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The polarization dependence of transient optical reflection, induced by nonequilibrium carriers isotropically distributed in momentum space, of graphene on substrate is experimentally and theoretically investigated. It is found that this transient optical reflection could be made greatly polarization dependent by using oblique incidence for light, and the characteristic of this polarization dependence could be flexibly altered with incident angle and incident direction (from graphene to substrate, or from substrate to graphene). Our results suggest that through polarization of incident beam is an efficient way of manipulating graphene transient optical reflection.
© 2015 Optical Society of America
1. Introduction
The outstanding optical properties of graphene (e.g., wavelength-independent linear absorption, ultrafast saturable absorption, etc) make it a promising material for, among others, a variety of photonic applications, including fast photodetectors, optical modulators, ultrafast mode-locked lasers and photosensing [1–16 ]. The capability of these photonic devices is closely related to the optical response (e.g., optical absorption, transmission and reflection) of graphene [2,12,17,18 ]. For many graphene-based optoelectronic and photonic applications, it is highly desirable to manipulate the optical response of graphene. Previous research has mainly focused on the graphene optical response at steady state [5,6,19,20 ]. However, manipulating the ultrafast optical response of graphene is important for the development/design of graphene-based high-speed photonic devices [2,12,17,18 ].
If the optical properties of materials are anisotropic, we could consecutively adjust the optical response of these materials by changing the polarization state of incident beam [21]. The graphene optical conductivity/extinction coefficient, which could accurately describe the optical behavior of graphene layers, is related to the carrier occupation at the energy state corresponding to the optical transition [6,19 ]. The carrier occupation could be modified by photoexcitation and carrier relaxation, resulting in the variation of charge density for optical transition and extinction coefficient [2,8,22 ]. It is showed that extinction coefficient change Δκ is polarization dependent only if the spatial distribution of non-equilibrium carriers’ momenta is anisotropic [9,20 ]. In photoexcited graphene, the initial distribution of photoexcited carriers centered at excited state is anisotropic [23], resulting in the transient anisotropic Δκ. However, this anisotropic distribution of nonequilibrium carriers in graphene is not stable and rapidly relaxes to fully isotropic in the whole energy band (within 200 fs) due to carrier-phonon scattering [8,23,24 ]. As a result, polarization independence dominates the Δκ over the nonequilibrium process. It is interesting to make the transient optical response induced by nonequilibrium carriers isotropically distributed in momentum space (i.e., isotropic Δκ) is polarization tunable, which would provide a potential method to modulate transient optical response for applications of graphene in high-speed photonic devices, yet such aspects have been overlooked so far.
In this paper, we report that the transient optical reflection of graphene, caused by isotropic Δκ, on substrate could be greatly polarization dependent by means of using obliquely incident beam. And, this polarization dependence characteristic could be manually adjusted through incident angle and incident direction (namely, from graphene to substrate, or from substrate to graphene). The polarization dependence is well explained with a theory based on a tri-layer structure model. It is suggested that tuning of polarization is an excellent method to manipulate transient optical reflection of graphene-based high-speed photonic devices.
2. Experiment
We employ nondegenerate pump-probe measurement [11,17 ] to study the transient optical reflection induced by nonequilibrium carriers. Linearly polarized 400 nm pump pulse is used to excite electrons from the valence band to the conduction band and create highly non-equilibrium carrier distribution. Immediately after optical excitation, these nonequilibrium carriers, centering at the excitation state of 1.55 eV, relax to the Dirac point via initial thermalization and subsequent cooling processes [9,24–32 ]. During relaxing to the Dirac point, these nonequilibrium carriers pass through the state to be optically probed (0.775 eV) and results in the change of extinction coefficient for 800 nm light [4,24 ]. Meanwhile, an 800 nm probe pulse is obliquely incident into the sample, the reflected beam is collected by a detector. We study the transient reflection signal resulting from Δκ, and concentrate on the dependence of this signal on polarization of probe beam. Since the nonequilibrium carriers at energy state of 0.775 eV are created by scattering from the excitation state [4,24,33 ], the distribution of these carriers is fully isotropic in the entire nonequilibrium process [4,23,24,33 ], i.e., Δκ is isotropic in the whole transient process. Thus, the potential contribution of initial anisotropic carrier distribution to our studied polarization dependence of transient optical reflection is excluded.
In the experiment, femtosecond laser source was a Ti: sapphire mode-locked amplifier (Spitfire Pro, Spectra Physics), which provided the laser pulses of 1 kHz repetition rate at wavelength of 800 nm. 400 nm pump pulses were obtained by frequency doubling of 800 nm pulses via a BBO crystal. Two λ/2 plate-Glan Taylor prism-λ/2 plate combinations control the linear polarization orientation [Fig. 1(d) ] and energy of probe and pump beams, making it possible to determine the reflection change as a function of the probe polarization. After passing the optical components before the sample, the durations τ FWHM of 800 nm and 400 nm pulses were 300 ± 30 and 230 ± 30 fs, respectively.
Fig. 1 (a)-(c) Pump-probe experiment configurations of G1, G2 and G3. (d) Polarization orientation of linearly polarized pump and probe beams. ϕ is the incident angle, θ is the angle of linear polarization with respect to the (S) polarization direction. Refractive index of substrates n quartz = 1.46, n prism = 1.5168.
As illustrated in Fig. 1, three typical experimental configurations were used to separately study the influence of incident angle and incident direction on the characteristic of the polarization dependence. In all the configurations, the probe beam was obliquely incident into the sample. To study the influence of incident angle, 800nm linearly polarized probe beam was incident into graphene film on fused quartz [sample A] at angles of 39 and 72 degrees [Figs. 1(a) and 1(b)], respectively. To explore the effect of incident direction, we apply a graphene film attached on the inclined plane of a right-angle prism [sample B, the influence of linear reflection from the air-substrate interface was excluded]. The probe beam was incident into graphene film from the prism, with an angle of 39 degrees relative to normal line of the inclined surface [Fig. 1(c), using this angle just to compare the experimental results between G1 and G3]. It is noted that the measured reflection change is from optically excited graphene in the three configurations. The probe spot size at sample (∼35 μm) was smaller than the pump spot (∼80 μm), the incident angles of pump pulse were 50 degrees. Since the strength of the measured transient reflection signal was independent of the polarization of the pump light [24], the polarization of pump beam can be randomly set (here the pump beam was s-polarized). And, owing to that the polarization dependence of transient reflection is focused, we did not use identical pump fluence for each configuration in the measurements.
Chemical vapor deposition (CVD)-grown multilayer graphene was used in the experiment (thickness changes with position, with an average of 4 monolayer thickness) [34]. The decoupled nature of the layers was evidenced by Raman spectroscopy [35]. To change the incident direction, we used two different samples: the first sample (sample A), graphene film was transferred onto a fused quartz substrate using the method described in [36]; the second sample (sample B), graphene film was on the inclined surface of a right-angle prism. We obtained consistent results at different sample position in the measurements.
3. Results and discussions
To study the transient optical reflection, we measured the differential reflection ΔR/R = (R' - R) / R, where R' and R are the reflectivities with and without pump excitation, respectively [14]. By varying the delay time between pump pulse and probe pulse, time-resolved ΔR(t)/R for different polarized probe beam was obtained. Figures 2-4 show the representative ΔR/R and linear reflection R for the three configurations. For all these ΔR/R curves, the differential reflection caused by Pauli blocking exhibits a fast initial change, of the order of the pulse duration, and a slower relaxation component [11,37,38 ]. The initial change of reflection is a result of carrier thermalization, and the subsequent decay of ΔR/R is due to carrier cooling [11,26–32,38 ].
Fig. 2 Probe polarization dependence of ΔR/R and R for G1. (a) time-resolved ΔR/R traces, zero time delay corresponds to peak pump-probe signal. (b) Linear reflection R and (c) the transient reflection signal at zero time delay max(ΔR/R) versus probe polarization, the symbols are experimental data, and the solid lines are theoretical fits (max(Δκ) = −0.29i, α ≈0.13). The pump fluence was ∼0.9 mJ/cm2, the carrier density of each graphene layer was ∼8 × 1013 cm−2.
Fig. 3 Probe polarization dependence of ΔR/R and R under G2. (a) time-resolved ΔR/R traces. (b) R and (c) max(ΔR/R) versus probe polarization. max(ΔR/R) is negative for s-polarized probe beam and could be successively altered to positive by changing the polarization of probe beam to (P). The parameters of max(Δκ) = −0.51i and α ≈0.13 are used for fitted curves. The pump fluence is estimated to be 1.8 mJ/cm2, the carrier density of graphene was 1.5 × 1014 cm−2.
Fig. 4 Probe polarization dependence of ΔR/R and R for G3. (a) time-resolved ΔR/R traces. (b) R and (c) max(ΔR/R) versus probe polarization. max(ΔR/R) is positive for s-polarized probe beam and negative for p-polarized probe beam. Parameters of max(Δκ) = −0.04i, and α ≈0.09 are used for the fitted curves. The pump fluence of 400 nm pulses was ∼0.079 mJ/cm2, the carrier density of each graphene layer was ∼6 × 1012 cm−2.
In contrast to the polarization-insensitive case of normal incidence [39], pronounced polarization dependence of transient reflection is observed when the beam is obliquely incident into the graphene layers. In the three configurations, the linear reflection R is always maximum/minimum when the incident beam is s-/p-polarized. However, the polarization dependence characteristic of ΔR/R differs greatly for the three configurations. As illustrated in Figs. 2-4 , the experiments show the following behaviors: For configuration G1 [Fig. 2], the ΔR/R is negative and the magnitude increases with polarization being changed from S to P. Under configuration G2 [Fig. 3], however, the ΔR/R curve features a transition from reduced to enhanced when the polarization is changed from S to P. For configuration G3 [Fig. 4], the form of ΔR/R also varies strongly with light polarization. Contrary to the cases of G1 [Fig. 2] and G2 [Fig. 3], the pump-induced reflection change of G3 [Fig. 4] exhibits a crossover from decreased to enhanced with changing light polarization from S to P. The prominent feature of G2 and G3 is that the pump-induced reflection change could be altered from enhanced to suppressed by polarization of incident beam. And, the polarization dependence feature of G2 is opposite to G3. These results indicate that the polarization dependence characteristics of transient reflection could be altered by incident angle and incident direction.
Now we turn to the explanation of the observed polarization dependence of the transient reflection. As we know, the transient reflection is a consequence of graphene extinction coefficient change induced by the nonequilibrium carriers in graphene due to Pauli blocking of inter-band transitions [18,37,38 ]. To analyze the transient reflection of graphene on substrate, we first calculate reflectivity of femtosecond laser pulses as a function of extinction coefficient of graphene.
For graphene film on a bulk dielectric substrate, we could simplify the area where graphene deposited on into a tri-layer structure [Fig. 5 ] consisting of graphene sandwiched between two semi-infinite dielectrics [40]. The graphene film could be regarded as a dielectric layer at optical frequency, and the optical behavior of graphene on substrate could be described by the optical constant [5,6,19 ]. The effect of multiple reflections from bulk substrate can be neglected since the incident angle of probe beam is large [21]. Thus, the optical response of graphene on a bulk dielectric substrate could be calculated based on a tri-layer structure model [40].
Fig. 5 (a) Optical reflection and transmission of a tri-layer structure. (b) Optical reflection and transmission of graphene on prism. Inset shows that the graphene on prism is a tri-layer structure.
For light incident into a tri-layer structure [Fig. 5(a)], the reflectivities of s- and p-polarized beams could be calculated based on Maxwell equations solving boundary conditions at the interfaces and are given by [21]
where ni, nt are the refractive indexes of incident and transmitted spaces, respectively. n 1 is complex refractive index of middle layer [Fig. 5(a)]. The other parameters are defined as: , , , , , , and is thickness of middle layer.For other polarized incident beam, we could divide the incident beam into s- and p-polarized components according to Malus' law. Corresponding, the reflectivity can be traced back to the collective reflectivity contributions of the s- and p-polarized beams.
For configurations G1 and G2, the reflectivity of graphene on substrate is given by
where na, ng, ns, are the complex refractive indexes of air, graphene and substrate, respectively. For graphene, the complex refractive index ng is the optical constant [6,19,20 ], i.e., ng = n0 + iκ. The preexcitation, room-temperature optical constant of graphene at 800 nm is ng = 3.0 + 1.45i [6].For configuration G3 [Fig. 5(b)], graphene on the incline surface of the prism is a tri-layer structure. Taking into account the transmissions at the right-angle sides of the prism, the reflectivity of s- and p-polarized beam is given by
where , , is the incident angle at the right-angle side of the prism [Fig. 5(b)]. Similarly, for any polarized incident beam, the reflectivity from the graphene on prism is given byClearly, the linear reflectivity depends on the polarization of incident beam in the three configurations. As shown in Figs. 2-4 , the measured polarization dependence of R could be well fit with Eqs. (3) and (6) . Therefore, the obtained reflectivity [Eqs. (3) and (6) ] is useful for quantitative optical analysis of graphene layers.As we know, the complex refractive index of graphene for 800 nm light is of if there is no optical pumping [6,39 ]. In optically excited graphene, the graphene extinction coefficient changes immediately after photoexcitation [39]. Subsequently, this extinction coefficient change decays with carrier relaxation in a bi-exponential function. So, we could describe the complex refractive index of optically excited graphene as [6,22,39 ], where τ 1 = 110 ± 40 fs and τ 2 = 900 ± 200 fs are the two decay times of the decay channels [16], and parameter α is the amplitude ratio between the slow decay channel and fast decay channel in the biexponential decay function [1,16 ]. The value of α indicates the relative contribution of the two channels to carrier relaxation [12]. The mechanisms of the two decay channels strongly depend on the excitation intensity [14–16,29 ]. At low pumping intensity, the fast and slow decay channel is ascribed to carrier-carrier scattering and carrier-phonon scattering, respectively [29]. However, at high pumping intensity, the fast decay channel is corresponding to the cooling process with optical phonon emission, and the slow decay channel is the cooling process by converting energy from optical phonons into acoustic phonons [16]. Δκ 0(1 + α)/2 is the maximum extinction coefficient change induced by optical excitation.
As discussed above, the fractional decrease in the reflection is given by the change in the extinction coefficient. We can therefore calculate the transient optical reflection in terms of the reflectivity of with and without optical pumping. By convolution calculation [22], the time-resolved pump-probe differential reflection measured by Gaussian femtosecond pulses is given by
For G1 and G2
For G3
According to Eqs. (7) and (8) , the ΔR(t)/R and max(ΔR/R) could be calculated for the three configurations. From fitting of time-resolved ΔR(t)/R (not shown), we could determine the values of α and max(Δκ) for each configurations (listed in the caption of each figure). The obtained value of α is within 0.11 ± 0.2, which does not change greatly with configuration within experimental errors. The change of max(Δκ) in each configuration is mainly caused by the pump fluence variation. As the solid lines shown in Figs. 2-4 , the calculated polarization-dependent max(ΔR/R) based on obtained α and max(Δκ) are in good agreement with the experimental results. We could explain the observed polarization dependence of transient optical reflection within the framework of Maxwell’s equations and extinction coefficient change Δκ induced by nonequilibrium carriers.
Now we focus on the process of generation of the secondary waves, i.e., reflected and transmitted light, at the air-glass interface (here, glass is fused quartz in G1 and G2, and BK7 glass in G3), to obtain a physical insight of the very different reflection behaviors affected by the graphene layer, in the three configurations. Since the graphene layer is extremely thin compared with substrate, the graphene layer can also be treated as an additive perturbation of the reflection from air-glass interface [41]. The secondary waves are physically generated by the electric dipoles at the glass surface, which are induced by the incident electric field [21]. In an isotropic medium such as glass, the dipole moment is in the same plane as the incident electric field. The emission directions of the dipole in the two media are governed by the Snell's law [21]. And the oscillation direction of the dipole is perpendicular to the transmitted light. The field strength of the reflected wave is determined by the component of the dipole moment that is perpendicular to the anticipated direction of reflection. The graphene layer on the glass has a two-fold effect on the secondary waves generation: (1) The charge carriers of graphene on glass can impose a greater electric force than the incident field on the dipole oscillation at the glass surface, increasing the tangential strength of dipole; (2) But when the dipole is induced before it interacts with the charges in graphene, some portion of the dipole oscillation energy will drive the charges in graphene to move, resulting in a reduced tangential dipole strength at the glass surface.
On the one hand, for light impinging on the glass surface from air, when the glass is covered by graphene, the light polarization density is enhanced in the graphene layer due to the increased charge density at the surface. Since graphene is a two dimensional material, this enhancement occurs only in the plane of graphene (i.e., the tangential direction). As a consequence, the induced dipole moment at the glass surface is also enhanced in the tangential direction. In general, this tangential change of the dipole moment modifies the reflection and transmission of light. [The only exception is p-polarized light incident at the Brewster's angle. In this case, the reflectivity is zero, because the dipole moment is aligned to the anticipated reflection direction, and it never emits energy to the parallel directions, see Fig. 6(a) .] Optical pumping results in the graphene extinction coefficient decreases due to Pauling blocking, therefore the charge density for optical transition decreases in graphene [24,26,39,42 ]. For S-polarized wave, whose electric field is always parallel to the air-glass interface, the dipole oscillation strength increases with increasing charge density, yielding the positive correlation between the light reflectivity and the charge density at the glass surface. Hence, the differential reflection is negative at any incident angle for s-polarized light (G1 and G2). But for p-polarized light, the tangential enhancement of the oscillating dipole can either increase or reduce the reflection. When the incident angle is less (or greater) than the Brewster's angle, the dipole component that is perpendicular to the reflection direction increases (or decreases) if the tangential component of the dipole moment increasing, see Figs. 6(c) and 6(e). Thus, the correlation between the reflectivity and the charge density at the glass surface is positive for incident angles less than the Brewster's angle, but negative for incident angles greater than the Brewster's angle. This is consistent with the experimental measurement that the differential reflection of p-polarized light is negative at 39 degrees incidence, and positive at 72 degrees incidence (see G1 and G2 [Figs. 2 and 3 ], the Brewster's angle at air-fused quartz interface is 55.6 degrees).
Fig. 6 Schematic graphs of oscillating dipoles at the glass surface for (P)-polarized light [black solid lines denote the incidence (with angle ϕ), reflection and transmission at interfaces]. When increasing the charge density at the glass surface, the tangential component of the dipole moment increases for light incident from air to glass [(a), (c), (e)], but decreases for the reverse configuration [(b), (d), (f)]. The dipoles for small and large surface charge densities are denoted by p and p', respectively. When light is incident at the Brewster's angle, the reflection vanishes [(a), (b)]. When light is incident at other angles [(c)-(f)], the reflection can ether increase or decrease, depending on the component of the dipole that is perpendicular to the reflection direction anticipated by the Snell's law.
On the other hand, for light impinging on the glass-air interface from glass to air, the induced dipole moment in the glass surface loses its energy when the glass is covered by a graphene layer. This is because the dipole oscillation in the glass surface drives the kinetic motions of the high-mobility charges in graphene, which carry away some portion of the dipole energy. Thus the tangential component of the dipole moment decreases when covered by graphene. In this case, we can also define an angle similar to the Brewster's angle for the p-polarized light, at which angle the reflection vanishes, see Fig. 6(b). When increasing the charge density at the glass surface, the light reflection will decrease for s-polarized light and p-polarized light at an incident angle less than the Brewster's angle [Fig. 6(d)], but will increase for p-polarized light at an incident angle greater than the Brewster's angle [Fig. 6(f)]. Because of the negative correlation for s-polarized light and positive correlation for p-polarized light at an incident angle greater than the Brewster's angle, the differential reflection is positive for the s-polarized light and negative for the p-polarized light at the incident angle of 39 degrees (the Brewster's angle at BK7 glass-air interface is 33.4 degrees in G3 [Fig. 1(c)]).
When the incident power of light is fixed, the s-polarized light has the largest tangential electric component, while the p-polarized light has the minimum tangential electric field. The graphene layer is a system of high-mobility charge carriers [28,33,42 ]. Like the case for noble metals, the graphene should also have the mirror effect for the tangential electric field. That is, the reflection of the incident light is positively correlated with the strength of the tangential field component. Hence, the s-polarized light always has the maximum reflectivity, while the p-polarized light always has the minimum reflectivity, which is consistent with the experiment [Figs. 2(b), 3(b) and 4(b) ].
Since other linearly polarized incident beams could be decomposed as their s- and p-components, the reflectivity and transient reflection of other polarized beams should vary within those of the s- and p-polarized beams [Eqs. (3), (6)-(8) ]. Therefore, as experimentally observed, the R and ΔR/R could be successively changed by light polarization. Since the transient reflection of s- and p-polarized light is opposite in G2 and G3, a transition from reduced to enhanced reflection could be observed when the polarization is changed between S and P.
Recent works reveals that population inversion appears at pump fluence of ∼2 mJ/cm2 for ultrashort pulses (<40 s) and its occurrence is accompanied with nonlinear saturation effect [14–16 ]. One direct consequence of the occurrence of population inversion at the optically probed state is photoinduecd extinction coefficient (or optical conductivity) change is larger than stable value. i.e., max(Δκ) < −1.45, or Δσ < −σ 0, at this moment, the optical conductivity is negative [14]. It is noted that the occurrence of population inversion does not affect the polarization dependence of transient reflection reported here. Within the pump fluence range used here, we observed linear dependence of max(ΔR/R) on pump fluence (not shown here). It indicates no occurrence of population inversion in our measurement although the pump fluence used here is close to the reported pump fluence threshold of population inversion, it may be due to the much larger pulse duration used here, as compared to that in [14]. The larger pulse duration results in lower peak intensity at same pump fluence (the peak intensity of pump fluence in configuration G2 is 7.4 GW/cm2, and the peak intensity of 2 mJ/cm2 in [14] is 54 GW/cm2). As result, the intensity used here was lower than the pump intensity threshold of population inversion [15].
4. Summary
Our results demonstrates that by using obliquely incident light, the graphene transient optical reflection induced by nonequilibrium carriers isotopically distributed in momentum space could be significantly and successively altered by light polarization. Moreover, this polarization dependence can be flexibly manipulated by means of incident angle and incident direction. We can understand this polarization dependence based on Maxwell’s equations solving boundary conditions and dipole moment perturbation from graphene layer. The realization of this polarization dependence, under isotropic distribution of noneqilibrium carriers’ momenta, opens the possibility of using this effect in manipulating transient optical response of graphene for applications in high-speed photonic devices.
Acknowledgments
This work was supported by the Chinese National Key Basic Research Special Fund (2011CB922003), National Natural Science Foundation of China (11304166, 11174159, 11374164, 11504265), the Fundamental Research Funds for the Central Universities (65145005), and International Science and Technology Cooperation Program of China (2013DFA51430). X-T Kong is supported by National Natural Science Foundation of China (11404075).
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