Transverse optical cavity mode

A transverse optical cavity mode is a resonant field pattern across the direction of propagation in an optical cavity. After propagation and reflection through a round trip, the mode reproduces its transverse shape up to a complex amplitude factor. Longitudinal order counts axial phase cycles; transverse order describes the pattern across the beam. Different transverse patterns can resonate at the same frequency, as in confocal transverse-mode degeneracy.

In an ideal paraxial cavity with suitable spherical mirrors, a Cartesian Hermite-Gaussian optical mode at its waist has field proportional towhere $H_m,H_n$ are Hermite polynomials, $m,n$ are nonnegative integers, and $w_0$ is the beam waist radius. The intensity is proportional to $|u_{mn}{|}^2$, not to the signed field. The primary photonics notes give these transverse solutions, their phase advance and lower-order patterns.^[ref]

For $(m,n)=(0,0)$, the intensity is a single Gaussian beam spot. For $(1,0)$, $H_1(s)=2s$, so intensity is proportional to $x^2{e}^{-2(x^2+y^2)/w_0^2}$. It vanishes on $x=0$ and its two maxima occur at $x=\pm w_0/\sqrt{2}$, $y=0$. The two lobes have opposite field signs even though both have positive intensity. The $(1,1)$ pattern has two crossing nodal lines and four intensity lobes.

For an ideal empty two-mirror cavity of length $L$, set $g_i=1-L/R_i$. On the near-planar stable branch with $g_1,g_2>0$, the resonance frequencies areThe transverse contribution is a Gouy phase correction to the round-trip phase. Other branches require a consistent phase choice; the square root alone should not erase that choice. For an original example, $L=0.10\mathrm{m}$ and $R_1=R_2=0.20\mathrm{m}$ give $g_1=g_2=1/2$ and a one-order transverse spacing $c/(6L)\approx 499.65\mathrm{MHz}$ at fixed $q$. Modes with the same $m+n$ are degenerate in this ideal symmetric model.

A proposed observation scans cavity length while recording both transmission peaks and a camera image of the transmitted beam. Translating or tilting the incident beam changes its overlap with different transverse modes; matching the incident wavefront and width favors the fundamental mode. A transmission peak alone cannot identify the spatial pattern, and an intensity image alone does not recover its phase. Finite apertures, astigmatism and mirror imperfections alter losses and split ideal degeneracies; state the cavity geometry and imaging plane when comparing observations.

A Hermite-Gaussian optical mode is a separable paraxial field pattern whose transverse dependence is a product of Hermite polynomials and a Gaussian beam envelope. In an ideal aligned optical resonator with suitable spherical mirrors, these patterns form a family of transverse optical cavity modes. The two nonnegative integer orders $m,n$ count the straight nodal lines along the chosen transverse axes. The primary photonics notes give the mode fields and their order-dependent phase advance.^[ref]

At a common waist plane, choose a unit-normalized scalar fieldHere $H_j$ uses the physicists' Hermite polynomial convention, with $H_0(s)=1$, $H_1(s)=2s$ and $H_2(s)=4s^2-2$. The normalization follows fromAfter the two coordinate substitutions, $\iint u_{mn}^{\ast }{u}_{pq}dxdy={\delta }_{mp}{\delta }_{nq}$. Thus coefficients in a common-plane expansion can be recovered by inner products; the modes are orthogonal field patterns, not independent bright spots.

For $(m,n)=(1,0)$, intensity is proportional to $x^2{e}^{-2(x^2+y^2)/w_0^2}$, so its maxima lie at $x=\pm w_0/\sqrt{2}$, $y=0$. At $w_0=100\mu \mathrm{m}$, the bright lobes are about $141.4\mu \mathrm{m}$ apart. The field changes sign across $x=0$, although an intensity camera records positive intensity on both sides. For $(2,0)$, the nodal lines instead lie at $x=\pm w_0/2$. These patterns are illustrated in transverse optical cavity mode using individually peak-normalized images; equal peak intensity is different from equal integrated power.

A second-moment width of a Hermite-Gaussian optical mode distinguishes these multi-lobed beams quantitatively. Applying the Hermite polynomial recurrence inside the normalized integrals givesFor order $(1,0)$, the RMS widths are $\sqrt{3}{w}_0/2$ horizontally and $w_0/2$ vertically, despite the same envelope parameter $w_0$. Reporting a fitted single-lobe radius without its definition can obscure this difference.

During propagation the beam radius and wavefront curvature evolve, and the Gouy phase is multiplied by $m+n+1$. Consequently, transverse order contributes to the resonator's frequency condition and to confocal transverse-mode degeneracy. A proposed experiment scans the resonator while imaging a specified plane, then interferes its output with a reference field to test the phase reversal across a node. An intensity fit alone does not establish a pure coherent mode: mixtures, astigmatism, finite apertures and degeneracy can alter what is observed.