CHPTR 35 Notes: Wave Interference

Overview of Wave Interference

• When two or more waves overlap, the resultant displacement at any point is obtained by summing the individual displacements of the waves.

  • This principle is fundamental in understanding wave behavior during interference events.In wave interference, there are two main types to consider: constructive interference, which occurs when waves combine to produce a wave of greater amplitude, and destructive interference, where waves combine to create a wave of reduced amplitude.

Constructive and Destructive Interference

• Constructive Interference

  • Results in bright fringes appearing at specific points where the superposition of waves leads to increased amplitude.

  • Occurs under the condition:

  • $ext{Path difference} = m imes ext{wavelength}$

    • Where $m = 0, 1, 2, ext{…}$

  • Mathematically expressed as:

  • $ext{sin}( heta) = rac{m imes ext{λ}}{d}$

• Destructive Interference

  • Leads to dark fringes where waves cancel each other out, resulting in reduced or zero amplitude.

  • Occurs under the condition:

  • $ext{Path difference} = (m + rac{1}{2}) imes ext{wavelength}$

    • Where $m = 0, 1, 2, ext{…}$

  • Mathematically expressed as:

  • $ext{sin}( heta) = rac{(m + rac{1}{2}) imes ext{λ}}{d}$

Fringe Position Calculations

• Given large distances, where the distance between sources ($d$) is significantly smaller than other distances involved, the position of fringes can be derived as:

  • For adjacent fringes:

  • $y_m = rac{m imes R imes ext{λ}}{d}$

    • Where $R$ is the distance to the screen from the slits.

  • $ext{tan}( heta) ext{ approximately equals } ext{sin}( heta)$ for small angles.

  • The calculation summarizes the position of bright and dark fringes along the screen.

Intensity and Electric Field Amplitude

• The intensity (brightness) of a fringe is directly proportional to the square of the electric field amplitude:

  • $I ext{ is proportional to } (E_p)^2$ where $E_p$ is the amplitude of the electric field.

Electric Field Representation

• Consider two electromagnetic waves with the same amplitude and wavelength arriving at a point, represented mathematically as:

  • $E_1 = E ext{cos}( ext{ω}t), ext{ } E_2 = E ext{cos}( ext{ω}t + φ)$

• The maximum electric field amplitude is:

  • $E_p = 2E ext{cos}( rac{φ}{2})$

Light Intensity Relationship with Phase Difference

• The light intensity at a point is maximum when the electric field amplitude is maximum, which occurs when the phase difference $φ = 0$ (in-phase) leading to maximum intensity.

• Conversely, intensity is zero when $E_p = 0$, which occurs if the phase difference is $φ = ext{π radians}$ (completely destructive interference).

Intensity of Bright Fringes (Two-source Interference)

• Given by the formula:

  • $I = rac{1}{2} ε_0 c E_p^2 = rac{1}{2} ε_0 c (2E)^2 ext{cos}^2 rac{φ}{2}$

  • Where:

  • $I_0 = 2ε_0 c E^2$ is the maximum intensity.

• The phase difference in relation to path length difference is expressed as:

  • $φ = rac{2π}{λ}(r_2 - r_1) = k(r_2 - r_1)$

  • Where $r_1$ and $r_2$ are the distances from the two sources to the point of interest.

Thin Film Interference

Reflection from Thin Layers

• When light reflects off thin layers of material (thickness = $t$), colorful bands may be observed. Examples include:

  • Colors seen in oil films on water or pavement.

  • The iridescent colors in soap bubbles.

Origin of Interference Patterns

• The interference patterns are formed by light reflected off the first surface (top layer) interfering with light that reflects off the second surface (bottom layer).

  • The second ray travels an additional distance of $2t$ compared to the first ray.

Conditions for Constructive and Destructive Interference

• Constructive Reflection

  • Occurs without a net phase shift:

  • $2t = mλ$ for $m = 0, 1, 2, …$

• Destructive Reflection

  • Also occurs without a net phase shift but follows:

  • $2t = (m + rac{1}{2})λ$ for $m = 0, 1, 2 …$

• If there is a net phase shift of $ext{π radians}$ (half cycle phase shift), the conditions reverse for constructive and destructive interference.

Wavelength in Thin Films

• The wavelength $λ$ employed in these equations must be the wavelength of light within the thin film, which can be represented as:

  • $λ = rac{λ_0}{n}$

  • Where $n$ is the index of refraction of the film medium.

Phase Shift in Reflected Waves

• The phase shift depends on the indices of refraction:

  • If n_a > n_b, there is no phase shift of the reflected wave relative to the incident wave.

  • If n_a < n_b, a phase shift of $ext{π radians}$ occurs (a half cycle phase shift).

• Example: If n_1 < n_2 < n_3:

  • This indicates:

  • Results in half cycle phase shift at the transition from $n_2$ to $n_3$, with no phase shift at $n_1 ext{ to } n_2$ (e.g., from air to a thin film).

Specific Applications of Interference

Newton’s Rings

• The combination of rays reflected from a flat plate and the curved surface of a lens creates an interference pattern known as Newton’s rings.

• Newton’s rings manifest as concentric circular bright and dark fringes.

• Applications include inspecting the manufacturing quality of lenses, where high-quality lenses require precision grinding to less than one wavelength.

Michelson’s Interferometer

• The Michelson's interferometer is a device that splits a light beam into two parts, then recombines them to produce an interference pattern.

• This apparatus can be utilized for precise wavelength measurements, showcasing the principles discussed regarding wave interference and light behavior.