Physics of Superpulses in Storage Ring Free-Electron Lasers

Abstract

Contradictory to the widespread perception, storage ring free-electron lasers with substantial net gain can generate peak lasing power reaching GW levels in the so-called superpulse mode. This power level is sufficient for studies of nonlinear processes and efficient intracavity harmonic generation. This letter describes the physics of the superpulses and a phase-space refreshment in the electron beam responsible for this phenomenon.

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All presently existing storage ring free-electron lasers (SR FELs) are oscillators with a two-mirror optical cavity and an optical klystron (OK) [1] comprising two wigglers and a buncher between them. SR FELs can operate in pulsed mode [2] with significant peak optical power. Early theoretical models [1] predicted peak intracavity optical power (discussed in this letter), 𝑃_opt, to be limited by𝑃_opt≤𝑃_𝑒-beam.𝜎_𝛾ind/𝛾_𝑜,𝑃_𝑒-beam=𝐼^𝑒.𝐸_𝑜𝑒,(1)where 𝐸_𝑜=𝛾_𝑜𝑚𝑐² is the central energy and 𝐼_𝑒 is the peak electron beam current, 𝜎_𝛾ind is RMS spread of 𝛾=𝐸/𝑚𝑐² induced by FEL interactions, and 𝑒 is the electron charge. For a typical SR FEL with 𝐸_𝑜~1 GeV, 𝐼_𝑒~10 A, and 𝜎_𝛾ind/𝛾_𝑜 ~ 10⁻³, (1) limits peak FEL power to about ten MW. Studies of selfconsistent SR FEL dynamics using the 3D uvfel code [3–5] revealed a mechanism of superpulses with peak power two-to-three orders of magnitude exceeding the estimate (1). Figure 1 shows the typical evolution of a superpulse simulated by the uvfel code [3, 4], which is known for reliably predicting the electron and the optical beam evolution in SR FELs [2, 7–9].

Figure 1 

Evolution of a superpulse in OK-4/Duke SR FEL predicted by uvfel code. Intracavity optical laser beam energy (black) and peak power (red) are plotted in log-scale (left), and relative RMS e-beam energy spread (blue) is plotted in linear scale (right). Operational parameters: Eo = 650 MeV, 𝐼𝑒 = 36.3 A, λ = 237 nm, Nw = 33.5, ND = 10.75, Qs = 0.0094, σ γo/γ =1.5.10−3, σ zo = 2.42 cm, starting peak gain Go = 16.5%, optical cavity losses are 2% per turn (see [8] for details). Inset shows peak power growth (in linear scale) for turns 150–200.

In this letter, we discuss phenomena responsible for superpulses in SR FEL. We present a self-consistent set of dimensionless nonlinear differential equation describing superpulse and the peak power dependencies on key parameters. We also discuss the physics of superpulses using results of computer simulations for illustration (~1000 runs of uvfel code with ~10⁶ macroparticles and total number of macroparticle passes through the FEL ~10¹¹ are used).

An ultrarelativistic electron is described in 6D phase space by its energy 𝐸_𝑒=𝐸_𝑜(1+𝜀), longitudinal coordinate 𝜉=𝑣_𝑒𝑡−𝑧, transverse coordinates and angles 𝑋^𝑇≡[𝑥,𝑥^,𝑦,𝑦^], where 𝑥^≡𝑑𝑥/𝑑𝑧 and z is the coordinate along the FEL axis. Without lasing, the electron beam obtains a natural distribution in 6D phase space, which is close to a Gaussian and is the product of the normalized transverse (t) and longitudinal (s) distribution functions 𝑓_6𝐷≅𝑁_𝑒.𝑓_𝑡(𝑋).𝑓_𝑠, where N_e is the number of electrons in the e-bunch [10]. This separation of variables is valid for an SR FEL located in the dispersion-free straight section and processes much shorter than that of radiation damping time, which we discussed in this letter. In the absence of FEL interactions, longitudinal motion is an oscillation [10]𝛿+𝑖𝜁=𝑎_𝑠.𝑒^𝑖^{𝑛Ω𝑠+𝜑𝑠},𝑓_𝑠(𝛿,𝜁)≅12𝜋𝑒⁻^𝛿2+𝜁2^/2,(2) where Ω_𝑠=2𝜋𝑄_𝑠, Q_s is the synchrotron tune, n is the turn number around the ring, and {𝛿=𝜀/𝜎_𝜀𝑜,𝜁=𝜉/𝜎_𝑧𝑜} are dimensionless coordinates scaled by the natural e-beam energy spread 𝜎_𝜀𝑜=𝑚𝑐²𝜎_𝛾𝑜 and the bunch length 𝜎_𝑧𝑜. Trajectories in {𝛿,𝜁} phase-space are simple clock-wise circular rotations (see Figure 3).

The FEL wigglers (with period λ_w and magnetic field B_w) provide a resonant interaction between electrons and the TEM optical wave at wavelength λ_o given by𝜆_𝑜=𝜆^𝑤2𝛾^2𝑜1+^→𝑎^2𝑤,^→𝑎_𝑤=𝑒^→𝐵_𝑤𝜆_𝑤2𝜋𝑚𝑐².(3) Passing through an OK, an electron radiates a wave packet with length equal to its total slippage Ξ=(2𝑁_𝑊+𝑁_𝐷+𝑂(𝜀,𝑥^2,𝑦^2))𝜆_𝑜, Ξ≪𝜎_𝑧𝑜, where N_w is the number of periods in one wiggler and N_D is the dimensionless slippage in the buncher [11]. When FEL interactions are present, the intracavity optical power builds up from spontaneous radiation, which is accumulated inside the optical cavity and amplified during consequent passes by the circulating e-bunch𝑑𝑃_opt(𝜁)𝑑𝑛=𝑃^opt(𝜁)𝐺(𝜁)−𝐺_th+𝑃^SR(𝜁),(4) where 𝐺_th=(𝑅₁𝑅₂)⁻¹−1 is the loss per turn in an optical cavity with mirror reflectivity R_1,2 and P_SR is the power of spontaneous radiation into the optical mode. At the beginning of the process (at n = 0 FEL interaction is turned on [2]), the longitudinal FEL gain profile is Gaussian, like the e-beam, with peak gain G_o𝐺_FEL(𝜁)=𝐺_𝑜𝑒^−𝜁2/2.(5) The complete set of dimensionless selfconsistent equations for a superpulse can be derived for a rather general SR FEL case, using Einstein’s relations between spontaneous and induced radiation [12–14]. The optical field in an FEL can be described as a wave packet with slowly varying complex amplitude and a normalized transverse mode function ^→𝑢(^→𝑟): 𝐸^→_opt=R∫e|||^→_→𝑢(^→𝑟)𝐴_𝑜(𝑐𝑡−𝑧)𝑒^{𝑖𝑘𝑜(𝑐𝑡−𝑧)},𝑃𝑢(^→𝑟)|||²𝑑𝑥𝑑𝑦=1,_opt(𝑐𝑡−𝑧)=𝑐8𝜋|||𝐴^𝑜|||𝑑𝐴_𝑜(𝑐𝑡−𝑧)|||²,(𝑧)𝑑𝑧|||≪|||𝐴^𝑜|||Ξ;𝑘^𝑜=2𝜋𝜆_𝑜.(6) Variation of electron energy in an FEL is the result of local interaction with a Ξ-long segment of the optical field and is given by an integral along the electron’s trajectoryΔ𝐸_𝑒(𝜉)=𝑒𝐸^→_opt^→𝑟_𝑒,𝜉(^→𝑟_𝑒,𝜀)𝑑^→𝑟_𝑒.(7) For a short wavelength FEL, the correlations between electrons are washed away every turn around the ring. Therefore, optical phases 𝜙 = k_o(ct_i − z_i) of electrons entering the FEL are random, resulting in the famous ratio between the energy loss and the energy diffusion [12, 13]⟨Δ𝛿⟩=𝑑𝐷^𝛿𝑑𝛿,𝐷^𝛿=12Δ𝛿².(8)

Using (7) and (8), one can write a set of dimensionless equations𝜕𝑓𝜕𝜈+𝑟𝛿𝜕𝜕𝑓𝜁−𝜁𝜕𝜕𝑓𝛿−𝑝^𝑜(𝜁).𝑔.𝜕𝑑𝑝_𝑜𝜕𝛿𝑞(𝛿)𝜕𝜕𝑓𝛿=0,(𝜁)𝑑𝜈=𝑝^𝑜(𝜁)(𝑔∫𝑓(𝜁,𝛿)−𝑑𝑞(𝛿)𝑑𝛿𝑑𝛿−1)+𝑝^SR(𝜁),(9) with 𝜈=𝑛.𝐺_th as the independent variable, and optical power is normalized as 𝑝_𝑜(𝜁)=𝑃_opt(𝜁)/𝑃_𝑛 by 𝑃_𝑛=√2𝜋𝑃_𝑒-beam𝜎_𝛾𝑜/𝛾_𝑜 which is of the same order as limit in (1). Equation (9) contains dimensionless parameters 𝑔=𝐺_𝑜/𝐺_th (gain parameter), 𝑟=Ω_𝑠/𝐺_th (refreshment parameter), and a dimensionless well-defined function𝑞(𝛿)=𝜋𝑁^𝑒𝑟_𝑒𝜎_𝑧𝑜𝜎_𝛾𝑜𝐺_𝑜.𝑑𝑋⁴𝑓_𝑡(𝑋)|||^→𝑢^→𝑟_𝑒𝑒^{𝑖𝑘𝑜𝜉1(𝑋,𝛿,𝑧)}𝑑^→𝑟_𝑒|||²(10) normalized as ∫𝑒^−𝛿2/2(𝑑𝑞(𝛿)/𝑑𝛿)𝑑𝛿=−1 to satisfy (5). Function ξ₁(X,δ,z) above is the electron’s slippage with respect to the wave packet𝜉₁=𝜉₁₀+^𝑧01+^→𝑎^2𝑤(𝑥,𝑦,𝑧̃)2𝛾^2𝑜1−2𝛿𝜎^𝜀𝑜+𝑥^2(𝑧̃)+𝑦^2(𝑧)̃2𝑑𝑧.̃(11)

For the optical beam to be amplified, the derivative of the diffusion coefficient q(δ) should be negative at and around δ = 0. It means that q(δ) reaches its maximum at negative δ and its minimum at positive δ (see Figure 3). N_D in the range1≤4𝜋𝑁_𝑤+𝑁_𝐷𝜎^𝜀𝑜≤√𝑔(12) provides for attainment of maximal peak power with |||𝑞(𝛿)|||∼1, |||𝑑𝑞(𝛿)/𝑑𝛿|||∼1 (the reduction of the peak power caused by finite transverse emittances of the e-beam is well described by F_x,y factors in [8]). Figure 2 shows dependencies of peak power on the gain and the refreshment parameters. The power is low close to the lasing threshold (𝑔−1≪1), grows very fast in 1.5 < g < 10 range, and saturates above 𝑔=10. Dependence on the refreshment parameter is less dramatic, but peak power increases about sixfold from 𝑟≪1, when the synchrotron motion is essentially frozen, to 𝑟≫1. Detailed studies show that optimal ratio between r and g is near π, that is, when electrons go through half of a synchrotron oscillation during one e-growth time ν_e = 1/g of the laser power. Enhancement comes from synchrotron oscillations moving electrons through the center region and allowing the short optical pulse to be amplified by entire electron bunch. Further increase of the refreshment parameter is not beneficial and can cause oscillations of optical power.

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(b)

Figure 2 

Dependencies of the normalized peak power po on: (a) the refreshment parameter for g = 8.14, (b) the gain parameter g for r = 10. Other parameters are the same as in Figure 1. Each point in the graphs shows the average value and standard error for eight to sixteen statistically independent runs of uvfel code.

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(b)

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Figure 3 

Four Poincaré plots (electrons are red dots) in (δ, ζ) phase space during the superpulse (parameters are as in Figure 1). The clips show the ζ-profiles of the optical power (blue) with power scales 10 W for turn 1, 10 MW for turn 100, and 1 GW for turns 200 and 250. The Turn 200 plot has additional clip on the right with the FEL energy diffusion function. The RMS length of optical pulse is σ opt = 0.35 at turn 200. The circle and arrow show the trajectory of synchrotron motion for a particle with as = 10.

Figure 3 shows details of the e-beam dynamics. During the superpulse, period of accumulation of spontaneous radiation is followed by a period of exponential growth. The correlation length of the optical field grows with the amplification Γ=∫𝐺_FEL(𝑛)𝑑𝑛 as 𝜉_corr≅Ξ√1+Γ, while the envelope of the optical pulse shrinks 𝜎_opt∝1/√1+Γ [6] to about 30% of the initial e-bunch length. The initial power growth by 5 to 6 orders of magnitude (see Figures 1 and 3, turns 1–100) does not affect significantly the e-beam distribution and the gain profile. The amplified optical pulse grown from the “white” noise consists of very short (Δζ = 1−2.10⁻³) spikes with peak power in the spikes exceeding the local average by approximately an order of magnitude, similar to that in SASE FELs [15, 16].

Further growth of the optical power affects the e-beam distribution and complicates the dynamics. Energy loss and diffusion in the FEL make the phase-space distribution of electrons asymmetric and time-dependent. The growth of the local energy spread reduces the FEL gain while the asymmetric gain profile and energy diffusion cause a drift of the optical pulse and the electron beam from the center [7, 18]—see changes in ⟨𝛿⟩,⟨𝜁⟩ in Figure 3. The optical power reaches maximum when net gain crosses zero at 𝐺_FEL=𝐺_th. After that, the optical power decays, and the electron beam distribution becomes more symmetric.

As shown in “Turn 200” plot in Figure 3, lower energy electrons (−3<𝛿<0) have maximum diffusion, while electrons with higher energy have a low diffusion energy corridor clearly noticeable at 1<𝛿<3. This diffusion asymmetry, which is fundamental to the amplification process in FEL [12, 13], produces the top-bottom asymmetry in the Poincaré plots. Electrons at large amplitude of synchrotron oscillations (a_s ≥ 1) interact with the intense optical pulse briefly (𝑛_int∝𝜎_opt/𝑎_𝑠Ω_𝑠) while passing near the ζ = 0. The diffusion imprints the level of the FEL power at that moment onto the distribution. Thus, the energy spread seen at angle ψ in Figure 3 shows that the FEL power level happened about Δ𝑛≅𝜓/Ω_𝑠 turns in the past. Thus, the segment 0 < ψ < π in the turn 200 graph shows the diffusion imprinted at negative δ between turns 147 and 200. Similarly, the segment 𝜋<𝜓<2𝜋 shows the diffusion imprinted at positive δ during the same turns. Thus, the snail-like dependence of the amplitude spread is a record of the evolution of the optical pulse power (see clip in Figure 1) as well as a clear indication of the diffusion asymmetry. Such snail-like distribution of electron beam was also observed experimentally [19].

The asymmetry in diffusion creates multipole moments in longitudinal phase space and residual quadrupole oscillations with frequency 2Ω_s can be seen in the oscillations of RMS energy spread in Figure 1.

The dimensionless nature of (8) provides for the scalability of the results presented in this letter: SR FELs with the same parameters {𝑟,𝑔,𝑞(𝛿)} would have the same average values and variations of the dimensionless peak power p_o. The peak superpulse power can be then simply obtained through multiplication by P_n. Thus, the simulation results presented in this letter can be used for predicting parameters in many SR FELs.

Overall, the phenomena of superpulses in SR FELs, which are presently verified both experimentally [18, 19] and confirmed by other FEL codes [20], open a range of new opportunities for generating coherent femtosecond X-ray pulses in SR FELS and their scientific applications. The OK-4 FEL at Duke produced superpulses at wavelengths of 240–270 nm for efficient coherent harmonic generation from 130 nm to 37 nm [9, 18]. These experiments were followed by successful harmonic generation in Elettra SR FEL [17]. These experiments also support predictions for superpulses presented in this letter [20].