Binary parity-time-symmetric nonlinear lattices with balanced gain and ...

2976

OPTICS LETTERS / Vol. 35, No. 17 / September 1, 2010

Binary parity-time-symmetric nonlinear lattices with balanced gain and loss Sergey V. Dmitriev,1 Andrey A. Sukhorukov,2,* and Yuri S. Kivshar2 1

2

Institute for Metals Superplasticity Problems, Russian Academy of Science, Ufa 450001, Russia Nonlinear Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra, Australian Capital Territory 0200, Australia *Corresponding author: [email protected]

Received June 22, 2010; accepted July 20, 2010; posted August 6, 2010 (Doc. ID 130425); published August 27, 2010 We study nonlinear binary arrays composed of parity-time-symmetric optical waveguides with gain and loss. We demonstrate that such nonlinear binary lattices support stable discrete solitons, which can be adiabatically tuned and switched through nonlinear symmetry breaking by varying gain and loss parameters. © 2010 Optical Society of America OCIS codes: 190.5940, 190.6135, 190.0190.

Photonic lattices combining loss and gain elements offer new possibilities for shaping optical beams and pulses compared to conservative structures [1–3]. It was shown that periodic lattices composed of waveguides where each waveguide combines gain and loss regions can play a role of complex parity-time (PT)-symmetric potentials [4–8]. In such lattices, the optical power can be conserved on average; yet the basic properties of diffraction and refraction may be strongly modified. Most recently, PT-symmetric properties in couplers composed of two waveguides were demonstrated experimentally [9,10]. In experiments, one waveguide exhibited zero [9] or adjustable [10] gain, and another waveguide was lossy. It is therefore an important practical question of how to realize PT-symmetric photonic lattices where gain and loss regions are separated between different waveguides, because such structures are easier to realize experimentally. In this Letter, we demonstrate that the PT-symmetry condition can be satisfied for lattices composed of alternating waveguides with gain and loss, where the separation between the waveguides is modulated according to a binary pattern, as illustrated in Fig. 1(a). We confirm that, similar to other types of PT-symmetric lattices [4,8], stable localized solitons can exist in the nonlinear regime. Moreover, we show that the solitons are robust and tend to remain localized if the gain and loss parameters are varying along the waveguides, whereas their shape either changes adiabatically or switches sharply to the region of gain due to nonlinear symmetry breaking. We describe the light propagation in a binary lattice composed of gain and loss waveguides by generalizing the coupled equations for two waveguides [10], with additional terms accounting for Kerr-type nonlinearity: daj þ iρa aj þ C 1 bj þ C 2 bj−1 þ Ga ðjaj j2 Þaj ¼ 0; dz dbj þ iρb bj þ C 1 aj þ C 2 ajþ1 þ Gb ðjbj j2 Þaj ¼ 0; i dz

ρ < 0); C 1;2 are the coupling coefficients; and the functions Ga;b characterize the nonlinear response. We note that in the framework of continuous evolution equations, the PT operator transforms an effective potential (refractive index profile for optical waveguides) as V ðxÞ → V
ð−xÞ [1–3], and for the discrete Eqs. (1), this corresponds to a transformation ρa;b → −ρb;a . Let us first analyze the linear case. Then the wave propagation in the framework of Eq. (1) can be fully defined by identifying the eigenmode solutions in the form of Floquet–Bloch modes: ðaj ; bj Þ ¼ ða0 ; b0 Þ expðiβz þ ikjÞ, where k is the Bloch wavenumber and β is the propagation constant. We obtain the following dispersion relation: β ðkÞ ¼ iρþ 

pffiffiffiffiffiffiffiffiffiffi DðkÞ;

DðkÞ ¼ C 21 þ C 22 þ 2C 1 C 2 cos K − ρ2− ;

ð2Þ

where ρ ¼ ðρa  ρb Þ=2. We find that the eigenmode amplitudes at waveguides with gain and loss are equal, ja0 j ≡ jb0 j, if and only if DðkÞ ≥ 0. Mathematically, this

i

ð1Þ

where j is the waveguide number; z is the propagation distance; aj and bj are the mode amplitudes at waveguides with gain and loss, respectively; ρa;b define the corresponding rates of loss (for ρ > 0) and gain (for 0146-9592/10/172976-03$15.00/0

Fig. 1. (Color online) (a) Schematic of a binary waveguide array with gain and loss. (b), (c) 3D plots of jaj ðzÞj2 (dots) and jbj ðzÞj2 (open circles) showing linear diffraction of smallamplitude Gaussian beam (initial amplitude ja0 ðz ¼ 0Þj2 ¼ 0:05 and inverse width 0.02) when PT-symmetry condition is (b) not satisfied (C 1 ¼ C 2 ¼ 3) and (c) satisfied (C 1 ¼ 3, C 2 ¼ 2). For both plots, −ρa ¼ ρb ¼ 0:8. © 2010 Optical Society of America

September 1, 2010 / Vol. 35, No. 17 / OPTICS LETTERS

is the PT-symmetry condition, generalized to the case of unbalanced gain and loss. For two waveguides, this condition reduces to the cases of previously studied couplers [9,10]. The physical meaning of the condition DðkÞ ≥ 0 is that all eigenmodes propagating through the lattice exhibit the same degree of gain or loss, according to the averaged value ρþ . In particular, when the rates of gain and loss are exactly balanced, ρa ¼ −ρb (which also satisfies the PT-symmetry condition), then ρþ ¼ 0 and the power of each mode is conserved. On the other hand, if DðkÞ < 0 at some k, then there will appear modes that exhibit stronger gain (or smaller loss) than other modes, and such imbalance underpins PT-symmetry breaking. To completely avoid PT-symmetry breaking for any input beam profiles, we need to satisfy DðkÞ ≥ 0 simultaneously for all possible real k. This leads to the requirement that jjC 1 j − jC 2 jj ≥ jρ− j ¼ jρa − ρb j=2:

ð3Þ

We note that Eq. (3) can be fulfilled only when jC 1 j ≠ jC 2 j (provided that the rates of gain and loss are nonzero), i.e., a binary modulation is necessary to have unbroken PT symmetry. We compare the linear diffraction of a Gaussian beam in lattices when Eq. (3) is not satisfied [Fig. 1(b)] and satisfied [with binary modulation, Fig. 1(c)], considering the case of balanced gain and loss coefficients, ρa ¼ −ρb . The numerical simulations are performed with the implicit Crank–Nicholson scheme for 300 lattice sites, and absorbing boundary conditions are used to exclude the influence of small-amplitude radiation reflected from the boundaries. In the first example, shown in Fig. 1(b), the binary modulation is absent and PT symmetry is broken, and the field amplitudes become amplified and concentrated in waveguides with gain. In the second example, shown in Fig. 1(c), when appropriate binary modulation is introduced to achieve PT symmetry, the total intensity is conserved on average due to effective compensation of gain and loss, and the field is equally spread out between the different waveguides. We now analyze the effect of nonlinearity and consider the Kerr-type response for all waveguides, Ga;b ðIÞ ¼ γI, where γ > 0 stands for self-focusing nonlinearity. Then, byffiffiffiffiffiintroducing the transformation ðaj ; bj Þ → ðaj ; bj Þ= p jγj, we can scale the nonlinear coefficient to unity, γ ¼ 1, and we use this value in numerical simulations. First, we consider a case when the rates of gain and loss in waveguides are balanced, −ρa ¼ ρb ¼ ρ, and the PTsymmetry condition is satisfied, such that the total intensity is conserved on average in the linear regime. Then we seek self-localized nonlinear solutions in the form of PT-symmetric solitons [4,8], faj ðzÞ; bj ðzÞg ¼ faj ðz ¼ 0Þ; bj ðz ¼ 0Þg expðiβzÞ;

2977

Fig. 2. (a) Approximate soliton profiles for ρ ¼ 0:8, shown at different propagation distances z ¼ 0 and z ¼ 18:3 to illustrate beating with internal mode. (b) Total intensity IðzÞ for approximate soliton solutions at different values of ρ, indicated for each curve. The values of C 1 and C 2 correspond to Fig. 1(c), and −ρa ¼ ρb ¼ ρ.

similar to conservative binary lattices [11]; however, we do not analyze here the gap solitons. We determine the approximate profiles of fundamental solitons using a shooting method. Then we use these profiles as initial conditions for numerical modeling of Eqs. (1). We consider the values of coupling coefficients corresponding to Fig. 1(c), C 1 ¼ 3 and C 2 ¼ 2, when the PTsymmetry condition is satisfied for jρj ≤ 1. In Fig. 2(a), we give an example of a soliton for ρ ¼ 0:8. The soliton bears an internal mode, so that its shape changes periodically, similar to conservative systems [12], and we show the two most asymmetric configurations in Fig. 2(a). The fundamental soliton is intersite-centered for any ρ, including ρ ¼ 0, and the latter case corresponds to a conservative binary lattice soliton [13]. As the numerical soliton solutions are not exact and exhibit breathing due to internal mode excitation, we notice the corresponding P periodic oscillation of the total intensity, IðzÞ ¼ j ðjaj j2 þ jbj j2 Þ, for all values of ρ with the exception of ρ ¼ 0; see Fig. 2(b). This confirms that in PTsymmetric lattices, the total intensity can be conserved on average, whereas in a conservative lattice (ρ ¼ 0), the total intensity is exactly constant. We also note that the period of internal mode oscillations very weakly depends on ρ. We now demonstrate that the solitons can be highly robust, and localization can be sustained when the gain

ð4Þ

where β is the soliton propagation constant. The localization conditions, fjaj j; jbj jg → 0 at j → ∞, can be satisfied when β is outside the band of linear waves. For fundamental solitons where propagation constants are above all linear bands, we have jβj > β ðkÞ for all k and, accordingly, jβj > βg ¼ ½ðjC 1 j þ jC 2 jÞ2 − ρ2− 1=2 . We note that there also appears a gap in the linear spectrum,

Fig. 3. Soliton tuning by balanced variation of gain and loss along the waveguides. (a) Introduced dependencies of ρ ¼ −ρa ¼ ρb on z; two different rates of increase of ρ are compared. (b) Corresponding evolution of total intensity IðzÞ, when the input profile corresponds to the soliton solution for a conservative lattice with ρ ¼ 0. The inset shows the intensity profiles at z ¼ 0 and z ¼ 400 corresponding to curve 1. For all the plots, the values of C 1 and C 2 correspond to Fig. 1(c).

2978

OPTICS LETTERS / Vol. 35, No. 17 / September 1, 2010

Fig. 4. Soliton tuning by gain-loss imbalance. (a) Total intensity versus z for ρa ¼ −ρ þ ρþ and ρb ¼ ρ þ ρþ , where the imbalance value ρþ for curves 1 to 4 is −1:6 × 10−4 , −0:8 × 10−4 , 0:8 × 10−4 , and 1:6 × 10−4 , respectively. (b) Soliton profiles for different z for ρþ ¼ 1:6 × 10−4 shown by curve 1 in (a). For all the plots, ρ ¼ 0:8, and the values of C 1 and C 2 correspond to Fig. 1(c).

and loss are varied along the waveguides. As the input condition, we choose the soliton solutions corresponding to a conservative lattice, for ρ ¼ 0. Then we consider an increase of ρ along the waveguides until it reaches a specified value. Two examples of gain/loss tuning are shown in Fig. 3(a), and the corresponding dependencies of the total intensity with selected intensity profiles are presented in Fig. 3(b). We note that the final value of the total intensity is the same in both cases, and the periodic oscillations are even weaker compared to Fig. 2(b), indicating that the soliton exhibits adiabatic reshaping and reaches a stable final state corresponding to ρ ¼ 0:8. Finally, we consider a case when the generalized PTsymmetry condition is satisfied; however, the gain and loss coefficients are not exactly balanced and ρþ ≠ 0. In the linear regime, the overall gain or loss can be formally removed from Eq. (1) using a gauge transformation [9]; however, this is not applicable in the nonlinear regime when the effect of gain/loss imbalance can be nontrivial. We perform a series of numerical simulations for soliton dynamics with different values of imbalance and show the corresponding dependencies of the total intensity IðzÞ in Fig. 4(a). Curves 3 and 4 in Fig. 4(a) correspond to the case of averaged loss (because ρþ > 0), and under such circumstances, the soliton reshapes adiabatically and remains localized as its intensity is gradually reduced. Completely different behavior is observed in the case of averaged gain imbalance (ρþ < 0), when after a gradual intensity growth, there occurs a sharp transition to a much faster amplification rate; see curves 1 and 2 in Fig. 4(a). To identify the origin of this transition, we plot in Fig. 4(b) the soliton profiles correspond-

ing to curve 1 at different propagation distances. We see that, as the intensity is increased, the soliton profile exhibits symmetry breaking. For asymmetric soliton profiles, nonlinearity breaks PT symmetry and energy becomes concentrated primarily in waveguides with gain. The resulting intensity amplification rate is then defined by the value of jρa j, which is much larger than the averaged gain jρþ j, and this explains the sharp transition in growth rates. In conclusion, we have studied nonlinear binary arrays of optical waveguides with gain and loss satisfying the PT-symmetry conditions. We have shown that such nonlinear lattices support stable solitons, which can be either adiabatically reshaped or sharply switched through nonlinear symmetry breaking by varying the gain and loss parameters. Our results can stimulate further experimental studies of linear and nonlinear beams in PT-symmetric lattices. S. V. D. thanks the Nonlinear Physics Center for the warm hospitality and acknowledges support of the Russian Foundation for Basic Research (RFBR) through grant 09-08-00696-a. This work was partially supported by the Australian Research Council. References 1. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005). 2. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007). 3. S. Klaiman, U. Guenther, and N. Moiseyev, Phys. Rev. Lett. 101, 080402 (2008). 4. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008). 5. M. V. Berry, J. Phys. A 41, 244007 (2008). 6. S. Longhi, Phys. Rev. Lett. 103, 123601 (2009). 7. S. Longhi, Phys. Rev. A 81, 022102 (2010). 8. K. G. Makris, R. El Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010). 9. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009). 10. C.E.Ruter,K.G.Makris,R.El-Ganainy,D.N.Christodoulides, M. Segev, and D. Kip, Nature Phys. 6, 192 (2010). 11. A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 91, 113902 (2003). 12. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003). 13. R. A. Vicencio and M. Johansson, Phys. Rev. A 79, 065801 (2009).