Optics
Microscope resolution and numerical aperture calculator
Enter the wavelength and the objective's numerical aperture to see how fine a detail it can separate, laterally and along the optical axis, for widefield and confocal imaging, and how small your pixels must be to capture it (Nyquist). Below: numerical aperture from the acceptance angle, and depth of field.
Resolution formulas, and when each applies
| Limit | Formula | Use it for |
|---|---|---|
| Abbe | d = wavelength ÷ (2 × NA) | The finest periodic detail the objective passes |
| Rayleigh | r = 0.61 × wavelength ÷ NA | Two point sources, objective and condenser matched (or fluorescence) |
| Brightfield with condenser | r = 1.22 × wavelength ÷ (NAobj + NAcond) | Transmitted light when the condenser NA is lower than the objective's |
| Axial, widefield | z ≈ 2 × wavelength × n ÷ NA² | How thin a slice a widefield image separates |
| Confocal, small pinhole | 0.37 × wavelength ÷ NA lateral; 0.64 × wavelength ÷ (n − √(n² − NA²)) axial | FWHM with the pinhole under 0.25 Airy units (Zeiss) |
Wavelength means the emission wavelength for widefield fluorescence and a mean of excitation and emission for confocal; NA is the objective's numerical aperture and n the refractive index of the immersion medium. Magnification appears nowhere: it decides how big the detail looks, not whether it is resolved. These are theoretical best cases; a misaligned condenser, low contrast, thick samples and refractive-index mismatch all make real resolution worse.
Worked example: 520 nm emission and a 1.4 NA oil objective: Rayleigh 0.61 × 520 ÷ 1.4 = 227 nm; Abbe 520 ÷ 2.8 = 186 nm; axial about 2 × 520 × 1.518 ÷ 1.96 = 805 nm.
Nyquist sampling: how small your pixels must be
To record detail the optics resolve, the pixel size at the sample must be at most half the smallest resolved period. SVI's Nyquist reference gives the critical sampling distance as emission wavelength ÷ (4 × NA) for widefield and excitation wavelength ÷ (8 × NA) for confocal; many facilities simply divide the resolution by 2.3 for a safety margin. Larger pixels lose detail for good (undersampling); much smaller ones only cost light, time and disk space, unless you plan to deconvolve.
Check your actual pixel size with the pixel size calculator: a 6.5 µm camera pixel behind a 60× objective is 108 nm at the sample, which samples 520 nm light at 1.4 NA (Nyquist 93 nm) slightly coarsely.
Numerical aperture
NA = n × sin(half-angle)
The half-angle is half the opening angle of the cone of light the objective accepts, and n the refractive index between the coverslip and the front lens. In air NA cannot reach 1; real dry objectives stop at about 0.95. Water immersion reaches about 1.27, silicone oil about 1.35, and oil immersion about 1.4 to 1.49. An objective's NA is engraved after its magnification, for example "60×/1.40 Oil".
Depth of field (MicroscopyU): d = wavelength × n ÷ NA² + n × e ÷ (M × NA), where e is the smallest distance the detector resolves (about a camera pixel) and M the objective magnification. High-NA objectives have a very shallow depth of field, which is why a thick section needs a Z-stack.
Questions
What is the resolution of a light microscope?
About 200–250 nm laterally with a high-NA oil objective in visible light (0.61 × 520 nm ÷ 1.4 ≈ 227 nm by the Rayleigh criterion), and roughly 500–800 nm along the optical axis.
What is the difference between the Abbe and Rayleigh limits?
Abbe (wavelength ÷ 2NA) is the finest periodic structure an objective transmits; Rayleigh (0.61 × wavelength ÷ NA) is the distance at which two point sources can just be told apart. Both scale with wavelength ÷ NA; Rayleigh is the more conservative number.
How do you calculate numerical aperture?
NA = n × sin(half-angle), where n is the refractive index of the immersion medium and the half-angle is half the opening angle of the light cone. A dry objective accepting a 64° half-angle has NA = sin 64° ≈ 0.90.
What pixel size do I need for Nyquist sampling?
At most wavelength ÷ (4 × NA) at the specimen for widefield (emission wavelength), wavelength ÷ (8 × NA) for confocal (excitation wavelength). For 520 nm and NA 1.4 that is about 93 nm (widefield) or 46 nm (confocal).
Does higher magnification give better resolution?
No. Resolution is set by NA and wavelength. Higher magnification without higher NA enlarges the same blur ("empty magnification").
Is this calculator free?
Yes. It runs in your browser and sends nothing anywhere. SlideScope, the desktop microscopy app made by the same team, is a paid app with a free preview.
Sources
- Nikon MicroscopyU — Resolution
- Nikon MicroscopyU — Depth of field and depth of focus
- ZEISS — Basics of confocal laser scanning microscopy (Wilhelm)
- SVI Huygens — The Nyquist rate
Results are theoretical or arithmetic; check them against your instrument's documentation before relying on them for clinical or regulated work.
Measure on your real images
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