1. DR MOHAMAD HALIM ABD. WAHID School of Microelectronic Engineering
CHAPTER 3
RAY OPTICS
Prepared by
DR MOHAMAD HALIM ABD. WAHID
School of Microelectronic Engineering

2. OBJECTIVES Studies about: 1) Postulates of Ray Optics
2) Definition of oblique rays
3) Understanding in Graphical Method for Ray Tracing
4) Ray-tracing Formula
5) Matrix Optic: For reflection & Reflection

3. POSTULATES OF RAY OPTICS
Light travels in form of rays. The rays are emitted by light sources and can be observed when they reach an optical detector.
An optical medium is characterized by a quantity n>1, called the reflective index. The reflective index n=C0 /C where C0 is the speed of light in free space and C is the speed of light in medium. Therefore, the time taken by light to travel a distance d is d/c=nd/C0

4. It is proportional to the product nd, which is known as the optical pathlength.
In an homogeneous medium the refractive index n(r) is a function of the position r= (x,y,z). The optical path length along a given path between two points A and B is therefore

5. Where ds is the differential element of length along the path
Where ds is the differential element of length along the path. The time taken by light travel from A to B is proportional to the optical pathlength.
OBLIQUE RAYS
All rays which lie in a plane through the principal axis and are not paraxial are called oblique rays.
When the law of refraction is accurately applied to the number of rays through one or more coaxial surfaces, the position of

6. the image point is found to vary with the obliquity of the rays.
Lens designers follow three general lines of approach to the problem of finding the optimum conditions.
The first is to use graphical methods to find the approximate radii and spacing of the surfaces that should be used for the particular problem at band.

7. The second is to use well-known aberration formulas to calculate the approx­imate shapes and spacings
If the results of these methods of approach do not produce image-forming systems of sufficiently high quality and better definition is required, the third method, known as ray tracing, is applied

8. The diagrams in Fig. 8A illustrate the construction for refraction at a single spherical surface separating two media of index n and n'. After the axis and the surface with a center at C are drawn, any incident ray like 1 is selected for tracing
FIGURE 8A
A graphical method fo. ray tracing through a single spherical surface. The method is exact and obeys Snell's law for all rays.

9. An auxiliary diagram is now constructed below, comparable in size, and with its axis parallel to that of the main diagram
With the point 0 as a center two circular arcs are drawn with radii proportional to the refractive indices.

10. Succeeding steps of the construction are carried out in the following order:
Line 2 is drawn through 0 parallel to ray 1.
Line 3 is drawn through points T and C
Line 4 is drawn through parallel to line 3 and extended to where it intersects the arc n' at Q

11. Line 5 connects 0 and Q, and line 6 is drawn through T parallel to line 5.
In this diagram the radial line TC is normal to the surface at the point T and corresponds to the normal NN' in Fig. I G. The proof that such construction follows Snell's law

12. RAY-TRACING FORMULAS
A diagram from which these formulas can be derived is given in Fig. 8D.
An oblique ray MT making an angle θ with the axis is refracted by the single spherical surface at T so that it crosses the axis again at M'.
FIGURE 8D
Geometry used in deriving the ray-tracing formulas

13. The line TC is the radius of the refracting surface and constitutes the normal from which the angles of incidence and refraction at T are measured.
As regards the signs of the angles involved, we consider that:

14. Slope angles are positive when the axis must be rotated counterclockwise through an angle of less than πl2 to bring it into coincidence with the ray.
Angles of incidence and refraction are positive when the radius of the surface must be rotated counterclockwise through an angle of less than nl2 to bring it into coincidence with the ray.

15. Accordingly, angles θ, ø, and ø' in Fig
Accordingly, angles θ, ø, and ø' in Fig. 8D are positive, while angle θ' is negative.
Applying the law of sines to the triangle MTC, one obtains

16. Since the sine of the supplement of an angle equals the sine of the angle itself,
Solving for sin ø, we find

17. Now by Snell's law the angle of refraction ø' in terms of the angle of incidence ø is given by
In the triangle MTM’ the sum of all interior angles must equal π. Therefore

18. Which upon solving for
θ’=ø’+ θ- ø (8c)
This equation allows us to calculate the slope angle of the refracted ray
To find where the ray crosses the axis and the image distance s', the law of sines may be applied to the triangle TCM', giving

19. The image distance therefore
An important special case is that in which the incident ray is parallel to the axis. Under this simplifying condition it may be seen from Fig. 8E that

20. where h is the height of the incident ray PT above the axis
where h is the height of the incident ray PT above the axis. For the triangle TCM', the sum of the two interior angles ø' and θ equals the exterior angle at C.
When the angles are assigned their proper signs, this gives

21. The six equations above which are numbered form an important set by which any oblique ray lying in a meridian plane can be traced through a number of coaxial spherical surfaces.
A meridian plane is defined as any plane containing the axis of the system. While most of the rays emanating from an extra axial object point do not lie in a meridian plane, the image-forming properties of an optical system can usually be determined from' properly chosen meridian rays

22. Example Ray-Tracing Formulas
A convex spherical surface of radius r = cm is ground and polished on the end of a large cylindrical glass rod of index Assume incident light parallel to the axis by using rays at heights of
(a) 3.0 em, (b) 2.0 cm,
(c) 1.0 cm, (d) 0 cm

23. ANS:

24. MATRIX OPTICS Technique for tracing paraxial rays
Rays assume travel only within a single plane.
So its applicable to systems with planar geometry and to meridional rays.
Optical system is describe by a 2X2 matrix called the ray transfer matrix.

25. The Ray-Transfer Matrix
Consider rays in plane containing the optical axes, y-z plane.
A ray crossing the transverse plane at z is completely characterized by coordinate y, crossing point at angle θ (Fig 1.4-1)
Optical component placed between two transverse plane Z1 and Z2
Arbitrary position and direction (y1,θ1) and another position (y2,θ2) at output plane.

27. In paraxial approximation, when all angles are sufficiently small, so sin θ~ θ
Relation between (y2, θ2) is linear and can generally written in forms
Where A,B,C and D are real number. That equation is write in matrix form as

28. Matrix M, whose elements are A,B,C,D characterizes the optical system completely permits (y2, θ2) to determined for any (y1,θ1).
Its known as ray-transfer matrix.
Radii that turn out negative indicate concave surface, while convex is positive.

29. Example Matrices of Simple Optical Components
Free space propagation
Rays travel along straight lines in a medium of uniform refractive index such as free space, a ray traversing a distance d is altered in accordance with y2= y1+θd
and θ2= θ1

30. Planar boundary between two media
Ray changes in accordance with Snell’s
Law n1 sin θ1= n’1 sin θ’1
Paraxial approximation n1θ1~ n2 sin θ2
Position of ray is not altered y2=y1

31. Reflection at Spherical Boundary
Relation between θ1 and θ2 for paraxial rays refracted at spherical boundary is provide in Eq
Ray high is not altered, y2~y1

32. Reflection from planar mirror
Ray position is not altered y2=y1
Z axis points in the general direction to travel of the rays, toward the mirror for incident rays and away from the selected rays, hence θ1= θ2

33. Reflection from Spherical Mirror
A mirror with radius of curvature R bends rays in a manner that is identical to that of a thin lens with focal length f=-R/2

34. Example to determine Matrix Optics
The Lens Matrix

35. Lens with centre C, thickness t1 between vertices V1 and V2
Ray arriving at point P1 with coordinates
(n1,α1,X1) while coordinate after crossing (n1’,α1’,X1’)
R1, R2 are radii at the centre C1 and C2
Angles of incidence and reflection (θ1 and θ1’) at the first interface are both positive as per sign convention.
Refracted ray is incident on the right interface at point P2.

36. Ray coordinate just before and just after the right interface (n2,α2,X2), (n2’,α2’,X2’) respectively
Assuming sharp boundaries between the media, refraction at each interface changes the direction of the ray, leaving the ray height unmodified.

37. For the angles at the first interface,
Where angle ø being negative is written as
Ø=-X1’/R since X1’ and R1 are positive
R1 of the first surface of the lens is to be measured from vertex V1.
Angle α1 is +ve since incident ray has been chosen.
Therefore drop the magnitude sign in Eq 4.5

38. Similarly
Substitute Eq 4.6 into Eq 4.3
Simplifies to

39. Re-writing Eq 4.4a, hence
Eq 4.7 can be expressed in the matrix notation as

40. where
The column matrices and
identify the incident and refracted rays, respectively, at the front surface of the lens. The 2 x 2 unimodular refraction matrix
Describe refraction at the front surface of the lens

41. SUMMARY
All rays which lie in a plane through the principal axis and are not paraxial are called oblique rays.
Ray tracing formulas

42. SUMMARY Ray tracing formulas θ’=ø’+ θ- ø (8c)
Optical system is describe by a 2X2 matrix called the ray transfer matrix.