The effective mode index (EMI) of a graphene-coated side-polished fiber (GSPF) is calculated numerically. Whereby, the influences of graphene atom layer number, residual radius of SPF, light frequency, scattering rate of graphene, and temperature on the EMI are investigated comprehensively. Two types of mechanisms for the electro-optical absorption modulation are found for such GSPF-based modulator. One mechanism is Pauli blocking effect (PBE) and the other is plasmonic attenuation effect (PAE). With the optimal design parameters, a PBE-based modulator is theoretically predicted to have a 0.0072 dB/μm modulation depth, 2.92 V driving voltage swing, 6.35 nJ/bit power consumption, and 56.2 THz optical modulation bandwidth. It is also predicted that a PAE-based modulator could have a 0.0056 dB/μm modulation depth, 0.6 V driving voltage swing, 0.27 nJ/bit power consumption, and 2.5 THz optical modulation bandwidth. By further optimization, the modulator performance such as the relatively high power consumption and the narrow operation bandwidth can be improved. Owing to their seamless connection to optical fiber networks, the GSPF-based modulators have great potential to be used in fast and high-capacity optical communication systems.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Graphene, a two dimensional carbon sheet with honeycomb atom structure, possesses unique electrical and optical properties [1, 2], and becomes hot points in several photonics and optoelectronics research fields. Graphene-based photodetectors [3, 4], optical sensors [5, 6], polarizers [7, 8], Plasmon devices [9, 10], and saturable absorbers in laser systems [11, 12] have been demonstrated. One of the fascinating features of graphene is that its carrier density can be tuned easily by electrical doping or photodoping, which result in its applications in optical modulators with large modulation depth, small footprint, broadband, and fast response due to its unique band structure and extremely high carrier mobility. The graphene-based modulators can be realized in two configurations: transverse modulator (radiation is incident on and transmitted through a graphene layer) and waveguide modulator (in which a waveguide mode is propagating along and confined near a graphene plane). Terahertz modulators often take transverse modulator configuration for electro-optical [13, 14] and all-optical [15, 16] modulation. Near-infrared and visible spectrum modulators often take waveguide modulator configuration, and the employed waveguides include Si waveguide [17–22], Si or Si3N4 microring [23, 24], Plasmonic waveguide , and optical fiber [26–34]. A graphene-covered Si waveguide was demonstrated to have a modulation depth of 0.1 dB/μm , and a double layer graphene-covered Si waveguide has a modulation depth of 0.16 dB/μm . A 20 μm-long Si waveguide with a double-layer graphene structure underneath it was demonstrated to have a modulation depth of 2 dB (that is 0.067dB/μm) . A graphene-based hybrid plasmonic waveguide modulator was demonstrated to have a modulation depth of >0.03dB/μm . A microfiber–based all-optical modulator with 1 cm-long bi-layer graphene-covered segment was demonstrated to have a modulation depth of 13 dB (that is 0.0013dB/μm) . A microfiber-based all-optical modulator with graphene-wrapped 2 mm-long segment was demonstrated to have a modulation depth of 38% (that is 0.001dB/μm) and response time of 2.2 ps . A GSPF-based all-optical modulator was also demonstrated to have a modulation depth of 9 dB in 5 mm-long graphene-covered segment (that is 0.0018dB/μm) . The first and sole up to now experimental demonstration of electro-optical modulation based on GSPF display that for 5 mm-long mono-, bi-, and quad-layer graphene-covered segment, the modulation depth were 44.2% (2.53 dB), 72.9% (5.67 dB), and 90.1% (10.04 dB) respectively .
Alone with the experimental approach, the theoretical investigation on graphene/waveguide-based modulators were also ongoing for different waveguide type: Si or Si3N4 waveguide [35–44], Si microring [45–48], plasmonic waveguide [49–53], optical fiber , etc. In these theoretical approaches, except 3D finite-difference time-domain (FDTD) method used in a few literatures [35, 49, 50], calculating the EMI of waveguide by using 2D finite element method (FEM) becomes a mainstream method. The vast majority of the theoretical approaches focus on Si, Si3N4, and plasmonic waveguides, only very few pay attention on graphene/optical fiber-based modulator .
In this paper the optical modulation characteristics of GSPF is investigated numerically by calculating its EMI. GSPF is chosen as an object of study for two reasons: (1) There exist some arguments in the investigation of GSPF-based optical devices. For example, the polarizer based on GSPF in , Bao et al, was found to have TE-pass feature (that is the polarization direction of the transmitted light parallel with graphene plane), while the polarization direction of transmitted light of GSPF in , Lee et al, was found to be perpendicular to graphene plane. So a further theoretical study on GSPF is necessary. (2) Although generally speaking modulators based on graphene/optical fiber possess lower modulation depth than that based on graphene/Si waveguide or graphene/hybrid plasmonic waveguide, they have a prominent advantages of joining in directly and conveniently existing fiber network, and have low insertion loss [32, 34], so it makes sense to make a comprehensive theoretical study on optical modulators based on GSPF. In part 2 the electro-optical model of graphene is presented. In part 3 the EMI of GSPF with air overlapping is calculated and two types of mechanisms for electro-optical absorption modulators are differentiated. The influences of residual radius of the fiber, graphene atom layer number, scattering rate of graphene, light frequency, and temperature on the EMI of GSPF are investigated. In part 4 the theoretical modulation depth of GSPF with ion liquid overlapping is found to agree well with the experimental results in , Lee et al, which indicates the validity and accuracy of the numerical method. With the optimal design parameters, both the two types of electro-optical absorption modulators based on GSPF are comprehensively studied in terms of modulation depth, driving voltage, power consumption, and bandwidth.
2. Electro-optical model of graphene
The change of charge carrier density ns will cause the change of chemical potential μc because of the following formula :55]:22]:
The optical conductivity of graphene σ is a function of chemical potential μc and is also dependent on the light frequency ω, scattering rate Γ, and temperature T. Their influences were calculated from Eq. (2) and depicted in Fig. 1. For different light frequency, σ-μc curves display similar shape (as an advantage of adopting the unit of ℏω ), however with the increase of frequency ω, the change of curve (including real and imaginary part) becomes steeper in thechemical potential interval of [0.4, 0.6] ℏω , as shown in Fig. 1(a). The influence of scattering rate Γ and temperature T on σ-μc curves are displayed in Figs. 1(b) and 1(c) respectively. Figure 1(d) shows the relative permittivity of graphene εg as a function of chemical potential for ω/2π = 1.94e14 Hz, T = 298 K, and Γ = 1e12 Hz. The absolute value of εg approaches to zero at μc = μc0 ≈0.62ℏω, which is also called epsilon-near-zero (ENZ) point. When μc<μc0, both the real and imaginary part of εg are positive, graphene acts as a dielectric material. When μc>μc0, the imaginary part of εg close to zero, the real part of εg becomes negative, graphene acts as a metallic material. The EMI of TM-mode will change dramatically in the vicinity of ENZ point . Since the degree of closeness between the absolute value of εg and zero at ENZ point would be influenced by light frequency, scattering rate, and temperature, therefore the change of EMI of TM-mode at ENZ point is also depends on these factors
3. Calculation and discussion of the EMI of GSPF
A sketch map of GSPF is shown in Fig. 2. A single mode fiber was polished from one side and then covered on top of it graphene. A stack of air-graphene-core-cladding is considered with refractive indices of 1-ng-1.468-1.463, respectively, where ng is potential dependent refractive index of graphene. The residual radius denotes the minimum distance between side- polished surface and the center of fiber and is abbreviated as R in the following. The EMI (denoted as neff below) of TE-mode (electric field parallel with graphene plane) and TM- mode (electric field perpendicular to graphene plane) can be calculated through 2D FEM simulation in COMSOL. The relative output optical power of TE- (or TM-) mode of GSPF can be calculated approximately through the following formula:
3.1 The influence of atom layer number of graphene N on the relationship of neff-μc
The relationships of neff-μc with different N for R = 1.5μm, T = 298K, Γ = 1e12Hz, and ω/2π = 1.94e14 Hz are depicted in Fig. 3. The curves for the real part of EMI in Figs. 3(b) and 3(d) are not smooth compared to the imaginary part of EMI in Figs. 3(a) and 3(c), the reason is as follows: the change of Re(neff) and the change of Im(neff) with chemical potential are similar in order of magnitude, whereas Re(neff) is much larger than Im(neff) (with the difference of 4 to 5 orders of magnitude), therefore the smaller change of Re(neff) with chemical potential may be “ate off” by Re(neff) due to rounding errors, which result in unsmoothed curves compare with the case of Im(neff).
It should be noted that the Im(neff) of TE-mode is larger than that of TM-mode for fixed N and fixed μc∈[0, 0.6], which indicates that the attenuation of TE-mode wave energy is larger than that of TM-mode, so the transmitted light possess TM-polarization feature. This calculation result is agree with the experimental result in , Lee et al, but opposite to the experimental result in , Bao et al. The possible reason of such a discrepancy is discussed in the appendix.
The variation of Im(neff) of TE-mode within the chemical potential interval of [0.42, 0.6] can be expressed as ΔIm(neff), as shown in Fig. 3(a), which concerns the modulation depth of the electro-optical absorption modulator based on the Pauli blocking effect. The variation of Re(neff) of TE-mode within the chemical potential interval of [0, 0.5] can be expressed as ΔRe(neff), as shown in Fig. 3(b), which concerns the modulation depth of the electro-optical refractive modulator. The same definitions also apply to TM-mode. ΔIm(neff) of TM-mode is much smaller than that of TE-mode for fixed N, therefore it is TE-mode (not TM-mode) play a leading role in PBE-based electro-optical absorption modulation, this calculation result is also agree with the experimental result in , Lee et al. Different from the case in graphene/fiber configuration, TM-mode play a leading role in modulators based on graphene/Si waveguide configuration  because either the Im(neff) or the ΔIm(neff) of TM-mode is larger than that of TE-mode. The peak of Im(neff) of TM-mode in the chemical potential interval of [0.6, 0.64]corresponds to the ENZ point and is also called ENZ peak. It is attributed to plasmonic mode associated with TM polarized light can be motivated and propagate alone the graphene sheet in the vicinity of ENZ point . The amplitude of ENZ peak (denoted as AENZ) concerns the modulation depth of electro-optical absorption modulator based on plasmonic attenuation effect.
Atom layer number of graphene has a remarkable impact on the modulation depth of modulators. With the increase of graphene layer number N, ΔIm(neff) and ΔRe(neff) of both TE- and TM-mode are all increase, as shown in Fig. 3, and AENZ also increases, as shown in Fig. 3(c). It can be calculated by using Eq. (5) that for N = 2, 5, 10, and 15, a PBE-based electro-optical absorption modulator possesses modulation depth of 1.9e-4, 4.8e-4, 9.6e-4, and 1.4e-3 dB/μm respectively; and a PAE-based electro-optical absorption modulator possesses modulation depth of 2.63e-5, 6.65e-5, 1.36e-4, 2.09e-4 dB/μm respectively. Obviously, for the configuration like GSPF and for working wavelength near 1550 nm, a PBE-based electro-optical absorption modulator has a larger modulation depth than a PAE-based electro-optical absorption modulator. While for other graphene/waveguide configuration, like graphene-embedded silicon waveguide , the modulation depth of PAE-based electro-optical absorption modulators (that is based on ENZ peak) is larger than that of PBE-based ones. As limited by the scope of the paper, the electro-optical refractive modulators are omitted from discussion.
It can be confirmed that when N≤5, the assumption of σN-layer = Nσmonolayer is valid, which is demonstrated experimentally in , Lee et al, and , Nair et al. Whereas it is not always valid for the larger graphene layer number that N>5, the limit for graphene layer number N is still should be further tested and verified through experiments. Under the assumption of σN-layer = Nσmonolayer, the changing rule of neff-μc curve with respect to the parameters (R, ω, Γ, T) are similar for different N. While choosing a larger N (for example, N = 10 as in section 3.2 and section 3.3) can be translated into tremendous saving in computing time and memory for the FEM simulation in COMSOL, which is necessary for dealing with the configuration with huge cross section like GSPF.
3.2 The influence of residual radius of SPF on ΔIm(neff) and AENZ
The relationships of neff-μc with different R for N = 10, T = 298 K, Γ = 1e12 Hz, and ω/2π = 1.94e14 Hz are calculated, ΔIm(neff) of both TE- and TM-mode, and AENZ as functions of R are depicted in Figs. 4(a) and 4(b), respectively. With the decrease of R, that is the polished surface is closer and closer to the center of fiber, ΔIm(neff) of both TE- and TM-mode as well as AENZ increase gradually, because the interaction between graphene and evanescent wave is stronger and stronger. However when R decrease further from 1.5 μm, ΔIm(neff) of both TE- and TM-mode as well as AENZ decrease rapidly. Field distribution picture of TE-mode corresponding to different R are depicted in Fig. 4(c) (the evolution of field distribution of TM-mode with R is similar). It is noted that with the decrease of R, graphene is closer and closer to the maximum intensity region in the core, nevertheless more and more fractional energy of mode field distributes outside of core. When R decrease further from 1.5 μm, much more fractional energy of mode field is pushed away from the core and is far from graphene. So there exist an optimized interval of R∈[1.2, 2.5] μm, in which ΔIm(neff) of TE-mode and AENZ of TM-mode possess larger value and hence the modulators possess higher modulation depth.
3.3 The influence of light frequency ω, scattering rate Γ, and temperature T on ΔIm(neff) and AENZ
Im(neff) of TE- and TM-mode as functions of μc with different ω for R = 1.5 μm, T = 298 K, Γ = 1e12 Hz, and N = 10 are depicted in Figs. 5(a) and 5(c) respectively. Color maps that display Im(neff) of TE- and TM-mode changing with both wavelength and chemical potential are placed in the appendix. ΔIm(neff) of TE-mode has a maximum value near λ = 1550 nm, as shown in Fig. 5(b). With the increase of frequency (that is decrease of wavelength), the amplitude of ENZ peak AENZ becomes larger and larger, especially when the frequency close to visible spectrum, AENZ is much larger than ΔIm(neff) of TE-mode, as shown in Fig. 5(d). It means that a PBE-based electro-optical absorption modulator is suitable for waveband near 1550 nm, a PAE-based electro-optical absorption modulator is suitable for waveband near visible spectrum.
Im(neff) of TE- and TM-mode as functions of μc with different Γ for R = 1.5 μm, T = 298 K, N = 10, and ω/2π = 1.94e14 Hz are depicted in Figs. 6(a) and 6(c) respectively. ΔIm(neff) of TE-mode decrease gradually with the increase of Γ, as shown in Fig. 6(b). The amplitude of ENZ peak AENZ decrease rapidly with the increase of Γ, and it disappear completely when Γ>5e13 Hz, as shown in Fig. 6(d). The considered carrier scatting rates 1e12 to 1e14 Hz correspond to charge carrier mobilities of 22563 to 226 cm2/Vs at 0.4 eV (calculated using μ = (evF2)/(Γμc) ). When Γ>5e13 Hz, the ENZ peak disappears completely and modulators that based on ENZ peak (PAE-based ones) would be failure.
Im(neff) of TE- and TM-mode as functions of μc with different T for R = 1.5 μm, N = 10, Γ = 1e12 Hz, and ω/2π = 1.94e14 Hz are depicted in Figs. 7(a) and 7(c) respectively. With the increase of temperature, ΔIm(neff) of TE-mode decrease slightly, as shown in Fig. 7(b), the amplitude of ENZ peak AENZ decreases rapidly, meanwhile the position of peak shifts to higher chemical potential, as shown in Fig. 7(c).
4. Design and optimization of electro-optical modulator based on GSPF
By using ion liquid to act as a highly efficient gating medium as well as a transparent over- cladding that greatly increases graphene-light interaction without significant optical loss, a field effect transistor on a side-polished fiber is demonstrated exhibiting significant optical transmission change (>90%) . Under the condition of R = 3.5μm T = 298K, Γ = 1e12 Hz, L = 5 mm, and ω/2π = 1.94e14 Hz, and the stack of ion liquid-graphene-core-cladding is considered with refractive index of 1.423-ng-1.468-1.463, the simulation result display that the optical transmission changes are 2.52, 5.02, and 9.98 dB for N = 1, 2, and 4 respectively, agree well with the experimental results in , Lee et al, which indicates that the numerical method is valid and accurate. Figure 8 display ΔIm(neff) of TE-mode versus residual radius R for the cases of N = 4 with ion liquid overlapping and N = 10 with air overlapping. We can see that the overlapping ion liquid play an important role to increase the modulation depth. The optimized interval of R for the case of ion liquid overlapping is about [0.8, 1.5] μm, a little different from [1.2, 2.5] μm for the case of air overlapping.
Although ion liquid can add gate voltage on graphene efficiently, its drawback is also obvious that the switching speed is limited by the low ionic mobility . To realize high speed electro-optical modulator based on GSPF, a double-layer graphene separated by a thin dielectric forms a parallel capacitor structure on the polished surface of fiber for adding electric gate, meanwhile ion liquid is reserved on the upper graphene layer for increasing graphene-light interaction due to its high refractive index, as shown in Fig. 9. Very thin and high refractive index dielectric is needed as it reduces the power consumption, at the meantime high breakdown voltage is also needed to avoid being broke down. Here a 5 nm-thick Al2O3 is chosen just for estimating driving voltage. Quad-layer graphene with scattering rate of Γ = 1e12 Hz is adopted for both the upper and the lower (connecting polished surface of fiber) graphene layer whose practical feasibility has been demonstrated experimentally . Other conditions can be taken as R = 1.2 μm, T = 298 K, ω/2π = 1.94e14 Hz, and L = 5 mm. By considering the stack of ion liquid-graphene-Al2O3-graphene-core-cladding have refractive index of 1.423-ng-1.64-ng-1.468-1.463, respectively, the modulation depth of a PBE-based electro-optical absorption modulator can reach to 35.78 dB (that is 0.0072 dB/μm), and the modulation depth of a PAE-based electro-optical absorption modulator can reach to 27.9 dB (that is 0.0056dB/μm). The driving voltage modifies carrier density of graphene ns as .Fig. 10(a), and the curves of TM- mode power versus chemical potential for wavelength of 1550, 1560, and 1540 nm are depicted in Fig. 10(b). For same driving voltage swing of 2.92 V within [3.45, 6.37] V, the TE-mode power variation for 1800 nm and 1350 nm are about half of that for 1550 nm, as shown in Fig. 10(a), indicating that the optical bandwidth of a PBE-based modulator reaches to ∼450 nm (that is 56.2 THz). For same driving voltage swing of 0.6 V within [6.37, 6.97] V, the TM-mode power variation for 1560 nm and 1540 nm are about half of that for 1550 nm, as shown in Fig. 10(b), indicating that the optical bandwidth of a PAE-based modulator reaches to ∼20 nm (that is 2.5 THz). The operation bandwidth of either the PBE-based or the PAE-based modulator is not likely limited by their optical bandwidth, but by the RC constant of the device. By taking R = 20 Ω , the 3dB bandwidth can be estimated as f3dB = 1/2πRC = 2.67 MHz. Due to the large capacitance of the double-layer graphene structure on the fiber polished surface, modulators based on GSPF possess relatively large power consumption and narrow operation bandwidth. However, these weaknesses can be overcome in some degree. For example, if the crossover region of the two graphene layers only covers the core (that is the area of the capacitor down to ∼10μm × 5mm), the capacitance can be reduced greatly without reducing the modulation depth.
Although the proposed GSPF-based modulator possesses lower modulation depth than that based on graphene/Si waveguide or graphene/hybrid plasmonic waveguide, it still can achieve easily a 20-dB extinction ratio typically required in telecommunications  with enough optical bandwidth. Furthermore, the advantages of low insertion loss and joining in directly and conveniently existing fiber network make it low-cost and robust in usage.
In summary, the EMI of GSPF as a function of chemical potential is calculated numerically, and two types of mechanisms for electro-optical absorption modulators are differentiated: one is Pauli blocking effect and the other is plasmonic attenuation effect. The modulation depths of PBE-based and PAE-based modulators are related to ΔIm(neff) of TE-mode and AENZ of TM-mode respectively. The influences of graphene atom layer number, residual radius of SPF, frequency of transmitted light, scattering rate of graphene, and temperature on ΔIm(neff) of TE-mode and AENZ of TM-mode are studied comprehensively. By calculating the EMI of GSPF with ion liquid overlapping, we find that the simulation result of modulation depth agree well with the experimental result in , Lee et al, which indicates the validity and accuracy of the numerical method. An electro-optical absorption modulator based on GSPF with optimized parameters is designed, and the modulation depth, driving voltage, power consumption, and modulation bandwidth of both the PBE-based and the PAE-based modulators are estimated.
Appendix 1 Explanation of the discrepancy on the polarization feature of transmitted light in GSPF
According to the numerical calculations in this paper, Im(neff) of TE-mode is larger than that of TM-mode in the chemical potential interval [0, 0.6] and [0.64, 1], only in the chemical potential interval [0.6, 0.64]the Im(neff) of TE-mode is smaller than that of TM-mode due to the plasmonic attenuation effect of TM-mode. The polarization feature of the transmitted light is determined by the doping level of graphene. If μc∈[0, 0.6] or μc∈[0.64, 1], the transmitted light possesses TM-polarization feature; if μc∈[0.6, 0.64], the transmitted light possesses TE-polarization feature. While [0.6, 0.64] represent different chemical potential interval for different wavelength: for wavelength of 1550 nm, [0.6, 0.64]→[0.48, 0.51] eV; for wavelength of 1310 nm, [0.6, 0.64]→[0.57, 0.61]eV, etc. Suppose the graphene used in , Bao et al, was on highly doping level and its chemical potential μc∈[0.48, 0.51] eV, the transmitted light of wavelength near 1550 nm will display TE-polarization feature. But the transmitted light of other wavelength (like 1310nm) will not display TE-polarization feature. That is to say, the broadband TE-polarization feature (from 488nm to 1550nm) described in , Bao et al, still can not be explained by the numerical calculations in this paper. So it is much more likely that TE-mode and TM-mode were confused in the experiment by mistake in , Bao et al.
Appendix 2 Color map corresponding to Figs. 5(a) and 5(c)
Figure 11 shows the color map corresponding to Figs. 5(a) and 5(c).
National Natural Science Foundation of China (NSFC) (61675092, 61575084, 61275046, 61705089, 61705087, 61475066, 61705086, 61775084, 61505069); Natural Science Foundation of Guangdong Province (2016A030313079, 2017A030313359, 2015A030313320); Science and Technology Projects of Guangdong Province (2017A010102006, 2017A010101013, 2016TQ03X962); Science and Technology Projects of Guangzhou (201707010396, 201707010500); Fundamental Research Funds for the Central Universities of China (21617333); Project of Jinan University Undergraduate Innovating and Pioneering Training Program (201710559003).
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