Frequency modulated free electron laser

: It is shown that the output frequency of a free electron laser may be modulated to generate a series of modes that span a bandwidth of at least an order of magnitude greater than the normal FEL bandwidth. This new method of frequency modulated FEL operation has close analogies to frequency modulation in conventional cavity lasers. The FM-FEL is analysed and described in the linear regime by a summation over the exponentially ampliﬁed frequency modes. Simulations using a 3D, broad bandwidth, numerical code also demonstrate FM-FEL operation for parameters typical of FEL facilities currently under construction. Harmonic bunching methods are used to seed the FM-FEL modes to generate a temporally correlated frequency modulated output over a large bandwidth. This new, FM-FEL mode of operation scales well for X-ray generation, oﬀering users a signiﬁcantly new form of high-power, short wavelength FEL output.


Introduction
Free Electron Laser (FEL) is currently the world's brightest source of X-rays by many orders of magnitude [1][2][3]. The FEL consists of a relativistic electron beam injected through a magnetic undulator with a co-propagating resonant radiation field. Initially, co-propagating radiation will occur due to incoherent spontaneous noise emission from the electron beam and may be supplemented by an injected seed laser. The electrons can interact cooperatively with the radiation they emit and become density modulated at the resonant radiation wavelength. This coherently modulated oscillating electron beam exponentially amplifies the co-propagating radiation field in a positive feedback loop. In the single-pass high-gain mode, the energy of the initial incoherent, spontaneous X-rays may be amplified by around ten orders of magnitude. With such an increase in brightness over other laboratory sources, the X-ray FEL has unique applications across a wide range of the natural sciences. FEL science is, however, still under development, and the creation of novel and improved output from the FEL is still an active topic of research.
For example, it has been shown via simulations that equally spaced frequency modes may be generated in a single-pass FEL amplifier [4,5] by introducing a series of delays to the electron beam with respect to the co-propagating radiation field (e.g. by using magnetic chicanes placed between undulator modules). These radiation modes are formally identical to those created in an oscillator cavity. Analogously with a mode-locked conventional laser oscillator, a modulation of the electron beam energy [4,5] or current [6] at the mode spacing can phase-lock the modes and amplify them to generate a train of short, high power pulses.
Multiple colours may also be excited by directly tuning each undulator module to switch between 2 (or more) distinct colours [7]. This colour switching may also excite and amplify modes via a resulting gain modulation [8].

Frequency modulated FEL model
The FEL resonant frequency ω r can be tuned via the electron beam energy γ 0 , the undulator period λ u and the undulator magnetic field strength B u , via the resonance relation ω r = 2γ 2 0 ck u /(1 +ā 2 u ), where k u = 2π/λ u and the RMS undulator parameterā u ∝ B u λ u . In the following, the resonant frequency of the FEL is periodically modulated by modulating the magnetic undulator field. This excites a Bessel-like modal radiation output with a corresponding temporal-frequency modulation and with the modal spacing being much greater than the normal FEL amplification bandwidth. The undulator parameter is modulated via its magnetic field so thatā u (z) =ā u0 (1 + ǫ(z)), wherez = z/l g is the propagation distance through the undulator scaled with respect to the FEL gain length l g = λ u /4πρ, and ρ is the FEL parameter [9]. Using ǫ(z) = κ sin(k Mz )/2, and in the limitā u0 ≫ 1 and κ ≪ 1, the resonant frequency is then sinusoidally modulated around the carrier frequency ω r0 as ω r = ω r0 (1 − κ sin(k Mz )).
As with frequency modulation (FM) in conventional cavity lasers or radio transmission [10], a modulation index µ = κ/(2ρk M ) may be defined. This is the ratio of the frequency modulation amplitude to the modulation frequency. The approximate number of modes within the modulation amplitude is then N M ≈ 2µ + 1 [10]. Cases of interest are when N M ≥ 3, and the relative mode spacing is greater than the nominal FEL bandwidth, ρ.
A coupled system of linearised equations is derived similar to those of [11] which describe a tapered undulator FEL, but for a periodically modulated helical undulator field, as described above. Unlike [11], the steady state approximation is not applied, as the position dependence within the electron pulse ofz 1 = (z − cβ z t)/β z l c [12] is retained, whereβ z =v z /c is the scaled mean electron z-velocity and l c = λ r0 /4πρ is the nominal FEL cooperation length. Using the collective, linearised FEL variables as defined in [9] for the electron phase θ =z 1 /2ρ, bunching b(z,z 1 ), scaled electron energy P(z,z 1 ), and the scaled, slowly varying complex radiation envelope A(z,z 1 ) (the electric field E x +iE y ∝ Ae iω r 0 (z/c−t) ), the following linerarised, fourier transformed system of equations can be derived which describe the FEL interaction for electron pulses with a modulated undulator in the linear regime [9]: Here, the Fourier transform is defined asx is the scaled radiation frequency in units of the amplification bandwidth ρ. The phase θ r0 (z) = ∫z 0 ǫ(z ′ )dz ′ /ρ describes the effect of the phase modulation of the electrons away from the carrier frequency ω r0 . For a sinusoidal modulation, the resonant phase modulation is θ r0 = µ(1 − cos(k Mz )) and, using the well-known Bessel identity, the resonant phase modulation terms in Eqs. (2)-(3) may be written: where J n is the n th order Bessel function, which defines the nth mode, and the coupling between the different modes is apparent. By decoupling the evolution of the electrons from the radiation, by setting and maintaining the spectral bunching at its initial valueb(z = 0,ω) =b 0 (ω) for all values ofz, the exact solution for the spontaneously radiated field envelope can be shown to be:   [16] of the output radiation field showing the temporalfrequency correlation of the radiation output; the spectrum of the electron beam bunching and the FM undulator modes; the radiation power spectrum. b) As a), but with an HGHG seed of λ HG = 104nm, corresponding tok H = 25 so that the harmonic bunching spectrum now overlaps only the n = 0, and ±1 undulator modes. c) As a), but now using an EEHG seed, with both lasers tuned to λ E E = 260nm, and with EEHG chicanes tuned to optimize the bunching around λ r0 = 5.2nm, the 50 th harmonic of λ E E .
where, defining sinc(x) = sin(x)/x: is the sum over what are defined as the FM undulator modes. For a single electron, or extremely short electron bunch length (l b ≪ λ r0 ) as a source, Eq. (5) yields a constant amplitude, sinusoidally frequency modulated signal similar to that generated by an conventional FM laser [10]. For long electron bunches (l b > λ r0 ), where the electrons are uniformly, randomly distributed over each radiation wavelength in the pulse, a clear, temporally correlated frequency modulated radiation output is obscured due to the electron beam shot-noise. The radiation has random, uncorrelated phases for each mode, generating a noisy broadband output.
Defining the envelope of the phase modulated field asĀ =Ãe iθ r 0 , and Laplace transforming Eqs.
where s n ≡ s + nk M ,Â n ≡Â(s n ,ω) andÃ 0 ≡Ã(z = 0,ω). For κ, n = 0, this modal system reduces to the cubic characteristic equation of the normal, un-modulated undulator FEL [9]. Numerical solutions of the linear system of Eqs. (1)-(3) (note that the linear system does not include saturation effects) are shown in Fig. 1, where multiple modes, corresponding to those of (6), are seen to be amplified over a greatly enhanced bandwidth from that of a normal FEL which has a (scaled) gain bandwidth of ∆ω ≈ 1. The gain length is seen to increase with the To generate FM output that possesses a well defined temporal-frequency correlation, the product of the beam bunchingb 0 and undulator modes M U of Eq. (5), should have correlated FM phases. This occurs when the beam is bunched coherently with a fixed-phase relationship between the undulator modes M U . The resultant product of the electron beam bunching spectrum with the undulator modes then possesses a well defined FM phase relationship.

FM-FEL simulations
In what follows, the broad bandwidth FEL simulation code Puffin [13] was used in 1D mode with periodic boundary conditions applied, to simulate both High-Gain Harmonic Generation (HGHG) [14] and Echo-Enabled Harmonic Generation (EEHG) [15] schemes that seed the electron beam bunchingb 0 (ω), at phase correlated frequencies corresponding to the undulator modes M U , of the subsequent modulated undulator system. Figure 2 gives examples of the resultant phase correlated FM output from the modulated undulators for both HGHG and EEGH bunched electron beams. It can be seen that the time correlated FM output bandwidth obtained is up to two orders of magnitude greater than the usual FEL bandwidth. At this stage, no attempt has been made to maximise the FM bandwidth.
A beam current I = 1.8kA and energy E 0 = 5GeV, with a homogeneous energy spread of σ E /E 0 = 1 × 10 −5 , was used with undulator period λ u = 4cm and mean undulator parameter a u0 = 5.0. This corresponds to a FEL parameter of ρ = 0.001, and carrier frequency ω r0 = 2πc/λ r0 with λ r0 = 5.2nm. The undulator has a sinusoidal modulation of the peak magnetic field of period 50 undulator periods, corresponding tok M = 10, and modulation index µ = 4. The beam is seeded in the HGHG mode using a laser wavelength λ H = 260nm, corresponding to scaled wavenumberk H = 10, so that the higher HGHG harmonics occur at the undulator modes of ω r0 − nck M , with ω r0 at the 50 th harmonic of this seed.
The beam energy spread in this example is small with σ E /E 0 = 10 −2 ρ. The primary aim here is to allow for the efficient up-conversion to the 50 th harmonic of the HGHG seed in a single stage and is not a requirement for subsequent FM-FEL operation. In a more realistic design, the requirements on the energy spread may mean that the harmonic up-conversion would require to be cascaded in 2 or more stages, as is frequently done at e.g. the FERMI facility [17]. In Fig. 2(a), the HGHG seed possesses higher harmonic bunching components overlapping all of the undulator modes with significant power. The FM temporal-frequency correlation is maintained until FEL saturation, at which point the phases of the modes are then seen to drift from the FM phases.
An HGHG example with a higher frequency seed at λ HG = 104nm is shown in Fig. 2(b). This beam has harmonic bunching components which coincide with only three of the undulator modes. The inherent tunability of the FEL and use of the HGHG seeding method also allows significant variation of the correlations made possible between the seeded beam bunching modes and those of the undulator modes in the same FM-undulator system.
It is noted that EEHG may be more advantageous to use to directly up-convert to a high harmonic in a single stage due to its improved harmonic conversion efficiency. Fig. 2(c) gives an example of an EEHG seed at 260nm, with harmonic bunching components optimized to around the 52 nd harmonic, overlapping a sequence of the undulator modes.
While the above examples of a sinusoidally modulated undulator allow for a more elegant analysis, a typical FEL facility undulator line is composed of many identical undulator modules, with free-space drifts between each module to allow for insertion of other beam optical elements such as diagnostic instruments and beam focusing quadrupoles. In this scenario, a more feasible form of the FM-FEL would be to e.g. linearly taper each module, resulting in a piecewise discontinuous modulation if the free-space drifts between undulator modules are taken into account. The harmonic seeding should then be at harmonics of the modes defined by the undulator module plus inter-module drift length. Note that the system of Eqs. (1)-(3) still remain valid for such a modulation in the 1D approximation, although in the following example we also model 3D effects to test the robustness of the scheme to effects such as diffraction and a finite electron beam emittance.
A full 3D simulation was performed using Puffin with parameters similar to the SwissFEL Athos undulator line [18], linearly tapering each module as described above, and including focusing quadrupole magnets and free space drifts between the modules. FEL lasing is at carrier wavelength λ r0 = 6.25Å , with each module being linearly tapered fromā u = 0.95 → 1.05, around undulator parameterā u = 1. The electron beam has mean energy E = 4.088GeV, energy spread of ∆γ/γ = 0.001%, a normalized emittance of ǫ n = 0.5µm, with a mean beam radius in the focusing channel of approximately 30µm, and a peak current of I pk = 1.44kA. The undulator modules are 100 periods long, of period λ u = 4cm, with a drift between undulator modules of 0.5m. This corresponds to a total relative slippage between electrons and radiation in the undulator-drift sections of 108λ r . HGHG modulation is made at the 108 th subharmonic of λ r0 . Figure 3 shows the simulation results with the output exhibiting a linear frequency chirp with a relatively small temporal modulation of the output. The scheme appears to be robust to the 3D effects of diffraction and electron beam emittance. There is a clear repetitive linear correlation in the temporal-frequency analysis, separated by small breaks corresponding to the drift sections between the undulator modules.

Conclusion
The FM-FEL has been shown to exhibit similar properties to that of its conventional laser counterpart. It is a more complex system to study due to the relative temporal slippage of the radiation field through the co-propagating electron beam. FM-FEL operation has favourable scaling into the X-Ray regime, where only relatively small changes of the undulator parameter within each module are needed for a significant extension of the mode-amplification bandwidth over that of a typical FEL. It has been demonstrated that it is possible to seed the FM-FEL interaction via harmonic lasing schemes e.g. HGHG, EEHG, or other methods where the bunching spectra exhibits a plateaux of harmonics/modes. The use of chicanes between the tapered undulator modules may also allow further control of the the modes [4], such as independent control of both the modal amplification bandwidth and spacing. These methods could lead to a powerful, tunable, broadband source of multi-coloured coherent radiation into the hard X-ray.
It is noted that the general concepts and methodology used here may also be valid and applicable within certain limits, to similar cooperative many-body systems, such as the Collective Atomic Recoil Laser (CARL) instability [19].

Disclosures
The authors declare that there are no conflicts of interest related to this article.