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We demonstrate tunable re-directing, blocking, and splitting of a light beam along defect channels based on spatial bandgap guidance in two-dimensional photonic lattices. We show the possibility for linear control of beam propagation and multicolor routing with specially designed junctions and surface structures embedded in otherwise uniform square lattices.
©2009 Optical Society of America
1. Introduction
One of the fascinating features of photonic band-gap structures is a fundamentally different way of guiding light by defects in otherwise uniformly periodic structures as opposed to guidance by total internal reflection [1]. In photonic crystals (PCs), for instance, splitting and routing of light along pre-designed paths has been highly touted and tested for optical communications, as lossless transmission of light around sharp bends is difficult to achieve in conventional optical fibers. Thus far, a variety of schemes have been proposed to achieve efficient switching and splitting based on linear time-domain frequency modes in PCs and cavity resonance waveguides [1–6]. On the other hand, bandgap guidance based on spatial frequency modes in closely-spaced waveguide arrays, or photonic lattices (PLs), represents another possibility for unconventional guidance of light. It has been proposed that blocking and routing of light can be achieved with discrete solitons in two-dimensional (2D) networks of nonlinear waveguide arrays [7,8].
Indeed, research with PLs as a discrete system has led to demonstrations of abundant linear and nonlinear phenomena of both fundamental interest and application potential [9–12]. These photonic waveguide arrays typically have a periodicity of tens of microns and a refractive index contract on the order of 10−3 or less, as achieved with different techniques from optical induction in nonlinear crystals [13–15] to semiconductor fabrication or fs-laser writing and ionic beam etching [16–20]. In such uniform waveguide arrays, much of the research has been focused on lattice solitons as nonlinear localized defect modes in optical periodic structures. However, introducing a defect into an otherwise perfect PL can substantially modify the bandgap properties of the lattice and allow for linear [21–26] as well as nonlinear [27–29] localized defect modes owing to either total internal or Bragg-type reflection. In particular, a missing or heterogeneous waveguide in the otherwise uniform waveguide arrays acts as a negative defect thereby providing beam guidance along the defect channel akin to guidance in PC fibers. In our previous work with reconfigurable PLs containing a single-site negative defect (or a point defect in 2D lattices), we have performed a series of experiments to demonstrate linear bandgap guidance in optically induced PCF-like structures [23–25]. These results represent the first step towards guiding light based on spatial photonic defect modes, and many defect-mediated phenomena including light routing along structured defect and surface channels [30–32] in PLs are yet to be explored.
In this paper, we demonstrate by numerical simulation that, along line defects (trains of missing or heterogeneous waveguides), a light beam with an initial input tilt (transverse momentum) can be guided and steered through the defect channel. By fine-tuning the defect strength at the intersection of appropriately designed “L”, “T” and “+” shaped defect channels, it is possible to achieve re-directing, blocking, and controllable power splitting of a light beam in the transverse directions while the beam propagates primarily along the longitudinal direction. Moreover, we propose light routing around the boundary of a finite waveguide arrays based on linear surface defect modes as well as multi-color routing around the corner of L-shaped defect channels.
2. Numerical model
Let us begin our analysis by considering a probe beam propagating through 2D PLs containing defects (see Fig. 1 ). To make the discussion more relevant to the experimental setting [25], we assume that the lattices are optically induced in a nonlinear photorefractive SBN crystal with unperturbed refractive index n 0=2.3. The induced index lattices have a spatial period of 13 μm and a refractive index modulation on the order of 10−4. The probe beam propagating in the lattices has a wavelength of 532 nm. Linear propagation of the probe beam in the PLs can be described by the following normalized equation [22,29]:
where U is the envelope of the optical field, z sets the propagation direction and (x, y) are the transverse coordinates, E 0 is the applied dc field, and IL=I 0cos2(x)cos2(y)D(x,y) is the normalized lattice intensity pattern with a peak intensity I 0. D(x,y) is used to structure the defects, and for the line defect shown in Fig. 1(e), D(x,y)={0.3, if −1/2≤x≤1/2; 1, if otherwise}. For all calculations, I 0=4, and E 0=8.4 corresponding to 4.5×105 V/m in real units.
Fig. 1 (a) Schematic drawing of a 2D photonic lattice containing a single-site defect; (b-d) Input of a Gaussian probe beam, its diffraction output in a uniform lattice, and localized output through the defect channel, respectively. (e) A line defect superimposed with an elliptical input beam tilting towards left; (f) output of the probe beam through the line defect in (e).
3. Results and discussion
By removing or decreasing the refractive index modulation in a lattice site, a negative single-site defect is introduced into the 2D lattice [Fig. 1(a)]. Such a defect can support a localized defect mode [21] in a photonic bandgap originating from repeated Bragg reflection, as demonstrated in our earlier work [25]. As a starting point, we illustrate such defect guidance in Figs. 1(a-d). Instead of undertaking discrete diffraction due to coupling between the evanescent waveguides as in uniform lattices [Fig. 1(c)], a focused 2D Gaussian beam [Fig. 1(b)] aimed into the defect can be confined in the defect channel [Fig. 1(d)] throughout the lattices under appropriate conditions, forming a defect mode. Our focus here is to show guiding and steering of a light beam along a line defect as illustrated in Fig. 1(e). With a line defect, light can be confined in the direction perpendicular to the defect path due to Bragg reflection, but can be routed along the defect path due to coupling/resonance of adjacent defect modes akin to that in resonance cavity waveguide [5]. As an example, we launch the 2D Gaussian probe beam into the line defect with an initial tilting angle (transverse momentum) toward left [Fig. 1(e)]. After propagating through the lattice, we can see clearly that the beam is guided and translated by the line defect [Fig. 1(f)]. For these results, the refractive index in the defect sites is decreased by 70% as compared with that in the surrounding lattice sites. The incident angle of the probe beam with respect to the longitudinal propagation direction z is tan−1(0.5π/kΛ) towards left (this angle is 0.25 degree with current parameters, which can be adjusted by changing the lattice spacing Λ). With such an incident angle, the probe beam aims to the zero diffraction direction initially, thus it is expected to experience less diffraction in the transverse y-direction..
We point out that such guidance is neither induced by nonlinear self-focusing since no self-action is taken into account here, nor by total internal reflection since the defect sites (guided region) has lower refractive index than the surrounding lattice sites. In fact, it is established by spatial bandgap guidance due to periodic refractive index modulation in the lattice. This also leads to the possibility of bending of light akin to that in PCs [2]. Some typical results for the guiding and steering of a light beam in L-shaped line defect with different designs of the corner defects are shown in Fig. 2 . The top row shows three lattices with different structured defects superimposed with the probe beam at input of the lattices, where the inserts are zoom-in pictures of different corner structures. For these three cases, the input beam has the same input tilt (as in Fig. 1) towards left. The bottom row shows the output profiles of the probe beam exiting the lattices after 6 cm of propagation. We can see clearly that, in each case, the input beam changes its direction of propagation around the corner of the L-shaped defect (Animation of the beam propagation can be seen in media file, bend.gif). Thus, the probe beam traveling to the left now ends up traveling in upward direction. It is noticeable that, at the output, there is a small portion of light turning backwards. If we consider the portion going upward is the “transmission”, then the portion going back will be the “reflection”. Clearly, different corner defect structures lead to different rate of transmission [estimated “transmission” is 90%, 82%, and 86% for Figs. 2(a-c), respectively].
Fig. 2 Re-directing of light in L-shaped line defects with different corner structures. Top panels show the lattice structures superimposed with a left-tilted probe beam at input (zoom-in pictures of the corner are shown in the insets for better visibility); Bottom panels show the output of the probe beam guided and steered by the line defect (Media 1).
In the present scheme, as the probe beam travels mainly along z-direction with an input tilt, the term “reflection” or “transmission” here refers merely to routing the probe beam to the lower or upper branch of the L-defect after propagating through the lattice. In fact, by properly design the defect structure at the corner, we can achieve blocking and controllable splitting of a light beam in the transverse directions while propagating through the lattices along z-direction. Some examples for blocking and splitting of a probe beam in L-shaped defects are shown in Figs. 3(a, b) , simply by fine-tuning the defect strength (index modulation) at the corner. In Fig. 3(a) the corner waveguide (indicated by the red arrow) has the same refractive index as in a normal lattice site. In this case, the traverse velocity of the probe beam reversed its direction (from –y to + y) at the corner [see Fig. 3(a), Media 2]. This means during its propagation the probe beam is “blocked” by the corner and “reflected” to opposite transverse direction. Intuitively, this results from the “anti-defect” at the corner that breaks the coupling of defect modes along two (vertical and horizontal) branches of the “L” shaped defects, so the probe beam traveling along the horizontal branch cannot be coupled into the vertical branch when hitting the corner. If we make the refractive index of the corner waveguide to be only 32.5% of that in a non-defect lattice site but still 2.5% more than that in the line defects, then the probe beam can split into two equal portions moving upward and backward [see Fig. 3(b), Media 3]. In such a case, the corner defect adjusts the coupling between horizontal and vertical branches, resulting in partial transmission and reflection. The ratio of beam power splitting can be changed by fine-tuning the refractive index of the corner waveguide. In addition to the L-shaped defect structure, splitting of a light beam by “T” and “+” shaped defect structures can also be realized, and some examples are shown in Figs. 3(c, d) obtained with same parameters as used for Fig. 2 and in Figs. 3(a, b).
Fig. 3 Blocking and splitting of light with different designs of line defects. Top panels show lattice structures superimposed with a left-tilted probe beam at input; bottom panels show the probe beam exiting the lattice. (a) and (b) show examples of blocking (Media 2) and 50/50 splitting (Media 3) by adjusting the refractive index of the corner waveguide in L-defect to be 100% and 32.5% with respect to that in the uniform waveguide arrays, respectively. (c) and (d) show splitting of a light beam by T and + shaped line defects in the lattices.
Next, we show that light routing around the boundary of a finite waveguide arrays based on linear surface defect modes is also possible provided that the surface line defects are properly designed. Nonlinear Tamm-like surface states and surface solitons [30] in waveguide arrays have been studies extensively, but demonstration of linear optical surface modes [31,32] requires specially designed surface defects. If a 2D PL is bounded by defects and a probe beam can evolve into a linear localized surface defect mode, then the probe beam can be routed anywhere in the lattice boundary with appropriate transverse momentum. Figures 4(a, b) show two results of proposed light routing along the surface of a semi-infinite square lattice. With appropriate design of the surface and corner defects, the probe beam can make a turn around surface corner [Fig. 4(a)], reverse its transverse travelling direction at the corner [Fig. 4(b)]. If the 2D waveguide lattice has a limited size or long enough propagation distance for the probe beam, then the probe beam can circle around the surface and make a round trip [see Fig. 4(c), Media 4 for routing around a 5x5 square lattice].
Fig. 4 Routing of a light beam around the surface of finite waveguide arrays. Top panels show the lattice structures superimposed with a left-tilted probe beam at input; and bottom panels show the output of the probe beam. (a) Steering of the probe beam around a corner along surface line defects, (b) blocking when the defect at the corner is missing, and (c) spiraling around the surface of a finite waveguide arrays bounded by surface line defects (Media 4).
Finally, we propose multi-color routing in the same setting of a L-shaped defect structure. Since bandgap guidance in PLs is based on spatial defect modes (different from that in PCs), it is possible to form defect modes in a defect channel with different wavelengths of light [25]. Typical results are shown in Fig. 5 , where in the same L-shaped defect structure [Fig. 5(a)], routing of a probe beam at 532nm [Fig. 5(b)], 488nm [Fig. 5(c)] and 633nm [Fig. 5(d)] is realized. We emphasize that these results were obtained under the same conditions (i.e. same lattice spacing, lattice index modulation, and same input tilt of the probe beam) as those in Figs. 2-4. Although the probe beam ended at different locations at the lattice output and the spatial broadening along the defect line is somewhat different due to different propagation velocities and diffraction along transverse x-direction for different wavelengths, routing along the defect channel is achieved simultaneously for all three colors. This could be promising for applications where switching and routing of white light or ultra-short pulses is desirable. Multi-color routing and polychromatic dynamic localization of light beams are also of fundamental interest, as has been demonstrated recently in curved photonic lattices [17].
Fig. 5 Routing of a light beam of different wavelengths in the same setting of L-shaped defect channel in lattice structure shown in (a). (b-d) show the output of the probe beam exiting the lattice. From (b) to (d), the wavelengths used are 532 nm, 488 nm, and 633 nm, respectively.
4. Summary
We demonstrated the possibility for re-directing, blocking, and tunable splitting of a light beam in 2D photonic lattices with structured defects. We show it is possible to route polychromatic as well as monochromatic light along predesigned paths inside or surrounding the lattices based on photonic bandgap guidance. These results might be useful for the development and implementation of microstructured devices where multi-color routing with low-index-contrast photonic structures is desirable. Our results might also be relevant to the Fano Resonance studied in different lattice systems [33,34].
This work was supported by NSF, AFOSR, and the 973 Program. We thank P. Zhang and J. Yang for assistance and discussion.
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