Figure 1.4: Relationship between HFOV, sensor size, and WD for a given
PMAG = 6,4 mm PMAG = 0,256X
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Once the required AFOV has been determined, the focal length can
be approximated using Equation 1.2 and the proper lens can be chosen
from a lens speci cation table or datasheet by nding the closest
available focal length with the necessary angular eld of view for the
sensor being used.
The 14.25° derived in Example 1 (see white box below) can be used
to determine the lens that is needed, but the sensor size must also be
chosen. As the sensor size is increased or decreased it will change how
much of the lens’s image is utilized; this will alter the AFOV of the system
and thus the overall FOV. The larger the sensor, the larger the obtainable
AFOV for the same focal length. For example, a 25 mm lens could be
used with a ½” (6,4 mm horizontal) sensor or a 35 mm lens could be used
with a ” (8,8 mm horizontal) sensor as they would both approximately
produce a 14,5° angular FOV on their respective sensors.
Alternatively if the sensor has already been chosen, the focal
length can be determined directly from the FOV and WD by substituting
Equation 1.2 in Equation 1.3, as shown in Equation 1.4,
FL = (h x WD)
where, h is the horizontal sensor dimension (number of horizontal
pixels multiplied by the pixel size) and FL is the focal length of the
lens, both in millimeters; the FOV and WD must be measured in the
same unit system. As previously stated, some amount of exibility
to the system’s working distance should be factored in, as the above
examples are only rst-order approximations and they also do not
take distortion into account.
HFOV Sensor Horizontal
Calculating FOV Using a Lens with a Fixed Magnifi cation
Generally, lenses that have xed magni cations have xed or limited
working distance ranges. While using a Telecentric or other Fixed
Magni cation Lens can be more constraining, as they do not allow for
di erent elds of view by varying the working distance, the calculations
for them are very direct, as shown in Equation 1.5.
Horizontal FOV (mm) = Horizontal Sensor Size (mm)
Since the desired FOV and sensor are often known, the lens selection
process can be simpli ed by restructuring Equation 1.5 into Equation 1.6.
PMAG = Horizontal Sensor Size (mm)
Horizontal FOV (mm)
If the required magni cation is already known and the working distance
is constrained, Equation 1.4 can be rearranged (replacing h/
FOV with magni cation) and used to determine an appropriate xed
focal length lens, as shown in Equation 1.7.
PMAG = FL
Be aware that Equation 1.7 is an approximation and will rapidly deteriorate
for magni cations greater than 0.1 or for short working distances.
For magni cations beyond 0.1, either a Fixed Magni cation
Lens or computer simulations (e.g. Zemax) with the appropriate lens
model should be used. For the same reasons, lens calculators commonly
found on the internet should only be used for reference. When
in doubt, consult a lens speci cation table.
Note: Horizontal FOV is typically used in discussions of FOV as
a matter of convenience, but the sensor aspect ratio (ratio of a sensor’s
width to its height) must be taken into account to ensure that the
entire object ts into the image where the aspect ratio is used as a
fraction (e.g. 4:3 = ⁄), Equation 1.8.
Horizontal FOV = Vertical FOV x Aspect Ratio
LENS FOCAL LENGTH EXAMPLES
Using WD and FOV to Determine Focal Length
Example 1: For a system with a desired working distance of
200 mm and a horizontal FOV of 50 mm, what is the Angular
Field of View (AFOV)?
2 x tan-1 50 mm
2 x 200 mm
AFOV = 14,25°
Calculating FOV Using a Lens with a Fixed Magni cation
Example 2: For an application using a ½” sensor, which has
a horizontal sensor size of 6,4 mm, a horizontal FOV of 25 mm
By reviewing a list of Fixed Magni cation or Telecentric
Lenses, a proper magni cation can be selected. Note: As the
magni cation increases, the size of the eld of view will decrease;
a magni cation that is lower than what is calculated
is usually desirable so that the full eld of view can be visualized.
In the case of Example 2, a 0,25X lens is the closest common
option, which yields a 25,6 mm FOV on the same sensor.
While most sensors are 4:3, 5:4 and 1:1 are alsoquite common.
This distinction in aspect ratio alsoleads to varying dimensions of
sensors of the samesensor format. All of the equations used in
this section can also be used forvertical FOV as long as the sensor’s
vertical dimension issubstituted in for the horizontal dimension
speci ed in the equations.