# 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 objective's numerical aperture and the condenser's 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)

Let's use this result to work through some practical examples.

Example 1: Given a camera with a Kodak KAF1401E CCD (pixel size 6.8 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 6.8) / (0.61 x 0.5 / 1.3) = 58

Thus, a 60x, 1.3 NA microscope objective provides a diffraction-limited image for a KAF1401E 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 camera with an e2v CCD37 (pixel size 15.0 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 15.0) / (0.61 x 0.5 / 1.3) = 128

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 CCDs 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.

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