# Resolution

Phaos Optic Science Educational Series

March 1, 2021
12:00 PM (GMT)

10 Minutes

## Resolution

#### Definition

The ability to distinguish two objects from each other. In other words, this is the minimum distance at which two distinct points of a specimen can still be seen – either by the observer or the microscope camera – as separate entities.

We will take a look at three mathematical concepts that are related to resolution.

##### 1. George Biddell Airy and ‘Airy Discs‘

An Airy Disc is the optimally focused point of light which can be determined by a circular aperture in a perfectly aligned system limited by diffraction. As shown in the figure below, the image was capture from above and it appears that there is a bright point of light around the concentric rings or ripples (aka Airy Pattern).

• Point-Spread Function: A three-dimensional representation of the Airy Pattern

The diffraction pattern is determined by the wavelength of light and the size of the aperture through which the light passes. The central point of the Airy Disc contains approximately 84% of the luminous intensity with the remaining 16% in the diffraction pattern around this point.

There are of course many points of light in a specimen as viewed with a microscope, and it is more appropriate to think in terms of numerous Airy Patterns as opposed to a single point of light as described by the term ‘Airy Disc’.

##### 2. Ernst Abbe and ‘Abbe’s Diffraction Limit’

As mentioned in the previous post, in order to increase the resolution (d=λ/2 NA), the specimen must be viewed using either shorter wavelength (λ)  light or through an imaging medium with a relatively high refractive index or with optical components which have a high NA (or a combination of all of these factors).

##### 3. John William Strutt and ‘The Rayleigh Criterion’

It built upon and expanded the work of George Airy and invented the theory of the ‘Rayleigh Criterion’.

The Rayleigh Criterion, as shown below, defines the limit of resolution in a diffraction-limited system, in other words, when two points of light are distinguishable or resolved from each other.

Using the theory of Airy Discs, if the diffraction patterns from two single Airy Discs do not overlap, then they are easily distinguishable, ‘resolved’ and are said to meet the Rayleigh Criterion, Figure (a).

When the center of one Airy Disc is directly overlapped by the first minimum of the diffraction pattern of another, they can be considered to be ‘just resolved’ and still distinguishable as two separate points of light, Figure (b).

Hence, if the Airy Discs were to get even closer, then they do not meet the Rayleigh Criterion and are ‘not resolved’ as two distinct points of light (or separate details within a specimen image, Figure (c).

By taking all the above-mentioned theories, it is evident that we will need to take in account a few factors when calculating the theoretical limits of resolution.

## Calculation

To express the relationship between numerical aperture, wavelength, and resolution,  several equations that have been derived as shown below.

Equation (1): Resolution (r) = λ/(2NA)

Equation (2): Resolution (r) = 0.61λ/NA

Equation (3): Resolution (r) = 1.22λ/(NA(obj) + NA(cond))

• NA : A general term for the microscope numerical aperture,
• λ : Imaging wavelength,
• NA(obj) : Objective numerical aperture
• NA(cond) : Condenser numerical aperture.

Hence, to achieve the maximum (theoretical) resolution in a microscope system, each of the optical components should be of the highest NA available (taking into consideration the angular aperture).

In addition, using a shorter wavelength of light to view the specimen will increase the resolution. Finally, the whole microscope system should be correctly aligned.