L25 · Capture, amplify, extract

Regenerative & Multipass Amplifier

Step through one prepared gain reservoir, miss the injection gate, or extract too late. Record the complete energy ledger and arrival-time sensitivity.

Optical architecture $\text{—}$

Completed gain traversals $\text{—}$

Extraction traversal $\text{—}$

Illustrative elapsed path delay $\text{—}$

Live seed-derived energy $\text{—}$

Live accepted background energy $\text{—}$

Remaining stored energy $\text{—}$

Accumulated path loss $\text{—}$

Captured input seed $\text{—}$

Rejected input seed $\text{—}$

Remaining logarithmic gain $\text{—}$

Seed-derived energy fraction $\text{—}$

Extracted useful signal $\text{—}$

Largest signal seen this cycle $\text{—}$

Fixed-shape peak-power proxy $\text{—}$

Signal fluence diagnostic $\text{—}$

Energy ledger residual $\text{—}$

Single-pass comparison $\text{—}$

Multipass comparison $\text{—}$

Regenerative comparison $\text{—}$

Captured timing response samples $\text{—}$

Timing-averaged extracted signal $\text{—}$

Relative timing-induced variation $\text{—}$

Cycle state $\text{—}$

L25

When should the trapped seed be extracted?

Architecture: single / multipass / regenerative

m

2

Seed energy

E_0

0.01\,\mathrm{\mu J}

Prepared logarithmic gain

g_0

1.5

Saturation energy

E_s

400\,\mathrm{\mu J}

Extraction gain traversal

N

12

Loss per gain traversal

\ell

3\,\mathrm{\%}

Seed arrival relative to capture gate

\delta t

0\,\mathrm{ns}

Arrival jitter standard deviation

\sigma_t

0.05\,\mathrm{ns}

Accepted spontaneous seed per traversal

E_{\rm sp}

0.0001\,\mathrm{\mu J}

Uniform beam area

A

1\,\mathrm{mm^2}

Stretched seed intensity FWHM

\tau

100\,\mathrm{ps}

Detail: switch / gain reservoir

j

0

Capture the early and late presets. More traversals can exhaust the reservoir and then dissipate a pulse already amplified. A missed seed can still produce background output.

A reduced single-shot, energy-only amplifier model. Each gain traversal applies the broadband Frantz–Nodvik fluence law to the signal and one accepted incoherent background channel sharing the same reservoir. The imposed spontaneous background seed per pass is scaled by remaining reservoir fraction and paid from that reservoir; it is an ASE surrogate, not spatial or spectral spontaneous-emission transport, quantum noise or a pulse-contrast spectrum. Signal and background share a lumped proportional gain, without their actual temporal ordering. All paths use one prepared saturation-energy reservoir, with no replenishment, recovery or upper-state decay during the cycle. A geometric multipass path is limited to twelve traversals and has no capture gate. The ring regenerative cavity captures a Gaussian stretched seed through an ideal half-nanosecond rectangular gate, traps it, then routes the entire remaining signal and background through an ideal lossless extraction switch after the selected gain traversal. One traversal per ring round trip is assumed; the six-nanosecond clock is an illustrative path delay. Real Pockels-cell rise times, voltage, extinction ratio, polarization dynamics, recirculated seeds and shot-to-shot gain recovery are omitted. Normal timing jitter is integrated with 321 deterministic samples over five standard deviations; these are simulated response samples, not 321 hardware shots. The Gaussian peak-power proxy holds the input duration fixed despite gain saturation; pulse reshaping, phase, spectral narrowing, diffraction and nonlinear phase are omitted. Uniform beam area sets a fluence diagnostic, not a damage prediction. Beam paths, aperture sizes and pulse lengths are schematic. Signal, accepted background, remaining reservoir and cumulative path loss close one energy balance; the rejected seed lies outside that balance and is reported separately.

Start with the essentials

Focus question: When should the trapped seed be extracted?
One-sentence intuition: Repeated gain traversal helps only while the remaining gain exceeds path loss.

Core mathematical model

Capture the arriving seed

\begin{aligned}E_{\rm cap}&=E_0\int_{-t_g/2}^{t_g/2}p(t-\delta t)\,dt\\t_g&=0.5\,\mathrm{ns}\end{aligned}

The normalized Gaussian seed enters only during the ideal gate. Geometric paths have no capture gate.

One saturated traversal

\begin{aligned}E_a&=E_s\ln(1+e^{g_n}u_n)\\u_n&=e^{E_{\rm in}/E_s}-1\end{aligned}

The integral Frantz–Nodvik law extracts finite energy from the prepared reservoir. The background shares a lumped gain with the seed.

Deplete the same reservoir

\begin{aligned}R_{n+1}&=R_n-(E_a-E_{\rm in})\\g_n&=R_n/E_s\end{aligned}

The accepted spontaneous seed is also paid from the reservoir before amplification. Stored energy is not reset at each pass.

Pay for every optical path

\begin{aligned}E_{n+1}&=(1-\ell)E_a\\\Delta E_{\rm loss}&=\ell E_a\end{aligned}

After depletion, extra traversals dissipate the already amplified pulse. The best extraction pass is finite when loss is nonzero.

Close the energy ledger

\begin{aligned}E_{\rm sig}+E_{\rm bg}+R+E_{\rm loss}\\=E_{\rm cap}+g_0E_s\end{aligned}

Rejected seed energy is outside this ledger. The balance does not certify actual material efficiency or damage safety.

Integrate timing sensitivity

\begin{aligned}\mu_E&=\int E_{\rm out}(t)p_j(t)\,dt\\\sigma_E^2&=\int(E_{\rm out}-\mu_E)^2p_j\,dt\end{aligned}

The recorded response uses deterministic normal quadrature over five standard deviations. These are model samples, not hardware shots.

Common difficulties

Background is the useful pulse

Typical misconceptionAny output energy proves successful seed amplification.

Better mental modelUse the seed-derived energy fraction. This model background is a reduced ASE surrogate, not a measured contrast spectrum.

The peak proxy predicts damage

Typical misconceptionFixed-shape peak power proves an optic survives.

Better mental modelPulse reshaping, beam hot spots, coatings and material thresholds are absent.

Run the experiment

01
Watch the reservoir disappear
Step the regenerative preset one traversal at a time, then extract and capture.
What to observe: Signal rises while stored energy falls. The captured traces come from this same cycle.
02
Find the extraction window
Compare early, ordinary and late extraction with the same reservoir.
What to observe: Too few passes leave gain unused; too many lose energy already extracted from the reservoir.
03
Miss the seed
Capture the missed-injection and background-takeover presets, then compare timing sensitivity.
What to observe: Nonzero output need not be the amplified seed. A narrow timing margin creates strong relative variation.