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Abstract
The radiation character of nonlinear Thomson scattering is investigated in the interaction of Laguerre-Gaussian circularly polarized laser pulses with a single electron in the angular plane. With theoretical analysis and numerical calculation, it is shown that the angular radiation distributions have annular structures with great fourfold or plane symmetry in pulses characterized by comparatively lower laser intensity (a0 < 6), prolonged pulse duration (τ > 50fs)or wide beam waist (b0 > 5μm). In other circumstances, a vortex radiation pattern is found for the first time on the basis of the electron dynamics. Further, by increasing the initial phase of laser pulse, the overall angular radiation has an interesting counter-clockwise rotating trend with a cycle of Δξ0 = 2π. These results would help the understanding of nonlinear Thomson scattering and push forward the research of twisted X/γ-ray generation in optical laboratory.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
26 April 2023: A typographical correction was made to the body of the paper.
1. Introduction
Thomson/Compton scattering is well-known as a scattering process between charged particles and electromagnetic (EM) field [1–8]. With the advent of Chirped Pulse Amplification (CPA) technology [9–14], there has been a renewed interest in Thomson scattering for its application as an attosecond pulse source. While the molecular/atomic dynamics [15–17] can yet be measured by femtosecond pulses, there exists much faster process like electron ionization [18], which necessitates deeper investigation into Thomson scattering. In the meanwhile, CPA also pushes research of Thomson scattering from the original petawatt level to a high-field domain with laser intensities up to 1022W/cm2. For electrons in a near infrared intense laser pulse, the onset threshold of relativistic motion has been defined as 1018W/cm2 by Pasto [19]. This indicates an ultra-relativistic motion of electron in the intense laser field, where the outer electrons are ionized and the released electrons are accelerated to extremely fast speed. Thus, more complex and nonlinear phenomena begin to appear in intense laser fields, among which high-order harmonic radiation [20–24] has the most practical application as twisted X-ray or even γ-ray sources. The twisted X/γ-ray can be used in multiple fields, including modern astrophysics [25], condensed matter physics [26] and quantum entanglement [27].
The radiation character of nonlinear Thomson scattering (NTS) has been studied by many authors for exploiting its further applications. Some works [28–29] concentrated on the symmetry features of radiation in linearly polarized laser pulses. For instance, Lan et. al. [28] found that the spatial distributions of radiation have fourfold or twofold symmetry when the electron is injected to the laser field at an initial phase of 0 or π, whereas in other phases a steady drift motion of electron could break the symmetry feature. Later, Li et al. [29] re-investigated this problem and made a novel discovery about the bifoliate radiation pattern of Thomson scattering. In more complicated cases of Thomson scattering, Liu et. al. [30] extended the radiation symmetry to circularly polarized pulses and investigated the annular shapes of high-order harmonics. Nevertheless, these works investigated the radiation character based on merely one factor and leave out discussion about asymmetric radiation. In this work, we theoretically and numerically investigate the evolvement of spatial radiation distributions in laser pulses with varied laser intensities, pulse widths, beam waists and initial phases. We also present the interconnection between the symmetric and non-symmetrical vortex radiation from the perspective of electrodynamics.
The remaining part of this paper is organized as follows: in Section 2, we first present the classic theory of a single electron’s orbit. Subsequently, the analytical expressions of spatial radiation in nonlinear Thomson scattering are derived on the basis of Lienard-Wiechert potentials. In Section 3, the angular radiation distributions are comprehensively investigated in varied Laguerre-Gaussian (LG) laser pulses and the transition between symmetric and vortex radiation characters is discussed based on the dynamics of electron. In Section 4, we conclude the spatial radiation features of nonlinear Thomson scattering and put forward its potential application in optical laboratories.
2. Theory and formula
2.1 Electron’s motion inside laser pulses
In the NTS model, we introduce a circularly polarized LG laser pulse which propagates along + z axis of the reference frame, with an incident angle ${\sigma _{in}} = 0$. Considering the Doppler effect under this ultrafast condition, the space and time coordinates are respectively normalized by the reciprocals of laser frequency ${\omega _0}$ and wave number ${k_0} = 2\pi /{\lambda _0}$. The field of incident laser pulse can be described by the vector potential as
For simplicity, we assume a stationary electron e- is set at the origin of the reference frame, with an initial distance of ${\eta _0} = 50\mu m$ to the pulse center. The motion of electron is governed by Lagrange function and the electron energy function as
2.2 Angular distribution of spatial radiation
In the previous discussion 2.1, we already work out the electron’s full-time momentum, displacement and acceleration utilizing electrodynamic theory in the rectangular coordinate. It is well known that relativistically accelerated electron can emit high order harmonic radiation, characterized by the electric field which can be calculated by the Lienard-Wiechert potentials
3. Numerical results
In the following section, we discuss the radiation characters of NTS. The incident circularly polarized laser pulse modeled by an envelope of Gaussian wings propagates along + z axis, which has a wavelength of ${\lambda _0} = 1\mathrm{\mu} \textrm{m}$. The charged electron with an initial Lorentz factor ${\gamma _0} = 1$ rests at the origin point, ${\eta _0} = 50\mathrm{\mu} \textrm{m}$ away from the pulse center. By projecting the electron’s time-integral spatial radiation to the transverse plane perpendicular to + z axis, more intuitive angular radiation distributions of NTS are displayed in Figs. 3–6 with each case’s radiated energy normalized by the maximum value. Figure 2 gives an illustration of several terms used in the following sections. In the direction of $\phi = {\phi _x}$, the angular gap of vortex radiation is defined as the distance between the closest local maximal radiation points, while the angular width of annular radiation is defined as the range with half of the local maximal radiated energy.
Fig. 2. Radiation components with regard to θ and ϕ in the angular plane. The central circle and the line crossing the center respective represents θx and ϕx radiation component. (a) The angular gap Δθ of the vortex radiation, defined as the distance between the closest local maximal radiation points in the direction of ϕ=ϕx. (b) The angular width of the annular radiation, defined as the range with half of the local maximal radiated energy in the direction of ϕ=ϕx. (c)
Fig. 3. Distributions of the electron’s time-integral angular radiation. The employed circularly polarized LG laser pulse has a peak amplitude of (a) a0 = 1, (b) a0 = 2, (c) a0 = 3, (d) a0 = 4, (e) a0 = 5, (f) a0 = 6, (g) a0 = 7, (h) a0 = 8, (i) a0 = 9. The pulse width L = 3μm and beam waist b0 = 3μm. Instead of applying a color bar, the radiated energy is respective normalized by the maximum value.
Fig. 4. Distributions of the electron’s time-integral angular radiation. The employed circularly polarized LG laser pulse has a pulse width L of (a) 1μm, (b) 2μm, (c) 3μm, (d) 4μm, (e) 5μm, (f) 6μm, (g) 7μm, (h) 8μm, (i) 15μm. Here, the peak amplitude a0 = 8 and beam waist b0 = 3μm.
Fig. 5. Distributions of the electron’s time-integral angular radiation. The employed circularly polarized LG laser pulse has a beam waist b0 of (a) 1μm, (b) 2μm, (c) 3μm, (d) 4μm, (e) 5μm, (f) 6μm, (g) 20μm, (h) 100μm, (i) Inf. Here, the peak amplitude a0 = 8 and pulse width L = 3μm.
Fig. 6. Distributions of the electron’s time-integral angular radiation. The circularly polarized LG laser pulse has an initial phase ξ0 of (a) 0, (b) π/4, (c) π/2, (d) 3π/4, (e) π, (f) 5π/4, (g) 3π/2, (h) 7π/4, (i) 2π. Here, the peak amplitude a0 = 8, pulse width L = 3μm and beam waist b0 = 1μm.
3.1 Pulses with varied laser intensities
Normally, the electron dynamic changes greatly within LG laser pulses of varied laser intensities I. The employed circularly polarized LG laser pulses share the same pulse width $L = 3\mathrm{\mu} \textrm{m}$ and beam waist ${b_0} = 3\mathrm{\mu} \textrm{m}$, whereas the peak amplitudes ${a_0}$ are different. Figure 3 presents the angular energy distributions from electron irradiation with ${a_0}$ changing from 1 $(I = 1.38 \times {10^{18}}\textrm{W/c}{\textrm{m}^\textrm{2}})$ to 8 $(I = 8.832 \times {10^{19}}\textrm{W/c}{\textrm{m}^\textrm{2}})$ with an interval of unit value. The relation between peak amplitude and laser intensity is presented in Section 2 as ${a_0} = 0.85 \times {10^{ - 9}}{\lambda _0}\sqrt I$.
Within comparatively less intensive LG laser pulses $({a_0} = 1 - 5)$, the electron radiates with a fourfold-symmetric pattern in Figs. 3(a)-3(e). The major radiated energy gathers around a complete circle and generates an annular shape, while the remaining plane barely receives any radiation. We know that in the angular plane, as is illustrated in Fig. 2(a), a central circle corresponds to a ${\vartheta _x}$ radiation component $(\phi = 0 - 2\pi )$ while a line crossing the center corresponds to a ${\phi _x}$ radiation component $(\vartheta = 0 - \pi )$. It is indicated by the annular distribution that angular radiation merely relates to the polar angle $\vartheta$. In other words, angular radiation is the same for directions with fixed polar angle ${\vartheta _x}$, where the energy distribution first rises to a peak at $\vartheta = {\vartheta _m}$ and then falls to zero soon after from $\vartheta = \pi$ to $\vartheta = 0$.
Further, by increasing the peak amplitude in Figs. 3(a)-3(e), the overall radiation converges towards the center of angular plane. This can be explained from the electron dynamics. Through restructuring the vector potential term $\boldsymbol{a}(\eta )$ in Eq. (1) into $\boldsymbol{a}({a_0},\eta ,\rho ) = {a_0}\boldsymbol{\tilde{a}}(\eta ,\rho )$, it can be derived that the EM field of laser pulse is directly proportional to the peak amplitude ${a_0}$. Nevertheless, with ${a_0}$ increasing, the longitudinal ponderomotive force is amplified several orders of magnitude greater than the transverse oscillating force. Under such circumstance, the increment of electron’s transverse velocity ${\boldsymbol{u}_ \bot }$ is negligible compared to that of longitudinal velocity ${\boldsymbol{u}_z}$. Thus, the trajectory, as well as spatial radiation, approaches + z axis with smaller deviation ${\vartheta _u} = {\tan ^{ - 1}}({u_ \bot }/{u_z})$, leading to a converged circle of radiation in the angular plane.
It is also noteworthy that the angular width of annular radiation, illustrated in Fig. 2(c), gets thinner in Fig. 3(e) in contrast to Fig. 3(a). This is caused by the electron velocity evolvement in above circumstances. We know that in Wiggler condition [31], angular radiation gathers inside a tiny cone of typical opening angle of $\Delta \zeta$ around the velocity vector, and $\Delta \zeta$ is inversely proportional to Lorentz factor $\gamma$ as $\Delta \zeta \sim 1/\gamma$. Intensive laser pulse energizes the electron with greater Lorentz factor throughout the interaction process. Thus, the angular width, with a time-integral relation of $\Delta \zeta$, gets smaller in pulses with higher peak amplitude due to the shrinkage of $\Delta \zeta$.
Interestingly, it has been discovered by Li. et. al. [29] that spatial radiation from electron is always characterized by a fourfold symmetric shape in the linearly polarized laser pulses, while we have extended Li’s findings into circularly polarized laser pulses with ${a_0} = 1 - 5$. However, the fourfold-symmetry radiation character vanishes and transforms abruptly into a vortex pattern in Figs. 3(f)-3(i). With the increase of peak amplitude, it is obvious that the radiation structure gets more diversified and changes much faster along a spiral in laser pulse with ${a_0} = 9$. This indicates that the electron’s transverse drift motion is highly unstable with the curvature radius changing frequently in intensive laser pulses, which in turn brings remarkable changes to angular radiation and breaks the symmetry of energy distributions.
3.2 Pulses with varied pulse widths/durations
In this section, we employ laser pulses with the same beam waist ${b_0} = 3\mathrm{\mu} \textrm{m}$ and peak amplitude ${a_0} = 8\textrm{ }(I = 8.832 \times {10^{19}}\textrm{W/c}{\textrm{m}^\textrm{2}})$, while the pulse widths L are different. The relation between pulse width L and pulse duration τ can be described by $\tau = L/c$, where one wavelength ${\lambda _0} = 1\mathrm{\mu} \textrm{m}$ corresponds to 10/3 fs. For more intuitive numerical results, we shall use L to describe the LG laser pulse hereinafter. Figure 4 presents the angular radiation distributions in pulses with L changing from 1μm to 8μm with an interval of $1\mathrm{\mu} \textrm{m}$.
As is shown in Fig. 4(a), within an ultrashort LG laser pulse L=1μm, the electron radiates in a semi-circle pattern which has obvious plane symmetry feature. This is due since the longitudinal vector potential term $exp ( - {\eta ^2}/{L^2})$ in Eq. (1) varies greatly in different parts of the incident laser pulse. We know that the electron starts off with ${\eta _0} = 50\mathrm{\mu} \textrm{m}$ away from the pulse center and the exponential term is $exp ( - {\eta ^2}/{L^2}) = exp ( - {(50{\lambda _0})^2}/\lambda _0^2) = exp ( - 2.5 \times {10^3})$. However, the longitudinal term soars to $exp ( - 0/\lambda _0^2) = 1$ in the middle of the pulse, over one thousand orders of magnitude larger than the initial value. The electron is highly excited merely in the central part of the laser pulse, $- {\lambda _0} < \eta < {\lambda _0}$ to be precise. Meanwhile, the pulse width $L = 1\mathrm{\mu} \textrm{m}$ is too short for the electron to travel several optical circles in the middle. Thus, the electron only emits high radiation outward within a small semi-circle part of the trajectory, leading to the apparent semi-circle in the angular plane.
With pulse width increasing in Figs. 4(b)-4(h), we find that the angular radiation evolves into a vortex pattern and the plane symmetry feature vanishes. From the angular regions away from + z axis $(\vartheta = \pi )$ to the nearby regions $(\vartheta = 0)$, the radiation shares a common trend of rising and then decreasing along a spiral. Moreover, compared to the $L = 3\mathrm{\mu} \textrm{m}$ case in Fig. 4(c), it is interesting to find the angular gap of vortex radiation narrows obviously in Fig. 4(h) with $L = 8\mathrm{\mu} \textrm{m}$, and the vortex structure is enriched with more closely distributed spirals. This indicates that, as pulse width L gets longer, the electron dynamics becomes more complicated within the laser pulse due to the prolonged period of relativistic laser-electron interaction.
In a plane pulse wave $({b_0} = Inf)$, Lan [28] has presented that the radiated energy distributions show fourfold or twofold symmetry in certain conditions. This finding can also be extended to tightly focused $({b_0} = 3\mathrm{\mu m)}$ laser pulse in a multi-cycle $(L \gg {\lambda _0})$ case. As is illustrated in Fig. 4(i), the angular radiation eventually evolves into a fourfold symmetric single-peak structure when $L = 15\mathrm{\mu} \textrm{m}$. This can be explained by the classical electrodynamics theory. We know that the total radiated energy of a specific direction is the time integration of instantaneous radiated power. Nevertheless, in a tightly focused LG pulse, it is the short peak-radiation period that dominates the whole radiation process in aspect of energy, whereas the electron barely radiates for the rest of time. This degrades the time-integral term in Eq. (10) $\int_{ - \infty }^\infty {dt[{\boldsymbol{n} \times (\boldsymbol{n} - \boldsymbol{u}) \times \boldsymbol{\dot{u}}} ]{e^{is(t - \boldsymbol{n} \cdot \boldsymbol{r})}}/[{1 - {{(\boldsymbol{n} \cdot \boldsymbol{u})}^2}} ]}$ into $\Delta t[{\boldsymbol{n} \times (\boldsymbol{n} - \boldsymbol{u}) \times \boldsymbol{\dot{u}}} ]{e^{is(t - \boldsymbol{n} \cdot \boldsymbol{r})}}/{[{1 - {{(\boldsymbol{n} \cdot \boldsymbol{u})}^2}} ]_{ra{d_m}}}$ with a time-product form, where the period $\Delta t$ is closely related to the pulse width L. According to Eq. (12), instantaneous radiation is determined by the electron’s velocity and acceleration, consequently the radiated power peaks at pulse center where both have high value. Yet, in a long-pulse-width case, the electron remains highly excited and radiates violently for a long period $\Delta t$. That is to say, long pulse width relieves the rapidly varying interaction process in tightly focused LG laser pulses, bringing about phenomenon characterized by plane wave pulses. Thus, the peak radiation only varies slightly at the period of $\Delta t$, which plumps up the asymmetric vortex angular radiation to the fourfold symmetric pattern that is common in plane wave pulses.
3.3 Pulses with varied beam waists
Beam waist largely determines the transverse energy distribution in LG pulses, which in turn affects the spatial radiation from NTS. In this section, we introduce LG laser pulses with the same pulse width $L = 3\mathrm{\mu} \textrm{m}$ and peak amplitude ${a_0} = 8$ $(I = 8.832 \times {10^{19}}\textrm{W/c}{\textrm{m}^\textrm{2}})$, whereas the beam waists ${b_0}$ are different. As is shown in Fig. 5(a), when the beam waist is narrowed down to the scale of wavelength ${\lambda _0} = 1\mathrm{\mu} \textrm{m}$, the energetic electron radiates in a perfect vortex pattern with long tail extending to the edge of the angular plane $(\vartheta = \pi )$. This is largely attributed to the limited regions of intense laser-electron interaction in ultra-tightly focused ${b_0} = {\lambda _0}$ pulses. According to Eq. (1), the transverse exponential term of vector potential $exp ( - {\rho ^2}/{b^2})$ determines that, once the electron’s quivering amplitude $\rho$ exceeds b, the oscillating component of ponderomotive force will shrink significantly. While the ponderomotive force peaks at the focus, the electron is expelled away transversely upon reaching the center of LG pulse. The subsequent weakening EM force can’t afford to maintain the circular propagation of electron. Thus, the electron then keeps moving away from + z axis transversely and radiates far less energy outwards, leading to the long-tail vortex angular radiation.
With beam waist increasing in Figs. 5(a)-5(d), we find the shrinking trend of angular gap in vortex radiation much similar to that in Fig. 4. However, more nonlinear phenomena appear this time. When the laser pulse becomes less focused in Fig. 5(d), the angular radiation covers much less area in contrast to Fig. 5(a). Moreover, instead of shrinking towards the center in a single-peak structure resembling Fig. 4(i), the angular radiation actually converges towards a circle with certain azimuth angle in Figs. 5(e)-5(i), approximately $\vartheta = 0.83$. When the LG laser turns into a non-tightly focused pulse $({b_0} = 12 - 100\mathrm{\mu m)}$ and even plane pulse wave $({b_0} = Inf)$, the angular radiation eventually distributes along a single circle with perfect plane symmetry.
We discuss the above phenomenon from two perspectives. With regard to the radiation pattern in Fig. 5, we already covered that the electron orbits spirally within a circularly polarized pulse, and the opening angle of radiated power [31] is inversely proportional to Lorentz factor $\Delta \zeta \sim 1/\gamma$. The eventual single-circle radiation structure in Figs. 5(f)-5(i), with negligible angular width, indicates that the electron is energized with ultrafast velocity inside the pulse center, during which the opening angle of radiation approaches zero $\Delta \zeta \to 0$. Therefore, the spatial radiation is highly collimated along the velocity vector u, forming a complete thin circle in the angular plane after the electron travels several optical circles within the pulse center. Furthermore, it is interesting that the radiation structure almost ceases to change ever since the beam waist increases to ${b_0} = 12\mathrm{\mu} \textrm{m}$. In this case, the electron’s transverse quivering amplitude $\rho$ is always much smaller than b, and the electron keeps propagating inside effective-interaction regions $\rho < b$. Therefore, the ponderomotive force, theoretically characterized by two exponential terms $exp ( - {\rho ^2}/{b^2})$ and $exp ( - {\eta ^2}/{L^2})$ in Eq. (1), degrades to the function of a single variable $\eta$ since $exp ( - {\rho ^2}/{b^2})$ is considered to be $exp ( - {\rho ^2}/\infty ) = 1$ when b is much greater than $\rho$. Such laser-electron interaction process eventually leads to the similar radiation structures in Fig. 5(f)-(i). This also brings us to an interesting topic of, on the basis of NTS from a single electron, how to determine LG laser pulse is a tightly focused, non-tightly focused or even plane pulse wave.
3.4 Pulses with varied pulse-initial phases
The initial phase ${\xi _0}$, as is presented in Eq. (5), is an additive phase term which determines the initial direction of EM field in the laser pulse. In this section, the employed laser pulses share the same parameters with Fig. 5(a). Figure 6 shows the angular radiation distributions under different initial-phase circumstances $({\xi _0} = 0 - 2\pi$ with an interval of $\pi /4)$. The angular radiation figures share similar vortex feature, rising and decreasing along a spiral from the edge to the center. We can see that the electron’s radiation depends sensitively on the initial phase ${\xi _0}$ of LG laser pulse, where the maximal angular radiation is respectively inclined to the direction of left back $({\xi _0} = 0,\textrm{ }2\pi )$, back $({\xi _0} = \pi /4)$, right back $({\xi _0} = \pi /2)$, right $({\xi _0} = 3\pi /4)$, right front $({\xi _0} = \pi )$, front $({\xi _0} = 5\pi /4)$, left front $({\xi _0} = 3\pi /2)$ and left $({\xi _0} = 7\pi /4)$. The counter-clockwise evolving trend of peak radiation indicates that the electron’s trajectory, as well as the angular radiation, rotates around the central + z axis with a certain angle determined by the initial phase of laser pulse.
Furthermore, comparing the first four figures with the second four ones in Fig. 6, we can see that the angular radiation distribution for ${\xi _0}$ is the inverse refection of that for ${\xi _0} + \pi$. This is because the laser field reverses its direction when the initial phase changes π. Consequently, the cycle period of the angular radiation is $\Delta {\xi _0} = 2\pi$ since the laser field is the same after the initial phase ${\xi _0}$ increases to ${\xi _0} + 2\pi$, as is illustrated in Fig. 6(a) and Fig. 6(i).
To better investigate the relation between angular radiation and ${\xi _0}$, we use ${\phi _m}$ and ${\vartheta _m}$ to denote the corresponding polar angles of peak angular radiation. Through increasing ${\xi _0}$ from 0 to 2π with an interval of π/90, the above polar angles, as well as the angular difference ${\phi _m} - {\xi _0}$, are presented in Fig. 7. The values of ${\phi _m}$ are increased by 2π at some point for better illustration. For the first time ever, we find that the polar angle ${\vartheta _m}$ has a constant value of 0.0175 regardless of initial phases ${\xi _0}$, whereas ${\phi _m}$ changes proportionally to the initial phase ${\xi _0}$. The linear relation between ${\phi _m}$ and ${\xi _0}$ is also demonstrated by the constant value of ${\phi _m} - {\xi _0} = 1.3439$, appearing as a horizontal line in Fig. 7. The numerical results perfectly explain the inverse refection phenomenon in Fig. 6, giving an intuitive proof to the $\Delta {\xi _0} = 2\pi$ cycle of the angular radiation.
Fig. 7. The polar angle ϕm (red line), θm (green line) of the peak angular radiation and the angular difference ϕm - ξ0 (blue line). The initial phases ξ0 of the LG pulse are sampled from 0 to 2π with an interval of π/90.
4. Conclusions
In conclusion, we theoretically and numerically study the radiation feature of nonlinear Thomson scattering in the interaction of varied circularly polarized laser pulses with a charged electron. On the basis of Lienard-Wiechert potentials, the analytical expressions for angular radiation distributions are derived from the electron dynamics. The simulation results show that LG pulses characterized by comparatively low laser intensity, prolonged pulse duration or wide beam waist lead to a fourfold or plane symmetric radiation pattern from electron irradiation. Nevertheless, in other circumstances the angular radiation is characterized by a vortex shape, where the steady drift motion of electron within the pulse center breaks the symmetry of energy distributions. Further, a novel phenomenon is discovered that, by increasing the initial phase of laser pulse, the angular radiation rotates counter-clockwise with a cycle of 2π. These results could help explore the nature of nonlinear Thomson scattering and generate twisted X/γ-rays in the laboratory frame.
Funding
Nanjing University of Posts and Telecommunications (NY2015154); Qinglan Project of Jiangsu Province of China; Natural Science Research of Jiangsu Higher Education Institutions of China (10KJB140006); National Natural Science Foundation of China (10947170/A05).
Acknowledgments
This work is part of the STITP Project under Grant No. SYB2020036.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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