= Solution
{c}
For a symmetric <confocal optical resonator>, the mirror separation equals either mirror's curvature, $R_2=L_2$, so its symmetric fundamental <Gaussian beam> has
$$
\boxed{z_B=L_2/2.}
$$
The same <optical mode matching> calculation applies, now with $f_{\min}=\sqrt{z_AL_2/2}$. Keeping the example's $L_2=10\,\mathrm m$ gives $z_B=5\,\mathrm m$ and $f_{\min}=0.707107\,\mathrm m$, compared with $0.930605\,\mathrm m$ for the nonconfocal target. With $f=1.5\,\mathrm m$, the two branches become
$$
\boxed{(u,v)\simeq(1.68708,10.85414)\,\mathrm m
\quad\hbox{or}\quad(1.31292,-7.85414)\,\mathrm m.}
$$
Again only the positive-$v$ branch fits the stated downstream geometry. The confocal target's smaller <Rayleigh range> means a smaller fundamental <beam waist> at the same <wavelength> and a less stringent minimum $f$.
A <confocal optical resonator> also has transverse <normal mode> degeneracies: in the ideal paraxial spectrum, increasing transverse order by two shifts the resonance by one longitudinal spacing. Such a cavity can transmit combinations of higher transverse <normal modes> at a fundamental resonance, so transmitted <power> alone is a weaker diagnostic of pure fundamental <optical mode matching>. The degeneracy does not remove the <complex beam parameter> matching requirement if the objective is specifically the symmetric fundamental <Gaussian beam>.
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