--- title: "X-ray Free-Electron Laser Interactive Lab | Sandbox Physics" description: "How can free electrons amplify light without atomic energy levels?" canonical_url: "https://sandboxphysics.com/labs/xray-free-electron-laser/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/xray-free-electron-laser/" last_modified: "2026-11-13" --- # 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. > Interactive simulation: open this page in a web browser to change the parameters and run the model. ### 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 1.2 Interaction length 8 Resonant seed field 0.02 Delay scan offset / fs 0 Pump energy and electron spread Optical pump energy / µJ 1 Scaled 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. ## From electron groups to a changing sample **Background** Alternating magnets make electrons wiggle as light overtakes them. A suitable wave first modulates their energies, then gathers them into tiny groups. **Why it matters** This interaction transfers electron kinetic energy into a short X-ray pulse. An experiment also needs to know the energy and sample arrival time of each pulse. - **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 misconception** The optical sample pump is the X-ray seed in the undulator. **Better mental model** The resonant seed participates in X-ray amplification. The optical pump follows another route and excites the sample. #### Sampling and resolution **Typical misconception** One-femtosecond pixels imply a one-femtosecond lifetime measurement. **Better mental model** Pixel pitch sets sampling. Pulse width, film response, blur and noise set resolving power, which known references must test. ### Run the experiment 1. 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. 2. 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. 3. 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. 4. 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. 5. 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. Physical conclusion Microbunching explains amplification; independent energy and timing references determine what a short pulse can measure. ## Related experiments and articles How physics instruments measure · Reading path ### How physics instruments measure Articles on this path that use this experiment, followed by the other experiments they suggest. 1. [Article **How a free-electron laser gathers electrons into a brighter beam**](https://sandboxphysics.com/stories/instruments/xray-free-electron-laser/) Fields & matter · Starter path ## From a field to an electron microscope Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface. [Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/) 1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/) 2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/) 3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/) 4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/) 5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/) 6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/) [View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)