Chapter 34. Thin Lenses
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1 Chapter 34 Thin Lenses
2 Thin Lenses Mirrors Lenses Optical Instruments MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/2013 2
3 Inversion A right-handed coordinate system becomes a left-handed coordinate system MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/2013 3
4 Mirrors Real Object Virtual Image Object and image distances measured from the plane of the mirror MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/2013 4
5 Object - Source of light Image - Destination of light Vitual Image - Can t be imaged on a screen Depth Inversion - R.H.S. => L.H.S. Lateral Distance = y /y Plane Mirrors (Perverted image) Magnification: m = y /y = - s /s m= -(-1)/1= +1 Object distance = s Image distance = s MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/2013 5
6 Multiple Images A complicated depiction of a fairly simple problem. The light rays here were drawn after the soluntion was determined by other means. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/2013 6
7 Multiple Images in Plane Mirrors The images are located behind the mirror at a distance equal to that by which the objects are located in front of the mirror MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/2013 7
8 Curved Mirrors Convention: The initial object is located to the left; the incident light rays travel from the left to the right. Concave mirror - center of curvature on the same side of the mirror as object. Convex mirror - center of curvature on the opposite side of the mirror as object MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/2013 8
9 Reflection from a Concave Mirror Object at Infinity- Light Ray Picture Construction method: incident angle = reflection angle Rays from an object at infinity will focus at a point in the Focal plane. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/2013 9
10 Reflection from a Concave Mirror Plane waves represent no wave front curvature. This is indicative of an object at infinity. Object at Infinity- Wave Front Picture The curved wave front converges to a point. Construction method: wave front construction using Huygens principle. The shape of the wave front changes due to the different arrival times of the flat wave front at the curved mirror surface. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
11 Reflection from Concave Mirrors The light ray method is easier to apply and can yield quantitative results. The wave front method is more difficult to apply and doesn t lend itself to quantitative results. The light ray method above used the incident angle = the reflected angle approach to construct the diagram. We will make some approximations to this approach to avoid the optical aberrations of curved surfaces. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
12 Reflection from a Concave Mirror Object outside center of curvature - Light Ray Picture Two rays are needed to determine the location of the image. Multiple rays are used for effect. Angle of incidence equal to angle of reflection violated. The radius of curvature R is equal to 2 x Focal Length = 2f MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
13 Conventions for Spherical Mirrors MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
14 Conventions for Spherical Mirrors Concave Convex R > 0 f > 0 R < 0 f < 0 MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
15 Making Sense of the Sign Conventions (For mirrors and lenses) The distances are positive if you are measuring where the light IS supposed to go. The distances are negative if the you are measuring where the light IS NOT supposed to go. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
16 Making Sense of the Sign Conventions Light can t go back here. Light is supposed to go here in front of the mirror. Distances positive in front of the mirror. Distances negative behind the mirror. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
17 Concave Mirror Parallel ray ray Focal ray ray Center ray Incident rays converge to an real, inverted image in front of the concave mirror. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
18 Concave Mirror 1. Parallel ray - This ray passes through the focal point after reflecting from the mirror surface. 2. Focal ray - This ray passes through the focal point and is parallel after reflecting from the mirror surface. 3. Center ray - This ray is passes through the center of curvature and is reflected back on itself. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
19 Convex Mirror Parallel ray ray Focal ray ray Center ray Incident rays diverge as though coming from a point behind the surface of the convex mirror. Image is virtual and erect. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
20 Convex Mirror 1. Parallel ray - This ray is reflected at the surface at a trajectory that can be traced back through the focal point behind the mirror surface. 2. Focal ray - This ray is reflected at the surface at a parallel trajectory that can be traced straight back behind the mirror surface. 3. Center ray - This ray is reflected at the surface at a trajectory that can be traced back through the center of curvature behind the mirror surface. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
21 Rays From the Focal Point Incident rays (a) diverge from the focal point of the concave mirror and reflect from the mirror as a parallel beam. Rays converging (b) on the focal point of the convex mirror reflect off the surface and form a parallel beam MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
22 Rays From Infinity Incident rays parallel to the axis (a) converge to the focal point F of a concave mirror, (b) diverge as though coming from the focal point F of a convex mirror. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
23 Concave Mirror Parallel ray ray Focal ray ray Center ray ray MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
24 Concave Mirror Parallel ray ray Focal ray ray Center ray ray MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
25 Approximation: Paraxial rays Some Equations We only consider rays that are approximately parallel to the axis of the lens system R + = + = f = s s' R s s' f 2 Concave: r > 0; f > 0 Convex: r < 0; f < 0 MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
26 It is useful to specify the object distance as a multiple of the focal length Some Equations This picture has s = 3f = s s' f = - s' f s fs s' = s - f 2 ( ) 3f f 3f s' = = = 1.5f 3f - f 2f R + = + = f = s s' R s s' f 2 Concave: r > 0; f > 0 s' 1.5f 1 m = - = - = - s 3f 2 MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
27 It is useful to specify the object distance as a multiple of the focal length This picture has s = 2f Some Equations = - s s' f = - - s' f s -fs s' = s + f 2 ( ) 2f f 2f s' = - = - = -0.67f 2f + f 3f R + = + = f = s s' R s s' f 2 Convex: r < 0; f < 0 (- 2f ) s' 3 1 m = - = - = + s 2f 3 MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
28 Transparent Materials An Introduction to Lenses MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
29 Basic Problem - Relate Image to Object We want to derive a relationship between the image and the object using ray optics to derive the mathematical relationships. We will then use a small angle approximation to simplify these equations. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
30 Magnification y -y' tanθ = ; tanθ = 1 2 s s' Small angle approx tanθ θ y -y' θ = ; θ = 1 2 s s' Snell's Law n θ = n θ y -y' n = n 1 2 s s' y' n s' 1 m = = - y n s MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
31 y n = n s y' m = = - y 1 2 -y' s' n1s' n s 2 Magnification s > 0 for an object on the incident side s > 0 for an image on the refracted side R > 0 if the center of curvature is on the refracred side of the glass. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
32 Magnification n1 n2 n2 - n1 + = s s' R For a single curved surface there isn t a simple relationship between the radius of curvature R and the focal length f. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
33 n 1 = 1.00 n 2 = 1.33 s = 10 cm R = 15 cm s =? Example Image n1 n2 n2 - n1 + = s s' R = = 10 s' = s' s' = - = ( 1.33) s' = - = -17.1cm 11.7 Magnification n1s' m = - = n s m = 1.29 ( )( ) ( )( ) Object The image is real, erect and magnified by 29% MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
34 Thin Lenses MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
35 The Lens Maker s Equation For surface 1 nair n n - n + = s s ' R 1 1 For surface air n nair nair - n + = -s ' s' R Add these together and let s =>. 1 n 1 1 = f nair R1 R2 MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
36 The Lens Maker s Equation 1 n 1 1 = f nair R1 R2 The Thin Lens Equation = s s' f MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
37 Conventions for Thin Lenses MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
38 Converging Thin Lens This is also called a positive lens or a convex lens MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
39 Object-Image Relationships - Convex Lens parallel rays meet at a principal focus object distance and image distance each equal to twice the focal length object at principal focus, image at infinity object inside principal focus, virtual image MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
40 Thin Converging (Positive) Lens Object distance outside the focal distance There are two focal points. F is the first focal point and F is the second focal point. It is important to distinguish between these two points. It will prove essential for treating the negative lenses. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
41 Thin Converging (Positive) Lens Object distance outside the focal distance Ray 1 = Parallel ray: This ray approaches the lens parallel to the axis. It is deflected at the lens center line and passes through the second focus. Ray 2 = Center ray: This ray passes undeflected through the center of the lens. Ray 3 = Focal ray: This ray passes through the first focal point, is deflected at the center line of the lens and travels parallel to the axis. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
42 Object at 3f Object at 2f Object at 1.5f Object at 1.0f MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
43 Object at 2f/3 As the object distance approaches zero so does the image distance. The magnification approaches +1. The physical meaning of this arrangement is questionable. Object at -f/2 MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
44 Diverging Thin Lens This is also called a negative lens or a concave lens MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
45 Thin Diverging (Negative) Lens Object distance outside the focal distance The first (F) and second (F ) focal points are reversed relative to the converging lens. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
46 Thin Diverging (Negative) Lens Object distance outside the focal distance Ray 1 = Parallel ray: This ray approaches the lens parallel to the axis. It is deflected at the lens center line and follows a trajectory that can be traced back through the second focus. Ray 2 = Center ray: This ray passes undeflected through the center of the lens. Ray 3 = Focal ray: This ray approaches the lens on a line that would pass through the first focal point. It is deflected at the center line of the lens and travels parallel to the axis. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
47 Thin Diverging (Negative) Lens Parallel ray Focal ray 1st Focal point 2nd Focal point Center ray MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
48 Two Convex Lens System MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
49 MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
50 Optical Instruments MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
51 Similar Optical Systems MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
52 The Human Eye MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
53 The Human Eye MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
54 The Human Eye The cornea, aquueous humor and the lens all affect the diffraction of the light rays. The combination of all three is considered to be the effective lens. The average length of the eyeball, defined as the lens to retina distance, is 2.50 cm. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
55 Object at Infinity - Imaged at the Focal Point Fixed Image Distance = ; + = = 40 Diopters s s' f f f = 2.5 cm This situation is described as the relaxed eye MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
56 Diopter Unit of Measure The lens equation deals with inverse length units of measure. The diopter unit (m -1 ) takes advantage of this and makes it easier to deal with the lens equation. The quantities measured in diopters are referred to as curvatures and they can be easily added and subtracted = s s' f = d d f o i MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
57 Convex Lens - Object Approaching As the object approaches from infinity the image grows in size and moves further back from the focal point MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
58 The near point x np is defined as the smallest object distance for which the human eye can focus an image on the retina. The Near Point - X np = ; + = = 44 Diopters s s' f f The near point is taken to be 25 cm. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
59 The Range of the Human Eye The range of the human eye is from 40D to 44D. Object at Infinity Object at X np Power = 40D Power = 44D MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
60 Lens Combinations For two thin lens that are in contact or spaced very close the inverse focal lengths can be added. Since the inverse focal lengths are called powers then behavior means that the power are additive = f f f f 1 2 eq eq f f 1 2 = f 1 + f 2 MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
61 The Hyperopic Eye - Far Sighted As the object moves farther away the focus point moves in. It is easier to see far away objects The lens is too weak - the power needs to be increased. The correction needs to be a positive lens MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
62 The Hyperopic Eye Corrected & Uncorrected A person with hyperopia is far sighted. In other words, it is easier for them to focus on objects that are far away. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
63 The Myopic Eye - Nearsighted As the object moves in closer the focus point moves back It is easier to see near objects The lens is too strong - the power needs to be reduced. The correction needs to be a negative lens. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
64 The Myopic Eye Corrected & Uncorrected A person with myopia is near sighted. In other words, it is easier for them to focus on objects that are close by. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
65 The Myopic Eye - Nearsighted MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
66 Eye Correction with Reading Glasses This person s uncorrected near point is 75cm. The use of reading glasses allows the person to move an object at 25cm out to 75cm. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
67 The Non-Spherical Lens - Astigmatism Due to different radii of curvature in the horizontal and vertical planes, the astigmatic eye focuses horizontal and vertical images in different planes MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
68 Apparent Size Apparent size is greater when the object is closer to the eye. Image size proportional to θ y' φ θ 2.5cm Snell's Law θ nφ y s nairsinθ = nsinφ y ny' 2.5cm y -y' -2.5cm 1 = nφ = y' = m = = s 2.5cm n s y n s MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
69 Apparent Size m = -2.5cm n * s Coompare with the general formula n1 s' m = - n s 2 This formula has been specialized for this particlar situation by placing the number 2.5 in it. The units have also been fixed by the choice of cm for the image distance. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
70 Angular Magnification Goal: Push the image to infinity to yield parallel rays and easy viewing with the eye. y θ 0 = ; θ = x np y f The lens allows the object to be moved in closer. Angular Magnification = Magnifying Power = M θ x M = = θ f 0 np Simple magnifiers are used as eyepieces (oculars) on microscopes and telescopes MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
71 Problems: (1) Ignores the convention that the object ray always starts from the left. (2) The angle θ i is not unique. (3) The distance D isn t neeeded. A Simple Magnifying Glass MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
72 Compound Microscope Goal: Make a small object appear larger while viewing it with a relaxed eye. Note: (1) The object is just outside the Objective lens focal length. (2) The focal lengths of the two lenses do not overlap. (3) The Eyepiece is placed so that the image from the Objective lens fall just inside its (the eyepiece) focal length. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
73 Compound Microscope The object is just outside the Objective lens focal length to maximize the size of the image. The Eyepiece is placed so that the image from the Objective lens falls just inside its (the eyepiece) focal length. This will give a virtual image at infinity. The rays from this image will be parallel and provide easy (relaxed lens) viewing by the eye. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
74 Compound Microscope y tanβ = = f np m e = f e 0 y' -L m 0 = = y f x np M = m0m e = f 0 f e 0 -y' L -Lx Compound microscope requirement: The image of the objective lens needs to be just inside the focal point of the eyepiece MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
75 The Telescope The goal of the objective lens is not magnification but to place an image inside the focal length of the eyepiece. Note: (1) The object is at infinity (the rays enter from the left and are parallel. (2) The distance between the lenses is just about the sum of the focal lengths. (3) The purpose of the eyepiece is again to place the final image at a convenient viewing distance. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
76 The Diagram for Calculating the Magnification of the Telescope With the telescope it is the angular magnification that is important. If you are looking at the moon it is obvious that you haven t magnified the size of the moon. What has happened is that the angular that the image subtends on the back of your eye has increased - the moon appears bigger. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
77 Magnification of the Telescope y tanθ0 θ 0 = = s y' tanθe θ e = f θ -f M = = θ f e 0 0 e e -y' f 0 Lens separation = f e + f o Object at infinity images at focal point. This point coincides with the focal point of the eyepiece. The image from the objective becomes the object of the eyepiece and is imaged out to infinty. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
78 Telescopes MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
79 The Evolution of the Telescope Once it was apparent that the telescopes allowed one to see other far away worlds, people wanted MORE. This meant more magnification and they wanted to see things still further away. The first telescopes were hand held and most of those used for navigation remained so. (Check the Pirates of the Caribbean.) But for astronomy there seems to be no end in the quest for MORE. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
80 Evolution of the Astronomical Telescope Larger meant an increase the magnification M = -f o /f e This meant an increase in f o and longer telescopes More distant objects meant larger diameters and bigger objective lenses. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
81 Refractor Telescope Astronomer 91.4 cm telescope at Lick Observatory MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
82 Longer and Bigger also Meant Heavier Reflector scope from the 1780 s used by Friedrich Wilhelm Herschel MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
83 The Reflecting Telescope Eliminates the Need for Large Lenses Folding the optical pathway to shorten the length. No lens requirements. Requirements moved to the objective mirror. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
84 Different Viewing Options MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
85 Cross Section of a Charge Coupled Device This device contributes one pixel to the final picture. An array of millions of these devices forms the actual CCD device. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
86 Charged Coupled Device Array MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
87 Quantum Efficiency Quantum efficiency (QE) is a quantity defined for a photosensitive device such as photographic film or a charge-coupled device (CCD) as the percentage of photons hitting the photoreactive surface that will produce an electron hole pair. Photographic film typically has a QE of much less than 10%, while CCDs can have a QE of well over 90% at some wavelengths. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
88 Quantum Efficiency MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
89 Telescope - CCD Camera System System Diagram MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
90 Hubble Image of M42, the Orion Nebula MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
91 No More Long Cold Nights in the Telescope Edwin Hubble in the observer s seat in the 5.08m Hale reflecting scope. MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
92 Appendix MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
93 Sample Problem MFMcGraw-PHY 2426 Chap34a-Lenses-Revised: 7/13/
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