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In this paper we report for the first time the presence of bistability in an acoustic-optic tunable filter (AOTF) operating with ultrashort (2 ps) optical light pulses. The results for the study of bistability has shown the dependence of the hysteresis curve with the product of the coupling constant (κ) by the length of the device (ξ
_{L}) and the conversion power-coupling constant factor (G). The range of bistability varies significantly with both G and with κξ
_{L} parameters. The variation of κξ
_{L} directly increases the size of the range of bistability hysteresis while the increase in G causes the bistability to occur at low powers. The phenomenon of optical bistability (OB) is the object of increasing interest due to its possibilities for important device applications. A bistable device is a device with a capability to generate two different outputs for a given input and the physical requirements for this are an intensity-dependence refractive index and an optical feedback mechanism.

The physical requirements for optical bistability (OB) are an intensity-dependence refractive index and an optical feedback mechanism [

Since its first discovery in late 1970’s, optical bistability has been found existing in many different optical systems. One of the simplest examples of bistable systems is a Fabry-Perot resonator with the cavity filled with a medium that presents saturable absorption or nonlinear dispersion. It has been established theoretically [

The transmission of information by optical way has known a considerable progress which led the scientists to wonder about the possibilities of creating systems with purely optical memories. Actually due the increasing communications supply, it is necessary fast, reliable and chippers services. In this context it is natural the development of devices capable of processing ultrafast signals, so the importance of all-optical devices. The acousto-optic tunable filter (AOTF) is an example of this. The acoustooptic effect has been successfully used since the early 1980’s in the design and construction of a variety of optical fiber devices such as frequency shifters [

The AOTF is an all solid state electronic dispersive device which is based on the diffraction of light in a crystal [19-22]. Light is diffracted by an acoustic wave because when an acoustic wave propagates in a transparent material, it produces a periodic modulation of the index of refraction (via the elasto-optical effect). This, in turn, will create a moving grating which diffracts portions of an incident light beam. The diffraction process can, therefore, be considered as a transfer of energy and momentum [

In our study we are considering the relative effects to the dispersion β^{(2)} and nonlinearity γ coefficients over propagated pulses in the AOTF. The two-input ultrashort soliton pulses (2 ps) are polarized in the TE and TM modes. The amplitude of TE and TM modes are A_{1} and A_{2} respectively, as we can see in Equations (1) and (2). In this investigation, the input pulses have a hyperbolic form and the initial potency will vary intensity in a way we will describe more precisely afterwards. The temporal full width at halt maximum is Δt_{PULSE} = 2ln (1 +) Δt_{0}.

The coupled differential equations describing the evolution of the slowly varying complex modal field (A_{1} and A_{2}) amplitudes of the pulse envelope in the AOTF [24,25] are:

The Equation (1) is for the TE mode and the Equation (2) is for TM mode. The α parameter is the optical loss, κ_{12} is the linear coupling coefficient and Δβ = β_{1} – β_{2} ± K (β_{1} and β_{2 } are the incident and diffracted light wave vectors components, respectively, along the direction of propagation of the acoustic wave with wave vector K is the momentum mismatch among the TE, TM modes and the acoustic wave). Further γ denotes the coefficient of self phase modulation (SPM), which is proportional to the nonlinear refractive index n_{NL} of the material and β^{(2)} represents the group-velocity dispersion parameter (GVD) of the optical medium. It is important to observer that in our study the sign of β^{(2)} is negative so we have the anomalous propagation regime. In a collinear interaction the momentum mismatch of the modes is proportional to optical birefringence (Δn = n_{1} – n_{2}) of the guide [

where λ_{0} is the pump wavelength, ƒ_{a} is the acoustic frequency and V_{a} is the velocity of sound in the optical medium. When the phase-matching condition (Bragg condition) is satisﬁed (Δβ = 0), one knows the acoustic frequency necessary for exact tuning of the pump wavelength λ_{0}. We will suppose an ideal AOTF normally operating at the condition when |κ_{12}|ξ_{L} = π/2 (ξ_{L} is the acousto-optic interaction length). So the power conversion is 100% (maximum efficiency in the conversion of energy among the coupled modes) when the phasematching condition is satisfied. Consequently, for a collinear interaction, the full bandwidth at half maximum of the AOTF (Δƒ_{aotf}) is inversely proportional to birefringence (Δn) and acoustic-optical interaction length (ξ_{L}) through of relation:

where c is velocity of light in the vacuum.

The manufacturing process of AOTF and the physical consequences of the choice of the material used to its construction can be found elsewhere [26-29]. In our study we are considering a centro-symmetric material, as the germanium crystal (Ge), to avoid the presence of the effects of second-order [χ^{(2)}] nonlinear susceptibility. For this choice, the principal refraction indexes could be found in [

To study OB one need to choose just one polarization in the output of the device and that same polarization will receive the feedback. Here the polarization that is studied is TE. The proposed model for the investigation of the performance of the AOTF as a bistable device possesses the architecture shown in the

device as a classical directional coupler, where the coupled modes are the TE and TM modes of the optical waveguide, and the coupling coefficient is proportional to the acoustic amplitude.

The TE and TM input modes go throughout the AOTF. After this the pulses pass in a polarization beam splitter where a small part of the TE transmitted beam is monitored by a photodiode detector whose output is proportional to the transmitted light intensity. Such beam is then amplified and the signal will pump the RF signal to be used as feedback to feeds the transducer. The proposed feedback can vary the acoustic wave intensity of frequency. It is possible to consider these effects separately. The radiofrequency amplitude, RF, controls the variation of the transmitted light intensity. This is equivalent to vary the product kx_{L} in the optic domain. However, the RF signal controls the acoustic-optic frequency and determines the optic frequency or wavelength.

Considering the proposed feedback, the change in the phase matching will be analyzed and interpreted by the following expression:

Here, is the initial phase matching (without feedback). _{0S} factor is associated to the fact that the change is proportional to the feedback intensity. The G parameter represents how much of the TE feedback will be used by the transducer to modify the acoustic wave inside the device. It is important to notice that this feedback structure will be responsible for the bistable response.

In Equations (1) and (2) the time t = t' – z/υ_{g} is measured in a frame of reference moving with the pulse in the group-velocity (υ_{g}). We have analyzed numerically the ultrashort pulse transmission in the anomalous propagation

regime through the AOTF. We are considering that the full temporal width at half maximum of the input pulses is Δt_{pulse} = 2 ps, corresponding to a full spectral bandwidth at half maximum Δƒ_{pulse} = 0.157 THz. The general form of the initials pulses at the AOTF input is given by:

The power for TE mode will suffer two process; in the first it will vary from 0 W to 30 W, and after this it will vary from 30 W to 0 W. For ultrashort soliton pulses of Δt_{pulse} = 2 ps, one has Δt_{0} = 1.135 ps. The AOTF length used was approximately ξ_{L} = 21.8 mm using the parameterized value Δn = 0.07 for the induced birefringence in the material. Starting this point one calculates the dispersive e nonlinear coefficients β^{(2)} = –0.127 ´ 10^{–27} ps^{2}/mm and γ = 0.098 ´ 10^{–3} (w·mm)^{–}^{1}, respectively, for the numeric study of the proposed model.

The system of coupled NLS Equations (1) and (2) was solved numerically using the 4th order Runge-Kutta method with 1024 temporal grid points taking in account the initial conditions given by Equation (6), in the situation without loss (α = 0). In order to solve the system of coupled NLSE with this method, used only to ordinary differential equations, was necessary replace the differential operator by –ω^{2}, where ω is the frequency in the Fourier domain. Since ω is just a number in Fourier space, the use of the FFT algorithm makes numerical evaluation of the last terms on the right side of (1) e (2) straightforward and relatively fast [_{1} and one phonon at Ω and simultaneous creation of a new (diffracted) photon at a frequency ω_{2} = ω_{1} ± Ω. Thus the converted light between the two modes is shifted in frequency by an amount equal to the sound frequency. Since the sound frequencies of interest are below 10^{10} Hz and those of the incident light are usually above 10^{13} Hz, one has that ω_{2} ≈ ω_{1} = ω = 2πc/λ_{0 }. This simplifies the computational numeric study of the Equations (1) and (2).

In our present configuration of study, we will use in the TM input (see

In _{L} = 1.2 and G = 100. The abscissa axis (I_{i}) represents the input power intensity at the AOTF and the ordinate axis (I_{o}) represents the output power intensity. The trajectory indicated by Up represents TE power increasing and the trajectory indicated by Down represents the case for TE power decreasing. A little analysis could be outlined to describe the phenomenon. For conditions in with the transmitted intensity increases faster than the incident intensity, the nonlinear response of the device can be used for differential gain in a manner similar to transistor amplifiers. In this situation, a small modulation on the incident light wave can be converted to a larger modulation on the transmitted wave.

It is also possible to observe in the same figure the range of the OB for the parameters chosen it is something about 20 - 22.4 W for input power intensity. _{L} values. For kx_{L} = 1.2, representing here by the solid line, the optical bistability range varies something about 6.4 - 7.6 W. Another curve represents the curve for kx_{L} = 1.4 as indicated in the figure. It now possible to compare the effect on the optical bistability range; the range for kx_{L} = 1.4 is something about 6.5 - 12.2 W; almost five times bigger.

_{L }value is sustained and G values varies. For this analysis we chose kx_{L} = 1.2 and the values for G are indicated in the figure. The first thing we could note is that with increasing the G value the critical powers of the hysteresis curve are decreasing. Another important fact to note is that when the increasing of G the hysteresis curve tends to decrease the internal area of the hysteresis

curve. This great difference for example between G = 100 and G = 200 would be very important in practical applications.

When the change occurs in the G parameter, the second term of the right side of Equations (1) and (2) are increasing this time. The coupling is not getting stronger and the energy transmission to another mode is smaller when G increases. The

In this paper we report for the first time the presence of bistability in an acoustic-optic tunable filter (AOTF) operating with ultrashort (2 ps) optical light pulses. The results for the study of bistability has shown the dependence of the hysteresis curve with the product of the coupling constant (κ) by the length of the device (x_{L}) and the conversion power-coupling constant factor (G). The range of bistability varies significantly with both G and with kx_{L} parameters. The variation of kx_{L} directly increases the size of the range of bistability hysteresis while the increase in G causes the bistability to occur at low powers. A bistable device is a device with a capability to generate two different outputs for a given input and the physical requirements for this are an intensity-dependence refractive index and an optical feedback mechanism.

A possible mechanism for this is behavior is a strong dependence of the optical intensity on the index of refraction as defined as a feedback function in the TE mode. The optical bistability has been used in a great range of applications like optical transistor, element, differential amplifier, etc. In this work we vary basically two kinds of parameters. We observe that the kx_{L} and G parameters will control the bistable behavior of the device.

The phenomenon of optical bistability (OB) is the object of increasing interest due to its possibilities for important device applications.

We thank CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), FINEP (Financiadora de Estudos e Projetos), FUNCAP (Fundação Cearense de Amparo a Pesquisa) for the financial support and CENTEC (Instituto Centro de Ensino Tecnológico).