We predict that a liquid crystal/silver nanoparticles mixture can be designed so that its effective ordinary and extraordinary permittivities have, in a frequency range, real parts of different signs. We exploit this result to design an optical switch obtained by sandwiching a few hundred nanometers thick slab of the proposed mixture between two silica layers. By resorting to full-wave simulations, we show that, by varying the direction of an externally applied electric field, the transmissivity of the structure can be switched between 0.02 and 0.4 at a wavelength close to the frequency range where the medium is indefinite. The device functionality physically stems from the fact that the orientation of the hyperbola characterizing extraordinary waves within the indefinite medium follows the applied electric field direction and therefore, if the hyperbola asymptote is nearly normal to the slab, full switch between evanescent and homogeneous propagating waves can be achieved within the medium.
©2011 Optical Society of America
Within the rapidly growing field of metamaterials, indefinite media , or extremely anisotropic metamaterials characterized by principal permittivities having different signs, are attracting a considerable research interest mainly for the novel potentialities they provide for achieving efficient optical steering and manipulation. The hyperbolic dispersion relation characterizing extraordinary plane waves through indefinite (or hyperbolic) media is the main physical ingredient leading to unusual optical effects. Examples are negative refraction [2–4], which is due to the fact that the hyperbola curvature is opposite to that of the circumference of standard media, and hyperlensing [5–8], arising from to the medium ability of converting vacuum evanescent waves into homogeneous propagating waves  (as a consequence of the lack of evanescent waves along the hyperbola major axis). Extreme anisotropy also fundamentally affects the excitation of waves, impinging from vacuum, into an indefinite medium to the point that, by suitably changing the mutual geometric orientation between the hyperbola and the metamaterial interface, novel interesting effects as cancellation of reflection and transmission can be predicted . Consequently, optical devices exploiting indefinite media have been proposed as, for instance, polarization beam splitters  and angular filters  which are based on the fact that transverse magnetic and transverse electric waves are characterized by different dispersive features (hyperbolic and standard behavior, respectively).
The condition of extreme anisotropy, where the principal permittivities have different signs, can be attained by resorting to suitable metal-dielectric nano-composites with a geometrical structure exhibiting at least one privileged direction. Exploiting the effective medium approach, it is possible to design the composite to show indefinite permittivity in a spectral range whose wavelengths are much greater than the nano-constituents size. Two relevant examples of such structured media are nanowire composites [13–15] (typically silver nanowires embedded in an allumina membrane) and layered media [16–19] (obtained by alternating metal and dielectric layers) whose privileged directions are, evidently, the nanowire and the stacking direction, respectively. Recently, it has been proposed that indefinite permittivity can even be observed in anisotropic natural materials as, for example, graphite  and strong anisotropic uniaxial crystals  which are shown to be indefinite in the ultraviolet (due to the hybrid electronic transitions) and the infrared region (due to the strong anisotropy of polar lattice vibrations), respectively.
Dynamic light manipulation is a key ingredient of photonics and therefore investigating tunable metamaterials is an essential task. Basically, it has been proposed that if the metamaterial constituents are immersed into a liquid crystal [22, 23] or the liquid crystal is used as one of the metamaterial constituents [24–26], the overall refractive index can be controlled through an external electric field, even allowing the full switch of the refractive index sign. Remarkable is the situation where the liquid crystal [27, 28] hosts dispersed structured nanoparticles since the negative-zero-positive transition of the refractive index sign can be more simply achieved by suitably designing the nanoparticle properties (filling fraction, radius, etc.). Due to its efficiency and manufacturing simplicity, the liquid crystal/nanoparticles mixture has also been investigated by means of refined approaches accounting for the nonlocal and spatial inhomogeneities of the refractive index . Although the present research interest in tunable metamaterials is rapidly growing, it should be stressed that, to the best of our knowledge, an externally driven metamaterial with indefinite permittivity has not been considered yet, not even in the wide and versatile research field of liquid crystal. An example of such an indefinite and tunable metamaterial is the main achievement of the present paper.
In this paper we consider a liquid crystal/silver nanoparticles mixture and we show that, by suitably choosing the volume filling fraction of the particles, the anisotropic effective medium theory predicts the existence of a spectral range where the permittivity tensor is indefinite. Therefore, to the best of our knowledge, we theoretically discuss the first example of indefinite optical metamaterial which 1) is obtained by dispersing metal nanoparticles within an anisotropic matrix and 2) is tunable since, due to the presence of the liquid crystal, an external electric field is able to change its optical response. These achievements allow us to consider an active optical device obtained by sandwiching a 300 nm thick slab of the considered tunable indefinite medium between two 70 nm thick silica layers and operating in the presence of an externally applied electric field. Exploiting the effective medium theory, we show that, at optical wavelengths where the inner medium is indefinite, the slab is completely opaque if the electric field is orthogonal to its interfaces whereas it becomes gradually more transparent if the angle is increased. Such a behavior is physically due to the fact that the orientation of the hyperbola characterizing the dispersion relation of extraordinary waves through indefinite medium follows the direction of the applied electric field and therefore, when the hyperbola asymptote is close to be normal to the device interface, a switch between evanescent and propagating waves within the medium is achieved thus producing the change of the slab transmissivity. In order to prove that the considered medium does actually exhibit indefinite permittivity and that the considered device operates the predicted optical switch, we have performed full-wave (FDTD) simulations and evaluated the device transmissivity as a function of the wavelength and the applied electric field direction. We show that a spectral region where the device operates the switch exists and it is very close to that predicted by the effective medium theory and, most importantly, that, at a specific wavelength, the actual device transmissivity can be switched between 0.02 and 0.4 through the externally applied electric field. It should be stressed that such a variation of the transmissivity could not be achieved through conventional liquid crystal electro-optical switches of the same thickness L = 300 nm since such length is too small, at optical frequencies, to produce the required modal phase difference.
2. Liquid Crystal/silver nanoparticles mixture as a tunable indefinite medium
Consider a slab of thickness L with its interfaces orthogonal to the z-axis, filled with a nematic liquid crystal (LC) and sandwiched between to silica layers (see. Fig.1). If the surface region of the LC is weakly anchored to the silica walls (condition that can be achieved through suitable surface  and/or bulk  preparation methods) and if the initial homeotropic director of the LC molecules are along the z axis, an externally applied electric field E S, lying in the xz plane and forming an angle ϕ with the z-axis, is able to align the LC homeotropic director along the unit vector n̂ = sinϕ ê x + cosϕ ê z. As a consequence the LC optically behaves as a uniaxially anisotropic crystal (with optical axis along the direction n̂) and its permittivity tensor is where and are the ordinary and extraordinary permittivities, respectively and32]. According to this approach, since silver nanoparticles are optically isotropic with scalar permittivity ɛ (Ag), the overall effective permittivity tensor is so that the homogeneous effective medium is a uniaxial crystal with optical axis along the direction n̂ and its ordinary (j = o) and extraordinary (j = e) effective permittivities are Eq.(2) are the anisotropic generalization of the standard Maxwell Garnett mixing rule (to which Eq.(2) reduces in the isotropic limit where Nj = 1/3), the depolarization factors accounting for the effect of the anisotropy. Note that, in analogy to the isotropic situation, the nanoparticles radius does not appear in Eq.(2) so that the mixture design have to be performed by tuning the volume filling fraction f.
In order to discuss a feasible example of mixture, we have chosen SG-1 as nematic LC  due to its high birefringence and we have reported its ordinary and extraordinary permittivities in Fig.2(a) as functions of the vacuum wavelength λ in the spectral range between 410 and 430 nm. For the silver permittivity we have used the Drude model with ɛ ∞ = 4.56, ωp = 1.38 · 1016 s−1 and Γ = 0.1 · 1015 s−1  and we have plotted the real and imaginary parts of ɛ (Ag) in Fig.2(b) for λ between 410 and 430 nm. Note that we have used a scattering rate Γ which is an order of magnitude higher than that of bulk silver in order to accounting for the surface scattering (size effect) occurring within the considered nanometric sized particles. After setting f = 0.05 for the nanoparticles volume filling fraction, we have evaluated from Eq.(2) the principal permittivities of the effective medium and we have plotted their real and imaginary parts in Fig.2(c). Note that for wavelengths between λ 1 = 418 nm and λ 2 = 427.5 nm, and so that the effective medium has indefinite permittivity. Combining this result with the fact that the direction of the optic axis can be varied through the externally applied electric field, we conclude that the proposed LC/silver nanoparticles mixture behaves, within the limits of validity of the effective medium theory, as a tunable hyperbolic medium.
The above described design of the anisotropic effective medium has been performed for the specific value f = 0.05 of the nanoparticles volume filling fraction. In order to complete the analysis and to elucidate the dependence of the whole phenomenology on the parameter f we have evaluated the wavelengths λ 1(f) and λ 2(f) characterizing the spectral region λ 1(f) < λ < λ 2(f) where the medium is indefinite for 0.02 < f < 0.1 and plotted them in Fig.3. Note that, a tolerance Δf = 0.005 for the volume filling fraction produces the uncertainties Δλ 1 = 2.3 nm and Δλ 1 = 1.4 nm, thus proving that the mixture design is not critical.
3. Optical switch
The intriguing optical properties of hyperbolic media combined with the tunability, allow the above described LC/nanoparticles mixture to be used for conceiving optical devices. As an example, consider the setup of Fig.1 where the LC slab with dispersed nanoparticles has thickness L = 300 nm and it is sandwiched between two d = 70 nm thick silica layers (of permittivity ɛ (S) = 2.088). If a monochromatic plane wave is made to impinge orthogonally, from vacuum, onto the device interface (as sketched in Fig.1), it is evident that we are considering an active device since its optical transmissivity can be altered by varying the direction of the externally applied electric field E S. Using the effective medium approach discussed in Sec.2 to model LC/nanoparticles mixture, the device transmissivity is easily evaluated and it is reported in Fig.4(a), as a function of both the vacuum wavelength λ and the angle ϕ defining the direction of the applied electric field. For clarity purposes, in Fig.4(a) we have reported two transparent gray planes corresponding to the two wavelengths 418 and 427.5 nm between which the effective medium predicts that the mixture is hyperbolic. The most striking feature of Fig.4(a) is the existence of three relevant spectral regions, I) λ < 418 nm, II) 418 nm < λ < 427.5 nm and III) λ > 427.5 nm, where the transmissivity is characterized by qualitatively different behaviors. Specifically within regions I and III, the slab is almost transparent and completely opaque, respectively whereas, within region II the transmissivity strongly depends on ϕ to the point that T ≃ 0 for ϕ ≃ 0 and, for increasing ϕ, it gradually increases eventually reaching its maximum (dependent on λ) for ϕ = π/2 (see Fig.4(b) where the level plot of the transmissivity of region II is plotted). Therefore, within region II, the considered structure can be regarded as a very thin optical switch which, being driven by the external electric field, is able to completely stop or allow the transmission of the light flow. Physically, such an overall behavior is readily grasped by noting that within region I and III (see Fig.2(c)) the effective medium behaves as a uniaxial crystal and an anisotropic conductor, respectively and therefore, within the same two regions, the plane wave impinging from vacuum exc