6.5 Physical Optics, Light Interference, Diffraction & Polarization

6.5 Physical Optics, Light Interference, Diffraction & Polarization

Physical optics examines the wave nature of electromagnetic radiation. Key phenomena including interference, diffraction, and polarization are major content areas tested in FSc Physics and the Pakistan Army Medical Cadet Initial Test.

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1. Nature of Light & Huygens' Wave Principle

Huygens' Principle

Huygens proposed that light travels as wave fronts:

1. Primary Wavefront: Locus of all points having identical phase.
2. Secondary Wavelets: Every point on a primary wavefront acts as a point source emitting secondary spherical wavelets.
3. New Wavefront: The forward envelope tangent to these secondary wavelets defines the new wavefront at a later time $t + \Delta t$.

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2. Light Interference & Young's Double Slit Experiment (YDSE)

Interference Conditions

For sustained observable interference:

1. Sources must be coherent (maintain constant phase difference).
2. Light must be monochromatic (single wavelength $\lambda$).
3. Amplitudes should be nearly equal for maximum fringe contrast.

Young's Double Slit Experiment (YDSE) Equations

• Slit separation: $d$, Distance to screen: $D \gg d$.
• Path difference: $\Delta s = d \sin\theta \approx d \left(\frac{y}{D}\right)$.

          Young's Double Slit Interference Setup
             |
       Slit 1+----\ 
             |     \  S_1 P
      (d)    |      \=========> P (Fringe Position y)
             |     /  S_2 P
       Slit 2+----/
             | 
       |<------ Distance D ------->|

• Constructive Interference (Bright Fringes / Maxima): $\Delta s = d \sin\theta = m \lambda \implies y_m = \frac{m \lambda D}{d} \quad (m = 0, \pm 1, \pm 2, \dots)$
• Destructive Interference (Dark Fringes / Minima): $\Delta s = d \sin\theta = \left(m + \frac{1}{2}\right) \lambda \implies y_m' = \left(m + \frac{1}{2}\right) \frac{\lambda D}{d} \quad (m = 0, \pm 1, \pm 2, \dots)$
• Fringe Spacing ($\Delta y$): Distance between consecutive bright or dark fringes: $\Delta y = \frac{\lambda D}{d}$

Thin Film Interference & Phase Reversals

When light reflects off a medium with higher refractive index ($n_{film} > n_{air}$), it undergoes a phase shift of $\pi$ radians (equivalent to path change of $\lambda/2$).

• Condition for bright reflection with top phase shift: $2 n t = \left(m + \frac{1}{2}\right) \lambda$
• Condition for dark reflection with top phase shift: $2 n t = m \lambda$ where $t$ is film thickness and $n$ is refractive index.

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3. Diffraction of Light & Diffraction Gratings

Single Slit Fraunhofer Diffraction

Bending of light around narrow slit aperture of width $a$:

• Minima Condition: $a \sin\theta = m \lambda \quad (m = \pm 1, \pm 2, \dots)$
• Central Maximum Width: Central bright fringe is twice as wide as secondary maxima.

Diffraction Grating

A diffraction grating consists of $N$ parallel rulings per unit length. The grating element is $d = \frac{1}{N}$.

• Grating Equation for Principal Maxima: $d \sin\theta = m \lambda \quad (m = 0, 1, 2, \dots)$
• Maximum observable order $m_{max} \le \frac{d}{\lambda}$ (since $\sin\theta \le 1$).

X-Ray Diffraction & Bragg's Law

X-rays diffracted by parallel atomic planes of crystal lattice with interplanar spacing $d$: $2 d \sin\theta = n \lambda$ where $\theta$ is the glancing angle of incidence and $n$ is diffraction order.

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4. Polarization of Light & Transverse Wave Proof

Transverse Wave Proof

Polarization restricts electric field vibrations of light waves to a single plane. Only transverse waves can be polarized; longitudinal waves cannot exhibit polarization. Polarization provides definitive proof that light is a transverse electromagnetic wave.

Methods of Polarization & Brewster's Law

1. Selective Absorption (Polaroids): Absorbs electric field vectors along one direction, transmitting orthogonal component.
2. Reflection & Brewster's Law: When light strikes a transparent surface at polarizing angle $\theta_p$, reflected light is 100% plane-polarized. Reflected and refracted rays are perpendicular: $\tan \theta_p = \frac{n_2}{n_1} = n$
3. Double Refraction (Birefringence): Calcite crystals split light into Ordinary ($O$) and Extraordinary ($E$) plane-polarized rays.

Malus's Law

Transmitted intensity $I$ when polarized light of intensity $I_0$ passes through an analyzer rotated by angle $\theta$ relative to polarizer: $I = I_0 \cos^2\theta$

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5. Worked AMC Numerical Examples

Example 1: Brewster's Angle Calculation

Question: Unpolarized light strikes a glass plate with refractive index $n = 1.732 = \sqrt{3}$ in air. Calculate Brewster's polarizing angle $\theta_p$.

Solution:

1. Apply Brewster's Law: $\tan \theta_p = n = \sqrt{3}$.
2. Solve for angle: $\theta_p = \tan^{-1}(\sqrt{3}) = 60^\circ$.

Example 2: YDSE Fringe Spacing Calculation

Question: Monochromatic light of wavelength $\lambda = 600\text{ nm}$ illuminates slits separated by $d = 0.3\text{ mm}$ placed $D = 1.5\text{ m}$ from a screen. Find fringe width $\Delta y$.

Solution:

1. Convert units: $\lambda = 600 \times 10^{-9}\text{ m}$, $d = 0.3 \times 10^{-3}\text{ m}$, $D = 1.5\text{ m}$.
2. Use formula: $\Delta y = \frac{\lambda D}{d} = \frac{600 \times 10^{-9} \times 1.5}{0.3 \times 10^{-3}} = \frac{900 \times 10^{-9}}{0.3 \times 10^{-3}} = 3.0 \times 10^{-3}\text{ m} = 3.0\text{ mm}$.

Example 3: Bragg X-Ray Diffraction

Question: First-order ($n=1$) Bragg diffraction occurs at glancing angle $\theta = 30^\circ$ for X-rays of wavelength $\lambda = 0.2\text{ nm}$. Find interplanar crystal spacing $d$.

Solution:

1. Apply Bragg's Law: $2 d \sin\theta = n \lambda$.
2. Substitute values: $2 d \sin(30^\circ) = 1 \times 0.2$.
3. Since $\sin(30^\circ) = 0.5$: $2 d (0.5) = 0.2 \implies d = 0.2\text{ nm} = 2.0 \times 10^{-10}\text{ m}$.