Numerical Analysis of Highly Sensitive Twin-Core, Gold-Coated, D-Shaped Photonic Crystal Fiber Based on Surface Plasmon Resonance Sensor

This research article proposes and numerically investigates a photonic crystal fiber (PCF) based on a surface plasmon resonance (SPR) sensor for the detecting refractive index (RI) of unknown analytes. The plasmonic material (gold) layer is placed outside of the PCF by removing two air holes from the main structure, and a D-shaped PCF-SPR sensor is formed. The purpose of using a plasmonic material (gold) layer in a PCF structure is to introduce an SPR phenomenon. The structure of the PCF is likely enclosed by the analyte to be detected, and an external sensing system is used to measure changes in the SPR signal. Moreover, a perfectly matched layer (PML) is also placed outside of the PCF to absorb unwanted light signals towards the surface. The numerical investigation of all guiding properties of the PCF-SPR sensor is completed using a fully vectorial-based finite element method (FEM) to achieve the finest sensing performance. The design of the PCF-SPR sensor is completed using COMSOL Multiphysics software, version 1.4.50. According to the simulation results, the proposed PCF-SPR sensor has a maximum wavelength sensitivity of 9000 nm/RIU, an amplitude sensitivity of 3746 RIU−1, a sensor resolution of 1 × 10−5 RIU, and a figure of merit (FOM) of 900 RIU−1 in the x-polarized direction light signal. The miniaturized structure and high sensitivity of the proposed PCF-SPR sensor make it a promising candidate for detecting RI of analytes ranging from 1.28 to 1.42.


Introduction and Literature Review
In recent decades, photonic crystal fiber (PCF)-based surface plasmon resonance (SPR) sensors [1][2][3][4][5][6][7] have been attractive to many sensor researchers due to their outstanding features, which include controllable dispersion and birefringence, low loss, high sensitivity, good sensor resolution, fast response time, label-free detection, real-time monitoring, reliability, operational flexibility, simple and miniaturized structure, and tunable structural parameters [8][9][10][11][12][13]. PCF-SPR sensors have shown potential for use in various fields, including bimolecular analyte detection, medical diagnosis, medical testing, blood group detection, virus detection, cancerous cell detection in the human body, drug testing, food quality control for safety, environmental monitoring, and other applications [14][15][16][17][18]. An SPR is the collective oscillation of free electrons at metal-dielectric interfaces that are stimulated by the photons of an evanescent optical field [19][20][21]. It occurs when the wavelength of free electrons and the wavelength of photons are in resonance with each other at the metal-dielectric interface [22,23]. However, the plasmonic effect refers to the interaction between the free electrons and the photons of evanescent optical field at the metal-dielectric interface [24][25][26][27]. The core of PCF has a higher refractive index than the cladding, which allows a small amount of light to penetrate into the cladding region [28]. This light interacts with the plasmonic material layer at the metal-dielectric interface, exciting the Figure 1 shows the experimental setup and operating principle of the proposed sensor. The setup includes various components, such as light amplification by stimulated emission of radiation (laser), an optical polarizer, a sensing unit, an optical spectrum analyzer (OSA), a laptop, a programmable pump, an analyte reservoir, and more. The laser light is coupled into the optical polarizer unit using an SMF fiber, which converts unpolarized light into polarized light and passes it through the sensing unit. Analyte is also introduced into the sensing unit using the PVC pipe, and the valve controls the flow of analyte with a standard pressure and temperature. The programmable pump circulates the analytes into the sensing unit step by step and finally discharges them into the waste reservoir. The interaction between the free electrons of plasmonic material and photons of the evanescent optical field takes place in the sensing unit. The OSA receives the polarized light signal from the sensing unit using SMF and quantifies the optical spectra. The laptop gathers the quantified optical spectra from OSA using SMF and displays the resonance peak loss curve as a monitor corresponding to each unknown analyte with a distinct resonance wavelength. By analyzing the resonance peak loss curve, the sensor is capable of detecting and identifying different analytes with high precision.

Block Diagram and Mode of Operation
focus on developing sensors with improved sensitivity and FOM while ke turing cost, sensor size, and fabrication complexity low. The paper pro gold-coated, D-shaped PCF-SPR sensor that overcomes the limitation ported sensors by determining four sensing parameters: wavelength sen sensitivity, sensor resolution, and figure of merit (FOM). The D-shape proves to be a more economical option in comparison to circular-shape due to its utilization of a smaller amount of plasmonic material and a re tween the plasmonic layer and the core. These features result in a hig evanescent field on the plasmonic material layer, allowing for more electrons in metal-dielectric interfaces and improved overall sensing p Figure 1 shows the experimental setup and operating principle of th The setup includes various components, such as light amplification by stim radiation (laser), an optical polarizer, a sensing unit, an optical spectrum laptop, a programmable pump, an analyte reservoir, and more. The laser the optical polarizer unit using an SMF fiber, which converts unpolarized light and passes it through the sensing unit. Analyte is also introduced in using the PVC pipe, and the valve controls the flow of analyte with a sta temperature. The programmable pump circulates the analytes into the s step and finally discharges them into the waste reservoir. The interactio electrons of plasmonic material and photons of the evanescent optical the sensing unit. The OSA receives the polarized light signal from the SMF and quantifies the optical spectra. The laptop gathers the quantifi from OSA using SMF and displays the resonance peak loss curve as a m ing to each unknown analyte with a distinct resonance wavelength. By onance peak loss curve, the sensor is capable of detecting and identif lytes with high precision.    Figure 2 shows a cross-sectional view of the D-shaped PCF-SPR sensor, which consists of a central air hole, two air hole rings, a gold layer, an analyte layer, and a PML layer. The first and second rings are made up of rectangular and circular air holes, respectively. The central air hole (A) is located in the core, while air holes B and C are located in the cladding to create a difference in RI between the core and cladding. The gold layer is positioned outside of the PCF structure, and two air holes are removed to create a D-shaped sensor. The analyte layer is also located outside the PCF structure, and analyte is circulated through it using a programmable pump. The PML layer is designed to improve sensing performance by absorbing unwanted light signals and canceling the reflection of light. All structural parameters were optimized step-by-step to achieve the best sensing performance. The optimum areas were found to be 0.0314 µm 2 , 0.6359 µm 2 , and 1.0 µm 2 for air holes A, B, and C, respectively. Correspondingly, the optimum thicknesses were found to be 26 nm, 580 nm, and 80 nm for the gold layer, analyte layer, and PML, respectively. sors 2023, 23, x FOR PEER REVIEW analyte layer is also located outside the PCF structure, and analyte is c using a programmable pump. The PML layer is designed to improve sens absorbing unwanted light signals and canceling the reflection of light. Al ters were optimized step-by-step to achieve the best sensing performance. were found to be 0.0314 µm 2 , 0.6359 µm 2 , and 1.0 µm 2 for air holes A, B, Correspondingly, the optimum thicknesses were found to be 26 nm, for the gold layer, analyte layer, and PML, respectively.  Figure 3 shows the maximum photon energy transfers from the core plasmon polariton (SPP) mode at resonance condition or phase matching and brick-colored dashed lines in Figure 3 characterize the loss and real RI of the core mode, respectively. Furthermore, the green dashed line ch tive RI of the SPP mode. From Figure 3, it can be observed that the loss creases with increasing wavelength up to the resonance point (B) and la with increasing wavelength. On the other hand, the real part of effective and SPP mode decrease with increasing wavelength. Additionally, the RI of the core mode suitable matches the effective RI of SPP mode and sa condition. In Figure 3, the resonance occurs at λ = 0.86 µm wavelength, ton energy transfers from the core mode to the SPP mode instantaneou condition is crucial for the operation of the proposed sensor because it e accurate detection of various analytes based on the interaction between  Figure 3 shows the maximum photon energy transfers from the core mode to the surface plasmon polariton (SPP) mode at resonance condition or phase matching condition. The blue and brick-colored dashed lines in Figure 3 characterize the loss and real part of the effective RI of the core mode, respectively. Furthermore, the green dashed line characterizes the effective RI of the SPP mode. From Figure 3, it can be observed that the loss of the core mode increases with increasing wavelength up to the resonance point (B) and later starts decreasing with increasing wavelength. On the other hand, the real part of effective RIs of the core mode and SPP mode decrease with increasing wavelength. Additionally, the real part of effective RI of the core mode suitable matches the effective RI of SPP mode and satisfies the resonance condition. In Figure 3, the resonance occurs at λ = 0.86 µm wavelength, and maximum photon energy transfers from the core mode to the SPP mode instantaneously. This resonance condition is crucial for the operation of the proposed sensor because it enables efficient and accurate detection of various analytes based on the interaction between the plasmonic material and the photons of the evanescent optical field.

Analysis of Mathematical Equations
A small amount of light signal transfers from the core mode to SPP mode in a PCF due to the difference in RI between the core and cladding. This phenomenon is translated as losses affecting the light. Loss is typically quantified in terms of the imaginary part of the effective RI of the modes involved. The loss profile of the PCF-SPR sensor can be stated using the following equation [57]: where = represents the wave number in free space, which is equal to 2π divided by the wavelength (λ) of the light. The represents the effective RI that is related to the attenuation or loss of the light signal as it propagates through PCF. The wavelength sensitivity is the ratio of the change in resonance wavelength (Δ ) to the change in RI of an adjacent analyte. It is a measure of how much the resonance wavelength shifts in response to a change in the surrounding RI. Typically, the resonance wavelength shifts towards a higher value as the RI increases. Wavelength sensitivity can be calculated using the wavelength interrogation method, which can be quantified using the following equation [58]: where Δ refers to the difference in wavelength between the peak positions of two spectral peaks and Δ refers to the effective RI difference between two adjacent analytes. The sensor resolution is a key performance parameter of a PCF-SPR sensor, which represents the ability of the sensor to resolve small changes in the RI of the analyte. The resolution of the PCF-SPR sensor can be defined as the minimum detectable change in the RI that can be calculated using the following equation [59]: where denotes the smallest detectable change in the sensor output, Δ denotes the difference in RI between two adjacent analytes, Δ denotes the difference in wavelength

Analysis of Mathematical Equations
A small amount of light signal transfers from the core mode to SPP mode in a PCF due to the difference in RI between the core and cladding. This phenomenon is translated as losses affecting the light. Loss is typically quantified in terms of the imaginary part of the effective RI of the modes involved. The loss profile of the PCF-SPR sensor can be stated using the following equation [57]: where K 0 = 2π λ represents the wave number in free space, which is equal to 2π divided by the wavelength (λ) of the light. The I m n e f f represents the effective RI that is related to the attenuation or loss of the light signal as it propagates through PCF. The wavelength sensitivity is the ratio of the change in resonance wavelength (∆λ peak to the change in RI of an adjacent analyte. It is a measure of how much the resonance wavelength shifts in response to a change in the surrounding RI. Typically, the resonance wavelength shifts towards a higher value as the RI increases. Wavelength sensitivity can be calculated using the wavelength interrogation method, which can be quantified using the following equation [58]: where ∆λ peak refers to the difference in wavelength between the peak positions of two spectral peaks and ∆n a refers to the effective RI difference between two adjacent analytes. The sensor resolution is a key performance parameter of a PCF-SPR sensor, which represents the ability of the sensor to resolve small changes in the RI of the analyte. The resolution of the PCF-SPR sensor can be defined as the minimum detectable change in the RI that can be calculated using the following equation [59]: where R denotes the smallest detectable change in the sensor output, ∆n a denotes the difference in RI between two adjacent analytes, ∆λ peak denotes the difference in wavelength between the resonance peaks of the sensor for two adjacent analytes, and ∆λ min is a constant value of 0.1 nm that is used in OSA to set the minimum detectable wavelength resolution. The amplitude sensitivity is a considerable sensing parameter of the sensor, which refers to the change in the amplitude of the output signal due to changes in the RI of the analyte. Amplitude sensitivity can be calculated using the amplitude integration method, which can be defined using the following equation [60]: where S A describes the change in the amplitude of the sensor signal per unit change in RI and δ a ( λ, n a ) describes the loss difference between two analytes, usually expressed in units of dB/cm. Wavelength sensitivity and full-width half maximum (FWHM) are two important parameters that are often used to analyze the performance of PCF-SPR sensors. However, the FOM is a term that is often used to characterize the overall performance of a PCF-SPR sensor. The sharp resonance curve and the utmost wavelength sensitivity collectively generate maximum FOM, which can be expressed using the following equation [61]: The Sellmeier equation is an important mathematical tool used to calculate the RI of a material. However, the Sellmeier equation is commonly used in the design and optimization of PCF-SPR sensors. Mathematically, the Sellmeier equation with its different parameters can be expressed as [62]: where n is the effective RI of the fused silica at a given operating wavelength λ, whereas B 1 , B 2 , B 3 and C 1 , C 2 , C 3 are the Sellmeier constants for fused silica; their numerical values are approximately 0.69616300, 0.407942600, 0.407942600, 0.897479400, 0.00467914826, 0.0135120631, and 97.9340025, respectively. The Drude-Lorentz model is utilized to calculate the permittivity of gold, which is a measure of how a material responds to electric fields. Mathematically, the Drude-Lorentz model can be expressed using the following equation [63]: Sensors 2023, 23, x FOR PEER REVIEW 6 of 18 between the resonance peaks of the sensor for two adjacent analytes, and Δ is a constant value of 0.1 nm that is used in OSA to set the minimum detectable wavelength resolution. The amplitude sensitivity is a considerable sensing parameter of the sensor, which refers to the change in the amplitude of the output signal due to changes in the RI of the analyte. Amplitude sensitivity can be calculated using the amplitude integration method, which can be defined using the following equation [60]: where describes the change in the amplitude of the sensor signal per unit change in RI and ( , ) describes the loss difference between two analytes, usually expressed in units of dB/cm. Wavelength sensitivity and full-width half maximum (FWHM) are two important parameters that are often used to analyze the performance of PCF-SPR sensors. However, the FOM is a term that is often used to characterize the overall performance of a PCF-SPR sensor. The sharp resonance curve and the utmost wavelength sensitivity collectively generate maximum FOM, which can be expressed using the following equation [61]: The Sellmeier equation is an important mathematical tool used to calculate the RI of a material. However, the Sellmeier equation is commonly used in the design and optimization of PCF-SPR sensors. Mathematically, the Sellmeier equation with its different parameters can be expressed as [62]: where n is the effective RI of the fused silica at a given operating wavelength λ, whereas , , and , , are the Sellmeier constants for fused silica; their numerical values are approximately 0.69616300, 0.407942600, 0.407942600, 0.897479400, 0.00467914826, 0.0135120631, and 97.9340025, respectively. The Drude-Lorentz model is utilized to calculate the permittivity of gold, which is a measure of how a material responds to electric fields. Mathematically, the Drude-Loren model can be expressed using the following equation [63]: where ɛ characterizes the permittivity of gold, ɛ also characterizes the permittivity of gold at the utmost frequency with a value of 5.9673, ω characterizes the angular frequency that is given by ω = 2πc/λ, c is the velocity of light in the medium, characterizes the plasma frequency, characterizes the damping frequency, ∆ɛ characterizes the weighting factor, characterizes the spectral width, and Ω quantifies the oscillator strength. Moreover, different constants of Lorentz oscillator are given as = 13,273.408 THz, = 100 THz, ∆ɛ = 1.09, = 658.53 THz, and Ω = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the length of the PCF used in the sensing element. In general, the sensor length is inversely proportional to the loss, which can be stated using the following equation [48]: where L indicates the length of sensor and ( , ) indicates the loss of PCF-SPR sensor.
6. Results and Performance Analysis 6.1. Loss and Amplitude Sensitivity Change with RI of Analyte In Figure 4a,b, the peak amplitude and resonance wavelength are shown as functions of the RI of the analyte. It can be observed that as the RI of the analyte increases, the peak between the resonance peaks of the sensor for two adjacent analytes, and Δ is a constant value of 0.1 nm that is used in OSA to set the minimum detectable wavelength resolution. The amplitude sensitivity is a considerable sensing parameter of the sensor, which refers to the change in the amplitude of the output signal due to changes in the RI of the analyte. Amplitude sensitivity can be calculated using the amplitude integration method, which can be defined using the following equation [60]: where describes the change in the amplitude of the sensor signal per unit change in RI and ( , ) describes the loss difference between two analytes, usually expressed in units of dB/cm. Wavelength sensitivity and full-width half maximum (FWHM) are two important parameters that are often used to analyze the performance of PCF-SPR sensors. However, the FOM is a term that is often used to characterize the overall performance of a PCF-SPR sensor. The sharp resonance curve and the utmost wavelength sensitivity collectively generate maximum FOM, which can be expressed using the following equation [61]: The Sellmeier equation is an important mathematical tool used to calculate the RI of a material. However, the Sellmeier equation is commonly used in the design and optimization of PCF-SPR sensors. Mathematically, the Sellmeier equation with its different parameters can be expressed as [62]: where n is the effective RI of the fused silica at a given operating wavelength λ, whereas , , and , , are the Sellmeier constants for fused silica; their numerical values are approximately 0.69616300, 0.407942600, 0.407942600, 0.897479400, 0.00467914826, 0.0135120631, and 97.9340025, respectively. The Drude-Lorentz model is utilized to calculate the permittivity of gold, which is a measure of how a material responds to electric fields. Mathematically, the Drude-Loren model can be expressed using the following equation [63]: where ɛ characterizes the permittivity of gold, ɛ also characterizes the permittivity of gold at the utmost frequency with a value of 5.9673, ω characterizes the angular frequency that is given by ω = 2πc/λ, c is the velocity of light in the medium, characterizes the plasma frequency, characterizes the damping frequency, ∆ɛ characterizes the weighting factor, characterizes the spectral width, and Ω quantifies the oscillator strength. Moreover, different constants of Lorentz oscillator are given as = 13,273.408 THz, = 100 THz, ∆ɛ = 1.09, = 658.53 THz, and Ω = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the length of the PCF used in the sensing element. In general, the sensor length is inversely proportional to the loss, which can be stated using the following equation [48]: where L indicates the length of sensor and ( , ) indicates the loss of PCF-SPR sensor.
6. Results and Performance Analysis 6.1. Loss and Amplitude Sensitivity Change with RI of Analyte In Figure 4a,b, the peak amplitude and resonance wavelength are shown as functions of the RI of the analyte. It can be observed that as the RI of the analyte increases, the peak between the resonance peaks of the sensor for two adjacent anal stant value of 0.1 nm that is used in OSA to set the minimum de lution. The amplitude sensitivity is a considerable sensing param refers to the change in the amplitude of the output signal due to analyte. Amplitude sensitivity can be calculated using the ampli which can be defined using the following equation [60]: where describes the change in the amplitude of the sensor sig and ( , ) describes the loss difference between two analytes, of dB/cm. Wavelength sensitivity and full-width half maximum (F parameters that are often used to analyze the performance of PCF-S FOM is a term that is often used to characterize the overall perform The sharp resonance curve and the utmost wavelength sensitivity c imum FOM, which can be expressed using the following equatio = The Sellmeier equation is an important mathematical tool u a material. However, the Sellmeier equation is commonly used zation of PCF-SPR sensors. Mathematically, the Sellmeier equa rameters can be expressed as [62]: where n is the effective RI of the fused silica at a given operating w , and , , are the Sellmeier constants for fused silica; the proximately 0.69616300, 0.407942600, 0.407942600, 0.897479400, 0.0 and 97.9340025, respectively. The Drude-Lorentz model is utilize tivity of gold, which is a measure of how a material responds to ically, the Drude-Loren model can be expressed using the foll where ɛ characterizes the permittivity of gold, ɛ also charac gold at the utmost frequency with a value of 5.9673, ω characterizes is given by ω = 2πc/λ, c is the velocity of light in the medium, frequency, characterizes the damping frequency, ∆ɛ character characterizes the spectral width, and Ω quantifies the oscillato ferent constants of Lorentz oscillator are given as = 13,273.408 1.09, = 658.53 THz, and Ω = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the length of ing element. In general, the sensor length is inversely proportio be stated using the following equation [48]: where L indicates the length of sensor and ( , ) indicates the 6. Results and Performance Analysis 6.1. Loss and Amplitude Sensitivity Change with RI of Analyte In Figure 4a,b, the peak amplitude and resonance waveleng of the RI of the analyte. It can be observed that as the RI of the a between the resonance peaks of the sensor for two adjacent analytes, and Δ is a constant value of 0.1 nm that is used in OSA to set the minimum detectable wavelength resolution. The amplitude sensitivity is a considerable sensing parameter of the sensor, which refers to the change in the amplitude of the output signal due to changes in the RI of the analyte. Amplitude sensitivity can be calculated using the amplitude integration method, which can be defined using the following equation [60]: where describes the change in the amplitude of the sensor signal per unit change in RI and ( , ) describes the loss difference between two analytes, usually expressed in units of dB/cm. Wavelength sensitivity and full-width half maximum (FWHM) are two important parameters that are often used to analyze the performance of PCF-SPR sensors. However, the FOM is a term that is often used to characterize the overall performance of a PCF-SPR sensor. The sharp resonance curve and the utmost wavelength sensitivity collectively generate maximum FOM, which can be expressed using the following equation [61]: The Sellmeier equation is an important mathematical tool used to calculate the RI of a material. However, the Sellmeier equation is commonly used in the design and optimization of PCF-SPR sensors. Mathematically, the Sellmeier equation with its different parameters can be expressed as [62]: where n is the effective RI of the fused silica at a given operating wavelength λ, whereas , , and , , are the Sellmeier constants for fused silica; their numerical values are approximately 0.69616300, 0.407942600, 0.407942600, 0.897479400, 0.00467914826, 0.0135120631, and 97.9340025, respectively. The Drude-Lorentz model is utilized to calculate the permittivity of gold, which is a measure of how a material responds to electric fields. Mathematically, the Drude-Loren model can be expressed using the following equation [63]: where ɛ characterizes the permittivity of gold, ɛ also characterizes the permittivity of gold at the utmost frequency with a value of 5.9673, ω characterizes the angular frequency that is given by ω = 2πc/λ, c is the velocity of light in the medium, characterizes the plasma frequency, characterizes the damping frequency, ∆ɛ characterizes the weighting factor, characterizes the spectral width, and Ω quantifies the oscillator strength. Moreover, different constants of Lorentz oscillator are given as = 13,273.408 THz, = 100 THz, ∆ɛ = 1.09, = 658.53 THz, and Ω = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the length of the PCF used in the sensing element. In general, the sensor length is inversely proportional to the loss, which can be stated using the following equation [48]: where L indicates the length of sensor and ( , ) indicates the loss of PCF-SPR sensor.
6. Results and Performance Analysis 6.1. Loss and Amplitude Sensitivity Change with RI of Analyte In Figure 4a,b, the peak amplitude and resonance wavelength are shown as functions of the RI of the analyte. It can be observed that as the RI of the analyte increases, the peak between the resonance peaks of the sensor for two adjacent analytes, and Δ is a constant value of 0.1 nm that is used in OSA to set the minimum detectable wavelength resolution. The amplitude sensitivity is a considerable sensing parameter of the sensor, which refers to the change in the amplitude of the output signal due to changes in the RI of the analyte. Amplitude sensitivity can be calculated using the amplitude integration method, which can be defined using the following equation [60]: where describes the change in the amplitude of the sensor signal per unit change in RI and ( , ) describes the loss difference between two analytes, usually expressed in units of dB/cm. Wavelength sensitivity and full-width half maximum (FWHM) are two important parameters that are often used to analyze the performance of PCF-SPR sensors. However, the FOM is a term that is often used to characterize the overall performance of a PCF-SPR sensor. The sharp resonance curve and the utmost wavelength sensitivity collectively generate maximum FOM, which can be expressed using the following equation [61]: The Sellmeier equation is an important mathematical tool used to calculate the RI of a material. However, the Sellmeier equation is commonly used in the design and optimization of PCF-SPR sensors. Mathematically, the Sellmeier equation with its different parameters can be expressed as [62]: where n is the effective RI of the fused silica at a given operating wavelength λ, whereas , , and , , are the Sellmeier constants for fused silica; their numerical values are approximately 0.69616300, 0.407942600, 0.407942600, 0.897479400, 0.00467914826, 0.0135120631, and 97.9340025, respectively. The Drude-Lorentz model is utilized to calculate the permittivity of gold, which is a measure of how a material responds to electric fields. Mathematically, the Drude-Loren model can be expressed using the following equation [63]: where ɛ characterizes the permittivity of gold, ɛ also characterizes the permittivity of gold at the utmost frequency with a value of 5.9673, ω characterizes the angular frequency that is given by ω = 2πc/λ, c is the velocity of light in the medium, characterizes the plasma frequency, characterizes the damping frequency, ∆ɛ characterizes the weighting factor, characterizes the spectral width, and Ω quantifies the oscillator strength. Moreover, different constants of Lorentz oscillator are given as = 13,273.408 THz, = 100 THz, ∆ɛ = 1.09, = 658.53 THz, and Ω = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the length of the PCF used in the sensing element. In general, the sensor length is inversely proportional to the loss, which can be stated using the following equation [48]: where L indicates the length of sensor and ( , ) indicates the loss of PCF-SPR sensor.
6. Results and Performance Analysis 6.1. Loss and Amplitude Sensitivity Change with RI of Analyte In Figure 4a,b, the peak amplitude and resonance wavelength are shown as functions where 3, 23, x FOR PEER REVIEW 6 of 18 between the resonance peaks of the sensor for two adjacent analytes, and Δ is a constant value of 0.1 nm that is used in OSA to set the minimum detectable wavelength resolution. The amplitude sensitivity is a considerable sensing parameter of the sensor, which refers to the change in the amplitude of the output signal due to changes in the RI of the analyte. Amplitude sensitivity can be calculated using the amplitude integration method, which can be defined using the following equation [60]: where describes the change in the amplitude of the sensor signal per unit change in RI and ( , ) describes the loss difference between two analytes, usually expressed in units of dB/cm. Wavelength sensitivity and full-width half maximum (FWHM) are two important parameters that are often used to analyze the performance of PCF-SPR sensors. However, the FOM is a term that is often used to characterize the overall performance of a PCF-SPR sensor. The sharp resonance curve and the utmost wavelength sensitivity collectively generate maximum FOM, which can be expressed using the following equation [61]: The Sellmeier equation is an important mathematical tool used to calculate the RI of a material. However, the Sellmeier equation is commonly used in the design and optimization of PCF-SPR sensors. Mathematically, the Sellmeier equation with its different parameters can be expressed as [62]: where n is the effective RI of the fused silica at a given operating wavelength λ, whereas , , and , , are the Sellmeier constants for fused silica; their numerical values are approximately 0.69616300, 0.407942600, 0.407942600, 0.897479400, 0.00467914826, 0.0135120631, and 97.9340025, respectively. The Drude-Lorentz model is utilized to calculate the permittivity of gold, which is a measure of how a material responds to electric fields. Mathematically, the Drude-Loren model can be expressed using the following equation [63]: where ɛ characterizes the permittivity of gold, ɛ also characterizes the permittivity of gold at the utmost frequency with a value of 5.9673, ω characterizes the angular frequency that is given by ω = 2πc/λ, c is the velocity of light in the medium, characterizes the plasma frequency, characterizes the damping frequency, ∆ɛ characterizes the weighting factor, characterizes the spectral width, and Ω quantifies the oscillator strength. Moreover, different constants of Lorentz oscillator are given as = 13,273.408 THz, = 100 THz, ∆ɛ = 1.09, = 658.53 THz, and Ω = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the length of the PCF used in the sensing element. In general, the sensor length is inversely proportional to the loss, which can be stated using the following equation [48]: where L indicates the length of sensor and ( , ) indicates the loss of PCF-SPR sensor.
6. Results and Performance Analysis between the resonance peaks of the sensor for two adjacent analytes, an stant value of 0.1 nm that is used in OSA to set the minimum detectable lution. The amplitude sensitivity is a considerable sensing parameter of refers to the change in the amplitude of the output signal due to chang analyte. Amplitude sensitivity can be calculated using the amplitude in which can be defined using the following equation [60]: where describes the change in the amplitude of the sensor signal per and ( , ) describes the loss difference between two analytes, usually of dB/cm. Wavelength sensitivity and full-width half maximum (FWHM) parameters that are often used to analyze the performance of PCF-SPR sen FOM is a term that is often used to characterize the overall performance of The sharp resonance curve and the utmost wavelength sensitivity collectiv imum FOM, which can be expressed using the following equation [61]:

=
The Sellmeier equation is an important mathematical tool used to a material. However, the Sellmeier equation is commonly used in the d zation of PCF-SPR sensors. Mathematically, the Sellmeier equation wit rameters can be expressed as [62]: where n is the effective RI of the fused silica at a given operating wavelen , and , , are the Sellmeier constants for fused silica; their nume proximately 0.69616300, 0.407942600, 0.407942600, 0.897479400, 0.00467914 and 97.9340025, respectively. The Drude-Lorentz model is utilized to cal tivity of gold, which is a measure of how a material responds to electric ically, the Drude-Loren model can be expressed using the following e where ɛ characterizes the permittivity of gold, ɛ also characterizes gold at the utmost frequency with a value of 5.9673, ω characterizes the angu is given by ω = 2πc/λ, c is the velocity of light in the medium, charac frequency, characterizes the damping frequency, ∆ɛ characterizes the characterizes the spectral width, and Ω quantifies the oscillator streng ferent constants of Lorentz oscillator are given as = 13,273.408 THz, 1.09, = 658.53 THz, and Ω = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the length of the PC ing element. In general, the sensor length is inversely proportional to th be stated using the following equation [48]: where L indicates the length of sensor and ( , ) indicates the loss of 6. Results and Performance Analysis 6.1. Loss and Amplitude Sensitivity Change with RI of Analyte ∞ also characterizes the permittivity of gold at the utmost frequency with a value of 5.9673, ω characterizes the angular frequency that is given by ω = 2πc/λ, c is the velocity of light in the medium,ω D characterizes the plasma frequency, Y D characterizes the damping frequency, ∆ Sensors 2023, 23, x FOR PEER REVIEW between the resonance peaks of the sensor for two adjacen stant value of 0.1 nm that is used in OSA to set the minimu lution. The amplitude sensitivity is a considerable sensing refers to the change in the amplitude of the output signal analyte. Amplitude sensitivity can be calculated using the which can be defined using the following equation [60]: describes the change in the amplitude of the sens and ( , ) describes the loss difference between two ana of dB/cm. Wavelength sensitivity and full-width half maxim parameters that are often used to analyze the performance of FOM is a term that is often used to characterize the overall pe The sharp resonance curve and the utmost wavelength sensit imum FOM, which can be expressed using the following e = The Sellmeier equation is an important mathematical a material. However, the Sellmeier equation is commonly zation of PCF-SPR sensors. Mathematically, the Sellmeier rameters can be expressed as [62]: where n is the effective RI of the fused silica at a given opera , and , , are the Sellmeier constants for fused silic proximately 0.69616300, 0.407942600, 0.407942600, 0.8974794 and 97.9340025, respectively. The Drude-Lorentz model is tivity of gold, which is a measure of how a material respon ically, the Drude-Loren model can be expressed using th where ɛ characterizes the permittivity of gold, ɛ also c gold at the utmost frequency with a value of 5.9673, ω charact is given by ω = 2πc/λ, c is the velocity of light in the mediu frequency, characterizes the damping frequency, ∆ɛ cha characterizes the spectral width, and Ω quantifies the os ferent constants of Lorentz oscillator are given as = 13,2 1.09, = 658.53 THz, and Ω = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the len ing element. In general, the sensor length is inversely prop be stated using the following equation [48]: where L indicates the length of sensor and ( , ) indicat characterizes the weighting factor, T L characterizes the spectral width, and between the resonance peaks of the sensor for two adjacent analytes, and Δ is a constant value of 0.1 nm that is used in OSA to set the minimum detectable wavelength resolution. The amplitude sensitivity is a considerable sensing parameter of the sensor, which refers to the change in the amplitude of the output signal due to changes in the RI of the analyte. Amplitude sensitivity can be calculated using the amplitude integration method, which can be defined using the following equation [60]: where describes the change in the amplitude of the sensor signal per unit change in RI and ( , ) describes the loss difference between two analytes, usually expressed in units of dB/cm. Wavelength sensitivity and full-width half maximum (FWHM) are two important parameters that are often used to analyze the performance of PCF-SPR sensors. However, the FOM is a term that is often used to characterize the overall performance of a PCF-SPR sensor. The sharp resonance curve and the utmost wavelength sensitivity collectively generate maximum FOM, which can be expressed using the following equation [61]: The Sellmeier equation is an important mathematical tool used to calculate the RI of a material. However, the Sellmeier equation is commonly used in the design and optimization of PCF-SPR sensors. Mathematically, the Sellmeier equation with its different parameters can be expressed as [62]: where n is the effective RI of the fused silica at a given operating wavelength λ, whereas , , and , , are the Sellmeier constants for fused silica; their numerical values are approximately 0.69616300, 0.407942600, 0.407942600, 0.897479400, 0.00467914826, 0.0135120631, and 97.9340025, respectively. The Drude-Lorentz model is utilized to calculate the permittivity of gold, which is a measure of how a material responds to electric fields. Mathematically, the Drude-Loren model can be expressed using the following equation [63]: where ɛ characterizes the permittivity of gold, ɛ also characterizes the permittivity of gold at the utmost frequency with a value of 5.9673, ω characterizes the angular frequency that is given by ω = 2πc/λ, c is the velocity of light in the medium, characterizes the plasma frequency, characterizes the damping frequency, ∆ɛ characterizes the weighting factor, characterizes the spectral width, and Ω quantifies the oscillator strength. Moreover, different constants of Lorentz oscillator are given as = 13,273.408 THz, = 100 THz, ∆ɛ = 1.09, = 658.53 THz, and Ω = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the length of the PCF used in the sensing element. In general, the sensor length is inversely proportional to the loss, which can be stated using the following equation [48]: between the resonance peaks of the sensor for two adjacent analytes, and Δ is a constant value of 0.1 nm that is used in OSA to set the minimum detectable wavelength resolution. The amplitude sensitivity is a considerable sensing parameter of the sensor, which refers to the change in the amplitude of the output signal due to changes in the RI of the analyte. Amplitude sensitivity can be calculated using the amplitude integration method, which can be defined using the following equation [60]: where describes the change in the amplitude of the sensor signal per unit change in RI and ( , ) describes the loss difference between two analytes, usually expressed in units of dB/cm. Wavelength sensitivity and full-width half maximum (FWHM) are two important parameters that are often used to analyze the performance of PCF-SPR sensors. However, the FOM is a term that is often used to characterize the overall performance of a PCF-SPR sensor. The sharp resonance curve and the utmost wavelength sensitivity collectively generate maximum FOM, which can be expressed using the following equation [61]: The Sellmeier equation is an important mathematical tool used to calculate the RI of a material. However, the Sellmeier equation is commonly used in the design and optimization of PCF-SPR sensors. Mathematically, the Sellmeier equation with its different parameters can be expressed as [62]: where n is the effective RI of the fused silica at a given operating wavelength λ, whereas , , and , , are the Sellmeier constants for fused silica; their numerical values are approximately 0.69616300, 0.407942600, 0.407942600, 0.897479400, 0.00467914826, 0.0135120631, and 97.9340025, respectively. The Drude-Lorentz model is utilized to calculate the permittivity of gold, which is a measure of how a material responds to electric fields. Mathematically, the Drude-Loren model can be expressed using the following equation [63]: where ɛ characterizes the permittivity of gold, ɛ also characterizes the permittivity of gold at the utmost frequency with a value of 5.9673, ω characterizes the angular frequency that is given by ω = 2πc/λ, c is the velocity of light in the medium, characterizes the plasma frequency, characterizes the damping frequency, ∆ɛ characterizes the weighting factor, characterizes the spectral width, and Ω quantifies the oscillator strength. Moreover, different constants of Lorentz oscillator are given as = 13,273.408 THz, = 100 THz, ∆ɛ = 1.09, = 658.53 THz, and Ω = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the length of the PCF used in the sensing element. In general, the sensor length is inversely proportional to the loss, which can be stated using the following equation [ between the resonance peaks of the sensor for two adjacent analytes, and Δ is a constant value of 0.1 nm that is used in OSA to set the minimum detectable wavelength resolution. The amplitude sensitivity is a considerable sensing parameter of the sensor, which refers to the change in the amplitude of the output signal due to changes in the RI of the analyte. Amplitude sensitivity can be calculated using the amplitude integration method, which can be defined using the following equation [60]: where describes the change in the amplitude of the sensor signal per unit change in RI and ( , ) describes the loss difference between two analytes, usually expressed in units of dB/cm. Wavelength sensitivity and full-width half maximum (FWHM) are two important parameters that are often used to analyze the performance of PCF-SPR sensors. However, the FOM is a term that is often used to characterize the overall performance of a PCF-SPR sensor. The sharp resonance curve and the utmost wavelength sensitivity collectively generate maximum FOM, which can be expressed using the following equation [61]: The Sellmeier equation is an important mathematical tool used to calculate the RI of a material. However, the Sellmeier equation is commonly used in the design and optimization of PCF-SPR sensors. Mathematically, the Sellmeier equation with its different parameters can be expressed as [62]: where n is the effective RI of the fused silica at a given operating wavelength λ, whereas , , and , , are the Sellmeier constants for fused silica; their numerical values are approximately 0.69616300, 0.407942600, 0.407942600, 0.897479400, 0.00467914826, 0.0135120631, and 97.9340025, respectively. The Drude-Lorentz model is utilized to calculate the permittivity of gold, which is a measure of how a material responds to electric fields. Mathematically, the Drude-Loren model can be expressed using the following equation [63]: where ɛ characterizes the permittivity of gold, ɛ also characterizes the permittivity of gold at the utmost frequency with a value of 5.9673, ω characterizes the angular frequency that is given by ω = 2πc/λ, c is the velocity of light in the medium, characterizes the plasma frequency, characterizes the damping frequency, ∆ɛ characterizes the weighting factor, characterizes the spectral width, and Ω quantifies the oscillator strength. Moreover, different constants of Lorentz oscillator are given as = 13,273.408 THz, = 100 THz, ∆ɛ = 1.09, = 658.53 THz, and Ω = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the length of the PCF used in the sensing element. In general, the sensor length is inversely proportional to the loss, which can be stated using the following equation [48]: L = 650.07 THz, respectively. The sensor length of a PCF-SPR sensor refers to the length of the PCF used in the sensing element. In general, the sensor length is inversely proportional to the loss, which can be stated using the following equation [48]: where L indicates the length of sensor and α(λ, n a ) indicates the loss of PCF-SPR sensor.

Loss and Amplitude Sensitivity Change with RI of Analyte
In Figure 4a,b, the peak amplitude and resonance wavelength are shown as functions of the RI of the analyte. It can be observed that as the RI of the analyte increases, the peak loss and resonance wavelength shift towards higher wavelengths. This is because a higher RI of the analyte allows for more efficient transfer of photon energy from the core mode to the SPP mode, leading to a stronger interaction between the plasmonic material and the analyte. In Figure 4a, the lowest RI of 1.28 provides a peak loss and resonance wavelength of 1.1256 dB/cm and 0.56 µm, respectively. On the other hand, in Figure 4b, the highest analyte RI of 1.42 provides a peak loss and resonance wavelength of 775.7314 dB/cm and 0.95 µm, respectively. This indicates that the peak loss and resonance wavelength increase with increasing RI of the analyte. 023, 23, x FOR PEER REVIEW loss and resonance wavelength shift towards higher wavelengths. This is because a RI of the analyte allows for more efficient transfer of photon energy from the core to the SPP mode, leading to a stronger interaction between the plasmonic material a analyte. In Figure 4a, the lowest RI of 1.28 provides a peak loss and resonance wavelen 1.1256 dB/cm and 0.56 µm, respectively. On the other hand, in Figure 4b, the highest a RI of 1.42 provides a peak loss and resonance wavelength of 775.7314 dB/cm and 0.9 respectively. This indicates that the peak loss and resonance wavelength increase w creasing RI of the analyte.   Figure 5a shows the relationship between the amplitude sensitivity and the RI analyte in the range of wavelengths from 0.5 µm to 0.75 µm, while Figure 5b shows th relationship in the range of wavelengths from 0.7 µm to 1 µm. In Figure 5a, it is ob that the amplitude sensitivity decreases as the RI of the analyte decreases. The lowe lyte RI of 1.29 offers the minimum amplitude sensitivity of −31.3674 RIU −1 , while the h analyte RI of 1.38 offers the maximum amplitude sensitivity of fe−236.8681 RIU −1 . In 5b, a similar trend is introduced in which the minimum amplitude sensitivity is ob for the lowest RI of 1.40, and the maximum amplitude sensitivity is obtained for the h RI of 1.42. However, the magnitude of the amplitude sensitivity is much higher com to Figure 5a, with the maximum amplitude sensitivity being −3746 RIU −1 . Overall r tion: it suggests that the sensing system is more sensitive to change in the RI of higher analytes at the specific wavelength. When the RI of the analyte is higher than 1.42, t no light signal detected in either the x-or y-polarized directions in the core area of th sor. This implies that the sensor is not able to detect the presence of the analyte at range. Conversely, when the RI of the analyte is lower than 1.27, the resonance wave  Figure 5a shows the relationship between the amplitude sensitivity and the RI of the analyte in the range of wavelengths from 0.5 µm to 0.75 µm, while Figure 5b shows the same relationship in the range of wavelengths from 0.7 µm to 1 µm. In Figure 5a, it is observed that the amplitude sensitivity decreases as the RI of the analyte decreases. The lowest analyte RI of 1.29 offers the minimum amplitude sensitivity of −31.3674 RIU −1 , while the highest analyte RI of 1.38 offers the maximum amplitude sensitivity of fe−236.8681 RIU −1 . In Figure 5b, a similar trend is introduced in which the minimum amplitude sensitivity is obtained for the lowest RI of 1.40, and the maximum amplitude sensitivity is obtained for the highest RI of 1.42. However, the magnitude of the amplitude sensitivity is much higher compared to Figure 5a, with the maximum amplitude sensitivity being −3746 RIU −1 .
Overall realization: it suggests that the sensing system is more sensitive to change in the RI of higher-index analytes at the specific wavelength. When the RI of the analyte is higher than 1.42, there is no light signal detected in either the x-or y-polarized directions in the core area of the sensor. This implies that the sensor is not able to detect the presence of the analyte at this RI range. Conversely, when the RI of the analyte is lower than 1.27, the resonance wavelength value of the analyte matches to the RI of 1.28, which means that the  Figure 6a shows that the peak loss varies with gold layer thickness and wavelength for an analyte RI of 1.41. For an RI of 1.41, the peak losses are 66.54 dB/cm, 68.73 dB/cm, and 67.34 dB/cm for gold layer thicknesses of 25 nm, 25.5 nm, and 26 nm, respectively. On the other hand, for an RI of 1.42, the peak losses are 400.08 dB/cm, 425.8 dB/cm, and 453.61 dB/cm for the same gold layer thicknesses, respectively. The highest RI of 1.42 offers the maximum peak loss compared to the lowest RI of 1.41, which is expected because maximum photon energy penetrates from the core to SPP modes at the higher RI value. The resonance wavelength also shifts towards higher wavelengths with increasing gold layer thickness and RI of analyte due to maximum damping loss happening at the highest gold layer thickness. The study finds tha the optimum gold layer thickness lies between 25 nm and 26 nm. Thicknesses less than 25 nm or greater than 26 nm do not produce any x-polarized direction or y-polarized direction light signals in the core area, which is necessary to obtain true data. In Figure 6b, the three thicknesses are examined; the 26 nm gold layer thickness provides the highest amplitude sensitivity of 3746 RIU −1 , while the sensitivities for 25 nm and 25.5 nm thicknesses are 3680 RIU −1 and 3710 RIU −1 , respectively. Based on these findings, the study concludes that a 26 nm gold layer thickness is the optimum value for this design.  On the other hand, for an RI of 1.42, the peak losses are 400.08 dB/cm, 425.8 dB/cm, and 453.61 dB/cm for the same gold layer thicknesses, respectively. The highest RI of 1.42 offers the maximum peak loss compared to the lowest RI of 1.41, which is expected because maximum photon energy penetrates from the core to SPP modes at the higher RI value. The resonance wavelength also shifts towards higher wavelengths with increasing gold layer thickness and RI of analyte due to maximum damping loss happening at the highest gold layer thickness. The study finds that the optimum gold layer thickness lies between 25 nm and 26 nm. Thicknesses less than 25 nm or greater than 26 nm do not produce any x-polarized direction or y-polarized direction light signals in the core area, which is necessary to obtain true data. In Figure 6b, the three thicknesses are examined; the 26 nm gold layer thickness provides the highest amplitude sensitivity of 3746 RIU −1 , while the sensitivities for 25 nm and 25.5 nm thicknesses are 3680 RIU −1 and 3710 RIU −1 , respectively. Based on these findings, the study concludes that a 26 nm gold layer thickness is the optimum value for this design.

Effect of Central Air Hole Area on Loss and Amplitude Sensitivity
Based on the simulation results reported in Figure 7a,b, it can be concluded that the peak loss and amplitude sensitivity of the sensor are affected by the central air hole area. Specifically, increasing the air hole area leads to higher peak loss, while decreasing the air hole area leads to lower peak loss and higher amplitude sensitivity. In this study, the air hole area of 0.0314 µm 2 is considered the best value for achieving superb sensing performance as it provides the highest amplitude sensitivity and the lowest peak loss. Overall, these findings suggest that careful optimization of the sensor design, including the central air hole area, is crucial for achieving optimal sensing performance.

Effect of Central Air Hole Area on Loss and Amplitude Sensitivity
Based on the simulation results reported in Figure 7a,b, it can be concluded that the peak loss and amplitude sensitivity of the sensor are affected by the central air hole area. Specifically, increasing the air hole area leads to higher peak loss, while decreasing the air hole area leads to lower peak loss and higher amplitude sensitivity. In this study, the air hole area of 0.0314 µm 2 is considered the best value for achieving superb sensing performance as it provides the highest amplitude sensitivity and the lowest peak loss. Overall, these findings suggest that careful optimization of the sensor design, including the central air hole area, is crucial for achieving optimal sensing performance.

Effect of Central Air Hole Area on Loss and Amplitude Sensitivity
Based on the simulation results reported in Figure 7a,b, it can be concluded that the peak loss and amplitude sensitivity of the sensor are affected by the central air hole area. Specifically, increasing the air hole area leads to higher peak loss, while decreasing the air hole area leads to lower peak loss and higher amplitude sensitivity. In this study, the air hole area of 0.0314 µm 2 is considered the best value for achieving superb sensing performance as it provides the highest amplitude sensitivity and the lowest peak loss. Overall, these findings suggest that careful optimization of the sensor design, including the central air hole area, is crucial for achieving optimal sensing performance.

Effect of Rectangular Air Hole Area of Cladding on Loss and Amplitude Sensitivity
According to the simulation results presented in Figure 8a,b, it can be observed that the peak loss and amplitude sensitivity vary with the area of the rectangular air holes for the analytes with RI values ranging from 1.41 to 1.42. In Figure 8a, the peak losses were found to be 79 dB/cm, 77.77 dB/cm, and 67.34 dB/cm for air hole areas of 0.57 µm 2 , 0.74 µm 2 , and 1.00 µm 2 , respectively. Where, the largest air hole area of 1.00 µm 2 exhibits the smallest peak loss. Therefore, it is considered the optimal value for effectively detecting the RI of analytes with values ranging from 1.41 to 1.42. Moreover, the amplitude sensitivity values were found to be 2805 RIU −1 , 3229 RIU −1 , and 3746 RIU −1 for air hole areas of 0.57 µm 2 , 0.74 µm 2 , and 1.00 µm 2 , respectively. As observed in Figure 8b, the largest air hole area of 1.00 µm 2 displays the highest amplitude sensitivity compared to the other two areas. Thus, it can be concluded that an air hole area of 1.00 µm 2 is the optimal value for both the loss and amplitude sensitivity of the designed sensor.

Effect of Rectangular Air Hole Area of Cladding on Loss and Amplitude Sensitivity
According to the simulation results presented in Figure 8a,b, it can be observed that the peak loss and amplitude sensitivity vary with the area of the rectangular air holes for the analytes with RI values ranging from 1.41 to 1.42. In Figure 8a, the peak losses were found to be 79 dB/cm, 77.77 dB/cm, and 67.34 dB/cm for air hole areas of 0.57 µm 2 , 0.74 µm 2 , and 1.00 µm 2 , respectively. Where, the largest air hole area of 1.00 µm 2 exhibits the smallest peak loss. Therefore, it is considered the optimal value for effectively detecting the RI of analytes with values ranging from 1.41 to 1.42. Moreover, the amplitude sensitivity values were found to be 2805 RIU −1 , 3229 RIU −1 , and 3746 RIU −1 for air hole areas of 0.57 µm 2 , 0.74 µm 2 , and 1.00 µm 2 , respectively. As observed in Figure 8b, the largest air hole area of 1.00 µm 2 displays the highest amplitude sensitivity compared to the other two areas. Thus, it can be concluded that an air hole area of 1.00 µm 2 is the optimal value for both the loss and amplitude sensitivity of the designed sensor.

Effect of Circular Air Hole Area of Cladding on Loss and Amplitude Sensitivity
The peak loss and amplitude sensitivity are dominated by the circular air hole area of the cladding for the RI range of the analyte from 1.41 to 1.42, and the thickness of the gold layer is 26 nm. The peak loss and amplitude sensitivity of the sensor vary with the circular air hole area of the cladding as presented in Figure 9a,b. In Figure 9a, the maximum air hole area of 0.6359 µm 2 displays the minimum peak loss, which is considered the best value for detecting the RI of analyte. In Figure 9b, a similar trend is found in which the maximum air hole area of 0.6359 µm 2 provides the maximum amplitude sensitivity, with a value of 3746 RIU −1 , compared to the minimum areas of 2943 RIU −1 and 3350 RIU −1 for air hole areas of 0.5024 µm 2 and 0.5806 µm 2 , respectively. Therefore, the maximum air hole area of 0.6359 µm 2 is selected as the optimal value for both peak loss and amplitude sensitivity.

Effect of Circular Air Hole Area of Cladding on Loss and Amplitude Sensitivity
The peak loss and amplitude sensitivity are dominated by the circular air hole area of the cladding for the RI range of the analyte from 1.41 to 1.42, and the thickness of the gold layer is 26 nm. The peak loss and amplitude sensitivity of the sensor vary with the circular air hole area of the cladding as presented in Figure 9a,b. In Figure 9a, the maximum air hole area of 0.6359 µm 2 displays the minimum peak loss, which is considered the best value for detecting the RI of analyte. In Figure 9b, a similar trend is found in which the maximum air hole area of 0.6359 µm 2 provides the maximum amplitude sensitivity, with a value of 3746 RIU −1 , compared to the minimum areas of 2943 RIU −1 and 3350 RIU −1 for air hole areas of 0.5024 µm 2 and 0.5806 µm 2 , respectively. Therefore, the maximum air hole area of 0.6359 µm 2 is selected as the optimal value for both peak loss and amplitude sensitivity.

Effect of PML Thickness on Loss and Amplitude Sensitivity
The peak loss and amplitude sensitivity change with the perfectly matched layer (PML) thickness, as displayed in Figure 10a,b. Figure 10a shows that the minimum PML thickness of 0.08 µm offers the lowest peak loss of 67.3408 dB/cm for the RI of 1.41, and the resonance wavelength does not shift towards higher wavelengths with increasing PML thickness. This is because a small amount of damping loss occurs with increasing PML thickness, which keeps the resonance wavelength in a fixed position. Figure 10b shows that the lower PML thickness of 0.08 µm provides the highest amplitude sensitivity of 3746 RIU −1 for the RI range of 1.41 to 1.42. The suggested structure is expected to display good sensing performance with a minimum PML thickness of 0.08 µm. It is noted that both peak loss and amplitude sensitivity slightly change with the PML thicknesses. However, the legend colors (red, blue, green) in Figure 10a,b are not able to display the loss and amplitude sensitivity results separately.

Effect of PML Thickness on Loss and Amplitude Sensitivity
The peak loss and amplitude sensitivity change with the perfectly matched layer (PML) thickness, as displayed in Figure 10a,b. Figure 10a shows that the minimum PML thickness of 0.08 µm offers the lowest peak loss of 67.3408 dB/cm for the RI of 1.41, and the resonance wavelength does not shift towards higher wavelengths with increasing PML thickness. This is because a small amount of damping loss occurs with increasing PML thickness, which keeps the resonance wavelength in a fixed position. Figure 10b shows that the lower PML thickness of 0.08 µm provides the highest amplitude sensitivity of 3746 RIU −1 for the RI range of 1.41 to 1.42. The suggested structure is expected to display good sensing performance with a minimum PML thickness of 0.08 µm. It is noted that both peak loss and amplitude sensitivity slightly change with the PML thicknesses. However, the legend colors (red, blue, green) in Figure 10a,b are not able to display the loss and amplitude sensitivity results separately.

Effect of PML Thickness on Loss and Amplitude Sensitivity
The peak loss and amplitude sensitivity change with the perfectly matched layer (PML) thickness, as displayed in Figure 10a,b. Figure 10a shows that the minimum PML thickness of 0.08 µm offers the lowest peak loss of 67.3408 dB/cm for the RI of 1.41, and the resonance wavelength does not shift towards higher wavelengths with increasing PML thickness. This is because a small amount of damping loss occurs with increasing PML thickness, which keeps the resonance wavelength in a fixed position. Figure 10b shows that the lower PML thickness of 0.08 µm provides the highest amplitude sensitivity of 3746 RIU −1 for the RI range of 1.41 to 1.42. The suggested structure is expected to display good sensing performance with a minimum PML thickness of 0.08 µm. It is noted that both peak loss and amplitude sensitivity slightly change with the PML thicknesses. However, the legend colors (red, blue, green) in Figure 10a,b are not able to display the loss and amplitude sensitivity results separately.

Effect of Analyte Layer Thickness (ALT) on Loss and Amplitude Sensitivity
The peak loss is regulated by the thickness of the analyte layer as shown in Figure 11a. In Figure 11a, the peak losses are found to be 67.8950 dB/cm, 67.6150 dB/cm, and 67.3408 dB/cm for analyte layer thicknesses of 0.54 µm, 0.56 µm, and 0.58 µm, respectively. The highest analyte thickness of 0.58 µm provides the least peak loss of 67.3408 dB, and the resonance wavelength does not shift towards the higher wavelength with increasing analyte layer thickness. This is because a small damping loss occurs with increasing analyte layer thickness, which stabilizes the resonance wavelength in a fixed position. In Figure 11b, a similar trend is realized in which the amplitude sensitivity of the sensor is affected by the thicknesses of the analyte layer. In particular, the amplitude sensitivity is found to increase with increasing analyte layer thickness, as denoted by the values of 3457 RIU −1 , 3596 RIU −1 , and 3746 RIU −1 for analyte layer thicknesses of 0.54 µm, 0.56 µm, and 0.58 µm, respectively. Moreover, when the analyte layer thickness is less than 0.54 µm or greater than 0.58 µm, the sensor cannot detect x-polarized direction and y-polarized direction light signals in the core area, which are necessary to calculate sensing parameters. The peak loss and amplitude sensitivity vary slightly with changing analyte layer thickness. Therefore, Figure 11a,b cannot display peak loss and amplitude sensitivity outcomes clearly according to their respective colors.

Effect of Analyte Layer Thickness (ALT) on Loss and Amplitude Sensitivity
The peak loss is regulated by the thickness of the analyte layer as shown in Figure 11a. In Figure 11a, the peak losses are found to be 67.8950 dB/cm, 67.6150 dB/cm, and 67.3408 dB/cm for analyte layer thicknesses of 0.54 µm, 0.56 µm, and 0.58 µm, respectively. The highest analyte thickness of 0.58 µm provides the least peak loss of 67.3408 dB, and the resonance wavelength does not shift towards the higher wavelength with increasing analyte layer thickness. This is because a small damping loss occurs with increasing analyte layer thickness, which stabilizes the resonance wavelength in a fixed position. In Figure 11b, a similar trend is realized in which the amplitude sensitivity of the sensor is affected by the thicknesses of the analyte layer. In particular, the amplitude sensitivity is found to increase with increasing analyte layer thickness, as denoted by the values of 3457 RIU −1 , 3596 RIU −1 , and 3746 RIU −1 for analyte layer thicknesses of 0.54 µm, 0.56 µm, and 0.58 µm, respectively. Moreover, when the analyte layer thickness is less than 0.54 µm or greater than 0.58 µm, the sensor cannot detect x-polarized direction and y-polarized direction light signals in the core area, which are necessary to calculate sensing parameters. The peak loss and amplitude sensitivity vary slightly with changing analyte layer thickness. Therefore, Figure 11a,b cannot display peak loss and amplitude sensitivity outcomes clearly according to their respective colors.  Table 1 presents a comparison of the sensing parameters of the proposed PCF-SPR sensor with those of previously published articles. Table 1 shows that some of the prior articles did not compute certain sensing parameters, such as sensor resolution, FOM, and amplitude sensitivity. In comparison, the designed sensor offers a good enough RI range of analytes, FOM, and amplitude sensitivity, suggesting that its sensing performance is better than that of the prior published articles. Table 2 displays the optimal values of various geometrical parameters for the PCF-SPR sensor, which include the air holes areas of A, B, and C, the gold layer thickness (GLT), the analyte layer thickness (ALT), and the perfectly matched layer (PML). The optimal areas determined for air holes of A, B, and C were 0.0314 µm 2 , 0.6359 µm 2 , and 1.0 µm 2 , respectively. Additionally, the optimal thicknesses for the gold layer, analyte layer, and PML were determined as 26 nm, 580 nm, and 80 nm, respectively.  Table 1 presents a comparison of the sensing parameters of the proposed PCF-SPR sensor with those of previously published articles. Table 1 shows that some of the prior articles did not compute certain sensing parameters, such as sensor resolution, FOM, and amplitude sensitivity. In comparison, the designed sensor offers a good enough RI range of analytes, FOM, and amplitude sensitivity, suggesting that its sensing performance is better than that of the prior published articles. Table 2 displays the optimal values of various geometrical parameters for the PCF-SPR sensor, which include the air holes areas of A, B, and C, the gold layer thickness (GLT), the analyte layer thickness (ALT), and the perfectly matched layer (PML). The optimal areas determined for air holes of A, B, and C were 0.0314 µm 2 , 0.6359 µm 2 , and 1.0 µm 2 , respectively. Additionally, the optimal thicknesses for the gold layer, analyte layer, and PML were determined as 26 nm, 580 nm, and 80 nm, respectively.

Correlation between Resonance Wavelength and Refractive Index (RI) of Analyte
The polynomial fitting curve presented in Figure 12 provides a useful tool for predicting the resonance wavelength of the PCF-SPR sensor in response to changes in the RI of analytes. The formula of polynomial curve fitting states as Y = 14,680x 4 − 80,650x 3 + 1641x 2 + 148,522x + 50,394, where Y represents the resonance wavelength and x represents the RI of the analyte. The researchers can estimate the expected resonance wavelength for a given change in RI using the formula: The high R 2 squared value of 0.9996 indicates that the polynomial fitting curve provides an accurate representation of the relationship between resonance wavelength and RI. This information is crucial for designing and optimizing sensors for particular applications.

Correlation between Resonance Wavelength and Refractive Index (RI) of Analyte
The polynomial fitting curve presented in Figure 12 provides a useful tool for predicting the resonance wavelength of the PCF-SPR sensor in response to changes in the RI of analytes. The formula of polynomial curve fitting states as Y = 14,680x 4 − 80,650x 3 + 1641x 2 + 148,522x + 50,394 , where Y represents the resonance wavelength and x represents the RI of the analyte. The researchers can estimate the expected resonance wavelength for a given change in RI using the formula: The high squared value of 0.9996 indicates that the polynomial fitting curve provides an accurate representation of the relationship between resonance wavelength and RI. This information is crucial for designing and optimizing sensors for particular applications. Figure 12. A polynomial fitting curve that illustrates how the resonance wave varies with respect to change in the RI of the analyte, specifically ranging from 1.28 to 1.42. Figure 12. A polynomial fitting curve that illustrates how the resonance wave varies with respect to change in the RI of the analyte, specifically ranging from 1.28 to 1.42.

Exploring the Correlation between Sensor Length and Loss
As shown in Figure 13, both the sensor length and loss depend on the RI of the analyte. Figure 13 indicates that the analyte with a minimum RI of 1.28 results in the maximum sensor length, whereas the analyte with a maximum RI of 1.42 results in the minimum sensor length. Correspondingly, the analyte with a minimum RI of 1.28 leads to the minimum loss, while the analyte with a maximum RI of 1.42 causes the maximum loss. In general, the sensor length and loss have an inverse relationship in which an increase in sensor length leads to a decrease in loss.

Exploring the Correlation between Sensor Length and Loss
As shown in Figure 13, both the sensor length and loss depend on the RI of the analyte. Figure 13 indicates that the analyte with a minimum RI of 1.28 results in the maximum sensor length, whereas the analyte with a maximum RI of 1.42 results in the minimum sensor length. Correspondingly, the analyte with a minimum RI of 1.28 leads to the minimum loss, while the analyte with a maximum RI of 1.42 causes the maximum loss. In general, the sensor length and loss have an inverse relationship in which an increase in sensor length leads to a decrease in loss.

Potential Fabrication Methods
The PCF-SPR sensor design proposed in the literature consists of microstructured circular and rectangular air holes, a gold layer, and an analyte layer. The accurate fabrication of the sensor's geometrical parameters is crucial for achieving optimum sensing performance. Several well-known fabrication methods can be used to fabricate these geometrical parameters, including sol-gel, standard stack-and-draw, stack-and-drilling, 3D printing, extrusion, capillary stacking, injection modeling, and more [74,75]. Sol-gel is a generalized fabrication method that can be used to form basic silica structures. This method involves several steps, including hydrolysis and polycondensation, gelation, aging, drying, densification, and crystallization [76]. The automatic machine-controlled stack-and-draw method can also be used to fabricate circular and rectangular air holes accurately [77]. This method involves using solid, thin-wall, and thick-wall rod tools to create the desired geometries. The fabrication of rectangular air holes in a PCF involves defining a rectangular pattern on a glass rod or preform, drilling the pattern with a laser or mechanical drill, and then drawing the preform into a fiber to elongate and compress the rectangular holes [78]. The resulting PCF has a periodic arrangement of rectangular air holes that can be used to achieve specific optical properties. Thin gold layers can be deposited on the outside of the PCF structure using various methods, including wheel polishing method (WPM), chemical vapor deposition (CVD), automatic layer deposition (ALD), and high-pressure microfluidic chemical deposition methods [79][80][81].

Promising Applications of PCF-SPR Sensors
The proposed PCF-SPR sensor design has potential applications in various fields due to its good enough sensitivity in detecting the RI of biological analytes within the range of 1.28 to 1.42. This RI range is particularly important as many crucial biochemical solutions and biological analytes have RIs that fall within this range. The suggested PCF-SPR sensor

Potential Fabrication Methods
The PCF-SPR sensor design proposed in the literature consists of microstructured circular and rectangular air holes, a gold layer, and an analyte layer. The accurate fabrication of the sensor's geometrical parameters is crucial for achieving optimum sensing performance. Several well-known fabrication methods can be used to fabricate these geometrical parameters, including sol-gel, standard stack-and-draw, stack-and-drilling, 3D printing, extrusion, capillary stacking, injection modeling, and more [74,75]. Sol-gel is a generalized fabrication method that can be used to form basic silica structures. This method involves several steps, including hydrolysis and polycondensation, gelation, aging, drying, densification, and crystallization [76]. The automatic machine-controlled stack-and-draw method can also be used to fabricate circular and rectangular air holes accurately [77]. This method involves using solid, thin-wall, and thick-wall rod tools to create the desired geometries. The fabrication of rectangular air holes in a PCF involves defining a rectangular pattern on a glass rod or preform, drilling the pattern with a laser or mechanical drill, and then drawing the preform into a fiber to elongate and compress the rectangular holes [78]. The resulting PCF has a periodic arrangement of rectangular air holes that can be used to achieve specific optical properties. Thin gold layers can be deposited on the outside of the PCF structure using various methods, including wheel polishing method (WPM), chemical vapor deposition (CVD), automatic layer deposition (ALD), and high-pressure microfluidic chemical deposition methods [79][80][81].

Promising Applications of PCF-SPR Sensors
The proposed PCF-SPR sensor design has potential applications in various fields due to its good enough sensitivity in detecting the RI of biological analytes within the range of 1.28 to 1.42. This RI range is particularly important as many crucial biochemical solutions and biological analytes have RIs that fall within this range. The suggested PCF-SPR sensor design can be used for real-time monitoring of various solutions and biological analytes, including silicone oil (RI = 1.403), acetone (RI = 1.

Conclusions
The proposed twin-core, D-shaped PCF-SPR sensor is a promising technology for detection RI of analytes due to its simple structure and high sensitivity. The use of COMSOL Multiphysics software and the finite element method (FEM) allows for the design and optimization of the sensor's structural parameters, resulting in optimal sensing performance. The proposed sensor can be fabricated using sol-gel, standard stack-and-draw, and automatic layer deposition (ALD) methods, which are widely used in the manufacturing of other PCF-SPR sensors. The results of the simulation show that the PCF-SPR sensor has a high maximum wavelength sensitivity of 9000 (nm/RIU), amplitude sensitivity of 3746 RIU −1 , FOM of 900 RIU −1 , and sensor resolution of 1 × 10 −5 RIU in the x-polarized direction light signal. These promising results suggest that the proposed PCF-SPR sensor could be a potential contender for detecting a wide RI range of biological agents, chemical solutions, and complex diseases in the human body. However, understanding the behaviors of the PCF-SPR sensors in different conditions and applications is crucial to modifying their designs and testing them in the future. Overall, the proposed PCF-SPR sensor could be used for health control, environmental monitoring, effective monitoring of air and water quality, ensuring the safety and quality of food products, and more.