14.9 Thin Film Interference

Thin film interference occurs when light interacts with layers as thin as its wavelength. This phenomenon explains the colorful patterns in soap bubbles and oil slicks, where varying film thicknesses create different interference effects.

Understanding thin film interference involves grasping concepts like transmission, reflection, and phase changes. These principles are crucial for applications such as antireflective coatings, which use destructive interference to reduce unwanted reflections.

Light Interaction with Thin Films

Transmission, Reflection and Absorption

When light encounters a boundary between two media, it can undergo three primary interactions 🌞

• Transmitted light passes through the new medium, continuing its journey but typically at a different angle due to refraction
• Reflected light bounces back from the boundary between the two media, following the law of reflection
• Absorbed light is taken in by the new medium and converted into other forms of energy (typically heat)

The relative proportions of these interactions depend on the properties of both media, particularly their indices of refraction.

Phase Change in Reflection

When light reflects from a boundary, it may undergo a phase change depending on the refractive indices of the materials involved.

• A 180-degree phase change (half-wavelength shift) occurs when light reflects from a medium with a higher index of refraction than the medium the ray is currently traveling through
• No phase change happens when light reflects from a medium with a lower index of refraction than the medium the ray is currently traveling through

For example:

• Light reflecting off water from air (lower to higher index of refraction) undergoes a 180-degree phase change
• Light reflecting off air from water (higher to lower index of refraction) experiences no phase change

This phase change is crucial in determining whether constructive or destructive interference will occur in thin films.

Phase in Refraction

Unlike reflection, refraction does not introduce phase shifts in the transmitted light.

• A wave's phase remains constant when it refracts and passes from one medium into another
• The wave's direction and speed change during refraction, but its phase stays the same
• This consistency in phase during refraction helps simplify the analysis of thin-film interference

Thin-Film Interference Concept

Thin-film interference occurs when a film's thickness is comparable to the wavelength of light interacting with it.

When light strikes a thin film, part of it reflects from the first surface while the rest enters the film. The light that enters the film then reflects from the second surface and exits back through the first surface. These two reflected waves—one from the first surface and one from the second surface—then interfere with each other 📊

The interference can be:

• Constructive: when waves are in phase, their amplitudes add, creating a brighter reflection
• Destructive: when waves are out of phase, their amplitudes subtract, potentially canceling each other out

Factors Affecting Interference

The nature of the interference (constructive or destructive) depends on several key factors:

• Film thickness: Determines the path length difference between the two reflected waves
• Wavelength of light: Different wavelengths (colors) will constructively or destructively interfere at different film thicknesses
• Indices of refraction: Determine whether phase shifts occur at each boundary
• Angle of incidence: Affects the path length through the film

For normal incidence (perpendicular to the surface), the path length difference is simply twice the film thickness.

Examples of Thin-Film Interference

Thin-film interference creates beautiful color patterns in everyday objects 🌈

• Soap bubbles: The varying thickness of the soap film creates a spectrum of colors as different wavelengths constructively interfere at different points
  • Thicker regions reflect longer wavelengths (reds, oranges)
  • Thinner regions reflect shorter wavelengths (blues, violets)
  • The colors shift as the bubble thins due to gravity and evaporation
• Oil slicks: When oil spreads on water, it forms a thin film that creates similar interference patterns
  • The thickness gradient of the oil film produces bands of different colors
• Antireflection coatings: These practical applications use thin-film interference to eliminate unwanted reflections
  • The coating thickness is carefully calculated to create destructive interference for reflected light
  • Typically, the coating thickness equals one-quarter of the wavelength in the coating material
  • The coating's index of refraction is ideally the square root of the product of the indices of the surrounding media
  • These coatings are common on camera lenses, eyeglasses, and solar panels

Practice Problem 1: Soap Bubble Interference

Solution

To determine why yellow light appears bright, we need to check if constructive interference occurs for this wavelength.

First, identify the phase changes:

• At the air-soap interface: Light goes from lower n (air, n=1) to higher n (soap, n=1.33), so there's a 180° phase change
• At the soap-air interface: Light goes from higher n (soap) to lower n (air), so there's no phase change

Next, calculate the wavelength in the soap film: $\lambda_{soap} = \lambda_{air}/n_{soap} = 580 \text{ nm}/1.33 = 436.1 \text{ nm}$

For constructive interference with these phase changes, the path difference (2t) must equal: $2t = (m + 1/2)\lambda_{soap}$ where m is an integer

Checking our thickness: $2(217.5 \text{ nm}) = 435 \text{ nm}$

This is very close to $\lambda_{soap}$, making $m = 1$ and satisfying the condition for constructive interference. Therefore, yellow light appears bright at this location.

Practice Problem 2: Antireflective Coating

Solution

For an antireflective coating to work effectively, we need destructive interference between light reflected from the air-coating interface and the coating-lens interface.

Phase changes:

• At air-coating interface: Light goes from lower n (air, n=1) to higher n (coating, n=1.38), so there's a 180° phase change
• At coating-lens interface: Light goes from lower n (coating) to higher n (lens, n=1.5), so there's another 180° phase change

Since both reflections have the same phase change (or effectively no relative phase difference), we need the path difference to create a half-wavelength shift for destructive interference.

For minimum thickness, we want: $2t = \lambda_{coating}/2$

First, find the wavelength in the coating: $\lambda_{coating} = \lambda_{air}/n_{coating} = 550 \text{ nm}/1.38 = 398.55 \text{ nm}$

Therefore: $t = \lambda_{coating}/4 = 398.55 \text{ nm}/4 = 99.64 \text{ nm}$

The minimum thickness should be approximately 99.6 nm.