Evolution of bright periodic lattices in negative nonlinear medium

Evolution of bright periodic lattices in negative nonlinear medium.E. AlvaradoMéndez, M. TrejoDurán, J. M. EstudilloAyala, J. A. Andrade Lucio, G. AnzuetoSánchez et al. Citation: AIP Conf. Proc. 992, 530 (2008); doi: 10.1063/1.2926922 View online: http://dx.doi.org/10.1063/1.2926922 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=992&Issue=1 Published by the AIP Publishing LLC. Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

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Evolution of bright periodic lattices in negative nonlinear medium.

E. Alvarado-Mendez"", M. Trejo-Duran'', J. M. Estudillo-Ayala^ J.A. Andrade Lucio^ G. Anzueto-Sanchez^, E. Vargas-Rodriguez^, I. Sukhoivanov^,

S. Chavez-Cerda . ^Facultad de Ingenieria Mecanica, Elecfrica y Electronica.

Universidad de Guanajuato. A.P. 215-A, 36730. Salamanca, Gto., Mexico. Instituto Nacional de Asfrofisica Optica y Elecfronica, Luis Enrique Erro # 1,

Tonantzintla, Pue. Mexico. Abstract

We study numerically the behavior of bright periodic lattices in negative nonlinear medium. We use two-dimensional nonlinear Schrodinger equation, and the solutions are found with split-step method. The critical parameters are the periodicity, the nonlinear constant, and the size of the each spot of the periodic lattice array. We found that bright spots interact with neighbors during the all distance of propagation. The negative refractive index of the medium, produce diffraction in bright spot of the light, in consequence, each spot of the light interact with the four close neighbors and the darks spots are self-focused. The initial period of the array selected, determine the behavior of the dark periodic arrays. We find two regions of interest of the dark periodic lattice pattern: circular dark lattices and linear dark periodic lattices. The evolution of the two-dimensional bright periodic lattice to one-dimensional periodic lattice is due to self-phase modulation of the beam. These numerical results are comparing with preliminary experimental results in nonlinear liquid materials and are a good agreement. For experimental conditions we use Ar laser 514 nm and the power was incremented. The periodic pattern is formed by the interference of four plane waves and the propagation distance of 1 cm.

Introduction Spatial pattern formation has been studied in nonequilibrium fluids systems such as Rayleigh-Benard convection or Taylor Couette flow, and seeks to compare and contrast these with other pattern forming systems encountered in solid state physics, chemistry, biology and nonlinear optics [1]. In particular, nonlinear propagation of light in periodic structures has become an attractive topic of research in recent years, holding sfrong promises for novel photonics applications. The periodic structures more known are photonic crystals: and represent periodic variations of refractive index of dielecfric media, and open the possibility of control of light in a way similar to the way semiconductors are used for manipulation of the flow of elecfrons. Several approaches for control of light propagation by engineered periodic sfructures have been predicted and demonstrated experimentally in recent years. These include manipulation of linear light propagation (refraction, diffraction) as well various nonlinear effect (harmonic generation, stimulated scattering, and nonlinear self-action). For example, the control of refraction and diffraction of light constitute the effects of

CP992, RIAO/OPTILAS 2007, edited by N. U. Wetter and J. Frejlich © 2008 American Institute of Physics 978-0-7354-0511-0/08/$23.00

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negative diffraction [2] and self-collimated [3] respectively. The nonlinear response of the material offers the opportunities for dynamic tenability of the structures sensitive to the light intensity [4]. The interplay between nonlinearity and periodicity represents a unique way to efficiently manipulate light by light for optical switching and signal processing applications. The principal line of research in periodic structures has been in study the propagation of spatial solitons in periodic structures with different nonlinearities of the medium [5]. However, the problem of propagation of bright periodic structures of beams in negative nonlinear medium has not been analyzed. In this work, we study numerically the propagation of bright periodic structures of light in negative refractive index of the medium. Ours preliminary results, show that bright spots of light form a continuum and the dark spots are self-focused due to the dependence with nonlinearity. In consequence, periodic sfructures with different geomefries are formed. Our numerical simulations are compare with experimental results obtained a 514 nm. of Ar laser in nonlinear liquid material. The results are in close agreement with the numerical simulations.

Physical Model

Our physical model is based in interference patterns. We can obtain interference of four plane waves induce a photonic lattice. The figure 1 shows the physical condition. In this system the periodic structures are propagated in nolinear medium type Kerr, in consequence, are governed by 2-D nonlinear Schrodinger equation (NLSE) [6],

dA ^\A\^ A (1)

Where X, Y and Z are normalized as: Z = Z/LD; X = x/xo; Y = y/yo; y = LD / LNL; 9 I 9 9

kouop ; p = '\IXQ +y^ ; LNL = l/|n2koPo|; with p being the width beam; no and ko are, respectively, the linear refractive index and the wavelength number, n2 is the Ken-refractive index of the medium and Po is the initial power of the laser beam. A negative nonlinearity was employed for the numerical simulations. Indeed, using two pairs of plane waves, 2D periodic patterns can be numerically generated, as in Figure 2a, where the initial pattern, when the initial condition is

^7rY^ (TIX A{XJ,Z = 0) = A^s,m cos + —

7y where A = i. shows that the lattice is

formed by dark and bright spots and, as they propagated through the nonlinear medium (with normalized nonlinear coefficient y = 6.7, figure 1, the negative nonlinearity of the medium produces bright diffraction spots; the physical effect of the medium over the lattice is the self phase modulation, which turns the bright spots into bright lines, the dark spots are self-focused and a dark periodic lattice in 2D is formed.

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I I I

I

I

i Figure 1. Photonic bright circular lattice in the input face.

Periodic photonic structures are showed in figure 1. We have in the input face of the negative nonlinear medium bright and dark periodic structures without distance propagation. The gray spots correspond to bright spots of light. Figure 2 show the dark periodic structures with /5 = 6.7, and Z = 1.5; bright spots of light are coupled due to self-phase modulation effect. The nonlinear lattice is diffracted in the bright spots. However, the dark spots are self-focused doe to negative nonlinearity of the medium.

I i

i i

Figure2. Dark photonic lattice after of 1.5 LD of distance of propagation. The nonlinear parameter is J3 =6.7

In the figure 1 and 2, we can observe the propagation of light in periodic optical lattices is determined by two effects: a) the coupling between neighboring sites of the lattice, and b) the specific lattice geometry. The geometry is of particular importance because affect the coupling of the lattice symmetry. In particular, photonic periodic lattice with the initial condition.

A(X, Z = 0) = A, sm ^TTY^

sm V ^ y

7r(X + 0A) 7i' + COS

^TTY^

cos ^7lX^

\ ^ J \ ^ J (2)

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The input face with equation 2, is show in figure 3. We can observe that the geometry is changed. Now elliptical patterns a 135 with respect to the X-axis is observed. The dark elliptical points correspond to intensity maxima.

in Figure 3. Elliptical patterns a 135° is rotated in the input face.

The propagation of photonic periodic lattice in nonlinear medium is simulated with 15 =1.1, and Z = 1.5; bright spots of light are coupled due to self-phase modulation effect. In other words, the phase of the photonic lattice is changed due to intensity of the light. Figure 4 show the photonic periodic lattice in the output face.

Figure 4. Lines periodic patterns with nonlinear coefficient /3 Z = 1.5 of propagation distance.

7.7.

The beams are coupled with the neighboring sites in the same direction of the incline elliptical beams. The evolution of elliptical beams (2+l)D, into the dark lines periodic is due to coupled phase by nonlinearity between the elliptical beams. This case we call evolution of (2+l)D beams into stripes periodic photonic patterns. Our preliminary experimental results were made with nonlinear liquid media, and use Ar laser 514 nm of wave length. The photonic periodic patterns were made by interference of four plane waves. The geometry is controlled by two mirrors as is showed in figure 5.

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Lheftdfle.

•Bra-

IC l -

5B-

r

^J%M^ j i

MoanVaJus:

Std DeviiEicn-

WimbertiJFtela:

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.(100

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Figure 5. Photonic lattice in the input face obtained by interference of four plane waves.

The figure 5 shows the photonic lattice geometry with 60 mW of power. The periodic arrays are displaced due to the refractive index. Bright periodic arrays are formed and the intensity distribution is showed in the left picture, we can see maximum and minimum disfributions are formed.

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HMf lV ik * SidDeiMtwv

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Figure 6. Photonic lattice in the output face with 80 mW of power.

Figure 6 shows the evolution of (2+l)D beams to sfripes photonic lattices due to self-phase modulation action by nonlinearity of the medium. In the right picture a continuum of bright of light is formed. This result is accord with numerical prediction obtained in figures 3 and 4.

Conclusions

In the present work we study numerically the propagation of bright photonic lattice in negative nonlinear medium. Our results show two very interesting cases, if the geometry and intensity of the photonic lattice is selected. Our preliminary experimental results show that in nonlinear liquid media, the propagation of photonic lattices can form stripes lines due to self-phase modulation of the periodic lattice. The power dependence induced can produce dark photonic lattices. These results have applications in the fields of optical communications and computing, beam shaping, and bio-sensing.

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Acknowledgements

This work was partially supported by DINPO (Propagacion de Estmcturas Periodicas Bidimensionales en Medios Nolineales), CONCYTEG 07-16-K662-061 A07, and CONACyT (Estudio de Patrones Faraday de Luz en Medios Nolineales);

References

[1]. M. C. Cross, P. C. Hohenberg, Rev. of Mod. Phys. 65 851 (1993). [2]. J. Wyller, Phys. D, 157, 90 (2000). [3]. L. Gurdev, T. Dreishuh, and D. Stoyanov, JOSA A, 10, 2296 (1993). [4]. A. Yakimenko, Y. Zaliznyak, and Y. Kivshar, Phys. Rev. E, 71, 065603 (2005). [5]. J. W. Fleisher, M. Seguev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 147 (2003). [6]. Eduardo Huerta-Mascotte, ''Numerical simulations in (2+l)D periodic lattices in nonlinear medium'', M.Sc. Thesis, FIMEE, Universidad de Guanajuato (2006).

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