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Cu2+-imprinted optical fiber SPR sensor for intelligent recognition

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Abstract

An optical fiber surface plasma resonance (SPR) sensor with MMF-TCF-MMF structure was designed to realize intelligent recognition of copper ions (Cu2+), and the selective adsorption sensitization was achieved by plating a layer of Cu2+-imprinted film on the surface of gold film excitation layer. Combining the principle of optical fiber interference and SPR, the proposed sensor realized the detection of the copper ions concentration through measuring the refractive index changes caused by ions adsorption on imprinted film. The Cu2+-imprinted optical fiber SPR sensor can realize the intelligent recognition and detection of copper ions in the complex environment and exhibits a detection sensitivity of -10.05 pm/ppm. The proposed sensor has tremendous development potential in practical application, and provides new ideas for the field of metal ions detection.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

With the expansion and development of modern cities, heavy metal pollution in soil, atmosphere, especially water environment is becoming more and more serious [1,2]. Owing to the continuous discharge of various pollutants in daily production and life, heavy metal ions pollution has become a key issue in the field of water environment. Generally, small amounts of heavy metals can be excreted from the human body through metabolism, while long-term excessive intake of heavy metal ions will lead to heavy metal ions poisoning, especially copper ions (Cu2+) [35]. This will put a burden on the functioning of the human organs, and then cause metabolic disorders, leading to cirrhosis, liver ascites, and other significant diseases [6]. One of the main ways for heavy metal ions to enter the human body is drinking water intake [1,4], so the detection of heavy metal ions concentration in drinking water is particularly important.

The traditional detection methods of heavy metal copper ions include atomic absorption spectrometry, inductively coupled plasma mass spectrometry (ICP-MS), spectrophotometry, chemiluminescence method, and electrochemical method [710]. The implementation of atomic absorption spectrometry and ICP-MS require expensive detection instruments, and samples need to be processed in advance, resulting in long detection time and complicated operation. Similarly, electrochemical methods also require chemical treatment of the sample, increasing the possibility of secondary contamination [11]. The spectrophotometric method not only cannot accurately identify similar metal ions but also has low detection accuracy [8]. The chemiluminescence method has poor ion selectivity and limited detection concentration [12]. In contrast, the optical fiber sensor has the unique advantages of convenient use, anti-electromagnetic interference, and high sensitivity, so the application of optical fiber sensor in the detection of heavy metal ions concentration has more excellent development prospect [1316]. Among different optical fiber sensors, the optical fiber surface plasma resonance (SPR) sensor stands out because of its strong anti-interference ability, real-time monitoring, and high sensitivity [1720]. Combining sensitive materials, optical fiber SPR sensor will occupy a larger position in the future development of environmental monitoring, application, and intelligent identification [12].

Over time, the researches of the optical fiber SPR sensors in the field of heavy metal ions detection become more and more in-depth and advanced. In 2014, R. Verma et al. used the metallic silver and indium tin oxide to prepare SPR probes, and coated the composite material of pyrrole and chitosan on the probe surface for the detection of metal ions [21]. The indium tin oxide on the surface of the sensor had the effect of protecting the silver layer from oxidation, while its effectiveness was limited, and the preparation of the composite material was complicated. In 2016, D. R. Raj et al. prepared an optical fiber SPR sensor by using gold nanoparticle-polyvinyl alcohol (PVA) heterojunction as the sensing material, and realized the detection of mercury ions with a detection limit of 1 × 10−6 M [22]. However, the hybrid results of gold nanoparticle and PVA were full of uncertainty, and the linear relationship of mercury ions detection was unstable in the high concentration range. In 2019, M. Pesavento et al. developed and tested a selective sensor for detecting the copper ions in drinking water. The sensor was composed of D-type plastic fiber and gold-penicillamine composite layer, which can realize the detection of copper ions in the concentration range of 4 × 10−6 M to 2 × 10−4 M [23]. Although the D-type plastic fiber enhanced the detection sensitivity, it also reduced the stability and mechanical strength of the sensor. In 2021, H. Z. Yuan et al. proposed a highly sensitive optical fiber SPR sensor, which detects mercury ions by using thymine-modified gold nanoparticles as signal amplification markers. The detection limit of the sensor for mercury ions can reach 9.98 nM [24], while the sensor was sensitive to various metal ions. Therefore, the environmental monitoring field urgently needs to design an excellent optical fiber SPR sensor with stable structure, simple fabrication, and high sensitivity for heavy metal ions detection.

In this work, we proposed a Cu2+-imprinted optical fiber SPR sensor for the specific detection of copper ions. The gold layer of the sensor will excite SPR to enhance the sensing sensitivity of inter-mode interference. The chitosan was deposited on the surface of the gold layer as a sensitive layer for copper ions sensing. The refractive index (RI) of chitosan film changes with ions adsorption, and the detection of metal ion concentration can be achieved indirectly through monitoring the change of interference wavelength. The selective adsorption sensitization of chitosan film to copper ions was further improved by introducing ions imprinting technology. It is theoretically and experimentally proved that the Cu2+-imprinted optical fiber SPR sensor has higher sensitivity and selectivity for copper ions detection.

2. Sensor fabrication, sensing principle, and simulation

The sensor designed in this paper is based on the surface plasmon resonance (SPR) and interference principle. The SPR is a phenomenon in which shear wave called surface plasmon polaritons is excited by incident light at a specific angle in the form of evanescent wave (EW) [2528]. The surface plasma wave (SPW) is stimulated at the interface. When the resonance matching condition is satisfied, partial energy conversion will occur between EW and SPW, which reduces the energy of reflected light and causes resonance absorption of the transmission spectrum. When the external RI changes, the resonance wavelength will move again to meet the new matching conditions [2931]. Therefore, the change of external RI can be obtained by the wavelength shift of the transmission spectrum, realizing the high sensitivity detection of the medium parameters to be measured on the metal surface [3235].

For the optical fiber SPR sensor, when the core mode and cladding mode of the fiber exist at the same time, due to the RI difference, they will interfere in the transmission process of the fiber, resulting in the fiber Mach-Zehnder interference. Since the effective depth of EW is greater than the thickness of the metal layer (generally about 50 nm), the EW produced by Mach-Zehnder interference still exists at the junction between the metal layer and the medium. The evanescent wave vector (Kz) and SPW wave vector (KSPW) in the optical fiber cladding along the interface Z-direction (radial direction) are respectively [17]

$${k_z} = \frac{\omega }{c}\sqrt {{\varepsilon _{cladding}}} \sin {\theta _p}$$
$${k_{SPW}} = {\textrm{Re}} \left[ {\frac{\omega }{c}\sqrt {\frac{{{\varepsilon_m}{\varepsilon_s}}}{{{\varepsilon_m} + {\varepsilon_s}}}} } \right]$$
where ω is the angular frequency of the light wave; c is the speed of light in the vacuum; εcladding is the dielectric constant of the optical fiber; θp is the incidence angle; εm and εs are the dielectric constants of the metal film and measured medium, respectively.

When the matching condition is satisfied, KZ = KSPW, the resonance absorption occurs between SPW and EW, and the SPR is generated. At this time, the incident angle θ is called resonance angle θP, and sinθP=ƒ(ω, εcladding, εm, εs) [3]. Owing to the corresponding conversion relationship between different parameters, the dielectric constant ε can be replaced by the RI of the sample n, and the angular frequency ω can be replaced by the wavelength of the incident light λ. Therefore, the relation can be transformed into θP=ƒ(λ, ncladding, nm, ns), where ncladding, nm, and ns are the RIs of the optical fiber, metal film, and measured medium, respectively. Since the material of the optical fiber is not magnetic, the relationship between the dielectric constant and RI of the optical fiber can be expressed as [20]

$${\varepsilon _{cladding}} = {({n_{cladding}})^2}$$

The resonance angle θP can be defined as

$${\theta _p} = \arcsin \sqrt {\frac{{{\varepsilon _m}{\varepsilon _s}}}{{({\varepsilon _m} + {\varepsilon _s}){\varepsilon _{cladding}}}}}$$

The value of θp is different at different wavelengths. In the optical fiber structure designed in this paper, namely the optical fiber SPR structure composed of single-mode fiber (SMF), multi-mode fiber (MMF), thin-core fiber (TCF), and gold (Au) film, the incidence angle can be calculated according to the dielectric constant and RI at different wavelengths. According to above theories and equations, the critical angle θp of 1525 nm, 1550 nm, and 1615 nm in different environments were calculated. The calculation results are shown in Table 1, and the angle calculation results prove that the designed structure does have the SPR effect [17,28].

Tables Icon

Table 1. Calculation results of angle

Under certain conditions, when the light is transmitted by total reflection inside the optical fiber, the light will have a difference in incident direction. In the MMF-TCF-MMF structure, when the beam is expanded in the MMF, multiple rays with different incident directions will be generated inside the three-dimensional fiber. These rays will be transmitted to the TCF in the cladding through the broken line path, and continue to transmit rays of various angles through the mismatched core nodes. The incident light in the propagation path has all the requirements of generating plasma wave excitation for a specific wavelength. The incident angle and θp meet the matching conditions, namely the theoretical resonance matching conditions. On the other hand, from the perspective of total reflection angle, when the total reflection propagates at the interface of TCF cladding and Au film, the total reflection angles in the wavelength range of 1525-1615 nm are calculated to be 20.62496° to 23.05669°. Compared with the angle results calculated in Table 1, θp is much larger than the total reflection angle, proving that there must be at least one angle matching θp among the generating angles of evanescent waves [2,3,7,17,20].

3. Simulation and fabrication

3.1 Optical fiber sensor

According to the principle of Mach-Zehnder interference, the optical fiber core mismatch structure can significantly change the transmission distribution of the beam without changing or destroying the fiber itself, which is stable and apt for sensor applications. Therefore, the MMF-TCF-MMF structure is selected as the sensor carrier in this paper, and the structure diagram is shown in Fig. 1.

 figure: Fig. 1.

Fig. 1. Structure diagram of optical fiber SPR sensor.

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The optical fiber sensing structures with different TCF lengths have been simulated, and the simulated optical fiber parameters were set to be consistent with the SMF (8.2/125 µm), MMF (60/125 µm), and TCF (5/125 µm). The optical field of the fiber sensing structure in the 3D state has been simulated, and the influence of TCF length on the distribution of evanescent field is shown in Fig. 2. It can be found that the optical field distribution in the cladding is optimal when the length of the fine core is set to 8 mm. The optical distribution of the cladding is sufficient, and the interference light can be coupled into the fiber core at the later stage of transmission to reduce transmission loss [19].

 figure: Fig. 2.

Fig. 2. Optical field distribution diagrams of optical fiber sensors with different TCF lengths (a) 6 mm, (b) 7 mm, (c) 8 mm, and (d) 9 mm.

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On this basis, sensors with different TCF lengths were fabricated according to the flow chart shown in Fig. 3 to verify the accuracy of the conjecture, and the interference spectra are illustrated in Fig. 4. By comparing the spectra of the sensors with different lengths of TCF, we can find that the optical fiber sensor with TCF length of 8 mm has better interference effect and spectral symmetry. Therefore, in this paper, the lengths of MMF1, TCF, and MMF2 are set to 5 mm, 8 mm, and 5 mm, respectively.

 figure: Fig. 3.

Fig. 3. Fabrication flow chart.

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 figure: Fig. 4.

Fig. 4. Spectra of optical fiber sensors with different TCF lengths.

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To intuitively observe the change process of the light field inside the structure, the light field distribution at 5 mm, 10 mm, 14 mm, 18 mm, and 23 mm in the 3D state are simulated, and the results are shown in Fig. 5. The light wave maintained the LP01 mode at 5 mm during the propagation of light. The beam expansion phenomenon occurred when the light wave was introduced into the MMF, and the high-order light mode was excited [19]. The light field distribution has changed significantly in the section of the 10 mm light field owing to the beam expansion effect and stimulated high-order light mode. At the connecting node of MMF and TCF, a significant core mismatch transmitted part of the light beam to the cladding of TCF, resulting in a strong evanescent field. From the cross-section distribution of the light field at 14 mm and 18 mm, we can find that the light wave mode diffused to varying degrees in the cladding, and the transverse field distribution was relatively complex. At 23 mm, the beam was coupled by the MMF to the core of the outgoing SMF, and the light intensity distribution in the fiber core was much larger than that in the cladding. It is concluded that the designed sensor with this structure has a better interference effect and transmission efficiency.

 figure: Fig. 5.

Fig. 5. Cross-section optical field distribution of optical fiber sensor with TCF length of 8 mm.

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3.2 Cu2+-imprinted optical fiber SPR sensor

The metals with high reflectivity and excellent chemical stability are usually selected as the metal film of SPR. In this paper, the gold (Au) was chosen as the excitation layer of SPR. The optical fiber SPR sensor was fabricated by sputtering a layer of 50 nm gold film on the surface of the sensor. The contrast diagram of interference spectrum before and after gold plating is shown in Fig. 6. According to the principle of SPR, the phase difference of the absorption spectrum of SPR is smaller than that of the interference spectrum. The spectral comparison diagram illustrates the existence of the SPR phenomenon in the optical fiber SPR sensor. As shown in Fig. 6, the spectrum of the optical fiber SPR sensor shows an obvious blue shift compared with that of the optical fiber sensor without Au film, indicating that the presence of Au film stimulates the SPR effect of the sensor.

 figure: Fig. 6.

Fig. 6. Comparison of interference spectra before and after sputtering Au film.

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Due to the poor sensitivity of bare optical fiber SPR sensors to trace copper ions in drinking water, it is necessary to combine sensitive materials to amplify the sensitivity of the sensor to copper ions (Cu2+). In this paper, the chitosan, which is not readily soluble in water, stable performance, and strong adsorption [8], is chosen as the sensitive material of the sensor surface. Since the ions imprinting technology can improve the selectivity of polymer to a great extent [13,33], this paper used ions imprinting technology to treat chitosan further to prepare Cu2+-imprinted chitosan film. Ions imprinting technology imprints copper ions on chitosan film, and ions form pores in the film with a similar size and shape to copper ions after elution, achieving the adsorption limitation of other ions except for copper ions [13,33]. Theoretically, this would make Cu2+-imprinted chitosan films insensitive to other heavy metal ions.

The 2 g chitosan powder was dissolved in 100 ml of 4% acetic acid solution and thoroughly stirred until completely dissolved. The quantitative solution containing Cu2+ was added to the solution according to the concentration ratio, and the Cu2+ was imprinted into chitosan as template ions after fully stirring. Then epichlorohydrin solution with 50% concentration was added for cross-linking polymerization, and continued to be thoroughly stirred to ensure the reaction was completed. To complete the experiment, the standard solutions of Cu2+, chromium ions (Cr3+), mercury ions (Hg2+), cadmium ions (Cd2+), and lead ions (Pb2+) with different concentrations were prepared, and the PH value was adjusted to the same value. Meanwhile, a certain concentration of the hydrochloric acid solution and Ethylene Diamine Tetraacetic Acid (EDTA) eluent were prepared for reserve.

The deionized water and alcohol were used to clean the surface of the optical fiber SPR sensor. The proposed sensor was coated with the Cu2+-imprinted chitosan film through the rotary impregnation method, and then it was dried at 60 °C for 2 hours in a vacuum drying oven. A 1 µm Cu2+-imprinted chitosan film was coated on the surface of the sensor. This is followed by rinsing and soaking with deionized water to remove unstable chitosan molecules and eliminate the water swelling effect of chitosan. A stable Cu2+-imprinted chitosan film was formed on the surface of the sensor after 24 hours. It is necessary to remove template ions in Cu2+-imprinted chitosan and eliminate the influence of anion in solution before use (the anion in this paper is Cl-). The spectra of optical fiber SPR sensor in the process of chitosan treating are shown in Fig. 7.

 figure: Fig. 7.

Fig. 7. Spectra of optical fiber SPR sensor in the process of chitosan treating.

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4. Experiments and discussion

Figure 8 shows the Cu2+ concentration sensing system. The sensor is placed in the metal ions solution, and connected with amplified spontaneous emission (ASE) source and optical spectrum analyzer (OSA). The concentration of metal ions in the surrounding environment of the sensor can be adjusted by regulating the concentration of metal ions solution.

 figure: Fig. 8.

Fig. 8. Schematic diagram of Cu2+ concentration sensing system.

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The chitosan is not only sensitive to copper ions but also has varying degrees of sensitivity to other heavy metal ions. The original intention of the sensor design is to achieve high sensitivity and intelligent detection of copper ions. To improve the selectivity of the sensor, the ions imprinting technology was introduced into the treat chitosan to prepare Cu2+-imprinted chitosan film. The sensing performance of the Cu2+-imprinted optical fiber SPR sensor was investigated, and the concentration range of metal ions solution ranged was 0-270 ppm in steps of 30 ppm. The sensing performance of the sensor is illustrated in Fig. 9. We can observe that the transmission spectrum of the sensor exhibits a blue shift as the concentration of the copper ions solution increases.

 figure: Fig. 9.

Fig. 9. Spectra of Cu2+-imprinted chitosan coated optical fiber SPR sensor for Cu2+ concentration detection.

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The variation of dips in the characteristic spectrum of the sensor versus the change of copper ion concentration are shown in Fig. 10. Owing to the unique adsorption characteristics of chitosan on ions [8], the spectrum of the sensor shifts with the change of Cu2+ concentration in the range of 0-270 ppm. The sensor reaches the saturation state after a certain concentration of adsorption, and the wavelength remains at a stable value.

 figure: Fig. 10.

Fig. 10. Characteristic spectra of Cu2+-imprinted optical fiber SPR sensor versus Cu2+ concentration (a) dip1, (b) dip2, (c) dip3, and (d) dip4.

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The linear detection range of dips in the transmission spectrum of the sensor was analyzed, and the linear detection range was 0-150 ppm. The relationship between spectral wavelength shift and Cu2+ concentration is shown in Fig. 11. We can find that the proposed sensor is sensitive to copper ions, and exhibits a detection sensitivity of -10.05 pm/ppm (dip3) for copper ions.

 figure: Fig. 11.

Fig. 11. Relationship between spectral wavelength shift and Cu2+ concentration in low Cu2+ concentration range (a) dip1, (b) dip2, (c) dip3, and (d) dip4.

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The detection experiment of copper ions standard solution proves that the proposed sensor is sensitive to copper ions. The detection experiments of other metal ions standard solutions were carried out to verify whether the proposed sensor is sensitive to other metal ions except for copper ions. In contrast experiments, in addition to changing the type and concentration of the ionic solution, other parameters in the experiment kept the same set value, such as hydraulic pressure, temperature, PH value, and so on. The characteristic spectra of the sensor in the detection of different ions standard solution are illustrated in Fig. 12. Compared with the initial wavelength difference of Cu2+, the wavelength shift of Cr3+, Hg2+, Cd2+, and Pb2+ are almost invisible within the same detection range. This indicates that the proposed sensor can not only realize the limitation of other metal ions except for copper ions in theory but also realize the design purpose in experimental detection.

 figure: Fig. 12.

Fig. 12. Characteristic spectra of Cu2+-imprinted optical fiber SPR sensor in the detection of different ions standard solution.

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The ultimate goal of sensor design is to realize the specific recognition and detection of copper ions. Therefore, it is necessary to create an experimental environment similar to the daily water environment for specific detection of the sensor. In this work, the solutions of Cr3+, Hg2+, Cd2+, and Pb2+ were mixed with different concentrations of the test solution to create a complex detection environment for the sensor, exploring the specific recognition and detection ability of the sensor for copper ions in a complex environment. Similarly, in this experiment, only the ionic concentration parameters need to be changed while the other parameters remain unchanged. The relationship between spectral wavelength shift and concentration of mixed ionic solution is shown in Fig. 13. By comparing the data of different detection environments, it can be found that the detection sensitivity are similar in Fig. 11 and Fig. 13. The detection of mixed ions is similar to that of copper ions standard solution, which proves that the sensor has realized the specific recognition and detection of copper ions in the complex environment.

 figure: Fig. 13.

Fig. 13. Relationship between spectral wavelength shift and concentration of mixed ionic solution (a) dip1, (b) dip2, (c) dip3, and (d) dip4.

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To explore the influence of ions imprinting technology on the sensing performance of the sensor, the comparative experiments were investigated. Figure 14 illustrates the comparison of Sensor1 (Detection of copper ions concentration by chitosan coated optical fiber SPR sensor), Sensor2 (Detection of copper ions concentration by Cu2+-imprinted chitosan coated optical fiber SPR sensor), and Sensor3 (Detection of mixed ions concentration by Cu2+-imprinted chitosan coated optical fiber SPR sensor). We can find that the Cu2+-imprinted chitosan coated optical fiber SPR sensor has an excellent improvement in both sensitivity and adsorption capacity. The addition of ions imprinting technology has improved the sensitivity and selectivity of the chitosan coated optical fiber SPR sensor to copper ions.

 figure: Fig. 14.

Fig. 14. Relationship between wavelength shift of different sensors and solution concentration.

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To maintain the accuracy of the sensing performance, all experiments were carried out on the basis of using the same sensor, and the EDTA eluent and hydrochloric acid were used to restore the sensor to the original state before each experiment. The elution effect and repeatability of the sensor were investigated by multiple elution experiments, and the results are shown in Fig. 15. In the multiple elution process, the wavelength of the sensor spectrum had no significant shift, only a slight intensity change, which proved the excellent repeatability of the sensor. The proposed sensor has excellent potential for practical application in the future.

 figure: Fig. 15.

Fig. 15. Numerical fluctuation diagrams of sensor spectrum in the elution experiments.

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5. Conclusion

In conclusion, we proposed and demonstrated a Cu2+-imprinted optical fiber SPR sensor to realize the intelligent detection of copper ions in the complex environment. The proposed sensor introduced ions imprinting technology to imprint copper ions onto chitosan film, which significantly improved the adsorption capacity, selectivity, and sensitivity of the sensor to copper ions. The experimental results show that the sensor has high sensitivity, specificity, and excellent repeatability for copper ions, and exhibits a detection sensitivity of -10.05 pm/ppm. Owing to its excellent properties, the Cu2+-imprinted optical fiber SPR sensor has exhibited potential applications in the field of copper ions detection. Besides, the proposed sensor provides a new idea for metal ions detection.

Funding

National Key Research and Development Program of China (2019YFC1804802); Natural Science Foundation of Heilongjiang Province (YQ2019A004); National Science Foundation for Post-doctoral Scientists of Heilongjiang Province (LBH-Z19070).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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35. T. Li and W. L. Feng, “Fiber-Optic Surface Plasmon Resonance Sensor for Trace Cadmium-Ion Detection Based on Ag-PVA/TiO2 Sensing Membrane,” IEEE Sensors J. 21(17), 18650–18655 (2021). [CrossRef]  

Data availability

No data were generated or analyzed in the presented research.

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Figures (15)

Fig. 1.
Fig. 1. Structure diagram of optical fiber SPR sensor.
Fig. 2.
Fig. 2. Optical field distribution diagrams of optical fiber sensors with different TCF lengths (a) 6 mm, (b) 7 mm, (c) 8 mm, and (d) 9 mm.
Fig. 3.
Fig. 3. Fabrication flow chart.
Fig. 4.
Fig. 4. Spectra of optical fiber sensors with different TCF lengths.
Fig. 5.
Fig. 5. Cross-section optical field distribution of optical fiber sensor with TCF length of 8 mm.
Fig. 6.
Fig. 6. Comparison of interference spectra before and after sputtering Au film.
Fig. 7.
Fig. 7. Spectra of optical fiber SPR sensor in the process of chitosan treating.
Fig. 8.
Fig. 8. Schematic diagram of Cu2+ concentration sensing system.
Fig. 9.
Fig. 9. Spectra of Cu2+-imprinted chitosan coated optical fiber SPR sensor for Cu2+ concentration detection.
Fig. 10.
Fig. 10. Characteristic spectra of Cu2+-imprinted optical fiber SPR sensor versus Cu2+ concentration (a) dip1, (b) dip2, (c) dip3, and (d) dip4.
Fig. 11.
Fig. 11. Relationship between spectral wavelength shift and Cu2+ concentration in low Cu2+ concentration range (a) dip1, (b) dip2, (c) dip3, and (d) dip4.
Fig. 12.
Fig. 12. Characteristic spectra of Cu2+-imprinted optical fiber SPR sensor in the detection of different ions standard solution.
Fig. 13.
Fig. 13. Relationship between spectral wavelength shift and concentration of mixed ionic solution (a) dip1, (b) dip2, (c) dip3, and (d) dip4.
Fig. 14.
Fig. 14. Relationship between wavelength shift of different sensors and solution concentration.
Fig. 15.
Fig. 15. Numerical fluctuation diagrams of sensor spectrum in the elution experiments.

Tables (1)

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Table 1. Calculation results of angle

Equations (4)

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k z = ω c ε c l a d d i n g sin θ p
k S P W = Re [ ω c ε m ε s ε m + ε s ]
ε c l a d d i n g = ( n c l a d d i n g ) 2
θ p = arcsin ε m ε s ( ε m + ε s ) ε c l a d d i n g
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