1. Chapter 1. Ray Optics

2. Postulates of Ray OpticsB ds

3. Reflection and Refraction

4. Fermat’s Principle: Law of Reflection
Light rays will travel from point A to point B in a medium
along a path that minimizes the time of propagation. x y
(x1, y1)
(0, y2)
(x3, y3) qr qi A B
: Law of reflection

5. Fermat’s Principle: Law of Refractionx y
(x1, y1)
(x2, 0)
(x3, y3) qt qi A ni nt
: Law of refraction
in paraxial approx.

6. Refraction –Snell’s Law :

7. Negative index of refraction : n < 0
RHM
N > 1
LHM
N = -1

8. Principle of reversibility

9. Coherent perfect absorbers(CPA): time-reversibility
Time-reversed lasers, Lasing in reverse, Suckers for light
silicon
Time-reversed lasing and interferometric control of absorption, Science , vol. 331, 889 ( )

10. Reflection in plane mirrors

11. Plane surface – Image formation

12. Total internal Reflection (TIR)

13. Imaging by an Optical System

14. Cartesian Surfaces A Cartesian surface – those which form perfect
images of a point object
E.g. ellipsoid and hyperboloid O I

15. Imaging by Cartesian reflecting surfaces

16. Imaging by Cartesian refracting Surfaces

17. Approximation by Spherical Surfaces

18. Reflection at a Spherical Surface

19. Reflection at Spherical Surfaces I
Reflection from a spherical convex surface gives rise to a virtual image.
Rays appear to emanate from point I behind the spherical reflector.
Use paraxial or small-angle approximation for analysis of optical systems:

20. Reflection at Spherical Surfaces II
Considering Triangle OPC and then Triangle OPI we obtain:
Combining these relations we obtain:
Again using the small angle approximation:

21. Reflection at Spherical Surfaces III
Image distance s' in terms of the object distance s and mirror radius R:
At this point the sign convention in the book is changed !
The following sign convention must be followed in using this equation:
1. Assume that light propagates from left to right.
Object distance s is positive when point O is to the left of point V.
2. Image distance s' is positive when I is to the left of V (real image)
and negative when to the right of V (virtual image).
3. Mirror radius of curvature R is positive for C to the right of V (convex),
negative for C to left of V (concave).

22. Reflection at Spherical Surfaces IV
The focal length f of the spherical mirror surface is defined as –R/2, where R is the radius of curvature of the mirror. In accordance with the sign convention of the previous page, f > 0 for a concave mirror and f < 0 for a convex mirror. The imaging equation for the spherical mirror can be rewritten as

23. Reflection at Spherical Surfaces VII
Real, Inverted Image
Virtual Image, Not Inverted

24. Refraction

25. Prisms

26. Beamsplitters

27. Spherical boundaries and lenses
At point P we apply the law of refraction to obtain
Using the small angle approximation we obtain
Substituting for the angles q1 and q2 we obtain
Neglecting the distance QV and writing tangents for the angles gives
n2 > n1

28. Refraction by Spherical Surfaces
Rearranging the equation we obtain
Using the same sign convention as for mirrors we obtain
P : power of the refracting surface
n2 > n1

29. Example : Concept of imaging by a lens

30. Thin (refractive) lenses

31. The Thin Lens Equation IC1 O C2 V1 V2
For surface 1: s1 t
s'1

32. The Thin Lens Equation II
For surface 1:
For surface 2:
Object for surface 2 is virtual, with s2 given by:
For a thin lens:
Substituting this expression we obtain:

33. The Thin Lens Equation III
Simplifying this expression we obtain:
For the thin lens:
The focal length for the thin lens is found by setting s = ∞:

34. The Thin Lens Equation IV
In terms of the focal length f the thin lens equation becomes:
The focal length of a thin lens is
positive for a convex lens,
negative for a concave lens.

35. Image Formation by Thin Lenses
Convex Lens
Concave Lens

36. Image Formation by Convex Lens
Convex Lens, focal length = 5 cm: F ho hi RI

37. Image Formation by Concave Lens
Concave Lens, focal length = -5 cm: hi VI ho F F

38. Image Formation: Two-Lens System I
60 cm

39. Image Formation: Two-Lens System II
7 cm

40. Image Formation Summary Table

41. Image Formation Summary Figure

42. Vergence and refractive power : Diopter
reciprocals
Vergence (V) : curvature of wavefront at the lens
Refracting power (P)
Diopter (D) : unit of vergence (reciprocal length in meter)

43. Two more useful equations

44. 2-12. Cylindrical lenses

45. Cylindrical lenses
Top view
Side view

46. D. Light guides

48. 1-3. Graded-index (GRIN) optics

49. Rays in heterogeneous media
The optical path length between two points x 1 and x 2 through which a ray passes is
Written in terms of parameter s ,
Because the optical path length integral is an extremum (Fermat principle),
the integrand L satisfies the Euler equations.
For an arbitrary coordinate system , with coordinates q1 , q2 , q3,
Lagrange’s equations

50. GRIN In Cartesian coordinates so the x equation is
Similar equations hold for y and z .
In Cartesian Coordinates with Parameter s = s .
: Ray equation
Paraxial Ray Equation
ds ~ dz

51. GRIN slab : n = n(y) : paraxial ray equation
% Derivation of the Paraxial Ray Equation in a Graded-Index Slab Using Snell’s Law
The two angles are related by Snell’s law,

52. Ex GRIN slab with
Assuming an initial position y(0) = yo, dy/dz = qo at z = 0,

53. GRIN fibers

54. 1.4 Matrix optics : Ray transfer matrix
In the par-axial approximation,

55. What is the ray-transfer matrix

56. How to use the ray-transfer matrices

57. How to use the ray-transfer matrices

58. Translation Matrix
( yo, ao )
( y1, a1 ) L

59. Refraction Matrix
y=y’

60. Reflection Matrix
y=y’

61. Thick Lens Matrix I

62. Thick Lens Matrix II

64. Thin Lens Matrix
The thin lens matrix is found by setting t = 0: nL

65. Summary of Matrix Methods

66. Summary of Matrix Methods

67. System Ray-Transfer Matrix
Introduction to Matrix Methods in Optics, A. Gerrard and J. M. Burch

68. System Ray-Transfer Matrix
Any paraxial optical system, no matter how complicated, can be represented by a 2x2 optical matrix. This matrix M is usually denoted
A useful property of this matrix is that
where n0 and nf are the refractive indices of the initial and final media of the optical system. Usually, the medium will be air on both sides of the optical system and

69. Significance of system matrix elements
The matrix elements of the system matrix can be analyzed
to determine the cardinal points and planes of an optical system.
Let’s examine the implications
when any of the four elements of the system matrix is equal to zero.
D=0 : input plane = first focal plane
A=0 : output plane = second focal plane
B=0 : input and output planes correspond to conjugate planes
C=0 : telescopic system

70. D=0
A=0
B=0
C=0

71. System Matrix with D=0 Let’s see what happens when D = 0.
When D = 0, the input plane for the optical system is the input focal plane.

72. Ex) Two-Lens System f1 = +50 mm f2 = +30 mm Input Output Plane r
q = 100 mm r s
Input
Plane
Output F1 F2 T1 R1 R2 T3 T2

73. < check! > ƒ1 ƒ2 d H H’ F F’ ƒ ƒ’ s’ s h r

74. System Matrix with A=0, C=0
When A = 0, the output plane for the optical system is the output focal plane.
When C = 0, collimated light at the input plane is collimated light at the exit plane but the angle with the optical axis is different. This is a telescopic arrangement, with a magnification of D = af/a0.

75. System Matrix with B=0
When B = 0, the input and output planes are object and image planes, respectively, and the transverse magnification of the system m = A.

77. Ex) Two-Lens System with B=0
f1 = +50 mm
f2 = +30 mm
q = 100 mm r s
Object
Plane
Image F1 F2 T1 R1 R2 T3 T2