Bragg Mirrors

Repeat two dielectric layers to form a finite reflection band. Compare quarter-wave design, index contrast, termination and pair count. This coherent, isotropic calculation uses planar electromagnetic boundary conditions.

Reflection from a periodic stack

A single interface between two ordinary dielectrics reflects a few per cent of the light. A Bragg mirror stacks many such interfaces, alternating a high index $n_H$ and a low index $n_L$ , and spaces them so that all the small reflections leave in step. A is the high-index layer ( $n_H$ ), B the low-index layer ( $n_L=1.5$ , the same as the glass). The examples below use six AB pairs on glass, with light arriving from air. Each internal step reflects 4.4% in the high-contrast example and 0.39% in the low-contrast one. Together they can reflect almost everything, but only over a limited band of wavelengths.

Why the reflections add up

Each layer adds a phase $\delta=2\pi q d/\lambda$ , where $d$ is the layer thickness and $q=n\cos\theta$ inside the layer is its normal wave number ( $q=n$ at normal incidence; see the calculation section). The design rule sets both layers to a quarter of the design wavelength $\lambda_0=550$ nm in optical thickness at normal incidence:

\begin{gathered}n_H d_H=n_L d_L=\frac{\lambda_0}{4}\\ \Rightarrow\ \delta_H=\delta_L=\frac{\pi}{2}\ \text{at}\ \lambda_0\end{gathered}

The interface reflections alternate in sign, going from low to high index and back. A reflection from one interface deeper also makes a round trip through one more layer, which adds half a wave. The two sign changes cancel, so every reflection returns in phase. Adding in phase sets where the peak is; multiple reflections keep $R$ below 1, which the closed form below includes. Away from $\lambda_0$ the phases slip, and the sum falls apart.

For this lossless quarter-wave stack at normal incidence, with the high-index layer facing the incident medium $n_0$ and the last low-index layer on a substrate $n_s$ , the reflectance of $N$ pairs at $\lambda_0$ has a closed form, derived here from the quarter-wave admittance recursion (see the derivation below):

\begin{gathered}R_N=\left(\frac{n_0-Y_N}{n_0+Y_N}\right)^2\\ Y_N=n_s\left(\frac{n_H}{n_L}\right)^{2N}\end{gathered}

It agrees with the solver to rounding error for 1 to 50 pairs in both contrast examples. The index ratio is raised to the power $2N$ , so a small contrast needs many more pairs. With Reverse the unit-cell order (BA) the low-index layer faces the air and $Y_N=n_s(n_L/n_H)^{2N}$ : six high-contrast pairs drop from 98.43% to 96.51%. Because $n_L$ equals the glass index here, the last B layer is optically invisible in AB order.

The infinite crystal

Now repeat the pair forever. A wave in a periodic medium changes from one cell to the next only by a factor $e^{iKa}$ , where $a=d_H+d_L$ is the pair thickness and $K$ is the Bloch wave number. $K$ follows from the half-trace of the 2×2 matrix $P$ that carries tangential $E$ and $H$ across one AB cell. That condition holds at any angle once $q$ and $Y$ replace $n$ ; only the Bloch bands tab restricts it to normal incidence. For a lossless, non-dispersive quarter-wave cell at normal incidence, the first-order gap has edges $\lambda_\pm$ , with $\lambda_+$ the long-wavelength edge:

\begin{gathered}\cos(Ka)=\tfrac12\operatorname{Tr}P\\ \frac{\lambda_0}{\lambda_\pm}=1\mp\frac{2}{\pi}\arcsin\frac{n_H-n_L}{n_H+n_L}\end{gathered}

The edge formula is equivalent to the gap–midgap ratio in Joannopoulos et al. (ch. 4, eq. 5, p. 52), whose midgap frequency corresponds to $\lambda_0$ , and is derived below. Where the half-trace exceeds 1 in magnitude, $K$ is complex: no wave propagates, and the field decays from cell to cell. That is the stop band. In its centre, $\operatorname{Im}K\,a=\ln(n_H/n_L)$ : each cell divides the field amplitude by $n_H/n_L$ , so transmittance at $\lambda_0$ falls roughly as $(n_L/n_H)^{2N}$ , the same power as in $Y_N$ .

The infinite crystal has no substrate, no first layer and no pair count. The finite mirror has all three. The Bloch bands tab shows the crystal, always at normal incidence; the Parameter dependence tab shows the spectrum of the actual device at the selected angle.

Derivation from Maxwell's equations

Start from the macroscopic Maxwell equations for a linear, isotropic, nonmagnetic medium with no free charge. Fields vary as $\exp(-i\omega t)$ , units are SI, $\varepsilon(\omega)$ is the relative permittivity, $k_0=\omega/c$ is the vacuum wave number and $Y_0=\sqrt{\varepsilon_0/\mu_0}$ is the vacuum admittance.

\begin{aligned}\nabla\times\mathbf E&=i\omega\mu_0\mathbf H, & \nabla\cdot(\varepsilon\mathbf E)&=0,\\ \nabla\times\mathbf H&=-i\omega\varepsilon_0\varepsilon\mathbf E, & \nabla\cdot\mathbf H&=0.\end{aligned}

In a uniform region $\varepsilon$ is constant, so $\nabla\cdot\mathbf E=0$ . Taking the curl of Faraday's law, substituting the Ampère–Maxwell law and using $\nabla\times\nabla\times\mathbf E=\nabla(\nabla\cdot\mathbf E)-\nabla^2\mathbf E$ gives the Helmholtz equation, with $n^2=\varepsilon$ :

\nabla^2\mathbf E+\varepsilon k_0^2\mathbf E=0

Light at an Interface derives the normal wave number $q$ , the admittance $Y$ and the continuity of tangential fields; Transfer and Scattering Matrices writes each layer's field as a forward and a backward wave. Below, $E$ and $H$ are the tangential components, with $H$ divided by $Y_0$ and signed so that a forward wave has $H=YE$ : $H=-H_x/Y_0$ for TE and $H_y/Y_0$ for TM. Here $Y=q$ (TE) or $\varepsilon/q$ (TM), so $Y=n$ at normal incidence.

One layer. With $z$ measured from the layer's entrance, $E=Ae^{iqk_0z}+Be^{-iqk_0z}$ and $H=Y\left(Ae^{iqk_0z}-Be^{-iqk_0z}\right)$ . Use $E(0)=A+B$ and $H(0)=Y(A-B)$ to eliminate $A$ and $B$ , then evaluate at $z=d$ :

\begin{gathered}\begin{pmatrix}E(d)\\ H(d)\end{pmatrix}=M\begin{pmatrix}E(0)\\ H(0)\end{pmatrix}\\ M=\begin{pmatrix}\cos\delta & \dfrac{i}{Y}\sin\delta\\ iY\sin\delta & \cos\delta\end{pmatrix}\\ \delta=qk_0d\end{gathered}

Because tangential $E$ and $H$ are continuous at every interface, the same pair enters the next layer, and the matrices of successive layers multiply. $\det M=\cos^2\delta+\sin^2\delta=1$ . This is the matrix used by the Bloch bands calculation.

The Bloch condition. Across one AB cell the fields are multiplied by $P=M_BM_A$ (A is crossed first), and $\det P=1$ . The infinite crystal is unchanged by a shift of $a$ , so by the Bloch–Floquet theorem it has solutions that repeat up to a factor: $P\mathbf v=e^{iKa}\mathbf v$ , where $\mathbf v=(E,H)^{\mathsf T}$ at the start of a cell; the other eigenvalue, $e^{-iKa}$ , is the backward Bloch wave. A 2×2 matrix with unit determinant has eigenvalues $\Lambda$ and $1/\Lambda$ whose sum is its trace, so $e^{iKa}+e^{-iKa}=\operatorname{Tr}P$ . Multiplying out, with $\sigma=\tfrac12(Y_A/Y_B+Y_B/Y_A)$ ,

\begin{aligned}\cos(Ka)&=\tfrac12\operatorname{Tr}P\\ &=\cos\delta_A\cos\delta_B\\ &\quad-\sigma\sin\delta_A\sin\delta_B\end{aligned}

The trace does not depend on the order, so AB and BA have the same bands (in AB order the code forms $M_AM_B$ ; BA order swaps the factors). This holds at any angle with $q$ and $Y$ . For lossless layers the right side is real; where its magnitude exceeds 1, $K$ is complex.

Quarter-wave cell. At normal incidence $Y=n$ . Write $x=n_H/n_L$ , so $\sigma=\tfrac12(x+x^{-1})$ . At $\lambda_0$ both phases are $\pi/2$ , so the cosines vanish. At the band centre the half-trace is below $-1$ , so $e^{iKa}$ is negative and real. Setting $Ka=\pi+i\kappa$ , with $\cos(\pi+i\kappa)=-\cosh\kappa$ and $\cosh(\ln x)=\sigma$ :

\begin{gathered}\cos(Ka)=-\sigma,\qquad \cosh\kappa=\sigma\\ \Rightarrow\ \operatorname{Im}K\,a=\kappa=\ln\frac{n_H}{n_L}\end{gathered}

Without dispersion, at normal incidence both layers keep equal phases $\delta=(\pi/2)\,\lambda_0/\lambda$ at every wavelength. With $\cos^2\delta=1-\sin^2\delta$ the half-trace is $1-(1+\sigma)\sin^2\delta$ . The first-order gap ends where it equals $-1$ :

\begin{gathered}\sin^2\delta=\frac{2}{1+\sigma}=\frac{4x}{(x+1)^2}\\ \Rightarrow\ |\cos\delta|=\frac{n_H-n_L}{n_H+n_L}\\ \Rightarrow\ \delta=\frac\pi2\mp\arcsin\frac{n_H-n_L}{n_H+n_L}\end{gathered}

Dividing by $\pi/2$ gives the edge formula above. The upper sign, the smaller phase, is the long-wavelength edge $\lambda_+$ .

Finite mirror. Follow the admittance $H/E$ from the substrate outwards. The substrate carries only a forward wave, so $H/E=n_s$ at the last interface. Inverting $M$ with $\delta=\pi/2$ gives $E(0)=-iH(d)/n$ and $H(0)=-inE(d)$ , so a quarter-wave layer of index $n$ turns the load admittance $\mathcal Y$ at its exit into $n^2/\mathcal Y$ at its entrance:

\begin{gathered}n_s\ \xrightarrow{\ B\ }\ \frac{n_L^2}{n_s}\ \xrightarrow{\ A\ }\ x^2n_s\\ Y_N=n_s\,x^{2N}\end{gathered}

In the incident medium $E=1+r$ and $H=n_0(1-r)$ . Setting $H/E=Y_N$ gives $r=(n_0-Y_N)/(n_0+Y_N)$ and $R_N=r^2$ . In BA order the substrate meets A first, $n_s\to n_H^2/n_s\to(n_L/n_H)^2n_s$ , so $Y_N=n_s(n_L/n_H)^{2N}$ .

Numerical check: reflectance from these matrices matches the solver within $2 imes10^{-15}$ for all three examples, both orders, 1 and 3 pairs, 400–900 nm at normal incidence; the half-trace matches the Bloch calculation within $3 imes10^{-16}$ ; the edge formula matches the computed edges; and $R_N$ matches the solver for 1 to 50 pairs in both orders.

Three things to try

Change the contrast. The high-contrast example uses $n_H=2.3$ and $n_L=1.5$ . At the page's default 30° TE, the spectrum in the Parameter dependence tab peaks at 99.0% near 528 nm and stays above 90% from 459 to 622 nm. The low-contrast example, with $n_H=1.7$ , peaks at only 62.8% (55.0% at normal incidence; TE contrast grows with angle). In the Bloch bands tab, which is always at normal incidence, the gap narrows from 484.6–635.9 nm to 528.9–572.8 nm, as the edge formula predicts.

Change the pair count. Set the angle to 0° and read the reflectance at 550 nm on the Power summary tab while changing AB pair count. For high contrast, 1, 2, 4, 6 and 10 pairs give 31.2%, 61.6%, 91.6%, 98.4% and 99.95%. With one pair, the low-index layer matches the glass, so it is just a coated surface. The low-contrast mirror reaches 98.2% with 20 pairs. The Bloch bands do not change at all.

Tilt it and switch polarization. Tilting reduces each layer's phase thickness at a given wavelength ( $q<n$ ), so the band moves to shorter wavelengths. The first-order Bragg estimate $2(q_Hd_H+q_Ld_L)$ gives 527.7 nm; the peak of the high-contrast mirror moves from 550 nm at 0° to 528 nm at 30°. At 30° TM reflects less at each interface than TE, so the band narrows: the 90% region becomes 473–597 nm and the peak falls to 97.4%. The detuned example is not quarter-wave ( $d_H=60$ nm, $d_L=130$ nm). Its Bloch gap lies between 590.0 and 766.7 nm, its midgap frequency corresponds to about 667 nm, close to $2(n_Hd_H+n_Ld_L)=666$ nm, and at normal incidence it reflects 5.5% at 550 nm.

UNIFORM LAYERS

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Stable scattering solution · revision 0

Finite mirrors and periodic media

The saved document keeps the AB definition (materials, design wavelength, pair count, order) alongside the expanded physical layers.

Calculation and evidence

Time dependence is $\exp(-i\omega t)$ . Amplitudes are tangential $E_y$ (TE) or $E_x$ (TM), with

\begin{aligned}q&=\sqrt{\varepsilon-n_0^2\sin^2\theta}\\ Y&=\begin{cases}q & \text{TE}\\ \varepsilon/q & \text{TM}\end{cases}\end{aligned}

Interfaces use Fresnel amplitudes; propagation uses $\exp(iq\,2\pi d/\lambda)$ . Reflection recursion composes scattering responses using only decaying exponentials in passive finite layers. Backward fields are anchored at each layer’s exit.

The exterior media must be transparent and positive-index. Internal layers can be passive and absorbing. All layers are coherent, uniform, isotropic and nonmagnetic. An exact zero longitudinal wave number is reported as a singular amplitude basis; nearby angles remain calculable. No roughness, incoherent thick-layer approximation or beam envelope is included.

Independent fixtures use Steven Byrnes’s tmm 0.2.0 for oblique TE/TM, absorbing films, evanescent layers and multilayers. Powers agree within $10^{-8}$ in the fixtures. Checks also cover boundary fields, antireflection and zero-thickness limits. The Python fixture generator is retained with the source.

History of the method

In 1887 Lord Rayleigh applied G. W. Hill's method for equations with periodic coefficients, developed in lunar theory and used in the same paper for vibrations maintained by forces of double frequency, to a laminated medium whose properties vary periodically along one direction. He wrote that the colours of thin plates led him there: old, superficially decomposed glass gives brighter reflected and purer transmitted tints than a single film can (Brewster had shown the glass was laminated). His simplest example was a stretched string with equal small loads at equal intervals. When the wavelength is about twice the spacing, the partial reflections concur in phase and add up to a strong reflection, even if one load does almost nothing. For a density that varies harmonically, he showed that waves are ultimately totally reflected over a finite range of wavelengths, not only at exact tuning.

In 1917 he returned to the problem with alternate strata, each uniform and of constant thickness. He cited the coloured reflection of potassium chlorate crystals described by Stokes, the colours of old decomposed glass, and probably those of some beetles and butterflies. He adapted Stokes's 1862 pile-of-plates calculation from intensities to complex amplitudes. He separated two cases: in one, adding plates drives the reflection towards completeness; in the other it keeps fluctuating however many plates are added. For glass plates of index 1.5 in air, spaced so that their reflections co-operate, his table shows reflection approaching unity at four or five plates. With quarter-wave plates and gaps at normal incidence, this solver reproduces his Table I to the four printed decimals. With very weak single reflections (single-surface amplitude reflection $\eta=0.02$ ), his second table gives 73% at 32 plates and 97.6% at 64.

In 1987 Eli Yablonovitch and Sajeev John published letters on three-dimensional dielectric structures. According to its abstract, Yablonovitch's letter proposed that a three-dimensionally periodic dielectric whose electromagnetic band gap overlaps an electronic band edge could rigorously forbid spontaneous emission. John's abstract describes strong localization of photons in disordered dielectric superlattices, with a pseudogap arising from remnant Bragg resonances. Joannopoulos and co-authors treat the multilayer film on this page as the simplest photonic crystal (chapter 4, p. 44). Joannopoulos and co-authors note that Yablonovitch's proposal came exactly a century after Rayleigh's 1887 paper (chapter 6, p. 96).

References

John D. Joannopoulos, Steven G. Johnson, Joshua N. Winn and Robert D. Meade, Photonic Crystals: Molding the Flow of Light, second edition, Princeton University Press (2008), chapter 4, printed pages 44–65: multilayer gaps, the quarter-wave gap width (eq. 5, p. 52), evanescent modes, defects and omnidirectional reflection. Open PDF on the authors’ book page, posted with Princeton University Press’s permission.

Steven J. Byrnes, Multilayer optical calculations, 2016, revision 5 (2020): field conventions, transfer matrices, flux and absorption. The browser implementation uses scattering recursion for numerical stability; the Python implementation supplies independent fixtures.

Lord Rayleigh (John William Strutt), On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 24(147), 145–159 (1887). Read in the reprint in Rayleigh's Scientific Papers, volume 3, Cambridge University Press (1902), article 142, pages 1–14; the original journal pages were not consulted.

Lord Rayleigh, On the reflection of light from a regularly stratified medium, Proceedings of the Royal Society of London A 93(655), 565–577 (1917); open scan, read in full.

Eli Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Physical Review Letters 58(20), 2059–2062 (1987). Only the abstract was read.

Sajeev John, Strong localization of photons in certain disordered dielectric superlattices, Physical Review Letters 58(23), 2486–2489 (1987). Only the abstract was read.

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