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Combination of an optical waveguide platform and ultra-thin spectrometer that enables increased surface plasmon resonance sensor compactness

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Abstract

A novel integrated surface plasmon resonance (SPR) sensor that combines an optical waveguide platform and an ultra-thin spectrometer is proposed. The core of the proposed method is a special-shaped optical waveguide structure that employs a wedge-shaped incident surface, which changes the position of the total reflection of the incident light on the sagittal plane without affecting the direction of propagation on the tangential plane. The parameters of the sensing module with the integrated SPR sensor and spectrometer module were designed and optimized to achieve higher performance in a compact optical waveguide platform. An experimental system was built based on the theoretical model, and the spectral sensitivity of the system was analyzed before sample detection, and the results showed that the spectral resolution in the working range could reach 9.9 nm. The refractive index sensitivity of this novel SPR sensor was 3186 nm/RIU with good stability by detecting different concentrations of sodium chloride samples. This new structure does not require an external spectrometer, thereby enabling an increase in the compactness of the SPR sensing system. The proposed method can provide a novel idea for the miniaturization of SPR sensors.

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

1. Introduction

Surface plasmon resonance (SPR) sensing technology is an important branch in the field of optical biosensors and has the advantages of high sensitivity, strong specificity, fast detection speed, and label-free detection. The role of SPR technology in biosensors has received increasing attention across several disciplines in recent years. It has been widely used in drug and biological preparation development, life science, medical research, environmental monitoring, and food safety evaluation. Traditional SPR sensors are well developed, stable, and have high detection accuracy; however, the system is bulky, expensive, and has poor integration, which cannot meet the needs of current development. In the last two decades, researchers have worked extensively on realizing the miniaturization of SPR sensors. The coupling structure that excites plasma resonance can be scaled down in essence; with the development of 3D printing technology and Micro-Electro-Mechanical System (MEMS) technology, it is completely possible to produce millimeter-level or even micro-nano level sensor devices from the process, such as the dove prism type portable SPR instrument for multi-channel detection [1,2], the SPR sensing chip with a hemispherical prism formed by adding a liquid optical adhesive (NOA61) drop on a PDMS cylindrical substrate [3], and the integration of a white organic light-emitting diode (OLED) on the leg side of an isosceles trapezoid [4]. In addition, the optical fiber structure, which has the advantage of being small in size, can be developed into different types, such as the D-type [5], H-shaped [6], and tapered fibers [7].

The above studies have focused on the miniaturization of the sensing probe, which still requires an external spectral detection device, such as a spectrometer after the sensor element. However, SPR technology can also be integrated with optical technology to eliminate the room for a spectrometer and thus build a compact sensing system. A representative commercial integrated SPR sensor, SPREETA, was developed by Jose Melendez et al. from Texas Instruments in 1996 [8]. In 2007, Chinowsky et al. developed a box-sized SPR semi-automatic detection system with 24 independent analysis areas based on the SPREETA sensing chip, with an external size of 28 cm × 22 cm × 13 cm and a refractive index resolution of 1–3 × 10−6RIU, which was the first integrated portable SPR sensor [9]. This commercial portable SPR sensor is angle-modulated, whereas the wavelength-modulated sensor does not require mechanical control for angle adjustment. Smartphones have grown rapidly over the past decade with the development of internet technology. With LED light sources and cameras, as well as the ability to develop adapted applications, smartphones can be integrated with SPR technology, providing a suitable platform for the portability of SPR sensors. Pakorn et al. first proposed an angle-resolved SPR detection system using screen illumination and a smartphone-based front camera [10]. Subsequently, mature SPR sensing parts have been integrated with smartphones, such as 3D-printed unibody microfluidic coupled SPR chips and optical fiber sensor structures, using silicon dioxide to excite long-range surface plasmons [11,12]. In 2006, Olga Telezhnikova and Ji1í Homola proposed a new structure (called surface plasmon resonance coupler and disperser, SPRCD) for the simultaneous excitation of surface plasmons and dispersion of light, which allows the direct analysis of SPR spectroscopy without the use of a spectrometer, thus significantly reducing the size of the sensor space [13]. In 2009, a high-performance SPR sensor based on the SPRCD structure with wavelength modulation was developed, which included four independent detection channels and a circuit device in addition to the optical part, with a footprint of less than 15 cm × 15 cm and a refractive index resolution of 3 × 10−7RIU [14]. Another high-performance compact SPR sensor was developed based on the angular interrogation of SPR on a gold-coated diffraction grating contained, which has a size of 29 cm × 22 cm × 12 cm and can detect 10 samples simultaneously; the refractive index resolution of the sensor is 6 × 10−7RIU after several experimental tests [15]. A variety of integrated SPR sensors are also available with gratings as key components, including a compact imaging spectroscopy system for high-throughput screening of biomolecular interactions [16], a centrifugal microfluidic Lab-on-a-Disc system for 5-fold multiplexed SPR-detection of IgG [17], and a grating-coupled SPR smartphone spectrometer for the detection of lipopolysaccharides [18].

In this study, we present a novel integrated SPR sensor by combining an optical waveguide platform and an ultra-thin spectrometer with our unique design, which employs a wedge-shaped incident surface that changes the position of the total reflection of the incident light on the sagittal plane without affecting the direction of propagation on the tangential plane. This method enables increased SPR sensor compactness and offers a new perspective for portable applications. The design and fabrication of the sensor are described, and its detection performance is represented by the refractive-index resolution.

2. Model and theory

2.1 Principle of operation

The proposed sensor is based on the wavelength interrogation of the SPR on the optical waveguide, as shown in Fig. 1. Broadband light waves from the light source are passed through the front optical system (collimating lens and polarizer) to obtain a collimated p-polarized beam, which is coupled into the optical waveguide platform through the wedge endface and transmitted forward in the form of total reflection within the optical waveguide platform. When the beam reaches the sensing area, the waveguide, gold film, and sample constitute a three-layer Kretschmann structure to excite the SPR phenomenon. The SPR reflected beam carrying the sample information continues to reach the grating in the form of total reflection, and the light of different wavelengths is diffracted in different directions and finally converges to the detector through the focusing mirror to obtain the SPR spectrum.

 figure: Fig. 1.

Fig. 1. Schematic of the proposed novel integrated SPR sensor by combining an optical waveguide platform and ultra-thin spectrometer.

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2.2 SPR sensing module

The SPR excitation model of the waveguide with a wedge endface structure is shown in Fig. 2(a), in which the light source and the optical waveguide platform are located on the same horizontal line. The wedge endface has two roles: (i) coupling the optical beams into the waveguide and (ii) adjusting the total reflection angle (θ) of the incident light in the waveguide by changing the wedge endface angles (α). The incident light undergoes attenuated total reflection at the interface between the medium and metal film, resulting in evanescent waves. The free electric charges on the metal surface collectively oscillate under external excitation, thus forming surface plasma waves (SPW) at the interface between the metal and sample. When the evanescent wave and the surface plasma wave meet the phase matching, part of the incident light energy is absorbed, resulting in a significant decrease in the intensity of the reflected light at a wavelength called the resonance wavelength (λSPR). According to Maxwell's equation and boundary conditions, the wave vector component kx and surface plasma wave vector kspw of the incident p-polarized light in the x-direction are

$${k_x} = \frac{{2\pi }}{\lambda }\sqrt {{\varepsilon _1}(\lambda )\sin \theta }$$
$${k_{spw}} = \frac{{2\pi }}{\lambda }\sqrt {\frac{{{\varepsilon _2}(\lambda ){\varepsilon _3}(\lambda )}}{{{\varepsilon _2}(\lambda ) + {\varepsilon _3}(\lambda )}}}$$
where λ is the incident light wavelength; θ is the incident angle at the interface between the waveguide dielectric and the metal film; ε1, ε2, and ε3 are the dielectric constants of the waveguide dielectric, metal film, and sample, respectively.

 figure: Fig. 2.

Fig. 2. (a) Schematic of the SPR excitation model of waveguide with wedge endface structure; (b) Cross-section plot of the total electric field along the direction perpendicular to the fused silica base; (c) Spectra of the proposed SPR sensor with varied external RIs, 60° wedge endface angle and gold thickness of 50 nm; (d) Relationship between the external RI and resonance wavelength at various wedge endface angles (α); (e) Minimum effective length and number of reflections with increasing thickness of waveguide.

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According to the multiple reflection theory and Fresnel principle, the optical reflection coefficient of p-polarized light can be expressed as follows:

$$R = r_p^N = {|{{r_{123}}} |^{2N}} = {\left|{\frac{{{r_{12}} + {r_{23}}{e^{2i{k_{z1}}{t_2}}}}}{{1 + {r_{12}}{r_{23}}{e^{2i{k_{z1}}{t_2}}}}}} \right|^{2N}}$$
$${r_{mn}} = \frac{{{\varepsilon _m}{k_{zn}} - {\varepsilon _n}{k_{zm}}}}{{{\varepsilon _m}{k_{zn}} + {\varepsilon _n}{k_{zm}}}},m = 1,2,n = 2,3$$
$${k_{zi}} = \sqrt {{{\left( {\frac{{2\pi }}{\lambda }} \right)}^2}{\varepsilon _i} - k_x^2} ,i = 1,2,3$$

Here, t2 is thickness of the Au film, N represents the number of reflections of light over the sensing area and Lmin represents the minimum effective length required for one reflection of light in the waveguide.

$$N = \frac{l}{{{L_{\min }}}}$$
$${L_{\min }} = D\tan \theta$$
where l represents the length of the sensing region, D represents the waveguide thickness, and the incidence angle (θ) at the interface between the waveguide medium and the metal film is given by
$$\theta = \alpha + {\sin ^{ - 1}}\left[ {\frac{{n0}}{{n1}}\sin ({{{90}^ \circ } - \alpha } )} \right]$$
where n0 and n1 represent the refractive indices of air and waveguide media, respectively.

In this study, the transmission characteristics of wedge-waveguide SPR sensors were investigated using a finite element analysis software. The metal layer was a gold film of 50 nm thickness, its dielectric constant was set to the experimental data from RG et al [19]. The waveguide medium is fused silica, and its Sellmeier dispersion model is given by

$${n_1}(\lambda ) = \sqrt {1 + \frac{{{A_1}{\lambda ^2}}}{{{\lambda ^2} - B_1^2}} + \frac{{{A_2}{\lambda ^2}}}{{{\lambda ^2} - B_2^2}} + \frac{{{A_3}{\lambda ^2}}}{{{\lambda ^2} - B_3^2}}}$$
where λ is the wavelength in µm, and A1, A2, A3, B1, B2, and B3 are the Sellmeier coefficients. The values of these coefficients are A1 = 0.696 1663, A2 = 0.407 9426, A3 = 0.897 4794, B1 = 0.068 4043, B2 = 0.116 2414, and B3 = 9.896 161.

We used finite element method (FEA) software to simulate the two-dimensional structure of the above model with the simulated structure of Fused silica/Au(50nm)/Sample layer. For this model, the Floquent periodicity periodic boundary conditions and periodic port conditions were applied, the extremely fine physics-controlled sized mapped mesh using triangle mesh. Meanwhile, to perform the wavelength interrogation technique, we varied the wavelength was simulated for 500 to 900 nm with 1 nm incremental deviation. By setting different θ and thus analyzing the relationship between the sample RI and resonance wavelength at various α. When the wedge endface angle is set to 60°, the corresponding θ is 80.08°, and the sample RI is 1.333, the electric field intensity distribution at the resonance wavelength of 643 nm is shown in the upper right corner of Fig. 2(b). Cross-section plot of the total electric field along the direction perpendicular to the fused silica base is shown in Fig. 2(b), the calculated penetration depth (PD) is 182nm. PD is defined as the distance from the interface at which the amplitude of the field decreases by a factor of 1/e [20].

The change in the SPR spectrum when the wedge endface angle is 60° and the refractive index of the sample changes from 1.333 to 1.379 is shown in Fig. 2(c). To obtain the appropriate wedge end angle, the change in the refractive index of the sensor for different samples with wedge angles of 50, 55, 60, 65, and 70° was simulated. The relationship between the external RI and resonant wavelength at various wedge end-face angles is illustrated in Fig. 2(d). From the results in the figure, we can observe that the sensor with a 50° wedge angle has the best sensitivity; however, the resonance wavelength is beyond the working wavelength range. The sensitivity of the sensor was similar for wedge angles of 55° and 60°. In Fig. 2(e), the blue color represents the relationship between the waveguide thickness D and the minimum effective length Lmin, and the red color represents the relationship between the waveguide thickness D and the number of reflections over the length l of the sensing area.

2.3 Spectrometer module

According to Section 2.1, the operating wavelength range of the sensor was 500–900 nm. Considering the angular dispersion of the grating and the compact space structure, a 600 l/mm planar diffraction grating was selected. To further reduce the processing difficulty, the collimated light is directly incident on the grating, and the diffraction grating is flush with the outgoing end of the optical waveguide platform. Figure 3(a) shows the schematic diagram of the design parameters of the spectrometer module, according to the grating equation,

$$\textrm{d}\sin {\varphi _0} = m{\lambda _C}$$
where φ0 is the diffraction angle of the grating, d is the grating constant, m is the diffraction order, and λC is the central wavelength of the working band. By substituting λC = 700 nm into Eq. (10), a diffraction angle φ0 = 24.78° can be obtained. The beam is re-incident to the optical waveguide platform after diffraction through the diffraction grating, and according to the law of refraction
$${\varphi _1} = \arcsin (\frac{{\sin {\varphi _0}}}{{{n_{01}}}})$$
where n01 is the refractive index of the waveguide dielectric relative to air, and the refracting angle φ1 = 16.7° at the incident waveguide endface can be obtained by substituting into the calculation. According to the geometric relationship, the inclination angle of the focusing mirror φ2 = 8.35°. The focal length of the focusing mirror is related to the sensing length of the selected line-array detector. In this experiment, the Sony ILX554B line array detector was used, which is a commonly applied detector. The detector's sensing length is 28.6 mm, and the focal length f of the focusing mirror is designed to be 70 mm to retain a certain design margin, such that the actual image plane length lD is 12.2 mm. These calculated parameters were input into the optical simulation software, and the optimized optical path is shown in Fig. 3(b). To evaluate the resolution of the designed ultra-thin spectrometer, 500, 700 and 900 nm were selected as the feature wavelengths, and 503, 703 and 897 nm adjacent to 3 nm were set as the references, respectively, and a large number of light rays were set for simulation of the optical system. As shown in Fig. 3(c) for the simulation in feature wavelengths, it can be seen that the two wavelengths of each group of feature wavelengths can be distinguished, and the resolution of the designed optical structure is better than 3 nm in the full wavelength range.

 figure: Fig. 3.

Fig. 3. (a) Schematic diagram of the parameters of spectrometer structure; (b) Optimized optical path; (c) Simulation in feature wavelengths.

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3. Experimental section

3.1 Experimental setup

A self-designed integrated SPR system based on a planar optical waveguide was constructed, and its structure is shown in Fig. 4, the size of optical waveguide platform is 67.3 mm × 40.0 mm × 2.5 mm. A halogen lamp (Zolix Instruments Co., Ltd.) was used as the light source. A fiber collimator (SMA905) and polarizer (GCL-050003) were purchased from Daheng Optics (Beijing, China). The large grating was cut using a nanosecond green laser to obtain a grating of the required size. The line array image sensor was selected from Sony ILX554B and contained 2048 pixels. Fused silica glass (JGS2, nD = 1.458) was chosen as the material for the optical waveguide platform, and the preparation process of the structure was simple and reproducible. First, the glass was cut using a picosecond laser-cutting machine to obtain a shaped glass sheet. To reduce the low reflectivity of the endface, which is caused by diffuse reflection, the glass sheet needs to be polished using the chemical mechanical polishing method, where the column surface of the focusing mirror is polished with a customized fixture. Finally, vacuum vapor deposition was used to apply a silver film as the reflective film at the location of the focusing mirror of the optical waveguide platform. Because the gold film is easy to peel off, direct deposition on the waveguide surface reduces the repeatability. In this study, a 50 nm thick gold film was deposited on the surface of a slide (JGS2, nD = 1.458) with a size of 15 mm × 15 mm × 1 mm using an evaporative coating instrument (Jusheng vacuum, DM250), and the slide with the gold film was coupled at the waveguide SPR sensing module position using refractive index matching fluids (nD = 1.458, Cargile).

 figure: Fig. 4.

Fig. 4. Experimental system of the integrated SPR sensor by combining an optical waveguide platform and ultra-thin spectrometer

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3.2 Samples and methods

The sensitivity, operating range, and refractive index resolution of the sensor were determined by detecting sample solutions with different refractive indices. In this study, a set of sample solutions with different refractive indices was obtained by preparing different concentrations of sodium chloride (Xilong Scientific Co., Ltd.) in deionized water, and their refractive indices were characterized using an Abbe refractometer (2WAJ, Shanghai Optical Instrument No. 6 Factory, China).

Sodium chloride solutions of 5.0, 10.0, 15.0, 20.0, and 25.0% concentrations were configured according to the volume ratio, and the refractive indices corresponding to different concentrations of solutions were 1.342, 1.351, 1360, 1.370, and 1.379, respectively, as measured using an Abbe refractometer. In the experiment, the sample solution was added dropwise to the surface of the gold film using a disposable dropper. After each measurement of a group of concentration samples, the gold film was cleaned with deionized water. To improve the accuracy of the experimental results, each group of concentration samples was repeated five times, and the mean and standard deviation of the corresponding resonance wavelengths were calculated.

4. Results and discussions

4.1 Spectral resolution

Spectral resolution is an important performance parameter for a spectrometer, where FWHM (the full width half maximum) is generally used to evaluate the spectral resolution [21]. The wavelength calibration using monochromator converts the horizontal coordinate from pixel point to wavelength, so that the value of FWHM represents the spectral resolution. As shown in Fig. 5(a), the spectral resolution of the corresponding wavelength is calculated every 25 nm in the spectral range of 500-900 nm. It can be seen that the spectrum of sampling wavelength is relatively uniform and the resolution size is relatively consistent. However, due to the defects of processing and polishing accuracy, there is stray light in the spectrometer, which affects the spectral resolution. As shown in Fig. 5(b), among all sampling wavelengths, there is the best resolution of 9.9 nm at 900 nm, and the worst resolution of 12.8 nm at 825 nm.

 figure: Fig. 5.

Fig. 5. (a) Normalized spectrum after wavelength calibration; (b) Spectral resolution at 500-900 nm.

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4.2 Stability and realtime measurements of the sensor

To test the sensor stability, we used deionized water (n = 1.333) as the sample and recorded the spectral data at an interval of 2 min. The eleven sets of measurement data and the analysis diagram of the stability are shown in Fig. 6(a), where the blue square in the figure indicates the shift of the resonance wavelength, and the red triangle indicates the change in the normalized transmitted intensity at the resonance wavelength.

 figure: Fig. 6.

Fig. 6. Results of stability testing (a) and realtime measurements(b).

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The measured data were processed, and the maximum fluctuation of the resonance wavelength was calculated to be approximately 0.491 nm. The mean value of the resonance wavelength was 679.993 nm, and the standard deviation was calculated to be 0.256. In addition, the normalized transmitted intensity at the resonance wavelength of each dataset was analyzed for a more comprehensive assessment of system stability. The maximum fluctuation of the normalized transmitted intensity was calculated as approximately 0.943%, the mean value of the normalized transmitted intensity was 48.623%, and the standard deviation was calculated as 0.273. Therefore, the experimental system exhibited good stability and repeatability.

Due to the limitations of the experimental conditions, we need to switch samples manually, and remove each sample after the test with blotting paper before performing another sample, so there is an empty period (no sample) on the surface of the gold film. At the same time, separate detection of each group of samples does not affect the realtime performance of the sensor. We performed realtime detection for samples with RI of 1.333,1.342, and 1.351, respectively, and the detection time of each group of samples was controlled at about 30s, and the time of sample switching was about 5s, and the results are shown in Fig. 6(b). The dashed line represents the empty period during sample switching, when there is no sample on the surface of the gold film and no resonance wavelength in the spectrum. In addition, there is a slight hysteresis phenomenon when each sample is switched, which is due to the residual little other liquid on the surface of the gold film.

4.3 Sensor response to refractive index

To investigate the refractive index sensitivity characteristics of the proposed integrated SPR sensor, the experimental test spectra corresponding to sample refractive indices of 1.333, 1.342, 1.351, 1.360, 1.370, and 1.379 are shown in Fig. 7(a), which shows that the response spectrum of the sensor is significantly red-shifted with an increase in the sample refractive index.

 figure: Fig. 7.

Fig. 7. (a) Refractive index measurement spectrum of the proposed SPR sensor; (b) Relation of resonance wavelength and the measured solution refractive index.

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The linear fitting curve of the resonance wavelength of the sensor and the refractive index of the sample to be measured can be obtained from Fig. 7(b); the fitted straight line shows that the resonance wavelength of the sensor has a good linear relationship with the refractive index of the sample to be measured, and the refractive index sensitivity of the sensor can be obtained as 3186 nm/RIU according to the slope of the expression. According to Homola's [22] definition of the refractive index resolution, the refractive index resolution of the sensor is 8.03 × 10−5RIU.

4.4 Discussions

In this paper, we focus on the integration of the SPR sensor structure and the practical fabrication of the proposed structure. Table 1 summarizes the condition of various integrated SPR sensors in terms of performance characteristics, size, and signal acquisition method. In this work, the sensor uses wavelength interrogation, which has performance limitations, but is more stable than angular interrogation because it does not require the use of mechanical parts. To the best of our knowledge, no report on spectrometer integrated SPR sensor is reported in literature. By the design of wedge-shaped endface, the sensing module and spectrometer module are fixed on an ultra-thin optical waveguide platform, which will provide a new structure for portable SPR sensors. At the same time, there is more room for development in sensor performance enhancement, such as using graphene and other 2D materials [23,24], because only one layer of gold is used in the sensing structure.

Tables Icon

Table 1. Summary of various integrated SPR sensor formats

5. Conclusion

We demonstrate a novel integrated SPR sensor, in which the combination of an optical waveguide platform and an ultra-thin spectrometer enables increased SPR sensor compactness. The thickness of this new structure is only 2.5 mm, and this structure allows incident light to excite SPs on the sagittal plane and obtain the diffraction spectrum on the tangential plane of the optical waveguide platform. It is designed and optimized in terms of the overall performance and size of both the sensing module and spectrometer module of the sensor. Finally, the sensor's experimental setup system was built, and its sensing performance was evaluated by testing samples with different refractive indices. The results showed that the spectral resolution in the working range could reach 9.9 nm and the RI sensitivity is 3186 nm/RIU with good stability. Based on this, better integration and higher performance will continue to be investigated in future work.

Disclosures

The authors declare no conflict of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)

Fig. 1.
Fig. 1. Schematic of the proposed novel integrated SPR sensor by combining an optical waveguide platform and ultra-thin spectrometer.
Fig. 2.
Fig. 2. (a) Schematic of the SPR excitation model of waveguide with wedge endface structure; (b) Cross-section plot of the total electric field along the direction perpendicular to the fused silica base; (c) Spectra of the proposed SPR sensor with varied external RIs, 60° wedge endface angle and gold thickness of 50 nm; (d) Relationship between the external RI and resonance wavelength at various wedge endface angles (α); (e) Minimum effective length and number of reflections with increasing thickness of waveguide.
Fig. 3.
Fig. 3. (a) Schematic diagram of the parameters of spectrometer structure; (b) Optimized optical path; (c) Simulation in feature wavelengths.
Fig. 4.
Fig. 4. Experimental system of the integrated SPR sensor by combining an optical waveguide platform and ultra-thin spectrometer
Fig. 5.
Fig. 5. (a) Normalized spectrum after wavelength calibration; (b) Spectral resolution at 500-900 nm.
Fig. 6.
Fig. 6. Results of stability testing (a) and realtime measurements(b).
Fig. 7.
Fig. 7. (a) Refractive index measurement spectrum of the proposed SPR sensor; (b) Relation of resonance wavelength and the measured solution refractive index.

Tables (1)

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Table 1. Summary of various integrated SPR sensor formats

Equations (11)

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k x = 2 π λ ε 1 ( λ ) sin θ
k s p w = 2 π λ ε 2 ( λ ) ε 3 ( λ ) ε 2 ( λ ) + ε 3 ( λ )
R = r p N = | r 123 | 2 N = | r 12 + r 23 e 2 i k z 1 t 2 1 + r 12 r 23 e 2 i k z 1 t 2 | 2 N
r m n = ε m k z n ε n k z m ε m k z n + ε n k z m , m = 1 , 2 , n = 2 , 3
k z i = ( 2 π λ ) 2 ε i k x 2 , i = 1 , 2 , 3
N = l L min
L min = D tan θ
θ = α + sin 1 [ n 0 n 1 sin ( 90 α ) ]
n 1 ( λ ) = 1 + A 1 λ 2 λ 2 B 1 2 + A 2 λ 2 λ 2 B 2 2 + A 3 λ 2 λ 2 B 3 2
d sin φ 0 = m λ C
φ 1 = arcsin ( sin φ 0 n 01 )
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