New results on inverse scattering from planar screens

Scientists at Tallinn University of Technology Prof. L. Päivärinta and S. Sadique together with their colleagues Prof. P. Ola at University of Helsinki and Assoc. Prof. E. Blåsten at Lappeenranta University of Technology investigate mathematical problems related to electromagnetic scattering from singular objects.

Recently P. Ola, L. Päivärinta and S. Sadique showed that the far–field pattern of a scattered electromagnetic field corresponding to a single incoming plane–wave uniquely determines a bounded super– conductive planar screen. This kind of research belongs to the field go Inverse Problems in applied mathematics. Although the demonstration is purely mathematical and completely rigorous the roots of the problem itself lie deeply in physics. It all started with a competition initiated by the Prussian Academy in 1879 to demonstrate the existence or non-existence of electromagnetic waves. James Clerk Maxwell’s predicted the existence of such waves by his mathematical theory 15 years earlier. In 1882, the competition was won by Heinrich Hertz, who used a dipole antenna to radiate and measure EM waves, thereby confirming Maxwell’s theory. This experiment, along with Maxwell’s theory, has had a profound impact on modern society.

In the mentioned study the scattering of waves from thin and large objects is rooted in the field of antenna theory. Recently, there has been a growing interest in solving the inverse scattering problem with reduced measurements. Inverse scattering problems involves determining the properties of an object, such as its shape, size, and composition, from the scattered field produced when it is illuminated with electromagnetic waves. One way to determine the shape of an object is by analysing the pattern of waves that are scattered by the object in the far field. 

Our scientists proved that single incident wave determines uniquely the exact shape of the screen and also the plane where it is supported. The result is contained in an article “Unique Determination of a Planar Screen in Electromagnetic Inverse Scattering” that will appear in SIAM Journal on Mathematical Analysis.