Saturable absorption and nonlinear refraction in free-standing carbon nanotube film: Theory and experiment

The nonlinear absorption and refraction of free-standing ﬁ lms made of single-walled carbon nanotubes (SWNTs) have been investigated experimentally and theoretically. By solving the quantum kinetic equations that take into account both the intra-and interband transitions, we obtain the analytical expression for the SWNT nonlinear conductivity. The nonlinear absorption coef ﬁ cient and saturation intensity of the ﬁ lm comprising randomly orientated SWNTs have been calculated in a broad spectral range spanning over M11, S11, and S22 absorption bands. The effects of the laser pulse duration and dynamic Burstein (cid:1) Moss shift on the saturation intensity have been revealed. We demonstrate in the experiment that, under irradiation with femtosecond laser pulses, the absorption modulation depth of SWNT ﬁ lm at resonance wavelength 1375 nm is as high as 30%. The observed saturation intensity minimum is red-shifted with respect to the absorption maximum due to the dynamic Burstein (cid:1) Moss shift. The saturation intensity within the S22 band is 26 (cid:1) fold lower than that out of the band at 795 nm. The closed-aperture z-scan measurements reveal the negative nonlinear refractive index n 2 ¼ (cid:1) 3.1 (cid:3) 10 (cid:1) 12 cm 2 /W at 795 nm. © 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
Free-standing single-walled carbon nanotube films (FSCNFs) offer a unique opportunity to explore the electronic ensemble of individual single-walled carbon nanotubes (SWNTs) [1].Aerosol synthesis [2] allows one to obtain FSCNFs composed of virtually non-electromagnetically interacting SWNTs having prescribed diameters [3,4].The advantageous electronic and optical properties of such FSCNFs enable transparent capacitive touch sensors, thinfilm transistors, electrochemical sensors, bright organic lightemitting diodes, incident light polarization analyzers, and terahertz modulators [3,5e7].In future, the films made SWNTs can be employed as transparent flexible electrodes for different applications, e.g. for solar cells, and in flexible electronics, e.g. for flexible touch screens and keyboards.Moreover, they can be used as an active material for the field effect transistors.In photonics, one of the most promising applications of FSCNFs is low-cost broadband mode-locking for compact yet powerful femtosecond lasers [8].FSCNFs have reduced nonsaturable absorption which inevitably increases when carbon nanotubes are embedded into a polymer matrix or deposited onto a substrate [4,9].They have demonstrated high performance and robustness in passive mode-locking of fiber lasers at the wavelengths of 1.05, 1.32, 1.56, 1.9, and 1.99 mm [3,4,10e12].Moreover, FSCNFs can be applied to vary the light pulse duration at the telecom wavelengths from the femtosecond to the microsecond [13].Very recently, we have also shown that the linear and nonlinear optical properties of FSCNFs [14] can be drastically improved by the combined acid and laser treatment.Specifically, the enhancement of linear transmittance across the visible range [14,15] and by 26% increase in the saturable absorption (SA) coefficient at the wavelength of 795 nm have been demonstrated.
The SA theory based on the numerical solution of the quantum kinetic equations for individual SWNT has been presented in Refs.[16,17].Such an approach is convenient if one considers a single interband transition between two sub-bands.However, when many electron interband transitions occur between different sub-bands in SWNTs comprising the film, obtaining the intensitydependent absorption coefficient of the FSCNF requires knowledge of the analytical solution of the kinetic equations.The efficiency of such an approach has been recently demonstrated neglecting the intraband transitions [18], however, the frequency dependence of the absorption coefficient of the FSCNF has not been studied yet.Also, only a few papers report on the measurements of the nonlinear refraction in free-standing SWNT films [19e21] and not covering 795e1500 nm although.
In the present work, we combine the experiment and the theoretical modeling to quantitatively describe the SA in the FSCNF across the 400e5000 nm spectral range.In particular, we arrive at a handy analytical description of the nonlinear absorption coefficient of the film.Using femtosecond open-aperture z-scan measurements, we obtain the saturation intensity I s and the SA coefficient a 0 across the bands M11 of the metallic SWNT, and S11 and S22 of the semiconducting SWNTs.The ratio of saturable to nonsaturable losses R and the absorption modulation depth D have also been determined experimentally.With the closed-aperture z-scan measurements, we have estimated the nonlinear refraction of FSCNF in the range of 795e1500 nm.
This study is among only a few devoted to the investigation of the nonlinear optical properties of free-standing films of SWNTs and not those suspended in solvents, neither bound to the substrate nor embedded in a polymer matrix.

Nonlinear response of SWNT
We consider the interaction of the light wave with p-electrons in an infinitely long SWNT by using the tight-binding approximation [16,17,22,23] and considering the first Brillouin zone of the hexagonal graphene lattice.Periodicity in the circumferential direction of the SWNT leads to the quantization of the transverse quasi-momentum and the formation of one-dimensional subbands.The interband transitions are permitted between the subbands with the same transverse quasi-momentum.
When the SWNT is oriented along z-axis of the laboratory Cartesian frame, the z-component of the electric field E z ðtÞ of the incident electromagnetic wave induces in the SWNT the azimuthally symmetrical axial current having the surface density [24] are the current densities due to the intra-and interband transitions, respectively.In Eqs. ( 1) and ( 2), e is the electron charge, a is the SWNT radius, p B is the first Brillouin zone boundary.Summation is taken over subbands s ¼ 1; 2;…;m, ε с ðp z ; sÞ is the energy of the s-th subband in the SWNT conduction band, R cn ðp z ; sÞ is the interband matrix element for the p-electron belonging to the s-th subband and having transverse momentum p z , r cc ðt; p z ; sÞ and r cn ðt; p z ; sÞ are elements of the one-electron density matrix for the s-th subband.FðεÞ is the Fermi equilibrium distribution function where k B is the Boltzmann constant and T is the temperature.The temporal evolution of the density matrix elements is governed by the following system of kinetic equations [16,17].
r in À r eq t 1
Numerical solution of Eqs. ( 4) and ( 5) allows us to calculate the intra-and interband currents densities introduced in Eqs. ( 1) and (2) for any E z ðtÞ.The performed numerical simulation (see Section 2.2) showed that the second term in the left-hand side of (5) can be ignored because its contribution to the current density is less than 1%.
The numerical solution of equations ( 4) and ( 5) is stable only in the vicinity of the interband transitions and the calculation time increases as the field amplitude increases.In order to reveal the main features of the nonlinear response of the FSCNF, we solve Eqs.
(4) and ( 5) for the monochromatic light wave at frequency u by assuming where E 0 is the wave amplitude.
In the steady-state approximation, we will present r in ðt; p z ; sÞ and r cv ðt; p z ; sÞ in the following form: After substitution of Eqs. ( 6), ( 8) and (9) in Eq. ( 4) and taking into account only resonance terms (rotating-wave approximation) we arrive at where I ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffi ffi ε 0 =m 0 p jE 0 j 2 is the intensity of the light pulse.The obtained solution of kinetic equations allows us to present the SWNT current density in the following form jðtÞ ¼ sðu; IÞE 0 expfÀiutg þ c:c: ; (12) where sðu; IÞ ¼ s intra ðu; IÞ þ s inter ðu; IÞ is the nonlinear conductivity comprising intra-and interband contributions s intra ðu; One can observe from Eqs. 13 and 14 that for a weak light wave r ðu; p z ; sÞ ¼ r eq ðp z ; sÞ and sðu; IÞ reduces to the linear conductivity [25].In such a low-intensity regime, longitudinal and transverse relaxation times t 1 and t 2 are associated with relaxation times of the intra-and interband transitions, respectively.We will consider a FSCNF having a thickness much smaller than the typical length of SWNTs.The in-plane conductivity of the film comprising identical randomly oriented SWNTs can be obtained by averaging sðu; IÞ over the random orientation of the nanotubes in the plane of the film aðu; 0Þ: (17) Our calculations show that the saturation intensity is 50% higher for the film with random orientations of SWNTs than that for the film with the SWNTs aligned along the light polarization direction.

Numerical simulation of the SWNT nonlinear response
In this Section, we shall solve Eqs. ( 4) and ( 5) numerically to describe the interaction of the SWNT with an intense Gaussian light pulse with where t p is the full width at half maximum of light intensity.Such a pulse is shown in Fig. 1a at u ¼ 1:3 Â 10 15 s À1 and t p ¼ 160 fs.
In our further calculations, we will use t 1 ¼ 100 fs [25] and t 2 ¼ 10 fs [26] and consider achiral zigzag and armchair SWNTs having chiral indexes ðm; 0Þ and ðm; mÞ, respectively.Table 1 shows the dependence of parameters of these SWNTs on the transverse quasimomentum p z and subband index s.Fig. 1c shows the time dependence of the population inversion r in ¼ r cc À r nn at several quasi-momentum p z for zigzag (23,0) SWNT.Pulse frequency coincides with bandgap energy, i.e.Zu ¼ 2εðp z ¼ 0; s ¼ 16Þ (transition number 1 in Fig. 1b).One can see in Fig. 1c that the temporal evolution of the population inversion depends on the quasi-momentum p z ; the smaller the difference 2εðp z ; 16Þ À Zu, the stronger the deviation of the inversion from -1.
Interband transitions at all p z contribute to the nonlinear response of the carbon nanotube.
For quasi-monochromatic incident pulse, the conductivity of SWNT at frequency U can be defined as sðUÞ ¼ jðUÞ=EðUÞ, where jðUÞ and EðUÞ are the relevant Fourier components of the current density j z ðt) and an incident field E z ðtÞ introduced in Eqs. ( 1) and ( 2) and Eq. ( 18), respectively.In Fig. 2, we demonstrate how the transitions between two subbands ±εðp z ; 16Þ shown in  In the linear regime, the interband electron transitions S22 manifest themselves as a broad resonance in the conductivity Reðs 22 Þ centered at about 1450 nm (see Fig. 2a).At the light pulse intensity of I 0 ¼ 640 MW/cm 2 , the absorption maximum is suppressed and the resonance frequency increases.Such a blue shift known as a dynamic BursteineMoss shift [27] originates from the saturation of the absorption just above the band edge.This phenomenon has not been studied in SWNTs before.At low intensity, the contribution of the interband transitions dominates over the intraband transitions.In the saturation regime, the intraband transitions significantly contribute to the imaginary part of the conductivity; whereas their contribution into the real part of the conductivity remains small (see Fig. 2b).It is worth noting that the results presented in Fig. 2 consider only transitions within two subbands of a single SWNT shown in Fig. 1b.

Nonlinear conductivity of the FSCNF
In order to describe the nonlinear response of the model film comprising 5 semiconducting zigzag SWNTs with indexes (20,0), (22,0), (23,0), (25,0) and (26,0) and 4 metallic SWNTs (21,0), (24,0), (13,13) and (14,14), we shall use the steady-state solution of the kinetic equations assuming that the SWNTs are randomly oriented in the plane of the film.The volume fraction of each type of semiconducting SWNTs is the same.This is also true for metallic nanotubes.The ratio of the volume fraction of all the semiconducting tubes to that of all the metallic ones is 2:1.The conductivity of each tube is calculated using Eqs.13 and 14.Film thickness is 38 nm and the total volume fraction of SWNTs in the film is taken to be 0.17.
Fig. 3a shows the measured (see Section 3) and calculated absorptance spectra of the FSCNF.One can observe that the calculated and measured absorptance spectra are well correlated in the range 800e2500 nm.There is no full coincidence between the spectra below 800 nm because the developed theory does not take into account the p e plasmon resonance at 215 nm.p e plasmon contributes to the nonsaturable absorption, which is described only empirically [28].It is worth noting that the difference between the  measured and the calculated spectra in the visible spectral range is because the modelling was performed for the film comprising only 9 types of SWNTs, whereas in our experiment, FSCNF is composed of much more types of SWNTs.Overlapping of their electron transitions at l < 800 nm smoothes the absorption spectrum of the FSCNF.The contributions to the linear absorption from the interband transitions in semiconducting SWNTs (S11 and S22) and metallic SWNTs (M11) are shown by dashed lines in Fig. 3a.It is worth noting that the interband transitions are inhomogeneously broadened due to the presence of SWNTs with different radii.The broadening is the weakest for the S11 peak.Moreover, the interband resonances have a high-frequency tail due to the curvature of the energy bands (see Fig. 1b).This leads to overlapping of the neighbouring resonances and explains the observation of the SA between the peaks [29].
As shown in Fig. 3a, the calculated saturable absorbance of the film at I ¼ 0 varies slightly near 0.07 in a wide wavelengths range of 400e2500 nm.The absorbance of the film at I ¼ 4:8 GW/cm 2 demonstrates a strong wavelength dependence (see Fig. 3a).The dipole moments of p-electrons are higher for the lower frequency electron transitions resulting in a stronger nonlinear response at lower frequencies.This tendency has been noticed also in e.g.Ref. [30].
Fig. 3b shows the calculated saturation intensity I s of the model FSCNF.One can see that I s has strong wavelength dependence demonstrating a dip near the transition S11.Due to the dynamic BursteineMoss shift, the central wavelength of the dip is slightly larger than the central wavelength of the absorption peak.Such behaviour can be seen in the spectra presented in Ref. [29], though the authors did not pay attention to this.
In order to estimate the saturation intensity from experimental data, the measured in the experiment nonlinear absorption coefficient is fitted with [27,31].
where a 0 and a ns are the saturable and nonsaturable absorption coefficients, respectively.Fig. 4 shows the calculated dependence of the normalized absorption coefficient on the intensity I for the SWNT film at the peaks S11 and S22 and above M11 peak; calculation was done for the same model film as in Fig. 3 assuming a ns ¼ 0 (solid lines); for comparison, an approximate formula aðIÞ ¼ a 0 =ð1 þI =I s Þ was also used (dashed lines).One can see that Eq. ( 19) well corresponds to the result of the numerical solution of Eq. ( 17) only at low intensities (I < I s ).At high intensities (I > I s ), Eq. ( 19) gives overestimated value of the saturation intensity.Such a departure from the simplified two-level model originates from both the curvature of the energy bands and the inhomogeneous broadening of the absorption bands.
Since the pulse duration used in the experiments (100e150 fs, see Refs.[29,30]) is comparable with the electron relaxation time t 1 z100 fs, the measured saturation intensity depends on the pulse duration.For example, our calculations made for the SWNT (23,0) at the central frequency of the interband transition S22 (l ¼ 1450 nm, see Fig. 2a) gives the saturation intensity of 0.44, 0.37, 0.32, and 0.27 GW/cm 2 for a pulse duration of 100, 160, 250, and 400 fs, respectively.We also found that in our model film, the inhomogeneous broadening of the S11 band is not strong enough to modify the value of I s .However, a stronger broadening of the band S22 leads to a 2Àfold increase in the saturation intensity.The heating of the SWNT film during the measurements can lead to a decrease in the electron relaxation time resulting in an increase of the saturation intensity.The relaxation time for intraband electron transitions t 1 is proportional to the absolute temperature T of the sample [25].The SWNT oxidation starts at T !600 K that is two times higher than the room temperature.If we assume that t 1 fT and t 2 fT then one can expect that heating up to 600 K can lead to a 2Àfold increase in I s .

FSCNF fabrication
The SWNTs are synthesized by the floating catalyst (aerosol) chemical vapour deposition (CVD) method [2], which is based on the high-temperature decomposition of ferrocene in a CO atmosphere.An advantage of this method is that the nanotubes are downstream from the reactor on membrane filters and by adjusting the collection time and the CO flow rate one can control the film thickness and the diameter of the tubes, respectively [2,3,5].At the outlet of the reactor, the SWNTs form a film with randomly oriented networks of metallic and semiconducting tubes.In contrast to SWNTs synthesized by arc-discharge [32,33] or by laser ablation [26,34], which require laborious processes of purification and dispersion in liquids before they can be transferred [35], the obtained network of nanotubes can be easily transferred to almost any material by an easy dry transfer technique [3,5].
In the experiments, the SWNT film is transferred by pressing it towards the surface [5] of a small glass plate with an opening in its middle similar to Ref. [14].The area of the opening provides a freestanding film of nanotubes for optical measurements.The mean diameter of the tubes is about 1.5 nm so that the second van Hove optical transition S22 covered the telecom range.At the same time, the film is adjusted to have a thickness of L ¼ 38 nm [13] and linear transmittance T 0 ~80% at 1.5 mm [10,12] similar to those used in fiber lasers at telecom wavelengths of 1.32 and ~1.56 mm.The film thickness is calculated according to an empiric equation L (nm) ¼ 417 Â A 550nm , where A is the absorbance at 550 nm [36].
The fabricated FSCNFs are characterized by optical absorption and scanning electron microscopy (Fig. 5).The dashed arrows in Fig. 5 mark the experimental laser excitation wavelength of 795 nm close to the band M11, and the resonance excitation wavelengths of 1200, 1300, 1375, 1440, 1500 nm within the band S22.While the theoretical modeling is performed in the range of 400e5000 nm, the filled area in Fig. 5 marks corresponding available measured spectra data 400e2500 nm (0.5e3.1 eV).The inset in Fig. 5 shows the morphology of the dry transferred network of randomly oriented SWNTs.The Raman spectra of the studied film that have been presented in Ref. [14] are dominated by characteristic for SWNTs' G-, D-, G 0 -peaks and the "radial breathing" modes.The transmission electron microscopy image of the individual single-walled carbon nanotube has been presented in Ref. [37].The atomic force microscopy and the X-ray photoelectron spectroscopy analysis presented in Ref. [15] identifies the morphology of the studied films and the presence of carbon, oxygen, and iron represented by C1s, O1s, Fe2p peaks, respectively.

Z-scan measurements
In the experiment, we employ the CDP 2017 optical parametrical amplifier (OPA) to get tunable radiation within the telecom range [38].In the OPA, the pulses of the Ti: sapphire laser with the energy of ~0.3 mJ and pulse duration of t ¼ 150 fs at a central wavelength of l ¼ 795 nm generate a femtosecond white-light continuum in a 2-mm thick sapphire slab.The spectrally selected part of the white light is used as a seed pulse to achieve the parametrical amplification in beta barium borate (BBO) crystal.Two passes through BBO crystal allow one to get the required pulse energy and beam quality in the broad spectral range [39].We use the fundamental mode for OPA pumping as it has less group-velocity dispersion in the IR range [40] than the second harmonic.Considering the low value of group velocity dispersion and the optical path length that the laser pulse travels, we have evaluated the duration of laser pulses at each converted excitation wavelength.In the experiments, the OPA radiation is tuned in the range of 1200e1500 nm.Since the conversion efficiency at each wavelength is different, we have found pulse energy of 40 nJ (~8.5 GW/cm 2 for 1375 nm) as an optimum to provide a comparison of SA amplitudes within the telecom range.In order to define the evolution of SA at the excitation wavelength of 795 nm, the incident pulse energy is varied from 4 nJ up to the damage threshold (~100 nJ) and above.
The linearly polarized laser beam is focused on the FSCNF using a F ¼ 75 mm lens.The beam waist radius measured at 1/e 2 of the maximal intensity at 795 nm is w 0 ¼ 18.5 mm corresponding to the Rayleigh length of z 0 ¼ pw 0 2 /l ¼ 1.35 mm.The Rayleigh length for excitation wavelengths of 1200, 1300, 1375, 1440, 1500 nm (marked in Fig. 5) is 2.03, 2.12, 2.35, 2.41, 2.51 mm, accordingly.The measurements of SA and nonlinear refraction are performed using open-and closed-aperture z-scan setups, respectively, described elsewhere [41e43].The transmittance E out /E in , where E in and E out are the energies of the incident and transmitted laser pulses, is measured as a function of the distance z between the FSCNF and the laser beam waist.The energies of the incident pulse are measured using Thorlabs S122C 700e1800 nm energy head calibrated with the Ophir PD10-C energy meter.

Open-aperture z-scan
We perform open-aperture z-scan measurements in the wavelength range spanning from 1200 to 1500 nm, where the nonlinear response is mainly governed by semiconducting SWNTs (see Fig. 5).The normalized transmittance T n as a function of the FSCNF position measured at the incident pulse energy of 40 nJ (see Fig. 6 and Fig. S1 in Supplementary) demonstrate a pronounced SA for all resonant wavelengths.
In order to explain the z-scan experimental data, we used a conventional two-level model [27,31].In this model, the evolution of the light intensity I inside the medium is described by the system of Eq. ( 19) and equation dI dz 0 ¼ À aðIÞ Â I; (20) where z 0 is the propagation coordinate.A solution to this system yields [44]: where I in and I out are the input and output intensities, respectively.The former depends on the FSCNF position z with respect to the beam waist, i.e.
the beam radius.Considering that I out z T 0 Â I in , we arrive at [14]: For each wavelength, the value of z 0 was determined separately.Solid lines correspond to approximations with Eq. ( 22).(A colour version of this figure can be viewed online.) T n ðzÞ ¼ where E s ¼ tpw 2 0 I s .The fitting with Eq. ( 22) matches well with the symmetrical experimental data points indicating a pure SA effect [45].The fitting returns a 0 /a ns and I s .By using the obtained value of a 0 /a ns and relation lnT 0 ¼ À(a ns þa 0 ) Â L, we get a 0 , a ns , and the modulation depth D ¼ a 0 /(a 0 þa ns ) [29].Large D at small a ns is desirable for a saturable absorber in laser mode-locking [9].Therefore, the ratio of saturable/nonsaturable losses is an important parameter [9].
The values of a 0 , a ns , R, D, and I s , which were revealed using Eqs.
( 22) and ( 23), are summarized in Table 2.One can observe that within the S22 band, a 0 , R, and D are enhanced as expected [29] and that maximum values are achieved at 1375 nm.However, the saturation intensity I s minimum occurs at wavelength of 1440 nm.Such a red shift with respect to the position of the absorption maximum may be associated with a BursteinÀMoss shift of the absorption band.It is worth noting that even within the S22 absorption band a 0 is lower than a ns .In Fig. 7, we compare the absorption modulation depth D ¼ a 0 / (a 0 þa ns ), the ratio of saturable to nonsaturable losses R z a 0 /a ns , and the saturation intensity I s with literature data [29,31,46e51] for the SWNT-based SA at the wavelengths of 1500e1560 nm.It is worth noting that we used larger-diameter SWNTs and exploited the S22 band, whereas authors of [31,47e49] dealt with thinner tubes thus exploiting the S11 band.This explains the higher saturation intensity for our samples (see Fig. 7).The highest (apart from presented in Ref. [46]) values of the modulation depth and ratio of losses are probably inherent to tens of nanometer-thin freestanding film; other authors studied tens of micrometer-thick composite films (except [46] where SWNTs was deposited on silica microtoroids) where the matrix could give an extra contribution to the nonsaturable absorption coefficient a ns .The difference between the experimentally obtained I s ~1 GW/cm 2 (or saturation fluence of F s ¼ 150 mJ/cm 2 ) and the theoretically estimated I s 0.6 GW/cm 2 (see Fig. 3b) may be partly explained by the usage of high-intensity radiation (I > I s ) and short pulse duration, as well as possible sample heating (see Sec. 2.2).
The nonlinear optical response of FSCNF at a non-resonance wavelength of 795 nm has been also studied.Similar measurements at the incidence pulse energy of 20 nJ (12.4 GW/cm 2 ) have been recently reported [14].We performed the measurements at pulse energies varied from 4 nJ (2.5 GW/cm 2 ) up to 130 nJ (80.6 GW/cm 2 ).The damage threshold of SWNT film is observed at the laser pulse energy of 100 nJ, which corresponds to the intensity of ~62 GW/cm 2 .Fig. 8 illustrates the data for the pulse energy range of 4e90 nJ, i.e. below the damage threshold.In these energies, the z-scan curves are wave-shaped symmetrical and reproducible indicating an increase in the transmittance T n originating from the saturation of light absorption in the film rather than from the film damage caused by either heating or high peak intensities.At the energies exceeding 100 nJ, the wave-shaped open-aperture z-scan curves are replaced with cone-like ones having a sharp apex z/ z 0 ¼ 0 because a hole with a diameter of ~7 mm is formed in the film.Z-scan data for the incident pulse energies of 100e130 nJ (i.e.above the damage threshold) are presented in Fig. S2 (see Supplementary materials).
The fitting of experimental data with (22) gives a 0 /a ns ¼ 0.39 and I s ¼ 25.6 GW/cm 2 at 795 nm above the absorption band M11 (see Table 2).The theoretical calculations made for the model film predicts the saturation intensity to be about 3.5 GW/cm 2 just above the M11 band (Fig. 3b).Such a difference can be explained by the  dependence of the saturation intensity on the carrier relaxation time that can depend on the temperature [25], incident power [52], frequency [53,54], crystalline quality, and chirality of the SWNTs [54].Since we cannot take into account all these effects, in calculations, we use the same frequency-independent relaxation time for all SWNTs comprising the film in Fig. 3. Another possible explanation is the presence of different impurities including iron nanoparticles.Their oxidation into ferric chloride (FeCl 3 ) may result in a significant decrease in saturation intensity [14].One may assume that the vanishing of impurities will further decrease the experimental saturation intensity and the nonsaturable absorption coefficient.The observed modulation depth of 28% is much higher than that obtained for SWNT composite films (~5%) at a nonresonant frequency close to the band M11 [29].Comparing the experimental results, it is seen that the amplitude of T n for the resonance band S22 is up to three times bigger compared with that for the non-resonant wavelength 795 nm, which is in good agreement with the reported values for SWNT composites [29,55,56].
Although the modulation depth is relatively constant, the boosted SA for the band S22 is accompanied by ~26Àfold lower saturation intensity than that at 795 nm (see Table 2).Note that the theory predicts at least a 10Àfold difference in the value of I s (see Fig. 3b).

Closed-aperture z-scan
The results of the closed-aperture z-scan (see Fig. S3a in Supplementary) just above the M11 band are similar to those collected in the open-aperture z-scan (Fig. 8) up to the energies of 50 nJ.This indicates the significant contribution of SA in the nonlinear response.With an increase of the incident pulse energy from 50 to 90 nJ, we record a significant deviation of closed-with the openaperture data by the presence of two peaks in film transmittance (see Fig. S3b in Supplementary) indicating the strengthening of nonlinear refraction.In order to visualize the contribution of SA and to obtain the nonlinear refraction coefficient, we plot the closedaperture/open-aperture data ratio [57].Fig. 9 presents the results obtained for 35 and 90 nJ pulse energy.The results obtained for pulse energies of 50, 55, and 70 nJ are presented in Fig. S3 in Supplementary materials.
The experimental data are normalized to T 0 and present the valleys and peaks with the symmetrical amplitudes relative to the z/z 0 ¼ 0 [58].With the relative positions of the peaks and valleys to the scanning direction (from a lens to aperture, from -z to þ z [58]) one can evaluate the film negative nonlinear refraction, i.e. the defocusing by the sample [59].
The solid lines in Fig. 9 show fitting of the experimental data with the following equation [60,61]: Here n 2 is the nonlinear refractive index, L eff is the effective length of the sample L eff ¼ [1eexp(ÀaL)]/a; where a is the linear absorption coefficient.Visible deviation of the experimental data from the theoretical approximation in Fig. 9 is caused by asymmetries relative to z ¼ 0 and can be explained by misalignment, which may essentially influence the experimental data at low intensity [62].The fitting of the nonlinear refractive index gives n 2 ¼ À3.1 Â 10 À12 cm 2 /W.The obtained negative sign of the n 2 is inherent to the SA effect [27].The negative value of n 2 has been also reported for free-standing films of DWNTs [19].To the best of our knowledge, this is the first-time report on the n 2 for FSCNFs for femtosecond pulse at 795 nm.The obtained n 2 is the same order of magnitude as for free-standing films of double-walled carbon nanotubes (DWNTs) [19] and is one order of magnitude lower than for SWNT film on substrate [63] and suspended SWNTs [64] and DWNTs [65].Since the noise in the measurements of T n within the S22 band is 10 times stronger than that at 795 nm (see Supplementary Fig. S4), we do not record the effect of nonlinear refraction within this band.
Table 3 summarizes the comparison of the obtained results with the nonlinear optical parameters of some promising saturable absorbers based on perovskites, metal-organic frameworks (MOF), and MXenes.The comparison analysis shows that the studied FSCNFs exhibit relatively higher saturation intensity than, e.g., perovskites, while it is lower than that for the MOFs or MXenes, especially at the wavelength of 1500 nm.The nonlinear refractive index of FSCNFs lies between that of the MOF and MXenes.The modulation depth of FSCNF (defined in Table 3 in accordance with [66e68] as the maximum possible value of T n at I / ∞) is relatively lower at 800 nm (with comparable saturation intensity) and approximately the same at 1550 nm (with significantly lower saturation intensity) in comparison with that for perovskites, MOFs, and MXenes [66e68].These results suggest that the FSCNF offers a good compromise between modulation depth and saturation intensity and has great application potentials as a saturable absorber.Moreover, further suppression of the unsaturated absorption in FSCNFs similar to Ref. [14] can make FSCNFs a champion in terms of the SA performance.
In comparison with nonlinear crystals using in the modelocking, e.g.LiF with F 2 À colour centres [74], films made of SWNTs can be easily embedded in the core of fibers similarly to nanodiamond films [75].Another advantage of the SWNT films is their broad spectral response, which may allow FSCNTs to outperform semiconductor saturable absorber mirrors (SESAMs) in femtosecond lasers mode-locking.SESAMs enable mode-locking in a narrow wavelength range determined by the semiconductor bandgap and quantum well thickness (5e20 nm).In our work, we demonstrate that the SWNT film is capable to provide absorbance change in the wavelength range spanning from 400 to 5000 nm.

Conclusion
In conclusion, we derived the analytical formula for the surface conductivity of SWNT in the presence of an intensive, monochromatic, steady-state field.The developed approach allows obtaining the intensity-dependent absorption coefficient of the film comprising different types of randomly oriented SWNTs.
The nonlinear absorption coefficient and the saturation intensity have been calculated in a wide spectral range (400e5000 nm) for the model FSCNF having linear absorbance spectra similar to that for realistic CVD synthesized SWNT film.Nonsaturable absorption has not been taken into account in calculations.It has been found that the saturation intensity is 1.5Àfold higher for the films with random orientations of the nanotubes than for the film with the SWNTs aligned along the incident polarization direction.The inhomogeneous broadening of the absorbance resonance due to distributions of the radius of SWNTs is stronger for S22 than for the S11 band.As a result, for model FSCNF, the broadening practically does not modify the saturation intensity at the S11 band but leads to a 2Àfold increase in the saturation intensity within the band S22.The dynamic BursteineMoss blue shift of the absorbance bands has been shown for the model FSCNF.
With ultrafast femtosecond experiments for thin free-standing SWNT films, we reveal strong SA above the absorption band M11 (795 nm) and enormously strong within the absorption band S22 (1200e1500 nm).The SA at 795 nm shows a pronounced dependence on the excitation pulse energy, saturating at the pulse energies exceeding 50 nJ (31 GW/cm 2 ) up to the damage threshold of ~100 nJ (62 GW/cm 2 ).Within the S22 band, we reveal, to the best of our knowledge, the largest ratio of saturable to nonsaturable losses and the modulation depth defined for free-standing SWNT films.With the closed-aperture z-scan, the negative nonlinear refraction near the first metallic transition n 2 ¼ À 3.1 Â 10 À12 cm 2 /W has been found.
This study shows that FSCNFs have strong and broad saturable absorption across the 400e5000 nm spectral range and suggesting that they have great application potential in pulse-shaping and optical modulation [76].These findings together with the doubledigit percent improvement of nonlinear optical performance [14] demonstrate that the FSCNF is a nanomaterial with tunable and scalable nonlinearity capable to open new and improve already existing nonlinear optical devices and technologies.Specifically, the obtained results open avenues towards the drastic improvement of the performance of the SWNT-based mode-locked lasers and optical switchers operating in the optical communication window.

F
is a volume fraction occupied by SWNTs.Our analysis shows that s eff ðu; IÞzF 0 ðpaÞ À1 sðu; IÞ at the moderate intensities of the incident light.Equation (15) can be easily generalized for the case of the film comprising a mixture of SWNTs having different radii and structures.At normal incidence, the intensity-dependent absorption coefficient a of the film can be introduced as aðu; intensity I s for the FSCNF can be found from the following equation aðu;I s Þ ¼ 1 2

s ðintraÞ 22 ,
Fig. 1b contribute to the conductivity of the (23,0) zigzag SWNT.Fig. 2a presents this contribution (s 22 ) in the linear regime and for the light pulse peak intensity of I 0 ¼ 640 MW/cm 2 at t p ¼ 400 fs.Here, we use t p [t 1;2 in order to satisfy the condition of the steady-state incident field.The relevant inter-and intraband conductivities, s ðinterÞ 22 and are shown in

Fig. 2 .Fig. 3 .
Fig. 2. (a) Wavelength dependence of the real and imaginary parts of the axial surface conductivity s 22 of (23,0) SWNT caused by electron transitions within two sub-bands depicted in Fig. 1b in the linear regime (solid line) and the regime of SA at I ¼ 640 MW/cm 2 (dashed line); (b) the contributions from the intra-and interband transitions into the conductivity shown in (a) at I ¼ 640 MW/cm 2 .(A colour version of this figure can be viewed online.)

Fig. 5 .Fig. 6 .
Fig. 5.The absorbance spectrum of FSCNF.M11 is the first energy transition of metallic tubes, S11 and S22 are the first and second energy transitions of semiconducting tubes, respectively.The inset presents a scanning electron microscopy image of FSCNF.The filled area indicates the part (up to 2500 nm) of the wavelength region where the theoretical calculations were performed.The vertical dashed lines correspond to the wavelengths of the laser excitation in the experiments.(A colour version of this figure can be viewed online.)

Fig. 7 .
Fig.7.A comparison of (a) the modulation depth D ¼ a 0 /(a 0 þa ns ), (b) the ratio of saturable to nonsaturable losses R z a 0 /a ns , and saturation intensity determined in this and other studies for SWNT-based saturable absorbers at the wavelengths of 1500e1560 nm.Our study is the only one at 1500e1560 nm that reports SA characteristics in FSCNF, while other authors report in SWNT composite films.The exclusion is Ref.[46] where SWNTs was deposited on silica microtoroids.(A colour version of this figure can be viewed online.)

Table 1
The Brillouin zone boundary p B , interband matrix element Rcnðpz; sÞ, and the energy of the s-th subband εсðpz; sÞ of the zigzag and armchair SWNTs.b ¼ 0:142 nm is the interatomic distance in graphite, g 0 y2:7 eV is the overlap integral.

Table 2
Spectral parameters characterizing the performance of FSCNF-based saturable absorber.

Table 3
Comparison of the nonlinear parameters of FSCNFs and other promising saturable materials.l, nm T n , % I s , GW/cm 2 n 2 , cm 2 Â W À1 Ref.