We develop a new concept for optical sub-wavelength discrimination of nanostructures positioned on a hyperbolic metamaterial (HMM) surface using only far-field measurements of the spectral scattered intensity. The device demonstrating our approach consists of an HMM structure, one strong scatterer on top of the HMM at an a priori known location, and the target nanostructures to be sensed on the opposite interface of the HMM. Under far-field illumination from the top, highly localized beams are generated within the HMM by the top scatterer, with wavelength-dependent propagation directions. If the incident wavelength is varied, the localized beam will sweep across the target(s) and the far-field scattered intensity versus wavelength from the bottom target(s) can be measured. We show that this far-field measurement can be used to determine the location of the target(s) with deep-subwavelength accuracy. In addition, discrimination of targets made from different materials can be achieved.
Metamaterials are artificial inhomogeneous structured media with the scale of inhomogeneity that is much smaller than the wavelength of interest; through engineering their structure, special optical properties not existing in natural materials can be realized.1 In the effective medium approximation, the response of most metamaterials can be characterized by macroscopic effective permittivity ε and permeability μ tensors. The hyperbolic metamaterial (HMM) is one type of metamaterial that exhibits hyperbolic dispersion, i.e., when the real parts of two different primary components of either the dielectric permittivity (electric HMM) or magnetic permeability (magnetic HMM) tensors have opposite signs. Electric HMM (magnetic HMM) can be either classified as type I if ε⊥ > 0 and ε∥ < 0 (μ⊥ > 0 and μ∥ < 0) or type II if ε⊥ < 0 and ε∥ > 0 (μ⊥ < 0 and μ∥ > 0),2 where ε⊥, μ⊥ represent the tensor component perpendicular to the optical axis and ε∥, μ∥ represent the tensor component parallel to the optical axis. HMM can support high-k modes3 and forms the basis for many potential applications, from enhanced spontaneous emission4 to subwavelength imaging via a hyper-lens5 and more recently to hyperstructured illumination.6
If the incident field illuminates an object present near an HMM, the scattered field excites volume plasmon polariton (VPP) modes7 inside the HMM (with dispersion relation as provided in supplementary material Sec. I); these modes have a highly localized field pattern and propagate along a well-defined direction, as illustrated in Fig. 1(c). This highly localized beam has subwavelength width, and its propagation direction is strongly wavelength dependent. These properties together have enabled a new approach to subwavelength imaging known as hyperstructured illumination:6 the localized beam leads to a far-field scattering pattern E(r, ω) from the object, which could be used to reconstruct the Motti projection6,8 of the object with deep subwavelength resolution (see discussion in supplementary material Sec. II). Reconstruction of the images of sub-wavelength objects using hyperstructured illumination in HMMs, however, requires coherent detection since both phase information and amplitude information are needed for complete image reconstruction. It is thus interesting to inquire to what extent it is possible to resolve subwavelength structures using intensity-only measurements in the far field.
In this paper, we show that, indeed, a spectral scan of the scattered intensity can be used to determine the positions of sub-λ objects (in this work, specifically dielectric and metallic scatterer objects are considered). Similar to hyperstructured illumination,6 we take advantage of the deep subwavelength resolution from these highly localized beams; however, our goal is not imaging but instead obtaining the nanoscale object’s spatial and material information by matching the measured spectral characteristics to known records, which we refer to as “fingerprinting.” We use a 2D finite element analysis (COMSOL 5.2) to demonstrate the feasibility of nanoscale fingerprinting (see supplementary material Sec. III for COMSOL implementation details).
In our simulations, the type II HMM device consists of 8 pairs of alternating 20.4-nm Ta2O5 and 6.5-nm Al-doped Ag layers.9,10 The device structure without the target object is shown in Fig. 1(a). An Al-doped Ag scatterer is placed 1 nm above the HMM and is illuminated with a normally incident plane wave. The small distance between the scatterer and the HMM enables coupling of the scattered field into VPP modes inside the HMM. The layered structure is uniaxially anisotropic, and under the effective medium theory (EMT) approximation,11 the structure has Re(ε⊥) < 0 and Re(ε∥) > 0 for wavelengths larger than 647 nm and behaves as the type II HMM (a calculation of the effective dielectric constants within the EMT approximation is given in supplementary material Sec. IV). We note that the use of a type II HMM is necessary in this scheme: for the type I HMM, the direct transmission of the incident field is high and will act as background noise in the experimental measurement. On the contrary, for the type II HMM, most of the incident field is reflected back toward the light source and does not contribute to the measurement noise. This is evident by considering the iso-frequency curve shown in Fig. 1(b): there is no mode with k⊥ = 0 (normal incident) supported by HMM. Figure 1(c) shows a typical scattered field distribution when the HMM device is illuminated by a plane wave (with an out-of-plane magnetic field) at a wavelength of 1200 nm. Note in the figure that two localized beams (VPP modes) with subwavelength beam width are generated in the HMM. The angle between the beam propagation direction and the x axis in the EMT limit is given by
For a small scatterer, the scattered light will have mostly large k⊥ components, so we may approximate k⊥ as , where ω is the angular frequency and c is the speed of light; in that case, Eq. (1) can be derived by noting that the group velocity direction is perpendicular to the iso-frequency curve, as illustrated in Fig. 1(b). (Further details are provided in supplementary material Sec. I.) In Fig. 1(d), the dashed lines show the beam angle obtained from EMT and a strong wavelength dependence is clearly apparent. The exact simulation [asterisks in Fig. 1(d)] also shows a similar wavelength dependence, and the difference between the two will be discussed below.
This wavelength dependence, as we will see, is central to our nanoscale fingerprinting concept. The physical processes illustrated in Fig. 1(c) are summarized as follows: The top scatterer generates a scattered field under excitation. The propagating components and some evanescent components () of the scattered field are totally reflected at the top HMM/air interface (see supplementary material Sec. I), while the remaining evanescent components () of the scattered field from the top scatterer are coupled to the propagating VPP modes in the HMM and form the highly localized beams. The propagation and reflection of the beams at the HMM/air interfaces leads to a zig-zag field pattern. At the bottom HMM/air interface, the beams couple to evanescent modes in air, which gives rise to a highly localized field distribution and can be scattered by nanoscale objects positioned nearby. This highly localized field distribution is wavelength dependent and enables nanoscale fingerprinting; two possible configurations are shown in Figs. 2(a) and 2(b).
In the configuration of Fig. 2(a), a target object to be identified (with refractive index n = 1.73) is placed below the HMM device, with a certain spacing relative to the top scatterer. By sweeping the wavelength of the incident light, the beam propagation direction θ will change, as indicated by asterisks in Fig. 1(d), and the target scatters strongly only at the wavelength for which the beam is localized close to the scatterer. Figure 3(a) shows the scattering strength, defined as , measured as a function of wavelength for different spacing (see supplementary material Sec. V for calculation of the scattering strength). It can be seen in the curve that targets located at different spacing differ in their peak scattered wavelength and the peak position shifts to longer wavelength monotonically as the spacing is increased, as expected from the wavelength dependence of the beam angle. Since objects at different spacing differ in their peak position, this spectral curve serves as the “fingerprint” for us to identify the target’s location.
Next consider the case in which two targets (again with the refractive index n = 1.73) are placed below the HMM with a gap between them, as shown in Fig. 2(b). Figure 3(b) shows the scattering strength versus wavelength for three different gap sizes. For a gap size of 60 nm, two peaks are clearly visible and the peak at shorter wavelength is due to the scattering from the target object closer to the top scatterer. For a gap size of 20 nm, two targets are close enough that they are just barely resolved. For a gap size of 0 nm, only one peak is present as expected. Again, the spectral shape encodes the bottom targets’ spatial position and serves as the “fingerprint” for us to determine the gap, which is deep-subwavelength in size.
We next consider the case in which two bottom targets in the configuration of Fig. 2(b) consist of different materials. Since the scattered power increases as the refractive index contrast of the target to the air is increased (see additional discussion in supplementary material Sec. VI), the magnitude of the spectral peak contains material information about the targets and helps to identify the target’s material composition, in addition to their spatial information during the “fingerprinting” process. Four material combinations using Ag and Si are simulated at a fixed gap size of 100 nm to show that a calibrated measurement can in principle distinguish their material composition information in addition to the spatial information, as shown in Fig. 3(c). Two peaks can be seen for each material combination in the plot, indicating that there is a resolvable gap (100 nm) between two bottom targets. In addition, each curve has its unique spectral shape, which can be used for “fingerprinting” to determine material information. Finally, we show in the supplementary material that changing the bottom target shape does not change the scattering strength significantly. As a result, the proposed “fingerprinting” process is robust to unintended small structure variations of the bottom target.
We will now return to the cause of the beam angle difference between the EMT calculation and the exact simulation, as observed in Fig. 1(d). This difference is due to the finite thickness of the metal/dielectric layers composing the HMM, which leads to a deviation in the exact iso-frequency curve from the EMT case, where the structure is assumed to be homogeneous. This deviation due to the finite thickness of the composing layers has been previously reported.12 As the ratio wavelength/unit cell size of the HMM structure becomes smaller, the EMT approximation should improve. This may be seen in Fig. 1(d), in which the beam angle versus wavelength/unit cell size using exact simulation at different wavelengths is plotted (solid lines). Note that the beam angle obtained via exact simulation (solid lines) approaches the value obtained from the EMT (dashed lines) as the unit cell size is reduced, and it is very close to the EMT result when an extremely small unit cell size (wavelength/unit cell size = 400) is used. This also reveals that the device operates in a very large k⊥ regime: the excited high k-modes in HMM require a very small unit cell size for the EMT to be accurate. It is worth noting that in the proposed 2D structure, the scattered field is transverse-magnetic (TM) polarized and couples to the extraordinary wave (e-wave) of hyperbolic dispersion in HMM under TM excitation. There is no transverse electric (TE) polarized component coupling to the ordinary wave (o-wave) in HMM. However, in a more realistic 3D geometry, the scattered field will have both TE and TM components and the transmission of TE components through HMM may reduce the signal-to-noise ratio on the measurement due to an increased background.13 We give an estimate of the noise magnitude compared to the signal in the 3D geometry in supplementary material Sec. VIII.
In summary, we have demonstrated a novel concept based on HMM that is able to discriminate nanostructures. The proposed “fingerprinting” process can be used to determine the location of a single target with deep-subwavelength accuracy, resolve a nanoscale gap between two targets, and obtain information on the target material. Similar devices working in other wavelength ranges can be achieved by appropriate modifications of the HMM design.
This work could potentially find applications in metrology or in biomolecular measurements. For example, determining the separation between two biomolecules with nanoscale separation is important and can be used to study interaction between proteins, such as dimerization of motor proteins14 or association of regulator proteins.15 Foster resonance energy transfer16,17 (FRET) methods are commonly used for this purpose; however, they are typically applicable when the separation is in the range 1-10 nm, and it is challenging to provide an absolute distance estimation between molecules in FRET because the energy transfer and fluorescence process are sensitive to the environment and orientation of the dyes18 and photobleaching of the dye molecules hinders distance estimation for a long period of time. On the other hand, our proposed method does not have the aforementioned limitations and works best in the separation range of tens of nanometers, which is hard to reach using FRET.
See supplementary material for information about volume plasmon polariton modes in HMM, localized beam angle calculation, hyperstructured illumination and Motti projection, COMSOL implementation, EMT calculation of HMM permittivity tensor components, calculation of the scattering strength, target material/shape dependence of the scattering strength, and 3D geometry noise analysis.
This work was partially supported by the National Science Foundation through NSF Grant No. DMR 1120923, the National Science Foundation Grant No. 1629276-DMR, the Army Research Office Grant No. W911NF-14-1-0639, and the Gordon and Betty Moore Foundation.
We acknowledge Dr. L. J. Guo and Dr. C. Zhang for kindly providing the permittivity values of Al-doped Ag and Ta2O5.