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Abstract
We theoretically and numerically investigate the ligh-matter interaction in a classic topological photonic crystal (PhC) heterostructure, which consists of two opposite-facing 4-period PhCs spaced by a dielectric layer. Due to the excitation of topological edge mode (TEM) at the interface of the two PhCs, the strong coupling between incident light and TEM produces a high quality resonance peak, which can be applied to many optical devices. As a refractive index sensor, it achieves a sensitivity of 254.5 nm/RIU and a high figure of merit (> 250), which is superior to many previously reported sensors. We further study the coupling between photons and excitons by replacing the pure dielectric layer with the J-aggregates doped layer. By tuning the thickness of the doped layer and the angle of incident light, the dispersive TEM can efficiently interact with the molecular excitons to form a hybrid mode with TEM-like or exciton-like components, showing interesting energy transfer characteristics and flexible modulation characteristics. This work may be helpful for a better understanding of light-matter interactions in a topological PhC heterostructure, and achieve potential applications in related optical devices.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The manipulation of the light-matter interactions in the weakly coupled and the strongly coupled nanostructures has aroused great interest from both theoretical and practical perspectives [1–6]. In the coupled system, when the characteristic interaction time is longer than the dephasing time, strong coupling will occur, resulting in hybrid resonance modes with light-like and matter-like components. In addition, the properties of the hybrid modes can be controlled through the light-like components. The strong coupling in hybrid systems (e.g., optical microcavities [7–9], plasmonic nanostructures [10–14], multilayered structures [15–19], and metallic nanostructures [20–22].) is conducive to exploring the light-matter interactions due to its unique energy-transfer and greater modulations.
Recently, photonic topological insulators have attracted much attention in integrated optics because of their unique properties of topologically protected edge states [23]. Many schemes for constructing topological photonic modes have been proposed, including optical waveguides [24,25], plasmonic nanoparticles [26], one-dimensional (1D) photonic crystal (PhC) [27], two-dimensional PhC [28], and three-dimensional PhC [29,30], etc. Among these structures, 1D topological PhC structure is favored for its advantages of simple design and easy fabrication. For example, Xiao et al. theoretically studied the effect of the relation between the surface impedance and bulk band geometric phases on the excitation of topological edge mode (TEM) in the 1D topological PhC [27]. Wang et al. realized multiband perfect absorption in a graphene-based 1D topological PhC heterostructure [31]. Gao et al. combined topology with Fano resonance through the weak coupling between TEM and Fabry-Perot cavity mode [32]. Wang et al. realized the capture and enhancement of the plasmon waves through the topological plasmonic Tamm states in a periodic metal-insulator-metal waveguide [33]. Inspired by these works, a 1D PhC heterostructure with robust topological properties can be designed to excite TEM, thereby enhancing light-matter interaction.
In this paper, we realize the manipulation of the strong light-matter interaction in a 1D topological PhC heterostructure. With proper design, TEM can be excited at the interface between the two opposite-facing PhCs, so a perfect transmission peak is observed when the incident light is perfectly coupled to the TEM. We show that the perfect transmission can be tuned not only by changing the angle of incident light but also by adjusting the thickness and dielectric constant of the dielectric layer. The high-quality resonance induced by TEM is conducive to high-performance refractive index sensing, narrowband filtering, and optical switching, etc. In addition, we further study the coupling between photons and excitons by doping the J-aggregates in the dielectric layer. As a result, two hybrid polariton bands emerge due to the strong TEM-exciton coupling. All polariton bands are composed of the two basic components, i.e., TEM with strongly dispersive photonic feature, and excitons acting as matter. We also find the coupling strength can be strongly affected by the doped layer thickness. This work may inspire related studies about hybrid light-matter interactions in a topological PhC heterostructure.
2. Structure and theory
The proposed heterostructure consists of two opposite-facing 4-period PhCs and a dielectric layer is sandwiched between them, as shown in Figs. 1(a) and 1(b). The PhC is composed of Si and SiO$_{2}$ alternately, whose refractive indices are $n_{a}$ = 2.82 and $n_{b}$ = 1.46 at the visible band [31]. The thickness of the unit cell is $\Lambda$ = $\lambda _{c}/4n_{a}$ + $\lambda _{c}/4n_{b}$, where the central wavelength is chosen as $\lambda _{c}$ = 700 nm. The thicknesses of Si and SiO$_{2}$ are set to $d_{a}$ = 0.4$\Lambda$ and $d_{b}$ = 0.6$\Lambda$, respectively. For the convenience of calculation, the TM polarized (i.e., magnetic field is parallel to $y$-axis) light with normal incidence is considered. The reflection ($R$) and transmission ($T$) of the multilayer structure can be calculated using the transfer matrix method (TMM).
Fig. 1. (a), (b) Schematic diagrams of the proposed structure with or without J-aggregates. (c) Transmission spectrum of the the PhC (with 4 unit cells)-symmetry PhC without dielectric spacer (i.e., $d$ = 0) under the normal illumination with TM polarization. (d), (e) The band structures (black curves) of PhC and symmetry PhC. The magenta strip represents the gap with positive topological property (i.e., $\xi$ > 0), while the cyan trip represents the gap with negative topological property (i.e., $\xi$ < 0). The corresonding Zak phases are shown in green.
The band structure of a binary PhC can be determined by [27]:
The band structure of the symmetry PhC is exactly the same as that of the PhC, as depicted in Figs. 1(d) and 1(e). However, due to the inverse arrangement of a unit cell, the two PhCs have different topological properties. In Fig. 1(d), $\xi ^{(1)}$ > 0 and $\xi ^{(3)}$ < 0. However, the sign of $\xi$ at first and third band gaps is opposite in Fig. 1(e). Thus, the TEM can be excited in these two gaps simultaneously by constructing an interface with PhC on one side and symmetry PhC on the other side. Figure 1(c) clearly shows the transmission spectrum of the PhC (with 4 unit cells)-symmetry PhC structure without dielectric layer (i.e., $d$ = 0). As expected, the sharp peaks appear in the first and third photonic band gaps, indicating the existence of TEM.
Fig. 2. (a), (b) Reflection and transmission spectra of the system with different element widths $d$. (c), (d) The value of $r_{L}r_{R}\textrm {exp}(2i\Phi )$ as a function of wavelength. (e), (f) Normalized electric field profile distributions at 700.6 nm and 729.9 nm when $d$ = 0 and 20 nm.
3. Results and analysis
3.1 Manipulating the interaction between incident light and TEM
As is well known, TEM can be excited at the interface of topological PhC heterostructure when an incident light irradiates the composite multilayer structure. The incident light and TEM can be strongly coupled to produce a sharp peak in the reflection (or transmission) spectrum. In addition, the manipulation of the resonant point can be realized by changing the thickness of the dielectric layer. Here the refractive index of dielectric layer is fixed as $n_{d}$ = 1.5. Figures 2(a) and 2(b) show the variations of reflection and transmission with incident wavelengths under different dielectric thicknessnes. High quality resonance is observed, whose resonant wavelength can be tuned continuously from 700.6 to 759.4 nm, with a small change in the thickness $d$ from 0 to 40 nm. The reflection (or transmission) peak generated by the strong coupling of incident light and TEM almost reaches 100% (or 0%), and the full width at half-maximum (FWHM) of the resonant peak is less than 2 nm. These advantages may have potential applications in narrowband selective filters, refractive index sensors, and optical switches.
To further demonstrate the physical condition for exciting the TEM in the 1D PhC heterostructure, we can calculate the eigenmodes of the multilayer structure. As shown in the inset of Fig. 2(c), the proposed multilayer structure can be regarded as a symmetric cavity, enclosed by two opposite-facing 4-period PhCs. The reflection coefficients of the left propagating wave at left interface $r_{L}$ and the right propagating wave at the right interface $r_{R}$ are introduced in the PhC heterostructure. Then, we can obtain the field of the eigenmode:
Figures 2(c) and 2(d) show the real and imaginary parts of $r_{L}r_{R}\textrm {exp}(2i\Phi )$ at different dielectric thicknesses. Taking $d$ = 0, 10, 20, 30, and 40 nm as examples, the theoretical resonant wavelengths are 700.7, 715.3, 730, 744.5, and 759.2 nm, which are consistent with the numerical results (i.e., 700.6, 715.2, 729.9, 744.7, and 759.4 nm) shown in Figs. 2(a) and 2(b). The theoretical and numerical calculations are nearly identical, indicating that the existence of the TEM. To better illustrate the TEMs excited in the multilayer system, we show the normalized electric field intensity distributions at 700.6 nm and 729.9 nm when $d$ = 0 and 20 nm, as shown in Figs. 2(e) and 2(f). It is observed that the most of the energy of the electric field is concentrated near the interface between the two opposite-facing PhCs, which makes it possible to achieve the strong light-matter interaction. The fine-tuning of the structure (changing the thickness of the dielectric layer) does not affect the reflection (transmission) performance, and the electric field distribution is the same as before. These results show that the topological properties of the structures and their associated edge states are very robust to the structural disturbances.
Next, we discuss the dynamic manipulation of the TEM. Figure 3(a) presents the transmission spectra as a function of wavelength and the incident angle ($\theta$) for TM polarization and TE polarization (i.e., electric field is parallel to $y$-axis). It can be seen that the increase of $\theta$ leads to a shift in the transmission peak to a shorter wavelength. For the case of TM polarization, the transmission peak blue-shifts from 700.1 nm to 650 nm when $\theta$ is increased from 0$^{\circ }$ to 42.6$^{\circ }$, and the maximum transmission is almost unaffected. Similarly, for the case of TE polarization, the transmission peak undergoes the same blueshift (from 700.1 nm to 650 nm) when $\theta$ is increased from 0$^{\circ }$ to 38.9$^{\circ }$. It can be concluded that the TEM has strong angle dependence, which provides a simple method to effectively adjust the resonant wavelength without affecting the performance (i.e., nearly 100% transmission and narrow bandwidth). Interestingly, the transmission responses of TE and TM polarizations are almost the same when the angle of light varies, which makes possible applications under multidirectional illumination.
Fig. 3. (a) Transmission spectra of the system ($d$ = 0) as a function of wavelength and incident angle for TM polarization. The inset shows the transmission spectra for TE polarization. (b) Transmission spectra of the system as a function of wavelength and thickness of the dielectric layer for TM polarization. (c) Normalized electric field profile distributions of first and second order resonant modes at different dielectric layer thicknessnes. (d) Transmission spectrum of the system ($d$ = 0) for different number of periods of PhC and symmetry PhC.
The existence of the dielectric layer makes the manipulation more diversified. According to the analysis in Figs. 2(a) and 2(b), the resonant wavelength can be manipulated by changing $d$. In the system, when $d$ is far less than the lowest thickness of the Fabry-Parot (FP) cavity mode (half wavelength in the dielectric material) [34], the FP mode cannot be excited. As $d$ continues to increase, the FP mode occurs, as shown in Fig. 3(b). Therefore, the zeroth resonant mode is a typical TEM, while the first and second order resonant modes are FP mode. The green round marks are the predictions from the theory, which are consistent with the simulation results of TMM. Although the resonance peak varies with $d$, the interface state always exists in the band gap we considered. Specially, when $\Delta \Phi$ = $k\pi$ ($k$ is an integer), the resonance wavelength will come back to the starting position. The normalized electric field profile distributions of the FP modes at different $d$ are shown in Fig. 3(c). Similar to TEM, the most of the energy of the electric field is concentrated in the dielectric layer, indicating strong field localization phenomenon. In addition, as seen in Fig. 3(d), with the number ($N$) of periods of PhC and symmetry PhC increasing, the bandwidth (i.e., FWHM) decreases significantly, while the resonance wavelength changes slightly, which greatly improves the $Q$-factor. It must be pointed out that increasing $N$ will increase the difficulty and cost of manufacturing, so we use smaller $N$ = 4 under the premise of good performance.
3.2 Sensing performance based on topological photonic crystal heterostructure
As stated above, the majority of the field energy is concentrated in the dielectric layer when TEM (or FP mode) is excited, and any change in the dielectric (e.g., thickness and refractive index) affects the resonant wavelength. Therefore, the proposed structure with ultra-narrow bandwidth can be used as a refractive index sensor because resonant mode is highly sensitive to the change of the dielectric refractive index. For this purpose, we evaluate the sensing capability by calculating sensitivity (i.e., it is defined as the shift in the resonant wavelength per a unit change in the dielectric refractive index, $S$ = $\Delta \lambda$/$\Delta n_{d}$), $Q$-factor (i.e., it is defined as the ratio of resonant wavelength to FWHM, $Q$ = $\lambda$/FWHM) and figure of merit (i.e., it is defined as the ratio of sensitivity to FWHM, FOM = $S$/FWHM). Figure 4(a) shows the transmission spectra of the system ($d$ = 500 nm) for different values of the dielectric refractive index. We can see that the resonant wavelength is positively correlated with $n_{d}$ and linearly increases with the increase of $n_{d}$. The slope of the fitted curve shows the sensitivity $S$ reaches 254.5 nm/RIU. In addition, $Q$-factor larger than 700 and FOM larger than 250 are also achieved, which can be attributed to the ultranarrow bandwidth (less than 1 nm), as shown in Fig. 4(b). It should be noted that many previous nanostructure-based sensors have only achieved a FOM that are less than 25 [35–37]. The sensing capability of this device is superior to many sensors reported in previous works [35–37], indicating that the system has great potential in high-performance refractive index sensing.
Fig. 4. Transmission spectra of the system ($d$ = 500 nm) for different values of the dielectric refractive index. The green quare marks are the predictions from the linear fit. (b) Calculated FWHM, $Q$-factor and FOM values of the resonant mode as a function of the dielectric refractive index.
3.3 Manipulating the interaction between molecular exciton and TEM
To introduce molecular excitons in the system, the pure dielectric layer can be replaced by the dielectric layer doped with J-aggregates, as shown in Fig. 1(b). The dielectric function of the dielectric layer doped with J-aggregates can be expressed as [12]:
Fig. 5. (a) Sketch of the interaction between molecular exciton and TEM. (b), (c), (d) TMM-simulated and COM-fitted absorption spectra of the system ($d$ = 20 nm) for different incident angles. The parameters of COM used for fitting are: (b) $c_{b}$ = 0.085, $\gamma _{b}$ = 3.427 THz, $\theta _{b}$ = 1.366, $c_{d}$ = 0.339, $\gamma _{d}$ = 1.661 THz, and $\theta _{d}$ = 1.674, respectively. (c) $c_{b}$ = 0.304, $\gamma _{b}$ = 2.262 THz, $\theta _{b}$ = 1.364, $c_{d}$ = 0.336, $\gamma _{d}$ = 2.189 THz, and $\theta _{d}$ = 1.771, respectively. (d) $c_{b}$ = 0.476, $\gamma _{b}$ = 1.315 THz, $\theta _{b}$ = 1.484, $c_{d}$ = 0.1061, $\gamma _{d}$ = 3.027 THz, and $\theta _{d}$ = 1.795, respectively.
Considering the loss of the dielectric layer doped with J-aggregates, the absorption of the system can be expressed as $A$ = 1 - $R$ - $T$. Figures 5(b), 5(c) and 5(d) show the absorption spectra of the new system ($d$ = 20 nm) at different incident angles. Interestingly, two absorption peaks are observed in the spectra due to the strong TEM-exciton coupling. As $\theta$ = 25$^{\circ }$, a low one with absorption of 11.2% at 688 nm and a high one with absorption of 48.3% at 713.8 nm can be observed. As $\theta$ increases to 33.4$^{\circ }$, two peaks with similar absorption are observed, corresponding to 35.1% at 682.8 nm and 32% at 702.9 nm, respectively. As $\theta$ is further increased to 45$^{\circ }$, the absorption peak on the left (34.3% at 664.3 nm) is higher than that on the right (8.9% at 695.8 nm). The exciton mode and the TEM can be considered as two oscillators, and the complex absorption coefficient can be approximated by fitting a sum of oscillator response functions in a coupled oscillator model (COM) [4,10,13]:
To better understand the coupling progress, the interaction can be described by the classic COM [38]:
Here $E_{X}$ and $E_{T}$ are the energies of excitons and TEMs, $\Gamma _{X}$ and $\Gamma _{T}$ are the damping losses of the two modes, $\Omega$ is the coupling energy between exciton and TEM, $\left | \alpha _{X} \right |^{2}$ and $\left | \alpha _{T} \right |^{2}$ represent the relative weightings of exciton and TEM in the polariton band ($\left | \alpha _{X} \right |^{2}$ + $\left | \alpha _{T} \right |^{2}$ = 1), and $E$ is the energy of the hybrid mode. The dispersion curves of the hybrid modes (green lines) match well with the numerical results, as shown in Fig. 6(a). As $\theta$ increases, the exciton (white line) and TEM (red line) couple together, leading to an anticrossing at the points where the exciton overlaps with the TEM, with mode splitting energy of $\Omega$ = 52 meV. In addition, $\Gamma _{X}$ = 52 meV can be obtained from Eq. (7), and $\Gamma _{T}$ < 10 meV can be obtained by extracting the FWHM of the TEM from Fig. 3(a). Obviously, the criterion for strong coupling ($\Omega$ > ($\Gamma _{X}$ + $\Gamma _{T}$)/2) is rigorously satisfied. For simplicity, the damping losses can be ignored in the strong coupling regime due to their weak effects.
Fig. 6. (a) Absorption spectra of the new system ($d$ = 20 nm) with different incident angles. The triangle marks indicate the dispersion of the hybrid modes. (b) Mixing fractions of the TEM and exciton in the polariton bands for the TEM-exciton coupling. (c), (d) Electric field profile distributions of the different hybrid modes. (e) Mode splitting energy as a function of doped layer thickness. (f), (g) Absorption spectra of the new system when $d$ = 50 and 250 nm.
Figure 6(b) shows the mixing fractions of the exciton and TEM in the two polariton bands. Obtained from the COM, the weighting efficiencies vary monotonically with the TEM-exciton detuning ($\Delta E$ = $E_{T}$ - $E_{X}$). Taking LPB as an example, a negative detuning ($E_{T}$ < $E_{X}$ for $\theta$ < 33.4$^{\circ }$) corresponds to the smaller $\left | \alpha _{T} \right |^{2}$ and larger $\left | \alpha _{X} \right |^{2}$ in LPB. It seems that the LPB mainly comes from excitons. As $\theta$ increases ($\theta$ > 33.4$^{\circ }$), $\left | \alpha _{T} \right |^{2}$ prevails over $\left | \alpha _{X} \right |^{2}$, $\Delta E$ becomes positive ($E_{T}$ > $E_{X}$), indicating that the TEMs contribute more to LPB. Specially, a zero detuning ($E_{T}$ = $E_{X}$ at $\theta$ = 33.4$^{\circ }$) corresponds to an intermediate state with same fraction ( $\left | \alpha _{T} \right |^{2}$ = $\left | \alpha _{X} \right |^{2}$). For the UPB, the trend of weighting efficiencies is contrary to that of the LPB. The electric field profile distributions (corresponding to six resonance peaks in Fig. 5(b)) at different detuning conditions are shown in Figs. 6(c) and 6(d). For the hybrid modes in LPB, the electric field intensity in the middle position increases with the increase of $\theta$, and finally presents a TEM-like field distribution. For the hybrid modes in UPB, a localized mode (TEM-like mode) gradually becomes a nonlocalized mode as $\theta$ increases, showing the opposite change with LPB. Clearly, the components in the LPB (or UPB) can be actively modulated by varying the incident angle, thus making the hybrid mode exhibits TEM-like or exciton-like characteristics.
Finally, we investigate the strong coupling with different thicknesses of the doped layer. As mentioned above, we can manipulate the strong coupling by adjusting the incident angle at a certain doped layer thickness (here we only consider the angle range from 0$^{\circ }$ to 70$^{\circ }$). For different doped layer thicknessns, the resonance wavelength of TEM is different, so the intersection point of TEM and exciton mode is also different. In particular, the dispersion curve of TEM (red line) may not pass through the nondispersion curve of exciton (white line) under a certain thickness, which means that TEM will not interact with exciton. As depicted in Fig. 6(e), the mode splitting energy increases with the doped layer thickness in general, but for some thickness, the mode splitting energy decreases to zero, indicating that there is no coupling in the system. Figures 6(f) and 6(g) show absorption spectra of the new system when $d$ = 50 and 250 nm. For $d$ = 50 nm, two polariton bands exhibit anticrossing feature near the intersection point (about 49$^{\circ }$) where the TEM overlaps with exciton mode, with mode splitting energy of 70 meV. As $d$ is increased to 250 nm, the mode splitting energy is increased to 124 meV and the intersection point is about 25$^{\circ }$. Especially, when $d$ increases to a certain thickness, FP-exciton coupling occurs in the system. In brief, the variation of doped layer thickness can make the manipulation of strong coupling more diversified.
4. Conclusion
We investigated the light-matter interactions in the classic topological PhC heterostructure theoretically and numerically. As a result of the strong coupling of light and TEM at the interface of the two PhCs, a high quality resonance peak (high transmission, narrow bandwidth) occurs in the configuration. We can achieve the manipulation of the perfect transmission peak by changing the incident angle and dielectric layer properties (thickness and dielectric constant). The system has an excellent transmission response, so it can be used for high-performance refractive index sensing, narrowband filtering, and optical switching, etc. We further studied the TEM-exciton coupling by doping the J-aggregates in the dielectric layer. As the incident angle changes, the dispersive TEM can interact with the nondispersive exciton, resulting in two polariton bands with anticrossing feature. The numerical results match well with the theoretical prediction of COM. In addition, we found that the mode splitting energy could be tuned by the doped layer thickness. These results may inspire related studies about hybrid light-matter interactions in a topological PhC heterostructure.
Funding
Scientific Research Foundation of the Graduate School of Southeast University (YBPY2014); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20-0103); Fundamental Research Funds for the Central Universities; 333 Projection of Jiangsu Province; Six Talent Peaks Project in Jiangsu Province (XCL-077); Higher Education Discipline Innovation Project (111-2-05); National Natural Science Foundation of China (61631007, 61701108, 61831006); National Key Research and Development Program of China (2017YFA0700201, 2017YFA0700202, 2017YFA0700203).
Disclosures
The authors declare no conflicts of interest.
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