L-4: The optics of the eye and resolving power

OBJECTIVES:

To study eye optics and how resolving power depends on aperture.

APPARATUS:

PART I: Eye model, (fill with distilled water to simulate the aqueous & vitreous humors); light source, lens set, 6`` plastic ruler.

PART II: (2) Optical benches, Na lamps, transparent mm slides, vernier calipers, L-2 telescopes; (1) 15 meter tape.

Figure 1: Various schematics for the eye

You may fix the moveable retina (cylindrical) in one of three positions:

R for the normal eye,

R_h for the hypermetropic (farsighted) eye,

R_m for the myopic (nearsighted) eye.

(An eye retina is actually spherical, but for small images the results are similar.)

The eye model has a fixed cornea C and racks for other lenses.

A lamp box serves as source for most of the experiment.

Set of lenses: (Strength in diopters = 1/f where f is in meters.)

Diopters Focal lengths in air

1. Converging spherical + 7 . 0 . 14 m

2. Converging spherical +20 . 0 . 05 m

3. Converging spherical + 2 . 00 0 . 50 m

4. Diverging spherical - 1 . 75 -0 . 57 m

5. Diverging cylindrical - 5 . 50 -0 . 182 m

6. Converging cylindrical + 1 . 75 0 . 57 m

7. Diaphragm with small hole . .

PART I - EXPERIMENTS WITH THE EYE MODEL

1. ACCOMMODATION: With the retina in the normal position, point the eye model at a window or other bright object 4 to 5 meters away. Insert the +7 diopter lens in the water at the inside mount farthest from the cornea. An image of the bright object should be in focus on the retina.

   Next use as object the lamp box and place it about 30 cm from the cornea. The image is blurred until one replaces the +7 diopter lens by the +20 diopter lens. This change illustrates accommodation. Eye muscles make the lens thicker for close vision. (Under relaxed conditions the ligaments supporting the lens are in radial tension and hence thin the lens. The ciliary muscle contracts against these ligaments allowing the lens to thicken for viewing near objects. See Fig. 1.)

2. NEAR and FARSIGHTEDNESS (MYOPIA and HYPERMETROPIA): With lamp box at 30 cm and the 20 diopter lens at L, place the retina in the position R_m. Decide which lens placed in slot S₁ or S₂ will bring the image into focus. Try it.

   Make the eye farsighted by moving the retina to R_h, and pick the proper lens to place in front of the eye to bring the image into focus.

   Focus again on a window 4 to 5 meters away when the +7 diopter lens is in the normal position L and the retina in position R.

   Make the eye nearsighted by moving the retina to R_m. Decide which lens will bring the image into focus. Try it.

   Remove the correcting lens; the image again blurs. At S₂ place the diaphragm with hole and note image improvement. With sufficient light try a smaller hole (e.g. a hole thru masking tape on the diaphragm). Explain the effect of the reduced aperture.

3. ASTIGMATISM: With the +20 diopter lens at L in the eye and the retina in the normal position, adjust the distance to the lamp box to get a sharp image on the retina. Then produce astigmatism by placing the strong diverging cylindrical lens in mount G₁. (In the human eye astigmatism generally results from a cornea with different curvatures in the horizontal and vertical directions.) Turn this lens and note the effect on the image.

4. COMPOUND DEFECTS: Combine tests 2 and 3, i.e. make a nearsighted astigmatic eye. Correct by using both a cylindrical and spherical lens in front of the cornea. How is this double correction managed in actual spectacles?
5. CATARACTS and LENS REMOVAL: A cataract results from a cloudy lens. A cataract operation involves removal of the lens. An implanted plastic lens or a strong convergent lens in front of the eye restores vision. Remove the eye lens (L) of the model and place the +7 diopter lens in front of the cornea. A clear image reappears for nearby objects.

6. Action of a SIMPLE MAGNIFIER: With the lamp box at 35 cm, the +20 diopter lens and the retina in the normal position, measure the size of the image. Now place the +7 diopter lens in front of the cornea and move the lamp box toward the eye until the image becomes clear. Record the increased size of the image. With the lamp box at 35 cm, the 20 diopter lens and the retina in the normal position, note the size of the image. Now make the eye nearsighted and bring the lamp box toward the eye until the image becomes clear. Note the increased size of the image in the nearsighted eye.

NOTE: For a real eye and lens the index of refraction, $\mu$, of the lens varies from 1.42 at its nucleus to 1.36 at its edges, and these values are not so different from those of the adjoining aqueous and vitreous humor ($\mu$ = 1.330 and $\mu$ = 1.337). Hence the real eye lens is very weak and is primarily for fine focus adjustment. Most of the refraction in the eye actually occurs at the cornea where $\mu$ changes from 1.0 (in air) to 1.37 in the cornea.

PART II - RESOLVING POWER
Before doing this part, read in your text about resolving power.

SUGGESTED EXPERIMENTS:

1. Use as a source a slide with mm divisions and illuminated by a sodium discharge lamp. The wavelength of the sodium yellow light is 589.3 nm.

2. Back away from the slide until the mm lines become indistinct. Then advance far enough so that they are just clearly visible; this procedure insures that eye defects or the density of rods and cones do not (for this measurement) limit the resolving power.

   Observing now with only one eye, place a vernier caliper⁵in front of the eye pupil with the caliper jaws parallel to the mm marks and open about 3 mm. Slowly close the jaws until the mm marks disappear. Note the separation of the jaws, w, and measure the distance, D, to the slide from the jaws.

   Compare your measurements of the angular separation $\alpha$ of the mm marks at your observing distance,

   $$\alpha$$ = $${\frac{{1~\mbox{mm}}}{{D~\left( \mbox{in~mm}\right) }}}$$,
   with the expression $\theta$ = $\lambda$/w where $\theta$ is the angular limit of resolution, (Rayleigh's criterion).
3. Mount a telescope so its objective is at the same distance from the slide. Repeat the experiment with the caliper in front of the objective.⁶How do the results compare to your measurement without the telescope?
4. Move the telescope $\geq$ 4 m from the slide and repeat the experiment.
5. Turn the vernier jaws perpendicular to the mm lines and repeat experiment #1. Comment on the result.

QUESTIONS:

1. Would an eye without any defects of vision have better resolving power at dusk or in bright sunlight? Explain.
2. Discuss how a contact lens functions. Would a ``soft'' contact lens correct well for astigmatism?