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Two dimensional spatial solitons in parity-time symmetric potential
Accepted Manuscript Title: Two Dimensional Spatial Solitons in Parity- Time Symmetric Potential Author: K Aysha Muhsina P.A. Subha PII: DOI: Reference:
S0030-4026(16)30471-5 http://dx.doi.org/doi:10.1016/j.ijleo.2016.05.038 IJLEO 57664
To appear in: Received date: Accepted date:
29-2-2016 10-5-2016
Please cite this article as: K Aysha Muhsina, P.A. Subha, Two Dimensional Spatial Solitons in Parity- Time Symmetric Potential, (2016), http://dx.doi.org/10.1016/j.ijleo.2016.05.038 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
K Aysha Muhsina1 and P. A. Subha2 1
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Department of physics, University of Calicut, Kerala, India. Department of physics, Farook College, University of Calicut, Kerala, India. email: 1 [email protected], 2 [email protected]
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Two Dimensional Spatial Solitons in Parity- Time Symmetric Potential
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Abstract
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This work analyzes the propagation of (2+1) dimensional spatial solitons in Parity-Time (PT ) symmetric potential. The stationary solution of the system has been studied. The beam dynamics has been analyzed using variational and numerical methods. The soliton beam propagation is stable when the coefficient of imaginary potential is less than a threshold, which is called the phase transition point. Above the transition point, the imaginary component of the solution starts to evolve and the solution becomes unstable. When the coefficient of imaginary potential exceeds this critical value, the power of the beam increases and results in the unstable beam propagation. The stability of the stationary solution against small perturbation has been studied using linear stability analysis. The imaginary eigen value is zero when the coefficient of the imaginary potential is low. Above the phase transition point, the imaginary eigen value becomes comparable with the real eigen value and hence the solution becomes linearly unstable. Keywords: Two dimensional spatial solitons, Non-Hermitian Hamiltonian with real eigen values 1. Introduction
The study of dissipative systems with non Hermitian Hamiltonians have attracted a lot of attention in recent years. Such non Hermitian operators posses real spectra, provided that they obey Parity- Time (PT ) symmetry. This concept was introduced by Bender and Boettcher in 1998 [1]-[3]. According to their view, those non- Hermitian Hamiltonian share common set of Preprint submitted to Optik
May 13, 2016
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eigen functions with the PT operator posses real spectra. The PT operations can be stated as follows[4]-[7]. Under the action of parity operator, pˆ → −ˆ p, xˆ → −ˆ x (ˆ p and xˆ are the momentum and position operators respectively) and the action of time reversal operator results pˆ → −ˆ p, xˆ → xˆ, i → −i. The ˆ = pˆ2 +V (x), where m is the mass and Hamiltonian of a physical system is H 2m ˆ = pˆ2 +V ∗ (x). V is the complex potential. Under time reversal operation, TˆH 2m ˆ = pˆ2 +V ∗ (−x) and H ˆ Pˆ Tˆ = pˆ2 +V (x). For After the parity operation Pˆ TˆH 2m 2m ˆ =H ˆ Pˆ Tˆ. That is possible only the Hamiltonian to be PT symmetric, Pˆ TˆH if the condition V (x) = V ∗ (−x) is satisfied [2]. In order to satisfy the above condition, the real part of the complex potential must be an even (symmetric) function of position, where as the imaginary part should be an odd (antisymmetric) function. The dissipative systems, which include exactly balanced linear gain and loss, are described by non-Hermitian Hamiltonians, whose Hermitian and non-Hermitian parts are spatially even and odd, respectively. It was also demonstrated that when the non-Hermitian Hamiltonians are PT symmetric, they undergo a phase transition (PT symmetry breaking) above a critical threshold, above which the eigen value spectrum becomes partially complex [8]. This concept of PT symmetry of a non-Hermitian Hamiltonian generalizes quantum mechanics in a complex domain [5, 9]. PT symmetry finds applications in various areas of physics ranging from PT symmetric quantum oscillators to linear and nonlinear optics [1, 4]. PT symmetric structures have been realized in optics by Christodoulides etal [6, 10]. Such systems can be realized through the inclusion of gain/loss regions in guided wave geometries. In the optical systems the complex refractive index distribution, n(x) = nr (x) + ini (x) plays the role of the optical potential [10]-[13]. The PT symmetry condition implies that the index wave guiding profile nr (x) is an even function in the transverse direction while the loss or gain term ni (x) is an odd function. PT symmetric optical systems have been experimentally demonstrated in AlGaAs [14], photorefractive materials [6], silicon, fiber optics, and light-written guides in glass [15]-[17] etc. The observation of PT symmetry in linear optical systems led to its generalization to the nonlinear case, which provides several interesting predictions. It was also demonstrated that PT symmetric nonlinear optical systems can support soliton solutions [12, 18]. Recently, soliton beam propagation in PT symmetric optical media has been a subject of intense research because the beam dynamics in such systems can exhibit unique characteristics such as double refraction, power oscillations, nonreciprocal diffraction patterns, etc. 2
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[19],[20]. The stability of the optical solitons in nonlinear PT symmetric systems is investigated in a number of complex potentials like PT symmetric periodic potential, hyperbolic Scarf [21]-[23] potential etc. Studies of PT symmetry have also been extended from nonlinear lattice to double channel coupled wave guides [24]-[26], double-well potentials and asymmetric optical amplifier [27]. Spatial solitons in self focusing and defocusing Kerr media and nonlocal media with PT symmetric potentials have also been investigated [28],[30]. This work analyzes the soliton beam propagation in a PT symmetric system, which is characterized by the nonlinear Schr¨odinger equation with a complex potential with competing gain and loss profile. The paper is organized as follows. In section (2), the stationary solutions of the system have been studied in different imaginary potentials. In section (3), the beam dynamics has been analyzed using variational and numerical methods. In section (4), the stability of the stationary solution against small perturbation has been analyzed using linear stability analysis. Section (5) concludes the paper. 2. Stationary Solution
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The beam evolution in a (2+1) D self focusing Kerr media is governed by the normalized nonlinear Schr¨odinger equation[29]- [31].
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iψz + ψxx + ψyy + V ψ + β|ψ|2 ψ = 0.
(1)
The suffixes z, x and y stand for the partial derivatives with respect to z, x and y respectively, V is the complex potential and β = +1 corresponds to the self focusing nonlinearity. For radial symmetry, x2 + y 2 = r2 and ∂ ∂2 + ∂r ∇2 = 1r ∂r 2 . Then equation(1) becomes 1 iψz + ψrr + ψr + V ψ + β|ψ|2 ψ = 0. r
(2)
A PT symmetric potential can be implemented through the the judicious inclusion of lumped amplification, Vr (r), which is associated with index guiding and a loss/gain distribution term, Vi (r). Then the beam dynamics in a PT symmetric graded index Kerr media is governed by 1 iψz + ψrr + ψr + Vr (r) + iVi (r) ψ + β|ψ|2 ψ = 0. (3) r 3
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PT symmetry demands that Vr (−r) = Vr (r) and Vi (−r) = −Vi (r). The complex PT symmetric potential is chosen in the form V (r) = V2r r2 + iVi r, where Vr and Vi are the coefficients of real and imaginary potential. Stationary solution of equation (3) can be of the form ψ(r, z) = φ(r)e−iµz [28], where φ(r) is the nonlinear eigen mode which is a complex function of r and µ is the corresponding propagation constant. The nonlinear eigen value equation satisfied by φ(r) is given by 1 φrr + φr + Vr r2 + iVi r φ + β|φ|2 φ = −µφ. (4) r The differential equation (4) has been studied to analyze the beam intensity in different imaginary potentials. The modulus, real and imaginary components of the stationary solution are shown in figure 1. Figure shows that the imaginary part of the solution starts to evolve when the coefficient of imaginary potential exceeds a critical threshold, which is referred as the phase transition or PT symmetry breaking. At µ = 0.5, the imaginary component of the solution is negligible when Vi < 0.6 (as shown in the upper row of figure. Figure 1(a)-1(c)). When Vi ≥ 0.6, the solutions possess real and imaginary components. Similarly, at µ = 1.5, the solution is complex when Vi ≥ 1.2 (lower row of figure 1). The phase transition point increases with the propagation constant. The variation of Vi with µ at constant Vr (Vr = 1) is shown in figure 2.
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3. Beam dynamics in PT symmetric potential The semi-analytical results of equation (3) are given by the variational analysis [25, 28],[29]-[36] for the solution of the form, ψ(r, z) = φ(r, z)exp(−iµz),
(5)
where φ(r, z) is the nonlinear eigen mode which is also a complex function and µ is the corresponding propagation constant. The eigenvalue equations satisfied by these solutions are: 1 (6) iφz + φrr + φr + Vr (r) + iVi (r) φ + β|φ|2 φ = −µφ. r The variational results are obtained from the Lagrangian density which generates equation (6). Since the system is a dissipative one, the total Lagrangian density consists of a conservative part and a non-conservative part [25]-[31]. L(φ, φ∗ , z, r, φz , φr , φ∗z , φ∗r ) = LC + LN C .
(7)
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(e) Vr = 1, Vi = 0.6
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(b) Vr=1, Vi=0.3
(c) Vr = 1, Vi = 0.6
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(f) Vr = 1, Vi = 1.2
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Figure 1: The real (dashdotted curve), imaginary components (solid red curve), and |φ(r)| (dotted blue curve) of the stationary solution at µ = 0.5 (upper row) and at µ = 1.5 (lower row) with V (r) = 1 and β = 1.
Figure 2: The phase transition point versus propagation constant, µ
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Then the conservative Lagrangian density is r βr ir ∗ (φz φ − φ∗ φz ) + |φr |2 − Vr r3 |φ|2 − |φ|2 |φ∗ |2 − µr|φ|2 . 2 2 2
(8)
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LC =
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The Euler-Lagrange equations describing the dynamics of the system is given by, ∂ ∂LC ∂ ∂LC ∂LC + − = Qi . (9) ∗ ∗ ∂z ∂φiz ∂r ∂φir ∂φ∗i
The function Qi consists of all dissipative processes described by the corresponding terms in the Lagrangian and is given by ∂ ∂LN C ∂ ∂LN C ∂LN C − − . ∗ ∂φi ∂z ∂φiz ∂r ∂φ∗ir
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Qi =
(10)
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From equation (6), Qi = −iVi rφ(r, z). The reduced Lagrangian for the conservative system is Z ∞ < LC >= CD LC dr, (11)
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where CD = 2π in the two dimensional case [31, 32]. The variational ansatz is chosen in the form −r2 b(z)r2 +i + iθ(z) , (12) φ(r, z) = A(z)exp a(z)2 2 where A(z) corresponds to the field amplitude, a(z) is the width of the soliton, b(z) is the wavefront curvature, and θ(z) is the amplitude of the phase profile. Substituting for the variational ansatz in equation (8), the reduced Lagrangian density corresponding to the conservative system is π 2 4 π π π A a bz +A2 a2 πφz −πµA2 a2 + A2 + A2 b2 a4 − βA4 a2 −Vr A2 a4 2 2 2 4 (13) The standard variational approach for the two dimensional dissipative system is given by Z ∞ ∂ ∂ < LC > ∂ < LC > ∂φ∗ − = 2Re Qi rdr, (14) ∂z ∂(ηiz ) ∂ηi ∂ηi −∞ < LC >=
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where η corresponds to all parameters free to vary in the variational ansatz (η = A, a, b and θ). The variational equations for the soliton parameters are: √ dA Vi aA π = − Ab, (15) dz 4 √ da −Vi a2 π = + ab, (16) dz 2 db 1 1 βA2 = 2 2µ(1 − a) − 2a2 b2 + 2 − − 2Vr a2 − b2 , (17) dz a a 2 dφ 1 3βA2 = µ(a − 2) − 2 + a2 b2 + . (18) dz a 4 Variational analysis gives that the total optical power is not conserved and evolves as Z ∞ dP (z) = −2 rVi (r)|ψ(r, z)|2 dr, (19) dz ∞
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where the power P = A2 a2 π, which is a constant in real potential (Vi = 0). The coupled differential equations (15)-(18) have been studied numerically and the variation of amplitude and width of the beam with propagation distance in different imaginary potentials are shown in figure 3(a)-3(d). Figures 3(a), 3(b) and 3(c) show that the beam amplitude increases and width decreases with propagation distance when the coefficient of imaginary potential, Vi < 2.1, which results in the beam propagation with constant power. When Vi ≥ 2.1, the amplitude and width of the beam increases, resulting in the rise in power and hence in the unstable beam propagation. The variation of power with propagation distance in different imaginary potentials are shown in figure (5.a). Figure shows that the power variation is small when the coefficient of imaginary potential is small. When Vi > 2.1, power increases faster and results in the unstable beam propagation. The partial differential equation (6) has been studied numerically using finite difference method. The intensity evolution of the soliton beam at µ = 0.5 in different imaginary potentials are shown in figure 4. The figure shows |φ(r)|, real and imaginary components of the solution. The imaginary part of the solution is negligible when Vi < 1 as shown in figure 4(a) and 4(b). At Vi = 1, imaginary components start to evolve as shown in figure 4(c) and the solution becomes complex. The magnitude of the imaginary component increases with Vi and it is shown in figure 4(d). The power of the beam in different imaginary potentials is shown in figure 5(b). Figure shows that 7
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(b) Vr = 1, Vi = 2
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(a) Vr = 1, Vi = 1
(c) Vr = 1, Vi = 2.1
(d) Vr = 1, Vi = 2.3
Figure 3: The amplitude and width of the beam in different imaginary potentials, plotted for V (r) = 1, µ = 0.5 and β = 1 (Result from the variational analysis).
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(b) Vr = 1, Vi = 0.5
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Figure 4: The intensity evolution of soliton solutions in different imaginary potentials, plotted for V (r) = 1, µ = 0.5 and β = 1 (Result from the numerical analysis)
the power variation is small when Vi = 0.1 and 0.5. When the coefficient of imaginary potential exceeds the value Vi = 1, the beam power rises faster and results in the unstable beam propagation. This is in agreement with the result from variational analysis. 4. Linear stability analysis The stability of steady state solutions of the NLSE is analyzed by considering small perturbations to the stationary solitonic solutions in the form ψj = [φj + a(r, z)] exp (−iµz) with a(r, z) = c(r) exp (iλz) + d(r) exp (−iλ∗ z). Assuming a(r, z) is very small, the NLSE in Eq. (1) is linearized. This results in a set of homogeneous equations satisfied by c and d, which is represented 9
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(a) Vr = 1, Vi = 0.1, 0.3, 0.6 (Variational (b) Vr = 1, Vi = 0.1, 0.3, 0.6 (Numerical result) result)
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Figure 5: Power verses propagation distance in different imaginary potentials.
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in matrix form as follows: c c L + iVi r φ2 λ = φ2 −L + iVi r d∗ d∗
(20)
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where L = ∇2 + Vr r2 /2 + µ + 2φ2 . This set of equations has a nontrivial solution only when the 2 × 2 determinant formed by the coefficient matrix vanishes [25, 28, 37]. The linear stability of a soliton is decided by the nature of the eigenvalues. If any eigenvalue has an imaginary part, the perturbed solution would grow exponentially with z, and the corresponding solitons become linearly unstable. The solutions are completely stable, if all imaginary eigen vales are equal to zero. i.e., when the system possesses solely real eigenvalues. Numerical calculations give the real and imaginary eigen values of the two dimensional PT symmetric system in different imaginary potentials and they are shown in figure (6). Figure shows that the imaginary eigen values are zero when Vi = 0.1 and Vi = 0.4, as shown in figure (6.a) and (6.b). The magnitudes of real and imaginary eigen values are comparable when Vi ≥ 0.5 as shown in figure (6.c) and (6.d), which shows that the eigen value spectrum is complex above Vi = 0.5 due to PT symmetry breaking and hence the corresponding solitonic solutions are linearly unstable. 5. Conclusion This work analyzes the dynamics of (2+1)D spatial solitons in PT symmetric potential. The stationary solutions of the system have been studied 10
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(b) Vr = 1, Vi = 0.4
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(a) Vr = 1, Vi = 0.1
(c) Vr = 1, Vi = 0.5
(d) Vr = 1, Vi = 0.8
Figure 6: The real and imaginary eigen values versus µ in different imaginary potentials with Vr = 1, β = 1.
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numerically. The stationary solutions are real when the coefficient of imaginary potential, Vi < 0.6. Above this transition point, the imaginary solutions are present and the solution is complex. The transition point increases with propagation constant. The beam dynamics has been analyzed using variational and numerical methods. Variational analysis gives the variation of width and amplitude with propagation distance. When Vi > 2.1, the width and amplitude of the beam increases and hence the power of the soliton also increases. This results in unstable soliton propagation. Numerical analysis shows that the soliton beam propagation is stable when Vi < 1. When Vi ≥ 1, the imaginary component of the solution starts to evolve and the solution becomes unstable. The stability of the stationary solution against small perturbation has been studied using linear stability analysis. The imaginary eigen value is zero when the coefficient of imaginary potential is small. When Vi ≥ 0.5 the imaginary eigen value evolves and it is comparable to the real eigen value and hence the solution becomes linearly unstable. Acknowledgments
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References
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KAM would like to thank UGC for providing financial assistance through MANF scheme (grant number F1-17.1/2012-13/MANF-2012-13-MUS-KER/15657) for doing the research work.
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S0030-4026(16)30471-5 http://dx.doi.org/doi:10.1016/j.ijleo.2016.05.038 IJLEO 57664
To appear in: Received date: Accepted date:
29-2-2016 10-5-2016
Please cite this article as: K Aysha Muhsina, P.A. Subha, Two Dimensional Spatial Solitons in Parity- Time Symmetric Potential, (2016), http://dx.doi.org/10.1016/j.ijleo.2016.05.038 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
K Aysha Muhsina1 and P. A. Subha2 1
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Department of physics, University of Calicut, Kerala, India. Department of physics, Farook College, University of Calicut, Kerala, India. email: 1 [email protected], 2 [email protected]
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Two Dimensional Spatial Solitons in Parity- Time Symmetric Potential
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Abstract
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This work analyzes the propagation of (2+1) dimensional spatial solitons in Parity-Time (PT ) symmetric potential. The stationary solution of the system has been studied. The beam dynamics has been analyzed using variational and numerical methods. The soliton beam propagation is stable when the coefficient of imaginary potential is less than a threshold, which is called the phase transition point. Above the transition point, the imaginary component of the solution starts to evolve and the solution becomes unstable. When the coefficient of imaginary potential exceeds this critical value, the power of the beam increases and results in the unstable beam propagation. The stability of the stationary solution against small perturbation has been studied using linear stability analysis. The imaginary eigen value is zero when the coefficient of the imaginary potential is low. Above the phase transition point, the imaginary eigen value becomes comparable with the real eigen value and hence the solution becomes linearly unstable. Keywords: Two dimensional spatial solitons, Non-Hermitian Hamiltonian with real eigen values 1. Introduction
The study of dissipative systems with non Hermitian Hamiltonians have attracted a lot of attention in recent years. Such non Hermitian operators posses real spectra, provided that they obey Parity- Time (PT ) symmetry. This concept was introduced by Bender and Boettcher in 1998 [1]-[3]. According to their view, those non- Hermitian Hamiltonian share common set of Preprint submitted to Optik
May 13, 2016
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eigen functions with the PT operator posses real spectra. The PT operations can be stated as follows[4]-[7]. Under the action of parity operator, pˆ → −ˆ p, xˆ → −ˆ x (ˆ p and xˆ are the momentum and position operators respectively) and the action of time reversal operator results pˆ → −ˆ p, xˆ → xˆ, i → −i. The ˆ = pˆ2 +V (x), where m is the mass and Hamiltonian of a physical system is H 2m ˆ = pˆ2 +V ∗ (x). V is the complex potential. Under time reversal operation, TˆH 2m ˆ = pˆ2 +V ∗ (−x) and H ˆ Pˆ Tˆ = pˆ2 +V (x). For After the parity operation Pˆ TˆH 2m 2m ˆ =H ˆ Pˆ Tˆ. That is possible only the Hamiltonian to be PT symmetric, Pˆ TˆH if the condition V (x) = V ∗ (−x) is satisfied [2]. In order to satisfy the above condition, the real part of the complex potential must be an even (symmetric) function of position, where as the imaginary part should be an odd (antisymmetric) function. The dissipative systems, which include exactly balanced linear gain and loss, are described by non-Hermitian Hamiltonians, whose Hermitian and non-Hermitian parts are spatially even and odd, respectively. It was also demonstrated that when the non-Hermitian Hamiltonians are PT symmetric, they undergo a phase transition (PT symmetry breaking) above a critical threshold, above which the eigen value spectrum becomes partially complex [8]. This concept of PT symmetry of a non-Hermitian Hamiltonian generalizes quantum mechanics in a complex domain [5, 9]. PT symmetry finds applications in various areas of physics ranging from PT symmetric quantum oscillators to linear and nonlinear optics [1, 4]. PT symmetric structures have been realized in optics by Christodoulides etal [6, 10]. Such systems can be realized through the inclusion of gain/loss regions in guided wave geometries. In the optical systems the complex refractive index distribution, n(x) = nr (x) + ini (x) plays the role of the optical potential [10]-[13]. The PT symmetry condition implies that the index wave guiding profile nr (x) is an even function in the transverse direction while the loss or gain term ni (x) is an odd function. PT symmetric optical systems have been experimentally demonstrated in AlGaAs [14], photorefractive materials [6], silicon, fiber optics, and light-written guides in glass [15]-[17] etc. The observation of PT symmetry in linear optical systems led to its generalization to the nonlinear case, which provides several interesting predictions. It was also demonstrated that PT symmetric nonlinear optical systems can support soliton solutions [12, 18]. Recently, soliton beam propagation in PT symmetric optical media has been a subject of intense research because the beam dynamics in such systems can exhibit unique characteristics such as double refraction, power oscillations, nonreciprocal diffraction patterns, etc. 2
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[19],[20]. The stability of the optical solitons in nonlinear PT symmetric systems is investigated in a number of complex potentials like PT symmetric periodic potential, hyperbolic Scarf [21]-[23] potential etc. Studies of PT symmetry have also been extended from nonlinear lattice to double channel coupled wave guides [24]-[26], double-well potentials and asymmetric optical amplifier [27]. Spatial solitons in self focusing and defocusing Kerr media and nonlocal media with PT symmetric potentials have also been investigated [28],[30]. This work analyzes the soliton beam propagation in a PT symmetric system, which is characterized by the nonlinear Schr¨odinger equation with a complex potential with competing gain and loss profile. The paper is organized as follows. In section (2), the stationary solutions of the system have been studied in different imaginary potentials. In section (3), the beam dynamics has been analyzed using variational and numerical methods. In section (4), the stability of the stationary solution against small perturbation has been analyzed using linear stability analysis. Section (5) concludes the paper. 2. Stationary Solution
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The beam evolution in a (2+1) D self focusing Kerr media is governed by the normalized nonlinear Schr¨odinger equation[29]- [31].
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iψz + ψxx + ψyy + V ψ + β|ψ|2 ψ = 0.
(1)
The suffixes z, x and y stand for the partial derivatives with respect to z, x and y respectively, V is the complex potential and β = +1 corresponds to the self focusing nonlinearity. For radial symmetry, x2 + y 2 = r2 and ∂ ∂2 + ∂r ∇2 = 1r ∂r 2 . Then equation(1) becomes 1 iψz + ψrr + ψr + V ψ + β|ψ|2 ψ = 0. r
(2)
A PT symmetric potential can be implemented through the the judicious inclusion of lumped amplification, Vr (r), which is associated with index guiding and a loss/gain distribution term, Vi (r). Then the beam dynamics in a PT symmetric graded index Kerr media is governed by 1 iψz + ψrr + ψr + Vr (r) + iVi (r) ψ + β|ψ|2 ψ = 0. (3) r 3
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PT symmetry demands that Vr (−r) = Vr (r) and Vi (−r) = −Vi (r). The complex PT symmetric potential is chosen in the form V (r) = V2r r2 + iVi r, where Vr and Vi are the coefficients of real and imaginary potential. Stationary solution of equation (3) can be of the form ψ(r, z) = φ(r)e−iµz [28], where φ(r) is the nonlinear eigen mode which is a complex function of r and µ is the corresponding propagation constant. The nonlinear eigen value equation satisfied by φ(r) is given by 1 φrr + φr + Vr r2 + iVi r φ + β|φ|2 φ = −µφ. (4) r The differential equation (4) has been studied to analyze the beam intensity in different imaginary potentials. The modulus, real and imaginary components of the stationary solution are shown in figure 1. Figure shows that the imaginary part of the solution starts to evolve when the coefficient of imaginary potential exceeds a critical threshold, which is referred as the phase transition or PT symmetry breaking. At µ = 0.5, the imaginary component of the solution is negligible when Vi < 0.6 (as shown in the upper row of figure. Figure 1(a)-1(c)). When Vi ≥ 0.6, the solutions possess real and imaginary components. Similarly, at µ = 1.5, the solution is complex when Vi ≥ 1.2 (lower row of figure 1). The phase transition point increases with the propagation constant. The variation of Vi with µ at constant Vr (Vr = 1) is shown in figure 2.
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3. Beam dynamics in PT symmetric potential The semi-analytical results of equation (3) are given by the variational analysis [25, 28],[29]-[36] for the solution of the form, ψ(r, z) = φ(r, z)exp(−iµz),
(5)
where φ(r, z) is the nonlinear eigen mode which is also a complex function and µ is the corresponding propagation constant. The eigenvalue equations satisfied by these solutions are: 1 (6) iφz + φrr + φr + Vr (r) + iVi (r) φ + β|φ|2 φ = −µφ. r The variational results are obtained from the Lagrangian density which generates equation (6). Since the system is a dissipative one, the total Lagrangian density consists of a conservative part and a non-conservative part [25]-[31]. L(φ, φ∗ , z, r, φz , φr , φ∗z , φ∗r ) = LC + LN C .
(7)
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(e) Vr = 1, Vi = 0.6
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(c) Vr = 1, Vi = 0.6
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(f) Vr = 1, Vi = 1.2
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Figure 1: The real (dashdotted curve), imaginary components (solid red curve), and |φ(r)| (dotted blue curve) of the stationary solution at µ = 0.5 (upper row) and at µ = 1.5 (lower row) with V (r) = 1 and β = 1.
Figure 2: The phase transition point versus propagation constant, µ
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Then the conservative Lagrangian density is r βr ir ∗ (φz φ − φ∗ φz ) + |φr |2 − Vr r3 |φ|2 − |φ|2 |φ∗ |2 − µr|φ|2 . 2 2 2
(8)
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LC =
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The Euler-Lagrange equations describing the dynamics of the system is given by, ∂ ∂LC ∂ ∂LC ∂LC + − = Qi . (9) ∗ ∗ ∂z ∂φiz ∂r ∂φir ∂φ∗i
The function Qi consists of all dissipative processes described by the corresponding terms in the Lagrangian and is given by ∂ ∂LN C ∂ ∂LN C ∂LN C − − . ∗ ∂φi ∂z ∂φiz ∂r ∂φ∗ir
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(10)
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From equation (6), Qi = −iVi rφ(r, z). The reduced Lagrangian for the conservative system is Z ∞ < LC >= CD LC dr, (11)
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where CD = 2π in the two dimensional case [31, 32]. The variational ansatz is chosen in the form −r2 b(z)r2 +i + iθ(z) , (12) φ(r, z) = A(z)exp a(z)2 2 where A(z) corresponds to the field amplitude, a(z) is the width of the soliton, b(z) is the wavefront curvature, and θ(z) is the amplitude of the phase profile. Substituting for the variational ansatz in equation (8), the reduced Lagrangian density corresponding to the conservative system is π 2 4 π π π A a bz +A2 a2 πφz −πµA2 a2 + A2 + A2 b2 a4 − βA4 a2 −Vr A2 a4 2 2 2 4 (13) The standard variational approach for the two dimensional dissipative system is given by Z ∞ ∂ ∂ < LC > ∂ < LC > ∂φ∗ − = 2Re Qi rdr, (14) ∂z ∂(ηiz ) ∂ηi ∂ηi −∞ < LC >=
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where η corresponds to all parameters free to vary in the variational ansatz (η = A, a, b and θ). The variational equations for the soliton parameters are: √ dA Vi aA π = − Ab, (15) dz 4 √ da −Vi a2 π = + ab, (16) dz 2 db 1 1 βA2 = 2 2µ(1 − a) − 2a2 b2 + 2 − − 2Vr a2 − b2 , (17) dz a a 2 dφ 1 3βA2 = µ(a − 2) − 2 + a2 b2 + . (18) dz a 4 Variational analysis gives that the total optical power is not conserved and evolves as Z ∞ dP (z) = −2 rVi (r)|ψ(r, z)|2 dr, (19) dz ∞
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where the power P = A2 a2 π, which is a constant in real potential (Vi = 0). The coupled differential equations (15)-(18) have been studied numerically and the variation of amplitude and width of the beam with propagation distance in different imaginary potentials are shown in figure 3(a)-3(d). Figures 3(a), 3(b) and 3(c) show that the beam amplitude increases and width decreases with propagation distance when the coefficient of imaginary potential, Vi < 2.1, which results in the beam propagation with constant power. When Vi ≥ 2.1, the amplitude and width of the beam increases, resulting in the rise in power and hence in the unstable beam propagation. The variation of power with propagation distance in different imaginary potentials are shown in figure (5.a). Figure shows that the power variation is small when the coefficient of imaginary potential is small. When Vi > 2.1, power increases faster and results in the unstable beam propagation. The partial differential equation (6) has been studied numerically using finite difference method. The intensity evolution of the soliton beam at µ = 0.5 in different imaginary potentials are shown in figure 4. The figure shows |φ(r)|, real and imaginary components of the solution. The imaginary part of the solution is negligible when Vi < 1 as shown in figure 4(a) and 4(b). At Vi = 1, imaginary components start to evolve as shown in figure 4(c) and the solution becomes complex. The magnitude of the imaginary component increases with Vi and it is shown in figure 4(d). The power of the beam in different imaginary potentials is shown in figure 5(b). Figure shows that 7
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(b) Vr = 1, Vi = 2
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(a) Vr = 1, Vi = 1
(c) Vr = 1, Vi = 2.1
(d) Vr = 1, Vi = 2.3
Figure 3: The amplitude and width of the beam in different imaginary potentials, plotted for V (r) = 1, µ = 0.5 and β = 1 (Result from the variational analysis).
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(b) Vr = 1, Vi = 0.5
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Figure 4: The intensity evolution of soliton solutions in different imaginary potentials, plotted for V (r) = 1, µ = 0.5 and β = 1 (Result from the numerical analysis)
the power variation is small when Vi = 0.1 and 0.5. When the coefficient of imaginary potential exceeds the value Vi = 1, the beam power rises faster and results in the unstable beam propagation. This is in agreement with the result from variational analysis. 4. Linear stability analysis The stability of steady state solutions of the NLSE is analyzed by considering small perturbations to the stationary solitonic solutions in the form ψj = [φj + a(r, z)] exp (−iµz) with a(r, z) = c(r) exp (iλz) + d(r) exp (−iλ∗ z). Assuming a(r, z) is very small, the NLSE in Eq. (1) is linearized. This results in a set of homogeneous equations satisfied by c and d, which is represented 9
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(a) Vr = 1, Vi = 0.1, 0.3, 0.6 (Variational (b) Vr = 1, Vi = 0.1, 0.3, 0.6 (Numerical result) result)
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Figure 5: Power verses propagation distance in different imaginary potentials.
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in matrix form as follows: c c L + iVi r φ2 λ = φ2 −L + iVi r d∗ d∗
(20)
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where L = ∇2 + Vr r2 /2 + µ + 2φ2 . This set of equations has a nontrivial solution only when the 2 × 2 determinant formed by the coefficient matrix vanishes [25, 28, 37]. The linear stability of a soliton is decided by the nature of the eigenvalues. If any eigenvalue has an imaginary part, the perturbed solution would grow exponentially with z, and the corresponding solitons become linearly unstable. The solutions are completely stable, if all imaginary eigen vales are equal to zero. i.e., when the system possesses solely real eigenvalues. Numerical calculations give the real and imaginary eigen values of the two dimensional PT symmetric system in different imaginary potentials and they are shown in figure (6). Figure shows that the imaginary eigen values are zero when Vi = 0.1 and Vi = 0.4, as shown in figure (6.a) and (6.b). The magnitudes of real and imaginary eigen values are comparable when Vi ≥ 0.5 as shown in figure (6.c) and (6.d), which shows that the eigen value spectrum is complex above Vi = 0.5 due to PT symmetry breaking and hence the corresponding solitonic solutions are linearly unstable. 5. Conclusion This work analyzes the dynamics of (2+1)D spatial solitons in PT symmetric potential. The stationary solutions of the system have been studied 10
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(b) Vr = 1, Vi = 0.4
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(a) Vr = 1, Vi = 0.1
(c) Vr = 1, Vi = 0.5
(d) Vr = 1, Vi = 0.8
Figure 6: The real and imaginary eigen values versus µ in different imaginary potentials with Vr = 1, β = 1.
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numerically. The stationary solutions are real when the coefficient of imaginary potential, Vi < 0.6. Above this transition point, the imaginary solutions are present and the solution is complex. The transition point increases with propagation constant. The beam dynamics has been analyzed using variational and numerical methods. Variational analysis gives the variation of width and amplitude with propagation distance. When Vi > 2.1, the width and amplitude of the beam increases and hence the power of the soliton also increases. This results in unstable soliton propagation. Numerical analysis shows that the soliton beam propagation is stable when Vi < 1. When Vi ≥ 1, the imaginary component of the solution starts to evolve and the solution becomes unstable. The stability of the stationary solution against small perturbation has been studied using linear stability analysis. The imaginary eigen value is zero when the coefficient of imaginary potential is small. When Vi ≥ 0.5 the imaginary eigen value evolves and it is comparable to the real eigen value and hence the solution becomes linearly unstable. Acknowledgments
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References
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KAM would like to thank UGC for providing financial assistance through MANF scheme (grant number F1-17.1/2012-13/MANF-2012-13-MUS-KER/15657) for doing the research work.
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