NUMERICAL STUDIES OF OPTICAL BISTABILITY OF ABSORPTIVE TYPE NUMERICAL STUDIES OF OPTICAL BISTABILITY OF ABSORPTIVE TYPE

Optical bistable devices have a large number of applications in all-optical communication systems. They are building blocks of optical logic, memories, switches, light power limiters etc. that are crucial parts of optical network nodes [1–3]. Optical bistability is a phenomenon in which the bistable device exhibits two stable output states under one input state. The optical bistability arises from the energy exchange between nonlinear optical medium and electromagnetic field. The system is said to be bistable when it can assume two output stable states for the same value of an applied input signal over some range of input signal values, Optical bistability requires a nonlinear medium in presence of feedback. When the nonlinearities are optically-induced, and the feedback is also provided in the optical domain, an all-optical device can be fabricated [4]. The phenomenon of absorptive optical bistability was first theoretically predicted in 1969 by Szoke [5]. Third-order nonlinear materials are mainly metal particles, semiconductors including those of quantum confinement and organics [6–8]. Among organic materials, π-conjugated compounds attract attention due to unique electrical and optical properties including large thirdorder optical nonlinearities, low-cost manufacturing and flexibility of properties modifications by chemical synthesis as well as deposition techniques. There has been a prolonged research of all-optical switching in organic devices based on optical bistabilities [9]. Parameters of nonlinear medium should be properly chosen to achieve the optical bistability. For the construction of optical bistable devices of absorptive type, saturable absorbers and positive feedback are necessary. A lot of studies report on nonlinear absorption in a great variety of organic compounds [10, 11]. Therefore, the prediction of their bistable behaviour based on theoretical calculations with real experimental data gathered on specific organics is of interest for designing optically bistable devices. This contribution presents numerical studies of purely absorptive optical bistabilities in saturable absorbers in the Fabry-Perot cavity. Organic dyes with spectral sensitivities such as xanthene, polymethene or coumarin dyes are known for their nonlinear optical properties including saturable absorption. In this paper, we predict and numerically demonstrate the optical bistability in two organic dyes of the xanthenes type dispersed in glass and polymethene dyes dissolved in organic solvents. We simulated their bistable behaviour considering absorption parameters and positive feedback conditions with experimental data from [8, 11].


Introduction
Optical bistable devices have a large number of applications in all-optical communication systems. They are building blocks of optical logic, memories, switches, light power limiters etc. that are crucial parts of optical network nodes [1][2][3]. Optical bistability is a phenomenon in which the bistable device exhibits two stable output states under one input state. The optical bistability arises from the energy exchange between nonlinear optical medium and electromagnetic field. The system is said to be bistable when it can assume two output stable states for the same value of an applied input signal over some range of input signal values, Optical bistability requires a nonlinear medium in presence of feedback. When the nonlinearities are optically-induced, and the feedback is also provided in the optical domain, an all-optical device can be fabricated [4]. The phenomenon of absorptive optical bistability was first theoretically predicted in 1969 by Szoke [5]. Third-order nonlinear materials are mainly metal particles, semiconductors including those of quantum confinement and organics [6][7][8]. Among organic materials, π-conjugated compounds attract attention due to unique electrical and optical properties including large thirdorder optical nonlinearities, low-cost manufacturing and flexibility of properties modifications by chemical synthesis as well as deposition techniques. There has been a prolonged research of all-optical switching in organic devices based on optical bistabilities [9]. Parameters of nonlinear medium should be properly chosen to achieve the optical bistability. For the construction of optical bistable devices of absorptive type, saturable absorbers and positive feedback are necessary. A lot of studies report on nonlinear absorption in a great variety of organic compounds [10,11]. Therefore, the prediction of their bistable behaviour based on theoretical calculations with real experimental data gathered on specific organics is of interest for designing optically bistable devices.
This contribution presents numerical studies of purely absorptive optical bistabilities in saturable absorbers in the Fabry-Perot cavity. Organic dyes with spectral sensitivities such as xanthene, polymethene or coumarin dyes are known for their nonlinear optical properties including saturable absorption. In this paper, we predict and numerically demonstrate the optical bistability in two organic dyes of the xanthenes type dispersed in glass and polymethene dyes dissolved in organic solvents. We simulated their bistable behaviour considering absorption parameters and positive feedback conditions with experimental data from [8,11].

Theory
A simple purely absorptive bistable device can be constructed by placing nonlinearly absorbing medium inside a resonator, e.g. Fabry-Perot cavity (Fig.1 The cavity is formed by two parallel identical flat highly reflective mirrors and demonstrated to provide optical bistability.
The absorption coefficient α is an important wavelength-dependent optical parameter expressing the decrease dI of the light intensity I in the length dz of the absorbing medium Eq. (1) is the well-known differential form of the Lambert-Beer law. It is easy to integrate if α is intensity-independent. However, in a nonlinear absorber, α depends on the light intensity. If the intensity of the input radiation is intense enough, the absorption of the medium decreases the consequence of what is the increase of the medium transmittance up to the saturation. If the saturation does not occur and at high intensities decreases again, the so-called reverse saturable absorption occurs. This effect can be used as optical power limiting [12]. According to the absorptive optical bistability model of Szoke [5], the amplitude of electric field entering the Fabry-Perot cavity at normal incidence, electric fields inside the cavity and the output field ( Fig.1) obey the following equations where l is the absorber thickness, k is the wavenumber, ρ and τ are Fresnel amplitudes of mirror reflectance and transmittance. If e Ϫ2ikl ϭ 1 (the Fabry-Perot resonance condition), the relationship between the electric field inside the Fabry-Perot cavity and the input electric field can be expressed as [5,13] Then for the light intensities (I ϳ E 2 ) the following is valid (7) where mirror transmittance T ϭ τ 2 , and reflectance R ϭ ρ 2 (R ϩ ϩ T ϭ 1, R ϾϾ T). Moreover, if αl ϽϽ 1, the equation (7) can be expresses as (8) If there is a nonlinear absorber in the Fabry-Perot cavity, α ϭ α(I 2 ). From Eq. (5) we can see that in case of αl ϽϽ 1 the output electric field E 3 ϭ τE 2 . Then the output intensity is simply I 3 ϭ TI 2 . Therefore it is sufficient to analyze Eq. (8) to judge if the absorber of the specific absorptive nonlinearity α ϭ α(I 2 ) will exhibit the bistable output intensity I 3 as a function of I 1 .
The simulations of Eq. (8) were performed with real experimental data for nonlinear organic absorbers and the prospects to achieve bistable performance were modelled. The constant spatial profile of the propagating pulses was supposed in our calculations and the optical pulse durations were considered to be larger than the lifetimes of the excited levels. Under this assumption the saturable absorbers act as fast absorbers and the fast relaxation approximation enables to expect their steady state behaviour.

Numerical Simulation
We numerically investigated the optical bistability in organic dyes of molecules that contain conjugated bonds with delocalized π -electrons that are responsible for large third-order nonlinear susceptibilities [9]: -The xanthene dye of fluorescein dispersed in boric acid glass film (BAG) (with 10 Ϫ4 M concentration) of optical parameters as in [11]. Fluorescein is a bright green fluorophore commonly used in fluorescence measurements. The maximum of the absorption spectrum is at ϳ 460 nm. -The xanthene dye of rhodamine 6G dispersed in boric acid glass film (BAG) (with 10 Ϫ4 M concentration) of optical parameters taken from [11]. Rhodamine 6G is chromophore used often as a tracer or a laser dye. The maximum of the absorption spectrum is at ϳ 515 nm. -Polymethine dyes of cationic type dissolved in ethanol or polyurethane. Polymethine dyes are known as organic colorants and media laser technique. The maximum of the absorption spectrum is at ϳ 1100 nm. The optical parameters for this study were taken from [8].
All investigated organic dyes dispersed in BAG or dissolved in organic solvents have energy level structure of singlet and triplet states typical for organic media (Fig. 2) with the singlet S 0 as the ground state.
After quick excitation to S 1 the molecule may relax to S 0 with the probability A 21 ϭ 1/τ 21 or to T 1 with the probability A 32 ϭ ϭ 1/τ 32 (τ 21 , τ 32 are the relaxation times). The transition from T 1 to S 1 can be neglected. In liquid state, the dye molecules decay very quickly to from T 1 to S 0 . On the contrary, in dye-doped solids,

Fig. 2 Energy level model of singlets (S) and triplets (T) typical for organic dyes under study. The model shows optical excitations (upwardpointing arrows) of probabilities given by Einstein's coefficients B 13 , B 24 . A 31 , A 32 , A 21 are Einstein's coefficients for the relaxations from excited states (downward-pointing arrows)
the T 1 → S 0 decay is on the millisecond scale. Moreover, by adjusting the excitation wavelength only to the transition S 0 → S 1 , the triplet-triplet absorption T 1 → T 2 does not occur.
With these assumptions, the absorption coefficient depends on the light intensity I2 in the cavity as follows (9) In Eq. (9) α 0 is the absorption coefficient at low intensities of light, β is the so-called parameter of nonlinearity. For the energy level model in Fig. 2 the parameter β is [11] (10) where τ 31 ϭ 1/A 31 , σ 13 is the absorption cross-section for S 0 → S 1 , h is the Planck's constant and ν 13 is the frequency corresponding to the S 0 → S 1 excitation. After substituting Eq. (9) into Eq. (8), we have the following transcendental relationship between I 2 and I 1 (11) The product α 0 l of the linear absorption coefficient α 0 and the medium length l is the so-called linear absorbance that is related to the initial linear transmittance T 0 of the sample as T 0 ϭ exp(Ϫα 0 l).
The values of α 0 l and the parameter of nonlinearity β for ten samples of fluorescein, four samples of rhodamine 6G doped BAG [11] and for three solutions of polymethine dye structures, namely cationic [7] used for the calculations are in Table 1.
In the first course of simulations the absorbers were settled in the Fabry-Perot cavity of the mirror reflectance of R ϭ 0.9 and the mirror transmittance T ϭ 1 Ϫ R.

Results and Discussion
The numerical study of the relationship between I 1 and I 2 according to Eq. (11) was performed under typical input powers from [10]. Results show that the optical bistability depends not only on nonlinear parameter β but also on linear absorbance α0l (Fig. 3). For the values of α 0 l Ͼ 1.39 for fluorescein the output intensity I 2 is a two-valued function of I 1 and the hysteresis loops increase. The simulation shows that fluorescein under these conditions would exhibit bistable behaviour even with relatively low input intensities of light. From Fig. 3a) for fluorescein dispersed in BAG we also see that onset of the optical bistability for increasing α 0 l requires greater incident light intensities I 1 .
The similar results are in Fig. 3b) for rhodamine 6G. The optical bistability could be achieved at input intensities Ͼ 20 W/cm 2 at α 0 l Ͼ 1.79. The input intensities necessary for the optical bistability are higher than those for fluorescein which could be explained by lower values of parameter β representing the nonlinearity of rhodamine 6G in comparison with fluorescein.
The optical bistability is evidenced in solutions of polymethine dyes shown in Fig. 3c) and 3d). However, the input intensities necessary for the bistable performance are much higher in comparison with fluorescein and rhodamine 6G because the nonlinearity represented by the parameter β is very small (Table 1).
In the next course of simulations we investigated the influence of the Fabry-Perot mirror reflectance on the bistable behaviour. Several different values of the mirror reflectance from the range of 0.6 Ϭ 0.98 were used (Figs. 4 and 5).
From Fig. 4a we can see that fluorescein requires mirrors reflectance at least 0.85 to exhibit slight bistability in the Fabry-Perot cavity. In Fig. 4b the numerical analysis for rhodamine 6G with the same mirror reflectance is shown. We found out that a potential bistable device based on rhodamine 6G of the above α 0 l should be placed in the resonator of the mirror reflectance R ϭ 0.9 at least. To improve the data visualisation, the numerical results in 3D are illustrated in Fig. 4 as well. The results proved the important role of the positive feedback in an absorptive bistable device.
The bistable behaviour was numerically investigated in solutions of polymethine dyes (Fig. 5). We see that increasing reflectance R facilitates the optical bistability occurrences, although at very high input light intensities.
The absorption lines of organic dyes are often inhomogeneously broadened inside a host material [6].The inhomogeneous broadening of optical transitions originates from the dye -matrix interaction, especially from different positions of molecules in the matrix. This could occur in case of dye molecules dispersion in glass matrix. On that occasion the intensity-dependent absorption coefficient is given by the following modification of Eq. (9)  Due to inhomogeneous absorption line broadening expressed by Eq. (12) the following relationship between I 2 and I 1 as a result of substituting Eq. (12) into Eq. (8) is valid (13) The results of numerical studies of the relationships between I 1 and I 2 according to Eq. (13) are in Fig. 6 for BAG doped by fluorescein and for solutions of polymethine dyes. We found out that the inhomogeneous broadening of the optical transitions results in the decay of possible bistable behaviour. Similar results were obtained for rhodamine 6G. Then either the inhomogeneous broadening should remain small or a thorough choice of parameters of the nonlinear medium with inhomogeneous broadening is neces-

Conclusion
We numerically analyzed the optical bistable behaviour of organic dyes. It is clearly shown that the bistable behaviour is influenced by the linear absorbance, the parameter of medium nonlinearity and by the Fabry-Perot mirror reflectance as a parameter of positive feedback of a optical bistable device. We numerically established linear absorbance thresholds for the optical bistability in fluorescein and rhodamine 6G doped glass and for polymethine dyes dissolved in organic solvents. At specific values of linear absorbance of all organic dyes, we numerically determined the optical bistability threshold of the cavity mirror reflectance. Additionally we found out that in case of inhomogeneous broadening of spectral lines related to the optical excitations and relaxations, the optical bistability is quenched in all investigated organic dyes.