Foundational Knowledge

# Numerical Aperture and Light Cone Geometry

2 May 2024 · 2 min read
• Foundational Knowledge
• Widefield Light Microscopy

## Abstract

Are you curious about how microscope objectives capture finer object structures to produce higher-resolution images? This foundational knowledge article on Numerical Aperture and Light Cone Geometry will give you a sound understanding of the light gathering ability of microscope objectives and how it is expressed through the numerical aperture (NA). An interactive tutorial allows you to visualize changes in the illumination cone as you vary NA values. You will also learn about the role of the refractive index and the limitations of the maximum achievable NA values.

Key Learnings:

• Numerical aperture (NA) is a crucial factor in microscopy as it determines the brightness and resolution of an image formed by an objective.
• The NA is affected by the front lens properties and refractive index of the imaging medium.
• Choosing objectives with higher NAs results in better resolution and specimen detail but comes with trade-offs (e.g. reduced free working distance, increased sensitivity to spherical aberration).

## What is Numerical Aperture in Microscopy?

The light gathering ability of a microscope objective is quantitatively expressed in terms of the numerical aperture (NA). The objective’s NA is a measure of its ability to capture image-forming light rays: Higher NA values allow increasingly oblique rays (representing finer object structures) to enter the front lens of the objective, producing a higher-resolution image with greater specimen detail. This interactive tutorial demonstrates the change in numerical aperture light cones displayed by a microscope objective with corresponding changes in numerical aperture. The angular aperture value corresponding to a given NA-value is also depicted here.

### Tutorial Guide

The tutorial displays a schematic drawing of a microscope objective. The actual angular aperture of the light cone and the corresponding NA value are indicated in the tutorial window. To operate the tutorial, use the Numerical Aperture slider to change the NA value from low (left) to high (right). As you vary the numerical aperture value with the slider, the size and shape of the illumination cone entering the objective’s front lens is altered. The adjustable NA for this tutorial is 0.03 to 0.95. The approximate objective magnification has also been assigned to each NA value.

### Relationship between Numerical Aperture and Resolution

The brightness and resolution of an image formed by an objective at a given magnification increases with its NA value, respectively the diameter of the angular aperture (the angle of the light cone collected by the objective). Light rays emanating from the specimen pass through air (or a liquid-based immersion medium) located between the cover glass and the objective’s front lens. The angular aperture is expressed as the angle between the microscope’s optical axis and the direction of the most oblique light rays captured by the objective (see the tutorial figure). Mathematically, the NA is expressed as:

Numerical Aperture (NA) = n · sin(θ) (1)

n is the refractive index of the media in the object space (between the cover glass and the objective’s front lens) and θ is half the full angular aperture. The value of n varies between 1.0 for air and 1.58 for most immersion media used in optical microscopy. The angular aperture, which varies with the objective focal length, is the maximum angle of image-forming light rays diffracted by the specimen that the front lens of the objective can capture when the specimen is in focus. As the objective focal length decreases, the maximum angle between the specimen and the outer diameter of the objective front lens increases, causing a proportional increase in the angular aperture. From the above equation, it is obvious that the NA increases with both the angular aperture and the refractive index of the imaging medium.

Limitations of Numerical Aperture

Theoretically, the maximum angular aperture achievable with a dry (air) microscope objective would be 180 degrees, resulting in a value of 90 degrees for the half angle used in the NA equation. The sine of 90 degrees is one, indicating that the numerical aperture is limited not only by the angular aperture but also by the refractive index of the imaging medium. Most microscope objectives are designed to operate with air (refractive index= 1.0) as the imaging medium between the cover glass and the front lens of the objective. This yields a theoretical maximum NA of 1.00. For practical reasons (available working distance), the highest desirable value for the NA of a dry objective is 0.95 (the half angle of the aperture is approximately 72 degrees). Immersion objectives achieve much higher NAs at the expense of free working distance and spherical aberration sensitivity.