Introduction

The terahertz (THz) wavelength, covering a broad frequency range from 0.1 to 10 THz, is utilized in various applications such as biotechnology, telecommunication, imaging, and biomedical sensing [1,2,3]. Recent advancements in THz sources and detectors demonstrated various THz modulators based on metamaterials. Metamaterials, as artificially engineered and consisting of optical meta-atoms, are subwavelength resonators with highly confined and enhanced electrical fields that can manipulate light in unconventional ways [4, 5]. However, the modulators reported are limited to operating within a specific frequency range. In practical scenarios, it is essential to possess a robust adjustable operating frequency for dynamically modulating the performance without reconstructing the system. Graphene, a two-dimensional (2D) carbon-based material, has recently garnered significant attention due to its exceptional optoelectronic properties. One fascinating aspect of graphene is its capability to easily adjust carrier density through electrical control, enabling flexible alteration of performance with bias voltage rather than necessitating changes in the geometries [6,7,8]. Moreover, different from acting like an absorption material in the near-infrared range, graphene can support surface plasmon polaritons (SPP) in THz frequency like a metallic material. Therefore, graphene becomes a perfect candidate for designing a plasmonic modulator with dynamic tunability in the THz frequency [9,10,11].

Note that the free space electromagnetic waves cannot directly interact with the graphene plasmon (GP) due to the vector mismatch. Various approaches have been presented to enhance the light-graphene interaction. A typical way is to pattern the graphene layer into periodic arrays forming a metasurface, such as strips [1, 12] and circles, to form a highly localized terahertz electromagnetic field. However, the carrier mobility of patterned graphene may be significantly reduced, and the precision of lithography can greatly affect the sensing performance [13, 14]. Coupling the evanescent wave with SPP in a metallic slip for optical modulators and sensing detectors has been widely reported and applied in the visible and infrared frequencies. Thus, similar plasmonic modulators and sensing devices can be constructed, as graphene can support SPP in the THz range. Pickwell-MacPherson et al. implemented a total internal reflection (TIR) arrangement to generate evanescent wave coupling with the graphene layer. They experimentally presented a THz light modulator with high modulation depth (> 90%) in broadband from 0.15 to 0.4 THz [15]. Similarly, using a higher refractive index (RI) prism, Hwang presented a design of an evanescent wave biosensor incorporating the surface plasmon resonance on a periodic graphene ribbons array [16]. Therefore, integrating a higher RI waveguide offers a practical strategy to improve vector matching, and exciting the plasmon [17, 18].

Similar to the TIR method, fiber functions as a high-caliber optical waveguide. Specifically, the photonic crystal fiber (PCF), with its adjustable and versatile structural characteristics, presents a fascinating approach to generating graphene plasmon. Consequently, the fusion of PCF with the graphene surface plasmon has attracted considerable interest and has been extensively studied in light modulation and sensors [19, 20]. Liu et al. experimentally constructed a half-meter-length graphene PCF modulator with strong light-matter interaction (~ 8 dB·cm−1 attenuation), broadband response (1150–1600 nm), and significant modulation depth (~ 20 dB·cm−1 at 1550 nm) [21]. With monolayer graphene located at the inner surface of the air hole near the core, Wang et al. designed a PCF sensor based on graphene plasmon. The frequency sensitivity is about 208.14 GHz/RIU, and the loss spectrum can be dynamically tuned by adjusting the Fermi energy within a sensitivity of 767.5 GHz/eV [22]. Here, RIU is short for refractive index unit, a property of the analyte measured with an Abel instrument [23]. In 2023, they theoretically designed a graphene plasmonic sensor based on D-shaped PCF, where the graphene layer is directly coated at the D-shaped surface plane. The average wavelength sensitivity is 305.5 µm/RIU while obtaining a maximum amplitude sensitivity of 30.85 RIU−1 [24]. Besides, Yao et al. proposed a tunable THz fiber modulator based on graphene-coated D-shaped fiber. The modulation depths of the two polarization states are 40.51% and 24.54% at 0.64 THz [25]. In these studies, the analyte is in contact with the graphene surface, thus modulating the graphene plasmon mode. The primary mechanism of these modulators and sensors involves the interaction between the core-guided light beam and the graphene surface plasmon. Moreover, the enormous structural flexibility of the PCF enables novel approaches to fill materials into its micro holes, which in turn can modulate the core-guided mode. This filling strategy has many applications, including mode-locked fiber lasers, surface plasmon generation, in-fiber Raman scattering, and light modulation devices [26, 27]. Consequently, if the detecting analyte is concurrently located in the core and sensing channel, new sensing and modulation phenomena may be achieved.

In this work, we theoretically demonstrate a graphene plasmonic resonance sensor based on liquid-core PCF, and the sensing performance is investigated by the finite element method (FEM). As the analyte is filled at the fiber core and graphene sensing channel, the analyte modulates both the GP and the core-guided mode. A negative RI sensitivity is observed for the first time in the THz plasmonic sensing setup. The detailed modulations of dispersion relations on analyte RI are studied to analyze the original mechanism. Additionally, the unique tunability property is also presented.

Theoretical Sensing Analysis

Structure Design and Simulation Model

The cross-sectional view of the proposed graphene plasmonic photonic crystal fiber is shown in Fig. 1a. It is an index-guiding PCF consisting of three layers of air holes arranged in a hexagonal way. A relatively big center hole with graphene coated in the inner wall carrying plasmon is used as the detecting channel filled with the analyte. Then, six liquid cores are formed by extending the corresponding air holes in the second layer with a diameter of 1.2 Λ and infilling the analyte with RI higher than the substrate [28]. Therefore, the index-guiding mechanism is guaranteed for each individual core, forming a multi-core graphene (MC-Gr) PCF sensor. With this arrangement, the light transmitted in each core can excite the plasmon coupling on the graphene/analyte intersurface, and the signal-to-noise ratio (SNR) can be improved consequently. Note that the detecting analyte fills in both the multi-core and center channel marked with blue blocks. Therefore, the analyte can modulate both the graphene plasmon and core-guided mode. The red solid circle indicates the graphene nanosheet with extra-thin structure.

Fig. 1
figure 1

a Cross-section of the designed multi-core PCF-based graphene plasmonic sensor. b FEM mesh and boundary conditions for the calculation. PML is short for a perfectly matched layer

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The center-to-center distance between air holes is called lattice constant with a value of Λ = 600 µm, and the air hole diameter is d = 0.5 Λ. The radius of the center sensing channel is 0.6 Λ. For our investigation in a THz range, TOPAS cyclic olefin copolymer (COC) can be used to construct the fiber because of its near-zero material dispersion, low absorption, and high chemical resistivity [29]. Thus, the PCF’s RI is assumed to be 1.5 RIU without considering the dispersion. The cladding hole is filled with air (nair = 1). A full-vectorial FEM method is used to analyze its light propagation property, especially the mode coupling between the multicore and surface graphene plasmon. With the structure featuring a C6V rotational symmetry, only a quarter of the PCF cross-section needs to be calculated. The perfect magnetic conductor (PMC) and perfect electric conductor (PEC) conditions are applied in the vertical and horizontal boundary to maintain the propagating mode [30]. The whole structure is discretized with triangular for calculation shown in Fig. 1b.

For the sheet structure of graphene with an extremely thin layer, a conductivity (σg) parameter is used to describe its property. According to the Kubo-equation, σg is the sum of two electron transitions called the intra-band term (σintra) and the inter-band term (σinter) [31, 32]. Because the low THz photon energy hardly induces the inter-band electron transition, only the intra-band electron transition (σintra) is considered:

$$\sigma_g\approx\sigma_{\mathrm{int}ra}=\frac{{ie}^2k_BT}{\pi{\hbar^2}\left(\omega+\frac i\tau\right)}\left(\frac{E_f}{k_BT}+2In\left(1+\text{exp}(-\frac{E_f}{k_BT})\right)\right)$$
(1)

Here, e, kB, and ℏ are universal constants, which symbolize the charge of an electron (1.6 × 10−19 C), the Boltzmann’s constant (1.3806 × 10−23 m2kgs−2K−1), and the reduced Planck constant [33]. Ef is the Fermi energy of graphene and has a trace of critical drive property by the bias voltage, which enables the sensor with dynamic tunability. As the graphene with low concentration doping ratio, |Ef|> > KBT and ℏω < 2|Ef| are considered. Thus, the contribution of intra-band can be simplified as follows [34]:

$${\upsigma }_{g}\approx \frac{{e}^{2}{E}_{f}}{\pi {\hbar}^{2}}\frac{i}{\omega +i/\tau }$$
(2)

Here, τ is the relaxation time. Therefore, the graphene permittivity can be expressed as εg = 1 + iσg/(ωε0tg), where ε0 is the vacuum permittivity. The graphene layer is ultra-thin with a tg value of 0.34 nm.

Mode Analysis and Coupling Mechanism

The designed MC-PCF sensor is a superstructure and supports multiple super modes [30, 35]. Each mode has a specific electric field distribution with a corresponding effective refractive index, as shown in Fig. 2. As the real part of the effective RI decreases in a specific sequence, they are identified as super mode one, two, and three (SP1, SP2, and SP3 for short), respectively. For every individual mode, the distribution of the field exhibits a Gaussian profile without any splitting features, implying that these three modes are super modes rather than being of a higher mode [35]. In the case of the SP1 mode, the electric field orientations in adjacent cores are uniform (indicated by a light arrow), whereas for the SP2 and SP3, the field vectors exhibit variations. It is worth mentioning that the imaginary part of SP3 mode is much higher than the other two, suggesting that the SP3 exhibits significant loss properties when propagating over the same PCF length. On the other hand, the real components exhibit disparity as well. Consequently, these dispersion relationships have the potential to intersect with graphene plasmon modes within varied terahertz ranges, thereby inducing resonance coupling.

Fig. 2
figure 2

Electric field distributions and corresponding effective RI (neff) for the super modes supported by the MC-PCF (analyte RI = 1.52, f = 0.6 THz, and Ef = 0.6 eV). The arrows represent the electric field direction

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The spectral loss, which relies on the imaginary component of the effective index of the core mode, can be represented by the following relationship [36, 37]

$$a\left(dB/m\right)=8.686\cdot{k}_{0}\cdot\mathrm{ Im}\left({n}_{eff}\right)$$
(3)

Here, k0 is the wavelength number calculated as k0 = 2π/λ. The dispersion relations of the core-guided SP3 and the GP mode, along with the loss spectra, are depicted in Fig. 3. An evident loss peak emerges at the intersection of the two dispersion lines, signifying the fulfillment of phase-matching conditions between the GP and SP3 mode. Consequently, a portion of the core-guided mode is transferred to the analyte/graphene intersurface, generating a graphene plasmon resonance. As a result, a loss peak manifests in the core transmission mode.

Fig. 3
figure 3

Dispersion lines of core-guide and GP mode with the loss spectrum. The inserts a, b, and c are the optical field distribution at different ranges. Insert i is the electric field distribution of the whole cross-section at the resonance frequency

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To clearly demonstrate the modulation mechanism, Fig. 3a–c provide a visual representation of the field distributions of the core mode across various frequency ranges. At lower frequencies, the core mode predominantly concentrates within the core area. As the frequency increases, the dispersion of GP mode rises faster than that of core-guided mode, and an intersection appears between these two. It indicates that a high matching degree is achieved. At the cross point, a greatly enhanced local field is generated at the analyte/graphene intersurface, as depicted in Fig. 3b. Simultaneously, the intensity of the core field is relatively diminished, indicating the transfer of core-guided mode to GP mode, resulting in a loss peak. Moreover, at higher frequencies, the electric field refocuses within the core due to phase mismatch. The utilization of the mirroring process reveals the full field distribution of the fiber cross-section at resonance, shown as insert (i). It suggests that the GP mode splits around the boundary. These field distribution characteristics indicate that the generated GP mode is a higher order. Furthermore, the core energy being partially transferred into the GP mode implies an incomplete coupling between graphene plasmon and core mode [35]. In addition, the mode coupling also causes the twist of the dispersion curve near the resonance [38, 39].

Sensing Performance

Negative Wavelength Sensing

The plasmonic sensor’s detecting mechanism is ensured by the dependence of the graphene plasmonic mode on the analyte’s RI. As the dispersion relations varying with analytes, here we focus on the RI dependence response. Figure 4a shows the corresponding loss spectra when the analyte varies from 1.51 to 1.58, taking the RIs of sensing liquid covering harmful drugs, oils, cholesterol, nicotine, and bacillus bacteria [40]. The spectra shift towards a higher frequency, and at the same time, the loss peak decreases dramatically. In contrast to the general trend where a higher RI corresponds to a longer wavelength, the designed PCF sensor exhibits a negative wavelength sensitivity. To understand this unique feature in detail, the dispersion relations of both the GP and SP3 modes are calculated and presented in Fig. 4b. When the analyte RI increases from 1.51 to 1.53, both the GP and the SP3 mode experience modulation and move upwards. Assuming that the core mode (SP3) remains unaffected by varying analyte RI, the original phase matching point [i] moves to a lower frequency point [ii] due to the upward movement of the GP mode dispersion. However, because the core-guided dispersion also moves upwards with the increase in RI, the phase matching point [ii] undergoes a right shift and eventually moves to point [iii]. By defining the increment of an effective index as the modulation degree marked in Fig. 4b, the modulation degree of SP3 on the analyte (Δneff.core) is found to be 1.7075 × 10−2, which is much higher than that of the graphene plasmonic mode (Δneff.GP = 0.7081 × 10−2). This results in the resonance moving towards a shorter wavelength (larger frequency) from point [i] to [iii]. Therefore, a larger upward movement corresponds to an enhanced modulation degree, meanwhile Δneff.GP and Δneff.core indicate the left and right frequency shift directions, respectively.

Fig. 4
figure 4

a Resonance loss spectrum at different RIs. b Dispersion relations of both the core-guided SP3 mode and graphene plasmon at analyte RI of 1.51 (solid lines) and 1.53 (dotted lines)

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Positive Wavelength Sensitivity Based on Electric Tunability

Since the modulation relation of Δneff.core > Δneff.GP leads to a negative wavelength sensitivity, a counter situation can be obtained when Δneff.core < Δneff.GP. Fortunately, graphene’s chemical potential (Fermi energy) can be easily controlled by adjusting its bias voltage, which allows us to increase Δneff.GP conveniently. Here is an example that demonstrates positive wavelength sensitivity. Figure 5 illustrates two loss spectra with dispersion lines, one at 1.52 RIU with Ef = 0.7 eV (solid lines) and the other at 1.54 RIU with Ef = 0.65 eV (dotted lines). A noticeable left shift is observed in the resonance. This is mainly due to the higher modulation degree of the GP mode on the Fermi energy, resulting in Δneff.core < Δneff.GP. To be more specific, assuming that the core-guided mode is maintained, we only consider the GP mode’s modulation by Ef. The original phase matching point [i] undergoes a left shift to the new position [ii] due to the decrease in Ef from 0.7 to 0.65 eV. Then, with an increase in RI, the intersection performs a relatively small right shift to point [iii]. In this case, Δneff.GP is 2.2302 × 10−2, which is higher than Δneff.core = 1.6171 × 10−2. Additionally, the detailed tunability of the loss spectra on Ef is illustrated as the insert (a). The spectrum center can be efficiently adjusted from 0.510 to 0.612 THz, resulting in a tuning sensitivity of 510 GHz/eV. It is important to note that during the tuning, the peak intensity varies slightly. However, compared to RI = 1.52 (Ef = 0.7 eV), the loss spectrum of RI = 1.54 (Ef = 0.65 eV) is noticeably damped. Thus, this reduction is mainly caused by the RI modulation rather than the Ef. With an increase in RI, the core-cladding index contrast becomes enhanced. Then, the light is more restricted in the core area, leading to less energy transfer. As a result, the coupling degree weakens, and the loss spectrum decreases.

Fig. 5
figure 5

Loss spectrum with dispersion lines of core-guided SP3 and GP mode at 1.52 RIU (Ef = 0.7 eV) and 1.54 RIU (Ef = 0.65 eV). Insert figure: loss spectrum at Fermi energies from 0.5 to 0.7 eV

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Geometry Dependence and Tunability of the Sensor

Geometry Dependence

Due to the flexible design of PCF’s structure and to facilitate optimized sensing, we investigate the dependences of the loss spectrum on main structure parameters, i.e., central analyte diameter d0, liquid-core diameter dc, and cladding air hole (d), as illustrated in Figs. 6a, b and 7a, respectively. Among these parameters, the loss spectrum can be efficiently adjusted by d0. With varying d0 from 0.55 to 0.65 Λ, we observe a significant left shift (from 0.586 to 0.544 THz) and a substantial increase in the loss peak intensity (from 168.9 to 562.7 dB/m). This can be attributed to the fact that the GP mode dispersion moves upward with an increase in d0. On the other hand, the core-guided SP3 mode remains unaffected by the changes in the sensing channel, resulting in a left shift of the dispersion intersection. For dc in the range of 0.4–0.6 Λ, the loss spectrum exhibits a slight variation, allowing much flexibility in the sensor design.

Fig. 6
figure 6

Loss curves for PCF-based graphene plasmonic sensor with different parameters: center detecting channel d0 (a), and cladding liquid core dc (b). Simulation resolution: 0.002 THz

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Fig. 7
figure 7

The dependence of loss spectrum on cladding air hole diameter d (a) and carrier relaxation time τ (b)

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Setting d as 0.4 Λ, 0.5 Λ, and 0.6 Λ, a right shift in resonance is observed, starting from 0.558 THz, then 0.562 THz, and finally reaching 0.572 THz. The intensity of the primary peak gradually decreases from 392.1 to 302.2 dB/m and then to 173.7 dB/m. This can be attributed to the reduction in the cladding effective index due to the increase in air hole diameter, which results in an enhancement of the core-cladding index contrast. Additionally, the dispersion relation of the core mode moves upwards. In contrast, the GP mode remains unaffected by the variation in d, leading to the rightward movement of the phase matching point. Further calculation shows that increasing lattice weakens the resonance intensity without any frequency shift. Moreover, the carrier mobility, i.e., the relaxation time τ, serves as a critical parameter for evaluating the quality of graphene. As reported in unsuspended devices, the carrier mobility ranges from 2000 to 25,000 cm2/V/s. For suspended systems, this value can reach up to 230000 cm2/V/s [41]. Figure 7b illustrates the impact of τ on the loss spectrum, where the resonance frequency remains relatively unchanged, but the peak intensity significantly decreases with decreasing τ. Simultaneously, the full width at half maximum (FWHM) broadens. In practical applications, a smaller FWHM is preferred to achieve a higher SNR, highlighting the importance of the graphene quality in sensing performance.

Tunability Based on the Fermi Energy (E f)

The distinctive advantage of a graphene-based plasmonic sensor lies in its tunable ability, achieved through the adjustment of the Fermi energy (Ef). The specific details of this capability are elucidated in Fig. 8. This depiction reveals that the detection frequency, for a given RI range (1.51–1.54 RIU), can be artificially manipulated. As the Ef value increases within the range of 0.6 to 0.9 eV, the loss spectrum shifts from 0.554–0.574 THz to 0.686–0.724 THz, with a tuning sensitivity of 0.44 THz/eV. Furthermore, for each Ef value, the resonance spectrum behaves similarly, shifting towards the right and gradually decreasing with RI increase. The comprehensive sensing parameters can be concluded in Table 1. Both the frequency (changing from 0.6667 to 1.2667 THz/RIU) and intensity (ranging from 5.2933 to 7.4350 × 103 dB/m/RIU) sensitivities are enhanced at higher Ef values, while the FWHM remains relatively stable. Supposing the spectral resolution to be 0.01 THz (Terasys 4000, Rainbow Photonics), the detecting limitation may reach up to 0.0079 RIU [42]. It is worth noting that, for a specific RI value with increased Ef, the resonance intensity diminishes. Therefore, there is a need to carefully consider the trade-off between sensitivity and resonance depth in the detection process.

Fig. 8
figure 8

Turnability of loss spectrum on Fermi energy from 0.6 to 0.9 eV, and corresponding RI sensing

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Table 1 Comparison of the sensing performance at different Ef
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Different from conventional plasmonic sensors, the tunability of graphene plasmon can be flexibly controlled by manipulating its bias voltage. Thus, to detect specific analyte RIs, the measurement frequency can be shifted to a desired range, making full use of the limited bandwidth optical source or the narrowband spectrometer. As a result, a multi-series-channel system can be constructed by solely manipulating the Ef. In comparison to alternative methods for achieving an adjustable detection range, such as incorporating nano-porous silica coatings [43], introducing materials into a hollow fiber [44, 45], or reconfiguring the components of sensors [46, 47], the work presented here provides a convenient and feasible design that holds great potential for a wide range of applications. Moreover, with the tunability of the THz resonance spectrum via controlling its bias voltage, this sensing probe can also work as a switch for specific frequencies [48, 49]. Besides the tunability, the high sensitivity makes the designed sensor stand out from the comparison in Table 2.

Table 2 Performance comparison with relative work
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For the fabrication process, the graphene layer can be obtained using the chemical vapor deposition (CVD) method. Then, with liquid etchants and solvents to delaminate graphene from its growth substrate, transfer the layer into the target channel in a liquid environment. Furthermore, the TOPAS COC can be fabricated using stack and draw-type techniques to ensure the smoothness of inner holes [54, 55].

Conclusion

This paper investigates a graphene plasmonic sensor based on a core-filled PCF to demonstrate the dispersion analysis method and modulation properties. The analyte is filled at both the core and sensing material surface, resulting in simultaneous modulation of the core-guided mode and plasmon mode by the RI. The modulation degree of Δneff.core > Δneff.GP (or Δneff.core < Δneff.GP) produces a negative (or positive) wavelength sensitivity. Furthermore, the impact of the geometry on the sensor performance is analyzed. Utilizing graphene as a tunable plasmonic material, the detection frequency can be dynamically controlled within desired ranges. This enables the full utilization of a specific frequency range and presents a new flexible approach to constructing cascaded sensing channels. This innovative design and analysis method can be widely applied in other similar plasmonic sensing scenarios.