Tool · Optical coatings

Thin-Film Reflectance Calculator

How much light does a stack of thin dielectric layers on a substrate reflect and transmit, and how does that change with wavelength, angle and polarization? The calculator applies the characteristic-matrix method for s and p polarization to any layer sequence, from single-layer and two-layer anti-reflection coatings to quarter-wave Bragg mirrors, laser facet coatings and free-standing etalons, and draws the spectrum and the angular response live. Background: characteristic matrix method, anti-reflection coating, and distributed Bragg reflector.

Media and geometry

Incident medium n₀

Substrate nₛ

Design λ₀ (nm) sets quarter- and half-wave thicknesses

Wavelength λ (nm)

Angle of incidence (deg)

measured in the incident medium, from the normal

Layers (incident side first)

#	n	d (nm)	n·d / λ₀

QW and HW set a layer to a quarter or half wave of optical thickness at λ₀. Indices are taken as constant over the plotted range; dispersion is not included.

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Readouts at λ, θ

Reflectance versus wavelength at the set angle

range to nm

spaverage

Reflectance versus angle at the set wavelength

Checked against

These checks run in your browser on every load. Each compares the matrix computation with an independent closed form: the Fresnel equations, the Brewster and critical angles, the quarter-wave and half-wave layer formulas, the quarter-wave stack admittance, the Airy etalon formula, energy conservation, and reciprocity.

Check	Expected	Computed	Tolerance

The table compares the transfer-matrix result with an independent closed form for each case (the Fresnel equations, the Brewster and critical angles, the quarter-wave and half-wave layer formulas, the quarter-wave stack admittance and the Airy formula) and checks energy conservation and reciprocity. Both columns are rounded to six decimals, and a row passes only when they agree in every digit shown.

The model

Each homogeneous layer of index $n_{j}$ and thickness $d_{j}$ has a phase thickness and a tilted optical admittance

δ_{j} = 2 π n_{j} d_{j} \cos θ_{j} / λ

η_{s} = n \cos θ, η_{p} = n / \cos θ

with the angle in each medium fixed by Snell’s law. The layer acts on the tangential fields through its characteristic matrix, and the stack is the product of the matrices taken from the incident side, applied to the substrate:

M_{j} = [\begin{array}{cc} \cos δ_{j} & i \sin δ_{j} / η_{j} \\ i η_{j} \sin δ_{j} & \cos δ_{j} \end{array}]

[\begin{array}{c} B \\ C \end{array}] = M_{1} M_{2} \dots M_{q} [\begin{array}{c} 1 \\ η_{s u b} \end{array}]

The reflection coefficient and the power ratios follow as

r = \frac{η_{0} B - C}{η_{0} B + C}, R = ∣ r ∣^{2}

T = \frac{4 η_{0} R e (η_{s u b})}{∣ η_{0} B + C ∣^{2}}

The unpolarized values are the averages of s and p. The layers are taken as lossless, isotropic, non-dispersive and perfectly smooth, and the substrate as semi-infinite: the reflection from the back face of a real substrate, 4.3 % for uncoated glass of index 1.52, is not added. Interference inside the layers is fully coherent, which is correct for thin films but not for a thick plate under a broadband source. For a single quarter-wave layer at normal incidence the matrix reduces to

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

which vanishes when $n_{1} = \sqrt{n_{0} n_{s}}$ .

Worked example

The default preset is a quarter-wave MgF₂ layer ( $n_{1} = 1.38$ ) on glass ( $n_{s} = 1.52$ ) in air, designed for 550 nm, so its thickness is 550/(4 × 1.38) = 99.6 nm. At normal incidence the bare glass reflects $(0.52 / 2.52)^{2} = 4.258 %$ . With the layer, $n_{1}^{2} = 1.9044$ and $R = (0.3844 / 3.4244)^{2} = 1.260 %$ , which is the value the tool shows at 550 nm; a zero would need $n_{1} = \sqrt{1.52} = 1.233$ . Moving the wavelength slider to 400 nm and 800 nm gives 2.205 % and 1.942 %. The two-layer V-coat preset adds a quarter wave of index 1.70 under the MgF₂; since ${1.38}^{2} \times 1.52 = 2.895$ is close to ${1.70}^{2} = 2.890$ , its reflectance at 550 nm falls to $6.6 \times 10^{- 7}$ , which the readout rounds to 0.00007 %. Coating design is treated in the article on anti-reflection coating design.

Related: for a single quarter-wave layer, the AR coating thickness calculator returns the film thickness and the ideal film index directly.

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