Letter
Vol. 40, No. 22 / November 15 2015 / Optics Letters
5235
Nonlinear competition in nematicon propagation URSZULA A. LAUDYN,1 MICHAŁ KWASNY,1 ARMANDO PICCARDI,2 MIROSŁAW A. KARPIERZ,1 ROMAN DABROWSKI,3 OLGA CHOJNOWSKA,3 ALESSANDRO ALBERUCCI,4 AND GAETANO ASSANTO2,4,* 1
Faculty of Physics, Warsaw University of Technology, PL-00662 Warsaw, Poland NooEL–Nonlinear Optics and OptoElectronics Laboratory, University Roma Tre, I-00146 Rome, Italy 3 Institute of Chemistry, Military University of Technology, PL-00908 Warsaw, Poland 4 Optics Laboratory, Tampere University of Technology, FI-33101 Tampere, Finland *Corresponding author: [email protected] 2
Received 29 September 2015; revised 15 October 2015; accepted 16 October 2015; posted 16 October 2015 (Doc. ID 251012); published 5 November 2015
We investigate the role of competing nonlinear responses in the formation and propagation of bright spatial solitons. We use nematic liquid crystals (NLCs) exhibiting both thermo-optic and reorientational nonlinearities with continuous-wave beams. In a suitably prepared dye-doped sample and dual beam collinear geometry, thermal heating in the visible affects reorientational self-focusing in the near infrared, altering light propagation and selftrapping. © 2015 Optical Society of America OCIS codes: (190.6135) Spatial solitons; (190.5940) Self-action effects; (190.4710) Optical nonlinearities in organic materials. http://dx.doi.org/10.1364/OL.40.005235
Nonlinear optical media are often described and employed on the basis of the dominant nonlinearity. In several instances, however, their behavior results from the superposition of different nonlinear processes [1]. Great attention has been devoted to the combination/competition of nonlinear mechanisms as it can potentially improve their tunability and control for optical signal manipulation. Among the early studies of such interplay was the case of an intensity-dependent refractive index in a Kerr-like material with higher than cubic order terms [2]. To date, various nonlinear mechanisms have been identified and investigated in conjunction with a focusing cubic response as a means to stabilize soliton propagation, including quintic terms in conservative [3,4] and dissipative [5,6] systems, as well as parametric processes [7–9]. The Kerr nonlinearity in quadratic media has been exploited for conversion enhancement in harmonic generation with self-confined waves [10,11]; light bullets have been predicted by the competition of nonlinearities acting on different temporal/spatial scales [8,12,13]. In recent years, liquid crystals have been employed as an ideal material platform for the investigation of solitons and their interactions, thanks to their optical properties and tunability by means of external stimuli and the coexistence/competition of different classes of nonlinearities [14] leading, for instance, to enhanced third-harmonic generation in nonlocal solitons [11], the possibility of generating light bullets based on the interplay 0146-9592/15/225235-04$15/0$15.00 © 2015 Optical Society of America
of electronic and molecular responses [13,15], and the formation of self-confined waves via both absorptive and reorientational responses [16]. In this Letter, with reference to the excitation and propagation of bright spatial solitons in dyedoped nematic liquid crystals (NLCs), termed nematicons, we address the competition and interplay of the two dominant nonlinearities when excited by continuous wave inputs, namely the thermo-optic and the reorientational responses. NLCs usually consist of elongated molecules with high anisotropy and large optical birefringence, with refractive indices n⊥ and n∥ for electric fields, respectively, perpendicular and parallel to the average orientation of the molecule main axis, the molecular director n. The low intermolecular forces associated with the liquid state prevents a crystalline structure; hence, no positional order on long ranges is present, albeit a finite degree of orientational order yields a positive uniaxial behavior (i.e., ϵa n2∥ − n2⊥ > 0) at optical frequencies, with the optic axis locally parallel to n [14]. Thus, for input wavevectors parallel to axis z and director lying in the plane yz at an angle θ with zˆ , the electric field of the extraordinary (e-) eigenwave oscillates in theqplane yz and perceives a θ-dependent refractive index ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ne
cos2 θ∕n2⊥ sin2 θ∕n2∥ ; moreover, an e-beam propa-
e gates with Poynting vector at an angle δ n1e ∂n ∂θ (walk-off ) with respect to zˆ [17]. In NLC, the nonlinear response can stem from either resonant or nonresonant interactions with electromagnetic fields, including electronic, electrostrictive, gain, molecular reorientation, and Janossy-enhanced reorientation, as well as photorefractive and ferroelectric effects in the presence of suitable dopants in the NLC mixture [14]. When optical radiation is absorbed near a resonance, thermal fluctuations cause a temperature dependent reduction in the orientational order, resulting in a net decrease (increase) of n∥ (n⊥ ) and thermo-optic light self-action [18–20]; in the nonresonant limit, the reorientational response stems from angular realignment of the director due to Coulomb forces (reorientational torque) on light-induced dipoles [14]. Hereby, we limit our analysis to noninstantaneous and spatially nonlocal nonlinearities obtainable with (low power) continuous wave beams. In this limit, the high coupling between light and molecular
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dipoles enables reorientational and thermal nonlinearities, even at mW powers, where other effects are comparatively negligible. A bell-shaped light beam can then be launched in an NLC volume and propagate in the nonuniform refractive index profile resulting from self-focusing/defocusing depending on polarization and wavelength of the excitation, as well as orientation and spectral resonance(s) of the material. Such self-action effects are at the basis of beam self-confinement, and the formation of bright/dark, as well as discrete solitons (nematicons) [21–23], their control, and the implementation of all-optical devices for signal routing and processing [24]. Both thermo-optic and reorientational nonlinearities can be modeled by Poisson-like equations [25]; therefore, the corresponding nonlinear index wells share the same spatial configuration. When they act simultaneously in response to a given beam profile, their contributions maintain their relative weights in transverse space. To investigate the interplay and competition of these two nonlinearities, we employed a host nematic liquid crystal doped with a specific guest dye, the latter resonant in the visible range. The sample, sketched in Fig. 1, was a planar cell composed of two parallel glass slides at a separation of h ≈ 50 μm, with inner surfaces rubbed to obtain planar anchoring of the director at 45° with respect to z. The cell was filled with the nematic liquid crystal 6CHBT [-4-(trans-4’-hexylcyclohexyl)isothiocyanatobenzene] (n⊥ 1.5021, n∥ 1.6314 at λ 1064 nm and n⊥ 1.52, n∥ 1.6746 at λ 532 nm, at room temperature [26]) doped with 0.1% of Sudan blue II dye. The latter exhibits an absorption peak at λ ≈ 604 nm [27]. We used two Gaussian beams of different wavelengths to decouple and investigate the contribution of distinct nonlinear responses. The input beams were coupled in the NLC cell after focusing with a microscope objective to a waist of w0 ≈ 3 μm; their evolution in the propagation plane yz was observed by acquiring the outof-plane scattered light with a CCD camera and a microscope. The beam at λ 1064 nm, far from the dye resonance, could preferably excite the reorientational response; the beam at λ 532 nm, well inside the absorption band of Sudan blue, but sufficiently removed from the resonance peak, enabled
Fig. 1. (a) Sketch of the NLC sample and (b) nonlinear competition in the guest–host material. (a) Two extraordinarily polarized beams are launched collinearly with a wavevector at θ 45° with the optic axis and exhibit different walk-off angles δ at their respective wavelengths. The blue ellipses represent the elongated molecules; the black arrow represents the director. (b) Calculated nonlocal Kerr-like coefficient n2 verus temperature in a planar 6CHBT sample with θ 45° and (black dashed line) e-wave light at λ 532 nm: reorientational nreor 2 j (solid lines) for Sudan blue concentrations yieldand thermal jnthermal 2 ing optical absorption of 1 × 102 (blue line), 3 × 103 (green line), and 6 × 103 m−1 (red line), respectively. The overall nonlinear response is defocusing when a solid line is above the dashed line.
Letter the dye-mediated thermal response while avoiding the detrimental effects of large attenuation. The relative role of the two nonlinearities can be addressed by using a single scalar figure, namely a nonlocal equivalent of the Kerr coefficient n2 [17]. For the reorientational nonlinearity, it is d ne ϵ0 ϵa Z 0 sin2θ−δ , with Z 0 being the vacuum nreor 2 θ; T d θ 2ne K cos2 δ impedance and K the effective elastic constant [14]. For the θ; T ddTn ακ (with n being thermal nonlinearity, it is nthermal 2 either n⊥ or ne for ordinary and extraordinary eigenwaves, respectively), with α and κ being the absorption coefficient and the thermal conductivity, respectively. For ordinary waves, nthermal is usually positive. For e-waves, the sign of nthermal 2 2 depends on θ: in 6CHBT for θ 45° the thermal nonlinear response is defocusing (nthermal < 0). As anticipated, nthermal is 2 2 proportional to optical absorption and can be enhanced by the use of suitable dopants in the right concentration [see Fig. 1(b)]. In our sample, to obtain an overall defocusing response to green e-beams, the dye concentration was chosen such that, at θ 45°, it was jnthermal j > nreor at room temper2 2 ature. We first studied the two nonlinear responses individually, waiting for the system to reach steady state before acquiring the data. Figure 2 shows the propagation of a low-power visible probe beam with P vis 100 μW, unable to excite any appreciable nonlinear effects, co-polarized and collinear (same Poynting vector directions) with a near-infrared (NIR) beam. To assess the nonlinear properties of the NIR beam through the self-induced refractive index potential, we monitored the visible light evolution when varying the NIR beam power (a pump–probe arrangement): the ordinary wave was not affected by changes in the NIR power (up to P NIR 50 mW) due to
Fig. 2. Propagation of a low power (≈100 μW) visible beam, λ 532 nm, collinear and co-polarized with the NIR beam, λ 1064 nm, as (a), (b) ordinary and (c), (d) extraordinary eigenwaves, at low (a), (c) and high (b), (d) NIR powers, respectively. In the ordinary polarization, no reorientation is observed. When both beams are e-polarized, for P NIR 1 mW (c) the green beam propagates with walk-off and is subject to linear diffraction; reorientational self-focusing for P NIR 6 mW (d) allows the formation of a selfconfined NIR beam that can guide the green light. (e) Green beam trajectory at P NIR 1 mW (blue line) and P NIR 6 mW (red line) for the extraordinary (solid) and ordinary (dashed) polarizations, and (f) input normalized width at P NIR 1 mW (blue line) and P NIR 6 mW (red line) for the e-polarization, respectively.
Fig. 3. Propagation of the visible beam, λ 532 nm, as (a), (b) ordinary and (c), (d) extraordinary eigenwave, for low (a), (c) and high (b), (d) powers. At low power, P vis 100 μW, nonlinear effects are not observable, whereas the thermal nonlinearity mediates self-confinement of the ordinary wave for P vis 6 mW (b). (e) Green beam trajectory and (f) input normalized width at low (blue lines) and high (red lines) power for ordinary (dashed) and extraordinary (solid) polarizations, respectively.
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smallness of thermal effects for powers below the Freedericksz threshold; the extraordinary beam traveled in yz along the walk-off direction δ45° ≈ 4.5° and, for P NIR > 1 mW, experienced the refractive index potential induced by the NIR beam through reorientation. For P NIR 4 mW, a NIR spatial soliton (a NIR nematicon waveguide) was formed leading to the confinement of the visible light. We then studied the thermal response associated with the presence of the dye-dopant guest by inputting only the green beam: opposite to the reorientational case discussed above, the heating through absorbed radiation produced negative and positive refractive index changes corresponding to extraordinary and ordinary polarizations, respectively. As illustrated in Fig. 3, the natural diffractive spreading of the extraordinary beam at P vis 100 μW was enhanced at powers above P vis 3 mW, as a result of selfdefocusing; the dye-mediated heating was accompanied by a lowered birefringence, in turn yielding a reduction of the walk-off angle localized close to the interface due to the presence of losses, as apparent in the beam trajectories plotted in Fig. 3(e). Conversely, when using the ordinary polarization, thermal self-focusing was observed up to P vis 6 mW [Fig. 3(b)]; the appearance of isotropic bubbles prevented further power increases. Having characterized the two different nonlinear responses individually, we studied their competition by co-launching an extraordinarily y-polarized beam at 1064 nm and a linearly polarized (ordinary or extraordinary) beam at 532 nm, varying the latter polarization as defined by the angle γ of the electric field with the axis x. We maximized the spatial overlap of the two beams for γ 90° (e-green) in the linear regime by introducing a slight wavevector tilt on the NIR component to compensate for their walk-off mismatch due to material dispersion. The effect of the NIR self-induced waveguide on the visible light was verified, at constant powers, by checking beam propagation versus angle γ, as shown in Fig. 4; when the two polarizations were orthogonal [γ 0, Fig. 4(a)], extraordinary for the NIR and ordinary for
Vol. 40, No. 22 / November 15 2015 / Optics Letters
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Fig. 4. Green beam at P vis 6 mW propagating for various input polarizations, in the presence of a collinear y-polarized NIR spatial soliton at P NIR 4 mW. Panels (a)–(e) correspond to γ 0° (purely ordinary), 30°, 45°, 75°, and 90° (purely extraordinary), respectively. The width of the ordinary component is not appreciably affected by the other beam, while the extraordinary component is guided by the NIR induced waveguide. Panel (f ) corresponds to e-green without the NIR waveguide. (g) Normalized intensity profiles (in z 350 μm) corresponding to the images in panels (a)–(e). Black to bright red lines refer to polarizations from ordinary (γ 0°) to extraordinary (γ 90°) as in (a)–(f ), respectively.
the green, the latter beam self-confined due to thermal effects and did not interact significantly with the NIR. By varying the polarization of the green beam, its extraordinary (growing with γ) component was gradually excited and perceived the NIRgenerated refractive index profile, eventually getting confined into it, i.e., guided by the soliton waveguide. Thus, the selffocusing molecular perturbation due to the NIR overcame the green-induced defocusing, i.e., the reorientational index well dominated over the thermal one. The results in Fig. 4 for an electric field polarized between the ordinary and extraordinary limits demonstrate how trapping occurs for the two green components; the ordinary wave confinement is a selfaction, whereas the extraordinary wave is nearly nondiffractive owing to the dominant NIR-induced reorientation over the green-defined thermo-optic index well. In addition, the two beams slightly interact via heating, with their extraordinary components getting closer to the ordinary as the temperature increases and the birefringence (walk-off ) reduces; the ordinary beam also moves toward the extraordinary one due to the thermal gradient associated with higher temperatures near the green e-beam. Next, we studied the NIR beam evolution for various green beam powers, for collinear excitation in the extraordinary polarization. In this case, the nonlinear responses at the two different wavelengths have opposite signs. We launched a NIR nematicon by setting the NIR power at P NIR 4 mW and varied the visible beam power in the interval P vis 0–6 mW. The role of the competing nonlinearities
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Letter in dye-doped nematic liquid crystals (NLCs). The propagation of either diffracting or self-confined beams is mutually affected by input polarization and power according to the nature of the response. In particular, the thermal response associated with the green beam can alter the propagation properties of the NIR beam, i.e., its trajectory and confinement. We believe that these findings, albeit preliminary and qualitative, open a variety of yet unexplored possibilities for light controlling light phenomena, particularly in the case of beam self-localization and soliton-based signal routing. Funding. Narodowe Centrum Nauki (NCN) (DEC-2012/ 06/M/ST2/00479); Suomen Akatemia (Academy of Finland) (282858). REFERENCES
Fig. 5. Propagation of collinear e-wave NIR and green beams in the presence of competing reorientational and thermal nonlinearities. (a) Acquired images of the NIR beam at P NIR 4 mW for various green beam powers, from 0 to 6 mW. (b) Trajectory of the NIR beam affected by the green beam: the thermal response lowers the material birefringence and reduces the NIR walk-off. (c) NIR beam width (normalized to the input value) versus visible beam power: the nonlinear competition increases the NIR breathing period. Black to bright red lines refer to increasing input powers as in (a).
was assessed by analyzing the NIR beam trajectory and width, as extracted from the acquired images. Figure 5 summarizes the main results. The green beam influenced the NIR beam trajectory through heating in two ways: first, a reduction of birefringence at higher temperatures and, consequently, a smaller walk-off; second, a different dependence of the walk-off at the two wavelengths on the local temperature, limiting the beam overlap and giving rise to an index gradient effectively pulling the NIR beam toward the z axis (wavevector direction). Accordingly, a gradual deflection of the NIR soliton could be observed from P vis 1 mW up to P vis 6.5 mW (above the latter value detrimental effects occurred close to the nematic-isotropic NLC transition). Furthermore, the NIR nematicon varied its width versus propagation due to green-induced defocusing: the latter reduced the available birefringence and, in turn, the magnitude of the reorientational response, thereby increasing the period of the excited NIR breather [28,29]. A quantitative evaluation of the competition between the two nonlinear responses can be carried out by considering their mutual effects on trajectory and breathing period, comparing experimental and theoretical results. Such a task requires extensive simulations based on the effective strength of both reorientational and thermal contributions in the presence of linear losses due to in-coupling and scattering at both wavelengths, as well as inevitable wavelength-dependent beam astigmatism, waist, walk-off, and diffraction. The corresponding model and numerical results will appear in a forthcoming publication. In conclusion, we investigated the interplay of nearinfrared and visible beams exciting competing nonlinearities
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Nonlinear competition in nematicon propagation URSZULA A. LAUDYN,1 MICHAŁ KWASNY,1 ARMANDO PICCARDI,2 MIROSŁAW A. KARPIERZ,1 ROMAN DABROWSKI,3 OLGA CHOJNOWSKA,3 ALESSANDRO ALBERUCCI,4 AND GAETANO ASSANTO2,4,* 1
Faculty of Physics, Warsaw University of Technology, PL-00662 Warsaw, Poland NooEL–Nonlinear Optics and OptoElectronics Laboratory, University Roma Tre, I-00146 Rome, Italy 3 Institute of Chemistry, Military University of Technology, PL-00908 Warsaw, Poland 4 Optics Laboratory, Tampere University of Technology, FI-33101 Tampere, Finland *Corresponding author: [email protected] 2
Received 29 September 2015; revised 15 October 2015; accepted 16 October 2015; posted 16 October 2015 (Doc. ID 251012); published 5 November 2015
We investigate the role of competing nonlinear responses in the formation and propagation of bright spatial solitons. We use nematic liquid crystals (NLCs) exhibiting both thermo-optic and reorientational nonlinearities with continuous-wave beams. In a suitably prepared dye-doped sample and dual beam collinear geometry, thermal heating in the visible affects reorientational self-focusing in the near infrared, altering light propagation and selftrapping. © 2015 Optical Society of America OCIS codes: (190.6135) Spatial solitons; (190.5940) Self-action effects; (190.4710) Optical nonlinearities in organic materials. http://dx.doi.org/10.1364/OL.40.005235
Nonlinear optical media are often described and employed on the basis of the dominant nonlinearity. In several instances, however, their behavior results from the superposition of different nonlinear processes [1]. Great attention has been devoted to the combination/competition of nonlinear mechanisms as it can potentially improve their tunability and control for optical signal manipulation. Among the early studies of such interplay was the case of an intensity-dependent refractive index in a Kerr-like material with higher than cubic order terms [2]. To date, various nonlinear mechanisms have been identified and investigated in conjunction with a focusing cubic response as a means to stabilize soliton propagation, including quintic terms in conservative [3,4] and dissipative [5,6] systems, as well as parametric processes [7–9]. The Kerr nonlinearity in quadratic media has been exploited for conversion enhancement in harmonic generation with self-confined waves [10,11]; light bullets have been predicted by the competition of nonlinearities acting on different temporal/spatial scales [8,12,13]. In recent years, liquid crystals have been employed as an ideal material platform for the investigation of solitons and their interactions, thanks to their optical properties and tunability by means of external stimuli and the coexistence/competition of different classes of nonlinearities [14] leading, for instance, to enhanced third-harmonic generation in nonlocal solitons [11], the possibility of generating light bullets based on the interplay 0146-9592/15/225235-04$15/0$15.00 © 2015 Optical Society of America
of electronic and molecular responses [13,15], and the formation of self-confined waves via both absorptive and reorientational responses [16]. In this Letter, with reference to the excitation and propagation of bright spatial solitons in dyedoped nematic liquid crystals (NLCs), termed nematicons, we address the competition and interplay of the two dominant nonlinearities when excited by continuous wave inputs, namely the thermo-optic and the reorientational responses. NLCs usually consist of elongated molecules with high anisotropy and large optical birefringence, with refractive indices n⊥ and n∥ for electric fields, respectively, perpendicular and parallel to the average orientation of the molecule main axis, the molecular director n. The low intermolecular forces associated with the liquid state prevents a crystalline structure; hence, no positional order on long ranges is present, albeit a finite degree of orientational order yields a positive uniaxial behavior (i.e., ϵa n2∥ − n2⊥ > 0) at optical frequencies, with the optic axis locally parallel to n [14]. Thus, for input wavevectors parallel to axis z and director lying in the plane yz at an angle θ with zˆ , the electric field of the extraordinary (e-) eigenwave oscillates in theqplane yz and perceives a θ-dependent refractive index ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ne
cos2 θ∕n2⊥ sin2 θ∕n2∥ ; moreover, an e-beam propa-
e gates with Poynting vector at an angle δ n1e ∂n ∂θ (walk-off ) with respect to zˆ [17]. In NLC, the nonlinear response can stem from either resonant or nonresonant interactions with electromagnetic fields, including electronic, electrostrictive, gain, molecular reorientation, and Janossy-enhanced reorientation, as well as photorefractive and ferroelectric effects in the presence of suitable dopants in the NLC mixture [14]. When optical radiation is absorbed near a resonance, thermal fluctuations cause a temperature dependent reduction in the orientational order, resulting in a net decrease (increase) of n∥ (n⊥ ) and thermo-optic light self-action [18–20]; in the nonresonant limit, the reorientational response stems from angular realignment of the director due to Coulomb forces (reorientational torque) on light-induced dipoles [14]. Hereby, we limit our analysis to noninstantaneous and spatially nonlocal nonlinearities obtainable with (low power) continuous wave beams. In this limit, the high coupling between light and molecular
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dipoles enables reorientational and thermal nonlinearities, even at mW powers, where other effects are comparatively negligible. A bell-shaped light beam can then be launched in an NLC volume and propagate in the nonuniform refractive index profile resulting from self-focusing/defocusing depending on polarization and wavelength of the excitation, as well as orientation and spectral resonance(s) of the material. Such self-action effects are at the basis of beam self-confinement, and the formation of bright/dark, as well as discrete solitons (nematicons) [21–23], their control, and the implementation of all-optical devices for signal routing and processing [24]. Both thermo-optic and reorientational nonlinearities can be modeled by Poisson-like equations [25]; therefore, the corresponding nonlinear index wells share the same spatial configuration. When they act simultaneously in response to a given beam profile, their contributions maintain their relative weights in transverse space. To investigate the interplay and competition of these two nonlinearities, we employed a host nematic liquid crystal doped with a specific guest dye, the latter resonant in the visible range. The sample, sketched in Fig. 1, was a planar cell composed of two parallel glass slides at a separation of h ≈ 50 μm, with inner surfaces rubbed to obtain planar anchoring of the director at 45° with respect to z. The cell was filled with the nematic liquid crystal 6CHBT [-4-(trans-4’-hexylcyclohexyl)isothiocyanatobenzene] (n⊥ 1.5021, n∥ 1.6314 at λ 1064 nm and n⊥ 1.52, n∥ 1.6746 at λ 532 nm, at room temperature [26]) doped with 0.1% of Sudan blue II dye. The latter exhibits an absorption peak at λ ≈ 604 nm [27]. We used two Gaussian beams of different wavelengths to decouple and investigate the contribution of distinct nonlinear responses. The input beams were coupled in the NLC cell after focusing with a microscope objective to a waist of w0 ≈ 3 μm; their evolution in the propagation plane yz was observed by acquiring the outof-plane scattered light with a CCD camera and a microscope. The beam at λ 1064 nm, far from the dye resonance, could preferably excite the reorientational response; the beam at λ 532 nm, well inside the absorption band of Sudan blue, but sufficiently removed from the resonance peak, enabled
Fig. 1. (a) Sketch of the NLC sample and (b) nonlinear competition in the guest–host material. (a) Two extraordinarily polarized beams are launched collinearly with a wavevector at θ 45° with the optic axis and exhibit different walk-off angles δ at their respective wavelengths. The blue ellipses represent the elongated molecules; the black arrow represents the director. (b) Calculated nonlocal Kerr-like coefficient n2 verus temperature in a planar 6CHBT sample with θ 45° and (black dashed line) e-wave light at λ 532 nm: reorientational nreor 2 j (solid lines) for Sudan blue concentrations yieldand thermal jnthermal 2 ing optical absorption of 1 × 102 (blue line), 3 × 103 (green line), and 6 × 103 m−1 (red line), respectively. The overall nonlinear response is defocusing when a solid line is above the dashed line.
Letter the dye-mediated thermal response while avoiding the detrimental effects of large attenuation. The relative role of the two nonlinearities can be addressed by using a single scalar figure, namely a nonlocal equivalent of the Kerr coefficient n2 [17]. For the reorientational nonlinearity, it is d ne ϵ0 ϵa Z 0 sin2θ−δ , with Z 0 being the vacuum nreor 2 θ; T d θ 2ne K cos2 δ impedance and K the effective elastic constant [14]. For the θ; T ddTn ακ (with n being thermal nonlinearity, it is nthermal 2 either n⊥ or ne for ordinary and extraordinary eigenwaves, respectively), with α and κ being the absorption coefficient and the thermal conductivity, respectively. For ordinary waves, nthermal is usually positive. For e-waves, the sign of nthermal 2 2 depends on θ: in 6CHBT for θ 45° the thermal nonlinear response is defocusing (nthermal < 0). As anticipated, nthermal is 2 2 proportional to optical absorption and can be enhanced by the use of suitable dopants in the right concentration [see Fig. 1(b)]. In our sample, to obtain an overall defocusing response to green e-beams, the dye concentration was chosen such that, at θ 45°, it was jnthermal j > nreor at room temper2 2 ature. We first studied the two nonlinear responses individually, waiting for the system to reach steady state before acquiring the data. Figure 2 shows the propagation of a low-power visible probe beam with P vis 100 μW, unable to excite any appreciable nonlinear effects, co-polarized and collinear (same Poynting vector directions) with a near-infrared (NIR) beam. To assess the nonlinear properties of the NIR beam through the self-induced refractive index potential, we monitored the visible light evolution when varying the NIR beam power (a pump–probe arrangement): the ordinary wave was not affected by changes in the NIR power (up to P NIR 50 mW) due to
Fig. 2. Propagation of a low power (≈100 μW) visible beam, λ 532 nm, collinear and co-polarized with the NIR beam, λ 1064 nm, as (a), (b) ordinary and (c), (d) extraordinary eigenwaves, at low (a), (c) and high (b), (d) NIR powers, respectively. In the ordinary polarization, no reorientation is observed. When both beams are e-polarized, for P NIR 1 mW (c) the green beam propagates with walk-off and is subject to linear diffraction; reorientational self-focusing for P NIR 6 mW (d) allows the formation of a selfconfined NIR beam that can guide the green light. (e) Green beam trajectory at P NIR 1 mW (blue line) and P NIR 6 mW (red line) for the extraordinary (solid) and ordinary (dashed) polarizations, and (f) input normalized width at P NIR 1 mW (blue line) and P NIR 6 mW (red line) for the e-polarization, respectively.
Fig. 3. Propagation of the visible beam, λ 532 nm, as (a), (b) ordinary and (c), (d) extraordinary eigenwave, for low (a), (c) and high (b), (d) powers. At low power, P vis 100 μW, nonlinear effects are not observable, whereas the thermal nonlinearity mediates self-confinement of the ordinary wave for P vis 6 mW (b). (e) Green beam trajectory and (f) input normalized width at low (blue lines) and high (red lines) power for ordinary (dashed) and extraordinary (solid) polarizations, respectively.
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smallness of thermal effects for powers below the Freedericksz threshold; the extraordinary beam traveled in yz along the walk-off direction δ45° ≈ 4.5° and, for P NIR > 1 mW, experienced the refractive index potential induced by the NIR beam through reorientation. For P NIR 4 mW, a NIR spatial soliton (a NIR nematicon waveguide) was formed leading to the confinement of the visible light. We then studied the thermal response associated with the presence of the dye-dopant guest by inputting only the green beam: opposite to the reorientational case discussed above, the heating through absorbed radiation produced negative and positive refractive index changes corresponding to extraordinary and ordinary polarizations, respectively. As illustrated in Fig. 3, the natural diffractive spreading of the extraordinary beam at P vis 100 μW was enhanced at powers above P vis 3 mW, as a result of selfdefocusing; the dye-mediated heating was accompanied by a lowered birefringence, in turn yielding a reduction of the walk-off angle localized close to the interface due to the presence of losses, as apparent in the beam trajectories plotted in Fig. 3(e). Conversely, when using the ordinary polarization, thermal self-focusing was observed up to P vis 6 mW [Fig. 3(b)]; the appearance of isotropic bubbles prevented further power increases. Having characterized the two different nonlinear responses individually, we studied their competition by co-launching an extraordinarily y-polarized beam at 1064 nm and a linearly polarized (ordinary or extraordinary) beam at 532 nm, varying the latter polarization as defined by the angle γ of the electric field with the axis x. We maximized the spatial overlap of the two beams for γ 90° (e-green) in the linear regime by introducing a slight wavevector tilt on the NIR component to compensate for their walk-off mismatch due to material dispersion. The effect of the NIR self-induced waveguide on the visible light was verified, at constant powers, by checking beam propagation versus angle γ, as shown in Fig. 4; when the two polarizations were orthogonal [γ 0, Fig. 4(a)], extraordinary for the NIR and ordinary for
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Fig. 4. Green beam at P vis 6 mW propagating for various input polarizations, in the presence of a collinear y-polarized NIR spatial soliton at P NIR 4 mW. Panels (a)–(e) correspond to γ 0° (purely ordinary), 30°, 45°, 75°, and 90° (purely extraordinary), respectively. The width of the ordinary component is not appreciably affected by the other beam, while the extraordinary component is guided by the NIR induced waveguide. Panel (f ) corresponds to e-green without the NIR waveguide. (g) Normalized intensity profiles (in z 350 μm) corresponding to the images in panels (a)–(e). Black to bright red lines refer to polarizations from ordinary (γ 0°) to extraordinary (γ 90°) as in (a)–(f ), respectively.
the green, the latter beam self-confined due to thermal effects and did not interact significantly with the NIR. By varying the polarization of the green beam, its extraordinary (growing with γ) component was gradually excited and perceived the NIRgenerated refractive index profile, eventually getting confined into it, i.e., guided by the soliton waveguide. Thus, the selffocusing molecular perturbation due to the NIR overcame the green-induced defocusing, i.e., the reorientational index well dominated over the thermal one. The results in Fig. 4 for an electric field polarized between the ordinary and extraordinary limits demonstrate how trapping occurs for the two green components; the ordinary wave confinement is a selfaction, whereas the extraordinary wave is nearly nondiffractive owing to the dominant NIR-induced reorientation over the green-defined thermo-optic index well. In addition, the two beams slightly interact via heating, with their extraordinary components getting closer to the ordinary as the temperature increases and the birefringence (walk-off ) reduces; the ordinary beam also moves toward the extraordinary one due to the thermal gradient associated with higher temperatures near the green e-beam. Next, we studied the NIR beam evolution for various green beam powers, for collinear excitation in the extraordinary polarization. In this case, the nonlinear responses at the two different wavelengths have opposite signs. We launched a NIR nematicon by setting the NIR power at P NIR 4 mW and varied the visible beam power in the interval P vis 0–6 mW. The role of the competing nonlinearities
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Letter in dye-doped nematic liquid crystals (NLCs). The propagation of either diffracting or self-confined beams is mutually affected by input polarization and power according to the nature of the response. In particular, the thermal response associated with the green beam can alter the propagation properties of the NIR beam, i.e., its trajectory and confinement. We believe that these findings, albeit preliminary and qualitative, open a variety of yet unexplored possibilities for light controlling light phenomena, particularly in the case of beam self-localization and soliton-based signal routing. Funding. Narodowe Centrum Nauki (NCN) (DEC-2012/ 06/M/ST2/00479); Suomen Akatemia (Academy of Finland) (282858). REFERENCES
Fig. 5. Propagation of collinear e-wave NIR and green beams in the presence of competing reorientational and thermal nonlinearities. (a) Acquired images of the NIR beam at P NIR 4 mW for various green beam powers, from 0 to 6 mW. (b) Trajectory of the NIR beam affected by the green beam: the thermal response lowers the material birefringence and reduces the NIR walk-off. (c) NIR beam width (normalized to the input value) versus visible beam power: the nonlinear competition increases the NIR breathing period. Black to bright red lines refer to increasing input powers as in (a).
was assessed by analyzing the NIR beam trajectory and width, as extracted from the acquired images. Figure 5 summarizes the main results. The green beam influenced the NIR beam trajectory through heating in two ways: first, a reduction of birefringence at higher temperatures and, consequently, a smaller walk-off; second, a different dependence of the walk-off at the two wavelengths on the local temperature, limiting the beam overlap and giving rise to an index gradient effectively pulling the NIR beam toward the z axis (wavevector direction). Accordingly, a gradual deflection of the NIR soliton could be observed from P vis 1 mW up to P vis 6.5 mW (above the latter value detrimental effects occurred close to the nematic-isotropic NLC transition). Furthermore, the NIR nematicon varied its width versus propagation due to green-induced defocusing: the latter reduced the available birefringence and, in turn, the magnitude of the reorientational response, thereby increasing the period of the excited NIR breather [28,29]. A quantitative evaluation of the competition between the two nonlinear responses can be carried out by considering their mutual effects on trajectory and breathing period, comparing experimental and theoretical results. Such a task requires extensive simulations based on the effective strength of both reorientational and thermal contributions in the presence of linear losses due to in-coupling and scattering at both wavelengths, as well as inevitable wavelength-dependent beam astigmatism, waist, walk-off, and diffraction. The corresponding model and numerical results will appear in a forthcoming publication. In conclusion, we investigated the interplay of nearinfrared and visible beams exciting competing nonlinearities
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