# Resolving Disputes

Super-resolution has been receiving a lot of press in the last 6 months, largely given the 2014 Nobel prize to Betzig, Hell and Moerner “for the development of super-resolved fluorescence microscopy“.

In this post, we’re going to take a look at one way to accurately measure the size of a point emitter which is helpful when used as a metric for resolving power.

### The Basics

Resolution is the ability to tell things apart. If two objects are below the resolution limit of your system, they will appear as one:

This is equally true for single fluorophores being examined by a microscope as it is for looking at two people standing side by side from a great distance. At some distance, you won’t be able to tell the two things apart; it will just look like one object.

The resolving power of a microscope system is well established by the Abbé Limit:

Which tells us that you can’t resolve things if they are closer than the wavelength of light (λ) divided by twice the numerical aperture (NA) of the microscope.

In practical terms, for a high-end imaging system this means that objects closer than ~220nm cannot be resolved.

### Improving resolution

There are many ways to improve resolving power, that I’m not going to go into here. If you’re trying to improve resolution however, you need a way to measure the resolving power of your system (or at least the relative change). In this instance, we’re going to do that by imaging a point emitter.

For example, if we image a Quantum Dot at 250x magnification (diameter of ~20nm) it looks something like this:

Each of those pixels represents ~100nm so an otherwise 20nm object appears closer to 400nm because of the diffraction of light.

In microscopy, anything smaller than the resolution limit of your microscope (remember, that’s about 220nm) will act as an infinitely small point in space. The diffraction of light means that by the time the light gets to your detector, it will have spread out. This is a fundamental concept in optics and the reason why there is a resolution limit at all.

### Measuring the diffraction pattern

Measuring the diffraction pattern, will give you some idea of the resolving power of your microscope (the higher the resolution, the sharper the diffraction pattern will appear). For the purposes of this post, we’re going to use Fiji to characterise the size of the emission profile. Here’s how:

Open the image in Fiji and using the Line Tool draw a line across the emission profile:

Use [Analyze > Plot Profile] (available with the shortcut Ctrl+k), to measure and plot the intensity profile along the line:

From here, hit the “Copy…” button to copy the plot data to the clipboard. Open the [Analyze > Tools > Curve Fitter…] tool and paste the values into the window:

We’re going to fit the emission profile using a Gaussian function, to estimate the major peak of an Airy pattern (it’s a recognised technique). A Gaussian function is defined by three parameters, the alpha (height of the curve), mu (the mean) and sigma (the standard deviation, which is related to the width of the curve) related according to the following equation (which also has a beta term acting as a ‘y offset’ taking into account any background signal):

`y = β + (α-β)*exp(-(x-μ)*(x-μ)/(2*σ*σ))`

This is a built-in function in Fiji, so pull down the box and select “Gaussian”. Hit fit and you’ll be presented with the results of the fit, as well as a plot of the curve on your data.

This is pretty, but we’re only really interested in the width of the curve, which can be calculated as the Full Width at Half Maximum, Calculated from the sigma (σ) value above.

```FWHM = σ * 2 * √ (2 * ln(2) )
≈ σ * 2.36```

This then gives you a value that you can compare after image processing for instance with Stochasic Optical Fluctuation Imaging although that’s definitely another post.