The Optical Low-Pass Filter Gain of a Ring Cavity

We have developed the equations describing the interaction of light and matter. The MaxwellSchrödinger equations were modified to be The Maxwell-Bloch equations by means of homogeneity and slowly varying wave approximation. The boundary conditions of a unidirectional ring cavity were performed to complete the analytical calculation. From this study, we observed the existence of a low-pass filter gain, parallel with the phenomenon which exists in opamp circuit. Introduction The interaction of light and matter in a ring cavit y in steady state condition has been studied since 1970s [e.g. 1-9] If we inject continuous wave laser to an empty optical system, e.g. a Fabry-Perrot or a ring cavity resonator, we observe output intensity which behaves proportionally to in put intensity. The interesting feature occurs if we put nonlinear media inside such system. The output intensity now becomes nonlinear function of the input intensity, which allows two (or more) stable outputs occur during th e process [14]. This phenomenon is called optical bistability . Study of optical bistability was begun from theoret ical prediction by Szöke, Daneu, Goldhar, and Kurnit [3] . They stated that absorptive bistability could occur on n onlinear medium exposed by laser light. The same problem in different media was considered also by Seidel [4], Austin and Deshazer [5] and Spiller [6]. Some years later, McCall [8] proved that under suitable condition the same o ptical system produce differential gain similar with trans istor principle. He also developed the numerical analysis of absorptive bistability in Fabry-Perrot cavity from MaxwellBloch equation. These results stimulated Gibbs, McC all and Venkatesan [9] to improve their experiment of bista bility phenomena in Sodium, in which both transistor opera tion and bistability were observed. Those results became a corner stone of hundreds of theoretical and experimental a ctivity concerning optical bistability. Now researches are continuously working on bistabil i y in electro-optical devices since such system has great potential to be an optical transistor. The invention of trans istor in 19 century has revolutionized the face of world's tech nology as nowadays. If optical transistor can be embodied, th e optical memory elements, cw to pulsed light converter, clip pers, discriminators, and limiters of light signal will a lso be realized, (see e.g. [10] and [11]). Thus the smarte r and smaller optical computer will someday transforms ou r electronic computer which meets its limited dimensi o . This paper is aimed to study the phenomenon of opti cal bistability in unsteady state condition. The behav ior of light and matter will be described in semiclassical appro ximation. The quantized Maxwell equations are used to describ e the light field, while the Schrödinger equation describ es the matter [e.g. 12]. Under the assumption of intense l a r light, the continuous field remains valid instead of the q uantized field. Solving Maxwell and Schrödinger equations will lead us to the microscopical view of matter. Since we deal wit h macroscopical process, we have to convert the micro scopic properties to macroscopic one. The assumption of st atistical homogeneity will be employed for simplicity. Homoge neity means the electric field affects the same way to po larization in each atom In the dynamical process of light-matter interactio n, spontaneous emission occurs. The consequence of spontaneous emission is the instability of the atom ic level [13,16]. Schrödinger equation merely deals with sta ble atomic level. Since the effect of unstable atomic l eve is significant, Schrödinger equation is no longer vali d. Therefore set of light-matter equations must be mod ified with valid assumptions. It deals with Bloch equatio n.Thus the working equation is the set of Maxwell-Bloch eq uations modified from Maxwell-Schrödinger equation. Mathematical Formulation Equation of Light Let us consider atomic system interacting with inte nse laser. Intense means we do not need to quantize electromag netic field. Electromagnetic field can be assumed continu ous. We use simplified model by reducing only one dimension space dependence and neglecting character of vectorial fi eld [13, 14]. / − / = (1 ) ⁄ / (1) In this equation is macroscopic atomic polarization of medium induced by interaction with the field, = International Journal of Basic & Applied Sciences I JBAS-IJENS Vol: 12 No: 01 64 125701-9494 IJBAS-IJENS © Febr uary 2012 IJENS I J E N S 1/ is the speed of light, and are magnetic permeability and dielectric constant in vacuum, res pectively. Equation of Matter Schrödinger equation well describes the evolution o f matter. But it will bring up a very complicated problem if we employ directly the time-dependent potential to Sch rödinger equation. In order to comprehend the dynamic respon d of the atoms during its interaction with intense laser , we use time dependent perturbation theory which is approac hed from Schrödinger equation (see e.g. [12]). ħ / = , (2) = + ( ), (3) ( ) = (4) where = ( ) is the projection of the electric moment in the direction of the polarization. Assuming the ato ms are two-level and steady-state system, Ψ = + ! " (5) #/ = 0, % = 0,1 (6) in the orthonormal basis | | + |! | = 1 (7) ' #( )* = +)# . (8) Applying Schrödinger equation to each state,