Phys. 281B Geometric Optics This Chapter 3 Physics Department Yarmouk University 21163 Irbid Jordan 1- Images Formed by Flat Mirrors 2- Images Formed by Spherical Mirrors 3- Images Formed by Refraction 4- Thin Lenses 5- Lens Aberrations Chapter 2: Image Formation Dr. Nidal Ershaidat http://ctaps.yu.edu.jo/physics/courses/phys281/chapter2 Part 1 Chapter 2 : Image Formation 1- Images Formed by Flat Mirrors 1
Reflection and Refraction Consider a (point) source of light and the rays incident on a flat surface of glass Fig. 1 shows several rays leaving the sources and their reflections and refractions P 5 Flat Mirror The simplest possible mirror is the flat mirror. Consider a point source of light placed at O in Figure 2, a distance p in front of a flat mirror. The (black) straight line represents the separating surface, i.e. the mirror s surface. 7 Fig. 2 Fig. 1 The distance p is called the object distance. What is a Mirror? A mirror is a surface separating two media which have different indices of refraction. The surface is perfectly polished.(مصقول) Almost 100% of incident light on this surface are reflected. We generally represent a mirror by a straight line and inclined lines in the nonpolished side. 6 Image Light rays leave the source and are reflected from the mirror. The rays continue to diverge. The dashed lines in Fig. 2 are extensions of the diverging rays back to a point of intersection at I. The diverging rays appear to the viewer to come from the point I behind the mirror. Point I is what we call the image of the object at O. 8 2
Image Distance Images are always located by extending diverging rays back to a point at which they intersect. Images are located either at a point from which rays of light actually diverge or at a point from which they appear to diverge. Because the rays in Figure 2 appear to originate at I. This is the location of the image. I is a distance q behind the mirror The distance q is called the image distance. 9 Real and Virtual Images The image formed by the mirror in Fig. 2 is virtual. The image of an object seen in a flat mirror is always virtual. Real images can be displayed on a screen (as at a movie), but virtual images cannot be displayed on a screen. We shall see an example of a real image in Section 2. 11 In the following sections we shall refer to the object distance by p and to the image distance by q. Real and Virtual Images Images are classified as real or virtual. A real image is formed when light rays pass through and diverge from the image point; A virtual image is formed when the light rays do not pass through the image point but only appear to diverge from that point. 10 Lateral Magnification 3
Animation If we repeat the process all other points of the object we get the image of the whole object. p 15 Properties of the Image formed by a flat mirror R O O A Standard Procedure - The Ray Diagram We choose two rays leaving the object*. The first one leaves the object at P, follows a horizontal path to the mirror, and reflects back on itself. The second ray follows the oblique path PR and reflects according to the law of reflection. Fig. 3 There are an infinite number of choices of direction in which light rays could leave each point on the object 14 p = q and h = h An observer in front of the mirror would trace the two reflected rays back to the point at which they appear to have originated, which is point P behind the mirror. The triangles PQR and P QR are congruent, i.e. PQ = P Q PO = P O p = q h = h طول الصورة = طول الجسم بعد الصورة = بعد الجسم O O 16 4
Lateral Magnification h' The ratio M = h is called the lateral magnification. This definition is valid for all types of mirrors and lenses. We see that in the case of a flat mirror M = 1 In general M could be larger or smaller than 1, depending on the instrument used to form the image 17 Summary Characteristics of the image of an object in a flat mirror: - Virtual It s formed behind the mirror - Upright or erect. - Its distance from the mirror equals that of the object. - The magnification here = 1 - Front to back reversal is observed 19 The left-right reversal One more feature of images in flat mirror is the socalled left-right reversal. Actually it is a front-back reversal. See Fig. 4 18 Multi-images Two flat mirrors are perpendicular to each other, as in Figure. 5, and an object is placed at point O. In this situation, multiple images are formed. Locate the positions of these images. 20 See the thumb! Fig. 4 See Example 36-1 Fig. 5 5
Multi-images Solution: Following the procedure we used previously, one can see that there are actually 3 images in this case! 21 Images Formed by Spherical Mirrors See Examples 36-2 and 36-3 Fig. 6 Further Reading For more details see: http://ctaps.yu.edu.jo/physics/courses/phys151 /Chapter6.pdf 22 Spherical Mirror A Spherical Mirror is simply a piece cut out of a reflective sphere. Its center of curvature, C, is the center of the sphere it was cut from. Each point of the mirror surface is, therefore, equidistant from C. R, the mirror's radius of curvature is the radius of the sphere. V is the vertex. 24 6
Concave and Convex Mirrors We have two types of spherical mirrors. Convex Concave mirrors ( ): ( ): light light is reflected is reflected from from the outer the concave or convex surface surface.. 25 Image in a Concave Mirror Consider a light source placed at point O, which lies on the principal axis Fig. 7 Shows two rays from O reflected by the mirror. They all meet at point I. 27 Rays arriving at I continue as if I itself is an object! The image I here is real, i.e. it can be collected on a screen Silvered reflecting surface Fig. 7 Principal Axis - Focal length The line extending through the center of curvature and the vertex of the mirror is the principal axis. Rays parallel to the principal axis are all reflected in such a way that they meet at a point on it lying halfway between the center of curvature and the vertex. This point is called the principal focus or the focal point. The focal length, f, is: f = R / 2 26 Paraxial Rays Rays diverging from an object which make small angles with the principal axis are called paraxial rays. All paraxial rays reflect through the image point. 28 7
Spherical Aberration Rays that are far from the paraxial axis converge to other points on the principal axis. The resulting image is a blurred image. 29 This effect is called spherical aberration. It is present to some extent for any spherical mirror. Images Formed by a Concave Mirror Construction of an Image in a Spherical Mirror Image in a Concave Mirror Consider an object of height h placed at some distance p > R of the mirror. In order to construct its image we do the following: 1) We first draw a ray from A passing through C. This ray reflects on itself since its perpendicular to the surface of the mirror: 2) We draw a 2 nd ray from A passing through the vertex V. This ray is reflected with respect to the law of reflection. The intersection of these rays form the image of A (Point A ) 32 8
Concave Mirror Image Step 1 1) We first draw a ray from A passing through C. This ray reflects on itself since its perpendicular to the surface of the mirror: A 1 33 Concave Mirror Image The intersection of these rays form the image of A (Point A ) A 35 A Fig. 8 Concave Mirror Image Step 2 2) We draw a 2 nd ray from A passing through the vertex V. This ray is reflected with respect to the law of reflection. 2 34 The Mirror Equation h is negative h p tanθ p M is given by: M = ' = = h q tanθ q Look now at the congruent triangles OAC and IA C h h' tanα = = p R R q h' R q q = = h p R p R p q p = pq q R ( p + q) = pq R 2 1 1 2 + = M p q R A A 36 9
Study of The Mirror Equation 1 1 2 For R + = 0 p q R R And we retrieve the relation for a flat mirror. This relation is valid for small angles. R For p q 2 p means that the object is very far from the mirror. In this case the image point is halfway between the center of curvature and the center point on the mirror. See Fig. 9. 37 Mirror Equation The mirror equation can be rewritten as: 1 1 1 + = M p q f This is a more familiar form of the mirror equation 39 Focal length The incoming rays from the object are essentially parallel in this figure since we assumed the source to be very far from the mirror. We call the image point in this special case the focal point F and the image distance the focal length f, where R f = Fig. 9 2 38 Images Formed by Convex Mirrors 10
Convex Mirrors If the reflecting surface of the mirror is the outer surface then the mirror is said to be convex. Here the center of curvature, C, is on the other side of the surface. We say that it is virtual The focal point, F, is halfway between C and the vertex V. It is also said to be virtual Silvered reflecting surface 41 Using the Ray Diagram Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected away from the focal point, F 1 Fig. 10-1 43 Image in a Convex Mirror We follow the same procedure in order to construct the image of an object in a convex mirror as for the concave mirror. Front Back 42 Image in a Convex Mirror Ray 2 is drawn through the center of curvature, C, on the back side of the mirror and is reflected back on itself 44 Fig. 10 2 In general, the image formed by a convex mirror is upright, virtual, and smaller than the object. Fig. 10-2 11
Image in a Convex Mirror For checking purposes we draw a 3 rd ray from the top of the object toward the focal point. This ray is reflected parallel to the principal axis 45 47 Convex Mirror = Diverging Mirror A convex mirror is sometimes called a diverging mirror. The rays from any point on the object diverge after reflection as though they were coming from some point behind the mirror The image is virtual because the reflected rays only appear to originate at the image point Fig. 10-3 Image in a Convex Mirror For checking purposes we draw a 3 rd ray from the top of the object toward the focal point. This ray is reflected parallel to the principal axis 46 Image in a Convex Mirror 48 1 3 2 Fig. 10-3 In general, the image formed by a convex mirror is upright, virtual, and smaller than the object. 12
Sign Conventions Sign Conventions Remember, of course, that what really happens, i.e., the physics of the situation, is entirely independent of what sign conventions you choose to use; 51 Locations and Sizes We study optical surfaces, i.e., mirrors, curved refracting surfaces, and lenses to understand how and why they produce images of objects; Therefore, we must agree on how to describe the locations and sizes, of the objects and their images created by the various surfaces. In addition, we need parameters to describe the size and/or shape of the optical surface, and an agreement as to which kind it is. A system containing more than one such surface will be studied one surface at a time, sequentially, following the actual path of a light ray through the system. 50 The Real is Positive Convention Considering a single optical surface, we first use it to divide all the space around it into two parts: the space where the light ray travels as it is incident on the surface; refer to this as Incoming Space; the space where the light ray travels as it leaves the optical surface; refer to this as Outgoing Space. The front: the space where the light ray travels as it is incident on the surface; The back: the space where the light ray travels as it leaves the optical surface. We shall adopt the so-called real is positive convention. The equations used for the concave mirror also apply to the convex mirror. 52 13
Sign Conventions Sign conventions apply to both concave and convex mirrors. Real is Positive Virtual is Negative 53 Summary With a concave mirror, the image may be either real or virtual When the object is outside the focal point, the image is real When the object is at the focal point, the image is infinitely far away When the object is between the mirror and the focal point, the image is virtual. With a convex mirror, the image is always virtual and upright. As the object distance decreases, the virtual image increases in size 55 Sign Conventions 54 3- Images Formed by Refraction 14
Image by Refraction Consider two transparent media having indices of refraction n 1 and n 2 The boundary between the two media is a spherical surface of radius R Rays originate from the object at point O in the medium with n = n 1 57 Image by Refraction Let s follow the (paraxial) ray OP, according to Snell s law we have:. n1 sinθ1 = n2 sin θ2 For small angles (θ 1 and θ 2 ) we can write: n1 θ1 = n2 θ2 59 S Fig. 11 Fig. 12 Image by Refraction In order to construct the image of the (point) source we shall follow a similar procedure as we did in the case of reflection, but here we have to apply Snell s law instead of the law of reflection, i.e. 58 Image by Refraction For small angles (θ 1 and θ 2 ) we can write: n1 θ1 = n2 θ2 We can easily see that: θ 1 = α + β and β = θ2 + γ ( α + β) = ( β γ) n 1 n2 n1 α + n2 γ = ( n2 n1 )β 60 S Fig. 12 Fig. 12 15
Geometry! d d d d tan α = = α, tanβ = = β, OS p SC R tan γ = d SI = d q γ 61 d d n1 + n2 = 2 1 p q d ( n n ) R Sign Conventions for Refracting Surfaces S Fig. 12 q is independent of θ 1 n1 n2 n2 n + = 1 L p q R This equation tells us that for a fixed object distance p, q does not depend on the angle on incidence. This also means that all paraxial rays from O will refract in I! Although we derived equation L in the case where we assumed that n 1 < n 2, this assumption is not necessary, however. Equation L is valid regardless of which index of refraction is greater. 62 Front and Back As with mirrors, we must use a sign convention if we are to apply this equation to a variety of cases. We define the side of the surface in which light rays originate as the front side. The other side is called the back side. Real images are formed by refraction in back of the surface, in contrast with mirrors, where real images are formed in front of the reflecting surface. 64 16
Comparing with Reflection Because of the difference in location of real images, the refraction sign conventions for q and R are opposite the reflection sign conventions. As an example, q and R are both positive in Figure 12. 65 Summary 67 A Flat Refracting Surface If a refracting surface is flat, then R is infinite and from equation L we have q = -(n 2 / n 1 )p The image formed by a flat refracting surface is on the same side of the surface as the object A virtual image is formed. 66 4- Thin Lenses 17
Lenses Lenses are commonly used to form images by refraction. Lenses are used in optical instruments Cameras Telescopes Microscopes 69 Locating the Image Formed by a Lens Lenses Light passing through a lens experiences refraction at two surfaces. The image formed by one refracting surface serves as the object for the second surface. 70 Locating the Image Formed by a Lens The lens has an index of refraction n and two spherical surfaces with radii of R 1 and R 2 R 1 is the radius of curvature of the lens surface that the light of the object reaches first R 2 is the radius of curvature of the other surface 72 The object is placed at point O at a distance of p 1 in front of the first surface. 18
Image From Surface 1 There is an image formed by surface 1 Since the lens is surrounded by the air, n 1 = 1 and n + p 1 n2 n2 n1 1 n n 1 q = R + p q If the image due to surface 1 is virtual, q 1 is negative, and it is positive if the image is real. 1 1 = R 1 73 Image Formed by a Thick Lens If a virtual image is formed from surface 1, then p 2 = -q 1 + t q 1 is negative t is the thickness of the lens If a real image is formed from surface 1, then p 2 = -q 1 + t q 1 is positive Then we have: 1 1 + p1 q2 = 1 R1 1 R2 ( n 1) Tl 75 Image From Surface 2 For surface 2, n 1 = n and n 2 = 1 The light rays approaching surface 2 are in the lens and are refracted into air Use p 2 for the object distance for surface 2 and q 2 for the image distance n + p 1 n2 n2 n1 n 1 1 q = R p 2 + q 2 = n R 2 74 Thin Lenses A thin lens is one whose thickness is small compared to the radii of curvature. For a thin lens, the thickness, t, of the lens can be neglected In this case, p 2 = -q 1 for either type of image. Then the subscripts on p 1 and q 2 can be omitted 76 19
Thin Lenses: p, q, n, R 1 and R 2 If we drop the subscripts on p 1 and q 2 in equation Tl then we have: 1 1 1 1 ( ) + = n 1 p q R1 R2 This expression is valid only - for paraxial rays and - when the lens thickness is much less than R 1 and R 2. 77 Lens Makers Equation Equation TL can be written as: 1 1 + = p q ( n 1) 1 1 R1 R2 This is what we call the lens makers equation 1 = f 79 Lens Makers Equation As for a mirror, the focal length of a thin lens is the image distance that corresponds to an infinite object distance. If p approaches and q approaches f in the previous equation, i.e. 1 1 1 + p q f q p f Then we see that the inverse of the focal length for a thin lens is given by: 1 f 1 R1 ( n 1) = 1 R2 78 Focal Length 20
Notes on Focal Length and Focal Point of a Thin Lens 81 Focal Length of a Diverging Lens 83 Because light can travel in either direction through a lens, each lens has two focal points One focal point is for light passing in one direction through the lens and one is for light traveling in the opposite direction However, there is only one focal length. Each focal point is located the same distance from the lens. The parallel rays diverge after passing through the diverging lens. The focal point is the point where the rays appear to have originated Focal Length of a Converging Lens 82 Determining Signs for Thin Lenses 84 The front side of the thin lens is the side of the incident light. The back side of the lens is where the light is refracted into. The parallel rays pass through the lens and converge at the focal point The parallel rays can come from the left or right of the lens This is also valid for a refracting surface. 21
85 Sign Conventions for Thin Lenses (Converging) Thin Lens Shapes These are examples of converging lenses They have positive focal lengths They are thickest in the middle 87 Magnification of Images Through a Thin Lens The lateral magnification of the image is M = h' h = When M is positive, the image is upright and on the same side of the lens as the object When M is negative, the image is inverted and on the side of the lens opposite the object q p 86 (Diverging) Thin Lens Shapes These are examples of diverging lenses They have negative focal lengths They are thickest at the edges 88 22
Converging Lenses - Ray 2 Ray 2 is drawn through the center of the lens and continues in a straight line. 91 Ray Diagram for Thin Lenses 5- Lens Aberration Converging Lenses - Ray 1 Ray 1 is drawn parallel to the principal axis and then passes through the focal point on the back side of the lens 90 Converging Lenses - Ray 3 Ray 3 is drawn through the focal point on the front of the lens (or as if coming from the focal point if p < ƒ) and emerges from the lens parallel to the principal axis. 92 23
Case: p > f The image is real (q>0) The image is inverted (h <0) The image is on the back side of the lens 93 Diverging Lenses For a diverging lens, the following three rays are drawn: Ray 1 is drawn parallel to the principal axis and emerges directed away from the focal point on the front side of the lens Ray 2 is drawn through the center of the lens and continues in a straight line Ray 3 is drawn in the direction toward the focal point on the back side of the lens and emerges from the lens parallel to the principal axis 95 p < f 94 Ray Diagram for Diverging Lens 96 The image is virtual (q<0) The image is upright (h >0) The image is larger than the object The image is on the front side of the lens The image is virtual The image is upright The image is smaller The image is on the front side of the lens 24
Image Summary For a converging lens, when the object distance is greater than the focal length (p > ƒ), the image is real and inverted For a converging lens, when the object is between the focal point and the lens, (p < ƒ) the image is virtual and upright. For a diverging lens, the image is always virtual and upright. This is regardless of where the object is placed. 97 Aberrations Assumptions have been: Paraxial Rays (Rays make small angles with the principal axis). The lenses are thin. The rays from a point object do not focus at a single point The result is a blurred image The departures of actual images from the ideal predicted by our model are called aberrations 99 Image Summary For a converging lens, when the object distance is greater than the focal length (p > ƒ), the image is real and inverted 98 Spherical Aberration This results from the fact that focal points of light rays far from the principal axis are different from the focal points of rays passing near the axis 100 For a converging lens, when the object is between the focal point and the lens, (p < ƒ) the image is virtual and upright. For a diverging lens, the image is always virtual and upright. This is regardless of where the object is placed. 25
Aberration Correction For a camera, a small aperture allows a greater percentage of the rays to be paraxial. For a mirror, parabolic shapes can be used to correct for spherical aberration. 101 Next Lecture Optical Instruments End of Chapter 2 Chromatic Aberration Different wavelengths of light refracted by a lens focus at different points. Violet rays are refracted more than red rays. The focal length for red light is greater than the focal length for violet light. Chromatic aberration can be minimized by the use of a combination of converging and diverging lenses made of different materials. 102 26