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The Diffraction Limit and f/#
imaging resource guide imaging lenses filters microscopy cameras illumination targets
f/# and Numerical Aperture (NA) – ADVANCED
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 defi ned as the sine of the marginal
ray angle in image space, and is shown in Figure 2.4.
It is important to remember that f/# and NA are inversely related.
NA = 1
Table 2.4 shows both a typical f/# layout on a lens (each successive
fi gure increasing by a factor of √ 2) along with its relationship with
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 specifi ed for microscope objectives are specifi
ed in object space, since light collection is often more easily thought
of there. The other interesting parallel is that infi nite conjugate microscope
objectives can be thought of as machine vision objectives
(focused at infi nity) in reverse.
More on f/# eff ects on resolution can be found in the sections on
MTF, the Diff raction Limit, and the Airy Disk. Details on f/# and DOF
can be found Section 4.4.
f/# greatly impacts all of the following sections and is a very important
concept to understand.
Section 2.5 Limitations on Resolution
and Contrast: The Airy Disk – ADVANCED
When light passes through any size aperture (every lens has a fi nite
aperture), diff raction occurs. The resulting diff raction pattern, a bright
region in the center together with a series of concentric rings of decreasing
intensity around it, is called the Airy Disk (see Figure 2.5).
The diameter of this pattern is related to the wavelength (λ) of the
illuminating light and the size of the circular aperture, which is important
since the Airy Disk is the smallest point to which a beam of light
can be focused. As focused Airy patterns from diff erent details on the
object come close together, they begin to overlap (refer to Section 2.3
for more information on contrast). When the overlapping patterns create
enough constructive interference to reduce contrast, as in Figure
2.3, they eventually become indistinguishable from each other. Figure
2.5 shows the diff erence in spot sizes between a lens set at f/2,8 and a
lens set at f/8.
As pixels continue to reduce in size, this eff ect becomes more of an
issue and eventually is very diffi cult to overcome. The Airy Disk, or minimum
spot size can be calculated using the f/# and wavelength in μm:
minimum spot size (Airy Disk diameter) in μm = 2,44 x λ (μm) x f/#
Table 2.5 shows the calculated Airy Disk diameter for diff erent f/#s
using green light (520 nm or 0,520 μm). The smallest achievable spot
size can quickly exceed the size of small pixels. This leads to diffi -
culties when trying to yield the full resolution capacities of a sensor
with any usable level of contrast. Additionally, this does not take into
account any lens design limitations and manufacturing errors associated
with the fabrication of lens elements and optical assemblies,
which can lead to reductions in the ability to produce the smallest
physically achievable spot and thus reduced levels of resolution and
contrast. Note: This is all theoretical and is the starting point for limitations
in an optical system.
Projection of image space marginal ray angle to edge of exit pupil
Exit Pupil Entrance Pupil
Figure 2.4: Visual representation of f/#, both for a simple lens (a)
and a real-world system (b).
f/# 1,4 2 2,8 4 5,6 8 11 16
NA 0,36 0,25 0,18 0,13 0,09 0,06 0,05 0,03
Table 2.4: Relationship between f/# and numerical aperture.
Figure 2.5: Diff raction increases as the imaging lens iris is closed
f/# Airy Disk Diameter (μm) at a Wavelength of 520 nm
Table 2.5: The minimum spot size, or Airy Disk, increases with f/#
and can quickly surpass pixel size. See Table 2.1 for sample pixel sizes.