Rayleigh Criterion Resolution Calculator

Calculate the minimum resolvable separation using the Rayleigh criterion for telescopes and microscopes.

Oil immersion: 1.4, Dry: 0.95
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Diffraction Sets a Resolution Scale

A finite aperture does not image a point source as a perfect point; diffraction spreads it into a pattern, placing a fundamental scale on how closely two sources can be distinguished. For a circular aperture, the image of a point is an Airy pattern. Rayleigh's conventional criterion says two equal point sources are just resolved when the central maximum of one Airy pattern lies at the first minimum of the other, giving θR≈1.22λ/D.

The formula shows two direct routes to finer angular resolution: use a larger aperture D or a shorter wavelength λ. For a microscope, a related lateral-resolution expression is d≈0.61λ/NA, where numerical aperture NA=n sinα combines the refractive index and collection angle. A smaller value of θ or d means better resolving ability.

Circular aperture: θR≈1.22λ/D,   microscope: d≈0.61λ/NA
SymbolMeaningWhy it appears / units
θRRayleigh angular separationrad; smaller is better angular resolution.
λWavelengthm; shorter wavelengths diffract less for the same aperture.
DCircular aperture diameterm; larger diameter narrows the Airy pattern.
NANumerical apertureDimensionless; larger NA improves microscope lateral resolution.
dMinimum lateral separation scalem; smaller d corresponds to finer microscope detail.

Rayleigh's criterion is a convention for two equal incoherent point sources, not an absolute statement that no information exists below that separation. Detector sampling, aberrations, atmospheric turbulence, signal-to-noise ratio, image processing, source contrast, and prior knowledge can all affect practical resolving performance.

Worked Examples

Example 1: Hubble: D=2400mm, λ=550nm
θ=1.22×550e-9/2.4
Result: 0.0575 arcsec — better than ground telescopes
Space avoids atmospheric seeing
Example 2: Oil immersion microscope: NA=1.4, λ=400nm
d=0.61×400/1.4
Result: 174 nm resolution
Near diffraction limit of optical microscopy
Example 3: 100 mm telescope in green light
D=0.100m, λ=550nm → θ=1.22λ/D
Result: θ≈6.71×10−6rad ≈1.38 arcsec
Doubling the aperture to 200mm would halve this diffraction-limited angular separation.
Example 4: Dry microscope objective
λ=550nm, NA=0.95 → d=0.61λ/NA
Result: d≈353nm
Higher numerical aperture improves resolution because the objective collects a wider cone of diffracted light.

Common Mistakes

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Thinking a larger resolution number means a better instrument

Resolution here is a minimum distinguishable separation. A smaller angular or linear value means finer detail can be separated.

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Mixing aperture diameter in millimeters with wavelength in nanometers

Convert both to compatible units before calculating λ/D. Otherwise the result can be wrong by factors of thousands or millions.

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Applying the 1.22 circular-aperture factor to every aperture shape

The 1.22 factor belongs to the first Airy minimum of a circular aperture. A one-dimensional slit has a different diffraction condition, with its first minimum at sinθ=λ/a.

Frequently Asked Questions

Why 1.22 factor?
Comes from the first zero of the Airy disk pattern (J₁ Bessel function). Two point sources are just resolved when the central maximum of one falls on the first minimum of the other.
Breaking the Rayleigh limit?
Super-resolution microscopy (STED, STORM, PALM) can resolve 20-50 nm — well below the diffraction limit. They use fluorescence physics to bypass the Rayleigh criterion.
Why does a larger telescope aperture improve resolution?
A larger circular aperture produces a narrower Airy diffraction pattern. Since θR is proportional to λ/D, doubling D halves the ideal angular separation associated with the Rayleigh criterion at the same wavelength.
What is the difference between magnification and resolution?
Magnification makes an image appear larger, while resolution determines whether neighboring details can be distinguished. Increasing magnification without improving resolution creates a larger blurred image rather than revealing new spatial information.
Why can real telescopes perform worse than the diffraction limit?
Atmospheric turbulence, optical aberrations, imperfect alignment, tracking errors, detector sampling, and finite signal-to-noise can broaden images beyond the ideal diffraction pattern. Adaptive optics and space-based observing can reduce some of these limitations.
Is the Rayleigh criterion an absolute physical cutoff?
No. It is a useful conventional benchmark for two equal point sources under a particular imaging model. Modern estimation and super-resolution methods can localize or distinguish features below that nominal scale when the measurement contains sufficient information and the physical assumptions are appropriate.

Formula Explorer connections

Interpretation: This relationship connects light propagation, geometry, wavelength, refraction, interference or image formation. Assumption: Use a consistent sign convention and units. Paraxial rays, thin elements, coherent light, vacuum wavelength or ideal optical components may be assumed.

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