Simultaneous sensing of refractive index and temperature based on a three-cavity-coupling photonic crystal sensor

Zheng Wang; ZhongYuan Fu; FuJun Sun; Chao Wang; Jian Zhou; HuiPing Tian

doi:10.1364/OE.27.026471

1. Introduction

In recent decades, optical sensors have been demonstrated with superior characterizations for detections in various fields. Considering practical optical sensing environment, several factors may influence the resonant wavelength simultaneously, such as temperature (T) and pressure [1,2], displacement [3], strain [4], humidity [5], liquid concentration [6], or refractive index (RI) [7–21], thickness and permittivity [22], tilt angle and vibration acceleration [23], RI and flow rate [24], and so on. Accordingly, to ensure the accuracy of the detections, multi-parameter sensors have been demonstrated as satisfactory settlements.

For biosensing, both RI and T have a great impact on the resonant wavelength in the spectrum. To detect RI and T simultaneously, different structural sensors are reported, such as photonic crystals (PhCs) [7–10], fibers [11–13], plasmons [14–16], Mach–Zehnder interferometer (MZI) [17], microring [18], metasurface [19], microdisks [20], liquid core fiber interferometer [21] and so on. So far, the dual-parameter sensing has been established by using different modes in one test or carrying out two measurements under different initial conditions. Shi et al. proposed a PhC sensor of two PhC nanobeam cavities (PCNCs), with different mode distributions in the air-mode cavity and dielectric-mode cavity, side-coupled to a waveguide. With a single measurement, a RI sensitivity of 254.6 nm/RIU (refractive index unit) and a T sensitivity of 30.1 pm/K for air-mode cavity were obtained. While for the dielectric-mode cavity, the RI sensitivity and the T sensitivity reached 105.5 nm/RIU and 56.4 pm/K, respectively [9]. Luo et al. provided a prism-based surface plasmon resonance (SPR) sensor and described a new double-incident angle technique to simultaneously measure changes in RI and T. As a result, SPR reflected signal shifts showed 8786.1 nm/RIU for RI sensitivity and -1131.8 pm/K for T sensitivity with a left fixed incident angle of 72.8252°. Meanwhile, for the right fixed incident angle of 73.9027°, a RI sensitivity of 6902.6 nm/RIU and a T sensitivity -790.5 pm/K were achieved, respectively [16]. Shi et al. reported an all-dielectric metasurface sensor with four holes into each silicon nanoblock, exhibiting a magnetic and an electric dipole dips in the transmission spectrum in one test. For the magnetic dipole dip, the RI and T sensitivities were 306.71 nm/RIU and 35.45 pm/K, respectively. In the meantime, the RI sensitivity of 204.27 nm/RIU and T sensitivity of 66.89 pm/K were achieved for the electric dipole dip, respectively [19]. Compared with other structure-based sensors, PhC sensors exhibit high accuracy [7], broad free spectral range (FSR) [8], and compact footprint [10]. Dual-parameter sensings in PhC structure have been achieved by different order modes in a single cavity [8,10] and two independent modes in two individual cavities [7,9]. Nonetheless, different modes in one cavity may account for decoupling inaccuracies and using a splitter or cascaded design will lead to a large footprint.

Moreover, because of the sharp transition from transmission to reflection [25], Fano resonance is utilized in many optical systems, such as optical switches [26,27], nonlinear optics [28,29], lasers [30,31], optical sensors [32–37], etc. For sensing applications, Fano-resonance-based devices exhibit high performance. Meng et al. studied the sensing application of the analog PCNC. After several attempts, an air-suspended silicon PhC nanobeam (PCN) is side-coupled with a PCNC and maximal figure of merit (FOM), calculated as RI sensitivity divided by bandwidth of the resonant peak, reaches 2095 due to the deep transmission valley [35]. Wang et al. reported an ultra-compact surface-normal optofluidic refractometric sensor based on a two-dimensional (2D) silicon photonic crystal, which realized a detection limit (DL) of 1.3×10⁻⁶ RIU [37]. Since Fano resonance is generated by the coupling between a continuum (or a wide resonance) and a discrete state (or a narrow resonance) [25], multi-cavity-coupling is a possible way to create Fano resonances, thereby improving the sensing performance.

In this paper, we theoretically present a three-cavity-coupling PhC geometry for simultaneous RI and T sensing in the aqueous environment. The three-cavity structure is proposed to weaken the interference between the side-coupled cavities and form double Fano resonances, improving the sensing performance and decoupling RI and T variations. The three-cavity-coupling modeling is established and the transmission expression is derived in Section 2 by temporal coupled-mode theory (TCMT). The correlation between the transmission and the resonant frequency, quality factor (Q-factor), as well as detuning frequency of each cavity are expounded. Section 3 is made up of the structure design and dual-parameter sensing theory in detail. The cavities are designed by rotating the air holes to form Gaussian attenuation. When the profiles of the cavities are selected, the transmission is mainly decided by the coupling distances. Therefore, the coupling strength between each cavity is discussed according to each resonant wavelength shifts by adjusting coupling distances. The simulated sensing and decoupling capabilities are exhibited in Section 4, with a comparison between our work and other sensors for simultaneous RI and T sensing reported lately. With a compact footprint of 14 × 2.6 (length × width) µm², our design shows high performance and accuracy. For the air mode cavity, RI and T sensitivity are 523 nm/RIU and 2.5 pm/K, respectively. For the dielectric mode cavity, RI and T sensitivity are 145 nm/RIU and 60.0 pm/K, respectively. Also, the decoupling deviations are approximately 0.6% and 0.4% for RI and T, respectively.

2. Modeling and transmission expression

In Fig. 1(a), the modeling of a three-cavity-coupling system is proposed. Since the coupling strengths between cav₁ and the side-coupled cavities are rather weaker than the damping coefficients of each cavity, weak coupling condition is satisfied [38]. Additionally, the whole system abides by linearity, time-invariance, conservation of energy and time-reversal invariance [39], thus the rate equations for the coupled-cavities are written as Eqs. (1)–(4):

(1)$$\frac{{d{a_1}}}{{dt}} = - i{\omega _1}{a_1} - ({\gamma _{1e}} + {\gamma _{1i}}){a_1} - i{\kappa _{12}}{a_2} - i{\kappa _{13}}{a_3} - 2{\gamma _{wg}}{a_1} + \sqrt {{\gamma _{wg}}} {s_{in}}$$

(2)$$\frac{{d{a_2}}}{{dt}} = - i{\omega _2}{a_2} - ({\gamma _{2e}} + {\gamma _{2i}}){a_2} - i{\kappa _{12}}{a_1}$$

(3)$$\frac{{d{a_3}}}{{dt}} = - i{\omega _3}{a_3} - ({\gamma _{3e}} + {\gamma _{3i}}){a_3} - i{\kappa _{13}}{a_1}$$

(4)$${s_{out}} = {\gamma _{wg}}{a_1}$$

where a_m and ω_m (m = 1, 2, and 3), respectively, represent the intracavity mode electric field and resonant frequency of each cavity. γ_wg denotes the decay constant to the waveguide of cav₁. γ_1e (γ_2e, γ_3e) and γ_1i (γ_2i, γ_3i), respectively, are the external coupling loss and intrinsic loss rate of cav₁ (cav₂, cav₃). κ₁₂ (κ₁₃) is the coupling strength between cav₁ and cav₂ (cav₃). s_in and s_out mean the incoming and outcoming electric fields, respectively. Moreover, the electric field distribution can be substituted by its amplitude and phase in Eqs. (5)–(6):

(5)$${a_\textrm{m}}={A_\textrm{m}}{e^{ - i\omega t}}$$

(6)$$\frac{{d{a_\textrm{m}}}}{{dt}} = - i\omega {A_\textrm{m}}{e^{ - i\omega t}} + \frac{{d{A_\textrm{m}}}}{{dt}}{e^{ - i\omega t}}$$

Thus, the rate equations take the form of Eqs. (7)–(10):

(7)$$\frac{{d{A_1}}}{{dt}} = (i{\Delta _1} - {\gamma _1}){A_1} - i{\kappa _{12}}{A_2} - i{\kappa _{13}}{A_3} - 2{\gamma _{wg}}{A_1} + \sqrt {{\gamma _{wg}}} {S_i}_n$$

(8)$$\frac{{d{A_2}}}{{dt}} = (i{\Delta _2} - {\gamma _2})A_2 - i{\kappa _{12}}{A_1}$$

(9)$$\frac{{d{A_3}}}{{dt}} = (i{\Delta _3} - {\gamma _3}){A_3} - i{\kappa _{13}}{A_1}$$

(10)$${S_{out}}={\gamma _{wg}}{A_\textrm{1}}$$

where Δ_m is the frequency detuning Δ_m = ω-ω_m, and γ_m (γ_m = γ_me + γ_mi) means the total radiation of each cavity. A_m represents the intracavity mode field amplitude of each cavity. S_in and S_out mean the amplitude of the incoming and outcoming fields, respectively. Here, considering the steady state of the entire system, the amplitude of electric field component approaches a certain constant, which means Eqs. (7)–(9) will equal to 0. Starting with the expressions given in Eqs. (1)–(10) and the input-output relation of the system, the steady-state normalized power transmission of the entire system in Fig. 1(a) is derived as Eq. (11):

(11)$$\textrm{T (}\omega \textrm{) = |}\frac{{{s_{out}}}}{{{s_{in}}}}{\textrm{|}^2}=\frac{{\gamma _{_{wg}}^3}}{{{{[i{\Delta _1} - {\gamma _1} + \frac{{\kappa _{12}^2}}{{i{\Delta _2} - {\gamma _2}}} + \frac{{\kappa _{13}^2}}{{i{\Delta _3} - {\gamma _3}}} - 2{\gamma _{_{wg}}}]}^2}}}$$

As seen in Eq. (11), the resonant frequency of all cavities, the decay constant of each component and coupling coefficient between each component influence the transmission together. Moreover, because of the correlations between Q-factors and decay or coupling listed in Eq. (12)–(14) [40], the steady-state normalized power transmission of the entire system can also take the form as Eq. (15):

(12)$${\gamma _\textrm{m}} = \frac{\omega }{{{Q_{r\textrm{m}}}}}$$

(13)$${\gamma _{wg}} = \frac{\omega }{{{Q_{wg}}_1}}$$

(14)$${\kappa _{_{1\textrm{m}}}} = \sqrt {\frac{\omega }{{2{Q_{wg}}_\textrm{m}}}}$$

(15)$$\textrm{T(}\omega \textrm{) = }\frac{{{{(\frac{\omega }{{{Q_{wg1}}}})}^3}}}{{{{[(i{{\Delta}_1} - \frac{\omega }{{{Q_{r1}}}}) + \frac{\omega }{{2{Q_{wg}}_2(i{{\Delta }_2} - {\omega \mathord{\left/ {\vphantom {\omega {{Q_{r2}}}}} \right.} {{Q_{r2}}}})}} + \frac{\omega }{{2{Q_{wg}}_3(i{{\Delta }_3} - {\omega \mathord{\left/ {\vphantom {\omega {{Q_{r3}}}}} \right.} {{Q_{r3}}}})}} - 2(\frac{\omega }{{{Q_{wg}}_1}})]}^2}}}$$

where Q_r_m are the quality factors of cav_m corresponding to radiation loss. Q_wg_m are channel quality factors of cav_m coupling to the bus waveguide. In Fig. 1(a), the butt-coupled cavity, cav₁, is coupled to the bus waveguide directly. Therefore, Eq. (13) represents the interaction between cav₁ and the waveguide. However, the side-coupled cavities, cav₂ and cav₃, are coupled to the bus waveguide indirectly by coupling to cav₁. Accordingly, the interactions between the side-coupled cavities and the waveguide are described as the interactions between cav₂ or cav₃ and cav₁, as exhibited in Eq. (14). Here, Q_i is calculated by Eq. (16) [40]:

(16)$${Q_i} ={\omega _0}U/{P_i}$$

where ω₀ is the resonant frequency of the localized mode, U and P_i are the optical energy localized in the cavity and the outgoing power through some channel i, respectively. Besides, for each cavity, the total quality factor Q_m can be linked with Q_r_m and Q_wg_m as Eq. (17) [41]:

(17)$$\frac{1}{{Q_{\textrm{m}}}} = \frac{1}{{Q_{r\textrm{m}}}} + \frac{1}{{Q_{wg\textrm{m}}}}$$

Fig. 1. (a) Modeling for the three-cavity-coupling system. The decays γ (cav₁ to the waveguide γ_wg (black) and each total radiative field γ_m (pink)), and couplings κ (side-coupled cavities to cav₁ (green)) are included. The transmitted coefficient between the input and cav₁ is in brown, with the incidence s_in (yellow) and outcoming filed s_out (purple). (b) A suspended PhC realization of the modeling. (c) The schematic of the proposed sensor from xy perspective.

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Moreover, the resonant peaks of cav₂ and cav₃ are used for dual-parameter sensing, and cav₁ is to establish Fano resonances and balance the coupling strengths of the side-coupled cavities. Therefore, the optimizations of Q₂ and Q₃ will be discussed in the following parts.

3. Structure design and dual-parameter sensing theory

3.1 Structure design

Here, the modeling is realized by replacing the cavity models with 220nm-thick 1D silicon PCNCs (n_Si = 3.46) insulated into a silica layer (n_SiO2 = 1.44) of 2 µm, as illustrated in Figs. 1(b)-(c). To reduce the transmission loss in the optical fiber, all the resonant frequencies of the three cavities are in the vicinity of 193.5 THz. Here, 191 THz for cav₃, 193 THz for cav₁, and 195 THz for cav₂ are chosen as target frequencies. In terms of the cavity design, a method of manipulating the edge state of PhC by rotating air holes is proposed [42]. Some examples of different rotated angles (θ) are shown in Fig. 2(a). For cav₁, a₁ = 400 nm, l₁ = 275 nm, and θ₁ = 0. The cavity length between the two middle air holes (s) is s = 85 nm. For cav₂, a₂ = 500 nm, l₂ = 238 nm, N₂ = 14, θ_2center = 0, and θ_2end= 45°, with θ₂ (i) = θ_2center+i²×(θ_2end – θ_2center)/$i_{\max }^2$, where i denotes the number of the hole increases from the center to the end, from 0 to i_max, i_max = N₂. For cav₃, a₃ = 350 nm, l₃ = 165 nm, N₃ = 20, θ_3center= 45°, and θ_3end= 0, from the center to the end with θ₃ (i)= θ_3center-i^1/2×(θ_3center -θ_3end)/$i_{\max }^{{1 / 2}}$, where i increases from 0 to i_max, i_max = N₃. All parameters of the cavities are given in Table 1. The fabrication of ∼100 nm square hole has been demonstrated to be experimentally achievable and yield high performances [43,44]. Besides, hole profiles influence the resonant wavelength and transmission line shape of each cavity under small fabrication inaccuracy, with little influence upon the sensing performance.

Fig. 2. (a) Examples of different rotated angles (θ) from 0 to 45°. (b) TE band diagram of cav₂ with θ₂ = 0 (red) and 45° (black). The red star indicates cav₂ resonant state. (c) TE band diagram of cav₃ with θ₃ = 0 (black) and 45° (blue). The blue star indicates cav₃ resonant state. The linear fitting of mirror strength via changing air hole number of cav₂ (d) and cav₃ (e). The Gaussian envelopes of the electric field distribution of cav₂ (f) and cav₃ (g).

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Table 1. The geometric and resonant parameters of cav₁, cav₂, and cav₃

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TE band diagrams of cav₂ and cav₃ are studied by three-dimensional finite-difference time-domain (3D-FDTD) method with Bloch boundary conditions in Figs. 2(b)-(c), respectively. Unlike the deterministic design proposed by Quan et al. [45], the design has air holes rotated instead of changing filling factors. Nonetheless, the mirror strengths of both cavities exhibit approximately linear increases in Figs. 2(d)-(e), respectively. The mirror strength is calculated by $\sqrt {{{{{({\omega _{abe}} - {\omega _{dbe}})}^2}} \mathord{\left/ {\vphantom {{{{({\omega_{abe}} - {\omega_{dbe}})}^2}} {{{({\omega_{abe}} + {\omega_{dbe}})}^2} - {{{{({\omega_{tar}} - {\omega_{mid}})}^2}} \mathord{\left/ {\vphantom {{{{({\omega_{tar}} - {\omega_{mid}})}^2}} {\omega {{_m^2}_{id}}}}} \right.} {\omega {{_m^2}_{id}}}}}}} \right.} {{{({\omega _{abe}} + {\omega _{dbe}})}^2} - {{{{({\omega _{tar}} - {\omega _{mid}})}^2}} \mathord{\left/ {\vphantom {{{{({\omega_{tar}} - {\omega_{mid}})}^2}} {\omega {{_m^2}_{id}}}}} \right.} {\omega {{_m^2}_{id}}}}}}}$, where ω_tar is the target frequency, ω_abe, ω_dbe, and ω_mid are the air band edge, dielectric band edge, and midgap frequency of each segment, respectively. For air and dielectric modes, the target frequency means the air and dielectric band edge frequency of the central cell, respectively [46]. Finally, Gaussian attenuations come into being in Figs. 2(f)-(g), respectively. Hitherto, only the coupling distances between cav₂ (d₁₂) or cav₃ (d₁₃) and the bus waveguide are undecided.

The coupling strength between each cavity can be described as each resonant wavelength shift by adjusting the coupling distances. Here, the coupling strength between each cavity is illustrated by the resonant wavelength shift of the other cavity. The transmissions of only cav₂, only cav₃ and both cavities side-coupled to cav₁ are investigated. The resonant wavelengths of cav₁ are different because when the gaps between each cavity vary, κ₁₂ and κ₁₃ are unequal. Nonetheless, even if the resonant wavelengths of cav₁ does not overlap, the resonant wavelengths of side-coupled cavities remain unchanged, as shown in Fig. 3(a). Therefore, the coupling strength between cav₂ and cav₃ (κ₂₃) turns out to be negligible, which means the interaction between cav₂ and cav₃ are rather weak and the coupling process will not lessen the sensing performance. However, different coupling strengths may lead to an ambiguous resonant wavelength of cav₁, thereby influencing the sensing process. To eliminate the perturbation by different coupling strengths, the correlations between the resonant wavelength shift of cav₁ and the coupling distances of d₁₂ as well as d₁₃ are investigated, respectively. As captured in Fig. 3(b), d₁₂ is set as 800 nm while d₁₃ is 300 nm so that κ₁₂ equals κ₁₃. Ultimately, Fig. 3(c) exhibits the transmission of the entire structure, where Q₂ is 15405, and Q₃ is 3633.9, respectively.

Fig. 3. (a) Transmission comparison between the structures of both cavities and one cavity side-coupled to cav₁. (b) The resonant wavelength shifts via different coupling strengths of cav₂ (red) and cav₃ (blue). (c) Transmission of the sensor. Electric field distributions of cav₁ (e), cav₂ (d) and cav₃ (f) at resonant wavelength (λ₁ = 1555.54 nm, λ₂ = 1539.29 nm, and λ₃ = 1576.37 nm).

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The air cavity mode is fundamental mode while the dielectric mode is 1^st order mode, because the extinction ratio (ER) of the fundamental dielectric mode is too weak for 20 air holes design, as shown in Fig. 3(c). Afterward, the transmission of the entire geometry and electric field distributions of the cavities are exhibited in Figs. 3(d)-(f). As illustrated in Figs. 3(d)-(f), each cavity is excited at different wavelength and the electromagnetic wave can be only localized in the resonant cavity at the corresponding wavelength.

3.2 Dual-parameter sensing theory

Considering that the ambient T has an impact upon RI of the analytes and structure materials, the resonant wavelength shifts are generated by the T variations and analytes together. Therefore, the influences of T and analytes upon the resonant wavelengths should be considered separately. Here, a sensor matrix S_n,_T is defined and the correlations between ambient RI changes, T variations, and wavelengths shifts are described as Eq. (18) [9,10] :

(18)$$\left( {\begin{array}{{c}} {\Delta \lambda_\textrm{2}}\\ {\Delta \lambda_\textrm{3}} \end{array}} \right)={S_{n,T}}\left( {\begin{array}{{c}} {\Delta n}\\ {\Delta T} \end{array}} \right) = \left( {\begin{array}{{cc}} {S_{n,2}}&{S_{T,2}}\\ {S_{n,3}}&{S_{T,3}} \end{array}} \right)\left( {\begin{array}{{c}} {\Delta n}\\ {\Delta T} \end{array}} \right)$$

where S_n,₂ and S_T_,2 represent the RI and T sensitivities for cav₂, respectively. S_n_,3 and S_T_,3 represent the RI and T sensitivities of cav₃, respectively. Δλ₂ and Δλ₃ are the resonant wavelength shifts of cav₂ and cav₃, respectively.

As far as the decoupling process is concerned, so long as the rank of the sensor matrix |S_n,T|≠0, the RI and T variations can be calculated as follows:

(19)$$\left( {\begin{array}{{c}} {\Delta n}\\ {\Delta T} \end{array}} \right) = S_{n,T}^{ - 1}\left( {\begin{array}{{c}} {\Delta \lambda_\textrm{2}}\\ {\Delta \lambda_\textrm{3}} \end{array}} \right) = {\left( {\begin{array}{{cc}} {S_{n,2}}&{S_{T,2}}\\ {S_{n,3}}&{S_{T,3}} \end{array}} \right)^{ - 1}}\left( {\begin{array}{{c}} {\Delta \lambda_\textrm{2}}\\ {\Delta \lambda_\textrm{3}} \end{array}} \right) = \frac{1}{{|{{S_{n,T}}} |}}\left( {\begin{array}{{cc}} {S_{T,3}}&{\textrm{ - }S_{T,2}}\\ {\textrm{ - }S_{n,3}}&{S_{n,2}} \end{array}} \right)\left( {\begin{array}{{c}} {\Delta \lambda_\textrm{2}}\\ {\Delta \lambda_\textrm{3}} \end{array}} \right)$$

To evaluate the sensing performance of simultaneous detections for RI and T, the relative deviation ratio (δ) is introduced here, defined as δ = |Δ_set – Δ_decoupled| / Δ_set. Besides, the maximum external interference ranges (a_n, a_T) [7] are investigated, to check the anti-external interference ability, where the maximum external interference ranges (a_n, a_T) are defined as:

(20)$$({{a_n},{a_T}} )= \left( {\left|{\frac{{|{{S_{T,2}}} |+ |{{S_{T,3}}} |}}{{|{{S_{n,T}}} |}}} \right|,\left|{\frac{{|{{S_{n,2}}} |+ |{{S_{n,3}}} |}}{{|{{S_{n,T}}} |}}} \right|} \right)$$

Both low δ, indicating the detecting precision, and small external interference ranges (a_n, a_T), denoting the maximal sensing deviation with small monitoring error, should be achieved.

4. Simultaneous sensing of RI and T

Here, the RI and T sensitivities of the proposed sensor are investigated, respectively. As shown in Fig. 4(a), when T is constant (T = 300 K), with the increasing of RI change from 0.000 to 0.004, both resonant wavelengths of cav₂ and cav₃ exhibit redshifts. However, the wavelength shifts of the two cavities are different with the same RI change. As seen in Fig. 4(b), the correlations between resonant wavelength shifts and RI changes are illustrated. RI sensitivities, as the slope of the fitting lines, of cav₂ and of cav₃ are 523 nm/RIU and 145 nm/RIU, respectively. Likewise, with a fixed RI (n = 1.33), the T sensitivity of both cavities are investigated. Note that the thermo-optic coefficient of silicon is ∼1.8 × 10⁻⁴ RIU/K, and the thermo-optic coefficient of water is -0.8 × 10⁻⁴ RIU/K, which are much greater than the thermal expansion of Si (∼10⁻⁶ RIU/K). Hence, T variations are investigated by exerting thermal index change to the silicon strips and the aqueous surrounding [9]. Accordingly, Fig. 4(c) shows the transmission spectrum of the resonant wavelength with the ambient T change varying from 0 K to 40 K. And the linear fittings of the wavelength shifts are illustrated in Fig. 4(d). As shown in Fig. 4(d), T sensitivities of cav₂ and cav₃ are 2.5 pm/K and 60.0 pm/K, respectively. Eventually, Eq. (18) and Eq. (19) take the forms:

(21)$$\left( {\begin{array}{{c}} {\Delta \lambda_\textrm{2}}\\ {\Delta \lambda_\textrm{3}} \end{array}} \right)=\left( {\begin{array}{{cc}} {523\textrm{nm/RIU}}&{2.5\textrm{pm/K}}\\ {145\textrm{nm/RIU}}&{\textrm{60}\textrm{.0pm/K}} \end{array}} \right)\left( {\begin{array}{{c}} {\Delta n}\\ {\Delta T} \end{array}} \right)$$

(22)$$\left( {\begin{array}{{c}} {\Delta n}\\ {\Delta T} \end{array}} \right) = \left( {\begin{array}{{cc}} {0.001934}&{ - 0.00008059}\\ { - 4.6747}&{16.8614} \end{array}} \right)\left( {\begin{array}{{c}} {\Delta \lambda_\textrm{2}}\\ {\Delta \lambda_\textrm{3}} \end{array}} \right)$$

Fig. 4. (a) The simulated transmission spectra varying the surrounding RI changes (Δn) from 0.000 to 0.004 when T change (ΔT) remains 0 (T = 300 K). (b) The linear fitting of the corresponding resonant wavelength shifts via varying different Δn. (c) The simulated transmission spectra varying the ambient ΔT from 0 K to 40 K when Δn remains 0 (n = 1.33). (d) The linear fitting of the corresponding resonant wavelength shifts via varying different ΔT.

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Therefore, high sensitivity ratio contrast between RI sensitivity ratio ratio_n = S_n_,3 / S_n_,2 = 0.27 and T sensitivity ratio ratio_T = S_T_,3 / S_T_,2 = 24 is realized, which is the reason for designing air mode and dielectric mode cavity. Besides, high sensitivity ratio contrast implies the potential for simultaneous sensing of RI and T. After some variation values ((Δn, ΔT)_set) are set, the simulated resonant wavelength shifts (Δλ₂, Δλ₃), and the decoupled values ((Δn, ΔT)_decoupled) calculated by Eq. (21) and Eq. (22) according to 3D-FDTD are compared in Table 2. The differences between the set and the decoupled values attribute to the interactions between T and RI, and the approximately linear fittings when calculating the sensitivities. However, the relative deviation ratios (δn, δT) of all these conditions are approximately 0.6% and 0.4% for RI and T, respectively, enhancing the detecting accuracy. Besides, the performance is compared with other dual-parameter sensors in Table 3. As seen in Table 3, compared with 1D PhC structures, without a splitter and other material covering layer, equivalent satisfying decoupling results are achieved compared with [7]. Opposite side-coupled design also contributes to the decrease of footprint compared with [9]. Although the structure in [8] and [10] are much smaller in footprint by establishing the dual-parameter sensor with different order modes in one cavity, the maximum external interference ranges of our sensor are much better, indicating the improvement of the anti-external interference ability. Compared with other types of sensors, high detecting accuracy and compact footprint are our strength. In terms of plasmonic sensors, although the sensitivities and maximum external interference ranges are much superb to those in our work numerically, a much compact footprint is realized by our work. As far as the fiber-based sensors are concerned, the diameters of the fiber are 125 µm in most cases. Therefore, the footprints are much larger than our work [13,17,21]. Besides, the maximum external interference ranges of our sensor are improved. The metasurface structure is of infinite period [19], and no precise structural parameters are claimed in the microring [18] and microdisks sensor [20], therefore, in terms of sensing accuracy only, our work exhibits better behavior.

Table 2. Simultaneous measurements of RI and T using the three-cavity-coupling sensor

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Table 3. Comparisons with other structures for simultaneous RI and T sensing

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5. Conclusion

In summary, we have proposed a theoretical model of a three-cavity-coupling system and derived the transmission expression. In the expression, when the profiles of the cavities are set, the transmission is correlated to the coupling distance between each cavity. Then the model is established in a PhC structure, composed of a cavity of wide linewidth, an air mode cavity, and a dielectric mode cavity. The independence between each cavity is demonstrated and the dependence between the Q-factors and the coupling distances are studied. Afterward, the dual-parameter sensing feasibility is investigated. Moreover, the entire footprint is only 14 × 2.6 (length × width) µm². Compared with the researches lately published, high detecting accuracy and compact footprint have been achieved. Therefore, all the model, transmission expression and sensor structure are feasible and the proposed method is promising in the future compact lab-on-chip detection system.

Funding

National Natural Science Foundation of China (61372038, 61431003); Fund of Joint Laboratory for Undersea Optical Networks.

Acknowledgments

The authors would like to thank Xuepei Li, Zekun Xiao, Jiawen Wang and Zhiqiang Liu for discussion.

References

1. H. Gao, Y. Jiang, Y. Cui, L. Zhang, J. Jia, and J. Hu, “Dual-Cavity Fabry–Perot Interferometric Sensors for the Simultaneous Measurement of High Temperature and High Pressure,” IEEE Sens. J. 18(24), 10028–10033 (2018). [CrossRef]

2. L. Zhang, Y. Jiang, H. Gao, J. Jia, Y. Cui, S. Wang, and J. Hu, “Simultaneous Measurements of Temperature and Pressure With a Dual-Cavity Fabry–Perot Sensor,” IEEE Photonics Technol. Lett. 31(1), 106–109 (2019). [CrossRef]

3. K. Tian, G. Farrell, W. Yang, X. Wang, E. Lewis, and P. Wang, “Simultaneous Measurement of Displacement and Temperature Based on a Balloon-Shaped Bent SMF Structure Incorporating an LPG,” J. Lightwave Technol. 36(20), 4960–4966 (2018). [CrossRef]

4. A. Wada, S. Tanaka, and N. Takahashi, “Fast and High-Resolution Simultaneous Measurement of Temperature and Strain Using a Fabry–Perot Interferometer in Polarization-Maintaining Fiber With Laser Diodes,” J. Lightwave Technol. 36(4), 1011–1017 (2018). [CrossRef]

5. Y. Wang, Q. Huang, W. Zhu, and M. Yang, “Simultaneous Measurement of Temperature and Relative Humidity Based on FBG and FP Interferometer,” IEEE Photonics Technol. Lett. 30(9), 833–836 (2018). [CrossRef]

6. Y. Zhang, Y. Zhao, and H. Hu, “Miniature photonic crystal cavity sensor for simultaneous measurement of liquid concentration and temperature,” Sens. Actuators, B 216, 563–571 (2015). [CrossRef]

7. X. Li, C. Wang, Z. Wang, Z. Fu, F. Sun, and H. Tian, “Anti-External Interference Sensor Based on Cascaded Photonic Crystal Nanobeam Cavities for Simultaneous Detection of Refractive Index and Temperature,” J. Lightwave Technol. 37(10), 2209–2216 (2019). [CrossRef]

8. C. Wang, Z. Fu, F. Sun, J. Zhou, and H. Tian, “Large-Dynamic-Range Dual-Parameter Sensor Using Broad FSR Multimode Photonic Crystal Nanobeam Cavity,” IEEE Photonics J. 10(5), 1–14 (2018). [CrossRef]

9. P. Liu and Y. Shi, “Simultaneous measurement of refractive index and temperature using cascaded side-coupled photonic crystal nanobeam cavities,” Opt. Express 25(23), 28398–28406 (2017). [CrossRef]

10. L. Zhang, F. Sun, Z. Fu, C. Wang, and H. Tian, “Ultra-compact dual-parameter sensing based on a photonic crystal rectangular holes nanobeam multimode microcavity,” in Proceedings of IEEE Conference on Lasers and Electro-Optics Pacific Rim (IEEE, 2017), pp. 1–2.

11. X. Li, L. V. Nguyen, M. Becker, D. Pham, H. Ebendorff-Heidepriem, and S. C. Warren-Smith, “Simultaneous measurement of temperature and refractive index using an exposed core microstructured optical fiber,” IEEE J. Sel. Top. Quantum Electron. Print ISSN: 1077-260X (Date of Publication: 01 April 2019, in press).

12. B. Yin, S. Wu, M. Wang, W. Liu, H. Li, B. Wu, and Q. Wang, “High-sensitivity refractive index and temperature sensor based on cascaded dual-wavelength fiber laser and SNHNS interferometer,” Opt. Express 27(1), 252–264 (2019). [CrossRef]

13. Y. Li, G. Yan, and S. He, “Thin-Core Fiber Sandwiched Photonic Crystal Fiber Modal Interferometer for Temperature and Refractive Index Sensing,” IEEE Sens. J. 18(16), 6627–6632 (2018). [CrossRef]

14. W. Luo, S. Chen, L. Chen, H. Li, P. Miao, H. Gao, Z. Hu, and M. Li, “Dual-angle technique for simultaneous measurement of refractive index and temperature based on a surface plasmon resonance sensor,” Opt. Express 25(11), 12733–12742 (2017). [CrossRef]

15. F. Xiao, D. Michel, G. Li, A. Xu, and K. Alameh, “Simultaneous Measurement of Refractive Index and Temperature Based on Surface Plasmon Resonance Sensors,” J. Lightwave Technol. 32(21), 4169–4173 (2014). [CrossRef]

16. W. Luo, R. Wang, H. Li, J. Kou, X. Zeng, H. Huang, X. Hu, and W. Huang, “Simultaneous measurement of refractive index and temperature for prism-based surface plasmon resonance sensors,” Opt. Express 27(2), 576–589 (2019). [CrossRef]

17. F. Yu, P. Xue, X. Zhao, and J. Zheng, “Simultaneous Measurement of Refractive Index and Temperature Based on a Peanut-Shape Structure In-Line Fiber Mach–Zehnder Interferometer,” IEEE Sens. J. 19(3), 950–955 (2019). [CrossRef]

18. P. Liu and Y. Shi, “Simultaneous measurement of refractive index and temperature using a dual polarization ring,” Appl. Opt. 55(13), 3537–3541 (2016). [CrossRef]

19. J. Hu, T. Lang, and G. Shi, “Simultaneous measurement of refractive index and temperature based on all-dielectric metasurface,” Opt. Express 25(13), 15241–15251 (2017). [CrossRef]

20. T. Ma, J. Yuan, F. Li, L. Sun, Z. Kang, B. Yan, Q. Wu, X. Sang, K. Wang, H. Liu, F. Wang, B. Wu, C. Yu, and G. Farrell, “Microdisk Resonator With Negative Thermal Optical Coefficient Polymer for Refractive Index Sensing With Thermal Stability,” IEEE Photonics J. 10(2), 1–12 (2018). [CrossRef]

21. S. Liu, H. Zhang, L. Li, L. Xiong, and P. P. Shum, “Liquid Core Fiber Interferometer for Simultaneous Measurement of Refractive Index and Temperature,” IEEE Photonics Technol. Lett. 31(2), 189–192 (2019). [CrossRef]

22. S. Lim, C. Kim, and S. Hong, “Simultaneous Measurement of Thickness and Permittivity by Means of the Resonant Frequency Fitting of a Microstrip Line Ring Resonator,” IEEE Microw. Wireless Compon. Lett. 28(6), 539–541 (2018). [CrossRef]

23. Y. Yang, E. Wang, K. Chen, Z. Yu, and Q. Yu, “Fiber-Optic Fabry–Perot Sensor for Simultaneous Measurement of Tilt Angle and Vibration Acceleration,” IEEE Sens. J. 19(6), 2162–2169 (2019). [CrossRef]

24. R. Gao, D. Lu, J. Cheng, and Z. Qi, “Simultaneous measurement of refractive index and flow rate using graphene-coated optofluidic anti-resonant reflecting guidance,” Opt. Express 25(23), 28731–28742 (2017). [CrossRef]

25. U. Fano, “Effects of Configuration Interaction on Intensities and Phase Shifts,” Phys. Rev. 124(6), 1866–1878 (1961). [CrossRef]

26. D. A. Bekele, Y. Yu, H. Hu, L. K. Oxenløwe, K. Yvind, and J. Mork, “Fano Resonances for Realizing Compact and Low Energy Consumption Photonic Switches,” in Proceedings of IEEE International Conference on Transparent Optical Networks (IEEE, 2018), pp. 504–509.

27. G. Dong, Y. Wang, and X. Zhang, “High-contrast and low-power all-optical switch using Fano resonance based on a silicon nanobeam cavity,” Opt. Lett. 43(24), 5977–5980 (2018). [CrossRef]

28. D. L. Sounas and A. Alù, “Fundamental bounds on the operation of Fano nonlinear isolators,” Phys. Rev. B 97(11), 115431 (2018). [CrossRef]

29. A. Krasnok, M. Tymchenko, and A. Alù, “Nonlinear metasurfaces: a paradigm shift in nonlinear optics,” Mater. Today 21(1), 8–21 (2018). [CrossRef]

30. A. R. Zali, M. K. Moravvej-Farshi, and M. H. Yavari, “Small-Signal Equivalent Circuit Model of Photonic Crystal Fano Laser,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–8 (2019). [CrossRef]

31. T. S. Rasmussen, Y. Yu, and J. Mork, “Modes, stability, and small-signal response of photonic crystal Fano lasers,” Opt. Express 26(13), 16365–16376 (2018). [CrossRef]

32. Y. Zhang, W. Liu, Z. Li, Z. Li, H. Cheng, S. Chen, and J. Tian, “High-quality-factor multiple Fano resonances for refractive index sensing,” Opt. Lett. 43(8), 1842–1845 (2018). [CrossRef]

33. G. Liu, X. Zhai, L. Wang, Q. Lin, S. Xia, X. Luo, and C. Zhao, “A High-Performance Refractive Index Sensor Based on Fano Resonance in Si Split-Ring Metasurface,” Plasmonics 13(1), 15–19 (2018). [CrossRef]

34. Z. Wang, C. Wang, F. Sun, Z. Fu, Z. Xiao, J. Wang, and H. Tian, “Double-layer Fano resonance photonic-crystal-slab-based sensor for label-free detection of different size analytes,” J. Opt. Soc. Am. B 36(2), 215–222 (2019). [CrossRef]

35. Z. Meng and Z. Li, “Control of Fano Resonances in Photonic Crystal Nanobeam Side-Coupled with Nanobeam Cavities and their Applications to Refractive Index Sensing,” J. Phys. D: Appl. Phys. 51(9), 095106 (2018). [CrossRef]

36. M. Mesch, T. Weiss, M. Schäferling, M. Hentschel, R. S. Hegde, and H. Giessen, “Highly Sensitive Refractive Index Sensors with Plasmonic Nanoantennas−Utilization of Optimal Spectral Detuning of Fano Resonances,” ACS Sens. 3(5), 960–966 (2018). [CrossRef]

37. S. Wang, Y. Liu, D. Zhao, H. Yang, W. Zhou, and Y. Sun, “Optofluidic Fano resonance photonic crystal refractometric sensors,” Appl. Phys. Lett. 110(9), 091105 (2017). [CrossRef]

38. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]

39. M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017). [CrossRef]

40. P. Yu, H. Qiu, H. Yu, F. Wu, Z. Wang, X. Jiang, and J. Yang, “High-Q and High-Order Side-Coupled Air-Mode Nanobeam Photonic Crystal Cavities in Silicon,” IEEE Photonics Technol. Lett. 28(20), 2121–2124 (2016). [CrossRef]

41. Y. Zou, S. Chakravarty, D. N. Kwong, W. Lai, X. Xu, X. Lin, A. Hosseini, and R. T. Chen, “Cavity-Waveguide Coupling Engineered High Sensitivity Silicon Photonic Crystal Microcavity Biosensors with High Yield,” IEEE J. Sel. Top. Quantum Electron. 20(4), 171–180 (2014). [CrossRef]

42. S. Hu and S. M. Weiss, “Design of Photonic Crystal Cavities for Extreme Light Concentration,” ACS Photonics 3(9), 1647–1653 (2016). [CrossRef]

43. J. Wei, F. Sun, B. Dong, Y. Ma, Y. Chang, H. Tian, and C. Lee, “Deterministic aperiodic photonic crystal nanobeam supporting adjustable multiple mode-matched resonances,” Opt. Lett. 43(21), 5407–5410 (2018). [CrossRef]

44. S. Kim, J. E. Fröch, J. Christian, M. Straw, J. Bishop, D. Totonjian, K. Watanabe, T. Taniguchi, M. Toth, and I. Aharonovich, “Photonic crystal cavities from hexagonal boron nitride,” Nat. Commun. 9(1), 2623 (2018). [CrossRef]

45. Q. Quan and M. Loncar, “Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities,” Opt. Express 19(19), 18529–18542 (2011). [CrossRef]

46. F. Sun, J. Wei, B. Dong, Y. Ma, Y. Chang, H. Tian, and C. Lee, “Coexistence of air and dielectric modes in single nanocavity,” Opt. Express 27(10), 14085–14098 (2019). [CrossRef]

(Δn, ΔT)_set / (RIU,K)	(Δλ₂, Δλ₃) / (nm,nm)	(Δn, ΔT)_decoupled / (RIU,K)	(δn, δT) / (%, %)
(0.001, 40)	(0.62,2.55)	(0.0010,40.09)	(0.62,0.24)
(0.002, 30)	(1.12,2.09)	(0.0020,30.00)	(0.09,0.01)
(0.003, 20)	(1.62,1.64)	(0.0030,20.07)	(0.05,0.39)
(0.004, 10)	(2.11,1.18)	(0.0040,10.03)	(0.33,0.32)

References	Structure type	RI sensitivities (nm/RIU)	T sensitivities (pm/K)	Footprint(µm²)	(a_n, a_T) / (RIU, K)**
[7]	1D PhC	0;354.6	-107.9;46.7	>10.4 × 4	(4.04×10⁻⁶, 9×10⁻³)
[8]	1D PhC	390.9;413.6	60.7;62.9	10 × 0.55	(2.38×10⁻⁴, 1.55)
[9]	1D PhC	105.5;254.6	56. 4;30.1	>13.68 × 1.52	(7.73×10⁻⁶, 0.03)
[10]	1D PhC	248.9;270	20.9;19.2	10 × 0.7	(4.64×10⁻⁵, 0.60)
[13]	photonic crystal fiber	48.9;283.9	1.7;-354.1	>π × 1.5²*	(2.00×10⁻⁵, 0.02)
[16]	surface plasmon	6902.6;8786	-790.5;1131.8	----	(1.30×10⁻⁷, 1×10⁻³)
[17]	MZI	-38.5;-47.3	62.6;70.9	π × 4.1²*	(5.77×10⁻⁴, 0.37)
[18]	microring	104;319	78.7;34.1	----	(5.23×10⁻⁶, 0.02)
[19]	metasurface	204.3;306.7	66.89;35.45	∞	(7.71×10⁻⁶, 0.04)
[20]	microdisks	25.1;110.6	75.2;51.3	----	(1.80×10⁻⁵, 0.02)
[21]	fiber interferometer	101.3;141.4	-24.3;-36.4	π × 2.5²*	(2.42×10⁻⁴, 0.97)
Our work	1D PhC	145; 523	60.0; 2.5	14 × 2.6	(2.00×10⁻⁶, 0.02)

(Δn, ΔT)_set / (RIU,K)	(Δλ₂, Δλ₃) / (nm,nm)	(Δn, ΔT)_decoupled / (RIU,K)	(δn, δT) / (%, %)
(0.001, 40)	(0.62,2.55)	(0.0010,40.09)	(0.62,0.24)
(0.002, 30)	(1.12,2.09)	(0.0020,30.00)	(0.09,0.01)
(0.003, 20)	(1.62,1.64)	(0.0030,20.07)	(0.05,0.39)
(0.004, 10)	(2.11,1.18)	(0.0040,10.03)	(0.33,0.32)

References	Structure type	RI sensitivities (nm/RIU)	T sensitivities (pm/K)	Footprint(µm²)	(a_n, a_T) / (RIU, K)**
[7]	1D PhC	0;354.6	-107.9;46.7	>10.4 × 4	(4.04×10⁻⁶, 9×10⁻³)
[8]	1D PhC	390.9;413.6	60.7;62.9	10 × 0.55	(2.38×10⁻⁴, 1.55)
[9]	1D PhC	105.5;254.6	56. 4;30.1	>13.68 × 1.52	(7.73×10⁻⁶, 0.03)
[10]	1D PhC	248.9;270	20.9;19.2	10 × 0.7	(4.64×10⁻⁵, 0.60)
[13]	photonic crystal fiber	48.9;283.9	1.7;-354.1	>π × 1.5²*	(2.00×10⁻⁵, 0.02)
[16]	surface plasmon	6902.6;8786	-790.5;1131.8	----	(1.30×10⁻⁷, 1×10⁻³)
[17]	MZI	-38.5;-47.3	62.6;70.9	π × 4.1²*	(5.77×10⁻⁴, 0.37)
[18]	microring	104;319	78.7;34.1	----	(5.23×10⁻⁶, 0.02)
[19]	metasurface	204.3;306.7	66.89;35.45	∞	(7.71×10⁻⁶, 0.04)
[20]	microdisks	25.1;110.6	75.2;51.3	----	(1.80×10⁻⁵, 0.02)
[21]	fiber interferometer	101.3;141.4	-24.3;-36.4	π × 2.5²*	(2.42×10⁻⁴, 0.97)
Our work	1D PhC	145; 523	60.0; 2.5	14 × 2.6	(2.00×10⁻⁶, 0.02)

Simultaneous sensing of refractive index and temperature based on a three-cavity-coupling photonic crystal sensor

Abstract

1. Introduction

2. Modeling and transmission expression

3. Structure design and dual-parameter sensing theory

3.1 Structure design

3.2 Dual-parameter sensing theory

4. Simultaneous sensing of RI and T

5. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (4)

Tables (3)

Equations (22)

Optics Express

parameters	cav₁	cav₂	cav₃
lattice constant (a_i) / nm	400	500	350
square hole side length (l_i) / nm	275	238	165
strip thickness (t_i) / nm	220	220	220
rotated region hole number (N_i)	----	14	20
central hole rotated angle (θ_icenter) / °	0	0	45
end hole rotated angle (θ_iend) / °	0	45	0
resonant wavelength (λ_i) / nm	1547.14	1539.13	1576.06