Presentation: corrective optics

In the previous presentation we considered vision problems caused by defects in the curvature (and focal length) of the eye. By augmenting the eye with a second lens, here either using a contact lens...

...or glasses, we can compensate for these common vision defects.

In general, we could use a two-lens approach, where light passes through the first lens 1, and then subsequently through the second lens 2.

So for any two-lens system, the main idea is to just take it one lens at a time. The object 1 in front of lens 1 will produce an (intermediate) image 1, and the thin lens equation is applied to to just lens 1.

This image 1 will then become the object 2 for lens 2, which will produce the (final) image 2, and the thin lens equation is again applied to just lens 2.

Keep in mind that the intermediate image 1 from the first lens is subsequently "fed" to the second lens as its object 2. We will be using this general two-step approach to two-lens systems for a magnifying glass (lens 1) held in front of an eye (lens 2); or a microscope (objective lens 1 and eyepiece lens 2); or a telescope (objective lens 1 and eyepiece lens 2).

However, for the specific case of a two-lens system of a contact lens and an eye, we can simplify the two-step approach and instead use a "combinatorial" equation, as a contact lens is placed directly on the eye, and as a result these two lenses can behave together as a single "stacked" lens. As a result, we will compress four years of post-graduate optometry school into this presentation, and be able to prescribe corrective optics for common vision defects.

First, applying the combinatorial model to contacts and eyes.

If you have a prescription from your optometrist, or the box that your contact lenses came in, check out the refractive power P ("diopters") listed there, but note that the focal length is nowhere to be found.

The focal length of contact lenses or glasses are typically not specified, instead they are rated in terms of refractive power P in "diopters," which are merely the inverse of the focal length f (measured in meters). Wait, why do optometrists do that?

The sign convention for refractive power is the same as for focal lengths: positive values for converging contact lenses, and negative values for diverging contact lenses.

For two lenses that are held together in a "stack," the inverse of their combined total focal length will be the addition of the inverse of their individual focal lengths. Note that this is only strictly true for contact lenses and eyes, as they are "in contact" with each other. For glasses and eyes, the space between them means that we can't really consider them a "stack," and must resort to the general two-step two-lens approach mentioned earlier. So for the purposes of prescribing corrective optics, we'll be only looking at contact lenses using the "stack" equation.

Since the refractive power is the reciprocal of the focal length, note that for two lenses held together in a "stack," their combined total refractive power is just the simple addition of their individual refractive powers. This is why optometrists use refractive powers instead of focal lengths, as the math is easier (after you have done the conversions from focal lengths into refractive powers.)

Second, applying this "stacking" equation of combining two lenses to prescribe contacts to correct common vision defects.

We only need to measure your actual far point and your actual near point. If the measured far point is less than the nominal value of infinity (i.e., some measurably finite value), and/or the measured near point is greater than the nominal value of 25 cm...

...then you will need corrective optics, which we can solve for using thin lens equations.

Let's set things up for a Physics 205B student with myopia (nearsightedness). This student has an uncorrected far point (say, 5.0 m), which is the farthest distance that this student's unaided eye can focus on (instead of the nominal farthest distance of ∞). So light from an object at the far point will go through this student's eye, and form an image on the retina.

When the student is wearing contacts to correct for this defect, the student can now focus on normal distant objects at ∞. So now light from an object at ∞ will go through the "stacked" contact lens (lens 1) and student's eye (lens 2), and form an image on the retina. This is all conceptual set-up, the actual math to find the prescription for these contact lenses using thin lens equations is discussed here.

When you do solve for the focal length of contact lenses that correct for myopia, you should find a negative value for the focal length (and refractive power), which means that contact lenses (and glasses) to correct for myopia are diverging lenses. This should make sense because diverging lenses will take light from a distant object to make a closer, upright virtual image that the eye can look at. (Which ray tracing(s) best match this?)

Now let's set things up for a Physics 205B student with hyperopia (farsightedness). This student has an uncorrected near point (say, 0.45 m), which is the nearest distance that this student's unaided eye can focus on (instead of the nominal nearest distance of 0.25 m). So light from an object at the near point will go through this student's eye, and form an image on the retina.

When the student is wearing contacts to correct for this defect, the student can now focus on normal nearby objects at 0.25 m. So now light from an object at 0.25 m will go through the "stacked" contact lens (lens 1) and student's eye (lens 2), and form an image on the retina. This is all conceptual set-up, the actual math to find the prescription for these contact lenses using thin lens equations is discussed here.

When you do solve for the focal length of contact lenses that correct for hyperopia, you should find a positive value for the focal length (and refractive power), which means that contact lenses (and glasses) to correct for myopia are converging lenses. This should make sense because converging lenses will take light from a nearby object to make a farther away, upright virtual image that the eye can look at. (Which ray tracing(s) best match this?)

Now what happens if a Physics 205B student is myopic (nearsighted, can see near, but can't see far) due to defects in curvature of the eye, who as a result of aging develops presbyopia, losing the ability to accommodate and see nearby objects? You would need to prescribe separate diverging and converging lenses to correct for myopia and presbyopia...

...or these two lenses could be combined into bifocal glasses, with the compromise that looking down (at reading distances) would have a diverging lens which only corrects for presbyopia, and looking straight (for far distances) would have a converging lens which only corrects for myopia.