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Section 2: Understanding Lens Specifications
Section 2.1: System Throughput,
f/#, and Numerical Aperture
The f/# (pronounced “F-number”) setting on a lens controls overall
light throughput, depth of field (DOF), and the ability to produce contrast
at a given resolution. Fundamentally, f/# is the ratio of the focal
length (f) of the lens to the effective aperture diameter (ØEA):
In most lenses, the f/# is set by the turning the iris adjusting ring
(see Section 1.1). This movement opens and closes the iris diaphragm
inside. The numbers labeling the ring denote light throughput associated
with the aperture diameter and usually increase by multiples of
√2. Increasing the f/# by a factor of √2 will halve the aperture area,
decreasing the light throughput by a factor of 2. Lenses with lower
f/#s are called fast and allow more light to pass through the system,
while lenses of higher f/#s are called slow and feature reduced light
throughput. Table 2.1 shows an example of f/#s, aperture diameters,
and effective opening sizes for a 25 mm 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/# and Effects on a Lens’s Theoretical Resolution,
Contrast, DOF:
The f/# impacts more than just light throughput. Specifically, f/# is directly
related to the theoretical resolution, contrast limit, and the depth
of field (DOF) and depth of focus of the lens (see Section 3.4 for more
information about DOF). The f/# also influences aberrations of a specific
lens design. As pixel size decreases, f/# becomes one of the most
important factors of system performance because f/# drives DOF and
resolution in opposite directions. As shown in Table 2.2, the requirements
are often in direct conflict and compromises must be made.
These tradeoffs are discussed in more detail later on in this section.
f/# Diffraction
Depth of
Field
Light
Throughput
Numerical
Aperture
10 +44 (0) 1904 788600 | Edmund Optics® targets f/# Change with Working Distance Change:
The definition of f/# in Equation 2.1 is limited; f/# is defined at an
infinite working distance (WD) where magnification is effectively zero.
In most machine vision applications, the object is located much closer
to the lens than infinitely far away. As such, working F-number,
(f/#)w, seen in Equation 2.2, is a more useful representation of f/# in
most applications.
In the equation for (f/#)w, m represents the magnification (ratio of image
to object height) of the lens. Note that as m approaches zero (as
the object approaches infinity), (f/#)w approaches f/#. It is especially
important to keep (f/#)w in mind at smaller WDs. For example, an
f/2,8, 25 mm focal length lens operating with a magnification of 0,5X
will have an effective (f/#)w of f/4,2. This impacts image quality as
well as the lens’s ability to collect light.
f/# and Numerical Aperture:
It can often be easier to talk about the overall light throughput as the
cone angle, or the numerical aperture (NA), of a lens. The NA of a
lens is defined as the sine of the angle made by the marginal ray and
the optical axis in image space, shown in Figure 2.1.
f
ØEA
f/# = 2.1
(f/#)w ≈ (1 + m) × f/# 2.2
ØEA
f
θ
Figure 2.1a
Projection of image space marginal ray angle to edge of exit pupil
Exit Pupil Entrance Pupil
θ
Figure 2.1b
Figure 2.1: Visual representation of f/#, both for a simple lens (a),
and a real-world system (b).
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 2.1: 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.
Limited Resolution
p q p q q
q p q p p
Table 2.2: Lens performance changes as the f/# changes.