Chapter 3 Laser coating
3.1 Dielectric thin films …
3.1.1 Fundamentals …
The discussion of optical thin films is largely the discussion of interference effects caused by partial reflections at boundaries between films and recombination of the resulting beams in which their phase differences are significant [64]. In this section, we will bring up a framework of conventions and definitions about these effects.
We deal with electromagnetic radiation in the form of plane, monochromatic waves of the simplest kind, where the electric vector, magnetic vector and direction of propagation form, in that order, a right-handed set.
There are two important constants of optical media, the refractive index, n, and the characteristic optical admittance, Y. Both are defined with respect to a harmonic wave propagating through the medium.
(
3.1.1)
The high frequencies of the optical region permits a simplification,
(
3.1.3)
n Y= η
where η (1/377 siemens) is the admittance of free space. By changing the units of characteristic admittance to free space units, we can use the same number for both n and Y. But at microwave frequencies, this simplification, which is due to the absence of direct magnetic influence on the electrons, is no longer possible.
We write the expressions for the waves in the complex form because this affords the simplest technique for the manipulation of phase shifts. The scalar form of the expression for such a wave, propagating along the positive direction of the z-axis (or say d direction), is,
( )
[
2 nd] (
3.1.4)
exp i ωt − π λ E
nd is known as the optical path as distinct from the physical length d. E is the complex amplitude and can also be written as E exp
( )
iϕ where ϕ is the relative phase of the wave. The part,exp[
i(
ωt−2πnd λ) ]
, of the wave expression is known as the phase factor. E can actually represent either magnetic or electric field amplitude but, since in the optical region only the electric vector produces significant effects on a medium, the amplitude of the wave is normally considered to be the electric amplitude. H , the magnetic amplitude can be derived by relationship (3.1.2),E H =Y .
Absorption in a medium can be included by permitting n, and Y, to become complex. Then n is replaced by n i− k where k is known as the extinction coefficient, and Y by
(
n i− k)
η, or(
n i− k)
free space units.(
n i− k)
is known as the complex index of refraction. In thin film optics we mainly deal either with materials where k is very small compared with n, the dielectric materials, or where k is large compared with n, the metallic materials. Dielectric materials are transparent while metals are opaque, unless very thin.The intensity of the plane wave, that is the mean rate of flow of energy carried
Usually it is sufficient to simplify this asI ∝nE2. This expression permits us to derive the relationship between absorption coefficient and extinction coefficient as
(
3.1.6)
k 4π λ α =
Finally, we need to set up a sign convention for reflection at an interface. We know, by experience, that the effect on an incident wave, of an interface between two media, is to split the wave into two parts. One part is transmitted into the emergent medium while the other is reflected in the incident medium. For the moment we consider normal incidence only. The various waves can be described by expressions of the form, relative phases. Comparison of the relative phases is difficult, because one of the phase factors has a different form from the two others. We therefore fix the coordinate system so that the interface is the origin. Then d (along z-axis) is zero at the interface.
There the phase factors reduce to exp
( )
iωt and the relative amplitudes and phases can be compared by comparing the complex amplitudes, E. To avoid ambiguities we must first define positive directions for the electric fields. The usual convention is shown in Fig. 3.2. We choose the simplest possible convention for the electric fields but, because the vectors form a right-handed set, the implied convention for themagnetic vectors is less simple. Figure 3.2 also contains the notation for the indices of refraction and the admittances of the two media.
Figure 3.2. Convention for positive directions of E and H for incident, reflected, and transmitted waves*.
Given this convention, we define the amplitude reflection coefficient as
(
Er Ei) (
3.1.10)
ρ=
and the amplitude transmittance coefficient as
(
Et Ei) (
3.1.11)
τ =
The corresponding ratios of intensities are known as reflectance and transmittance respectively and given by
(
3.1.12)
It can readily be derived that
(
3.1.13)
These expressions, for a single interface, are derived from the boundary conditions that the total electric and magnetic fields are continuous across a boundary.
n
0n
1Y
0Y
1In this respect, Y1 can be considered as the admittance of the emergent medium or as the ratio of the total magnetic field to the total electric field at the interface. In the first case it is the characteristic admittance of a medium, in the second case it is the admittance of the interface. This concept of the admittance of an interface is a particularly powerful one in optical thin film work. Take any optical thin film multilayer system. If the optical admittance presented to the incident medium is γ , then the amplitude reflection coefficient of the system will be given by
(
3.1.14)
and the reflectance by
(
3.1.15)
ρ 2
= R
The calculation of the reflecting properties of the multilayer then reduces to the change produced by the multilayer in the admittance of the system. The multilayer can be considered as an admittance transformer. In fact, it is extremely productive to consider first the admittance of the substrate and its transformation by each layer in turn. Transmittance, unfortunately, cannot be dealt with a simply, but in dielectric systems it is just the complement of reflectance [65-68].