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Vol. 25, No. 20 | 2 Oct 2017 | OPTICS EXPRESS 24400

Mode suppression of 53 dB and pulse repetition rates of 2.87 and 36.4 GHz in a compact, mode-locked fiber laser comprising coupled Fabry-Perot cavities of low finesse ( F = 2) H. CHENG,1,2,3,5 Y. ZHOU,1,2,3 A. E. MIRONOV,4 W. WANG,1,2,3 T. QIAO,1,2,3 W. LIN,1,2,3 Q. QIAN,1,2,3 S. XU,1,2,3 Z. YANG,1,2,3,6 AND J. G. EDEN4 1

State Key Laboratory of Luminescent Materials and Devices and Institute of Optical Communication Materials, South China University of Technology, Guangzhou 510640, China 2 Guangdong Engineering Technology Research and Development Center of Special Optical Fiber Materials and Devices, South China University of Technology, Guangzhou 510640, China 3 Guangdong Provincial Key Laboratory of Fiber Laser Materials and Applied Techniques, South China University of Technology, Guangzhou 510640, China 4 Laboratory for Optical Physics and Engineering, Department of Electrical and Computer Engineering, University of Illinois, Urbana, Illinois 61801, USA 5 [email protected] 6 [email protected]

Abstract: Multiplication of the pulse repetition frequency (PRF) of a compact, mode-locked fiber laser by a factor as large as 25 has been achieved with two coupled Fabry-Perot (FP) resonators of low finesse (F = 2). Reducing the FP finesse by at least two orders of magnitude, relative to previous pulse frequency multiplication architectures, has the effect of stabilizing the oscillator with respect to pulse-to-pulse amplitude, dropped pulses, and other effects of cavity detuning. Coupling two Fabry-Perot cavities, each encompassing a 3.3-3.6 cm length of fiber, in a hybrid geometry resembling that of the coupled-cavity laser interferometer has yielded side mode suppressions ≥ 50 dB while simultaneously doubling the laser PRF to 2.87 GHz. Pulses approximately 3.9 ps in duration (FWHM) are emitted at intervals of 27.5 ps, and in groups (bursts) of pulses separated by 350 ps. Thus, the PRF within the pulse bursts is 36 GHz, a factor of 25 greater than the free spectral range for a conventional mode-locked cavity having a length of 6.9 cm. Experimental data are in accord with simulations of the phase coherence and temporal behavior of the mode-locked pulses. © 2017 Optical Society of America OCIS codes: (140.7090) Ultrafast lasers; (190.7110) Ultrafast nonlinear optics; (140.4050) Mode-locked lasers.

References and links 1. 2. 3. 4. 5. 6.

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#297561 Journal © 2017

https://doi.org/10.1364/OE.25.024400 Received 9 Jun 2017; revised 26 Aug 2017; accepted 29 Aug 2017; published 26 Sep 2017

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Commun. 3, 765 (2012). Y. L. Qi, H. Liu, H. Cui, Y. Q. Huang, Q. Y. Ning, M. Liu, Z. C. Luo, A. P. Luo, and W. C. Xu, “Graphenedeposited microfiber photonic device for ultrahigh-repetition rate pulse generation in a fiber laser,” Opt. Express 23(14), 17720–17726 (2015). D. Mao, X. Liu, Z. Sun, H. Lu, D. Han, G. Wang, and F. Wang, “Flexible high-repetition-rate ultrafast fiber laser,” Sci. Rep. 3(1), 3223 (2013). R. S. Fodil, F. Amrani, C. Yang, A. Kellou, and Ph. Grelu, “Adjustable high-repetition-rate pulse trains in a passively-mode-locked fiber laser,” Phys. Rev. A 00(0), 003800 (2016). G. T. Harvey and L. F. Mollenauer, “Harmonically mode-locked fiber ring laser with an internal Fabry-Perot stabilizer for soliton transmission,” Opt. Lett. 18(2), 107–109 (1993). D. E. T. F. Ashby and D. F. Jephcott, “Measurement of plasma density using a gas laser as an infrared interferometer,” Appl. Phys. Lett. 3(1), 13–16 (1963). J. B. Gerardo, J. T. Verdeyen, and M. A. Gusinow, “Spatially and temporally resolved electron and atom concentrations in an afterglow gas discharge,” J. Appl. Phys. 36(11), 3526–3534 (1965). S. H. Xu, Z. M. Yang, T. Liu, W. N. Zhang, Z. M. Feng, Q. Y. Zhang, and Z. H. Jiang, “An efficient compact 300 mW narrow-linewidth single frequency fiber laser at 1.5 µm,” Opt. Express 18(2), 1249–1254 (2010). H. Kotb, M. Abdelalim, and H. Anis, “Generalized analytical model for dissipative soliton in all-normaldispersion mode-locked fiber laser,” IEEE J. Sel. Top. Quantum Electron. 22(2), 1100209 (2016). V. L. Kalashnikov and A. Apolonski, “Chirped-pulse oscillators: A unified standpoint,” Phys. Rev. A 79(4), 043829 (2009). H. Cheng, W. Lin, T. Qiao, S. Xu, and Z. Yang, “Theoretical and experimental analysis of instability of continuous wave mode locking: Towards high fundamental repetition rate in Tm3+-doped fiber lasers,” Opt. Express 24(26), 29882–29895 (2016). W. H. Renninger, A. Chong, and F. W. Wise, “Giant-chirp oscillators for short-pulse fiber amplifiers,” Opt. Lett. 33(24), 3025–3027 (2008). X. Liu, “Numerical and experimental investigation of dissipative solitons in passively mode-locked fiber lasers with large net-normal-dispersion and high nonlinearity,” Opt. Express 17(25), 22401–22416 (2009). A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25(2), 140–148 (2008). F. Amrani, A. Haboucha, M. Salhi, H. Leblond, A. Komarov, P. Grelu, and F. Sanchez, “Passively mode-locked erbium-doped double-clad fiber laser operating at the 322nd harmonic,” Opt. Lett. 34(14), 2120–2122 (2009). J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34(9), 1749–1757 (1998). D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multisoliton formation and soliton energy quantization in passively mode-locked fiber lasers,” Phys. Rev. A 72(4), 043816 (2005). P. Grelu, F. Belhache, F. Gutty, and J.-M. Soto-Crespo, “Phase-locked soliton pairs in a stretched-pulse fiber laser,” Opt. Lett. 27(11), 966–968 (2002).

1. Introduction The development of novel methods for multiplying the pulse repetition frequency (PRF) of mode-locked lasers has been driven primarily by the introduction of optical frequency combs and their applications in, for example, optical communications and the precision calibration of

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astronomical spectrograms in the search for Earth-like exoplanets [1–3]. First reported by Sizer in 1989 [4] and subsequently confirmed by additional experiments [5–10], laser PRF multiplication has generally been realized through Fabry-Perot (FP) etalons installed outside or within the optical cavities of mode-locked lasers. Both active amplitude and phase modulation have been demonstrated [11–14], but the drawbacks of existing approaches include the complexity of the feedback control mechanism (and associated circuitry) and the variations in pulse-to-pulse amplitude that result from frequency detuning. Furthermore, fourwave mixing is also capable of initiating an ultrahigh PRF laser pulse train [15–17] but its appearance is accompanied by a poor supermode suppression ratio, particularly as the length of the master oscillator is increased [18,19]. Several of the limitations of conventional PRF multiplication techniques are traceable to the FP optical cavities that underlie most architectures. Specifically, FP resonator finesse values in the 102-104 range have typically been required for the purpose of ensuring minimally-acceptable levels of sidemode suppression [1–14,20]. Although spurious mode suppression is enhanced by multiple passes through an FP cavity, a lower bound of F = 400 has been deemed to be necessary in order to suppress off-resonant modes by at least 50 dB [6]. An unfortunate consequence of employing F ≥ 400 FP cavities as spectral filters, however, lies in the difficulty of maintaining the coincidence in wavelength between an FP cavity transmission peak and that of a laser resonator longitudinal mode. Coupling of the FP and laser cavities is particularly sensitive to cumulative phase walk-off. Prior to the work described here, low finesse FP cavities had received little attention in PRF multiplication laser systems because the spectral widths of the individual longitudinal modes were believed to be too large to achieve the spectral discrimination required for most applications of the PRF multiplication process. We demonstrate here that incorporating FP cavities having a finesse of < 2 into a modelocked fiber laser is capable of suppressing off-resonant modes by more than 50 dB while multiplying the PRF of the laser to 2.87 and 36.4 GHz. Perhaps not surprisingly, one benefit of decreasing the FP finesse (as well as the cavity length) is the stabilization of the pulse-topulse amplitude, thereby reducing the intensity variabilities and pulse dropout rates characteristic of detuning in conventional pulse multiplication systems. The optical arrangement adopted for the present experiments comprises two FP cavities coupled through a shared planar mirror which is simply a spliced connection of two short fiber sections. One of the fibers is doped so as to provide gain over a wavelength region near 1.5 µm whereas the second fiber section is undoped and, therefore, passive. If two reflectors are then installed so as to encompass both fibers, a mode-locked fiber laser having an internal reference cavity is realized. Such a device closely resembles the design of the coupled cavity laser interferometer, introduced in 1963 [21], in which the interaction of the longitudinal modes in the two coupled cavities provides precise measurements of a change in the effective refractive index of either cavity [22]. In the present situation, a similar principle is responsible for stabilizing the performance of a fiber laser mode-locked by a semiconductor saturable absorbable mirror (SESAM) substituted for one of the two conventional mirrors of an interferometer. 2. Resonator design and operating characteristics Panel (a) of Fig. 1 is a diagram illustrating the design of the laser resonator adopted for the present experiments. The overall cavity length of 6.9 cm is defined by mechanically splicing a section of Er-Yb:glass fiber, 3.3 cm in length and fabricated at the South China University of Technology [23], to a 3.6 cm long portion of commercially-available, single-mode fiber. Construction of the laser resonator is completed by depositing a multilayer dielectric coating directly onto the exposed face of the passive fiber section and affixing a SESAM to the opposite end of the two joined fibers. Figures 1(b) and 1(d) illustrate, respectively, the measured reflectivities of the dielectric mirror and the SESAM over the 1525-1575 nm and

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1400-1700 nm wavelength intervals. The former is essentially flat over the 50 nm region of interest, having a reflectivity of 99.3% at 1564 nm, whereas the SESAM reflectivity decreases monotonically from 95.5% at 1460 nm to 90.5% at 1590 nm.

Fig. 1. Design and optical characteristics of the hybrid, mode-locked fiber laser: (a) Diagram of the laser structure, comprising two coupled Fabry-Perot cavities, a dielectric mirror, and a SESAM; (b) Reflectivity of the dielectric mirror of (a) between 1525 nm and 1575 nm. The inset is an end-on image of the dielectric mirror deposited onto the exposed face of the silica fiber segment; (c) Calculated reflectivity of the FP cavity comprising the mirror, silica fiber, and fiber interface (FP1); (d) Measured reflectivity of the SESAM between 1350 nm and 1750 nm; (e) Calculated reflectivity of the FP cavity comprising the SESAM, the gain fiber, and the fiber interface (FP2); (f) Product of the spectra in (c) and (e), indicating the calculated FSR of 2.86 GHz and an envelope period of 36.19 GHz.

In effect, the laser cavity of Fig. 1(a) comprises two coupled Fabry-Perot cavities characterized by a common mirror of low reflectivity (e.g., the fiber union). Having a reflectivity of only 4% at 1564 nm, the fiber splice yields the calculated longitudinal mode spectra shown in Figs. 1(c) and 1(e) for the dielectric mirror-passive fiber FP cavity (denoted FP1) and the gain fiber-SESAM cavity (FP2), respectively. The FP finesse, defined as F = π(R1R2)1/4/(1-(R1R2)1/2) where R1 and R2 are the mirror and SESAM reflectivities, is found to be 1.6 and 1.8, respectively, for FP1 and FP2. As expected, such low values of F have the result of modulating the reflectance (and transmission) of FP1 and FP2 by only 3% and 6%, respectively (note the suppression of zero in both panels (c) and (e) of Fig. 1). Not surprisingly, therefore, the calculated transmission of the entire laser structure of Fig. 1(a) reflects the interference between the longitudinal modes of the coupled cavities, resulting in the spectral modulation evident in Fig. 1(f). Notice that the interaction between the two FP resonators yields a mode separation (free spectral range, FSR) of 2.86 GHz, which is twice the FSR for a single resonator having a length (L) of 6.9 cm (~1.44 GHz). It is also apparent in Fig. 1(f) that the mode spectrum is modulated in intensity with a period of 36.19 GHz. This unique feature of the laser output in the spectral domain is responsible (as discussed in Sect. 4) for a 36.4 GHz PRF in the modulated output of the fiber laser. The principle of operation of the dual FP cavity structure of Fig. 1 is similar to that of the coupled-cavity interferometer introduced in 1963 [21]. Ashby and Jephcott coupled two optical cavities by means of a partially transmitting mirror, and the presence of a time-varying refractive index in either cavity had the effect of sweeping (chirping) the longitudinal mode spectrum of one cavity with respect to that of the second cavity. In the present experiments, the interference between the fixed longitudinal mode spectra of the two optical cavities is manifested in a factor of two or 25 increase in the PRF of an L = 6.9 cm cavity. Of equal importance is the observation that the sinusoidal modulation of the longitudinal mode

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intensities in the spectral domain (Fig. 1(f)) results, owing to the Fourier transform, in the laser generating bursts of pulses in a periodic manner, as described in detail in Sect. 4. 3. Simulation model The temporal and spectral behavior of the mode-locked laser system of Fig. 1 is welldescribed by the master Ginzburg-Landau equation (GLE) [24,25]. Because this governing equation includes both dissipative and gain bandwidth terms, it is effective in reproducing experimental results but difficulties arise when a periodic filtering function (such as that required to describe pulse trains or longitudinal mode spectra) is considered. In order to overcome this limitation and investigate in detail the temporal and spectral dynamics of the mode-locked pulses, a lumped model of the linear cavity of Fig. 1 was adopted, thereby allowing for the neglect of the standing wave effect [26]. Consequently, the temporallyresolved propagation of the optical field in both fiber segments was obtained by solving numerically the generalized, nonlinear Schrӧdinger equation [27,28]:

β ∂ 2u g g ∂ 2u ∂u 2 = −i 2 2 + iγ u u + u + (1) ∂z 2 ∂t 2 2Ω 2 ∂t 2 where u represents the electric field amplitude, z is the spatial coordinate coincident with the axis of both fibers, t denotes retarded time, and ω is the angular frequency. Also, the constants β2, γ, and g represent (respectively) the coefficients for second-order dispersion, nonlinearity, and gain in the fibers. The parameter Ω in Eq. (1) denotes the gain bandwidth. 2

Note that g = 0 for the segment of passive fiber whereas g = g 0 e− (  |u| dt )/ ES for the active fiber segment, where g0 is the small-signal gain coefficient and ES is the gain saturation energy. Full simulations of Eq. (1), applied to Fig. 1, also require descriptions of the SESAM, the spliced fiber interface, and the dielectric mirror by the relations: Tsat = Ra −

TFP1 =

TFP 2 =

l0 1 + P / Psat

(2)

RS − (1 − RS ) R f e −2iω ns ls / c

(3)

1 − R f RS e −2iω ns ls / c RP − (1 − RP ) Ra e 1 − Ra RP e

−2 iω n p l p / c

−2 iω n p l p / c

.

(4)

where Eq. (2) describes the SESAM and l0 is the modulation depth for the reflectivity spectrum of Fig. 1(d), and Ra accounts for the reflectance of the SESAM at saturation [29]. Also, P and Psat are the instantaneous (peak) mode-locked pulse power and the saturation power, respectively. Equations (3) and (4) describe the two coupled cavities, FP1 and FP2, respectively, where Rs and Rp represent the reflectance of the fiber-fiber interface from the silica (undoped) fiber direction and the phosphate (gain) fiber direction, Rf is the reflectivity of the dielectric mirror (at left in Fig. 1(a)), ns,p are the refractive indices of the silica and phosphate fibers, and ls,p are the lengths of the respective fibers. Numerical simulations were performed by first introducing Gaussian white noise to the left end of the laser resonator in Fig. 1. The generalized, nonlinear Schrodinger equation was solved with a split-step Fourier algorithm which discretizes the optical spectrum into frequency samples separated by ∆f = 1/T, where T is the width of the temporal window chosen for the calculations. This feature of the split-step Fourier method provides a convenient vehicle for introducing longitudinal modes into the simulations. For all of the results to be presented in the next Section, ∆f was chosen to be 1.43 GHz which corresponds to 16384 mesh points dispersed over the time interval of −350 ps to + 350 ps. This value for

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∆f was chosen to approximate the FSR of 1.44 GHz for the full optical cavity of Fig. 1. With regard to the two fiber segments of Fig. 1(a), β2 was taken to be 30 ps2/km and −20 ps2/km for the phosphate (gain region) fiber and the passive fiber, respectively. The other parameters used for the simulations are: Ω = 12 THz, Ra = 0.94, l0 = 0.06, Rf = 0.993, Rp = 0.044, Rs = 0.034, ns = 1.40, np = 1.53, ls = 3.6 cm, lp = 3.3 cm, γs = 1.3 W−1km−1, γp = 3 W−1km−1, g0 = 1 cm−1, ES = 94 pJ, Psat = 100 W. 4. Results and discussion Both experimental and theoretical waveforms for the hybrid mode-locked laser are presented in Fig. 2. Measurements of the pulse train temporal behavior with an oscilloscope and a photodiode having bandwidths of 25 GHz and 12.5 GHz, respectively, show (cf. Figure 2(a)) output pulses separated by ~348 ps which corresponds to the intermode spacing (FSR) of 2.87 GHz in Fig. 1(f). Because of the limited bandwidth available with the oscilloscope/photodiode detection system, the laser output was examined more closely with an autocorrelator. As shown in Fig. 2(b), the intensity maxima of Fig. 2(a) were found to consist of 6-7 pulses separated by 27 +/- 1.0 ps which is consistent with the value expected from the period of the longitudinal mode envelope of 36.2 GHz (Fig. 1(f)). The inset to panel (b) of Fig. 2 is a magnified view of the most intense member of the pulse train and its temporal width (FWHM) is observed to be 3.9 ps, assuming the intensity profile to be Gaussian. Simulations consistently match the experimental results, as exemplified by the calculations presented in Figs. 2(c) and 2(d). Specifically, the mode-locked laser is predicted to generate bursts of approximately 4 ps pulses separated by 27.5 ps, and the pulse bundles are emitted at a repetition frequency of 2.88 GHz which represents a factor of two increase with respect to the FSR of 1.44 GHz for a mode-locked laser having a resonator length of 6.9 cm. Of greater significance is the PRF of 1/27.5 ps = 36.4 GHz within the pulse bursts, a value that is a factor of 25 larger than 1.44 GHz. As discussed later, the operation of this laser was, from a technological perspective, found to be quite robust and, in particular, insensitive to small changes in the FSR of the FP1 and FP2 Fabry-Perot cavities.

Fig. 2. Experimental and theoretical waveforms for the mode-locked laser: (a) Laser waveform measured with an oscilloscope and a photodiode having bandwidths of 25 GHz and 12.5 GHz, respectively. The interval between intensity peaks is 348 ps; (b) Autocorrelation trace of one of the maxima of (a) showing that each pulse burst consists of 6–7 individual pulses separated by approximately 27.5 ps. The inset is an expanded view of the most intense pulse; (c) Numerical simulation of a portion of the mode-locked laser waveform; (d) Calculated autocorrelator waveform.

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Figure 3 summarizes the results of spectral and output power measurements when the fiber laser was pumped at 976 nm with a launched power of 850 mW. In this range of pump power, the laser operates in the CW mode-locked mode and a panoramic view of the laser spectrum (blue curve, Fig. 3(a)) shows laser modes extending from below 1559 nm to beyond 1568 nm. Panel (a) of Fig. 3 also compares the calculated laser spectrum (red trace) with experiment. Both spectra peak at 1564 nm and exhibit a mode periodicity of approximately 0.35 nm which is in agreement with the 36.19 GHz envelope period of Fig. 1(f). The predicted wavelengths of the mode positions also match experiments well, but the relative mode intensities for the theoretical spectrum are shifted to longer wavelengths by approximately 1 nm, with respect to the measured spectrum.

Fig. 3. Spectral and power output measurements: (a) Comparison of the experimental output spectrum of the laser (blue curve) with the calculated version (red); (b) Spectrum of the photodiode signal in the 2.87205-2.87705 GHz region acquired with an RF spectrum analyzer; (c) RF spectrum of the laser in the 0-10 GHz frequency interval; (d) Measured variation of the laser average output power with the launched pump power (976 nm).

More precise measurements of various aspects of the mode-locked pulse train are afforded by examining the photodiode output signal with an RF spectrum analyzer, and Fig. 3(b) is representative of the spectra recorded between 2.87205 GHz and 2.87705 GHz. A single peak at 2.8746 GHz is observed and it must be emphasized that, as indicated in Fig. 3(b), the background noise is suppressed by 67 dB, despite the simple configuration of the laser architecture reported here. A broader scan of the spectral region below 10 GHz is shown in Fig. 3(c) in which the fundamental and two harmonics of the PRF are visible. Supermode suppression in this case is at least 53 dB, an extraordinary value when one recalls that the finesse of FP1 and FP2 are less than 2. The spectra of panels (b) and (c) also illustrate the stability of the laser when operating continuously and, specifically, they demonstrate that the laser output is free of Q-switching instabilities. The dependence of the average output power of the laser on launched pump power is illustrated in Fig. 3(d) and, as indicated, the hybrid fiber laser of Fig. 1 operates in a Q-switched, mode-locking regime when the pump power is decreased below approximately 600 mW. Note, too, that laser threshold lies at approximately 150 mW and the variation of output power with input power remains linear above 600 mW of launched power but the slope efficiency drops by a factor of three. Several previous studies of harmonically mode-locked lasers have reported the splitting of a single pulse circulating within the cavity [30,31]. Generally attributed to soliton energy quantization [32], this effect is observed when the pump power is raised above a threshold

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value. In contrast, the hybrid mode-locked laser reported here maintains a PRF of 2.87 GHz over the full range in pump power investigated to date. For a launched pump power of 850 mW, for example, the laser operates CW (as noted earlier) and is robust, producing a PRF of 2.87 GHz as shown by the pulse train of Fig. 4(a). Reducing the CW pump power to 350 mW (Fig. 4(b)) transitions the laser operation to Q-switched mode-locking but the PRF remains at 2.87 GHz, as demonstrated by the small portion of Fig. 4(b) (indicated by the arrow) that is expanded in panel (c) of Fig. 4. Consequently, the stability of the laser repetition frequency, despite large variations in pump power, shows unambiguously that the observed frequency doubling of the mode-locked fiber laser is the result of the interaction of two FP cavities, as opposed to the splitting of a single pulse or other mechanisms.

Fig. 4. Laser waveforms recorded for two levels of 976 nm pump power: (a) Mode-locked pulse train recorded for 850 mW of pump power over a time window of more than 80 ns; (b) Laser waveform observed over a 2 µs time interval for 350 mW of pump power, showing Qswitched mode locking of the fiber laser; (c) Expanded view of the temporal region indicated by the arrow in part (b). The frequency of the pulse train is 2.87 GHz.

The dynamics of phase coherence in the hybrid mode-locked fiber laser have also been investigated theoretically, and the left-hand portion of Fig. 5 illustrates the accumulation of phase (φ) during a sequence of two bursts of mode-locked pulses. Expressed in radians, the phase is indicated by the right ordinate of the figure. The key point to be made is that the phase remains essentially constant during the generation of the most intense pulses, whereas φ increases linearly with time during the interval between the pulse bundles. A magnified view of the phase and intensity of the most intense pulse in the first burst of pulses (at left) is shown by the inset to Fig. 5(a). In an effort to explore the potential of these mode-locked pulse trains for information storage on a time scale of at least 30 ps, the phase difference between the two most intense pulses was extracted from the calculations. The results, displayed in Fig. 5(b) for as many as 104 round trips, depict the rapid onset of dephasing. Of particular interest is the observation that the temporal nature of the gradual loss of coherence demonstrates that the reproducibility of the pulse envelopes of Figs. 2, 4, and 5, as well as the pulse period, are the product of interference between the longitudinal mode spectra of FP1 and FP2. If one considers the experimental bandwidth of the laser spectrum (≥ 1 THz, Fig. 3(a)) and the measured 3.9 ps duration of the individual pulses, it is also evident that the mode-locking process utilizes virtually all of the available bandwidth and the SESAM is effective in maintaining the relative phase between the longitudinal modes of the resonator.

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Therefore, the data of Figs. 2, 4, and 6 demonstrate that the frequency multiplication process is not the result of soliton generation [33]. Further numerical investigations of the stability of this mode-locked fiber architecture have been conducted by calculating the temporal behavior of the repetition frequency. Figure 6(a) demonstrates that, if ES is assumed to be 94 pJ and the product of the fiber refractive index and its length are nsls = 5.04 cm for the silica fiber and nplp = 5.05 cm for the gain fiber, then the predicted PRF of the laser is 2.97 GHz and stable operation is reached in 1000 GHz) but the output often suffers from excessive supermode noise because of the length of the master oscillator [18, 19]. Inspired by the complementarity of the two design philosophies, we offer the present design because of its combination of a saturable absorber (to provide intensity discrimination) and two coupled cavities (for frequency discrimination). The synergy between the these resonator components is responsible for suppressing spurious modes by > 50 dB while simultaneously increasing pulse intensity so as to realize stable phase locking of the longitudinal modes over the full gain bandwidth of the Er-Yb:glass fiber. 5. Conclusion Multiplying the repetition frequency of a mode-locked fiber laser by as much as a factor of 25 has been achieved by coupling two FP cavities of low finesse (F = 2) through a fiber-fiber interface. Doubling of the laser PRF occurs through the interference between the longitudinal modes of both FP resonators. Laser pulse trains comprising bursts of 6-7 pulses are produced in which adjacent pulses are separated by approximately 27 ps and the periodicity of the bursts is approximately 350 ps. Consequently, an additional benefit of this fiber mode-locking architecture is the realization of a PRF of 36.4 GHz within each burst of pulses. Supermode suppression levels >50 dB have been achieved and the operation of the laser has been found to be exceptionally stable despite its simplicity. Calculations of the temporal variation of the phase and intensity of each pulse, based on a lumped model of the resonator, are in agreement with experiment. Funding The support of this work by the China Postdoctoral Science Foundation (2016M602462), the China National Fund for Distinguished Young Scientists (61325024), the National Key Research and Development Program of China (2016YFB0402204), the High-Level Personnel Special Support Program of Guangdong Province (2014TX01C087), the Fundamental Research Fund for the Central Universities (2017BQ110), the Science and Technology Project of Guangdong Province (2015B090926010), and the U.S. Air Force Office of Scientific Research (FA9550-14-1-0002) is gratefully acknowledged.

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