Negative phase advance through a single layer of near-IR negative index metamaterial (NIM) is identified through interferometric measurements. The NIM unit cell, sub-wavelength in both the lateral and light propagation directions, is comprised of a pair of Au strips separated by two dielectric and one Au film. Numerical simulations show that the negative phase advance through the single-layer sample is consistent with the negative index exhibited by a bulk material comprised of multiple layers of the same structure. We also numerically demonstrate that the negative index band persists in the lossless limit.
© 2010 OSA
Negative index metamaterials (NIMs) were demonstrated, first in the microwave regime , and subsequently at near-infrared or higher optical frequencies [2, 3]. While there have been a number of reports of optical NIMs  including a recent prism refraction experiment at a wavelength of λ = 1.5μm , interferometric measurements of the phase advance of light through optical metamaterials [3, 6–8] have been rare. In the few such reports on optical NIMs [3, 6, 8], the relationship between the measured phase advance through samples consisting of only a single layer of unit cells and that expected from transmission through hypothetical multi-layer bulk NIMs was not investigated. Moreover, most of the reported optical NIMs [2–6] have a unit cell size approaching λ/2 in at least one dimension. This relatively large unit cell makes their description as effective media problematic [9, 10].
Recently, we reported a significantly subwavelength unit cell (~λ/7 in both the lateral and light propagating directions) optical NIM structure  that exhibits negative index effects at λ ≈1μm. Through photonic band calculations, this structure was shown to display a bulk negative effective index . Furthermore, a single layer of unit cells was also theoretically shown through the S-parameter method to possess a negative effective index . Structural characterization and optical transmission measurements of a fabricated single-layer structure  indicate that its physical dimensions are consistent with the design of the NIM structure. In this work, we present experimental evidence for negative phase advance through this sample using a polarization interferometer. Numerical scattering simulations further suggest that the negative phase advance though the sample is consistent with that exhibited by a bulk, multi-layer material. Moreover, using numerical simulations that include artificially low metallic losses, we confirm that the negative effective index exhibited by this structure is not the result of high losses at optical frequencies .
2. Experiment: sample and interferometric characterization
A schematic of the near-infrared NIM unit cell , extending infinitely along the direction , is shown in Fig. 1 . It consists of a pair of Au strips, separated from a continuous Au layer by polymer (cyclotene, BCB) dielectric spacers. The unit cell has a y-period of Ly = 150nm. For transverse magnetic (TM) polarized (H = Hz ) electromagnetic waves, the unit cell supports even- and odd-Hz resonant modes with respect to the central plane. Currents flow in opposite (same) directions along the strips in even (odd) resonance, which is associated with an effective magnetic (electric) dipole response that is weakly (strongly) dependent on the presence of the central metallic layer . Near λ = 1μm, this structure therefore has a negative effective index for the TM polarization propagating in the x-y plane along the direction . The orthogonal (TE) polarization is cut off inside the metamaterial which serves as a wire grid polarizer. However, transmission of the TE polarization  is sufficient for conducting polarization interferometry experiments described below.
The NIM used for the interferometric measurement was fabricated using a combination of optical and electron beam lithography in 100μm x 100μm sample blocks on a glass substrate . The dimensions of the NIM obtained by fitting the transmission spectra  give an Au strip and central layer thickness of t = 22nm, strip width W = 105nm, and thicknesses ht = 15nm and hb = 9nm for the top and bottom dielectric spacer layers, respectively. The refractive indices used for theoretical fitting are 1.45 for the glass substrate, and 1.56 for the dielectric layers, as obtained from ellipsometry. The optical constants for Au are obtained from the literature .
Our approach to experimentally characterize the negative phase advance of the TM-polarized light is to interfere it with the TE-polarized light propagating through the same sample, and then to measure the relative phase advance () using a polarization interferometer. There are two important features of this approach. First, both polarizations propagate through the same sample, so there is no need to account for their geometrical path difference. Second, it can be analytically shown (see below) that, because of the cutoff nature of the TE mode, . Therefore, is negative whenever
A polarization interferometer (Fig. 2 ) is used to measure the phase shift between orthogonally polarized states of light following a technique similar to that in . We employ a tunable Ti:Sapphire ultrafast oscillator (~150-fs duration pulses at a repetition rate of 80 MHz) to perform measurements from λ = 700 nm to 980 nm.
The laser beam is passed through polarizer P1, oriented at 45° relative to the vertical. The polarization after P1 is the vector sum of horizontal and vertical components with identical phase. A microscope objective focuses this beam to a ~10μm diameter spot centered on the sample block. The sample is oriented such that vertically (TM) polarized light is expected to undergo a negative phase advance. Transmitted light is re-collimated by a second microscope objective. The two co-propagating polarization states then pass through a liquid crystal variable phase retarder (LC) that introduces a controllable phase shift between the two polarizations. A compensator plate bonded to one window of the LC ensures this shift passes through zero at sufficiently high LC drive voltages. The light then passes through a second polarizer, P2, oriented parallel to P1, and is finally incident on a Si photodiode.
The photodiode voltage is proportional to the induced photocurrent and the incident optical power. Taking a time average over the period of optical oscillations, the measured photodiode voltage, , can be written as:
The extrema voltages of the photodiode are given by:Eqs. (1-2),
Figure 3a plots the photodiode voltage as a function of the LC drive voltage. These data are taken at a given wavelength for a reference sample consisting of an unstructured gold layer on the same substrate as the NIM. Using Eq. (3), we plot the wrapped phase () as a function of LC drive voltage in Fig. 3b. The periodicity of the cosine function introduces ambiguity in extracting the phase advance. However, the LC phase shift monotonically decreases with increasing drive voltage. Since is within the domain (-π, π), this is sufficient to accurately determine the phase advance, obtaining the calibration result in Fig. 3c. As the LC phase advance is wavelength dependent, it is necessary to obtain calibration data for every wavelength of interest.
Equation (1) shows that the maximum photodiode voltage occurs when the phase shift due to the NIM is the additive inverse of the LC applied phase, which may be obtained from the calibration curve in Fig. 3c. We thus obtain the relative phase shift due to the NIM by finding the LC drive voltage that produces the maximum of the photodiode voltage in Fig. 3d. Note that the relative phase shift is determined only up to an additive constant of (m is an integer. For sufficiently thin samples, however, the effective index determined for m = 0 corresponds to the photonic band structure of the unit cell within the first Brillouin zone, which is consistent with Snell’s Law in a refraction experiment .
3. Results and discussion
Numerically simulated , , andbetween the two polarization states using the sample dimensions in Fig. 1 are plotted as continuous lines in Fig. 4a , where the TM phase advance dominates the relative phase shift, as expected . In these simulations, plane electromagnetic waves are assumed to be normally incident on, and scattered by the sample. The measured relative phase advances obtained for two sample blocks on the same substrate, are also plotted in Fig. 4a (open symbols). The random error in the photodiode voltage is <1%, as determined by repeated measurements. The uncertainty in LC drive voltage that corresponds to the maximum photodiode voltage is more significant due to the zero slope of the data at this point, resulting in an error of approximately 0.2 rad in phase retrieval. Finally, the uncertainty in the actual sample dimensions may also be a significant source of error, as indicated by the differences between data taken on different sample blocks with identically designed structures.
To examine whether the negative TM phase advance exhibited by a single layer is consistent with the properties of a bulk medium with multiple layers of similar unit cells, the band structure was calculated for TM polarized electromagnetic modes propagating in the -direction, assuming a periodic array of complete square unit cells (inset of Fig. 4b)  capped with polymer dielectric layers with the same parameters as the fabricated sample in Fig. 1 . When the number of layers is sufficiently large, the propagating electromagnetic wave is a Bloch mode, and the phase advance per unit cell, , is the bulk property , as is shown in Fig. 4b. The negative between λ = 880nm and 1100nm, therefore, indicates a negative index band for light transmitted through a similarly structured periodic bulk medium. The negative index band of the bulk NIM (black line in Fig. 4b) resembles that simulated for a single layer (blue line in Fig. 4a). This is due to the single cell resonance  that is responsible for the negative index band and the weak inter-layer interaction, which is necessary for inferring the bulk optical properties from single layer measurements . There is, however, a substantial shift between the TM phase advances from scattering simulations in Fig. 4a and the calculated band structure in Fig. 4b. In particular, the node and discontinuity (due to the band structure of the NIM that determines its effective index ) on the frequency axis that delimit the negative index band are shifted.
To understand the shift of the node, we exclude the influence of the substrate by numerically simulating the phase advance by a single layer of the complete unit cells, as shown in Fig. 4b. It is significantly different from the phase advance derived from band structure due to surface effects of the single layer. Indeed, the simulated phase advance per unit cell by ten stacked, repeating layers is almost identical with that obtained from the band structure.
The discontinuity in the phase advance at λ = 1100nm for both the single layer and ten stacked layers in Fig. 4b agrees with that of the band structure. The experimental sample in Fig. 1 is on top of glass substrate but without the capping polymer dielectric layers. To ascertain whether the shifted discontinuity at λ = 970nm of the TM phase advance in Fig. 4a is due to the presence of the glass substrate or the absent capping polymer dielectric layers, we simulated the phase advance using a complete unit cell on a substrate. In this case, the discontinuity remains at around λ = 1100nm. Hence, we infer that the frequency shift of the discontinuity in Fig. 4a is due to the missing capping dielectric layers in the fabricated sample.
Note that the phase advance for the tunneling TE mode is also negative. This phase advance occurs at the entrance and exit interfaces, and is a consequence of the sub-barrier tunneling amplitude  across a barrier of thickness D which presents a purely imaginary refractive index to the tunneling wave:4]. For our NIM structure, even assuming a much reduced Au absorption from its actual value (e.g. 10% of the imaginary part of its actual permittivity), the calculated phase advance per unit cell (green line in Fig. 4b) still indicates a persistent negative index band, although it is narrower than that calculated taking into account the actual loss of the Au strips (black line). In fact, the unit cell structure in Fig. 4b with two different spacings between the central continuous film and the metallic strips can be described as a bi-anisotropic medium, due to its lack of inversion symmetry . Still assuming 10% of the imaginary part of Au permittivity and using the approach in Ref . for extracting effective parameters from a reciprocal  bi-anisotropic medium, we calculated the effective index (nx), permittivity (εy), permeability (μz), and magnetoelectric coupling coefficient (ξ0) for our structure. The results are shown in Fig. 5 . The sign of Re(nx) is determined by requiring Im(nx)>0. In the (shaded) majority of the negative index band extending from λ = 1000nm to 1070nm, the imaginary parts of all of the effective parameters are negligible compared with their real parts. Within this low-loss region, the dispersion relation of a propagating mode in a bi-anisotropic negative index medium , , requires simultaneously negative real parts of both the effective permittivity and permeability, regardless of the sign of Re(ξ0). Indeed, Re(εy) and Re(μz) are both negative in this wavelength region. We caution, however, that although the negative effective index is capable of predicting the phase advance and the refraction angle in a prism experiment , these local homogeneous material parameters are approximate, and predictions based on their values may still deviate from the macroscopic electromagnetic behavior of the composite, as we have shown previously .
The relative phase advance by a single layer of a near-infrared NIM structure has been measured, and is consistent with numerical simulations. The simulated absolute negative phase advance of TM waves by the sample resembles the expected phase advance due to the bulk properties of an infinite periodic medium calculated from the band structure of the NIM unit cell. Apart from fabrication and experimental uncertainties, the disagreement between experiment and the predictions of phase advance from the band structure model is likely due to surface effects introduced by employing only a single-layer unit cell sample lacking capping dielectric layers. Simulations also indicate that a sample comprising a few layers of unit cells exhibits a phase advance more similar to the bulk property.
We thank the AFOSR MURI contract FA 9550-06-01-0279 for financial support. Experimental work at UT-Austin is supported by: NSFDMR-0747822, ONRN00014-08-1-0745, AFOSR FA9550-08-1-0058, AFOSR FA9550-10-1-0022, and the Sloan Foundation.
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