1. Introduction

Graphene, one-atom-thick semiconductor with unique linear band structure near Dirac point, exhibits peculiar optical and electrical properties and has recently attracted significant attention [1–8 ]. The fascinating optical properties of graphene (such as its universal optical absorbance [9] and electrostatic gating-modified optical absorbance [10]) make it an ideal material for many photonics applications, including optical sensors [11, 12 ], optical modulators [13–15 ], and photodetectors [16, 17 ]. In addition, extinction coefficient k and refractive index n (i.e., optical constants) are two important and useful optical parameters for accurately describing the optical characteristics of graphene in these photonics devices/structures [11, 13, 14, 16–18 ]. Extinction coefficient k is especially important because most graphene-based optical devices utilize the graphene’s optical absorption response properties [11–13, 16–18 ]. Precise estimation of graphene’s extinction coefficient k is important for designing and optimizing graphene-based photonics and optoelectronics devices.

The extinction coefficient/absorbance of graphene, arising from the optical transition within the linear energy band structure, is related to the energy state occupancy corresponding to the optical transition [1, 4, 19 ]. The optical constants of graphene have been extensively characterized by several groups using spectroscopic ellipsometry [2–4, 19, 20 ]. These reports demonstrated that the extinction coefficient is isotropic in the graphene’s plane [3, 4, 16 ]. However, previous studies concentrated on the steady-state extinction coefficient. It is known that the energy state occupancy is modified in optically excited graphene owing to the injection of photo-excited carriers and subsequent carrier relaxation [7, 21–26 ], changing the extinction coefficient. Recent pump-probe studies reported by Mittendorff and Malic et al show that the initial spatial occupancy of momentum states by photo-excited carriers is anisotropic at optically pumped state [7, 21 ], and this occupancy immediately evolves to fully isotropic with carrier relaxation [7, 21, 27 ]. We can deduce that the transient change of extinction coefficient induced by optical pumping is polarization dependent. However, the polarization dependence of optical pump-induced change of graphene extinction coefficient has not been unambiguously investigated. A clear understanding of optical pump-induced changes of extinction coefficient and the dependence of this change on polarization can help predict graphene’s transient optical response characteristics (such as, transient reflection, transient transmission and transient absorption). This knowledge is likely to be important for developing graphene-based high-speed photonic devices, such as saturable absorbers and optical modulators.

In this work, we performed degenerate and nondegenerate pump-probe measurements to explicitly study the dependence of optical pump-induced changes of graphene extinction coefficient on polarization of incident light. The degenerate pump-probe experiment presented here is similar to that reported by Mittendorff et al, which was used to study distribution of photo-excited carriers in momentum space [7]. During the initial stage of photo-excited carrier relaxation, the dependence of pump-induced change of extinction coefficient on polarization was observed for light with photon energies identical to that of the excitation light. Spatial distributions of non-equilibrium carriers’ momenta determined the dependence of extinction coefficient change on polarization. Our results suggest an efficient way for controlling graphene’s transient extinction coefficient by manipulating the distribution of hot carriers. This work is closely related and in certain respect repeats the work of Mittendorff et al [7].

1.A Relationship between Δk and ΔT/T

For beams that are normally incident onto graphene placed on a substrate, the normalized optical transmission can be written as [28]:

For a single graphene layer (N = 1) on quartz substrate and incident wavelength of λ = 800 nm, the change Δk of the extinction coefficient can be directly related to the differential transmittance by $\Delta k=-81.92\Delta T/T$. Therefore, a fundamental material property of graphene can be obtained from the differential transmittance. We performed pump-probe measurements to study the dependence of optical pump-induced change of Δk on polarization. In this method, the change Δk_p-p(t_delay) (obtained from the pump-probe differential signal ΔT(t)/T using Eq. (5)) equals to the convolution of Δk(t) and Gaussian intensity distribution of the probe pulse, i.e., $\Delta k_{p-p}(t_{delay})=\frac{2\sqrt{\ln 2}}{\sqrt{\pi }{\tau }_{\text{FWHM}}}\text{[}\Delta k(t)\ast \exp (-{(\frac{\sqrt{2\ln 2}}{{\tau }_{FWHM}}t)}^2)\text{]}$. The parameter τ_FWHM is the pulse width of the probe beam. Clearly, the dependence of Δk_p-p(t_delay) on polarization is determined by Δk(t). Thus, in what follows we study the dependence of Δk_p-p(t_delay) on polarization.

2. Experiment and analysis

2.A. Experimental setup

A Ti: sapphire mode-locked laser provided the laser pulses (repetition rate of 1 KHz, central wavelength of 800 nm). The pump-probe configuration used in our experiments is shown in Fig. 1(a) . To reduce the noise arising owing to the scattering of the pump beam, the pump and the probe beams were incident onto the sample at different angles. After passing through a λ/2 plate-Glan Taylor prism-λ/2 plate combination, the 800 nm linearly polarized probe beam was incident normally onto the graphene sample (the probe beam was incident normally onto the sample for avoiding any potential influence of oblique incidence of the probe beam on the Δk(t_delay) measurement [19, 20 ]). The probe beam’s linear polarization orientation (Fig. 1(b)) was controlled by the corresponding λ/2 plate of the combination. For the data presented here, the pump beam was incident onto the sample at an angle of 35° relative to the sample surface normal. It should be noted the measured polarization dependence of Δk is independent on the pump beam’s incident angle. i.e., the $\frac{\max {(\Delta k_{p-p})}_{\left|\right|}}{\max {(\Delta k_{p-p})}_{\perp }}$ is ∼1.4 and 1 for degenerate and non-degenerate measurement, respectively. The pump beam’s linear polarization was controlled by using another λ/2 plate-Glan Taylor prism-λ/2 plate combination.

 

Fig. 1 (a) Schematic of the pump-probe experiment. (b) Polarization orientation of linearly polarized probe and pump beams. (c) Raman spectra of graphene on quartz substrate. The 2D peak is described by a single Lorentzian.

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Two combinations of pump/probe frequency had been used for studying the changes of extinction coefficient induced by optical pumping. The 800 nm and 400 nm pulses were used separately for optical pumping. To reveal the relationship between the dependence of Δk(t) on polarization and the carrier distribution, the dependence of $\Delta k_{\lambda =800nm}$ on polarization was compared for two types of optical excitation (λ_pump = 800 and 400 nm). For pump beams of 400 nm the pulses were obtained by frequency doubling of 800 nm pulses in a BBO crystal. Pump-induced transmittance change was measured using a balanced detector in conjunction with a lock-in amplifier referenced to a 383 Hz mechanically chopped pump. Pulse durations τ_FWHM at the sample were 300 ± 30 fs and 230 ± 30 fs, obtained by two-photon absorption autocorrelation in ZnSe for 800 nm pulses [29] and in CS₂ for 400 nm pulses [30], respectively. The spot size of the probe beam at the sample was approximately 35 μm, the probe beam’s fluence was ∼0.13 mJ/cm² (the density of carriers was ∼1.2 × 10¹³ cm⁻²). The spot size of the pump beam at the sample was ∼80 μm. The measured ΔT/T was converted into Δk_p-p using Eq. (5).

We studied a chemically vapor-deposited (CVD) grown graphene monolayer that was transferred onto a quartz substrate. Figure 1(c) shows the typical Raman spectrum at 514 nm, which is similar to graphene monolayers spectra reported in literature (the 2D-peak is located at ∼2695 cm⁻¹ and the G-peak is located at ∼1581 cm⁻¹) [31]. This spectral distribution confirms that the sample studied by us was a graphene monolayer. Measurements performed at different locations on the sample yielded similar results.

2.B. Experimental results

First, we present the experimental results obtained from degenerate pump-probe measurements. Figures 2(a) and 2(b) show the probe polarization dependent Δk_p-p time trace for pumping with s- and p-polarized pulses, respectively. Owing to optical absorption bleaching caused by Pauli blocking, optical excitation yielded strongly reduced extinction coefficient around zero-delay time [21, 22, 25, 26, 32 ]. Immediately following the photo-excitation, the change of extinction coefficient, Δk, recovered to zero owing to carrier thermalization and subsequent carrier cooling [21].

 

Fig. 2 Probe polarization-independent Δk_p-p time traces for the pump wavelength of 800 nm. (a) s- and (b) p-polarized pulses were used in the measurements. The shaded area accentuates the significant dependence of Δk_p-p on probe polarization. The longer decay time extracted from bi-exponential fitting of the Δk_p-p data at positive delays (corresponding to the carrier cooling process) is 1 ± 0.1 ps, in broad agreement with previous measurements (0.8 (−0.3/+1.5) ps in [34], 0.7 ± 0.2 ps in [8]). The faster decay time is on the order of the pulse width and is, therefore, not accurately resolvable. (c), (b) Peak Δk_p-p as a function of probe polarization. The solid lines are the cosine fits. The pump fluence was ∼2.7 mJ/cm² and the excited carrier density was ∼2.5 × 10¹⁴ cm⁻².

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As Fig. 2 shows, the Δk_p-p around the delay of 0 (i.e., the photo-excitation process and the initial stage of carrier relaxation) is a function of probe polarization. For either S or P pump polarizations the peak Δk_p-p is maximal when the probe polarization is parallel to the pump polarization and monotonically decreases as the probe polarization is varied to cross with pump polarization. In other words, the pump-induced Δk is maximal for the light that has the same polarization as the excitation light. With carrier relaxation, this dependence of Δk on polarization decreases rapidly, and no dependence on polarization is observed for long delays (above 0.7 ps).

According to previous reports, the photo-excited carriers are primarily located in the excited state, and the spatial distribution of momenta of these carriers is maximized/vanishes in the direction normal/parallel to the polarization of the excitation light [7, 21, 22, 33 ]. Immediately following the photo-excitation, non-equilibrium carriers are scattered into energetically higher and lower states through two competing processes: carrier-carrier scattering and carrier-phonon scattering. This reduces the excitation state occupancy by the carriers [7, 21, 22, 34 ]. At the same time, this anisotropic distribution centered at the excitation state rapidly evolves into fully isotropic in the entire energy band [22, 27 ]. In energy states away from the excitation state, the distribution of hot carriers is fully isotropic throughout the entire non-equilibrium process [22, 27 ]. Because the optical pump-induced reduction of extinction coefficient indicates the inter-band transition bleaching caused by Pauli blocking, the observed dependence of Δk_p-p on polarization originates from anisotropic distribution of non-equilibrium carriers (Fig. 3(a) ). If the distribution of carriers in the optical coupling state is isotropic, the pump-induced Δk should be polarization-independent (Fig. 3(b)).

 

Fig. 3 Schematic of optical transition and carrier distribution in momentum space. Blue arrows denote the polarization of incident light.

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2.C. Analysis of the dependence of Δk and max(Δk_p-p) on polarization

In what follows we analyze the dependence of Δk on polarization, based on the distribution of carriers as reported in literature [7, 21, 22 ]. S polarization direction was set as the angular coordinate in the momentum space. According to the works of Mittendorff and Malic et al. [7, 21 ], the spatial occupancy of momentum states by the carriers photo-excited by light with polarization angle of ${\theta }_l$ is proportional to ${\left|\sin (\phi -{\theta }_l)\right|}^2$, where $\phi$ is the angle between the momentum vector of the photo-excited carriers and the polarization of excitation light. Clearly, the initial occupancy of the photo-excited carriers is anisotropic and peaks in the direction normal to the polarization of excitation light (Fig. 3(a)) [22]. As discussed above, the change of extinction coefficient caused by Pauli blocking arises owing to the interaction between the non-equilibrium carriers from the excitation light and the carriers that are photo-generated by the studied incident light [7]. If the momentum spatial occupation of nonequilibrium carriers (photo-generated by the excitation light) in the energy state corresponding to the studied wavelength is denoted by $f(\phi ,t)$, the resulting Δk can be written as:

Only if $f(\phi ,t)$ is a function of $\phi$ the quantity $\Delta k({\theta }_l)$ depends on the polarization of light ${\theta }_l$. If the distribution of carriers could be described as $f(\phi )={\left|\sin (\phi -{\theta }_m)\right|}^2$ (e.g., the initial distribution of carriers photo-excited by the excitation light with polarization angle of ${\theta }_m$), one would obtain $\Delta k\propto [2+\cos 2({\theta }_l-{\theta }_m)]$ [7].

Because the laser pulses used in the present study were not extremely narrow, the distribution of carriers changes during probing owing to carrier relaxation [22]. As a result, the value of Δk is different for different parts of the probe pulse (Fig. 4 ). In what follows, we use the methods of differential calculus to derive the dependence of optical pump-induced Δk_p-p on polarization, given the degenerate pump-probe measurements. As shown in Fig. 4, the probe pulse lags the pump pulse by $t_{delay}$. We assume the pump polarization and probe polarization are characterized by the angles ${\theta }_{pump}$ and ${\theta }_{probe}$, respectively, relative to the S polarization direction. For randomly chosen segments of the probe and pump pulses (closed segments in Fig. 4), the momentum spatial occupation numbers of the carriers generated by the two segments are:

 

Fig. 4 Pump and probe pulses partitioned into several segments.

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Former pump-probe experiments showed that the relaxation of the population of carriers could be satisfactorily described by a bi-exponential function [8, 34 ]. We used the functional form of $\exp (-\frac{t_{pump}-t_{probe}}{{\tau }_1})+A\exp (-\frac{t_{pump}-t_{probe}}{{\tau }_2})$ for describing the population relaxation of those carriers photo-excited by the pump pulse segment. Here, τ₁ = 110 ± 40 fs and τ₂ = 900 ± 200 fs are the two decay times [8, 23 ] and A is a free parameter used to describe the weight of the two relaxation pathways. Same as for silicon [35], the relaxation of anisotropy in the distribution of carriers could be assumed to be described by a single exponential function $[{\left|\sin ({\Phi }_k-\theta )\right|}^2{e}^{-\frac{t_{pump}-t_{probe}}{{\tau }_{aniso}}}+0.5(1-e^{-\frac{t_{pump}-t_{probe}}{{\tau }_{aniso}}})]$. The parameter τ_aniso is the anisotropy decay time. Because this relaxation function is used for describing the ratio distribution of non-equilibrium carriers for different momenta vectors, the term $[1-\exp (-\frac{t_{pump}-t_{probe}}{{\tau }_{aniso}})]/2$ is used for guaranteeing that the integration over $\phi$ does not change after carrier relaxation. Thus, when these carriers interact with the probe pulse segment at time $t_{probe}$, the distribution of the carriers in the excitation state can be written as:

According to Eq. (6), Δk is proportional to ${\displaystyle {\int }_0^{2\pi }{n}_{pump,decay}(t_{pump})\cdot n_{probe}(t_{probe})d\phi }$ . By integrating over all segments of pump and probe pulses, the measured Δk _p-p can be written as:

Thus, the measured peak Δk_p-p (i.e., Δk_p-p(t_delay = 0)) can be described as a cosine function of the probe polarization (angle ${\theta }_{probe}$). The measured dependence of peak Δk_p-p on polarization, induced by s- and p-polarized pulse pumping, is shown in Figs. 2(c) and 2(d), respectively. As the solid lines demonstrate, this dependence is satisfactorily described by a cosine function. From this fit, we estimated that ${\tau }_{aniso}$≈45 fs. This value broadly agrees with the reported anisotropy relaxation time in ΔT/T (i.e., 150 fs, corresponding to ${\tau }_{aniso}$≈30 fs, exp(−150/30)≈0) [7]. Clearly, the time ${\tau }_{aniso}$ is much smaller than the pulse width used in our experiment. Figure 3 and the integration in Eq. (9) can help explain why a clear dependence of Δk _p-p on polarization could be observed when pulses with duration longer than ${\tau }_{aniso}$ were used. We suggest that the anisotropy in the distribution of carriers photo-excited by the pump pulse segment could be monitored by the probe pulse segment which is temporally very close to (and possibly overlaps with) this pump pulse segment.

According to Eq. (9), the $\frac{\max {(\Delta k_{p-p})}_{\left|\right|}}{\max {(\Delta k_{p-p})}_{\perp }}=\frac{M+N}{M-N}$ ratio is always above 1 owing to the non-zero value of N. In principle, the dependence of max(Δk_p-p) on polarization could be observed for any pulse. Detailed calculations show that $\frac{\max {(\Delta k_{p-p})}_{\left|\right|}}{\max {(\Delta k_{p-p})}_{\perp }}$ decreases with pulse width and increases with ${\tau }_{aniso}$ (data not shown). This implies that the effective Δk decreases with the pulse width owing to the rapid relaxation of anisotropy in the carrier distribution. The analysis presented above is only valid for Δk corresponding to the optically pumped states. For Δk corresponding to other energy states, obtaining an explicit expression on the dependence of Δk on polarization is challenged by the complicate carrier relaxation process in graphene band structure [21].

To further study the optical pump-induced Δk and confirm the relationship between the dependence of optical pump-induced Δk on polarization and distribution of non-equilibrium carriers, we performed the measurements with the pump of 400 nm and probe of 800 nm [36]. The non-equilibrium carriers in the optically probed state (0.775 eV) were created by scattering of photo-excited carriers in the energy state of 1.55 eV (pump: 400 nm) [37]. As discussed above, the distribution of these non-equilibrium carriers was isotropic during the non-equilibrium process because the optically probed state was away from the excitation state [22, 27 ], and no dependence of Δk on polarization was expected. The obtained Δk_p-p is shown in Fig. 5 vs. the delay time.

 

Fig. 5 Probe polarization-independent Δk_p-p time traces for the pump wavelength of 400 nm. The resulting curves overlap (the Δk_p-p time traces for p-polarized pulse pumping are not shown here). The pump fluence was ∼1.9 mJ/cm² (the carrier density was ∼1.6 × 10¹⁴ cm⁻²).

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Following the photo-excitation, the extinction coefficient decreased owing to carrier thermalization, and Δk_p-p reached a peak value when a distribution of hot carriers was established [21, 22 ]. As expected, the Δk_p-p time traces for different probe beam polarizations were the same.

Figure 3 shows the schematic of the optical transition and carrier distribution. If the distribution of non-equilibrium carriers in the energy state matching the incident light energy is anisotropic (Fig. 3(a)), the optical pump-induced Δk is a function of light polarization. Conversely, if the distribution is isotropic, the optical pump-induced Δk is polarization-independent. Thus, for avoiding anisotropy of transient absorption in the spectral range, graphene can be excited using shorter-wavelength light.

3. Summary

The dependence of optical pump-induced change of graphene extinction coefficient on polarization was determined by the spatial distribution of momenta of non-equilibrium carriers. For the light with energy identical to that of the excitation light, the optical pump-induced change of graphene extinction coefficient was maximal/minimal for the light with polarization parallel/normal to the excitation light. This polarization dependence decreased rapidly with increasing carrier relaxation. For the light with energy away from that of the excitation light, the optical pump-induced change of graphene extinction coefficient was polarization-independent. Manipulating the excitation light is an efficient way for controlling the polarization characteristics of graphene extinction coefficient. This, in turn, allows manipulating transient optical response characteristics of graphene.