1. Introduction

The recording of volume holograms in polar oxide crystals by means of optical formation of small bound polarons has been established as a novel type of hologram recording mechanism, with LiNbO₃ as an example, only recently [1]. An incoming fringe pattern is transferred to a population density hologram by exploiting the optical absorption, transport and strongly localized lattice distortion related with small bound polarons [2]. The concept of small-polaron based hologram formation has been developed to accelerate the hologram formation time in polar oxide crystals to the sub-μs-time regime as well as to enable a spectrally broad optical response from the visible to the near-infrared spectrum. Appropriate oxide crystals are inevitably required as hologram recording media for a variety of modern holographic applications like real-time holographic displays [3], volume holographic optical correlators [4] or tunable Bragg-filters for photonic networks [5]. In LiNbO₃, small polarons have formation times in the sub-ps time regime [6], thus, hologram recording with fs-, ps-, and ns-laser pulses can be expected. The respective small-polaron absorption features are broad (≈ 1 eV) and are positioned in the visible and near-infrared spectrum (2.5 eV, 1.6 eV and 1.0 eV) depending on the particular small polaron type [7, 8]. Thereby, an optical response for recording and reconstruction from 3.0 to 0.5 eV is likely.

So far, we have probed the polaron-based hologram features in the near-infrared spectrum (λ = 785 nm). This allowed us to uncover the dominating role of small bound ${\text{Nb}}_{\text{Li}}^{4+}$ polarons (GP) in the appearance of mixed absorption and index volume gratings.

In this work, we focus on hologram features at a probing light wavelength of λ = 488 nm, i.e., a spectral range that is of utmost importance for a variety of modern holographic applications of LiNbO₃ such as real-time holographic displays [9]. The interest in the blue spectrum is particularly driven by our findings that hologram recording can be performed within a single 8 ns laser pulse while thermally-driven hologram self-decay takes place in the range of a few milliseconds at room temperature. Thus, thermally reduced LiNbO₃ represents a photosensitive, re-recordable hologram medium that is updatable at kHz frequencies.

From the point of view of small polarons, the blue spectrum is dominated by the presence of ${\text{Nb}}_{\text{Li}}^{4+}:{\text{Nb}}_{\text{Nb}}^{4+}$ electron bipolarons (BP, absorption maximum at λ ≃ 500 nm) [7] and small O⁻ hole polarons (HP, λ ≃ 500 nm) [8]. These two kinds of polarons exhibit essentially different features with respect to light-matter-interaction: Bipolarons stable at room temperature, are dissociated (gated) optically by light exposure within a one-photon absorption process. The maximum addressable BP number density is determined by the respective number density in the ground state. In contrast, short-lived small bound hole polarons are generated via two-photon absorption. Their number density grows until all possible O²⁻ lattice sites, one in the vicinity of each Li vacancy, are saturated [8]. These processes can be easily distinguished experimentally by the study of the grating efficiency as a function of the pump-beam intensity.

We show that the hologram read-out in the visible spectrum can be comprehensively explained in the frame of a complex interplay of BP, HP and GP. In particular, we find an abnormal dispersive behavior that is due to the characteristic positions of the absorption maxima of the small polarons involved. The dependence of the diffraction efficiency on the pump-beam intensity is mono-exponential for the case of probing the population saturation behavior of BP. In contrast, it follows a quadratic dependence without saturation behavior for the recording configuration that allows for a predominant electro-optic contribution. This finding can be explained by considering the presence of a small-polaron based bulk photovoltaic effect that is particularly related to the intrinsic defect structure. An analogous model to Fe-doped LiNbO₃ [10], that explains the build-up of localized electric space charge fields, is discussed.

With these results and our earlier findings [1], the recording of polaron-based holograms with ns-laser pulses in LiNbO₃ is demonstrated over a broad range in the visible spectrum. Further impact of the work is revealed by considering the possibility of recording with fs- and ps-laser pulses, thus enabling fs-holography. Also, we like to point to the possible transfer of the polaron concept for hologram recording to other oxide crystals like KNbO₃ [11].

2. Samples and experimental setup

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.). The sample under study (cf. Ref. [1]) with aperture (a × c) = (6.54 ± 0.01) × (5.69±0.01) mm² and thickness d = (1.23±0.03) mm has been thermally pre-treated by heating it for 6 hours at T = (970 ± 10) K in a reducing atmosphere of p < 10⁻⁴ mbar. Thus, high densities of ${\text{Nb}}_{\text{Li}}^{4+}:{\text{Nb}}_{\text{Nb}}^{4+}$ bipolarons (BP) and of small bound ${\text{Nb}}_{\text{Li}}^{4+}$-polarons (GP) [12, 13] were generated that are stable at room temperature. We have calculated the number densities N_BP and N_GP using the steady-state absorption for extraordinary light polarization at λ = 488 nm and λ = 785 nm and the absorption cross sections published in Refs. [1, 14]. All relevant parameters are summarized in Table 1.

Table 1. Absorption features and polaron number densities of the reduced lithium niobate sample under study in the steady state at room temperature. The sample is identical to the one used in Ref. [1].

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Hologram recording and time-resolved detection of the hologram decay were performed in a two-beam interferometer setup. A single pulse of a frequency-doubled YAG:Nd-laser (Innolas Spitlight 600, λ_p = 532 nm and average pulse duration τ_FWHM ≈ 8 ns) was used for recording an unslanted volume grating (equal intensities I_R = I_S, parallel light polarization e_R = e_S, modulation depth m = 1, Bragg angle Θ_B = 6.3°). The decay of the grating was probed in the blue-green spectral range with the Bragg-matched beam of a continuous-wave laser at λ_t = 488 nm (Coherent Sapphire, I₀ = 10 kW/m²). Time-dependent data collection via a Si-PIN diode and digital storage oscilloscope was limited to the range 1μs – 100 s in order to suppress unwanted signal contributions from thermal gratings [15]. The sample temperature was adjusted by a PID-controlled thermoelectric element from room temperature up to 410 K.

3. Experimental results

Figure 1 depicts the temporal dynamics of the intensity of the first order diffracted probe beam obtained upon hologram recording with a single 8 ns-laser pulse. The intensity I^(1st)(t), plotted on a logarithmic time scale, is normalized to the intensity of the incoming probe beam I₀ taking into account Fresnel reflection and thus corresponds to the efficiency η of the diffraction process. The light polarizations of pump and probe beams were aligned parallel to the polar c-axis of the sample corresponding to s-polarization in configuration (a) and p-polarization in configuration (b). The grating vector of the recorded hologram K was chosen either perpendicular (a) or parallel (b) to the c-axis as sketched in the insets. Thereby, in configuration (a) the corresponding elements of the electro-optic (r₃₃₁ = 0) and photovoltaic tensor (β₁₃₃ = 0) equal zero, whereas in configuration (b) they are non-zero (r₂₂₃ ≠ 0, r₃₃₃ ≠ 0, β₃₁₁ ≠ 0, β₃₃₃ ≠ 0) [16]. All experiments were performed for a Bragg-matched read-out beam. Higher diffraction orders were not observed. This accords with our previous findings on the rocking curve that unambiguously verified the presence of a volume grating for this type of hologram recording mechanism and set of recording parameters [1].

 

Fig. 1 Semilogarithmic plot of the temporal dynamics of the normalized intensity of the first order diffracted beam I^(1st)/I₀ for (a) K ⊥ c-axis (s-polarization) and (b) K || c-axis (p-polarization). Recording conditions: λ_p = 532 nm, e_p || c-axis and Bragg angle Θ_B = 6.3°. I_p = I_R + I_S = 380 GW/m². Bragg-matched probing conditions: λ = 488 nm, e || c-axis. The solid lines correspond to fits of Eq. (1) to the data. The insets sketch the respective recording and probing configurations.

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Qualitatively, both gratings decay with a similar temporal dependence. The striking observation in the two spectra is the severely different starting amplitude at t = 1μs of the diffraction efficiency that is by a factor of approximately 500 larger in configuration (b) compared with (a). Fitting a stretched exponential function:

Table 2. Parameters obtained from fitting Eq. (1) to the experimental data depicted in Fig. 1 and of Eq. (6) to the data in Fig. 2.

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In the next step, we have studied the dependence of the starting amplitude on the intensity of the recording beams, see Fig. 2. We find that the diffraction efficiency saturates at a value of about 1.8 · 10⁻⁴ and pump beam intensities larger than ≈ 200 GW/m² using the recording configuration (a). In contrast, I^(1st)(t = 1μs)/I₀ increases as a function of I_p in a superlinear way for configuration (b), i.e., saturation is not observed up to the maximum available pump beam intensity of 380 GW/m².

 

Fig. 2 Normalized intensity of the first order diffracted beam I^(1st)/I₀ at t = 1μs as a function of pump intensity I_p for (a) K ⊥ c-axis (s-polarization) and (b) K || c-axis (p-polarization) using the same recording and probing condition as in Fig. 1. The solid line corresponds to a fit of a saturation function Eq. (6) while the dashed line represents a fit of a quadratic intensity dependence to the data.

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According to these findings, we need to verify the relation between the recorded gratings and the optically-induced formation of small polarons for configuration (b). For this purpose, the activation energy related to the grating decay has been determined by means of temperature dependence measurements τ(T) in a range of 300 K to 410 K. Performing an Arrhenius plot a linear dependence is found allowing for the determination of the activation energy to E_A = (0.57 ± 0.07) eV. This value can be compared with literature values for the activation energies of different types of small polarons in LiNbO₃ and points to the activation energy of the small bound electron polaron, ${\text{Nb}}_{\text{Li}}^{4+}$ for which Schirmer et al. obtained a value of 0.62 eV from conductivity measurements [18].

4. Discussion

Our results highlight the prominent features of polaron-based holograms in the visible spectrum in thermally reduced lithium niobate. For the field of applications, the striking features are the diffraction efficiency of η = (0.108 ± 0.01) by means of single 8 ns-laser pulse recording (I_p = 380GW/m²) and the hologram self-decay within a few miliseconds at room temperature. Taking into account the crystal thickness of d = 1.23 mm this corresponds to a photosensitivity [19] of:

From the point of view of small polarons, there are two outstanding findings associated with the visible spectrum:

The further properties have already been reported for the near-infrared spectral range [1] and give strong evidence for the polaronic origin of the hologram recording and read-out process:

We should like to note the assumptions made in the following discussion: First, the action of extrinsic defect centers, that foster hologram recording via the photorefractive effect, is not considered [21]. This is justified because the number density of extrinsic photorefractive/transition metal centers, such as Fe, is far below 5 ppm in our nominally undoped LiNbO₃ samples [14]. Furthermore, the thermal reduction pre-treatment results in a considerable transfer of the valence state from acceptor to donor levels. For instance, the number density of Fe³⁺ is considerably damped c(Fe³⁺) ≪ 1, but it is decisive for the photorefractive response as the amplitude of the index change is linearly dependent on c(Fe³⁺) [22]. Second, the role of small free ${\text{Nb}}_{\text{Nb}}^{4+}$ electron polaron densities is neglected because of their sub-μs-lifetime [23] and the chosen limitation of the temporal dynamics to the time range t ≥ 1μs.

4.1. Hologram recording by spatially modulated polaron densities

Exposure to a spatially modulated intensity pattern I(x) = I_p[1 + cos(|K|x)] results in the appearance of spatially modulated polaron densities. Densities N_GP,HP of small bound electron and small hole polarons are increased in the bright region of the fringe pattern while the bipolaron density N_BP is diminished in the same regions. In the dark regions there are no changes of polaron densities. The formation time of the small polaron density is limited by a sequence of processes starting from the optical excitation of a charge-carrier by absorption and ending with self-localization at a ${\text{Nb}}_{\text{Li}}^{5+}$-site via electron-phonon coupling. The period in-between is characterized by coherent electron transport in the conduction band. Because of its coherence in the presence of electron-lattice coupling this intermediate state can be regarded as a large polaron. The formation time of small free polarons at room temperature has been reported for Mg-doped LiNbO₃ to ≈ 110 fs [6] while small bound polarons appear within 400 fs [24] after the optical pulse. If the duration of illumination exceeds the polaron formation time, a repetitive excitation of charge carriers from localized states becomes possible. Thereby, the distance of the small polaron with respect to the initial polaron site can be increased significantly. This may affect the appearance of polaron-based photovoltaic currents (cf. section (4.4)), or the dynamics of the hologram recording process.

As the polaron densities are related to the polaron absorptions α_GP,HP,BP of the respective types by constant factors (the absorption cross sections σ_GP,HP,BP), the polaron absorptions become spatially modulated and together yield the overall absorption modulation α(x) with amplitude α₁ around a mean value (α₀ +α₁) in a crystal with ground state absorption α₀ of the unexposed sample (see Fig. 3(a)). This absorption modulation is causally related to a spatial modulation of the index of refraction n₁(x) (see Fig. 3(b)), i.e., by considering the interplay of the different polaron density modulations, we end up with a population density grating. Its diffraction efficiency is determined from the maximum amplitude of the change of the complex susceptibility arising from the optical generation of small polarons.

 

Fig. 3 (a) Spatial modulation of the absorption coefficient α(x) with amplitude α₁ and average value of α₀ +α₁. The overall absorption change in the maximum of the fringe pattern α_li is assembled from absorption changes of the individual polaron types: α_GP,HP,BP. All absorption contributions are related to λ = 488 nm and extraordinary light polarization. (b) Sinusoidal intensity pattern I(x) applied for exposure with average intensity I_p = I_R + I_S and modulation depth unity resulting in a modulated density of polarons and, therefore, a modulated change of absorption α(x). This modulated absorption change is linked to a modulated change of the index of refraction n(x) via the Kramers-Kronig relation as shown in figure 7 in Ref. [1].

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4.2. Dispersion of diffraction efficiency

The energetic positions and bandwidths of the absorption features of the small polarons involved [7, 8], their absorption cross sections [14] and dependence on the pump beam intensity [12] are well documented in literature. Hence, it is possible to estimate the maximum value of the absorption change (≡ 2α₁ = α_li) for the sample under study and applied pump beam intensity as a function of wavelength (see figure 7 in Ref. [1]). As a result, we find α₁(488nm) = (−40 ± 15) m⁻¹ at λ = 488 nm, so that the light fringe pattern I(x) and the spatial absorption modulation α(x) are phase-shifted by π with respect to each other, as depicted in Fig. 3(b). At the same time, the knowledge of the dispersive absorption features allows for the calculation of the causally related changes n₁(λ) of the index of refraction via Kramers-Kronig-relation [25]. It yields n₁(488nm) = (−3.4 ± 1.0) · 10⁻⁶ and a spatial modulation of the index of refraction n(x) that is in phase with I(x) (Fig. 3(b)). The modulations α(x) and n(x) resemble a mixed diffraction grating with efficiency η that is given by Kogelnik’s coupled wave theory [26]:

In Fig. 4(a), we have plotted the dependence η^est.(λ) (black line). The experimentally determined values for probing wavelengths 488 nm (△) and 785 nm (□) and K ⊥ c-axis have been added for comparison. A good agreement between the estimate and experimental data is found. Moreover, we should like to point out the considerable decrease of the diffraction efficiency from the near-infrared to the blue spectral range in the estimated spectrum. It is a result of the rather different positions of the absorption features of the small polarons involved. In detail, both the index grating with n_1,BP and the absorption grating with α_1,GP related to bipolaron and bound polaron density alterations, respectively, likewise contribute to the diffraction efficiency at 488 nm. The hole polaron density can be neglected. In contrast, it was shown that the diffraction efficiency at 785 nm is mainly due to the action of the absorption amplitude related to the small bound ${\text{Nb}}_{\text{Li}}^{4+}$ polaron (center of absorption feature at 1.6 eV), i.e., the contribution of n_1,GP and of susceptibility changes related to bipolarons and hole polarons could be neglected [1]. This fact is highlighted in Fig. 4(b) by the dispersion of the ratio of the diffraction efficiency for a pure absorption grating and a pure index grating η(n₁ = 0)/η(α₁ = 0). As the estimate of the diffraction efficiency only slightly depends on the direction of the grating vector, with a correction of cos(2θ_B) ≈ 0.96 for p-polarization and probing wavelength 488 nm [26], it is valid for both recording configurations under study.

 

Fig. 4 (a) Dispersion of the diffraction efficiency η^est.(λ) (solid line) that has been estimated according to Eq. (3) and the parameters published in Ref. [1]. The grey area denotes the error for η^est.(λ). The experimentally determined efficiencies at a probing wavelength of 488 nm (△, this work) and 785 nm (□, Ref. [1]) have been added for comparison. (b) Dispersion of the ratio of the diffraction efficiency for a pure absorption grating and a pure index grating. A predominant contribution of the absorption grating is found at 785 nm while amplitude and index grating likewise contribute to the overall efficiency at 488 nm.

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4.3. Intensity dependence of diffraction efficiency

The key in understanding the saturation behavior of the diffraction efficiency at 488 nm as a function of pump beam intensity (Fig. 2(a)) is the intensity dependence of the small polaron densities involved in the hologram recording process, i.e., of bipolarons and bound polarons. According to our model, the bipolaron density is reduced via optical gating processes in the bright regions of the fringe pattern and yields an increase of the number density of small bound polarons by N_li,GP with

4.4. Dependence on the recording configuration

Understanding the dependence of the diffraction efficiency on the recording configuration K ⊥ c-axis and K || c-axis remains a challenging task. While the situation for K ⊥ c-axis is completely explained in the model of a spatially modulated polaron density, further physical processes need to be taken into account to model the results for K || c-axis. It is reasonable to assign the pronounced increase of the diffraction efficiency to the action of the linear electro-optic effect because r₃₃₁ = 0, r₂₂₃ ≠ 0, and r₃₃₃ ≠ 0. Moreover, a predominant contribution of the index grating over the absorption grating is inferred from the efficiency exceeding the theoretical limit 3.7% of pure lossy gratings. As a consequence, the build-up of an electric field with light exposure is required. At the same time, the stretched-exponential grating decay and its temperature dependence with an activation energy of E_A = (0.57±0.07) eV unambiguously assign the temporal behavior of the diffraction signal to the decay of a spatially modulated density of small bound ${\text{Nb}}_{\text{Li}}^{4+}$ polarons. Thus, a relation between the small polaron density and the electric field, both spatially modulated along the direction of K, must be postulated. The electric field strength that modulates the index with amplitude n₁ = (−2.0 ± 0.5) · 10⁻⁴ can be estimated via $E=-2n_1/\left(n_{\text{e}}^3{r}_{333}\right)$ (cf. Ref. [1]). With n_e(488nm) = (2.2556 ± 0.0005) [27] and r₃₃₃(488nm) = 34.4 pm/V [28] we get E ≈ 10 kV/cm that is much larger than the saturation field for diffusion transport mechanisms E_diff = (k_BT/e)(2π/Λ) ≈ 0.67 kV/cm. Hence, a photovoltaic transport mechanism can be concluded, similar to the small polaron based model approach that explains the bulk photovoltaic effect in Fe-doped LiNbO₃ [10].

As a consequence of this conclusion, the electric field E and thereby n₁ must show a dependence on the pump beam intensity I_p. In fact, the photovoltaic current density is given by j_phv ∼ βI_p with the effective element of the photovoltaic tensor β. The dependence n₁(I_p) is reflected by the data in Fig. 2(b), where a pronounced increase of the diffraction efficiency η on the pump beam intensity is depicted. Here we note, that η can be approximated by $\eta ~n_1^2$ for small amplitudes n₁ according to Eq. (3), i.e., $\eta =a\cdot I_{\text{p}}^2$ for a linear dependence of n₁ on I_p. Therefore, we have fitted a quadratic function to the experimental data (dashed line in Fig. 2(b)) obtaining a = (8±1)·10⁻²⁵ m⁴/W². Good agreement between the fit and the experimental data is found over a broad range of intensities, thus additionally supporting the conclusion of a bulk photovoltaic effect related to the optical formation of small polarons that is at the origin of the hologram recording mechanism for the configuration K || c-axis.

We finally note that nonlinearities of the diffraction efficiency are reported in photorefractive-recording of holograms with ns-laser pulses in transition-metal doped LiNbO₃, as well [29]. Contrary to solely polaron-based hologram recording, the simultaneous interplay of carriers excited from intrinsic and extrinsic defect centers needs to be considered. Thereby, an intensity limit of the nonlinearity results that represents a striking difference to our findings.

5. Conclusion

In conclusion, polaron-based recording of holograms in LiNbO₃ is verified in the visible spectrum. It obeys characteristic features that distinguish the recording process clearly from laser pulse recording via the photorefractive effect involving extrinsic traps [30, 31, 32], but can be closely related to the early findings of Schirmer and von der Linde [33, 34]. A remarkable hologram recording sensitivity of S = 8.4cm/J being nonlinearly dependent on the pump beam intensity is verified. Our results clearly indicate the impact of small bound polarons as optically addressable quasi-particles that have, so far, been identified as metastable intermediate centers in photorefractive two-step recording mechanisms [35, 36, 37]. All results underline the validity of our small polaron approach for the recording of volume holograms in LiNbO₃. Taking into account the formation time of small polarons in the sub-ps regime [6, 24, 38], the small polaron approach may be also applied for fs-holography.

From the materials’ point of view, nominally undoped LiNbO₃ becomes more attractive as hologram material for modern holographic applications, including real-time holographic displays. It can be expected that equivalent recording mechanisms are present for a variety of other oxide crystals, as well. For instance, pronounced small polaron densities can also be generated by single laser pulses in KNbO₃ and BaTiO₃ [11].