1. Introduction

Optical surface modes resonance [1,2] has been widely adopted in the designs of low-cost label-free bio-sensors [3–8]. During the sensing process, aqueous solution of bio-molecules is generally directly put on the sensor surface that sustains the surface mode, and the surface modes are strongly localized at the interface between the bio-solution and the sensors [1, 2, 9]. When the refractive index (RI) of the bio-solution is modulated by the different concentration of bio-molecules, the optical surface mode will well be monitored. The concentration information of the bio-molecules can be found by interrogating the shifts of the resonant spectrum either in wavelength or angular domain. Because the bio-solution locates directly at the surface of the sensors, it is very convenient to clean and use for the next time, which well reduces the cost of sensing.

Generally, the RI changes induced by the concentration of bio-molecules in the solutions are tiny. To get a better sensing performance, it is expected that the scattering peaks (or dips in the reflection) caused by the surface mode resonance should be sharp and sensitive to RI modulations. The figure of merit (FOM) of a sensor is generally judged by the ratio between the sensitivity and the full width at half maximum (FWHM) of the resonant spectrum [5]. In the traditional sensor designs based on prism-coupled Bloch surface wave (BSW) resonance or surface plasmon polaritons (SPP), the excitation of the resonant mode is marked by the reflection dip resulted from the enhanced absorption near the interface [10–12]. However, the presence of the absorption makes that the FWHM of the resonance cannot be made too small. The FOMs of the corresponding sensors are restricted by the absorption. In the sensor designs based on guided BSW resonance (GBR) (grating-coupled BSW resonance) [8, 13], the excitation of the BSW well adjusts the scattering of the propagating diffraction mode (e. g., the zeroth order diffraction) even when there is no absorption. One can directly detect the GBR by the transmission spectrum of the propagating mode. The absorption is then not as indispensable as in the cases of sensor designs based on prism-coupled BSW resonance or SPP. The FWHM of the GBR and thus the FOMs of the corresponding sensor designs are no longer restricted by absorption. By simply increasing the number of the periods of the photonic crystal (PhC) slab under the gratings or utilizing shallow grating or grating with very narrow grooves [14, 15], GBR with very large quality factor and thus sensors with extremely large FOM can be realized [8].

Although the FOM of a GBR sensor can be made very high by narrowing down the FWHM of the resonance, the increasing of the sensitivity is still of special importance. Generally, a large sensitivity can well reduce the detecting difficulties and thus the cost of corresponding designs. In the designs of SPP sensors where metal components provide strong dispersion near the working wavelength, large wavelength sensitivity can be realized, especially when metal grating is introduced in the designs [16–20]. In the case of sensors based on BSW, the shift of the resonance spectrum in the angular domain is often interrogated. Recent studies [8, 13] show that, the angular sensitivity of GBR sensors in the case of non-azimuthal illuminations is about 100-300 degree per RI unit (Deg/RIU). The sensitivity is larger than that of sensors based on prism-coupled BSW resonance which is no larger than 100 Deg/RIU [6, 7, 10–12]. More recently [21], the angular sensitivities of sensors based on a kind of grating-coupled leaky BSW resonance under azimuthal illuminations [21, 22] is studied. By fixing the polar angle to a small value (θ = 5.4°), the azimuthal sensitivity of the leaky BSW reaches nearly 2000 Deg/RIU when it works near the azimuthal angle of 5°. The azimuthal sensitivity is almost an order higher than the polar angle sensitivity of sensors based on GBR excited by non-azimuthal illuminations. The leakage BSW mode is however a kind of defect mode which is mostly localized inside the coupling grating whose grooves are filled with bio-solutions. The high sensitivity should arise from the fact that the changes of the RI of the bio-solution filled into the grating well adapt the diffraction of the detecting signals in the gratings [16–20, 23]. In this paper, the azimuthal sensitivity of the GBR excited by transverse electric (TE) polarized illumination at the wavelength of λ₀ = 632.8 nm is studied and the criterion for large sensitivities is provided. The rest of this paper is organized as following. In Sec. 2, the principles for the TE polarized BSW mode are discussed. The azimuthal sensitivity of sensors based on GBR is theoretically analyzed in Sec. 3. In Sec. 4, the numerical studies about the azimuthal sensitivity of GBR sensor designs are given. The discussion and conclusion are given as Sec. 5.

2. The excitation of TE polarized BSW mode at the interface between a one-dimensional PhC slab and a bio-solution layer

Inside one dimensional PhCs, the periodicity of the PhC in the normal (z-) direction makes that electromagnetic (EM) field inside the PhCs obeys Bloch-Floquet theorem. The EM field then takes the form of Bloch modes propagating (decaying when lying in a bandgap) towards two contrary directions [2, 24]. The in-plane EM field components of the Bloch mode at the lth interface of the PhCs unit takes

In the case of BSW excited at the interface between a finite PhC slab and a bio-solution layer, the BSW mode lies in the bandgap of the PhC slab. When the bio-solution layer lies in the backward side of the interface, the BSW mode takes the form of the Bloch mode decaying along the –z direction (increasing along the z direction) inside the PhC slab [24]. We express the EM field of the BSW mode at the PhC side of the interface as $E_F\left[\begin{array}{c}1\\ {\xi }_B^{-}\end{array}\right]$ where $E_F$ is the tangential electric field component and ${\xi }_B^{-}$ should then be the ratio between the tangential components of respectively the magnetic and the electric field of the Bloch mode evanescent along the –z direction. By simple algebraic calculations, one can get the ${\xi }_B^{-}$ by solving the eigen vector of the transfer matrix of one PhC unit [24]. Here, the ${\xi }_B^{-}$ has the dimension of optical admittance and, in the following, it is named as Bloch admittance and is normalized to the admittance of vacuum ($\sqrt{{\epsilon }_0/{\mu }_0}$). For a bandgap eigen mode, the ${\xi }_B^{-}$ is purely imaginary.

Inside the bio-solution layer with the RI being n, the tangential wave vector of the BSW mode, $k_{\parallel }^b$ takes (normalized to k₀)

The Eq. (5) is however a simple guidance to the design of hetero-structures which can sustain BSW [24]. Generally, the Bloch admittance ranging from minus to positive infinity can all be realized by only adjusting the thicknesses of the composing layer of the one-dimensional PhCs. In the following, we will focus on BSW with a small and imaginary $k_z$. A small $k_z$ makes that the BSW decays slow and thus extends far inside the bio-solution layer. The corresponding designs can then detect not only the RI modulations in the immediate vicinity of the interface, but also that occurs farther away from the PhC slab.

3. Sensor designs based on guided Bloch surface wave resonance interrogated in the azimuthal angle domain and the sensitivity

A GBR sensor under azimuthal illumination is schematically shown in Fig. 1. It is designed that the BSW should be localized at the interface between the PhC slab and the thick bio-solution layer. At the other side of the sensor, optical gratings are designed to couple the incident EM illuminations to the BSW mode and thus the GBR is excited. There is an additional buffer layer set between the gratings and the PhC slab. The grating/multilayer/bio-solution hetero-structure is placed in air with the RI being 1. A platform which can be rotated azimuthally should be utilized to fix the hetero-structure and to make the sensing. The polar angle θ and the azimuthal angle φ are respectively measured according to the surface normal and to the plane perpendicular to the ridges of the grating.

 

Fig. 1 The schematic of the grating-coupled BSW resonance sensor under azimuthal illuminations.

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In the design of the grating/multilayer GBR hetero-structure, we firstly set a small $k_z^b$ which is the normal wave vector component of the target BSW inside the bio-solution layer. The tangential wave vector component of the BSW in the bio-solution layer is then

According to the conclusions made in the former part (Eq. (5)), the normal wave vector component of the BSW mode in the bio-solution layer, $k_z^b$ is the most important parameter for the BSW. Careful considerations on $k_z^b$ are required for the design of GBR sensor with better performance. When the mth diffraction order of the incident EM wave is coupled to the BSW mode, there is then

When the shifts of the GBR in the azimuthal angle domain are interrogated in the sensing practice, the polar angle of the incident EM wave is fixed. The $k_z^b$ is then a function of the azimuthal angle and the RI of the bio-solution layer. When there are small changes in these two parameters ($\Delta n$ and $\Delta \phi$), there is then

By the constant Bloch admittance approximation, according to Eq. (5), there should be

Comparing with the angular sensitivity of the GBR sensors designed to work under non-azimuthal illuminations [8, 13], the Eq. (13) shows that the current scheme of sensor designs provides an additional adjusting dimension of the azimuthal angle φ for the modulation of the sensitivity. One should especially notice that, the sine functions in the denominator of the expression on the right-hand side of Eq. (13) approach 0 for small θs and φs. Extremely large sensitivity can then be realized when the two angles are near zero. One can then detect very tiny RI modulations in the bio-solution layer.

4. Numerical simulations

In the numerical calculations, a four-period PhC slab (ABC)₄ is used where the A and C layers are dielectrics with the RI being 2.584 (TiO₂ at the working wavelength of λ = 632.8nm [25]) and the B layers are designed to be made of SiO₂ with the RI being 1.457 [26]. The thicknesses of the corresponding layers are respectively d_A = 40 nm, d_B = 100 nm and d_C = 30 nm. The bio-solution layer with the RI being near 1.333 (water) is placed adjacent to the outmost C layer. The BSW is designed to be excited at the interface between this PhC slab and the bio-solution layer with the normal wave vector component of the BSW mode in the bio-solution layer being small ($k_z^b\approx 0.23$). All through this paper, the thickness of the region to be probed is set to be 3μm. However, when the bio-solution layer is thick enough, the outer surface of the probed region affects little the excitation of the BSW mode [8] because the BSW mode mainly localized near the inner (front) surface of the corresponding region. Thus, one does not need to strictly control the thickness of this layer in the sensing process. At the other side of the PhC slab, there is a 160 nm buffer layer which is made of SiO₂ (n_B = 1.457) lying between the PhC slab and the diffraction gratings. The gratings are designed to couple the propagating illuminations to the BSW mode. The RIs of the ridges and the grooves of the grating are respectively 2.0 and 1.0 (air). In the following numerical simulations, only the physical dimensions of the gratings (the period Λ, the filling factor f, and the thickness d_O) are adjusted to excite BSW under different illumination conditions and to optimize the transmission profiles. By the optimized gratings, when BSW is excited, the transmission from the grating-multilayer configuration takes typical Fano resonant profiles [27] with sharp peaks. The locations of the peaks are then used to indicate the RI of the region to be probed.

The Rigorous coupled-wave analysis (RCWA) [28, 29] method dealing with conical diffraction of one-dimensional gratings is used in the following simulations [29]. The RCWA is well-adopted in the numerical simulations of the EM wave scattered by micro-structures containing optical gratings. In [21], hetero-structures made of one dimensional grating and flat layers under azimuthal illuminations are simulated by both RCWA and full-field computation methods such as finite differential time domain method (FDTD) and finite element method (FEM). It is shown that the RCWA provides results that fit precisely with that by the compared methods.

The simulated transmission of the GBR sensor designed to work near θ = 5° and φ = 5° are provided in Fig. 2(a). The thickness and the period of the gratings are respectively d_O = 116nm and Λ = 439.7 nm and the filling factor of the gratings is f = 0.34. One can find that the resonance peak is very sensitive to the azimuthal angle changes and quite insensitive to that of the polar angle. The BSW mode in the hetero-structure and the typical Fano-resonance transmission profile are respectively demonstrated in Fig. 2(b) and the inset. One should notice that, due to the small $k_z^b$, the BSW mode decays slowly and extends several wavelengths deep into the bio-solution.

 

Fig. 2 (a) The transmission of a GBR sensor versus the azimuthal and the polar angles. The structure is designed to excite GBR by illuminations near θ = 5° and φ = 5°. (b) The electric field of the BSW mode and the refractive index distribution inside the sensing configurations. The inset is the transmission in the azimuthal angle domain with the polar angle fixed at θ = 5°.

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The sensing performance of the GBR sensors designed for azimuthal interrogation are numerically studied and the results are provided in Fig. 3. In the simulation, the RI of the region to be probed are supposed to be evenly shifted from 1.3328 to 1.3332 by a step of 0.0001. In Fig. 3(a), we studied the shift of the GBR excited by the structure studied in Fig. 2. One can find that the azimuthal angle interval between the neighboring resonance peaks being approximately ${\text{Δφ}}_1=0.508°$. Concerning the 0.0001 RI changes, the sensitivity is about 5080 Deg/RIU, which is, to the best of our knowledge, well larger than the angular sensitivities that can be found in literatures.

 

Fig. 3 The shift of the GBR with the changing of the refractive index of the bio-solution. The sensors in the four cases are designed for: (a) θ = 5° and φ≈5°; (b) θ = 10° and φ≈5°; (c) θ = 5° and φ≈10°; (d) θ = 10° and φ≈10°.

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In Fig. 3(b), the shift of GBR which is designed to be excited by illuminations of $\text{θ}\approx 10°$, $\text{φ}\approx 5°$ is studied. Here, the polar angle $\text{θ}$ is two times of that studied in Fig. 3(a). Comparing with the structure studied in Fig. 3(a), only the physical dimension of the grating is modulated (here, d_O = 116 nm, Λ = 440 nm and f = 0.37). One can clearly find that, the sensitivity of the GBR studied in Fig. 3(b) is nearly half of that in the case of Fig. 3(a) (the azimuthal angle interval between two neighboring peaks ${\text{Δφ}}_2=0.23°$ is nearly half of the ${\text{Δφ}}_1$ in Fig. 3(a)). The result however is well consistent with our theoretical analysis (Eq. (13)). In Eq. (13), the sines of the polar and the azimuthal angle are in the denominator of the fraction. When the polar angle changes from $5°$ to $10°$, the sin(θ) in the denominator of Eq. (13) nearly gets doubled, which in turn leads to the halved sensitivity. In Fig. 3(c), the GBR is excited by illuminations with the polar angle being equal to that in Fig. 3(a) ($\text{φ}\approx 10°$) and the azimuthal angle being doubled ($\text{φ}\approx 10°$). The physical parameters of the gratings utilized are d_O = 115 nm, Λ = 414.8 nm and f = 0.4, respectively. The azimuthal angle interval of ${\text{Δφ}}_3=0.24°$ is very close to the ${\text{Δφ}}_2$ in Fig. 3(b) and is also approximately half of ${\text{Δφ}}_1$ in Fig. 3(a). Clearly the numerical result again fits well the conclusion of Eq. (13). In Fig. 3(d), the shifts of GBR excited by illuminations with $\text{θ}=10°$, $\text{φ}\approx 10°$ are studied. The azimuthal angle interval ${\text{Δφ}}_4=0.114°$ is nearly half of that in Fig. 3(b) and 3(c) and a quarter of that in Fig. 3(a), which also fits the conclusions from Eq. (13).

The azimuthal sensitivity versus the azimuthal angle around which the sensor works is studied. As the polar angle is fixed in the sensing process, the theoretical analysis (Eq. (13)) predicts that the BSW sensor is approximately inversely proportional to the sine of the azimuthal angle of the incident EM wave exciting the GBR mode. Firstly, we fix the polar angle of the incident EM wave to be $5°$ and design sensors working respectively around the azimuthal angle of $5°$, $10°$, $15°$, $25°$, $35°$ and $45°$. We numerically studied the shift of the GBR spectrum when the RI of the region to be probed changes from 1.3329 to 1.333 and get the numerically simulated sensitivity for each case. In Fig. 4(a), the numerically simulated sensitivities for the six cases are provided as upward triangles linked with a dashed line. As a comparison, the theoretical sensitivities calculated by Eq. (13) are provided by the downward triangles linked with solid line. In Fig. 4(b), similar results are provided for the cases that the polar angle is fixed to be 10° and the GBR are also excited near the same six azimuthal angles, respectively. Clearly, the tendency of the numerically simulated azimuthal sensitivity as the azimuthal angle changes is in good agreement with that predicted by Eq. (13) and the discrepancy exists only in the magnitude. In each case studied in Fig. 4, the sensitivity which is numerically simulated is approximately (slightly larger than) 2/3 of that theoretically predicted by Eq. (13). This discrepancy should arise from the fact that, in the theoretical analysis, the constant Bloch admittance approximation is made. Despite this discrepancy, the Eq. (13) explicitly discloses the origin of the huge azimuthal sensitivity (the small polar and azimuthal angle). As has discussed in the third part, the discrepancy will get smaller when the normal wave vector components of the BSW mode inside the bio-solution layer ($k_z^b$ in Eq. (11)) gets nearer to zero.

 

Fig. 4 The theoretically predicted (downward triangles linked with solid lines) and the numerically simulated (upward triangles linked with dashed lines) azimuthal sensitivities for GBR sensors designed to work around the azimuthal angle of 5°, 10°, 15°, 25°, 35°, 45°, respectively. In the subplots (a) and (b), the polar angles are respectively fixed at θ = 5° and 10°.

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5. Conclusions and discussion

The scheme of bio-sensors based on guided Bloch surface wave resonance (GBR) excited by azimuthal illumination is studied. Both theoretical analysis and numerical simulations show that the azimuthal sensitivity of the GBR sensor is simultaneously inversely proportional to the sine of the polar angle and the sine of the azimuthal angle of the illuminations. As the sine of a near zero angle approaches zero, the azimuthal sensitivity gets extremely large when the polar angle of the illumination is fixed to a small value and the azimuthal angle also sweeps around a small value. In the numerical simulation, the azimuthal sensitivity reaches as large as 5080 Deg/RIU for sensors with the polar angle being fixed at 5° and working around the azimuthal angle of 5°. Such an angular sensitivity is an order higher than that of the sensors based on GBR excited by non-azimuthal illuminations where the angular sensitivity is generally not larger than 300 Deg/RIU [10, 11, 13]. The azimuthal sensitivity can be made even larger by designing sensors with GBR excited by illuminations with smaller polar or smaller azimuthal angles. At the same time, one can easily enlarge the quality factor of the GBR by simply increasing the numbers of the periods of the PhC slab or by utilizing shallower gratings or gratings with narrow grooves [8, 14, 15]. Both the one-order-higher sensitivity and the easily realized high quality factors make that extremely high detecting of limit can be realized. We believe that this scheme of GBR bio-sensors may find its application in future low-cost label-free bio-sensing applications.