1. Introduction

Recently, the control of group velocity of light has attracted much interest due to its potential applications, such as tunable optical buffers [1, 2], optical memories [3, 4], the enhancement of the sensitivity of interferometers [5, 6], and the improvement of nonlinear effects [7]. In order to slow down the group velocity of a propagating pulse several schemes with different kind of materials have widely demonstrated [8–10]. Among the applied methodologies, good results were obtained by electromagnetically induced transparency (EIT) [11–13], coherent population oscillations (CPO) [14, 15], atomic double resonances [16, 17], photonic crystal waveguides [18–20], coupled resonator optical waveguides (CROWs) [2, 21, 22], holographic induced transparency [23,24] and stimulated Brillouin scattering (SBS) in optical fibers [25,26].

Furthermore, in relation to the employed materials, it has been reported slow and fast light in biological thin films and solutions of Bacteriorhodopsin (bR) [27–29]. In this sense, due to the photoisomerization processes, in polymeric films group velocities of 0.091 mm/s were obtained at a wavelength of 568 nm with modulation frequencies on the Hz-range [27]. Moreover, it was also demonstrated the possibility to slow-down the group velocity by the use of a second beam at 442 nm [27]. In the case of aqueous solutions of bacteriorhodopsin, due to the higher thermal diffusion and mobility, longer slow light velocities were reached but with higher modulation frequencies [28]. In relation to the theoretical analysis of these results in bR systems, there is a controversy about the mechanism of the resulting slow light process in this system because it can also be equally explained by a temporal variation of the absorption (saturable absorption) [30–33] or by coherent population oscillations [27, 28]. In the case of CPO model, a narrow spectral hole is formed when a strong beam and a weak beam, slightly frequency detuned, copropagate through this biological material (saturable absorber). The quantum interference of the two monochromatic beams causes an oscillation of the ground-state population at the beat frequency which results in a reduction of the absorption of the probe beam and to a rapid spectral variation of refractive index and, consequently, to a group velocity reduction. On the other hand, the saturable absorption theory is based on the same assumptions, the pump saturates the homogeneously broadened absorption band, resulting in a modification of the time transmission compared to the incident optical pulse. Both approaches are equivalent, but the saturable absorption theory only takes into account the absorption and it provides analytical results in much more general situations and it justifies the effect of the mutual coherence and polarization state of the beams [33].

Usually, theoretical analyses performed by saturable absorption and CPO models take into account two or three states [34–36]. However, in the case of Bacteriorhodopsin when it is illuminated by yellow-green radiation undergoes several structural transformations in a complex photocycle that generates a great number of intermediate states. In a previous work we performed an analysis of the dynamic and steady-state photoinduced processes of thick bacteriorhodopsin bR films, taking into account all the physical parameters, the coupling of rate equations with the energy transfer equation and the effect of temperature change [37]. As a result, the temporal variation of the transmittance as a function of the pump intensity was well described and inhomogeneous profile of the population densities can be obtained depending on the intensity and wavelength of the pump beam. The aim of this work is to theoretically and experimentally analyze the propagation of sinusoidal pulses in thick bacteriorhodopsin bR films and to analyze peak delay effects. For this, instead of the two or three level approximations commonly used, we are going to the take into account the physical parameters of six states of the photocycle and the temperature effect of the intensity.

2. Theoretical background

The theoretical procedure and the corresponding parameters are the same as the previously described in reference [37], but considering a sinusoidally modulated incident beam instead of a constant intensity beam. The analysis is based on the fact that bR films are composed by an inert matrix (usually a polymer or gelatin) and a large number of photochromic protein bacteriorhodopsin. Regarding the description of the bR molecules, we have to distinguish two different frames attending to the optical properties. On the one hand, a chain of aminoacids forms a helicoidal superstructure shown in Fig. 1, whose main function is to serve as a rigid support for the light active core inside the bacterial membrane. This core is composed by few aminoacids and the retinal chromophore whose function is to act as light-driven proton pump. This mechanism is composed by six states, 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. Apart from the normal evolution of the photocycle the protein can also return directly to the B state from K, L, M and N states upon photon absorption [38, 39].

 

Fig. 1 Schematic representation of the bacteriorhodopsin structure and its photocycles.

Download Full Size | PDF

2.1. Six level model

In order to describe the photoinduced changes in a photochromic material we employ the radiative energy transfer along an inhomogeneous medium which is given by:

Where I_a is the intensity of the pump and signal beam both polarized along the a direction, ζ is the propagation coordinate of the electromagnetic field, β_aa and ${\beta }_{aa}^R$ are the macroscopic parameters related to absorption and scattering cross sections respectively. Furthermore, according to statistical mechanics, the expressions of these macroscopic magnitudes are related to the corresponding microscopic properties by:

Where the B and n_ia vectors are $\mathbf{B}=\left\{\partial {\text{n}}_{ia}^B/\partial \text{t},\partial {\text{n}}_{ia}^K/\partial \text{t},\partial {\text{n}}_{ia}^L/\partial \text{t},\partial {\text{n}}_{ia}^M/\partial \text{t},\partial {\text{n}}_{ia}^N/\partial \text{t},1\right\}$ and ${\mathbf{n}}_{ia}=\left\{{\text{n}}_{ia}^B,{\text{n}}_{ia}^K,{\text{n}}_{ia}^L,{\text{n}}_{ia}^M,{\text{n}}_{ia}^N,{\text{n}}_{ia}^O\right\}$, while the coefficient matrix A is:

Where the terms ${\mathrm{\Phi }}_{ia}^{\kappa }$ are given by:

In this system ${\text{n}}_{ia}^{\kappa }$ are the normalized population densities for the κ state ( ${\text{n}}_{ia}^{\kappa }={\text{N}}_{ia}^{\kappa }/{\text{N}}_{ia}^{bR}$), the parameters ϕ_κ are the quantum efficiencies of the photoinduced reactions, k_κκ′ are the rate constants of the transition κ → κ′ (for simplicity, in previous equations we have omitted the dependency on ζ and time of all the population densities and the light intensity of the pump beam). Moreover, it is assumed that only the states B, K, L and N have absorption at the radiation frequency (532 nm). The last line of the coefficient matrix (A) accounts for the conservation of the total purple membrane units oriented in a fixed direction ( ${\text{N}}_{ia}^{bR}={\text{N}}_T/3$ for isotropic non-oriented films).

The resulting set of equations that describe the photoinduced processes in thick bR films has seven unknown functions (the six bR state population densities and the intensity), a large number of parameters (rate constants, quantum efficiencies and microscopic optical properties) and two variables (position and time). This system has no analytical solutions for time and position, and therefore numerical methods must be used to solve it. So, regarding to the rate constants and quantum efficiencies of the different transitions of the photocycle, these set of parameters are greatly influenced by the environment, i.e. the transmembrane nature of bR, mainly the water content and the pH, but also the chemical additives [40,41] in polymer matrix. We have employed the corresponding values of quantum efficiencies and rate constants that we previously obtained in reference [37]. In the case of the rate constants, their values depend on the temperature, which is not constant during the pump stage, so it has to be taken into account. The temperature dependency of rate constants is given by the Eyring relationship [42], which allows us to represent the rate constant at a given temperature T₁ as a function of the k₀ at another temperature T₀:

Where K_B and h are the Boltzmann and Planck constants respectively. We have assumed that the temperature is given by Eq. 8, where the temperature change is related to the equilibrium between the increase due to the light absorption and a decrease produced by the thermal dissipation, which occurs with different time constants. Moreover, it is taken into account that the maximum temperature is not higher than 340 K for light intensities of about 2000–3000 W/m², since no degradation of the protein is observed in the experiments at these intensities.

The coefficients a_in and a_de are related to the total temperature increase produced by light absorption and decrease due to the thermal dissipation respectively, and both depend on the incident intensity. The temporal behavior is described as a combination of ascending saturation curves for the heating process and descending saturation curves for the dissipation process, where c_i and g_i are the time constants of these processes weighted with the coefficients b_i and h_i (∑_i b_i=1 and ∑_i h_i=1). We have employed the numerical values of all of these parameters in the bR film that we previously obtained in reference [37] from the fitting of the temporal variation of the pump beam transmission.

Regarding to the microscopic optical properties of all the elements of the system, which are not affected by the environment (pH and water content), the values of the microscopic absorption cross sections of the different states of the system and the microscopic scattering cross sections for the working wavelength are given by [43, 44]:

Where ${\mu }_{eg}^{\kappa }$ is the ground to excited state transition dipole moment of the κ state of bR, ${\omega }_{eg}^{\kappa }\left({\omega }_{eg}^{\kappa }=2\pi \text{c}/{\lambda }_{eg}^{\kappa }\right)$ is the frequency of this transition, γ denotes the width of the absorption curve, while α_ii(ω) is the polarizability of the κ element at ω, c is the light speed and ε₀ is the dielectric permittivity of the vacuum. In order to assign reliable numerical values to these microscopic parameters we follow two different strategies. On one hand, we use experimental values for ${\mu }_{eg}^{\kappa }$, ${\omega }_{eg}^{\kappa }$ and γ [45, 46]. On the other hand the polarizability components, which can not be easily measured experimentally, were evaluated using quantum mechanical simulations by means of a procedure described in a previous work [47, 48] and the calculations were performed by the Gaussian 98 package [49]. For this, we consider that the polarizability of the different states of the Bacteriorhodopsin can be separated into the contribution of the light active core and that of the structural chain of aminoacids (α₀, α̂^Cκ):

Where α₀ denotes the sum of the traces of static polarizability of the 296 aminoacids that form the bR protein and α̂^Cκ is the polarizability matrix of the light active core of the protein (retinal chromophore and surrounding aminoacids). The elements of the polarizability matrix are given by Eq.:

Where ${\alpha }_{ii,e}^{C\kappa }(0)$ and ${\mathrm{\Omega }}_1^{\kappa }(\omega )$ are:

The polarizabilities of K and N states of bR were not calculated since their population densities were two orders of magnitude lower than those of the other states, so their contribution to the scattering losses was negligible. We have used the same values of the microscopic optical properties of the different states of bR, damping factor (Γ) and α₀ as our previous work [37] as it can be seen in table 1.

Table 1. Values for the microscopic optical properties of the different states of bR (μ_1g and λ_1g from references [45, 46]). γ=Γ/2.355=2×10¹⁴s⁻¹, α₀=2.76×10⁻²⁶m³ and rate constants at 293 K are: k_kl=2300 s⁻¹, k_lm=169 s⁻¹, k_mn=0.98 s⁻¹, k_no=110 s⁻¹, k_ob=5.2 s⁻¹, k_lk=230 s⁻¹, k_ml=64.6 s⁻¹, k_nm=40 s⁻¹, k_on=0.1 s⁻¹.

View Table

At this point, the analysis of the beam propagation is described by a set of coupled differential equations (Eqs. 1 – 14) which depends on two variables, and the numerical approach was also a difficult task. We solve the system by decoupling the differential equation of the ζ variable from those of the time variable, which can be done if we discretize the material in the light propagation direction. For an adequate number of layers (or step size (Δζ)), we considered that the population densities do not depend on the ζ variable, so Eq. 1 can be solved analytically for each layer, and the transmitted intensity of the j layer is:

2.2. Two level model

As it has been stated, the theoretical description realized in the previous section corresponds to the analysis of the photoinduced processes of thick bacteriorhodopsin bR films widely analyzed in reference [37]. In this work, we have used this treatment by employing the same values of the parameters that we previously obtained (table 1). Since, in order to analyze the temporal variation of absorption, instead of using a constant intensity we have employed a sinusoidally modulated intensity given by:

3. Experimental procedure

In this study, the propagation of sinusoidal pulses have been studied by using the experimental arrangement shown in Fig. 2 with a commercially available bacteriorhodopsin film (MIB), whose main characteristics are an optical density of 2.8 at 560 nm and a thickness of 100 μm. The linearly p-polarized pump beam was obtained from a frequency doubled Nd:VO₄ laser operating at 532 nm, that was splitted into two beams. One beam is the pump beam with intensity of 2650 W/m² and the other the signal beam which consists of two frequencies that was achieved by the use of a phase electro-optic modulator (PEM). Due to the temporal beating of light a sinusoidally modulated beam of light can be obtained; to do this, the signal beam is splitted into two beams where one of them passes through a phase electro-optic modulator which is driven by a function generator. Both beams are collimated and recombined by mirrors and a beam splitter, resulting in two sinusoidally modulated beams of light at a modulation frequency driven from the function generator mentioned above. One of the combined beam is directed toward the sample with an intensity of 84 W/m² and the other is used as a reference beam for measuring peak delay. The probe beam reaches normal to the surface of the film, whereas the angle of incidence of the pump is near to the normal. Finally, the pump beam, the transmitted probe beam and the reference beam are detected by three photodetectors (D) and all of them are connected to an oscilloscope.

 

Fig. 2 The experimental setup used to analyze the propagation of sinusoidal pulses in bacteriorhodopsin film.

Download Full Size | PDF

4. Results and discussion

Taking into account the previous theoretical and experimental descriptions, Fig. 3 shows the experimental and theoretical temporal variation of the intensity of the pump beam. The good concordance between theoretical and experimental data confirms that the model was a good approximation of the events that take place when the material is illuminated. It is important to remark that the theoretical curve does not result from a fitting procedure and all the parameters used for the theoretical results of Fig. 3 were the corresponding ones obtained in reference [37] and showed in table 1. Regarding to the obtained results based on the two-level model, there is good concordance between both theories, but better results are obtained by using the rigorous model. Furthermore, as it can be seen, higher differences are obtained at the initial stage of the process. So, this result implies that stationary condition used for obtaining the two level model is not correct at initial the stage. In any case, this result demonstrates that the two-level approximation qualitatively describes this process.

 

Fig. 3 Temporal variation of the pump beam with an intensity of 2650 W/m² and a wavelength of 532 nm. The points are the experimental data, the blue line represents the results obtained by the rigorous theory and the red one corresponds to the two level approximation, where it has been used the parameters shown in table 1

Download Full Size | PDF

The experimental and theoretical transmitted sinusoidally modulated beam are shown in Fig. 4. As it can be seen, as the pump beam (showed in Fig. 3), good agreement between the theoretical and experimental responses is shown. Therefore, the spatial and temporal variation of intensity is explained by the radiative energy transfer along an inhomogeneous medium (Eq. 1) and the effect of the index modulation is not taken into account as it is stated in the CPO model. In this sense, in Fig. 5 the reference beam and the corresponding experimental and theoretical signal intensity are shown at different times, i.e. different sine-peaks. As it can be seen, at the first sinusoidal-peak the maximal time delay is reached, and after that the time delay decreases. As it has been previously analyzed, the rigorous model gives better concordance with experience than the two level model.

 

Fig. 4 Temporal variation of the sinusoidal modulated beam (signal) with an intensity of 84 W/m² and a wavelength of 532 nm. The points are the experimental data, the blue line represents the results obtained by the rigorous theory and the red one corresponds to the two level approximation, where it has been used the parameters shown in table 1

Download Full Size | PDF

 

Fig. 5 Temporal variation of different peaks of the sinusoidal modulated beam (signal) with an intensity of 84 W/m² and a wavelength of 532 nm. The yellow line is the experimental reference beam, the blue line represents the results obtained by the rigorous theory and the red one corresponds to the two level approximation, where it has been used the parameters shown in table 1

Download Full Size | PDF

Regarding to the population densities of the different states of bacteriorhodopsin, Fig. 6 shows the temporal variation of B, K, L, M, N and O states at different layers of the thickness of the film when the system is illuminated at the experimental conditions shown at Figs. 3 and 4. As it has been previously described [37], from the analysis of Figs. 6(a–c) an inhomogeneous distribution of the normalized population densities along the light propagation direction is obtained. At the initial layer (Fig. 6(a)) the population density of the initial B state decreases while the population of the other states increases. At the first layer, at these illumination conditions, the M state saturates at the highest population density while the B and L states reach similar values. On the other hand, the population densities of the other states (K, N, O) are significant lower. At the half of the thickness (Fig. 6(b)), the similar behavior is observed, but due to attenuation of the pump beam intensity the rate of is lower than the previous case and as a result B and M states are the highest population density, while L state reaches lower values than those ones. The population densities of states K, N, O present the same tendency with lower values. At the last layer, due to attenuation, the diminution of the population density of B state are low being the value reached higher as the other states.

 

Fig. 6 Temporal variation of the normalized population densities (n_X) at different layers of the bacteriorhodopsin film obtained by the rigorous model. It has been analyzed the case of a pump beam intensity of 2650 W/m² and sinusoidal modulated beam (signal) with an intensity of 84 W/m². As before, the wavelength of both beams were 532 nm and it has been used the parameters shown in table 1.

Download Full Size | PDF

By comparing Figs. 5(a) and 6(a–c), it can be deduced that the highest delay obtained at the first peak is justified by the highest change of population densities (mainly B, M and L) produced at the whole thickness. On the other hand, the other peaks present lower or null delays (Figs. 5(b–d)), which it corresponds to the time interval where the population densities have reached the stationary state nearly at the whole thickness.

Furthermore, it has been previously reported that in aqueous solution of bR, the delay decreases as the modulation frequency increases [28]. These results have been well explained by the saturable absorber theory and CPO model [28, 32]. In this sense, by using the rigorous theoretical analysis of photoinduced processes described in this work at pump intensity of 2650 W/m² and signal intensity of 84 W/m², Fig. 7 shows the theoretical variation of the delay as a function of the modulation frequency. The delay was obtained by comparing the maximum of the transmitted pulse at the first cycle and the input one. As it can be seen, the typical response of a saturable absorber is obtained [32, 50], being the theoretical behavior curve the same as the experimental data previously measured in aqueous bR [28]. Regarding to the effect of the modulation frequency on the population densities of the different states, we have analyzed all the cases studied in Fig. 7 and we have not observed significant differences on the temporal behavior of the population densities of the bR states. According to these results, by this rigorous treatment, the delay of the pulse is explained by the fact that the leading edge of the pulse will thus experience more absorption than the trailing edge of the pulse, and that consequently the peak of the pulse will be shifted to later times.

 

Fig. 7 Theoretical variation of the delay of sinusoidal modulated beam (signal) as a function of the modulation frequency obtained by the rigorous model. It has been analyzed the case of a signal intensity of 84 W/m² and pump of 2650 W/m² with wavelength of 532 nm both. As before, it has been used the parameters shown in table 1.

Download Full Size | PDF

The effect of intensity has been analyzed in Figs. 8 and 9, where it is shown the delay as a function of the pump intensity and modulation frequency. Experimental and theoretical values are compared in Fig. 8 where it can be seen a good concordance between theory and experience, in particular as the higher frequencies studied. The corresponding theoretical predictions of the rigorous model for a higher number of cases are shown in Fig. 9. For almost intensities, as it was previously reported [27, 28], at fixed pump intensity the delay decreases as the frequency detuning is raised. At lower frequencies than 4 Hz, the dependence of the delay with the pump intensity presents a minimum value (Fig. 9(a)). In these cases, the maximum delay is given at low pump intensity instead of higher ones. However, as it can be seen in Fig. 9(b) at higher frequencies as 4 Hz the predicted variation of the delay as a function of pump intensity corresponds (with different values) to the previously reported with bR film [27]. In this case, the maximum delay that can be obtained depends on the modulation frequency. However, at lower frequencies the maximum delay is given at low pump intensity instead at higher ones.

 

Fig. 8 Theoretical and experimental variation of the delay of sinusoidal modulated beam (signal) as a function of the pump intensity obtained by the rigorous model. It has been analyzed the case of a signal intensity of 5 % of the pump beam and the wavelength of both beams were 532 nm. As before, it has been used the parameters shown in table 1.

Download Full Size | PDF

 

Fig. 9 Theoretical variation of the delay of sinusoidal modulated beam (signal) as a function of the pump intensity obtained by the rigorous model. It has been analyzed the case of a signal intensity of 84 W/m² and the wavelength of both beams were 532 nm. As before, it has been used the parameters shown in table 1.

Download Full Size | PDF

Furthermore, the corresponding temporal variation of density populations of different states is analyzed in Fig. 10. Since the results showed that the rate of diminution of the initial B state increases as the intensity of the pump beam raises. While for M and L state the rate increases with intensity. Although, as it can be seen at higher intensities at the initial layer the density population of M state reaches higher values than B state. On th other hand, at higher depths due to the attenuation this effect is not produced. Another interesting result, is given by the analysis of the density population in the whole depth, so by increasing the incident intensity causes a smaller difference between the population densities of the first and the last layers.

 

Fig. 10 Temporal variation of the normalized population densities (n_X) at different layers of the bacteriorhodopsin film obtained by the rigorous model. It has been analyzed the case of a modulation frequency of 0.3 π Hz and sinusoidal modulated beam (signal) with an intensity of 5% of the pump beam. As before, the wavelength of both beams were 532 nm and it has been used the parameters shown in Table 1.

Download Full Size | PDF

Finally, regarding to the polarization of the pump and signal beams, this rigorous model is valid when the polarization state is the same in both cases. Moreover it could be analyzed another kind of polarization states, for example linear and circular polarizations could be used with an arbitrary optical axis.

5. Conclusions

We have performed a study of the dynamic photoinduced processes of thick bacteriorhodopsin films, taking into account all the physical parameters, the coupling of rate equations with the energy transfer equation, and the effect of temperature change for the analysis of the propagation of sinusoidal pulses. Good concordance between theoretical and experimental data have been obtained. By this analysis arbitrary signal and pump beams can be used in order to explain delays or advancements of pulses in bacteriorhodopsin films. Furthermore, due to this model takes into account six states of the photocycle, the equations of the two level model have been obtained by applying the stationary state condition to the rigorous six level theory. It has been observed that good results are also obtained (but worse than the rigorous theory), therefore it is demonstrated the validity of the two-level approximation in thick bacteriorhodopsin films in a qualitative form.