Schroeder diffuser using a metasurface: new idea?

Posted on June 1, 2017 by Trevor Cox

There is great interest in acoustic metamaterials at the moment – you may have seen papers about sonic invisibility cloaks or extremely efficient absorbers. There is a paper on the preprint archive arxiv.org[1] that makes an acoustic diffuser from a metasurface. But I’m not sure it is as new as it sounds.

 What is an acoustic diffuser?
These are used to treat acoustic defects in performance spaces and studios that can arise from large flat surfaces. When a diffuser is applied to a large flat wall it is a bit like applying a frosting to a mirror. The visual image in a frosted mirror is blurred, similarly the acoustic image when sound reflects off a diffuser is less distinct. This can help with sonic aberrations such as echoes [2].

The catalyst for much research in this area was Schroeder’s revolutionary diffusers from the 1970s – an example is shown in Figure 1. When sound reflects from the surface, different parts of the wave are delayed by different amounts depending on the depth of each well. This breaks up the reflected wavefront creating dispersion. As the delays relate to changing the phase of the wave, these are often called phase grating diffusers.

Is this a metasurface diffuser? Metamaterials are usually made up from a large number of small repeating elements, and that is certainly true. Metamaterials also should give properties that are not found in nature. As the depths are based on mathematic sequences, this is also true. (See also this blog on what is an optical metasurface). So Schroeder invented a metamaterial, several decades before people started to use the term!

Figure 1, Omni diffuser from RPG with 64 wells

Ultra-thin
Figure 2 shows the proposed new diffuser from the paper on arxiv.org. To impose phase changes on the reflected waves a series of resonators are formed. These are Helmholtz resonators with a constriction at the entrance of width w opening up to a cavity behind of width D. A clever idea but is it new?

Figure 2, Metasurface diffuser [2]

Here is a quote from my and Peter D’Antonio’s book on diffusers[2]:

‘An alternative regime to gain more low frequency performance, is to use perforated sheets to add mass to the impedance of the wells, to lower the resonant frequencies and hence lower the design frequency. An example of such a device is shown in Figure 10.30, where the use of perforated sheets has enabled the longest wells to be shortened.’

Figure 10.30 from reference [1]

A perforated sheet over a cavity is a Helmholtz resonator – the same thing as found in the new metasurface diffuser. There is a constriction and a cavity behind. Peter and I shouldn’t take credit for this idea, however, because it was first tried by Hunecke in 1997 [3]. So the idea of forming a diffuser from Helmholtz resonators instead of 1/4 wavelength wells should be credited to Hunecke. He should also be credited with the design method of matching the phase of the resonator to that of the normal diffuser wells (as is done in the paper on arxiv.org).

So why has no-one taken up Hunecke’s idea before? The thesis being in German is one factor, but then Peter and I wrote a description in English in 2004. The real problem is that the performance at high frequency suffer if every well is replaced by a Helmholtz resonator. When the wavelength gets small compared to the spacing between the holes in Figure 2, then you get a strong reflection from the flat front surface. You can see the decreased performance at the highest frequencies tested in the paper on arxiv.org (see Figure 5 for example). I can think of no application where it would be good for a diffuser to stop working at high frequency, because that is the bandwidth where perceptual aberrations are most prominent. That is why when we sketched Figure 10.30 we kept some of the wells as conventional 1/4 wave resonators.

You might be looking at Figure 2 and Figure 10.30 and thinking they look quite different even if they are both Helmholtz resonators. So let’s look at another design first published in a patent in 2006 and available to buy as a product for a decade. Figure 3 shows a surface that incorporates an inverted T-shaped well in the middle of the diffuser. This middle well is very similar to the cross-section of one of the metasurface elements shown in Figure 2. Again, the reason for only using one T-shaped well per period is to maintain high frequency performance.

Figure 3, RPG Flutterrfree-T

Epilogue
I note from one of the author’s publication lists that the paper has now been accepted for publication in Physical Review X [4]. The quality of the modelling and measurements in the paper is very good, but the basic ideas have been done before. What do you think? Am I being unfair?

Notes

[1] https://arxiv.org/abs/1701.08908

[2] Cox, T.J. and D’Antonio, P., 2016. Acoustic absorbers and diffusers: theory, design and application. CRC Press.

[3] J. Hunecke, “Schallstreuung und Schallabsorption von Oberflächen aus Mikroperforierten Streifen”, PhD thesis, University of Stuttgart (1997).

[4] Yifan Zhu, Xudong Fan, Bin Liang*, Jianchun Cheng* and Yun Jing*, Ultra-thin acoustic metasurface-based Schroeder diffuser, Physical Review X, accepted, 2017