In many applications and in a more realistic description of the electromagnetic interaction, the response of a medium immersed in an electromagnetic field cannot be described as a linear response:

(i.e., by considering that the dielectric susceptibility of the material is not dependent on the amplitude of the electric field). A more complete description of the electric polarization can be expressed in a general way as [17,21]:

Such dependence can be expanded in a power series of the electric field, and in a general case the electric polarization vector can be written as [17]:

where the nonlinear susceptivities χ⁽ⁿ⁾ are used (the χ⁽¹⁾ term is responsible for the linear refractive index). In centrosymmetric materials are present only the even term of Equation (3), otherwise in non-centrosymmetric materials the second order term χ⁽²⁾ can be dominant with respect to the other nonlinear terms.

We limit our attention only at the second order term that is responsible of the linear electro-optic (EO) effect (or Pockels effect) as well as other interesting phenomena and applications such as second harmonic generation (SHG), frequency mixing, photorefractivity [17,22].

In Equation (3) if we consider that the driving field oscillates at one frequency ω, the only possible effects in the polarization term P are second harmonic generation (where P oscillates at 2ω) and optical rectification (where the solution with P that oscillates at ω = 0 means that a static field results to be applied at the material when it is shined by the light).

If the exciting field is composed by an electromagnetic field oscillating at ω₁ = ω and an external static electric field that is applied to the material (ω₂ = 0), the resulting polarization P oscillates with the same frequency of the incident electromagnetic field, like in the case of the linear term χ⁽¹⁾, but the linear term responsible for the refractive index is modified by the presence of the static field:

This phenomenon is the EO effect of our interest.

Organic conjugated molecules and polymers show interesting and peculiar physical properties, given by electron delocalization of the π-bonds [1]. If we consider benzene (C₆H₆), it shows complete delocalization of π-electrons on the aromatic ring (Figure 1(a)). As we saw in the previous chapter, as benzene is a nonpolar molecule and it is centrosymmetric, it does not show any second order nonlinear contribution to the polarization term.

If we substitute some hydrogen atoms with an electron acceptor A (for example NO₂) or electron donor group D (for example NH₂) (Figure 1(b,c)), the benzene ring allows the electrical charges to flow from the donor to the acceptor, generating a dipole moment along the conjugation direction. We can expect that non-centrosymmetric molecules, with large mobility of charges will show a large degree of polarization and nonlinear optical terms, especially of the second order.

For a general molecule, using a description similar to that used in the previous paragraph for macroscopic nonlinear media, we can write down the expression of the induced dipole moment as:

where μ_0i are the static electric dipole moment components, α_ij are the polarizability tensor components, β_ijk and γ_ijkl are the components of the hyperpolarizabilities tensors of the first and second order, respectively, and finally F_i represent the i components of the local electric field that is related to the macroscopic electric field by the local field factor f^ω via the following relation:

The local field factor in the range of the optical frequencies is given by the Lorenz-Lorentz formula:

where n(ω) is the refractive index at ω. For static fields, the field factor is given by the Onsager expression:

where ε_r0 is the static relative dielectric constant and ε_r∞ is the relative dielectric constant in the range of optical frequencies [12,23].

One of the best known electro-optic chromophore used in applications is the 4-[N-ethyl-N-(2-hydroxyethyl)]amino-4’-nitroazobenzene (Disperse Red 1; DR1), whose structure is shown in Figure 2(a).

The main characteristics are reported in Table 1 [12,24,25,26,27,28,29]. A comparison between electrostatic units (esu) and International System units (SI), is given in [30].

Up to now, we described the nonlinear optical properties of a single molecule. However we are generally interested to the nonlinear optical response of bulk macroscopic materials made out of a large number of such molecules.

In order to evaluate the nonlinear response of the whole macroscopic material it is not sufficient to know the nonlinear properties of the single molecule, but we need to consider the number of chromophores per volume unit and their orientation inside the macroscopic structure. In particular, the macroscopic nonlinear optical properties increase with the density of oriented chromophores. Considering a system constituted by molecules with large β, if they are isotropically oriented in the space, the macroscopic second order dielectric susceptibility χ⁽²⁾ is zero because the material, as a whole, is centrosymmetric.

Since we are interested in electro-optic applications, we need to achieve a high χ⁽²⁾ value and consequently we need to fabricate a structure in which the nonlinear molecules are oriented in a polar order. One way is to realize a solid solution of highly nonlinear molecules in a high optical quality passive polymer matrix. The matrix, in order not to interfere with the optical properties of the chromophore, is in general transparent in the wavelength used for applications and can be considered as optically linear; it is just a host material that gives mechanical stability to the ensemble of nonlinear molecules. One of the most used polymer matrices is the poly-methyl-metha-acrylate (PMMA) [12] whose chemical structure is shown in Figure 2(b). Usually, the material is deposited onto different types of substrates by spin-coating, producing amorphous thick films (1~2 μm). The chromophores, randomly oriented at the end of the film deposition procedure, must be aligned in a non-centrosymmetric fashion after deposition by means of a poling procedure, as we shall describe later.

There are several ways of introducing nonlinear optical chromophores into the polymer matrix. The two most common methods are dissolving (guest-host systems) and covalently attaching (functionalized system) the chromophores to the polymer.

In the guest-host system, nonlinear optic (NLO) chromophores are dissolved, in the polymer host. The principal disadvantages of such a structure are: (i) low nonlinear activity, that it is due to the dopant maximum solubility (largest concentration is about 10% of the host weight), because greater concentrations can lead to chromophores’ aggregation or crystallization; (ii) the presence of optical losses due to scattering originated by the non homogeneity of the structure; (iii) the thermal and temporal instability, in fact, the chromophores are not attached at the polymer matrix and they can move with a higher degree of freedom. In the functionalized systems, the chromophores are attached to the polymer matrix, and the relative dopant concentration can be much higher than in guest-host systems. Moreover functionalized systems are more stable since their orientational mobility is significantly hindered and the order of such systems therefore relaxes more slowly.

In the functionalized systems nonlinear optical chromophores can be incorporated in the polymer matrix by chemically attaching them into the polymer main-chain (or backbone) or as a pendant side-group. In the side-chain systems, the chromophores can rotate around the attaching point and it is easier to orient them in the poling procedure. Nevertheless, the orientation is maintained longer than in the guest-host systems. For all these reasons, side-chain copolymers are widely used in optical application but also guest-host systems are used because of the relatively easy fabrication process.

In order to induce non-centrosymmetry after the films deposition, several poling techniques can be used. The most common ones are:

Contact Poling or Electrode Poling. Chromophores such as DR1 show a large static dipole moment μ₀, due to their rod shape with donor and acceptor at the opposite sides. As a result, it is possible to use a large static electric field to orient these polar molecules along the field and thus induce an asymmetric configuration. For contact poling, the sample must be prepared by spinning the polymeric solution on a substrate that was previously covered by an electrode. Then, after spinning and drying, the polymeric film must be topped with another electrode, via sputtering deposition or other techniques, in order to obtain a sandwich structure with the copolymer sandwiched between the two electrodes. Then, the sample is ready for the poling procedure; the temperature of the sample is increased to the glass transition one (T_g), where the polymer matrix reaches a large mobility. A large static electric field is then applied between the two electrodes for a sufficient time, in order to align the nonlinear molecules; then with the field still on, the temperature is slowly decreased to Room Temperature and the field is subsequently removed. At Room Temperature the orientational mobility of the molecules is sufficiently low to maintain the chromophores alignment for a long time and consequently giving rise to a long lasting electro-optic coefficient. With such a method it is not possible to apply a very large field, because unavoidable small imperfections in the polymer film can lead to a decrease of the dielectric rigidity of the material and, as a consequence to electrical discharge with complete damage of the sample.

Corona Poling. In such a technique [23] it is not necessary to apply electrodes on the polymeric material, as a consequence, it is possible for example to deposit, by spin coating, a polymeric film directly on a bare substrate. In the corona poling procedure the sample is put on a metal plate that plays simultaneously the role of heater and bottom electrode. The top electrode is given by a layer of ions which are deposited on the film surface. In order to produce the ions, a tungsten needle is placed near the surface of the polymer film at a potential of about 1–10 kV with respect to the substrate. The needle generates an intense electric field that ionizes the surrounding atmosphere (usually an inert gas, nitrogen, for example) and the generated ions are pushed on the film surface where they accumulate. As a consequence, the charges produce a quite uniform high intense field that orients the nonlinear dipoles. As in the case of electrode poling, the temperature is increased close to T_g for a suitable time interval during the poling procedure, then it is lowered to Room Temperature, still maintaining the voltage drop between needle and substrate. Corona poling is more efficient than electrode poling because it can generate very intense fields. Also during corona poling procedure, there is the possibility of having electrical discharge due to the high field, but in this case, since the ions are not mobile on the surface of the material, the discharges are localized and the sample does not damage the whole surface.

The method is, however, the subject of criticism for material characterization purposes: in fact, the real effective poling field remains unknown and it is not well reproducible.

Photoassisted Poling. In the previously described techniques it is necessary to pole the sample at a high temperature in order to have sufficient chromophore mobility. Nevertheless, at high temperatures, there is a competition between the alignment effect of the electric field and, on the other side, the thermal motion of the molecules. For high-T_g copolymers, poling samples with such techniques can lead to low efficiency. The advantage of photoassisted poling [31,32] is that this procedure can be executed also at room temperature. The sample is prepared as in the electrode poling, with the polymer sandwiched between two electrodes, but in this case at least one of the electrodes must be transparent. During poling, a voltage drop is applied between the two electrodes. Meanwhile, the sample is shined, from the transparent electrode side, by a light beam which wavelength is tuned in the region of the absorption resonance of the chromophore. In conjugated molecules with a N=N bridge along the principal axis, as DR1, the light, with a suitable polarization state, is absorbed by the chromophores and produces a trans-cis-trans isomerization (Figure 3). After several trans-cis-trans cycles, a net centrosymmetric orientation of the dipoles perpendicular to the pump light field can be achieved. With additionally applied dc electric poling fields, the process permits non-centrosymmetric orientation of chromophores giving rise to a macroscopic electro-optic coefficient even at room temperature.

In all the three cases described above, the electro-optic coefficient induced by poling will relax on a long-time scale, because the chromophores’ orientational mobility is not zero, even at room temperature.

In this paragraph, we show a theoretical model describing the nonlinear optical response of a medium constituted by nonlinear units (chromophores) dispersed in a linear matrix and oriented along a preferential axis [33,34]. We suppose that each molecule is identified by an intrinsic reference system with the axes named 1, 2, 3, and this system can be rotated in respect to the system that describes the whole macroscopic structure with the axes named x, y, z (laboratory axis). Let us consider that the orientation of the nonlinear molecules is described by a statistical angular distribution function f(Ω) = f(θ,φ) of the molecular axes, where θ and φ are shown in Figure 4. f(Ω) must fulfill the normalization condition . In the following we consider that z is the direction of the poling field and 3 is the direction of the long axis of the molecule.

The second order nonlinear dipole moment results in:

Considering the cylindrical and elongated shape of the chromophores, we can assume that the only nonvanishing component of β_ijk is β₃₃₃.

The macroscopic electro-optic polarization induced by an external dc field and by an optical field , both of them along the z direction, is:

where

and cos(,) is the) cosine of the angle formed by the two directions , .

In order to find another component of the macroscopic nonlinear optical susceptibility, we consider the case where the external dc field is along the z direction, and the external optical field is along the x direction, . In this case, the electro-optic susceptibility can be obtained by considering the induced polarization along the x direction:

From the above relations we obtain the following relations for the macroscopic electro-optic susceptibility:

During poling, two competing mechanisms determine the final steady-state equilibrium distribution. The electric dipole interaction of the molecular dipoles with the dc field favors the alignment of molecular dipoles along the z direction, while the thermal agitation tries to disorient the dipoles, leading to the maximum entropy configuration. For a sufficiently high temperature (even room temperature is high enough), the statistical distribution follows, as we said before, a Boltzmann distribution Equation (11): . The dipole interaction energy is given in the weak poling approximation by where with .

This expression allows to estimate directly the second order susceptibility terms from Equations (13) and (14):

The Equation (15) give, also, a first order approximation of the ratio, that results in:

By applying the relation between the electro-optic tensor with the tensor [16], it is possible to apply the previous relations (15–16) to the electro-optic coefficient too.

The calculations previously exposed neglect the chromophore–chromophore intermolecular electrostatic interactions. In this regime the electro-optic coefficient will increase in a linear manner with chromophore number density N (loading) in the host polymer matrix. More realistically, if the chromophore density is high enough, the reciprocal average distance between two nonlinear molecules, is so close to not allow neglecting of electrostatic interaction between the chromophores. Since the chromophores are polar molecules with a static dipole moment μ₀, they will interact, thereby minimizing their energy and forming an anti-ferroelectric state, where neighbouring chromophores are preferentially oriented in opposite directions, leading to a decrease of the effective electro-optic coefficient [27]. In order to avoid the negative interaction between chromophores, non-polar second order nonlinear molecule were proposed [35]. Unfortunately, in these molecules, since they have no μ₀, it is not possible to align them using the standard poling procedure shown previously. A more complicated all-optical poling was proposed [35].

At the end of the poling procedure, when temperature has been lowered to the ambient one and the poling field has been turned off, the orientational distribution f(Ω) is no longer at thermodynamic equilibrium and will relax via an orientational diffusion process, slowly decreasing the magnitude of the induced electro-optic properties. The distribution function is then time-dependent f(Ω, t) = f(θ, t). The same happens if we consider the inverse problem, when an isotropic material is subject, since a certain time, to a poling field.

In order to describe these time-dependent phenomena, we introduce the rotational diffusion equation (Smoluchowski-Einstein) for a molecular dipole [33]:

where D is a diffusion constant, related to the orientational mobility of the nonlinear dipoles in the polymer matrix.

However the behavior expected from the model showed above is not always in strong agreement with experimental data, because in the model, it is assumed that the material homogeneously acts on the mobility of the chromophores via a determined diffusion constant D. In reality, the polymer matrix is an amorphous inhomogeneous medium and each chromophore feels a different environment with a own diffusion constant D_i. The values of the diffusion constants are statistically distributed with a mean value and variance. For example in the guest-host system the chromophores manly feel two time constants [36], the faster one due to the rotation of the single chormophores around their original position, while the slowest one is due to the relaxation of the polymer host matrix. In side-chain polymers, usually we do not observe a single or double time constant exponential relaxation, but a stretched exponential with a homogeneous distribution of time constants with a given spread. In this case, in order to describe the time dependence of the electro-optic coefficient for an inhomogeneous matrix, the following expression is often used:

where r_zz(∞) is the steady-state electro-optic response, Δr_zz(E_p) is the induced electro-optic coefficient for a poling field E_p and Φ(t) is the Kohlrausch-Williams-Watts (KWW) stretched exponential function given by [37,38,39]:

Where β is a stretching parameter describing the deviation from the simple exponential law. The value of β is always between 0 and 1, where 0 represents a wide distribution of time constants and 1 represents a Dirac delta distribution leading to a single exponential relaxation, representative of the Debye model. For β ≠ 1, the τ time constant alone is not meaningful and its average must be taken into account:

where Γ(x) is the Euler gamma-function .

The average time constant <τ> depends on temperature with different laws:

For temperature above or equal to T₀, a characteristic temperature of the material that is close to, but lower than, the glass transition, the matrix is in a quasi-fluid state and the chromophores can move easily. In this case, the time constant follows the Vogel-Fulcher-Tamann-Hesse (VFTH) or Williams-Landel-Ferry (WLF) law that is typical of semi-fluid glassy systems:

where B is a suitable constant.

For temperatures below T₀, the average time constant follows an Arrhenius behavior:

where E_a is the activation energy of the process and R is the universal gas constant (R = K_BN_a = 8.31447 J·mol⁻¹·K⁻¹).

Also β depends on temperature, as it is related to dipole mobility. It was shown that its dependence on temperature can be heuristically described by the expression [31]:

Where β₀, T_β and A are suitable constants.

Nonlinear ellipsometry (NLE) is based on a single wavelength reflection configuration proposed, independently, by Teng and Man [13] and Schildkraut [14] in 1990. Since its introduction, the Teng and Man technique (TMT) has been widely used for the determination of the electro-optic susceptibility of organic polymer films and it demonstrated to be sensitive to a number of contributions to the measured signals that can lead to the evaluation of several nonlinear parameters [1,40,41]. The TMT permits the measurement of both and [42] complex susceptibilities. Moreover, it is sensitive to space distributions of charges trapped in the polymer films [1,16,43]. In order to be applied, the TMT needs to operate on samples in which the polymer film is sandwiched between two electrodes. In the case of polymers for χ⁽²⁾ applications, the two electrodes are used either to pole the film or to apply a modulating voltage on the polymer film itself in order to measure the χ⁽²⁾ values. As the TMT is based on the measurement of the electric field induced change of the optical phase difference experienced by the s (perpendicular to the incidence plane) and p (parallel to the incidence plane) components of a laser which propagates through the polymer film, at least one (reflection operation) of the electrodes must be transparent to the probe light.

Usually, a semiconductive transparent oxide thin layer, deposited on a glass substrate is used as transparent electrode. A polymer film is then deposited by spin coating it and then coating it with a metal electrode. Some researchers used polymer films sandwiched between two transparent electrodes and performed the experiments in a transmittance configuration [44,45]; in this case the samples were produced by spin coating pairs of polymer films onto oxide coated glass slides and then sandwiching them face to face; elimination of the air gap was guaranteed by baking at glass transition temperature under mechanical pressure.

The experimental configuration in reflection mode for TMT is usually composed by a cw laser that impinges on the sample at an incident angle α after passing through a polarizer, which sets its polarization at 45° with respect to the incidence plane, and a phase compensating device that changes the relative phase between the s and p components of the laser beam by an amount ψ_c. After reflection at the polymer-metal interface, the beam is analyzed by a polarizer that is crossed with respect to the input one. The power of the beam transmitted by the whole system is detected by a photodiode. Between the electrodes it is applied either a dc voltage V_p during the poling procedure, or a modulated voltage V(t) during the measurement of the electro-optic coefficient, given by:

The optical power at the output of the whole system is modulated, due to the response of the polymer film, both at Ω and 2Ω through the linear χ⁽²⁾ and quadratic χ⁽³⁾ electro-optic response.

The average output power can be written as:

where P₀ is the input beam power after the first polarizer, ψ_ps and ψ_c are the phase differences between s and p polarizations originated by the propagation of the light in the polymer and in the phase compensator, respectively. Φis defined by the ellipsometry relation

where and r_p and r_s are the Fresnel field reflection coefficients for the p and s polarization, respectively.

In the low birefringence approximation, with the assumption of Δ|r_s|² ≈ 0, the amplitude of the P_ac component at Ω is given by:

with in the weak poling approximation.

By varying the phase difference ψ_c introduced by the compensator from 0 to 2π, Equations (26) and (28) give the parametric equations of an ellipse.

There are two working points (labeled 1 and 2) where the P_dc reach half of its maximum value: . Those conditions can be found when: Ψ = Ψ_ps + Ψ_c = ± π/2.

By using a lock-in amplifier, it is possible to measure the effective P_ac signal in these two working points and then it is possible to retrieve the electro-optic coefficient by the relation:

We point out that the measurements in the two working points 1 and 2 are faster than the measurement based on the sampling of the whole ellipse, but suffer from lower accuracy. The faster measurements can be usefully adopted in monitoring the time decay of the electro-optic coefficient, as we will show later.

Usually, when analyzing experimental data in an ellipsometric measurement, one makes the assumption that the laser beam is passing just twice through the polymer layer. However, if the reflectance of the semiconductor electrode is not negligible at the measurement wavelength, the electrodes system can behave as a Fabry-Perot resonator and the evaluation of the electro-optic coefficient obtained via the application of the simplified single pass model is wrong [46]. As most researchers are using indium tin oxide (ITO) as the semitransparent electrode, this means that, unless a deeper theoretical approach is used [45,47,48,49], all the measurements obtained for wavelengths larger than about 1μm are affected by large errors due to the fact that ITO has got its plasma resonance in near-infrared and becomes reflective. Better results can be obtained in the near-infrared region by using aluminium doped zinc oxide (ZnO:Al) or, alternatively, more transparent electrodes [50,51].

Since the EO coefficient r is related with the second order susceptibility term χ⁽²⁾ (see Equation (14)) it is possible to evaluate it by measuring the efficiency of the second harmonic generation process in polymeric films. Second harmonic generation was the first all-optical phenomenon shown in an experiment in nonlinear optics [52]. The experiment consisted in focusing a ruby laser beam on a crystalline quartz plate and revealing the second harmonic with a prism spectrometer on a photographic plate. The conversion efficiency has greatly improved with respect to the first experiments and second harmonic generation is now ordinarily used in photonic devices and applications. Moreover, due to the fact that the χ⁽²⁾ coefficient is present only in non-centrosymmetric materials, SHG was widely used in order to study symmetry properties of bulk materials, thin films [53,54] or surfaces [55,56] with high sensitivity in terms of linear measurements [57,58].

Usually, the polymeric samples used in SHG measurements are film deposited by spin coating on glass substrates. In order to prevent influencing the measurement, the substrate was an amorphous glass slide (no χ⁽²⁾ contribution) without electrodes on the polymeric film. Since, after deposition, the orientation of the nonlinear chromophores is isotropic, the material must be poled before measurement. In this case, the poling technique used is the corona poling, as it does not require any contact electrode. An alternative poling procedure is the in-plane poling or lateral poling [59] where the electric field is applied between two coplanar electrodes that are deposited on the polymeric surface beside the region where the SHG measurements are performed. In the case of using an electrode for contact poling, the analysis of the multilayer structure should be performed as in [60].

A standard technique used in order to measure the electro-optic coefficient by SHG from poled samples is the Maker fringes scheme [18,19,20] where second harmonic signal is detected as a function of the incidence angle α. The fundamental beam at the angular frequency ω is generated by a pulsed laser that can be suitably polarized (e.g., in s or p polarization state); likewise an analyzer selects either s or p polarization for the signal generated at 2ω. The power generated by the sample can be calculated by means of the theoretical expression for the SHG power [19] given by:

where A is the area of the transverse section of the fundamental beam, Τ_2ω is the intensity transmittance of the output film/air interface at 2ω, t_ω is the field transmission coefficient of the input air/glass interface at ω (neglecting the glass/film interface due to the small refractive index contrast), W_ω represents the input power of the fundamental beam, n_ω(α) and n_2ω(α) are the polarization dependent refractive indices at ω and 2ω and d_eff is the effective nonlinearity. The phase factor Ψ(α) is given by:

L being the sample thickness, and the internal refraction angles corresponding to ω and 2ω.

In the Equation (30), d_eff(α) is the effective nonlinearity term depending on the tensor components, the optical axes orientation with respect to the sample surface and the fundamental beam incidence angle α, where the d tensor is is related with the χ⁽²⁾ tensor by the relation

The expression for d_eff(α) is generally rather complicated but it becomes significantly simplified for particular symmetries. Specifically, the tensor corresponding to a poled polymer presents only two nonzero different components, d₃₃ and d₃₁ = d₃₂ = d₂₄ = d₁₅. Depending on the polarization of both fundamental and generated beams, it is possible to retrieve separately the value of these components. The d₃₁ component can be directly evaluated from SHG measurements obtained in the s_ωp_2ω configuration, since in this case the expression of d_eff is given by:

Where α_ω is the fundamental beam propagation angle in the nonlinear film. The d₃₃ coefficient can be obtained from a p_ωp_2ω SHG measurement, by using the d₃₁ evaluated with an independent s_ωp_2ω measurement, since d_eff in the p_ωp_2ω case is given by:

Where α_2ω is the SH beam propagation angle in the nonlinear film.

The previous procedure can be used to find the susceptibility coefficient via the Equation (32). Then, in order to recover the electro-optic coefficient a modified expression of Equation (17) can be used [61,62]:

where the field factors Equations (7) and (8) are present. We point out that in the two-index simplified notation, d_ij is proportional to r_ji. SHG measurement evaluates independently d₃₁ and d₃₃, so the technique can be used to check the ratio and, indirectly, verify if the assumption r₃₃ ≈ 3r₁₃, used in the nonlinear ellipsometric technique is valid. By continuously measuring the SHG signal, it is possible to perform faster measurements that can be usefully used in monitoring the time decay of the electro-optic coefficient as we will show later.

In order to retrieve preliminary information on the stability of the induced electro-optic coefficient, temperature scanning measurements were performed. This type of measurement consists of the continuous measurement of the electro-optic coefficient of a poled sample while the temperature is increased from room temperature up to temperatures above T_g at a fixed rate, typically ranging between 1 °C/min to 5 °C/min.

During the temperature scanning procedure, the polymer electro-optic coefficient quickly relaxes and it is not stable enough to allow the use of the complete ellipsometric technique or the complete Maker fringe technique. In this case, the faster measurements shown previously (at the end of the Section 3.1, Section 3.2 paragraph) are preferred. The sample must be poled as usual and cooled down to room temperature, then the sample is gradually heated to the glass transition temperature (T_g), while the electro-optic coefficient is continuously monitored (by using either fast TMT or fast SHG). At the end of the measurements, the r₃₃ is evaluated as a function of the temperature. Typical graphs are given by normalizing the r₃₃ value with respect to the r₃₃ coefficient measured at the beginning in room temperature condition. As an example, Figure 5 reports the measurement performed on a fluorinated electro-optic polymer [62], the HFIP-DR1AF that we will investigate more extensively in the fourth chapter. It is clear that the electro-optic coefficient remains stable up to a certain temperature, then decreases down to zero when the sample is heated close to T_g. It is possible to define a depolarization temperature (T_dep) as the temperature at which r₃₃ gets half of the initial maximum value at room temperature. In this case, T_dep represents an indication of the stability of chromophore orientation in the polymer matrix.

It is evident that it is possible to compare different polymer types with this technique only if all the measurements are performed under the same conditions, i.e., with the same temperature scanning rate.

A more reliable and useful measurement technique to probe stability of the EO properties is the series of isothermal relaxation measurements. In this technique, the sample must be poled, as usual, at the poling temperature T_p for a time interval ∆t_p with an applied poling voltage V_p, then the sample is cooled down to the measurement temperature T_m maintaining the applied poling field. As soon as the T_m is reached, the poling field is switched off and the EO coefficient is continuously monitored by using either fast TMT or fast SHG measurements. In this case, the value of r₃₃ is evaluated as a function of time for the given temperature T_m. As an example, Figure 6 reports the isothermal measurements at 90 °C, 110 °C, 130 °C (in log scale in this example) performed on HFIP-DR1AF [62].

Aligning the behaviour of the EO coefficient with the Kohlrausch-Williams-Watts (KWW) stretched exponential law Equation (20), it is possible to obtain the value of the time constant τ, of the stretching constant β and, from Equation (21), the value of the average time constant <τ> at a given temperature.

By comparing the curves for different temperatures, it is possible to retrieve the dependence of βand <τ> as a function of temperature. The β dependence is usually described by the Equation (24), while <τ> is well described by the Equation (22) for temperatures close or greater than T₀, and by the Equation (23) for temperatures below T₀. In Figure 7, we report the <τ> values obtained in the previous measurements in logarithmic scale as a function of the reciprocal of the absolute temperature (multiplied by 1,000), as is usually done when describing temperature activated processes [62].

For temperatures below T₀ ≈ 110 °C, (1/T × 1,000 > 2.6 K⁻¹), the average time constant follows an Arrhenius behaviour Equation (23). The Arrhenius behavior is described by a straight line in such type of plot. an activation energy E_a = 58 kcal/mol is then obtained by fitting the measurements below 110 °C with the Equation (23). For temperatures above 110 °C the relaxation is faster than that predicted by the Arrhenius model and follows the Vogel-Fulcher-Tamann-Hesse (VFTH) behavior (or Williams-Landel-Ferry (WLF)) described in the Equation (22). The activation energy is a parameter that represents the stability of the chromophores orientation in a more complete way than the depolarization temperature reported in the previous paragraph. Moreover, the activation energy can be recovered by performing measurements at relatively high temperatures, and can be used to extrapolate time constants at a lower temperature (room temperature) that cannot easily be measured (because it would take years).

We point out that, for this purpose, it is necessary to be sure that the measurements are performed in the Arrhenius regime and not in the VFTH regime. This last observation shows that it is not possible to perform relaxation measurements at high temperatures, near T_g, (appealing for their convenience, as they are rather quick measurements) in order to extrapolate the relaxation time constants at a low temperature (e.g., 50 °C) by using an Arrhenius type model. From the activation energy E_a, by using Equation (23), it is possible to estimate another stability parameter, the one-year stability working temperature (T^1yr) that is the working temperature corresponding at an average time constant of one year.

The copolymer described here is a fluorinated side-chain copolymer, the HFIP-DR1AF [62], with Tg = 130 °C, whose structure is described in Figure 8(a). It is an evolution of the well-known poly(methylmethacrylate) bearing the DR1 chromophore (DR1PMMA), with structure shown in Figure 8(b), where a part of the C-H groups was substituted with C-F.

The synthesis of the copolymers was carried out with the hexafluoroisopropyl alphafluoroacrylate monomer (HFIPAF) and the alphafluoroacrylate monomer bearing the Disperse Red 1 (DR1) chromophore (DR1AF). The polymer present 46 % of DR1 substituted group, the electro-optic coefficient of the sample was measured at λ = 1,550 nm, resulting to be r₃₃ = (4.6 ± 0.5) pm/V, for a poling field of E_p = 43 V/μm.

Thermal stability of the poling induced electro-optic coefficient was measured by means of both temperature scanning and isothermal relaxation measurements as shown in Figure 5, Figure 6 and Figure 7. We scanned the temperature from the ambient one up to 140 °C, above the T_g, at the rate dT/dt = 5 °C/min. From Figure 5 it results a depolarization temperature T_dep = 124 °C, close to, but lower than, the polymer glass-transition temperature T_g.

We then performed isothermal relaxation measurements (Figure 6) for different temperatures ranging between 90 °C and 130 °C. The retrieved activation energy is E_A ≈ 58 kcal/mol.

The copolymer described here is a fluorinated side-chain copolymer, the FATRIFE-DR1AF [64] with T_g = 112 °C, whose structure is similar to the one described in Figure 8(a) except that the side group –CH(CF₃)₂ is substituted by the side group –CH₂CF₃. It is an evolution of DR1PMMA, with structure shown in Figure 8b, where a part of the C-H groups was substituted with C-F. The polymer constitute 30% of the DR1 substituted group; the electro-optic coefficient of the sample was measured at λ = 1550 nm, resulting to be r₃₃ = (4.7 ± 0.5) pm/V, for a poling field of E_p = 48 V/μm.

Thermal stability of the poling induced electro-optic coefficient was measured by means of both temperature scanning and isothermal relaxation measurements [64]. We scanned the temperature from the ambient one up to 140 °C, above the T_g, at the rate dT/dt = 5 °C/min. The resulting depolarization temperature T_dep = 114 °C, close to, but lower than, the polymer glass-transition temperature T_g.

We then performed isothermal relaxation measurements for different temperatures ranging between 60 °C and 110 °C. The retrieved activation energy is E_A ≈ 60 kcal/mol.

The copolymer described here is a fluorinated side-chain copolymer, the ADAMANTANE-DR1AF [11] with T_g = 148 °C. The synthesis of the copolymer was carried out with a molar fraction x = 73% of the α-fluoroacrylate monomer bearing a heavy stabilizing diamond-like side group, the adamantane (–CH₂-C₁₀H₁₅), and a molar fraction y = 27% of the α-fluoroacrylate monomer bearing the Disperse Red 1 (DR1) chromophore (DR1AF). The general structure is given in Figure 9(a). Also the ADAMANTANE-DR1AF is an evolution of the DR1PMMA (Figure 9(b)) [11], where a part of the C-H groups was substituted with C-F and with the adamantane as a supplementary side group.

The electro-optic coefficient of the sample fabricated with ADAMANTANE-DR1AF was measured at λ = 1,550 nm and it becomes r₃₃ = (2.8 ± 0.3) pm/V, for a poling field of E_p = 43 V/μm.

Thermal stability of the poling induced electro-optic coefficient was measured by means of both temperature scanning and isothermal relaxation measurements [11].

We scanned the temperature from the ambient one up to 150 °C, above the T_g, at a constant rate ∆T/∆t = 5 °C/min. In Figure 10, the results obtained for the ADAMANTANE-DR1AF copolymer is shown. From the measurements, the results show T_dep = 136 °C, close to, but lower than, the polymer glass-transition temperature T_g.

We then performed isothermal relaxation measurements on the ADAMANTANE-DR1AF sample for different temperatures ranging between 110 °C and 140 °C. In Figure 11 we show the isothermal relaxation of the EO coefficient normalized to the initial value r₃₃(t)/r₃₃(0), for the three different fluorinated compounds presented so far, performed at the same temperatures T_m= 110 °C. The increase of the thermo-stability of the ADAMANTANE-DR1AF copolymer with respect to the other species is evident.

In Figure 12, we plot the values of log<τ> vs. 1000/T for the ADAMANTANE-DR1AF copolymer. The graph shows that relaxation of the chromophores orientation can be described by the Arrhenius law with activation energy E_A≈ 105 kcal/mol. This result must be compared with the value E_A≈ 58 kcal/mol obtained for the HFIP-DR1AF [62] and E_A≈ 60 kcal/mol for the FATRIFE-DR1AF [64]). This high activation energy, together with the high T_dep, leads to a high thermo-stability of the EO properties even at a high operative temperature, compared with other similar fluorinated DR1 based compounds.

The copolymer described here [65] is a fluorinated side-chain copolymer formed by a backbone of polyimides, whose synthesis was carried out by following the procedure in [65] with a 43% concentration in weight of 2-[4’-(N-ethyl-N-2-hydroxyethyl)-amino-phenylazo]-5-nitrothiazole (EHNT) of the chromophore side group [65]. The EHNT chromophore show a larger second order hyperpolarizability with respect to the DR1 chromophore shown previously. The Polyimides-EHNT shows a very high glass transition temperature of T_g = 200 °C.

The electro-optic coefficient of a sample fabricated with Polyimides-EHNT was measured at λ = 1,550 nm, resulting in r₃₃ = 18 pm/V, after a corona poling procedure of one hour with an applied voltage of 8.5 kV at 190 °C.

Thermal stability of the poling induced electro-optic coefficient was measured by means of isothermal relaxation measurements performed at 80 °C and 120 °C, showing that at 120 °C the poled polymer maintains the 84% of its original r₃₃ value after 200 h [65].

In [66], three different types of acrylate functionalized dendritic nonlinear optical chromophores were dispersed in high-Tg temperature anthracene crosslinkable matrix. The synthesis of the guest-host system was carried out by following the procedure in [66] with a chormophore concentration of about 17% in weight. The glass transition temperature of the host matrix (labelled P2 in [66]) is exceptionally high T_g = 215 °C, meanwhile the Tg of the chromophores (labeled C1, C2, C3) ranges from 84 °C to 102 °C.

The electro-optic coefficient of the P2/C3 sample is the highest one with a value of r₃₃ = 126 pm/V (measured at λ = 1,310 nm) after a poling procedure at 185 °C.

Thermal stability of the poling induced electro-optic coefficient was measured by means of TMT isothermal relaxation measurements performed at 150 °C, showing that the poled polymer maintains 88% of its original r₃₃ value after 500 h [66].

The polymer described in [10] is a side-chain copolymer formed with 38% of the weight of the chromophore denoted in FTC [10] linked to a crosslinkable matrix formed by methylmethacrylate (MMA), dimetyl-azobisisobuthlate (MAIB) and methacryloyloxyyetyl isocyanata (MOI). The glass transition temperature results show T_g = 170 °C, and the electro-optic coefficient measured at λ = 1,310nm is 139 pm/V. By performing isothermal relaxation measurements at 85 °C, the poled polymer maintains 83% of its original r₃₃ value after 500 h.