Optics 093 · Waveguides, structured light, and modern optics

Optical Vortex & OAM

Can a bright petal pattern carry zero average orbital angular momentum? Yes. Orbital angular momentum is set by how many whole turns the phase winds around the axis, and in which direction. Beams with opposite windings show the same dark-cored ring but carry opposite momentum. Mix them coherently in equal parts and they interfere into bright petals, while their momenta cancel to zero on average. The brightness pattern alone cannot reveal the winding.

Normalized Laguerre–Gaussian beams show their radial rings, the sign of their phase winding and a coherent fork interference pattern. Superpose opposite charges to make petals, read the angular Fourier spectrum, test an ideal sorter with a finite angular window and freeze detector records.

Mean orbital angular momentum per photon

\langle L_z\rangle/\hbar

$0.50$

Gaussian envelope width at detector

w(z)

$50\%$

Sampled integrated signal power

P_{\rm sample}

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Optical Vortex & OAM

First mode signed charge

\ell

3

Gaussian envelope waist; not donut radius

w_0

0.60\,\mathrm{mm}

Radial mode order

p

0

Detector distance from prepared waist

z/z_R

1

Opposite-charge power fraction; ignored at zero charge

q

0

Relative modal phase; not global phase

\delta

0^{\circ}

Reference cycles across detector window

N_f

18

Reference power divided by signal input power

P_{\rm ref}/P_0

0.70

A bright ring alone cannot tell you which way its phase winds. Reverse the topological charge and the intensity stays the same, but the phase winding and the fork pattern flip. Mix opposite charges and you get bright petals that can carry zero average orbital angular momentum. Every view comes from one normalized complex field propagated to the detector.

Light at 633 nm propagates through air in the scalar paraxial approximation. The input beam is ideal, and the adaptive sampling adds no noise. The sorter is a separate, ideal angular transform; no real sorter device is modeled. Phase colors and the unequal display scales are diagnostic aids and do not show photon paths.

Physics tutorial

Audit the physics of Optical Vortex & OAM

BackgroundAn optical vortex is a complex phase that winds around the beam axis, not a spiral path traced by photons. The Lab prepares a normalized field: $u=\sqrt{1-q}\,u_p^{\ell}+\sqrt q\,e^{i\delta}u_p^{-\ell}$ . Opposite charges give the same radial intensity but different phases. A coherent superposition can turn the ring into petals without adding any net orbital angular momentum.

Why it mattersCan a bright petal pattern carry zero average orbital angular momentum?

Start with the essentials

Focus question: Can a bright petal pattern carry zero average orbital angular momentum?
One-sentence intuition: For two distinct modes of opposite charge, $\langle L_z\rangle/\hbar=(1-2q)\ell$ . The Lab computes this mean from the angular Fourier coefficients of the sampled field; it does not report the control setting as a measurement. At zero charge there is only one mode, so the mixing fraction is ignored.

Core mathematical model

Normalized Laguerre–Gaussian propagation

\begin{aligned}u_p^\ell&=\sqrt{\frac{2p!}{\pi(p+|\ell|)!}}\frac{1}{w}\left(\frac{\sqrt2r}{w}\right)^{|\ell|}L_p^{|\ell|}\!\left(\frac{2r^2}{w^2}\right)e^{-r^2/w^2}e^{i\Phi}\\\Phi&=\ell\phi+\frac{kr^2}{2R}-(2p+|\ell|+1)\arctan(z/z_R)\end{aligned}

For the forward-going carrier, the axial phase is positive and the time phase is negative; the carrier is removed from the numerical envelopes. All modes share one uniform linear polarization. The radial polynomial and the Gouy order matter: putting a spiral phase on a Gaussian beam does not by itself make a pure LG mode. The input is prepared ideally, and no phase plate is modeled.

Waist, rings and phase curvature are different quantities

\begin{aligned}z_R&=\pi w_0^2/\lambda,\quad w(z)=w_0\sqrt{1+(z/z_R)^2}\\r_{\rm peak}&=w\sqrt{|\ell|/2}\quad(p=0),\quad\langle r^2\rangle=\tfrac12(2p+|\ell|+1)w^2\end{aligned}

Lengths are in millimeters and the wavelength is 633 nanometers. The ring-peak formula holds only for radial order zero and nonzero charge. At zero charge, the fundamental mode is brightest on the axis. Changing the waist changes the shape of the field and the physical Rayleigh range. The axial direction is drawn in Rayleigh units, so its scale differs from the transverse millimeters.

Interference is complex addition

I_{\rm fork}=|u+v_{\rm ref}|^2,\quad v_{\rm ref}=\sqrt{P_{\rm ref}}\,\frac{\sqrt{2/\pi}}{w}e^{-r^2/w^2}e^{i2\pi N_fx/(12w)}

The reference beam is defined at the detector, with a Gaussian envelope and a set transverse tilt. Its power is given relative to a signal input of unit power. The total integrated power includes a coherent overlap term, so it need not equal the sum of the separate powers. The reference is specified ideally at the detector; no interferometer that would prepare it is traced.

An OAM spectrum is not a bright-ring count

c_m(r)=\frac{1}{2\pi}\int_{-\pi}^{\pi}u(r,\phi)e^{-im\phi}\,d\phi,\quad P_m=2\pi\int_0^\infty|c_m(r)|^2r\,dr

The spectrum shown is the radial integral of the angular coefficients of the field. For a pure mode, the mean equals the charge. For a superposition, it is a power-weighted mean, and a single overall topological charge may not be defined. The radial index changes the rings and the spreading, but not the OAM eigenvalue of a pure mode. Spin and total angular momentum lie outside this scalar model.

A finite angular aperture gives sinc overlap

\widetilde u(r,s)=\sum_m c_m(r)\operatorname{sinc}(s-m),\quad J(s)=2\pi\int|\widetilde u(r,s)|^2r\,dr

An ideal angular unwrapper turns the winding phase into a straight phase gradient, and a Fourier lens turns that gradient into a position. One angular cycle has finite width, so each pure mode gives a sinc-squared response. Cross terms are kept for coherent superpositions. The plot covers a finite range, and the tails outside it are not renormalized back in. The model has no fan-out, no real mask aberrations and no experimental data, and it makes no claim about sorting efficiency.

Common difficulties

Does a donut prove orbital angular momentum?

Typical misconceptionAny dark-centered ring has a unique nonzero orbital angular momentum.

Better mental modelNo. Intensity throws away the phase. Pure modes with opposite charges have the same intensity, and a coherent mixture can have zero mean OAM. You need the complex field or a suitable phase-sensitive measurement; the shape alone is not enough.

Run the experiment

01
Scene 1: A normalized vortex field
Reverse the signed charge. Compare the intensity, which stays the same, with the phase winding and fork direction, which reverse. Then set the charge and the radial order to zero: the center should turn bright.
What to observe: Intensity alone cannot tell opposite charges apart. Interference with a reference and the phase diagnostics answer different questions.
02
Scene 2: Read the fork with a reference
Change the envelope waist, radial order and detector distance. The beam volume, the physical sampling plane and the data all change together. Note that the waist is not the radius of the bright ring.
What to observe: A bright, structured field can have zero mean OAM. Petals do not mean the beam is a single OAM eigenmode.
03
Scene 3: Opposite charges and ideal sorting
Choose opposite charges and ideal sorting, then change the mixture and the relative phase. Rotate the petals while the OAM weights stay fixed, and look at the overlapping spots from the sorter. Freeze a detector frame before you change the controls again.
What to observe: The 3D volume colored by phase is a diagnostic view of the field. Animating the global phase does not change the intensity, the OAM weights or any recorded frame.