Theoretical and experimental analysis of pulse delay in bacteriorhodopsin films by a saturable absorber theory

Salvador Blaya; Manuel Candela; Pablo Acebal; Luis Carretero; Antonio Fimia

doi:10.1364/OE.22.011600

1. Introduction

In recent years, slow and fast light in biological thin films and solutions of Bacteriorhodopsin (bR) has been reported [1–3]. 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) [4–7] or by coherent population oscillations (CPO) [1, 2]. In the case of the 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 a rapid spectral variation of refractive index and, consequently, 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 providing analytical results in much more general situations and justifying the effect of the mutual coherence and polarization state of the beams [5, 7].

Regarding the saturable absorption theory, these systems are well reproduced by using a two-level model with a very short coherence relaxation time compared to the population relaxation time, since, the propagation of laser pulses in the medium is simply described by two equations coupling the light intensity and population difference [8, 9]. In this sense, Macke et al [5] obtained general analytical expressions of the transmitted pulse which permit analysis of the delay with respect to the incident pulse for an arbitrary saturable absorber. In the case of bacteriorhodopsin, we recently performed a rigorous 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 [10]. This numerical analysis took into account six states of the photocycle and the corresponding equations of the two level model were also obtained, observing that this approximation in thick bacteriorhodopsin films can describe the propagation of sinusoidal pulses in a qualitative form.

Previous experimental studies of slow-light in saturable absorbers such as Bacteriorhodopsin have been performed by two coherent beams, a strong beam (pump) and a signal beam or a beam with high background with respect to the temporal modulated signal [1–3]. Basically, the effects of signal time delay are controlled by this strong beam. In this paper, we propose the analysis of sinusoidal pulses propagating in bacteriorhodopsin without the use of a pump beam (near zero background intensity) which could be interesting for different applications. To do so, analytical expressions of the propagation of sinusoidal pulse have been obtained by means of a modification of the theoretical treatment developed by Macke et al [5] for the particular case of bacteriorhodpsin. The delay respect of the incident pulse, distortion and fractional delay have been experimentally and theoretically analyzed in bacteriorhodpsin films using this model.

2. Theoretical procedure

The absorption of light by bacteriorhodopsin initiates a photocycle that accompanies the transportation of protons. Previously, we analyzed the six states mechanism, 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 [10]. 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 [11, 12]. In some cases, due to the short life-time of K, L, N and O states, this process can be theoretically modeled as a two-level system by taking into account only B and M states [13, 14]. Therefore, in order to analyze the propagation of an arbitrary function or/and intensity modulation, we will modify the theoretical model described by Macke et al [5]. So, a light beam propagating in the z direction through bacteriorhodopsin modeled as a two-level system according to the rate equation for the populations (B and M) (molecules/cm³) is given by [13]:

\frac{\partial M}{\partial t} = ϕ_{B} σ_{B} I B - ϕ_{M} σ_{M} I M - \frac{M}{τ_{M}}

where σ_i is the cross section for the i-specie (cm²/molecule), ϕ_i the quantum yield (molecules/photon), I the radiation flux (photon/s cm²) at angular frequency ω and τ_M the thermal lifetime of M → B transformation. Moreover, the radiation flux variation as function of depth is given by:

\frac{\partial I}{\partial z} = - (σ_{B} I B + σ_{M} I M)

Furthermore, due to the two level approximation, it is possible to relate the concentration of both species to the total concentration of bacteriorhodopsin (N₀) according to:

N_{0} = B + M

By using equation 3 and defining the normalized concentration of level M as N (N = M/N₀), β₁ = ϕ_B σ_B τ_M, β₂ = (ϕ_B σ_B + ϕ_M σ_M)τ_M y β₃ = ϕ_B(σ_M − σ_B)τ_M, equations 1 and 2 can be written as:

τ_{M} \frac{\partial N}{\partial t} = β_{1} I - N (β_{2} I + 1)

\frac{τ_{M} ϕ_{B}}{N_{0}} \frac{\partial I}{\partial z} = - I (β_{3} N + β_{1})

Note at this point, that equations 4 and 5 are not the same as the Bloch-Maxwell equations employed by Macke et al for an arbitrary saturable absorber [5]. In any case the same procedure can be performed for obtaining the transmission equations in bacteriorhodopsin films. This is done by combining equation 4 and 5 and assuming that ϕ_Bσ_B >> ϕ_Mσ_M (approximation valid for 532 nm and longer wavelengths) [13] and therefore β₂ ≈ −β₃ it is obtained the corresponding nonlinear wave equation for bacteriorhodopsin:

\frac{\partial}{\partial z} (τ_{M} \frac{\partial ln I}{\partial t} + ln I + β_{2} I + \frac{β_{1}}{ϕ_{B}} \frac{N_{0}}{τ_{M}} z) = 0

By integrating respect to z taking into account that the linear absorption coefficient is defined as α₀ = σ_B N₀, the thickness of the bacteriorhodopsin film is L, I_out the radiation flux at (z = L) and I_in at z = 0, the resulting transmission equation is given by:

τ_{M} \frac{\partial ln I_{out}}{\partial t} + ln I_{out} + β_{2} I_{out} + α_{0} L = τ_{M} \frac{\partial ln I_{in}}{\partial t} + ln I_{in} + β_{2} I_{in}

It is important to note that this equation corresponds to the transmission equation obtained by Macke et al [5] by identifying β₂ I to the normalized intensity to the saturation intensity. Introducing the transmittance T = I_out /I_in, Eq. 7 can be rewritten as:

τ_{M} \frac{\partial ln T}{\partial t} + ln T + β_{2} I_{in} (T - 1) + α_{0} L = 0

At steady-state this equation reduces to:

ln T + β_{2} I_{in} (T - 1) + α_{0} L = 0

Taking into account the high value of α₀ L for bacteriorhodopsin films and the low transmission at the wavelength of radiation, Eq. 9 can be approximated to ln T + α₀ L ≈ β₂ I_in and consequently Eq. 7 can be given by:

τ_{M} \frac{\partial ln T}{\partial t} + ln T + α_{0} L \approx β_{2} I_{in} (t)

In order to obtain analytical expressions for arbitrary signals, the radiation fluxes are given by [5]:

I_{in} = C_{in} + S_{in} (t)

I_{out} = C_{out} + S_{out} (t)

where C_in and C_out correspond to constant radiation fluxes at z = 0 and z = L and S_in(t) and S_out (t) are the temporal modulated radiation fluxes (signal) at z = 0 and z = L respectively.

Defining Z(t) as:

Z (t) = ln T (t) + α_{0} L - β_{2} C_{in}

and using equations 11 and 12, Eq. 10 transforms to:

τ_{M} \frac{\partial Z (t)}{\partial t} + Z (t) = β_{2} S_{in} (t)

whose analytical solution is:

Z (t) = β_{2} \frac{e^{\frac{- t}{τ_{M}}}}{τ_{M}} \int_{- t_{0}}^{t} S_{in} (θ) e^{\frac{θ}{τ_{M}}} d θ

where t₀ corresponds to the time that S_in(t₀) = 0 is accomplished. Therefore, by using Eq. 13, taking into account equations 11 and 12, the following equation is obtained:

(C_{out} + S_{out} (t)) = (C_{in} + S_{in} (t)) e^{Z (t) - α_{0} L + β_{2} C_{in}}

Furthermore, when no-temporal modulated signal is used, Eq. 16 is written as:

C_{out} = C_{in} e^{- α_{0} L + β_{2} C_{in}}

By substituting this expression in 16, the modulated transmitted signal (S_out (t)) is given by:

S_{out} (t) = (C_{in} (e^{Z (t)} - 1) + S_{in} (t) e^{Z (t)}) e^{- α_{0} L + β_{2} C_{in}}

It is important to point out that equation 17 determines the intensity range where the model is valid, because the approximated transmission equation (Eq. 10) is accomplished when α₀L >> β₂C_in. Moreover, arbitrary signals can be analyzed with analytical expressions if the integral given at Eq. 15 can be analytically solved.

Time delay effects can be analyzed by the difference between the maximum of the reference modulated signal (S_in(t)) and that corresponding to the output of the material (S_out (t)). t_smax being the corresponding time where (S_in(t)) is maximum (dS_in(t)/dt|_{t_smax} = 0), so if there is no significant deformation on the output signal, dS_out (t)/dt|_{t_smax} will be zero if the output signal is identical to the output, dS_out (t)/dt|_{t_smax} < 0 if an advancement on the pulse is produced and the signal will be delayed when dS_out (t)/dt|_{t_smax} > 0. By deriving Eq. 18 with respect to time and evaluating at t_smax, the following is obtained:

{\frac{d S_{out} (t)}{d t} |}_{t_{smax}} = {\frac{d Z (t)}{d t} |}_{t_{smax}} (C_{in} + S_{in} (t_{smax})) e^{Z (t_{smax}) - α_{0} L + β_{2} C_{in}}

Therefore, it follows that the sign of dS_out (t)/dt|_{t_smax} is determined by the sign of ${\frac{d Z (t)}{d t} |}_{t_{smax}}$ which can be obtained from Eq. 14:

{\frac{d Z (t)}{d t} |}_{t_{smax}} = τ_{M}^{- 1} (β_{2} S_{in} (t_{smax}) - Z (t_{smax}))

By using Eq. 15, Z(t_smax) is given by:

Z (t_{smax}) = β_{2} \frac{e \frac{- t_{smax}}{τ_{M}}}{τ_{M}} \int_{- t_{0}}^{t_{smax}} S_{in} (θ) e^{\frac{θ}{τ_{M}}} d θ

and taking into account that t₀ + t_smax > 0, it follows that:

\begin{array}{l} Z (t_{smax}) = β_{2} \frac{e^{\frac{- t_{smax}}{τ_{M}}}}{τ_{M}} \int_{- t_{0}}^{t_{smax}} S_{in} (θ) e^{\frac{θ}{τ_{M}}} d θ \leq β_{2} e^{\frac{- t_{smax}}{τ_{M}}} S_{in} (t_{smax}) (e^{\frac{t_{smax}}{τ_{M}}} - e^{\frac{- t_{0}}{τ_{M}}}) \leq \\ \leq β_{2} S_{in} (t_{smax}) (1 - e^{\frac{- (t_{0} + t_{smax})}{τ_{M}}}) \leq β_{2} S_{in} (t_{smax}) \end{array}

It is, therefore, demonstrated that dZ(t)/dt|_{t_smax} > 0 and consequently the signal, will always be delayed under these conditions. It is important to mention that this result does not contradict the superluminal light obtained at a wavelength, where the system is a reverse saturable absorber instead of a saturable absorber i.e it does not accomplish Eq. 6 [3].

In this paper, we are going to analyze sinusoidal signals, specifically S_in(t) will be:

S_{in} (t) = I_{0} {sin}^{2} (\frac{π (t - t_{0})}{τ_{in}})

Introducing Eq. 23 into Eq. 15, we obtain that Z(t) is:

Z (t) = \frac{β_{2} I_{0} (τ_{in}^{2} + 4 (1 - e^{- \frac{(t - t_{0})}{τ_{M}}}) π^{2} τ_{M}^{2} - τ_{in} (2 π τ_{M} sin (\frac{2 π (t - t_{0})}{τ_{in}}) + τ_{in} cos (\frac{2 π (t - t_{0})}{τ_{in}})))}{2 (τ_{in}^{2} + 4 π^{2} τ_{M}^{2})}

Finally, the distortion of the pulse has been analyzed by means of the following equation [15, 16]:

D = {(\frac{\int_{- t_{0}}^{t_{1}} | {Sn}_{out} (t) - {Sn}_{in} (t - τ_{D}) | d t}{\int_{- t_{0}}^{t_{1}} {Sn}_{in} (t) d t})}^{\frac{1}{2}}

where τ_D is the delay of the pulse, t₁ is time where the input signal is null (end of first cycle), Sn_out (t) is the normalized output intensity signal and Sn_in(t) corresponds to the normalized reference intensity signal. Note that D is equal to zero when the pulse is undistorted (the best possible result) [16].

3. Experimental procedure

In this study, the propagation of sinusoidal pulses has been studied by using the experimental setup shown in Fig. 1 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 is obtained from a frequency doubled Nd:VO₄ laser operating at 532 nm. The beam is split into two beams where one of them passes through a phase electro-optic modulator (PEM) 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 (signal beam) and the other is used as a reference beam for measuring peak delay. The signal beam reaches normal to the surface of the film. Finally, the transmitted signal beam and the reference beam are detected by two photodetectors (D) both of which are connected to an oscilloscope. Both photodetectors were previously calibrated for obtaining the equivalence between Volts and W/cm² by using, for measuring the beam area, a knife-edge detector from Coherent.

Fig. 1 The experimental setup used to analyze the propagation of sinusoidal pulses in bacteriorhodopsin film.

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The area of the beam were measured by knife-edge detector from Coherent with precision of 0.1 μm.

4. Experimental results

According to the previous theoretical and experimental descriptions, in this paper we are going to analyze the propagation of sinusoidal beams without a pump beam in bacteriorhodopsin films. The study has been performed by the corresponding non-linear fit of the experimental data to the developed model. Firstly, the reference modulated intensity has been fitted by means of equation 23, obtaining the parameters I₀, C_in and t₀ and τ_in with regression coefficients (r²) close to 0.999. Following the same procedure, the output signal intensity has been fitted by Eq. 18 where equations 23 and 24 have been introduced. For this non-linear fit, the numerical value of (ϕ_B σ_B + ϕ_M σ_M) is taken from reference [17] (β₂ = 158.71τ_M cm²/W) and the previously obtained parameters of the reference signal (I₀, C_in, t₀ and τ_in) have been used, obtaining α₀ L and τ_M with regression coefficients higher to 0.99. In figure 2 the experimental and fitted curves of the normalized signal and reference beams to the maximum intensity are shown (for clarity, only the first cycle is shown), where the predicted delay of the signal is measured. As shown, according to Eq. 19 and inequality 22 a delayed output signal is observed. So, in order to analyze these results, we are going to study the obtained parameters obtained for the model as a function of the total intensity of the pulse.

Fig. 2 Temporal variation of the experimental and fitted sinusoidal modulated beam (signal) and the corresponding experimental and fitted reference beam, where C_in = 0.4 mW/cm², I₀ = 5.5 mW/cm² and the regression coefficients of the reference and signal beams were 0.999 and 0.992 respectively.

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In Fig. 3, the fitted parameters (τ_M, and α₀L) of the signal output intensity curves are analyzed as a function of the total intensity for different signal frequencies. Each of the values shown correspond to the mean of the fitted parameters obtained in similar intensity conditions. Meanwhile the error bars correspond to the standard deviation associated to the mean of those parameters. As can be seen in Fig. 3(a) τ_M can be considered nearly constant (0.6 ± 0.1), being the observed variations justified by thermal effects. In contrast to the previous analysis of the rigorous theory [10], we have assumed that τ_M does not vary during the signal propagation. Finally, the variation of the initial absorption α₀L as a function of the total intensity is analyzed in Fig. 3(b). As can be seen, the values reached are constant (4.6 ± 0.1) around the corresponding previously reported value [10, 17]. In this case, continuous lines are the mean value of all α₀L-parameters.

Fig. 3 Variation of the fitted parameters of the signal intensity curves as a function of the total intensity: τ_M (a) and α₀L (b). The ratio C_in/I₀ oscillates between 0.05 to 0.12. Orange line correspond to the mean value of all the obtained parameters.

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In Fig. 4 the obtained experimental and theoretical time delay, fractional delay and distortion are analyzed as a function of total intensity for different signal frequencies, where the values of time delay were obtained from the difference of the corresponding maximum of the fitted curves (signal and reference) and the fractional delay (F) by τ_D/τ_in. As can be seen from Fig. 4(a) as total intensity rises, time delay increases, with higher values being obtained at low frequencies. Continuous lines were obtained by using Eq. 18 taking into account Eq. 24 and the mean value of α₀L and τ_M obtained from fitted curves (orange lines at Figure 3(a) and 3(b)). As it can be seen, there is a good concordance between theory and experimental values. Moreover, for the fractional delay, due to the pulse width, the differences between the analyzed frequencies are lower. Similar to time delay, a good agreement between theory and experience is obtained. Finally, distortion is studied by using Eq. 25. As can be seen distortion is related to time delay, increasing as a function of the total intensity reaching a saturation value. Therefore, higher delays are obtained as total intensity increases, which implies an increase on the pulse distortion. Furthermore, at low frequencies, the distortion of the pulse increases. Also, a good agreement between the simulated distortion and the experimental one is observed. It is important to note, that these values are in the same range as the corresponding obtained by Bigelow et al in a different material [16].

Fig. 4 Variation of the time delay of a sinusoidal modulated beam (a), fractional delay (b) and distortion of the pulse as a function of the total intensity. The ratio C_in/I₀ oscillates between 0.05 to 0.12. Theoretical simulations by using Eq. 18 taking into account Eq. 24, the mean value of α₀L and τ_M are shown for each frequency (blue line 0.2 Hz, red line 0.7 Hz, green line 1 Hz and black line 1.5 Hz).

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5. Conclusions

We have performed an analysis of the time delay of transmitted pulses with respect to the incident pulse in bacteriorhodopsin films. To do so, analytical expressions of the transmittance for the particular case of bacteriorhodopsin have been obtained, showing that a delay always exists on the pulse propagated in the material. Through the non-linear fit of the experimental data, the effect of the intensity of the pulse has been analyzed, observing a good agreement between theory and experiences. As a result, time delay, distortion and fractional delay have been analyzed for sinusoidal pulses with a low background.

Acknowledgments

The authors acknowledge support from project FIS2009-11065 of Ministerio de Ciencia e Innovación of Spain and ACOMP/2012/151 from the Consellería d’Educació, Formació i Ocupació de la Generalitat Valenciana.

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Theoretical and experimental analysis of pulse delay in bacteriorhodopsin films by a saturable absorber theory

Abstract

1. Introduction

2. Theoretical procedure

3. Experimental procedure

4. Experimental results

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (25)

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