Diffraction efficiency, relaxation behavior and dependence on pump-beam intensity of small-polaron based holograms are studied in thermally reduced, nominally undoped lithium niobate in the visible spectrum (λ = 488 nm). The pronounced phase gratings with diffraction efficiency up to η = (10.8 ± 1.0)% appeared upon irradiation by single ns-laser pulses (λ = 532 nm) and are comprehensively assigned to the optical formation of spatially modulated densities of small bound electron polarons, electron bipolarons, and O− hole polarons. A remarkable quadratic dependence on the pump-beam intensity is discovered for the recording configuration K || c-axis and can be explained by the electro-optic contribution of the optically generated small bound polarons. We discuss the build-up of local space-charge fields via small-polaron based bulk photovoltaic currents.
© 2012 OSA
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 . 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 . 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 , volume holographic optical correlators  or tunable Bragg-filters for photonic networks . In LiNbO3, small polarons have formation times in the sub-ps time regime , 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 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 . 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 electron bipolarons (BP, absorption maximum at λ ≃ 500 nm)  and small O− hole polarons (HP, λ ≃ 500 nm) . 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 . 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 , that explains the build-up of localized electric space charge fields, is discussed.
With these results and our earlier findings , 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 .
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. ) 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 bipolarons (BP) and of small bound -polarons (GP) [12, 13] were generated that are stable at room temperature. We have calculated the number densities NBP and NGP 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.
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, λp = 532 nm and average pulse duration τFWHM ≈ 8 ns) was used for recording an unslanted volume grating (equal intensities IR = IS, parallel light polarization eR = eS, modulation depth m = 1, Bragg angle ΘB = 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 λt = 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 . 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) . 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 .
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. We note that Eq. (1) accords with the empirical dielectric decay function introduced by Kohlrausch, Williams and Watts (KWW) .
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 Ip 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.
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 EA = (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, for which Schirmer et al. obtained a value of 0.62 eV from conductivity measurements .
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 (Ip = 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  of:19, 20].
From the point of view of small polarons, there are two outstanding findings associated with the visible spectrum:
- The amplitude of the diffraction efficiency at 488 nm is reduced by a factor of two in comparison with probing at 785 nm, i.e. an abnormal dispersion behavior is found.
- The intensity dependence of the diffraction efficiency is qualitatively different for the recording geometries K ⊥ c and K || c. We will discuss these two findings below along the expectations for polaron-based hologram recording in the visible.
The further properties have already been reported for the near-infrared spectral range  and give strong evidence for the polaronic origin of the hologram recording and read-out process:
- ⊳ a stretched-exponential decay of the diffraction efficiency,
- ⊳ a hologram lifetime in the ms-range,
- ⊳ a thermally activated hologram decay,
- ⊳ an activation energy of the latter that corresponds to the activation energy of the small bound polaron and
- ⊳ a pronounced dependence of the diffraction efficiency on the recording geometry.
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 . 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 . 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+) . Second, the role of small free electron polaron densities is neglected because of their sub-μs-lifetime  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) = Ip[1 + cos(|K|x)] results in the appearance of spatially modulated polaron densities. Densities NGP,HP of small bound electron and small hole polarons are increased in the bright region of the fringe pattern while the bipolaron density NBP 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 -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  while small bound polarons appear within 400 fs  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 αGP,HP,BP of the respective types by constant factors (the absorption cross sections σGP,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.
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  and dependence on the pump beam intensity  are well documented in literature. Hence, it is possible to estimate the maximum value of the absorption change (≡ 2α1 = αli) for the sample under study and applied pump beam intensity as a function of wavelength (see figure 7 in Ref. ). 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 . 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 :1]. We have taken into account absorption losses in the crystal volume (d − dh) due to the reduced effective grating thickness. This corresponds to an additional absorption factor not included in Eq. (3) that results in an estimated diffraction efficiency of ηest.(λ) = η(λ) · [exp(−2(α0 +α1)(d − dh)/ cosΘB)]. Deviations from a sinusoidal intensity pattern due to scattering, beam profile inhomogeneities and the reported nonlinear response of αli(Ip) on the pump intensity are neglected.
In Fig. 4(a), we have plotted the dependence ηest.(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 n1,BP and the absorption grating with α1,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 polaron (center of absorption feature at 1.6 eV), i.e., the contribution of n1,GP and of susceptibility changes related to bipolarons and hole polarons could be neglected . 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(2θB) ≈ 0.96 for p-polarization and probing wavelength 488 nm , it is valid for both recording configurations under study.
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 Nli,GP with12]. Because of the relation given by Eq. (4), the intensity dependence of small bound polarons shows saturation with the same characteristic intensity Ic, but an amplitude by a factor of two larger compared with Eq. (5). Furthermore, the absorption amplitude and the increase of polaron number density are directly linked via the absorption cross section. Hence, saturation as a function of pump beam intensity also appears for the amplitudes α1,BP,GP,HP and, taking into account Kramers-Kronig relation, for n1,GP,BP,HP. The latter enter equation Eq. (3), so that the diffraction efficiency inevitably shows saturation, as well. For small amplitudes α1, n1 and α1 ≪ α0, we can approximate Fig. 2. The resulting fitting parameters are summarized in Table 2, the characteristic intensity Ic is of the same order of magnitude as the characteristic intensity of the light-induced absorption as reported in Ref. .
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 EA = (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 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 (cf. Ref. ). With ne(488nm) = (2.2556 ± 0.0005)  and r333(488nm) = 34.4 pm/V  we get E ≈ 10 kV/cm that is much larger than the saturation field for diffusion transport mechanisms Ediff = (kBT/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 .
As a consequence of this conclusion, the electric field E and thereby n1 must show a dependence on the pump beam intensity Ip. In fact, the photovoltaic current density is given by jphv ∼ βIp with the effective element of the photovoltaic tensor β. The dependence n1(Ip) 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 for small amplitudes n1 according to Eq. (3), i.e., for a linear dependence of n1 on Ip. 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 . 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.
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 .
The authors thank Gerda Cornelsen and Werner Geisler for sample preparation and the Deutsche Forschungsgemeinschaft (projects IM37/5, INST 190/137-1) and the Deutscher Akademischer Austausch Dienst in cooperation with the Hungarian Scholarship Board Office (projects 50445542, 54377942 and P-MB/840 and 29696) for financial support.
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