**1. Introduction**

Since their early implementation in the 1920s [1,2] and subsequent optimization in the 1970s [3] optical fibers have become the most widespread technological platform for telecommunications, mainly due to their relatively low losses and huge bandwidths which greatly exceed the performances of any other system for the transmission of information [4]. For example, it is possible to dope optical fibers with rare earth ions such as erbium to obtain optical amplifiers [5], with ytterbium or neodymium to create fiber lasers, or to embed Bragg grating mirrors and filters into them [6].

Optical fibers are typically exploited as a reliable technology for non-linear photonic devices based on their higher order intrinsic non-linear susceptibility χ(3). This, by definition, requires high laser pump intensities and appropriate phase matching conditions to operate efficiently. Third harmonic generation (THG), optical Kerr effect, self-focusing, intensity dependent refractive index, four-wave mixing (FWM) are some of these χ(3) -related effects exploited in all-fiber non-linear devices such as, for example, supercontinuum sources [7]. Nevertheless, the absence of intrinsic second order properties in centrosymmetric materials, such as silicate glasses, does not in the first instance allow for their exploitation in creating parametric effects related to this lower order optical non-linearity [8].

However, in 1991 Myers et al. developed a technique, called thermal poling [9], to permanently create effective second-order susceptibility χ(2) *eff* inside glasses. The method consists in the concomitant heating process of a piece of glass and application of a relatively high static electric field through it. When the glass reaches the temperature where some alkali impurity ions (already included inside the glass matrix) have a non-negligible diffusion and drift mobility, the alkali ions start to electromigrate consequently forming a static electric field which is later frozen-in the glass after it is cooled down and the external electric field is removed. The thermal poling technique, at first adopted for bulk glasses, was later used for optical fibers [10] with the main motivation of overcoming some of the issues typical of the classical approach for the realization of non-linear optical devices, based on the interaction between intense light beams and non-linear crystals (such as for example lithium triborate (LBO),

beta-barium borate (BBO) or lithium niobate (LiNbO3)). These issues can include thermal instabilities of non-linear crystals when illuminated by very high pump powers [11,12], relatively short interaction lengths between light waves involved in the non-linear process, high costs and low damage thresholds of the non-linear crystals and coupling losses due to the presence of air/non-linear crystal interfaces as well as the onerous requirement for continuous optical alignment necessary in free-space optical setup. The appeal represented by the idea of a new technological platform for the realization of efficient and all-fiber non-linear devices produced a significant scientific effort towards the complete exploitation of the thermal poling technique. Since its first appearance, many papers have been published where continuous improvements of the experimental thermal poling technique are presented. In this work we focus our attention on the chronological development of the theoretical models implemented in the last 25 years to explain the glass chemistry and physics behind thermal poling, with the final aim of shedding more light on the mechanisms involved in the creation of the second order non-linearity and ideally understanding how eventually to push the features of the technique beyond its current limits.

#### **2. Early Evidence of Second-Order Non-Linearities in Silica Fibers and Thermal Poling**

An amorphous dielectric medium can be considered macroscopically isotropic and centrosymmetric and consequently invariant by parity inversion [13]. This means that a glass, as an amorphous medium, lacks any second order non-linear susceptibility χ(2) in the electric dipole approximation, because of the parity invariance [8]. For example, silica optical fibers possess a zero χ(2) as evidenced by the absence of any quadratic non-linear effect. However, in the 1980s some quadratic nonlinear phenomena were observed in silica optical fibers excited by the radiation generated by high power lasers, for example, the generation of wavelengths corresponding to sum-frequency radiation. The source adopted was a Q-switched and mode-locked neodymium-doped yttrum aluminum garnet (Nd:YAG) laser at 1.064 μm while the sum-frequency light was generated mixing the light of fundamental wavelength and the Stokes wavelengths generated via Raman inelastic scattering [14–16]. It is due to the work of Gabriagues et al. the first ever reported observation of second harmonic generation (SHG) in optical fibers [17], while a few years later Osterberg et al. studied the SHG process produced in a silica fiber with laser pulses characterized by a time duration of 100-130 psec and peak power of 70 kW. They observed that, after constantly illuminating a silica fiber, some SH light was collected at its output and the intensity of the light generated grew after a certain time [18,19]. In a later work, it was reported the possibility of reducing this "preparation" time from hours to minutes by illuminating the fiber not only with fundamental wavelength, but also with the SH one [20].

The SHG produced in optical fibers was explained in two different ways. Farries et al. considered that the existence of a non-linear electric quadrupole susceptibility causes the generation of a feeble SH radiation when elevated intensities of the pump light are used [21]. This process produces the formation of color centers (created where fundamental and SH radiation are in phase) in an axially periodic arrangement [22]. Stolen et al., instead, attribute the SHG to a sort of photoinduced phenomenon forming the χ(2). Basically, they assume that the origin of the SHG process is the creation of a DC polarization due to the mix of the fundamental and the SH wavelength (already present inside the fiber or even fed from outside). The polarization is characterized by a certain periodicity and is capable of orienting defects and consequently create a phasematched χ(2) [20].

Finally, Kashyap created an experimental setup to produce phase-matched electric field-induced second harmonic (EFISH) in single-mode Germania-doped silica fibers [23] by applying a periodic electric field across the core of an optical fiber. The static field created in the fiber's core generates a periodic <sup>χ</sup>(2) <sup>∝</sup> <sup>χ</sup>(3)*E*. It was possible to tune the period of the electric field simply rotating the electrode by an appropriate angle.

As previously discussed, significant permanent effective second order non-linear susceptibility (≈1 pm/V) in centrosymmetric media such as bulk silica glass was demonstrated by Myers et al. [9]. The technique is defined thermal poling and consists in the application of high electric potentials (3–5 kV) through a piece (thickness of 1.6 mm) of fused silica kept at a temperature between 250

and 325 ◦C for a temporal interval in the range 15–120 min. After the heating phase, the glass is cooled down to room temperature while the voltage is still maintained. The non-linearity is created permanently only in the first few microns of the sample close to the surface where the anodic potential is applied. The χ(2) <sup>33</sup> value for fused silica was found to be 20% of the typical value of the <sup>χ</sup>(2) <sup>22</sup> measured for LiNbO3. A relevant experimental result obtained by Myers et al. is the strict relationship between the value of χ(2) obtained and the concentration of the impurities present in the glass the fiber is made of. This observation suggested that the presence of the impurities is of critical importance to make thermal poling an efficient process.

## *2.1. First Theoretical Explanation of Thermal Poling: Single-Carrier Model*

In 1994 Mukherjee et al. presented a first model to explain the thermal poling process dynamics [24]. The model is based on transport of ionic species together with bonds reorientation and states that, by applying an external electric field, the impurity ions already included in the glass matrix create locally static electric fields capable to orient the bonds (related to impurities or Si-O bonds). The induced χ(2) is expressed by:

$$
\chi^{(2)} \approx \chi^{(3)} E\_{\rm DC} + \frac{np\beta}{5k\_B T} E\_{\rm DC} \tag{1}
$$

where χ(3)*EDC* is the term representing the optical rectification process of third order and *EDC* is the local field due to the non-uniform charge distribution. The other term of Equation (1) represents the electric-field-induced orientation of the molecular second order hyperpolarizability β, with *kB* the Boltzmann constant, *T* the absolute temperature of the sample, *p* the permanent dipole moment associated with the bond, *N* the number of dipoles involved in the process and a uniaxial molecular system is assumed for the sake of simplicity (the direction of *EDC* is fixed). Mukherjee et al. introduced for the first time the concept of depletion region formation. The latter consists in the creation of a space-charge zone, situated in proximity of the anodic electrode, which is emptied of impurities. This portion of the glass includes the negatively charged non-bridging oxygen (NBO−) centers. By applying high electric fields at high temperature it is possible to move away the ions originally electrostatically linked to them. Applying the Poisson's equation it is possible to obtain the electric field in the depletion region, which is given by [24]:

$$E\_{DC} = \frac{qn}{\varepsilon}(a - \mathbf{x}),\ 0 < \mathbf{x} < a,\ \mathbf{a} = \left(\frac{2\varepsilon V}{qn}\right)^{\frac{1}{2}}\tag{2}$$

where *a* is the depletion width, *n* is the concentration of ionized impurities, *q* is the magnitude of the electronic charge, ε the dielectric constant and *V* the difference of potential externally applied between the two electrodes. This result has been obtained assuming that the depletion layer width on the anodic side is much greater than the corresponding cathode accumulation layer. This model is based on the assumption that there is only one type of carrier involved in the formation of the depletion region, but a later work of Alley et al. [25] highlighted a series of experimental observations which are incompatible with the single-carrier model, including in particular the observation of multiple time scales for the poling, and the dependence of the non-linearity on the sample thermal poling history.

#### *2.2. Multiple-Carrier Model for Space-Charge Region Formation*

Although the early experimental results seemed to confirm the formation of a negatively charged region underneath the anodic surface of the bulk silica sample [26], as predicted by the single carrier model described in the Section 2.1, other observations indicated that in a thermal poling process of silica there is something more complex than a simple uniformly negatively charged region. In particular, Kazansky et al. found regions of alternating charge below the anode [27] while Myers et al. found that the depth of the non-linearity generated in poled bulk samples was greater for samples poled for 2 h than for 15 min [9]. If the depletion region was a uniformly negatively charged region and the

electric field frozen into the glass was expressed by Equation (2), according to the Equation (1), the χ(2) induced would be peaked in the region closest to the anode and not at a certain distance as commonly observed in the early poling experiments [28].

After the work of Alley et al. [25], published in 1998, where the important experimental observation of multiple time scales for the formation of the SH signal was reported, in 2005 Kudlinski et al. realized a more exhaustive description of space-charge region formation and induced second order non-linearity in bulk silica glasses [29]. The samples used in their work were disks of fused silica (InfrasilTM) characterized by the presence of some types of impurity carriers (typically Na+, Li+, K+, ...) located in the glass matrix with a concentration value of 1 ppm. Disks of different thickness, sandwiched between two Si electrodes, were heated at 250 ◦C, and poled at 4 kV for different temporal durations. When the impurity carriers become mobile, a high electric field is applied through them, producing their electromigration toward the cathode of the system. As a consequence, a negative space charge is created underneath the anodic surface, due to the fact that negative charges are motionless in the glass matrix (NBO− centers). A huge electric field similar to the dielectric breakdown field is consequently established within the depletion region and a second order susceptibility is then created:

$$
\chi^{(2)} = \mathfrak{Z}\chi^{(3)} E\_{\mathcal{DC}} \tag{3}
$$

where we are assuming a system unidimensional with the electric fields involved all linearly polarized along the same direction for the sake of simplicity. For the first few seconds of the electromigration process the single-carrier model can be still used to describe the time evolution of the depletion region formation [30], while after a certain time, defined optimal time (topt) [31], it is necessary to use a multiple carrier model to describe the temporal evolution of the poling process. If we consider the fast carriers (impurity charges) and the slow carriers (hydrogenated species) and both the migration and the diffusion phenomena, the equation of continuity and the Poisson's equation can be written as [29]:

$$\frac{\partial p\_i}{\partial t} = -\mu\_i \frac{\partial (p\_i E)}{\partial \mathbf{x}} + D\_i \frac{\partial^2 p\_i}{\partial \mathbf{x}^2} \tag{4}$$

$$\frac{\partial E}{\partial \mathbf{x}} = \frac{q}{\varepsilon} \left| \sum\_{i} (p\_i - p\_{0,i}) \right| \tag{5}$$

where *pi*, *p*0,*<sup>i</sup>* and μ*<sup>i</sup>* are respectively the instantaneous concentration (ions/m3), the initial (at t = 0) concentration and the mobility (at the temperature where the poling experiment is realized) of the i th species, *q* is the electron charge, ε = 3.8ε<sup>0</sup> is the permittivity of the medium and *Di* = *kBT*μ*i*/*q* is the diffusion constant of the ith species, with *kB* the Boltzmann constant and *T* the temperature of the medium. The system of Equations (4) and (5) gives the spatial distribution of the electric field in the sample as function of the poling duration. The assumptions related to the voltage applied are that the potential at the anodic surface (*x* = 0) is *Vapp*, while the potential at the cathodic surface (*x* = *l*) is zero. Therefore, the first boundary condition is:

$$\int\_{0}^{l} Edx = V\_{app} \tag{6}$$

While the impurity charges (such as Na+) are already present into the sample with the initial uniform concentration *p*0,*Na*<sup>+</sup> , the hydrogenated species possess an initial density *p*0,*H*<sup>+</sup> = 0 and are injected into the glass with an injection rate which depends on the electric field strength at the anodic surface. Therefore, the second boundary condition can be written as:

$$\left(\frac{\partial p\_{H^{+}}}{\partial t}\right)\_{x=0} = \sigma\_{H^{+}}E(x=0) \tag{7}$$

where σ*H*<sup>+</sup> is an adjustable parameter used to describe the charge injection into the glass of the hydrogenated species.

In order to describe the dynamical evolution of the space-charge region, we can assume that, as a consequence of the application of the voltage (*Vapp*) throughout the whole sample of length *l*, an electric field is created equal to *Vapp*/*l* and, because μ*Na*<sup>+</sup> μ*H*<sup>+</sup> , at first a depleted layer close to the anodic surface of the glass is formed due to the Na ions migrating toward the cathode. The induced electric field at the surface increases and screens the external electric field in the part of the sample placed outside the depletion region. When the space charge region is completely created, the maximum value of *EDC* <sup>≈</sup> 109*V*/*<sup>m</sup>* is obtained. At this time the concentration of the injected carriers per second increases rapidly to the value of 7.5 <sup>×</sup> <sup>10</sup>−22*m*−3*s*−<sup>1</sup> (according to the equation that governs the injection into the glass of the hydrogenated species, which affirms that the concentration of those species at the anodic surface is linearly proportional to the value of the electric field at the same surface). At the same time, the drift velocity of the injected hydrogenated species ν*H*<sup>+</sup> = μ*H*<sup>+</sup> *EDC* reaches the same order of magnitude of to the velocity of the Na ions, which are outside the depletion region, where the external electric field is reduced because it is screened by the formation of the space charge. For poling durations longer than few minutes, these injected ions move deeper and deeper into the glass replacing slowly the Na ions removed previously, consequently neutralizing the NBO− centers (refer to [29] and figures in that paper).
