Ultrabroadband 3D invisibility with fast-light cloaks

An invisibility cloak should completely hide an object from an observer, ideally across the visible spectrum and for all angles of incidence and polarizations of light, in three dimensions. However, until now, all such devices have been limited to either small bandwidths or have disregarded the phase of the impinging wave or worked only along specific directions. Here, we show that these seemingly fundamental restrictions can be lifted by using cloaks made of fast-light media, termed tachyonic cloaks, where the wave group velocity is larger than the speed of light in vacuum. On the basis of exact analytic calculations and full-wave causal simulations, we demonstrate three-dimensional cloaking that cannot be detected even interferometrically across the entire visible regime. Our results open the road for ultrabroadband invisibility of large objects, with direct implications for stealth and information technology, non-disturbing sensors, near-field scanning optical microscopy imaging, and superluminal propagation.


Derivation of Eqs. (1) and (2) of the main text
A lightfield polarized across the x-axis ( ) is incident from free space on a double-shell tachyonic cloak hiding a spherical object of radius r1, as shown in Supplementary Fig. 1 below. For certainty, and for the sake of obtaining exact analytic expressions, let us assume that the magnetic field H0 (but not the electric field E0) of the incident lightfield varies slowly in time [39]), with the results being fully extendable to faster variations of H0 provided that the group velocity vg of the lightwave at the outer shell (3) increases correspondinglyso that the bandwidth BW = vg/(Q·Δℓ) may remain large (see discussions in the main text). Owing to the azimuthal symmetry of the problem, the electric potential in the four regions, φe,i (i = 1-4), will be of the form and φe,4 = (A4r + B4 r -2 )cosθ, the electric field in the region (4) outside the cloaked object will be given by the following exact (to all multipole orders) expression: which is Eq. (1) in the main text. The solution to the above system of equations involves tedious calculations, and because of Eq. (S9) (and the associated discussions in the main text) below we provide the final, compact expression for (only) the coefficient B4: where τ1 = (ε2 -ε3)(2ε2 + ε1), τ2 = (ε1 -ε2)(2ε2 + ε3), τ3 = (ε2 -ε1)(ε2 -ε3), and τ4 = (2ε2 + ε1)(2ε3 + ε2). Equation (S10) above is used as Eq. (2) in the main text.

Superluminal light propagation in the tachyonic cloak and maintenance of relativistic causality
To attain the broadband cloaking response shown in Fig. 3 of the main text, a short pulse of bandwidth B must travel superluminally, with a group velocity vg > c, inside the cloak (and around the object 22 ) so that there is no appreciable delay compared with a pulse that travels straightly in vacuum and reaches the exit side of the cloak. Both, the amplitude and the phase of the two pulses must (ideally) be exactly the same, so that the object and the cloak cannot be detected using interferometric or time-of-flight measurements. In this section we shall theoretically show that such a feat is, indeed, possible (the corresponding numerical corroboration, obtained from full-wave causal simulations, is presented in Figs [22]). As explained herein in this section, only the shaded parts of the pulses are causally connected. Note that the shaded part of the input pulse does not contain the peak of that pulse (at t = t2), whereas the shaded part of the exit pulse does contain the peak of that pulse (at t = t3)i.e., the two peaks, the time-distance between which determines the group velocity, are not causally connected; hence, the group velocity of the pulse can be superluminal without violation of relativistic causality.
The characteristics of the input (incident on the cloak) and output (exiting the cloak) pulses are shown in Supplementary Fig. 2 above. We shall now prove these shown characteristics in the following, specifically showing that: -The sharp/sudden front discontinuity of the input pulse, associated with the pulse's signal velocity, propagates with a velocity c; -Only the shaded parts of the two pulses, as shown in Supplementary Fig. 2, are causally connected; -The shadowed part of the input pulse does not contain the peak of that pulse, whereas the shadowed part of the exit pulse does contain the peak of that (exit) pulse.
To prove the first point, let us assume that E(z = 0, t) is the electric-field value of the pulse entering the tachyonic cloak (at an arbitrary point z = 0)cf. Suppl. Fig. 2 above. The input pulse is "turned on" at a time instant t = t0, as shown in Suppl. Fig. 2, i.e. it is: Then, the electric-field value, E(z, t), of the pulse exiting the cloak will simply be given by: where k is the wavevector of the pulse inside the cloak, and (S12) We shall, now, show that E(z, t) = 0 for t < t0 + tf = t0 + z/c, as shown in Suppl. Fig. 2. To this end, we note that the permittivity model describing the inverted (gain) medium in the outer shell of the tachyonic cloak (in its most general form that model is given by: ε(ω) = ε0 + ε0fωp 2 /(ω0 2 -ω 2 + jωγ), with f < 0 being the oscillator strength, and ωp the "plasma" frequency) leads to the wavevector k(ω) having poles only in the upper complex ω-plane, i.e. being analytic in the lower half plane, as shown in Supplementary  Fig. 3 below. Thus, from complex integration we have: The crucial point, now, is that on the semicircle HRit is |ω| = R → ∞, hence the wavevector k will equal its free-space value, k0 = ω/c, because . ) ( lim 0 ε ω ε ω =  → Equation (S13) may, therefore, be re-written as: (S14) Since, as mentioned above, we wish to check the value of E(z, t) for t < t0 + z/c, i.e. since t -t0 + z/c < 0, and under the reasonable assumption that on HRit is 0 ) (0, e 0 j → ω E ωt Supplementary Fig. 3. Complex ω-plane and associated contour of integration for the evaluation of the integral of Eq. (S13). ωR R -R HR -ωI 6 (for instance, for a causal signal , which indeed tends to zero for |ω| → ∞), it immediately follows from the Jordan lemma that the integral in Eq. (S14) vanishes, i.e. for t < t0 + tf it is indeed E(z, t) = 0 as Suppl. Fig. 2 shows. Thus, the velocity with which the front discontinuity propagates (i.e., the signal velocity) is not larger than the speed of light in vacuum, c (i.e., relativistic causality is fully respected in the tachyonic cloak), since the signal-front delay is not larger than tf = z/c.
Next, we assume that q(z, t) is the impulse response of the cloak, given (as usual) by: Based on similar arguments as before for the behaviour of the wavevector k on the complex ω-plane (cf. Suppl. Fig. 3), we may successively write: from where we see that the tachyonic cloak satisfies the causality condition: q(z, t) = 0 for We now note that, by the convolution theorem, the electric-field value of the exit pulse that has propagated through the cloak (and around the object) can, in addition to Eq. (S11), be written as: In this integral, it should be: t -t´ ≥ z/c (causality condition for q(z, t)) and also: t´ ≥ t0 (since E(0, t´) is "turned on" at t = t0; cf. Suppl. Fig. 2), i.e. overall it should be: t0 ≤ t´≤ t z/c, leading to the following expression for E(z, t): with t > t0 + tf = t0 + z/c. Equation (S18), which has been obtained on the basis of the causality conditions dictating the propagation of a pulse in the tachyonic cloak, shows that the electric-field amplitude of the exiting pulse at times t > t0 + z/c, i.e. the shaded part of the E(z, t) pulse in Suppl. Fig. 2, is causally determined by the shaded part of the input E(0, t) pulse. Thus, only the shaded parts of the two pulses shown in Suppl. Fig. 2 are causally connected, as we were intending to show. Furthermore, since in the tachyonic cloak it is vg > c, the group delay tg = z/vg will be smaller than the signal-front delay tf = z/c (tg < tf). Therefore, if the peak of the exiting pulse occurs at a time point t = t3 (cf. Suppl. Fig. 2), it will be: i.e. E(z, t3) will be causally determined by values of the input E(0, t) pulse in the time interval [t0, t3z/c], which does not include the time-point t = t3tg = t3z/vg (the time point when the peak of the input E(0, t) pulse occurs) because vg > c. Thus, the peak of the input E(0, t) pulse is not causally connected to the peak of the exiting E(z, t) pulseas a result of which, the time interval between the two peaks (group delay tg) can indeed be "superluminal" without violation of relativistic causality.

Recursive scattering transfer matrix formulation
An exact methodology for the scattering problem at hand can be obtained through a recursive T matrix analysis. Particularly, let us initially assume an incoming plane electromagnetic (EM) wave of frequency , expressed in a spherical-wave basis around the center of the scatterer as follows: (S20) where is the wavenumber, and are the usual angular momentum indices, are the vector spherical harmonics, with being the spherical Bessel function and are appropriate expansion coefficients with index . A similar expansion can be written for the outgoing scattered wave by replacing the spherical Bessel function with outgoing Hankel functions: The spherical expansion coefficients of the scattered waves, denoted , can be connected to the coefficients of an incoming incident wave through the scattering T matrix: For a spherical scatterer, the T matrix actually takes the form where: 8 We now, for our purposes, assume a multilayer spherical particle which consists of a spherical core and N-1 concentric spherical shells. Particularly, let the radius of the central spherical core be and the outer radii of each additional concentric spherical shells be . Clearly, . The permittivities and permeabilities of the successive media are assumed to be and , respectively, and it is , for the host medium. The corresponding wavenumbers are and , where is the velocity of light in vacuum. Again, fully analytical solutions are here feasible; however, it is preferable to use a recursive formulation owing to its simplicity. By requiring the continuity of the tangential components of the EM field at each spherical interface between the consecutive media, we can write , i.e., the matrix for a sphere that consists of the first shells, embedded in the ( th medium which, one assumes, extends to infinity. The matrix is written recursively in terms of , which describes a sphere consisting of the first shells in the th host medium, . In that case, we may obtain: A similar expression is also obtained for , with the magnetic permeabilities , used in place of the dielectric functions, , , respectively. For non-spherical scatterers, the T matrix is not diagonal in the spherical-wave basis and the elements can be evaluated numerically by employing the extended-boundary-condition method [1], properly modified [2]. -layered sphere (cf. Fig. 2) For a two-layered sphere, the total scattering cross section t Q due to plane wave scattering can be computed exactly, without even requiring a recursive T matrix formulation, using analytic theory. Suppose that the spherical core has radius a , and its material is described by constitutive parameters 1  where 0  is the free-space wavelength. The general expressions for the expansion coefficients  Supplementary Figure 4 below presents the normalized radar cross section (RCS) of a tachyonic cloak for different radii of an uncloaked (red) and cloaked (blue) spherical object, starting with an initial radius r0. In all cases the incident wavelength is 600 nm. We may note that, as expected from the discussion in the main body of the work (bandwidth BW = vg/(Q·Δℓ)), for the mildly (realistic) superluminal group velocities used in all cases herein (vg| outer-shell ~2.236c), i.e., with the group velocity in the outer layer being approximately twice larger than the speed of light in vacuum, the ultrabroadband (across the visible regime) bandwidth performance of the cloak can be maintained for sizes of the object that can, accordingly, approximately double before discriminable scattering ensues (i.e., with Δℓ doubling, the superluminal group velocity vg in the outer layer should also be, approximately, double the speed of light in vacuum, so that BW can be maintained large).

Performance of the tachyonic cloak for different object sizes and shapes
Using this methodology, one may also cloak an arbitrary shaped objectin this case (Suppl. Fig. 5 below), a randomly oriented Si spiral. In particular, as shown in Suppl. Fig.  5(b), the method can be used for cloaking, over extremely broad bandwidths, hollow metallic/plasmonic 3D spheres, inside which one may place a completely arbitrary object. We note that this manner for cloaking arbitrarily shaped objects was first proposed in the