Optics 063 · Diffraction, Fourier optics, and computational imaging

Abbe Resolution Lab

Why can tilting the light reveal stripes a microscope misses, and why can blocking some light fake finer ones? Fine stripes diffract light steeply, and the objective images them only if at least two orders enter its pupil and interfere. Tilted illumination shifts the orders sideways so the zero order and one first order both get in, and the stripes reappear. Keep only the two first orders and block the zero order, and they make stripes at half the real period: detail the specimen does not have.

Select diffraction orders on an Abbe–Porter bench. Recover a hidden grating with oblique illumination, open the pupil and see how false half-pitch detail appears.

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 $\text{valid model regime}$

Model checked against analytic results

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 (names on the buttons below)

o

0

Pupil mask index (names on the buttons below)

h

0

Binary opening width / pitch

D

0.50

Specimen normal translation / pitch

\delta/p

0.20

Linear display gain (raw data unchanged)

G

1

In the first preset the stripes disappear, because only one diffraction order enters the pupil. In the second, tilt the illumination without widening the aperture: the zero order and one neighbouring order now interfere and bring back the original period. In the third, blocking the zero order makes the pattern look twice as fine, but the signal at the true period vanishes. High contrast does not mean faithful resolution. Drag the zero-order spot sideways in the computed pupil to steer the illumination, or rotate a crossed grid behind the horizontal slit to erase one direction. The iris, the passed orders, the raw line cut and the image all come from the same coherent field. Enlarged records stay outside the 3D apparatus.

The specimen is ideal, scalar, coherent and infinitely periodic, imaged in air through a sine-coordinate pupil; the unit-magnification 4f hardware is schematic. The image is the exact sum of the accepted harmonics. Orders pass or are blocked at hard thresholds, values are point samples, and the scene uses unequal scales. Finite edges, vector effects, aberrations and camera noise are not modeled. The result is not a universal two-point criterion and not a ptychographic reconstruction.

Physics tutorial

Audit the physics of Abbe Resolution Lab

BackgroundA periodic specimen splits a coherent plane wave into separate diffraction orders. An order passes through the circular pupil when $\left\lVert\boldsymbol s_i+\lambda\boldsymbol\nu_{mn}\right\rVert\le\mathrm{NA}$ . Tilting the source slides all the orders together. Their spacing is set by the specimen's full period.

Why it mattersWhy can tilting the light reveal stripes a microscope misses, and why can blocking some light fake finer ones?

Start with the essentials

Focus question: Why can tilting the light reveal stripes a microscope misses, and why can blocking some light fake finer ones?
One-sentence intuition: The image is the interference of the orders that get through the pupil: $I_{\rm img}=\left|\sum c_{mn}P_{mn}e^{-2\pi i(\boldsymbol\nu_{mn}\cdot\boldsymbol r+m\delta/p)}\right|^2$ . With one order left, the intensity is uniform. Two neighbouring orders can reproduce the original period, but two opposite orders without the zero order give stripes at half the period.

Core mathematical model

Grating frequencies and positions in the pupil

\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}

The transverse axes point horizontally and vertically downward. The rotation angle turns the first grating normal from horizontal toward the downward vertical. Wavelength and pitch are in micrometres; pupil positions are in millimetres. The pupil map is an ideal sine mapping, not the vector design of a real high-NA lens.

Cutoff pitch for axial and aligned oblique illumination

\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 periods, not half-periods and not the Rayleigh separation of two points. The last limit needs the illumination at the pupil edge, lined up with the grating frequency. The live readout of the supported pitch also accounts for grating rotation, but it ignores any extra pupil mask.

Blocking the zero order halves the stripe period

\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}

Take smooth, aligned stripes with both first orders collected. Blocking the zero order removes the fundamental frequency from the intensity, yet the line-cut contrast can still be one. The simulation keeps this artifact in the image.

Mean intensity over one period and the 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}

Because the orders are orthogonal, the sum gives the scalar mean-square transmission over one full period. It is not the total power in the displayed crop, nor a calibrated flux for oblique light. The receiving image is a 192 by 192 grid of point samples across eight micrometres. The raw line cut has 512 points spanning one period along the first grating normal. Its vertical range is labelled and may rescale, but the data never include display gain.

Common difficulties

Finer-looking stripes can be the wrong stripes

Typical misconceptionIf the contrast goes up or the stripes double in number, the resolution must have improved.

Better mental modelBlock the zero order of smooth stripes and only the opposite first orders remain. They interfere into stripes at half the original period, while the amplitude at the original frequency drops to zero. The extra stripes come from which orders were let through; they carry no new information about the specimen. Check the labelled amplitude at the original frequency, not just the contrast.

Run the experiment

01
Scene 1: Axial: invisible stripes
In the axial preset, open the pupil slowly until the first pair of orders enters. The switch is abrupt because the model treats the grating as infinitely long.
What to observe: Changing the wavelength moves the orders in the pupil, but the physical NA stop stays put. Changing NA resizes the physical aperture; with the normalized tilt held fixed, the source moves as well.
02
Scene 2: Tilt: recover the period
In the tilt preset, drag the zero-order spot in the pupil view sideways. Then rotate the grating: the same tilt does not help a perpendicular frequency as much.
What to observe: Binary openings contain many harmonics. Collecting more of them sharpens the edges, but the ringing and false detail are real consequences of which orders pass.
03
Scene 3: Block zero: false half-pitch
Compare an open pupil, a pupil that passes only the zero order and one that blocks it. Then send a crossed grid through the horizontal slit and compare the raw line cut along the grating normal.
What to observe: One tilted exposure recovers information along the chosen direction only. It does not produce a high-resolution image that is equally sharp in every direction, and it cannot recover phase from intensity.