L25 · Capture, amplify and extract a pulse

Regenerative & Multipass Amplifier

When should a pulse circulating in a regenerative amplifier be let out? Around the pass where the gain left in the medium no longer beats the loss along the path. Until then, every extra pass adds energy to the pulse. After that, each pass costs more than it gains, so the pulse only gets weaker.

Step a pulse through the gain one pass at a time as it drains a single store of energy. Try missing the injection gate or extracting too late, and record the full energy balance and how sensitive the output is to arrival time.

Optical architecture $\text{—}$

Completed gain traversals $\text{—}$

Extraction traversal $\text{—}$

Illustrative elapsed path delay $\text{—}$

Recorded final seed-derived energy $\text{—}$

Recorded final accepted background $\text{—}$

Stored energy after extraction $\text{—}$

Total cycle path loss $\text{—}$

Captured input seed $\text{—}$

Rejected input seed $\text{—}$

Remaining logarithmic gain $\text{—}$

Extracted seed-derived fraction $\text{—}$

Extracted useful signal $\text{—}$

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

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

Signal fluence diagnostic $\text{—}$

Energy balance 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 a pulse circulating in a regenerative amplifier be let out?

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 Extract early and Extract late presets. Extra passes can use up the stored energy and then drain a pulse that is already amplified. A missed seed can still give background output.

This is a simplified single-shot amplifier model that tracks energy only. On each pass through the gain, the broadband Frantz–Nodvik fluence law is applied to the signal and to one accepted incoherent background channel, and both draw on the same stored energy. The spontaneous background seed added on each pass is scaled by the fraction of stored energy left and is taken from that store. It stands in for ASE; it does not model spatial or spectral transport of spontaneous emission, quantum noise or a pulse-contrast spectrum. Signal and background share one lumped gain in proportion to their energies, and their actual order in time is ignored. All paths start from one prepared store of energy, set in units of the saturation energy, with no refilling, recovery or upper-state decay during the cycle. The geometric multipass path allows at most twelve passes and has no capture gate. The regenerative ring cavity captures a stretched Gaussian seed through an ideal rectangular gate half a nanosecond long and traps it. After the selected pass, an ideal lossless extraction switch sends out all the remaining signal and background. Each round trip of the ring is assumed to pass the gain once, and 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 left out. Normally distributed timing jitter is averaged with 321 fixed samples spanning five standard deviations; these are samples of the simulated response, not 321 hardware shots. The Gaussian peak-power proxy keeps the input pulse duration even when the gain saturates; pulse reshaping, phase, spectral narrowing, diffraction and nonlinear phase are left out. The uniform beam area gives a fluence diagnostic, not a damage prediction. Beam paths, aperture sizes and pulse lengths are schematic. Signal, accepted background, remaining stored energy and accumulated path loss add up to one energy balance. The rejected part of the seed lies outside that balance and is reported separately.

Start with the essentials

Focus question: When should a pulse circulating in a regenerative amplifier be let out?
One-sentence intuition: Another pass through the gain helps only as long as the remaining gain outweighs the loss on that pass.

Core mathematical model

Seed energy captured by the gate

\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}

Only the part of the normalized Gaussian seed that arrives while the ideal gate is open gets in. The geometric paths have no capture gate.

Energy after one saturated pass

\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 integrated Frantz–Nodvik law gives the energy after one pass, and a pass can only take a finite amount from the stored energy. The background shares the lumped gain with the seed.

Every pass drains the same store

\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 taken from the stored energy before it is amplified. The stored energy is not reset between passes.

Loss on every pass

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

Once the stored energy is used up, extra passes only drain the amplified pulse. With any nonzero loss, the best pass for extraction comes after a finite number of passes.

Energy balance for the whole cycle

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

Seed energy rejected at the gate is not part of this balance. The balance says nothing about the efficiency of a real material or whether the optics are safe from damage.

Averaging over timing jitter

\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 averages over normally distributed arrival times, using fixed quadrature points that span five standard deviations. These are model samples, not hardware shots.

Common difficulties

Background is not the useful pulse

Typical misconceptionIf energy comes out, the seed must have been amplified.

Better mental modelLook at the fraction of energy that came from the seed. The background in this model is a simplified stand-in for ASE, not a measured contrast spectrum.

The peak-power proxy does not predict damage

Typical misconceptionThe fixed-shape peak power tells me whether an optic will survive.

Better mental modelThe model leaves out pulse reshaping, beam hot spots, coatings and material thresholds.

Run the experiment

01
Watch the stored energy run down
Choose Regenerative and watch the replay. A complete pass-by-pass cycle is recorded automatically; use Pause replay to inspect the pulse on its route.
What to observe: The signal grows as the stored energy falls. The captured traces come from this same cycle.
02
Find the extraction window
Compare Extract early, Regenerative and Extract late, then drag the extraction dial between them.
What to observe: Too few passes leave gain unused. Too many passes lose energy that was already pulled out of the store.
03
Miss the seed
Choose Background takeover, then open Advanced controls for Miss injection and Timing sensitivity. Inspect the separate timing response below the apparatus.
What to observe: Output that is not zero is not necessarily the amplified seed. When the timing margin is narrow, the output varies strongly in relative terms.