Optics 063 · Diffraction, Fourier optics, and computational imaging

Abbe Resolution Lab

An Abbe–Porter order-selection bench: recover a hidden grating by oblique illumination, open a real pupil and expose false half-pitch detail.

Axial / current unmasked pitch support

p_{\rm axial},p_{\rm support}

$0.50$

Transmitted orders / unit-cell throughput

N_{\rm pass},T_{\rm cell}

$50\%$

Normal-cut contrast / true-period amplitude

C_{\rm cut},A_{1/p}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Abbe Resolution Lab

Effective objective NA (air)

\mathrm{NA}

0.45

Full grating pitch (not half-pitch)

p

800\,\mathrm{nm}

Coherent monochromatic wavelength

\lambda

550\,\mathrm{nm}

Horizontal incidence sine / objective NA

\beta

0

Grating normal angle from horizontal

\alpha

0^{\circ}

Object index (named buttons below)

o

0

Pupil mask index (named buttons below)

h

0

Binary opening width / pitch

D

0.50

Specimen normal translation / pitch

\delta/p

0.20

Linear display gain (not raw data)

G

1

First the stripes disappear: only one order enters the pupil. In the second preset, tilt the illumination without widening the aperture—the zero order and one neighbour now interfere and reveal the original period. In the third, blocking zero makes the pattern look twice as fine, but its true-period signal vanishes: high contrast does not mean faithful resolution. Drag the zero-order spot horizontally in the computed pupil to steer illumination; rotate a crossed grid through the horizontal slit to erase one direction. The physical iris, order support, raw line cut and receiving image all use the same coherent field. Enlarged records remain outside the 3D apparatus.

Ideal scalar coherent infinite-periodic specimen and sine pupil in air, with schematic unit-magnification 4f hardware. Exact accepted harmonic sum; hard order thresholds, point samples, unequal scene scales. No finite-edge, vector, aberration or camera-noise model; not a universal two-point criterion or ptychographic reconstruction.

Physics tutorial

Audit the physics of Abbe Resolution Lab

BackgroundA periodic specimen diffracts a coherent plane wave into discrete orders. An order fits the circular pupil when $\left\lVert\boldsymbol s_i+\lambda\boldsymbol\nu_{mn}\right\rVert\le\mathrm{NA}$ . Tilting the source shifts all orders together; the spacing between them is set by the full specimen pitch.

Why it mattersWhy can tilting the illumination reveal invisible stripes—and why can blocking one order fake twice the detail?

Start with the essentials

Focus question: Why can tilting the illumination reveal invisible stripes—and why can blocking one order fake twice the detail?
One-sentence intuition: The image is an interference measurement, not a blur painted onto a reference: $I_{\rm img}=\left|\sum c_{mn}P_{mn}e^{-2\pi i(\boldsymbol\nu_{mn}\cdot\boldsymbol r+m\delta/p)}\right|^2$ . One surviving order gives constant intensity. Two neighbouring orders can encode the period, but two opposite orders without zero encode half the period.

Core mathematical model

Frequency and physical pupil coordinates

\begin{aligned}\boldsymbol\nu_{mn}&=\frac1p\mathsf R_\alpha(m,n)\\\boldsymbol s_i&=(\beta\mathrm{NA},0)\\\boldsymbol r_P&=f(\boldsymbol s_i+\lambda\boldsymbol\nu_{mn})\\R_P&=f\mathrm{NA}\end{aligned}

Coordinates are transverse horizontal and downward vertical. The rotation angle turns the first grating normal toward downward vertical. Wavelength and pitch use micrometres; pupil positions use millimetres. The ideal sine-pupil map is not a high-NA vector lens prescription.

Axial versus aligned oblique pitch

\begin{aligned}p_{\rm axial}&=\frac\lambda{\mathrm{NA}}\\p_{\rm aligned}&=\frac\lambda{\mathrm{NA}+|\sin\theta_i|}\\p_{\rm edge}&\to\frac\lambda{2\mathrm{NA}}\end{aligned}

These are full grating pitches, not half-pitches or Rayleigh point separation. The last limit needs the illumination at the pupil edge and aligned with the grating frequency. The live support readout also accounts for grating rotation and ignores any extra pupil mask.

False half-pitch without new information

\begin{aligned}t(x)&=\tfrac12+\tfrac12\cos(2\pi x/p)\\U_{\rm no\,DC}&=\tfrac12\cos(2\pi x/p)\\I_{\rm no\,DC}&=\tfrac18[1+\cos(4\pi x/p)]\end{aligned}

For aligned smooth stripes with both first orders collected, blocking zero removes the fundamental intensity frequency. Its line-cut contrast can still be one. The simulation retains this artifact instead of replacing it with a sharp specimen image.

Unit-cell intensity and detector display

\begin{aligned}\langle I_{\rm out}\rangle&=\sum|c_{mn}P_{mn}|^2\\T_{\rm cell}&=\frac{\langle I_{\rm out}\rangle}{\langle|t|^2\rangle}\\B&=\min(1,G I/I_{\rm inc})\end{aligned}

Orthogonality gives scalar mean-square transmission over a full periodic cell, not total power over the displayed crop or calibrated oblique flux. The 192-square receiving image is point sampled over eight micrometres. The raw 512-point cut spans one period along the first normal; its vertical plot range is labelled and may rescale, but its data never include display gain.

Common difficulties

A sharper-looking pattern can be the wrong pattern

Typical misconceptionHigher contrast or twice as many stripes proves improved resolution.

Better mental modelRemoving zero from smooth stripes leaves opposite first orders. Their intensity has half the original pitch while the original-frequency amplitude is zero. This is selective-order interference, not new specimen information. Inspect the labelled original-frequency amplitude rather than contrast alone.

Run the experiment

01
Scene 1: Axial: invisible stripes
In axial mode, open the pupil slowly until the first pair enters. The hard transition belongs to this infinite-periodic idealization.
What to observe: Changing wavelength moves order positions while the physical NA stop stays fixed. Changing NA changes the physical aperture radius; fixed normalized tilt also moves the source.
02
Scene 2: Tilt: recover the period
In the tilt preset, drag the pupil zero order horizontally. Then rotate the grating: the same tilt no longer helps a perpendicular frequency equally.
What to observe: Binary openings have many harmonics. Collecting more can sharpen edges, but ringing and false detail remain physical consequences of selection.
03
Scene 3: Block zero: false half-pitch
Compare open, zero-only and zero-blocked pupils. Try crossed grids with the horizontal slit and compare the raw normal-direction cut.
What to observe: Source tilt recovers selected directional information in one exposure. It does not synthesize an isotropic high-resolution image or recover phase from intensity.