The fraction of the PSF in z direction that is transmitted by a confocal device is controlled by the pinhole diameter as long as the diameter is comparably large. This is the range of "geometrical optics" shown in Figure 1. When the diameter is in the range of the diffraction pattern, diffraction effects become more relevant and the diameter of the pinhole does not control the slice thickness. The smallest achievable optical section is ruled by diffraction (diffraction limit), i.e. by wavelength and numerical aperture (Figure 1).
A possibility to qualitatively describe the dependence on the pinhole diameter is to merge the geometrical dependence (true for large diameters) and the diffraction dependence (true for diameter zero) as done in the formula in Figure 2. In this formula, the diffraction-limited value (left summand) and the geometrical function (right summand) are merged applying the Pythagorean theorem (an empirical approach).
There are different opinions about the amplitude that should be used for both contributions; the formula given is a practical compromise, where the amplitude for the diffraction-limited term is 1. It is vital to keep in mind that these theoretical formulae are (usually rough) approximations. A correct (theoretical) value would require an explicit convolution of illumination and detection beams. And more importantly: the real diffraction patterns depend on lens design and system concepts. Therefore, all theoretical considerations are a rule of thumb – in the best case. For practical work, it is advisable not to overstate the theoretical derivations. If the actual performance of a confocal system is requested, it is necessary to measure the intensity profile in z direction and analyze the full width half maximum for varying pinhole diameters, lens adjustments, colors and samples.
How to measure optical section profiles
To measure intensity profiles in axial direction, the microscope must be equipped with a motorized focus drive. Usually, the software will provide a tool for acquisition of a 3D image stack or – more elegant – a quick profile section, called xz scan. Profile sectioning is performed by scanning the illumination point on a single straight line while simultaneously incrementing the focus position. The displayed image is an xz cut through the sample. If the sample contains structures that are sufficiently smaller than the diffraction limited resolution, the image of the structure is the actual PSF. Software packages for measuring intensity profiles through such an xz cut are widely available.
To monitor the performance of a system, a planar mirror is usually used as sub-diffraction structure (which is true in z dircetion only, of course). Reflected light confocal microscopy yields thinner optical sections as compared to fluorescence, but sample preparation and execution of the measurement are somewhat simpler. These measurements allow system performance to be monitored over long time periods or for comparison with other systems. It is vital, though, to ensure proper sample preparation. For instance, oil-immersion lenses corrected for covered samples need mirrors covered with a coverslip and mounted with immersion oil on both sides of the coverslip. The z positioning must be better than the z resolution and must be calibrated. The pinhole diameter must be calibrated and should be varied between 0.5 and 2 Airy-disc diameter (Airy Unit, AU).
As most applications imply fluorescence, the measurement of actual section sizes should use fluorescent beads. For classical true confocal systems, beads of diameter 50 … 100 nm are appropriate. Preparations of these samples are a bit more cumbersome and fluorescence markers tend to bleach with time.
The measurement is very sensitive to any kind of sample imperfection, refractive index mismatch, temperature and other parameters. In order to avoid unnecessary frustration, only experienced microscope operators should do these tests.
What is an appropriate pinhole diameter?
As indicated above, large pinhole diameters give smooth images, but do not perform optical sections. A fully opened pinhole resembles a widefield microscope (more or less). If the pinhole is very small, the sectioning performance does not improve, but approaches the diffraction limit. As the number of photons passing the pinhole decreases by the square radius when closing the pinhole, the images become unnecessarily noisy. A good setting is around the transition from geometrical to diffraction-limited. This is when the diameter just covers the inner structure of the diffraction pattern, which is also called the Airy disc. This diameter is easily calculated as AU = 1.21 × l/NA and resembles the area inside the first zero of the diffraction pattern generated by a circular aperture. Modern true confocal scanning microscopes automatically set the pinhole diameter to 1 AU, if system parameters are modified.