L24 · Stretch, amplify, compress

Chirped Pulse Amplification

Can a short pulse be amplified without its peak power getting dangerously high? It can, if you stretch the pulse first. Dispersion spreads it out in time: same energy, much lower peak power. The amplifier can then add energy without the peak getting dangerously high, and a compressor afterwards undoes the stretch to give back a short, intense pulse.

Follow one complex pulse through four probe planes as it is stretched, amplified and compressed. Record its energy and peak power, then spoil the compression with leftover dispersion or a narrow spectral acceptance.

Selected physical plane $\text{—}$

Recorded pulse energy $\text{—}$

Recorded main-peak FWHM $\text{—}$

Selected-plane peak power $\text{—}$

Recorded 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 a short pulse be amplified without its peak power getting dangerously high?

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

The four pulse volumes are recorded snapshots, replayed together with a common nonlinear time scale and fixed logarithmic brightness. Their radius is schematic, not the beam area. Time plots below use linear physical units. The phase modules have equivalent transmission; no grating-ray geometry is solved.

Compare Matched CPA with Bypass stretcher. Check the peak intensity and nonlinear phase in the amplifier, not just the final energy. Setting the second-order mismatch to zero 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, a gain-weighted Gaussian spectral acceptance filter, an effective nonlinear plate and a phase-only compressor. The spectral acceptance is a separate lossy step that stands in for gain narrowing; it is not a joint frequency-dependent inversion or Maxwell–Bloch solution. Rejected energy is reported separately. The amplifier uses the cumulative pulse energy and a saturation energy and initial log gain that you set; the pump does not replenish the gain during the pulse. A uniform beam area converts energy and power into fluence and intensity. Changing the area keeps the saturation energy fixed, so it does not model a fixed material cross-section. The effective plate at the output plane has a one-centimetre path and an illustrative nonlinear index; its phase is not the nonlinear phase integrated through the real gain medium. Turning the plate off gives a linear comparison. Damage thresholds, thermal fracture, ASE, compressor throughput, diffraction and spatial hot spots are not simulated. Stretching keeps the incident energy and fluence while lowering the peak power; it does not guarantee that a real optic escapes damage. The 32768-point Fourier grid spans 65.536 picoseconds with two-femtosecond spacing. Edge fractions show when the pulse is cut off in time or strains the frequency grid. The main-peak FWHM does not capture all the satellite pulses. Capturing the four planes copies the computed fields; it is not hardware detector data. The dashed output curve is an ideal pulse with zero spectral phase built from the output spectrum, not a reconstruction of the captured pulse. The dispersive blocks are schematic and do not trace rays through grating angles.

Start with the essentials

Focus question: Can a short pulse be amplified without its peak power getting dangerously high?
One-sentence intuition: Stretching lowers the peak intensity, amplification adds energy, and compression removes the spectral phase again.

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 uses the positive sign convention. With positive group-delay dispersion, frequencies above the carrier arrive later.

Analytic stretching of a Gaussian pulse

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

The Lab checks the propagated field against this analytic intensity FWHM. Stretching leaves the energy and the spectral magnitude unchanged.

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]

The Frantz–Nodvik map is applied to each time bin using the energy that has already passed. The energy taken from the initial reservoir can never exceed the saturation energy times the log gain.

Spectral acceptance filter

\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 spectrum that gets through. It stands in for gain narrowing as a separate step in the chain and does not resolve inversion depletion by frequency. The rejected energy is reported.

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}

Cancelling the second-order dispersion leaves any third-order or nonlinear phase in place, and it cannot bring back spectral energy that was 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 lowers the peak intensity but not the incident fluence. The phase in the effective plate is a teaching diagnostic: it does not certify that an optic survives, and it is not integrated along the amplifier path.

Common difficulties

A short main peak is not the whole story

Typical misconceptionIf the main peak is short, the pulse must be good.

Better mental modelAlso check the satellite pulses, the RMS duration, the energy near the edges of the numerical grid and the separate ideal reference.

Run the experiment

01
Record every plane
Choose Matched CPA and move Physical probe plane through the four positions. All four fields are recorded automatically.
What to observe: The stretcher and the compressor each pass on all the energy they receive.
02
Skip the stretcher
Compare the Matched CPA and Bypass stretcher presets.
What to observe: The amplifier holds about the same stored energy in both cases, but the peak intensity and the plate phase are very different.
03
Spoil the compression
Drag the compressor dial, then compare Dispersion mismatch, Cubic phase and Narrow acceptance. Check the short output pulse against its dashed ideal reference below.
What to observe: Cancelling one order of dispersion is not enough to restore the pulse fully.

Chirped Pulse Amplification

Carry energy through a lower peak

Core mathematical model

Phase-only stretching

Analytic stretching of a Gaussian pulse

Broadband saturated amplification

Spectral acceptance filter

Compression and residual phase

Peak intensity is different from fluence

Common difficulties

A short main peak is not the whole story

Run the experiment

Record every plane

Skip the stretcher

Spoil the compression

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