Round and Round

Lord Rayleigh studied a curiosity of the gallery that runs beneath the dome of St Paul's Cathedral, where a whisper spoken against the wall can be heard clearly on the opposite side, more than a hundred feet away. He showed in 1910 [1] that the sound is not crossing the open space. It follows the curved wall, reflecting at shallow angles the whole way around, and he called these whispering-gallery waves.

The same effect works for light. If light is confined in a small disk of glass, it can circle the rim thousands or millions of times before it leaks out. The corresponding resonances, the whispering-gallery modes, are the highest-quality resonances a dielectric cavity supports, and they are the basis for microlasers, sensitive optical biosensors, and the frequency combs used in optical clocks [2].

A uniform disk is also one of the few cavity shapes whose resonances can be written down exactly, in terms of Bessel functions. Everything in this notebook is that exact solution, computed live. In Part II we deform the disk, the exact solution no longer exists, and a numerical solver takes over.

The modes of the disk

The image below is a whispering-gallery mode of the glass disk. Inside the disk the field is a Bessel mode, matched to an outgoing Hankel wave outside,

$$ \psi(r,\phi) = \begin{cases} J_m(nkr)\,e^{im\phi}, & r < R, \\ \alpha\,H_m^{(1)}(kr)\,e^{im\phi}, & r > R, \end{cases} $$

and the resonant wavenumbers $k$ are the complex roots of the boundary-matching condition

$$ n\,J_m'(nkR)\,H_m^{(1)}(kR) \;-\; J_m(nkR)\,{H_m^{(1)}}'(kR) = 0. $$

Two integers label each mode:

• $m$ is the azimuthal mode number, the number of wavelengths that fit around the rim. A standing-wave mode has $2m$ bright spots around the ring.
• $p$ is the radial mode number, the number of intensity nodes between the caustic and the wall. The $p = 0$ mode is a single bright ring and has the highest quality factor.

Adjust the two sliders and watch the field. The quality factor Q in the title is roughly the number of optical cycles the light survives before it escapes.

Why the light stays near the wall

The field is not spread across the disk. It occupies a thin annulus just inside the boundary. The inner edge of that annulus is a caustic, at radius

$$ r_c = \frac{m}{nk}. $$

Inside the caustic the field is evanescent and decays toward the center, because a ray carrying this much angular momentum cannot reach that region. The intensity rises to its first maximum near the caustic and oscillates out to the wall. Increasing $p$ adds more radial oscillations across the annulus.

The underlying ray

Each mode corresponds to a classical ray. For a whispering-gallery mode the ray travels around the rim, stays tangent to the caustic circle, and strikes the wall at a fixed angle of incidence $\chi$,

$$ \sin\chi = \frac{r_c}{R} = \frac{m}{nkR}. $$

Total internal reflection sets in when $\chi$ exceeds the critical angle,

$$ \sin\chi_c = \frac{1}{n}, $$

which is $\chi_c = 30^\circ$ for a refractive index $n = 2$. Above it, almost no light escapes at each bounce — the disk confines light with no reflective coating at all. The confinement comes from the geometry of the rays, not from mirrors.

How Q scales with m

As $m$ increases the ray strikes the wall at a steeper angle and the mode is confined more tightly, so it leaks less. The quality factor grows roughly exponentially with $m$,

$$ Q \sim e^{\alpha m}, $$

a straight line on a logarithmic scale. This is a characteristic feature of whispering-gallery modes: even a small disk can reach a $Q$ of millions or billions. For a given $m$, the fundamental $p = 0$ mode always has the highest $Q$.

Why the circle is special

All of this rests on one property of the circular disk: it is integrable. Angular momentum is conserved, each ray stays on its own caustic, and the wave equation separates into radial and angular parts that reduce to Bessel functions. That is why the modes can be written down in closed form.

Once the boundary is deformed away from a circle, angular momentum is no longer conserved. Some rays stay regular, but others become chaotic, and the caustics break up. The clean Q ladder no longer holds, and there is no exact solution to fall back on. This is the situation Einstein first identified in 1917 [3], and that A. Douglas Stone later placed at the center of quantum chaos [4]. It is where Part II begins.

Next: Part II, Quantizing Chaos.