Human eye and its focal length range....

[fog37]

Hello,

The human eye has an average diameter of $2.5 cm$ and an optical power between $60D$ and $64D$. An object can be brought as close as the near point which is about $25 cm$. If $d_{o}=0.25m$ (object distance) and $d_{i}= 0.025m$ (distance to the retina), the optical power becomes $44D$ when the object is located at the near point using the lens equation. But $44D$ is much less than stated maximum optical power $64D$. What is going on? I know that ~70% of the lensing is done by the cornea and the remaining ~30% by the crystalline...

Thanks!

 

[Ibix]

You appear to be treating the eye as a thin lens in air. It isn't - the humours on the inside of your eye have a refractive index significantly different from that of air.

I'd suggest taking a look at this Hyperphysics page. It'll get you started, at least.

 

[Andy Resnick]

The human eye has an average diameter of $2.5 cm$ and an optical power between $60D$ and $64D$. An object can be brought as close as the near point which is about $25 cm$. If $d_{o}=0.25m$ (object distance) and $d_{i}= 0.025m$ (distance to the retina), the optical power becomes $44D$ when the object is located at the near point using the lens equation. But $44D$ is much less than stated maximum optical power $64D$. What is going on? I know that ~70% of the lensing is done by the cornea and the remaining ~30% by the crystalline...

My go-to reference for this is Atchison and Smith "Optics of the Human Eye"; it's quite comprehensive. What you are asking about is the effect of accommodation; all schematic eyes (Gullstrand, Le Grand, etc) use 'fixed accommodation forms'. The book provides some data comparing relaxed and fully accommodated schematic eyes, the accommodated whole-eye power goes up to about 70D; my suspicion is that your calculation kept the same focal length, which isn't the case of an accommodated eye.

 

[fog37]

Thanks. I read that the human eye has optical power between 60D (when it is relaxed, no accomodation) up to 65D (when it accommodates the most).

For a healthy eye, the near point $NP$ is about $25 cm$. This is the closest distance we can clearly see an object. That is our object distance $d_{o}=0.25$. The image distance must be equal to the diameter of the eye, i.e. the distance from the cornea to the retina. This distance is about $2.5 cm$ and represents $d_i$.

Using the lens equation, we get that the required optical power in this case is about $44D$ which is much less than the optical power of the eye when it is relaxed ($60D$)

How do we justify that?

 

[Ibix]

How do we justify that?

You don't. As I said in #2, you are applying a formula based on a thin lens approximation to something that isn't remotely a thin lens, and that is not justifiable. The formula for the optical power of a single surface in the Hyperphysics page I linked does apply, and gives a much more sensible figure. It's still low, but it's neglecting the lens entirely, so not unexpectedly so.

The full calculation would involve working out both focal planes of the whole system and measuring from there. You can probably treat it as a series of thin lenses if you get the powers of the individual surfaces, but I don't recall off the top of my head what complications there are from not being in air at the back of the system.

 

[fog37]

Thank you ibix. I did treat the whole system as a single converging lens with optical power of 60D. I felt like this was a fine approach since we can always simplify a complex system to a much simpler one. However, I think what you say is correct in the sense that the various materials (with different index of refraction) that the light rays crosse to reach the retina are such that the overall optical power is $60D-65D$. Modeling all this as a single lens located a distance $2.5cm$ from the retina is clearly an incorrect oversimplification. I am sure it is possible to reduce the entire optical system to a single positive lens but its distance from the retina would probably not be equal to $2.5cm$.

Thank you for the the discussion!

 

[Andy Resnick]

Thank you ibix. I did treat the whole system as a single converging lens with optical power of 60D. I felt like this was a fine approach since we can always simplify a complex system to a much simpler one.

But this contradicts the facts: when eyes accommodate, the focal length changes! That's what the ciliary muscles do.

 

[fog37]

Sure, I see how the focal length changes when the eye accommodates. This is necessary since the image distance is fixed and approximately equal to the eyeball diameter...

I was specifically focusing on the near point situation when the eye has to focus the most (smallest focal length and largest optical power) and the case of objects at infinity where the focal length is the smallest and equal to the image distance and the eye is fully relaxed (no accommodation).

 

[Ibix]

I felt like this was a fine approach since we can always simplify a complex system to a much simpler one.

You can. But, as you say, the thin lens in air that (approximately) replaces the cornea, lens, and humors does not lie at the same location as the cornea. You could attempt to model it by a thin lens if you look at the object-to-image distance and determine where a lens of the specified power must go in this system. I don't know if this is a valid approximation for what you are doing, and I'd be inclined to model it as a single surface interface between air and humor (or as multiple surfaces) unless you are very certain that the thin lens will do.