A laser begins oscillating only when the amplification provided by the gain medium is large enough to overcome all cavity losses.
This condition is called the threshold condition.
Below threshold, spontaneous emission may still exist inside the cavity, but it is not amplified strongly enough to produce sustained laser action.
Above threshold, stimulated emission dominates and the cavity field can build up rapidly.
In a simple one-dimensional model, the threshold gain is obtained by balancing the distributed internal loss against the loss associated with imperfect mirror reflectivity.
Using the formula specified in this calculator, the threshold gain is
\[
g_{\text{th}}=\alpha+\frac{1}{L}\ln\!\left(\frac{1}{R}\right),
\]
where \(L\) is the cavity length in centimeters, \(\alpha\) is the distributed loss coefficient in \(\text{cm}^{-1}\), and \(R\) is the mirror reflectivity parameter used in the simplified model.
The first term represents absorption, scattering, and other distributed losses inside the resonator.
The logarithmic term represents loss caused by incomplete reflection at the cavity mirrors.
The most important physical idea is that the gain medium must provide at least this amount of gain for laser oscillation to start.
If the available gain is smaller than \(g_{\text{th}}\), the intracavity field decays instead of growing.
If the available gain is larger, the field can increase from noise or spontaneous-emission seeds.
To connect threshold to population inversion, this calculator uses the small-signal relation
\[
g_0=\sigma_e N,
\]
where \(\sigma_e\) is the emission cross section and \(N\) is the inversion density.
The threshold inversion density is therefore
\[
N_{\text{th}}=\frac{g_{\text{th}}}{\sigma_e}.
\]
This is a very convenient engineering form because it tells you directly how much inversion is required before the laser can switch on.
Once \(N\) exceeds \(N_{\text{th}}\), the device is above threshold.
For the sample values in the prompt,
\(L=50\,\text{cm}\),
\(R=0.98\),
and
\(\alpha=0.01\,\text{cm}^{-1}\),
the mirror-loss term is
\[
\frac{1}{L}\ln\!\left(\frac{1}{R}\right)
=
\frac{1}{50}\ln\!\left(\frac{1}{0.98}\right)
\approx 4.04\times10^{-4}\,\text{cm}^{-1}.
\]
So the threshold gain becomes
\[
g_{\text{th}}
\approx
0.01+4.04\times10^{-4}
=
0.0104\,\text{cm}^{-1}.
\]
This means the numerical result implied by the stated formula is about
\(0.0104\,\text{cm}^{-1}\),
not
\(0.04\,\text{cm}^{-1}\).
That difference is significant, because the threshold inversion density depends directly on this value.
Above threshold, real lasers do not let the gain increase without limit.
Instead, gain saturation and carrier depletion clamp the effective gain near threshold while the excess pump power is converted into useful output.
This calculator therefore reports a normalized above-threshold output estimate using a slope-efficiency factor rather than pretending to provide a complete absolute power model.
It is meant to show the turn-on behavior clearly:
once the pump ratio
\(r=N/N_{\text{th}}\)
exceeds 1, output begins to rise.
A helpful way to visualize this is with a gain-versus-loss plot.
The gain medium produces a linearly increasing small-signal gain as inversion increases,
while the cavity loss remains fixed at the threshold-loss level.
The intersection of those two defines the onset of lasing.
After crossing threshold, the startup transient can be very fast because stimulated emission amplifies the intracavity field exponentially until nonlinear saturation becomes important.
At a more advanced university level, one includes unequal mirror reflectivities, round-trip gain,
gain saturation,
relaxation oscillations,
spatial hole burning,
rate equations,
and output coupling in absolute power units.
One can also derive threshold conditions from the full round-trip field-amplification requirement.
But even in that more detailed theory, the central message remains unchanged:
lasing begins only when gain balances or exceeds cavity loss.
\[
g_{\text{th}}=\alpha+\frac{1}{L}\ln\!\left(\frac{1}{R}\right),
\qquad
g_0=\sigma_e N,
\qquad
N_{\text{th}}=\frac{g_{\text{th}}}{\sigma_e}.
\]
These formulas are the core of the threshold problem and explain how cavity design, mirror reflectivity, and inversion density determine whether a laser will start or remain below threshold.
