Tool · Optical coatings

AR Coating Thickness Calculator

What thickness should a single anti-reflection layer have, and how much reflectance remains with the film materials available? The tool gives the quarter-wave thickness and the ideal film index for a chosen substrate and design wavelength, the reflectance at that wavelength with and without the film, the band over which the coated surface stays below a chosen reflectance, and the reflectance spectrum around it. Background: anti-reflection coatings, the Fresnel equations, and the characteristic matrix method.

Design

Examples

Substrate

Substrate index n_s

Coating material

Typical thin-film indices; deposited values depend on the process.

Film index n₁

Ambient index n₀

Design wavelength λ₀ (nm)

Band threshold R (%)

sets the reflectance used for the band readout and the green line on the plot

Design result

Reflectance vs wavelength

coateduncoatedλ₀band threshold

Beyond one layer

A single layer reaches zero reflectance only when a film with the ideal index exists. Two-layer and V-coat designs, which reach zero with available materials, and multilayer stacks at any angle of incidence can be computed with the Thin-Film Reflectance Calculator. The companion article Anti-reflection coating design works through the quarter-wave conditions, and Laser diode facet coatings covers what the facet reflectivity does to a laser. Runs entirely in your browser; nothing you enter is uploaded or stored.

Checked against

These checks run in your browser on every load. The closed forms are compared with hand-computed values, the characteristic-matrix spectrum with the two-interface Airy summation and with energy conservation, and the band edges with a numerical root search.

Check	Expected	Computed	Tolerance

The table sets each value this page computes beside a hand calculation from the closed forms or an independent evaluation (the Airy summation of Born and Wolf, section 1.6, and a scipy root search on the spectrum); the tolerance is the largest difference from Expected, in the same units, that still passes.

The model

A single film of index $n_{1}$ and thickness $t$ lies between an ambient medium of index $n_{0}$ and a substrate of index $n_{s}$ . Light reflected from the two interfaces, with Fresnel amplitudes $r_{01} = (n_{0} - n_{1}) / (n_{0} + n_{1})$ and $r_{12} = (n_{1} - n_{s}) / (n_{1} + n_{s})$ , adds with the phase accumulated across the film, and the reflectance at wavelength $λ$ is

r = \frac{r_{01} + r_{12} e^{- 2 i δ}}{1 + r_{01} r_{12} e^{- 2 i δ}}

δ = \frac{2 π n_{1} t}{λ}, R = ∣ r ∣^{2}

The tool sets the thickness to a quarter wave at the design wavelength, $t = λ_{0} / 4 n_{1}$ , so that the two reflections return in antiphase. The reflectance at $λ_{0}$ then takes the closed form

R (λ_{0}) = {(\frac{n_{0} n_{s} - n_{1}^{2}}{n_{0} n_{s} + n_{1}^{2}})}^{2}

which is zero when $n_{1} = \sqrt{n_{0} n_{s}}$ . The uncoated value is $((n_{0} - n_{s}) / (n_{0} + n_{s}))^{2}$ . The spectrum evaluates the same reflectance, written as the characteristic matrix of the layer, with the thickness fixed and the wavelength varied; the page checks the two forms against each other.

With $c = \cos 2 δ$ , the reflectance of the quarter-wave film is

R = \frac{r_{01}^{2} + r_{12}^{2} + 2 r_{01} r_{12} c}{1 + r_{01}^{2} r_{12}^{2} + 2 r_{01} r_{12} c}

which moves monotonically from the quarter-wave value at $c = - 1$ (at $λ_{0}$ ) to the bare-substrate value at $c = + 1$ , where the film is a half wave and drops out. Because $2 δ = π λ_{0} / λ$ , solving $R = R_{t}$ for $c$ gives the edges of the band in which the coated reflectance stays below a threshold $R_{t}$ :

λ_{hi} = \frac{π λ_{0}}{\arccos c}, λ_{lo} = \frac{π λ_{0}}{2 π - \arccos c}

A film whose index does not lie between $n_{0}$ and $n_{s}$ reflects at least as much at $λ_{0}$ as the bare surface, and the tool says so instead of reporting a band.

The model assumes normal incidence, lossless films, indices that do not change with wavelength, and a substrate thick enough that its back surface can be ignored. At oblique incidence the phase thickness falls as the cosine of the internal angle and the s and p reflectances separate; dispersion shifts the minimum; and deposited film indices depend on the process, so the preset values are typical rather than exact. A single layer reaches zero only when a material with the ideal index exists. Two-layer and V-coat designs, which can reach zero with available materials, can be computed with the thin-film reflectance tool.

Worked example, from the default inputs (silicon, n = 3.48; Si₃N₄ film, n = 1.99; air; 1550 nm). The quarter-wave thickness is $1550 / (4 \times 1.99) = 194.7 nm$ and the ideal index is $\sqrt{3.48} = 1.865$ . The bare surface reflects $(2.48 / 4.48)^{2} = 30.64 %$ . With the film, $n_{1}^{2} = 3.960$ and $R = (0.480 / 7.440)^{2} = 0.416 %$ . Because the nitride index is above the ideal value, the reflectance does not reach zero, and 0.416 % is the lowest a single Si₃N₄ layer on silicon can reach at this wavelength. The coated reflectance stays below 1 % from 1442.9 to 1674.2 nm, a band 231.3 nm wide, or 14.9 % of the design wavelength.

References: M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), sec. 1.6. H. A. Macleod, Thin-Film Optical Filters, 4th ed. (CRC Press, 2010).