Polarization independent silicon on calcium fluoride-based MIR optical modulator

Efficient mid-infrared (MIR) optical modulator is reported and numerically analyzed for both the transverse electric (TE) and transverse magnetic (TM) polarized modes. The proposed design is based on the silicon-on-calcium-fluoride platform with vanadium dioxide (VO2) as a phase changing material. Due to the attractive property of its phase transition between dielectric (ON) and metallic (OFF) states under the effect of an applied electric field, VO2 is utilized to enable the modulation process. At an operating wavelength of 3.5 μm, the reported modulator realizes an extinction ratio (ER) of 10.9 dB/μm with an insertion loss (IL) of 0.24 dB/μm for the TE polarized mode. However, for the TM polarized mode, an ER, and IL of 9.5 dB/μm, and 0.19 dB/μm, respectively are achieved. Additionally, the suggested design has a good fabrication tolerance of ± 10% where the ER is better than 10.4 dB/μm and 8.6 dB/μm for the TE and TM polarized modes with IL less than 0.26 dB/ μm. Therefore, the suggested modulator can play a pivotal role in different MIR applications including imaging, sensing, security, and communications.


Introduction
Mid infrared (MIR) based applications have been growing rapidly in recent years, such as free-space communications, medical diagnostics, absorption spectroscopy, gas sensing, and thermal imaging (Lavchiev and Jakoby 2016;Zou et al. 2018). Much appreciation goes to silicon photonics as an emerged platform in the MIR region due to its unique properties such as low loss and compatibility with complementary metal-oxide-semiconductor (CMOS) technology (Zou et al. 2018). Moreover, in the MIR region, the two-photon absorption effect (TPA) and the associated free carriers become negligible due to the drop of the TPA to zero beyond 2 μm (Abd-Elkader et al. 2022). So, it is possible to exploit nonlinear optical effects for device applications without any limitations from absorption of TPA-generated free carriers (Abd-Elkader et al. 2022). Silicon on insulator (SOI) (Abd-Elkader et al. 2019a, b) is the most prominent silicon photonics platform owing to the high index contrast between silicon and silicon dioxide. However, the traditional SOI platform is unsuitable for MIR based applications because the transparent waveband of the silica (SiO 2 ) does not exceed wavelengths beyond 3.6 µm (Pi et al. 2020). To circumvent this limitation, other suitable materials to substitute SiO 2 are required. Therefore, materials such as sapphire (Al 2 O 3 of refractive index (n) ∼ 1.7), silicon nitride (Si 3 N 4 of n ∼ 1.9), and calcium fluoride (CaF 2 of n ∼ 1.4) have been utilized as they are optically transparent in a wide wavelength ranges up to ∼ 5.5 μm, ∼ 7 μm, and ∼ 9 μm, respectively (Kumari et al. 2019). Consequently, due to its wider transparency window as well as its higher index contrast in the MIR, CaF 2 appears a suitable substrate material for many devices in the MIR region including polarization splitters (Kumari et al. 2019;Wang et al. 2022), polarization rotators (Kumari et al. 2018;Wang et al. 2021), optical absorbers (Bakan et al. 2018;Peng et al. 2020), and optical modulators (Alrayk et al. 2021).
The global increasing need for exchanging large amounts of information in today's communications makes it necessary to develop communication systems with higher transmission and processing capacities. Therefore, optical communication systems with ultra-compact and high-speed photonic integrated circuits (PICs) based optical devices are in a great demand. Optical modulator is a vital component in optical communication systems that is used in the modification of the amplitude, phase, or polarization of an optical carrier (Heidari et al. 2021). Moreover, the development of high-speed optical modulators would also have a great impact on free space optics (FSO) communication applications in the MIR range that contains two atmospheric transparency windows (i.e., 3-5 and 8-14 μm) (Luzhansky et al. 2015;Montesinos-Ballester et al. 2022;Su et al. 2018). Several techniques have been reported for realizing optical modulators depending on the electro-opticeffect (Liu et al. 2015;Phatak et al. 2016), electro-absorption (EA) effect (Mastronardi et al. 2018;Tahersima et al. 2019), thermo-optic effect (Nedeljkovic et al. 2014), and magneto-optic effect (Ross et al. 1983). Silicon (Si) based modulators were initially dependent on changing the refractive index of Si under the influence of electrical bias through carrier accumulation, injection, or depletion in either a metal-oxide semiconductor capacitor or a PN diode (Zhu et al. 2014). Hence, the phase variations of the transmitted light are converted into amplitude modulation usually using Mach-Zehnder interferometer (MZI) (Dong et al. 2012;Obayya et al. 2003) or a ring resonator (Marris-Morini et al. 2009). However, Si based MZI modulators suffer from low modulation efficiency and large footprint due to the weak plasma dispersion effect of Si and the diffraction limit of the Si waveguides (Streshinsky et al. 2013;Zhu et al. 2014). On the other hand, ring modulators have a reduced footprint and low power consumption, but they suffer from narrow optical bandwidth and high temperature sensitivity. Therefore, efficient active materials including transparent conductive oxides (TCOs), graphene, and phase changing materials (PCMs) have been introduced to combine with silicon photonics to produce high performance, compact, and fast optical modulators.
Recently, TCOs based optical modulators (Abdelatty et al. 2018;Das and Arya 2021a;Lu et al. 2012;Shah et al. 2018) have gathered a considerable attention due to the distinctive epsilon-near zero (ENZ) property of such materials. Meanwhile, the permittivity of TCOs, such as indium tin oxide (ITO) and aluminum-doped zinc oxide (AZO), can be tuned electrically. In this regard, an optical modulator based on AZO slot waveguide has been reported in (Lu et al. 2012). Moreover, Shah et al. (Shah et al. 2018) have proposed and theoretically analyzed an ITO electro-absorption modulator based on sidewall angled silicon waveguide.
To design highly efficient and compact optical modulators, PCMs are interesting alternative due to its exceptional characteristics. Vanadium dioxide (VO 2 ) is one of the most important PCMs that has received an increasing attention in the past few years. Advantages of using VO 2 include the ability of controlling the reversal phase transition between dielectric and metallic states through the application of an external electric field or temperature variations (Janjan et al. 2017;Shibuya et al. 2019). The large change of order of unity in complex refractive index during the phase transition help diminish the footprint of the optical modulators compared to other reported modulators. Moreover, compatibility of VO 2 with CMOS technology facilitates the integration with other PICs based devices in optical communication systems (Younis et al. 2019). In this context, an optical modulator is suggested in (Alrayk et al. 2021) to operate in the MIR region depending on a silicon-on-calcium-fluoride (SOCF) platform with a VO 2 top layer. This design has an ER of 14.99 dB with IL of 0.431 dB at a device length of 2 µm and a wavelength of 3.95 µm (Alrayk et al. 2021). It is noteworthy that the low speed of phase transition of VO 2 is a drawback, which limits the modulation speed and constrains the use of VO 2 based modulators to the sub-Gbps band (Alrayk et al. 2021). In response to the application of an electric field, the process of VO 2 phase transition from the dielectric to the metallic state may occur in a very short time i.e., few nanoseconds (Markov et al. 2015). However, the opposite transition from metallic to dielectric state is a much slower process, accordingly the switching time of VO 2 is quite large (Miller et al. (2018). This transition delay is essentially due to the low thermal conductivity of VO 2 that causes a limitation in the heat dissipation rate of the device (Markov et al. 2015). To solve this restriction, current-limiting resistance (Markov et al. 2015) and PAM4 (Mohammadi-Pouyan et al. 2018) based devices are introduced to increase the speed of VO 2 phase transition. However, such structures suffer from their fabrication complexity with relatively low ERs. Recently, combination between VO 2 based modulators and graphene is suggested as a suitable solution to improve the phase transition speed and reduce the modulation time delay Sadeghi et al. 2020). Accordingly, graphene is utilized as an efficient microheater due to its high thermal conductivity that improves the heat transfer capability and consequently the modulation speed . In ), a MIR optical modulator is proposed using a vertically embedded VO 2 layer in a Si slot waveguide with graphene electrodes. Graphene electrodes accelerates the heat dissipation in VO 2 layer, then the response time of reverse phase transition is reduced. The theoretical analysis of such structure (Sadeghi et al. 2020) at a wavelength of 3.8 µm showed a modulation speed of 23 ns with relatively low ER of 4.4 dB/µm and IL of 0.1 dB/µm. Light in optical communication systems usually has a random polarization state due to optical birefringence (Abd-Elkader et al. 2019b). Therefore, a degradation in the signal-to-noise ratio occurs when light is coupled with sensitive polarization modulators. Diversity scheme that can be implemented via polarization handling devices (Abd-Elkader et al. 2019a) may be suggested to solve this problem, however, using this technique will increase the complexity of the system with high propagating and coupling losses. On the other hand, most of the reported optical modulators are polarization sensitive (Alrayk et al. 2021;Heidari et al. 2021). Accordingly, it is important to develop optical modulators with similar performance for both TE and TM polarizations. In this context, Jin et al. (Jin et al. 2018) have introduced a polarization insensitive EA modulator based on a silicon waveguide cladded with Au/SiO 2 /ITO in both x and y directions. In such structure (Jin et al. 2018), the MOS capacitor formed by the Au/SiO 2 /ITO stack was used to tune the carrier dispersion effect in the ITO layer, which was used for the modulation process. At λ = 1.55 µm, a low ER of 1.43 dB/μm is obtained with ΔER (difference between the ERs of the two polarization states) equals 0.009 dB/μm (Jin et al. 2018).
In this paper, a polarization independent optical modulator based on SOCF platform with VO 2 as an active material is introduced and analyzed for MIR applications. The modulation process in the proposed design depends on the phase transition of VO 2 material between dielectric and metallic states. The refractive index change between the two states is induced through the application of an external electric field (~ 6.5 × 10 7 V/m) at a temperature of 300° (Alrayk et al. 2021). The VO 2 layers are added in both vertical and horizontal directions to couple with both the TE (x-polarized) and TM (y-polarized) polarization modes, respectively. When the VO 2 material is in its dielectric state, the induced IL is very low, and the modulator is in the ON-state. On the other hand, changing the VO 2 into metallic phase maximizes the mode optical loss and accordingly the modulator is in the OFF-state. The modal characteristics of the suggested modulator are studied using the fullvectorial finite element approach (Obayya et al. 2000(Obayya et al. , 2002. Additionally, the 3D finite difference time domain (FDTD) (Kunz and Luebbers 1993) simulation is employed to test the light propagation characteristics of the presented waveguiding structure. In this context, the reported Si/VO 2 MIR waveguide has excellent propagating properties for both the quasi-TE and quasi-TM polarizations during the ON sate showing very low optical loss with strong optical confinement. The geometrical parameters of different waveguide layers are studied to control the electromagnetic (EM) field losses occur due to the metallic layers (metallic VO 2 and Ag). The optical loss increases dramatically when the VO 2 material is switched to the metallic phase. The numerical results, at λ = 3.5 μm, show a high ER of 10.9 dB/μm with low IL of 0.24 dB/μm and FOM of 45 for the TE polarized mode. However, for the TM polarized mode, the obtained ER, IL, and FOM are 9.5 dB/μm, 0.19 dB/ μm, and 49, respectively at λ = 3.5 μm. In addition, the introduced modulator is simple in design and can be easily fabricated. The suggested modulator offers higher ER compared to those reported in (Abdelatty et al. 2018;Alrayk et al. 2021;Jin et al. 2018;Sadeghi et al. 2020). Moreover, it operates for the two polarized modes with relatively similar high performance in the MIR region counter to those demonstrated in (Alrayk et al. 2021;Sadeghi et al. 2020) which operates only for the TM mode.

Design concepts
The 3D schematic diagram and the 2D cross sectional view of the proposed optical modulator are depicted in Fig. 1. The suggested structure consists of a CaF 2 substrate on which a Si core is deposited and surrounded by three thin layers of VO 2 , CaF 2 , and Silver (Ag), respectively. Thus, the reported structure is based on the SOCF platform that emerges in the MIR waveguiding applications (Alrayk et al. 2021;Bakan et al. 2018;Kumari et al. 2019Kumari et al. , 2018Peng et al. 2020;Wang et al.2021Wang et al. , 2022. The Si core is a rectangle with a width; w s and a height; h s . Further, the Si core is directly surrounded by a layer of VO 2 material with a thickness; w v to maximize the interaction between the core-guided light and the added VO 2 layer which enhances the modulation characteristics of the proposed modulator. The Calcium-Fluoride is selected as a low-index spacer to act as a thin cladding layer. Additionally, Silver is utilized as a metal electrode to apply the external electric field that induces the VO 2 dielectricmetallic phase transition. Table 1 summarizes the initial geometrical parameters of the suggested modulator, which were determined based on a specified criterion to attain high device performance. The refractive indices of the VO 2 material for both metallic and dielectric phases are taken from (Wan et al. 2019). Further, the refractive index of CaF 2 layer is taken from (Kumari et al. 2019). Additionally, the wavelength dependent Drude model of Ag in MIR regime (Yang et al. 2015) is considered in all simulations in this work. It is worth noting that Si has considerable dispersive characteristics especially in the mid-IR regime (Osgood et al. 2009). In this context, the wavelength-dependent refractive index of Si material in the mid-IR regime can be calculated through the Sellmeier equation reported in (Osgood et al. 2009): where λ is the wavelength in microns, ε, λ 1 , A, and B are the Sellmeier coefficients and equal to 11.6858, 1.1071 μm, 0.939816 μm 2 , and 8.10461 × 10 −3 , respectively. An efficient optical modulator should have low optical loss at the ON (dielectric VO 2 ) state. Thus, the obtained IL is minimum. On the other hand, the optical loss in OFF (metallic VO 2 ) state (1) Fig. 1 a 3D schematic diagram and b 2D cross section of the proposed optical MIR modulator should be maximized to get the highest ER. In this study, ER for both TE and TM modes is defined as the difference in optical loss between the OFF and ON states. In addition, the structural parameters are investigated to attain the best performance of the proposed modulator.
The presented electro-optical modulator has two operating states. The first one is the ON state which can be induced at the dielectric phase of the VO 2 material. In the ON state, high optical transmission occurs through the reported design due to the ultra-low losses introduced by the dielectric VO 2 layer. Otherwise, switching the VO 2 phase into metallic phase induces the OFF state where low optical transmission occurs due to the high losses introduced by the metallic VO 2 layer. It is worth noting that the extinction coefficient (κ; imaginary part of the refractive index) of VO 2 in its dielectric state is very small (κ ~ 0.09 at λ = 3.5 µm) (Wan et al. 2019). However, VO 2 material in its metallic phase (OFF state) has a very large κ of about 5.08 at λ = 3.5 µm (Wan et al. 2019). On the other hand, optical waveguides constructed using the SOCF have a large index contrast between Si and CaF 2 and are polarization sensitive. Thus, the operation of the reported modulator can be easily controlled through the aforementioned properties with high degree of flexibility. In addition, the standard fabrication techniques of SOI based structures could help in the fabrication process of the reported waveguide structure. In this regard, the silicon waveguide can be built using electron beam (e-beam) lithography followed by dry etching. Then, the deposition of the thin CaF 2 , VO 2 , and Ag layers can be performed layer-by-layer. The deposition of the VO 2 layer can be performed using the well-developed pulsed laser deposition (PLD) technique at 450 °C (Shibuya and Sawa 2015). Additionally, the molecular beam epitaxial deposition technique (Filimonova et al. 2017;Sata et al. 2000) followed by dry etching using H 2 O plasma (Matsutani et al. 2008) can be utilized to formulate the CaF 2 layer. However, sputtering and/or E-beam evaporation deposition techniques can be utilized to attach the Ag layer .
Optical control of VO 2 's phase transition at room temperature has great potential for investigating the intrinsic physical mechanism and realizing optical modulation devices. Currently, optical pumping is used to induce the photoexcitation insulator-metal phase transition, which promises to allow vital insights into the nature of each state and may lead to metastable new phases under non-equilibrium conditions (Cavalleri et al. 2001;Liu et al. 2012;Sood et al. 2021;Del Valle et al. 2021). However, being an ultrafast excitation, such optical means cannot introduce stable phase transition; instead, a transient process on the picosecond scale is induced. Several works show that the electrical properties of VO 2 can be modulated through various means of irradiation, such as electron beams (Zhang et al. 2017), X-ray (Dietze et al. 2014), and even UV light (Zhang et al. 2016). These results indicate the possibility of photo-controlled phase transition in VO 2 . In this context, a novel neuromorphic sensor based on the optical control of phase transition in VO 2 films with UV light is presented in . The proportion of monoclinic phase in the film decreases with UV radiation dose, indicating the tunability of the phase transformation introduced by the optical stimulation. Based on this mechanism, the optoelectronic synaptic functions including optical modulation, integrated sensing and nonvolatile multilevel storage features can successfully be realized in VO 2 based structures.
A hybrid plasmonic TE and TM modulator should have at least two metal-dielectric interfaces. One interface should be in the x direction while the other in the y direction to ensure the modulation of the two polarized (TE and TM) modes. In this case, the waveguide core is surrounded by a lossy material (metallic VO 2 layer) from both vertical and horizontal directions. Thus, the optical modulation process can be polarization independent by optimizing the mode field distributions and the position of the active layer. Then, by tailoring the structural parameters, both TE and TM modes can be modulated with a relatively similar IL and ER. The main goal of this research is to maximize the ER of both TE and TM polarized modes. This aim can be achieved by reduce the IL of both polarizations in the ON (dielectric VO 2 ) state as well as induce a significant increase in the optical losses of the TE and TM modes during the OFF (metallic VO 2 ) state. Unfortunately, a trade-off between increasing the ER and decreasing the IL during the process of tuning the structural parameters occurs. Therefore, a figure-of-merit (FOM) parameter that is the ratio between the obtained ER and IL i.e., FOM = ER/IL is introduced.

Analysis and numerical results
The full-vectorial finite element method (FVFEM) via COMSOL Multiphysics software package (https:// www. comsol. com) is employed to analyze the modal characteristics of the reported optical waveguide/modulator for both polarizations i.e., TE and TM. Additionally, scattering boundary conditions and minimum mesh element size of 8 × 10 -5 μm are applied for sufficient accuracy and simulation time. Further, the FDTD method using Lumerical software package (https:// www. lumer ical. com) is utilized to numerically study the propagation characteristics. A uniform mesh is used with a step size of 0.01 nm in transverse directions (x and y directions) while the structure is discretized in the longitudinal direction with 0.1 nm step size. Further, scattering boundaries are applied at all transverse directions to attenuate the field and reject any back reflections. In addition, the device is designed to operate at a wavelength of 3.5 μm that lies in the MIR wavelength regime. Figure 2 depicts the major components of the electric field (E-field); E x and E y that correspond to the fundamental TE and TM polarized modes, respectively in both dielectric, Fig. 2a, b, and metallic, Fig. 2c, d, VO 2 phases. It can be seen from Fig. 2a, b that, E x component of the TE polarized mode and E y component of the TM polarized mode are well confined in the silicon core. This would keep the light propagation loss at low levels for both polarizations at the ON state. However, after applying a suitable external voltage across the electrodes (Fig. 1b), the phase transition is induced and the VO 2 material is switched to its metallic phase "OFF state". In this case, due to the metallic behavior of VO 2 , the light is highly attenuated with weak confinement in the Si core, as shown in Fig. 2c, d, where most of the field is attracted to the edges of the VO 2 layer with maximum confinement of E x and E y in the CaF 2 layer.
In order to achieve the best modulation characteristics, all structural parameters are considered to study their variations effect on the performance of the suggested modulator. In this study, it is aimed to balance between realizing the best performance parameters, ER, IL, and FOM for each polarization mode and minimizing the modulator footprint. In addition, achieving a relatively close performance for the two polarized modes i.e., attaining minimum ΔER and ΔIL. Current analysis is performed at λ = 3.5 µm while the Si core height h s and width w s are fixed to their initial values in Table 1. First, the effect of the thickness of VO 2 layer is investigated for the two polarized modes. Figure 3 indicates the ER, IL, and FOM dependence on the thickness of VO 2 layer (w v ) for the TE polarized mode; Fig. 3a, b, and for the TM polarized mode, Fig. 3c, d. Such investigation is carried out while the thicknesses of the CaF 2 and Ag layers are fixed at w f = 60 nm and w a = 100 nm, respectively. It may be noted from Fig. 3a that as w v increases from 70 to 100 nm, the ER for the TE mode increases from 6.8 dB/µm to 10.2 dB/µm. However, if w v is further increased to 130 nm, the ER decreases to 9.3 dB/µm as may be seen in Fig. 3a.
Additionally, IL increases from 0.2 to 0.3 dB/µm by increasing w v from 70 to 130 nm. Moreover, the FOM variation with w v is shown in Fig. 3b where the maximum FOM of 41.5 is achieved at w v = 90 nm. On the other hand, for the TM polarization mode, it may be observed from Fig. 3c that when w v increases, the ER and IL increase gradually. It may be also seen from Fig. 3d that the FOM has a very similar behavior as that of the IL and ER (Fig. 3c). Thus, in the subsequent simulations, w v will be fixed to 100 nm to achieve the highest ER in case of TE mode and keep the IL as low as possible. It is worth noting that at w v = 100 nm, w f = 60 nm, and w a = 100 nm, the ER, IL, and FOM for the TE polarized mode are 10.2 dB/µm, 0.25 dB/µm, and 41.5, respectively, however, they are 6 dB/µm, 0.21 dB/µm, and 28.7 for the TM polarized mode.
Next, the thickness of the CaF 2 layer (w f ) is studied and analyzed to show its effect on the modulator behavior, for the two polarized modes. In this study, w f is tuned to obtain the best performance while all other structural parameters are fixed to their initial values in Table 1 and w v is fixed to 100 nm. Figure 4 shows the variation of the ER, IL, and FOM with w f for the TE polarization mode, Fig. 4a, b, and for the TM polarization mode, Fig. 4c, d. It may be seen from Fig. 4a that, for the TE polarized mode, ER increases from Fig. 2 Electric field profiles correspond to the E x (x-polarized mode) and E y (y-polarized mode) of the reported modulator structure for a, b dielectric VO 2 phase, and c, d metallic VO 2 phase, respectively where the geometrical parameters are fixed to their initial values depicted in Table 1 4.9 dB/µm to 10.7 dB/µm as w f increases from 30 to 70 nm. If w f is further increased to 150 nm the ER is slightly increased to 10.9 dB/µm. However, the IL decreases to approximately 0.24 dB/µm when w f is increased to 100 nm. Then IL is relatively constant for w f values greater than 100 nm as evident from Fig. 4a. The obtained FOM for the TE polarization mode has the same behavior of the ER as shown in Fig. 4b. On the other hand, it can be noted from Fig. 4c, d for the TM polarized mode that the ER and FOM increase with increasing w f and they tend to be stable after w f = 100 nm. Moreover, the IL decreases with the increase in w f within the range from 30 to 150 nm as shown in Fig. 4c. At w f = 100 nm, w v = 100 nm, and w a = 100 nm, the modulator achieves an ER, IL, and FOM of 10.9 dB/ µm, 0.24 dB/µm, and 45, respectively for the TE polarized mode, however, it achieves an ER of 9.5 dB/µm with IL of 0.19 dB/µm, and FOM of 28.7 for the TM polarized mode.
It is worth noting that it may be possible to achieve a closer performance for the two polarizations, but this will come at the expense of other performance factors. To elucidate this, the following study is performed as an example. Figure 5 shows the variation of the ERs of the two polarized modes and the difference between them ΔER with the thickness Fig. 3 Effect of the thickness of VO 2 layer (w v ) on a the IL, ER, b FOM for the quasi-TE (x-polarized) mode and c the ER, IL, d FOM for the quasi-TM (y-polarized) mode at wavelength of 3.5 μm. The thicknesses of the CaF 2 and Ag layers are set as w f = 60 nm and w a = 100 nm, respectively Fig. 4 Effect of the thickness of CaF 2 layer (w f ) on a the IL, ER, b FOM for the quasi-TE (x-polarized) mode and c the ER, IL, d FOM for the quasi-TM (y-polarized) mode at wavelength of 3.5 μm. The thicknesses of the VO 2 and Ag layers are set as w v = 100 nm and w a = 100 nm, respectively Fig. 5 Effect of the thickness of CaF 2 layer (w f ) on the ERs of the quasi-TE and quasi-TM modes and the difference between the ER values of two polarized modes, i.e., the ΔER at λ = 3.5 μm. In this study, w v and w a are fixed to 100 nm of the CaF 2 layer (w f ). It can be observed that the minimum values of ΔER are achieved at w f = 30 nm and 150 nm. However, at w f = 30 nm the values of ER for TE and TM modes is very low (~ 4.5 dB/μm), while, at w f = 150 nm the footprint is increased with a little improvement in the ER of the TM mode from 9.5 to 10.5 dB/μm. Therefore, a tradeoff between these factors is considered and w f = 100 nm is chosen to achieve a relatively small ΔER with high ERs for the TE and TM modes. Further, the effect of the thickness of the Ag layer in the range from 80 to 120 nm is also investigated, however, the numerical results reveal that it has a very slight (negligible) effect on the ER, IL and hence the FOM. From the numerical analysis, the optimum dimensions of the suggested modulator are taken as w v = w f = w a = 100 nm.
To reveal the modulation behavior of the proposed modulator, 3D FDTD simulations are carried out assuming an initial device length of 7-μm. Additionally, MIR light source of λ = 3.5 μm is used to launch the TE and TM polarizations into a Si input waveguide of length 0.5 μm with the same cross section of the modulator's silicon core (w s × h s ). Then, the light is coupled to the proposed modulator, and finally coupled into the output Si waveguide which is typical to the input Si waveguide. It may be seen from Fig. 6a that the TEmode propagates along the proposed design with low losses at the ON state during the dielectric phase of the VO 2 material. However, the TE-mode is highly attenuated in the OFF state when VO 2 is switched to its metallic phase as may be noted in Fig. 6b. Similarly, the Fig. 6 Field evolution through the proposed modulator with input and output waveguides, excited by the quasi a, b TE and c, d TM input modes. a-c are for dielectric VO 2 phase. b-d are for metallic VO 2 phase. The operating wavelength is 3.5 μm and the thickness of VO 2 , CaF 2 , and Ag are set as w v = w f = w a = 100 nm TM-mode is well propagated with very low optical loss at the ON state, Fig. 6c. However, at the OFF state, the same TM mode suffers from very high optical loss with no propagation through the device as shown Fig. 6d. Due to the similar performance for both the TE and TM modes, the light waves carried by the two modes at the output are balanced out with a 7-μm-modulation length.
The wavelength-dependent ER and IL of the reported modulator are also investigated through the MIR wavelength range from 2.5 to 4.5 μm. Figure 7a, b show the variation of the ER and IL with the wavelength for the TE-and TM-polarized modes, respectively. It is revealed from these figures that the presented modulator offers relatively broad bandwidth of operation for the two polarized modes. It may be also seen from Fig. 7a that for the TE-polarized mode, the modulator achieves an ER higher than 9 dB/µm over the whole wavelength range. On the other hand, the TM-polarized mode has an ER higher than 8 dB/ µm throughout the wavelength range of 3-4.5 μm as depicted in Fig. 7b. Further, the IL is below 0.32 dB and 0.23 dB for the TE-and TM-polarization modes, respectively over the whole studied wavelength range.
The fabrication tolerances of the structural parameters of the reported modulator are then investigated at λ = 3.5 μm. Table 2 summarizes the variation in ER and IL according to a change of ± 10 nm from the optimized thicknesses of the VO 2 (w v ), CaF 2 (w f ) and Ag (w a ) layers for the TE and TM polarizations. It can be noticed from this table that the proposed modulator has a large fabrication tolerance in terms of the performance indicators i.e., ER and IL, which ensures the robust of the reported modulator to the fabrication errors.
In addition to the performance parameters of the optical modulators discussed above (ER, IL, and FOM), modulation speed and energy consumption are important parameters to examine the modulation efficiency of optical modulators. The ultimate speed of the proposed modulator is determined by VO 2 phase transition speed and resistance-capacitance (RC) response time of the structure. According to (Joushaghani et al. 2015;Miller et al. 2018;Mohammadi-Pouyan et al. 2018), different techniques can be employed to induce the phase transition process of VO 2 . These techniques, known as thermo-optic, electro-optic, and all optic (photo-induced) techniques have different phase transition times depending on the technique itself (Miller et al. 2018). According to the literature (Joushaghani et al. , the required phase transition in photo-induced technique occurs faster than the other reported response times for the electrical actuation of the VO 2 . In this work, the capacitance (C) can be approximately estimated from a simple parallel plate capacitor model (formed by the VO 2 layer as a dielectric sandwiched between two electrodes acting as two conductive plates). However, experimental capacitance may vary due to additional parasitic from disorder in film deposition, wiring, and making metal contacts. Here, the modulation speed can be calculated as (Das and Arya 2021a); where R is the system resistance and assumed to be 500 Ω (Das and Arya 2021a;Shah et al. 2018). The capacitance of the device can be estimated as C = εε 0 A/d where A is the area of the active material, ε 0 is the permittivity of free space, ε is the permittivity of the dielectric material (CaF 2 ), and d is the thickness of the dielectric CaF 2 layer. Hence, the capacitance of the proposed structure is calculated to be C ∼ 0.521 fF. This value is used in Eq. (2) to calculate a modulation speed of ∼ 611 GHz. According to the utilized RC model, the response time of the introduced modulator is estimated as τ = 0.26 ps. This indicates that the RC response time is in the range of the picosecond scale and therefore does not restrict the modulation speed considering the nanosecond range of VO 2 transition time (Sadeghi et al. 2020). Moreover, the energy consumption per bit (E bit ) can be determined by Phatak et al. 2016); where ∆V is the required voltage to switch between ON and OFF States of the modulator, which is assessed to be ∆V = 6.5 V for a 100 nm thick VO 2 layer, achieving only an E bit = 5.5 fJ/bit.
A comparison between the suggested modulator and those reported in literature is presented in Table 2. Such comparison is introduced in terms of the basic platform employed, active material used, operating wavelength, IL, ER, FOM, E bit , type of work (theoretical or experimental), and the ability to work with the two polarized modes. The polarization independent operation in the MIR is the most important feature of the suggested modulator based on the SOCF platform with VO 2 as an active material. In addition, the proposed modulator achieves a better ER than those reported in (Abdelatty et al. 2018;Alrayk et al. 2021;Das and Arya 2021a, b;Janjan et al. 2020;Jin et al. 2018;Phatak et al. 2016;Sadeghi et al. 2020;Shah et al. 2018) with comparable IL as summarized in Table 3.

Conclusion
Highly efficient polarization independent MIR optical modulator is proposed and studied. The suggested structure is based on the SOCF platform with VO 2 material which is used to control the modulation process owing to its phase transition between dielectric (ON) and metallic (OFF) states under the application of external electric field. For the TE polarization mode, the presented modulator achieves an ER of 10.9 dB/μm with IL of 0.24 dB/μm and FOM of 45, at the operating wavelength of 3.5 μm. On the other hand, an ER, IL, and FOM of 9.5 dB/μm, 0.19 dB/μm, and 49, respectively are obtained for the TM polarization mode. Additionally, an average energy per bit (E bit ) of 5.5 fJ/bit as well as modulation speed up to 611 GHz, are attained. In comparison with the reported modulators in literature, the proposed optical modulator has a compact device length with high ER and low IL over a wide MIR wavelength range.