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Lecture 25 1 Friday, November 13, 15 Recall: Focal lengths converging lens f diverging lens f 2 Friday, November 13, 15 Recall: Images with Convex Lenses (2) § We start with a ray along the optical axis of the lens that passes straight through the lens that defines the bottom of the image. § A second ray is then drawn from the top of the object parallel to the optical axis • This ray is focused through the focal point on the other side of the lens § A third ray is drawn through the center of the lens that is not refracted in the thin lens approximation § A fourth ray is drawn from the top of the object through the focal point on the same side of the lens that is then directed parallel to the optical axis 10 Friday, November 13, 15 Now, let’s study concave Lenses (1) § A concave lens is shaped such that parallel rays will be caused to diverge by refraction such that their extrapolation would intersect at a focal distance from the center of the lens on the same side of the lens as the rays are incident § Assume that a light ray parallel to the optical axis is incident on a concave glass lens § At the surface of the lens, the light rays are refracted toward the normal § When the rays leave the lens, they are refracted away from the normal as shown § The extrapolated line shown as a red and black dashed line that points to the focal point on the same side of the lens as the incident ray 14 Friday, November 13, 15 Concave Lenses (2) § Let us now study several horizontal light rays incident on a concave lens § After passing through the lens, the rays will diverge such that their extrapolations intersect at a point a distance f from the center of the lens on the same side of the lens as the incident rays § To the right is a concave lens with five parallel lines of light incident in the surface of a concave lens from the left § In the second panel we have drawn red lines representing light rays 15 Friday, November 13, 15 Concave Lenses (3) § We can see that the light rays diverge after passing through the lens § We have drawn red and black dashed lines to show the extrapolation of the diverging rays § The extrapolated rays intersect a focal length away from the center of the lens § In the third panel we draw the diverging rays using the thin lens approximation where the incident rays are drawn to the center of the lens 16 Friday, November 13, 15 Images Formed with Concave Lenses (1) § Here we show the formation of an image using a concave lens § We place an object standing on the optical axis represented by the green arrow § This object has a height ho and is located a distance do from the center of the lens such that do > f § We again start with a ray along the optical axis of the lens that passes straight through the lens that defines the bottom of the image 17 Friday, November 13, 15 Images Formed with Concave Lenses (2) § A second ray is then drawn from the top of the object parallel to the optical axis • This ray is refracted such that its extrapolation of the diverging ray passes through the focal point on the other side of the lens § A third ray is drawn through the center of the lens that is not refracted in the thin lens approximation • This ray is extrapolated back along its original path § The image formed is virtual, upright, and reduced 18 Friday, November 13, 15 The Lens Equation (1) § The images formed by lenses are described by the lens equation § This equation is the same relationship between focal length, image distance, and object distance that we had found for mirrors § To treat all possible cases for lenses we must define some conventions for distances and heights • We define the focal length f of a convex lens to be positive and the focal length of a concave lens to be negative 20 Friday, November 13, 15 Power of Lenses § Often, the power of a lens is quoted rather than its focal length § The power of a lens, D (diopters), is given by the equation § For example, common reading glasses have a power of D = 1.5 diopters § The focal length of these glasses is 24 Friday, November 13, 15 Lens Maker Formula § If the front surface of the lens is part of a sphere with radius R1 and the back surface is part of a sphere with radius R2, then we can calculate the focal length f of the lens using the lens-makers formula R1 R2 positive negative positive negative 2 Friday, November 13, 15 Lens Maker Formula (1) § If the front surface of the lens is part of a sphere with radius R1 and the back surface is part of a sphere with radius R2, then we can calculate the focal length f of the lens using the lens-makers formula R1 R2 positive negative positive negative 2 Friday, November 13, 15 Lens Maker Formula (2) R1 R2 positive negative positive negative § The curvatures R1 and R2 have different signs depending on whether they point to the positive side (same side as object) or negative side • For a convex lens, R1 is positive and R2 is negative • For a concave lens, R1 is negative and R2 is positive § If we have a lens with the same radii on the front and back of the lens so that R1 = R2 = R, we get 3 Friday, November 13, 15 The Human Eye (1) § The human eye “sees” by absorbing light § Refraction at the cornea and lens surfaces produces a real image on the retina of the eye § For an object to be seen clearly, the cornea image must be formed at the location of retina the retina as shown to the right § The shape of the eye cannot be changed so shaping the lens must control the distance of the image § The lens is held in place by ligaments that connect it to the ciliary muscle that allows the lens to change shape and thus change the focus of the lens § The index of refraction of the two fluids in the eye are close to that of water with a value of 1.44; the index of refraction of the material making up the lens is 1.34 § Thus most of the refraction occurs at the air/cornea boundary. 33 Friday, November 13, 15 The Human Eye The extremes over which distinct vision is possible are called the far point and near point • The far point of a normal eye is infinity • The near point of a normal eye depends on the ability of the eye to focus § 15 Friday, November 13, 15 The Human Eye (3) § Several common vision defects result from incorrect focal distances § In the case of myopia (near-sightedness), the image is produced in front of the retina § In the case of hypermetropia (far-sightedness), the image is produced behind the retina § 35 Friday, November 13, 15 Reading glasses Myopia can be corrected using convex (converging) lenses 17 Friday, November 13, 15 Systems of Lenses § Now we will look at images formed by systems of lenses rather a single lens § We use the first lens to image an object § We use the image of the first lens as the object for the second lens § Thus we can produce various optical instruments with combinations of lenses 30 Friday, November 13, 15 The Telescope § First we will discuss the refracting telescope and then reflecting telescopes § The refracting telescope consists of two lenses • The objective lens and the eyepiece § In our example we represent the telescope using two thin lenses § However, an actual refracting telescope will use more sophisticated lenses 42 Friday, November 13, 15 20 Friday, November 13, 15 Geometry of the Telescope § Because the object to be viewed is at a large distance, the incoming light rays can be thought of as being parallel (the object is at infinity) § The objective lens forms a real image of the distance object at distance fo § The eyepiece is placed so that the image formed by the objective is at distance fe from the eyepiece § The eyepiece forms a virtual, magnified image of the image formed by the objective § The image is at infinity, again producing parallel rays 43 Friday, November 13, 15 Problems with Refracting Telescopes § The objective lens of a refracting telescope is large and heavy • The 40-inch refractor at Yerkes weighed 500 pounds § Supporting a large glass lens is difficult • Must be supported by its edges § Constructing large glass lenses is difficult § Glass lenses are thick and absorbed light § A glass lens has chromatic aberration • Different focal lengths for different colors § Solution: Replace the objective lens with a mirror 48 Friday, November 13, 15 The Reflecting Telescope § Most large astronomical telescopes are reflecting telescopes with the objective lens being replaced with a concave mirror § Large mirrors are easier to fabricate and position than large lenses § The eyepiece is still a lens § Various types of reflecting telescopes have been developed § We will discuss three examples of the geometries of reflecting telescope • Reflector • Newtonian • Cassegrain 49 Friday, November 13, 15 Basic Reflecting Telescope § Basic reflector § Replace the objective lens with a parabolic mirror § This design is impractical because the observer must be in the line of the incident light 50 Friday, November 13, 15 Newtonian Reflecting Telescope § In 1670 Newton presented his design for a reflecting telescope to the Royal Society • The idea for a reflecting telescope came from James Gregory § Newton solved the observer problem by placing a small mirror that reflect the light out to an eyepiece § This mirror is small compared with the objective mirror and causes only a small loss of light from the image 51 Friday, November 13, 15 Cassegrain Geometry for Reflecting Telescope § A further improvement on the geometry of the reflecting telescope is the Cassegrain geometry (named for the French sculptor Sieur Guillaume Cassegrain) first proposed in 1672 § Here a small mirror is used to reflect the image through a hole in the center of the objective mirror § This design and many improvements to this basic idea are the basis of modern astronomical telescopes 52 Friday, November 13, 15 The Microscope § Microscopes exist in many forms § The simplest microscope is a system of two lenses Objective Lens Eyepiece Object § The first lens is a converging lens of short focal length, fo, called the objective lens § The second lens is another converging lens of greater focal length, fe, called the eyepiece § The object to be magnified is placed just outside the focal length of the objective lens 39 Friday, November 13, 15