1. Introduction

Three dimensional (3D) photonic structures have been highly touted and proposed for many applications due to their capability of controlling and manipulating the flow of light in all dimensions [1], yet for decades it has been a challenge to create or fabricate 3D structures in bulk media at microscopic length scale. Thus far, various techniques have been proposed and developed for making 3D microstructures and photonic crystals, from traditional mechanical hole drilling to semiconductor microfabrication, from self-assembly and colloidal crystallization to advanced layer-by-layer fabrication, and from laser-induced direct-write deposition to multibeam holographic lithography or interference lithography [2–8]. Such photonic crystals are designed as bandgap materials mainly for time-domain frequency modes. To manipulate light flow with spatial frequency, closely-spaced waveguide arrays (photonic lattices) have been used as an effective platform for spatial bandgap related studies [9]. Apart from the above sophisticated fabrication techniques, a more convenient way for reconfiguration of photonic lattices is the so-called “optical induction” [10], in which a variety of fascinating characteristics of discrete light propagation has been uncovered [10–13]. However, to the best of our knowledge, all previous experimental work on spatial light manipulation is based on either fabricated or optically induced 1D or 2D photonic structures, as a 3D photonic lattice has not been demonstrated in experiment. Although in 1D and 2D domains, lattice modulation along the longitudinal propagation direction (such as using zigzag or curved waveguide lattices) has led to observations of a host of intriguing wave behavior (e.g., diffraction management, soliton steering, Rabi oscillation, and multicolor dynamic localization [14–23]), we expect new phenomena are yet to be explored in 3D photonic lattices. For example, it has been proposed that 3D photonic structures can be used for novel spatial filters, self-collimators and slow-light generators, in addition to the study of a new type of extended waveforms and vortex structures in a 3D setting [24,25].

In this paper, we experimentally demonstrate 3D reconfigurable photonic lattices by employing the optical induction method in a biased bulk photorefractive crystal. As illustrated

in Fig. 1 , the 3D lattices are established by sending two 2D square-lattice beams into the crystal from two orthogonal directions (each perpendicular to the crystalline c-axis). Simply by switching the polarity of the bias field, different 3D lattice structures [Figs. 1(b, c)] form under a self-focusing or -defocusing nonlinearity. The induced 3D periodic index structures are monitored by plane-wave guidance and Brillouin zone (BZ) spectroscopy from the two orthogonal directions. Enhanced discrete diffraction of a focused probe beam in the induced 3D lattice structures is clearly observed, as compared with beam diffraction in 2D lattices. These 3D photonic lattices can be reconfigured at ease to other structures such as 3D hexagonal or ionic-type lattices.

 

Fig. 1 (a) Schematic drawing of the optical induction method for 3D photonic lattices. The photorefractive strontium barium niobate (SBN) crystal is a uniaxial crystal, with its crystalline c-axis oriented along y-direction. The two lattice-inducing beams are launched along two crystalline a-axes (oriented along x- and z-directions). (b, c) Illustration of induced lattice structures under self-focusing and self-defocusing nonlinearity, respectively.

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2. Experimental setup

The experimental setup used for generation of 3D lattices with the optical induction method is illustrated in Fig. 2 . Two sets of square lattices are induced in the biased photorefractive crystal (SBN:60; a × a × c = 5mm × 10mm × 5mm), each by an ordinarily-polarized partially coherent light beam (diffused light from an argon ion laser @ 488nm) sent through an amplitude mask, as used in our earlier work [11,13]. After imaging the mask onto two input facets of the crystal, two square lattice patterns are created and sent through the crystal from two orthogonal directions. By employing a proper spatial band-pass filter at the Fourier plane of the mask, the Talbot self-imaging effect can be totally killed, so that the partially coherent periodic pattern remains invariant during the propagation throughout the crystal [26]. With a dc bias field, these periodic intensity patterns induce periodic index changes in the otherwise uniform crystal, forming waveguide arrays or index lattices. The spatial filter can be adjusted so that a uniform plane wave can be generated to probe the induced lattice structure and obtain near field pattern of plane-wave guidance. Another partially coherent beam (generated by a third rotating diffuser) is focused and split into two parts to detect the BZ spectrum from the two orthogonal directions using the method proposed in Ref [27]. In addition, a coherent beam from the same laser (without passing through any diffuser) is used as a probe beam to test the discrete diffraction properties of the induced 3D lattices. We mention that when inducing the lattice structures ordinarily-polarized light beams are used, but for probing the induced lattices (including plane-wave guidance, BZ spectrum, and discrete diffraction), the polarization of the probe beam are adjusted to be extraordinarily-polarized with a half-wave plate. The near-field and far-field patterns of the lattice beams and the spatial Fourier spectrum of the probe beam are monitored with imaging lenses and CCD cameras. A dc field is applied along the y-direction (corresponding to the crystalline c-axis, which is oriented perpendicular to the propagation directions of all beams in the crystal). For all experiments, the crystal is illuminated when necessary with an incoherent background beam from a cold white light source to fine-tune the nonlinearity [11–13].

 

Fig. 2 Experimental setup for inducing and monitoring 3D photonic lattices in a biased SBN crystal. The left two beam paths are for two lattice-inducing beams, the third path is for Brillouin zone spectrum measurement, and the forth (far right) path is for testing the diffraction of a focused probe beam. RT: Reversed telescope; BS: Beam splitter; RD: Rotating diffuser; L: Lens; MS: Amplitude Mask; M: Mirror; F: Fourier-plane Filter; λ/2: half-wave plate.

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3. Induced lattice structures

Typical experimental results of optically induced 3D photonic lattices are shown in Fig. 3 . The two 2D square lattice beams with the same periodicity (20 μm) and intensity are fine-tuned to overlap from two orthogonal directions to generate the 3D periodic intensity pattern in the nonlinear photorefractive crystal. To show clearly an invariant 3D structure through the crystal, images of each 2D square lattice pattern are taken at both input and output, along with its far-field pattern or k-space spectrum. In Fig. 3, the top and bottom rows are images taken from the 10-mm propagation along z-direction and those from the 5-mm propagation along y-direction, respectively. Evidently, the square lattices remain nearly the same at input and output in both directions. The four bright spots from each side of the k-space spectrum mark the edges of the first BZ of the square lattices [27,28].

 

Fig. 3 Lattice inducing beams and their Fourier spectra for propagation along z-direction (10 mm crystal length, top) and x-direction (5mm crystal length, bottom). (a, b) shows the intensity patterns of the lattice beams at crystal input (a) and output (b) facets, and (c) shows the Fourier spectra of the lattice beams at crystal output.

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With a bias field of 1.8kV/cm applied appropriately along the crystalline c-axis, the crystal turns into a self-focusing medium, and the induced lattice structures monitored under this condition are displayed in Fig. 4 . The near-field patterns of the index lattices are monitored by employing a uniform plane-wave probe beam, while the lattice from the same direction can be turned off [Fig. 4(a)] for 2D or on [Fig. 4(b)] for 3D configuration. Far-field patterns of the 3D lattices along with BZ spectra are shown in Figs. 4(c, d). Figure 4(a) shows results of guided plane-wave when the lattice along the same direction of the probe beam is absent

 

Fig. 4 Experimental results for induced lattices with self-focusing nonlinearity. (a) Plane-wave guidance from the perpendicular direction of the induced 2D lattices; (b) Plane-wave guidance of the induced 3D lattices; (c) Fourier transformation of (b); (d) Brillouin zone spectroscopy of (b). The upper and lower rows correspond to results obtained along 10mm and 5mm of crystal lengths.

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absent (so the induced structure is 2D arrays of index rods in perpendicular direction). It can be seen that these induced 2D index rods have nearly uniform structure across the crystal. (Such a side-view near field pattern plays a nontrivial role for the alignment of two lattice beams in our experiment.) When both lattices are present, the near field pattern possesses 2D periodic modulation in both orthogonal directions, as shown in Fig. 4(b). The 2D square-like lattice structures are further visualized by the far field pattern [Fig. 4(c)] and the BZ spectrum [Fig. 4(d)]. It is evident that 2D index lattices are established in two orthogonal directions, thus forming a 3D lattice in the bulk crystal. Noted that the spots in Fig. 3(c) mark the boundary (corners) of the first BZ, whereas the spots in Fig. 4(c) correspond to the centers of the first and extended BZs, similar to the X-ray diffraction pattern for atomic lattices.

Next, solely by reversing the polarity of the bias field (i.e., change it to −1.8kV/cm), 3D photonic lattices under self-defocusing nonlinearity are also established. Figure 5 depicts our experimental results, where (a, b) and (c, d) correspond to lattice near-field patterns and BZ spectra obtained from two orthogonal directions. From the near-field patterns of Figs. 5(a) and 5(c), it is clear that the induced index lattices possess a “backbone” or “fishnet” structure, quite different from those of Fig. 4(b) obtained under self-focusing nonlinearity, although the BZ spectra is somewhat similar. We expect that waveguide coupling, bandgap structure and beam dynamics in such lattices will be quite different from those in self-focusing lattices. In addition, it is apparent that the index modulation (contrast) of the lattices is lower than that created under self-focusing nonlinearity. Indeed, under a negative bias field, the SBN crystal is susceptible to depolarization, so it is a challenge to achieve a high index contrast for opening the higher bandgaps in the “backbone” photonic lattices.

 

Fig. 5 Near field patterns (a, c) and BZ spectra (b, d) of induced 3D “backbone” lattices with self-defocusing nonlinearity. (a, b) and (c, d) correspond to results obtained along 10mm and 5mm side of the crystal, respectively.

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4. Enhancement of discrete diffraction

Finally, we present experimental and numerical results on discrete diffraction of a probe beam propagating along z-direction through the 3D lattice induced under self-focusing nonlinearity.

To assure linear propagation of the probe beam, its intensity is adjusted to be very low and its output patterns are taken instantaneously. Typical experimental results are summarized in the top panels of Fig. 6 . For these results, the period of the two lattice beams is about 40 μm, the intensity of lattice beam2 (see Fig. 1) is fixed, but the intensity of lattice beam1 is varied. Three examples of discrete diffraction patterns in x-y output plane from the crystal are shown in Figs. 6(c-e), where the intensity ratio between lattice beam2 and beam1 is varied from 1:0, to 1:1, and to 1:2. Thus, discrete diffraction is enhanced dramatically in 3D lattices [Figs. 6(d, e)] as compared to that in the 2D lattices [Figs. 6(c)], and it is apparent that the enhancement increases as the lattice modulation gets stronger in the propagation direction of the probe beam. Intuitively, we can understand that such diffraction enhancement results from increased coupling among waveguides oriented along z-direction when they are joining by waveguides oriented along the transverse x-direction. Indeed, these experimental results are also corroborated with our numerical simulations. Numerical results obtained with parameters close to those from experiment are presented in the bottom panels of Fig. 6. Clearly, there is good agreement between experimental and numerical results.

 

Fig. 6 Experimental observation (top) and numerical simulation (bottom) of diffraction enhancement in a 3D photonic lattice. (a, b) Input and output of a probe beam propagating along z-direction without lattice; (c-e) its output discrete diffraction patterns after propagating through the lattice when the intensity of the lattice-inducing beam along x-direction (see Fig. 1) is gradually increased.

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Further numerical simulation indicates that discrete diffraction properties such as zero-diffraction and Bragg-reflection are significantly different from their counterparts in 2D square lattices or curved waveguide lattices [22,23]. In addition, it might be possible to realize a new type of all-optical beam steering and orientation-selective self-trapping in 3D lattices. The study of wave dynamics in 3D photonic lattices may prove to be relevant to similar phenomena in other 3D discrete systems including solid state physics and BECs trapped in 3D optical lattices [29].

5. Summary

In summary, we have demonstrated for the first time the formation of 3D photonic lattices by optical induction [30] with both self-focusing and -defocusing nonlinearity. Due to additional periodic modulation along the propagation direction, discrete diffraction of a light beam can be significantly enhanced. Such induced reconfigurable 3D photonic lattices may provide a new setting for exploring discretizing light behavior and bandgap control in three dimensions.