Abstract

Nonlinear absorption is studied in presence of small polaron formation in lithium niobate using the z-scan technique and ultrashort laser pulses with pulse durations of 70 – 1,000 fs. A model for the analysis of the transmission loss as a function of pulse duration is introduced that considers (i) the individual contributions of two-photon and small polaron absorption, (ii) the small polaron formation time and (iii) an offset time between the optical excitation of free carriers by two-photon absorption and the appearance of small polarons. It is shown that the model allows for the analysis of the experimentally determined z-scan data with high precision over the entire range of pulse durations using a two-photon absorption coefficient of β = (5.6 ± 0.8) mm/GW. A significant contribution by small polaron absorption to the nonlinear absorption is uncovered for pulse durations exceeding the characteristic small polaron formation time of ≈ 100fs. It can be concluded that the small polaron formation time is as short as (70 – 110) fs and the appearance of small polaron formation is delayed with respect to two-photon absorption by an offset of about 80 fs.

© 2015 Optical Society of America

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 ( NbNb4+) placed at a regular lattice site and, due to the non-stoichiometric crystal structure bounded by an antisite defect, the small bound polaron ( NbLi4+) 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 Eph = 2.5 eV, electron-phonon cooling process with time constant τS, relaxation to the ground state (τR) and subsequent formation of small polarons (τFC–P) in lithium niobate. Absorption cross section σ and number density of polarons NP 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 Eph = 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 τTPA ≈ 5 fs has been deduced from degenerate recording of TPA gratings for LN crystals [21], such that τTPAτ 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 τS ≈ 100 fs [22]. For LN, τS = 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 τS = 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 τR. Or carrier localization occurs followed by strong vibrational coupling, i.e. a small polaron is formed with characteristic time constant τFC–P (small polaron formation time). In LN, τFC–P ≈ 100 fs for small free NbNb4+ polarons (room temperature) [6, 7] and τFC–P < 400 fs for small bound NbLi4+ 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

T(z)=1q(z)πln(1+q(z)exp(s2))ds
with q(z)=βIdeff/(1+z2/z02), the peak intensity I, the two-photon absorption coefficient β, the effective sample thickness deff and integration constant s. The additional impact of free-carrier absorption to the transmission was derived in Ref. [19]; Ogusu et al. [30] introduced the appearance of a transient free carrier absorption with silicon as an example. Based on these theoretical concepts, we need to add the characteristic time of electron-phonon relaxation τS, i.e. a temporal offset prior to small polaron formation, as a particular feature of small polaron dynamics. Then, the change of the pulse intensity through a sample as a function of the propagation depth L, radius r and time t is given by:
I(L,r,t)L=[α+βI(L,r,t)+σNP(L,r,t)]I(L,r,t).
Here, α is the one-photon absorption coefficient, and σ and NP are the absorption cross section and number density of small polarons. The temporal evolution of the latter is modeled according to Fig. 1 by considering the number density of free carriers NFC, the subsequent electron-phonon cooling (τS), carrier recombination (τR) and/or small polarons formation (τFC–P) during pulse duration (τ) via:
NP(L,r,t)t=NFC(L,r,tτS)τFCP
with the temporal evolution of free carriers:
NFC(L,r,t)t=αI(L,r,t)hν+βI2(L,r,t)2hνNFC(L,r,tτS)τRNFC(L,r,tτS)τFCP
Differential equation (2) is solved numerically with spatial and temporal Gaussian distribution of the input intensity for data analysis; Fig. 2 highlights the impact of pulse duration on the transmission of a z-scan measurement. For numerical solution, the following model parameters are used: α = 0 m−1, β = 5 mm/GW, τS = 100 fs, τFC–P = 100 fs, τR = 100 fs and the pulse duration is varied from 100 fs to 1,000 fs while the peak intensity I at z = 0 is kept constant at 8.6 PW/m2. The plots show the characteristic drop of the transmission while scanning along the z-coordinate with a minimum transmission at z = 0, and are mirror symmetric to z = 0. The additional contribution of non-instantaneous absorption processes results in a pronounced increase of the transmission loss in the order of several tens of percentage, i.e. the increase of nonlinear absorption by small polaron formation, with increasing pulse duration is obvious. For comparison, the result of Eq. (1) considering the action of TPA, only, is depicted in addition (red curve). The numerical solution Eq. (2) converges to the result of Eq. (1) for pulse durations equal to the electron-phonon cooling rate, ττS, or below. Thus, it is possible to estimate the characteristic time constants of small polaron formation from z-scan experiments as a function of pulse duration, if τ is in the order of τS. It is important to note, that the analysis of nonlinear absorption using Eq. (1) and pulse durations exceeding τS will result in an overestimate of the TPA coefficient β. This is due to the fact, that the numerical solutions not only show an increase in the transmission loss, but also a change in the shape of T(z) (cf. inset of Fig. 2). A limit of the numerical analysis is that it is not possible to distinguish between several types of small bound polarons; this, however, may be solved by performing a systematic study at different photon energies of the incident pulse.

 

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, τS = 100 fs, τFC–P = 100 fs, τR = 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 NP = 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 (ec) 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 τS = 80 fs, for interband relaxation of τR = 100 fs and for small polaron formation of τFC–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 τS = 80 fs, for interband relaxation of τR = 100 fs and for small polaron formation of τFC–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 MSEmin = (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

MSE(t)=MSE(t=)×[1exp((ttoffset)τexp)]+MSEmin
to the data set yielding the following parameters: saturation amplitude MSE(t = ∞) = (4 ± 0.1) × 10−4, characteristic time constant τexp = (137 ± 8) fs, temporal offset of toffset = (86 ± 5) fs, and minimum value of mean squared error MSEmin = (1.5 ± 0.1) × 10−4.

 

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 MSEmin = (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 τexp = (137 ± 8) fs, temporal offset of τ = (86 ± 5) fs, and minimum value of mean squared error MSEmin = (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 τS. 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 τS is equal to the pulse duration where the transmission trace starts to alter its shape. Therefore, the temporal offset τS = 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 τS falls much below the pulse duration (τS ≪ (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 τFC–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 τS = 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 τFC–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 τFC–P is only related to small bound polarons ( σNbLi4+=4.01022m2, σO = 4.1 · 10−22m2 at 2.5 eV [5]). Due to the negligible absorption of small free polarons at our probing wavelength ( σNbLi4+=0.81022m2at 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 τS. 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 (τR = ∞ 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 NbLi4+ polarons, although the maxima of the respective absorption features are fairly different (2.5 eV for O and 1.6 eV for NbLi4+). Due to two-photon interband excitation it is reasonable to assume that the number density of optically generated electron Np,e and hole polarons Np,h is identical, Np,e = Np,h. As a consequence, it is necessary to exchange the factor σNp in Eq. (2) by the sum (σp,eNp,e + σp,hNp,h) = 2 σp,eNp,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 σp,e needs to be reduced by a factor of two. Still, however, σp,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 τFC–P and τR is not independent from other model parameters and directly impacts the density numbers NFC and NP 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 τS can be varied between 70 fs and 90 fs while the polaron formation time is limited to 70 fs < τFC–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 τS yielding τS = toffset = (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.

Acknowledgments

The authors gratefully acknowledge discussion from G. Corradi, L. Kovács (SZFKI, Hungary), and from A. Büscher. Financial support by the Deutsche Forschungsgemeinschaft, DFG (project numbers: IM 37/5-2, INST 190/137-1 FUGG, INST 190/165-1) is gratefully acknowledged.

References and links

1. M. Fejer, G. Magel, D. H. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992). [CrossRef]  

2. O. A. Louchev, H. H. Hatano, N. Saito, S. Wada, and K. Kitamura, “Laser-induced breakdown and damage generation by nonlinear frequency conversion in ferroelectric crystals: Experiment and theory,” J. Appl. Phys. 114, 203101 (2013). [CrossRef]  

3. O. F. Schirmer and D. von der Linde, “Two–Photon and X–Ray–Induced Nb4+ and O Small Polarons in LiNbO3,” Appl. Phys. Lett. 33, 35 (1978). [CrossRef]  

4. D. Emin, Polarons (Cambridge University Press, Cambridge, 2013).

5. C. Merschjann, B. Schoke, D. Conradi, M. Imlau, G. Corradi, and K. Polgar, “Absorption cross sections and number densities of electron and hole polarons in congruently melting LiNbO3,” J. Phys. Chem. Sol. 21, 015906 (2009).

6. Y. Qiu, K. B. Ucer, and R. T. Williams, “Formation time of a small electron polaron in LiNbO3: measurements and interpretation,” Phys. Stat. Sol. C 2, 232 (2005). [CrossRef]  

7. S. Sasamoto, J. Hirohashi, and S. Ashihara, “Polaron dynamics in lithium niobate upon femtosecond pulse irradiation: Influence of magnesium doping and stoichiometry control,” J. Appl. Phys. 105, 083102 (2009). [CrossRef]  

8. O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals,” Phys. Rev. E 71, 056603 (2005). [CrossRef]  

9. O. Beyer, D. Maxein, T. Woike, and K. Buse, “Generation of small bound polarons in lithium niobate crystals on the subpicosecond time scale,” Appl. Phys. B 83, 527–530 (2006). [CrossRef]  

10. H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999). [CrossRef]  

11. M. Imlau, H. Badorreck, and C. Merschjann, “Optical nonlinearities of small polarons in lithium niobate,” Appl. Phys. Rev. 2, 040606 (2015). [CrossRef]  

12. D. Maxein and K. Buse, “Interaction of Femtosecond Laser Pulses with Lithium Niobate Crystals: Transmission Changes and Refractive Index Modulations,” J. Holography Speckle 5, 1–5 (2009). [CrossRef]  

13. A. Seilmeier and W. Kaiser, “Generation of tunable picosecond light pulses covering the frequency range between 2700 and 32,000 cm−1,” Appl. Phys. A 23, 113 (1980). [CrossRef]  

14. R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324 (1996). [CrossRef]  

15. H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659 (1997). [CrossRef]  

16. R. Ganeev, I. Kulagin, A. Ryasnyansky, R. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004). [CrossRef]  

17. O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Femtosecond time-resolved absorption process in lithium niobate crystals,” Opt. Lett. 30, 1366 (2005). [CrossRef]   [PubMed]  

18. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive Measurement of Optical Nonlinearities Using A Single Beam,” IEEE J. Quantum Electron. 26, 760 (1990). [CrossRef]  

19. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of Bound-electronic and Free-carrier Nonlinearities In ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405 (1992). [CrossRef]  

20. S. Redfield and W. J. Burke, “Optical Absorption Edge of LiNbO3,” J. Appl. Phys.45 (1974). [CrossRef]  

21. H. Badorreck, School of Physics, Osnabrueck University, Barbarastr. 7, 49076 Osnabrueck, Germany, A. Shumelyuk, S. Nolte, M. Imlau, and S. Odoulov are preparing a manuscript to be called “Selfdiffraction from moving gratings recorded in LiNbO3 with ultra-short laser pulses of different colors.”

22. A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. 83, 1789–1830 (1998). [CrossRef]  

23. M. Garcia-Lechuga, J. Siegel, J. Hernandez-Rueda, and J. Solis, “Imaging the ultrafast Kerr effect, free carrier generation, relaxation and ablation dynamics of Lithium Niobate irradiated with femtosecond laser pulses,” J. Appl. Phys. 116, 113502 (2014). [CrossRef]  

24. O. F. Schirmer, O. Thiemann, and M. Wöhlecke, “Defects in LiNbO3 — I. Experimental Aspects,” J. Phys. Chem. Solids 52, 185 (1991). [CrossRef]  

25. O. F. Schirmer, “O bound small polarons in oxide materials,” J. Phys. Condens. Matter 18, R667 (2006). [CrossRef]  

26. O. F. Schirmer, M. Imlau, C. Merschjann, and B. Schoke, “Electron small polarons and bipolarons in LiNbO3,” J. Phys. Condens. Matter 21, 123201 (2009). [CrossRef]  

27. B. Faust, H. Müller, and O. F. Schirmer, “Free small polarons in LiNbO3,” Ferroelectrics 153, 297 (1994). [CrossRef]  

28. Y. Li, W. G. Schmidt, and S. Sanna, “Intrinsic LiNbO3 point defects from hybrid density functional calculations,” Phys. Rev. B 89, 094111 (2014). [CrossRef]  

29. Y. Li, S. Sanna, and W. G. Schmidt, “Modeling intrinsic defects in LiNbO3 within the Slater-Janak transition state model,” The Journal of Chemical Physics 140, 234113 (2014). [CrossRef]  

30. K. Ogusu and K. Shinkawa, “Optical nonlinearities in silicon for pulse durations of the order of nanoseconds at 1.06 μm,” Opt. Express 16, 14780–14791 (2008). [CrossRef]   [PubMed]  

31. P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-Beam Z-Scan: Measurement Techniques and Analysis,” J. Nonlinear Opt. Phys. Mater. 06, 251–293 (1997). [CrossRef]  

32. M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990). [CrossRef]   [PubMed]  

33. O. F. Schirmer, M. Imlau, and C. Merschjann, “Bulk photovoltaic effect of LiNbO3:Fe and its small-polaron-based microscopic interpretation,” Phys. Rev. B 83, 165106 (2011). [CrossRef]  

References

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  1. M. Fejer, G. Magel, D. H. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [Crossref]
  2. O. A. Louchev, H. H. Hatano, N. Saito, S. Wada, and K. Kitamura, “Laser-induced breakdown and damage generation by nonlinear frequency conversion in ferroelectric crystals: Experiment and theory,” J. Appl. Phys. 114, 203101 (2013).
    [Crossref]
  3. O. F. Schirmer and D. von der Linde, “Two–Photon and X–Ray–Induced Nb4+ and O− Small Polarons in LiNbO3,” Appl. Phys. Lett. 33, 35 (1978).
    [Crossref]
  4. D. Emin, Polarons (Cambridge University Press, Cambridge, 2013).
  5. C. Merschjann, B. Schoke, D. Conradi, M. Imlau, G. Corradi, and K. Polgar, “Absorption cross sections and number densities of electron and hole polarons in congruently melting LiNbO3,” J. Phys. Chem. Sol. 21, 015906 (2009).
  6. Y. Qiu, K. B. Ucer, and R. T. Williams, “Formation time of a small electron polaron in LiNbO3: measurements and interpretation,” Phys. Stat. Sol. C 2, 232 (2005).
    [Crossref]
  7. S. Sasamoto, J. Hirohashi, and S. Ashihara, “Polaron dynamics in lithium niobate upon femtosecond pulse irradiation: Influence of magnesium doping and stoichiometry control,” J. Appl. Phys. 105, 083102 (2009).
    [Crossref]
  8. O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals,” Phys. Rev. E 71, 056603 (2005).
    [Crossref]
  9. O. Beyer, D. Maxein, T. Woike, and K. Buse, “Generation of small bound polarons in lithium niobate crystals on the subpicosecond time scale,” Appl. Phys. B 83, 527–530 (2006).
    [Crossref]
  10. H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
    [Crossref]
  11. M. Imlau, H. Badorreck, and C. Merschjann, “Optical nonlinearities of small polarons in lithium niobate,” Appl. Phys. Rev. 2, 040606 (2015).
    [Crossref]
  12. D. Maxein and K. Buse, “Interaction of Femtosecond Laser Pulses with Lithium Niobate Crystals: Transmission Changes and Refractive Index Modulations,” J. Holography Speckle 5, 1–5 (2009).
    [Crossref]
  13. A. Seilmeier and W. Kaiser, “Generation of tunable picosecond light pulses covering the frequency range between 2700 and 32,000 cm−1,” Appl. Phys. A 23, 113 (1980).
    [Crossref]
  14. R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324 (1996).
    [Crossref]
  15. H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659 (1997).
    [Crossref]
  16. R. Ganeev, I. Kulagin, A. Ryasnyansky, R. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
    [Crossref]
  17. O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Femtosecond time-resolved absorption process in lithium niobate crystals,” Opt. Lett. 30, 1366 (2005).
    [Crossref] [PubMed]
  18. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive Measurement of Optical Nonlinearities Using A Single Beam,” IEEE J. Quantum Electron. 26, 760 (1990).
    [Crossref]
  19. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of Bound-electronic and Free-carrier Nonlinearities In ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405 (1992).
    [Crossref]
  20. S. Redfield and W. J. Burke, “Optical Absorption Edge of LiNbO3,” J. Appl. Phys.45 (1974).
    [Crossref]
  21. H. Badorreck, School of Physics, Osnabrueck University, Barbarastr. 7, 49076 Osnabrueck, Germany, A. Shumelyuk, S. Nolte, M. Imlau, and S. Odoulov are preparing a manuscript to be called “Selfdiffraction from moving gratings recorded in LiNbO3 with ultra-short laser pulses of different colors.”
  22. A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. 83, 1789–1830 (1998).
    [Crossref]
  23. M. Garcia-Lechuga, J. Siegel, J. Hernandez-Rueda, and J. Solis, “Imaging the ultrafast Kerr effect, free carrier generation, relaxation and ablation dynamics of Lithium Niobate irradiated with femtosecond laser pulses,” J. Appl. Phys. 116, 113502 (2014).
    [Crossref]
  24. O. F. Schirmer, O. Thiemann, and M. Wöhlecke, “Defects in LiNbO3 — I. Experimental Aspects,” J. Phys. Chem. Solids 52, 185 (1991).
    [Crossref]
  25. O. F. Schirmer, “O− bound small polarons in oxide materials,” J. Phys. Condens. Matter 18, R667 (2006).
    [Crossref]
  26. O. F. Schirmer, M. Imlau, C. Merschjann, and B. Schoke, “Electron small polarons and bipolarons in LiNbO3,” J. Phys. Condens. Matter 21, 123201 (2009).
    [Crossref]
  27. B. Faust, H. Müller, and O. F. Schirmer, “Free small polarons in LiNbO3,” Ferroelectrics 153, 297 (1994).
    [Crossref]
  28. Y. Li, W. G. Schmidt, and S. Sanna, “Intrinsic LiNbO3 point defects from hybrid density functional calculations,” Phys. Rev. B 89, 094111 (2014).
    [Crossref]
  29. Y. Li, S. Sanna, and W. G. Schmidt, “Modeling intrinsic defects in LiNbO3 within the Slater-Janak transition state model,” The Journal of Chemical Physics 140, 234113 (2014).
    [Crossref]
  30. K. Ogusu and K. Shinkawa, “Optical nonlinearities in silicon for pulse durations of the order of nanoseconds at 1.06 μm,” Opt. Express 16, 14780–14791 (2008).
    [Crossref] [PubMed]
  31. P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-Beam Z-Scan: Measurement Techniques and Analysis,” J. Nonlinear Opt. Phys. Mater. 06, 251–293 (1997).
    [Crossref]
  32. M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
    [Crossref] [PubMed]
  33. O. F. Schirmer, M. Imlau, and C. Merschjann, “Bulk photovoltaic effect of LiNbO3:Fe and its small-polaron-based microscopic interpretation,” Phys. Rev. B 83, 165106 (2011).
    [Crossref]

2015 (1)

M. Imlau, H. Badorreck, and C. Merschjann, “Optical nonlinearities of small polarons in lithium niobate,” Appl. Phys. Rev. 2, 040606 (2015).
[Crossref]

2014 (3)

M. Garcia-Lechuga, J. Siegel, J. Hernandez-Rueda, and J. Solis, “Imaging the ultrafast Kerr effect, free carrier generation, relaxation and ablation dynamics of Lithium Niobate irradiated with femtosecond laser pulses,” J. Appl. Phys. 116, 113502 (2014).
[Crossref]

Y. Li, W. G. Schmidt, and S. Sanna, “Intrinsic LiNbO3 point defects from hybrid density functional calculations,” Phys. Rev. B 89, 094111 (2014).
[Crossref]

Y. Li, S. Sanna, and W. G. Schmidt, “Modeling intrinsic defects in LiNbO3 within the Slater-Janak transition state model,” The Journal of Chemical Physics 140, 234113 (2014).
[Crossref]

2013 (1)

O. A. Louchev, H. H. Hatano, N. Saito, S. Wada, and K. Kitamura, “Laser-induced breakdown and damage generation by nonlinear frequency conversion in ferroelectric crystals: Experiment and theory,” J. Appl. Phys. 114, 203101 (2013).
[Crossref]

2011 (1)

O. F. Schirmer, M. Imlau, and C. Merschjann, “Bulk photovoltaic effect of LiNbO3:Fe and its small-polaron-based microscopic interpretation,” Phys. Rev. B 83, 165106 (2011).
[Crossref]

2009 (4)

O. F. Schirmer, M. Imlau, C. Merschjann, and B. Schoke, “Electron small polarons and bipolarons in LiNbO3,” J. Phys. Condens. Matter 21, 123201 (2009).
[Crossref]

C. Merschjann, B. Schoke, D. Conradi, M. Imlau, G. Corradi, and K. Polgar, “Absorption cross sections and number densities of electron and hole polarons in congruently melting LiNbO3,” J. Phys. Chem. Sol. 21, 015906 (2009).

S. Sasamoto, J. Hirohashi, and S. Ashihara, “Polaron dynamics in lithium niobate upon femtosecond pulse irradiation: Influence of magnesium doping and stoichiometry control,” J. Appl. Phys. 105, 083102 (2009).
[Crossref]

D. Maxein and K. Buse, “Interaction of Femtosecond Laser Pulses with Lithium Niobate Crystals: Transmission Changes and Refractive Index Modulations,” J. Holography Speckle 5, 1–5 (2009).
[Crossref]

2008 (1)

2006 (2)

O. F. Schirmer, “O− bound small polarons in oxide materials,” J. Phys. Condens. Matter 18, R667 (2006).
[Crossref]

O. Beyer, D. Maxein, T. Woike, and K. Buse, “Generation of small bound polarons in lithium niobate crystals on the subpicosecond time scale,” Appl. Phys. B 83, 527–530 (2006).
[Crossref]

2005 (3)

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals,” Phys. Rev. E 71, 056603 (2005).
[Crossref]

Y. Qiu, K. B. Ucer, and R. T. Williams, “Formation time of a small electron polaron in LiNbO3: measurements and interpretation,” Phys. Stat. Sol. C 2, 232 (2005).
[Crossref]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Femtosecond time-resolved absorption process in lithium niobate crystals,” Opt. Lett. 30, 1366 (2005).
[Crossref] [PubMed]

2004 (1)

R. Ganeev, I. Kulagin, A. Ryasnyansky, R. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

1999 (1)

H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
[Crossref]

1998 (1)

A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. 83, 1789–1830 (1998).
[Crossref]

1997 (2)

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-Beam Z-Scan: Measurement Techniques and Analysis,” J. Nonlinear Opt. Phys. Mater. 06, 251–293 (1997).
[Crossref]

H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659 (1997).
[Crossref]

1996 (1)

R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324 (1996).
[Crossref]

1994 (1)

B. Faust, H. Müller, and O. F. Schirmer, “Free small polarons in LiNbO3,” Ferroelectrics 153, 297 (1994).
[Crossref]

1992 (2)

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of Bound-electronic and Free-carrier Nonlinearities In ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405 (1992).
[Crossref]

M. Fejer, G. Magel, D. H. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

1991 (1)

O. F. Schirmer, O. Thiemann, and M. Wöhlecke, “Defects in LiNbO3 — I. Experimental Aspects,” J. Phys. Chem. Solids 52, 185 (1991).
[Crossref]

1990 (2)

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive Measurement of Optical Nonlinearities Using A Single Beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

1980 (1)

A. Seilmeier and W. Kaiser, “Generation of tunable picosecond light pulses covering the frequency range between 2700 and 32,000 cm−1,” Appl. Phys. A 23, 113 (1980).
[Crossref]

1978 (1)

O. F. Schirmer and D. von der Linde, “Two–Photon and X–Ray–Induced Nb4+ and O− Small Polarons in LiNbO3,” Appl. Phys. Lett. 33, 35 (1978).
[Crossref]

Ashihara, S.

S. Sasamoto, J. Hirohashi, and S. Ashihara, “Polaron dynamics in lithium niobate upon femtosecond pulse irradiation: Influence of magnesium doping and stoichiometry control,” J. Appl. Phys. 105, 083102 (2009).
[Crossref]

Badorreck, H.

M. Imlau, H. Badorreck, and C. Merschjann, “Optical nonlinearities of small polarons in lithium niobate,” Appl. Phys. Rev. 2, 040606 (2015).
[Crossref]

Beyer, O.

O. Beyer, D. Maxein, T. Woike, and K. Buse, “Generation of small bound polarons in lithium niobate crystals on the subpicosecond time scale,” Appl. Phys. B 83, 527–530 (2006).
[Crossref]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals,” Phys. Rev. E 71, 056603 (2005).
[Crossref]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Femtosecond time-resolved absorption process in lithium niobate crystals,” Opt. Lett. 30, 1366 (2005).
[Crossref] [PubMed]

Burke, W. J.

S. Redfield and W. J. Burke, “Optical Absorption Edge of LiNbO3,” J. Appl. Phys.45 (1974).
[Crossref]

Buse, K.

D. Maxein and K. Buse, “Interaction of Femtosecond Laser Pulses with Lithium Niobate Crystals: Transmission Changes and Refractive Index Modulations,” J. Holography Speckle 5, 1–5 (2009).
[Crossref]

O. Beyer, D. Maxein, T. Woike, and K. Buse, “Generation of small bound polarons in lithium niobate crystals on the subpicosecond time scale,” Appl. Phys. B 83, 527–530 (2006).
[Crossref]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals,” Phys. Rev. E 71, 056603 (2005).
[Crossref]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Femtosecond time-resolved absorption process in lithium niobate crystals,” Opt. Lett. 30, 1366 (2005).
[Crossref] [PubMed]

Byer, R.

M. Fejer, G. Magel, D. H. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Chapple, P. B.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-Beam Z-Scan: Measurement Techniques and Analysis,” J. Nonlinear Opt. Phys. Mater. 06, 251–293 (1997).
[Crossref]

Conradi, D.

C. Merschjann, B. Schoke, D. Conradi, M. Imlau, G. Corradi, and K. Polgar, “Absorption cross sections and number densities of electron and hole polarons in congruently melting LiNbO3,” J. Phys. Chem. Sol. 21, 015906 (2009).

Corradi, G.

C. Merschjann, B. Schoke, D. Conradi, M. Imlau, G. Corradi, and K. Polgar, “Absorption cross sections and number densities of electron and hole polarons in congruently melting LiNbO3,” J. Phys. Chem. Sol. 21, 015906 (2009).

Denks, V.

H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
[Crossref]

DeSalvo, R.

R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324 (1996).
[Crossref]

Emin, D.

D. Emin, Polarons (Cambridge University Press, Cambridge, 2013).

Faust, B.

B. Faust, H. Müller, and O. F. Schirmer, “Free small polarons in LiNbO3,” Ferroelectrics 153, 297 (1994).
[Crossref]

Fejer, M.

M. Fejer, G. Magel, D. H. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Ganeev, R.

R. Ganeev, I. Kulagin, A. Ryasnyansky, R. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

Garcia-Lechuga, M.

M. Garcia-Lechuga, J. Siegel, J. Hernandez-Rueda, and J. Solis, “Imaging the ultrafast Kerr effect, free carrier generation, relaxation and ablation dynamics of Lithium Niobate irradiated with femtosecond laser pulses,” J. Appl. Phys. 116, 113502 (2014).
[Crossref]

Grigorjeva, L.

H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
[Crossref]

Hagan, D.

R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324 (1996).
[Crossref]

Hagan, D. J.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of Bound-electronic and Free-carrier Nonlinearities In ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405 (1992).
[Crossref]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive Measurement of Optical Nonlinearities Using A Single Beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

Hatano, H. H.

O. A. Louchev, H. H. Hatano, N. Saito, S. Wada, and K. Kitamura, “Laser-induced breakdown and damage generation by nonlinear frequency conversion in ferroelectric crystals: Experiment and theory,” J. Appl. Phys. 114, 203101 (2013).
[Crossref]

Hermann, J. A.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-Beam Z-Scan: Measurement Techniques and Analysis,” J. Nonlinear Opt. Phys. Mater. 06, 251–293 (1997).
[Crossref]

Hernandez-Rueda, J.

M. Garcia-Lechuga, J. Siegel, J. Hernandez-Rueda, and J. Solis, “Imaging the ultrafast Kerr effect, free carrier generation, relaxation and ablation dynamics of Lithium Niobate irradiated with femtosecond laser pulses,” J. Appl. Phys. 116, 113502 (2014).
[Crossref]

Hirohashi, J.

S. Sasamoto, J. Hirohashi, and S. Ashihara, “Polaron dynamics in lithium niobate upon femtosecond pulse irradiation: Influence of magnesium doping and stoichiometry control,” J. Appl. Phys. 105, 083102 (2009).
[Crossref]

Hsieh, H. T.

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals,” Phys. Rev. E 71, 056603 (2005).
[Crossref]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Femtosecond time-resolved absorption process in lithium niobate crystals,” Opt. Lett. 30, 1366 (2005).
[Crossref] [PubMed]

Imlau, M.

M. Imlau, H. Badorreck, and C. Merschjann, “Optical nonlinearities of small polarons in lithium niobate,” Appl. Phys. Rev. 2, 040606 (2015).
[Crossref]

O. F. Schirmer, M. Imlau, and C. Merschjann, “Bulk photovoltaic effect of LiNbO3:Fe and its small-polaron-based microscopic interpretation,” Phys. Rev. B 83, 165106 (2011).
[Crossref]

O. F. Schirmer, M. Imlau, C. Merschjann, and B. Schoke, “Electron small polarons and bipolarons in LiNbO3,” J. Phys. Condens. Matter 21, 123201 (2009).
[Crossref]

C. Merschjann, B. Schoke, D. Conradi, M. Imlau, G. Corradi, and K. Polgar, “Absorption cross sections and number densities of electron and hole polarons in congruently melting LiNbO3,” J. Phys. Chem. Sol. 21, 015906 (2009).

Ji, W.

H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659 (1997).
[Crossref]

Jundt, D. H.

M. Fejer, G. Magel, D. H. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Kaiser, W.

A. Seilmeier and W. Kaiser, “Generation of tunable picosecond light pulses covering the frequency range between 2700 and 32,000 cm−1,” Appl. Phys. A 23, 113 (1980).
[Crossref]

Kitamura, K.

O. A. Louchev, H. H. Hatano, N. Saito, S. Wada, and K. Kitamura, “Laser-induced breakdown and damage generation by nonlinear frequency conversion in ferroelectric crystals: Experiment and theory,” J. Appl. Phys. 114, 203101 (2013).
[Crossref]

Kotomin, E.

H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
[Crossref]

Kulagin, I.

R. Ganeev, I. Kulagin, A. Ryasnyansky, R. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

Li, H.

H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659 (1997).
[Crossref]

Li, Y.

Y. Li, W. G. Schmidt, and S. Sanna, “Intrinsic LiNbO3 point defects from hybrid density functional calculations,” Phys. Rev. B 89, 094111 (2014).
[Crossref]

Y. Li, S. Sanna, and W. G. Schmidt, “Modeling intrinsic defects in LiNbO3 within the Slater-Janak transition state model,” The Journal of Chemical Physics 140, 234113 (2014).
[Crossref]

Louchev, O. A.

O. A. Louchev, H. H. Hatano, N. Saito, S. Wada, and K. Kitamura, “Laser-induced breakdown and damage generation by nonlinear frequency conversion in ferroelectric crystals: Experiment and theory,” J. Appl. Phys. 114, 203101 (2013).
[Crossref]

Magel, G.

M. Fejer, G. Magel, D. H. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

Maxein, D.

D. Maxein and K. Buse, “Interaction of Femtosecond Laser Pulses with Lithium Niobate Crystals: Transmission Changes and Refractive Index Modulations,” J. Holography Speckle 5, 1–5 (2009).
[Crossref]

O. Beyer, D. Maxein, T. Woike, and K. Buse, “Generation of small bound polarons in lithium niobate crystals on the subpicosecond time scale,” Appl. Phys. B 83, 527–530 (2006).
[Crossref]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals,” Phys. Rev. E 71, 056603 (2005).
[Crossref]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Femtosecond time-resolved absorption process in lithium niobate crystals,” Opt. Lett. 30, 1366 (2005).
[Crossref] [PubMed]

Mcduff, R. G.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-Beam Z-Scan: Measurement Techniques and Analysis,” J. Nonlinear Opt. Phys. Mater. 06, 251–293 (1997).
[Crossref]

Mckay, T. J.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-Beam Z-Scan: Measurement Techniques and Analysis,” J. Nonlinear Opt. Phys. Mater. 06, 251–293 (1997).
[Crossref]

Merschjann, C.

M. Imlau, H. Badorreck, and C. Merschjann, “Optical nonlinearities of small polarons in lithium niobate,” Appl. Phys. Rev. 2, 040606 (2015).
[Crossref]

O. F. Schirmer, M. Imlau, and C. Merschjann, “Bulk photovoltaic effect of LiNbO3:Fe and its small-polaron-based microscopic interpretation,” Phys. Rev. B 83, 165106 (2011).
[Crossref]

O. F. Schirmer, M. Imlau, C. Merschjann, and B. Schoke, “Electron small polarons and bipolarons in LiNbO3,” J. Phys. Condens. Matter 21, 123201 (2009).
[Crossref]

C. Merschjann, B. Schoke, D. Conradi, M. Imlau, G. Corradi, and K. Polgar, “Absorption cross sections and number densities of electron and hole polarons in congruently melting LiNbO3,” J. Phys. Chem. Sol. 21, 015906 (2009).

Millers, D.

H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
[Crossref]

Müller, H.

B. Faust, H. Müller, and O. F. Schirmer, “Free small polarons in LiNbO3,” Ferroelectrics 153, 297 (1994).
[Crossref]

Nagirnyi, V.

H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
[Crossref]

Ogusu, K.

Othonos, A.

A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. 83, 1789–1830 (1998).
[Crossref]

Polgar, K.

C. Merschjann, B. Schoke, D. Conradi, M. Imlau, G. Corradi, and K. Polgar, “Absorption cross sections and number densities of electron and hole polarons in congruently melting LiNbO3,” J. Phys. Chem. Sol. 21, 015906 (2009).

Psaltis, D.

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals,” Phys. Rev. E 71, 056603 (2005).
[Crossref]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Femtosecond time-resolved absorption process in lithium niobate crystals,” Opt. Lett. 30, 1366 (2005).
[Crossref] [PubMed]

Qiu, Y.

Y. Qiu, K. B. Ucer, and R. T. Williams, “Formation time of a small electron polaron in LiNbO3: measurements and interpretation,” Phys. Stat. Sol. C 2, 232 (2005).
[Crossref]

Redfield, S.

S. Redfield and W. J. Burke, “Optical Absorption Edge of LiNbO3,” J. Appl. Phys.45 (1974).
[Crossref]

Ryasnyansky, A.

R. Ganeev, I. Kulagin, A. Ryasnyansky, R. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

Said, A.

R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324 (1996).
[Crossref]

Said, A. A.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of Bound-electronic and Free-carrier Nonlinearities In ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405 (1992).
[Crossref]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive Measurement of Optical Nonlinearities Using A Single Beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

Saito, N.

O. A. Louchev, H. H. Hatano, N. Saito, S. Wada, and K. Kitamura, “Laser-induced breakdown and damage generation by nonlinear frequency conversion in ferroelectric crystals: Experiment and theory,” J. Appl. Phys. 114, 203101 (2013).
[Crossref]

Sanna, S.

Y. Li, S. Sanna, and W. G. Schmidt, “Modeling intrinsic defects in LiNbO3 within the Slater-Janak transition state model,” The Journal of Chemical Physics 140, 234113 (2014).
[Crossref]

Y. Li, W. G. Schmidt, and S. Sanna, “Intrinsic LiNbO3 point defects from hybrid density functional calculations,” Phys. Rev. B 89, 094111 (2014).
[Crossref]

Sasamoto, S.

S. Sasamoto, J. Hirohashi, and S. Ashihara, “Polaron dynamics in lithium niobate upon femtosecond pulse irradiation: Influence of magnesium doping and stoichiometry control,” J. Appl. Phys. 105, 083102 (2009).
[Crossref]

Schirmer, O. F.

O. F. Schirmer, M. Imlau, and C. Merschjann, “Bulk photovoltaic effect of LiNbO3:Fe and its small-polaron-based microscopic interpretation,” Phys. Rev. B 83, 165106 (2011).
[Crossref]

O. F. Schirmer, M. Imlau, C. Merschjann, and B. Schoke, “Electron small polarons and bipolarons in LiNbO3,” J. Phys. Condens. Matter 21, 123201 (2009).
[Crossref]

O. F. Schirmer, “O− bound small polarons in oxide materials,” J. Phys. Condens. Matter 18, R667 (2006).
[Crossref]

B. Faust, H. Müller, and O. F. Schirmer, “Free small polarons in LiNbO3,” Ferroelectrics 153, 297 (1994).
[Crossref]

O. F. Schirmer, O. Thiemann, and M. Wöhlecke, “Defects in LiNbO3 — I. Experimental Aspects,” J. Phys. Chem. Solids 52, 185 (1991).
[Crossref]

O. F. Schirmer and D. von der Linde, “Two–Photon and X–Ray–Induced Nb4+ and O− Small Polarons in LiNbO3,” Appl. Phys. Lett. 33, 35 (1978).
[Crossref]

Schmidt, W. G.

Y. Li, S. Sanna, and W. G. Schmidt, “Modeling intrinsic defects in LiNbO3 within the Slater-Janak transition state model,” The Journal of Chemical Physics 140, 234113 (2014).
[Crossref]

Y. Li, W. G. Schmidt, and S. Sanna, “Intrinsic LiNbO3 point defects from hybrid density functional calculations,” Phys. Rev. B 89, 094111 (2014).
[Crossref]

Schoke, B.

O. F. Schirmer, M. Imlau, C. Merschjann, and B. Schoke, “Electron small polarons and bipolarons in LiNbO3,” J. Phys. Condens. Matter 21, 123201 (2009).
[Crossref]

C. Merschjann, B. Schoke, D. Conradi, M. Imlau, G. Corradi, and K. Polgar, “Absorption cross sections and number densities of electron and hole polarons in congruently melting LiNbO3,” J. Phys. Chem. Sol. 21, 015906 (2009).

Seilmeier, A.

A. Seilmeier and W. Kaiser, “Generation of tunable picosecond light pulses covering the frequency range between 2700 and 32,000 cm−1,” Appl. Phys. A 23, 113 (1980).
[Crossref]

Sheik-Bahae, M.

R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324 (1996).
[Crossref]

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of Bound-electronic and Free-carrier Nonlinearities In ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405 (1992).
[Crossref]

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive Measurement of Optical Nonlinearities Using A Single Beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

Sheldon, P.

H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
[Crossref]

Shinkawa, K.

Siegel, J.

M. Garcia-Lechuga, J. Siegel, J. Hernandez-Rueda, and J. Solis, “Imaging the ultrafast Kerr effect, free carrier generation, relaxation and ablation dynamics of Lithium Niobate irradiated with femtosecond laser pulses,” J. Appl. Phys. 116, 113502 (2014).
[Crossref]

Solis, J.

M. Garcia-Lechuga, J. Siegel, J. Hernandez-Rueda, and J. Solis, “Imaging the ultrafast Kerr effect, free carrier generation, relaxation and ablation dynamics of Lithium Niobate irradiated with femtosecond laser pulses,” J. Appl. Phys. 116, 113502 (2014).
[Crossref]

Staromlynska, J.

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-Beam Z-Scan: Measurement Techniques and Analysis,” J. Nonlinear Opt. Phys. Mater. 06, 251–293 (1997).
[Crossref]

Sturman, B.

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals,” Phys. Rev. E 71, 056603 (2005).
[Crossref]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Femtosecond time-resolved absorption process in lithium niobate crystals,” Opt. Lett. 30, 1366 (2005).
[Crossref] [PubMed]

Thiemann, O.

O. F. Schirmer, O. Thiemann, and M. Wöhlecke, “Defects in LiNbO3 — I. Experimental Aspects,” J. Phys. Chem. Solids 52, 185 (1991).
[Crossref]

Tugushev, R.

R. Ganeev, I. Kulagin, A. Ryasnyansky, R. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

Ucer, K.

H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
[Crossref]

Ucer, K. B.

Y. Qiu, K. B. Ucer, and R. T. Williams, “Formation time of a small electron polaron in LiNbO3: measurements and interpretation,” Phys. Stat. Sol. C 2, 232 (2005).
[Crossref]

Usmanov, T.

R. Ganeev, I. Kulagin, A. Ryasnyansky, R. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

Van Stryland, E.

R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324 (1996).
[Crossref]

Van Stryland, E. W.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of Bound-electronic and Free-carrier Nonlinearities In ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405 (1992).
[Crossref]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive Measurement of Optical Nonlinearities Using A Single Beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

von der Linde, D.

O. F. Schirmer and D. von der Linde, “Two–Photon and X–Ray–Induced Nb4+ and O− Small Polarons in LiNbO3,” Appl. Phys. Lett. 33, 35 (1978).
[Crossref]

Wada, S.

O. A. Louchev, H. H. Hatano, N. Saito, S. Wada, and K. Kitamura, “Laser-induced breakdown and damage generation by nonlinear frequency conversion in ferroelectric crystals: Experiment and theory,” J. Appl. Phys. 114, 203101 (2013).
[Crossref]

Wang, J.

Wei, T. H.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of Bound-electronic and Free-carrier Nonlinearities In ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405 (1992).
[Crossref]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive Measurement of Optical Nonlinearities Using A Single Beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

Williams, R.

H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
[Crossref]

Williams, R. T.

Y. Qiu, K. B. Ucer, and R. T. Williams, “Formation time of a small electron polaron in LiNbO3: measurements and interpretation,” Phys. Stat. Sol. C 2, 232 (2005).
[Crossref]

Wöhlecke, M.

O. F. Schirmer, O. Thiemann, and M. Wöhlecke, “Defects in LiNbO3 — I. Experimental Aspects,” J. Phys. Chem. Solids 52, 185 (1991).
[Crossref]

Woike, T.

O. Beyer, D. Maxein, T. Woike, and K. Buse, “Generation of small bound polarons in lithium niobate crystals on the subpicosecond time scale,” Appl. Phys. B 83, 527–530 (2006).
[Crossref]

Yochum, H.

H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
[Crossref]

Young, J.

Zhang, X.

H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659 (1997).
[Crossref]

Zhou, F.

H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659 (1997).
[Crossref]

Appl. Phys. A (1)

A. Seilmeier and W. Kaiser, “Generation of tunable picosecond light pulses covering the frequency range between 2700 and 32,000 cm−1,” Appl. Phys. A 23, 113 (1980).
[Crossref]

Appl. Phys. B (2)

H. Li, F. Zhou, X. Zhang, and W. Ji, “Picosecond Z-scan study of bound electronic Kerr effect in LiNbO3 crystal associated with two-photon absorption,” Appl. Phys. B 64, 659 (1997).
[Crossref]

O. Beyer, D. Maxein, T. Woike, and K. Buse, “Generation of small bound polarons in lithium niobate crystals on the subpicosecond time scale,” Appl. Phys. B 83, 527–530 (2006).
[Crossref]

Appl. Phys. Lett. (1)

O. F. Schirmer and D. von der Linde, “Two–Photon and X–Ray–Induced Nb4+ and O− Small Polarons in LiNbO3,” Appl. Phys. Lett. 33, 35 (1978).
[Crossref]

Appl. Phys. Rev. (1)

M. Imlau, H. Badorreck, and C. Merschjann, “Optical nonlinearities of small polarons in lithium niobate,” Appl. Phys. Rev. 2, 040606 (2015).
[Crossref]

Ferroelectrics (1)

B. Faust, H. Müller, and O. F. Schirmer, “Free small polarons in LiNbO3,” Ferroelectrics 153, 297 (1994).
[Crossref]

IEEE J. Quantum Electron. (3)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive Measurement of Optical Nonlinearities Using A Single Beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324 (1996).
[Crossref]

M. Fejer, G. Magel, D. H. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[Crossref]

J. Appl. Phys. (4)

O. A. Louchev, H. H. Hatano, N. Saito, S. Wada, and K. Kitamura, “Laser-induced breakdown and damage generation by nonlinear frequency conversion in ferroelectric crystals: Experiment and theory,” J. Appl. Phys. 114, 203101 (2013).
[Crossref]

S. Sasamoto, J. Hirohashi, and S. Ashihara, “Polaron dynamics in lithium niobate upon femtosecond pulse irradiation: Influence of magnesium doping and stoichiometry control,” J. Appl. Phys. 105, 083102 (2009).
[Crossref]

A. Othonos, “Probing ultrafast carrier and phonon dynamics in semiconductors,” J. Appl. Phys. 83, 1789–1830 (1998).
[Crossref]

M. Garcia-Lechuga, J. Siegel, J. Hernandez-Rueda, and J. Solis, “Imaging the ultrafast Kerr effect, free carrier generation, relaxation and ablation dynamics of Lithium Niobate irradiated with femtosecond laser pulses,” J. Appl. Phys. 116, 113502 (2014).
[Crossref]

J. Holography Speckle (1)

D. Maxein and K. Buse, “Interaction of Femtosecond Laser Pulses with Lithium Niobate Crystals: Transmission Changes and Refractive Index Modulations,” J. Holography Speckle 5, 1–5 (2009).
[Crossref]

J. Nonlinear Opt. Phys. Mater. (1)

P. B. Chapple, J. Staromlynska, J. A. Hermann, T. J. Mckay, and R. G. Mcduff, “Single-Beam Z-Scan: Measurement Techniques and Analysis,” J. Nonlinear Opt. Phys. Mater. 06, 251–293 (1997).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. Sol. (1)

C. Merschjann, B. Schoke, D. Conradi, M. Imlau, G. Corradi, and K. Polgar, “Absorption cross sections and number densities of electron and hole polarons in congruently melting LiNbO3,” J. Phys. Chem. Sol. 21, 015906 (2009).

J. Phys. Chem. Solids (1)

O. F. Schirmer, O. Thiemann, and M. Wöhlecke, “Defects in LiNbO3 — I. Experimental Aspects,” J. Phys. Chem. Solids 52, 185 (1991).
[Crossref]

J. Phys. Condens. Matter (2)

O. F. Schirmer, “O− bound small polarons in oxide materials,” J. Phys. Condens. Matter 18, R667 (2006).
[Crossref]

O. F. Schirmer, M. Imlau, C. Merschjann, and B. Schoke, “Electron small polarons and bipolarons in LiNbO3,” J. Phys. Condens. Matter 21, 123201 (2009).
[Crossref]

Opt. Commun. (1)

R. Ganeev, I. Kulagin, A. Ryasnyansky, R. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (2)

O. F. Schirmer, M. Imlau, and C. Merschjann, “Bulk photovoltaic effect of LiNbO3:Fe and its small-polaron-based microscopic interpretation,” Phys. Rev. B 83, 165106 (2011).
[Crossref]

Y. Li, W. G. Schmidt, and S. Sanna, “Intrinsic LiNbO3 point defects from hybrid density functional calculations,” Phys. Rev. B 89, 094111 (2014).
[Crossref]

Phys. Rev. E (1)

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, “Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals,” Phys. Rev. E 71, 056603 (2005).
[Crossref]

Phys. Rev. Lett. (1)

M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

Phys. Stat. Sol. C (1)

Y. Qiu, K. B. Ucer, and R. T. Williams, “Formation time of a small electron polaron in LiNbO3: measurements and interpretation,” Phys. Stat. Sol. C 2, 232 (2005).
[Crossref]

Rad. Eff. Def. Sol. (1)

H. Yochum, K. Ucer, R. Williams, P. Sheldon, V. Nagirnyi, V. Denks, L. Grigorjeva, D. Millers, and E. Kotomin, “Short-pulse excitation and spectroscopy of KNbO3, LiNbO3 and KTiOPO4,” Rad. Eff. Def. Sol. 150, 271–276 (1999).
[Crossref]

The Journal of Chemical Physics (1)

Y. Li, S. Sanna, and W. G. Schmidt, “Modeling intrinsic defects in LiNbO3 within the Slater-Janak transition state model,” The Journal of Chemical Physics 140, 234113 (2014).
[Crossref]

Other (3)

S. Redfield and W. J. Burke, “Optical Absorption Edge of LiNbO3,” J. Appl. Phys.45 (1974).
[Crossref]

H. Badorreck, School of Physics, Osnabrueck University, Barbarastr. 7, 49076 Osnabrueck, Germany, A. Shumelyuk, S. Nolte, M. Imlau, and S. Odoulov are preparing a manuscript to be called “Selfdiffraction from moving gratings recorded in LiNbO3 with ultra-short laser pulses of different colors.”

D. Emin, Polarons (Cambridge University Press, Cambridge, 2013).

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Figures (5)

Fig. 1
Fig. 1 Potential diagram of the band-to-band excitation by one-photon (α) and two-photon (β) absorption with photon energies of Eph = 2.5 eV, electron-phonon cooling process with time constant τS, relaxation to the ground state (τR) and subsequent formation of small polarons (τFC–P) in lithium niobate. Absorption cross section σ and number density of polarons NP determine the absorption triggered by optically induced transport of small polarons [11].
Fig. 2
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, τS = 100 fs, τFC–P = 100 fs, τR = 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 NP = 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.
Fig. 3
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.
Fig. 4
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 τS = 80 fs, for interband relaxation of τR = 100 fs and for small polaron formation of τFC–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.
Fig. 5
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 MSEmin = (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 τexp = (137 ± 8) fs, temporal offset of τ = (86 ± 5) fs, and minimum value of mean squared error MSEmin = (1.5 ± 0.1) × 10−4.

Equations (5)

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T ( z ) = 1 q ( z ) π ln ( 1 + q ( z ) exp ( s 2 ) ) d s
I ( L , r , t ) L = [ α + β I ( L , r , t ) + σ N P ( L , r , t ) ] I ( L , r , t ) .
N P ( L , r , t ) t = N FC ( L , r , t τ S ) τ FC P
N FC ( L , r , t ) t = α I ( L , r , t ) h ν + β I 2 ( L , r , t ) 2 h ν N FC ( L , r , t τ S ) τ R N FC ( L , r , t τ S ) τ FC P
MSE ( t ) = MSE ( t = ) × [ 1 exp ( ( t t offset ) τ exp ) ] + MSE min

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