L24 · Stretch, amplify, compress

Chirped Pulse Amplification

Follow one complex pulse through four probe planes. Capture its energy and peak power, then break compression with residual dispersion or a narrow spectral acceptance.

Selected physical plane $\text{—}$

Live pulse energy $\text{—}$

Live main-peak FWHM $\text{—}$

Live peak power $\text{—}$

Live RMS duration $\text{—}$

Stretched FWHM $\text{—}$

Compressed main-peak FWHM $\text{—}$

Output zero-phase reference FWHM $\text{—}$

Effective plate peak nonlinear phase $\text{—}$

Amplified pulse fluence $\text{—}$

Amplified peak intensity $\text{—}$

Broadband reservoir extraction fraction $\text{—}$

Spectral acceptance rejected energy $\text{—}$

Actual compressor dispersion $\text{—}$

Maximum plane edge-energy fraction $\text{—}$

Captured physical planes $\text{—}$

Captured energy at selected plane $\text{—}$

Numerical window status $\text{—}$

Spectral edge energy fraction $\text{—}$

L24

Can the same energy cross the amplifier with a much lower peak?

Seed intensity FWHM

\tau_0

40\,\mathrm{fs}

Seed energy

E_0

0.02\,\mathrm{\mu J}

Stretcher dispersion

\phi_{2,s}

80000\,\mathrm{fs^2}

Broadband log gain

g_0

8

Saturation energy

E_s

20\,\mathrm{\mu J}

Spectral acceptance width

B

35\,\mathrm{THz}

Residual second-order dispersion

\Delta\phi_2

0\,\mathrm{fs^2}

Residual third-order dispersion

\phi_3

0\,\mathrm{fs^3}

Uniform beam area

A

1\,\mathrm{mm^2}

Effective nonlinear plate strength

\eta_{\rm NL}

1

Physical probe plane

j

3

Capture matched CPA, then bypass the stretcher. Compare amplifier peak intensity and nonlinear phase, not only final energy. Zero second-order mismatch does not remove cubic phase.

A scalar 800-nanometre Gaussian seed passes through an ideal phase-only stretcher, a broadband Frantz–Nodvik saturated-gain map, an explicit gain-weighted Gaussian spectral acceptance, an effective nonlinear plate, and a phase-only compressor. The spectral acceptance is an ordered lossy surrogate for gain narrowing; it is not a joint frequency-dependent inversion or Maxwell–Bloch solution. Rejected energy is reported separately. The amplifier uses cumulative pulse energy, a user-set saturation energy and initial log gain, with no pump replenishment during the pulse. A uniform beam area converts energy and power to fluence and intensity; changing area holds saturation energy fixed and therefore does not model a fixed material cross-section. The effective output-plane plate uses a one-centimetre path and an illustrative nonlinear index; its phase is not an integrated nonlinear phase through the actual gain medium. Disabling the plate gives a linear comparison. No damage threshold, thermal fracture, ASE, compressor throughput, diffraction or spatial hot spots are simulated. Stretching preserves incident energy and fluence while reducing peak power; it does not guarantee that a real optic avoids damage. The 32768-point Fourier grid spans 65.536 picoseconds with two-femtosecond spacing. Edge fractions expose temporal truncation and frequency-grid stress; main-peak FWHM does not measure all satellites. Four-plane acquisition copies the computed fields, not hardware detector data. The dashed output is a zero-spectral-phase ideal made from the output spectrum, not a reconstruction of the captured pulse. The dispersive blocks are schematic, without grating-angle ray tracing.

Start with the essentials

Focus question: Can the same energy cross the amplifier with a much lower peak?
One-sentence intuition: Stretching changes peak intensity; amplification changes energy; compression removes phase.

Core mathematical model

Phase-only stretching

\widetilde A_s(\Omega)=\widetilde A_0(\Omega)\exp(-i\phi_{2,s}\Omega^2/2)

The Fourier transform uses a negative exponent and the carrier convention is positive. A positive group-delay dispersion delays positive frequency offsets.

Independent Gaussian stretching limit

\tau_s=\tau_0\sqrt{1+\left(\frac{4\ln 2\,\phi_{2,s}}{\tau_0^2}\right)^2}

This analytic intensity FWHM is compared with the propagated field. Energy and spectral magnitude remain constant.

Broadband saturated amplification

E_{\rm out}(t)=E_s\ln\!\left[1+e^{g_0}\left(e^{E_{\rm in}(t)/E_s}-1\right)\right]

Apply the cumulative Frantz–Nodvik map to temporal bins. Energy extracted from the initial reservoir cannot exceed saturation energy times log gain.

Explicit spectral acceptance

\widetilde A_a\longmapsto\widetilde A_a\exp\!\left[-2g_0\ln 2\left(\frac{\nu}{B}\right)^2\right]

This lossy step narrows the accepted spectrum. It is an ordered surrogate, not frequency-resolved inversion depletion; report the rejected energy.

Compression and residual phase

\begin{aligned}\phi_{2,c}&=-\phi_{2,s}+\Delta\phi_2\\ \widetilde A_c&=\widetilde A_a\exp\!\left[-i\left(\frac{\phi_{2,c}\Omega^2}{2}+\frac{\phi_3\Omega^3}{6}\right)\right]\end{aligned}

Matching second-order dispersion does not cancel third-order phase or nonlinear phase, and cannot recreate spectral energy already rejected.

Peak intensity is different from fluence

\begin{aligned}F&=E/A\\ I_{\rm peak}&=P_{\rm peak}/A\\ B_{\rm plate}&=\eta_{\rm NL}\frac{2\pi n_2L}{\lambda_0}I_{\rm peak}\end{aligned}

Stretching reduces peak intensity but not incident fluence. The effective plate phase is diagnostic teaching physics, not a damage certificate or an amplifier-path integral.

Common difficulties

A short peak proves success

Typical misconceptionMain-peak width alone establishes pulse quality.

Better mental modelInspect satellites, RMS duration, numerical boundaries and the separate ideal reference.

Run the experiment

01
Capture the chain
Capture matched CPA and move the probe through all four planes.
What to observe: Stretching and compression preserve their respective input energy.
02
Bypass stretching
Compare matched and unstretched presets.
What to observe: The gain step has similar stored energy but very different peak intensity and plate phase.
03
Break compression
Compare second-order mismatch, cubic phase and narrow acceptance.
What to observe: Matching one dispersion order cannot recover all pulse quality.