Mathematics in Optics!

It may not be surprising to hear, as in most subject areas, that there is a substantial amount of mathematics in Optics. Luckily for most of us (or unluckily if you like maths) this maths is ‘hidden’ by using rule of thumb systems, data tables or computer software that does all the work for us.

This is also largely true of Optics, though there are some purists out there who prefer the aid of mathematical equations in deriving specific results. I am naturally one of those purists, proud of it too. However, there are some equations that are routinely used in practice, both by Dispensing Opticians, and Optometrists.

Firstly let us consider something fundamental, the power of a correcting lens. In times of old, lenses were given in terms of their focal lengths – the point at which parallel light comes to a focus behind the lens. This was not an ideal system. You may have noticed that when we test eyes, sometimes we need to use a combination of lenses. In the focal length system, if we wanted to add a one metre lens to a two metre lens, the resultant lens is a 66.67cm lens. How does that work?

The total lens power in focal lengths for thin lenses (I think we should avoid thick lens theory at this stage) is:

ftotal = (1/f1 + 1/f2)-1

But there is a better way! The French Ophthalmologist Ferdinand Monoyer pioneered the use of the dioptre to measure spectacle lenses in 1872. The dioptre is the reciprocal of the focal length (F = 1/f) meaning that if we wanted to add a one dioptre lens to a three dioptre lens, the resultant power is four dioptres. Although Maths is fun (citation needed), this additive method is clearly simpler than the more error prone method of focal lengths.

There are also expressions used to calculate the expected lens thickness. They are very accurate and I use them a lot. If we perturb some parameters, we can see how the overall result changes. Doing so indicates some expected relations, like an increase in the prescription relates to an increase in lens thickness. Frame dimensions make a big difference too as well as physiology, with patients with a larger distance between their eyes (up to a point) having a more favourable case to keep the thickness down.

A number of equations relate to the effective power of a spectacle lens. This means that although a lens has a specific power, the power that reaches the eye is not necessarily the same. Some of the factors that contribute to this include the angle of the lens before the eye and the distance between the back surface of the lens and the eye. Let us look at this last case;

FEff = F/(1-dF)

Wear FEff is the effective power of the lens, F is the measured power of the lens and d is the distance in metres between the back surface of the lens and the eye. It can be observed that in small powers, or for small values d that there is barely a change to the effective power. The case where d = 0 very accurately describes the case of contact lenses. Interestingly, the equation correctly predicts that if one holds a positive powered lens far enough away, it behaves like a negative powered lens. The image through the lens is also reversed. To see this, let us imagine a lens of power +2.00 held a metre away. In this case +2.00/(1-1*2.00) = -2.00. Luckily we don’t routinely fit our spectacle lenses a metre away...

That’s all I have time for today, there are many more equations we could have looked at, some would be a lot more unkind to inflict you with!