Thin-Film Optical Coating Calculator

Related Calculators

Refractive index (n): A unitless number that tells how much light slows down in a material and how much the ray bends when it enters the material.[2]

Incident angle (degrees): The angle between the incoming ray and the surface normal (a line straight out of the surface).

Refraction angle inside film: The angle the ray travels at inside the film, found with Snell's law.[2]

Optical path difference (OPD): The extra optical distance between the two main reflected rays in the simple two-ray picture; in a film it depends on film thickness, refractive index, and the inside-film angle.[2]

Phase flip: A 180 degree (pi radians) phase shift that can happen on reflection when a wave reflects from a lower-index side to a higher-index side (lossless shortcut rule).[2]

s polarization and p polarization: Two ways the electric field can be oriented compared to the plane of incidence (the plane made by the incoming ray and the surface normal). s (TE) is perpendicular to that plane; p (TM) is parallel. They can reflect differently at nonzero angles.[2]

Reflectance (R): The fraction of incoming light power that is reflected (example: R = 0.04 means 4% reflected).

Transmittance (T): The fraction of incoming light power that makes it through the film into the substrate side.

Quarter-wave coating thickness: A film thickness chosen so the film adds about a quarter-wavelength of phase (in the film) at the design wavelength, often used for simple single-layer anti-reflection at one wavelength.[1]

Extinction coefficient (k2): A unitless number that models absorption in the film at the chosen wavelength. If k2 is not 0, some energy is absorbed, so R + T can be less than 1.

What this calculator models: A single thin film between two media: incident medium (n1) to film (n2, and optional k2 for absorption) to substrate (n3). It reports (1) a learning-friendly interference summary using optical path difference and phase flips (lossless case) and (2) single-wavelength reflectance/transmittance for s and p polarization using a coherent one-layer thin-film amplitude formula.[2]

1) Angles from Snell's law: The calculator converts the incident angle to radians and finds the refraction angle in the film using Snell's law.[2]

θ1 (incident angle in radians) = θ1_deg * (π/180)

sin(θ2) = (n1/n2) * sin(θ1)

If |sin(θ2)| is greater than 1, the film angle is not real (total internal reflection at the first interface). In that case, θ2 and any outputs that depend on it are shown as N/A and a warning is shown.

2) Optical path difference (two-ray approximation): For the two main reflected rays (top reflection plus one down-and-back pass in the film), the optical path difference is:[2]

OPD = 2 * n2 * d * cos(θ2)

d is the physical film thickness you enter in nm, so OPD is also reported in nm (because n2 is unitless).

3) Phase flip flags (lossless shortcut): When absorption is off (k2 forced to 0), the calculator marks a phase flip at an interface when the reflection is from lower n into higher n at that interface.[2] When absorption is on, n2 is complex, so this Yes/No shortcut is not reliable; phase flip outputs are shown as N/A and you should rely on the reflectance numbers instead.

4) Interference label for reflected light: The calculator combines OPD with the phase flips to label the reflected light as constructive, destructive, or in-between at the chosen wavelength. This label is meant to match the common homework-style two-ray explanation; it is not a guarantee that reflectance is at its absolute minimum for every setup.[2]

5) Coherent single-layer reflectance (core model): For each polarization (s and p), it computes Fresnel amplitude reflection coefficients at the top and bottom interfaces (r12 and r23), and a phase thickness δ (delta) for one pass through the film, then combines them into a total amplitude reflection coefficient r:[2]

δ (phase thickness) = (2*π/λ) * n2 * d * cos(θ2)

r = (r12 + r23 * exp(2*i*δ)) / (1 + r12*r23*exp(2*i*δ))

R = |r|^2

If absorption is enabled, the film refractive index is treated as complex (n2 - i*k2) inside the Fresnel coefficients and inside δ. This can make R + T less than 1 because some power is absorbed in the film.

6) Transmittance and absorption estimate: The calculator reports transmittance into the substrate side (Ts and Tp). If absorption is enabled, it also estimates absorbed fraction as:

A_est = 1 - R - T

If A_est is negative by more than a tiny rounding amount, the calculator shows N/A and a warning (this usually means invalid inputs or numerical issues).

AR design outputs: Quarter-wave optical thickness (QWOT) uses the inside-film angle:[1]

d_qw = λ / (4 * n2 * cos(θ2))

At normal incidence, cos(θ2) = 1 so d_qw = λ/(4*n2). The ideal single-layer AR refractive index at normal incidence (lossless) is:[1]

n2_ideal = sqrt(n1 * n3)

Mini example (normal incidence): Let λ = 550 nm, θ1 = 0 degrees, n1 = 1.00, n2 = 1.38, and d = 100 nm. Then θ2 = 0 degrees and OPD = 2*1.38*100 = 276 nm, which is about 0.50 wavelengths (276/550) of optical path difference. Whether that leads to constructive or destructive reflected light depends on whether there is a phase flip at one interface or both.[2] The quarter-wave thickness at 550 nm is d_qw = 550/(4*1.38) = 99.6 nm.[1]

Assumptions and limits: This is a single-wavelength, coherent, single-layer model with flat interfaces. Real coatings can be multilayer, refractive index can change with wavelength (dispersion), surfaces can be rough, and thick substrates can create extra reflections not included here.[2] At larger angles, s and p can differ a lot, so do not expect one polarization result to match the other unless you are near normal incidence.