Thin Films
10.3 Optical Coating Design
Now we’ll consider a few of the simplest optical coating designs in common use.
10.3.1 Single-Layer Antireflection Coating
Consider a single-layer film at normal incidence. The transfer matrix is just the single-layer matrix from Eq. (10.30)
Putting the matrix elements into the expression from Eqs. (10.37) for the film reflection coefficient, we find
r =
10.3 Optical Coating Design 177
and niis the refractive index of the incidence material (say, air), n1is the index of the coating and ns is the index of the substrate (say, glass).
We can then calculate the reflectance (intensity reflection coefficient) of the film as
R = |r|2= n12(ni− ns)2cos2δ + (nins− n12)2sin2δ n12(ni+ ns)2cos2δ + (nins+ n12)2sin2δ.
(reflectance, single-layer film) (10.43) This is still a bit messy, but we’ll now make it simpler.
An important but simple case is the quarter-wave film (still at normal incidence). The thickness is
d =λ 4 = λ0
4n1, (10.44)
where it is important to note that the layer is a quarter wavelength for the wavelength inside the medium.
This thickness leads to a phase shift of
δ = 2π
(reflectance, single quarter-wave film) (10.46) Notice that the reflectance vanishes if the film index is chosen such that
n1=√
nins. (10.47)
(single-layer antireflection condition) As an aside, let’s quickly compare this to the Fabry–Perot etalon discussion from the first part of this chapter.
The reflectance vanishes even though there is only λ/2 of propagation over one full round trip. Normally, a Fabry–Perot cavity requires an integer multiple of λ of propagation distance per round trip, but now we have to be careful about phase changes on reflection. For example, if ni < n1, then the antireflection condition (10.47) implies that n1 < ns (and n1 > ns in the case that ni > n1). Thus, exactly one of the reflections inside the film will produce a π phase shift, playing the role of an extra λ/2 of propagation distance. This is different from the etalon that we considered early on in this chapter, which was for the case ni = ns, which requires a thickness of λ/2 for R to vanish.
Let’s now consider the example of a real, single-layer antireflection (AR) coating. For an air–glass interface, we have ni= 1 and ns= 1.52 for optical crown glass. Ideally, n1= 1.23 for an AR coating (where R vanishes). The closest material suitable for making an optical coating (with good optical properties and durability) is MgF2, which has n = 1.38 (for visible wavelengths). Putting this into the formula (10.46), we find R = 1.3%, which is much better than the uncoated reflectance of R = 4.3% but not really that close to zero. The cheapest optical coatings in the visible are single-layer MgF2.
10.3.2 Two-Layer Antireflection Coating
The basic problem with single-layer antireflection coatings is that there is a limited choice of suitable coating materials, and so it is difficult to match the optimum index condition very precisely. One solution to this problem is to add a second layer, in the hope that an extra free parameter will make the choice of materials more flexible.
178 Chapter 10. Thin Films
Let’s assume a stack of two wave films at normal incidence. The matrix for a single quarter-wave layer is
The transfer matrix for the two-layer stack is the product of two of these matrices:
F = F1F2=
Again, putting the matrix elements into the expression from Eqs. (10.37) for the film reflection coefficient, we find allowing us to pick the ratio of two refractive indices rather than a single absolute index.
As an example, again for crown glass (ni= 1 and ns= 1.52), the ideal AR ratio is n2/n1= 1.23. We can choose ZrO2(n2= 2.1) and CeF3(n1= 1.65), which gives n2/n1= 1.27, a value that is pretty close to the ideal value. Working out the reflectance, we find R = 0.1% for this coating (about as good as you can expect given fabrication tolerances). This kind of coating is called a “vee” coating due to the shape of the reflectance plot as a function of wavelength: there is a relatively narrow range (say, 10 nm or so) of good antireflection for a coating in the visible. In the example plotted here, the design wavelength is λ0= 532 nm, which is marked in the plot, and the uncoated air–glass reflectance is also marked in the plot.
l0
350 400 500 532 600 700 750
reflectance
0.17
0 0.043
10.3 Optical Coating Design 179
It is possible to make a more broadband AR coating by adding a third layer, but we won’t go into that here.
10.3.3 High Reflector: Quarter-Wave Stack
From the analysis of the two-layer AR coating, we note that if the ratio n2/n1is very different frompns/ni, then the reflectance is relatively large. Thus, the idea behind making a high reflectance (HR) coating is to essentially reverse the order of the layers in the antireflection coating: n2must have a larger index than n1
for good reflection.
As an example, let’s take the same materials as for the AR coating [ZrO2 (n2 = 2.1) and CeF3
(n1= 1.65)], but reverse the order of the layers. In this case, the reflectance turns out to be R = 18%. This is a much higher reflectance than the AR coating, although it is not as high as, say, a good metallic coating.
Of course, it is possible to get higher reflectances by choosing other materials with a greater index contrast. But these improvements will be marginal. How can we get a reflectance approaching unity? The general idea is to stack many of these double layers to form a quarter-wave stack.
Consider N double layers. The transfer matrix for one of the double layers at normal incidence is
F =
Thus, the transfer matrix for the whole stack is
FN =
(reflectance of dielectric stack, normal incidence) (10.56) For the same materials as above, ni/ns= 1/1.52 and n2/n1= 1/1.27, so for
Thus, we see that as the number of double layers increases, the reflectance rapidly (exponentially) converges to unity. We could use fewer layers with a better choice of materials (higher index contrast), although we should note that the intermediate reflectances shown here are also useful as beam splitters. The periodic structure of these quarter-wave stacks gives a simple example of a photonic crystal or Bragg reflector.
180 Chapter 10. Thin Films