Nonlinear mode coupling in graphene-buried optical waveguides

The phenomenon of power coupling between two modes propagating along a waveguide structure, such as a periodic waveguide or two parallel waveguides,^1–3 has found numerous applications in optical communications, optical signal processing, optical sensing, etc. In most practical applications where the Kerr nonlinearity of the waveguide material is negligible, the mode-coupling effect is independent of the input power. However, interesting nonlinear optical functions that depend on the input power can be achieved by exploring the Kerr nonlinearity in the mode-coupling devices.

Over the years, nonlinear mode-coupling effects induced by the Kerr nonlinearity in different waveguide structures, which include microresonators,^4–6 parallel waveguides,^7–16 long-period gratings,^17–20 and Bragg gratings,^21,22 have been investigated extensively for achieving different nonlinear functions, such as all-optical switching,^7–20 optical pulse compression,^19,21,22 and soliton control.^4,5,21,22 As the Kerr nonlinearity in common optical materials is weak, high optical powers are required for the observation of significant mode-coupling effects and such high powers are available only with expensive pulsed lasers. The use of short laser pulses, however, can induce non-uniform nonlinearity across the pulse and also generate unwanted nonlinear effects (e.g., multiphoton absorption^11,14), which may obscure the observation of nonlinear mode-coupling effects. As such, most of the studies of nonlinear mode coupling are theoretical and the experimental results are relatively scarce. Here, we highlight a few pioneering experiments. In an early study of nonlinear mode coupling in a 5-mm long dual-core silica fiber with 100-fs pulses, a peak power as high as 32 kW is required¹³ and the output pulses are highly distorted, which is likely caused by the large intermodal dispersion in the fiber.²³ In a recent study that employs a 4.8-cm long chalcogenide dual-core fiber, the nonlinear switching power is lowered to 126 W.⁷ By using two parallel AlGaAs waveguides, the power required is 42 W.¹⁶ Similar to the case of a nonlinear dual-core fiber, all-optical switching in a nonlinear long-period grating written in a silica fiber requires a pulse peak power of several kilowatts^17,19 and the use of a chalcogenide fiber can reduce the power to ∼50 W.^18,20 Gap solitons in a fiber Bragg grating were observed with a pulse intensity of 12 GW/cm².²¹ The high-power requirement imposes serious limitations on experimental investigation of nonlinear mode coupling, let alone its practical applications. In this paper, we propose a waveguide platform for the study of nonlinear mode-coupling effects, where the photothermal effect of graphene is employed to produce Kerr-like nonlinearity and the power required can be as low as several tens of milliwatts, which is available with inexpensive continuous-wave (CW) low-power lasers, in particular, semiconductor lasers developed for different optical communication bands.

Our proposed waveguide platform is simply a waveguide structure with graphene buried at a specific location that depends on the function to be achieved. When light is launched into such a structure, it is partially absorbed by graphene, and the absorbed power is converted into heat by the photothermal effect of graphene.^24–26 The heat so generated then changes the refractive index of the waveguide through the thermo-optic effect and affects the propagation of the light wave, which, in turn, can change the amount of light to be absorbed. The output characteristics reach a steady state at a given input optical power. Such a self-generated thermo-optic effect via graphene’s absorption of light depends on the input optical power and provides a physical mechanism for the study of nonlinear mode coupling. To demonstrate the features of our proposed platform, we present three waveguide structures as examples: (i) a graphene-buried long-period waveguide grating (LPWG), (ii) a symmetric directional coupler (DC) with graphene buried in two cores, and (iii) a symmetric DC with graphene buried in one core. We establish physical models for these structures based on the coupled-mode theory and fabricate these structures with polymer waveguides. The first structure serves to show nonlinear coupling between a guided core mode and a cladding mode, while the second and the third structure serve to show nonlinear mode switching between two waveguides. We should note that the third structure is actually an asymmetric structure, since nonlinearity exists only in the core buried with graphene. Such a structure is difficult to realize for the demonstration of genuine Kerr effects, as it would demand that the two cores have identical linear refractive indices but highly different Kerr nonlinearities. Nevertheless, theoretical analyses of nonlinear asymmetric DCs are available.^9,15

Our proposed waveguide platform is not limited to any specific material system. In our study, we choose polymer as the waveguide material for several reasons. First, the large thermo-optic coefficient and the low heat conductivity of polymer can help reduce the input power for the observation of the photothermal effect of graphene. Second, the low-index-contrast waveguide formed with polymer can help reduce the fiber-coupling loss and, more importantly, can offer strong polarization-dependent absorption,^27,28 which provides a means of controlling the strength of nonlinearity by adjusting the polarization state of the input light. In fact, graphene’s polarization-dependent absorption has been explored for the realization of a wide range of optical waveguide devices, such as polarizers,^27,28 mode filters,^28,29 mode switches,³⁰ all-optical switches,^31,32 and tunable filters.³³ Last but not the least, the polymer waveguide technology based on the spin-coating process allows graphene to be buried at desired locations in a waveguide to ensure proper operation of the device.^27–33 

In Secs. II–V, we present our theoretical models and experimental results for the three graphene-buried waveguide structures, together with discussions on their transmission characteristics, and then conclude our study.

We first consider nonlinear mode coupling in an LPWG buried with graphene. An LPWG is a periodic structure formed along a waveguide with a pitch designed for coupling between two forward propagating modes.³⁴  Figure 1(a) shows a schematic diagram of the proposed LPWG, where a corrugated grating of length L_g is formed at the bottom of the waveguide core and a graphene film of the same length is placed on the surface of the core above the grating. The waveguide supports a single core mode and a number of cladding modes.

In general, the effective index (or the propagation constant) of a mode in a graphene-buried waveguide is a complex number, where the imaginary part accounts for graphene’s absorption loss as well as the waveguide loss. The real part of the effective index, however, is not significantly affected by the presence of graphene.²⁷ Therefore, to simplify our notations, all the effective indices and propagation constants referred to in this study should be understood as those for lossless waveguides, i.e., they are all real numbers.

The transmission characteristics of the nonlinear LPWG depend on the operation wavelength and the input power. When the core mode at a wavelength λ is launched into the grating, it is partially absorbed by the graphene film and the heat generated by graphene’s photothermal effect increases the temperature of the waveguide. As the graphene film is placed on the core, the core mode experiences a larger average temperature increase and, hence, a larger decrease in its effective index than the cladding mode (note that the thermo-optic coefficient of polymer is negative). As a result, the effective-index difference between the two modes N_co – N_cl decreases with the input power and the resonance shifts toward shorter wavelengths, i.e., experiences a blue shift, as shown in Fig. 1(b). Whether the resonance shift brings the input light closer to resonance or moves it away from resonance, which affects the strength of mode coupling, depends on the operation wavelength. In the case that the operation wavelength is the same or longer than the intrinsic resonance wavelength λ₀ (i.e., the resonance wavelength at a low power), i.e., λ ≥ λ₀, the resonance shift caused by an increase in the input power weakens the coupling from the core mode to the cladding mode and thus results in a monotonic increase in the output power, as shown in Fig. 1(c). In the case that the input wavelength is shorter than the intrinsic resonance wavelength, i.e., λ < λ₀, as the input power increases initially, the small blue shift in resonance, though bringing the input light closer to the resonance, is not sufficient to lower the output power. The output power increases with the input power and reaches a peak value, where the blue shift in resonance is significant enough (or the mode-coupling effect becomes strong enough) to compensate for the increased input power. As the input power increases further, the output power starts to drop and reaches the minimum value when the shifted resonance wavelength coincides with the operation wavelength. As the input power continues to increase, the resonance is shifted away from the operation wavelength, which weakens the mode-coupling effect and results in a monotonic increase in the output power. The transmission of the core mode for the case λ < λ₀ is illustrated in Fig. 1(d).

The coupled-mode equations (2) and (3) for our graphene-buried LPWG differ from those for long-period gratings in fibers with Kerr nonlinearity^19,20 in the loss terms and the nonlinear terms, where the Kerr effect is replaced by the self-generated thermo-optic effect.

To demonstrate our idea, we design an LPWG to achieve coupling between the fundamental core mode (the E₁₁ mode) and a low-order cladding mode (the E₁₂ mode). Figure 2(a) shows a schematic diagram of the cross section of the graphene-buried waveguide. We use polymer materials EpoCore and EpoClad (Micro Resist Technology)^27–33 for the core and the cladding, respectively, whose refractive indices measured at 1530 nm are 1.569 and 1.560, respectively. The grating is formed on a SiO₂-on-Si substrate, where the SiO₂ layer is 5.0 μm thick. We calculate the effective indices for different core sizes with the mode solver in COMSOL. To ensure a single-mode core, we choose a core size of 5.0 × 5.2 μm². The thickness of the cladding is 20.0 μm. Each mode in the polymer waveguide has two approximately linear polarizations: the quasi-TE polarization with the major electric field parallel to the surface of graphene and the quasi-TM polarization with the major electric field perpendicular to the surface of graphene. For simplicity, they are referred to as the TE polarization and the TM polarization, respectively. The intensity profiles of the core mode and the cladding mode calculated for the TE polarization at 1550 nm are shown in Fig. 2(b). The corresponding effective indices are 1.5635 and 1.5587, respectively. To determine the grating pitch Λ from Eq. (1), we calculate the effective indices of the two modes at different wavelengths and choose a grating pitch of 326 μm for our experimental work, which corresponds to an intrinsic resonance wavelength of ∼1580 nm. To ensure maximum mode coupling at resonance, we use a corrugation depth of 0.50 μm and a grating length of 18 periods, i.e., ∼5.8 mm. The graphene film has a width of 4.0 μm and the same length as the grating. Figure 2(c) shows an optical microscope image of an end-face of the fabricated LPWG. The total length of the fabricated LPWG is ∼9 mm, which includes the lengths of the waveguide leads at the two ends. The fabrication process is described in Sec. VI.

The experimental setup and the procedures for the characterization of the fabricated LPWG are described in Sec. VI. Figure 2(d) shows the normalized transmission spectra of the fabricated LPWG measured for the TE and TM polarizations with a low-power broadband source. As shown in Fig. 2(d), the resonance dip is at 1573.6 nm (1569.2 nm) for the TE (TM) polarization, which is close to the theoretical value 1579.6 nm (1578.4 nm). The small discrepancies can be attributed to fabrication errors. The contrast at the resonance wavelength is larger than 30 dB for both polarizations, which implies a coupling efficiency larger than 99.9%. The output near-field images taken at different wavelengths, which are also shown in Fig. 2(d), confirm the coupling between the E₁₁ core mode and the E₁₂ cladding mode. As graphene’s absorption of the TE-polarized light is larger than that of the TM-polarized light,^27,28 the transmission for the TE polarization is weaker than that for the TM polarization. The graphene-induced losses to the core mode and the cladding mode for the TE polarization, measured with reference waveguides without gratings at 1550 nm, are ∼0.76 dB/mm (α_gco ≅ 0.17/mm) and ∼0.40 dB/mm (α_gcl ≅ 0.09/mm), respectively, while those for the TM polarization are negligible. The fiber-waveguide coupling loss is ∼2.4 dB for one end, and the propagation loss of the core mode is ∼0.30 dB/mm (α_wco ≅ 0.07/mm).

To study nonlinear mode coupling with our graphene-buried LPWG, we launched TE-polarized light into the core mode. The dependence of the output power on the input power measured at three different operation wavelengths is shown in Fig. 3. Here and thereafter, the input power refers to the output power from the input single-mode fiber coupled to the waveguide.

When the operation wavelength is set at the intrinsic resonance wavelength of the grating, i.e., 1573 nm, the output power increases monotonically with the input power, as shown in Fig. 3(a). The growth of the output power is slow initially, when the shift in the resonance wavelength with the input power is small and most of the input power is coupled to the cladding mode. The growth of the output power becomes rapid, when the shift in the resonance wavelength with the input power becomes more and more significant and less and less input power is coupled to the cladding mode. The measured characteristic is consistent with that depicted in Fig. 1(c).

When the operation wavelength is set at 1550 nm, which is shorter than the intrinsic resonance wavelength but within the bandwidth of the rejection band, the output power changes with the input power in a complicated way, as shown in Fig. 3(b). At a low input power, the small shift in the resonance wavelength toward the operation wavelength is not large enough to lower the output power. The output power increases with the input power until it reaches a peak value at an input power of ∼7 mW, where the mode-coupling effect is strong enough to compensate for the increased input power. As the input power increases further, the output power decreases and reaches the minimum value at an input power of ∼15 mW, where the resonance is shifted to the operation wavelength. A further increase in the input power moves the resonance away from the operation wavelength and thus results in a monotonic increase in the output power. The measured characteristic is consistent with that depicted in Fig. 1(d).

When the operation wavelength is set at 1530 nm, which is near the edge of the rejection band of the grating, a much higher input power is required to shift the resonance to the operation wavelength. As shown in Fig. 3(c), the output power increases with the input power in the beginning and reaches a peak value at an input power of ∼12 mW. It then decreases with an increase in the input power. An input power as large as 35 mW is not high enough to shift the resonance to the operation wavelength.

The simulation results obtained by solving the coupled-mode equations (2) and (3) are also shown in Fig. 3 for comparison, where the loss coefficients are taken from the measurement data and the propagation loss of the cladding mode is assumed to be the same as that of the core mode (i.e., α_wcl = α_wco). The values of Δ for the operation wavelengths 1573, 1550, and 1530 nm are 0, 286.0, and 541.7/m, respectively, and the values of n_pco and n_pcl, calculated at 1550 nm, are −2.8 × 10⁻⁴ and −1.9 × 10⁻⁴ mm/mW, respectively, which are assumed to be the same for the three operation wavelengths. As shown in Fig. 3, the experimental results and the simulation results agree well, considering the approximate nature of the physical model and the parameters used in the simulation. As expected, the discrepancies are more significant at high input powers. Our model based on Eqs. (2) and (3) is useful for the analysis of nonlinear mode coupling in graphene-buried LPWGs. Thanks to the strong self-generated thermo-optic effect in our grating platform, we can study distinct nonlinear mode-coupling effects in a grating with low-power CW light over a broad range of wavelengths, which is not feasible with nonlinear gratings based on the Kerr effect.^17–20 

The measurement results for the TM polarization are shown in Fig. 4. The output power increases linearly with the input power, regardless of the operation wavelength, which is consistent with the fact that graphene’s absorption of TM-polarized light in a graphene-buried polymer waveguide is negligible.^{27,28,30–33} For the TM polarization, the grating functions as a linear device. These results confirm that the observed nonlinearity for the TE polarization is, indeed, caused by graphene’s photothermal effect.

We next consider nonlinear mode coupling in a symmetric DC buried with graphene. Figure 5(a) shows the structure of the DC, which consists of two parallel identical single-mode cores (core 1 and core 2) with graphene buried in both. Light coupling between the two cores in a symmetric DC is characterized by a coupling coefficient κ, which is a measure of the spatial overlap of the two modes in one of the cores and, hence, the strength of the mode-coupling effect.² The shortest length of the parallel section of the DC required for complete light coupling between the two cores is equal to L_c = π/(2κ), which is referred to as the coupling length. Here, we assume that our DC has a length of L_c and graphene films are buried in the two cores along the entire parallel section, as shown in Fig. 5(a).

The coupled-mode equations (10) and (11) for our graphene-buried DC differ from those for a DC with Kerr nonlinearity^7–9,11,15 in the loss terms and the nonlinear terms, where the Kerr effect is replaced by the self-generated thermo-optic effect.

We design and fabricate two symmetric graphene-buried DCs to demonstrate nonlinear mode switching at 1550 nm. Figure 5(b) shows a schematic diagram of the cross section of the coupling region of the DC. We employ the software library COMSOL to calculate the thermal and optical characteristics of the DC and a 3D beam propagation method (RSoft) to tune the physical parameters of the final design that includes the waveguide bends at the two ends of the DC. To ensure single-mode operation at 1550 nm, we choose a core size of 4.0 × 4.0 μm². Each core is covered with a graphene film, which has the same width as the core. We choose two values of core separation for the experimental study: g = 6.0 and 5.2 μm, and label the corresponding DCs as DC1 and DC2, respectively. The coupling lengths of DC1 and DC2 are L_c = 2.1 and 1.5 mm, respectively. We fabricate the DCs with the polymer materials EpoCore and EpoClad on a Si substrate. More details about the fabrication process are given in Sec. VI. Figure 5(c) shows an end-face of the fabricated DC1. Both DCs have the same waveguide bends and the same total length of ∼6 mm.

The experimental setup and the procedures for the characterization of the fabricated DCs are described in Sec. VI. The dependences of the output powers from the bar port and the cross port of the DC on the input power measured at 1550 nm for the TE polarization are shown in Figs. 6(a) and 6(b) for DC1 and DC2, respectively. At a low input power, the input light is mostly coupled to the cross port and, as the input power increases, less and less light is coupled to the cross port. The output powers from the two cores become equal at a specific input power, which is referred to as the critical power.^7,13 A further increase in the input power continues to weaken light coupling to the cross port. Light can be switched from the cross port to the bar port by increasing the input power beyond the critical power. The critical power thus serves as a measure of the effectiveness of the nonlinear switching effect. As shown in Fig. 6(a), DC1 has a critical power of 32.3 mW and achieves a high contrast of 13.9 dB between the two output ports at 49.0 mW. Meanwhile, as shown in Fig. 6(b), DC2 has a larger critical power of 44.7 mW and a switching power exceeding 50 mW. The measured waveguide loss coefficient and graphene-induced loss coefficient are ∼0.30 dB/mm (α_w ≅ 0.07/mm) and ∼1.04 dB/mm (α_g ≅ 0.24/mm) for the fabricated DCs, respectively. The corresponding graphene-induced losses for DC1 and DC2 are ∼2.2 and ∼1.6 dB, respectively. The stronger nonlinearity of DC1 is attributed to the larger absorption by graphene. The fiber-waveguide coupling loss for both DCs is ∼2.0 dB for one end.

The simulation results obtained from solving Eqs. (10) and (11) based on the measured parameters of the DCs are also shown in Fig. 6, which agree well with the experimental results. The value of n_pd for both DCs is −3.1 × 10⁻⁴ mm/mW, while that of n_pc is −1.5 × 10⁻⁴ mm/mW for DC1 and −1.6 × 10⁻⁴ mm/mW for DC2. Our graphene-buried DC shows a much lower critical power and a much larger contrast for power-dependent switching, compared to DCs based on the Kerr effect.^7,11–14,16

The measurement results for the TM polarization are shown in Fig. 6(c). As graphene’s absorption of TM-polarized light is negligibly small, an increase in the input power does not affect the mode coupling between the two cores and the output power increases linearly with the input power. The graphene films play a little role in the TM polarization. Our results confirm that the nonlinear characteristics of our DC are caused by graphene’s large photothermal effect on the TE polarization.

We finally consider a DC with graphene buried in one core. As shown in Fig. 7(a), the structure of this DC is the same as that discussed in Sec. III. The only difference is that, in the present case, graphene is buried only in core 1. Both the lengths of the parallel section of the DC and the graphene film are equal to the coupling length L_c so that, in the linear regime, light is completely coupled from core 1 to core 2 and vice versa. As graphene exists only in one core, the power-dependent characteristics of the DC should depend on which core light is launched into the DC. In the case that light is launched into core 1, much light is absorbed by graphene in core 1 and then converted into heat, which results in a large change in the refractive index in core 1 and thus leads to strong nonlinear characteristics. Meanwhile, in the case that light is launched into core 2, only that part of light coupled into core 1 is absorbed by graphene and converted into heat; the resultant nonlinear effect should be weak. In a DC with graphene buried in both cores, light is absorbed by graphene in both cores, which results in refractive-index changes in both cores and thus some cancellation effect in terms of breaking the symmetry of the DC. In a DC with graphene buried in one core, however, there is no such cancellation effect and the resultant nonlinear characteristics can be stronger.

The design used in our experimental study has a core size of 5.0 × 5.0 μm² and a core separation of 4.7 μm, which corresponds to a coupling length of L_c = 5.7 mm at the operation wavelength 1550 nm. The graphene film, which is placed symmetrically on the surface of core 1, has a width of 8.0 μm. Figure 7(b) shows the cross section of the DC, together with the waveguide materials and the dimensions, and Fig. 7(c) shows an end-face of the fabricated DC. The fabricated DC has a total length of ∼10 mm.

The dependences of the output powers from the bar port and the cross port of the DC on the input power measured at 1550 nm for the TE polarization are shown in Fig. 8(a) for the case with core 1 as the input core and in Fig. 8(b) for the case with core 2 as the input core. When light is launched into core 1, the self-generated thermo-optic effect is strong, as shown in Fig. 8(a). The critical power is only 22.7 mW, and a high contrast of 11.5 dB between the two output ports is achieved at 31.4 mW. The nonlinear characteristics are stronger than those observed for the DCs with graphene buried in both cores (Fig. 6). When light is launched into core 2, the self-generated thermo-optic effect is much weaker, as shown in Fig. 8(b). The critical power is larger than 35 mW. The measured waveguide loss coefficient and graphene-induced loss coefficient are ∼0.24 dB/mm (α_w ≅ 0.05/mm) and ∼0.65 dB/mm (α_g ≅ 0.15/mm), respectively, and the fiber-waveguide coupling loss is ∼1.5 dB for one end. The simulation results obtained by solving Eqs. (18) and (19) based on the above-mentioned parameters are also shown in Fig. 8. The values of n_pd and n_pc are −3.0 × 10⁻⁴ and −1.6 × 10⁻⁴ mm/mW, respectively. The experimental results agree well with the simulation results, considering the approximate nature of the physical model and the parameters used in the simulation.

The measurement results for the TM polarization with core 1 as the input core are shown in Fig. 8(c). Again, the results show that the graphene film plays a little role in the TM polarization. In fact, the DC functions as a linear coupler for the TM polarization, regardless of which core is the input core.

We have presented three examples of graphene-buried polymer waveguide structures to demonstrate nonlinear mode coupling effects. Compared to conventional waveguide/fiber platforms based on Kerr nonlinearity, our waveguide platform shows several distinct features.

1. Low-power operation: Thanks to graphene’s photothermal effect, together with polymer’s large thermo-optic effect, our structures allow for the observation of strong nonlinear mode-coupling effects with input powers in the range of several tens of milliwatts, which are at least three or four orders of magnitude lower than those required in conventional Kerr-type nonlinear devices. Inexpensive CW semiconductor lasers with many choices of wavelengths can be used to operate our nonlinear structures. Our platform offers many new opportunities for the study of nonlinear guided-wave optics at low powers and over a broad range of wavelengths. More importantly, the use of CW lasers, instead of high-power pulse lasers, allows for the generation of clean nonlinear effects, as evident by the high switching contrasts achieved in our DCs. It is possible to further reduce the operation power by reducing the insertion loss of the structure through improving the waveguide material, the waveguide design, and the fabrication process. We should mention, however, that graphene’s saturable absorption³⁷ could ultimately limit the maximum power for the generation of nonlinear mode-coupling effects.

2. Flexible nonlinearity design: The strength and the characteristics of the nonlinear mode-coupling effect to be investigated depend on the location(s) of the graphene film(s) placed in the waveguide structure. Much flexibility is available for burying graphene in a polymer waveguide^27–33 and, thus, tailoring the nonlinearity of the waveguide structure. As shown by our DCs, burying graphene in one core or two cores can lead to different nonlinear switching characteristics. In fact, an asymmetrically nonlinear DC, such as the one with graphene buried in one core, is difficult to realize with the conventional Kerr nonlinearity. With the graphene-buried waveguide platform, it is possible to study nonlinear mode-coupling effects in practically any few-mode waveguide structures.

3. Polarization-dependent nonlinearity: Thanks to graphene’s strong polarization-dependent absorption in a polymer waveguide,^27–33 graphene’s photothermal effect works effectively only for the TE polarization, as demonstrated by all of our examples. It is possible to balance between the insertion loss and the nonlinearity by controlling the strength of the overall nonlinearity with a proper mix of the TE and TM polarizations. Such polarization-dependent nonlinearity control does not seem to be available with other nonlinear systems (definitely not available with Kerr-type nonlinearity).

While our graphene-buried waveguide platform can generate strong Kerr-like nonlinear effects, the thermally generated nonlinearities are intrinsically slow. The response times of the devices demonstrated in this study are of the order of milliseconds, which are typical for thermo-optic polymer-waveguide devices.^31–33 

In summary, we have experimentally demonstrated the nonlinear mode coupling generated by graphene’s photothermal effect in three different graphene-buried polymer waveguide structures. These structures can produce Kerr-like nonlinearity with distinct features, as discussed above. We have developed physical models based on the coupled-mode theory for these structures, which successfully explain the experimental results. Our idea could be extended to other mode-coupling structures and waveguide systems,^37–40 such as Si waveguides, which can provide a strong graphene–light interaction and also have a large thermo-optic coefficient.^39,40 The small size of Si waveguides and the large thermal conductivity of Si could reduce the response time to microseconds.^39,40 Our platform can be employed to construct fiber-compatible nonlinear devices, such as optical power limiters and power-dependent optical switches, for applications that involve low optical powers and do not require ultra-fast response, for example, self-directed circuit switching. It should also be possible to generate optical bistability⁴¹ with a graphene-buried waveguide Bragg grating. Our study opens up a new and effective approach to the discovery and the exploration of nonlinear mode-coupling effects in different waveguide structures for practical applications.

The graphene-buried polymer waveguide devices were fabricated with our in-house microfabrication facilities.^27–33 First, a thick EpoClad film was spin-coated on the substrate as the lower cladding. To fabricate the LPWG, periodic corrugations were formed on the lower cladding by photolithography and reactive ion etching (RIE). An EpoCore film was then spin-coated on the lower cladding as the core layer, which was patterned into channel waveguides by using photolithography and wet-etching. To fabricate the DCs, the core layer was directly etched into the DC patterns by photolithography and wet-etching on the lower cladding. Another EpoClad film slightly thicker than the core layer was next spin-coated onto the sample as the mid-cladding, which was then etched down to the core by RIE to form a flat surface for graphene transfer. The RIE process also helped trim the core layer to the desired thickness. A commercial monolayer graphene film with a polymethyl methacrylate (PMMA) buffer (Hefei Vigon Material Technology) was transferred onto the sample by the wet-transfer method, and the PMMA buffer was removed by acetone. The graphene film was then patterned into stripes by photolithography and O₂ RIE. Finally, another EpoClad film was spin-coated onto the sample as the upper cladding. To facilitate device characterization, reference waveguides with and without graphene were also fabricated on the same substrate.

To measure the transmission spectrum of the LPWG, light emitted from a broadband source (Leukos, Samba) was launched into the LPWG via a single-mode fiber (SMF) and detected at the output end with an optical spectrum analyzer (OSA) (Agilent, 86140B) via another SMF. The polarization state of the broadband light was controlled with a fiber polarizer and a polarization controller (PC). The fiber-coupling conditions at the two ends were carefully adjusted to ensure that only the core mode was excited and detected. The spectrum of the LPWG was normalized with respect to that measured for a reference waveguide without graphene. To measure the power-dependent characteristics of the LPWG at different wavelengths, a tunable laser (Keysight, 8163B), together with an erbium-doped fiber amplifier (EDFA) (May-ray photonics, EDFA-C-BA-25-SM-B), was used as the light source and the output light was detected with the OSA via an SMF. The polarization state of the input light was controlled with a PC. The output near-field images of the LPWG were captured using an infrared camera (Hamamatsu, C2714).

To estimate the graphene-induced loss to the TE polarization, the output powers for both the TE and TM polarizations from a reference waveguide with graphene were measured and compared, where the graphene-induced loss to the TM polarization was assumed negligible. To estimate the graphene-induced loss to the TE-polarized cladding mode, the cladding mode (the E₁₂ mode) of the waveguide was excited by carefully adjusting the alignment between the input SMF and the waveguide for a reference waveguide with graphene and the output powers for the two polarizations were measured and compared. The uncertainty in the loss measurements is ±0.3 dB. The measured graphene absorption losses for TE-polarized light are comparable to those of graphene-buried polymer-waveguide devices reported in our recent studies.^32,33

To characterize the DCs, light emitted from the tunable laser with amplification by the EDFA was launched into a core of the DC via an SMF and the output powers from the two cores were measured separately using a power meter (Newport, 1830-C) via a 40× objective lens. The polarization state of the input light was controlled using a PC.

To calculate the photothermal coefficient (in mm/mW), we solve the heat conduction equation for the temperature change distribution δT(x, y) by using the heat transfer module in COMSOL, where the graphene film is modeled as a uniform heat source without thickness with a length of 1 mm and the thermal power generated over this heat source is 1 mW. The thermal conductivities of polymer, SiO₂, and Si are set at 0.12, 1.38, and 130 W/m/K, respectively. The corresponding refractive-index change profile is then given by C_tδT(x, y), where C_t is the thermo-optic coefficient of the polymer material with an assumed value of −1.0 × 10⁻⁴/K. Figure 9 shows the refractive-index change profiles calculated with 1-mW thermal power generated over a 1-mm long graphene film for the graphene-buried LPWG and the DC with graphene buried in one core, respectively. The effective indices of the modes with and without the induced refractive-index change are calculated by solving the full-vector wave equation available in the electromagnetic wave module in COMSOL, from which the photothermal coefficients necessary for solving the coupled-mode equations in our physical models are determined. We should note that the values for the graphene length and the thermal power used in the simulation are not important, as long as the final result is properly scaled. The photothermal coefficient depends only on the transverse position of the graphene film and the waveguide mode. The photothermal coefficient in the characterization of the photothermal effect plays a similar role as the nonlinear coefficient in the characterization of the Kerr effect.

The coupled-mode equations in our study are numerically solved by the Runge–Kutta method. In the coupled-mode equations, the input power P_in refers to the power at z = 0, i.e., the beginning location of the graphene film, and the output power refers to the power at the ending location of the graphene film. In the experiments, however, the input power refers to the output power from the input SMF and the output power refers to the power measured at the output end of the device. For a valid comparison between experimental results and simulation results, the power readings in the simulation results are calibrated with measured fiber-coupling losses and additional waveguide losses beyond the region where graphene is located. As graphene’s absorption loss is insensitive to the wavelength,^28,29 the measured value at 1550 nm is assumed to be the same for all operation wavelengths in the study of LPWG.

This work was supported by the Research Grants Council, University Grants Committee (RGC, UGC), Hong Kong (CityU 11212621).

The authors have no conflicts to disclose.

Lianzhong Jiang: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Wenfan Jiang: Conceptualization (supporting); Data curation (supporting); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (lead); Writing – review & editing (equal). Kin Seng Chiang: Conceptualization (equal); Data curation (supporting); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (lead); Supervision (lead); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.