Lambda Physics

Optical Coatings Research Laboratory

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Bragg Mirror Calculator

Design and analyze an ideal quarter-wave dielectric mirror (Bragg mirror / distributed Bragg reflector) at normal incidence. Enter the refractive indices, design wavelength and number of layer pairs to calculate reflectance, transmittance, layer thicknesses, total coating thickness and the spectral response.

Mirror parameters

Optical media
Bragg stack
Target and spectral graph

Bragg mirror calculation results

Reflectance at λB
–
%
Transmittance at λB
–
%
Minimum pairs for target
–
pairs
Index contrast nH/nL
–
ratio
High-index layer dH
–
nm
Low-index layer dL
–
nm
Total coating thickness
–
nm
Total number of layers
–
layers
Bare substrate R
–
%
Reflection increase
–
percentage points
Approx. lower band edge
–
nm
Approx. upper band edge
–
nm
Stack: –
Model: ideal quarter-wave, lossless, isotropic dielectric layers at normal incidence. Each H and L layer has optical thickness λB/4. The displayed spectrum is calculated with the full characteristic-matrix method for the finite stack. Refractive indices are treated as constant over the displayed spectral range; real coatings may require dispersion n(λ), absorption k(λ), thickness errors, substrate absorption and oblique-incidence effects.
Calculation method
δj = 2π njdj / λ
Mj = [[cosδ, i·sinδ/n], [i·n·sinδ, cosδ]]
M = M1 M2 ··· M2N
Y = (C + Dns) / (A + Bns)
r = (n0 − Y) / (n0 + Y)
R = |r|²
T = (ns/n0) |t|² = 1 − R   (lossless case)

Quarter-wave thicknesses: dH = λB/(4nH),   dL = λB/(4nL)
At λB for an H/L-start stack: Y = ns(nH/nL)2N

Bragg mirror and quarter-wave dielectric mirror calculation

This free Bragg mirror calculator models a finite stack of alternating high- and low-refractive-index dielectric layers. The standard quarter-wave design makes each layer an optical quarter wavelength thick at the selected Bragg wavelength, producing constructive interference of the reflected waves.

Enter the refractive indices of the high-index material, low-index material and substrate, then choose the design wavelength and number of layer pairs. The calculator reports the layer thicknesses, total coating thickness, design-wavelength reflectance and transmittance, and plots the complete spectral response.

Typical applications

  • Estimate the reflectance of dielectric laser mirrors and Bragg reflectors.
  • Compare the effect of refractive-index contrast and number of layer pairs.
  • Calculate quarter-wave layer thicknesses for a selected wavelength.
  • Visualize the reflection band of a multilayer optical coating.
  • Estimate the number of pairs needed to reach a selected target reflectance.

Quarter-wave Bragg mirror reference

A Bragg mirror, also called a distributed Bragg reflector (DBR) or quarter-wave mirror, consists of alternating layers with different refractive indices. For a selected wavelength and number of pairs, the quarter-wave condition is the standard ideal high-reflectance design at normal incidence.

The reflection bandwidth depends strongly on the refractive-index contrast. Higher contrast generally produces a broader high-reflectance band and reduces the number of pairs needed for a specified reflectance.

Lambda Physics works with optical coatings, dielectric mirrors and optical components. Visit the General Stock catalog for available optical components, mirrors, beam splitters, filters, prisms, lenses and coating materials.