Optics 091 · Waveguides, structured light, and modern optics

Self-Healing Beam Arena

How do Bessel and Airy beams rebuild themselves after part of them is blocked? A Bessel beam is built from a cone of plane waves converging on the axis, so when the core is blocked, light arriving from around the obstacle refills it farther on. An Airy beam follows a curved caustic and bends sideways as it travels. Both heal only over a limited distance, and the price is the energy carried in their sidelobes, which feed the main lobe.

Move an opaque disk into finite Bessel and Airy beams, with a Gaussian beam as control, and compute the 2D diffraction of that same field. Read the absolute local recovery, the surviving power that stays constant, the straight-line path of the centroid and frozen complex detector data.

Absolute local intensity agreement

Q_{\rm local}

$0.50$

Remaining integrated power

P_{\rm out}/P_0

$50\%$

Full-field normalized fidelity

F_{\rm full}

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Self-Healing Beam Arena

Opaque disk radius; zero removes it

r_o

75\,\mu\mathrm m

Bessel cone angle; Gaussian core matching

\theta

0.14^{\circ}

Disk horizontal offset from the brightest lobe of the reference

\Delta x_o

0\,\mu\mathrm m

Obstacle position from prepared source plane

z_o

10\,\mathrm{mm}

Detector distance downstream of obstacle

d

75\,\mathrm{mm}

Bessel envelope radius; Airy tail scale

w_{\rm reserve}

300\,\mu\mathrm m

Block the brightest part of a beam. Can the pattern behind the block recover even though the blocked power never comes back? Move an opaque disk through finite Bessel, Airy and Gaussian beams. The shadow comes from propagating the field, and an unblocked copy of each beam serves as the reference.

The calculation is scalar and paraxial at 532 nm, on a finite periodic window, with an ideal source and a thin opaque mask. Brightness is fixed and the 3D axes use unequal scales. Scattering from the mask material is not modeled, no measured device is reproduced, and self-healing is not claimed to be universal.

Physics tutorial

Audit the physics of Self-Healing Beam Arena

BackgroundA self-healing beam does not pass through a mirror or an opaque object. The obstacle really removes part of the complex amplitude: $u_d=\mathcal P_d[(1-\chi_o)\mathcal P_{z_o}u_0]$ . The field that survives then diffracts, and some of its light spreads into the shadow. The finite side lobes act as a reserve; they are not an endless supply of power.

Why it mattersHow do Bessel and Airy beams rebuild themselves after part of them is blocked?

Start with the essentials

Focus question: How do Bessel and Airy beams rebuild themselves after part of them is blocked?
One-sentence intuition: A local bright spot can come back while the total power stays reduced for good: $\int |u_d|^2\,dA=\int |(1-\chi_o)u_o|^2\,dA<P_0$ . Judge the recovery against the unobstructed intensity in absolute terms, not by comparing two pictures that were each brightened separately. The normalized overlap over the whole field cannot quietly climb back to perfect.

Core mathematical model

Paraxial propagation past a real obstacle

i\partial_z u=-\frac{\nabla_\perp^2u}{2k},\quad\widetilde{\mathcal P_z u}=\widetilde u\exp\!\left[-\frac{iz(k_x^2+k_y^2)}{2k}\right],\quad k=\frac{2\pi}{\lambda_0}

The forward carrier is removed, with its positive spatial phase and negative time phase. Internal lengths are in metres, and the axis points from the source to the detector. The disk is a binary amplitude mask at the chosen axial position. The finite numerical window is periodic, not absorbing; warnings about power at the edge and comparisons at different resolutions mark how far the results can be read.

A Bessel beam with a finite reserve

u_0(r)=C J_0(k\sin\theta\,r)e^{-r^2/w^2},\quad z_{\rm geom}\sim\frac{r_o}{\tan\theta},\quad z_{\rm reserve}\sim\frac{w}{\tan\theta}

The cone angle sets the radial wave number. The Gaussian envelope keeps the power finite. The two geometric distances are rough estimates, not the computed recovery or a sharp cutoff, and they are least reliable for an off-center obstacle or an exhausted envelope. The real field is propagated numerically. The Gaussian control starts with roughly the same central half-maximum width, before the Bessel envelope correction, and the same unit input power; its peak brightness is not matched.

The bright lobe curves, but the centroid does not

u_0(x,y)=C\operatorname{Ai}(x/x_0)\operatorname{Ai}(y/x_0)e^{a(x+y)/x_0},\quad a=\frac{x_0}{w_{\rm reserve}},\quad x_c\simeq\frac{z^2}{4k^2x_0^3}

The Airy source is built from its Fourier spectrum, which has a Gaussian amplitude and a cubic phase, with a positive exponential apodization and the same scale in both transverse directions. The parabolic expression describes the ideal caustic, not the exact brightest point for a finite apodization. The plotted peak and the full-field centroid are both sampled from the propagated unobstructed field. A finite beam in free space conserves transverse momentum even while its bright feature bends.

What the recovery scores compare

Q_{\rm local}=\max\!\left(0,1-\frac{\int_{\rm ROI}|I-I_{\rm ref}|\,dA}{\int_{\rm ROI}I_{\rm ref}\,dA}\right),\quad F_{\rm full}=\frac{|\langle u_{\rm ref},u\rangle|^2}{P_{\rm ref}P}

The comparison disk follows the brightest lobe of the reference beam, and its radius is shown below the plot. It compares absolute intensity, without renormalizing to the surviving power. In this lossless propagation model, the full-field fidelity and the surviving power stay constant after the obstacle; for an initially binary mask and the same reference, the two are equal. The square-root contrast of the images is only a display mapping. A captured record stores both complex arrays, not an ideal template.

Common difficulties

Does self-healing restore absorbed light?

Typical misconceptionIf the central spot comes back, the whole beam and its energy must have come back too.

Better mental modelNo. The outer structure that survives interferes inside the shadow and fills it in. Power that was removed stays removed, and the full-field overlap still shows the damage even when a local feature looks familiar.

Run the experiment

01
Scene 1: Bessel shadow recovery
Begin with the Bessel beam. Set the downstream distance to zero to see the central shadow. Then move the detector away and compare the lobe that recovers locally with the remaining-power trace, which stays flat. A disk radius of zero gives the exact unobstructed case.
What to observe: Local diffraction fringes can rise above the reference intensity. The recovery is not monotonic, and it is not guaranteed for every disk, distance or finite envelope.
02
Scene 2: Airy bends without a force
Move the disk sideways or along the bench. Its offset is measured from the brightest lobe of the unobstructed beam at the obstacle plane, and it moves the real obstacle. Then shrink the envelope reserve and watch the recovery fail after a finite distance.
What to observe: The source port delivers an ideal field that is already prepared. The housing, the disk thickness and the transverse magnification are schematic. This is not an axicon design or a calibrated apparatus.
03
Scene 3: Gaussian control experiment
Select Airy, then Gaussian. The brightest Airy lobe follows a curved path, but the centroid of the whole field moves in a straight line. The Gaussian is a control, and it too can diffract light into a shadow. Freeze the detector fields before you change settings.
What to observe: All detector maps, trajectories and frozen records below come from the computed fields. Inside the 3D apparatus, only the outgoing sampling plane carries an image.