1. Introduction

Periodical optical systems such as waveguide arrays (or photonic lattices) and photonic crystals have demonstrated many novel phenomena, for example, discrete diffraction [1, 2, 3] and self-collimation [4, 5, 6, 7, 8], for which no counterparts exist in the homogeneous media. This provides new opportunity to manipulate light propagation through structure design. The periodical system becomes more attractive if optical nonlinearity is introduced, and it can support spatially self-localized optical states named as spatial solitons [9] when the nonlinear confinement effect balances the tunnelling effect among the adjacent potential wells. The nonlinear periodical system enables light controlling light and novel potential applications such as optical router, switching and navigator where spatial solitons serve as one of the basic building block [10]. Various kinds of spatial solitons such as discrete/gap solitons [11, 12, 13, 14, 15, 16, 17], dipole-like solitons [18, 19], multi-band solitons [20, 21], soliton trains [22, 23], surface solitons [24, 25, 26, 27] and vortex solitons [28, 29, 30] were demonstrated in nonlinear periodical system. Analogies between wave optics in a periodically curved optical waveguide and quantum mechanical dynamics of electrons in an atomic potential interacting with an intense electromagnetic field were also studied recently [31]. In this paper, we demonstrated that a weakly modulated two-dimensional (2D) photonic lattice slab (PLS) offers versatile possibility to control light propagation through structure design and optical nonlinearity but with a relatively simple fabrication technology and easy chip-level integration for various optical devices. Linear discrete diffraction and nonlinear formation of discrete spatial solitons were observed in such a large-area 2D PLS exhibiting a saturable self-defocusing nonlinearity. The corresponding beam propagation dynamics in the weakly modulated large-area PLS were also studied numerically.

2. Fabrication and characterization of 2D PLS

The weakly modulated large-area 2D PLS was fabricated by use of optical induction technique. A LiNbO₃:Fe crystal with dimensions of 20×1×27 mm³ (X×Y×Z) was used as the virginal material exhibiting saturable self-defocusing nonlinearity due to the photorefractive photovoltaic effect. The crystalline C-axis was set along the X-coordinate direction. An amplitude mask was deposited directly on the surface parallel to the crystalline C-axis (X-Z plane), and the periodical structure on the amplitude mask was fabricated through photo-etching process. For simplicity but without loss of generality, we chose an amplitude mask with a square lattice consisting of tiny pinholes. The places with pinholes were transparent, and the pinhole diameter was 10 µm and the pinhole lattice spacing d was 20 µm. The 2D PLS was fabricated by launching an e-polarized and collimated 532 nm beam onto the X-Z surface of the crystal with the amplitude mask on it, as shown in Fig. 1, in which the experimental setup is depicted. This e-polarized and collimated 532 nm beam was called lattice beam and illuminated the whole X-Z surface of the crystal. A weakly modulated large-area 2D PLS with a backbone structure was generated due to the saturable self-defocusing nonlinearity of the LiNbO₃:Fe crystal after an appropriate amount of light exposure. Note that the refractive index at the places with pinholes was reduced and that it was homogeneous along the Y-coordinate.

 

Fig. 1. Experimental setup. S. F., spatial filter; L₁ and L₂, lens; M, mirror; λ/2, half-wavelength plate; A. T., attenuator; CCD, charge coupled device camera. The inset shows the mask on the X-Z plane of the LiNbO₃:Fe crystal, each pinhole has a diameter of 10 µm and the spacing between the two nearest pinholes is 20 µm; C denotes the crystalline C-axis.

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The structure parameters of the weakly modulated large-area 2D PLS was characterized by means of the Bragg-matched light diffraction technique. In the experiment, a collimated He-Ne laser beam operating at 632.8 nm was launched into the 2D PLS from the X-Y plane side and propagated paraxially along the 27-mm Z-coordinate direction (without the lens L₁ in Fig. 1). The intensity of the He-Ne laser beam was attenuated to be of micro-watt level in order not to induce any noticeable optical nonlinearity or erasure effect on the PLS. The light intensity distribution on the output surface of the PLS was monitored by a CCD camera through a lens L₂. Light diffraction was observed only at Bragg-matched incident angles obeying the relationship 2d sinθ_B=kλ_p, where θ_B is the Bragg-matched incident angle in air, k is an integer and λ_p is the wavelength of the He-Ne laser beam in vacuum. Figure 2(a) shows the output intensity profile for the normal incidence. Figures 2(b)–2(e) show the Bragg-matched diffraction which could be assigned to the 2D square lattice with Miller indices (h, k) integers [32]. The refractive index modulation amplitude Δn ₀ of the PLS can be estimated by measuring the Bragg-matched diffraction efficiency η10 with Miller indices (1, 0) through a formula ${\mathrm{\Delta }}_{n0}={\lambda }_p{\sin }^{-1}\left(\sqrt{{\mathit{\eta }}_{10}}\right)\cos {\theta }_B/\left(\mathit{\pi d}\right)$ [32, 33]. For a 2D PLS generated by a lattice beam of 1.38 mW/mm² and an exposure time of 40 minutes, the refractive index modulation amplitude of the PLS was estimated to be ~10^-4. It is evident that 2D PLS with various refractive index modulation amplitude can be fabricated by controlling the lattice beam exposure fluence.

 

Fig. 2. Bragg-matched diffraction of a collimated He-Ne laser beam from the 2D PLS. The red spots in (a)–(e) correspond to output positions of the incident beam and the diffracted beam at the incident angles of 0°, 0.92°, 1.87°, 2.86° and 3.75° in air, respectively. The Bragg-matched diffractions in (b)–(e) are associated with lattice faces with Miller indices (1, 0), (2, 1), (3, 1) and (4, 1), respectively.

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Fig. 3. The fine intensity distribution structure (the bottom image) on the X-Y output surface of an optically induced 2D PLS when one launches a collimated He-Ne laser beam onto the 2D PLS along the Z-coordinate, which indicates the effective waveguide array property of the photonic structure. The top image shows the profile of the amplitude mask on the X-Z plane.

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3. Linear and nonlinear discrete light propagation in 2D PLS

It is interest to note that such an optically-induced 2D PLS can also be viewed as an effective one-dimensional (1D) waveguide array. Figure 3 shows the fine structure of intensity distribution of the light spot at the output surface of the 2D PLS when one launches a collimated He-Ne laser beam normal to the X-Y surface of the crystal and propagates along the Z-coordinate direction. Here the 2D PLS was induced by a lattice beam of 1.38 mW/mm² with an exposure time of 40 minutes. It is seen that the incident light was coupled into and guided along the refractive index backbones of the 2D PLS. This is because the LiNbO₃:Fe crystal is of self-defocusing nonlinearity and the refractive index at the pinholes will be reduced under the illumination of the lattice beam, therefore the refractive index is high at the backbones of the PLS, forming effective 1D waveguide array along the Z-coordinate direction.

 

Fig. 4. The intensity distribution images (left column) and the corresponding intensity distribution profiles (right column) on the output X-Y surface of the 2D PLS at different exposure time t when a probe He-Ne laser beam was coupled into one backbone waveguide. The exposure time of the probe beam was t=0 minutes for (a) and (d), t=200 minutes for (b) and (e), and t=530 minutes for (c) and (f), respectively.

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It is not surprising that, by coupling a He-Ne probe beam into one backbone waveguide of 2D PLS employing an additional lens L₁, discrete diffraction typical for a waveguide array can be observed in such a 2D PLS. But more interestingly, we demonstrated that discrete solitons could also be generated in such a 2D PLS when the light tunnelling effect among the adjacent waveguides is balanced by the nonlinear confinement effects. Figure 4 shows the temporal evolution of the light intensity pattern at the output surface of the PLS when a He-Ne probe beam was coupled into one backbone waveguide of the 2D PLS using the lens L₁. In the experiment, the intensity ratio between the probe beam and the lattice beam was set to be 560:1. It is seen that linear discrete diffraction is clearly demonstrated in Figs. 4(a) and 4(d) when the nonlinearity is not arose at the very beginning of the probe beam incidence. As the time goes on and the nonlinearity increases, the light begins to contract gradually (see Figs. 4(b) and 4(e) for an exposure time of 200 minutes), and is finally confined mainly in the original waveguide (Figs. 4(c) and 4(f) at time t=530 minutes), indicating the formation of discrete soliton.

Note that the intensity ratio between the probe and the lattice beams was set to be very high as compared to those in literature reported for rigid waveguide arrays [34]. This is because there is relatively large energy leakage between the adjacent backbone waveguides due to the imperfect waveguide structure of the backbone PLS, therefore the nonlinearity necessary to form a discrete soliton should be much stronger than that for a rigid waveguide array even with the same virginal nonlinear material. It is well known that the wave coupling strength between the adjacent waveguides in a rigid waveguide array is mainly determined by the distance between the adjacent waveguides and the refractive index modulation amplitude of the individual waveguide [2]. Interestingly, for a backbone waveguide array in a 2D PLS, besides the distance between the adjacent waveguides and the refractive index modulation amplitude of the waveguides, it is also possible to finely tune the wave coupling strength between the adjacent waveguides by designing the longitudinal periodical distance of the pinholes, providing an additional way to manipulate the wave propagation behavior in such a 2D PLS.

 

Fig. 5. Numerical simulation of the light propagation dynamics in the 2D PLS. Output intensity profiles (top row) and the corresponding beam propagation(bottom row) for different values of the normalized input intensity 0 (a), 0.69 (b) and 1.69 (c), respectively.

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For comparison, we also simulated the beam propagation dynamics within the PLS using beam propagation method (BPM). The refractive index distribution profile Δn(x, z) of our 2D PLS can be approximately described as Δn(x, z)=Δn ₀ cos[(x+z)/d]cos[(x-z)/d], where Δn ₀ is the refractive index modulation amplitude of the PLS, d=20 µm is the lattice spacing, x and z are the transverse and longitudinal coordinates, respectively. Figure 5 shows the numerical simulations of the light propagation dynamics within the PLS corresponding to the cases shown in Fig. 4. From the results we can see that the light intensity distribution on the output surface (X-Y surface) experiences a transition from the very beginning discrete diffraction pattern to a final discrete soliton state with the increment of the optical nonlinearity. The nonlinear refractive index change induced by the soliton is ~2×10^-4. The numerical simulations are in good agreement with the experimental results, indicating that the 2D PLS indeed can be used to control the light propagation dynamics and to support a discrete soliton state. In addition, due to the relatively larger energy leakage between the neighboring effective waveguides in the 2D PLS than that in traditional 2D photonic crystal and waveguide arrays [8], the coupling effect between the neighboring lattice structure is prominently important. More important, it is relatively easy to embed lattice defects in the lattices, to fabricate lattice interfaces among different lattice structure, to integrate functional lattice units on chip level, and finally to achieve fine tuning of the light circuit in such a large-area 2D PLS.

4. Conclusion

We fabricated large-area weakly modulated two-dimensional photonic lattice slabs using the optical induction method in LiNbO₃:Fe crystal exhibiting a saturable self-defocusing nonlinearity. We observed experimentally the linear and nonlinear discrete light propagation dynamics in such a photonic lattice slab. The experimental results show that the weakly modulated two-dimensional photonic lattice slab supports the Bragg-matched light diffraction, the discrete light diffraction, and the formation of discrete optical spatial soliton under appropriate conditions. However, higher optical nonlinearity is required to form a discrete soliton due to the stronger energy leakage between the adjacent effective waveguides in the lattice slab. The numerical simulation confirms the experimental observation.