Tutorial: Nonlinear optics in carbon nanotube, graphene, and related 2D materials

Transcription

1 Tutorial: Nonlinear optics in carbon nanotube, graphene, and related 2D materials Shinji Yamashita Research Center for Advance Science and Technology (RCAST), The University of Tokyo Komaba, Meguro-ku, Tokyo , Japan (Phone): , (Fax): , Abstract: One- and two-dimensional forms of carbon, carbon nanotube and graphene, have attracted great attention by researchers in many fields for their interesting and useful electrical, optical, chemical and mechanical properties. In this tutorial, we will introduce the basic physics and the linear optical properties of these 1D/2D materials. We then focus on their nonlinear optical properties, saturable absorption, electro-optic effect, and nonlinear Kerr effect. We will also review and discuss a few key applications using the ultrafast nonlinear phenomena possessed by these 1D/2D materials: (1) short-pulse fiber lasers using saturable absorption, (2) electro-optic modulators, and (3) all-optical signal processing devices. 1

2 1. Introduction Nonlinear (NL) devices are crucial components in various systems as they enable the manipulation of physical quantities with control signals. In electronics, we have an excellent nonlinear device, the transistor. Initially, transistors were used as a single discrete device, but soon they were integrated in a large scale to become the standard building blocks of modern electronics, mostly in the form of complementary metal-oxide-semiconductor (CMOS) integrated circuit (IC) chips, whose density and speed have been accelerated ever since in accordance to the Moore s Law. Nowadays billions of transistors can be integrated into a tiny chip, which serves as the key component in the modern information age. In optics, conventional systems, such as a solid-state laser illustrated in Fig. 1(a), are usually constructed using bulk-optic components in a free-space-optics configuration, whereby rays of light beams are reflected by mirrors, refracted by lenses, diffracted by gratings, and amplified by gain crystals. These free-space-optics systems have shown good performances in the laboratories; however, they suffer from the requirement of stringent optical alignment, sensitivity to environmental perturbations, large footprint, and high energy consumption. The typical dimensions of these bulk-optic systems are in the order of meters by meters, and the typical beam size is around 1mm. NL optics was initiated shortly after the invention of lasers, the first demonstration was the second-harmonic generation (SHG), followed by various other NL optical phenomena [1], which will be discussed later in this paper. NL optical devices used in bulk-optic systems are usually in the form of bulk crystals or semiconductors, for example, Barium Borate (BBO), Lithium 2

3 Niobate (LN), and the semiconductor saturable absorber mirror (SESAM) [2]. The typical sizes of these devices are in the order of centimeters by centimeters (Fig. 1(b)). Optical fiber- and optical-waveguides-based systems started in the 1970 s with the advent of lowloss optical fibers. The demands for low-lost and low-cost optical fibers in the telecommunication industry have resulted in technological advancement to mass production of high-quality optical fibers with attenuation and price close to the minimum values. The introduction of rare-earth-doped optical amplifiers and the wavelength-division-multiplexing (WDM) technologies driven by the explosive growth of in bandwidth requirement for the internet in the 1990 s further accelerated the technological advancement of functional optical fiber/waveguide components, such as fiber couplers, gain fibers, optical isolators and electro-optic modulators [3]. Optical fiber/waveguide technology has soon been applied to other non-telecommunication fields with the availability of specialty optical fibers/waveguides in various optical systems, such as fiber lasers (Fig. 1(a)) and fiber sensors. The length of such fiber-optic systems are typically a few tens to hundreds of meters, but owing to the flexibility of optical fibers, it can be coiled to sizes in the order of ~10cm in coiling diameter whilst guiding lights with beam sizes comparable to the fiber core diameter of ~10μm. NL optical devices used in the fiber-optic systems are either waveguide-based or fiber-based, such as the waveguide-based LN electro-optic waveguide modulators or switches and the fiber-based nonlinear loop mirrors (NOLM), which utilizes the nonlinear effects of the waveguide/fiber materials [4]. These devices have the beam size compatible to that of the fiber-optic systems, as shown in Fig. 1(b). 3

4 In parallel with the development of fiber-optic technologies, on-chip optical technologies emerged with silicon as a guiding medium. Integrated-optic system, or photonic integrated circuit (PIC), using the silicon-on-insulator (SOI) platform started in 1985, and the research field have become very active since the 2000 s, as silicon photonics [5]. Silicon photonics is compatible with the matured industrial standard CMOS technology, which allows the fabrication of optical waveguide structures with sizes down to 10nm at a low cost. The strong optical confinement (typically 100nm) in silicon waveguides allows integration of many optical devices, such as directional couplers, bandpass filters, electro-optic modulators and optical switches, into a small (<1mm) chip, similar to that of the electronic CMOS IC chips (Fig.1(a)). Silicon waveguides can also be used as NL optical devices, as the third-order (3) optical nonlinearity of silicon is 2 order of magnitude higher than that of silica. However, silicon also possesses a high two-photon absorption (TPA) nonlinearity which causes loss at high optical intensities [6]. Therefore, silicon photonics devices would benefit from the incorporation of NL optical materials without TPA but still compatible with the nano-scale waveguide structures. The nano-size NL 1D/2D materials such as carbon nanotube (CNT), graphene, and related 2D materials, like transition metal dichalcogenides (TMDs) (Fig. 1(b)) [7]-[10] are potential candidate for these the integrated-optic system. In particular, the atomically-thin 2D materials are compatible with the layer-by-layer fabrication processes of silicon photonics. Nevertheless, fiber-optic systems remain to be an important technology platform, not only for telecommunications but also for high-power and/or short pulsed lasers and fiber-optic sensing applications, and these NL 1D/2D materials are also highly suitable for application in these fiber/waveguide-optic systems [11]-[15]. 4

5 In Sec.2 of this tutorial, we describe the fundamental physical properties and the linear/nl optical properties of these 1D/2D materials. Since the NL phenomena in 1D/2D materials reported so far are too broad to be covered extensively in this tutorial paper (see Ref.[8] for extensive report on NL optics in 2D materials), we will restrict our scope to focus on the three main NL phenomena, namely (1) saturable absorption (SA), (2) electro-optic effect, and (3) nonlinear Kerr effect; which we believe are the most important NL effects for many photonic applications. In Sec.3, we will discuss short-pulse fiber lasers using the SA effects in 1D/2D materials. The electro-optic effect in graphene, which is a NL effect in a broader sense, and the works on graphene modulators will be reviewed in Sec.4. Finally, in Sec.5, we will discuss all-optical signal processing devices using NL phenomena, such as four-wave-mixing (FWM), self-phase modulation (SPM), and cross-phase modulation (XPM). We would like to briefly outline the current state of the art in this field. Figure 2(a) shows the number of publications by year searched at the Google Scholar for CNT, graphene, and other 2D materials with a keyword saturable absorption, as of the of June Note that the numbers may include papers which do not contain the keyword but cited papers with the keyword. CNT came first after the initial proposal of mode-locked fiber lasers using CNT-SA in 2003 [16]-[18]. Graphene was also demonstrated to have the SA effect applicable for laser mode-locking in 2009 [19][20]. It is rather difficult to determine which paper was the first among many of the other 2D NL materials, but it is believed that that the topological insulator (TI), Bi 2Te 3, demonstrated in 2012, was the first among the other 2D materials to be used as a NL optical materials [21][22]. It is evident that the number of publications on CNT-SA has been increasing but saturating, 5

6 whereas the number of publications on graphene and the other 2D materials are steadily growing, with no sign of slowing down. Figure 2(b) is the number of publications searched in the same way with keywords graphene, electro-optic and modulator, as of June A steady growth in the number of publications since the first demonstration of graphene electro-optic modulator in 2011 [23], very similar to the trend shown in Fig. 2(a). (Note that some part of graphene papers in Fig.2 is for microwave/terahertz waves.) Because of so vast number of publications to date, and the tutorial and review nature of this paper, we have to restrict the number of our references to a selected number of representative papers in the field. 2. Fundamentals of graphene, CNT, and related 2D materials 2.1 Electrical and linear optical properties (a) Graphene Single-layer graphene is the flat monolayer of carbon atoms tightly packed into a 2D honeycomb lattice by sp 2 hybridization (inset of Fig.1(b)). Due to its 2D honeycomb lattice structure, it has been known to possess a unique band structure. In ordinary materials, an electron has an effective mass, a momentum and an energy ; expressed as and /2, respectively, where is the velocity of the electron. As the result, the relation between and (dispersion relation) is /2, resulting in a parabolic band shape. In graphene, it is known that the dispersion relation is not parabolic, but a linear function, expressed as,, (1) 6

7 where /2π is the reduced Planck constant,, is the 2D wave vector around K points in the hexagonal Brillouin zone, and ~10 m/s (~ /300 where is the light speed) is the Fermi velocity [24][25]. Eq.(1) represents that the valence and conduction bands form a pair of cones, and touches at the vertexes (= K point), as shown in Fig. 3(a). The zero-gap linear band structure shown in Fig. 3(a) is referred as Dirac cones, and the K point as Dirac point. The density of states (DOS) of the graphene is calculated to be also linear around the Dirac point, which is identical to the Fermi level in the undoped pristine graphene, as shown in Fig.3(b). Thus graphene has a semi-metallic, or zero-gap semiconducting nature. Eq.(1) also implies that, which is similar to the linear dispersion relation of photon,. Therefore, electrons on the graphene behave like mass-less Dirac fermions, or like photon, at the velocity of, which is 1/300 of the light speed. Due to this linear band structure, graphene exhibits a variety of unique and unusual electric transport phenomena, such as extremely high mobility (μ) of up to 10 6 cm 2 V 1 s 1 owing to ballistic transport of electrons in graphene, minimum DC conductivity of ~4 / ( is the electron charge) even with zero carrier concentration, anomalous quantum Hall effects, etc. [26]. Graphene also has interesting and useful optical properties. Despite being only one atom thick, graphene absorbs a significant fraction of incident light, because of graphene s unique electronic properties. The transmissivity and reflectivity of a freestanding single-layer graphene in case of normal incidence are calculated to be 1 1 ~1 πα~0.977, ~ , (2) 7

8 where σ is the optical conductivity of graphene (σ /4 ~60μS), is the vacuum permittivity, and α is the fine structure constant (α /4 ~1/137) [27][28]. Equation (2) implies that the rest of the transmitted and reflected light, 1 ~πα σ ~ 2.3%, is absorbed by the single-layer graphene. Because of the linear dispersion relation and DOS in graphene (Fig.3(a)(b)), the absorption of 2.3% is wavelength independent (Fig.3(c)) [27]. Also, absorption increases linearly proportional to the number of layers (inset of Fig.3(c)), which enables the estimation of the number of the layer by observing the optical contrast in the sample. There are several graphene derivatives, such as graphene oxide (GO) or reduced GO (rgo), which can be viewed as a graphene with various oxygen containing functionalities. They possess linear dispersion relations, but their bandgap is proportional to the concentration of the oxygen atoms. As the result, they are known to have electrical and optical properties similar but not the same as that of graphene s [29]. (b) CNT CNTs can be classified into multi-wall and single-wall nanotubes (MWNT and SWNT). A SWNT is a rolled tubule of single-layer graphene with end-caps and a tube diameter of typically ~1nm and a tube length of ~1 m, thus it can be considered as a long 1D material (inset of Fig.1(b)). The CNT structure is determined by how the single-layer graphene is folded, and the characteristic is expressed by a single parameter, its chirality, expressed by a chiral vector, with two integers and, that specifies the connecting points in folding of the 2D graphene sheet [30]. 8

9 When a 2D graphene sheet is folded to a 1D CNT, the additional quantization arising from electron confinement around the CNT circumference must be accounted for. The periodic boundary condition of 2, where is an integer, must be satisfied [30]. As a result, the electronic band structure of a specific CNT is given by the superposition of the cuts of the graphene electronic energy bands along the corresponding allowed lines (cutting lines). When one of these cuts contains the Dirac (K) point, the CNT is metallic (Metallic CNT: m-cnt, Fig.4(a)). Otherwise, it will become semiconducting and possesses a bandgap (Semiconducting CNT: s-cnt, Fig.4(b)). It is easy to show that 3 ( : integer) for m- CNTs, and 3 for the s-cnts. Figures 4(c) and (d) show examples of the DOS of (9,0) m-cnt and (10,0) s-cnt [31]. Many sharp peaks appear (c 1, c 2, at the conduction band, and v 1,v 2, at the valence band) corresponding to the cuts that originate from the additional level of quantization and are known as Van Hove singularities. It should be noted that the DOS near Femi-level in the m-cnt is constant, in contrast to linear relation in graphene in Fig.3(b). This is because of the 1D structure of the m-cnt, which makes it purely metallic rather than semi-metallic. In the case of s-cnt, optical transitions can occur between the bandgaps v 1 - c 1, v 2 - c 2,, labeled as,,, and the first bandgap energy (in ev) is inversely proportional to the tube diameter (in nm), which can be expressed as: ev, (3) where (=0.144nm) is the C-C distance, is the transfer integral between first-neighbor orbitals (~2.9eV) [30]. Note that this physical picture of the CNT is called a single-particle model, where only the 9

10 electrons are under consideration. In 1D CNTs, however, we must consider the bound state of the electron and hole pair (e-h), that is, the exciton [32]. Because of the binding energy of the exciton, the optical bandgap is smaller than that derived from the single-particle model. Nevertheless, the excitonic optical bandgap remains inversely proportional to the tube diameter. The s-cnt has a peak absorption wavelength corresponding to its optical bandgap. Figure 4(e) shows a typical transmission spectrum of a CNT sample [18]. The S1 and S2 peaks correspond to the absorption between the bandgap transitions v 1 - c 1 and v 2 - c 2. respectively. The peak absorption wavelength (in m) is proportional to the tube diameter (in nm), which can be derived from Equation (3), ignoring the excitonic effect, as: μm. (4) The tube diameter of a typical single-walled CNT is in the range of nm, corresponding to a peak absorption wavelength of 1 2 m. The absorption wavelength of a CNT sample can be engineered by choosing a proper tube diameter, however, in practice, selective growth of one type of s-cnt with a single chirality is proven extremely difficult. In most CNT fabrication processes, the sample obtained is a mixture of several kinds of s-cnts and m-cnts with a variety of tube diameters. The chirality distribution is determined by the fabrication method. Thus, the absorption peak wavelength of the fabricated CNT sample is determined by the mean tube diameter, whilst the absorption spectral bandwidth is dependent on the tube diameter distribution. Note that optical absorption in a single CNT is anisotropic, that is, a CNT absorbs light with optical polarization parallel to the axial direction of the tube. Therefore, an aligned CNT sample 10

11 will exhibit a polarization dependency [33]. Most CNT samples have a randomly oriented tube bundles and therefore are polarization independent. (c) Other 2D materials There are many other 2D materials, besides the graphene family. Topological insulators (TIs), such as Bi 2Te 3 or Bi 2Se 3, are 3D crystals with bandgaps and their surface states possess graphene-like linear dispersion relations (Dirac cones) [29][34], as shown in Fig.5(a). Thus, they have electronic and photonic properties similar to that of graphene, such as high electron mobility and broadband optical absorption [35]. Transition metal dichalcogenides (TMDs) are the most studied 2D layered materials. TMDs are semiconductor materials with the formula of MX 2, whereas M refers to a transition metal element, such as Mo, W, etc., and X refers to a chalcogen element, such as S, Se, or Te. The structure of a TMD is shown in Fig.1(b). The properties of TMD can be altered from those in bulk crystal form to those in single-atomiclayer. For example, a bulk crystal of MoS 2 has an indirect band gap of 1.29 ev, while a single-layer MoS 2 has a direct band gap of 1.8 ev [29][36], as shown in Fig.5(b). It is confirmed that the single-layer MoS 2 has an absorption peaks wavelength around 670 and 627 nm, corresponding to the direct bandgap [36]. Black phosphorus (BP) is a layered semiconductor material consisting of phosphorus atoms [37]. Its single- and few-layer structure, phosphorene, has its own unique optical properties. For example, its direct bandgap is dependent on the number of layer and can be tailored from 0.3 to 2eV (corresponding to the wavelength range from 4 to 0.6μm) [38]. This has properties similar to that of a s-cnt, and is unique among 11

12 2D semiconductor materials. Phosphorene is expected to bridge the gap between the zero-bandgap graphene and the relatively large bandgap TMDs [9]. Other 2D semiconducting materials, such as the hexagonal boron nitride (h-bn), InSe, GaSe, etc., are summarized in Ref.[9] with their respective bandgap regions. In this tutorial, we mainly focus on 2D materials with similar nonlinear optical properties (SA, electro-optic effect, and nonlinear Kerr effect) as CNT and graphene, that is, TIs, TMDs and BP. 2.2 Nonlinear optical properties In general, assuming an instantaneous dielectric response in an isotropic material, the relation between an induced polarization (, scalar) and an electric field (, scalar) is expressed by..., (5) where is the permittivity of vacuum, is the linear susceptibility, and and are the second- and third-order nonlinear susceptibilities, respectively [1][6]. The second-order nonlinearity is responsible for second harmonic generation (SHG), and sum- and difference-frequency generations (SFG and DFG). It is nonzero only for the material that lacks an inversion symmetry at the molecular level [1]. Since the honeycomb carbon structure has the inversion symmetry, both CNT and graphene do not possess the second-order nonlinearity, unless the symmetry is disturbed. In contrast, some 2D semiconductors are known to have a large second-order nonlinearity [8]. is also related to the electro-optic effect, such as the Pockels effect [1]. The third-order nonlinearity is responsible for third harmonic generation (THG), nonlinear refractive index change (nonlinear Kerr effect) and nonlinear absorption change (saturable absorption and 12

13 multi-photon absorption). The nonlinearity has been shown to be very large in CNT, graphene, and other 2D materials [8][11]. The changes of refractive index and optical absorption are dependent on the incident optical intensity, and the complex refractive index can be expressed as [1][6], 4, (6) where is the optical intensity, is the nonlinear refractive index (Kerr coefficient), and is the nonlinear absorption coefficient. Both and are interrelated with the real and complex part of as, Re, 3 2 Im. (7) It should be noted that the changes in the optical absorption will have a strong effect on the refractive index for wavelengths near the absorption edge, due to the Kramers-Kronig relation [1]. (a) Saturable absorption Saturable absorption (SA) is a phenomenon related to the imaginary part of (3) as expressed in Eq.(7), where high intensity light bleaches the material, and reduces the optical absorption, which can be expressed as, 1 /, (8) where is the linear absorption coefficient, and is the saturation intensity [1]. In the linear regime, where the incident optical intensity is relatively weak, the SA absorbs the incident light, resulting in the attenuation of the light. When the incident optical intensity is high, the lower energy states are depleted whereas the upper energy states are filled, thus saturation of absorption occurs, resulting in a decreasing 13

14 attenuation. Note that Eq.(8) can be approximated to ~ / for small value of. The black curve in Fig. 6 shows an example of the SA property. Here we define the normalized intensity as /, the normalized absorption as 1 / 1 since the non-absorbed light after transmitted through a sample of thickness is, and 0.1. The saturation intensity is given by the absorption cross section and the recovery time as:, (9) where is the optical angular frequency, and is the saturation fluence [39]. Saturable absorption is a universal phenomenon in any material that exhibits optical absorption due to the electronic transition between two energy levels. However, it is uncommon to find a saturable absorber with a fast recovery time suitable for generating ultrashort pulses at time scales of a few picoseconds to a few hundred femtoseconds. Both CNT and graphene have inherently fast saturable absorption responses [11]-[15]. The optical absorption of CNT and graphene is depicted in Fig.7(a)(c). Saturation takes place when all the allowed states in the conduction band are fully populated and the valence band emptied due at high optical intensities (Fig.7(b)(d)). The SA effect of a CNT has a recovery time consisting of a fast- and a slow-component, as shown in Fig.8(a) [41]. The typical recovery time of the transition in bundled CNT samples is in the order of 1ps, and that of the transition is even faster, in the order of 0.1ps, as shown in Fig.8(a) [40]. It is known that isolated CNTs have a slower recovery time of ~30ps (Fig.8(b)) [41] and are luminescent, whereas bundled CNTs are not. The prevalent theory [42] suggests that the fast recovery is due to the transition of 14

15 the free carriers from s-cnt into m-cnts (tube-tube interaction) in bundled CNT sample, which also explains the luminescence only in isolated CNTs, while the slow recovery time is due to the inter-band transition (Fig.8(b)). Figure 9(a) shows the saturation characteristics of a bundled CNT sample in thin-film form [18]. The 10% saturation fluence is measured to be 3 J/cm 2 at the wavelength of 1.55 m, which corresponds to an estimated saturation intensity of 12.5 MW/cm 2, a value very similar to that of a typical semiconductor-based SA, SESAMs [39]. At other wavelength regions, similar saturation intensities around 10 MW/cm 2 have been reported, as long as the peak matches with the operating wavelength [43]. It is also possible to use the transition for saturable absorption, at which a much higher saturation intensity of ~ 200 MW/cm 2 has been reported [44]. This is a reasonable value since is inversely proportional to as in shown in Eq.(9). The recovery time of graphene is even faster, in the range between ps, which is attributed to the fast intraband transition associated with carrier thermalization. The slow component of its recovery time of a few picoseconds range is due to the slow inter-band transition [45]. Figure 9(b) shows an example of the saturation characteristics of a single-layer graphene sample at an operating wavelength of 1.55 m [46]. The estimated saturation intensity I S of ~ 250 MW/cm 2 is an order of magnitude higher than that of the CNT. The saturation intensities reported so far in various publications are not consistent; varying from ~ 0.7 MW/cm 2 [19][47]-[49] to MW/cm 2 [20][46][50]. This might be explained by the difference of the inelastic collision time of electrons in the graphene samples (Fig.10) [51]. 15

16 As shown in Fig.3(c), graphene has a wavelength-independent linear optical absorption, however, its nonlinear SA is not wavelength-independent. The saturation intensity becomes lower at longer wavelengths, as predicted theoretically in several literatures [51][52]. Figure 10 shows the predicted interband saturation intensity as a function of the wavelength and the inelastic collision time [51]. This prediction seems to be reasonable since it is harder for the absorption of graphene to be saturated at higher photon energies as the result of its linear band structure, as shown in Fig.7(d). Indeed, higher saturation intensities have been reported at shorter wavelengths at 800nm [53]. Therefore, at longer wavelengths, it is more favourable to use graphene SA for laser mode-locking. Graphene can work as saturable absorbers at Terahertz or microwave frequencies. At 100GHz, the saturation intensity is reported to be very low ~0.04 mw/cm 2 [54]. It has been reported that GO and rgo have a fast ( ps) saturable absorption [55] and a low saturation intensity of around 1 MW/cm 2 [56]-[58]. On the other hand, TI materials such as Bi 2Te 3 is reported to possess saturable absorption with both a fast (~0.5ps) and a slow (~10ps) recovery time [59] and a higher saturation intensity of >100 MW/cm 2 [22][60]. The saturation intensity has a similar wavelength dependency to that of graphene ( becomes lower at longer ) [60]. There are an increasing number of papers recently on saturable absorption in TMD materials, such as MoS 2 [61]-[63], MoSe 2 [64], WS 2 [65][66], and BPs [67]- [71], as shown in Fig.2(a). The reported saturation intensity for TMD materials are rather diverse, from a few to several hundreds of MW/cm 2 [61]-[66]. A fast recovery time of 0.15ps is reported for TMDs [64]. 16

17 BPs are reported to have a relatively lower saturation intensity of a few MW/cm 2 [67]-[71] and a very fast recovery time of 24fs [71]. It should be noted that these materials may have the opposite effect to saturable absorption, namely reverse saturable absorption (RSA) or optical limiting. The main cause of RSA is multiphoton absorption, such as two-photon absorption (TPA) [72]. SA and RSA can coexist; in this case, Eq.(8) becomes 1 /, (10) where is the TPA coefficient. The red curve in Fig. 6 shows the absorption property with only RSA, whilst the blue curve is that when both SA and RSA coexist, assuming 10. It is reported that RSA is dominant in GO whereas SA dominates when GO is reduced to rgo [73]. (b) Electro-optic effects Due to the inversion symmetry of graphene lattice system, it lacks the second-order nonlinearity, whereas it possesses an electro-optic effect on its optical absorption, namely the electro-absorption effects. As discussed in Sec.2.1(a), the absorption of graphene can be written as πα σ ~2.3%, and is wavelength independent because of the linear dispersion relation. This is valid for an undoped intrinsic graphene whose Fermi level is at the Dirac point ( 0), and the lower cone (valance band) is filled with electrons whilst the upper cone (conduction band) is empty, as shown in Fig.11(a). In contrast, p- or n-doped graphene (chemically or electrically doped) has a Fermi level shifted from the Dirac point to the lower or upper cone region. In this case, the absorption is no longer wavelength independent due to Pauli blocking, as shown in Fig.11(b). The optical conductivity σ can be written as [74][75]: 17

18 σ σ 2 tanh 2 4 tanh 2 4 σ 2 log σ, (11) where is the Boltzmann constant, is the temperature, and is the intraband scattering time constant. The first and second terms in Eq.(11) correspond to the real and the imaginary parts of the interband transition, and the third term to intraband transition. The real part of the conductivity corresponds to the absorption, whilst the imaginary part corresponds to the change in refractive index. Figure 12 shows the calculated real and imaginary parts of the optical conductivity σ, for 1.55 m ( ~0.8eV) and 300K. It is found that the absorption vanishes at 2, around which the imaginary part exhibits a large deviation, resulting in a phase change of the incident light. Thus, the absorption of graphene can be switched from a high value to virtually zero, theoretically, by shifting the Fermi level more than half of the input photon energy. The most straight forward way to shift the Fermi level is through a capacitor-like structure, as presented in Fig.13. By applying a voltage to a capacitor, where at least one electrode is graphene, graphene is effectively doped with electrons (or holes, depending on the polarity of voltage), thus enabling the shift of the Fermi level. The graphene's Fermi level dependency on voltage can be derived as [76],, (12) where is the thickness of the planar capacitor, is the dielectric constant of the oxide spacer, is the applied voltage, and is the voltage that corresponds to initial doping of graphene. With this equation, we 18

19 can observe that the graphene's conductivity is dependent on the applied voltage of the capacitor-based device, therefore graphene's absorption can be switched or modulated by applying a voltage. (c) Nonlinear Kerr effects It is also known that the real part of in Eq.(7), which is responsible for the nonlinear refractive index change or the nonlinear Kerr effect, can be rather large in CNTs and graphene. The main cause of the nonlinear effects in nano-carbon materials is the nonlinear polarization of π electrons in the carbon honeycomb network, which may be enhanced by excitons in a 1D/2D structure and also by the change in absorption through the Kramers-Kronig relation. In CNTs, it has been predicted theoretically that Re under the resonant condition can be as high as ~ esu, which corresponds to a nonlinear refractive index of ~2 10 m 2 /W [77], 8 orders of magnitudes higher than that of silica ( ~3 10 m 2 /W). This has been experimentally confirmed (Re ~ esu) [78]. Note that in the esu unit can be converted to the SI unit as [1], m V esu esu, (13) and can be obtained using Eq.(7). In graphene, higher measured values of ~10 m 2 /W [79] -[81] have been reported, which is consistent with the theoretical prediction [82]. It has also been reported that GO and rgo have similar values of, of ~10 m 2 /W [83][84]. Such high values are attractive for nonlinear photonic applications, such as all-optical switching and wavelength conversion, which will be discussed later. 19

20 There are also many reports on measurement of in other 2D materials, TIs [85][86], TMDs [87]-[89] and BP [90][91]. TIs was reported to have a high of ~10 m 2 /W [85][86], whereas the reported values for TMDs and BP are somewhat smaller, 10 m 2 /W [87]-[91]. Nevertheless, these materials are still attractive candidates for nonlinear photonic applications. 2.3 Production The first single-wall CNTs were synthesized using the arc-discharge method [92], followed by the laser-ablation method [93]. Nowadays, most of the commercially available single-wall CNTs are produced in a large scale by various chemical vapor deposition (CVD) methods, such as the HiPCO (High pressure carbon monooxide) [94] and the CoMoCAT (Cobalt-Molybdenum catalyst) [95] methods from carbon monoxide, and the ACCVD (Alcohol catalytic CVD) method from alcohol [96]. Most commercially available CNTs are supplied in powder form or in dispersed solution. They commonly have tube diameters of ~1 nm and tube length from a few hundred nm to a few m. Normally, a typical CNT sample contains a mixture of s-cnts and m-cnts with various chiralities and tube lengths. To date, there are not yet any fabrication method capable of selective production of a single type of s-cnts or m-cnts, although selecting and sorting of CNTs are somehow possible as a post-process [97]. Single-wall CNTs with an aligned orientation can be fabricated with the alcohol catalytic CVD method [98]. Graphene was first produced using mechanical exfoliation (or micromechanical cleavage) of graphite with an adhesive tape, often called the Scotch-tape method. With this technique, small pieces of few-layer or even single-layer graphene is easily obtained with small number of defects [25]. Another 20

21 mechanical exfoliation method is the liquid-phase exfoliation (LPE) method, in which graphite flakes are dispersed in a solvent, followed by ultra-sonification and centrifugation to obtain a dispersed solution with small flakes of few-layer and single-layer graphene [29][99]. Mass production of large-area graphene, for applications in transparent electrodes of solar cells and displays, relies mostly on CVD methods [100][101]. A record large 30-inch continuous graphene film has been reported [101]. CVD-grown graphene films are now readily available from several commercial sources. As for the other 2D materials, mechanical exfoliation and LPE from bulk materials are often used as is the case of graphene [29][102][103]. CVD growth of TMDs has also been developed for mass production [102]. 3. Short pulse fiber lasers using saturable absorption 3.1 Pulse generation with saturable absorber By including a proper saturable absorber (SA) in a laser cavity, ultra-short optical pulse train can be generated through passive mode-locking, as illustrated in Fig.14 [104]-[106]. The laser action starts from the spontaneous emission noise of the gain medium of the laser. When a SA is placed in the laser cavity, the spike components of the spontaneous emission noise tends to survive through the SA, these components are then amplified by the gain medium and reshaped by the SA. This process is iterated repeatedly to form a stable pulse train so that a single (fundamental mode-locking) or an integer number of pulses (harmonic mode locking) circulating in the laser cavity. In the frequency domain, pulse formation is achieved when the 21

22 optical phase of each longitudinal cavity resonance mode stays constant or locked to each other, hence it is also called mode locking. In fundamental mode locking, which is crucial for a stable operation, the repetition frequency is equals to the free spectral range (FSR) of the laser cavity, which can be expressed as:, (14) where is the laser cavity length, is the effective refractive index of the laser cavity. Note that we assume a ring cavity here, and shall be replaced with 2 in the case of a laser with a Fabry-Perot (linear cavity). The typical repetition rate is for a fiber ring laser is a few 10 s of MHz. The pulse width is determined by the number of longitudinal modes contributing to the pulse formation, written as: ~ 1 1, (15) where is the lasing spectral width [104]. The value of is called the time-bandwidth product (TBP), and it is known that =0.315 for ideal sech 2 -shaped soliton pulses, and =0.441 for ideal Gaussian-shaped pulses. The TBP value becomes larger when the phase or frequency of the optical carrier is changed in the pulse (chirping). Passively mode-locked lasers can generate ultra-short pulses in the picosecond or femtosecond regimes, and even attosecond pulses are possible with an extremely broadband gain media. It should be noted that the recovery time of the SAs do not have to be as fast as the pulse-width itself. It has been shown that the recovery time can be more than an order of magnitude longer than that of the laser pulse-width [107][108]. Another pulse formation mechanism, Q-switching, can also be achieved using SAs [104]. Passively Q-switched lasers can generate very high pulse energies, although the pulse widths are much broader (ns to 22

23 μs) and the repetition rate is much slower (1-100kHz) than that of the passively mode-locked lasers. Sometimes both mechanisms can occur at the same time (Q-switched mode locking) depending on the cavity condition and the saturation depth of the SA [105]. Q-switching is sometimes an obstacle for applications when a stable and pure mode-locked pulses (continuous-wave, CW mode-locking) are desired. It is known that the SA should have slower recovery time for stable CW mode locking, because the longer time constant results in a reduced saturation intensity for a part of the absorption, which facilitates self-starting of the mode locking process and enhances the stability [105]. As discussed in the previous section, CNT and graphene have both a fast and a slow saturable absorption component, which is ideal for CW mode locking. 3.2 SAs using CNT, graphene, and related 2D materials Graphene, CNT and related 2D materials need to be incorporated in an optical fiber or waveguide systems so that they can interact with light efficiently. One way is to form a thin film using these nonlinear materials (or in a polymer composite) and launching light directly through the film. Another way is to place the thin film along the core of optical waveguides or fibers so that the light interacts with these nonlinear materials via the evanescent field coupling, as shown in Fig.15. The film does not necessarily have to be formed by aligned CNTs or continuous 2D sheets but can be of randomly bundled CNTs or stacks of 2D flakes. Several kinds of fiber-type devices have been demonstrated for fiber laser applications. The simplest and the most common fiber-type device is realized by sandwiching a thin layer of the nonlinear materials between two optical fiber connectors [18], as show in Fig.16(a). The film can either be pristine CNT/2D 23

24 material film formed by, for example, direct CVD synthesis [109], the spray method [110], optical deposition [111]-[113], or polymer composite [114][115]. These devices are very easy to fabricate; however they are prone to optical damage when used in high power lasers. We demonstrated the enhancement of optical damage threshold by sealing the device with nitrogen gas [116]. For high power lasers or nonlinear device applications, the evanescent-coupling devices as shown in Fig.15 are preferred. The nonlinear materials interact with the evanescent field which has a much lower power whilst the interaction length is extended profoundly to achieve a large amount of nonlinear effects. Figures 16(b) and (c) are two representative examples. Figure 16(b) is the tapered fiber device with a tapered waist of a few m, surrounded by the CNT/2D material film for evanescent field coupling [117]. Again, the film can either be pristine CNT/2D material or in the form of a polymer composite. We have demonstrated optical deposition of CNTs around tapered fibers to form an evanescent-field-coupled nonlinear device [118]. Figure 16(c) shows a side-polished (or D-shaped) fiber device, with a polished flat surface separated a few m from the fiber core, and the CNT/2D material film is formed on the flat surface for evanescent field coupling [119][120]. 3.3 Mode-locked fiber lasers using CNT, graphene, and related 2D materials Figure 17 shows the typical fiber laser configurations we have used for mode-locking with CNT- or G-SA. Figure 17(a) is the fiber Fabry-Perot (FP) laser configuration, where the CNT/2D thin film is formed on the output mirror, such as a fiber ferrule mirror. The rare-earth-doped gain fiber can be pumped from outside of the cavity as shown in Figure 17. The gain fiber can also be pumped in the cavity through a 24

25 wavelength-division-multiplexing (WDM) fiber coupler. This configuration allows the realization of a short cavity length. Figure 17(b) is the most popular fiber ring laser configuration and the SA is placed inside the ring cavity. Any type of CNT/2D-SA can in principle be used in this configuration and achieve laser modelocking with ease with an appropriate design of the cavity fiber dispersion and nonlinearity. CNT/2D-SA can also be used in combination with another mode-locking mechanism such as the fiber-type artificial SA using nonlinear polarization rotation (NPR) or the nonlinear optical loop mirror (NOLM). In a mode-locked fiber laser, the fiber dispersion and nonlinearity play a greater role than the SA. The most common mode-locked fiber lasers are based on the soliton mode locking mechanism, in which soliton pulse-shaping is realized through the fine balance between the anomalous chromatic dispersion and the Kerr nonlinearity in the fiber (Fig. 18(a)) [106]. This process, by itself, is capable of producing transformlimited, chirp-free pulses with pulse durations much shorter than the recovery time of the SA [107][108]. Nevertheless, the SA is needed to initiate or self-start the mode locking process and to stabilize it by preventing the buildup of noise in the intervals between the pulses. However, for a given laser cavity design, single-pulse fundamental mode-locking is only stable for up to a specific power level, beyond that with a higher pump power would lead to multiple solitons in the cavity. In a soliton laser design, there is a limit to the highest achievable pulse energy, which is typically less than 100pJ. The red curves in Fig. 19(a) are the spectrum and the SHG autocorrelation trace of the soliton pulses generated from a simple Er-doped fiber ring laser at 1.5μm (Fig. 17(b)) using a tapered fiber coated with CNT-polymer composite (Fig. 16(b)) [121]. The red output spectrum in Fig. 19(a1) has several side peaks called Kelly sidebands caused by the periodic 25

26 perturbation of the pulses in the cavity coupled with the dispersive wave, which is characteristic in most soliton mode locked fiber lasers [122]. Here the spectral width is estimated to be ~7nm ( ~930GHz), pulse width ~490fs, giving a TBP of ~0.45, which is larger than the ideal transform-limited value of indicating that the pulses are chirped. The repetition rate is ~7MHz, and the pulse energy is limited to ~40pJ. By enhancing the perturbation of pulses in the cavity using of segments of normal and anomalous dispersion, higher pulse energy up to ~1nJ has been realized. This is called stretched pulse (SP) mode locking, or dispersion managed (DM) mode locking. In the stretched pulse mode locking regime, the pulse changes its temporal shape and spectra inside the cavity as shown in Fig. 18(b). The blue curves in Fig. 19(a) are the spectrum and the SHG autocorrelation trace of the stretched pulses generated from the same Er-doped fiber ring laser used for previous soliton mode locking except that the net cavity dispersion is adjusted to a much smaller value [121]. Here the spectral width is broadened further to ~ 14nm and the Kelly sidebands disappear. The pulse width is reduced to < 200fs, and the pulse energy is enhanced to ~100pJ. By further decreasing the anomalous dispersion in the cavity, the laser enters another operating regime called dissipative soliton (DS) mode locking [123]. In the DS mode locked laser cavity, the pulse is constantly broadened in the normal dispersion fiber and reshaped by the SA and the spectral filter, as shown in Fig.18(c). The output pulse has a high pulse energy of over 1nJ and is linearly chirped, which can be easily dechirped after the laser output. Fig. 19(b) shows the spectrum and the SHG autocorrelation trace of an allnormal-dispersion Yb-doped fiber ring laser at 1μm wavelength mode locked using a tapered fiber coated 26

27 with CNT-polymer composite [124]. High energy chirped pulses with a pulse width of 1.5 ps, and a pulse energy of 3 nj are generated. The output spectrum has a spectral bandwidth of ~15nm and a square flat-top shape, which is the characteristic of a DS mode locked fiber laser. The output pulses is eventually dechirped with pulse width compressed down to a 235fs. Typical mode-locked fiber lasers operate in the fundamental mode at a repetition rate of a few MHz to a few tens of MHz. This is determined by the laser cavity length of a few tens to a few hundreds of meters, as described by Eq.(14). High-repetition-rate mode-locked lasers with repetition rates of a few hundred MHz to a few tens of GHz are useful for many applications, such as optical communications, microscopy and metrology [125]. To realize fundamental mode locking, for example, at 10GHz, the fiber ring laser cavity length must be reduced to ~ 2cm, which is not practical for a ring laser configuration. The fiber FP laser configuration shown in Fig.17(a) makes such high repetition rate possible in conjunction with a highlydoped gain fiber and a compact CNT/2D-SA. Figure 19(c) shows the spectrum and the SHG autocorrelation trace of a 1cm-long, 10GHz mode-locked FP laser using a CNT-SA [126]. The CNT thin film is formed onto the fiber ferrule s highly-reflective (HR) mirror (R~99%) and butt-coupled to a 1-cm-length high-gain Er:Yb co-doped fiber placed in a ferrule as shown in the inset of Fig.19(c2). The gain fiber is pumped through another fiber ferrule HR mirror on the opposite side. Stable and transform-limited mode-locked pulses with a pulse width of ~1ps is achieved. We have also realized a 1cm-long, 10GHz mode-locked FP laser using a graphene SA [127]. However, the performances, such as the mode-locking threshold and stability, are not as 27

28 good as the one using a CNT-SA. This is possibly due to the higher saturation intensity of the graphene at 1.5 m, as shown in Fig.10. The most frequently used gain media for fiber lasers have been Er-doped fiber at 1.5 m (e.g. Fig.19(a)(c)) and Yb-doped fiber at 1 m (e.g. Fig.19(b)). Recently, there are growing interests in fiber laser sources at the mid-infrared (mid-ir) wavelength region from 2-15 m for various applications such as gas spectroscopy, remote sensing, materials processing, medical surgery, etc. [128]. Tm- and Ho-doped silica fibers are commonly used for the wavelength region around 2 m, rare-earth elements such as Tm, Ho, Er, Pr, Dy can also provide gain for wavelength regions beyond 2 m in different host glasses [128]. It should be noted that the ordinary silica-glass fibers are usable only up to around 2 m wavelength because they are not transparent beyond 2 m. Therefore, fiber lasers beyond 2 m have to be composed of the fibers made from non-silica host materials, for example chalcogenide and fluoride glass such as ZBLAN (ZrF 4-BaF 2- LaF 3-AlF 3-NaF) glass [128]. There have been many reports on 2 m fiber lasers mode locked by CNT [129][130] and graphene [131][132]. Figure 20(a) shows an example of the output spectra from a Tm-doped mode-locked fiber ring laser using a CNT-polymer composite sandwiched between fiber connectors [130]. We have demonstrated using the same laser cavity 3 different mode-locking regimes from soliton, stretched pulse, to dissipative soliton, by changing the intra cavity dispersion from net anomalous to net normal dispersion. The resulting pulse widths after pulse compression is around 1ps. We have also achieved similar results in a Tm-doped mode-locked fiber ring laser using a graphene-covered tapered fiber [132]. 28

29 In contrast, there are only a limited number of reports on mode locked fiber lasers beyond 2 m using CNTs and graphene, possibly because of the limited availability of optical components at longer mid- IR wavelengths and the difficulties in the handling of fluoride glass and the management of the intracavity dispersion. Figure 20(b) shows the output spectra from an Er-doped fiber ring laser at 2.8 m using a graphene SA [133]. They reported a low saturation intensity of graphene at 2.8 m, ~ 2MW/cm 2, which coincides with our discussion on the wavelength dependence of the saturation intensity of graphene at Sec.2.2(a) and Fig. 10. An interesting mode of operation of the fiber ring laser using CNT/graphene-SA is the bidirectional mode-locking operation [134][135]. It has been known that continuous-wave bidirectional ring lasers are unstable due to the gain competition between the clockwise (CW) and the counter-clockwise (CCW) lasing modes. The same was believed to be true for bidirectional mode-locked ring lasers, but it turns out that it was not the case [134]. More importantly, in bi-directional mode-locked lasers, the repetition rates of the CW and CCW pulses are slightly different due to the anisotropies in the fiber cavity birefringence and the nonlinearity between the CW and the CCW modes. Differences of repetition rates of 2kHz at 1.5 m [134] and 0.6kHz at 2 m [135] have been demonstrated. This stable mode-locking with slightly different repetition rates is ideal for applications such as dual-comb spectroscopy, which will be discussed in the next section. There are a number of publications recently on mode-locked fiber lasers using other 2D-SAs, for example, GO/rGO [57][58], MoS 2 [61][62], WS 2 [65][66], and BP [68][70]. Most of them reported soliton mode-locked fiber lasers at 1.5 m, in which the role of the SA is not significant. There is an interesting study 29

30 on short pulse generation from a visible fiber laser at 635nm, although the operation is still limited to passive Q-switching [136]. 3.4 Applications Mode-locked fiber lasers using CNT-SA have been applied as a compact and robust short-pulse source for a non-synchronous optical sampling scope [137] and a label-free multi-photon microscopy [138]. An important application of mode-locked fiber lasers is the generation of coherent broadband spectra (frequency combs) through supercontinuum (SC) processes in highly nonlinear fibers (HNLFs). Fiber-based frequency combs offer many applications in metrology, spectroscopy, microscopy, imaging, etc. [139]. Especially, octave-spanning SC source having and 2 components in its spectrum is essential for the realization of a true frequency comb without the carrier envelope offset frequency [140], which is highly desirable for absolute frequency measurement in metrology applications. A typical fiber-based SC source is depicted in Fig.21(a). Short pulses, preferably fs pulses, are amplified by fiber amplifier(s), and then launched into a length of HNLF to broaden the spectrum through the nonlinearity in the HNLF (mainly SPM). Note that the amplification of fs pulses is not a trivial process, requiring a careful design, for example, by using large-core gain fibers and/or chirped pulse amplification (CPA) to avoid unnecessary pulse distortions and chirping due to fiber nonlinearity [139]. All-fiber SC sources have been realized using a CNT-SA [141][142]. Octave spanning SC spectrum from 1μm to >2μm has been generated using a CNT-based mode-locked fiber laser [141]. The laser generates 220fs, 46MHz pulses with an average power of 2mW and the pulses are amplified and compressed to 65fs, before they are 30

31 launched to a 79cm-long HNLF. The SC spectrum spans from 1μm to more than 2 μm at the maximum, as shown in Fig.21(b). Such broadband SC have been used, not only for laser stabilization using the -and-2 components, but also for generating fs pulses at the 1μm wavelength region [143], mid-ir frequency comb through difference frequency generation [144], synchronized ps pulses at 1μm and 1.5μm for stimulated Raman scattering (SRS) microscopy [145]. Figure 21(c) depicts the SC spectrum generated by the aforementioned 1cm-long 10GHz modelocked FP laser using a CNT-SA [126]. The 1ps pulses are amplified to an average power of ~300mW, then input to a 30m-long HNLF. We could obtain a flat SC spectrum spanning from 1.4μm to >1.7μm. The mode structures in the SC spectrum is visible in the inset of Fig. 21(c), showing that it is a coherent SC frequency comb. There is a strong demand for such a sparse frequency comb source especially in the metrology field [125]. Recently, dual-comb spectroscopy (DCS) is attracting a fair amount of attention as a rapid spectroscopic tool [146][147]. By employing two mutually coherent frequency combs having a slightly different comb spacings, it allows the spectral response of a sample to be measured in the radio frequency (RF) domain, rather than measuring in the optical domain using a spectrometer. Bidirectional mode-locked fiber lasers mentioned in the previous section are ideal for DCS, since CW and CCW pulses are naturally synchronized as they share the same cavity for the cancellation of common-mode drift whilst maintaining slightly different repetition rates (= comb spacings). There are reports on real-time DCS using a free-running 31

32 bidirectional mode-locked fiber laser with a CNT-SA [148], and also on octave-spanning DCS using a similar bidirectional fiber laser followed by SC generation [149]. 4. Graphene electro-optic modulators There are several high-speed electro-optic modulator technologies already in use for optical fiber communication systems for decades, such as the LN-based and the semiconductor-based modulators [3]. However, they do not fit well in the integrated-optic system as we have discussed in Sec.1. There is strong demand for high-speed and low-energy-consumption modulators which can be integrated into the silicon photonic circuits, and the graphene-based modulator is one of the options. The most straightforward realization of a graphene electro-optic modulator is the direct use of the capacitor structure as shown in Fig.13. It has been realized as a normal-incidence graphene electro-optic modulator [76], in which a modulation bandwidth of 154MHz has been achieved by optimizing the structure, as shown in Fig.22. However, the modulation depth is limited, to a few % in this case, since the single layer graphene has merely 2.3% of absorption for normal incidence. This low modulation depth is still useful for the stabilization of mode-locked fiber lasers and the consequential frequency combs [150][151]. Recently, wideband active Q-switching and active mode locking of fiber lasers have been demonstrated with a similar graphene electro-optic modulator [152][153]. Nevertheless, a higher modulation depth is required for applications such as data communications. One way to achieve a higher modulation depth is to deploy graphene in the evanescent coupling device configuration as shown in Fig

33 Figure 23(a) is the first demonstrated graphene-based modulator on a silicon waveguide [23]. In this realization, the graphene layer is placed on the top of the silicon waveguide via a thin aluminum oxide layer serving as a dielectric spacer, forming a graphene/dielectric/silicon capacitor structure for electrical doping. The following report from the same group introduced the two-graphene-layer capacitor as shown in Figure 23(b) [154]. In this case, instead of using doped silicon as one of the electrodes of the capacitor, another layer of graphene is utilized, which. This structure helps to increase the absorption of the graphene device by a factor of ~2 and is very attractive because of the small footprint (~25μm 2 ), low drive voltage (~5V), and a broad operation wavelength. However, the performance of these modulators is limited by their operating speed of around 1 GHz, as shown in Figure 23(c). Since the introduction of these graphene modulators in 2011, there has been a lot of researches on graphene-based modulators on silicon waveguides, as show in Fig.2(b), and their operating speed has been improved significantly. For example, 10Gb/s operation has been reported in 2016 [155], and the highest speed reported so far reaches ~30 Gb/s [156][157], although further improvement is needed due to their complex structure and high drive voltages. Even with these advances, we have not yet reached the limit of graphene which has a potential to operate at >100GHz. For this technology to be truly competitive with an all-silicon-photonic structures, the operating speed should cross the 100 Gb/s boundary. In addition, it is also important to operate at such speeds whilst maintaining a low energy consumption, in the order of 10fJ/bit. The main limiting factor in the operating speed and energy consumption of graphene-based modulators is the response of the device, where is the sum of the sheet resistance and the contact resistance, and 33

34 is the capacitance of the capacitor structure for electrical doping. The -limited bandwidth can be express as: 1 2. (16) One way to decrease the constant is to simply increase the capacitor dielectric spacing, since the capacitance of the structure is given by:, (17) where and are the width and the length of the capacitor, respectively. However, a larger dielectric spacing would increase the required drive voltage (Eq.(12)) and thus a higher energy consumption. An alternative way is to decrease the width and the length, although decreasing has a limited influence on the constant, it will affect the capacitance strongly, as will be explained in more detail later. The waveguide structure needs to be optimized in order to maximize the absorption of the graphene layer and to reduce the length. We have explored the absorption dependency of the graphene-covered silicon waveguide design, by varying the thickness and width ( in this case) of the waveguide structure, using a modified 2D finite difference method (FDM) [158]. Figure 24(a) shows the simulated absorption curves for the TE and the TM modes at λ 1.55μm. We have found that there exists the reversal of the dominantly absorbed mode at around 190nm, and the peak absorption of 0.14dB/μm can be achieved for the TM mode at 240nm. These curves coincide well with previously reported experimental data [23][155][159]. Figure 24(b) shows the simulated distributions of the electric field 34

35 component tangential to the graphene layer at = 240nm and =600nm, which clearly indicates the concentration of the field overlapped with the graphene region for the TM mode. We have also proposed a novel structure of a two-layer graphene modulator [160], where the two layers are partially overlapped only around the central region of the silicon waveguide via a thin dielectric spacer to make <. We discover that by using the optimum waveguide thickness of 240nm, the absorption is enhanced by a factor of ~ 2 to 0.28dB/μm, with a device bandwidth 20GHz and an energy consumption < 100 fj/s. We assume a constant resistance of graphene based on previous reports, but in reality, the reduction of also leads to an approximately equal amount of increment in the sheet resistance, making the influence on speed limited. Nevertheless, the reduction of is needed in general for the reduction in power consumption, which is only dependent on the capacitance and the insertion loss. It is difficult to reduce the width further as far as ordinary silicon waveguide structure is concerned. One way is by the use of the slot waveguide structure [161], the light field can be confined in the low-index region (air) sandwiched in between high-index region (silicon). A new device structure as shown in Figure 25(a) is proposed to realize an ultra-fast graphene slot waveguide modulator [162]. The device consists of a silicon slot waveguide, covered with two layers of graphene, spaced by a thin aluminum-oxide dielectric, partially overlapped over the slot region of the waveguide to form a graphene capacitor. Figure 25(b) shows the calculated dependency of the absorption and real part of the effective refractive index on the applied voltage, with 10nm, 210nm, and 50nm ( is equal to the slot width.) The device characteristic is simulated using the finite element method (FEM) on COMSOL, under the same 35

36 assumptions as the previous FDM simulation. With the proposed slot-waveguide structure, we discover that a modulation depth of 0.2dB/ m is achievable with a low drive voltage (~3V) and a low insertion loss (~1.5 db). Figure 25(c) shows the calculated response bandwidth and energy consumption of the device with respect to different slot widths. At 50nm, the bandwidth is ~120GHz, and the energy consumption is ~12 fj/bit. The calculation assumes that and vary with respect to the changes in the capacitor structure whilst using conservative values for the graphene sheet and the contact resistance. Apart from intensity modulators, there has also been a great interest in the development of graphene-based phase modulators on silicon photonic waveguides [163]. There is significant change in the refractive index at the band edge of the device where the absorption drops, as shown in Fig. 12(b). This means that a single device can potentially be utilized as both an absorption, and as a phase modulator, depending on the applied voltage, which opens a pathway to the development of novel modulation techniques. There is a report on demonstrating a fiber-based electrically-tunable graphene SA with a structure similar to that of the graphene absorption modulator [164]. In this device structure, a D-shaped fiber is used as the waveguide, and graphene is placed on the waveguide with ion-liquid as the dielectric medium to form a field-effect transistor (FET) structure. The optical absorption and the SA properties of the device can be controlled by changing the applied gate voltage, for mode locking of fiber lasers. Despite a slow operating speed, limited by the mobility of the ion liquid, this kind of device is promising for active control of fiber lasers. 36

37 5. All-optical signal processing devices The speed and power of electronic signal processing is constantly increasing, driven by the availability of integrated electronics in accordance to Moore s Law. Nowadays, signal processing beyond 40Gb/s is possible with high-speed A/D, D/A converters and digital signal processors (DSPs). The availability of high-speed DSP has revolutionized the optical fiber communication industries with the advent of the digital coherent technology [165]. Nevertheless, all-optical signal processing is still advantageous in yet faster speed with lower energy consumption and will be deployed as optically-assisted electrical signal processing in optical fiber networks [166]. Optical signal processing exploiting NL optical processes we discussed in Sec.2.2, mostly nonlinear Kerr effect and consequential NL effects, such as FWM, SPM, and XPM. Amongst them, FWM is regarded as the most important NL Kerr-based process which allow the generation of new light by mixing pump light(s) and signal light. In the degenerate case where a single pump light is used, the electric field of the newly generated (converted) light can be expressed as: exp 2, (18) where and are the complex amplitudes of the pump and the signal fields, respectively; and are the angular frequencies of the pump and signal fields, respectively, * denotes the complex conjugate, and 2 / is the nonlinear coefficient, is the length of the device, is the wavelength and is the effective cross-sectional area [4][166]. In a case when the pump light is a continuous wave, Eq.(18) implies that the converted light has new frequency at 2, whilst preserving the amplitude of the modulated data signal. This is called wavelength conversion whereby the converted signal is 37

38 spectrally inverted with respect to the original spectrum because of the complex conjugate nature of the FWM process. In another case when both pump and signal is modulated, FWM serves as an optical multiplier, which is essential for various signal processing [166]. Note that, for an efficient FWM generation, the new light has to be generated constructively, for which the phase-matching conditions should be satisfied [1][4]. As we have discussed in Sec. 2.2, CNT, graphene, and other 2D materials possess a high nonlinear refractive index. We have reported a wideband wavelength conversion of 10Gb/s Non-Return-to-Zero (NRZ) signal using a CNT-deposited D-shaped fiber in a scheme based on nonlinear polarization rotation [167]. The nonlinear coefficient of the CNT-deposited D-shaped fiber is estimated to be as high as ~ 500W m. We have also demonstrated FWM-based wavelength conversion on 10Gb/s NRZ signal using CNT-deposited D-shaped fibers [168], tapered fibers [169][170], and planar lightwave circuit (PLC) waveguides [171]. Figure 26 is the experimental setup, FWM spectra, and the bit-error rate (BER) performances with 10Gb/s eye-diagrams of the input and the converted signals using the CNT-deposited PLC waveguide. With the device length of 5cm, a wavelength tuning range of 8nm and a peak conversion efficiency of ~21dB has been achieved, and a wavelength conversion power penalty of the around 3dB at 10 BER level is obtained. FWM in CNT coated optical fiber Bragg grating has also been reported [172]. A recent report on photon-pair generation for quantum information processing though the FWM process in a 100nm-thick CNT film, which is 1000 times thinner than the smallest existing devices [173]. There have also been a number of reports on FWM generation [174]-[180] and FWM-based signal processing [181]-[183] in graphene, since the first report on wideband FWM generation in single-layer 38

39 graphene [79]. They are either fiber-based devices [174][177][180]-[183] or silicon waveguide-based devices [175][176][178][179]. It has been reported recently that the nonlinear coefficient of a graphenecovered SiN waveguide can be varied with an applied gate voltage to detune the Fermi energy level. A peak value of ~ 6400W m is measured through FWM in the vicinity of the interband absorption edge, ~ 2 [184]. There is also a numerical study on wideband wavelength conversion via FWM in a foundry-compatible silicon waveguide covered with graphene, with a conclusion that conversion efficiencies exceeding 30 db can be achieved over a 3.4THz-wide signal bandwidth situated as much as 58 THz away from the pump frequency [185]. There have also been several works on all-optical signal processing using other nonlinear phenomena other than FWM. We have demonstrated waveform regeneration of 10Gb/s, 1.8ps Return-to- Zero (RZ) signal using spectral spread by the SPM in CNT-deposited D-shaped fiber followed by an offset spectral filtering, sometimes called the Mamyshev regenerator [186]. All-optical modulation has been reported using cross saturable absorption [187] as well as XPM [188] in a tapered fiber wrapped by graphene. A good review paper on all-optical modulation using 2D materials was published recently [189]. There has been no report on all-optical signal processing using other 2D materials, until a recent demonstration of FWM-based wavelength conversions using BP deposited on a D-shaped fiber [190] and a BP coated device around a tapered fiber [191]. 39

40 6. Conclusion In this tutorial, we first described the basic physical properties, the linear and the NL optical properties of CNT, graphene, and other related 2D materials. We focused mainly on three NL phenomena, namely the SA, the electro-optic effect, and the nonlinear Kerr effect. We then discussed short-pulse fiber lasers using SA, graphene electro-optic modulators, and all-optical signal processing devices based on the nonlinear Kerr effect. Mode locking using CNT-SAs is now a well-established technology, particularly suitable for short-pulse fiber lasers where the lasing wavelength matches to the absorption wavelength of the CNT. Graphene has a broad wavelength coverage and a wavelength-independent absorption, however its SA is wavelength-dependent, which makes it more attractive for short-pulse fiber lasers at longer wavelengths, such as the mid-ir wavelength regions. There are an increasing number of reports on short-pulse fiber lasers using new SAs based on related 2D materials. These materials may have similar or better physical and optical properties than CNT or graphene as we have discussed, however the question remains which the best SA for short-pulse fiber lasers is, in terms of fabrication cost, robustness, high-power endurance, etc. So far, none of these papers has given an answered the question. In our opinion, there is currently no strong reason to replace the well-established CNT- or graphene-sa with new materials, unless there are specific advantages, such as SA for visible wavelengths [136]. In some reports on short-pulse fiber lasers, there seems to be a misunderstanding that the generation of very short pulses from the fiber laser is the proof of the excellence of the SA used for mode 40

41 locking, but this is not exactly true since the fiber dispersion and nonlinearity of the laser cavity plays greater roles in the mode-locking of fiber lasers, especially in the case of soliton mode locking. At the wavelength of 1.5μm, due to the availability of telecom components, we can easily control the intracavity dispersion and nonlinearity at low cost and with low loss, which facilitates the ML operation, even when using a SA with a high saturation intensity. This is not true for mid-ir wavelengths, so the role of the SA tends to be more important. Graphene and related 2D materials are very attractive for electro-optic and all-optical signal processing devices. We believe that their real values will be appreciated much more in integrated optic systems, but perhaps not in the fiber/waveguide optic systems, since we will have many other choices of NL fibers/waveguides. However, there are still controversies whether graphene (and other 2D materials as well) is the worthwhile choice as a nonlinear material for integrated optics [192]-[194]. We do hope that integrated electro-optic and all-optical signal processing devices based on these 2D materials can be realized to becoming a material of choice for practical applications in the near future. Acknowledgements The author would like to express great appreciations to Prof. Sze Y. Set and Dr. Goran Kovacevic of the University of Tokyo, and Prof. Zhipei Sun of Aalto University, for their valuable comments and suggestions to improve this article. 41

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69 (a) Large Output ~ m Filter M Large OC System size M Gain crystal M Bulk-optic system Beam diameter: ~ mm Pump Pump SA ~ 10cm (coiled) Gain fiber NL, disp. Fiber/waveguide-optic system Beam diameter: ~ 10μm Output < mm Output Beam size Small Input Integrated-optic system Beam diameter: ~ 100nm Small Year Fig.1 (a) Evolution of optical systems, (b) Evolution of nonlinear optical devices. 69

70 (b) Large ~ cm ~ a few cm Fiber length: 1m - 1km NL fiber Size Bulk NL crystal NL waveguide ~ a few cm ~ nm C M: Mo, W, etc. X: S, Se, Te 0.65nm 0.35nm Small CNT Graphene 1D/2D NL materials TMD (MX 2 ) TI, BP, etc. Year Fig.1 (a) Evolution of optical systems, (b) Evolution of nonlinear optical devices. 70

71 (a) (b) Fig.2 Number of publications by year searched at the Google Scholar with a keyword, (a) saturable absorption, and (b) graphene, electro-optic and modulator. 71

72 (a) E (b) E (c) Conduction band x DOS Dirac Point (~ Fermi level E F ) y Valence band Fig.3 (a) Band structure, (b) DOS, and (c) linear optical absorption spectra of graphene. Fig.3(c) is reproduced with permission from Ref.[27]. Copyright 2008 The American Association for the Advancement of Science. 72

73 (a) E (b) E (c) E (9,0) (d) E (10,0) Valence band c 2 x Dirac Point x Fermi level DOS c 1 E 11 v 1 E 22 DOS y Conduction band y v 2 Metallic Semiconducting (e) Fig.4 (a)(b) Band structures and (c)(d) DOSs of m- and s-cnt, and (e) linear optical transmission spectrum of a CNT sample. Fig.4(e) is reproduced with permission from Ref.[18]. Copyright 2004 IEEE. 73

74 (a) (b) E k Fig.5 Band structures of (a) Bi 2 Te 3 (b) MoS 2 and. For Bi 2 Te 3 the blue regions represent bulk states while the red dashed lines are surface states. Reproduced with permission from Ref.[29]. Copyright 2014 OSA. 74

75 1.0 Normarized Absorption SA only RSA only SA and RSA Normarized Intensity Fig.6 Calculated SA and RSA properties. 75

76 (a) E Weak light (b) Slow interband transition (a few 10s ps) Strong light Fast transition via m-cnt (~1ps) (c) E DOS (d) Weak light Fast intraband transition ( ps) Strong light DOS Slow interband transition (a few ps) Fig.7 Optical absorption and saturation in (a)(b) CNT and (c)(d) graphene. 76

77 (a) (b) E E (c) Fig.8 Transmittivity transients for (a) bundled CNTs, (b) isolated CNTs, and (c) graphene. Fig.8(a) is reproduced with permission from Ref.[40]. Copyright 2003 APS. Fig.8(b) is reproduced with permission from Ref.[41]. Copyright 2005 APS. Fig.8(c) is reproduced from Ref.[45], with the permission of AIP Publishing 77

78 (a) (b) Fig.9 Saturable absorption properties of (a) a CNT thin film, and (b) a single-layer graphene. Fig.9(a) is reproduced with permission from Ref.[18]. Copyright 2004 IEEE. Fig.9(b) is reproduced with permission from Ref.[46]. Copyright 2012 Springer Nature. 78

79 Ref.[19] Ref.[47] Ref.[48] Ref.[50] Ref.[49] Ref.[53] Ref.[20] Fig.10 Predicted interband saturation intensity as a function of wavelength and inelastic collision time. Reproduced with permission from Ref.[51]. Copyright 2017 APS. Reference numbers are changed so as to correspond to those in this paper. 79

80 (a) E (b) E E F DOS DOS E F Fig.11 Tuning of the optical absorption of graphene by shifting Fermi level. 80

81 (a) Re σ (b) Im σ E [ev] E [ev] Fig.12 Real (a) and Imaginary (b) parts of graphene s optical conductivity with respect to the Fermi level for wavelength of 1550nm. 81

82 ħω Graphene V ε d Fig.13 Graphene capacitor structure for shifting Fermi level. 82

83 Saturable Absorber Loss Pulse Intensity Δτ Mirror Gain medium Output coupler t 1/f FT Δf f f Fig.14 Passively mode-locked laser using saturable absorber (SA). 83

84 B. Evanescent coupling A. Normal incidence CNT/2D thin film Substrate/Fiber/ Waveguide Fig.15 Interaction between the light and CNT/graphene thin film. 84

85 Optical fiber Holder Tapered fiber CNT/2D film CNT/2D film Ferrule CNT/2D film ~a few m Core (b) Core (c) N 2 sealing (optional) (a) Side-polished D-shaped fiber Fig.16 Fiber-type CNT/2D-SA devices. (a) CNT/2D film sandwiched in between fiber connectors (b) A tapered fiber coated with CNT/2D film at the taper waist (c) A side-polished D-shaped fiber with CNT/2D film on the flat surface 85

86 Gain fiber Pump (a) WDM mirror Gain fiber Output mirror with SA Pump (b) WDM coupler SA Output coupler Fig.17 Typical fiber laser configurations mode-locked by CNT/2D-SA. (a) Fiber Fabry-Perot (FP) laser (b) Fiber ring laser 86

87 Pulse width (a) Anomalous dispersion Cavity position O C S A (b) Anomalous dispersion O C S A Normal dispersion (c) Normal dispersion O C S F S A Fig.18 Pulse evolution in fiber laser cavity. (a) Soliton mode locking (b) Stretched-pulse (SP) mode locking (c) Dissipative soliton (DS) mode locking OC: Optical coupler, SF: Spectral filter 87

88 (a1) Δλ~7nm (Soliton) (b1) (c1) Δλ~14nm (SP) (a2) Δτ~480fs (Soliton) (b2) (c2) Δτ~300fs (SP) Fig.19 Output spectra and autocorrelation traces from CNT-based mode-locked fiber lasers. (a) Soliton and stretched pulse mode locking [121] (b) Dissipative soliton mode locking [124] (c) High repetition rate mode locking [126] Fig.19(b) is reproduced with permission from Ref.[124]. Copyright 2008 OSA. 88

89 (a) 20 Power [dbm] 40 Soliton SP DS (b) Wavelength [nm] Fig.20 Output spectra from mode-locked fiber lasers at mid-ir wavelengths. (a) Tm-doped fiber laser mode-locked by CNT [130] (b) Er-doped ZBLAN fiber laser mode-locked by graphene [133] Fig.20(b) is reproduced with permission from Ref.[133]. Copyright 2016 IEEE. 89

90 (a) ML fiber laser Fiber amplifier HNLF (b) (c) 5 Power [dbm] SC Input Wavelength [nm] Fig.21 Broadband frequency comb generation through SC in HNLF. (a) Typical setup (b) Octave spanning SC spectrum from 1μm to >2μm using 220fs, 46MHz pulses [141] (c) SC spectrum from 1.4μm to >1.7μm using 1ps, 10GHz pulses [126] Fig.21(a) is reproduced with permission from Ref.[141]. Copyright 2010 IEEE. 90

91 (a) (b) (c) Fig.22 Normal-incidence graphene electro-optic modulator [76]. (a) Device structure (b) Frequency response (c) Modulation depth Reproduced with permission from Ref.[76]. Copyright 2012 OSA. 91

92 (a) (b) (c) Al 2 O 3 Fig.23 Graphene electro-optic modulators on silicon waveguide. (a) Single graphene layer [23] (b) Double graphene layer [154] (c) Frequency response [154] Fig.23(a) is reproduced with permission from Ref.[23]. Copyright 2012 Springer Nature. Fig.23(b)(c) are reproduced with permission from Ref.[154]. Copyright 2012 American Chemical Society. 92

93 (a) (b) [159] [155] [23] [159] [155] Fig.24 FDM simulation of graphene-coated silicon waveguide [158]. (a) Absorption for TE and TM modes (b) Field distributions at d 240nm and w 600nm 93

94 (a) (b) (c) Fig.25 Graphene-coated silicon slot waveguide modulator [162]. (a) Device structure (b) Dependences of the absorption and real part of the effective refractive index on the applied voltage (c) Response bandwidth and energy consumption of the device with respect to different slot widths 94

95 (a) ECL1 ECL2 10 Gb/s pattern generator EDFA 3 db coupler ` CNT-deposited PLC waveguide optical filter CNT-deposited PLC waveguide Receiver/ BERT Spectral Intensity (10 db/div) (b) Converted signal Pump Input signal (c) Wavelength (nm) Fig.26 FWM wavelength conversion of 10Gb/s NRZ signal using CNT-deposited PLC waveguide [171]. (a) Experimental setup (b) Output spectrum (c) BER curves of back-to-back and converted 10Gb/s signals 95