4. Reflection at Normal Incidence
Fig.3 A wave of unit amplitude incident from a
medium of
index n1
on a substrate of index n2 covered by a film of index
n.
The application of Eq.3 to reflection from a film-covered substrate is illustrated in Fig.3. A wave of unit amplitude is incident from the left on the surface of the film and a wave of amplitude r is reflected. A wave of amplitude t is transmitted into the substrate. Note that the z-axis origin is placed at the film surface, and that the symbol t is used both for time and for the transmitted amplitude. Time is eliminated from the equation when the common factor exp(iwt) is cancelled, and Eq.3 becomes:
(6a)
The equation can be simplified by noting that k is 2p n/l 0, and the common factor can be cancelled except in the product kd for which the symbol b will be used
In addition, a number of minus signs can be eliminated by bringing the i's into the matrix, and writing the equation
(6b)
The two forms of Eq.6 are completely equivalent. Either can be solved for the complex numbers r and t that give the amplitude and phase of the reflected and transmitted waves. Some special cases illustrate the use of the equation. If the surface is bare,
This is the familiar result that tells us that light has its amplitude reduced by a factor of 5 and its phase changed by 180° when it is reflected from glass. When
and if
and the film acts as an anti-reflection coating. For an arbitrary value of b, both r and t are complex, and a calculator (such as an HP48) that works directly with complex numbers is very useful.
Eq.6 has much broader application than the above examples indicate. Wave propagation through an optically absorbing medium such as a metal can be dealt with by writing the refractive index as a complex number, n - ik . (It is more usual to use n - ik, but k already has one meaning, and will also be used as the unit vector in the z direction.) If we substitute a complex refractive index into Eq.4 and use k0 for 2p/l0,
(7)
Eq.7 describes a wave propagating in the +z direction with exponentially decreasing amplitude. All that is necessary to deal with a metallic substrate or an absorbing film is to substitute a complex value for the refractive index in Eq.6b. Barrier tunneling in quantum mechanics can be dealt with similarly - an imaginary value is substituted for k in Eq.6a.