Optics 095 · Waveguides, structured light, and modern optics

Photonic Nanojet & Hook

How can a tiny transparent particle squeeze light into a very narrow jet, or bend it into a hook? Scattered and incident waves interfere near the back of a transparent particle. Suitable sizes and refractive indices can produce a jet narrower than half the wavelength in air. Different materials in the two halves can bend the bright region into a hook; asymmetry alone does not guarantee one.

Inspect the vector Mie field of a sphere and the outgoing Helmholtz field of uniform and Janus cylinders. Shaded 3D particles connect to their physical field slices. Measure near-field width, inspect phase and Poynting directions, refine the cylinder grid and freeze complex samples for CSV export.

Width at selected section

w_{1/2}/\lambda_0

$0.50$

Exterior peak squared electric amplitude

I_{\rm peak}/I_0

$50\%$

Mie orders or cylinder linear residual

N_{\rm Mie},r_{\rm lin}

$0.00\pi$

Model regime $\text{valid model regime}$

Air wavelength is 532 nm; materials are lossless, nondispersive and nonmagnetic. The sphere uses an exterior vector Mie series on the polarization meridian; its gray interior is not computed. Cylinders use an infinite axial-electric model with finite-grid volume integration and complex-field Richardson extrapolation. Finite cylinder extrusion is schematic. Sampling is nonperturbing. No noise, trapping forces or calibrated experiment is included; near-field width is not an imaging resolution test.

Physics tutorial

Audit the physics of Photonic Nanojet & Hook

BackgroundA transparent particle scatters an incident plane wave. Near its shadow side the fields interfere: $E_z=E_{\rm inc}+E_{\rm scat}$ . A narrow bright region can form outside the surface. Its shape depends on size, polarization and refractive index. Compare a sphere, computed with vector Mie harmonics, with an infinite circular cylinder whose electric polarization is along its axis. The cylinder halves can have different indices.

Why it mattersHow can a tiny transparent particle squeeze light into a very narrow jet, or bend it into a hook?

Start with the essentials

Focus question: How can a tiny transparent particle squeeze light into a very narrow jet, or bend it into a hook?
One-sentence intuition: Intensity and energy flow answer different questions: $\langle\mathbf S\rangle=\frac12\Re(\mathbf E\times\mathbf H^*)$ . Intensity locates bright regions. Time-averaged Poynting vectors give local energy-flow directions. A sequence of transverse maxima is neither a photon trajectory nor necessarily a streamline; its curvature must be inspected over a connected bright region.

Core mathematical model

Maxwell wave equation

\nabla\times\nabla\times\mathbf E=k_0^2n^2\mathbf E

Propagation points right and the displayed transverse direction points up. The sphere is shown on the plane containing its incident electric polarization; its vector Mie solution matches electromagnetic boundary conditions, and the gray interior is not evaluated. The cylinder axis is perpendicular to the slice. Its axial electric polarization reduces Maxwell to a scalar Helmholtz equation with continuous field and normal derivative. These are different geometries and polarizations.

Cylinder outgoing scattering field

\begin{aligned}E&=E_{\rm inc}+k_0^2\int G\chi E\,\mathrm dA\\G(r)&=\frac{i}{4}H_0^{(1)}(k_0r),\quad\chi=n^2-1\end{aligned}

The integral couples all material cells, including repeated scattering. Two-grid Richardson extrapolation combines the complex fields before forming intensity. It accelerates smooth exterior convergence; material discontinuities remain approximate. A small linear-system residual proves only that the discrete equations were solved. Compare sampling densities to assess grid error. The circular-cylinder series provides an independent exterior reference.

Squared amplitude and power flow

\begin{aligned}I/I_0&=\mathbf E^*\cdot\mathbf E/|E_0|^2\\\langle\mathbf S\rangle&=\tfrac12\Re(\mathbf E\times\mathbf H^*)\end{aligned}

The width is measured across the connected peak at half its maximum, with linear interpolation and no background subtraction. It is divided by the wavelength in air. Some settings have no bounded half-maximum width. A subwavelength near-field width does not establish far-field imaging resolution.

Common difficulties

Does a narrow bright spot beat every diffraction limit?

Typical misconceptionA half-wavelength hotspot proves that any two objects at that separation can be resolved.

Better mental modelNo. Local near-field concentration and imaging resolution are different measurements. Resolution depends on the objects, collection optics, transfer of spatial frequencies and noise. This Lab measures fields near dielectric particles.

Run the experiment

01
Scene 1: Mie sphere meridian
Compare the sphere and uniform cylinder. Change the diameter and drag the intensity map to move the gold sampling line. The sphere uses both electric components in its polarization meridian; the cylinder has one axial electric component.
What to observe: Diameter changes the physical outline and the wave solution. The Janus contrast changes material composition, not geometric radius.
02
Scene 2: Uniform cylinder
Choose Janus and reverse the signed half-index contrast. With a horizontal interface the pattern mirrors vertically. Rotate the interface and compare intensity maxima with white Poynting directions.
What to observe: Equal indices restore vertical symmetry when the interface is horizontal. In the index-matched limit the field becomes the incident plane wave.
03
Scene 3: Janus near field
Increase cells per wavelength and compare width and peak intensity at the same coordinates. Freeze one result, change the controls and download its complex-field CSV.
What to observe: The cylinder extrusion illustrates an invariant cross-section; its finite ends are not computed. The sphere is solved separately, with the field sampled on its physical polarization meridian. Neither model includes loss or particle forces.