A spatio-temporal analysis of non-degenerate two-wave mixing in a saturable absorber, such as bacteriorhodopsin (bR) film, is performed. To do this, a theoretical model describing the temporal variation of the intensities is developed by taking into account the dielectric constant as a function of bR population. A good agreement between theory and experimental measurements is obtained. Thus, the dependence of the optical gain and the main dielectric constant parameters are studied at different intensities and frequencies. As a result, the best intensity-frequency zones where higher coupling is reached are proposed, and it is also demonstrated that non-uniform patterns, which evolve over time as a function of frequency difference, can be observed.
© 2016 Optical Society of America
Due to its ideal properties, such as fast response, high spatial resolution, excellent stability and high quantum efficiency [1–3] and for being one of the most promising photon-functional materials for electronic and photonic applications, the photosensitive protein bacteriorhodopsin contained in the purple membrane of the archea Halobacterium salinarium has been widely studied in recent years. In this sense, the use of bR has been proposed and demonstrated for a variety of technological applications in optics such as data storage [2,4,5], real-time holography [6, 7], optical display and spatial light modulation [8, 9], optical image processing  or slow light .
For applications such as those mentioned above, two-wave mixing (TWM) has been used as a building block for such applications, including optical image processing, optical amplification based on wave mixing, holographic data storage, dissipative holographic solitons and slow light [12–17]. Two-wave mixing is an interesting area in nonlinear optics and in the last three decades it has been studied intensively in many different nonlinear media, such as second-order nonlinear media like photorefractive materials [12,18–20], third-order nonlinear material like Kerr media [12, 21, 22], gain media like semiconductor amplifiers [23–25] and saturable absorbers [26–32]. Moreover in relation to two-wave mixing, phase-modulated holography technique has been used to analyze the photochemistry and photophysics of molecules in the condensed phase [33, 34]. With this method, amplitude and phase gratings generated during the recording process can be monitored separately and simultaneously, obtaining, as a result, information related to the photoinduced matrix effects and direct measurements of photochemical quantum yields [33, 34]. An extension of this method, by introducing a phase modulation of one input beam, is the dynamical-phase-modulated holography technique which, additionally, permits the measurement of relaxation time constants without using pulsed lasers .
The two-wave mixing process is based on the intersection of two coherent beams that are incident on a media forming an interference pattern. This pattern is characterized by a periodic spatial variation of the intensity; thus, due to the linear or nonlinear response of the material, a refractive index and/or absorption periodic variation will be induced. This refractive index and/or absorption variation (modulation of the optical susceptibility) form a periodic volume grating which deflects the incident waves through Bragg scattering . In photorefractive materials, the grating is formed out of phase with the intensity pattern, and one beam is enhanced at the expense of the other [13, 14]. Alternatively, in Kerr media, in which the refractive index is intensity dependent, the grating is in phase with the interference pattern, and no net energy exchange results .
However, an energy transfer can be produced by a frequency shifting in one of the beams, (non-degenerate TWM), resulting in a phase shift between modulation and interference. In saturable absorbers, through nonlinear absorption, both phase and absorption gratings are created, which have been studied in systems such as ruby , Cr3+ : YAlO3-based compositions [36–38] or dye doped solids [27, 30, 39–43]. Theoretically, different alternatives were used for explaining the experimental results obtained. On one hand, these materials were analyzed by two beam coupling in Kerr media in terms of the modulation of the medium refractive index which was assumed real [12, 21, 22] and complex [27, 28, 39, 44, 45]. On the other hand, in order to take into account the saturation intensity (steady-state populations), the two beam coupling was interpreted as self-diffraction by the induced population grating in the direction of the input beams [29,30,41,42,46,47]
The aim of this study is to dynamically analyze non-degenerate TWM in saturable absorbers. To do this, we will perform the dynamical study of non-degenerate Two Wave Mixing in bacteriorhodopsin films at different intensities and frequencies. In contrast to the above-mentioned models, the results obtained will be explained through our study of two beam coupling in terms of the modulation of the complex dielectric constant, taking into account its dynamical behaviour by means of the temporal variation of the populations analyzed by using a two-level model. Due to the change of the refractive index will be proportional to the incident light intensity with a constant nonlinear coefficient, the photoinduced changes in the populations of the isomers in the BR photocycle will imply a nonlinearity.
2. Theoretical procedure
The theoretical description of the non-degenerate two wave mixing process will consist of two parts: one is the corresponding differential equations system that will describe the intensities and phases variations on the propagation direction (z); and the other is the temporal variation of populations which will be related to the refractive index and absorption coefficient variations. Thus, firstly, we will describe the two wave mixing process, taking into account uniform refractive index and absorption changes and the corresponding modulation on the refractive index and absorption coefficient. Then, the corresponding temporal variations of the bateriorhodopsin population will follow, which will be related to the previous non-degenerate wave mixing differential equations.
So, the TWM process is described by the propagation of two beams through a non-linear medium and their interaction with the medium. For this, we consider two copropagating plane waves that interfere in a bacteriorhodopsin film with electric fields given by:48], so due to the principal axes of the anisotropic media match to the polarization state of the incident beams, no anisotropic effects in light propagation will be observed.
In a saturable absorber, such as bacteriorhodopsin, the interference pattern of these two beams will produce a variation in the relative dielectric constant of the material (due to the photoisomerization of the chromophore), which will be spatially shifted with respect to the interference pattern. We will assume that the change in the dielectric constant is proportional to the square of the electric field (Δεr = k|E1 + E2|2). According to these assumptions the permittivity is given by:Eq. (3)), εia is the uniform change in the imaginary part of the relative dielectric constant (defined by Eq. (4)), krm, kim, kr and ki are real proportional constants that will depend on time, ϕp is the phase that indicates the spatial shift between the generated phase grating and the light pattern, ϕa is the phase that indicates the spatial shift between the absorption grating and the light pattern, K⃗ is the grating vector given by the difference in the wave vector of the beams (k⃗2 − k⃗1) with magnitude K = 2π/Λ, where Λ is the period of the grating, and Ω0 is the frequency difference of both beams (ω2 − ω1).
In Eq. (5), α is the material absorption coefficient (related to εi0 by ), λ is the wavelength, θ is the angle inside the material formed by one of the beams and the normal to the surface. Parameters κ0, κ1, γ0 and γ1 are given respectively by:Eqs. (3) and (4)), and nwm(t) and αwm(t) correspond to the modulated refractive index and absorption change during the wave mixing process respectively. Note at this point, that parameters nw(t), nwm(t), αw(t) and αwm(t) will be the connection to the temporal response of the material as we will describe. However, we can express the complex amplitudes A1(z) and A2(z) as function of the wave intensities (I1(z) and I2(z)) and phases (Ψ1(z) and Ψ2(z)) as follows:
It is important to note that, unlike other non-degenerate two wave mixing descriptions [12,25,50], in this study, we have taken into account the uniform and modulated refractive index and absorption coefficient changes. Furthermore, the phase differences between the generated refractive index and the absorption coefficient patterns with respect to the light interference are also considered in this theory. So, once the non-degenerate two wave mixing equations have been described, as previously stated, we will now analyze the temporal response of the bacteriorhodopsin film. To do so, we will follow the methodology previously described in reference  so as to obtain the dynamical behaviour absorption and refractive index response. Thus, the absorption of light by bacteriorhodopsin initiates a photocycle, starting from the B state, which upon illumination is converted into the M state via K and L states, returning to the B state via N and O states [6, 52]. According to previous results, from experimental analysis for the bR film used in this work at 532 nm, an inhomogeneous distribution of the normalized population densities along the light propagation direction is obtained being the population densities of the K, L, N, and O states significant lower than B and M [48, 53]. Thus, as an approximation, due to the short life-time of K, L, N and O states, this process can be theoretically modeled in a qualitative manner [53, 54] as a two-level system by taking into account only B and M states [55,56], resulting in:
Note at this point that, instead of the temporal solution, the previous saturable absorber treatments [21,27,29,30,39,42,44,46,47] have taken into account the stationary state solution of an equation similar to Eq. (13). Thus, as described in photorefractive materials , in order to solve the non-linear differential equation (Eq. (13)), we are going to make the assumption that this equation can be written as:Eq. (13)) can be written according to:
Assuming that νMp(x, t)Ip(x, t), is negligible the differential equation (Eq. (15)) can be linearized, and as a result, the space-independent and space-dependent terms in this equation can be separated as follows:Eqs. (16)–(17)), as in other theoretical methodologies in photorefractive materials [14, 57], a quasi-stationary approximation to the intensity is assumed, ie. I0(t) ≈ I0. Therefore, assuming that Ip(x, t) = Ip exp(−i(Kx + Ω0t + φ0)) and νMp(x, t) = νMp(t) exp(−iKx), (where φ0 is the initial phase difference produced due to the frequency difference of one of the beams), and Ip = 2 (I1 I2)1/2, it can be analytically solved by introducing these expressions in Eq. (17). So, by taking the real part, we finally obtain that νM0(t) and νMp(t) are given by:
So, in order to connect the non-degenerate two wave mixing differential equations (Eqs. (11) and (12)) to the temporal variations of the uniform and modulated normalized concentrations, we will assume that nw(t) nwm(t), αw(t) and αwm(t) are proportional to the corresponding populations, and due to the dependency on the intensity of the populations, they are inversely proportional to the intensity. Therefore, we approximate that the uniform variations of the refractive index and absorption coefficient, (nw(t) (|A1(z)|2 + |A2(z)|2) and αw(t) (|A1(z)|2 + |A2(z)|2)), in Eqs. (6) and (8) are given respectively by:
Thus, the quasi-stationary state approximation used in this work is given by numerically solving the differential equations (Eqs. (16) and (17)) at each time as a function of the thickness of the material. To do so, the temporal dependence is introduced on definitions (6) and (8) by Eqs. (22)–(25) where Δnw is the uniform refractive index change constant, Δnwp the modulated refractive index change constant, αw the absorption coefficient change constant, and αwp the modulated absorption coefficient change constant. Moreover, it is important to note, that due to the depth dependency on the intensity, νM0 and νMp will also be a function of z.
3. Experimental procedure
In this study, the typical non-degenerated two wave mixing setup was used (Fig. 1), where the pump and probe beams were derived from a linearly p-polarized frequency doubled Nd:VO4 laser operating at 532 nm. The ratio of their intensities was fixed in all experiments, 1.8:1 (pump:probe) and the incidence angle of both beams was 45° in air with respect to the normal. The pump beam was Doppler-shifted by reflection from a mirror mounted on a piezoelectric transducer model PI P-753 LISA (PZT), which was driven by an amplified linear voltage derived from a function generator. The motion of the mirror was previously calibrated by placing it in one arm of a Mach-Zehnder interferometer and the applied frequency was varied to achieve different pump-probe detunings resulting in a temporal phase shift. The wave mixing process was detected by two photodetectors (D) both of which were connected to an oscilloscope. Both photodetectors were previously calibrated to obtain the equivalence between Volts and W/cm2 by using, a knife-edge detector from Coherent with precision of 0.1 μm for measuring the beam area. Regarding to the bacteriorhodopsin film, was a wild-type commercially available from MIB, with an optical density of 2.8 at 560 nm and a thickness of 100 μm.
4. Experimental results
As it was previously described in the above sections, we are going to study the two wave mixing in bacteriorhodopsin films. To do this, the normalized experimental temporal variations of the intensities of both beams were compared to the corresponding theoretical ones obtained by numerically solving the coupled differential equations (Eqs. (11)–(12)) at different times by taking into account Eqs. (18)–(25) through Eqs. (6)–(9). In Fig. 2 the corresponding analysis of a single experience is showed. As seen in Fig. 2(a), where the experimental temporal variations of both beams are shown, the typical bleaching behaviour of a saturable absorber is obtained for both beams. Coupled beatings, produced by the frequency difference and the amplification of the lower intensity beam, can also be observed. In this sense, the obtained theoretical description of the wave mixing process in these conditions is shown in Fig. 2(b), where bleaching, coupling and amplification of the lower intensity beam is also observed. For clarity, the theoretical and experimental results are compared in Figs. 2(c)–2(d), which show there is a good agreement between theory and experiment, regression coefficients being (r2) of 0.96 and 0.97.
Following the same methodology used in Fig. 2, we are going to study the effect of the intensity of the beams (with the same beam ratio) and the frequency difference between both beams. To do so, 250 experiments were analyzed by searching the combination of parameters that give the best description of the results obtained. In all the cases studied, the parameters directly related to the absorption coefficient, had the same value, to be exact, Δαw = 6.0×10−3, Δαwp = 6.0 × 10−4 and α = 450cm−1. First of all, in order to measure the energy interchange, in Fig. 3 the optical gain given by Eq. (26) is analyzed as a function of the total intensity and the frequency difference (Ω0). As seen from Fig. 2, note that this magnitude depends on time due to the frequency difference. Furthermore, theoretically, as deduced from Eqs. (19), (24) and (11), an oscillatory behaviour on the gain may be observed due to the temporal dependence of the coupling parameter, κ1.
In Fig. 2, in the saturation region, the temporal variation of the gain is observed to symmetrically oscillate around zero reaching a stationary state. Figure 3 shows the maximum gain obtained at the end of the non-degenerate wave mixing process in the mentioned saturation region, and an optimal frequency and intensity region where the maximum energy exchange is obtained. When there is no frequency difference, the beam coupling is close to null and rises to frequencies lower than 8 (Hz), being the maximum values beyond 1/τM. From this frequency range the gain significantly diminishes at all the intensities. Regarding the intensity, the energy exchange is higher at values lower than 0.1 W/cm2 in the mentioned frequency difference region. In this sense, similar results were previously observed in different saturable absorbers such as acridine-yellow-doped boric acid glass .
Regarding the parameters obtained, Fig. 4 shows the analysis of the variation of τM parameter as a function of the total intensity at different frequencies, where the error bars correspond to the standard deviation associated to the mean of these parameters. As can be seen, there is no effect of the frequency difference between the beams on τM, and the higher values of this parameter are obtained at lower intensities decreasing toward a saturation value as intensity rises. Note at this point, that these values are close to those previously obtained, when the propagation of sinusoidal pulses were analyzed in bacteriorhodopsin , and also that the amplitude of the error bars is too small.
Figure 5 shows the analysis of the variation of β1 and β2 parameters as a function of the total intensity at different frequencies are analyzed. As observed with τM parameter, there is no effect from the frequency difference between the beams, so for clarity we show the results at a fixed frequency. Both parameters present a decreasing tendency as a function of the total intensity, since, β1 approximates to a linear behaviour between 250 and 100 cm2/J, and on the other hand, β2 also decreases with values ranging between 280 and 380 cm2/J. According to the definition of β1, the variation of the product of the quantum yield and cross section of B specie is given by Fig. 5. Thus, in this case, the result is that the quantum yield of this specie decreases as intensity rises. Furthermore, the product of the quantum yield and cross section of M specie can also be analyzed by subtracting β2 and β1. In this sense, Fig. 6 shows the analysis of the variation of quantum yield and the cross section of M specie is analyzed as a function of the total intensity. As with the B specie, the quantum yield depends on the intensity, the behaviour being similar to an exponential saturation function of intensity.
The phase differences between the light pattern and the absorption coefficient (ϕa) and refractive index (ϕp) are nearly constant as a function of the intensity. As a function of the frequency difference, both parameters increase as a saturation function, ϕp between 0.2 to 1 rad and ϕa between 3.3 to 4.1 rads. After the analysis of the parameters related to the two-level model of the photoresponse of bacteriorhodopsin, we are going to show the corresponding results of the uniform and modulated index change. In this sense, Δnw is nearly constant as a function of the intensity and frequency detuning, presenting values ranging between 0.07 to 0.095. In Fig. 7, the parameter related to refractive index modulation change (Δnwp) is analyzed as a function of the total intensity. This parameter is observed to decrease as a function of the total intensity, but, in contrast to the others, it depends on the frequency difference of the beams. As the frequency difference is raised the modulated refractive index change constantly increases, this effect being more important at high intensities.
5. Analysis of the modulated and uniform refractive index change
After describing the parameters of the model as a function of the intensity and frequency detuning, we are going to analyze the temporal and depth variations of the corresponding uniform refractive index change and refractive index modulation change. Taking into account Eqs. (22)–(25), the corresponding parameters γ0, γ1, κ0, κ1, and, being Ip(z, t) = 2 (I1(z, t) I2(z, t))1/2, the uniform refractive change and absorption coefficient are given by:
Note at this point that the temporal and depth dependency on these parameters are given by νM0(z, t) and νMp(z, t), since the temporal and depth variation of the uniform change of the refractive index and absorption will be the same (with opposite signs), as also occurs with the modulation refractive index and absorption. In this sense, in Fig. 8 we show the temporal and depth variation of the uniform refractive index as a function of the total intensity, where, as stated, the uniform absorption coefficient variation is similar. As it can be seen, the uniform change does not vary in the whole thickness, for clarity, the represented depth being half the thickness of the film (100 μm). Due to the high absorption, the maximum uniform change is close to the surface where higher intensity is given. At a given depth, typical exponential saturation behaviour is observed, where the magnitude of the Δn0 decreases as depth increases. Regarding the effect of intensity, the increase of this variable occurs as a result of an extension of the depths of the refractive index change. Thus, at lower intensities uniform refractive changes close to 25 μm are obtained and at higher intensities they are around 40 μm.
In the case of the modulation of the refractive index, the corresponding temporal and depth dependence is shown in Fig. 9 at Ω0/(2π) = 1Hz, where the frequency difference between the beams is seen to produce temporal beatings resulting in a change in the sign of the modulation. The highest modulations are obtained at a specified temporal and depth region and depend on the initial phase difference. In the case shown in Fig. 9, the highest index modulation change is obtained at short times and as intensity rises these values extend to higher depths. On the other hand, the negative change regions are nearly elliptical and as the intensity increases, the center of these ellipses goes toward higher thickness. Moreover, it is important to note that the modulation change of negative to positive is not symmetric with respect to zero, with the positive refractive index modulation change absolute magnitude being nearly three times higher than the negative. Finally, by comparing the scales, the maximum and minimum values are obtained at the lowest intensity, but it is important to mention that the spatial extension in these conditions is the closest to the surface (Fig. 9(a)). Regarding the modulation of the absorption coefficient, despite the phase difference between the absorption grating and index grating, by comparing both modulations they are complementary. When the refractive index is positive absorption is negative and they extend temporally and spatially in similar manner.
Finally, in Fig. 10, the dependence on the frequency difference for the refractive index modulation change is shown at a fixed total intensity. When both beams have the same frequency (Fig. 10(a)), the typical temporal saturation curve is obtained, with the maximum value of the refractive index modulation change reached being at 30 μm, a value that can be increased at higher intensities. When the frequency difference between both beams differs, the modulation change are temporally and spatially localized in approximate elliptical regions. In this sense, the case shown in Fig. 10(b) corresponds to an initial phase difference which in the initial stages of the wave mixing process results in a negative refractive index modulation change. The maximal values are obtained at 0.5 seconds close to 30 μm, but at 1.5 seconds lower values are reached at the same depth. However, at higher frequencies (beyond 1/τM) a significant response of bacteriorhodopsin is only observed at short times. Thus, as shown in Fig. 10(c), higher modulations are obtained at 100 ms reaching 10 μm. At longer times, beatings of the refractive index modulation changes of around zero are obtained with an amplitude one magnitude order lower than the maximal value. At the highest frequency studied, (Fig. 10(d)), similar behaviour to other cases is observed, but the values reached are one magnitude order lower.
An analysis of non-degenerated two wave mixing in saturable absorbers such as bacteriorhodopsin films has been performed. A two-level model of the population variation in combination with the two coupled wave theory has been developed in order to analyze the temporal variation of the intensities. To do this, the uniform and the corresponding modulated real and imaginary part of the dielectric constant have been taken into account. The experimental temporal variation of the intensities of both beams have been compared to the theoretical ones, observing a good agreement between them. The effect of the initial intensity of both beams (with the same beam ratio) and the frequency difference between both beams (phase shift) have been analyzed. First of all, the gain has been studied as a function of these parameters, observing that when the phase difference is zero the coupling is close to null. Moreover, higher gain is obtained at low intensities and lower frequencies. Furthermore, from the theoretical study of these experiments, all the parameters of the model have been analyzed as a function of the intensity and frequency. Finally, the temporal and spatial variation of the uniform refractive index and absorption coefficient change, and those corresponding to modulation have been analyzed as a function of the intensity and frequency. As a result, it was observed that the corresponding photoinduced patterns are non-homogeneous in depth and vary as function of time. Only at low frequencies are significant values obtained close to the surface of the material at the initial stages of the wave mixing process.
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