Optics 096 · Waveguides, structured light, and modern optics

Superoscillation Designer

How can band-limited waves add up to a hotspot smaller than the diffraction limit? Carefully chosen amplitudes and phases let band-limited waves oscillate locally faster than any component. Their central lobe can be narrower than a conventional focus with the same bandwidth. Strong exterior sidebands hold most of the energy; the narrow spot alone does not establish an imaging resolution.

Design a band-limited TE field with a minimum-norm angular spectrum. Explore its 3D propagation slice, move the sampling plane, compare a uniform-spectrum reference and measure width, sidebands, per-period energy fraction, passive transmission and usable field of view.

Central half-maximum width

w_{1/2}/\lambda_0

$0.50$

Central-lobe electromagnetic energy

\eta_u

$50\%$

Exterior maximum / central intensity

I_{\rm side}/I_c

$0.00\pi$

Model regime $\text{valid model regime}$

Ideal coherent, y-invariant air field at 532 nm; exact propagating-wave TE synthesis on a transverse period. The complex-amplitude channel filter is not a binary mask or fabricated lens. Electric intensity, electromagnetic energy and axial power are separate quantities. The uniform-spectrum line is a reference, not an imaging resolution bound. Finite frames and vertical thickness are schematic; oscillation is slowed. No evanescent waves, noise, material scattering, object reconstruction or experimental calibration.

Physics tutorial

Audit the physics of Superoscillation Designer

BackgroundInterference can squeeze a small bright lobe between two closely spaced zeros, even when every incident wave obeys $|k_x|\le k_0\mathrm{NA}$ . Here the field is periodic across the beam and constant vertically. An ideal angular-channel filter sets the complex coefficients; all waves propagate forward in air. The design plane lies at the center of the propagation section.

Why it mattersHow can band-limited waves add up to a hotspot smaller than the diffraction limit?

Start with the essentials

Focus question: How can band-limited waves add up to a hotspot smaller than the diffraction limit?
One-sentence intuition: At the error-free design plane the two central zeros satisfy $2b<\pi/k_{x,\max}$ . They are closer than the half-period of the highest transverse spatial frequency. The coefficients still contain no higher frequency. Bright exterior sidebands and a small central energy fraction accompany the compressed lobe.

Core mathematical model

Forward TE angular spectrum

\begin{aligned}E_y&=\sum_{m=-M}^{M}c_m e^{i(k_{xm}x+k_{zm}z)}\\k_{xm}&=2\pi m/L,\quad L=M\lambda_0/\mathrm{NA}\\k_{zm}&=\sqrt{k_0^2-k_{xm}^2}>0\end{aligned}

Propagation is along positive axial distance, the transverse coordinate runs across the frames, and the electric field points vertically. Lengths use the air wavelength. Magnetic components follow from each wavevector crossed with the electric field. The square root is exact, so this TE construction does not need a paraxial approximation.

Minimum-norm zero placement

\begin{gathered}\min\sum_m|a_m|^2,\quad E(0,0)=1\\E(\pm b,0)=0,\quad b=\frac{\lambda_0}{4a\,\mathrm{NA}}\\c_m=a_m/\max_j|a_j|\end{gathered}

Real symmetric coefficients satisfy two independent linear constraints. Orthogonal projection gives the smallest squared coefficient norm for this finite mode set; it does not optimize a binary mask or a global imaging criterion. Scaling makes every channel passive. The compression parameter places zeros; it is not a measured local phase gradient. The uniform reference sets every transmission coefficient to one.

Energy concentration and power transmission

\begin{aligned}\eta_u&=\frac{\int_{-x_{\min}}^{x_{\min}}u\,\mathrm dx}{\int_{-L/2}^{L/2}u\,\mathrm dx}\\u&=\tfrac12(|E_y|^2+|\widetilde H_x|^2+|\widetilde H_z|^2)\\T_P&=\frac{\sum_m|c_m|^2k_{zm}}{\sum_m k_{zm}}\end{aligned}

The magnetic field is impedance-normalized. Common dimensional energy factors cancel in the fraction. Central-lobe limits are the first intensity minima, not half-maximum points. Transmission instead counts axial Poynting power with equal incident electric amplitude in each channel. One-period integration makes both definitions finite; these periodically repeated plane waves are not a finite-energy isolated beam.

Common difficulties

Does a narrow illumination lobe resolve every nearby object?

Typical misconceptionA narrow central intensity peak proves unlimited imaging resolution.

Better mental modelNo. Objects outside the small useful window also receive the bright sidebands. Object structure, collected spatial frequencies, detector response and noise determine imaging performance. This Lab designs illumination, not an object reconstruction.

Run the experiment

01
Scene 1: Narrow central lobe
Compare the narrow-lobe preset with the uniform reference. The central plot normalizes each curve to its own center; check the absolute center ratio in the energy panel.
What to observe: No spectral bar moves beyond the chosen angular cutoff when the central zeros move closer. The coefficient signs and amplitudes change.
02
Scene 2: Uniform-spectrum reference
Choose the smaller-spot preset, then increase the zero compression. Inspect the whole-period sideband plot and the electromagnetic energy fraction together.
What to observe: At the design plane, tighter zeros increase exterior contrast and reduce the central-lobe energy fraction. Passive transmission and concentration are different efficiencies.
03
Scene 3: Smaller spot, stronger sidebands
Drag the gold sampling frame away from the design plane, or add alternating phase error. Download the current coefficients and complex field to reproduce the profiles.
What to observe: Defocus and phase errors change the width and fill the dark minima. Width is assigned only when the axis is a local maximum; every computed field remains available for export.