## Resolution and numerical aperture

The numerical aperture (NA) is related to the refractive index (n) of a medium through which light passes as well as the angular aperture (α) of a given objective (NA= n x sin α). The resolution of a microscope is not solely dependent on the NA of an objective, but the NA of the whole system, taking into account the NA of the microscope condenser. More image detail will be resolved in a microscope system in which all of the optical components are correctly aligned, have a relatively high NA value and are working harmoniously with each other. Resolution is also related to the wavelength of light which is used to image a specimen; light of shorter wavelengths are capable of resolving greater detail than longer wavelengths.

There are three mathematical concepts which need to be taken into consideration when dealing with resolution: ‘Abbe’s Diffraction Limit’, ‘Airy Discs’ and the ‘Rayleigh Criterion’. Each of these are covered below in chronological order.

## George Biddell Airy and ‘Airy Discs’ (1835)

George Biddell Airy (1801-1892) was an English mathematician and astronomer. By the 1826 (aged 25) he was appointed Professor of Mathematics at Trinity College and two years later, he was appointed Professor of Astronomy at the new Cambridge Observatory. From 1835 to 1881he was the ‘Astronomer Royal’ and he has a lunar and Martian crater named in his honour.

Also in the year 1835, he published a paper in the Transactions of the Cambridge Philosophical Society entitled ‘On the Diffraction of an Object-Glass with Circular Aperture’. Airy wrote this paper very much from the view of an astronomer and in it he describes “the form and brightness of the rings or rays surrounding the image of a star as seen in a good telescope”. Despite writing in a different scientific field, these observations are relevant to other optical systems and indeed, the microscope

An **Airy Disc** is the optimally focussed point of light which can be determined by a circular aperture in a perfectly aligned system limited by diffraction. Viewed from above (Figure 1), this appears as a bright point of light around which are concentric rings or ripples (more correctly known as an **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’.

The three-dimensional representation of the Airy Pattern as illustrated in the lower half of Figure 1 is also known as the ‘Point-Spread Function’.

## Ernst Abbe and ‘Abbe’s Diffraction Limit’ (1873)

Ernst Karl Abbe (1840-1905) was a German mathematician and physicist and in 1866, he met Carl Zeiss and together they founded what was known as the ‘Zeiss Optical Works’, now known as Zeiss. In addition, he also co-founded Schott Glassworks in 1884. Abbe was also the first person to define the term **numerical aperture**. In 1873, Abbe published his theory and formula which explained the diffraction limits of the microscope. Abbe recognised that specimen images are composed of a multitude of overlapping, multi-intensity, diffraction-limited points (or Airy Discs).

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, indeed, a combination of all of these factors).

However, even taking all of these factors into consideration, the limits in a real microscope system are still somewhat limited due to the complexity of the whole system, transmission characteristics of glass at wavelengths below 400 nm and the achievement of a high NA in the complete microscope. Lateral resolution in an ideal light microscope is limited to around 200 nm, whereas axial resolution is around 500 nm (for examples of resolution limits, please see below).

## John William Strutt and ‘The Rayleigh Criterion’ (1896)

John William Strutt, 3rd Baron Rayleigh (1842-1919) was an English physicist and a prolific author. During his lifetime, he wrote an astonishing 466 publications including 430 scientific papers. He wrote on a huge range of topics as diverse as bird flight, psychical research, acoustics and in 1895, he discovered argon (for which he was later awarded the Nobel Prize in Physics in 1904).

Rayleigh built upon and expanded the work of George Airy and invented the theory of the ‘Rayleigh Criterion’ in 1896. The Rayleigh Criterion (Figure 2) 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, ‘well resolved’ and are said to meet the **Rayleigh Criterion** (Figure 2, left). When the centre 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 2, mid). If the Airy Discs are closer than this, 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 2, right).

## How to calculate the resolution of a microscope

Taking all of the above theories into consideration, it is clear that there are a number of factors to consider when calculating the theoretical limits of resolution. Resolution is also dependent on the nature of the sample. Let’s look at calculating resolution using Abbe’s diffraction limit and also using the Rayleigh Criterion.

Firstly, it should be remembered that:

**NA= n x sin α**

Where n is the refractive index of the imaging medium and α is half of the angular aperture of the objective. The maximum angular aperture of an objective is around 144º. The sine of half of this angle is 0.95. If using an immersion objective with oil which has a refractive index of 1.52, the maximum NA of the objective will be 1.45. If using a ‘dry’ (non-immersion) objective the maximum NA of the objective will be 0.95 (as air has a refractive index of 1.0).

Abbe’s diffraction formula for lateral (i.e. XY) resolution is:

**d= λ/2 NA**

Where λ is the wavelength of light used to image a specimen. If using a green light of 514 nm and an oil immersion objective with an NA of 1.45, then the (theoretical) limit of resolution will be 177 nm.

Abbe’s diffraction formula for axial (i.e. Z) resolution is:

**d= 2 λ/NA ^{2}**

Again, if we assume a wavelength of 514 nm to observe a specimen with an objective of NA value of 1.45, then the axial resolution will be 488 nm.

The Rayleigh Criterion is a slightly refined formula based on Abbe’s diffraction limits:

**R= 1.22 λ/NAobj+NAcond**

Where λ is the wavelength of light used to image a specimen. NAobj is the NA of the objective. NAcond is the NA of the condenser. The figure of ‘1.22’ is a constant. This is derived from Rayleigh’s work on Bessel Functions. These are used for calculating problems in systems such as wave propagation.

Taking the NA of the condenser into consideration, air (with a refractive index of 1.0) is generally the imaging medium between the condenser and the slide. Assuming the condenser has an angular aperture of 144º then the NAcond value will equal 0.95.

If using a green light of 514 nm, an oil immersion objective with an NA of 1.45, condenser with an NA of 0.95, then the (theoretical) limit of resolution will be 261 nm.

As stated above, the shorter the wavelength of light used to image a specimen, then the more detail will be resolved. So, if using the shortest visible wavelength of light of 400 nm, with an oil immersion objective with an NA of 1.45 and a condenser with an NA of 0.95, then R would equal 203 nm.

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.