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Abstract
Using photonic structures resonating at the characteristic absorption frequency of the target molecules is a widely-adopted approach to enhance the absorption and improve the sensitivity in many spectral regions. Unfortunately, the requirement of accurate spectral matching poses a big challenge for the structure fabrication, while active tuning of the resonance for a given structure using external means like the electric gating significantly complicates the system. In this work, we propose to circumvent the problem by making use of quasi-guided modes which feature both ultra-high Q factors and wavevector-dependent resonances over a large operating bandwidth. These modes are supported in a distorted photonic lattice, whose band structure is formed above the light line due to the band-folding effect. The advantage and flexibility of this scheme in terahertz sensing are elucidated and exemplified by using a compound grating structure on a silicon slab waveguide to achieve the detection of a nanometer scale α-lactose film. The spectral matching between the leaky resonance and the α-lactose absorption frequency at 529.2 GHz by changing the incident angle is demonstrated using a flawed structure which exhibits a detuned resonance at normal incidence. Based on the high dependence of the transmittance at the resonance on the thickness of α-lactose, our results show it is possible to achieve an exclusive detection of α-lactose with the effective sensing of thickness as small as 0.5 nm.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Terahertz (THz) waves, usually defined with the frequency ranging from 0.1 to 10.0 THz, have attracted broad research interests in recent decades [1–6]. Due to their unique properties including high transparency in most dielectrics and the nonionizing radiations, THz technology has demonstrated broad application prospects in both fundamental research and applied sciences [7]. In this band, a large number of chemicals, especially organic molecules, show strong absorption and dispersion characteristics, resulting from molecule’s rotation, intra- and inter-molecular vibrations. These spectral features can be employed as molecular fingerprints in biological and chemical sensing applications [6,8–10]. However, most molecules are much smaller than the THz wavelengths (nanometer-scale versus tens to hundreds of microns), leading to the requirement of a large volume of samples to achieve observable absorption to identify the molecules. This limits the applications of THz detection in biomedical areas where samples with volume as small as possible are preferred. One effective approach to overcoming this limitation is to make use of the enhanced local electric field provided by artificial resonating structure to amplify the effective absorption cross section and thus improve the THz absorptivity response. In this context, a local enhancement of the electric field at the same incident power is important. This idea has been widely utilized in other frequency regimes, e.g. surface-enhanced Raman scattering [11], surface-enhanced infrared absorption spectroscopy [12], and has also been extended to the THz band. For example, B. Han et al. used a square split-ring resonator sensor and demonstrated a successful detection of a 8 µm-thick of lactose layer [13]. Dong-Kyu Lee et al. also investigated slit-form nano-antennas in the metal film and effectively detected carbohydrate molecules with low concentrations at the mmol/L scale [14]. Since the highest local electric field enhancement happens roughly at the resonance, to have most pronounced absorption enhancement one has to make sure the optical structures resonate at the target chemical's fingerprint frequency. Furthermore, the level of nearfield enhancement is intrinsically associated with the Q factor of the far field resonance [15]. So people aspires for resonances with higher Q factor to get a higher sensitivity. This requirement poses significant challenges to the device manufacturing if one relies on the nanofabrication techniques to tune the structure dimension and the corresponding resonance, especially when a narrow bandwidth resonance is needed. People have tried various external means of resonance tuning by using e.g. electrical gating of the Fermi energy level of nanostructured graphene to dynamically tune the plasmon resonance [16]. However, these kinds of external tuning using other physical fields like the electric field or stress field require much research effort and also complicates the whole sensing system.
In this paper, we propose a novel THz fingerprint sensing scheme based on the use of a new type of leaky resonances, the quasi guided modes (QGMs) supported by elaborately designed periodic structures. We note that although the structure used here to support the QGM may seem similar to that in the well-known symmetry-protected bound states in the continuum (BIC), the underlying physics is fundamentally different. Here the structure must have a period-doubling perturbation and the relations between guided modes (GM), BICs, quasi-BICs, and radiation are elucidated in a recent work of us [17]. The QGMs are transitioned from true guided modes (GMs), and the transition happens as a result of the band-folding effect, when a geometric perturbation is introduced into the original structure supporting the GMs to double the period. Since the GMs have no external leakage (infinite Q factors), the derivatives of QGMs possess ultrahigh Q factors which can be controlled by the level of perturbation, quite similarly to the quasi-bound states in the continuum (QBIC) modes. Furthermore and more importantly, the QGMs inherit the spatial dispersion property of the GMs. As a result, one can tune the frequency of the QGMs over a large band by simply changing the wavevector or the incident angle. In contrast, the QBIC modes are derivatives of BICs, which only exist at very few discrete points in the ω-k space. The operation bandwidth of the quasi-BIC is thus limited into a very narrow bandwidth containing the original frequency of BIC. This is more obvious if one aspires for QBIC modes with high Q factors when weak perturbation is required. As a result, the QBIC can not provide the large spectral tuning as we will demonstrate in this work by simply changing the incident angle. Using a silicon-based compound grating structure which supports the QGMs, we demonstrate that even when the structure has a resonance detuned from the target absorption frequency at normal incidence, it is still quite feasible to tune the QGM resonance by choosing the proper incident angle to match the absorption. Based on the enhanced absorption of THz radiations using the high Q factor QGMs, our results show that it is possible to achieve an effective detection of α-lactose thickness as small as 0.5 nm. The maintaining of the substance identification capability from THz spectroscopic techniques is further illustrated by using another chemical with a different absorption frequency. These results show important applications in enhanced THz sensing of the scheme by employing the wavevector-dependent QGMs, eliminating the needs of complicated external means to achieve the required spectral matching.
2. Structure and simulations
The structure of the silicon compound grating on a slab waveguide is schematically shown in Fig. 1. As shown by an enlarged view of a unit cell in the inset of Fig. 1, the special feature of the compound grating is that it is composed of two alternatingly aligned ridge arrays with the same pitch and different ridge width characterized by the difference δ. We assume both the grating ridges and the slab are made from high-resistivity silicon, whose refractive index is 3.418 in the THz regime [18]. Other geometric parameters include w = 10 µm, t = 100 µm, h = 10 µm. The whole structure can be made on a mechanically polished silicon wafer with photolithography and reactive ion etching. When the compound grating is considered as a periodic structure with a period of P2, it supports the regular guided mode resonance (GMR) [19] and the value of P2 is determined by the following equation:
Fig. 1. (a) Schematic of the compound grating structure on a silicon slab waveguide, which support the QGMs for enhanced terahertz sensing. The inset illustrates the geometric parameters for a unit cell. (b) The schematic diagram of THz sensing for α-lactose, by applying a thin layer of α-lactose to the top of the composite grating.
3. Result and discussion
3.1 THz sensing at normal incidence
We first present some results to demonstrate the enhanced absorption of α-lactose with an ideal compound grating whose resonance at normal incidence matches exactly the α-lactose absorption peak of 529.2 GHz. The transmission spectrum of the compound grating structure without α-lactose is plotted in Fig. 2(a), which exhibits a sharp resonance dip of the Fano type at 529.2 GHz. The Fano resonance is attributed to an interference between the QGM and the Fabry-Perot resonance supported by the Si slab. The inset in Fig. 2(a) shows the real part of the Ez component of the QGM resonance, which shows an out-of-phase distribution within the slab under the two ridges. This kind of phase distribution usually happens at the boundary of the larger FBZ of the regular grating structure with the same ridge width supporting the GMs. We further find with detailed calculations that the electric field magnitude in the compound grating is enhanced by a factor of 652 compared to that of the incident plane wave. Although it is illustrated by the inset of Fig. 2(a) that the maximum electric field is located within the slab layer, a significant electric enhancement is also present above the slab, which is the accommodation position for the coated α-lactose. According to the absorption of electromagnetic waves in a certain area $A = {\int {|E |} ^2}\varepsilon ^{\prime\prime}dV$ where ε” is the imaginary part of the permittivity at the resonance frequency, one can still expect a significantly enhanced absorption considering the huge |E| enhancement at the QGM. We further illustrate the molecular level sensitivity of the compound grating structure. When a thin layer of 4 nm α-lactose is deposited on the top surface of the compound grating, a distinct decrease in the transmission peak from 0.90 to 0.55 at the frequency of 529.2 GHz is observed, together with a slight red shift in the resonance peak, as shown in Fig. 2(b). The red shift is attributed to a large refractive index of the α-lactose, whose introduction into the cladding of the slab waveguide leads to an increase of the GM effective index. With a larger increase of the α-lactose thickness, more pronounced decrease of the on-resonance transmittance and further red shift of the resonance are both seen. Figure 2(c) presents the dependence of the resonance transmittance as a function of the coated α-lactose thickness and it is seen that based on the enhanced absorption of THz radiations using the high Q factor QGMs (the Q factor can reach up to 3 × 106), an effective detection of α-lactose thickness as small as 0.5 nm can be achieved in principle. This sensitivity is about 1000 times higher than that using a photonic crystal cavity [22]. Furthermore, the thickness of the compound grating structure is much smaller than that of a photonic crystal cavity, which needs at least wavelength-scale dimension of the cavity to enable the back and forth bouncing of the photonic mode.
Fig. 2. (a) The transmission spectrum for an ideal compound grating at normal incidence, which exhibits a resonance exactly at 529.2 GHz. (b) The transmission spectra with different thicknesses of α-lactose coated on the compound gratings. (c) The dependence of resonance transmittance at different α-lactose thicknesses. The inset presents that a comparison of the transmission spectra for three cases: the bare compound grating (black line), the compound grating with 1 nm of another sample with the absorption frequency at 0.75 THz (blue line), and the compound grating with 1 nm of α-lactose.
To further demonstrate the substance selectivity of this scheme, we use another chemical which is assumed to have a different absorption frequency at ωP = 750 GHz and thus can be considered a lossless dielectric at 529.2 GHz. When this chemical is coated onto the same compound grating which was originally designed to have a resonance of 529.2 GHz, it only works as a regular dielectric with a refractive index larger than unity. As a result, it exhibits no influence to the resonance transmittance close to 529.2 GHz and only leads to a slight red shift of the spectrum. The calculated transmission spectra for three cases, the bare compound grating, the compound grating with 1 nm of α-lactose, and the compound grating with 1 nm of the new sample with absorption at 0.75 THz, are shown in the inset in Fig. 2(c). The capability of substance identification is clearly seen. For the same structure, 1 nm thickness of α-lactose leads to a significant decrease of the resonance transmittance while the spectrum only sees a slight red-shift for the different sample.
3.2 Active THz sensing with QGMs
The scheme of using high Q resonances to improve THz sensing performances with significantly enhanced sensitivity and retained substance identification capability seems to be ideal for practical applications. However, we should note that it is vital in this scheme to achieve the resonance matching between the photonic structures and the absorption frequency of the target sample. In practical applications, a slight deviation in the fabricated structure is quite common and it is challenging to use fabrication techniques to precisely control the resonance. For example, when the original period is reduced by only 260 nm (the new period is now P2 = 193.48 µm), there is a significant blue shift in the transmission of its resonance peak (from 529.2 GHz to 529.78 GHz), as will be shown in Fig. 4 (a). This spectral shift leads to a negligible local electric field enhancement at the original target frequency of 529.2 GHz and annihilates all the fascinating features of the sensing applications demonstrated above.
We next reveal that the dependence of the QGM resonance supported by the compound grating on the slab waveguide on the incident angle can be employed to provide an additional freedom to achieve the resonance matching and thus can address the above problem, without the need of using external means of resonance tuning. This is extremely important when a manufactured photonic structure has a resonance detuned from the target frequency. The compound grating can be considered as a perturbed result of a regular grating by introducing symmetry-breaking perturbation of non-zero δ when the width of every second grating element changes. Then the period of the original grating will be doubled and the first Brillouin zone (FBZ) of it will be halved. As a result, the dispersion diagrams of the GMs close to the boundary of the FBZ in the original grating will be folded to be around the Γ point in the FBZ of the compound grating, and the GM will switch to QGM with finite yet ultra-high Q factors controlled by the level of perturbation. The spatial dispersion of the GMs will be retained for the QGMs, leading to their occurrence over a large band and a high dependence of the frequency on the lateral wavevector. As a result, their resonances can be tuned by the lateral wavevector, or by the incident angle of the excitation wave. The dispersion properties of the QGMs supported by the compound grating are calculated by using the eigenfrequency analysis at different lateral wavevectors and presented in Fig. 3(a). The black circled lines indicate the dispersion of light in air, above which the modes can be accessed by free space excitations and is usually referred to as the continuum. When the width difference δ in the compound grating decreases to 0, the compound grating restores to a regular grating with the period halved to P = 0.5P2. All the QGMs will change to GMs with infinite Q factors and no access to free space radiations any more. The dispersion of the GMs supported by the regular grating is illustrated by the blue circled lines in the shaded area in Fig. 3(a), which extends from kx = -π/P to π/P. A small gap at the boundary of this FBZ (kx=±π/P) can be seen due to the coupling between two counter-propagating GMs, which is a typical behavior in corrugated waveguides and this effect has been utilized in many applications, e.g. in distributed feedback (DFB) lasers. Since the dispersion of the GMs is located below the light line, all the GMs have no coupling to free-space excitations. When a symmetry-breaking perturbation (by using a non-zero δ in this work, but one can also shifts the lateral position of every second ridge) is introduced, the period is doubled and the FBZ of the compound grating shrinks from kx = -π/2P to π/2P. Due to the translational symmetry along kx, the original dispersion of the GMs at kx=±π/P will appear above the light line at the Γ point in the compound grating, and thus the GMs will switch to QGMs with the possibility of free space excitations. The red circled lines in Fig. 3(a) present the dispersion for the QGMs, which are composed of two branches similar to those of the GMs at kx = ±π/P. Due to the mirror symmetry across the central plane of either ridge, an ideal symmetry-protected BIC is also supported by the composite grating. The Γ point on the upper branch corresponds to a resonance with infinite Q factor, which is a BIC resonance of the symmetry-protected type. However, because it cannot be excited with free space radiations, this resonance can not be used for sensing applications. All other resonances along the two branches exhibit ultra-high Q factors, which are reflected by the color of circles in Fig. 3(a). Since the QGMs are derivative of the GMs, they possess ultrahigh Q factors which can be controlled by the level of perturbation, similar to the QBIC modes. For any resonance along the two branches, the Q factor increases as the perturbation (characterized by δ) decreases. A typical evolution of the Q factor as a function of δ is presented in Fig. 3(b). As δ decreases from 2 to 0.5, the Q factor changes from 7.50 × 105 to 1.21 × 107. As δ vanishes, the Q factor of the QGM diverges. In that case, the QGM restores to the QM with the dispersion curve below the light line, as shown by the blue circled line in Fig. 3(a). In this paper, all the calculations adopt the value of δ = 1 µm, to keep a balance between fabrication challenge and the relative high value of the Q factor.
Fig. 3. (a) Dispersion curves for the QGMs (solid circles of red color, the depth of color represents the strength of the Q factor) supported by the compound grating and the GMs (blue solid circles) by the regular grating. The black circled lines are for light in free space. b) A typical evolution of the QGM Q factor as a function of δ.
To demonstrate that spatial dispersion of the QGMs can be employed to tune the resonance for the spectral matching without the need of external stimulus, we reuse the same flawed structure with the period P2 = 193.48 µm. As shown by the red line in Fig. 4(a), this structure exhibits a resonance at normal incidence away from 529.2 GHz and is not suitable for the enhanced detection of α-lactose based on the THz absorption. However, the resonance can be tuned by using different incident angle. For this structure, an incident angle of 0.44° will help move the resonance to 529.2 GHz. The black line in Fig. 4(a) presents the calculated transmission spectrum at this incident angle and the inset shows the field distribution of the real part of Ez within a single periodic unit. A similar level of local electric field enhancement is found, due to the Q factor of the resonance at the incident angle of 0.44° is comparable to that at normal incidence. To show the enhanced sensing performances at inclined incidences, we shown in Fig. 4(b) the transmission spectrum when a 1 nm lactose is coated on this compound grating. It is seen that the transmittance at resonance decreases from 0.9 to 0.6. Therefore, with the structure proposed in this paper to support QGMs, we demonstrate that it is very feasible to tune the resonance of QGMs by choosing the appropriate angle of incidence to match the absorption of α-lactose at 529.2 GHz. An angular change of 0.44 degrees is quite easy to achieve since it is far from the angular resolution of 0.01 degrees in many commercial ellipsometry systems.
Fig. 4. (a) The transmittance spectrum with periods of P2 = 193.48 µm (0°) and P2 = 193.48 µm (0.44°), respectively. (b) Simulated transmission spectrum of 1 nm lactose with the angle of incidence is 0.44°.
4. Conclusions
In summary, a novel terahertz fingerprint sensing scheme based on the dependence of the QGM resonances on the incident angle is proposed and its application is exemplified by using the detection of nanometer thickness of α-lactose. Using a flawed compound grating which has a resonance at normal incidence detuned from the absorption frequency of α-lactose, our results demonstrate that it is quite feasible to tune the resonance back to the target frequency by choosing the proper incident angle. The enhanced sensitivity, i.e. the decrease of the on-resonance transmittance, is shown as a function of the sample thickness and the efficient detection of α-lactose layer as low as 0.5 nm can still be easily achieved. For other target chemicals, one should first use numerical tools to design the grating structure so that its resonance at normal incidence matching the absorption of the sample. However, it is always challenging to fabricate the structure with the precise spectral matching. This scheme of tuning of spectral resonance by changing the incident angle eliminates the requirement of external means to tune the resonance when an imperfect structure with the resonance slightly detuned from the target frequency is used, and is found to be a flexible and reliable approach for enhanced sensing applications in chemical and biological fields based on the infrared and THz spectroscopic techniques.
Funding
National Natural Science Foundation of China (11974221, 12274269, 11974218); Local science and technology development project of the central government of China (YDZX20203700001766).
Disclosures
The authors declare no conflicts of interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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