Distributed Bragg reflector (DBR mirror)

A distributed Bragg reflector (DBR) is a mirror made of alternating layers of two materials, of high index $n_H$ and low index $n_L$, each a quarter of a wavelength thick in its material at the design wavelength ${\lambda }_0$:

$$d_H=\frac{{\lambda }_0}{4n_H},d_L=\frac{{\lambda }_0}{4n_L}\mathrm{.}$$

Each interface reflects only a little, but at ${\lambda }_0$ the reflections from all of them return in phase (Bragg condition) and add, so a stack of enough pairs reflects almost all the light. Away from ${\lambda }_0$ the reflections fall out of step, partly cancel (interference), and the reflectivity drops. The reflection is distributed over many layers rather than made at one surface, which is where the name comes from. Dielectric DBRs absorb far less than metal mirrors, which lose a few percent at every reflection, and they are the high-reflectivity mirrors of VCSELs, of laser cavities and of cavities for precision measurement.

Peak reflectivity

At the design wavelength a stack of quarter-wave layers presents an optical admittance that follows from the indices alone. For $N$ pairs whose high-index layers face the incident medium (index $n_0$) and whose low-index layers face the substrate (index $n_s$), the admittance is $Y=(n_H\mathrm{/}{n}_L{)}^{2N}{n}_s$, and with the order reversed it is $Y=(n_L\mathrm{/}{n}_H{)}^{2N}{n}_s$. The reflectance is

$$R={\left(\frac{n_0-Y}{n_0+Y}\right)}^2\mathrm{.}$$

Once $R$ is high, each further pair cuts the transmitted fraction by the factor $(n_H\mathrm{/}{n}_L{)}^2$: 5.8 for Si/SiO₂, 2.5 for TiO₂/SiO₂ and 1.36 for a pair of AlGaAs alloys with indices 3.5 and 3.0, which is why low-contrast mirrors need many more pairs. Halving the index step roughly doubles the number of pairs for the same reflectivity. The characteristic matrix method derives the admittance, and the Thin-Film Reflectance Calculator computes the full spectrum of any quarter-wave stack, with a Ta₂O₅/SiO₂ mirror at 1550 nm as a preset.

The values assume abrupt interfaces and lossless layers. With the same AlGaAs pair, 18 pairs reach 99.5% toward air, the arrangement of a VCSEL's top mirror, where the light leaves; the VCSEL entry explains why its mirrors must exceed 99%.

Stop band

High reflection holds over a band of frequencies centered on the design frequency, of relative width

$$\frac{\mathrm{\Delta }f}{f_0}=\frac{4}{\pi }\arcsin \text{\hspace{0.17em}⁣}\left(\frac{n_H-n_L}{n_H+n_L}\right),$$

so the band edges lie at ${\lambda }_0\mathrm{/}[1\pm \frac{2}{\pi }\arcsin \text{\hspace{0.17em}⁣}\left(\frac{n_H-n_L}{n_H+n_L}\right)]$, as in the table. For small contrast the width in wavelength is close to ${\lambda }_0\mathrm{\Delta }f\mathrm{/}{f}_0$, 83 nm for the AlGaAs pair at 850 nm; for Si/SiO₂ the band spans 900 nm and is lopsided toward long wavelengths. Outside the band the reflectance falls to a series of side lobes. Adding pairs steepens the edges and raises the reflectance inside the band but does not widen it; wider bands are made by chirping the layer thicknesses or stacking mirrors designed for neighboring wavelengths. At oblique incidence the band moves to shorter wavelengths and splits between the two polarizations, the effect that dichroic mirrors are designed around.

Penetration depth

Because the reflection builds up over many layers, a DBR behaves like a mirror located some distance inside the stack. For a stack of many pairs with a small index step $\mathrm{\Delta }n=n_H-n_L$, that distance is about

$$L_{\text{pen}}\approx \frac{{\lambda }_0}{4\mathrm{\Delta }n},$$

the $\tanh (\kappa L)\mathrm{/}2\kappa$ of a Bragg grating in the limit of a strong grating, with a coupling coefficient $\kappa =2\mathrm{\Delta }n\mathrm{/}{\lambda }_0$ (DBR laser). For the AlGaAs pair at 850 nm it is 425 nm, about 3.2 pairs. The effective length of a VCSEL cavity is therefore about 1.1 µm, although the spacer between its mirrors is only one wavelength (243 nm) thick, and the mode spacing and photon lifetime follow from the longer length. Babic and Corzine give analytic expressions for the delay, penetration depth and absorption of quarter-wave mirrors.

Real mirrors

The table's pair counts are lower limits. Semiconductor mirrors grade the composition at each interface to lower the electrical resistance of the many heterojunctions that the current of a VCSEL crosses, which weakens each reflection, and doped layers absorb light through their free carriers, so real VCSEL mirrors use more pairs than the abrupt estimate. Dielectric mirrors are limited by absorption and scattering in the layers once their transmission is small: ion-beam-sputtered dielectric mirrors have reached total losses of a few parts per million, reflecting more than 99.999% (Rempe and colleagues), and cavities built from them serve precision measurement and laser stabilization. An anti-reflection coating uses the same materials with the reflections phased to cancel instead of add.

Other Bragg reflectors

The same principle works along a waveguide or a fiber, where the index is modulated periodically along the direction of propagation. A fiber Bragg grating with a sinusoidal index modulation of amplitude $\mathrm{\Delta }n$ reflects ${\tanh }^2(\kappa L)$ at its Bragg wavelength, with $\kappa =\pi \mathrm{\Delta }n\mathrm{/}\lambda$ when the modulation fills the mode: a modulation of ${10}^{-4}$ at 1550 nm reaches 99% in 15 mm, about 28,000 periods, since the index contrast per period is thousands of times smaller than in a thin-film stack. Etched gratings in semiconductor waveguides are the mirrors of DBR lasers, and periodic structures in two and three dimensions extend the idea to photonic crystals.

Common questions

What is a distributed Bragg reflector?

A mirror made of alternating quarter-wave layers of high and low refractive index. The weak reflections from all the interfaces add in phase at the design wavelength, so a stack of enough pairs reflects nearly all the light within a band around that wavelength.

How many layer pairs does a DBR need?

It depends on the index contrast and the reflectivity required. With abrupt interfaces and no loss, Si/SiO₂ pairs reach 99.9% in four pairs and TiO₂/SiO₂ in eight, while AlGaAs pairs with indices of 3.5 and 3.0 need 27 between a cavity and a GaAs substrate. Real semiconductor mirrors, with graded interfaces, use more.

What determines the stop band of a Bragg mirror?

The index contrast alone: the relative width in frequency is $(4\mathrm{/}\pi )\arcsin [(n_H-n_L)\mathrm{/}(n_H+n_L)]$. Adding pairs raises the reflectance and steepens the edges but does not widen the band.

What is the difference between a DBR mirror and a fiber Bragg grating?

Both reflect by adding many weak reflections in phase. A DBR mirror is a stack of thin films with a large index step between layers, so a few to a few tens of pairs suffice; a fiber Bragg grating modulates the index of a fiber core by ${10}^{-5}$ to ${10}^{-3}$ and needs thousands to tens of thousands of periods.

References: H. A. Macleod, Thin-Film Optical Filters, 4th ed. (CRC Press, 2010). L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012). D. I. Babic and S. W. Corzine, "Analytic expressions for the reflection delay, penetration depth, and absorptance of quarter-wave dielectric mirrors," IEEE Journal of Quantum Electronics 28, 514 (1992). G. Rempe, R. J. Thompson, H. J. Kimble and R. Lalezari, "Measurement of ultralow losses in an optical interferometer," Optics Letters 17, 363 (1992). R. Michalzik (ed.), VCSELs: Fundamentals, Technology and Applications of Vertical-Cavity Surface-Emitting Lasers (Springer, 2013). R. Kashyap, Fiber Bragg Gratings, 2nd ed. (Academic Press, 2010).