1. GEOMETRICAL OPTICS

2. Laws of Reflection Laws of Refraction

3. Law of reflection

4. Law of refraction

5. Huygen’s Principle All point on a given wavefront are taken as point sources for the production of spherical secondary waves (wavelets). These propagate outward. After some time the new position of the wavefront is the surface tangent tot the wavelets.

6. Huygen’s Principle

8. Fermat’s Principle – least time When light travels between two points, its path is the one that requires the least time.

9. Imaging by optical system O and I are conjugate points If O and I are points – perfect imagery Non ideal images due to (in practice) (i) light scattering (ii) aberrations (iii) diffraction Some rays leaving O do not arrive at I because of (i) reflection losses at refracting surfaces (ii) diffuse reflection from reflecting surfaces (iii) scattering by inhomogeneities in transparent media - thus diminishing of brightness

10. Cartesian surfaces form perfect images - reflecting and refracting

11. Sign Convention Assume light is from left to right Reflection (i) s> 0 if object is on same side as incoming light (ii) s′ > 0 if image is on same side as incoming light (iii) R > 0 if centre of curvature is away from incoming light Refraction (i) and (iii) same (ii) s′ > 0 if image is on the other side of vertex V

12. Reflection at a spherical surface

13. In paraxial approx

14. Observe if R →∞, plane mirror How do you describe the image (i) image distance (ii) magnification (iii) orientation

15. From diagram

16. + sign if O and I have same orientation - sign if O and I have different orientation

17. Refraction at a spherical interface Snell’s law of refraction Paraxial approximation nθ 1 ≈ n′θ 2

18. For an object at infinity, Image distance = second focal length

19. n′- n is called the power of the refraction R surface If R→∞ (plane surface) s′ apparent depth

21. Refraction by a thin lens

22. If n′′ = n i.e. same medium on both sides of lens (ii) becomes …… (iii) Subs. s 2 = -s 1 ′ in ( iii ) and adding (i) and (iii)

23. If object at infinity i.e. s = ∞ then s′ = f lensmaker’s equation NB (1) Convex lens – converging lens Rays refracted by lens bend towards axis (2) Concave lens – diverging Rays bend away from axis Magnification m = -s′ s

24. Sample ray diagram for converging (convex)

25. diverging (concave) lens

26. A lens thicker in the middle than at the ends - convex A lens thicker at the ends than in the middle – concave Magnification m = -s′ s The image is formed in the same medium as the object

27. Vergence and Refracting power

28. Newtonian equation for thin lens

29. Newtonian Equation for a Thin Lens Object and image distances measured relative to F Similar ∆’s and using + for height above and – for height below

31. Matrix methods for complex optical systems Complex optical systems, involving a number of refracting and/or reflecting elements may be analyzed using system matrices. A spherical thick lens is a lens whose thickness along the optical axis cannot be ignored without leading to serious errors when analyzed. Just when a lens moves from a thin to thick depends on the accuracy required.

32. Matrix methods for complex optical systems The thick lens can be treated by the methods discussed earlier where the glass medium is bounded by two spherical refracting surfaces. The thick lens can also be described in a way that allows graphical determination corresponding to arbitrary objects. This description, in terms of the cardinal points of the lens, is useful because it can be applied to more complex optical systems.

33. Cardinal points and Cardinal planes There are 6 cardinal points on the axis of a thick lens from which its imaging properties can be deduced. F 1 and F 2 – First and Second focal points P 1 and P 2 – First and Second principal points N 1 and N 2 – First and Second nodal points Planes normal to the axes at these six cardinal points are called Cardinal planes, focal planes, principal planes and Nodal planes.

34. Cardinal points and Cardinal planes NB 1. Between principal planes – unit magnification 2. At nodal planes rays laterally displaced 3. Distances to right of reference planes – positive and to the left – negative

35. Cardinal points and Cardinal planes

37. Translation matrix

38. Matrix Method One relates output (y′,α′) to input (y,α) 1. Translation matrix – translation through a medium

39. Refraction Matrix Refraction occurs at a point on the refracting surface

41. In refraction matrix, if surface is concave, R is negative; if surface is plane R = ∞ For a plane surface the refraction matrix is

42. Reflection Matrix A concave mirror is used, so R < 0 Sign convection is used Angles positive if rays point upwards and negative if downward y′ = y since reflection occurs ie y′ = (1)y + (0)α …(i)

44. Thick Lens Matrix Refraction at first surface, R 1

45. For a thick lens, where there is a refraction, followed by a translation, then a refraction, the system matrix is

46. Thin Lens But for a thin lens, lensmaker’s formula is

48. Element of the composite ray transfer matrix as The input plane was at the left surface (first refraction) and output plane at the right surface (second refraction) The determinant of the matrix given

50. Physical significances of system Matrix If m 22 = 0, then α f = m 21 y 0 (from (ii)) Now y 0 is fixed, so rays leaving a point on input plane emerge with same α f at the output plane (independent of α 0. input plane coincides with the First Focal Plane and parallel rays emerge.

51. Physical significances of system Matrix If m 11 = 0, then y f = m 12 α 0 from (i) y f is independent of y 0.All rays leaving input plane at same α 0 (parallel ray) arrive at the same point in the output plane ie at the Second Focal Plane. m 21 = 0 then from (ii), α f = m 22 α 0 So α f independent of y 0. Parallel input rays produce parallel output rays in another direction. is the angular magnification. This is called a “telescopic system.”

52. Physical significances of system Matrix If m 12 = 0 then from (i) y f = m 11 y 0 Independent of α 0. Case of conjugate planes. Input plane is object plane and output plane is image plane. is the linear magnification

53. Location of Cardinal Points

54. P is towards the left and using paraxial approx. p gives the location of the first focus relative to the input plane

55. Location of Cardinal Points From the figure r = - (f 1 – p) …(iv) the location of the first principal plane relative to input plane.

57. Location of Cardinal Points v is the location of the nodal plane relative to the input plane

58. Cardinal point locations in terms of system matrix elements

61. Useful Generalizations (1)r – s = v – w Note : If then and so (1)r = v and s = w ie Principal and Nodal points coincide (2) are equal in magnitude

64. Two thin lenses in air Let input and output planes be located at lenses’ centre System matrix involves (1) refraction by f 1 (2) translation through d (3) refraction by f 2

66. i.e. 1 st principal point and 1 st nodal point coincide Also 2 nd principal point and 2 nd nodal point coincide