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# f/# (Lens Iris/Aperture Setting)

This is Section 2.4 of the Imaging Resource Guide

The f/# setting on a lens controls many of the lens’s parameters: overall light throughput, depth of field, and the ability to produce contrast at a given resolution. Fundamentally, f/# is the ratio of the effective focal length (EFL) of the lens to the effective aperture diameter (DEP):

(1)$$f/ \# = \frac{\text{EFL}}{D_{\text{EP}}}$$

In most lenses, the f/# is set by the turning the iris adjusting ring, thereby opening and closing the iris diaphragm inside. The numbers labeling the ring denote light throughput with its associated aperture diameter. The numbers usually increase by multiples of √2. Increasing the f/# by a factor of √2 will halve the area of the aperture, effectively decreasing the light throughput of the lens by a factor of 2. Lenses with lower f/#s are considered fast and allow more light to pass through the system, while lenses of higher f/#s are considered slow and feature reduced light throughput.

Table 1 shows an example of f/#, aperture diameters, and effective opening size for a 25mm focal length lens. Notice that from the setting of f/1 to f/2, and again for f/4 to f/8, the lens aperture is reduced by half and the effective area is reduced by a factor of 4 at each interval. This illustrates the reduction in throughput associated with increasing a lens’s f/#.

f/#Lens Aperture Diameter (mm)Aperture Opening Area (mm2)
1 25.0 490.8
1.4 17.9 251.6
2 12.5 122.7
2.8 8.9 62.2
4 6.3 31.2
5.6 4.5 15.9
8 3.1 7.5

Table 1: The relationship between f/# and effective area for a 25mm singlet lens. As the f/# increases, the area decreases, leading to a slower system with less light throughput.

## f/# and Effects on a Lens’s Theoretical Resolution, Contrast, and DOF

The f/# impacts more than just light throughput. Specifically, f/# is directly related to the theoretical resolution and contrast limits and the Depth of Field (DOF) and depth of focus of the lens. Additionally, it will influence the aberrations of a specific lens design.

As pixels continue to decrease in size, f/# becomes one of the most important limiting factors of a system’s performance because its effects on DOF and resolution move in opposite directions. As shown in Table 2, the requirements are often in direct conflict and compromises must be made.

f/#Diffraction Limited ResolutionDepth of FieldLight ThroughputNumerical Aperture

Table 2: Lens performance changes as the f/# varies.

## f# Changes with Working Distance Change

The definition of f/# in Equation 1 is limited in the sense that it is defined at an infinite working distance where the magnification is effectively zero. Most often in machine vision applications, the object is located much closer to the lens than an infinite distance away, and f/# is more accurately represented by the working f/#, Equation 2.

In the equation for working f/#, m represents the paraxial magnification (ratio of image to object height) of the objective. Note that as m approaches zero (as the object approaches infinity), the working f/# is equal to the infinite f/#. It is especially important to keep working f/# in mind at smaller working distances. For example, an f/2.8, 25mm focal length lens operating with a magnification of -0.5X will have an effective working f/# of f/4.2. This impacts image quality as well as the lens’s ability to collect light.

(2)$$\left( f/ \# \right)_w \approx \left( 1 + \left| m \right| \right) \times \left( f/ \# \right)$$

## f/# and Numerical Aperture (NA)

It can often be easier to talk about overall light throughput in a lens in terms of the cone angle, or the numerical aperture (NA), of a lens. The numerical aperture of a lens is defined as the sine of the marginal ray angle in image space, and is shown in Figure 1.

It is important to remember that f/# and NA are inversely related.

(3)$$\text{NA} = \frac{1}{2 \cdot \left( f/ \# \right)}$$
##### Figure 1: Visual Representation of f/#, both for a Simple Lens (a) and a Real-World System (b)

Table 3 shows both a typical f/# layout on a lens (each successive figure increasing by a factor of √2) along with its relationship with numerical aperture.

f/#Numerical Aperture
1.4 0.36
2 0.25
2.8 0.18
4 0.13
5.6 0.09
8 0.06
11 0.05
16 0.03

Table 3: Relationship between f/# and numerical aperture.

Notation in terms of numerical aperture as opposed to f/# is especially common in microscopy, but it is important to keep in mind that the NA values that are specified for microscope objectives are specified in object space, since light collection is often more easily thought of there. The other interesting parallel is that infinite conjugate microscope objectives can be thought of as machine vision objectives (focused at infinity) in reverse.

More on f/# effects on resolution can be found in the sections on MTF, the Diffraction Limit, and the Airy Disk. Details on f/# and DOF can be found in Sensor Relative Illumination, Roll Off and Vignetting.

f/# greatly impacts all of the following sections and is a very important concept to understand.

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