.b.sup.2/ε is the beta function provided by the guiding field (undulator plus external focusing) and ε.sub.n is the normalized emittance defined as ε.sub.n≡γε assuming that a beta function is constant along the length of the undulator.

(78) A e-folding gain length L.sub.gain over which the power grows exponentially according to exp(2s/L.sub.gain) is given by

(79) $L_{\mathrm{gain}}=\frac{{\lambda }_u}{4\pi \sqrt{3}{\rho }_{\mathrm{FEL}}}\left(1+{\Lambda }^2\right).$

(80) In order to minimize the gain length, one needs a large Pierce parameter ρ.sub.FEL and a normalized longitudinal velocity spread Λ sufficiently low compared to 1 that means a sufficiently small energy spread σ.sub.γ/γ and ε. This expression applies to moderately small beam size σ.sub.b such that the diffraction parameter B 1 where B is defined as

(81) $B={\frac{16{\pi }^2{A}_u{\sigma }_b^2}{{\lambda }_X{\lambda }_u}\left[\frac{K_u^2\text{/}2}{\gamma \left(1+K_u^2\text{/}2\right)}\frac{I_b}{I_A}\right]}^{1\text{/}2}.$

(82) A saturation length L.sub.sat required to saturate the amplification can be expressed as

(83) $L_{\mathrm{sat}}=L_{\mathrm{gain}}\ln \left[\left(\frac{{\Lambda }^2+3\text{/}2}{{\Lambda }^2+1\text{/}6}\right)\frac{P_{\mathrm{sat}}}{P_{\mathrm{in}}}\right],$

(84) where P.sub.in and P.sub.sat are an input and a saturated power.

(85) The input P.sub.in and saturated power P.sub.sat are related to an electron beam power P.sub.b according to
P.sub.b=γI.sub.bm.sub.ec.sup.2=I.sub.bE.sub.b,
P.sub.sat≅1.37ρ.sub.FELP.sub.bexp(−0.82Λ.sup.2),
P.sub.in≅3√{square root over (4π)}ρ.sub.PEL.sup.2P.sub.b[N.sub.λ.sub.X ln(N.sub.λ.sub.X/ρ.sub.FEL)].sup.−1/2,

(86) where N.sub.λ.sub.X is the number of electrons per wavelength given by N.sub.λ.sub.X=I.sub.bλ.sub.X/(ec).

6.4. Embodiment of a Free-Electron Laser Used as an Extreme UltraViolet Source at 13.5 nm Wavelength

(87) A fiber laser driven Laser Plasma Accelerator (LPA) based Free-Electron Laser produced Extreme UltraViolet radiation source at λ.sub.X=13.5 nm wavelength using the undulator with period λ.sub.u=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 gap-period ratio g/λ.sub.u=0.2, e.g. g=1 mm (Case A), 2 mm (Case B), 3 mm (Case C), 4 mm (Case D) and 5 mm (Case E), respectively. A hybrid undulator comprising NdFeB materials with grade N52, e.g., VACODYM® 722HR, and ferromagnetic materials such as tempered Co—Fe alloys (vanadium permendur), e.g., VACOFLUX® 50, provide the peak magnetic field B.sub.u [T]=3.694exp(−5.068×0.2+1.520×0.2.sup.2)=1.425. The corresponding undulator parameter becomes K.sub.u=0.1331λ.sub.u [mm]=0.6655, 1.331, 1.9965, 2.662, 3.3275□ for λ.sub.u [mm]=5, 10, 15, 20, 25□.

(88) The electron beam energy E.sub.b required for producing the Extreme UltraViolet radiation at the wavelength λ.sub.X=13.5 nm is given by γ=192.45λ.sub.u.sup.1/2(1+0.008858λ.sub.u.sup.2).sup.1/2, i.e., E.sub.b [MeV]=98.45λ.sub.u.sup.1/2(1+0.008858λ.sub.u.sup.2).sup.1/2. For Case A to E, γ=475.6, 835.7, 1290, 1834, 2460□ and E.sub.b [MeV]=243, 427, 659, 937, 1257□.

(89) The Laser Plasma Accelerator (LPA) can provide a high-peak current bunched beam, e.g., I.sub.A=50 kA for electron charge Q.sub.b=0.5 nC and bunch duration τ.sub.b 10 fs. A fiber laser pulse with wavelength λ.sub.L=1 μm after compression is focused on the entrance of gas cell at the normalized laser field a.sub.0=2 corresponding to the laser intensity I=5.5×10.sup.18 Wcm.sup.−2. Self-guided propagation of such laser pulse in the gas cell requires the group velocity correction factor κ.sub.self=1.19 and the matched spot radius R.sub.m≡k.sub.pr.sub.m=3.2. The wakefield reduction factor α due to loaded charge Q.sub.b is calculated from α.sup.2+Cα.sup.3/2−1=0 for the electron beam radius k.sub.pσ.sub.b=1, where the coefficients are C=9.0, 6.8, 5.5, 4.6, 4.0□ as α=0.223, 0.267, 0.302, 0.335, 0.364□, respectively, for Case A to E.

(90) The important Laser Plasma Accelerator (LPA) parameters are provided as follows:

(91) (1) The operating plasma density; n.sub.e[10.sup.17 cm.sup.−3]=8.3, 5.6, 4.2, 3.2, 2.6

(92) (2) The accelerator length; L.sub.acc [mm]=18, 32, 51, 74, 102

(93) (3) The required pulse duration; τ.sub.L [fs]=46, 56, 65, 73, 82

(94) (4) The matched spot radius; r.sub.m [μm]=19, 23, 27, 30, 34

(95) (5) The matched power; P.sub.L [TW]=29, 43, 59, 75, 93

(96) (6) The required laser pulse energy; U.sub.L [J]=1.34, 2.40, 3.79, 5.52, 7.57

(97) For the Free-Electron Laser operation, the coupling factor A.sub.u(ξ) are A.sub.u=0.9527, 0.8696, 0.8083, 0.7711, 0.7486 with ξ=0.09065, 0.2349, 0.3329, 0.3899, 0.4235 for Case A to E, respectively. The root mean square (r.m.s) transverse size of the electron bunch is set to σ.sub.b=25 μm in the undulator and is usually much larger than the normalized transverse emittance ε.sub.n of the order of 1 μm for the Laser Plasma Accelerator (LPA) produced electron beam. For the peak current I.sub.b=50 kA with the number of electrons per wavelength N.sub.λ.sub.N=1.4×10.sup.7 and the diffraction parameter B 1, the important Free-Electron Laser parameters are given as follows according to the one-dimensional Free-Electron Laser theory:

(98) (1) The Pierce parameter; ρ.sub.FEL [%]=1.117, 1.507, 1.597, 1.596, 1.572

(99) (2) The longitudinal velocity spread; Λ≈1 for setting σ.sub.γ/γ≈ρ.sub.FEL

(100) (3) The e-folding gain length; L.sub.gain [mm]=41, 61, 86, 115, 146

(101) (4) The saturated power; P.sub.sat [GW]≅0.6ρ.sub.FELP.sub.b=82, 194, 317, 451, 596

(102) (5) The input power; P.sub.in [MW]=0.94, 3.03, 5.26, 7.48, 9.72

(103) (6) The saturation length; L.sub.sat [mm]=499, 721, 1016, 1355, 1723

(104) (7) The total number of periods; N.sub.u=100, 72, 68, 68, 69.

(105) (8) The spectral bandwidth; Δλ.sub.X/λ.sub.X [%]˜1/N.sub.u≈1.0, 1.4, 1.5, 1.5, 1.5

(106) (9) The r.m.s. radiation cone angle;

(107) ${\theta }_{\mathrm{rms}}\left[\mu \mathrm{rad}\right]=\frac{1}{2\gamma }{\left(\frac{1+K_u^2\text{/}2}{N_u}\right)}^{1\text{/}2}=116,97,82,71,63$

(108) (10) The average power at the repetition frequency f.sub.rep [MHz];
P.sub.av [kW]˜P.sub.satτ.sub.Xf.sub.rep=(0.82,1.94,3.17,4.51,5.96)×f.sub.rep [MHz],

(109) assuming the radiation duration τ.sub.X≈τ.sub.b˜10 fs.

(110) The repetition rate f.sub.rep to be required for generating the average Extreme UltraViolet power of P.sub.EUV, =1 kW yields f.sub.rep [MHz]≈P.sub.EUV/(P.sub.satτ.sub.X)=1.22, 0.515, 0.315, 0.223, 0.168. For the production of 1 kW Extreme UltraViolet radiation, the average fiber laser power yields P.sub.Lav [MW]≈U.sub.Lf.sub.rep=1.63, 1.24, 1.19, 1.22, 1.27

(111) Consequently, the minimum average laser power takes place for Case C with the undulator period 15 mm. The average beam power yields P.sub.bav [kW]=Q.sub.bf.sub.repE.sub.b≈148, 110, 104, 104, 105. The efficiency of the electron beam acceleration is η.sub.laser.fwdarw.beam [%]=P.sub.bav/P.sub.Lav≈9.1, 8.9, 8.7, 8.5, 8.3. The efficiency of the production of Extreme UltraViolet radiation yields η.sub.laser.fwdarw.EUV [%]=0.061, 0.081, 0.084, 0.082, 0.079

(112) Said Laser Plasma Accelerator (LPA) and Free-Electron Laser parameters for Case A to E producing the Extreme UltraViolet radiation of 1 kW at 13.5 nm wavelength are summarized as shown in Table 1.

(113) TABLE-US-00001 TABLE 1 Examnles of the fiber laser driven Laser Plasma Accelerator (LPA) based Free-Electron Laser Extreme UltraViolet light source at 13.5 nm Case A B C D E Fiber laser parameters Laser wavelength [μm] 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 [μm] 19 23 27 30 34 LPA parameters Electron beam energy [MeV] 243 427 659 937 1257 Plasma density [10.sup.17 cm.sup.-3] 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 α 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 Normalized emittance [mm mrad] ~1 ~1 ~1 ~1 ~1 Transverse beam size [μm] 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 FEL parameters 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 K.sub.u 0.666 1.33 2.00 2.66 3.33 Pierce parameter [%] 1.117 1.507 1.597 1.596 1.572 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 [μrad] 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

6.4. Embodiment of a Free-Electron Laser Used as an Extreme UltraViolet Source at 6.7 nm Wavelength

(114) A fiber laser driven Laser Plasma Accelerator (LPA) based Free-Electron Laser produced Extreme UltraViolet radiation source at λ.sub.X=6.7 nm wavelength using the undulator with period λ.sub.u=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 gap-period ratio g/λ.sub.u=0.2, e.g. g=1 mm (Case A), 2 mm (Case B), 3 mm (Case C), 4 mm (Case D) and 5 mm (Case E), respectively. A hybrid undulator comprising NdFeB materials with grade N52, e.g., VACODYM® 722HR, and ferromagnetic materials such as tempered Co—Fe alloys (vanadium permendur), e.g., VACOFLUX® 50, provide the peak magnetic field B.sub.u [T]=3.694exp(−5.068×0.2+1.520×0.2.sup.2)=1.425. The corresponding undulator parameter becomes K.sub.u=0.1331λ.sub.u [mm]=0.6655, 1.331, 1.9965, 2.662, 3.3275 for λ.sub.u [mm]=5, 10, 15, 20, 25.

(115) The electron beam energy E.sub.b required for producing the Extreme UltraViolet radiation at the wavelength λ.sub.X=6.7 is given by γ=273.18λ.sub.u.sup.1/2(1+0.008858λ.sub.u.sup.2).sup.1/2, i.e., E.sub.b [MeV]=98.45λ.sub.u.sup.1/2(1+0.008858λ.sub.u.sup.2).sup.1/2. For Case A to E, γ=675.1, 1186, 1830, 2604, 3492 and E.sub.b [MeV]=345, 606, 935, 1331, 1784.

(116) The Laser Plasma Accelerator (LPA) can provide a high-peak current bunched beam, e.g., I.sub.A=50 kA for electron charge Q.sub.b=0.5 nC and bunch duration τ.sub.b 10 fs. A fiber laser pulse with wavelength λ.sub.L=1 μm after compression is focused on the entrance of gas cell at the normalized laser field a.sub.0=2 corresponding to the laser intensity I=5.5×10.sup.18 Wcm.sup.−2. Self-guided propagation of such laser pulse in the gas cell requires the group velocity correction factor κ.sub.self=1.19 and the matched spot radius R.sub.m≡k.sub.pr.sub.m=3.2. The wakefield reduction factor α due to loaded charge Q.sub.b is calculated from α.sup.2+Cα.sup.3/2−1=0 for the electron beam radius k.sub.pσ.sub.b=1, where the coefficients are C=7.55, 5.70, 4.59, 3.84, 3.32 as a=0.249, 0.295, 0.335, 0.369, 0.400, respectively, for Case A to E.

(117) The important Laser Plasma Accelerator (LPA) parameters are provided as follows:

(118) (1) The operating plasma density; n.sub.e[10.sup.17 cm.sup.−3]=6.5, 4.4, 3.2, 2.5, 2.0

(119) (2) The accelerator length; L.sub.acc [mm]=26, 47, 74, 109, 150

(120) (3) The required pulse duration; τ.sub.L [fs]=52, 63, 73, 83, 93

(121) (4) The matched spot radius; r.sub.m [μm]=21, 26, 30, 34, 38

(122) (5) The matched power; P.sub.L [TW]=37, 55, 75, 97, 120

(123) (6) The required laser pulse energy; U.sub.L [J]=1.92, 3.47, 5.51, 8.06, 11.1

(124) For the FEL operation, the coupling factor A.sub.u(ξ) are A.sub.u=0.9527, 0.8696, 0.8083, 0.7711, 0.7486 with ξ=0.09065, 0.2349, 0.3329, 0.3899, 0.4235 for Case A to E, respectively. The root mean square (r.m.s) transverse size of the electron bunch is set to σ.sub.b=25 μm in the undulator and is usually much larger than the normalized transverse emittance ε.sub.n of the order of 1 μm for the Laser Plasma Accelerator (LPA) produced electron beam. For the peak current I.sub.b=50 kA with the number of electrons per wavelength N.sub.λ.sub.X=7×10.sup.6 and the diffraction parameter B 1, the important Free-Electron Laser parameters are given as follows according to the one-dimensional Free-Electron Laser theory:

(125) (1) The Pierce parameter; ρ.sub.FEL [%]=0.787, 1.061, 1.125, 1.125, 1.107

(126) (2) The longitudinal velocity spread; Λ≈1 for setting σ.sub.γ/γ≈ρ.sub.FEL

(127) (3) The e-folding gain length; L.sub.gain [mm]=58.4, 86.6, 123, 163, 207

(128) (4) The saturated power; P.sub.sat [GW]≅0.6ρ.sub.FELP.sub.b=82, 194, 317, 451, 596

(129) (5) The input power; P.sub.in [MW]≈0.94, 3.05, 5.3, 7.5, 9.8

(130) (6) The saturation length; L.sub.sat [mm]=709, 1024, 1441, 1923, 2445

(131) (7) The total number of periods; N.sub.u=142, 102, 96, 96, 98.

(132) (8) The spectral bandwidth; Δλ.sub.X/λ.sub.X [%]˜1/N.sub.u≈0.71, 0.98, 1.04, 1.04, 1.02

(133) (9) The r.m.s. radiation cone angle; θ.sub.rms [μrad]=69, 57, 48, 42, 37

(134) (10) The average power at the repetition frequency f.sub.rep [MHz];
P.sub.av [kW]˜P.sub.satτ.sub.Xf.sub.rep=(0.82,1.94,3.17,4.51,5.96)×f.sub.rep [MHz],

(135) assuming the radiation duration τ.sub.X≈τ.sub.b˜10 fs. The repetition rate f.sub.rep to be required for generating the average EUV power of P.sub.EUV=1.5 kW yields f.sub.rep [MHz]≈P.sub.EUV/(P.sub.satτ.sub.X)=1.83, 0.773, 0.473, 0.332, 0.252. For the production of 1.5 kW EUV radiation, the average fiber laser power yields P.sub.Lav [MW]≈U.sub.Lf.sub.rep=3.52, 2.68, 2.60, 2.68, 2.80

(136) Consequently, the minimum average laser power takes place for Case C with the undulator period 15 mm. The average beam power yields P.sub.bav [kW]=Q.sub.b f.sub.repE.sub.b≈316, 234, 221, 221, 225. The efficiency of the electron beam acceleration is η.sub.laser.fwdarw.beam [%]=P.sub.bav/P.sub.Lav=8.97, 8.73, 8.49, 8.26, 8.03. The efficiency of the production of Extreme UltraViolet radiation yields η.sub.laser.fwdarw.EUV [%]=0.043, 0.056, 0.058, 0.056, 0.054.

(137) Said Laser Plasma Accelerator (LPA) and Free-Electron Laser parameters for Case A to E producing the Extreme UltraViolet radiation of 1 kW at 6.7 nm wavelength are summarized as shown in Table 2.

(138) TABLE-US-00002 TABLE 2 Examples of the fiber laser driven Laser Plasma Accelerator (LPA) based Free-Electron Laser Extreme UltraViolet light source. Case A B C D E Fiber laser parameters Laser wavelength [μm] 1 1 1 1 1 Average laser power [MW] 3.52 2.68 2.60 2.68 2.80 Repetition rate [MHz] 1.83 0.773 0.473 0.332 0.252 Laser energy per pulse [J] 1.92 3.47 5.51 8.06 11.1 Peak power [TW] 37 55 75 97 120 Pulse duration [fs] 52 63 73 83 93 Matched spot radius [μm] 21 26 30 34 38 LPA parameters Electron beam energy [MeV] 345 606 935 1331 1784 Plasma density [10.sup.17 cm.sup.-3] 6.5 4.4 3.2 2.5 2.0 Accelerator length [mm] 26 47 74 109 150 Charge per bunch [nC] 0.5 0.5 0.5 0.5 0.5 Field reduction factor α 0.249 0.295 0.335 0.369 0.400 Bunch duration [fs] 10 10 10 10 10 Energy spread [%] ~0.8 ~1.1 ~1.1 ~1.1 ~1.1 Normalized emittance [mm mrad] ~1 ~1 ~1 ~1 ~1 Transverse beam size [μm] 25 25 25 25 25 Peak current [kA] 50 50 50 50 50 Average beam power [kW] 316 234 221 221 225 Efficiency of laser to beam [%] 9.0 8.7 8.5 8.3 8.0 FEL parameters Radiation wavelength [nm] 6.7 6.7 6.7 6.7 6.7 Undulator period [mm] 5 10 15 20 25 Gap [mm] 1 2 3 4 5 Peak magnetic field [T] 1.425 1.425 1.425 1.425 1.425 Undulator parameter K.sub.u 0.666 1.33 2.00 2.66 3.33 Pierce parameter [%] 0.787 1.06 1.125 1.125 1.107 Gain length [mm] 58.4 86.6 123 163 207 Saturation length [mm] 709 1024 1441 1923 2445 Number of periods 142 102 96 96 98 Spectral bandwidth [%] 0.71 0.98 1.04 1.04 1.02 r.m.s. Radiation cone angle [μrad] 69 57 48 42 37 Input power [MW] 0.94 3.05 5.3 7.5 9.8 Saturated power [GW] 82 194 317 451 596 Duration of EUV pulse [fs] 10 10 10 10 10 Average EUV power [kW] 1.5 1.5 1.5 1.5 1.5 Efficiency of EUV generation [%] 0.043 0.056 0.058 0.056 0.054

6.5 Embodiment of Tuning a Free-Electron Laser for the Wavelength Range from 5 nm to 15 nm

(139) The FEL amplification takes place in the undulator with the undulator period λ.sub.u at the resonant wavelength given by

(140) ${\lambda }_X=\frac{{\lambda }_u}{2{\gamma }^2}\left(1+\frac{K_u^2}{2}\right),$
where γ=E.sub.b/m.sub.ec.sup.2 is the relativistic factor of the electron beam energy E.sub.b, K.sub.u=0.934B.sub.u [T]λ.sub.u [cm]. Setting the peak magnetic field of the undulator to be B.sub.u=1.425 T, the corresponding undulator parameter becomes K.sub.u=0.1331λ.sub.u [mm]=1.9965 for λ.sub.u=15 mm (CASE C). The electron beam energy E.sub.b required for producing the EUV radiation at the wavelength λ.sub.X is given by

(141) $E_b\left[\mathrm{MeV}\right]=659{\left(\frac{13.5\mathrm{nm}}{{\lambda }_X}\right)}^{1\text{/}2}$
The important Laser Plasma Accelerator (LPA) parameters are provided as a function of FEL wavelength λ.sub.X: (1) The operating plasma density;

(142) $n_e\approx 4.2\times {10}^{17}\left[{\mathrm{cm}}^{-3}\right]{\left(\frac{{\lambda }_X}{13.5\mathrm{nm}}\right)}^{1\text{/}2}$ (2) The accelerator length;

(143) $L_{\mathrm{acc}}\approx 51\left[\mathrm{mm}\right]{\left(\frac{13.5\mathrm{nm}}{{\lambda }_X}\right)}^{3\text{/}4}$ (3) The required pulse duration;

(144) 0${\tau }_L=65\left[\mathrm{fs}\right]{\left(\frac{13.5\mathrm{nm}}{{\lambda }_X}\right)}^{1\text{/}4}$ (4) The matched spot radius;

(145) $r_m\approx 27\left[\mu m\right]{\left(\frac{13.5\mathrm{nm}}{{\lambda }_X}\right)}^{1\text{/}4}$ (5) The required laser peak power;

(146) $P_L\approx 59\left[\mathrm{TW}\right]{\left(\frac{13.5\mathrm{nm}}{{\lambda }_X}\right)}^{1\text{/}2}$ (6) The required laser pulse energy;

(147) $U_L=3.8\left[J\right]{\left(\frac{13.5\mathrm{nm}}{{\lambda }_X}\right)}^{3\text{/}4}$
The important Free-Electron Laser parameters are all given as a function of FEL wavelength λ.sub.X: (1) The Pierce parameter;

(148) ${\rho }_{\mathrm{FEL}}=1.597\left[\%\right]{\left(\frac{{\lambda }_X}{13.5\mathrm{nm}}\right)}^{1\text{/}2}$ (2) The longitudinal velocity spread; Λ≈1 for setting σ.sub.γ/γ≈ρ.sub.FEL (3) The e-folding gain length;

(149) $L_{\mathrm{gain}}=86\left[\mathrm{mm}\right]{\left(\frac{13.5\mathrm{nm}}{{\lambda }_X}\right)}^{1\text{/}2}$ (4) The saturated power; P.sub.sat=317 [GW] (5) The input power; P.sub.in≈5.3 [MW] (6) The saturation length;

(150) $L_{\mathrm{sat}}=1016\left[\mathrm{mm}\right]{\left(\frac{13.5\mathrm{nm}}{{\lambda }_X}\right)}^{1\text{/}2}$ (7) The total number of periods;

(151) $N_u=L_{\mathrm{sat}}\text{/}{\lambda }_u=68{\left(\frac{13.5\mathrm{nm}}{{\lambda }_X}\right)}^{1\text{/}2}$ (8) The spectral bandwidth; Δλ.sub.X/λ.sub.X 1/N.sub.u≈1.5[%] (9) The r.m.s. radiation cone angle;

(152) ${\theta }_{\mathrm{rms}}=\frac{1}{2\gamma }{\left(\frac{1+K_u^2\text{/}2}{N_u}\right)}^{1\text{/}2}=82\left[\mu \mathrm{rad}\right]{\left(\frac{{\lambda }_X}{13.5\mathrm{nm}}\right)}^{1\text{/}4}$ (10) The average power at the repetition frequency f.sub.rep [MHz];
P.sub.avP.sub.satτ.sub.Xf.sub.rep=3.17 [kW]f.sub.rep
for the radiation duration τ.sub.X≈τ.sub.b˜10 fs. (11) The repetition rate f.sub.rep to be required for generating the average Extreme UltraViolet power of P.sub.EUV=1 kW;
f.sub.rep≈P.sub.EUV/(P.sub.satτ.sub.X)=0.315 [MHz] (12) The average fiber laser power for the production of 1 kW Extreme UltraViolet radiation;

(153) $P_{\mathrm{Lav}}\approx U_L{f}_{\mathrm{rep}}=1.19\left[\mathrm{MW}\right]{\left(\frac{13.5\mathrm{nm}}{{\lambda }_X}\right)}^{3\text{/}4}$ (13) The average electron beam power;

(154) 0$P_{\mathrm{bav}}=Q_b{f}_{\mathrm{rep}}{E}_b=104\left[\mathrm{kW}\right]{\left(\frac{13.5\mathrm{nm}}{{\lambda }_X}\right)}^{1\text{/}2}$ (14) The efficiency of the electron beam acceleration;

(155) ${\eta }_{\mathrm{laser}.fwdarw.\mathrm{beam}}=P_{\mathrm{bav}}\text{/}{P}_{\mathrm{Lav}}=8.7\left[\%\right]{\left(\frac{{\lambda }_X}{13.5\mathrm{nm}}\right)}^{1\text{/}4}$ (15) The efficiency of the production of Extreme UltraViolet radiation;

(156) ${\eta }_{\mathrm{laser}.fwdarw.\mathrm{EUV}}=P_{\mathrm{EUV}}\text{/}{P}_{\mathrm{Lav}}=0.084\left[\%\right]{\left(\frac{{\lambda }_X}{13.5\mathrm{nm}}\right)}^{3\text{/}4}$
For undulator period λ.sub.u=15 mm, the average Extreme UltraViolet power of P.sub.EUV=1 kW, the electron beam energy E.sub.b, the operating plasma density n.sub.e, the accelerator length L.sub.acc, the required laser peak power P.sub.L, the required laser pulse energy U.sub.L, the Pierce parameter ρ.sub.FEL, the saturation length L.sub.sat, the average fiber laser power P.sub.Lav and the efficiency of the production of Extreme UltraViolet radiation η.sub.eff are shown as a function of the radiation wavelength λ.sub.X for the range from 5 nm to 15 nm in FIG. 14, obtained for a given peak magnetic field and a given undulator period, which have been previously set (e.g. B.sub.u=1.425 T and λ.sub.u=15 mm.)

(157) Other examples have been given respectively in table 1 and 2 of section 6.4 for other cases A, B, D, E, F related to other examples of ondulator period with B.sub.u=1.425 T).

(158) An exemplary embodiment of the present disclosure provides a new embodiment of a Free Electron Laser, which is more compact and efficient, cheaper and has a higher repetition rate and a higher average power than the prior art Free Electron Lasers.

(159) An exemplary embodiment provides an efficient Free-Electron Laser-based Extreme UltraViolet light source, usable for industrial lithography technology.

CITATION LIST

(160) Patent Literature

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(162) Although the present disclosure has been described with reference to one or more examples, workers skilled in the art will recognize that changes may be made in form and detail without departing from the scope of the disclosure and/or the appended claims.