## Abstract

The observation of discrete spatial solitons in fs laser written waveguide arrays in fused silica is reported for the first time. The fs writing process permits the specific setting of the linear and nonlinear guiding properties of the waveguides. The results in this paper reveal a new avenue for the fabrication of various nonlinear optical devices.

© 2005 Optical Society of America

Recently it was demonstrated that one can write optical waveguides along arbitrary paths by tightly focusing ultrashort femtosecond laser pulses into fused silica bulk material [1]. Due to the high energy densities in the focal region multi-photon absorption occurs. As a result a very localized change of the material’s properties can be achieved. By moving the sample transversely to the beam a continuous change of the refractive index can be created. With this powerful technique it is possible to fabricate three-dimensional optical elements [2] and waveguide arrays of arbitrary shape [3].

The linear propagation in one-dimensional (linear) arrays of evanescently coupled waveguides is well understood [4], so the next consequential step is the investigation of nonlinear effects. Discrete nonlinear self-focusing was first suggested in 1988 [5] and observed in 1998 in etched ridge waveguides on AlGaAs substrates [6]. So far nonlinear results were obtained only in arrays of etched ridge waveguides, in optically induced lattices in compliant media such as in photorefractives [7], [8] and periodic voltage biasing in liquid crystals [9] and fiber waveguide arrays [10]. Etched arrays are stable but limited to planar configurations while arrays induced in photorefractives do not need manufacturing to modify the structural geometry but are sensitive to real time conditions [11]. Up to now fiber waveguide arrays exhibit strong disorder due to the fabrication process which eliminates homogeneous coupling.

In this paper we demonstrate nonlinear discrete localization in a planar fs laser written waveguide array for the first time, to the best of our knowledge, which lays the foundation for future work in two-dimensional arrays. The use of fs laser written waveguide arrays exhibits several advantages. All structures induced in the bulk material are permanent and therefore non-sensitive to external distortions. Furthermore all devices are of arbitrary shape and not limited to planar geometries. Up to now only the use of fs written waveguide arrays allows the fabrication of three dimensional nonlinear devices such as nonlinear two-dimensional switches [12] and soliton routers [13]. The investigation of the nonlinear properties of such structures is therefore the base for future nonlinear applications.

For the fabrication of the waveguides we used a Ti:Sapphire laser system (RegA/Mira, Coherent Inc.) with a repetition rate of 100 kHz, a pulse duration of about 150 fs and 0.3 μJ pulse energy at a wavelength of 800 nm. The beam was focused into a polished fused-silica sample by a 20x microscope objective with a numerical aperture of 0.45 (Fig. 1). In the focal area multi-photon absorption occurs due to the high energy density, inducing a permanent refractive index change in the material. The focal plane inside the sample was about 150 μm deep. The high repetition rate of the laser system allowed a transverse movement of the sample as fast as 1000 μm/s, performed by a positioning system (ALS 130, Aerotech).

The resulting index changes were determined by measuring the near-field profile at a wavelength of 800 nm (Fig. 2(a)), and solving the Helmholtz-Equation [14]. The maximum index change obtained was Δ*n* ≈ 1 × 10^{-3} with a size of 3 μm × 14 μm (Fig. 2(b)). The transmission losses of a single waveguide, measured by a cut-back method, were < 0.4 dB/cm.

Our arrays consist of 9 waveguides with a separation of 48 μm and a length of 74 mm (Fig. 3). In order to avoid damage of the device when exciting with high power laser pulses, the waveguides are buried 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. The additional coupling losses are measured to be only 15 %.

For the investigation of the spatial nonlinear propagation we used a Ti:Sa CPA laser system (Spitfire, Spectra-Physics) with a pulse duration of about 100 fs, a repetition rate of 1 kHz and pulse energies of up to 3 μJ at 800 nm. The light was coupled into the center waveguide with a 4x microscope objective (NA = 0.10), coupled out by a 10x objective (NA = 0.25) and projected onto a CCD-camera.

The propagation in discrete systems with a cubic nonlinearity can be modeled by a coupled mode approach under the assumption that field shapes are constant along the propagation direction and that only the modal amplitudes *E _{n}* evolve. This leads to a system of coupled ordinary differential equations [5]

where *E _{n}* is the amplitude in the

*n*th waveguide,

*β*is the propagation constant,

*c*is the coupling constant between adjacent waveguides and

*κ*is a measure for the waveguide’s third-order non-linearity. This so-called Discrete Nonlinear Schroedinger Equation (DNLSE) yields the possibility of the formation of discrete localization in waveguide arrays.

For small input power propagation in a waveguide array without boundary effects can be adequately modeled by Eq. (1) neglecting the nonlinear term (*κ* = 0). Under the condition that only one waveguide is excited (*E*
_{0} = *A*
_{0}, *E*
_{n≠0} = 0 at *z* = 0) the field propagation corresponds to the Green’s function of the system. The energy distribution for the nth waveguide is then [15]

where *J _{n}*(

*ξ*) is a Bessel function of order

*n*. Comparison of the experimental data with the numerical results yields a coupling length of

*l*= 116 mm at 800 nm, according to a coupling constant of

_{c}With increasing pulse energy the nonlinearity of the system becomes important so that for a valid description of the light propagation the nonlinear term in Eq. (1) can no longer be neglected (*κ* ≠ 0). The predicted localization with increasing peak power is experimentally observed in Fig. 4. The output pattern is shown as a function of the peak power.

Whereas in the lower peak power range almost all of the guided energy is coupled to the adjacent waveguides due to linear coupling, at a peak power of 1000 kW the output intensity pattern is localized in the center waveguide which has been excited at the entrance. A comparison of the experimental data with a numerical evaluation of Eq. (1) is shown in Fig. 5, which exhibits an excellent qualitative agreement in the evolution of the amplitudes in the individual waveguides.

However, evaluating Eq. (1) with the nonlinear refractive index of unprocessed fused silica (*n*
_{2} = 2.7·10^{-20}
*m*
^{2}/*W*) yields a peak power of only 250 kW necessary for the generation of a localization. An explanation of that unexpected behavior is the value of the coefficient *κ* which is a measure of the waveguide’s effective third-order nonlinearity. It is calculated as

where *ω* is the optical frequency and *v* the vacuum speed of light. In usual approximations the material’s nonlinearity *n*
_{2}(*r*) is assumed to be constant and therefore taken out of the integral [16]. In the present case this simplification does not apply. Due to the waveguide-writing process not only the linear but also the nonlinear index of refraction seem to be modified. Since the dimensions of the modes do not highly exceed the waveguide area this yields a considerable change of the integral in the numerator of Eq. (4). An approximation of Eq. (4) is, considering the mean value theorem,

A reduction of *n*
_{2} due to fs laser structuring has been reported recently [17]. This behavior can be explained by the fact, that the illumination with fs laser pulses changes the inner molecular structure of the bulk material in the focal region. Hence, not only the linear but also the nonlinear refractive index, which are determined by the inner structure, are changed by the writing process. Therefore the *κ*-coefficient in Eq. (1) is reduced, which leads to an increasing required peak power necessary for the generation of a nonlinear localization. Hence, compared to the *n*
_{2} of the bulk material, the effective nonlinear refractive index of the waveguides is reduced to *n*
_{2(eff)} = 0.25*n*
_{2(bulk)}. However, since the change in *n*
_{2} strongly depends on the writing parameters this allows to adapt the nonlinear refractive index to the experimental requirements as an additional degree of freedom in the fabrication of fs laser written waveguide arrays.

In conclusion we have demonstrated the ability to induce nonlinear localization in fs laser written waveguide arrays in fused silica bulk material for the first time. It turned out that due to the writing process the nonlinear coefficient *n*
_{2} of the material was locally changed which has to be taken into account for the design of such discrete nonlinear systems. The use of fs laser written waveguide structures provides a flexible technique to fabricate permanent structures for linear and nonlinear applications. The next step in development is the fabrication of two-dimensional nonlinear waveguide arrays which are the basis for new integrated optical devices such as nonlinear switches and routers.

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, ZIK) and the Thüringer Kultusministerium.

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