It is well-known that nonlinear phase shifts (ΦNL
) can lead to distortion of short optical pulses [1
1. G. P. Agrawal, Nonlinear Fiber Optics 3rd (Academic, San Diego, California, 2001).
]. In chirped-pulse amplification (CPA) [2
2. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]
], a pulse is stretched to reduce the detrimental nonlinear effects that can occur in the gain medium. After amplification, the pulse is dechirped, ideally to the duration of the initial pulse. The stretching is typically accomplished by dispersively broadening the pulse in a segment of fiber or with a diffraction-grating pair. For pulse energies of microjoules or greater, the dechirping is done with gratings, to avoid nonlinear effects in the presence of anomalous group-velocity dispersion (GVD), which are particularly limiting. The magnitude of the dispersion of a grating stretcher can exactly equal that of the gratings used to dechirp the pulse, to all orders [3
3. M. Pessot, P. Maine, and G. Mourou, “1000 times expansion/compression of optical pulses for chirped pulse amplification,” Opt. Commun. 62, 419–421 (1987). [CrossRef]
4. O. E. Martinez, “Design of high-power ultrashort pulse amplifiers by expansion and recompression,” IEEE J. Quantum. Electron. 23, 1385–1387 (1987). [CrossRef]
]. At low energy, the process of stretching and compression can thus be perfect. At higher energy, some nonlinear phase ΦNL
will be accumulated and this will degrade the temporal fidelity of the amplified pulse. For many applications, ΦNL
must be less than 1 to avoid unacceptable structure on the amplified pulse [5
5. M. D. Perry, T. Ditmire, and B. C. Stuart, “Self-phase modulation in chirped pulse amplifiers,” Opt. Lett. 19, 2149–2151 (1994). [CrossRef] [PubMed]
6. A. Braun, S. Kane, and T. Norris, “Compensation of self-phase modulation in chirped-pulse amplification laser system,” Opt. Lett. 22, 615–617 (1997) [CrossRef] [PubMed]
The total dispersion of a fiber stretcher differs from that of a grating pair, and this mismatch results in uncompensated third-order dispersion (TOD), which will distort and broaden the pulse, at least in linear propagation. At wavelengths where the fiber has normal GVD (such as 1 µm, which will be our main focus here), the TOD of the fiber adds to that of the grating pair. Stretching ratios of thousands are used in CPA systems designed to generate microjoule and millijoule-energy pulses, in which case the effects of TOD would limit the dechirped pulse duration to the picosecond range. It has thus become “conventional wisdom” that fiber stretchers are unacceptable in CPA systems, and as a consequence, grating stretchers have become ubiquitous in these devices.
Apart from the difficulty of compensating the cubic phase, a fiber offers major advantages as the pulse stretcher in a CPA system. Coupling light into a fiber is trivial compared to aligning a grating stretcher. The grating stretcher includes an imaging system [7
7. O. E. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in the 1.3–1.6 µm region,” IEEE J. Quantum Electron. 23, 59–64 (1987). [CrossRef]
] that can be misaligned, and when misaligned will produce spatial and temporal aberrations in the stretching. A fiber stretcher cannot be misaligned. The fiber is also less sensitive to drift or fluctuations in wavelength or the pointing of the beam that enters the stretcher. Beam-pointing fluctuations may reduce the coupling into the fiber to below the optimal level, whereas they translate into changes in dispersion with a grating pair. Finally, the spatial properties of the beam can influence the stretching with a grating pair, while the pulse stretched in a fiber cannot have spatial chirp, experiences the same stretching at all points on the beam, and exits the fiber with a perfect transverse mode. With fiber amplifiers, there is naturally strong motivation to employ fiber stretchers - grating stretchers detract substantially from the benefits of fiber.
Here we show that the nonlinear phase shift accumulated by a chirped pulse in amplification can actually be compensated to some degree by the cubic spectral phase. (Or vice-versa
-- the cubic spectral phase can be compensated by nonlinearity.) Thus, the performance of a CPA system can be improved by employing mismatched stretcher and compressor dispersions, and allowing the pulse to accumulate a significant nonlinear phase shift. This conclusion has important consequences for short-pulse fiber amplifiers. First, the performance of a CPA system with a fiber stretcher and grating compressor actually improves with nonlinearity
. This behavior contrasts with that of a CPA system with matched stretcher and compressor, where the pulse fidelity decreases monotonically with nonlinear phase shift. Second, the peak power achievable with a fiber stretcher can exceed the highest peak power that can be reached with a grating stretcher
. This compensation of TOD by nonlinear phase shift will not offer much advantage in solid-state amplifiers, in which the spatial consequences of the nonlinear phase shift will be limiting: small-scale or whole-beam self-focusing will distort the beam or, eventually, damage the gain medium. Liu et al
. reported that TOD can be compensated by nonlinear phase shifts accumulated by a pulse with an asymmetric spectrum [12
12. Z. Liu, L. Shah, I. Hartl, G. C. Cho, and M. E. Fermann, “The Cubicon Amplifier”, oral presentation at Photonics West 2005, Conference 5709 - Fiber Lasers II: Technology, Systems, and Applications, late breaking developments, January 24, 2005.
]. The effects that we describe here occur with symmetric spectra, although they can be enhanced by spectral asymmetry. In this report we focus on ordinary single-mode fiber, but the approach can be used in any implementation of CPA, such as those employing diffraction gratings, fiber Bragg gratings, or microstructure fibers.
2. Principle and numerical modeling
It is perhaps remarkable that nonlinear phase shifts and TOD should compensate each other. After all, TOD acting alone produces an anti-symmetric phase, while self-phase modulation alone produces a symmetric phase. We can offer a qualitative rationale for the compensation by considering pulse propagation in a fiber stretcher, followed by a nonlinear segment with negligible dispersion, and a grating compressor. For wavelengths below 1.3 µm the TOD from the fiber stretcher and the grating compressor add, so the net TOD can be large, while the net GVD is relative small. (The optimal GVD is only nonzero in the presence of some nonlinear phase shift.) Under these conditions, the spectral phase can be flattened by an appropriate choice of ΦNL
(see Fig. 1
). As a result, a better-quality pulse is obtained with nonzero ΦNL
. This can be interpreted as the compensation of net TOD by nonlinearity because the net TOD dominates, even though the net GVD plays an important role in the process. This explanation is consistent with the numerical results presented below, but more work is needed to understand this process thoroughly.
Numerical simulations were employed to study CPA with a fiber stretcher, a fiber amplifier and a grating compressor (the key elements of the experimental setup shown in Fig. 4
). The parameters of the simulations were taken as those of the experiments described below, to allow comparison of theory and experiment. All the fiber is single-mode fiber (SMF), with mode-field diameter of 6 µm. The input pulses to this system represent the output of an Yb fiber oscillator [13
13. H. Lim, F. Ö. Ilday, and F. W. Wise “Generation of 2 nJ pulses from a femtosecond Yb fiber laser,” Opt. Lett. 28, 660–662 (2003). [CrossRef] [PubMed]
14. H. Lim, J. Buckley, and F. W. Wise, “Wavelength tunability of femtosecond Yb fiber lasers,” Conference on Lasers and Electro-Optics (Optical Society of America, San Francisco, Calif., 2004), presentation CThK3.
] and were taken to be 150-fs gaussian pulses with 10.4-nm bandwidth at 1060 nm. The stretcher consists of 100 m of SMF, and analogous results for a 400-m stretcher will be summarized below. The amplifier consists of 1 m of Yb-doped gain fiber, and is followed by 3 m of SMF, where most of the nonlinear phase shift is accumulated. The magnitude of the nonlinear phase shift is adjusted by varying the gain of the amplifier. The compressor is a pair of gratings with 1200 lines/mm, used in a double-pass configuration. The GVD and TOD of the fibers and grating pairs are included in the simulations (the numerical values are listed in the caption of Fig. 2
). The nonlinear Schrödinger equations that govern propagation in each section are solved by the standard split-step technique.
Fig. 1. Illustration of compensation of nonlinearity by TOD. (a) Pulses in time domain for the indicated values of ΦNL (b) Phase spectra for the same values of ΦNL.
After propagation through 100 m of stretcher fiber, the pulse duration is 46 ps. The compressor grating separation is optimized to produce the shortest output pulse at each pulse energy; as expected, the grating separation decreases with increasing ΦNL. All numerical and experimental results reported here are based on the optimal grating separations.
As an approximation to linear propagation, the amplifier gain was adjusted to produce a low-energy (4 nJ) pulse. The resulting ΦNL=0.4π
. The pulse shape after compression (Fig. 2
(a)) exhibits the signature asymmetric broadening and secondary structure from TOD. The full-width at half-maximum (FWHM) pulse duration has increased to 290 fs, and the peak power is 9.5 kW. The envelopes of the interferometric autocorrelation of the output pulse are shown in Fig. 2
(b). The autocorrelations are provided because they will be compared to experimental results below.
Increasing the nonlinear phase shift improves the quality of the output pulse. Best results are obtained with amplified pulse energy of 19 nJ, which produces ΦNL≈1.9π
. With this nonlinear phase shift, the compressed pulse (Fig. 2
(c)) duration is reduced to 191 fs, which is within ~25% of the original pulse width. Equally significant is the suppression of the trailing “wing” of the pulse. The resulting peak power is 74 kW; the pulse energy is 5 times larger than in Fig. 2
(a), but the peak power is 8 times larger owing to the improved pulse quality. The corresponding autocorrelation is shown in Fig. 2
(d). The power spectrum broadens by less than 5% at the highest energy. Spectral broadening is roughly proportional to the nonlinear phase shift divided by the stretching ratio, so small broadening is expected for ΦNL
and a stretching ratio of 300. For larger nonlinear phase shifts, the pulse quality degrades. Thus, for a given amount of cubic phase, there is an optimal value of ΦNL
. Simulations with the signs of the TOD of the fiber and grating pair reversed produce identical results; a positive (self-focusing) nonlinearity can compensate the effects of either sign of TOD.
For comparison, we show the best results that can be obtained with a grating stretcher and compressor at the same value of ΦNL
., the same pulse energy). The compressed pulse (Fig. 2
(e)) has FWHM duration 214 fs and peak power 67 kW. Thus, for this low stretching ratio and pulse energy, the fiber stretcher offers ~10% improvement over the grating stretcher. However, we emphasize that the advantage of the fiber stretcher increases with increased stretching ratio and nonlinear phase shift.
Fig. 2. Intensity profiles (a, c, e) and autocorrelation envelopes (b, d, f) obtained from numerical simulations. (a) and (b): fiber stretcher, ΦNL=0.4π. (c) and (d): fiber stretcher, ΦNL=1.9π. (e) and (f) grating stretcher, ΦNL=1.9π. Parameters used in the simulations: nonlinear coefficient γ=4.3 kW-1m-1; GVD coefficient β2=230 fs2/cm and TOD coefficient β3=254 fs3/cm. For the grating-pair compressor, β2=-1.2×105 fs2/cm and β3=4.5×105fs3/cm.
The results of a series of similar calculations with varying nonlinear phase shift are summarized in Fig. 3
(a). We define the relative peak power as the ratio of the peak power of the output pulse to the ideal peak power that would be obtained in the absence of TOD and nonlinear phase shift. That is, the ideal peak power is that of the output pulse energy with the input pulse intensity profile. With a grating stretcher, the relative peak power decreases monotonically with ΦNL
. This trend is well-known and is the reason why ΦNL
is typically limited to 1 in CPA systems. The relative peak power with a fiber stretcher increases until ΦNL~2π
, and then decreases. For ΦNL>1.5π
the fiber stretcher achieves higher relative peak power than the grating stretcher, albeit by a small margin. The inset shows the variation of the FWHM pulse duration with ΦNL
Fig. 3. (a) Variation of relative peak power with ΦNL for CPA with grating (square symbols), and 100-m fiber (round symbols) stretchers. The pulse is stretched to ~46 ps in each case. The inset shows the variation of the pulse duration with ΦNL. (b) Same as a) but with 400-m stretcher, which stretches to ~140 ps. The lines are only to guide the eye.
Analogous results for a 400-m stretcher are shown in Fig. 3
(b) and, together with Fig. 3
(a), illustrate the scaling of the compensation. The optimal value of ΦNL
increases roughly linearly with the magnitude of the TOD. The benefit of nonlinearity is larger with greater TOD: with a 100-m stretcher the relative peak power is ~50% larger than that obtained in linear propagation (ΦNL
), while with a 400-m stretcher the relative peak power increases by nearly a factor of two. The advantage of a fiber stretcher over a grating stretcher at the optimal value of ΦNL
also increases with increasing TOD. On the other hand, the maximum value of the relative peak power decreases with increasing TOD.
An important point of Fig. 3
is that larger peak power may be obtained with a fiber stretcher than with a grating stretcher, even if the grating stretcher produces higher relative peak power. For example, we will assume that ΦNL
is limited to π with the grating stretcher, which determines the maximum pulse energy. A pulse that accumulates ΦNL≈6π
(near the optimum of Fig. 3
(b)) will have ~6 times larger energy, so even with relative peak power of 0.4, the peak power will be 6*0.4≈2.4 times larger with the fiber stretcher. It is possible for the increased pulse energy to more than offset the reduction of relative peak power when SPM and TOD compensate each other.
3. Experimental results
A schematic of the experimental setup is shown in Fig. 4
. The Yb fiber laser generates 140-fs pulses with ~12-nm bandwidth at 1060 nm. The parameters of the experiment are dictated by the fact that we were limited to 400-mW pump power in our laboratory, which impacts the range of nonlinear phase shifts that can be reached conveniently in a controlled experiment. The pulse is stretched in 100 m of fiber. After a preamplifier stage, the repetition rate can be cut from 40 MHz to 3 MHz with an acousto-optic modulator (AOM). The amplified and compressed pulses are characterized with an interferometric autocorrelator. Fig. 2
shows that the autocorrelation obscures the dramatic variation in the pulse shape and asymmetry as ΦNL
is varied. However, the variation in the pulse duration is readily observable.
Fig. 4. Schematic of the experimental setup.
The variation of the output pulse with ΦNL
is shown in Fig. 5
. The lowest pulse energy was limited by the desire to record the autocorrelation with adequate signal-to-noise ratio. With the AOM turned off, the pulse energy was 3 nJ, which produced ΦNL=0.4π
. (The ΦNL
obtained with a given pulse energy is slightly larger than that of the simulations, which neglect ~2 m of SMF that couple light into and out of the AOM.) The autocorrelation (Fig. 5
(a)) implies a pulse duration of 240 fs. A similar result is obtained with lower signal-to-noise ratio when the AOM is turned on to reduce the repetition rate. At 3 MHz, the pulse is amplified to 15 nJ, and ΦNL=1.8π
. The pulse duration (Fig. 5
(b)) decreases to 180 fs, and the secondary structure that arises from TOD diminishes. When the pulse energy increases to 17 nJ (ΦNL=2.1π
), the pulse duration increases again to 195 fs (Fig. 5
(c)) and the secondary structure begins to increase as well. The experimental trend agrees qualitatively and semi-quantitatively with the numerical simulations, and exhibits a clear minimum in the pulse duration near the expected optimum value of ΦNL
. As expected, the power spectrum broadens slightly at the highest pulse energies.
Fig. 5. Autocorrelations of amplified and dechirped pulses with (a) ΦNL=0.4π, (b) ΦNL=1.8π, and (c) ΦNL=2.1π. The corresponding pulse duration is shown in each panel.
Initial results obtained with the 400-m stretcher (Fig. 6
) demonstrate the scaling to higher energies. The pulse is stretched by a factor of 1200, to 180 ps. In linear propagation, the dechirped pulse is broadened to 450-fs duration. Amplification to nearly the microjoule level impresses a nonlinear phase shift of 10π to 15π on the pulse, and results in ~150-fs dechirped pulses. More careful and systematic experiments are in progress, and there are some quantitative discrepancies with the numerical results, but these experiments are clearly consistent with the expected compensation of TOD by SPM.
Fig. 6. Autocorrelations of amplified and dechirped pulses with (a) ΦNL=0.2π (8 nJ pulse energy), (b) ΦNL=15π (0.8 µJ pulse energy). The corresponding pulse duration is shown in each panel.
4. Discussion and conclusion
The demonstration that TOD and nonlinear phase shifts can compensate each other seems likely to change the way future short-pulse fiber amplifiers are designed. The agreement between the experimental results and the numerical simulations allows us to extrapolate to higher pulse energies than employed here. There is interest in construction of fiber amplifiers for millijoule-energy pulses, but nonlinearity presents a major challenge in the design of such devices. In the approach presented here, nonlinearity is desirable
. Amplifiers can be designed to operate with a certain nonlinear phase shift, instead of attempting to avoid nonlinearity entirely. The stretcher and compressor can be designed to operate with a certain TOD, instead of attempting to minimize it. Work is currently in progress to design a fiber amplifier for 1-mJ and 100-fs pulses, with multimode fiber [11
11. M. E. Fermann, A. Galvanauskas, and M. Hofer, “Ultrafast pulse sources based on multi-mode optical fibers,” Appl. Phys. B 70, 1, (2000). [CrossRef]
] or large-mode-area photonic-crystal fiber [10
10. J. Limpert, A. Liem, T. Schreiber, M. Reich, H. Zellmer, and A. Tunnerman
, “High-performance ultrafast fiber laser systems,” in Fiber Lasers: Technology, Systems, and Applications, ed.
L. N. Durvasula, Proceedings of SPIEvol. 5335 (SPIE, Bellingham, WA, 2004), pp. 245–252.
]. Additional effects such as gain-narrowing will have to be assessed as part of this design. Scaling to shorter pulses also seems to be possible. For some applications, the overwhelming practical benefits of a fiber stretcher would recommend its use if the resulting performance was equal to, or even slightly worse than, that obtained with a grating stretcher.
This report has focused on the mismatched dispersions that arise with a fiber stretcher and grating compressor. However, we emphasize that the compensation of nonlinearity by TOD is general. As Fig. 3
shows, the highest relative peak power is obtained by avoiding nonlinearity. This has led prior workers to design matched stretcher and compressor with diffraction gratings, in which case the pulse energy is limited by nonlinearity in the amplifier. One can argue that further stretching of the pulse is possible, but this approach encounters the serious practical limitations mentioned in the Introduction for stretching ratios around 10,000 or pulse durations around 1 ns. The pulse energy and peak power might then be increased by allowing the pulse to accumulate a nonlinear phase shift and designing a mismatched pair of stretcher and compressor gratings to compensate the nonlinearity.
Even considering only fiber-based devices, the compensation of nonlinearity by TOD can be implemented in various ways. Fiber Bragg gratings offer an appealing combination of compactness and dispersion control and efforts exist to demonstrate customized chirping of the grating period to compensate higher orders of dispersion. Disadvantages of FBGs include an inherent tradeoff between bandwidth and dispersion, the requirement of expensive (often custom) fabrications, and limited adjustability in dispersion for a given design. These properties contrast with the simplicity and availability of SMF. Custom SMF can also be designed to optimize the ratio of TOD to GVD, for example. Of course, the compensation of nonlinearity by TOD described here can be implemented with FBGs: the FBG would be designed to provide a certain amount of TOD, depending on the desired pulse energy. Similar arguments can be made for dechirping the pulse with photonic-bandgap fiber. We have not investigated the effects of nonlinear phase accumulation in a fiber stretcher. Thus, the results presented above are directly relevant to sub-nanojoule oscillator pulses. If nonlinearity in the stretcher turns out to be detrimental (and it is not clear that it will be), with high-energy oscillators it may be more sensible to stretch and compress with mismatched grating pairs as described above.
To summarize, we have demonstrated that nonlinearity and TOD can compensate each other to a large degree in fiber CPA systems. For a given magnitude of TOD, there exists an optimum value of the nonlinear phase shift, for which the output pulse duration is minimized. The output pulse can be significantly shorter and cleaner than in the absence of nonlinearity, and the peak power is correspondingly increased. Initial experiments with nanojoule- and microjoule-energy femtosecond pulses clearly exhibit the main features and trends of the theoretical predictions. Extension of this approach to higher pulse energies appears to be straightforward. Compensation of dispersion beyond third-order by nonlinear phase shifts may also be expected. Finally, the concept described here can be combined with other devices such as fiber Bragg gratings or photonic-bandgap fibers, and we expect it to find significant use in future high-energy short-pulse fiber amplifiers.