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Surface dark solitons in photovoltaic nonlinear media are reported. Taking advantage of diffusion and photovoltaic nonlinearities we demonstrated the surface dark solitons and their behaviors near surface theoretically and experimentally in LiNbO3 crystal. It is very interesting that surface dark soliton is just half of dark soliton in bulk. Another interesting thing is that transverse modulation instability can be perfectly suppressed by surface dark soliton in virtue of surface. In addition, surface waveguides were written successfully utilizing surface dark soliton.
©2013 Optical Society of America
1. Introduction
Self-guided surface waves along the surface of a non-linear medium are among the most intriguing phenomena in nonlinear optics and may result in very strong enhancement of nonlinear surface optical phenomena (such as surface adsorbed molecular luminescence, Raman scattering, and surface second-harmonic generation) [1, 2]. From one aspect, these can be attributed to the natural line path supported by surface, which provides the fine situation for the phase-matching condition [2]. Nonlinear surface waves have been demonstrated to solve the phase-mismatch problem in bulk due to beam self-bending and to support giant enhancement of second-harmonic generation [2].
Since the prediction [3] and experimental observation [4, 5] of photorefractive solitons (PR solitons), their existence at low power and in both transverse dimensions has attracted much fundamental interest, driven also by potential applications such as optical-switching, optical interconnects, enhancement of second harmonic generation (SHG), and nonlinear optical devices [6–8]. Surface provides a natural line path, where the so called surface solitons can propagate along surface straightly. Similar to nonlinear surface waves the phase-matching condition may be satisfied taking advantage of surface solitons and the enhancement of various nonlinear effects may be expected. The earlier surface solitons were achieved in nonlinear optical lattice and were named surface lattice solitons [9–12], which utilized periodic structures and near surface defects induced by nonlinearity to confine light beam propagating along surface. Surface solitons in uniform nonlinear medium were advanced by Barak Alfassi et al. in 2007 utilizing nonlocal self-focusing type thermal nonlinearity, where prefabrication of periodical structures was not needed [13]. In 2009, we demonstrated surface solitons in virtue of the cooperation of nonlocal diffusion and local drift PR nonlinearities in uniform photorefractive (PR) medium [14]. In 2010 Jassem Safioui et al. demonstrated surface-wave pyroelectric photorefractive solitons in LiNbO3 [15].
In contrast to bright spatial soliton in self-focusing medium, the so called dark spatial soliton can be formed in self-defocusing media [16]. Recently the surface dark screening soliton with self-defocusing nonlinearity is put forward [17]. It is very interesting that a surface dark soliton is just like half of a dark soliton in bulk, but not a whole dark soliton propagating along surface. The typical photovoltaic nonlinear photorefractive crystal, e.g. LiNbO3, is also a typical self-defocusing medium and is easily to be operated, where the additional external applied electric field is not needed. Moreover plenty of researches reveal that transverse modulation instability (MI) may occur in dark soliton strip and external repulsive potential barrier may suppress MI, and even render the dark-soliton stripe completely stable [18]. Can a surface play a similar role to suppress MI?
In this paper, we report the observation of surface dark solitons and demonstrate their behaviors experimentally for the first time. It is very interesting that surface dark soliton is just like half of dark soliton in bulk, but not a whole dark soliton propagating along surface. The suppression of transverse modulation instability in virtue of surface was also observed for the first time to our knowledge. This will further arouse the attentions on the significance of surface in nonlinear optics domain.
2. The model
Considering an e-polarized slit light beam propagating along the boundary of a photorefractive crystal (PRC), taking into account the diffusion and photovoltaic components of PR nonlinearity simultaneously, the complex amplitude E(x,y,z) of the light field satisfies the scalar wave equation:
where k = k0[n + ∆n], k0 = 2π/λ0, λ0 is the wavelength in vacuum, n is the refractive index of e-polarized beam in the PRC, ∆n is the nonlinear refractive index change, [n + ∆n]2 = n2−n4reffEsc, reff is the effective electro-optical coefficient, Esc is the space-charge field, which can be written as:where kB is Boltzman constant, T is the temperature, q is the charge of carriers, (negative for the electrons and positive for the holes); I is the light intensity of the PR surface soliton, Ib is the intensity of background illumination and Id is the equivalent dark irradiance; βeff is the effective photovoltaic coefficient, γR is the recombination coefficient, s is the photoionization cross section, μ is the mobility of charge carriers; ND and ND+ are the total donor density and ionized donor density, respectively. The first and the second terms in the right side of Eq. (2) describe the effects of diffusion and the photovoltaic components of PR nonlinearity, respectively.For convenience we look for the (1 + 1)-dimensional [(1 + 1)D] stationary PR surface dark soliton solution as E(x,z) = A(x)exp(ißz), where β is the propagation constant and A(x) = [I(x)/(Ib + Id)]1/2 is the normalized amplitude. Equation (1) can be rewritten as:
here, Ep = βeffγRND+/qμs(ND− ND+).3. Numerical results
Taking LiNbO3 as a sample, based on Eq. (3), the surface dark solitons can be numerically solved. In the calculation the parameters are as follows: λ0 = 532nm, n = 2.2, reff = 30.8 × 10−12m/V γR = 1.0 × 10−15m−3/s, s = 6.2 × 10−5m2/J, μ = 1.0 × 10−5m2/Vs, βeff = 5.2 × 10−8A/W, ND+ = 3.3 × 1024m−3, ND = 6.6 × 1024m−3, T = 300K, q = −1.6 × 10−19C. Figures 1(a) –1(c) show the stationary surface photovoltaic soliton solutions with different β. Based on Eq. (3), n4reffEp + k02n2 < β2 < k02n2 should be satisfied. The local photovoltaic nonlinearity ensures the beam maintaining a dark soliton shape just like photovoltaic dark solitons in bulk. The effect of nonlocal diffusion nonlinearity is asymmetrical along c axis, and confines surface dark soliton near –c surface. It is very interesting that surface dark soliton is just a half of dark soliton in bulk. That is very different to the case of surface bright solitons, which is a whole soliton just like that in bulk. To satisfy the continuous condition at interface surface dark solitons will present an asymmetrical “half a dark soliton” profile. As a result, the size of surface soliton and the consequently induced optical waveguide can be reduced by half. The Full Width at Half Maximum (FWHM) of surface dark soliton depends on Imax/(Id + Ib), their relationship is sketched in Fig. 1(d). A required size of surface dark soliton can be obtained by adjusting the ratio of the soliton intensity to the background illumination.
Fig. 1 The (1 + 1)D stationary surface photovoltaic dark soliton solutions, stabilities and their intensity FWHM dependence on Imax/(Id + Ib). (a), (b) and (c) the profiles of surface photovoltaic dark solitons with β = 2.5982634 × 107m−1, 2.5976282 × 107m−1 and 2.5974742 × 107m−1, respectively; (d) the intensity FWHM of stationary surface photovoltaic dark solitons vs. Imax/(Id + Ib); (e) simulated propagation of stationary surface photovoltaic dark soliton corresponding to (b) with 10% noise; (f) the input (in red) and output (in black) light fields corresponding to e).
To investigate the stability of surface dark photovoltaic solitons, we used the beam propagation method (BPM) to simulate the evolution of the stationary surface dark photovoltaic soliton solution with a random noise (10%). As shown in Figs. 1(e) and 1(f), the perturbed surface dark soliton evolves into stable surface dark soliton immediately within a very short propagation length (~3.5mm). This indicates that the surface dark soliton is very stable and propagates smoothly without distortion or deviations in its trajectory.
Based on Eqs. (1) and (2), the excitation and evolution of surface dark photovoltaic soliton can be simulated using the beam propagation method (BPM). For convenience, the (1 + 1)D circumstance is considered and a Gaussian notched light beams with π phase jump is launched into the PRC. Figure 2 shows the excitation and evolution of surface dark photovoltaic solitons in LiNbO3 crystal. When the Gaussian notched light beam is launched into the bulk of crystal, the incident dark notch transforms into dark soliton immediately and is deflected towards - c direction due to diffusion nonlinearity as shown in Fig. 2(a). The self-bending distance is about 11 μm for 10mm propagation length.
Fig. 2 The excitation of dark solitons and their propagation. (a) the self-bending of dark soliton in bulk; (b) the reflection of dark soliton at the boundary of PRC; (c) the excitation and straightly propagation of surface dark soliton along PRC boundary; (d) and (e) the profiles of light field at input face (in solid line) and at output face (in dashed line) corresponding to (b) and (c), respectively. The crystal surface is located at x = 0μm.
When the launched point is at x = 23μm near the boundary, the self-bended dark soliton experiences a total reflection at the boundary and is deflected again towards −c direction as shown in Fig. 2(b). When the launched point is at x = 0μm, i.e. the center of incident dark notch is exactly at the boundary, a surface dark soliton is excited, which can straightly propagate along surface, as shown in Fig. 2(c). Meanwhile the deflection from diffusion nonlinearity and the repulsion from the boundary are exactly balanced by each other.
4. Experiments and results
The experimental setup is sketched in Fig. 3 . We performed the behavior of surface dark solitons in Fe:LiNbO3 (9 × 5 × 10 mm3) doped with 0.02% of iron. The c axis is oriented along the 5mm transverse dimension. An e-polarized laser beam (λ = 532nm) with 5.6 mw output power and 1mm diameter passes through a coverslip to form a dark notch with necessary π phase jump in its center. Then the beam is focused (f = 8.5 cm) into Fe:LiNbO3 crystal. The pattern on the input or output face of the crystal is imaged by a CCD.
Firstly, a dark notch was launched into the crystal far away from the boundary. At the beginning the width of narrow dark notch expended due to diffraction, as shown in Figs. 4 (a1) and 4(b1). Then the diffracted beam immediately scatted due to Light-induced scattering effect (fanning effect) within 1 minute as shown in Fig. 4(c1). 1 minute later it began to shrink as shown in Fig. 4(d1). After 90 minutes, the beam reached its steady state and a dark soliton formed as shown in Fig. 4(e1). At the same time, the dark notch deflected towards −c surface about 10μm. The green dashed line shows the center of dark notch at the beginning. To verify the formation of dark soliton, the experiment of guiding waves was carried out. Shut off the input beam and moved the mask away, then turned on the laser again with lower intensity as probe light beam. At the location of dark soliton, there was a bright stripe whose shape is just like that of dark soliton, as shown in Fig. 5(a) . That is to say the dark soliton wrote a waveguide and the bright stripe was just the guide wave.
Fig. 4 The formation of dark solitons and their guided waves in bulk and near surface. (a) The incident dark notch lase beam at input face; (b)–(e) the evolvement of the pattern at the output face along with time; (a1)–(e1) in bulk; (a2)–(e2) d = 6μm; (a3)–(e3) d = 0μm.
Fig. 5 The guided wave of the waveguide written by photovoltaic dark solitons. (a), (b) and (c) are the guided waves of the waveguides written by photovoltaic dark solitons corresponding to Fig. 4(e1), 5(e2) and 5e(3), respectively.
Secondly, we launched the dark notch near the boundary at d = 6μm and 0μm, respectively. Here, d is the distance from the center of dark notch to the boundary of the crystal. Figures 4(b2)–4(e2) and 4(b3)–4(e3) describe the entire evolvement processes of the images at the output face for 0, 1, 5 and 90 minutes, respectively. The green dashed line shows the boundary of the crystal. For d = 6μm, the evolvement process is similar to that in bulk, but at the output face there is a distance between the formed dark soltion and boundary and there was still some light located at the boundary as shown in Figs. 4(b2)–4(e2). From the above experimental result, the dark soliton should not straightly propagate. The deflection distance is larger than d = 6μm within 10mm propagating distance, it should reach the boundary and undergoes reflection at the boundary. The circumstance is just like that in Figs. 2(b) and 2(d). The corresponding experimental result of guided wave is shown in Fig. 5(b). When d = 0μm, i. e. a half of dark notch was launched close to the boundary, there was no light located at the boundary after a similar evolvement process, as shown in Figs. 4(b3)–4(e3). The circumstance is just like that in Figs. 2(c) and 2(e). The half of dark notch evolved into a fine shape of a half dark soliton and located at boundary and a steady surface dark soliton was excited, which can straightly propagate along surface. The corresponding experimental result of guided wave is shown in Fig. 5(c). At the location of surface dark solitons, there was also a bright stripe (circled by green line) without redundant light at boundary, which indicated that a surface dark soliton was excited and its induced waveguide was written successfully. The experimental results meet with our theoretical analysis very well.
The transverse modulation instability is one of the disadvantages for dark-soliton. For a dark soliton strip in bulk, a snakelike transverse deformation will occur and is named snake instability. In the above experiment the snake instability was observed, as shown in Figs. 4(b1) and 4(e1). At the beginning the dark notch in the output face is straight, while in the end the dark notch deformed into snakelike shape along with the formation of dark soliton. It is very interesting that the snake instability did not appear in the formation of surface dark soliton, as shown in Figs. 4(e2) and 4(e3).
Manjun Ma et al. has studied the possibilities to suppress the transverse modulation instability (MI) of dark-soliton stripes in two-dimensional Bose-Einstein condensates (BEC’s) and self-defocusing bulk optical waveguides by means of quasi-one-dimensional structures. When an external repulsive barrier potential (which can be induced in Bose-Einstein condensates by a laser sheet, or by an embedded plate in optics) is added, it is possible to reduce the MI wave number band, and even render the dark-soliton stripe completely stable [18]. For air the refractive index is n = 1, which is corresponding to a high potential region for photon. Consequently the surface provides a straight potential barrier, under which the snaking instability can be suppressed perfectly. To our knowledge this is the first time to utilize a natural structure, surface, to solve the problem of transverse modulation instability. It indicates that more problems about instability induced by nonlinearity maybe solved and fine nonlinear phenomena maybe expected take advantage of surface.
5. Summary
In conclusion, we observed surface dark photovoltaic solitons experimentally and theoretically for the first time to our knowledge. The photovoltaic nonlinearity ensures the beam maintaining a dark soliton shape, just like the formation of photovoltaic dark solitons in bulk; while the effect of nonlocal diffusion nonlinearity is to confine dark soliton near boundary and permit it propagates along surface. The suppression of transverse modulation instability take advantage of surface was also observed for the first time. These merits are promised to be applied to the surface waveguide, nonlinear effect enhancement and design of surface photonic devices etc.
Acknowledgments
This work was supported by the Chinese National Key Basic Research Special Fund (2011CB922003), the National Natural Science Foundation of China (NSFC, 61078014, 61178005, J1103208), the Specialized Research Fund for the Doctoral Program of Higher Education (20100031110007, 20120031110030), the Program for New Century Excellent Talents in University, the 111 Project (B07013), and the research funding for undergraduate students of Nankai University.
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