An ABCD formalism is identified to characterize a seeded Free Electron Laser (FEL) with three chirps: an initial frequency chirp in the seed Laser, an energy chirp in the electron bunch, and an intrinsic frequency chirp due to the FEL process. A scheme of generating attosecond few-cycle pulses is proposed by invoking an FEL seeded by high-order harmonic generation (HHG) from an infrared laser. The HHG seed has generic attosecond structure. It is possible to manipulate these three chirps to maintain the attosecond structure via post-undulator chirped pulse compression.
© 2007 Optical Society of America
Intense, coherent, attosecond (few-cycle) light pulses are very important for many scientific research fields . Schemes have been proposed to generate attosecond Free Electron Laser (FEL) pulses [2, 3, 4]. In this paper, we propose an FEL scheme that is seeded by high-order harmonic generation (HHG) driven by an ir laser [5, 6], in which three chirps are manipulated in the HHG seeded FEL . In a seeded FEL, the seed can have a frequency chirp, and the electron bunch can have an energy chirp. Besides these two externally imposed chirps, the FEL process introduces an intrinsic frequency chirp on the FEL pulse. In this paper, we identify an ABCD formalism [8, 9, 10] to characterize the interplay of the chirps and the evolution of the temporal duration, the spectral bandwidth, and the chirp of the FEL pulse. We consider a seeded FEL starting from a frequency chirped Gaussian laser pulse, i.e., Es(t,z) = E 0 e i(ksZ-ωst)e-(αs+iβs)ω2s(t-z/vg,0)2 with E 0, ks, ωS, αs, βs, and V g,0 = αs/(ks + 2kw/3) characterizing the peak field, the pulse wavenumber, frequency, duration, chirp, and group velocity respectively. The chirp (2βs ω 2 s ≡ dω/dt) is independent of the pulse duration . The FEL electric field in the (t,z) coordinates is ,
where the initial conditions are E 0,FEL (z = 0) = E 0, αs,f (z = 0) = αs, βs,f (z = 0) = βs, and the centrovelocity  vc(z = 0) = (〈t〉/z)-1∣z=0 = v g,0 In Eq. (1), ρ is the Pierce parameter [12, 13], kw = 2π/λw with λw being the undulator period, αs,f(z) = [4σ 2 t,s,f(z)ω 2 s]-1, β 2 s,f(z) = αs,f(z)σ 2 ω,s,f(z)/ω 2 s - α 2 s,f with σt,s,f(z) being the FEL pulse rms temporal duration and σωs,f(z) being the FEL pulse rms frequency bandwidth. The ABCD matrix can be used to transform the complex Gaussian pulse parameter, p(z) ,
to obtain the new chirp and pulse duration after the FEL interaction. For a seeded FEL, the canonical transformation is a symplectic ABCD matrix,
due to symplecticity. In Eq. (4),
being the rms bandwidth of the FEL Green function for a coasting electron beam without energy chirp , and μ characterize the energy chirp in the electron bunch. Notice that, P > 0 and R > 0, but Q can be negative or positive. For μ = 0, we have C = 0 and D = 1 . In this case, the form of the ABCD matrix (A = D = 1,C = 0) and B complex is characteristic of a system with group velocity dispersion (ReB) and gain (ImB). To illustrate the concept, imaging that a light pulse is represented by an ellipse in the t-ω space with t as the horizontal-axis and ω the vertical-axis. The group velocity dispersion is responsible for the “horizontal shearing” of the phase space ellipse. Since for μ = 0, we have C = 0; there is no inherent time lensing or temporal chirping effect (“vertical shearing”) in the FEL  without electron bunch energy chirp. An input pulse changes its overall chirp due to the group velocity dispersion and bandwidth reduction. It is possible to start with a negatively chirped seed and exit the FEL with no chirp. Contrarily, the energy chirp in the electron bunch provides a temporal chirping effect (“vertical shearing”). This leads to the complicated interplay of the three chirps and optimization of chirped pulse compression . Indeed, the energy chirp in the electron bunch is the key to maintain a large bandwidth in the FEL pulse, so that an effective chirped pulse compression after the undulator is possible to generate an attosecond pulse train. The ABCD analysis is not limited to a simple chirped Gaussian seed pulse. Arbitrary seed pulses can be constructed from a complete set of chirped Hermite-Gaussian functions characterized by a single p-parameter and propagated through the system with the ABCD matrix formalism . Indeed, this ABCD matrix in Eq. (3) is identified after the follow derivation by introducing the Wigner function; however, in order to highlight the physics first, we present it before the derivation.
To explore the importance of the energy chirp in the electron bunch, recall that the the ABCD matrix is applied to the following coordinate transformation
with the group velocity of the FEL light being v -1 g,0 = dk/dω∣ω=ωs = (ks + 2kw/3)/ωs with or without energy chirp in the electron bunch. The vector (τ,dτ/dξ)T describes a time ray with τ standing for the delayed time and ξ the normalized propagation distance. Its position τ represents time deviation from a reference time, whereas its slope (dτ/dξ) represents frequency sweep. A nonzero B in the system stands for a horizontal shearing along the t-axis, while a nonzero C stands for a vertical shearing along the ω-axis. It is now clear how important a nonzero C is. It is only with increment in the frequency bandwidth via a nonzero C, can a net temporal pulse compression be possible. For a seeded FEL with three chirps, the ABCD matrix is given in Eq. (3), we find that the only possibility to have a nonzero C element is to have a nonzero μ, i.e., there has to be an energy chirp in the electron bunch. Indeed, a seeded FEL is represented by an integral Eq. 
where the slow varying envelope function A(θ,Z) is introduced by E(t,z) = A(θ,Z)e i(θ-Z) with dimensionless variables as Z = kwz and θ = (ks + kw)z-ωst. This integral Eq. (8) which propagates an input seed A(θ,0) through the FEL via the Green function is of the general form of an integral representation of an ABCD canonical transformation, as such the phase space area and longitudinal coherence is preserved [9, 10]. Readers may refer to Ref.  for details.
We have emphasized the importance of an energy chirp in the electron bunch, and highlighted the physics of a seeded FEL in a compact ABCD formalism. We now study a HHG seeded FEL. We demonstrate that with properly introducing an energy chirp in the electron bunch, a few-cycle attosecond FEL pulse train can be retained by post-undulator chirped pulse compression. We will also provide the derivations which leads to this ABCD matrix in Eq. (3).
We study an HHG seed with central frequency (carrier) being a single harmonic order s 
where E s,0 is the peak amplitude of the electric field. Even though, the HHG electric field consists of multiple orders, s with wavenumber, ks and angular frequency, ωs, and multiple pulse-lets, n as a double summation, only the resonant harmonic is relevant, since FEL is a narrow bandwidth filter . The temporal structure is a sequence of 2N + 1 ultrashort pulselets with attosecond structure which is referred as the attosecond pulse train (APT). Pulselet peaks occur at times, tn = nτ/2 where τ is the ir laser period. The duration of a single attosecond pulselet (SAP) can be less than one femtosecond. The temporal envelope of the entire HHG pulse has an rms duration σ t,0. We define the much shorter rms duration, σt,s for the single attosecond pulselet with αs ω s 2 ≡ 1/(4σ 2 t,s); and βs characterizes the single harmonic seed chirp. The APT in Eq. (9) is simply the amplitude modulation of a single carrier frequency, ωs. We usest, σ t,0 = 10 fsec, σt,s = τ/10 ≈ 267 attoseconds in the Fourier transform limit, and the ir laser wavelength λir = 800 nm in this paper. According to the study in Ref. , the attosecond structure will be temporally smeared out quickly. However, if the energy chirp in the electron bunch is large enough, the pulselets will have a large chirp and a post-undulator chirped pulse compression will restore the attosecond few-cycle pulselets. For simplicity, we provide derivation for a single pulselet: the middle pulselet, i.e., for n = 0, so that tn = 0. For this single pulselet, the electric field in the (t,z) coordinates is given in Eq. (1). To compute the second moments, especially, the chirp, we introduce a Wigner function ,
where δt = t-z/vc, δω = ω - ωs, and
With the Wigner function in Eq. (10), we can compute the second moments to have
With the second moments given in Eq. (12), the ABCD matrix in Eq. (3) can be verified. The centrovelocity vc  is . Equation (29) of Ref.  has a formatting typo in generating the final production version.
We now consider how the chirp can affect the system. We take parameters of those in Ref. : ωs = 2πc/λs with λs as the 27th harmonic of their laser, ρ = 6.3 × 10-3, undulator period λw = 3 cm, and z ∈ [0,2] m (the saturation length Lsat ≈ 4 m). Following the resonance condition, initially the seed and the electron bunch are matched, i.e.,
According to Eq. (6), μ ≈ 2βs. It is easy to check that with this set of parameters, we have , and for ∣μ∣ ≪ 1 which is usually satisfied. With this set of parameters, it is difficult to manipulate the FEL pulse duration via chirps during the FEL process; however, the residual frequency chirp is apparently inherited from the initial energy chirp in the electron bunch. Hence, post-undulator compression will be possible to generate FEL pulse with attosecond duration. In the following, let us choose, ω(t = σt,s) = ωs + 0.1σω,s ⇒ βs ∼ 8.7 × 10-5. This amount of chirp for the seed of rms duration of σ t,0 = 10 fs translates into about 3.3 nm rms bandwidth on the 27th harmonic (i.e., 30 nm) of the ir laser. For the energy chirp, we take ∣μ∣ = 2βs ≈ 1.7 × 10-4 ≪ 1, which translates to ∣Δδ∣t=σt,s = ∣Δγ/γ 0∣t=σt,s ≈ 1.5 × 10-3. Forthis parameter set, we show the FEL pulse rms duration σt,s,f as a function of location z into the undulator in the upper left subplot of Fig. 1. The solid (red) curve is for μ = 2βs, the dashed (green) curve is for μ = -2βs, and the dash-dotted (blue) curve is for μ = 0. For all these three cases, βs ≈ 8.7 × 10-5. The dotted (purple) curve is for μ = βs = 0. However, for this set of parameters, there is very little difference for σt,s,f, as we expect from the above mentioned limiting case of | μ| ≪ 1, i.e., . The frequency rms bandwidth omsf is shown as the upper right subplot in Fig. 1, and the cross moment 〈(t - 〈t〉)(ω - 〈ω〉)〉 is shown as the lower right subplot in Fig. 1. With the energy chirp in the electron bunch, the final FEL pulse inherits a frequency chirp, which is crucial for an effective post-undulator chirped pulse compression. Shown as the lower left subplot in Fig. 1 is the FEL pulse rms duration after post-undulator chirped pulse compression as a function of the undulator length. Such a post-undulator compression is a final “horizontal shearing”, which results in a pulselet rms duration below the femtosecond. To illustrate this explicitly, we show as the upper row in Fig. 2 for the case of three pulselets in an HHG seed with the frequency chirp βs ≈ 8.7 × 10-5 and the energy chirp to be μ = 2βs. Even though, the attosecond pulselets are temporally smeared out  in the undulator as shown in the left subplot; the inherited large frequency chirp enables the post-undulator compression to regenerate attosecond few-cycle pulselets as shown in the right subplot. In contrast to this, shown as the lower row in Fig. 2 are three puslelets in the HHG seed with the same chirp βs ≈ 8.7 × 10-5, but there is no energy chirp in the electron bunch; the post-undulator chirped pulse comprssion is not adequate to restore an attosecond pulse train. Indeed, μ can either be negative or positive so that the FEL frequency chirp exiting the undulator can either be negative or positive.
In this paper, we study the possibility of generating an attosecond pulse train by manipulating the three chirps in a seeded FEL. An useful ABCD formalism is identified. This compact formalism illustrates clearly the importance of an energy chirp in the electron bunch. With a proper energy chirp in the electron bunch, it is possible to keep large frequency bandwidth, so that a post-undulator pulse manipulation can restore an attosecond pulse train inherited from the HHG seed. The work of JW and PRB was supported by the US Department of Energy under contract DE-AC02-76SF00515. The work of JBM was supported by the US Department of Energy under contract DE-AC02-98CH10886 and the Office of Naval Research.
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