Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Aboites, V.; Barmenkov, Y.; Kiryanov, A.; Wilson, M. Bidimensional dynamic maps in optical resonators Revista Mexicana de Física, vol. 60, núm. 1, enero-junio, 2014, pp. 13-23 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57048160002 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative EDUCATION REVISTA MEXICANA DE FISICA´ 60 (2014) 13–23 JANUARY–JUNE 2014 Bidimensional dynamic maps in optical resonators V. Aboitesa, Y. Barmenkova, A. Kiryanova and M. Wilsonb aCentro de Investigaciones en Optica,´ Loma del Bosque 115, Leon,´ 37150 Mexico.´ e-mail: [email protected] bUniversite´ des Sciences et Technologies de Lille 1, CNRS UMR 8523-UFR de Physique-Batimentˆ P5, 59655 Villeneuve d’Ascq Cedex, France. Received 29 August 2012; accepted 24 February 2014 In this article an introduction to the dynamical behavior of a beam within a ring phase-conjugated optical resonator is presented and modeled using two dimensional iterative maps. Three well known iterative maps are described: Duffing, Tinkerbell and Henon,´ and are applied to the description of optical resonators. It is explicitly shown how the difference equations of these maps can be used to describe the dynamic behavior of what we call Tinkerbell, Duffing and Henon´ beams i.e. beams that behave according to these maps. The matrix of a map generating device are found in terms of the specific map parameters, the state variables and the resonator parameters for each of the three named maps. Keywords: Resonator; chaotic maps; phase conjugation. En este art´ıculo se presenta una introduccion´ al comportamiento dinamico´ de un haz dentro de un resonador optico´ de anillo de conjugacion´ de fase el cual es modelado usando mapas iterativos bidimensionales. Tres bien conocidos mapas iterativos son descritos: Duffing, Tinkerbell y Henon,´ y son utilizados para la descripcion´ de resonadores opticos.´ Se muestra expl´ıcitamente como´ las ecuaciones de diferencia de los mapeos anteriores pueden ser utilizados para describir el comportamiento dinamico´ de lo que llamamos, Haces de Tinkerbell, Duffing y Henon´ i.e. haces que se comportan siguiendo dichos mapas. La matriz de un dispositivo generador del mapeo se encuentra en terminos´ de los parametros´ espec´ıficos del mapa, de las variables de estado y de los parametros´ del resonador para cada uno de los tres mapas anteriores. Descriptores: Resonador; mapeo caotico;´ resonador de anillo. PACS: 42.15.-I; 42.60.Da; 42.65.Hw 1. Introduction tion areas: such as high-brightness laser oscillator/amplifier systems, cavity-less lasing devices, laser target-aiming sys- Recently, optical phase conjugation (OPC) has been an im- tems, aberration correction for coherent-light transmission portant research subject in the field of lasers and nonlin- and reflection through disturbing media, long distance op- ear optics. As it is known, OPC defines a link between tical fiber communications with ultra-high bit-rate, optical two coherent optical beams propagating in opposite direc- phase locking and coupling systems, and novel optical data tions with reversed wave front and identical transverse am- storage and processing systems (see Ref. 1 and references plitude distributions. The unique characteristic of a pair of therein). The power performance of a phase conjugated laser phase-conjugate beams is that the aberration influence im- oscillator can be significantly improved introducing intracav- posed on the forward beam passed through an inhomoge- ity nonlinear elements, e.g. Eichler et al., [2] and O’Connor neous or disturbing medium can be automatically removed et al., [3] showed that a stimulated-Brillouin-scattering (SBS) for the backward beam passed through the same disturbing phase conjugating cell placed inside the resonator of a solid- medium. There are three leading approaches that are effi- state laser reduces its optical coherence length, because each ciently able to generate the backward phase-conjugate beam. axial mode of the phase conjugated oscillator experiences a The first one is based on the degenerate (or partially de- frequency shift at every reflection by the SBS cell resulting generate) four-wave mixing processes (FWM), the second is in a multi-frequency lasing spectrum, that makes the laser in- based on a variety of backward simulated (e.g. Brillouin, sensitive to changing operating conditions such as pulse rep- Raman or Kerr) scattering processes, and the third is based etition frequency, pump energy, etc. This capacity is very on one-photon or multi-photon pumped backward stimulated important for many laser applications including ranging and emission (lasing) processes. Among these different methods, remote sensing. The intracavity cell is also able to compen- there is a common physical mechanism in generating a back- sate optical aberrations from the resonator and from thermal ward phase-conjugate beam, which is the formation of the effects in the active medium, resulting in near diffraction lim- induced holographic grating and the subsequent wave-front ited output [4], and eliminate the need for a conventional restoration via a backward reading beam. In most experi- Q-switch as well, because its intensity-dependent reflectiv- mental studies, certain types of resonance enhancements of ity acts as a passive Q-switch, typically producing a train of induced refractive-index changes are desirable for obtaining nanosecond pulses of diffraction limited beam quality. One higher grating-refraction efficiency. OPC-associated tech- more significant use of OPC is a so-called short hologram, niques can be effectively utilized in many different applica- 14 V. ABOITES, Y. BARMENKOV, A. KIRYANOV AND M. WILSON which does not exhibit in-depth diffraction deformation of and its angle with the same axis as y(z) and θ(z), when the the fine speckle pattern of the recording fields [5]. A thermal ray undergoes a transformation as it travels through an optical hologram in the output mirror was recorded by two speckle system represented by the matrix [A; B; C; D], the resultant waves produced as a result of this recording a ring Nd:YAG values of y and θ are given by [18]: laser [6]. Phase conjugation by SBS represents a fundamen- µ ¶ µ ¶ µ ¶ tally encouraging approach for achieving power scaling of y AB y n+1 = n : (1) solid-state lasers [7,8] and optical fibers [9]. There are several θn+1 CD θn theoretical models to describe OPC in resonators and lasers. One of them is to use the SBS reflection as one of the cavity For any optical system, one may obtain the total mirrors of a laser resonator to form a so-called linear phase [A; B; C; D] matrix, by carrying out the matrix product of conjugate resonator [10], however ring-phase conjugate res- the matrices describing each one of the optical elements in onators are also possible [11]. The theoretical model of an the system. OPC laser in transient operation [12] considers the temporal and spatial dynamic of the input field the Stokes field and the acoustic-wave amplitude in the SBS cell. On the other hand Constant ABCD elements the spatial mode analysis of a laser may be carried out using transfer matrices, also know as ABCD matrices, which are a In passive optical elements, such as lenses, interfaces be- useful mathematical tool when studying the propagation of tween two media, reflections, propagation, and many others, light rays through complex optical systems. They provide a the elements A, B, C, D are constants and the determinant simple way to obtain the final key characteristics (position Det[A; B; C; D] = nn/nn+1, where nn and nn+1 are the re- and angle) of the ray. As an important example we could fraction index before and after the optical element described mention that transfer matrices have been used to study self- by the matrix. Since typically nn and nn+1 are the same, it adaptive laser resonators where the laser oscillator is made holds that Det[A; B; C; D] = 1. out of a plane output coupler and an infinite nonlinear FWM medium in a self-intersecting loop geometry [13]. Non constant ABCD elements In this article we put forward an approach where the in- tracavity element is presented in the context of an iterative Nevertheless, for active or non-linear optical elements the A, map (e.g. Tinkerbell, Duffing and Henon)´ whose state is de- B, C, D matrix elements are not constant but may be func- termined by its previous state. It is shown that the behavior tions of various parameters. The following three examples of a beam within a ring optical resonator may be well de- are worth mentioning. scribed by a particular iterative map and the necessary con- ditions for its occurrence are discussed. In particular, it is shown that the introduction of a specific element within a ring Curved interface with a Kerr electro-optic material phase-conjugated resonator may produce beams described by a Duffing, Tinkerbell or Henon´ map, which we call “Tinker- Due to the electro-optic Kerr effect the refraction index of bell, Duffing or Henon´ beams”. an optical media n is a function of the electric field strength The idea of introducing map generating elements in opti- E [19]. The change of the refraction index is given by cal resonators from a mathematical viewpoint was originally ¢n = ¸KE2, where ¸ is the wavelength and K is the Kerr explored in [14-17] and due to its pedagogical purposes this constant of the media.