SCIENCE · INVESTIGATE AND EXPLAIN

Integrate a circular-aperture response over detector samples

A uniformly illuminated circular aperture has a different transform from a rectangular slit. In the scalar far field, the normalized amplitude is 2J1(v)/v, with v=πDθ/λ in the small-angle approximation. J1 is the first-order Bessel function. Its limiting amplitude at v=0 is one, and the intensity response is the squared amplitude.

Learning outcome: Reconstruct the two shifted responses, integrate central and source-coordinate pixels independently and compare widths. Retain the one-dimensional averaging boundary and identify noise or unknown brightness as additional inference requirements.

Starting recap

Retain complex amplitude before intensity, definite-integral boundaries, normalization, calibration covariance and unused evidence.

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Try

At separation 1.2 λ/D and pixel width 0.4 λ/D, the point midpoint is about 0.761284. Pixel averaging gives midpoint 0.781156 and source-coordinate value 0.974841, hence contrast about 0.198684. The ideal first-zero scale 1.21967 is unchanged by this detector averaging.

Open the experiment

Predict

Does a fixed optical first-zero scale imply that every detector samples the same profile?

Your prediction and reason

Experiment

Reconstruct the two shifted responses, integrate central and source-coordinate pixels independently and compare widths. Retain the one-dimensional averaging boundary and identify noise or unknown brightness as additional inference requirements.

Integrate a sampled two-source response

Worked teaching model · synthetic values · Original two equally bright mutually incoherent points; scalar circular-aperture PSF [2J1(πu)/(πu)]² in a one-dimensional cut, u=Dθ/λ. Each displayed pixel averages that cut over its width using 64-panel Simpson integration; no two-dimensional detector-area model, noise or guaranteed resolution.

Equal-source angular separation: 1.2 λ/D Pixel angular width: 0.4 λ/D

Reset keeps your prediction, captures and practice history.

A conventional aperture criterion and a sampled profile are different objects; pixel integration, noise and inference strategy affect distinguishability.

Original two equally bright mutually incoherent points; scalar circular-aperture PSF [2J1(πu)/(πu)]² in a one-dimensional cut, u=Dθ/λ. Each displayed pixel averages that cut over its width using 64-panel Simpson integration; no two-dimensional detector-area model, noise or guaranteed resolution.

The vertical axis is rescaled to this case’s actual range; equal display height does not imply equal optical power. The finite curve is illustrative; the ledger preserves the exact case values and stated numerical quadrature.

Original conditional optical ledger
Quantity	Value	Unit
Point-sampled midpoint	0.761284	single-source peak units
Pixel-averaged midpoint	0.781156	single-source peak units
Pixel average at source coordinate	0.974841	single-source peak units
Source-coordinate versus midpoint contrast	0.198684	ratio
Approximate first-zero separation criterion	1.21967	λ/D
Pixel width	0.400000	λ/D

Virtual original records only. No laser, intense-light source, instrument alignment, physical apparatus operation or completed practical competence is supplied.

A conventional aperture criterion and a sampled profile are different objects; pixel integration, noise and inference strategy affect distinguishability.

Observation to attach to a capture

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

Understand

1. Establish the optical mechanism

The supplied response can be evaluated from the series 2J1(v)/v=Σ from k=0 to infinity of (−v²/4)^k/[k!(k+1)!]. This series and the aperture-transform interpretation are introductory calculus extensions, not a requirement to memorize a special-function table. The first zero occurs near Dθ/λ=1.21967.

The original image cut adds intensities from two equal mutually incoherent point sources at u=±s/2, where u=Dθ/λ. A detector pixel then averages this summed one-dimensional profile over its own width w: (1/w)∫I(u)du. The tool uses 64-panel Simpson integration. This is a one-dimensional cut average, not a two-dimensional rectangular-pixel area integral.

The conventional first-zero separation criterion describes one ideal aperture response. It is not a guarantee that two sources are distinguishable in a noisy, sampled record, nor a universal impossibility statement below that separation for every inference strategy. The source-coordinate/midpoint contrast shown here is a descriptive statistic, not a validated detection threshold; it can be negative for a blended profile.

2. Work the original evidence

3. Predict and investigate

Does a fixed optical first-zero scale imply that every detector samples the same profile?

Transfer the derivation to a changed case

If both sources became mutually coherent with a stable relative phase, could this intensity-sum model be reused unchanged?

Retain your prediction, intermediate reasoning and assumptions before opening public feedback.

Compare a worked response

No. One must add their complex image fields with the relative phase and then square; an interference cross term can alter the profile. Keeping the same aperture does not preserve the incoherent-source inference.

Primary mechanism context. Checked 31 October 2026. Original prose, records and diagrams; no copied source exercises or physical light-source operation. Retain first reasoning and public-feedback exposure in the course evidence workspace when registered.

Conventional resolution context. The original detector calculation is a one-dimensional average and is not a two-dimensional camera simulation.

Apply

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

If both sources became mutually coherent with a stable relative phase, could this intensity-sum model be reused unchanged?

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: Derive an image plane by composing ray transformations Next: Separate diffraction scale degeneracy from an unused prediction

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