**Matching
Resolution**

Researchers using a CCD camera in conjunction
with a microscope desire to work at the maximum
possible spatial resolution allowed by their system.
In order to accomplish this, it is necessary to
properly match the magnification of the microscope
to the CCD.

The first step in this process is to determine
the resolving power of the microscope. The ultimate
limit on the spatial resolution of any optical system
is set by light diffraction; an optical system that
performs to this level is termed diffraction limited.
In this case, the spatial resolution is given by:

**d = 0.61 x lambda / NA**

where d is the smallest resolvable distance,
lambda is the wavelength of light being imaged,
and NA is the numerical aperture of the microscope
objective. This is derived by assuming that two
point sources can be resolved as being separate
when the center of the airy disc from one overlaps
the first dark ring in the diffraction pattern of
the second (the Rayleigh criterion).

It should be further noted that, for microscope
systems, the NA to be used in this formula is the
average of the objectives numerical aperture and
the condensers numerical aperture. Thus, if the
condenser is significantly underfilling the objective
with light, as is sometimes done to improve image
contrast, then spatial resolution is sacrificed.
Any aberrations in the optical system, or other
factors that adversely affect performance, can only
degrade the spatial resolution past this point.
However, most microscope systems do perform at,
or very near, the diffraction limit.

The formula above represents the spatial resolution
in object space. At the detector, the resolution
is the smallest resolvable distance multiplied by
the magnification of the microscope optical system.
It is this value that must be matched with the CCD.

The most obvious approach to matching resolution
might seem to be simply setting this diffraction-limited
resolution to the size of a single pixel. In practice,
what is really required of the imaging system is
that it be able to distinguish adjacent features.
If optical resolution is set equal to single-pixel
size, then it is possible that two adjacent features
of like intensity could each be imaged onto adjacent
pixels on the CCD. In this case, there would be
no way of discerning them as two separate features.

Separating adjacent features requires the presence
of at least one intervening pixel of disparate intensity
value. For this reason, the best spatial resolution
that can be achieved occurs by matching the diffraction-limited
resolution of the optical system to two pixels on
the CCD in each linear dimension. This is called
the Nyquist limit. Expressing this mathematically
we get:

**(0.61 x lambda / NA) x Magnification
= 2.0 x (pixel size)**

Les use this result to work through some practical
examples.

**Example 1:** Given a camera with
a Sony ICX 285 CCD (pixel size 6.45 µm), visible
light (lambda = 0.5 µm), and a 1.3 NA microscope
objective, we can compute the magnification that
will yield maximum spatial resolution.

**M = (2 x 6.45) / (0.61 x
0.5 / 1.3) = 55**

Thus, a 60x, 1.3 NA microscope objective provides
a diffraction-limited image for a Sony ICX 285 CCD
camera without any additional magnification. Keep
in mind, however, that this assumes that the condensing
system also operates at a NA of 1.3. This high NA
means the condenser must be operated in an oil-immersion
mode, as well as the objective.

**Example 2:** Given a EMCCD camera
with an e2v CCD97 (pixel size 16.0 µm), visible
light (lambda = 0.5 µm), and a 100x microscope objective
with a NA of 1.3, we can compute the magnification
that will yield maximum spatial resolution.

**M = (2 x 16.0) / (0.61 x
0.5 / 1.3) = 136**

Since the microscope objective is designed to
operate at 100x, we would need to use an additional
projection optic of approximately 1.25x in order
to provide the optimum magnification.

It should be kept in mind that as magnification
is increased and spatial resolution is improved,
field of view is decreased. Applications that require
both good spatial resolution and a large field of
view will need CCD's with greater numbers of pixels.
It should also be noted that increasing magnification
lowers image brightness on the CCD. This lengthens
exposure times and can limit the ability to monitor
real-time events.