A one-dimensional dielectric mirror is built from alternating non-absorbing layers of…

PHYSICS · Subjective · Fermi University Physics · Optics & Photonics · Wave propagation and light in media · Interference and diffraction

What an Instrument Can See, Ch Media that are designed: plasmons, metamaterials and photonic crystals

Problem

A one-dimensional dielectric mirror is built from alternating non-absorbing layers of index

n_H

and

n_L

, each exactly a quarter-wave thick at a design angular frequency

\omega_0

(centre wavelength

\lambda_0

), lit at normal incidence. Each layer has characteristic matrix

\mathbf M_j(\delta_j)=\begin{pmatrix}\cos\delta_j & i\sin\delta_j/n_j\\ in_j\sin\delta_j & \cos\delta_j\end{pmatrix}

with

\delta_j

its phase thickness. (a) Show that at a general frequency

\omega

near

\omega_0

the phase thickness of both layers is the same,

\delta_H=\delta_L=\delta=(\pi/2)(\omega/\omega_0)

, and that the period matrix

\mathbf M_\Lambda=\mathbf M_H(\delta)\mathbf M_L(\delta)

has half-trace

\Delta(\delta)=1-\sin^2\delta\left[1+\tfrac12\left(\frac{n_H}{n_L}+\frac{n_L}{n_H}\right)\right].

(b) Using the fact that a genuine Bloch wave exists only while

|\Delta|\le1

, show that the fractional width of the stop band nearest

\omega_0

\frac{\Delta\omega}{\omega_0}=\frac{4}{\pi}\arcsin\left(\frac{n_H-n_L}{n_H+n_L}\right).

n_H=2.40

) and magnesium fluoride (

n_L=1.38

) quarter-wave layers is designed for a centre wavelength

\lambda_0=550\ \mathrm{nm}

. Find the wavelengths of the two edges of the stop band.