Nearly copropagating sheared laser pulse FEL undulator for soft x-rays

A conceptual design for a soft x-ray free-electron laser (FEL) is proposed which is an order of magnitude lower in cost than now standard designs. This new design utilizes angle-dependent Doppler tuning of a table-top terawatt (T³) laser pulse in the 1 µm wavelength range and an electron beam of modest energy (∼170 MeV). Shearing of the T³ or pump laser pulse is proposed to optimize the FEL interaction and reduce the energy requirement for the pump laser. Parameters are suggested for a FEL with x-ray output in the longest wavelength water window of ∼2.35 to ∼4.4 nm or ∼530 to ∼280 eV (e.g. Rymell et al 1995, De Stasio et al 2000).

The operation of the Free electron LASer in Hamburg⁵ (FLASH), the world's first x-ray FEL, and the Linear Coherent Light Source⁶ (LCLS) at SLAC, a harder x-ray FEL, has demonstrated x-ray FEL technology in an impressive fashion. These successes have established the FEL as the technology of choice for fourth generation light source facilities over a range of x-ray photon energies. The ultimate scientific impact of these coherent x-ray sources will depend in part on how many beam lines can be made available worldwide.

Laser undulators for FELs have been analysed in many papers (e.g. Gallardo et al 1988, Gordon et al 2001, Bonifacio et al 2007, Sprangle et al 2009). Numerous reviews, tutorials, and articles on FEL theory have been published (e.g. Colson 1985, Colson et al 1985, Xie 2000, Huang and Kim 2007, Pellegrini 2010). At this time no FEL with an optical or electromagnetic undulator has been operated. Spontaneous Compton scattering from a laser undulator and a relativistic electron beam has been studied extensively since shortly after the invention of the laser (e.g. Milburn 1963, Federici et al 1980, Sandorfi et al 1983, Weeks et al 1997) and is of particular interest for nuclear spectroscopy and medical radiotherapy. There is continuing interest in the development of new types of spontaneous Compton scattering sources employing short-pulsed 'table-top' lasers (e.g. Chung et al 2011). These authors proposed generating femtosecond keV x-ray pulses using a tightly focused laser copropagating with a relativistic electron beam. The advances in laser technology, particularly in table top or university-scale laser technology, have been so dramatic that revisiting ideas for laser undulators is worthwhile.

Design studies for soft x-ray FELs often propose a CO₂ laser undulator (e.g. Pantell et al 1968, Gallardo et al 1988, Dattoli et al 2012). Both high average power and high energy pulses are available from CO₂ lasers. Clearly CO₂ lasers represent a well-developed technology at a mid-infrared wavelength. A long wavelength undulator laser is needed because traditional FEL designs involve directing the laser undulator beam 'head-on', or nearly head-on into the electron beam. The head-on geometry is analogous to the first FEL which used a magnetic undulator fixed in the lab (Deacon et al 1977). In this traditional geometry the magnetic undulator period or laser undulator wavelength must be a factor of ∼2γ² or ∼4γ², respectively, longer than the desired soft x-ray wavelength. The Lorentz transform γ is the ratio of total electron energy over rest energy mc². For a soft x-ray of ∼3 nm wavelength and a laser undulator (hereafter called pump) wavelength of λ_pump = 10.6 µm, the requirement is γ ∼ 30. Such low electron beam energies are difficult to produce with the low geometric emittance, small energy spread, and high current which are necessary for FEL gain. Most recently Chang et al (2013) proposed a laser undulator using 'tilted optical wave' pulses from a CO₂ laser propagating at right angles to a relativistic electron bunch. The conceptual design we explore in this work is more general than the design described by Chang et al and our design is compatible with shorter wavelength T³ pump lasers.

Short-pulsed optical lasers in the visible to 1 µm wavelength range are very highly developed in the form of Ti : sapphire lasers with amplifier chains. Newer and relatively inexpensive fibre laser systems in the ∼1 to ∼2 µm wavelength range also have great promise (e.g. Hecht 2012). Pulse energies ∼1 J are now available from university-scale table-top Ti : sapphire systems with amplifier chains and our conceptual design employs a typical Ti : sapphire wavelength. Pulse durations ranging from as short as ∼25 fs to very long (e.g. continuous lasing) are available from Ti : sapphire lasers due to their very broad gain bandwidth from ∼650 nm to ∼1100 nm. Such laser systems are appropriately described as table-top terawatt or T³ lasers. Focusing of such a laser yields a higher energy density than can be achieved with a longer wavelength pump laser and a similar pulse energy. Maximum intensities at the waist of a laser beam scale as (λ_pump)⁻² and the energy density for a pulse of 'n' cycles scales as (nλ_pump)⁻¹. The far larger gain bandwidth of a Ti : sapphire laser in comparison to a CO₂ laser is essential for producing pulses that are ∼10 or fewer wavelengths long. Unfortunately, these high energy, short pulses from Ti : sapphire lasers are not suitable for soft x-ray FEL pump lasers in the traditional head-on design. It becomes more difficult to get FEL gain at a shorter x-ray wavelength λ_x-ray due to variation of electromagnetic mode density as (λ_x-ray)⁻² and resulting gain variation as (λ_x-ray)² of the FEL (Madey et al 1973). Even though Ti : sapphire laser systems routinely achieve electromagnetic field energy densities six decades higher than conventional magnetic undulators, it is nevertheless difficult to use them as laser undulators in the traditional head-on geometry.

In this paper we propose using a low emittance, modest-energy (∼170 MeV) electron beam with a nearly copropagating sheared laser pulse from a Ti : sapphire or similar laser system. Gain estimates are presented which indicate that self-amplified spontaneous emission (SASE) lasing will occur at soft x-ray wavelengths. Seeding such a FEL with a soft x-ray pulse from a high harmonic generation (HHG) system should yield additional advantages (Lambert et al 2008). Soft x-rays in the 1 to 2 keV range from HHG have been demonstrated using a mid-infrared 3.9 µm wavelength pump laser system (Popmintchev et al 2012). Operation of a nearly copropagating soft x-ray FEL using a pump laser pulse in an optical enhancement cavity (e.g. Anderson and Lalezari 2012 for a recent review of the technology) would dramatically lower the average power requirement on the pump laser and would yield a high repetition rate and attractive average output power from the soft x-ray FEL. Ultimately, we anticipate that a fibre laser would be used as a pump laser for a soft x-ray FEL facility. Fibre lasers have very favourable ratios of amplifier surface area to volume. These favourable ratios are enabling designers to develop relatively low-cost high average power fibre laser systems (e.g. Hecht (2012) for a recent review of the technology).

This section defines a FEL geometry and introduces symbols needed for a gain estimate. Angle-dependent Lorentz transforms are reviewed here for the gain estimation. Figure 1 is a schematic of the geometry envisioned in which a relativistic electron bunch of diameter D, length L^lab, and density n^lab, all lab frame parameters, is moving from left to right. This electron bunch is crossed by a nearly copropagating optical laser pulse at some small angle θ^lab. The laser pulse has a width of W^lab in the plane of the figure and duration T^lab observed from the lab frame. This laser waist at the intersection with the electron beam is astigmatic with a width D_laser normal to the plane of the figure. Matching the electron bunch diameter in this fashion can somewhat reduce the required undulator or pump laser pulse energy E^lab for some desired gain. It is possible to substantially reduce the transverse intensity gradient of a TEM₀₀ laser beam by using higher order off axis modes (e.g. Yariv 1989).

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The wavelength in the electron rest frame is

$$\begin{equation} \lambda^{{\rm elec}}=\frac{\lambda_{{\rm pump}}}{\gamma [1-\beta \cos (\theta^{{\rm lab}})]}, \end{equation} \tag{ 1 }$$

where

$$\begin{equation} \gamma =\frac{1}{\sqrt {1-\beta^{2}}} \end{equation} \tag{ 2 }$$

and β is the ratio of the axial (+z direction) electron velocity to the speed of light c. This wavelength in the electron frame is variable between (1 + β)γλ_pump and λ_pump/[(1 + β)γ] as θ^lab is varied between 0 and π. In the lab frame the wavelength, neglecting recoil, of the scattered radiation parallel to the +z-axis is

$$\begin{equation} \lambda_{{\rm x\mbox{-}ray}} =\frac{\lambda^{{\rm elec}}}{(1+\beta )\gamma}=\frac{\lambda_{{\rm pump}} (1-\beta )}{1-\beta \cos (\theta^{{\rm lab}})}. \end{equation} \tag{ 3 }$$

This (soft x-ray) wavelength in the lab frame is continuously variable between λ_pump and ∼λ_pump/(4γ²) depending on θ^lab. Of course the gain vanishes if θ^lab = 0 and λ_x-ray = λ_pump. Equation (3) reduces to the more familiar λ_x-ray = λ_pump/(2γ²) if θ^lab = π/2, and for the head-on case reduces to λ_x-ray = λ_pump/(4γ²) with θ^lab = π. It is convenient to introduce a dimensionless quantity

$$\begin{equation} M=\frac{\lambda_{{\rm pump}}}{\lambda_{{\rm x\mbox{-}ray}}}=\frac{1-\beta \cos (\theta^{{\rm lab}})}{1-\beta}\cong (\theta^{{\rm lab}}\gamma )^{2} \end{equation} \tag{ 4 }$$

to represent the approximate factor or 'multiplier' by which pump laser photon energy is increased to the energy of soft x-ray photons via Doppler shifts in the spontaneous and proposed stimulated scattering. The approximate expansion is for small θ^lab and for β near unity. (The ponderomotive impulse to the electrons when the pump laser pulse first reaches them is not included.) In the rest frame of the electron bunch the cosine of the angle between the +z-axis and the direction of the laser pulse is given by the stellar aberration formula from undergraduate E&M texts (e.g. Barger and Olsson 1987)

$$\begin{equation} \cos (\theta^{{\rm elec}})=\frac{\cos (\theta^{{\rm lab}})-\beta}{1-\beta \cos (\theta^{{\rm lab}})}. \end{equation} \tag{ 5 }$$

For a γ ⩾ 100 the corresponding value of β is so close to 1.0 that over a wide range of θ^lab the pump laser pulse appears to be nearly 'head-on' in the electron frame with cos (θ^elec) ≈ −1. The stellar aberration formula is typically introduced to explain the small angular displacement of stars as the earth orbits the sun first observed by James Bradley in 1725 (see the discussion in Barger and Olsen). The extreme FEL application for relativistic electrons is correct because equation (5) is an exact Lorentz transform. The pump laser intensity (power/area) in the lab frame is

$$\begin{equation} I^{{\rm lab}}=\frac{E^{{\rm lab}}}{W^{{\rm lab}}DT^{{\rm lab}}}, \end{equation} \tag{ 6 }$$

where E^lab is the undulator or pump laser pulse energy in the lab. The laser intensity in the electron rest frame is

$$\begin{equation} I^{{\rm elec}}=I^{{\rm lab}}\gamma^{2}[1-\beta \cos (\theta^{{\rm lab}})]^{2}. \end{equation} \tag{ 7 }$$

A combination of equations (1) and (7) reveals that Iλ² is Lorentz invariant. The electron bunch diameter D is also Lorentz invariant, but the length of the bunch is transformed

$$\begin{equation} L^{{\rm elec}}=L^{{\rm lab}}\gamma \end{equation} \tag{ 8 }$$

and the bunch density in the electron rest frame is

$$\begin{equation} n^{{\rm elec}}=n^{{\rm lab}}/\gamma . \end{equation} \tag{ 9 }$$

Although it is sometimes said that the magnetic force 'nearly cancels' the electric force from a laser pulse in a perfectly copropagating direction with an electron bunch, this angle-dependent cancellation is captured by the Lorentz transforms. An electromagnetic wave in the lab frame is still an electromagnetic wave in the electron frame (or any other frame).

Since the wave or phase fronts of the pump laser pulse are at angle π/2−θ^lab with respect to the direction of the electron beam axis, they have a projected velocity on the electron beam axis of V_p = c/ cos (θ^lab) which is greater than c. Phase fronts move through the electron bunch from left to right viewed from the lab frame as the pump laser pulse moves to the right at c cos (θ^lab) and towards the top of the figure at c sin (θ^lab). These advancing phase fronts 'wiggle' the electrons to yield spontaneous Compton scattering and, with a sufficient length of interaction, stimulated scattering and lasing.

A simple way of describing the primary difficulty of a non-head-on laser undulator can be based on the interaction time of the pump laser pulse and electron bunch. Viewed from the lab frame the available interaction time τ^lab for amplification of spontaneous emission is limited by the beam crossing time or light travel time from point A to point B

$$\begin{equation} \tau^{{\rm lab}}\le \frac{W^{{\rm lab}}}{c\sin (\theta^{{\rm lab}})}. \end{equation} \tag{ 10 }$$

The important quantity is the number of cycles of interaction between the moving electron beam and the fields in the pump pulse, which is more easily viewed in the electron rest frame. There the interaction time is

$$\begin{eqnarray} &&\fl \tau^{{\rm elec}}\le \gamma\left( {\frac{W^{{\rm lab}}}{c\sin (\theta^{{\rm lab}})}-\beta \Delta z^{{\rm lab}}/c}\right)\nonumber\\ &&=\frac{W^{{\rm lab}}}{c\sin (\theta^{{\rm lab}})}\sqrt {\frac{1-\beta}{1+\beta}} \cong \frac{W^{{\rm lab}}}{2\gamma c\sin (\theta^{{\rm lab}})}, \end{eqnarray} \tag{ 11 }$$

where $\Delta z^{{\rm lab}}=z_{B}^{{\rm lab}} -z_{A}^{{\rm lab}} =W^{{\rm lab}}/\sin (\theta^{{\rm lab}})$. Even if W^lab is 100's of λ_pump the upper limit for the interaction time in the electron frame is only a few cycles. Many cycles of interaction, of order 100 or more, are needed for significant gain and lasing. The interaction time τ^elec with a reasonable choice for W^lab is unacceptably short and must be substantially lengthened to have any possibility of significant gain and ultimate lasing. Any attempt to increase τ^elec to a useful value by simply increasing W^lab and T^lab yields an unreasonably high value for E^lab, the pump laser pulse energy. A lengthening of τ^elec is much more feasible using a sheared laser pulse as described in the next section. Shearing the wave fronts effectively widens the pump laser beam by including only the part of the pump laser pulse which interacts with the electron bunch.

An analogy can be made to an electromagnetic wave reflecting up and down between the top and bottom of a square hollow wave guide. The axial phase velocity of such a wave is V_p = c/ cos (θ^lab), the axial group velocity is V_g = c cos (θ^lab), and the waveguide expression V_pV_g = c² applies (e.g. Jackson 1999). The possibility of using a superconducting waveguide or cavity at microwave frequencies as an electromagnetic undulator is being explored (e.g. Yeddulla et al 2009). Although the use of a hollow optical fibre might appear to be an attractive alternative to a sheared pump laser pulse, the difficulties due to laser damage of the fibre and due to wake field effects from the fibre on the electron bunch are formidable.

In transforming parameters to or from the electron frame it is useful to note that D, Iλ² and some other quantities are Lorentz invariant. For example, Lorentz invariance of the laser pulse length in wavelengths and number of photons in the pump laser pulse imply invariance of Iλ². Although D is Lorentz invariant only for frames moving along the z-axis, the pump laser pulse quantities including Iλ² are invariant for any frame independent of θ^lab. The invariant Iλ² is directly proportional to the undulator parameter $a_{0}^{2}$ as used by Gordon et al (2001) and defined between equations (3) and (4) of their paper. Identical $a_{0}^{2}$ notation is used by Bonifacio et al (2007). Other authors including Gallardo et al (1988) use K² instead of $a_{0}^{2}$ and we shall follow the currently more popular notation and use K² as the undulator parameter in subsequent discussions. For circularly polarized radiation

$$\begin{equation} K^{2}=\frac{r_{{\rm e}} I\lambda^{2}}{mc^{3}\pi}, \end{equation} \tag{ 12 }$$

where r_e is the classical radius of the electron.

Figure 2 illustrates electrons moving from left to right as viewed from the lab frame while interacting with a sheared pump laser pulse. The laser pump pulse envelope propagates in the direction of the Poynting vector which is tipped, along with the wave vector, by an angle θ^lab with respect to the direction of the electron beam axis. Electrons would gain on the pump laser pulse envelope because β is much closer to 1.0 than cos (θ^lab) except for the shearing of the wave fronts by an angle of ∼θ^lab/2 with respect to the pulse envelope. A derivation of the desired shear angle ∼θ^lab/2 is conveniently based on an isosceles triangle and a realization that the electron bunch moves at very nearly c (e.g. >0.999 95c for γ > 50). This isosceles triangle vertex is on the z-axis ahead of the bunch in figure 2. The base of the isosceles triangle is on the leading edge of the pump laser pulse envelope. The two equal legs of the triangle are then along the z-axis and along the wave vector or Poynting vector which are always parallel in vacuum. The desired amount of shearing is then ∼θ^lab/2 so that the leading edge of the electron bunch and the pump laser pulse envelope arrive simultaneously at the vertex.

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Somewhat similar sheared wave fronts are familiar solutions of Maxwell's equations in birefringent crystals, but in such crystals the Poynting vector and wave vector are not parallel. Although the Poynting vector and wave vector are always parallel in vacuum, it is possible to construct a laser pulse in vacuum like those of figure 2 with sheared wave fronts, if the laser amplifier has sufficient optical bandwidth. Commercially available Ti : sapphire lasers produce rather high energy pulses of ∼10 waves or less duration and thus such sheared pulses can be constructed. This shearing maintains an overlap of the laser pulse envelope and electron bunch for an extended time determined by the width of laser pulse. The width of the laser pulse can be made far greater than the width of a single wave front.

This idea of using wave front shearing to extend the interaction time of a laser pulse with a relativistic electron bunch has been proposed by others. For example, Rosenzweig et al (1995) described a 'proposed dielectric-loaded resonant laser accelerator' using a 'dielectric or metallic 'staircase' echelon to produce the necessary 45° angle in the laser intensity profile'. Plettner and Byer (2008) described a 'proposed dielectric-based microstructure laser-driven undulator' using an ultrashort pulse with an envelope 'tilted to maintain overlap with the electron travelling inside the vacuum channel'. Subsequently Plettner et al (2009) proposed 'photonic-based laser-driven electron beam deflection and focusing structures' and noted that 'To maintain extended overlap with the electron beam along the vacuum channel, the laser beam has to be pulse-front tilted'. The wavelength scale dielectric structures proposed by these authors provide additional control and enhancement of optical fields for both acceleration and deflection of relativistic electrons. However, the dielectric structures introduce a damage threshold in the ∼1 J cm⁻² fluence range and have the potential for wake field problems from the ∼1 µm scale structures. Such problems may be manageable with sufficiently small electron bunches and high energy, ∼GeV, electrons envisioned in 'FEL on a chip' concepts. The damage and wake field problems from dielectric structures appear to be more formidable in an attempt to make a soft x-ray FEL using modest-energy (∼170 MeV) electrons. The use of a sheared laser pulse as an undulator for 'high-yield' spontaneous scattering sources in the extreme ultraviolet and x-ray regions has been proposed and discussed (e.g. Debus et al 2010). Most recently Chang et al (2013) proposed using 'tilted optical wave' pulses from a CO₂ laser at right angles to a relativistic electron bunch as an undulator of a FEL. All of the above authors have recognized both the availability and some of the potential of a wave front sheared laser pulse although they have used different terms, such as envelop tilted or pulse-front tilted to describe the laser pulse. The technology of short-pulsed lasers available today is essential to these laser undulator concepts and the technology is vastly superior to that available in the early years of FELs.

Figures 3 and 4 illustrate the system of figure 2 shortly before the edge of the sheared pump laser pulse envelope reaches the electron bunch. The difference in these two figures is due to the choice of a lab reference frame in figure 3 and the electron rest frame in figure 4. The view in figure 4 illustrates again that the system is quite similar to a traditional head-on laser undulator as predicted using the stellar aberration formula. Note that the amount of shear in the pump laser pulse is frame dependent and the shear is essentially absent in the electron rest frame.

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The next section has a discussion of the production and delivery of a sheared pump laser pulse to the relativistic electron bunch. In most situations it will be necessary to image the surface of the diffraction grating used to shear the pump laser pulse in order to avoid diffractive spreading and curvature of the phase fronts. The subsequent section 5 explores the off axis FEL interaction in the electron rest frame. Section 6 has a gain estimate which suggests that SASE lasing in the soft x-ray region is achievable.

There may be several ways to produce the needed sheared laser pump pulse. Figure 5 is a schematic of a simple approach which is similar to the echelon scheme suggested by Rosenzweig et al (1995). A modest amount of shear is needed in a nearly copropagating geometry. A short, 30 fs = 12λ_pump/c, duration pulse from a Ti : sapphire laser with λ_pump = 750 nm is expanded laterally to a width of ∼600λ_pump/θ^lab for 300 cycles of interaction or # = 300 as shown in figures 2–4. The vertical expansion (out of the plane of the figure) can be much larger to maintain an energy fluence significantly less than the damage threshold (<1 J cm⁻²) on the grating. The desired wave front shear is then introduced using a properly blazed grating.

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It is important to deliver the sheared pump laser pulse to the electron bunch with flat phase fronts and uniform field amplitude in the interaction region. The focusing out of the plane of the figure is best accomplished with a cylindrical optic, for example a low loss dielectric mirror, because a lens introduces undesired dispersion. Consider a case with θ^lab = 0.0576 rad and λ_pump = 750 nm which is discussed in more detail in section 6. The ∼600λ_pump/θ^lab = 7.8 mm wide phase fronts incident on the grating are 'sheared' into phase fronts of width 24 λ_pump/θ^lab = 312 µm. The near field region extends ∼(312²/0.75) µm = 130 mm from the grating surface. A simple proximity focus might be possible in this case.

In other cases, for example in the right-angle illumination described by Chang et al (2013), it is necessary to image the surface of the grating onto the electron bunch. The flat phase fronts at the grating must be transferred to the electron bunch with minimal aberrations and minimal phase front curvature. A very similar imaging problem exists in a spatial heterodyne spectrometer (SHS) where it is necessary to image fringes formed at the crossing of a diffraction grating and a virtual image of another grating onto a detector array (e.g. Harlander and Roesler 1990). Since a SHS is an interferometric spectrometer the imaging must be done with sub-wavelength fidelity over the entire area of the diffraction grating. If unity magnification is acceptable then the afocal Offner (1973) system constructed from three spherical mirrors is ideal (e.g. Harlander et al 2010). An afocal system, such as any telescope focused for infinity, yields a plane wave output for an input plane wave. The afocal Offner system cancels all five of the familiar (third order) Siedel aberrations for unity magnification. A slightly earlier design provides non unity magnification but requires non spherical mirrors (Offner 1972). Undesirable phase front curvature and diffractive spreading of the relatively short sheared wave fronts can be eliminated by properly imaging the grating surface on the oblique electron bunch trajectory. An afocal image transfer system (infinite focal length) yields an 'in focus' image of the grating surface onto an oblique electron bunch trajectory. To achieve this near perfect image of the grating surface on the oblique electron bunch trajectory, it is essential to have the grating surface parallel to the electron bunch trajectory and hence the grating must be used in a non Littrow orientation as shown in the figure. The primary disadvantage of the Offner image transfer system in this application is the intermediate focus of the laser pulse on the middle convex mirror. An Offner system would need to be used before final amplification of the pump laser pulse.

Many articles have discussed intensity and frequency/phase uniformity requirements for head-on laser and magnetic undulators (e.g. Tanaka 2004, Huang and Kim 2007). For a SASE or high gain FEL the constraints on intensity detuning via variations in K² (see equation (12)) are tied to the e-folding gain length in wavelength units (e.g. equation (117) of Huang and Kim). Gain per wavelength is proportional to the FEL or Pierce parameter evaluated in section 6. Shearing of pump laser pulse with an initial length of 12 waves to yield a 300 wave long pulse effectively produces 25 overlapping replicas of the initial pulse. Thus a constraint must be imposed on the transverse uniformity of the pump laser pulse intensity on the grating to avoid gross detuning during the 300 wave pump pulse. Fortunately, small values of K² are less sensitive to detuning. For the K² ∼ 0.2 as discussed in section 6, there is a five-fold reduction of the frequency shift for a given fractional intensity fluctuation compared to the effect with a large K². Numerical simulations are being pursued. However simple considerations suggest that for an exponential gain length of 10 to 20 waves as discussed in the section 6, gross detuning over the 300 wave long pulse will be tolerable if the fractional transverse variation of the pump laser intensity on the grating is maintained within a per cent or so. There is also a possibility of K² detuning within the 12 wave long pump pulse. Such errors are replicated by the shearing and yield periodic intensity fluctuations in the 300 wave long pump laser pulse. Again a gain length of 10 to 20 waves yields a fractional gain bandwidth of 0.1 to 0.05 and suggests that the performance loss will be small if the pump laser intensity during the 12 wave long pulse is constant within a per cent or so.

Frequency or phase errors are also a source of concern. As discussed above shearing replicates the initial 12 wave long pump pulse to produce 25 overlapping copies, and thus any frequency/phase errors are periodic in the 300 wave long pump pulse. Numerical simulations, currently underway, are needed to fully explore the effect of frequency/phase errors in the initial 12 wave long pump pulse. Simple arguments can be used to explore the frequency content in the spectral profile of spontaneous scattering with a chirp in the form of a quadratic phase error in the initial 12 wave long pump pulse. A chirp which yields a 2.7% frequency shift at the ends of a 12 wave long pump pulse reduces the spontaneous spectral profile by 20% and reduces its derivative, related to the gain, by 25%. This suggests it will be necessary to limit fractional frequency variations in the 12 wave long pump pulse to values less than 1%.

To some extent it should be possible to offset frequency variations using intensity variations within the initial 12 wave long pump laser pulse since K² depends both on intensity and wavelength. Corrections of the transverse intensity distribution and/or phase on the grating are relatively simple to implement. Although one cannot now purchase an 'off the shelf' T³ laser with the needed pulse energy and which satisfies the above constraints, it is likely that engineering and fabrication costs of such a laser would be a tiny fraction of the savings from use of lower energy electrons from a shorter linac.

This section discusses the FEL process in the electron rest frame, where additional insights can be obtained. Repeated reference will be made to figure 4. In the electron beam frame, the pump laser approaches the head of the electron bunch a few degrees off axis. In figure 4, it is clear that θ^elec < π although it is close to π. In this beam rest frame, the analysis of Bosch (2003) applies. In particular, if one considers the amplification of a wave propagating 180° (back scattering) with respect to the pump laser, the configuration is essentially identical to the standard collinear FEL. Together with the usual analysis identifying optical undulators with magnetostatic undulators, this theoretical approach yields expressions that match the gain formulas that have been presented earlier. The 180° amplified wave has the highest gain in the fundamental (Colson et al 1985) since harmonic generation is minimized, assuming that sufficient electron bunch medium is available.

It might be argued that in the rest frame of the beam, the pump laser propagates across the electron bunch at an angle and has insufficient electron medium to reach saturation or that one is now in a short bunch regime. For example, for the case discussed, the number of periods to reach saturation results in a transverse motion exceeding the transverse dimension of the beam during the interaction length. One must distinguish, however, the direction of maximum overall gain and lasing from the direction of the pump laser pulse, while taking into account the shape of the electron bunch. We argue below that, for the case being discussed, the former is essentially aligned with the bunch axis and so the misalignment of the pump beam will not preclude saturation. Still, as a conceptual proof of principle, a crab cavity (Palmer 1988, Oide and Yakoya 1989) could be introduced that would align the electron bunch axis with the pump laser. In the beam frame, this would result in a tilt of only a few degrees. In the lab frame, this tilt is substantial, nearly 90°. However, given the low beam energy, a deflection system on the order of that used in B factories (Yamamoto et al 2010) would be sufficient. Further beam optics optimization, for example, by carrying out the deflection before full bunch compression, could reduce requirements significantly. In any case, the crab configuration establishes the conceptual validity of the overall approach.

However, such a system would not be required. The back scattered amplified wave at a few degrees in the electron rest frame propagates in the forward direction at an angle well within the cone of intense spontaneously scattered radiation in the lab frame. Following the arguments of Colson et al (1985) it can be shown that for the nearly copropagating laser pulse configuration discussed here nearly optimal total gain is achieved in this cone of intense spontaneously scattered radiation which has an angular width of order 1/γ in the lab frame. The gain along the bunch axis, therefore, is essentially the same as at the optimal direction. Moreover, it does not suffer from limitations of transverse beam dimensions over a saturation length since with propagation along the bunch axis the full bunch length is available for amplification. Likely, the best SASE amplification will be obtained somewhere between these two directions.

Although it is helpful for establishing conceptual validity, this crab configuration does not eliminate shearing of the x-ray beam in the lab frame. One can easily imagine a series of Gaussian beam phase fronts being emitted along the new symmetry axis in the electron rest frame and then imagine the effect of a Lorentz transform back to the lab frame. The wavelength of the perfectly backscattered radiation along the new symmetry axis in the electron rest frame is shifted to the x-ray region and the Poynting or wave vector direction is transformed much closer to the +z-axis, but there is some shear (sideways slippage) of the phase fronts as the electron bunch lases and moves in the direction of the +z-axis. There is also a small correction to the relativistic Doppler shift formulae of section 1. Of course the Poynting and wave vector are well within the cone of angular width of order 1/γ where there is intense spontaneous scattering and thus gain. The overall shearing will be mitigated because the FEL interaction is growing exponentially, and only the last few periods of the amplification provide the output x-rays. The small shear of the soft x-ray pulse might be exploited in some experiments. For example, an aperture might be used to cleanly select a sub fs part of the soft x-ray pulse which is only ∼10 cycles long. Another possibility is to use a grating at grazing incidence to remove the shear in the soft x-ray output pulse.

An estimate of the gain with a nearly copropagating pump laser pulse can be based on slightly modified versions of well-known FEL formulae for a 1D model (e.g. Pellegrini 2010). This 1D approximation is applicable if constraints on the electron beam emittance, electron beam energy spread, and gain are met. These constraints are discussed in more detail below. It is useful to identify Lorentz invariant quantities in this discussion. Figure 4 illustrates that the geometry of the nearly copropagating pump laser pulse is only slightly different than the geometry of the traditional head-on case when viewed from the electron rest frame. The consistency of analytic gain estimates and numerical simulations for head-on optical undulators has been demonstrated in multiple studies (e.g. Maroli et al 2007, Petrillo et al 2008, Sprangle et al 2009).

In section 1 we noted that Iλ² is Lorentz invariant and directly proportional to now standard undulator parameter K². (The angular factors in equation (1) and (7) cancel when the equations are combined to evaluate Iλ².) The polarization of x-rays in the direction of the electron beam will very nearly follow the polarization of the circularly polarized pump laser pulse with only a small amount of ellipticity introduced by the non-head-on interaction in the electron rest frame. A linearly polarized pump laser pulse will yield x-rays in the direction of the electron beam with the same linear polarization.

It is helpful to write the FEL (Pierce) parameter in a form which yields a gain per wavelength or cycle of interaction in the electron rest frame. One can combine equations (9) and (10) from Pellegrini's (2010) FEL review article

$$\begin{equation} \rho =\left( {\frac{K}{4}\frac{\Omega_{{\rm p}}}{\omega_{{\rm u}}}} \right)^{2/3}=\left( {\frac{K^{2}}{16}\frac{4\pi c^{2}r_{{\rm e}} n^{{\rm lab}}}{\gamma^{3}\omega_{{\rm u}}^{2}}} \right)^{1/3}, \end{equation} \tag{ 13 }$$

where Ω_p is the beam plasma frequency and ω_u = 2πc/λ_u is the frequency associated with a helical magnetic undulator period λ_u. The powers of γ in equation (13) can be absorbed for small K² by rewriting ρ in terms of electron rest frame quantities

$$\begin{equation} \rho =\left( {\frac{K^{2}(\lambda^{{\rm elec}})^{2}r_{{\rm e}} n^{{\rm elec}}}{16\pi}} \right)^{1/3}. \end{equation} \tag{ 14 }$$

In the electron rest frame the FEL interaction can have no dependence on γ and the stimulated scattering is at nearly the same wavelength as the incident electromagnetic wave at λ^elec (e.g. Bosch 2003). It is also possible to rewrite ρ in terms of lab frame quantities as

$$\begin{equation} \rho =\left( {\frac{K^{2}\gamma (\lambda_{{\rm x\mbox{-}ray}} )^{2}r_{{\rm e}} n^{{\rm lab}}}{4\pi}} \right)^{1/3}. \end{equation} \tag{ 15 }$$

This result is confirmed in calculations of the FEL gain in the rest frame of the electron beam (e.g. Bosch 2003). The Lorentz invariant gain per wavelength or per cycle of interaction is

$$\begin{equation} \frac{\lambda^{{\rm elec}}}{L_{{\rm G}}^{{\rm elec}}}=4\pi \sqrt 3 \rho \end{equation} \tag{ 16 }$$

and saturation of a SASE-FEL occurs when

$$\begin{equation} 4\pi \sqrt 3 \rho \# \cong 20 \end{equation} \tag{ 17 }$$

where # is the number of cycles of interaction. The fact that the incident wave in the electron rest frame is not quite parallel to the axis of the 'long' electron bunch in the electron frame has been neglected. The head-on geometry is not essential because the FEL will work along a direction providing a long path through the interaction region with the maximum total gain. The situations of interest have a small θ^lab with θ^elec ∼ π and $\theta_{{\rm scatt}}^{{\rm elec}} \sim \pi$ in the electron rest frame yielding FEL output nearly along the +z-axis. The coherent x-ray beam will be well within the cone of spontaneously scattered radiation of angular width of order 1/γ viewed from the lab frame.

Table 1 includes some electron beam and pump laser parameters needed for a gain estimate based on 1D approximation (Pellegrini 2010). The normalized emittance in table 1 is specified to satisfy the FEL 1D model emittance constraint

$$\begin{equation} \varepsilon_{n} /\gamma \le \lambda_{{\rm x\mbox{-}ray}}/(4\pi), \end{equation} \tag{ 18 }$$

and it is smaller than the typical normalized emittance of 3 × 10⁻⁷ m achieved at FEL facilities today. We note that the violation of the 1D model emittance constraint is substantial in the scheme described by Chang et al (2013). A violation of the emittance constraint for a 1D gain estimation does not eliminate gain; such a violation reduces the gain (e.g. Xie 1996). We suggest that a factor of five reduction in normalized emitttance from values in use today at LCLS² is a reasonable research goal. Progress is being made in reducing electron gun emittance values (e.g. Legg et al 2012, Gulliford et al 2013). Very innovative studies are exploring the use of photo-ionization of laser-cooled atoms or of a Bose Einstein Condensate to produce extremely small emittance electron beams (Luiten et al 2007). The second of Pellegrini's constraints for use of the 1D model involves the fractional energy spread of the electron bunch and the FEL parameter

$$\begin{equation} \Delta \gamma /\gamma \le \rho. \end{equation} \tag{ 19 }$$

This constraint is satisfied by a factor of three in table 1. The third constraint is that the gain must be larger than loss from diffractive spreading. This constraint can be written in terms of a gain per wavelength and an inverse Rayleigh range in wavelength units in the electron rest frame

$$\begin{equation} \frac{(\lambda^{{\rm elec}})^{2}}{\pi (D/2)^{2}}<4\pi \sqrt 3 \rho =\frac{\lambda^{{\rm elec}}}{L_{{\rm G}}^{{\rm elec}}}. \end{equation} \tag{ 20 }$$

The diffraction constraint is satisfied in table 1. The length of the interaction region in the lab frame is a useful parameter when thinking about 1D model constraints. This lab frame length is the width of the laser pulse ∼600λ_pump/θ^lab = 7.8 mm divided by θ^lab or 136 mm.

Table 1. Parameters for a gain estimate with a nearly copropagating sheared pump laser pulse.

Electron energy Lorentz factor	γ = 330
Fractional energy spread	Δγ/γ = 1 × 10−3
Normalized emittance	εn = 6 × 10−8 m
Electron bunch charge	20 pC
Electron bunch duration in the lab frame	25 fs
Electron bunch length in the lab frame	Llab = 7.5 µm
Electron bunch diameter	D = 7 µm
Electron density in the lab frame	nlab = 4.3 × 1023 m−3
Pump laser wavelength	λpump = 750 nm
Laser beam angle in the lab frame	θlab = 0.0576 rad
Number of waves of interaction	# = 300
Total width of sheared pump laser pulse in direction of shear (see figure 2)	∼600λpump/θlab
Sheared pump laser pulse thickness (see figure 2)	∼12λpump = 9 µm
Pump laser pulse width normal to direction of shear (normal to figure 2)	Dlaser = 7 µm
Pump laser pulse energy in the lab frame	Elab = 15 J
Pump laser intensity in the lab frame	Ilab = 9.1 × 1021 W m−2
Cosine of laser beam angle in the lab frame	cos(θlab) = 0.9983
Cosine of laser beam angle in e− rest frame	cos(θelec) = −0.9945
Undulator parameter	K2 = 0.189
X-ray wavelength	λx-ray = 2.46 nm
FEL (Pierce) parameter	ρ = 0.0033
$4\pi \sqrt 3 \rho \#$	22
Number of x-ray photons per electron at saturation	Nph,sat = 1.1 × 103
X-ray pulse energy	11 µJ

A pump laser pulse energy of 15 J is possible but the repetition rate would be low unless the pump laser pulse is stored in an enhancement cavity. The primary effect of the ponderomotive impulse which occurs when the pump laser pulse starts to overlap with the electron bunch is to decrease the axial velocity of the bunch. This effect has been included in table 1 using equation (3) with β replaced by β^* where

$$\begin{equation} \gamma^{\ast}=\frac{1}{\sqrt {1-(\beta^{\ast})^{2}}}=\frac{\gamma}{\sqrt {1+K^{2}}}. \end{equation} \tag{ 21 }$$

This expression yields an approximate correction for the axial ponderomotive impulse which occurs when the pump laser pulse first overlaps the electron bunch (see figure 4), but the small transverse part of the impulse is not included. The transverse part of the impulse slightly tilts the electron bunch axis a small amount and slightly deflects the electron bunch. The tilting effect can be eliminated by reshaping the edge of the laser pulse envelope that first reaches the electron bunch (see figure 3) but it is likely not worthwhile because the beam deflection persists. This deflection and possible tilting of the electron bunch produce a slightly sheared x-ray pulse that is unlikely to be a problem in experiments using the x-rays. These effects and other effects from asymmetric illumination yield a FEL x-ray pulse at a slight angle (e.g. tens of micro radian in the lab frame) with respect to the +z-axis and yield small fractional corrections (e.g. <0.01) to the relativistic Doppler shift formulae of section 1.

In principle all of the above transverse effects can be eliminated by using a symmetric pump laser pulse delivery system similar to that described by Chang et al (2013). Although Chang et al proposed using mid-infrared CO₂ laser pulses at right angle to the electron beam direction, a symmetric and nearly copropagating pump laser pulse could be used to illuminate the electron bunch from above the z-axis in figures 2, 3 and 4. This modified conceptual design employs a near infrared T³ laser undulator and it is actually identical to that of Chang et al (2013) when viewed from a selected reference frame between the lab frame and electron rest frame. Indeed, there is a frame which shifts the pump laser beam's directions from nearly copropagating with respect to the electron bunch to orthogonal with respect to the z-axis, and also shifts pump laser wavelength from the near infrared to the mid-infrared. A symmetric pump laser delivery system will yield a pattern of nodes and anti-nodes (standing waves with right-angle illumination) in the transverse direction at the interaction region. These cosine squared intensity modulations have a Lorentz invariant transverse spatial period

$$\begin{equation} \Lambda =\frac{\lambda_{{\rm pump}}}{2\sin (\theta^{{\rm lab}})}. \end{equation} \tag{ 22 }$$

The standing wave pattern limits the electron bunch diameter, the number of cycles of interaction, and forces changes in several other parameters in table 1. The primary difficulty in the symmetric design may be that the reduced bunch diameter, D ⩽ 1 µm, yields diffraction losses which substantially reduce the FEL gain. We are studying the possibility that 'optical or gain guiding' might mitigate diffraction losses (Scharlemann et al 1985), but the diffraction losses are serious with such a small bunch diameter.

The electron energy of ∼170 MeV in table 1 is chosen to yield soft x-ray photons with ∼500 eV. The x-ray wavelength of 2.5 nm is well within the longest wavelength water window of ∼2.35 to ∼4.4 nm or ∼530 to ∼280 eV. The number of photons per electron at saturation given in the second last line of table 1 is

$$\begin{equation} N_{{\rm ph,sat}} =\rho \frac{\gamma mc^{2}}{(hc/\lambda_{{\rm x\mbox{-}ray}} )}, \end{equation} \tag{ 23 }$$

where h is Planck's constant (Pellegrini 2010). The modest electron energy results in a modest number of x-ray photons per electron at saturation given by the product of the FEL parameter and the ratio of the electron energy to x-ray photon energy. The resulting 11 µJ pulse of x-rays is small compared to the pulse which can be generated using a 1 to 2 GeV electron beam and conventional magnetic undulator, however a repetition rate in the MHz range is possible with a superconducting linac (e.g. Legg et al 2012) and an enhancement/storage cavity for the pump laser pulse. A soft x-ray laser operating in the water window with an average power in the 10 W range would be an ideal facility for a great variety of bio-experiments. The conversion efficiency of ∼10⁻⁶ is similar to that achieved in HHG using 10 mJ pulses of 80 fs duration centred at a wavelength of 3.9 µm (Popmintchev et al 2012). An FEL facility with the parameters of table 1 would have a far higher, ∼×1000, x-ray pulse energy.

The preceding gain estimate does not consider the energy spread or the beam divergence due to the emittance of the electron bunch. Based on the fitting formulae of Xie (1996), these effects are expected to only cause a ∼33% decrease in the FEL parameter ρ. Modification of gain by transverse and longitudinal nonuniformity of the laser undulator is not addressed in the above estimates. It is likely though that the pump laser pulse could be optimized to reduce both transverse and longitudinal inhomogeneities. Effects from the transverse and longitudinal inhomogeneities of the electron bunch are not yet included. One can use the formulae of Xie (1996) to estimate the decrease in the FEL parameter ρ from a normalized emittance larger than the desired value of 6 × 10⁻⁸ m, for example ρ is decreased by a factor of two using a currently available normalized emittance of 2 × 10⁻⁷ m. Timing jitter is a concern, however the timing problems may not be significantly different than those which have already been solved to use conventional laser pulses to seed a FEL (Lambert et al 2008). There are other small effects which are not yet included. We are studying an adaptation of a numerical code such as Genesis for FEL modelling of the nearly copropagating sheared laser pulse undulator (e.g. Petrillo et al 2008).

The estimates in this section are sufficiently favourable to continue research on these concepts. The major research challenges are likely: (1) the reduction of the normalized emittance of the electron bunch to a value at least somewhat below 1 × 10⁻⁷ m and (2) the development of an enhancement/storage cavity for the sheared pump laser pulse. A good enhancement/storage cavity for the sheared pump laser is essential for an attractive repetition rate and average x-ray output power. The recent demonstration of a fibre laser system with an enhancement cavity for HHG is encouraging (Cingöz et al 2012), but the pump laser pulse energy required to cross threshold in HHG is far lower than needed for a FEL.

The conceptual design for a nearly copropagating laser undulator described in the preceding sections has some attractive features and some research challenges. Tunability is an attractive feature. It is not necessary to vary the K parameter. The device could be tuned by varying the electron energy, by varying the angle θ^lab, or by varying λ_pump. It may be simplest to vary the angle θ^lab by deflecting the electron beam. If a pump laser pulse is 10 or more cycles long, then some tunability is still available due to the very broad gain bandwidth of Ti : sapphire.

The emphasis in this paper is on soft x-ray FEL systems of modest cost, but there are other applications of the concepts. The combination of angle tuned Doppler shifts with a sheared pump laser pulse presented in this work is also applicable to spontaneous Compton scattering x-ray sources. Advantages include more efficient use of pump laser pulse energy, flexibility in tuning x-ray wavelength, control over pulse duration, and control over bandwidth (e.g. Debus et al 2010).

The 6 × 10⁻⁸ m normalized emittance used in the 1D gain estimate at modest electron energy is not currently available from electron guns. Use of an electron gun with a larger normalized emittance results in some gain reduction and only the 'central' part of the bunch participates in lasing. The asymmetric illumination of the electron bunch may produce some undesirable effects, e.g. the FEL x-ray laser pulse may be sheared. A scheme for alleviating such effects using a grazing incidence grating is proposed. An enhancement/storage cavity for the pump laser pulse is necessary for high repetition rate operation. The conceptual design in this work uses a Ti : sapphire wavelength because such laser systems are already highly developed and do deliver the needed short (∼25 fs) duration pulses with substantial pulse energy. We suggest that a high repetition rate facility requiring a high average power pump laser system will be built using a fibre laser system. Fibre laser systems currently in development scale favourably to high average output powers.

There are many possibilities for further development of this nearly copropagating sheared laser pulse FEL undulator. Seeding of the FEL using soft x-rays from HHG is a possibility. Another possibility is the use of a chirped pump laser pulse to extract more energy from electron bunches as is now done at LCLS. Performance degradation from diffractive spreading of FEL x-rays could be mitigated by appropriate curving of phase fronts of the pump laser pulse. A variety of techniques are available to control and manipulate the transverse phase variation and longitudinal phase or frequency variation of lasers in the near infrared spectral range. The extraordinary energy density in the electromagnetic fields of near infrared laser pulses can be angle tuned into resonance with a modest-energy relativistic electron bunch to produce a soft x-ray FEL. The 2.5 nm wavelength or ∼500 eV x-ray case study is by no means exhaustive of possibilities. The case is chosen in part because of the water window and possible bio-science applications. A shorter wavelength x-ray FEL is possible with higher electron energy, although systems costs are roughly proportional to electron energy and linac length. Operation of a FEL at a longer x-ray wavelength using a nearly copropagating laser undulator will be easier and less expensive.

The soft x-ray pulse energy or number of x-ray photons in the pulse will be lower than what can be achieved using a ∼1 to 2 GeV energy electron beam and a conventional magnetic undulator, but for many experiments a lower pulse energy is preferred. More detailed numerical modelling beyond the simple analytic estimates presented herein is desirable to explore a variety of effects neglected in the analytic estimates. Some realistic cost versus performance estimates for nearly copropagating laser undulators in soft x-ray FELs are also desirable. The major cost savings in comparison to a soft x-ray FEL design using an ∼1 to 2 GeV electron beam and conventional magnetic undulator is due to the use of much lower energy electrons. This reduces linac, infrastructure, and civil construction costs.

This research is supported by the University of Wisconsin Graduate School, the US Department of Energy under Grant DE-FG02-07ER46383 (TCC), and NASA under Grant NNX10AN93G (JEL). Helpful discussions on the ponderomotive impulse with Emeritus Professor Charles Goebel are acknowledged.