Ultra-sensitive terahertz sensing based on Rayleigh anomaly in hyperbolic metamaterial gratings

Figure 1 illustrates the proposed THz sensing configuration based on hyperbolic metamaterial grating of width w and period Λ. The hyperbolic metamaterial is composed of four pairs of metal gold and insulator polyimide (PI) multilayers (denoted as 'MIMIMIMI'), of which the heights are $h_\textrm{M} = 20\,\mu$m and $h_\textrm{I} = 20\,\mu$m, respectively. The grating sits on a high-resistivity silicon substrate which is transparent in the terahertz regime, and is immersed in liquid.

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The proposed hyperbolic metamaterial grating can be fabricated using state-of-the-art micro-fabrication techniques. The gold film is first deposited on the high-resistivity silicon substrate, followed by the spin-coating of the PI film. By repeating these procedures four times, we can obtain the four pairs of gold-PI multilayers. The periodic slits in the as-deposited multilayers can be fabricated by means of femtosecond laser writing.

All the calculations were performed with a home-built package for rigorous coupled wave analysis method, which is developed following [39, 40], and which is shown to be efficient for calculating the reflectance and transmittance spectra, R and T of periodic structures. The absorbance spectra A is then obtained through $A = 1-R-T$. In the simulations, we adopted 301 retained orders, which are large enough to reach the convergence regime. The relative permittivity of the high-resistivity silicon substrate is set to be $\varepsilon_\textrm{sub} = 11.6$, the refractive index of the PI is set to be $n_\textrm{I} = 1.8 + 0.04i$, and the relative permittivity of gold is modeled using the Drude model with the plasma frequency $\omega_\textrm{p} = 1.367\times 10^{16}$ rad s⁻¹ and the damping coefficient $\gamma = 4.072\times 10^{13}$ s⁻¹ [41]. Unless otherwise specified, the terahertz plane wave has TM polarization (polarized along x direction) and unitary amplitude ($|E_0| = 1$), and normally impinges onto the hyperbolic metamaterial from the liquid superstrate side, and the grating has ridge width $w = 25\,\mu$m and period $\Lambda = 91\,\mu$m, and is immersed in liquid with refractive index of $n_{\textrm{sup}} = 1.45$. As references, the same-sized gratings in other multilayered films, including a single layer metal film (denoted as 'M'), a bi-layered metal-dielectric film ('MI'), a metal–insulator–metal film ('MIM'), and similarly 'MIMI', 'MIMIM', 'MIMIMI' and 'MIMIMIM' films will also be investigated.

Figure 2 shows the reflectance, transmittance and absorbance spectra for various multilayered gratings. Figure 2(a) shows that for the M grating, both the reflectance and transmittance spectra exhibit abrupt changes around the (1,0) order RA frequency, which is expressed as

$$\begin{align} f_\textrm{RA} = c/(n_{\textrm{sup}} \Lambda), \end{align} \tag{ 1 }$$

and which is calculated to be $f_\textrm{RA} = 2.272$ THz for $\Lambda = 91\,\mu$m with c the speed of light in vacuum. The absorbance spectrum is negligible, indicating little absorption loss due to the metal absorption. For the MI grating, figure 2(b) shows that there exists a small peak locating at $f_\textrm{RA}$ in the absorbance spectrum. This absorbance peak becomes pronounced for the MIM grating, and meanwhile a narrow and shallow Fano-type dip appears at $f_\textrm{RA}$ within the broad resonance peak of the reflectance spectrum, as shown by figure 2(c). For the MIMI grating, the absorbance peak and the reflectance dip, both of which locate at $f_\textrm{RA}$, become sharp with large modulation depths and narrow linewidths, as shown by figure 2(d).

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However, by adding a metal grating on top, i.e. for the MIMIM grating, the absorbance peak and reflectance dip locating at $f_\textrm{RA}$ are greatly suppressed, whereas a new resonance with broad linewidth appears around $f_1 = 2.360$ THz, as shown by figure 2(e). By further adding an insulator grating, i.e. for the MIMIMI grating, figure 2(f) shows that the resonance at $f_\textrm{RA}$ becomes sharp again, whereas the resonance at f₁ is almost unchanged. Similar behaviors can be found by further adding more metal and insulator gratings on top in sequence. For example, figure 2(g) shows that for the MIMIMIM grating the resonance at $f_\textrm{RA}$ is suppressed again, and meanwhile another resonance locating at f₂ appears; figure 2(h) shows that for the MIMIMIMI grating the resonance at $f_\textrm{RA}$ becomes sharp again, whereas the resonances at f₁ and f₂ remain unchanged and broad.

Therefore, we discover that for the resonances at $f_\textrm{RA}$, the absorbance peak and reflectance dip become sharp only for at least two pairs of MI films. The more pairs of the MI films, the larger modulation depth for the resonance at $f_\textrm{RA}$, and the more high-order resonances with broad linewidths. Strikingly, figure 2(h) shows that, for the MIMIMIMI grating, the Fano-shaped reflectance dip locating at $f_\textrm{RA}$ has an extremely narrow asymmetric linewidth, i.e. FWHM of $\delta f = 4.4$ GHz. This corresponds to an exceptionally high quality factor of $Q\equiv f_\textrm{RA}/\delta f = 516$.

However, by removing the top insulator grating from the grating in at least two pairs of MI films, the Fano-shaped resonance at $f_\textrm{RA}$ is broadened with smaller modulation depth, whereas the high-order resonances (if applicable) are unchanged; by adding a metal grating on top, the resonance at $f_\textrm{RA}$ is also broadened, and meanwhile another high-order resonance appears.

We emphasize that the above phenomena can be observed only for the TM polarization under normal incidence. Under oblique incidence or for the TE polarization, although the RA phenomenon also occurs, the FWHM of the resonance is very large (not shown for clarity), corresponding to limited quality factor. This is why we restrict ourselves to TM polarization under normal incidence in this work.

We also note that, replacing gold with silver or aluminum in the multilayered gratings leads to similar results (not shown for clarity). This is because these metals behave similarly in the terahertz regime. Therefore, in fabrications one can replace gold with aluminum, which is cheaper and chemically stable.

In order to understand physics underlying the far-field spectral features, we plot the near-fields associated with the resonances at $f_\textrm{RA}$ in figure 3, and with high-order resonances at f₁ and f₂ in figure 4. From the M grating to the MIMI grating, figures 3(a)–(d) show that the local electric field amplitude $|E|/|E_0|$ is gradually enhanced over an increasing area in the liquid region. Correspondingly, the reflectance dip and the absorbance peak at $f_\textrm{RA}$ in figures 2(a)–(d) become more pronounced and sharper. Remarkably, for the MIMI grating, the electric field amplitude is enhanced by an order of magnitude over an extremely large and long-ear shaped area in the liquid region, as shown by figure 3(d). This greatly enhanced near-field, which is partially confined to the lossy PI layers, can explain the sharp reflectance dip and absorbance peak at $f_\textrm{RA}$ in figure 2(d).

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By adding a metal grating on top, i.e. for the MIMIM grating, figure 3(e) shows that the region of enhanced electric field is reduced from the long-ear-shaped large area to a much smaller area around the top insulator layer. This corresponds to a less sharper resonance at $f_\textrm{RA}$ in figure 2(e). However, by further adding an insulator grating on top, i.e. for the MIMIMI grating, the greatly enhanced electric field changes back to the large and long-eared shape in the liquid, as shown in figure 3(f). Correspondingly, the resonance at $f_\textrm{RA}$ becomes sharp again in figure 2(f). Similar behaviors in the near-field distributions and far-field spectra can be observed by sequentially adding more metal and insulator gratings on top, as shown by figures 3(g), (h) and 2(g), (h), respectively.

Figures 3(b), (d), (f) and (h) also show that the more pairs of metal-dielectric layers, the larger portion of the electric field is confined to the lossy PI layers. As a result, the absorption peak at $f_\textrm{RA}$ increases with the metal-dielectric layer, as shown by figures 2(b), (d), (f) and (h).

Therefore, we discover that at least two pairs of MI layers with gratings are unique in having a sharp dip (or peak) in the reflectance (or absorbance) spectra at $f_\textrm{RA}$, and that this unique characteristic originates from the extremely large and long-ear-shaped region of greatly enhanced electric field in the liquid at the RA frequency. Since at least two pairs of MI thin films can be equivalently modeled as hyperbolic metamaterials with effective dielectric tensor parallel ($\varepsilon_{\parallel}$) and perpendicular ($\varepsilon_{\perp}$) to the x–y plane given by [42]

$$\begin{align} \varepsilon_{\parallel}&=\frac{h_\textrm{M}\varepsilon_\textrm{M}+h_\textrm{I}\varepsilon_\textrm{I}}{h_\textrm{M}+h_\textrm{I}}, \end{align} \tag{ 2 }$$

$$\begin{align} \varepsilon_{\perp}&=\frac{\varepsilon_\textrm{M}\varepsilon_\textrm{I}(h_\textrm{M}+h_\textrm{I})}{h_\textrm{M}\varepsilon_\textrm{I}+h_\textrm{I}\varepsilon_\textrm{M}}, \end{align} \tag{ 3 }$$

our results suggest that the sharp reflectance dip and absorbance peak, and the large and long-ear-shaped area of greatly enhanced electric field at the RA frequency are unique characteristics of the hyperbolic metamaterial gratings.

For high-order resonances at f₁ and f₂ with broad linewidth in figures 2(e)–(h), we find that the corresponding electric field enhancement is very weak or is moderate but over limited area, as shown by figure 4. These results further highlight the strong correlation between the sharp absorbance peak (reflectance dip) and the greatly enhanced near-fields over a large area.

Hereafter, we focus on the MIMIMIMI hyperbolic metamaterial grating. Taking advantage of the exceptionally high quality factors and the associated greatly enhanced near-fields over an extremely large area in the liquid region, we expect that the hyperbolic metamaterial grating can act as an excellent platform for ultra-sensitive terahertz sensing. In order to investigate the changes in the reflectance spectra as the liquid environment is modified, the grating is immersed in various fluids, namely, gasoline ($n_{\textrm{sup}} = 1.41$), diesel ($n_{\textrm{sup}} = 1.45$), and liquid paraffin ($n_{\textrm{sup}} = 1.49$).

Figure 5(a) shows that, as the refractive index of the liquid n_sup increases, the reflectance dip with smaller modulation depth shifts to lower resonance frequency, which locates exactly at the RA frequency determined by equation (1), as indicated by the vertical dotted lines. As a result, the resonance frequency decreases linearly as n_sup increases, as shown by figure 5(b). The sensitivity, defined as $S \equiv \Delta f_\textrm{RA}/ \Delta n_{\textrm{sup}}$, is calculated to be S = 1.56 THz RIU⁻¹, or $S = 9.06\times 10^4$ nm RIU⁻¹. In order to compare the sensitivity S of various terahertz sensors operating at different resonance frequencies f₀, Hu et al [29] recently proposed the concept of normalized sensitivity, which is defined as

$$\begin{equation} S^*\equiv S/f_{0}. \end{equation} \tag{ 4 }$$

With equation (4), the normalized sensitivity of the proposed MIMIMIMI hyperbolic grating is calculated to be $S^* = 0.69\,\textrm{RIU}^{-1}$.

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Additionally, figure 5(a) shows that the linewidth of the reflectance dip $\delta f$ decreases as n_sup increases: for $n_{\textrm{sup}} = 1.41$, 1.45 and 1.49, $\delta f = 10.2$ GHz, 4.4 GHz and 4.0 GHz, corresponding to quality factors of Q = 229, 516 and 552, respectively. Therefore, the FOM defined by

$$\begin{equation} \textrm{FOM}\equiv S/\delta f, \end{equation} \tag{ 5 }$$

are 153, 355 and 390 RIU⁻¹ for $n_{\textrm{sup}} = 1.41$, 1.45 and 1.49, respectively, as shown by figure 5(c).

We summarize the sensing performance of our proposed structure and those of the reported terahertz sensors in table 1. The comparison clearly shows that for different terahertz sensor configurations and even for the same configuration, the operation frequency f₀, the quality factor Q, the sensitivity S, the normalized sensitivity $S^*$ and the FOM are distinct. Among these configurations, we find that the quality factor of the proposed hyperbolic metamaterial grating is one of the largest, reaching Q = 516. In terms of the sensitivity, our proposed structure is among the best with S = 1.56 THz, an order of magnitude larger than most other structures. Remarkably, our structure has the largest normalized sensitivity of $S^* = 0.69\,\textrm{RIU}^{-1}$. With the largest quality factor and the superhigh sensitivity, the proposed structure has very large FOM of $355\,\textrm{RIU}^{-1}$, which is one or two orders of magnitude larger than most of previously reported structures.

Table 1. Comparison of various terahertz sensors. Superscripts 'a' and 'b' stand for simulation and experiment results, respectively.

Year	References	f0 (THz)	Q	S (THz RIU−1)	$S^*$ (RIU−1)	FOM (RIU−1)
2011	[36]$^{a,b}$	0.77	20	0.15	0.19	3.8
2013	[12]b	1.7	43	0.52	0.31	13
2014	[32]b	0.5	28	0.05	0.08	2.8
2015	[43]a	0.52	90	0.06	0.12	10.7
2015	[33]a	11.5	39	1.91	0.17	6.6
2015	[44]a	2.89	48	1.48	0.51	24.6
2016	[45]a	0.85	283	0.18	0.21	59
2016	[29]b	0.7	3.6	0.22	0.31	1.2
2016	[29]a	6.4	6.3	3.5	0.55	3.3
2017	[28]a	1.5	20	0.79	0.53	10.5
2017	[35] a	0.89	19	0.08	0.09	1.8
2019	[46]b	1.87	5.3	0.07	0.55	0.2
2021	[47]a	4.3	3	0.99	0.23	0.69
2021	[37]a	1.816	15230	0.11	0.06	922
2022	[38]a	2.46	1016	0.78	0.31	284
2022	[48]a	1.49	93	0.27	0.18	16.6
–	This worka	2.272	516	1.56	0.69	355

By comparing the near-field distributions of all these terahertz sensor structures, we find that our proposed structure is unique in having greatly enhanced local fields over an extremely area in the sensing region. The greatly enhanced near-fields and the extremely large area significantly strengthen the effective light–matter interactions in the sensing region, resulting in exceptionally high sensitivity and FOM.

According to equation (1), we now show that the resonance frequency of sharp reflectance dip and absorbance peak can be conveniently tuned by varying the grating period. Figure 6(a) shows that for different grating periods, the red-shifted resonance frequencies locate exactly at the RA frequencies determined by equation (1). This red-shift scales linearly with the period, as shown by figure 6(b). Figure 6(c) shows that the quality factor first increases dramatically as the period increases from $\Lambda = 89\,\mu$m, and then keeps as high as more than Q > 500.

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On the other hand, figures 6(d)–(f) show that, if the grating width varies whereas the period keeps constant, the resonance frequency keeps almost constant because the RA frequency determined by the period does not change. However, the linewidth first decreases and then increases, corresponding to first increasing and then decreasing quality factor. In other words, the proposed grating is resonant at exactly the RA frequency that is determined by the grating period, and given the grating period, there exists an optimized grating width for achieving the optimal quality factor.

Figure 7 shows three typical optimized sets of grating width and period (w, p) illustrating the tuning of the resonance frequency while keeping high quality factors. For various parameters, the resonance occurs at exactly the RA frequency that is determined by equation (1), as indicated by the vertical dotted lines.By increasing the grating period from 80 µm to 105 µm, the resonance frequency is blueshifted from 2.584 THz to 1.969 THz. By optimizing the grating width, the obtained quality factors can be as high as Q > 400 for these resonances.

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Figure 8 depicts the calculated reflectance spectra of the MIMIMIMI hyperbolic metamaterial grating with different metal or insulator thicknesses while keeping the other parameters fixed. It is clear that there exists a best set of metal–insulator thicknesses, $(h_\textrm{M} = 20\,\mu\textrm{m}, h_\textrm{I} = 20\,\mu\textrm{m})$ for achieving the narrowest FWHM and the largest quality factor. For too thin metal or insulator, the RA phenomenon is weak. On the other hand, as the thickness of metal or insulator increases, the resonance is broadened and is slightly shifted to lower frequency.

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