1. Introduction

The nonlinear optical properties of different metallic nanoparticles have been investigated extensively [1-4]. These materials are being improved continuously to exhibit a stronger and faster optical response, controlled shape and size, easy fabrication, durability and thermo-mechanical stability [5]. Both the optical properties and the spatial distribution of metallic nanoparticles prepared by ion implantation in silica exhibit a very good stability [6]. We present results for silica samples containing Au nanoparticles and showing a considerable nonlinear optical response. This property makes them attractive for potential applications such as optical limitation, optical coupling, optical modulation and waveguiding.

For most of the optical materials the refractive index can be expressed as n=n0+n₂I, where n₀ represents the linear refractive index in a weak field, I is the light intensity in the media and n₂ is the nonlinear refractive index. In a similar way, the intensity dependent absorption coefficient is defined as: α(I)=α₀+βI, where α₀ and β represent the linear and nonlinear absorption coefficients, respectively [7].

The magnitude of the third-order nonlinear optical susceptibility χ⁽³⁾, can be expressed as,

A relationship of n₂ and β with χ⁽³⁾ (esu) has been established:

where c is the light velocity in the vacuum, λ is the wavelength of the light, and n₂ and β are expressed in m²/W and m/W, respectively.

The parameters n₂ and β depend on the different components of the tensor of third-order nonlinear optical susceptibility χ⁽³⁾. The search of n₂ and β has generated an important number of works. Different techniques have been developed to measure them, such as Z-scan [8], four-wave mixing [9], third harmonic generation [10], nonlinear interferometry [11], ellipse rotation [12], nonlinear ellipsometry [13], Raman spectroscopy [14], holography [15], measure of beam distortion [16], photoacoustic experiments [17] or spectrally resolve two-beam coupling [18]. In this work we present a method based on a FWM technique to measure the sign and magnitude of n₂ and β in a nonlinear optical material and we identified the physical mechanisms of optical nonlinearity exhibited by our samples.

2. Theory of coupling and self-diffraction of waves

We consider a thin isotropic nonlinear media in interaction with two optical waves associated with the amplitudes E₁ and E₂, with the same optical frequencies, linear polarization, and mutually coherent, but with arbitrary intensity. The vectors of the incident and self-diffracted waves are represented in Fig. 1, where z is the propagation direction, ±θ is the geometric incident angle for the propagation vectors k⃗₁ and k⃗₂ ; k⃗₃ and k⃗₄ correspond to the selfdiffracted waves; Δk is the difference in propagation constants between the zero and the first order of diffraction.

 

Fig. 1. Diagram of the propagation vectors of the incident and self-diffracted beams.

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Consider the two circular components of polarization of the electric field E⃗+ and E⃗- :

for an optically isotropic sample the index of refraction for each component of polarization is:

where A=χ⁽³⁾₁₁₂₂ and B=χ⁽³⁾1212, are components of the tensor χ⁽³⁾ [7].

We write the equation describing the amplitude of the electric field E along the nonlinear absorptive media with length D as:

with:

where,

The amplitude of transmitted and self-diffracted waves is calculated considering E(x,D) as a Fourier series, using the expression exp(iΨcosKx)=∑i^mJ_m(Ψ)exp(iKx), where K=2π/Λ is the wave number of the induced grating, J_m(Ψ) is a Bessel function of order m, and a property of the Bessel functions J-_m=(-1)m J_m. We take in account that the intensity of the self-diffracted waves is much lower than the intensity of the incident waves. We obtained:

where E _1± and E _2± are the amplitudes of the circular components of the incident beams; E _3± and E _4± are the amplitudes of the self-diffracted waves (see Fig. 1).

3. Experiment

For the preparation of the nanoparticles, a high-purity silica glass plate was implanted at room temperature with 2 MeV Au²⁺ ions at a fluence of around 2.8×10¹⁶ ions/cm². After implantation the sample was thermally annealed in an oxidizing atmosphere (air) at a temperature of 1100°C for 1 hr [19]. The concentration depth distribution of Au and the ion fluences were determined by Rutherford Backscattering Spectrometry (RBS) measurements using a 2.54 MeV ⁴He⁺ beam. Ion implantation and RBS analysis were performed at the IFUNAM ’s 3MV Tandem accelerator (NEC 9SDH-2 Pelletron). The maximum of the Au depth profile distribution is located approximately 0.5 µm beneath the surface sample.

For the FWM experiment we use a frequency doubled Q-switched laser, with pulses of 7 ns FWHM and linear polarization. The maximum pulse energy in our experiment was 4 mJ with a wavelength λ=532 nm. The intensity rate I₁:I₂ was 1.5:1. The radius of the beam waist at the focus in the sample was measured to be 0.15 mm. Figure 2 shows the experimental set up, where L represents the focusing system of lens, BM is a beam splitter, P is a half-wave plate, M1 and M2 are mirrors, MNL is the sample, and FD1-4 are photodetectors.

 

Fig. 2. Experimental set up.

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4. Results and discussion

We observe that our samples behave like waveguides in the near infrared spectrum. However, the linear absorption is high at the UV region and at the surface plasmon resonance band associated with the presence of Au nanoparticles in the silica. Figure 3 shows the linear optical absorption spectrum for our sample and one can clearly observe the surface plasmon resonance band (520 nm) associated with the Au nanoparticles in the silica sample.

 

Fig. 3. Linear optical absorption of a silica sample containing Au nanoparticles as a function of the wavelength.

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To increase the accuracy of the measurements of nonlinear parameters of the silica with Au nanoparticles, we first calibrated our measurement system using CS₂ with D=1 mm, as nonlinear medium with well known third-order nonlinear susceptibility |χ⁽³⁾|=1.9×10^-12 esu [7]. The calibration allows us to reduce the measurement uncertainty associated mainly with the determination of the beam diameter. The intensities of the diffracted beams behind the silica sample containing Au nanoparticles with D=1 µm were measured when the incident waves E⃗₁ and E⃗₂ had parallel linear polarizations. The intensities measured by the four photodetectors behind the sample were I₁=300 MW/cm², I₂=200 MW/cm², I₃=90 W/cm², and I₄=60 W/cm².

Also we obtained the intensity I₃ of the self-diffracted wave E⃗₃ in the FWM experiment, but in this case the polarization of the wave E⃗₁ was fixed, and the polarization of the wave E⃗₂ was rotated by the half-wave plate P. Each point in Fig. 4 corresponds to an average of 20 shots in the same experimental conditions, when the polarization of the incident wave was controlled by P (see Fig. 2).

 

Fig. 4. Intensity of I₃ as a function of the angle ϕ between planes of polarization of the incident waves.

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When the nature of the sample presents an isotropic mechanism of nonlinearity of index, then B=0 in Eq. (4). In this case, according to Eq. (4), we can notice that in a FWM experiment with incident waves with orthogonal linear polarizations, there is no induced circular birefringence, neither intensity fringes, so nonlinear absorption can be considered isotropic [20]. Under these circumstances, as we can observe in Fig. 4, there is no intensity of self-diffraction when the angle of planes of polarization of the incident waves is 90°.

With Eqs. (11-14) and our experimental data we get for our sample: |χ⁽³⁾|=2.2×10^-9 esu, n₂=-2×10^-15 m²/W, β=-5×10^-8 m/W. Using these results and Eq. (2) we obtain: Reχ⁽³⁾=-1.5×10^-9 esu and Imχ⁽³⁾=-1.6×10^-9 esu. We noticed that the error bars for the magnitude of the nonlinear parameters obtained decrease from ±20% to ±2.5% in our results when we obtained higher self-diffracted intensities in the experiment, but in the best case it is necessary to be careful with the occurrence of optical damage in the sample.

From our calculations using Eqs. (11-14) we obtained a negative sign for n₂ in our samples. Thus, we can say that a thermal effect is responsible for their nonlinearity of index, because that means that the change of index is produced by a positive isotropic change of density. The sign of β leads us to identify the absorption mechanism in the samples, being in this case saturable absorption. We expected this result taking in account the magnitude of the intensities of transmitted and self-diffracted waves in our experiments and our observations of laser ablation in similar samples. To have a comparative result we measured the transmittance of the sample that can be expressed as:

The data obtained are shown in Fig. 5. The adjusted curve was calculated from Eq. (15) using the parameters obtained from calculations by using Eq. (2). As we can see there is good agreement between the experimental and theoretical results.

 

Fig. 5. Transmitted intensity as a function of the incident intensity in the sample.

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5. Conclusions

Using a four-wave mixing experiment we obtained the sign and a significant magnitude of n₂ and β for silica samples containing Au nanoparticles. To discriminate the physical mechanism of n₂ from a straightforward measure of |χ⁽³⁾|, we consider the two cases of induced birefringence generated: when the polarizations of the incident waves are parallel and when they are mutually orthogonal. We indicated the presence of a saturable absorption and a thermal effect responsible for the mechanism of nonlinearity in our samples. The method presented here can be useful for optical measurements of several materials, and with some modifications it can also detect the time of response of the third order nonlinearity.