Invisibility cloaks in relativistic motion

We consider an ideal invisibility cloak which is illuminated by monochromatic light and which moves in vacuum at constant relativistic velocity with respect to the common inertial frame of light source and observer. We show that, in general, the moving cloak becomes detectable by image distortions and by generating a broad frequency spectrum of the scattered light. However, for many special combinations of incident light frequency, wave vector of light, and cloak velocity, ideal cloaking remains possible. It becomes nonreciprocal though. This means that light rays emitted by the light source arrive at the observer as though they have traveled through vacuum, but they take completely different paths after being retroreflected at the observer position.

Figure 1

Characteristic light rays in a frame ($x^{\prime }{y}^{\prime }$ plane) comoving with a dispersive cylindrical invisibility cloak moving with constant velocity $v$ along the positive $x$ direction in the laboratory system of light source and observer. The cloak radii $R_1$ and $R_2$ are indicated by the black circles, and ray colors indicate light frequency with ${\omega }_0$ as green. The invariance of the cylindrical cloak along the $z^{\prime }$ axis allows all optics to be captured in the $x^{\prime }{y}^{\prime }$ plane. The monochromatic light rays with frequency $\omega ={\omega }_0$ are emitted along the positive $x$ direction in the laboratory frame. ${\omega }_0$ is the cloak operation frequency, i.e., if the cloak is at rest with respect to the light source, the condition $\omega ={\omega }_0$ leads to perfect cloaking. With increasing velocity $\beta =v/c_0$ with respect to the vacuum speed of light $c_0$, increasing aberrations are found. Light rays for which the emerging direction (or wave vector of light ${\vec{k}}^{\prime }$) includes an angle $\varphi$ with the incident rays in the $x^{\prime }{y}^{\prime }$ plane are received with a frequency $\tilde{\omega }\ne \omega$ by an observer at rest in the laboratory frame. This leads to the relative frequency shift $\mathrm{\Delta }\omega /\omega \equiv (\tilde{\omega }-\omega )/\omega =(cos\varphi -1)\beta /(\beta +1)$, which is a simplified version of Eq. (4) for the geometry shown here.

Figure 2

Same as Fig. 1, but for light rays with $\omega ={\omega }_0$ impinging along the positive $y$ direction (rather than the positive $x$ direction) in the laboratory frame. In the shown comoving frame, the incident rays at the bottom are inclined with respect to the $y^{\prime }$ axis due to the Lorentz transformation, although they are along the positive $y$ direction in the laboratory frame. Again, with increasing cloak velocity $v/c_0$, more pronounced aberrations are observed. Some rays end within the cloak because they hit the vicinity of local resonances at a radius indicated by the magenta circle. For arbitrarily small yet nonzero damping, these rays will be absorbed. As in the longitudinal case, there will also be a frequency spread due to deflections in the direction of the wave vectors with respect to their incidence directions upon interacting with the cloak.

Figure 3

As in Fig. 1, but for the light frequency $\omega ={\omega }_0\sqrt{(1-\beta )/(1+\beta )}$ in the laboratory frame with $\beta =v/c_0=0.1$ such that the Doppler blue-shifted frequency in the comoving frame obeys ${\omega }^{\prime }={\omega }_0$, leading to perfect cloaking (green color) for light propagating with frequency $\omega$ along the negative $x$ direction in the laboratory frame. In sharp contrast, upon retroreflecting these rays at the observer [where again $\omega ={\omega }_0\sqrt{(1-\beta )/(1+\beta )}$], strong aberrations are observed because ${\omega }^{\prime }={\omega }_0(1-\beta )/(1+\beta )$ due to a Doppler red shift from the laboratory to the comoving frame. This means that the cloaking behavior is nonreciprocal. Infinitely many such special configurations exist for the different possible laboratory-frame frequencies $\omega$.

Figure 4

Nonreciprocal cloaking for light with frequency $\omega ={\omega }_0$ (green rays) and wave vector $\vec{k}$ in the laboratory frame obeying Eq. (19) and impinging on the cylindrical invisibility cloak while the latter is moving at relative velocity $v/c_0=0.5$ in the positive $x$ direction. This light has the same frequency ${\omega }^{\prime }={\omega }_0$ in the comoving frame as well. Hence, the cloaks works perfectly. However, upon retroreflection of the light in the laboratory frame, its wave vector becomes $-\vec{k}$, and this light will no longer have a frequency ${\omega }^{\prime }={\omega }_0$ in the comoving frame as it acquires a Doppler blue shift, leading to imperfect cloaking. A few rays end within the cloak as they hit the vicinity of a local resonance at a radius indicated by the magenta circle. For arbitrarily small yet nonzero damping, these rays are absorbed there. Finally, note that even though $\vec{k}$ and $-\vec{k}$ are retroreflections of one another in the laboratory frame, their Lorentz boosts in the comoving frame are not. In other words, if $\vec{k}$ is boosted to ${\vec{k}}^{\prime }$, that does not mean that $-\vec{k}$ will get boosted to $-{\vec{k}}^{\prime }$, i.e., retroreflection in the laboratory frame appears a bit unintuitive if represented in the comoving frame.