## Abstract

We propose a simple technique based on electric field induced second harmonic (EFISH) generation of femtosecond pulses to measure the third order susceptibility *χ*
^{(3)}(2ω,0,ω,ω) in glasses. First we present the principle of the method, then we validate our experimental set-up and develop unexpected aspects of EFISH technique. Finally, we give a numerical value of the third order susceptibility in various glasses and discuss these results.

© Optical Society of America

## 1. Introduction

The development of high-speed communication networks relies on optical fibers technology. To optimise the signal transmission, one needs to know accurately the linear dispersion, and the third order non-linearity on the glass material. Several techniques have already been used to determine the *χ*
^{(3)} coefficient: third harmonic generation [1], four-wave mixing [2], collinear pump-probe experiment [3], Z-scan [4], and femtosecond chirped pulses [5]. These experiments enable to determine the third order susceptibility in glass samples by measuring odd order polarizations intensities: *χ*
^{(3)}(3ω,ω,ω,ω), or *χ*
^{(3)}(ω,ω,-ω,ω).

However only few experiments have been performed – to our knowledge – to determine the *χ*
^{(3)}(2ω,0,ω,ω) of glasses. Owing to dispersion in the glass material, this term can be different from *χ*
^{(3)}(3ω,ω,ω,ω) or *χ*
^{(3)}(ω,ω,-ω,ω). It is involved in Electric Field Induced Second Harmonic generation (EFISH) [6] and in Second Harmonic Generation (SHG) experiments in poled glasses [7]. During the thermal poling of glasses or fibers, one induces an effective second order susceptibility ${{\chi}_{\text{eff}}}^{\left(2\right)}$ (2ω,ω,ω)~ *χ*
^{(3)}(2ω,0,ω,ω) E_{ind} which depends on *χ*
^{(3)}(2ω,0,ω,ω) and on the built-in electric field E_{ind}. The measurement of *χ*
^{(3)}(2ω,0,ω,ω) is therefore important to develop and optimize fiber integrated frequency converters.

Here, we propose a simple method based on EFISH of femtosecond pulses to measure the *χ*
^{(3)} (2ω,0,ω,ω) coefficient in glasses. We discuss its validity and use it to evaluate some non-linear coefficients in pure fused silica and in commercial lead silicates glasses.

## 2. Principles and experiment

#### 2.1 EFISH in the femtosecond regime

The EFISH technique sketched in Fig.1 consists in applying an intense DC-field (>1 kV/mm) to a glass sample (dimensions : L_{x}, L_{y}, L_{z}) and to detect the harmonic signal.

Here we consider a monochromatic wave E^{ω} polarized linearly, travelling along the z direction in a non absorbing glass sample where a constant static electric field ${{E}^{0}}_{\mathrm{y}}$=U/L_{y} is applied. The harmonic polarization generated at z position is given by [8] :

where - i, j, k can be x or y,

- Δk is the phase-mismatch between fundamental and harmonic waves Δk=k(2ω)-2k(ω)

- ${\mathrm{\chi}}_{\text{ijkl}}^{\left(3\right)}$ (2ω,0,ω,ω) is the third order susceptibility tensor of the glass. (It satisfies to Kleinman’s relations in isotropic media [8]). To clarify the text, we simply write ${\mathrm{\chi}}_{\text{ijkl}}^{\left(3\right)}$.

Introducing I^{ω} the intensity of the fundamental wave, the harmonic intensity detected is :

As the harmonic intensity is modulated at the coherence length, (L_{c}=π/Δk~12 µm in silica at 800 nm) the value of the harmonic signal is very sensitive to the thickness of the sample, which must be known with a sub-micrometric precision. Maker’s fringes technique [9] can overcome this drawback but it requires long acquisition times. To avoid this, it is convenient to use femtosecond laser sources whose wide spectrum moderates directly the modulation effect. In this case, the harmonic intensity is given by [10] :

where : - Ω=2ω_{0}-ω, ω_{0} is the central wavelength of the fundamental pulse

- E^{ω}(Ω) is the spectral Fourier transform of the envelope of the laser pulse

- Δk(Ω)=Δk(ω_{0})+Δu^{-1}Ω=Δk_{0}+Δu^{-1}Ω

- Δu^{-1} corresponds to the group velocity dispersion : Δu^{-1}=${\mathrm{u}}_{2\mathrm{\omega}}^{-1}$-${\mathrm{u}}_{\mathrm{\omega}}^{-1}$,u_{ω} and u_{2ω} are the group velocities of the fundamental and harmonic pulses respectively.

The integration of Eq. (3) yields :

where s^{2ω}(Ω)=|∫${\mathrm{E}}_{\mathrm{j}}^{\mathrm{\omega}}$(Ω-Ω^{′})${\mathrm{E}}_{\mathrm{k}}^{\mathrm{\omega}}$(Ω^{′})dΩ′|^{2}

The integrated intensity is equal to:

${I}_{i}^{2\omega}=\int {I}_{i}^{2\omega}\left(\Omega \right)d\Omega \propto {\lceil {\chi}_{\mathrm{iyjk}}^{\left(3\right)}.{E}_{y}^{0}\rceil}^{2}{L}_{z}^{2}\int \mathrm{sin}{c}^{2}\left(\frac{1}{2}\left(\Delta {k}_{0}+\Delta {u}^{-1}\Omega \right){L}_{z}\right){s}^{2\omega}\left(\Omega \right)d\Omega $

Since Δk_{0}≫Δu^{-1} Ω (in silica at 800 nm, Δk_{0}=10^{5} m^{-1}≫Δu^{-1} Ω=1.8 10^{3} m^{-1}), thus :

${I}_{i}^{2\omega}\propto {\left[{\chi}_{\mathrm{iyjk}}^{\left(3\right)}.{E}_{y}^{0}\right]}^{2}\frac{4}{{\left(\Delta {k}_{0}\right)}^{2}}\int {\mathrm{sin}}^{2}\left(\frac{1}{2}\left(\Delta {k}_{0}+\Delta {u}^{-1}\Omega \right){L}_{z}\right){s}^{2\omega}\left(\Omega \right)d\Omega $

Moreover, if the spectral density s^{2ω} varies slower than the sin^{2} term (i.e. the width of the harmonic spectrum is larger than 1/Δu ^{-1}L_{z}) then :

Note that ${\mathrm{I}}_{\mathrm{i}}^{2\mathrm{\omega}}$ is independent from the thickness L_{z}. Therefore, by measuring the harmonic signal and controlling the experimental parameters s^{2ω}(Ω), Δk_{0} and ${{\mathrm{E}}^{0}}_{\mathrm{y}}$, one can simply deduce the value of the third-order susceptibility of the glass. We have previously verified that the assumptions leading to Eq. (5) were fulfilled in all our experiments.

## 2.2 Experimental set-up

First we perform the technique of EFISH of femtosecond pulses in a commercial fused silica polished plate (Herasil II, Heraeus, L_{x}=L_{y}=5 mm, L_{z}=4 mm). The sample is inserted between two plane steel electrodes. A carbon ink layer improves the electric contact between the electrode and the side of the glass. The top electrode is linked to a high voltage generator (U=0 to 5 kV). The bottom electrode is linked to the ground. The DC-field is directed along the y direction (${{\mathrm{E}}^{0}}_{\mathrm{y}}$=U/L_{y}).

The linearly polarized beam delivered by a Ti:Sapphire femtosecond oscillator (2 nJ, 76 MHz, 800 nm, 120 fs) is slightly focused in the glass sample by a 100 mm focal length lens. A l/2 plate sets the polarization. The confocal parameter is evaluated to 5 mm, and is longer than L_{z}, the thickness of the sample. Thus we consider that the spatial profile of the laser beam is almost constant along its propagation in the sample. When the beam emerges from the sample, it is collimated by a second 100 mm focal length lens. A blue filter cuts the 800 nm component of the beam. The harmonic component (400 nm) generated through EFISH process in the glass sample is detected by a photomultiplier linked to a photon counting chain. We have the possibility as well to monitor the harmonic spectrum using a spectrograph and a cooled CCD camera.

## 3. Experimental validation and discussion

#### 3.1 Influence of the polarization, dependence on the applied DC-voltage

If the fundamental wave is polarized along i=x or y, then the non-vanishing components of the susceptibility tensor are ${\chi}_{\text{yyii}}^{\left(3\right)}$. According to Eq. (5) ${\mathrm{I}}_{\mathrm{y}}^{2\mathrm{\omega}}$∝[${\chi}_{\text{yyii}}^{\left(3\right)}$${\mathrm{E}}_{\mathrm{y}}^{0}$]^{2}. We check in Fig. 2 the quadratic dependence with respect to U, i.e. to ${\mathrm{E}}_{\mathrm{y}}^{0}$ for both polarizations. We have previously observed that the harmonic signal is homogeneous in the whole sample.

In Fig. 2, note that the intensity ratio between both polarizations is equal to 9, according to Kleinmann’s relation: ${\chi}_{\text{yyyy}}^{\left(3\right)}$=3${\chi}_{\text{yyxx}}^{\left(3\right)}$

## 3.2 Spectral fringes discussion

The sinc^{2} term in Eq. (4) is responsible for the modulation of the harmonic spectrum. The modulation step can be written in the wavelength domain as $\Delta \lambda =\frac{{\lambda}_{0}^{2}}{4.\Delta {u}^{-1}.c.{L}_{z}}=0.83\phantom{\rule{.2em}{0ex}}\mathrm{nm}$ for our experiment in fused silica. Dots in Fig. 3 are an experimental harmonic spectrum, while the solid line represents the fitting function written in Eq. (4) considering the former value of Δ*λ*. Note the excellent adjustment between the theoretical model and our data.

## 3.3 Influence of the design of the electrodes

Here we suppose an impulsion centered at ω0 propagating in a glass sample (0<z<L_{z}) where a non constant profile of effective second order non-linearity ${{\chi}_{\text{eff}}}^{\left(2\right)}$ (z) is induced. According to Eq. (3) the harmonic field generated at the end of the sample is :

If we consider now that ${\chi}_{\text{eff}}^{\left(2\right)}$(z) can be extended from -∞ to +∞ by : ${\chi}_{\mathrm{eff}}^{(2)}(z)=0\phantom{\rule{.5em}{0ex}}\mathrm{if}\phantom{\rule{.4em}{0ex}}\text{z}<0\phantom{\rule{.5em}{0ex}}\mathrm{or}\phantom{\rule{.5em}{0ex}}\text{z}>\phantom{\rule{.3em}{0ex}}{\text{L}}_{\text{z}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\mathrm{then}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}{I}^{2\omega}\left(\Omega \right)\propto {\mid \underset{-\infty}{\overset{+\infty}{\int}}{\chi}_{\mathrm{eff}}^{\left(2\right)}\left(z\right){e}^{i\Delta k\left(\Omega \right)z}dz\mid}^{2}{s}^{2\omega}\left(\Omega \right)$

Introducing ${\stackrel{\mathit{~}}{\chi}}_{\text{eff}}^{\left(2\right)}$(q), the spatial Fourier transform of ${\chi}_{\text{eff}}^{\left(2\right)}$(z), we obtain :

This demonstrates that one detects the value of the Fourier transform of the second order non-linearity profile at a wave number equal to the phase-mismatch.. Then a harmonic signal is generated when ${\chi}_{\text{eff}}^{\left(2\right)}$(z) varies on spatial scales around the coherence length (~12µm).

As *χ*
^{(3)} is constant in the material, the *χ*
^{(2)} profile is determined by the profile of the DC-field seen by the laser wave along its propagation in the glass sample. This profile results from the design of the electrodes : if they cover the whole surface of the glass, then the field lines are parallel and the electric field is constant along z (fig.4a). The effective second-order nonlinearity is a step profile, and its Fourier transform is a sinc function (fig. 4b), thus I^{2ω} (Ω) is spectrally modulated.

If the electrodes do not cover the whole surface of the sample (fig. 5a), then the field lines are no more parallel and nor is the DC field constant along z. In this case, the effective *χ*
^{(2)} has a smooth (assumed gaussian) profile : consequently its Fourier transform has a gaussian shape (fig.5b), and I^{2ω} (Ω) is very small.

To illustrate this, we performed an EFISH experiment on a wedged shaped glass sample. A rectangular carbon electrode is painted on it (Fig.6). For various positions of the laser beam in the sample, we measure the harmonic spectrum. For each position, the length of the electrode is the same. The length of the light path not covered by the electrode is equal to 2 mm, 200 µm and 0 for positions (3), (2) and (1) respectively. Fig.6 presents the harmonic spectra measured at each position.

The harmonic intensity is clearly linked to the pattern of the electrodes. As soon as the *χ*
^{(2)}(z) differs from a step profile, the SHG signal decreases and vanishes. This experiment confirms that the design of electrodes is a key parameter for EFISH measurements.

## 4. *χ*^{(3)} measurements: results and comments

The *χ*
^{(3)}(2ω,0,ω,ω) measurements are performed by measuring the EFISH signal for a fundamental wave polarized along y. Thus the deduced third order susceptibility is ${{\chi}^{\left(3\right)}}_{\text{yyyy}}$. The glass samples we use are the previous silica sample and lead silicates (SF1, SF4, SF57 ; Schott) with the same dimensions. The electrode covers the whole sample. We first calibrate the set-up by measuring under the same experimental conditions the SHG signal generated in a phase-matched 10 µm long BBO crystal and comparing it with its theoretical expression. Their ratio gives the detection efficiency of our set-up. According to this, we calculate the theoretical EFISH signal in the glass as a function of the *χ*
^{(3)} term and compare it with the recorded signal. SHG intensities generated in the BBO and in the glass are respectively :

${I}^{2\omega}={T}_{2\omega}\frac{2{\omega}^{2}{\left({\chi}_{\mathrm{BBO}}^{\left(2\right)}\right)}^{2}}{{n}_{\omega}^{2}{n}_{2\omega}{c}^{3}{\epsilon}_{0}}{L}_{y}^{2}{T}_{\omega}^{2}{\left({I}^{\omega}\right)}^{2}\phantom{\rule{.9em}{0ex}}\text{and}\phantom{\rule{.9em}{0ex}}{I}^{2\omega}={T}_{2\omega}\frac{2{\omega}^{2}{\left({\chi}^{\left(3\right)}{E}^{0}\right)}^{2}}{{n}_{\omega}^{2}{n}_{2\omega}{c}^{3}{\epsilon}_{0}}\frac{2}{\Delta {k}_{0}^{2}}{T}_{\omega}^{2}{\left({I}^{\omega}\right)}^{2}$

where T_{2ω} and T_{ω} are the Fresnel coefficients and n_{2ω} and n_{ω} the indexes of the material for the harmonic and fundamental polarizations respectively.

The values we obtain are presented in the following table and compared with the values of ${{\chi}^{\left(3\right)}}_{\text{yyyy}}$(ω,-ω,ω,ω) given in a time resolved optical Kerr effect experiment at 800 nm [11].

Note that these values of *χ*
^{(3)}(2ω,0,ω,ω) are close to *χ*
^{(3)}(ω,-ω,ω,ω). However it is important to remark that as the glass absorbs the laser wavelength by a two photon absorption process the harmonic signal is found to decrease due to the screening of the electric field through photoexcitation of charges [12]. This reduces the precision of the measurement. However, far away from the absorption band of the material, EFISH generation of femtosecond pulses is a suitable technique for the determination of the third-order susceptibility of glasses.

## Acknowledgments

We thank the Région Aquitaine and the Ministère de l’Education Nationale, de la Recherche et de la Technologie for their financial support.

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