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We report a new type of thermal nonlinear media (m-cresol/nylon solutions) that exhibits a giant tunable self-defocusing nonlinearity. The measured Kerr coefficient in such thermal nonlinear solutions is orders of magnitude higher than that of most previously known thermal materials. In addition, we demonstrate the generation of dark spatial solitons in these isotropic nonlocal nonlinear media, and observe to our knowledge the strongest effect of dark-soliton attraction ever reported in thermal defocusing media.
© 2014 Optical Society of America
1. Introduction
Nonlinear media with self-focusing or defocusing properties have been the subject of intensive studies for decades due to both fundamental scientific interest and their potential applications. For instance, self-defocusing media have served as platforms for many experimental demonstrations of interesting nonlinear optical phenomena such as dark spatial solitons [1–9] and dispersive shock waves [10–14]. An optical soliton is a self-trapped beam that does not change its profile during propagation [1]. Contrary to a bright soliton exhibiting an intensity peak, a dark soliton corresponds to an intensity trough in an otherwise uniform bright background that maintains its shape throughout a nonlinear defocusing medium [2,3]. As light defocuses due to the action of nonlinearity, the diffraction of a dark “notch” (in the one-dimensional case) [2,4–6] or a dark “hole” (in the two-dimensional case) [7–9] can be balanced by the self-defocusing nonlinearity, leading to the formation of optical spatial dark solitons. Similar to bright solitons which can behave as self-induced waveguides, dark solitons can also act as straight, Y-splitting, or circular waveguides [4,6,9], which have applications in all-optical switches and couplers.
Solitons also exhibit particle-like behavior, exerting attractive or repelling forces when colliding or propagating at close vicinity [1,15,16]. Dark solitons were traditionally known to only repel from each other [3,17], while bright solitons can attract or repel depending on their relative phases [15,18]. However, over the past several years, studies have shown that attraction can also occur between dark solitons in nonlocal defocusing materials due to their unique properties [19–21]. The nonlocal effect saturates the change in refractive index due to formation of dark solitons in close proximity, merging the index-change in the central overlapping area to create a broad waveguide that squeezes the dark solitons together [21].
When light propagates through a thermal nonlinear material, absorption occurs and the material gains heat. The increase in the temperature causes thermal expansion, which, for most materials, reduces the index of refraction [22–24]. This phenomenon is known as the thermo-optic effect. If the beam is Gaussian, it will create an index gradient opposite to the temperature gradient, which behaves as a concave lens, thus leading to self-defocusing of an optical beam. Since heat is the main source of the nonlinearity, diffusion causes a nonlocal response of the material to the beam. It has been shown that non-locality can prevent a beam from collapsing, thus supporting stable self-trapped solitons [25].
Since the temperature change (hence the amount of refractive index change) increases with the light intensity for most of the thermal materials, it is a common practice to quantify the strength of this nonlinearity using a Kerr model [26]. This model assumes a Kerr-type intensity-dependent change in the refractive index, Δn = n2I, along the directions transverse to beam propagation, where n2 is known as the Kerr coefficient [26], and is negative for self-defocusing-type nonlinear media. Please note that the non-local thermal effect studied in this paper is different from the optical Kerr effect, in which the nonlinear index change is due to the instantaneous third order nonlinear polarization, χ(3). In general, an optical Kerr effect has a much shorter response time than thermal nonlinear responses, and its Kerr coefficient magnitude is less than that of common thermal nonlinear media.
In this paper, we report a new type of thermal nonlinear media: dilute solutions consisting of nylon dissolved in m-cresol. m-Cresol is an organic liquid, which has been previously used as a solvent for nonlinear optical studies [27]. The m-cresol/nylon solutions exhibit a giant self-defocusing nonlinearity, which can be tuned easily by varying the concentration of nylon. Furthermore, we demonstrate the generation of spatial dark solitons in such media, and observe, to our best knowledge, the strongest attraction between two interacting dark solitons ever reported in thermal defocusing media.
2. Characterization of the self-defocusing nonlinearity in m-cresol/nylon solutions
Our experimental setup is similar to what has been described in [28]. A collimated CW laser beam (λ = 532 nm, Coherent Verdi) with linear polarization in the vertical direction is launched into a glass cuvette containing the m-cresol/nylon solutions. Both the linear and nonlinear propagation dynamics are observed by monitoring the input and output transverse intensity patterns using an imaging lens and a CCD camera. Input and output beam profiles are taken at the inner front and back surfaces of the glass cuvette, respectively.
First, we demonstrate the giant defocusing effect displayed by our nonlinear solutions in Fig. 1(a)-1(c). The Gaussian beam is focused to 12μm (FWHM) at the front surface of a 2mm-long cuvette [Fig. 1(a)], containing an m-cresol/nylon solution with 0.58% mass concentration. When the power is less than 1mW, the beam experiences linear diffraction and its width increases to 33μm (FWHM) after propagating through the 2-mm-long sample [Fig. 1(b)]. As the power is increased to 30mW, the beam experiences remarkable self-defocusing, expanding to 219μm after only 2-mm of propagation, as shown in Fig. 1(c).
Fig. 1 Transverse intensity patterns imaged at the input or output surfaces, demonstrating the giant and isotropic self-defocusing effect in the m-cresol/nylon solution. (a) The focused Gaussian beam (12μm FWHM) at the input of the sample. (b) The output (33μm FWHM) due to normal diffraction after propagating 2mm in our sample at a power less than 1mW. (c) The output (219μm FWHM) due to strong defocusing nonlinearity at a power of 30mW. (d) The input dark cross (38 μm stripe width) created by using an amplitude mask of two thin wires. (e) Linear diffraction output after 5mm propagation in the sample at a power less than 1mW. (f) At a power of 100mW, the output profile shows that Y-junctions of diverging solitons are created with little variation between the horizontal and vertical directions.
Next, a cross of thin wires is used as an amplitude mask to create dark stripes, which is then imaged on the front input surface of a 5mm long cuvette containing the same solution mentioned above [Fig. 1(d)]. At a low power (~1mW), the dark cross linearly diffracts as shown in Fig. 1(e). When the power is increased to 100mW, each dark stripe splits into two distinct stripes, forming Y-junctions of diverging solitons [Fig. 1(f)], as expected for nonlinear defocusing media [2,4,6]. We also found that there is less than 5% difference (less than the experimental uncertainties) between the vertical and horizontal stripe widths as well as the stripe separation, indicating the isotropic feature of the nonlinear responses in these thermal nonlinear solutions, which may be different in other liquid crystal [29] or photorefractive polymer [30] based samples.
More importantly, we found that the strength of the nonlinear response of the m-cresol/nylon solutions can be easily controlled by varying the concentration of nylon. We focus the same Gaussian beam with 12μm FWHM into the m-cresol/nylon solutions with different mass concentrations and measure the output profiles at 2mm propagation distance. Figure 2(a) shows the output beam diameter (measured at 86.5% of maximum power) as a function of input power for samples of different concentrations. As the nylon concentration is increased, the beam experiences a stronger self-defocusing effect. Using the z-scan method [26], the Kerr coefficient is determined for these samples with different concentrations of nylon. The absolute value of the measured Kerr coefficient, |n2|, as a function of the nylon concentrations is plotted in Fig. 2(b). By varying the nylon mass concentration from 0 to 3.5%, the measured |n2| increases over two orders of magnitude from 9 × 10−8 cm2/W (pure m-cresol) to 1.6 × 10−5 cm2/W (3.5% nylon). Table 1 presents a comparison of the n2 values between our solutions and those of other strongly-defocusing nonlinear media reported previously. More data of previously reported nonlinearity of typical defocusing media can be found in [26]. The Kerr coefficient of our m-cresol/nylon solution is orders of magnitude higher than that of most previously known thermal nonlinear materials [26].
Fig. 2 Quantifying the defocusing nonlinearity of the m-cresol/nylon solutions with different concentrations. (a) The output beam diameter (after 2mm propagation) as a function of input power for different concentrations of nylon. (b) The measured |n2|, absolute values of the Kerr coefficient, as a function of nylon concentrations using the z-scan method. As seen from these figures, the strength of the nonlinearity in our thermal media can be easily turned across a large range, simply by varying the nylon concentration.
Table 1. Comparison of measured n2 values for various known thermal defocusing media.
3. Generation of 1D dark solitons in m-cresol/nylon solutions
Next we experimentally demonstrate the formation of spatial dark solitons in the above synthetic defocusing media. The 532nm laser beam is collimated to provide bright background illumination. To generate a dark stripe, half of the beam is passed through a thin glass plate, whose orientation is adjusted to provide a π-phase jump at its edge [2]. This π-phase jump creates a dark stripe, due to destructive interference. The dark stripe is imaged onto the inner front surface of a 10 mm long cuvette as the input for our samples, and the output beam profile is recorded when exiting the cuvette.
Figure 3 shows the experimental results obtained with the 10 mm long sample of 0.58% m-cresol/nylon solution. For this concentration, the Kerr coefficient is measured to be n2 = −6 × 10−6 cm2/W, using the z-scan method. The width of the input stripe [Fig. 3(a)] is 39 ± 2μm. As the beam propagates through the sample at low power (<1mW), the beam experiences linear diffraction and the dark stripe widens to 100 ± 3μm at the output, as shown in Fig. 3(b). As the power is increased, the dark stripe experiences less diffraction. At a power of 50mW, the output dark stripe has approximately reached its initial input size within the experimental uncertainty range, forming a dark soliton [Fig. 3(c)]. The decrease in output stripe width saturates as input power is further increased.
Fig. 3 Transverse intensity patterns imaged at the input or output surfaces demonstrating formation of a dark soliton observed in m-cresol/nylon solution. (a) Input: a dark stripe with a width of 39 ± 2 μm created using a π-phase mask. (b) Linear output beam profile after 10 mm of propagation at power less than 1 mW. The dark stripe diffracts to a width of 100 ± 3 μm. (c) Nonlinear output beam profile after 10 mm of propagation at 50mW power. The stripe width decreases to 41 ± 2 μm.
4. Dark soliton attraction in m-cresol/nylon solutions
Finally, we demonstrate the interaction between dark solitons employing the giant self-defocusing nonlinearity shown in the above media. Two closely overlapping glass plates were used to create two dark stripes due to two separate π-phase-shifts across the otherwise uniform beam [Fig. 4(a)].Figure 4(b) shows the input stripes (41 ± 2 μm each) separated by a distance of 100 ± 3μm (trough-to-trough). At powers less than 1mW, linear diffraction causes the widths of the stripes to widen after propagating 5mm through 0.58% m-cresol/nylon solution, while the separation remains at 100 ± 3μm [Fig. 4(c)]. Figures 4(d)-4(f) show the output intensity profile at a power of 1.0W, 2.0W, and 3.0W, respectively. As the input power is increased gradually, not only do the widths of the dark stripes decrease, but their separation also decreases as a direct consequence of the nonlocal nonlinear response of the solution. At 3.0W, the widths of the stripes decreases to about 25μm, and the separation decreases to 64 ± 3μm. Figure 4(g) shows the output separation as a function of input power. As the power is increased, the strength of attraction caused by dark soliton interaction in this nonlocal nonlinear medium also increases [21], pulling the dark solitons closer than the input separation. The 36% separation distance reduction at 3.0 W demonstrates a dark-soliton attraction much stronger than what has been reported previously [19].
Fig. 4 Demonstration of dark soliton attraction. Transverse intensity patterns imaged at the input or output surfaces showing: (a) Interference indicating the two separate π-phase-jumps; (b) The two dark stripes at the input, separated by a distance of 100μm; (c) Linear output at power less than 1mW after 5mm of propagation. (d)-(f) Output patterns taken at different powers of 1.0, 2.0 and 3.0 W respectively. The stripe separation decreases to 64 μm in (f). (g) Plot of the stripe separation as a function of input power.
5. Summary
In summary, we have demonstrated that m-cresol/nylon solutions exhibit a strong tunable nonlocal self-defocusing nonlinearity that can be used to study dark soliton interactions. The nonlinear Kerr coefficient in such isotropic thermal solutions can be easily tuned across a large range, from n2 = −9 × 10−8 cm2/W to n2 = −1.6 × 10−5 cm2/W, which is orders of magnitude larger than that of most previously reported for other thermal nonlinear media [26]. The strength of the nonlinearity can be easily controlled by varying the nylon concentration in the solutions. In these thermal defocusing media, we generated stable spatial dark solitons with milli-Watt power levels. In addition, we demonstrated strong dark soliton attraction in these nonlocal defocusing media: At 3.0W of laser power, the soliton separation reduced from 100μm to 64μm after 5mm of propagation, showing the strongest dark soliton attractions ever reported. Our results bring about many possibilities of using these thermal solutions as extraordinary nonlinear optical materials for studying nonlinear wave dynamics, including vortex dynamics, modulation instability, and spatial shock waves. Results on shock wave generation in these media will be reported elsewhere [33].
Acknowledgment
This work was supported by the ACS (grant PRF# 52644-UR6), the AFOSR (FA9550-12-1-0111), and by NSF (grants ECCS-1128520, PHY-1100842). The authors would like to thank Jai Prakash and Shima Fardad for assistance and discussion.
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