How to Build a Maxwell’s Fishpond

Back in 1854, the Scottish physicist James Clerk Maxwell explored the theoretical properties of spherical lenses with the special property that their refractive index changed as a function of the radius. 

One mathematical solution stuck out as being particularly special: a lens in which all rays of light follow circular arcs. In this lens, light from any point in or on the sphere is always focused to another point on the other side of the sphere.   

The amazing promise of such a lens is that it would focus light from all regions of space at the same time, a bit like a fisheye lens. It consequently became known as Maxwell’s fisheye lens.

However, Maxwell’s fisheye lens suffers from such severe aberrations that nobody is sure it can ever work. For example, the lens can only sharply focus light from point sources on its surface and within itself. Any object larger than a point suffers extreme aberrations. 

In recent years, some physicists have suggested that these problems can be fixed using metamaterials to bend light in ways that are not normally possible but the jury is still out on this.

Today, Paul Kinsler and pals at Imperial College London take a different approach to the problem. These guys have built a two dimensional version of Maxwell’s fisheye lens which works with water waves rather than electromagnetic ones. Naturally, they call their device a Maxwell’s fishpond.  

The device is remarkably simply–a shallow dish with a curved bottom that rises towards the middle (see below).

When filled with water, waves on the surface behave in a remarkable way. The depth of the water determines the refractive index of waves on the surface. So the trick here is to design the shape of the bottom of the dish in a way that follows Maxwell’s plan.

That turns out to be pretty straightforward with the result that a circular wave pattern does not simply spread out and disperse. Instead, it always reconverges to a point on the other side of the dish, regardless of where it formed in the first place.  

In fact, Kinsler and co say they can watch waves reform many times. “To the eye, our Maxwell’s Fishpond was capable of reforming a disturbance up to five times, although such a feat required taking considerable care, close observation and a little luck,” they say.

That’s an interesting little aquatic device. Kinsler and co say they made it partly out of friendly competition with other teams struggling to make electromagnetic versions of the lens.   

It’s uses are a little harder to pin down. It certainly shows the power of transformation optics–the ability to bend waves in bizarre ways. The Imperial team built this one as part of a third year undergraduate project. 

But perhaps there’s a bigger market. Maxwell’s fishponds look very cheap to make–the dishes themselves could be manufactured in high volume for peanuts. Which is why the most obvious application of a Maxwell fishpond is as a toy that could provide hours of fun for any child and many adults.  

If these these guys have tied up the intellectual property, they could now have a bit of fun learning about sales and marketing too. If they haven’t, expect to see these toys in the shops and learning centres sooner rather than later.

Ref: arxiv.org/abs/1206.0003: Maxwell’s Fishpond