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Abstract
Light modulation has been recognized as one of the most fundamental operations in photonics. In this paper, we theoretically designed a Bloch surface wave assisted modulator for the active modulation of graphene electro-absorption. Simulations show that the strong localized electrical field generated by Bloch surface waves can significantly enhance the graphene electro-absorption up to 99.64%. Then by gate-tuning the graphene Fermi energy to transform graphene between a lossy and a lossless material, electrically switched absorption of graphene with maximum modulation depth of 97.91% can be achieved. Meanwhile, by further adjusting the incident angle to tune the resonant wavelength of Bloch surface waves, the center wavelength of the modulator can be actively controlled. This allows us to realize the active modulation of graphene electro-absorption within a wide near-infrared region, including the commercially important telecommunication wavelength of 1550 nm, indicating the excellent performance of the designed modulator via such mechanism. Such Bloch surface waves assisted wavelength-tunable graphene electro-absorption modulation strategy opens up a new avenue to design graphene-based selective multichannel modulators, which is unavailable in previous reported strategies that can be only realized by passively changing the structural parameters.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since its discovery, graphene [1], a two-dimensional material, has attracted considerable attention due to its unique optoelectronic properties. Especially, the advantages of broad operation bandwidth, gate-tunable conductivity and high-density integration etc., make graphene a good candidate for high-performance electro-optic modulation [2,3]. Based on different optical responses of graphene in different wavelength regions, various kinds of graphene-based optical modulators have been designed. For example, in the mid-infrared and terahertz regions, doped graphene acts like a metal that supports highly confined surface plasmonic resonances (SPR) [4–9] with enhanced absorption value up to nearly 100% [10–14], enabling one to realize efficient plasmonic modulators with high modulation depth [15–17]. While in the visible and near-infrared regions, it behaves more like an absorptive dielectric and cannot support plasmonic resonances. In this case, the optical absorption of graphene is due to the interband transitions and can be modulated by gate-tuning graphene Fermi energy due to Pauli-blocking [18–22]. However, the small optical absorption (∼2.3%) of graphene limits the performance of the devices and restricts their relevant applications in the near-infrared region, especially in the commercially important telecommunications waveband. To address this issue, various kinds of strategies using resonant photonic structures have been proposed to improve the graphene absorption, such as Fabry-Perot resonant cavity [23,24], metallic resonant structures [25], dielectric guided mode resonance structure [26–28], Bragg reflector [29], magnetic polaritons structures [30,31]. Despite of this progress, the center wavelengths of these devices cannot be changed once they are fabricated, because the graphene absorption wavelength is dominated by the structural parameters of the resonant structures.
Bloch surface waves (BSWs) [32,33] is a kind of surface photonic wave that propagates at the interface between an one-dimensional photonic crystal (PC) and an adjacent dielectric medium. It can confine the electromagnetic energy into a sub-wavelength region very close to the surface that can enhance the light-matter interaction [34,35], and thus may be also considered as a dielectric analog to SPR. Compared to SPR, BSWs exhibits outstanding advantages benefiting from its metal-free structure. For example, the metal-free feature ensures long propagation lengths [36,37] and high-quality resonances [38] of BSWs. Additionally, because of the simple geometry, easy fabrication and great freedom in the choice of dielectric materials, BSWs can solve some issues that SPR structures are facing, such as metallic heating and incompatibility with CMOS fabrication processes. More importantly, the BSWs resonant wavelength can be tailored within a ultrawide wavelength ranges from UV [39] to visible [40] and mid-infrared [41] region by simply controlling the bandgap and the defect layer of the PC, which can be further actively modulated by adjusting the incident angle. These exciting properties render BSWs a promising all-dielectric photonic platform, which have triggered extensive research interests for the design of various BSWs-based chip-integrated optoelectronic devices [42–46]. Recently, a BSWs-based graphene modulator had been experimentally demonstrated to realize the ultrafast optical reflection modulation of graphene at 785 nm [47], showing great potential for the realization of all-optical switches using 2D materials. However, the systematical studies for the illustration on the active modulation of graphene electro-absorption still remain unexplored, and great efforts also need to be made to optimize the spectral response of device for better modulation performance, especially in the commercially important telecommunications waveband.
Inspired by this idea, we theoretically present a BSWs-assisted graphene electro-absorption modulator consisting of a graphene monolayer and one-dimensional PC separated by a defected layer. By exciting the BSWs mode, the induced localized electric field can significantly improve the graphene electro-absorption, of which the center wavelength can be actively controlled by adjusting incident angle. Then by further tuning the graphene Fermi energy to transform graphene between a lossy and a lossless material, the modulator demonstrates excellent performance with modulation depth of 97.91% within a wide near-infrared region, including the commercially important telecommunication wavelength of 1550 nm.
2. Results and discussion
2.1 Modulation principle of the modulator
The schematic diagram of the proposed modulator is shown in Fig. 1(a). The monolayer graphene is placed on the top of the photonic crystal (PC). The one-dimensional PC consists of alternating low-index (ZnSe, na = 2.46, da = 163 nm) and high-index (Ge, nb = 4.30, db =93 nm) materials. Upon the one-dimensional PC is a buffer layer (ZnS, nc =2.28 + 0.00001*i, dc =615 nm). The prism is chosen as fused silica (np = 1.445). The period number of the one-dimensional PC is optimized to be 4 to achieve a high-quality resonant BSWs mode and compact configuration (discussed in the next section). Additionally, the modulator is designed to operate in both TE (electric field is parallel to y-axis) and TM (magnetic field is parallel to y-axis) polarizations. In such configuration, the excitation of the BSWs resonant mode can produce a strong localized electric field around graphene, resulting in a great improvement of graphene electro-absorption. Then by electrically tuning the graphene Fermi energy to transform graphene between a lossy and a lossless material, the graphene electro-absorption can be effectively modulated. Meanwhile, by adjusting incident angle to tune the resonant wavelength of BSWs, the center wavelength of graphene electro-absorption can be dynamically modulated within a wide near-infrared region.
To investigate the modulation mechanism of the designed modulator, simulations are carried out using finite element analysis software employing COMSOL Multiphysics. The graphene monolayer is modeled as an ultrathin dielectric layer with permittivity as [26]
where σg is graphene conductivity, ω is the angular frequency of the incident wave, ɛ0 is vacuum permittivity, and d =0.34 nm is the monolayer graphene thickness. The σg consisted the contributions of both intraband term σintra and interband term σinter in the considered near-infrared region, which can be characterized by the Kubo formula as [48]The simulated absorption spectra (red lines) of the modulator for TE and TM polarizations are shown in Fig. 2(a) and 2(c), respectively, when the incident angle is 49. 65 deg. The absorption spectra (black lines) of the structure without graphene are also provided for comparison. Two absorption peaks can be observed at 1500 nm and 1550 nm for TE and TM polarizations, which are associated with the excitation of BSWs modes. Additionally, by comparing the absorption spectra of the structures with and without graphene, it is clear that graphene makes a major contribution to the absorption of the modulator. Another interesting phenomenon is that the modulator exhibits completely different absorption feature at two wavelength regions divided by λ= 1512 nm for different polarizations. For TE polarization, the absorption of the modulator is only 10% at 1550 nm. While for TM polarization, nearly perfect absorption (99.64%) can be achieved at 1500 nm. It should be mentioned that, the absorption of graphene is due to direct interband transitions determined by its Fermi energy Ef in the near-infrared wavelength region. By plotting the real and imaginary parts of graphene permittivity in Fig. 2(c), it is found that there is a sharp peak in the real part and a sudden fall in the imaginary part at 1512 nm (corresponding to the optical energy of 0.41 eV, i.e., 2Ef). Such permittivity property results in the different absorption property of graphene at different wavelength region. For the wavelength region where the optical energy is larger than 2Ef, graphene possesses a large imaginary part of the permittivity and can be regarded as a lossy material. In this case, when BSWs is excited in this region, the induced strong localized electrical field (Fig. 1(d)) greatly enhances the graphene-light interaction. Due to the interband transition of electrons, the electric current in graphene significantly enlarges and thus promotes the improvement of graphene electro-absorption. While for the wavelength region where the optical energy is smaller than 2Ef, the imaginary part of graphene permittivity rapidly decreases to nearly zero and graphene is approximately close to a lossless material. Thus, the absorption of graphene is limited even when graphene is placed near the strong electric field induced by BSWs, as shown in Fig. 1(e). Using such different absorption property of graphene, we can modulate the light absorption in graphene to realize the switched absorption effect.
Fig. 2. (a) The BSWS absorption peak of the structure whether with graphene in TE mode, the graphene Fermi energy Ef = 0.41eV. (b) The absorption of the structure whether with graphene in TM mode, the graphene Fermi energy Ef = 0.41eV. (c) Real and imaginary part of the graphene permittivity, the graphene Fermi energy Ef = 0.41eV. Simulated intensity distribution of electric field components for (d) TE and (e) TM polarizations. The blue line in the figure shows the electric field intensity in z-direction. It's obvious that the electric field is constrained in the interface between the structure and the surrounding medium.
2.2 Effect of the geometric structure
For the proposed modulator, the enhanced graphene electro-absorption is induced by the strong electric field generated by BSWs. Its center wavelength is determined by the BSWs resonant wavelength, which is originally determined by the geometric structure of the modulator. Taking TE polarization as an example, we then investigate the effect of the structural parameters on the spectral response of the modulator. Figure 2(a) and 2(b) show the absorption spectra of the thickness of ZnSe layer (dZnSe) and Ge layer (dGe), respectively. The Fermi energy of graphene is fixed at Ef = 0.4eV. One can see that dZnSe and dGe play a critical role in the spectral response of the modulator since they together determine the band structure of one-dimensional PC. Multiple absorption bands can be observed due to the excitation of multi-order BSWs modes, and their center wavelengths red shift towards the long-wavelength region as dZnSe and dGe increase. Different from dZnSe and dGe, the period number Npair does not affect the resonant wavelength or the band structure but has a strong influence on the absorption of the modulator, as shown in Fig. 3(c). To be specific, when Npair is too small, most of light is reflected back due to the total internal reflection and the PC effect is too weak for the whole system to reach an admittance matching. When Npair is too large, the light is reflected directly by PC without entering the system since an over-length PC leads to the over-change of the admittance loci and consequently the mismatched admittance. Thus, there is a trade-off between the period number and the graphene absorption. In this work, the period number of 4 is chosen to achieve a high-quality resonant absorption and compact configuration of the modulator.
Fig. 3. Effects of structural parameters on the spectral response of the modulator for TE polarization. (a) The thickness of ZnSe dZnSe. (b) The thickness of Ge dGe. (c) The period number of the PC Npair. (d) The thickness of defect layer ddefect. The white dash line is the absorption boundary at 1550 nm when graphene Fermi energy Ef = 0.4eV. The incident angle is 49. 65 deg.
We further investigate the influence of the thickness of defect layer ddefect on the spectral response of the modulator, and the result is provided in Fig. 2(d) when graphene Fermi energy Ef = 0.4eV. Compared to controlling the thickness of ZnSe and Ge layers, modulating the thickness of defect layer is a much more direct way to manipulate the spectral response of the modulator because it does not change the geometry of barrier or the band structure of the one-dimensional PC. It can be clearly seen from the figure that the center wavelength (λ0) of the modulator continuously red shifts for a long distance as ddefect increases. A common phenomenon observed from all these figures is that the resonant absorption of the modulator remains strong until reaching the absorption boundary at 1550 nm. Such phenomenon can be easily understood by the above analysis that the graphene electro-absorption strongly depends on the comparison between λ0 and Ef. When λ0 < ℏπc/Ef = 1550 nm, the interband transition of electrons dominates the optical response and hence the modulator keeps a high absorption with a maximum value of 99.64%. While when λ0 > ℏπc/Ef = 1550 nm, disorder-induced trapping overwhelms the interband transition, graphene is closed to a lossless material, resulting in the rapid decrease of absorption with a maximum value of only 2.1%.
2.3 Modulation performance
The actively controlled spectral response of optoelectronic devices is highly desirable for their practical applications. For the designed modulator, the electro-absorption feature of graphene can be actively modulated by gate-tuning graphene Fermi energy. While the BSWs resonant wavelength can be dynamically controlled by adjusting the incident angle, which allows us to shift the center wavelength of graphene electro-absorption in a wide wavelength region. Hence, taking TE polarization for an example, we will investigate the active modulation of graphene electro-absorption by adjusting graphene Fermi energy and incident angle in this section.
The absorption mapping of the modulator as a function of incident wavelength and graphene Fermi energy Ef is shown in Fig. 4(a) for TE polarization. It is obvious that the modulator exhibits distinct absorption features in two wavelength regions with a clear boundary (the position of graphene interband transition). To gain a clearer view of such difference, we further extract several typical absorption spectra with varied graphene Fermi energy from 0.36 to 0.44 eV in a step of 0.02 eV, as shown in Fig. 4(b). When Ef = 0.36 eV and 0.38 eV, the absorption of the modulator remains almost the same with maximum value of 99.64%. As Ef increases to 0.4 eV, the absorption suddenly drops to 60.6% at position of the interband transition and then rapidly drops to only 2.1% when Ef further increase to 0.44 eV. Such evolution of spectral response can be explained by the change of graphene permittivity as Fermi energy varied, as indicated by Eq. (2). The real and imaginary parts of wavelength-dependent graphene permittivity are shown in Fig. 4(c) and 4(d), respectively, with Fermi energy increased from 0.36 to 0.44 eV in a step of 0.02 eV. The graphene interband transition is reflected by the narrow-band spectral feature in its complex permittivity. For every Ef, a sharp peak in the real part of ɛg and a quick fall in the imaginary part of ɛg can be observed. Moreover, the position of the sharp peak in the real part of ɛg blueshifts from 1722 to 1409 nm as Ef increases from 0.36 to 0.44 eV. While the imaginary part of ɛg exhibits a sudden change and drops to be almost zero for the wavelengths larger than the interband transition. This results in effective modulation of absorption loss in graphene around the interband transition, which allows us to realize the electrically switched effect of graphene absorption in a wide near-infrared region, including the commercially important telecommunication wavelength of 1550 nm.
Fig. 4. (a) Simulated absorption spectra of the modulator with varying graphene Fermi energy for TE polarization. The incident light angle is fixed at 49.65 deg. (b) Extracted absorption spectra with varying graphene Fermi energy from 0.36 eV to 0.44 eV in a step of 0.02 eV. The inset shows the enlarged spectral region. (c) Real and (d) imaginary parts of graphene permittivity with varying graphene Fermi energy. (e) Resonant absorption of graphene and calculated modulation depth of the modulator with varying Fermi energy for λ = 1550 nm.
To evaluate the modulation performance of the device, the graphene absorption at λ = 1550 nm with varying Ef from 0.3 to 0.5 eV are calculated and shown as the red line in Fig. 4(e). For the telecommunication wavelength of 1550 nm, graphene is a lossy material when Ef is smaller than 0.4 eV and its absorption remains at a high value (larger than 80%). As Ef increases to a critical value of 0.4 eV, the absorption falls rapidly and decreases to only 2.1% since graphene is now turned into an almost lossless material. Therefore, by gate-tuning Fermi energy, graphene can be transformed from a lossy material to a lossless material, which enables us to realize the electrically switched absorption of graphene with high modulation depth. The modulation depth can be defined as 1 - A/Amax, where A is the corresponding absorption as a function of Ef, and Amax is to the maximum absorption of graphene. The calculated modulation depth of graphene electro-absorption is shown as blue line in Fig. 4(e). It shows that the modulation depth increases rapidly from almost zero to nearly 97.91% by a small variation of Ef near the interband transition, showing excellent modulation of graphene electro-absorption.
We further demonstrated that the graphene electro-absorption can be dynamically modulated at a specific desired wavelength in a wide near-infrared region benefiting from the tunable BSWs. For the proposed modulator, its center wavelength is dominated by the BSWs resonant wavelength, which can be actively manipulated by tuning the incident angle. As an illustration, the absorption spectra with varying incident angle θ for the modulators with and without graphene are simulated and compared in Fig. 5(a) and 5(b), respectively. The figures indicate that the center wavelength of the modulator covers a wavelength region from 1300 to 1630 nm as θ increases from 44 to 85 deg for both cases. However, the resonant absorption of the modulator without graphene is almost undetectable from absorption mapping in Fig. 5(b). By covering a graphene monolayer, the absorption of the modulator can be significantly improved. Especially for the wavelength region from 1300 to 1550 nm, nearly perfect absorption can be achieved for the modulator covered with graphene, since graphene (with Fermi energy of 0.5 eV) can be considered as a lossy material at this wavelength region.
Fig. 5. (a) Absorption spectra of the modulator with varying incident angle, the white dash line is the absorption boundary when the graphene Fermi energy Ef = 0.4eV. (b) Absorption spectra of the modulator with varying incident angle without graphene. (c) Absorption spectra of the modulator for several specific incident angles. (d) Resonant absorption of the modulator with varying graphene Fermi energy for λ=1330 nm, θ=83.36 deg (pink line), λ=1440 nm, θ=61.98 deg (yellow line), λ=1550 nm, θ=49.65 deg (cyan line), λ=1620 nm, θ=44.1 deg (purple line). (e) Resonant absorption of the modulator with varying graphene Fermi energy for λ=1550 nm for different carrier density of 103 cm2/(V·s) and 104 cm2/(V·s). (f) Real and imaginary parts of graphene permittivity with varying graphene Fermi energy for different carrier density.
The absorption spectra of the proposed modulator are further extracted from Fig. 5(a) and displayed in Fig. 5(c). The center wavelength of graphene absorption can be actively controlled within a wide wavelength region by tuning the incident angle. Additionally, the resonant absorption of graphene varied in a range between 90% and 100% in the wavelength region from 1300 to 1550 nm. While the resonant absorption abruptly drops to be almost zero as the center wavelength is shifted towards the wavelength region that is larger than 1550 nm, since graphene is regarded as the lossless material in this region. This allows us to selectively realize the modulation of graphene electro-absorption at a desired wavelength with high modulation contrast and modulation depth. As shown in Fig. 5(d), by further optimizing the incident angle, ithe absorption of graphene can reach a maximum value for all the wavelengths and then drops abruptly to almost zero as Ef increases to their corresponding critical values. This results in the effective modulation of graphene electro-absorption with high modulation depth of 97.91% in a wide waveband in the near-infrared region. Such wavelength-tunable graphene electro-absorption strategy based on BSWs provides a promising method for the design of graphene-based selective multichannel switches or modulators, which is not possible for previous reported strategies that based on metallic plasmons [25], magnetic polaritons [30,31], guide-mode resonance [26,27]or F-P cavity resonance [23,24].
We finally show that the graphene electron relaxation time, determined by the carrier density, shows great influence on the graphene electro-absorption and thus the modulation efficiency. Figure 5(e) compared the graphene effective permittivity when the graphene carrier mobility is 103 cm2/(V·s) and 104 cm2/(V·s). One can see that, for graphene film with lower carrier density, the imaginary part of permittivity did not decrease to zero for the wavelength region where the optical energy is smaller than 2Ef, instead, it still remains a relatively large value. We calculated the graphene absorption at 1550 nm with varying Fermi energy for different relaxation time, as compared in Fig. 5(f). One can see that the carrier density almost does not affect the graphene absorption for Ef < ℏπc/λ0, but significantly influences the graphene absorption for Ef > ℏπc/λ0. The graphene absorption still maintains larger than 12% as graphene Fermi energy is larger than 0.4 eV for the graphene carrier mobility of 103 cm2/(V·s), resulting in the decrease of modulation depth from 98% to 87%. The result suggests that high-quality graphene monolayer with larger relaxation time is highly desirable to achieve the better modulation performance.
3. Conclusion
In summary, we theoretically designed a specific modulator for the active modulation of graphene electro-absorption, which consists of a continuous graphene monolayer and one-dimensional photonic crystal separated by a defected layer. In such a configuration, Bloch surface wave can be excited under appropriate incident angle, which can produce a strong localized electrical field that significantly improves the graphene electro-absorption. By gate-tuning the graphene Fermi energy to transform graphene between a lossy and a lossless material, the graphene electro-absorption can be effectively modulated with modulation depth of 97.91%, indicating the excellent performance of the designed modulator via such mechanism. We further show that the center wavelength of graphene electro-absorption can be tuned by adjusting the incident angle, which enables us to actively modulate the graphene electro-absorption within a wide near-infrared region, including the commercially important telecommunication wavelength of 1550 nm. Such wavelength-tunable graphene electro-absorption modulator with impressive performance offers a novel way to design graphene based electro-optical modulators that would find potential applications in active photonic and optoelectronic devices.
Funding
National Natural Science Foundation of China (61805165, 61905147, 61935013, 61975128, 62105210, U1701661); China Postdoctoral Science Foundation (2021M692175); Natural Science Foundation of Guangdong Province (2019TQ05X750, 2020A1515010598); Science, Technology and Innovation Commission of Shenzhen Municipality (20200803150227003).
Acknowledgments
The authors would like to acknowledge the Photonics Center of Shenzhen University for technical support.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that supports the findings of this study are available within the article.
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