1. Introduction

Nonlinear optical beam coupling in photorefractive materials [1] gives rise to a variety of exciting phenomena as light filamentation (fanning) [2,3] and self-pumping phase conjugation [4] among others. However, they are also the origin of a major drawback in many photonic devices, namely the photorefractive optical damage (OD). This phenomenon is a major scientific and technological issue in nonlinear optical materials [5,6] as most of them present significant photorefractive effect, i.e., a light induced refractive index change generated by the combined action of photoconductivity and electrooptic activity [1]. In particular, optical damage (OD) seriously limits the highest light intensity which can be used in LiNbO₃, a reference photonic material, mainly because above a certain threshold, a chaotic beam distortion (catastrophic damage) appears which rules out any control of the propagating light wavefront [5,7]. This is particularly serious in harmonic generation devices, due to the high light intensity required for efficient frequency conversion [8]. In LiNbO₃, damage is firstly reduced by decreasing as much as possible the Fe concentration because of its central role in photorefraction. A further damage reduction has been experimentally well confirmed by doping with Mg, Zn and other ions [9–11] which displace the Nb_Li intrinsic defect (i.e. Nb⁵⁺ in Li⁺-sites, also called anti-sites) to its regular lattice positions [6]. This has been attributed to an increase of the electronic conductivity. It has also been reported a substantial increase of the damage resistance by raising the crystal temperature above room temperature [11–13], supposedly due to a carrier mobility increase and to a decrease of the bulk photovoltaic contribution [1] to the carrier transport. In spite of the mechanisms invoked to explain the above data (and some others), a model describing all these features in a comprehensive and useful way is still lacking. Particularly relevant is the appearance of intensity thresholds for chaotic OD which remains reluctant to any satisfactory explanation.

Some previously reported works [7,11] associates the phenomenon of catastrophic OD in LiNbO₃ to amplification of light noise through beam coupling [3]. In this process, photorefractive gratings are randomly recorded by the interference of the strong incident beam with weak beams scattered from crystal defects. When these gratings exhibit high optical gain, scattered beams are strongly amplified at the expense of the incident beam which almost completely converts into chaotic optical noise. Apart from the need for some initial scattering [14] (always present in ordinary crystals), the heart of the model is the gain coefficient Γ for beam amplification in the random gratings. However, there are no experimental or theoretical available reports on the intensity dependence of Γ, which turns out to be dependent on the photorefractive index grating amplitude, Δn, and the phase mismatch ϕ between the index grating and the light pattern, namely Γ∝Δn sin ϕ. Then, mechanisms for strong increasing Γ with the beam intensity should provide the sharp damage threshold experimentally observed.

Hints for thresholding are the superlinear dependence on the intensity of the photovoltaic current j_pv observed in bulk [15] and planar LiNbO₃ waveguides [16], and the intensity dependence of the saturating Δn _sat value also reported for bulk [17] and planar waveguides [18]. Moreover, the appearance of the photovoltaic current superlinearity has been recently correlated with observed OD thresholds [14]. Since in ordinary photorefractive models operating with a single defect (namely Fe²⁺↔Fe³⁺) the function j_pv(I) is linear and Δn _sat does not depend on I [1], it was proposed [19] that a second (unknown) shallow trap contributed to the charge transport. Using this two-center scheme, a model has been developed for bulk [17] and extended to waveguides [16] that successfully explains the j_pv superlinearity and predicts an intensity dependence of Δn _sat. Aside from the ordinary photovoltaic center (Fe²⁺↔Fe³⁺), a second strongly photovoltaic shallow center, namely Nb_Li ⁴⁺↔Nb_Li ⁵⁺, is assumed to contribute to electron transport [20]. Because of the low activation energy for thermal electron detrapping, this secondary center only gives a significative contribution to charge transport and, particularly to the photovoltaic current, at high light intensities.

In this letter we develop and apply the above quoted two-center model for studying the influence of the secondary center Nb_Li on the intensity dependence of Γ, the key parameter in catastrophic OD. The model is expected to provide an explanation for intensity thresholds and to predict the main trends of reported experimental results on Mg-doping and temperature dependences of threshold values. In addition, the model should explain other features such as the influence of the sample reduction state, i.e. [Fe²⁺]/[Fe³⁺] ratio, recently reported [21].

2. Model

The rate and transport equations for the two center model write [16]:

Here, subscript i=1,2 stands for primary and secondary centers, N_i and N _i0 for the donor and initial donor concentrations respectively and S_r is the electron recombination coefficient, which, for the sake of simplicity, is assumed the same for both types of traps. Finally, S_ti=S _ti0exp(-ε _ti/KT) is the donor thermal ionization probability, μ the electron mobility and D the diffusion coefficient (μ=De/KT, D=D ₀exp(-ε/KT)). Note that, to simplify, direct electron transfer from Fe²⁺ to Nb_Li ⁵⁺ proposed in Ref. [17] has been neglected taking into account the low Fe-concentration of undoped LiNbO₃. Any contribution from absorption gratings likely appearing at very high intensities has also been disregarded here for simplicity.

The solution of the above equations in steady state has been analytically obtained for a sinusoidal light pattern, assuming a small modulation depth i.e. taking only the zero and first harmonics of the charge pattern [1], which most often provide the essential physical information of the phenomenon. For simplicity, accumulation of charge at the boundaries of the light beam is considered negligible. As the solutions are lengthy and cumbersome to be written here, the results will be illustrated through some significant plots in the following.

Table I. Parameters used for the simulations in I.S. units excepting ei in eV.

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3. Simulation results and discussion

Numerical values for the material parameters appearing in Eqs. (1) have been taken from references [16,22,23], and they are listed in Table I. Although the accuracy of some of these values is very poor, they have proved good semi-quantitative agreement with a variety of experimental data in α-phase proton exchanged planar waveguides.

 

Fig. 1. (a) Model simulation for the photorefractive gain coefficient Γ (continuous line) and the zero order of the photovoltaic current j_pv (dashed line) as a function of the light intensity for an undoped congruent LiNbO₃ sample at room temperature. (A thin dashed line has been drawn for easy recognition of the superlinear region of j_pv). Inset: simulated dependences of the saturating refractive index Δn _sat and phase mismatch ϕ. (b) The continuous line of (a) has been plotted together with the same dependences for a quasi-stoichiometric sample ([Nb^Li]=10²⁴ m^-3) at 295 K (dashed line), and at 395 K (dotted line). Either in (a) or (b) [Fe²⁺]/[Fe³⁺]=0.05.

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Using the analytical solutions of (1a)–(1e), the zero and first order approximation of the basic quantities, namely Γ (gain coefficient), Δn_sat (saturating index grating amplitude), ϕ (grating phase mismatch), j_pv (photovoltaic current) and σ (the photoconductivity) are easily obtained as functions of the beam intensity, sample temperature, defect concentration and reduction state. Regarding Γ, the key quantity for optical noise amplification, it has been plotted in logarithmic scale as continuous line in Fig. 1(a), for room temperature, within the significant intensity range 1–10⁸ W/cm². The curve shows a strong increase of about two orders of magnitude at intermediate intensities ~10²–10⁴ W/cm² joining two almost intensity independent regions at both sides of the plotted range. The low intensity behaves as predicted by the one-center (Fe) model. The onset of the fast increasing region (at ~100 W/cm²) strongly suggests the occurrence of a threshold for catastrophic optical damage, as we will further discuss along the paper. The intensity dependence of the zero order solution of j_pv is also shown in Fig. 1(a) in dashed line. It can be seen that the superlinear region exactly coincides with the increasing region for Γ. On the other hand, the dependence of Δn _sat on the intensity is parallel to that of Γ, as shown in the inset (continuous line) of Fig. 1(a), although the increasing region appears at somewhat lower intensity than in the Γ-plot. In contrast, the same region appears at greater intensities in the ϕ plot (dotted line in the inset) which shows a minimum before increasing. Finally, the model predicts a sublinear intensity dependence for the photoconductivity that also starts at the threshold region but is less pronounced than the superlinearity of j_pv. This sublinear behavior has been observed in proton-exchanged guides at similar intensities [24]. It is worthwhile remarking that all intensity dependences are due to the increasing number of electrons trapped in Nb_Li [16]. This gives rise to a superlinear photovoltaic current and a sublinear photoconductivity that both contribute to the threshold behavior although according to the simulations the first mechanism is clearly dominant.

Let us make a simple estimate of the amplification gain. According to the fanning model mentioned above, the interference of a weak-intensity beam I_s scattered from some crystal defect with the strong-intensity incident beam I can write a grating which couples both beams, amplifying the weak one to an intensity I_d. For the undepleted pump approximation [1] both quantities are related by the expression

where z is the length parallel to the grating planes. Assuming a plausible grating effective size of z~0.5 cm and Γ~1 cm^-1 (i.e. below threshold in Fig. 1), expression (2) gives exp(Γz)~1 and no appreciable OD exists. However, for Γ~200 cm^-1 (i.e. above the increasing region), exp(Γz)~10⁴³, i.e. much greater than needed for efficient energy transfer from the incident beam to chaotic optical noise, justifying the appearance of dramatic (catastrophic) OD in the transition region. With the parameters used here (Table I), corresponding to α-phase proton exchange waveguides, experimental thresholds are between 10 and 100 W/cm² [7,14], i.e. in reasonable semi-quantitative agreement with the onset of Fig. 1 at ~100 W/cm². It is worth noticing that, in typical low intensity photorefractive applications, iron-doped LiNbO₃ shows quite low beam amplification mainly due to diffusion carrier transport. In the undoped samples we are simulating, strong beam amplification comes from the great photovoltaic transport length of Nb_Li ⁴⁺ defects operating at high light intensity, where ϕ may easily approach π/2 (see inset in Fig. 1).

The model simulations also show good agreement when considering two basic features of the behavior of OD thresholds, which are relevant for practical usage in many LiNbO₃ applications. One of these is the threshold increase observed with Mg-doping (and similar ions) [6, 10] and in quasi-stoichiometric crystals (where NbLi is nearly absent) [6]. It is well documented and accepted that Mg-doping implies a strong reduction of the Nb_Li concentration [6]. The model prediction for this case is illustrated in Fig. 1(b), where the intensity dependence of the gain coefficient Γ has been plotted for a congruent (undoped) crystal, [Nb_Li]=3×10²⁶ m^-3, as continuous line, and for [Nb_Li]=3×10²⁴ m^-3 (a situation roughly similar to 6% Mg-doping) as dashed line. The intensity threshold for Γ is seen to increase by about two orders of magnitude. The dependences predicted by the model for Δn _sat, ϕ and j_pv show a similar shift of two orders of magnitude. Now, much more photoelectrons are needed for producing the same Nb_Li ⁴⁺ concentration which is the last responsible for the superlinearity. In fact, a similar shift of the onset of the superlinear behavior of the j_pv has been experimentally measured in nearly stoichiometric samples with lower Nb_Li concentration [15]. This is in good accordance with the model predictions, i.e. the threshold increase has to be attributed to the Nb_Li decrease, and so mainly to the lower photovoltaic current.

A second feature explained by the model is the increase of the OD threshold obtained by the increase of the sample temperature [12,13]. Particularly in optical waveguides where the beam power is tightly confined with a much greater intensity than in bulk, this is an important, additional procedure to substantially improve device capabilities [11,25]. Although dark conductivity arising from thermal excitation of Fe²⁺ is negligible at room temperature in LiNbO₃ [26], it has been included in order to apply the model at higher temperatures. The model prediction for the intensity dependence of the gain coefficient Γ in this case is also illustrated in Fig. 1(b), as dotted line. This curve corresponds to the same Nb_Li concentration as the previous dashed curve, i.e. like a quasi-stoichiometric crystal or [Nb_Li]=3×10²⁴ m^-3, but the temperature has been raised ΔT=100 K above room temperature. The intensity threshold for Γ is seen to increase by about two additional orders of magnitude, i.e. four orders of magnitude with regard to the congruent sample at room temperature, continuous line. The lower Γ-values observed on the low intensity region of this curve are due to the dark conductivity competition with photoconductivity at this temperature. Once again, a similar threshold shift is observed in the other studied quantities. Now much more free carriers, photo-excited from Fe2+, are needed for compensating the much shorter lifetime of Nb_Li ⁴⁺ ions. This last prediction on the temperature role was previously advanced [16] and experimentally measured in [12] where more than one order of magnitude increase was found for ΔT≈100 K, in good semi-quantitative agreement with Fig. 1(b). Again this indicates the crucial role of Nb_Li-defects on intensity thresholds.

 

Fig. 2. (a) Simulations of the gain coefficient Γ as a function of the light intensity I ₀ in undoped congruent LiNbO₃ for several [Fe²⁺]/[Fe³⁺] ratios. (b) Dependences of the threshold intensity (taken at Γ=10 cm^-1) as a function of [Nb_Li] (bottom horizontal axis) and T (top horizontal axis) with [Fe²⁺]/[Fe³⁺]=0.05.

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The model predictions for threshold changes with the reduction state of Fe are particularly interesting. Fig. 2(a) shows the dependence of Γ on the light intensity for the [Fe²⁺]/[Fe³⁺] ratios 0.05, 0.5, 1, 2 (indicated in the figure). About one order of magnitude increase in the threshold is obtained when increasing the [Fe²⁺]/[Fe³⁺] ratio from 0.05 to 2. This is also in semi-quantitative agreement with a similar threshold increase recently measured in proton exchanged planar waveguides between the same reduction states [21], providing a satisfactory explanation of that apparently surprising result. In order to better illustrate the dependences of the intensity threshold (conveniently taken at Γ=10 cm^-1), Fig. 2(b) shows how it increases as a function of the Nb_Li concentration and the temperature.

Finally, although the analysis has focused here on thresholding induced by beam amplification in random gratings, i.e. periodic light patterns, it provides some hints for channel waveguides where gratings can not be written. In this case, Δn _sat instead of Γ is the key parameter controlling the OD. As mentioned above, the intensity dependences of Δn _sat on varying the Nb_Li-concentration and the temperature are very similar to those of Γ. These dependences are in good qualitative agreement with those experimentally reported in [11].

In summary, the appearance of light intensity thresholds for catastrophic photorefractive damage in bulk and planar waveguides on LiNbO₃ has been satisfactorily explained. By using a two-centre model based on the Fe²⁺↔Fe³⁺ and Nb_Li ⁴⁺↔Nb_Li ⁵⁺ defect pairs, light intensity dependences of the gain coefficient Γ (the key quantity for noise amplification) show a clear threshold behavior. A reasonable semi-quantitative agreement is obtained with most experimental results reported for the damage behaviour on varying the Mg and Zn doping, the temperature or the [Fe²⁺]/[Fe³⁺] ratio.