Geometrical distortion: the Fishtank Effect
A geometrical distortion happens whenever distances in the sample under our microscope as we record them are actually different in reality.
This happens for example in the presence of a Refractive Index mismatch (which also provokes Spherical Aberration!!!). The objective moves along the optical axis a certain distance (a z step that is recorded in the data file), but the focus shifts inside the sample an actual different step. Therefore objects will appear shortened (as in a fish tank when the objects in the water are viewed from the air) or elongated in the microscope, depending on the ratios of the indices.
In most cases the z-sampling as specified in the raw datafile is the nominal sampling distance, i.e. the distance the table or objective actually moved in Z without taking foreshortening due to Refractive Index Mismatch into account. The Huygens Software will automatically adapt the PSF to this situation, but it will not modify the image geometry (see Focal Shift Correction).
After deconvolution the remaining geometric distortion can be corrected by multiplying the (nominal) z-Sampling Distance by the ratio of the medium and immersion refractive indices, in most of the cases a number < 1.
Lateral (XZ) views of a 3D confocal image deconvolved with Huygens. Left: the image is still elongated in a factor of ~ 1.13 along the optical axis due to a refractive index mismatch between the oil lens and the water medium (1.51 / 1.33). Right: after correcting the distortion by dividing the Z sampling by 1.13 (the elongation factor). So assuming that the nominal z-sampling is 300 nm the corrected for geomatric distortion z-sampling = 264 nm (calculation: 300 nm * (1.33/1.51) = 264 nm)
See Focal Shift Correction.
What happens to the aperture angle?
If distances change, then also the aperture angle (the angle subtended by the lens at the focus, but not the Numerical Aperture) will change!!! It is like the apparent bending of a stick half-inmmersed in water...
Image from plus magazine
Let's consider a set up with Numerical Aperture NA = 1.2 and Lens Immersion Medium with Refractive Index nl = 1.44. In the graph below, the x axis is the refractive index nm of the Specimen Embedding Medium, that is mismatched except in the point 1.44.
The red curve is the half-aperture angle (degrees) as it happens in the medium: amc = arcsin(NA/nm). The green curve is what is actually observed, in a Paraxial Simplification, due to the geometry distortion caused by the refractive mismatch: amo = atan( (nm/nl) * tanx? ). The blue line is a reference: the angle that should be observed in a total match situation: alc = arcsin(NA/nl).
When the sample medium is "denser", we are in the case of the fish tank effect: objects are collapsed on the axial direction, thus angles look wider. The green curve goes above the red one.
When it is the other way around objects look elongated, and the apparent angles are narrower.
This effect is much enhanced it the real case where not only paraxial rays are taken into account. Then the apparent angle (green curve), on the left side of the graph, can even go below the blue line!!!