blaya s_2014_pulse delay br films

8/12/2019 Blaya S_2014_pulse Delay BR Films

http://slidepdf.com/reader/full/blaya-s2014pulse-delay-br-films 1/10



16. M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of smooth and discontinuous pulses

through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18, 3117–3126 (2006).

17. P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomo-

geneous anisotropy in bacteriorhodopsin films,” Phys. Rev. E 76, 016608 (2007).

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 sys-

tems, 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 re-

duction. 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 equa-

tions 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 bacte-

riorhodopsin, 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 accountsix 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 trans-

portation of protons. Previously, we analyzed the six states mechanism, starting from the B

#207294 - $15.00 USD Received 26 Feb 2014; revised 3 Apr 2014; accepted 14 Apr 2014; published 6 May 2014

(C) 2014 OSA 19 May 2014 | Vol. 22, No. 10 | DOI:10.1364/OE.22.011600 | OPTICS EXPRESS 11601



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]:

∂ M

∂ t = φ B σ B I B−φ M σ M I M −

M

τ M (1)

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:

∂ I

∂ z=−(σ B I B +σ M I M ) (2)

Furthermore, due to the two level approximation, it is possible to relate the concentration of

both species to the total concentration of bacteriorhodopsin ( N 0) according to:

N 0 = B + M (3)

By using equation 3 and defining the normalized concentration of level M as N ( N = M / N 0),

β 1 = φ Bσ B τ M , β 2 = (φ Bσ B + φ M σ M )τ M y β 3 = φ B(σ M −σ B)τ M , equations 1 and 2 can be

written as:

τ M ∂ N

∂ t = β 1 I − N (β 2 I + 1) (4)

τ M φ B

N 0

∂ I

∂ z= − I (β 3 N +β 1) (5)

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 pro-

cedure 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 M σ M (approxima-

tion valid for 532 nm and longer wavelengths) [13] and therefore β 2 ≈ −β 3 it is obtained the

corresponding nonlinear wave equation for bacteriorhodopsin:

∂

∂ z

τ M

∂ ln I

∂ t + ln I +β 2 I +

β 1 N 0

φ B τ M z

= 0 (6)

By integrating respect to z taking into account that the linear absorption coefficient is defined

as α 0 = σ B N 0, 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 ∂ ln I out

∂ t + ln I out +β 2 I out +α 0 L = τ M

∂ ln I in

∂ t + ln I in +β 2 I in (7)

It is important to note that this equation corresponds to the transmission equation obtained

by Macke et al [5] by identifying β 2 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 ∂ ln T

∂ t + ln T + β 2 I in (T −1) +α 0 L = 0 (8)

At steady-state this equation reduces to:

ln T + β 2 I in (T −1) +α 0 L = 0 (9)

Taking into account the high value of α 0 L for bacteriorhodopsin films and the low trans-

mission at the wavelength of radiation, Eq. 9 can be approximated to ln T + α 0 L ≈ β 2 I in and

consequently Eq. 7 can be given by:

τ M ∂ ln T

∂ t + ln T +α 0 L ≈ β 2 I in(t ) (10)

In order to obtain analytical expressions for arbitrary signals, the radiation fluxes are given

by [5]:

I in = C in + S in(t ) (11)

I out = C out + S out (t ) (12)

where C in and C out correspond to constant radiation fluxes at z = 0 and z = L and S in(t ) andS 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 (13)

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

τ M ∂ Z (t )

∂ t + Z (t ) = β 2 S in(t ) (14)

whose analytical solution is:

Z (t ) = β 2e−tτ M

τ M

t

−t 0

S in(θ ) eθ τ M dθ (15)

where t 0 corresponds to the time that S in(t 0) = 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 (16)

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 (17)

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 (18)

It is important to point out that equation 17 determines the intensity range where the modelis valid, because the approximated transmission equation (Eq. 10) is accomplished when

α 0 L >> β 2C 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:

dS out (t )

dt

t smax

= dZ (t )

dt

t smax

(C in + S in(t smax)) e Z (t smax)−α 0 L+β 2 C in (19)

Therefore, it follows that the sign of d S out (t )/dt |t smax is determined by the sign of dZ (t )

dt |t smax

which can be obtained from Eq. 14:

∂ Z (t )

∂ t

t smax

= τ −1 M (β 2 S in(t smax)− Z (t smax)) (20)

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

Z (t smax) = β 2e−tsmaxτ M

τ M t smax

−t 0 S in(θ ) e

θ τ M dθ (21)

and taking into account that t 0 + t smax > 0, it follows that:

Z (t smax) = β 2e−tsmaxτ M

τ M

t smax

−t 0

S in(θ ) eθ τ M dθ ≤ β 2 e

−tsmaxτ M Sin(tsmax)

e

tsmaxτ M − e

−t0τ M

≤

(22)

≤ β 2 S in(t smax)

1− e

−(t 0+t smax)τ M

≤ β 2 S in(t smax)

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 saturableabsorber 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 sin2

π (t − t 0)

τ in

(23)

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

Z (t ) =

β 2 I 0

τ 2in + 4

1− e

−(t −t 0)τ M

π 2τ 2 M − τ in

2πτ M sin

2π (t −t 0)

τ in

+ τ in cos

2π (t −t 0 )

τ in

2τ 2in

+ 4π 2τ 2 M

(24)

Finally, the distortion of the pulse has been analyzed by means of the following equation

[15, 16]:





D =

t 1

−t 0

|Snout (t )−Snin(t − τ D)|dt

t 1

−t 0

Snin(t )dt

12

(25)

where τ D is the delay of the pulse, t 1 is time where the input signal is null (end of first cycle),Snout (t ) is the normalized output intensity signal and Snin(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 µ m. The

linearly p-polarized pump beam is obtained from a frequency doubled Nd:VO 4 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 beamsof 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.

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

0.1 µ m.

(D)

(D)

(PEM)

(bR)

(Oscilloscope)

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1

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

bacteriorhodopsin film.





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 I 0, C in and t 0 and τ in with regression coefficients (r 2

)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] (β 2 = 158.71τ M cm2/W ) and the previously

obtained parameters of the reference signal ( I 0, C in, t 0 and τ in) have been used, obtaining α 0 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.

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.0

0.2

0.4

0.6

0.8

1.0

Time s

I n t . a . u

Sout Fitted

Sout Exp.

Sin Fitted

Sin Exp.

Fig. 2. Temporal variation of the experimental and fitted sinusoidal modulated beam

(signal) and the corresponding experimental and fitted reference beam, where C in = 0.4mW /cm2, I 0 = 5.5 mW /cm2 and the regression coefficients of the reference and signal

beams were 0.999 and 0.992 respectively.

In Fig. 3, the fitted parameters (τ M , and α 0 L) of the signal output intensity curves are an-

alyzed 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 α 0 L 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 correspond-

ing previously reported value [10, 17]. In this case, continuous lines are the mean value of all

α 0 L-parameters.

In Fig. 4 the obtained experimental and theoretical time delay, fractional delay and distortion





0.004 0.006 0.008 0.010 0.012 0.0140.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

I 0C in Wcm2

Τ M

s

a

0.2 Hz

0.7 Hz

1 Hz

1.5 Hz

0.004 0.006 0.008 0.010 0.012 0.0140

1

2

3

4

5

I 0C in Wcm2

Α L

b

0.2 Hz

0.7 Hz

1 Hz

1.5 Hz

Fig. 3. Variation of the fitted parameters of the signal intensity curves as a function of the

total intensity: τ M (a) and α 0 L (b). The ratio C in/ I 0 oscillates between 0.05 to 0.12. Orange

line correspond to the mean value of all the obtained parameters.

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 fre-

quencies. Continuous lines were obtained by using Eq. 18 taking into account Eq. 24 and the

mean value of α 0 L 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 de-

lays 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].





0.004 0.006 0.008 0.010 0.012 0.0140.00

0.05

0.10

0.15

0.20

0.25

0.30

I 0C in Wcm2

Τ D

s

a

0.2 Hz

0.7 Hz

1 Hz

1.5 Hz

0.004 0.006 0.008 0.010 0.012 0.0140.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

I 0C in Wcm2

F

b

0.2 Hz

0.7 Hz

1 Hz

1.5 Hz

0.004 0.006 0.008 0.010 0.012 0.0140.0

0.1

0.2

0.3

0.4

0.5

I 0C in Wcm2

D

c

0.2 Hz

0.7 Hz

1 Hz

1.5 Hz

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/ I 0 oscillates

between 0.05 to 0.12. Theoretical simulations by using Eq. 18 taking into account Eq. 24,the mean value of α 0 L 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).





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 agreementbetween 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 acknowledgesupport from project FIS2009-11065of Ministerio de Ciencia e Inno-

vacion of Spain and ACOMP/2012/151 from the Consellerıa d’Educacio, Formacio i Ocupacio

de la Generalitat Valenciana.



blaya s_2014_pulse delay br films

Documents