Thin-Film Interference

Calculating an Air-Backed Thin Film

Let's trace a concrete example for a film floating in air, to see how to predict whether a specific wavelength will be strongly reflected.

1. Assume a film with refractive index n = 1.5, thickness d = 100 nm, floating in air (Substrate Air, n = 1).
2. Calculate the round-trip optical path length: 2 × n × d = 2 × 1.5 × 100 = 300 nm.
3. For a vacuum wavelength of λ = 600 nm, the propagation phase shift is 300 / 600 = 0.5 cycles.
4. Determine reflection shifts:
   - Top surface (Air to Film, 1 → 1.5): Higher index, so 0.5 cycle shift.
   - Bottom surface (Film to Air, 1.5 → 1): Lower index, so 0 cycle shift.
   - Relative reflection shift = 0.5 - 0 = 0.5 cycles.
5. Total relative phase = Propagation + Reflection = 0.5 + 0.5 = 1.0 cycles.

Since 1.0 is an exact integer, the two reflected rays are perfectly in phase. The normalized intensity is 1.0 (a bright reflection for 600 nm light).

Experiment Simulation

Use the Thin-Film Interference module to verify the calculation and explore other conditions.

1. Set Film refractive index n to 1.5, Substrate to Air, Film thickness d to 100, and Vacuum wavelength lambda to 600. Read the round-trip optical length 300.0 nm, relative phase 1.0 cycles, reflection shift 0.5 cycles, and Normalized Intensity 1.00000000000.
2. Change Film thickness d to 200. The round-trip optical length becomes 600.0 nm. The propagation shift is now 600 / 600 = 1.0 cycle. Added to the 0.5 reflection shift, the total phase is 1.5 cycles, giving an intensity of 0 (a dark reflection).
3. Change Substrate to Glass and Film refractive index n to 1.25. Now the bottom reflection goes from n=1.25 to Glass (n=2.5), causing a second phase inversion. The relative reflection shift is 0. Set Vacuum wavelength lambda to 500 and Film thickness d to 100. Read relative phase 0.50 cycles and intensity 0.
4. Set Film thickness d to 0 with Substrate Air. The propagation path is 0, leaving only the 0.5 reflection shift. The relative phase is 0.5 cycles (dark).
5. Set Film thickness d to 0 with Substrate Glass. Both surfaces cause phase inversions, so the relative shift is 0. The relative phase is 0 cycles (bright).

Limitations and Constraints

The model calculated here uses an explicitly equal-amplitude two-ray contrast model to determine intensity: (1+cos(δ))/2. This demonstrates pure phase interference cleanly, ranging exactly from 0 to 1. However, real thin films (like anti-reflective coatings) exhibit multiple internal reflections and differing reflection coefficients (Fresnel reflectance) at each boundary. A full multiple-reflection model (like the Airy formula) would yield sharper interference fringes and varying maximum intensities, especially when the refractive indices of the three media differ greatly. The model also assumes strictly normal incidence (perpendicular to the surface); at oblique angles, the path length difference depends on the angle of refraction.

External References

For a detailed mathematical treatment of phase changes upon reflection and path length differences:

• OpenStax University Physics Volume 3, 3.4 Interference in Thin Films