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Experiments on Gaussian beams and vortices in optically induced photonic lattices Zhigang Chen, Hector Martin, and Anna Bezryadina Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132
Dragomir Neshev and Yuri S. Kivshar Nonlinear Physics Centre and Centre for Ultrahigh-Bandwidth Devices for Optical Systems, Research School of Physical Sciences and Engineering, Australian National University, Canberra 0200, Australia
Demetrios N. Christodoulides School of Optics/CREOL, University of Central Florida, Orlando, Florida 32816 Received October 28, 2004; accepted January 3, 2005 We investigate experimentally the propagation of fundamental Gaussian beams and vortices in a twodimensional photonic lattice optically induced with partially coherent light. We focus on soliton–lattice interactions and vortex–lattice interactions when the lattice is operated in a nonlinear regime. In this case a host of novel phenomena is demonstrated, including soliton-induced lattice dislocation–deformation, soliton hopping and slow-down, and creation of structures akin to optical polarons. In addition, we observe that the nonlinear interaction between a vortex beam and a solitonic lattice leads to lattice twisting due to a transfer of the angular momentum carried by the vortex beam to the lattice. Results demonstrating a clear transition from discrete diffraction to the formation of two-dimensional, discrete fundamental and vortex solitons in a linear lattice are also included. © 2005 Optical Society of America OCIS codes: 190.5940, 270.5530, 190.4420.
1. INTRODUCTION Recently there has blossomed an interest in the study of light propagation in periodic structures including intrinsic localized modes and discretizing light behavior in linear and nonlinear waveguide lattices.1,2 Closely spaced waveguide arrays represent a typical example of optical periodic structures that have a strong link with the emerging science and technology of nonlinear photonic crystals. In such array structures the collective behavior of wave propagation exhibits many fascinating phenomena that are also encountered in other discrete nonlinear systems. For example, optical structures with a periodic change in the material refractive index can profoundly affect the diffraction properties of light. Even in a linear waveguide array, due to evanescent coupling between nearby waveguide sites, a light beam experiences discrete diffraction. If the waveguide array is embedded in a nonlinear medium, more interesting wave behavior is expected to occur. A typical example is the balance between discrete diffraction and nonlinear self-focusing that leads to optical self-trapped states better known as discrete solitons.3 In nonlinear optics, discrete solitons were first demonstrated in one-dimensional (1D) AlGaAs semiconductor waveguide arrays in 1998,4,5 and they have since attracted considerable research interest. In part, this is because such discrete solitons are expected to exist also in a variety of other discrete nonlinear systems, including those in biology, solid-state physics, and Bose–Einstein condensates.6,7 Apart from fundamental interest, these 0740-3224/05/071395-11/$15.00
discrete solitons have been also proposed as good candidates for all-optical blocking and switching of signals in two-dimensional (2D) waveguide networks.8 Yet in spite of the exciting prospects for applications, the design and manufacture of such 2D waveguide arrays (with bandgaps in k space) is a difficult, expensive, and time-consuming task. Additionally the materials used for implementing such periodic systems normally do not possess strong properties for nonlinear self-action of light. Therefore a new approach is needed to realize strongly nonlinear periodic structures in higher dimensions. Recently a theoretical study suggested that it was possible to create reconfigurable periodic structures by use of optical induction in photorefractive crystals with strong anisotropic nonlinearity.9 This soon led to the experimental observation of 1D and 2D discrete solitons in such optically induced waveguide lattices, as first reported in Refs. 10,11 and then in Refs. 12,13. Unlike AlGaAs waveguide arrays, the optically induced lattices can be readily made in two or even three dimensions. This allows the study of lattice solitons in higher dimensions. Meanwhile photonic lattices optically induced by pixellike spatial solitons have been created with amplitude modulation of partially spatially incoherent light14,15 as well as with phase engineering of coherent light,16,17 leading to study of optical control and manipulation of individual channels in a large array of solitonic lattices. By exploiting the anisotropic photorefractive nonlinearity, the lattices could be conveniently operated in either the © 2005 Optical Society of America
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linear or the nonlinear regime. If the probe beam and the lattice beam are extraordinarily polarized (e polarized), they both experience high nonlinearity as they propagate together and thus interact strongly. In this configuration the lattice beam itself tends to form arrays of spatial solitons, so the soliton-induced waveguide lattice is said to be in the “nonlinear” regime, which becomes considerably more susceptible to soliton-induced deformation.15 This in turn brings about the interesting possibility of studying optical soliton-lattice interactions that might exhibit many of the basic characteristic features of other physical processes such as those encountered in polaron excitation–formation in solid-state physics. In particular, soliton-induced lattice dislocation–deformation,13 lattice compression waves,18 and a novel type of composite bandgap solitons19 are expected to occur in such nonlinear lattices. On the other hand if the lattice beam is ordinarily polarized (o polarized) while the probe beam is e polarized, the probe would experience strong nonlinearity but the induced lattice would remain nearly invariant during propagation due to the anisotropic property of the nonlinear crystal. In this case the waveguide lattice is said to be in the “linear” regime. This latter configuration can be used as a test bench for demonstration of many nonlinear wave propagation phenomena in periodic systems, including discrete solitons. When the lattice beam is created with amplitude modulation and partial coherence13 rather than by coherent multibeam interference,10–12 an enhanced stability of the lattice structure can be achieved in both the linear and nonlinear regimes as a result of suppression of incoherent modulation instabilities, as well as reduced sensitivity to ambient fluctuations. In fact it is in such optically-induced, partially coherent photonic lattices that a clear transition from 2D discrete diffraction to formation of 2D discrete solitons has been successfully demonstrated.13 This convenient way of creating reconfigurable photonic structures and of successfully demonstrating discrete self-trapping of light has opened the door for the observation of many distinctive phenomena in 2D periodic media that have no counterpart in the bulk. These include, for example, discrete vortex solitons,20,21 discrete soliton trains,22 and discrete dipole and vector solitons in 2D lattices.23,24 In addition, other interesting phenomena that have been predicted or observed in 1D waveguide lattices, such as discrete random-phase solitons, discrete modulation instability, grating-mediated waveguiding, discrete soliton interaction, and generation and control of spatial gap solitons,25–30 are expected to occur in 2D lattices as well, with even much richer dynamics yet to be explored. In this paper we present our experimental work on the study of 2D fundamental Gaussian beams and vortices propagating in optically induced linear and nonlinear photonic lattices induced by partially coherent light. We shall focus mainly on the nonlinear interaction between a Gaussian–vortex beam and a 2D solitonic lattice. When the lattice is operated in the nonlinear regime, the simultaneous propagation and interaction between the Gaussian–vortex beam and the lattice leads to the observation of new phenomena. This includes, for example,
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soliton-induced lattice dislocation–deformation, vortexinduced lattice twisting, and the creation of optical structures that are somewhat analogous to polaron states in solid-state physics. The Gaussian beam itself tends to form a soliton in the self-focusing nonlinear medium, but its propagation is strongly affected by the interaction with the nonlinear lattice, leading to its transverse velocity slowdown in the lattice. The vortex beam with unit topological charge tends to break up into four filaments when launched appropriately into the nonlinear lattice, but the filaments are somewhat confined by the lattice potential so they cannot rotate freely. Vortices with higherorder charges are less confined by the nonlinear lattice, and instead they often cause stronger deformation of the lattice. For completeness, we will also include our recent experimental results for Gaussian–vortex beams propagating as probes in linear waveguide lattices. Conversely when the lattice is operated in the linear regime so that it does not itself experience strong nonlinearity during propagation, we observe that the probe beam evolves from discrete diffraction to 2D fundamental and vortex discrete solitons as the level of nonlinearity for the probe is increased. Our experimental results for discrete solitons are in good agreement with the theoretical analysis of these effects.
2. EXPERIMENTAL SETUP AND CREATION OF 2D LATTICES The experimental setup for our study is illustrated in Fig. 1. A partially spatially incoherent light beam 共 = 488 nm兲 is generated by converting an argon-ion laser beam into a quasi-monochromatic light source with a rotating diffuser. Such a diffused laser source has the advantage of providing an incoherent beam with a controllable degree of spatial coherence and intensity suitable for experiments with partially coherent light.31,32 A biased photorefractive crystal (SBN:60, 5 ⫻ 5 ⫻ 8 mm3, r33 = 280 pm/ V and r13 = 24 pm/ V) is employed to provide a saturable, self-focusing noninstantaneous nonlinearity. To generate a 2D waveguide lattice we use an amplitude mask to spatially modulate the otherwise uniform incoherent beam after the diffuser. The mask is then imaged onto the input face of the crystal, thus creating a pixellike input intensity pattern.15 This lattice beam can be turned into either extraordinarily or ordinarily polarized light with a half-wave plate as necessary. A Gaussian beam split from the same laser but without passing through the
Fig. 1. Experimental setup. PBS, polarizing beam splitter; SBN, strontium barium niobate. The vortex mask is used only for experiments with a vortex beam.
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Fig. 2. Output intensity patterns of a 2D square lattice as created by optical induction. (a) Stable o-polarized linear lattice at 20 m spacing, (b) its corresponding output when the lattice beam is turned to e polarized. (c) Stable e-polarized nonlinear lattice at 37 m spacing, (d) its corresponding output when the nonlinearity is too high.
diffuser is used as the probe beam propagating along with the lattice. For the vortex experiment the probe beam is sent through a transmission vortex mask (bottom path) and then focused onto the crystal input face, propagating collinearly with the lattice. In addition a uniform ordinarily polarized background beam (top path) as created with another rotating diffuser is used as incoherent background illumination for fine-tuning the nonlinearity. The input and output faces of the crystal are imaged onto a CCD camera. The beam separation in a coupled or interacting state is achieved by blocking one of the components and quickly recording the other one, taking advantage of the noninstantaneous response of the crystal. In our experiment the probe beam is e polarized and “fully” coherent, while the lattice beam is partially coherent with either extraordinary or ordinary polarization as needed. In general, in an anisotropic photorefractive crystal, the nonlinear index change experienced by an optical beam depends on its polarization as well as its intensity. Under appreciable bias conditions, i.e., when the photorefractive screening nonlinearity is dominant,9 this index change is given approximately by ⌬ne = 关n3e r33E0 / 2兴共1 + I兲−1 and ⌬no = 关n3o r13E0 / 2兴共1 + I兲−1 for e polarized and o polarized beams, respectively. Here E0 is the applied electric field along the crystalline c axis (x direction), and I is the intensity of the beam normalized to the background illumination. As a result of the difference between the nonlinear electro-optic coefficients r33 and r13 in the SBN:60 crystal we used, ⌬ne is more than 10 times larger than ⌬no under the same experimental conditions. Thus when it is e polarized, the lattice beam experiences a nonlinear index change comparable to that of the probe beam, whereas the lattice beam evolves almost linearly when it is o polarized. In this way by exploiting the anisotropic photorefractive nonlinearity, the lattices can be conveniently operated in either the linear or nonlinear regime. Typical examples of 2D square lattices created by optical induction in our crystal with both o polarized and e polarized beams are shown in Fig. 2. Our experiment shows that the linear square lattices generated with an o polarized partially coherent beam are stable and robust, even at a small lattice spacing of 20 m or less. However, to create a nonlinear (solitonic) square lattice with the e polarized beam, it is a challenge to obtain a stable lattice without distortion at such small lattice spacing. In addi-
tion because of the anisotropic self-focusing nonlinearity, the pixels (or waveguides) tend to merge in the y direction if the square lattice is oriented in horizontal–vertical directions. The diagonal orientation of the square lattice (i.e., its principal axes oriented in the 45° directions relative to the x and y axes) favors stable lattice formation as a result of the enlarged y separation between pixels, especially when the lattice spacing is small. Figure 2(a) shows a stable 2D linear lattice with 20 m spacing created with an o polarized beam, whereas Fig. 2(b) shows the corresponding pattern when the same lattice beam was changed to e polarized with all other conditions unchanged. Clearly, the lattice is strongly distorted at this spacing in the nonlinear regime. The reason for this is quite intuitive. The e polarized lattice experiences a strong nonlinear self-focusing effect, thus each lattice site tends to form a pixel-like soliton itself. When the lattice spacing is small, the size of each soliton pixel is also small. With the photorefractive screening nonlinearity, the formation of solitons must satisfy certain conditions better described by the soliton-existence curve.33 In essence, forming a smaller soliton requires a stronger nonlinearity, which is usually achieved by increasing the bias field across the nonlinear crystal. However, with a stronger nonlinearity, an e polarized lattice beam tends to suffer from stronger modulation instability as driven by noise such as defects and striations in the crystal, which afflicts the formation of a stable solitonic lattice. This is why we introduce partial coherence in the lattice beam. As has been previously predicted34 and experimentally demonstrated,14,35,36 partial coherence can result in suppression of modulation instabilities. Therefore, to create a nonlinear lattice with enhanced stability often requires fine-tuning of such experimental parameters as the bias field, lattice spacing, lattice beam intensity, and spatial coherence. If the spacing of a nonlinear lattice is too small (20 m or less), it is difficult to obtain a stable, densely packed solitonic lattice in our photorefractive crystal even with a partially coherent beam. In fact the lattice beam would go haywire [as shown in Fig. 2(b)] because of strong modulation instability and interaction between neighboring lattice sites. At larger lattice spacings, nonlinear lattices of pixel-like spatial solitons can indeed be established by fine-tuning the nonlinearity. Figure 2(c) shows a typical example of a stable nonlinear lattice (at 37 m
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spacing) obtained in our experiment, and Fig 2(d) shows the corresponding lattice when the background illumination and–or the partial coherence of the beam is removed so that the nonlinearity becomes too strong. In the following the square lattice will be oriented diagonally for the experiments on discrete solitons and vortex–lattice interaction, while it will remain in a horizontal–vertical orientation for the experiments on soliton-lattice interaction.
3. GAUSSIAN BEAMS IN LINEAR AND NONLINEAR PHOTONIC LATTICES A. Formation of Discrete Solitons in Linear Lattices First we present our experimental results of a 2D Gaussian beam used as a probe in a 2D linear waveguide lattice. In this case the lattice beam is o polarized while the Gaussian beam is e polarized. As discussed earlier, the lattice in this configuration “sees” only a weak nonlinearity as compared with that experienced by the e polarized probe beam and therefore remains nearly invariant as the bias field increases. At the same time, the partially coherent lattice provides linear waveguiding for the probe beam. Typical experimental results are presented in Fig. 3, for which a lattice with a small spacing of 20 m [as shown in Fig. 2(a)] is first generated. The probe beam (whose intensity is 4 times weaker than that of the lattice) is launched into one of the lattice sites, propagating collinearly with the lattice. As a result of weak coupling between closely spaced waveguides, the probe beam undergoes discrete diffraction when the nonlinearity is low [Fig. 3(c)], which clearly shows that most of the energy flows from the center toward the diagonal directions of the lattice. However, the probe beam forms a 2D discrete soliton at an appropriate (high) level of nonlinearity [Fig. 3(d)], so that most of its energy is concentrated in the center and the four neighboring sites along the principal axes of the lattice. These experimental results are in good agreement with expected behavior from the theory of 2D discrete systems.11,13 We note that in this case the lattice
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itself behaves just like a fabricated linear waveguide array, and it is not considerably affected by the weak probe or the increased bias field. B. Interaction of a Soliton with a Nonlinear Lattice Next we present the results of interaction of a Gaussian beam (as a control–probe beam) with a nonlinear lattice induced by 2D pixel-like spatial solitons. In this case both the Gaussian beam and the lattice beam are e polarized. Under appropriate conditions a stable soliton lattice is established, and then the control–probe beam that tends to form a soliton itself is launched into the lattice. If the intensity of the Gaussian beam is much weaker than that of the lattice, the Gaussian beam behaves more as a probe and is simply guided by the soliton channel into which it is targeted without much influence on the propagation of the solitonic lattice. However, once the intensity of the Gaussian beam becomes comparable with that of the lattice, it behaves more like a control beam and thus plays a role in the interaction process. For such a configuration, optical control and manipulation of individual soliton channels by off-site interaction at relatively large lattice spacing (70 m or more) have been established previously either by use of a partially coherent lattice15 or by phase engineering of a coherent lattice.16,17 Here we focus on novel aspects of behavior concerning soliton-lattice interactions when the input soliton is aimed at one of the soliton pixels (on-site interaction) and when the lattice spacing is relatively small (less than 70 m). Of special interest is the interaction resulting from a probe–control soliton that carries an initial transverse momentum. Typically when the Gaussian beam is launched at one of the lattice sites with a small tilt angle relative to the propagation direction of the lattice, we observe lattice dislocation due to soliton dragging. Figure 4 shows an example of such an interaction. For this experiment the coherence length of the partially coherent lattice is 40 m, and its intensity is about 3 times higher than that of the background illumination. The lattice spacing is 65 m, and the intensity FWHM of each pixel is 17 m. The 2D solitonic nonlinear lattice was created at a bias field of
Fig. 3. Gaussian beam propagating in optically-induced linear lattices, showing formation of a 2D discrete soliton. Top, 3D intensity plots, bottom, 2D transverse patterns of (a) the probe beam at crystal input, (b) output of normal diffraction without the lattice, (c) output of discrete diffraction at low nonlinearity (bias field, 900 V / cm), (d) output of discrete soliton at high nonlinearity (bias field, 3000 V / cm).
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Fig. 4. Top, soliton-induced lattice dislocation when a probe soliton was launched into a nonlinear lattice toward the (a) right, (b) left, (c) up, (d) down directions. The launching angle relative to the direction of lattice propagation is 0.5° for (a), (b) and 0.8° for (c), (d). Bottom, superimposed images of the probe beam taken with (denoted by “A”) and without (denoted by “B”) the lattice. (a), (b) show a slowdown in the transverse velocity of the probe soliton, (c), (d) show soliton hopping from site to site under the influence of the lattice. The lattice spacing is 65 m in all cases.
2400 V / cm. At this bias field, the Gaussian beam with an intensity approximately equal to that of the lattice formed a soliton itself. When this control soliton was aimed into one of the lattice sites at a shallow angle in four different directions, the lattice overall remained uniform except for a dislocation created by the soliton (Fig. 4, top panels). In all cases the nonlinear lattice and the Gaussian beam have interacted for a long period of time and the crystal has reached a steady state. By capitalizing on the noninstantaneous response of the crystal, the control beam and the lattice were monitored separately. The photographs in the top panel of Fig. 4 were taken immediately after the Gaussian beam was blocked in an interacting state. The crosshair marker serves as a point of reference to indicate how much the “target” pixel has shifted from its original location at the output of the crystal as a result of soliton dragging. This soliton-induced dislocation in the lattice disappeared after the Gaussian beam was blocked for more than 5 min and the crystal had reached a new steady state; so the lattice was restored in the absence of the control beam. Clearly, the control beam had the tendency to drag the interacting soliton pixel to any direction at will. Meanwhile the control beam itself retained its soliton identity, but its lateral shift (at the crystal output) was reduced significantly because of the interaction with the lattice, indicating a slowdown in its transverse velocity (Fig. 4, bottom panels). In the bottom panel of Figs. 4(a) and 4(b), the control beam traveled 68 m in the horizontal (left and right) directions without the lattice (the spot far away from the center denoted B), but it traveled only about 26 m when interacting with the lattice (the spot close to the center denoted A). These two photographs were taken separately (in two separate experiments with and without lattice while keeping all other conditions unchanged) and then superimposed in the same figure. Slowdown of the soliton transverse velocity as caused by the lattice is evident. Interestingly enough, when the control beam was aimed in
vertical (up and down) directions at a 0.8° launching angle, the slowdown of its transverse velocity was less significant. However, in this case, soliton hopping might have occurred since the probe soliton jumped from the “target” lattice site to the next closest lattice site. In the bottom panels of Figs. 4(c) and 4(d) it can be seen that the control soliton (denoted A) ended up at the output location one site above the “target” lattice site. By comparing with Figs. 4(a) and 4(b) where no hopping has occurred at a 0.5° launching angle, we expect that this kind of soliton hopping might be more evident at larger launching angles as the control–probe soliton jostles its way through the nonlinear lattice. The above-mentioned soliton-induced lattice dislocation became even more pronounced when the lattice spacing was decreased further and–or the probe intensity was increased. In addition other interesting phenomena were observed as the initial conditions for soliton-lattice interaction were varied, including generation of lattice defects, lattice deformation, and lattice compression. Figure 5 shows an example of how the intensity of the probe soliton plays its role in the dynamics of interaction. For these experiments the spacing of the lattice was kept at ⬇50 m, and the intensity of the probe was increased gradually with all other experimental conditions unchanged. The results in Fig. 5 were obtained when the probe beam was aimed into one of the lattice sites but with a tilt angle of 0.8° toward the right (+x direction). In the top panels of Fig. 5 it can be seen clearly that the lattice dislocation caused by the control soliton became stronger as the intensity of the soliton was increased. In fact, besides the pixel initially targeted by the control soliton (the central one as referenced by the cross-hair), other pixels in close proximity (the right one and bottom one) were also dislocated from their initial positions. Pixels farther away in the direction of the initial transverse momentum of the control soliton have also been affected [see Fig. 5(c)] as if the lattice were compressed slightly by the
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control soliton. This kind of lattice compression (or excitation-lattice compression waves) has been observed also during a process of interaction between a stripe soliton and a 2D nonlinear lattice.18 For comparison the lattice restored to steady state after removing the control beam is shown in Fig. 5(d). Clearly, the increased lattice distortion shown in Figs. 5(a)–5(c) results from the interaction with the control soliton. In the bottom panels of Fig. 5, the location of the control soliton at crystal output is shown. When the lattice was absent [bottom panels of Figs. 5(a)–5(c)] the control soliton experienced an increasing self-bending toward the crystalline c axis (+x direction) as its intensity was gradually increased. However, when interacting with the nonlinear lattice, such diffusion-induced self-bending effect37 was strongly suppressed, and the soliton propagated through the lattice with a much lower transverse velocity, as shown in the bottom panel of Fig. 5(d).
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For comparison the experiments of Fig. 5 were repeated under the same conditions except that the control beam was launched with a tilt angle of 0.8° toward the left (−x direction). The corresponding results are presented in Fig. 6 where, from (a) to (c), the top panel shows the lattice dislocation–distortion as the intensity of the control soliton was increased gradually, and the bottom panel shows the corresponding free propagation (no lattice present) of the control soliton. Clearly, the central pixel that was targeted moved away, and other pixels nearby (the left one and bottom one) were also dislocated as compared with the unperturbed solitonic lattice [Fig. 6, top]. When the intensity of the control soliton was strong, pixel merging and pixel annihilation was achieved. A difference was observed between launching the control beam with and against the crystalline c axis that may be attributed to the anisotropic soliton self-bending effect. We note that the pixel annihilation was reproduced in several
Fig. 5. Top, soliton-induced lattice dislocation when the intensity of a probe beam was increased gradually. The launching angle relative to the direction of lattice propagation is ⬇0.8° toward the right. The intensity of a probe beam normalized to that of the lattice beam is (a) 0.9, (b) 1.0, (c) 1.1; (d) shows the restored lattice after the probe beam is turned off. Bottom, the corresponding intensity patterns of the probe soliton when the lattice beam is turned (a)–(c) off, (d) on. The lattice spacing is 50 m in all cases.
Fig. 6. Same as in Fig. 5, except that the launching angle relative to the direction of lattice propagation is ⬇0.8° toward the left, and the intensity ratio between the probe soliton and the lattice is (a) 0.9, (b) 1.1, (c) 1.3.
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Fig. 7. Formation of soliton-induced polaronlike structures as the intensity of the control soliton was increased gradually. The intensity ratio between the probe soliton and the lattice is (a) 0.1, (b) 0.6, (c) 1.0, (d) 1.5. The lattice spacing is about 55 m for the top panels and 40 m for the bottom panels. The images in the bottom panels have been enlarged for better visualization of the central part of the lattice.
other experiments when the control soliton had an initial transverse momentum in the −x direction. While the output location of the control soliton was strongly dependent on its intensity in the absence of the lattice [closer to the center owing to stronger self-bending toward x direction at a higher intensity; see Figs. 6(a)–6(c), bottom], it stayed nearly the same when the control soliton propagated through the nonlinear lattice with three different intensities [Fig. 6(d), bottom]. As a result of interaction and energy exchange with more than one soliton pixel in the lattice, the shape of the control soliton was also slightly distorted. We have also investigated the case when the control soliton was launched into one of the lattice sites with no initial transverse momentum, i.e., the soliton beam and the lattice beam propagate along the same direction. In this case, strong lattice deformation, including generation of polaronlike structures, was also observed due to nonlinear soliton–lattice interaction. Typical examples are shown in Fig. 7, where from (a) to (d) the intensity of the probe soliton was increased gradually while keeping all other conditions unchanged. The lattice spacing is ⬇55 m for the top panels and ⬇40 m for the bottom panels. As the intensity of the probe soliton was increased to a point comparable with that of the nonlinear lattice, a polaronlike induced structure was observed in which the probe soliton dragged toward it some of the neighboring sites while pushing away the other [Fig. 7(c)]. When the probe beam intensity was increased to be much higher than that of the lattice, the lattice structure became strongly deformed in such a way that the site dislocations extended beyond the immediate neighborhood [Fig. 7(d)]. The polaronlike structure was reproduced with different lattice spacing, and the effect became more pronounced
when the lattice spacing was reduced to ⬇40 m (Fig. 7, bottom panel). The observed process is quite similar to that caused by a polaron in solid-state physics in which an electron drags and dislocates heavy ions as it propagates through an ionic crystal.38 Since our lattice itself is partially coherent and its spacing is not smaller than the coherence length, nearby pixels in the lattice have no, or only weak, initial phase correlation. In addition the lattice is mutually incoherent with the probe. Thus one would expect only attraction between nearby lattice sites rather than repulsion from incoherent soliton interactions. Furthermore, since the closest four neighboring solitons are initially equally spaced around the central one at which the probe beam was aimed, the observed behavior cannot be attributed simply to the anomalous interaction between photorefractive solitons in which attraction or repulsion depends merely on soliton mutual separation.39 Instead the observed polaronlike structure suggests that the probe beam might have induced a certain degree of coherence in the neighboring lattice sites with different phase correlation through interaction. This issue certainly merits further investigation.
4. OPTICAL VORTICES IN LINEAR AND NONLINEAR PHOTONIC LATTICES A. Formation of Discrete Vortex Solitons in Linear Lattices An important nonlinear phenomenon in 2D lattices is the propagation of optical beams with complex internal structure on the lattice, e.g., the propagation of optical vortices carrying orbital angular momentum. Here we present our experimental results of a vortex beam as probe propagating in a 2D linear waveguide lattice. In
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this case, the lattice beam is o polarized while the vortex beam is e polarized. Typical experimental results are shown in Fig. 8, for which a waveguide lattice with 28 m spacing is created. The vortex beam with unit topological charge [normal diffraction shown in Fig. 8(a)] is then launched straight into the middle of the four lattice sites, so that the vortex center sits right in the middle of four lattice sites (off-site excitation20,21). As a result of waveguide coupling, the vortex beam undergoes discrete diffraction when the nonlinearity is low [Fig. 8(b) and 8(c)], but it evolves into a discrete vortex soliton at an appropriate level of high nonlinearity [Fig. 8(d)] with most of the energy concentrated at the central four sites along the principal axes of the lattice. To confirm the nontrivial phase of the vortex soliton, a weak reference beam is introduced to interfere with the discrete vortex after it exits the crystal. We use a piezotransducer mirror in the reference beam path to control its phase relative to the vortex beam. As we move the piezotransducer mirror, a series of interferograms is recorded to reconstruct the phase of the vortex. Examples of the interferograms are presented in Figs. 8(f)–8(h), which show that one of the four lobes increases its intensity, whereas the corresponding diagonal lobe decreases its intensity. Furthermore, the lobe with the strongest intensity is alternating among the four spots. These interferograms confirm that the four lobes of the output vortex have a nontrivial phase relation—a / 2 phase ramp—as expected for the discrete vortex soliton. By removing the reference beam, the discrete vortex soliton is restored in steady state [Fig. 8(e)]. B. Interaction of a Vortex with a Nonlinear Lattice Finally, we present our experimental results on interaction of a vortex beam with a nonlinear lattice induced by 2D, pixel-like spatial solitons. In this case, both the probe and the lattice beams are e polarized. When the lattice is operated in the nonlinear regime, it can acquire linear or angular momentum when interacting with other beams. The deformation of the lattice caused by soliton-lattice in-
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teraction as shown in Figs. 4–6 can be considered examples of transferring linear transverse momentum from a Gaussian beam to the lattice. Here we address the issue of interaction between a solitonic lattice and beams carrying angular momentum. This is a nontrivial issue, as in periodic structures or lattices the continuous rotational symmetry of homogeneous media is broken, thus angular momentum is not generally conserved. In the case of a discrete vortex-ring soliton presented in Fig. 8, it is the combination of “discrete” dynamics and nonlinearity that allows the soliton to maintain its vortex phase.20,21,40,41 However, higher-order vortices may not conserve their topological charges even in the linear lattice.42 It is thus natural to ask: Can the angular momentum carried by a vortex beam be transferred to a nonlinear lattice? To answer this question, we launched a vortex beam of opposite topological charges into a nonlinear lattice. The experimental results are summarized in Fig. 9, for which a solitonic lattice of about 40 m spacing was first created. The input size of the vortex ring was adjusted by proper imaging to cover the central four lattice sites [see Fig. 9(a)]. After linear propagation through the 8 mm-long crystal the vortex ring expanded to about 2.5 times in its diameter, while the propagation direction of the vortex core was unchanged (collinear with the lattice propagation). After a bias field of 2.2 kV/ cm was applied to the crystal, the vortex beam broke up into two major filaments as a result of azimuthal instability, each of which rotated slightly in direction corresponding to the sign of topological charge. Such breakup can be seen clearly in Fig. 9(b), where the two filaments rotated counterclockwise and clockwise for vortices of opposite topological charge m = + 1 and m = −1, respectively. The simultaneous propagation of the lattice and the vortex, however, drastically influenced the vortex dynamics. Because of the refractive index modulation as induced in a nonlinear lattice, the vortex ring broke up into four filaments [Fig. 9(c)] rather than two, similar to the vortex propagation in a linear lattice [Fig. 8]. In contradistinction to that in a
Fig. 8. Vortex beam propagating in an optically-induced linear lattice showing formation of a 2D discrete vortex soliton. Top panels show the 3D intensity plots of the vortex beam at crystal output of (a) normal diffraction without the lattice (b), (c) discrete diffraction at low nonlinearity, (d) discrete soliton at high nonlinearity (bias field, 3000 V / cm). The lattice spacing is 28 m. Bottom panels show (e) discrete vortex soliton reproduced in a lattice of 20 m spacing, (f)–(h) a series of interferograms between the vortex soliton and a plane wave confirming the nontrivial spiral phase structure of the vortex.
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Fig. 9. Vortex beam of unit topological charge interacting with an optically-induced nonlinear lattice showing lattice twisting driven by the angular momentum carried by the vortex beam. (a) Input intensity patterns of the lattice (top) and the vortex (bottom). Nonlinear output of the vortex propagating (b) alone, (c) with the lattice. (d) Nonlinear output of the lattice corresponding to (c) when the vortex has a positive (top) or a negative (bottom) charge. The lattice spacing is 40 m in all cases.
Fig. 10. Doubly-charged vortex beam 共m = + 2兲 interacting with a nonlinear solitonic lattice. (a) Input intensity pattern of the vortex. Nonlinear output of the vortex propagating (b) alone, (c) with the lattice. (d) Nonlinear output of the lattice propagating with the vortex; the corresponding unperturbed lattice is shown in Fig. 2(c). The lattice spacing is 37 m in all cases.
linear lattice, the four spots resulting from the vortex breakup rotated from their initial orientation, which also caused lattice twisting in the area of interaction. As seen in Fig. 9(d) the direction of twisting is opposite for vortices with opposite topological charges. Through these experiments we believe that the lattice acquired some angular momentum from the vortex beam, which was then redistributed to all the lattice sites. This behavior was more clearly visible in the transient dynamical evolution by taking advantage of the slow response of the photorefractive nonlinearity. Right after launching the vortex beam into the lattice, we observed that the corresponding lattice sites were strongly twisted, but the twisting was later reduced through interaction with the neighboring lattice sites. While it is possible to confine the high-order vortices in a linear lattice to form localized states of quasi-vortices,42,43 the nonlinear interaction between a high-order vortex and a solitonic lattice often leads to disordered structures. Experiments were performed with a charge-2 and a charge-4 vortex in a manner similar to that with the charge-1 vortex. Vortex-induced twisting of
the interacting lattice sites was observed, but the highorder vortices broke up into more filaments that tended to escape from the four lattice sites at which the vortex was initially aimed. Figure 10 shows the preliminary results obtained with a charge-2 vortex. Although vortex-induced lattice twisting was clearly observed in a transient state, the outcome of vortex-lattice nonlinear interaction in steady state is very sensitive to the initial symmetry of the vortex ring, the uniformity of the lattice, and the alignment of the vortex–lattice beams. We note that angular-momentum-carrying beams or optical vortices are known to be useful for manipulation of microparticles.44,45 Since a nonlinear lattice can be considered a lattice of solitonic particles, it is reasonable to expect that the orbital angular momentum can be correspondingly transferred from a vortex beam to a solitonic lattice. However, solitons as formed in a nonlinear lattice are not free to move, and the lattice typically breaks the rotational symmetry of homogeneous media. Thus angular momentum is not generally conserved in the process of vortex–lattice interaction. We believe that the interaction could lead to vortex-induced defect states in a nonlinear
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lattice, and it might also be possible to form a mutually trapped bound state between the high-order vortices and the lattice. Further experimental and numerical studies are currently under way to better understand the dynamics of vortex–lattice interaction.
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5. SUMMARY We have demonstrated experimentally a number of novel phenomena associated with soliton–lattice and vortex– lattice interactions in a 2D, optically-induced, nonlinear photonic lattice, including lattice dislocation and creation of structures akin to optical polarons. We have observed soliton-induced lattice deformation–compression when the interacting soliton has an initial linear transverse momentum, as well as vortex-induced lattice twisting due to the transfer of angular momentum carried by the vortex. In addition 2D fundamental discrete solitons or discrete vortex solitons have been successfully demonstrated when a Gaussian beam or a vortex beam is launched appropriately into an optically induced linear lattice. Our results may pave the way for observation of similar phenomena in other types of nonlinear periodic systems, such as Bose–Einstein condensates loaded onto an optical lattice.
ACKNOWLEDGMENTS Z. Chen is also with the Applied Physical School, Nankai University, Tianjin, China. This work was supported by Air Force Office of Scientific Research, Army Research Office, the Overseas Excellent Young Researcher Award from National Natural Science Foundation of China (No. 60328406), the Pittsburgh Supercomputing Center, and the Australian Research Council. We thank J. Yang, E. D. Eugenieva, M. Segev, P.G. Kevrekidis, T.J. Alexander, E. A. Ostrovskaya, I. Makasyuk, J. Young, W. Krolikowski, and J. Xu for assistance and numerous discussions. Corresponding author Z. Chen’s e-mail address is
[email protected].
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