AkCalculators

Optical resolution — Rayleigh criterion, unit conversions, and practical examples for telescopes and microscopes

Optical resolution determines the smallest angular or linear separation at which two point sources (stars, beads, or fine details) can be distinguished. In systems limited by diffraction — e.g., perfect lenses or telescopes without aberrations — the Rayleigh criterion gives a widely used estimate of the diffraction-limited angular resolution for a circular aperture:

θ = 1.22 · λ / D

Here θ is the angular resolution (in radians), λ is the wavelength of light, and D is the aperture diameter. The factor 1.22 arises from the position of the first minimum of the Airy diffraction pattern produced by a circular aperture (the first zero of the Bessel function). The Rayleigh criterion is a convention: two point sources separated by θ such that the central maximum of one lies on the first minimum of the other are considered "just resolved."

Units and converting θ

θ computed from the Rayleigh formula is in radians if λ and D are in the same linear units. Common output units used in astronomy and imaging are degrees and arcseconds because angular values are often tiny. Useful conversions:

• Radians ↔ degrees: 1 rad = 180/π degrees.
• Degrees ↔ arcseconds: 1 degree = 3600 arcseconds.
• Radians ↔ arcseconds: 1 rad = (180/π)×3600 ≈ 206265 arcseconds.

So to convert θ (rad) to arcseconds: θ_arcsec = θ_rad × 206265. To go from arcseconds to radians: θ_rad = θ_arcsec / 206265.

Linear resolution on a target

Often you need the linear size R on a target at distance L corresponding to angular resolution θ. For small angles (typical optical cases) the small-angle approximation applies:

R ≈ θ · L

Here R and L must use consistent units (e.g., meters). If θ is in radians and L = 1000 m, then R = θ × 1000 gives meters. If you used arcseconds for θ, convert to radians first: θ_rad = θ_arcsec / 206265.

Why wavelength matters

Because θ ∝ λ, resolution improves when observing at shorter wavelengths. For example, a telescope with fixed aperture will resolve smaller details in blue light (shorter λ) than in red light. In microscopy, using shorter-wavelength illumination extends resolution limits.

Telescope example — astronomical resolution

Consider a 200 mm (0.2 m) amateur telescope observing at λ = 550 nm (green light). Convert λ to meters: 550 nm = 550×10⁻9 m = 5.5×10⁻7 m. Rayleigh θ = 1.22 × (5.5×10⁻7) / 0.2 ≈ 3.355×10⁻6 rad. In arcseconds: θ_arcsec ≈ 3.355×10⁻6 × 206265 ≈ 0.692 arcsec. Real atmospheric seeing usually limits ground-based telescopes to ~0.5–2 arcsec, so a 200 mm telescope is near the diffraction limit only under excellent seeing.

If observing a target at L = 1 astronomical unit (not practical for linear resolution), converting angular to linear is unnecessary — astronomers use angular measures. For ground-based remote sensing, linear resolution is useful: e.g., from an aircraft at L = 10000 m, R = θ·L ≈ 3.355×10⁻6 × 10000 ≈ 0.0336 m (≈3.36 cm). That indicates the smallest resolvable features at that distance under diffraction-limited optics.

Microscope example — resolving small features

Microscopes are commonly limited by both diffraction and the numerical aperture (NA) of the objective and condenser. A related formula for lateral (in-plane) resolution in microscopy (Abbe diffraction limit) is:

d = 0.61 · λ / NA

NA = n · sin(α), where n is refractive index of the medium (air ≈ 1.0, oil ≈ 1.515) and α is half the acceptance angle. For example, using λ = 550 nm and NA = 1.4 (oil-immersion), d ≈ 0.61×550e-9/1.4 ≈ 2.4×10⁻7 m ≈ 240 nm — so sub-micron features are resolvable.

Note the different prefactor (0.61 vs 1.22/π etc.) reflects differences between Rayleigh, Abbe and other criteria and whether one discusses full-width or Airy disk behavior. For practical microscopy, Abbe’s criterion and numerical aperture are the standard descriptors.

Practical considerations: atmosphere, aberrations and detector sampling

Real optical systems rarely reach the theoretical diffraction limit. For telescopes, atmospheric turbulence (seeing) usually dominates; adaptive optics or space telescopes overcome this. For microscopes and cameras, lens aberrations, alignment, and detector pixel size (sampling) limit performance. Nyquist sampling requires at least two detector pixels per resolution element to avoid undersampling; match pixel size to optical resolution for best results.

Worked examples summary

1. Telescope (200 mm aperture, λ = 550 nm): θ ≈ 0.69 arcsec.
2. Linear sensing from 10 km: R ≈ 0.69 arcsec → θ_rad ≈ 3.35×10⁻6 → R ≈ 33.6 m ? (Note: convert units carefully; in earlier aircraft example we used 10 km = 10000 m giving R≈33.5 m; for 10 km altitude the resolvable ground feature would be tens of meters for such small aperture.)
3. Microscope (NA=1.4, λ=550 nm): d ≈ 240 nm — submicron resolution.

Unit-conversion quick reference (useful formulas)

• λ: 1 nm = 1×10⁻9 m ; 1 µm = 1×10⁻6 m.
• D: 1 mm = 1×10⁻3 m ; 1 cm = 1×10⁻2 m.
• Radians ↔ arcseconds: θ(arcsec) = θ(rad) × 206265 ; θ(rad) = θ(arcsec) / 206265.
• Degrees ↔ arcseconds: 1° = 3600 arcsec ; θ(deg) = θ(arcsec) / 3600.
• Linear: R = θ(rad) × L (ensure L in meters if R in meters).

When the Rayleigh criterion is appropriate

Use Rayleigh for diffraction-limited circular apertures and for order-of-magnitude angular resolution estimates. For extended objects, complex imaging systems, or detector-limited cases, consider more detailed point-spread-function (PSF) analysis, modulation transfer function (MTF), and practical sampling considerations.

This calculator provides immediate conversions and worked steps to help you check designs, quick-estimate telescope or microscope performance, or compute required aperture sizes to achieve a target resolution. Always check real-world constraints (seeing, NA, aberrations) in detailed designs.