Investigation of the nonlinear absorption spectrum of all-trans retinoic acid by using the steady and transient two-photon absorption spectroscopy

Abstract

This work investigates the two-photon absorption (2PA) spectrum of all-trans retinoic acid (ATRA) in DMSO solution, employing the wavelength-tunable Z-scan and white-light pump-probe techniques with femtosecond pulses. Our results showed that ATRA presents two 2PA allowed bands at 280 nm (2hν = 560 nm) and 365 nm (2hν = 730 nm) with 2PA cross-section values of 34 GM and 40 GM, respectively. The 2PA band at 280 nm was ascribed to the transition from ground to high energy excited states with contribution from the real intermediate excited state (1¹B_u⁺-like). The 2PA band at 365 nm was attributed to an overlap of 1¹B_u⁺-like and 2¹A_g⁻-like excited states, which are essential to describe the photochemical processes that this class of organic materials play in nature. Through solvatochromic measurements and by using the two-energy level approach, we found that the 1¹B_u⁺-like and 2¹A_g⁻-like states contribute, respectively, 48% and 52% of the lowest energy 2PA band of ATRA in DMSO solution. From femtosecond transient spectroscopy we verified that when ATRA is excited at 775 nm (2PA excitation) its excited state absorption (ESA) spectrum presents a red-shift of ∼5.5 nm in comparison to the same spectrum excited at 387.5 nm (1PA excitation), corroborating the interpretation about the 2PA spectrum for the ATRA in DMSO.

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I Introduction

Organic compounds with biological relevance such as retinoids, carotenoids and flavonoids have emerged as potential candidates for applications in bio-photonic devices due to their capacity to convert optical signals into biochemical events, serving as ultrafast electro-optical devices.^1–5 For instance, carotenoids act as light-harvesting pigments and efficiently transfer energy to bacteriochlorophyll,^6–8 flavonoids, due to their antioxidant activity, have been used for the development of biological and pharmaceutical products^9–11 and retinal molecules are key chromophores found in the opsin,^12,13 being responsible for visible light absorption and subsequent conversion into nervous impulses via photoisomerization.^14–18

Among the retinyl polyenes, the all-trans retinoic acid (ATRA) has been the least studied from the photophysical and nonlinear optical response points of view. However, ATRA presents important biological functions, such as the activating gene transcription through their chemical binding to heterodimers of the retinoic acid receptor (RAR).¹⁹ Nevertheless, retinyl polyenes, in general, are almost insoluble and chemically unstable in the aqueous medium.²⁰ Consequently, ATRA is found bound to specific retinoid-binding proteins to be protected and transported in body fluids.²¹ On the other hand, in polar organic solvents like dimethyl sulfoxide (DMSO) and ethanol, it presents high solubility and good chemical stability.²² As it can be seen in Fig. 1, the ATRA has five C=C double bonds in its polyene backbone connected by a carboxylic acid and a β-ionone group in its extremities (linear structure); a structural analogue of carotenoids with a fewer number of conjugated C=C bonds.

Fig. 1 Chemical structure for the all-trans retinoic acid.

Retinyl polyenes are well-known because of their biological role and, therefore, they have been the subject of numerous theoretical and spectroscopic studies in the last decades. One of the first studies in this class of materials using two-photon absorption (2PA) excitation was performed by Birge et al.,^23,24 using fluorescence technique with nanosecond pulses. According to that study, the all-trans retinal (ATR), analogous to the ATRA, presents three low-lying excited singlet states with nπ* (1PA allowed weakly), 2¹A_g⁻-like (ππ*-2PA allowed strongly) and 1¹B_u⁺-like (ππ*-1PA allowed strongly) symmetries, respectively. It is important to mention that the 2¹A_g⁻ state is forbidden by 1PA excitation due to the dipole electric selection rules, since retinyl polyenes derivatives present ground state with 1¹A_g⁻ symmetry.^24,25 Therefore, to describe the origin and order of these low-lying states in retinyl polyenes, Birge et al.^23,24 compared the 2PA spectrum for the ATR dissolved in ethyl ether–isopentane–alcohol (EPA) at 77 K with their one-photon absorption (1PA) spectrum and found a red-shift of ∼2800 cm⁻¹. The authors attributed this result to the presence of a two-photon allowed state near to the 1¹B_u⁺-like state. Based on quantum chemical calculations,²⁴ the authors concluded that this state should have the 2¹A_g⁻-like symmetry. Forbidden states like the 2¹A_g⁻-like in ATR are extremely important to describe the photochemical properties of these molecules.¹² In this same context and more recently, Yamaguchi and Tahara²⁶ have employed femtosecond pump-probe technique to measure the 2PA spectrum of ATR in hexane at room temperature. It was also observed a red-shift (∼7 nm) in the 2PA peak with respect to the one-photon absorption, which also was attributed to presence of 2¹A_g⁻-like state.

Regarding to the ATRA, although some works have been published on their photophysical properties, we must mention the recent work of Presiado et al.²⁷ They measured the time-resolved fluorescence spectra dynamics of ATRA in several solvents using femtosecond pulses at 390 nm as excitation. They showed that the time-resolved emission signal is composed of three decay components, arising from three different excited-states. They attributed the ultrafast component (80 fs), dominant at λ ≤ 500 nm, to the 1¹B_u⁺-like → 2¹A_g⁻-like relaxation, while the intermediate component (from 1 to 4 ps, dominant at longer wavelengths) was ascribed to forbidden transition from the excited 2¹A_g⁻-like state to the ground-state. Another interesting study about the excited state dynamics for the ATRA was performed by Lei Zhang, et al.²⁸ In that work, they investigated the excited state dynamics of ATRA in hexanol purged with Ar with excitation at 400 nm by using the femtosecond time-resolved difference absorption spectroscopy. The authors used the single value decomposition (SVD) method together with global fitting to decompose the excited state absorption spectra and found three fast components, which, according to them, are associated with the S₁ (2¹A_g⁻), S₂ (nπ*) and S₃ (1¹B_u⁺) states. However, both studies were performed employing 1PA excitation that is responsible to populate only the strongly 1PA allowed excited state (1¹B_u⁺-like). Therefore, a complete study about the transient excited state dynamics induced by 2PA, as well as the 2PA cross-section spectrum for ATRA has not been carried out so far, to best of our knowledge.

Nowadays, there are yet a great interest on the low-lying excited states of carotenoids and linear polyenes like ATRA^29,30 due to its relevance in some biological functions. In this context, the aim of this present study is contribute to the better understanding about the photochemical and nonlinear optical properties of ATRA. For that, we investigated its degenerate 2PA spectrum in DMSO, using the wavelength tunable femtosecond Z-scan technique. In an effort to further comprehend the nonlinear spectrum, we carried out femtosecond transient absorption and solvatochromic shift measurements.

II Experimental

A Linear optical measurements

For the ground-state absorption, Stokes-shift and excitation anisotropy measurements, ATRA was dissolved in different solvents (toluene, chloroform, tetrahydrofuran (THF), dimethyl sulfoxide (DMSO), ethanol) in a concentration of ∼1.0 × 10⁻⁵ mol L⁻¹. In these experiments, the samples were placed in 1 cm thick quartz cuvette. The steady-state absorption and fluorescence spectra were recorded using a Shimadzu UV-1800 spectrophotometer and a PerkinElmer LS55 fluorimeter, respectively. Linear and nonlinear optical measurements were performed at a temperature of 20 °C. We measured the linear absorption and fluorescence spectra before and after each nonlinear optical measurement and no degradation was observed for the temperatures and intensities used.

B Z-scan experiments

The 2PA cross-section spectra were measured using the Z-scan technique in the open-aperture configuration. The Z-scan measurements were performed with 160 fs pulses at 1 kHz repetition rate, delivered by a tunable optical parametric amplifier pumped at 775 nm by a Ti:sapphire chirped pulse amplifier (CPA). For each wavelength, the pulse energy was kept at approximately 100 nJ for 2PA. A Gaussian beam profile was obtained by spatial filtering the excitation beam before the Z-scan setup. A silicon detector was employed to monitor the laser beam intensity in the far-field. To improve the signal to noise ratio, a lock-in amplifier was used to integrate 1000 shots for each point of the Z-scan signature. For nonlinear measurements, the samples were prepared in a concentration of 1 × 10⁻³ M and placed in a 2 mm fused silica cell. For an absorptive non-linearity, the light field induces an intensity dependent absorption, α = α₀ + βI, in which I is the laser beam intensity, α₀ is the linear absorption coefficient and β is the 2PA coefficient. Therefore, once an open aperture measurement is carried out, the nonlinear absorption coefficient can be unambiguously determined by fitting the experimental data. The two-photon absorption cross-section can be obtained through the expression σ_2PA = hνβ/N, where N is the number of molecules per cm³, and hν is the photon energy.

C Femtosecond pump-probe measurements

Femtosecond time-resolved ESA experiment was implemented using 160 fs pulses (775 nm) from a regenerative Ti:sapphire CPA (Chirped Pulse Amplifier) system operating at 1 kHz repetition rate (Clark 2001-MXR). A 90/10 beam-splitter was used to divide the beam into pump and probe pulses. The stronger beam was doubled (387.5 nm) using a 300 μm thick BBO crystal cut for type-I second harmonic generation. 1PA induced ESA measurements were performed using the pulse at 387.5 nm as pump pulse after the reminiscent fundamental beam is filtered by dichroic mirrors. For 2PA induced ESA measurements, the fundamental pulse at 775 nm was used as pump pulse. A small portion of the weaker beam was focused with a 50 mm focal length lens in a 1 mm-thick sapphire plate to generate the white-light continuum (WLC) probe. The fundamental beam intensity used to generate the WLC was carefully controlled adjusting the angle between two thin wire grid polarizers in order to achieve a stable WLC. Time-resolved ESA measurements were performed monitoring the probe intensity spectral distribution as a function of the time delay τ between pump and probe pulses, which was varied by a computer controlled translation stage that provides a resolution of 75 fs. For each time delay, the probe pulse intensity spectral distribution is recorded with and without the pump pulse irradiating the sample. In this way, we can calculate the spectra of differential absorbance (time-resolved ESA signal), ΔA(λ, τ), and separate the signal related to the excited states or photo-induced species from the ground state signal. The differential absorbance signal is described by eqn (1):in which, I^{pump on}_probe and I^{pump off}_probe correspond, respectively, to the probe pulse spectra with and without the pump pulse irradiating the sample for each time delay τ. The WLC probe pulse intensity spectral distribution was monitored by means of a fast spectrometer (Ocean Optics-USB2000). The chirp of WLC probe pulse was measured to be <1 ps in the 400–700 nm spectral region. Pump and probe pulses energies were adjusted to be smaller than 1 μJ and 1 nJ, respectively, and their polarizations relative angle was set to the magic angle (54.7°) to eliminate orientational components from the signal.

III Results and discussions

Fig. 2 presents the normalized linear absorption (solid line) and excitation fluorescence anisotropy (diamonds) spectra of the ATRA in DMSO. The lowest energy one-photon allowed absorption band located at 358 nm for the ATRA dissolved in DMSO is related to their charge distribution along the polyene chain (1¹A_g-like → 2¹B_u-like transition).³¹ Such band exhibits a considerable molar absorptivity of ca. 2.80 ± 0.20 × 10⁴ M⁻¹ cm⁻¹ at 358 nm, which is related to the significant numbers of π-electron and planarity of polyene chain.^24,32 The diamonds represents the excitation anisotropy spectrum. As it can be seen between 400 and 335 nm, a constant value on the excitation anisotropy is attributed to transition from the ground-state to the first strongly 1PA allowed singlet excited state (1¹A_g⁻-like → 1¹B_u⁺-like). At 290 nm, a great change in the excitation anisotropy value indicates that the angle between the absorption and emission transitions is being altered and, as consequence, another excited state should is being reached (most probably, this state is related with the 1¹A_g⁻-like → 2¹B_u⁺-like transition).³³

Fig. 2 Ground-state (solid line – left axis) and excitation fluorescence anisotropy (diamonds, right axis) spectra of ATRA dissolved in DMSO at room temperature.

The 2PA cross-section spectrum for the ATRA in DMSO is depicted in Fig. 3 (diamonds), which was determined by performing open-aperture Z-scan measurements (the 2PA cross-section were plotted as a function of half excitation wavelength). The 2PA spectrum presents two 2PA allowed bands located at 280 nm (2hν = 560 nm) and 365 nm (2hν = 730 nm) with 2PA cross-section values of about 34 GM and 40 GM, respectively. The relatively small 2PA cross-section values observed for ATRA are probably related to the weak push–pull character of the chromophore. In fact, the β-ionone ring and carboxylic acid are, respectively, weak electron donor and moderated acceptor groups, and, therefore, the charge redistribution along the polyene chain is small at the excited state.^22,34 In the same figure, we present the 1PA spectrum to allow a direct comparison between the energy levels involved in 1PA and 2PA allowed bands. One can observe that the 2PA band is red-shifted by approximately 7 nm in comparison to the 1PA band, indicating that the state accessed by 2PA does not correspond, necessarily, to the state accessed by one-photon. As previously reported, some studies have shown this same behavior for another retinyl polyene, in special, for the all-trans retinal.^{24,32,35–38}

Fig. 3 The black solid line represent the 1PA spectrum, while the diamonds represent the 2PA spectrum of ATRA dissolved in DMSO at room temperature. The shades curves represent the individual contribution to the lowest energy 2PA band of the 2¹A_g-like and 1¹B_u-like states found in this study. The red line along the diamonds represents the sum of both individual contributions.

For molecules such as ATRA, in which it is not possible to define precisely the parity of excited states because of the low molecular symmetry, the same electronic state may be accessed via one and two-photon absorptions. Therefore, the 1¹A_g⁻-like → 1¹B_u⁺-like transition should contribute to the lowest-energy allowed 2PA band. In order to quantify this contribution, we performed Stokes-shift solvatochromic measurements and sum-over-essential states approach. For the lowest energy 2PA band at 730 nm (2hν), we assumed a two-energy level system. Taking into account the average over all possible molecular orientations in an isotropic medium and considering excitation with linearly polarized light, the 2PA cross-section can be written as:

in whichandin these equations, c is the speed of light, ω₀₁ is the transition angular frequency, and L = 3n²/(2n² + 1) is the Onsager local field factor introduced to take into account the medium effect³⁹ with the refractive index n = 1.479 for DMSO at 20 °C. [small mu, Greek, vector]₀₁ is the transition dipole moment, Δ[small mu, Greek, vector]₀₁ is the difference between the permanent dipole moments vectors of the excited ([small mu, Greek, vector]₁₁) and ground ([small mu, Greek, vector]₀₀) states, and θ is the angle between the dipole moments [small mu, Greek, vector]₀₁ and Δ[small mu, Greek, vector]₀₁. By measuring the 2PA with linearly and circularly polarized light,⁴⁰ we determined that θ ≈ 0°. N_A is the Avogadro's number, and g_gf(2ω) = (1/π){Γ_gf/[(ω_gf − 2ω)² + Γ_gf²]} represents the normalized line width function (Lorentzian line-shape), in which, Γ_gf is the damping constant describing half width at half-maximum (FHWM) of the final state line width. υ = υ_abs − υ_em is the difference between the peak of the absorption and fluorescence emission in cm⁻¹. F(n, ξ) = 2[(ξ − 1)/(2ξ + 1) − (n² − 1)/(2n² + 1)] is the Onsager polarity function, in which, ξ is the dielectric constant of the solvent. Vol is the volume of the molecular cavity. To find the molecular cavity volume, we performed quantum chemical calculations based on the density function theory, taken into account the DMSO solvent effect by using the polarization continuum model within the integral equation formalism variant (IEF-PCM).⁴¹ This model is based on creating the solute cavity within the dielectric continuum via a set of overlapping spheres centered on the heavy atoms of the molecule (solute).⁴¹ With such approach, we determined a hydrodynamics volume for ATRA of approximately 515.4 ų.

Fig. 4(a) shows the steady-state absorption and fluorescence spectra of the ATRA in five different solvents (toluene, chloroform, THF, DMSO and ethanol), while Fig. 4(b) illustrates the result of the Stokes-shift (υ) as a function of the Onsager polarity parameter (F(n, ξ)). We calculated a positive linear slope indicating that the permanent dipole moment for the first excited state is higher than the ground-state one ([small mu, Greek, vector]₁₁ > [small mu, Greek, vector]₀₀).⁴²

Fig. 4 (a) Normalized absorption and fluorescence spectra for ATRA in five different solvents (toluene, chloroform, dichloromethane, THF, DMSO and ethanol). (b) Solvatochromic Stokes shift (υ) measurements obtained as a function of the Onsager polarity function (F(n, ξ)).

Using this result, the hydrodynamic volume calculated through DFT-PCM calculations and eqn (2), we obtained a maximum 2PA cross-section of 2.0 ± 0.7 × 10¹ for the lowest energy 2PA allowed band of ATRA in DMSO, which is approximately half of the value experimentally observed. Using eqn (2), we simulated the contribution of this state to the lowest-energy 2PA band as shown in Fig. 3 (black shadow Lorentzian curve). Since considering only the 1¹A_g⁻-like → 1¹B_u⁺-like transition, we could not observe a good agreement between the experimental and theoretical result. This important outcome indicates that another state, which is not observed in the 1PA spectrum, is contributing to the lowest-energy 2PA band. As previously mentioned, the retinyl polyenes present two main low-lying excited singlet states, named ¹A_g⁻-like and ¹B_u⁺-like respectively. The 1¹B_u⁺-like state is the one responsible for the considerable molar absorptivity (2.80 ± 0.10 × 10⁴ M⁻¹ cm⁻¹) at 358 nm, and, therefore, it is the state that presents 2PA cross-section of about 20 GM. In this context, the 2¹A_g⁻-like state, which presents the same parity of the ground state (1¹A_g⁻-like) of retinyl polyenes, should contribute to the lowest-energy 2PA band observed in Fig. 3. We have used an analogous equation as eqn (2) to adequately fit the experimental 2PA data. The red shaded Lorentzian curve shows the effective contribution of the 2¹A_g⁻-like state to the lowest energy band of ATAR in DMSO. By integrating the Lorentzian curves, we found that the 1¹B_u⁺-like and 2¹A_g⁻-like states contribute, respectively, with 48% and 52% to the lowest energy 2PA band.

To shed more light about the electronic structure of ATRA, in special to the 2¹A_g⁻-like state, we have measured the excited-state absorption for ATRA in DMSO solution by using the white-light femtosecond pump-probe technique. This experiment was performed in order to observe if there is or not a significant difference between the relaxation times from the both involved 2PA states to the ground state. Fig. 5 shows the results of transient absorption for ATRA in DMSO with the pump at 387.5 nm (1PA Fig. 5(a)) and 775 nm (2PA Fig. 5(b)). In this figure, the left hand side shows the colormap representing the time- and wavelength-resolved dynamics of transient absorption spectrum; the central part displays the ESA spectra for different times, while the right hand side illustrates the decay curves for probe pulse at 540 nm (a) and 545.5 nm (b), corresponding respectively, to the peak of the 1PA and 2PA ESA spectra.

Fig. 5 Excited state dynamics for ATRA in DMSO solution (a) 1PA and (b) 2PA excitations. ESA colormap representing the time- and wavelength-resolved dynamics of transient absorption spectrum (left hand). ESA spectra for different times (central part). Decay curves for probe pulse at (a) 540 nm and 545.5 nm (b) corresponding, respectively, to the 1PA and 2PA ESA peak (right hand).

The inset in Fig. 5 (right side) depicts the decay time as a function of the excitation wavelength. As can be seen in Fig. 5, the spectral and temporal dynamics can be considered the same, given our experimental error, for ATRA when excited by one- or two-photon absorption.

More specifically, when ATRA is excited by one-photon (387.5 nm), we observed an electronic relaxation time of approximately 3.2 ps at 540 nm (ESA-1PA peak). For the probe at 545.5 nm (ESA-2PA peak), when the sample is excited by two-photon of 775 nm the electronic relaxation time is 3.3 ps. The difference observed between the 1PA and 2PA excited decay curves for ATRA are within the experimental error (laser pulse with FWHM line width of ∼160 fs). In Fig. 6 we show a comparison between the ESA spectra for the pump laser at 387.5 nm (1PA) and 775 nm (2PA).

Fig. 6 Comparison between the ESA spectra excited by one- and two-photon absorption. It is observed a red-shift to the ESA spectrum excited by two-photons (775 nm) in comparison to the one-photon excited one (387.5 nm). Both curves correspond to the same time.

It is observed in Fig. 6 that the ESA spectrum excited by 2PA present a red-shift of ca. 5.5 nm when compared to the ESA spectrum excited by 1PA. This result indicates the pump pulses, which have the same total energy (one photon of 387.5 nm and two photons of 775 nm have energy of about 3.18 eV), promote electrons to different energy levels, analogously to what was observed in Fig. 3. This result indicates that although the pump pulses at 387.5 nm and 775 nm induces transitions at the same total energy (∼3.18 eV), by 1PA and 2PA respectively, they promote electrons to different electronic states whose central energy are slightly displaced, analogously to what was observed in Fig. 3.

Finally, in order to model the higher energy 2PA allowed band at 280 nm (see Fig. 3), we used a three-level energy diagram, consisting of a ground-state (1¹A_g⁻-like), one intermediate 1PA allowed excited state (1¹B_u⁺-like) and the final excited state (|S_f〉). For this system, the 2PA cross-section can be written as (assuming linearly polarized light and that the dipole moments are parallel):⁴⁰

in which, [small mu, Greek, vector]₁₂ is the transition dipole moment between the excited states |S₁〉 → |S_f〉, and R(ω) = ω²/[(ω₀₁ − ω)² + Γ₀₁²(ω)] is the resonance enhancement factor. In general, for non-centrosymmetric molecules as ATRA, eqn (5) presents additional terms related with the permanent dipole moment change and the interference term between the two distinct excitation channels.⁴⁰ However, in molecular system with a weak push–pull character as observed in ATRA, the factor R(ω)|[small mu, Greek, vector]₀₁|²|[small mu, Greek, vector]_1f|² dominates the 2PA allowed transition. Considering the experimental data obtained from the Z-scan measurements at 280 nm (2ν = 560 nm), σ^(2PA-max) = 34 GM, we found |[small mu, Greek, vector]_1f1| = 5.5 D. Such value corresponds to a molar absorptivity of approximately 1.74 × 10⁴ M⁻¹ cm⁻¹. In order to observe this band along the ESA, it is necessary to use a probe with wavelength higher than 850 nm. However, in our experiment, the filter used to remove the strong 775 nm that generates the white-light probe pulses also remove wavelengths longer than 770 nm. In addition, as the molar absorptivity to this transition is smaller than those to the ground state, this region should be associated with the saturable absorption effect.⁴³ However, to the resonance enhancement effect region (shorter than 280 nm, data not shown), we observed a 2PA cross-section about 140 GM (at 250 nm or 2ν = 500 nm), which, using eqn (5), gives a |[small mu, Greek, vector]_1f2| = 12.0 D. Such value corresponds to a molar absorptivity of ∼9.5 × 10⁴ M⁻¹ cm⁻¹, which is similar to the one estimated for the ATRA (∼1.25 × 10⁵ M⁻¹ cm⁻¹ at 540 nm) using rhodamine B as reference material⁴⁴ through the pump-probe experiments.⁴⁵ This result indicates that the same states are being accessed by one- and two-photon absorption at the resonance enhancement region.

IV Final remarks

In this study we have investigated the 2PA spectrum of ATRA, a chromophore with interesting optical and biological properties, by using the steady and transient 2PA spectroscopy. We observed that ATRA presents a 2PA spectrum composed by a 2PA band of high energy (280 nm) that has contribution of an intermediate excited state and a lowest-energy 2PA band. The latter band was ascribed to the overlap between the 2¹A_g-like and 1¹B_u-like states due the lower molecular symmetry of the ATRA, which allows electrons to be excited by 2PA to states with different parity in respect to the ground state. By means of solvatochromic measurements and using a two-energy level model, we were able to quantify the individual contribution of the 2¹A_g-like and 1¹B_u-like states to the lowest energy 2PA band. We found similar behavior between the 2PA and ESA-2PA spectra as compared to their correspondent 1PA and ESA-1PA spectra, more specifically, a red-shift of about 5.5 nm was observed to the 2PA spectra. In summary, here we reported for the first time the 2PA cross-section spectrum and the excited state dynamic induced by 2PA for the ATRA dissolved in DMSO. We believe that these results contribute to a better understanding of the photochemical and nonlinear optical features of polyenes as ATRA.

Acknowledgements

Financial support from FAPESP (Fundação de Amparo à Pesquisa do estado de São Paulo, processo no. 2011/12399-0 and no. 2009/11810-8), FAPEMIG (Fundação de Amparo à Pesquisa do estado de Minas Gerais), CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) and the Air Force Office of Scientific Research (FA9550-12-1-0028) are acknowledged.

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