Revista Facultad de Ingeniería ISSN: Universidad de Tarapacá Chile

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1 Revista Facultad de Ingeniería ISSN: Universidad de Tarapacá Chile Zamorano L., Mario; Torres S., Héctor Solution of the Schrödinger Equation for a Nonlinear Kerr Medium with Linear Chiral Activity and Cubic Anisotropy using the Split-Step Fourier Method Revista Facultad de Ingeniería, núm. 8, julio-diciembre,, pp. -9 Universidad de Tarapacá Arica, Chile Available in: How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative

2 REVISTA FACULTAD DE INGENIERIA, U.T.A. (CHILE), VOL 8, SOLUTION OF THE SCHRÖDINGER EQUATION FOR A NONLINEAR KERR MEDIUM WITH LINEAR CHIRAL ACTIVITY AND CUBIC ANISOTROPY USING THE SPLIT-STEP FOURIER METHOD 1 Mario Zamorano L. Héctor Torres S. ABSTRACT The problem of wave propagation in nonlinear Kerr medium with linear optical or microwave activity and cubic anisotropy is considered. The chirality effect is characteried through the Born-Fedorov formalism. The split step method was used as a computational technique, to solve the Schrödinger equation. The analysis of this study has shown that the balance between the non linearly and linear girotropic phenomena results in the existence of spatial polaried solitons with fixed states of polariation. The main theoretical expression, equation 17, shows modifications of the attenuation and nonlinear coefficients compared with the typical coefficents in a nonlinear Schrödinger equation for a non chiral medium. Numerical simulations and results by using the split-step Fourier method are given for fundamental, second and third -order solitons. INTRODUCTION Chiral materials - also known as optically active materials - can be characteried as reciprocal materials with a different response to the right (RCP) and the left (LCP) circularly polaried electromagnetic waves. The reason is that the phase velocities of the RCP and LCP waves differ, which results in rotation of the wave plane through the chiral medium. The macroscopic properties have their origin in the right-handedness or left handedness of the microscopic entities constituting the chiral materials (for instance, right/left-handed molecules and metal micro-helices in the materials that are optically active in the visible light and the microwave region, respectively). Phenomenological studies by Drude [1] indicated that the rotation of the plane of polariation can be predicted from Maxwell s equations, provided the vector P has an additional term proportional to E These considerations lead Born [] to the proposal that D ε( E+ ζ E) withε ε (1 + χ e ) and B µ H. However, a medium following the above relations is non reciprocal and fails to satisfy some boundary conditions, so the Born s proposal for the isotropical chirality was further modified by Fedorov [][4] to D ε( E + ζ E) (1) B µ ( H + ζ H) () Which are symmetric under the time-reversal and duality transformations. The pseudoscalar ζ represents the chirality of the material and it has the unit of a length [5]. In the limit ζ, the constitutive relationship for a standard linear isotropic lossless dielectric with permittivity ε and permeability µ are recovered. Following the Drude-Born -Fedorov approach we can characterie a nonlinear chiral medium at optical and microwave frequencies, through the proposed equations 1 This work has been financed by the project FONDECYT N 11, Chile and projects N s 87-1 and of the Universidad de Tarapacá. Universidad de Tarapacá, Facultad de Ingeniería, Departamento de Electrónica, amorano@uta.cl, htorres@uta.cl.

3 Zamorano, M., Torres, H. Solution of the Schrödinger equation for a... D ε E+ εζ E () n * The electric field E can be represented by a located wave propagating in the direction, (RCP wave) B µ ( H + ζ H) (4) where the chiral medium has a Kerr type non-linearity characteried by the refraction index such that the total permittivity is n + E [6], where ε ε ε ε is the linear part and ε is the non linear part, respectively of ε. n BASIC PROPAGATION EQUATION Using the above constitutive equations, the corresponding Maxwell s equations are [7] B E t H ( H) E µ µζ (5) D H + σ E E E H εn + εζ + σe (6) Ert (,) ( xˆ + jyˆ) Ψ(,) rte φe jk ( ωt) jk ( ωt) where φ represents the complex envelope. * The conditions of slowly variant envelope must be φ φ jk jωφ φ φφ φ A jω φ jωφφ * The phenomenon of dispersion is included in heuristic form through the relation 1 k v k 1 k 1 k k j j ω ω 6 ω k k ω (1) finally, we obtain the following wave equation (8) (9) and the resulting wave equation is E E + µεζ E µε + E E µε E + µεζ + E E E µεζ E + µζσ + µσ (7) A 1 A 1 A j + + k '' Z vg 1 A ωα ''' + ( 1 ζ ) 6 k j k k j A βω α ζk 1 k A A+ 1 k A ( ζ ) where ζ ( k) (11) If we make the follow considerations [7] 4

4 REVISTA FACULTAD DE INGENIERIA, U.T.A. (CHILE), VOL 8, A kφ, Z 1 ζ k, v ω α µσ, k, β µε v 1 1 v g k' k/ ω, k k '' ω, 1 εµ k ''' k Eq. (11) describes, by example, the propagation of pulses in a chiral dispersive nonlinear optical or microwave fiber. The analysis of each term is as follows [7]: The first term represents the evolution of pulse with distance. The second, third and fourth terms represent the dispersion of the medium. k' 1/ v indicates that g the pulses are moving with the group velocity, while the dispersion of the group velocity (GVD) is represented by k '' and corresponds to the chromatic dispersion, which alters the relative phases of the frequency components of pulses producing its temporal widening. At optical frequencies k '' is null in the region of 1.µ m, For values of λ less than 1.µ m, k '' it is positive (normal dispersion region) and for values higher than 1.µ m, is negative (anomalous dispersion region). The term k ''' represents the slope of the group velocity dispersion, it is also denominated cubic dispersion and corresponds to a higher order dispersion; it is important in ultra short pulses and in the second optical window '' where k is null 1.µ m. The cubic dispersion, besides, is important in optical fiber with shifted dispersion to the region of 1.5µ m. At microwave frequencies like a bioplasma, this parameters must be studied carefully. The fifth term is associated with the attenuation of the fiber (α), in this case those losses are weighed by the chirality of the fiber. The term associated to A ω A represents the nonlinear effect, and it is due to the Kerr effect, which is characteried by having a refraction index depending on the intensity of the applied field. This term also depends on the chirality of the fiber. The last term is highly associated to the chirality of the fiber and may be eliminated by a unitary transformation. In order to ease up the solution of the propagation equation the following changes of variables are introduced: t' t v reference system will be the Eq. (11) takes the form g and * t t' + v, thus the original g and * A 1 A 1 A ωα C j + k'' j k''' + j A ' ' 6 ' k βω C k + (1 ) ( k ) where C 1 kζ A A C and (1) Equation (1) describes the propagation of pulses in a chiral, dispersive and nonlinear optical fiber. A simplified version of this equation is obtained considering that the fiber is working in the third optical windows, is null λ 1.55 µ m, where k ''' A 1 A '' ωαc j + k + j A ' ' k βω C k + (1 ) ( k ) A A C (1) This equation also characteries the solitonic propagation with loss, so it accepts the standard initial condition for this kind of propagation, so [9] where P must be t ' A(, t') Psenh (14) T Nk'' k T βω P (15) with N being the order of soliton and T is the wide of the pulse. 5

5 Zamorano, M., Torres, H. Solution of the Schrödinger equation for a... NUMERICAL TECHNIQUE The propagation equation, Eq. (1), is a nonlinear partial differential equation that does not generally lend itself to analytic solutions except for some specific cases in which the inverse scattering method can be employed. A numerical approach is therefore often necessary for understanding of the nonlinear effects in chiral materials. A large number of numerical methods can be used for this purpose. These can be classified into two categories known as (i) the finite-difference (FD) methods and (ii) the pseudospectral methods. Generally speaking, pseudospectral methods are faster by up to an order of magnitude to achieve the same accuracy. The one that has been used extensively to solve the pulse propagation problem in nonlinear dispersive media is the split -step Fourier method. The relative speed of this method compared with most FD methods can be attributed, in part, to the use of finite-fourier-transform (FFT) algorithm. This section describes the FFT numerical technique used to study the pulsepropagation problem in chiral nonlinear media. SLIT-STEP FOURIER METHOD To understand the philosophy behind the split-step Fourier method, we write the chiral nonlinear spatiotemporal wave equation, Eq. (1), in the following notational form [9] A jqa ˆ (16) here, all linear and nonlinear effects are lumped into the differential operator ˆQ. The solution of Eq. (16) as a function of propagation distance is written A j Q d A ( ) exp ˆ ( ) ( ) (17) where A ( ) is the initial value in space and time of the x-y polaried envelope at. Equation (17) can be examined more closely by writing the operator in terms of linear homogeneous and inhomogeneous parts Qˆ Dˆ + Nˆ. Choosing a small propagation distance, and noting that ˆD can be integrated directly, Eq. (17) reduces to [9] ( ) ( ) A( ) exp j Dˆ + Nˆ ( ' ) d A (18) The integral can be approximated to second-order accuracy in so that the propagation equation becomes ( ) ( ) ( ) ˆ + ˆ ( ) A exp j D N A (19) Even though Eq. (19) is a second-order approximation to the integral of the nonlinear operator, there is no guarantee that the actual implementation is second order accurate, since self-consistent envelopes at are unknown. The approximation used in the split-step method is that these values are determined by a linear propagation step, thus neglecting the nonlinear contribution. The result is that the nonlinear step is nominally first order accurate. The exponential can be rewritten in the well-known symmetried form A( ) exp ˆ exp ˆ j D j N exp ˆ j D A ( ) () which is second-order accurate in and requires three operations for one longitudinal propagation step. This separation into linear and inhomogeneous steps is the main characteristic of the split-step method. It is clear that the linear operator is most naturally applied in the Fourier domain, where time derivates are converted into temporal frequency and space derivates are converted into spatial frequencies. As a result, the linear propagation steps are evaluated in the Fourier domain as follows 6

6 REVISTA FACULTAD DE INGENIERIA, U.T.A. (CHILE), VOL 8, where exp j Dˆ A( ) { exp j { Dˆ } { A( )}} 1 I I I 1 I is the inverse-fourier transform operation. (1) In general, linear and nonlinear parts act together along the length of the material. The split-step method obtains an approximate solution by assuming that in propagation the optical field over a small distance, is carried out in two steps. In the first step, from to +, the linear part acts alone, and ˆN is ero B exp ˆ L j D + A ( ) () In the second step, the nonlinearity acts alone in the point, ( ˆD is null) ˆ D T τ L D where L D k U U τ T k The nonlinear operator is is the dispersion length. ( k ) LNL (6) ˆ βω C C N PU U Γ CP U U U U(7) βω C ( k ) where Γ is the nonlinear coefficient and L NL 1 is the nonlinear length. Γ P ˆ BNL exp j N BL + + () Finally the pulse, given by the Eq. (), is propagated only with linear part to the end of distance. Finally, considering the changes between the time and frequency domains, the propagation over all distance is determined by [9] { { { }}} ( ) I I I I{ ( ) } A R S T A (4) where { ˆ} { ˆ} R exp j I D ; S exp j I N exp { ˆ T j I D} For the numerical calculation, we considered a lossless media, so the Eq. (1) will be U 1 k U βω C j + PU U Z T ( k ) τ which allows us to define the linear operator as (5) ANALYSIS OF RESULTS The Eq. (5) represents the basic modeling of the pulse propagation in a chiral fiber, which is dispersive and nonlinear. The soliton pulses obtained are normalied so they can be at optical or microwave frequencies. At optical frequencies, it is applicable both in the second and third optical windows. For the numerical calculation we use k 17.4 ps / km, Γ, which correspond to the anomalous region for a fiber length equal to.9km. Figs. 1 and correspond to one-order solitons with input power peak P.87W and C.85 and 1.15 respectively. Fig. shows an increase of the intensity when the pulse propagates. ζ k is negative so if the This effect appears when losses are included the chirality factor can compensate the typical decrease of the power pulse of the normal optical fiber. Figs. and 4 correspond to the second order solitons. Here we put P.49W ; this peak power is required to support the second order soliton. If we compare Figs. and 4, we see that with ζ k positive the signal is les s distorted. Finally Figs. 5 and 6 shows the behavior of the third order solitons, with 7

7 Zamorano, M., Torres, H. Solution of the Schrödinger equation for a... P 7.86W. Similar results are obtained at microwave frequencies by scaling the linear and the nonlinear operators (Eqs. 6, 7) respectively. Fig..- Second order solitons Fig. 1.- One order solitons Fig. 4.- Second order solitons Fig..- One order solitons 8

8 REVISTA FACULTAD DE INGENIERIA, U.T.A. (CHILE), VOL 8, Fig. 5.- Third order soliton [5] Torres Silva H.; Chiro-Plasma Surface Waves, Advances in complex Electromagnetic Materials, Kruwer Academic Publishers, Netherlands, 49, [6] Agrawal G.; Non linear fiber optics, Academic Press, [7] Zamorano M. H. et al.; Ecuación de Schrödinger para una fibra óptica quiral, Revista Mexicana de Física, 46(1), 6-66, Feb.. [8] Torres H. et al.; Efecto quiral en la ecuación de Schrödinger para una fibra óptica, Revista de la Facultad de Ingeniería, UTA, vol. 7, 5-4,. [9] Flores R.; Simulación de la propagación solitónica en fibra óptica, Memoria de Titulación Ing. Civil Electrónica, Universidad de Tarapacá, Arica-Chile, Oct.. Fig. 6.- Third order soliton CONCLUSION In this work we have obtained the Schrödinger equation for a media with temporal dispersion and chirality and have nonlinear behavior. The effect of the chirality is shown over the term associate to medium lossy and to the nonlinear coefficient. The most important result in our work it the possibility to use the chirality of the medium to cancel out losses and non linearity of the medium, using the optical fiber as communication channel, would allow to modify radically their behavior. REFERENCES [1] Drude P.; Lehrbuch der Optik, Leipig, 19. [] Born M.; Optik, Heidelberg, 197. [] Fedorov F. I.; On the theory of optical activity in crystals. The law of conservation of energy and the optical activity tensors, Opt. Spectrosc. 6, 49, [4] Fedorov F. I.; On the theory of optical activity in crystals. Crystals of cubic symmetry and plane classes of central symmetry, Opt. Spectrosc. 6, 7,