We commonly encounter the idea of negative frequencies while analyzing electromagnetic waves and signals, which are often helpful as a mathematical concept to help us keep track of signals that vary over time. We can safely ignore negative frequencies in situations where incoming and outgoing electromagnetic radiation would be equivalent in the event that time was reversed; for example, in the case of a simple ray tracing problem of imaging with lenses. However, we cannot regard negative frequencies as simply a mathematical concept in cases where electromagnetic radiation behaves differently when time is reversed.
One such situation is a medium that is moving. In this case, electromagnetic waves that are propagated in the direction of motion of the medium and in the opposite direction undergo Doppler shifts of different sizes. This means the reflection co efficient of a light wave moving in the direction of the medium is red-shifted in frequency compared with when the medium is at rest. This Doppler effect has to be taken into account in scenarios such as laser measurements of the velocity of a moving fluid. If the medium is moving at a high enough velocity, this red shift in the frequency can be so large as to make the frequency zero or even negative. Such Doppler shift to negative frequencies only occurs if the velocity of motion is greater than the phase velocity of light in the medium, similar to the well-known Cherenkov radiation condition.
Incident evanescent electromagnetic waves in a moving medium seem to possess frequencies of different signs in the laboratory and in a moving frame of reference: see Figure 1. A regular decaying evanescent photonic mode in the laboratory can be Doppler-shifted to a negative frequency in a moving frame. The reflection coefficient, which describes the interaction of light with matter when bodies are at rest and in motion, has to be evaluated in the same frame of reference as a body in motion. For modes with negative frequency, carefully calculated reflection coefficients show the existence of instabilities and growing evanescent waves. This phenomenon has been studied in the case of moving plasmas, in which the presence of waves with negative energy implies that the moving plasma would enable a photonic mode to be generated from noise to decrease its net energy.
Figure 1. (a) An evanescent wave that is incident on a stationary plate is absorbed. (b) An evanescent wave that is incident on a moving plate can be amplified after reflection if the frequency of the wave is negative after Doppler shifting. ω: Angular frequency of incident wave. ω′ : Angular frequency in co-moving frame.
We have pointed out an extremely interesting case where coupling can occur between modes with positive and negative frequency,1 which leads to a unique resonance in moving photonic media. A simple practical configuration where this can occur is if two plates separated by a small gap are in motion relative to each other: see Figure 2(a). A single wave can appear as a positive photonic mode on one plate and be Doppler-shifted to a mode of equal—but negative frequency—on the other plate if the velocity of the plates and the size of the gap between them both have critical values. This situation gives rise to perfect coupling of waves of positive and negative frequency in the near-field region and a resonance with an infinite quality factor, which is fundamentally different from any resonance in stationary systems.
Figure 2. (a) Diagram shows the plates of a Fabry-Perot interferometer, which are in relative motion with a fixed gap and evanescent waves bouncing between them. Perfect coupling can occur between waves with positive and negative frequency modes, which leads to a unique resonance with infinite quality factor. (b) Between moving metallic plates a spectrum of thermal energy (right) is exchanged. The y-axis represents frequency and the x-axis the lateral momentum of the waves normalized according to the equation β=v/c. The gap d is close to the critical gap distance d0. A large enhancement is seen (red region), which originates from the resonance with negative frequency. v: Velocity of motion. kx: Momentum of wave. c: Speed of light.
One major implication of this resonance is with regard to the phenomenon of vacuum friction.2, 3 Vacuum fluctuations exert a lateral force, referred to as the Casimir force, on the plates of a Fabry-Perot interferometer, which pushes them close together. Once the plates are set in motion relative to each other with a fixed gap, these fluctuations also exert a drag force, which acts to slow the plates down. This interaction takes place through all possible modes that exist in the moving plate system and we have shown that this force takes very high values, owing to this unique resonance with negative frequency: see Figure 2(b).4
The velocity of motion that is required to observe such a resonance with negative frequency is of the order of the Fermi velocity of electrons in the metallic plates. The gap needs to be maintained at roughly 10nm. This type of mechanical motion is very difficult to maintain and further research in nanoscale optomechanical systems will be necessary to achieve this. We could use techniques such as simulating a moving medium using pulses in a non-linear optical fiber. However, we may be able to demonstrate resonances with negative energy in an experiment using light-induced potentials or rapidly spinning nanoparticles.
University of Alberta
Zubin Jacob is an associate professor of electrical and computer engineering at the University of Alberta. He is also a visiting faculty member at the International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore, India.
1. Y. Guo, Z. Jacob, Singular evanescent wave resonances in moving media, Opt. Express 22, p. 26193-26202, 2014.
2. A. I. Volokitin, B. N. J. Persson, Quantum friction, Phys. Rev. Lett. 106, p. 94502, 2011.
3. J. B. Pendry, Shearing the vacuum - quantum friction, J. Phys. Condens. Matter 9, p. 10301, 1997.
4. Y. Guo, Z. Jacob, Giant non-equilibrium vacuum friction: role of singular evanescent wave resonances in moving media, J. Opt. 16, p. 114023, 2014.