Mathematics in Optics!
- Jun 29, 2017
- 3 min read
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!
SEO Company in Toronto discussions often include topics like technical SEO, page speed, content quality, and backlink development. Every area contributes to overall performance.
Mathematics in optics, now that's where numbers actually make sense of the visible world. As a PhD student balancing research and my part-time role at Last-Minute Assignments, I've come to appreciate how formulas reveal hidden beauty. My college years were a haze of confusion; I'd stare at equations, desperate to write my assignment, feeling like I was missing the light. That struggle taught me that clarity comes gradually, like a lens slowly focusing. Whether it's light bending through a prism or a student finding their path, precision matters, but so does patience. Thanks for showing how math isn't cold; it's illuminating.
I found this post very informative and enjoyable to read. The article explains the subject clearly and provides useful details that are easy to understand. The structure is well planned, and the content flows naturally from one section to the next. It is obvious that the writer has put thought and research into this work. Thank you for creating such valuable content.
Raja game login
Spikes, launch pads, whirling gears, TNT, and wrecking balls exist in every ragdoll hit arena. Hazards kill faster than most weapons. Grab and hurl your opponent into a hazard to avoid a long struggle. Always assess your surroundings before attacking because you can lose in one grab if you're near a threat.
My younger brother showed me eggy car last week and now the whole family is trying to beat each other’s distance records. Funny concept, but the physics make it genuinely fun.