Numerical aperture (NA) refers to the cone of light that is made from a focusing lens and describes the light gathering capability of the lens (similar to f/# ). NA is defined by the following equation, where n is the index of refraction of the medium (often n=1 for air), and α is the half angle of the cone of light exiting the lens pupil.

### Equating to f/#

Now that we have briefly explained what numerical aperture is, we can equate it to f/#. As explained here, f/# is also a measure of how much light can get through a lens. f/# of a simple lens is defined by the following equation, where f is the focal length of the lens and D is the diameter (or more specifically the entrance pupil diameter for more complex lens systems).

###### Using Geometric Relationships

In order to equate NA and f/#, we can use simple geometric relationships. Figure 3.1 shows a simple lens focusing light rays (blue lines) from infinity to a point. This creates a cone of light that can be described by numerical aperture using the previous equation. The half angle, α, can now be defined by the following equation:

Now if we insert this definition for α into the equation for NA we get:

Now if we remember that the f/# = f/D, the NA and f/# can be equated a shown below:

This is an exact equation relating the NA to the f/#, but it is often convenient to have an approximation for this. When n = 1 (medium is air) and if we use a small angle approximation (sin α ≈ tan α) then:

The small angle approximation above is the most common equation for relating NA and f/#.

If the medium is not air, as is common for some microscope objectives, the approximation above can be multiplied by the index of refraction of the medium as shown below.

### Why Use Numerical Aperture?

There are a few main reasons to use NA instead of f/#.

- NA is widely used to specify microscope objectives.
- NA allows for a precise calculation of the point spread function (PSF) and the modulation transfer function (MTF)
- The approximation for NA and f/# significantly breaks down at speeds faster than f/1.2.

### Application Example: Conflicting System Specifications

Often times when starting the design process one can inadvertently request conflicting specifications. This example will show how easy this is to do and how to avoid it when specifying a lens.

###### Requested Lens Specifications: (Refer to Figure 3.2)

- NA = 0.25
- FOV = ±10°
- Image Height h = 10mm
- Entrance Pupil Diameter D = 20mm

These seemingly reasonable specifications will result in a lens that is impossible to design.

###### Finding Compatible Specifications

To determine whether lens specifications are compatible, we need to find the resulting numerical aperture from the other three specifications. To do this we will first need to use the equation below to relate the image height h, focal length f and the half field of view Θ. This equation can be derived using simple geometry using the relationships shown in the red triangle in figure 3.2.

We can now find the focal length of the lens.

Now that we know the focal length, we can calculate the f/# since we have also been given the entrance pupil diameter.

We can now calculate the NA using the approximation derived above.

This conflicts with the initial specification of NA=0.25. The three other specifications have determined the NA!

So what if the customer needs a numerical aperture of 0.25? To get this, at least one of the other specifications need to change. To do this, lets start with the initial specification for NA=0.25 and find what the f/# would be using this spec.

Now we can find the focal length of the lens by assuming that the customer wants entrance pupil diameter to stay at the specified 20mm.

Now lets say that the customer wants to keep the field angle Θ=10°. We can find what the new image height would be.

So then, allowing the image size to change, the non-conflicting lens specification would be:

- NA = 0.25
- FOV = ±10°
- Image Height h = 7.05mm
- Entrance Pupil Diameter D = 20mm.

Below are three sets of lens specifications that would result in the desired system NA. There is an infinite number of specifications that will give the desired NA if one is allowed to change more than one spec.

###### 1. Image Size Flexible

- NA = 0.25
- FOV = ±10°
- Image Height h = 7.05mm
- Entrance Pupil Diameter D = 20mm

###### 2. FOV Flexible

- NA = 0.25
- FOV = ±14.03°
- Image Height h = 10mm
- Entrance Pupil Diameter D = 20mm

###### 3. Entrance Pupil Flexible

- NA = 0.25
- FOV = ±10°
- Image Height h = 10mm
- Entrance Pupil Diameter D = 28.36mm