X-ray Free-Electron Laser

How can free electrons amplify light without atomic energy levels? Light modulates electron energies and gathers electrons into tiny groups that transfer kinetic energy to the wave. Amplification eventually saturates. Measuring a pumped sample also requires per-shot energy and timing references to distinguish its response from pulse width.

Follow energy modulation and microbunching through an undulator. Use the energy and timing record of each pulse to measure a pumped sample.

Bunch, radiate, probe

Drag a gold handle. Each acquisition scans nine delay positions; this offset shifts the whole scan. Demonstrations use numerically checked sources; other source settings begin as previews.

Undulator strength 1.2Interaction length 8Resonant seed field 0.02Delay scan offset / fs0

Pump energy and electron spread

Optical pump energy / µJ1Scaled electron energy spread 0.15

Preparing a fresh default measurement…

Model assumptions

A finite, seeded, one-dimensional FEL teaching model with a given electron beam and effective transverse area. Complex radiation and electron phase/energy evolve together, with radiation slipping toward the bunch head. Transverse diffraction, emittance, harmonics, wakefields and facility performance are outside its scope. The unseeded demonstration is only a short uniform cooperation-cell startup limit, not a finite SASE pulse or a sample-lifetime measurement. The resonant X-ray seed is distinct from the 800 nm optical sample pump. Longitudinal and transverse dimensions have separate scene scales; apertures, electron wiggle and actuator travel are further enlarged. Slowed markers do not count photons. Detector pixels, energy ADCs, encoder readings and blurred timing traces are simulated observations. Field, phase-space and photon-account displays are separate forward-model references. Each acquisition generates fresh finite readouts. Inference does not use the random seed, hidden arrival shift or unknown sample lifetime. One-femtosecond sampling is not one-femtosecond resolution. A time constant is reported only within the actual calibration, reference, temporal-coverage and numerical limits; unresolved fast responses retain their bounds.

Light made by the electrons

Current source model reference. Cyan: north; rust: south.

Growing radiation and electron bunching · simulated reference

Inspect station

How to read the source plots

What this pulse measured

Saved measurement

Same-shot spatial timing trace · actual ADC codes. Pixel pitch is sampling, not resolution.

Finite pump-off/on sample observations · references and unsupported results remain visible.

Pulse reference, detector pixels and timing limits

Current live source expectation · separate from the selected shot. This is not a reconstructed complex field.

Actual detector exposure · acquired pixels, separately from source diagnostics.

The blurred optical timing trace constrains a pulse kernel. It does not measure the X-ray phase or prove temporal coherence. An unresolved fast sample is not a measured zero lifetime.

Raw observations and independent references

The table condenses each frame; JSON and CSV retain every detector and timing pixel. Scroll sideways for calibrated results.

Frame	Role	Pump	Trigger tick	Encoder code	Energy ADC	Pump ADC	Detector ADC sum	Timing first ADC	Stage delay / fs	Transmission	Standard uncertainty

Source account and model limits

The apparatus shows a representative pump-on optical path. The protocol also includes pump-off controls.

Terminal	Count or energy

A prescribed optical relay is an external momentum reservoir. The account does not claim an isolated apparatus four-momentum balance.

\theta_j'=p_j,\qquad p_j'=-2\operatorname{Re}\!\left(Ae^{i\theta_j}\right)

\left(\partial_\zeta+\partial_s\right)A=n(s)\sum_jw_je^{-i\theta_j}

Save or reopen the complete measurement

Import

Eight immutable records; 64 MiB decoded and 96 MiB text limits. Exports retain raw readings, units and independent calibrations. Invalid imports leave the session unchanged.

Reopen an archive in an empty session

Focus question: How does light gather electrons and help measure a changing sample?
One-sentence intuition: Electron and field evolution produce gain. A sample response must be interpreted together with the width and timing of the pulse that measured it.

Core mathematical model

Forward resonant wavelength

\lambda_r=\frac{\lambda_u}{2\gamma^2}\left(1+\frac{K^2}{2}\right)

Undulator period, electron energy and magnetic strength set resonance in this supplied one-dimensional paraxial model.

Coupled electron and field evolution

\frac{d\theta_j}{d\zeta}=p_j,\qquad\frac{dp_j}{d\zeta}=-2\operatorname{Re}(Ae^{i\theta_j}),\qquad(\partial_\zeta+\partial_s)A=n\sum_j w_j e^{-i\theta_j}

Phase locates an electron relative to the wave. Scaled energy controls its advance, and the field changes that energy. Radiation slips toward the bunch head; energy leaving the finite window remains in the source account.

A carriage changes the optical route

\Delta D=-\frac{2d}{c},\qquad D=t_{\mathrm{probe}}-t_{\mathrm{pump}}

Positive carriage travel lengthens two path segments and delays the pump. An independent reference supplies time zero. Enlarged scene travel does not enter flight-time calculations.

Finite illumination and causal recovery

r_i(D,\tau)=\int_0^\infty k_i(D-u)e^{-u/\tau}\,du

The pulse-overlap curve and causal recovery determine each transmission. The measured blurred timing trace constrains that shot’s overlap; film response and optical blur retain their calibration uncertainty.

Common difficulties

Seed and pump have different jobs

Typical misconceptionThe optical sample pump is the X-ray seed in the undulator.

Better mental modelThe resonant seed participates in X-ray amplification. The optical pump follows another route and excites the sample.

Sampling and resolution

Typical misconceptionOne-femtosecond pixels imply a one-femtosecond lifetime measurement.

Better mental modelPixel pitch sets sampling. Pulse width, film response, blur and noise set resolving power, which known references must test.

Run the experiment

01
Follow the tiny groups
Choose Seeded pulse and compare electron phase and energy distributions at different positions below the apparatus.
What to observe: Energy modulation develops into bunching. Field growth and electron energy loss come from the same interaction.
02
Lengthen the interaction
Choose Into saturation and inspect the later part of the growth curve.
What to observe: Detuning and trapping slow radiation growth as the electron distribution continues to change.
03
Compare a different start
Choose Noise starts the field, then return to a seeded pulse.
What to observe: Physical electron number fixes the startup fluctuations. This short uniform-cell comparison reports mode power and bunching.
04
Send the probe through first
Choose Probe arrives first or drag the delay carriage. Compare mirrors, path and transmission.
What to observe: Travel adds twice its distance to the optical route. Negative probe-minus-pump delay sends the probe through an unexcited sample.
05
Test temporal resolving power
Choose Too fast to resolve and expand the timing trace, references and sample curve.
What to observe: Pulse and timing-film widths can blur a fast recovery. Insufficient resolution retains a bound and the original exposures.

X-ray Free-Electron Laser

Light made by the electrons

What this pulse measured

From electron groups to a changing sample

Core mathematical model

Forward resonant wavelength

Coupled electron and field evolution

A carriage changes the optical route

Finite illumination and causal recovery

Common difficulties

Seed and pump have different jobs

Sampling and resolution

Run the experiment

Follow the tiny groups

Lengthen the interaction

Compare a different start

Send the probe through first

Test temporal resolving power

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