1. Introduction

Graphene, an atomic layer of conjugated sp² carbon atoms arranged in a two dimensional hexagonal lattice, possesses many exceptional electrical and optical properties, due to its unique linear electronic band [1,2]. Its Dirac point type of energy band-gap structure endows that graphene can readily absorb photons ranging from the visible to the infrared with the record strong inter-band transition efficiency. The wideband optical absorption property provokes the realization of an ultrafast graphene photo-detector with bandwidth exceeding 500 GHz [3], a broadband graphene optical modulator [4] and a broadband graphene polarizer [5]. Under intensive illumination, optical absorbance of graphene decreases with increasing the light intensity and becomes saturated once the incident light exceeds a threshold power, as a consequence of Pauli blocking principle, that is, valence band depletion and conductance band filling blocks further absorption [6,7]. This property already makes graphene widely applicable in the mode-locking of different types of lasers [8–18]. Worthy of mentioning are graphene mode-locked Nd:yttrium aluminum garnet laser at 1064 nm [8], Ytterbium-doped fiber lasers at 1069.8 nm [9], Cr:forsterite laser at 1250 nm [10], erbium doped fiber laser with wavelength tunable from 1570 to 1600 nm [11], Tm:YAlO₃ laser near 2 µm [12], CO laser at 5 µm and CO₂ laser at 10.6 µm [13]. Those results demonstrate that graphene has important applications at optical band.

Equally fascinating, graphene also shows unusual electromagnetic response of Dirac quasi-particles with several anomalous properties even in the absence of magnetic field [19]. Recently, microwave response of graphene at weak power regime had already been investigated, for example, propagation of microwave in graphene [20], microwave switching of top-gate field effect graphene transistor [21] and microwave frequency multiplier [22], demonstrating that graphene may also have attractive application at microwave band. Naturally, one fundamental question arises: in view that graphene shows nontrivial optical absorption feature, how about its microwave response at high power regime?

Moreover, strong nonlinear electromagnetic response of graphene in microwave and THz region was also theoretically predicted in Refs [23–26]. However, direct experimental evidence on the nonlinear microwave response of graphene was not yet provided. In this contribution, we aimed at the experimental investigation of nonlinear absorption in graphene and uncovered that graphene also exhibits broadband saturable absorption at microwave band for the first time. Regardless of the microwave frequency from 96 GHz to 100 GHz, saturable absorption feature can be always observed, with saturable power varied from 28.8 µW to 79.6 µW and modulation depth of 4.58%~12.77%. Based on an open-aperture Z-scan technique by one laser wavelength with central wavelength tunable from 1525 nm to 1570 nm and another laser at 1053 nm, broadband optical saturable absorption of the same graphene sample was also characterized, with saturable intensity of about 7.89 MW/cm² at the wavelength of telecommunication band and 10.32 MW/cm² at wavelength at 1053 nm, respectively.

2. Results and discussion

2.1. Characterization of graphene

The graphene dispersion has been prepared by liquid-phase exfoliation of graphite [27,28]. Graphene dispersions were prepared by adding graphite at an initial concentration of 5 mg/mL to 100 mL PVP solution (0.1 mg/mL). Ultrasonication was carried out in a table-top ultrasonic cleaner for 10 hours. Gravity sedimentation was standing for more than 10 days after sonication, and then stable graphene dispersion up to 6 months. Then the graphene dispersion was diluted and drop cast on the quartz substrate using a dropper. Raman spectroscope was used to evaluate the crystallininity and scanning electron microscopy (SEM) was used to characterize the morphology of graphene, as shown in Fig. 1 .

 

Fig. 1 (a) The graphene sample on quartz plate for microwave and optical saturable absorption measurement. (a) Raman spectra of the graphene sample. (b) The SEM image of the graphene sample.

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Figure 1(a) shows the Raman spectrum with two peaks located at 1348 cm^–1 and 1581 cm^–1, respectively. The peak at 1581 cm^–1 (G band) contributes to an E_2g mode of graphite and is related to the in-plane vibration of sp²-bonded carbon atoms, while the peak at 1348 cm^–1 (D band) is associated with vibrations of carbon atoms with sp³ electronic configuration of disordered graphite. The intensity ratio of the D and G bands (I_D/I_G) of graphene sheets is about 0.4, indicating the defects of the graphene sample.

2.2. Characterization of microwave source

Optical frequency multiplication (OFM) technique based on external modulation to generate the high-frequency microwave has the advantage of being stable, high-spectral-purity and cost-effective [29–31]. In our previous work, the OFM technique was used for photonic generation of 40 GHz millimeter-waves and delivery of wireless signals to remote antennas in radio-over fiber systems [32, 33]. Figure 2 shows the experimental setup for the generation of a 100 GHz continuous microwave based on external modulation.

 

Fig. 2 Experimental setup for 100 GHz microwave generation and microwave saturable absorption characterization system. ECL: external cavity laser, LN-MZM: LiNbO₃ Mach–Zehnder modulator. IL: 50/100 GHz optical interleaver. EDFA: erbium-doped fiber amplifier. ATT: optical attenuator. PD: photodiode. EA: electrical amplifier. Sample: graphene. The resolution for all optical spectra is 0.01 nm.

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An external cavity laser (ECL) was used to achieve a continuous-wave light at 1565.3 nm. The optical carrier was modulated by a single arm Mach–Zehnder LiNbO₃ intensity modulator (Fujitsu, 3dB bandwidth >25 GHz) which is driven by a 25 GHz microwave signal. Here, an electrical frequency doubler (Doubler) is used to generate 25 GHz microwave signal by doubling a 12.5 GHz radio-frequency (RF). Since the driving voltage is 0.34 V, the odd-order optical sidebands are sufficiently suppressed in order that the extinction ratio of the second-order sidebands can be larger than 40 dB, as shown in Fig. 2(a). In order to obtain two well-distinguished second-order sidebands, a 50/100 GHz two-output interleaver is used to eliminate the optical carrier. We can observe that the first-order sidebands have an intensity of 30 dB lower than the second-order sideband, as shown in Fig. 2(b). After being amplified through a commercial EDFA, optical signal still exhibits high quality with a wavelength separation between the two second-order sidebands of 0.8 nm (corresponding to 100 GHz) as shown in Fig. 2(c). An optical attenuator is used to alter the PD input power, which therefore makes the output microwave power widely adjustable. The 100 GHz electrical signal is generated by beating two second-order sidebands with a high-speed photodiode (U2t, 100-GHz PD). Then, the electrical signal is amplified by the narrow-band electrical amplifier (work frequency, 96 GHz ~100 GHz) and finally the microwave is radiated from a W-band antenna with a gain of 25 dB (for details, refer to Appendix A).

We can tune the as-generated microwave with a frequency interval of 0.8 GHz through changing the RF frequency with a frequency interval 0.1 GHz (for details, refer to Appendix A). Due to bandwidth limitation of the narrow-band electrical amplifier (EA), the frequency tuning range is confined to be only 4 GHz. The 100 GHz microwave radiation from antenna is amplitude modulated by using a 30 Hz chopper (TTI, c-995), and its output power is then detected by an absolute THz power meter (Thoumas Keating Instruments THz Power Meter). The aperture diameter of chopper is 15 mm with a slot width of 4.5 mm and the distance between the antenna and power meter is 5 cm. The graphene sample deposited on the glass substrate has a diameter of is 25 mm and a thickness of 1 mm. The glass substrate is placed on the horizontal translation stage perpendicular to the aperture of chopper.

2.3. Microwave saturable absorption

The above-mentioned microwave source was used to investigate the microwave response of graphene. In the system, the microwave power can be adjusted from 30 μW to 450 μW by varying the PD input power. The microwave power (starting from the minimum power of 30 μW to the maximum power of 450 μW) could be automatically measured by an absolute terahertz power meter. Over 5000 spots were detected and averaged, and no significant intensity variation was found. The microwave source showed excellent long term stability, and its output power kept at a reasonably constant value, as shown in Fig. 3 . By moving the horizontal translation stage, the microwave power without passing through the graphene sample was measured by the power meter. In order to sufficiently reduce experimental errors, 200 data points are averaged as the input power at one PD input power. Moreover, by continuously varying PD input power, a series of microwave powers under different PD input power are recorded as input power P₁. By further moving horizontal translation stage and thereby placing the graphene sample close at the center of the chopper aperture, microwave power transmitted through graphene sample could be measured. Similarly, microwave powers under different PD input power could be also recorded as input power P₂. Finally, upon dividing the power data of P₂ by P₁, change of microwave transmittance T against different input powers could be characterized.

 

Fig. 3 The minimum and maximum power curves of 96~100 GHz microwave sources, with respect to different time.

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Similar to the optical absorbance model in Ref [7], the microwave absorbance of graphene as a function of increasing microwave intensity could be described by:

The corresponding transmittance T can be fitted by

By continuously varying the microwave frequency from 96 GHz to 100 GHz, with a frequency interval of 0.8 GHz, saturable absorption under different microwave frequencies have been summarized in Fig. 4 . The inferred saturable intensity is found to be slightly wavelength dependent, which slightly decreases towards longer wavelengths as shown in Fig. 5(a) . The modulation depth is 4.58%~12.77% as shown in Fig. 5(b).

 

Fig. 4 Power dependent microwave saturable absorption in graphene at different frequencies. Circles: experimental results; solid curve: fitting results.

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Fig. 5 Relations between the inferred microwave frequency and (a) saturation intensity, (b) modulation depth.

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2.4. Optical saturable absorption

Optical saturable absorption of graphene could be measured by using an open-aperture Z-scan technique. The experimental setup is schematized in Fig. 6(a) similar to Ref [34], which has been adapted to thin-film measurement. A pico-second fiber laser with center wavelength tunable from 1525 nm to 1570 nm is used as the laser source. These pulses emitted from the laser are amplified through an EDFA, and then focused by a 20 times microscope objective. The beam waist was measured and fitted to be 3 μm (for details, refer to Appendix C, D). A portion of the input laser beam is picked off by the beam splitter and measured by detector D1 as the reference of the optical power.

 

Fig. 6 (a) Schematic of the Z-scan setup. EDFA: Erbium doped fiber amplifier; BS: beam splitter; D1 and D2: power meters. (b) A Z-scan curve at 1550 nm. (c) The corresponding saturable absorption curve at 1550 nm. (d) Wavelength dependent saturable absorption curve. (e) Wavelength dependent saturable intensity curve.

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The same graphene sample is placed perpendicularly to the beam axis. By translating the sample through the focal plane with a Newport ESP301 linear motorized stage along its propagation (Z) axis, output power was measured by detector D2 continuously. Dividing the data from D2 by the data from D1, a Z-scan curve with a strong peak near the focus point was obtained in Fig. 6(b). Taking account of the change of the beam waist and the relative position with respect to the focusing objective, the transmittance as a function of the incident laser fluence was shown in Fig. 6(c). Fitting this curve by the Eq. (2) yields a saturable intensity of about 7.89 MW cm^–2 at wavelength of 1550 nm.

Then, by continuously shifting the center wavelength of the pico-second fiber laser from 1525 nm to 1570 nm, with a wavelength separation of about 5 nm, saturable absorption curves against laser fluencies under different laser wavelengths are shown in Fig. 6(d). By fitting those curves, saturation depth of the graphene sample is found to about 6%, which slightly decreases towards longer wavelengths. The inferred saturable intensity is slightly wavelength dependent, as shown in Fig. 6(e).

The optical saturable absorption of graphene at 1053 nm has also been measured using the similar method in Ref [7]. The nonlinear absorption measurement apparatus is shown in Fig. 7(a) . The laser source is the seed pico-second oscillator mode-locked by the broadband semiconductor saturable absorber of High-Q pico-second regenerative amplifiers (RA) system (Pico-REGEN, High Q Laser, Watertown, MA). The pulse duration is 75 ps and repetition rate is 80 MHz. By placing the graphene sample near the focus point of the 500 mm lens and adjusting the input laser power, the transmittance can be measured, as shown in Fig. 7(b). Fitting this curve yields a saturable intensity of about 10.32 MW cm^–2 at the wavelength of 1053 nm.

 

Fig. 7 (a) Schematic experimental setup for measuring nonlinear power dependent absorption of graphene samples. (b) Power-dependent nonlinear absorption properties at 1053 nm.

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To take the advantage of the broadband optical saturable absorption, the same graphene sample is employed to passively mode lock an erbium-doped fiber laser at 1564 nm (for details, refer to Appendix E). It is a weakly birefringent cavity fiber laser, where an artificial birefringence filter is induced by the cavity birefringence, with tunable filter bandwidth. As transmittance of the artificial birefringence filter varies with the cavity birefringence, therefore, in our laser simply adjusting the orientation of the intra cavity polarization controller the peak wavelength of the mode-locked pulses can be tuned.

2.5. Discussions

Graphs of the valence band and the conduction band in graphene are smooth-sided cones that almost meet at one point, called the Dirac point. Graphene has a small band gap in the range of several meV [35], indicating that graphene can absorb microwave photons at an arbitrary frequency around 100 GHz. Figure 8 shows schematic of microwave saturable absorption in graphene at different microwave frequencies. Through the absorption of microwave photons with energy of ħω, electrons at the valence band with energy of E_F–ħω/2 can be excited to the corresponding conduction band with energy of E_F + ħω/2. Despite of broadband microwave response, the absorbance is intensity dependent. Under weak irradiation, microwave photons can be continuously depleted through the excitation of the electrons from the valence band to the conduction band. However, under sufficiently strong microwave irradiation, owing to the Pauli blocking principle, the newly generated carriers fill the valence bands, preventing further excitation of electrons at valance band and therefore allowing microwave transmitted without absorption, which interprets the mechanism of graphene saturable absorption. The threshold fluence to saturate the absorption of graphene is termed as saturable fluence, which is proportional to the total amount of electrons at the valance band covering from E_F–ħω/2 to E_F. Correspondingly, graphene shows weaker saturable fluence at the microwave frequency ω₁ than that at ω₂ because larger amount of electrons are filled at the valance band at the microwave frequency at ω₁. The higher the microwave frequency, the higher the saturable fluence, which can explain why saturable intensity decreases towards longer wavelength in Fig. 5. Concerning the Z-scan measurement, if a continuous wave at the same wavelength and mean power as the pulse laser is used as the input laser source, the typical peak Z-scan curve is never found while only featureless flat curve is observed. This indicates that a CW is unable to saturate the absorption of graphene because the peak power of CW is significantly weak and below the saturation threshold. In view that microwave frequency (100 GHz) is three orders of magnitude smaller than optical frequency (1550 nm corresponds to 193 THz), saturable intensity at microwave band should be much lower than that at optical band and therefore a CW microwave with power higher than 80 µW is able to saturate the absorption of graphene.

 

Fig. 8 Schematic of microwave saturable absorption in graphene under different incident frequencies.

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Based on the microwave saturable absorption behavior of graphene, we anticipate that several novel graphene microwave devices could be eventually realized through borrowing concepts from the well-developed graphene optics research. 1) The broadband microwave saturable absorption renders graphene suitable for the mode-locking of Microwave Amplification by Stimulation Emission of Radiation (MASER), which was theoretically proposed by L. Mertz in 1974 [36] but not yet experimentally verified. By placing a graphene microwave saturable absorber inside a MASER cavity, frequency-tunable and ultra-short pulses at microwave band could be produced from a graphene mode-locked MASER. 2) Owing to one advantage that work function of graphene can be controlled by either chemical doping, applying an external electrical or magnetic field, one can expect the controlling, tuning and tailoring of microwave saturable absorption of graphene. Correspondingly, it may enable the generation of graphene microwave modulator, in which modulation is achieved by actively tuning the Fermi level of graphene, similar to the structure of graphene-based broadband optical modulator [4], which may be comparable to, if not better than, the traditional microwave modulator in terms of speed, cost and broad bandwidth. 3) Broadband graphene polarizer at microwave band, which replies on the coupling and interaction between micrometer electromagnetic-wave and graphene, could be realized as well by following the geometry of broadband graphene optical polarizer [5].

3. Conclusion

In summary, we uncover that graphene also shows saturable absorption at microwave band for the first time. This unique microwave property renders graphene as a promising broadband saturable absorber with potential applications at both optical and microwave band. Further exploration on nonlinear microwave property of graphene may lead to new graphene microwave devices (microwave saturable absorber, modulator, polarizer, etc) and also pave the way for applications of graphene based microwave communications: such as microwave signal processing, broad-band wireless access networks, sensor networks, radar, satellite communications, and so on.

Appendix

A. Others on microwave source

An optical carrier via an intensity modulator (IM), which is driven by a microwave signal with $2f_{RF}$frequency, the IM output field can be expressed as:

(3)$$\begin{array}{l}{E}_{out1}(t)=\frac{E_0(t)\cos (2\pi f_{0}t)}{{10}^{\alpha /20}}\gamma \cdot \exp \left[j\pi \frac{V_{RF}}{V_{\pi }}\cos \left(2\pi \cdot 2f_{RF}t+\theta \right)\right]+\\ \frac{E_0(t)\cos (2\pi f_{0}t)}{{10}^{\alpha /20}}(1-\gamma )\cdot \exp \left[j\pi \frac{V_{RF}}{V_{\pi }}\cos \left(2\pi \cdot 2f_{RF}t\right)+j\pi \frac{V_{bias}}{V_{\pi }}\right]\end{array}$$

where $E_0$ is the amplitude of the optical field, $V_{\pi }$is the half-wave voltage of the modulator, $V_{RF}$, $f_{RF}$ and $\theta$ are the voltage amplitude, frequency, and phase of the microwave signal, respectively, $V_{bias}$ is the dc bias voltage applied to IM, $\gamma$ is the power splitting ratio of the two IM arms, and $\alpha$ is the IM insertion loss.

For an ideal IM, $\gamma$ is about 0.5 and $\alpha$ is approximately zero. The field of the optical signal can be written as:

(4)$$E_{out1}=\frac{E_0}{2}\left\{\cos \left[2\pi f_{0}t+\beta \cos (4\pi f_{RF}t+\theta )\right]+\cos \left[2\pi f_{0}t+\beta \cos (4\pi f_{RF}t)+\phi )\right]\right\}$$

where$\beta =\pi V_{RF}\text{/}{V}_{\pi }$ is modulation depth of the modulator and $\phi =\pi V_{bias}\text{/}{V}_{\pi }$ is constant phase shift.

When expanding Eq. (4) by using Bessel Functions and set $\phi \text{=2}k\pi \text{,(}k\text{=0,1,2}\text{)}$ and $\theta =\pi$, the odd-order optical sidebands are suppressed, the optical signal can be written as:

(5)$$E_{out1}=E_0\left[\cos (2\pi f_{0}t)J_0(\beta )+2\cos (2\pi f_{0}t){\displaystyle \sum _{n=1}^{\infty }{(-1)}^n{J}_{2n}(\beta )\cos (8n\pi f_{RF}t)}\right]$$

where $J_n$ is the Bessel function of the first kind of order n.

Eliminating the carrier by an optical interleaver and ignoring higher than second-order Bessel functions, the signal only has two second-order optical sidebands, which can be approximately expressed as:

(6)$$E_{out2}(t)\cong -E_0\cdot J_2(\beta )\cdot [\cos (2\pi f_{0}t+8\pi f_{RF}t)+\cos (2\pi f_{0}t-8\pi f_{RF}t)]$$

Beating using a square-law PD, the generated microwave signal is:

(7)$$I_{microwave}=\mu {\left|E_{out2}(t)\right|}^2\approx \mu \cdot E_0^2\cdot J_2^2(\beta )\cdot \cos \left[(2\pi \cdot (8f_{RF}t)\right]$$

where μ is the responsivity of the PD. Setting $f_{RF}$ = 12~12.5 GHz, the 96~100 GHz microwave signal is obtained.

In order to measure the generated the microwave signal, the sub-harmonically pumped conversion mixer system is used to down conversion the microwave signal as shown in Fig. 9(a) . The system is comprised of three parts: microwave signal, the local oscillator signal and intermediate frequency (IF) signal. The generated 100 GHz microwave signal under test is received by an antenna, and then the signal is inputted to the subharmonically pumped mixer, which is driven by a local oscillator (LO) signal. The output signal of mixer is the intermediate frequency (IF) signal and the frequency relationship between them can use the following formula,

(8)$$f_{microwave}=f_{IF}+2\cdot f_{LO}$$

where the local oscillator (LO) frequency is 49.48 GHz, it is obtained by using a frequency multiplier (FM). The spectrum of IF is shown in Fig. 9(b) and the frequency is 0.98 GHz. According to the Eq. (8), the frequency of microwave is about 100 GHz.

 

Fig. 9 (a) The setup for detecting the generated microwave signal. (b) The spectrum of IF signal. ESA: electrical spectrum analyzer.

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B. Microwave saturable absorption

Graphene as a function of increasing microwave power can be described by:

(9)$$\alpha \left(P\right)=\frac{{\alpha }_s}{1+P/P_s}\text{=}{\alpha }_s\cdot \exp (-P/P_s)$$

where $P_s$ is the saturable power, defined as the optical power required in a steady state to reduce the absorption to half of its unbleached value, ${\alpha }_s$ and ${\alpha }_{ns}$ are the saturable and non-saturable loss. The corresponding transmittance fitting equation of graphene can be written by:

(10)$$T\left(P\right)=1-\alpha -{\alpha }_{ns}=\text{1}-{\alpha }_s\cdot \exp (-P/P_s)-{\alpha }_{ns}$$

where ${\alpha }_S$, $P_s$,${\alpha }_{ns}$ is the fitting parameters.

The fitted saturable absorption parameters, ${\alpha }_S$, $P_S$and ${\alpha }_{ns}$ are summarized in Table 1 .

Table 1. Saturable absorption properties of graphene

View Table

C. Z-scan modeling

Saturable absorption can be modeled by the following equation:

(11)$$\frac{dI}{d\zeta }=-\frac{{\alpha }_0}{1+I/I_S}I-{\beta }_0{I}_0^2-\sigma \Delta NI$$

where α₀ is the one photon absorption (including intrinsic free carrier absorption) coefficient, β₀ is the fundamental three-photon absorption (TPA) coefficient, and σ₀ is free carrier absorption (FCA) cross section. I_s characterizes the saturable absorption intensity; the intensity I at the radial position r, time t, the position ς in the sample, and the location of the sample z is denoted as I(z, ς, r, t).

For graphene, we only consider the one photon absorption.

(12)$$\frac{dI}{d\zeta }=-\frac{{\alpha }_0}{1+I/I_S}I$$

The boundary condition required to solve the above equation is the input intensity which is assumed to be a Gaussian:

(13)$$I(z,0,r,t)=I_0{\left[\frac{w_0}{w(z)}\right]}^2\exp \left[-\frac{2r^2}{w{(z)}^2}\right]\exp \left[-\frac{t^2}{{\tau }_0^2}\right]$$

where $w_0$ is the beam waist at the focus, $w(z)=w_0\sqrt{1+{(z/z_0)}^2}$ is the beam radius at z, $z_0=\pi w_0^2/\lambda$ is the diffraction length of the beam, τ₀ is the half width at of the maximum of the pulse, and λ is the wavelength.

From Eq. (12), we can get the following conclusion

(14)$${\displaystyle {\int }_{I(\varsigma =0)}^{I(\varsigma =L)}\frac{dI}{I}}=-{\displaystyle {\int }_{\varsigma =0}^{\varsigma =L}\frac{{\alpha }_0}{1+I/I_s}}d\varsigma$$

where L is the thickness of graphene sample.

From Eq. (14), we can get

(15)$$\ln \left[\frac{I(\varsigma =L)}{I(\varsigma =0)}\right]=-\frac{{\alpha }_{0}L}{1+I/I_s}$$

(16)$$T^{real}=\exp \left(-\frac{{\alpha }_{0}L}{1+I/I_s}\right)\approx 1-\frac{{\alpha }_{0}L}{1+I/I_s}$$

where $T^{\text{real}}$ is the transmittance of laser passing through the graphene sample. However, to analyze the Z-scan data of graphene saturable absorption, normally, we use the normalized transmittance:

(17)$$T=\frac{T^{real}}{T^{real}(I=0)}=\left(1-\frac{{\alpha }_{0}L}{1+I/I_s}\right)\cdot \frac{1}{1-{\alpha }_{0}L}$$

We can use Eq. (15) to determine the intensity I:

(18)$$I(z)=\frac{I_0}{1+z^2/z_0^2}$$

In conclusion,

(19)$$T(z)=\left[1-\frac{{\alpha }_{0}L\cdot (1+z^2/z_0^2)}{1+z^2/z_0^2+I_0/I_s}\right]\cdot \frac{1}{1-{\alpha }_{0}L}$$

However, in the measurement, the focus point is not always at z=0, we have used the following modified equation to fit the saturable absorption curve.

(20)$$T(z)=\left\{1-\frac{{\alpha }_{0}L\cdot I_s\cdot \left[z_0^2+{(z-z_c)}^2\right]}{I_s\cdot \left[z_0^2+{(z-z_c)}^2\right]+I_0\cdot z_0^2}\right\}\cdot \frac{1}{1-{\alpha }_{0}L}$$

here, $z_c$ is the position for the maximum transmittance.

D. Laser beam characterization

A beam profiler was used to measure the beam waist from a Pritel picosecond laser. Firstly, we ensure that after the focusing objective, the laser beam axis is perpendicular to the pin hole of the beam profiler, which is mounted upon the translation stage. Then, finely adjusting the position of the beam profiler through the Newport ESP301 linear motorized stage, the laser beam intensity distribution before and after the focus point can be captured by the beam profiler. Figure 10 shows a typical laser spectrum, the relation between the laser beam waist and the relative position of the focusing objective and the beam intensity profile near the focus point. Based on the relation between the measured laser beam waist and limited by the detection resolution of the beam profiler, the beam waist near the focus point is measured to be as broad as 10 μm. The exact beam waist can be further inferred through fitting a typical Z-scan curve, which gives an estimated beam waist of 3 μm.

 

Fig. 10 (a) Typical laser optical spectrum. (b) Relation between the laser beam waist and the relative position with respect to the focusing objective. (c) Beam intensity profile near the focus point. (d) Relation between laser beam intensity and the relative position of the focusing objective for an optical input power of about 3 mW.

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For the 1053 nm laser source, the optical spectrum and the focal spot have been measured, as shown in Fig. 11 . The full width at half maximum (FWHM) of the laser is about 0.1 nm and the beam waist is about 75 μm, respectively.

 

Fig. 11 (a) The optical spectrum of 1053 nm laser. (b) The image of the focal spot by CCD.

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E. Laser mode locking for the same graphene sample

Using the same the grapheme sample, we have implemented a laser mode locking experiment. Figure 12(a) shows the sketch of the experimental setup of the fiber laser. It is an erbium-doped fiber ring laser made of pure abnormal dispersion fibers. The total cavity length is about 169 m, which contains a 1.1 m EDF. The GVDs of EDF and SMF are –20 ps²/km and –23 ps²/km at 1550 nm, respectively. A polarization independent isolator (PII) is used to force the unidirectional operation of the ring cavity. An intra-cavity polarization controller (PC) is used to adjust the cavity birefringence. A 980/1550 nm WDM coupler is used to couple the pump light and a 10% fiber coupler is used to output the laser emission. The laser is pumped by a 980 nm pump laser. Finally, a pigtailed fiber bench spacing about 9 cm is added to this cavity. To integrate the graphene sample into the fiber laser cavity, we insert the sample into the fiber bench in the ring cavity.

 

Fig. 12 (a) Schematic of the experimental setup. (b) soliton spectra. (c) Oscilloscope trace of the soliton. (d) An autocorrelation trace of the laser emission.

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If the graphene sample is absent in the ring cavity, there are stable continuous wave output. When the graphene sample is inserted into the cavity, the mode-locking pulse can be generated by adjusting PC. Self-started mode locking of the laser occurred at the incident pump power of about 70 mW. The optical spectrum of the mode locked pulses is centered at 1564.4 nm and has a 3dB bandwidth of 1.1 nm as shown in Fig. 12(b). We have observed the Kelly sidebands on the spectrum, indicating the soliton operation. Figure 12(c) shows the measured oscilloscopes trace within nanosecond time scale. The pulse circulated in the cavity with the fundamental cavity repetition rate 1.21 MHz. Figure 12(d) is the measured autocorrelation trace of the mode locked pulses. It has a FWHW width of 3.81 ps. If a Sech² pulse is assumed, the pulse duration is 2.47 ps.

Acknowledgment

The authors thank Dr. Encai Ou (College of Chemistry and Chemical Engineering, Hunan University) for the help with the graphene preparation. This work is partially supported by the National 973 Program of China (Grant No. 2012CB315701), the National Natural Science Foundation of China (Grant No. 61025024), the National 863 Program of China (Grant No. 2011AA010203), Program for New Century Excellent Talents in University of China (Grant No. NCET 11-0135), and Hunan Provincial Natural Science Foundation of China (Grant No. 12JJ7005).

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