= Solution
{c}
The source <optical resonator> has <Rayleigh range>
$$
z_A=\sqrt{L_1(R_1-L_1)}=\sqrt{1\times0.010}\,\mathrm m=0.100\,\mathrm m.
$$
The symmetric target <optical resonator> has its fundamental <Gaussian beam> <beam waist> halfway between the mirrors. At distance $\ell=L_2/2$ from that <beam waist>, <wavefront> curvature is $R_2=\ell+z_B^2/\ell$. Thus
$$
z_B=\sqrt{\frac{L_2}{2}\left(R_2-\frac{L_2}{2}\right)}
=\sqrt{75}\,\mathrm m\simeq8.66025\,\mathrm m.
$$
Here $f^2=2.25\,\mathrm{m^2}>z_Az_B=0.866025\,\mathrm{m^2}$, so <2/b> gives two real branches:
$$
\boxed{(u,v)\simeq(1.62642,12.44786)\,\mathrm m
\quad\hbox{or}\quad(1.37358,-9.44786)\,\mathrm m.}
$$
Only the first branch has the target <beam waist> downstream of the <optical lens>. It is also geometrically accessible: the <optical lens> is $u-L_1\simeq0.62642\,\mathrm m$ beyond the source output mirror, and the target input mirror is $v-L_2/2\simeq7.44786\,\mathrm m$ beyond the <optical lens>. The total separation between <beam waists> is $u+v\simeq14.07428\,\mathrm m$.
Back to article page
