1 mm
1000 μm
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introduction fundamentals lens specifications real world performance telecentricity lens mechanics lens selection guide
It is important to remember that f/# and NA are inversely related.
1
NA = 2 × (f/#) 2.3
Table 2.3 shows a typical f/# layout on a lens with each f/# value
increasing by a factor of √2 along with its NA.
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 2.3: Relationship between f/# and numerical aperture.
Light throughput is typically referred to as NA instead of f/# in microscopy,
but it is important to note that the NA values that are specified
for microscope objectives are specified in object space.
More information about how f/# affects resolution can be found in
the sections on MTF, the diffraction limit, and the Airy disk. Details
on f/# and DOF can be found Section 3.4.
Section 2.2: Resolution
Resolution is an imaging system’s ability to reproduce object detail.
It can be influenced by factors such as the type of lighting used, the
sensor pixel size, and the capabilities of the optics. The smaller the
object detail, the higher the required resolution.
Dividing the number of horizontal or vertical pixels on a sensor into
the size of the object one wishes to observe will indicate how much
space each pixel covers on the object and can be used to estimate
resolution. However, this does not truly determine if the information
on the pixel is distinguishable from the information on any other pixel.
It is important to understand what can limit system resolution. Figure
2.2 shows a pair of squares on a white background. If the squares are
imaged onto neighboring pixels on the camera sensor, they will appear
to be one larger rectangle (a), rather than two separate squares
(b). To distinguish the squares, at least one pixel must be between
them. This minimum distance is the limiting resolution of the system.
The absolute limitation is defined by the size of the pixels on the sensor
and number of pixels on the sensor.
The Line Pair and Sensor Limitations
Resolution is often reported as a dimension. This concept of resolution
being a minimum size that is resolvable by a lens or system is not
a complete description of resolution. Resolution is more accurately
described as a frequency, measured in line pairs per millimeter (lp/
mm). A line pair is a pair of black and white squares in object space.
Lens resolution is unfortunately not absolute. At a given resolution, the
ability to see the two squares as separate entities will be dependent
on grey scale level. A system can more easily resolve a line pair if the
difference in the grey scales of the squares and space between them
is greater (Figure 2.2b). This grey scale difference is known as contrast.
Resolution is thus defined as a spatial frequency, given in lp/mm, at
which a specific contrast is achieved. Contrast is set by convention
for different lens and camera manufacturers but is usually specified
for lenses to be 20%. For this reason, calculating resolution in terms
of lp/mm is extremely useful when comparing lenses and for determining
the best choice for given sensors and applications. Contrast is
explained in more detail in Section 2.3. The sensor is where the system
resolution calculation begins. By starting with the sensor, it is easier
to determine the lens performance required to match the sensor or
other application requirements. The highest frequency resolvable by a
sensor, the Nyquist frequency, is effectively two pixels or one line pair.
Table 2.4 shows the Nyquist limit associated with pixel sizes found on
some common sensors. The resolution of the sensor (ξSensor), referred
to as the system’s image space resolution (ξImage Space ), is calculated by
multiplying the pixel size (s), usually in units of microns, by 2 (to create
a pair), and dividing that into 1000 to convert from μm to mm:
Sensors with larger pixels will have lower limiting resolutions. Sensors
with smaller pixels will have higher limiting resolutions. How this
information is used to determine necessary lens performance can be
found in Section 3. The limiting resolution on the object to be viewed
can be calculated using the relationships between the sensor size, the
field of view (FOV), and the number of pixels on the sensor.
Sensor size refers to the size of a camera sensor’s active area, specified
by the sensor format size (page 145). However, the exact sensor
proportions will vary depending on the aspect ratio, so the nominal
sensor formats should be used only as a guideline, especially for telecentric
lenses and high magnification objectives. The sensor size (H);
horizontal, vertical, or diagonal; can be directly calculated from the
pixel size and number of active pixels on the sensor (p).
Object Space Resolution
To determine the absolute minimum resolvable spot viewable on the
object, the ratio of the FOV to the sensor size must be calculated. This
is also known as the system magnification.
Figure 2.2 Camera Resolution Limit
SENSOR
LENS
OBJECT
SENSOR
Pixels
(a) (b)
Line-Pair
Figure 2.2: Resolving two squares. If the space between the squares
is too small (a) the camera sensor will be unable to resolve them as
separate objects.
2.4
1 1000 μm
ξ 2 × s 1 mm Sensor = ξImage Space = ×
H = s × p × 2.5
Pixel Size (μm) Associated Nyquist Limit (lp/mm)
1,67 299,4
2,2 227,3
3,45 144,9
4,54 110,1
5,5 90,9
Table 2.4: As pixel size increases, the associated Nyquist limit in lp/mm
decreases proportionally.
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