Estimation of soft X-ray and EUV transition radiation power emitted from the MIRRORCLE-type tabletop synchrotron

Toyosugi, N.; Yamada, H.; Minkov, D.; Morita, M.; Yamaguchi, T.; Imai, S.

doi:10.1107/S0909049507003007

research papers

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 14| Part 2| March 2007| Pages 212-218

doi:10.1107/S0909049507003007

Estimation of soft X-ray and EUV transition radiation power emitted from the MIRRORCLE-type tabletop synchrotron

N. Toyosugi,^a H. Yamada,^a,^b,^c ^* D. Minkov,^a M. Morita,^a T. Yamaguchi ^c and S. Imai ^c

^aPhoton Production Laboratory Ltd, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu-shi, Shiga 525-8577, Japan, ^b21st Century COE Synchrotron Light Life Science Center, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu-shi, Shiga 525-8577, Japan, and ^cRitsumeikan University, 1-1-1 Nojihigashi, Kusatsu-shi, Shiga 525-8577, Japan
^*Correspondence e-mail: hironari@se.ritsumei.ac.jp

(Received 22 August 2006; accepted 19 January 2007)

The tabletop synchrotron light sources MIRRORCLE-6X and MIRRORCLE-20SX, operating at electron energies E_el = 6 MeV and E_el = 20 MeV, respectively, can emit powerful transition radiation (TR) in the extreme ultraviolet (EUV) and the soft X-ray regions. To clarify the applicability of these soft X-ray and EUV sources, the total TR power has been determined. A TR experiment was performed using a 385 nm-thick Al foil target in MIRRORCLE-6X. The angular distribution of the emitted power was measured using a detector assembly based on an NE102 scintillator, an optical bundle and a photomultiplier. The maximal measured total TR power for MIRRORCLE-6X is P_max ≅ 2.95 mW at full power operation. Introduction of an analytical expression for the lifetime of the electron beam allows calculation of the emitted TR power by a tabletop synchrotron light source. Using the above measurement result, and the theoretically determined ratio between the TR power for MIRRORCLE-6X and MIRRORCLE-20SX, the total TR power for MIRRORCLE-20SX can be obtained. The one-foil TR target thickness is optimized for the 20 MeV electron energy. P_max ≅ 810 mW for MIRRORCLE-20SX is obtained with a single foil of 240 nm-thick Be target. The emitted bremsstrahlung is negligible with respect to the emitted TR for optimized TR targets. From a theoretically known TR spectrum it is concluded that MIRRORCLE-20SX can emit 150 mW of photons with E > 500 eV, which makes it applicable as a source for performing X-ray lithography. The average wavelength, $[\overline\lambda]$ = 13.6 nm, of the TR emission of MIRRORCLE-20SX, with a 200 nm Al target, could provide of the order of 1 W EUV.

Keywords: MIRRORCLE-type tabletop synchrotron; transition radiation; total soft X-ray power; total EUV power.

1. Introduction

A MIRRORCLE-type synchrotron (Yamada, 1996 ) emitting transition radiation (TR) represents an important source of soft X-ray and extreme ultraviolet (EUV) radiation. Such radiation can be used for performing deep-edge lithography for the LIGA process micro-fabrication (Ehrfeld & Munchmeyer, 1991 ), X-ray lithography (XRL) (Piestrup et al., 1985 ) and possibly EUV lithography (EUVL) (Shriever, 2004 ) for manufacturing next-generation semiconductor chips. Therefore, the development of MIRRORCLE-type sources emitting larger TR power is of significant practical interest.

Linear accelerators (LINACs) have been mostly used for emission of TR, and the maximum reported TR power emitted from a LINAC is 15.2 mW (Piestrup et al., 1991 ). LINACs operate at very high electron energies, E_el = 100–10000 MeV; their electrons pass only once along linear trajectories which leads to a relatively low beam current, I_B = 1–100 µA. More recently, storage rings have been used for emission of TR. Storage rings operate at lower electron energies, E_el ≃ several tens of MeV, and their electrons pass many times along quasi-circular orbits, which results in a much larger I_B. Although we are not aware of any experimental data regarding TR power emitted from storage rings, it has been concluded that the expected emitted TR power should be ∼100 mW (Piestrup et al., 1998 ).

In Japan, the compact storage ring Aurora (Yamada, 1990 ), which uses a superconducting magnet, has not gained sufficient recognition owing to the inconvenience of using liquid He and an insufficient I_B. A theory for the emission of bright hard X-rays by MIRRORCLE-type synchrotrons (Yamada, 1996) has laid the foundation for development of the low-energy storage ring MIRRORCLE, which can operate at a very large beam current, I_B > 1 A. After resolving the injection problem (Yamada et al., 2001 ), the tabletop storage ring MIRRORCLE-6X, operating at E_el = 6 MeV (Photon Production Laboratory, https://www.ppl-xray.com ), has been developed (Hasegawa, 2004 ) and used for the generation of bright hard X-rays (Yamada, 2004 ).

MIRRORCLE-6X is suitable for performing TR experiments owing to its large beam current and compactness. Problems for its practical application as a TR source are insufficient beam lifetime and insufficient emission of TR power per electron owing to its low electron energy. Therefore, at present, we are commissioning the storage ring MIRRORCLE-20SX with E_el = 20 MeV (Toyosugi et al., 2006 ).

To find out the present limitations of the applicability of MIRRORCLE as an EUV and soft X-ray source, we need to evaluate the maximum emitted TR power for both MIRRORCLE-6X and MIRRORCLE-20SX. In this paper the emitted TR power is determined in an experiment using MIRRORCLE-6X and an Al stripe TR target. Based on this experimental observation, the available total TR power is estimated for both MIRRORCLE-6X and MIRRORCLE-20SX.

2. Experimental set-up

Our experiment was performed on MIRRORCLE-6X, which contains a 6 MeV microtron injector and storage ring (Fig. 1). The microtron injects electron pulses with a peak injected current I_P ≤ 100 mA, an injection pulse time T ≤ 150 ns and an injection rate R ≤ 400 Hz (Hasegawa et al., 2004). The dimensions of MIRRORCLE-6X are 1 m × 2 m × 1.5 m (https://www.ppl-xray.com ). A schematic diagram of the set-up of our experiment is shown in Fig. 2.

Figure 1
Photograph of the storage-ring synchrotron MIRRORCLE-6X at SLLS Ritsumeikan University.

Figure 2
Set-up of our experiment for measurement of the emitted power from MIRRORCLE-6X with a TR target.

The electron beam in the storage ring has an elliptical cross section, with a length of 20 mm along its horizontal axis and a height of 3 mm. To ensure a small square emitting area, our TR target represented a vertical stripe of one Al foil with a width of 2.5 mm. When one electron hits the target, it emits TR with a conical shape (Ter-Mikaelyan, 1961 ) (Fig. 3).

Figure 3
Illustration of the cross section of the electron beam in the storage ring, and the positioning of the TR target. TR is emitted with a conical spatial distribution, and θ ≅ Δθ.

The detector includes a plastic scintillator (PS) made of NE102, an optical bundle and a photomultiplier (PMT). The PS is driven to points x_j located along a horizontal line, which is perpendicular to the axis of the TR emission cone and crosses it at x = 0. The PS is kept under the storage-ring vacuum, and there are no obstacles between it and the target. The distance between the target and the PS is 483 mm, at the point x = 0. The current from the PMT output is recorded for each PS location.

The PS is a rectangular piece of NE102 with a surface area of 3 mm × 3 mm and a thickness of 8.5 µm. The PS is glued to a quartz substrate of dimensions 8 mm × 8 mm × 1 mm. An optical bundle of diameter 6 mm, made of 0.3 mm-diameter optical fibres, is glued to the other side of the quartz substrate. The PMT (H3165-10, Hamamatsu Photonics) is greased to the other end of the optical bundle. When illuminated by soft X-rays, NE102 emits visible light with a peak at 423 nm and a pulse width of ∼2.7 ns. H3165-10 operates at 1 kV, and its response time is ∼2.5 ns. The charge generated at the PMT output is supplied to an I–F converter, whose current output is recorded in counts, where 1 count = 10 pA.

The TR emitted from the TR target includes photons with a continuous spectrum of energies E from the EUV to the soft X-ray regions. In this paper the photon energy is counted in eV. Our detector was designed to have a maximum efficiency at E ≃ 1000 eV. We used the detector calibration service offered by the AIST storage-ring source to calibrate the detector in the spectral range E = 500–1150 eV.

3. Analytical method

The total current I_PT measured at the PMT output is

$[I_{\rm{PT}}=\left(i_{\rm{c}}/2S_0\right)\textstyle\sum\limits_j{Y_jS_j},\eqno(1)]$

where Y_j is the PMT output current, in counts, at the jth measurement point x_j, i_c = 10 pA is the current value for one count of the PMT output, S₀ = 9 × 10⁻⁶ m² is the surface area of our PS,

$[S_j={{\pi{d_{\rm{meas}}^{\,2}}}\over4}\,{\rm{Abs}}\left\{{\left[{\tan(\theta_j)+\tan(\theta_{j+1})}\right]^2-\left[{\tan(\theta_j)+\tan(\theta_{j-1})}\right]^2}\right\}]$

is the surface area from the measurement plane limited between the circles with a centre at the point 0, where the emission cone axis crosses the measurement plane and radii $[d_{\rm{meas}}{\rm{Abs}}\{{{[{\tan(\theta_j)+\tan(\theta_{j+1})}]/2}}\}]$ and $[d_{\rm{meas}}{\rm{Abs}}\{{{[{\tan(\theta_j)+\tan(\theta_{j-1})}]/2}}\}]$ , and $[\theta_j]$ = arctan(x_j/d_meas). It is assumed that, if the PS is located at any point on this surface area, the PMT output would be Y_j counts. The term 1/2 in (1) discounts the use of measurement points x_j located on both sides of the TR axis.

The measured emitted TR power for our reference experiment with MIRRORCLE-6X is

$[P_{\rm{r}}=I_{\rm{PT}}\textstyle\int\limits_{E_{\min}}^{E_{\max}}\left[\,f(E)/\eta(E)\right]\,\,{\rm{d}}E,\eqno(2)]$

where

$[f(E)={{\int\limits_{\theta=0}^{\theta_{\max}}{{{{\rm{d}}^2N}\over{{\rm{d}}E\,{\rm{d}}\Omega}}}\,\,{\rm{d}}\Omega}\over{\int\limits_{E_{\min}}^{E_{\max}}{\left({\int\limits_{\theta=0}^{\theta_{\max}}{{{{\rm{d}}^2N}\over{{\rm{d}}E\,{\rm{d}}\Omega}}\,\,{\rm{d}}\Omega}}\right){\rm{d}}E}}}={{f_0(E)}\over{\Delta{N_{{\rm{ph}}/{\rm{el}}}}}},]$

and η(E) [A W⁻¹] is the detector efficiency. The TR cross section, d²N/dEdΩ, represents the number of photons emitted within the photon energy interval dE, and the axially symmetrical spatial angle dΩ = 2πsinθ dθ, for one pass of one electron through the target, where θ is the photon emission angle with respect to the direction of the electron. The denominator ΔN_ph/el is the number of emitted photons within the measured spatial angle for one pass of one electron through the target. The analytical approximation of η(E) for our detector is derived in Appendix A.

The calculated emitted TR power is

$[P_{\rm{c}}=N_{{\rm{el}}/{\rm{s}}}\,E_{{\rm{TR}}/{\rm{el}}},\eqno(3)]$

where N_el/s is the number of electrons passing through the TR target for 1 s and E_TR/el is the emitted TR energy for one pass of one electron through the TR target. Formulae for calculating N_el/s and E_TR/el are presented in Appendices B and C, respectively. The calculated ratio R_P of the TR power P_c and P_c2 emitted in two experiments, performed using different synchrotron operation parameters I_P, T and R, as well as different TR targets, is

$[R_{\rm{P}}=P_{\rm{c}}/P_{{\rm{c}}2}.\eqno(4)]$

If the second of the above experiments is considered to be identical to our reference experiment with MIRRORCLE-6X, the TR power P observed in the first experiment is estimated to be

$[P=P_{\rm{r}}\,R_{\rm{P}}.\eqno(5)]$

4. Results

4.1. Experiment using MIRRORCLE-6X

Our reference experiment was performed using MIRRORCLE-6X with operating conditions I_P = 40 mA, T = 100 ns and R = 200 Hz. The TR target was one Al foil stripe of width 2.5 mm, height 15 mm and thickness d = 385 nm. Two experiments were conducted, with and without a target, by moving the TR target in and out of the electron beam. The measured dependence of the PMT output current count Y as a function of the emission angle θ is shown in Fig. 4 for both cases.

Figure 4
Measured dependence of the PMT output current count Y as a function of the emission angle θ, with and without a target (background). The experiment was performed using MIRRORCLE-6X at I_P = 40 mA, T ≅ 100 ns and R = 200 Hz. The TR target was a 2.5 mm-wide Al foil stripe of thickness d = 385 nm. The PMT was operated at 1 kV.

4.2. Comparison between experimental and calculated results for MIRRORCLE-6X

The difference between the above PMT output current counts with and without a target represents the measured angular dependence of the target emission. The measured and calculated angular dependencies for both TR and bremsstrahlung (BR) are shown in Fig. 5 for the above experimental conditions.

Figure 5
Comparison between the measured and calculated angular dependencies for emission of TR and BR

. The measured dependence is the difference between the PMT output without and with a target from Fig. 4

. The calculations are performed for the same experimental conditions of MIRRORCLE-6X.

The shape of the measured angular dependence is similar to that of TR, with two similar magnitude maxima at ±θ_max = 40 mrad. The significant difference between the measured and the calculated results is ascribed to the presence of a magnetic field B = 0.1282 T at the target in MIRRORCLE-6X. This magnetic field provides motion of the relativistic electrons along an almost circular orbit. It also limits the motion of atomic electrons in the target during the generation of TR, and leads to a smaller measured value of θ_max compared with its calculated value. The theoretical study of TR under a magnetic field is necessary in order to clarify this difference for further investigation, but is not reported here. Whether TR is enhanced or suppressed by the magnetic field, and whether the TR distribution is compressed by the magnetic field are interesting points to be studied.

It is assumed that approximately 15% of TR is emitted at angles θ > θ_max for this experiment. Therefore, the total emitted TR power is P_r ≅ 394 µW, and the total emitted BR power is P_BR ≅ 2.2 µW. At full-power operation (at 100 mA and 400 Hz injection rate), P_r ≅ 2.95 mW should be obtained.

4.3. Estimation of the practically available TR power emitted from MIRRORCLE-6X and MIRRORCLE-20SX

The full-power operating conditions of MIRRORCLE-6X and MIRRORCLE-20SX are I_P = 100 mA, T = 150 ns and R = 400 Hz. The design of TR targets optimized for performing XRL using MIRRORCLE was investigated by Minkov et al. (2006a ). It was concluded there that the best one-chemical element TR target for performing XRL using MIRRORCLE-6X contains one C foil of thickness d = 35 nm. The best one-chemical element TR target for performing XRL using MIRRORCLE-20SX contains one foil of (a) Be with d = 240 nm, (b) C with d = 220 nm or (c) Al with d = 200 nm.

The ratio R_P is calculated next from equation (4), for these optimum TR experiments with MIRRORCLE-6X and MIRRORCLE-20SX. The maximum emitted TR power is estimated from (5). In spite of the significant difference between the experimental and the theoretical angular distributions of the target emission for our MIRRORCLE-6X experiment, we use (5) for estimation of emitted TR power from MIRRORCLE-20SX because our first TR experiments with the 20 MeV machine show that the ratio between the measured θ_max and its theoretical value is almost the same for the MIRRORCLE-6X and MIRRORCLE-20SX experiments. Since only photons with energy E > 500 eV can be used for performing XRL, the emitted power of such photons is also calculated. All obtained results are included in Table 1.

Table 1
Characteristics of the experimental conditions, and the calculated emitted power

The first data line shows the results from our experiments. The asterisks (*) in the fifth column indicate the maximum possible total TR power which can be emitted from the present scheme MIRRORCLE-6X and 20SX.

Storage-ring synchrotron	Injected beam current I_Bi (µA)	One-foil TR target	Average beam current I_B (A)	Emitted TR power P (mW)	Emitted BR power P_Br (mW)	Emitted TR power P_E>500eV (mW)	Average photon wavelength $[\overline\lambda]$ (nm)
MIRRORCLE-6X experiment	0.8	385 nm Al	0.012	0.394	0.0022	0.0128	26.7
MIRRORCLE-6X	6.0	35 nm C	2.09	65.6*	0.0092	1.42	37.9
MIRRORCLE-20SX	6.0	200 nm Al	0.69	288.3	0.1377	63.9	13.6
MIRRORCLE-20SX	6.0	220 nm C	1.63	606.9	0.0846	132.0	12.1
MIRRORCLE-20SX	6.0	240 nm Be	2.32	811.6*	0.0642	151.2	10.9

The spectral dependence

$[E_{\rm{S}}(E)=E\left({\int\limits_\theta{{{{\rm{d}}^2N}\over{{\rm{d}}E\,{\rm{d}}\Omega_{\left|E\right.}}}\,{\rm{d}}\Omega}}\right)\times1\,{\rm{eV}}]$

of the emitted TR energy from MIRRORCLE-20SX for one 240 nm Be foil TR target and for one 220 nm C foil TR target is shown in Fig. 6. E_S(E) is the number of emitted TR photons with energies [E − 0.5 eV, E + 0.5 eV] for one incidence of one electron multiplied by E.

Figure 6
Spectral dependence E_S(E) (see text) of the TR energy emitted within 1 eV around the photon energy E for one pass of one electron through one 220 nm-thick C foil (dashed line) and one 240 nm-thick Be foil (solid line) in MIRRORCLE-20SX.

5. Discussion

The introduction of an explicit analytical expression for the beam lifetime τ [see equation (7) in Appendix B] allows calculation of the emitted TR power P by a storage ring for the first time. This cannot be done using the theory of TR emission by low-energy storage rings (Minkov et al., 2006b ) alone, where τ depends on an unknown constant.

The results obtained from this work show that a TR power of P = 65 mW can be emitted using a 35 nm-thick C-foil target from MIRRORCLE-6X, and P = 810 mW can be emitted using a 240 nm-thick Be foil from MIRRORCLE-20SX. Since at least 10 mW power of photons with E > 500 eV is needed for performing XRL (Piestrup et al., 1985), MIRRORCLE-6X cannot be used, but MIRRORCLE-20SX can be used comfortably. An EUV source with $[\overline\lambda]$ = 13–14 nm is required for EUVL (Shriever, 2004). Therefore, MIRRORCLE-20SX, with a 200 nm-thick Al target, can be considered as a possible source for performing EUVL.

The expected emission of such a large TR power from MIRRORCLE-20SX is due to the large lifetime τ and the small circulation radius r₀ of the electrons injected into the storage ring. Furthermore, the very large dynamic aperture of MIRRORCLE leads to a long lifetime τ due to the 1 cm vertical half-chamber aperture (Yamada, 1996).

The significant difference between the measured and the calculated angular dependencies of the PMT output current count for our MIRRORCLE-6X experiment indicates that the storage-ring magnetic field strongly influences the TR emission, decreasing the emitted power. Since there has been no study of the influence of the magnetic field on the emission of TR, such a study must be of interest for our future work.

APPENDIX A

Detector efficiency

The quantum efficiency of our detector, η₀(E) = η(E)E × 1.602 × 10⁻¹⁹ (C photon⁻¹), is determined by the spectral response of the NE102 PS, and is approximated analytically over the entire emitted TR spectrum to fit to its calibration data. The low-energy region of NE102 is modelled by analogy with that of La_1.996Tm_0.004O₂S resulting from their similar energy conversion efficiencies (3% for NE102 and 3.6% for La_1.996Tm_0.004O₂S). The large diffusion lengths of generated carriers, ∼45 nm, are due to a lower concentration of scintillating centres and a medium level of surface recombination, ∼10. La_1.996Tm_0.004O₂S has a plasma energy E_pl ≅ 19.4 eV. Its quantum efficiency at E ≤ 25 eV has a small slope limited by surface recombination, and η₀ (E ≤ 25 eV) ≅ η₀ (100 eV)/13 × E/25 (Benitez et al., 1991 ). For NE102, E_pl ≅ 16.15 eV and non-radiation surface recombination should dominate at E ≤ 16.15 × 25/19.40 = 20.8 eV. According to the definition of η₀(E), it should also increase linearly for E > 20.8 eV but with a larger slope. For much higher photon energies, only a decreasing part of the incident TR is absorbed in NE102, which leads to decreasing η₀(E) with E. Therefore, our detector efficiency is approximated as follows,

$[\eqalign{\eta(E)\,\left[{{A\over{W}}}\right]=&\left|\matrix{{{\eta_0\,(100\,{\rm{eV}})}\over({13\,\times\,20.8\,\times\,1.602\,\times\,10^{-19}})} \hfill {\rm{for}}\,\,E\le20.8\,{\rm{eV}} \cr\cr {\scriptstyle{{\rm{max}}}}\left\{{\scriptstyle{\eta\,(20.8\,{\rm{eV}})}},{{0.2764(E\,-\,20.8)\left\{1\,-\,\exp\left[-\mu_{\rm{PS}}(E)d_{\rm{PS}}\right]\right\}}\over{E}}\right\} }\right. \cr& \vphantom{\Big\{}\hskip106pt{\rm{for}}\,\,E\,\,\gt\,20.8\,{\rm{eV}}}\eqno(6)]$

where μ_PS(E) and d_PS = 8.5 µm are the linear absorption coefficient and the thickness of our NE102 PS, respectively, and the term in the large brackets describes the absorption of TR in the NE102 PS. The coefficient 0.2764 in the expression for E > 20.8 eV is introduced in order to fit the analytical dependence to the calibration data. Both the calibration data and the analytical approximation of the quantum efficiency of our detector, η₀(E), are shown in Fig. 7.

Figure 7
Spectral dependence of the quantum efficiency η₀(E) of our detector. Circles: calibrated data; line: our analytical approximation of these data. The detector includes an 8.5 µm-thick NE102 plastic scintillator, an optical bundle of diameter 6 mm, and a H3165-10 photomultiplier (Hamamatsu Photonics) operated at 1 kV.

APPENDIX B

Lifetime of the injected electrons and their number of electron passes through the target for 1s

In both MIRRORCLE-6X and MIRRORCLE-20SX the injected electrons move along almost perfectly circular orbits with a radius r₀ = 155.8 mm. The beam width is 20 mm in MIRRORCLR-6X and 10 mm in MIRRORCLE-20SX, and the beam height is 3 mm for both. The lifetime τ of the electron beam in the storage ring is given by (Yamada, 1996)

$[\eqalignno{\tau={}&{{137}\over{4r_{\rm{e}}^2Z^2}}\left\{{{{2\gamma^2}\over{137\pi}}\left({{{b_\bot}\over{\beta_\bot}}}\right)^2+{3\over{4\left[{\ln\left({{{183}\over{Z^{1/3}}}}\right)}\right]\left[{\ln\left({{1\over{\left({\Delta{p/p}}\right)_{\max}}}-{5\over8}}\right)}\right]}}}\right\} \cr& \times{{\Delta{tA}}\over{R_sN_{\rm{A}}\rho{d}}},&(7)}]$

where $[b_\bot]$ and $[\beta_\bot]$ are the half-chamber aperture and the average β-function of the storage ring in either the horizontal or the vertical transverse direction, whichever is more lifetime limiting. (Δp/p)_max is the half-momentum aperture of the ring, γ is the relativistic factor, and d, ρ, A and Z are the thickness, density, atomic weight and the atomic number of the foil material. R_s = S_T/S_B is the ratio between the area of the target hit by the beam and the beam cross section, Δt = 2πr₀/c is the time for completion of one turn of an electron in the storage ring, r_e is the classical electron radius and N_A is the Avogadro constant. (Δp/p)_max = 0.333 for MIRRORCLE-6X and (Δp/p)_max = 0.2 for MIRRORCLE-20SX, while β_z = 0.1835 m, and b_z = 0.01 m for both of them (Yamada, 2005 ). For light material targets, τ is limited by elastic scattering along the vertical direction, described by the first addend in (7).

The average beam current is given by (Yamada, 1996)

$[I_{\rm{B}}={{I_{\rm{p}}RT\tau}\over{\Delta{t}}}\left[{1-\exp\left({-{1/{R\tau}}}\right)}\right],\eqno(8)]$

while the beam current injected in the storage ring is I_Bi = I_PRT. The number of electrons passing through the target for 1 s is given by (Yamada, 2003 )

$[N_{{\rm{el}}/{\rm{s}}}={{I_{\rm{B}}R_s}\over{1.6022\times10^{-19}}}.\eqno(9)]$

APPENDIX C

Emitted TR energy for one pass of one electron through the target

The theory of TR emission from low-energy storage rings was developed (Minkov et al., 2006b) based on the theory for TR emission from LINACs (Okazaki et al., 2004 ). It points out that thin foil TR targets emit most TR power, and more TR power is emitted from one-foil TR targets compared with multi-foil targets. The TR cross section for a low-energy storage ring and a one-foil TR target is (Minkov et al., 2006b)

$[\eqalignno{{{{\rm{d}}^2 N}\over{{\rm{d}}\Omega\,{\rm{d}}E}}& = F_{\rm{s}} \, F_{\rm{i}} \cr& = \left\{ {{{4\alpha \sin ^2 \theta } \over {\pi ^2 E}}{{(\delta ^2 + \beta ^2)} \over {[(1/\gamma ^2 + \theta ^2 + 2\delta)^2 + (2\beta)^2] (1/\gamma ^2 +\theta^2)^2}}} \right\} \cr& \quad\times \left\{ {\left[{1 - \exp({ - {E\beta d / {\hbar c}}}) } \right]^2 + 4\exp({ - {E\beta d / {\hbar c}}})}\right. \cr& \left.\quad\,\times \sin^2\left[\left({{Ed}/{4\hbar{c}}}\right)\left(1/\gamma^2+\theta^2+2\delta\right)\right] \right\} &(10)}]$

where F_s gives the TR contribution from one interface foil/vacuum, and F_i formulates the TR interference between the two emitting interfaces of the foil. The dielectric constant of the foil, ∊(E) = 1 − 2[δ(E) − iβ(E)], is calculated from the known complex scattering factor of the material (Attwood, 1999 ).

The emitted TR energy for one incidence of one electron on the TR target is

$[\eqalignno{E_{{\rm{TR/el}}}&= \int\limits_{E_{\min } }^{E_{\max } } {E\left({\int\limits_\theta ^{} {{{{\rm{d}}^2 N} \over {{\rm{d}}E\,{\rm{d}}\Omega }}\,{\rm{d}}\Omega } } \right)}\,{\rm{d}}E \cr& = \sum\limits_{k = 1}^{k = E_{\max } - E_{\min } } {E_k } \left({\int\limits_\theta ^{} {{{{\rm{d}}^2 N} \over {{\rm{d}}E\,{\rm{d}}\Omega }}_{\left| {E = E_k } \right.} {\rm{d}}\Omega } } \right) \times (1\,{\rm{eV}}) \cr& = \sum\limits_{k = 1}^{k = E_{\max } - E_{\min } } {E_{\rm{S}} (k)},&(11)}]$

where E_S(E) is the spectral dependence of the emitted TR energy for one incidence of one electron on the target.

Taking into account that the TR cross section decreases strongly at higher energy of the emitted photons, the upper limit of the photon energy integrals in (2) and (11) is chosen to be E_max = 4000 eV. It is known that radiation with energy below the plasma energy E_pl of a given material is absorbed strongly in that material (Attwood, 1999). For the light TR target materials of our interest, E_pl(Al) = 15.8 eV, E_pl(C) = 22.3 eV and E_pl(Be) = 18.5 eV. Nevertheless, the lower limit of the photon energy integrals in (2) and (11) is chosen to be the minimum energy for which there are literature data about the complex scattering factor of these materials (National Institute of Standards and Technology, https://physics.nist.gov/PhysRefData/FFast/html/form.html ). Therefore, we use E_min(Al) = 8.5 eV, E_min(C) = 6.5 eV and E_min(Be) = 8.5 eV.

The average wavelength of the emitted TR power is calculated as (Attwood, 1999)

$[\overline\lambda\,({\rm{nm}})=1239.84/{\overline{E}}\,({\rm{eV}}),]$

where

$[{\overline{E}}={{\int\limits_{E_{\min}}^{E_{\max}}{E\left({\int\limits_\theta{{{{\rm{d}}^2N}\over{{\rm{d}}E\,{\rm{d}}\Omega}}\,{\rm{d}}\Omega}}\right)}\,{\rm{d}}E}\Big/{\int\limits_{E_{\min}}^{E_{\max}}{\left({\int\limits_\theta{{{{\rm{d}}^2N}\over{{\rm{d}}E\,{\rm{d}}\Omega}}\,{\rm{d}}\Omega}}\right)\,{\rm{d}}E}}}\,.\eqno(12)]$

APPENDIX D

Bremsstrahlung cross section

Bremsstrahlung (BR) emission from the TR target is also accounted for. It is known that, for electrons with E_el = E_el0 = 0.511 MeV, BR emission is described by the `non-screening approximation' of the target nucleus potential, as well as that, for electrons with E_el ≥ E_{el_cs} = (192/Z^1/3)E_el0, BR emission is described by the `compete screening approximation' of the nucleus potential (Jackson, 1999 ). Therefore, the BR cross section is approximated as (Yamada, 1996)

$[\eqalignno{{{{\rm{d}}^2N_{{\rm{BR}}}}\over{{\rm{d}}E\,{\rm{d}}\Omega}}&={{E_{{\rm{el}}\_{\rm{cs}}}-E_{{\rm{el}}}}\over{E_{{\rm{el}}\_{\rm{cs}}}-E_{{\rm{el0}}}}}{{{\rm{d}}^2N_{{\rm{BR}}\_{\rm{ns}}}}\over{{\rm{d}}E\,{\rm{d}}\Omega}}+{{E_{{\rm{el}}}-E_{{\rm{el0}}}}\over{E_{{\rm{el}}\_{\rm{cs}}}-E_{{\rm{el0}}}}}{{{\rm{d}}^2N_{{\rm{BR}}\_{\rm{cs}}}}\over{{\rm{d}}E\,{\rm{d}}\Omega}} \cr& =\left[{\left({{{E_{{\rm{el}}\_{\rm{cs}}}-E_{{\rm{el}}}}\over{E_{{\rm{el}}\_{\rm{cs}}}-E_{{\rm{el}}0}}}}\right)\ln\left({{{2\gamma{E}_{{\rm{el}}}}\over{E}}-{1\over2}}\right)+\left({{{E_{{\rm{el}}}-E_{{\rm{el0}}}}\over{E_{{\rm{el}}\_{\rm{cs}}}-E_{{\rm{el}}0}}}}\right)}\right. \cr& \left.\quad\times \ln\left({{{233}\over{Z^{1/3}}}}\right)\right]{{8\alpha}\over{137\pi{E}}}\left({r_{\rm{e}}Z\gamma}\right)^2{{1+\gamma^4\theta^4}\over{\left({1+\gamma^2\theta^2}\right)^4}},&(13)}]$

where N_BR, N_{BR_ns} and N_{BR_cs} are the number of BR emitted photons, number of BR photons emitted by a non-screened nucleus potential and number of BR photons emitted by a completely screened nucleus potential, respectively, for one pass of one electron through the target. For the Al target, E_{el_cs} = 41.7 MeV.

References

Attwood, D. (1999). Soft X-rays and Extreme Ultraviolet Radiation, pp. 52–61. Cambridge University Press. Google Scholar
Benitez, E. L., Husk, D. E., Schnatterly, S. E. & Tarrio, C. (1991). J. Appl. Phys. 70, 3256–3260. CrossRef CAS Web of Science Google Scholar
Ehrfeld, W. & Munchmeyer, D. (1991). Nucl. Instrum. Methods, A303, 523–531. CrossRef Google Scholar
Hasegawa, D., Yamada, H., Kleev, A. I., Toyosugi, N., Hayashi, T., Yamada, T., Tohyama, I., Ro, Y. D. (2004). Portable Synchrotron Light Sources and Advanced Applications, AIP Conference Proceedings 716, pp. 116–119. Melville, NY: AIP Press. Google Scholar
Jackson, J. D. (1999). Classical Electrodynamics. New York: John Wiley and Sons. Google Scholar
Minkov, D., Yamada, H., Toyosugi, N., Yamaguchi, T., Kadono, T. & Morita, M. (2006a). Appl. Phys. B, 86, 19–23. Web of Science CrossRef Google Scholar
Minkov, D., Yamada, H., Toyosugi, N., Yamaguchi, T., Y., Kadono, T. & Morita, M. (2006b). J. Synchrotron Rad. 13, 336–342. Web of Science CrossRef CAS IUCr Journals Google Scholar
Okazaki, Y., Toyosugi, N., Yamada, H., Narazaki, Y., Takashima, T. & Imai, S. (2004). Portable Synchrotron Light Sources and Advanced Applications, AIP Conference Proceedings 716, pp. 124–127. Melville, NY: AIP Press. Google Scholar
Piestrup, M. A., Boyers, D. G., Pincus, C. I., Harris, J. L., Caplan, H. S., Silzer, R. M. & Skopik, D. M. (1991). Appl. Phys. Lett. 59, 189–191. CrossRef CAS Web of Science Google Scholar
Piestrup, M. A., Kephart, J. O., Park, H., Klein, R. K., Pantell, R. H., Ebert, P. J., Moran, M. J., Dahling B. A. & Berman, B. L. (1985). Phys. Rev. A, 32, 917–927. Google Scholar
Piestrup, M. A., Powell, M. W., Cremer, J. T., Lombardo, L. W., Kaplin, V. V., Mihal'chuk, A. A., Uglov, S. R., Zabaev, V. N., Skopik, D. M., Silzer, R. M. & Retzlaff, G. A. (1998). Proc. SPIE, 3331, 450–463. CrossRef Google Scholar
Shriever, G. (2004). XUV Technologies and Applications, 326th Heraeus Seminar, Bad Honnef, Germany, 7–9 June 2004. Google Scholar
Ter-Mikaelyan, M. L. (1961). Nucl. Phys. 24, 43–61. CrossRef CAS Web of Science Google Scholar
Toyosugi, N., Yamada, H., Noh, Y. D., Hasegawa, D., Hayashi, T. & Minkov, D. (2006). Ninth International Conference on Synchrotron Radiation Instrumentation (SRI 2006), Daegu, Korea. Abstract 21. Google Scholar
Yamada, H. (1990). J. Vac. Sci. Technol. B, 8, 1628–1631. CrossRef CAS Web of Science Google Scholar
Yamada, H. (1996). Jpn. J. Appl. Phys. 35, L182–L185. CrossRef CAS Web of Science Google Scholar
Yamada, H. (2003). Nucl. Instrum. Methods, B199, 509–516. CrossRef Google Scholar
Yamada, H. (2004). Portable Synchrotron Light Sources and Advanced Applications, AIP Conference Proceedings 716, pp. 12–17. Melville, NY: AIP Press. Google Scholar
Yamada, H. (2005). Oyo Butsuri, 74, 462–471. CAS Google Scholar
Yamada, H., Kitazawa, Y., Kanai, Y., Tohyama, I., Ozaki, T., Sakai, Y., Kleev, A. I. & Bogomolov, G. D. (2001). Nucl. Instrum. Methods, A467–468, 122–125. CrossRef Google Scholar

© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For more information, click here.

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 14| Part 2| March 2007| Pages 212-218

doi:10.1107/S0909049507003007

Format		BIBTeX
		EndNote
		RefMan
		Refer
		Medline
		CIF
		SGML
		Plain Text
		Text

Format		BIBTeX
		EndNote
		RefMan
		Refer
		Medline
		CIF
		SGML
		Plain Text
		Text

Search IUCr Journals		doi		Advanced search
Author		volume	page

research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Estimation of soft X-ray and EUV transition radiation power emitted from the MIRRORCLE-type tabletop synchrotron

1. Introduction

2. Experimental set-up

3. Analytical method

4. Results

4.1. Experiment using MIRRORCLE-6X

4.2. Comparison between experimental and calculated results for MIRRORCLE-6X

4.3. Estimation of the practically available TR power emitted from MIRRORCLE-6X and MIRRORCLE-20SX

5. Discussion

APPENDIX A

Detector efficiency

APPENDIX B

Lifetime of the injected electrons and their number of electron passes through the target for 1s

APPENDIX C

Emitted TR energy for one pass of one electron through the target

APPENDIX D

Bremsstrahlung cross section

References

research papers