Past exam of the laser science and quantum information processing course of the University of Oxford / 2022 / 2

Use the complex beam parameter conventionHere $R_b$ is the wavefront curvature, distinguished from the mirror curvature $R$. In an unfolded propagation coordinate, a concave mirror has the ray transfer matrix $\left(\begin{array}{cc}1& 0\\ -2/R& 1\end{array}\right)$: a ray at height $x$ changes direction by $-2x/R$. The return to the original plane is thereforeThe Gaussian beam transformation $q^{\prime }=(Aq+B)/(Cq+D)$ givesFor $R=2L$, $M=\left(\begin{array}{cc}0& L\\ -1/L& 0\end{array}\right)$ and $q^{\prime }=-L^2/q=i{L}^2/z_R$. The returning Gaussian beam consequently has a beam waist at the same plane, withThis alone does not make it the same Gaussian beam; equal Rayleigh ranges require $z_R=L$.

Adding the plane mirror requires a self-consistent optical resonator normal mode, $q^{\prime }=q=i{z}_R$. Since $A=D$, the fixed-point equation reduces to $C{q}^2=B$, givingFor a finite stable Gaussian beam, $0<L<R$. The same formula remains valid at $R=2L$, where self-consistency selects $z_R=L$. At the geometrical hemispherical limit $R=L$ it predicts $z_R=0$; that marginal boundary has no finite-width stable fundamental Gaussian beam. Thus the source's word “hemispherical” is used for the plano-concave family, including the near-hemispherical case below.

Propagate from one beam waist to the other, using the optical lens ray transfer matrix:Optical mode matching requires $q_A=i{z}_A$ to transform to $q_B=i{z}_B$. Multiplying the denominator across givesEquating real and imaginary parts and multiplying by $f$ yieldsSet $\delta =u-f$. The second equation gives $v=f+(z_B/z_A)\delta$. Substituting this into the first cancels the terms $f\delta$:Consequently the two algebraic branches areEquivalently, $u=f+\delta$. The signs in $u$ and $v$ are the same. Real solutions require $\boxed{f^2\ge z_A{z}_B}$; a converging optical lens therefore needs $f\ge \sqrt{z_A{z}_B}$. At equality $u=v=f$. For the stated arrangement, additionally require $u,v\ge 0$ and that the chosen optical lens position is accessible outside both optical resonators. A negative $v$ is an algebraic virtual-waist solution, not a downstream beam waist.

The source optical resonator has Rayleigh rangeThe 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 $R_2=\ell +z_B^2/\ell$. ThusHere $f^2=2.25{\mathrm{m}}^2>z_A{z}_B=0.866025{\mathrm{m}}^2$, so b gives two real branches:Only the first branch has the target beam waist downstream of the optical lens. It is also geometrically accessible: the optical lens is $u-L_1\simeq 0.62642\mathrm{m}$ beyond the source output mirror, and the target input mirror is $v-L_2/2\simeq 7.44786\mathrm{m}$ beyond the optical lens. The total separation between beam waists is $u+v\simeq 14.07428\mathrm{m}$.

For a symmetric confocal optical resonator, the mirror separation equals either mirror's curvature, $R_2=L_2$, so its symmetric fundamental Gaussian beam hasThe same optical mode matching calculation applies, now with $f_{\min }=\sqrt{z_A{L}_2/2}$. 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 becomeAgain 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.