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Coherence and shot-to-shot spectral fluctuations in noise-like ultrafast fiber lasers Antoine F. J. Runge,* Claude Aguergaray, Neil G. R. Broderick, and Miro Erkintalo Physics Department, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand *Corresponding author: [email protected] Received June 26, 2013; revised September 16, 2013; accepted September 20, 2013; posted September 23, 2013 (Doc. ID 192934); published October 21, 2013 We report on experimental studies of coherence and fluctuations in noise-like pulse trains generated by ultrafast fiber oscillators. By measuring the degree of first-order coherence using a Young’s-type interference experiment, we prove the lack of phase coherence across the seemingly regular array of pulses. We further quantify the pulse-to-pulse fluctuations by recording the single-shot spectra of the megahertz pulse train, and experimentally demonstrate the existence of spectral fluctuations that remain unresolved in conventional time-averaged ensemble measurements. Phase incoherence and spectral fluctuations are contrasted with quantified coherence and spectral stability when the laser is soliton mode-locked. © 2013 Optical Society of America OCIS codes: (140.3510) Lasers, fiber; (140.7090) Ultrafast lasers; (060.5530) Pulse propagation and temporal solitons. http://dx.doi.org/10.1364/OL.38.004327
Mode-locked lasers can be regarded as epitomes of stability, with the produced pulse trains being sufficiently regular to facilitate frequency metrology with unprecedented precision [1]. Despite their reputation, numerous passively mode-locked laser architectures can also be operated at the other end of the spectrum, in a regime characterized by a seemingly regular train of pulses, closer scrutiny of which reveals what resemble bursts of chaotic, noise-like (NL) fluctuations. While such unstable mode-locking has been extensively studied in solid-state and gas lasers [2–4], it has more recently aroused renewed interest in the context of fiber-based devices. NL pulses were first reported from a fiber laser in 1997 by Horowitz et al. who used an Er-doped laser mode-locked by nonlinear polarization evolution (NPE) [5]. Subsequent studies have evidenced the generality of the regime, with similar outputs observed using different mode-locking mechanisms [6], dispersion landscapes [7,8], and active fibers [9–11]. Although most past studies on NL pulses have revolved around practical applications, the regime has recently translated into the focal point of fundamental physics, owing to possible connections with rogue waves [12–14]: freak events observed in seemingly disjoint systems such as hydrodynamics and supercontinuum generation [15–19]. So far, characteristics of fiber-based NL pulses have been predominantly assessed only indirectly by means of numerical simulations or multi-shot measurements. This deficiency stems from the typical megahertz repetition rate of the pulse train, which makes shot-to-shot measurements intractable for conventional diagnostic techniques. As a result, characteristics of NL pulse trains, such as their shot-to-shot spectral stability and phase coherence, remain unconfirmed by means of direct experiments. In this Letter, we address this deficiency and report on direct experiments investigating the single-shot spectral stability and phase coherence of a megahertz repetition rate NL pulse train. We use two complementary experimental techniques: (i) in our first set of experiments we adopt a Young’s-type interference technique, widely used to quantify the phase stability of fiber supercontinua [20–23], and highlight the complete lack of 0146-9592/13/214327-04$15.00/0
pulse-to-pulse phase coherence across the NL pulse train; (ii) in our second set of experiments, we use the photonic time-stretch technique [24–30] to record roundtrip-toroundtrip spectra of NL pulses, providing experimental evidence how pulse-to-pulse fluctuations are smoothed in conventional multi-shot spectral measurements. Because our laser allows for simple switching between NL and soliton operation, we are able to quantitatively compare and contrast the two regimes with minimal changes in the laser configuration. In addition to direct quantitative insights on NL pulses provided by our experiments, the techniques themselves are expected to become established tools for the characterization of ultrafast fiber lasers. The laser source used in our experiments is a 180 m long erbium-doped NPE-mode-locked laser with a repetition rate of 1.1 MHz. It is similar to that used in [5] and [14]. This type of cavity is the simplest possible system displaying both soliton mode-locking and NL pulses. When the intracavity polarization controller is suitably aligned and the pump power sufficiently low, the laser self-mode-locks into the soliton regime. In this regime, the output displays a narrow spectrum (∼1.2 nm bandwidth) with strong Kelly sidebands, as shown in Fig. 1(a). The temporal pulse duration is approximately 2.1 ps as deconvolved from the autocorrelation trace shown in Fig. 1(c), assuming a sech2 pulse shape. Transition into the NL regime is achieved by adjusting the polarization controller and/or increasing the pump power, and results in a clear change in the output pulse characteristics. As evidenced in Fig. 1(b), this regime is identified by a substantially broader (∼15 nm) spectrum compared with the soliton regime; it is also noteworthy that the Kelly sidebands visible in the solitonic spectrum [cf. Fig. 1(a)] are completely absent in the NL regime. The transition is apparent also in the time-domain: the autocorrelation trace shown in Fig. 1(d) can be seen to be two orders of magnitude broader than in the soliton regime, and in fact to exceed the temporal span of our autocorrelator (200 ps). The trace is also seen to contain a narrow (450 fs) peak riding atop the broad background, and we remark that this feature, in conjunction with a broad and © 2013 Optical Society of America
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Fig. 1. Characteristics of the laser in the soliton (a), (c), (e) and NL (b), (d), (f) regimes. (a), (b) Output spectrum; (c), (d) autocorrelation trace, and (e), (f) output pulse trains measured with a photodiode (PD).
smooth spectrum, is an identifying feature common to all reported NL configurations. Note that the narrow peak should not be confused with the duration of the pulse envelope. It is a coherent artifact that arises from the substructure of NL pulses [4]. In both regimes, the laser outputs a train of pulses separated by the cavity repetition rate [see Figs. 1(e) and 1(f)]. We first assess the phase stability of both operating regimes. To this end, we adopt a common technique used to quantify the stability of supercontinuum sources by performing spectral interferometry between consecutive pulses from the laser output [20]. The experimental coherence setup is schematically shown in Fig. 2. It corresponds to a Michelson interferometer with a path difference of one cavity roundtrip between the arms of the device. A 50∕50 fiber coupler is used to divide the laser output, and the shorter interferometer arm contains a free-space variable delay line that allows for precision tuning of the temporal delay between consecutive pulses. The longer arm comprises a segment of dispersionshifted fiber (DSF) whose length (∼89 m) is adjusted to match half the laser repetition rate. Following the DSF the optical signal is reflected by means of an optical loop
mirror [31]. Signals from both interferometer paths are then combined in the 50∕50 coupler and the resultant spectral interferogram is recorded with an optical spectrum analyzer (OSA). By scanning the free-space variable delay line we are able to decrease the temporal separation of consecutive pulses in order to resolve the spectral modulation with the OSA. The visibility of the modulation is a direct measure of the wavelength-dependent magnitude of the complex degree of first-order coherence of the electric field [20]. In Fig. 3(a), we plot the spectral interference pattern at the output of the interferometer when the laser is operating in the soliton regime. We can clearly identify the expected spectral modulation with a fringe visibility V ∼ 0.98 across the whole spectrum, indicating almost total pulse-to-pulse phase coherence when the laser is operating in the soliton regime. When the laser is switched into the NL regime the interference pattern vanishes, as shown in Fig. 3(b), where we plot the spectrum at the interferometer output immediately post-transition. This result directly reveals the absence of phase correlations among pulses in a NL pulse train. Note that the vanishing of the fringe visibility cannot be explained by changes in the interferometer delay upon transition. Indeed, we find that the repetition rate is precisely the same in both regimes, and that the 3 nm shift in central wavelengths of the regimes translates to a group delay difference of only ∼0.3 ps, thanks to the low-dispersion of the used DSF. Such a small temporal translation cannot explain the vanishing of the fringe pattern. We have, of course, verified that no fringe pattern can be observed, even through extensive scans of the relative pulse delay. The lack of phase correlations, as quantified above, indirectly suggests that the NL pulse train exhibits significant pulse-to-pulse fluctuations in the spectral intensities. This is also in accordance with past numerical simulations [8]. To gain direct experimental verification, we have performed additional measurements that allow us to record the roundtrip-to-roundtrip spectral variations at the laser output. Specifically, we use a photonic time-stretch technique so as to overcome the slow acquisition time of conventional spectrum analyzers [24–30]. This technique relies on the fact that any signal transforms into its Fourier transform when subjected to a sufficiently large quadratic spectral phase. Here, we stretch the laser output pulses in a single-mode fiber (SMF) with length L 10 km and group-velocity dispersion β2 −24.6 ps2 km−1 . We then acquire the 1.2
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resulting roundtrip-to-roundtrip power spectral densities with a 12 GHz photodetector and a 12.5 GHz oscilloscope. In order to avoid nonlinear effects in the SMF, we attenuate the laser output using a 10% coupler prior to stretching. We have also verified that no spectral distortions take place by comparing the optical spectrum before and after the SMF. To show that the amount of dispersive stretching is sufficient for our signals, we note that the wavelength-to-time technique associates to each point τ along the input temporal envelope an error jδλτj ∼ λ2 τ∕2πcjβ2 jL, where λ is the center wavelength of the signal. The maximum error originates from the extreme wing of the pulse, and can be estimated as maxfjδλjg ∼ jδλτ0 j, where τ0 is the temporal width of the envelope (before stretching). We can obtain an estimate for sufficient stretching by requiring the maximum error to be much smaller than the signal bandwidth. This yields jβ2 jL ≫ λ2 τ0 ∕2πcB ≡ jβ2 jLlim , where B is the spectral width of the signal. In our case, the temporal e−2 width of the NL burst is τ0 ∼ 230 ps. With the bandwidth B ∼ 15 nm this yields jβ2 jLlim 20 ps2 . For soliton operation we have jβ2 jLlim 4 ps2 . Both limits are more than an order of magnitude smaller than the stretching provided by our SMF jβ2 jL 240 ps2 . This confirms the validity of our configuration. In Figs. 4(a) and 4(b) we show the recorded single-shot spectra over one thousand consecutive pulses (density maps in the bottom of the graph) when the laser is operating in the soliton and in the NL regimes, respectively. The wavelength axis is obtained from the oscilloscope timebase t as λ 2πc∕Δω ω0 , where Δω t∕β2 L, and ω0 is the center (angular) frequency of the spectrum. We can clearly see that the spectra in the soliton regime
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are nearly indistinguishable from one another, resulting in insignificant variation between the ensemble-averaged mean (solid curve at the top of the graph) and an arbitrarily chosen single realization (dashed curve). Note that, as expected, the solitonic single-shot spectra are in excellent agreement with the time-averaged spectrum recorded with the OSA [cf. Fig. 1(a)]. The Kelly sidebands appear diminished and broadened in the single-shot measurements, but we attribute this discrepancy to the finite spectral resolution of the system. Indeed, the response time of the photodetector is approximately Δt ∼ 80 ps, which maps into a wavelength resolution of Δλ Δtλ2 ∕2πcjβ2 jL ∼ 0.4 nm. This resolution is not quite sufficient to fully resolve the sidebands whose bandwidth, inferred from Fig. 1(a), is approximately 0.1 nm. Single-shot measurements for the NL regime are shown in Fig. 4(b). Here the results can be seen to be markedly different compared to the soliton regime, with the roundtrip-to-roundtrip spectra exhibiting substantial fluctuations. In this regime each of the individual spectra are found to be highly structured (an arbitrarily picked sample is shown as the dashed curve at the top of the graph), which is to be contrasted with the smoothness apparent in the corresponding OSA measurement [see Fig. 1(b)]. In fact, when an ensemble average is computed from the fluctuating constituents, the fine-structure is washed out resulting in a smooth spectral density (solid curve at the top of the graph), which is in good agreement with the spectrum measured with the OSA. Indeed, we remark that the 15.5 nm width of the reconstructed spectrum differs by less than 1 nm from that measured with the OSA. Our results, therefore, constitute the first direct experimental evidence that the smooth spectrum characteristic for NL pulses arises precisely from the ensemble averaging of countless highly structured elementary events. To complete our study, we characterize the spectral fluctuations more quantitatively by extracting the standard deviation at the wavelength corresponding to the full width at half-maximum of the average spectra. We obtain σ s 0.0065 and σ NL 0.1624 for the soliton and the NL measurements, respectively. Because both ensembles are normalized such that the mean value at the FWHM is precisely μ 0.5, we can conclude that the NL pulses fluctuate 25 times more at the FWHM than the solitons. Finally, we emphasize that the results presented above are not limited to the specific laser configuration utilized, but reflect ubiquitous characteristics of fiber lasers operating in the NL regime. Indeed, we have repeated both experiments using an all-normal dispersion Yb-doped fiber laser mode-locked with a nonlinear amplifying loop mirror. The specific architecture is based on those introduced in [32] and [33], and the NL regime is achieved via significant cavity lengthening [34]. Despite the entirely different cavity design and mode-locking mechanism, the results obtained using this architecture are qualitatively indistinguishable from those reported above; total lack of phase-coherence and significant shot-to-shot spectral fluctuations in the NL regime, contrasted by phase-coherence and spectral indistinguishability in the dissipative soliton regime. To conclude, we have reported on an experimental study of coherence and fluctuations in NL fiber oscillators. Using a Young’s-type interference experiment we
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have shown that NL pulses, while manifesting as a seemingly regular train of pulses, exhibit significant pulse-to-pulse phase fluctuations. By using a photonic time-stretch technique we have recorded the single-shot spectra of the 1.1 MHz pulse train and presented direct experimental evidence that the smooth spectrum characteristic for NL pulses arises from the ensemble averaging of a multitude of highly structured and fluctuating elementary spectra. To our knowledge, these are the first real-time spectral measurements of an ultrafast fiber laser operating in the NL regime. We expect that the experimental techniques adopted here will become widespread in the context of fiber lasers, providing significant insights on the coherence and single-shot characteristics of a wide variety of laser outputs. Finally, the possibility for simple real-time measurements is further anticipated to spur advances in theoretical modeling, leading to a better understanding of the formation dynamics of NL pulses and related dissipative structures. We thank G. Steinmeyer for useful discussions. References 1. Th. Udem, R. Holzwarth, and T. W. Hänsch, Nature 416, 233 (2002). 2. M. A. Duguay, J. W. Hansen, and S. L. Shapiro, IEEE J. Quantum Electron. QE-6, 725 (1970). 3. J. M. Dudley, C. M. Loh, and J. D. Harvey, Quantum Semiclass. Opt. 8, 1029 (1996). 4. M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, Laser Photon. Rev. 7, 557 (2013). 5. M. Horowitz, Y. Barad, and Y. Silberberg, Opt. Lett. 22, 799 (1997). 6. O. Pottiez, R. Grajales-Coutiño, B. Ibarra-Escamilla, E. Kuzin, and J. Hernández-García, Appl. Opt. 50, E24 (2011). 7. L. M. Zhao, D. Y. Tang, J. Wu, X. Q. Fu, and S. C. Wen, Opt. Express 15, 2145 (2007). 8. S. Kobtsev, S. Kukarin, S. Smirnov, S. Turitsyn, and A. Latkin, Opt. Express 17, 20707 (2009). 9. S. Smirnov, S. Kobtsev, S. Kukarin, and A. Ivanenko, Opt. Express 20, 27447 (2012). 10. Q. Wang, T. Chen, B. Zhang, A. P. Heberle, and K. P. Chen, Opt. Lett. 36, 3750 (2011).
11. A. Boucon, B. Barviau, J. Fatome, C. Finot, T. Sylvestre, M. W. Lee, P. Grelu, and G. Millot, Appl. Phys. B 106, 283 (2012). 12. J. M. Soto-Crespo, Ph. Grelu, and N. Akhmediev, Phys. Rev. E 84, 016604 (2011). 13. A. Zaviyalov, O. Egorov, R. Iliew, and F. Lederer, Phys. Rev. A 85, 013828 (2012). 14. C. Lecaplain, Ph. Grelu, J. M. Soto-Crespo, and N. Akhmediev, Phys. Rev. Lett. 108, 233901 (2012). 15. M. Onorato, A. R. Osborne, M. Serio, and S. Bertone, Phys. Rev. Lett. 86, 5831 (2001). 16. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007). 17. M. Erkintalo, G. Genty, and J. M. Dudley, Opt. Lett. 34, 2468 (2009). 18. B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, Nat. Phys. 6, 790 (2010). 19. A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, Phys. Rev. X 2, 011015 (2012). 20. J. M. Dudley and S. Coen, Opt. Lett. 27, 1180 (2002). 21. X. Gu, M. Kimmel, A. Shreenath, R. Trebino, J. M. Dudley, S. Coen, and R. Windeler, Opt. Express 11, 2697 (2003). 22. F. Lu and W. Knox, Opt. Express 12, 347 (2004). 23. J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006). 24. Y. C. Tong, L. Y. Chang, and H. K. Tsang, Electron. Lett. 33, 983 (1997). 25. P. Kelkar, F. Coppinger, A. S. Bhushan, and B. Jalali, Electron. Lett. 35, 1661 (1999). 26. D. R. Solli, J. Chou, and B. Jalali, Nat. Photonics 2, 48 (2008). 27. D. R. Solli, G. Herink, B. Jalali, and C. Ropers, Nat. Photonics 6, 463 (2012). 28. B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, Sci. Rep. 2, 882 (2012). 29. T. Godin, B. Wetzel, T. Sylvestre, L. Larger, A. Kudlinski, A. Mussot, A. Ben Salem, M. Zghal, G. Genty, F. Dias, and J. M. Dudley, Opt. Express 21, 18452 (2013). 30. K. Goda and B. Jalali, Nat. Photonics 7, 102 (2013). 31. N. J. Doran and D. Wood, Opt. Lett. 13, 56 (1988). 32. C. Aguergaray, N. G. R. Broderick, M. Erkintalo, J. S. Y. Chen, and V. Kruglov, Opt. Express 20, 10545 (2012). 33. M. Erkintalo, C. Aguergaray, A. Runge, and N. G. R. Broderick, Opt. Express 20, 22669 (2012). 34. C. Aguergaray, A. Runge, M. Erkintalo, and N. G. R. Broderick, Opt. Lett. 38, 2644 (2013).
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Coherence and shot-to-shot spectral fluctuations in noise-like ultrafast fiber lasers Antoine F. J. Runge,* Claude Aguergaray, Neil G. R. Broderick, and Miro Erkintalo Physics Department, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand *Corresponding author: [email protected] Received June 26, 2013; revised September 16, 2013; accepted September 20, 2013; posted September 23, 2013 (Doc. ID 192934); published October 21, 2013 We report on experimental studies of coherence and fluctuations in noise-like pulse trains generated by ultrafast fiber oscillators. By measuring the degree of first-order coherence using a Young’s-type interference experiment, we prove the lack of phase coherence across the seemingly regular array of pulses. We further quantify the pulse-to-pulse fluctuations by recording the single-shot spectra of the megahertz pulse train, and experimentally demonstrate the existence of spectral fluctuations that remain unresolved in conventional time-averaged ensemble measurements. Phase incoherence and spectral fluctuations are contrasted with quantified coherence and spectral stability when the laser is soliton mode-locked. © 2013 Optical Society of America OCIS codes: (140.3510) Lasers, fiber; (140.7090) Ultrafast lasers; (060.5530) Pulse propagation and temporal solitons. http://dx.doi.org/10.1364/OL.38.004327
Mode-locked lasers can be regarded as epitomes of stability, with the produced pulse trains being sufficiently regular to facilitate frequency metrology with unprecedented precision [1]. Despite their reputation, numerous passively mode-locked laser architectures can also be operated at the other end of the spectrum, in a regime characterized by a seemingly regular train of pulses, closer scrutiny of which reveals what resemble bursts of chaotic, noise-like (NL) fluctuations. While such unstable mode-locking has been extensively studied in solid-state and gas lasers [2–4], it has more recently aroused renewed interest in the context of fiber-based devices. NL pulses were first reported from a fiber laser in 1997 by Horowitz et al. who used an Er-doped laser mode-locked by nonlinear polarization evolution (NPE) [5]. Subsequent studies have evidenced the generality of the regime, with similar outputs observed using different mode-locking mechanisms [6], dispersion landscapes [7,8], and active fibers [9–11]. Although most past studies on NL pulses have revolved around practical applications, the regime has recently translated into the focal point of fundamental physics, owing to possible connections with rogue waves [12–14]: freak events observed in seemingly disjoint systems such as hydrodynamics and supercontinuum generation [15–19]. So far, characteristics of fiber-based NL pulses have been predominantly assessed only indirectly by means of numerical simulations or multi-shot measurements. This deficiency stems from the typical megahertz repetition rate of the pulse train, which makes shot-to-shot measurements intractable for conventional diagnostic techniques. As a result, characteristics of NL pulse trains, such as their shot-to-shot spectral stability and phase coherence, remain unconfirmed by means of direct experiments. In this Letter, we address this deficiency and report on direct experiments investigating the single-shot spectral stability and phase coherence of a megahertz repetition rate NL pulse train. We use two complementary experimental techniques: (i) in our first set of experiments we adopt a Young’s-type interference technique, widely used to quantify the phase stability of fiber supercontinua [20–23], and highlight the complete lack of 0146-9592/13/214327-04$15.00/0
pulse-to-pulse phase coherence across the NL pulse train; (ii) in our second set of experiments, we use the photonic time-stretch technique [24–30] to record roundtrip-toroundtrip spectra of NL pulses, providing experimental evidence how pulse-to-pulse fluctuations are smoothed in conventional multi-shot spectral measurements. Because our laser allows for simple switching between NL and soliton operation, we are able to quantitatively compare and contrast the two regimes with minimal changes in the laser configuration. In addition to direct quantitative insights on NL pulses provided by our experiments, the techniques themselves are expected to become established tools for the characterization of ultrafast fiber lasers. The laser source used in our experiments is a 180 m long erbium-doped NPE-mode-locked laser with a repetition rate of 1.1 MHz. It is similar to that used in [5] and [14]. This type of cavity is the simplest possible system displaying both soliton mode-locking and NL pulses. When the intracavity polarization controller is suitably aligned and the pump power sufficiently low, the laser self-mode-locks into the soliton regime. In this regime, the output displays a narrow spectrum (∼1.2 nm bandwidth) with strong Kelly sidebands, as shown in Fig. 1(a). The temporal pulse duration is approximately 2.1 ps as deconvolved from the autocorrelation trace shown in Fig. 1(c), assuming a sech2 pulse shape. Transition into the NL regime is achieved by adjusting the polarization controller and/or increasing the pump power, and results in a clear change in the output pulse characteristics. As evidenced in Fig. 1(b), this regime is identified by a substantially broader (∼15 nm) spectrum compared with the soliton regime; it is also noteworthy that the Kelly sidebands visible in the solitonic spectrum [cf. Fig. 1(a)] are completely absent in the NL regime. The transition is apparent also in the time-domain: the autocorrelation trace shown in Fig. 1(d) can be seen to be two orders of magnitude broader than in the soliton regime, and in fact to exceed the temporal span of our autocorrelator (200 ps). The trace is also seen to contain a narrow (450 fs) peak riding atop the broad background, and we remark that this feature, in conjunction with a broad and © 2013 Optical Society of America
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Fig. 1. Characteristics of the laser in the soliton (a), (c), (e) and NL (b), (d), (f) regimes. (a), (b) Output spectrum; (c), (d) autocorrelation trace, and (e), (f) output pulse trains measured with a photodiode (PD).
smooth spectrum, is an identifying feature common to all reported NL configurations. Note that the narrow peak should not be confused with the duration of the pulse envelope. It is a coherent artifact that arises from the substructure of NL pulses [4]. In both regimes, the laser outputs a train of pulses separated by the cavity repetition rate [see Figs. 1(e) and 1(f)]. We first assess the phase stability of both operating regimes. To this end, we adopt a common technique used to quantify the stability of supercontinuum sources by performing spectral interferometry between consecutive pulses from the laser output [20]. The experimental coherence setup is schematically shown in Fig. 2. It corresponds to a Michelson interferometer with a path difference of one cavity roundtrip between the arms of the device. A 50∕50 fiber coupler is used to divide the laser output, and the shorter interferometer arm contains a free-space variable delay line that allows for precision tuning of the temporal delay between consecutive pulses. The longer arm comprises a segment of dispersionshifted fiber (DSF) whose length (∼89 m) is adjusted to match half the laser repetition rate. Following the DSF the optical signal is reflected by means of an optical loop
mirror [31]. Signals from both interferometer paths are then combined in the 50∕50 coupler and the resultant spectral interferogram is recorded with an optical spectrum analyzer (OSA). By scanning the free-space variable delay line we are able to decrease the temporal separation of consecutive pulses in order to resolve the spectral modulation with the OSA. The visibility of the modulation is a direct measure of the wavelength-dependent magnitude of the complex degree of first-order coherence of the electric field [20]. In Fig. 3(a), we plot the spectral interference pattern at the output of the interferometer when the laser is operating in the soliton regime. We can clearly identify the expected spectral modulation with a fringe visibility V ∼ 0.98 across the whole spectrum, indicating almost total pulse-to-pulse phase coherence when the laser is operating in the soliton regime. When the laser is switched into the NL regime the interference pattern vanishes, as shown in Fig. 3(b), where we plot the spectrum at the interferometer output immediately post-transition. This result directly reveals the absence of phase correlations among pulses in a NL pulse train. Note that the vanishing of the fringe visibility cannot be explained by changes in the interferometer delay upon transition. Indeed, we find that the repetition rate is precisely the same in both regimes, and that the 3 nm shift in central wavelengths of the regimes translates to a group delay difference of only ∼0.3 ps, thanks to the low-dispersion of the used DSF. Such a small temporal translation cannot explain the vanishing of the fringe pattern. We have, of course, verified that no fringe pattern can be observed, even through extensive scans of the relative pulse delay. The lack of phase correlations, as quantified above, indirectly suggests that the NL pulse train exhibits significant pulse-to-pulse fluctuations in the spectral intensities. This is also in accordance with past numerical simulations [8]. To gain direct experimental verification, we have performed additional measurements that allow us to record the roundtrip-to-roundtrip spectral variations at the laser output. Specifically, we use a photonic time-stretch technique so as to overcome the slow acquisition time of conventional spectrum analyzers [24–30]. This technique relies on the fact that any signal transforms into its Fourier transform when subjected to a sufficiently large quadratic spectral phase. Here, we stretch the laser output pulses in a single-mode fiber (SMF) with length L 10 km and group-velocity dispersion β2 −24.6 ps2 km−1 . We then acquire the 1.2
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Fig. 3. Spectral interference pattern of consecutive pulses when the laser is operated in the (a) soliton and (b) NL regime. Solid curves show the spectrum at the output of the interferometer and dashed curves indicate the laser output spectrum for comparison.
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resulting roundtrip-to-roundtrip power spectral densities with a 12 GHz photodetector and a 12.5 GHz oscilloscope. In order to avoid nonlinear effects in the SMF, we attenuate the laser output using a 10% coupler prior to stretching. We have also verified that no spectral distortions take place by comparing the optical spectrum before and after the SMF. To show that the amount of dispersive stretching is sufficient for our signals, we note that the wavelength-to-time technique associates to each point τ along the input temporal envelope an error jδλτj ∼ λ2 τ∕2πcjβ2 jL, where λ is the center wavelength of the signal. The maximum error originates from the extreme wing of the pulse, and can be estimated as maxfjδλjg ∼ jδλτ0 j, where τ0 is the temporal width of the envelope (before stretching). We can obtain an estimate for sufficient stretching by requiring the maximum error to be much smaller than the signal bandwidth. This yields jβ2 jL ≫ λ2 τ0 ∕2πcB ≡ jβ2 jLlim , where B is the spectral width of the signal. In our case, the temporal e−2 width of the NL burst is τ0 ∼ 230 ps. With the bandwidth B ∼ 15 nm this yields jβ2 jLlim 20 ps2 . For soliton operation we have jβ2 jLlim 4 ps2 . Both limits are more than an order of magnitude smaller than the stretching provided by our SMF jβ2 jL 240 ps2 . This confirms the validity of our configuration. In Figs. 4(a) and 4(b) we show the recorded single-shot spectra over one thousand consecutive pulses (density maps in the bottom of the graph) when the laser is operating in the soliton and in the NL regimes, respectively. The wavelength axis is obtained from the oscilloscope timebase t as λ 2πc∕Δω ω0 , where Δω t∕β2 L, and ω0 is the center (angular) frequency of the spectrum. We can clearly see that the spectra in the soliton regime
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Fig. 4. Single-shot spectral results for (a) soliton and (b) NL regime. (Top) The solid curves show the mean spectrum computed from the single-shot measurements and the dashed curve shows an example of a single spectral realization. (Bottom) Linear scale density maps of the single-shot spectra over 1000 consecutive roundtrips. The spectral axis is derived from the oscilloscope timebase.
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are nearly indistinguishable from one another, resulting in insignificant variation between the ensemble-averaged mean (solid curve at the top of the graph) and an arbitrarily chosen single realization (dashed curve). Note that, as expected, the solitonic single-shot spectra are in excellent agreement with the time-averaged spectrum recorded with the OSA [cf. Fig. 1(a)]. The Kelly sidebands appear diminished and broadened in the single-shot measurements, but we attribute this discrepancy to the finite spectral resolution of the system. Indeed, the response time of the photodetector is approximately Δt ∼ 80 ps, which maps into a wavelength resolution of Δλ Δtλ2 ∕2πcjβ2 jL ∼ 0.4 nm. This resolution is not quite sufficient to fully resolve the sidebands whose bandwidth, inferred from Fig. 1(a), is approximately 0.1 nm. Single-shot measurements for the NL regime are shown in Fig. 4(b). Here the results can be seen to be markedly different compared to the soliton regime, with the roundtrip-to-roundtrip spectra exhibiting substantial fluctuations. In this regime each of the individual spectra are found to be highly structured (an arbitrarily picked sample is shown as the dashed curve at the top of the graph), which is to be contrasted with the smoothness apparent in the corresponding OSA measurement [see Fig. 1(b)]. In fact, when an ensemble average is computed from the fluctuating constituents, the fine-structure is washed out resulting in a smooth spectral density (solid curve at the top of the graph), which is in good agreement with the spectrum measured with the OSA. Indeed, we remark that the 15.5 nm width of the reconstructed spectrum differs by less than 1 nm from that measured with the OSA. Our results, therefore, constitute the first direct experimental evidence that the smooth spectrum characteristic for NL pulses arises precisely from the ensemble averaging of countless highly structured elementary events. To complete our study, we characterize the spectral fluctuations more quantitatively by extracting the standard deviation at the wavelength corresponding to the full width at half-maximum of the average spectra. We obtain σ s 0.0065 and σ NL 0.1624 for the soliton and the NL measurements, respectively. Because both ensembles are normalized such that the mean value at the FWHM is precisely μ 0.5, we can conclude that the NL pulses fluctuate 25 times more at the FWHM than the solitons. Finally, we emphasize that the results presented above are not limited to the specific laser configuration utilized, but reflect ubiquitous characteristics of fiber lasers operating in the NL regime. Indeed, we have repeated both experiments using an all-normal dispersion Yb-doped fiber laser mode-locked with a nonlinear amplifying loop mirror. The specific architecture is based on those introduced in [32] and [33], and the NL regime is achieved via significant cavity lengthening [34]. Despite the entirely different cavity design and mode-locking mechanism, the results obtained using this architecture are qualitatively indistinguishable from those reported above; total lack of phase-coherence and significant shot-to-shot spectral fluctuations in the NL regime, contrasted by phase-coherence and spectral indistinguishability in the dissipative soliton regime. To conclude, we have reported on an experimental study of coherence and fluctuations in NL fiber oscillators. Using a Young’s-type interference experiment we
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have shown that NL pulses, while manifesting as a seemingly regular train of pulses, exhibit significant pulse-to-pulse phase fluctuations. By using a photonic time-stretch technique we have recorded the single-shot spectra of the 1.1 MHz pulse train and presented direct experimental evidence that the smooth spectrum characteristic for NL pulses arises from the ensemble averaging of a multitude of highly structured and fluctuating elementary spectra. To our knowledge, these are the first real-time spectral measurements of an ultrafast fiber laser operating in the NL regime. We expect that the experimental techniques adopted here will become widespread in the context of fiber lasers, providing significant insights on the coherence and single-shot characteristics of a wide variety of laser outputs. Finally, the possibility for simple real-time measurements is further anticipated to spur advances in theoretical modeling, leading to a better understanding of the formation dynamics of NL pulses and related dissipative structures. We thank G. Steinmeyer for useful discussions. References 1. Th. Udem, R. Holzwarth, and T. W. Hänsch, Nature 416, 233 (2002). 2. M. A. Duguay, J. W. Hansen, and S. L. Shapiro, IEEE J. Quantum Electron. QE-6, 725 (1970). 3. J. M. Dudley, C. M. Loh, and J. D. Harvey, Quantum Semiclass. Opt. 8, 1029 (1996). 4. M. Rhodes, G. Steinmeyer, J. Ratner, and R. Trebino, Laser Photon. Rev. 7, 557 (2013). 5. M. Horowitz, Y. Barad, and Y. Silberberg, Opt. Lett. 22, 799 (1997). 6. O. Pottiez, R. Grajales-Coutiño, B. Ibarra-Escamilla, E. Kuzin, and J. Hernández-García, Appl. Opt. 50, E24 (2011). 7. L. M. Zhao, D. Y. Tang, J. Wu, X. Q. Fu, and S. C. Wen, Opt. Express 15, 2145 (2007). 8. S. Kobtsev, S. Kukarin, S. Smirnov, S. Turitsyn, and A. Latkin, Opt. Express 17, 20707 (2009). 9. S. Smirnov, S. Kobtsev, S. Kukarin, and A. Ivanenko, Opt. Express 20, 27447 (2012). 10. Q. Wang, T. Chen, B. Zhang, A. P. Heberle, and K. P. Chen, Opt. Lett. 36, 3750 (2011).
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