1. Introduction

Coupling the air with a metamaterial that exhibits optically opposite properties, they annihilate each other acting for light propagation as an optical antimatter [1]. In such a very particular environment the light has very special properties. Indeed alternating thousands slabs of air with equal length slabs of metamaterial, with effective refractive index equal to - 1, such complementary media transmits the light in plane without diffraction, preserving the source profile of the entrance side. Using a macroscopic sample length of 4 mm, we experimentally verify [2] this theoretical prediction that intuitively can be described as a generalization of the superlens effect [3]. However in such a very special electromagnetic medium, that has a zero or quasi-zero average refractive index, other very interesting phenomena are expected. In particular a divergent beam, or an internal point source, is transformed in an extremely directive beam in medium with refractive index close to zero. We demonstrate here that, using very accurate experimental measurements, the diffracted beam along diffraction order of the grating composed by air and anti-air metamaterial has an angular dispersion as small as Δθ = 0.06°, whereas the input beam is strongly divergent $\left(\Delta \theta \sim 20°\right)$, due to the focused incident beam from the lensed fiber. We also determine that, with a great accuracy, the wavenumber of the beam propagating without diffraction in the QZAI plane is the wavenumber in the air: $2\pi /\lambda$. Then also the vertical y-component of the beam propagating within the QZAI metamaterial is practically zero, analogously to the lateral x-component, and the beam strongly confined propagates only along the z-direction, even if the incident beam is highly divergent in lateral and vertical direction [x and y in Fig. 1(b) ]. Analogously, fixing the angle of the detector and scanning the input wavelength, we obtain a very narrow diffracted peak with an extremely well defined Gaussian shape. These features allow to determine the peak location with an high accuracy and could find application in lab-on-chip sensing.

 

Fig. 1 A sketch of the experimental set-up (a) and of the grating sample (b).

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2. Quasi-Zero-Average-Index Metamaterials

Let consider a silicon (n = 3.45 for λ = 1.55 µm) air holes Photonic Crystal (PhC) slab arranged in a hexagonal lattice in the (x,z) plane with hole radius r, lattice parameter a and a ratio r/a = 0.385, Fig. 1(a). This particular PhC shows at the normalized frequency ω_n = 0.305 an almost isotropic effective index n_eff = –1 for TM polarization (the electric field directed along the holes axis). For λ = 1.55 µm, the corresponding parameters are r = 180 nm and a = 472 nm (see Fig. 1 Ref. 2 for details). In a previous paper [2] we demonstrate that a Quasi-Zero-Average-Index (QZAI) structure, obtained alternating a negative index metamaterial with air, is able to propagate radiation strongly confined along the z axis for long distance, without diffraction spread of the beam profile along x transverse direction. In this experiment it has been shown that a fundamental step to obtain such a result is the managing of the PhC layer terminations [4], in order to slightly break the zero average index condition [5]. The grating period is $\Lambda =3\sqrt{3}a=2.453\mu m$, given by the distance between two air layers (or equivalently two PhC layer, see Fig. 1 for a sketch of the grating structure). The design requires a high precision nanofabrication over large scale and involves high-voltage electron beam lithography and a gas-chopping inductively-coupled plasma etching process. An excellent agreement has been demonstrated between the Finite Difference Time Domain (FDTD) simulations and experiments on a 4 mm long sample fabricated on a SOI (Silicon On Insulator), where the oxide substrate provides the low index medium for the vertical light confinement inside the higher index silicon layer on the top. The beam propagates, along the z-direction, preserving an extremely well collimated shape in the absence of any lateral waveguide structure confining the beam, resulting in a strong macroscopic super-self-collimation effect [2]. However the half part of the QZAI metamaterial is simply air. Therefore in this portion of metamaterial there is not a simple physical mechanism that can assure the vertical confinement. At a first insight, the radiation should not be vertically confined in the air part of the grating metamaterial and propagating the radiation should be completely scattered out after few periods. This is not our experimental evidence in [2], where the beam propagates for thousands wavelength and across thousand interfaces preserving both a strongly confined lateral shape and practically the same intensity for the whole 4 mm propagation length. It appears clear that also for the surprisingly efficient vertical confinement the explanation has to be find in the coupling of air with an optically anti-air metamaterial, which has an effective refractive index opposite to the air. Using a metamaterial with permittivity close to zero, Enoch et al. produced an extremely narrow antenna pattern for microwave [6]. In general it is expected that in an epsilon-near-zero (ENZ) metamaterial a complex wavefront is transformed in a plane wave [7] and in such ENZ metamaterial a tunneling of electromagnetic waves has been predicted [8] and experimentally proven [9], [10]. The same apply also in our case even if the structure has an average refractive index close to zero whereas the local index medium is ± 1 in air and metamaterials, respectively. For such a reason the wavenumber inside is not vanishing and we cannot consider that the entire medium radiates coherently in phase [11], which is a physical mechanism that in ENZ metamaterials explains the narrow pattern in far field and the tunneling effect. It is then quite surprising that in QZAI metamaterial the beam is not only laterally confined but also very well confined in vertical direction. In such a way the light can propagate for long distances in QZAI, diffracting in far field an extraordinarily collimated beam, in correspondence of a divergent input beam.

3. Experiments

Experimental investigation has been performed in near infrared using a CW tunable laser that provides light source at λ = 1.52 ÷ 1.62 µm connected to a lensed input fiber on a 3-axis nanopositioning stages (NanoMax by Thorlabs) with a spatial resolution of 20 nm (in piezo-electrical retroaction mode) in order to align with high accuracy the injected light. A fiber coupler and collimator (resolution 0.0004 degree) mounted on a precision rotation stage (resolution 0,001 degree) in order to achieve high angular resolution on the detection system. An IR camera Xenics Xeva 185 and a high numerical aperture (NA = 0.42) objective with a long working distance was used to determine the optimal alignment and to observe the light propagating in the structure, as in [2]. See the Fig. 1(a) for a sketch of experimental setup and Fig. 1(b) for a sketch of the grating sample. The focus size of the input beam is 3.0 μm, in excellent agreement with target value of the commercial lensed fiber, and has been measured using a knife-edge technique. Measuring the angular and the spectral width of the diffracted peaks we obtain a direct quantification of the propagation length and of the angular spreading of the mode in this virtual waveguide, which has not a physical structure for the lateral and vertical confinement. In correspondence of a propagating monochromatic plane wave, the spectral resolution (or resolving power) R of the m^th–diffracted order from a grating, is directly proportional to the number of periods N that compose such a grating:

Using a tunable laser, (Ando AQ4321D, spectral range 1520-1620 nm, wavelength accuracy ± 10 pm) we measure the spectral dispersion Δλ, fixing the observation angle of the diffracted beam to θ_-1 = 21.55°, which corresponds to the m = −1 peak of the wavelength λ = 1.55 μm, see Eq. (2). The Full Width at Half Maximum (FWHM) results Δλ = (3.08 ± 0.06) nm.

Citing Born and Wolf [12] “the resolving power is equal to the number wavelengths in the path difference between rays that are diffracted in the direction θ from the two extreme ends … of the grating”. From Eq. (1) we determine that the beam propagates inside the grating for, at least, N = 516 periods i.e. for a length of NΛ = 1.265 mm. This is an underestimation of the propagation length. Indeed apart the considerations on the experimental limitations inherent in our experimental set-up, the previous formula (1) is derived within the fundamental assumption that the propagating wave in the grating is a plane wave. Eq. (1) is strictly connected with the classical grating equation that is obtained within the same hypothesis

In order to quantify the expected broadening, if we consider that a Gaussian beam appropriately describes the incident wave produced by the lensed fiber, we get the minimum divergence value. Transversally to the propagation direction an estimation of the angular beam spread in both x and y-direction is given by the apex angle of the asymptotic cone of the Gaussian beam ${\theta }_{beam}=2{\tan }^{-1}\left(\raisebox{1ex}{$\lambda $}\!\left/ \!\raisebox{-1ex}{$\pi w_0$}\right.\right)\simeq 18.5°$, that is a manifestation of the diffraction of a wave confined in the transverse direction [14]. The real value of θ_beam is probably in the middle respect to such value and the worst case arising from simple geometrical construction where the input beam from the lensed fiber has a minimum waist w = 3 μm at z = 13 μm from the fiber (core diameter W = 9 μm), with a light cone aperture ${\theta }_{beam}=2actg\left(\frac{W-w}{z}\right)\simeq 50°$. In order to explain that, in an ordinary grating, an angular divergence of the incidence wave implies a spread of diffracted peaks in Eq. (2), let consider that for a divergent incident beam the x and y components are non-zero, k_ix≠0 and k_iy≠0, and consequently the incident wavevector component along z is reduced. For instance in the limit case of a pure grazing incidence θ_i = 90° (k_iy = 0), the spread in the propagation plane, k_ix≠0, imply that z-component of incident wavevector is consequently reduced $k_{iz}=\sqrt{{\left(2\pi /\lambda \right)}^2-k_{ix}^2}\le 2\pi /\lambda \equiv {k_{iz}|}_{\max }$respect to the maximum value,${k_{iz}|}_{\max }$. Clearly the same is true as a consequence of the divergence in the vertical direction, k_iy≠0. Then at first insight a spread of incident k-wavevector means a spread of k-wavevector interacting with the grating and finally a spread of diffracted peaks. This is not the experimental evidence, where spectral and angular peaks are extremely narrow and for such a reason we defined such a radiation as extraordinarily collimated: an ordinary grating cannot give such a high collimation. We can expect that x-components propagates in QZAI metamaterial only on the small scale where the periodic re-focusing is produced, whereas globally the beam is strongly collimated and in the grating plane (x,z) only the z-component of the wavevector transport the energy, as we previously determine by imaging measurements. Such a self-collimating effect is explained by the coupling of materials with opposite refractive index, optically annihilating each other the light propagation. In terms of transformation optics this annihilation is equivalent to a space folding of the optically opposite regions, then both regions in the transformed space optically disappear. The folding applies in the (x,z) plane without considering the y-direction. On the one hand, due to the thickness of the SOI structure, the negative refraction properties of the photonic crystal are correctly derived for a 2D structure considered as infinite along the y-direction, putting k_iy = 0, being the effective index of the fundamental mode supported by the SOI waveguide practically coincident with the index of the bulk silicon [2]. On the other hand the same cannot be straightforwardly applied, in the vertical direction, to the air part of the structure. Experiments prove that the mode supported in the 2D PhC with k_iy = 0 propagates in the whole structure, shrinking the light in x- and y-direction. Indeed we measure the wavevector y-component propagating in QZAI measuring the spread of angular peaks. Considering the divergence θ_beam into the sin θ_i term in grating Eq. (2) around the central value θ_i = 90° the diffracted angle θ_m spread of few degrees. Then an accurate measurement of the diffracted angle is also an accurate measurement of the k_0z propagating in the grating. Limiting to a grazing incidence study, sin θ_i = 90°, and considering λ = 1.55 μm only three orders (m = –1, –2, –3) in Eq. (2) satisfy the condition $\left|\sin {\theta }_m\right|\le 1$ and can propagate in air. The corresponding diffracted angle are θ_-1 = 21.55°, θ_-2 = −15.38°, θ_-3 = −63.89°. In Fig. 3 the peaks position θ_-1 and θ_-2 are located in excellent agreement with expected values for sin θ_i = 90°. From one side this confirms that modulus of the propagating wavevector in z-direction is equal to the free space value of the wavenumber: k_hz = 2π/λ. Moreover, a fine angular scan around θ_-1 (inset of Fig. 3) provides a measurement of extraordinary directivity of the scattered beam Δθ_{m = −1} = (0.061 ± 0.003)° in excellent agreement with our previous estimation based on the spectral resolution. From former discussion we know that the spread of diffracted peaks, in angular or in spectral domain, is due to both finite propagation length in the grating and from the deviation of the coupled z-component from its maximum value, as a consequence of the beam spreading in vertical and lateral direction. In absence of the self-collimation and vertical confinement effects, the strong beam aperture would be projected along the grating direction and finally would broad the diffraction peaks.

 

Fig. 3 The angular scan at λ = 1550 nm and the zoom in the range around θ_-1 (inset) . The full width half maximum is FWHM = (0.061 ± 0.003)°.

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These measurements definitely show that a metamaterial with average refractive index close to zero produces an extremely collimated beam in correspondence of a divergent excitation [9] an extraordinary result that cannot be explained without the unusual properties of metamaterials. We underline also that peaks are not only very narrow but they also fit extremely well a Gaussian shape: in the normalized spectral dispersion of Fig. 2 the coefficient of determination $R^2=1-S{S}_{err}/S{S}_{tot}=0.99061$ is very close to unity, being very small the ratio of sum of squares of residuals $S{S}_{err}={\displaystyle {\sum }_i^N{\left(o_i-g_i\right)}^2}$ with the total sum of square $S{S}_{tot}={{\displaystyle {\sum }_i^N\left(o_i-\overline{o}\right)}}^2$, where o_i are the observed values, g_i are the values of the fitted Gaussian function and $\overline{o}=1/N{\displaystyle {\sum }_i^N{o}_i}$is the mean of observed data over the N observations [15]. The goodness of such fit means that Gaussian model likely predicts future observations in particular if, after a suitable calibration, the presence of the interesting analyte to the surface of this device produces a small shift of the resonance frequency. It is outside the scope of the present paper a detailed analysis of performances of this device as an integrated spectrometer. Anyway the very high goodness of fit open to future exploration of this open structure as integrated spectrometer, with a very small lateral dimensions (few microns) and an high accuracy at least comparable with that of a compact spectrometer integrating negative refracting elements [16]. Finally we underline that we do not expect a Gaussian shape for the diffracted peaks and also the finite size of the grating would produce a sinc function, instead of a Gaussian peak. Then Gaussian shape are determined from the optical set-up, in particular from the collimator element mounted over the rotational stage, which has a larger Gaussian shape that convolutes the diffracted peaks, smaller compared to the peaks of Fig. 2 and Fig. 3.

 

Fig. 2 Measured spectral dispersion at θ_-2 = –20,7, (black square) fit perfectly with a Gaussian curve (red line).

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3. Conclusions

The extraordinary collimation of the diffracted peaks and equivalently the extremely narrow spectral resolution, demonstrate the special properties of the beam propagation in a quasi zero refractive index metamaterial on a macroscopic scale. Together with later confinement, we prove the extremely singular propagation in this metamaterial environment, where the beam propagates as in a conventional waveguide even without lateral and vertical structures that ensure such a confinement. Finally we underline that this metamaterial has an open configuration, where the air that strongly interact with external solicitations composes roughly half part of the structure. High precision sensing spectroscopy is then possible over a macroscopic region in a sample exhibiting a very large contact area with environment. The strong lateral confinement allows a probe of a very narrow region of few microns close to the interface due to the strong vertical confinement.