Refraction: Swimming goggles, short-sightedness, and underwater vision

Glasses which correct for nearsightedness produce a virtual image nearer to the eye than the actual object:

I'm also nearsighted: with my glasses off, I can only focus about a foot in front of my face. My glasses take objects that are infinitely far away and diverge the light coming from them, so that there are virtual images less than a foot from my face; that's how I'm able to focus on objects through my glasses.

I have also observed that, underwater, I can see all the way across the pool with my swim goggles on. Most swim goggles are bubble-shaped, so we can model the swim goggles as a plano-convex lens:

(source)

The curved surface of the "goggle-lens" is the water-air interface; the flat "surface" is the air-air interface, where there isn't any refraction. Now if you build a plano-convex lens out of glass and use it in the air, it's a converging lens. That's because the speed of light is slower in glass than in air, so the light bends towards the fat part of the lens. However your goggles are acting as a lens made of air and submerged in water. Since the relative indices of refraction are reversed, light moving from the water into the "air lens" is bent away from the fat part of the lens. The goggles, when underwater, therefore act like diverging lenses, which is the way to correct for nearsightedness.

I wonder if people who don't need glasses, or people who are farsighted, find things blurrier underwater with goggles on? Perhaps such a swimmer will comment.

But what is instead the mechanism at work here that renders air-glasses useless under water?

The index of refraction is different between air (~1) and glasses (~1.3). This difference is what makes "air" glasses function. The problem is that water has a refractive index of ~1.3 as well. Now there's hardly any difference in index of refraction between the glasses and water so light hardly bends.

How does this work? [on vision being better with goggles below water vs nothing above water]

I believe I've seen the same results you talk about when wearing goggles. The only thing I can think of is that the goggles themselves are putting pressure on the eyes and slightly changing the shape of them temporarily reducing myopia. If this is actually happening, a normal visioned person can just compensate with the lens within their eye. A far-sighted person would have more difficulty seeing though because they're already past the range that their internal lens can compensate.

do contact lenses have an effect under water (ignoring the risk of them swimming away quickly)?

Contact lenses will have virtually no effect under water because the materials they use for them also have a refractive index near 1.3. Light won't bend much when passing from water to contact to eye.

Finally, what would be the dimensions of a pair of glasses that actually work under water

It's basically impossible to have glasses correct any sort of vision underwater because the refractive index is very similar to water. They simply won't bend the light much. Due to the same issue, contact lenses wouldn't work either because they would basically act like another layer of water. The only way to see clearly would be to have a water/air-gap/eye interface which is what goggles provide.The reason this works is because air's refractive index is very different than water.

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Lastly, it's a terrible image, but it shows that underwater, our eyes hardly bend light which results in us effectively becoming extremely far-sighted (no matter how near-sighted we are above water).

From an optometrist, via email:

This is a great question. I have wondered about something similar to this for years - what glasses prescription would one need in order to see clearly underwater without glasses. I just had a patient who scuba dives ask me about wearing glasses underwater. The optical principals others have posted here are correct. I reached out to an optics professor at Nova Southeastern University school of Optometry in Fort Lauderdale with this question: I have a theoretical optics question for you: Is there a certain glasses Rx that would see clearly under water? I would assume highly myopic patients would have better acuity under water.

Here was her answer: The refractive power of the cornea is due to both its curvature and the difference in index of refraction between the outside air and inside aqueous fluid inside the eye. If an eye is under water, there will no longer be a difference in the refractive index on either side of the cornea (water and aqueous), so the cornea wouldn’t have any refractive power. The normal refractive power of the cornea is around +43 diopters, so if we eliminate that, a patient with zero glasses prescription would become a +43 diopters farsighted! Or, a -43 diopter nearsighted patient would see clearly under water. I’ve seen some pretty high myopic patients, but not quite that high :)

I just spent a minute brushing up on how refractive error is usually corrected for scuba/snorkeling/swimming. The corrective lens is made with a flat front surface to fit inside of the mask. All of the correction comes from the back surface of the corrective lens which is inside the mask in the air. This way, the mask will provide the same correction both outside the water and inside the water.

So the bottom line is that if the refractive index is similar (water, cornea and aqueous liquid inside the eye...the eye has no focusing power). What happens with glasses above water is the light entering the lens slows down because the refractive index of the lens is higher than air. The curvature or diopter power of the lens alters the way the light focuses. Theoretically if you had a high index lens material that was curved enough - one could see clearly under water. FYI refractive index of air is about 1.0, water is 1.33, cornea is about 1.4, standard glasses lens material is 1.5 and high index lenses (often used for high glasses Rxs is around 1.74). One last point is the two main factors that determine ones refractive error is curvature of the cornea and length of the eye. So if someone had a very long eye they would see better under water. A cool experiment would be to take an eye chart under water with a few patients of varying axial lengths. That should show this principal - corneal curvature shouldn't make much of a difference.