Refractive index sensing device based on titanium dioxide nanopore array and testing method thereof
By constructing a composite structure of gold and silver films on a titanium dioxide nanopore array, the problems of complex structure and high cost of traditional sensor devices are solved, realizing a refractive index sensor device with high sensitivity and high quality factor, which is suitable for food safety and environmental monitoring.
Patent Information
- Application Number
- CN202511120390.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-08-12
- Publication Date
- 2025-11-11
AI Technical Summary
Existing metal grating structures are complex and costly to manufacture, resulting in low sensor sensitivity and quality factor.
A composite structure is constructed by using a titanium dioxide nanopore array as a substrate, with a gold film layer covering the sidewalls and bottom of the nanopores, a silver film layer covering the upper surface of the titanium dioxide substrate, and a silicon dioxide protective layer covering the outer surface of the metal layer. The gold film excites the surface plasmon resonance effect, the silver film promotes the leakage of the light field, and the silicon dioxide protective layer provides protection.
It achieves synergistic optimization of high sensitivity and quality factor, reduces the amount of precious metals used, adapts to the detection needs of liquid and gas scenarios, has low cost and multi-scenario adaptability, and is suitable for food safety and environmental monitoring.
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Figure CN120927616A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of micro-nano optoelectronics technology, and in particular to a refractive index sensor based on a titanium dioxide nanopore array and its testing method. Background Technology
[0002] Optical refractive index sensors, with their advantages of high sensitivity, calibration-free operation, and real-time monitoring, have demonstrated significant application value in fields such as biomedical detection, environmental monitoring, and food safety. In the field of optoelectronics, refractive index sensing technology based on surface plasmon resonance (SPR) has become a research hotspot. SPR exhibits significant environmental sensitivity; even small fluctuations in the refractive index of the analyte can alter its coupling conditions, leading to a shift in the resonance peak. This characteristic perfectly matches the functional requirements of refractive index sensors. Sensors based on the SPR principle can detect a variety of analytes, including gases, liquids, and biofilms, and have broad application prospects.
[0003] Currently, surface plasmon resonance sensors mainly employ bound states in the continuous (BIC) metasurface structures or attenuated total internal reflection prism structures. Although these two types of structures can improve detection sensitivity, there is still room for optimization in terms of miniaturization integration and material innovation.
[0004] Plasma resonance sensors based on metal nanoparticles achieve sensing functionality by shifting the resonance peak due to the interaction between particles on the nanoparticle surface. For example, in 2022, Xiuhong L et al. theoretically verified a high-sensitivity refractive index sensor based on an integrated double resonant metagrating (DRMG). This structure consists of two layers of grating with supergrooves and a planar waveguide layer, achieving a simulated sensitivity of up to 930 nm / RIU. In 2024, Xiangxian Wang et al. proposed a plasma structure based on a two-dimensional grating of nano-bow-shaped particles, which can achieve refractive index sensing of different analytes in the same wavelength domain, with a liquid sensing sensitivity of 950 nm / RIU. Although the above studies have made some progress in the field of refractive index measurement, the related sensors generally suffer from problems such as complex structure, cumbersome fabrication process, and high cost. Moreover, from the perspective of sensing performance, their quality factor still needs to be improved.
[0005] The SPR effect on the surface of gold nanoparticles can generate a localized strong electromagnetic field, which enhances the Raman signal of molecules adsorbed on their surface by 10. 6 —10¹ 5Titanium dioxide exhibits advantages over other solid coating materials in nanoscale structure design, including high electron mobility, mature preparation processes, low raw material costs, and environmental friendliness. The unique surface plasmon resonance effect of gold, combined with the excellent processability of titanium dioxide, offers possibilities for cutting-edge applications such as high-performance refractive index sensing and highly sensitive detection of biomolecules. Therefore, a thorough analysis of the current research status in this field, both domestically and internationally, is of great significance for promoting the development of related technologies. Summary of the Invention
[0006] To address this issue, this invention provides a refractive index sensor based on a titanium dioxide nanopore array and its testing method, which solves the problems of complex metal grating structures, high manufacturing costs, and low sensor sensitivity and quality factor in the prior art.
[0007] To address the above problems, embodiments of the present invention provide a refractive index sensor based on a titanium dioxide nanopore array, the sensor comprising: A titanium dioxide substrate, wherein the surface of the titanium dioxide substrate is provided with a tetragonal periodically arranged array of nanopores; A gold film layer, wherein the gold film layer covers the sidewalls and bottom surface of the nanopore; A silver film layer is provided, which covers the upper surface of the titanium dioxide substrate and is connected to the gold film layer at the edge of the nanopore opening. A silicon dioxide protective layer is applied to the exposed surfaces of the gold and silver films, forming a protective enclosure around the metal layers.
[0008] Preferably, the period of the nanopore array is 900nm-1200nm.
[0009] Preferably, the diameter of the nanopore is 500nm-900nm.
[0010] Preferably, the depth of the nanopore is 600nm-1000nm.
[0011] Preferably, the thickness of the gold film on the sidewall is 60nm-150nm.
[0012] Preferably, the thickness of the gold film at the bottom of the hole is 60nm-150nm.
[0013] Preferably, the thickness of the silver film layer is 40nm-100nm.
[0014] Preferably, the thickness of the silicon dioxide protective layer is 100nm-500nm.
[0015] This invention also provides a testing method based on the refractive index sensor device of the titanium dioxide nanopore array described above, comprising the following steps: S1: Construct the sensor device; S2: A plane wave light source is incident perpendicularly on the surface to be measured of the sensor, and the horizontal direction is set as a periodic boundary condition, while the upper and lower surfaces are set as perfect absorption boundary conditions. S3: Obtain the reflection spectrum curves under different refractive indices by wavelength scanning; S4: Calculate the refractive index sensitivity based on the wavelength shift and refractive index change corresponding to the minimum value of the reflection spectrum curve; The formula for calculating the refractive index sensitivity is as follows: ; In the formula, Indicates refractive index sensitivity. Indicates the wavelength offset. It represents the change in refractive index.
[0016] Preferably, the quality factor is calculated using the full width at half maximum (FWHM) of the minimum reflectance spectrum curve. The calculation formula is as follows: ; In the formula, The quality factor is represented by FWHM, which stands for Full Width at Half Peak.
[0017] As can be seen from the above technical solutions, this invention application has the following beneficial effects: (1) Synergistic optimization of high sensitivity and quality factor: The composite structure designed in this invention achieves a breakthrough in sensing performance. The gold film covers the sidewalls and bottom of the nanopore, exciting the surface plasmon resonance effect, which concentrates the light field in the metal grating region and ensures the quality factor; the silver film modifies the upper surface of the nanopore, driving the light field to leak to the low refractive index analyte, and strengthening the refractive index coupling with the external medium. The synergistic effect of the two metal films makes the sensor more sensitive to changes in the refractive index of the medium, achieving high sensitivity of 830nm / RIU and 1001nm / RIU in liquid and gas scenarios, respectively. Furthermore, by compressing the peak half-width (e.g., 6.5nm in liquid scenarios), the quality factor is improved, solving the problem of "difficulty in balancing sensitivity and quality factor" in traditional sensors.
[0018] (2) Integration of low cost and scenario adaptability: In terms of material and structural design, this invention combines economy and scenario specificity. Titanium dioxide is selected as the substrate, and the metal film adopts a partitioned design of "local modification of the gold film (pore wall / pore bottom) + surface coverage of the silver film". Compared with the full precious metal coating solution, the amount of precious metal used is reduced, balancing performance and cost. At the same time, for different application scenarios such as liquids (refractive index 1.33-1.41) and gases (refractive index 1.01-1.05), by adjusting parameters such as the nanopore period and the thickness of the metal film, the detection requirements can be accurately adapted. It has practical value of "low-cost deployment + multi-scenario coverage" in the fields of food safety and environmental monitoring. Attached Figure Description
[0019] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly described below. Referring to the drawings will make the features and advantages of the present invention clearer. The drawings are illustrative and should not be construed as limiting the present invention in any way. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. Wherein: Figure 1 A schematic diagram of the structure of a refractive index sensor based on a titanium dioxide nanopore array provided by the present invention; Figure 2 This is a cross-sectional view of the sensor device of the present invention; Figure 3 The present invention provides the reflection, transmission, and absorption spectral response curves under the condition of perpendicular incidence and a refractive index of 1.37 for the analyte. Figure 4 This is a simulated electric field distribution diagram under the condition of perpendicular incidence and a refractive index of 1.37 for the test object. Figure 5 The following are the reflection spectrum curves corresponding to different refractive indices in Example 1 of the present invention; Figure 6 This is the linear fitting curve of refractive index and resonant wavelength in Example 1 of the present invention; Figure 7 The following are the reflection spectrum curves corresponding to different refractive indices in Example 2 of the present invention; Figure 8 This is the linear fitting curve of refractive index and resonant wavelength in Example 2 of the present invention.
[0020] Explanation of the markings in the accompanying drawings: 1. Titanium dioxide substrate; 2. Gold film layer; 3. Silver film layer; 4. Silicon dioxide protective layer. Detailed Implementation
[0021] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0022] To address the problems of complex metal grating structures, high manufacturing costs, and low sensor sensitivity and quality factor in existing technologies. For example... Figure 1 As shown, this invention proposes a refractive index sensor based on a titanium dioxide nanopore array, the sensor comprising: Titanium dioxide substrate 1, the surface of titanium dioxide substrate 1 is provided with a tetragonal periodically arranged array of nanopores; Gold film layer 2 covers the sidewalls and bottom surface of the nanopores; Silver film layer 3 covers the upper surface of titanium dioxide substrate 1, and silver film layer 3 and gold film layer 2 are connected at the edge of the pore opening of the nanopore. The silicon dioxide protective layer 4 covers the exposed surfaces of the gold film layer 2 and the silver film layer 3, forming a protective enclosure for the metal layers.
[0023] As can be seen from the above technical solution, this invention proposes a refractive index sensor based on a titanium dioxide nanopore array. Using a titanium dioxide substrate as the base, a tetragonal periodically arranged array of nanopores is formed on its surface. A gold film layer is coated on the sidewalls and bottom of the nanopores, and a silver film layer is coated on the upper surface of the titanium dioxide substrate (with the silver and gold film layers connecting at the edges of the nanopore openings). A silica protective layer is then applied to the exposed surfaces of the gold and silver film layers to form a protective enclosure, thus constructing a layered composite structure. This technical solution achieves light field concentration through surface plasmon resonance effect induced by the gold film layer to ensure the quality factor, while the silver film layer promotes light field leakage to the analyte to improve sensitivity. These two elements synergistically solve the problem of low sensitivity and quality factor in traditional sensors. Simultaneously, the low cost of the titanium dioxide substrate, combined with the localized coverage of the gold film and the partial replacement of the gold film with the silver film, reduces the amount of precious metals used and the manufacturing cost, simplifies the metal grating structure, and the silica protective layer effectively delays metal film loss, improving device stability. This allows the sensor to achieve both high performance and cost-effectiveness in multi-scenario detection.
[0024] The refractive index sensor of this invention mainly consists of a titanium dioxide substrate 1, a nanopore array, a gold film layer 2, a silver film layer 3, and a silicon dioxide protective layer 4. These components work together to achieve accurate sensing of the refractive index.
[0025] Specifically, the titanium dioxide substrate 1 serves as the fundamental support structure for the entire sensor device. It possesses advantages such as high electron mobility, mature fabrication technology, low raw material cost, and environmental friendliness. Its surface is processed with a regularly arranged array of nanopores, providing a specific microstructure for the subsequent construction of functional layers. Nanopore arrays: Nanopore arrays are periodic arrays arranged in a square pattern. This periodic structure plays a crucial role in the modulation of the light field and the excitation of surface plasmon resonance effects. Parameters such as the diameter D, depth h, and array period P of the nanopores can be optimized and adjusted according to actual application requirements. In some embodiments, the period P of the nanopore array can be set in the range of 900nm-1200nm, the diameter D of the nanopores is 500nm-900nm, and the depth h is 600nm-1000nm. Gold film layer 2: Gold film layer 2 covers the sidewalls and bottom surface of the nanopore. Gold's unique surface plasmon resonance effect allows this gold film layer 2 to interact strongly with incident light. When light irradiates gold film layer 2, it excites surface plasmon waves, forming a localized strong electromagnetic field. The thickness of gold film layer 2 also affects the sensor's performance. In practical applications, the thickness δ of the gold film on the nanopore sidewalls and the thickness of the gold film at the bottom of the pore can be selected within the range of 60nm-150nm. Silver film layer 3: Silver film layer 3 covers the upper surface of the titanium dioxide substrate 1, including the opening region of the nanopores. Silver has excellent optical reflectivity, and the presence of silver film layer 3 further optimizes the light reflection and absorption characteristics of the sensor device, enhancing the surface plasmon resonance effect. Silver film layer 3 and the gold film layer 2 on the sidewalls of the nanopores achieve a good connection at the pore openings, ensuring the integrity and functionality of the entire metal layer structure. The thickness δ of silver film layer 3 can generally be set between 40 nm and 100 nm. It should be noted that a silver film is used instead of a gold film on the upper surface of the titanium dioxide substrate 1. On the one hand, the silver film has advantages such as high reflectivity and relatively lower coating cost, and can achieve higher sensing sensitivity when surface plasmon resonance is excited; on the other hand, replacing the gold film with a silver film can effectively compress the peak half-width of the spectral response, thereby helping to improve the sensor's quality factor.
[0026] Silica protective layer 4: Covering the exposed surfaces of the gold film layer 2 and the silver film layer 3, the silica protective layer 4 provides comprehensive protection for the metal layers. It effectively prevents the metal layers from being oxidized and corroded, while also avoiding interference from external environmental factors on the plasmonic resonance characteristics of the metal layer surface, thus ensuring the long-term stability and reliability of the sensor. The thickness of the silica protective layer 4 is typically in the range of 100nm-500nm.
[0027] Furthermore, the present invention also provides a testing method based on the above-mentioned titanium dioxide nanopore array refractive index sensor device, comprising the following steps: S1: Construct sensor devices; S2: A plane wave light source is incident perpendicularly on the surface to be measured of the sensor device, and the horizontal direction is set as a periodic boundary condition, while the upper and lower surfaces are set as perfect absorption boundary conditions. S3: Obtain the reflection spectrum curves under different refractive indices by wavelength scanning; S4: Calculate the refractive index sensitivity based on the wavelength shift and refractive index change corresponding to the minimum value of the reflection spectrum curve; The formula for calculating refractive index sensitivity is: ; In the formula, Indicates refractive index sensitivity. Indicates the wavelength offset. It represents the change in refractive index.
[0028] Furthermore, this invention calculates the quality factor using the full width at half maximum (FWHM) of the minimum reflectance spectrum curve. The calculation formula is as follows: ; In the formula, The quality factor is represented by FWHM, which stands for Full Width at Half Peak.
[0029] To further illustrate the advantages of this invention, the following description is based on specific simulations.
[0030] Example 1:
[0031] I. Structural Design and Modeling This invention discloses a high-performance refractive index sensor. A geometric model is constructed and materials are selected using Lumerical FDTD Solutions simulation software based on the finite-difference time-domain (FDTD) method. For applications where the analyte is a liquid (refractive index approximately 1.33), the following three-dimensional periodic structure is designed: a periodic array of nanopores arranged in a square pattern (e.g., Figure 1 , Figure 2 As shown), the period P is 900 nm, the inner diameter D is 740 nm without coating, and the depth h from the bottom to the edge of the pore is 660 nm; the nanopore walls and bottom are covered with a gold film of uniform thickness (e.g., Figure 1 , Figure 2 As shown in the figure), the thickness δ of the gold film on the hole wall and bottom is 70 nm; the upper surface of the titanium dioxide substrate 1 is uniformly covered with a silver film (as shown in the figure). Figure 1 , Figure 2 As shown), the silver film thickness is 50 nm; both metal films are uniformly covered with a silicon dioxide film (as shown). Figure 1 , Figure 2 As shown, the silicon dioxide film is 100 nm thick. A periodically arranged array of gold-titanium dioxide pores was designed to be generated.
[0032] II. Simulation Parameter Settings Complete the light source settings in the Sources module of the FDTD Solutions software. Configure a plane wave light source on the test surface of the sensor metasurface. There are no specific requirements for the polarization direction of the light source. In the FDTD settings, set the four sides as periodic boundary conditions and the top and bottom sides as perfect absorption boundary conditions (PML) to achieve a periodic arrangement on the horizontal plane.
[0033] When solving for system transmission or reflection, based on the set physical parameters, set a Frequency-domain field and power monitor above the light source and below the absorption boundary. Call the transmission function on this monitor in the command bar to obtain the reflectivity R (unit: %, the same below). Similarly, set a Frequency-domain field and power monitor below the substrate, call the transmission function on this monitor in the command bar and take the negative value to obtain the transmittance T. When solving for the system's light absorptivity A, calculate A = 1 − T − R.
[0034] Wavelength scanning calculations were performed in the 1100nm-1300nm band to solve for the spectral response (including transmission, reflection, and absorption spectra) within this band. Figure 3 As shown in the figure, the resonance wavelength is subsequently obtained mainly through the reflection spectrum. When the above structural parameters are used, the optimal spectral response can be obtained (the minimum value of the reflection spectrum is close to 0).
[0035] III. Analysis of Light Field and Sensing Principles To obtain a high-sensitivity, high-quality refractive index sensor, it is necessary to enhance the constraint of the light field by the two-dimensional grating, concentrating the light field in the high-RI grating region. For example... Figure 4 As shown in the simulated electric field distribution diagram, the gold-titanium dioxide coupling structure introduced at the edge of the titanium dioxide nanopores excites the surface plasmon resonance (SPR) effect of the gold film, and the light field is more concentrated in the metal grating region, ensuring the quality factor of the refractive index sensing; at the same time, the silver film on the upper surface allows more light field to leak to the low refractive index analyte, ensuring the high sensitivity of the refractive index sensing.
[0036] IV. Liquid Refractive Index Sensing Test Figure 5This invention provides a method for testing the refractive index sensor of a liquid background by detecting its reflection spectrum. Based on the reflection spectrum curve obtained from the aforementioned simulation steps, different liquid background refractive indices were set (gradually increasing from 1.33 to 1.41, with each increase of 0.02). At a refractive index of 1.33, a wavelength scan was performed in the 1100nm-1400nm band, yielding... Figure 5 The first reflection spectrum curve in the middle; when the refractive index increases to 1.35, the wavelength scan calculation yields... Figure 5 The second reflection spectrum curve; and so on, when the refractive index reaches 1.41, we obtain... Figure 5 The fifth reflection spectrum curve in the middle.
[0037] Observations show that the reflectance spectrum curve shifts towards longer wavelengths as the refractive index of the liquid increases. (From...) Figure 5 When the gas refractive index increases from 1.33 to 1.41, the wavelength corresponding to the minimum value of the reflection spectrum curve increases from 1205.5 nm to 1272.0 nm, and the peak half-width of the sensor is 6.5 nm.
[0038] When studying the refractive index sensing of a device, based on the obtained optimal spectral response, the background refractive index variation is simulated to explore the device's refractive index sensitivity (S), which depends on the coupling degree between the light free-space mode and the plasma propagation mode. Plasma resonance refractive index sensing utilizes the resonance peak wavelength. The sensitivity S is adjusted according to the refractive index change of the surrounding environment during the offset detection. Calculate (where This is due to the wavelength shift of the resonance peak. (This represents the change in refractive index around the surface, in nm / RIU).
[0039] Figure 6 To detect the relationship between the minimum wavelength of the refractive index and the reflection spectrum curve of different test liquids, a linear fitting was performed to form a straight line. The slope of the straight line is the refractive index sensitivity, which is 830 nm / RIU. The sensor quality factor (FOM) is 127.7.
[0040] Example 2:
[0041] I. Structural Design For applications where the analyte is a gas (refractive index 1.01-1.05), a three-dimensional periodic structure was designed: titanium dioxide nanopores are arranged in a square periodic array with a period P of 1200 nm; titanium dioxide is used as the substrate and nanopore material, and the surface, sidewalls, and bottom of the nanopores are covered with a gold film. The diameter D of the nanopores is 700 nm, the depth h is 800 nm, and the thickness δ of the gold film on the sidewalls and bottom of the nanopores is 150 nm; a silver film is uniformly covered on the upper surface of the titanium dioxide substrate 1, and the silver film layer 3 and the gold film layer 2 are connected at the edge of the nanopore opening with a thickness of 80 nm; a silicon dioxide film is uniformly covered on the upper surface of the two metal films with a thickness of 100 nm.
[0042] II. Spectral Scanning and Performance Testing Reflectance spectral scanning was performed in the wavelength range of 1300nm-1500nm (scanning and processing results are shown in [link]). Figure 7 , Figure 8 The sensor was tested to have a sensitivity of 1001 nm / RIU, a peak half-width of 11 nm, and a quality factor (FOM) of 91 in a gaseous environment.
[0043] In summary, the refractive index sensor based on titanium dioxide nanopore array proposed in this invention has outstanding advantages such as low peak half-width, high sensitivity, low consumable cost, real-time dynamic monitoring capability, and no need for labeling, and has broad application prospects in food safety testing, environmental monitoring, product inspection and other fields.
[0044] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.
Claims
1. A refractive index sensor device based on a titanium dioxide nanopore array, characterized in that, include: Titanium dioxide substrate (1), the surface of which is provided with a square periodically arranged array of nanopores; A gold film layer (2) is applied to the sidewalls and bottom surface of the nanopores. A silver film layer (3) is applied to the upper surface of the titanium dioxide substrate (1), and the silver film layer (3) is connected to the gold film layer (2) at the edge of the pore opening of the nanopore. A silicon dioxide protective layer (4) is applied to the exposed surfaces of the gold film layer (2) and the silver film layer (3) to form a protective coating on the metal layer.
2. The refractive index sensor based on a titanium dioxide nanopore array according to claim 1, characterized in that, The period of the nanopore array is 900nm-1200nm.
3. The refractive index sensor based on a titanium dioxide nanopore array according to claim 1, characterized in that, The diameter of the nanopore is 500nm-900nm.
4. The refractive index sensor based on a titanium dioxide nanopore array according to claim 1, characterized in that, The depth of the nanopores is 600nm-1000nm.
5. The refractive index sensor based on a titanium dioxide nanopore array according to claim 1, characterized in that, The thickness of the gold film on the sidewall is 60nm-150nm.
6. The refractive index sensor based on a titanium dioxide nanopore array according to claim 1, characterized in that, The thickness of the gold film at the bottom of the hole is 60nm-150nm.
7. The refractive index sensor based on a titanium dioxide nanopore array according to claim 1, characterized in that, The silver film thickness of the silver film layer (3) is 40nm-100nm.
8. The refractive index sensor based on a titanium dioxide nanopore array according to claim 1, characterized in that, The thickness of the silicon dioxide protective layer (4) is 100nm-500nm.
9. A testing method for a refractive index sensor device based on a titanium dioxide nanopore array, characterized in that, The method, based on the refractive index sensor device based on a titanium dioxide nanopore array as described in any one of claims 1 to 8, includes the following steps: S1: Construct the sensor device; S2: A plane wave light source is incident perpendicularly on the surface to be measured of the sensor, and the horizontal direction is set as a periodic boundary condition, while the upper and lower surfaces are set as perfect absorption boundary conditions. S3: Obtain the reflection spectrum curves under different refractive indices by wavelength scanning; S4: Calculate the refractive index sensitivity based on the wavelength shift and refractive index change corresponding to the minimum value of the reflection spectrum curve; The formula for calculating the refractive index sensitivity is as follows: ; In the formula, Indicates refractive index sensitivity. Indicates the wavelength offset. It represents the change in refractive index.
10. The test method according to claim 9, characterized in that, The quality factor is calculated using the full width at half maximum (FWHM) of the minimum reflectance spectrum curve. The formula is as follows: ; In the formula, The quality factor is represented by FWHM, which stands for Full Width at Half Peak.