In 1678, Christiaan Huygens stipulated that ‘…light beams traveling in different and even opposite directions pass though one another without mutual disturbance’1 and in the framework of classical electrodynamics, this superposition principle remains unchallenged for electromagnetic waves interacting in vacuum or inside an extended medium.2 Since the invention of the laser, colossal effort has been focused on the study and development of intense laser sources and nonlinear media for controlling light with light, from the initial search for optical bistability3 to recent quests for all-optical data networking and silicon photonic circuits. However, interactions of light with nanoscale objects provide some leeway for violation of the linear superposition principle. This is possible through the use of coherent interactions, which have been successfully engaged in applications ranging from phased array antennas to the manipulation of light distributions and quantum states of matter.4,5,6,7,8,9,10,11
Consider a thin light-absorbing film of sub-wavelength thickness: the interference of two counter-propagating incident beams A and B on such a film is described by two limiting cases illustrated in Figure 1: in the first, a standing wave is formed with a zero-field node at the position of the absorbing film. As the film is much thinner than the wavelength of the light, its interaction with the electromagnetic field at this minimum is negligible and the absorber will appear to be transparent for both incident waves. On the other hand, if the film is at a standing wave field maximum, an antinode, the interaction is strong and absorption becomes very efficient. Altering the phase or intensity of one beam will disturb the interference pattern and change the absorption (and thereby transmission) of the other. For instance, if the film is located at a node of the standing wave, blocking beam B will lead to an immediate increase in loss for beam A and therefore a decrease in its transmitted intensity. Alternatively, if the film is located at an antinode of the standing wave, blocking beam B will result in a decrease of losses for beam A and an increase in its transmitted intensity. In short, manipulating either the phase or intensity of beam B modulates the transmitted intensity of beam A.
To optimize the modulation efficiency, the film should absorb half of the energy of a single beam passing through it. Under such circumstances, 100% light-by-light modulation can be achieved when signal A is modulated by manipulating the phase of beam B and 50% modulation can be achieved if control is encoded in the intensity of beam B. Moreover, one will observe that when the intensities of the two beams are equal and the film is located at an antinode, all light entering the metamaterial will be absorbed, while at a node, light transmitted by the film will experience no Joule losses.
Here, it should be noted that for fundamental reasons, an infinitely thin film can absorb not more than half of the energy of the incident beam.12,13 At the same time, a level of absorption of 50% is difficult to achieve in thin unstructured metal films: across most of the optical spectrum, incident energy will either be reflected or transmitted by such a film. Recently reported much higher absorption levels have only been achieved in layered structures of finite thickness14,15,16,17,18 that are unsuitable for implementation of the scheme presented in Figure 1. However, in the optical part of the spectrum, a very thin nanostructured metal film can deliver strong resonant absorption approaching the 50% target at a designated wavelength. Such metal films, periodically structured on the sub-wavelength scale, are known as planar plasmonic metamaterials.