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Relation between pinhole size and sampling rate in confocal systems

As it is defined, the Nyquist Rate is independent of the pinhole size. This is due to the choice to relate the Nyquist rate to the theoretical Band Width of the system (the spatial frequency beyond which nothing is transferred by the microscope) and not to Spatial Resolution.

The bandwidth is determined by the Point Spread Function (PSF) and its Fourier Transform the Optical Transfer Function (OTF). It is a good reference point if only because it is independent from factors like noise, pinhole size, optical quality, and others that can effectively reduce the resolution.

It would be a different story if we would have used a criterion based on attenuation of spatial frequencies below a certain factor. Larger pinholes attenuate higher frequencies more while they increase the PSF's Half Intensity Width (HIW), reducing the resolution.

Bandwidth and resolution

Bandwidth doesn't necessarily increase when the HIW of the PSF becomes more narrow. This means that while Spatial Resolution improves, bandwidth not necessarily does.

Changing the illumination has indeed a noticeable effect on both: the conventional (widefield) system's OTF is almost triangle shaped laterally, the confocal OTF, being the convolution of the triangle with itself, looks more like a Gaussian. That means that it is twice as wide but tapers off quickly as it approaches the bandlimit. Increasing the confocal pinhole widens the HIW, and makes the tapering off more rapid. Still, for different pinhole sizes, the OTF becomes truly zero at exactly the same frequency, leaving the bandwidth unchanged.

Defining the ideal Sampling Density in terms of an attenuated frequencies criterion, although a pragmatic approach, involves an arbitrary choice. Therefore we base the Nyquist rate on the well defined theoretical bandwidth. Still, about one third of the band has so low intensities on confocal systems that they can be considered zero in most practical cases. The Huygens Software takes this into account by not reporting about UnderSampling when sampling is ~40% larger than the theoretical Nyquist one (see the Nyquist Calculator for more details).

Note however that with single photon excitation confocals even a very large pinhole will still have a noticeable effect on the blur contributions of far off focus regions, thus improving Spatial Resolution along the optical axis. See Pinhole And Resolution.

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