1. Introduction

Lithium niobate, LiNbO₃ (LN), acts as a reference material in the field of nonlinear optics and experiences increasing attention as frequency converter due to its sound second-order nonlinearity χ⁽²⁾, chemical and mechanical stability as well as availability [1]. For efficient conversion, LN is exposed to very intense, short and ultra-short laser pulses; at high intensities, however, the problem of nonlinear absorption arises due to a predominant contribution of third order nonlinearities χ⁽³⁾, in particular two-photon absorption (TPA). Besides limiting the conversion efficiency, TPA plays a major role in irreversible bulk laser-induced damage mechanisms of ferroelectric crystals by heating as well as via the generation of free carriers [2]. In LN the subsequent formation of small polarons from optically excited free carriers must be considered [3], as well, i.e. carriers that become trapped with strong coupling within a self-induced distortion of the surrounding unit cell [4]. As we are dealing with lithium niobate grown from the congruently melting composition, the term small polarons applies to the small free polaron ( ${\text{Nb}}_{\text{Nb}}^{4+}$) placed at a regular lattice site and, due to the non-stoichiometric crystal structure bounded by an antisite defect, the small bound polaron ( ${\text{Nb}}_{\text{Li}}^{4+}$) as well as the small hole polaron (O⁻). The contribution of small polarons to the nonlinear absorption is determined by their broad band (≈ 1 eV) and pronounced absorption features [5] and characterized by a transient absorption with lifetimes in the regime from μs to ms at room temperature, thus commonly exceeding the pulse duration. Nonlinear interactions of small polarons with short, intense laser pulses in LN have been already addressed by different research groups [6–9]. However, the dependency of nonlinear absorption on incident laser pulses with different pulse durations has not been studied, so far, which is the topic of this work. The pulse duration dependency is of particular interest for nonlinear optical applications using sub-ps laser pulse durations that fall in the time regime of small polaron formation; the latter has been estimated to values much below 400 fs [6,7,9–11]. Without this knowledge, nothing is known about the individual contribution of TPA and small polaron absorption to the transmission loss of a propagating (ultra-)short laser pulse. Furthermore, it remains unclear to what extent pulse durations exist where the contribution of small polaron absorption becomes negligible.

In LN, first efforts have been made to separate optically excited carriers from the Kerr third order nonlinearity by means of grating recording [12] or transient absorption [8], demonstrating the impact of carrier dynamics in the time regime subsequent to the incident ultrashort laser pulse (240 fs). Furthermore, the two-photon absorption coefficient β has been determined using various pulse durations in the range of 80 fs – 55 ps yielding values of β between 1.5 and 5.2 mm/GW in the green spectral range (Refs. [7, 13–17], cf. review [11]).

We here study the dependency of nonlinear absorption on the duration of incident (ultra-)short laser pulses in lithium niobate by systematically scanning the transmission loss over the time regime of 70 fs < τ < 1,000 fs by means of z-scan technique. For analysis, the differential equation for the intensity decrease of a propagating laser pulse along the crystal coordinate is derived. The temporal interplay of non-instantaneous processes with rise/relaxation times in the sub-ps time regime are considered: TPA, free carrier relaxation, small polaron formation and cascaded carrier excitation. The model extensions are motivated by a significant deviation – increasing for longer pulse durations – between experimental data and data analysis based on TPA (cf. original work from Sheik-Bahae [18]), but also TPA with free-carrier absorption [19]. Using our model for analysis, the experimental findings are described over the entire range of pulse durations with high precision. A deconvolution of the individual contributions – particularly of small polaron and two-photon absorption – to nonlinear absorption along the time coordinate becomes possible. We discuss the obtained temporal evolution of the small polaron impact from the viewpoint of a more precise estimate for the small polaron formation time, the existence of an offset of small polaron appearance upon the incident laser pulse as proposed by Qiu et al. [6] and pulse durations that are insignificantly affected by small polaron formation. It can be concluded, that the pulse duration dependency of nonlinear absorption yields important information for the area of nonlinear applications with intense, (ultra-)short laser pulses of LN, but also of the variety of nonlinear optical materials showing small polaron formation in general.

2. Modeling

2.1. Temporal evolution of optically excited carriers in lithium niobate

For our study, we refer to the potential scheme depicted in Fig. 1, particularly describing the interplay of two-photon and small polaron absorption. In what follows, the configuration coordinate q will be identified as spatial displacements of carriers in the real crystal lattice in the nanoscopic regime as well as the time axis covering the sub-ps regime.

 

Fig. 1 Potential diagram of the band-to-band excitation by one-photon (α) and two-photon (β) absorption with photon energies of E_ph = 2.5 eV, electron-phonon cooling process with time constant τ_S, relaxation to the ground state (τ_R) and subsequent formation of small polarons (τ_FC–P) in lithium niobate. Absorption cross section σ and number density of polarons N_P determine the absorption triggered by optically induced transport of small polarons [11].

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We focus our considerations on the absorption of incident (ultra-)short sub-ps laser pulses with a photon energy of E_ph = 2.5 eV via one-photon and two-photon absorption in a nominally undoped LN crystal. The probability of the former is small because of the large band gap of lithium niobate (α = 20 cm⁻¹ at ≈ 3.8 eV [20]), i.e. the LN crystal is transparent for this photon energy at low incident laser intensities. At the contrary, two-photon absorption dominates the nonlinear absorption at elevated intensities. For the regime of sub-ps laser pulses, the TPA coefficient β is reported to values of β = 5.2 mm/GW (80 fs, 400 nm, Mg-doped LN) [7] and β = 3.5 mm/GW (240 fs, ≈ 480 nm, nominally undoped LN) [17]. The response time of TPA being related to bound electrons is nearly instantaneous; an estimate of τ_TPA ≈ 5 fs has been deduced from degenerate recording of TPA gratings for LN crystals [21], such that τ_TPA ≪ τ for all pulse durations (70 – 1,000 fs) in our study. The temporal dynamics of TPA will thus be neglected in the differential equations given in the theoretical subsection below.

With TPA, free carriers far from the thermal equilibrium (hot carriers) are generated by means of valence-to-conduction band excitation. Energy relaxation to the lowest levels of the conduction band occurs preliminary via the emission of phonons; for semiconductors, this electron-phonon cooling process is characterized by a relaxation time of τ_S ≈ 100 fs [22]. For LN, τ_S = 80 fs has been calculated for relaxation from an energetic level 0.5 eV above the conduction band minimum (T=296 K) [6]. By experimental means, ultrafast pump-probe-spectroscopy with LN did not show indications of an energy relaxation of hot carriers, so far [6,8], whereas the time-resolved measurement of the reflectivity of a lithium niobate surface reveals a delayed signal with τ_S = 600 fs that has been attributed to free electron generation [23].

In the next step, the carrier may either relax to the ground state by recombination with a hole and a characteristic time constant τ_R. Or carrier localization occurs followed by strong vibrational coupling, i.e. a small polaron is formed with characteristic time constant τ_FC–P (small polaron formation time). In LN, τ_FC–P ≈ 100 fs for small free ${\text{Nb}}_{\text{Nb}}^{4+}$ polarons (room temperature) [6, 7] and τ_FC–P < 400 fs for small bound ${\text{Nb}}_{\text{Li}}^{4+}$ polarons [9].

If the pulse duration exceeds the temporal regime until small polaron formation, the optical excitation from a small polaron must be considered, as well, and the carrier relaxes repeatedly into a small polaron. We assume that this process can be repeated several times over the incident pulse duration, i.e. cascaded excitation occurs [11].

In a similar way, the remaining holes in the valence band become localized as O⁻ hole polarons in the vicinity of lithium vacancies [3, 24, 25]. In the following, however, we will not distinguish between the particular types of small electron and hole polarons [24,26,27], because of probing at a single photon energy and the overlap of the absorption features of small free and bound electron and small hole polarons [5]. All types of small polarons result from the intrinsic defect structure of LiNbO₃ that – although complex – has been modeled from first principles in very recent articles by Li et al. [28, 29]. We, thus, use LN crystals grown from the congruently melting composition in our experimental study.

2.2. Transfer to z-scan technique

The starting point for the determination of the nonlinear absorption composed by TPA and small polaron absorption is the z-scan technique. It has been developed originally for the analysis of near-instantaneous third order nonlinearities, particularly of the two-photon coefficient β and the nonlinear index of refraction n₂ [18]. A nonlinear optical sample is shifted (scanned) along the z-coordinate through the focus of a laser pulse with spatial and temporal Gaussian intensity profile. The transmission as a function of z is expressed by

 

Fig. 2 Numerical solution of Eq. (2) as a function of pulse duration (100 fs – 1,000 fs) and the following model parameters: α = 0 m⁻¹, β = 5 mm/GW, τ_S = 100 fs, τ_FC–P = 100 fs, τ_R = 100 fs, σ = 100 · 10⁻²² m², and peak intensity I = 8.6 PW/m² at z = 0. E.g., this results in a maximum polaron number density of N_P = 1.7 · 10¹⁷ mm⁻³ at the center of the pulse with maximum intensity for τ = 1, 000 fs. For comparison the red dotted graph representing Eq. (1) is also shown. The inset highlights the change in the shape of the transmission traces exemplarily for a pulse duration of 1,000 fs and a fitted graph using the original z-scan theory Eq. (1) with an TPA-coefficient increased by a factor of 2.2 in comparison to the main figure.

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3. Experimental section

3.1. Setup and lithium niobate samples

The systematic scan of the nonlinear absorption over the time regime of 70 fs – 1,000 fs has been performed using the common z-scan technique described in Ref. [18] with an incident laser pulse of spatial and temporal Gaussian profile that – as an original feature – has been compressed/stretched to the desired pulse duration. The extended setup, as depicted schematically in Fig. 3, is composed by (i) a prism pulse stretcher/compressor, (ii) a spatial frequency filter, and (iii) the common z-scan configuration. An optical parametric amplifier (Coherent Inc., model: OPerA solo) pumped by a regeneratively amplified Ti³⁺:Sapphire laser (Coherent Inc., model: Libra-F HE) serves as the source for ultrashort laser pulses (pulse energy maximum: 150 μJ at 2.5 eV, center wavelength: 488 nm). A neutral density filter is used for intensity adjustment. The repetition rate of 250 Hz is reduced to 12.5 Hz using a Chopper wheel, in order to avoid cumulative absorption from pulse to pulse due to long-lived small polarons (maximum characteristic lifetime ≈ 3 ms at room temperature, see e.g. Ref [11]). The pulses first enter a prism pulse stretcher/compressor that allows for tuning the pulse duration by means of adjustment of prism P2 and mirror M using a linear stage (LS). A spectral width of Δλ = 5 nm is obtained, equal with a bandwidth limited pulse duration of τ ≈ 70 fs. For the purpose of our study, the pulse duration is varied up to 1,000 fs and characterized by means of a scanning autocorrelator (APE, model pulseCheck 15). After the stretcher/compressor, the pulses enter a spatial frequency filter that consists of two concave mirrors (focal length: 500 mm), avoiding chromatic abberation, and a pinhole (diameter d = 100μm). Astigmatism is minimized by a small angle of incidence of approximately 2 degree. The pulse spatial radius is determined to (r = 2.0 ± 0.1) mm. Both, M² and r, are required for the calculation of beam waist and intensity of each pulse as a function of position z. A nearly spatial (M² = 1.1 ± 0.1) and temporal Gaussian beam profile is verified using beam profile measurements and an autocorrelator, thus, fulfilling the experimental conditions of the z-scan technique to a great extent [31]. The as-prepared pulses are focused by lens L1 (f = 150 mm) and propagate through the lithium niobate crystal LN. The position of LN can be shifted along the direction of pulse propagation (z-coordinate) by means of a motorized linear stage (MLS). Incident and transmitted pulse energies are detected using biased Si-PIN detectors D1-D3 (Thorlabs, DET10a). The detectors D2 and D3 are equipped with opened and closed apertures, respectively, thus allowing for the determination of both nonlinear absorption and nonlinear index of refraction. Lenses L2-L4 focus the pulses to a spot size less than the photosensitive area of the detector.

 

Fig. 3 Sketch of the optical setup composed by a prism stretcher/compressor (PSC) (P1; P2 on a linear stage LS), a spatial frequency filter (SFF) (CM: concave mirrors with f = 500 mm, PH: pinhole with diameter of 100 μm) and a common configuration for z-scan technique: L1: lens (f = 150 mm), LN: lithium niobate crystal, MLS: motorized linear stage, L2–L4: lenses (f = 50 mm), D1–D3: Si-PIN detectors (photosensitive area ≫ beam spot), A: aperture with diameter of 4 mm. Incident pulses obey a maximum pulse energy of 150 μJ at 2.5 eV (center wavelength: 488 nm) and are adjusted in intensity by a neutral density filter. The repetition rate of 250 Hz is reduced to 12.5 Hz using a Chopper wheel. The pulse duration can be varied with PSC from 70 fs – 1,000 fs.

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All studies were performed with thin a-cut plates (aperture 8×6 mm², thickness d = (260 ± 10)μm) of nominally pure LN grown from a congruently melting composition (Crys-Tec GmbH). The thickness ensures that the sample is much thinner than the Rayleigh length, avoiding asymmetric z-scan traces and effects of the group velocity dispersion (GVD). Front and back surfaces are carefully polished to optical grade nearly plane parallel (wedge below 5 arcmin). The one-photon absorption coefficient is determined to α = (0.16 ± 0.1) cm⁻¹ for extraordinary (e ‖ c) light polarisation and λ = 488 nm.

3.2. Experimental results

The experimentally determined z-scan traces are plotted in Fig. 4, exemplarily, for four pulse durations: (a) (70 ± 10) fs, (b) (220 ± 10) fs, (c)(430 ± 10) fs and (d) (840 ± 30) fs, all for a constant pulse energy of (270 ± 30) nJ and at a center wavelength of λ = 488 nm. The upper parts of the figures show the transmission T obtained from the signal ratio of diodes D2 and D1 as a function of z-coordinate from −15 mm to +15 mm. The travel range is chosen such that one-photon absorption dominates the transmission at ±15 mm; the transmission is normalized to unity at |z| > 15 mm and shows a pronounced drop by scanning over the focus of the incident pulse. The minimum of transmission is used to define the position z = 0; all data sets are almost mirror-symmetric to z = 0. Qualitatively, the shape of the z-scan traces are comparable with each other for all pulse durations, and are characterized by a very low noise. A closer inspection reveals that the shape becomes narrower with increasing τ, which is a sign of higher order nonlinearities as discussed above (cf. inset of Fig. 2). In addition, and because of the adjustment of a constant pulse energy for all pulse durations, the varying peak intensity yields an increase of the transmission with increasing τ from T ≈ 20% at 70 fs to T ≈ 45% at 840 fs.

 

Fig. 4 (Upper parts): Experimentally determined transmission as a function of scanning coordinate z for four pulse durations: (a) (70±10) fs, (b) (220±10) fs, (c)(430±10) fs and (d) (840±30) fs, all for a constant pulse energy of (270±30) nJ and at a center wavelength of λ = 488 nm. The results of our numerical fitting procedure according to Eqs. (2) to (4) are shown as green lines with the following model parameters: a two-photon absorption coefficient of β = (5.6 ± 0.8) mm/GW, a small polaron absorption cross section of σ = (210 ± 70) × 10⁻²² m², and characteristic times for electron-phonon relaxation of τ_S = 80 fs, for interband relaxation of τ_R = 100 fs and for small polaron formation of τ_FC–P = 100 fs. For comparison, fitting of Eq. (1) to the experimental data is shown as red dashed line. The error of a single measuring point is indicated by the errorbars for selected points. (Lower parts): squared error of the fits with respect to the experimental data as a function of z. It is noteworthy, that the amount of polaronic absorption, e.g. in (d) is about 45% at z = 0; the fit with Eq. (1) would result in an overestimate for β of about 15 mm/GW.

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The data plots are analyzed by numerical solution of Eqs. (2) to (4). Fitting was performed iteratively by optimizing the squared error between fit and experimental data and the same model parameters for all data sets; the fitting results are shown as green curves in Fig. 4(a)–(d) with the following model parameters: a two-photon absorption coefficient of β = (5.6 ± 0.8) mm/GW, a small polaron absorption cross section of σ = (210 ± 70) × 10⁻²² m², and characteristic times for electron-phonon relaxation of τ_S = 80 fs, for interband relaxation of τ_R = 100 fs and for small polaron formation of τ_FC–P = 100 fs. A high degree of agreement is obvious from the inspection by eye over the entire range of scanning. The lower plots of Fig. 4(a)–(d) highlight the squared error of the fits with respect to the experimental data as a function of z with values much below 0.1% throughout the data set, typically below 0.03%. For comparison, the data sets have been also analyzed using Eq. (1), i.e. the z-scan theory considering two-photon absorption, only, yielding the red curves in Fig. 4(a)–(d). For the shortest pulse duration of 70 fs, the data can be modeled by the z-scan theory with very good coincidence, expressed by a squared error of below 0.1%. From this data set, the fit yields a two-photon absorption coefficient of β = (5.8 ± 0.8) mm/GW (for the nonlinear refractive index n₂ we obtained a value of about 5 × 10⁻²⁰ m²/W). For longer pulse durations, a deviation of the original TPA theory to the experimentally determined traces becomes obvious and is more pronounced with increasing pulse duration (exceeding a squared error of ≈ 0.2%). The evolution of mean squared errors (MSE) between fit and data set is plotted for both theoretical approaches in Fig. 5 for all measured pulse durations. Again, the plot highlights the excellent agreement of our model approach with the experimental data over the entire regime of pulse durations; a minimum value of mean squared error MSE_min = (1.5 ± 0.1) × 10⁻⁴ is reached throughout the pulse durations. The original z-scan theory describes the data with high precision in the regime of the shortest pulse durations (below 100 fs), as well. However, a characteristic rise of MSE of Eq. (1) becomes obvious for pulse durations ≥ 100 fs with a development that can be best described by a single-exponential growth function. We thus have fitted the function

 

Fig. 5 Mean squared error between fit and experimental data for both, the numerical solution of our model approach according to Eqs. (2) to (4) (green), and the original z-scan theory using Eq. (1) (red). The dashed lines represent best fits with constant minimum value of the mean squared error MSE_min = (1.5±0.1)×10⁻⁴ (green) and a fit with Eq. (5) to the data points (red) with saturation amplitude MSE(t = ∞) = (4 ± 0.1) × 10⁻⁴, characteristic time constant τ_exp = (137 ± 8) fs, temporal offset of τ = (86 ± 5) fs, and minimum value of mean squared error MSE_min = (1.5 ± 0.1) × 10⁻⁴.

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4. Discussion

From the experimental viewpoint, a very high quality of transmission traces are collected using the applied optical setup that represents an extension of the common z-scan technique. In particular, the important conditions of a Gaussian spatial profile is obtained, using a spatial frequency filter, and a Gaussian temporal profile is verified by autocorrelation measurements. A diode-pumped, regeneratively amplified Ti:Sa-laser system obeys an excellent beam profile and pointing stability and, thus, is ideally suited for generation of harmonic, Gaussian pulses by means of an optical parametric amplifier. The RMS noise of the pulse energy at the OPAs’ output (≪ 5%) could be reduced successfully by selection of incident pulses with equal pulse energy using diode D1. The disadvantage of this procedure is the extended measurement time, which is further enlarged by the low pulse repetition rate of 12.5 Hz and the two dimensional parameter range (z, τ) of our particular study. However, all these efforts are justified considering the outmost marginal deviations between the model approaches typically of below 0.03% (cf. Fig. 4).

As a superior result, a more detailed insight to the underlying physics of nonlinear absorption in lithium niobate is obtained, and clear evidence for small polaron formation and its contribution to nonlinear absorption is found. From the dependency of the shape of the z-scan trace as a function of pulse duration and its analysis, it is possible to derive a set of important material parameters that will be discussed in the following. First, the two-photon absorption coefficient β = (5.6 ± 0.8) mm/GW is obtained from the complex numerical fitting procedure, that uses a single set of parameters for all pulse durations. Thus, the transmission is described throughout the regime of pulse durations from 70 fs – 1,000 fs with the same TPA coefficient – a result that is to be expected as β is a material parameter independent on τ. Considering the high quality of the numerical fits, a reliable value of β with an error of ≈ 15% occurs for the LN crystal under investigation. Nearly the same value (β = (5.8 ± 0.8) mm/GW) is obtained using the original z-scan theory at the shortest pulse duration of (70 ± 10) fs, that underlines the reliability of the obtained value. Moreover, it is an important (first) anchor for proving the action of our model approach as nonlinear absorption persists solely from TPA for pulse durations much below the small polaron formation time. The TPA value itself is much larger than the state-of-the-art knowledge in literature. For comparable pulse durations and photon energies, values of β = 5.2 mm/GW (80 fs, 400 nm, Mg-doped LN) [7] and β = 3.5 mm/GW (240 fs, ≈ 480 nm, nominally undoped LN) [17] are reported. Besides the possibility, that this difference may be attributed to the impact of small polaron absorption according to our model approach, we like to emphasize the role of stoichiometry on the TPA coefficient. It is well established, that the position of the band gap energy is strongly dependent on the crystals’ stoichiometry. Being an absorption process of higher order, the transition probability for TPA is expected to change in accordance with the linear absorption feature [32]. Therefore, differences in the TPA coefficient may be due to differences in stoichiometry of the studied LN crystals, which seems to be very likely to us.

New insights are revealed to the relaxation dynamics of optically excited free carriers, characterized by τ_S. First evidence for this intermediate process was presented by Qiu et al. [6] performing time-resolved fs-absorption spectroscopy in Mg-doped LN crystals. As a remarkable characteristic of the small polaron formation dynamics, a temporal offset was discovered prior to the rise of small polaron absorption. No change of the transmission was observed during this temporal offset. A similar observation is reported in the work of Beyer et al. [8] using 240 fs laser pulses. Here, we also need to take into account a temporal offset prior to small polaron formation. In advance of the results of Qiu et al. [6], the time of the incident pulse does not need to be determined which is a particular feature of the approach of pulse duration scanning; the time constant τ_S is equal to the pulse duration where the transmission trace starts to alter its shape. Therefore, the temporal offset τ_S = 80 fs can be determined with appealing precision. The lack of free-carrier absorption prior to carrier-phonon relaxation may be attributed either to the fact that the photon energy of the probing pulse is insensitive to the free-carrier absorption cross section or the carrier-phonon relaxation time τ_S falls much below the pulse duration (τ_S ≪ (70 ± 10) fs in our case). However, the latter is unlikely because the carrier-phonon relaxation time exceeds 10⁻¹³ s in most semiconductors and for common optical phonon modes as pointed out in Ref. [22]. Further studies, particularly with shorter pulse durations need to be performed for clarification.

From the rise of the nonlinear absorption as a function of pulse duration we further obtain the small polaron formation time of τ_FC–P = 100 fs in the numerical modeling procedure. This value is in good coincidence with the formation times obtained by transient absorption measurements [6, 7, 9]. Combining, however, our result on the presence of an offset time of τ_S = 80 fs and the transient data from Sasamoto et al. using also very short, 80 fs pump and probe laser pulses [7], it is possible to conclude an upper limit of τ_FC–P ≈ 100 fs. It is because the transmission loss due to two-photon absorption and due to small polaron absorption can not be resolved on the temporal axis in pump-probe experiments and overlap with each other.

It should be noted, that the small polaron formation time τ_FC–P is only related to small bound polarons ( ${\sigma }_{{\text{Nb}}_{\text{Li}}^{4+}}=4.0\cdot {10}^{-22}{\text{m}}^2$, σ_O⁻ = 4.1 · 10⁻²²m² at 2.5 eV [5]). Due to the negligible absorption of small free polarons at our probing wavelength ( ${\sigma }_{{\text{Nb}}_{\text{Li}}^{4+}}=0.8\cdot {10}^{-22}{\text{m}}^2$at 2.5 eV [5]), this type of polarons is (nearly) not detected in this experiment. Assuming, that the lattice relaxation of small free polarons is part of the small bound polaron formation path, the corresponding time constant must be considered as a part of τ_S. However, for a deeper insight a study at different wavelengths including the spectral range of small free polaron absorption is required.

Another parameter obtained from our study is the small polaron absorption cross section σ that is by more than one order of magnitude higher than the values published by Merschjann et al. [5]. We note that it is not possible to find a converging numerical solution using Merschjann’s value as fixed fitting parameter, even by neglecting relaxation of excited carriers to the ground state (τ_R = ∞ results in a value of σ ≈ 110 · 10⁻²²m², still exceeding Merschjann’s value by more than one magnitude). One explanation for the striking difference is an oversimplification of our model approach presented in section 2. In particular, the dynamics of holes with the possibility of O⁻ small hole polaron formation in LN [25] has been disregarded. As stated above, such holes obey nearly the same absorption cross section at a photon energy of 2.5 eV as it is the case for small bound ${\text{Nb}}_{\text{Li}}^{4+}$ polarons, although the maxima of the respective absorption features are fairly different (2.5 eV for O⁻ and 1.6 eV for ${\text{Nb}}_{\text{Li}}^{4+}$). Due to two-photon interband excitation it is reasonable to assume that the number density of optically generated electron N_p,e and hole polarons N_p,h is identical, N_p,e = N_p,h. As a consequence, it is necessary to exchange the factor σN_p in Eq. (2) by the sum (σ_p,eN_p,e + σ_p,hN_p,h) = 2 σ_p,eN_p,e. This means, that by considering small hole polarons, our numerical analysis is running with twice the polaron density. To counterbalance this value to correctly describe the experimentally determined contribution of small polaron absorption, the absorption cross section σ_p,e needs to be reduced by a factor of two. Still, however, σ_p,e exceeds the literature value. It is thus likely, that the larger value of σ may be attributed to a larger variety of different small polaron species. As individual types of small polarons are not resolved by our study, further pulse duration dependencies using different photon energies are required for a more quantitative analysis of this aspect. We like to add, that there is a severe difference in the boundary conditions of small polaron absorption between the study of Merschjann et al. [5] and the present one: Merschjann’s cross sections were determined using pump-probe experiments, i.e. under the conditions of no light, whereas small polarons are inspected during the presence of a strong pump in our study, i.e. with light. Considering the presence of specific charge transport phenomena in LN, particularly of bulk photovoltaic currents [33], a difference in the absorption features of small polarons with and without light illumination can not be excluded.

It is noteworthy that all parameters discussed above and obtained from the rather complex numerical solution of Eqs. (2) to (4) show a correlation with each other. Particularly, the ratio between τ_FC–P and τ_R is not independent from other model parameters and directly impacts the density numbers N_FC and N_P as well as the polaron absorption coefficient σ. As a consequence, the characteristic times can be varied in a limited regime, only. The offset time τ_S can be varied between 70 fs and 90 fs while the polaron formation time is limited to 70 fs < τ_FC–P < 110 fs. At the same time, it is very reasonable that the increase of the number of fitting parameters in general results in the optimization of fitting functions. However, in the present case, the extension of the original z-scan theory by small polaron absorption is justified by the striking deviation of the transmission traces between theory and experimental data with increasing pulse duration. As this deviation rises after a characteristic offset time, the data plot of Fig. 5 showing the mean squared error as a function of pulse duration, can be applied also for the determination of the offset time τ_S yielding τ_S = t_offset = (86±5) fs – in accordance with the numerical results of our model approach.

5. Summary and conclusion

The pulse duration dependency of nonlinear absorption has been studied in lithium niobate by means of z-scan technique over the time regime of 70 – 1,000 fs and has been analyzed from the viewpoint of small polaron formation using a numerical approach. It is shown, that the transmission loss of (ultra-)short laser pulses propagating through LN crystals can be described with remarkable precision and can be attributed to the complex interplay of near-instantaneous nonlinearities, particularly of two-photon and small-polaron absorption. Surprisingly, very minor alterations in the shape of the well-known transmission traces of the z-scan technique result in severe changes in the set of nonlinear optical coefficients; for instance, using the original z-scan theory, the TPA coefficient β may be overestimated by a factor of three in the regime of long pulse durations.

Besides the precise determination of β or the absorption cross section of small polarons under illumination σ, the amount of information on the underlying photophysical processes obtained from our study is unexpectedly high: the proposed offset of transient absorption features during the time regime of electron-phonon cooling has been verified by experimental means and could be determined to (80 ± 10) fs. Furthermore, a more accurate regime of the small polaron formation time of 70 – 110 fs has been obtained.

These results are of utmost importance for the physics and microscopic modeling of the small polaron approach in LN [25, 26] and the further understanding of the small-polaron based bulk photovoltaic effect [33]. But also for the field of applications in nonlinear photonics as frequency converter – particularly in the growing field of ultrafast laser systems. In more general, the presented theoretical model and the experimental approach of pulse duration scanning can be transferred to the wide field of oxide materials showing small polaron formation and can be applied in the area of ultrafast lasers.