To generate and study the interactions of self-trapped laser beams, we synthesized a pH-responsive poly(acrylamide-co-acrylic acid) (p[AAm-co-Aac]) hydrogel containing covalently attached dangling SP moieties (Fig. 1). Details of the synthesis and gel composition are provided in Materials and Methods, and SI Appendix, section S1 describes the synthesis of the chromophore monomer (SI Appendix, section S1.2.1) and its incorporation into cross-linked hydrogels (SI Appendix, section S1.2.2.1), non-cross-linked polymer (SI Appendix, section S1.2.2.2), and solution samples (SI Appendix, section S1.2.3) (21, 22, 40, 41). When immersed in water, the tethered chromophores exist predominantly in the protonated ring-open merocyanine form due to the presence of vicinal acrylate anions (40). When this SP-modified p(AAm-co-AAc) gel is irradiated with visible light, the isomerization of merocyanine to its closed-ring SP (1) form triggers a cascade of events, which culminates in a local increase in the refractive index (Δn) of the hydrogel. Specifically, the conversion of merocyanine to SP results in an increase in hydrophobicity, which in turn triggers the local expulsion of water and ultimately, contraction of the hydrogel matrix along the irradiated path (22, 33). Detailed theoretical treatment of the light-induced contraction of the hydrogel due to changes in hydrophobicity is provided in SI Appendix, section S2.3.1; coupling of the photoisomerization process to gel dynamics is detailed in SI Appendix, sections S2.3.2. Evidence for photoisomerization is shown in Fig. 1C, where the depletion of the absorbance band (λ max = 420 nm), attributed to the π-π* electronic transition of the protonated merocyanine, is accompanied by growth of an absorbance band in the UV region (λ max = 320 nm), associated with the π-π* electronic transition of the chromene moiety in the SP isomer (40). Isomerization is reversible, so that in the absence of visible radiation SP thermally relaxes back to the merocyanine isomer.

SP-modified hydrogels. (A) Photoisomerization scheme of chromophore substituents from the protonated merocyanine (MCH + , Left) to SP (Right) forms in the methylenebis(acrylamide) cross-linked p(AAm-co-AAc) hydrogel. (B) Photographs of chromophore-containing p(AAm-co-AAc) hydrogel monoliths employed in experiments. (C) UV-visible absorbance spectra demonstrating reversible isomerization of MCH + (absorption λ max = 420 nm) to SP (λ max = 320 nm) in solution. (D) Experimental setup (Top) to probe laser self-trapping due to photoinduced local contraction of the hydrogel, schematically depicted on the Bottom (see also Movie S1 ). A laser beam is focused onto the entrance face of the hydrogel while its exit face is imaged onto a CCD camera.

Self-Trapping of Visible Laser Light.

We exploit the light-triggered contraction and corresponding increase in refractive index, Δn, of the SP-modified p(AAm-co-AAc) hydrogel to elicit rapid, efficient, and reversible self-trapping of a visible laser beam. We deliberately selected the 532-nm wavelength for our studies; the relatively low absorbance at this wavelength ensures that the laser beams could propagate through the sample without significant attenuation but also possess sufficient intensity to initiate the ring-closing reaction of the chromophore moieties. The significance of striking a balance in this way between the transparency of the medium and eliciting photoisomerization is detailed in SI Appendix, section S2.1.1. In all cases herein, the beam diameter is defined in the conventional way, as twice the axial distance at which the beam intensity drops to 1/e2 of its maximum value, and is set to be 20 µm at z = 0 mm. Under linear conditions, where there are no photoinduced changes along its path (i.e., at low powers or in the absence of isomerizable chromophores), the beam divergence is calculated to be ∼130 μm for a propagation distance of 3.9 mm through the gel (greatly exceeding the Rayleigh length of 0.6 mm) (4). We anticipated that this natural optical divergence would be strongly suppressed when the beam initiates isomerization of protonated merocyanine moieties and in turn, contraction of the pH-responsive hydrogel along its propagation path (Fig. 1 D, Bottom). Because this contracted region contains a greater volume fraction of polymer, its refractive index is greater compared to its immediate surroundings, which now contains an increased proportion of water [refractive indices of the polymer and water are n ∼ 1.49 and n ∼ 1.33, respectively (42)]. This densified region serves as a cylindrical microscopic waveguide (19, 43⇓⇓–46)—a self-induced optical fiber—that entraps the laser beam as its fundamental optical mode and guides it through the medium without diverging.

Experimental results confirming our hypothesis are shown in Fig. 2. Under linear conditions—in the absence of photoinduced changes—a visible laser beam indeed diverges along the 3.9-mm pathlength to a width of ∼120 μm (in agreement with the calculated value of 130 μm; SI Appendix, Fig. S1D). When launched through the SP-modified p(AAm-co-AAc) hydrogel, the beam self-traps and propagates without diverging indicative of unique nonlinear conditions generated in the gel. The temporal dynamics of the beam is contained in plots of peak intensity and width and corresponding spatial intensity profiles (Fig. 2A and SI Appendix, section S2.1). Within 50 s, the beam undergoes an ∼20-fold increase in peak intensity from ∼10 to 200% with a concurrent ∼threefold decrease in width from 120 to 40 μm (Fig. 2A). With time, the beam continues to increase in relative intensity to 390% with a corresponding decrease in width to 22 µm, which is comparable to its width of 20 μm at the entrance face. This signifies that the self-trapped beam now propagates from the entrance to the exit face with negligible divergence. As detailed in SI Appendix, section S2.1.2, these results also show that despite the absorbance inherent to the medium, self-trapping is an efficient process.

Fig. 2. Evolution of self-trapping in the SP-modified hydrogel; experiments and simulations. (A) Experimentally measured temporal evolution of peak intensity (blue) and effective width (red) of a laser beam (532 nm, 6.0 mW, with a width of 20 μm––corresponding peak intensity = 3.77 kW cm−2) acquired at the sample exit face; the beam is turned on at t = 0. Breaks in plots are time lapses between image logs. The experimental plots (dotted lines) are compared to numerical simulations (solid lines); the dashed black box above provides a zoomed-in view from 0 to 50 s, emphasizing the match between the experimental results and simulations. (B) Two-dimensional (2D) spatial intensity profiles experimentally acquired at select times. (C) Temporal evolution of beam width during self-trapping experiments at different optical powers. (D) Comparison of calculated and experimental values of minimum self-trapped beam width as a function of beam power.

We find that self-trapping dynamics depends strongly on optical intensity (Fig. 2C and SI Appendix, section S2.2 and Fig. S2). Self-trapping efficiency, defined as the greatest percent change in beam width relative to the initial diverged width, increases monotonically from roughly 30 to 80% (with a concomitant decrease in minimum beam width from ∼110 to ∼20 µm) when optical power was increased from 0.37 to 6.0 mW. However, the efficiency decreased to ∼54% when the power was increased further to 9 mW. These trends were observed in at least nine repeat experiments at each intensity.

To describe and provide insight into the self-trapping process, we developed a numerical model that couples the photoisomerization of chromophores with localized volume changes in the hydrogel; details are provided in Materials and Methods and SI Appendix, section S2.3. Briefly, the model couples the photoinduced isomerization of SP to the local swelling and contraction of the hydrogel (SI Appendix, sections S2.3.1 and S2.3.2) and calculates the resulting impact on light propagation in the medium (SI Appendix, sections S2.3.3–S2.3.5) (47⇓–49). As the covalently tethered SP moieties cannot diffuse freely, they are transported with their host polymer chains upon swelling and contraction of the hydrogel. The amount of isomerized SP within a given volume—i.e., its concentration—therefore depends on the flux of polymer chains as well as the optical intensity-dependent rate constant associated with merocyanine-to-SP isomerization and the intensity-independent rate of thermal relaxation of SP to merocyanine. The osmotic pressure induced by the isomerization process leads to a local change in the polymer volume fraction. The corresponding changes in refractive index (Δn) and isomerization-dependent light absorption are calculated and employed in the nonlinear paraxial wave equation to determine the intensity distribution of light within the gel. This distribution of optical intensity is then employed to calculate the isomerization dynamics and associated Δn in the gel. This cycle is repeated iteratively until the desired time is reached.

Consistent with experiments, the model accurately captures the short-term (<500 s) self-trapping dynamics along with the intensity-dependent self-trapping efficiency of a single laser beam in the hydrogel (Fig. 2A and SI Appendix, Fig. S6). As in experiments, the simulated self-trapping efficiency increases monotonically with beam power spanning 0.37 to 6.0 mW (Fig. 2D). This trend originates from the intensity dependence of the photoisomerization process: at lower intensities, the proportion of protonated merocyanine isomerized to SP does not extend far enough into the hydrogel to create the Δn needed for appreciable self-focusing; as the intensity increases, the concentration of isomerized SP molecules rapidly saturates to a plateau in self-trapping efficiency. At the higher intensities, the SP-rich region surrounding the beam leads to contraction of the hydrogel that is large enough to prevent significant divergence of the beam, enabling a greater proportion of the optical energy to extend further into the gel. This triggers more isomerization and thus additional focusing of the beam in a nonlinear feedback loop. At greater powers and longer times, the material exhibits a decrease in self-trapping efficiency due to the excitation of high-order modes (m > 0) (43) as the saturation of isomerization at large intensities forms a flat-top concentration profile that extends beyond the beam width, creating a wider waveguide (SI Appendix, Fig. S7).