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P H Y S I C A L R E V I E W L E T T E R S Eliminating the Transverse Instabilities of Kerr Solitons
Charalambos Anastassiou,1,2 Marin Soljaˇcic´,1,3 Mordechai Segev,1,2 Eugenia D. Eugenieva,4 Demetrios N. Christodoulides,4 Detlef Kip,1,5 Ziad H. Musslimani,6 and Juan P. Torres1,7 1Physics Department and the Solid State Institute, Technion, Haifa 32000, Israel 2Electrical Engineering Department, Princeton University, Princeton, New Jersey 08544 3Physics Department, Princeton University, Princeton, New Jersey 08544 4Electrical Engineering and Computer Science Department, Lehigh University, Bethlehem, Pennsylvania 18015 5Physics Department, Universität Osnabrück, 49069 Osnabrück, Germany 6Mathematics Department, Technion, Haifa 32000, Israel 7Department of Signal Theory and Communications, Polytechnic University of Catalonia, Jordi Girona 1-3, Campus Nord D3, 08034 Barcelona, Spain We show analytically, numerically, and experimentally that a transversely stable one-dimensional ͓͑1 1 1͒D͔ bright Kerr soliton can exist in a 3D bulk medium. The transverse instability of the solitonis completely eliminated if it is made sufficiently incoherent along the transverse dimension. We derivea criterion for the threshold of transverse instability that links the nonlinearity to the largest transversecorrelation distance for which the 1D soliton is stable.
Research on optical spatial solitons has made much Thus far, in order to avoid TI, experiments with 1D progress during the past decade: new systems that support solitons were conducted in either planar waveguides [5,6] solitons have been identified, solitons of more than one or nonlinearities in which TI was greatly suppressed [8].
transverse dimension have been demonstrated, and a whole Here we demonstrate how to produce a truly stable stripe range of soliton interactions was explored [1]. Despite Kerr soliton propagating in a 3D bulk medium without the diversity of the physical systems that support them, suffering from transverse instability. We show that if the solitons are a universal phenomenon and share many com- soliton is made “sufficiently” incoherent in its transverse mon features [1], one of which is transverse instability (TI) dimension y, then TI is completely eliminated.
[2 – 10]. TI is a symmetry breaking instability: almost all First, recall incoherent solitons made of partially inco- solitons [11] of a particular dimension that propagate in a herent light [12]. They are multimode (speckled) beams of higher dimension system [by having a uniform wave func- which the instantaneous amplitude varies randomly with tion in the additional dimension(s)] are unstable to pertur- time. If such beams are launched into a noninstantaneous bations in the dimension(s) in which they are uniform. TI self-focusing medium, so that the response time of the non- occurs because perturbations in the dimension of unifor- linearity greatly exceeds the fluctuation time, then self- mity have nothing to restrain them from growing (driven focusing is driven solely by the average intensity. Then, by the nonlinearity) and breaking the soliton up.
the incoherent beam induces a multimode waveguide and In the particular case of a spatial optical ͑1 1 1͒D soli- guides itself in it by properly populating the guided modes, ton that is self-trapped in one dimension x, is uniform thus forming an incoherent soliton [12 – 18].
in the transverse dimension y, and is propagating along A clue that TI could be completely eliminated for soli- z, TI causes the soliton to break up along y into an ar- tons was given by two recent discoveries: modulation ray of 2D filaments [2 – 10]. The transverse wavelength instability (MI) of incoherent light [19] and elliptical in- of these perturbations is usually much larger than the soli- coherent solitons [14,20]. MI belongs to the same family ton width [2 – 4]. Transverse instability is especially severe of symmetry breaking instabilities as TI does, and it oc- for Kerr nonlinearities and prohibits spatial 1D Kerr soli- curs when a plane wave (or a very broad beam or pulse) is tons in a bulk medium. This is why spatial ͑1 1 1͒D Kerr launched into a self-focusing medium. If this plane wave solitons have to be launched in a planar waveguide con- is fully coherent, it breaks up into a train of filaments due figuration, in which the y confinement is much narrower to MI. Recently, it has been shown theoretically and ex- than the self-trapped (soliton) width in x [5,6]. TI actually perimentally [20] that MI does exist also for incoherent occurs for solitons in any nonlinearity, including, for ex- light, but it occurs only if the nonlinearity exceeds a well- ample, quadratic solitons [7] and photorefractive solitons [8,9]. Interestingly, saturation arrests transverse instabil- the coherence of the light. If the nonlinearity is below ity [10] but never completely eliminates it. In fact, it is threshold, then MI is eliminated and the wave is stable.
the suppression of TI due to saturation that facilitates the This generic idea has enabled the observation of anti- observation of stable 1D solitons in a bulk photorefractive dark solitons [21], which were thought to be unstable in crystal for more than ten diffraction lengths [8].
conservative nonlinear systems [22]. The new finding of P H Y S I C A L R E V I E W L E T T E R S incoherent elliptical solitons [14,20] is based on having tion, E͑x, y, z, t͒ being the slowly varying amplitude. The different coherence function for the two transverse dimen- ͗ ͘ denote averaging over the response time of the medium sions of self-trapping. Combing these ideas, one can gen- t, which is much larger than tcoh. From the paraxial wave erate a 1D soliton that is fully coherent in x (direction of trapping), partially incoherent but uniform in y, and propa- gating along z. The intimate relation between TI and MI ෇ ik ͕dn͑x1, y1, z͒ suggests that TI of incoherent beams should also exhibit a threshold for its existence. Therefore, if the degree of 2 dn͑x2, y2, z͖͒B , (1) coherence in y is such that TI is below the threshold, alltransverse perturbations are suppressed and TI is elimi- where z is the propagation direction, k is the carrier wave nated. This is the core idea of our Letter. The idea of number, n0 is the bias refractive index, dn is the nonlin- using the threshold to eliminate TI applies to any type of ear contribution to the refractive index, x ෇ ͑x1 1 x2͒͞2, nonlinearity, yet we will concentrate on the Kerr nonlinear- y ෇ ͑ y1 1 y2͒͞2 are the middle point coordinates, and ity for two reasons. First, wave propagation in Kerr media rx ෇ x1 2 x2, ry ෇ y1 2 y2 are the difference coordi- is described by the cubic nonlinear Schrödinger equation nates. When rx ෇ ry ෇ 0, B͑x, y, rx, ry, z͒ is the time- (NLSE) which is one of the most general soliton equations averaged intensity I͑x, y, z͒. Let BS͑x, y, rx, ry͒ ෇ u͑x 1 [23]. Generally speaking, the NLSE describes envelope rx͞2͒ uء͑x 2 rx͞2͒An͑ry͒ be a z-independent solution solitons in dispersive wave systems with weak symmetric It represents a 1D soliton stripe, which is anharmonicity. Second, the effect of TI for Kerr solitons self-trapped and fully coherent in x, while being uni- is very strong and we can demonstrate a convincing dif- form and incoherent along y with an angular spectrum of ference between having TI and eliminating TI by making An͑ry͒. u͑x͒ is determined by the nonlinearity and can be the soliton incoherent along y.
taken to be real without loss of generality. To study TI, An incoherent beam can be represented as a series we add a small perturbation B1 to BS where B1 ø BS.
of coherent speckles that change, on average, every The nonlinear index change in Kerr media is dn͑I͒ ෇ gI, We define B͑x1, y1x2, y2, z͒ ෇ where g is the nonlinear coefficient ͑n2͒. Linearizing ͗Eء͑x2, y2, z, t͒E͑x1, y1, z, t͒͘, the spatial correlation func- , rx ෇ 0, ry ෇ 0, z͒ 2 B1͑x 2 , rx ෇ 0, ry ෇ 0, z͔͒BS͑x, y, rx, ry͒ x ෇ 0, ry ෇ 0͒ 2 BS ͑x 2 2 x ෇ 0, ry ෇ 0͔͒B1͑x, y, rx , ry , z͒ We seek solutions in the form B1͑x, y, rx, ry, z͒ ෇ exp͑gz͒ exp͑iay͒L͑x, rx͒Af͑ry͒ 1 exp͑gءz͒ exp͑2iay͒ 3 Lء͑x, 2rx͒Aءf͑2ry͒, where a is the transverse wave number, g is the TI growth rate (gain), and Af͑ry͒ is the angularspectrum. The necessary condition B1͑x, y, rx, ry, z͒ ෇ Bء1͑x, y, 2rx, 2ry, z͒ is satisfied [17,19]. Substituting B1 intoEq. (2) gives gL͑x, rx͒Af͑ry͒ 2 ͓L͑x 1 x , 0͒A x ͒u͑x 2 x ͒A f ͑0͒eiary͞2 2 L͑x 2 2 f ͑0͒e2iary͞2͔u͑x 1 2 1 ͓u͑x 1 rx͞2͒2 2 u͑x 2 rx͞2͒2͔L͑x, rx͒Af͑ry͒An͑0͒ We are interested in determining the threshold condi- the “cutoff wavelength” [2 – 4]. This means that for a co- tion: to find the conditions under which the growth rate herent soliton, the growth rate is positive (and TI exists) g͑a͒ goes from a positive value to a negative value for all for a band of wave numbers a between zero and the cutoff a. The procedure of determining the threshold applies to wave number. For a soliton that is partially coherent in y, any form of spatial coherence (angular power spectrum), we expect that for u0 small enough (a beam that is coherent but for simplicity, we consider an initial Gaussian angu- enough), g͑a͒ will be positive in a band of wave numbers, lar power spectrum An͑ry͒ ෇ exp͓2͑ryu0k͞2͒2͔, where just as the coherent case. But, as u0 increases, this band u0 defines the degree of coherence (correlation distance).
becomes narrower until it completely disappears at some The higher u0 the more incoherent the soliton is. For a value u0T . If u0 is larger than this value, then TI is elimi- fully coherent soliton, if we were to calculate the growth nated. We therefore expect that, at the threshold u0 ෇ u0T , rate g as a function of transverse wave number a, then g the two boundary points at which g͑a͒ ෇ 0 (one at a ෇ 0 starts from 0 (at a ෇ 0), increases and reaches a maxi- and the other at the cutoff wave number) coincide. Thus, mum positive value (at the wave number with the largest we seek the value of u0 at which (i) g͑a ෇ 0͒ ෇ 0 and growth rate), and then drops back to 0 at a associated with (ii) g0͑a ෇ 0͒ ෇ 0. We solve Eq. (3) by expansion while P H Y S I C A L R E V I E W L E T T E R S retaining up to O͑a͒. This eliminates the first term on theleft-hand side (LHS) because g ෇ O͑a2͒. The growth rateof the transverse instabilities, g, is independent of x [2 –4]even though the actual shape of the perturbations dependson x. We can therefore seek solutions of Eq. (3) under theconditions (i) and (ii) at the center of the soliton, i.e., atx ෇ rx ෇ 0, and assume that the threshold we find is thesame everywhere on the soliton. It can be easily shown(by expanding into derivatives with respect to x1 and x2)that the second term of the LHS is zero for x1 ෇ x2, i.e.,for rx ෇ 0. Thus, from Eq. (3) we get Simulations of a 1D Kerr soliton with a Gaussian spec- 3 exp͓2͑ku0T ry͞2͒2͔ , (4) trum, for various degrees of coherence. The predicted thresh-old is u0 ෇ 0.549±. (a) Input intensity. ( b) Output beam after where I0 ෇ u͑0͒2 is the peak intensity of the soliton. It is 0.8 cm of propagation for a fully coherent beam u0 ෇ 0±; the unlikely that a small perturbation will alter the coherence soliton is destroyed by TI. (c) Output beam close to the threshold statistics of the soliton (especially here that propagation (for u0 ෇ 0.5±), after 4.5 cm of propagation: As the thresholdis approached, TI gain is reduced and it takes a longer propa- effects, given by g, are of the order of a2 and are ignored).
gation before TI is evident. (d ) Output beam for u0 ෇ 0.56±, Thus, we assume that Af ͑ry͒ ෇ An͑ry͒. Equation (4) which is below the threshold, after 4.5 cm of propagation. TI is Dn0 ෇ gI0 is the maximum change in the refractive in-dex. One can actually calculate, using numerical methods diffuser. Then, by moving the focal point of this lens closer similar to [10], the function g͑a͒ and from it obtain the (farther away) from the diffuser, we increase (decrease) threshold for any angular distribution function [24].
the coherence in y. The x coherence is not affected by To verify the analytic predictions, we perform simula- the translation of this lens. After the diffuser, the beam tions using the coherent density approach [12]. We launch is collimated (to ϳ2 cm) and passed through a narrow a 1D Kerr soliton with a Gaussian angular power spec- (along x) slit. The slit is made narrower than the speckle trum, A͑ ry͒ ෇ exp͓2͑ryu0k͞2͒2͔, for various values of size in x, and it effectively creates a 1D beam that is u0. In this example, n0 ෇ 2.3, l ෇ 0.5 mm in vacuum, narrow and coherent in x and “infinitely” long (uniform) FWHM ෇ 9 mm, which yields a Dn0 ෇ 0.000 105 6 and and incoherent in y. The slit is then imaged to the input an analytic prediction of the threshold of u0T ෇ 0.55±. Our face of the crystal. We get a reasonable estimate of the results are displayed in Fig. 1, where we show images of correlation distance by stopping the diffuser and measur- the intensity distribution of the soliton and cross sections ing the average speckle size at the crystal input plane.
of the intensity along y for x ෇ 0. Figure 1(a) shows the Finally, we use an orthogonally polarized background input soliton at z ෇ 0, and Fig. 1(b) shows a fully coherent beam that covers the crystal uniformly as necessary for soliton ͑u0 ෇ 0±͒ after 0.8 cm of propagation. As clearly photorefractive screening solitons [8].
depicted there, TI breaks the soliton up into a train of 2D output faces of the crystal are imaged onto a CCD camera.
filaments. As we approach the threshold, the TI gain is The photorefractive nonlinearity is in general saturable getting smaller: As we set u0 to 0.5±, it takes a 4.5 cm but resembles the Kerr nonlinearity when the peak inten- propagation to exhibit signs of TI [Fig. 1(c)]. To show that sity of the soliton normalized to the background intensity TI is completely eliminated when the nonlinearity is below is much smaller than unity [8,25]. In our case, this ratio threshold, we increase u0 to 0.56±. As shown in Fig. 1(d), is ϳ0.1. At this normalized intensity, a soliton that is after 4.5 cm of propagation there are absolutely no signs fully coherent in both x and y exhibits strong transverse instability [8]. We then gradually increase the incoherence Our experiments are conducted in a photorefractive in y (decrease the speckle size) until the soliton becomes SBN:75 crystal in a setup similar to that of [12]. The beam transversely stable, while keeping all other parameters is made spatially incoherent by passing it through a rotat- (applied field, intensity) constant. Our results are shown in ing diffuser. The rotating diffuser provides a new phase Fig. 2. The 12 mm FWHM input beam [Fig. 2(a)] linearly and amplitude distribution every tcoh ϳ 1 ms, which diffracts to a 60 mm output after 6 mm propagation in the is much shorter than the response time of the medium crystal [Fig. 2( b)]. The nonlinearity is turned on with the t ϳ 1 s. Unlike all previous experiments with incoherent application of 2.7 kV͞cm and the beam self-traps forming solitons, here we need to generate a beam which is very a soliton in x. When the beam is fully coherent, the soliton narrow and fully coherent in x, yet uniform and partially suffers from TI and breaks up into filaments [Fig. 2(c)].
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and close to it. We compared the threshold found here (an- This work was supported by the Israeli Science Foun- alytically) to the value we find using numerical methods dation, the U.S. Army Research Office, the NSF, and similar to [10], and to the value found from beam prop- AFOSR. It is part of the MURI program on optical spa- agation (Fig. 1). All three values are in excellent agree- tial solitons. E. E. acknowledges support by NSF-NATO.
ment. For the purpose of calculating the threshold, the This research was supported in part by the Pittsburgh above ansatz is the most convenient means.
Supercomputing Center, Pittsburgh, PA.
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