Diffraction limited resolution (theoretical)
- Microscopy Cluster (Unlicensed)
The ability to spatially resolve areas of interest in heterogeneous samples is often a major aim of many Users on the IRM beamline. However, even we are governed by the laws of physics.
Two of the major players in understanding spatial resolution in microscopy were Ernst Abbe and Lord Rayleigh. Thanks to these brilliant physicists, we can calculate the smallest theoretical distance two objects need to be separated to be resolved when viewed under a microscope.
For simplicity, we will just consider the Rayleigh equation:
where
and n = refractive index of the medium between the front lens and the sample
θ = one-half the angular aperture of the objective or condenser
The following graph shows the minimum spatial resolution across the typical wavelengths measured in mid-IR. There are three assumptions in this graph:
The condenser and objective NAs are the same (or matched)
The refractive index between lens and sample is air, or n = 1
There are no other optical aberrations in the system that would make the resolution worse*
*Note: Achievable spatial resolution is often quoted as 2r to account for optical aberrations
Figure: Theoretical spatial resolution (r) achievable for various NAs across the mid-IR wavenumber range for a refractive index through air (n = 1). The conversion to wavelength, in microns, is shown in green on the x-axis.
For our “hybrid” macro-ATR technique using a Ge hemisphere (n = 4), the resolution is as follows:
Figure: Theoretical spatial resolution (r) for the two objectives currently used with our “hybrid” macro-ATR technique. Calculation factors in the refractive index of the Ge hemisphere crystal and the doubling back of the light path through the objective.
Note: For the macro-ATR technique, the NA of the objective needs to be halved in the Rayleigh equation due to the light returning back through the same objective.