1. Introduction

Recently, topological photonics have attracted tremendous attentions [1–3]. Various artificial micro-structures have been proposed to realize topological edge states [4–7]. As a one-dimensional case in photonic systems, the edge states (also called Tamm plasmon polaritons) can be excited at the interface of two kinds of one-dimensional photonic crystals (1DPCs) [8–12]. Similarly, heterostructures composed of metal layers and 1DPCs can also support edge states [13,14]. The electromagnetic fields of the edge states are strongly localized at the interface between the metal layer and the 1DPC. Different from the surface plasmon polaritons, such edge states can be directly excited without prism or grating coupling because their dispersion relations lie inside the light cone of air. The edge states in such heterostructures can be utilized in a variety of applications, such as solar cells [15], lasers [16], enhancement of Faraday rotation [17], perfect optical absorbers [18,19], and nonlinear optical devices [20,21]. Very recently, researchers exploited the singularity of the ellipsometric phase (i.e., reflection phase difference between two orthogonal polarizations) to design ultrasensitive optical sensors based on heterostructures composed of metal layers and all-dielectric 1DPCs [22,23]. At oblique incidence, the edge state wavelengths for TM and TE polarizations are different. As a result, around the edge state wavelength for TM polarization, the ellipsometric phase will change dramatically because the reflection phase for TM polarization changes dramatically (resonance) while that for TE polarization changes smoothly (off resonance). Similarly, around the edge state wavelength for TE polarization, the ellipsometric phase will also change dramatically. This phase-singularity property can be used in ultrasensitive optical sensing. However, the band gaps in the all-dielectric 1DPCs will shift toward short wavelengths (i.e., blueshift) as the incident angle increases both for TM and TE polarizations [24], which indicates that the edge states in the heterostructures composed of metal layers and all-dielectric 1DPCs will also be blueshited both for TM and TE polarizations [13]. At small incident angles (e.g., $20°$), the edge state wavelengths for TM and TE polarizations are very near, leading to a tiny ellipsometric phase. As a result, such sensors can only work at large incident angles (e.g., $65°$) in which the edge state wavelengths for two polarizations can be well separated [22,23], leading to a narrow operating angle range as we will discuss in Sec. 2. For the sensor in integrated optical system [25], the input beam with finite width has angular divergence [26,27]. This angular divergence will introduce a distribution of the incident angle, leading to the reduction of the sensing resolution [28]. Therefore, sensors which can work with high resolution in a broad angle range are desirable in integrated optical systems.

Recently, hyperbolic metamaterials (HMMs) with hyperbolic iso-frequency surfaces have been utilized in a variety of applications, such as ultrasensitive sensing [29–31], super-resolution imaging [32], wavefront manipulation [33], plasmonic waveguide [34], and angle-independent band gap [35,36]. In 2014, researchers found that unconventional surface wave and volume plasmon polariton can be realized in 1DPC containing layered HMMs [37,38]. Very recently, researchers achieved redshift gap for TM polarization and blueshift gap for TE polarization in 1DPC composed of layered HMMs and isotropic dielectrics [39]. For TM polarization, the propagating phase within the HMM layer will increase as the incident angle increases because the iso-frequency curve of HMM is hyperbola. In the isotropic dielectric layer, the propagating phase will decrease with the increase in the incident angle because the iso-frequency curve is circle. Therefore, if the increasing rate of the propagating phase within the HMM layer is larger than the decreasing rate of the propagating phase within the isotropic dielectric layer, a redshift gap can be realized according to the Bragg condition. However, for TE polarization, the iso-frequency curves of HMM and isotropic dielectric are both circle, leading to a blueshift gap. As a result, the gap edges of such gap can be utilized in high-efficient polarization selectivity [39].

In this paper, we realize the edge states in the heterostructure composed of a metal layer and a 1DPC with HMMs and demonstrate that the edge state for TM polarization is redshifted while that for TE polarization is blueshifted. With the increase in the incident angle, the edge state will shift toward opposite directions for TM and TE polarizations. Different from conventional heterostructures composed of metal layers and all-dielectric 1DPCs, the edge state wavelengths for TM and TE polarizations can be well separated even at a small incident angle (e.g., $20°$) in the proposed heterostructure. As a result, the phase-singular property can be maintained in a wide angle range. Based on this wide-angle phase-singular property, we design a ultrasensitive biosensor with the resolution equal to or lower than the order of ${10}^{-5}$RIU which can work in a broad angle range from $5°$ to $84°.$

2. Wide-angle ellipsometric phase singularity

Firstly we design a redshift gap (i.e., the gap shifts toward long wavelengths as the incident angle increases) in the 1DPC composed of alternating HMMs and dielectrics for TM polarization according to the theory in [39]. Here the HMM layer A is mimicked by a subwavelength metal-dielectric multilayer (CD)^S and the isotropic dielectric layer is denoted by B. Therefore, the 1DPC can be denoted by [(CD)^SB]^N. We choose S = 2 and N = 9. The thicknesses of C, D and B layers are $d_C,$ $d_D$ and $d_B,$ respectively. The thickness of the HMM layer A is $d_A=S(d_C+d_D).$ For the subwavelength dielectric layer C and the isotropic dielectric layer B, we choose Si with a refractive index $n_C=n_B=3.48$ [40]. For the subwavelength metal layer D, we choose indium tin oxide (ITO) because it is a kind of low-loss plasmonic materials in infrared wavelengths [41]. The relative permittivity of ITO can be described by the Drude model $\epsilon ={\epsilon }_{\infty }-{\omega }_P^2/({\omega }^2+i\omega \gamma ),$ where ${\epsilon }_{\infty }$ represents the high-frequency permittivity, ${\omega }_P$ and $\gamma$ represent the plasma frequency and damping frequency, respectively [42]. The values of the parameters can be fitted by the experimental measurement, which are ${\epsilon }_{\infty }=4,$ $\hslash {\omega }_P=2.03eV$ and $\hslash \gamma =0.0827eV$ [42]. The filling ratio of ITO $f=d_D/(d_C+d_D)$ is chosen to be 0.23. According to the effective medium theory [43], the multilayers (CD)^S can be viewed as a type-I HMM within the wavelength ranging from 1233 to 1677 nm. Here, we choose the Bragg wavelength as ${\lambda }_{Bragg}=1430nm.$ Then two thickness conditions of redshift gap can be determined by [39]

Then we calculate the reflectance spectra of the 1DPC [(CD)²B]⁹ versus incident angle for TM and TE polarizations based on the transfer matrix method [44], as shown in Fig. 1. The incident and exit media are air and BK7 with a refractive index of $n_S=1.515$ [45], respectively. The blue dashed lines represent the gap-edges extracted from the reflectance dips. One can see that the gap redshifts for TM polarization as we predicted. However, this gap still blueshifts for TE polarization because the iso-frequency curve of HMM is a circle rather than a hyperbola for TE polarization [39].

 

Fig. 1 Reflectance spectra of [(CD)²B]⁹ versus incident angle for TM and TE polarizations. Blue dashed lines represent the gap-edges.

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Next we put a metal layer M on the top of 1DPC and constitute a heterosturecture denoted by M[(CD)²B]⁹, as shown in Fig. 2(a). We choose Cu as the metal layer. The relative permittivity of Cu can also be described by the Drude model. The experimental fitting parameters are ${\epsilon }_{\infty }=1,$ $\hslash {\omega }_P=7.38eV$ and $\hslash \gamma =0.0910eV$ [46]. The thickness of the metal layer is set to be $d_M=8nm.$ Suppose that the incident and exit media are air and BK7, respectively. Based on the transfer matrix method, the reflectance spectra versus incident angle for TM and TE polarizations are calculated, as given in Fig. 2(b). The blue dashed line represents the gap-edges. One can see that there is always a reflectance dip within the gap, which indicates the formation of the edge state [13]. The Variation of the edge state wavelength with the incident angle is shown by the red dotted line. From Fig. 2(b), it can be seen that the variation tendency of the edge state wavelength with the incident angle is the same as that of the gap-edge. In appendix, we explain this property based on the reflection phase canceling condition of the edge state [13] and also demonstrate that the variation tendency of the edge state wavelength is robust as the layer thicknesses $d_M,d_C,d_D$and $d_B$ change. For TM polarization, the edge state shifts from 1448 to 1464 nm as the incident angle increases from $0°$ to $60°.$ While for TE polarization, the edge state shifts from 1448 to 1397 nm. As a result, when the incident angle is larger than $18°,$ the wavelength difference of the edge state for two different polarizations can be larger than 10 nm. This phenomenon will lead to wide-angle ellipsometric phase singularity property.

 

Fig. 2 (a) Schematic of the heterostructure M[(CD)²B]⁹. (b) Reflectance spectra of M[(CD)²B]⁹ versus incident angle for TM and TE polarizations. Red dotted and blue dashed lines represent the edge state wavelength and the gap-edge wavelength, respectively.

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Then we choose two typical incident angles $\theta =20°$ and $\theta =40°$ to show the reflection property of the proposed structure. Figures 3(a) and 3(c) give the reflectance and the reflection phase spectra of M[(CD)²B]⁹ at $\theta =20°$ and Figs. 3(b) and 3(d) give those at $\theta =40°.$ The red and blue solid lines represent TM and TE polarizations, respectively. P1-P4 represent the edge states for TM and TE polarizations at $\theta =20°$ and $\theta =40°.$ One can see that the wavelength differences between the edge states for TM and TE polarizations are 13 and 42 nm at $\theta =20°$ and $\theta =40°,$ respectively. Besides, the reflection phase changes dramatically around the edge state wavelength both for TM and TE polarizations because of the resonance of the edge state [22]. However, the reflection phase changes smoothly in the non-resonance wavelength range. Suppose a linearly polarized light composed of TM and TE components is incident onto the structure, the reflected light will become an elliptically polarized light, which can be described by two ellipsometric angles: the magnitude of the ellipsometric reflectivity ratio $\Psi =\arctan (\left|r_{TM}/r_{TE}\right|)$ and the ellipsometric angles $\Delta ={\phi }_{TM}-{\phi }_{TE}$ [22]. Here, $r_{TM,TE}=\left|r_{TM,TE}\right|e^{i{\phi }_{TM,TE}}$ and ${\phi }_{TM,TE}$ represent the complex reflection coefficient and the reflection phase for TM or TE polarization, respectively. We calculate the ellipsometric phase spectra at $\theta =20°$ and $\theta =40°,$ as shown in Figs. 3(e) and 3(f). Around the edge state wavelength for TM polarization ${\lambda }_{TM},$ the ellipsometric phase will change dramatically because the reflection phase for TM polarization ${\phi }_{TM}$ changes dramatically while that for TE polarization ${\phi }_{TE}$ changes smoothly. Similarly, around the edge state wavelength for TE polarization ${\lambda }_{TE},$ the ellipsometric phase will also change dramatically. As a result, one can see that the ellipsometric phase changes dramatically around the dual edge state wavelengths for TM and TE polarizations in Figs. 3(e) and 3(f). This phase-singular phenomenon around dual wavelengths can be utilized to design ultrasensitive reflective index sensors [47]. For the proposed structure M[(CD)²B]⁹, the phase-singular property can be maintained in a broad angle range.

 

Fig. 3 (a) Reflectance, (c) reflection phase and (e) ellipsometric phase spectra of M[(CD)²B]⁹ at $\theta =20°.$ (b) Reflectance, (d) reflection phase and (f) ellipsometric phase spectra of M[(CD)²B]⁹ at $\theta =40°.$ Red and blue solid lines represent TM and TE polarizations, respectively. (g)-(j) Normalized magnetic or electric field intensity distributions of M[(CD)²B]⁹ for edge states P1-P4. The thickness of the metal layer is ten times of its real thickness for better visibility. Black solid line represents the boundary of the structure. Green solid line represents the interface between the metal layer and the 1DPC.

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In order to show the field localization property of the edge state, we also do the full-wave simulation based on the finite element method. Figures 2(g)-2(j) give the normalized magnetic (electric) field intensity distributions at the edge states P1-P4 for TM (TE) polarization. Note that the thickness of the metal layer is ten times of its real thickness for better visibility. The black solid line represents the boundary of the structure. The green solid line represents the interface between the metal layer and the 1DPC. One can see that the magnetic (electric) field of the edge state is localized around the interface between the metal layer M and the 1DPC for TM (TE) polarization. Note that the maximal magnetic or electric field is not exactly located at the interface between the metal layer and the 1DPC because the metal layer is thin (so the electromagnetic wave can slightly permeate into the 1DPC). Besides, the maximal electric field position for TE polarization is closer to the interface between the metal layer and the 1DPC compared to the maximal magnetic field position for TM polarization. The reason is that the reflectance of the metal layer for TM polarization is lower than that for TE polarization.

Without loss of generality, we just focus on the edge state for the TE branch. In order to quantify the degree of the phase singularity at different incident angles, we calculate the corresponding absolute value of the derivative of the ellipsometric phase to the wavelength $K=\left|d\Delta /d\lambda \right|$ at the TE edge state wavelength, as shown by blue five-pointed stars in Fig. 4. For comparison, we also calculate $K$ at the TE edge state wavelength for different incident angles in the conventional heterostructure composed of a metal layer and an all-dielectric 1DPC, as shown by green five-pointed stars. The conventional heterostructure can be denoted by M(AB)⁹, where M, A and B represent Cu layer, Si layer with a refractive index of $n_A=3.48$ [40] and SiO₂ layer with a refractive index of $n_B=1.44$ [40,48]. The thicknesses of M, A and B layers are chosen to be $d_M=13nm,$ $d_A=79nm$ and $d_B=191nm,$ respectively. The incident and exit media are also chosen to be air and BK7, respectively.

 

Fig. 4 Absolute value of the derivative of the ellipsometric phase to the wavelength $K=\left|d\Delta /d\lambda \right|$ at the edge state wavelength for TE polarization as a function of incident angle. Blue and green five-pointed stars represent the cases of the proposed structure M[(CD)²B]⁹ and the conventional structure M(AB)⁹, respectively. Black dashed line represents $K=\text{30}\text{.}$

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Usually, as the degree of the phase singularity becomes greater, the singular-phase-based reflective index sensor becomes more sensitive [23,47]. We plot the black dashed line for $K=\text{30}$ in Fig. 4 to mark the effective operating angle range. Here the reference value of $K=30$ is chosen according to [22]. One can see that the operating angle of the proposed structure ranges from $\text{24}°$ to $\text{45}°,$ with an angle range of 21 degrees. However, the operating angle of the conventional structure ranges from $\text{62}°$ to $\text{67}°,$ with an angle range of only 5 degrees. In integrated optical systems, vertical-cavity surface-emitting laser (VCSEL) is often used as the light source. However, such light source with finite width has angular divergence. For the conventional oxide-confined 850 nm VCSEL, the angle range of divergence is about 20 degrees [49,50]. The operating angle range of the proposed structure can cover this angle range of divergence while that of the conventional structure cannot. At small incident angles (e.g., $10°$), the degree of the phase singularity in the conventional structure is far lower than that in the proposed structure because the edge state wavelengths for TM (1442 nm at $10°$) and TE polarizations (1445 nm at $10°$) are very close together in the conventional structure. As a result, the phase singularity can only maintained at large incident angles (e.g., $65°$) in the conventional structure. Compared to the conventional structure, the proposed structure has wider operating angle range.

3. Wide-angle ultrasensitive biosensor assisted by ellipsometric phase singularity

Figure 5(a) shows the schematic of the proposed biosensor. On the bottom, there is a thick liquid cell filled with bio-solution, whose refractive index is $n_{Bio}.$ A linearly polarized light composed of TM and TE components is incident onto the top of the biosensor and the ellipsometric phase of the reflected light can be calculated as $\Delta ={\phi }_{TM}-{\phi }_{TE}.$ Here we choose a small incident angle $\theta =20°$ as the operating angle to assess the performance of the biosensor. From Fig. 3(a), one can see that the edge state wavelength for TE polarization is $\lambda =1438nm,$ which is chosen to be the operating wavelength. We calculate the ellipsometric phase spectra around $\lambda =1438nm$ for $n_{Bio}=1.33,$ as shown by black solid line in Fig. 5(b). For comparison, the ellipsometric phase for $n_{Bio}=1.34$ [0.01RIU (Refractive Index Units) deviating from $n_{Bio}=1.33$] is also given by black dashed line. It can be seen that the ellipsometric phase changes from 68.41 to 67.54 degrees ($\left|\delta \Delta \right|=0.87$deg) at the wavelength $\lambda =1438nm,$ As the refractive index of the bio-solution increases from 1.33 to 1.53, the ellipsometric phase almost linearly decreases from 68.41 to 50.82 degrees ($\left|\delta \Delta \right|=17.59$deg), which indicates that the proposed biosensor has near-linear response in a relative wide range of refractive index. Besides, the sensitivity of the proposed biosensor can be defined by the change rate (absolute value) of the ellipsometric phase with the refractive index $S(n_{Bio})=\left|d\Delta /d{n}_{Bio}\right|$ [23,51], which is shown in Fig. 5(d). The sensitivity remains almost constant in the refractive index range from 1.33 to 1.53. For $n_{Bio}=1.409,$ the sensitivity reaches the maximum $S_{\max }=89.6$deg/RIU. Under the current ellipsometric phase measuring technique, a realistically achievable limit for the ellipsometric phase noise can be low as $\alpha =0.001\mathrm{deg}$ [52]. Therefore, the minimal refractive index resolution of the biosensor at the incident angle $\theta =20°$ can be determined by $\alpha /S_{\max }=1.1\times {10}^{-5}$RIU, indicating that the proposed biosensor can work with a high refractive index resolution even at a small incident angle.

 

Fig. 5 (a) Schematic of the proposed biosensor. (b) Ellipsometric phase spectra for $n_{Bio}=1.33$ and $n_{Bio}=1.34$ at $\theta =20°.$ (c) Ellipometric phase as a function of refractive index at $\theta =20°$ and $\lambda =1438nm.$ (d) Sensitivity as a function of refractive index at $\theta =20°$ and $\lambda =1438nm.$

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Finally, we plot the minimal refractive index resolution as a function of the incident angle to assess the angle-dependent performance of the biosensor, as shown in Fig. 6. It can be seen that the resolution of the singular-phase-based reflective index sensor is strongly relate to the degree of the phase singularity (see Fig. 4). As the degree of the phase singularity becomes greater, the reflective index resolution becomes lower. The black dashed line represents the resolution ${10}^{-4}$RIU. One can see that the resolution can be equal to or lower than the order of ${10}^{-5}$RIU in a broad angle range from $5°$ to $84°.$ At the incident angle $\theta =39.90°,$ the resolution reaches minimum $2.8\times {10}^{-8}$RIU. Therefore, our biosensor can work with high refractive index resolutions in a broad angle range.

 

Fig. 6 Minimal refractive index resolution as a function of the incident angle. Black dashed line represents the resolution ${10}^{-4}$RIU.

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4. Conclusion

In summary, we realize redshift edge states for TM polarization and blueshift edge states for TE polarization in heterostructures composed of metal layers and 1DPCs containing HMMs. Different from the conventional heterostructures composed of metal layers and all-dielectric 1DPCs, such edge states shift toward opposite directions for two orthogonal polarizations as the incident angle increases. This property will lead to the wide-angle ellipsometric phase singularity, which can be utilized for designing wide-angle ultrasensitive biosensors.

Appendix Reflection phase analysis and robustness of the edge state

Here we calculate the reflection phases of the metal layer M (shown by solid line) and the 1DPC [(CD)²B]⁹ (shown by dashed line) at different incident angles for TM and TE polarizations, as given in Figs. 7(a) and 7(b), respectively. The black, green and red lines represent the reflection phases at the incident angles $\theta =0°,$ $20°$ and $40°,$ respectively. It should be noted that we multiply the reflection phase of the metal layer by minus one. Therefore, the crossing point between the solid and dashed lines represents the reflection phase canceling point. One can see that the phase canceling wavelength is 1439 nm at normal incidence, which is approximately equal to the edge sate wavelength 1448 nm extracted from the reflectance spectrum. Owing to the absorptive loss of the metal layer and the 1DPC, the reflectance cannot reach unity, leading to a slight deviation between the phase canceling wavelength and the edge state wavelength extracted from the reflectance spectrum [13]. For TM polarization, the phase canceling point shifts from 1439 to 1449 nm when the incident angle increases from $0°$ to $40°,$ leading to the redshift edge state. While for TE polarization, the phase canceling point shifts from 1439 to 1414 nm, leading to the blueshift edge state.

 

Fig. 7 Reflection phase of the metal layer (multiple by minus one) and the 1DPC [(CD)²B]⁹ at different incident angles for (a) TM and (b) TE polarizations. Black, green and red solid (dashed) lines represent the reflection phase of the metal layer (the 1DPC) at the incident angles $\theta =0°,$ $20°$ and $40°,$ respectively. Crossing point between solid and dashed lines represents the reflection phase canceling point.

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Then we change the thickness of the metal layer $d_M$ and give the corresponding dependence of the edge state wavelength (extracted from the reflectance spectrum) on the incident angle for TM and TE polarizations, as shown in Fig. 8. It can be seen that although the edge state wavelength changes with the metal layer thickness $d_M,$ the edge state always remains redshift property for TM polarization while always remains blueshift property for TE polarization. Therefore, the phenomenon that the edge state wavelength shifts toward opposite directions for two orthogonal polarizations as the incident angle increases is robust against the thickness of the metal layer $d_M.$

 

Fig. 8 Dependences of the edge state wavelength on the incident angle for TM and TE polarizations with different metal layer thickness $d_M.$

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At last, we also give the robust analysis of the influence of the thicknesses $d_C,d_D$and $d_B$ on the gap-edge and the edge state wavelengths (fix the thickness of the metal layer $d_M=8nm$), as shown in Figs. 9(a)-9(f). The solid and dashed line in Figs. 9 (a), 9(c) and 9(e) represent the upper and bottom gap-edges, respectively. From the results in Figs. 9(a), 9(c) and 9(e), one can see that two gap-edge always remain redshift property for TM polarization while always remain blueshift property for TE polarization. Therefore, the variation tendency of the bandgap wavelength is robust as the layer thicknesses $d_C,d_D$ and $d_B$ change. From the results in Figs. 9(b), 9(d) and 9(f), one can see that the edge state also remains redshift property for TM polarization and remains blueshift property for TE polarization, indicating that the variation tendency of the edge state wavelength is also robust as the layer thicknesses $d_C,d_D$ and $d_B$ change.

 

Fig. 9 Dependences of the gap-edge and the edge state wavelengths on the incident angle for TM and TE polarizations with different layer thicknesses $d_C,d_D$ and $d_B.$ Solid and dashed lines in Figs. 9 (a), (c) and (e) represent the upper and bottom gap-edges, respectively.

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Funding

National Key Research Program of China (2016YFA0301101); National Natural Science Foundation of China (61621001, 11774261, 11234010, 61661007 and 91850206); Science Foundation of Shanghai (17ZR1443800); Shanghai Science and Technology Committee (18JC1410900); Natural Science Foundation of Guangxi Province (2016GXNSFAA380198).

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