## Areas of various geometries

The amount of light that can pass through an aperture obviously depends on the size of that aperture. A circle, for example, will transmit more light if the radius is increased. The amount then depends on the area A_{c} that is defined by this radius r:

If the aperture was a square, the calculation of the area A_{s} would depend on the edges a:

Finally, the area A_{h} of a hexagon is given through the edge s:

All these dependencies are known from secondary school. Yet, there is freedom to compare the different sizes: for example, one can compare the inscribed hexagons that fit into the circle. In this case, the areas will have the ratio A_{c }: A_{h }: A_{s} = 100 : 83 : 64. The size of the square is ca. 30 % less than that of the hexagon. This result was used to claim that "30 % higher brightness is achieved" when employing a hexagonal pinhole as compared to a square pinhole. This statement was officially offered, even on public websites.

Very similarly, one could use polygons which have the corresponding circle inscribed. In that case the ratios of the areas would be A_{c }: A_{h }: A_{s} = 100 : 110 : 127, and one could conclude that one achieves a higher brightness by ca. 15 % when employing a square pinhole.

It is very obvious that both approaches are scientifically indefensible and will confuse the reader – which might be the purpose of such statements.

As the amount of light that passes through the aperture depends on the area, the correct parameter for comparing different sizes is the area of the various geometries. One can derive by very basic mathematical operations:

to obtain polygons that have the same area as a circular aperture with the radius r. This agreement has been accepted in the confocal community since the very beginning in the late 1980s (for it is the only sensible approach).

As a consequence, pinholes of different geometries will transmit identical amounts of light, if the comparison is done correctly, and the light is distributed homogeneously.

## Convolution of pinhole and PSF

As a matter of fact, the intensity distribution that is projected to the pinhole is by no means homogeneous, but described by the Airy diffraction pattern. In order to find out the actual transmission efficiency, one must superimpose the Airy pattern with the different aperture geometries and sizes. This is for example done for circular and square pinholes, where the independent parameter is a length v_{d} which is presumably adapted to the circle radius in order to yield identical areas [3].

We have calculated the amount of light from an Airy pattern passing through apertures of various geometries and sizes as a function of the area itself, which is easier to understand and gives an appropriate comparison. The simplest case is the dependence of a circular aperture, which allows the intensity distribution of the incoming (fixed) Airy pattern to be integrated:

To calculate the intensities for polygonal apertures, the superposition was done numerically.

Figure 2 shows very clearly that the focal intensity is sufficiently independent of the geometry, at least for circles and equilateral polygons. The differences vary in a range of some 2 %, and the hexagonal distribution is somewhere between the circular and square – as expected.

Any conclusion on the efficiency for transmitting the focal plane signal (referred to as "brightness" in (1)) of the shape of the detection pinhole is therefore not valid. If there are differences, then the comparing parameter was chosen incorrectly.

Fig. 2: Comparison of signal transmission through apertures of different shapes and sizes. Currently used geometries are circles, squares and hexagons. When calculated for illumination by an Airy pattern, all geometries show almost identical dependencies on the size, which is defined by the area of the aperture.

## Pinhole geometries in spectral confocals

As shown above, the number of edges in a regular polygon has no effect on the transmission of the focus signal. There is no reason to assume that this is equally true for defocus signals. It seems that circular, square and hexagonal pinholes are identical in performance. For classical, filter-based confocal microscopes this might be a sensible conclusion. There are differences when applied in confocal microscopes that use dispersive elements as separation devices for various emission bands.

The spectral resolution of a setup that contains dispersive elements does not only depend on the performance of the element but also on the size and geometry of the incident beam. Obviously, a large beam diameter will reduce the spectral resolution. We call the intermediate image plane where the spectrum is recorded (the position of the photometer sliders etc) the spectral plane. Larger objects will cause larger images in the spectral plane. At any given position in the spectral plane, more colors will contribute to the local intensity when the size of the object is large. An inherent result of spectral detection is therefore the dependence of the spectral resolution upon the pinhole size, because it is the pinhole that is imaged in the spectral plane!

The geometry of the image in the spectral plane is therefore the (magnified) diffraction pattern of the pinhole. And these diffraction patterns differ very significantly, depending on the geometry. A circular aperture will cause the well-known Airy pattern, which is rotationally symmetric. A rotation of the pinhole in the spectral plane will cause no differences. The diffraction pattern of a square is not rotationally symmetric and features intense lobes at the edges, while the intensity drops very steeply in the direction from the center to the corners. This effect was used to improve the spectral resolution in true confocal scanning microscopes [4]: when the pinhole is rotated by 45° in the spectral plane, most of the intensity outside of the center disk is led away from the detection range. The overlap of colors in the spectral plane is much reduced – in this case by ca. 1.5-fold.

In a system that uses a hexagonal pinhole, this effect is much less prominent, and the gain would be negligible.

The best spectral confocal setup therefore employs an appropriately oriented square pinhole.

## References

- Light VJ, Maverick G: Paradigm shift in laser scanning confocal microscopy: Resonant real time live spectral imaging for cell dynamics. Advanced Biotech. March 36–40 (2008).
- www.nikon.com/products/instruments/lineup/bioscience/confocal/singlephoton/a1/ (May 13
^{th}, 2013) - Sheppard CRJ: Signal Level in Confocal Microscopes, in: Pawley J (ed.): Handbook of Biological Confocal Microscopy, 3
^{rd}edition, p. 444 (2006). - Engelhardt J: Optical arrangement provided for a spectral fanning out of a light beam. US Pat. 6801359 B1, filed Jan 29
^{th}, 1998.

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