^{*}

Edited by: Arya Fallahi, Foundation for Research on Information Technologies in Society, ETH Zurich, Switzerland

Reviewed by: Ilia L. Rasskazov, University of Rochester, United States; Masoud Mehrjoo, Helmholtz Association of German Research Centers (HZ), Germany

This article was submitted to Optics and Photonics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This paper examines the impact of non-linear longitudinal phase distortions on the spectral bandwidth in echo seeded free electron lasers (FELs). It extends the existing theory developed in Hemsing [

Externally seeded FELs use lasers to produce coherent high harmonic density modulations (bunching) in relativistic electron beams that are then used for the emission of coherent radiation at short wavelengths. Seeding is useful for overcoming the otherwise noisy output of SASE (self-amplified spontaneous emission) from these beams to generate narrowband FEL radiation. Besides improved longitudinal coherence, external seeding also enables control over the character of the FEL output pulses [

Several different external seeding schemes have been proposed, and a few demonstrated (see e.g., [

A primary goal of seeding schemes is to minimize the FEL bandwidth and produce transform-limited pulses. In principle, the narrowest obtainable bandwidth is given by the inverse electron beam bunch length. In practice however, electron beams in modern FELs have distortions in the longitudinal phase space that mix additional frequencies into the harmonic up-conversion that spoil the purity of the final output spectrum [

The magnitude, shape, and location of these distortions, within both the electron beam and the laser pulse, is an important factor in the optimization of external seeding schemes. The analysis in Hemsing [

Here, using an extension of the formalism in Hemsing [

The paper is arranged as follows. We first present an extension to the theory in Hemsing [^{−1/3} harmonic compression effect, and are able to calculate the transform-limited bandwidth. Arbitrary phase distortions are then included as a Taylor series, and their impacts to each order on the total bandwidth and time-bandwidth product (TBP) are then calculated analytically. We then derive the conditions on the laser pulse length and e-beam length to minimize the bandwidth, and study a few examples.

Notation closely follows that of Xiang [

where the normalized laser modulations are _{1, 2} = Δ_{1, 2}/σ_{E}, normalized dispersions are _{E}, and _{2}(_{1}), as well as a longitudinally dependent phase ψ_{2}(_{1}). The first laser, _{1}, is assumed to be ideal and infinite in length. Additional energy distortions in the electron beam, Δ_{1} and Δ_{2}, are again modeled as occurring alongside the laser modulations.

The bunching spectrum near the harmonic peak _{E} = _{E}_{1} = (_{1} is given by

where ξ_{E} = _{1} + _{E}_{2} is the EEHG scaling parameter, and _{1}_{1}/_{1} and are small enough so that they do not lead to large changes in the phase space distribution after the first chicane. We also assume that the longitudinal variations in Δ_{2}, _{2}, and ψ_{2} are sufficiently slowly-varying that the full integral can be simplified by replacing _{1} in the function arguments. It is also assumed that the system is far from the minimum pulse duration limit [

With these approximations, the phase φ(_{1, 2}, and the phase variation ψ_{2} in the second laser,

The _{2}(_{m} Bessel function, which complicates simple analytic solutions. We therefore search for an ansatz of the form,

where the imprint of the slowly-varying laser envelope is captured by the function _{2} = _{2}(0) is the peak of the modulation.

With

where

The average frequency is then

and the spectral bandwidth of the harmonic bunching spike is

where the transform-limited bunching bandwidth is [

This gives the bandwidth due to the combination of the laser modulation

We consider the effect of a Gaussian laser pulse in the second echo modulator on the longitudinal profile of the bunching. This will enable us to calculate an approximate form for

The optimum amplitude of the second laser from time-independent echo theory is _{m}(_{E}_{2}_{2}) is peaked, with

where _{L}/

To lowest order near the

where ^{2} is, to a good approximation,

(Top) Bunching envelope and harmonic compression effect for

Thus we recover the ^{−1/3} scaling of the harmonic compression effect of the initial laser pulse length [

Normalized RMS and FWHM lengths of the bunching envelope from exact numerical calculations (solid lines) and from the super-Gaussian approximation (dashed lines).

Harmonic compression is the result of the high harmonics being increasingly more sensitive to the optimal modulation amplitude to produce bunching, so the longitudinal region of the Gaussian modulation that matches this condition becomes narrowed. The flattened, super-Gaussian form of

With an analytic form for _{z}, and properly normalized,

The rms transform-limited bandwidth of the bunching spectrum then has simple analytic solution that is approximately (see

Similarly, the rms length of the bunching envelope

_{kE} varies with σ_{L} for different harmonics. Two limiting regimes can be identified. If the laser is much longer than the beam _{kE} is independent of ^{1/3} because of the harmonic compression effect. Note that the relative bandwidth decreases in this limit like ^{2/3},

assuming _{E} = _{E}_{1} ≈ _{2}, and σ_{νL} = 1/2σ_{L}_{2} is the relative bandwidth of the laser.

Transform-limited bandwidth σ_{kE} as a function of laser pulse length σ_{L} from Equation (15). Because of harmonic compression, the laser pulse length needs to increase slightly with increasing harmonic number to maintain a fixed bandwidth.

In general, the transform-limited bunching bandwidth σ_{kE} is set by the length of the laser on the electron beam. However, nonlinear phase structures φ(_{kE}, either with the laser pulse length σ_{L} or the electron beam length σ_{z}. In other words, what combination of σ_{L} or σ_{z} gives the smallest total bandwidth of the bunching spike σ_{k}? Consider the example of an electron beam with a quadratic chirp, as shown in

Example electron beam phase space distribution (from FERMI [Allaria, personal communication]) with dominant quadratic structure and higher order structure near the head and tail. The optimized laser pulse length to produce a minimum bandwidth depends on the amplitude of the non-linearities in the beam.

Without regard to the origin of the nonlinear phase structure (i.e., laser phase ψ_{2} or electron beam energy structures Δ_{1, 2}), let us first calculate the impact of generalized phase structures on the bunching bandwidth σ_{k}. We will then use the results to derive conditions for obtaining the minimum bandwidth.

Arbitrary continuous phase distortions can be expanded in a Taylor series about

where each _{0} is ignored, as it does not affect the frequency or bandwidth. With Equation (6), the instantaneous frequency is

The analysis is simplified by isolating a single term in the phase expansion,

The total bunching bandwidth is then written in terms of the nonlinear coefficient ϕ_{N}, as

The second term is the excess bandwidth from the phase nonlinearity. The analytic expression for the numerical coefficient _{N} is given in the

Numerical values of _{N}.

_{1} |
0 |

_{2} |
0.34 |

_{3} |
0.035 |

_{4} |
0.0073 |

_{5} |
0.00046 |

_{6} |
0.000033 |

The time-bandwidth product (TBP) of the bunching spectrum is the dimensionless product of the rms envelope length (16) and the rms bandwidth (20),

Time-bandwidth product as a function of the phase change over the rms bunching envelope

With the expression for the bunching bandwidth in (20), it is straightforward to find the values of the laser pulse length or the electron beam length that minimize σ_{k} in the presence of a phase nonlinearity. In either case this amounts to finding the optimal value of σ_{kE} for a given ϕ_{N}. Assuming that ϕ_{N} is fixed with respect to the parameter being changed, the minimum bandwidth occurs when σ_{kE} can be adjusted to satisfy,

The minimum total bandwidth then scales directly with optimal σ_{kE},

An example of how the minimum bandwidth varies in general for different amplitude phase distortions is shown in

Total bandwidth vs. σ_{kE} for different _{N} is large enough (i.e., if Equation 22 is satisfied). Otherwise the minimum is set by the minimum possible value of σ_{kE} (σ_{kE} = 1, in this example for ϕ_{N} = 0 and 1).

The full expressions for the optimal electron beam length and laser pulse length are given in (60) and (62) the

Scaled bunching bandwidth vs. relative laser duration for phase distortions of

Alternatively, if the laser is held fixed and

Note that this is independent of the harmonic, in contrast to the optimal laser pulse length.

The TBP at the minimum bandwidth is,

It has the same dependence on the order of the phase distortion as σ_{k}; both approach their transform-limited values as

Inspection of the full expression for the TBP shows that it cannot be minimized simultaneously with the total bandwidth by adjustment of σ_{kE} alone, though it is possible if ϕ_{N} is not held fixed.

In modern FELs, it is common that the electron beam has some residual energy-time correlation in the phase space. These correlations can be the result of wakefields or other collective effects during compression and transport. They may be difficult to remove completely, and may be present at the entrance to the EEHG seeding system, or can develop within the EEHG beam line from collective effects [

Consider a purely quadratic energy chirp on the electron beam of the form,

The chirp amplitude is characterized by the dimensionless factor _{2}. Assuming the second seed laser is transform-limited (ψ_{2} = 0), the bunching phase can be written generically as [

where η = ξ_{E} or _{E}_{2} depending on where the energy distortion occurs, as expressed in Equation (3). The nonlinear coefficient in (19) for

Plugging this into Equation (22) and optimizing the laser pulse length, one can show that if the amplitude of the quadratic beam chirp strongly satisfies

then the laser pulse length that minimizes the bandwidth is, from Equation (24),

The corresponding relative bandwidth from Equation (23) is then

Consider the realistic case of electron beams at the FERMI FEL [^{2} upstream of the 260 nm seeding sections. This corresponds to _{E} = 0.15 MeV. In this case Equation (30) is satisfied for beams longer than σ_{z} = 30 μm (100 fs), which is easily the case for the ps-scale beams at FERMI (assuming η = 1/2). The minimum bandwidth setting from (31) requires a laser pulse duration of 250 fs at the 36th harmonic and produces

Similar to quadratic structure on the electron beam, a linear laser chirp in the second seed laser introduces frequency-time correlations that mix into the harmonic up-conversion process and impact the final bunching bandwidth. If the electron beam phase space is flat, the bunching phase φ(

It is useful to separate laser chirps according to whether they are defined in terms of a fixed pulse length as in Siegman [

A linear frequency chirp corresponds to a quadratic phase of the form [

Assuming the pulse length is fixed, α only affects the laser bandwidth. The quadratic coefficient of the bunching phase is then ϕ_{2} = 2

On the other hand, for a fixed-bandwidth laser pulse with rms intensity σ_{kL}, a linear frequency chirp that changes only the pulse length σ_{L} corresponds to a phase,

where the chirp is

and

is the pulse stretch factor. The laser bandwidth corresponds to transform-limited pulse length, σ_{L0} = 1/2σ_{kL}.

Note that the chirp grows from zero to a maximum of |

Chirp as a function of stretch factor, _{L}/σ_{L0}. From Equation (36).

Wigner distributions of a chirped laser pulse with fixed bandwidth with

The quadratic coefficient of the bunching phase is,

The chirp is multiplied by the harmonic. Because the laser pulse length changes, the bandwidth σ_{kE} is also a function of the stretch factor

Inserting (38) into (22), one can see that if the pulse is stretched such that

The optimum electron beam length is the smallest at the maximum chirp, _{L0} unless the harmonic chirp is smaller than one,

_{z}.

Scaled bandwidth vs. electron beam length for different values of a fixed-badwidth laser stretched with a linear chirp at

Note that if the electron beam also has quadratic curvature, the laser chirp can be set to negate the bandwidth broadening if _{E} = 0.45 MeV) beam at LCLS-II using EEHG to reach harmonic _{z} = 15 μm (50 fs), and quadratic electron beam curvature

Finally, it is useful to look at a practical example in which the electron beam is long compared to the length of the second laser, irrespective of the chirp (e.g., FERMI). The total relative bandwidth from Equation (20) is then

Inspection reveals that σ_{ν} grows with increasing

Relative bandwidth as the fixed-bandwidth second laser is stretched for different harmonics, assuming σ_{z} → ∞. Exact solutions are solid lines, and approximate solutions from Equation (42) are dashed lines. The disagreement with exact solutions is attributed to the super-Gaussian approximation for

Finite laser pulse length effects in EEHG FEL seeding are investigated. We have derived an approximate super-Gaussian form for the laser modulation

The author confirms being the sole contributor of this work and has approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by the U.S. Department of Energy Contract No. DE-AC02-76SF00515 and the U.S. DOE Office of Basic Energy Sciences under award number 2017-SLAC-100382. The author would like to thank L. Giannessi for helpful discussions.

The Supplementary Material for this article can be found online at: