+44 (0)1424 225228 sales@torrscientific.co.uk
AR Coating Curves

Figure 1: AR coatings are sometimes classified according to the shape of the reflectance curve.

Torr Scientific can design and deposit AR (anti-reflective) coatings on their optical components, including viewports and glass vacuum cells. Uncoated glass reflects about 4% of incident visible light from each face, so losses from a typical window will be around 8% in total. These reflections could pose a hazard in the lab, or cause artefacts in optical systems, as well as reducing light throughput.

AR coatings can sometimes be conveniently classified according to the shape of the curve when reflectance is plotted as a function of wavelength (see Figure 1). If reflectance is minimized for a single wavelength, the curve will typically be V-shaped, so single wavelength coatings are known as VAR. For two wavelengths, there are two dips in the curve like a W, so these are WAR. For a continuous range of wavelengths, they are called BBAR – not for the shape, but for broadband. For a more complex collection of different wavelengths, we just call them SPAR for special. These can sometimes be very challenging to design.

Torr uses thin layers of transparent dielectric materials to fabricate AR coatings. A single face of uncoated glass gives a single reflection. If we deliberately introduce another reflection, we can take advantage of a phenomenon known as destructive interference. Adding a single thin, transparent coating on a face gives two reflections, one from the top surface of the coating and one from the coating-substrate interface. Thinking of light as propagating electromagnetic waves, if we have two reflections from a coated face then we can tune one wave so that its peaks coincide with the troughs of the other, and the troughs of the second reflection coincide with the peaks of the first – their phases are opposed.

Figure 2: The graphic on the left shows how opposing troughs and peaks cancel each other. The graphic on the right shows how two reflected waves from a coated face can be made to interfere destructively in this way. Because R2 has travelled further before reflection, its phase (peak/trough position) is changed relative to R1 such that the troughs and peaks are opposed.

The additive nature of this interaction leads to the reflected peaks and troughs cancelling each other out and so reducing the level of reflection (Figure 2). When the reflectance is decreased, transmittance is increased. The trick to designing the coating is to choose the material and thickness of the dielectric layers so that the peaks and troughs are opposed and cancel out at the wavelengths of interest. It can sometimes (though not always!) be a relatively easy task to tell the coating design software the required performance and leave it to crunch the numbers. However, giving the software a reasonable starting point can increase the chances that an appropriate design can be created. In addition, a level of insight as to how the coating achieves the performance can help the designer to judge whether the design is optimal in terms of materials used and time spent depositing.

A graphical tool that provides such insight is the admittance diagram. This is a two-dimensional plot with real admittance on the horizontal axis and imaginary admittance on the vertical. Onto these are plotted a path through this ‘admittance space’, starting at a point on the real axis that is defined by the optical properties of the uncoated substrate at the wavelength of interest. It is possible to achieve some useful insight into the coating without even worrying about what on Earth real and imaginary admittance are, as long as some basic properties of these plots are known. The path proceeds by a series of clockwise arcs, each joined to the other head‑to‑tail. An example is shown in Figure 3.

Figure 3: An admittance plot for a two-layer design. The starting point at real admittance 1.5 is defined by the properties of the uncoated substrate. The larger the distance from real admittance 1 on the real axis, the higher the reflectance. The design should travel in clockwise arcs and arrive at point (1, 0), jumping from one material locus to another (starting a new layer) as needed.

The path can be switched to the locus of a different material at any point; this represents the start of a new layer. Each arc corresponds to a layer in the coating. The length of each arc defines the thickness of that layer. The first arc (layer) starts at 1.5 on the real axis (according to the optical properties of the uncoated substrate), then proceeds clockwise. At a certain layer thickness the path hops onto a new arc, so beginning a new layer with a new material. The perfect anti-reflective performance is achieved at the point on the real axis where real admittance has a value of 1, so the design needs to ensure that the last arc finishes here.

The underlying paths of these arcs, known as loci, are defined by the optical properties of the coating material at the wavelength of interest. A set of loci has been calculated and plotted in Figure 4 for the hypothetical coating materials of the design of Figure 3. Each material has an infinite number of loci, though for clarity only a selected few are shown. Notice that from the starting point at 1.5 (similar to that for glass and fused silica), there is no available high-index locus that passes through the target at real admittance 1 This is why we need more than one layer to achieve a design with near-zero reflectance. When designing for multiple wavelengths, extra layers may be needed. Notice also that the loci are circular for transparent materials – making a layer thicker and thicker brings you back repeatedly to the same point. If any arc on the plot covers more than 360° then the designer might well infer that this particular layer could be thinner and still achieve the same performance. This can save on deposition time and materials, and also reduce the effects of absorption and the chance of laser damage

Figure 4: a set of selected loci for two arbitrary coating materials with refractive indices 1.4 and 2.2, respectively. For each material, there are an infinite number of loci for any particular wavelength and they do not intersect. The design of Figure 3 is superimposed in bold: notice how the design jumps from one locus to another (starts a new layer) to allow intersection with point (1, 0).

Look again at Figure 4, representing the two-layer design and some underlying loci. There is a second route from the starting point to the target position, just by terminating the first layer in a different position and hopping onto a locus of the other material. Can you find this second route? There is a hint below. If you can find your way from (1.5, 0) to (1, 0) then you have just designed a two-layer AR coating.

Hint: There is only one red locus that passes through (1, 0) – you need to find the place where you can hop onto it at an earlier stage of layer 1 than in the design of Figure 4.