#### Diffraction patterns and resolution

In the previous post we traced the passage of light through a roof prism, which required decomposing vectors into components, accounting for phase shifts upon reflection, and then looking at how the waves will be recombined. Nothing was quantified from any of that, because frankly, the calculations are difficult and they need to involve a whole range of polarizations and incoming phases. But we can at least now see that there will be ample opportunity for interference effects to degrade the performance of an optical system containing the prism, producing an image that is dimmer and less sharp.

Even without optical elements, the passage of light is affected by phase differences and the resulting interference. These are of course the well-known *diffraction* effects, intrinsic to all waves and which cannot be eliminated. Diffraction stems from the fact that any source of waves must in fact produce an ensemble of wavelets that interfere with one another as they expand out. This is illustrated in figure 1(a), which shows waves emanating from an aperture. Here we show just two sets of wavelets arising from the source, each of which spreads out in a uniform circular pattern. When we look at different locations beyond the aperture, we see that these wavelets generally travel different distances, meaning that they will arrive with different phases. We can map out the intensity of the light by projecting it onto a screen, and thereby capture all this in terms of a diffraction pattern, as shown in figure 1(b). The light will form a bright central spot that is wider than the aperture itself, surrounded by lightless areas as well as much fainter regions. This pattern represents the fundamental limit to resolution, because no detail smaller than the width of the main spot can be resolved.

When the light from such a narrow aperture passes through an optical system such as a telescope, the very best that we can hope for is that the instrument doesn’t degrade (that is, broaden) the pattern further, in which case the instrument is said to be *diffraction limited*. Our goal now is to examine the effect of placing a roof prism between a light source and the projected diffraction pattern.

For an aperture, it is convenient to use a very thin slit illuminated from behind. From a great distance, it might appear immeasurably thin, but if we were to magnify it, we’d find an image that consists of a central line with some finite width, surrounded by fainter bands (it will appear like figure 1(b) but stretched into a second dimension). Placing a simple Amici roof prism in front of the source will direct the light downward, where we will project it upon a screen, as shown in figure 2. There are two orientations to consider, as shown by diagrams (a) and (b), with the slit parallel or perpendicular to the roof line orientation, respectively. We will focus on the parallel case, and later discuss why the perpendicular one has no effect.

With this orientation, half of the light from the slit takes one path through the prism, and the remaining light goes the other route. In the previous post we focused on two parallel rays that arrive at the prism with the same phase. There is a lot more to manage now, as we must consider the interference corresponding to all the wavelets emanating from the slit. We must account for the phase shifts due to the different wavelet path lengths *and* the shifts due to the prism surfaces. And that is just for one polarization; we will need to do all of this for all incoming electric field orientations and then add up the results. (See section 3 of the technical article).

It should not be surprising that when we combine the two sources of phase shifts – those native to the diffraction pattern and those stemming from the prism TIR – that the destructive interference effects are only going to get worse. It would be incredibly fortuitous if one set of shifts somehow undid the other. After all, there is only one way to add waves to get constructive interference, but many ways to get some amount of amplitude loss. What we find, when the calculation dust settles, is that the net result of the prism is the introduction of a *modulating function* (denoted in the technical article) which multiplies the normal diffraction pattern and distorts its shape, depending on the relative s- and p-component phase shifts. This is animated in figure 3. On the top left we show the p- and s-components changing in relative phase, and the top right shows how the modulating function responds. The figures on the bottom show the interference pattern, both in terms of the image itself (bottom left) and the intensity versus position (right).

The point of these animations is to depict not only the effect of the phase shifts on resolution, but to show what would happen if we could “dial in” any amount of relative phase shift between the p- and s-components. When these are in phase, the modulating function becomes highly extended, but when the components are out-of-phase, it becomes narrow, with a low maximum and high minimum. Multiplying this against the nominal diffraction pattern pushes the peak down while bringing the off-peak portions up, which has a net effect of broadening the peak while causing it to lose about 40% of its amplitude as a worst case scenario. The resulting appearance of the slit, as seen in the bottom left, noticeable degrades. But if we can bring the phase shifts together, we can recover the normal diffraction pattern.

A readable account of how early workers, who were using roof prism for military application during the Second World War, came to understand the cause of these effects, is given by A.I. Mahan (“Some Newly Solved and Some Unsolved Problems in Optics.” *Journal of the Washington Academy of Sciences* 44, no. 6 (1954): 165–94, which can be read online via a free JSTOR account, here. Since they had no recourse to computer animations then, the author constructed a model using strings and matchsticks, which is shown in a photograph). There were uses for roof prisms which had angles other than 90 degrees between the two reflecting faces, and Mahan describes how for some of them, the effect of phase shift was much worse than what an uncoated binocular prism would produce; some degraded diffraction patterns even appeared as two parallel lines, causing true “image doubling.”

Everything so far has been for the parallel case – what happens if the slit is rotated by 90 degrees, for the perpendicular orientation shown in figure 2(b)? We need not do any calculations, because we can appeal to symmetry considerations, The light leaving any one portion of the slit is no longer confined to pass through only one of the two paths in the prism. Instead it gets channeled equally through both, so there is no way that any net difference in phase can be acquired. In the Mahan paper, this result was made clear by viewing cross-shaped targets, which produced images of a narrow line bisected by a widened, blurrier one.

#### How coatings can mitigate the problem

Now that we’ve seen how phase shifts collude with the roof prism geometry to cause a degraded image, we turn to the matter of how to fix it. The goal is to somehow eliminate the *difference* in the p- and s-polarization phase shifts without ruining the TIR. We will only consider what we might accomplish by adding one or more dielectric, or insulating, materials between the glass and air. A metal coating would solve the phase problem, but the TIR is then lost.

We will roughly follow the approach of Mauer (“Phase Compensation of Total Internal Reflection,” J. Opt. Soc. Am. 56, 1219-1221 (1966)), who demonstrated that by using layers with carefully chosen refractive indices and thicknesses, it is possible to alter the phase difference adequately, at least for a range of incident angles and wavelengths. This was an early effort, and modern coatings are certainly more sophisticated. That being said, even Mauer’s three layer approach is far from trivial. Since we are eschewing the mathematics here, we’ll aim for a descriptive overview of what Mauer was trying to do.

#### A single dielectric layer

The simplest way to modify the prism is to add a single, uniform layer (we use terms such as “layer” and “coating” interchangeably) having a refractive index that differs from that of the prism. If the index is less than that of the glass (we will continue to use 1.517 as a nominal value for the index of glass), there will be some range of angles at which we have TIR at the glass/coating interface, and a different range for which it occurs at the coating/air boundary. The transmission angle will be larger than the incidence angle, leading to a reflection at the coating/air interface that is also at a larger angle. As we saw in the first post, the more that an angle tends towards the “glancing” case, the smaller the difference in the p- and s-component phase shifts.

If we intend to leverage this, we will need to address with the fact that not all of the light will make it to the coating/air boundary; there will first be some partial reflection at the glass/coating interface, with the remainder of the light transmitted into the layer and on towards TIR. The situation is significantly more complicated now. The light is again being split into different paths, but here, the paths involve different distances. This means that various different phase offsets will result, with potential interference effects when all the light is added back together. The only knob we will have for tuning such effects, besides the refractive index of the layer, is the layer thickness. But the shifts will also depend on exactly how many wavelengths fit into the path length, so we will be introducing chromatic effects that we didn’t have to deal with before.

It is a promising strategy, but with consequences that need to be addressed. The wavelength dependence is a big one, as is the limited range of angles over which the phase offset can be kept sufficiently low. In the Mauer paper, the aim was to show that in using three layers instead of one, there would be greater flexibility in finding an adequately large solution space.

#### A multiple layer coating

Figure 4(a) illustrates Mauer’s three layer approach. Note that there are multiple reflections which must be accounted for (there are secondary reflections and beyond as well, which would be difficult to track “by hand” but which the transfer matrix method used in the technical article handles easily). Diagram (b) shows the resulting s- (red) and -p-polarization (blue) phase shifts. There is a fairly broad range of thicknesses for which the phase shifts are very nearly equal, pointing to a potential design space. These curves replicate those in figure 2 of the Mauer paper.

Figure 4 (a) Schematic of the three layer phase coating proposed by Mauer. Diagram (b) shows the final phases for the reflected light having p- (blue) and s-polarizations (red). There is a range of thicknesses over which the phase offset can be tuned very close to zero.

#### Conclusion

Our goal here was to try to make the origin of the roof prism problem a bit more intuitive, and to show what a phase coating strategy can look like. We note that Mauer’s approach was an early effort, and the question of how to optimize from this basic design, and better account for wavelength dependence and so on, lies beyond the scope of this introduction (and would seem to reside in the world of trade secrets). There seems to be a paucity of published information on modern phase coatings. This is expected, if inconvenient, since optics manufacturers would be understandably loathe to share proprietary information in such a highly competitive market.

I solicit any and all comments, complaints, and questions via hurbenm at gmail.com. I am especially interested in any technical content pertaining to modern roof prism phase coatings, as I have found very little detail on this.

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