Abstract

Ultrafast lasers with high repetition rates are of considerable interest in applications such as optical fiber telecommunications, frequency metrology, high-speed optical sampling, and arbitrary waveform generation. For fiber lasers mode-locked at the cavity round-trip frequency, the pulse repetition rate is limited to tens or hundreds of megahertz by the meter-order cavity lengths. Here we report a soliton fiber laser passively mode-locked at a high harmonic (2GHz) of its fundamental frequency by means of optoacoustic interactions in the small solid glass core of a short length (60 cm) of photonic crystal fiber. Due to tight confinement of both light and vibrations, the optomechanical interaction is strongly enhanced. The long-lived acoustic vibration provides strong modulation of the refractive index in the photonic crystal fiber core, fixing the soliton spacing in the laser cavity and allowing stable mode-locking, with low pulse timing jitter, at gigahertz repetition rates.

© 2015 Optical Society of America

Optical fibers provide an excellent platform for ultrafast lasers, offering many advantages compared to bulky, free-space counterparts, including high beam quality, efficient heat dissipation, and compact and simple configurations with no need for alignment [14]. A limitation of ultrafast fiber lasers, however, is that it is difficult to generate pulses at gigahertz repetition rates [1,4]. Typical fiber lasers [57] have round-trip frequencies of tens or hundreds of megahertz, limited by the long cavity lengths [4]. To increase the round-trip frequency up to several gigahertz, ultrashort (centimeter-scale) cavity lengths are necessary [811]. In order to increase the pulse repetition rate of conventional fiber lasers, several active [1214] and passive [1519] harmonic mode-locking schemes have been proposed, allowing equally spaced multipulses to circulate in the laser cavity. In active schemes with either amplitude or phase modulation, expensive radio-frequency (RF) sources and modulators are used, which increases the laser complexity [4]. For passive schemes, one solution is to incorporate subcavities to fix the pulse repetition rate at a harmonic of the fundamental cavity frequency [15,16], but this requires sophisticated stabilization electronics to balance the optical phase instabilities between the major and subsidiary cavities. Another possible passive scheme makes use of temporally long-range interactions between pulses [1720]. Such long-range pulse interactions are, however, very week in the conventional single-mode fiber (SMF) [5,1720], resulting in erratic repetition rates and large pulse timing jitter.

Recently, enhancement of optomechanical interactions by tight field confinement has been reported in photonic crystal fibers (PCFs) [2123]. In a silica–glass solid-core PCF with high air-filling fraction, simultaneous confinement of light and mechanical vibrations within a small core area leads to high optical and acoustic energy intensities and a large optoacoustic overlap, resulting in an enhancement of the optoacoustic effect by around two orders of magnitude [21,22]. Moreover, PCFs with core diameters of 1μm support gigahertz acoustic resonances, and can be easily integrated into conventional fiber lasers [24].

In this Letter, we report that the pulse repetition rate of a soliton fiber laser can be passively locked to a gigahertz acoustic resonance in the solid core of a 60-cm-long silica PCF, corresponding to a very high-order harmonic N of the cavity round-trip frequency fRT (a few megahertz). We achieve hyperbolic secant pulses with subpicosecond durations—much shorter than in previous work [2527]. The tunability of the repetition rate, pulse amplitude noise, and timing jitter are also explored.

The fiber laser configuration and diagnostic setup are shown in Fig. 1. In the unidirectional ring cavity, a 0.6 m length of Er-doped fiber (EDF) with peak absorption of 110dB/m at 1530 nm is used as the gain medium, and two 976 nm laser diodes (LDs) provide continuous-wave pump light. Two polarization controllers (PC 2 and PC 3), a polarizer, and a 0.2 m length of highly nonlinear fiber were necessary to realize a fast saturable absorber [28,29]. A tunable delay line was used to alter the cavity length, and a 9010 fiber coupler provided the laser output. A 0.6 m length of solid-core silica PCF [see the scanning electron micrograph (SEM) in Fig. 1] was used. The core diameter (dcore) was 1.8μm, and the air-filling fraction (rair) 0.53. A polarization controller (PC 1) was used in front of the PCF so as to ensure that linearly polarized light was launched into one of the principal polarization axes of the fiber. The measured birefringence was 1.5×104 at 1560 nm, which corresponds to a beat-length of 10mm and ensures efficient suppression of polarization mode coupling. All the other components in the laser cavity were made from conventional SMF. In order to reduce the splicing loss between PCF and SMF, a 1cm length of ultrahigh-numerical-aperture fiber was used to form a transition [24]. The total SMF–PCF–SMF loss was 3dB, including the intrinsic PCF loss (<0.05dB/m measured by the cut-back method). The total cavity loss was 6dB, and the cavity mode spacing was 16.8 MHz, corresponding to a total cavity length of 12.2 m.

 

Fig. 1. Experimental setup and SEM of the solid-core PCF. EDF, Er-doped fiber; LD, laser diode; WDM, wavelength division multiplexer; PC, polarization controller; HNLF, highly nonlinear fiber; DCF, dispersion compensating fiber; OSA, optical spectrum analyzer; PD, photodetector; ESA, electrical spectrum analyzer.

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In the diagnostic setup, an autocorrelator was used to measure the pulse duration. In front of the autocorrelator, a section of 0.9m dispersion compensating fiber (DCF), with normal dispersion of 55ps2/km at 1560 nm, was used to compensate the anomalous dispersion of the 2m length of conventional SMF patch cord. The pulse spectrum was measured using an optical spectrum analyzer with a 0.01 nm resolution, and the time-domain pulse train and its RF spectrum were recorded using a 33 GHz photodetector, a 33 GHz oscilloscope, and a 26.5 GHz electrical spectrum analyzer (ESA).

Lasing commenced at 60mW, noisy pulsations appearing just above this pump power. When the pump power was increased further to 550mW, and all three PCs and the tunable delay line were carefully adjusted, a clear transition from a noisy signal to a stable high-harmonic mode-locked pulse train was observed. At these higher pump powers, the laser always started up in high-harmonic mode. After turning off and restarting the laser, only fine-tuning of the polarization controllers was required for the stable high-harmonic mode-locking to be recovered. At a pump power of 670 mW, the time-domain pulse train was recorded over 30 min using an oscilloscope working in infinite persistence mode; a typical measurement is shown in Fig. 2(a). A stable gigahertz pulse train was observed with a long-term amplitude instability of <2.5% root-mean-square (rms) and a pulse timing jitter <2psrms (the measurement was limited by the 1.7ps intrinsic timing jitter of the oscilloscope). The RF spectrum of the pulse train measured by an ESA is shown in Fig. 2(b). The peak at 2.1221 GHz and its integer multiples correspond to mode-locking at the 126th harmonic of the fundamental cavity frequency. All other cavity harmonics are suppressed with a side-mode suppression ratio greater than 50 dB.

 

Fig. 2. Experimental results. (a) typical pulse train recorded over 30 min using an oscilloscope operating in the infinite persistence mode. (b) RF spectrum of the pulse train measured by the ESA. (c) left-hand axis: measured autocorrelation of the output pulses (blue circles), plotted against the delay time of the autocorrelator (bottom axis), and the hyperbolic secant fit (red solid line) of the measured data. Right-hand axis: measured optical spectrum of the laser, plotted against wavelength (top axis). Kelly sidebands are clearly visible.

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The pulse autocorrelation function is shown in Fig. 2(c). The full width half-maximum (FWHM) pulse duration is 540fs, estimated by fitting the data to a hyperbolic secant function (autocorrelation function width 830 fs). The optical spectrum of the laser output is also shown in Fig. 2(c). It has an optical bandwidth of 4.3nm, and the time-bandwidth product of 0.3 is close to the transform limit for a hyperbolic secant pulse. Strong Kelly sidebands [30] in the optical pulse spectrum indicate that the fiber laser is operating in the soliton regime.

The laser shows good long-term stability. Monitoring the 2.1221 GHz peak using the ESA for 1 h revealed that it drifted slowly within a 5 kHz range, corresponding to a relative long-term fluctuation in repetition rate of less than 2.5 parts per million. We also ran the laser continuously over 10 h without observing any pulse degradation.

The short-term pulse amplitude noise and timing jitter were estimated by measuring the single-sideband (SSB) noise spectra for the baseband and different harmonics of the pulse repetition rate [31,32] (see Supplement 1). By integrating the baseband noise of the laser, we estimated the relative amplitude noise of the laser to be 0.1% over a bandwidth of 1 Hz to 1 MHz [31]. The SSB phase noise spectra of the first, fourth, and eighth harmonics, measured in the frequency range from 100 Hz to 1 MHz, were used to estimate the short-term pulse timing jitter, giving a value of 40fs over a bandwidth of 100 Hz to 1 MHz [31] (see Supplement 1).

In the experiments we investigated the dependence of the pulse duration and optical bandwidth on the intracavity pulse energy. By varying the pump power level, the FWHM pulse duration and 3 dB optical bandwidth were measured for different intracavity pulse energies. The results are plotted in Fig. 3. When the pulse energy was increased from 21 to 38 pJ, the pulse duration decreased from 670 to 450 fs while the optical bandwidth increased from 3.6 to 4.9 nm. As also shown in Fig. 3, the time-bandwidth product remained almost constant at 0.3, indicating that the laser was operating in the soliton regime. The deviation at high pulse energies between data and fitted curves (based on the fundamental soliton assumption [33]) is mainly due to increasing conversion to Kelly sidebands [30] as shown in Fig. 2(c).

 

Fig. 3. Measured dependence of pulse parameters on intracavity pulse energy. Upper left-hand axis: deviation of pulse time-bandwidth product (ΔTBP) from 0.31 (black triangles), plotted against estimated pulse energy in laser cavity. Lower left-hand axis: measured FWHM pulse duration (black squares) fitted to Eq. (2) (full black line). Right-hand axis: measured 3 dB optical bandwidth of the pulses (blue circles) fitted to Eq. (2) using TBP=0.31 (full blue line).

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By changing the length of the delay line in the laser cavity (shown in Fig. 1), the repetition rate could be varied continuously over a range of 9.6MHz (from 2.1161 to 2.1257 GHz, corresponding to a cavity length tuning of 80mm), while the laser remained stably mode-locked. During this process the pulse duration remained almost constant at 540 fs (the deviation is within the ±30fs measurement accuracy of the autocorrelator).

We measured the comb structure of the laser using a heterodyne method, employing a fiber laser with a 2 kHz linewidth as the local oscillator. The results (see Supplement 1) show a comb of frequencies spaced by the cavity round-trip frequency fRT. The nth comb line is optoacoustically coupled to every (n+mN)th comb line, where NfRT is the harmonic mode-locking frequency and m the order of the mth high-harmonic comb line. Thus the spectrum consists of many interleaved but independent gigahertz combs, each of which alone would produce a coherent train of mode-locked soliton pulses. Because, however, each comb is spaced apart from the others by multiples of the cavity round-trip frequency, and the combs have random relative phases, the result is a train of uncorrelated soliton pulses. Although there is no particular phase relationship among the 126 solitons within the laser cavity, the acoustic resonance in the PCF core forces the temporal spacing between the pulses to remain constant.

The principle of passive mode-locking to an acoustic resonance in the core of a PCF can be simply explained as follows: in the steady state, the gigahertz pulse train, propagating in the solid-core PCF, drives a trapped acoustic wave through electrostriction. The index modulation produced by the vibration acts in turn on the driving pulses. The enhanced optoacoustic effect in the small area of the PCF core allows successive pulses to interact, efficiently stabilizing the pulse spacing in the fiber laser cavity and suppressing pulse timing jitter.

Optically driven acoustic vibrations, tightly guided in the PCF core, have been studied in detail by Kang et al. [22]. In practice, only the fundamental radial (R01) acoustic mode is considered because this acoustic mode is most efficiently excited by the fundamental optical mode. In experiments the repetition frequency of the driving pulses (Ω/2π=NfRT) is equal to an integer multiple N of the round-trip frequency fRT, which is determined by the cavity length, while the resonant frequency of the R01 acoustic mode (Ω01) is mainly determined by the PCF structure. We denote the offset between them as δ=ΩΩ01. In the steady state a train of pulses with energy EP and duration much shorter than the acoustic period can drive a modulation of the relative permittivity of the glass (in the form of an acoustic wave ρ; see Supplement 1) as

Δεr(z,t,r,θ)=γeρρ0=γe2|Q|Epρ01(r,θ)ei(ΩtqzΔφ)4πneffcAeffρ04δ2+ΓB2+c.c.,
Δφ=arccot(2δ/ΓB),0Δφπ,
where γe and ρ0 are the electrostrictive coefficient and density of silica, neff and Aeff are the effective refractive index and mode area of the fundamental optical mode in the PCF, c is the speed of light in vacuum, ρ01(r,θ) is the dimensionless acoustic profile of the R01 acoustic mode, q is its propagation constant along the PCF axis, and ΓB is its Brillouin linewidth. The overlap integral Q between the fundamental optical mode and the R01 acoustic mode is defined in Supplement 1. As shown in Eq. (1a), the acoustic wave frequency equals the pulse repetition frequency, while the phase matching condition requires that the propagation constant of the acoustic wave equals that of the driving pulses [22,34]. Acoustic gain appears when the phase shift Δφ between the driving pulse train and the acoustic wave [Eq. (1b); see Supplement 1] lies within the range (0, π) [34].

The light-driven acoustic core resonance, operating as a phase modulation, acts back on the light, fixing the soliton spacing inside the laser cavity. The passive pulse-spacing stabilization can be understood most intuitively as an enhancement of temporally long-range pulse interactions by the coherent excitation of an acoustic vibration tightly confined in the PCF core, and is somewhat similar to active mode-locking by regenerative feedback [35,36], except that no high-frequency electronic components such as photodiodes, amplifiers, or band-pass filters are required.

The acoustic gain bandwidth of the PCF was measured to be 8MHz [22], corresponding to an acoustic quality factor of 250 (see Supplement 1). For stable high-harmonic mode-locking, the tunable delay line must be adjusted so as to place one harmonic of the cavity mode spacing within the acoustic gain bandwidth. Continuous tuning of the pulse repetition rate can be achieved by adjusting the cavity length, but only if the cavity harmonic remains within the acoustic gain bandwidth, ensuring that the light-driven acoustic resonance in the PCF remains strong enough to lock the pulse positions.

Since the pulse duration (subpicosecond) is much shorter than the period of acoustic oscillation (470ps), it is much more strongly affected by Kerr-related self-phase modulation than by the optoacoustic effect. Note also that self-amplitude modulation in the saturable absorber is much weaker than self-phase modulation in the cavity (see Supplement 1). In a laser cavity with anomalous average dispersion, the pulse duration is thus dominated by the formation of sech2 fundamental solitons [33] of energy EP and FWHM pulse duration τFWHM:

EPτFWHM=3.52β2γKerr,
where β2 is the average group-velocity dispersion and γKerr is the average Kerr nonlinearity coefficient in the cavity (estimated to be 21.9ps2/km and 3.4km1W1; see Supplement 1). Using Eq. (2) we calculate the product of pulse energy and FWHM duration to be 22.7 pJ·ps, which is quite close to the measured value of 14.1 pJ·ps (calculated from Fig. 3). The disparity is most likely due to an underestimation of the intracavity pulse energy in the experiments.

Soliton fiber ring lasers can be stably mode-locked at a high harmonic of the round-trip frequency using optoacoustic interactions at few-gigahertz frequencies in the small glass core of a PCF. Pulse repetition rates of 2GHz, durations of 500fs, and energies of tens of picojoules are typically achieved with short-term pulse amplitude noise of 0.1% over a bandwidth of 1 Hz to 1 MHz, and pulse timing jitter 40fs over a bandwidth of 100 Hz to 1 MHz.

ACKNOWLEDGMENT

The authors thank T. Roethlingshoefer and B. Stiller from the Leuchs Division at MPL for providing some components and equipment for the experiments.

 

See Supplement 1 for supporting content.

REFERENCES

1. M. E. Fermann, A. Galvanauskas, and G. Sucha, Ultrafast Lasers Technology and Applications (Dekker, 2003).

2. M. E. Fermann and I. Hartl, Nat. Photonics 7, 868 (2013). [CrossRef]  

3. L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, and E. P. Ippen, Appl. Phys. B 65, 277 (1997). [CrossRef]  

4. M. E. Fermann, Appl. Phys. B 58, 197 (1994). [CrossRef]  

5. I. N. Duling III, Opt. Lett. 16, 539 (1991). [CrossRef]  

6. K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, Opt. Lett. 18, 1080 (1993). [CrossRef]  

7. Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. Ferrari, ACS Nano 4, 803 (2010). [CrossRef]  

8. J. J. McFerran, L. Nenadovic, W. C. Swann, J. B. Schlager, and N. R. Newbury, Opt. Express 15, 13155 (2007). [CrossRef]  

9. A. Martinez and S. Yamashita, Opt. Express 19, 6155 (2011). [CrossRef]  

10. H. Chen, Z. Haider, J. Lim, S. Xu, Z. Yang, F. X. Kaertner, and G. Chang, Opt. Lett. 38, 4927 (2013). [CrossRef]  

11. H. Byun, M. Y. Sander, A. Motamedi, H. Shen, G. S. Petrich, L. A. Kolodziejski, E. P. Ippen, and F. X. Kaertner, Appl. Opt. 49, 5577 (2010). [CrossRef]  

12. H. Takara, S. Kawanishi, M. Saruwatari, and K. Noguchi, Electron. Lett. 28, 2095 (1992). [CrossRef]  

13. T. F. Carruthers, I. N. Duling III, M. Horowitz, and C. R. Menyuk, Opt. Lett. 25, 153 (2000). [CrossRef]  

14. M. W. Phillips, A. I. Ferguson, G. S. Kino, and D. B. Patterson, Opt. Lett. 14, 680 (1989). [CrossRef]  

15. E. Yoshida, Y. Kimura, and M. Nakazawa, Appl. Phys. Lett. 60, 932 (1992). [CrossRef]  

16. M. L. Dennis and I. N. Duling III, Electron. Lett. 28, 1894 (1992). [CrossRef]  

17. B. C. Collings, K. Bergman, and W. H. Knox, Opt. Lett. 23, 123 (1998). [CrossRef]  

18. K. S. Abedin, J. T. Gopinath, L. A. Jiang, M. E. Grein, H. A. Haus, and E. P. Ippen, Opt. Lett. 27, 1758 (2002). [CrossRef]  

19. A. B. Grudinin and S. Gray, J. Opt. Soc. Am. B 14, 144 (1997). [CrossRef]  

20. A. N. Pilipetskii, E. A. Golovchenko, and C. R. Menyuk, Opt. Lett. 20, 907 (1995). [CrossRef]  

21. P. Dainese, P. St. J. Russell, G. S. Wiederhecker, N. Joly, H. L. Fragnito, V. Laude, and A. Khelif, Opt. Express 14, 4141 (2006). [CrossRef]  

22. M. S. Kang, A. Nazarkin, A. Brenn, and P. St. J. Russell, Nat. Phys. 5, 276 (2009). [CrossRef]  

23. M. S. Kang, A. Brenn, and P. St.J. Russell, Phys. Rev. Lett. 105, 153901 (2010). [CrossRef]  

24. L. Xiao, M. S. Demokan, W. Jin, Y. Wang, and C. L. Zhao, J. Lightwave Technol. 25, 3563 (2007). [CrossRef]  

25. M. S. Kang, N. Y. Joly, and P. St. J. Russell, Opt. Lett. 38, 561 (2013). [CrossRef]  

26. M. Pang, X. Jiang, G. K. L. Wong, G. Onishchukov, N. Y. Joly, G. Ahmed, and P. St. J. Russell, Advanced Photonic 2014 (Optical Society of America, 2014), paper NTh4A.5.

27. B. Stiller and T. Sylvestre, Opt. Lett. 38, 1570 (2013). [CrossRef]  

28. M. Hofer, M. E. Fermann, F. Haberl, M. H. Ober, and A. J. Schmidt, Opt. Lett. 16, 502 (1991). [CrossRef]  

29. M. Salhi, H. Leblond, and F. Sanchez, Phys. Rev. A 67, 013802 (2003). [CrossRef]  

30. S. M. J. Kelly, Electron. Lett. 28, 806 (1992). [CrossRef]  

31. D. von der Linde, Appl. Phys. B 39, 201 (1986). [CrossRef]  

32. H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993). [CrossRef]  

33. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

34. R. W. Boyd, Nonlinear Optics (Academic, 2008).

35. M. Nakazawa, E. Yoshida, and K. Tamura, Electron. Lett. 32, 1285 (1996). [CrossRef]  

36. K. K. Gupta, D. Novak, and H. Liu, IEEE J. Quantum Electron. 36, 70 (2000). [CrossRef]  

References

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  1. M. E. Fermann, A. Galvanauskas, G. Sucha, Ultrafast Lasers Technology and Applications (Dekker, 2003).
  2. M. E. Fermann, I. Hartl, Nat. Photonics 7, 868 (2013).
    [Crossref]
  3. L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, E. P. Ippen, Appl. Phys. B 65, 277 (1997).
    [Crossref]
  4. M. E. Fermann, Appl. Phys. B 58, 197 (1994).
    [Crossref]
  5. I. N. Duling, Opt. Lett. 16, 539 (1991).
    [Crossref]
  6. K. Tamura, E. P. Ippen, H. A. Haus, L. E. Nelson, Opt. Lett. 18, 1080 (1993).
    [Crossref]
  7. Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, A. Ferrari, ACS Nano 4, 803 (2010).
    [Crossref]
  8. J. J. McFerran, L. Nenadovic, W. C. Swann, J. B. Schlager, N. R. Newbury, Opt. Express 15, 13155 (2007).
    [Crossref]
  9. A. Martinez, S. Yamashita, Opt. Express 19, 6155 (2011).
    [Crossref]
  10. H. Chen, Z. Haider, J. Lim, S. Xu, Z. Yang, F. X. Kaertner, G. Chang, Opt. Lett. 38, 4927 (2013).
    [Crossref]
  11. H. Byun, M. Y. Sander, A. Motamedi, H. Shen, G. S. Petrich, L. A. Kolodziejski, E. P. Ippen, F. X. Kaertner, Appl. Opt. 49, 5577 (2010).
    [Crossref]
  12. H. Takara, S. Kawanishi, M. Saruwatari, K. Noguchi, Electron. Lett. 28, 2095 (1992).
    [Crossref]
  13. T. F. Carruthers, I. N. Duling, M. Horowitz, C. R. Menyuk, Opt. Lett. 25, 153 (2000).
    [Crossref]
  14. M. W. Phillips, A. I. Ferguson, G. S. Kino, D. B. Patterson, Opt. Lett. 14, 680 (1989).
    [Crossref]
  15. E. Yoshida, Y. Kimura, M. Nakazawa, Appl. Phys. Lett. 60, 932 (1992).
    [Crossref]
  16. M. L. Dennis, I. N. Duling, Electron. Lett. 28, 1894 (1992).
    [Crossref]
  17. B. C. Collings, K. Bergman, W. H. Knox, Opt. Lett. 23, 123 (1998).
    [Crossref]
  18. K. S. Abedin, J. T. Gopinath, L. A. Jiang, M. E. Grein, H. A. Haus, E. P. Ippen, Opt. Lett. 27, 1758 (2002).
    [Crossref]
  19. A. B. Grudinin, S. Gray, J. Opt. Soc. Am. B 14, 144 (1997).
    [Crossref]
  20. A. N. Pilipetskii, E. A. Golovchenko, C. R. Menyuk, Opt. Lett. 20, 907 (1995).
    [Crossref]
  21. P. Dainese, P. St. J. Russell, G. S. Wiederhecker, N. Joly, H. L. Fragnito, V. Laude, A. Khelif, Opt. Express 14, 4141 (2006).
    [Crossref]
  22. M. S. Kang, A. Nazarkin, A. Brenn, P. St. J. Russell, Nat. Phys. 5, 276 (2009).
    [Crossref]
  23. M. S. Kang, A. Brenn, P. St.J. Russell, Phys. Rev. Lett. 105, 153901 (2010).
    [Crossref]
  24. L. Xiao, M. S. Demokan, W. Jin, Y. Wang, C. L. Zhao, J. Lightwave Technol. 25, 3563 (2007).
    [Crossref]
  25. M. S. Kang, N. Y. Joly, P. St. J. Russell, Opt. Lett. 38, 561 (2013).
    [Crossref]
  26. M. Pang, X. Jiang, G. K. L. Wong, G. Onishchukov, N. Y. Joly, G. Ahmed, P. St. J. Russell, Advanced Photonic 2014 (Optical Society of America, 2014), paper NTh4A.5.
  27. B. Stiller, T. Sylvestre, Opt. Lett. 38, 1570 (2013).
    [Crossref]
  28. M. Hofer, M. E. Fermann, F. Haberl, M. H. Ober, A. J. Schmidt, Opt. Lett. 16, 502 (1991).
    [Crossref]
  29. M. Salhi, H. Leblond, F. Sanchez, Phys. Rev. A 67, 013802 (2003).
    [Crossref]
  30. S. M. J. Kelly, Electron. Lett. 28, 806 (1992).
    [Crossref]
  31. D. von der Linde, Appl. Phys. B 39, 201 (1986).
    [Crossref]
  32. H. A. Haus, A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
    [Crossref]
  33. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
  34. R. W. Boyd, Nonlinear Optics (Academic, 2008).
  35. M. Nakazawa, E. Yoshida, K. Tamura, Electron. Lett. 32, 1285 (1996).
    [Crossref]
  36. K. K. Gupta, D. Novak, H. Liu, IEEE J. Quantum Electron. 36, 70 (2000).
    [Crossref]

2013 (4)

2011 (1)

2010 (3)

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, A. Ferrari, ACS Nano 4, 803 (2010).
[Crossref]

H. Byun, M. Y. Sander, A. Motamedi, H. Shen, G. S. Petrich, L. A. Kolodziejski, E. P. Ippen, F. X. Kaertner, Appl. Opt. 49, 5577 (2010).
[Crossref]

M. S. Kang, A. Brenn, P. St.J. Russell, Phys. Rev. Lett. 105, 153901 (2010).
[Crossref]

2009 (1)

M. S. Kang, A. Nazarkin, A. Brenn, P. St. J. Russell, Nat. Phys. 5, 276 (2009).
[Crossref]

2007 (2)

2006 (1)

2003 (1)

M. Salhi, H. Leblond, F. Sanchez, Phys. Rev. A 67, 013802 (2003).
[Crossref]

2002 (1)

2000 (2)

T. F. Carruthers, I. N. Duling, M. Horowitz, C. R. Menyuk, Opt. Lett. 25, 153 (2000).
[Crossref]

K. K. Gupta, D. Novak, H. Liu, IEEE J. Quantum Electron. 36, 70 (2000).
[Crossref]

1998 (1)

1997 (2)

A. B. Grudinin, S. Gray, J. Opt. Soc. Am. B 14, 144 (1997).
[Crossref]

L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, E. P. Ippen, Appl. Phys. B 65, 277 (1997).
[Crossref]

1996 (1)

M. Nakazawa, E. Yoshida, K. Tamura, Electron. Lett. 32, 1285 (1996).
[Crossref]

1995 (1)

1994 (1)

M. E. Fermann, Appl. Phys. B 58, 197 (1994).
[Crossref]

1993 (2)

K. Tamura, E. P. Ippen, H. A. Haus, L. E. Nelson, Opt. Lett. 18, 1080 (1993).
[Crossref]

H. A. Haus, A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[Crossref]

1992 (4)

S. M. J. Kelly, Electron. Lett. 28, 806 (1992).
[Crossref]

E. Yoshida, Y. Kimura, M. Nakazawa, Appl. Phys. Lett. 60, 932 (1992).
[Crossref]

M. L. Dennis, I. N. Duling, Electron. Lett. 28, 1894 (1992).
[Crossref]

H. Takara, S. Kawanishi, M. Saruwatari, K. Noguchi, Electron. Lett. 28, 2095 (1992).
[Crossref]

1991 (2)

1989 (1)

1986 (1)

D. von der Linde, Appl. Phys. B 39, 201 (1986).
[Crossref]

Abedin, K. S.

Agrawal, G. P.

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ACS Nano (1)

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, A. Ferrari, ACS Nano 4, 803 (2010).
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Appl. Opt. (1)

Appl. Phys. B (3)

L. E. Nelson, D. J. Jones, K. Tamura, H. A. Haus, E. P. Ippen, Appl. Phys. B 65, 277 (1997).
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M. E. Fermann, Appl. Phys. B 58, 197 (1994).
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D. von der Linde, Appl. Phys. B 39, 201 (1986).
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Appl. Phys. Lett. (1)

E. Yoshida, Y. Kimura, M. Nakazawa, Appl. Phys. Lett. 60, 932 (1992).
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Electron. Lett. (4)

M. L. Dennis, I. N. Duling, Electron. Lett. 28, 1894 (1992).
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H. Takara, S. Kawanishi, M. Saruwatari, K. Noguchi, Electron. Lett. 28, 2095 (1992).
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S. M. J. Kelly, Electron. Lett. 28, 806 (1992).
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M. Nakazawa, E. Yoshida, K. Tamura, Electron. Lett. 32, 1285 (1996).
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IEEE J. Quantum Electron. (2)

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H. A. Haus, A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
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J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Nat. Photonics (1)

M. E. Fermann, I. Hartl, Nat. Photonics 7, 868 (2013).
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Nat. Phys. (1)

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Opt. Express (3)

Opt. Lett. (11)

Phys. Rev. A (1)

M. Salhi, H. Leblond, F. Sanchez, Phys. Rev. A 67, 013802 (2003).
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Phys. Rev. Lett. (1)

M. S. Kang, A. Brenn, P. St.J. Russell, Phys. Rev. Lett. 105, 153901 (2010).
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Other (4)

M. Pang, X. Jiang, G. K. L. Wong, G. Onishchukov, N. Y. Joly, G. Ahmed, P. St. J. Russell, Advanced Photonic 2014 (Optical Society of America, 2014), paper NTh4A.5.

M. E. Fermann, A. Galvanauskas, G. Sucha, Ultrafast Lasers Technology and Applications (Dekker, 2003).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

R. W. Boyd, Nonlinear Optics (Academic, 2008).

Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1. Experimental setup and SEM of the solid-core PCF. EDF, Er-doped fiber; LD, laser diode; WDM, wavelength division multiplexer; PC, polarization controller; HNLF, highly nonlinear fiber; DCF, dispersion compensating fiber; OSA, optical spectrum analyzer; PD, photodetector; ESA, electrical spectrum analyzer.
Fig. 2.
Fig. 2. Experimental results. (a) typical pulse train recorded over 30 min using an oscilloscope operating in the infinite persistence mode. (b) RF spectrum of the pulse train measured by the ESA. (c) left-hand axis: measured autocorrelation of the output pulses (blue circles), plotted against the delay time of the autocorrelator (bottom axis), and the hyperbolic secant fit (red solid line) of the measured data. Right-hand axis: measured optical spectrum of the laser, plotted against wavelength (top axis). Kelly sidebands are clearly visible.
Fig. 3.
Fig. 3. Measured dependence of pulse parameters on intracavity pulse energy. Upper left-hand axis: deviation of pulse time-bandwidth product ( Δ TBP ) from 0.31 (black triangles), plotted against estimated pulse energy in laser cavity. Lower left-hand axis: measured FWHM pulse duration (black squares) fitted to Eq. (2) (full black line). Right-hand axis: measured 3 dB optical bandwidth of the pulses (blue circles) fitted to Eq. (2) using TBP = 0.31 (full blue line).

Equations (3)

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Δ ε r ( z , t , r , θ ) = γ e ρ ρ 0 = γ e 2 | Q | E p ρ 01 ( r , θ ) e i ( Ω t q z Δ φ ) 4 π n eff c A eff ρ 0 4 δ 2 + Γ B 2 + c.c. ,
Δ φ = arccot ( 2 δ / Γ B ) , 0 Δ φ π ,
E P τ FWHM = 3.52 β 2 γ Kerr ,

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