A Ti:sapphire cavity mirror at nm is coated as

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Interference and diffraction · Interference and diffraction

What an Instrument Can See, Ch Multiple beams: the stack as a matrix and the etalon as a filter

Problem

A Ti:sapphire cavity mirror at

\lambda=800

nm is coated as

(\mathrm{HL})^8\mathrm{H}

, eight pairs plus one extra high-index layer, on a fused-silica substrate (

n_s=1.46

) from air (

n_0=1

), using silica itself as the low-index layer (

n_L=1.46

). The brief specifies

\mathcal R\ge0.999

. ((a)) The stock high-index material is hafnia,

n_H=2.05

. Using the closed-form admittance

Y=\dfrac{n_H^2}{n_0n_s}\left(\dfrac{n_H}{n_L}\right)^{2N}

, find the reflectance eight pairs of hafnia and silica actually deliver, and show it falls short. ((b)) Because the substrate and the low-index layer here share the same index,

n_s=n_L

, the admittance collapses to a single power of

n_H

alone. Use that simplification to find the minimum

n_H

, with

n_L=1.46

and

N=8

held fixed, that would meet the

0.999

specification with a higher-index coating material.

Related theory

Interference and diffraction

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