Acronym: DBR = distributed Bragg reflector
Definition: mirror structures based on Bragg reflection at a period structure
More general term: distributed mirrors
Author: Dr. Rüdiger Paschotta
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A Bragg mirror (also called distributed Bragg reflector) is a mirror structure which consists of an alternating sequence of layers of two different optical materials. The most frequently used design is that of a quarter-wave mirror, where each optical layer thickness corresponding to one quarter of the wavelength for which the mirror is designed. The latter condition holds for normal incidence; if the mirror is designed for larger angles of incidence, accordingly thicker layers are needed.
The principle of operation of such a reflector can be understood as follows. Each interface between the two materials contributes a Fresnel reflection. For the design wavelength, the optical path length difference between reflections from subsequent interfaces is half the wavelength; in addition, the amplitude reflection coefficients for the interfaces have alternating signs. Therefore, all reflected components from the interfaces interfere constructively, which results in a strong reflection. The reflectance (reflectivity) achieved is determined by the number of layer pairs and by the refractive index contrast between the layer materials. The reflection bandwidth is determined mainly by the refractive index contrast; if that is too small, even using a very large number of coating layers will not work.
Figure 1 shows the field penetration into a Bragg mirror made of eight layer pairs of TiO2 and SiO2. The blue curve shows the intensity distribution of a wave with the design wavelength of 1000 nm, incident from the right-hand side. Note that the intensity is oscillating outside the mirror due to the interference of the counterpropagating waves. The gray curve shows the intensity distribution for 800 nm, where a significant part of the light can get through the mirror coating.
Figure 2 shows the reflectance and the group delay dispersion as functions of the wavelength. The reflectance is high over some optical bandwidth, which depends on the refractive index contrast of the materials used and on the number of layer pairs. The dispersion is calculated from the second derivative of the reflection phase with respect to the optical frequency. It is small near the center of the reflection band, but grows rapidly near the edges.
Figure 3 shows with a color scale how the optical field penetrates into the mirror. It can be seen that there is little field penetration well within the reflection band.
Figure 4 shows reflectance spectra of a Bragg mirror for different angles of incidence. The larger that angle, the more the reflection spectrum is shifted towards shorter wavelengths.
Types of Bragg Mirrors
Bragg mirrors for applications in optics can be fabricated with different technologies:
- Dielectric mirrors based on thin-film coating technology, fabricated for example with electron beam evaporation or with ion beam sputtering, are used as laser mirrors in solid-state bulk lasers. The mirror structure then consists of amorphous materials.
- Fiber Bragg gratings, including long-period fiber gratings, are often used in fiber lasers and other fiber devices. They can be fabricated by irradiating a fiber with spatially patterned ultraviolet light. Similarly, volume Bragg gratings can be made in photosensitive bulk glass.
- Semiconductor Bragg mirrors can be produced with lithographic methods. They are used, for example, in surface-emitting semiconductor lasers and in semiconductor saturable absorber mirrors, but also as separate optical components (→ crystalline mirrors).
- There are various types of Bragg reflectors used in other waveguides, based on, e.g., corrugated waveguide structures which can be fabricated via lithography. Such kind of gratings are used in some distributed Bragg reflector or distributed feedback laser diodes.
There are other multilayer mirror designs which deviate from the simple quarter-wave design. They generally have a lower reflectance for the same number of layers, but can be optimized e.g. as dichroic mirrors or as dispersive chirped mirrors for dispersion compensation.
|||Analysis of a Bragg mirror with the RP Coating software|
|||R. Paschotta, case study on a Bragg mirror|
See also: mirrors, Bragg gratings, fiber Bragg gratings, dielectric mirrors, crystalline mirrors, chirped mirrors, dispersive mirrors, laser mirrors, distributed Bragg reflector lasers, distributed feedback lasers
Questions and Comments from Users
Does the reflectance spectrum of a Bragg mirror depend on whether the light is incident from the air side or from the glass substrate side? For example, could a 'hot mirror' be laminated between two sheets of glass and still have the same reflectance spectrum as one where the light is incident from air? Or would the alternating sequence of layers need to be changed to obtain the same reflectance spectrum?
The author's answer:
The reflectance of such a device cannot depend on the direction of light propagation. So if you take the same device both times and only come with light from different sides, you get the same reflectance.
However, the reflectance can change if you modify the setup, for example by putting a layer of glass on top of a mirror.
Why do the spectral features shift to shorter wavelengths when the mirror is tilted? Shouldn't the wavelength grow since the optical distance between the films also grows?
The author's answer:
It is surprising indeed, but I have explained that in a blog article.
What is the origin of the reflection bandwidth of a Bragg mirror?
The author's answer:
The Bragg condition is still approximately fulfilled for wavelengths which are close enough to the Bragg wavelength. More precisely, the optical phase changes caused by the wavelength deviation are small enough not to cause substantial changes of the reflectance. If the refractive index contrast is strong, a relatively small number of layers is relevant, so that the involved phase changes are relatively small for a given wavelength deviation.
Why does the thickness need to be 1/4 of the wavelength, not 1/2 the wavelength?
The author's answer:
The λ/4 thickness implies λ/2 for the way back and forth through one layer, or a <$\pi$> phase shift. That looks like the wrong condition at a first glance. However, in addition there is a <$\pi$> phase difference between subsequent reflections because it makes a difference whether you go from higher refractive index to a lower one or vice versa. As a result, the phases of reflected contributions from different interfaces are the same, so that they can add up constructively.
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For a Bragg mirror, why are there smaller reflectivity peaks outside the main one?
The author's answer:
That's not easy, but best understood in cases with few layer pairs and a low reflectivity per layer pair, where we can neglect effects of multiple reflections. The reflected light amplitude is then the superposition of one amplitude contribution per layer pair. The wavelength offset from the wavelength of peak reflectivity determines the phase change between two such contributions. If you had only two of those, you would get a purely oscillatory behavior of reflectivity vs. wavelength. With more of them, it gets more complicated, but certainly you shouldn't expect the reflectivity to monotonously decay for increasing phase changes.
Mathematically, you can approach that with a Fourier transform, e.g. applied to a rectangular function resembling amplitude contributions from some finite length in which you have the refractive index oscillations.