In the last decade ultrafast laser science and technology has undergone revolutionary growth. Femtosecond sources are now used routinely as probes of ultrafast processes in biological, chemical, and physical systems, as well as for applications in communications and medicine. All femtosecond lasers rely on intensity dependent optical response – that is on optical nonlinearity, to produce and shape the pulses. The formalism of nonlinear dynamics and chaos explains the behavior of complex systems with such nonlinear interactions, and can predict their instabilities. Far from being a mere mathematical curiosity, the nonlinear dynamics approach to lasers has yielded a wealth of basic knowledge pertinent to laser development, as for example in controlling semiconductor laser instabilities. Despite their general utility, these approaches have only recently been applied to the study of femtosecond laser systems. Here we briefly describe our work on two such systems, the additive pulse modelocked laser (APM) operating at 1.55 microns, and the Kerr lens modelocked Titanium Sapphire laser (KLM), operating at 800nm.
APM is a passive modelocking technique, in which light from the laser cavity containing the gain medium is coupled to an external cavity which contains a material with an intensity dependent refractive index (usually an optical fiber.) Pulses returning from the external cavity combine interferometrically with those in the main cavity, and the relative phase of the two cavities is adjusted to optimize the pulse shortening which results. There are several possible topologies for the coupled cavities, two of which are the Michelson (M-APM) and Fabry-Perot (FP-APM), shown in Figure 1. Although it has been known empirically that the M-APM and FP-APM lasers have different stability characteristics, recent experiments and theory reveal that these differences originate in topology dependent nonlinear dynamics. [1,2,3] Our experiments were performed with an NaCl F-center laser, configured in either the FP-APM or M-APM geometry. Both cavities are 2 m in length, corresponding to a repetition rate of 76-MHz, and synchronous pumping is provided by a mode-locked Nd:YLF laser. The lasers produce pulses of approximately 100fs duration.
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| Figure 1: a) Schematic of the Fabry Perot-APM laser cavity. The main cavity contains the gain (NaCl), a birefringent tuner plate (BTP) and an output coupler (OC). The control cavity contains 10 cm of single mode fiber (SMF), and a 50% beamsplitter (BS). The cavity length is adjusted by means of an end mirror (M2) which is mounted on a piezoelectric transducer (PZT). b) Michelson APM laser cavity. |
Instabilities in ultrafast lasers are often monitored using a photodiode with ns response and a GHz digital oscilloscope. Because the photodiode is slow compared to the sub-ps width of the optical pulses, each peak observed on the oscilloscope reflects the integrated energy in a pulse. Variations from normal “period-1” modelocking, in which all pulses are identical in energy, will thus be reflected in the pulse train on the oscilloscope. In the FP-APM, an increase of power in the main cavity over 240mW results in a shift from a period 1 (P1) to a period doubled (P2) pulse sequence, in which energy oscillates in alternate pulses between the main and control cavities, as shown in figure 2. These dynamics do not result in a reshaping of the average pulse spectrum or autocorrelation. The pulse dynamics of the M-APM are very different. An increase of laser power in this topology results in significant temporal distortion, but no period doubling. We have performed numerical simulations of the pulse evolution in each APM laser topology (Figure 3). These support the experimental data, showing that the FP-APM undergoes a period doubled route to chaos at a much lower nonlinearity than does the M-APM, and thus that the M-APM is significantly more resistant to instabilities. Morgner and Mitschke have performed a detailed theoretical analysis of the nonlinear dynamics of this system [2], and have found that the Lyapunov exponents of the dynamic maps for the FP-APM and M-APM are very different. Lyapunov exponents are a primary measure of nonlinear dynamics, revealing the manner in which nearby points in phase space separate over time. Thus the tools of nonlinear dynamics and chaos clearly reveal the underlying cause of differences in stability of these lasers. The topological approach we describe here is not limited to APM lasers, since many technologically important fiber laser systems are also based on coupled cavity modelocking. For example, it can be shown that, with certain restrictions on the polarization and the coupling, a figure-8 laser that employs two unidirectional rings can be described by the same equations as for the FP-APM. The M-APM equations are similar to those for a figure 8 laser with one unidirectional and one bi-directional ring. Generalizing from our experiments and simulations, we conclude that any fast saturable absorber (FSA) modelocked laser that is topologically equivalent to the FP-APM laser will have a lower threshold for pulse train instablilities such as subharmonic bifurcations and chaos, and that any FSA modelocked laser that is topologically equivalent to the M-APM laser will be be more resistant to such pulse train instabilities, even when it is subjected to large nonlinearities.
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| Figure 2: Pulse trains from the FP-APM laser under period doubled mode-locking conditions. Note the antiphase behavior in the main and control cavities, indicating energy exchange between cavities. |
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| Figure 3. Calculated bifurcation diagrams of output pulse energy versus nonlinearity for a) FP-APM and b) M-APM configurations. Pulse intensity profiles are shown to indicate predicted pulse shapes at various points on the bifurcation curve. |
Unlike APM, KLM requires coupling between temporal and spatial variables, thus giving the possibility of spatio-temporal nonlinear dynamics. In KLM the intensity dependent index of refraction in the gain medium results in self focusing of the modelocked pulse train. This intensity dependent, or Kerr, lens can be combined with either a physical aperture (hard aperturing) or with increased overlap with the gain profile (soft aperturing) to yield a net increased gain for the pulsed field over the CW field in the laser cavity, leading to stable pulsing. Our experiments are performed in a soft-apertured Titanium Sapphire laser with “X” cavity configuration as shown in figure 4. The laser typically produces 50fs pulses centered at 800nm. The coupling between temporal and spatial degrees of freedom in the KLM laser means that instabilities in pulse intensity are frequently accompanied by spatial modulations. In addition, the soft apertured KLM laser can support two or more transverse cavity modes simultaneously, and this superposition of modes yields a new set of dynamical regimes. For a nonuniform pulse train, such as that produced by dynamical instabilities, measurements in the spatial, temporal, and spectral domains require the development of novel pulse resolved techniques. Ordinarily, pulse temporal, spectral, and spatial profiles are measured with slow detectors which average over thousands of pulses. Such averaging detection systems do not have the ability to reveal instabilities which manifest themselves in differences among pulses. We have developed pulsed resolved detection techniques for each of these variables, and used them to analyze the KLM dynamics in both the single and the multi-transverse mode regimes. The relevance of these techniques was demonstrated in our initial study of subharmonic oscillations.[4] Just as in the APM, the pulse train of the KLM laser undergoes subharmonic (P2 or P3) oscillations under certain conditions. However, if the KLM laser is operating with multiple transverse modes, these oscillations may be accompanied by spatial sweeping of the beam.[4,5] Figure 5 shows a two-dimensional spatial map of pulse energies measured in the P3 regime. This map was obtained by translating a small area, ns response, photodiode across the transverse profile of the beam, and recording on a 4GHz digital oscilloscope the pulse train at each position. The oscilloscope is triggered from a second photodiode so that the pulses can be assigned unambiguously as the map is created. The observed spatial oscillations can be explained by phase locking of two Hermite-Gaussian transverse modes of the cavity, resulting in spatial sweeping of the beam at the difference frequency of the two transverse modes. For this multiple transverse mode regime, analysis of the variation of the P3 oscillation frequency with intracavity power reveals frequency locking. That is, as the intracavity power is varied smoothly, the oscillation frequency sticks at a particular value, and then undergoes a sudden jump. Such frequency locking dynamics are characteristic of nonlinear interactions among coupled oscillators. To understand the manifestations of these nonlinear dynamics in the temporal regime, we have created a pulse resolved intereferometric autocorrelation technique. The system is based on a standard autocorrelator, but incorporates 1ns detection of the upconverted light and a computer pulse picking algorithm to retrieve full pulse trains for each autocorrelator delay. Using this method we are able to measure the interferometric autocorrelation of each pulse in the P3 train, and thus to demonstrate that the spatial modulations of the beam are not accompanied by changes in the temporal profile.
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| Figure 4. Schematic of KLM Ti:Sapphire laser cavity. Mirror M1 is a 10% output coupler. M2 and M3 have 10 cm radius of curvature. Two Brewster angle fused silica prisms (P1 and P2) are placed 62 cm apart for dispersion compensation. |
Even without the presence of multiple locked transverse modes, the KLM laser exhibits instabilities which can be analyzed in terms of nonlinear dynamics. We have, for example, observed period doubled, quasi –periodic, and potentially chaotic regimes of oscillation which are not accompanied by multiple transverse modes. Such behavior was predicted theoretically for the KLM with hard aperturing by Kalashnikov et. al.[6], and for soft aperturing by Kovalsky et. al. [7], and is the subject of our current studies. The APM and KLM studies are examples of the fruits of the nonlinear dynamics approach to fs laser behavior. With continuation of this work we look forward to ongoing improvements in our ability to analyze, predict, and control the instabilities of technologically important laser systems.
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| Figure 5. Two dimensional spatial map of pulse profiles in the P3 regime. |
[1] G. Sucha, D.S. Chemla, and S.R. Bolton, JOSA B, 15, p.2847, 1998
[2] U. Morgner and F. Mitschke, Physical Review A, 55, p. 3124, 1997.
[3] G. Sucha, S.R. Bolton, S. Weiss, and D.S. Chemla, Optics Letters, 20, p. 1794, 1995.
[4] S.R. Bolton, R. A. Jenks, C.N. Elkinton, and G.Sucha, JOSA B, 16, p.339, 1999.
[5] D. Cote and H. van Driel, Optics Letters, 23, p. 715, 1998.
[6] V. Kalashnikov, I. G. Poloyko, V. P. Mikhailov, and D. von der Linde, JOSA B, 14, p. 2691, 1997.
[7] M. Kovalsky, A. Hnilo, and G. Inchauspe, Optics Letters, 24, p. 1638, 1999.
