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Metal-printing directly defined polymer optical waveguide sensors for detecting effective anti-inflammatory concentrations of peimine as fritillaria alkaloid drugs

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Abstract

Novel polymer optical sensors for detecting effective anti-inflammatory concentrations of peimine as fritillaria alkaloid drugs are successfully designed and fabricated by the metal-printing directly defined active waveguide technique. The erbium-containing cross-linked copolymer and the organic–inorganic grafting PMMA material are used as a sensing core layer and cladding window, respectively. Multi-hydrogen-bonding affinity interactions between the tested drug and the sensing polymer layer are analyzed by the molecular docking model. The multimode interference (MMI) waveguide sensor structure is optimized to design. The sensitivity included of the bulk and surface parts for the waveguide optical sensor is analyzed. The optical transmission field of the device is simulated based on the refractive indices of different peimine concentrations. The sensitivity of the device is measured as 2 × 103 RIU−1. The resolution and detection limit of the sensor are characterized to be 2.5 × 10−4 and 1.3 × 10−7 RIU, respectively. The effective anti-inflammatory concentration range (10-25 mg/L) of the peimine solution could be efficiently detected in the scope of 5 dB corresponding to the maximum output optical power change. Relative optical gain of the chip is obtained as 3 dB for achieving calibration of the sensor. The technique could realize a simple, rapid and controlled process with high sensitivity for detecting effective anti-inflammatory concentration of alkaloid drugs in traditional Chinese medicine.

© 2017 Optical Society of America

1. Introduction

Bulbus Fritillariae (BF) known by the Chinese name “Beimu”, which is widely used as antitussive and antiasthma drug with positive therapeutic effects in Traditional Chinese Medicine (TCM) for more than 2000 years. As the main bioactive ingredient, Peimine, a fritillaria alkaloid, is extracted from BF and has remarkable antitussive, expectorant and antiasthmatic activities [1,2]. The molecular structure of Peimine is shown as Fig. 1(a). Now it has been demonstrated that Peimine has the effect of anti-inflammatory, which can be used to treat the inflammatory diseases of the respiratory tract such as cough, asthma and chronic bronchitis [3,4]. To realize scientific application of BF in TCM, it is important for selecting effective anti-inflammatory concentrations of Peimine solution.

 figure: Fig. 1

Fig. 1 Molecular structures of (a) Peimine, (b) GETPM, and (c) organic–inorganic grafting PMMA. (d) Molecular docking model simulated between GETPM and Peimine.

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Currently, the main method for detecting drugs monomers is carried out by high-performance liquid chromatography [5,6], but this detection technique is limited due to high cost, long processing period and complicated operation. Integrated optical waveguide sensors with high sensitivity, small size and immunity to electromagnetic interference are the most promising low-cost, high precision detection chip [7,8]. Most of integrated optical waveguide sensors are designed based on the evanescent wave working principle. Different structures of the integrated optical waveguide sensors have been developed to date, such as interferometers, ring resonators and grating couplers. Interferometric devices have shown the high sensitivity around 1 × 10−7 or 1 × 10−8 RIU, and have become one of the preferred options for achieving truly point-of-care tools with label-free and fast detection [9,10]. Especially, polymer optical waveguide interferometric sensors can have a cheap mass-production process to manufacture the functional devices [11–14]. Furthermore, compared with inorganic material, organic copolymer material system is easier to form specific adsorption to detect organic small molecules in traditional Chinese medicine by multi-hydrogen-bonding interactions [15,16].

In this work, a new type of cross-linked polymer multimode interferometer (MMI) optical sensor for detecting the effective anti-inflammatory concentration of Peimine was designed and fabricated by metal-printing directly defined active waveguide technique. The copolymer with Er3+ organic complexes and the organic–inorganic hybrid material were used as sensing core layer and window cladding materials, respectively. Waveguide structures and optical properties of the active MMI sensor were designed and simulated. The sensitivity included of bulk and surface parts for the proposed waveguide optical sensor was analyzed. Hydrogen-bonding affinity, hydrophilic-hydrophobic and adhesion characteristics on liquid/solid interface between tested drug solution and polymer sensing layer were studied. The sensitivity, resolution and detection limit of the device were obtained. The effective detection for anti-inflammatory Peimine concentration range can be achieved corresponding to the output optical power changed with refractive indices. Loss compensation was realized through Er3+ amplification by pump light for calibration of the sensor. This technique would be suitable for ensuring drug security analysis and dosage monitoring of Chinese herbal ingredients efficiently and accurately.

2. Experiment

2.1 Polymer sensing materials

The copolymer with Er3+ organic complex and the organic–inorganic grafting PMMA material were used as sensing core layer and window cladding materials, respectively. As sensing core layer material, the molecular structure of the copolymers poly(GMA-co-Er(TTA)2(Phen)(MA)) (GETPM) [17] is given in Fig. 1(b). The Er3+ complex is polymerized with MMA and GMA by free-radical polymerization of dilute monomer solutions. GETPM could form a highly cross-linked epoxy matrix structure, which exhibits good chemical stability and corrosion resistance. The 2.5 wt% concentration of Er3+ in the cross-linkable copolymers under the role of pump light could realize infrared signal light power amplification for calibration role of the sensor. As sensing window cladding material, the hybrid material shown in Fig. 1(c) is synthesized by hydrolysis and polycondensation of 3-methacryloxy-proyltrimethoxysilane (MAPTMS, KH570), methylmethacrylate (MMA), epoxypropylmethacrylate (GMA) and tetraethylorthosilicate (TEOS) [18]. The transparent and stable organic–inorganic network grafting PMMA films are resistant to organic solvents.

The GETPM copolymer with a weakly polar matrix and hydrogen bonding groups could selectively adsorb the organic molecule of the alkaloid Peimine with phenol groups based on hydrogen-bonding affinity. Furthermore, the rigid and porous structure of the GETPM copolymer with a high specific surface area is also helpful for enhancing multi-hydrogen-bonding interactions between the tested drug and the sensing core film. Meanwhile, the organic–inorganic grafting PMMA as the sensing window cladding material can effectively avoid the erosion from the solvents of the Peimine. Molecular docking model between GETPM and Peimine is simulated by autodock software, which is given as Fig. 1(d). The binding energy is obtained as about −12.6. It can be obviously shown that hydrogen bonding forces are found and remain stable adsorption structure, which is advantageous to obtain high sensitivity for the optical sensor.

2.2 Sensor design and fabrication

To realize the optical sensor, the metal-printing directly defined active waveguide technique is used. Figure 2(a) shows the cross-sectional structure of the metal-printing defined waveguide sensor. The SiO2 layer grown on Si wafer substrate was used as the lower cladding and the thickness of the oxidized film was 5 μm. The Er-containing copolymer as sensing core layer with 3 μm thickness and 5 μm width was cured on the lower cladding. Au-printing strips for defining waveguide structure were formed by photolithography process. The thickness and width of the Au strip were 0.1 μm and 10 μm, respectively. The organic–inorganic grafting PMMA was spin-coated as the upper cladding. The sensing window structure with 10 μm depth on the upper cladding was realized by lithography and ICP etching technology. Peimine ethanol solution dropped into the sensing window was used as the tested drug. The refractive indices of SiO2, Er-containing copolymer, Au and organic-inorganic grafting PMMA are 1.450, 1.505, 0.559 + 11.5i and 1.475 around 1550-nm wavelength, respectively. The single-mode distribution of the Au-printing defined waveguide structure calculated by COMSOL multiphysics software is shown as Fig. 2(b).

 figure: Fig. 2

Fig. 2 Structural analysis (a) cross-sectional profile of the metal-printing defined waveguide sensing region and (b) single-mode distribution calculated based on the effective index method.

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The modal equations based on effective index method [19] are defined as

k0(n12N12)1/2b=nπ+arctann12(N12n22)1/2n22(n12N12)1/2+arctann12(N12n32)δ3n32(n12N12)1/2(n=0,1,2,3,...),δ3=1+ε3(N12n42)1/2ε4(N12n32)1/2ε3(N12n42)+ε4(N12n32)1/2exp[2k0(N12n32)1/2h)1ε3(N12n42)1/2ε4(N12n32)1/2ε3(N12n42)1/2+ε4(N12n32)1/2exp[2k0(N12n32)1/2h),
k0(n12N22)1/2b=nπ+arctann12(N22n22)1/2n22(n12N22)1/2+arctann12(N22n42)1/2n42(n12N22)1/2(n=0,1,2,3,...),
k0(N12N2)1/2a=mπ+2arctanN12(N2N22)1/2N22(N12N2)1/2(m=0,1,2,3,...).
where n1 is the refractive index of the core layer, n2 is the refractive index of the lower cladding, n3 is the refractive index of the metal strip, n4 is the refractive index of the upper cladding, a is the width of the core waveguide, h is the height of the metal strip, b is the thickness of the core waveguide, N1 and N2 are equivalent refractive indices of side regions, respectively.

According to the modal equations, the relative modal curves between the core waveguide thickness b and effective refractive index Neff are shown in Fig. 3(a) and (b). It can be found that when the core thickness b is defined as 3 μm, single-mode propagation of the waveguide can be realized and the birefringenceΔNeff between TE and TM mode is about 6 × 10−4.

 figure: Fig. 3

Fig. 3 Waveguide mode analyzed (a) Relations between the core thickness b and the effective refractive Neff and (b) varied curves between the core thickness b and birefringenceΔNeff.

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When the tested Peimine ethanol solution is filled into the sensing window, the alkaloid drug can be selectively adsorbed on the waveguide surface of the MMI sensing region. According to previous studies [20], anti-inflammatory and cytotoxicity of Peimine concentrations were analyzed. It can be found that Peimine concentrations in the range (10-25 mg/L) have effective anti-inflammatory activity and don’t display any cellular toxicity. Different concentrations of Peimine ethanol solution were formulated as analyte for the proposed optical waveguide sensor. Refractive indices of the tested drug solution were measured by Abbe refractometer and given as Table 1. As given in Fig. 4(a), it can be found that the fundamental mode effective index Neff of core waveguide becomes large when Peimine concentration is increased. The slope of Linear fit is 0.9992.

Tables Icon

Table 1. Refractive indices of Peimine ethanol solution with different concentrations

 figure: Fig. 4

Fig. 4 (a) The relation between the mode effective index Neff and refractive index ns of different Peimine concentration. (b) The schematic view of the structural model of the MMI sensor.

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Based on the proposed waveguide structure and the self-imaging theory [21], the 1 × 1 MMI optical sensor was designed and constructed, as shown in Fig. 4(b). The coupling equation of the MMI device is defined as

L=4NeffW23λ.
For the MMI sensor designed, 20 mg/L of the effective drug concentrations is designed as the reference point. The length L and width W of MMI waveguide sensing region are defined as 1300 μm and 30 μm, respectively. When the concentration of Peimine is different from 20 mg/L, the effective refractive index of the MMI waveguide area is modified. The self-imaging point formed in the interference pattern of modes would be shifted at output waveguide channel according to the change of the effective refractive index. Through the Rsoft simulation of optical field transmission, the output optical power of the device would be modulated by radiating away high-order modes with different Peimine concentrations, as given in Fig. 5.

 figure: Fig. 5

Fig. 5 The simulation of the optical field transmission for the device with the change of the Peimine concentrations.

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The sensitivity included of bulk and surface parts [22,23] for the proposed waveguide optical sensor was analyzed. The bulk sensitivity depends on the effective mode variation among the interfered modes related to the change of refractive index for the tested drug at the sensing area of the device. The intrinsic bulk sensitivity can be calculated by

ηbulk=(Δneff)ns,
Where ns is the refractive index of the tested Peimine solution and Δneff is the difference of the effective refractive indices for the MMI waveguide sensing region with respect to the change of Peimine concentrations.

The bulk sensitivity of the device due to the variation of the effective refractive index difference directly affects the phase change ΔΦ between the modes at the end of the interferometric region, which determines the power coupled to the single-mode waveguide output channel

Δφ=2πΔneffLλ,
Sbulk=(Δφ)ns2πLληbulk.
where λ is the operating wavelength, L is the length of sensing region and Sbulk is the bulk sensitivity of the optical sensor.

The surface sensitivity is defined with respect to the thickness of a receptor layer. When a bimolecular interaction takes place on top of the sensing area, the mode properties are affected by a refractive index variation at their evanescent tail. For the interferometric sensor, it is key role to previously functionalize the sensor surface with affinity receptors that will capture the specific analyte. Owing to the evanescent profile of the guided modes in the sensing region of the sensor, the sensitivity may reach a maximum value asymptotically as the receptor layer thickness increases. The intrinsic surface sensitivity is defined as

ηsurf=(Δneff)b,
where b is the thickness of the receptor layer, which is equal to the thickness of the waveguide core layer in this study. The surface sensitivity is defined as
Ssurf=(Δφ)b=2πLληsurf.
The high penetration depth δ is important for obtaining the high surface sensitivity. It is inversely proportional to the evanescent tail decay α, defined as

α=1δ=2πλ(neff2ns2)1/2.

The large penetration depth means that the evanescent field can penetrate more easily into the drug sensing layer to create the interferometric response.

Based on the relevant theory and experimental parameters, both the bulk and surface sensitivities of the sensor for detecting Peimine effective concentration were simulated in Fig. 8, respectively. As shown in Fig. 6(a), it is observed that compared with the other defined thickness of sensing layers, when the thickness of the sensing core is close to 3 μm, the bulk sensitivity of the device reaches the maximum value. As given in Fig. 6(b), it is obviously found that when the thickness of the sensing layer is 3 μm, the surface sensitivity of the sensor comes to the extreme value.

 figure: Fig. 6

Fig. 6 The bulk and the surface sensitivity of the sensor simulated (a) the relation between the bulk sensitivities and the refractive indices of the Peimine solutions with the different thickness of sensing layers; (b) the correlation between the surface sensitivities and the thickness of sensing layers with the refractive indices of the Peimine solutions.

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2.3 Sensor measurement and discussion

Adhesion effect on liquid/solid interface between Peimine ethanol solution and polymer GETPM sensing layer is analyzed. As shown in Fig. 7 (a), 7(b) and 7(c), when the sensor sample with droplets of Peimine solution in horizontal, vertical and inverted states, there is almost no sliding angle for the analyte. The high adhesion is helpful to effectively ensure the detection accuracy and reliability of the device.

 figure: Fig. 7

Fig. 7 Adhesion characteristics on liquid/solid interface between Peimine ethanol solution and polymer GETPM sensing layer in (a) horizontal, (b) vertical and (c) inverted states. Contact angles of Peimine ethanol solution on the surfaces of (d) GETPM sensing layer, (e) grafting PMMA layer and (f) Si layer.

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Hydrophilic and hydrophobic characteristics between the sensing polymer layers and Peimine ethanol solution were measured and analyzed. Peimine ethanol solution contact angles (CAs) on the surfaces of the GETPM sensing layer, grafting PMMA layer and Si layer were shown in Fig. 7 (d), 7(e) and 7(f). Compared to hydrophobic properties (90°<CA<150°) on Si layer, hydrophilic behavior (CA≤90°) is observed for the drug droplets on the surfaces of the GETPM sensing layer and grafting PMMA cladding layer. The Hydrophilic characteristic is helpful to increase the effective operating area between drug droplets and sensing region and enhance the interaction between the drug and polymer molecules.

The surface morphology from the Au-printing strip of the MMI waveguide sensor fabricated was measured by AFM and shown in Fig. 8(a). The structural pattern of Au strips and waveguide section were given by microscope ( × 500) in Fig. 8(b). The thickness of Au layer is about 170 nm and the surface roughness less than 1.5 nm. It was found that the parameters designed of the metal-printing directly defined active waveguide could be realized well and the process enables precise control of the core size. Furthermore, the Au strip structure with good stability and uniformity could be formed on the core copolymer without any adhesion layer. There are no cracking for Au films or high roughness on the surface of the copolymer.

 figure: Fig. 8

Fig. 8 (a) Surface morphology from the Au strip measured by AFM and (b) structural patterns of Au strips and waveguide section by microscope ( × 500).

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Figure 9(a) schematically shows the sensing measurement system. A tunable laser (Santec TSL-210) with wavelength ranges from 1510 nm to 1590 nm was used as the signal source and a 980 nm laser diode with a maximum output power of 400 mW was used as the pump light source. The role of the pump light source is to achieve the sensor loss-compensation characteristics and adjust the signal output power reference point. TM polarized signal light driven from a polarization controller at 1546 nm wavelength with an output optical power as 1 mW and the pump light were launched into the channel waveguides by a wavelength division multiplexing (WDM) coupler. The drug solution was injected into sensing chamber by microfluidic channels. The output light from the MMI sensor was coupled into an optical power meter which was remotely manipulated through a computer interface developed using the LabVIEW program. For flow control, an acrylic well designed for the attachment of the tubes was adhered onto the surface of the chip prior to applying drug solution. The volume of the PMMA flowing chamber was 50 μL. Liquid flow was controlled by a peristaltic pump. The ethanol solution was first injected into the liquid chamber. The refractive index of the ethanol solution is 1.3572. When the optical power was stabilized, the Peimine ethanol solution of different concentrations was flown onto the sensor at a flow rate of 6.0 μL/min, respectively. The output optical intensity was measured at 1546 nm wavelength was shown in Fig. 9(b). The average response time of output optical power change for every single measurement is less than 60 s. It could be observed that once the ethanol solution flushed into the sensing chamber by the micro-channels, the output power came to the same value again.

 figure: Fig. 9

Fig. 9 (a) Schematic overview of sensing measurement system. (b) The output optical intensity with different Peimine concentrations at 1546 nm wavelength. (c) The relationship between the output optical power and refractive indices of different Peimine concentrations at 1546 nm wavelength. (d) The transmission spectra change with the wavelength range from 1530 nm to 1590 nm.

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Figure 9(c) gives that the relationship between the output optical power and refractive indices of different Peimine concentrations at 1546 nm wavelength and Fig. 9(d) shows that the transmission spectra change with the wavelength range from 1530 nm to 1590 nm. The sensitivity of the sensor was characterized by injecting Peimine ethanol solution of different concentrations (5, 10, 20, 25 and 50 mg/L) into the MMI sensing region. The sensitivity S of the sensor is 2 × 103 RIU−1, which is the gradient of the relationship between the measured intensity and the refractive index. The limit of detection (LOD) for the MMI waveguide sensor is defined as L = R/S, where R is sensor resolution related to system noise and S is the sensitivity. The resolution of the MMI sensor system can be estimated from R = 3σ, where σ is the standard deviation of the measured intensity for the blank sample as the system output noise. The resolution is 2.5 × 10−4 and the detection limit is estimated to be 1.3 × 10−7 RIU. The minimum insertion loss of the sensor in the spectral range was obtained as 3 dB, which may be mainly caused by mode mismatching loss between fiber and waveguide and mode coupling loss between MMI region and input/output waveguides.

As shown in Fig. 9(c) and 9(d), when the maximum output peak with Peimine concentration as 20 mg/L was used as benchmark, it can be obtained that the drug concentrations corresponding to the output optical power change in the scope of 5 dB could form effective detection for anti-inflammatory Peimine concentration range (10-25 mg/L). Furthermore, the Er-containing copolymer sensing core layer could generate 3 dB relative gain by 980 nm pumping light. By adjusting optical pumping power, the maximum signal output optical power of the sensor at the benchmark point (20 mg/L Peimine solution concentration) can be set at fixed value to achieve the calibration process for each measurement, which guarantees the accuracy and the stability of the MMI sensor.

As given in Table 2, compared with other reported 1 × 1 MMI waveguide sensors [24–26], including our work, the sensing lengths were around 15 mm. The values of the detection limit were put into perspective by comparison with the performance of the MMI waveguide sensor published in the literature. It can be observed that the proposed 1 × 1 MMI device could achieve stable operation well with a smaller size and a lower detection limit. The advantages of the performances from the overall device can be obviously noted.

Tables Icon

Table 2. Comparison with other publish results for 1 × 1 MMI waveguide sensor.

3. Conclusion

In summary, the polymer optical waveguide sensors proposed for detecting effective anti-inflammatory concentrations of Peimine as fritillaria alkaloid drugs were achieved by metal-printing directly defined active waveguide technique. The hydrogen-bonding affinity, hydrophilic and adhesion characteristics on liquid/solid interface between tested drug solution and polymer sensing layer obtained are advantageous to obtain high sensitivity for the optical sensor. The sensitivity of the device was measured as 2 × 103 RIU−1. The resolution and detection limit of the sensor were characterized to be 2.5 × 10−4 and 1.3 × 10−7 RIU, respectively. The effective anti-inflammatory concentration range (10-25 mg/L) of Peimine solution could be efficiently detected in the scope of 5 dB corresponding to the output optical power. Relative optical gain of the chip was obtained as 3 dB for achieving calibration of the sensor. The proposed technique are not only applicable to detect single drug Peimine effective anti-inflammatory concentration range, but also adjust to other fritillaria alkaloid drugs as Chinese herbal ingredients anti-inflammatory concentration analysis. The polymer waveguide sensor will be suitable for integrated optical lab-on-chip technique and be widely applied in effective detection for the anti-inflammatory alkaloid drug concentration range and efficient environmental monitoring for drug cytotoxicity concentrations.

Funding

The Key Project of Chinese National Programs for Research and Development (No. 2016YFD0501005); the National Key Research and Development Plan of China (No. 2016YFB0402502); the National Natural Science Foundation of China (No. 61575076); the Science and Technology Development Plan of Jilin Province, China (No. 20140519006JH).

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Figures (9)

Fig. 1
Fig. 1 Molecular structures of (a) Peimine, (b) GETPM, and (c) organic–inorganic grafting PMMA. (d) Molecular docking model simulated between GETPM and Peimine.
Fig. 2
Fig. 2 Structural analysis (a) cross-sectional profile of the metal-printing defined waveguide sensing region and (b) single-mode distribution calculated based on the effective index method.
Fig. 3
Fig. 3 Waveguide mode analyzed (a) Relations between the core thickness b and the effective refractive Neff and (b) varied curves between the core thickness b and birefringenceΔNeff.
Fig. 4
Fig. 4 (a) The relation between the mode effective index Neff and refractive index ns of different Peimine concentration. (b) The schematic view of the structural model of the MMI sensor.
Fig. 5
Fig. 5 The simulation of the optical field transmission for the device with the change of the Peimine concentrations.
Fig. 6
Fig. 6 The bulk and the surface sensitivity of the sensor simulated (a) the relation between the bulk sensitivities and the refractive indices of the Peimine solutions with the different thickness of sensing layers; (b) the correlation between the surface sensitivities and the thickness of sensing layers with the refractive indices of the Peimine solutions.
Fig. 7
Fig. 7 Adhesion characteristics on liquid/solid interface between Peimine ethanol solution and polymer GETPM sensing layer in (a) horizontal, (b) vertical and (c) inverted states. Contact angles of Peimine ethanol solution on the surfaces of (d) GETPM sensing layer, (e) grafting PMMA layer and (f) Si layer.
Fig. 8
Fig. 8 (a) Surface morphology from the Au strip measured by AFM and (b) structural patterns of Au strips and waveguide section by microscope ( × 500).
Fig. 9
Fig. 9 (a) Schematic overview of sensing measurement system. (b) The output optical intensity with different Peimine concentrations at 1546 nm wavelength. (c) The relationship between the output optical power and refractive indices of different Peimine concentrations at 1546 nm wavelength. (d) The transmission spectra change with the wavelength range from 1530 nm to 1590 nm.

Tables (2)

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Table 1 Refractive indices of Peimine ethanol solution with different concentrations

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Table 2 Comparison with other publish results for 1 × 1 MMI waveguide sensor.

Equations (10)

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k 0 ( n 1 2 N 1 2 ) 1 / 2 b = n π + arc tan n 1 2 ( N 1 2 n 2 2 ) 1 / 2 n 2 2 ( n 1 2 N 1 2 ) 1 / 2 + arc tan n 1 2 ( N 1 2 n 3 2 ) δ 3 n 3 2 ( n 1 2 N 1 2 ) 1 / 2 ( n = 0, 1, 2, 3, ... ), δ 3 = 1 + ε 3 ( N 1 2 n 4 2 ) 1 / 2 ε 4 ( N 1 2 n 3 2 ) 1 / 2 ε 3 ( N 1 2 n 4 2 ) + ε 4 ( N 1 2 n 3 2 ) 1 / 2 exp [ 2 k 0 ( N 1 2 n 3 2 ) 1 / 2 h ) 1 ε 3 ( N 1 2 n 4 2 ) 1 / 2 ε 4 ( N 1 2 n 3 2 ) 1 / 2 ε 3 ( N 1 2 n 4 2 ) 1 / 2 + ε 4 ( N 1 2 n 3 2 ) 1 / 2 exp [ 2 k 0 ( N 1 2 n 3 2 ) 1 / 2 h ) ,
k 0 ( n 1 2 N 2 2 ) 1 / 2 b = n π + arc tan n 1 2 ( N 2 2 n 2 2 ) 1 / 2 n 2 2 ( n 1 2 N 2 2 ) 1 / 2 + arc tan n 1 2 ( N 2 2 n 4 2 ) 1 / 2 n 4 2 ( n 1 2 N 2 2 ) 1 / 2 ( n = 0, 1, 2, 3, ... ),
k 0 ( N 1 2 N 2 ) 1 / 2 a = m π + 2 arc tan N 1 2 ( N 2 N 2 2 ) 1 / 2 N 2 2 ( N 1 2 N 2 ) 1 / 2 ( m = 0, 1, 2, 3, ... ) .
L = 4 N e f f W 2 3 λ .
η b u l k = ( Δ n e f f ) n s ,
Δ φ = 2 π Δ n e f f L λ ,
S b u l k = ( Δ φ ) n s 2 π L λ η b u l k .
η s u r f = ( Δ n e f f ) b ,
S s u r f = ( Δ φ ) b = 2 π L λ η s u r f .
α = 1 δ = 2 π λ ( n e f f 2 n s 2 ) 1 / 2 .
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