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The complex beam parameter combines a Gaussian beam's wavefront curvature $R_b$ and spot radius $w$: $1/q=1/R_b-i\lambda /(\pi w^2)$. With $q=z+i{z}_R$, the beam waist is $q=i{z}_R$. A real paraxial ray transfer matrix maps it by $q^{\prime }=(Aq+B)/(Cq+D)$. Positive imaginary part selects the physical finite-width branch in this convention.

Optical mode matching makes the spatial field of an incident beam equal to a desired optical cavity normal mode. For fundamental circular Gaussian beams, matching the complex beam parameter fixes the beam waist size and location; the optical axis, polarization and frequency must also match. A single converging optical lens connects waist Rayleigh ranges $z_A,z_B$ only when $f^2\ge z_A{z}_B$. With $s=\sqrt{f^2/(z_A{z}_B)-1}$, waist-to-lens distances obey $u=f\pm z_{A}s$, $v=f\pm z_{B}s$ with correlated signs, followed by physical distance constraints.

For a paraxial Gaussian beam, choose the complex beam parameter withHere $R_b$ is the wavefront radius and $w$ the intensity $1/e^2$ radius. The ABCD law isFor a real, lossless round-trip ray transfer matrix $M$ in free space, $AD-BC=1$. A reproducing cavity mode requires $q=(Aq+B)/(Cq+D)$, henceExcept for a scalar identity matrix, a finite physical solution requires a complex-conjugate root pair, one with positive imaginary part. The generic strict stability condition is thereforeEquivalently, the ray eigenvalues are distinct unit-modulus numbers $e^{\pm i\mu }$, so repeated round trips do not cause unbounded ray excursions. Adequate mirror apertures and longitudinal resonance are also needed for a low-loss resonant mode.

The strict inequality has a degenerate exception relevant to this paper. If $M=-I$, then $q^{\prime }=q$ for every physical $q$ even though $A+D=-2$. A symmetric confocal resonator has exactly this matrix and supports Gaussian modes; it is not excluded as a physical cavity. Thus the printed strict condition should be understood as the generic, nondegenerate stability criterion, not a universal necessary condition covering the confocal case in c.

Put $s=z-z_0$. In the source's complex beam parameter convention,Substitution separates the transverse wave amplitude and phase:Thus the Gaussian beam is circularly symmetric, has a Gaussian function transverse light intensity, and narrows to a beam waist at $z_0$ before spreading by diffraction. Its wavefront is planar at the beam waist, converging before it and diverging after it. The quadratic transverse phase identifies the signed radius of curvature:Define spot size as the wave amplitude $1/e$ radius, equivalently the light intensity $1/e^2$ radius. ThenThe Rayleigh range is where $w=\sqrt{2}{w}_0$; the far-field beam divergence half-angle is $\lambda /(\pi w_0)$. The factor $1/q$ also produces a longitudinal phase shift relative to a plane wave, changing by $\pi$ across the beam waist; its sign here follows the given $e^{i(kz-\omega t)}$ convention. The on-axis light intensity decreases as $1/(s^2+z_R^2)$ while transverse integration leaves the power constant. These are scalar paraxial approximation results.

A thin lens has ray transfer matrixThe complex beam parameter transformation gives $q_2=q_1/(1-q_1/f)$, henceThe sign is also obtained by multiplying the incident wave amplitude by the thin lens phase $\exp [-ik{\rho }^2/(2f)]$.

Choose the thin lens plane as $z=0$ and put $a=\pi w_0^2/\lambda$, the incident Rayleigh range. Since the incident beam waist lies there, $q_1=-ia$. Immediately after the thin lens,Subsequent free propagation adds $z$, so write $q(z)=z-z_f-i{z}_R^{\prime }$. Comparing real and imaginary parts givesFor a positive focal length, the new beam waist lies downstream but closer than the geometrical focus. At fixed $f,\lambda$, as $w_0\to \infty$,This is the formal unbounded-aperture scalar limit; eventually finite aperture and the paraxial approximation limit the extrapolation.

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.

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.