Introduction (Read Before Lab up to Experimental Procedure) Snell s Law

Transcription

1 GEOMETRIC OPTICS Introduction (Read Before Lab up to Experimental Procedure) In this lab you will measure the index of refraction of glass using Snell s Law, study the application of the laws of geometric optics to systems of thin lenses, and examine the operation of various optical instruments: the human eye, a microscope and telescope. Snell s Law The index of refraction n of a material is defined to be n = speed of light in vacuum speed of light in medium = c v. (1) For air, n is very close to 1. Note that greater index of refraction implies a smaller speed of light. (Some meticulous physicists want us to note that there is no single index of refraction, since it s really wavelength dependent. We ll only be working with visible light today, so you can think of our n as an average value for the 400 to 700 nm region of the spectrum.) For a light beam passing from medium 1 to medium 2 as in Figure 1, Snell s law is n 1 sin θ 1 = n 2 sin θ 2, (2) where n 1 and n 2 are the indices of refraction of the two materials, and θ 1 and θ 2 are the angles of the light ray from the perpendicular to the interface. Figure1: A ray of light at an interface Thin Lenses The basic equation we will use is the thin lens equation: 1 s + 1 = 1, (3) s' f where s is the distance from the object to the center of the lens, s' is the distance from the center of the lens to the image, and f is the focal length of the lens. An illustration for a converging lens is given in Figure 2.

2 Figure 2: A converging lens with real image I and object O. In this illustration, s, s', and f are all positive. The rays converge at the image point s', forming an image: a region of space from which light rays emanate as though there were an object there. Note that this image is inverted it is upside down compared to the original object. Right-side up images are said to be erect. Images also can be different sizes in general from the original object (an effect called magnification even though the image can be larger or smaller than the object). If we place a screen at s', we will observe an image of the object. The image is thus said to be real, meaning the rays of light actually converge to those points in space and hence the image can be displayed on a screen. Examples of real images are those created by a computer projector or the lens and cornea of the human eye. The magnification is defined to be the ratio of the image height to the object height. If the image is inverted, then the magnification is negative. For a thin lens, the magnification M is given by M = s' s. (4) Figure 3: A converging lens with a virtual image. Images can also be virtual: even though the rays of light appear to emanate from a region of space, there is no point at which the rays of light actually converge and at which a real image can be captured. To see how this can be true, look at the illustration of a virtual image in figure 3. In this example, s and f are positive, but s' is negative. If you were to look from the right-hand side through the lens, the image would appear to be at s', even though the light rays are not converging at s'. If we were to place a screen at s', we would not observe an image of the object on the screen. (Note that this image is also erect.) Virtual images are formed by magnifying glasses, binoculars, and microscopes and telescopes with eyepieces. The refractive power P of a lens is commonly given in diopters, P = 1, (5) f

3 where f is the focal length expressed in meters. Thus a lens with P = 20 has a focal length of f = 1/20 m = 0.05 m, or 5 cm. The power P is positive for a converging lens, and negative for a diverging lens. A typical prescription for a person who is mildly nearsighted would be a diverging lens with P = 2, or f = 50cm. Combinations of Thin Lenses Combinations of thin lenses can be handled by successive application of the thin lens equation to each of the lenses. A real image formed by one lens can serve as the object for another. We examine two useful lens combinations below. The arrangement in figure 3 above corresponds to a simple magnifier. An object placed just inside the focal length of the converging lens gives rise to a magnified, virtual image. A similar principle is used in the compound microscope. We first use a converging lens to form a real, inverted image, as in figure 2. This lens is called the objective, and typically has a short focal length. We then use a simple magnifier, called the eyepiece, to enlarge this image. The best results are obtained when the object is placed just outside the focal distance of the objective, and the image formed by the objective is just inside the focal distance of the eyepiece. This means that the distance between the two lenses is much greater than the sum of the two focal lengths. (Once you have your microscope built, notice whether it s really magnifying or not.) Do you have a real or inverted image? Which do you want? The principles behind the refracting telescope are similar to those behind the compound microscope. However, the object is now assumed to be very far away, so light rays from the object are essentially parallel. The objective typically has a long focal length. The eyepiece is again used to magnify the image formed by the objective. The distance between the two lenses is approximately equal to the sum of the two focal lengths. Anatomy of the Human Eye The human eye forms an optical image that stimulates special nerve cells, creating the sensation of sight. The eye consists of an aperture and lens system at the front, and a light-sensitive surface at the back. Light enters the eye through the aperture-lens system, and is focused on the back wall. The lens system consists of two lenses: the corneal lens on the front surface of the eye, and the crystalline lens inside the eye. The space between the lenses is filled with a transparent fluid called the aqueous humor. Also between the lenses is the iris, an opaque, colored membrane. At the center of the iris is the pupil, a muscle-controlled, variable-diameter hole, or aperture, which controls the amount of light that enters the eye. The interior of the eye behind the crystalline lens is filled with a colorless, transparent material called the vitreous humor. On the back wall of the eye is the retina, a membrane containing light-sensitive nerve cells known as rods and cones. Rods are very sensitive to low light levels, but provide us only with low-resolution black-and-white vision. Cones allow us to see in color at higher resolution, but they require higher light levels. The fovea, a small area near the center of the retina, contains only cones and is responsible for the most acute vision. Surrounding the fovea is the sensitive region called the macula, which is

4 responsible for central vision. Signals from the rods and cones are processed in a neural network in the retina before being carried by nerve fibers to the optic nerve, which leads to the brain. The optic nerve connects to the back of the eye; there are no light-sensitive cells at the point where it attaches, resulting in a blind spot. cameras. Still and video cameras use the same basic optical design as the human eye. Cameras of course have lenses that move mechanically back and forth rather than being fixed in place like the crystalline lens and cornea, and they sense light with electronic sensors or film rather than a retina, but the optical design issues you learn here will apply to Optics of the Eye The corneal lens and crystalline lens together act like a single convergent lens. Light entering the eye from an object passes through this lens system and forms an inverted, real image on the retina. The eye focuses on objects at varying distances by accommodation, or the use of muscles to change the curvature, and thus the focal length, of the crystalline lens. In its most relaxed state, the crystalline lens has a long focal length, and the eye can focus the image of a distant object on the retina. The farthest distance at which the eye can accommodate is called the far point for distinct vision. For a normal eye, the far point is infinity. When muscles in the eye contract and squeeze the lens, the center of the lens bulges, causing the focal length to shorten, and allowing the eye to focus on closer objects. The nearest distance at which they eye can accommodate is called the near point for distinct vision. The near point of a normal eye is about 25 cm.

5 Experimental Procedure Experiment 1: Snell s law The configuration for determining the index of refraction of the glass is shown in Figure 4. Figure 4: Determining n from Snell s law The light is refracted upon entering the semicircle of glass. However, since the light exits the glass perpendicular to the interface, it is not refracted a second time. Thus, you need only consider refraction at the first interface. 1) For at least 3 different incident angles, determine the index of refraction of the glass from measurements of the angles of incidence and refraction of the laser beam. Take the average of your measurements as the best estimate of n. 2) Determine how to measure the angle of total internal reflection using this apparatus. Find out what the critical angle is using your design. Experiment 2: The thin lens equation For each of these experiments, be sure to draw the configuration of the lenses, light source, and screen in your notes. Record whether the image is real or virtual, and whether it is inverted or erect. Measure the actual focal length of each of your lenses, as it may differ from the nominal value. 1. Single converging lens Real Image. Use either the P = 10 or P = 20di lens. Place the object light source outside the focal length of the lens, as in Figure 2 above. Find the image point s such that the image is focused on the screen. For at least 3 different object distances, determine s and the magnification M. Compare your answers with the expected values. 2. Single converging lens Virtual Image. Use the P = 10di lens. Place the object light source inside the focal length of the lens, as in Figure 3 above. Record whether the virtual image is inverted or erect.

6 Object Lens 1 Lens 2 Image d d d Figure 5: The arrangement and notation for parts 3 and 4 3. Compound Systems. Use the P = 10 and P = 20di converging lenses to form a real erect image of the arrow. Make sure that the lenses are separated by a distance greater than the sum of their focal lengths. Find the location of the image. Measure the distances from the object to the first lens, between the lenses, and the distance from the second lens to the image. Compare the image distance to the predicted value. Measure the magnification M of the image and compare to the predicted value. 4. Refractor telescope. Build a refractor telescope. Use the large P = 2di lens for the objective and the P = 20di lens for the eyepiece. Try your telescope on several distant objects out the window. Record the configuration that works best for you. Record whether the image is real or virtual, and whether it is erect or inverted. Your human eye model The PASCO Human Eye Model consists of a sealed plastic tank shaped roughly like a horizontal cross section of an eyeball. A permanently mounted plano-convex glass lens on the front of the eye model acts as the cornea. The tank is filled with water, which models the aqueous and vitreous humors. The crystalline lens of the eye is modeled by a changeable lens behind the cornea. A movable screen at the back of the model represents the retina.

7 The lenses of the eye model are equipped with handles, which allow them to be easily inserted into the water. The handles of the plastic lenses are marked with their focal lengths in air. Two of the lenses are cylindrical lenses for causing and correcting astigmatism in the model; these can be identified by notches on their edges that mark the cylindrical axes. WARNING: DO NOT wipe or rub the lenses with a cloth or tissues to dry or clean them. They are plastic and easily scratched. We will provide blowoff guns you can use to dry them off. The crystalline lens, which is supported in the slot labeled SEPTUM, can be replaced with different lenses to accommodate, or focus, the eye model at different distances. Two other slots behind the cornea, labeled A and B, can hold additional lenses to simulate changing the power of the crystalline lens. A cylindrical lens can be placed in slot A or B to give the eye astigmatism. The pupil aperture can also be placed into slot A or B to demonstrate the effect of a round or cat-shaped pupil. Two slots in front of the cornea, labeled 1 and 2, can hold simulated eyeglasses lenses to correct for near-sightedness, far-sightedness, and astigmatism. The retina screen can be placed in three different positions (labeled NORMAL, NEAR, and FAR) to simulate a normal, near-sighted, or farsighted eye.

8 Experiment 3: Image formation 1. Put the retina screen in the NORMAL slot and the +120 mm lens in the SEPTUM slot. 2. Fill the model with water. This simulates the effect of the aqueous and vitreous humors that actually fill the eyeball. 3. Aim the eye at a bright, distant object such as objects outside a window. A real image should be formed on the retina screen. Observe and record the orientation of the image relative to the object (up/down, left-right). Why are you not aware of any reversals of the retinal image compared to the real-life object in your own eyes? This experiment simulates what happens for normal vision, when the eye muscles are fully relaxed, and the eye can image distant objects using its longest focal length configuration. Experiment 4: Focusing at different distances: Accommodation In the process of accommodation, muscles in the eye change the shape of the crystalline lens to change its focal length so as to image close objects. Accommodation in the eye model is simulated by changing the lens or lenses that represent the crystalline lens. You will now simulate what happens for normal vision when the eye muscles exert force on the lens, causing it to curve and shorten its focal length to allow closer objects to be in focus. 1. Place the eye model about 35 cm from the light source you used in the first lens experiments and object. Replace the +120 mm lens with the shorter focal length +62 mm lens in the SEPTUM slot. 2. Is the image in focus now? Describe the image on the retina screen as you move the eye model back and forth. Move the eye model as close as possible to the light source while keeping the image in focus. 3. Measure the object distance, o, from the screen of the light source to the top rim of the eye model, as pictured below. (The front of the rim is a convenient place to measure to and marks the center of the eye model s two-lens system.) Record this distance, which is the near point of the eye model when equipped with the +62 mm lens. Measure and record the near point for your eyes and those of your lab partner by finding the shortest distances at which you can focus. (Keep your glasses on if you wear them. Note that although we ve given a typical value above, your near point may be much smaller or larger than this value!) 4. Far-sightedness & reading glasses With age, your eye muscles atrophy and you may become unable to focus close enough to read. We model this now. Increase the ability of the eye model to focus on a close object by adding the +400 mm lens to slot B. This new combination of lenses models a different focal length for the crystalline lens, due either to accommodation (if the eye muscles are working properly) or wearing reading glasses (if not!) How close can the eye focus now?

9 5. Remove both lenses and place the +62 mm lens in the SEPTUM slot. Adjust the eye-source distance to the near point distance for this lens (which you found in step 3) so that the image is in focus. While looking at the image, place the round pupil in slot A. What changes occur in the brightness and clarity of the image? Move the light source closer to and farther from the eye model. Is the image still in focus? Remove the pupil and observe the change in clarity of the image. Both with and without the pupil, how much can you change the eye-source distance and still have a sharp image? What are the tradeoffs for image formation in having the pupil in place and removed? Experiment 5: Refractive errors: near sightedness A person affected by myopia (near-sightedness) has a longer-than-normal eye ball, making the retina too far away from the lenses. This causes the image of a distant object to be formed in front of the retina. By contrast, a person affected by far-sightedness (hyperopia) has a shorter-than-normal eye ball, making the retina too close to the lens system. This causes images of near objects to be formed behind the retina. (As discussed earlier, aging can cause a decline in accommodation that results in similar optical effects, a condition called presbyopia.) These refractive errors of course can be compensated for with eyeglasses or with laser surgery to change the curvature of the cornea. We will explore only the first condition here. 1. Set the eye model to normal, far vision (put the +120 mm lens in the SEPTUM slot, remove other lenses, and put the retina screen in the NORMAL position). Turn the eye model to look at a very distant object, as in the first eye experiment above. Make sure the image is in focus. Now put the retina screen in the NEAR position. Describe the image. This is what a near-sighted person sees when trying to look at a far-away object. The lens in the SEPTUM slot represents the crystalline lens in its most relaxed, flattest state, with its longest-possible focal length. Can an eye compensate for myopia by accommodation? Explain. 2. You will now correct the myopia by putting an eyeglass lens on the model. Find a lens that brings the image into focus when you place it in front of the eye in slot 1. Record the focal length of this lens. Laser surgery for myopia corrects near-sightedness by changing the shape of the cornea. Does it do so by making the cornea flatter or more curved? Explain. Experiment 6: Astigmatism In a normal eye, the lens surfaces are spherical and rotationally symmetrical; but an eye with astigmatism has lens surfaces that are not rotationally symmetrical that is to say, they are thicker along one direction rather like a rugby ball or football. This makes the eye able to focus sharply only on lines of certain orientations, while lines at other orientations look blurred. Astigmatism can be corrected with a cylindrical eyeglass lens one that is also thicker along one direction--that is oriented to cancel out the defect in the eye. Each cylindrical lens included with the eye model has its cylindrical axis marked by two notches in the edge. (You can see this effect if you need eye glasses for astigmatism. Take off your glasses and look at the chart below while rotating it to see how only one orientation lines is in focus at a time.)

10 1. Set the eye model to normal, near vision (put the +62 mm lens in the SEPTUM slot, remove other lenses, and put the retina screen in the NORMAL position). With the eye model looking at the nearby light source, adjust the eyesource distance so that the image is in focus. 2. Place the -128 mm cylindrical lens in slot A. The side of the lens handle marked with the focal length should be towards the light source. Describe the image formed by the eye with astigmatism. 3. Rotate the cylindrical lens. What happens to the image? This shows that astigmatism can have different directions depending on how the defect in the eye s lens system is oriented. 4. Eyeglasses can be outfitted with compensating cylindrical lenses to correct astigmatism. Now place the other astigmatic lens in the eyeglass holder slot and move it until the image is again focused. What manipulation did you have to do to bring the image into focus again? Describe how you could find out experimentally if a pair of eye glasses have astigmatic corrections.