1. Introduction

The discovery of self-trapped incandescent light 15 years ago [1], revealed a new facet of nonlinear light propagation, which previously had been restricted to coherent optical waves. The spatial and temporal incoherence as well as the polychromaticity of incandescent provided insight into the spatial correlation [2], shape [3], frequency distribution [4] and modulation instability (MI) [5] associated with the nonlinear propagation of light beams. We demonstrated that the prerequisites for nonlinear propagation of incandescent light, a non-instantaneous photoresponse, saturable refractive index and self-consistent propagation of the multiple modes of white light, was satisfied by refractive index changes associated with a free-radical polymerization reaction. This enabled studies of a range of nonlinear phenomena excited by incandescent light including the dynamics [6, 7] and interactions [8] of self-trapped incandescent beams and spontaneous pattern formation due to modulation instability [9]. In this article, we show how the simultaneous modulation instability of three orthogonally propagating incandescent beams generates a cubic lattice of incandescent light and the corresponding lattice of cylindrical waveguides.

We previously demonstrated that a broad, uniform beam of incandescent light suffers modulation instability and spontaneous filamentation as it propagates through a photopolymerizable organosiloxane medium [9]. Random noise in the system (adventitious variations in intensity and refractive index) becomes greatly amplified due to the relatively large (0.006) changes in refractive index (Δn) associated with polymerization. Rendered unstable by resulting strong modulations in intensity, the optical field stabilizes by collapsing into multiple filaments. Each filament self-traps and propagates through the medium (typically with pathlength = 10.0 mm) without diverging. Because Δn due to polymerization are irreversible, these self-trapped filaments permanently inscribe a densely but randomly packed array of cylindrical waveguides in the medium.

We then showed that by deliberately introducing noise in the form of weak spatial modulations across the beam, it is possible to direct the spatial organization of MI-induced filaments. For example, in the case of a beam bearing a 1-D periodic pattern of grey and bright stripes, filaments formed selectively within the bright stripes and in this way, organized into a 2-D array [9]. Similarly, when beams were patterned with hexagonal array of grey spots or rings, filaments organized into hexagonal lattices [10]. Significantly, this approach was successful only when the periodicity of the modulation was commensurate with the characteristic filament diameter.

We extended this approach to 3-D by employing an orthogonal pair of white light beams, which simultaneously underwent modulation instability in the photopolymerizable medium [10, 11]. The resulting orthogonal pair of filament arrays intersected to generate a 3-D lattice of white light. Each filament-constituent of the lattice underwent >100 intersections with its orthogonal counterparts as it propagated through the medium, which typically had a pathlength of 10.0 mm. Remarkably, these intersections, which are reminiscent of the elastic collisions between solitons, neither attenuated nor altered the propagation-direction of the filaments. The intersecting filament arrays permanently inscribed the corresponding 3-D lattice of cylindrical waveguides in the medium. By varying the spatial modulation that was introduced on the beam, it was possible to generate 3-D waveguide arrays with woodpile (interleaved FCC [10]), BCC [10] and near-cubic [11]) symmetry.

The above-described approach to generating optical and microstructural lattices is termed optochemical self-organization [10]. The term emphasizes the collaborative roles of the optical field and the photochemical reaction as well as the spontaneity of modulation instability. Specifically, MI produces filaments in photopolymer systems, which possess a self-selected characteristic diameter that is determined through the coherence of the optical field [12, 13]. In other words, spatial modulations imposed on the optical field by amplitude masks dictate neither the shape nor size of filaments. Instead, they seed the spatial organization of filaments into specific geometries. This method differs from holographic lithography, which until now was the only route to 3-D optical lattices and is based on the interference of at least four, mutually coherent, quasi-monochromatic laser beams.14 Optochemically organized 3-D lattices can be generated with two mutually incoherent, polychromatic beams, each of which is itself incoherent in time and space. Furthermore, these lattices originate from the self-action of filaments, which propagate throughout the medium without diverging (as they would under linear conditions). Resulting lattices therefore span very large volumes (e.g. 2 cm x 1 cm x 1 cm).10 By contrast, microstructures generated through passive techniques including holographic lithography are limited by the natural attenuation (Beer-Lambert law) and diffraction of light, which blur and gradually erase any spatial modulation in the incident optical field; pattern transfer therefore occurs only to typical depths of 10 s of microns [14]. Most significant to the current study, optochemically organized lattices permanently inscribe the corresponding microstructural lattice comprising tens of thousands of cylindrical, multimoded, polychromatic waveguides.

In this Letter, we report what is to our knowledge the first example of a 3-D periodic waveguide lattice comprising 3 mutually orthogonal and intersecting arrays of cylindrical, multimode waveguides. The lattice is generated with 3 mutually orthogonal incandescent beams, which simultaneously undergo MI to form three intersecting arrays of white light filaments. This white light lattice is a 3-D Bravais lattice with primitive cubic symmetry and inscribes the corresponding lattice of waveguides into the photopolymer medium. We then demonstrate that this cubic lattice exhibits a nonlinear photoresponse, which enables it to host a self-trapped incandescent beam that is akin to a lattice soliton [15].

2. Experimental

2.1. Generation of 3-D cubic optical and waveguide lattices

Figure 1 is a scheme of the optical assembly employed to generate 3-D optical and waveguide lattices. The assembly comprised three mutually perpendicular quartz-tungsten-halogen (QTH) lamps, each of which emitted a beam of incandescent light (400 nm – 800 nm; ~3.5 mW). Each of the three beams was collimated with a planoconvex lens (L1); at least one beam was passed through a binary amplitude mask with a 1-D periodic pattern (periodicity = 80 μm) before being launched into a transparent glass cuvette (10 mm x 10 mm x 10 mm) containing the photopolymerizable organosiloxane sol (Fig. 1(b)). The modulation imposed on the beam by the mask was sufficiently weak that it rapidly decayed with propagation distance; the optical field therefore appeared uniform at the sample exit face (pathlength = 10.0 mm). We classify this modulation as a controlled form of noise, which cannot be detected under linear conditions but under nonlinear conditions, can seed MI. The spatial intensity profile of beam Z at the exit face was imaged through a planoconvex lens pair (L2) onto a CCD camera and monitored over time.

 

Fig. 1 (a) Scheme of assembly used to generate optical and waveguide lattices within a photopolymer. (b) Inset of sample region, depicting configuration X_z + Y + Z.

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We use the following notation to describe the experimental configuration employed to generate lattices: each beam launched into the sample cell is denoted by a capital letter (X, Y or Z) corresponding to its propagation axis with a subscripted letter (x, y, or z) to denote the axis of 1-D periodic modulation imposed by an amplitude mask on the beam. For example, configuration X_z + Y + Z depicted in Fig. 1(b) corresponds to three orthogonal white light beams propagating along the x-, y- and z-axes where beam X is periodically modulated along the z-axis; beams Y and Z are not modulated and are uniform in intensity.

2.2 Generation of self-trapped white light beams in the lattice

White light emitted by a QTH lamp was passed through a pinhole and focused through a biconvex lens onto the entrance face of the cubic waveguide lattice contained in a transparent cuvette; the sample was mounted on a micrometer stage to enable translation along the direction transverse to the beam propagation axis. At the entrance face, the beam had a diameter of 100 μm (FWHM) and power of ~15 mW. The spatial intensity distribution at the exit of the lattice (pathlength = 10.0 mm) was imaged through biconvex lens (F. L. = 8.83 cm) onto a CCD camera and monitored over time.

2.3 Scanning electron microscopy (SEM) of waveguide lattices

Samples for SEM were extracted from the glass cuvette immediately after optical experiments and cleaved to expose the desired cross-section. Samples were sputter-coated with Au and mounted with Ag-paste onto SEM stubs. Micrographs were acquired with a Philips 515 scanning electron microscope.

3. Results and discussion

3.1 Coaxing white light filaments induced by modulation instability into ordered lattices

To understand the significance of the generation of a cubic waveguide lattice, it is useful to summarize how the spatial organization of MI-induced filaments can be controlled in 1 [9] - and 2 [11]-dimensions. We previously showed that MI of incandescent light (with spatial coherence length = 300 nm [9]) in the photopolymerizable medium yields filaments with a characteristic diameter (d_f) of ~80 μm [9]. In the case of an unmodulated, homogeneous optical field, the filaments were closely packed but randomly arranged in space [9]. The sequence of images in Fig. 2 show how different degrees of spatial ordering of filaments can be achieved by introducing a weak, 1-D periodic modulation on the optical field. The single and double-beam experiments represented in Fig. 2 were extensively described elsewhere [9–11] and therefore briefly summarized below.

 

Fig. 2 Introducing increasing degrees of spatial order of MI-induced filaments. Lattice formation achieved with configurations (a) Z_y, (b) X_z + Z_y and (c) X_z + Y_x + Z_y Each scheme in (a-c) traces the evolution of the lattice from the initial (linear) propagation of the beam(s) to the formation of bright lamellae (Stage 1) and finally, MI and emergence of stable filaments (Stage 2). Experimentally acquired spatial intensity profiles of the (001) face of the final lattice (Stage 2) are included in each case. For intensity profile, 1 pixel = 9 μm.

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Figure 2(a) depicts the behavior of a single incandescent beam that was patterned at the entrance face with a 1-D modulation of bright and grey stripes along the y axis. This corresponds to the experimental configuration, Z_y (vide supra). The intensity contrast of the pattern was sufficiently weak that it disappeared rapidly with propagation distance (along z) and the beam appeared uniform at the exit face of the medium (z = 10.0 mm). As the beam initiated polymerization along its propagation path, Δn maximize along the bright stripes. This leads to Stage 1 of the process and the emergence of a 1-D periodic stack of high-refractive index planes. These serve as planar waveguides and selectively trap intensity from the incidence beam to form a stack of bright planes (lamellae) [9, 11]. In Stage 2, the continued increase of Δn within each lamella elicits MI and filamentation. The result is a stable, 1-D stack of planes, each comprising an array of co-planar filaments.

As observed by the spatial intensity profile in the (001) plane (Fig. 2(a)), filament planes induced by beam Z_y are periodically stacked along y; filaments within each plane however are randomly positioned along x. The degree of spatial order is quantified by parameters, σ_a and σ_p, which correspond respectively to the average standard deviations of filament alignment within planes, and the periodicity with which the planes are stacked. Based on a previously established scale, values < 0.15 typically indicate excellent alignment or periodicity [11]. In the case of the filament lattice in Fig. 2(a), σ_a and σ_p are typically 0.11 and 0.04 [11]. These values denote the strong confinement of filaments within the (xz) planes and the highly periodic stacking of these planes along y. By contrast, typical values of σ_a and σ_p for filament confinement within (yz) planes and the periodicity of these planes along x are 0.26 and 0.36, respectively. This indicates that similar co-planar confinement of filaments and periodic stacking of planes do not occur along these axes [11].

Figure 2(b) shows that a lattice of two intersecting filament populations can be generated by simultaneously eliciting MI of an orthogonal pair of beams; the experimental configuration is X_z + Z_y. In this case, the ordering of filaments in the (001) face is comparable to the case of the single-beam experiment. However, along the x axis, filamentation is not only confined to lamellae induced by beam X but also to the intersections with the orthogonally stacked lamellae induced by beam Y. The intersections comprise superposed intensity from both beams and therefore are the highest intensity regions generated in Stage 1. During Stage 2, filamentation is restricted to these 2-D periodic intersections resulting in a square array of filaments in the (100) face. This is consistent with values of excellent alignment of filaments in both (xy) and (xz) planes (typical σ_a = 0.09 and σ_a = 0.12, respectively). There is also highly periodic stacking of both sets of planes along the z and y axes (typical σ_p = 0.06 and σ_p = 0.06, respectively).

Although the double-beam experiments represented in Fig. 2(b) yield 3-D lattices, they possess near-cubic symmetry; there is no double-beam configuration that could generate a simple cubic lattice, where the unit cell comprises a cube with equal sides. In this case, all faces of the waveguide lattice must be equivalent and display square symmetry. Figure 2(c) shows how such a simple cubic lattice can be generated by introducing a third orthogonal incandescent beam corresponding to configuration X_z + Y_x + Z_y. In Stage 1, each beam-pair induces two mutually orthogonal stacks of lamellae and in this way, generates high-intensity intersections with 2-D periodicity (square symmetry) along each of the three axes. In Stage 2, filamentation is therefore restricted to the intersections and generates a simple cubic lattice. Details of this triple-beam experiment are provided in the following Section.

3.2 Optochemical organization of simple cubic lattices

Typical experimental results confirming the generation of a simple cubic lattice with configuration X_z + Y_x + Z_y are presented in Fig. 3 (See Media 1 in Supporting Information). Spatial intensity profiles of the (001) face acquired at increasing times trace the three distinct stages of lattice formation.

 

Fig. 3 Optochemical organization of an incoherent white light lattice and waveguide lattice with primitive cubic symmetry (see also Media 1 in Supporting Information). (a) Spatial intensity distributions of the (001) face of the lattice (after a propagation distance = 10.0 mm) trace the three stages of lattice formation from linear divergence, to the intersections of lamellae (Stage 1) to the formation of stable 2-D periodic filaments (Stage 2) (1 pixel = 9 μm). Scanning electron micrographs of the corresponding waveguide lattice show that (b) (100), (c) (010) and (d) (001) faces of the lattice show a square array of filaments, which confirmed by FFTs that are included as insets.

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At early times, the spatial intensity profile is nearly uniform with a very weak 1-D modulation of intensity along y. This is because the large divergence characteristic of incandescent light [16] causes rapid decay of the 1-D modulation imposed on the beam at the entrance face of the sample. As polymerization and corresponding Δn are initiated in Stage 1, the three beams simultaneously induce a periodic stack of bright lamellae along their respective propagation paths. Orthogonal intersections between lamellae induced by beams Z_y and Y_x lead to high-intensity rectangular prisms with long-axes along Z (Fig. 2(c)). These high-intensity regions are projected onto the (001) face as a 2-D periodic array of bright spots. MI in Stage 2 is restricted to the high intensity regions, leading to a square array of circular self-trapped filaments in the (001) face (Fig. 3(a)).

The critical role of beam Y in eliciting 2-D ordering of filaments becomes evident when comparing the experiments depicted in Fig. 2(c) with 2(b); both experiments are identical except for the absence of beam Y_x in the latter. Square filament arrays in the double-beam experiment can only be generated in the (100) plane (Fig. 2(b)). In the triple-beam experiment with configuration X_z + Y_x + Z_y, orthogonal intersections of lamellae leads to square arrays of filaments along all three axes, i.e., lamellae intersections induced by Y_x + Z_y, X_z + Z_y, X_z + Y_x creates square arrays in (001), (100) and (010), respectively. These three mutually orthogonal, square filament arrays intersect each other to generate a simple cubic lattice of white light. Here, each lattice point is represented by the intersection of 3 mutually orthogonal filaments.

The white light lattice induces the corresponding waveguide lattice with cubic symmetry in the photopolymer medium. Scanning electron micrographs in Fig. 3(b-d) confirm its simple cubic symmetry; each lattice face comprised a square array of filaments confirmed by FFT, which generated the corresponding reciprocal (square) lattice. Notably, the lattices are different from those fabricated through passive photolithography including holographic lithography [14]. In the latter, a predetermined intensity distribution is transferred onto a photoresponsive system; the depth of the recorded pattern is limited by absorbance of the optical field by the medium (Beer-Lambert law). Our approach by contrast relies on self-action and nonlinear propagation of the beam. For example, under linear conditions, the modulations imposed on the beam decay rapidly so that it is nearly uniform at the sample exit-face. Under nonlinear conditions however, filaments generated through MI self-trap and propagate without diverging over long distances and in this way, inscribe waveguide lattices over large volumes. This is strikingly evident in the photograph of a cubic waveguide lattice contained in a cuvette of dimensions 10 mm x 10 mm x 10 mm; such a lattice is assembled from three intersecting square arrays, each comprising ~10, 000 cylindrical waveguides. The photograph in Fig. 4(b) shows light from a hand-held HeNe laser (beam width ~1 mm) that was coupled into the waveguide array and projected onto a far-field screen as a square array of bright spots. This is consistent with the cubic symmetry of the lattice and specifically, the square arrangement of filaments along each axis.

 

Fig. 4 Photographs of (a) optochemically organized cubic lattice in a 1 cm³ cuvette. The streak of light propagating through the lattice corresponds to a self-trapped incandescent beam (vide infra) and (b) output of the lattice (from (001) face) from a 632.8 nm laser beam.

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3.3 Generation of a self-trapped white light beam in the optochemically organized cubic lattice

The cubic waveguide lattice generated through optochemical organization was not a passive dielectric structure. This is because lattice formation takes place before the large maximum change in Δn in the medium (0.006) saturates. As a result, the structure can undergo further increases in Δn; at the molecular level, this originates from residual polymerizable methacrylate groups in the medium, which undergo additional free-radical reactions upon irradiation to cause an increase in the macroscopic Δn. We exploited this property to demonstrate the behavior of the structure as a nonlinear photonic lattice and its ability to generate self-trapped lattice beams of incoherent white light.

To generate the self-trapped lattice beam, light from an incandescent QTH lamp (400 nm – 800 nm) was spatially filtered, focused to 100 μm (FWHM) and coupled into a central waveguide at the entrance face of the cubic lattice. Spatial intensity profiles in Fig. 5 trace the evolution of the narrow beam after it had propagated through the 10.0 mm-long waveguide lattice (see also Media 2 in Supporting Information). Under linear conditions (t = 0 s), the beam diverged with propagation distance and tunnelled into a large population (~90) of neighboring waveguides. (Under linear conditions but in the absence of the lattice, the beam diverged to 216 μm over the same propagation distance.) With time, refractive index changes induced by the beam along its propagation path in the lattice led to the formation of a self-trapped lattice beam. At 170 s, this was indicated by a nearly 3-fold in the relative peak intensity of the beam from 35% to 92%; intensity of the self-trapped beam was strongly localized to a single waveguide and accompanied by significant decrease of light intensity in vicinal lattice sites. The lattice self-trapped beam was stable at long-times; although it underwent a decrease in relative intensity to 71%, it never reverted to its original divergent form.

 

Fig. 5 Self-trapped incoherent white light beam in a primitive cubic lattice. 3-D and 2-D spatial intensity profiles tracing the temporal evolution of a narrow white light beam propagating through an optochemically organized primitive cubic lattice. Intensity profiles were acquired of the (001) face at a propagation distance = 10.0 mm. The intensity scale in the 3-D images is the same; the scale in the 2-D image in (a) was normalized to the maximum peak intensity for clarity (1 pixel = 9 μm). (Media 2)

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6. Conclusions

We demonstrated that optochemical organization is an efficient, inexpensive method to generate the first example of a waveguide array that is a 3-D Bravais lattice. In particular, the method exploits the soliton-like, inert intersections of orthogonal self-trapped filaments to generate microstructures built entirely from a 3-D grid of intersecting waveguides. (Remarkably, each self-trapped beam undergoes ~125 intersections as it propagates through the typical 10.0 mm pathlength of the sample.) To our knowledge, such intersecting waveguide lattices would be impossible to achieve with conventional photolithographic techniques. Importantly, this technique is extremely inexpensive requiring only three incandescent lamps at relatively low intensities and therefore could be achieved with miniature LED sources, which also emit incoherent visible light.

In the current study, the periodicity of the waveguide lattice is defined by the characteristic diameter of filaments (~80 μm) generated through MI in this particular system. We have shown elsewhere that this parameter can be varied through the coherence of the optical field [13], which should enable generation of lattices with tuneable periodicity. Furthermore, we have demonstrated that symmetries of waveguide lattices obtained through single- and double-beam experiments can be varied using incandescent beams that are spatially modulated with dark regions [10, 17]. This could now be applied to the triple-beam experiments to access other 3-D Bravais lattices.

We showed the potential of the primitive cubic waveguide lattice to serve as a nonlinear photonic crystal. This was achieved by eliciting a self-trapped beam of incoherent white light in the lattice. While significant advances have been made in the generation of lattice solitons in 2-D waveguide lattices [18–22]. this to our knowledge is the first example of the observation of a self-trapped beam in a 3-D waveguide lattice.

The findings described in this article therefore provide easily accessible experimental routes to 3-D waveguide lattices and self-trapped incoherent beams. This will create fundamentally new opportunities to complement and verify theoretical predictions of incandescent lattice solitons [15] and more generally, provide insight into the nonlinear propagation of waves in natural systems, which are expected to be incoherent [18]. These findings also motivate our future studies, which will probe the existence of bandgaps in the 3-D waveguide lattices, examine the evolution of the self-trapped beam in reciprocal space, map its behavior along different lattice axes and probe for self-trapped lattice beams in 3-D waveguide lattices with different symmetries. The waveguide architectures also have potential for applications in the linear regime; for example, multiple waveguide lattices that intersect at angles < 90° and thus are oriented over a range of angles that span 180° could serve as efficient, angle-independent light-collection and delivery structures for optical devices including solar cells.