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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 7, Iss. 12 — Dec. 19, 2012
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Rigorous analysis of the propagation of sinusoidal pulses in bacteriorhodopsin films

Pablo Acebal, Salvador Blaya, Luis Carretero, R. F. Madrigal, and A. Fimia  »View Author Affiliations


Optics Express, Vol. 20, Issue 23, pp. 25497-25512 (2012)
http://dx.doi.org/10.1364/OE.20.025497


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Abstract

The propagation of sinusoidal pulses in bacteriorhodopsin films has been theoretically analyzed using a complete study of the photoinduced processes that take into account all the physical parameters, the coupling of rate equations with the energy transfer equation and the temperature change during the experiment. The theoretical approach was compared to experimental data and a good concordance was observed. This theoretical treatment, can be widely applied, i.e when arbitrary pump and/or signal is used or in the case of the pump and signal beams have different wavelengths. Due to we have performed a rigorous analysis, from this treatment the corresponding two level approximation has also been analyzed for these systems.

© 2012 OSA

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

1. J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc Am B 22, 1062–1074 (2005). [CrossRef]

, 2

2. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007). [CrossRef]

], optical memories [3

3. M. A. Anton and F. Carreno, “Quantum memory and all-optical switching in positive charged quantum dots via Zeeman coherent oscillations,” J. Opt. 12, 104006 (2010). [CrossRef]

, 4

4. N. Akopian, L. Wang, A. Rastelli, O. G. Schmidt, and V. Zwiller, “Hybrid semiconductor-atomic interface: slowing down single photons from a quantum dot rid f-4017-2010,” Nat. Photonics 5, 230–233 (2011). [CrossRef]

], the enhancement of the sensitivity of interferometers [5

5. Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J. C. Howell, “Slow-light fourier transform interferometer.” Phys Rev Lett 99, 240801(1–4) (2007). [CrossRef]

, 6

6. J. F. Wang, Y. D. Zhang, X. N. Zhang, H. Tian, H. Wu, Y. X. Cai, J. Zhang, and P. Yuan, “Enhancing the sensitivity of fiber Mach-Zehnder interferometers using slow and fast light,” Opt. Lett 36, 3173–3175 (2011). [CrossRef] [PubMed]

], and the improvement of nonlinear effects [7

7. Y. Dumeige, “Quasi-phase-matching and second-harmonic generation enhancement in a semiconductor microresonator array using slow-light effects,” Phys Rev A 83, 045802 (2011). [CrossRef]

]. In order to slow down the group velocity of a propagating pulse several schemes with different kind of materials have widely demonstrated [8

8. R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56, 1908–1915 (2009). [CrossRef]

10

10. G. S. Agarwal and T. N. Dey, “Non-electromagnetically induced transparency mechanisms for slow light,” Laser Photonics Rev 3, 287–300 (2009). [CrossRef]

]. Among the applied methodologies, good results were obtained by electromagnetically induced transparency (EIT) [11

11. A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, “Electromagnetically induced transparency - propagation dynamics,” Phys Rev Lett 74, 2447–2450 (1995). [CrossRef] [PubMed]

13

13. L. V. Hau, “Optical information processing in Bose-Einstein condensates,” Nat. Photonics 2, 451–453 (2008). [CrossRef]

], coherent population oscillations (CPO) [14

14. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003). [CrossRef] [PubMed]

, 15

15. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys Rev Lett 90, 113903 (2003). [CrossRef] [PubMed]

], atomic double resonances [16

16. R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys Rev A 73, 063812 (2006). [CrossRef]

, 17

17. R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-optical delay of images using slow light,” Phys Rev Lett 98, 043902 (2007). [CrossRef]

], photonic crystal waveguides [18

18. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005). [CrossRef] [PubMed]

20

20. M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73, 096501 (2010). [CrossRef]

], coupled resonator optical waveguides (CROWs) [2

2. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007). [CrossRef]

, 21

21. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett 24, 711–713 (1999). [CrossRef]

, 22

22. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys Rev A 69, 063804(1–6) (2004). [CrossRef]

], holographic induced transparency [23

23. L. Carretero, S. Blaya, P. Acebal, A. Fimia, R. Madrigal, and A. Murciano, “Coupled wave analysis of holographically induced transparency (HIT) generated by two multiplexed volume gratings,” Opt Express 19, 7094–7105 (2011). [CrossRef] [PubMed]

,24

24. L. Carretero, S. Blaya, A. Murciano, P. Acebal, A. Fimia, and R. Madrigal, “Coupled-wave theory analysis of holographic structures for slow-light applications,” Holography: Advances and Modern Trends II 8074, 807417 (2011).

] and stimulated Brillouin scattering (SBS) in optical fibers [25

25. Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201–206 (2007). [CrossRef]

,26

26. L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008). [CrossRef]

].

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

27. P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys Rev Lett. 95, 253601 (2005). [CrossRef] [PubMed]

29

29. C. S. Yelleswarapu, S. Laoui, R. Philip, and D. V. G. L. N. Rao, “Coherent population oscillations and superluminal light in a protein complex,” Opt. Express 16, 3844–3852 (2008). [CrossRef] [PubMed]

]. 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

27. P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys Rev Lett. 95, 253601 (2005). [CrossRef] [PubMed]

]. Moreover, it was also demonstrated the possibility to slow-down the group velocity by the use of a second beam at 442 nm [27

27. P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys Rev Lett. 95, 253601 (2005). [CrossRef] [PubMed]

]. 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

28. C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007). [CrossRef] [PubMed]

]. 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

30. V. S. Zapasskii and G. G. Kozlov, “A saturable absorber, coherent population oscillations, and slow light,” Opt. Spectrosc. 100, 419–424 (2006). [CrossRef]

33

33. A. C. Selden, “Practical tests for distinguishing slow light from saturable absorption,” Opt. Express 18, 13204–13211 (2010). [CrossRef] [PubMed]

] or by coherent population oscillations [27

27. P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys Rev Lett. 95, 253601 (2005). [CrossRef] [PubMed]

, 28

28. C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007). [CrossRef] [PubMed]

]. 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

33. A. C. Selden, “Practical tests for distinguishing slow light from saturable absorption,” Opt. Express 18, 13204–13211 (2010). [CrossRef] [PubMed]

].

2. Theoretical background

The theoretical procedure and the corresponding parameters are the same as the previously described in reference [37

37. P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys Rev E 76, 016608 (2007). [CrossRef]

], 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

38. N. Hampp, A. Popp, C. Bruchle, and D. Oesterhelt, “Diffraction efficiency of bacteriorhodopsin films for holography containing bacteriorhodopsin wildtype BRWT and its variants BRD85E and BRD96N,” J. Phys. Chem. pp. 4679–4685 (1992). [CrossRef]

, 39

39. J. D. Downie and D. T. Smithey, “Measurements of holographic properties of bacteriorhodopsin films,” Appl. Opt. 35, 5780–5789 (1996). [CrossRef] [PubMed]

].

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

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:
Ia(ζ,t)ζ=(βaa(ζ,t)+βaaR(ζ,t))Ia(ζ,t)
(1)

Where the B and nia vectors are B={niaB/t,niaK/t,niaL/t,niaM/t,niaN/t,1} and nia={niaB,niaK,niaL,niaM,niaN,niaO}, while the coefficient matrix A is:
A=(ΦiaBΦiaKΦiaL0ΦiaNkobΦiaB(ΦiaK+kkl)klk0000kkl(ΦiaL+klk+klm)kml0000klm(kml+kmn)knm0000kmn(ΦiaN+knm+kno)kon111111)
(5)

Where the terms Φiaκ are given by:
Φiaκ=ϕκh¯ωσiiκIa
(6)

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

40. Q. W. Song, C. Zhang, R. Blumer, R. B. Gross, Z. Chen, and R. Birge, “Chemically enhanced bacteriorhodopsin thin-film spatial light modulator,” Opt. Lett. 18, 1373–1375 (1993). [CrossRef] [PubMed]

,41

41. E. Korchemskaya, D. Stepanchikov, and T. Dyukova, “Photoinduced anisotropy in chemically-modified films of bacteriorhodopsin and its genetic mutants,” Opt. Mater 14, 185–191 (2000). [CrossRef]

] in polymer matrix. We have employed the corresponding values of quantum efficiencies and rate constants that we previously obtained in reference [37

37. P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys Rev E 76, 016608 (2007). [CrossRef]

]. 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

42. H. Eyring, “The activated complex and the absolute rate of chemical reactions,” Chem. Rev. 17, 65–77 (1935). [CrossRef]

], which allows us to represent the rate constant at a given temperature T1 as a function of the k0 at another temperature T0:
k1=KBT1h(hk0KBT0)T0/T1
(7)

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

43. V. May and O. Khn, Charge and energy transfer dynamics in molecular system (Wiley-VCH, 2000).

, 44

44. C. Penney, “Light scattering in term of oscillator strenghts and refractive indices,” J. Opt. Soc. Am. 59, 34–42 (1969). [CrossRef]

]:
σiiκ(ω)=ω(μeg,iκ)2ch¯ε0Exp[(ωωegκ)22γ2]2πγ
(9)
σii,Rκ(ω)=24π(αiiκ(ω))2ω49c4
(10)

Where α0 denotes the sum of the traces of static polarizability of the 296 aminoacids that form the bR protein and α̂ 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.:
αiiCκ(ω)e,e1αii,eCκ(0)+αii,1Cκ(0)Ω1Cκ(ω;ω)
(12)

Where αii,eCκ(0) and Ω1κ(ω) are:
αii,eCκ(0)=2(μeg,iκ)2h¯ε0ωegκ
(13)
Ω1Cκ(ω;ω)=(ω1gκ)2((ω1gκ)2ω2)((ω1gκ)2ω2)2+Γ2ω2
(14)

Table 1. Values for the microscopic optical properties of the different states of bR (μ1g and λ1g from references [45, 46]). γ=Γ/2.355=2×1014s−1, α0=2.76×10−26m3 and rate constants at 293 K are: kkl=2300 s−1, klm=169 s−1, kmn=0.98 s−1, kno=110 s−1, kob=5.2 s−1, klk=230 s−1, kml=64.6 s−1, knm=40 s−1, kon=0.1 s−1.

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At this point, the analysis of the beam propagation is described by a set of coupled differential equations (Eqs. 114) 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:
Ia[j]=Ia[j1]e(κi3Niaκ(j,t)(σiiκ(ω)+σii,Rκ(ω))+i3Niamσii,Rm)Δζ
(15)
Where the Niaκ(j,t) were solved by the rate equations, which only depend on the time variable using Eqs. 214 and the parameters given in table 1, taking into account that the intensity function needed to solve the j layer was that of the j-1 layer.

2.2. Two level model

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.

4. Results and discussion

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

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
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

Fig. 6 Temporal variation of the normalized population densities (nX) 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.

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

28. C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007). [CrossRef] [PubMed]

]. These results have been well explained by the saturable absorber theory and CPO model [28

28. C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007). [CrossRef] [PubMed]

, 32

32. A. C. Selden, “Slow light and saturable absorption,” Opt. Spectrosc. 106, 881–888 (2009). [CrossRef]

]. 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

32. A. C. Selden, “Slow light and saturable absorption,” Opt. Spectrosc. 106, 881–888 (2009). [CrossRef]

, 50

50. V. S. Zapasskii and G. G. Kozlov, “On two models of light pulse delay in saturable absorber,” Opt. Spectrosc. 109, 407–412 (2010). [CrossRef]

], being the theoretical behavior curve the same as the experimental data previously measured in aqueous bR [28

28. C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007). [CrossRef] [PubMed]

]. 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.

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

27. P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys Rev Lett. 95, 253601 (2005). [CrossRef] [PubMed]

, 28

28. C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007). [CrossRef] [PubMed]

], 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

27. P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys Rev Lett. 95, 253601 (2005). [CrossRef] [PubMed]

]. 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.
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.

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 (nX) 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.

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.

Acknowledgment

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.

References and links

1.

J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc Am B 22, 1062–1074 (2005). [CrossRef]

2.

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007). [CrossRef]

3.

M. A. Anton and F. Carreno, “Quantum memory and all-optical switching in positive charged quantum dots via Zeeman coherent oscillations,” J. Opt. 12, 104006 (2010). [CrossRef]

4.

N. Akopian, L. Wang, A. Rastelli, O. G. Schmidt, and V. Zwiller, “Hybrid semiconductor-atomic interface: slowing down single photons from a quantum dot rid f-4017-2010,” Nat. Photonics 5, 230–233 (2011). [CrossRef]

5.

Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J. C. Howell, “Slow-light fourier transform interferometer.” Phys Rev Lett 99, 240801(1–4) (2007). [CrossRef]

6.

J. F. Wang, Y. D. Zhang, X. N. Zhang, H. Tian, H. Wu, Y. X. Cai, J. Zhang, and P. Yuan, “Enhancing the sensitivity of fiber Mach-Zehnder interferometers using slow and fast light,” Opt. Lett 36, 3173–3175 (2011). [CrossRef] [PubMed]

7.

Y. Dumeige, “Quasi-phase-matching and second-harmonic generation enhancement in a semiconductor microresonator array using slow-light effects,” Phys Rev A 83, 045802 (2011). [CrossRef]

8.

R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56, 1908–1915 (2009). [CrossRef]

9.

R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science 326, 1074–1077 (2009). [CrossRef] [PubMed]

10.

G. S. Agarwal and T. N. Dey, “Non-electromagnetically induced transparency mechanisms for slow light,” Laser Photonics Rev 3, 287–300 (2009). [CrossRef]

11.

A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, “Electromagnetically induced transparency - propagation dynamics,” Phys Rev Lett 74, 2447–2450 (1995). [CrossRef] [PubMed]

12.

M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev Mod Phys 75, 457–472 (2003). [CrossRef]

13.

L. V. Hau, “Optical information processing in Bose-Einstein condensates,” Nat. Photonics 2, 451–453 (2008). [CrossRef]

14.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301, 200–202 (2003). [CrossRef] [PubMed]

15.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys Rev Lett 90, 113903 (2003). [CrossRef] [PubMed]

16.

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys Rev A 73, 063812 (2006). [CrossRef]

17.

R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-optical delay of images using slow light,” Phys Rev Lett 98, 043902 (2007). [CrossRef]

18.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005). [CrossRef] [PubMed]

19.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008). [CrossRef]

20.

M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73, 096501 (2010). [CrossRef]

21.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett 24, 711–713 (1999). [CrossRef]

22.

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys Rev A 69, 063804(1–6) (2004). [CrossRef]

23.

L. Carretero, S. Blaya, P. Acebal, A. Fimia, R. Madrigal, and A. Murciano, “Coupled wave analysis of holographically induced transparency (HIT) generated by two multiplexed volume gratings,” Opt Express 19, 7094–7105 (2011). [CrossRef] [PubMed]

24.

L. Carretero, S. Blaya, A. Murciano, P. Acebal, A. Fimia, and R. Madrigal, “Coupled-wave theory analysis of holographic structures for slow-light applications,” Holography: Advances and Modern Trends II 8074, 807417 (2011).

25.

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25, 201–206 (2007). [CrossRef]

26.

L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008). [CrossRef]

27.

P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys Rev Lett. 95, 253601 (2005). [CrossRef] [PubMed]

28.

C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett. 32, 1788–1790 (2007). [CrossRef] [PubMed]

29.

C. S. Yelleswarapu, S. Laoui, R. Philip, and D. V. G. L. N. Rao, “Coherent population oscillations and superluminal light in a protein complex,” Opt. Express 16, 3844–3852 (2008). [CrossRef] [PubMed]

30.

V. S. Zapasskii and G. G. Kozlov, “A saturable absorber, coherent population oscillations, and slow light,” Opt. Spectrosc. 100, 419–424 (2006). [CrossRef]

31.

B. Macke and B. Segard, “Slow light in saturable absorbers,” Phys Rev A 78, 013817 (2008). [CrossRef]

32.

A. C. Selden, “Slow light and saturable absorption,” Opt. Spectrosc. 106, 881–888 (2009). [CrossRef]

33.

A. C. Selden, “Practical tests for distinguishing slow light from saturable absorption,” Opt. Express 18, 13204–13211 (2010). [CrossRef] [PubMed]

34.

A. C. Selden, “Pulse transmission through a saturable absorber,” Brit. J. Appl. Phys. 18, 743–748 (1967). [CrossRef]

35.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Ultra-slow and superluminal light propagation in solids at room temperature,” J. Phys-Cond Mat 16, R1321–R1340 (2004). [CrossRef]

36.

G. S. Agarwal and T. N. Dey, “Sub- and superluminal propagation of intense pulses in media with saturated and reverse absorption,” Phys Rev Lett 92, 203901 (2004). [CrossRef] [PubMed]

37.

P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys Rev E 76, 016608 (2007). [CrossRef]

38.

N. Hampp, A. Popp, C. Bruchle, and D. Oesterhelt, “Diffraction efficiency of bacteriorhodopsin films for holography containing bacteriorhodopsin wildtype BRWT and its variants BRD85E and BRD96N,” J. Phys. Chem. pp. 4679–4685 (1992). [CrossRef]

39.

J. D. Downie and D. T. Smithey, “Measurements of holographic properties of bacteriorhodopsin films,” Appl. Opt. 35, 5780–5789 (1996). [CrossRef] [PubMed]

40.

Q. W. Song, C. Zhang, R. Blumer, R. B. Gross, Z. Chen, and R. Birge, “Chemically enhanced bacteriorhodopsin thin-film spatial light modulator,” Opt. Lett. 18, 1373–1375 (1993). [CrossRef] [PubMed]

41.

E. Korchemskaya, D. Stepanchikov, and T. Dyukova, “Photoinduced anisotropy in chemically-modified films of bacteriorhodopsin and its genetic mutants,” Opt. Mater 14, 185–191 (2000). [CrossRef]

42.

H. Eyring, “The activated complex and the absolute rate of chemical reactions,” Chem. Rev. 17, 65–77 (1935). [CrossRef]

43.

V. May and O. Khn, Charge and energy transfer dynamics in molecular system (Wiley-VCH, 2000).

44.

C. Penney, “Light scattering in term of oscillator strenghts and refractive indices,” J. Opt. Soc. Am. 59, 34–42 (1969). [CrossRef]

45.

J. Y. Huang, Z. Chen, and A. Lewis, “Second-harmonic generation in purple membrane-poly(vinyl alcohol) films: probing the dipolar characteristics of the bacteriorhodopsin chromophore in bR570 and M412,” J. Phys. Chem. 93, 3314–3320 (1989). [CrossRef]

46.

J. A. Stuart, D. L. Marcy, K. J. Wise, and R. R. Birge, “Volumetric optical memory based on bacterirhodopsin,” Synth. Met. 127, 3–15 (2002). [CrossRef]

47.

P. Acebal, L. Carretero, S. Blaya, R. F. Madrigal, A. Murciano, and A. Fimia, “Simulation of diffraction efficiency in oriented bacteriorhodopsin films,” Advances in Computational Methods in Sciences and Engineering 2005, Vols 4 A & 4 B 4A–4B, 1–4 (2005).

48.

P. Acebal, S. Blaya, L. Carretero, and A. Fimia, Upper limits of dielectric permittivity modulation in bacteriorhodopsin films “Upper limits of dielectric permittivity modulation in bacteriorhodopsin films,” Phys Rev E 72, 011909 (2005). [CrossRef]

49.

M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Menucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petereson, P. Y. Ayala, Q. Cui, K. Morokuma, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P. M. W. Gill, B. G. Johnson, W. Chen, M. W. Wong, J. L. Andres, M. Head-Gordon, E. S. Replogle, and J. A. Pople, GAUSSIAN 98, Revision A.7, Gaussian, Inc, Pittsburg PA, 1998.

50.

V. S. Zapasskii and G. G. Kozlov, “On two models of light pulse delay in saturable absorber,” Opt. Spectrosc. 109, 407–412 (2010). [CrossRef]

OCIS Codes
(190.4710) Nonlinear optics : Optical nonlinearities in organic materials
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(160.1435) Materials : Biomaterials

ToC Category:
Nonlinear Optics

History
Original Manuscript: August 27, 2012
Revised Manuscript: October 1, 2012
Manuscript Accepted: October 1, 2012
Published: October 25, 2012

Virtual Issues
Vol. 7, Iss. 12 Virtual Journal for Biomedical Optics

Citation
Pablo Acebal, Salvador Blaya, Luis Carretero, R. F. Madrigal, and A. Fimia, "Rigorous analysis of the propagation of sinusoidal pulses in bacteriorhodopsin films," Opt. Express 20, 25497-25512 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-23-25497


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References

  1. J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc Am B22, 1062–1074 (2005). [CrossRef]
  2. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics1, 65–71 (2007). [CrossRef]
  3. M. A. Anton and F. Carreno, “Quantum memory and all-optical switching in positive charged quantum dots via Zeeman coherent oscillations,” J. Opt.12, 104006 (2010). [CrossRef]
  4. N. Akopian, L. Wang, A. Rastelli, O. G. Schmidt, and V. Zwiller, “Hybrid semiconductor-atomic interface: slowing down single photons from a quantum dot rid f-4017-2010,” Nat. Photonics5, 230–233 (2011). [CrossRef]
  5. Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J. C. Howell, “Slow-light fourier transform interferometer.” Phys Rev Lett99, 240801(1–4) (2007). [CrossRef]
  6. J. F. Wang, Y. D. Zhang, X. N. Zhang, H. Tian, H. Wu, Y. X. Cai, J. Zhang, and P. Yuan, “Enhancing the sensitivity of fiber Mach-Zehnder interferometers using slow and fast light,” Opt. Lett36, 3173–3175 (2011). [CrossRef] [PubMed]
  7. Y. Dumeige, “Quasi-phase-matching and second-harmonic generation enhancement in a semiconductor microresonator array using slow-light effects,” Phys Rev A83, 045802 (2011). [CrossRef]
  8. R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt.56, 1908–1915 (2009). [CrossRef]
  9. R. W. Boyd and D. J. Gauthier, “Controlling the velocity of light pulses,” Science326, 1074–1077 (2009). [CrossRef] [PubMed]
  10. G. S. Agarwal and T. N. Dey, “Non-electromagnetically induced transparency mechanisms for slow light,” Laser Photonics Rev3, 287–300 (2009). [CrossRef]
  11. A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, “Electromagnetically induced transparency - propagation dynamics,” Phys Rev Lett74, 2447–2450 (1995). [CrossRef] [PubMed]
  12. M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev Mod Phys75, 457–472 (2003). [CrossRef]
  13. L. V. Hau, “Optical information processing in Bose-Einstein condensates,” Nat. Photonics2, 451–453 (2008). [CrossRef]
  14. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science301, 200–202 (2003). [CrossRef] [PubMed]
  15. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys Rev Lett90, 113903 (2003). [CrossRef] [PubMed]
  16. R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys Rev A73, 063812 (2006). [CrossRef]
  17. R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-optical delay of images using slow light,” Phys Rev Lett98, 043902 (2007). [CrossRef]
  18. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature438, 65–69 (2005). [CrossRef] [PubMed]
  19. T. Baba, “Slow light in photonic crystals,” Nat. Photonics2, 465–473 (2008). [CrossRef]
  20. M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys.73, 096501 (2010). [CrossRef]
  21. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett24, 711–713 (1999). [CrossRef]
  22. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys Rev A69, 063804(1–6) (2004). [CrossRef]
  23. L. Carretero, S. Blaya, P. Acebal, A. Fimia, R. Madrigal, and A. Murciano, “Coupled wave analysis of holographically induced transparency (HIT) generated by two multiplexed volume gratings,” Opt Express19, 7094–7105 (2011). [CrossRef] [PubMed]
  24. L. Carretero, S. Blaya, A. Murciano, P. Acebal, A. Fimia, and R. Madrigal, “Coupled-wave theory analysis of holographic structures for slow-light applications,” Holography: Advances and Modern Trends II8074, 807417 (2011).
  25. Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol.25, 201–206 (2007). [CrossRef]
  26. L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics2, 474–481 (2008). [CrossRef]
  27. P. F. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys Rev Lett.95, 253601 (2005). [CrossRef] [PubMed]
  28. C. S. Yelleswarapu, R. Philip, F. J. Aranda, B. R. Kimball, and D. V. G. L. N. Rao, “Slow light in bacteriorhodopsin solution using coherent population oscillations,” Opt. Lett.32, 1788–1790 (2007). [CrossRef] [PubMed]
  29. C. S. Yelleswarapu, S. Laoui, R. Philip, and D. V. G. L. N. Rao, “Coherent population oscillations and superluminal light in a protein complex,” Opt. Express16, 3844–3852 (2008). [CrossRef] [PubMed]
  30. V. S. Zapasskii and G. G. Kozlov, “A saturable absorber, coherent population oscillations, and slow light,” Opt. Spectrosc.100, 419–424 (2006). [CrossRef]
  31. B. Macke and B. Segard, “Slow light in saturable absorbers,” Phys Rev A78, 013817 (2008). [CrossRef]
  32. A. C. Selden, “Slow light and saturable absorption,” Opt. Spectrosc.106, 881–888 (2009). [CrossRef]
  33. A. C. Selden, “Practical tests for distinguishing slow light from saturable absorption,” Opt. Express18, 13204–13211 (2010). [CrossRef] [PubMed]
  34. A. C. Selden, “Pulse transmission through a saturable absorber,” Brit. J. Appl. Phys.18, 743–748 (1967). [CrossRef]
  35. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Ultra-slow and superluminal light propagation in solids at room temperature,” J. Phys-Cond Mat16, R1321–R1340 (2004). [CrossRef]
  36. G. S. Agarwal and T. N. Dey, “Sub- and superluminal propagation of intense pulses in media with saturated and reverse absorption,” Phys Rev Lett92, 203901 (2004). [CrossRef] [PubMed]
  37. P. Acebal, L. Carretero, S. Blaya, A. Murciano, and A. Fimia, “Theoretical approach to photoinduced inhomogeneous anisotropy in bacteriorhodopsin films,” Phys Rev E76, 016608 (2007). [CrossRef]
  38. N. Hampp, A. Popp, C. Bruchle, and D. Oesterhelt, “Diffraction efficiency of bacteriorhodopsin films for holography containing bacteriorhodopsin wildtype BRWT and its variants BRD85E and BRD96N,” J. Phys. Chem. pp. 4679–4685 (1992). [CrossRef]
  39. J. D. Downie and D. T. Smithey, “Measurements of holographic properties of bacteriorhodopsin films,” Appl. Opt.35, 5780–5789 (1996). [CrossRef] [PubMed]
  40. Q. W. Song, C. Zhang, R. Blumer, R. B. Gross, Z. Chen, and R. Birge, “Chemically enhanced bacteriorhodopsin thin-film spatial light modulator,” Opt. Lett.18, 1373–1375 (1993). [CrossRef] [PubMed]
  41. E. Korchemskaya, D. Stepanchikov, and T. Dyukova, “Photoinduced anisotropy in chemically-modified films of bacteriorhodopsin and its genetic mutants,” Opt. Mater14, 185–191 (2000). [CrossRef]
  42. H. Eyring, “The activated complex and the absolute rate of chemical reactions,” Chem. Rev.17, 65–77 (1935). [CrossRef]
  43. V. May and O. Khn, Charge and energy transfer dynamics in molecular system (Wiley-VCH, 2000).
  44. C. Penney, “Light scattering in term of oscillator strenghts and refractive indices,” J. Opt. Soc. Am.59, 34–42 (1969). [CrossRef]
  45. J. Y. Huang, Z. Chen, and A. Lewis, “Second-harmonic generation in purple membrane-poly(vinyl alcohol) films: probing the dipolar characteristics of the bacteriorhodopsin chromophore in bR570 and M412,” J. Phys. Chem.93, 3314–3320 (1989). [CrossRef]
  46. J. A. Stuart, D. L. Marcy, K. J. Wise, and R. R. Birge, “Volumetric optical memory based on bacterirhodopsin,” Synth. Met.127, 3–15 (2002). [CrossRef]
  47. P. Acebal, L. Carretero, S. Blaya, R. F. Madrigal, A. Murciano, and A. Fimia, “Simulation of diffraction efficiency in oriented bacteriorhodopsin films,” Advances in Computational Methods in Sciences and Engineering 2005, Vols 4 A & 4 B4A–4B, 1–4 (2005).
  48. P. Acebal, S. Blaya, L. Carretero, and A. Fimia, Upper limits of dielectric permittivity modulation in bacteriorhodopsin films “Upper limits of dielectric permittivity modulation in bacteriorhodopsin films,” Phys Rev E72, 011909 (2005). [CrossRef]
  49. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Menucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petereson, P. Y. Ayala, Q. Cui, K. Morokuma, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P. M. W. Gill, B. G. Johnson, W. Chen, M. W. Wong, J. L. Andres, M. Head-Gordon, E. S. Replogle, and J. A. Pople, GAUSSIAN 98, Revision A.7, Gaussian, Inc, Pittsburg PA, 1998.
  50. V. S. Zapasskii and G. G. Kozlov, “On two models of light pulse delay in saturable absorber,” Opt. Spectrosc.109, 407–412 (2010). [CrossRef]

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