Gain medium
A thin sheet of gain with a Lorentzian line. Put it in a cavity: if its gain beats the round-trip loss the cavity lases, and above threshold it settles at the power where the saturated gain balances the loss, and its output leaves each partly transmitting mirror as a beam.
The model
Small-signal gain
The sheet amplifies the power of light crossing it by its single-pass gain, \(g_0\) at the line centre \(\lambda_c\), falling to 1 off the line with a Lorentzian of width \(\Delta\lambda\):
\[G_0(\lambda) = 1 + \frac{g_0 - 1}{1 + u^2}, \qquad u = \frac{2(\lambda - \lambda_c)}{\Delta\lambda}.\]The pump sets \(g_0\). The sheet multiplies each ray’s field by \(\sqrt{G}\).
In a cavity
The cavity is found from the light’s own path (see Free-space optics: cavities). Its round-trip passive loss is what a pilot beam lost round the loop, less what the gain gave it. The small-signal gain against that loss says whether it lases. A weak seed source at the lasing wavelength, aimed along the cavity, is what lets the cavity be found.
Saturation
Above threshold, each gain sheet saturates as a homogeneous medium does with every pass at once (Rigrod, 1965). Through it \(I_+ I_-\) is constant, so every pass sees the same gain \(G\):
\[\ln G = \ln G_0 - \frac{(G - 1)\,S}{P_\text{sat}}, \qquad P_\text{sat} = \frac{I_\text{sat}\,\pi w^2}{2},\]with \(S\) the power entering it summed over its passes, and \(w\) the cavity mode’s radius at the sheet. Walking the loop with the mirrors’ reflectances gives \(S\) for a power \(X\) on the output coupler, and \(X\) is where the round trip is one. The saturated gain then applies across the whole line, as the log of the small-signal gain scaled down: a homogeneous line.
What leaves each partly transmitting mirror is the power arriving there times its transmission. It leaves as a beam, with the cavity’s own mode.
The saturation intensity is a property of the medium: 2.9 kW/cm² for Nd:YAG, about 10 for Yb:YAG, about 200 for Ti:sapphire. Set it to 0 for gain that does not saturate.
Settings
| Field | Default | Meaning |
|---|---|---|
length | 6 | mm: its aperture across the beam (it is a thin sheet). |
g0 | 1.2 | small-signal single-pass power gain at the line centre (×); the pump sets it. |
isat | 2.9 | kW/cm² saturation intensity (Nd:YAG 2.9, Ti:sapphire 200, Yb:YAG 10); 0: no saturation. Above threshold the cavity settles at the power where the saturated gain balances the loss, and the output leaves each partly transmitting mirror as a beam. |
centre | 1064 | nm line centre. |
fwhm | 30 | nm line width. |
From the tool's own tables, file format version 1: the names are those in a .jekray file.
Assumptions
- The gain is a thin sheet: no thermal lens, no gain guiding, and no spatial hole burning along a standing wave.
- The line is homogeneous, so a laser runs on one line. Inhomogeneous broadening and multi-line competition are not modelled.
- The laser’s output is its steady state: no relaxation oscillations, Q-switching or mode locking from the gain itself.
Related
- Mirror, for the cavity’s mirrors and output coupler.
- Free-space optics: gain.