Use the parameters in the text for Schematic eye 1 to...

Use the parameters in the text for Schematic eye 1 to determine how far past the anterior surface of an optical element incident parallel rays converge to the optic axis to form a focus when the element is assumed to have a finite thickness and compare this to that assuming the thin lens approximation. Do this for the: (a) Cornea. (b) Crystalline lens. (c) Does this suggest that the correction for a thick lens is significant?

Use the parameters in the text for Schematic eye 1 to determine how far past the anterior surface of an optical element incident parallel rays converge to the optic axis to form a focus when the element is assumed to have a finite thickness and compare this to that assuming the thin lens approximation. Do this for the:
(a) Cornea.
(b) Crystalline lens.
(c) Does this suggest that the correction for a thick lens is significant?

Key Concepts

Thin Lens Approximation

The thin lens approximation simplifies optical calculations by assuming that the lens has negligible thickness. This means that all the refraction is presumed to occur at a single plane, allowing for easier computation of the focal point and image formation without accounting for the internal distances or separations between the lens surfaces.

Thick Lens Analysis

Thick lens analysis takes into account the finite thickness of the lens, which means that the refraction occurs at more than one surface and the distance between these surfaces becomes significant. This approach uses more complex formulas to account for the physical separation of refracting surfaces, and can lead to a different effective focal point compared to the thin lens approximation.

Principal Planes

In a thick lens, the concept of principal planes is used to describe the effective locations where refraction appears to occur. These are imaginary planes that simplify the analysis by allowing the thick lens to be modeled in a way similar to a thin lens, but with adjusted positions for the incoming and outgoing rays. The positions of the principal planes are critical in determining where parallel rays will converge.

Focal Length and Ray Convergence

The focal length of an optical element is a measure of how strongly it converges or diverges light. In the context of thick lenses, the computed focal length takes into account the lens’s actual physical dimensions and the refractive index variations across its thickness. This calculation often results in a different convergence point for parallel rays compared to that predicted by the thin lens approximation.

Schematic Eye Model

The schematic eye model is a simplified representation of the human eye that includes its main optical components, such as the cornea and crystalline lens. It is used to analyze how light is refracted through the eye and to assess the impact of lens thickness on focusing properties. This model helps in comparing theoretical predictions with real physiological behavior, highlighting the significance of corrections due to finite lens thickness.

Transcript

00:01 Okay, so we're doing chapter 34, problem 35 here.

00:05 So this one talks about the cornea as a simple lens, and it says it behaves as a thin lens of approximate focal length of 1 .8 centimeters, even though this varies a bit.

00:18 We can assume that the material is made out of as an index of a fraction of 1 .38 here, and its front surface is convex with a radius of curvature, of 5 millimeters.

00:34 Okay.

00:36 So for part a, it asks if this focal length is an air, what is the radius of curvature of the backside of the cornea? so we're going to use the lens maker equation for this, where we know that 1 over f equals n minus 1 times 1 over r1 minus 1 over r2.

00:57 Cool.

00:58 So let's go ahead and just plug stuff in here.

01:01 Actually, let's simplify things first.

01:03 So we have 1 over f times n minus 1 plus 1 over r2 equals 1 over r1.

01:16 So now we can say 1 over r2 equals 1 over r1 minus 1 over f minus and minus 1 minus 1 minus 1 minus 1.

01:38 Inverted.

01:41 Cool.

01:42 So let's plug in our numbers for that and we should have one over 0 .5 because that's half centimeter minus 1 over 1 .8 times 1 .38 minus 1 .3 minus 1.

01:59 Cool.

02:01 That should be everything now.

02:03 So let's plug that in our calculator and we should see this comes out to being 1 .86 centimeters.

02:14 And this is r2.

02:17 Cool.

02:19 So now part b asks, the closest distance at which a typical person can focus on an object is about 24 centimeters.

02:33 Where would the cornea focus the image of an 8 millimeter tall object at that point? okay...