1. Introduction

In the optical telecommunications field, the evolution of transmission networks requires the use of active components. They should be able to carry out new functions, such as wavelength conversion or channel switching. Glass-based materials are of great interest in that prospect, because of their low cost and excellent optical properties. But in the dipole approximation, even-order nonlinear (NL) coefficients χ ⁽²ⁿ⁾ (n positive integer) of amorphous media vanish because of the macroscopic inversion symmetry of the material. As a consequence, second-order NL optical effects are forbidden in such media. Nevertheless, it is now known that a large second-order nonlinearity (SON) can be induced in glasses by breaking their centrosymmetry with appropriate treatments. The most practical method to create such a SON in glasses is the thermal poling technique [1], that allows to induce χ ⁽²⁾ coefficients of typically ~1 pm/V in silica glasses. This method consists in applying a high DC bias to a glass sample previously heated to a temperature of ~300°C. After a poling duration t _p of several minutes (or tens of minutes), the sample is cooled down to room temperature and finally the voltage is switched off.

Usually, the origin of the induced SON in silica glasses is assumed to be mainly related to migration of positive ions [2]. The most mobile charge carriers in silica glasses are alkali ions (Li⁺, Na⁺, K⁺,…), present as impurities in Infrasil^TM glass (typically with a concentration of about 1 ppm). When these ions are thermally activated, the application of an external electric field makes them migrate toward the cathode of the system. As a consequence a negative space charge is created under the anodic surface, due to the fact that negative charges are motionless in the glass matrix. The thickness of this depleted layer is typically in the range 3–80 μm in silica glasses. A high electric field E _DC close to the dielectric breakdown field is then induced within the depletion region. This E _DC field coupled to the χ ⁽³⁾ susceptibility of the glass is at the origin of the effective SON χ ⁽²⁾, via Eq. (1) :

From an experimental point of view, Faccio et al. have shown experimentally that there is an optimum t _opt in the poling duration for the second-order coefficient χ ⁽²⁾, depending on the sample thickness [3]. In a previous paper, we have shown that a single charge carrier model is sufficiently accurate to describe the creation of the depleted layer, for poling durations shorter than t _opt [4]. Such a model offers the advantage to solve the problem analytically for short poling durations, and thus to describe the space charge creation with simple equations [5], leading to the determination of the mobility and the concentration of the charge carrier mainly involved in the χ ⁽²⁾ creation [4]. However, for poling durations higher than t _opt, this single carrier model does not fit the experimental results anymore, because other phenomena such as charge injection or ionization become predominant [4]. In this case, a multiple carrier model has to be used to describe the long-time evolution of the SON [6, 7, 8]. The aim of the present paper is to compare the χ ⁽²⁾ profiles predicted by such a model with the experimental observations, for various sample thicknesses and poling durations.

2. Model

2.1. General equations

The single-carrier model used to describe the SON creation for poling durations shorter than t _opt is inadequate to explain the different time scales experimentally observed in Refs. [3, 4]. In particular, this model doesn’t predict the decrease of the induced χ ⁽²⁾ susceptibility and the increase of the NL layer width experimentally observed for poling durations higher than t _opt [3, 4, 9]. The experimental behavior of the SON dynamics can be understood if we assume that another positive ionic species is injected at the anodic surface, so that the poling efficiency is affected for long poling durations [4]. It is usually admitted that these injected positive charges are hydrogenated species coming from the surrounding atmosphere after ionization [6, 3], or ions diffusing from the electrode [10]. If both migration and diffusion processes are taken into account, the local equation of continuity and the Poisson’s equation lead to the following system of partial differential equations [5] :

where the index i represents one of the positive ionic species (i = 1 corresponds to the sodium ions and i = 2 to the injected charges), p_i = p_i(x,t) is the instantaneous concentration of the charge i (in ions/m³), μ_i is its mobility, p _0,i depicts its initial concentration within the glass, e is the electronic charge, and ε = 3.78 ε ₀ is the dielectric permittivity of fused silica. D_i is the diffusion constant, which is related to the mobility through the Nernst-Einstein relation D_i = k _B Tμ_i/e, k _B being the Boltzmann’s constant and T the temperature.

The system of Eqs. (2) and (3) allows to determine the spatial distribution of the electric field within the sample as a function of the poling duration. Eq. (1) is then used to deduce the χ ⁽²⁾ spatial distribution within the sample.

2.2. Boundary conditions

We assume that the potential at the anodic surface (x = 0) is V _app (V _app being the applied voltage during poling), and that the electrode at the cathodic side (x = ℓ) is grounded (V(x = ℓ) = 0, ℓ being the sample thickness) during the poling process. In this situation, the first boundary condition can be written in the form of Eq. (4) :

Furthermore, we assume that positive charge carriers are able to be injected through the anodic surface of the sample (non blocking anode condition) [6]. The initial density of these injected charges p _0,2 is taken equal to zero at t = 0 in the whole material (for 0 ≤ x ≤ ℓ). We assume in the present model that the injection rate depends on the electric field magnitude at the anodic side (x = 0) [7]. This choice can be explained by (i) the fact that injection of positive charges would require an electric field different from zero at the anodic surface and (ii) the fact that this condition is necessary to fit correctly the experimental results presented in section 3. An adjustable parameter σ ₂ is used to describe the charge injection into the glass. Therefore, the variation of injected carrier density per unit of time at the anodic surface can be written as Eq. (5):

where E(x = 0) is the electric field magnitude at the anodic surface of the sample (x = 0). Note that once the positive injected carrier is in the material, its movement in the glass matrix is also described by Eqs. (2) and (3).

3. Experiments

3.1. Poling conditions

The samples under investigation consist of 200 μm Infrasil^TM silica disks cut from the same rod, to avoid chemical composition variations and thus to ensure a better reproducibility of the poling experiments. After a careful cleaning, the samples were sandwiched between two p-doped Si electrodes, and heated to a temperature of 250°C for 30 minutes to ensure a good thermalization of the system. A voltage of 4 kV was then applied for 1, 3, 5, 15 and 45 minutes across the samples. Note that the typical poling temperature reported in the literature is generally higher (around 300°C). The low temperature of 250°C in the present experiments is used to slow down the dynamics of the SON creation.

3.2. Characterization

After the poling treatment, all samples have been characterized with the “layer peeling” method [11]. It consists in reconstructing step by step the SON profile, providing the knowledge of the second harmonic (SH) power as a function of the thickness under the anodic surface. To this end, the beam of a Q-switched Nd:YAG laser is focused onto the sample for a fixed incident angle θ (θ ≠ 0). The power of the generated SH wave is measured while the sample is etched by a hydrofluoric solution. The removed thickness of glass is deduced from a real-time interferometric measurement [12], that allows the knowledge of the SH power as a function of the NL layer thickness. An iterative algorithm is then used to reconstruct the SON profile with a spatial resolution in the order of 50 nm. Readers can refer to Ref. [11] for more details about this characterization method.

 

Fig. 1. (a) SON profiles experimentally obtained with the “layer peeling” method, for 200 μm-thick samples poled for 1, 3, 5, 15 and 45 minutes, and (b) the corresponding χ ⁽²⁾ spatial distributions obtained with the two-charge carrier model.

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3.3. Results

In a previous paper, we have reported the experimental time-evolution of the χ ⁽²⁾ spatial distributions obtained in 200 μm-thick samples [4]. For the sake of convenience, these results are displayed in Fig. 1(a). In particular, we have shown that the shape of the profiles is triangular for poling durations shorter than t _opt, and that the shape tends to flatten for higher poling durations. The experimental time-evolution of both the χ ⁽²⁾ coefficient maximum value and the NL thickness are plotted respectively in Figs. 2(a) and 2(b) (full squares). As we can see, the NL width increases for increasing poling durations, and there is an optimum duration t _opt of 5 minutes for the χ ⁽²⁾ susceptibility creation. These results confirm those previously obtained by Faccio et al. in Ref. [3].

 

Fig. 2. Time evolution of the χ ⁽²⁾ maximum value and of the nonlinear layer width for sample thicknesses of 200 μm (respectively (a) and (b)) and of 500 μm (respectively (c) and (d)). Squares corresponds to experimental data and solid lines represent numerical simulations performed with the two carrier model.

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4. Simulations results

For the most mobile ionic species (assumed to be Na⁺), it has been demonstrated in Ref. [4] that the initial Na⁺ concentration p _0,1 and the sodium mobility μ ₁ can be deduced from the above experiments :

In the present work, these two parameters p _0,1 and μ ₁ deduced from experimental results in 200 μm-thick samples are included in the system of Eqs. (2) and (3) to perform the simulations. The two missing parameters μ ₂ and σ ₂ corresponding respectively to the mobility and the injection rate of injected charges are adjusted to get the best fit of the experimental data. Indeed, the model presented in Ref. [4] does not provide any information about these two parameters. Simulations are performed to determine them by a trials and errors method. The best fit gives μ ₂ = 2 × 10^-18 m².V^-1.s^-1, and σ ₂ = 5 × 10¹² m^-2.V^-1.s^-1. Note that the order of magnitude of μ ₂ is in good agreement with the hydrogen mobility, as shown in Ref. [6].

The theoretical SON profiles are displayed for each poling duration of the experimental study [see Fig. 1(b)]. As we can see, the NL layer thickness and the absolute χ ⁽²⁾ magnitude obtained by simulations match very well with the experimental values. Moreover, the shapes of the simulated profiles are close to the spatial distributions experimentally obtained. Indeed, the simulated profiles are nearly triangular for poling durations shorter than t _opt (full squares, open squares and full circles in Fig. 1(b)), and they flatten more and more for increasing poling times (open circles and crosses in Fig. 1(b)). It is stressed here that the model allows to calculate the induced electric field, the χ ⁽²⁾ coefficient being deduced from E _DC through Eq. (1). The good agreement between the experimental measurements and the simulations confirms that charge migration phenomena are mainly at the origin of the induced χ ⁽²⁾ in silica glasses. Note that the difference between the simulated and experimental χ ⁽²⁾ profiles in the first micrometers under the anode surface can be attributed to a change in the χ ⁽³⁾ of the glass, as reported in Ref. [15]. This phenomenon is not taken into account in the model and is not crucial for the comparison between the experimental and simulated profiles in this work.

In Fig. 2(a), the experimental maximum values of the χ ⁽²⁾ coefficient (full squares) are compared with the simulated ones (solid line). The poling duration is theoretically found to be optimal after 4’40 s, what is in good accordance with the experimental t _opt value of 5 min. The theoretical χ ⁽²⁾ magnitude increases rapidly to reach a value of about 1 pm/V for a poling duration of t _opt. Then it decreases slowly to 0.4 pm/V for a 100 min-long poling process, for example. The total NL layer width experimentally measured and theoretically determined are also compared in Fig. 2(b). In both cases, it increases quickly until t _opt to reach a value of about 4 μm, and the increase for longer poling durations is slower. For example, after 100 min of poling, the NL width is theoretically equal to 8.4 μm.

5. Influence of the sample thickness

In order to confirm the validity of the two-charge carrier model presented here, the influence of the sample thickness ℓ has been investigated, all other parameters of the simulations found previously being fixed. Indeed, a change of the sample thickness would give rise to a change in the time-evolution of the χ ⁽²⁾ susceptibility since the initial electric field applied to the sample (V _app/ℓ) has changed.

5.1. Experiments

We have used 500 μm-thick Infrasil^TM disks which have been poled with the same conditions as for the 200 μm-thick samples. The poling durations have been chosen between 1 and 100 minutes (1, 5, 10, 30 and 100 minutes). In Fig. 3(a) are displayed the SON profiles obtained experimentally with the “layer peeling” method. The profile obtained for the 1 min poled sample is approximately triangular, and the χ ⁽²⁾ magnitude is very low χ ⁽²⁾ = 0.025 pm/V, with a NL layer width of 500 nm). The 5 min and 10 min poled samples exhibit also a quasi-triangular shape, and the χ ⁽²⁾ magnitude as well as the NL layer thickness increase for increasing poling durations χ ⁽²⁾ maximum magnitude of respectively 0.95 and 1.15 pm/V, and NL width of 2.3 and 4 μm). For higher poling durations (30 and 100 min), the NL layer continues to spread until respectively 5.5 and 7.8 μm, whereas the χ ⁽²⁾ magnitude decreases, to reach respectively 0.8 and 0.6 pm/V. But in these two last cases, the shape of the profile is no more triangular. Indeed, the profile tends to flatten more and more when the poling duration increases, as expected when the poling duration is higher than t _opt. Full squares of Fig. 2(c) and 2(d) correspond respectively to the experimental values of the maximum χ ⁽²⁾ susceptibility and of the NL layer width.

 

Fig. 3. (a) SON profiles experimentally obtained with the “layer peeling” method, for 500 μm-thick samples poled for 1 [see insert], 5, 10, 30 and 100 minutes, and (b) the corresponding χ ⁽²⁾ spatial distributions obtained with the two-charge carrier model.

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5.2. Simulations results

Since all experiments have been performed at the same temperature of 250° C, the mobility of different species and the injection rate parameter are fixed to the same values as for the simulations of 200 μm-thick samples, presented above. the SON profiles obtained by the simulations are presented in Fig. 3(b), for all experimental poling durations, in the case of 500 μm-thick samples. The shape of the profiles proves to be in good accordance with the experimental ones. In Figs. 2(c) and 2(d) are shown respectively the theoretical time-evolution of the maximum value of the χ ⁽²⁾ susceptibility and of the NL layer width (solid lines). Note once again that there is a very nice agreement between these values measured experimentally and the results of the simulations performed with only a change of the ℓ parameter. The maximum χ ⁽²⁾ coefficient of 0.86 pm/V is theoretically obtained after 9’30 s of poling (experimental t _opt of 10 min). As in the 200 μm-thick samples case, the NL thickness increase is fast for poling durations shorter than t _opt, and it becomes slower for higher poling times.

6. Discussion

When the voltage V _app is applied at t = 0 (i.e. when the charge migration phenomenon starts), the electric field is equal to V _app/ℓ across the sample. Since the drift velocity of each species is given by the product of its mobility times the electric field magnitude, the drift velocity of sodium ions (v ₁ = μ ₁.V _app/ℓ) is much higher than the one of the injected ions (v ₂ = μ ₂.V _app/ℓ) because μ ₁ ≫ μ ₂. Consequently, one initially expects the creation of the depletion layer beneath the anodic surface due to sodium ion migration toward the cathode. The induced electric field E_DC at the surface increases thus significantly to reach a value of ~ 10⁹ V/m and tends to screen the applied electric field in the bulk of the sample (outside the depletion region). During this phase, injected ions remain localized within a thin layer under the surface [see the movie], and the spatial profile of the induced electric field in the NL layer is almost triangular. The maximum electric field is obtained once the space charge has been fully established, and the shape of the electric field is not significantly modified at this stage of the process [4]. The characteristic time of the space charge relaxation can be described by the τ parameter introduced in Ref. [5]. An estimation of this relaxation time can be calculated via Eq. 6:

 

Fig. 4. Results of simulations in a 200 μm-thick sample for a poling duration of 100 minutes. (a) Representation of the charge distribution (the black line corresponds to the sodium density and the red line represents the injected carrier density). (b) Schematization of the charge distribution (regions I and III are negatively charged, regions II and IV are neutral). (c) Resulting electric field distribution. The movie represents the time-evolution of the charge distribution and the resulting electric field for poling durations between 0 and 100 minutes (618 KB).

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Using the parameters of the simulations, we obtained a τ value 2.5 times lower for a sample thickness of 200 μm than for a 500 μm-thick sample (respectively 68 s and 170 s). Since the τ parameters linearly depends on the sample thickness ℓ the theoretical optimal poling duration t _opt is found to decrease when the sample thickness decreases (Figs. 2(a) and 2(c)). Note that these simulations are in good agreement with experimental results reported by Faccio et al. in Ref. [3].

After this space charge creation, the concentration of injected carrier per second grows quickly to reach a value of about 7.5 × 10²² m^-3.s^-1 (according to Eq. 5) when the E _DC field reaches its maximum magnitude (E _DC ~ 10⁹ V/m). The drift velocity of these ions v ₂ = μ ₂.E _DC, which is initially low, becomes comparable to the one of sodium ions.

For longer poling times (several tens of minutes), injected ions are driven deeper and deeper into the sample because of the high electric field present within the depletion region. They tend to substitute little by little for the sodium ions in the glass matrix. This phenomenon is illustrated in the movie where the sodium and injected charges distributions are plotted respectively in black and red lines.

For poling durations in the order of several hours, the depletion region is almost filled up with injected carriers, as shown in the simulation of a 100 min-long poling for a sample thickness of 200 μm (Fig. 4(a)). The NL layer is made of three different regions, written I to III in the Fig. 4(b). Region I is a slightly negatively charged zone under the anodic surface. It is followed by a large neutral zone because injected positive carriers replace sodium ions (region II). Finally, region III corresponds to a thinner negatively charged depleted zone. In this case, the shape of the electric field E _DC is nearly flat [see Fig. 4(c)], as experimentally observed for long poling durations. The region IV of Fig. 4(b) corresponds to the bulk of the sample (outside the NL region) and is consequently neutral.

Contrary to a model taking into account the migration of one single charge carrier [5], an electric field different of zero is induced outside the NL layer during the poling process for long poling durations (region IV). This field is due to positive charge injection phenomena occurring at the anodic surface of the sample. In this situation, the total electric field E _DC induced within the sample can be written as follows [see Fig. 4(c)] :

where w is the depletion region width.

Although the bulk electric field E _bulk is weak compared to the E _layer field within the depleted layer (E _bulk ~ 4×10⁵ V/m according to the simulations, as can be seen in the movie), it allows the sodium ions to move towards the cathode. However, at this stage of the process, the velocity drift of the injected charges cloud is higher than the one of sodium ions (μ ₂.E _layer > μ ₁.E _bulk). Thus the thickness of region III in Fig. 4(b) decreases for increasing poling durations. This phenomenon can be observed in the movie for a sample thickness of 200 μm : region III is 3 μm-thick for 10 min of poling, and it decreases until 1.25 μm for a poling duration of 100 min. As the number of negative charges in region III decreases, the magnitude of the electric field within the NL layer (E _layer) decreases, leading to a reduction of the χ ⁽²⁾ coefficient.

7. Conclusion

We have studied experimentally and theoretically the dynamics of the SON creation in thermally poled Infrasil^TM samples. We have shown that the experimental χ ⁽²⁾ profiles exhibit a triangular shape for poling durations shorter than the optimal one, and that they tend to flatten for longer poling times. A two-charge carrier model taking into account an electric field dependant charge injection has been used to report on these experimental results. A good agreement between the experimental measurements and numerical simulations is reported, even for long poling durations. This is the first time to our best knowledge that such a good agreement is observed for the χ ⁽²⁾ magnitude, the NL width and the χ ⁽²⁾ profile shape. Consequently, the creation of a SON a Infrasil^TM samples can be attributed without any ambiguities to charge migration mechanisms and the observation of an optimal poling duration can be attributed to positive charge injection occurring through the anode during the poling process.

During the space charge creation, injected ions can be neglected, since they remain confined in a thin layer under the surface, because of their low mobility. After this stage, the high electric field present within the depletion region drives injected positive ions into the sample so that they substitute to sodium ions little by little in the glass matrix. The consequence is a continuous flattening of the SON profile shape, and thus a decrease of the χ ⁽²⁾ magnitude for increasing poling durations. The model presented here provide valuable information about the design of twin-holes optical fibers for manufacturing poled fibered components.