We theoretically propose a multilayered polar-dielectric superlens system capable of sub-diffraction limited imaging simultaneously at different wavelengths. Our theory and simulation results show that this multilayered lens can fulfill a superlensing condition at multiple different wavelengths due to phonon resonances of polar dielectrics, and the number of superlensing wavelengths of the lens can be easily tuned by controlling the number of polar dielectrics. Ideally, by suitably choosing polar dielectrics, our lens can cover wavelengths ranging from infrared to THz frequencies.
©2012 Optical Society of America
Metamaterials composited of subwavelength artificial structures have sparked great interest because of their special electromagnetic properties and already introduced many different applications like negative refraction leading to high-resolution microscopy and lithography [1,2], electromagnetic cloaking [3,4], and transformation optics [5,6]. Among those interesting applications, the superlens is a very promising device to overcome the diffraction limit and resolve subwavelength features [7–12]. Usually, the spatial resolution of conventional optical microscopy is limited to about one-half of the illuminating wavelength because the information carried by evanescent waves is lost in the far field. However, the resolution of a superlens is not limited by diffraction since near-field information can be recovered by using a thin negative permittivity slab to enhance evanescent waves .
This superlensing effect has been demonstrated by different material systems at near-UV  and infrared (IR) wavelengths [9–12]. In particular for the case using a scattering-type scanning near-field optical microscopy (s-SNOM) , one can image an object with both amplitude and phase information, due to the fact that a sharp tip of the s-SNOM can locally collect near-field information of the image transferred by a silicon carbide (SiC) superlens. An obvious advantage of this new superlens-based microscopy system is that the probing tip does not have to be very close to the sample, and therefore opens a door for mapping otherwise inaccessible objects.
The s-SNOM is a powerful scanning probe technique [13–17], overcoming the diffraction limit from visible  to terahertz (THz) spectral regime . Due to its frequency-independent character, the s-SNOM can probe IR vibration absorption spectroscopy of chemical composition  and THz mobile carrier contrast in doped semiconductors . Yet, the superlens has only a very narrow bandwidth, since superlensing requires a matching condition εlens(λ) = −|εhost(λ)|, where εhost is the permittivity of the host medium interfacing the lens . For dispersive materials, this condition limits operation to a small wavelength range for each material system. For example, the condition is met at a wavelength of around 365 nm for silver–polymer combination , and 11 μm [9,10] for SiC-SiO2 case. Therefore, to cover a wider wavelength range, a series of superlenses consisting of different material systems must be used together. Recently, Kehr et al. showed that multiple different perovskite-based superlenses could cover a frequency range from 18.6 to 22.2 THz [11,12]. However, considering the convenience of practical applications, broadening the operation wavelength range of a single superlens is favorable.
In this paper we propose a possibility of increasing the number of superlensing wavelengths (SWs) based on a multilayered (ML) design, which can lead to subwavelength imaging at different wavelengths with only one single lens. Our theory and simulation results show that the matching condition of the superlens can be fulfilled at multiple distinct wavelengths by exploiting different phonon resonances of polar dielectrics. Considering the huge abundance of suitable material candidates in different wavelength ranges  (for examples, doped oxide in near-infrared, III-V semiconductors in mid- and far-infrared) and current fabrication techniques (deposition techniques , doping methods  or film-growth approaches [11,12]), fabrication of a device capable of superlensing at several different wavelengths simultaneously may indeed be a realistic possibility.
2. Basic idea
Since a three-layered superlens  must match as εlens (ω) = −|εhost(ω)|, such a lens usually consists of noble metals or polar dielectrics, the permittivity of which can be negative in certain wavelength range due to plasmon resonance (or phonon resonance). Considering their frequency dispersions, usually one suitable wavelength for superlensing can be found around a phonon resonance [9–12]. From this point, designing a multi-wavelength (MW) superlens requires to choose a material with multiple resonances. However, this is not feasible because of the limited number of resonances in real materials.
In this paper our design is based on a ML superlensing theory, which has been discussed recently [21–23]. Since adding different polar dielectric layers can provide additional degrees of freedom for more resonances, a MW superlensing can be easily achieved in a single superlens. According to that theory, a ML system (see Fig. 1 ) consists of two different materials and can be seen as an effective anisotropic material with22, 23], a superlensing effect can be found in this design. Considering a simple case of d1 = d2, this condition reduces to å1 = −å2. Different from a simple three-layered superlens [7,10–12], this condition can be fulfilled in two cases (see Fig. 1): (1) å1>0, å2<0 and (2) å1<0, å2>0. In most of ML superlensing designs, permittivities of investigated materials will not change sign in the wavelength range of interest, meaning that only one matching wavelength can be obtained [21–23]. In fact, the sign of permittivities in many real materials will be changed in distinct wavelength ranges. Polar dielectrics, such as SiC, indium phosphide (InP), indium antimonide (InSb) and so on, would be typical examples. As shown in Fig. 2 , their dielectric permittivities (obtained using oscillator models from ) are negative in the so-called reststrahlen band between the transverse and longitudinal optical-phonon frequencies, which gives rise to the existence of so-called surface phonon polaritons . When the wavelength is out of the reststrahlen band, their permittivities become positive (see also Fig. 1), and thus the material just acts as a “dielectric filler” material. Exploiting different phonon resonances of multiple polar dielectrics can easily provide a possibility of superlensing simultaneously at multiple wavelengths. It is clearly seen in Fig. 2 that the conditions for a SiC-InSb system are matched at two distinct wavelengths: Re[åInSb] = −Re[åSiC] = 15.6 at λ1 = 11.7 μm, and Re[åInSb] = −Re[åSiC] = −9.9 at λ2 = 53.7 μm. We also mention that the respective absorption of the two materials is small at the matching wavelengths (Im[åInSb]≈0.1, Im[åSiC]≈0.5 at λ1 and Im[åInSb]≈0.8, Im[åSiC]≈0.1 at λ2), which will significantly improve the resolution of our superlens. Moreover, thanks to the abundance of polar dielectrics, required wavelengths can be easily obtained with different composites: for example, λ1 = 11.7, λ2 = 30.3 μm for SiC-InP case, and λ1 = 30.3, λ2 = 50.3 μm for InP-InSb lens. Based on these very promising dispersion properties, we will present detailed results of these MW sub-diffraction imaging systems below, starting with the SiC-InSb system.
To confirm the MW superlensing effect, we first perform numerical simulations using commercial finite-element software COMSOL. Our simulated structure is a simple four-layered structure composed of alternately InSb and SiC with a thickness of d = 200 nm for each layer. We assume that an Au layer of 60 nm thickness is deposited on the object side of the lens, and a double slit of 500 nm width with 1 μm center-center separation is cut through the Au film. In all the simulated cases the structure is excited by a uniform plane wave impinging from the top of the figure. Figure 3 (top plots) shows the simulated distribution of magnetic field H for the designed structure at two different SWs (a) λ1 = 11.7 μm and (b) λ2 = 53.7 μm. Strong field enhancement can be found at the InSb-SiC interface at both wavelengths, indicating the enhancement of evanescent waves. The simulated electric-field intensity at the image plane is also illustrated in Fig. 3 (bottom). For the case without our lens, only a single intensity peak is obtained, which means that two slits are not resolved (red dashed lines) because of the diffraction limit. However, when our lens is applied (blue solid lines), two peaks for the double slit with 1 μm separation are observed at both wavelengths. These results definitely confirm the MW superlensing effect of our lens and suggest that at least λ/10 resolution at λ1 = 11.7 μm and λ/50 resolution at λ2 = 53.7 μm are obtained. We also note that two peaks are not so well resolved at λ = 53.7 μm, although the performance of our lens is already good enough considering the fact that the double slit here is a deep subwavelength structure compared with the operation wavelength. One major reason is that at this wavelength, the permittivity of InSb is negative so that a surface impendence mismatch at the object plane (air-InSb interface) would cause a large reflection to affect the performance of our lens . In order to overcome this problem, by inserting an additional thin layer with positive permittivity material (for an example, a Si layer of 50 nm thickness) between the object and our lens, the two peaks are well resolved (results are not shown here).
The imaging ability and bandwidth performance of a superlens can be evaluated by its transfer function, which is the transmittance |T|2 through the lens as a function of wave vector kx and the wavelength. In Fig. 4 , we calculate the transmittance of our four-layered structure using a transfer matrix method [22,23] in two different ranges near our SWs. For a perfect lens, |T|2 = 1 is for all kx and thus a perfect image can be produced. However, in free space the transmittance decays exponentially for evanescent waves (kx/k0>1). Compared with that case, it can be seen that the transmittance of our four-layered superlens decays slowly at both SWs. It still maintains unity up to kx = 15k0 for λ1 = 11.7 μm and kx = 36k0 for λ1 = 53.7 μm. Here the lens works better at λ = 53.7 μm because it is much thinner compared with the wavelength. These good performances confirm that our lens at both operation wavelengths can enhance evanescent waves carrying subwavelength information, which is very crucial for our proposed MW superlensing. The bandwidth of our lens is also seen in Fig. 4. For kx = 10k0, it is about 0.13 μm when åSiC<0 and 0.7 μm when åInSb<0. Although the bandwidth of each superlensing mode is still narrow, the total working wavelength range is successfully broadened due to the additional superlensing wavelength in a single superlens.
We already showed intrinsic bandwidths of our lens under the condition that the SiC layer and InSb layer have the same thickness. In fact, from the Eq. (1), we know that SWs of our lens can be also tuned by changing the thickness ratio f (defined as d1/(d1 + d2)) of those two different layers. When f is changing from 0.2 to 0.8 (considering the fabrication feasibility), we can cover two relatively large ranges: 11−12.3 μm and 53−55 μm for the SiC-InSb case. More examples of the possible tuning ranges using other material combinations are shown in Table 1 . From these figures, we obtain a practical guide for future superlens designs: 1) choosing a suitable material system to determine general positions of SWs; 2) fine tuning SWs with different thickness ratios.
3. Extension to multiple different layers for spectroscopic applications
So far, we showed a way to a double-wavelength superlens based on the design of alternating InSb-SiC layers. An extension to a larger number of different layers, in order to achieve superlensing simultaneously at a larger number of wavelengths, is feasible following a similar design. We rewrite Eq. (1) to a more general form,Fig. 5 , the matching conditions of dielectric permittivities for the case that each layer has the same thickness are fulfilled at three different wavelengths (with low material absorption): λ1 = 11.2 μm, λ2 = 30.4 μm and λ3 = 53.3 μm. By changing thickness ratios, this lens can also cover three ranges: 11~12.3 μm, 30~32 μm and 53~55 μm. Thus, just a single lens can work in multiple different wavelength ranges, which is an obvious advantage over other superlenses. We show the imaging ability of a six-layered InSb-InP-SiC superlens with a total thickness 800 nm (include a 50 nm thickness Si layer) in Fig. 6 . Our investigated object is the same double slit shown in Fig. 3. As expected, the two peaks of the double slit are well resolved with our lens for three cases, which suggests that resolutions of at least λ/10, λ/30 and λ/50 at each wavelength can be obtained with our lens.
As mentioned before, previous reported superlenses can resolve only “one point” of the samples spectroscopic information due to their narrow bandwidths. Now, our MW lens provides a promising way to perform the superlens-based spectroscopy. A possible application sample is the subsurface mapping of free-carrier concentrations of doped semiconductors. Different doping concentrations in doped semiconductors will lead to a large shift of the free-carrier plasma-resonance [17,25], which requires our superlens to cover a wide wavelength range from mid-infrared to THz. As shown in Fig. 7 , using the same finite-dipole model as in , we calculate the infrared s-SNOM amplitude of n-doped InAs  for three different doping concentrations. The gray bars correspond to three covered ranges (bandwidths and tuning ranges) of the InSb-InP-SiC superlens. It can be seen that our lens can cover the resonances of three different cases with low, middle and high doping concentrations. For each doped InAs sample, our proposed MW lens also has “three points” to record its spectral information. Especially for the case with doping concentration n = 1.2 × 1018 cm−3, an almost complete spectroscopy including the information from the resonance, lower and upper branches of the s2 curve, can be obtained together by using a single MW superlens. Based on this information, our lens will be very useful for identifying doped semiconductors and characterizing their free-carrier concentrations.
In conclusion, we have shown a possibility for subwavelength imaging at multiple different wavelengths by a single superlens. The proposed scheme is based on the fact that additional degrees of freedom for superlensing at multiple wavelengths can be provided by increasing the number of phonon resonant dielectrics in a ML system. In other words, a subwavelength image can be achieved at many different wavelengths by only one single lens, which is a large advantage over other recently proposed techniques for superlensing [11,12]. Considering the abundance of polar dielectrics, our lens can easily cover a wavelength range from IR to THz by choosing suitable materials. Applying our MW lens to the spectroscopy of doped semiconductors, we present an example of how a novel superlens-based spectroscopy for mapping free-carrier concentrates of doped semiconductors could be realized. Based on these good performances of our lens, future applications at both IR and THz frequencies simultaneously may be also envisioned in other fields, such as organic structure identification  and vibrational spectroscopy . Moreover, we also note that our idea is not only restricted to superlensing, but should be also useful for other multilayer-based metamaterial concepts, especially for a hyperlens  and for negative refraction  at multiple wavelengths.
We acknowledge financial support from the Ministry of Innovation NRW and the German Excellence Initiative. We are grateful to B. Hauer, J. M. Hoffmann, and A. P. Engelhardt for valuable discussions and proofreading of the manuscript.
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