Study of optical bistability in a fully optimized laser Fabry-Perot system

The analytical study of optical bistability is concerned in a fullyoptimized laser Fabry-Perot system. The related phenomena ofswitching dynamics and optimization procedure are also included.From the steady state of optical bistability equation can plot theincident intensity versus the round trip phase shift (φ) for differentvalues of dark mistuning 12,6,3,1.50 , o    or finesse (F= 1, 5, 20,100). In order to obtain different optical bistable loops. The inputoutputcharacteristic for a nonlinear Fabry-Perot etalon of a differentvalues of finesse (F) and using different initial detuning (φ0) are usedin this research. When the cavity finesse values increase, the switch-ON intensity increases for the same value of (φ0), because thebistable loop width increases with increasing the cavity finessevalues.

The analytical study of optical bistability is concerned in a fully optimized laser Fabry-Perot system. The related phenomena of switching dynamics and optimization procedure are also included. From the steady state of optical bistability equation can plot the incident intensity versus the round trip phase shift (φ) for different values of dark mistuning or finesse (F= 1, 5, 20, 100). In order to obtain different optical bistable loops. The inputoutput characteristic for a nonlinear Fabry-Perot etalon of a different values of finesse (F) and using different initial detuning (φ 0 ) are used in this research. When the cavity finesse values increase, the switch-ON intensity increases for the same value of (φ 0 ), because the bistable loop width increases with increasing the cavity finesse values.

Introduction
In recent years considerable attention has been devoted to the perspectives of optical digital computers. A major factor and incentive for the rapid developments in this area has been the possibility of using light beams for parallel processing and the expectation of achieving a significant increase in computing speed as compared with conventional electronic computers. In the near future the development of simple digital optical signal processor is expected for applications in telecommunication.
The rapid processing of large amounts of data is of particular importance in this field, e.g. for high resolution television and for optical information transmission. Analog optical devices are already being used for similar applications such as image processing and signal filtering. Universal digital optical computers superior in performance to electronic systems are expected at later stages of future developments. New computer architectures, such as neutral networks structured in analogy to the living brain, require a very large number of interconnections. Optical beam offers a solution to this problem.
An important prerequisite for the technical realization of digital optical processors is the development suitable switching elements, which have also been dubbed "optical transistor". In electronic transistor switches an electrical output signal is controlled by a likewise electrical input signal. In analogy to this, in a photonic switch, a light wave is controlled by another light wave whose phase, amplitude and polarization are modulated.
In conventional transistor the term "electronic" means that the input signals regulate the output by means of electrons in the switching elements. This is also the case for photonic switches, which, also in this connection, can be regarded as an extension of electronic elements to optical frequencies [1].
Dawes et al. [2] report an all optical switch that operates at low light level, and consists of laser beams counter propagating through a warm rubidium vapor that induces an off axis optical pattern. A switching laser beam causes this pattern to rotate even when the power in the switching beam is much lower than the power in the pattern. The observed switching energy density is very low, suggesting that the switch might operate at the single photon level. This approach opens the possibility of realizing a single photon switch for quantum information network.
In narrow bandgap semiconductors, the discovery of giant third order optical susceptibility [3] and optical bistability in InSb [4] and GaAs [5] opened up many possibilities for all optical data manipulation such as signal processing and optical computing [6,7]. In all these processes the bistable switching mode can be considered as the most important optical function.
Switching experiments in GaAs were first demonstrated by   [8]. In this investigation, the GaAs etalon was cooled to a temperature of ~10 K (liquid helium temperature) and obtained switch-ON time was ~1ns using an external pulse (200 Ps pulse duration, 590nm wavelength and Energy = 0.6nJ). Other interesting results were achieved by Tang et al. (1982) [9] on a GaAs sample held at 80 K (liquid Nitrogen temperature) using pulse (10Ps pulse duration, 600 nm wavelength and Energy = 1nJ) for which the observed switch -ON time was ~200Ps limited by the detector time resolution. For an InSb etalon, bistable switching was achieved [10] using a pulse from a Nd:YAG mode locked laser. The work of this field has been extended to cover different kinds of semiconductors such as CuCl. For this etalon the switch-ON of ~250Ps and switch-OFF of ~450Ps were measured [11] and it was shown that the switch time depends on the absorption.
The optical bistability (O.B) in photonic multilayers doped by grapheme sheets are studied by Dong Zhao et al in 2017, stacking two Bragg reflectors with a defect layer between the reflectors. O.B stems from the nonlinear effect of grapheme, so the local field of defect mode (D.M) could enhance the nonlinearity and reduce the thresholds of bistability achieves the tenability of bistability due to that the D.M frequency and transmittance could be modulated by the chemical potential. Bistability thresholds and interval of the two stable states could be remarkable reduced by decreasing the chemical potential. The study suggests that the tunable bistability of the structure could be used for alloptical switches in optical communication systems [12].
The dynamical hysteresis of a semiconductors microcavity as a function of the sweep time illustrated by Rodriguezet et al. in 2017. The hysteresis area exhibits a double power law decay due to the influence of fluctuations, which trigger switching between metastable states. Upon increasing the average photon number and approaching the thermodynamics limit, the double power law evolves into a single power law. This algebraic behavior characterizes a dissipative phase transition [13].
Jeremy Butet and Olivier J.F.Martin introduced the influence of Fano resonances on the nonlinear response of hybrid plasmonic nanostructures, i.e., nanoantennas loaded with a nonlinear optical material, is theoretically investigated using the combination of a surface integral equation method and an analytical model [14].
It has been established that the switch-ON and OFF in optically bistable devices will occur by superimposing an external pulse on a continuous wave holding the system close to one of the critical points of the bistability curve. The role of the external pulse in this process is to generate more carrier in order to change the refractive index and thereby change the optical path length of the beam. However, if the difference between the critical power and the holding power is small compared with the power change involved by the switching pulse, the switching is ruled by a pulse energy scaling law [15]. This characteristic is pointed out in a numerical study of absorptive bistability in a good cavity limit (where the cavity lifetime is more than the material life time, τ c >>T 1 ) and has been analytically proved by Mandel [10] by a asymptotic analysis. The switch-ON of intrinsic optical bistable devices with external pulse was demonstrated [16,17] and to get the switch-OFF time, the input intensity must be reduced below the switch-OFF intensity. Obviously, the switching of the devices in either direction ON and OFF with external pulses and keeping the input intensity constant is very desirable in many applications.
Optical bistability is a phenomenon which arises in the transmission of light by an optical cavity filled with a resonant medium. Precisely, we consider a c-w laser beam which is injected into a resonant cavity tuned or nearly tuned to the incident light. The incident beam is partially transmitted partially reflected and partially scattered by the cavity. When the cavity is empty the transmitted intensity is proportional to the incident intensity, where the proportionality constant depends on the cavity detuning and the finesse of the cavity. On the other hand, when the cavity is filled with material resonant or nearly resonant with the incident field the transmitted intensity is a nonlinear function of the incident intensity. Therefore, the transmitted power and the system behavior can be determined by the factor , where α is the absorption coefficient, L is the sample thickness, and T is the mirror transmissivity. In particular, under suitable conditions, the transmitted intensity varies discontinuously and exhibits a hysteresis cycle. This is socalled optical bistability (OB). This phenomenon was predicted by Szoke and Coworkers in 1969 [18]. Some years later performed a numerical treatment of OB in a Fabry -Perot cavity, which suggested experiment works as illustrated in Fig.1. The main requirements to get such phenomenon are the nonlinearity of the medium and the optical feedback action. Optical bistability can be classified as absorptive or dispersive depending on whether the feedback occurs by way of an intensity dependent absorption or by way of an intensity-dependent refractive index. These experiments have shown that this system is the basis for quite a number of device application as optical memory, optical transistor, clipper, …etc. These crucial results stimulated a very active, theoretical research which bifurcated into two direction. The first channel was mainly interested in the device aspects of the phenomenon and in particular investigated the feasibility of electro-optical system to produce bistability and related phenomena. The second channel studies optical bistability as a fundamental field of the interaction between multiatomic system and radiation. Precisely, it considers the optical bistable system as the passive counterpart of the laser.

Theory of optical bistability
Two requirements are needed to observe O.B.: the nonlinearity of the medium and cavity feedback. The medium may be material with an intensity dependent refractive index, n = n 0 + n 2 I (1) where n 0 is the linear refractive index and n 2 the nonlinear refractive index. The feedback is produced by placing material in a F-P. etalon. Let us now consider the case of linear F-P. (length = D) of refractive index = n 0 and mirror reflectivity R, as shown in Fig.1. When this cavity is illuminated with input intensity I int , the transmission (T) of this cavity is described by as [19]:

Fig.1: Fabry-Perot interferometer of length (D) and mirror reflectivity (R α ), I in and I T are the incident and transmitted intensities respectively.
As the linear refractive index (n 0 ) is changed to a nonlinear (n), Eq. (1) becomes: where I int is the internal intensity inside the cavity. The nonlinear phase shift is described by (5) The general form of the transmission (Airy function) which includes the linear and nonlinear phase shift is: I int related to the internal phase thickness is: where R B is the reflectance of the output mirror of the F-P, and I T = T . I in (8) By using Eqs. (7) and (8) (9) so that scaling: (10) and appropriately scaling I in to be dimensionless, eq.(9) becomes (11) Different bistable loops obtained when the loops plotted between  0 . The transmission of a F-P cavity is also given as

Results of optical bistability
From the steady state optical bistability Eq. (11) one can plot the incident intensity versus the round trip phase shift (φ) for different values of dark mistuning    Table 1 shows the results of each state and gives the values of the switch-ON and switch-OFF intensity for each value of initial detuning and finesse values. Fig. 5(a) shows the relation between the dark mistuning and the switch-ON intensity for different values of cavity finesse. Fig. 5(b) shows the relation between the cavity finesse and the switch-ON intensity for different values of dark mistuning, each of which concludes that as the cavity finesse values increase, the switch-ON intensity increases for the same value of (φ 0 ), because the bistable loop width increases with increasing the cavity finesse values, also the switch-ON intensity decreases as the dark mistuning increases for the same value of finesse. This is also because the bistable loop width decreases as the dark mistuning value increases.

Conclusions
An analytical study for the design of a bistable optical system was made. First, we present a theory of optical bistability for a Fabry-Perot etalon containing a nonlinear refraction material which gives various nonlinear relationship between the input and output intensity (in transmission and reflection mode of operation) starting from differential gain and critical switching to various widths of bistable loops which leads to a different switch ON & OFF points depending on the initial detuning and the cavity finesse values.
The O.B. method is supported by the graphical solution theory in calculating these switching points which shows agreement with the above results.