Angular Resolution Calculator

Diffraction Limit & Rayleigh Criterion

Calculate the minimum resolvable angle for optical systems using wavelength and aperture size. Supports multiple units and result formats.

Examples

See how the calculator works with real-world scenarios.

Astronomical Telescope (Visible Light)

telescope

Calculate the angular resolution for a telescope with a 200 mm aperture using 550 nm light.

λ: 550 nm, D: 200 mm

Result Unit: arcsec

Optical Microscope (Blue Light)

microscope

Find the angular resolution for a microscope with a 0.95 mm aperture using 450 nm light.

λ: 450 nm, D: 0.95 mm

Result Unit: deg

Radio Telescope (21 cm Hydrogen Line)

radio

Calculate the angular resolution for a radio telescope with a 25 m aperture using 0.21 m wavelength.

λ: 0.21 m, D: 25 m

Result Unit: rad

Human Eye (Green Light)

eye

Estimate the angular resolution of the human eye (7 mm pupil) using 550 nm light.

λ: 550 nm, D: 7 mm

Result Unit: arcmin

Other Titles
Understanding Angular Resolution: A Comprehensive Guide
Learn the science, math, and real-world impact of angular resolution in optics and astronomy.

What is Angular Resolution?

  • Definition and Importance
  • Rayleigh Criterion Explained
  • Diffraction Limit in Practice
Angular resolution is the smallest angle between two points that an optical system can distinguish as separate. It is a fundamental concept in optics, astronomy, and microscopy, determining the level of detail visible in images.
Rayleigh Criterion
The Rayleigh criterion provides a formula to calculate the minimum resolvable angle based on the wavelength of light and the aperture diameter: θ = 1.22 × λ / D.
Diffraction Limit
No optical system can resolve details smaller than its diffraction limit, which is set by the wavelength and aperture size.

Practical Examples

  • A large telescope can resolve stars that are closer together than a small telescope.
  • Microscopes with larger objectives can distinguish finer details.
  • Radio telescopes use long wavelengths, requiring huge apertures for fine resolution.

Step-by-Step Guide to Using the Angular Resolution Calculator

  • Input Selection
  • Unit Conversion
  • Result Interpretation
Input Selection
Enter the wavelength and aperture diameter. Choose the correct units for each input to match your optical system.
Unit Conversion
The calculator automatically converts all units to SI before calculation, ensuring accurate results regardless of input units.
Result Interpretation
The result is shown in your selected angular unit: radians, degrees, arcminutes, or arcseconds. Smaller values mean better resolution.

Usage Scenarios

  • Input 550 nm and 200 mm for a typical telescope.
  • Try 0.21 m and 25 m for a radio telescope.
  • Switch result units to see the same value in different formats.

Real-World Applications of Angular Resolution

  • Astronomy and Telescopes
  • Microscopy and Biology
  • Photography and Imaging
Astronomy and Telescopes
Astronomers use angular resolution to distinguish close stars, planets, and galaxies. Higher resolution reveals more detail in celestial images.
Microscopy and Biology
Biologists rely on microscopes with high angular resolution to observe tiny structures within cells and tissues.
Photography and Imaging
Camera lenses are rated by their resolving power, which affects image sharpness and detail.

Application Examples

  • Hubble Space Telescope's high resolution enables deep space discoveries.
  • Super-resolution microscopy breaks traditional limits using advanced techniques.
  • Professional photographers choose lenses with high resolving power for crisp images.

Common Misconceptions and Correct Methods

  • Aperture vs. Magnification
  • Wavelength Effects
  • Practical Limitations
Aperture vs. Magnification
A larger aperture improves resolution, not magnification. Magnification without resolution leads to blurry images.
Wavelength Effects
Shorter wavelengths (blue light) provide better resolution than longer wavelengths (red light or radio waves).
Practical Limitations
Atmospheric turbulence, lens quality, and detector sensitivity can limit real-world resolution below the theoretical value.

Misconception Examples

  • A small telescope with high magnification cannot resolve fine details.
  • Blue filters improve microscope resolution.
  • Adaptive optics help telescopes overcome atmospheric blurring.

Mathematical Derivation and Examples

  • Rayleigh Criterion Formula
  • Unit Conversions
  • Worked Examples
Rayleigh Criterion Formula
θ = 1.22 × λ / D, where θ is in radians, λ is the wavelength, and D is the aperture diameter.
Unit Conversions
1 radian = 57.2958 degrees, 1 degree = 60 arcminutes, 1 arcminute = 60 arcseconds.
Worked Example
For λ = 550 nm and D = 200 mm: θ = 1.22 × 550e-9 / 0.2 = 3.355e-6 radians ≈ 0.69 arcseconds.

Math Examples

  • Convert 3.355e-6 radians to degrees: 3.355e-6 × 57.2958 ≈ 0.000192°.
  • 0.000192° × 60 = 0.0115 arcminutes; 0.0115 × 60 = 0.69 arcseconds.
  • Try your own values in the calculator!