Distributed Bragg Reflector

A distributed Bragg reflector, or dielectric mirror, is a reflector used in waveguides and optical fibers. A distributed Bragg reflector has extremely low losses at optical and infrared frequencies compared to ordinary metallic mirrors. Its structure is formed from periodic thin layers of alternating materials with high and low refractive indices. Typically the stack would be made up of an odd number of layers where the first and last layers are chosen to have high refractive index.

Each layer boundary causes a partial reflection of an optical wave. When the wavelength is close to four times the optical thickness of the layers, the many reflected waves tend to interfere constructively, causing the layers to act as a high-quality reflector. The range of wavelengths in which most of the incident intensity is reflected is called the photonic stopband. In the limit in which the reflector contains a very large number of layers, radiation in this range of wavelengths cannot propagate into the structure.

Distributed Bragg reflectors are critical components in vertical cavity surface emitting lasers and other types of narrow-linewidth laser diodes such as distributed feedback lasers.

Figure 1: Bragg reflector with 9 layers (N = 4).

Model Definition

The model consists of a single domain containing a substrate with refractive index ns = 1.5. The exterior of the modeling domain is air with refractive index na = 1.0. At the default boundary condition on the surfaces of the substrate, a number of features are added to represent the alternating layers.

The layers of greater refractive index are made of ZnS with nH = 2.32, and the layers of lower refractive index contain MgF2 with nL = 1.38. The thicknesses of the layers are calculated such that nHtH = nLtL = λ0/4.

To show the response of the mirror across a span of wavelengths, a range of wavelengths (or frequencies) can be specified using the feature.

Of particular interest in this device is the reflectance R of the device and what range of wavelengths it is effective over, Δλ. The reflectance for the distributed Bragg reflector is given by:

(1)

where N is the number of pairs of dielectric layers; for example, N = 5 implies that the reflector consists of 11 layers (five pairs plus an additional layer of high refractive index on top).

The bandwidth Δλ of the photonic stopband is given by:

(2)

where λ0 is the central wavelength of the band.

Results and Discussion

Figure 2 shows the response of the dielectric mirror across a range of wavelengths from 400 nm to 800 nm. The vacuum wavelength λ0 used in the specification of the layers is 550 nm. At this wavelength, a configuration with 2 unit cells (5 total layers) gives roughly 87% reflectance whereas a configuration with 5 unit cells (11 total layers) gives roughly 99.5% reflectance. The computed values of R agree with Equation 1.

The calculated stopband is 180 nm using Equation 2. As shown in Figure 2 the reflectance approaches 100% within the stopband as the number of layers increases. For the maximum number of layers the reflectance is about 100% for a range of free-space wavelengths from 475 nm to 655 nm for a stopband of about 180 nm.

Figure 2: Response of the distributed Bragg grating for different numbers of layers, from a minimum of 5 total layers to a maximum of 41.

Ray_Optics_Module/Prisms_and_Coatings/distributed_bragg_reflector

Modeling Instructions

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Model Wizard

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Geometry 1

Add some parameters for the refractive indices of the materials and the central wavelength.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
ns	1.5	1.5	Refractive index of substrate
nh	2.32	2.32	Refractive index of ZnS
nl	1.38	1.38	Refractive index of MgF2
lam0	550[nm]	5.5E-7 m	Vacuum wavelength
Nc	2	2	Number of unit cells