Abstract
We have designed and realized a threedimensional invisibilitycloaking structure operating at optical wavelengths based on transformation optics. Our blueprint uses a woodpile photonic crystal with a tailored polymer filling fraction to hide a bump in a gold reflector. We fabricated structures and controls by direct laser writing and characterized them by simultaneous high–numericalaperture, farfield optical microscopy and spectroscopy. A cloaking operation with a large bandwidth of unpolarized light from 1.4 to 2.7 micrometers in wavelength is demonstrated for viewing angles up to 60°.
As today’s nanofabrication capabilities continue to improve, we are better able to address the inverse problem of electromagnetism with respect to what nanostructure will perform a requested functionality. In this regard, transformation optics (1–14) is a unique and intuitive scientific tool that allows for the mathematical mapping of desired distortions of space onto an actual distribution of optical material properties in normal Cartesian space. Tailored inhomogeneous metamaterials enable us to approximate these target distributions. Invisibilitycloaking structures (1–16) can serve as benchmark examples for the much broader ideas of transformation optics.
So far, invisibilitycloaking experiments at microwave (5, 10) and optical frequencies (11–14) have been performed exclusively in twodimensional (2D) waveguide geometries. In other words, these structures are immediately visible from the third dimension. Cloaking works only in the plane; the viewing angle is effectively zero in one direction. Nevertheless, these structures have supported the validity of the concepts of both transformation optics and metamaterials.
We designed, fabricated, and characterized 3D invisibilitycloaking structures using tailored, dielectric facecentered–cubic (fcc) woodpile photonic crystals. We studied the behavior of these structures from wavelengths near the woodpile rod spacing (where the onset of diffraction of light leads to the Wood or Rayleigh anomaly in transmittance and reflectance) up to wavelengths much larger than this spacing (the effectivemedium limit).
In the carpetcloak geometry (8, 10–13), a bump in a metallic mirror is hidden by adding a tailored refractiveindex distribution on top. This distribution can be calculated using the rules of transformation optics (8, 10). Though originally designed for two dimensions, it has been shown (by numerical rendering of photorealistic images via ray tracing) that the carpetcloak concept should also work in a truly 3D setting and also for very large viewing angles (17). In our 3D blueprint (Fig. 1A), the bump is translationally invariant along the z direction and follows y(x) = hcos^{2}(πx/w) for x ≤ w/2 and zero otherwise. Here, h = 1 μm is the height of the bump, and w = 13 μm is its full width. For the quasiconformal mapping of the cloak, we choose a width of 26 μm in the x direction and 10 μm in the y direction. This cloak is surrounded by a homogeneous woodpile structure, which is characterized in fig. S5 (18). Inside the cloak, the local effective refractive index is controlled via the volume filling fraction (f) of the polymer serving as constituent material for a usual woodpile photonic crystal (19, 20). The diamondsymmetry woodpile geometry is chosen because it is expected to lead to nearly isotropic optical properties. The effective refractive index becomes n = 1.52 for f = 1 (bulk polymer) and n = 1.00 for f = 0 (air void). For intermediate values of f, we used the Massachusetts Institute of Technology PhotonicsBands package (21) to evaluate the effective local refractive index (Fig. 1B) on the basis of usual photonicband–structure calculations. We found that the calculated 3D isofrequency contours are very nearly spherical in the longwavelength limit (see inset in Fig. 1B).
Figure 2 shows the target refractiveindex distributions obtained from the quasiconformal mapping (8) and corresponding electron micrographs of some of our structures made by standard direct laser writing lithography (20, 22). Fabrication details and sample dimensions are reported in the supporting online material (SOM) (18). To reveal their interiors, the structures are cut by means of focused ion beam (FIB) milling. This destructive measure is of utmost importance, as inspection of sample edges can be highly misleading due to the proximity effect.
For what wavelengths do we expect reasonable cloaking? On one hand, it is sometimes argued that the wavelength of light needs to be at least one order of magnitude larger than the period or lattice constant to truly reach the effectivemedium limit. This very conservative estimate would lead to operation wavelengths larger than 8 μm for a = 800 nm or a wavelength of 11 μm for the fcc lattice constant of 1.131 μm. On the other hand, the most aggressive and optimistic approach is to argue that the effectivemedium method can work up to the Wood anomaly. Once diffraction occurs, the periodic structure can no longer be considered a homogeneous effective material. For normal incidence, diffraction of light becomes possible if the material wavelength is equal to or smaller than the lattice constant. For a = 0.8 μm and a glasssubstrate refractive index of n = 1.5, the Wood anomaly is expected to occur at a vacuum wavelength of 1.2 μm. These conservative and aggressive considerations obviously differ by about one order of magnitude in wavelength.
To evaluate the performance of our fabricated samples, we start by discussing the results obtained by brightfield optical microscopy and spectroscopy. The numerical aperture (NA) of the used microscope lens of NA = 0.5 is equivalent to a full opening angle of the cone of light of 60° (Fig. 1A). Therefore, light is propagating not only in the xy plane (which would merely be 2D), but also in oblique directions (3D). Details of the homebuilt setup are described in the SOM (18). Corresponding spatially and spectrally resolved normalincidence data (normalized) show that the bump is immediately visible by two pronounced spatial minima (Fig. 3A) resulting from light that is reflected by the two slopes of the bump toward the sides and is not collected by the lens. The light reflected from the top of the bump (where the tangent is horizontal) leads to the narrow bright stripe in the middle. The spectral oscillations are FabryPerot fringes. Their free spectral range agrees well with the expectation based on the total 10μm thickness of the woodpile structure. In presence of the cloak and bump (Fig. 3B), the visibility of the bump is strongly suppressed, whereas cloaking is not quite perfect. This recovery works well in the depicted wavelength range of 1.5 to 2.6 μm in Fig. 3 (data over a larger spectral interval are shown in fig. S1). For wavelengths shorter than ~1.2 μm, the image becomes dimmer. Furthermore and most important, in this regime, no recovery due to the presence of the cloak is observed. We interpret this shortwavelength dark region as being due to the Wood anomaly. Indeed, an ideal polymer woodpile can diffract as much as 50% of the incident light for wavelengths smaller than the Wood anomaly (see fig. S5). These levels explain the findings of our present work. For wavelengths shorter than that of the Wood anomaly, the light field can no longer effectively average over the nanostructure. Hence, the structure is not expected to act like a locally homogeneous dielectric, and no cloaking action is expected—consistent with our above observations. For the present conditions, the effectivemedium approximation turns out to be much more forgiving than one might be tempted to believe at first sight.
Control samples with only (i) the highindex region or (ii) the two lowindex regions are shown in figs. S2 and S3, respectively. For case (ii), the bump appears wider than in Fig. 3A. For case (i), cloaking is worse than for the complete cloak in Fig. 3B. These two observations show that one really needs the complete refractiveindex profile derived from the quasiconformal mapping in transformation optics to obtain good invisibilitycloaking performance.
Finally, Fig. 4 depicts data taken in darkfield mode from 1.5 to 2.6μm wavelengths (data over a larger spectral interval are shown in fig. S4). Here, the same sample as in Fig. 3 is tilted such that the optical axis lies within the xy plane and includes an angle of 35° with the y axis. As usual for the darkfield mode, the collected light results from scattering by the sample. These data are normalized with respect to a normalincidence reflection spectrum taken on the gold film. The bump without cloak in Fig. 4A is immediately visible. We assign this finding to enhanced scattering from the illuminated side of the bump. The visibility is again drastically reduced for the case of bump with cloak in Fig. 4B.
Supporting Online Material
www.sciencemag.org/cgi/content/full/science.1186351/DC1
SOM Text
Figs. S1 to S5

↵* These authors contributed equally to this work.
References and Notes
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 ↵See supporting material available on Science Online.
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 We thank K. Busch, G. von Freymann, S. Linden, and M. Thiel for discussions and help regarding sample fabrication and photonic bandstructure calculations. We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) and the State of BadenWürttemberg through the DFG–Center for Functional Nanostructures within subprojects A1.4 and A1.5. We also thank the Future and Emerging Technologies (FET) program within the Seventh Framework Programme for Research of the European Commission (FET open grant number 213390) for financial support of the project PHOME. The project METAMAT is supported by the Bundesministerium für Bildung und Forschung. The Ph.D. education of T.E. is embedded in the Karlsruhe School of Optics and Photonics (KSOP); N.S. is supported as a mentor in the KSOP.