Calculate Numerical Aperture

Calculator for NA, refractive index, and aperture angle

NA Calculator (JavaScript)

Core formula

Numerical aperture is defined as NA = n·sin(α) with refractive index n and half aperture angle α.

°
Result

Example calculations

Example 1: Water immersion objective

Given: n = 1.33, α = 60°

\[NA=n\cdot\sin(\alpha)=1.33\cdot\sin(60^\circ)=1.15\]

Result: NA ≈ 1.15

Example 2: Refractive index from NA

Given: NA = 0.95, α = 70°

\[n=\frac{NA}{\sin(\alpha)}=\frac{0.95}{\sin(70^\circ)}\approx1.01\]

Result: n ≈ 1.01 (near air)

Example 3: Angle from NA and n

Given: NA = 1.25, n = 1.52

\[\alpha=\arcsin\left(\frac{NA}{n}\right)=\arcsin\left(\frac{1.25}{1.52}\right)\approx55.3^\circ\]

Result: α ≈ 55.3°

Formulas and comprehensive description

Numerical aperture is a key optical parameter describing light-gathering ability and resolving potential of objective lenses. Higher NA enables better detail resolution but requires stricter optical conditions. In microscopy, NA directly affects lateral resolution and depth of field.

Numerical aperture
\[NA=n\cdot\sin(\alpha)\]
Refractive index
\[n=\frac{NA}{\sin(\alpha)}\]
Aperture angle
\[\alpha=\arcsin\left(\frac{NA}{n}\right)\]
Physical limit
\[NA\le n\]
Note
If NA is entered larger than n, the angle is physically impossible. Immersion media with higher n allow larger NA values and therefore higher resolution.

Comprehensive Description

What is Numerical Aperture?

Numerical Aperture (NA) is a fundamental optical property of objectives and lenses. It describes a lens's ability to gather and focus light. Higher NA means better resolving power (finer details visible) and greater light gathering power (brighter image). NA is a dimensionless quantity defined by the formula NA = n·sin(α).

Components of Numerical Aperture
Parameter Symbol Meaning Typical Values
Refractive index n Optical density of medium between objective and specimen 1.00 (air), 1.33 (water), 1.52 (oil)
Half aperture angle α Half-angle of light cone from objective lens 30° to 70° (typically)
Numerical aperture NA Light-gathering property of the lens 0.1 to 1.4 (air microscopy to oil immersion)
The Basic Formula

Numerical aperture is calculated using:

\[NA = n \cdot \sin(\alpha)\]
  • n is the refractive index of the medium between objective and specimen
  • sin(α) is the sine of the half aperture angle
  • The multiplication shows that NA depends on both factors
Immersion Media and Their Refractive Indices
Medium Refractive Index n Typical NA Values Use
Air 1.00 up to 0.95 Standard microscopy, cost-effective
Water 1.33 up to 1.20 Living specimens, reduced aberrations
Immersion oil 1.515 up to 1.40 Highest resolution, standard in microscopy
Cedar oil 1.52 up to 1.42 Historical, rarely used today
Special oil 1.56 up to 1.50 Research applications, high-end systems
NA and Resolving Power (Rayleigh Criterion)

The Rayleigh resolution limit – the smallest distance between two points that can still be distinguished as separate – is directly related to numerical aperture:

\[d = \frac{\lambda}{2 \cdot NA}\]
  • d – minimum resolvable distance (resolving power)
  • λ – wavelength of light (~0.5 µm for visible light)
  • NA – numerical aperture of the objective

Higher NA means smaller d, which means better resolution!

Depth of Field and NA

Depth of field (DOF) is the thickness of the layer that appears in focus. It is strongly influenced by NA:

\[DOF = \frac{\lambda}{2 \cdot NA^2} \approx \frac{0.5}{NA^2}\,\mu m\]
  • High NA → small depth of field (thin focused layer, good layer discrimination)
  • Low NA → large depth of field (thick focused layer, good overview)
Practical NA Values for Different Objectives
Objective Type Magnification NA (Air) NA (Oil) Resolution (Air)
Low power 4×, 10× 0.10–0.25 – ~2.0–1.0 µm
Medium power 20×, 40× 0.40–0.65 0.65–0.80 ~0.6–0.4 µm
High power 63×, 100× 0.70–0.95 1.25–1.40 ~0.35–0.2 µm
Oil immersion 100×, 150× – 1.30–1.40 ~0.2 µm (optimal)
NA and Light-Gathering Power

An objective with higher NA collects more light and produces a brighter image:

  • Light intensity is proportional to NA²
  • Doubling NA → four times the light intensity
  • This is especially important for fluorescence microscopy and dark-field illumination
Mathematical Limits

There is a fundamental physical limit to numerical aperture:

\[NA \le n\]

This is because sin(α) can at most equal 1 (when α = 90°). Therefore, maximum NA is:

  • In air: NAmax = 1.00
  • In water: NAmax = 1.33
  • In oil immersion: NAmax ≈ 1.40–1.50
How to Determine NA of an Objective

The NA is usually engraved on the objective barrel, typically in the following format:

Example: 40× / 0.65 or 100× / 1.40 Oil
  • First number: Magnification of the objective
  • Second number: Numerical aperture
  • Label "Oil": Indicates that oil immersion is required
Practical Tips
  • Use oil immersion: For highest resolution, use the correct oil type and keep the objective clean
  • Consider wavelength: Shorter wavelengths (blue, UV) enable better resolution
  • Condenser NA: Condenser NA should be ≥ objective NA for optimal illumination
  • Aperture diaphragm: Too large an aperture can reduce contrast despite high NA
  • Corrections: High-quality objectives are corrected for chromatic and spherical aberrations (apochromats)
Comparison: Different Objective Types
Objective Type Correction NA Range Aberration Cost
Achromat 2 colors (red, blue) 0.1–0.95 Moderate €–€€
Fluorite/Semi-apochromat Better color correction 0.1–1.20 Low €€€
Apochromat All colors (ideal) 0.1–1.40 Minimal €€€€
Summary: Why NA Matters
Higher NA enables:
  • ✓ Better resolution – finer details visible
  • ✓ Greater light-gathering power – brighter images
  • ✓ Improved contrast – better distinction
  • ✗ Smaller depth of field – thin focused layer (sometimes disadvantageous)
  • ✗ Higher aberrations – produces distortion without correction
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