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 [13]. 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 108s. This means that even detection with 107 images per second qualifies as static. For comparison, commercially available high-speed cameras work only up to 105 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 (D1=0) core with diameter 2R1=32.1mm. This structure forms the “obstacle.” For the “cloak,” this core is additionally surrounded by a shell with outer diameter 2R2=39.8mm and diffusivity D2. 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 D0, 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=60mm. The white-paint concentration, and thus the diffusivity of the surrounding, is adjusted to provide good cloaking. Without absorption, Kerner’s formula [911] allows to directly calculate the ratio D2/D0>1 from the ratio R2/R1. With increasing absorption, the optimum ratio D2/D0 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 <500ps 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 (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, 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 70mm to +70mm 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: Fig. 1.

Fig. 1. Illustration of the experimental setup for time-resolved diffusive light transmission experiments. The tank (not filled here for illustration) with the cylindrical core–shell structure inside is illuminated by a pulsed divergent red laser. The diffusively transmitted light is collected point-wise by a horizontally moveable multimode fiber and is fed into a time-correlated single-photon counting unit.

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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.

 figure: Fig. 2.

Fig. 2. Measured spatially and temporally resolved diffusive light transmission through the water–paint mixture in the tank (a) without any sample, (b) with the obstacle, and (c) with the cloak centered in the tank. Each panel shows a color-coded contour plot of the spatially and temporally resolved photon count rate per time bin (=4ps). The curve on top shows the photon count rate integrated over all time bins. The black data points on the left-hand side are a vertical cut through the center of the contour plot (see dashed white line), with the extracted decay times τ indicated in red. The gray data correspond to ballistic light transmission through the empty tank. Comparison of panels (a) and (c) reveals the failure of the core–shell cloaking in the transient regime, as the cloak’s maximum photon count is higher by about a factor of 2 and occurs earlier than for the reference.

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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.

 figure: Fig. 3.

Fig. 3. Numerical solutions of the time-dependent diffusion equation corresponding to the experimental results shown in Fig. 2. Parameters of the numerical model as explained in Supplement 1 are tank thickness L=60mm; diameters 2R1=32.1mm and 2R2=39.8mm of the core and shell, respectively; diffusivities D0=7.58×108cm2/s, D1=0cm2/s, and D2=158.72×108cm2/s of the surrounding, core, and shell, respectively; photon lifetimes τ0=9.5ns and τ12=0.2s in the surrounding and on the cylinder surface, respectively; and photon loss velocity K=0.743×109cm/s.

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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 [1217] 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 R2R1, 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.

 figure: Fig. 4.

Fig. 4. Calculated photon density distributions on the tank’s rear surface for (a) a lossless core–shell and (b) a transformation-optics (TO)-based cloak under homogeneous illumination of the front surface. Parameters are R2/R1=1.25 (hence D2/D0=4.56 for the core–shell cloak) and L/(2R2)=1.5. The solid curves are horizontal cuts of the normalized photon density. The first row shows static results where both cloaks work perfectly. The remaining rows show snapshots of the transient behavior after an illumination pulse at t=0, with the photon density of the reference case of a homogeneous medium added as dashed curves for comparison. The times given are normalized to the diffusive time constant τdiff=L2/(π2D0). Both cloaks show a similar transient behavior: they appear too bright for early times and cast a shadow for later times, similar to the experiments shown in Fig. 2.

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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 R1 [12] (also see Supplement 1). To allow direct comparison, the ratio R2/R1=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]  

References

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  1. A. Fick, Ann. Phys. 170, 59 (1855).
    [Crossref]
  2. F. Martelli, S. Del Biance, A. Ismaelli, 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, 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, Q. Zhang, IEEE Signal Process. Mag. 18(6), 57 (2001).
    [Crossref]
  6. T. Durduran, R. Choe, W. B. Baker, 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, 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, N. Engheta, Phys. Rev. E 72, 016623 (2005).
    [Crossref]
  12. J. B. Pendry, D. Schurig, 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, X. Zhang, Nanoscale 4, 5277 (2012).
    [Crossref]
  16. J. P. Pendry, A. Aubry, D. R. Smith, S. A. Maier, Science 337, 549 (2012).
    [Crossref]
  17. M. Kadic, T. Bückmann, R. Schittny, M. Wegener, Rep. Prog. Phys. 76, 126501 (2013).
    [Crossref]
  18. S. Guenneau, T. M. Puvirajesinghe, J. R. Soc. Interface 10, 20130106 (2013).
    [Crossref]
  19. S. Guenneau, C. Amra, D. Veynante, Opt. Express 20, 8207 (2012).
    [Crossref]
  20. R. Schittny, M. Kadic, S. Guenneau, M. Wegener, Phys. Rev. Lett. 110, 195901 (2013).
    [Crossref]
  21. H. Xu, X. Shi, F. Gao, H. Sun, B. Zhang, Phys. Rev. Lett. 112, 054301 (2014).
    [Crossref]
  22. T. Han, X. Bai, D. Gao, J. T. L. Thong, C. W. Qiu, Phys. Rev. Lett. 112, 054302 (2014).
    [Crossref]

2014 (3)

R. Schittny, M. Kadic, T. Bückmann, M. Wegener, Science 345, 427 (2014).
[Crossref]

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

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

2013 (3)

M. Kadic, T. Bückmann, R. Schittny, M. Wegener, Rep. Prog. Phys. 76, 126501 (2013).
[Crossref]

S. Guenneau, T. M. Puvirajesinghe, J. R. Soc. Interface 10, 20130106 (2013).
[Crossref]

R. Schittny, M. Kadic, S. Guenneau, M. Wegener, Phys. Rev. Lett. 110, 195901 (2013).
[Crossref]

2012 (3)

S. Guenneau, C. Amra, D. Veynante, Opt. Express 20, 8207 (2012).
[Crossref]

Y. Liu, X. Zhang, Nanoscale 4, 5277 (2012).
[Crossref]

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

2010 (1)

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

2008 (1)

V. M. Shalaev, Science 322, 384 (2008).
[Crossref]

2006 (3)

J. B. Pendry, D. Schurig, D. R. Smith, Science 312, 1780 (2006).
[Crossref]

U. Leonhardt, Science 312, 1777 (2006).
[Crossref]

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

2005 (1)

A. Alù, N. Engheta, Phys. Rev. E 72, 016623 (2005).
[Crossref]

2003 (1)

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

2001 (1)

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

1956 (1)

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

1855 (1)

A. Fick, Ann. Phys. 170, 59 (1855).
[Crossref]

Alù, A.

A. Alù, N. Engheta, Phys. Rev. E 72, 016623 (2005).
[Crossref]

Amra, C.

Aubry, A.

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

Bai, X.

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

Baker, W. B.

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

Boas, D. A.

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

Brooks, D. H.

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

Bückmann, T.

R. Schittny, M. Kadic, T. Bückmann, M. Wegener, Science 345, 427 (2014).
[Crossref]

M. Kadic, T. Bückmann, R. Schittny, M. Wegener, Rep. Prog. Phys. 76, 126501 (2013).
[Crossref]

Calderón, A. P.

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

Choe, R.

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

Del Biance, S.

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

DiMarzio, C. A.

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

Durduran, T.

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

Engheta, N.

A. Alù, N. Engheta, Phys. Rev. E 72, 016623 (2005).
[Crossref]

Fick, A.

A. Fick, Ann. Phys. 170, 59 (1855).
[Crossref]

Gao, D.

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

Gao, F.

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

Gaudette, R. J.

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

Greenleaf, A.

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

Guenneau, S.

S. Guenneau, T. M. Puvirajesinghe, J. R. Soc. Interface 10, 20130106 (2013).
[Crossref]

R. Schittny, M. Kadic, S. Guenneau, M. Wegener, Phys. Rev. Lett. 110, 195901 (2013).
[Crossref]

S. Guenneau, C. Amra, D. Veynante, Opt. Express 20, 8207 (2012).
[Crossref]

Han, T.

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

Ismaelli, A.

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

Kadic, M.

R. Schittny, M. Kadic, T. Bückmann, M. Wegener, Science 345, 427 (2014).
[Crossref]

M. Kadic, T. Bückmann, R. Schittny, M. Wegener, Rep. Prog. Phys. 76, 126501 (2013).
[Crossref]

R. Schittny, M. Kadic, S. Guenneau, M. Wegener, Phys. Rev. Lett. 110, 195901 (2013).
[Crossref]

Kerner, E. H.

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

Kilmer, M.

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

Lassas, M.

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

Leonhardt, U.

U. Leonhardt, Science 312, 1777 (2006).
[Crossref]

Liu, Y.

Y. Liu, X. Zhang, Nanoscale 4, 5277 (2012).
[Crossref]

Maier, S. A.

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

Martelli, F.

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

Miller, E. L.

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

Milton, G. W.

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

Pendry, J. B.

J. B. Pendry, D. Schurig, D. R. Smith, Science 312, 1780 (2006).
[Crossref]

Pendry, J. P.

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

Puvirajesinghe, T. M.

S. Guenneau, T. M. Puvirajesinghe, J. R. Soc. Interface 10, 20130106 (2013).
[Crossref]

Qiu, C. W.

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

Schittny, R.

R. Schittny, M. Kadic, T. Bückmann, M. Wegener, Science 345, 427 (2014).
[Crossref]

M. Kadic, T. Bückmann, R. Schittny, M. Wegener, Rep. Prog. Phys. 76, 126501 (2013).
[Crossref]

R. Schittny, M. Kadic, S. Guenneau, M. Wegener, Phys. Rev. Lett. 110, 195901 (2013).
[Crossref]

Schurig, D.

J. B. Pendry, D. Schurig, D. R. Smith, Science 312, 1780 (2006).
[Crossref]

Shalaev, V. M.

V. M. Shalaev, Science 322, 384 (2008).
[Crossref]

Shi, X.

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

Smith, D. R.

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Supplementary Material (1)

» Supplement 1: PDF (432 KB)     

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Figures (4)

Fig. 1.
Fig. 1. Illustration of the experimental setup for time-resolved diffusive light transmission experiments. The tank (not filled here for illustration) with the cylindrical core–shell structure inside is illuminated by a pulsed divergent red laser. The diffusively transmitted light is collected point-wise by a horizontally moveable multimode fiber and is fed into a time-correlated single-photon counting unit.
Fig. 2.
Fig. 2. Measured spatially and temporally resolved diffusive light transmission through the water–paint mixture in the tank (a) without any sample, (b) with the obstacle, and (c) with the cloak centered in the tank. Each panel shows a color-coded contour plot of the spatially and temporally resolved photon count rate per time bin ( = 4 ps ). The curve on top shows the photon count rate integrated over all time bins. The black data points on the left-hand side are a vertical cut through the center of the contour plot (see dashed white line), with the extracted decay times τ indicated in red. The gray data correspond to ballistic light transmission through the empty tank. Comparison of panels (a) and (c) reveals the failure of the core–shell cloaking in the transient regime, as the cloak’s maximum photon count is higher by about a factor of 2 and occurs earlier than for the reference.
Fig. 3.
Fig. 3. Numerical solutions of the time-dependent diffusion equation corresponding to the experimental results shown in Fig. 2. Parameters of the numerical model as explained in Supplement 1 are tank thickness L = 60 mm ; diameters 2 R 1 = 32.1 mm and 2 R 2 = 39.8 mm of the core and shell, respectively; diffusivities D 0 = 7.58 × 10 8 cm 2 / s , D 1 = 0 cm 2 / s , and D 2 = 158.72 × 10 8 cm 2 / s of the surrounding, core, and shell, respectively; photon lifetimes τ 0 = 9.5 ns and τ 12 = 0.2 s in the surrounding and on the cylinder surface, respectively; and photon loss velocity K = 0.743 × 10 9 cm / s .
Fig. 4.
Fig. 4. Calculated photon density distributions on the tank’s rear surface for (a) a lossless core–shell and (b) a transformation-optics (TO)-based cloak under homogeneous illumination of the front surface. Parameters are R 2 / R 1 = 1.25 (hence D 2 / D 0 = 4.56 for the core–shell cloak) and L / ( 2 R 2 ) = 1.5 . The solid curves are horizontal cuts of the normalized photon density. The first row shows static results where both cloaks work perfectly. The remaining rows show snapshots of the transient behavior after an illumination pulse at t = 0 , with the photon density of the reference case of a homogeneous medium added as dashed curves for comparison. The times given are normalized to the diffusive time constant τ diff = L 2 / ( π 2 D 0 ) . Both cloaks show a similar transient behavior: they appear too bright for early times and cast a shadow for later times, similar to the experiments shown in Fig. 2.

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