Self-Similarity in Ultrafast Nonlinear Optics

John M Dudley Département d’Optique P. M. Duffieux, Institut FEMTO-ST, CNRS UMR 6174 Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France

Christophe Finot, David J. Richardson Optoelectronics Research Centre (ORC), University of Southampton Southampton SO171BJ, United Kingdom

Guy Millot Laboratoire de Physique de l’Université de Bourgogne (LPUB) 9 avenue A. Savary, 21078 Dijon, France

ABSTRACT

Recent research in nonlinear optics has led to the discovery of a new class of ultrashort pulse that is generated in optical fibre amplifiers as a result of the self-similar amplification of an arbitrary injected input pulse. Such pulses have been termed “similaritons” and represent a universal form of asymptotic attractor towards which any input pulse of given energy will converge, irrespective of its specific intensity profile. Self-similar fibre amplifiers can be straightforwardly constructed using readily-available components, and provide convenient testbeds with which to study numerous fundamental aspects of dynamical self-similarity. Our objective here is to provide a succinct overview of current progress in this field. In addition to their intrinsic scientific interest, self-similar amplifiers are of much practical significance, and we also review their application in fields such as high power ultrashort pulse source-development, pulse synthesis and all-optical regeneration.

INTRODUCTION Many natural phenomena exhibit the remarkable property of self-similarity, reproducing themselves on different temporal and/or spatial scales. Although similarity and scaling laws in physics have been studied since the time of Galileo [1], their application in the modern era dates to the early years of the twentieth century, with an influential correspondence in Nature initiated by Lord Rayleigh [2] and the development of formal dimensional analysis by Buckingham [3]. The fundamental premise of dimensional analysis is that physical laws cannot be dependent on any particular choice of units, and that it must be possible to express them using dimensionless parameters.

Dimensional analysis is

particularly powerful in reducing the number of degrees of freedom needed to describe the physics of a particular system, and in providing a systematic procedure to derive precise scaling relations between the key physical parameters involved. It thus provides a general technique for analyzing phenomena across very different fields of physics and, indeed, even the brief report by Rayleigh considered an impressively diverse range of examples ranging from optical resolving power to the acoustic properties of the Æolian harp. The search for scaling laws forms an essential part of mathematical physics, and selfsimilarity has become a key concept in interdisciplinary research revealing the presence of internal structure and symmetry in many different systems [4]. The basic concept of similar triangles is of course very well known, but more sophisticated examples of geometrical selfsimilarity are also widespread and can be found in many different settings ranging from natural branching patterns and coastlines [5], to the nodal properties of complex networks such as the world-wide web [6]. As well as these examples involving spatial geometry, self-similarity also occurs in many dynamical problems as a natural stage in the temporal evolution of a system from a particular set of initial conditions. One of the most famous illustrations of this type concerns

the evolution of the radius of a blast wave of a nuclear explosion, which was first analysed by the British physicist G. I. Taylor in the 1940s [7]. Although a nuclear weapon is a very complex device, Taylor’s insight was to realize that the huge energy release from the explosion would result in the formation of a spherical shock wave whose self-similar expansion (see Figure 1) could be described in terms of only four dimensional physical quantities: the time-dependent shock wave radius R(t), the time after the explosion t, the ambient air density ρ and the energy released in the explosion E. The application of dimensional analysis to this problem seeks to combine these four quantities to form dimensionless “similarity parameters,” and it is easy to see how they can be combined into one such parameter: θ = ρR5/Et2. Once this relationship is derived, it follows immediately that the blast wave radius must expand according to the simple scaling law R(t) = θ1/5 (Et2/ρ)1/5, where we see how the similarity variable θ here plays the role of a proportionality constant.

In fact, numerical computation yields a specific value for

θ (approximately unity) and Taylor himself was able to use declassified images of the Trinity explosion in New Mexico in 1945 to confirm this scaling hypothesis and calculate the energy of the blast [8].

SELF SIMILAR DYNAMICS The blast-wave example is one where simple dimensional analysis works particularly well, but more sophisticated techniques also exist to determine self-similar solutions for more complex systems.

Such formal similarity techniques extend the toolbox available to

mathematical physicists, and are of particular importance in analysing nonlinear problems described by partial differential equations – well known to be notoriously difficult to solve exactly. For cases such as these, the general approach is to reduce the degrees of freedom of the system through a reformulation of the problem in terms of similarity variables (where

they exist) so that the original problem of solving partial differential equations can be recast into a simplified problem involving ordinary differential equations [9, 10]. The application of similarity techniques to physical problems is now very common, and important results have been obtained in solving many important problems in hydrodynamics, mechanics and plasma physics [4]. Somewhat surprisingly, the study of selfsimilarity in modern optics has perhaps not been as widespread as in other fields of physics, but a number of important results have nonetheless been obtained. For example, previous important research includes studies of the nonlinear temporal dynamics of a range of different lasers and optoelectronic devices [11-13], nonlinear self-action and collapse processes [14-18], stimulated Raman scattering [19], the excitation of fractal structures in solitonsupporting systems [20-21], and spatial fractal pattern formation [22-25]. Other studies with more of a materials emphasis have demonstrated self-similarity characteristics in the growth of Hill gratings [26] and the evolution of self-written waveguides [27, 28]. A particular area of optics research where self-similar dynamical effects have attracted much recent interest has been in the study of nonlinear ultrashort pulse propagation in optical fiber amplifiers.

Fiber amplifiers are key components in optical

telecommunications systems and also find applications in high power optical source development, but they are generally configured to operate in an amplification regime where nonlinear effects are negligible. However, recent results have demonstrated a fundamentally new operating regime of fiber amplifiers where nonlinear propagation effects are in fact exploited to generate a particular class of ultrashort parabolic pulse that evolves selfsimilarity as it is amplified [29-32]. As is often the case, optical systems provide ideal testbeds with which to study physical processes of widespread interest and the study of dynamical self-similarity in optical pulse propagation has now developed into an important area of research in ultrafast photonics.

OPTICAL “SIMILARITONS” - CHARACTERISTICS AND SCALING The propagation of high power ultrashort pulses in optical fibers is well-known to be associated with distortions and break-up effects due to the interaction of the fiber nonlinearity and dispersion [33]. Although in the anomalous dispersion regime of a fiber these effects can balance and give rise to soliton propagation [34, 35], fundamental soliton stability exists at only one particular power level, and propagation at higher power excites higher-order solitons which are sensitive to perturbation and break-up through soliton fission [36, 37]. Propagation in the normal dispersion regime at high-power is also subject to instability through the appearance of optical wave-breaking on the temporal pulse envelope [38]. These distortion effects limit the energy of ultrashort pulses able to propagate in an optical material, and are particularly detrimental for the development of high-gain optical fiber amplifiers. In 1993, however, it was demonstrated by Anderson et al. that high power pulse propagation in the normal dispersion regime was not inevitably associated with the effects of optical wave-breaking and in fact it could be completely avoided for a particular class of pulse possessing a parabolic intensity profile and a linear frequency chirp [39]. This crucial physical insight was followed up in 1996 by Tamura and Nakazawa [40], who used numerical simulations to show that ultrashort pulses injected into a normal dispersion optical fiber amplifier appeared to naturally evolve toward the parabolic pulse profile with amplification and, moreover, retained their parabolic shape even as they continued to be amplified to high power. Tamura and Nakazawa attempted to verify these results experimentally in an erbium-doped fiber amplifier, but the available pulse diagnostic techniques did not allow the generation of parabolic pulses to be conclusively confirmed. The unambiguous experimental observation of parabolic pulse generation was first reported by Fermann et al. in 2000 in an Ytterbium-doped fiber amplifier with normal dispersion at 1.06 μm [29]. These measurements were compelling because the use of the

ultrashort pulse measurement technique of frequency-resolved optical gating (FROG) invented during the 1990’s provided experimental access to the complex field of the ultrashort parabolic pulse profile [41]. As well as representing a major experimental advance, the results in Fermann et al. also applied theoretical analysis based on symmetry reduction to the nonlinear Schrödinger equation (NLSE) with gain, formally demonstrating the selfsimilar nature of the generated parabolic pulses. A major result of the theoretical analysis was to demonstrate that the self-similar parabolic pulse is a rigorously asymptotic solution to the NLSE with gain, representing a type of nonlinear “attractor” towards which any arbitrarily-shaped input pulse of given energy would converge with sufficient distance. By analogy with the well known solitary-wave behaviour of solitons, these self-similar parabolic pulses have been termed “similaritons” [30], and it is interesting to note that solitons themselves can also be interpreted as an example of self-similarity [4]. Further application of similarity techniques also yielded a quantitative description of the important intermediate asymptotic regime, where the fine structure due to the initial conditions has disappeared yet the system has not yet attained its ultimate asymptotic state. This was a particularly significant advance from a fundamental viewpoint as it demonstrated the presence of similarity characteristics on different scales in this system. The formal application of similarity analysis to propagation in an optical fiber amplifier yields closed form expressions of particular simplicity for the self-similar parabolic pulse characteristics [31]. For completeness, we summarise these results in Fig. 2. The results are obtained assuming that a fibre amplifier can be described by the NLSE with gain expressed in the following (dimensional) form: i

∂Ψ β 2 ∂ 2 Ψ g = − γ | Ψ |2 Ψ + i Ψ 2 ∂z 2 ∂T 2

(1)

Here, Ψ ( z , T ) [W1/2] is the slowly varying pulse envelope in a frame comoving at the envelope group velocity, β2 [ps2m-1] and γ [W-1m-1] are the fiber dispersion and Kerr nonlinear coefficients respectively, and g [m-1] is the distributed gain coefficient of the amplifier. The left panel in Fig. 2 (a) shows generic parabolic pulse characteristics plotted on both linear and logarithmic scales, with the former illustrating the parabolic nature of the central core, and the latter showing the presence of low amplitude wings. Writing the field as

Ψ ( z, T ) = A( z, T ) exp(iΦ( z, T )) , the right panel Fig 2 (b) gives the corresponding closed-form analytic expressions in terms of the amplitude and phase A and Φ respectively.

The

instantaneous frequency chirp across the pulse envelope is Ω( z , T ) = − dΦ ( z , T ) / dT . The asymptotic characteristics correspond to a strictly parabolic pulse core with compact support.

Although presented here for the case of a constant gain, these

characteristics are, in fact, also observed under more arbitrary conditions when the amplifier gain varies longitudinally along the amplifier length [30, 42-43]. The asymptotic nature of the solution is reflected in the fact that the amplitude and width scaling (in both the time and frequency domains) depends only on the amplifier parameters and the input pulse energy, and are completely independent of the input pulse shape. We also note that the pulse chirp is completely independent of propagation distance, a property that has particular significance for optical compressor design. In the intermediate asymptotic regime, the reshaping of any arbitrary input pulse generates exponentially decaying low amplitude wings, but the scaling of the intermediate characteristics with distance is more complex, and does depend on the exact input pulse shape used. The scaling constants in this case must be determined numerically. Figure 3 illustrates explicitly the attractive nature of the asymptotic parabolic pulse solution. Here we show the results of numerical simulations calculating the evolution of input pulses with durations varying over the range 100 fs–1 ps in a 5 m amplifier with

β2 = + 25 ps2km-1, γ = 5 W-1km-1 and g = 1.92 m-1. The pulse energy in each case is identical at 100 pJ. The convergence to the asymptotic solution with propagation can be conveniently examined in a phase-space representation in terms of the ratios of the root-mean square (rms) temporal and spectral widths relative to the widths expected for an asymptotic similariton pulse at the same propagation distance (see caption for definitions).

Using such a

representation, we clearly see that, although different input pulses do follow different evolution trajectories, they nonetheless are all attracted to the asymptotic parabolic pulse solution with sufficient propagation. In contrast to other forms of nonlinear propagation in fibers, a particular feature of the self-similar regime is that both the temporal and spectral width increase exponentially and monotonically with propagation.

This is illustrated explicitly in Figure 4 which uses

numerical simulations to compare the evolution of a 300 fs pulses in fiber amplifiers with (a) normal and (b) anomalous dispersion. The simulation parameters are provided in the caption, with the values used corresponding to realistic current fiber amplifier technology. We also note that the simulations are based on the NLSE with gain Eq. (1) with the addition of stimulated Raman scattering in the fused silica fiber host material to model realistic perturbative effects [33]. The figure clearly shows the difference between amplification with normal and anomalous dispersion. With normal dispersion and self-similar dynamics, the amplification of the pulse energy is associated with the simultaneous increase in both the temporal and spectral widths of the amplified pulses, and the absence of any wave-breaking or pulse distortion. In contrast, for an amplifier with the same gain yet with anomalous dispersion, as the pulse energy increases with amplification, higher-soliton instabilities become apparent and the pulse breaks up from the effect of soliton fission. In this case both the pulse temporal and spectral characteristics exhibit a very complex structure and the output characteristics

would be undesirable for many applications. Note that the soliton fission observed in the anomalous dispersion regime case is induced by stimulated Raman scattering in the fiber, but this has negligible effect on the evolution in the normal dispersion regime. Indeed, it is even possible to exploit Raman scattering in fibers as a gain mechanism to generate parabolic pulses [44].

EXPERIMENTAL ASYMPTOTICS We consider the practical application of self-similar amplifiers in the next section, but at this stage we present a brief review of recent experiments where we have been able to examine the fundamental self-similar scaling properties using fiber amplifiers operating around the telecommunications wavelength of 1550 nm. Particular aims of these experiments were to develop convenient self-similar amplifier configurations using only readily-available and relatively inexpensive fibre components, and to use a high dynamic range FROG setup for detailed pulse characterization. In one experiment using a self-similar amplifier based on Raman gain in fused silica, the asymptotic nature of the generated parabolic pulses was explicitly verified by injecting a range of different input pulses, and examining the output pulse characteristics in each case [44-46]. The output pulse characteristics were found to be invariant with input pulse duration and profile, being determined only by the amplifier parameters and input pulse energy. In another experiment, the evolution towards the asymptotic regime was explicitly examined in an Erbium fibre amplifier [47]. Figure 5(a) shows simulations illustrating the expected amplification and reshaping of a 1.4 ps hyperbolic secant input pulse to a parabolic similariton for this system when operated at a gain of 13.6 dB. Corresponding experiments were then performed by constructing such an amplifier and cutting it back in 50 cm segments in order to directly measure the pulse evolution. A feature of these results of particular

experimental interest is the characterization of the pulse electric field over an intensity dynamic range of 40 dB, and with the possibility to resolve sub-100 fs structure over a 20 ps timebase.

The measurements are amongst some of the most demanding ever made in

ultrafast optics, and the results obtained have allowed the quantitative comparison of measured similarity characteristics with numerical modelling of the self-similar evolution process. Figure 5(b) shows the remarkable agreement between experiments and simulations obtained.

At the intermediate distance of 7 m, Figure 5(c) compares experiment and

simulation on a logarithmic scale to explicitly show the presence of intermediate asymptotic wings about the central parabolic pulse core. Additional experiments have shown how the relative energy in the wings decreases as the pulse enters the asymptotic regime, and represent what is, to our knowledge, the first experimental observation of intermediate asymptotic self-similar dynamics in optics.

PRACTICAL SELF-SIMILAR AMPLIFIERS In parallel with the fundamentally-oriented studies described above, the remarkable scaling properties of parabolic pulses have been applied to the development of a new generation of optical fiber amplifier. From a technological viewpoint, self-similar amplifiers possess a number of very attractive features. In common with the well-known technique of chirped pulse amplification (CPA), catastrophic pulse break up due to excessive nonlinear phase shifts is avoided [48].

However, in contrast to CPA where the aim is to avoid

nonlinearity by dispersive pre-stretching before amplification, a self-similar amplifier actively exploits nonlinearity, allowing for the very attractive possibility of obtaining output pulses after recompression that are actually shorter than the initial input pulse. The existence of analytic design criteria for self-similar amplifiers makes it straightforward to tailor amplifier design to a very wide range of input pulses and amplifier

types. Moreover, the fact that the asymptotic pulse duration and chirp depend only on the input pulse energy makes the amplification process insensitive to a wide class of seed pulse instabilities. As a result, practical self-similar amplifiers have been demonstrated using Ytterbium, Erbium and Raman gain media, seed pulses in the range 180 fs–10 ps, fiber lengths in the range 1.2 m–5.3 km and gains varying from 14–32 dB [29, 49-54]. The possibility to obtain amplified pulse energies exceeding 1 uJ in an environmentally stable and polarization-maintaining configuration has been a significant recent demonstration [55]. Another important recent result has been the use of passively-modelocked vertical-externalcavity surface emitting semiconductor lasers (VECSELs) as the primary master oscillator source of seed pulses [56]. The fact that the output pulse chirp depends only on the amplifier gain and dispersion considerably simplifies the post-compressor design, and high-quality compressed pulses in the 100 fs regime with megawatt peak powers have been obtained [51, 55]. An exciting development has been the application of the novel dispersion properties of photonic bandgap optical fiber [57] to replace the use of bulk gratings in the compression stage, allowing the realisation of an all-fiber format source of ~200 fs pulses around the technologicallyimportant wavelength of 1550 nm [47]. Fig. 6 illustrates results obtained with such a system, showing (a) an electron-micrograph of a typical bandgap fiber used and (b) the high quality compressed pulses obtained. An even more recent experiment has developed this system further, combining self-similar dynamics with higher-order soliton propagation to develop a hybrid similariton-soliton amplifier-compressor system yielding pulses as short as 20 fs [58]. At the operating wavelength of 1550 nm, this pulse duration represents only 4 optical cycles, and experimental FROG characterisation allows the reconstruction of the electric field to explicitly illustrate their few-cycle nature as shown in Fig. 6(c).

THE “SIMILARITON LASER” The development of any new optical amplifier technology invariably suggests application in a laser, and self-similar dynamics are no exception. The combination of selfsimilar propagation in normal dispersion fiber with optical feedback and loss has been studied both theoretically and experimentally in the context of fiber laser systems, promising to lead to a new generation of fiber lasers that overcome existing power limitations of soliton modelocking [59-64]. Although self-similar evolution naturally presents many advantages in the design of fiber lasers, the possibility to observe self-similar dynamics in solid-state modelocked lasers such as Ti:Sapphire has also been considered [65]. Significantly, this has motivated exciting work on the more general question of exploiting normal GVD propagation dynamics to extend solid state oscillator systems to the microjoule regime, and on more general considerations of pulse-shaping mechanisms in this regime [66-68].

GHz SIMILARITONS In parallel with the research described above where self-similar amplification has been exploited to generate ultrashort pulses at high power levels, extensive research has also considered their application at the much lower powers associated with telecommunications systems. For this purpose, the interest is not so much in the power scalability of a similariton amplifier, but in the use of the unique asymptotic properties of parabolic pulses for the manipulation and shaping of ultrafast pulses generated at GHz repetition rates. The crucial first step here was the generation of GHz-repetition rate parabolic pulse trains around 1550 nm, and the demonstration of their potential for multi-wavelength source development [53]. This research has since opened the doors to exploiting the remarkable properties of parabolic

pulses for the ultrafast signal processing applications that are essential for an all-optical network. For example, the invariance of the output profile to input pulse fluctuations is of great interest for pulse shaping and pulse synthesis applications, enabling highly stable output that can be applied to applications such as spectral slicing [69, 70]. Applications in the domain of regeneration and retiming are also promising and can benefit from the perfectly linear chirp induced through self-phase or cross-phase modulation [71-73].

WIDER IMPLICATIONS As will be clear from the above discussion, self-similar propagation effects in optical amplifiers have already found important applications in a wide range of photonic technologies. In addition, this work has also highlighted to the nonlinear optics community the power of self-similarity techniques in yielding analytic solutions to nonlinear propagation equations, motivating wider research in a number of other areas aside from amplifier technologies.

Notable results that have been obtained include the study of self-similar

propagation and compression in passive fibres with longitudinally varying parameters, and the establishment of important connections with soliton propagation [74-80].

Recent

experiments have also reported the generation of parabolic pulses in dispersion decreasing fiber [81] or have studied other novel approaches to passive parabolic pulse generation and shaping dynamics [82-83].

These represent developments which might be expected to

considerably simplify applications in telecommunications where power consumption is a paramount consideration. Other work of more general interest has considered self-similar effects in the spatial propagation of beams and guided waves [84, 85] and the interaction of self-similar dynamics and gain-limiting to induce novel solitary wave propagation [86]. For amplifiers with low dispersion, analogies with the nonlinear Whitham equation of hydrodynamics has allowed the development of specific analytic models describing the

intermediate self-similar dynamics [87]. In this regard, related studies of interest also include the study of a modified Gross-Pitaevskii gain equation to reveal the presence of self-similar behaviour in the evolution of Bose-Einstein condensate dynamics [88, 89].

CONCLUSIONS It is clear that nonlinear optical fibre amplifiers that exploit self-similar evolution dynamics have developed into a mature alternative to other ultrafast pulse generation and shaping techniques, complementing other approaches to high power pulse generation such as CPA. From a more general perspective, it is also interesting to remark that current research in many areas of nonlinear photonics increasingly relies on sophisticated simulation and modelling, and whilst such numerical treatments are indispensable, the underlying physics is sometimes difficult to readily interpret. In our opinion, one of the most attractive features of the extensive recent interest in self-similarity is that it has reminded the nonlinear optics community of an extensive array of mathematical tools that can be used to find analytic solutions to complex dynamical problems. As modelocked lasers generate ever shorter and more intense pulses, such nonlinear propagation problems will become of increasing importance, and a complete understanding of the propagation dynamics will be required both for source optimisation and the many potential multidisciplinary applications. To this end, we believe that the search for universal patterns in these phenomena, as pioneered centuries ago by Galileo, will continue to form a very profitable direction of research.

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ACKNOWLEDGEMENTS Our own contributions to this field have benefited from valuable collaborations and discussions with numerous colleagues and friends. We extend thanks to Cyril Billet, Neil Broderick, Martin Fermann, John Harvey, Nicolas Joly, Bertrand Kibler, Jonathan Knight, Vladimir Kruglov, Rainer Leonhardt, Anna Peacock, Periklis Petropoulos, and Benn Thomsen. JMD extends particular thanks to Moti Segev for sparking his interest in the wider aspects of optical self-similarity.

COMPETING FINANCIAL INTERESTS

The authors declare that they have no competing financial interests.

FIGURES

Fig. 1: Self-similar dynamical evolution is ubiquitous in physics. Some of the most important work during the 20th century involved the application of similarity techniques to study blastwave expansion during nuclear explosions. See for example, G. I. Taylor: Proc. Royal Soc. London A 201 175-196 (1950).

(a)

Asymptotic Solution |T| ≤ Tp(z)

(b)

Ψ ( z, T ) = A( z, T ) exp(iΦ( z, T )) 2

T ⎛g ⎞ A( z, T ) = A0 exp⎜ z ⎟ 1 − 2 Tp ( z ) ⎝3 ⎠ 3γA02 g 2 ⎛2 ⎞ Φ( z, T ) = ϕ0 + exp ⎜ gz ⎟ − T 2g ⎝ 3 ⎠ 6β 2

A0 =

1 ⎛⎜ gU in ⎞⎟ 2 ⎜⎝ γβ 2 / 2 ⎟⎠

1 3

Tp ( z) =

6 γβ 2 / 2 ⎛g ⎞ A0 exp ⎜ z ⎟ g ⎝3 ⎠

Intermediate Asymptotic Solution |T| > Tp(z) Ψ ( z , T ) = Aw ( z , T ) exp(iΦ w ( z , T ))

Aw ( z, T ) =

⎛ T Aw0 ⎛g ⎞ exp⎜ z ⎟ exp⎜⎜ − Λ z z ⎝2 ⎠ ⎝

Φ w ( z, T ) = ϕ w0 +

Fig. 2.

β 2 Λ2 2z

−

⎞ ⎟ ⎟ ⎠

T2 2β 2 z

(a): Generic similariton characteristics. Top: parabolic intensity profile (left axis) and linear chirp (right axis). Bottom: the intensity profile on a logarithmic scale. (b): Analytic expressions for the amplitudes A(z,T) and Aw(z,T), and the phases Φ(z,T) and Φw(z,T) of the asymptotic and intermediate-asymptotic solutions respectively. Here φ0 and φw0 are arbitrary constants, and the intermediate asymptotic scaling parameters Λ and Aw0 are determined numerically for particular initial conditions.

Fig. 3.

Phase-space portrait of different evolution trajectories in a 5 m normaldispersion fiber amplifier for input pulses with durations (FWHM) in the range 100 fs-1 ps as indicated. With fixed input pulse energy, all trajectories are attracted to the asymptotic sink.

The phase space variables X and Y are

respectively calculated from the ratio of the evolving rms temporal and spectral widths relative to the corresponding rms widths of the expected asymptotic parabolic pulse solution at the same distance. Specifically, for a propagation distances

(1 / 5 )

z,

the

rms

temporal

width

is

given

by:

Δτasym(z)

=

γβ 2 / 2 (6 / g ) A0 exp( gz / 3) and the rms spectral width is given by:

(

2πΔνasym(z) = 1 / 5

)

2γ / β 2 A0 exp( gz / 3) .

(a) Self similar evolution in normal dispersion amplifier

(b) Soliton fission in anomalous dispersion amplifier

Fig. 4.

Simulated pulse evolution for propagation in fiber amplifiers with (a) normal and (b) anomalous dispersion using parameters typical of realistic Yb and Er doped gain media respectively. The figures are false-color representations of the pulse temporal intensity and power spectrum. The dispersion, nonlinearity and gain are (a) β2 = + 25 ps2km-1, γ = 5 W-1km-1, g = 1.92 m-1 (integrated gain 25 dB) and (b) β2 = - 25 ps2km-1, γ = 5 W-1km-1, g = 1.54 m-1(integrated gain 20 dB). The input pulse had a hyperbolic

secant profile and 12 pJ energy. These results illustrate both the fundamental properties and practical advantages of self-similar amplification.

Fig. 5

(a) and (b): Simulation and experimental results showing similariton evolution in a 9 m long Erbium fiber amplifier. (c) Shows a detail of the similariton intensity and chirp after 7 m propagation. The shading is used to distinguish the intermediate asymptotic wings from the parabolic similariton core. Results taken from Ref. 47.

(a)

Fig. 6

(b)

(c)

Few cycle pulse generation around 1550 nm through the combination of nonlinear similariton and soliton dynamics. Fig. 3(a) shows an electron micrograph of a typical hollow core photonic bandgap fiber suitable for linear chirp compensation around 1550 nm. Fig 3(b) shows high quality pulses of around 200 fs duration after using this fiber to compress the linearly chirped parabolic similaritons generated in an EDFA. Fig 3(c) shows how these pulses can be further nonlinearly compressed to the few cycle regime by exploiting nonlinear soliton compression to obtain 20 fs pulses, representing around 4 optical cycles at this wavelength. The figure shows the reconstructed electric field of these pulses obtained from experimental FROG measurements. Taken from Ref. 58.

Figure of General Interest: Galileo discussed aspects of scaling and dimensionality in Dialogues Concerning Two New Sciences. These pages are facsimiles taken from the English translation by Henry Crew and Alfonso de Salvio, MacMillan, New York (1914).