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Thin Films and Multilayer Coatings

Build a stack of coherent thin layers, each with a thickness and a complex index n + iκ, and compute its reflectance, transmittance and absorptance against wavelength and angle for s and p light. The page also plots the standing-wave field inside the stack. Presets cover anti-reflection coatings, Bragg mirrors, soap films and metal films.

Edit layers in the table • Click a plot to set λ, θ or thickness • Every slider has a number box
R — Reflectance
T — Transmittance
A — Absorptance
λ — Probe wavelength
⚙ — Solver used

Layer stack light enters from the top row

Layers are listed in the order light meets them. N = n + iκ; thickness in nm; QWOT is the optical thickness n·d in units of λ_d/4 (normal incidence). On narrow screens the table scrolls sideways.

# Name d (nm) n κ QWOT Actions

R, T, A against wavelength θ₀ = 0°

Power fractions of the incident beam: solid R, dashed T, dotted A = 1 − R − T (absorbed in the films and, with an absorbing substrate model, in the substrate). The colour strip marks the visible band. Click to set the probe wavelength. Dashed vertical line: design wavelength λ_d.

R and T against angle λ₀ = 550 nm

Cyan: s (TE); pink: p (TM). Solid R, dashed T. Click to set θ₀. Markers show the bare-substrate Brewster angle and, when n₀ > nₛ, the critical angle.

Field inside the stack |E|²/|E_inc|²

Time-averaged |E|² at the probe λ₀ and θ₀, normalised to the incident wave (the incident medium shows the incident + reflected standing wave). The shading marks the layers: taller bands have higher n, and hatched bands absorb (κ > 0). For p light the normal component E_z jumps at each interface, so |E|² does too. The substrate is semi-infinite here.

Thickness scan and interference colour layer 1

R at the probe λ₀ against the thickness of the chosen layer, with every other layer fixed. The strip underneath is the approximate reflected colour of each thickness under ≈6500 K white light at θ₀. It uses CIE 1931 colour-matching fits, is scaled so the brightest thickness is full white, and is white-balanced. Treat it as a display aid, not calibrated colour. Click to set the thickness.

R (s / p)
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T (s / p)
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A (s / p)
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r_s (|r| ∠ φ)
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r_p (|r| ∠ φ)
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Σ |Im δ| (at probe)
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Abelès vs stable: max |ΔR|, |ΔT| over the spectrum
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Stop band (analytic)
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Stop band (Bloch edges |½ Tr M_HL| = 1, numerical)
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Half-max width of the computed R (finite N: wider, → Bloch band as N grows)
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Peak R (analytic / computed)
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Reflected colour (unpolarised)
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Transmitted colour
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Where the power goes at the probe λ₀ and θ₀ for the selected polarisation (unpolarised = s/p mean). Layer absorption comes from the drop in Poynting flux across each layer. Semi-infinite substrate.
Layer Absorbed Peak |E|²
💡 How to use

Learn with this tool

Learning objectives

  • Predict the design of a single-layer and two-layer anti-reflection coating from the quarter-wave admittance rule, and explain its wavelength and angle dependence.
  • Explain what sets the peak reflectance (number of periods) and the stop-band width (index contrast only) of a quarter-wave Bragg mirror.
  • Tell when a layer must be treated coherently and when incoherently, and why thick absorbing layers need a numerically stable algorithm.

Prerequisites

Model

The model has plane, isotropic, nonmagnetic, homogeneous layers with sharp interfaces. The light is a monochromatic plane wave, and every film is treated as coherent. Fields use E(r, t) = Re{E₀ exp[i(k·r − ωt)]}. An absorbing medium has N = n + iκ with κ ≥ 0, and each layer's normal wavenumber is qj = √(Nj² − n₀² sin²θ₀), taking the root with Im q ≥ 0. The layer phase thickness is δj = (2π/λ₀) qj dj. Tilted admittances are η = q (s) and η = N²/q (p). The characteristic matrix of a layer maps the tangential fields at its back face to its front face:

[E; H]front = [[cos δ, −i sin δ/η], [−i η sin δ, cos δ]] · [E; H]back,   [B; C] = M₁M₂…ML[1; ηs],   Y = C/B

r = (η₀ − Y)/(η₀ + Y),   R = |r|²,   T = Re(ηs) |2η₀/(η₀B + C)|² / η₀,   A = 1 − R − T

The −i signs follow from the exp(−iωt) convention. Macleod's textbook uses exp(+iωt) and writes +i. For p light the returned rp uses the Fresnel tool's basis êp = ŷ × k̂, so rp = −(η₀ − Y)/(η₀ + Y) and rp = −rs at normal incidence. The power coefficients do not depend on that sign.

λ₀
vacuum wavelength (the medium wavelength is λ₀/n)
N = n + iκ
complex refractive index (constant, i.e. non-dispersive, in this tool)
θ₀, n₀
angle and (real) index of the incident medium; n₀ sin θ₀ is conserved in every layer
δ
phase thickness k₀ q d; Im δ > 0 means attenuation or evanescence across the layer
η, Y
tilted admittance of a medium; input admittance of the stack
QWOT
quarter-wave optical thickness: n d = λ_d/4
Derivation, and why a second (stable) algorithm is needed

Inside layer j the tangential field is E(z) = A eik₀qz + B e−ik₀qz, and H = η(A eik₀qz − B e−ik₀qz). Solving for A and B at z = d and substituting at z = 0 gives the matrix above. Tangential E and H are continuous, so the layer matrices simply multiply. In the incident medium E = 1 + r and H = η₀(1 − r) at the first interface. Their ratio fixes r = (η₀ − Y)/(η₀ + Y).

For an absorbing or evanescent layer, cos δ and sin δ grow like eIm δ/2. The matrix product mixes that growing part with a decaying part that is 10−16 smaller. Precision is lost, and once Σ Im δ ≳ 700 the product overflows to NaN. (A 1 mm silver film has Im δ ≈ 4×10⁴ at 550 nm.) The recursive Airy/Rouard method starts at the substrate and carries the reflection coefficient of everything behind each interface:

ρj = (rj,j+1 + ρj+1 e2iδj+1) / (1 + rj,j+1 ρj+1 e2iδj+1),   rab = (ηa − ηb)/(ηa + ηb)

The only exponential that appears is eiδ, with |eiδ| ≤ 1, so the result stays finite for any thickness. This is the 2×2 form of a scattering-matrix method. The same forward amplitudes give the internal field. In Auto mode the tool uses Abelès matrices while Σ|Im δ| < 20 and the recursion otherwise. The “Abelès vs stable” readout compares the two algorithms over the whole spectrum. They agree to about 10⁻¹⁵ wherever both are stable.

Coherent or incoherent? Interference between the two faces of a layer survives only if the round-trip path 2nd cos θ is shorter than the source's coherence length, and only if the thickness is uniform to a fraction of λ over the beam. A 1 mm glass window illuminated with a lamp or a spectrometer of 1 nm resolution fails both tests. Its fringes, with period λ²/(2nD) ≈ 0.1 nm, are averaged away. The correct model then adds the multiply reflected substrate beams in intensity: R = Rf + Tf² τ² Rb/(1 − Rf′Rbτ²), with τ = exp(−2k₀ Im qs D). This formula is the exact phase average of the coherent result. Mixing the two, for example by adding amplitudes for some beams and intensities for others inside one film, is not a valid model.

Exercise 1: Quarter-wave anti-reflection

  1. For glass nₛ = 1.52 in air, what film index and thickness make R = 0 at 550 nm? What is R at 275 nm?
  2. Load Ideal QW AR, then set the probe to 275 nm (widen λ min).
  3. Read R at 550 nm and at 275 nm. Then switch to MgF₂ AR and read the residual R at 550 nm.
  4. Why is the 275 nm value exactly the bare-glass value?
Show answer

n = √(1 · 1.52) = 1.233 and d = λ/(4n) = 111.5 nm give R = 0 (the preset rounds n and d, so the solver reads ≈10⁻¹¹). At 275 nm the film is a half-wave layer: its matrix is −I, so it is “absentee” and R = 4.26 %, the bare-glass value. MgF₂ (n = 1.38) is the lowest-index durable coating material. It leaves R = [(1.52 − 1.904)/(1.52 + 1.904)]² = 1.26 %.

Exercise 2: Bragg mirror bandwidth versus reflectance

  1. Doubling the number of HL periods: does the stop band get wider, narrower or stay the same? What happens to peak R?
  2. Load Bragg (HL)ᴺH (6 periods). Then add or remove H/L pairs with the table, or duplicate rows.
  3. Compare the measured stop band with the analytic Δλ, and the peak R with the admittance formula.
  4. Which parameter would you change to double the bandwidth?
Show answer

Peak R → 1 as Y = (n_H/n_L)^{2N} n_H²/nₛ grows (99.82 % for N = 6, nH = 2.35, nL = 1.38). The width tends to a fixed value set only by the contrast: Δg = (2/π) asin[(n_H − n_L)/(n_H + n_L)] = 0.167 in g = λ₀/λ, so the edges sit at 514 and 721 nm. The common estimate Δλ/λ₀ ≈ (4/π) asin(…) = 0.335 gives 201 nm. Only a larger n_H/n_L widens the band. With few periods the band edges are rounded and the half-maximum width is wider (about 250 nm for N = 6, 214 nm for N = 20). More periods sharpen the edges towards the 514–721 nm Bloch edges and raise R.

Exercise 3 (limiting cases): What does the stack reduce to?

  1. (a) With Remove all films, what should the angle plot be? (b) As a silver film becomes 1 µm, 1 mm, 1 m thick, what are R and T? Can the Abelès product compute it?
  2. Remove all films. Then load Thick Ag (stable) and switch the solver between the three methods.
  3. Compare with the Fresnel tool at the same n₀, nₛ, θ₀. Read R, the solver and the Abelès-vs-stable readout for the thick film.
  4. Why does the stable recursion survive where the matrix product fails?
Show answer

(a) With no films, Y = ηₛ and the result is exactly the single-interface Fresnel formula, including Brewster's zero for p and TIR (the unit tests check this to 10⁻¹²). (b) T → 0 and R → the bulk silver value |(1 − N)/(1 + N)|² = 98.2 % at 550 nm. For 1 mm, Im δ ≈ 3.8×10⁴, and cosh(Im δ) overflows. The forced Abelès solver returns NaN (the plot is blank and a warning appears). The recursion only multiplies by e^{iδ}, whose modulus is ≤ 1.

Exercise 4: Why soap films turn black before they burst

  1. Air | water (n = 1.33) | air. Is a film much thinner than λ bright or dark in reflection? At what thickness is it brightest at 550 nm?
  2. Load Soap film and study the thickness scan and colour strip.
  3. Locate the first R maximum and the thickness range of first-order colours.
  4. Account for the reflection phases at the two faces.
Show answer

The two faces have opposite reflection amplitudes (r and −r), so as d → 0 the two reflections cancel and the film is black. Maxima occur where 2nd = (m + ½)λ, first at d = 103 nm for 550 nm. Colours are vivid only for the first few orders (d ≲ 1 µm). Beyond that, many orders overlap across the visible band and the colour washes out to a pale white.

Worked example: a six-period dielectric mirror

A mirror for 600 nm uses TiO₂ (n_H = 2.35) and MgF₂ (n_L = 1.38) quarter-wave layers on glass (nₛ = 1.52): air | (HL)⁶ H | glass. The physical thicknesses are d_H = 600/(4·2.35) = 63.8 nm and d_L = 600/(4·1.38) = 108.7 nm.

  1. Each quarter-wave layer transforms an admittance Y into η²/Y. Working from the substrate: Y = (n_H/n_L)^{12} n_H²/nₛ = 594 × 3.633 = 2.16×10³.
  2. R = [(1 − Y)/(1 + Y)]² = 0.99815, so 99.8 % is reflected. There is no absorption, so T = 0.19 %.
  3. Stop band: Δg = (2/π) asin(0.97/3.73) = 0.1675. The edges are λ = 600/(1 ± 0.1675) = 514 nm and 721 nm, a width of 207 nm (34 % of λ₀).
  4. At 45° the stop band shifts to shorter λ, and the p band is narrower than the s band, because η_p = n/cos θ and η_s = n cos θ change the effective contrast in opposite ways. Check this on the angle and spectrum plots.

Load the Bragg (HL)ᴺH preset. The readouts show these numbers computed by the solver, which agree with the formulas to about 10⁻¹².

When the model fails

  • Dispersion: every index here is constant. Real MgF₂, TiO₂ and especially metals (Ag, Al) change with λ. The metal presets quote values near 550 nm, so spectra far from there are only qualitative.
  • Thick layers and finite bandwidth: films much thicker than the coherence length (or than λ²/Δλ of the detector) are not coherent. Use the incoherent back-side model or average over bandwidth.
  • Roughness, interdiffusion and inhomogeneous or anisotropic films are not modelled. Nor are scattering, nonlocal or quantum-size effects in metal films only a few nm thick, or islands in ultrathin metal.
  • Finite beams: plane waves only. Beams narrower than the lateral shift in a thick stack, or strongly focused light (a spread of angles), need an angular average.
  • The incoherent back-side formula treats substrate absorption as weak (the reverse stack uses Re nₛ). The interference colours use an approximate illuminant and colour-matching functions.

References

  • H. A. Macleod, Thin-Film Optical Filters, 5th ed., CRC Press (2018), chs. 2–3 and 6.
  • M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge (1999), §1.6 (stratified media).
  • E. Hecht, Optics, 5th ed., Pearson (2017), §9.7 (thin-film interference and multilayers).
  • C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt. 41, 3978 (2002).
  • C. Wyman, P.-P. Sloan and P. Shirley, “Simple analytic approximations to the CIE XYZ color matching functions,” JCGT 2(2), 1 (2013).
  • P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370 (1972). This is the source for the Ag value n ≈ 0.055 + 3.32i at 550 nm.