Distributed Bragg reflector

Image:DBRREFL.png

The DBR's reflectivity, $R$, for intensity is approximately given by {{cite journal |first=C.J.R. |last=Sheppard |authorlink=Colin Sheppard |title=Approximate calculation of the reflection coefficient from a stratified medium |year=1995 |journal=Pure and Applied Optics: Journal of the European Optical Society Part A |volume=4 |issue=5|page=665 |doi=10.1088/0963-9659/4/5/018 |bibcode=1995PApOp...4..665S }}

:$R\; =\; \backslash left[\backslash frac\{n\_o\; (n\_2)^\{2N\}\; -\; n\_s(n\_1)^\{2N\}\}\{n\_o\; (n\_2)^\{2N\}\; +\; n\_s\; (n\_1)^\{2N\}\}\backslash right]^2,$

where $n\_o,\backslash \; n\_1,\backslash \; n\_2$ and $n\_s\backslash ,$ are the respective refractive indices of the originating medium, the two alternating materials, and the terminating medium (i.e. backing or substrate); and $N$ is the number of repeated pairs of low/high refractive index material. This formula assumes the repeated pairs all have a quarter-wave thickness (that is $n\; d\; =\; \backslash lambda\; /\; 4$, where $n$ is the refractive index of the layer, $d$ is the thickness of the layer, and $\backslash lambda$ is the wavelength of the light).

The frequency bandwidth $\backslash Delta\; f\_0$ of the photonic stop-band can be calculated by

:$\backslash frac\{\backslash Delta\; f\_0\}\{f\_0\}\; =\; \backslash frac\{4\}\{\backslash pi\}\backslash arcsin\backslash left(\backslash frac\{n\_2\; -\; n\_1\}\{n\_2\; +\; n\_1\}\backslash right),$

where $f\_o$ is the central frequency of the band. This configuration gives the largest possible ratio $\backslash frac\{\backslash Delta\; f\_0\}\{f\_0\}$ that can be achieved with these two values of the refractive index.{{cite book |last=Orfanidis |first=Sophocles J. |title=Electromagnetic Waves and Antennas |publisher=ECE Department, Rutgers University |year=2016 |url=http://eceweb1.rutgers.edu/~orfanidi/ewa/}}{{cite journal | first=B. |last=Osting |title=Bragg structure and the first spectral gap |year=2012 |journal=Applied Mathematics Letters |volume=25 |issue=11|pages=1926–1930 |doi=10.1016/j.aml.2012.03.002|doi-access=free }}

Increasing the number of pairs in a DBR increases the mirror reflectivity and increasing the refractive index contrast between the materials in the Bragg pairs increases both the reflectivity and the bandwidth. A common choice of materials for the stack is titanium dioxide (n ≈ 2.5) and silica (n ≈ 1.5).{{cite encyclopedia |url=http://www.rp-photonics.com/bragg_mirrors.html |title=Bragg Mirrors |encyclopedia=Encyclopedia of Laser Physics and Technology |publisher=RP Photonics |first=Rüdiger |last=Paschotta|accessdate=May 1, 2009}} Substituting into the formula above gives a bandwidth of about 200 nm for 630 nm light.

Distributed Bragg reflectors are critical components in vertical cavity surface emitting lasers and other types of narrow-linewidth laser diodes such as distributed feedback (DFB) lasers and distributed bragg reflector (DBR) lasers. They are also used to form the cavity resonator (or optical cavity) in fiber lasers and free electron lasers.

File:TE TM bragg map.png showed as a white dashed line, right half represents TE reflection.]]

This section discusses the interaction of transverse electric (TE)

and transverse magnetic (TM) polarized light with the DBR structure, over several

wavelengths and incidence angles. This reflectivity of the DBR structure (described below)

was calculated using the transfer-matrix method (TMM), where

the TE mode alone is highly reflected by this stack, while the TM modes are passed

through. This also shows the DBR acting as a polarizer.

For TE and TM incidence we have the reflection spectra of a DBR stack, corresponding

to a 6 layer stack of dielectric contrast of 11.5, between an air and dielectric layers.

The thicknesses of the air and dielectric layers are 0.8 and 0.2 of the period, respectively.

The wavelength in the figures below, corresponds to multiples of the cell period.

This DBR is also a simple example of a 1D photonic crystal. It has a complete TE band gap, but only a pseudo TM band gap.

File:Bio-inspired-colorimetric-film-based-on-hygroscopic-coloration-of-longhorn-beetles-(Tmesisternus-srep44927-s3.ogv

Bio-inspired Bragg reflectors are 1D photonic crystals inspired by nature. Reflection of light from such a nanostructured matter results in structural colouration. When designed from mesoporous metal-oxides{{Cite journal |last=Bertucci |first=Simone |last2=Megahd |first2=Heba |last3=Dodero |first3=Andrea |last4=Fiorito |first4=Sergio |last5=Di Stasio |first5=Francesco |last6=Patrini |first6=Maddalena |last7=Comoretto |first7=Davide |last8=Lova |first8=Paola |date=2022-05-04 |title=Mild Sol–Gel Conditions and High Dielectric Contrast: A Facile Processing toward Large-Scale Hybrid Photonic Crystals for Sensing and Photocatalysis |journal=ACS Applied Materials & Interfaces |language=en |volume=14 |issue=17 |pages=19806–19817 |issn=1944-8244 |pmc=9073830 |pmid=35443778 |doi=10.1021/acsami.1c23653}}{{Cite journal |last1=Guldin |first1=Stefan |last2=Kolle |first2=Mathias |last3=Stefik |first3=Morgan |last4=Langford |first4=Richard |last5=Eder |first5=Dominik |last6=Wiesner |first6=Ulrich |last7=Steiner |first7=Ullrich |date=2011-07-06 |title=Tunable Mesoporous Bragg Reflectors Based on Block-Copolymer Self-Assembly |journal=Advanced Materials |language=en |volume=23 |issue=32 |pages=3664–3668 |doi=10.1002/adma.201100640 |pmid=21732558 |s2cid=10065931 |issn=0935-9648 |url=http://discovery.ucl.ac.uk/1446958/1/Mesoporous_Bragg_reflectors_AdvMat_final_prod.pdf}}{{Cite journal |last1=Ghazzal |first1=Mohamed N. |last2=Deparis |first2=Olivier |last3=De Coninck |first3=Joel |last4=Gaigneaux |first4=Eric M. |date=2013 |title=Tailored refractive index of inorganic mesoporous mixed-oxide Bragg stacks with bio-inspired hygrochromic optical properties |journal=Journal of Materials Chemistry C |language=en |volume=1 |issue=39 |pages=6202 |doi=10.1039/c3tc31178c |issn=2050-7526}} or polymers,{{Cite journal |last1=Lova |first1=Paola |last2=Manfredi |first2=Giovanni |last3=Boarino |first3=Luca |last4=Comite |first4=Antonio |last5=Laus |first5=Michele |last6=Patrini |first6=Maddalena |last7=Marabelli |first7=Franco |last8=Soci |first8=Cesare |last9=Comoretto |first9=Davide |date=2015-03-10 |title=Polymer Distributed Bragg Reflectors for Vapor Sensing |journal=ACS Photonics |language=EN |volume=2 |issue=4 |pages=537–543 |doi=10.1021/ph500461w |issn=2330-4022|hdl=11696/32604 |hdl-access=free }} these devices can be used as low-cost vapor/solvents sensors.{{Cite journal |last1=Wang |first1=Hui |last2=Zhang |first2=Ke-Qin |last3=Wang |first3=Hui |last4=Zhang |first4=Ke-Qin |date=2013-03-28 |title=Photonic Crystal Structures with Tunable Structure Color as Colorimetric Sensors |journal=Sensors |language=en |volume=13 |issue=4 |pages=4192–4213 |doi=10.3390/s130404192 |pmc=3673079 |pmid=23539027 |doi-access=free}} For example, colour of this porous multi-layered structures will change when the matter filling up the pores is replaced by another, e.g. replacing air with water.

• {{annotated link|Bragg's law}}
• {{annotated link|Bragg diffraction}}
• {{annotated link|Diffraction}}
• {{annotated link|Diffraction grating}}
• {{annotated link|Dielectric mirror}}
• {{annotated link|Fabry–Pérot interferometer}}
• {{annotated link|Fiber Bragg grating}}
• {{annotated link|Photonic-crystal fiber}}
• {{annotated link|Vertical-cavity surface-emitting laser}}
• Photonic crystal sensor

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