Optics Summary

Reflection, Refraction, Lenses, Wave Optics, and Polarization

Reflection
Refraction
Lenses
Wave Optics
Polarization

1. Reflection

1.1 Plane Mirrors

Properties of Plane Mirror Images:

  • Image is always virtual, erect, and of same size as object
  • Image is at same distance behind mirror as object is in front
  • If object moves towards mirror at speed v, image approaches at same speed v
  • Image suffers lateral inversion (left-right reversal)

Deviation by Plane Mirror

\[ \delta = 180^\circ - (i + r) = 180^\circ - 2i \]

Maximum deviation occurs at normal incidence: δmax = 180° when i = 0°

1.2 Spherical Mirrors

Mirror Formula
\[ \frac{1}{u} + \frac{1}{v} = \frac{1}{f} \]

Where u = object distance, v = image distance, f = focal length

Lateral Magnification
\[ m = \frac{-v}{u} \]
Newton's Formula
\[ x_1 x_2 = f^2 \]

Where x₁ and x₂ are object and image distances measured from focus

Applications:

  • Convex mirrors: Used as rear-view mirrors (increased field of view)
  • Concave mirrors: Used by dentists (enlarged image when object between F and P)

2. Refraction

Snell's Law
\[ \frac{\sin i}{\sin r} = \text{constant} = \mu \]

2.1 Apparent Shift

\[ h' = \frac{h}{\mu} \] \[ s = h - h' = h\left(1 - \frac{1}{\mu}\right) \]

2.2 Total Internal Reflection

Total internal reflection occurs when light travels from denser to rarer medium and angle of incidence exceeds critical angle.

\[ \mu_2 \sin \theta_c = \mu_1 \] \[ \theta_c = \sin^{-1}\left(\frac{\mu_1}{\mu_2}\right) \]

If rarer medium is air (μ₁ = 1):

  • For glass: θc ≈ 42°
  • For water: θc ≈ 49°

2.3 Prisms

A prism is an optical medium bounded by two plane refracting surfaces inclined at an angle.

\[ \delta = A(\mu - 1) \quad \text{(for small angles)} \] \[ \mu = \frac{\sin\left(\frac{\delta_{\min} + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

2.4 Dispersion

Dispersion is the splitting of white light into its constituent colors due to wavelength-dependent refractive index.

\[ \mu = A + \frac{B}{\lambda^2} \quad \text{(Cauchy's formula)} \]

Angular Dispersion

\[ \theta = \delta_v - \delta_r \]

Dispersive Power

\[ \omega = \frac{\theta}{\delta_y} = \frac{\delta_v - \delta_r}{\delta_y} = \frac{\mu_v - \mu_r}{\mu_y - 1} \]

Achromatic Prism (Deviation without Dispersion)

\[ (\mu_v - \mu_r)A = (\mu'_v - \mu'_r)A' \]

Direct Vision Prism (Dispersion without Deviation)

\[ (\mu - 1)A = (\mu' - 1)A' \]

3. Lenses

3.1 Refraction at Curved Surfaces

\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]
\[ m = \frac{\mu_1 v}{\mu_2 u} \]

3.2 Lens Formula

Lens Formula
\[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \]
Magnification
\[ m = \frac{v}{u} \]
Lensmaker's Formula
\[ \frac{1}{f} = \left(\frac{\mu_2}{\mu_1} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]

3.3 Lens Power and Combinations

Power of a lens: P = 1/f (in diopters, when f is in meters)

\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} + \ldots + \frac{1}{f_n} \] \[ P = P_1 + P_2 + \ldots + P_n \]

3.4 Optical Instruments

Simple Microscope

\[ M = 1 + \frac{D}{f} \]

Compound Microscope

\[ M_{\min} = -\frac{v}{u} \frac{D}{f_e} \quad \text{(image at ∞)} \] \[ M_{\max} = -\frac{v}{u} \left(1 + \frac{D}{f_e}\right) \quad \text{(image at D)} \]

Astronomical Telescope

\[ M = \frac{f_o}{f_e} \]

4. Wave Optics

4.1 Huygen's Principle

Key Points:

  • Every point on wavefront vibrates in same phase with same frequency
  • Every point acts as new source of secondary waves
  • Wavefronts move with wave velocity in the medium

4.2 Coherent Sources

Two sources are coherent if they have same frequency and constant phase difference.

4.3 Young's Double Slit Experiment

Fringe Width
\[ \omega = \frac{\lambda D}{d} \]
Optical Path
\[ L = \mu d \]

Intensity Distribution

\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos\phi \]

For equal intensities (I₁ = I₂ = I₀):

\[ I = 4I_0 \cos^2(\phi/2) \]

Maxima: Imax = 4I₀ when φ = 2nπ

Minima: Imin = 0 when φ = (2n-1)π

4.4 Diffraction

Diffraction is the spreading of waves when passing through openings comparable to wavelength.

\[ a \sin\theta = n\lambda \quad \text{(diffraction minima)} \]
\[ I = I_0 \left(\frac{\sin\phi}{\phi}\right)^2 \] \[ \phi = \frac{\pi a \sin\theta}{\lambda} = \frac{\pi}{\lambda}\left(\frac{ay}{D}\right) \]

5. Polarization

Polarization is the phenomenon of restricting light oscillations to a particular plane perpendicular to direction of propagation.

5.1 Types of Light

  • Unpolarized: Oscillations in all possible directions
  • Plane polarized: Oscillations restricted to single direction

5.2 Methods of Polarization

  • Refraction
  • Reflection
  • Dichroism (Nicol prism)
  • Scattering

5.3 Brewster's Law

Brewster's Law
\[ \mu = \tan i_p \]

At polarizing angle, reflected and refracted rays are perpendicular: ip + r = 90°

5.4 Nicol's Prism

Based on double refraction in calcite crystal, producing ordinary and extraordinary rays (both plane polarized).

5.5 Malus Law

Malus Law
\[ I = I_0 \cos^2\theta \]

For unpolarized light: I = I₀/2

5.6 Polaroids

Artificial polarizing devices using quinine iodosulphate crystals in plastic sheets.