Low Insertion Loss Plasmon-Enhanced Graphene All-Optical Modulator.

Graphene has emerged as an ultrafast optoelectronic material for all-optical modulators. However, because of its atomic thickness, it absorbs a limited amount of light. For that reason, graphene-based all-optical modulators suffer from either low modulation efficiencies or high switching energies. Through plasmonic means, these modulators can overcome the aforementioned challenges, yet the insertion loss (IL) of plasmon-enhanced modulators can be a major drawback. Herein, we propose a plasmon-enhanced graphene all-optical modulator that can be integrated into the silicon-on-insulator platform. The device performance is quantified by investigating its switching energy, extinction ratio (ER), IL, and operation speed. Theoretically, it achieves ultrafast (<120 fs) and energy-efficient (<0.6 pJ) switching. In addition, it can operate with an ultra-high bandwidth beyond 100 GHz. Simulation results reveal that a high ER of 3.5 dB can be realized for a 12 μm long modulator, yielding a modulation efficiency of ∼0.28 dB/μm. Moreover, it is characterized by a 6.2 dB IL, which is the lowest IL reported for a plasmon-enhanced graphene all-optical modulator.

Introduction

The incessant demand for bandwidth-intensive applications necessitates the quest for ultrahigh-speed optoelectronic devices. Global data center traffic is growing faster by more than 25% a year.[1] To sustain this enormous growth, ultrafast and energy-efficient integrated optical transceivers are indispensable.[2] In these data centers, integrated silicon-based photonic interconnects serve as the communication link between servers, racks, and digital logic chips, where the transmission loss in Si waveguides can be as low as 0.1 dB/cm using the silicon-on-insulator (SOI) platform.[3] Integrated optical interconnects consist of a transmitter, a waveguide, and a receiver. The modulator imposes a data stream on the optical waveguide mode.

Most modulators employed in silicon photonics (SiPh) are based on silicon.[4] These high-speed modulators operate based on the plasma dispersion effect and are limited by an intrinsic bandwidth of ∼60 GHz.[5,6] Plasma dispersion modulators with bit rates exceeding 100 Gbps have been demonstrated by introducing additional fabrication steps or incorporating reactive components in the driver circuit.[7,8] Nevertheless, these devices are based on large-footprint Mach–Zehnder modulator structures, and their energy consumption is on the order of pJ/bit or more. Modulators based on thin-film lithium niobate (LiNbO3) which is heterogeneously integrated into silicon waveguides were demonstrated with a bandwidth of up to 110 GHz.[5,9] However, these devices operate with stringent phase-matching conditions which complicate their design and require unconventional fabrication processes, making them unfavorable for mass production. Two-dimensional materials on silicon recently emerged as active materials for electro-optic modulation, with graphene on the lead due to its exceptionally fast response.[10−13] These materials are characterized by a favorable set of inherent features, which include high-speed, small footprint, low-cost manufacturing, low-power consumption, and compatibility with complementary metal–oxide semiconductor processes.[4,14] Nonetheless, the bandwidth of graphene-based electro-optic modulators is limited by device parasitics to a few tens of GHz, which hinders the adoption of these devices in ultrahigh-speed transceiver links.

In recent years, on-chip all-optical modulators stood out for their extraordinary high-speed performance.[15−19] These modulators exhibit ultrafast switching speeds (<1 ps) but consume picojoules of energy.[18,20] Low-energy all-optical switching (<1 pJ) has been demonstrated in refs,[21−23] yet the operation speed of these devices is limited to tens of GHz. On-chip all-optical graphene modulators were lately investigated in refs.[17,24] These modulators suffered from relatively high switching energies or low modulation efficiencies, and both limitations can be attributed to the intrinsically low optical absorption of graphene. Introducing plasmonic metals to these modulators to plasmonically enhance the effective absorption of graphene is a promising solution to overcome these limitations. Indeed, plasmonic all-optical graphene switches and modulators were experimentally studied in refs,[25,26] where an ultrafast switching time (260 fs) at an ultra-low energy (35 fJ) was reported in ref (25). Both plasmonic devices utilized plasmonic gap waveguides, which in addition to their high Ohmic losses require plasmonic mode converters to achieve decent coupling efficiencies with Si waveguides. Therefore, despite the remarkable performance of these devices, they are characterized by an excessive insertion loss (IL), for example, 19 dB in ref (25), making them impractical for future communication links.

Here, we propose a plasmon-enhanced graphene all-optical modulator featuring a low IL, an ultrafast operation, a low-energy switching, and a high modulation efficiency. The device structure is based on a silicon rib waveguide, which facilitates its integration into the SOI platform. The device structure is presented, and its operation principle is explained. Its performance is quantified by investigating its switching energy, modulation efficiency, IL, and operation speed. Following that, a comparison is made with state-of-the-art devices. Finally, this report is concluded by pointing out the major findings of this study.

Results and Discussion

Device Structure

The structure of the on-chip modulator is illustrated in Figure a. A silicon rib waveguide on top of a bottom oxide (BOX) layer guides the incoming light to the modulator section. Silicon dioxide (SiO2) is placed on the silicon ridge sides to facilitate the placement of the graphene monolayer on top of the waveguide. A gold (Au) stripe is placed on top of graphene to plasmonically enhance its interaction with the optical mode, which in turn boosts its effective optical absorption. The waveguide supports a quasi-transverse magnetic (quasi-TM) mode, which is chosen for its superior interaction with the graphene sheet (see Supporting Information Section 1).

Figure 1

(a) All-optical modulator based on a silicon rib waveguide structure. (b) Electric field profile of the propagating quasi-TM mode in the waveguide-integrated modulator. The dashed white line represents the graphene sheet plane. λ = 1550 nm. Si: silicon, SiO2: silicon dioxide, BOX: bottom oxide, Gr: graphene, and Au: gold.

Figure a presents the coupling efficiency between the SOI waveguide and the modulator section. The coupling efficiency is taken as the power overlap between the fundamental waveguide and modulator TM modes that yield the highest coupling efficiency, at each waveguide height and width, for a fixed 160 nm thick slab and 2 μm BOX layer. The Au stripe is 20 nm thick, and its width was initially set equal to the waveguide width. A commercially available 500 nm SOI waveguide height was chosen to achieve a decent coupling efficiency (70%). For heights below 340 nm, the coupling efficiency is <30%, which results in a high IL. On the other hand, the coupling efficiency increases for heights beyond 500 nm, yet these heights are not common in SOI-based SiPh. The waveguide width was chosen to be 460 nm, which is a standard width in SiPh. All-optical modulators are employed for ultrafast information processing and photonic computing systems.[15,16] These systems are likely to contain heterogeneously integrated III–V lasers on Si. In such a case, a 500 nm thick waveguide would be a favorable choice because thick SOI waveguides are essential to achieve decent coupling efficiencies with III–V lasers.[27−30]

Figure 2

(a) Coupling efficiency between the SOI waveguide and the modulator section as a function of the waveguide height and width and (b) coupling efficiency as a function of the taper length. λ = 1550 nm.

The Au stripe width was later changed from its initial 460 nm width to 380 nm, in order to achieve high absorption efficiency in graphene (see Supporting Information Section 2). The coupling efficiency was also improved as a result of this modification to become ∼77%. Furthermore, tapered Au stripes that are 600 nm long were incorporated into the design, which boosted the coupling efficiency to ∼88% (see Figure b). For longer taper lengths, the coupling efficiency is limited by the Ohmic losses induced by the Au stripe. The power overlap between the two modes was computed using Lumerical MODE.[31,32] The coupling efficiency with a taper was computed using the mode expansion monitor,[33,34] where the fundamental SOI TM-mode was imported into Lumerical FDTD to compute its power overlap with the modulator waveguide after tapering the Au stripe.

The waveguide satisfies the single-mode condition for a deep-etched sub-micron SOI rib at λ = 1550 nm, which is determined by the following relations[35]where W is the waveguide width, H is the waveguide height, and h is the slab thickness. A deep-etched silicon waveguide is one, where h < H/2 is satisfied. We propose some fabrication methods and discuss the impact of some potential variations on the device performance in Supporting Information Section 5. In the next section, we explore the operation principle of the device.

Operation Principle

Graphene can be utilized to build extinction or phase modulators. In ref (36), it was reported that graphene-based phase all-optical modulators exhibit a higher modulation efficiency than extinction modulators. However, this conclusion is only applicable for graphene modulators that are based on a Mach–Zehnder structure. In another report,[37] it was concluded that higher modulation efficiencies can be achieved based on extinction modulators, and for a smaller device footprint. These modulators operate based on the principle of Pauli-blocking,[38−40] which occurs when photogenerated electrons fill the conduction band states of graphene following a sufficiently intense pump excitation, thereby blocking the interband transition of other electrons.

Graphene is unintentionally doped when placed on a substrate, giving a chemical potential (μ) in the range of 0.1–0.2 eV.[37,41] For telecom wavelengths, a photon has an energy ℏω > 2μ, which induces an interband transition following its absorption by graphene, thereby creating an electron–hole pair.[42]Figure a illustrates the interband absorption of a pump photon with an energy ℏωpump > 2μ. Upon absorbing a sufficiently intense pump excitation, electrons fill up the conduction band states, resulting in a chemical potential μ′. Consequently, incoming pump photons cannot induce an interband transition because ℏωpump ≤ 2μ′ and are thus transmitted through graphene (see Figure b). Similarly, a probe photon with an energy ℏωprobe ≤ ℏωpump is also transmitted because ℏωprobe ≤ 2μ′. As such, all-optical amplitude modulation can be realized; the probe signal is transmitted when the pump signal is HIGH, or absorbed when the pump signal is LOW.

Figure 3

(a) Interband absorption of a pump photon with energy ℏωpump. (b) Pump photon Pauli-blocked (transmitted) after applying a sufficiently high pump intensity. (c) Transmission of the probe signal, as determined by the pump signal amplitude. Black and white circles represent electrons and holes, respectively. Filled energy states are represented by darker shades.

From Figure b, it can be inferred that μ′ = ℏωpump/2.[25] The chemical potential of graphene is related to the carrier density (n0) by the following relation[42]where vF is the Fermi velocity. The interband absorption is Pauli-blocked when graphene gains a chemical potential energy that is given by

Δn can be calculated by reordering eq

For a graphene sheet with an area A = WL, the number of carriers that are needed to reach μ′ is m = ΔnWL. Because each absorbed photon generates an electron–hole pair, the energy that is required to saturate the absorption of graphene (Usw) can be expressed as[25]where Usw is the switching energy, and ℏω is the pump photon energy. We are interested in the graphene area that interacts with the optical mode. For the modulator structure of Figure a, the interaction area is the waveguide width (W) times the modulator length (L). Figure a shows the calculated switching energy for a 1550 nm pump signal. The switching energy increases at low chemical potentials and is maximized when the charge neutrality point coincides with the Dirac point (μ = 0). As illustrated in Figure , more electrons are needed to fill the conduction band states up to μ′ when μ is low and that necessitates higher switching energies. Similarly, longer modulators require a higher pump energy to saturate the absorption of graphene, which is expected because the number of carriers (m) is proportional to L. The modulation efficiency is also a function of the modulator length, as is going to be elaborated in the next section. Because graphene is effectively doped by the substrate, the switching energy is practically considered for chemical potentials in the range of 0.1–0.2 eV. In that range, the switching process is inherently efficient, where the optical absorption saturates when graphene absorbs energies <60 fJ. However, in reality, the optical mode experiences coupling and Ohmic losses as it propagates in the modulator waveguide. In addition, graphene absorbs only a small fraction (AGr) of the total mode energy (see Methods). Moreover, a part of graphene absorption is nonsaturable and does not contribute to the modulation function.[25,43−45] Thus, the power fraction that is effectively absorbed by graphene (AGr) can be expressed aswhere As and Ans are the saturable and nonsaturable fractions of AGr. For the all-optical modulator that is presented in ref (44), it has been reported that monolayer graphene absorbs 69.12% of light, of which 65.88% is saturable and 3.24% is nonsaturable. Based on these experimental values, the nonsaturable absorption of monolayer graphene is ∼5% of the total effective absorption: (3.24/69.12) × 100%, which is used to quantify Ans in this study. Monolayer graphene has the least nonsaturable absorption; Ans is higher for thicker graphene.[44−46] The higher Ans reduces the modulation efficiency and contributes to a higher IL. In addition to graphene thickness, surface defects, for example, wrinkles, result in a higher Ans.[44] To calculate the effective switching energy (Ueff), we consider the limited absorption of graphene and compensate for the coupling, Ohmic, and nonsaturable absorption losseswhere Γ is the coupling loss and AAu is the Ohmic loss that is induced by the Au stripe (see Methods). On the left hand side of eq , we added the amount of switching energy that is lost to coupling and ohmic losses. On the right hand side, we subtracted the nonsaturable component of the effective absorption of graphene because it does not contribute to the modulation function. Γ is accounted for only once in eq because the optical mode couples out of the modulator after switching takes place. The calculated effective switching energy for λpump = 1550 nm is shown in Figure b. As expected, Ueff is higher than Usw, and by a factor of ∼11×. Despite that, energy-efficient switching is attainable, where pump signals with energies <600 fJ can saturate the absorption of graphene for λpump = 1550 nm, considering the aforementioned practical range of μ. It is possible to tune the chemical potential of graphene on a substrate by thermal annealing.[41,47,48] For a reasonable μ = 0.2 eV and 12 μm long modulator, Ueff is as low as 242 fJ for a 1550 nm pump signal. In the next section, we investigate the modulation efficiency and the IL of the device.

Figure 4

(a) Switching energy (Usw) and (b) effective switching energy (Ueff) as a function of the chemical potential (μ) and modulator length (L). λ = 1550 nm.

Modulation Efficiency

The modulation efficiency of the modulator is quantified by its extinction ratio (ER),[24−26,39,40] which is given bywhere Ton and Toff represent the transmitted power of the probe signal when the pump signal is turned on and off, respectively. Graphene’s absorption is maximized when the pump signal is turned off. By considering the fractions of the power absorbed by Au and graphene, and the coupling losses that the guided mode experiences as it couples in and out of the modulator, Toff can be expressed as

Graphene is transparent when the pump signal is turned on, except for the nonsaturable fraction (Ans) of AGr, as was explained in the previous section. The maximum graphene transparency and, consequently, the maximum ER of the modulator are obtained when the pump signal has an energy U ≥ Ueff. By considering the nonsaturable fraction of AGr, the maximum transmitted power of the probe signal (Tmax) can be expressed as

The IL is given by[49]

Figure a presents the maximum ER and Tmax, and Figure b shows the corresponding IL and ER/IL ratio as a function of the modulator length. By linearly fitting the ER curve that is shown in Figure a, we obtain a modulation efficiency of ∼0.28 dB/μm. To maximize the ER and the ER/IL ratio, we set the modulator length to 12 μm. At this length, a 3.5 dB ER is achievable for a relatively low IL (6.2 dB). The slight offset in ER that is seen at L = 0 is due to the graphene portion that is placed beneath the taper, where its effective absorption has been incorporated into the calculations (see Supporting Information Section 3).

Figure 5

(a) Maximum transmitted power of the probe signal (Tmax) and the maximum ER (ERmax) as a function of the modulator length (L). (b) IL and ER/IL ratio as a function of the modulator length (L). λ = 1550 nm.

The energy of a pulse is a function of the pulse duration and power: U = tpulseP. Considering a constant pulse duration, the saturable absorption of graphene can be expressed as a function of the pump signal energy[25]where Usat is the energy at which As is equal to half of its maximum value, that is, As(Usat) = AGr(1 – Ans)/2. By incorporating eqs into 12 and 13, the transmitted power of the probe signal (Ton) can be expressed as a function of the pump signal energy

We estimate Usat by setting it equal to Usw/2 because the losses are incorporated into Ton. Figure shows Ton as a function of the pump energy. The modulator length and the chemical potential are varied in Figure a,b, respectively. It can be seen in both figures that Ton is minimized when U = 0, where it is equal to Toff. Ton rises as the pump energy increases and becomes nearly flat as U → Ueff, indicating the onset of saturable absorption. A higher Ton/Toff ratio, and hence, a higher ER, is achieved for longer modulators. This is consistent with the results that are presented in Figure a. A greater pump energy is required to saturate the absorption of graphene at lower μ (see Figure b). This observation is consistent with the calculated Ueff, as is presented in Figure b.

Figure 6

(a) Transmitted power of the probe signal (Ton) as a function of the pump signal energy at multiple modulator lengths for a fixed μ = 0.2 eV. (b) Transmitted power of the probe signal (Ton) as a function of the pump energy (U) at multiple chemical potential (μ) values and fixed L = 12 μm.

Atotal, AAu, and Γ are functions of the wavelength (see Supporting Information Secion 4). Consequently, ER, IL, and Ueff are also functions of the wavelength. Figure a shows the ER and the IL of the modulator as a function of the probe signal wavelength. The IL follows the trend of the coupling efficiency, which decreases at longer wavelengths, whereas the ER is more affected by the propagation loss as a function of wavelength (see Supporting ). In Figure a, we assume that a pump signal with an energy U ≥ Ueff has been applied with a wavelength λpump < λprobe. Figure b presents the calculated Usw and Ueff as a function of the pump signal wavelength at a fixed μ = 0.2 eV. Interestingly, Usw is higher for pump signals of shorter wavelengths (see eq ). This result agrees with previous experimental studies, where it has been reported in ref[45,50] that a lower pump intensity is required to saturate graphene at longer pump wavelengths. This can be explained by the unique conical dispersion of graphene, which becomes wider at higher energies. Consequently, a greater number of high-energy photons would be needed to fill the high-energy states in the conduction band with photoexcited electrons, which results in a higher overall switching energy. Therefore, a higher pump energy would be required to saturate the absorption of graphene, so that a probe signal with a wavelength λprobe > λpump would be transmitted through graphene. We also observe that Ueff is minimized at longer wavelengths. However, it is noted that this trend is not as linear as it is the case for Usw at shorter wavelengths because for this device, external factors such as the coupling efficiency, Ohmic losses, and the effective absorption of graphene dominate the trend of Ueff at shorter wavelengths. More specifically, the absorption efficiency of graphene and the coupling efficiency are both maximized at shorter wavelengths (see Supporting Information Section 4), and therefore, both contribute to a smaller Ueff based on eq . The resultant Usw and Ueff at other chemical potentials are presented in Supporting Information Section 4. In the next section, we calculate the device bandwidth and switching speed.

Figure 7

(a) Maximum ER (ERmax) of the modulator and its IL as a function of the probe signal wavelength (λprobe). (b) Switching energy (Usw) and effective switching energy (Ueff) as a function of the pump signal wavelength (λpump) for fixed L = 12 μm and μ = 0.2 eV.

Operation Speed

The operation speed of the modulator is determined by the carrier cooling dynamics in monolayer graphene. Graphene has a unique conical dispersion, where its density of states fades away at the Dirac point. As a result, electrons in the vicinity of the Dirac point have a relatively low heat capacity. Upon photoexcitation, these electrons immediately scatter with one another, creating a fleeting Fermi–Dirac distribution of hot thermalized electrons within a few tens to 150 fs.[51,52] As such, photon energy is converted to electron heat. Eventually, hot electrons cool down in a few picoseconds by emitting optical and acoustic phonons, coupling with surface optical phonons, and most importantly through disorder-assisted scattering which dominates at room temperature.[47,53−55] Hence, electrons first heat up through intraband electron–electron scattering, then they cool down through phonon- and disorder-assisted scattering.

The Boltzmann transport theory can be used to calculate the timescale in which electron–electron scattering events take place.[56] First, the electrical conductivity of graphene can be expressed as[57−59]where σ0 is the minimum conductivity taken from,[57]h is the Planck’s constant, and Δ is the minimum conductivity plateau. Δ is a parameter that quantifies the amount of disorder in a graphene sample. For graphene-on-SiO2, the minimum conductivity plateau can be as low as Δ ≈ 55 meV,[57,60] or as high as Δ ≈ 100 meV.[58] The Drude mobility (η) of graphene is given by[61]

Now the electron–electron scattering time (τscat) is calculated according to the Boltzmann transport theory[62]

The electron cooling time (τcool) is the inverse of the electron cooling rate (γcool), which can be expressed as[53,57,63]where Tk = 300 K is the temperature, g is the electron–phonon coupling constant, ϱ is the density of states, kF is the Fermi wave vector, is the mean free path, TBG is the Bloch–Grüneisen temperature, D = 20 eV is the deformation potential constant,[47] ρ = 7.6 × 10–7 kg/m2 is the mass density of graphene, and s = 2 × 104 m/s is the speed of longitudinal acoustic phonons.[42]

Figure a shows τscat and Figure b shows τcool for low-disorder (Δ = 55 meV) and high-disorder (Δ = 100 meV) graphene. It is observed that electron–electron scattering occurs in 30–120 fs, which is consistent with the experimentally reported electron heating times.[51,52] The electron cooling time is 4–10 ps, which is similar to what has been reported in the literature.[47,53] Moreover, the scattering and cooling events occur on a shorter timescale for the highly disordered graphene, which is consistent with experimental findings.[64] The calculated τscat and τcool significantly increase as μ → 0 and thus were not shown in its vicinity. As was previously explained, graphene is effectively doped by the substrate, and the μ → 0 case is therefore not applicable. For graphene, the time evolution of the electron temperature, that is, ΔTe(t), can be described by a two-exponential function[51,64,65]where τ1 and τ2 are analytically fitted time constants. Based on the calculated τscat and τcool values, we consider the case of 100 fs heating time and 4 ps cooling time, which are taken as the rise (τr) and fall (τf) times of ΔTe, respectively. For the same τr and τf, the resultant ΔTe is shown in Figure , where the corresponding time constants are τ1 = 0.06 ps and τ2 = 1.8 ps (see Supporting Information Section 7); both are in agreement with experimentally fitted time constants.[44,51,64−66] As seen in Figure , electrons heat up within a sub-ps duration and subsequently cool down in a few picoseconds. Therefore, this device can operate as an ultrafast switch, with a switching time that is determined by electron–electron scattering events, that is, 30–120 fs. As a modulator, this device is limited by the electron cooling time, that is, <10 ps, depending on the chemical potential and sample disorder. This corresponds to an ultra-high bandwidth beyond 100 GHz.

Figure 8

(a) Electron–electron scattering time (τscat) and (b) electron cooling time (τcool) for low-disorder (Δ = 55 meV) and high-disorder (Δ = 100 meV) graphene as a function of chemical potential (μ).

Figure 9

Time evolution of the electron temperature (ΔTe) in graphene following a pumping event, and the corresponding normalized transmittance of the probe signal as a function of time.

In practice, the modulator can impose a pump pulse on the probe signal, even if electrons are not totally cooled down. However, this comes at the cost of a lower modulation efficiency. A high ΔTe indicates that many electrons are photoexcited and are thus occupying the conduction band states. When ΔTe → 0, graphene is reverted to its steady state, where electrons occupy the valence band states. Consequently, the transmittance of the probe signal (Ton) increases with the increase of ΔTe (see Figure ). Based on eq , a higher Ton/Toff ratio results in a higher ER. Thus, the modulation efficiency is also a function of the time delay between two consecutive pump pulses; it is possible to modulate a probe signal when the time delay between two consecutive pulses is less than τcool, but this may result in a poor ER because the Ton/Toff ratio would be reduced.

Table presents the performance metrics of on-chip graphene all-optical switches and modulators that were previously reported. The device presented in this work achieves the fastest switching speed because it is based on monolayer graphene. Besides offering a higher modulation efficiency, fewer layer graphene is characterized by faster carrier heating and cooling mechanisms.[44,65] The device presented in ref (25) is based on bilayer graphene and has demonstrated a switching time of 260 fs. It is noted in Table that plasmonic devices achieve a decent performance at a small footprint, with the most compact footprint reported in ref (25). For scenarios where the device footprint and the switching energy are not a concern, nonplasmonic devices are the favorable choice for their negligible IL, for example refs (17) and (24). Though this device is longer than other plasmonic ones reported in literature, it can achieve a similar modulation efficiency at a significantly lower IL.

Table 1

Performance Metrics of On-Chip Graphene All-Optical Switches/Modulators

structure	IL	ER (dB)	L (μm)	ER/μm	IL/μm	tswitch	Ueff
dielectric WG[24]	negligible	2.75	100	0.0275 dB/μm	negligible	n/a	n/aa
SOI WG[17]	negligible	1.1b	30	0.0367 dB/μm	negligible	1.2 psc	2.1 pJ
plasmonic WG[26]	n/a	2.1	10	0.21 dB/μm	n/a	n/a	n/ad
plasmonic WG[25]	19 dB	3.5	4	0.875 dB/μm	4.75 dB/μm	260 fs	35 fJ
this work	6.2 dB	3.5	12	0.28 dB/μm	0.517 dB/μm	30–120 fs	<0.6 pJ

Input light power is 60 mW.

Modulation depth is 22.7%.

Limited by the resolution time of the asynchronous pump–probe system.

Input light power is 46 mW. n/a: not available (not reported).

Conclusions

To sum up, a waveguide-integrated plasmon-enhanced graphene all-optical modulator is proposed. The device structure is based on a thick Si rib waveguide, which enables its integration into the SOI platform. It operates based on the principle of Pauli-blocking, where the optical absorption of graphene saturates by absorbing sufficiently intense pump radiation. A plasmonic gold (Au) stripe was incorporated into the modulator waveguide to enhance the effective absorption of graphene by a factor of ∼4×. Furthermore, the Au stripe was tapered to improve the coupling efficiency between the SOI and modulator waveguides to be as high as 88%. We investigated the effective switching energy, modulation efficiency, and IL of the device as a function of the modulator length. The modulator length was set to 12 μm to achieve a high ER (3.5 dB) at a relatively low IL (6.2 dB), which is the lowest IL reported for a plasmon-enhanced graphene all-optical modulator. The modulator achieves efficient sub-picojoule switching, where the effective switching energy is <600 fJ for λpump = 1550 nm. The switching time of the device is limited by intraband electron–electron scattering, where the switching time was calculated to be 30–120 fs. As a modulator, its operation bandwidth is determined by the hot carrier cooling dynamics, leading to ultrahigh bandwidths beyond 100 GHz. This device is characterized by very promising performance metrics that are expected to serve the needs of ultrahigh-speed links in next-generation data centers.

Methods

The total propagation loss of the computed mode that is shown in Figure b is αtotal = 0.43 dB/μm (see Figure a). Removing graphene yields a propagation loss αAu = 0.34 dB/μm (see Figure b), which is attributed to Au. The propagation loss (α) is defined as[67]where Ei and Ef represent the electric field intensity before and after propagating a distance L in the modulator waveguide, respectively. The total fraction of power absorbed in the waveguide (Atotal) can be calculated using the Beer–Lambert law

Figure 10

(a) Propagation loss of the waveguide mode with graphene and (b) without graphene. The white line represents the graphene sheet plane. λ = 1550 nm.

The propagation loss that is related to graphene is given bywhich results in αGr = 0.09 dB/μm. Thus, the Au stripe enhances the absorption of graphene by a factor of ∼4× (see Supporting Information Section 2). The fraction of power that is absorbed by graphene (AGr) can be taken as[68]

Then, the remainder constitutes the Ohmic absorption that is introduced by Au