## Abstract

An ideal invisibility cloak makes any object within itself indistinguishable from its surrounding—for all colors, directions, and polarizations of light. Nearly ideal cloaks have recently been realized for turbid light-scattering media under continuous-wave illumination. Here, we ask whether these cloaks also work under pulsed illumination. Our time-resolved imaging experiments on simple core–shell cloaks show that they do not: they appear bright with respect to their surrounding at early times and dark at later times, leading to vanishing image contrast for time-averaged detection. Furthermore, we show that the same holds true for more complex cloaking architectures designed by spatial coordinate transformations. We discuss implications for diffuse optical tomography and possible applications in terms of high-end security features.

© 2015 Optical Society of America

In many situations like in fog, milk, or white wall paint, the flow of light cannot be described by ballistic light propagation according to the macroscopic Maxwell equations for continua, but rather by Fick’s diffusion equation for light [1–3]. The latter is adequate if the transport mean free path length of light in the disordered light-scattering medium is (much) shorter than the relevant system size and (much) larger than the wavelength of light.

Recently, we have demonstrated omnidirectional broadband invisibility cloaking of macroscopic three-dimensional objects throughout the entire visible range in the quasi-static diffusive regime [4]. “Quasi-static” refers to temporal variations of the illumination on timescales longer than the underlying time constants. For example, for the centimeter-size structures in Ref. [4], the diffusion time constant and the absorption time constant were estimated to be around ${10}^{-8}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{s}$. This means that even detection with ${10}^{7}$ images per second qualifies as static. For comparison, commercially available high-speed cameras work only up to ${10}^{5}$ images per second. Thus, under typical every-day-life conditions, omnidirectional broadband macroscopic invisibility cloaking works for diffusive light propagation.

In this Letter, we ask whether these core–shell light-diffusion cloaks also work under transient illumination conditions. More generally, we ask whether any light-diffusion cloak, including more complex structures designed by coordinate transformations, would work in the transient regime.

These questions are not only relevant in the context of cloaking and coordinate
transformation physics, but are also of practical importance for time-of-flight
tomography, e.g., of light-scattering human tissue [5,6]. There, tissue is illuminated by
short optical pulses and one aims at reconstructing the tissue distribution by
time-resolving the scattered light. In the static case, the generalized Laplace
equations for diffusion and electrical conduction are equivalent. For the latter,
Greenleaf *et al.* proved mathematically that the Calderón
tomography problem is not unique [7,8]. This means that one cannot unambiguously
reconstruct the tissue distribution from a static diffusion experiment. If cloaking also
worked in the transient regime, the same would hold true for time-of-flight experiments:
the cloaked object would appear in tomography just like a homogeneous light-scattering
medium.

Our experiments are based on a core–shell cloak identical to the one we have characterized previously under quasi-static conditions [4] (also see corresponding Supplement 1). A hollow metal cylinder coated with white acrylic paint approximates a zero-diffusivity (${D}_{1}=0$) core with diameter $2{R}_{1}=32.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$. This structure forms the “obstacle.” For the “cloak,” this core is additionally surrounded by a shell with outer diameter $2{R}_{2}=39.8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ and diffusivity ${D}_{2}$. It is made of polydimethylsiloxane doped with melamine resin microparticles that act as randomly distributed scattering centers. The structures are immersed in a homogeneous surrounding with light diffusivity ${D}_{0}$, composed of a mixture of de-ionized water and a plain white wall paint (which mainly consists of scattering microparticles) in a flat tank of thickness $L=60\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$. The white-paint concentration, and thus the diffusivity of the surrounding, is adjusted to provide good cloaking. Without absorption, Kerner’s formula [9–11] allows to directly calculate the ratio ${D}_{2}/{D}_{0}>1$ from the ratio ${R}_{2}/{R}_{1}$. With increasing absorption, the optimum ratio ${D}_{2}/{D}_{0}$ tends to increase.

We have previously considered cylindrical as well as spherical core–shell cloaks [4]. Their behavior has been qualitatively similar, but the effects are more pronounced for the cylindrical case. Thus, we study only the cylindrical case in the present work.

Figure 1 shows an illustration of our experimental setup. We illuminate the tank with laser pulses of $<500\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$ duration with a repetition rate of 10 MHz at around 640 nm wavelength. The laser (PicoQuant LDH-D-C-640) is coupled to an optical fiber ($\mathrm{NA}=0.1$). The light diverging from this fiber leads to a spot diameter of 3.2 cm (FWHM) on the front surface of the tank. The light transmitted diffusively to the other side of the tank is collected by a multimode optical fiber (200 m core diameter, $\mathrm{NA}=0.12$, separated by 6 mm from the tank), the far end of which is connected to a fast photodetector (PerkinElmer SPCM-AQR-14), connected to a time-correlated single-photon counting unit (PicoQuant PicoHarp 300). The overall time resolution of this setup is determined by jitter, leading to a decay time constant of 750 ps. By scanning the optical fiber along a straight line parallel to the tank and perpendicular to the cylinder axis, we obtain the spatial resolution (see white double-arrow in Fig. 1). We scan the detection fiber in 1 mm steps from $-70\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ to $+70\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ around the center of the tank. At each detection point, we collect photons for 30 s, leading to a total measurement time of about 70 min.

Figure 2 exhibits the measurement results for the homogeneous surrounding, i.e., without any sample in the tank [“reference,” Fig. 2(a)], and for the obstacle [Fig. 2(b)] and the cloak structure [Fig. 2(c)] centered inside the tank. For each case, we first show the photon count rate integrated over all time bins versus the horizontal fiber position at the top of the figure panel. Integration over time is equivalent to static illumination. In the reference case, we find a bell-shaped curve that reflects the spatially inhomogeneous illumination by the laser. For a perfectly homogeneous static illumination, we would expect a straight horizontal line instead (compare [4]). In the same plot in Fig. 2(b), the obstacle blocks the biggest part of the diffusive transmission, leading to a pronounced diffusive shadow. In Fig. 2(c) for the cloak, however, the result almost perfectly resembles the reference curve, showing once again that the diffusive-light core–shell cloaking is working in the quasi-static limit.

Below each time-integrated plot, we show the spatially as well as temporally resolved photon count rate as a color-coded contour plot, with $t=0$ corresponding to the time the laser pulse maximum reaches the tank. Additionally, we display another graph left of this contour plot, showing the temporal evolution at a single detection fiber position of 0 mm, i.e., a vertical cut through the center of the contour plot (see dashed white line). Comparing obstacle and reference, we again observe how the metal cylinder in the tank disturbs the diffusive light transmission, as expected from the discussion above. Importantly, reference and cloak behave differently in the transient regime. In particular, the maximum photon count rate for the cloak is higher and occurs earlier in time than for the reference. Furthermore, the cloak’s photon count rate drops quickly to values significantly below those of the reference for later times. Intuitively, the high-diffusivity cloaking shell transports the incoming light too quickly, leading to an overshooting of intensity in the beginning, followed by a lack of photons toward later times.

For comparison, we show numerical solutions of the time-dependent diffusion equation in Fig. 3, corresponding to our experiments outlined above. The numerical model is identical to the one used in Ref. [4], adapted for a time-resolved calculation. Further details are given in Supplement 1. Clearly, the numerical calculations are in very good agreement with the experimental results shown in Fig. 2.

Is the failure of the diffusive-light core–shell cloak in the transient regime merely a result of the experimental imperfections, i.e., of the finite absorption or the inhomogeneous illumination conditions? Or would a more complex gradually inhomogeneous and anisotropic structure designed by coordinate transformations [12–17] work in the transient regime? To address these questions, we have performed additional calculations depicted in Fig. 4. For zero absorption everywhere, the time-dependent diffusion equation is scalable. Hence, we represent its solutions in terms of dimensionless ratios (rather than absolute quantities) to emphasize the general character. In Fig. 4(a), we consider a lossless core–shell cloak under homogeneous illumination. The other conditions are as above. Under static conditions, this device cloaks exactly. Yet, under transient conditions, it again behaves very similarly to the experiment discussed above, being too bright at early and too dark at later times. The same holds true for other parameter combinations of diffusivity and radius of the shell (not depicted). For very large shells with ${R}_{2}\gg {R}_{1}$, the failure under transient conditions becomes less pronounced. However, for radii that large, even in the static case there is not much to cloak to start with.

Mathematically, the time-dependent diffusion equation (which has only one material parameter, namely, diffusivity) is generally not form-invariant under coordinate transformations [18], whereas the related time-dependent heat conduction equation (which has two material parameters, namely, heat conductivity and specific heat) has been shown to be form-invariant under coordinate transformations [19,20]. In Fig. 4(b), we study a coordinate-transformation-based cloak designed for the static version of the diffusion equation, which is mathematically identical to the static heat-conduction equation and is thus form-invariant [18]. We use a Pendry-type transformation of a point to a circle with radius ${R}_{1}$ [12] (also see Supplement 1). To allow direct comparison, the ratio ${R}_{2}/{R}_{1}=1.25$ is chosen to be the same as for the core–shell cloak in Fig. 4(a). Obviously, this more sophisticated cloak behaves closely similar to the simple core–shell cloak [compare Fig. 2(a)]. In particular, it also fails to cloak in the transient regime despite the fact that, by construction, it works perfectly under all illumination conditions in the static case.

Finally, we connect to recent experiments that reported successful transient thermal cloaking of core–shell geometries like ours for light diffusion [21,22]. These reports seem to contradict our findings. To resolve this issue, we have performed additional calculations corresponding to their parameters (not depicted). We find that these cloaks also fail under truly transient conditions, while they work nicely under the quasi-static conditions investigated in Refs. [21] and [22]. Indeed, it is experimentally more difficult to realize the counterpart of short incident optical pulses in heat conduction.

In conclusion, omnidirectional broadband core–shell invisibility cloaking of macroscopic three-dimensional objects throughout the entire visible range works nicely within the static and quasi-static regimes of diffusive light propagation, but inherently fails under truly transient conditions. Likewise, more complex light-diffusion cloaks designed by coordinate transformations also fail under transient conditions because the time-dependent diffusion equation is not form-invariant under such transformations. It contains only a single material parameter, diffusivity, whereas two parameters would be needed (also see Supplement 1). This failure is good news for time-of-flight tomography of light-scattering media like human tissue, which would otherwise not be unique. As a possible application, we suggest using miniaturized versions of the diffusive core–shell cloak as a high-end security feature. Under usual illumination, one would only see an unsuspicious diffusively scattering homogeneous area, e.g., on a banknote or on some package. Under femtosecond pulsed illumination and time-gated detection, hidden signatures (e.g., a bar code) would become visible.

See Supplement 1 for supporting content.

## REFERENCES

**1. **A. Fick, Ann. Phys. **170**, 59 (1855). [CrossRef]

**2. **F. Martelli, S. Del Biance, A. Ismaelli, and G. Zaccanti, *Light Propagation through Biological Tissue and Other
Diffusive Media: Theory, Solutions, and Software*
(SPIE, 2010).

**3. **C. M. Soukoulis, ed., *Photonic Crystals and Light Localization in the
21st Century* (Springer,
2001).

**4. **R. Schittny, M. Kadic, T. Bückmann, and M. Wegener, Science **345**, 427 (2014). [CrossRef]

**5. **D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, IEEE Signal Process. Mag. **18**(6), 57
(2001). [CrossRef]

**6. **T. Durduran, R. Choe, W. B. Baker, and A. G. Yodh, Rep. Prog. Phys. **73**, 076701 (2010). [CrossRef]

**7. **A. P. Calderón, Comput. Appl. Math. **25**, 133 (2006).

**8. **A. Greenleaf, M. Lassas, and G. Uhlmann, Math. Res. Lett. **10**, 685 (2003).

**9. **E. H. Kerner, Proc. Phys. Soc. London Sect. B **69**, 802 (1956).

**10. **G. W. Milton, *The Theory of Composites*
(Cambridge University,
2002).

**11. **A. Alù and N. Engheta, Phys. Rev. E **72**, 016623 (2005). [CrossRef]

**12. **J. B. Pendry, D. Schurig, and D. R. Smith, Science **312**, 1780 (2006). [CrossRef]

**13. **U. Leonhardt, Science **312**, 1777 (2006). [CrossRef]

**14. **V. M. Shalaev, Science **322**, 384 (2008). [CrossRef]

**15. **Y. Liu and X. Zhang, Nanoscale **4**, 5277 (2012). [CrossRef]

**16. **J. P. Pendry, A. Aubry, D. R. Smith, and S. A. Maier, Science **337**, 549 (2012). [CrossRef]

**17. **M. Kadic, T. Bückmann, R. Schittny, and M. Wegener, Rep. Prog. Phys. **76**, 126501 (2013). [CrossRef]

**18. **S. Guenneau and T. M. Puvirajesinghe, J. R. Soc. Interface **10**, 20130106 (2013). [CrossRef]

**19. **S. Guenneau, C. Amra, and D. Veynante, Opt. Express **20**, 8207 (2012). [CrossRef]

**20. **R. Schittny, M. Kadic, S. Guenneau, and M. Wegener, Phys. Rev. Lett. **110**, 195901 (2013). [CrossRef]

**21. **H. Xu, X. Shi, F. Gao, H. Sun, and B. Zhang, Phys. Rev. Lett. **112**, 054301 (2014). [CrossRef]

**22. **T. Han, X. Bai, D. Gao, J. T. L. Thong, and C. W. Qiu, Phys. Rev. Lett. **112**, 054302 (2014). [CrossRef]