On control of chaos and synchronization in the vibronic laser

doi:10.1364/OE.17.014166

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On control of chaos and synchronization in the vibronic laser

B. Ratajska-Gadomska and W. Gadomski »View Author Affiliations

Optics Express, Vol. 17, Issue 16, pp. 14166-14171 (2009)
http://dx.doi.org/10.1364/OE.17.014166

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– Abstract
1. Introduction
2. Theory
3. Numerical results
4. Synchronization of two alexandrite lasers
5. Summary
– References and links

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Abstract

It is shown theoretically that the method of time delayed incoherent optical feedback ensures control of chaotic dynamics in the vibronic alexandrite laser. The numerical solutions of the laser equations including the optical delayed feedback term are presented and the conditions for stabilization of the laser output are discussed. The possibility of synchronization of two chaotic vibronic lasers is reported when one of them is driven by the output of the other, thus giving the basis for secure communication.

1. Introduction

Since the first paper of E. Ott, C. Grebogi and J.A. Yorke [1

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990). [CrossRef] [PubMed]

], who have shown that given a chaotic system one can convert it into the system exhibiting regular periodic motion, the interest in the field called control of chaos is not weakening. The problem of stabilizing the unstable dynamics concerns many nonlinear systems, among them lasers. Chaotic dynamics and the possibility of its control has been investigated, both theoretically and experimentally, in few lasers; e.g. CO₂ laser [2–5

F. T. Arecchi, W. Gadomski, and R. Meucci, “Generation of chaotic dynamics by feedback on a laser,” Phys. Rev. A 34(2), 1617–1620 (1986). [CrossRef] [PubMed]

], solid-state Nd:YAG laser [6

R. Roy, T. W. Murphy Jr., T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68(9), 1259–1262 (1992). [CrossRef] [PubMed]

], ¹⁵NH₃ laser [7

R. Dykstra, D. Y. Tang, and N. R. Heckenberg, “Experimental control of a single-mode laser chaos by using continuous, time-delayed feedback,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(6), 6596–6598 (1998). [CrossRef]

], semiconductor lasers [8–10

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]

], and recently the Kerr lens mode locked Ti:Sapphire laser [11

M. G. Kovalsky, “On control of Chaos in the Kerr lens mode locked Ti:Sapphire laser,” Opt. Commun. 260(1), 265–270 (2006). [CrossRef]

]. As to our knowledge the solid state transition metal ion lasers have not been widely studied from this point of view. Nowadays the alexandrite laser (chromium doped chrysoberyl crystal) has gained high interest due to its wide applicability in medicine, cosmetics and as a main part of a lidar in investigations of the atmospheric pollution. This is the reason why the control of its stability and the type of its output appears to be a crucial problem.

In our recent papers we have shown the peculiar dynamics of the alexandrite laser in its vibronic mode of operation [12–15

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

]. Depending on frequency and power of the pumping laser it exhibits completely different types of output; e.g., CW operation, self-pulsations, homoclinic dynamics and, in the region of two latter ones, the chaotic dynamics. We have provided the theoretical explanation of the observed phenomenon assuming that the nonequilibrium phonons of the host crystal actively participate in the laser action [16

B. Ratajska-Gadomska and W. Gadomski, “Quantum theory of the vibronic solid-state laser,” J. Opt. Soc. Am. B 16(5), 848–860 (1999). [CrossRef]

]. The laser model presented by us [12–16

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

] appeared to describe well the experimental results obtained by us [12–15

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

] and other authors [17

H. Ogilvy, M. J. Withford, R. P. Mildren, J. A. Piper, M. J. Withford, R. P. Mildren, and J. A. Piper, “Investigation of the pump wavelength influence on pulsed laser pumped Alexandrite lasers,” Appl. Phys. B 81(5), 637–644 (2005). [CrossRef]

]. It has been shown experimentally [12–15

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

, 17

] that, when pumped by a short wavelength laser (<600nm), both by a cw-laser [12–15

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

] and by a pulsed laser [17

], the alexandrite laser operates as a train of pulses, whereas long wavelength pump (>600nm) [12–15

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

, 17

] yields the stable cw output. Thus, the pump wavelength can be a control parameter. Moreover we have demonstrated that, for short wavelength pump and certain values of other control parameters such as pump intensity or cavity losses [13

W. Gadomski and B. Ratajska-Gadomska, “Homoclinic orbits and chaos in the vibronic short-cavity standing-wave alexandrite laser,” J. Opt. Soc. Am. B 17(2), 188–197 (2000). [CrossRef]

, 14

W. Gadomski, B. Ratajska-Gadomska, and R. Meucci, “Homoclinic dynamics if the vibronic laser,” Chaos Solitons Fractals 17(2–3), 387–396 (2003). [CrossRef]

], the alexandrite laser output exhibits homoclinic or chaotic character. We have shown that in this range of parameters the vibronic laser dynamics can be well described in terms of the homoclinic orbits and Shil’nikov chaos [13

W. Gadomski and B. Ratajska-Gadomska, “Homoclinic orbits and chaos in the vibronic short-cavity standing-wave alexandrite laser,” J. Opt. Soc. Am. B 17(2), 188–197 (2000). [CrossRef]

, 14

W. Gadomski, B. Ratajska-Gadomska, and R. Meucci, “Homoclinic dynamics if the vibronic laser,” Chaos Solitons Fractals 17(2–3), 387–396 (2003). [CrossRef]

In this paper we present the theoretical study on the possibility of control of the alexandrite laser chaotic dynamics. Our numerical results prove that using the time delayed incoherent optical feedback, according to the Pyragas method [18

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170(6), 421–428 (1992). [CrossRef]

], assures the expected result. Moreover, we show that in the system of two nearly identical vibronic lasers, where one is driven by the output of the other the ideal synchronization can be achieved. Thus, the system might be used for secure message transmission.

2. Theory

In the theoretical model proposed by us [12–16

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

] the dynamics of the nonequilibrium host lattice phonons plays the crucial role. The peculiarity of the vibronic alexandrite laser lies in the fact that the upper lasing level has the energy higher than the storage level [19

J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable CW Alexandrite Laser,” IEEE J. Quantum Electron. 16(2), 120–121 (1980). [CrossRef]

]. Thus, the pumping process takes place with participation of phonons of energy fitting to the energy gap between the storage level and the upper lasing level. Moreover these are not the thermal phonons [16

B. Ratajska-Gadomska and W. Gadomski, “Quantum theory of the vibronic solid-state laser,” J. Opt. Soc. Am. B 16(5), 848–860 (1999). [CrossRef]

], but the coherent nonequilibrium phonons obtained in the process of nonradiative depopulation of the pumped level to the storage level. Although the alexandrite laser is a multimode laser, we have proved [12

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

, 15

W. Gadomski and B. Ratajska-Gadomska, “Chaotic dynamics of the vibronic laser” in Recent Advances in Laser Dynamics: Control and Synchronization , A.N. Pisarchik, ed. (Research Signpost, Kerala, India, 2008)

, 16

B. Ratajska-Gadomska and W. Gadomski, “Quantum theory of the vibronic solid-state laser,” J. Opt. Soc. Am. B 16(5), 848–860 (1999). [CrossRef]

] that the one mode model is sufficient to visualize the variety of laser outputs for different conditions. The laser dynamics is connected with the competition between phonons and photons produced in the system and not with its multimode character. The central mode of the laser is much stronger than the other ones [16

B. Ratajska-Gadomska and W. Gadomski, “Quantum theory of the vibronic solid-state laser,” J. Opt. Soc. Am. B 16(5), 848–860 (1999). [CrossRef]

]. We have shown [15

, 16

B. Ratajska-Gadomska and W. Gadomski, “Quantum theory of the vibronic solid-state laser,” J. Opt. Soc. Am. B 16(5), 848–860 (1999). [CrossRef]

] that the dynamics of the laser, observed by us experimentally [12–14

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

], can be well described by the rate equations for the following variables: W is the population inversion between the lasing levels, n₃ is the population of the storage level, I is the number of laser photons and N is the number of nonequilibrium phonons active in the pumping process. In order to introduce the control of chaotic dynamics we apply the Pyragas method [18

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170(6), 421–428 (1992). [CrossRef]

] of the time delayed feedback control. It is a convenient method in continuous dynamical systems. Thus, we introduce the optical intensity feedback in the set of equations derived by us previously for one laser mode [12

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

, 15

, 16

B. Ratajska-Gadomska and W. Gadomski, “Quantum theory of the vibronic solid-state laser,” J. Opt. Soc. Am. B 16(5), 848–860 (1999). [CrossRef]

\frac{d}{dt} n_{3} = - Γ_{3} n_{3} + K_{3 W} W + K_{30} n_{0}

(1a)

\frac{d}{dt} W = - Γ_{W} W - 2 \frac{CIW}{1 + CI / Γ_{P}} + K_{W 3} n_{3} + K_{W 0} n_{0}

(1b)

\frac{d}{dt} I (t) = - 2 κI (t) + 2 \frac{CI (t)}{1 + CI (t) / Γ_{P} + BN / Γ_{P}} + 2 ε κ I (t - τ)

(1c)

\frac{d}{dt} N = - Γ_{P} (N - 〈 N (0) 〉) - K_{N 3} n_{3} + K_{NW} W + K_{N 0} n_{0}

(1d)

where n ₀ is the Cr-ion concentration density, κ = cT /2L (c is the light velocity, T is the transmission of the mirror, L is the cavity length) denotes the cavity losses, T_p is the phonon decay rate, C is the electron-field coupling coefficient, B is the electron-phonon coupling coefficient, I ₀ is the number of pumping photons, and the other coefficients are N -dependent:

Γ_{W} = [2 γ_{4} + BN + A I_{0}] / 2 Γ_{3} = γ_{3} + [A I_{0} + 3 BN] / 2

K_{W 3} = [3 BN - A I_{0}] / 2 + γ_{4} - γ_{3} K_{3 W} = (BN - A I_{0}) / 2

K_{W 0} = (A I_{0} - gBN - 2 γ_{4}) / 2 K_{30} = (A I_{0} + BN) / 2

K_{N 3} = [3 BN + f A I_{0}] / 2 K_{N W} = (BN - f A I_{0}) / 2

K_{N 0} = (f A I_{0} + BN) / 2

The parameter _f is the result of the energy conservation [12

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

, 15

, 16

B. Ratajska-Gadomska and W. Gadomski, “Quantum theory of the vibronic solid-state laser,” J. Opt. Soc. Am. B 16(5), 848–860 (1999). [CrossRef]

]. It has the sense of the mean order of nonradiative multiphonon transition from the pumped level to the storage level. It is a very important parameter, because it includes the information about the wavelength of the pumping laser [12

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

, 16

B. Ratajska-Gadomska and W. Gadomski, “Quantum theory of the vibronic solid-state laser,” J. Opt. Soc. Am. B 16(5), 848–860 (1999). [CrossRef]

]. The higher pump wavelength the larger value of the parameter f.

The delayed term in Eq. (1)c) is characterized by parameters: ε as the control weight parameter, ∣ε∣ < 1, and τ as a time delay.

3. Numerical results

According to the Pyragas method [18

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170(6), 421–428 (1992). [CrossRef]

] stabilization of a particular unstable periodic orbit can be achieved if τ = T_i , where T_i is the period of the i-th orbit. In our system we are dealing with the periodicity connected with repetition of pulses when the laser output has the form of the regular pulse train [12–14

W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]

], whereas the shape and the height of pulses depends on the pump wavelength [14

W. Gadomski, B. Ratajska-Gadomska, and R. Meucci, “Homoclinic dynamics if the vibronic laser,” Chaos Solitons Fractals 17(2–3), 387–396 (2003). [CrossRef]

]. The chaotic output, obtained in the period doubling route, has the form of irregular pulsations [13

W. Gadomski and B. Ratajska-Gadomska, “Homoclinic orbits and chaos in the vibronic short-cavity standing-wave alexandrite laser,” J. Opt. Soc. Am. B 17(2), 188–197 (2000). [CrossRef]

, 14

W. Gadomski, B. Ratajska-Gadomska, and R. Meucci, “Homoclinic dynamics if the vibronic laser,” Chaos Solitons Fractals 17(2–3), 387–396 (2003). [CrossRef]

]. Thus it consists of the set of unstable periodic pulse trains (the unstable periodic orbits in our laser). Numerical calculations have been performed for different values of the delay time τ , putting it equal to T or T/n, T-being the period of the unstable orbit, n-denotes the natural number. The results of the calculations are shown in Fig. 1 for ε > 0. The case ε > 0 describes the experiment of optical delayed feedback, in which the delayed output laser signal is injected into the laser cavity. It can be seen in Fig. 1 that we get evident control of the chaotic behavior of the system also in the case when the period of the i-th orbit is the high multiplicity of the delay time. This information is important from the point of view of the experimental realization of the predicted here theoretical picture, which yields shorter delay path.

Fig. 1. Laser output vs. time in the regime of self-pulsations; a) chaotic dynamics in the absence of the control term ε = 0; b) and c) stabilization of the orbit of the period T = 1.7μs at ε = 5 10^-5 and the delay time: (b) τ = T or (c) τ = T/40; d) and e) stabilization of the orbit of the period T = 2.2μs at ε = 5 1031 and the delay time: (d) τ = T or (e) τ = T/5.

4. Synchronization of two alexandrite lasers

Herewith we consider two identical lasers in a driving system- response system configuration [20

L. M. Pecora and T. L. Carroll, “Driving systems with chaotic signals,” Phys. Rev. A 44(4), 2374–2383 (1991). [CrossRef] [PubMed]

]. The delayed output of the first laser is injected into the cavity of the second one, what is described by Eqs. (2).

\frac{d}{dt} n_{13} = {- Γ}_{13} n_{13} + K_{31 W} W_{1} + K_{310} n_{0}

(2a)

\frac{d}{dt} W_{1} = - Γ_{1 W} W_{1} - 2 \frac{C I_{1} W_{1}}{1 + C I_{1} / Γ_{P}} + K_{W_{1} 3} n_{13} + K_{W 10} n_{0}

(2b)

\frac{d}{dt} I_{1} (t) = - 2 κ I_{1} (t) + 2 \frac{C I_{1} (t)}{1 + C I_{1} (t) / Γ_{P} + B N_{1} / Γ_{P}}

(2c)

\frac{d}{dt} N_{1} = - Γ_{P} (N_{1} - 〈 N (0) 〉) - K_{N 13} n_{13} + K_{N 1 W} W_{1} + K_{N 10} n_{0}

(2d)

\frac{d}{dt} n_{23} = - Γ_{23} n_{23} + K_{32 W} W_{2} + K_{320} n_{0}

(2e)

\frac{d}{dt} W_{2} = - Γ_{2 W} W_{2} - 2 \frac{C I_{2} W_{2}}{1 + C I_{2} / Γ_{P}} + K_{W 23} n_{23} + K_{W 20} n_{0(2f)}

\frac{d}{dt} I_{2} (t) = - 2 κ I_{2} (t) + 2 \frac{C I_{2}}{1 + C I_{2} (t) / Γ_{P} + B N_{2} / Γ_{P}} + 2 εκ (I_{1} - I_{2})

(2g)

\frac{d}{dt} N_{2} = - Γ_{P} (N_{2} - 〈 N (0) 〉) - K_{N 23} n_{23} + K_{N 2 W} W_{2} + K_{N 20} n_{0}

(2h)

where the parameters Γ_i3,Γ_iW,K _3iW,K _3i0,K _wi3,K _Wi0,K_NiW ,K _Ni3,K _Ni0 are defined in Eq. (1), whereas N is substituted by Ni, for i = 1 or 2 respectively.

The numerical solutions of Eqs. (2) are shown in Fig. 2 for ε = 0 and ε = 0.0006 . It can be seen in Fig. 2a,b,c that in the absence of coupling, ε = 0, the outputs of two lasers with identical parameters but different initial conditions, remain uncorrelated. On the other hand when the small amount, ε = 0.0006, of the I₁ output (called the transmitter signal) is injected into the cavity of the I₂ laser (called the receiver laser) we observe high synchronization of both signals, what becomes apparent if one compares Fig. 2a and Fig. 2d. Thus the system of two alexandrite lasers may be used for secure transmission of the encoded message. We follow the idea realized experimentally in the system of two Nd: YVO₄ microchip lasers by U. A. Uchida et. al [21

A. Uchida, S. Yoshimori, M. Shinozuka, T. Ogawa, and F. Kannari, “Chaotic on off keying for secure communications,” Opt. Lett. 26(12), 866–868 (2001). [CrossRef]

]. The chaotic output of the driving laser, treated as transmitter, is modulated by the encoded signal of the amplitude much smaller than the amplitude of the carrying signal. In our case we put the coupling parameter in Eq. (2g) to be time dependent:

ε (t) = {\begin{array}{c} ε_{0} + Δ ε for t_{i} < t < t_{i} + Δt \\ ε_{0} - Δ ε for t_{i} + Δ t < t < t_{i + 1} \end{array} .

Thus, the slave laser, called the receiver, is synchronized to the modulated signal. The encoded message can be read as a difference between the receiver and the unmodulated transmitter signals. The results of our calculations are shown in Fig. 3.

5. Summary

Herewith we have presented the possibility of control of chaos in the vibronic laser by use of the delayed optical feedback. We have also shown that two vibronic lasers in chaotic regime of operation can be synchronized and thus applied for secure communication.

Fig. 2. Numerical solutions of Eqs. (2). Laser outputs of two unsynchronized lasers in chaotic regime: a), b), c) and for two synchronized lasers: a), d), e).

Fig. 3. Numerical solutions of Eqs. (2) with time dependent coupling parameter ε(t) : a) signal from the transmitter laser, b) signal from the transmitter laser modulated by encoded message for ε ₀ = 1, ∆ε = 0.008, c) signal from the receiver laser synchronized to the signal in (b), d) difference signal between (c) and (a).

Acknowledgement

This work has been supported by the grant No N202.115.31/1463 of the Polish Ministry of Scientific Research and Higher Education.

References and links

1.	E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990). [CrossRef] [PubMed]
2.	F. T. Arecchi, W. Gadomski, and R. Meucci, “Generation of chaotic dynamics by feedback on a laser,” Phys. Rev. A 34(2), 1617–1620 (1986). [CrossRef] [PubMed]
3.	F. T. Arecchi, R. Meucci, and W. Gadomski, “Laser dynamics with competing instabilities,” Phys. Rev. Lett. 58(21), 2205–2208 (1987). [CrossRef] [PubMed]
4.	R. Meucci, W. Gadomski, M. Ciofini, and F. T. Arecchi, “Experimental control of chaos by means of weak parametric perturbations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 49(4), R2528–R2531 (1994). [CrossRef] [PubMed]
5.	S. Bielawski, D. Derozier, and P. Glorieux, “Controlling unstable periodic orbits by a delayed continuous feedback,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 49(2), R971–R974 (1994). [CrossRef] [PubMed]
6.	R. Roy, T. W. Murphy Jr., T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68(9), 1259–1262 (1992). [CrossRef] [PubMed]
7.	R. Dykstra, D. Y. Tang, and N. R. Heckenberg, “Experimental control of a single-mode laser chaos by using continuous, time-delayed feedback,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(6), 6596–6598 (1998). [CrossRef]
8.	R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
9.	G. Hek and V. Rottschafer, “Semiconductor laser with filtered optical feedback: bridge between conventional feedback and optical injection”, ENOC-2005, Eindhoven, Netherlands, 7–12 August (2005).
10.	S. Schikora, P. Hövel, H. J. Wünsche, E. Schöll, and F. Henneberger, “All-optical noninvasive control of unstable steady states in a semiconductor laser,” Phys. Rev. Lett. 97(21), 213902 (2006). [CrossRef] [PubMed]
11.	M. G. Kovalsky, “On control of Chaos in the Kerr lens mode locked Ti:Sapphire laser,” Opt. Commun. 260(1), 265–270 (2006). [CrossRef]
12.	W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]
13.	W. Gadomski and B. Ratajska-Gadomska, “Homoclinic orbits and chaos in the vibronic short-cavity standing-wave alexandrite laser,” J. Opt. Soc. Am. B 17(2), 188–197 (2000). [CrossRef]
14.	W. Gadomski, B. Ratajska-Gadomska, and R. Meucci, “Homoclinic dynamics if the vibronic laser,” Chaos Solitons Fractals 17(2–3), 387–396 (2003). [CrossRef]
15.	W. Gadomski and B. Ratajska-Gadomska, “Chaotic dynamics of the vibronic laser” in Recent Advances in Laser Dynamics: Control and Synchronization , A.N. Pisarchik, ed. (Research Signpost, Kerala, India, 2008)
16.	B. Ratajska-Gadomska and W. Gadomski, “Quantum theory of the vibronic solid-state laser,” J. Opt. Soc. Am. B 16(5), 848–860 (1999). [CrossRef]
17.	H. Ogilvy, M. J. Withford, R. P. Mildren, J. A. Piper, M. J. Withford, R. P. Mildren, and J. A. Piper, “Investigation of the pump wavelength influence on pulsed laser pumped Alexandrite lasers,” Appl. Phys. B 81(5), 637–644 (2005). [CrossRef]
18.	K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170(6), 421–428 (1992). [CrossRef]
19.	J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable CW Alexandrite Laser,” IEEE J. Quantum Electron. 16(2), 120–121 (1980). [CrossRef]
20.	L. M. Pecora and T. L. Carroll, “Driving systems with chaotic signals,” Phys. Rev. A 44(4), 2374–2383 (1991). [CrossRef] [PubMed]
21.	A. Uchida, S. Yoshimori, M. Shinozuka, T. Ogawa, and F. Kannari, “Chaotic on off keying for secure communications,” Opt. Lett. 26(12), 866–868 (2001). [CrossRef]

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.1540) Lasers and laser optics : Chaos
(140.3490) Lasers and laser optics : Lasers, distributed-feedback
(140.5680) Lasers and laser optics : Rare earth and transition metal solid-state lasers
(190.3100) Nonlinear optics : Instabilities and chaos

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: June 2, 2009
Revised Manuscript: July 24, 2009
Manuscript Accepted: July 27, 2009
Published: August 3, 2009

Citation
B. Ratajska-Gadomska and W. Gadomski, "On control of chaos and synchronization in the vibronic laser," Opt. Express 17, 14166-14171 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14166

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References

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990). [CrossRef] [PubMed]
F. T. Arecchi, W. Gadomski, and R. Meucci, “Generation of chaotic dynamics by feedback on a laser,” Phys. Rev. A 34(2), 1617–1620 (1986). [CrossRef] [PubMed]
F. T. Arecchi, R. Meucci, and W. Gadomski, “Laser dynamics with competing instabilities,” Phys. Rev. Lett. 58(21), 2205–2208 (1987). [CrossRef] [PubMed]
R. Meucci, W. Gadomski, M. Ciofini, and F. T. Arecchi, “Experimental control of chaos by means of weak parametric perturbations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 49(4), R2528–R2531 (1994). [CrossRef] [PubMed]
S. Bielawski, D. Derozier, and P. Glorieux, “Controlling unstable periodic orbits by a delayed continuous feedback,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 49(2), R971–R974 (1994). [CrossRef] [PubMed]
R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68(9), 1259–1262 (1992). [CrossRef] [PubMed]
R. Dykstra, D. Y. Tang, and N. R. Heckenberg, “Experimental control of a single-mode laser chaos by using continuous, time-delayed feedback,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(6), 6596–6598 (1998). [CrossRef]
R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
G. Hek and V. Rottschafer, “Semiconductor laser with filtered optical feedback: bridge between conventional feedback and optical injection”, ENOC-2005, Eindhoven, Netherlands, 7–12 August (2005).
S. Schikora, P. Hövel, H. J. Wünsche, E. Schöll, and F. Henneberger, “All-optical noninvasive control of unstable steady states in a semiconductor laser,” Phys. Rev. Lett. 97(21), 213902 (2006). [CrossRef] [PubMed]
M. G. Kovalsky, “On control of Chaos in the Kerr lens mode locked Ti:Sapphire laser,” Opt. Commun. 260(1), 265–270 (2006). [CrossRef]
W. Gadomski and B. Ratajska-Gadomska, “Self-pulsations in the phonon assisted lasers,” J. Opt. Soc. Am. B 15(11), 2681–2688 (1998). [CrossRef]
W. Gadomski and B. Ratajska-Gadomska, “Homoclinic orbits and chaos in the vibronic short-cavity standing-wave alexandrite laser,” J. Opt. Soc. Am. B 17(2), 188–197 (2000). [CrossRef]
W. Gadomski, B. Ratajska-Gadomska, and R. Meucci, “Homoclinic dynamics if the vibronic laser,” Chaos Solitons Fractals 17(2-3), 387–396 (2003). [CrossRef]
W. Gadomski and B. Ratajska-Gadomska, “Chaotic dynamics of the vibronic laser” in Recent Advances in Laser Dynamics: Control and Synchronization, A.N. Pisarchik, ed. (Research Signpost, Kerala, India, 2008)
B. Ratajska-Gadomska and W. Gadomski, “Quantum theory of the vibronic solid-state laser,” J. Opt. Soc. Am. B 16(5), 848–860 (1999). [CrossRef]
H. Ogilvy, M. J. Withford, R. P. Mildren, J. A. Piper, M. J Withford, R. P Mildren, and J. A Piper, “Investigation of the pump wavelength influence on pulsed laser pumped Alexandrite lasers,” Appl. Phys. B 81(5), 637–644 (2005). [CrossRef]
K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170(6), 421–428 (1992). [CrossRef]
J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable CW Alexandrite Laser,” IEEE J. Quantum Electron. 16(2), 120–121 (1980). [CrossRef]
L. M. Pecora and T. L. Carroll, “Driving systems with chaotic signals,” Phys. Rev. A 44(4), 2374–2383 (1991). [CrossRef] [PubMed]
A. Uchida, S. Yoshimori, M. Shinozuka, T. Ogawa, and F. Kannari, “Chaotic on off keying for secure communications,” Opt. Lett. 26(12), 866–868 (2001). [CrossRef]

Appl. Phys. B

H. Ogilvy, M. J. Withford, R. P. Mildren, J. A. Piper, M. J Withford, R. P Mildren, and J. A Piper, “Investigation of the pump wavelength influence on pulsed laser pumped Alexandrite lasers,” Appl. Phys. B 81(5), 637–644 (2005). [CrossRef]

Chaos Solitons Fractals

W. Gadomski, B. Ratajska-Gadomska, and R. Meucci, “Homoclinic dynamics if the vibronic laser,” Chaos Solitons Fractals 17(2-3), 387–396 (2003). [CrossRef]

IEEE J. Quantum Electron.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable CW Alexandrite Laser,” IEEE J. Quantum Electron. 16(2), 120–121 (1980). [CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

M. G. Kovalsky, “On control of Chaos in the Kerr lens mode locked Ti:Sapphire laser,” Opt. Commun. 260(1), 265–270 (2006). [CrossRef]

Opt. Lett.

A. Uchida, S. Yoshimori, M. Shinozuka, T. Ogawa, and F. Kannari, “Chaotic on off keying for secure communications,” Opt. Lett. 26(12), 866–868 (2001). [CrossRef]

Phys. Lett. A

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170(6), 421–428 (1992). [CrossRef]

Phys. Rev. A

F. T. Arecchi, W. Gadomski, and R. Meucci, “Generation of chaotic dynamics by feedback on a laser,” Phys. Rev. A 34(2), 1617–1620 (1986). [CrossRef] [PubMed]
L. M. Pecora and T. L. Carroll, “Driving systems with chaotic signals,” Phys. Rev. A 44(4), 2374–2383 (1991). [CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

R. Meucci, W. Gadomski, M. Ciofini, and F. T. Arecchi, “Experimental control of chaos by means of weak parametric perturbations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 49(4), R2528–R2531 (1994). [CrossRef] [PubMed]
S. Bielawski, D. Derozier, and P. Glorieux, “Controlling unstable periodic orbits by a delayed continuous feedback,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 49(2), R971–R974 (1994). [CrossRef] [PubMed]
R. Dykstra, D. Y. Tang, and N. R. Heckenberg, “Experimental control of a single-mode laser chaos by using continuous, time-delayed feedback,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(6), 6596–6598 (1998). [CrossRef]

Phys. Rev. Lett.

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990). [CrossRef] [PubMed]
R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68(9), 1259–1262 (1992). [CrossRef] [PubMed]
F. T. Arecchi, R. Meucci, and W. Gadomski, “Laser dynamics with competing instabilities,” Phys. Rev. Lett. 58(21), 2205–2208 (1987). [CrossRef] [PubMed]
S. Schikora, P. Hövel, H. J. Wünsche, E. Schöll, and F. Henneberger, “All-optical noninvasive control of unstable steady states in a semiconductor laser,” Phys. Rev. Lett. 97(21), 213902 (2006). [CrossRef] [PubMed]

Other

W. Gadomski and B. Ratajska-Gadomska, “Chaotic dynamics of the vibronic laser” in Recent Advances in Laser Dynamics: Control and Synchronization, A.N. Pisarchik, ed. (Research Signpost, Kerala, India, 2008)
G. Hek and V. Rottschafer, “Semiconductor laser with filtered optical feedback: bridge between conventional feedback and optical injection”, ENOC-2005, Eindhoven, Netherlands, 7–12 August (2005).

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