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Study of some laser signals emergent from nonlinear optical media
Computer Physics Communications 147 (2002) 633–636 www.elsevier.com/locate/cpc
Study of some laser signals emergent from nonlinear optical media Mihaela Ghelmez ∗ , Cristian I. Toma, Paul E. Sterian Politehnica University of Bucharest, 77 206 Bucharest, Romania
Abstract The paper presents an experimental and computer study of laser signals emergent from thin layer samples of fatty acids and fatty acid-cholesterol mixtures, while in the mesomorphic state. In correlation with the amount of cholesterol in mixtures and with their response in external electric fields, we studied some nonlinear optical effects and the appropriate conditions for their measurement. Computer-based methods for estimating the cholesterol amount in mixtures, and for taking into account the presence of fluctuations are presented. 2002 Elsevier Science B.V. All rights reserved. PACS: 42.62.Hk; 83.80.Lz Keywords: Laser signal; Fatty acids; Cholesterol; Liquid crystal; Nonlinear optics; Computer processing; Test function
1. Introduction Fatty acids (FA) and cholesterol (Ch) are important substances for the living matter, especially for the biological membrane [1]. Since the liquid crystal (LC) state of these substances can give information on some membrane mechanisms [2], we studied their answer to some external stimuli within the mesomorphism interval. It is known [3] that the relative motion of molecules varies considerably in terms of membrane type, from highly mobile arrangements to rigid structures. FA–Ch mixtures undergo a melting process by which molecules turn from a close-packed crystalline or a gel-type structure into a disordered liquid phase, via a LC phase. * Corresponding author.
E-mail address: [email protected] (M. Ghelmez).
The possibility of inducing a non-linearity in such systems could lead to a radical change of their dynamics. Interesting non-linear optical laser based answers were obtained in different FA and FA–Ch thin film samples [4]. We analyzed these effect answer and the measurement procedures. Taking advantage of the TableCurve3D program, it was used to fit the experimental dependencies of the output signals on different input physical amounts and to forecast the best experimental conditions. A method to estimate the cholesterol percentage in a mixture with fatty acids was found. Then we analyzed the processing of the signal received by the measuring stage by using Runge–Kutta equations in MATLAB. As it is known, in averaging procedures the user is interested in the mean value of the received signal over a certain time interval. Since the methods usually used for sampling the received signal based on an integration are very sensi-
0010-4655/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 2 ) 0 0 3 5 4 - 5
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M. Ghelmez et al. / Computer Physics Communications 147 (2002) 633–636
tive at random variations of the integration period— generated by the switching phenomena at the end of the integration—a multiplication of the received signal by a test function is recommended. We present some invariance properties of differential equations, which can be used for generating a “practical” test function (PTF), at this time interval. We were looking for truncated test functions that differ from zero only at a certain interval and have only some continuous derivatives on the real axis. We also present the properties of second order processing systems, considered to generate PTF, in filtering and sampling procedures [5]. Numerical simulations used Runge–Kutta equations of order 4–5 in MATLAB.
2. Materials and methods Some components or forerunners of the biological membrane, namely: (AR) arachidic acid CH3 (CH2 )18 · COOH; (LA) lauric acid CH3 (CH2 )10 COOH; (BA) butyric acid CH3 CH2 CH2 COOH; (AA) arachidonic acid (all-cis-5,8,11,14-eicosatetraenoic), and Ch (C27 H45 OH) mixtures in different molar percentages were studied. The following mixtures: (0.5% AR; 0.25% LA; 0.25% BA); (0.45% AR; 0.25% LA; 0.15% BA; 0.15% Ch); (0.25% AR; 0.25% LA; 0.25% BA, 0.25% Ch) were subjected to external electric fields and to c.w. and pulsed laser beams. The substances were encapsulated by melt drawing into 20–25 µm thickness cells with transparent flat electrodes of SnO2 . At room temperature (T = 300 K), thin films of such mixtures exhibited a thermotropic smectic-type liquid crystal (TSLC) state [4]. Under an external applied voltage, these systems have a dielectric feature, with weak conduction, the electric current dependence versus the applied voltage being hysteretical and depending on the amount of Ch. Considering the system equivalent with a flat condenser, we estimated the dielectric constant εr of the sample. The experimental results were processed with the TableCurve3D program by using the Surface-Fit All Equations option. Then the Numeric Summary option displayed all numerical information, including goodness of fit criteria, coefficient standard errors and confidence limits for the fitted parameters, function extreme, the fitting method, an analysis of variance, and
Fig. 1. Dielectric constant versus applied voltage and Ch amount in mixtures.
Fig. 2. Laser pulse width D versus the supplied voltage U and Ch amount in mixture.
data table statistics. The Precision Summary option can be used to determine how much precision is preserved in the current equation. Residuals Graph is a separate window displaying the residuals for the current surface fit. The data table can be weighted, if necessary. Fig. 1 presents the dielectric constant εr change in terms of the applied voltage and Ch percentage. Thus, the program gives the possibility to choose other experimental conditions in terms of the future experimental requirements and purposes. A helium-neon laser beam with λ = 6328 Å and 20 mW optical power was focused on the samples.
M. Ghelmez et al. / Computer Physics Communications 147 (2002) 633–636
The ring pattern obtained in far-field showed the occurrence of a self-phase modulation and an external self-focusing of the beam. The dependence of the output beam power in terms of the input one and the Ch percentage is generally hysteretical showing a memory effect of the samples. A refractive index change with optical intensity, in connection with the dielectric constant modification with the Ch percentage, occurred upon interaction with the laser light. Using an Nd3+ glass laser we emphasized the pulse width modification [6] in the time domain, displayed on a sampling oscilloscope and correlated with the Ch percentage (Fig. 2). Based on this, a method to estimate the amount of Ch by means of the output pulse width was developed [4].
635
in MATLAB a function f having a form similar to ϕ, but with an amplitude of about 10−12 smaller. We continue the analysis by studying a second order differential equation without free term, which has as possible solution the function ϕ: (2) f (2) = (6τ 4 − 2)/(τ 2 − 1) f. As initial conditions for f , f (1) we chose the values of ϕ, ϕ (1) at the moment τ0 = −1 + 0.01. The numerical simulation gave as a solution a PTF function similar to ϕ, but still with small amplitude, only four times greater than that previously obtained. Then we use now a test function having the form ϕa = exp[0.1/(τ 2 − 1)], and as second order differential equation without free term: f (2)(τ ) = (0.6τ 4 − 0.36τ 2 − 0.2)/(τ 2 − 1)4 f (τ ) (3)
3. Averaging procedures of the measured signal ϕa , ϕa(2) ,
The averaging procedure starts by writing a test function similar to a Dirac pulse ϕ = exp[1/(τ 2 − 1)] where τ = t − tsym , tsym being the middle of the time interval. Such a function has non zero values only for τ ∈ [−1, 1] and the derivatives ϕ (1) , ϕ (2) and ϕ (3) . We are looking for a differential equation with the function ϕ as a solution. This would imply the necessity for a derivative of certain order n—noted f (n) —to make a jump at the initial moment from the zero value to a non zero value, in contradiction with the property of the test functions to have continuous derivatives of any order on the whole real axis. Therefore, a test function could not be generated by a differential equation, but it would be quite possible for such an equation to possess as a solution a PTF function, i.e. a function with non zero values at the interval τ ∈ [−1, 1] and a certain number of continuous derivatives on the whole time axis. Therefore, we try to study such evolutions depending only on the values f, f (1) , f (n) equal to the values of ϕ, ϕ (1), . . . , ϕ (n) at a moment very close to τ = −1. The simplest differential equation satisfying these requirements has the form: f (1) = −2τ/(τ 2 − 1) f. (1) We chose as initial moment τ0 = −1 + 0.01 and as initial condition for f the value ϕ(τ0 ). By numerical simulation using the 4–5 order Runge–Kutta equations
and with suggested by the expressions of initial conditions for f, f (1) equal to 0.0002 and 0.02. The solution is a function very close to a rectangular unitary pulse and with the amplitude close to unity for more than 2/3 of the integration period. In our experiments with laser, a Gaussian A exp[−(τ − τm )2 /σ 2 ] signal must be processed. In this case, we consider null initial conditions for the system and add a free term in the differential equation. The working interval was chosen about 10 times greater than the pulse width. The time interval between the beginning of the working period and the moment corresponding to the maximum of the pulse was considered equal to 10σ , which means that the processing system was activated by the front of the pulse. The differential equation can be written as: f (2) = (0.6τ 4 − 0.36τ 2 − 0.2)/(τ 2 − 1)4 f (4) + A exp −(τ + 0.9)2 /(0.01)2 . The numerical simulations led to a generated pulse f with the width of 2 units, that can be more easily integrated by the electronic device, the result of the integration being proportional to the amplitude of the laser pulse. At the end of the integrating period, the integrating signal f is still high, about 10% of the peak obtained at the working interval. The best way of using a PTF f is by multiplying the expression of any received signal intensity “u” by f at the working interval (Fig. 3). Since f is similar to a step function,
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M. Ghelmez et al. / Computer Physics Communications 147 (2002) 633–636
Fig. 3. Function f generated by the equation without free term.
the result of the integration represents a measure for the average intensity of the laser beam. It must be also noticed that this result does not depend on the small stochastic variations of the integration period, usually generated by the switching phenomena, since the signal and its derivative have null values at the end of the integrating period. The properties of f are transmitted to the product f × u. Practically, using operational amplifiers with higher accuracy one can build a system generating the function f and working at a frequency of about 10 kHz. A linear system generating an oscillating signal at the frequency of 1 kHz has already been obtained. The multiplication of the optical signal received by the photodetector by f can be performed by supplying the photodetector with a voltage corresponding to the sum of a certain constant value f0 and the function f . f0 must be chosen so that the photodetector is at the limit of its sensitivity for f = 0; the photoelectric current has to appear for f > 0 and to disappear when f becomes zero again.
4. Conclusions FA and their mixtures with Ch present under some conditions the LC state, and a nonlinear feature while
in this state. The study of such effects can bring valuable information on some biological mechanisms, and open up the perspective of using some specific computer-based mathematical procedures. Our study concentrates on the computer processing of the experimental data, following the finding of new information on the studied systems, and of better working conditions. Numerical simulations using Runge–Kutta equations of order 4–5 in MATLAB helped us in improving the use of PTF for optical measurements. Thus, the influence of switching phenomena upon averaging methods usually used in sampling procedures has been practically rejected. The TableCurve3D program was used to fit the experimental dependencies of the output-input physical amounts. The program gives useful information on the fitting criteria, and the possibility to choose other experimental conditions, in terms of the requirements and purposes of future experiments. A method to estimate the Ch percentage in a mixture with FA was developed. Since FA and Ch are important substances for the living matter and especially for the biological membrane, we consider this study a promising bridge to biophysical studies by computer use.
References [1] A.C. Guyton, J.E. Hall, Textbook of Medical Physiology, 9th edn., Saunders, 1996. [2] G.H. Brown, J.J. Wolken, Liquid Crystals and Biological Structures, Academic Press, New York, 1979. [3] P.J. Quinn, D. Chapman, Biochem. Soc. Trans. 8 (1980) 38. [4] M. Dumitru (Ghelmez), M. Honciuc, M.C. Piscureanu, C. Gheorghe, Optical Engin. 35 (5) (1996) 1372. [5] S. Gong, H. Hentzell, S. Peterson, H. Hesselbon, B. Lofstedt, M. Hause, in: 8th Annual Intl. ASIC Conference, 18–21 Sept. 1995. [6] A. Adolf, D. Chatrefou, C. Guedard, J. Opt. (Paris) 15 (2) (1984) 83.
Study of some laser signals emergent from nonlinear optical media Mihaela Ghelmez ∗ , Cristian I. Toma, Paul E. Sterian Politehnica University of Bucharest, 77 206 Bucharest, Romania
Abstract The paper presents an experimental and computer study of laser signals emergent from thin layer samples of fatty acids and fatty acid-cholesterol mixtures, while in the mesomorphic state. In correlation with the amount of cholesterol in mixtures and with their response in external electric fields, we studied some nonlinear optical effects and the appropriate conditions for their measurement. Computer-based methods for estimating the cholesterol amount in mixtures, and for taking into account the presence of fluctuations are presented. 2002 Elsevier Science B.V. All rights reserved. PACS: 42.62.Hk; 83.80.Lz Keywords: Laser signal; Fatty acids; Cholesterol; Liquid crystal; Nonlinear optics; Computer processing; Test function
1. Introduction Fatty acids (FA) and cholesterol (Ch) are important substances for the living matter, especially for the biological membrane [1]. Since the liquid crystal (LC) state of these substances can give information on some membrane mechanisms [2], we studied their answer to some external stimuli within the mesomorphism interval. It is known [3] that the relative motion of molecules varies considerably in terms of membrane type, from highly mobile arrangements to rigid structures. FA–Ch mixtures undergo a melting process by which molecules turn from a close-packed crystalline or a gel-type structure into a disordered liquid phase, via a LC phase. * Corresponding author.
E-mail address: [email protected] (M. Ghelmez).
The possibility of inducing a non-linearity in such systems could lead to a radical change of their dynamics. Interesting non-linear optical laser based answers were obtained in different FA and FA–Ch thin film samples [4]. We analyzed these effect answer and the measurement procedures. Taking advantage of the TableCurve3D program, it was used to fit the experimental dependencies of the output signals on different input physical amounts and to forecast the best experimental conditions. A method to estimate the cholesterol percentage in a mixture with fatty acids was found. Then we analyzed the processing of the signal received by the measuring stage by using Runge–Kutta equations in MATLAB. As it is known, in averaging procedures the user is interested in the mean value of the received signal over a certain time interval. Since the methods usually used for sampling the received signal based on an integration are very sensi-
0010-4655/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 2 ) 0 0 3 5 4 - 5
634
M. Ghelmez et al. / Computer Physics Communications 147 (2002) 633–636
tive at random variations of the integration period— generated by the switching phenomena at the end of the integration—a multiplication of the received signal by a test function is recommended. We present some invariance properties of differential equations, which can be used for generating a “practical” test function (PTF), at this time interval. We were looking for truncated test functions that differ from zero only at a certain interval and have only some continuous derivatives on the real axis. We also present the properties of second order processing systems, considered to generate PTF, in filtering and sampling procedures [5]. Numerical simulations used Runge–Kutta equations of order 4–5 in MATLAB.
2. Materials and methods Some components or forerunners of the biological membrane, namely: (AR) arachidic acid CH3 (CH2 )18 · COOH; (LA) lauric acid CH3 (CH2 )10 COOH; (BA) butyric acid CH3 CH2 CH2 COOH; (AA) arachidonic acid (all-cis-5,8,11,14-eicosatetraenoic), and Ch (C27 H45 OH) mixtures in different molar percentages were studied. The following mixtures: (0.5% AR; 0.25% LA; 0.25% BA); (0.45% AR; 0.25% LA; 0.15% BA; 0.15% Ch); (0.25% AR; 0.25% LA; 0.25% BA, 0.25% Ch) were subjected to external electric fields and to c.w. and pulsed laser beams. The substances were encapsulated by melt drawing into 20–25 µm thickness cells with transparent flat electrodes of SnO2 . At room temperature (T = 300 K), thin films of such mixtures exhibited a thermotropic smectic-type liquid crystal (TSLC) state [4]. Under an external applied voltage, these systems have a dielectric feature, with weak conduction, the electric current dependence versus the applied voltage being hysteretical and depending on the amount of Ch. Considering the system equivalent with a flat condenser, we estimated the dielectric constant εr of the sample. The experimental results were processed with the TableCurve3D program by using the Surface-Fit All Equations option. Then the Numeric Summary option displayed all numerical information, including goodness of fit criteria, coefficient standard errors and confidence limits for the fitted parameters, function extreme, the fitting method, an analysis of variance, and
Fig. 1. Dielectric constant versus applied voltage and Ch amount in mixtures.
Fig. 2. Laser pulse width D versus the supplied voltage U and Ch amount in mixture.
data table statistics. The Precision Summary option can be used to determine how much precision is preserved in the current equation. Residuals Graph is a separate window displaying the residuals for the current surface fit. The data table can be weighted, if necessary. Fig. 1 presents the dielectric constant εr change in terms of the applied voltage and Ch percentage. Thus, the program gives the possibility to choose other experimental conditions in terms of the future experimental requirements and purposes. A helium-neon laser beam with λ = 6328 Å and 20 mW optical power was focused on the samples.
M. Ghelmez et al. / Computer Physics Communications 147 (2002) 633–636
The ring pattern obtained in far-field showed the occurrence of a self-phase modulation and an external self-focusing of the beam. The dependence of the output beam power in terms of the input one and the Ch percentage is generally hysteretical showing a memory effect of the samples. A refractive index change with optical intensity, in connection with the dielectric constant modification with the Ch percentage, occurred upon interaction with the laser light. Using an Nd3+ glass laser we emphasized the pulse width modification [6] in the time domain, displayed on a sampling oscilloscope and correlated with the Ch percentage (Fig. 2). Based on this, a method to estimate the amount of Ch by means of the output pulse width was developed [4].
635
in MATLAB a function f having a form similar to ϕ, but with an amplitude of about 10−12 smaller. We continue the analysis by studying a second order differential equation without free term, which has as possible solution the function ϕ: (2) f (2) = (6τ 4 − 2)/(τ 2 − 1) f. As initial conditions for f , f (1) we chose the values of ϕ, ϕ (1) at the moment τ0 = −1 + 0.01. The numerical simulation gave as a solution a PTF function similar to ϕ, but still with small amplitude, only four times greater than that previously obtained. Then we use now a test function having the form ϕa = exp[0.1/(τ 2 − 1)], and as second order differential equation without free term: f (2)(τ ) = (0.6τ 4 − 0.36τ 2 − 0.2)/(τ 2 − 1)4 f (τ ) (3)
3. Averaging procedures of the measured signal ϕa , ϕa(2) ,
The averaging procedure starts by writing a test function similar to a Dirac pulse ϕ = exp[1/(τ 2 − 1)] where τ = t − tsym , tsym being the middle of the time interval. Such a function has non zero values only for τ ∈ [−1, 1] and the derivatives ϕ (1) , ϕ (2) and ϕ (3) . We are looking for a differential equation with the function ϕ as a solution. This would imply the necessity for a derivative of certain order n—noted f (n) —to make a jump at the initial moment from the zero value to a non zero value, in contradiction with the property of the test functions to have continuous derivatives of any order on the whole real axis. Therefore, a test function could not be generated by a differential equation, but it would be quite possible for such an equation to possess as a solution a PTF function, i.e. a function with non zero values at the interval τ ∈ [−1, 1] and a certain number of continuous derivatives on the whole time axis. Therefore, we try to study such evolutions depending only on the values f, f (1) , f (n) equal to the values of ϕ, ϕ (1), . . . , ϕ (n) at a moment very close to τ = −1. The simplest differential equation satisfying these requirements has the form: f (1) = −2τ/(τ 2 − 1) f. (1) We chose as initial moment τ0 = −1 + 0.01 and as initial condition for f the value ϕ(τ0 ). By numerical simulation using the 4–5 order Runge–Kutta equations
and with suggested by the expressions of initial conditions for f, f (1) equal to 0.0002 and 0.02. The solution is a function very close to a rectangular unitary pulse and with the amplitude close to unity for more than 2/3 of the integration period. In our experiments with laser, a Gaussian A exp[−(τ − τm )2 /σ 2 ] signal must be processed. In this case, we consider null initial conditions for the system and add a free term in the differential equation. The working interval was chosen about 10 times greater than the pulse width. The time interval between the beginning of the working period and the moment corresponding to the maximum of the pulse was considered equal to 10σ , which means that the processing system was activated by the front of the pulse. The differential equation can be written as: f (2) = (0.6τ 4 − 0.36τ 2 − 0.2)/(τ 2 − 1)4 f (4) + A exp −(τ + 0.9)2 /(0.01)2 . The numerical simulations led to a generated pulse f with the width of 2 units, that can be more easily integrated by the electronic device, the result of the integration being proportional to the amplitude of the laser pulse. At the end of the integrating period, the integrating signal f is still high, about 10% of the peak obtained at the working interval. The best way of using a PTF f is by multiplying the expression of any received signal intensity “u” by f at the working interval (Fig. 3). Since f is similar to a step function,
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M. Ghelmez et al. / Computer Physics Communications 147 (2002) 633–636
Fig. 3. Function f generated by the equation without free term.
the result of the integration represents a measure for the average intensity of the laser beam. It must be also noticed that this result does not depend on the small stochastic variations of the integration period, usually generated by the switching phenomena, since the signal and its derivative have null values at the end of the integrating period. The properties of f are transmitted to the product f × u. Practically, using operational amplifiers with higher accuracy one can build a system generating the function f and working at a frequency of about 10 kHz. A linear system generating an oscillating signal at the frequency of 1 kHz has already been obtained. The multiplication of the optical signal received by the photodetector by f can be performed by supplying the photodetector with a voltage corresponding to the sum of a certain constant value f0 and the function f . f0 must be chosen so that the photodetector is at the limit of its sensitivity for f = 0; the photoelectric current has to appear for f > 0 and to disappear when f becomes zero again.
4. Conclusions FA and their mixtures with Ch present under some conditions the LC state, and a nonlinear feature while
in this state. The study of such effects can bring valuable information on some biological mechanisms, and open up the perspective of using some specific computer-based mathematical procedures. Our study concentrates on the computer processing of the experimental data, following the finding of new information on the studied systems, and of better working conditions. Numerical simulations using Runge–Kutta equations of order 4–5 in MATLAB helped us in improving the use of PTF for optical measurements. Thus, the influence of switching phenomena upon averaging methods usually used in sampling procedures has been practically rejected. The TableCurve3D program was used to fit the experimental dependencies of the output-input physical amounts. The program gives useful information on the fitting criteria, and the possibility to choose other experimental conditions, in terms of the requirements and purposes of future experiments. A method to estimate the Ch percentage in a mixture with FA was developed. Since FA and Ch are important substances for the living matter and especially for the biological membrane, we consider this study a promising bridge to biophysical studies by computer use.
References [1] A.C. Guyton, J.E. Hall, Textbook of Medical Physiology, 9th edn., Saunders, 1996. [2] G.H. Brown, J.J. Wolken, Liquid Crystals and Biological Structures, Academic Press, New York, 1979. [3] P.J. Quinn, D. Chapman, Biochem. Soc. Trans. 8 (1980) 38. [4] M. Dumitru (Ghelmez), M. Honciuc, M.C. Piscureanu, C. Gheorghe, Optical Engin. 35 (5) (1996) 1372. [5] S. Gong, H. Hentzell, S. Peterson, H. Hesselbon, B. Lofstedt, M. Hause, in: 8th Annual Intl. ASIC Conference, 18–21 Sept. 1995. [6] A. Adolf, D. Chatrefou, C. Guedard, J. Opt. (Paris) 15 (2) (1984) 83.