Resonators and Gaussian Beams

This is the highest-value calculation page for a laser-design interview. Draw the cavity, define the sign convention, multiply the matrices, check stability, solve the eigenmode, and then test apertures, gain overlap, thermal-lens range, and alignment sensitivity.

ABCD method

For paraxial propagation in air and a thin lens,

\[\begin{split}T(d)=\begin{bmatrix}1&d\\0&1\end{bmatrix}, \qquad L(f)=\begin{bmatrix}1&0\\-1/f&1\end{bmatrix}.\end{split}\]

A spherical mirror of radius \(R\) acts like a thin lens of focal length \(R/2\) on reflection,

\[\begin{split}M_R=\begin{bmatrix}1&0\\-2/R&1\end{bmatrix}.\end{split}\]

Use the convention of the chosen ray vector consistently. Reduced-angle vectors change the translation and refraction matrices inside a dielectric.

For a round-trip matrix

\[\begin{split}M_{\rm rt}=\begin{bmatrix}A&B\\C&D\end{bmatrix},\end{split}\]

the paraxial stability condition is

(1)\[\left|\frac{A+D}{2}\right|<1.\]

Equality is a stability boundary, not a comfortable design point. Manufacturing tolerances, thermal lensing, mirror motion, and refractive-index drift can push a nominally marginal cavity unstable.

Two-mirror shortcut

For two mirrors separated by \(L\), define

(2)\[g_1=1-\frac{L}{R_1}, \qquad g_2=1-\frac{L}{R_2}.\]

The cavity is stable when

\[0<g_1g_2<1.\]

This shortcut is excellent for intuition but does not replace the full ABCD model when a gain rod, thermal lens, Brewster plate, telescope, nonlinear crystal, or nonuniform index lies inside the resonator.

Complex beam parameter

Define

(3)\[\frac1{q(z)}=\frac1{R(z)}-i\frac{\lambda}{\pi w^2(z)}, \qquad q_2=\frac{Aq_1+B}{Cq_1+D}.\]

The resonator eigenmode reproduces itself after a round trip:

\[q=\frac{Aq+B}{Cq+D},\]

so

\[Cq^2+(D-A)q-B=0.\]

Choose the root with a physically positive beam-radius solution. Once \(q\) is known at one plane, propagate it to every optic and record \(w\), \(R\), and Gouy phase.

For an ideal Gaussian beam,

(4)\[z_R=\frac{\pi w_0^2}{\lambda}, \qquad w(z)=w_0\sqrt{1+(z/z_R)^2}, \qquad \theta=\frac{\lambda}{\pi w_0}.\]

For a real beam, a convenient second-moment model is

(5)\[w^2(z)=w_0^2+ \left[\frac{M^2\lambda(z-z_0)}{\pi w_0}\right]^2, \qquad \theta=\frac{M^2\lambda}{\pi w_0}.\]

State the beam-radius convention. A \(1/e^2\) intensity radius, a D4σ second-moment diameter, and a camera threshold diameter are not interchangeable.

Mode spacing and transverse modes

For an empty two-mirror cavity, one useful mode-frequency form is

(6)\[\nu_{qmn}=\frac{c}{2L} \left[ q+\frac{m+n+1}{\pi}\cos^{-1}\!\left(\sqrt{g_1g_2}\right) \right].\]

The first term is the longitudinal comb; the Gouy-phase term separates transverse families. Degeneracies become likely in symmetric or special geometries and can make transverse-mode control harder.

Design margins that matter

Check	Calculation	Engineering interpretation
Thermal stability	Sweep the thermal-lens dioptric power through cold, nominal, and hot states	The entire operating range should remain stable with useful mode-size margins
Aperture clearance	Compare every clear radius with local \(w\) or measured second-moment radius	A common first pass is several beam radii, then refine from permitted clipping loss
Gain overlap	Compare cavity intensity with absorbed-pump distribution throughout the medium	Oversized cavity modes waste inversion; undersized modes raise intensity and thermal sensitivity
Coating loading	Convert circulating power to peak irradiance on each optic	Account for angle, polarization, standing-wave enhancement, pulse shape, and hot spots
Alignment	Perturb each mirror in the ray/misalignment model	Near-boundary and high-magnification cavities can be extremely sensitive
Tolerance	Monte Carlo radii, spacing, focal power, decenter, tilt, and index	Report yield or worst credible margin, not only the nominal solution

Thermal lens as a design variable

Model the pumped gain element initially as a thin lens \(f_{\rm th}\) at its principal plane, but treat that as a range, not a fixed catalog value:

\[\Phi_{\rm th}=\frac1{f_{\rm th}} =\Phi_{dn/dT}+\Phi_{\rm bulge}+\Phi_{\rm photoelastic}+\cdots.\]

Its power depends on absorbed pump, pump radius and distribution, cooling boundary conditions, geometry, material properties, and polarization. Higher order aberration and stress birefringence are not captured by a perfect thin lens.

A credible resonator answer therefore says:

1. how the cold cavity was chosen;

2. what thermal-lens interval is expected;

3. how eigenmode radii move over that interval;

4. where stability boundaries lie;

5. which optic or aperture becomes limiting; and

6. how the prediction will be measured and updated.

Common interview traps

• Stable does not mean robust; it only passes the ideal eigenmode criterion.

• A small waist improves nominal focusability but raises divergence and local irradiance.

• \(M^2\) is not a fixed spot-size multiplier at every plane; propagate the second-moment beam consistently.

• Cavity mode matching concerns a resonator eigenmode; external pump focusing is a separate overlap problem.

• A beam that looks circular on one camera plane may still be astigmatic.

• Clipping can make a camera fit look deceptively clean while power and diffraction loss degrade.