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.

Numerical apertureunit

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Numerical Aperturecalculator

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.

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.

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:

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Numerical apertureformula

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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).

Numerical apertureof optical fiber

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Numerical aperture nacalculator

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.

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.

Numerical apertureof microscope

Numerical apertureof lens

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.

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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.

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:

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