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

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 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 1 mm. 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 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-waveguide-based systems were started in the 1970s with the advent of low-loss optical fibers. The demands for low-lost and low-cost optical fibers in the telecommunication industry have resulted in technological advancement of the mass production of high-quality optical fibers with attenuation and price close to the minimum values. The introduction of the rare-earth-doped optical amplifiers and the wavelength-division-multiplexing (WDM) technologies driven by the explosive growth of bandwidth requirement for the internet in the 1990s 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 is 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 ∼10 cm in coiling diameter whilst guiding lights with beam sizes comparable to the fiber core diameter of ∼10 µm.

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 with that of the fiber-optic systems, as shown in Fig. 1(b).

In parallel with the development of fiber-optic technologies, on-chip optical technologies emerged with silicon as a guiding medium. The integrated-optic system, or photonic integrated circuit (PIC), using the silicon-on-insulator (SOI) platform started in 1985, and the research field has become very active since the 2000s, 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 10 nm at a low cost. The strong optical confinement (typically 100 nm) in silicon waveguides allows the integration of many optical devices, such as directional couplers, bandpass filters, electro-optic modulators, and optical switches, into a small (<1 mm) 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 χ⁽³⁾ optical nonlinearity of silicon is 2 orders 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 photonic 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 candidates for 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 

In Sec. II 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. III, 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 studies on graphene modulators will be reviewed in Sec. IV. Finally, in Sec. V, 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 June 2018. 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 the topological insulator (TI), Bi₂Te₃, demonstrated in 2012, was the first among the other 2D materials to be used as a NL optical material.^21,22 It is evident that the number of publications on CNT-SA has been increasing but saturating, 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 2018. A steady growth in the number of publications since the first demonstration of the 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.

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 10⁶ cm² V⁻¹ s⁻¹ owing to the ballistic transport of electrons in graphene, “minimum” DC conductivity of ∼4e²/h (e is the electron charge) even with zero carrier concentration, anomalous quantum Hall effects, etc.²⁶ 

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 a result, they are known to have electrical and optical properties similar but not the same as that of graphene’s.²⁹ 

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 ∼1 nm and a tube length of ∼1 µm; thus, it can be considered as a long 1D material [the inset of Fig. 1(b)]. The CNT structure is determined by how the single-layer graphene is folded; the characteristic is expressed by a single parameter, and its chirality is expressed by a chiral vector C_h = (n, m) with two integers n and m that specifies the connecting points in folding of the 2D graphene sheet.³⁰ 

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 C_h·κ = 2qπ, where q 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 κ 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 n − m = 3k (k: integer) for m-CNTs, and n − m ≠ 3k for the s-CNTs. Figures 4(c) and 4(d) show the examples of the DOS of (9, 0) m-CNT and (10, 0) s-CNT.³¹ Many sharp peaks appear (c₁, c₂, … at the conduction band, and v₁, v₂, … 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.

The tube diameter of a typical single-walled CNT is in the range of 0.7–1.5 nm, corresponding to a peak absorption wavelength of 1–2 µm. 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 will exhibit a polarization dependency.³³ Most CNT samples have randomly oriented tube bundles and therefore are polarization independent.

There are many other 2D materials, besides the graphene family. Topological insulators (TIs), such as Bi₂Te₃ or Bi₂Se₃, 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 MX₂, 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 a single-atomic-layer. For example, a bulk crystal of MoS₂ has an indirect bandgap of 1.29 eV, while a single-layer MoS₂ has a direct bandgap of 1.8 eV,^29,36 as shown in Fig. 5(b). It is confirmed that the single-layer MoS₂ has an absorption peak 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 layers and can be tailored from 0.3 to 2 eV (corresponding to the wavelength range from 4 to 0.6 µm).³⁸ This has properties similar to those of a s-CNT and is unique among 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) such as CNT and graphene, that is, TIs, TMDs, and BP.

The second-order nonlinearity χ⁽²⁾ 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. By contrast, some 2D semiconductors are known to have a large second-order nonlinearity.⁸  χ⁽²⁾ is also related to the electro-optic effect, such as the Pockels effect.¹ 

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 Figs. 7(a) and 7(c). Saturation takes place when all the allowed states in the conduction band are fully populated and the valence band emptied at high optical intensities [Figs. 7(b) and 7(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 E₁₁ transition in bundled CNT samples is in the order of 1 ps, and that of the E₂₂ transition is even faster, in the order of 0.1 ps, as shown in Fig. 8(a).⁴⁰ It is known that isolated CNTs have a slower recovery time of ∼30 ps [Fig. 8(b)]⁴¹ and are luminescent, whereas bundled CNTs are not. The prevalent theory⁴² suggests that the fast recovery is due to the transition of the free carriers from s-CNT into m-CNTs (tube-tube interaction) in the 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 µJ/cm² at the wavelength of 1.55 µm, which corresponds to an estimated saturation intensity I_S of 12.5 MW/cm², a value very similar to that of a typical semiconductor-based SA, SESAMs.³⁹ At other wavelength regions, similar saturation intensities around 10 MW/cm² have been reported, as long as the E₁₁ peak matches with the operating wavelength.⁴³ It is also possible to use the E₂₂ transition for saturable absorption, at which a much higher saturation intensity of ∼200 MW/cm² has been reported.⁴⁴ This is a reasonable value since I_S is inversely proportional to τ as shown in Eq. (9).

The recovery time of graphene is even faster, in the range between 0.1 and 0.2 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 µm.⁴⁶ The estimated saturation intensity I_S of ∼250 MW/cm² 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^219,47–49 to 100–300 MW/cm².^20,46,50 This might be explained by the difference of the inelastic collision time of electrons in the graphene samples (Fig. 10).⁵¹ 

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 literature studies.^51,52 Figure 10 shows the predicted inter-band 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 a result of its linear band structure, as shown in Fig. 7(d). Indeed, higher saturation intensities have been reported at shorter wavelengths at λ = 800 nm.⁵³ Therefore, at longer wavelengths, it is more favorable to use graphene SA for laser mode-locking. Graphene can work as a saturable absorber at Terahertz or microwave frequencies. At 100 GHz, the saturation intensity is reported to be very low ∼0.04 mW/cm².⁵⁴ 

It has been reported that GO and rGO have a fast (0.2-0.4 ps) saturable absorption⁵⁵ and a low saturation intensity of around 1 MW/cm².^56–58 On the other hand, TI materials such as Bi₂Te₃ are reported to possess saturable absorption with both a fast (∼0.5 ps) and a slow (∼10 ps) recovery time⁵⁹ and a higher saturation intensity of >100 MW/cm².^22,60 The saturation intensity has a similar wavelength dependency to that of graphene (I_S becomes lower at longer λ).⁶⁰ There are an increasing number of papers recently on saturable absorption in TMD materials, such as MoS₂,^61–63 MoSe₂,⁶⁴ WS₂,^65,66 and BPs,^67–71 as shown in Fig. 2(a). The reported saturation intensity for TMD materials is rather diverse, from a few to several hundreds of MW/cm².^61–66 A fast recovery time of 0.15 ps is reported for TMDs.⁶⁴ BPs are reported to have a relatively lower saturation intensity of a few MW/cm^267–71 and a very fast recovery time of 24 fs.⁷¹ 

There are also many reports on measurement of n₂ in other 2D materials, TIs,^85,86 TMDs,^87–89 and BP.^90,91 TIs were reported to have a high n₂ of ∼10⁻¹² m²/W,^85,86 whereas the reported values for TMDs and BP are somewhat smaller, n₂ < 10⁻¹³ m²/W.^87–91 Nevertheless, these materials are still attractive candidates for nonlinear photonic applications.

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 μm. 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 selectively producing a single type of s-CNT or m-CNT, 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 a small number of defects.²⁵ Another mechanical exfoliation method is the liquid-phase exfoliation (LPE) method, in which graphite flakes are dispersed in a solvent, followed by ultra-sonication 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-in. 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 

Passively mode-locked lasers can generate ultra-short pulses in the picosecond or femtosecond regimes, and even attosecond pulses are possible with extremely broadband gain media. It should be noted that the recovery time of the SAs does 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 μs) and the repetition rate is much slower (1-100 kHz) 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 Sec. II, CNT and graphene have both a fast and a slow saturable absorption component, which is ideal for CW mode locking.

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 launch 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 shown in Fig. 16(a). The film can either be pristine CNT/2D 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 16(c) are two representative examples. Figure 16(b) is the tapered fiber device with a tapered waist of a few μm, surrounded by the CNT/2D material film for evanescent field coupling.¹¹⁷ Again, the film can either be a pristine CNT/2D material or in the form of a polymer composite. We have demonstrated the 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 μm from the fiber core, and the CNT/2D material film is formed on the flat surface for evanescent field coupling.^119,120

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 Fig. 17. The gain fiber can also be pumped in the cavity through a 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 mode-locking 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 transform-limited, 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 100 pJ. 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 1.5 µm [Fig. 17(b)] using a tapered fiber coated with the 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 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 Δλ is estimated to be ∼7 nm (Δf ∼ 930 GHz), pulse width ∼490 fs, giving a TBP of ∼0.45, which is larger than the ideal transform-limited value of 0.315 indicating that the pulses are chirped. The repetition rate $frep$ is ∼7 MHz, and the pulse energy is limited to ∼40 pJ.

By enhancing the perturbation of pulses in the cavity using the segments of normal and anomalous dispersion, higher pulse energy up to ∼1 nJ 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 Δλ ∼ 14 nm and the Kelly sidebands disappear. The pulse width is reduced to <200 fs, and the pulse energy is enhanced to ∼100 pJ.

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 1 nJ and is linearly chirped, which can be easily dechirped after the laser output. Figure 19(b) shows the spectrum and the SHG autocorrelation trace of an all-normal-dispersion Yb-doped fiber ring laser at 1 µm wavelength mode locked using a tapered fiber coated with the 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 ∼15 nm and a square flat-top shape, which is the characteristic of a DS mode locked fiber laser. The output pulses are eventually dechirped with pulse width compressed down to 235 fs.

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 10 GHz, the fiber ring laser cavity length must be reduced to ∼2 cm, which is not practical for a ring laser configuration. The fiber FP laser configuration shown in Fig. 17(a) makes such a high repetition rate possible in conjunction with a highly doped gain fiber and a compact CNT/2D-SA. Figure 19(c) shows the spectrum and the SHG autocorrelation trace of a 1 cm-long, 10 GHz 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 ∼1 ps is achieved. We have also realized a 1 cm-long, 10 GHz mode-locked FP laser using a graphene SA.¹²⁷ However, the performances, such as the mode-locking threshold and stability, are not as good as the one using a CNT-SA. This is possibly due to the higher saturation intensity of the graphene at 1.5 µm, as shown in Fig. 10.

The most frequently used gain media for fiber lasers have been Er-doped fiber at 1.5 µm [e.g., Figs. 19(a) and 19(c)] and Yb-doped fiber at 1 µm [e.g., Fig. 19(b)]. Recently, there is growing interest in fiber laser sources at the mid-infrared (mid-IR) wavelength region from 2 to 15 µm 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 µm, rare-earth elements such as Tm, Ho, Er, Pr, and Dy can also provide gain for wavelength regions beyond 2 µm in different host glasses.¹²⁸ It should be noted that the ordinary silica-glass fibers are usable only up to around 2 µm wavelength because they are not transparent beyond 2 µm. Therefore, fiber lasers beyond 2 µm have to be composed of the fibers made from non-silica host materials, for example, chalcogenide and fluoride glass such as ZBLAN (ZrF₄-BaF₂-LaF₃-AlF₃-NaF) glass.¹²⁸ 

There have been many reports on 2 µm fiber laser 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 intracavity dispersion from net anomalous to net normal dispersion. The resulting pulse widths after pulse compression are around 1 ps. We have also achieved similar results in a Tm-doped mode-locked fiber ring laser using a graphene-covered tapered fiber.¹³² 

By contrast, there are only a limited number of reports on mode locked fiber lasers beyond 2 µm 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 µm using a graphene SA.¹³³ They reported a low saturation intensity of graphene at 2.8 µm, ∼2 MW/cm², which coincides with our discussion on the wavelength dependence of the saturation intensity of graphene at Sec. II B 1 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 2 kHz at 1.5 µm¹³⁴ and 0.6 kHz at 2 µm¹³⁵ 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 Sec. III D.

There are a number of publications recently on mode-locked fiber lasers using other 2D-SAs, for example, GO/rGO,^57,58 MoS₂,^61,62 WS₂,^65,66 and BP.^68,70 Most of them reported soliton mode-locked fiber lasers at 1.5 µm, in which the role of the SA is not significant. There is an interesting study on short pulse generation from a visible fiber laser at 635 nm, although the operation is still limited to passive Q-switching.¹³⁶ 

Mode-locked fiber lasers using CNT-SA have been applied as a compact and robust short-pulse source for a non-synchronous optical sampling scope¹³⁷ 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, an octave-spanning SC source having f and 2f components in its spectrum is essential for the realization of a true frequency comb without the carrier envelope offset frequency $fCEO$⁠,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 The octave spanning SC spectrum from 1 µm to >2 µm has been generated using a CNT-based mode-locked fiber laser.¹⁴¹ The laser generates 220 fs, 46 MHz pulses with an average power of 2 mW, and the pulses are amplified and compressed to 65 fs, before they are launched to a 79 cm-long HNLF. The SC spectrum spans from 1 µm to more than 2 µm at the maximum, as shown in Fig. 21(b). Such broadband SC has been used, not only for laser stabilization using the f-and-2f components but also for generating fs pulses at the 1 µm wavelength region,¹⁴³ mid-IR frequency comb through difference frequency generation,¹⁴⁴ synchronized ps pulses at 1 µm and 1.5 µm for stimulated Raman scattering (SRS) microscopy.¹⁴⁵ 

Figure 21(c) depicts the SC spectrum generated by the aforementioned 1 cm-long 10 GHz mode-locked FP laser using a CNT-SA.¹²⁶ The 1 ps pulses are amplified to an average power of ∼300 mW, then input to a 30 m-long HNLF. We could obtain a flat SC spectrum spanning from 1.4 µm to >1.7 µm. The mode structures in the SC spectrum are 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 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 Sec. III C 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 bidirectional mode-locked fiber laser with a CNT-SA¹⁴⁸ and also on octave-spanning DCS using a similar bidirectional fiber laser followed by SC generation.¹⁴⁹ 

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. I. 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 154 MHz 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. 15.

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 Fig. 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 (∼25 µm²), low drive voltage (∼5 V), and a broad operation wavelength. However, the performance of these modulators is limited by their operating speed of around 1 GHz, as shown in Fig. 23(c).

The waveguide structure needs to be optimized in order to maximize the absorption of the graphene layer and to reduce the length L_c. We have explored the absorption dependency of the graphene-covered silicon waveguide design, by varying the thickness d and width w (=w_c 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 = 1.55 µm. We have found that there exists the reversal of the dominantly absorbed mode at around d = 190 nm, and the peak absorption of 0.14 dB/μm can be achieved for the TM mode at d = 240 nm. These curves coincide well with previously reported experimental data.^23,155,159 Figure 24(b) shows the simulated distributions of the electric field component tangential to the graphene layer at d = 240 nm and w = 600 nm, 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 w_c < w. We discover that by using the optimum waveguide thickness of d = 240 nm, the absorption is enhanced by a factor of ∼2 to 0.28 dB/μm, with a device bandwidth B > 20 GHz and an energy consumption <100 fJ/s. We assume a constant resistance of graphene based on previous reports, but in reality, the reduction of L_c also leads to an approximately equal amount of increment in the sheet resistance R, making the influence on speed limited. Nevertheless, the reduction of L_c 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 w_c further as far as an 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 regions (silicon). A new device structure as shown in Fig. 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 d_c = 10 nm, d = 210 nm, and w_c = 50 nm (w_c is equal to the slot width w_Slot). The device characteristic is simulated using the finite element method (FEM) on COMSOL, under the same assumptions as the previous FDM simulation. With the proposed slot-waveguide structure, we discover that a modulation depth of 0.2 dB/μm is achievable with a low drive voltage (∼3 V) 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 w_Slot = w_c = 50 nm, the bandwidth is ∼120 GHz, and the energy consumption is ∼12 fJ/bit. The calculation assumes that C and R 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 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 both as 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.

The speed and power of electronic signal processing are constantly increasing, driven by the availability of integrated electronics in accordance to Moore’s law. Nowadays, signal processing beyond 40 Gb/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 

As we have discussed in Sec. II B, CNT, graphene, and other 2D materials possess a high nonlinear refractive index n₂. We have reported a wideband wavelength conversion of the 10 Gb/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 γ of the CNT-deposited D-shaped fiber is estimated to be as high as γ ∼ 500 W⁻¹ m⁻¹. We have also demonstrated FWM-based wavelength conversion on the 10 Gb/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 10 Gb/s eye-diagrams of the input and the converted signals using the CNT-deposited PLC waveguide. With the device length of 5 cm, a wavelength tuning range of 8 nm and a peak conversion efficiency of ∼21 dB has been achieved, and a wavelength conversion power penalty of the around 3 dB at the 10⁻⁹ 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 100 nm-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 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 γ of a graphene-covered SiN waveguide can be varied with an applied gate voltage to detune the Fermi energy level. A peak value of γ ∼ 6400 W⁻¹ m⁻¹ is measured through FWM in the vicinity of the interband absorption edge, $ħω∼±2EF$⁠.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 3.4 THz-wide signal bandwidth situated as much as 58 THz away from the pump frequency.¹⁸⁵ 

There have also been several studies on all-optical signal processing using other nonlinear phenomena other than FWM. We have demonstrated waveform regeneration of the 10 Gb/s, 1.8 ps 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 absorption¹⁸⁷ as well as XPM¹⁸⁸ 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 fiber¹⁹⁰ and a BP coated device around a tapered fiber.¹⁹¹ 

In this tutorial, we first described the basic physical properties, the linear properties, 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 with 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 have 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 locking, but this is not exactly true since the fiber dispersion and nonlinearity of the laser cavity play greater roles in the mode-locking of fiber lasers, especially in the case of soliton mode locking. At the wavelength of 1.5 µm, 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 become a material of choice for practical applications in the near future.

The author would like to express great appreciations to Professor Sze Y. Set and Dr. Goran Kovacevic of the University of Tokyo and Professor Zhipei Sun of Aalto University for their valuable comments and suggestions to improve this article.