Fresnel-Fizeau drag: Invisibility conditions for all inertial observers

It was recently shown [J. C. Halimeh et al., Phys. Rev. A 93, 013850 (2016)] that as a result of the Doppler effect, inherently dispersive single-frequency ideal free-space invisibility cloaks in relative motion to an observer can only cloak light whose frequency in the cloak frame coincides with the operational frequency of the cloak, although an infinite number of such rays exist for any cloak motion. In this article, we show analytically and through ray-tracing simulations that even though this relationship can be relaxed by simplifying the ideal invisibility cloak into a broadband amplitude cloak, Fresnel-Fizeau drag uncloaks the phase of light in the inertial frame of the cloak thereby compromising its amplitude cloaking in all other inertial frames. In other words, only an invisibility device that perfectly cloaks both the amplitude and the phase of light in its own inertial frame will also (perfectly) cloak this light in any other inertial frame. The same conclusion lends itself to invisible objects that are not cloaks, such as the invisible sphere.

Figure 1

In the Minkowski vacuum, concentric cylinders indicate the timelike world-tube of a two-dimensional slice of a cylindrical cloak. Outside the cloak, lightlike rays travel on straight lines at ${45}^{\circ }$. Inside the cloak their paths vary from timelike to spacelike (indicating regions of faster than light propagation) but their average behavior is null. Through-going rays exit the cloak at the same space-time events as would uncloaked, unobstructed rays.

Figure 2

(a) In the cloak frame, outgoing rays suffer a time delay but no spatial shift relative to an uncloaked ray. (b) A ray in the laboratory frame is Fresnel-Fizeau dragged by the moving cloak. Lorentz transforming from cloak to laboratory, the time interval in the cloak frame is mixed into a combination of time and space intervals. For this specific illustration, the light ray is Doppler red shifted going from the laboratory frame to the cloak frame as the cloak velocity $\vec{\beta }$ and the the light spatial wave vector $\vec{k}$ share the same direction along the $x$ axis in the laboratory frame. However, the same deterioration in cloaking would occur even if the cloak were traveling in the opposite direction.

Figure 3

In the cloak frame, a light ray (solid curve) traverses the world tube (pink-shaded region) of an amplitude cloak such that the ray tangent is always timelike in the cloak. The detailed behavior of the ray through the cloak may be replaced by the average ray behavior with tangent vector $w^{{\mu }^{\prime }}$ (dashed line). The image of the cloaked ray on imaging plane $y_d^{\prime }$ is shifted in time relative to the uncloaked ray (dotted line). This time delay manifests as a combined time delay and spatial shift in any other inertial frame.

Figure 4

A perfect cloak moving at different speeds in the positive $x$ direction of the laboratory frame is illuminated by light from the top left of each panel. All panels carry the same length scale. The ingoing wave four-vector $k_{\mu }$ is adjusted (blue shifted) for each speed to ensure that Eq. (1) is satisfied and the frequency ${\omega }^{\prime }$ observed in the cloak frame always corresponds to the green color, which denotes the cloak operational frequency, in the top left panel, where the cloak is at rest in the laboratory frame. It is also adjusted such that the ray always enters the cloak at the same angle in the cloak rest frame. The black parts of the light traces are its paths inside the cylindrical cloak, where the wave four-vector is not null. Perfect cloaking is exhibited at all relative speeds.

Figure 5

Same as Fig. 4 but for a cloak moving in the negative $x$ direction of the laboratory frame. The incident wave vector is adjusted (red shifted) for each speed to ensure Eq. (1) is satisfied and to maintain constant green color frequency ${\omega }^{\prime }$ coincident with the operational frequency of the cylindrical cloak, and entry angle in the cloak frame. Perfect cloaking is again exhibited at all relative speeds.

Figure 6

As in Fig. 4 but for an amplitude device–an invisible sphere–rather than a single-frequency perfect cloak. For all non-zero speeds, light rays are displaced by Fresnel-Fizeau drag, which becomes increasingly pronounced at higher speeds. The invisible device becomes increasingly detectable at higher relative velocities.

Figure 7

As in Fig. 6 but for an invisible sphere moving in the negative $x$-direction of the laboratory frame. Incident light is red-shifted as in Fig. 5. The detailed behavior is somewhat different from Fig. 6, but is qualitatively the same: Fresnel-Fizeau drag distorts the image in the laboratory frame, revealing the presence of the invisible sphere.