Comprehensive Study Notes on Wave Optics and the Nature of Light

NATURE AND THEORIES OF LIGHT (5/6/26)

Newton's Corpuscular Theory (1675)

• Assumptions:

  • Light sources emit a large number of tiny, discrete particles called 'corpuscles'.
  • When these corpuscles strike our retina, the sensation of vision is produced.
  • Different colors of light are due to the different sizes of the corpuscles.
  • Due to their high speed and extremely low mass, corpuscles are not significantly affected by Earth's gravitational field.

• Drawbacks:

  • The theory could not explain the phenomena of partial reflection and refraction, Interference, Diffraction, or Polarization.
  • Newton incorrectly theorized that the speed of light in a denser medium is greater than the speed of light in a rarer medium ($C_{denser} > C_{rarer}$).
  • The theory suggests that as particles are emitted from the light source, its mass must decrease. However, practically, the mass of a light source is found to remain constant.

Chronology of Light Theories

1. Newton's Corpuscular Theory (1675)
2. Huygens' Wave Theory (1678)
3. Maxwell's Electromagnetic Theory (1873)
4. Planck's Quantum Theory (1900)
5. De-Broglie's Dual Nature Theory (1974)

Fundamental Formulas in Optics

• Refractive Index ($\text{RI}$):
  • ${_1\text{RI}_2} = \frac{1}{\text{sin}(i_c)}$
  • ${_1\text{RI}_2} = \frac{\text{Velocity}_1}{\text{Velocity}_2} = \frac{C_1}{C_2}$
  • ${_1\text{RI}_2} = \frac{\text{Wavelength}_1}{\text{Wavelength}_2} = \frac{\lambda_1}{\lambda_2}$
  • ${_1\text{RI}_2} = \frac{\text{Real Depth}}{\text{Apparent Depth}} = \frac{\text{RD}}{\text{AD}}$
  • ${_1\text{RI}_2} = \text{tan}(i_p)$

INTERFERENCE OF LIGHT (TYPE 4)

Definitions and Principles

• Interference of Light: The formation of bright and dark bands when light waves superimpose.
• Constructive Interference (Bright Band): Occurs for brightness when a crest of one wave meets the crest of another, or a trough of one wave meets the trough of another.
  • Phase Difference ($\phi$): $0, 2\pi, 4\pi, 6\pi, \dots$
  • Path Difference ($\text{PD}$): $0, \lambda, 2\lambda, 3\lambda, \dots = n\lambda$ (where $n = 0, 1, 2, \dots$).
• Destructive Interference (Dark Band): Occurs for darkness when a crest of one wave meets the trough of another.
  • Phase Difference ($\phi$): $\pi, 3\pi, 5\pi, \dots$
  • Path Difference ($\text{PD}$): $\frac{\lambda}{2}, \frac{3\lambda}{2}, \frac{5\lambda}{2}, \dots = (2m - 1)\frac{\lambda}{2}$ (where $m = 1, 2, 3, \dots$).
• Visual Shape: Interference bands are conjugate hyperbolas in shape.

Conditions for Steady-State Interference Pattern

1. The two light sources must be coherent.
2. The sources must be monochromatic.
3. The sources must be equally bright (emit light of equal amplitude and intensity).
4. The sources must be sufficiently narrow.
5. The sources must be close to each other.
6. The sources should be at a sufficiently large distance from the screen.
7. The two interfering waves must be in the same state of polarization.
8. The waves must travel in the same direction.

Analytical Treatment (Derivation of Distance of Bands)

• Given: Two sources $S_1$ and $S_2$ separated by distance $d$. Screen is at distance $D$. Point $P$ is at distance $x$ from central bright band ($CBB$).

• Triangle calculations for path from sources to point $Q$ on screen:

  • $S_1Q^2 = D^2 + (x - \frac{d}{2})^2$
  • $S_2Q^2 = D^2 + (x + \frac{d}{2})^2$
  • $S_2Q^2 - S_1Q^2 = (D^2 + x^2 + xd + \frac{d^2}{4}) - (D^2 + x^2 - xd + \frac{d^2}{4}) = 2xd$
  • $(S_2Q - S_1Q)(S_2Q + S_1Q) = 2xd$
  • Since $S_2Q \approx S_1Q \approx D$, then $(S_2Q - S_1Q)(2D) = 2xd$
  • Path Difference ($PD$): $\frac{xd}{D}$

• Distance of $n^{\text{th}}$ Bright Band (BB): $\frac{xd}{D} = n\lambda \rightarrow x_n = \frac{n\lambda D}{d}$

• Distance of $m^{\text{th}}$ Dark Band (DB): $\frac{xd}{D} = (2m - 1)\frac{\lambda}{2} \rightarrow x_m = \frac{(2m - 1)\lambda D}{2d}$

Concept of Bandwidth (Fringe Width)

• Definition: The distance between any two successive bright bands or any two successive dark bands. Represented by $\beta$ or $X$.
• Derivation:
  • For Bright Bands: $X = x_{n+1} - x_n = \frac{(n+1)\lambda D}{d} - \frac{n\lambda D}{d} = \frac{\lambda D}{d}$
  • For Dark Bands: $X' = x_{m+1} - x_m = \frac{[2(m+1)-1]\lambda D}{2d} - \frac{(2m-1)\lambda D}{2d} = [\frac{2m+1-2m+1}{2}]\frac{\lambda D}{d} = \frac{\lambda D}{d}$
  • Conclusion: Bright and dark bands are equally spaced.

Variations and Shortcuts

• Immersion of Setup: If the entire interference setup is dipped in a medium of refractive index $\mu$
  • $\beta_{medium} = \frac{\lambda_{medium} D}{d} = \frac{\lambda_{air} D}{\mu d} = \frac{\beta_{air}}{\mu}$
• Coincident Bands: If the $n_1^{\text{th}}$ band of wavelength $\lambda_1$ coincides with the $n_2^{\text{th}}$ band of wavelength $\lambda_2$, then: $n_1 \lambda_1 = n_2 \lambda_2$
• Interposing a Glass Slab: If a glass slab of thickness $t$ and refractive index $\mu$ is placed in front of one slit, the whole pattern shifts by distance $X_s$:
  • One Slab: $X_s = \frac{\beta}{\lambda}(\mu - 1)t = \frac{D}{d}(\mu - 1)t$
  • Two Slabs: $X_s = \frac{\beta}{\lambda}(\mu_1 - \mu_2)t = \frac{D}{d}(\mu_1 - \mu_2)t$

INTENSITY AND AMPLITUDE IN INTERFERENCE

• Intensity ($I$) is proportional to the square of the amplitude ($a$): $I \propto a^2$
• Resultant Amplitude ($a_R$): $a_R = \sqrt{a_1^2 + a_2^2 + 2a_1 a_2 \text{cos}(\phi)}$
  • $a_{max} = a_1 + a_2$ (when $\phi = 0$
  • $a_{min} = a_1 - a_2$ (when $\phi = 180^{\circ}$
• Resultant Intensity ($I_R$): $I_R = I_1 + I_2 + 2\sqrt{I_1 I_2} \text{cos}(\phi)$
  • $I_{max} = (a_1 + a_2)^2 = (\sqrt{I_1} + \sqrt{I_2})^2$
  • $I_{min} = (a_1 - a_2)^2 = (\sqrt{I_1} - \sqrt{I_2})^2$
• Special Case ($I_1 = I_2 = I_0$): $I_R = 4I_0 \text{cos}^2(\frac{\phi}{2})$
• Fringe Visibility ($V$): $V = \frac{I_{max} - I_{min}}{I_{max} + I_{min}}$

DIFFRACTION OF LIGHT (27/6/26)

Classification and Definitions

• Diffraction: The bending of light near the edges and corners of an obstacle or slit and spreading into the region of geometrical shadow.
• Fraunhofer Diffraction:
  • Source and screen are effectively at infinite distance from the diffracting system.
  • Pattern is obtained using a convex lens.
  • We consider plane wavefronts.
• Fresnel Diffraction:
  • Source and screen are at a finite distance.
  • We consider cylindrical or spherical wavefronts.

Single Slit Diffraction Experiment

• Slit width = $a$.
• Screen distance = $D$.
• Secondary Minima:
  • Path Difference ($PD$) = $a \text{sin}(\theta) = n\lambda$
  • Linear spread/width: $x_n = \frac{n\lambda D}{a}$
  • Angular spread/width: $\theta = \frac{n\lambda}{a}$
• Secondary Maxima:
  • Path Difference ($PD$) = $a \text{sin}(\theta) = (2m + 1)\frac{\lambda}{2}$
  • Linear width: $x_m = \frac{(2m + 1)\lambda D}{2a}$
  • Angular width: $\theta = \frac{(2m + 1)\lambda}{2a}$
• Central Maxima:
  • Linear spread ($X_{cm}$): $2x_1 = \frac{2\lambda D}{a}$
  • Angular spread ($\theta_{cm}$): $2\theta_1 = \frac{2\lambda}{a}$

RESOLVING POWER OF OPTICAL INSTRUMENTS

Microscope

• Non-Luminous Objects:
  • Limit of Resolution ($d$): $d = \frac{\lambda}{2\mu \text{sin}(\theta)}$
  • Resolving Power ($RP$): $RP = \frac{1}{d} = \frac{2\mu \text{sin}(\theta)}{\lambda}$
• Luminous Objects:
  • Limit of Resolution ($d$): $d = \frac{1.22 \lambda}{2\mu \text{sin}(\theta)} = \frac{0.61 \lambda}{\mu \text{sin}(\theta)}$
  • Resolving Power ($RP$): $RP = \frac{\mu \text{sin}(\theta)}{0.61 \lambda}$
• Numerical Aperture ($NA$): $NA = \mu \text{sin}(\theta)$
  • $\mu =$ Refractive index of the medium.
  • $\theta =$ semi-vertical angle subtended by the lens at the object.
  • Average $\lambda = 5500\,A^{\circ}$

Telescope

• Angular Limit of Resolution ($d\theta$): $d\theta = \frac{1.22 \lambda}{D}$
  • $D =$ diameter/aperture of lens.
• Resolving Power ($RP$): $RP = \frac{1}{d\theta} = \frac{D}{1.22 \lambda}$

POLARIZATION OF LIGHT (TYPE 2) (8/6/26)

Basics of Polarization

• Definition: The process of restricting light to vibrate in one particular plane perpendicular to the direction of propagation.
• Nature of Waves:
  • Transverse waves can be polarized.
  • Longitudinal waves cannot be polarized.
• Polarizer: Uses a crystallographic axis (Pass axis) to filter unpolarized light into polarized light.

Mathematical Laws of Polarization

• Malus's Law:
  • After passing through the first polarizer, intensity becomes exactly half: $I = \frac{I_0}{2}$
  • After passing through subsequent polarizers: $I = I_{\text{incident}} \text{cos}^2(\theta)$, where $\theta$ is the angle between successive polarizers.
• Brewster's Law:
  • Refractive index $\mu$ of the denser medium is equal to the tangent of the polarizing angle ($i_p$).
  • Formula: $\mu = \text{tan}(i_p)$
  • Proof: At the polarizing angle, the reflected and refracted rays are perpendicular ($i_p + r = 90^{\circ}$. Using Snell's law: $\mu = \frac{\text{sin}(i_p)}{\text{sin}(r)} = \frac{\text{sin}(i_p)}{\text{sin}(90 - i_p)} = \frac{\text{sin}(i_p)}{\text{cos}(i_p)} = \text{tan}(i_p)$.

WAVEFRONT AND HUYGENS' PRINCIPLE

Wave Theory Concepts

• Wavefront: The locus of all points in a medium which receive light at the same time and which are in the same phase.
• Types of Wavefronts:
  1. Spherical: Source is a point at a finite distance.
  2. Plane: Source is at an infinite distance.
  3. Cylindrical: Source is linear at a finite distance.
• Wave Normal: The perpendicular drawn at any point to the surface of a wavefront, showing the direction of light propagation.

Huygens' Principle

1. Each point on a wavefront acts as a secondary source of light, emitting secondary light wavelets in all directions.
2. Wavelets travel with the speed of light in that medium.
3. The new wavefront is the envelope of these secondary wavelets travelling in the forward direction.
4. Wavelets travelling in the backward direction are ineffective.

Proofs with Plane Wavefronts

• Reflection: Proven by showing the angle of incidence equals the angle of reflection ($i = r$) using congruency of triangles ($\triangle ABC \cong \triangle ADC$) based on common side $AC$ and travel time $ct$.
• Refraction: Proven using Snell's Law. In $\triangle ABC$, $\text{sin}(i) = \frac{c_1 t}{AC}$. In $\triangle ADC$, $\text{sin}(r) = \frac{c_2 t}{AC}$. Thus, $\frac{\text{sin}(i)}{\text{sin}(r)} = \frac{c_1}{c_2} = \mu$, which is constant.

NUMERICAL EXAMPLES AND CASE STUDIES

Interference Problems

• Case 1: Path Difference Calculation

  • Optical path difference = $0.24\,mm = 0.24 \times 10^{-3}\,m$. Band number $n = 371$.
  • Calculation: $\lambda = \frac{2.4 \times 10^{-4}}{371} = 6.469 \times 10^{-7}\,m$.

• Case 2: Nature of Illumination

  • Distances: $L_1 = 6.5\,cm$, $L_2 = 6.65\,cm$. $\lambda = 5000\,A^{\circ}$.
  • $PD = L_2 - L_1 = 0.15\,cm = 0.15 \times 10^{-2}\,m$.
  • $n = \frac{PD}{\lambda} = \frac{0.15 \times 10^{-2}}{5 \times 10^{-7}} = 3000$.
  • Result: Since $n$ is a whole number, it is the $3000^{\text{th}}$ Bright Band.

• Case 3: Biprism Wavelength Change

  • Red light ($\lambda_1 = 6000\,A^{\circ}$, $n_1 = 4$) is replaced by Blue light ($n_2 = 5$).
  • $n_1 \lambda_1 = n_2 \lambda_2 \rightarrow 4 \times 6000 = 5 \times \lambda_2 \rightarrow \lambda_2 = 4800\,A^{\circ}$.

• Case 4: Setup Immersion Calculation

  • Air fringe width = $0.6\,mm$. RI of water = $4/3$.
  • New width $X_{med} = \frac{0.6}{4/3} = 0.6 \times \frac{3}{4} = 0.45\,mm$.

Polarization Calculations

• Solar Angle Example: Angle sun rays must make with surface of lake ($\mu = 1.33$) for complete polarization of reflected rays.
  • $\mu = 1.33 = 4/3$.
  • $i_p = \text{tan}^{-1}(1.33) = 53.1^{\circ}$.
  • Angle with surface $\theta = 90 - 53 = 37^{\circ}$.