Chapter 34: The Wave Nature of Light, Interference, and Polarization

Chapter 34: The Wave Nature of Light, Interference, and Polarization

34.1 Waves vs Particles; Huygens’ Principle and Diffraction

The nature of light has been debated as either a wave or a particle. The wave theory is supported by phenomena such as interference and diffraction. Huygens’ Principle provides a powerful model for understanding wave propagation and diffraction.

• Huygens’ Principle: Every point on a wave front acts as a point source of spherical wavelets. The new wave front is the surface tangent to these wavelets.

• Diffraction: When light waves encounter a barrier or pass through a narrow opening, they bend around the edges and spread out. This behavior is explained by Huygens’ Principle.

• Application: Diffraction is observed when light passes through slits or apertures, producing patterns of constructive and destructive interference.

• Example: The spreading of light waves after passing through a slit demonstrates diffraction.

34.2 Huygens’ Principle and the Law of Refraction

Huygens’ Principle can be used to derive the law of refraction (Snell’s Law), which describes how light bends when passing from one medium to another. See Section 15.10 for details.

• Snell’s Law: $n_1 \sin \theta_1 = n_2 \sin \theta_2$

• Application: Used to predict the angle of refraction when light enters a new medium.

34.3 Interference — Young's Double-Slit Experiment

Interference is a hallmark of wave behavior. Young’s double-slit experiment provides direct evidence for the wave nature of light by producing an interference pattern.

• Interference: Occurs when two or more waves overlap, resulting in regions of constructive (bright) and destructive (dark) interference.

• Coherence: To observe clear interference patterns, the light must be coherent:

  • Spatial coherence: Light travels in the same direction (collimated).

  • Temporal coherence: Light has a constant phase difference and single frequency (monochromatic).

• Sources of Coherent Light: Sunlight and incandescent light are incoherent, but can be made coherent by passing through a pinhole or single slit.

• Young’s Double-Slit Setup: Light passes through two closely spaced slits, producing an interference pattern on a screen.

• Path Difference: The difference in distance from each slit to a point on the screen determines whether interference is constructive or destructive.

• Conditions for Interference:

  • Constructive (bright fringes): $d \sin \theta = m \lambda$, $m = 0, 1, 2, \ldots$

  • Destructive (dark fringes): $d \sin \theta = (m + \frac{1}{2}) \lambda$, $m = 0, 1, 2, \ldots$

• Fringe Spacing: The distance between consecutive bright or dark fringes is called fringe spacing.

• Example: White light produces colored fringes due to wavelength dependence; violet and red light appear at different positions in the pattern.

34.7 Polarization

Polarization describes the orientation of the oscillations of the electric field in an electromagnetic wave. Light can be polarized by reflection, transmission through certain materials, or by filters.

• Electromagnetic Waves: Light is a transverse wave with electric and magnetic fields oscillating perpendicular to the direction of propagation.

• Polarized Light: The electric field oscillates in a single plane. Unpolarized light consists of waves with random polarization directions.

• Polarizing Filters: Materials (such as Polaroid) transmit only the component of light polarized parallel to their axis. The transmitted amplitude is $E \cos \theta$.

• Malus’s Law: The intensity $I$ of transmitted light through a polarizer is:

  • $I = I_0 \cos^2 \theta$

  • For unpolarized light, intensity after one polarizer is $I_0/2$

• Multiple Polarizers: If two polarizers are crossed (axes at 90°), no light passes. If a third is placed at 45°, some light is transmitted.

• Brewster’s Angle: Light reflected at Brewster’s angle is completely polarized:

  • $\tan \theta_p = \frac{n_2}{n_1}$

  • For water ($n_2$) and air ($n_1$), $\theta_p = 53.1^\circ$

• Applications: Polarizing sunglasses reduce glare from surfaces like water and snow.

34.8 Liquid Crystal Displays (LCD)

Liquid crystals are materials that can change their optical properties in response to an electric field. This property is used in display technologies.

• Unpolarized State: Without voltage, liquid crystals transmit light easily.

• Polarized State: When voltage is applied, crystals become polarized and block light, appearing dark.

• Applications: Used in calculators, digital watches, and screens.

34.9 Scattering of Light by the Atmosphere

Scattering occurs when light interacts with molecules in the atmosphere, affecting the color and polarization of the sky.

• Partial Polarization: Skylight is partially polarized due to scattering; the degree of polarization depends on the angle relative to the Sun.

• Application: Polarization of skylight is used in photography and navigation.

Summary Table: Coherence Types

Type	Spatially Incoherent	Spatially Coherent
Temporally Incoherent	Multiple directions, random phase	Single direction, random phase
Temporally Coherent	Multiple directions, constant phase	Single direction, constant phase

Summary Table: Polarizer Transmission

Configuration	Transmitted Intensity
Single polarizer (unpolarized light)	$I_0/2$
Two polarizers, axes at angle $\theta$	$I_0/2 \cos^2 \theta$
Two polarizers, axes at 90°	0
Three polarizers (0°, 45°, 90°)	$I_0/8$

Key Equations

• Constructive Interference: $d \sin \theta = m \lambda$

• Destructive Interference: $d \sin \theta = (m + \frac{1}{2}) \lambda$

• Malus’s Law: $I = I_0 \cos^2 \theta$

• Brewster’s Angle: $\tan \theta_p = \frac{n_2}{n_1}$

Additional info: These notes expand on the provided slides and text, adding definitions, examples, and tables for clarity and completeness.