Optics of the Human Eye References: Equipment: Ford, Kenneth W., Classical and Modern Physics Vol2 Xerox College Publishing 1972 pp. 900-922. Pasco Human Eye Model Instruction Manual (OS-8477) pp. 1-34. a) Human Eye Model an eye-shaped plastic vessel, which is filled with ~1 liter of water to help simulate the aqueous and vitreous humors. It contains a permanent corneal lens towards the front. The crystalline lens is modeled by changeable lenses (see part b) behind the corneal lens which can be placed in a series of slots labeled A, B and SEPTUM. The movable screen towards the back of the eye models the eye's retina. It can be adjusted to three positions, FAR, NORMAL or NEAR. This allows us to simulate various visual defects. Defects can be corrected by placing lenses in front of the cornea to simulate eyeglasses in the slots labeled 1 or 2. b) Lenses Each lab setup has 6 lenses, which are fashioned with long handles to allow easy insertion and removal from the slots when the water is filled into the model. On each handle is written the focal length of the lens in air. Two of the lenses are cylindrical lenses and can be identified by notches along the plastic rim which indicate the cylindrical axis. Do not wipe or rub the lenses to clean them or they will get scratched. Let them air dry on a paper towel or in their foam holder. c) Pupil Aperture The amount of light entering the eye model can be limited by using the round or cat-shaped pupil. d) Optics Bench To keep all optical components at the same level and to make distance measurements easier, we will be employing an optics bench. There is a ruler along the length of the bench to help measure distances between objects. It also provides a magnetic base for the light source to be attached to. e) Light Source Every optics bench is also equipped with a light source. This light source will allow you to project images onto the eye. Make sure the bulb inside is pointed towards the human eye model by adjusting the knob on its top. 140516 1
Introduction: A lens is merely an optical system which includes two or more refracting surfaces. From the geometry of light rays, it turns out that spherical refracting surfaces are most interesting and most practical. We define the first focal point of a lens F! to be the point such that if a point source of light is placed at it all rays will be parallel after passing through the lens. Similarly the second focal point F! is the common point which parallel light rays pass through after passing through the lens. In the case of a lens which is relatively thin compared to the distance from one surface to F! or F! the first and second focal lengths are the respective distances from F! and F! to the center of the lens. If the refractive index of the materials on either side of the lens is the same (e.g. if the lens is in air), the first and second focal lengths are equal to a common value f. The human eye is an incredible biological lens system with amazing optical abilities. The Pasco Human eye model is a scaled up model of the human eye, the experiments listed in this manual will help you understand many of its optical properties. Theory: The Human Eye as a Single-Lens System Let us first try to model how the eye works by imagining what a simple convex lens does when light passes through it. Figure 1 Notice how the parallel light rays coming from infinity on the left side pass through the lens and focus at a point called the focal point. The distance from the center of the lens to where the light focuses is called the focal length. We say that the lens is convex when it causes the light to converge on a single point. The position of the focal point and the value of the focal length are specific to a given lens and do not change no matter what the lighting conditions are. 140516 2
Light, however, does not always originate at infinity as in the scenario above. In most cases light diverges off an object and focuses through the lens at a distance greater than the lens' focal length, as shown below. We therefore define the distance from the object to the lens to be called the object distance. The distance from the lens to where the image is formed is called the image distance. Figure 2 FACT CHECK: The image distance is not always the same distance as the focal length. Remember, the focal length is defined as the distance from the lens to the point where the lens focuses light from infinity. If the light is coming from an object at a finite distance (as above) the image will not form at the focal point. For all single lens systems there is an equation which governs the relationship between focal length (f), image distance (s') and object distance (s). It is called the thin lens formula: 1 s + 1 s! = 1 f Distances to the left of the lens are defined as negative and distances to the right are positive. We can see that if the light comes from infinity (i.e. s = ), then 1/s = 0 and we arrive at the result f = s', or that the focal distance is equal to the image distance. Recall this is the definition of the focal length. For many cases, if the object we're looking at is very far away (compared to the focal length) we can make the assumption that 1/s = 0. 140516 3
In some cases, the size of the image is different than the size of the object. This is known as the magnification and is defined to be: M = s ' s If M is positive, the image is upright (or the same orientation as the object). If M is negative the image is inverted (opposite orientation of the object). In the example below the object has been magnified larger and is inverted. Figure 3 A More Realistic Model of the Eye As you have probably noticed, modeling the eye as just a single lens has many limitations. The main source of limitations is that the eye itself is of a fixed size, and so the distance from the eye's lens to where the image forms is always the same. How, then, is it possible for the eye to form focused images from objects that are located at many distances from the eye? What makes the eye such a unique optical system is that it has the ability to change its focal length in order to focus objects at varying distances away from it. The mechanism by which the eye changes its focal length is called accomodation. To understand how the eye accomodates we must first look at what components make up the human eye. Figure 4 shows a cross section of the human eye and the relavant anatomy. Notice how the lens of the eye is actually a compound lens composed of two lenses, the corneal lens and the chrystalline lens. 140516 4
These lenses focuses incoming light through a water-like substance called the vitreous humor onto the back wall of the eye, the retina. Figure 4 How the Human Eye Accomodates: The corneal and chyrstalline lenses act together as a single lens, which converges the light towards the back of the eye (Figure 5). The eye uses muscles attached near the chrystalline lens to change its shape - more specifically, its curvature. By changing the curvature the focal length of the lens is changed, therfore allowing the eye to accommodate. The furthest point at which a healthy eye can accommodate for is at infinity. This point is called the far point. In this state the eye is in its most relaxed position, when the muscles are not contracted. When the muscles do contract, they cause the lens to bulge out, shortening the focal length and allowing us to focus on objects closer to us. This nearest distance for which a healthy eye can accommodate for is called the near point and is usually about 25cm. 140516 5
Figure 5 Procedure: Experiment 1: Images Formed in the Eye 1. Do not fill the eye with water yet. Put the retina screen in the slot labeled NORMAL. Put the +400mm lens in the SEPTUM slot. 2. Place your hand in front of the eye model, about 50cm from the cornea. Use the light source to shine on your hand. Record your observations of what you see on the retina. Do you notice anything unusual about the image? When you move your hand, how does the image move? Record these observations. 3. Draw an asymmetrical picture on a piece of paper. Illuminate it with the light source. What do you notice about the image on the retina? Experiment 2: Accomodation Recall that when the eye accomodates it is using muscles in the eye to change the shape of the chrystalline lens to change its focal length. To simulate this changing focal length, we will use a series of different lenses to help bring objects into focus. 1. Do not fill the eye model with water yet. Leave the +400mm lens in the SEPTUM slot. 140516 6
Position the eye model about 35cm from the light source. Point the eye model at the light source. Can you see an image on the retina? If not, adjust the position of the eye model until the images focuses well on the retina. Note this position for later use. 2. Now fill the eye model up with water from the sink. Fill it up to 2cm from the top. Return it to the position you found in Step 1. Is the image still focused? If not, try to get it in focus by adjusting the position of the eye model. 3. Place the eye model about 35cm from the light source. Replace the +400mm lens with the +62mm lens into the SEPTUM slot. Is the image in focus now? Move the eye model as close to the light source while keeping the image in focus. Record what you observe. 4. Measure the object distance (i.e. the distance from the light source to the eye model). It is convenient to measure to the rim of the eye model as that is the position in between the eye's two lenses. 5. Measure the image distance (i.e. the distance from the eye model's rim to the retina screen). Use the thin lens formula to calculate the overall focal length of the eye. You can record your work below. 6. We are now going to increase the focusing capacity by adding another lens (+400mm) to slot B. Move the eye model as close to the light source as possible such that it's still in focus. Record this distance. 7. Keeping the +400mm lens where it is, replace the lens in the SEPTUM slot with the +120mm lens. What distance does the eye model focus at now? What process are we simulating by swapping out lenses? 8. Remove both lenses and place the +62mm lens in the SEPTUM slot. Set the eye model at the near point such that the image is in focus. While looking at the image, place the round pupil into slot A. Describe below how the brightness and clarity of the image changes when you insert the pupil. Describe any change in focus. Predict what would happen if you put the cat's eye pupil in slot A. 140516 7
Try it this and compare it to your prediction. Where you right? 9. Remove the pupil. Turn the eye model towards a distant object in the room. Is the image on the retina still focused? Replace the lens in the SEPTUM slot with one that makes a clear image on the retina screen. This is the far vision lens. Record this focal length. Visual Defects and Their Correction As anyone who wears corrective lenses will tell you, the human eye is not without its flaws. There are many scenarios which cause the eye to create blurred images or, in some case, no image at all. These defects are typically a result of a misshapen eye and/or chrystalline lens. Here we investigate several common visual defects and how they can be corrected for. a) Myopia (near-sightedness): if a person's eye is too elongated it will cause the light from a distant object to be formed in front of the retina because the distance from the lens to the retina is longer than necessary. To correct this, a concave divergent lens (i.e. eyeglasses) is used to cause the image to focus on the retina instead of in front of it. Quick Question: Would the far point of a mypoic eye be infinity or less than infinity? Explain. Figure 6 140516 8
Experiment 3: Myopia 1. Set the eye model to normal near vision. Adjust the eye model's distance from the eye source to be in focus. 2. Move the retina screen to the NEAR position. Record what happens to the image. Decrease the pupil size by placing the round pupil into slot A. What happens to the clarity of the image? Record your observations. Remove the pupil. 3. Now, correct the myopia by placing eyeglasses on the eye model. Record which lens does the correcting, and calculate its power below. Does rotating the lens affect the image? 4. Remove the eyeglasses. Adjust the position of the eye model such that the image is focused. Is this eye-source distance different than that found in step 1? If so, record its value and justify why it makes sense. 5. Turn the eye model to look at a distance object. Record your observations of the image. Replace the lens in the SEPTUM slot with the far vision lens (Part 2, Step 9). Is the image in focus? This is what a near-sighted person would see looking at distant objects. The lens in the SEPTUM slot represents the chrystalline lens in its most relaxed state. Can an eye compensate for myopia via accomodation? FACT CHECK: We say that the eye is in its most relaxed state because it is the state in which the eyes muscles have to do the least amount of straining in order to focus the image of an object. b) Hypermetropia (far-sightedness): Converse to the case of myopia, hypermetropia is the case where the eyeball is too short and causes light from a distant object to form beyond the retina. To correct this defect an eyeglass must be used which helps converge the light earlier onto the surface of the retina. 140516 9
Figure 7 There is a form of hypermetropia called presbyopia, which is not a defect due to the length of the eye, but instead due to the hardening of the chrystalline lens as one ages. If the lens hardens it is less capable of changing shape and therfore causes blurred vision. Quick Question: Would the near point of a hypermetropic eye be less or greater than normal? Explain. Experiment 4: Hypermetropia 1. Place the +62mm lens in the SEPTUM slot and place the eye model at the near point (from now one we may refer to this as setting the eye model for normal near vision ). 2. To simulate a shorter eye, move the retina screen to the FAR position. Describe what happens to the image. This is what a far-sighted person see when looking at a distant object. 3. Point the eye model at a distant object in the room. Does a person with hypermetropia have difficulty seeing distant objects? Explain why below. 4. Now we will correct the hypermetropia by placing eyeglasses in front of the eye. Find a lens that brings the image into focus when placed into slot 1. Record the focal length of the lens. Does rotating the lens affect the image or how well it focuses? 5. Corrective lenses are not usually described by their focal length, but rather by their light bending power (measured in units of diopters). To calculate the power (P) we take the reciprocal of the focal length: P= 1 f 140516 10
Calculate the power of the lens that corrected the hypermetropia. Show your work below. 6. Remove the eyeglasses. Add the +120mm lens into slot B. What do you observe? What do your observations tell you about how the eye can accommodate? c) Astigmatism: When the chrystalline lens of the eye is not symmetric it will focus light from an object differently because it bends different amounts because the thickness of the lens is not the same throughout. In effect, the lens will have multiple focal lengths. As a result the eye will only be able to focus on lines/objects of a certain orientation. Notice how in Figure 8 the astigmatic eye focuses lines of different orientation differently and an different distances from the lens system. To correct this condition a cylindrical lens must be used, which has curvature in one plane but not another. If the orientation of the cylindrical lens is changed it will not effectively correct the astigmatism. Figure 8 140516 11
Experiment 5: Astigmatism 1. The figure below is a test chart for astigmatism. All of the lines are of the same thickness. Someone with an astigmatism would see some of these lines thicker/darker than others. Cover one of your eyes and look at the astigmatism chart. Do some lines look thicker/darker than others? If so, rotate the chart 90º to convince yourself the lines are all of the same. If you wear glasses try this test both with and without glasses. Figure 9 2. Set the eye model to normal near vision. While looking at the light source, adjust the eye model's position to be in focus. 3. Place the -128mm cylindrical lens into slot A. The side of the lens handle with the focal length on it should be facing the light source. Draw a picture of how the image changes below. 4. Rotate the cylindrical lens (while keeping it in its slot). What happens to the image? What is the lens doing to the light traveling through it? 5. Now we will procede to correct the astigmatism by fitting the eye model with special eyeglasses. Place the +307mm cylidrical lens into slot 1. Make sure the front/back orientation of the lens is correct. 140516 12
Rotate the corrective cylindrical lens (keeping it in its slot). Describe, in detail, what is happening to the image on the retina as the lens is rotated. Find the orientation of the corrective lens that produces the sharpest image. What is the angle between the cylindrical axes of the corrective lens and the chrystalline lens? Why is this relationship the way it is? Make an argument for your reasoning. d) Blind Spot: a small area of the retina where the optic nerve is attached to the eye. The eye lacks light-sensitve material there and so there is a gap in our vision. We do not realize this blind spot because our brain automatically fills in the missing pieces to give us the perception of a seamless visual field. We can now paint a more complete picture of the human eye, Figure 10. Figure 10 Experiment 6: The Blind Spot 1. Cover you left eye and look at Figure 11 below with only your right eye. Hold the paper at arm's length and stare at the + sign with your right eye. Keep staring at the + sign and you should be able to see the dot in your peripheral vision. While continuing to stare at the + sign, slowly move the paper closer and closer to your eyes. At a distance of about 30cm the dot symbol should disappear. Keep moving the figure closer. Does the dot re-appear? 140516 13
Figure 11 3. As you can see from the above experiment, your brain uses past experiences and information to fill in the missing gap created by the blind spot. Repeat the exercise with Figure 11 below. Adjust the paper s distance from your eye such that the dot in the middle of the cross disappears. Do you see a white dot where the dot was, or do you see an interection of the two lines of the cross? If you try to create your own patterns on a separate piece of paper, you will notice that our brains are not very good at making up the missing details. Try using different colors. Figure 12 140516 14
3. Set the eye model to normal near vision. 4. Use the large copy of the above symbols (Appendix B). Hold it about 30cm from the front of the eye model and shine the light source from behind. Adjust the eye model such that the image is in focus. The blind spot of the eye is simulated by a hole in the retina screen. In a trial and error process, adjust the position of the paper such that the image on the retina has the + sign on the center of the retina and the dot falling in the blind spot. You should now clearly see why our eyes have blind spots. Experiment 7: Perform Your Own Diagnosis In this section, each member of your group is going to perform the role of an optometrist. When it's your turn, you will observe, diagnos and correct a visual defect. 1. Set up the model to normal near vision. 2. Choose a member of your group to be the optometrist. Have that person step aside for a few minutes. This person should not look at what your lab partners do (i.e. they should be in the blind, no pun intended). 3. The remaining person or persons should now create a mock scenario for a visual defect of their choice. This should involve adjusting the retina screen, eye-source distances, etc. in order to simulate one of the visual defects you've learned about. 4. Once the defect has been created, the group member which left should re-join the group and play the role of optometrist. Based off of his/her observations, the optometrist should observe the conditions of the setup, create a diagnosis and formulate a procedure to correct the problem. Record each of the following steps in your lab write-up. I. Observations describe characteristics of what you see in the eye model setup II. Diagnosis write a few sentences about what condition you think your patient has and use your observations to justify your conclusions. III. Corrective Measures list a series of corrective measures you can take to reverse the defect you've observed. Mention specific equipment/instruments you would use and how. 140516 15
Experiment 8: The Telescope In the final part of this laboratory, you will construct a telescope, which will allow you to magnify an image and study it. 1. Place the light source at one end of the optics bench and call the position of the light 0cm. 2. Place a +200mm glass lens on the optics bench 51cm away from the light source and a +100mm glass lens at 93cm. These two lenses compose the optical system of a telescope and the setup should resemble the arrangement below. Figure 13 3. Place your head at the end of the optics bench and look through both lenses towards the light source. Compare the image that you see through the lenses with the image you see looking directly at the light source from the same distance. Make an estimate of the magnification. Does the magnification depend on how far away your eyes are from the object? Record your answers. 4. Now take the eye model (filled with water) and place the +120mm lens in the SEPTUM slot, the +400mm lens in slot B and the retina screen at normal. This is the set-up for normal medium distance vision and arranges the eye to accommodate at a distance of 1m. 5. Use a stack of books to prop the eye model to the proper height of the light source and position it in such a way as to have the corneal lens be at the 100cm (1m) mark. 6. Observe the image on the retina. Is it in focus? What happens to the image when you adjust the positions of the two lenses? Move the two lenses back to their original positions. Record the width of the image. Is the image inverted? 7. Now remove the two telescope lenses. Does the image still focus on the retina? Is it still inverted? Is it larger or smaller than with the two telescope lenses? Record the width of the 140516 16
image. Explain your observations. 8. Now, we will perfom some analysis of the telescope system. Divide the size of the retina image as viewed through the telescope by the size of the image without the telescope. Record this value, which is the telescope's angular magnification. 9. The focal length of the first lens is f 1 = 20cm. The object distance is o = 20cm. Use the thin lens formula to calculate the image distance i 1 of the image formed by this lens. Call this image Image 1. Show your calculations below. 10. Is Image 1 in front our behind lens 1? Is it real or virtual? Calculate the magnification of the first lens using the magnification equation. Call this value M 1. Show your work. 11. Place a screen/paper at the position you calculated for Image 1. Can you see the image? What does this tell you about what type of image it is? Is Image 1 smaller or larger than the actual object? Is it inverted? 12. Create a scaled-down ray diagram of the entire telescope system such that it will fit below (or on a separate piece of paper to be handed in). Try your best to keep all horizontal distances in proportion. You can exaggerate the vertical dimension for clarity. Be sure to label all of the focal points, all parts and mark all of the horizontal distance measurements. 13. Use ray tracing to find the position and height of Image 1. Extend the rays after that until they meet Lens 2. 14. Image 1 is the same thing as Object 2. Meaning, from Lens 2's point of view the object it sees is actually the image that Lens 1 has formed. However, the obejct distance for Image 2 is not the same as the image distance for Image 1. Why? What is the distance bewteen the lenses? What is the distance between Lens 1 and Object 2? Use the distances to calculate the object distance (o 2 ) between Object 2 and Lens 2. Mark o 2 on your diagram. Is o 2 positive or negative? 15. Use the thin lens formula to calculate the image distance for Lens 2, i 2. Show your work below. 16. On what side of Lens 2 is this image formed? If you place a screen at the location of Image 2, what do you see? What do you expect to see? Is Image 2 real or virtual? 140516 17
17. Using the formula below and the image and object distance for both lenses, calculate the telescope's total magnification. Show your work below. How does this compare with the measurements/observations you took earlier? M tot = M 1 M 2 = i 1i 2 o 1 o 2 18. Finish your ray diagram by adding Image 2 at the position you calculated. Use the magnification you found to draw in the height of Image 2. Extend the rays from Lens 2 to show how Image 2 is formed. Appendix A: Questions Below are questions you must answer in your lab write-up. The numbers denote which experiment that the question refers to. Question 2A: What effect does the aqueous and vitreous humors (i.e. the water) have on the focal length of the eye's lenses? Question 2B: You've just measured the closest distance at which this eye/lens system will focus an image. What is this position called? How does it compare to the value for the real human eye? Question 2C: Draw below a ray diagram of the eye model in focus with the +62mm lens in place. Label all relavant distances and positions. You can treat the lens system as a single simple lens. Question 2D: In a real human eye, the eye accomodates by changing the shape of the chrystalline lens. When the lens changes shape to focus from a distant object to a near object, does the curvature of the lens increase or decrease? Question 3A: Does the correction of myopia require a convex or concave lens? Explain why. Question 3B: On the eye model, the retina screen has three positions labeled. Why is the NEAR position furthest from the lens? What do the words NORMAL, FAR and NEAR refer to? 140516 18
Question 4A: Why does reducing the pupil size make the image clearer? Question 4B: Look carefully at the +62mm and +400mm lenses. Which one has a greater curvature? How does the curvature effect the power? Question 4C: We showed in the last part of this section that the eye can accomdate to correct hypermetropia. Why might this accomodation not be sufficient to allow someone to read without reading glasses? Question 5A: As you should have seen in the previous step, astigmatism is orientationsensative. What, anatomically, is the reason that this is true? Question 5B: Why does rotating the corrective lens for astigmatism affect the image, but not so in the case of myopia and hypermetropia? What quick test could you perform in order to determine if someone's eyeglasses are designed to correct for astigmatism or not? Question 6A: In the first blind spot test you did, why wouldn't the test work if you used both eyes? Question 6B: Does the screen on the eye model represent the retina of the left or right eye? Explain your reasoning. Question 8A: In your model telescope, the positions of the two lenses were chosen to make Image 2 form at about the same location as the object on the light source. If you adjusted the telescope to form an image at infinity, what would you have to do to the eye model to allow it to clearly see the image? Question 8B: What does the sign of a telescope's magnification tell you? What's the difference between having a magnification of 4 versus -4? Question 8C: The simple telescope you constructed forms an image that is inverted from the object. So anything you look at through the telescope appears inverted. Look again at the image on the retina screen. Is it inverted from the object? Why not? 140516 19
Appendix B 140516 20