1. Introduction

It is possible to induce second-order nonlinear response in optical fibers by means of thermal poling [1–3]. Until now, however, the coefficients χ⁽²⁾_eff demonstrated in practical applications of poled fibers in optical switching [4,5], E-field detection [6] and frequency doubling [7] fall significantly short of the 1 pm/V achieved near the anode-side surface of bulk samples. Modeling poling increases the understanding the physical processes that take place and guides optimization in a large parameter space. For instance, one may consider optimizing the temperature [8], the poling duration [9,10] and the applied voltage [11,12]. One can in principle also control the ionic content of the glass that incorporates various mobile cations known to be important, such as Na⁺, Li⁺ and hydrogen species (e.g., H₃O⁺) [13]. Besides, depending on poling conditions, the recorded electric field and nonlinear coefficient varies rapidly from the surface of the hole to inside the material [14,15]. Since the core of the fiber is located a few microns away from the surface to minimize loss, the interaction of the optical waveguide and the recorded electric field is strongly dependent on the exact distances involved. This is complicated by the rounded shape of the holes that accommodate the electrodes. Any reasonably accurate model of poled fibers, therefore, should not leave out the fiber’s complex geometry.

Modeling thermal poling with at least two cations, one of high mobility (e.g., Na + ) and one of low mobility (e.g., a hydrogen species) is required to describe adequately the ion-exchange that takes place [16,17]. Excellent agreement is obtained in this way for bulk samples, reproducing the shape and extent of the depletion region, and the time evolution and magnitude of the induced χ⁽²⁾_eff in fused silica slabs [17]. Other similar models in the literature include a description of an ionizing and a shielding field created by moving charges [18], and an ion-exchange model that takes into account a third mobile ion such as Ca²⁺, considering the anode to be a blocking or a non-blocking electrode [19,20] and a model where the negative ions are also considered to be mobile [21]. Except for the latter, all poling models in the literature treat bulk glass and consider a one-dimensional (1D) problem, not addressing the specific and complex geometry of poled fibers with internal electrodes.

In this letter, we describe results of simulations based on a 2D implementation of an ion-exchange model developed for poled fibers using a COMSOL Multiphysics platform. The model is validated through comparisons with experimental data obtained from the induced electrooptic effect. Two techniques for inducing an effective χ⁽²⁾ are considered, traditional thermal poling [1] and electrostatic charging with both internal electrodes at the same high voltage (HV) potential [22]. The simulations take into account the presence of a faster cation (Na⁺) and a slower cation (Li⁺) that are initially uniformly distributed in the glass matrix, and the injection of a hydrogen-related species from the electrodes (H₃O⁺). The main motivation for including Li⁺ in the simulation is the report [13] using secondary-ion mass spectroscopy (SIMS) that the mobile ion consistent with the space-change region measured typically (<10 µm) is lithium [13]. It is found that the inclusion of the Li⁺ ion in the model affects the poling dynamics. After the 2D model is implemented, the poling process is studied numerically. The distribution of various carriers is examined on the fiber cross-section and the recorded field evolution is evaluated along the poling process. It is found that even small changes in the fiber design, in poling voltage or carrier concentration can significantly affect the dynamics of the creation of the second-order nonlinearity. Finally, the model is used to scan various poling parameters and to determine their impact on the expected χ⁽²⁾.

2. Physical processes and the model

The physics of poling as understood and implemented in the model here is summarized below. Thermal poling [1] is based on the ability to charge the glass semi-permanently. The charge distribution creates an electric field that breaks the symmetry of silica, and the recorded field acting on the third-order nonlinear susceptibility results in an effective χ⁽²⁾ [1]. The charges arise from the displacement of cations repelled from the anode towards the cathode. The existence of cations in sufficient numbers (~ppm concentration) is assumed, a necessary condition for the creation of a significant χ⁽²⁾_eff. Sodium is the cation of highest mobility in silica and its displacement towards the cathode results in a cation-depleted region of high electrical resistivity near the anode. The unpaired electrons left behind have very low mobility at 300 °C and are responsible for the appearance of the recorded field. The ions reaching the cathode are assumed to recombine on the surface, rather than to accumulate [23]. If poling is continued long enough, the width of the depletion region increases and other slower cations are driven into the glass (e.g., H₃O⁺) to neutralize the electrons [16]. Total cancellation of the field is never possible, but the neutrality near the anode can be re-established [24]. It has been pointed out [25] that in twin-hole fibers, when the HV poling voltage is switched off, positive charge gathers on both anode and cathode surfaces. The field then points inwards from both electrodes to the trapped electrons inside the glass. Approximately only half of the field lines remain across the core after the poling bias is removed to create the second order nonlinearity [26]. The use of two anodes remedies this to a large extent [22], and the maximum field recorded after the bias is switched off is approximately twice as large than in conventional poling. It should be remarked here that the positive charge referred to above is attracted to the glass surface and should not be confused with the injected H₃O⁺ that is inside the glass as part of the ion-exchange process.

The physics of the numerical model here is based on the COMSOL Multiphysics solution of the transport of diluted species problem, assuming that low concentration ions (~1 ppm) move in the presence of diffusion and drift due to an electric field [16,17]. For the implementation of the model, two fibers fabricated for poling purposes are used as illustration, with dimensions specified in Table 1 below. The holes are assumed to be entirely filled by metal [27], and to provide a perfect equipotential. A dense two-dimensional mesh is established with COMSOL over the cross-section of the fiber studied, defined with the help of a scanning electron microscopy (SEM) picture of the fiber. The problem solved in x, y and t for the concentration of the j-th ion (Na⁺, Li⁺ and the hydrogen species, such as H₃O⁺) is [17]:

Table 1. Parameters of the fibers used for model validation and in variational studies.

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The boundary conditions used in the model are that: (1) before poling starts, the fiber is electrically neutral, i.e., the same combined concentration of cations and stationary anions are uniformly distributed in the fiber; (2) the potential at the surfaces of the holes is the applied voltage during poling and zero after the high-voltage is switched off [25]; (3) the outer surface of the fiber is at zero volts [22]; and (4) cations exit the outer surface and do not return [23].

The uniformly distributed mobile cations are assumed to be Na⁺ and Li⁺ [13]. The hydronium species is assumed to be injected from the surface of the holes and move driven by the field with low mobility. Two conditions were investigated, one where the injection rate of H₃O⁺ is constant, i.e., assuming the existence of ions already at the surface of the hole (in the glass) [16], and one when the injection is proportional to the electric field on the surface of the hole, as implemented in [17]. The numerical model easily allows for inclusion of other ions, such as K⁺ and Ca²⁺. Although increasing the number of mobile ions allows reproducing the time evolution of the recorded electric field accurately, it is considered here that unnecessarily increasing the degrees of freedom reduces the usefulness of the model, and therefore only a few simulations included a fourth mobile cation.

Typical initial carrier concentrations c(ions) assumed for fiber 1 in the simulations are: c(Na⁺) = 1 ppm uniformly distributed in the glass at t = 0 sec; c(Li⁺) = 1 ppm uniformly distributed in the glass at t = 0 sec; c(H₃O⁺) = up to 2 ppm injectable from the holes, initially zero in the entire fiber, supplied at a rate that is either constant, linearly dependent on the field at the electrode edge, or decaying exponentially as the ion supply is exhausted; c(non-bridging oxygen centers NBO^-) = 2 ppm uniformly distributed in the glass at t = 0 sec for initial charge neutrality. For fiber 2, the corresponding values assumed are in most cases 0.35 × those above. A finer estimate of the concentrations is used in section 8.

The actual cation concentrations in fibers available for experiments are actually unknown. Although the values used here are roughly consistent with the contamination level of the ILMASIL PN fused quartz [28] used as inner cladding in preform fabrication, adjustment of cation content affects the maximum χ⁽²⁾ recorded, and must be regarded as a weakness of the model used here and by others.

The mobility of Na⁺ at a given temperature depends strongly on type of silica glass sample considered and its OH and cation content, leading to a large spread of mobility values in the literature. Here, the mobility of the cations expressed in m²/Vs is derived from the diffusivity using the steady-state condition when the current is zero and the drift of carriers cancels diffusion, in which case µ_ion = z F D_ion /RT, where z = 1, F = 9.65 × 10⁴ C/mol and R = 8.31 J mol⁻¹ K⁻¹. The diffusion coefficients are calculated from the value for sodium D_Na = D₀ (Na⁺) exp (-H/k_BT),, where the activation energy for sodium ions is assumed to be H = 1.237 eV. Except when specified otherwise, the mobility values in the simulations at 265 °C are 2∙10⁻¹⁵ m²/Vs for sodium, 0.15∙µ_Na for Li⁺ [29] and µ_Na/1000 for H₃O⁺. These values differ relatively little from those reported in [16, 17, 21, 30, 31], correcting for the temperature.

The recorded electric field experienced by the light propagating in the core is determined in the model from the weighted average of the field value at five points across the core (i.e., a discrete overlap integral). The second-order nonlinearity value is obtained from the expression . The magnitude of χ⁽³⁾ used in the simulations affects the value of Except for in Section 8, the estimates of the induced second-order nonlinearity here use the value of χ⁽³⁾ measured through the ${\chi }_{eff}^{(2)}=3{\chi }^{(3)}\cdot E_{rec}$ Kerr effect in fibers before poling [32], 2.0 × 10⁻²² m²/V². In the ${\chi }_{eff}^{(2)}$ validation section, the value of χ⁽³⁾ for every poling time deviates slightly from this value. Two fiber types were used more extensively to validate the model here. The Ge-doped core of fiber F1 (single-mode at 1.5 µm) is relatively large to allow for low loss splicing and the distance between the core center and the edge of the nearest hole is also large (8.2 µm) to minimize optical loss. As a consequence, poling of F1 takes a relatively long time. Fiber F2 is single-mode at 1 µm and its Ge-doped core is nearer the hole to maximize the E-field recorded. Although this would imply in higher loss due to the interaction of the optical field and the metal electrode, the poled fiber is intended for use in frequency doubling, when the electrodes are removed. Both fiber preforms were assembled with the inner cladding made of ILMASIL fused quartz glass, which provides the necessary contamination for poling. Poling of F2 requires a relatively short time. The separation between holes is named D.

3. Electrical configurations

Two-dimensional maps of the electric field and of the potential distribution at the start of poling are illustrated in Fig. 1, for the internal anode/cathode (+/−) configuration (Figs. 1(a) and 1(b)) and anode/anode (+/+) configuration (Figs. 1(c) and 1(d)). The applied voltage potential is 5 kV. A large field develops between the anode(s) and the grounded fiber surface. The field between electrodes is non-uniform because of the holes’ curvature, and peaks at the surface of the electrode. For the (+/+) configuration the initial field at the midpoint between electrodes indicated by the arrow is zero. This value is still relatively low (1.5 × 10⁷ V/m) in the middle of the core.

 

Fig. 1 Map of electric field (a) and potential (b) across a fiber at the start of poling. The left electrode is the anode biased at 5 kV and the right electrode is the grounded (Gnd) cathode. The field has a wedged-shape [27]. Map of electric field (c) and potential (d) when both electrodes are anodes at + 5 kV. The arrows indicate direction and field strength.

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4. Ion dynamics

The dynamics of the charge movement is illustrated in Fig. 2, which shows the distribution of each mobile cation considered, their sum and the potential map at various times for fiber F1. In this simulation, the initial uniform concentration of Na⁺ and Li⁺ is 1 ppm (indicated by the cyan color), and the 2-anode configuration is used. The bias voltage is 5 kV and the temperature 265 °C. After 32 s, the depletion region of Na⁺ near the anodes begins to be noticeable (dark blue indicates concentration zero), wider where the field is strongest [9].

 

Fig. 2 Time evolution of the distribution of mobile cations, their sum and the potential map at four representative times for fiber F1 with both electrodes biased at positive high-voltage. Here, the injection of H₃O⁺ is assumed to be inexhaustible and capable of neutralizing the unpaired electrons created by the displacement of Na⁺ and Li⁺.

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At 90 s, the Na⁺ concentration is depleted even between the electrodes and begins to reach the fiber core. While the diffusion of H⁺ is imperceptible, one notices that the Li⁺ ions are depleted from the region closest to the anodes (dark blue) and accumulate at the outer boundary of the region depleted of sodium (yellow). This piling-up has been reported in various experimental and theoretical studies for bulk samples [13, 19, 33] but not been evidenced in fibers. The H₃O⁺ species has not began to be injected significantly (remaining dark blue). The map of the total cation concentration shows that the extent of the cation-depleted region is narrower than that depleted of Na⁺ because of the Li⁺ ions pile-up behind the Na⁺ ions . This is in agreement with the experimental studies by SIMS. The high resistivity rings around the holes brings the potential (U, bottom row) at the mid-point between electrodes much closer to zero volts than at the beginning of poling (red at first, now yellow) and most potential drop happens around the electrode holes. For this reason, the growth of the depletion region between electrodes becomes more accentuated.

At 4000 s, the Na⁺ depletion regions around the holes become adjacent to each other, the charge accumulation in the Li⁺ map is very obvious, and hydronium ions have moved from the electrodes, driven by the field to occupy the vacancies left by Na⁺ and Li⁺. The negatively charged region is now only a thin ring around the electrodes. Because of the large distance between the core and the electrodes in this fiber, at 4000 seconds the core still does not experience a large field – the potential across the core (uniform green) and its gradient is approximately zero.

With the expansion of the depletion region, the gradient of the potential increases. It reaches its maximum in the core center at 8800 s, when the recorded field is maximum. At this time, the region depleted of sodium has a figure-of-eight shape consistent with etching experiments shown in [34]. The Li⁺ ions accumulate near the periphery of the sodium-depleted layer, and result in a circular all-cation depleted region. Hydronium ions occupy the inner sites left by Na⁺ and Li⁺, and the resulting net distribution of charges is a thin circular layer of unpaired electrons around each electrode. By design, the core of the fiber has both rings of charges located on the same side of the core (right side, here) and the electric field contributions from both rings add. In the configuration of conventional poling with one anode and one cathode, one obtains only one ring of negative charges around the anode only (approximately half of the field, once the poling bias is removed).

The ion dynamics in the case of (+/−) poling is similar to the one illustrated in Fig. 2 and is not repeated here. Only one ring of charges results, around the anode electrode. Since the initial field strength between electrodes is much higher than in the configuration with two anodes, the initial growth of the Na⁺ depletion region is faster.

5. Field recorded at the end of poling

Figure 3 shows a 2D map for the conventional (+/−) configuration of the electric field and the potential at the end of poling when the recorded χ⁽²⁾ at the core center is maximized. The optimum poling time differs by more than 1000 s at the nearest and furthest core-edge. In Figs. 3(a) and 3(b) it is assumed that the high voltage potential (5 kV) is still applied to the anode. The field is entirely concentrated around the anode electrode. Figures 3(c) and 3(d) illustrate the significant change that takes place when the voltage bias is switched off after poling and the potential at the electrodes becomes zero (e.g., electrodes grounded). One sees that the field points to the narrow ring of negative charge around the anode. The cathode electrode also gives rise to field lines pointing towards to electrons, resulting in a reduction of the field magnitude across the center of the core. The potential map shows a negative value in the core region, guaranteeing the zero value of the line integral of the field, when going from on electrode to the other [25].

 

Fig. 3 2D map of electric field (a) and potential (b) across a fiber poled in a conventional (+/−) configuration when the field recorded at the center of the core reaches its maximum (8800 s). When the 5 kV voltage bias is switched off, the electric field distribution (c) and the potential (d) change significantly and the maximum recorded field is E_rec = 1.6 × 10⁸ V/m .

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Figure 4 shows a similar map for the two-anodes configuration (+/+) of the electric field and the potential when the recorded χ⁽²⁾ at the core center is maximized. In Figs. 4(a) and 4(b) it is assumed that the high voltage potential (5 kV) is still applied to the anodes. Here too, a significant change takes place when the voltage bias is switched off after poling and the potential at the electrodes becomes zero. Almost twice as much negative charge is recorded and the reduction of the field magnitude across the core center is small when the bias is removed. A straight comparison between Figs. 3(c) and 3(d) and Figs. 4(c) and 4(d) is possible because the color-scale used is the same. The values of the recorded field are shown in Fig. 7 below.

 

Fig. 4 2D map of electric field (a) and potential (b) across a fiber poled in a two-anode (+/+) configuration when the field recorded at the center of the core reaches its maximum (8800 s). When the 5 kV voltage bias is switched off, the electric field distribution (c) and the potential (d) change significantly. The maximum recorded field at the center of the core after the process is completed is E_rec = 3.1 × 10⁸ V/m .

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6. Dynamics for various cations

The model allows studying the dynamics of poling in the presence of various cations. Such a study implemented for the geometry of fiber 2 is illustrated in Fig. 5 for the conventional poling configuration (anode on the left) and in Fig. 6 for the two-anode arrangement. The same time-dependence plot of the field-induced second-order optical nonlinearity (χ⁽²⁾) during poling determined at the core center is shown on a linear-scale in minutes (left) and on a log-scale with the time expressed in seconds (right). The value of χ⁽³⁾ is assumed to be invariant and equal to 2.0 × 10⁻²² m²/V² . Six ionic combinations are examined. Referring to Fig. 5, the effective nonlinearity is non-zero already at t = 0 because of the large applied poling field.

 

Fig. 5 Simulation of the dynamics of poling associated to different cations in a fiber poled with internal electrodes on a linear time-scale (left) and a log time-scale (right). The apparent second-order nonlinearity here is determined at the core center and in the presence of the HV poling bias, i.e., it is partly induced by the applied voltage. The poling voltage assumed is 5 kV and the temperature 265 °C.

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Fig. 6 Simulation of the dynamics of electrostatic charging with two anodes associated to different cations in a fiber with internal electrodes on a linear time-scale (left) and on a log time-scale (right). The apparent second-order nonlinearity here is determined at the core center and in the presence of the HV poling bias.

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In the presence of 0.35 ppm Na⁺ alone (red trace), the slight reduction of the apparent nonlinearity seen between 2 and 3 min comes from the initial creation of the high-resistivity Na⁺-depleted layer around the anode, which initially reduces the bias field at the core center. The depletion region expands to 15 µm within ~1 min, after which the nonlinearity becomes stationary at ~0.19 pm/V. In this situation, all applied voltage falls across the depletion region. Since most unpaired electrons are on the right side of the core, when the poling voltage bias is switched off the field recorded in the middle of the core is positive, i.e., points from the anode to the cathode.

It is seen that when the slow cation H₃O⁺ is also available (orange trace), the recorded nonlinearity can grow slightly, above the value obtained with Na⁺ (or Li⁺) alone. This takes place on a longer time-scale [16]. If only 0.35 ppm Li⁺ are available (blue trace), the formation of the depletion region is slower than with Na⁺ alone. The onset of the recorded high field happens after 3.5 min and the maximum field is developed after ~7 minutes. When Na⁺ and Li⁺ are available and the total cation content is 0.7 ppm (grey trace), the recorded nonlinearity does not increase from the one-carrier case. Here, the addition of the second slower ion results in lower nonlinearity because the pile-up region neutralizes a large fraction of the unpaired electrons that establish the field across the core. Besides, like in the presence of all three ions (black trace), the growth of the recorded field as a function of time is no longer nearly monotonic, as seen between 2.5 and 3.0 min. The passage of the pile-up region by the core center can cause a momentary reduction of the field. This “bump” in the nonlinearity curve happens later if lower mobility ions are involved. Finally, in the case of the three mobile ions (again black trace), for longer poling times (>20 min), the diffusion of H₃O⁺ first causes an increase in effective nonlinearity, when the electrons trapped between the anode and the core are neutralized. Later (>70 min), the fiber becomes overpoled, i.e., H₃O⁺ neutralizes the electrons on the right of the core, and this causes a reduction in the recorded field.

From the results shown in Fig. 5 for fiber 2, the dynamics for < 0.5 minute is dominated by the displacement of Na⁺, for poling times ~1-10 minutes by the movement of Li⁺, and for times in the neighborhood of 1 hour by the slow cation H₃O⁺. However, in the presence of more than one ion, the growth of the nonlinearity starts with the movement of the fastest ion and ends when the slowest reaches the center of the core.

Simulations were also carried out for various concentrations of Na⁺ only. As mentioned above, for 0.35 ppm the final recorded field is positive, pointing from the anode to the cathode. For 2 ppm, the depletion region formed is much thinner, only 6.5 µm wide. In this case, since most trapped electrons are located on the left-side of the core, after the poling voltage is switched-off the resulting field at the core center is negative, pointing from the cathode to the anode. For a concentration 0.5 ppm Na⁺, the depletion region at steady-state has length 13 µm. The amount of unpaired electrons located on the left and on the right of the core center is approximately the same, and the recorded field when the poling bias is switched-off is nearly zero. This example illustrates how poling of fibers is sensitive to the various parameters – here the ionic concentration – in an unparalleled way compared to poling of bulk samples. The outcome of poling in fibers can vary greatly for an apparently small change in the experimental conditions.

The black curve shown in Fig. 5 describes qualitatively the experimental results from groups that measured the evolution of fiber poling in real time (e.g., [18]), keeping in mind that the amplitude and time of occurrence of various features depend on the relative amounts of cations and fiber geometry. One can identify the initial fast growth due to Na⁺ (sometimes) with a bump due to Li⁺, the local saturation when the Li⁺ pile-up region passes the core, a long and weaker rise when H₃O⁺ starts to neutralize the depletion region and finally the subsequent the fall when H₃O⁺ starts to cancel the electrons past the core.

The simulation of the dynamics of electrostatic charging of the fiber with two anodes is shown in Fig. 6. Many of the features seen are similar to the case of (+/−) poling discussed above. Here, the applied field is initially weak and correspondingly, the effective nonlinearity is close to zero at t = 0. The low value of the initial field also causes an extra delay in the dynamics of formation of the depletion region. For example, with Na⁺ alone, the field reaches a maximum only after 4 minutes, and with Li⁺ alone, after ~10.5 minutes. Overpoling here is accentuated, since the unpaired electrons are neutralized by H₃O⁺ coming from both fiber holes.

Figure 7 shows the simulation of the time evolution of the recorded field for fibers 1 and 2 and for both configurations (+/− and +/+). The voltage is switched off the the nonlinearity reaches its maximum, resulting in a step-like reduction of the field. The cation levels and mobility values are those listed in section 2.

 

Fig. 7 Time evolution of the absolute value of the recorded field at the centre of the core for fiber 1 (left) and fiber 2 (right). The bias is 5 kV and the temperature 265 °C. The cation concentrations are those in section 2, and the mobility values from ref [17]. Both configurations are studied, (+/−) in blue and (+/+) in red. At the end of poling, the fiber is assumed to be cooled off and then the voltage switched off, when the residual electric field becomes constant. The onset of the nonlinearity in fiber 1 is slow because of the long distance from the electrodes to the core.

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The poling temperature is 265 °C and the bias voltage 5 kV. While the field increases significantly after <10 minutes in fiber 2, due to the larger core-hole separation of fiber 1 this growth only occurs after 90 minutes here. For both fiber designs, when the process is interrupted the field drops only marginally in the (+/+) case and almost halves in the (+/−) case. These two final values are those illustrated in Figs. 4(c) and 5(c) above. It is worthwhile noticing that for the (+/−) case the recorded field after the bias V_appl is removed is E_rec ~V_appl/D. For the (+/+) case, the recorded field is E_rec ~2V_appl/D.

7. Experimental validation

In order to validate the model here we performed a number of experiments. Figure 8 shows a qualitative identification of the various rings associated with Na⁺, Li⁺ and H₃O⁺ in a piece of fiber electrostatically charged at 4.3 kV for 33 minutes with two AuSn internal electrodes. The temperature of the hot-plate used here was 255 °C. The use of three carriers in the model is motivated by many similar pictures where three distinct anular regions can be identified around the anode electrodes.

 

Fig. 8 Qualitative identification of the rings associated with the three cations considered in the model. After electrostatic charging at 4.3 kV, the sample was cleaved and etched for 45 seconds in 40% HF and photographed with a phase-contrast microscope with 500 × magnification. In consonance with the model here, the red arrows indicate the boundary of the depletion region of Na⁺, the turquoise arrows point to the edge of the accumulation region of Li⁺ and the yellow arrows show the extent of the region with injection of H₃O⁺.

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Figure 9 shows a quantitative comparison between the numerical simulations for the recorded electric field at different times and experimental measurements. For the experimental data, pieces of fiber F2 are electrostatically charged each for a different time interval at a 5 kV bias and a temperature of 265 °C. Every experimental data point is obtained by measuring the voltage-induced phase-shift with a ± 2.3 kV voltage ramp [22]. The phase-shift vs applied voltage graph is the Kerr parabola, which gives the effective χ⁽²⁾, the recorded field E_rec and χ⁽³⁾. The value of χ⁽³⁾ increases in these experiments up to 35%, from 2.0 × 10⁻²² m²/V² to 2.7 × 10⁻²² m²/V², as reported by several groups previously [18, 35, 36]. The experimental electric field values used in the comparison are affected by this variation in χ⁽³⁾, since E_rec = χ⁽²⁾_eff/ 3χ⁽³⁾ . Including the modification in χ⁽³⁾ value here where the data are available gives a better estimate of the field recorded than using a fixed value 2.0 × 10⁻²² m²/V².

 

Fig. 9 Comparison of experimental data to the results of the simulation for fiber F2. Individual pieces of fiber are electrostatic charged at 5 kV and 265 °C.

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For the simulation, the ionic concentration values used are c_Na+ = 0.7 ppm and c_Li+ = 0.1 ppm. The hydronium species is injected from a finite supply with a maximum concentration c_H3O+ = 0.7 ppm. For the black curve in Fig. 10, the mobility values are μ_Na+ = 2.0 × 10⁻¹⁵ m²/Vs,µ_Li+ = 0.3 × 10⁻¹⁵ m²/Vs and µ_H3O+ = 3.0 × 10⁻¹⁸ m²/Vs.The fit to the experimental data is good. Figure 10 also shows the sensitivity of the fitting to the assumed activation energy. A ± 5% uncertainty (red and blue traces) creates a large deviation in the onset of poling, and it is apparent that the error in the value of activation energy used here (H = 1.237 eV) is small, in the range ± 0.01 eV.

 

Fig. 10 Simulation of the time evolution of the effective nonlinearity for different applied voltages for (a) the poling configuration; (b) electrostatic charging; and (c) bias voltage dependence of the value of the nonlinearity recorded after the voltage is switched off. The recorded field is the residual field when the voltage is switched off at the maximum of poling/electrostatic charging, when the field becomes constant in time.

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8. Variational study

Having implemented the 2D model to describe poling with Na⁺, Li⁺ and H₃O⁺ as mobile species, one can now carry out a variational analysis of the impact of various parameters on the resulting recorded nonlinearity. The dependence on poling voltage, ion concentration and poling temperature are studied below.

8.1. Dependence on applied voltage

There are a few reports in the literature on the use of higher poling bias to increase the recorded field in bulk samples. The field recorded in isolators with a single mobile ion is calculated to be proportional to the square-root of the applied bias [37]. The second harmonic intensity in bulk samples was found to depend on the square of the poling voltage [38] Additionally, it was remarked in [12] that an increased effective second-order nonlinearity could be induced by poling at a lower bias and at the end of process increasing the voltage. Although only increasing the voltage was not found to lead to higher χ⁽²⁾_eff in bulk samples, when both temperature and voltage were raised, the resulting χ⁽²⁾ was increased [11]. The configuration with two anodes was not studied by those groups.

The simulations here are performed with the doping concentrations: 1 ppm Na⁺, 1 ppm Li⁺ and up to 2 ppm H₃O⁺, and temperature 265 °C. The bias voltage is assumed to be 2-10 kV. Possible electrical breakdown limitations are not considered and the results are illustrated in Fig. 10.

It is found that the value of the bias voltage makes a significant difference to the maximum value of the recorded field. In the range considered here, this increase exceeds the square root dependence of a single ion model and is approximately linear. The time taken to reach the maximum χ⁽²⁾ is voltage dependent and follows a 1/V_appl dependence.

The dynamics of poling is sensitive to the ionic concentration. Figure 11 illustrates the evolution of the χ⁽²⁾ as a function of time when the concentration of all ions is halved from that considered in Fig. 10. In the presence of less ions, quite distinct curves are obtained at lower and higher voltage bias. Point B in the nonlinearity growth in time (see Fig. 5) is reached earlier for higher voltages. The increase in χ⁽²⁾ obtained is smaller as the bias is increased, i.e. a saturation is observed. This is likely to be the regime investigated experimentally by the Stanford group for bulk samples [11]. With even lower ionic concentration, increasing the poling voltage alone does not increase the nonlinearity. The number of available charges would have to increase too.

 

Fig. 11 Simulation of the time evolution of the effective nonlinearity for different applied voltages when the carrier concentration is halved from that in Fig. 10.

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8.2. Dependence on ion concentration

The graphs below, Fig. 12, illustrate the result of changing the concentration of doping ions for a constant bias voltage. As above, the relative ratios of ion concentrations is kept constant (N_Na = N_Li = N_H3O/2). The poling voltage assumed here is kept constant at 5 kV and the temperature 265 °C. From the three traces of the graph on the left one sees that although the increase in recorded χ⁽²⁾ becomes more rapid at higher concentrations, the absolute value of the field recorded and of the nonlinearity saturates, as reported in the literature [1]. “Very clean” synthetic silica or Suprasil produces a weaker nonlinearity than other types of silica with higher dopant concentrations. Here, without adjusting any parameters except the concentration level, it is found that below about 0.2 ppm of Na⁺ the recorded field becomes very weak, but for values higher than 1 ppm, it does not help to have more doping ions.

 

Fig. 12 Simulation of the time evolution of the effective nonlinearity for different carrier concentrations. The χ⁽²⁾ induced saturates for higher concentrations (~0.5-1 ppm).

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8.3. Dependence on temperature

Increasing the poling temperature has potentially two effects, that of raising the ionic mobility and of increasing the number of free species. With higher mobility, the faster moving ions should indeed accelerate the process of poling, something easily observed experimentally. Besides, since the field is recorded as a result of the ionic displacement, one could intuitively expect that more displaced charges could lead to a larger recorded field. From experimental studies in bulk [8] and in fibers [39], it is known that the nonlinearity induced by thermal poling is lower for temperatures below 250 °C and above 280 °C. One can estimate the increase in carrier concentration with temperature, by comparing the increase in electrical conductivity with the increase in mobility as the temperature is raised. For silica GE214 with contamination levels similar to the preforms used for fiber fabrication, the activation energy for electrical conduction is E_c = 29.9 kcal/mol, i.e., 1.297 eV [40] and the ratio between conductivities is calculated to be σ (280 °C)/ σ (265 °C) = 2.13. The ratio of mobilities for Na⁺ at 280 °C and at 265 °C is µ_Na+(280 °C)/ µ_Na+(265 °C) = 2.03. Therefore, when the temperature of the fiber is raised from 265 °C to 280 °C, the glass conductivity increases 2.13 times, while the mobility increases only 2.03 times. This difference may be assigned to an increase in carrier concentration of ~4.8%. A similar procedure for a raise from 250 °C to 265 °C gives a calculated increase in concentration of ~6.3%. However, the uncertainty in the value of activation energy in the literature is large, nontheless because each glass sample is different from another. The minor (~10%) increase in cation concentration estimated when raising the temperature from 250 °C to 280 °C is well within the uncertainty of the calculation. Consequently, in the simulation below it is assumed that the concentration of Na⁺ ions is constant when the temperature is raised in this limited range.

The result illustrated in Fig. 13 of the simulations of electrostatic charging with 5 kV bias at 250 °C, 265 °C and 280 °C with 1 ppm Na⁺, Li⁺ and a maximum 2 ppm H₃O⁺ shows that the optimum poling interval can be reduced 3.6 times by increasing the temperature in this range, consistent with numerous experimental studies (see for instance [39],).

 

Fig. 13 Simulation of the time evolution of the effective nonlinearity for different temperatures. The time can be shortened with a temperature increase but the nonlinearity induced does not change if the carrier concentrations are assumed to be constant.

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9. Conclusions

The dynamics of poling (and electrooptical charging) was investigated in this paper for the specific geometry of optical fibers with internal electrodes. The tool used was a two-dimensional model based on the solution of the transport of diluted species in the COMSOL platform, which takes into account the movement of Na⁺, Li⁺ and H₃O⁺ ions, believed to be important in the process of poling of silicate fibers. The signature of each cation is identified and the time evolution of the nonlinearity recorded is related to the movement of each cation. The simulations are validated against experimental data on the growth in time of the nonlinear coefficient induced. The model allows performing a variational analysis of the importance of various parameters to optimize poling of fibers and confirms the usefulness in increasing the bias voltage during poling. It agrees with reports for bulk samples that show that the nonlinear coefficient induced saturates for cation concentrations in the range of 1 ppm [1], and that the main effect of temperature in the range 250-280 °C is to reduce the poling time, not of increasing the χ⁽²⁾ [11].

Some of the limitations of the model here are that (1) the exact concentration and initial distribution of various cations is unknown; (2) the values of activation energy and mobility of various ions available from the literature has an uncertainty that affects the simulations; (3) the barrier created at the core-cladding interface makes it more difficult for ions to enter and exit the core, but this is neglected; (4) the initial nonuniform distribution of species containing hydrogen, such as H₂O and H₃O⁺ is neglected; (5) the nonuniform distribution in the value of χ⁽³⁾ is neglected; (6) the junction between metal electrode and glass is implicitly assumed to be ohmic for both polarities and independent of bias voltage. These simplifications in the model may obscure new ways of increasing the nonlinearity, for example with multi-layers [41], and should be dealt with in a more complete description of poling.

Although it is assumed that cations reaching the cladding-air interface recombine, simulations show that accumulation does not affect the results dramatically for the fiber geometries considered here. Also, simulations show that in most cases the field at the anode-glass boundary reaches a limiting value ~10⁹ V/m in a few minutes and becomes relatively constant. Since the injection of H₃O⁺ only becomes important for longer poling times, it is found here that the field-dependence of hydrogen injection does not give significantly different results from injection at a constant rate.

One can consider optimization of the linear electrooptic effect induced in fibers poled with two internal electrodes. It is assumed that the diameter of the fiber core and its minimum distance to the nearest electrode define a minimum value for the distance D between electrodes. In the case of electrostatic charging, most of the final trapped electrons are located between the region with pile-up of Li⁺ ions and the region neutralized by H₃O⁺, near the midpoint between electrodes, i.e., D/2 away from the holes. Optimization then implies in recording all the applied bias across half the separation between electrodes. This can mathematically be expressed as E_{max rec} = V_appl/(D/2). For example, for fiber F1, one obtains that under optimal conditions of electrostatic charging with two anodes the maximum recorded field is E_{max rec} = 5 kV / 14.2 µm = 3.5 x 10⁸ V/m. Since the field recorded is typically in excess of 3 x 10⁸ V/m, it can be concluded that presently the conditions are already close to optimal. It is not expected that changing the poling temperature or ionic concentrations one could increase significantly the recorded field, and consequently the value of χ⁽²⁾. In order to increase the recorded field, one could conceivably reduce the separation D between the holes, but this brings about a large increase in the optical loss, if the application requires the use of electrodes. The conclusion is then that efforts should be put in increasing the bias voltage during electrostatic charging. The configuration is compatible with high voltage bias because the two internal electrodes should be at the same potential, minimizing the risk for breakdown. The difficulty lies in preventing breakdown to the outside of the fibre, while effectively grounding its outer surface.

When the fiber is poled with an anode and a cathode, the qualitative conclusion is similar although the amount of charges displaced is half from the electrostatic charging case, i.e., E_{max rec} = V_appl/D. The main strategy for increasing the recorded field is also to increase the the voltage bias.