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Superposition overview

Topic 9 of 9

Resolve two sources

An aperture spreads each point source into a diffraction pattern. Two nearby source images can overlap, so their separation must be compared with the pattern width.

State the resolution criterion

Under the Rayleigh criterion, two equal sources are just resolved when one individual pattern's central maximum coincides with the other's first minimum. For the aperture-width model used here:

Limiting angular separation θR ≈ λ/b

Use wavelength λ and aperture width b in matching length units; the resulting small angle is in radians. The actual source angular separation and the limiting diffraction angle are different quantities.

This comparison uses equal, mutually incoherent sources. Their time-averaged image intensities add. That differs from adding coherent field contributions to produce an interference pattern.

Estimate the angular scale

Visible wavelength of order half a micrometre divided by an aperture width of a few millimetres gives a resolution angle of order 10-4 rad. This is a rough scale check; it is not an angle of order one degree.

Worked aperture comparison

Keep the source separation fixed

Two sources emit light of wavelength 550 nm and have angular separation 0.20 mrad. For aperture width 2.0 mm:

θR ≈ (550 × 10-9)/(2.0 × 10-3)
= 2.75 × 10-4 rad = 0.275 mrad

The actual 0.20 mrad separation is smaller, so the sources fall below this resolution criterion. Widen the aperture to 4.0 mm:

θR ≈ (550 × 10-9)/(4.0 × 10-3)
= 0.1375 mrad

The unchanged 0.20 mrad separation now exceeds the limit, so the sources are resolved by the criterion. At the boundary between these cases:

b = λ/θ
= (550 × 10-9)/(0.20 × 10-3)
= 2.75 × 10-3 m = 2.75 mm

Keep the source separation fixed while changing aperture width

Two equal, mutually incoherent sources remain at -0.10 and +0.10 mrad, separated by 0.20 mrad. Brown long dashes and green short dashes show the individual intensity profiles; solid blue is their intensity sum. All panels have the same axes.

Each individual profile is normalised to unit peak to compare widths. This does not claim that different apertures pass the same absolute power or give the same physical peak intensity.

b = 2.0 mm: Below the Rayleigh criterion

Below the Rayleigh criterion with aperture width 2 millimetresIndividual incoherent-source intensity profiles and their sum share an angular axis from negative 0.50 to positive 0.50 milliradians and a normalised-intensity axis from zero to two. The source centres stay at negative 0.10 and positive 0.10 milliradians. Each individual profile has unit peak and first minimum 0.2750 milliradians from its own centre. Intensities are added, not coherent field amplitudes. The normalisation compares widths without asserting equal physical throughput between apertures. Positions of the summed profile's maxima need not coincide exactly with the source centres.-0.50-0.250.000.250.50012Normalised intensitySource centres: -0.10, +0.10 mradAngular position / mrad

First-minimum offset from either source centre: 0.2750 mrad. The source separation remains 0.20 mrad.

b = 2.75 mm: Just resolved by the criterion

Just resolved by the criterion with aperture width 2.75 millimetresIndividual incoherent-source intensity profiles and their sum share an angular axis from negative 0.50 to positive 0.50 milliradians and a normalised-intensity axis from zero to two. The source centres stay at negative 0.10 and positive 0.10 milliradians. Each individual profile has unit peak and first minimum 0.2000 milliradians from its own centre. Intensities are added, not coherent field amplitudes. The left source's first minimum coincides with the right source's central maximum, marked at the same angular coordinate. This is the stated just-resolved criterion. The normalisation compares widths without asserting equal physical throughput between apertures. Positions of the summed profile's maxima need not coincide exactly with the source centres.-0.50-0.250.000.250.50012Normalised intensitySource centres: -0.10, +0.10 mradAngular position / mrad

First-minimum offset from either source centre: 0.2000 mrad. The brown square marks one profile's first minimum directly below the other profile's central peak.

b = 4.0 mm: Resolved by the criterion

Resolved by the criterion with aperture width 4 millimetresIndividual incoherent-source intensity profiles and their sum share an angular axis from negative 0.50 to positive 0.50 milliradians and a normalised-intensity axis from zero to two. The source centres stay at negative 0.10 and positive 0.10 milliradians. Each individual profile has unit peak and first minimum 0.1375 milliradians from its own centre. Intensities are added, not coherent field amplitudes. The normalisation compares widths without asserting equal physical throughput between apertures. Positions of the summed profile's maxima need not coincide exactly with the source centres.-0.50-0.250.000.250.50012Normalised intensitySource centres: -0.10, +0.10 mradAngular position / mrad

First-minimum offset from either source centre: 0.1375 mrad. The source separation remains 0.20 mrad.

Source centres stay at -0.10 and +0.10 mrad on identical angular axes. Each individual profile is normalised to unit peak for a width comparison; their incoherent intensities add. This normalisation does not imply equal absolute throughput at different aperture widths or fixed locations of the summed peaks.
Optional check Two equal incoherent sources of wavelength 550 nm have angular separation 0.20 mrad. Using the stated Rayleigh criterion theta approximately lambda/b, what aperture width just resolves them?
Two equal incoherent sources of wavelength 550 nm have angular separation 0.20 mrad. Using the stated Rayleigh criterion theta approximately lambda/b, what aperture width just resolves them?

Explain changes in resolving power

A larger aperture or shorter wavelength reduces λ/b and narrows the diffraction patterns. It can improve resolution without changing the actual angular separation of the sources.

Use the stated criterion consistently. It is a model for judging optical resolution, not a universal claim about every observer or image-processing method. Do not substitute a different aperture-shape formula when the width model θR ≈ λ/b is specified.