Multiple Parallel Interfaces
6.8 Multilayer Coatings
In this section, we generalize our previous analysis of a double interface to an arbitrary number of parallel interfaces (i.e. multilayer coatings). As we saw in section 6.3, a single coating applied to an optical surface is often insufficient to accomplish the desired effect, especially if the goal is to make a highly reflective mirror. For example, if we want to make a mirror surface using a dielectric coating (with the advantage of being less fragile and more reflective than a metal coating), a single layer is insufficient to reflect the majority of the light, even if a relatively high index is used. In P 6.3 we compute that a single dielectric layer deposited on glass can reflect at most about 46% of the light. We would like to do much better (e.g. >99%), and this can be accomplished with multilayer dielectric coatings which can have considerably better reflectivities than metal surfaces such as silver.
We now proceed to develop the formalism of the general multi-boundary problem.
Rather than incorporate the single-interface Fresnel coefficients into the problem as we did in section 6.2, we return to the basic boundary conditions for the electric and magnetic fields at each interface between the layers.
We examine p-polarized light incident on an arbitrary multilayer coating (all interfaces parallel to each other). We leave it as an exercise to re-derive the formalism for s-polarized light (see P 6.11). The upcoming derivation is valid also for complex refractive indices, although our notation suggests real indices. The ability to deal with complex indices is very important if, for example, we want to make mirror coatings work in the extreme ultraviolet wavelength range where virtually every material is absorptive. Consider the diagram of a multilayer coating in Fig. 6.14 for which the angle of light propagation in each region may be computed from Snell’s law:
n0sin θ0 = n1sin θ1 = · · · = nNsin θN = nN +1sin θN +1 (6.53) where N denotes the number of layers in the coating. The subscript 0 represents the initial medium outside of the multilayer, and the subscript N + 1 represents the final material, or the substrate on which the layers are deposited.
6.8 Multilayer Coatings 151
Figure 6.14 Light propagation through multiple layers.
In each layer, only two plane waves exist, each of which is composed of light arising from the many possible bounces from various layer interfaces. The subscript i indicates plane wave fields in individual layers that travel roughly in the incident direction, and the subscript r indicates plane wave fields that travel roughly in the reflected direction. In the final region, there is only one plane wave traveling with a forward or transmitted direction.
We will re-label it as Et(p)
N +1 ≡ Ei(p)
N +1 since it is the overall transmitted field.
As we have studied in chapter 3 (see (3.9) and (3.13)), the boundary conditions for the parallel components of the E field and for the parallel components of the B field lead respectively to
cos θ0 E0(p)
+ E0(p) = cos θ1 E1(p)
+ E1(p)
(6.54) and
n0 E0(p)− E(p)0 = n1 E1(p)− E1(p)
(6.55) These equations are applicable only for p-polarized light. Similar equations give the field connection for s-polarized light (see (3.8) and (3.14)).
We have applied these boundary conditions at the first interface only. Of course there are many more interfaces in the multilayer. For the connection between the jth layer and the next, we may similarly write
cos θj
Ej(p)eikj`jcos θj+ Ej(p)e−ikj`jcos θj
= cos θj+1 Ej+1(p) + Ej+1(p)
(6.56) and
nj
Ej(p)eikj`jcos θj− Ej(p)e−ikj`jcos θj
= nj+1 Ej+1(p) − Ej+1(p)
(6.57) Here we have set the origin within each layer at the left surface. Then when making the connection with the subsequent layer at the right surface, we must specifically take into account the phase kj· (`jˆz) = kj`jcos θj. This corresponds to the phase acquired by the plane wave field in traversing the layer with thickness `j. The right-hand sides of (6.56) and (6.57) need no phase adjustment since the (j + 1)thfield is evaluated on the left side of its layer.
At the final interface, the boundary conditions reduce to These equations are the same as (6.56) and (6.57) when j = N . However, we have written them here explicitly since they are unique in that EN +1(p) ≡ 0.
At this point we are ready to solve (6.54)–(6.59). We would like to eliminate all fields besides E0(p), E0(p), and E(p)N +1. Then we will be able to find the overall reflectance and transmittance of the multilayer coating. In solving (6.54)–(6.59), we must proceed with care, or the algebra can quickly get out of hand. Fortunately, most students have had training in linear algebra, and this is a case where that training pays off.
We first write a general matrix equation that summarizes the mathematics in (6.54)–
(6.59), as follows:
Then we solve (6.60) for the incident fields as follows:
Ej(p) We can use (6.63) to connect the fields in the initial and final layers. If we write (6.63) for the j = 0 case, and then substitute using (6.63) again with j = 1 we find
E0(p)
where we have grouped the matrices related to the j = 1 layer together via M1(p)≡ By repeating this procedure for all N layers, we connect the fields in the initial medium with the final medium as follows:
E0(p)
6.8 Multilayer Coatings 153 where the matrices related to the jth layer are grouped together according to
Mj(p) ≡
The matrix inversion in the first line was performed using (0.44). The symbol Π signifies the product of the matrices with the lowest subscripts on the left:
N
Y
j=1
Mj(p)≡ M1(p)M2(p)· · · MN(p) (6.68)
As a finishing touch, we divide (6.64) by the incident field E0(p) and perform the matrix inversion using (0.44) to obtain
1
In the final matrix after the product in (6.70) we have replaced the entries in the right column with zeros. This is permissable since the column vector that A(p) operates on in (6.69) has a zero in the bottom component. (Having zeros in the matrix can save computation time when calculating with large N .)
Equation (6.69) represents two equations, which must be solved simultaneously to find the ratios E0(p)/E0(p) and EN +1(p) /E0(p). Once the matrix A(p) is computed, this is a relatively
The convenience of this notation lies in the fact that we can deal with an arbitrary number of layers N with varying thickness and index. The essential information for each layer is contained succinctly in its respective 2 × 2 matrix. To find the overall effect of the many layers, we need only multiply the matrices for each layer together to find A, and then we can use (6.71) and (6.72) to compute the reflection and transmission coefficients for the whole system.
The derivation for s-polarized light is similar to the derivation for p-polarized light. The equation corresponding to (6.69) for s-polarized light turns out to be
1
where We can then compute the transmission and reflection coefficients in the same manner that we found the p-components
In general high-reflection coatings are designed with alternating high and low refractive indices. For high reflectivity, each layer should have a quarter-wave thickness. That is, we need
βj = π
2 (high reflector) (6.78)
This amounts to the condition on the thickness of
`j = λvac
4njcos θj
(high reflector) (6.79)
Since the layers alternate high and low indices, at every other boundary there is a phase shift of π upon reflection from the interface. Hence, the quarter wavelength spacing gives maximum reflectivity since the reflected wave in each layer meets the wave in the previous layer in phase. In this situation, the matrix for each layer becomes
Mj(p)=
0 −i cos θj/nj
−inj/ cos θj 0
(high reflector, p-polarized) (6.80) The matrices for a high and a low refractive index layer are multiplied together in the usual manner. Each layer pair takes the form
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