- Email: [email protected]
Observation of a period-doubling sequence in a nonlinear optical fiber ring cavity near zero dispersion
OPTICS COMMUNICATIONS
Optics Communications 104 (1994) 379-384 North-Holland
Observation of a period-doubling sequence in a nonlinear optical fiber ring cavity near zero dispersion G. Steinmeyer, D. Jaspert and F. Mitschke Institut fiir Quantenoptik, Universitlit Hannover, Welfengarten 1, D-30167 Hannover, Germany Received 14 June 1993; revised manuscript received 19 August 1993
The dynamics of a ring cavity containing an optical fiber are reexamined. A train of picosecond pulses from an additive pulse mode-locked laser synchronously drives the system. The fiber introduces a fast Kerr-type nonlinearity and limits the cavity to a single transverse mode. The experiment is performed near the zero dispersion wavelength of the fiber. A clear route to chaos up to period 8 is observed.
1. Introduction Ikeda et at. [ l ] predicted the first all optical system that demonstrated self-pulsing, period-doubling, and optical chaos. The system is quite simple, an optical cavity containing a material with a nonlinear index of refraction. We consider the cleanest case, a fast Kerr-type nonlinearity. Although theoretically straightforward, experimental realizations have met several obstacles that are traceable back to two assumptions in Ikeda's analysis. First there is the assumption of cw input intensity, second that of plane waves. In any experiment there is a transverse beam structure, for example a gaussian beam. Since the effect relies on an intensity-dependent index of refraction, self-focusing and other more complicated transverse phenomena compound the dynamics [ 2 ]. One way around this difficulty is to put the nonlinearity into a waveguide. This is the idea behind the suggestion [ 3 ] to do the experiment in an optical fiber. However, given the rather small nonlinearity coefficient of fused silica fiber, the intensities required for instabilities to occur are attainable only in pulsed operation, in conflict with the first assumption that a stationary (cw) input intensity be used. To use short pulses of light even has the additional benefit of ruling out any instabilities due to Brillouin scattering in the fiber which could considerably complicate the situation. However, the disadvantage is
that pulse propagation in fibers is subject to group velocity dispersion which was not considered in the original model. Adopting the theoretical analysis of ref. [ 1 ] and taking dispersion not into account, one could decompose the pulse propagating in the fiber ring into infinitesimally small temporal slices, each of which would evolve independently of the others. In the chaotic regime, adjacent slices would evolve in a totally different way because of their different initial amplitudes [4]. This would give the pulse circulating in the resonator an extremely ragged envelope: points infinitesimally apart in time would be macroscopically different in amplitude. It is quite obvious that dispersion, through creating an interaction between neighboring slices, will prevent this from happening. We have numerically studied this simplified situation in which there is no dispersion and an arbitrarily fast fiber nonlinearity. The calculation consists of the repeated iteration of two steps: (i) propagating of the pulse around the ring, and (ii) interference with the incoupled light at the input coupler. The result is an envelope modulation of the pulse that can be arbitrarily fast in the sense that after a few round trips it contracts to whatever temporal grid is used. This situation is, of course, utterly unphysical. It only highlights the crucial role played by dispersion. Vall6e [ 5 ] recently argued the same point
0030-4018/94/$ 07.00 © 1994 Elsevier Science B.V. All rights reserved.
379
Volume 104, number 4, 5,6
OPTICS COMMUNICATIONS
convincingly. We may point out that the finite response time of the nonlinearity or a less than exact match of resonator round trip time and pulse repetition frequency would have a similar effect. The first experiment on a synchronously driven ring resonator with a fiber in it [ 3 ] used a Q-switched mode-locked laser as a source of a short train of intense light pulses; consequently steady parameter values were not obtained in the experiment, and no detailed analysis of the results was possible. Nevertheless, a period-doubled oscillation (period 2) and a somewhat irregular pulse sequence were found, suggestive of a route to chaos. Unfortunately, this experiment had no control over the cavity phase, which is an all-important parameter due to the interference involved. Later experiments [6,7,8 ] employed continuous trains of pulses from a mode-locked pump laser to assure steady conditions, but only ref. [ 8 ] had control over the cavity phase. All these experiments were done under conditions of strong group velocity dispersion (silica fiber in the visible). We report here a similar experiment performed near the zero dispersion wavelength of the fiber, i.e. close to that wavelength in the infrared at which the leading order of group velocity dispersion goes through zero. Note, though, that when the leading order of dispersion vanishes, there still is higher order dispersion. By performing the experiment at this particular wavelength, we minimize the interaction between different temporal parts of the pulses which tends to wash out much of the nonlinear effects. It would be false to conclude, of course, that for an experiment at the zero dispersion wavelength, a cw model without dispersion would be an adequate description. Our experiment is distinguished in that we (i) had a particular laser light source at our disposal that was uniquely suited to the purpose, (ii) found a very simple trick how to determine the cavity phase for each measurement, and (iii) had access to the latest in data acquisition hardware. In their combination, these features allow us to get significantly cleaner data so that theories can now be put to much more specific tests.
1 January 1994
2. Experiment
The setup of our experiment is shown in fig. 1. The cw mode-locked pulse train is generated by an additive pulse mode-locked Nd: YAG laser at 2 = 1.3188 Ixm [ 9,10 ]. This laser produces practically chirp-free sech 2 shaped pulses with durations of ~ 12 ps at a repetition rate of 82.4 MHz. Average output powers up to 1.5 W and peak powers exceeding 1 kW are achieved so that the necessary nonlinear phase shift o f the order of Trcan be produced in just a few meters of fiber. Back reflections from the experiment into the laser are suppressed by an optical diode. A half wave retarder, together with the first polarizer in the diode, serves to suitably attenuate the optical power going into the experiment. Part of the laser beam is split off into both a fast photo diode and an autocorrelator so that power and pulse width of the laser source are permanently monitored throughout our experiments. None of the instabilities described below occurs at this position. The fiber ring resonator basically consists of a R = 30% input coupling mirror and L = 9.1 m of polarization-maintaining single mode fiber (York HB 1250). The fiber ring has an optical length of 15 m so that the round trip time is exactly four times the laser pulse repetition frequency. Fiber ends are angle-cut to reduce spurious reflections. Coupling efficiencies of 60% are obtained with G R I N lenses. The resonator has a finesse of 3.5. Light is linearly polarized along the slow axis of
Fig. 1. Experimental set-up. APM: additive pulse mode locked Nd:YAG laser, 2/2: half wave retarder, OD: optical diode (Faraday rotator and polarizers), LD: laser diagnostics, consistingof autocorrelator, fast photo diode, and power monitor C: custommade chopper wheel (see text), PZT: piezoelectric ring transducer, F: 9.1 m polarization-maintaining fiber, RD: ring diagnostics, consisting of scanning Fabry-P~rot interferometer and photodetectors, M~: mirrors (M1 has 30%, M2 has 31%, and M3 has 97% reflectivity), Li: lenses.
380
Volume 104, number 4,5,6
OPTICS COMMUNICATIONS
the fiber throughout. The fiber was measured to have an effective mode field area of 36 I.tm 2. The zero dispersion wavelength for the same polarization is 20= 1323.8 nm according to information provided by the manufacturer [ 11 ]; this is only 5 nm from our operating wavelength of 2 = 1318.8 nm. Therefore we have a residual second-order dispersion r2 = 0.45 ps2/ km and a third-order dispersion r3 = - 0.08 psa/krn. Part of the power emerging from the fiber end is split o f f b y a R = 3% mirror for analysis. We monitor the optical spectrum with a scanning Fabry-P6rot interferometer (free spectral range: 820 GHz, finesse: 100) and the temporal dynamics with a fast Ge photo diode (~3~B=2.3 G H z ) which is connected to either a radio frequency spectrum analyzer or to an ultrafast digital storage oscilloscope (Hewlett Packard HP-54720A) with a maximum sampling rate of 4 GSa/s. The cavity phase is subject to drift on a time scale of seconds mostly due to thermal fluctuations. It is therefore desirable to measure the resonator phase along with any data taken. We use the following very simple procedure: before being fed into the fiber ring the laser beam is modulated by a specially shaped chopper blade. Once every revolution, the beam is fully unblocked for a 2 ms interval. For the rest of the revolution, the edge of the chopper blade clips the beam such that only at most 5% of the total intensity are transmitted, allowing "small signal" transmission measurement. (Note that beam shape distortion is not a problem here since the fiber restores the modal shape. ) At the same time, the fiber length is slowly ramped (25 ms for a free spectral range) by means of a PZT stretcher. Interpolation of the Airy peaks measured at low intensity across the "high power" interval where data acquisition took place directly gives the phase pertaining to the acquired data. Note that the actual data acquisition takes only 8 gs out of 2 ms so that we can let transients die out; also, the ramp can be considered stationary in this short time. Drift of parameters apparently was not a problem over the 8 gs data acquisition time. Only in a few cases, we noticed a change in the types of behavior during the acquisition interval; consistently, this corresponded with a power fluctuation from the pump laser (relaxation oscillation). Such data sets were disregarded in the data analysis.
1 January 1994
3. Data analysis The optical pulses are much faster than the temporal resolution of the electronics; therefore we can only measure an integrated value of the pulse energy. Note that any dynamical processes taking place at the high intensities close to the pulse peak will be nearly swamped in this integral value by contributions from the pulse wings. We must therefore expect that a large signal-to-noise ratio will be required for the detection of the instabilities. This is indeed the case. In reconstructing the pulse energies from the recorded data files, two technical points need to be addressed: (i) At a sampling interval of 250 ps, there are only very few data points in each electrical pulse of about 1 ns width. It turns out to be advantageous to purposely low-pass filter the signal at 600 MHz so that each pulse is represented by several more data points while overlap of successive pulses remains negligible. (ii) The ring resonator is driven at its fourth harmonic, which means that at any given time there are four pulses traveling in it. In a manner of speaking, these represent four independent experiments at virtually identical conditions. In the recorded data files, these pulse sequences are interleaved and need to be sorted apart properly. The evaluation procedure is straightforward: F r o m the recorded file (which contains ~ 656 pulses), we first obtain the precise repetition rate and average pulse shape. Note that the latter is represented by 656 points per 250 ps sampling interval; this corresponds to an effective sampling rate of 2.6 THz. Next, each individual pulse is compared with the so-obtained average pulse as a reference; a vertical scaling factor is assigned to each pulse to describe its relative energy. It is this step where the low-pass filter mentioned above is most helpful. Finally, the time series is split into four by extracting every fourth pulse. Fourier spectra can then be calculated for each of the partial series, and then be averaged together. Actually, for the data shown in fig. 2 the four partial time series were first concatenated into one long data set to enhance spectral resolution. Doing so poses no problem in the case of periodic data, as long as one takes care to do so with the correct phase of the subharmonics. While the concatenation is not strictly justifiable for the irregular data, we saw to it that the 381
Volume 104, number 4,5,6
OPTICS COMMUNICATIONS
1 January 1994
/"
0
-20 -40 -60
0
•b
-20 -40 -GO
.c C
•
4
5t
-2O
7~
-4o
"~
-60
o o.
0
+
-2o ~
-4(I -60
0
.e ,~
.
.
.
-20 -40 -60
0 •
• ..'.t~,:'...
-20 -40 -60 i
~
i
i
i
i
i
0.5
i
i
i
1.0
Pulse # N
Frequency Fig. 2. Power spectra (left column) and first return maps (right column) of experimental data. Frequencies are normalized to the repetition rate of 82.4 MHz. (a): period 1 at P=220 W and ~offi -0.87~, (h): period 2 at/~--220 W and ~o=0.25~, (c): period 4 at P=230 W and ~o= -0.37r, (d): period 8 at/~=240 W and ~Oo=-0.6~, (e): weak chaos at P=300 W and ~o=0.757~, (f): full chaos at P = 380 W and ~Oo= - 0.95n. Comparisons of the return maps (c) and (d) reveals that there are two distinct arrangements of points which appear to be mirror images of each other. Examples of either arrangement were found for both period four and period eight data.
Volume 104, number 4,5,6
OPTICS COMMUNICATIONS
1 January 1994
spectrum of the combined data sets was closely similar to the spectra of the individual data sets.
X
40° 4. D i s c u s s i o n of experimental results In fig. 2 examples of our experimental results are displayed. The power spectra (left column) were calculated through Fourier transformation of the corresponding time series. They clearly resolve a period doubling sequence up to period 8. Note that in the power spectra a dynamic range of 70 dB is obtained; only this excellent signal-to-noise ratio allows the detection of the weak period 8 modulation. The return maps generated from the same data (right column) are not too similar to typical return maps from a period-doubling system; we attribute this to the swamping process mentioned above. In the case of the period 8 signal, for which the power spectrum still allows clear identification, the return map is blurred to appear as period 4. In the spectra shown in figs. 2e and 2f the sharp needles that are characteristic for periodic behavior are increasingly replaced by a broad irregular structure. Concomittantly the period 2 structure in the return map contracts and gives way to a single irregularly shaped cloud of points. Figure 2f will be referred to as fully developed chaos while fig. 2e is designated as weak chaos. Our knowledge of the resonator phase for each measurement allows us to map out the parameter space experimentally. Figure 3 shows how various types of behavior are distributed in the go-P-plane, where g0 is the (small signal) resonator phase (defined such that go = 0 corresponds to resonance), and /~ the peak input power. Domain boundaries between various types of behaviour were sketched using the results from a total of 220 data sets distributed over the entire plane more or less uniformly. Period 1 behaviour (i.e. all pulses having the same energy) is found below an upper limit of about 250 W peak intensity (dashed line). This threshold shows only a weak dependence on phase and corresponds to a peak single pass nonlinear phase shift g ~ 1.2n in the fiber. Subharmonics occur for - 0.7n < go < + 0.Sn and 0 . 7 n < g , l < l . S n . Period 4 exists for g o < 0 and gm~ 1.2n; period 8 was observed for the same value
....® .........J:~'
•..............
o 300 __A~).
Sta'fic Resonat.or Phase ~'0
Fig. 3. Tentative sketch of the parameter space indicating the positions of the various periodic and irregular regimesas obtained from 220 experimentaldata fries. P 1: period one ( exists up to the dashed line), PI: period i, with i--2, 4, or 8. (X): weak chaos, ft. fully developed chaos. Encircled letters identify the parameters of the correspondingpanel in fig. 2. of g~ at go-- - 0 . 5 ~ . Outside these periodic regimes and above a threshold corresponding to about g~, = n there is weak chaos; as g~ approaches 2~, this gives way to fully developed chaos. As we compiled the data that went into fig. 3, we encountered several cases where different types of behavior occurred for essentially the same parameters. This phenomenon can at best partly be explained by measurement errors ( A P ~ + 1 0 % , Ag0~ _+0.05x); there were even data files in which the four interleaved time series differed in the type of behavior. O f course, the four copropagating pulses are practically guaranteed to be subject to the same physical conditions. The obvious conclusion is that there must be a coexistence of attractors, or generalized bistability, in some parts of the parameter space. At present, our data do not yet allow a full mapping out of the hysteretic regions, and thus fig. 3 should be taken as tentative. It is certainly unfortunate that the expected modulation of the pulse envelopes can not temporally be resolved by electronic detectors. We might even mistake for a period 1 solution what really is a period 2. For example, there might be an intensity modulation on top of a smooth pulse that flips phase every other round trip. To go beyond pulse energy measurements, it might be feasible to use a cross-correlation technique. However, that requires a reference pulse of much shorter duration. Here, we resort to an analysis of the optical spectra recorded along with the temporal data. 383
Volume 104, number 4,5,6
OPTICS COMMUNICATIONS
The spectra show a dramatic increase in width as soon as the irregular pulsing regime is entered. (However, we convinced ourselves by checking at the appropriate wavelengths with a m o n o c h r o m a t o r that stimulated R a m a n scattering plays no role.) It is instructive to break the feedback loop o f the resonator under these conditions by blocking the beam path. What remains is the spectrum o f the pulses broadened by self phase modulation during a single pass through the fiber. The width o f that kind of spectrum is dramatically reduced from the case with feedback ( 3 0 - 5 0 GHz, as opposed to hundreds of G H z with the resonator closed). We take this as an indication that indeed a rapid modulation of the pulse envelope, too fast to be detected directly, is taking place in the resonator. It appears reasonable to estimate that an upper limit for the spectral width &o imposed by dispersion would be given by 509 ~ 1 / x/fl2L. Similarly, for third order dispersion, one might estimate 8o9~1/'~/-fl~3L. For the dispersion values in our fiber, these expressions indicate roughly equal contributions from either order of dispersion amounting to about 2.5 T H z and 1.8 THz, respectively. The smaller of these numbers is about three times more than we observe. This, however, is not unreasonable because both our rather low resonator finesse and a possible slight mismatch between pulse repetition rate and the fourth harmonic o f the resonator round trip frequency would prevent us from reaching the ultimate limit.
5. Conclusion The system discussed here is structurally so simple that it can serve as the paradigm of an optical system prone to instabilities, Beyond that, there are strong implications for various techniques of generating short pulses, like synchronously driven lasers. Also, in the context o f more recent mode-locking schemes like additive pulse mode locking [ 10 ], an optical fiber with its fast intensity-dependent phase is often employed to simulate a fast saturable absorber. That
384
1 January 1994
requires a transformation of the nonlinear phase shift into an amplitude effect, which is easily done with an interferometer. The similarity between such a nonlinear cavity coupled to a laser and the nonlinear fiber ring discussed here is striking. Astonishingly, while instabilities of the type discussed here do occur in coupled cavity lasers [ 12 ], they have not yet enjoyed the attention they deserve.
Acknowledgements We thank Hewlett Packard G e r m a n y for giving us temporal access to their ultrafast digitizing oscilloscope, HP-54720A. We also gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft.
References [ 1] K. Ikeda, Optics Comm. 30 (1979) 257; K. Ikeda, H. Daido and O. Akimoto, Phys. Rev. Lett. 45 (1980) 709; K. Ikeda, K. Kondo and O. Akimoto, Phys. Rev. Lett. 49 (1982) 1467; K. Ikeda, J. Phys. 44 (1983) C2-183. [2]See, e.g., J.V. Moloney, in: Instabilities and chaos in quantum optics II, eds. N.B. Abraham, F.T. Arecchi and L.A. Lugiato (Plenum, 1988) p. 193-218. [ 3 ] H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda and M. Matsuolta, Phys. Rev. Lett. 50 (1983) 109. [4] D.W. McLaughlin, J.V. Moloney and A.C. Newell, Phys. Rev. Lett. 54 (1985) 681. [ 5 ] R. Vall6e, Optics Comm. 93 ( 1992) 389. [6] M. Nakazawa, K. Suzuki and H.A. Haus, Phys. Rev. A 15 (1988) 5193. [7] R. Vall6e, Optics Comm. 81 (1991) 419. [ 8 ] M.B. van der Mark, J.M. Schins and A. Lagendijk, Optics Comm. 98 (1993) 120. [9] E.P. Ippen, H.A. Haus and L.Y. Liu, J. Opt. Soc. Am. B 6 (1989) 1736. [ 10] F. Mitschke, G. Steinmeyer, M. Ostermeyer, C. Fallnich and H. Welling, Appl. Phys. B 56 (1993) 335. [ 11 ] York Fiber Ltd., HB 1250 fiber data sheets. [12]G. Sucha, S.R. Bolton, S. Weiss and D.S. Chemla, Conference on Lasers and Electro-Optics, 1993, Vol. 11, OSA Technical Digest Series 146.
Optics Communications 104 (1994) 379-384 North-Holland
Observation of a period-doubling sequence in a nonlinear optical fiber ring cavity near zero dispersion G. Steinmeyer, D. Jaspert and F. Mitschke Institut fiir Quantenoptik, Universitlit Hannover, Welfengarten 1, D-30167 Hannover, Germany Received 14 June 1993; revised manuscript received 19 August 1993
The dynamics of a ring cavity containing an optical fiber are reexamined. A train of picosecond pulses from an additive pulse mode-locked laser synchronously drives the system. The fiber introduces a fast Kerr-type nonlinearity and limits the cavity to a single transverse mode. The experiment is performed near the zero dispersion wavelength of the fiber. A clear route to chaos up to period 8 is observed.
1. Introduction Ikeda et at. [ l ] predicted the first all optical system that demonstrated self-pulsing, period-doubling, and optical chaos. The system is quite simple, an optical cavity containing a material with a nonlinear index of refraction. We consider the cleanest case, a fast Kerr-type nonlinearity. Although theoretically straightforward, experimental realizations have met several obstacles that are traceable back to two assumptions in Ikeda's analysis. First there is the assumption of cw input intensity, second that of plane waves. In any experiment there is a transverse beam structure, for example a gaussian beam. Since the effect relies on an intensity-dependent index of refraction, self-focusing and other more complicated transverse phenomena compound the dynamics [ 2 ]. One way around this difficulty is to put the nonlinearity into a waveguide. This is the idea behind the suggestion [ 3 ] to do the experiment in an optical fiber. However, given the rather small nonlinearity coefficient of fused silica fiber, the intensities required for instabilities to occur are attainable only in pulsed operation, in conflict with the first assumption that a stationary (cw) input intensity be used. To use short pulses of light even has the additional benefit of ruling out any instabilities due to Brillouin scattering in the fiber which could considerably complicate the situation. However, the disadvantage is
that pulse propagation in fibers is subject to group velocity dispersion which was not considered in the original model. Adopting the theoretical analysis of ref. [ 1 ] and taking dispersion not into account, one could decompose the pulse propagating in the fiber ring into infinitesimally small temporal slices, each of which would evolve independently of the others. In the chaotic regime, adjacent slices would evolve in a totally different way because of their different initial amplitudes [4]. This would give the pulse circulating in the resonator an extremely ragged envelope: points infinitesimally apart in time would be macroscopically different in amplitude. It is quite obvious that dispersion, through creating an interaction between neighboring slices, will prevent this from happening. We have numerically studied this simplified situation in which there is no dispersion and an arbitrarily fast fiber nonlinearity. The calculation consists of the repeated iteration of two steps: (i) propagating of the pulse around the ring, and (ii) interference with the incoupled light at the input coupler. The result is an envelope modulation of the pulse that can be arbitrarily fast in the sense that after a few round trips it contracts to whatever temporal grid is used. This situation is, of course, utterly unphysical. It only highlights the crucial role played by dispersion. Vall6e [ 5 ] recently argued the same point
0030-4018/94/$ 07.00 © 1994 Elsevier Science B.V. All rights reserved.
379
Volume 104, number 4, 5,6
OPTICS COMMUNICATIONS
convincingly. We may point out that the finite response time of the nonlinearity or a less than exact match of resonator round trip time and pulse repetition frequency would have a similar effect. The first experiment on a synchronously driven ring resonator with a fiber in it [ 3 ] used a Q-switched mode-locked laser as a source of a short train of intense light pulses; consequently steady parameter values were not obtained in the experiment, and no detailed analysis of the results was possible. Nevertheless, a period-doubled oscillation (period 2) and a somewhat irregular pulse sequence were found, suggestive of a route to chaos. Unfortunately, this experiment had no control over the cavity phase, which is an all-important parameter due to the interference involved. Later experiments [6,7,8 ] employed continuous trains of pulses from a mode-locked pump laser to assure steady conditions, but only ref. [ 8 ] had control over the cavity phase. All these experiments were done under conditions of strong group velocity dispersion (silica fiber in the visible). We report here a similar experiment performed near the zero dispersion wavelength of the fiber, i.e. close to that wavelength in the infrared at which the leading order of group velocity dispersion goes through zero. Note, though, that when the leading order of dispersion vanishes, there still is higher order dispersion. By performing the experiment at this particular wavelength, we minimize the interaction between different temporal parts of the pulses which tends to wash out much of the nonlinear effects. It would be false to conclude, of course, that for an experiment at the zero dispersion wavelength, a cw model without dispersion would be an adequate description. Our experiment is distinguished in that we (i) had a particular laser light source at our disposal that was uniquely suited to the purpose, (ii) found a very simple trick how to determine the cavity phase for each measurement, and (iii) had access to the latest in data acquisition hardware. In their combination, these features allow us to get significantly cleaner data so that theories can now be put to much more specific tests.
1 January 1994
2. Experiment
The setup of our experiment is shown in fig. 1. The cw mode-locked pulse train is generated by an additive pulse mode-locked Nd: YAG laser at 2 = 1.3188 Ixm [ 9,10 ]. This laser produces practically chirp-free sech 2 shaped pulses with durations of ~ 12 ps at a repetition rate of 82.4 MHz. Average output powers up to 1.5 W and peak powers exceeding 1 kW are achieved so that the necessary nonlinear phase shift o f the order of Trcan be produced in just a few meters of fiber. Back reflections from the experiment into the laser are suppressed by an optical diode. A half wave retarder, together with the first polarizer in the diode, serves to suitably attenuate the optical power going into the experiment. Part of the laser beam is split off into both a fast photo diode and an autocorrelator so that power and pulse width of the laser source are permanently monitored throughout our experiments. None of the instabilities described below occurs at this position. The fiber ring resonator basically consists of a R = 30% input coupling mirror and L = 9.1 m of polarization-maintaining single mode fiber (York HB 1250). The fiber ring has an optical length of 15 m so that the round trip time is exactly four times the laser pulse repetition frequency. Fiber ends are angle-cut to reduce spurious reflections. Coupling efficiencies of 60% are obtained with G R I N lenses. The resonator has a finesse of 3.5. Light is linearly polarized along the slow axis of
Fig. 1. Experimental set-up. APM: additive pulse mode locked Nd:YAG laser, 2/2: half wave retarder, OD: optical diode (Faraday rotator and polarizers), LD: laser diagnostics, consistingof autocorrelator, fast photo diode, and power monitor C: custommade chopper wheel (see text), PZT: piezoelectric ring transducer, F: 9.1 m polarization-maintaining fiber, RD: ring diagnostics, consisting of scanning Fabry-P~rot interferometer and photodetectors, M~: mirrors (M1 has 30%, M2 has 31%, and M3 has 97% reflectivity), Li: lenses.
380
Volume 104, number 4,5,6
OPTICS COMMUNICATIONS
the fiber throughout. The fiber was measured to have an effective mode field area of 36 I.tm 2. The zero dispersion wavelength for the same polarization is 20= 1323.8 nm according to information provided by the manufacturer [ 11 ]; this is only 5 nm from our operating wavelength of 2 = 1318.8 nm. Therefore we have a residual second-order dispersion r2 = 0.45 ps2/ km and a third-order dispersion r3 = - 0.08 psa/krn. Part of the power emerging from the fiber end is split o f f b y a R = 3% mirror for analysis. We monitor the optical spectrum with a scanning Fabry-P6rot interferometer (free spectral range: 820 GHz, finesse: 100) and the temporal dynamics with a fast Ge photo diode (~3~B=2.3 G H z ) which is connected to either a radio frequency spectrum analyzer or to an ultrafast digital storage oscilloscope (Hewlett Packard HP-54720A) with a maximum sampling rate of 4 GSa/s. The cavity phase is subject to drift on a time scale of seconds mostly due to thermal fluctuations. It is therefore desirable to measure the resonator phase along with any data taken. We use the following very simple procedure: before being fed into the fiber ring the laser beam is modulated by a specially shaped chopper blade. Once every revolution, the beam is fully unblocked for a 2 ms interval. For the rest of the revolution, the edge of the chopper blade clips the beam such that only at most 5% of the total intensity are transmitted, allowing "small signal" transmission measurement. (Note that beam shape distortion is not a problem here since the fiber restores the modal shape. ) At the same time, the fiber length is slowly ramped (25 ms for a free spectral range) by means of a PZT stretcher. Interpolation of the Airy peaks measured at low intensity across the "high power" interval where data acquisition took place directly gives the phase pertaining to the acquired data. Note that the actual data acquisition takes only 8 gs out of 2 ms so that we can let transients die out; also, the ramp can be considered stationary in this short time. Drift of parameters apparently was not a problem over the 8 gs data acquisition time. Only in a few cases, we noticed a change in the types of behavior during the acquisition interval; consistently, this corresponded with a power fluctuation from the pump laser (relaxation oscillation). Such data sets were disregarded in the data analysis.
1 January 1994
3. Data analysis The optical pulses are much faster than the temporal resolution of the electronics; therefore we can only measure an integrated value of the pulse energy. Note that any dynamical processes taking place at the high intensities close to the pulse peak will be nearly swamped in this integral value by contributions from the pulse wings. We must therefore expect that a large signal-to-noise ratio will be required for the detection of the instabilities. This is indeed the case. In reconstructing the pulse energies from the recorded data files, two technical points need to be addressed: (i) At a sampling interval of 250 ps, there are only very few data points in each electrical pulse of about 1 ns width. It turns out to be advantageous to purposely low-pass filter the signal at 600 MHz so that each pulse is represented by several more data points while overlap of successive pulses remains negligible. (ii) The ring resonator is driven at its fourth harmonic, which means that at any given time there are four pulses traveling in it. In a manner of speaking, these represent four independent experiments at virtually identical conditions. In the recorded data files, these pulse sequences are interleaved and need to be sorted apart properly. The evaluation procedure is straightforward: F r o m the recorded file (which contains ~ 656 pulses), we first obtain the precise repetition rate and average pulse shape. Note that the latter is represented by 656 points per 250 ps sampling interval; this corresponds to an effective sampling rate of 2.6 THz. Next, each individual pulse is compared with the so-obtained average pulse as a reference; a vertical scaling factor is assigned to each pulse to describe its relative energy. It is this step where the low-pass filter mentioned above is most helpful. Finally, the time series is split into four by extracting every fourth pulse. Fourier spectra can then be calculated for each of the partial series, and then be averaged together. Actually, for the data shown in fig. 2 the four partial time series were first concatenated into one long data set to enhance spectral resolution. Doing so poses no problem in the case of periodic data, as long as one takes care to do so with the correct phase of the subharmonics. While the concatenation is not strictly justifiable for the irregular data, we saw to it that the 381
Volume 104, number 4,5,6
OPTICS COMMUNICATIONS
1 January 1994
/"
0
-20 -40 -60
0
•b
-20 -40 -GO
.c C
•
4
5t
-2O
7~
-4o
"~
-60
o o.
0
+
-2o ~
-4(I -60
0
.e ,~
.
.
.
-20 -40 -60
0 •
• ..'.t~,:'...
-20 -40 -60 i
~
i
i
i
i
i
0.5
i
i
i
1.0
Pulse # N
Frequency Fig. 2. Power spectra (left column) and first return maps (right column) of experimental data. Frequencies are normalized to the repetition rate of 82.4 MHz. (a): period 1 at P=220 W and ~offi -0.87~, (h): period 2 at/~--220 W and ~o=0.25~, (c): period 4 at P=230 W and ~o= -0.37r, (d): period 8 at/~=240 W and ~Oo=-0.6~, (e): weak chaos at P=300 W and ~o=0.757~, (f): full chaos at P = 380 W and ~Oo= - 0.95n. Comparisons of the return maps (c) and (d) reveals that there are two distinct arrangements of points which appear to be mirror images of each other. Examples of either arrangement were found for both period four and period eight data.
Volume 104, number 4,5,6
OPTICS COMMUNICATIONS
1 January 1994
spectrum of the combined data sets was closely similar to the spectra of the individual data sets.
X
40° 4. D i s c u s s i o n of experimental results In fig. 2 examples of our experimental results are displayed. The power spectra (left column) were calculated through Fourier transformation of the corresponding time series. They clearly resolve a period doubling sequence up to period 8. Note that in the power spectra a dynamic range of 70 dB is obtained; only this excellent signal-to-noise ratio allows the detection of the weak period 8 modulation. The return maps generated from the same data (right column) are not too similar to typical return maps from a period-doubling system; we attribute this to the swamping process mentioned above. In the case of the period 8 signal, for which the power spectrum still allows clear identification, the return map is blurred to appear as period 4. In the spectra shown in figs. 2e and 2f the sharp needles that are characteristic for periodic behavior are increasingly replaced by a broad irregular structure. Concomittantly the period 2 structure in the return map contracts and gives way to a single irregularly shaped cloud of points. Figure 2f will be referred to as fully developed chaos while fig. 2e is designated as weak chaos. Our knowledge of the resonator phase for each measurement allows us to map out the parameter space experimentally. Figure 3 shows how various types of behavior are distributed in the go-P-plane, where g0 is the (small signal) resonator phase (defined such that go = 0 corresponds to resonance), and /~ the peak input power. Domain boundaries between various types of behaviour were sketched using the results from a total of 220 data sets distributed over the entire plane more or less uniformly. Period 1 behaviour (i.e. all pulses having the same energy) is found below an upper limit of about 250 W peak intensity (dashed line). This threshold shows only a weak dependence on phase and corresponds to a peak single pass nonlinear phase shift g ~ 1.2n in the fiber. Subharmonics occur for - 0.7n < go < + 0.Sn and 0 . 7 n < g , l < l . S n . Period 4 exists for g o < 0 and gm~ 1.2n; period 8 was observed for the same value
....® .........J:~'
•..............
o 300 __A~).
Sta'fic Resonat.or Phase ~'0
Fig. 3. Tentative sketch of the parameter space indicating the positions of the various periodic and irregular regimesas obtained from 220 experimentaldata fries. P 1: period one ( exists up to the dashed line), PI: period i, with i--2, 4, or 8. (X): weak chaos, ft. fully developed chaos. Encircled letters identify the parameters of the correspondingpanel in fig. 2. of g~ at go-- - 0 . 5 ~ . Outside these periodic regimes and above a threshold corresponding to about g~, = n there is weak chaos; as g~ approaches 2~, this gives way to fully developed chaos. As we compiled the data that went into fig. 3, we encountered several cases where different types of behavior occurred for essentially the same parameters. This phenomenon can at best partly be explained by measurement errors ( A P ~ + 1 0 % , Ag0~ _+0.05x); there were even data files in which the four interleaved time series differed in the type of behavior. O f course, the four copropagating pulses are practically guaranteed to be subject to the same physical conditions. The obvious conclusion is that there must be a coexistence of attractors, or generalized bistability, in some parts of the parameter space. At present, our data do not yet allow a full mapping out of the hysteretic regions, and thus fig. 3 should be taken as tentative. It is certainly unfortunate that the expected modulation of the pulse envelopes can not temporally be resolved by electronic detectors. We might even mistake for a period 1 solution what really is a period 2. For example, there might be an intensity modulation on top of a smooth pulse that flips phase every other round trip. To go beyond pulse energy measurements, it might be feasible to use a cross-correlation technique. However, that requires a reference pulse of much shorter duration. Here, we resort to an analysis of the optical spectra recorded along with the temporal data. 383
Volume 104, number 4,5,6
OPTICS COMMUNICATIONS
The spectra show a dramatic increase in width as soon as the irregular pulsing regime is entered. (However, we convinced ourselves by checking at the appropriate wavelengths with a m o n o c h r o m a t o r that stimulated R a m a n scattering plays no role.) It is instructive to break the feedback loop o f the resonator under these conditions by blocking the beam path. What remains is the spectrum o f the pulses broadened by self phase modulation during a single pass through the fiber. The width o f that kind of spectrum is dramatically reduced from the case with feedback ( 3 0 - 5 0 GHz, as opposed to hundreds of G H z with the resonator closed). We take this as an indication that indeed a rapid modulation of the pulse envelope, too fast to be detected directly, is taking place in the resonator. It appears reasonable to estimate that an upper limit for the spectral width &o imposed by dispersion would be given by 509 ~ 1 / x/fl2L. Similarly, for third order dispersion, one might estimate 8o9~1/'~/-fl~3L. For the dispersion values in our fiber, these expressions indicate roughly equal contributions from either order of dispersion amounting to about 2.5 T H z and 1.8 THz, respectively. The smaller of these numbers is about three times more than we observe. This, however, is not unreasonable because both our rather low resonator finesse and a possible slight mismatch between pulse repetition rate and the fourth harmonic o f the resonator round trip frequency would prevent us from reaching the ultimate limit.
5. Conclusion The system discussed here is structurally so simple that it can serve as the paradigm of an optical system prone to instabilities, Beyond that, there are strong implications for various techniques of generating short pulses, like synchronously driven lasers. Also, in the context o f more recent mode-locking schemes like additive pulse mode locking [ 10 ], an optical fiber with its fast intensity-dependent phase is often employed to simulate a fast saturable absorber. That
384
1 January 1994
requires a transformation of the nonlinear phase shift into an amplitude effect, which is easily done with an interferometer. The similarity between such a nonlinear cavity coupled to a laser and the nonlinear fiber ring discussed here is striking. Astonishingly, while instabilities of the type discussed here do occur in coupled cavity lasers [ 12 ], they have not yet enjoyed the attention they deserve.
Acknowledgements We thank Hewlett Packard G e r m a n y for giving us temporal access to their ultrafast digitizing oscilloscope, HP-54720A. We also gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft.
References [ 1] K. Ikeda, Optics Comm. 30 (1979) 257; K. Ikeda, H. Daido and O. Akimoto, Phys. Rev. Lett. 45 (1980) 709; K. Ikeda, K. Kondo and O. Akimoto, Phys. Rev. Lett. 49 (1982) 1467; K. Ikeda, J. Phys. 44 (1983) C2-183. [2]See, e.g., J.V. Moloney, in: Instabilities and chaos in quantum optics II, eds. N.B. Abraham, F.T. Arecchi and L.A. Lugiato (Plenum, 1988) p. 193-218. [ 3 ] H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda and M. Matsuolta, Phys. Rev. Lett. 50 (1983) 109. [4] D.W. McLaughlin, J.V. Moloney and A.C. Newell, Phys. Rev. Lett. 54 (1985) 681. [ 5 ] R. Vall6e, Optics Comm. 93 ( 1992) 389. [6] M. Nakazawa, K. Suzuki and H.A. Haus, Phys. Rev. A 15 (1988) 5193. [7] R. Vall6e, Optics Comm. 81 (1991) 419. [ 8 ] M.B. van der Mark, J.M. Schins and A. Lagendijk, Optics Comm. 98 (1993) 120. [9] E.P. Ippen, H.A. Haus and L.Y. Liu, J. Opt. Soc. Am. B 6 (1989) 1736. [ 10] F. Mitschke, G. Steinmeyer, M. Ostermeyer, C. Fallnich and H. Welling, Appl. Phys. B 56 (1993) 335. [ 11 ] York Fiber Ltd., HB 1250 fiber data sheets. [12]G. Sucha, S.R. Bolton, S. Weiss and D.S. Chemla, Conference on Lasers and Electro-Optics, 1993, Vol. 11, OSA Technical Digest Series 146.