Chaotic dynamics in erbium-doped fiber ring lasers Henry D. I. Abarbanel,* Matthew B. Kennel, † Michael Buhl, ‡ and Clifford Tureman Lewis ‡ Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402 Received 5 November 1998; revised manuscript received 25 January 1999 Chaotically oscillating rare-earth-doped fiber ring lasers DFRLs may provide an attractive way to exploit the broad bandwidth available in an optical communications system. Recent theoretical and experimental investigations have successfully shown techniques to modulate information onto the wide-band chaotic oscil- lations, transmit that signal along an optical fiber, and demodulate the information at the receiver. We develop a theoretical model of a DFRL and discuss an efficient numerical simulation which includes intrinsic linear and nonlinear induced birefringence, both transverse polarizations, group velocity dispersion, and a finite gain bandwidth. We analyze first a configuration with a single loop of optical fiber containing the doped fiber amplifier, and then, as suggested by Roy and VanWiggeren, we investigate a system with two rings of optical fiber—one made of passive fiber alone. The typical round-trip time for the passive optical ring connecting the erbium-doped amplifier to itself is 200 ns, so 10 5 round-trips are required to see the slow effects of the population inversion dynamics in this laser system. Over this large number of round-trips, physical effects like GVD and the Kerr nonlinearity, which may appear small at our frequencies and laser powers via conventional estimates, may accumulate and dominate the dynamics. We demonstrate from our model that chaotic oscilla- tions of the ring laser with parameters relevant to erbium-doped fibers arises from the nonlinear Kerr effect and not from interplay between the atomic population inversion and radiation dynamics. S1050-29479908607-2 PACS numbers: 42.65.Sf, 42.55.Wd, 05.45.-a I. INTRODUCTION Rare-earth-doped fibers 1,2 appear very attractive for use as optical amplifiers in various stages of a communica- tions system and as inline active elements in ring lasers used as transmitters and receivers in such a system. Erbium-doped lasers are especially attractive for long haul communications over optical fibers as the lasing wavelength, about 1550 nm, is near the minimum attenuation and dispersion point of stan- dard single-mode optical fiber. The goal of using the base- band communications bandwidth of tens of terahertz THz around the carrier frequency of a few hundreds of THz might well be realized using a chaotic communications scheme in which the transmitter and receiver work over a very broad Fourier power spectrum while retaining sufficient state space structure to allow modulation and demodulation of informa- tion on the chaotic ‘‘carrier’’ or courier of information. The use of chaos as a ‘‘courier’’ of information is primarily for 1 effective utilization of the enormous bandwidth of these systems, and 2 the autosynchronization of the transmitter and receiver in the communications scheme we discuss. This paper develops the theory of the operation in chaotic regimes of doped fiber ring lasers, especially those with pa- rameters corresponding to doping with erbium. A communi- cations window near 1300 nm is of some interest as well 3. In addition to studying ring lasers with parameters associated with those of Er 3 + doping as we do in this paper, in our future work we plan to analyze structurally similar ring la- sers choosing parameters associated with Nd 3 + and Pr 3 + and doping where lasing at 1300 nm is seen 2. We investigate models of doped fiber ring lasers DFRLs. Such model equations have been discussed by Roy, Williams, and their collaborators 4,5, and they have shown, experimentally and theoretically, that chaotic field oscillations can be observed in these DFRLs. Roy and Van- Wiggeren 6 and our group 7 have shown experimentally and theoretically, respectively, that such ring laser systems can be used for communicating information from a transmit- ter laser to an open loop optical receiver. In work subsequent to that just cited, VanWiggerenm and Roy 8,9 extended their results to both higher symbol transmission rates and richer modulation schemes. The main themes of this paper are the origin of chaos in the operation of erbium-doped fiber ring lasers and effective methods for solution of the relatively well known equations of motion for these lasers. These two issues are raised by the experiments of Roy and VanWiggeren 6 and possess sub- stantial interest in themselves whether or not these laser sys- tems turn out to be utilized in communications systems. The experiments of Roy and VanWiggeren demonstrate the pres- ence of chaotic regimes in this kind of laser system, and a consideration of the time scales of the atomic lifetime 10 ms and the ring round-trip time 100 ns alone would suggest that in standard models of fiber lasers chaos would be absent. The system, as we recall below, is basically that of a delay differential equation where the delay is the ring round- trip time and enters a map for the complex electric field amplitude and the differential equation describes the popula- tion inversion dynamics which is dictated by the fluores- *Also at Department of Physics, and Marine Physical Laboratory, Scripps Institution of Oceanography, UC San Diego. Electronic ad- dress: hdia@hamilton.ucsd.edu † Electronic address: mkennel@ucsd.edu ‡ Also at Department of Physics, UC San Diego. Electronic ad- dresses: mbuhl@click.ucsd.edu MB, ctl@click.ucsd.edu CTL. PHYSICAL REVIEW A SEPTEMBER 1999 VOLUME 60, NUMBER 3 PRA 60 1050-2947/99/603/236015/$15.00 2360 ©1999 The American Physical Society