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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 5 — Mar. 1, 2010
  • pp: 5041–5054
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Timing jitter and phase noise of mode-locked fiber lasers

Rüdiger Paschotta  »View Author Affiliations


Optics Express, Vol. 18, Issue 5, pp. 5041-5054 (2010)
http://dx.doi.org/10.1364/OE.18.005041


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Abstract

The noise properties of mode-locked fiber lasers differ in various respects from those of bulk lasers. The reasons for this are both quantitative and qualitative differences concerning the pulse formation. The underlying theoretical aspects are discussed in detail. It is found that the achievable noise level and the limiting effects depend strongly on the type of fiber laser. Depending on the pulse formation mechanism, noise levels may be much higher than predicted by simplified models.

© 2010 OSA

1. Introduction

In recent years, various kinds of picosecond and femtosecond fiber lasers have been developed. The substantial progress in terms of performance parameters such as pulse duration and output power was partly a consequence of improved components and optimized parameters, but also partly became possible because novel mechanisms for the pulse shaping have been utilized. There is now a range of mode-locked fiber lasers using fairly different internal mechanisms: examples are soliton fiber lasers [1

1. L. F. Mollenauer and R. H. Stolen, “The soliton laser,” Opt. Lett. 9(1), 13–15 (1984). [CrossRef] [PubMed]

], stretched-pulse lasers [2

2. K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18(13), 1080–1082 (1993). [CrossRef] [PubMed]

], similariton lasers and other wavebreaking-free fiber lasers [3

3. F. Ö. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nJ pulses at 1.03 μm from a wave-breaking-free fiber laser,” Opt. Lett. 28(15), 1365–1367 (2003). [CrossRef] [PubMed]

5

5. 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). [CrossRef]

]. The pulse dynamics in such fiber lasers differ substantially, and most mode-locked fiber lasers differ very much both quantitatively and qualitatively from mode-locked bulk lasers.

Such differences must be expected to affect not only basic performance parameters such as pulse energy and duration, but also the noise properties, in particular in terms of timing jitter and optical phase noise. Such noise properties are of high interest for some applications, in particular in the context of optical frequency metrology [6

6. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002). [CrossRef] [PubMed]

]. For example, there is the still not finally settled question to what extent fiber lasers can compete with Ti:sapphire lasers as sources of ultra-stable frequency combs.

Only the noise properties of some simpler kinds of fiber lasers, in particular soliton fiber lasers, have been analyzed in some depth; section 2 reviews some earlier results and adds new considerations on phase noise. In section 3, this paper presents a more comprehensive analysis of how typical properties of fiber lasers influence their noise properties. Section 4 addresses some specific types of fiber lasers to which simple analytical models are not directly applicable. The concluding section 5 summarizes the main results.

Throughout this paper, the emphasis is on quantum noise influences, which are more fundamental than various sources of technical noise and have in several cases been demonstrated to actually limit the noise performance of various fiber lasers [7

7. S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13(12), 2817–2823 (1996). [CrossRef]

9

9. O. Prochnow, R. Paschotta, E. Benkler, U. Morgner, J. Neumann, D. Wandt, and D. Kracht, “Quantum-limited noise performance of a femtosecond all-fiber ytterbium laser,” Opt. Express 17(18), 15525–15533 (2009), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-17-18-15525. [CrossRef] [PubMed]

]. This is particularly the case for all-fiber setups, which can be mechanically very stable.

2. Fundamental theoretical results

In this section, fundamental theoretical expectations for simple kinds of mode-locked lasers are discussed. Much of this is a compact review of earlier results, but some new considerations on phase noise are added.

Timing jitter

An early analysis by Haus and Mecozzi [10

10. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993). [CrossRef]

] was based on soliton perturbation theory. Essentially, the temporal evolution of various pulse parameters under the influence of quantum noise was investigated based on a simplified analytical model, which allowed to calculate correlation functions and thus to derive the noise spectra. A central result of this analysis is that there are two different quantum noise influences on the timing jitter. First, there is a direct influence of quantum noise on the pulse timing. The corresponding power spectral density (PSD) depends on the pulse energy, the pulse duration, the round-trip time and the inverse square of the noise frequency. The second influence is based on a different mechanism: quantum noise causes a jitter of the mean position of the optical spectrum, and this is translated into timing jitter via group delay dispersion (GDD). That second influence depends not only on the magnitude of GDD, but also on factors like the spectral width and the gain bandwidth.

A later analysis [11

11. R. Paschotta, “Noise of mode-locked lasers. Part II: Timing jitter and other fluctuations,” Appl. Phys. B 79(2), 163–173 (2004). [CrossRef]

] based on a more general model, which does not require the assumption of soliton pulse shaping effects, essentially yielded the same results (after correction of some trivial errors in Ref [10

10. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993). [CrossRef]

].) but for a much wider range of mode-locked lasers, also including bulk lasers. It was realized that in these lasers the pulse shaping mechanism, which may or may not include soliton effects and saturable absorption, has no direct influence on the noise properties. Quantum noise is introduced by gain and losses, and its strength is influenced by the resonator losses as well as by pulse parameters like the pulse duration and repetition rate. However, the particular pulse shaping mechanism can have substantial indirect effects, for example by limiting the possible pulse energy and pulse repetition rate. Also, there are other pulse shaping mechanisms which also have a substantial direct effect. Such effects have indeed been identified, as discussed in section 4, but are ignored in this section 2.

Although we will see that the validity of the model of Ref [11

11. R. Paschotta, “Noise of mode-locked lasers. Part II: Timing jitter and other fluctuations,” Appl. Phys. B 79(2), 163–173 (2004). [CrossRef]

]. has its limitations in the area of fiber lasers, the model and its key results are briefly reviewed in the following. The model assumes a single circulating pulse in the laser resonator, although a generalization to the case with multiple pulses (harmonic mode locking) is easily possible, as shown in section 3.2 of Ref [12

12. R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier–envelope offset noise of mode-locked lasers,” Appl. Phys. B 82(2), 265–273 (2006). [CrossRef]

]. Further assumptions are that the dominant noise influence comes from the gain and linear losses in the resonator, whereas the pulse shaping mechanism itself does not contribute directly (except as far as it contributes power losses) and also does not provide a “restoring force” for the pulse timing. Such a restoring force, as it occurs in actively mode-locked lasers, could easily be included, however. Essentially, it forces the PSD of the timing noise to level off at low frequencies, where it otherwise exhibits a divergence.

The indirect contribution of quantum noise to the timing jitter is
SΔt,indirect(f)=(D2fTrt)2Sνc(f)
where D2 is the total group delay dispersion per resonator round trip and Sνc(f) is the PSD of the fluctuations of the “center of gravity” of the optical frequency spectrum, or in short the mean optical frequency. Chromatic dispersion couples such frequency fluctuations to timing fluctuations, as it describes a frequency-dependent group delay. The frequency fluctuations can be described by a quantum noise influence in the frequency domain, which leads to a PSD proportional to the square of the spectral width Δνp. Also, there is a filtering factor, arising from the fact that the limited gain bandwidth (or some other kind of spectral filter) provides a “restoring force” for the mean frequency:
Sνc(f)0.531(2πf)2+τνc2θhνEpltotTrtΔνp2
where the filter time constant τνc is given by

τνc0.47Trtg(ΔνgΔνp)2.

Here, Δνg is the gain bandwidth. In the original equation for τνc in Ref [10

10. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993). [CrossRef]

], the pulse bandwidth was replaced via Δνp=0.315/τp according to the known time−bandwidth product of an unchirped sech2-shaped pulse. That replacement was reversed here in order to apply the equation also to non-transform-limited pulses. What matters for the filtering effect is of course the bandwidth, not the pulse duration.

If the spectral filtering is done not by the limited gain bandwidth but by a different filtering element, we have to modify the equation for the filter constant. For Gaussian pulse spectra and a Gaussian filter transmission spectrum, one finds

τνc=Trt(ΔνfΔνp)2.

For other spectral shapes and filter shapes, one may require an additional numerical factor of the order of unity.

Optical phase noise

We first recall a central result for continuous-wave single-frequency lasers, as it turns out to be relevant also for mode-locked lasers. Here, the quantum limit for the linewidth is given by the Schawlow−Townes formula [13

13. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958). [CrossRef]

], which in a modified form [14

14. R. Paschotta, H. R. Telle, and U. Keller, Noise of Solid State Lasers (CRC Press, Boca Raton, 2007), Chap. 12.

] reads

ΔνST=θhνltot4πTrt2Pint.
(2)

This modified version of the original equation from Ref [13

13. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958). [CrossRef]

]. has been adapted to our notation and uses the total resonator losses ltot instead of the resonator bandwidth. The result indicates the FWHM (full width at half maximum) optical bandwidth, and Pint is the intracavity power.

Interestingly, it has been shown [15

15. P.-T. Ho, “Phase and amplitude fluctuations in a mode-locked laser,” IEEE J. Quantum Electron. QE 21(11), 1806–1813 (1985). [CrossRef]

] that the same formula can be applied to all lines in the spectrum of an actively mode-locked laser, using the total average power (rather than the power in a particular line) for Pint. The reason for that is essentially that the mode-locking mechanism couples together the phases of all lines, preventing independent phase drifts of the lines.

From Eq. (2), we obtain the linewidth for a given pulse energy Ep:
ΔνST=14πθhνEpltotTrt
(3)
Here, we have used Pint=EpTrt. This equation, not containing the number of pulses, should be applied even when multiple pulses circulate in the resonator of a harmonically mode-locked laser; what counts is the average power corresponding to a single pulse.

Sφopt,ST(f)=18π2θhνEpltotTrt1f2.
(4)

That equation, however, is valid only for low noise frequencies. Its validity ends about 1 to 3 orders of magnitude below the pulse repetition rate. For higher noise frequencies, the phase noise is stronger than predicted, because the mode-locking mechanism is effective only on a longer time scale.

A new result is that even for mode locking with a fast absorber, the phase noise can be strongly increased by the Kerr nonlinearity, which is substantial in nearly every fiber laser and couples intensity fluctuations to phase fluctuations. We must assume the intensity noise to be at least at the shot noise level; in fact, higher intensity noise is not unlikely given the variety of effects which couple the intensity to other pulse parameters. A short calculation shows in the following that even intensity noise at the shot noise level leads to substantial phase noise well above the Schawlow–Townes limit. If the average nonlinear phase shift per round trip is φnl, the additional phase fluctuations caused by intensity noise evolve according to

Trtφt=φnlδEpEp.

This relation can be used to relate intensity noise (i.e., fluctuations of Ep) to phase noise. What follows for the PSDs is the contribution
Sφ,nl(f)=(φnl2πfTrt)2SI(f)
to the phase noise, where SI(f) quantifies the relative intensity noise. This would add to phase noise of other origins, if it is not correlated with the latter, and in any case allows to estimate the influence via the Kerr nonlinearity. At the shot noise level, we have
SI(f)=θhνEp/Trt
so that

Sφ,nl(f)=(φnl2πf)2θhνEp1Trt.

We can compare this to the Schawlow–Townes limit:
Sφ,nl(f)Sφopt,ST(f)=2φnl2ltot.
From this we see that for example in a soliton laser and indeed in most fiber lasers, where the nonlinear phase shift cannot be made very small, the phase noise is easily pushed to a level well above the Schawlow–Townes limit, even if the intensity noise is at the shot noise level. In fact we should expect that the intensity noise can easily be higher, as various mechanisms couple different quantities such as pulse energy, duration, spectral position and chirp to each other. Therefore, for most mode-locked fiber lasers it would be unrealistic to expect phase noise near the Schawlow–Townes limit even if technical noise influences could be fully suppressed.

It is also important to realize that at least for lasers emitting longer pulses (with many optical cycles), technical noise influences make it substantially harder to reach the quantum limit for the phase noise, compared with the quantum limit for the timing jitter. This can be understood by considering how resonator length changes and quantum noise affect the pulse timing and phase. The quantum-limited timing jitter, but not the phase noise, scales in proportion to the squared pulse duration. It can be shown using Eqs. (1) and (4) that for very short (few-cycle) pulses the quantum-limited jitter and phase noise correspond to a similar (very small) magnitude of resonator length changes, whereas for example for 1-ps pulses the quantum-limited timing jitter is higher by many tens of decibels, and can thus be much more easily reached in practice.

Noise of the carrier–envelope offset

Noise of the carrier–envelope offset frequency νceo can be estimated by extrapolating the optical phase noise within the pulse spectrum to zero frequency [12

12. R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier–envelope offset noise of mode-locked lasers,” Appl. Phys. B 82(2), 265–273 (2006). [CrossRef]

]. That kind of extrapolation is intrinsic in the definition of νceo, and is also involved when νceo is measured with an f–2f interferometer. If a laser produces pulses with a duration far above the optical period and a spectral width far smaller than the center frequency, that noise extrapolation to zero frequency yields strong noise of the carrier–envelope offset. This effect can also be understood by considering the carrier–envelope offset in the time domain: a long pulse duration leads to increased timing jitter, i.e., to increased jiggling of the pulse envelope, whereas the optical phase noise is not increased for long pulse durations. The PSD of the carrier–envelope offset phase is higher than that of the optical phase noise at the center frequency νc by a factor (2πνcτp)2, which is proportional to the squared number of optical cycles in the pulse. We thus see that lasers emitting few-cycle pulses (with octave-spanning spectrum) have the best potential for producing frequency combs with ultralow noise of the carrier–envelope offset. In principle, one may suppress the stronger timing jitter of a fiber laser (emitting longer pulses) with a feedback scheme. However, this improves the carrier–envelope offset noise only if a control element is used which also has its “fix point” [16

16. H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74(1), 1–6 (2002). [CrossRef]

] in the optical spectrum. The latter condition may be difficult to meet. It may require the combination of two control elements with different “fix points”.

It must be emphasized that all the equations presented here have been derived based on various assumptions, which are fulfilled only in sufficiently “simple” lasers, and certainly not in some fiber lasers where the pulse evolution is strongly influenced by nonlinear and dispersive effects. In section 4, such lasers are investigated.

3. Analysis of various influences

This section addresses the influence of various typical properties of fiber lasers on the quantum-limited noise performance. Some specific types of mode-locked fiber lasers are treated in section 4.

Pulse energy

A high pulse energy, or in fact a high number of photons per pulse, is essential for keeping quantum noise influences small. Mode-locked fiber lasers, however, typically exhibit a fairly low intracavity pulse energy – it is often by orders of magnitude smaller than in bulk lasers. The reason is essentially that the strong nonlinearities of fibers, resulting from a small mode area and a long length of the medium, prevent the realization of higher pulse energies. (The thermal power handling capability of fibers can normally by far not be fully exploited for that reason.)

For a soliton fiber laser, the overall nonlinear phase shift per round trip must be kept below some limit, which depends on various factors (for example, the strength of the saturable absorber) but is at most of the order of a few radians. That introduces a limit on PpTrt/Aeff, where Pp is the intracavity peak power and Aeff the effective mode area of the fiber. The longer the resonator, the smaller the limit for the peak power and thus (for a given pulse duration) the pulse energy. Some other kinds of mode-locked lasers such as stretched-pulse lasers and similariton lasers are more tolerant in this respect, and are discussed in section 4.

Mode-locked bulk lasers and fiber lasers exist in quite different forms, so that a general comparison is difficult. Nevertheless, an example shows that the difference in pulse energy is often substantial. Consider a simple 1064-nm bulk laser with a pulse repetition rate of 50 MHz and a pulse duration of 10 ps, using a single high-brightness pump diode and emitting an average power of 1 W. The output pulse energy is then 20 nJ. With 5% output coupling, the intracavity pulse energy is 400 nJ, and the intracavity peak power is roughly 40 kW. A fiber ring laser for 50 MHz has a fiber length of ≈4 m. With a mode area of 100 μm2, a reduced intracavity peak power of 10 kW (for 20% output coupling) would result in a nonlinear phase shift of >60 rad, which is more than an order of magnitude higher than acceptable, e.g., for a soliton fiber laser. Possible solutions are to use a fiber with very large mode area (possibly >1000 μm2), which however restricts the choice of fiber-optic components and the scope for dispersion control, or to use more sophisticated pulse shaping mechanisms. Obviously, shorter pulse durations such as 1 ps or 100 fs would further tighten the limits for the average power. On the other hand, mode-locked bulk lasers can easily generate substantially higher powers than in our example, even for much shorter pulses.

Laser gain and losses

Now we consider the effect of the laser gain on the noise. Fiber lasers typically use a higher degree of output coupling, compared with bulk lasers. This is essentially because the low saturation powers of active fibers make it easy to obtain a high gain, and a high output coupler transmission lowers the intracavity pulse energy and thus reduces challenges from the high nonlinear phase shifts.

However, higher gain and losses increase quantum noise influences on the circulating pulse. Besides the tentatively lower pulse energy, larger laser gain implies a stronger noise input. Equations (1) and (4), for example, contain the round-trip losses, which must be balanced by the (saturated) gain. Furthermore, a high gain implies that the pulse energy at the beginning of the amplifying fiber is substantially lower than at the output, further increasing the sensitivity to quantum noise. The previously used equations contain the term ltot/Ep, which obviously assumes a constant pulse energy. For cases with substantial gain, we can generalize this. First, we use g/Ep with the exponential gain coefficient g, and then generalize this to
0gdxEp,finex=1egEp,fin=eg1Ep,fout
where Ep,fin and Ep,fout are the pulse energy at the input and output, respectively, of the active fiber.

For a numerical example, assume a bulk laser with 10 nJ output pulse energy and 5% output coupling. The intracavity pulse energy is then 200 nJ, and the term ltot/Ep amounts to 2.5⋅105 J−1. For a fiber laser with 10 nJ output pulse energy and 5 dB output coupling directly after the active fiber and no other resonator losses, we would have Ep,fout ≈14.6 nJ, Ep,fin = 4.6 nJ, and (eg1)/Ep,fout = 1.5⋅108 J−1. The quantum noise influence is thus ≈590 times (≈28 dB) higher than in the bulk laser, despite the same output pulse energy, if we assume the same θ values for both lasers.

In conclusion, quantum noise influences are strongly increased for fiber lasers with high gain and loss. This particularly affects some of those kinds of fiber lasers which allow for higher pulse energies, such as similariton lasers. Therefore, fiber laser designs for higher pulse energies will not necessarily exhibit better noise properties.

Resonator length and pulse repetition rate

Essentially, the resonator length has two effects on the quantum noise inputs. The direct effect is that for a long resonator round-trip time Trt, the noise influences from gain and losses act less frequently on the circulating pulse; what counts are the gain and losses per second, not per round trip. This aspect is reflected by the factor Trt in the denominator of Eqs. (1) and (4), for example.

An indirect effect of the resonator length is that it can affect the possible pulse energy. In bulk lasers, a longer resonator normally means that a higher pulse energy is possible, because the pulse energy is limited by the average power possible with the given pump source. Only in a few cases, the pulse energy is limited by nonlinearities. In fiber lasers, however, it is the usual case that the pulse energy is limited by nonlinearities, rather than by the available pump power. As the overall strength of the nonlinearity in a fiber setup increases with the resonator length, a longer resonator often implies an even lower pulse energy. Thus we can see that whereas a longer resonator for a bulk laser substantially lowers the quantum noise impact, this is often not the case for fiber lasers.

Another aspect is harmonic mode locking, which is often used in high-repetition-rate fiber lasers, because their resonators cannot be made sufficiently short for fundamental mode locking. For a given pulse energy, harmonic mode locking introduces less quantum noise than fundamental mode locking, simply because of the Trt dependence as discussed above. On the other hand, the longer resonator for harmonic mode locking may reduce the pulse energy because of the higher nonlinearity. Also, harmonic mode locking requires additional care to reach the quantum noise limit, as there are challenges arising from supermode noise.

Because of these essential aspects, specific comparisons of fiber and bulk lasers depend on the pulse repetition rate regime. For low pulse repetition rates of e.g. <50 MHz, fiber lasers have much lower pulse energies than bulk lasers, and thus much stronger quantum noise. That difference becomes smaller for higher repetition rates of e.g. 200 MHz, as the pulse energy of a bulk laser is then reduced while that of the fiber laser may be even larger. For very high pulse repetition rates of e.g. 10 GHz, a bulk laser will usually be a compact low-power laser [17

17. L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38(5), 225–227 (2002). [CrossRef]

] with an intracavity pulse energy of e.g. a few hundred pJ. A fiber laser for 10 GHz will usually be harmonically mode-locked and have a resonator length of several meters, with a few hundred circulating pulses. The pulse energy may even be somewhat higher than for the bulk laser, at least in the picosecond pulse duration regime. Generally, we see that fiber lasers have much stronger quantum noise influences than bulk lasers for low pulse repetition rates, but that difference may disappear for very high pulse repetition rates.

Pulse duration

Equation (1) shows that the PSD of the timing jitter scales with the square of the pulse duration. For a given pulse energy, a longer pulse has a lower peak power. Also, the power changes induced by quantum noise in a wider temporal range can more strongly influence the temporal “center of gravity” of the pulse. For these reasons, a pulse duration as short as possible leads to a low quantum limit for the timing jitter, whereas such a dependence does not exist for the optical phase noise.

Mode-locked fiber lasers often have pulse durations of a few hundred femtoseconds, but sometimes even well below 100 fs. A short pulse duration increases the peak power for a given pulse energy, so that nonlinearities tend to limit more severely the achievable pulse energy. This reduces or even fully neutralizes the benefit of shorter pulses for the timing jitter, while increasing the optical phase noise. Furthermore, even the shortest pulses from fiber lasers are still much longer than from certain bulk lasers, particularly Ti:sapphire lasers, which can reach roughly 5 fs [18

18. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24(6), 411–413 (1999). [CrossRef]

,19

19. D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, “Semiconductor saturable-absorber mirror assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime,” Opt. Lett. 24(9), 631–633 (1999). [CrossRef]

], and this even combined with much higher pulse energies than are possible with fiber lasers.

Note that although for some fiber lasers the emitted pulses may externally be compressed to much shorter pulse durations, that compression cannot be expected to reduce the timing jitter as well. What counts for the jitter is the pulse duration within the laser, not the externally compressed pulsed duration.

4. Specific types of mode-locked fiber lasers

Simple soliton mode-locked fiber lasers can be treated with analytical models as discussed in section 2. In this section, we discuss some types of mode-locked fiber lasers to which simple analytical models cannot be applied, since the pulse evolution is more complicated.

Stretched-pulse fiber lasers

The basic idea behind a stretched-pulse fiber laser is to use fiber spans of alternating signs of chromatic dispersion, so that the pulses are periodically stretched and recompressed in every resonator round trip. As the pulses are strongly chirped and accordingly long at most locations in the resonator, the overall nonlinear phase shift per round trip for a given pulse energy is strongly reduced, comparing with a soliton fiber laser, for example. For that reason, substantially higher pulse energies are possible. In addition, the generated pulses can be rather short, or at least can be substantially compressed outside the laser.

The question to be addressed in the following is whether the more complicated pulse formation process in a stretched-pulse laser introduces timing noise beyond that predicted by a simple model, disregarding the details of pulse formation. This is investigated with a numerical model of a test case, which has been constructed such that it exhibits the typical pulse formation of a stretched-pulse laser. The schematic setup is shown in Fig. 1
Fig. 1 Schematic setup of a stretched-pulse laser as assumed for the numerical model.
. It contains two pieces of passive (non-amplifying) fiber with a negative group velocity dispersion of −20000 fs2/m and an active fiber with a positive group velocity dispersion of + 20000 fs2/m. The active fiber is 1 m long, and the two passive fibers are 0.5 m long each, such that the total group delay dispersion (GDD) is zero. Higher-order chromatic dispersion is assumed to be absent. All fibers have an effective mode area of 100 μm2. The active fiber provides wavelength-independent saturable gain with a simplified saturation behavior (instantly reacting to the pulse energy). Between the two pieces of passive fiber, there is a 10% output coupler, a fast saturable absorber (with a saturation power several times lower than the peak power), and a spectral filter with a FWHM transmission bandwidth of 60 nm (mimicking the limited gain bandwidth of the active fiber). In the steady state, as reached after a few tens of round trips, pulses with ≈0.3 nJ internal pulse energy exhibit the breathing behavior which is typical for such lasers: the pulse duration is at a minimum (≈66 fs) near the output coupler, much longer (≈400 fs) and down-chirped at the end of the passive fiber, shorter again (≈160 fs) in the middle of the active fiber, longer again (310 fs) and up-chirped after the active fiber. The temporal pulse shape is approximately Gaussian at all locations. For substantially higher pulse energies, the nonlinear influence becomes strong, and the pulse formation deviates substantially from the simple behavior as described above.

Of course, there are somewhat different operation regimes, found when parameters such as the magnitude of total GDD, the pulse energy, the mode areas in both fibers etc. are varied. Here, however, the intention is not a comprehensive study on stretched-pulse lasers, but rather the investigation of a test case concerning the noise properties in a typical situation.

The noise properties are calculated with statistical methods as described in Ref [20

20. R. Paschotta, “Noise of mode-locked lasers. Part I: Numerical model,” Appl. Phys. B 79(2), 153–162 (2004). [CrossRef]

]. Essentially, various pulse parameters (pulse energy, duration, temporal position, spectral width, etc.) are recorded over thousands of round trips, and the power spectral densities are estimated from these data.

For the analytical estimate, an important question is which pulse duration to insert into the equation. The pulse duration varies substantially during each resonator round trip, and even within the active fiber alone. Certainly, the expected timing noise should be between the values calculated inserting the minimum and maximum pulse duration, respectively. Figure 2
Fig. 2 Simulated timing phase noise for a stretched-pulse fiber laser (dots), compared with analytical estimates (lines) based on the minimum and maximum pulse duration occurring within the laser resonator. The simulated data have been averaged over 8 simulation runs.
shows that the numerically simulated timing phase noise indeed lies between these two levels. This result suggests that the application of the simple analytical model provides a reasonable lower and upper estimate for the timing jitter. In other words, the substantial breathing of the pulse duration during each round trip does lead to a somewhat stronger timing jitter (compared to a laser where the pulse duration would stay at the minimum level), but it does not appear to introduce an unexpected level of excess noise.

Of course, this result may not apply to all stretched-pulse lasers, as various parameters can be varied in large ranges. For example, cases with substantially stronger nonlinear action, where the dynamics can be more complicated, or cases with a substantial imbalance between positive and negative dispersion, might be different. We learn, however, that the stretched-pulse mode-locking technique does not necessarily introduce strong excess noise.

The simulated optical phase noise (not shown in the graph) is also ≈30 dB above the Schawlow–Townes limit. This is not surprising, though, as this is already the case for a simple soliton fiber laser, for example, as discussed in section 2.

Wavebreaking-free fiber lasers

Another important concept is that of wavebreaking-free lasers, often with only normal chromatic dispersion in the resonator. Here, the interplay of normal dispersion and Kerr nonlinearity leads to the formation of approximately linearly up-chirped pulses, which typically exhibit a stable and “clean” spectrum (often with a steep decay of PSD in the wings) and no tendency for pulse break-up in the time domain. The spectral width is increased during the passage through a fiber and periodically reset to a smaller value by the effect of an optical bandpass filter. In addition, a saturable absorber can also reduce the spectral width, as it suppresses the temporal wings, which carry the extreme spectral portions. Similarly, the bandpass filter can help to reduce the pulse duration, which is also increased in the fiber.

A special variant is the similariton laser [4

4. F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902–213905 (2004). [CrossRef] [PubMed]

], where the spectral broadening occurs in an amplifying fiber. In that fiber, the pulse evolves toward a similariton pulse, but may actually not always get close to a real similariton, as that asymptotic solution may be approximated only after a longer length of fiber. Therefore, the actual dynamics can be considered to be not too different from those in lasers where part of the spectral broadening occurs in passive (non-amplifying) fibers. The common essential feature is the periodic nonlinear spectral broadening and subsequent spectral compression by some filter.

A numerical model has been set up for an example case, where in each round trip the pulse passes a 10% output coupler, a fast saturable absorber, a bandpass filter 15 nm FWHM bandwidth, a passive fiber with a GDD of + 0.1 ps2, and an amplifying element without dispersion and nonlinearity. The circulating pulse with an energy of ≈5 nJ after the active fiber exhibits the typical evolution in a wavebreaking-free laser, as described above.

Figure 3
Fig. 3 Simulated timing phase noise of a wavebreaking-free fiber laser. For noise frequencies below 5 MHz, the timing jitter is substantially stronger than according to estimates based on a simplified analytical model. The solid and dashed lines show the jitter with and without the effect of center frequency fluctuations, respectively, according to the analytical model.
shows the simulated timing jitter. It is compared with theoretical estimates, with and without the indirect effect via fluctuations of the center frequency. The latter effect would be expected to be very weak, essentially because the bandpass filter provides a strong “restoring force” for the center frequency. The simulated timing jitter, however, exhibits a strong increase at lower frequencies. This largely results from center frequency noise, which is much stronger than expected from the simple analytical model (see Fig. 4
Fig. 4 Numerically simulated noise of the optical center frequency (normalized to the mean frequency) of a wavebreaking-free fiber laser. For noise frequencies below 5 MHz, this noise is much stronger than expected from a simple analytical model (solid curve). This largely explains why the timing jitter is also stronger.
). Because of the large positive GDD, this center frequency noise can be effectively coupled to timing noise.

Again, this paper is not meant to provide a comprehensive study of wavebreaking-free fiber lasers. It is conceivable that the noise performance can vary substantially depending on various parameters. However, it becomes clear from the investigated example case that simplified analytical models, not taking into account the complicated nonlinear dynamics, do not correctly describe the noise properties of such lasers. The mentioned dynamics can lead to substantial excess noise. In addition, one should note that such lasers exhibit relatively high resonator round-trip losses, and that high pulse energies are only possible because of the long intracavity pulse duration, which also increases the noise. We can thus conclude that although wavebreaking-free fiber lasers allow for much higher pulse energies than other mode-locked fiber lasers, it appears that this cannot be translated into superior quantum noise properties.

So far, there are not many experimental results on the noise properties of wavebreaking-free fiber lasers. A comprehensive set of measurements has been presented in Ref [9

9. O. Prochnow, R. Paschotta, E. Benkler, U. Morgner, J. Neumann, D. Wandt, and D. Kracht, “Quantum-limited noise performance of a femtosecond all-fiber ytterbium laser,” Opt. Express 17(18), 15525–15533 (2009), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-17-18-15525. [CrossRef] [PubMed]

]. for a laser with high normal net dispersion and no dispersion compensation. Here, the timing jitter has been found to be far above the results of simple analytical estimates, but in good agreement with numerical simulations of the same kind as discussed in this article. It became clear that although technical noise influences appeared to be weak, the quantum noise obtained an increased impact on the timing jitter through the complicated pulse formation process. The carrier−envelope offset noise has also been measured, and was found to be in good agreement with an estimate based on the measured timing jitter and the assumption of a quasi-fix point of the phase fluctuations in the optical spectrum (which is well compatible with the low measured optical phase noise). Due to the increased timing jitter, the carrier−envelope offset noise was not particularly low. All aspects of the noise performance of this laser are in good agreement with the theoretical results presented in this article.

On the other hand, Ref [21

21. T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008). [CrossRef]

]. reported a very low carrier−envelope linewidth of a simpler type of wavebreaking-free fiber laser, where dispersion compensation was done with a linearly chirped fiber Bragg grating having a reflection bandwidth of 60 nm. The laser emitted pulses with a bandwidth of ≈40 nm, which could be dispersively compressed to 60 fs duration. The short-term carrier−envelope linewidth of the free-running (not stabilized) laser was reported to be only a few kHz. This result is quite surprising. Numerical simulations with the parameters of that laser indicated that the short-term carrier−envelope linewidth should be of the order of 200 kHz. Furthermore, the low measured noise of the carrier−envelope offset indicates that the timing jitter must be extremely low. For example, if the optical phase fluctuations have a quasi-fix point in the optical spectrum (as it is the case in numerical simulations), the r.m.s. timing jitter has to be ≈1 fs for integration over noise frequencies starting at 1 kHz, if the carrier−envelope linewidth is 10 kHz. On the other hand, even considering only the direct effect of quantum noise (without the Gordon−Haus term, which should be small here), we obtain nearly 4 fs already, and the linewidth is proportional to the square of the r.m.s. timing jitter. A quasi-fix point closer to zero frequency is also not a viable explanation, since the measured phase noise in the optical spectrum is also very low, so that an extremely low timing jitter (below the expected direct effect of quantum noise) would still be required. Discussions with the authors of Ref [21

21. T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008). [CrossRef]

]. have not resolved this anomaly so far; further research will be required. In particular, improved measurements of the timing jitter, not limited by electronic noise, would be helpful. Also, it would be worthwhile to check carefully whether the experimental laser is influenced by any additional effects with a potential for reducing the timing jitter. For example, there might be some kind of weak optical feedback which is not known yet and therefore not taken into account in the numerical model. Note that a fiber-based pulse stretcher and power amplifier were connected to the laser without an optical isolator.

5. Conclusions

The noise properties of mode-locked fibers lasers of various types have been examined and compared with those of mode-locked bulk lasers, using both analytical calculations and (for more complicated cases) some numerical simulations. There are substantial differences between different fiber lasers and bulk lasers, which arise from different parameter values, from different limiting design factors, and from different pulse formation mechanisms. Despite this complexity, some relatively general conclusions can be drawn.

For simple types of mode-locked fiber lasers, employing pulse formation mainly with some type of saturable absorber or with the additional aid of soliton formation in the fiber, the noise properties can be reliably predicted with a simple analytical model. The pulse energy is fairly limited by the strong fiber nonlinearity, and the pulse duration cannot be very short. This leads to a high level of quantum-limited timing jitter, particularly comparing with bulk lasers in the regime of low pulse repetition rates. For higher pulse repetition rates, that difference has been shown to become substantially smaller. Particularly concerning noise of the carrier–envelope offset, however, such fiber lasers cannot compete with bulk lasers exhibiting a combination of very short pulse durations and high pulse energies.

The principle of stretched-pulse mode locking allows one to increase the pulse energy substantially, and it has been shown that the breathing of pulse parameters during each round trip does not necessarily introduce excess noise, although the quantum-limited timing jitter is somewhat stronger than from a simple estimate based on the minimum pulse duration.

A different result has been obtained for wavebreaking-free fiber lasers. Here, the complicated nonlinear dynamics can introduce substantial excess noise, and simplified analytical models not taking into account these dynamics cannot correctly predict the noise properties. Even the underestimated jitter from a simple model is not at a low level, as the intracavity pulse duration is long. Although allowing substantially increased pulse energies, such lasers appear not to offer a potential for operation with particularly low noise.

For simple types of mode-locked lasers, optical phase noise can be close to the Schawlow–Townes limit, calculated with the average power per pulse in the resonator. That is no longer true, however, if there is a substantial Kerr nonlinearity, which is nearly always the case for mode-locked fiber lasers. For comparison, at least for longer pulse durations a bulk laser might reach the Schawlow–Townes limit as long as technical noise can be suppressed. Generally, reaching the Schawlow–Townes limit of the optical phase noise is substantially more difficult than reaching quantum-limited timing jitter, if the pulse duration amounts to many optical cycles.

Acknowledgements

The author thanks O. Prochnow, E. Benkler and H. R. Telle for useful discussions and comments on the manuscript.

References and links

1.

L. F. Mollenauer and R. H. Stolen, “The soliton laser,” Opt. Lett. 9(1), 13–15 (1984). [CrossRef] [PubMed]

2.

K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18(13), 1080–1082 (1993). [CrossRef] [PubMed]

3.

F. Ö. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nJ pulses at 1.03 μm from a wave-breaking-free fiber laser,” Opt. Lett. 28(15), 1365–1367 (2003). [CrossRef] [PubMed]

4.

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902–213905 (2004). [CrossRef] [PubMed]

5.

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). [CrossRef]

6.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002). [CrossRef] [PubMed]

7.

S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13(12), 2817–2823 (1996). [CrossRef]

8.

M. E. Grein, L. A. Jiang, H. A. Haus, E. P. Ippen, C. McNeilage, J. H. Searls, and R. S. Windeler, “Observation of quantum-limited timing jitter in an active, harmonically mode-locked fiber laser,” Opt. Lett. 27(11), 957–959 (2002). [CrossRef]

9.

O. Prochnow, R. Paschotta, E. Benkler, U. Morgner, J. Neumann, D. Wandt, and D. Kracht, “Quantum-limited noise performance of a femtosecond all-fiber ytterbium laser,” Opt. Express 17(18), 15525–15533 (2009), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-17-18-15525. [CrossRef] [PubMed]

10.

H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993). [CrossRef]

11.

R. Paschotta, “Noise of mode-locked lasers. Part II: Timing jitter and other fluctuations,” Appl. Phys. B 79(2), 163–173 (2004). [CrossRef]

12.

R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier–envelope offset noise of mode-locked lasers,” Appl. Phys. B 82(2), 265–273 (2006). [CrossRef]

13.

A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958). [CrossRef]

14.

R. Paschotta, H. R. Telle, and U. Keller, Noise of Solid State Lasers (CRC Press, Boca Raton, 2007), Chap. 12.

15.

P.-T. Ho, “Phase and amplitude fluctuations in a mode-locked laser,” IEEE J. Quantum Electron. QE 21(11), 1806–1813 (1985). [CrossRef]

16.

H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74(1), 1–6 (2002). [CrossRef]

17.

L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38(5), 225–227 (2002). [CrossRef]

18.

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24(6), 411–413 (1999). [CrossRef]

19.

D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, “Semiconductor saturable-absorber mirror assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime,” Opt. Lett. 24(9), 631–633 (1999). [CrossRef]

20.

R. Paschotta, “Noise of mode-locked lasers. Part I: Numerical model,” Appl. Phys. B 79(2), 153–162 (2004). [CrossRef]

21.

T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008). [CrossRef]

OCIS Codes
(140.3510) Lasers and laser optics : Lasers, fiber
(140.4050) Lasers and laser optics : Mode-locked lasers
(270.2500) Quantum optics : Fluctuations, relaxations, and noise

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: January 11, 2010
Revised Manuscript: February 18, 2010
Manuscript Accepted: February 19, 2010
Published: February 25, 2010

Citation
Rüdiger Paschotta, "Timing jitter and phase noise
of mode-locked fiber lasers," Opt. Express 18, 5041-5054 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-5041


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References

  1. L. F. Mollenauer and R. H. Stolen, “The soliton laser,” Opt. Lett. 9(1), 13–15 (1984). [CrossRef] [PubMed]
  2. K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18(13), 1080–1082 (1993). [CrossRef] [PubMed]
  3. F. Ö. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nJ pulses at 1.03 μm from a wave-breaking-free fiber laser,” Opt. Lett. 28(15), 1365–1367 (2003). [CrossRef] [PubMed]
  4. F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902–213905 (2004). [CrossRef] [PubMed]
  5. 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). [CrossRef]
  6. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002). [CrossRef] [PubMed]
  7. S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13(12), 2817–2823 (1996). [CrossRef]
  8. M. E. Grein, L. A. Jiang, H. A. Haus, E. P. Ippen, C. McNeilage, J. H. Searls, and R. S. Windeler, “Observation of quantum-limited timing jitter in an active, harmonically mode-locked fiber laser,” Opt. Lett. 27(11), 957–959 (2002). [CrossRef]
  9. O. Prochnow, R. Paschotta, E. Benkler, U. Morgner, J. Neumann, D. Wandt, and D. Kracht, “Quantum-limited noise performance of a femtosecond all-fiber ytterbium laser,” Opt. Express 17(18), 15525–15533 (2009), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-17-18-15525 . [CrossRef] [PubMed]
  10. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993). [CrossRef]
  11. R. Paschotta, “Noise of mode-locked lasers. Part II: Timing jitter and other fluctuations,” Appl. Phys. B 79(2), 163–173 (2004). [CrossRef]
  12. R. Paschotta, A. Schlatter, S. C. Zeller, H. R. Telle, and U. Keller, “Optical phase noise and carrier–envelope offset noise of mode-locked lasers,” Appl. Phys. B 82(2), 265–273 (2006). [CrossRef]
  13. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. 112(6), 1940–1949 (1958). [CrossRef]
  14. R. Paschotta, H. R. Telle, and U. Keller, Noise of Solid State Lasers (CRC Press, Boca Raton, 2007), Chap. 12.
  15. P.-T. Ho, “Phase and amplitude fluctuations in a mode-locked laser,” IEEE J. Quantum Electron. QE 21(11), 1806–1813 (1985). [CrossRef]
  16. H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74(1), 1–6 (2002). [CrossRef]
  17. L. Krainer, R. Paschotta, G. J. Spühler, I. Klimov, C. Y. Teisset, K. J. Weingarten, and U. Keller, “Tunable picosecond pulse-generating laser with a repetition rate exceeding 10 GHz,” Electron. Lett. 38(5), 225–227 (2002). [CrossRef]
  18. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24(6), 411–413 (1999). [CrossRef]
  19. D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, “Semiconductor saturable-absorber mirror assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime,” Opt. Lett. 24(9), 631–633 (1999). [CrossRef]
  20. R. Paschotta, “Noise of mode-locked lasers. Part I: Numerical model,” Appl. Phys. B 79(2), 153–162 (2004). [CrossRef]
  21. T. R. Schibli, I. Hartl, D. C. Yost, M. J. Martin, A. Marcinkevičius, M. E. Fermann, and J. Ye, “Optical frequency comb with submillihertz linewidth and more than 10 W average power,” Nat. Photonics 2(6), 355–359 (2008). [CrossRef]

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