# Thin-Film Optical Coating Calculator

URL: https://speedycalculator.com/calculators/physics/thin-film-optics/
Last updated: 2026-10-30

> The thin-film optical coating calculator finds the reflectivity of a single-layer coating for s- and p-polarised light, the optical path difference and the thickness of a quarter-wave anti-reflection coating.

## Inputs

- Incident angle θ₁ (°, rad, arcmin, arcsec, mrad)
- Optical film thickness d (nm, µm, mm, m, pm, Å)
- Wavelength λ (nm, µm, mm, m, pm, Å)
- Refractive index of the first medium n₁
- Refractive index of the film n₂
- Refractive index of the substrate n₃

## Results

- Optical path difference
- s-polarised reflectivity
- p-polarised reflectivity
- Bare substrate reflectivity (normal incidence)
- Minimum anti-reflection coating thickness
- Ideal coating index √(n₁n₃)

## Formula

```
r_ij from the Fresnel equations,  δ = 2π·n₂·d·cos θ₂ / λ
R = (r₁₂² + r₂₃² + 2r₁₂r₂₃cos 2δ) / (1 + r₁₂²r₂₃² + 2r₁₂r₂₃cos 2δ),  d_AR = λ/(4n₂)
```

## Explanation

Anti-reflection coatings on glasses and camera lenses are thin films about a quarter wavelength thick. Light reflected from the top and bottom of the film interferes destructively, cutting reflections from about 4% to under 1%.

### How to use the Thin-Film Optical Coating Calculator

- **Incident angle θ₁**
- **Optical film thickness d**
- **Wavelength λ**
- **Refractive index of the first medium n₁**
- **Refractive index of the film n₂**
- **Refractive index of the substrate n₃**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Enter the incidence angle, film thickness, wavelength and the three refractive indices (incident medium, film, substrate). A quarter-wave film of index √(n₁n₃) gives zero reflection at normal incidence.

### Inputs explained

FieldWhat to enterIncident angle θ₁Units: °, rad, arcmin, arcsec, mradOptical film thickness dUnits: nm, µm, mm, m, pm, ÅWavelength λUnits: nm, µm, mm, m, pm, ÅRefractive index of the first medium n₁A numberRefractive index of the film n₂A numberRefractive index of the substrate n₃A number

### What each result means

- **Optical path difference** = 2×[Refractive index of the film n₂]×[Optical film thickness d]×[c2]
- **s-polarised reflectivity** = ([rs12]^2 + [rs23]^2 + 2×[rs12]×[rs23]×cos(2×[delta]))÷(1 + [rs12]^2×[rs23]^2 + 2×[rs12]×[rs23]×cos(2×[delta]))×100
- **p-polarised reflectivity** = ([rp12]^2 + [rp23]^2 + 2×[rp12]×[rp23]×cos(2×[delta]))÷(1 + [rp12]^2×[rp23]^2 + 2×[rp12]×[rp23]×cos(2×[delta]))×100
- **Bare substrate reflectivity (normal incidence)** = (([Refractive index of the first medium n₁] - [Refractive index of the substrate n₃])÷([Refractive index of the first medium n₁] + [Refractive index of the substrate n₃]))^2×100
- **Minimum anti-reflection coating thickness** = [Wavelength λ]÷(4×[Refractive index of the film n₂])
- **Ideal coating index √(n₁n₃)** = sqrt([Refractive index of the first medium n₁]×[Refractive index of the substrate n₃])

## Worked example

Inputs — Incident angle θ₁: 0 °; Optical film thickness d: 99.6 nm; Wavelength λ: 550 nm; Refractive index of the first medium n₁: 1; Refractive index of the film n₂: 1.38; Refractive index of the substrate n₃: 1.52.

Results — Optical path difference = 274.896 nm; **s-polarised reflectivity = 1.2601 %**; p-polarised reflectivity = 1.2601 %; Bare substrate reflectivity (normal incidence) = 4.258 %; Minimum anti-reflection coating thickness = 99.63768 nm; Ideal coating index √(n₁n₃) = 1.2329.

## Key terms

- **Quarter-wave layer**: A film with optical thickness λ/4.

## FAQ

**Why are AR coatings a quarter wavelength thick?**
Reflections from the two surfaces then cancel by destructive interference.

**What material is used for single-layer coatings?**
Often magnesium fluoride (n ≈ 1.38).

**Why do coated lenses look purple or green?**
Reflection is lowest in the middle of the spectrum, so the ends reflect slightly.

**What about multilayer coatings?**
Stacks of layers reduce reflection over a broad range of wavelengths.


## Sources

- [OpenStax – University Physics Volume 3](https://openstax.org/details/books/university-physics-volume-3)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)