Next: Diffraction phenomena
Up: Gratings
Previous: Gratings
If one visualizes the material where the light is propagating as
continuous, one can use Maxwell´s Equations to solve the
propagation of waves and this leads to equation 1

(1) 
This equation is in orthogonal Cartesian coordinates and
stands for electric field, for time and
is material conductivity. The last term
is a first temporal order
derivative. The time rateofchange of
generates
a voltage, currents circulate and since the material is resistive,
the part of light energy is converted to thermal energy and so
absorption occurs. If the permittivity is reformulated as a
complex quantity, the expression can be reduced to the
unattenuated wave equation. This leads to a complex index of
refraction , which is tantamount to
absorption [1]. It can be expressed as

(2) 
where
is index of refraction usually defined
as the ratio where is the speed of electromagnetic
waves in vacuum( m/s) and the speed of waves in
the material. The is called imaginary part of refractive
index. Note that both and are real numbers.
As the wave progresses trough the medium, its amplitude is
exponentially attenuated. It can be summarized by following
equation

(3) 
where stands for intensity at the interface of our absorbing
material and is called linear absorbtion
coefficient or attenuation coefficient. This is called
Lambert law of classical optics.^{1} It may be also expressed like this

(4) 
where the is absorbtion index and is
wavelength of light in the vacuum. The absorbtion constant
used in next sections is proportional to linear
absorbtion coefficient . The values of and are
different for many orders of magnitude in various spectral
regions.
The flux density will drop by factor of
after
the wave has propagated a distance , known as skin or
penetration depth. For the material to be transparent the
penetration depth must be large in comparison to its thickness.
The penetration depth for semiconductors and metals is
exceptionally small. For example, copper in the ultraviolet region
( nm) has a miniscule penetration depth, about 0.6
nm, while it is still only about 6 nm at the infrared region
(
m). It explains generally observed opacity of
metals[1]. For semiconductor materials, values are of
course much higher.
Now should be checked how the reflectance is affected by this
complex index of refraction. It can be proven using Fresnel's
equations that if a plane wave incidents normal (i.e.
perpendicular) to a plane surface between vacuum () and
material of complex refractive index the reflectance
will be

(5) 
Just for illustration at nm can be calculated this
set of values for a gallium single crystal ,
and for normal incidence[3].
Next: Diffraction phenomena
Up: Gratings
Previous: Gratings
root
20020523