**Propagation of waves through sinusoidally varying permittivity media [4]**A plane electromagnetic wave with an electric field

, propagating through an isotropic and uniform medium, obeys the equation

, where

, (1)

and has a general solution composed of forward- and backward-propagating plane waves

. (2)

This wave impinges on a rugate filter with a sinusoidally varying permittivity

,

,

. (3)

This results in Mathieu's differential equation [1],

, (4)

whose analytic solution is a linear combination of Mathieu

functions [2]

. (5)

The equation in the emerging side is also (1), but with different integration constants

. (6)

If the output medium is semi-infinite to the right, there is no reflected wave, therefore

is equal to zero.

From Maxwell equations, it can be shown that

and

have to be continuous at the input and output boundaries. The integration constants

to

are calculated from these conditions.

The reflected power

is thus defined as

. (7)

Such a filter reflects the forward-propagating wave strongly at Bragg wavelengths (

for

), due to interference of incoming and reflected waves, creating reflection peaks.

, (8)

,

,

and the period in

is

.

,

where

are the eigenvalues, and

has period

:

and

.

The eigenvalues

can be calculated with the following Wolfram Language function:

MathieuCharacteristicExponent[a, b]. (9)

Reflection takes place in the

ranges where

is complex, and transmission where

is real. The forbidden or reflection bands calculated with (9) are the same as those calculated with (7).

, defined by

, is the permittivity of the filter base material as well as the permittivity of the input and output sections.

defines, together with

, the amplitude

of the sinusoidal function of the filter.

(nm) is the period of

, that is,

;

is the wavelength of reflectance peak, according to a quarter wavelength Bragg reflection approximation.

cycles is the number of complete cycles of the sinusoid. Reflectance

increases with the number of cycles.

"reflectance" is reflectance of the filter.

increases with cycles and

.

"transmittance" is transmittance of the filter,

.

"Mathieu characteristic exponent": identifies the forbidden transmission bands, or equivalently the reflective bands. They do not depend on boundary conditions.

Snapshot 1: if

, contrast is low, as is reflectance

Snapshot 2: if

as before, but the number of cycles is increased to 200, reflectivity increases to

Snapshot 3: Reflecting peaks occur at

,

,

, … with decreasing amplitude. Only the first two peaks are shown; they do not depend on specific boundary conditions.

[4] P. Yeh,

*Optical Waves in Layered Media*, New York: John Wiley, 1988.