Understanding Ocular Wavefront aberrations

Understanding wavefront aberrations and how they apply to the eye is not essential knowledge for eye care practitioners in day to day practice, however when it comes to myopia management, ocular aberrations, particularly spherical aberration and coma, are increasingly being referred to or reported on in published research papers. In this article, I’ll help you understand how to interpret the literature on ocular aberrations and their relevance in myopia management.

Wavefront aberrations can appear complex to understand, however when broken down to core concepts a single wavefront is nothing more than a three dimensional shape that is frozen in time. The complexity comes down to how to describe the shape, and you already do this to some degree in your everyday practice! Sphere and cylinder are corrections for lower order aberrations and both of these are just shapes. A lens providing refractive power of -0.25D is cut from a very large sphere that is almost flat on its surface, while a -10.00D lens is cut from a much smaller sphere to provide a more curved surface. Imagine these as upturned bowls and the -0.25 is like a very large mixing bowl and the -10.00 is like a small dessert bowl. In both cases a complex spherical shape is being described by just one number!

Cylinder powers are the same, except when describing a basic cylinder there is now a flat edge and a curve in just one meridian. Again this seemingly complex shape can be described by a single number, the radius of curvature along the curved meridian or its power if a refracting surface. By combining sphere and cylinder we can define an even more complex shape with different curvature along two meridians described by two numbers. The overall steepness of the surface is described by the sphere value, with a higher number describing an overall steeper surface, and the cylinder power defining how one meridian deviates from the spherical surface.

Higher order aberrations when described in numerical form are nothing more than shapes of progressively increasing complexity that follow a similar system. For each wavefront snapshot the only thing that needs to be done is to calculate the amount, or weighting, of each shape that exists within the wavefront, with the weights usually described in units of wavelength of the wavefront being measured. OK, I admit that this is beginning to sound as difficult at trying to learn a foreign language, and that calculating the weighting of individual wavefront shape components is highly complex, however computer software takes care of this complexity and will happily punch out a series of wavefront numbers to any surface you throw at it. You only need to be able to interpret the numbers, and in this case all you need to remember is that a weighting value of zero means that none of the shape is present - just like cylinder power of zero would indicate that no astigmatism is present, and that an increasing higher weighting indicates that increasing more of the shape is present in the wavefront.

Zernike representation

There are numerous mathematical functions that can be used to describe complex surfaces, but when it comes to wavefront analysis an often used series of shapes are represented by what are known as Zernike polynomials. Fortunately, you don’t need to understand anything about the mathematical constructs of Zernike polynomials to be able to understand how they can represent the shape of a wavefront. They are just a series of shapes, simple as that, and all you need to visualise is the shapes that each individual Zernike polynomial represents.

Conveniently, each polynomial can be identified by a single number, and the good news is that when it comes to eyes, and particularly myopia control, you only really need to concentrate on two of them - spherical aberration (Z12) and coma (Z7 & Z8).

Spherical aberration (Z12)

Spherical aberration can be thought of as representing a shape like a bowl with a lip in comparison to the spherical shape previously described that has no lip (see images below). You’ll also notice by looking at the left image that the bowl is upside down, which is where negative weighting values are introduced. Adding a negative sign indicates that the shape is inverted and not that there is less of the shape. So a spherical aberration with a weight of 0.5 will have the same dimensions as a -0.5 weight spherical aberration except that in the latter case the image will be inverted.

Positive spherical aberration

Positive spherical aberration

Negative spherical aberration

Negative spherical aberration

Like a sphere, spherical aberration is what’s classed as a rotationally symmetrical aberration - if you image a slice (section) through the shape and then spin it around its central axis by any amount and take a second section image, both sections will look the same. A cylinder on the other hand changes shape as you spin it around, meaning that it is a non-rotationally symmetrical surface. More in this shortly when we dive into coma, but before we concede this aspect, the measurement aperture (also known as pupil size) of the wavefront needs to be understood.

The importance of aperture (pupil size)

Zernike wavefront shapes are calculated over a unit circle (aperture/pupil). In the case of spherical aberration this is important, because the pupil diameter determines where the spherical aberration bowl type lip just described falls. Imagine a dessert bowl with a big lip that is say 10cm in diameter that you have been asked to describe using Zernike terms. You would start by working out how much of the bowl could be represented by a sphere - lets say arbitrarily that the bowl is around 70% like a sphere - it’s sphere weighting is 70% - now I know I talked about wavelength units earlier, but there we were considering wavefronts. Here we are instead talking about a dessert bowl, so 70% spherical weighting, bear with me.

Next up we work out how much of the bowl looks like the spherical aberration (Z12) shape. In this case the bowl has a big lip, so maybe it has 60% weighting of spherical aberration, but remember what we have already said about shape. To make spherical aberration look like a bowl we need to invert the shape, so in this case the weighting would be recorded as -60%. Now we are in a position to be able to call any dessert bowl making factory that understands Zernike notation and ask them to make a bowl that is 10cm in diameter, with 70% spherical weight and -60% spherical aberration weight, and they would do a reasonable job of re-creating the bowl just from telling them these numbers, while you quietly reflect on the amazingness and wonder of Zernike polynomials! Well not quite because you still need to get your head around how pupil size affects these measurements.

Spherical aberration bowl comparison 10cm

Let’s now take the same bowl, except we are now going to only consider the central 6cm diameter - imagine you are looking at the bowl from the top through a piece of card with a 6 cm hole. Same bowl, we’ve just reduced the shape assessment aperture from 10cm to 6cm. Now when we apply our Zernike shape descriptors we might still say that the bowl is even more like a sphere so has a sphere weighting of 90%, but now when it comes to assessing weighting of the spherical aberration shape we are missing the lip that has been obscured from assessing over the smaller 6cm aperture. Now we have a smaller amount of the spherical aberration as there is no lip, for arguments sake let’s say just -10% weighting (remember that the negative value just means that the shape has been inverted).

Spherical aberration bowl comparison 6cm

Same bowl and same shape descriptors, just that when measured over a 10cm diameter there is -60% weight of the spherical aberration shape and when measured over a 6cm diameter there is only -10% weighting. Also the sphere rating has increased from 70% at 10cm diameter to 90% at 6cm diameter. The really really important point I am getting to here is that aberrations, at least when described using Zernike polynomials, are meaningless descriptors if the pupil size is not stated. If we called up our fictitious bowl maker from earlier and just told them the Zernike numbers without telling them the diameter, they wouldn’t know how large to make the bowl.

Spherical aberration is arguably the most important ocular aberration to understand in myopia management, but this post is starting to get way too long. Dive into the following posts to find out more - Understanding Spherical Aberration and Spherical aberration, accommodation and multifocal soft contact lenses.

Coma (Z7 & Z8)

Coma is represented by two Zernike terms because coma describes a non-rotationally symmetrical shape (a slice taken through the shape changes as the shape is rotated around its central axis) , which should be easy to get your head around because you use a non-rotationally symmetrical shape descriptor every time you prescribe a cylinder power. You define the weight of the cylinder (its power) and its orientation (axis) - in other words, you are already adept at using two numbers to define the shape of a non-rotationally symmetrical surface. When it comes to wavefronts this is just done slightly differently - bear with me as we take another side track, this time into vector representation of refraction cylinder power.

Vector representation of refraction

As eye care practitioners we are used to axis notation when defining cylinder orientation, but a different way that cylindrical surfaces can be defined is by how much cylinder power exits in the horizontal/vertical meridian and how much cylinder power exists in the oblique (45°/135°) meridians. These meridians in vector refraction representation are given the terms J180 and J45. Using this notation a cylinder with axis 180° would all be weighted in the J180 term with no weight in the J45 term, and a cylinder with axis 45° would all be weighted in the J45 term with no weight in the J180 term. Cylinder axes in-between have proportional weighting, so a cylinder with axis 22.5° would have identical weight in J180 and J45, and a cylinder with axis 30° would have 2/3 of weight in J45 and 1/3 in J45.

Clearly this notation is not practical for describing cylinder in a refraction, but it does make for an easier descriptor when comparative consistency is needed, for example when wanting to apply statistical analysis, because J180 and J45 use the same descriptor. In this case both are measured in diopters, while the notation used in practice uses diopters (power) and degrees (axis). It’s easier to make comparisons across different people if the same descriptors are used, which is why vector representation is typically described in research papers, but I digress.

Back to coma

Coma is like astigmatism in that is is a non-rotationally symmetrical shape - Z7 describes how much weight of the shape exists in the vertical meridian and Z8 describes how much weight of the coma shape exists in the horizontal meridian. The coma shape itself is like an offset pimple - see below, and the weight determines how much of this pimple is apparent in the wavefront being described. A coma value of zero indicates that the shape is not apparent in the wavefront at all, and the pimple shape becomes more pronounced with increasing weight of coma.

Z7 Coma

Z7 Coma

Z8 Coma

Z8 Coma

Just like spherical aberration, coma is also affected by pupil size - assessing a wavefront over a smaller pupil size will move the coma ‘pimple’ closer to the optical axis, and assessing over a larger pupil size will move the coma ‘pimple’ further from the optical axis. This further illuminating why Zernike defined wavefront terms are meaningless without know the pupil diameter they have been assessed across.

Total aberrations

A weighting value for total aberrations is simply achieved by summing the absolute values (any negative signs removed) of the individual Zernike polynomial weights - a higher number indicating greater complexity in the overall shape of the wavefront. For this reason wavefronts measured across larger pupil diameters tend too, though not always, exhibit higher amounts of aberration. Higher total wavefront aberrations with increasing pupil diameter can also be explained by larger pupils containing a larger surface area - so more area over which deviation can occur. Once again this highlights the importance of indicating the pupil sized that was used during measurement when displaying wavefront aberration values. Sorry that I keep banging this drum, but I’ve sadly read too many scientific posters and papers reporting on wavefront aberrations in some form that have not included the pupil diameter that was used thereby rendering the reported analysis meaningless.

Ocular aberrations

Eyes are very powerful refracting devices so as a result, wavefront surfaces as they refract through the eye are generally very steep - they look roughly look like a sphere regardless of how much aberration (deviation) is present. This means that a large part of their overall shape can be described as being like a sphere, and as we’ve already discussed, you already good at visualising this because you do it every working day when considering a spherical refraction.

There is also a reasonable amount of spherical aberration present in human eyes, which, just to make things more interesting, varies with accommodation. Some multifocal contact lenses deliberately manipulate spherical aberration as a way to increase depth of focus for presbyopia correction. Here there is a delicate balance though as is the case for most higher order wavefront aberrations, higher weighting tends to lead to greater deviation from the ideal focus. Induce too much spherical aberration and the benefit of increased depth of focus is lost to too much image distortion. Read more about spherical aberration in Understanding Spherical Aberration and Spherical aberration, accommodation and multifocal soft contact lenses.

Coma is typically less apparent than spherical aberration, but varies across individuals and tends more towards distorting quality of vision. Some people have a slightly displaced corneal apex meaning that their line of sight crosses to the side of the corneal apex. These register as coma because they partially fit the displaced ‘pimple’ effect I described earlier, where the pimple is the apex of the cornea. In more extreme cases, like keratoconus, the ‘pimple’ is more pronounced and more displaced leading to higher amounts of coma. Understanding this association with keratoconus hopefully helps visualise why coma tends create a negative effect on quality of vision, with higher amounts of coma creating greater distortion to vision.

Summary

If you have read this far you should now have sufficient understanding on how to interpret wavefront aberrations and be able to form an understanding of their interaction when mentioned in research papers. Throughout this post I have included links that will direct you to posts that dive deeper into understanding higher order aberrations. Otherwise the main take home messages are that wavefront aberration measurements are just shape descriptors and always keep in mind that the pupil diameter over which the wavefront was assessed is an essential characteristic that can not be omitted.

Further reading in our science series on aberrations

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About Paul

Dr Paul Gifford is a research scientist and industry innovator based in Brisbane, Australia, and co-founder of Myopia Profile.

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