## Abstract

We report on the measurement of the nonlinear refractive index of fs-laser written waveguides in fused silica by analyzing self-phase modulation of the propagating light. The nonlinear index is reduced considerably compared to the bulk material by the writing process and is furthermore highly dependant on processing parameters.

© 2006 Optical Society of America

## 1. Introduction

Material processing using infrared femtosecond laser pulses has been of great interest for several years. When intense ultra-short laser pulses are focused inside transparent materials nonlinear absorption will occur in the focal volume leading to optical breakdown and the formation of a microplasma. This localized energy deposition induces permanent structural and refractive index changes [1]. By moving the sample with respect to the focus of the laser beam waveguides can be written along arbitrary paths. Since the material is transparent for the processing laser beam this technology is applicable like no other for the fabrication of three-dimensional integrated optical devices [2]. The direct laser writing approach has been used to fabricate a variety of photonic devices like waveguides [3], 3D-couplers [4], evanescently coupled waveguide arrays [5, 6], or waveguide lasers [7].

However, the experimental investigation has been limited to the linear guiding properties of these structures. Only most recently the first investigations of the nonlinear behavior of fs-written waveguides have been reported using third harmonic generation microscopy [8] and analyzing the localization of light in arrays of fs-laser written waveguides [9]. However, the utilization of nonlinear effects in fs-laser written waveguides and devices would increase the potential for applications considerably. Especially devices for opto-optical switching are of great interest [10]. In this paper we report on the nonlinear properties of fs-written structures and their dependence on processing parameters. This knowledge is essential for the design of novel nonlinear integrated optical devices.

## 2. Experimental details and results

For the fabrication of the waveguides laser pulses at a wavelength of 800 nm with an energy of 0.3 μJ and a duration of about 150 fs have been focused into polished fused silica bulk material. The pulses were generated by an amplified Ti:sapphire laser system (Coherent RegA) with a repetition rate of 100 kHz. The beam was focused by a 20× microscope objective with a numerical aperture (NA) of 0.45 which was corrected for a cover-glass thickness of 0.17 mm. The focal depth inside the sample was 200 μm. To allow for a precise movement the sample was mounted on a computer controlled three axes positioning system (Aerotech ALS 130). A schematic of the setup is given in Fig. 1.

To determine the linear guiding properties we coupled light at a wavelength of 800 nm into the waveguides using a 4× microscope objective with a NA of 0.1 and measured the near field profile by imaging the transmitted light with a 10× microscope objective (NA = 0.25) onto a CCD camera (DataRay Inc., Wincam). Based on the measured near-field profile the refractive index profile can be calculated by solving numerically the Helmholtz equation [11]:

In Fig. 2 (a) measured near-field profile and the corresponding index profile are shown for a waveguide written at a speed of 500 μm/s. The dimensions are 3 μm × 12 μm which corresponds to the size of the focus of the writing beam. The maximum change of the refractive index was 1.3 × 10^{-3}. These results agree with measurements using a Refracted Near-Field profilometer (Rinck Elektronik, Germany) and an interference microscope.

In order to investigate the influence of the writing velocity on the properties of the waveguides we varied the sample movement between 500 and 1500 μm/s. The refractive index change was determined for each waveguide in the way mentioned above, and the losses of the waveguides were measured using the cut back method. The results are summarized in Fig. 3.

The index change is a function of the writing speed and thus on the deposited energy, as expected. Furthermore the experimental results propose an exponential dependence. This behavior is in agreement with recent experiments [12]. In contrast the loss of the waveguides does not depend significantly on the writing velocity.

For the nonlinear characterization we used a Ti: Sapphire CPA laser system (Spectra Physics, Spitfire) with a pulse duration of about 150 fs, a repetition rate of 1 kHz and pulse energies up to 0.3 μJ. The pulses were coupled into the waveguides by a 4× microscope objective (NA 0.1). The spectrum of the transmitted light was measured with a commercial spectrometer (Ando, AQ-6315A) for different pulse peak powers. Due to the less tight focus no further refractive index modifications were induced by the coupled-in pulses. This was confirmed by measuring the intensity-dependant transmittance of the waveguides which exhibits a linear behavior. In addition, no different results were obtained by repeating the measurement several times. In order to avoid surface damage when investigating the nonlinear guiding properties the waveguides were buried about 1.5 mm away from the incoupling facet. This reduces the applied fluence at the surface which has a significantly lower damage threshold than the bulk material. Therefore pulses with a higher peak power can be coupled in the waveguides.

The propagation of ultrashort pulses through a waveguide can be described by the Generalized Nonlinear Schrödinger Equation (GNLSE) [13]

In this equation a retarded timeframe is used and the field amplitude *A* is normalized such that |*A*|^{2} is equal to the optical power. *α* is the loss of the waveguide, *β*
^{2} is the group velocity dispersion and *T*_{R}
accounts for the delayed Raman response. *γ* is a measure for the third order nonlinearity and is connected to the nonlinear refractive index n_{2} by

where *A*_{eff}
is the effective mode-area [13], *ω* is the frequency of the propagating light and c denotes the vacuum speed of light.

Since the GNLSE cannot be solved analytically we used a symmetric fourier split-step method to solve Eq. (2) numerically [13]. For these simulations we assumed the pulses to be gaussian and transform-limited. The energy of the pulses was calculated from the measured average power, considering Fresnel losses and the overlap integral of the focal spot of the incoupling microscope objective and the shape of the propagating mode in every waveguide. The pulse duration was measured by a second order autocorrelation and the losses of the waveguides were determined as mentioned above. The group velocity dispersion is taken to be equal as in the unprocessed bulk material while the Raman effect was neglected due to the low influence on the shape of the propagating pulse. The nonlinear refractive index was used as a parameter to fit the calculated spectra to the measured ones. Typical measured and simulated spectra after propagation through a waveguide are shown in Fig. 4.

With respect to the shape of the spectra experiment and theory are in good agreement. The actual broadening of the spectrum, however, is smaller than the theory predicts. This may be due to the assumptions made for modeling the propagation such as transform limited or the Gaussian shape of the pulses. Nevertheless, this analysis allows an identification of the accumulated nonlinear phase shift in the experimentally obtained spectra. Since the number and amplitude of the peaks in the spectra is characteristic for the nonlinear phase shift these can be used for a precise estimation of the nonlinear phase shift [13]. The obtained values for the nonlinear refractive index, summarized in Fig. 5 as a function of the writing velocity, are in good agreement with results from other nonlinear experiments performed with waveguides written with the same parameters [9].

It is important to note that the values obtained here for the nonlinear refractive index are effective values, i.e. mean values weighted by the intensity profile *I(x,y)* of the mode. Since the mode extends in the bulk material surrounding the waveguide the values for the effective nonlinearity are influenced by both, the *n*_{2}
within the waveguide region and that of the bulk. Therefore the ${n}_{\mathit{2}}^{\mathit{\text{eff}}}$
is calculated as

where *I(x,y)* is the intensity distribution of the propagating mode and *n*_{2}*(x,y)* is the local value of the nonlinear refractive index. Following this one could expect an even stronger decrease of *n*_{2}
in the waveguide region in order to obtain the observed values of the effective nonlinear refractive index. However, to calculate the amplitude of *n*_{2}
based on Eq. (4) requires the detailed knowledge of the spatially resolved shape of *n*_{2}
. Localized measurements of *n*_{2}*(x,y)* are in preparation.

As Fig. 5 shows the effective nonlinearity is strongly reduced due to the writing process. The changes in the nonlinearity (approximately to 25 % of the original bulk value at a writing velocity of 500 μm/s) are much stronger than the changes of the linear refractive index. However, both changes are dependent on the writing parameters. This offers the possibility to adjust not only the linear but also the nonlinear optical properties of the material according to the experimental requirements.

## 3. Conclusion

In conclusion we have investigated the nonlinear properties of fs-laser written waveguides analyzing self phase modulation. The nonlinear refractive index is strongly reduced by the writing process. Since the reduction is highly dependent on writing speed this allows adjusting the nonlinearity of the material to the experimental requirements. Thus, these results will have an important impact on the future design of fs-written nonlinear integrated optical devices.

## Acknowledgment

We acknowledge support by the Deutsche Forschungsgemeinschaft (Research Group “Nonlinear spatial-temporal dynamics in dissipative and discrete optical systems”, FG 532), the Federal Ministry of Education and Research (Innoregio, 03ZIK051) and the Thüringer Kultusministerium.

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