1. Introduction

The resolution of conventional optical imaging systems is limited to about half wavelength due to the loss of high spatial frequency evanescent components, which contain the fine information of the object. Numerous attempts have been proposed to overcome this limit, the one which clearly stands out is the “superlens”, based on the negative index materials first proposed by Pendry in 2000 [1]. While the true “double negative” superlens has not been implemented in practice, its simplified version was realized using a silver slab with a negative real part of the dielectric permittivity and resolution measured to be about one-sixth of the illumination wavelength [2]. After that, many similar “not-quite-superlens” had been realized in experiments [3–6]. Later, a more logical extension of superlens called the “hyperlens” was proposed [7] and subsequently realized. It was based on the multilayer metal/dielectric assembly structured in a cylindrical geometry. The “hyperlens” not only increases the distance between the object and image, but is also capable of transforming the evanescent wave into the propagating one, thereby providing magnification. However, due to the inevitable metal loss, the total amount of light passing through the hyperlens decreases with the increase in number of layers. After that, many similar superlens and hyperlens based on the hyperbolic metamatrials (HMMs) have been realized [8–15]. Both the superlens and hyperlens have been modeled extensively [16–18], however, most of the models are numerical and do not reveal the physical reasons that eventually limit the resolution.

Previously we have developed the “eigenmode” model of superlens [19] in which the whole phenomenon of “superlensing” was explained as the coupling of light scattered (or emitted) by the sub-wavelength features of the object into the surface plasmon polaritons (SPPs) modes of the metal slab and their subsequent combining into the image on the other side of the slab. The destructive interference between symmetric and antisymmetric modes was shown to suppress the optical transfer function (OTF) at large spatial frequencies and thus limiting the resolution of superlens. More recently, we have considered hyperbolic materials and shown that their eigenmodes are nothing but coupled SPPs modes [20] which exist in any multilayer metal-dielectric structure, irrespective of their classification as hyperbolic or non-hyperbolic as per the effective medium approximation..

Armed with this knowledge, we can now extend our eigenmode model to the image-forming multilayer structures, which for the sake of generality we shall refer to as “plasmonic lenses” and explore their limitations. Although in this work we only consider the flat structures, the main conclusions can also be applied to the cylindrical hyperlens. While the eigenmode approach is indispensable for revealing the physics behind performance of the plasmonic lens, the same outcomes can be essentially achieved using several numerical methods. One such technique is the Transition Matrix Method (TMM) that we have employed in numerous examples later explained using the eigenmode theory.

2. Eigen-mode approach

The eigenmode approach is built upon a simple principle, viz. the energy of the object, when treated here as a superposition of oscillating dipoles, can only be coupled into the physical eigen-modes of the system. Hence, if the system is capable of supporting the “sub-wavelength” modes with wavevectors (spatial frequencies) k exceeding the diffraction limit 2πn/λ, then a sub-wavelength image can be formed by superposition of these modes. The structures incorporating metal layers are obviously capable of supporting such large k-vector modes, called surface plasmon polaritons (SPPs). In the multilayer structures, such as one shown in Fig. 1, these modes couple with each other to give rise to the so-called supermodes extending throughout the entire structure.

 

Fig. 1 Symmetric and antisymmetric eigen modes in multilayers hyperbolic metamaterials contain N metal layers with thickness t_m and separated by dielectric with distances t_d. d_o and d_i are the distance between object, image and hyperbolic metamaterials.

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The multilayer lens in Fig. 1 contains N metal layers with complex relative permittivity ∊_m = ∊′_m + i∊″_m where ∊′_m and ∊″_m are the real and imaginary parts respectively, and thickness t_m separated by dielectric with permittivity ∊_d and thickness t_d, and is capable of supporting N + 1 coupled SPPs eigenmodes ${\mathit{f}}_k^n(z)\mathit{exp}(i\mathit{k}\cdot \mathit{r}-{\omega }_{n}t)$ (n = 1, 2, 3 · · · N + 1) where k is the in-pane wave vector. Each eigenmode contains in-plane and out of plane contributions and has a dispersion relation ω̃_n(k) = ω_n(k) + iγ_n(k) which connects the complex eigenmode resonant frequency ω̃_n with the real in-plane wave vector k; and the imaginary part of frequency γ_n(k) is the effective loss rate estimated via the ratio of energy in metal to the total energy in the multilayer structure [21]. Each of the eigenmodes is a solution of the homogeneous wave equation,

A dimensionless point object’s image, which can describe the response of the imaging system is called point spread function (PSF) and can be easily calculated by using OTF,

3. Results and discussion

3.1. Effect of number of metal layers and metal loss

First, we solve numerically using TMM to obtain the SPP eigenmode profiles and the dispersion as shown in Fig. 2. We use an example of the standard hyperbolic metamaterial consisting of alternating 15 nm layers of silver and aluminum oxide [23], which means the metal fill ratio (FR = t_m/(t_m + t_d)) is 50%, and the length of one period is 30nm (P = 30nm). The permittivity of Ag, is described by the Drude model ${∊}_m=1-f_p^2/\left(f^2+i\gamma f\right)$, where the plasma frequency is f_p = 2.166 × 10¹⁵s⁻¹ and the damping is γ = 2.02 × 10¹³s⁻¹ [24]. Al₂O₃ is assumed to have a constant permittivity ∊_d = 3.61 [23] within the spectral region of interest. Figure 2(a) shows the dispersion curves for N = 5 periods metamaterial, while in Fig. 2(b) are the profile of the in-plane component of electric field (real part) for the six eigen modes. Note that the first two modes, labeled as “1” and “2” are the “surface” (in the sense of the surface of whole metamaterial as first introduced in [25]) SPP modes that peak at the interfaces between the metamaterial and surrounding dielectric, extending significantly into the latter. Since the absolute values of ${\mathit{f}}_k^n{\left(z_0\right)}^{*}\cdot {\mathit{f}}_k^n\left(z_i\right)$ for these modes are larger, these two modes are expected to have larger contribution to OTF. However, from Fig. 2(b) we can also see that one of them is symmetric and the other is antisymmetric, which means that their contributions to the OTF have opposite signs. As the number of layers increases, the dispersion curves of the two eigen modes get closer and become degenerate which according to [25] the other modes (“3”–“6”) are the “bulk” (in the sense of being inside the whole metamaterial) plasmon polariton (BPP) modes, and compared to the first two modes, they have stronger field confinement, which means that the BPP modes have smaller contribution to the process of imaging. Actually, half of the N + 1 eigen modes are symmetric and the other half are antisymmetric, so due to the opposite symmetries, their contribution to OTF tends to cancel each other. The only factor preventing complete cancellation is the difference of resonant frequencies, so as N increases, the resonant frequencies get closer and the cancellation between symmetric and antisymmetric eigen modes gets progressively stronger, especially at the large wavevector k, and resolution is expected to decrease. The absolute value of OTF for the imaging using the multilayer structure (calculated using TMM) is shown in Fig. 2(c) at the wavelength λ ≈ 600nm (which corresponds to half of surface plasmon frequency) for N increasing from 1 to 17. The OTF goes through some peaks and valleys for small spatial frequencies due to resonances with eigenmodes and then decays steadily. The key feature of Fig. 2(c) is the fact that as N increases, the OTF curves not only shift downward (this is simply the result of absorption by larger number of layers) but also experience progressive rapid fall-off at large spatial frequencies, shifting leftward. This a the clear manifestation of the increased cancellation in the structures with large N.

 

Fig. 2 (a) The dispersion curves of the eigen modes when the superlens contains five metal slabs; (b) The profile of electric in-plane component of each eigen modes inside the superlens; (c) The absolute value of OTF when N=1, 5, 9, 13, 17, respectively; (d) The cut-off lateral wavevectors versus the number of metal layers for three different object and image distances.

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This cancellation effect can be further gauged by first normalizing the OTF curve to its value at the second (broadest) peak and introducing the cut-off spatial frequency k_cut–off such that |OTF(k_cut–off)| = 0.1, which can be construed as an admittedly arbitrary yet reasonably appropriate resolution limit. The plot of k_cut–off as a function of N is shown in Fig. 2(d) for different values of object and image distances. After sharp jump for N = 3 associated with the “merging” of the dispersion curves of the first two modes (and hence their nearly complete cancellation), the resolution of the “lens” experiences a steady decrease with the number of metal layers. This is an important result, relevant for any multi-layered imaging contraption, including the hyperlens [7]-as the number of layers increases not only the overall amount of light getting to the image is reduced (this can be mitigated in principle using optical amplifiers and/or more sensitive low noise detectors) but the resolution of the scheme also decreases which cannot be undone by any amplification or post-processing.

In addition to OTF we also calculated the point spread function (PSF) to estimate the specific resolution of the multilayer superlens. As shown in Fig. 3(a), increase in the loss from γ₂ = 1.01 × 10¹⁴s⁻¹ to γ₁ = 2.02 × 10¹⁴s⁻¹ causes not only a decrease in the height of PSF but also its spread, i.e. the resolution of the “lens”. This effect is similar to the changes in PSF with the number of layers as shown in Fig. 3(b). Figure 3(c) demonstrates the impact of the loss in metal on the OTF-at small (admittedly unrealistic) values of loss, the OTF shows multiple peaks associated with excitation of the individual SPP modes, but as the loss increases to realistic values the peaks disappear and the whole OTF gets reduced by different values of metal loss.

 

Fig. 3 (a) Comparison of PSF with different metal loss when the superlens contains four metal slabs; (b) Comparison of PSF when the superlens contain four and five metal slabs respectively; (c) Change of OTF with different metal loss, when N = 5, FR = 50%, P = 30nm, d_i = d_o = 30nm. λ₀ is the wavelength of signal.

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3.2. Cancellation effect

Before proceed further, we want to present an illustration of how increase in number of layers suppresses the OTF at large spatial frequencies thus reducing resolution, and the different contributions of modes to image. In Figs. 4(b) and 4(c) we show the evolution of OTF for the N = 10 layered structure at two different values of k as we keep adding the individual terms in Eq. (9). As one can see the sum shows large swings with addition of each SPP mode since each two adjacent modes have different parities, until the sum converges. Moreover, from Figs. 4(b) and 4(c) we can see that the contribution of each eigen mode to the OTF are different, higher order modes have smaller contribution because of their stronger field confinement [25], hence the total OTF will not converge to zero, even though it contains a large number of metal/dielectric period, the real main limit of OTF or resolution is metal loss.

 

Fig. 4 (a) Dispersion relations of the eigen modes when N=10; (b, c) The “OTF oscillations” as a function of the number of layers in the sum Eq. (9) for two values of special frequencies k = 5k₀ and k = 10k₀.

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One can also see how the cancellation gets stronger with increasing spatial frequency k from the dispersion curves shown in Fig. 4(a). As spatial frequency increases the dispersion curves practically merge and the resonant denominators in the coupling coefficients in Eq. (7) become equal, ensuring more complete cancellation. Indeed, the oscillations in Fig. 4(c) converge much faster and to lower steady value. It is important to note that this steady value is low yet not zero because cancellation is never complete when one of the mode is in resonance, or, when the frequency is between the eigen frequencies of two adjacent modes of different parities. Indeed according to Eq. (9) these two modes will be added with opposite signs which means that the fields in the image space will actually add up.

3.3. Impact of granularity

Besides the number of layers and metal loss, the performance of the multilayer lens gets affected by the geometrical parameters, such as the size of one metal/dielectric period and its composition, i.e. the fill ratio of metal (FR = t_m/(t_m + t_d)). The effect of different periods for the FR=1/2 is shown in Fig. 5. The first three (a)–(c) panels in this figure display the dispersion relations of eigen modes when the period is increased from P = 10nm to 30nm and 50nm. Rather naturally, as the period is increased and the separation between the interfaces grows the individual surface plasmon polaritons becomes less and less coupled to each other and the eigen modes dispersion curves become denser, with the first two dispersion curves merging. As explained in the section 3.2 above, the cancellation gets stronger and the OTF gets drastically reduced with increase of the period as evidenced from Fig. 5(d). The impact on the resolution is even more pronounced as seen from Fig. 5(e) where cut-off spatial frequency k_cut–off is displayed. Reducing granularity clearly benefits imaging but using metal structures in the visible range reducing the period beyond λ/10 is difficult. Better results maybe attained with the semiconductor based multilayers operating in the mid-IR region [29], although the loss in them can be quite large.

 

Fig. 5 (a)–(c) The dispersion relations of eigen modes when the period P=10nm, 30nm, 50nm respectively; (d) Change of the OTF with increase in period when the metal fill ratio is set at 50%; (e) Change of the cut-off wavevector with different period when the fill ratio is set at 50%.

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3.4. Impact of metal fill ratio

The fill ratio can also play an important role in determining the “lens” performance. We have calculated the OTF for different metal fill ratios ranging from 10% to 80% for the fixed period P = 30nm, and N = 5. Figs. 6(a)–(c) display dispersion curves for the filling ratio of 10%, 50% and 80% respectively. One can see that the curves for small fill ratio of 10% (Fig 6(a)) are those of coupled slab SPPs with the first mode being a “long range SPPs” [26, 27] and the rest are closely spaced short range SPPs [28]. In the opposite extreme of 80% fill ratio (Fig. 6(c)) the closely spaced dispersion curves correspond to weakly coupled “gap SPPs” [20]. For 50% fill ratio (Fig. 6(b)), the distribution of dispersion is more uniform. From Fig. 6(d) we can see that even though, 10% fill ratio has smaller metal loss, due to the denser dispersion curves and larger resonance spatial frequency, the OTF is still smaller than other fill ratios. For 80% fill ratio, due to both larger metal loss and denser dispersion curves, the OTF is much smaller. Fig. 6(e) shows the change of cut-off spatial wavevector with the number of period for different metal fill ratios. Due to larger resonance spatial frequency, after we normalized the OTF to its second peak to obtain the cut-off wavevector, extra small and large fill ratios have large cut-off spatial frequency, practically however, from Fig. 6(d), we can tell that large OTF (better resolution) can be obtained around metal fill ratio 33%.

 

Fig. 6 (a)–(c) are the dispersion relation of eigen modes with metal fill ratio 10%, 50%, 80% respectively; (d) The absolute value of OTF for different metal fill ratios; (e) The cut-off wavevector for different fill ratios.

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3.5. Multilayered imaging at the extreme

Next, we explore the limitation of imaging with multilayered structures that contains large number of metal layers with different granularity. As the number of metal/dielectric periods increases and granularity decreases the multilayered structure approaches the limit in which it can be characterized by effective permittivity and continuous hyperbolic (or elliptical) dispersion. In other words, the eigenfunctions f becomes the Bloch [30] functions. However, the Bloch function method does not describe the previously mentioned “surface modes” [25] well and these modes play an important role in imaging with hyperlens. Even though these structures would be impractical from the fabrication point of view at the present time, it is worthwhile to consider what limits they can reach in the future. Also, while as far as we know it is next to impossible to reduce substantially the ohmic losses in metals, let alone completely cancel them, it is also interesting to consider what happens when the number of layers increases and the loss becomes very small. When N becomes large, while the loss gets small, let us say N > 50, the OTF under monochromatic will contain a number of sharp resonant peaks, in practical situation when the light is not strictly monochromatic these peaks will be averaged out and it is these averaged OTF that presents practical interest. In Figs. 7(a)–(c) we show an example of OTF for N = 100 and three different values of loss (γ = 10¹⁰s⁻¹, 10¹²s⁻¹, and 10¹⁴s⁻¹) which shows rapid oscillation as loss decreases to admittedly unrealistic values. Next to it in Fig. 7(d) we show “incoherent OTF” obtained as $\overline{{\mathit{OTF}}_n}={\left[\sum _{k_{n-l}}^{k_{n-r}}{\left|\mathit{OTF}{(k)}_n\right|}^{2}dk/\left(k_{n-r}-k_{n-l}\right)\right]}^{1/2}$, where k_n−l and k_n−r are the valley wavevectors on the left and right side of the n-th OTF peak. The “smoothed” OTF of Figs. 7(a)–(c) shows that reduction of loss increases the OTF and the cut-off wavevector but with diminishing returns. Two orders of magnitude reduction of loss from 10¹⁴s⁻¹ to 10¹²s⁻¹ bring 12-fold improvement in OTF but the further reduction to 10¹⁰s⁻¹ (highly unrealistic) value only yields a 10-fold. In other words cancellation is still at work at low loss. To compare the effects of granularity and number of layers we now plot the averaged OTF for N = 50 and N = 100 with two different granularities in Fig. 8 for three aforementioned values of loss. Comparing Figs. 7(a) with 7(c) and Figs. 7(b) with 7(d) reveals that increase in the number layers from 50 to 100 still causes the OTF deterioration and fall in k_cut–off-result of the cancellation, but the reduction is small as discussed in the end of section 3.2. Comparison of Figs. 7(a) with 7(b) and Figs. 7(c) with 7(d) shows the improvement achieved with lower granularity as expected from the effective hyperbolic medium, yet the benefit from lower granularity is quite insignificant for the large number of layers. It is possible that these advantages can be put into productive use with all-semiconductor structures in mid-IR range.

 

Fig. 7 (a)–(c) The OTF of hyperlens contains 100 metal/dielectric period for different metal loss when the thickness of meta and dielectric are t_m = t_d = 5nm; (d) The averaged OTF of hyperlens contains 100 metal/dielectric period for different metal loss when the thickness of meta and dielectric are t_m = t_d = 5nm.

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Fig. 8 (a) and (b) The averaged OTF of hyperlens contains 50 metal/dielectric periods for different metal loss when the thickness of meta and dielectric are t_m = t_d = 5nm and t_m = t_d = 15nm respectively; (c) and (d) The averaged OTF of hyperlens contains 100 metal/dielectric periods for different metal loss when the thickness of meta and dielectric are t_m = t_d = 5nm and t_m = t_d = 15nm respectively.

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4. Conclusions

In this work, we investigated performance of the multilayer metal/dielectric near field imaging structures. We have demonstrated that the imaging process of these multilayer structure can be best understood as coupling of object signal into the SPP eigenmodes supported by the structure and subsequent formation of the image on the other side. Due to the ensuing strong cancellation of the contributions of symmetric and antisymmetric eigen modes, the resolution decreases with more metal layers thus limiting the number of layers to less than 20 and the object to image distance to only a fraction of a micrometer-much less than what can be expected from the simple metal loss considerations. We have also investigated and explained the impact of metal loss, granularity, and fill ratio on the near field imaging performance of the multilayer structure. Our main conclusion, which is also relevant to the cylindrical hyperlenses is that multilayer metal dielectric structures do not offer a performance superior to the single layer superlenses, but may find limited range of niche applications where one needs to have object to image distance increased beyond tens of nanometers. These results will hopefully be useful to the researchers active in the area.

Appendix

Here we offer a concise description of the TMM applied to the multilayer structure that contains M total layers (N metal layers and M − N dielectric layers) as shown in Fig. 9, it has 2N metal/dielectric interfaces. Due to the existence of reflection at every interfaces, there are both right and left going waves inside the multilayer superlens, indicated by ‘+’ and ‘−’, respectively. For transverse magnetic (TM) mode, we use the magnetic field component H_y, whose polarization is along the interface and is continuous inside the structure, to help us build the matrices. As shown in Fig. 9, the subscripts show the layer number and the interface in the present layer, the superscripts show the direction of the wave. For example, $H_{21}^{+}$ indicates the right going wave at the first interface in the second layer. Inside each layer, the tangential magnetic fields are the superposition of the right and left going plane waves, i.e. H = H⁺ + H⁻. The propagating waves inside every layer and at each interface are connected by 2 × 2 matrices. The defined dynamic matrix D relates the fields on two sides of each interface. The dynamic matrices for propagating wave from dielectric to metal (D_d→m) and from metal to dielectric (D_m→d) are shown in Eq. (11) and Eq. (12):

(11)$$D_{d\to m}=\left[\begin{array}{cc}\left(\frac{1}{2}+\frac{{∊}_m(f)K(k,f)}{2{∊}_{d}Q(k,f)}\right)& \left(\frac{1}{2}-\frac{{∊}_m(f)K(k,f)}{2{∊}_{d}Q(k,f)}\right)\\ \left(\frac{1}{2}\right)-\frac{{∊}_m(f)K(k,f)}{2{∊}_{d}Q(k,f)}& \left(\frac{1}{2}+\frac{{∊}_m(f)K(k,f)}{2{∊}_{d}Q(k,f)}\right)\end{array}\right]$$

(12)$$D_{m\to d}=\left[\begin{array}{cc}\left(\frac{1}{2}+\frac{{∊}_{d}Q(k,f)}{2{∊}_m(f)K(k,f)}\right)& \left(\frac{1}{2}-\frac{{∊}_{d}Q(k,f)}{2{∊}_m(f)K(k,f)}\right)\\ (\frac{1}{2}-\frac{{∊}_{d}Q(k,f)}{2{∊}_m(f)K(k,f)})& \left(\frac{1}{2}+\frac{{∊}_{d}Q(k,f)}{2{∊}_m(f)K(k,f)}\right)\end{array}\right]$$

where ∊_m and ∊_d are the permittivities of metal and dielectric; $K={\left(k^2-{∊}_0{k}_0^2\right)}^{1/2}$ and $Q={\left(k^2-{∊}_m{k}_0^2\right)}^{1/2}$ are the decay constants in dielectric and metal, respectively. k₀ is the wavevector in vacuum.

 

Fig. 9 Configuration of multilayer metal/dielectric superlens.

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For the fields inside each layer, we write the propagation matrices for dielectric (P_d) and metal (P_m) respectively, as shown in Eq. (13) and Eq. (14),

(13)$$P_d=\left[\begin{array}{cc}\mathit{exp}\left(K(k,f)t_d\right)& 0\\ 0& \mathit{exp}\left(-K(k,f)t_d\right)\end{array}\right]$$

(14)$$P_m=\left[\begin{array}{cc}\mathit{exp}\left(Q(k,f)t_m\right)& 0\\ 0& \mathit{exp}\left(-Q(k,f)t_m\right)\end{array}\right]$$

where t_m and t_d are the thickness of metal and dielectric. With the dynamic and propagation matrices, we can connect the electric field in the object and image space:

(15)$$\left[\begin{array}{c}{E}_{xn}^{+}\\ E_{xn}^{-}\end{array}\right]=\underset{D_{m\to d}(k,f)P_m(k,f)\cdots P_m(k,f)D_{d\to m}}{\mathit{Nmetallayers}}(k,f)\left[\begin{array}{c}{E}_{x1}^{+}\\ E_{x1}^{-}\end{array}\right]=\left[\begin{array}{cc}{M}_{11}(k,f)& M_{12}(k,f)\\ M_{21}(k,f)& M_{22}(k,f)\end{array}\right]\left[\begin{array}{c}{E}_{x1}^{+}\\ E_{x1}^{-}\end{array}\right]$$

One can obtain the eigen-solutions $f_n^k$ from Eq. (15) by imposing the condition of no input wave ( $E_{xn}^{+}=E_{x1}^{-}=0$, ∊″_m = 0), i.e. M₁₁(k, f) = 0 to find the dispersion relations and then eigenfunctions themselves for using these solutions to explore the physics of imaging in the multilayer structure as described in the main text. At the same time, one can obtain OTF directly as (∊″_m ≠ 0),

(16)$$\mathit{OTF}(k,f)=\frac{E_{\mathit{img}}}{E_{\mathit{obj}}}=\frac{1}{M_{11}(k,f)}$$

Fig. 10 demonstrates that the two approaches do give similar results which only start deviating from each other at large wave-vectors where the SPPs are highly damped.

 

Fig. 10 Comparison of the calculated OTF with eigenmode model and TMM.

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Funding

National Science Foundation (NSF) (1507749); Army Research Office (W911NF-15-1-0629).

Acknowledgments

We thank the indispensable help of Dr. P. Noir.

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