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## Light Propagation and Index of Refraction

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 rate-of-change 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 . 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. 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.   Next: Diffraction phenomena Up: Gratings Previous: Gratings
root 2002-05-23