Microstructured optical fiber for multichannel sensing based on Fano resonance of the whispering gallery modes

Wei Lin; Hao Zhang; Shih-Chi Chen; Bo Liu; Yan-Ge Liu

doi:10.1364/OE.25.000994

1. Introduction

In recent years, whispering-gallery modes (WGMs) have been extensively investigated and used in fundamental studies due to its unique features of high Q-factor and ultra-small mode volume [1, 2]. To guide WGMs, various resonators have been proposed including microring [3, 4], microdisk [5], microtoroid [6], microsphere [7], microcapillary [8] and microbottle [9] etc. Compared with solid resonators [3–7], resonators with open cavities, e.g., microcapillary [8] and microbottle [9], are more versatile; for example, they can be used as bio-/chemical sensors, localized laser excitation devices, or field-dependent photonic devices. Traditionally, open-cavity resonators are based on the single channel design, which often prolongs and complicates the sequential measurement processes. To perform multichannel sensing, arrays of resonators have to be used simultaneously to measure multiple samples [10], which calls for a complex system of high cost.

Since the introduction of microstructured optical fibers (MOFs) in the 1990s [11, 12], MOFs have been extensively researched and used. Good examples include fiber sensors [13, 14], dispersion management components [15, 16], nonlinear optical components [17, 18] and tunable photonic devices [19, 20]. These devices are operated by controlling the eigenmodes propagating along the axial direction of the MOF. Besides the axial eigenmodes, MOFs also guide the WGMs that travel in the circumferential direction of the fiber. In other words, MOFs may be used to generate WGM resonance for various applications, e.g., magnetic field sensing [21] and signal processing [22]. Accordingly, MOF-based WGM resonators have become increasingly attractive, especially for those with multiple hollow channels inside that enables the transport of fluid or gas samples. Although designs in [21] and [22] employ similar structures, they cannot guide WGMs in the hollow channels due to the low refractive index (RI) inside.

In this work, we propose a new MOF for multichannel sensing based on the Fano resonance among the different WGMs propagating in the MOF. The proposed MOF consists of a number of capillary channels of different diameters inside a tubular frame. To illustrate the working principle, we analyze a dual-channel MOF and demonstrate how it can be used for measuring the RI in biological samples. First, a theoretical model for the dual-channel MOF resonator is established to predict the excitation of the Fano resonance and the output spectrum. Then, the RI-sensing characteristics of the MOF is investigated in detail through numerical modeling. The results indicate our new MOF design are compact, flexible and low-cost and may find important applications in the multichannel bio/chemical sensing, multi-microcavity lasers, and tunable photonics devices.

2. Design of the MOF

Figure 1 presents conceptual designs of the multichannel MOF, where the blue regions represent glass, and the white regions represent the hollow parts in fibers. An n-channel MOF design consists of n capillary channels of different diameters inside a tubular frame. The different channel diameters ensure the Fano resonant peaks will be separated in the output spectrum. Also, each channel is appropriately spaced with others in order to avoid cross-talk. During operations, gases or fluids can fill different capillary channels to alter the phases of the WGMs in the corresponding channels, which effectively shifts the resonant peaks. The tubular frame then collects signals from the fluidic channels via Fano resonance and makes them observable in the output spectrum. A typical structure of MOF with dual-channel is taken as an example in further investigation.

Fig. 1 Cross-sections of the proposed MOF

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3. Modeling

3.1 WGMs in a MOF

In this section, we model and analyze a dual-channel MOF design. According to Smith’s work [23], the dual-channel MOF is a morphology-dependent resonator which can be mathematically described as three independent capillary resonators. The radial distribution of the WGM in the resonators can be expressed as [24]:

Ψ z (r) = {\begin{array}{l} A J_{m} (k_{0} n_{1} r) \\ B J_{m} (k_{0} n_{2} r) + C N_{m} (k_{0} n_{2} r) \\ D H_{m}^{(1)} (k_{0} n_{3} r) \end{array} \begin{matrix} \begin{matrix}  \end{matrix} & \begin{matrix} r > R_{1} \\ R_{1} \leq r \leq R_{2} \\ r > R_{2} \end{matrix} \end{matrix}

where J_m, N_m and H_m⁽¹⁾ are the m^th order Bessel function, Neumann function and Hankel function of the first kind, respectively. A, B, C and D are constants. R₁ and R₂ are the inner and outer radii of the capillary tubes respectively. n₁, n₂ and n₃ are the RI of the fiber core, wall, and the surrounding medium. And k₀ refers to the wavenumber. According to the boundary conditions, the eigen-equation of the WGM resonator can be expressed as:

N \frac{J'_{m} (k_{0} n_{1} R_{1})}{J_{m} (k_{0} n_{1} R_{1})} = \frac{(B / C) J'_{m} (k_{0} n_{2} R_{1}) + N'_{m} (k_{0} n_{2} R_{1})}{(B / C) J_{m} (k_{0} n_{2} R_{1}) + N_{m} (k_{0} n_{2} R_{1})}; N = {\begin{matrix} n_{1} / n_{2}; TM \\ n_{2} / n_{1}; TE \end{matrix}

where B/C can be expressed as:

\frac{B}{C} = \frac{M H_{m}^{(1)}' (k_{0} n_{3} R_{2}) N_{m} (k_{0} n_{2} R_{2}) - H_{m}^{(1)} (k_{0} n_{3} R_{2}) N_{m}' (k_{0} n_{2} R_{2})}{H_{m}^{(1)} (k_{0} n_{3} R_{2}) J'_{m} (k_{0} n_{2} R_{2}) - M H_{m}^{(1)}' (k_{0} n_{3} R_{2}) J_{m} (k_{0} n_{2} R_{2})}; M = {\begin{matrix} n_{3} / n_{2}; TM \\ n_{2} / n_{3}; TE \end{matrix}

By the solving the eigen-equation, the wavenumber k₀, the azimuthal quantization number m and the radial quantization number l can be obtained. Accordingly, the WGMs can be determined.

3.2 Mode coupling, transmittance and Fano resonance

Considering a single WGM resonator, when a continuous light wave is sent into the waveguide, the light is coupled into the resonator through the interaction between the evanescent field of the waveguide mode and WGM. When stable optical oscillations are established, the transmittance amplitude can be expressed as [21]

t_{o u t} = t_{w} + \frac{{| κ |}^{2} α}{α t - e^{i φ}}

where t_w and t are the self-coupling coefficients of the waveguide mode and the WGMs respectively; κ is the coupling coefficient between the waveguide mode and WGMs. When the loss during the coupling process is neglected,

| t_{w} | = | t |

and

{| κ |}^{2} + {| t_{w} |}^{2} = 1

. α and φ represent the transmittance and phase, respectively, of the WGMs propagating for one cycle, where

φ = k_{0} n_{e f f} L = \frac{2 π}{λ} \cdot n_{e f f} \cdot (2 π \cdot R_{e f f})

. n_eff and R_eff are the effective refractive index and radius of the excited WGM respectively.

As illustrated in Fig. 2(a), in a dual-channel MOF, the coupling between the WGMs among the channels and frame may also occur through the WGMs’ evanescent field when the light is coupled into the tubular frame. Hence, each channel constitutes a single WGM resonator system coupled to the frame. The transfer functions t_ch_,1 and t_ch_,2 can be expressed in Eq. (4). Accordingly, the mode coupling can be represented by an equivalent model, shown in Fig. 2(b). Equation (5) presents a matrix for describing the mode coupling in an MOF

[\begin{matrix} E_{o u t} \\ E_{f 2} \end{matrix}] = [\begin{matrix} t_{w} & κ_{0} \\ - κ_{0}^{*} & t \end{matrix}] [\begin{matrix} E_{i n} \\ E_{f 1} \end{matrix}]

where t_w and t are the self-coupling coefficients of the waveguide mode and the WGM of the frame resonator respectively; κ₀ is the coupling coefficient between the waveguide mode and WGM of the frame. E_in and E_out represent the electric field in the waveguide before and after coupling into the microcavity respectively. The electric field of the light before and after traveling around the resonator for one cycle can be expressed as Eqs. (6) and (7) respectively.

Fig. 2 (a) Mode coupling in the dual-channel MOF; (b) an equivalent model for (a)

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\begin{array}{l} E_{f 1} = (α_{f 3} e^{- i φ_{f 3}}) t_{c h, 1} (α_{f 2} e^{- i φ_{f 2}}) t_{c h, 2} (α_{f 1} e^{- i φ_{f 1}}) E_{f 2} \\ \begin{matrix} = α_{f} e^{- i φ_{f}} t_{c h, 1} t_{c h, 2} E_{f 2} \end{matrix} \end{array}

E_{f 2} = \frac{κ_{0}^{*}}{α_{f} t t_{c h, 1} t_{c h, 2} e^{- i φ_{f}} - 1} E_{i n}

According to Eq. (4), $t_{c h, i} = t_{w, i} + \frac{{| κ_{c h, i} |}^{2} α_{c h, i}}{α_{c h, i} t_{i} - e^{i φ_{c h, i}}}$ (i = 1, 2 for channel 1, 2). α_f = α_f₁α_f₂α_f₃ and φ_f = φ_f₁ + φ_f₂ + φ_f₃ represent the transmittance and the phase, respectively, of the WGMs propagating for one cycle. The electric field of the output light can be expressed as:

E_{o u t} = (t_{w} + \frac{{| κ_{0} |}^{2} α_{f} t_{c h, 1} t_{c h, 2}}{α_{f} t t_{c h, 1} t_{c h, 2} - e^{i φ_{f}}}) E_{i n}

The ratio of the output electric field to the input electric field as well as the transmittance of the MOF micro-resonator can be expressed as Eqs. (9) and (10) respectively.

t_{o u t} = \frac{E_{o u t}}{E_{i n}} = t_{w} + \frac{α_{f} t_{c h 1} t_{c h 2} {| κ_{0} |}^{2}}{α_{f} t t_{c h 1} t_{c h 2} - e^{i φ_{f}}}

T_{o u t} = {| t_{o u t} |}^{2}

where

φ_{i} = \frac{2 π}{λ} \cdot n_{e f f, i} \cdot (2 π \cdot R_{e f f, i}) = \frac{{(2 π)}^{2}}{λ} L_{e f f, i}

; and

L_{e f f, i} = n_{e f f, i} R_{e f f, i}

(i = f, 1, 2 for frame, channel 1, channel 2 respectively). When the phases of the WGMs in the channel and frame satisfy φ_f = φ_i + 2kπ (k is integer), the morphology-dependent resonances (MDRs) will occur. However, due to the differences of the WGMs’ Q factors, the MDRs may express different characteristics. When the WGMs’ Q factors are close, one can observe mode splitting in the output spectrum [23, 25]. When the Q factor of the frame resonator is much lower than those of the channels, the MDRs will appear as the Fano resonance [25, 26]. According to Eqs. (9) and (10), the transmittance are calculated based on the aforementioned conditions; the results are presented in Fig. 3.

Fig. 3 Transmittance of channel 1 (red), channel 2 (green), frame (blue), and the output spectrum (black) of the dual-channel WGM resonator system under the conditions of: (a) mode splitting (b) Fano resonance

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The parameters used in Fig. 3(a) include α = α_ch,₁ = α_ch,₂ = 0.998; κ₀ = κ_ch,₁ = κ_ch,₂ = 0.02; L_eff_, _f = 120 μm; L_eff_{, 1} = 60 μm; and L_eff_{, 2} = 48μm. The Q-factors for the frame, channel 1, and channel 2 are calculated to be Q_f = 6.75 × 10⁵, Q₁ = 4.32 × 10⁵, and Q₂ = 2.83 × 10⁵ respectively. The parameters used in Fig. 3(b) include α = 0.800; κ₀ = 0.20; the rest of the parameters remain the same as in the Fig. 3(a) case. The Q-factors are found to be Q_f = 6.38 × 10³, Q₁ = 4.32 × 10⁵, and Q₂ = 2.83 × 10⁵ respectively. Figures 3(a) and 3(b) confirm that (1) when Q_f is close to Q₁ and Q₂, mode splitting can be observted in the output spectrum; and (2) when Q_f is much lower than Q₁ or Q₂, Fano resonance can be obseved. Considering multichannel sensing, although high Q factors can be achieved in the mode splitting condition, it is difficult to demodulate the sensing signals due to the avoided-crossing effect between the two splitting modes [25, 27], and thus Fano resonance is a better operating condition for multichannel sensing applications. Accordingly, the Q factor of the frame should decrease to obtain the Fano resonance; this can be achieved by controlling the coupling efficientcy κ₀ and the transmittance α_f.

4. MOF for multichannel bio-sensing

In this section, we present the analyses of multichannel fluidic sensing based on a dual-channel MOF. As most biological samples have an RI close to water, we set the RI in this analysis to be from 1.330 to 1.345. The microstructures in the dual-channel MOF has a constant wall thickness of 2 μm; the outer radii of channel 1, channel 2 and the frame are 40 μm, 35 μm and 85 μm respectively. The RI of the channels and frame is 1.444. Table 1 lists the detailed parameters in the calculation.

Table 1. Design parameters of the MOF.

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According to Eqs. (2) and (3) and the parameters in Table 1, the wavenumber k₀ and the quantization number l and m of the WGM can be determined. The phase of the light travelling around the resonator for a cycle will satisfy $φ = k_{0} n_{e f f} (2 π \cdot R_{e f f}) = 2 m π$ when any WGM is excited. Following that, we can obtain L_eff(λ,n₁) = m/k₀ under different wavelengths (λ) and core RI (n₁). Table 2 presents the λ- and n₁-dependent parameter, L_eff, calculated via Eq. (2) and Eq. (3) and linear fitted in the wavelength ranging from 1500 nm to 1600 nm and RI ranging from 1.330 to 1.345.

Table 2. Expressions of L_eff.

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Based on Eq. (9) and the parameters in Tables 1 and 2, we can calculate the output spectrum of the first order TE mode (l = 1) with different media (of different RI) filled in each channel; the results are shown in Fig. 4. As a reference, we first analyze the case where both channels are filled with water, shown Fig. 4(a), where the output spectrum of the first order TE mode is calculated. From the results, we can observe that the Fano resonance is achieved between channel 2 and the frame at ~1543 nm, and between channel 1 and the frame at ~1574 nm. Accordingly, we use these two spectral windows to discuss the sensing charactersitics of the TE mode. As shown in Fig. 4(b), when channel 1 is filled with bio-samples and channel 2 is filled with water, the Fano resoance for channel 1 shifts linearly to the longer wavelengths with increasing sample RI, while the Fano resonance for channel 2 and the WGM resonant peaks of frame remain unchanged. As shown in Fig. 4(c), when channel 1 is filled with water and channel 2 is filled with bio-samples, the two Fano resonant peaks shows opposite behavior as compared with the case in Fig. 4(b). Note that the WGM resonant peaks of the frame will not shift with the variation of the sample RI. For the ease of demodulation of the Fano resonant peaks, the spectrum of the frame WGM is filterd and their central wavelengths are measured. The detail wavelength response of the Fano resonant peaks are presented in Fig. 5.

Fig. 4 Output spectrum of the TE mode with different media in the channels: (a) both channels filled with water; (b) channel 1: bio-samples with different RI and channel 2: water; and (c) channel 1: water and channel 2: bio-samples with different RI. The color changes from red to blue indicating Fano resonance shifts towards longer wavelengths with increasing RI.

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Fig. 5 Responses of the Fano resonant peaks as a function of sample RI in the TE mode: (a) channel 1 filled with bio-samples and channel 2 filled with water; and (b) channel 1 filled with water and channel 2 filled with bio-samples. For (a) and (b), the corresponding transmission of channel 1 and 2 is plotted in the middle and bottom rows respectively.

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As shown in Fig. 5(a), when channel 1 is filled with bio-samples, the Fano resonance peak shifts linearly to the longer wavelengths with increasing sample RI, i.e., middle plot of Fig. 5(a), and the sensitivity is calculated to be 29.0557 nm/RIU (refractive index unit). At the same time, the Fano resonance of channel 2 remains unchanged, i.e., bottom plot in Fig. 5(a). As shown in Fig. 5(b), when channel 2 is filled with bio-samples, its resonance peak also moves linearly with increasing RI, i.e., bottom plot in Fig. 5(b), and the sensitivity is calculated to be 22.9160 nm/RIU; meanwhile the Fano resonance peak of channel remains unchanged, i.e., middle plot in Fig. 5(b). These results have proved that there is no coss-sensitivity between the two channels; and the dual-channel MOF can work independently to perform multichannel sensing.

Next we analyze the sensing characteristics of the first order TM mode; the results are presented in Fig. 6. As shown in Fig. 6(a), when the two channels are both filled with water, the Fano resonances for channel 1 and 2 are achieved at ~1561 nm and 1558 nm respectively. Accordingly, we choose these two spectral windows for further analyses. Similar to the TE mode, when channel 1 is filled with bio-samples, its Fano resonant peak shifts while the resonant peak for channel 2 remains unchanged. As shown in Fig. 6(c), opposite behavoir is observed for the case when channel 2 is filled with bio-samples.

Fig. 6 Output spectrum of the TM mode with different RI in the channels: (a) both channels filled with water; (b) channel 1: bio-samples with different RI and channel 2: water; and (c) channel 1: water and channel 2: bio-samples with different RI. The color changes from red to blue indicating Fano resonance shifts towards longer wavelengths with increasing RI.

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Figures 7(a) and 7(b) present the responses of the Fano resonant peaks as a function of the sample RI for the aforementioned two cases respectively. The sensitivities of the TM mode Fano resonance for channel 1 and 2 are 16.0694 nm/RIU and 13.3181 nm/RIU respectively. The sensitivities of the TM mode are lower those of the TE mode. The reason is that the overlaping of the mode field and liquid area for the TM mode is less than those for the TE mode.

Fig. 7 Responses of the Fano resonant peaks as a function of the sample RI for the TM mode: (a) channel 1 filled with bio-samples and channel 2 filled with water; and (b) channel 1 filled with water and channel 2 filled with bio-samples. For (a) and (b), the corresponding transmission of channel 1 and 2 is plotted in the middle and bottom rows respectively.

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The excitation ratio (ER) of Fano resonace, i.e., the ratio between the peak and dip value of the Fano resonance, will vary with the sample RI due to the change of the phase diferences between the frame and individual channels, i.e., Δφ = φ_f - φ_ch,i, as shown in Fig. 4 and Fig. 6. When the phase is matched (Δφ = 0), a minimal T_out and maximal ER can be obtained according to Eq. (10). When the phase is not matched (Δφ ≠ 0), the ER of Fano resonance will decrease with increasing Δφ. It will be difficult to observe the Fano resonance when the value of Δφ is too large, which is mainly determined by the Q factor of the frame. As such, the Q factor of the frame determines the measuring range of the RI. When the Q factor is low, the frame WGM resonant dips will have a wider bandwidth; and a larger RI mearsuring range can be achieved.

5. Discussions on fabrication and sensitivity of MOF

With the recent advances in MOF fabrication technologies, the proposed MOF can be produced by direct fiber pulling processes in combination with the caplillary-stacking technique; the channel dimension and wall thicknesses can be typically controlled to within a few hundred nanometers precision. Best results can be achieved by fine-tuning the pulling speed, temperature and atmospheric pressure etc [28–30]. Post-processing steps, such as CO₂ laser heat treatment, electric arc cleaning, and hydrofluoric acid etching, can be applied to further improve the precision and surface quality of the MOF.

Next, we investigate the effect of varying (1) the radius and (2) wall thickness of the capillary channel. The study helps to understand how the channel geometry should be selected and optimized as well as how fabrication errors may affect the sensor performance. As the TE and TM modes have similar sensing properties, we only calculate the sensitivities of the first order TE mode as a function of the channel radius and channel wall thickness. The results are presented in Fig. 8.

Fig. 8 (a) Sensitivity as a function of the channel radius with a wall thickness (t) of 1.5 μm, 2 μm and 2.5μm; (b) sensitivity as a function of wall thickness when the channel radius is 40 μm.

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In Fig. 8(a), it is observed that when the channel wall thickness is constant, the sensitivity will increase as the channel radius increases. In Fig. 8(b), it is observed that when the radius is constant (40 µm), the sensitivity will increase as the wall thickness decreases. Thus, to achieve a high sensitivity, one should increase the channel radius and reduce the channel wall thickness. It is worth to note that, in Fig. 8(b), the sensitivity will change rapidly when the wall thickness varies at low values, e.g, 1.2 µm. Accordingly, one should consider choosing a larger wall thickness to avoid fabrication error-induced sensitivity variation. In our design, the wall thickness is selected to be 2 µm, which falls on a reasonable value in terms of both fabrication practicality and sensor repeatability.

6. Conclusion

In this paper, we present the design and theoretical modeling of a MOF for multichannel sensing based on the Fano resonance among the different WGMs. The MOF design consists of a number of capillary channels inside a tubular frame, all with different diameters. A typical structure of the MOF with a dual-channel configuration has been investigated. First, a theoretical model is established to analyze the Fano resonance among the WGMs. Next, we demonstrate the multichannel bio-sensing via the dual-channel MOF, where the sensing characteristics and the output spectrum of the two channels are calculated under different sample RI. The results show that there is no cross-sensitivity between the two channels. The new MOF presents a compact, flexible, and low-cost solution for bio/chemical multichannel sensing, multi-microcavity laser, and tunable photonics devices etc.

Funding

National Natural Science Foundation of China (NSFC) (Grant 11274182, and 61322510); Tianjin Natural Science Foundation (16JCZDJC31000); the HKSAR Innovation and Technology Commission (ITC), Innovation and Technology Fund (ITF), ITS/007/15FP; and HKSAR Research Grants Council (RGC), General Research Fund (GRF/ECS), (Grant CUHK 439813).

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	*L_eff* (TE mode)	*L_eff* (TM mode)
Frame	122.99700028 - 0.00300054 λ	122.591014 - 0.002465 λ
Channel 1	55.87192005 - 0.00154895 λ + 1.07999803 n₁	56.33764010 - 0.0012770 λ + 0.6007594 n₁
Channel 2	49.01018366 - 0.00138888 λ + 0.76119901 n₁	49.30054572 - 0.0011487 λ + 0.4370257 n₁

	*L_eff* (TE mode)	*L_eff* (TM mode)
Frame	122.99700028 - 0.00300054 λ	122.591014 - 0.002465 λ
Channel 1	55.87192005 - 0.00154895 λ + 1.07999803 n₁	56.33764010 - 0.0012770 λ + 0.6007594 n₁
Channel 2	49.01018366 - 0.00138888 λ + 0.76119901 n₁	49.30054572 - 0.0011487 λ + 0.4370257 n₁

Microstructured optical fiber for multichannel sensing based on Fano resonance of the whispering gallery modes

Abstract

1. Introduction

2. Design of the MOF

3. Modeling

3.1 WGMs in a MOF

3.2 Mode coupling, transmittance and Fano resonance

4. MOF for multichannel bio-sensing

5. Discussions on fabrication and sensitivity of MOF

6. Conclusion

Funding

References and links

Cited By

Figures (8)

Tables (2)

Equations (10)

Optics Express

	R₁ (μm)	R₂ (μm)	n₃	n₂	α	κ
Frame	83	85	1	1.444	0.800	0.20
Channel 1	38	40	1	1.444	0.998	0.02
Channel 2	33	35	1	1.444	0.998	0.02