# What force can bend light waves

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A beam of light can make a curve in a vacuum without fanning out, even if the electromagnetic field is not subject to any external influences. This is what researchers in Israel have calculated - and experiments confirm this.

Light normally propagates in a straight line, if one disregards diffraction effects. "Bessel rays", which are generated approximately with ring-shaped apertures, maintain their direction of propagation without being bent and diverging. But there are also other "non-diffractive" rays of light that surprisingly break out of the original direction by themselves, as theoretical studies have shown. They are based on solutions to Maxwell's equations, which do not change their shape and move at an accelerated rate without any forces acting on them.

Fig.: The light beam coming from below bends to the left. In doing so, it "heals" itself, as the dotted line shows: If the beam is initially faded out, it forms again after a short distance. (Image: I. Kaminer et al., PRL)

However, the solutions constructed up to now had to be paraxial, i.e. the angle between the wave vector of the light and the optical axis had to be sufficiently small - and it had to remain so. But now researchers working with Moti Segev have constructed solutions to Maxwell's equations that correspond to non-paraxial monochromatic light rays that can deviate from their original direction by almost 90 degrees.

The key to this unusual behavior of the light rays lies in an observation made by Michael Berry and Nandor Balazs in 1979. They had constructed time-dependent solutions of the one-dimensional, potential-free Schrödinger equation that move without changing their shape and diverging . In addition to plane waves, they also found solutions in the form of an Airy function, which move at an accelerated rate, although no force acts on them.

Berry and Balazs showed that this behavior does not contradict the Ehrenfest theorem, according to which the center of gravity of a force-free wave packet must move without acceleration. The theorem cannot be applied because the airy function is not square-integrable and cannot be interpreted as a probability density for a single, force-free particle. Rather, it represents an infinite number of particles, as is the case with a plane wave.

In the Airy function, these particles are for t= 0 in the form of a parabola x = –p2 distributed over the phase space. Although each of the particles moves without any force, the entire parabola moves with constant acceleration along the xAxis (and consequently at constant speed along the pAxis) without changing its shape. The wave function belonging to this parabola also moves accelerated along the x-Axis.

Not only the Schrödinger equation has such accelerated solutions, but also other linear wave equations, such as the Helmholtz equation, which results from the Maxwell equations that apply in a vacuum. Demetrios Christodoulides and his co-workers had calculated this in 2007 for the one-dimensional case, and they had experimentally proven such accelerated Airy rays.

In two or three dimensions, the accelerated movement of the light waves opens up the possibility of generating light rays that leave their initial direction. However, it turned out that the "Airy trick" fails as soon as a ray deviates noticeably from its initial direction and is no longer paraxial. That is why Moti Segev and his team have chosen a more general approach. You have constructed ray-shaped solutions of the two-dimensional Helmholtz equation, in which the ray makes a curve in the form of a quarter circle. They were able to trace these solutions back to Bessel functions.

The researchers found both transverse electrical (TE) and transverse magnetic (TM) solutions in which the electrical field or the magnetic field is perpendicular to the direction of propagation. They followed the development of the ray by looking for the areas of space in which the electric field was particularly intense. With the TM solution, the direction of polarization changed along the beam. At first the beam pointed in z-Direction and was in x-Direction polarized, then the beam bent so that it went in x-Pointed towards and now in zDirection was polarized. So the beam was not only bent but also twisted.

The designed solutions proved to be stable against disturbances, they even showed "self-healing": If the most intense area of ​​a beam was masked out at the beginning of the curve described by it, the areas of the electromagnetic field surrounding the beam contributed to the fact that the beam stood up again and could continue its curved path.

In the meantime, French researchers working with Francois Courvoisier have generated curved beams with femtosecond laser pulses for the first time. However, these light fields eventually diverged because, strictly speaking, they were not "non-diffractive" rays - in contrast to the rays that Segev and his colleagues had constructed. But the Israeli scientists are confident that these rays can also be realized experimentally.

The curved rays open up fascinating possibilities - and not only for light waves but e.g. B. also for sound waves or surface waves in liquids. Radial areas with a high wave amplitude could be guided around obstacles in a targeted manner. In connection with the non-linear optics, further interesting possibilities would then arise.

Rainer Scharf

OD