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Definition: pulses with a parabolic intensity profile
A parabolic pulse is an optical pulse (usually an ultrashort pulse) that has a temporal intensity profile with a parabolic shape – not only near the pulse center, but also well towards the wings up to the point where the intensity goes to zero. In the ideal case, the pulse would have an intensity profile according to
Figure 1 shows an approximately parabolic pulse, as it can result from pulse propagation in a fiber amplifier under certain circumstances. Parabolic pulses have a number of remarkable properties, which have turned out to be particularly important in the context of fiber amplifiers for generating high-energy ultrashort pulses.
Figure 1: Temporal profile of a nearly parabolic pulse, such as can be generated in a fiber amplifier fed with Gaussian pulses. The data are from a simulation with the software RP ProPulse.
Up-chirped parabolic pulses can be generated in a fiber amplifier where the amplified pulses not only experience optical gain, but also the Kerr nonlinearity and normal chromatic dispersion. In this situation, a chirped parabolic pulse is the asymptotic solution for long propagation lengths [3, 8]. After a sufficiently long propagation length, associated with a sufficiently large amplification factor, the shape of the resulting pulses no longer depends on the shape or duration of the seed pulses, and the peak power and duration after a long propagation length depends only on the seed pulse energy. In that regime, the time–bandwidth product of the pulses is large and grows further in proportion to E2/3.
With increasing propagation length, the peak power rises in proportion to E2/3 and the pulse duration in proportion to E1/3, where E is the pulse energy. This is a case of self-similar pulse propagation, since the parabolic pulse shape, although not the pulse duration, is preserved. Therefore, such pulses are sometimes called similaritons. The pulses have a very nearly linear up-chirp, i.e., a linearly rising instantaneous frequency. The chirp rate (in THz/ps) approaches a constant value, which is determined only by the fiber dispersion and gain per unit length. The spectral bandwidth rises in proportion to the pulse duration, i.e., in proportion to E1/3.
Figure 2: Evolution of the pulse duration in a parabolic fiber amplifier. The solid curves show the pulse duration at the fiber output (blue) and after a compressor with optimized GDD (red). The dashed curve shows the asymptotic values, which in this case are reached after ∼ 20 dB of amplification.
Depending on the input pulse parameters, in particular on the pulse duration and energy, the parabolic evolution may set in only after some amount of linear amplification. Therefore, the pulse duration and bandwidth do not necessarily rise by orders of magnitude even if the pulse energy is increased by several orders of magnitude. This is shown in Figure 2, where the seed pulses had 10 pJ energy and 500 fs duration. Initially, the amplification is nearly linear, but then both the bandwidth and pulse duration begin to rise significantly. Due to the increased bandwidth, dispersively compressed pulses can be significantly shorter than the initial pulses.
Attractions of Parabolic Pulse Amplification
The remarkable properties of parabolic pulses lead to a number of technologically attractive features of fiber amplifiers for ultrashort pulses, when they are designed as “parabolic amplifiers”:
- Wave breaking is safely avoided. The pulses generated have no significant side lobes in the temporal or spectral shape.
- Such an amplifier does not require a seed source with carefully chosen pulse parameters, as these are not important provided that the amplification factor is large enough. The seed pulses may be significantly longer than the output pulses. They should not be too long, however, as otherwise the asymptotic solution is only slowly approached.
- The fiber parameters (chromatic dispersion, nonlinearity and gain) are also not critical. (This is remarkably different to schemes based on soliton effects.)
- As the amplified pulses have a nearly linear chirp, it is relatively easy to obtain nearly transform-limited pulses by dispersive pulse compression. However, higher-order chromatic dispersion of the compressor should not be excessive.
- The chirp during amplification can be an important advantage, because the amplification of transform-limited pulses could lead to excessive nonlinear phase shifts. The situation is therefore similar to that of chirped-pulse amplification, but with a lower amount of pulse stretching.
Limitations arise from the following effects:
- The limited gain bandwidth becomes important for very high amplification factors.
- If the fiber parameters including the gain and a spectral bandwidth of the output are given, the pulse energy of a parabolic pulse is fixed. Its value depends on the third power of the spectral bandwidth. It is thus not possible to obtain a high output pulse energy when the optical bandwidth is limited to a small value.
- Higher-order chromatic dispersion of the fiber and a compressor should not be too strong.
- The amplified pulse generates Raman gain at some longer wavelengths, so that eventually a strong Stokes pulse can be formed. Once this takes over a significant fraction of the energy, the original pulse can be severely distorted.
- Amplifier noise may also be relevant under certain circumstances.
To find an optimized design of a parabolic amplifier, it is advisable to study its behavior with numerical pulse propagation modeling, including the potentially disturbing effects mentioned above.
The principle of parabolic pulse amplification has also been applied to mode-locked fiber lasers [6] (similariton fiber lasers). Here, the spectral width of the circulating pulse is strongly oscillating: it increases during amplification but is reset in every round trip by some optical filter. This approach makes it possible to reach significantly higher pulse energies with potentially cheap fiber laser sources.
Bibliography
| [1] | D. Anderson et al., “Wave-breaking-free pulses in nonlinear-optical fibers”, J. Opt. Soc. Am. B 10 (7), 1185 (1993) |
| [2] | K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers”, Opt. Lett. 21 (1), 68 (1996) |
| [3] | M. E. Fermann et al., “Self-similar propagation and amplification of parabolic pulses in optical fibers”, Phys. Rev. Lett. 84 (26), 6010 (2000) |
| [4] | V. I. Kruglov et al., “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers”, Opt. Lett. 25 (24), 1753 (2000) |
| [5] | T. Hirooka and M. Nakazawa, “Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion”, Opt. Express 29 (5), 498 (2004) |
| [6] | F. Ö. Ilday et al., “Self-similar evolution of parabolic pulses in a laser”, Phys. Rev. Lett. 92 (21), 213902 (2004) |
| [7] | C. K. Nielsen et al., “Self-starting self-similar all-polarization maintaining Yb-doped fiber laser”, Opt. Express 13 (23), 9346 (2005) |
| [8] | V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters”, J. Opt. Soc. Am. B 23 (12), 2541 (2006) |
| [9] | T. Schreiber et al., “Microjoule-level all-polarization-maintaining femtosecond fiber source”, Opt. Lett. 31 (5), 574 (2006) |
| [10] | A. Ruehl et al., “Dynamics of parabolic pulses in an ultrafast fiber laser”, Opt. Lett. 31 (18), 2734 (2006) |
| [11] | J. M. Dudley et al., “Self-similarity and scaling phenomena in nonlinear ultrafast optics”, Nat. Phys. 3, 597 (2007) |
| [12] | D. N. Papadopoulos et al., “Generation of 63 fs 4.1 MW peak power pulses from a parabolic fiber amplifier operated beyond the gain bandwidth limit”, Opt. Lett. 23 (17), 2520 (2007) |
See also: fiber amplifiers, nonlinearities, chirped-pulse amplification, pulse compression, pulse propagation modeling
Categories: nonlinear optics, pulses
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