Quarter-Wave Bragg Mirror Reflectivity Calculator
Laser Thin-Film Optics: Compute multi-layer dielectric high-reflector ($HR$) reflectivity, transmission loss ($1-R$ in ppm), stopband reflection bandwidth, and layer thicknesses.
Dielectric Stack Materials & Wavelength
Reflectivity & Stopband Performance
Quarter-Wave Bragg Dielectric Mirror Principles
Distributed Bragg Reflectors (DBRs) form the backbone of laser resonators, cavity ring-down spectroscopy, and high-energy physics optics.
1. Mathematical Formulations
d_H = λ₀ / (4 · n_H) [Physical thickness of high-index layer] d_L = λ₀ / (4 · n_L) [Physical thickness of low-index layer] R = [ (1 - (n_s/n₀)·(n_L/n_H)^(2N)) / (1 + (n_s/n₀)·(n_L/n_H)^(2N)) ]² Δλ₀ = (4 · λ₀ / π) · arcsin( (n_H - n_L) / (n_H + n_L) )
2. Coating Design Tradeoffs
- Layer Count ($N$): Every additional $4\sim 5$ pairs adds another "9" of reflectivity (e.g., $99.9\% \to 99.99\% \to 99.999\%$).
- Material Selection: $\text{SiO}_2$ ($n=1.45$) is the universal low-index oxide; $\text{Ta}_2\text{O}_5$ ($n=2.10$), $\text{Nb}_2\text{O}_5$ ($n=2.25$), or $\text{HfO}_2$ ($n=1.95$) are standard high-index layers.
Frequently Asked Questions
What is a quarter-wave dielectric Bragg mirror?
A quarter-wave dielectric mirror (also known as a Distributed Bragg Reflector or DBR) is a periodic optical coating composed of alternating high ($n_H$) and low ($n_L$) refractive index thin dielectric films. Each layer has an optical thickness equal to one quarter of the design wavelength ($n \cdot d = \lambda_0 / 4$). Partial Fresnel reflections from each interface interfere constructively in reflection, producing near-perfect reflectivity ($R > 99.999\%$) with virtually zero absorption loss compared to metallic mirrors.
What determines the reflection bandwidth of a dielectric mirror?
The stopband reflection bandwidth ($\Delta\lambda_0$) is governed by the refractive index contrast between the two materials: $\Delta\lambda_0 = \frac{4\lambda_0}{\pi} \arcsin\left(\frac{n_H - n_L}{n_H + n_L}\right)$. A larger index difference (e.g. $\text{TiO}_2/\text{SiO}_2$ or $\text{ZnS}/\text{ThF}_4$) yields a broader high-reflection zone, while low contrast (e.g. $\text{AlGaAs}$ semiconductor stacks) requires dozens of additional layer pairs and yields a narrow reflection band.
Why are dielectric mirrors preferred over silver or gold in high-power lasers?
Metallic mirrors absorb between $0.5\%\sim 3\%$ of incident light, which in high-power continuous-wave (CW) or pulsed industrial lasers causes severe thermal lensing, substrate warping, and laser-induced damage. Low-loss dielectric mirrors made by Ion Beam Sputtering (IBS) achieve transmission losses below $10\,\text{ppm}$ ($R > 99.999\%$) and absorption below $1\,\text{ppm}$, tolerating megawatt-scale circulating cavity powers.