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Abstract
The effect of surface phonon resonance (SPhR) and long range SPhR (LRSPhR) on the Goos-Hänchen shift (GHS) in the mid-infrared wavelength region are investigated. The GHS is significantly enhanced around the resonant angles of SPhR and LRSPhR with the p-polarized incident light. A highly sensitive refractive index sensor based on the enhanced GHS is proposed. The LRSPhR shows higher GHS and sensitivity than those of SPhR. The GHS and refractive index sensitivity can be further enhanced by engineering the damping rate of the phononic material. These results provide a potential route toward the large GHS and high refractive index sensitivity, thus opening up new opportunities for high sensitivity optical sensors based on GHS at the mid-infrared wavelength range.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical sensing has been widely used in medical diagnostics [1], environmental monitoring [2], and food safety [3] in the last few decades. Various optical sensing mechanisms have been proposed and demonstrated [4–6], one of the most common sensing mechanisms is the surface plasmon resonance (SPR) due to its superior characteristics, such as simple structure, label-free detection, rapid response, and high sensitivity. Noble metals (e.g., gold and silver) are the most commonly used materials for the generation of SPR that operating in the visible and near-infrared wavelength regime.
Biomedical applications [7,8] , environmental monitoring [9], and bacteria detection [10] in the mid-infrared regime have attracted considerable attention in recent years, thus optical sensors operating in this regime are of great interest and show promising applications in diagnostics, biosensing and chemical sensing [11,12]. However, it is difficult to get SPR with noble metals at the mid-infrared wavelengths due to the large intrinsic losses of noble metals [13,14]. Polar dielectric crystals supporting surface phonon polariton (SPhP) modes have been identified as promising low-loss alternatives [15–18]. Optical sensors based on SPhP or surface phonon resonance (SPhR) operating in the mid-infrared regime have been proposed and demonstrated in recent years [19–22].
Goos-Hänchen shift (GHS), first experimentally demonstrated by Goos and Hänchen in 1947 [23], is a lateral displacement of the reflected light beam. However, the GHS is usually small, and therefore limits its practical applications. In the past few years, various structures and materials are proposed and demonstrated to enhanced the GHS [24–29]. For example, the SPR enhanced GHS of 50$\lambda$ with the incident wavelength $\lambda =632.8$ nm is obtained [24]. The enhanced GHS is then used for various applications, such as optical waveguide switch [30], biosensing [31], refractive index sensing [32], and temperature sensing [33,34].
The enhancement effect of SPR and long range SPR (LRSPR) on the GHS has been demonstrated in recent years [24,25,34,35]. It is then expected to achieve the SPhR enhanced GHS due to the similarity between SPhR and SPR. Here, in this work, we investigate the effect of SPhR on the GHS in the reflection of a light beam from the structure consists of prism, silicon carbide (SiC), and surrounding material. With the generation of SPhR, the GHS is significantly enhanced around the resonant angle of SPhR. The GHS is further enhanced within the long range SPhR (LRSPhR) structure consisting of prism, SiC, calcium fluoride (CaF$_2$), and surrounding material. The SPhR and LRSPhR enhanced GHS is then used for refractive index sensing. Moreover, the effect of damping rate of SiC on the GHS and refractive index sensitivity are also studied.
2. Theoretical model
Similar to the SPR, SPhR can be generated with the Kretschmann-Raether configuration [5,36], as shown in Fig. 1(a). In this structure, SiC film is deposited on a germanium (Ge) prism, and kept in contact with the surrounding material (e.g., water solution). Here, SiC is the phononic material, and its complex permittivity can be obtained with the Drude-Lorentz model [37]:
The longitudinal optical phonon frequency is $\omega _{\textrm {LO}}=972\;\textrm{cm}^{-1}$, the transverse optical phonon frequency is $\omega _{\textrm {TO}}=796\; \textrm{cm}^{-1}$, the damping rate due to vibrational harmonicity is $\gamma =3.75\; \textrm{cm}^{-1}$, and the high-frequency dielectric constant is $\epsilon _\infty =6.5$. The refractive index of Ge prism is given by [22,38]
Here, $\mu =p,s$ represents the $p$- and $s$-incident light. $k_i=2\pi n_{\textrm {Ge}}/\lambda$ is the incident wave number within the Ge prism, and $\lambda$ is the wavelength of the incident light. $\phi _\mu$ is the phase of the reflection coefficient $r_\mu =|r_\mu |\textrm {exp}(i\phi _\mu )$. The reflection coefficient $r_\mu$ can be obtained with the transfer matrix method [40–43].
3. Results and discussion
To generate the SPhR in the Kretschmann-Raether configuration, the real part of the permittivity of the phononic material (i.e., SiC in this work) has to be negative [44]. Here, the wavelength of the incident light is selected as 10.8 $\mu$m, at where the permittivity of SiC is $\varepsilon _{\textrm {SiC}}=-2.54+0.14i$. The refractive index of the surrounding material was set as $n_s=1.33$. The reflectivity as the function of incident angle and SiC thickness for the SPhR structure with $p$- and $s$-polarized incident light are shown in Fig. 2(a) and (b), respectively. A resonance dip was observed in the reflectivity for the $p$-polarized light, whereas no resonant dips are obtained with the $s$-polarized light. Thus, the SPhR can only generated with the $p$-polarized light in Kretschmann-Raether configuration, which is similar to the SPR [5,34,36]. The minimum reflectivity for $p$-polarized light is obtained with the $0.94\ \mu m$ thick SiC at the incident angle of $\theta =32.39^\circ$.
Fig. 2. Reflectivity as a function of the incident angle and SiC thickness for the SPhR and LRSPhR structures with $p$- and $s$-polarized light. (a) SPhR structure with $p$-polarized light, (b) SPhR structure with $s$-polarized light, (c) LRSPhR structure with $p$-polarized light, and (d) LRSPhR structure with $s$-polarized light. Here, the wavelength of incident light is 10.8 $\mu m$, and the refractive index of the surrounding material is $n_s=1.33$. The thickness of CaF$_2$ film in the LRSPhR structure is 5.5 $\mu m$.
Besides SPR, the LRSPR [43,45,46] has also attracted much attention during the last few decades. Similarly, LRSPhR can also be obtained [47], for example, by inserting a CaF$_2$ film with the refractive index of $n_{\textrm {CaF}_2}=1.3$ between the Ge prism and SiC in the conventional Kretschmann-Raether configuration, as shown in Fig. 1(b). The CaF$_2$ is replaceable by other materials as long as the index matches the surrounding material. Similar to the SPhR structure, resonance dip in reflectivity was only observed with the $p$-polarized light [see Figs. 2(c-d)]. Compared to the SPhR case, the LRSPhR structure has narrower resonance dip and smaller resonant angle. In addition, the required SiC thickness to get the resonance dip with the LRSPhR structure is smaller than that with the SPhR structure. The minimum reflectivity with the LRSPhR structure was achieved at the incident angle of $\theta =20.79^\circ$ with 0.62 $\mu$m thick SiC.
The normalized GHSs (${D_\mu }/ {\lambda }$ ) as the function of incident angle for the SPhR and LRSPhR structures are shown in Fig. 3(a) and (b), respectively. The SiC thickness at where the minimum reflectivity of $p$-polarized light is achieved in Fig. 2 was selected in the calculation of GHS. The GHS of $p$-polarized light with the SPhR structure is significantly enhanced around the resonant angle of SPhR [see Fig. 3(c)], specifically, a negative GHS of −353$\lambda$ is achieved. In contrast, the GHS of $s$-polarized light is almost zero. Besides the negative GHS, positive GHS can also be obtained, which depends on the SiC thickness. For example, a positive GHS of 17$\lambda$ is achieved with the 0.92 $\mu$m thick SiC. The positive (negative) GHS indicates the displacement of the reflected light beam in forward (backward) direction. For the LRSPhR structure, the $p$-polarized light shows enhanced positive GHS around the resonant angle [see the reflectivity in Fig. 3(d)], while the GHS of $s$-polarized light is negligible, as shown in Fig. 2(b). The maximum GHS of 1808$\lambda$ is obtained for the $p$-polarized light, which is about four times of magnitude higher than the SPhR case (−353$\lambda$). By selecting different thicknesses of SiC and CaF$_2$, the negative GHSs for the LRSPhR structure can be achieved. For example, a negative GHS of −136$\lambda$ is obtained when the 0.65 $\mu$m thick SiC and 5.5 $\mu$m thick CaF$_2$ is used in the LRSPhR structure (figure not shown here). For the LRSPhR structure with 0.62 $\mu$m thick SiC and 5.7 $\mu$m thick CaF$_2$, a negative GHS −535$\lambda$ is achieved.
Fig. 3. Normalized GHSs (${D_\mu }/ {\lambda }$) and reflectivity as the function of incident angle for the SPhR and LRSPhR structures. (a) GHS for the SPhR structure, (b) GHS for the LRSPhR structure, (c) reflectivity for the SPhR structure, and (d) reflectivity for the LRSPhR structure. Here, the SiC thicknesses in the SPhR and LRSPhR structures are 0.94 $\mu$m and 0.62 $\mu$m, respectively. The thickness of CaF$_2$ is 5.5 $\mu$m.
The GHS for $p$-polarized can also obtained with the intrinsic damping $\textrm {Im}(\beta ^{0})$ and radiative damping $\textrm {Im}(\Delta \beta ^{rad})$ [35,48,49]:
Here, $\beta ^{0}$ is the eigenpropagation constant of a surface phonon polariton in the SPhR or LRSPhR structures without the prism. $\Delta \beta ^{rad}$ is the difference between $\beta ^{0}$ and the eigenpropagation constants of the prism coupling structures. When the radiative damping is smaller than the intrinsic damping, a negative GHS can be obtained. For the LRSPhR structure, the intrinsic and radiative dampings can be roughly approximated by [48]
The SPhR and LRSPhR enhanced GHS then can be used for sensing applications, such as refractive index sensing. Optical sensing based on GHS has been theoretically investigated and experimentally demonstrated in the last few decades [32,34,50–52]. Here, the refractive index sensitivity is defined as
where $\Delta D_{ps}=D_{ps}(n_s+\Delta n_s)-D_{ps}(n_s)$, $D_{ps}=D_{p}-D_{s}$. $D_{ps}(n_s)$ is the GHS difference between the $p$- and $s$-polarized light with the refractive index $n_s$ of the surrounding material. The refractive index variation $\Delta n_s=10^{-3}$ is used in the following calculations. Both the positive and negative GHSs can be used for sensing applications [33,34]. The normalized refractive index sensitivity $S_{GHS}/\lambda$ as a function of the incident angle is shown in Fig. 4. The sensitivity for the SPhR structure shows a positive peak of $3.46\times 10^5\lambda \ 1/\textrm {RIU}\sim 3.74\times 10^6$ $\mu$m/RIU (RIU: refractive index unit) and a negative peak of $-1.15\times 10^5\lambda \ 1/\textrm {RIU}\sim 1.24\times 10^6$ $\mu$m/RIU, as shown in Fig. 4(a). The positive and negative sensitivities are also achieved with the LRSPhR structure. Specifically, the refractive index sensitivity for the LRSPhR structure exhibits a positive peak of $3.63\times 10^6\lambda \ 1/\textrm {RIU}$ ($\sim 3.92\times 10^7$ $\mu$m/RIU), which is about ten times that of SPhR structure. The negative sensitivity of $-1.78\times 10^6 \lambda \ 1/\textrm {RIU}$ ($\sim 1.92\times 10^7 \mu$m/RIU) is obtained with the LRSPhR structure, as shown in Fig. 4(b), which is about nine times higher than that with the SPhR structure. It should be noted that both the positive and negative peaks of the refractive index sensitivity appear around the resonant angle of SPhR or LRSPhR.Fig. 4. Refractive index sensitivity as a function of the incident angle for the (a) SPhR and (b) LRSPhR structures. The other parameters are the same as those in Fig. 3.
The damping rate of SiC $\gamma =3.75\; \textrm{cm}^{-1}$ is used in the studies above, which is obtained from the SiC epitaxially grown on a Si (001) substrate. However, the damping rate will change if the SiC is grown on a different substrate [22,53]. It is therefore necessary and interesting to investigated the effect of damping rate of SiC on the GHS and its sensing performance. The GHS as a function of the incident angle with different damping rates is shown in Fig. 5. It is found that the GHS of $p$-polarized light is strongly influenced by the damping rate. In contrast, the damping rate almost has no effect on the GHS of $s$-polarzied light with both the SPhR and LRSPhR structures. The sign and magnitude of GHS for the $p$-polarized incident light depend on the value of damping rate. For example, for the LRSPhR structure, a positive GHS of $74\lambda$ is obtained with the damping rate $\gamma =2\; \textrm{cm}^{-1}$. However, a negative peak of $-518\lambda$ appears when the damping rate changes to $\gamma =4\; \textrm{cm}^{-1}$, as shown in Fig. 5(c). In addition, the incident angle corresponding to the maximum magnitude of GHS of $p$-polarized light deceases with the damping rate, which is a result of the decreased resonant angle of SPhR or LRSPhR (figures not shown here).
Fig. 5. Normalized GHSs (${D_\mu }/ {\lambda }$) as the function of incident angle for the SPhR and LRSPhR structures with different damping rates. (a) GHS for the SPhR structure with $p$-polarized light, (b) GHS for the SPhR structure with $s$-polarized light, (c) GHS for the LRSPhR structure with $p$-polarized light, (d) GHS for the LRSPhR structure with $s$-polarized light. The other parameters are the same as those in Fig. 3.
In Figs. 6, the refractive index sensitivity is plotted as the function of the incident angle with different damping rates. Apparently, the refractive index sensitivity based on GHS can be engineered with the damping rate of SiC. For example, with the damping rate $\gamma =4\; \textrm{cm}^{-1}$, the refractive index sensitivity shows a positive peak of $8728\lambda$ and a negative peak of $- 6890\lambda$ for the SPhR structure, as shown in Fig. 6(a). The sensitivities are further improved with the LRSPhR structure, $440693\lambda$ and $-368189\lambda$, which are about 50 times those with SPhR structure.
Fig. 6. Refractive index sensitivity as a function of the incident angle for the (a) SPhR and (b) LRSPhR structures with different damping rates. The other parameters are the same as those in Fig. 3.
The sensitivities for various GHS-based refractive index sensors are shown in Table 1. Various mechanisms, such as SPR [50], Bloch surface wave (BSW) [54], bound state in the continuum (BIC) [55], are proposed and demonstrated to enhanced the GHS. The enhanced GHS is then used for refractive index sensing. For example, the refractive index sensitivity of $1.5\times 10^7$ $\mu$m/RIU is obtained with the BIC-enhanced GHS in an all-dielectric metasurface [55], which is about 4 times the sensitivity of the proposed SPhR structure ($3.74\times 10^6$ $\mu$m/RIU). However, the refractive index sensitivity achieved with the LRSPhR enhanced GHS is improved to $3.92\times 10^7$ $\mu$m/RIU, which is about 2.6 times that of the BIC enhanced GHS sensor. Compared to the conventional SPR enhanced GHS sensor [50], 2 and 3 orders of magnitude improvements are achieved with the SPhR and LRSPhR enhanced GHS sensors, respectively. It should be noted that the proposed GHS sensors here have simpler structures compared to the 2D materials incorporated SPR structure [56] and the all-dielectric metasurface [55], which is a priority for the fabrication process. In addition, the proposed SPhR and LRSPhR enhanced GHS refractive index sensors operate in the mid-infrared regime, which is different from the SPR, BSW, and BIC enhanced GHS sensors [50,54–58]. Therefore the proposed GHS sensors demonstrate the superiority of high sensitive refractive index sensors, especially operating in the mid-infrared regime.
Table 1. Comparison of the refractive index sensitivities of the proposed GHS sensor with those of previously reported research
4. Conclusion
In summary, we have studied the effect of SPhR and LRSPhR on the GHS. The GHS is significantly enhanced around the resonant angle of SPhR and LRSPhR with the $p$-polarized incident light. A negative GHS of $-353\lambda$ is obtained with the SPhR structure, which is then further improved to $1808\lambda$ with the LRSPhR structure. In contrast, no enhancement effect on the GHS are found with the $s$-polarized light in both the SPhR and LRSPhR structures. Refractive index sensors based on the SPhR and LRSPhR enhanced GHS are proposed, and the sensitivities of $3.74\times 10^6$ $\mu$m/RIU and $3.92\times 10^7$ $\mu$m/RIU can be achieved, respectively. In addition, both the GHS and refractive index sensitivity can be engineered with the damping rate of SiC. The SPhR and LRSPhR structures offer a potential route towards large positive and negative GHSs. The high refractive index sensitivities obtained with the SPhR and LRSPhR enhanced GHS provide a promising approach for biosensing and chemical sensing, and may find potential applications in medicine diagnostics, environmental monitoring, and food safety at the mid-infrared wavelength range.
Funding
Yantai School Local Integration Development Project in 2020 (2020XDRHXMXK09); China Postdoctoral Science Foundation (2021M690235).
Acknowledgments
Yi Xu would like to acknowledge the Project funded by China Postdoctoral Science Foundation (Grant No. 2021M690235). Jing Zhang and Yibin Song would like to acknowledge the Project funded by Yantai School Local Integration Development Project in 2020 (Grant No. 2020XDRHXMXK09).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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