Dec 16, 2021
Resolution of diffraction-limited imaging systems using the point spread function
Characterizing the resolution of a diffraction-limited imaging system, such as a microscope, can be done in different ways. In this blog post, the point spread function (PSF) was calculated in OpticStudio to obtain an objective measure of the resolution of these imaging systems. Two methods, which overlap the PSFs of two field points on the image (detector) plane are introduced. The first method uses the multi-configuration editor and the second is the image simulation tool. Both methods are compared, and their pros and cons are discussed.
The performance of imaging systems relates to their resolution, but the definition of resolution varies. In super-resolution microscopy, Fourier Ring Correlation  is used to evaluate the resolution. In diffraction-limited microscopes, the resolution is estimated with the Rayleigh, or Sparrow criterion . In practice, the resolution of those systems can also be measured with microbeads, chosen significantly smaller than the expected resolution, assuming one of the above criteria. These microbeads act as point emitters forming a PSF, which size gives an estimate of the resolution in the image, once again, this size varies depending on its definition. In this blog, the PSF in OpticStudio is used to evaluate the resolution of diffraction-limited imaging systems more objectively.
Method 1: Multi-Configuration Editor (coherent imaging)
Throughout this blog post a microscope design based on a TL4X-SAP objective lens (4X, 0.2 NA), and a TTL200 tube lens, as depicted in Figure 1. Both lenses are available as black boxes from the THORLABS website.
Figure 1 - Microscope design composed of black box elements from THORLABS. The magnification is 4X, and the numerical aperture (NA) is 0.2.
We start by removing all but the on-axis field (Field 1), and converting it to Object Height, as depicted in Figure 2.
Figure 2 - Field setup for the multi-Configuration method to analyse the microscope resolution. Only the on-axis field is kept, and it has been converted to Object Height.
Then, we create two configurations with a single YFIE operand and specify a value of 1.8e-3 mm in the second configuration, as shown in Figure 3.
Figure 3 - Multi-Configuration setup for the PSF overlap analysis. The two-point sources are separated by 1.8 um in the object plane.
Finally, we use a Huygens PSF, and Huygens PSF Cross Section to analyse the overlap of the two PSFs in the image plane. Those two analyses can perform a coherent sum of the individual PSFs across the two configurations (see the Help File for more details). The analyses settings are displayed in Figure 4, and the peculiar multi-Configuration setting is shown with a red box and arrow (this option is not available with the FFT PSF).
Figure 4 - Huygens PSF settings. By checking all configurations from the menu bar, a coherent sum is performed of the individual PSFs.
We focus the resolution analysis on the On-axis Field, but the same analysis can be conducted in every part of the field.
The results of the Huygens PSF are shown in Figure 5.
Figure 5 - Results of the Huygens PSF, and PSF Cross Section overlap with an object plane Y-field separation of 1.8 um (Rayleigh criterion) in multiple configurations. The two-point sources are hardly distinguishable by eye in this microscope design.
As one can see, the two field points are severely overlapped in the image plane, and their respective PSFs are nearly indistinguishable. Two reasons can explain this result. First, by performing a coherent sum of the PSFs, the incoherent illumination assumption of the Rayleigh criterion is violated and causes a degradation of the resolution. Second, the OPD Fan shows aberrations in the order of 0.25 waves, and this microscope sits at the edge of the diffraction limit, meaning it is sufficiently diffraction-limited to allow for analyses such as the Huygens PSF, but it still presents some geometric aberrations, which alters the diffraction-limited performance of the system. From experience of microscope designs, which maximize both fields of view, and resolution tend to fall in that category of near-diffraction-limit systems and are often difficult to characterize based solely on the Rayleigh criterion.
From the Rayleigh criterion, we can increase the separation distance of the fields, and re-evaluate the results. This has been done in Figure 6 with a separation of 2.3 um in the object plane.
Figure 6 - Results of the Huygens PSF, and PSF Cross Section overlap with an object plane Y-field separation of 2.3 um in multiple configurations. By increasing the separation distance between the field points, the PSFs start to separate in the image plane, and one can observe two distinct peaks.
With a greater field separation, the resulting PSFs become distinguishable. The peak separation in the Huygens PSF Cross Section is almost 10 um, which agrees with the system magnification (4X). When we say "distinguishable", it is a qualitative assessment of what we see in Figure 6. However, this criterion can be made more objective if one defines how the peaks should be separated in terms of post-processing. For example, a criterion could be "I want to be able to apply a threshold at 80% and detect two separate spots", in which case, one can use OpticStudio to optimize the peak inter distance to correspond to 80% of the maximum relative irradiance (this is out of the scope of this article).
Lastly, we can also account for the physical pixel size of our detector to get an image as seen from the microscope. The PSFs have a full-width at half maximum of approximately 12 um, and our hypothetic detector has a physical pixel size of 6.5 um, clearly violating Nyquist-Shanon's sampling theorem, which is yet another limitation of the microscope design. Figure 7 shows the Huygens PSF results when the image sampling is changed to 32 by 32 pixels with an Image Delta (the physical pixel size) of 6.5 um.
Figure 7 - PSFs overlap when accounting for the physical pixel size of the detector. Too few pixels compose the PSFs overlap and further degrade the resolution of the microscope.
As one can see, the inadequate physical pixel size further degrades the resolution of the microscope, and while the two peaks were distinguishable in Figure 6, they are now overlapping again in Figure 7. In this case, the microscope resolution is said to be pixel-limited and is given by, at least, twice the pixel size scaled by the magnification, meaning 3.25 um (two times 6.5 divided by 4). The result of a 3.25 um separation distance between the fields in the object plane is depicted in Figure 8.
Figure 8 - PSFs overlap when accounting for the physical pixel size of the detector. A separation of 3.25 um allows separating the close fields again. This distance corresponds to twice the pixel size divided by the magnification, a consequence of Nyquist-Shanon's sampling theorem.
By accounting for the detector pixel size, greater separation is needed to avoid aliasing of the PSF, and ensure it is represented by at least 2 pixels. The fields separation of 3.25 um, is quite different from the 1.8 um Rayleigh criterion and shows just how ambiguous the definition of resolution can be, and yet we have not considered tolerancing of the microscope in this blog, which would further reduce this metric.
To learn about the second method using OpticStudio's Image Simulation, please click here to read the full knowledgebase article.
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