A Gaussian beam propagating in a photopolymer undergoes self-phase modulation to form diffraction rings and then transforms into a single ring, which in turn ruptures into a necklace of stable self-trapped multimode filaments. The transitions of the beam between the three distinct nonlinear forms only occur at intensities where the beam-induced refractive index profile in the medium slowly evolves from a Gaussian to a flattened Gaussian.
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Self-trapped light filaments that are arranged in a circle show intriguingly different behaviour from solitary self-trapped beams propagating under the same conditions. For example, the mutual stabilization of necklace filaments enables their quasi-solitonic propagation in a Kerr medium, which by contrast does not support individual 3-D solitons . Circular arrays of filaments also exhibit rich and varied dynamics with respect to optical power and the presence of defects , mutual incoherence  or phase difference  between filaments, propagation in photonic lattices , topological charge of the parent vortex [5,6] and interactive forces .
In both experimental and theoretical studies, circular filament arrays have been predominantly generated through modulation instability and consequent filamentation of optical vortices [4–7] while filamentation of uniform phase, ring beams has also been reported [8,9]. Here we report that an entirely unmodified and relatively weak Gaussian, continuous wave beam spontaneously transforms into a stable circular array of filaments as it propagates through a photopolymer. Under the nonlinear conditions originating from free-radical polymerization , the beam first undergoes self-phase modulation and excites a series of diffraction rings before transforming into a distinct single ring, which stabilizes upon collapse into a circular array of self-trapped multimode filaments. Because refractive index changes due to polymerisation are irreversible, the cylindrical waveguides induced by the self-trapped filaments remain permanently inscribed and provide an indelible imprint of filament trajectories. Experimental observations of the seamless and relatively slow transitions (spanning seconds to minutes) of the beam between three different forms of nonlinear propagation, i.e., diffraction rings, single ring and ring of multimode filaments are to our knowledge unprecedented.
Our findings are relevant to but different from a recent report of the transformation of a super-Gaussian beam with its characteristic flat-top profile into a ring while propagating through a nonlinear Kerr medium . Seeded by noise, the ring subsequently collapsed into filaments. This spontaneous and dramatic change in beam profile was considered unique to the super-Gaussian beam. Until the current report, Gaussian beams propagating under similar nonlinear conditions were assumed to collapse to the Townes profile .
We previously examined nonlinear propagation of the Gaussian beam across a broad range (3 x 10−5 Wcm−2 to 1.3 x 104 Wcm−2) of intensities that spanned over 10 orders of magnitude.10 There were significant intensity-dependent variations in the rate, magnitude and shape of refractive index changes induced by the beam at the entrance face of the sample. The different spatial profiles of refractive index changes in turn elicited a diverse collection of nonlinear propagation phenomena: at small intensities, the beam induced an approximately Gaussian refractive index profile, which initiated self-trapping of the beam. At mid-level intensities, the rapid increase in magnitude of the same Gaussian index profile led to spatial self-phase modulation of the beam and consequent excitation of diffraction rings . At extremely high intensities, the almost instantaneous saturation of refractive index over the entire cross-section of the beam immediately led to its filamentation.
The transitions of the same Gaussian beam described in this report between different forms of nonlinear propagation originate from the specific way in which the refractive index profile evolves in the time-scale of the experiment. We located these transitions within an extremely narrow range (27 Wcm−2 to 111 Wcm−2) of the incident beam. Such a spontaneous evolution of the refractive index profile and consequent modulation of the beam profile are due to the relative large, slow and saturable refractive index changes that are unique to the photopolymerisation process  and cannot be achieved in nonlinear materials such as Kerr media and photorefractive crystals.
2. Results and discussions
2.1 Experimental observations
Preparation of the photopolymer employed in this study has been detailed elsewhere . Briefly, the photopolymer consisted of a sol of oligomeric siloxanes with photopolymerizable methacrylate substituents. The sol was sensitized to visible wavelengths through the addition of 0.05 wt% of a titanocene free-radical photoinitiator (λmax = 393 nm, 460 nm). The sample was contained in a cylindrical cell (diameter = 12 mm, pathlength (z) = 6.0 mm) with optically flat and transparent windows. The sol was partially photopolymerised and transformed into a non-free flowing gel through uniform illumination with white light emitted by a quartz-tungsten halogen lamp. In a typical experiment, a linearly polarized, continuous wave, 532 nm laser beam (TEM00 mode) was focused to a diameter (1/e2) of 20 µm and intensity of 80 Wcm−2 onto the entrance face (z = 0.0 mm) of the gel. The spatial intensity profile of the beam at the exit face (z = 6.0 mm) of the sample was imaged through a pair of planoconvex lenses onto a CCD camera and monitored over time.
Typical experimental results presented in Fig. 1 showed that the beam initially diverged as it propagated through the medium from a width of 20 µm at z = 0.0 mm to 319 µm at z = 6.0 mm (Fig. 1a). Within the next few seconds, concentric rings emerged around the beam and rapidly increased in number (Fig. 1b-d). Up to 5 diffraction rings, the outermost of which was the most intense, could be resolved. The beam then developed a dark central spot that gradually widened, effectively transforming the beam at 37 s into a single ring with a thickness of 190 µm and outer diameter of 420 µm. At 61 s, the dark spot had widened sufficiently to decrease the thickness of the single ring to 27 µm; narrowing of the ring was accompanied by a 2-fold increase in its relative intensity. It is important to note that the thick single ring was distinct from the thin diffraction rings, which were faintly visible during this time but underwent no further significant changes and eventually disappeared.
The single ring, which maintained its diameter of 420 µm from this point, was unstable and spontaneously divided into multiple filaments. Although up to 17 filaments were observed at 100 s (Fig. 1g), they suffered rapid and random fluctuations in number and intensity until they stabilized at 253 s into a set of 7 azimuthally positioned filaments. Each filament underwent strong self-trapping as indicated by a 20-fold increase in intensity (relative to the single ring at 61 s). The circular array of filaments remained stable for as long as it was monitored (509 s) (Fig. 1j, k). The sequential transitions of the Gaussian beam from diffraction rings to a single-ring structure, which stabilized upon filamentation were reproducible in all 5 experiments performed under identical conditions.
2.2. Evolution of beam-induced refractive index profiles in the organosiloxane
The transitions of the beam observed in Fig. 1 correlate directly to the evolution of the refractive index profile that it induces at z = 0.0 mm. Because they originate from a free-radical polymerization process, photoinduced index changes in the organosiloxane are large (Δn = 0.006), saturable and slow, spanning seconds to minutes . As a result, the spatial profile of the index changes induced by the Gaussian beam evolves significantly over this timescale.
The temporal variation of the refractive index profile induced by the beam at z = 0.0 mm is presented in Fig. 2 ; index profiles were calculated from the expression for index changes due to photoinitiated free-radical polymerization .
Because the rate of polymerization is intensity-dependent, the maximum index change is first induced by the axial region of the beam with a cylindrically symmetric decay from this point. As a result, the beam initially induces an index profile that corresponds to its own Gaussian shape (step 1 in Fig. 2). With time, polymerization and corresponding index changes increase and then saturate first in the axial region and gradually in the less intense regions surrounding the axis. The index profile therefore transforms from Gaussian to an increasingly flattened Gaussian shape (step 20 Fig. 2).
It is important to note that according to Equation , the evolution of the refractive index profile from Gaussian to flattened Gaussian plotted in Fig. 2 would occur at all incident intensities. However, the rate of this evolution varies significantly with incident intensity. For example, at small intensities, the Gaussian shape of the index profile leads to self-trapping (thus precluding self-phase modulation and single-ring formation) while at very large intensities, the rapid, almost instantaneous saturation of the index profile causes filamentation. The transitions described in the current report are found within a narrow range of intensities within which the evolution of the refractive index profile from Gaussian to flattened Gaussian and the consequent transformation of the beam occur at a measurable timescale. As detailed below, this spontaneous and relatively slow evolution of the refractive index profile underlies the transitions of the beam observed in Fig. 1.
2.3. Self-phase modulation due to a Gaussian index profile
The Gaussian refractive index profile induced by the beam at early times (step 1 in Fig. 2) leads to self-phase modulation [13,16], which gives rise to the diffraction rings observed in Fig. 1b-d. Self-phase modulation occurs when the Gaussian refractive index profile imposes a transverse phase shift on the beam according to16:Eq. (3), the profile of the phase shift Δψ(r) corresponds to the Gaussian profile of the refractive index change (Δn(r, z)). Radiation from any two points along Δψ(r) with the same wavevector will undergo constructive (destructive) interfere when Δψ(r1) – Δψ(r2) = mπ, where m is an even (odd) integer. Multiple rings form when the maximum phase shift Δψo > 2π. The cylindrically symmetric profile of Δψ(r) produces an array of interference cones that propagate through the medium and are projected as the concentric diffraction rings at z = 6.0 mm (Fig. 1b). Consistent with the theory of self-phase modulation , the outermost ring, which originates from radiation with the greatest about the inflection point, was the most intense (Fig. 1b).
2.4. Single ring formation due to a flattened refractive index profile
The transition of the beam from diffraction rings into a single ring (Fig. 1c-f) takes place when the Gaussian index profile originally induced by the beam flattens over time (Fig. 2). Under nonlinear conditions, the beam encountering the flattened-Gaussian index profile at z = 0.0 mm is focused into a ring, which then continues to propagate along z. We confirmed this through beam propagation simulations in which a Gaussian beam was launched into the photopolymer medium that had a flattened Gaussian refractive index profile introduced at z = 0.0 mm (Fig. 3 ). In accordance with the experimental system, the simulated beam was made to propagate under nonlinear conditions by inducing index changes along its propagation path according to Eq. (1). Simulations were carried out using the commercially available software package BeamPROP (RSoft Design Group, Inc.) where refractive index updates were iteratively provided through an external subroutine.10
Results of simulations presented in Fig. 3 show the temporal evolution of the spatial intensity profile of the initially Gaussian beam at z = 6.0 mm. Consistent with experimental findings, the beam gradually developed a dark central depression, effectively transforming it into a ring. (It is important to note that consistent with experiment, simulations that show ring formation were carried out when the refractive index profile induced at z = 0.0 mm had already evolved to a flattened Gaussian profile. By contrast, similar beam propagation calculations carried out at relatively smaller intensities lead to self-trapping of the beam, also in accordance with experiment10).
Our findings of single-ring formation are consistent with previous findings of a flat-top beam with a super-Gaussian profile that transformed into a single ring while propagating under nonlinear conditions in a Kerr medium . Here, the super-Gaussian beam induced refractive index changes with the corresponding flat profile. As confirmed through phase and ray plots, this flattened refractive index profile focused the beam into a ring that evolved into a self-consistent G-profile. The flattened Gaussian profiles of the index changes induced in the organosiloxane are in fact close approximations of the super-Gaussian profile, which is described by:
Equation (4), which reduces to the Gaussian form when n = 2, increases in flatness with n. The excellent overlap between the super-Gaussian and flattened Gaussian profiles is displayed in Fig. 4 ; the strong correspondence between the two spatial profiles indicates that the origin of single ring formation is the same in both Kerr and organosiloxane systems. However, the critical difference is that while single rings were obtained with a super-Gaussian beam in Kerr media, they were elicited with an entirely unmodified Gaussian beam in the organosiloxane. In our system, the flattened refractive index profile does not correspond to the original spatial intensity profile of the beam but instead, spontaneously evolves over time as the laser induced refractive index profile reaches saturation (Fig. 2).
2.5. Modulation instability of single ring to azimuthally arranged multimode filaments
The single ring induced by the beam was unstable to random noise in the organosiloxane system and collapsed into a ring of filaments (Fig. 1). We have previously shown that broad, uniform beams propagating in the organosiloxane under similar conditions suffer modulation instability . During polymerization, weak amplitude perturbations (noise) in the medium that are negligible under linear conditions became amplified triggering the spontaneous division of the beam into filaments that were randomly positioned in space. In the current study, modulation instability of the single ring led to an azimuthal arrangement of filaments that initially fluctuated in number but rapidly stabilised into 7 filaments. Each filament underwent strong self-trapping as indicated by a decrease in average filament width (1/e2) from 50 µm at 173 s to 21 µm at 211 s and a 20-fold increase in intensity relative to the initial intensity of the beam.
Each self-trapped filament behaved like an individual self-trapped waveguide and was able to support more than one optical mode. Although their spatial positions and number did not change further, careful scrutiny revealed that the intensity distribution within each filament did continue to change. Figure 5 traces the behaviour of a single self-trapped filament in the ring.
Until 207 s, the filament retained the tightly focused profile characteristic of a self-trapped beam; this corresponds to the fundamental mode (LP01) supported by a circular waveguide . At 313 s, the intensity distribution within the filament changed into the two lobed profile characteristic of the first order waveguide mode, LP11. The filament reverted back to LP01 at 318 s. Such oscillations between LP01 and LP11 continued until 393 s. Only the first and the last oscillation of the beam between LP01 and LP11 are presented in Fig. 5.
We recently demonstrated that under certain conditions, a single self-trapped beam evolved from single mode to multimode propagation in the organosiloxane . We detected the sequential appearance of up to 5 high order waveguide modes, which underwent oscillations similar to those observed between the LP01 and LP11 modes in the filaments (Fig. 5). Self-trapped beams in most other media including Kerr materials and photorefractive crystals induce only a single mode waveguide and propagate as its fundamental mode . Relative to these materials, the maximum index change in the organosiloxane is greater by at least 2 orders of magnitude. Here, the beam initially self-traps as the fundamental mode but continues to increase the refractive index of its own waveguide and in this way, sequentially excite high order modes. Theoretical simulations showed that the oscillations between modes originate from the slowly increasing refractive index of the self-induced waveguide . This causes a continual variation in both the number and propagation constants of modes, which is observed as the complex oscillations between modes at the exit face of the medium.
The remarkable similarity in behaviour of the filaments in Fig. 5 to that of a single self-trapped beam confirms, as has been proposed by Chiao , Campillo and associates , that the filaments resulting from modulation instability are a microscale example of self-trapping. In contrast to the self-trapped beam however, only one high order mode was observed in the self-trapped filaments. As intensity of the beam must be distributed between multiple filaments, the intensity and therefore the refractive index change within a single filament is probably insufficient to support multiple modes. Due to the relatively small dimensions of each self-trapped filament, it was moreover difficult to detect high order modes in all of the repeat experiments. However, we observed them at least twice for each of the intensities studied.
2.6. Intensity dependence of filamentation due to modulation instability of single rings
In addition to 80 Wcm−2, the three transitions of the beam were observed at 5 different intensities spanning a narrow range (27 Wcm−2, 40 Wcm−2, 64 Wcm−2, 95 Wcm−2 and 111 Wcm−2). Images of the single ring as well as the corresponding circular array of filaments induced at different intensities are collected in Fig. 6 , associated parameters, tabulated (Table 1 ) and complete results for each intensity, presented as supplementary information
Although the widths of single rings were statistically identical (Table 1), the number of filaments that formed upon ring collapse varied with intensity. Rings induced at 27 Wcm−2 and 40 Wcm−2 generally ruptured into a coplanar pair of filaments. A significant increase in their number was found at greater intensities. 16 and 17 filaments were observed at 64 Wcm−2 and 80 Wcm−2, respectively while 14 and 16 filaments were induced at 95 Wcm−2 and 111 Wcm−2. This trend is consistent with the previous study of the propagation of a super-Gaussian beam in a Kerr medium, in which the number of filaments increased in proportion to the square-root of the input power .
Interestingly, the filaments were preferentially positioned along the polarization direction of the incident (linearly polarized) beam. Although particularly evident in the case of the two smallest intensities where few filaments formed, this bias in orientation was also noted at some of the greater intensities where the two most intense filaments within the ring were collinear and coincided with the polarization direction.
2.7. Self-induced channel waveguides
Because refractive index changes due to polymerization are permanent, the self-trapped filaments permanently inscribed a corresponding ring of channel waveguides within the organosiloxane. Optical micrographs revealed that the self-induced channels were azimuthally arranged with each channel projecting outwards from the original propagation path of the beam (Fig. 7 ). This broadening of the ring of filaments is consistent with the conical trajectory of the single ring; the filaments that form upon collapse of the single ring then trace the surface of a cone.
The number of channel waveguides inscribed in the organosiloxane was in agreement with the number of filaments observed during optical experiments. For example, two channel waveguides were induced at 27 Wcm−2 while four were induced in the case of 40 Wcm−2 (in the latter, the number of filaments changed from two to four towards the end of the experiment). At greater intensities, the number of self-induced channels increased according to the number of filaments observed by the end of the optical experiment (Fig. 7).
We have shown that a Gaussian beam propagating in a photopolymerisable organosiloxane transitions between three distinct forms of nonlinear behaviour, self-phase modulation, single-ring formation and filamentation. The ring of filaments, which is generated at intensities that are considerably smaller compared to other systems, does not require any prior modification of the Gaussian beam. This behaviour is attributed to the temporal evolution of the refractive index profile that is unique to beam-initiated polymerisation, which evolves from a Gaussian to a flattened Gaussian shape. The former elicits diffraction rings due to self-phase modulation while the latter leads to the formation of a single optical ring, which ultimately collapses due to modulation instability into an array of azimuthally positioned self-trapped filaments. The filaments inscribed corresponding rings of cylindrical waveguides within the organosiloxane medium, which enabled direct visualization of the 3-D trajectories of the self-trapped filaments.
Apart from its significance as a fundamentally new nonlinear process that spontaneously transforms a Gaussian beam into a stable circular array of self-trapped filaments, these findings have potential application as an optics-free method for beam shaping. The process also opens route for mask free lithography of 3-D cylindrical waveguides with non-planar trajectories that are impossible to achieve through conventional mask-based lithographic techniques.
Funding from the Natural Sciences and Engineering Research Council of Canada, Canadian Foundation for Innovation, Ontario Institute of Technology and McMaster University is gratefully acknowledged. We thank CIBA GEIGY, Canada for generous donation of the photoinitiator IRGACURE-784®.
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