1 Introduction

Thanks to the THz-applications, the gap between microwave and infrared frequency bands is minimized. This frequency band which extends from 0.1 to 10 THz attracted the attention of numerous scientists (Liang et al. 2015). Such a band covers wide range of applications such as sensing (Jacobsen et al. 1996), biotechnology (Nagel et al. 2002), imaging (Chan et al. 2007; Mittleman et al.1999; Chen et al. 2003), spectroscopy (Zhang et al. 2004), astronomy (Ho et al. 2008) and non-invasive imaging (Chen et al. 2000). The material loss in the THz regime is large (Kersting et al 2000; Chan et al. 2009). Therefore, it is aimed to obtain a low loss medium and a convenient THz detection technique (Kersting et al 2000; Chan et al. 2009). Recently, several waveguides are proposed for THz regime such as metallic wire (Wang et al. 2004), metal coated dielectric tubes (Bowden et al. 2007), plastic fibers (Chen et al. 2006), Bragg fibers (Skorobogatiy et al. 2007), and polystyrene foam (Zhao et al. 2002). However, these waveguides still suffer from high material loss, coupling and bending losses to the surrounding environment (Wang et al. 2004; Bowden et al. 2007; Chen et al. 2006; Skorobogatiy et al. 2007; Zhao et al. 2002). Additionally, hollow core fibers (Hossain et al. 2014) and porous core photonic crystal fibers (PCFs) (Zhao et al. 2002; Hossain et al. 2014; Pristinski et al. 2006) have been recently employed in THz applications. The PCFs exhibit many unique properties that cannot be achieved using the traditional optical fibers such as endlessly single mode guidance, large effective modes and anomalous dispersion (Birks. et al. 1997; Hoang et al. 2018). Further, the optical properties of PCFs can be controlled by filling the air holes with different materials such as liquid crystals (Pan et al. 2008(, solvents, gases and liquid crystals as presented in (Sabbah et al. 2023; Le et al. 2021; Van et al. 2023). Additionally, the geometrical parameters in PCF structures including the hole pitch, hole diameter, and core shape can be tuned to control the guiding properties of the PCF (Bulbul et al. 2020). Therefore, the PCF has been used for different applications including polarization rotator (Hameed et al. 2010, Hameed et al. 2011b, Hameed et al. 2013), polarization splitter (Younis et al. 2018), multiplexer–demultiplexer (Hameed et al. 2009b).

Chromatic dispersion is one of the basic limitations in optical communication systems (Zsigri et al. 2004). In fiber optic communication systems, dispersion limits the maximum transmission distance and the bit rate (Zsigri et al. 2004). The dispersion causes the broadening of optical pulses when transmitted through the fiber. When an incident light travels through a waveguide like optical fiber it disperses as each source of light has a certain spectral width (Yadgeer et al. 2023). Within the same waveguide, every dispersed spectral component travels at a different group velocity where the phase velocity changes depending on the material and wavelength (Yadgeer et al. 2023). As a result, the pulse broadens in time because each spectral component arrives at the receiver at a different time. Thus, it must be compensated in the long-distance optical data transmission systems to nullify the pulse broadening (Zsigri et al. 2004). In this context, dispersion limits the optical bandwidth of operation and spreads the propagating optical pulses through the optical channels i.e., optical waveguides. The management of the chromatic dispersion has been reported in previous literatures (Dupuis et al. 2011; Nielsen et al. 2009). In a long-haul optical link, chromatic dispersion increases with the increase in the propagation distance. In recent years, some PCF based designs for dispersion compensation have been reported (Aliramezani et al. 2010). Most of the reported PCF designs were based on dual-concentric-cores where a central core and a ring-core are used (Aliramezani et al. 2010). In the presence of ring-core, if the geometrical parameters of the PCF are well selected, large negative dispersion can be obtained (Lu¨ et al. 1997). In a dual-concentric core fiber, the fundamental mode is confined in the core region as the inner mode at wavelengths shorter than the phase-matching wavelength (Han et al. 2014). However, the fundamental mode switches to the cladding defect area as the outer mode at longer wavelengths than the phase-matching wavelength (Han et al. 2014). The dual-concentric core fiber (DCF) shows a large negative waveguide dispersion (D) around the phase-matching wavelength because of the fundamental mode’s switch from the inner to the outer core (Han et al. 2014). A DCF with larger negative dispersion compensates for the dispersion of single-mode fiber (SMF) with shorter length, which can reduce the losses and nonlinearity of the conventional compensating fibers (Han et al. 2014). Therefore, studying DCF with large negative dispersion is substantial. Therefore, a strategy should be present to mitigate the dispersion effect in long distance optical communications. In this regard, dispersion compensation fibers (DCFs) (Koshiba et al. 2003) are employed to oppose or cancel the bad effects of dispersion. Dispersion compensating fiber (DCF) high negative dispersion is used to nullify the accumulated positive dispersion of single mode fibers (SMFs) (Koshiba et al. 2003, Abdelaal et al. 2020). Various DCFs were optimized to compensate for the dispersion in a single band, e.g., the S-band (1460–1530 nm), C-band (1530–1565 nm), L-band (1565–1625 nm), and U-band (1625–1675 nm) (Gruner-Nielsen et al. 1999, Varshney et al. 2005). It has also been reported that combining two or more DCFs can realize broadband compensation from the C-band to the L-band (Nielsen et al. 2002). In THz regime, Depuis et al. (2010) have used a porous core with background of Polystyrene (PE) with dispersion less than 1 ps/THz/cm over a frequency range of 0.1–0.5 THz. Additionally, (Nielsen et al. 2009) have utilized a PCF with cyclic olefin copolymer (COC) as a background material and reported a dispersion < 1 ps/THz/cm over the 0.35–0.65 THz range. Further, (Bao et al. 2015) have utilized anti-resonant HC fiber where the guiding mechanism is based on the combination of an anti-resonant effect and inhibited coupling between the core and the cladding modes with a background of Polymethylmethacrylate (PMMA). The obtained dispersion is less than 10 ps/THz/cm over the 0.3–1.0 THz range.

Flat dispersion is also highly desirable in the THz waveguides where several types of designs have been reported in (Atakaramians et al. 2009; Islam et al. 2015; Wu et al. 2016; Luo et al. 2017; Hasan et al. 2014; Hasan et al. 2015; Ali et al. 2016a; Ali et al. 2016b; Alice et al. 2021; Jibon et al. 2021; Wei et al. 2008). In this context, Atakaramians et al. (2009) have proposed a porous core PCF with rectangular and circular configurations to control the dispersion properties. The suggested structure with a slotted core PCF has been presented with a low dispersion of ± 0.5 ps/THz/cm through the frequency range 1–1.3 THz (Islam et al. 2015). Wu et al. (2016) have introduced an oligo-pc-PCF with low dispersion of 0.2 − 1 ps/THz/cm in a wide frequency range of 2.5–4 THz. Further, Luo et al. (2017) have presented a square porous-core PCF with low flattened dispersion of ± 0.12 and ± 0.08 ps/THz/cm over frequency range of 0.85–1.9 THz for the TE and TM modes, respectively. Ultra-low material loss PCF with TOPAS as a background material has been reported with flattened dispersion of ± 0.18 ps/THz/cm over the frequency range of 1.0–1.8 THz (Hasan et al. 2014). A Kagome PCF structure has been also presented with a low dispersion of 0.61 ± 0.14 ps/THz/cm (Hasan et al. 2015). Additionally, dispersion coefficients of 0.40 ± 0.04 ps/THz/cm (Ali et al. 2016a) and 0.95 ± 0.05 ps/THz/cm (Ali et al. 2016b) have been reported by using a hybrid design. Such hybrid core contains circular and triangular designs as a central porous region. Furthermore, a PCF with a cladding region of circular air-holes with a porous core region has been introduced to show flat zero dispersion of 1.85 × 10–33 ps/THz/m through the frequency range of 0.1- 0.6 THz (Alice et al. 2021). Further, a PCF with slotted porous core and rectangular cladding has been used to achieve dispersion of ± 0.004 ps/ THz/cm from 0.9 to 1.5 THz (Jibon et al. 2021).

In this paper, a TOPAS background DC-LCPCF structure is proposed and analyzed to control the dispersion in the THz band. Dual-porous core regions are implemented where the coupling between the supported modes is studied to achieve high negative dispersion. In order to have tunable design, the porous holes are selectively infiltrated by nematic LC (NLC) material of type 5CB. The full vectorial finite element method (FVFEM) (Obayya 2004) via COMSOL Multiphysics software (https://www.comsol.com/) is used to analyze the dispersion characteristics of the two polarized modes supported by the suggested design. The reported DC-LC PCF shows a high negative dispersion of −30.59 ps/THz/cm at f = 0.4027 THz for the TM mode and -44.57 ps/THz/cm for the TE mode at f = 0.386 THz. Therefore, the suggested design has a good potential for THz communication applications.

2 Design considerations

The 2D cross section view of the reported DC-LCPCF structure is depicted in Fig. 1. The TOPAS (n = 1.5258) is used as a background material with low material loss (Nielsen et al. 2009). The air holes in the cladding region of the suggested design are arranged in a hexagonal lattice with diameter (d) of 250 µm, and a hole pitch (Λ) of 398 µm. The background material of the suggested design is TOPAS material with low material dispersion and constant refractive index of 1.5258. Further, the TOPAS material has relatively low material loss where it increases with a rate of 0.36 \({cm}^{-1}\)/THz from 0.06 \({cm}^{-1}\) at 0.4 THz in the 0.1–1.5 THz range. Furthermore, it has low water absorption (~ 1/10 of the PMMA) and has a chemical resistance to common solvents (Khanarian 2001; Nielsen et al. 2009; Yang et al. 2018a, b).

Fig. 1
figure 1

Cross sectional view of the proposed PCF

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To control the dispersion of the DC-LCPCF, inner and outer porous cores are used. The inner core has a porous region that consists of three rings of air holes arranged in hexagonal lattice with diameter (dc) of 91 µm and a hole pitch (Λc) = 394 µm. Additionally, the porous holes are also selectively infiltrated by nematic LC (NLC) material of type 4′-n-pentyl-4 cyanobiphenyl (5CB to obtain high tunability. The refractive indices of the LC materials can be tuned by applying an external electric field (Wei et al. 2008; Zografopoulos et al. 2006; Ren et al. 2008). The refractive indices and transmission losses have been measured for many NLCs including 5CB, in far infrared and THz range (Nose et al. 1997; Pan et al. 2008). It has been reported that in the frequency range of 0.2–1 THz, 5CB doesn’t show any resonant absorption and clear dispersion (Nose et al. 1997; Pan et al. 2008).

The NLCs are anisotropic materials consisting of rod-like molecules, which are characterized by an ordinary refractive index (no), and an extraordinary refractive index (ne). The relative permittivity tensor of the NLC material is given as:

$$\varepsilon _{r} = \left( {\begin{array}{*{20}l} {no2sin2\phi + ne2cos2\phi ~} \hfill & {\left( {~ne2~ - ~no2} \right)sin\phi cos\phi } \hfill & 0 \hfill \\ {ne2~ - ~no2)~sin\phi cos\phi ~} \hfill & {no2cos2\phi + ne2sin2\phi } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {no2} \hfill \\ \end{array} } \right)$$
(1)

where ϕ is the rotation angle of the director of the NLC molecules around the x-axis as shown in the inset of Fig. 1. The Cauchy models of no and ne in the THz band of 0.219–0.9 THz at 25 °C have been reported in (Pan et al. 2008). Liquid crystal are ideal candidates for THz tunable devices due to their electrical control, large modulation depth and feasible integration with other materials/devices. The optical properties of the NLC-E7 in THz range have been experimentally studied in (Lei et al. 2017; Nico et al. 2010; Chan et al. 2010). In (Lei et al. 2017), an active terahertz metamaterial absorber based on LC has been studied under different DC bias directions and different incident wave polarization directions using THz time domain spectroscopy. Most of the LC devices operate with AC voltages. Nevertheless, the applied voltage to the LC is RF signal, however, its frequency is so lower than the propagating wave optical frequency. Therefore, the RF appears as a “DC”. In addition, when a DC bias voltage is applied to the NLC, the molecules tend to be reoriented parallel to the direction of external electric field. In this regard, Hou et al. (2014) have studied the effect of the applied voltage on the rotation angle of the NLC molecules. It has been shown that when the applied voltage increases from 1 to 2.3 kV, the rotation angle increases rapidly (Hou et al. 2014). If the voltage is further increased beyond 3 kV, the rotation angle will be approximately flat where the molecules are oriented along the E-field direction. In this paper, the 5 kV DC bias voltage is applied to the NLC to achieve a rotation angle of φ = 90°. Therefore, it is easy to apply and already studied (Lei et al. 2017; Nico et al. 2010; Chan et al. 2010; Hou et al. 2014).

The most common way to fill a PCF with a liquid is by capillarity force (Algorri et al. 2018). One PCF end is dipped into the fluid while the other is left open at room pressure. The necessary condition for the infiltration of the liquid through the holes is the affinity with the PCF material, namely the cohesive forces between the molecules of the liquid must be lower than the adhesion forces of the liquid with the channel material (Algorri et al. 2018). Extra care is needed in cases where the inner surfaces of the microcapillaries have to be functionalized by coatings or alignment layers (Algorri et al. 2018). Often the selected liquid for infiltration in a PCF can have an excessive viscosity or the time necessary to infiltrate by capillary is too long. In other cases, one end of the PCF is blocked due to splicing to a standard fiber. Under such circumstances, it is necessary to apply an external positive or negative pressure to fill the fiber in (Algorri et al. 2018). When combined with infiltration in vacuum, these methods become very efficient and enable the infiltration of viscous materials such as liquid crystals or polydimethylsiloxane (PDMS) (Markos et al. 2010). When the infiltration of only a certain pattern of microcapillaries is needed, most employed methods rely on the selective blocking of the holes. This can be achieved by directly blocking one by one the selected holes with some other material, such as a polymerizable glue (Wu et al. 2009; Milen´ko et al. 2013), or by milling a microchannel into the end facet of a PCF (Wang et al. 2011). Another commonly used method is by selectively collapsing the smaller holes with a fusion splicer (Xiao et al. 2005). Another approach takes advantage of the difference in infiltration speed by capillary action among holes with different diameter (Huang et al. 2004; Nielsen et al. 2005). As the bigger holes are infiltrated faster than the smaller, the difference between the infiltrated lengths for holes of different size increases with time, creating two fronts of infiltrated material within the PCF. By cleaving the fiber at a distance between the two fronts only the bigger holes remain infiltrated with the liquid. This technique is simpler, but it is limited by the geometry of the PCF and hence does not allow for arbitrary infiltration patterns. More advanced methods were also developed, carving aside-access, or using micromachining with a femtosecond laser to expose their inner holes for material infiltration (Wang. et al. 2011; Martelli et al. 2007). This method only allows for a limited number of infiltration patterns, contrary to the unrestricted, but more time-consuming method of blocking the capillaries one by one.

3 Numerical results and discussion

Dispersion is one of the most crucial properties that limit the quality of signal transmission in THz based optical waveguides. Thus, dispersion should be treated in optical communications and specifically in PCFs. Dispersion is defined as the optical pulse broadening that occurs during the propagation process through the optical waveguides (including PCFs). It is important to reduce the problem that occurs because of the signal overlap due to the dispersion problem. Therefore, an ultra-flat dispersion and/or dispersion compensating fibers are mandatory for effective transmission of broadband waves (Hasan et al. 2016). There are different types of dispersion that may include but are not limited to material dispersion and waveguide dispersion. Material dispersion results from the wavelength or frequency dependence of the material’s refractive index (Ghataket al. 2000). However, the waveguide dispersion occurs due to the different group velocities of waveguide modes which doesn’t depend only on the wavelength but also on the structure or dimensions of the optical waveguide (Ghatak et al. 2000). Since the refractive index of TOPAS is constant between 0.1 and 2 THz (Bao et al. 2012), the induced material dispersion can be neglected. Therefore, the pulse spreading (dispersion) occurs in the reported structure is only due to the waveguide dispersion. The waveguide dispersion coefficient (β2) can be calculated as a function of the real part of the mode effective index (neff) and is given by Eq. (2) (Islam et al. 2017):

$$\beta_{2} = \frac{2 }{c}\frac{{d_{neff} }}{d\omega } + \frac{\omega }{c} \frac{{d^{2}_{neff} }}{{d\omega^{2} }} {\text{s}}/{\text{THz}}/{\text{cm}}$$
(2)

where ω is the angular frequency, neff is the real part of the effective index of the studied mode, and c is the speed of the light in vacuum.

The operation of the proposed device depends on the coupling process between one of the two fundamental inner core modes (TE or TM) and the corresponding outer core mode to obtain high negative dispersion. The strong coupling can be obtained by achieving the phase matching condition between the two modes supported by the two cores; inner core and outer core. Phase matching between any two modes can be obtained when the two modes have the same propagation constant where they travel through the waveguide structure with the same velocity. In order to achieve the phase matching condition, the porous air holes through the inner and outer cores are infiltrated by NLC material of type 5CB as shown in Fig. 1. Further, the phase matching can be controlled through the liquid crystal properties by the application of an external voltage (Das et al. 2016; Al-Mamun et al. 2020a, b). A suitable homeotropic anchoring conditions (Zografopoulos et al. 2006) should be used to produce the proposed alignment of the NLC molecules. Moreover, the external voltage should exceed the Fredrick’s threshold (Hameed et al. 2015) through pair of metal electrodes (Haakestad et al. 2005) to achieve good alignment of the molecules along the applied electric field as may be seen in Fig. 1. In this context, (Tasolamprou et al. 2009) have studied the influence of the applied voltage on the NLC molecules orientation at V = 10, 40, and 120 V. It has been shown that the director tuning can be achieved above a saturating voltage of 150 V (Tasolamprou et al. 2009). Moreover, a good alignment of the director of the NLC with a constant rotation angle ϕ can be obtained by controlling the space between the electrodes and the PCF. In addition, (Lei et al. 2009) have utilized sets of electrodes to adjust the NLC molecules alignment. In our study, a uniform alignment along the x-axis with ϕ = 0° can be obtained with Vx = V0 and Vy = 0. However, at Vx = 0 and Vy = V0, a constant y-parallel alignment will be achieved with ϕ = 90° as shown in Fig. 1.

The excitation of the suggested THz PCF can be obtained using simple porous fibers similar to those reported in (Dupuis et al. 2009, 2010; Cruz et al. 2013). In this regard, the porous fiber fabricated in (Cruz et al. 2013) can be used for the excitation of the proposed PCF with slight larger dimensions (dc = 91 μm, Ʌc = 394 μm) and adding only 3 air holes rings to construct the porous core region. In this case, large overlap integral and the power coupling between the two fundamental TE modes of the two fibers are equal to 0.975191 and 0.975189, respectively. This means that approximately 97.5% of the power will be sufficiently coupled from the excitation THz fiber to the proposed THz PCF. It is worth noting that the fundamental mode is either the TE or the TM mode where the reported design may be employed to work with any of them. In our proposed work, TE mode means x- polarized mode where the major component is in the x-direction (Ex) and the minor component is in the y-direction (Ey). On the other hand, TM mode means y-polarized mode in which the major component is Ey, and the minor component is Ex.

First, the modal characteristics of the proposed PCF structure are studied. The supported modes in the PCF structure are inner modes (confined in the inner core region) and outer modes (supported by the outer core region infiltrated by the NLC material). Each mode is studied in terms of its neff, mode profile and dispersion coefficient (β2). The simulations for this design are carried out via COMSOL Multiphysics software package (https://www.comsol.com/) which is based on the well-known FEM (Obayya et al. 2004). Perfectly matched layer (PML) boundary conditions are applied to all transverse directions to truncate the simulation domain. The proposed structure is discretized with minimum element size of 0.596 µm while the number of degrees of freedom solved for is 1.3 × 106. It is worth mentioning that a perfectly matched layer (PML) (Yang et al. 2018) boundary condition is applied to truncate the simulation region from all transverse directions. After performing an optimization study, a PML radius of ~ 6.2 mm is used to ensure numerical stability of the obtained results.” The main goal of the reported PCF design is to obtain as much as high negative dispersion. Thus, the suggested PCF can be employed for dispersion compensation purpose. Further, it is important to study the influence of different parameters such as outer core diameter (d) and core porosity (P) on the dispersion characteristics of the reported waveguide design. The initial geometrical parameters taken from (Hossain et al. 2021) are listed in the Table1. These dimensions help analysis the suggested PCF in THz regime to achieve highly negative dispersion compensation by using Topas background material. Additionally, the design ensures feasible excitation process especially in THz frequencies. Further, the used dimensions will offer simple NLC infiltration process.

Table 1 Initial geometrical parameters of the proposed PCF structure depicted in Fig. 1
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Initially, the DC-LCPCF structure shown in Fig. 1 with the initial geometrical parameters summarized in Table 1 is considered. It is worth noting that the two fundamental polarized modes (TE and TM modes) are guided in the inner core of the reported PCF. In addition, the same types of modes are supported in the NLC-filled rings (outer core). Under certain conditions, strong coupling is obtained when only one mode from the two inner modes (TE or TM mode) is coupled to the corresponding outer mode (Abdelaal et al. 2020). However, the other inner mode propagates through the inner core region without any power transfer to the outer core. Figure 2 depicts the field distributions of the studied TE and TM modes at different THz frequencies, before the coupling point, exactly at the coupling point and after the coupling point between the inner and outer modes. It is worth noting that the rotation angle (ϕ) of the NLC molecules can be utilized to control the coupling process due to the phase matching between one of the modes supported by the inner core and the corresponding outer mode. At ϕ = 90°, the director of the 5CB material is normal to Ex and is parallel to Ey, so the dielectric permittivity tensor εr takes the diagonal form of [no2, ne2, no2] which can be verified via Eq. (1). At f = 0.3937, no and ne are equal to 1.722 and 1.613, respectively. So, at ϕ = 90˚, εxx = (no)2 = (1.722)2 while εyy is equal to (ne)2 = (1.613)2. Therefore, εyy is smaller than εxx that makes the effective index of the TE outer mode larger than that of the TM outer mode. Thus, the TM inner mode is coupled to the corresponding TM outer mode due to the phase matching between them at f = 0.368 THz as may be seen in Fig. 2(a). In this case (ϕ = 90°), the TE inner mode is well-confined in the inner core region with no power transfer to the outer core. At ϕ = 0°, following the same mechanism explained for the case at ϕ = 90°, the TE inner mode is coupled to the corresponding TE outer mode at f = 0.366 THz as shown in Fig. 2(b). However, the TM inner mode is confined in the inner core without any coupling with any of the outer core supported modes.

Fig. 2
figure 2

Field distributions of different inner/outer modes supported by the proposed DC-PCF structure at different operating frequencies including the phase matching frequency at a ϕ = 90° and b ϕ = 0°

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At frequencies below the coupling (phase matching) frequency (i.e., f = 0.361 THz) for a given polarization, the inner, or the fundamental, mode has no coupling with the corresponding outer, or the second-order, mode as may be seen in Fig. 2a and b. However, exactly at the coupling frequency (i.e., f = 0.368 THz), the power of the fundamental (inner) mode is fully coupled to the second order (outer) mode due to the phase matching condition. Additionally, at higher frequencies than the coupling frequency (i.e., f = 0.375 THz) for a given polarization, very weak coupling occurs. Furthermore, the uncoupled mode in each case is the one that propagates through the fiber without any coupling; TE inner mode at ϕ = 90° and TM inner mode at ϕ = 0° as shown in Fig. 2a and b, respectively.

It is worth mentioning that through the considered frequency range (0.3–0.45 THz), the values of ne and no are very close to each other with a difference that is approximately 0.1 (Hameed and Obayya 2011). Thus, both polarizations at different rotation angles are expected to have coupling (phase matching) points with the corresponding higher-order modes. To prove this concept, the frequency dependent effective indices of the supported inner and outer TE/TM modes are calculated and plotted in Fig. 3a and b with the initial geometrical parameters summarized in Table 1. It may be seen from Fig. 3a that at ϕ = 90°, inner TE mode has three phase matching points with three different higher order (outer) modes at different frequencies. The same operation is obtained for the inner TM mode at the same rotation angle as depicted in Fig. 3b. However, the phase matching points of the TM mode are obtained at three different frequencies than those of the TE mode; Fig. 3b. The same behavior is obtained at ϕ = 0° for the two polarized modes with different phase matching frequencies. Thus, the rotation angle of the NLC molecules can be used to control phase matching points for both TE and TM modes. It may be also seen from Fig. 3 that the coupling points are obtained for both polarizations through the studied frequency range (0.3–0.45 THz).

Fig. 3
figure 3

Frequency dependent effective indices of the a fundamental and higher-order TE polarized modes and b fundamental and higher-order TM polarized modes, of the proposed DC-PCF. This study is performed at ϕ = 90° and T = 25 °C while all geometrical parameters are fixed to their initial values summarized in Table 1

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Figure 4 shows the dispersion of the two polarized modes in the previous study (depicted in Fig. 3). As may be seen from Fig. 4a, the TM inner mode at ϕ = 90° is coupled to a corresponding outer mode with the same polarization at frequencies 0.3979 THz, 0.368 THz, and 0.3397 THz with induced negative dispersion values of − 26.36 ps/THz/cm, − 8.770 ps/THz/cm, and − 6.679 ps/THz/cm, respectively. However, the inner TE mode is coupled to the corresponding outer TE modes at f = 0.3937 THz, 0.3661THz, and 0.3366 THz with corresponding negative dispersion values of − 28.083 ps/THz/cm, − 6.922 ps/THz/cm, and − 7.634 ps/THz/cm, respectively at the same rotation angle i.e., ϕ = 90° as shown in Fig. 3b.

Fig. 4
figure 4

Frequency dependent dispersion calculations of different inner/outer modes supported by the proposed DC-PCF structure, a TM modes and b TE modes, at ϕ = 90° and c TM modes and d TE modes, at ϕ = 90°. This study is performed at T = 25 °C while all geometrical parameters are fixed to their initial values summarized in Table 1

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The same behavior is obtained for both TE and TM polarized modes at both ϕ = 0° as may be seen in Fig. 4c and d, respectively. The coupling frequencies and the corresponding negative dispersion values for both polarizations at ϕ = 90° and ϕ = 0° are summarized in Table 2. It may be noted that a maximum negative dispersion value of − 39.125 (Ps/THz/cm) is achieved for the TM polarized mode at f = 0.3945THz and ϕ = 0˚. However, for the TE mode, -37.47 (Ps/THz/cm) is achieved at f = 0.368 THz and ϕ = 90°. Thus, rotating the angle of the NLC molecules from 0˚ to 90˚ can control the operation of the DC-LC compensating PCF.

Table 2 Dispersion values for both polarizations at ϕ = 90˚ and ϕ = 0˚. In this study, the geometrical parameters d, dc, Λc and Λ are fixed to 250 µm, 90 µm, 394 µm, and 398 µm, respectively
Full size table

The effects of the different geometrical parameters on the dispersion characteristics of the reported design are studied. First, the impact of the diameter of the cladding air holes (d) on the dispersion behavior is investigated. Initially, d is taken as 250 µm where d is changed by ± 5 µm (± 2% of its original value) while the other geometrical parameters are fixed to their initial values summarized in table.1. Figure 5a and b show the frequency dependent dispersion of the TM and TE polarized modes, respectively at ϕ = 90˚ using different values of d. Additionally, Table 3 summarizes the dispersion values and the corresponding phase matching frequencies for the TM and TE modes at the studied d values. At ϕ = 90°, high negative dispersion of − 30.59 ps/THz/cm is achieved at f = 0.4027 THz at d = 255 µm using the TM mode as shown in Fig. 5a. Further, the TE mode has high negative dispersion of − 44.57 ps/THz/cm at f = 0.386THz at d = 245 µm as shown in Fig. 5b. Therefore, d is fixed to its original value of 250 µm through the subsequent simulations to keep high dispersion for the two polarized modes.

Fig. 5
figure 5

Frequency dependent dispersion of the a TM and b TE modes at different values of the cladding air hole diameters d while all other geometrical parameters dc, Λc, and Λ are fixed to 91 μm, 398 μm, and 394 μm, respectively. Additionally, ϕ and T are taken as 90° and 25 °C, respectively

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Table 3 Dispersion values and corresponding operating frequencies for both TE and TM modes at ϕ = 90° at different d values while all other parameters; dc, Λc, and Λ are fixed to 91 µm, 394 µm and 398 µm, respectively at T = 25 °C
Full size table

The influence of the diameter (dc) of porous holes on the dispersion and coupling characteristics is then considered. The studied values for dc are 90 µm, 91 µm and 92 µm while the other geometrical parameters are fixed to their initial values summarized in Table 1. Figure 6a and b depict the frequency dependent dispersions of the TM and TE polarized modes, respectively at ϕ = 90° using different values of dc. In addition, Table 4 summarizes the dispersion values and the corresponding operating frequencies for the two polarized modes at the studied dc values. At ϕ = 90°, high negative dispersion of − 27.20 ps/THz/cm is achieved at f = 0.4032 THz when dc = 92 µm for the TM mode as shown in Fig. 6a. However, the TE mode has high negative dispersion of − 37.74 ps/THz/cm at f = 0.3497 THz when dc = 91 µm as shown in Fig. 5b. It is worth noting that, dc is fixed to its original value of 91 µm through the subsequent simulations to keep high dispersion for the two polarized modes.

Fig. 6
figure 6

Frequency dependent dispersion of the a TM and b TE modes at different values of the inner and outer core diameters dc while all other geometrical parameters d, Λc, and Λ are fixed to 250 μm, 398 μm, and 394 μm, respectively. Additionally, ϕ and T are taken as 90° and 25 °C, respectively

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Table 4 Dispersion of the TE and TM modes at ϕ = 90° at different dc values at d = 250 µm, Λc = 394 µm and Λ = 398 µm
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Finally, the effect of the hole pitch (Λc) of the porous cores on the dispersion of the proposed design is studied while the other parameters are fixed to their optimum values (d = 250 µm, and dc = 91 µm). Figure 7a and b show the frequency dependent dispersion of the TM and TE polarized modes, respectively at ϕ = 90˚ at different values of Λc. Table 5 summarizes the dispersion values and the corresponding operating frequencies for the TM and TE modes at the studied Λc values. At ϕ = 90°, high negative dispersion of − 27.53 ps/THz/cm is achieved at f = 0.3989 THz when Λc = 395 µm for the TM mode as shown in Fig. 7a. Further, the TE mode has high negative dispersion of − 37.47 ps/THz/cm at f = 0.3497 THz when Λc = 394 µm as shown in Fig. 7b. Therefore, Λc is fixed to its original value of 394 µm through the subsequent simulations to keep high negative dispersion for the two polarized modes. It is worth noting that the effect of the hole pitch of the cladding Λ on the dispersion of the structure is also investigated. It is evident from the numerical results that Λ has a negligible effect on the dispersion characteristics of the supported modes.

Fig. 7
figure 7

Frequency dependent dispersion of the a TM and b TE modes at different values of hole pitch for the inner core ring (Λc) while all other geometrical parameters d, dc, and Λ are fixed to 250 µm, 398 μm, and 394 µm, respectively. Additionally, ϕ and T are taken as 90° and 25°C, respectively

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Table 5 Dispersion of the TE and TM modes at ϕ = 90˚ at different hole pitches Λc at d = 250µm, dc = 394µm and Λ = 398 µm
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The proposed design can be fabricated using the most common stack-and-draw method (Chow et al. 2012). Our proposed DC-LCPCF is assumed to be simple-to-stack as the whole structure enjoys a high degree of symmetry around the center of the core with relatively low core porosity percentages. Therefore, the stacking process will be easy and will provide a high degree of accuracy by the end of the drawing process. Further, the liquid crystal selective infiltration through small holes with diameter of 1.0 μm has been experimentally achieved with a hole pitch of 2.0 μm (Woliñski et al. 2005). Additionally, the PCF selective filling can be obtained using two photons direct laser writing technique with high flexibility (Vieweg et al. 2010). Further, the NLC filling is made at its isotropic case to reduce the filling time. In this context, NLC infiltration through multi-core PCF coupler filled has been made through central hole diameter 1.0 μm (Saitoh et al. 2008). At THz regime, the geometrical parameters of the proposed PCF are greater than those experimentally fabricated (Wojcik et al. 2010; Woliñski et al. 2005; Saitoh et al. 2008). Therefore, the reported THz PCF can be experimentally fabricated successfully.

The operation of the suggested THz PCF is performed at room temperature (\(T=25\)  C) as widely reported in (Sasaki et al. 2019; Vieweg et al. 2012). In order to control the temperature, thermo-electric module can be used as experimentally achieved in (Woliñski et al. 2006; Leon-Saval et al. 2005) where the temperature can be controlled in the 10–120 °C range with 0.1 °C long-term stability.

In order to ensure the robustness of the suggested device to fabrication errors, a fabrication tolerance study is performed for the TE and TM polarized modes and summarized in Table 6. In this study, one parameter is studied while the others are kept constant at their optimum values. Each parameter is studied within variation of ± 3 µm from its optimum values. It may be seen that the variation from the optimum values slightly affect the dispersion value and coupling frequency. This ensures the feasibility of the reported PCF for dispersion compensation in the THz applications.

Table 6 Dispersion of the quasi-TE and TM modes and the resonance frequency of the proposed design at ϕ = 90˚ using the optimum geometrical parameters and with a tolerance of ± 3 µm
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Table 7 shows a comparison between the suggested PCF relative to those presented in the literature (Depuis et al. 2010; Nielsen et al. 2009; Bao et al. 2015; Markov et al. 2013, Anthony et al. 2013, Rana et al. 2016, Rana et al. 2018, Sultana et al. 2018, Paul et al. 2018, Mei et al. 2019, Luo et al. 2018, Hasanuzzaman et al. 2018, Islam et al. 2015 and Islam et al. 2018) in terms of background materials, dispersions, and frequency range in THz regime. The comparison includes the utilized background material and the obtained dispersion values and the frequency range of each design. It is evident that the proposed PCF based on Topas background material and selectively infiltrated with NLC material has higher negative dispersion than those reported in (Depuis et al. 2010; Nielsen et al. 2009; Bao et al. 2015; Markov et al. 2013). The Cyclic-olefin copolymer also known as TOPAS is used as a background material of the suggested PCF. The TOPAS material has lower material absorption loss than Teflon, Polymethylmethacrylate (PMMA) (Nielson et al. 2009). Further, the TOPAS has a low material absorption loss of 0.2 cm−1 at f = 1 THz. Further, TOPAS material does not absorb water vapor (Emiliyanov et al. 2007) which is advantageous for manufacturing THz waveguides. It is also revealed form the comparison summarized in Table 7 that the proposed design achieves high negative dispersion for both TE and TM modes of − 44.57 ps/THz/cm and − 30.59 ps/THz/cm at f = 0.386 THz and 0.4027 THz, respectively. The obtained results surpass those reported in literature in the THz regime.

Table 7 Comparison between the dispersion for proposed design with other designs in THz regime which used different backgrounds as Polystyrene (PE), Cyclic olefin copolymer (COC), Polymethylmethacrylate (PMMA) and Topaz
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4 Conclusion

A novel design of DC-PCF-THz selectively filled with NLC for dispersion compensation is reported and analyzed. The dispersion control of the two polarized modes of the suggested PCF depends on the coupling between the inner and outer porous regions. Further, the dispersion properties of the TE and TM modes can be tuned by using an external electric field. It is found that that the TE and TM modes have large negative chromatic dispersions of − 44.57 ps/THz/cm and − 30.59 ps/THz/cm at f = 0.386 THz and 0.4027 THz, respectively. Therefore, the proposed design can be effectively used in THz communication systems.