X-Scheme: Terawatt-Level Coherent Hard X-rays

• X-Scheme is a technical framework that integrates cascade self-seeding with wake monochromators in a tunable-gap undulator, achieving complete coherence and a three orders of magnitude increase in peak brightness over conventional SASE methods.
• It employs step-wise tapering to maintain resonance as electrons lose energy, enabling pulse peak powers from 400–500 GW up to terawatt levels with exceptional spectral purity.
• The approach offers compact, low-cost implementation for advanced experiments such as biomolecular imaging and inelastic scattering, validated by simulation studies using GENESIS 1.3.

The term "X-Scheme" refers to a technical framework devised for the generation of fully coherent, terawatt-level (TW) power hard X-ray pulses at the European X-ray Free-Electron Laser (XFEL) facility. The scheme integrates a cascade self-seeding structure with wake monochromators within a tunable-gap undulator, fundamentally enhancing the brightness, coherence, and power of the XFEL output over traditional SASE (Self-Amplified Spontaneous Emission) methods (Geloni et al., 2010). The X-Scheme is particularly notable for its ability to deliver complete longitudinal and transverse coherence alongside three orders of magnitude increase in peak brightness—extending capabilities in fields such as biomolecular imaging, inelastic scattering, and nuclear resonant scattering.

1. Physical Principles and Resonance Management

In an XFEL, electrons are accelerated to high energies and pass through a periodic magnetic structure known as an undulator, emitting X-rays due to their oscillations. The resonant wavelength $\lambda$ of the emitted radiation is determined by: $$\lambda = \frac{\lambda_u}{2\gamma^2} \left[1+\frac{K^2}{2}\right]$$ where $\lambda_u$ is the undulator period, $\gamma$ is the Lorentz factor of the electrons, and $K$ is the undulator parameter, proportional to the magnetic field strength.

During the FEL process, as electrons radiate, they lose energy ($\gamma$ decreases), causing them to fall out of resonance. The X-Scheme employs step-wise tapering: the undulator magnetic field ($K$) is reduced incrementally along the undulator to match the evolving $\gamma$, maintaining resonance and maximizing energy extraction. This is formalized via a cell-to-cell taper law, typically with a $\sim2\%$ decrement across the tapered section.

2. Cascade Self-Seeding with Wake Monochromators

Unlike SASE, which starts from electron shot noise and produces partially coherent pulses, the X-Scheme introduces a cascade self-seeding process:

• After an initial undulator segment, a magnetic chicane spatially separates electrons and X-ray pulses.
• The X-rays are filtered through a diamond crystal acting as a wake monochromator, operating in Bragg reflection.
• The monochromator produces a narrow, highly monochromatic seed pulse.
• Electrons and filtered X-rays are recombined in a following undulator stage, where the seed is exponentially amplified: $E_{\text{out}}(z) \propto E_{\text{seed}}\,e^{gz}$ with $E_{\text{seed}} \gg E_{\text{noise}}$, and $g$ the FEL gain parameter.

Multi-stage cascades further improve seed purity, spectral narrowness, and pulse coherence.

3. Output Radiation Characteristics

The X-Scheme generates pulses with three distinct features:

• Full longitudinal and transverse coherence: Pulse bandwidth approaches $10^{-4}$ relative FWHM, near the transform limit. Spatial divergence is nearly diffraction-limited.
• Extreme peak brightness: Simulations predict up to $10^{37}$ ph/s/mm$^2$/mrad$^2$/0.1% BW, versus typical $10^{34}$ for SASE sources.
• TW-level peak and kW-average power: Compared to baseline saturation values ($\sim$20–30 GW), post-seeding and tapering achieves 400–500 GW routinely, with exceptional cases reaching TW levels.

The table below contrasts key output parameters:

Parameter	SASE XFEL (LCLS)	X-Scheme (European XFEL)
Longitudinal coherence	Partial	Complete
Transverse coherence	Partial	Complete
Peak brightness (ph/s/mm$^2$/mrad$^2$/0.1% BW)	$10^{34}$	$10^{37}$
Peak power	20–30 GW	400–500 GW to 1 TW
Pulse duration (short-bunch)	~5–50 fs	~5 fs

4. Feasibility, Compactness, and Implementation

The scheme’s upgrades are achieved by:

• Inserting weak chicanes and diamond crystals without major redesign.
• Adjusting the gap of baseline undulators for tapering.
• Minimal additional component footprint.
• Extremely compact (the monochromator and chicane are small relative to undulator length) and low-cost—feasible for installation during commissioning.

Simulation studies, performed with GENESIS 1.3, validated performance in both short-bunch (0.025 nC, 5 fs pulses, 2 mJ energy) and long-bunch (0.25 nC, up to 20 mJ energy) operational modes.

5. Applications and Scientific Impact

The increased coherence and peak power open new experimental domains:

• Single biomolecule imaging: Ultra-short, high-brightness pulses allow damage-free atomic-resolution structure determination.
• Inelastic X-ray scattering: High spectral resolution due to narrow bandwidth enhances sensitivity for solid-state physics and chemistry.
• Nuclear resonant scattering: High monochromaticity and coherence enable exploration of nuclear transitions and hyperfine interactions at unprecedented energy resolution.

Immediate deployment during XFEL commissioning provides access to these advanced capabilities from the outset.

6. Technical Trade-offs and Limitations

Key considerations for adoption:

• Tapering effectiveness is contingent on beam quality and undulator gap precision. Deviation from optimal taper profile can reduce achievable power and coherence.
• Crystal monochromator efficiency: Practical throughput limited by crystal reflectivity and induced losses, though simulations indicate seed-to-noise ratios remain robust ($>$10x minimum required).
• Pulse duration/spectral chirp: For certain experimental configurations, additional pulse shaping or stabilization may be desirable.

7. Future Directions

Broader application of the X-Scheme framework suggests:

• Extension to other XFEL facilities and undulator configurations.
• Replacement or augmentation of SASE-driven experiments with fully seeded, tapered systems.
• Exploration of multi-color seeding, adaptive tapering laws, or further compact monochromator designs.

The X-Scheme represents the convergence of advanced seeding optics and undulator tapering to enable next-generation hard X-ray sources with transformative capabilities for ultrafast and high-resolution scientific investigations (Geloni et al., 2010).