A detailed theoretical and experimental study of metal-microheater integrated silicon waveguide phase-shifters has been carried out. It has been shown that the effective thermal conductance gw and the effective heat capacitance hw evaluated per unit length of the waveguide are two useful parameters contributing to the overall performance of a thermo-optic phase-shifter. Calculated values of temperature sensitivity, SH = 1/gw and thermal response time, τth = hw/gw of the phase-shifter are found to be consistent with the experimental results. Thus, a new parameter ℱH = SH/τth = 1/hw has been introduced to capture the overall figure of merit of a thermo-optic phase-shifter. A folded waveguide phase-shifter design integrated in one of the arms of a balanced MZI switch is shown to be superior to that of a straight waveguide phase-shifter of the same waveguide cross-sectional geometry. The MZI switches were designed to operate in TE-polarization over a broad wavelength range (λ ∼ 1550 nm).
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Integrated optical metal-strip microheaters are widely used for reconfigurable silicon photonics devices because of large thermo-optic coefficient of bulk silicon crystal (dn/dT = 1.86 × 10−4 K−1)  and they can be easily implemented using CMOS front-end technology. Some attractive applications of such thermo-optic phase-shifters are reconfigurable wavelength filter [2–4], wavelength interleaver , tunable directional coupler , large scale optical switch matrix for WDM systems [7–10], etc. In all these examples, metallic microheater strip lines are suitably integrated close to waveguides, acting as waveguide thermo-optic phase-shifters. The thermo-optic phase shift is directly proportional to the temperature rise in the waveguide and its effective length . Again, the temperature rise in the waveguide is proportional to the electrical power dissipated (or consumed as Joule heating) in metallic microheater. Therefore, the power efficiency can be greatly improved by a clever design of waveguide phase-shifter system (waveguide + microheater).
In an integrated optical thermo-optic phase-shifter, the waveguide core is heated typically through the top cladding oxide using metallic (Ti, TiN, CrAu, NiCr, graphene etc.) microheater strip-line integrated directly above the waveguide [12–14]. An attractive meander-type metal microheater design over the oxide cladding of spiraled long waveguide was reported earlier exhibiting lower switching power (Pπ = 6.5 mW) and extremely small switching temperature (ΔTπ = 0.67K) . However, the reported switching time and optical insertion loss in spiraled waveguide section are relatively large (τth = 14 μs, IL = 6.5 dB). Other efforts in switching power reduction were reported by selectively removing the cladding oxide and providing undercuts to the waveguide [16–18]. But this undercut results into a large switching time (few hundreds of microseconds). In contrast, integration of a metallic microheaters directly on the waveguide slab are shown to be relatively faster (τth ∼ 5 μs), but at the cost of large switching power (Pπ ∼ 50 mW) [4, 11]. Therefore, it remains a challenging task to design a silicon photonics thermo-optic switch fulfilling all desired features, viz. lower on-state switching power and switching time, compact design for large scale integration, lower optical insertion loss, etc. There is a little effort seen so far for an optimum design and/or modeling of metallic microheater integrated waveguide phase-shifters. Very recently, Bahadori et al. reported a compact model describing detail dc and transient characteristics of microheater-ring resonator system in silicon photonics platform . Atabaki et al. investigated earlier with numerical simulations followed by experimental demonstration estimating some figure of merits (FOMs) of metallic microheaters used for silicon photonics applications . However, the authors in their model concentrated only on the width of microheater strip-line and the effect of cladding oxide thicknesses. Nevertheless, the FOMs of a thermo-optic phase-shifter (switching power, time and temperature, compactness, optical insertion loss, etc.) can be improved significantly by optimizing the waveguide and microheater design parameters as well as engineering the thermal properties (heat capacitance and thermal conductance) of the system. In this work, we carried out a detailed theoretical analysis of metallic microheater integrated waveguide phase-shifters for the optimizations of above mentioned FOMs. Besides cladding layer, the waveguide cross-sectional geometry and proximity of microheaters are considered for determining the FOMs. The background theory related to all FOMs and necessary numerical simulation results are presented in section 2. In section 3, we have presented the experimental demonstration of Mach-Zehnder interferometer (MZI) thermo-optic switch with a slab-integrated microheater in one of its interferometric arms and validated with theoretically predicted FOMs. The MZI switches were specially designed to operate in TE polarized guided mode over a broad wavelength range (1520 nm ≤ λ ≤ 1630 nm). Finally, concluding remarks comprising of a brief summary of this work and future scopes are given in section 4.
2. Theory and simulation results
For a given waveguide thermo-optic phase-shifter of length Lw, the rise in waveguide temperature for π phase-shift at an operating wavelength λ is given by :15]. Further, the steady-state temperature rise (ΔTs) in waveguide core is directly proportional to the Joule heating power consumption by a resistive/metallic microheater. The thermal sensitivity SH of the phase shifter can be defined as :
On the other hand, the transient rise in waveguide temperature ΔT(t) can be expressed as:Eq. (2) reduces to:
For performance analysis, we have considered two different waveguide microheater configurations, commonly used for integrated optical thermo-optic phase-shifter in SOI platform as shown in Fig. 1. According to Atabaki et al., the two configurations shown in Fig. 1(a) and Fig. 1(b) are categorized as Type-I and Type-II architectures, respectively . It was shown with detail analyses that the Type-II microheaters are superior than that of Type-I microheaters in terms of thermal response time of a micro-disk resonator. To analyze the performance of a Type-I or Type-II waveguide phase-shifter, we need to consider waveguide parameters like width W, slab height h, and the lateral distances of microheater to waveguide dI (dII) for Type-I (Type-II) as defined in the figures. The width (WH = 1 μm) and thickness (tH = 100 nm) of a Ti metallic microheater strip are assumed same for both types of architectures. Moreover, following the standards of silicon photonics foundries, we have considered device layer thickness of H = 220 nm and BOX layer (SiO2) thickness of tBOX = 2 μm.
For a given design of SOI waveguide geometry (W, H, and h) discussed above, the thermal properties (thermo-optic efficiency and response time) of the waveguide phase-shifter are mainly determined by the waveguide slab layer thickness h, lateral distance of microheater dI or dII and top cladding material SiO2 or Air. Therefore, it is important to estimate the valid range of W and h for single-mode guidance over a desired wavelength range (λ ∼ 1550 nm). We have simulated allowed guided modes and their polarizations by varying W (300 nm to 700 nm) and h (0 to 200 nm) using Lumerical’s MODE Solutions separately for SiO2 and air top-cladding layers (see Fig. 2).
The degree of polarization of the guided modes are defined by comparing the fraction of transverse electric field component along x-direction:Figs. 2(a) and 2(b) as a function of W and h for a given H = 220 nm. Besides single-mode guiding condition, we also observe that TE0 mode is supported for all values of h when W is kept below 375 nm (500 nm) for air (SiO2) top-cladding. Both TE0 and TM0 modes are supported for h ≤ 15 nm (30 nm) for oxide (air) top-cladding.
In order to investigate thermo-optic FOMs like SH and τth, we have considered TE0 mode guidance in waveguides with W = 350 nm and 0 ≤ h ≤ 200 nm for both Type-I and Type-II architectures. The important properties of core Si, cladding SiO2, and microheater Ti-strip used for numerical simulation are given in Table 1. The thickness dependent thermal conductivities  were appropriately considered for solving the heat transport equation  using COMSOL Multiphysics  with appropriate boundary conditions at the interfaces as described in Ref. . To start with, we have simulated the steady-state values of temperature rise (ΔTs) at the waveguide core (W = 350 nm, H = 220 nm, h = 100 nm) as a function of electrical power dissipation per unit length (pw) of waveguide due to positioning of Ti strip microheaters (WH = 1 μm and tH = 100 nm) at dI = dII = 1 μm. As expected, the ΔTs vs. pw plots shown in Fig. 3(a) are following the linear relationship as defined in Eq. (2). It is evident that the sensitivity SH for Type-I architecture (195 K·μm/mW) is reasonably higher than that of Type-II architectures (∼130 K·μm/mW). This is expected due to the large thermal conductivity of silicon slab (h = 100 nm) which increases the effective thermal conductance (gw) of the slab-heating configuration and hence SH reduces [Eq. (5)] compared to oxide-heating. However, it is also observed that the value of SH for Type-II (air cladding) is slightly higher than that of Type-II (oxide cladding). This is attributed to the lower thermal conductivity of air cladding than that of SiO2. The transient response of the temperature rise ΔT(t) in the waveguide core is evaluated by exciting the microheater with a step input voltage signal at t = 0. The simulated results (with ΔTs normalized to unity) for the above mentioned Type-I and Type-II architectures are shown in Fig. 3(b). The thermal response time τth estimated for Type-I, Type-II (oxide cladding), and Type-II (air cladding) are 12.7 μs, 8.5 μs, and 3.8 μs, respectively. Thus the above simulation results confirm that top oxide cladding results into an effective decrease of thermal conductance (gw) and/or effective increase of thermal capacitance (hw). The proximity of microheater to the waveguide core also play a role in reducing the value of τth as well as increasing SH.
A more detail simulation results for SH and τth as a function of both h and dI or dII are shown in color contour plots in Fig. 4. Note that, irrespective of the heater configuration (oxide-heating or slab heating), SH reduces with dI,II and h because of the increase in effective thermal conductance (gw). In contrast, τth increases with dI,II. Though gw of the system increases with h, the value of τth remains unaffected w.r.t. h for oxide-heating, since the waveguide core is heated right from the top at a constant distance of dI. We notice that the value of SH can be as high as > 350 K·μm/mW and τth is as low as < 4.5 μs for a Type-I configuration with h < 25 nm and dI < 250 nm. On the other hand, they are > 190 K·μm/mW (> 220 K·μm/mW) and < 8.0 μs (< 4 μs) for a oxide clad (air clad) Type-II configuration with h < 50 nm and dII < 500 nm. Thus the studies help for maximizing the figure of merit ℱH of the waveguide phase-shifter defined by SH/τth. In doing so, one needs to minimize the value of dI or dII. However, closer proximity of metal microheater to the waveguide introduces optical attenuation of the guided mode because of plasma dispersion effect through evanescent field overlap. We have numerically estimated (using Lumerical’s MODE Solutions) the optical loss coefficient (αh) for TE0 guided mode as a function of dI,II (considering h as a parameter) with complex refractive index of Ti-strip (WH = 1 μm, tH = 100 nm) as 3.6848 + j4.6088 at λ = 1550 nm . This has been shown in Figs. 5(a) and 5(b) for Type-I and Type-II (oxide cladding and air cladding), respectively. As expected, αh (expressed in dB/mm) increases as the value of dI or dII decreases. However, for type-II configuration (with a given dII), αh increases with slab height due to poor confinement of the optical mode inside waveguide core. Whereas, for Type-I (with a given dI), the evanescent field-strength along y direction reduces with h and hence αh reduces. It must be noted that the αh is nearly same for oxide cladding and air cladding in Type-II configuration. Nevertheless, the metallic losses are negligibly small (αh < 0.1 dB/mm) for dI,II > 0.5 μm.
It is worth mentioning here that we have restricted our above numerical studies for a straight waveguide phase-shifter where both waveguide and metallic microheater strip-line are parallel to each other for both Type-I and Type-II architectures. However, one can extend this study to spiraled waveguide geometry and meander-type metallic microheater in Type-I configurations as described in Ref. . From their experimental results, we estimate the value of SH as large as ∼ 640 K·μm/mW. Similarly, for a bend or folded waveguide (and microheater) design in Type-II configuration, the value of SH can be enhanced significantly. This has been validated with numerical simulations as well as experimental results described in following section.
3. Experimental results and discussion
The theoretical model discussed above has been validated with experimental results by integrating Type-II (air cladding) phase-shifters in one of the arms of balanced MZIs fabricated in SOI platform (device layer ∼ 220 nm, BOX layer ∼ 2 μm). The MZIs were designed with wavelength independent directional couplers (WIDCs) for switching operation over a broad wavelength range (operating at λ ∼ 1550 nm) as described in Ref. . The waveguides were designed with W = 350 nm and h = 160 nm, supporting only TE0-polarized guided mode for the above mentioned desired wavelength range. The passive MZI structures and reference waveguides were fabricated using electron beam lithography (EBL) followed by inductively coupled plasma reactive ion etching process (ICPRIE). Afterwards, contact pads (Al) and microheater (Ti) both were patterned one after another using EBL and subsequent lift-off process. Detailed fabrication process parameters of MZI and microheater can be found elsewhere [4, 27]. For device characterizations, the input/output waveguides are terminated with broadband grating couplers of bandwidth ∼ 70 nm, as described in .
For a comparative study, two different types of MZIs were fabricated; the first type has straight waveguide phase-shifters (S-MZI), whereas the second type has folded waveguide phase-shifters F-MZI). Both straight and folded waveguide phase-shifters have same length LH = 400 μm, dII ∼ 3 μm, WH ∼ 1 μm, and tH ∼ 100 nm. The microscopic images of both types of fabricated MZIs are shown in Fig. 6. The line resistance of fabricated microheaters is estimated as rH ∼ 70 Ω/μm and the resistance of two flaring regions including the probe contact resistance is extracted as ∼ 4 kΩ (RCH). Thus for a waveguide microheater of length LH used for above mentioned MZI switch, the actual electrical power consumption for thermo-optic switching experiments for all fabricated devices: Pe = I2rH LH, where I is the dc or rms value of current through the microheater. As the length of microheaters is kept relatively long (LH = 400 μm), the effective phase-shifter length Leff ≈ LH = Lw as it was modeled earlier in Ref. . Moreover, for a waveguide phase-shifter of length Lw ≈ 400 μm, the value of ΔTπ calculated using Eq. (1) is found to be small (∼ 11K) and hence the temperature dependent change in microheater resistance is assumed to be insignificant during switching operation.
For thermo-optic switching characterizations of fabricated devices, we used two fiber-optic probes for input and output light coupling via grating couplers (GCs) and simultaneously two electrical probes for driving the microheater. A high resolution spectrum analyzer (APEX 2043B) with inbuilt CW laser source (1520 nm ≤ λ ≤ 1620 nm) was used for the experiment. The microheaters were sourced by connecting terminal Al contact pads with a current source measuring unit (SMU). The experimental set-up for studying thermo-optic switching characteristics of fabricated devices may be found elsewhere . The wavelength dependent transmission characteristics for both ON-state and OFF-state at the bar ports (1520 nm ≤ λ ≤ 1620 nm) normalized with a reference waveguide transmission (to eliminate wavelength dependent fiber to input/output grating coupler losses) are shown in Figs. 7(a) and 7(b) corresponding to an S-MZI and an F-MZI, respectively. A wavelength independent insertion loss of ∼ 2.5 dB is recorded for both S-MZI and F-MZI switches, which is mainly attributed to the total insertion loss of two 3-dB power splitters. The observed oscillations at longer wavelengths may be attributed to the weaker coupling efficiency of the grating couplers resulting into noise level detection limit of the photodetector. Typical electrical power required for switching optical signal from cross port to the bar port (with an extinction of > 20 dB) is Pe ∼ 52.5 mW (∼ 36.7 mW) for fabricated S-MZI (F-MZI) switches. In case of folded waveguide microheater, heat is dissipated more effectively in the smaller volume, compared to that of straight waveguide microheater of same length. The steady-state transmitted optical signal (λ = 1550 nm) at the bar and cross ports of the above mentioned S-MZI and F-MZI as a function of electrical power dissipated/consumed by the microheaters are shown in Figs. 8(a) and 8(b), respectively. Though both types of MZIs were designed with balanced arms, a little mismatch occasionally observed for some fabricated devices. For example, an additional bias electrical power of Pe = 8.75 mW was required for achieving maximum (minimum) transmission at the cross (bar) port of the S-MZI. Typical transient response of optical transmission at both bar- and cross-ports of the MZI switch (straight or folded microheater) are shown in Fig. 8(c). This has been obtained for an input modulating voltage of 5 V peak to peak at a repetition rate of 12 kHz superimposed with a dc biasing voltage across the curved waveguide microheater. The recorded rise/fall time for both types of microheaters are measured to be nearly same (∼5 μs) which is higher than the theoretical calculation (4.2 μs) as shown in Fig. 4(f). This deviation may be due to the assumption of lower value of thermal conductivities and/or higher values of heat capacitances (for both core and claddings) in theoretical simulation.
The steady-state switching characteristics obtained for S-MZI and F-MZI are fitted with appropriate analytical transfer functions  and we have extracted the values of ΔTs (steady-state temperature in the waveguide core) vs. pw (electrical power consumed per unit length of a waveguide phase-shifter) for both straight and folded waveguide phase-shifters as shown in Fig. 9(a). The slope of ΔTs vs. pw is a measure of the sensitivity figure of merit SH of the waveguide phase-shifter as defined earlier. The value of SH extracted for an S-MZI is 82.4 K·μm/mW, which is slightly higher than that of theoretical prediction of 73.5 K·μm/mW. This may be attributed to the fact of our assumption Lw = LH; a correction term is required for Lw = Leff = LH + Δ, where Δ ∼ 10 μm for LH > 100 μm . In other words, the effective conductance (gw) of fabricated devices is slightly lower than that of theoretical calculations which is consistent with the observed higher values of thermal response time (τth) of the fabricated devices discussed earlier. Nevertheless, we observed a significantly enhanced value of SH = 119 K·μm/mW in case of folded waveguide phase-shifter as predicted earlier. The transient characteristics obtained for S-MZI and F-MZI are again fitted with the corresponding analytical functions to extract the temperature rise ΔT(t) in the waveguide core of the phase-shifters [see Fig. 9(b)]; both straight and folded waveguide phase-shifters are shown to be following Eq. (3) with τth = 5.1 μs. Thus the experimentally observed value of ℱH in a folded waveguide phase-shifter (23.3 K·μm/mW·μs) is about 1.5 times higher than that of a straight waveguide phase-shifter (16.2 K·μm/mW·μs) for our fabricated devices. It is possible to improve the value of ℱH further by designing more tightly folded waveguide phase-shifter with close proximity of microheater but one needs to take care of associated optical losses of the guided mode. Using Eqs. (5) and (6), we have extracted the thermal characteristic parameters gw and hw as 1.21 × 10−2 mW/K·μm (1.21×10−2 mW/K·μm) and 6.17×10−2 mW·μs/K·μm (4.29×10−2 mW·μs/K·μm), respectively, for the fabricated straight (folded) design waveguide phase-shifters in Type-II (air cladding) architecture.
In summary, a theoretical model for the performance analysis of a metallic strip-line microheater integrated waveguide phase-shifter has been developed. The model helps to define two important figure of merits such as temperature sensitivity SH and ℱH (= SH/τth) following thumb-rules of SH ·gw = 1 and ℱH · hw = 1, respectively, where gw and hw are characteristic line conductance and line heat capacitance of the waveguide phase-shifter. These figure of merits were calculated for SOI waveguides (supporting TE0 guided mode) integrated with Type-I and Type-II microheater architectures. It has been shown by numerical simulation that Type-II microheaters offer faster switching time than that of Type-I microheaters. It has been also shown that both SH and ℱH can be improved significantly by proper choices of waveguide design parameters, closer proximity of microheater to the waveguide core, and suitably folding the waveguide-microheater phase-shifter system. However, care must be taken to limit the bend induced waveguide loss and loss due to evanescent field overlap with metallic microheater. The theoretical model is further validated with experimental results by fabricating 2 × 2 MZI switches (wavelength independent) designed with Type-II microheaters in SOI platform. Though the demonstrated MZI switches were not designed with the best possible values of figure of merits (SH, τth and ℱH), the experimental technique described here helps to extract the characteristic parameters like gw and hw of a thermo-optic waveguide phase-shifter for its modeling and optimized design.
1. G. Cocorullo and I. Rendina, “Thermo-optical modulation at 1.5 μm in silicon etalon,” Electron. Lett. 28, 83–85 (1992). [CrossRef]
2. H. Shen, M. H. Khan, L. Fan, L. Zhao, Y. Xuan, J. Ouyang, L. T. Varghese, and M. Qi, “Eight-channel reconfigurable microring filters with tunable frequency, extinction ratio and bandwidth,” Opt. Express 18, 18067–18076 (2010). [CrossRef] [PubMed]
4. R. Sumi, R. K. Gupta, N. DasGupta, and B. K. Das, “Ultra-broadband add-drop filter/switch circuit using subwavelength grating waveguides,” IEEE J. Sel. Top. Quantum Electron. 25, 1–11 (2019).
5. L.-W. Luo, S. Ibrahim, A. Nitkowski, Z. Ding, C. B. Poitras, S. B. Yoo, and M. Lipson, “High bandwidth on-chip silicon photonic interleaver,” Opt. Express 18, 23079–23087 (2010). [CrossRef] [PubMed]
6. P. Orlandi, F. Morichetti, M. J. Strain, M. Sorel, A. Melloni, and P. Bassi, “Tunable silicon photonics directional coupler driven by a transverse temperature gradient,” Opt. Lett. 38, 863–865 (2013). [CrossRef] [PubMed]
7. K. Tanizawa, K. Suzuki, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, T. Sugaya, S. Suda, G. Cong, T. Kimura, K. Ikeda, S. Namiki, and H. Kawashima, “Ultra-compact 32 × 32 strictly-non-blocking Si-wire optical switch with fan-out LGA interposer,” Opt. Express 23, 17599–17606 (2015). [CrossRef] [PubMed]
8. K. Suzuki, K. Tanizawa, S. Suda, H. Matsuura, T. Inoue, K. Ikeda, S. Namiki, and H. Kawashima, “Broadband silicon photonics 8 × 8 switch based on double-Mach–Zehnder element switches,” Opt. Express 25, 7538–7546 (2017). [CrossRef] [PubMed]
9. C. Li, W. Zheng, P. Dang, C. Zheng, Y. Wang, and D. Zhang, “Silicon-microring-based thermo-optic non-blocking four-port optical router for optical networks-on-chip,” Opt. Quantum Electron. 48, 552 (2016). [CrossRef]
10. M. S. Dahlem, C. W. Holzwarth, A. Khilo, F. X. Kärtner, H. I. Smith, and E. P. Ippen, “Reconfigurable multi-channel second-order silicon microring-resonator filterbanks for on-chip WDM systems,” Opt. Express 19, 306–316 (2011). [CrossRef] [PubMed]
12. X. Wang, J. A. Martinez, M. S. Nawrocka, and R. R. Panepucci, “Compact thermally tunable silicon wavelength switch: modeling and characterization,” IEEE Photon. Technol. Lett. 20, 936–938 (2008). [CrossRef]
13. A. Atabaki, E. S. Hosseini, A. Eftekhar, S. Yegnanarayanan, and A. Adibi, “Optimization of metallic microheaters for high-speed reconfigurable silicon photonics,” Opt. Express 18, 18312–18323 (2010). [CrossRef] [PubMed]
14. D. Schall, M. Mohsin, A. A. Sagade, M. Otto, B. Chmielak, S. Suckow, A. L. Giesecke, D. Neumaier, and H. Kurz, “Infrared transparent graphene heater for silicon photonic integrated circuits,” Opt. Express 24, 7871–7878 (2016). [CrossRef] [PubMed]
15. A. Densmore, S. Janz, R. Ma, J. H. Schmid, D.-X. Xu, A. Delâge, J. Lapointe, M. Vachon, and P. Cheben, “Compact and low power thermo-optic switch using folded silicon waveguides,” Opt. Express 17, 10457–10465 (2009). [CrossRef] [PubMed]
16. P. Sun and R. M. Reano, “Free-standing silicon-on-insulator strip waveguides for submilliwatt thermo-optic switches,” in Frontiers in Optics, (Optical Society of America, 2010), p. FMH3. [CrossRef]
17. Q. Fang, J. F. Song, T.-Y. Liow, H. Cai, M. B. Yu, G. Q. Lo, and D.-L. Kwong, “Ultralow power silicon photonics thermo-optic switch with suspended phase arms,” IEEE Photon. Technol. Lett. 23, 525–527 (2011). [CrossRef]
18. Z. Lu, K. Murray, H. Jayatilleka, and L. Chrostowski, “Michelson interferometer thermo-optic switch on soi with a 50-μW power consumption,” in Photonics Conference (IPC), 2016 IEEE, (IEEE, 2016), pp. 107–110. [CrossRef]
19. M. Bahadori, A. Gazman, N. Janosik, S. Rumley, Z. Zhu, R. Polster, Q. Cheng, and K. Bergman, “Thermal rectification of integrated microheaters for microring resonators in silicon photonics platform,” J. Light. Technol. 36, 773–788 (2018). [CrossRef]
21. “COMSOL Multiphysics version 5.3,” https://www.comsol.com.
22. B. Singh and N. Surplice, “The electrical resistivity and resistance-temperature characteristics of thin titanium films,” Thin Solid Films 10, 243–253 (1972). [CrossRef]
23. M. Asheghi, M. Touzelbaev, K. Goodson, Y. Leung, and S. Wong, “Temperature-dependent thermal conductivity of single-crystal silicon layers in SOI substrates,” J. Heat Transf. 120, 30–36 (1998). [CrossRef]
24. P. Johnson and R. Christy, “Optical constants of transition metals: Ti, C, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9, 5056 (1974). [CrossRef]
25. F. Kreith, R. M. Manglik, and M. S. Bohn, Principles of Heat Transfer (Cengage Learning, 2012).
26. R. K. Gupta, S. Chandran, and B. K. Das, “Wavelength independent directional couplers for integrated silicon photonics,” J. Light. Technol. 22, 4916–4923 (2017). [CrossRef]
27. S. Chandran, R. K. Gupta, and B. K. Das, “Dispersion enhanced critically coupled ring resonator for wide range refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23, 424–432 (2017). [CrossRef]