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Light travels differently in water and air. Some of that is because of light's speed in a substance, and some is the light absorption that substance exhibits.

Is the difference in light scattering underwater enough to make a sea-dwelling race (like merfolk) be noticeably Nearsighted or Farsighted when compared to land-dwelling races (like humans) when on land/in our atmosphere?

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Interesting question!

First things first, let’s talk about how an eye works. Well, I don’t know. I’m no doctor. It’s probably crazy complex. But for our purposes, an eye is just gonna be made up of a converging lens in front of the retina. The job of the lens is to make light converge on the retina. If the focal point of the lens is in front, you’re nearsighted. If the focal point is somewhere behind the retina, you’re farsighted (all of this is very simplified, but that's enough to just grab the concepts we need for a qualitative answer)

nearsighted vs farsighted

Ok, well an eye is a {lens+retina} system. But how does a lens work? That I do know, but we’re still going to keep it simple. We’ll just say it uses the geometry of the interface to decide what to do with the light rays (concave=diverging lens, convex=converging), thanks to refraction. Refraction is what you mention in your question; when light goes from a material with refraction index $n_1$ to another with refraction index $n_2$, light bends according to the Snell-Descartes law that you might remember from your high school physic classes: $$sin(i_2)=\frac{n_1}{n_2}sin(i_1)$$

where $i_1$ and $i_2$ are the angles that it makes with the interface, but we don’t care much about the details of how the light bends actually. The only thing we need to notice is that how much it bends is all decided by the ratio $\frac{n_1}{n_2}$. This means that, the bigger the difference between $n_1$ and $n_2$, the more refraction you get (For example, if $n_1=n_2$, the angle would remain unchanged and you get no refraction at all, making your lens useless).

So! The refraction index of air is lower than that of water. That means that the difference in refraction index between the air and the lens will be larger than it is between lens and water. So, once you emerge, the lens in front of your retina suddenly bends light harder; the focal point is at a shorter distance. If your species is adapted to see clearly underwater, you would then be in the situation shown on the left of the image above.

Your amphibians would be near-sighted in the air, compared to what they are underwater

Also, they will of course get much brighter light above the surface than in the depths of the sea. If they have nothing to adapt to that, they might just be blinded too. Otherwise, they can simply have cat-like pupils to accommodate for both dark and bright conditions.

Important edit (estimation time):

Ok, I was actually quite curious to know how bad it would be. And boy, oh boy, would the little mermaid be blind.

We will assume that the cornea of your merfolks allow them to focus a point at infinity when they are underwater (just like us humans in air)

Without going into the details of the proof, you can establish that, for a given geometry of your lense, the refractive power varies proportionally to $\frac{n_{medium}}{n_{lens}}-1$. We will take $n_{air}=1$, $n_{water}=1.33$.

The ratio of the corrective powers of the cornea in air vs under water will then be given by:

$$\text{power ratio}=\frac{n_{air}-n_{lens}}{n_{water}-n_{lens}}$$

If we assume that the eyes of your merfolks are built similarly to ours (with just a difference in curvature of the lens), things get ridiculous real fast. If we take the human refractive index for cornea ($n_{cornea}=1.376$), the formula above gives you a ratio of about 8!! The power of the cornea will be multiplied by 8 as your merfolk emerge. With an eyeball diameter of 2.3cm, you need an optical power of about 43 dioptres to focus light correctly. So, out of the water, the cornea of your merfolks would now have a power of... 355 dioptes!!

Good luck finding -312 glasses. For geometrical reasons, it would actually be impossible for your merfolks to have such a good vision underwater in the first place with those conditions anyway

Ok, your merfolks obviously have to be built different. The highest refractive index used in pharmaceutical lenses seems to be around 1.7. Let's say your $n_{cornea}=1.7$ for your merfolks, this time the power of their cornea would only be about doubled when they emerge (according to the equation above). So, you'd be doing ok with -39 dioptre glasses. We're getting somewhere!

Double their eye size, and they only need their cornea to offer 22 dipotres under water. They would then have about 41 dioptres in air, which means they can get away with "only" a -19 correction this time. That's still super duper nearsighted but it seems to be at the extreme range of what can be corrected by glasses.

Conclusion: Your merfolks need at the very least a super-effective cornea and eyes twice as big if they wanna have the slightest hope of seeing anything out of the water. Even then, they will still need to wear the best glasses we can make

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    $\begingroup$ Good answer. Nearsighted humans can see with better focus underwater, so the reverse is logically the case, that a creature with good focus underwater would be nearsighted in air. $\endgroup$ Aug 9 at 16:49
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    $\begingroup$ @NuclearHoagie I'm nearsighted and I never noticed that! It's time for experiments. Wish my eyeballs luck. $\endgroup$ Aug 9 at 16:52
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    $\begingroup$ @BarbaudJulien If your vision is correctable, you aren't nearsighted enough to focus correctly under water. I wear -6.75 diopter lenses, and my vision is still blurry under water (opposite direction from in air). $\endgroup$
    – Zeiss Ikon
    Aug 9 at 18:07
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    $\begingroup$ Worth noting that there are fish with "bifocal" eyes -- each eye divided; top half has correct refraction in air, bottom half is right for water. An eye that can change shape pretty radically would also solve the problem. Or wearing "inverted goggles" -- like swim goggles but filled with water. $\endgroup$
    – Zeiss Ikon
    Aug 10 at 11:15
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    $\begingroup$ Of course if you're going with corrective glasses the easiest solution would be goggles just like humans use under water. A flat piece of glass on goggles filled with sea water would give perfect vision just like goggles filled with air allow us to see underwater $\endgroup$ Aug 10 at 14:20
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For a complementary thought experiment, what focal length do your merfolk need? Depending on depth and murkiness of water, it may be pointless to have long-distance focus anyway - Wikipedia tells me that optimal conditions still result in a max distance of 80m, and in most cases this would be a wildly optimistic overestimate. Since focal range is always a compromise between near- and long-sightedness, I am guessing your merfolk will be under selective pressure to be near-sighted, since long-sightedness brings them no advantage anyway. As Julien then calculates, this will be even more marked once they come out of the water.

The other aspect, which only you know, is how much they rely on sight (for example, if they use bioluminescence or other visual means of communication). They may simply have poor vision, since the medium they live in is not as conducive to the sense of sight as air. This may be somewhat story-based - do you want these merfolk to be able to interact easily with humans? How human-like do you want them to be? Will they be point-of-view characters (and therefore responsible for relaying descriptions to the reader)?

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  • $\begingroup$ That's a good point, but the value of 80m is true for a human eye. Most nocturnal animals require much less light than we do to see. I think an eye adapted for vision in dark conditions could push significantly further than that limit under water. But the deeper you go, the harder it gets $\endgroup$ Aug 10 at 12:08
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    $\begingroup$ I suspect the limit is turbidity, not light - water is nowhere as transparent as air, and a natural body of water has a lot more stuff in it. The 80m limit was recorded with optical instruments $\endgroup$
    – Ottie
    Aug 10 at 12:13
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Nearsighted or Blind

Even clean water absorbs half of the light in the first 10 metres. This is doubly bad for your merfolk as being 10m underwater and looking at something 10m away you only get at most 1/4 of the light you would otherwise. This is because the light travels 10m to hit the object and 10m again to hit your eyeballs.

More likely however the light travels diagonally for the first part of the journey you get less than 1/4 of the starting light.

Seeing distant objects is a lost cause. Seeing closeby objects only works in shallow waters.

This suggests your merfolk are nearsighted if they live in crystal clear shallow water. For example a tropical reef. But they still need other senses to detect things far away.

If they live in murky or deep water then their eyes are no use. With time they become almost blind. Like the Yangtze River Dolphins.

enter image description here

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  • $\begingroup$ Your eyeballs aren't transmitting the light, so the double distance idea is silly. $\endgroup$
    – JRE
    Aug 10 at 14:33
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    $\begingroup$ @JRE 10m from the surface down through the water to hit the object. Another 10m from the object to your eyes. 20m total. If we were 100m down and 10 away it would be 110m total. $\endgroup$
    – Daron
    Aug 10 at 14:36

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