By Mathew Guy Musladin, ABOM

Power, who wouldn’t want a little more power? But power has many forms. For instance, money is power. Knowledge is power. Regarding the field of optics, curvature is power. Since optics is our focus, let’s increase the power of knowledge in this field by exploring the power of curvature.

Curvature is used to describe many things in the field of optics. For instance, there are the concepts of vergence and lens design. Vergence is the idea that expresses how light propagates (moves) regarding optical systems. It can be light coming from an object, going through a visual system, or coming to a focus. Vergence can diverge, run parallel to, or converge in relation to an optical axis.

For our purposes, we will concentrate our study to geometric optics and basic lens design. The vergence needed to correct various forms of ametropia is why we have plus curves and minus curves when designing eyewear. Plus curves are usually described as convex. In other words, they are bulging curves whereas minus curves are described as concave.

When learning optics for the first time, things can be a little confusing. It’s common for the reality of geometric optics to be counterintuitive to what we think the result should look like. I’ve noticed this confusion consistently when teaching students to differentiate between radius of curvature and focal point. This can be problematic when studying single surface refracting areas in relation to the focal point and the radius of curvature, which are necessary concepts for understanding the “mechanics” of ophthalmic lenses.

Again, radius of curvature is the center of a spherical surface, whereas the focal point is defined as the ability of a refracting surface to focus 1cm of parallel light (light that originated at optical Infinity, which is at least 20 feet or 6 meters away) at a particular dioptric length.

Now, the diopter is defined in terms of the meter. A one diopter lens focuses light along the optical center (or the line of sight or the optical axis) one meter from the refracting surface.

The radius of curvature necessary to produce this one diopter focus can vary from one transparent lens material to another transparent lens material depending on the index of refraction. The index of refraction is a measurement of how light is displaced or slows down through a transparent medium. This affects the ability of that lens material to focus light. The higher the index the greater the ability to focus light. But there is a seemingly contradictory concept. The higher the index, the longer the radius! Intuitively, this may not make sense. A longer radius technically means a flatter curve, and flatter curves offer less power. But let’s think about this for a minute. Since a higher index material has more “power”, then the power from the curve would have to be “flatter” in relation to other lower index materials or the tooling index. This is why higher-powered Rx’s can have significantly flatter curves, and thus better cosmetics and lessened magnification or minification effects than an Rx in a lower index.

There is a way to determine any radius of curvature needed for a particular dioptric power for a specific material based on its index of refraction. It is very straightforward, once the radius for a one diopter lens is calculated! This is very simple for a transparent material in air. How, you say? Well, let’s pick an index. What about Trivex at 1.53, since it shares its index with conventional tooling? Take the index and subtract 1.00. In this case it leaves 0.53. Multiply this by 100 (because there is one-hundred centimeters in a meter and focal lengths are typically measured in centimeters) and you have 53 centimeters. That’s it. This is the radius of curvature of a one-diopter lens in Trivex. Whether this produces a plus or minus focal length depends on whether it is a convex plus-focus behind the lens from direction of propagation, or a minus concave-focus in front of the lens. For a two-diopter Trivex lens, divide by two, for a three-diopter lens divide by three, and so on, in order to get the appropriate radius.

Other examples for a one diopter lens are:

High index: 1.70 -1.00 = 0.70 x 100 = 70cm

Polycarbonate: 1.586 - 1.00 = 0.586 x 100 = 58.6cm

For CR39 1.498 -1.00= 0.498 x 100 = 49.8cm.

These examples would all produce a 1.00 diopter lens with a focal length of one meter, even though the radii are different.

The greater the dioptric requirements, the more dramatic the curvature differences between various indices. To get the radius, simply divide the radius for a one diopter lens of a particular index by the desired dioptric power. For instance, a 2.00 diopter lens has a focal length of 50 centimeters. The various radii for different indices to achieve this focal length, and thus the power must be divided by two. They are:

Trivex: 53cm/2 = 26.5cm

High index 1.70: 70cm/2 = 35cm

Polycarbonate: 58.6cm/2 = 29.3cm

CR39: 49.8cm/2 = 24.9cm

This works for any chosen dioptric power.

The moral of the story is this: Higher index lens materials are recommended for higher-powered Rx’s for a very good reason. Flatter curves for the same power equal better cosmetics and visual optics. This equals greater customer satisfaction, and that is an equation we can all be happy with!