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ξDiffraction Limit = ×
When the diffraction limit is reached, the lens is incapable of resolving
greater frequencies. Table 2.6 shows the diffraction limit for contrast at
0% at given f/#s. These numbers may seem large but are theoretical;
several other factors must also be considered. As a rule, and because
of inherent background noise, imaging sensors cannot reproduce information
at or near 0% contrast. The contrast generally needs to be
10% or greater to be detected on standard imaging sensors. To avoid
imaging complications, it is recommended to target 20% contrast or
higher at the application-specific critical resolution. Additionally, lens
aberrations and variations associated with manufacturing tolerances
also reduce performance.
Section 2.5: The Modulation Transfer Function
Understanding and calculating a lens's performance can be difficult.
There are many variables that affect lens performance, including diffraction,
optical aberrations, design criteria and philosophy, and manufacturing
tolerances and errors. To obtain optimal system performance, both
optical designers and end users must have access to the metrics used to
display measured lens performance; this measured lens performance is
provided in the form of modulation transfer function (MTF) curves.
Modulation Transfer Function (MTF)
The modulation transfer function (MTF) curve is an informationdense
metric that reflects how a lens reproduces contrast as a function
of spatial frequency (resolution). These curves offer a composite
view of how optical aberrations affect performance at a set of fundamental
parameters set by application need. Understand that changing
any setting on a vision system, including fundamental parameters,
will change the characteristics of the curve. Fundamental lens and
imaging parameters are defined in Section 1.1.
Figure 2.6 shows a common type of MTF curve, which describes the
modulus of the optical transfer function (contrast) vs. frequency (resolution).
How frequency is determined is covered in Section 2.2. This
curve provides a broad overview of a lens's performance at a specific
WD, f/#, sensor size, and wavelength range. Because MTF curves
show resolution and contrast information simultaneously, multiple
lenses can be evaluated based on application requirements and compared
to one another. If used correctly, MTF curves can be used to
determine if an application is feasible.
While there are several ways to display MTF (e.g. MTF v frequency,
MTF vs field), a common way of expressing it for MTF v frequency
f/# and working distance parameters for this MTF curve
Pixel Size:
14 +44(0) 1904 788600 | Edmund Optics® targets is with multiple colored curves (black, blue, green, and red). The solid
black line at the top is the diffraction limit of the lens and represents
the absolute limit of lens performance. No matter how advanced the
lens performance, it cannot be modified to rise above this line. The additional
colored lines on the curve below the diffraction limit represent
the actual MTF performance of the lens. They correspond to different
field heights (positions across the sensor). In this case, there are three
different field heights represented: on-axis (blue), which represents the
center of the image circle; 70% of the diameter of the image circle
(green), which represents about half the image area; and the full image
circle (red), which is the corner of the image sensor that is in use. Note
that some curves will contain more field points for analysis.
The other noteworthy feature on the curves is the difference between
solid and dashed lines, represented on the curve by the letters T and
S, which represent the tangential (T: yz) and sagittal, or “radial” (S: xz)
planes of focus, respectively. These fields are different due to aberrations
that are caused by asymmetry, such as astigmatism, which
is why there is not separate tangential and sagittal on-axis curves.
This variation causes field spots to blend together more quickly in
one direction than the other and produces different contrast levels
on different axes at the same frequency. It is important to consider
the implications of the lower of these two lines when evaluating a
lens for a given application. It is generally advantageous to maximize
the contrast level across the entire sensor to gain the highest levels
of performance in a system. If element tilts or decenters exist, as
they do in manufactured lenses, the asymmetry would cause there
to be different T and S curves on-axis as well, though this is never
illustrated in MTF curves.
10μm 5μm
Spatial Frequency in Cycles Per mm
Figure 2.6: An MTF performance curve illustrates contrast vs. frequency.
Contrast (%)
100
90
80
70
60
50
40
30
20
10
0
0.0 75.0
150.0
MTF: f/2.8, 218mm WD, #63-777
TS 5.50mm
TS 4.00mm
TS 0.00mm
Diff. Limit
Figure 2.6
Dashed Color Lines: Tangential plane MTF
Solid Color Lines: Sagittal plane MTF
MTF On-Axis
Diffraction Limit
pixel limited resolution by pixel size
(Nyquist Freq.)
15 - 20% contrast is typically the
minimum contrast for acceptable
image quality
field height (multiply by 2
for image circle diameter)
Lens Stock Number
2.11
1
(f/#) × λ
1000μm
1mm
f/# Airy Disk Diameter (μm) at a Wavelength of 520nm
1.4 1.78
2 2.54
2.8 3.55
4 5.08
5.6 7.11
8 10.15
11 13.96
16 20.30
Table 2.6: The minimum spot size, or Airy Disk, increases with f/#
and can quickly surpass pixel size. See Table 2.1 for sample pixel sizes.