SCIENCE · INVESTIGATE AND EXPLAIN

Separate slit spreading from circular-aperture resolution

A finite aperture spreads a wave even in a perfectly manufactured optical system. For a single slit of width w, the first dark directions satisfy w sinθ=λ. On a perpendicular screen at L, the central bright region extends between the two signs, so its width is2L tanθ. Confusing this full width with its half-width gives a factor-of-two error.

Learning outcome: Compare slit widths20,40,80 µm and circular diameters2,3,4 mm. Retain the two independent geometries and compare the stated source separation with the circular criterion. Explain which real-system observation could invalidate a claim based only on the ideal aperture formula.

Starting recap

Use normal-based angles, signed conjugate distances and explicit optical-path/reflection phases. Separate geometric direction, amplitude and energy; preserve focal and critical boundaries.

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Try

A40 µm slit has first sine .015. At L=2 m the full central width is4(.015)/√(1−.015²)≈.0600068 m. A4 mm circular aperture has Rayleigh angle1.22×600e−9/.004=.183 mrad. The stipulated .2 mrad source pair meets this criterion; a3 mm aperture gives .244 mrad and does not.

Open the experiment

Predict

For fixed wavelength, doubling a circular aperture diameter changes its Rayleigh angle by:

Your prediction and reason

Experiment

Compare slit widths20,40,80 µm and circular diameters2,3,4 mm. Retain the two independent geometries and compare the stated source separation with the circular criterion. Explain which real-system observation could invalidate a claim based only on the ideal aperture formula.

Distinguish slit minima from a circular resolution criterion

Worked teaching model · synthetic values · Wavelength600 nm in air; slit far-field screen at2 m. Separately two incoherent equal-brightness point sources have angular separation.2 mrad through an ideal circular aperture. Rayleigh angle1.22 lambda/D is a stated criterion, not a universal detector or eyesight limit.

Single slit width: 40 µm Circular aperture diameter: 4 mm

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Circular aperture criterion, not a binary detector law

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: Rayleigh criterion (solid).2: supplied source separation (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Circular aperture criterion, not a binary detector law

Rayleigh criterion

aperture diameter (mm)	angular separation (mrad)
1	0.732
1.05	0.697143
1.1	0.665455
1.15	0.636522
1.2	0.61
1.25	0.5856
1.3	0.563077
1.35	0.542222
1.4	0.522857
1.45	0.504828
1.5	0.488
1.55	0.472258
1.6	0.4575
1.65	0.443636
1.7	0.430588
1.75	0.418286
1.8	0.406667
1.85	0.395676
1.9	0.385263
1.95	0.375385
2	0.366
2.05	0.357073
2.1	0.348571
2.15	0.340465
2.2	0.332727
2.25	0.325333
2.3	0.318261
2.35	0.311489
2.4	0.305
2.45	0.298776
2.5	0.2928
2.55	0.287059
2.6	0.281538
2.65	0.276226
2.7	0.271111
2.75	0.266182
2.8	0.261429
2.85	0.256842
2.9	0.252414
2.95	0.248136
3	0.244
3.05	0.24
3.1	0.236129
3.15	0.232381
3.2	0.22875
3.25	0.225231
3.3	0.221818
3.35	0.218507
3.4	0.215294
3.45	0.212174
3.5	0.209143
3.55	0.206197
3.6	0.203333
3.65	0.200548
3.7	0.197838
3.75	0.1952
3.8	0.192632
3.85	0.19013
3.9	0.187692
3.95	0.185316
4	0.183
4.05	0.180741
4.1	0.178537
4.15	0.176386
4.2	0.174286
4.25	0.172235
4.3	0.170233
4.35	0.168276
4.4	0.166364
4.45	0.164494
4.5	0.162667
4.55	0.160879
4.6	0.15913
4.65	0.157419
4.7	0.155745
4.75	0.154105
4.8	0.1525
4.85	0.150928
4.9	0.149388
4.95	0.147879
5	0.1464

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

supplied source separation

aperture diameter (mm)	angular separation (mrad)
1	0.2
1.05	0.2
1.1	0.2
1.15	0.2
1.2	0.2
1.25	0.2
1.3	0.2
1.35	0.2
1.4	0.2
1.45	0.2
1.5	0.2
1.55	0.2
1.6	0.2
1.65	0.2
1.7	0.2
1.75	0.2
1.8	0.2
1.85	0.2
1.9	0.2
1.95	0.2
2	0.2
2.05	0.2
2.1	0.2
2.15	0.2
2.2	0.2
2.25	0.2
2.3	0.2
2.35	0.2
2.4	0.2
2.45	0.2
2.5	0.2
2.55	0.2
2.6	0.2
2.65	0.2
2.7	0.2
2.75	0.2
2.8	0.2
2.85	0.2
2.9	0.2
2.95	0.2
3	0.2
3.05	0.2
3.1	0.2
3.15	0.2
3.2	0.2
3.25	0.2
3.3	0.2
3.35	0.2
3.4	0.2
3.45	0.2
3.5	0.2
3.55	0.2
3.6	0.2
3.65	0.2
3.7	0.2
3.75	0.2
3.8	0.2
3.85	0.2
3.9	0.2
3.95	0.2
4	0.2
4.05	0.2
4.1	0.2
4.15	0.2
4.2	0.2
4.25	0.2
4.3	0.2
4.35	0.2
4.4	0.2
4.45	0.2
4.5	0.2
4.55	0.2
4.6	0.2
4.65	0.2
4.7	0.2
4.75	0.2
4.8	0.2
4.85	0.2
4.9	0.2
4.95	0.2
5	0.2

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Single-slit first minimum angle	0.859469	degrees
Central slit maximum width on screen	0.0600068	m
Circular-aperture Rayleigh angle	0.000183	rad
Source angular separation	0.0002	rad
Meets stated Rayleigh separation criterion	1	flag
Diameter needed for equality	3.66	mm

A narrower slit spreads its central pattern more, while a larger circular aperture lowers the stated resolution angle.

Single-slit first zeros obey w sin(theta)=lambda; the central screen width extends between both signs, giving 2L tan(theta). A circular aperture instead uses the supplied small-angle Rayleigh convention1.22lambda/D for two equal incoherent point sources. Detector noise, brightness imbalance, aberration and image processing can change actual distinguishability; this ideal criterion is not an eyesight or medical assessment.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

A narrower slit spreads its central pattern more, while a larger circular aperture lowers the stated resolution angle.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Define the optical model

Build the signed account

A circular aperture has a different diffraction pattern. The supplied Rayleigh criterion for two equal-brightness incoherent point sources uses angular separation1.22λ/D, where D is aperture diameter. It is a convention for comparing overlapping diffraction images, not an absolute law that every instrument must display a binary resolved/unresolved result. Detector noise, aberration, brightness imbalance and processing matter to actual distinguishability.

Test the limits and evidence

The model keeps wavelength at600 nm and compares a slit with a separate circular aperture. Narrowing the slit broadens its central pattern; increasing circular diameter reduces the Rayleigh angle. More collected light and finer ideal angular resolution are related to aperture but are not interchangeable claims. These are virtual calculations with no eyesight, health, laser-exposure or telescope-observation procedure. Each formula has its own geometry and assumptions.

Worked original case

Predict and investigate

Changed-case practice

Try the changed case before feedback

At λ=600 nm and circular diameter3 mm, find the Rayleigh angle in milliradians.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original text and numerical cases checked 30 October 2026 against the primary explanation. All optical activity is virtual: no laser, sunlight viewing, eye exposure, physical lens alignment or device construction is requested. Public feedback is not independent mastery; retain first calculations, critique and changed-case evidence in the separate course workspace.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which independent observation would reject the approximation, sign convention or calibration premise rather than merely refit it?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Separate interference order from finite-width brightness Next: Include reflection phase in the optical path ledger

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