9.1 Introduction

Optics is the branch of physics that deals with the study of light — its behavior, properties, and interactions with matter.

In physics, optics is broadly classified into two categories:
- Ray Optics (Geometrical Optics): Light is treated as rays; explains reflection, refraction, and image formation.
- Wave Optics (Physical Optics): Light is treated as a wave; explains interference, diffraction, and polarization.

Ray optics assumes rectilinear propagation of light and is valid when the dimensions of interacting objects are much larger than the wavelength of light.

9.2 Reflection of Light by Spherical Mirrors

The laws of reflection state that:
1. The angle of incidence equals the angle of reflection: $\angle i = \angle r$
2. The incident ray, reflected ray, and the normal to the surface lie in the same plane.

These laws apply to all reflecting surfaces, including spherical mirrors.

For spherical mirrors:
- The normal at the point of incidence is along the radius of the sphere from the centre of curvature to the point of incidence.

Important terms:
- Pole (P): Geometric center of the mirror’s surface
- Centre of Curvature (C): Center of the sphere from which the mirror is a part
- Principal Axis: Line joining P and C
- Focus (F): Point where parallel rays converge or appear to diverge
- Focal Length (f): Distance between P and F

 Sign Convention (Cartesian)
- Distances measured from the pole (P)
- Positive: along the direction of incident light (usually left to right)
- Negative: against the direction of incident light
- Heights above the principal axis: positive
- Heights below the principal axis: negative

Focal Length and Radius of Curvature

For spherical mirrors, using geometry and small-angle approximation:

$ \tan \theta = \frac{MD}{CD} = \frac{MD}{FD} = \tan 2\theta \Rightarrow FD = \frac{CD}{2} $

So,
$ f = \frac{R}{2} $

Where $f$ is the focal length and $R$ is the radius of curvature.

9.3 Refraction

When a ray of light travels from one transparent medium to another, it bends at the interface. This bending is called refraction.

Snell’s Law:
$ \frac{\sin i}{\sin r} = n_{21} $

Where:
- $i$ is the angle of incidence
- $r$ is the angle of refraction
- $n_{21}$ is the refractive index of medium 2 w.r.t. medium 1

Refractive index from vacuum to a medium: $ n = \frac{c}{v} $

Reciprocal relation: $ n_{12} = \frac{1}{n_{21}} $

Optical density is not the same as mass density.

9.4 Total Internal Reflection

Occurs when light travels from a denser to a rarer medium and the angle of incidence exceeds the critical angle $i_c$.

Critical Angle:
$ \sin i_c = \frac{1}{n_{21}} $

Conditions:
1. Light must travel from a denser to a rarer medium.
2. Angle of incidence $i > i_c$

Applications:
- Optical fibers
- Prisms
- Diamonds

9.5.1 Refraction at a Spherical Surface

For a spherical interface between two media:

$ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} $

Where:
- $u$ = object distance
- $v$ = image distance
- $R$ = radius of curvature
- $n_1, n_2$ = refractive indices

9.5.2 Refraction by a Lens

Lens Maker’s Formula:
$ \frac{1}{f} = \left( \frac{n_2}{n_1} - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) $

Thin Lens Formula:
$ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} $

Sign Convention:
- $f > 0$ for convex lens, $f < 0$ for concave lens
- $u < 0$ when object is to the left
- $v > 0$ for real images (right side), $v < 0$ for virtual (left side)

9.5.3 Power of a Lens

Power of a lens:
$ P = \frac{1}{f \text{(in metres)}} $

- Unit: Dioptre (D)
- $P > 0$ for convex lens (converging)
- $P < 0$ for concave lens (diverging)

9.5.4 Combination of Thin Lenses in Contact

If multiple lenses of focal lengths $f_1, f_2, \dots$ are placed in contact:

$ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \dots $

Power:
$ P = P_1 + P_2 + \dots $

Total magnification:
$ m = m_1 \cdot m_2 \cdot \dots $

9.6 Refraction Through a Prism

For a prism of angle $A$ and angle of minimum deviation $D_m$:

$ n = \frac{\sin\left( \frac{A + D_m}{2} \right)}{\sin\left( \frac{A}{2} \right)} $

Angle of deviation:
$ d = i + e - A $

At minimum deviation:
- $i = e$
- $r_1 = r_2$

9.7 Optical Instruments

Simple Microscope:
$ m = 1 + \frac{D}{f} $ or $ m = \frac{D}{f} $

Compound Microscope:
$ m = \left( \frac{L}{f_o} \right) \left( 1 + \frac{D}{f_e} \right) $

Astronomical Telescope (normal adjustment):
$ m = \frac{f_o}{f_e} $

If final image is at distance $D$:
$ m = \left( \frac{f_o}{f_e} \right) \left( 1 + \frac{f_e}{D} \right) $

Reflecting telescopes use mirrors instead of lenses.

9.8 Summary

Reflection: $ \angle i = \angle r $

Refraction (Snell's law): $ \frac{\sin i}{\sin r} = n $

Total Internal Reflection: $ \sin i_c = \frac{1}{n} $

Mirror Formula: $ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} $

Lens Formula: $ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} $

Power: $ P = \frac{1}{f} $

Prism: $ n = \frac{\sin\left( \frac{A + D_m}{2} \right)}{\sin\left( \frac{A}{2} \right)} $

Microscope: $ m = \left( \frac{L}{f_o} \right) \left( 1 + \frac{D}{f_e} \right) $

Telescope: $ m = \frac{f_o}{f_e} $ (normal adjustment)