Thinking in Waves Episode 35 ／ Part XIII · Frontier physics as waves

Phase was never only about time

Digging further into light ──
when phase becomes a function of angle, and when it becomes an area Make the phase a function of azimuthal angle and $\ell$ can only be an integer ── and that integer survives being moved around (topology).
★★ And take a polarization state once around the sphere ── a phase remains, equal to half the solid angle (agreeing to $10^{-16}$).
★ Phase turns out to have at least three origins: time, angle, and path.

Prerequisites: Episodes 1, 5, 9, 13, 21, 31Computation: kenshou/hikari2.js

From Episode 1 through 34, phase was always a function of time ── $\varphi(t)=\omega t$, with order $\alpha$ meaning "the phase turns by $90\alpha$ degrees".
★ This episode adds two new axes to phase.

01Phase as a function of angle ── a helical wavefront

Laguerre–Gauss ── a field whose phase is proportional to the azimuthal angle $\phi$ $$u(r,\phi)\propto r^{|\ell|}e^{-r^2/w^2}\,e^{i\ell\phi}$$

The wavefront becomes a spiral staircase

$\ell$	Intensity at $r=0$	Peak radius (numeric)	Predicted $w\sqrt{\|\ell\|/2}$
0	1.0000	0 (centre)	──
1	0.0000	0.70710	0.70711
5	0.0000	1.58110	1.58114
10	0.0000	2.23610	2.23607

The heart of §01

★ ① For $\ell\ne0$ the intensity at the centre is exactly zero (the phase is undefined there) ── a doughnut beam, with peak radius $w\sqrt{|\ell|/2}$.
★★ ② $\ell$ can only be an integer ── going once around must return the same value, so $e^{i\ell\cdot2\pi}=1$.
This is the angular version of Episode 5's "boundary conditions make discrete modes". There a transverse wall did it; here it is "come back after one turn".

02The winding number survives being moved

We count the winding number numerically ── go around a closed loop and add up the phase increments ($\frac{1}{2\pi}\oint d\varphi$). The testbed is $u=(x+iy)^2-a^2$ (two vortices at $x=\pm a$).

$a$	Big circle $R=10$	Around $+a$, $R=0.1a$	Around the origin, $R=0.1a$
0.001	2.000000	1.000000	0.000000
0.1	2.000000	1.000000	0.000000
3.0	2.000000	1.000000	0.000000

The heart of §02

The big circle always gives exactly 2 ── move the vortices anywhere and it does not change. A small circle gives 1 if a vortex is inside and 0 if not ── only integers appear.
★★★ The winding number cannot be varied continuously (a topological conserved quantity). Amplitude and phase can be moved as smoothly as you like, but the winding number jumps.

★ This is a different discreteness from Episode 13's discrete scale invariance ── there it was equal spacing on a log axis (a continuous symmetry broken to a discrete one), here it is "return after one turn" (topology). Both are "discrete", but they come from different places.

03★★ Phase as an area ── the geometric phase

Now polarization. A polarization state is a point on the Poincaré sphere ── $|\psi(\theta,\phi)\rangle=(\cos\frac\theta2,\ \sin\frac\theta2 e^{i\phi})$, with the poles circular and the equator linear.

The Pancharatnam phase against the solid angle $$\gamma=-\arg\big[\langle A|B\rangle\langle B|C\rangle\langle C|A\rangle\big]\qquad\text{vs}\qquad -\frac{\Omega}{2}$$

Triangle	$\gamma$ [rad]	$-\Omega/2$ [rad]	Difference
One octant	−0.78539816	−0.78539816	1.1e−16
A small triangle	0.01680306	0.01680306	6.9e−18
A large triangle	−2.13508004	−2.13508004	0
A nearly degenerate one	0.00000022	0.00000022	3.3e−19

The heart of §03

★★★ $\gamma=-\Omega/2$ ── phase has appeared as an area (a solid angle).

★ Episode 1's phase was a delay in time. This phase has nothing to do with time at all ── it is fixed by the shape of the path.

Does refining the path change it? We approximate a circle at colatitude $\theta$ by an $N$-gon.

$\theta$	$N$	Accumulated $\gamma$	$-\pi(1-\cos\theta)$	Difference
1.00	12	−1.41629693	−1.44418290	2.8e−2
1.00	100	−1.44378740	−1.44418290	4.0e−4
1.00	2000	−1.44418191	−1.44418290	9.9e−7

★ It converges to $-\pi(1-\cos\theta)$ ── the same value at any speed and any subdivision. Hence "geometric".

04★ Angle and $\ell$ are a Fourier pair ── but one of them is periodic

If the phase is $e^{i\ell\phi}$, then $\phi$ and $\ell$ are Fourier conjugates. But $\phi$ is periodic ($0$ to $2\pi$) ── so $\ell$ is discrete (an integer). The same structure as Episode 31.

The periodic one	The one that goes discrete	Where it shows up
Sampling time (period $T$)	frequency folds at $2\pi/T$	Episode 31, the unit circle
★ Angle $\phi$ (period $2\pi$)	★ orbital angular momentum $\ell$ is an integer	★ here
A crystal lattice (period $a$)	reciprocal lattice, Brillouin zone	solid-state physics

★ The angular version of the bandwidth theorem $$\ell\le\frac{2\pi r}{\lambda}\qquad\text{── "how many wavelengths fit around the circumference"}$$

Aperture radius $r$	Wavelength $\lambda$	★ Limit on $\ell$
1 mm	1.55 µm	4054	fibre scale
1 cm	1.55 µm	40537	small telescope
1 m	1.55 µm	4.05e+6	large aperture
1 cm	1 cm (30 GHz)	6.28	microwave

The heart of §04

"The number of usable $\ell$ ＝ how many wavelengths fit around the circumference." Episode 9's $\Delta t\cdot\Delta\omega\ge1/2$ has simply become $\Delta\phi\cdot\Delta\ell\ge1/2$. It also sets how many channels can carry information (Episode 21).

05Spin versus orbit, in numbers

Kind	Per photon	Allowed values	What it is
Spin (polarization)	$\pm\hbar$	★ only 2	the direction of the field vector
★ Orbital (OAM)	$\ell\hbar$	★ every integer	the twist of the phase

The heart of §05

Spin offers only two states while orbit has no ceiling (as far as the aperture allows). So orbit is the better place to carry information; $\ell=10010$ has been reported.

★ In wave language ── polarization ＝ which way the field points (an amplitude-side freedom, two of them); OAM ＝ how many times the phase winds (a phase-side freedom, infinitely many).
Episode 32 said "the Higgs is AM, the Goldstone is FM". The same split is at work here.

Exercises

Can a beam with two vortices be deformed continuously into one with a single vortex?
Show the answer
Not without changing the winding number outside. As §02 showed, the winding on a big circle is always 2 however the vortices move.
★ To reduce it you must bring in an opposite vortex ($\ell=-1$) from outside and annihilate ── and that changes the outer winding too.
★★ "A quantity that continuous deformation cannot change" ＝ a topological quantity, which is also why OAM is attractive for communication (mild disturbances do not flip an integer).
The geometric phase is said not to depend on time. So where did the phase accumulated while waiting go?
Show the answer
It is a separate account. The real phase is the sum of a dynamical phase ($-Et/\hbar$, proportional to time) and a geometric phase (fixed by the path alone).
★ §03 measured only the latter, and it was the same whether the loop was traversed slowly or quickly (the dynamical phase would not be).
★ Experiments must separate them ── the standard trick is to compare two paths with the same endpoints.
★★ In this series' language ── the dynamical phase lives on the time axis, the geometric phase in state space. Different spaces, so they add (the habit of Episodes 29–30).
Is §04's $\ell\le2\pi r/\lambda$ the same thing as Episode 9's uncertainty?
Show the answer
The same theorem stated on a different axis. The general Fourier rule $\Delta x\cdot\Delta k\gtrsim1$ with $x\to\phi$ and $k\to\ell$.
★ With one difference ── $\phi$ is periodic, so $\Delta\phi\le2\pi$, and that is exactly what makes $\ell$ discrete. Time and frequency have no such periodicity, so this does not happen there.
★★ Episode 31's "sampling makes frequency fold" was the same duality used in the opposite direction.

Summary　Phase has at least three origins

For 34 episodes phase was always a function of time. This episode added two axes.

★ ① Make phase a function of azimuthal angle (orbital angular momentum). The centre of $u\propto r^{|\ell|}e^{-r^2/w^2}e^{i\ell\phi}$ has exactly zero intensity and the peak sits at $w\sqrt{|\ell|/2}$ (5-digit agreement out to $\ell=10$). And $\ell$ can only be an integer ── the return-after-one-turn condition. The angular version of Episode 5's "boundary conditions make discrete modes".

That integer survives being moved. In $u=(x+iy)^2-a^2$, moving the vortices leaves the winding on a big circle always exactly 2; small circles give only 1 or 0. ★ The winding number cannot vary continuously ── a topological conserved quantity, a different kind of discreteness from Episode 13's.

★★ ② Take a polarization state once around the sphere and a phase remains, equal to half the solid angle.
$\gamma=-\Omega/2$ (agreeing to $10^{-16}$), and an $N$-gon path converges to $-\pi(1-\cos\theta)$.

The same value at any speed and any subdivision ── hence "geometric". Episode 1's phase was a delay in time; this one has nothing to do with time.

The structure came into view too ── angle and $\ell$ are Fourier conjugates, and because angle is periodic, $\ell$ is discrete (the same duality as Episode 31's unit circle). The bandwidth theorem has an angular version, $\ell\le2\pi r/\lambda$ ＝ "how many wavelengths fit around the circumference" (40,000 for a 1 cm aperture at telecom wavelength). And spin offers two states while orbit offers infinitely many ── an amplitude-side freedom versus a phase-side freedom, the same split as Episode 32.

Next episode Having looked at the freedoms of phase, ★ next comes the noise of phase.
★ Light has amplitude noise and phase noise, and their product has a floor (Episode 9's uncertainty).
★★ Which means one can be reduced by pushing it into the other. That is squeezed light, and LIGO actually uses it. Trading AM noise for FM noise ── a continuation of Episode 32.

This document is Episode 35 of the "Thinking in Waves" series. The computations are in kenshou/hikari2.js. On the numbers here: the Laguerre–Gauss intensity and peak radii, the numerical winding numbers, the Pancharatnam phase against the solid angle, and the $N$-gon convergence were all computed for this article. Laguerre–Gauss beams and phase singularities, the $\ell\hbar$ of orbital angular momentum, the Pancharatnam–Berry phase being half the solid angle, and the topological conservation of winding number are all well-known results and not discoveries of this article ── what this article did was place them on a single axis, "where phase comes from", and sort it into time, angle and path. The formula in §01 omits the radial Laguerre polynomial ($p\ne0$), which would add further rings. The bound $\ell\le2\pi r/\lambda$ in §04 is a naive Nyquist estimate for fringes around a circle; real channel capacity depends on aperture shape, propagation distance and turbulence. The $\ell=10010$ figure is a reported value, not verified here. "Two spin states" in §05 is the count for a transverse wave in free space. ── To print, use your browser's Print → Save as PDF.