Bragg mirrors: periodic stacks and stop bands

Alternate a high index $n_H$ and a low index $n_L$, each a quarter wave thick at $\lambda_0$, and the reflections from every interface return in phase: the wave reflected at an H→L boundary has the opposite sign to the one reflected at an L→H boundary, and the quarter-wave round trip adds another $\pi$, so all of them add. With enough periods the stack reflects nearly everything over a band around $\lambda_0$: a dielectric mirror with no absorption, the basis of laser mirrors, of distributed Bragg reflectors in VCSELs, and the one-dimensional photonic crystal. Yeh, Yariv and Hong's 1977 paper is the one that treats it as a crystal, and this section follows them.

The peak: quarter waves in series

At $\lambda_0$ each quarter-wave layer maps an admittance $Y$ onto $n^2/Y$ (see the transfer matrix). Starting from the substrate $n_s$, a pair L then H multiplies it by $(n_H/n_L)^2$, so $N$ pairs, high index outermost, present

$Y$ grows geometrically with $N$, so $1 - R$ falls geometrically, by the factor $(n_L/n_H)^4$ per pair once $Y \gg n_0$: eight pairs of TiO₂-like and SiO₂-like layers (2.3 and 1.45) already reach 99.84 %, and eleven pairs 99.99 %. Ending the stack with a low-index layer instead divides $Y$ by $n_L^2$ and lowers the peak; the outermost layer should be the one of higher admittance contrast with the incident medium. For a VCSEL mirror in GaAs/AlAs (3.5/2.9, a contrast of only 1.2) the same 99.9 % needs about thirty pairs, which is why such mirrors are grown rather than evaporated.

The band: Bloch waves and the dispersion relation

An infinite periodic stack is a one-dimensional crystal, and Bloch's theorem says its eigenwaves reproduce themselves from one period to the next up to a phase: $E(z + \Lambda) = e^{iK\Lambda}E(z)$, with $K$ the Bloch wavenumber. The characteristic matrix $M$ of one period (the product $M_H M_L$) must therefore have the eigenvalue $e^{iK\Lambda}$, and since $\det M = 1$ the two eigenvalues are $e^{\pm iK\Lambda}$, whose sum is the trace:

This is the dispersion relation of the crystal. Where the right-hand side lies between $-1$ and $1$, $K$ is real and the wave propagates; where it exceeds 1 in magnitude, $K$ is complex, the Bloch wave decays as $e^{-|\mathrm{Im}\,K| z}$, and a thick enough stack reflects it completely. That decaying range is the stop band; its edges are where $\cos K\Lambda = \pm 1$. For a quarter-wave stack at normal incidence, $\delta_H = \delta_L = \delta$ and the condition $|\cos 2\delta - \ldots| = 1$ gives a band centred on $\delta = \pi/2$ with a full width, in the relative wavenumber $g = \lambda_0/\lambda$, of

about 0.29 for 2.3/1.45, or 524–702 nm around 600 nm, and 0.11 for GaAs/AlAs. The width depends on the index contrast alone; the period count only decides how close to 1 the reflectance gets inside the band and how sharp its edges are. Inside the band the imaginary part of $K$ is largest at the centre, $|\mathrm{Im}\,K|\Lambda \approx 2(n_H - n_L)/(n_H + n_L)$ for small contrast, which gives the number of periods the field penetrates.

N periods exactly: Chebyshev polynomials

For a finite stack of $N$ periods Yeh, Yariv and Hong evaluate $M^N$ in closed form through the Chebyshev polynomials of the second kind, $U_N(\cos K\Lambda) = \sin[(N+1)K\Lambda]/\sin K\Lambda$:

with $A, B, C, D$ the elements of the single-period matrix. Outside the band $K\Lambda$ is real, $U_{N-1}$ oscillates, and the reflectance has $N - 1$ zeros between two bands, one fewer than the number of periods, which is the count the classic card below invites. Inside the band $U_{N-1}$ grows like $\sinh(N|\mathrm{Im}\,K|\Lambda)$, and $R_N \to 1$ exponentially in $N$, in agreement with the admittance result at the centre.

import numpy as np
from pystratify import planar

n = lambda wl: [1.0, 2.3, 1.45, 2.3, 1.45, 2.3, 1.45, 2.3, 1.45, 2.3, 1.45, 2.3, 1.45, 2.3, 1.45, 2.3, 1.45, 1.52]  # incident medium, layers, substrate
d = [np.inf, 0.065217, 0.103448, 0.065217, 0.103448, 0.065217, 0.103448, 0.065217, 0.103448, 0.065217, 0.103448, 0.065217, 0.103448, 0.065217, 0.103448, 0.065217, 0.103448, np.inf]  # thicknesses, µm
lam = (400 + 1 * np.arange(501)) * 1e-3  # wavelength, µm
th = np.radians(0)
R, T = {}, {}
for pol in ('s', 'p'):
    rows = [planar.coh_tmm(pol, n(wl), d, th, wl) for wl in lam]
    R[pol], T[pol] = np.array([r['R'] for r in rows]), np.array([r['T'] for r in rows])
R_avg, T_avg = (R['s'] + R['p']) / 2, (T['s'] + T['p']) / 2  # unpolarized

Outside the band the stack is nearly transparent, with ripples, one fewer minimum between bands than there are periods. Yeh, Yariv and Hong's 1977 paper treats the periodic stack as a one-dimensional crystal: the eigenvalues of the unit cell's transfer matrix give the Bloch wavenumber, real in the pass bands and complex in the stop bands, and the band edges, the number of zeros and the closing of the TM gap at the Brewster angle all follow from one dispersion relation.

Penetration depth and the phase of a mirror

A Bragg mirror does not reflect at its surface. The field decays into the stack over a few periods, and a cavity closed by such a mirror is effectively longer than its spacer by the penetration depth, which for a quarter-wave stack of small contrast is (Babic and Corzine 1992)

about 180 nm for TiO₂/SiO₂ at 600 nm and 0.4 µm for GaAs/AlAs at 1 µm. The same quantity sets how fast the mirror's reflection phase changes with wavelength, which decides the free spectral range and the mode spacing of a VCSEL or of a dielectric Fabry–Pérot cavity, and it is the reason a cavity between two Bragg mirrors resonates at a wavelength slightly off the naive $2nd = m\lambda$.

Higher bands, and the bands that vanish

The dispersion relation is periodic in $\delta$: the stop band repeats wherever the layers are an odd number of quarter waves thick, at $\lambda_0/3$, $\lambda_0/5$, …, with the same width in $g$ (so narrower in wavelength). At $\lambda_0/2$, where both layers are half waves, $\cos K\Lambda = 1$ exactly: the second-order band of a quarter-wave stack has zero width, and the stack is transparent there. In general the $m$-th band of a stack with optical-thickness ratio $n_H d_H : n_L d_L$ vanishes when $m\,n_H d_H/(n_H d_H + n_L d_L)$ is an integer, a selection rule that is used to suppress unwanted bands and, run backwards, to measure a stack's thickness ratio from which bands are missing.

At oblique incidence

Tilting the beam shortens every layer's phase thickness, $\delta_j = k_0 n_j d_j\cos\theta_j$, so the band moves to shorter wavelengths (a Bragg mirror seen at an angle turns bluer), and splits by polarization: the admittances $\eta_H/\eta_L$ of the two polarizations diverge, so the s band widens and the p band narrows, closing entirely at the Brewster angle of the H/L interface, $\tan\theta_B = n_L/n_H$ inside the stack, where successive interfaces no longer reflect p light at all. In a stack lit from air the internal angle can never reach that Brewster angle if

and with enough contrast the stop bands of all angles and both polarizations then overlap over a common range: Fink, Winn and co-workers (1998) built such an omnidirectional reflector from nine pairs of polystyrene and tellurium, a dielectric mirror for every angle. The classic card's Yeh stack, lit from inside GaAs, does reach the Brewster angle, which is where its TM gap closes.