The complex refractive index (CRI) of graphene waveguide (GW) is of great importance for modeling and developing graphene-based photonic or optoelectronic devices. In this paper, the CRI of the GW is investigated theoretically and experimentally, it is found that the CRI of the GW will modulate the intensity and phase of transmitting light. The phase alterations are obtained spectrally by a Microfiber-based Mach–Zehnder interferometer (MMZI), experimental results demonstrate that the CRIs of the GW vary from 2.91-i13.92 to 3.81-i14.64 for transmitting wavelengths ranging from 1510 to 1590 nm. This method cannot only be used to determine the CRI of the GW optically and provide one of the fundamental parameters for designing graphene-based optic devices for communication and sensing applications, but also is adoptable in graphene-based transformation optics for determination of the CRI of the GW at other wavelengths.
© 2013 Optical Society of America
Graphene with complex permittivity has been extensively investigated in recent years due to its attractive physical properties, such as strong conductivity , good transparency  and notable medium nonlinearity , etc. Recently, many physics phenomena, such as graphene-based Fermi effect  and saturable absorptions , have been discovered and a number of graphene-based photonic devices, such as mode locked laser , optical switch  and LED , have been reported. Especially, when graphene acts as an optical waveguide, it has great potential for applications to photonic integrated circuits , optical fiber communication and sensing, such as intensity modulator , polarizer , and gas sensor . The determination of the complex refractive index (CRI) of the graphene-waveguide (GW) becomes essential for designing of novel graphene-based waveguide devices. However, in many reports, graphene was only regarded as boundary conditions, because it is still a challenge to determinate the complex refractive index (CRI) of the GW effectively. So far, the CRI of G-films rather than GWs could be determined by using an approximate theory  or the Picometrology method proposed for visible wavelengths, e.g. for 532nm and 633nm, the CRI of graphene film is 2.4-i1.0 and 3.0-i1.4, respectively . As the permittivity of graphene is anisotropic and coupling light into and out the GW is still a challenge, it is very difficult to measure its CRI optically. This remains as a fundamental problem to be solved in graphene-based transformation optics and plasmonics [15,16].
In this paper, microfiber is used as an effective and convenient means to launch and collect evanescent light from the GW, and offers the flexibility of changing the contact length in a graphene/microfiber hybrid waveguide (GMHW). By this means, graphene induced optical phase alterations and attenuations are measured by microfiber-based Mach–Zehnder interferometer (MMZI) [17,18], thus the CRI of the GW in the wavelength window of 1510nm-1590nm are calculated from the measured effective refractive index of the GMHW. It is summarized when the light propagates along the GW, the CRI of the single layer GW ng ranges from 2.91-i13.92 to 3.81-i14.64 at room temperature for this wavelength window. Such a method provides a powerful way to investigate the optical properties of graphene or other photonic thin films by spectral measurement.
2. Modeling and theoretical analysis
Figure 1(a) schematically shows the configuration of the GMHW. A microfiber drawn from a single mode fiber (SMF) is attached onto the graphene film tightly. The graphene is deposited on a MgF2 substrate. The evanescent wave is launched to the GMHW from one taper of the microfiber and collected by the other taper at the output port. Here, the effective length of the GMHW would be adjusted conveniently via controlling the interaction distance between the microfiber and the graphene film. The geometry of the cross-section of the GMHW is shown in Fig. 1(b). The microfiber (blue) with diameter of 1.1μm and refractive index (RI) of n1 = 1.45 is located onto the graphene film (black), the permittivity of the GW is related to its conductivity σ as εg,eq = -σg,i/ω△ + iσg,r/ω△ . Here σg,i and σg,r are the imaginary and real part of σ, ω is the light’s angular frequency, △ is the thickness of graphene. Here, the graphene is regarded as an ultrathin bulk waveguide with △ = 0.5nm (single layer in physics) in this model rather than a boundary condition when only considering losses . Considering the pristine graphene at room temperature, the CRI of the GW is calculated in Fig. 1(c), by applying ω2μ0ε = n2k02. Here μ0 is the permeability, ε is the permittivity, and k0 is the wave number in free space [15,19]. The MgF2 substrate has a 3mm thickness and its RI is equal to 1.37 at ~1550nm . By adopting the Finite Element Method (FEM) to multilayer waveguides, the effective RI of the GMHW for the fundamental mode is numerically calculated and given in Fig. 1(d). For the wavelength range from 1300nm to 1600nm, the real part of the effective index decreases from 1.405 to 1.339, while the imaginary part increases from 3.1 × 10−4 to 3.8 × 10−4. For the microfiber (D = 1.1μm) without GW attached, the effective RI for the wavelength band 1510nm-1590nm is a constant 1.426 . We simulate the sectional field distribution of the transmitting light along the GMHW with λ = 1550nm, shown in Fig. 1(e). According to Fig. 1(e), we observe that the GW attracts a large portion of evanescent waves to transmit along it. Figure 1(f) shows the intensity distribution for y-axis according to Fig. 1(e). The curve is discontinuous at the graphene layer, which is caused by the inherent optical absorption of graphene . It is clear that once the neff of the GMHW could be measured, the CRI of the pure GW can be numerically calculated from the experimental results.
First, the process of evanescent wave couples from microfiber to the GW is investigated. The evanescent wave transmitting along microfiber is determined by its eigenequations [22,23], once the microfiber coupled the intrinsic graphene, the transmitting mode is not only HE11 any longer [24,25]. Considering boundary condition n × (H1-H2) = σE and adopting the model shown in Fig. 1(a), we write the light just influenced by the GW as E(x,y,z) = (exx + eyy + ezz)ejßzejωt, where we assume that the GW is located at the y-z flat and the wave vector is in the direction of z, here β = neff(ω)(ω/c). Referring neff(ω) = neffRE(ω) - jneffIM(ω), the transmitting wave could be written as E(x,y,z) = (exx + eyy + ezz) exp[ω(-neffIM × z/c)] exp[jω(neffRE × z/c) + t]. For transmitting waves, exp[ω(-neffIM × z/c)] is the attenuation coefficient and exp[jω(neffRE × z/c) + t] is the phase coefficient. Here ω = 2πc/λ. The neffRE induced phase change and the neffIM induced attenuation change (z = 20μm, λ = 1550nm) are simulated in Figs. 2(a) and 2(b). Figures 2(c)–2(e) show the simulated spatial distributions of electric fields transmitting along microfiber, microfiber attached MgF2 and the GMHW respectively, for z = 20μm and λ = 1550nm as well. Larger attenuation, less compressed phase distribution in the GMHW can be obtained in the simulations. Moreover, in Fig. 2(e), TE polarized surface plasmon on the GW is obvious, which has been predicted in .
3. Experimental verification based on specially designed MMZI
In order to measure the neff of the GMHW, a MMZI with one arm with the GMHW is constructed, shown in Fig. 3(a). A SMF with a length of ~50cm is used as the lead-in fiber, and a polarization controller is adopted to optimize the incident polarization mode to be TE-polarized for the graphene layer. Through a 50:50 Coupler, light is divided into 2 interfering arms. For the arm 1, the light is guided by SMF, here a tunable attenuator is used to match the intensities of the arm1 and arm 2. The length difference of the two arms L2-L1 is ~5cm (arm 2 is longer). For the arm 2, the SMF is drawn into a microfiber with a diameter of ~1.1μm and a length of ~17cm , the waist part of the microfiber is attached onto the graphene film supported by the MgF2 substrate (40mm × 40mm), shown in Fig. 5. The output light is collected by the SMF at the other end of the microfiber. Finally the arm 2 and arm 1 are combined via a 50:50 coupler and sent into an optical spectrum analyzer (OSA, Micron Si 720, USA). When broadband light lunches into the MMZI, the output is a resonant spectrum with interference fringes. The GMHW in experiment is shown in Fig. 3(b) schematically. The tapers of the microfiber are fixed on the docks and the waist of the microfiber is pendulous naturally. The graphene deposited on the MgF2 substrate (G/MgF2) is fixed on a translation stage. When the microfiber is close to graphene, under the effects of Van der Waals force and electrostatic force, they can touch tightly, forming a stable structure. Figure 3(c) shows the SEM photograph of the microfiber attached onto the GW.
In our experiment, graphene films were grown on Cu foils (Alfa Aesar, No. 13382) by chemical vapor deposition (CVD) in the way described in . To transfer the graphene onto the objective MgF2 substrate, PMMA was spin-coated on the surface of graphene/Cu foil and then the underlying Cu foil was etched with 1M FeCl3 solution. Subsequently, the PMMA/graphene was washed in DI water several times and transferred onto the MgF2 substrate which had been ultrasonically cleaned in sequence by acetone, ethanol and DI water. Then it was allowed to dry at room temperature for overnight and baked at 180°C for 10 min. Finally, the PMMA was removed by acetone. The Raman spectrum of the GW is shown in Fig. 3(d). A very weak D peak, a ~0.23 G-to-2D intensity ratio, and a 2D-peak with a full width at half maximum of ~36.3 cm–1, suggest that the graphene adopted in this experiment is monolayer with high quality.
The phase alteration of the microfiber by the GW in one arm of the MMZI would cause the spectral shift. For the MMZI with only microfiber in arm 2 (without graphene attached), the location of any resonance dip λd is shown in Eq. (1). Here L1 is the length of arm 1, L2 is the length of arm 2, N is a natural number related to neffRE and (L2-L1), vf is the phase velocity of light in fiber. Once the GW is attached, the phase velocity in the GMHW zone is vG rather than vf, so that the location of the spectral dip would shift to λd’, as shown in Eq. (2). As the phase velocity of electromagnetic waves is v = c/nRE, when the neffRE decreases, the dip location λd has a blue shift. The shift Λ = λd’-λd is shown in Eq. (3). Here △nRE = neffREGMHW -neffREMF <0 is the difference between the real part of the CRI for the GMHW and the microfiber, so that the spectra will have a blue shift. Figures 4(a) and 4(b) show the simulated |△nRE(λ)| and |△φ| = 2πLG|△nRE(λ)|/λ with the band of 1500nm-1600nm, according to Fig. 1(c). In Fig. 4(b), LG is assumed to be 1μm. It indicates that light with longer wavelength tends to couple out from the microfiber and transmit around the GW. Figure 4(c) shows the relationship between excitation wavelength λ, interacting length LG, and the resonant shift Λ in the MZI with L1 = 0.95m and L2 = 1m. In Eqs. (1)–(3), each dip’s N is different.
Figure 5 schematically shows the operation of adjusting the effective interacting length of the GMHW. Through tuning the height of the console shown in Figs. 5(a)–5(c), LG can be controlled easily. Figures 5(d)–5(f) are the photographs for different interacting lengths between the microfiber and the graphene, here the insertion loss of the microfiber is ~22dB.
The experimental results adopting the setup in Fig. 3 are shown in Figs. 6 and 7. First of all, we consider the influence brought by the MgF2 substrate. Figure 6 shows the output spectra of the MMZI whose microfiber in arm 2 is attached onto the MgF2 substrate without graphene. Because the RI of MgF2 is smaller than the microfiber, it is very hard to attract light to transmit along MgF2, so that both the phase and attenuation influences induced by the pure MgF2 substrate can be ignored: Fig. 6(a) shows the calculated RI of the microfiber located on MgF2 (RI of MgF2 is 1.3707 at 1510nm and 1.3703 at 1590nm, while the RI of the microfiber is 1.450 ). The △nRE is close to zero. As a result, there is a very small spectral shift caused by MgF2, as shown in Figs. 6(b)–6(d). Similarly, for the nIM of MgF2 is almost zero, the attenuation induced by MgF2 less than 0.2dB/mm also could be ignored here.
Figure 7 shows the spectra of the GMHW in the experiment, as the GW alters the polarization (to be TE mode ) and absorption of the evanescent waves, a longer LG brings a larger attenuation and a bigger extinction ratio [12,13]. Figures 7(a)–7(c) present the spectral shifts near 1510nm, 1550nm and 1590nm, respectively, for LG = 5mm, at most ~90, ~190 and ~290pm shifts are observed. Therefore, it is verified that the spectral alterations do come from the GW. The shifts are summarized and linearly fitted approximately in Fig. 7(d), here the error bars mark that the experiment is repeatable. Removing the MMZI, the spectral attenuations induced by the attached GW are sampled in Fig. 7(e).
According to Eq. (3), experimentally measured neff of the GMHW is shown in Fig. 8(a), here the dashed curves are the simulated results. The real part of the neff of the GMHW decreases from ~1.38 to ~1.34 as the wavelength of transmitting light increases from 1510nm to 1590nm, while the imaginary part of the GMHW increases from 3.1 × 10−4 to 3.6 × 10−4. The deviations between the experimental results and simulated results are mainly from: 1) The serious absorption of the GW for evanescent light ; 2) There may be a few defects or dirt on the graphene sample making it not so perfect as in the simulations . Reconsidering the effective index method for multilayer waveguide to obtain the neff from ng, the CRI of the single-layer graphene ng is calculated from the neff reversely. Interpolated by the MATLAB, the fitting function is shown in Eq. (4). Therefore, the CRI distribution of graphene ng is presented in Fig. 8(b). The experimental results correspond to the theoretical analysis (dashed curves). The measured ngRE is ~0.2 smaller than the calculated, while the measured ngIM is ~0.1 larger than the calculated. This deviation is mainly induced by the approximations.
In summary, we investigated the CRI of the GW, which is an essential parameter for study of the GW physics. The results of simulation and experiment indicate that when graphene acts as a part of waveguide, its CRI can be determined by the change in its transmission wavelengths. This method of using the microfiber to launch and collect light from the GW has been demonstrated to be not only simple but also an effective way to study the CRI of the GW and other optical film-based waveguides. This work would be useful for understanding the modulation mechanisms of GW and guiding the design of G-based photonic devices for various applications.
The authors thank for the helpful discussions and suggestions from Prof. K. S. Chiang, City University of Hong Kong. This work was supported by National Natural Science Foundation of China under Grant 61290312, 61107072, 61107073 and the Fundamental Research Funds for the Central Universities of China (ZYGX2010J005). This work was also supported by Program for Changjiang Scholars and Innovative Research Team in Universities of China (PCSIRT).
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