Geometric Optics: Laws & Refraction Details

1. GEOMETRIC OPTICS

2. Law of Reflection • The normal is a line perpendicular to the surface • It is at the point where the incident ray strikes the surface • The incident ray makes an angle of θ1 with the normal • The reflected ray makes an angle of θ1’with the normal

3. Law of Reflection, cont • The angle of reflection is equal to the angle of incidence • θ1= θ1’

4. Refraction of Light • When a ray of light traveling through a transparent medium encounters a boundary leading into another transparent medium, part of the ray is reflected and part of the ray enters the second medium • The ray that enters the second medium is bent at the boundary • This bending of the ray is called refraction

5. Refraction of Light, cont • The incident ray, the reflected ray, the refracted ray, and the normal all lie on the same plane • The angle of refraction, θ2, depends on the properties of the medium

6. Following the Reflected and Refracted Rays • Ray  is the incident ray • Ray  is the reflected ray • Ray  is refracted into the lucite • Ray  is internally reflected in the lucite • Ray  is refracted as it enters the air from the lucite

7. More About Refraction • The angle of refraction depends upon the material and the angle of incidence • The path of the light through the refracting surface is reversible

8. Refraction Details, 1 • Light may refract into a material where its speed is lower • The angle of refraction is less than the angle of incidence • The ray bends toward the normal

9. Refraction Details, 2 • Light may refract into a material where its speed is higher • The angle of refraction is greater than the angle of incidence • The ray bends away from the normal

10. The Index of Refraction • When light passes from one medium to another, it is refracted because the speed of light is different in the two media • The index of refraction, n, of a medium can be defined

11. Index of Refraction, cont • For a vacuum, n = 1 • For other media, n > 1 • n is a unitless ratio

12. Frequency Between Media • As light travels from one medium to another, its frequency does not change • Both the wave speed and the wavelength do change • The wavefronts do not pile up, nor are created or destroyed at the boundary, so ƒ must stay the same

13. Index of Refraction Extended • The frequency stays the same as the wave travels from one medium to the other • v = ƒ λ • The ratio of the indices of refraction of the two media can be expressed as various ratios

14. Snell’s Law of Refraction • n1 sin θ1 = n2 sin θ2 • θ1 is the angle of incidence • 30.0° in this diagram • θ2 is the angle of refraction

15. Total Internal Reflection • Total internal reflection can occur when light attempts to move from a medium with a high index of refraction to one with a lower index of refraction • Ray 5 shows internal reflection

16. Critical Angle • A particular angle of incidence will result in an angle of refraction of 90° • This angle of incidence is called the critical angle

17. Notation for Mirrors and Lenses • The object distance is the distance from the object to the mirror or lens • Denoted by p • The image distance is the distance from the image to the mirror or lens • Denoted by q • The lateral magnification of the mirror or lens is the ratio of the image height to the object height • Denoted by M

18. Types of Images for Mirrors and Lenses • A real image is one in which light actually passes through the image point • Real images can be displayed on screens • A virtualimage is one in which the light does not pass through the image point • The light appears to diverge from that point • Virtual images cannot be displayed on screens

19. Flat Mirror • Simplest possible mirror • Properties of the image can be determined by geometry • One ray starts at P, follows path PQ and reflects back on itself • A second ray follows path PR and reflects according to the Law of Reflection

20. Properties of the Image Formed by a Flat Mirror • The image is as far behind the mirror as the object is in front • q = p • The image is unmagnified • The image height is the same as the object height • h’ = h and M = 1 • The image is virtual • The image is upright • It has the same orientation as the object • There is an apparent left-right reversal in the image

21. a Normal line O O' Principal axis p q O O2 b c O1 O3 N = Numbers of image = The angle between two flat mirrors

22. Spherical Mirrors • A spherical mirror has the shape of a segment of a sphere • A concave spherical mirror has the silvered surface of the mirror on the inner, or concave, side of the curve • A convex spherical mirror has the silvered surface of the mirror on the outer, or convex, side of the curve

23. Concave Mirror, Notation • The mirror has a radius of curvature of R • Its center of curvature is the point C • Point V is the center of the spherical segment • A line drawn from C to V is called the principle axis of the mirror

24. Image Formed by a Concave Mirror • Geometry shows the relationship between the image and object distances • This is called the mirror equation

25. Image Formed by a Concave Mirror • Geometry can also be used to determine the magnification of the image • h’ is negative when the image is inverted with respect to the object

26. Focal Length • If an object is very far away, then p and 1/p  0 • Incoming rays are essentially parallel • In this special case, the image point is called the focal point • The distance from the mirror to the focal point is called the focal length • The focal length is ½ the radius of curvature

27. Focal Point and Focal Length, cont • The focal point is dependent solely on the curvature of the mirror, not by the location of the object • f = R / 2 • The mirror equation can be expressed as

28. Relationship between the image and object distances, Radius (R), focal length (f) a  v C    Principal Axis R p q

29. Convex Mirrors • 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 it lies behind the mirror at the point where the reflected rays appear to originate • In general, the image formed by a convex mirror is upright, virtual, and smaller than the object

30. Image Formed by a Convex Mirror

31. Ray Diagrams • A ray diagram can be used to determine the position and size of an image • They are graphical constructions which tell the overall nature of the image • They can also be used to check the parameters calculated from the mirror and magnification equations

32. Drawing A Ray Diagram (SKIP) • To make the ray diagram, you need to know • The position of the object • The position of the center of curvature • Three rays are drawn • They all start from the same position on the object • The intersection of any two of the rays at a point locates the image • The third ray serves as a check of the construction

33. The Rays in a Ray Diagram • Ray 1 is drawn parallel to the principle axis and is reflected back through the focal point, F • Ray 2 is drawn through the focal point and is reflected parallel to the principle axis • Ray 3 is drawn through the center of curvature and is reflected back on itself

34. Notes About the Rays (SKIP) • The rays actually go in all directions from the object • The three rays were chosen for their ease of construction • The image point obtained by the ray diagram must agree with the value of q calculated from the mirror equation

35. Ray Diagram for Concave Mirror, p > R • The object is outside the center of curvature of the mirror • The image is real • The image is inverted • The image is smaller than the object

36. Ray Diagram for a Concave Mirror, p < f • The object is between the mirror and the focal point • The image is virtual • The image is upright • The image is larger than the object

37. Ray Diagram for a Convex Mirror • The object is in front of a convex mirror • The image is virtual • The image is upright • The image is smaller than the object

38. Notes on Images • 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 increases, the virtual image gets smaller

39. Sign Conventions for Mirrors

40. Images Formed by Refraction • Rays originate from the object point, O, and pass through the image point, I • When n2 > n1, • Real images are formed on the side opposite from the object

41. Sign Conventions for Refracting Surfaces

42. Flat Refracting Surface • The image formed by a flat refracting surface is on the same side of the surface as the object • The image is virtual • The image forms between the object and the surface • The rays bend away from the normal since n1 > n2

43.    ‘ )  L' L   V   Principal Axis C R p q Normal line n tg = n' tg  ' n =n'  ' n( + )=n' (- ) n  + n= n' - n'  n (h/p) + n (h/R) = n' (h/R)- n' (h/q) n (h/p) + n' (h/q) =n' (h/R) - n(h/R) n/p + n'/q =(n' - n)/R  =  +   =  ' +  n sin  = n ' sin  ' if , , , , and ' very small, than

44. Thin Lenses • A thin lens consists of a piece of glass or plastic, ground so that each of its two refracting surfaces is a segment of either a sphere or a plane • Lenses are commonly used to form images by refraction in optical instruments

45. Thin Lens Shapes • These are examples of converging lenses • They have positive focal lengths • They are thickest in the middle

46. More Thin Lens Shapes • These are examples of diverging lenses • They have negative focal lengths • They are thickest at the edges

47. Focal Length of Lenses • The focal length, ƒ, is the image distance that corresponds to an infinite object distance • This is the same as for mirrors • A thin lens has two focal points, corresponding to parallel rays from the left and from the right • A thin lens is one in which the distance between the surface of the lens and the center of the lens is negligible

48. Lens Equations • The geometric derivation of the equations is very similar to that of mirrors

49. Lens Equations (SKIP) • The equations can be used for both converging and diverging lenses • A converging lens has a positive focal length • A diverging lens has a negative focal length

50. Sign Conventions for Thin Lenses