1. Introduction

Various manifestations of small polarons [1] affect strongly the linear and nonlinear optical properties in the oxide crystal lithium niobate, LiNbO₃: free electron ${\text{Nb}}_{\text{Nb}}^{4+}$ polarons, polarons bound to antisite defects, ${\text{Nb}}_{\text{Li}}^{4+}$, and ${\text{Nb}}_{\text{Nb}}^{4+}:{\text{Nb}}_{\text{Li}}^{4+}$ bipolarons [2]. In addition to these three types of electron polarons, also O⁻ hole small polarons bound to Li-vacancies can occur [3]. Such polarons show strong absorption bands peaked near 1 eV ( ${\text{Nb}}_{\text{Nb}}^{4+}$), 1.6 eV ( ${\text{Nb}}_{\text{Li}}^{4+}$) and 2.5 eV (both, ${\text{Nb}}_{\text{Nb}}^{4+}:{\text{Nb}}_{\text{Li}}^{4+}$ and O⁻). These peak energies are related to the respective polaron stabilization energies E _p and, in addition, to the binding defect potentials for the bound species [4, 5]. The absorption bands are rather asymmetric with pronounced high energy wings, having half-widths of typically 1 eV. Thus, they fall within the large band gap of lithium niobate (α = 20 cm⁻¹ at ≈ 3.8 eV [6]) over a wide spectral range.

A steady-state polaron absorption at room temperature in nominally undoped crystals can be created by raising the Fermi level, e.g. by chemical reduction. Then, ${\text{Nb}}_{\text{Nb}}^{4+}:{\text{Nb}}_{\text{Li}}^{4+}$ bipolarons are stable. Their optical and/or thermal dissociation leads to bound polarons ${\text{Nb}}_{\text{Li}}^{4+}$ and to free ${\text{Nb}}_{\text{Nb}}^{4+}$ polarons. The latter recombine very fast within several μs with Nb_Li antisite defects at room temperature. Hence, considerable ${\text{Nb}}_{\text{Nb}}^{4+}$-densities are present in thermal equilibrium only at rather elevated temperatures [7]. Besides, it is possible to generate a pronounced transient small polaron absorption by optical interband excitation of carriers and/or by carrier-release via photo-ionization of defect centers. Number densities of generated small polarons up to tens of 10²² m⁻³ strongly absorb light in the near-infrared and visible spectral range taking into account the typical polaron absorption cross sections exceeding 4 · 10⁻²² m² [8–10].

In recent years, this particular type of photochromic effect assigned to small polaron generation has been applied for purposes of spectroscopic studies of carrier localization phenomena in lithium niobate related to both electrons and holes [11, 12]. These studies already uncovered the impact of the intrinsic defect structure on the small polaron formation [13]. A time delay between free-carrier excitation and localization of ≈ 100 fs at room temperature was found [14] and a detailed model approach to the thermally-activated diffusive hopping of small polarons upon their formation could be deduced [12, 15].

Obviously, the optical generation of small polarons and their absorption features must be considered for the various fields of applications of lithium niobate in nonlinear photonics [16]. For instance, laser-induced damage mechanisms in lithium niobate may be triggered by small polaron absorption as verified for charged point defects in several prominent oxides widely applied for frequency conversion [17–19]. Furthermore, it has been shown that small bound polarons may act as intermediate shallow traps in a multi-step recording scheme for phase holograms featuring non-destructive read-out [20, 21]. Only recently, the optical features of small polarons were shown to play an important role in the prominent bulk photovoltaic effect in Fe-doped lithium niobate [22].

It is the aim of the present investigation to extend the area of applications based on the optical generation of small polarons and their pronounced absorption features in lithium niobate. We particularly focus on the recording of transient volume holograms in thermally reduced, nominally pure lithium niobate with ns-laser pulses by means of spatially modulated polaron densities. Mixed absorption and index gratings are experimentally verified with diffraction efficiencies up to 23% in the near-infrared spectral range. The polaronic nature of the gratings causes a non-exponential grating relaxation with lifetimes in the ms-time regime. Furthermore, the relaxation process is thermally activated with an activation energy (≈ 0.57 eV) pointing to the predominant contribution of ${\text{Nb}}_{\text{Li}}^{4+}$ small bound polarons. The results are discussed in a comprehensive model taking optically-induced density-alterations of small bound electron and hole polarons ( ${\text{Nb}}_{\text{Li}}^{4+}$, ${\text{Nb}}_{\text{Li}}^{4+}:{\text{Nb}}_{\text{Nb}}^{4+}$, and O⁻) into account. The presence of mixed volume gratings is ascribed to the Kramers-Kronig relation between optical absorption and refractive index changes as well as to an electro-optical response related to polaron formation. Results and model show that hologram recording by means of small polarons is promising for applications in nonlinear photonics that particularly require near-infrared sensitivity and (ultra-)fast recording (e.g. re-adjustable Bragg filters, phase conjugation mirrors, volume data storage or optically correlators).

2. Samples and experimental setup

2.1. Sample preparation and polaronic features

Our studies were performed with single crystals of thermally reduced lithium niobate grown from a congruent nominally undoped melt via Czochralski growth technique (Crystal Technology, Inc.). Dimension, orientation and concentration of residual Fe [23] for the sample under study as well as parameters for thermal pre-treatment are summarized in Table 1.

Table 1. Dimension, Orientation, Fe Content and Thermal Pre-Treatment Parameters of the Lithium Niobate Sample under Study

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The pre-treated sample features a grayish coloration due to the presence of ${\text{Nb}}_{\text{Li}}^{4+}:{\text{Nb}}_{\text{Nb}}^{4+}$-bipolarons (BP) and small bound ${\text{Nb}}_{\text{Li}}^{4+}$-polarons (GP) [23, 24] as a result of chemical reduction. The respective polaron number densities N _BP and N _GP can be simply determined from inspection of the steady-state absorption α and the values for the absorption cross sections σ at λ = 488 nm and λ = 785 nm, i.e. at the wavelengths that characterize the maxima of the bipolaron and bound polaron absorption bands (cf. Ref. [25] and references therein). Literature values and estimates for our sample are given in Table 2.

Table 2. Steady-State Absorption α^o/e, Polaron Absorption Cross Section σ^o/e at λ = 488 nm and λ = 785 nm, as well as Polaron Number Densities N _BP and N _GP

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2.2. Experimental setup

Hologram recording and time-resolved detection of the hologram decay is performed in a two-beam interferometer as schematically depicted in Fig. 1. A pulse of a frequency-doubled YAG:Nd-laser (Innolas Spitlight 600, |k _p| = 2π/λ _p, λ _p = 532 nm, maximum applied intensity $I_{\text{p}}^{\text{max}}\hspace{0.17em}=\hspace{0.17em}760\hspace{0.17em}{\text{GW}/\text{m}}^2$, 10 Hz repetition rate, and average pulse duration τ _FWHM ≈ 8 ns) was singled out by an electromechanical pulse picker (PP). The pulse was split by a 50% : 50% beam splitter (BS), i.e., equal intensities were adjusted along the two optical paths (I ^R = I ^S) yielding a modulation depth of $m\hspace{0.17em}=\hspace{0.17em}2\sqrt{I_{\text{R}}{I}_{\text{S}}}/(I_{\text{R}}\hspace{0.17em}+\hspace{0.17em}{I}_{\text{S}}){\mathbf{\text{e}}}_{\text{p}}^{\text{R}}\hspace{0.17em}\cdot {\mathbf{\text{e}}}_{\text{p}}^{\text{S}}\hspace{0.17em}=\hspace{0.17em}{\mathbf{\text{e}}}_{\text{p}}^{\text{R}}\hspace{0.17em}\cdot \hspace{0.17em}{\mathbf{\text{e}}}_{\text{p}}^{\text{S}}$ with the light polarization unit vectors ${\mathbf{\text{e}}}_{\text{p}}^{\text{R}}$ and ${\mathbf{\text{e}}}_{\text{p}}^{\text{S}}$. A symmetric angle of incidence Θ_B was chosen with respect to the sample’s normal thus enabling the recording of unslanted gratings. Beam superposition in the time domain was adjusted by an optical delay stage (DS). The Gaussian spatial beam profiles (2ω ₀ ≈ 4 mm) were masked to the sample aperture by a diaphragm (DP). Peak intensity and light polarizations were adjusted by a combination of Glan-Taylor prism (GP) and half-wave plate (λ/2). A fiber-coupled GaAlAs-diode laser (Coherent Cube, |k| = 2π/λ, λ = 785 nm) served for Bragg-matched as well as off-Bragg continuous wave (cw) probing. Its intensity was limited to I ₀ = 10 kW/m², thus not affecting the polaron-density dynamics. The probe-beam angle Θ was precisely controlled by mounting the fiber collimator (FC) onto a motorized rotation stage (Newport, URM 80CC, angular resolution of 0.001°) with an extension arm.

 

Fig. 1 Scheme of the experimental setup for hologram recording as described in the text. PP: electromechanical pulse picker, BS: 50% : 50% beam splitter, GP: Glan-Taylor prism, λ/2: half-wave plate, M1-M5: dielectric mirrors, DP: diaphragm, D1-D3: Si-PIN diodes, DS: optical delay stage, DSO: digital storage oscilloscope, FC: fiber collimator with wave plate and polarizer.

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The intensities of diffracted and transmitted probe beams were determined with Si-PIN diodes (D1, D2) equipped with optics that adjust the beam diameter to the photosensitive area. Small band-width optical filters served for suppression of the pump beam light. Time-dependent data collection in the range from 1μs – 100 s with a time resolution of up to 1 ns by a multi-channel digital storage oscilloscope (LeCroy, Waverunner LT584L) was triggered via a photodiode (D3) detecting pulse light scattered from the sample surface. The time regime 1 ns −1μs is excluded in our study in order to suppress unwanted signal contributions from thermal gratings recorded via the thermo-optic effect [26]. The sample temperature was adjusted by a PID-controlled thermoelectric element from room temperature up to 410 K.

Transient absorption was measured in a pump-probe-setup as described elsewhere (cf. [12]). Contrary to the two-beam interferometer, the pump pulse is directed normal to the sample surface while the beam path of the cw probe laser is marginally slanted. Two probing wavelengths at 488 nm and 785 nm were used.

3. Experimental results

3.1. Hologram recording: Bragg-matched probing

Figure 2 highlights the temporal dynamics of the first order diffracted probe beam obtained after hologram recording with a 8 ns-laser pulse.

 

Fig. 2 Temporal dynamics of the intensity of the first order diffracted beam I ^(1st) for (a) K ⊥ c-axis, e _p || c-axis, s-polarization and (b) K || c-axis, e _p || c-axis and p-polarization. Recording conditions: λ _p = 532 nm, Θ_B = 11.5°, spatial frequency Λ ≈ 1.3μm, I _p = I _R + I _S = 380 GW/m² and 230 GW/m², respectively. Bragg-matched probing conditions: λ = 785 nm, e || c-axis with (a) s- and (b) p-polarization. The data are normalized to the intensity of the incoming probe beam I ₀ and a logarithmic time scale is applied. The solid lines correspond to a fit of a stretched-exponential function Eq. (1) to the data set. The insets sketch the respective recording and probing configurations.

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The grating wavevector K was aligned (a) perpendicularly to the polar c-axis with light polarizations of pump and probe beams parallel to c and s-polarization. In (b), K was aligned parallel to the polar c-axis again with light polarizations of pump and probe beams parallel to c but p-polarization. Thereby the recording without (r ₃₃₁ = 0) and with (r ₂₂₃ ≠ 0, r ₃₃₃ ≠ 0) an electro-optic contribution according to the point symmetry 3m of lithium niobate was studied, respectively [27].

The intensity I ^(1st)(t) is normalized to the intensity of the incoming probe beam I ₀ and thus corresponds to the efficiency η of the diffraction process. It is plotted on a logarithmic time scale and characteristically shows a non-exponential decay to zero. Obviously, gratings with similar temporal characteristics but severely different efficiencies are recorded in (a) and (b). Fitting a stretched exponential function according to the empirical dielectric decay function by Kohlrausch, Williams and Watts (KWW) [28]

Table 3. Parameters Obtained by Fit of Eq. (1) to the Data of Fig. 1

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The temporal dynamics were determined for both directions of K as a function of crystal temperature T in the temperature regime 300 – 410 K. The analysis of the individual spectra by fitting Eq. (1) allowed to extract the dependence of the decay time on the sample temperature τ(T). Performing an Arrhenius plot, a linear dependence is found that allowed for the determination of the activation energies E _a (see Table 3) yielding an average value of Ē _a = (0.57 ± 0.07) eV. As a final note of this subsection we remark, that we were not able to detect any signal in the direction of second or higher order diffraction.

3.2. Hologram recording: off-Bragg probing

In order to verify the presence of volume gratings, the angular sensitivity of the Bragg condition was determined by scanning the diffraction efficiency as a function of the angular detuning ±δΘ. Exemplarily, Fig. 3 shows the spectrum for K || c-axis, commonly called rocking curve, that appeared for probing at 785 nm in an angular detuning range of ±0.25° and at 1 μs. Because of the grating lifetime in the ms-regime, the data points were deduced from a series of individual measurements of the temporal dynamics, each adjusted for a specific read-out angle. The rocking curve reflects a distinct maximum for Bragg-incidence (δΘ = 0°) and a full width at half maximum of δΘ_FWHM = (0.11 ± 0.02)°.

 

Fig. 3 Angular dependence of the intensity of the normalized first order diffracted beam I ^(1st)/I ₀ at t = 1μs after the pump pulse as a function of the deviation δΘ of the Bragg-angle Θ_B (external values). The solid line represents the result of fitting Eq. (6) to the experimental data.

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3.3. Transient absorption

For analysis of the hologram recording mechanism, the temporal dynamics of the light-induced absorption α _li was determined for the sample under study and for probing light at 785 nm (Fig. 4). The data were retrieved from the transmitted probe beam intensity as a function of time I _t(t) as sketched in the inset.

 

Fig. 4 The temporal dynamics of the light-induced absorption α _li determined for the sample under study and for probing light at λ = 785 nm and λ = 488 nm. The intensity of the pump beam at λ _p = 532 nm was I _p = 760 GW/m² with e _p || e || c-axis and s-polarization. The solid lines are the results of fitting Eq. (2) to the experimental data sets. The inset sketches the experimental arrangement of pump and probe beams.

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The data, which reflect the population density dynamics of ${\text{Nb}}_{\text{Li}}^{4+}$, were collected for an intensity of the pump beam of I _p = 760 GW/m² and light polarizations of pump and probing beams parallel to the polar c-axis and s-polarization. Again, a logarithmic scale is chosen for the time range 1 μs – 100 s.

Additionally to the data at 785 nm, the temporal dynamics were also collected for the probe wavelength λ = 488 nm, which contains information about the optical gating process of bipolarons and the optical generation of hole polarons in thermally reduced lithium niobate. Here, the signal decays to zero from a negative starting value, i.e. transient transparency is observed as previously reported and explained [23]. Both dynamics show a non-exponential decay behavior. Thus, the stretched exponential function

Table 4. Parameters Deduced from Fitting Eq. (2) to the Experimental Data in Fig. 4

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From the temperature dependence of the transients at 785 nm we get an activation energy of E _a = (0.53 ± 0.07) eV in accordance with literature data [12].

4. Discussion

Our experimental results clearly demonstrate the possibility to record volume holographic gratings in thermally reduced, nominally undoped lithium niobate with unique features: (i) a non-exponential relaxation behavior with stretching coefficient β < 1, (ii) a lifetime in the dark in the ms-regime, (iii) a temperature dependence pointing to a thermally activated relaxation process with activation energy of ≈ 0.57 eV and (iv) highly efficient recording with one single 8 ns-laser pulse at 532 nm. Because of these features, the underlying recording and relaxation mechanisms can not be understood in the frame of well-known photo-induced processes, such as Drude-Lorentz nonlinearities, the thermo-optic/pyroelectric effect [26], or the prominent photorefractive effect [29]. The latter can be excluded, as well, due to the absence of a significant number density of Fe³⁺ as a result of the thermal reduction procedure, due to the rather low Fe impurity content (< 5 ppm) itself and due to the fact that the photovoltaic as well as electro-optic tensor elements are zero from symmetry considerations.

In fact, the hologram dark-decay-features comply with the characteristics of the transients depicted in Fig. 4. Such transients have been studied extensively in lithium niobate (see e.g. Refs. [30, 31]), and other niobates (such as KNbO₃ in Ref. [32]), so far. It is well understood, that carrier-localization, i.e. small bound polaron formation, is at the origin of the optically induced absorption according to the small polaron theory [5]: the absorption band width extends over more than 1 eV, the absorption shape obeys a strong asymmetry and the energetic position of the absorption maximum directly relates to the polaron binding energy [2]. From the transients, the polaron formation time has been attributed to the 100 fs regime [14]. Hence, we assume that polaron formation is terminated for the time regime of our study whereas small-polaron-transport and mutual recombination processes are to be considered [4]. From the comparison of the activation energy related to hologram relaxation with the one of small bound polarons (≈ 0.58 eV, see Ref. [2] and references therein) we further conclude that we deal with the population and depopulation of ${\text{Nb}}_{\text{Li}}^{4+/5+}$ antisite defect centers.

In the following we will follow this small-polaron approach for hologram recording in more detail by first considering the grating recording mechanism within the state-of-the-art picture of small polarons in thermally reduced lithium niobate. It will be the starting point to invest available optical data of literature in order to end-up with a theoretical estimate for the hologram efficiency. The model further will be applied to predict the hologram relaxation dynamics. All estimates can be compared with those listed in the experimental section and allows to further evaluate the validity of the small polaron approach.

4.1. Grating recording mechanism

4.1.1. Homogeneous exposure

Figure 5 summarizes the optical population paths, defect sites and the respective energetic positions that are relevant for the analysis of thermally reduced, nominally undoped lithium niobate exposed to a single ns laser pulse at 532 nm, i.e. the case of spatial homogeneous light exposure, and probing at 785 nm (cf. Refs. [2, 3]).

 

Fig. 5 Polaron generation in thermally reduced lithium niobate upon exposure to ns laser pulses. Left: optical gating of ${\text{Nb}}_{\text{Li}}^{4+}:{\text{Nb}}_{\text{Nb}}^{4+}$ bipolarons into small bound ${\text{Nb}}_{\text{Li}}^{4+}$ and free ${\text{Nb}}_{\text{Nb}}^{4+}$ polarons. Right: two-photon excitation yielding O⁻ hole as well as bound and free polarons. The valencies of the individual centers correspond to the sketched location of electrons and holes at the trap centers (left) and in the valence band (right).

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In the blue-green spectral range, the groundstate absorption of the unexposed sample is determined by ${\text{Nb}}_{\text{Li}}^{4+}:{\text{Nb}}_{\text{Nb}}^{4+}$ bipolarons with absorption features exhibiting a maximum at 2.5 eV. ${\text{Nb}}_{\text{Li}}^{4+}$ polarons appear at 1.6 eV and thus affect the transmission of the probing light. The strengths of both absorption features are determined by the respective polaron number densities in our sample (see Table 2). We note that ${\text{Nb}}_{\text{Li}}^{5+}$ antisites do not contribute to the optical features.

The number density of ${\text{Nb}}_{\text{Li}}^{4+}$ polarons is altered upon light exposure with 2.5 eV via optical gating of bipolarons (left half of Fig. 5) and via two-photon-absorption processes (right half) [23]. The increase of the bound polaron density is inevitably accompanied with a depopulation of the bipolaron density as well as the appearance of small free ${\text{Nb}}_{\text{Nb}}^{4+}$ electron and O⁻ bound hole polarons (absorption maxima at 1.0 eV and 2.5 eV). Fortunately, we can neglect the contribution of small free ${\text{Nb}}_{\text{Nb}}^{4+}$ polarons in what follows, because of their comparably small lifetime in the sub-μs regime [33] and the limitation of our temporal dynamics to t ≥ 1 μs (cf. Fig. 2 and Fig. 4).

4.1.2. Inhomogeneous exposure

Taking into account these light-induced processes and the sinusoidal intensity pattern I(x) = I _p[1 + cos(|K|x)], that was applied for exposure in our experiments (K ⊥ c-axis), we expect the appearance of spatially periodic density modulations of small bound electron and hole polarons as well as of bipolarons as sketched in Fig. 6(a). Obviously, N _GP(x) and N _HP(x) appear in phase to I(x) while N _BP(x) is phase-shifted by π.

 

Fig. 6 (a) Sinusoidal intensity pattern I(x) applied for exposure in our experiments with average intensity I _p = I _R + I _S and modulation depth unity, spatially periodic density modulations of small bound polarons N _GP(x), hole polarons N _HP(x) and bipolarons N _BP(x). (b) Spatial modulation of the absorption coefficient α(x) with amplitude α ₁ and average value of α + α ₁. The overall absorption change in the maximum of the fringe pattern α _li = 2α ₁ is assembled from absorption changes of the individual polaron type: α _li,GP, α _li,HP and α _li,BP. All absorption contributions are related to λ = 785 nm and extraordinary light polarization.

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It is possible to probe these spatial density modulations via light diffraction by considering the related spatial modulations of absorption α(x) and index of refraction n(x) that are mutually linked via Kramers-Kronig relation. Figure 6(b) shows the overall spatial absorption modulation α(x) that can be expected from the individual contributions of bound α _GP(x), hole α _HP(x) and bipolarons α _BP(x) at our probing wavelength λ = 785 nm and for extraordinary light polarization. Here, we note that the pronounced band width of ≈ 1 eV of small polaron absorption features are considered, i.e. the tails of the absorption features of bipolarons and hole polarons can not be neglected at 785 nm although their absorption maxima are located in the blue-green spectral range. Equivalent considerations are valid for the overall index modulation n ₁(x), i.e. the overall light-induced index change is composed from individual index changes n _GP(x), n _HP(x), n _BP(x) related to light-induced changes of polaron number densities.

For the particular case of Bragg-incidence and under the assumption of an unslanted, mixed elementary holographic volume grating (mutual phase shift is 0 or π) the diffraction efficiency can be expressed via Kogelnik’s formula [34]:

We note that this type of hologram recording does not require any of the further crystal-physical effects present in lithium niobate, such as the electro-optic effect. Nevertheless, diffraction gratings recorded via other mechanisms may appear simultaneously to polaron-density modulations in various recording configurations in lithium niobate. For the particular experimental conditions of our experiments (c(Fe³⁺) < 5ppm, K ⊥ c-axis, s-polarization, t ≥ 1μs) this can be neglected, i. e. the experimental data can be analyzed in the framework of the presented hologram recording mechanism.

4.2. Turnover to data analysis

Let us estimate the diffraction efficiency given by Eq. (3) for the experimental conditions of Fig. 2(a) (probe light at λ = 785 nm, pump beam intensity I _p = 380 GW/m² and Bragg angle Θ_B = 11.5°, K ⊥ c-axis, extraordinary light polarization and s-polarization) and the groundstate absorption α(785) = (270 ± 10) m⁻¹ (cf. Table 2). For this purpose we need to determine the amplitudes α ₁(785) and n ₁(785) as well as the hologram thickness d _h.

The amplitude α ₁(785) can be deduced from the transients in Fig. 4: The initial value of the light-induced absorption α _li(785) represents the maximum absorption change in the intensity maximum of the fringe pattern 2 · I _p = 760 GW/m². According to Fig. 6(b), we get α ₁(785) = α _li(785)/2 = (130 ± 15) m⁻¹ from the values given in Table 4.

The amplitude n ₁(785) can be deduced from the application of the Kramers-Kronig relation to the spectral dependence α _li(λ). We model α _li(λ) as follows: The light-induced absorption is related to the changes of the number densities of light-induced bound polarons N _li,GP, bipolarons N _li,BP, and hole polarons N _li,HP via:

Equation (4) can be solved by considering the values of the light-induced absorption at the maximum of the individual polaron absorption bands, i. e. α _li(785) and α _li(488), that have been experimentally determined for the case of homogeneous exposure (cf. Fig. 4 and Table 4). Further, the values of the individual absorption cross sections are required which are given in Table 2 for bound polarons and bipolarons. For hole polarons σ _HP(488) = (4.0 ± 1.0)·10⁻²² m² was taken from Ref. [25] under the assumption that the absorption features do not differ for ordinary and extraordinary light-polarization. This assumption is justified by inspection of the absorption spectra published in Ref. [35] and Fig. 2 therein. As a result we get the light-induced alterations of the number densities listed in Table 5.

Table 5. Changes of the Number Densities of Light-Induced Bound Polarons N _li,GP, Bipolarons N _li,BP, and Hole Polarons N _li,HP

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Now, the individual polaron line shapes of the optically induced absorption features can be determined using the band shapes of small polarons. The overall spectral dependence α _li(λ) is depicted in Fig. 7 based on data given in Refs. [3, 24, 36, 37].

 

Fig. 7 Spectral dependence of α _li(λ) determined from the analysis of the experimentally determined light-induced absorption at 785 nm and 488 nm, the literature data on polaron absorption cross sections and the previously published experimental band shapes of small polarons. The dispersion n _li(λ) is calculated from α _li(λ) applying the Kramers-Kronig-relation, Eq. (6). For details see text.

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With the help of α _li(λ), the dispersion of the index change n _li(λ) is calculated via the Kramers-Kronig relation [38]:

Finally, Eq. (3) requires an estimate for the hologram thickness d _h ≤ d. We determine this value from the analysis of the rocking curve in the configuration K || c-axis (cf. Fig. 3). The pronounced diffraction efficiency ensured a sufficient signal-to-noise ratio for the entire range of angular detuning. This enabled fitting the Kogelnik formula that considers an angular Bragg-mismatch δΘ [34]

Knowledge of n, α and α ₁ allows to determine d _h. Here we find d _h = (0.62 ± 0.07) mm. We assume that the hologram thickness for K ⊥ c-axis and K || c-axis, both with the same polarization states for recording and read-out, either s- or p-polarization for each of the two experimental configurations, do not differ significantly.

Obviously, the hologram thickness d _h is smaller than the crystal thickness. It is therefore necessary to take into account the absorption loss of the diffracted beam within the crystal volume d – d_h, i.e. the estimate of the diffraction efficiency via Eq. (3) must contain the additional term exp[–2(α + α ₁)(d – d_h)/ cos Θ_B]. All values necessary for the calculation and the determined value for I ^{(1 st )}/I ₀ are summarized in Table 6.

Table 6. Parameters Determined for an Estimate of I ^(1st)/I ₀ via Eq. (3) for Hologram Recording with K ⊥ c-Axis, Light Polarization Parallel c-Axis and s-Polarization

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The comparison shows that the estimate η ≈ 6· 10⁻⁴ coincides well with the experimentally determined value of η = (4.2 ± 1.0) · 10⁻⁴ in the configuration K ⊥ c-axis. We note that we neglect fringe pattern deviations from a sinusoidal function due to light scattering and beam profile inhomogeneities. We further did not consider the reported nonlinear response of α _li(I _p) on the pump beam intensity [23], which affects the shape of the generated absorption pattern α(x).

4.3. Hologram relaxation

A second characteristic feature of the recorded holograms, that has to be analyzed in the framework of a spatially modulated bound polaron density, is the non-exponential relaxation of the diffraction efficiency. The holograms obey a relaxation time in the ms-regime and an activation energy of Ē_a = (0.57 ± 0.07) eV. According to our model approach, the hologram relaxation resembles the temporal decay of the amplitudes n ₁(785) and α ₁(785), i.e. the decay of the spatial density modulation of small polarons. The relaxation is terminated for n ₁ = α ₁ = 0, i. e., where the spatial homogeneous polaron density distribution of the groundstate (≡ unexposed sample) is recovered.

The temporal decay of optically generated polaron densities upon spatial homogeneous light exposure in lithium niobate has been studied intensively in recent years by means of transient absorption spectroscopy [12, 23, 30, 31, 40]. The decay is determined by thermally activated diffusive hopping transport of the individual small polarons and their mutual recombination. Hopping of polarons can be understood as the transfer of localized carriers between next-neighboring cation sites [4, 5]. The site-specific electronic potentials are mutually separated by an activation barrier W _h. According to Austin and Mott [41], it is essentially one half of the polaron binding energy E _p in the case of equivalent free small polarons and larger, if the initial site is further stabilized by the influence of an additional defect potential with energy W _d: W _h = 1/2E _p + 1/2W _d. Thus, for all small polarons under study in lithium niobate, thermally induced hopping transport prevails.

The hopping process is terminated by charge recombination which commonly takes place within the direct vicinity of an intrinsic defect. The mean hopping path length, given by the sum of the spatial charge displacements of the individual hopping events, thus is determined by the number densities of defects within the sample [12]. The recombination time depends on the type and number of polaronic centers that build the hopping path. We note that the inspection of the polaron number density by means of probing light yields the information on an ensemble of optically generated polarons, their hopping and recombination. The determined relaxation time thus resembles the average value of a very large number of individual recombination paths. The recombination path length of an individual polaron is given by the sum of the spatial charge displacements of all hopping events along this path. It is thus determined by the number density of defects in the specific spatial region of the sample.

The activation energy, that can be deduced from the relaxation kinetics, resembles the energy barrier of the polaron types dominating the recombination paths. Since the spatial length and temporal duration of the path depend on the defect density in a non-trivial way, the non-exponential relaxation behavior is a result of the presence of a spectrum of activation energies, disorder, and spatial recombination path lengths.

In our sample we expect two recombination paths of the optically generated polarons according to the generation mechanisms sketched in Fig. 5: (i) the decay of polarons for the recovery of bipolarons and (ii) the electron-hole-recombination by recombination of bound electron polarons with bound hole polarons.

From these considerations we deduce the following expectation for the relaxation kinetics of the recorded holograms: (i) The relaxation kinetics must resemble the activation energy of small bound polarons, because they dominate both recombination paths. As a consequence, the hologram relaxation time is expected in the ms-time regime at room temperature [12]. (ii) The relaxation kinetics must resemble the statistical superposition of non-exponential relaxation kinetics. At each position of the spatially periodic modulation of the polaron number density, we expect a characteristic non-exponential relaxation behavior. This is due to the dependence of the diffusive hopping transport on the local defect number density. Our probe beam diameter is orders of magnitude larger than the spatial frequency of the polaron modulation. Thus, non-exponential kinetics from different regions of the sample are superimposed on the detector. The situation is similar to the statistical superposition of transient absorption from different regions of the sample in periodically poled lithium niobate [40].

Both expectations are indeed verified by our experimental results: while the time-constants determined for homogeneous and inhomogeneous exposure do not differ, we find rather different values of the stretching coefficient β.

4.4. Hologram recording with K || c-axis

Let us now address the data analysis for K || c-axis, which revealed a diffraction efficiency of η = (0.23 ± 0.05) (see Fig. 2 and Table 3). In accordance with the pronounced value of η, fitting of Eq. (7) to the experimental data yields an amplitude of the index modulation up to n ₁(785) = (–2.7 ± 0.6) · 10⁻⁴.

These values are not in accord with η = (2.5 ± 1.0) · 10⁻⁴ and n ₁(785) = (–1.8 ± 1.0) · 10⁻⁶ that can be estimated from the light-induced absorption features and Kramers-Kronig relation (Eq. (6)) analogously to the considerations of the previous sections. Hence, an effect that contributes to a light-induced index change in excess of Kramers-Kronig relation must be considered. It further needs to be dependent on the direction of K with respect to the polar c-axis.

We first note, that for K || c-axis the linear electro-optic effect is active with non-zero tensor elements r ₂₂₃ ≠ 0 and r ₃₃₃ ≠ 0 in this recording geometry. Assuming that the electro-optic effect is the reason for the pronounced diffraction efficiency, an electric field being simultaneously present to the spatial density modulation of small polarons must be present. We may estimate the electric field strength E via

We now take into account that the light wave vectors propagate at a small angle with respect to the normal of the crystal and that the relation r ₃₃₃ ≈ 3·r ₁₁₃ (see e.g. Ref. [42]) holds. Thereby we approximate n _eff with the index of refraction for extraordinary light polarization n_e and r _eff ≈ r ₃₃₃. At wavelength of λ = 785 nm, we find n_e(785nm) = (2.1776 ± 0.0005) [39]. The electrooptic coefficient has been precisely determined to r ₃₃₃(632.8nm) = 31.4 pm/V in Ref. [43]. Together with the dispersive behavior published in Ref. [44] we extrapolate r ₃₃₃(785nm) = 30.8 pm/V. All values hold for lithium niobate grown from the congruent melt and at room temperature. Then, the estimate for the electric field strength is $E\hspace{0.17em}=\hspace{0.17em}-2\hspace{0.17em}\cdot \hspace{0.17em}{n}_1/(n_e^3\cdot r_{333})\hspace{0.17em}\approx \hspace{0.17em}17\hspace{0.17em}\text{kV}/\text{cm}$.

Such a high electric field strength can not be explained with the diffusion mechanism with a saturation field of E _diff = (k _B T /e)(2π/Λ) ≈ 1.25 kV/cm. Here, k _B is the Boltzmann constant and e the electron charge. We can also exclude drift mechanisms in the absence of an externally applied electric field. Instead, electric fields with a strength up to 100 kV/cm are common for Fe-doped lithium niobate induced via the bulk photovoltaic effect [45]. For Fe-doped lithium niobate, this effect has been accounted for by asymmetries in the Nb⁵⁺-Fe²⁺-distances along the ±c-direction. Only recently, a small-polaron-based microscopic interpretation has been established that succeeds in the explanation of the large variety of electrical and optical properties of the bulk photovoltaic effect [22].

Because of the absence of Fe-doping with significant concentration and a predominant valency 2+ upon thermal reduction, a photovoltaic current may be assigned to the optical generation of small polarons in the samples under study. Such approach on a bulk photovoltaic contribution of small bound polarons was already postulated by Schirmer et al. in 1987 [46]. Our results give evidence for the validity of bound-polaron-based bulk photovoltaic currents by considering the hologram relaxation dynamics, in particular in the view of lifetime and activation energy.

The non-exponential decay is a remarkable feature for holograms recorded in lithium niobate. Holograms recorded conventionally via the bulk photovoltaic effect and the linear electro-optic effect in doped lithium niobate, i. e. via the photorefractive effect, inevitably obey a relaxation behavior following a single-exponential function. Its lifetime is determined by the dielectric relaxation time [47]. However, taking an electric field related to optically generated small bound polarons into account, we must expect a field decay according to the relaxation behavior of bound polaron densities. The latter is well-known to exhibit a non-exponential decay behavior upon exposure to single ns-laser pulses (see e.g. Refs. [12, 23, 31]) as it is the case for the samples under study (see Fig. 4). Polaron densities further decay in the ms-range that is accelerated at elevated temperatures. Here, we find τ = (2.8 ± 0.4) ms and an Arrhenius behavior with E _a = (0.53 ± 0.07) eV. The coincidence with the corresponding data of the hologram relaxation dynamics in the same sample (τ = (3.0 ± 0.3) ms and Ē _a = (0.57 ± 0.07) eV, Table 3) is striking.

Taking all these aspects together, a relation between the optically generated small bound polaron density and the appearance of an electric field, that alters the index of refraction via the linear electro-optic effect, seems likely. The underlying currents that result in the build-up of the electric field may be assigned to the small-polaron-based bulk photovoltaic effect at ${\text{Nb}}_{\text{Li}}^{4+/5+}$ and ${\text{Nb}}_{\text{Nb}}^{4+/5+}$ defect centers according to the interpretation related to Fe^2+/3+ [22].

5. Conclusion

Concluding our results and analysis, we succeeded in recording volume holographic gratings in thermally reduced lithium niobate. Taking the dominant intrinsic defect structure and state-of-the-art knowledge on small polarons in lithium niobate into account, a hologram recording mechanism based on the optical generation of a spatially modulated small polaron density is proposed. Following this model, light diffraction is due to the presence of a mixed holographic grating, i.e. the simultaneous presence of an absorption and an refractive index grating. The absorption grating resembles the population density of small polarons and causally affects the index of refraction via Kramers-Kronig relation. Predicted grating features, such as a non-exponential relaxation behavior, a ms-lifetime in the dark, a thermally activated relaxation process and highly efficient ns-pulse-recording, are experimentally verified.

In a second recording configuration it is demonstrated that pronounced holograms with efficiencies up to 23% can be recorded. From symmetry considerations we argue that it is due to a dominant contribution of the linear electro-optic effect. The presence of an electric field is discussed in the frame of our polaron-based model approach and the bulk photovoltaic effect. Considering the polaron formation time of ≈ 100 fs, we deal with a tremendous grating recording sensitivity that is much larger than the well-known photorefractive sensitivity in lithium niobate [20].

From the polaronic nature of the recorded gratings we may deduce several interesting predictions. For instance, a great tuning ability of grating features via the thermal reduction process (≡ adjustment of c _BP) can be expected. Further, hologram recording and probing will be possible over a broad spectral range. In the blue spectral range, gratings may be probed via the spatial modulation of bipolarons and hole polarons while small free polarons allow the readout in the infrared spectral range. The latter will be characterized by lifetimes in the μs time regime related to the ${\text{Nb}}_{\text{Nb}}^{4+}$ activation energy of ≈ 0.25 eV [33].