We experimentally observe the formation of spatial dark gap solitons in higher bands in one-dimensional waveguide arrays possessing a saturable defocusing nonlinearity. By using the prism-coupler scheme, pure Floquet-Bloch modes of higher bands are excited and dark gap solitons are formed due to the counteraction of normal diffraction and the defocusing nature of the photovoltaic nonlinearity. The modulation of refractive index induced by the soliton formation is demonstrated by the guidance of a low-power probe beam in the waveguide array sample. Additionally, the phase structure of dark solitons formed in the second band is discussed.
©2010 Optical Society of America
Spatial solitons, as one of the most intriguing nonlinear wave phenomena, have been extensively investigated in various homogeneous media [1–4] for decades. This research direction has been extended further into different complex photonic geometries, for example Bose-Einstein condensates (BEC’s) in optical lattices , optically-induced photonic lattices , and nonlinear channel waveguide arrays [7–11]. For the simple case of one-dimensional (1D) nonlinear periodic systems such as waveguide arrays, the propagation of the scalar field can be modeled by the paraxial wave equation :
Here E represents the field amplitude and k = nsub k 0. Here k 0 = 2π/λ is the wave number in vacuum, λ is the wavelength, and nsub is the substrate refractive index. The transverse periodic modulation of refractive index is denoted by Δn(z), while Δnnl (z) represents the refractive index change induced by the nonlinearity. A self-focusing nonlinearity is described by Δnnl > 0 while Δnnl < 0 denotes the self-defocusing case. Without this nonlinear term, the solutions of Eq. (1) are extended Floquet-Bloch (FB) modes labeled by the band index and the Bloch momentum kz which varies in the Brillouin zone (BZ). The permitted FB modes form discrete bands separated by gaps, where wave propagation is forbidden. However, in a nonlinear medium, a localized excitation of the array can form a corresponding local deformation, i.e. the refractive index is either increased or decreased in the illuminated region when compared with the uniform lattice. Consequently, defect states are formed above or below allowed bands, which support the existence of discrete lattice solitons. Among these, gap solitons are one type of solutions enabled by the interplay between nonlinearity and discrete diffraction [11–13]. In a system possessing a self-defocusing nonlinearity both, bright and dark, gap solitons can be found. Bright gap solitons exist in regimes of anomalous diffraction. For bands with an odd index, including the 1st and 3rd band, bright solitons can be found at the edge of the BZ, whereas in bands with an even index, like the 2nd band, they exist at the center of the BZ [14,15]. In contrary dark gap solitons arise due to interaction of a defocusing nonlinearity and normal diffraction. For odd bands they locate at the BZ center, and for even bands they exist at the BZ edge. Vice versa, bright gap solitons exist also for focusing nonlinearity and normal diffraction, while their dark counterparts are formed in anomalous diffraction regions in such media. In this work, we report on the experimental formation of dark gap solitons originating from higher bands in a defocusing material, and compare our results by numerically solving the nonlinear paraxial wave equation.
2. Experimental methods
The 1D waveguide array (WA) used for our experiments was fabricated by high temperature in-diffusion of titanium into an iron-doped lithium niobate (LiNbO3) substrate. This nonlinear crystal exhibits a saturable defocusing nonlinearity originating from the bulk photovoltaic effect . The transverse direction z is parallel to the ferroelectric c-axis and the propagation direction is along the y-axis. The sample array consists of 250 parallel channels with a period Λ = 8.5 μm. One of the end facets of the sample is polished to optical quality to enable the direct observation of the out-coupled light from the array.
Due to the unique ability of selective excitation of FB modes in any desired band and the possibility of precise determination of band structures of waveguide arrays, the prism-coupling setup is employed in our experiment [17,18]. As shown in the schematic drawing of the experimental setup in Fig. 1 , the input light of wavelength λ = 532 nm from a Nd:YVO4 laser is expanded to a plane wave and then split into two beams. One beam passes firstly through a phase mask which provides a π phase shift by covering half of the input beam along the transverse direction z. In this way a dark notch is formed perpendicular to z. This signal beam is imaged by a lens onto the coupling point of the prism coupler, and by using different focal lengths of this lens we can control the input width of the dark notch on the WA. One part of the input signal beam is reflected at the prism base and leaves the prism from the input facet. This beam is monitored by a CCD camera combined with a microscopic objective. In this way the phase mask can be adjusted in order to optimize the position of the dark notch relative to the waveguides. The second beam is again divided into two beams. One of these beams is superimposed to the signal beam and serves as a broad probe beam for checking both, the guiding properties of the array and the phase profile of the signal light on the prism coupler (by using interference). Furthermore, the intensity of the not in-coupled probe beam leaving the prism can be detected by a photodiode to measure the sample’s band structure. The third beam is utilized to form an interferogram with the out-coupled signal light and thus allows for analyzing the phase structure of dark solitons. Finally, after propagating to the end facet, the output signal is imaged by another CCD camera with a microscopic objective, which allows the observation of the intensity (and phase) of an area of about 25 channels.
3. Experimental and numerical results
First, the linear band structure of the sample has been determined. As can be seen in Fig. 2 , the change of the effective refractive index neff , which is associated with the propagation constant β = neff k 0 of the permitted modes, depends on the transverse wave vector kz. Here nsub = 2.2341 is the refractive index of our LiNbO3 substrate for extraordinary polarized light.
The band structure in the left panel shows three guided bands of the sample. In addition, the increasing curvature from the 1st to the 2nd to the 3rd band demonstrates the overall increase of the diffraction coefficient for higher bands. As mentioned before, to generate dark gap solitons using the self-defocusing nonlinearity of LiNbO3, FB modes should be excited in the region of normal diffraction, which are, as marked in Fig. 2, for the 1st and 3rd band at Bloch momentum kz = 0 and for the 2nd band at kz = π /Λ. The measured intensity profiles of those modes are also presented in the right panels in Figs. 2(b) and (2d), and 2(c) respectively.
In our experiments, dark solitons from the 2nd and 3rd band are investigated. At the center of the 1st band, although dark solitons do exist , normal diffraction of our sample is too weak to clearly manifest the counteraction of the defocusing nonlinearity. For dark solitons originating from the 2nd and 3rd band, we studied the linear and nonlinear propagation of the input signal beam, and monitored the induced refractive index changes by launching a weak probe beam into the array before and after soliton formation.
To support our experimental results, we performed numerical simulations by solving Eq. (1) with the nonlinear beam propagation method. For this we used the parameters of our WA: The effective transverse modulation for the refractive index of the WA is approximated by Δn(z) = 0.015 cos2(πz/Λ), and the refractive index change induced by the saturable defocusing nonlinearity takes the form as Δnnl = Δne I / (I + Id), with Δne = −2.5 × 10− 4 and I / Id = 1.5, where I is the light intensity and Id is the saturation irradiance.
For the formation of a dark soliton from the 2nd band, as presented in Fig. 3 , the input dark notch has a FWHM (full width at half maximum) of about two periods of the waveguide array, and it is centered on one channel. In the linear case with input power below 2 nW per channel, the linear discrete diffraction broadens the output of the dark notch to a width of about 5 channels after 20 mm propagation length. As soon as the input power is increased to 160 nW per channel, the nonlinearity starts to build up and gives rise to a narrowing of the output width of the dark notch. After 15 minutes the diffraction is totally balanced by the nonlinearity and a steady state is reached. These experimental results are in good agreement with the numerical simulations presented in the bottom panels of Figs. 3(a) and 3(b). In the experiment, the guiding properties of a broad weak probe beam (about 5 nW per channel) before and after the dark soliton formation are investigated. As shown in the top row of Fig. 3(a) and 3(b), respectively, before soliton formation (i.e., in the case of a linear array), the guidance of the probe beam is homogenous. However, after the soliton is formed, light is mainly guided in the region which is covered by the dark soliton. Obviously, the induced dark soliton imprints an additional (positive) refractive index modulation on the lattice, which effectively forms a defect state where light of the probe beam is trapped in. We also successfully proved the stability of these dark-notch structures by adding up to 10% of random amplitude noise to the input profiles.
The same experiments are conducted for a dark soliton originating from the 3rd band. The results are shown in Fig. 4 . As already discussed in , the stronger linear diffraction of higher bands requires higher nonlinearity to compensate for. Because the available saturable nonlinear index change in our sample is limited, the possible width of soliton solutions increases from lower to higher bands. Therefore, for the 3rd band, a larger input width of the dark notch is used to form a dark soliton, covering roughly 7 channels. The corresponding numerical results again fit nicely to our experimental results. The increase of the soliton width is proven by the stable guidance of the probe beam after the buildup of the dark soliton.
The investigation of the phase structure of the output signal light is a well-adapted method to check the successful formation of dark solitons. In the 2nd band, extended FB modes have a non-staggered form (i.e., adjacent elements have identical phase profiles), but on each single element a dipole structure with equal magnitudes but opposite phases occur [see also the right panel in Fig. 2(c)]. Thus, from the input signal amplitude of the dark soliton used in the numerical simulation in Fig. 5(a) , we know that, when superimposing an additional π phase shift in the center of the dark notch, the resulting phase distribution has to be symmetric. With the interferometer shown in Fig. 1 we revealed the phase profile of the output light [Fig. 5(b)] after dark soliton formation, and the result is given in Fig. 5(c). It can be clearly seen from the interferogram that the phase structure of the output light takes a symmetric form, which agrees well with the conclusions from our numerical simulations. However, we also notice that the inner part of the phase profile of the dark notch (at channel “0”) contains some small asymmetries. This may be attributed to a non-perfect phase adjustment (small deviations of the generated phase offset of π of the used mask), and possible small misalignments when adjusting the dark notch onto one channel of the waveguide array.
In this work, we have presented the experimental formation of dark gap solitons in the second and third band of a one-dimensional waveguide array which possesses a saturable defocusing nonlinearity. Because of the fixed magnitude of the induced photorefractive nonlinearity, and due to the increasing normal diffraction for higher bands, the width of a dark soliton originating from the third band is larger than that for the second band. The guiding properties of the induced dark soliton structures are experimentally probed by an extended probe beam, and clearly monitor the positive index change in the region of the dark notch. Furthermore, we analyzed the output phase profiles of the formed solitons, which are in excellent agreement with our numerical modeling. This work opens the way for the study of more complicated soliton (interaction) scenarios, for example the interplay of dark and bright localized excitations originating from different bands.
This research was supported by the Deutsche Forschungsgemeinschaft (grant KI 482/11-2). J. X. and D. S. thank the Humboldt Foundation for financial support.
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