III. Graphene Nanophotonic Modulators for Mid-Infrared Applications -
3.2. Theoretical Investigation of Mid-Infrared Graphene Plasmonic Modulator based on
3.2.2. Design of the Modulator
The design of the modulator starts from the selection of the values of tH and tZ. Since HfO2 has a complex refractive index (1.7469 + i0.0125 at λ = 8 μm), the absorption by HfO2 as well as graphene affects the propagation loss of the GPWM. Under the assumption that the GPWM is completely confined in the slot region, an Al-HfO2-graphene-ZnS waveguide is analyzed rather than the whole hybrid plasmonic waveguide. As shown in the relation between the propagation loss of the GPWM and tH at λ = 8 μm (Figure 3.2.2(a)), the propagation loss becomes minimum for tH = 22 nm. If tH decreases from this value, the propagation loss increases due to the absorption by the Al layer. In contrast, if tH increases from 22 nm, it increases due to the absorption by the HfO2 layer. Therefore, tH is chosen to be 22 nm. The electric field distribution of the GPWM in the inset of Figure 3.2.2(a) indicates that the electric field is negligible more than 100 nm below the graphene-ZnS interface, and tZ is chosen to be 140 nm such that the assumption is satisfied. The next step of the design is the selection of the value of tG. With regard to the photonic waveguide, only the fundamental TM mode is supported in the 7-9 μm spectral range for tG
between 0.5 and 1.625 μm. Hence, tG should be determined in this interval, considering the efficient excitation of the HPWM by the fundamental TM mode. The excitation efficiency is calculated by analyzing the modulator, the hybrid plasmonic waveguide of which is 50 nm long and does not contain the graphene and the grating. Since the hybrid plasmonic waveguide is much shorter than the propagation distance of the HPWM (114.5 μm at λ = 8 μm), the excitation efficiency in dB is half the transmission of the modulator in dB. As shown in the relations of the excitation efficiency to λ for a few values of tG (Figure 3.2.2(b)), the efficiency increases with tG, especially at longer wavelengths, since the fundamental TM mode and the HPWM are more confined in the Ge layer for a larger value of tG. However, the overlap between the HPWM and the GPWM becomes less intense as tG increases.
Therefore, tG is chosen to be 1.25 μm. Figure 3.2.2(c) shows the electric field distributions of the HPWM in section 1 with the ZnS layer intact and section 2 with an air gap of thickness dg between the graphene and the ZnS layer (correspondingly, tZ is reduced by dg). The grating region is the alternation of sections 1 and 2. Tentatively, dg is chosen to be 70 nm. In addition, those of the GPWM in section 1 and section 2 are shown in Figure 3.2.2(d). We can confirm that the HPWM is strongly confined in the slot region and that the GPWM completely overlaps the HPWM there. The propagation constant and loss of the HPWM in section i, denoted by βHi and αHi, are shown as functions of λ in Figure 3.2.2(e). Those of the GPWM in sections i, denoted by βGi and αGi, are shown as functions of λ in Figure 3.2.2(f). βGi is about 25 times larger than βHi, and αGi is more than two orders of magnitude larger than αHi.
Figure 3.2.2. Design of the hybrid plasmonic waveguide. (a) Relation of the propagation loss of the GPWM to the HfO2
thickness tH. For convenience, an Al-HfO2-graphene-ZnS waveguide rather than the hybrid plasmonic waveguide was analyzed. tH is determined to be 22 nm such that the propagation loss is minimum. The inset shows the electric field distribution of the GPWM for tH = 22 nm. The field is completely confined in a ~100-nm-thick region below the graphene.
Therefore, the ZnS thickness tZ is determined to be 140 nm. (b) Relations between the efficiency of exciting the HPWM from the photonic waveguide mode and the wavelength. When the Ge thickness tG is determined to be 1.25 m, the excitation efficiency is larger than –0.35 dB in the 7-9 m spectral range. (c) Electric field distributions of the HPWM in
The final step of the design is to determine the grating period and the number of periods to make the transmission of the modulator as close to zero as possible at λ = 8 μm. The length of section i, denoted by Λi, is initially determined by Λi = π / (βG - βHi) at λc such that the HPWM and the GPWM have a phase difference of π after they propagate in section i. Then, Λ1 and Λ2 are 55 nm and 87 nm, respectively, and Λ1 + Λ2 is the period. The optimal number of periods is determined based on the transmission spectra calculated for various numbers of periods. The transmission spectra calculated by the mode matching method show that the transmission at λc = 8 μm becomes minimum when the number of periods is 26 (Fig. 3(a)). To confirm this result, the FDTD method was used to calculate the accurate transmission spectra. They are similar to the transmission spectra calculated by the mode-matching method, but they are centered at λc = 7.823 μm, and the minimum transmission is smaller than –17 dB for the number of periods larger than 40 (Figure 3.23(a)). Inaccuracy exists in the calculation based on the mode-matching method since only the guided modes are considered despite the fact that the HPWM and the GPWM undergo abrupt and rapid structural changes in the grating region. Adjustments of Λ1
and Λ2 are required to shift the transmission spectrum to 8 μm. The relations of λc to Λ1 and Λ2 were calculated and they are almost linear (Supplementary Information S4 in ref. [106]). The values of
∂λc/∂Λ1 and ∂λc/∂Λ2 are 40 nm/nm and 25 nm/nm, respectively, and they are used to adjust Λ1 and Λ2. For Λ1 = 58 nm and Λ2 = 90 nm, the transmission spectrum is centered at λc = 8.014 μm (Figure 3.23(b)), and the transmission at λc is –27 dB if the number of periods is 40. The negative of the transmission at a wavelength away from λc is considered as the insertion loss of the modulator, which is about –1.72 dB. It is about 1 dB larger than the sum of two times the coupling loss between the photonic waveguide and the hybrid plasmonic waveguide and the loss of the hybrid plasmonic waveguide of length 5.92 μm.
One inset of Figure 3.23(b) shows that the fundamental TM mode at λ = 7 μm passes through the modulator without noticeable loss. In contrast, the other inset shows that the HPWM is removed at λc
such that the fundamental TM mode does not exist in the output photonic waveguide. In addition to the major rejection band, the minor rejection band exists at λ = 8.61 μm in the transmission spectrum. This is caused by contra-directional coupling between the HPWM and the GPWM while the major rejection band is caused by the co-directional coupling. This can be confirmed from the electric field distribution in the modulator at λ = 8.014 μm (Figure 3.2.3(c)) and that at λ = 8.61 μm (Figure 3.2.3(d)). At λ = 8.014 μm, the electric field becomes stronger around the graphene as the distance from the left end of the grating increases up to ~2.2 μm. However, at λ = 8.61 μm, the electric field is strong around the graphene just at the left end. Figure 3.2.3(e) more clearly shows how the normalized power of the HPWM and that of the GPWM change at λ = 8.014 μm as the distance from the left end increases. The former monotonically decreases, but the latter first increases and then decreases. These trends accord with what can be expected from co-directional coupling of a low-loss mode to a high-loss mode. The normalized-power expressions of the low-loss and high-loss modes involved in such coupling are
power to the distance. There is good agreement between the relations and the fitted curves. The maximum power of the GPWM is 0.42 at the distance of 2.2 μm. In other words, 42 % of the power of the input fundamental TM mode is transferred to the GPWM via the grating with 15 periods. This value is 1.7 times larger than the previous efficiency of converting a surface phonon polariton into a GPP via a tapering structure [105]. The results in Figure 3.2.3 demonstrate that the transmission of the modulator can be made almost equal to zero at λ ≅ 8 μm when the grating region length is just 5.92 μm (corresponding to 40 periods) which is shorter than the wavelength. In addition, they show that the grating with 15 periods can be used to efficiently excite the GPWM.
Figure 3.2.3. Analysis of the grating-assisted coupling of the HPWM to the GPWM. (a) Transmission spectra of the modulator calculated by using the FDTD method and the mode-matching method (MMM) for different numbers of periods. They were calculated for 1 = 55 nm and 2 = 87 nm (1 and 2 denote the lengths of sections 1 and 2, respectively). (b) Transmission spectrum of the modulator calculated by using the FDTD method. To locate the rejection band at a wavelength of 8 m, 1 and 2 were adjusted to 58 nm and 90 nm, respectively. The number of periods was 40.
The insets show the magnetic field distributions in the modulator at wavelengths of 7 m and 8.014 m, respectively. The distribution at the coupling wavelength (8.014 m) confirms that the HPWM disappears traveling along the hybrid plasmonic waveguide such that the transmission becomes very small. (c) Electric field distribution in and around the slot region of the hybrid plasmonic waveguide at the coupling wavelength (8.014 m). (d) Electric field distribution in and around the slot region at the center wavelength (8.61 m) of the minor rejection band which is generated by the contra- directional coupling. (e) Changes of the normalized powers of the HPWM and the GPWM along the grating at the coupling wavelength. The symbols were obtained from the calculation based on the FDTD method. The lines calculated by using approximate expressions for the normalized powers were fitted to the symbols. The expressions were derived from the coupled mode theory (CMT). In the calculations related to all the panels, the chemical potential and mobility of graphene were set at 0.6 eV and 10,000 cm2/V/s, respectively.