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Abstract
We propose Tamm plasmon (TP) and surface plasmon (SP) hybrid modes for hemoglobin (Hb) detection in anisotropic graphene-photonic-crystal (GPC) structures. The proposed GPC sensor shows polarization-dependent responses due to the in-plane anisotropic property. The reflection profiles of the proposed sensor exhibit two reflectivity minima due to the simultaneous excitation of TP and SP modes. When used to detect Hb, the TP mode offers a greater figure-of-merit (FoM) than the SP mode. Using a Fourier mode spectral analysis, we observe energy coupling from the TP to the SP mode when the incident light’s polarization changes, providing an option to enhance the sensor’s sensitivity. We propose a double dips method (DDM) to detect Hb based on the simultaneous excitation of TP and SP modes. Using DDM, the proposed sensor offers a maximum sensitivity of 314.5 degrees/RIU and a FoM of 1746 RIU−1 when the Hb level is 189 g/L. The proposed anisotropic GPC sensor offers possible applications for highly sensitive bio-molecule detection with high FoM.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In surface plasmon resonance (SPR), transverse magnetic (TM) surface waves propagate at the dielectric-metal interface. Such SPR resonances show a broad reflected light intensity profile ($R$-profile) due to scattering and absorption losses in the noble metal [1]. Therefore, SPR optical sensors are limited in their performances and further development. Using two-dimensional (2-D) graphene in SPR sensors is promising to enhance performance. However, the light absorption by a graphene monolayer is only 2.3%, insufficient for optical sensing and modulation [2]. Increasing the graphene layer number can increase the light absorption and the SPR sensor’s sensitivity. However, the full-width at half-maximum (FWHM) of $R$-profiles broaden when the graphene layer number increases [3], limiting graphene’s applications in plasmonic sensors [4].
Different research groups have proposed graphene-photonic-crystal (GPC) structures to increase the light-graphene layer interaction [1,5–7]. GPCs, generally known as an artificial periodic array structure formed by graphene, dielectrics, and photonic crystals (PCs), are a form of photonic devices created through intermittent preparation of dielectric layers of different refractive indices [8]. Structures based on multi-layer dielectric on monolayer graphene have been explored previously [9,10]. The graphene monolayer in a GPC can enhance the photon absorption by Bragg fluctuation, and therefore, the light-graphene interaction [4]. GPCs are suitable for numerous sensing applications due to their fast response time and ultra-high sensitivity [11]. In addition, the electromagnetic wave propagation in GPCs can be controlled by exploiting the structure’s photonic bandgap (PBG) [12].
GPC structures have applications in optical modulators, polarizers, filters, and biosensors [13]. Surface plasmons (SPs) can be excited in GPCs with significant photon absorption, although SPs suffer scattering losses in GPCs [14]. GPCs also show properties of hyperbolic metamaterials and support the excitation of Tamm plasmons (TPs), critical coupling, and negative refraction [15–20]. TPs are surface modes excited between metal and distributed Bragg reflector (DBR) [21]. TPs can be stimulated directly by the TM or transverse electric (TE)-polarized incident light, and there is no need for dispersion regulation like exciting SPs. The $R$-profile of TPs is narrower than SPs [7]. In addition, TPs show low absorption loss and high quality factor, which is appropriate for optical modulation and sensing [22]. We need high quality factors for optical sensing as it improves the detection accuracy and signal-to-noise ratio.
Hybrid TP-SP modes can be excited in a metal-GPC structure [23]. The TP mode is excited at the metal-GPC interface, while the SP mode is excited at the metal-sensing layer interface. As a result, an anti-crossing effect is observed on their dispersion relation [23]. The anti-crossing effect alters the hybrid TP-SP plasmonic modes’ dispersion properties and decreases the SP mode’s absorption losses. Therefore, the $R$-profile due to the excitation of hybrid TP-SP modes becomes narrower than the traditional SPR. The anti-crossing effect shows a coupling state between TP and SP, allowing energy exchange between these modes [24].
Recently, different research groups have proposed TP-based refractive index and temperature sensors [7,25,26]. A graphene-porous silicon PC has been proposed, where figure-of-merit (FoM) and sensitivity are controlled by varying the incident light angle and the porosity and thicknesses of silicon layers [27]. Also, asymmetric graphene-DBR has been used to create TP modes, and the effects of the incident light’s polarization on sensor performances have been investigated [28]. In contrast, a refractive index sensor containing GPCs has been proposed as a defect layer [7]. However, the proposed refractive index sensor is polarization and phase-insensitive. Recently, a terahertz (THz) graphene-DBR refractive index sensor has been proposed where sensor performances are phase and polarization-dependent [29]. THz frequency cannot be used for complex bio-molecule detection due to non-ionizing behavior and low photonic energy radiation [30]. Besides, TP and SP modes can be combined to improve the sensor performances in the optical wavelength region [31].
Hemoglobin (Hb) is an essential part of blood found in the red blood cell (RBC). The oxygen supply to tissues depends on blood’s Hb levels, and a variation in the Hb level causes health concerns [32]. Normal blood Hb levels are 120 to 160 g/L for females and 140 to 180 g/L for males [33]. The refractive index of blood changes by 0.001 unit (RIU) when the Hb level increases by 6.1025 g/L [33]. Heparin is widely used to determine the Hb level from RBC, requiring extensive pre-processing [34]. Heparin-based Hb level detection suffers measurement errors. A whispering gallery-based optical method has been used to detect the Hb level, although the sensitivity of this sensor is low [33]. In addition, a graphene-based SPR sensor has recently been proposed for Hb detection, with the sensitivity fluctuating with the Hb level [34].
This work proposes GPC-based TP-SP hybrid modes for Hb detection in the visible wavelength region. We have optimized the silver (Ag) layer thickness and the number of the unit cell ($N$) of GPCs, considering the minimum $R$ and FWHM values. The energy coupling from TP to SP mode has been determined using the Fourier mode spectral analysis (FMSA). We show anti-crossing effects between TP and SP modes for the incident light’s different polarization angles ($\varphi$). We use Barer’s analytical model to calculate the optical properties of Hb in blood. The sensor’s change of resonance angle ($\Delta \theta _r$) and FWHM for both TP and SP modes are calculated using the finite difference time domain (FDTD)-based angular interrogation method. In addition, we apply the double-dips method (DDM) to determine the sensitivity ($S$) and FoM. The proposed sensor shows energy coupling from TP to SP modes, helping improve the sensitivity for Hb detection.
The rest of the paper is organized as follows: Sec. 2 presents and discusses the proposed sensor structure and optimization. Section 2 discusses the anti-crossing effect between TP and SP modes and the simulation approaches. Section 3 presents the analytical models applied to investigate the optical properties of different materials and discusses the TP resonance mode identification and the energy coupling from TP to SP modes. We show and discuss the calculated sensor performances in Sec. 4. Finally, we conclude the findings in Sec. 5.
2. Proposed sensor
2.1 Configuration
Figure 1 illustrates the proposed anisotropic GPC TP-SP hybrid Hb detection sensor. The graphene-based PC material layers are stacked on the BK7 prism substrate. The GPC comprises consecutive layers of graphene, SiO$_2$, and TiO$_2$ with thicknesses 1.02 nm, 110 nm, and 70 nm, respectively. The GPC structure is similar to a DBR structure [35]. Graphene monolayer thickness ($t_g$) is 0.34 nm. In the proposed sensor, we use three graphene monolayers (3$t_g$). SiO$_2$ and TiO$_2$ thicknesses are calculated by $d_{s}=\lambda /4 n_s$ and $d_{t}=\lambda /4 n_t$, where $\lambda$ is the operating wavelength and $n_s$ and $n_t$ are the refractive indices of SiO$_2$ and TiO$_2$, respectively [7]. We have set $\lambda =633$ nm as the operating wavelength, which is frequently used in SPR-based experiments [36]. SiO$_2$ and TiO$_2$ layers are chosen for wide PBG due to their significant refractive index contrast [1].
Fig. 1. Schematic illustration of the proposed anisotropic GPC-based TP and SP hybrid mode Hb detection sensor.
GPC is a periodic structure with a periodicity of 181.02 nm. The number of periods ($N$) is optimized as 9 for the minimum $R$ value. At the top of the last period, we place another graphene layer with 3$t_g$ thickness as the termination layer. The terminating graphene layer reduces the scattering loss of the SP mode [37]. Also, the graphene termination layer helps the Ag layer avoid oxidation. Followed by the graphene termination layer, the Ag layer has an optimized thickness ($d_{\text {Ag}}$) of 60 nm, which determines the strength of interaction between TP and SP modes in the sensing layer. Therefore, $d_{\rm Ag}$ plays a crucial role in the TP-SP mode coupling [26].
As Hb is a bio-marker of several diseases, fast and sensitive Hb detection is crucial. The human blood serum contains Hb as a sensing element that can flow on the top of the sensor surface by a flow channel [38]. We will need to bind Hb to immobilized antibodies on the graphene surface. Graphene has a rich $\pi$-$\pi$ structure, which can immobilize antibodies like antibody-enzyme complex (AEC) [39]. We use a 20-nm AEC layer on the terminating graphene layer as the bio-recognition element (BRE) AEC to detect Hb in human blood. AEC will capture specific Hb present in the sample.
We have used the optimum thicknesses for different layers. The proposed sensor structure is planar, and using present-day fabrication technologies, the fabrication of the proposed structure is not complex. In practice, the graphene layer can be deposited on the prism or GPC by dissolving atomic carbon in a vacuum using high temperature [40]. Dielectric materials can be grown on each other using evaporation techniques [41]. A glass flow cell can be used to inject analytes into the sensor [42].
2.2 Optimization of sensor structure
The Ag metal layer thickness and the GPC unit cell number $N$ will critically affect the overall performance of the proposed sensor. Therefore, optimizing the proposed sensor’s structural parameters is crucial for the best response. We have optimized $d_{\text {Ag}}$ using the approach discussed in Ref. [43]. In particular, we examine the effect of $d_{\text {Ag}}$ and the variation of $N$ on $R$-profiles as a function of the incidence angle ($\theta _i$). The optimization of $d_{\text {Ag}}$ and $N$ depends on the minimum reflected light intensity ($R_{\text {min}}$) and FWHM of the $R$-profile. While $R_{\rm min}$ occurs when the light absorption is maximum on the sensor surface, the FWHM represents the loss in the metal layer. Therefore, both $R_{\text {min}}$ and FWHM are crucial for the sensitivity enhancement of a sensor, and an optimized $d_{\rm Ag}$ and $N$ should produce both $R_{\text {min}}$ and FWHM as small as possible. Here, we consider both resonances of TP and SP modes when optimizing $d_{\text {Ag}}$ and $N$.
To optimize $N$, we vary it while keeping the layer thicknesses fixed. Here, we use sensing layer refractive index $n_{\text {Hb}}=1.32919$ and $d_{\text {Ag}}=60$ nm. Figure 2(a) shows $R$-profiles against $\theta _i$ for different $N$. We find two separate $R$ minimum values. The first dip at $\sim$57.52$^\circ$ is due to the excitation of TP between Ag and GPC, whereas the second dip at $\sim$63.97$^\circ$ is due to the excitation of SP at the Ag-sensing layer interface. A detailed explanation of the identification of TP and SP modes is given in Sec. 3.2. We find that $R_{\text {min}}$ and FWHM are minimum for the SP mode when $N=9$. However, the TP mode shows a minimum $R_{\rm min}$ and FWHM when $N=5$. When $N$ increases from 5 to 9, $R_{\text {min}}$ due to the TP mode increases from 0.2258 to 0.2718, indicating an absorption decrease of the incident light. Simultaneously, the $R_{\text {min}}$ due to the SP mode decreases from 0.0490 to 0.0029, indicating an absorption increase of the incident light. We choose $N=9$ for the proposed sensor.
Fig. 2. $R$-profiles of the proposed sensor with the GPC unit cell number (a) $N=5$, (b) $N=7$, and (c) $N=9$, when $d_{\rm Ag}=60$ nm. $R$-profiles of the proposed sensor with the Ag layer thickness (d) $d_{\rm Ag}=50$ nm, (e) $d_{\rm Ag}=60$ nm, and (f) $d_{\rm Ag}=70$ nm, when $N=9$. In each case, $n_{\rm Hb}=1.32919$, and $\lambda =633$ nm.
Next, we optimize $d_{\text {Ag}}$ from calculations shown in Fig. 2(d)-(f). We change $d_{\text {Ag}}$ while the other layer thicknesses are fixed, and $N=9$. When $d_{\text {Ag}}=60$ nm, both $R_{\text {min}}$ and FWHM show minimum values for the SP mode. However, the TP mode shows minimum value when $d_{\text {Ag}}=50$ nm. Increasing $d_{\text {Ag}}$ from 50 nm to 60 nm, the FWHM becomes narrower for both SP and TP modes. Furthermore, when $d_{\text {Ag}}=70$ nm, the $R$ value increases for both resonances. When $d_{\text {Ag}}$ increases from 50 nm to 60 nm, $R_{\text {min}}$ due to the TP mode increases from 0.088 to 0.2818, indicating an absorption decrease of the incident light. Simultaneously, $R_{\text {min}}$ due to the SP mode decreases from 0.076 to 0.0029, indicating an absorption increase of the incident light.
2.3 Anti-crossing behavior
The proposed sensor’s energy band can be altered using the anti-crossing effect between hybrid TP-SP modes [44]. The difference in bandgap positions between the TP and SP modes is exhibited by the corresponding modes’ $R$-minimum positions. Hence, the broadening or narrowing of the $R$-profile indicates the change in the coupling strength. The coupling between the TP and SP modes leads to the repulsion of their dispersion profile, which also depends on the metal layer thickness [23]. Likewise, the excitation of TP-SP hybrid modes depends on the total internal reflection conditions, and thus, these modes are sensitive to the incident light’s polarization [1]. The anti-crossing effect alters the hybrid TP-SP plasmonic modes’ dispersion properties and decreases the SP mode’s absorption losses. Therefore, the hybrid TP-SP system enhances the sensitivity if used as a sensor.
The coupling between the TP and SP modes against the incident light’s polarization angle is manifested in the $R$-profiles shown in Fig. 3. Figure 3 also shows the effects of $d_{\text {Ag}}$. When $d_{\text {Ag}}=50$ nm, as shown in Fig. 3(a), there is minimal splitting between TP and SP hybrid modes, $\sim$2 nm, at $\varphi =0$. TP and SP resonance spectra overlap and the modes do not interact with each other significantly. The results are similar when we increase $\varphi$. However, increasing $\varphi$ increases $R$, which may degrade the sensor performance.
Fig. 3. $R$-spectra of the proposed sensor vs. polarization angle ($\varphi$) when (a) $d_{\text {Ag}}=50$ nm, (b) $d_{\text {Ag}}=60$ nm, and (c) $d_{\text {Ag}}=70$ nm. In each case, $n_{\text {Hb}}=1.32919$.
Figure 3(b) shows a spectral separation of 30 nm between the TP and SP modes when $\varphi =0$ and $d_{\text {Ag}}=60$ nm. Therefore, a minimum gap is observed, which splits the SP mode from the TP mode and moves it far away from the Ag layer. Also, a minimum $R$ value is observed, which indicates a narrower $R$-profile. Thus, it enhances the sensor performance. However, increasing $\varphi$ increases $R$, indicating decreasing incident light absorption on the GPC-based sensor. As a result, the sensor performance decreases. Figure 3(c) shows a spectral separation of 8 nm when $\varphi =0$ and $d_{\text {Ag}}=70$ nm. The spectral separation decreases when $d_{\text {Ag}}$ increases from 60 to 70 nm. When the Ag layer thickness is $>60$ nm, SP and TP modes show weak inter-modal interaction [45]. Consequently, using $d_{\text {Ag}}=70$ nm in the proposed GPC-based sensor reduces the sensor performances. The sensor shows $\varphi$-dependence with $d_{\text {Ag}}=70$ nm as well.
2.4 Simulation method
In this work, we apply 2-D FDTD simulations to analyze the interaction of the sensor structure with the incident light. The simulation area is $9000\times 1000$ nm$^2$ in the $xy$ plane. We use the perfectly matched layers (PML) boundary condition in the $x$-direction while the Bloch boundary condition in the $y$-direction. We find solutions for a 2-D structure as the proposed sensor structure does not vary in the $z$ direction. We use a non-uniform meshing technique with ultra-fine mesh grids in FDTD simulations to limit the overall error to $<0.05$%. We have set a high mesh accuracy and used 1000 fs simulation time for the method. For convergence testing, we varied the PML layer number from 8 to 16 and found the minimum reflected light intensity variation of about 0.01% for TP and 0.028% for SP modes.
We used a TM- or TE-polarized light source with a 633 nm wavelength. The light incidence is oblique to enhance absorption by the GPC [46]. We vary the incidence angle from 55$^\circ$ to 70$^\circ$ to determine the $R$-profile. The light source is placed in the BK7 region at 4250 nm from the BK7-graphene interface. The $R$ value is also recorded in the BK7 region at 4400 nm from the same interface.
3. Theoretical modeling
3.1 Optical properties
The optical properties of different layers of the proposed sensor are dispersive. As a result, the interaction of light with the sensor structure depends on the incident light’s wavelength. The 633-nm wavelength of the incident light is often used in experiments [36]. The wavelength-dependent refractive index of the BK7 prism layer is calculated using the following expression [47]
Since graphene is an optically anisotropic uniaxial material due to its 2-D behavior, its permittivity ($\varepsilon _{g}$) tensor is given by [48]
The dispersion relation of anisotropic graphene can be expressed as [51]
The refractive index of SiO$_2$ and TiO$_2$ are 1.45702 and 2.30, respectively [54,55]. The Refractive index of Ag is determined by the Drude-Lorentz model [56]. We use AEC as an antibody made by fibronectin with a refractive index of 1.45 [57]. Human blood refractive index for Hb varies from 1.32919 to 1.36019, and the Hb level grows 6.1025 g/L for every 0.001 RIU variations in blood [33]. Therefore, the refractive index of the blood sample with different Hb levels is determined by Barer’s model [58]
where $n_0$ is the reference refractive index, $C=0.00276$ nm$^{-1}$ is the protein refraction increment constant, and $\Delta \lambda$ is the change of resonance wavelength. The Hb level is related to $n_{\text {Hb}}$ by [34] where $H_0$ is the reference Hb level, $H_r$ is the final Hb level, ${\rm A}=6102.50$ is a constant, $\Delta n$ is the change in refractive index.3.2 TP and SP resonances
TP modes are electromagnetic waves excited at the interface between a metal and GPC due to consecutive reflections [29]. In our proposed sensor, TP modes are created at the Ag-GPC interface. The dispersion curve of a TP mode lies within the light line, and thus, the TP mode can be excited by both oblique and normal incident light [59]. When TP modes are excited, $R_{\rm M}R_{\rm {GPC}}=1$, where $R_{\text M}$ and $R_{\text {GPC}}$ are the amplitude reflection coefficients of metal and GPC, respectively [60]. Notably, $R_{\text M}$, $R_{\text {GPC}}\simeq -1$ at the visible wavelength range [29]. Therefore, TP modes are confined between Ag, which has a negative real permittivity, and GPC bandgap. As light is incident on the sensor from the GPC side, as shown in Fig. 1, the energy confinement in TP modes will create a dip in the $R$-profile [60]. In addition, in such a GPC-based structure where Ag is present, TP and SP modes cannot be excited separately at a specific incidence angle [23]. There is a superposition of electromagnetic waves localized at the Ag-GPC interface due to TP and at the Ag-sensing layer interface due to SP, creating a hybrid state.
In Fig. 4, we show the $R$-profile of the proposed sensor structure when $n_{\text {Hb}}=1.32919$, $N=9$, and $\varphi =0^{\circ }$. We note two distinct $R$ minima. The first dip at 57.52$^{\circ }$ is due to the excitation of TP between Ag and GPC, whereas the second dip at 63.96$^{\circ }$ is due to the excitation of SP at the Ag-sensing layer interface. The FWHM due to TP is 0.1708$^{\circ }$, whereas it is 0.4730$^\circ$ due to the excitation of SP. In addition, several ripples in the $R$-profile are due to weak photonic modes excited in the various GPC layers.
Fig. 4. $R$-profile of the proposed sensor with and without the Ag layer, when $N=9$, $\lambda =633$ nm, $n_{\text {Hb}}=1.32919$, and $d_{\text {Ag}}=60$ nm. The dips in the solid red line at 57.52$^{\circ }$ and 63.96$^{\circ }$ within the PBG represent TP and SP resonance modes, respectively.
In Fig. 4, we show $R$-profile with and without the Ag layer. In the first case, we simulate the sensor without the Ag layer, i.e., the BK7-GPC-Sensing layer structure, to determine PBG [61]. The sensor without Ag shows a PBG when $\theta _i$ varies from 41.1$^{\circ }$ to 60$^{\circ }$, with no excited mode in this region. Next, we simulate the sensor with the Ag layer, i.e., the BK7-GPCs-Ag-Graphene-Antibody-Sensing layer structure. We observe a resonance dip at 57.52$^{\circ }$. This resonance is a TP mode located inside the PBG, created by the interference of electromagnetic waves at the interface between Ag and GPC layers [31].
3.3 Coupling energy to TP and SP modes
FMSA approach permits the examination of each mode’s characteristics, including spatial energy density profile and uncoupling of different scattering [62]. We follow the approach of Ref. [63] to determine how strongly energy couples to a particular mode and shifts from TP to SP modes. Using the FMSA method helps us to analyze the energy coupled to each mode separately. One-dimensional (1-D) fast Fourier transform (FFT) is performed on the coupled light to the sensor to find the wave vector or spatial frequency. Figures 5(a)-(c) presents a Fourier power profile vs. spatial frequency for different $N$. In addition, we present the same analysis when $\varphi$ changes to observe the $\varphi$-dependent energy coupling from TP to SP in Figs. 5(d)-(f).
Fig. 5. Fourier power spectrum vs. spatial frequency for (a) $N=5$, (b) $N=7$, and (c) $N=9$, when $\varphi =0$, and (d) $\varphi =0$, (e) $\varphi =30^{\circ }$, and (f) $\varphi =60^{\circ }$, when $N=9$. In each case, $n_{\text {Hb}}=1.32919$ and $d_{\text {Ag}}=60$ nm.
The first mode at 0.0173 $\mu$m$^{-1}$ in Fig. 5(a) represents the TP mode, whereas the second mode at 0.7983 $\mu$m$^{-1}$ represents the SP mode. We have also calculated the spatial frequency for SP modes using $k_{\text {SP}}=k_{0}\sqrt {{\varepsilon _{m}\varepsilon _{\text {Hb}}}/( {\varepsilon _{m}+\varepsilon _{\text {Hb}}})}$ and for TP modes using $k_{\text {TP}}=k_{0}\sqrt {{\varepsilon _{m}\varepsilon _{\text {eff}}}/({\varepsilon _{m}+\varepsilon _{\text {eff}}})}$, where $\varepsilon _{m}$ is the real part of the permittivity of Ag and $\varepsilon _{\text {eff}}$ is the effective permittivity of GPC. In Figs. 5(a) and (b), when $N=5$ and 7, maximum power is coupled to the sensor with the TP mode confining greater power than the SP mode. When $N$ increases from 7 to 9, the TP mode’s power decreases, with both modes confining power approximately equally, making the coupled energy transfer from the TP mode to the SP mode when $N$ increases.
Figures 5(d)–(f) show that the coupled power to TP and SP modes decreases as $\varphi$ increases, however, with power confining more to the SP mode than the TP mode. We aim to couple energy from TP to SP in the proposed sensor. As the maximum energy of TPs is confined in the GPC and near the Ag-GPC interface, TP electromagnetic fields barely relate to the outer sensor surface. Therefore, the TP mode features are less responsive to the changes of the refractive index of sensing medium [25].
3.4 Sensor performance parameters
Sensitivity ($S$) and FoM are the main performance parameters of GPC-based TP-SP hybrid resonance sensors. These parameters are determined using the $R$-profile. The sensitivity is determined as [64]
where $\Delta \theta _r(\text {SP/TP})$ is the change in resonance angle of $\theta _r$ for either SP or TP mode. On the other hand, FoM is determined as [65]4. Sensing performance
Figure 6(a) shows the sensing layer refractive index as a function of $C_{\rm Hb}$. We vary $C_{\rm Hb}$ from zero to 189 g/L. The sensing layer refractive index $n_{\text {Hb}}$, in the presence of Hb, is calculated using Eqs. (7) and (8). When $C_{\rm Hb}=0$, $n_{\text {Hb}}=1.32919$, and $n_{\text {Hb}}$ linearly increases to 1.36019 as $C_{\rm Hb}$ increases to 189 g/L. As $C_{\rm Hb}$ changes in the human blood sample, the sensing layer experiences a difference in $n_{\text {Hb}}$. Consequently, the index variation changes SP and TP wave vectors, ultimately changing the resonance angle $\theta _r$ for TP and SP modes.
Fig. 6. (a) Sensing layer refractive index vs. Hb concentration ($C_{\rm Hb}$), (b) $\Delta \theta _{r}$ vs. $C_{\rm Hb}$ for the TP resonance mode, and (c) $\Delta \theta _{r}$ vs. $C_{\rm Hb}$ for the SP resonance mode.
Figure 6(b) shows $\Delta \theta _{r}$ for the TP mode as a function of the $C_{\rm Hb}$ level for different $\varphi$. We note that $\Delta \theta _r$ increases with $C_{\rm Hb}$ for the TP mode. However, $\Delta \theta _r$ does not vary noticeably when $\varphi$ changes from $0^{\circ }$ to $30^{\circ }$. When $\varphi$ changes from $0^{\circ }$ to $60^{\circ }$, the change in $\Delta \theta _{r}$ is significant. Figure 6(c) shows a similar trend in the $\Delta \theta _{r}$ vs. $C_{\rm Hb}$ relationship for the SP mode. However, $\Delta \theta _{r}$ changes more significantly for the SP mode than the TP mode. We note that $\Delta \theta _{r}$ for the SP mode is $\sim$2 times greater than $\Delta \theta _{r}$ for the TP mode.
Figure 7 shows the sensor performances as a function of $C_{\rm Hb}$ using the TP resonance separately. Figure 7 also shows the polarization-dependent anisotropic behavior of the proposed sensor. We use $\varphi =0$, $30^{\circ }$, and $60^{\circ }$, where $\varphi =0$ means that the incident light is TM polarized, whereas $\varphi =30^{\circ }$ and $60^{\circ }$ represent the TE-polarized incident light. Figure 7(a) shows the sensitivity of the proposed sensor as a function of $C_{\rm Hb}$. Considering the TP mode, the sensitivity increases slightly and depends on $C_{\rm Hb}$ negligibly. Since the energy of the excited TP mode is confined at the Ag-GPC interface, the electromagnetic field due to the TP mode insignificantly interacts with the sensing layer. Hence, $\Delta \theta _r$ is linear for the TP mode and increases approximately at a constant rate, as shown in Fig. 6(b). Therefore, the sensitivity is approximately constant for a $C_{\rm Hb}$ from 0 to 189 g/L. The maximum sensitivity is observed at $\varphi =0$, and the sensitivity decreases when $\varphi$ increases since $\Delta \theta _r$ reduces gradually as $\varphi$ increases, as shown in Fig. 6(b). Therefore, we note that the TP mode shows low polarization-dependent sensitivity.
Fig. 7. (a) $S$, (b) FWHM, and (c) FoM as a function of $C_{\rm Hb}$ of the proposed GPCs-based sensor using TPs resonance mode.
Figure 7(b) shows the FWHM of the proposed sensor as a function of $C_{\rm Hb}$ for different $\varphi$ for TP resonances. The FWHM increases when $C_{\rm Hb}$ or $\varphi$ increases since the $R$-profile broadens when $C_{\rm Hb}$ or $\varphi$ increases. In Fig. 7(c), we show the FoM of the proposed sensor for the TP mode. Generally, FoM decreases when $C_{\rm Hb}$ or $\varphi$ increases. However, when $\varphi =0$ and $C_{\rm Hb}$ is zero, the FoM is maximum with a value of 545 RIU$^{-1}$. The value of FoM gradually decreases when $C_{\rm Hb}$ increases and ${\rm FoM}=517$ RIU$^{-1}$ when $C_{\rm Hb}$ is 189 g/L. As the increasing $C_{\rm Hb}$ raises the $n_{\text {Hb}}$ value, $\Delta \theta _r$ increases. However, the FWHM increases as $C_{\rm Hb}$ increases, and hence decreasing FoM. Similar tendency is observed when $\varphi =30^{\circ }$ and $\varphi =60^{\circ }$. However, FoM decrease more when $\varphi =60^{\circ }$ than when $\varphi =30^{\circ }$. Therefore, the proposed GPC-based sensor shows performances that depend on the polarization of the incident light.
Figure 8(a) shows the sensitivity of the proposed sensor as a function of $C_{\rm Hb}$ considering the SP mode. The sensitivity increases significantly as a function of $C_{\rm Hb}$ due to a large resonance angle shift. As energy confines in the Ag-sensing layer interface due to the SP resonance, the electromagnetic field interacts significantly with the sensing element. The sensitivity increases from 138 to 220 degrees/RIU when $C_{\rm Hb}$ increases from zero to 189 g/L. At $C_{\rm Hb}= 189$ g/L, the sensitivity using the SP mode is 57.60% greater than the TP mode. However, the sensitivity decreases with $\varphi$ and decreases maximum when $\varphi =60^{\circ }$. We note that when $C_{\rm Hb}$ is 189 g/L, the sensitivity decreases from 220 to 181 degrees/RIU as $\varphi$ increases from 0 to $60^{\circ }$. Thus, the proposed sensor’s sensitivity depends on the incident light’s polarization.
Fig. 8. (a) $S$, (b) FWHM, and (c) FoM as a function of $C_{\rm Hb}$ of the proposed GPCs-based sensor using SPs resonance mode.
Figure 8(b) shows the FWHM of the proposed sensor for the SP mode as a function of $C_{\rm Hb}$ for different $\varphi$. The FWHM increases when $C_{\rm Hb}$ or $\varphi$ increases. The FWHM increases by approximately three times the TP mode as the $R$-profile of SP resonances is much broader than TP resonances when $C_{\rm Hb}$ or $\varphi$ increases. The main reason for the broader FWHM of the SP mode is the losses associated with the Ag layer, which is essential for the plasmonic wave creation [1]. In Fig. 8(c), we show the FoM of the proposed sensor for the SP mode. Generally, FoM decreases when $C_{\rm Hb}$ increases from zero to 100 g/L. After that, it starts to increase. This tendency is observed for all the values of $\varphi$. Similar to the trend observed for the sensitivity, the FoM decreases more significantly when $\varphi =60^{\circ }$.
However, when $\varphi =0$ and $C_{\rm Hb}=0$, the FoM is maximum with a value of 290 RIU$^{-1}$. If we use the SP mode instead of the TP mode, FoM reduces by 46.77% when $\varphi =0$. FoM gradually decreases when $C_{\rm Hb}$ increases and ${\rm FoM}=228$ RIU$^{-1}$ when $C_{\rm Hb}= 100$ g/L. As $n_{\text {Hb}}$ increases with $C_{\rm Hb}$, $\Delta \theta _r$ also increases. However, the FWHM increases as $C_{\rm Hb}$ increases, decreasing FoM. A similar tendency is observed when $\varphi =30^{\circ }$ and $60^{\circ }$. FoM increases for $C_{\rm Hb}> 100$ g/L as FWHM gradually saturates when $C_{\rm Hb}>100$ g/L, as shown in Fig. 8(b). The SP mode shows a significant dependence of its performance on the polarization of the incident light.
In a hybrid TP-SP mode, the TP mode drives the SP mode away from the Ag layer, decreasing the loss associated with the SP mode. In TP resonance, the FWHM is narrower due to low losses, as only a part of the TP mode propagates at the Ag-GPC interface, offering higher FoM. However, the TP mode is excited at the inner side of the Ag layer; hence, does not directly access the sensing element [1]. Consequently, using the hybrid TP-SP mode for biosensing explores the TP mode in a strong coupling region, and applying one more mode provides the accuracy of sensing performances [5]. We can control the width of the $R$-profile by transferring the TP mode energy to the SP mode, helping the SP mode provide the minimum $R$ value with a narrower FWHM.
Next, we calculate the sensitivity and FoM using DDM to hybrid TP-SP modes in the proposed sensor. The sensitivity is calculated as [66,67]
Figure 9(b) shows the FoM determined using the TP mode. The FoM is considerably high, $\sim$1740 RIU$^{-1}$, for $\varphi =0$ and $C_{\rm Hb}=189$ g/L. The TP resonance provides a narrow FWHM due to a small loss of incident light at the Ag-GPC interface. The FoM using the TP mode also shows a polarization-dependent behavior. Figure 9(c) also shows the FoM of the proposed sensor for the SP resonance mode. Generally, FoM decreases when $C_{\rm Hb}$ increases from zero to 126 g/L and then increases for $C_{\rm HB}>126$ g/L. This tendency is observed for all $\varphi$. As usual, the FoM reduction rate in the sensitivity analysis is more severe when $\varphi =60^{\circ }$. Compared to the SP resonance mode, when $\varphi =0$ and $C_{\rm Hb}=0$, the FoM enhancement is $\sim$40% using DDM. Using the SP mode, FoM progressively decreases at a $C_{\rm Hb}$ is around 100 g/L, as shown in Fig. 8(c). However, using DDM, this position shifts to 126 g/L as DDM offers more sensitivity enhancement.
Fig. 9. (a) $S$, (b) FoM using TPs, and (c) FoM using SPs as a function of $C_{\rm Hb}$ of the proposed GPCs-based sensor applying DDM.
The proposed GPC-based sensor’s sensitivity and FoM are better than the state-of-the-art optical sensors. We compare the sensitivity and FoM of the proposed sensor in Table 1 with some recently reported sensors. $C_{\rm Hb}$ is 189 g/L, which corresponds to $n_{\text {Hb}}=1.36019$ with those reported in the literature, as given in Table 1. The compared sensors also operate at 633 nm incident wavelength and report results for a sample of index $\sim$1.36. We have considered whether or not the sensor uses the DDM technique to determine the sensitivity and FoM.
Table 1. Performance comparison of our proposed sensor with different recently proposed sensors with or without using DDM.
5. Conclusion
The proposed anisotropic GPC-based hybrid SP-TP mode sensor can sense the Hb level in a blood sample with high sensitivity and FoM. We find that the sensor performances depend on the hybrid plasmonic mode, which can be controlled through the energy coupling between TP and SP modes with an appropriate Ag layer thickness. This mode coupling decreases losses incurred in the Ag layer and narrows the plasmonic resonances. We utilize the anisotropic properties of the GPC and observe polarization-dependent sensor performance. Using DDM, a maximum of 314.5 degree/RIU sensitivity and 1746 RIU$^{-1}$ FoM are found when $C_{\rm Hb}$ is 189 g/L. Besides, we can employ the TP resonance mode as a secondary probe, which can provide extra information for analyzing the adsorption process at different interfaces of metal-GPC structure. The proposed TP-SP hybrid modes of the GPC-based sensors can be used for various types of protein detection.
Disclosures
The authors declare that they have no conflict of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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