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Chapter 24: Geometric optics M. C. Escher What will we learn in this chapter? Contents: Reflection at a plane surface Reflection at a spherical surface Mirrors Refraction at a spherical surface Thin lenses Graphical methods for lenses Reflection at a plane surface Several rays of light diverge from the object point P and are reflected at the mirror. The rays diverging from P are reflected as if their direction had come from the image point P’. We say that the mirror forms an image of point P’. The rays do not come from P’, but their directions are the same as though they had come from that point. Refraction at a plane surface A similar behavior can be found at a refracting surface. Rays coming from P are refracted at the interface. When the angles of incidence are small it seems as if the rays are coming from the image point P’. When na > nb ( na < nb ) P’ seems closer (farther) to the surface than P. plane mirror Geometric (optics) construction The ray PV is normal to the mirror and thus retraces its path. The distance s = PV is called object distance, the distance s’ = VP’ image distance. The ray PB makes an angle θ with PV and strikes the mirror at an angle of incidence θ and is reflected at the same angle. |s| = |s! | When both rays are extended backwards, they intersect at P’. Both triangles PVB and P’VB are congruent, thus PV = VP’. This can be repeated for all rays originating from P (real image). The direction of all outgoing rays are the same as though they had originated at P’ (virtual image). Sign rules Sign rules for object and image distances: Object distance: !When the object is on the same side of the reflecting ! ! ! ! ! or refracting surface as the incoming light, the object ! ! ! ! ! distance s is positive, otherwise negative. Image distance: ! When the image is on the same side of the reflecting ! ! ! ! ! or refracting surface as the outgoing light, the image ! ! ! ! ! distance s’ is positive, otherwise negative. Note: These rules seem unnecessarily complex at this point, but will be very handy later for more complex situations. For a plane mirror s = −s! . Sign rules illustrated In both cases the object distance s is positive because the object point P is on the incoming side of the reflecting/refracting surface. The image distance is not on the same side as the outgoing ray, hence s’ < 0. The object distance is on the same side as the incoming ray, i.e., s > 0. The image distance is on the other side of the outgoing ray, i.e., s’ < 0. Objects with finite size || mirror The object is represented by the arrow PQ and has height y. The image is represented by the arrow P’Q’ and has height y’. Because the triangles PQV and P’Q’V have equal angles it follows that object and image have the same size, i.e., y = y’. object image Lateral magnification: For an object of height y and image y’, the lateral magnification m is y! m= y For a plane mirror, m = 1. Objects with finite size || mirror contd. Conventions: Objects are represented by arrows in diagrams. When the image arrow points in the same direction, we call the image erect, otherwise it is inverted. Note: the image formed by a plane mirror is always erect. A negative value of the lateral magnification m means the image is inverted. 3D objects on plane mirrors: The size of 3D objects are the same, but they are reversed even though the image is erect. Multiple reflecting/refracting surfaces The image of P in mirror 1 (P1’) serves as “object” for the image in mirror 2 (P3’). Therefore more complex setups can be treated. Note: Later these concepts will be used for successive curved refraction in a lens. Reflection at a spherical surface ’ vertex Setup: Radius of curvature R at the center of curvature C. Concave side faces incident light. P is an object point. For now assume PV > R. Point V is called the vertex of the mirror. The line PVC is called the optic axis. The ray PB is reflected at the mirror with an angle θ and passes the point P’. Reflection at a spherical surface contd. One can show that all rays emitted from P and reflected in the mirror pass point P’. This means P’ is a real image point. ’ vertex Using plane geometry one can show that α + β = 2φ Reflection at a spherical surface contd. Sign rule for the radius of curvature: When the center of curvature C is on the same side as the outgoing (reflected) light, the radius of curvature R is positive; otherwise negative. For a convex [concave] surface R is negative [positive]. R>0 R<0 Calculation of image distance s’ (spherical) tan β = tan φ = s! h −δ h R−δ vertex Goal: ! compute the image ! ! distance s’. Using trigonometry one can show that: h tan α = s−δ Because these equations cannot be solved, we can use a smallangle approximation for tan(x) and neglect the distance δ : 1 h h h tan x = x + x3 + . . . α= β= ! φ= 3 s s R Calculation of image distance s’ contd. Using α + β = 2φ we obtain an expression for the image distance. Image distance for a spherical mirror: 1 1 2 + ! = s s R Note: This is an approximation valid only if the angle α is small (paraxial approximation). As the angle α increases, the point P’ moves closer to the spherical mirror and there is no precise point image any longer. This is called spherical aberration. If R = ! (plane mirror) we obtain s = –s’ as before. Focal point (spherical mirror) When P is very far from the mirror (s = !) the incoming rays can be treated as parallel. It follows: R s! = 2 The distance FV is called focal length: f= R 2 It follows: 1 1 1 + ! = s s f Note: the focal length will be of importance when we study optical instruments. V Focal point (spherical mirror) contd. When the source of the rays in a spherical mirror lies at the focal point, they are reflected parallel to the optical axis. This construction is generally used for flashlights. Focal point of a concave spherical mirror: 1. ! Any incoming ray parallel to the optical axis is reflected trough the ! focal point. 2.! Any incoming ray that passes trough the focal point is reflected ! parallel to the optical axis. For spherical mirrors this is only true for paraxial rays, for parabolic mirrors this is exactly true. Lateral magnification of a spherical mirror What happens when we have an object of finite size PQ = y (arrow)? One can show for the lateral magnification: s! m=− s A negative sign means the image is inverted w.r.t. the object. Convex mirrors Now the incident light faces a convex spherical mirror. Because the center of the curvature is on the other side of the incoming rays, R < 0. One can show that all rays along PB are reflected and, when projected backward intersect the optical axis at P’, provided that the angle α is small. P’ is thus the image point of P. Note: s > 0, s’ < 0. One can show: 1 1 1 + ! = s s f Determination of the size of an image Perform a similar construction as before: As with a concave spherical mirror we obtain s! s The derived relations are valid for both convex and concave spherical mirrors! m=− Convex mirrors contd. When R < 0, incoming parallel rays are reflected as if they would come from the focal point F behind at a distance f the mirror. In this case F is called a virtual focal point. Applications of spherical mirrors: Wide-angle view mirrors at intersections. Solar reactors. … Graphical methods for mirrors Idea: Graphical methods to determine the size an position of reflected images in mirrors by drawing a selection of principal rays. Strategy: Identify known quantities such as s, s’, R and f and determine the unknowns. Pay attention to the signs of the objects! Use a ruler and draw the lines carefully. If the lines do not converge, extend the optical axis further. If necessary, color code the rays. Graphical methods for mirrors (concave) Graphical methods for mirrors (convex) Refraction at a spherical surface Goal: ! Study refraction at a spherical interface between two optical ! ! materials with different indices of refraction na and nb. Setup: P is the object point, P’ the image. We use the same rules as for the spherical mirror. Refraction at a spherical surface contd. When the angle α is small, we can do a similar calculation than before. Use the law of refraction na sin θa = nb sin θb and the paraxial approximation to obtain na θa = nb θb . Result: na nb nb − na + ! = s s R Note: Since the equation does not contain the angle α it is valid for all paraxial rays from P. The derived equations can be applied to both convex and concave interfaces. Use the sign rules consistently! Magnification at a spherical surface contd. Study the triangles PQV and P’Q’V. It follows: y −y ! tan θa = tan θb = ! use also na sin θa = nb sin θb s s and the approximation that for small angles tan(x) ≈ sin(x) . We obtain na y/s = −nb y ! /s! y! na s! m= =− y nb s Special case: plane surface For a plane surface R = !. In this case na nb m=1 + ! =0 s s It follows that s! = − nb s na If nb > na then objects are closer than they appear. Example: ! 2m deep pool. How deep does it ! ! ! seem? s! = − nb 1.00 s=− 2m = −1.5m na 1.33 Thin lenses Definition: ! A thin lens is made from two spherical surfaces so close ! ! ! ! that their distance is negligible. optic axis second focal point Characteristics: The center of the lens defines the optic axis. When parallel beams pass the lens they converge at the second focal point F2. Similarly, one can invert the picture: focal length Rays coming from F2 (F1) exit parallel to the optic axis. F1 and F2 are called the first and second focal point, respectively. Regardless of the lens curvature, the focal length f is the same on both sides. Determining an image trough a thin lens Let s and s’ be the object and image distances, respectively, and y and y’ their heights. Study rays QOQ’ and QAQ’. PQO and PQ’O’ are similar triangles, it follows: y! s! =− y s equate The negative sign is because the image is below the optic axis. The triangles OAF2 and P’Q’F2 are similar, it follows: y ! s! − f =− y f Determining an image contd. We obtain for a thin lens: 1 1 1 + ! = s s f s! m=− s Note: The negative sign tells us that when both s and s’ are positive, the image is inverted and y and y’ have opposite signs. The equations are the same as for spherical mirrors. So are the sign rules. Unlike mirrors, the image is not reversed, but inverted: a left hand is a left hand, but the fingers point in opposite directions. Converging vs diverging lenses Converging lens: Studied so far. Diverging lens: The focal length is negative. The focal points are virtual and reversed w.r.t. the focal points of a converging lens. The same equations as before apply (!) General definitions Converging lens: !a lens that is thicker at the center than at the ! ! ! ! ! edges (f > 0). Samples: meniscus planoconvex double convex Diverging lens:! a lens that is thinner at the center than at the ! ! ! ! ! edges (f < 0). Samples: meniscus planoconcave double concave Thin-lens equation Goal: ! derive a relationship between the focal length f, the index of ! ! refraction n and the radii of curvature R1 and R2 of an ! ! arbitrary lens. Setup: Study the general problem of two spherical interfaces separating three materials with indices of refraction na, nb, nc. Assume the distance t between the interfaces is small enough that it can be neglected. The idea is to daisy-chain the different interfaces and compute relations between the distances. Thin-lens equation contd. The object and image distances are shown above. Note that for example s2 = –s’1 depending from which surface it is seen from. Use the single-surface equation twice to obtain: na nb nb − na + ! = s1 s1 R1 nb nc nc − nb + ! = s2 s2 R2 Thin-lens equation contd. In general, we study a lens in air/vacuum, i.e., na = nc = 1 and nb = n. We obtain by using s2 = –s’1: 1 n n−1 n 1 1−n + ! = − ! + ! = s1 s1 R1 s1 s2 R2 Add both equations: ! " 1 1 1 1 + ! = (n − 1) − s1 s2 R1 R2 Since we treat the lens as a single unit, we set s1 = s and s’2 = s… Lensmaker equation (thin-lens equation): 1 1 1 + ! = = (n − 1) s s f ! 1 1 − R1 R2 " Thin lens equation sign conventions Previous sign conventions still work. Example: s, s’ and R1 are positive, R2 is negative. Graphical methods for lenses Converging lens: Graphical methods for lenses contd. Diverging lens: Graphical methods for lenses contd.