- Email: [email protected]
Computer aided instruction in laser dynamics
Computer Physics Communications 121–122 (1999) 583–585 www.elsevier.nl/locate/cpc
Computer aided instruction in laser dynamics Mihaela Dumitru, V. Ninulescu, P.E. Sterian, M. Piscureanu 1 Politehnica University of Bucharest, Bucharest 77206, Romania
Abstract We describe some aspects of our activities with students of Quantum Electronics at the Advanced Studies level. A package of programs in MATLAB for Windows 4.0 was set up to simulate the dynamics of several types of lasers operating in various regimes.We illustrate the nonlinear evolution of some familiar gas lasers. 1999 Elsevier Science B.V. All rights reserved.
1. Computer study of the dynamics of a CO2 laser Physics is better taught by using experiments which illustrate and demonstrate the relevant information. In fact, teaching laser physics and its applications requires the use of both experimental setups and computer software. We discuss here software developed to help the understanding of Quantum Electronics and its applications in most domains. Let us assume a CO2 monomode laser as described by the following equations [1,2]: AD dI = 2kI − 1 , dt 1 + δ2 dD DI = γ1 1 − D − . (1) dt 1 + δ2 Here I and D are the normalized laser intensity and normalized population inversion (positively defined), respectively. A, δ, γ1 , and k are the pumping strength parameter, the cavity detuning from resonance in units of polarization relaxation rate, the molecular relaxation rate, and the cavity damping rate, respectively. Despite the apparent simplicity of this mathematical model, its analytical study provides only some lim1 E-mail: [email protected].
ited results and the dynamics cannot be predicted in most cases. We developed several programs written in MATLAB for Windows 4.0 which simulate the laser dynamics. We briefly present here some of our results, and compare them with results obtained by us by working in the traditional manner. System (1) has the stationary solution: I0 = A − 1 + δ 2 , (2) D0 = 1 + δ 2 /A from which we infer the threshold condition for the laser oscillation: A > 1 + δ 2 . When the laser action takes place, the system is dissipative, i.e. it contracts in the phase space. (I0 , D0 ) is the globally attracting fixed point for the system. By linearizing the system (1–10 ) at this point, the eigenvalue equation in λ is obtained: At low excitation level A/(1 + δ 2 ) ∼ 1, and k γ1 , and one finds: q γ1 A A ± i 2kγ1 1+δ (3) λ1,2 = − 2 −1 . 2 2 1+δ This shows that the initial disturbance from the stationary solution relaxes by an oscillation of frequency: 1 q A 2kγ1 1+δ (4) f0 = 2 −1 . 2π
0010-4655/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 9 9 ) 0 0 4 1 1 - 7
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M. Dumitru et al. / Computer Physics Communications 121–122 (1999) 583–585
Fig. 1. Bifurcation diagram of the laser with modulated losses (modulation index is increasing). Typical dynamics are shown in Fig. 2.
Fig. 2. Laser dynamics: temporal dynamics of the intensity (left) and I –D phase plane diagrams (right). Results are shown for (a) m = 0.025, (b) m = 0.030, (c) m = 0.033, (d) m = 0.037, (e) m = 0.040, and (f) m = 0.050.
The damped oscillation for typical parameters of a waveguide CO2 laser can be found in [2], and we obtain f0 = 2.86 × 105 Hz. Let us analyze some practical cases. One of them would be a laser with sinusoidal modulated losses (amplitude modulation): DI dI = −2k 1 + m sin(2πf t) I + 2kA , dt 1 + δ2 DI dD = γ1 1 − D − , dt 1 + δ2
(5) (6)
where f and m are the frequency and the losses modulation index. When f is close to the system frequency f0 , a nonlinear interaction of resonant type is expected, which limits the modulation. To study this situation we have chosen f = 250 kHz. For relatively low amplitudes of modulation, the system dynamics is presented in Fig. 1, where many types of temporal evolution can be observed: an oscillation at the period of the modulation, a route to chaos through subharmonic bifurcations, a window of regular dynamics, and a chaotic dynamics. The asymptotic evolution depends – in some regions – on the initial conditions. Details of the diagrams reveal small regions of regular evolution. To characterize a given dynamics one can represent also the Fourier spectrum of the intensity, the strange attractor in the form of a Poincaré section. Its details will indicate similar structures at any level, i.e. the attractor is of fractal type [4]. The chaotic attractor might also be characterized by the correlation func-
Fig. 3. Power spectrum at m = 0.031 near the bifurcation point from the period 2T to the period 4T : (a) noise-free laser; (b) noise added.
tion, Lyapunov exponents and dimension (capacity, correlation dimension). Taking into account the noise mechanism, even at low level, the dynamics can change completely and be concentrated around the noise-free trajectory. Another interesting feature at the same value of m is revealed by the spectral power density plot. In absence of noise, the power exhibits two peaks at the fundamental frequency f , and at f/2 (Fig. 3(a)). The noisy system has also two small peaks as a sign of the transition to a period 4T -cycle (Fig. 3(b)).
M. Dumitru et al. / Computer Physics Communications 121–122 (1999) 583–585
2. Analytical study of a FM He–Ne laser Let us consider now another familiar gas laser, the He–Ne laser. Typical modulation could then be the frequency modulation (FM), obtained by introducing a phase perturbation in the laser resonator; thus the study has to take into account the physical parameters of the active medium in the presence of the perturbation. The analytical mathematical treatment leads – in the time domain [5] – to a repetitive series of pulses, which is a proof that a mode-locking regime for the laser has been reached. In the frequency domain [6], the FM operation can be studied by using the equations from Ref. [7] for mode amplitudes, frequencies and phases. A coupling coefficient ℵ between the adjacent modes is defined and, for a sinusoidal phase perturbation, and in the linear approximation, a steady state solution is also obtained for this type of laser. This is a frequency modulated oscillation with amplitude expressed in terms of Bessel functions of the first kind Jn (m), where m is the modulation index. At the limit of small detuning δ and when m → ∞, the solution describes a mode-locking operation. The number of modes coupled by the nonlinearities can be estimated, too.This is possible when introducing the net saturated gain parameters ρn of the modes [7]: (7) ρn = αn − gn 1 − βEn2 , where αn is the single path loss coefficient for the nth mode, β the saturation parameter assumed equal for all the modes, and gn the single path gain coefficient for the nth mode. The equations describing in this case the laser operation are nonlinear [6]. Taking into account the relative phase angles and the conservation relation for the steady state operation [6], it results in a limitation of the number of modes with significant values: 1 (8) 0 < (ρn + ρn−1 + · · · + ρn−p+1 ) 6 1. ℵ The physical parameters of the system have a dominant role in this estimation. For example, for typical values of the above parameters: δ = 0.013, g0 = 0.75 Ku, β = 0.3, αn = 0.070, Ku being a measure for the Doppler broadening of the laser line, it results p = 5; therefore 5 coupled modes determine the mode-locking operation of the laser. Relation (8) is an analytical result, obtained in a traditional manner, without computer help. This is a
585
useful relation, but not a complete evaluation of the laser operation under different conditions. It is clear that the result we presented in the first part of this paper is a more general, complete, and elegant one, while obtaining it does not require much more effort. Following the method in [8], where circuit theory is exposed as a model for a mode-locked gas laser, we applied our computer generated results to a real simple system with a similar dynamics: an electric circuit composed by a diode (the nonlinear element), an inductance and a resistor, driven by a sinusoidal voltage. The students previously analyzed this system on a double spot oscilloscope.
3. Conclusions We studied two kind of gas lasers modulated in amplitude and in frequency, respectively. By using the characteristic equations of evolution for these lasers, with some concrete conditions, but solved in a different manner – by using the computer and without using it, respectively – we can compare the methods. The traditional one is applied only for a class of laser and gives a particular result; the other one can easily be used for any kind of laser and in various operating conditions. Moreover, the later offers lots of conclusions and results, and it allows forecasting the dynamical evolution of the system. The programs were written in MATLAB for Windows 4.0 and the signal processing has been performed by taking advantage of a graphic user interface. References [1] J.R. Tredice, F.T. Arecchi, G.P. Puccioni, A. Poggi, W. Gadomski, Phys. Rev. A 34 (1986) 2073. [2] D. Dangoisse, P. Glorieux, D. Hennequin, Phys. Rev. A 36 (1987) 4775. [3] G.L. Baker, J.P. Gollub, Chaotic Dynamics. An Introduction (Cambridge University Press, 1990). [4] K-H. Becker, M. Dorfler, Dynamical Systems and Fractals (Cambridge University Press, 1991). [5] D.J. Kuizenga, A.E. Siegmann, IEEE J. Quantum Electron. QE6 (11) (1970) 694. [6] I.M. Popescu, M. Dumitru, P.E. Sterian, A.Gh. Podoleanu, Rev. Roum. Phys. 28 (8) (1983) 699. [7] S.E. Harris, O.P. McDuff, IEEE J. Quantum Electron. QE-3 (2) (1967) 101. [8] J.R. Fontana, IEEE J. Quantum Electron. QE-8 (8) (1972) 699.
Computer aided instruction in laser dynamics Mihaela Dumitru, V. Ninulescu, P.E. Sterian, M. Piscureanu 1 Politehnica University of Bucharest, Bucharest 77206, Romania
Abstract We describe some aspects of our activities with students of Quantum Electronics at the Advanced Studies level. A package of programs in MATLAB for Windows 4.0 was set up to simulate the dynamics of several types of lasers operating in various regimes.We illustrate the nonlinear evolution of some familiar gas lasers. 1999 Elsevier Science B.V. All rights reserved.
1. Computer study of the dynamics of a CO2 laser Physics is better taught by using experiments which illustrate and demonstrate the relevant information. In fact, teaching laser physics and its applications requires the use of both experimental setups and computer software. We discuss here software developed to help the understanding of Quantum Electronics and its applications in most domains. Let us assume a CO2 monomode laser as described by the following equations [1,2]: AD dI = 2kI − 1 , dt 1 + δ2 dD DI = γ1 1 − D − . (1) dt 1 + δ2 Here I and D are the normalized laser intensity and normalized population inversion (positively defined), respectively. A, δ, γ1 , and k are the pumping strength parameter, the cavity detuning from resonance in units of polarization relaxation rate, the molecular relaxation rate, and the cavity damping rate, respectively. Despite the apparent simplicity of this mathematical model, its analytical study provides only some lim1 E-mail: [email protected].
ited results and the dynamics cannot be predicted in most cases. We developed several programs written in MATLAB for Windows 4.0 which simulate the laser dynamics. We briefly present here some of our results, and compare them with results obtained by us by working in the traditional manner. System (1) has the stationary solution: I0 = A − 1 + δ 2 , (2) D0 = 1 + δ 2 /A from which we infer the threshold condition for the laser oscillation: A > 1 + δ 2 . When the laser action takes place, the system is dissipative, i.e. it contracts in the phase space. (I0 , D0 ) is the globally attracting fixed point for the system. By linearizing the system (1–10 ) at this point, the eigenvalue equation in λ is obtained: At low excitation level A/(1 + δ 2 ) ∼ 1, and k γ1 , and one finds: q γ1 A A ± i 2kγ1 1+δ (3) λ1,2 = − 2 −1 . 2 2 1+δ This shows that the initial disturbance from the stationary solution relaxes by an oscillation of frequency: 1 q A 2kγ1 1+δ (4) f0 = 2 −1 . 2π
0010-4655/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 9 9 ) 0 0 4 1 1 - 7
584
M. Dumitru et al. / Computer Physics Communications 121–122 (1999) 583–585
Fig. 1. Bifurcation diagram of the laser with modulated losses (modulation index is increasing). Typical dynamics are shown in Fig. 2.
Fig. 2. Laser dynamics: temporal dynamics of the intensity (left) and I –D phase plane diagrams (right). Results are shown for (a) m = 0.025, (b) m = 0.030, (c) m = 0.033, (d) m = 0.037, (e) m = 0.040, and (f) m = 0.050.
The damped oscillation for typical parameters of a waveguide CO2 laser can be found in [2], and we obtain f0 = 2.86 × 105 Hz. Let us analyze some practical cases. One of them would be a laser with sinusoidal modulated losses (amplitude modulation): DI dI = −2k 1 + m sin(2πf t) I + 2kA , dt 1 + δ2 DI dD = γ1 1 − D − , dt 1 + δ2
(5) (6)
where f and m are the frequency and the losses modulation index. When f is close to the system frequency f0 , a nonlinear interaction of resonant type is expected, which limits the modulation. To study this situation we have chosen f = 250 kHz. For relatively low amplitudes of modulation, the system dynamics is presented in Fig. 1, where many types of temporal evolution can be observed: an oscillation at the period of the modulation, a route to chaos through subharmonic bifurcations, a window of regular dynamics, and a chaotic dynamics. The asymptotic evolution depends – in some regions – on the initial conditions. Details of the diagrams reveal small regions of regular evolution. To characterize a given dynamics one can represent also the Fourier spectrum of the intensity, the strange attractor in the form of a Poincaré section. Its details will indicate similar structures at any level, i.e. the attractor is of fractal type [4]. The chaotic attractor might also be characterized by the correlation func-
Fig. 3. Power spectrum at m = 0.031 near the bifurcation point from the period 2T to the period 4T : (a) noise-free laser; (b) noise added.
tion, Lyapunov exponents and dimension (capacity, correlation dimension). Taking into account the noise mechanism, even at low level, the dynamics can change completely and be concentrated around the noise-free trajectory. Another interesting feature at the same value of m is revealed by the spectral power density plot. In absence of noise, the power exhibits two peaks at the fundamental frequency f , and at f/2 (Fig. 3(a)). The noisy system has also two small peaks as a sign of the transition to a period 4T -cycle (Fig. 3(b)).
M. Dumitru et al. / Computer Physics Communications 121–122 (1999) 583–585
2. Analytical study of a FM He–Ne laser Let us consider now another familiar gas laser, the He–Ne laser. Typical modulation could then be the frequency modulation (FM), obtained by introducing a phase perturbation in the laser resonator; thus the study has to take into account the physical parameters of the active medium in the presence of the perturbation. The analytical mathematical treatment leads – in the time domain [5] – to a repetitive series of pulses, which is a proof that a mode-locking regime for the laser has been reached. In the frequency domain [6], the FM operation can be studied by using the equations from Ref. [7] for mode amplitudes, frequencies and phases. A coupling coefficient ℵ between the adjacent modes is defined and, for a sinusoidal phase perturbation, and in the linear approximation, a steady state solution is also obtained for this type of laser. This is a frequency modulated oscillation with amplitude expressed in terms of Bessel functions of the first kind Jn (m), where m is the modulation index. At the limit of small detuning δ and when m → ∞, the solution describes a mode-locking operation. The number of modes coupled by the nonlinearities can be estimated, too.This is possible when introducing the net saturated gain parameters ρn of the modes [7]: (7) ρn = αn − gn 1 − βEn2 , where αn is the single path loss coefficient for the nth mode, β the saturation parameter assumed equal for all the modes, and gn the single path gain coefficient for the nth mode. The equations describing in this case the laser operation are nonlinear [6]. Taking into account the relative phase angles and the conservation relation for the steady state operation [6], it results in a limitation of the number of modes with significant values: 1 (8) 0 < (ρn + ρn−1 + · · · + ρn−p+1 ) 6 1. ℵ The physical parameters of the system have a dominant role in this estimation. For example, for typical values of the above parameters: δ = 0.013, g0 = 0.75 Ku, β = 0.3, αn = 0.070, Ku being a measure for the Doppler broadening of the laser line, it results p = 5; therefore 5 coupled modes determine the mode-locking operation of the laser. Relation (8) is an analytical result, obtained in a traditional manner, without computer help. This is a
585
useful relation, but not a complete evaluation of the laser operation under different conditions. It is clear that the result we presented in the first part of this paper is a more general, complete, and elegant one, while obtaining it does not require much more effort. Following the method in [8], where circuit theory is exposed as a model for a mode-locked gas laser, we applied our computer generated results to a real simple system with a similar dynamics: an electric circuit composed by a diode (the nonlinear element), an inductance and a resistor, driven by a sinusoidal voltage. The students previously analyzed this system on a double spot oscilloscope.
3. Conclusions We studied two kind of gas lasers modulated in amplitude and in frequency, respectively. By using the characteristic equations of evolution for these lasers, with some concrete conditions, but solved in a different manner – by using the computer and without using it, respectively – we can compare the methods. The traditional one is applied only for a class of laser and gives a particular result; the other one can easily be used for any kind of laser and in various operating conditions. Moreover, the later offers lots of conclusions and results, and it allows forecasting the dynamical evolution of the system. The programs were written in MATLAB for Windows 4.0 and the signal processing has been performed by taking advantage of a graphic user interface. References [1] J.R. Tredice, F.T. Arecchi, G.P. Puccioni, A. Poggi, W. Gadomski, Phys. Rev. A 34 (1986) 2073. [2] D. Dangoisse, P. Glorieux, D. Hennequin, Phys. Rev. A 36 (1987) 4775. [3] G.L. Baker, J.P. Gollub, Chaotic Dynamics. An Introduction (Cambridge University Press, 1990). [4] K-H. Becker, M. Dorfler, Dynamical Systems and Fractals (Cambridge University Press, 1991). [5] D.J. Kuizenga, A.E. Siegmann, IEEE J. Quantum Electron. QE6 (11) (1970) 694. [6] I.M. Popescu, M. Dumitru, P.E. Sterian, A.Gh. Podoleanu, Rev. Roum. Phys. 28 (8) (1983) 699. [7] S.E. Harris, O.P. McDuff, IEEE J. Quantum Electron. QE-3 (2) (1967) 101. [8] J.R. Fontana, IEEE J. Quantum Electron. QE-8 (8) (1972) 699.