## Abstract

We investigate dynamic localization in curved femtosecond (fs) laser written waveguide arrays. The light propagation inside the array is directly observed by monitoring fluorescence of color centers induced during the fs writing process. In addition to monochromatic excitation the spectral response of the arrays is investigated by launching white light supercontinuum into the arrays.

© 2008 Optical Society of America

## 1. Introduction

Since the first experimental demonstration of discreteness in optics in waveguide lattices in 1975 using planar GaAs channel waveguides [1], the research in discrete optical systems has gained much interest in recent years, when a variety of effects in waveguide arrays were discovered and studied. Much attention was spent on nonlinear propagation effects [2], in particular after the prediction of discrete solitons in waveguide lattices [3]. The experiments dealing with such effects were not just limited to one-dimensional geometries [4–6], but also extended into the two-dimensional regime [7, 8]. Even broadband localization was achieved using supercontinuum radiation [9].

However, waveguide arrays also exhibit numerous linear peculiarities worth to be subject of intensive research. So, both the unique imaging [10, 11] and propagation [12] properties of waveguide lattices were theoretically analyzed and harmonic oscillation [13] and the propagation in finite lattices [14, 15] was experimentally studied. Furthermore, Zener tunneling [16], the discrete Talbot effect [17] and quasi-incoherent propagation [18] were experimentally investigated in detail. Due to the equivalent mathematical structure of the underlying equations, discrete optical phenomena exhibit a strong correspondence to similar effects in other fields of physics. One of the most prominent examples are electronic Bloch oscillations in solid state physics, which have an analogue in arrays of evanescently coupled optical waveguides [19,20]. Here, a linear gradient of the refractive index of the individual guides acts as a potential, yielding a periodic motion of the excitation. Since the experimental approach for the investigation of the optical Bloch oscillations is much simpler than for the analog effect in solids, this offers the unique possibility to investigate Bloch oscillations by using fundamental symmetries of the governing equations. Consequently, photonic Bloch oscillations were experimentally verified in one-dimensional [21, 22] as well as in two-dimensional [23] waveguide lattices.

As well as Bloch oscillations the Dynamic localization has its origin in solid state physics [24] and was an extensively investigated research topic in the following years (see e.g. [25]). In optical waveguide systems the external field is mimicked by a curvature of the waveguides itself. Dynamic light localization can be achieved in periodically bent waveguide arrays [26] due to the different effective paths in the individual guides caused by the bending. Iyer *et. al.* [27] investigated the spectral behavior of Dynamic localization at 1555 nm. These experiments were accomplished in arrays in LiNbO_{3} and AlGaAs using complex fabrication processes, which demand a careful sample preparation. In contrast to these methods, the fs writing technique [28] allows flexible and fast preparation of diverse high-precision samples with almost arbitrary curvatures and lengths.

In this paper we demonstrate Dynamic localization in curved fs laser written arrays at visible wavelengths. The localization is spectrally resolved using a white light supercontinuum as well as directly observed along the propagation in the samples using color center fluorescence.

## 2. Dynamic localization

In the continuum limit the propagation in a general one-dimensional waveguide array is physically equivalent to the propagation in a film waveguide with a transverse periodic potential, which is governed by the paraxial Schroedinger equation [29]

Here, Ψ(*x*-*x*
_{0}, *z*) represents the field envelope, *λ* the wavelength, *n*
_{0} the refractive index of the medium and *v*(*x*-*x*
_{0}(*z*)) the transverse refractive index profile.A periodic longitudinal bending of the waveguides is described by *x*
_{0}(*z*), where the periodicity yields *x*
_{0}(*z*)=*x*
_{0}(*z*+*z*
_{0}) with *z*
_{0} as the bending period. In this approximation, the bending is much larger than the the transverse period *d*, so that *z*
_{0}≫*d*. Applying the coordinate transformation *ẑ*=*z* and *x̂*(*z*)=*x*-*x*
_{0}(*ẑ*) resulting in ∂_{x}=∂_{x̂} and ∂_{z}=∂_{ẑ}-*ẋ*
_{0}∂_{x̂} where the point denotes the derivation by *z*, the field Ψ(*x̂*, *ẑ*) satisfies the equation

Applying the gauge transformation

Eq. 1 transforms into an expression which describes a straight array under the influence of an external ac field

Assuming the field to be a superposition of single modal fields of the individual waveguides [29], one arrives at the discrete approximation, whose coupled mode equations read

with *ω*=2*πn*
_{0}
*d*/*λ* as a normalized optical frequency. The coefficient *c* defines the coupling strength between adjacent guides in a straight array, where *x*
_{0}(*ẑ*)≡0. Additionally, *c* depends exponentially on wavelength and waveguide spacing [30]. The eigenfunctions of Eq. 5 are

with *κ* as the transverse wave number. From this expression it directly follows that after a full bending period *ẑ*→*ẑ*+*z*
_{0}, the diffraction is equivalent to that in a straight lattice with the effective coupling coefficient [31]

Dynamic localization occurs when *c*
_{eff}=0. This can be achieved for the bending profile

with *A*
_{0} as the bending amplitude. Inserting this in Eq. 7 and using the relation [32]

for the Bessel function *J _{α}*, one arrives at

Dynamic localization is obtained, if

holds, where *η* is any root of the equation *J*
_{0}(*η*)=0. It results from additional periodic phase shifts of the propagating modes caused by the bending. Hence, this effect is rather distinct from the well known Bloch oscillations [19], where localization is due to total internal and Bragg reflection.

## 3. Experimental results

In our experiments we used one-dimensional lattices composed of 15 waveguides fabricated by fs direct writing [28]. This technique provides the possibility to fabricate large one- [33] and even two-dimensional [34, 35] lattices. Writing waveguides using fs laser pulses provides several advantages. The variety of modified materials ranges from different glasses [36, 37] to crystals [38] and polymers [39]. The properties of every single guide can be precisely tuned by the fabrication parameters [30, 40] and, as a particular feature, the waveguides can be written along arbitrary paths [41]. Hence, sinusoidally curved lattices can be produced with high precision. The principal setup and a microscopy image are shown in Fig. 1. For the writing procedure we used a Ti:Sapphire laser system (Coherent Mira/RegA) at 800 nm with a repetition rate of 100 kHz at an average power of 22-28 mW and a pulse length of 180 fs FWHM. The beam was focused using a 20× microscope objective (NA=0.35) approximately 500 *µ*m below the surface of our fused silica samples resulting in waveguides with a height of 12 *µ*m and a width of 4 *µ*m. The writing velocity was chosen to be 1500 *µ*m/s. Our samples were 25 mm and 50 mm long respectively with a waveguide separations of 14 *µ*m or 16 *µ*m.

To visualize the periodic localization inside the sample, a special type of fused silica with a high content of OH was used (Suprasil 311). In this case, non-bridging oxygen hole centers (NBOHCs) are generated during the writing process due to breaking the ≡Si-O-Si≡-bond into ≡Si-O^{●} and _{●}Si≡ (positively charged oxygen vacancies, E’ center) [42]. The NBOHCs have broad absorption bands at 4.8 eV and 2.0 eV [43] and exhibit an absorption maximum at 620 nm [44], so that, when launching red light from a HeNe laser at *λ*=633 nm (corresponding to 1.95 eV) into the waveguides, these color centers are excited and the resulting fluorescence (*λ*=650 nm) can be directly observed [18]. Since the color centers are formed exclusively inside the waveguides, this technique yields a high signal-to-noise ratio. In contrast to fluorescent polymers (see e.g. [16]), the bulk material causes almost no background noise. The experimental setup is sketched in Fig. 2(a). The laser beam is launched into the waveguide array using a 10× microscope objective (NA=0.25). The fluorescence distribution representing the diffraction pattern is imaged onto a CCD camera (Basler sc1000-30fm) using a 5× objective (NA=0.13). Most of the scattered HeNe-laser light is blocked by passing through a 650 nm longpass filter which improves the image quality. The camera’s electronic noise is suppressed by averaging over 10 images. Since the lattice size is 50 mm×300 *µ*m (160:1 in terms of length-width aspect ratio) 125 images have to be recorded to visualize the whole array with high resolution. The single images are resized to 20 pixels in the propagation direction and stitched together yielding one image with a resolution of 40 *µ*m×0.5 *µ*m per pixel. Figure 2(b) demonstrates that the normalized light distribution inside the excited waveguide of a straight array and the fit with the theoretical result known as the zero-order Bessel function *J*
_{0}(2*cz*) [15] where the coupling constant c acts as the fitting parameter are coinciding very well. This verifies the linear dependence of the fluorescence on the intensity of the propagating light in waveguides and proves that the fluorescence intensity reproduces very exactly the behavior inside the waveguides. Therefore, the measured intensity pattern can be used to investigate the light evolution in more complex structures instead of reconstructing it from the intensity output pattern.

In addition, the fluorescence method allows to determine the propagation losses easily without cutting the sample or applying the intricate Fabry-Perot method. At first single waveguides with a particular bending were written with the fs-laser. To achieve a curved waveguide (e.g. a cosine profile) instead of a straight, one period was split into 100 steps of small straight lines. This fragmentation was chosen as a compromise between the limitations of the positioning system and the necessary resolution to minimize deviations from the intended path. The segments are processed continuously resulting in a smoothly curved waveguide without any defects at the conjunctions. The additional bending losses are measured for cosine profile waveguides with a period of 20 mm, where the sideshift represents the peak-to-peak amplitude *A _{PP}* (Fig. 3) of the bending profile. For laser powers of 22 mW and 28 mW corresponding in the straight case to the lower boundary of guiding in the fabricated structures and the upper limit of single mode behavior respectively, the amplitude was increased from 0

*µ*m to 90

*µ*m (Fig. 3). Straight waveguides exhibit low losses for low writing powers [45] while high power written waveguides have a higher refractive index profile Δ

*n*=4·10

^{-4}resulting in lower losses for transverse shifts larger than 40

*µ*m/cm. The graph shows that for optimized straight waveguides (green curve) the losses increase rapidly for larger transverse shifts due to the weaker guiding. This can be observed in the corresponding fluorescence images where the intensity inside the waveguides has almost completely vanished at the end of the sample for large bending amplitudes (Fig. 3 upper right image).

Based on these measurements of the propagtion losses, an intermediate power was chosen for working with reduced bending losses but operating in single-mode regime from 500 nm–800 nm. The array consists of 15 waveguides and the selected profile corresponding to Eq. 8 with a full amplitude *A*
_{0}=47.5 *µ*m and a periodicity of *z*
_{0}=25 mm inside a 50 mm long sample. The light evolution in such an array is shown in Fig. 4. After the first half-period the light spreads across 9 waveguides, accumulates its inverse phase in the second half-period and is localized after a full cycle. The discrete theory (Eq. 5) neglects the higher order coupling. This assumption is made because the ratio of second order coupling and next neighbor coupling *c*(28*µm*)/*c*(14*µm*)≈3% [30]. In contrast AlGaAs waveguide arrays [27] have much smaller waveguide spacings and the nearest neighbor tight binding approximation does not hold so that more generalized curvature profiles have to be applied [46, 47]. Derivations from the localization can be explained by small inhomogeneities inside the array itself, since the output profile is asymmetric.

The characterization of the arrays for differentwavelengths was performed using polychromatic light [48,49], which was generated by nonlinear processes inside a photonic crystal fiber (PCF). For that purpose laser pulses of about 1 ps from a Ti:Sapphire oscillator (Spectra Tsunami) were expanded using a telescope and then coupled into and out of the PCF (Crystal Fibre - NL-1.7-650) using a 20× microscope objective (NA=0.35). The spectrum of the generated supercontinuum was measured by a spectrometer (Ocean Optics, USB 2000). The light was launched into the waveguide array by a 10× microscope objective (NA=0.25). The outcoupling of the light was accomplished using a 4× microscope objective (NA=0.10). To generate chromatic dispersion the light passed a prism and was then projected onto a color CCD camera (Linos DFW V500). A sketch of the setup is shown in Fig. 5(a), the supercontinuum spectrum is depicted in Fig. 5(b).

For the polychromatic characterization an array with a slight larger spacing of 16 *µ*m and a length of 25 mm was used because the coupling constant increase strongly for wavelengths up to 800 nm. When launching polychromatic light into a straight waveguide lattice, discrete diffraction occurs, which increases with growing wavelength due to changes in the coupling strength [30]. The corresponding polychromatic output pattern for a straight array is depicted in Fig. 6(a) which shows an increasing broadening of the beam from left (smaller wavelengths) to the right (larger wavelengths). While in the blue regime the light spreads only to the immediate neighboring waveguides, in the infrared the light spreads almost over the entire lattice. In Fig. 6(b) a corresponding simulation using a wavelength dependent coupling *c*(*λ*)=*c*
_{0} exp(*γλ*) [30] with *c*
_{0}=1.4m^{-1} and *γ*=6.75 *µ*m^{-1} is shown. While the diffraction width of the beam is well described, in particular in the green and blue wavelength region the output pattern slightly differs in the experiment and the simulation. This is due to the onset of multimode behavior at wavelengths below 500 nm. Additionally, the CCD camera was slightly overdriven since the intensity in the red is weaker than in the green and blue Fig. 5(b). Nevertheless, the simulation gives a rather accurate picture of the varying coupling behavior at different wavelengths.

The diffraction behaviour changes dramatically in curved arrays. According to Eq. 11, for dynamic localization at 633 nm, the curving amplitude is *A*
_{633nm}=42 *µ*m for a period of *z*
_{0}=25 mm. In this case, the accumulated phase caused by the bending yields a vanishing effective coupling so that dynamic localization occurs as shown in Fig. 7(a). The white arrow indicates the design wavelength of 633 nm, at which the light does not spread into the adjacent waveguides. In contrast, at other parts of the spectrum discrete diffraction is clearly observed. Dynamic localization at 543 nm is obtained for an amplitude of *A*
_{543nm}=36 *µ*m for the same bending period. The resulting polychromatic output pattern is shown in Fig. 7(b).

## 4. Conclusion

In conclusion, we demonstrated dynamic localization in fs laser written curved waveguide arrays. The fluorescence imaging allows measuring the propagation losses and monitoring the light propagation in particular dynamic localization. In addition the spectral dependence was investigated by launching white light supercontinuum into the waveguide arrays. It could be shown that dynamic localization occurs only at the design wavelength and light at the wavelengths in the vicinity still experience discrete diffraction. These results may pave the way for the experimental investigation of more complex, in particular two-dimensional diffraction managed waveguide lattices.

## Acknowledgement

The authors wish to thank I. Garanovich from The Australian National University Canberra for fruitful discussions. A. Szameit was supported by a grant from the Jenoptik AG.We further acknowledge support by the Deutsche Forschungsgemeinschaft (Research Unit 532 “Nonlinear spatial-temporal dynamics in dissipative and discrete optical systems”) and the Federal Ministry of Education and Research (Innoregio, 03ZIK051).

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