Numerical study of transient absorption saturation in single-layer graphene for optical nanoscopy applications

Transient absorption, or pump–probe microscopy is an absorption-based technique that can explore samples ultrafast dynamic properties and provide fluorescence-free contrast mechanisms. When applied to graphene and its derivatives, this technique exploits the graphene transient response caused by the ultrafast interband transition as the imaging contrast mechanism. The saturation of this transition is fundamental to allow for super-resolution optical far-field imaging, following the reversible saturable optical fluorescence transitions (RESOLFT) concept, although not involving fluorescence. With this aim, we propose a model to numerically compute the temporal evolution under saturation conditions of the single-layer graphene molecular states, which are involved in the transient absorption. Exploiting an algorithm based on the fourth order Runge–Kutta (RK4) method, and the density matrix approach, we numerically demonstrate that the transient absorption signal of single-layer graphene varies linearly as a function of excitation intensity until it reaches saturation. We experimentally verify this model using a custom pump–probe super-resolution microscope. The results define the intensities necessary to achieve super-resolution in a pump–probe nanoscope while studying graphene-based materials and open the possibility of predicting such a saturation process in other light-matter interactions that undergo the same transition.

addition of a doughnut-shaped beam that saturates a specific transition at the periphery can significantly enhance imaging resolution.When the power of this beam is increased, saturation grows, leading to enhanced resolution.However, it is crucial to carefully assess the required illumination intensity to avoid potential photodamage to the sample during imaging 20,21 .This study presents a model that estimates and predicts the power necessary to achieve saturation conditions in the transient absorption process.We specifically focus on the transient absorption occurring in single-layer graphene (SLG) when using visible/near-infrared light, which involves interband transitions 17 .Through theoretical modelling, numerical simulations, and experimental validation, we demonstrate changes in signal intensity under saturation conditions at various pump and probe intensities.

Theoretical framework
Nonlinear optical processes can occur as a consequence of light-matter interaction through high laser intensity.The polarization ( P ) of the material is often described as a power series expansion of the total applied optical field ( E ).The nonlinearity results in higher order powers of E and, in particular, the third term that accounts for the third order susceptibility, χ 322 , which is the focus of this work.Such a tensor is involved in many phenom- ena exploited in the nonlinear optics field 23 , as it implies that three optical fields interact to produce a fourth field.Although the χ 3 interaction is thus a four-photon process; there can be up to three different input laser frequencies, but also as few as one.For example, the simultaneous absorption of two photons (2PA), the consecutive interaction of three photons followed by the generation of the final photon (THG), the absorption-emission-absorption-emission sequence (frequency mixing or CARS), and transient absorption are examples of third order nonlinear processes.In general, transient absorption involves a two-state (two energy levels) system, and it uses an ultra-short pump pulse to cause a transition from a certain state to a state at higher energy followed by an ultra-short probe pulse that records the variation 18,24 .If the first pulse produces a highly populated excited state, the absorption coefficient for the probe is reduced and the process is called ground state depletion (GSD) Fig. 1a 10,24 .Other mechanisms, such as stimulated emission (SE), excited-state absorption (ESA), and SRS Fig. 1b 25 , are also possible but are not the aim of this work.
In graphene, the excitation process near the Dirac point can be described by the Dirac Hamiltonian which has a linear energy dispersion dependent on the wavevector ( ε = υ F |κ| , where υ F = 10 6 m/s is the Fermi veloc- ity) 26,27 as shown in Fig. 1c.
Upon photoexcitation by laser pulses with a wavelength in the visible/near-infrared region and duration of few hundred femtoseconds, electrons in the conduction band (ε = ћω/2) and holes in the valence band (ε = − ћω/2) are created in a non-equilibrium state while conserving momentum.Triangular warping and other nonlinear effects are considered negligible, even at visible wavelengths of up to 400 nm 28 .Considering Fermi's golden rule, we can have a complete description of graphene-light interactions 27 .Under low optical excitation where the carriers undergo fast interband decay in graphene 29 , the linear optical transmittance is dependent on the fine structure constant but independent of the excitation wavelength and the material parameter ν F .On the other hand, high-intensity excitation leads to significant populations of carriers in the valence band (VB) and the conduction band (CB).This results in reduced absorption of photons with the same energy (within the pulse duration), causing a bleaching effect.At the same time, the non-equilibrium carrier distributions in the CB and VB undergo ultrafast relaxation through non-dissipative carrier-carrier scattering and, to a lesser extent, carrier-optical phonon coupling 21,29 .
In this study, we consider a time-dependent uniform electric field during light interaction.To analyze its dynamic, we employ a semiclassical description derived from the Liouville-von Neumann equation with phenomenological decay term 30,31 (Suppl.Appendix 1).Since we want to be as general as possible, we consider that nonlinear processes encompass various states such as ground state, vibration state, electronic state, and others.In the presence of intense laser excitation, overtone states may participate in transitions through nonlinear processes 31,32 .Figure 1a illustrates the specific energy level diagrams under investigation, where a sample is exposed to both pump (ω pu ) and probe (ω pr ).The status of population per each energy level state of the system can be presented by a density matrix description 32 ρ = |ψ��ψ|.
We take Ĥ = H 0 + H int + H r , where H 0 is the Hamiltonian for the matter system itself, H int is the Ham- iltonian describing the strength of the light-matter interaction, and H r incorporates the relaxation process, which is considered to be time-independent.By employing the eigenfunctions of H 0 as a basis, along with other approximations and simplifications detailed in Supplementary Appendix 1, we can derive the Liouville-von Neumann equations (Eqs.2, 3) 30 , allowing for the quantitative calculation of the temporal evolution of electron carrier populations in the system.This approach, adapted from Boyd's textbook, semiclassically describes the light-matter interaction.It can be expressed as: 30 these states encompass various possibilities such as ground, intermediate, vibrational, electronic, etc., denoted as |g�, |i�, |ν�, |e�, . . ., respectively.The selection of 'm' and 'n' depends on the specific transition being considered.
Here, Ĥint = μ • E , represents the Hamiltonian governing light-matter interaction, where μ is the electric dipole moment operator.This operator is derived from the absorption cross-section σ nm under the condition of zero population, with µ nm = �n| μ|m� denoting the matrix element of μ for a system with varying levels (n = m), Additionally, in the context of a linearly polarized incident electric field, the relationship µ mn = (µ nm ) * is valid 34,35 .

and the associated density matrix is
The electric field of incident light is expressed by: where A pu and A pr denote the field amplitude of the pump and probe beams, respectively 18,30 .
Using the above Eq.( 3) A pu and A pr are converted into the peak of intensity I pu and I pr Here, n denotes the refractive index of the sample, c is the speed of light in vacuum, and E 0 represents the vacuum permittivity 18,30 .The pulse width of both pump and probe beams are defined by τ (τ pu = τ pr ).
Within this quantum mechanical framework, our focus shifts towards super-resolution imaging in graphene.We synchronize the pump and probe pulses, tuning their wavelengths to 805 and 1030 nm, respectively, as these wavelengths are optimal for imaging using our custom microscope 17 .Given that the levels excited by the pump and probe pulses differs, we computed the model for a three levels system, namely the ground state in the valence band and two excited in the conduction band.However, since in graphene conduction band can be considered continuous and fast intraband relaxations occur, we expect that the populations of carriers excited by the two beams are the same, leading to an empty 'third state' , Fig. 1d.In a three levels system, there could be two or more possible quantum pathways that lead to the same final state: Therefore, by coupling the appropriate laser pulses, the different pathways will result in the same final state 36 .It is worth noting that the duration of the pulses has to be comparable to the relaxation time, and therefore we used it in the numerical derivation 36 .After rewriting Eqs.(2, 3) in a three levels picture, we derived all density matrix differential equations to study SLG 37,38 .We calculated the temporal evolution of carrier population in each state.We considered the pulse widths as τ = 0.5 ps, while the frequency of the pump and probe are set to be ω pu = 2.3887 × 10 3 THz (805 nm) and ω pr = 1.8278 × 10 3 THz (1030 nm).As shown in Fig. 1d, the probability of a third state |3� is close to zero, 6.2 × 10 −14 Suppl.Figure 1, and thus negligible as expected.Thus, we can defi- nitely assume the transient absorption in SLG as a two levels process where states |1� and |2� indicate the ground state, |g� , and excited state, |e� , are the ones involved in the process, and we modified density matrix accordingly: We consequently derived the equations 31 , from Eq. (2, 3), as Hence, we set Ŵ 12 = Ŵ 21 = Ŵ that we denote this quantity 1 T 1 , which is decay rate.Similarly, we denote the quantity γ 12 = γ 21 as 1 T 2 , which commonly is referred to decoherence rate between |1� and |2� (Suppl.Appendix 1).Then, the four Eqs. ( 5), ( 6), (7), and (8) (suppl.appendix 1 equations A5) were solved numerically using the fourth order Runge-Kutta algorithm 39 .The solution was computed over the time interval from 0 to 4 ps, with step size of 0.035 ps.Notably, the pulse was centered at t 0 = 2 ps, Suppl.Figure 1c.We specifically focused on t = 4 ps, a time point by which the light-matter interaction is nearly complete 18,40 .The algorithm, written in Python (available upon request), calculates the temporal evolution of carriers populating different interband levels of single-layer graphene.

Single-layer graphene (SLG)
Sample was purchased from Graphene Supermarket (Graphene Laboratories Inc., Calverton, NY).It consists of a monolayer graphene film grown by chemical vapor deposition (CVD) processing onto copper foil and then transferred onto a 0.17 mm thick glass substrate.This slide with the deposited SLG was mounted on a 0.17 mm cover glass to be placed under a water immersion objective.The graphene film is mostly continuous, with occasional holes and cracks, and has a polycrystalline structure made of grains with different crystallographic orientations.The supplier reports the sample transmission to be above 97% in the visible spectral window, which makes it suitable to collect the signal efficiently in transmission.

Pump-probe nanoscopy
In this work, pump-probe, and saturated pump-probe images were performed using a custom-built nearinfrared pump-probe nanoscope.Figure 2 shows the schematic illustration of the pump-probe microscopy system.Our pump-probe nanoscope consists of a tunable mode-locked femtosecond pulsed Ti: sapphire laser (680-1080 nm, 80 MHz, 140 fs, Chameleon Ultra II, coherent), a commercial laser scanning confocal Nikon A1 MP microscope, and the optical items depicted in Fig. 2 that allow realizing the three optical beams configuration.Two femtosecond pulsed laser beams generated by an OPO (Coherent Inc., Santa Clara, CA, USA) pumped by a laser source are used as the pump (tunable in the range 740-880 nm) and probe (tunable in the range 1000-1600 nm), respectively.The pump beam intensity is modulated at 1 MHz through an electro-optical modulator EOM (LM 0202, Qioptiq, Goettingen, Germany).A digital delay generator controls and generates the electronic synchronization between the pulse rate and the modulation (DG645, Stanford Research Systems, Sunnyvale, CA).A motorized delay line (M-521.PD, Physik Instrument, Karlsruhe, Germany), equipped with a retroreflector (UBBR2.5 − 2I, Newport Corp., Irvine, CA) delays up to 1.3 ns with 3 fs accuracy.For most of the experiments (imaging and saturation measurements), the pump beam's pulse and the probe beam's pulse have been synchronized.Also, the two beams (pump and probe) are spatially combined and overlapped through a dichroic mirror and delivered into a Nikon Confocal A1 (multiphoton) scanning microscope and are focused onto the sample with an objective lens (40 × 1.15 NA, water immersion).In the saturated case, the pump beam is taken before the EOM with a polarizing beam splitter (PBS, Thorlabs, Newton, NJ).Then the beam passes through another motorized delay line (LTS150/M, Thorlabs, Newton, NJ, USA) with a delay of 1 ns with 16 fs accuracy used to adjust its temporal alignment connecting pump and probe pulses.The beam passes through a vortex phase plate (VPP-1a, RPC Photonics, Rochester, NY, USA) to create a doughnut shape coupled with the

Result and discussion
Single-layer graphene (SLG) has a high nonlinear third order χ 3 susceptibility, which enables harmonic genera- tion, transient absorption, and frequency mixing.Such a property is due to the optical resonance among interband electronic transitions [41][42][43][44] .To estimate the energy transferred to the upper level in SLG, we quantitatively calculated the population temporal evolution for each molecular state at pump = 805 nm, probe = 1030 nm, and pulse width = 500 fs.As we discussed in appendix, in this model, off-diagonal elements of the electric dipole moment matrix have the same value, set to µ mn = 0.196e • nm when (n = m) , which are derived from the absorp- tion cross-section σ ij 32 , while all diagonal elements are set to zero (Suppl.Appendix 1).The refractive index of SLG was assumed to be n = 2.58 41 .The decay of off-diagonal elements of density matrix ρ nm is caused by decoherency between two-state, which the decoherent rate can be obtained with bandwidth of absorption spectrum 10,18,32 .Also, the decoherent rate between vibrational and ground state can be estimated from bandwidth of the spontaneous Raman spectrum 18,33,39 We measured the decoherent rate between ground and upper excited state by time-resolved spectrum 18 of pump-probe (Fig. 4c), and we set to be γ nm = 0 for (n = m), and γ nm = 1.824 ps −1 10,39 when (m ≠ n) (Suppl.Appendix 1), and ω pu + ω pr = 2744meV .The interband relaxation time of the carriers is about 150 fs while the following interband electron-hole recombination has a time decay of about 2 ps 45 Therefore, in our simulations, we considered different values of Ŵ = 0.01, 0.1, and 0.11 ps −1 (Table 1).Then, the algorithm was computed from 0 to 4 ps to completely simulate the entire light-matter interaction, which is completed in < 4 ps.In this model, we made the simplified assumption that all the carriers are in the ground state ( ρ gg = 1); after light interaction they undergo the electronic transition, |g� → |e� , and the population in the ground state, |g� decreases while the population in the upper excited state, |e� , increases until saturation occurs.We calculate ρ ee and ρ gg at different values of I pu and I pr to estimate the required laser intensity to reach the complete depletion of the ground state.In the PP process, the energy transferred to excited states can also be assessed by E exc = ρ nn | t=4ps ω nm , where the n and m terms describe all the possible lower and upper excited states, respectively.If we assume that the number of annihilated photons for both the pump and probe are equal, then the number of photons lost per molecule in the PP process is E exc (ωpu+ωpu) .Typically, the pump-probe signal, ( T ), exhibits a weak nonlinearity, and its intensity will vary proportionally to the analyte concentration N 0 , and the product of the pump and probe intensities I pu and I pr , respectively, T ∝ N 0 I pu I pr 46 .In the case of the GSD mechanism, the absorption of an intense pump beam causes a population inversion from the ground to the excited state, and consequently the decrease of the absorption coefficient.Thus, a probe beam in resonance with the absorption transition will exhibit a transmission increase.Such a behavior can be derived from the Lambert-Beer relation 47 : where N 0 is the molecular concentration of the ground state, σ pu and σ pr are the linear absorption cross-sections of the ground state for the pump and probe beams, respectively, t is the time delay between pump and probe pulses, and τ is the lifetime of the excited state.When t = 0 the strongest signal is achieved which exponentially decays at longer t , reflecting the characteristic relaxation time τ of the excited state 47 .
To study how the population changes at the interband level as a function of beam intensity, we assumed the pump power, I pu , increases while probe power, I pr , remains constant and vice-versa.Panels a and b in Fig. 3 show that the pump-probe signal intensity has a linear dependency in both conditions up to a specific limit where the initial intensities were assumed to be I pu = I pr = 10 GW/cm 2 .Moreover, we calculated the pump-probe signal intensity under different total powers ( I pp = I pu + I pr ), which fits a quadratic polynomial behavior, plotted in (Fig. 3c).The initial intensities were assumed to be I pu = I pr = 2.94 GW/cm 2 .It is worth noting that to achieve absorption saturation our simulation uses a slightly higher intensity.The theoretical findings can be essential to quantitatively predict the required excitation power to achieve saturation, thus resolving the parameters to maximize super-resolution imaging performance.In fact, the optical resolution, d, defined by Abbe's law as the shortest distance between two points that can be distinguished as separate units, is reduced by a factor that depends on saturation and can be written in the following equation: where c = ln2 2 , I s is the saturation factor defined as the intensity necessary to force 50% depopulation of the ground state, and I is the intensity distributed in a doughnut shaped beam at the periphery of the pump and probe beams (sat-PUMP in Fig. 2) 48 .In this context, we acquired pump-probe images of SLG using our custom-built near-infrared pump-probe nanoscope (Fig. 2) to experimentally validate our numerical simulation, verifying the real effect of power changes.The pump-probe intensity signal was acquired sequentially while imaging SLG and increasing I pu and I pr respectively.Figure 4a shows PP signal vs I pu I pr increasing I pu from ( 44 to 287 GW/cm 2 ), while I pr is kept constant at ( 33 GW/cm 2 ), and Fig. 4b shows the same but increasing Saturation intensities are defined in (a-c).R-square is the coefficient of determination, a value between 0 and 1 that statistically qualify the regression.
from 55 to 107 GW/cm 2 , while I pu is constant at (70 GW/cm 2 ).Also, we measured the time decay of the transi- tion (Fig. 4c) which resulted in line with the literature 10 .
The theoretical (Fig. 3a, b) and experimental (Fig. 4a, b) results show that the intensities required to reach saturation are approximately I sat ≈ 3I pu , 3I pr in simulation, and I sat ≈ 2I pu , 2I pr experimentally.These finding can be key to quantitatively predicting the required excitation power to achieve saturation, allowing optimisation of the parameters to maximize the performance of super-resolution imaging.Our numerical simulations and experimental data are in good agreement, showing that we can efficiently achieve super-resolution while imaging SLG.
In our PP microscope, the resolution is improved by using an unmodulated doughnut-shaped pump beam overlapped with the other two conventional beams, all focused on the sample.The beam is shaped using a vortex phase plate, and its polarization is made circular with two retarder plates, i.e., a half-and a quarter-wave plate 49 .A dedicated optical delay line is used to synchronize the pump and probe pulses, and all the beams are synchronized at time zero to maximize the signal-to-noise ratio.The doughnut-shaped pump beam, realized with the vortex phase plate (VPP in Fig. 2), saturating the depletion of the ground state, can avoid the probe absorption at the rim of the focal spot, leading to STED-like super-resolution 10,13 .We obtained a resolution improvement by tuning the wavelengths and power in the following manner: pump = 805 nm with 4 mW, probe = 1030 nm with 5 mW, and = 805 nm with 11 mW.To perform super-resolution imaging, we set the power of unmodulated saturation pump to I satpump ≈ 2I pr .As shown in Fig. 5, panels a and b, we acquired unsaturated and saturated pump-probe images of SLG after defining the conditions to achieve ground state depletion saturation and, therefore, super-resolution.From the insets in Fig. 5c, d, we highlight some details that are better resolved in saturated pump-probe condition.The spatial resolution is quantitatively measured by Fourier Ring Correlation 50,51 , Fig. 5e, resulting in being better than /4 and in agreement with the literature 10 , another example at lower saturation intensity is presented in suppl.Figure 2.
It is worth noting that such an improvement is not the best achievable resolution but is what can be already obtained at a power above the one we predicted saturation happens.Since saturation occurs, the resolution will improve at higher powers.

Conclusions
We have proposed a theoretical model and an experimental study on the temporal behavior of single-layer graphene absorption states under strong illumination in the NIR spectral window.We numerically and experimentally presented how the pump-probe signals of SLG depend upon increasing illumination intensity varying I pu or I pr .We successfully calculated the required peak intensity to achieve the saturation of the transient absorption process in graphene, and validated our simulation with experimental data and super-resolution imaging.Thus, our theoretical model can have the potential to predict the light intensity required to reach nanoscale optical resolutions on a diverse range of materials.

Figure 1 .
Figure 1.Energy level diagrams of two nonlinear optical processes, illuminating the sample with two incident beams at frequency ω pu and ω pr (or ω s ).(a) Ground-state depletion when both the incident beams induce ground-state absorption.(b) Stimulated Raman scattering when ω pu − ω s equals the vibrational frequency of sample, .(c) Band diagram and mechanism of TA process of graphene, E F is the Fermi energy (d).Temporal evolution of the probability of each state in SLG considering a three levels system where τ = 0.5 ps, λ pu = 805 nm, and λ pr = 1030 nm.

Figure 3 .
Figure 3. Numerical simulations of PP signal in SLG as function of the illumination powers.(a, b)The PP signal as a function I pu I pr , for eg = 805nm , τ = 500fs , while I pr is kept constant in (10 GW/cm 2 ) (a), and I pu is constant in (10 GW/cm 2 ) (b). (c) PP signal of SLG as I pu I pr , for eg = 805 nm , τ = 500 fs , while depends on total intensities ( I pu + I pr ), where the initial intensities were assumed to be I pu = I pr = 2.94 GW/cm 2 .Saturation intensities are defined in (a-c).R-square is the coefficient of determination, a value between 0 and 1 that statistically qualify the regression.

Figure 4 .
Figure 4. PP microscopy experiments on SLG.The PP signal versus I pu I pr , for pu = 805 nm , pr = 1030 nm , while I pr is kept constant at (33 GW/cm 2 ) (a), and I pu is constant at 70 GW/cm 2 (b).(c)Time- resolved spectra of SLG defects obtained at different delays of the probe pulses with respect to pump pulses.The pump-probe signal was fitted with an exponential decay which retrieved a fast component with (1.38 ± 0.04) ps for lifetime.The R-square is the coefficient of determination, a value between 0 and 1 that statistically qualifies the regression.

Figure 5 .
Figure 5. Normalized pump-probe (pp) (a) and saturated pump-probe (spp) (b) images of SLG foldings and defects.(c, d) Zoomed region of (a, b) respectively the arrows highlight the resolution improved obtained in spp.(e) is the plot of the Fourier Ring Correlation (FRC) of the two images: (a) in black and (b) in red.The FRC has been obtained using two sequential images, (a) and (b) are the sum of such images.FRC calculation has been carried on using carried out by the FRC_plugin of ImageJ available from the PTBIOP Update Site (https:// www.epfl.ch/ resea rch/ facil ities/ ptbiop/).Therefore, the lateral resolution in (a, b) is better than 530 nm and 250 nm, respectively.The pump and probe wavelengths used are pu = 805 nm, pr = 1030 nm , respectively.

Table 1 .
It shows the values of the intensities required to reach saturation when ρ 11 = ρ 22 ≈ 1/2 for different values of Ŵ.