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(Continued from page 37)
Zoom lenses not only are optimized for performance at a single focal
length, but they are required to function over a broad range of focal
lengths, which lowers their overall performance.
Other interesting optical properties may arise as a direct result of
the complex movements in a zoom lens; depending on how the lens
was designed, the f/# can change as the focal length changes. This
type of design is typically avoided for photography or videography
lenses, but for machine vision lenses this is often not the case. It is
also important to recall from Section 2.1: System Throughput, f/#, and
Numerical Aperture, that the working f/# will still change as the magnification
changes, resulting in different exposures.
How to Choose a Variable Magnification Lens
Section 6.1 explained some of the more common types of imaging
lenses to choose from based on the application. In order to narrow
down to a more exact choice for an imaging lens, the fundamental
system parameters of the imaging system must be known (see Section
1: Lens and Imaging Fundamentals on pages 7-9 for more details).
At a minimum, the WD, FOV, and resolution are constraints required
before the proper selection of a lens can occur. In this section, it is
assumed that a camera has already been chosen, as it narrows down
the selection criteria and makes the lens selection easier.
For this discussion, fixed focal length lenses and zoom lenses can
be assumed to operate on the same principles and can be chosen in
the same fashion. This assumes that zoom lenses are being specified
at individual focal lengths and that their zoom functionality has been
locked down.
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All variable magnification lenses come to a point of focus based upon
Equation 6.1. where z' is the image distance (which can be thought of
as the distance between the image plane and the last element), z is
the object distance (or the distance between the object being focused
on and the front lens element), and f is the focal length of the lens.
In this version of this equation, the values of z', z, and f should all be
positive. Equation 6.1 is an approximate equation that assumes that
the lens has no thickness; it is included here to show the relationship
between image and object distances. For a given focal length, as the
object distance (WD) increases, the image distance will decrease.
For a single lens element, such as a plano-convex or bi-convex lens,
this equation is useful in determining which focal length is proper,
given an object and image distance. However, in a machine vision system
that uses objectives with many elements (such as those in Figure
6.1), the equation falls short in several ways: it does not describe the
FOV, and since measuring the image distance in a machine vision lens
is impractical, solving for focal length becomes impossible.
By definition, as zoom lenses change FOV, they remain in focus. If
a lens is defocused as its focal length is changed, it is more accurately
referred to as a varifocal lens, not a zoom lens.
A zoom lens is chosen in much the same way as a fixed focal length
lens, with the additional caveat that the focal length is a variable as
opposed to a fixed parameter, which will alter the AFOV.
Types of Fixed Magnification Lenses:
Telecentric Lenses
Telecentric lenses are highly specialized fixed-magnification lenses
with many powerful optical capabilities and should be used when
high-accuracy measurements are needed. The working principles
and advantages of telecentric lenses are discussed in detail in Section
4: Telecentricity and Perspective Error on pages 29-33.
The selection of a telecentric lens is often thought of as more challenging
than that of a fixed focal length lens, though this is almost
always not the case, as will be seen in Section 6.2.
Microscope Objectives
Microscope objectives are used to image very small objects, generally
with magnifications much greater than 1X. They are fixed magnification
optics that only function properly at a single WD, which is
generally quite small relative to other imaging lenses. Microscope
objectives should be used when a high-magnification image
is required and there are no strict minimum WD constraints.
z'
z
H'
=H
Using the equation for magnification, Equation 6.2, where H' and H are
the size of the image plane (most often a sensor size) and FOV respectively,
Equation 6.1 can be rearranged into a more useful form, shown
in Equation 6.3.
z
Equation 6.3 provides a quick and easy way to solve for which focal
length lens is required to solve an application, given fundamental parameters
such as FOV and sensor size. Often, Equation 6.3 is shown
with the “-1” term dropped, as it is small compared to the rest of
the quantity.
The key assumption made in the application of Equation 6.3 to aid in
lens selection is that the camera has already been chosen, and the only
variable being solved for is the focal length (f) of the appropriate lens.
If this is the case, then lens selection becomes much easier. For example,
assume that FOV of 175 mm needs to be achieved with a 500
mm WD on an IMX250 sensor (2/3”, 5MP). Using Equation 6.3, a 25 mm
lens is the best choice.
In general, when lens calculators are used online, they are using
some form of Equation 6.3 to generate their answer. Note that these are
all first-order calculations and will deviate from ideal when lenses with
large distortion values are used (such as fisheye lenses), and generally
assume that lenses have zero thickness.
Figures 6.3 and 6.4 show Equation 6.3 plotted for several lenses with
different focal lengths on different sensors (corresponding to separate
y-axes).
These plots are useful to visually determine the proper focal length
for a machine vision lens if a camera has already been chosen: simply
follow along the x-axis to the required WD and use the corresponding
y-axis (depending on the sensor that is being used), to find where the
points meet on the coordinate plane. The closest lens to the intersect-
Figure 6.2: A zoom lens at multiple optical magnifications.
Section 6.2: Basic Lens Selection
m = 6.2
H = 1 6.3
H f – '
1
f = 6.1
1
z
1
z' +