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Journal ArticleDOI

Self-similar propagation and amplification of parabolic pulses in optical fibers.

26 Jun 2000-Physical Review Letters (American Physical Society)-Vol. 84, Iss: 26, pp 6010-6013
TL;DR: Self-similarity analysis of the nonlinear Schrödinger equation with gain results in an exact asymptotic solution corresponding to a linearly chirped parabolic pulse which propagates self-similarly subject to simple scaling rules.
Abstract: Ultrashort pulse propagation in high gain optical fiber amplifiers with normal dispersion is studied by self-similarity analysis of the nonlinear Schrodinger equation with gain. An exact asymptotic solution is found, corresponding to a linearly chirped parabolic pulse which propagates self-similarly subject to simple scaling rules. The solution has been confirmed by numerical simulations and experiments studying propagation in a Yb-doped fiber amplifier. Additional experiments show that the pulses remain parabolic after propagation through standard single mode fiber with normal dispersion.

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Summary

  • Ultrashort pulse propagation in high gain optical fiber amplifiers with normal dispersion is studied by self-similarity analysis of the nonlinear Schrödinger equation with gain.
  • An exact asymptotic solution is found, corresponding to a linearly chirped parabolic pulse which propagates self-similarly subject to simple scaling rules.
  • The solution has been confirmed by numerical simulations and experiments studying propagation in a Yb-doped fiber amplifier.
  • Additional experiments show that the pulses remain parabolic after propagation through standard single mode fiber with normal dispersion.

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VOLUME
84, NUMBER 26 PHYSICAL REVIEW LETTERS 26J
UNE
2000
Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers
M.E. Fermann
IMRA America, 1044 Woodridge Avenue, Ann Arbor, Michigan 48105
V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey
Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand
(Received 22 February 2000)
Ultrashort pulse propagation in high gain optical fiber amplifiers with normal dispersion is studied by
self-similarity analysis of the nonlinear Schrödinger equation with gain. An exact asymptotic solution
is found, corresponding to a linearly chirped parabolic pulse which propagates self-similarly subject
to simple scaling rules. The solution has been confirmed by numerical simulations and experiments
studying propagation in a Yb-doped fiber amplifier. Additional experiments show that the pulses remain
parabolic after propagation through standard single mode fiber with normal dispersion.
PACS numbers: 42.81.Dp, 05.45.Yv, 42.65.Re
The establishment of self-similarity is a key element
in the understanding of many widely differing nonlinear
physical phenomena, including the propagation of thermal
waves in nuclear explosions, the formation of fractures in
elastic solids, and the scaling properties of turbulent flow
[1]. In particular, the presence of self-similarity can be
exploited to obtain asymptotic solutions to partial differ-
ential equations describing a physical system by using the
mathematical technique of symmetry reduction to reduce
the number of degrees of freedom [2]. Although the pow-
erful mathematical techniques associated with the analysis
of self-similar phenomena have been extensively applied
in certain areas of physics such as hydrodynamics, their
application in optics has not been widespread. However,
some important results have been obtained, with previous
theoretical studies considering asymptotic self-similar be-
havior in radial pattern formation [3], stimulated Raman
scattering [4], and in the nonlinear propagation of ultra-
short pulses with parabolic intensity profiles in optical
fibers with normal dispersion [5]. This latter case has
also been studied using numerical simulations, with results
suggesting that parabolic pulses are generated in the ampli-
fication of ultrashort pulses in nonlinear optical fiber am-
plifiers with normal dispersion [6]. To date, however, there
has been no experimental demonstration of self-similar
parabolic pulse propagation either in optical fibers or in
nonlinear optical amplifiers.
In this Letter we present results of calculations using
self-similarity methods [14] to analyze pulse propagation
in an optical fiber amplifier described by the nonlinear
Schrödinger equation (NLSE) with gain and normal
dispersion. These calculations show that parabolic pulses
are, in fact, exact asymptotic solutions of the NLSE with
gain, and propagate in the amplifier self-similarly subject
to exponential scaling of amplitude and temporal width.
In addition, the pulses possess a strictly linear chirp. Our
theoretical results are confirmed both by numerical simu-
lations and by experiments which have taken advantage of
the current availability of high gain optical fiber amplifiers
and of recent developments in methods of ultrashort pulse
characterization. In particular, the use of the pulse charac-
terization technique of frequency-resolved optical gating
(FROG) [7] has allowed us to measure the intensity and
chirp of parabolic pulses generated in a Yb-doped fiber
amplifier, and to compare these experimental results
directly with theoretical predictions. Additional experi-
ments have demonstrated that the pulses remain parabolic
in profile during propagation in normally dispersive
fiber, confirming the self-similar nature of propagation in
this regime.
These asymptotic self-similar parabolic pulses are of
fundamental interest since they represent a new class of so-
lution to the NLSE with gain and, from a practical point of
view, their linear chirp facilitates efficient pulse compres-
sion. In particular, the asymptotic pulse characteristics are
found to be determined only by the incident pulse energy
and the amplifier parameters, with the initial pulse shape
determining only the map toward this asymptotic solution.
In addition, all of the incident pulse energy contributes to
the output parabolic pulse. This is in contrast to the bet-
ter known soliton solutions of the NLSE in the absence
of gain [8], which require accurate control of the input
pulse energy and where a given input pulse develops into a
soliton of fixed amplitude shedding the remaining energy
into a continuum. “Parabolic fiber amplifiers” therefore
have potential wide-ranging applications in many areas
of current optical technology, allowing the generation of
well-defined linearly chirped output pulses from an optical
amplifier, even in the presence of input pulse distortions.
High power linearly chirped parabolic pulses can be effi-
ciently compressed and indeed, after compression of the
parabolic pulses generated in our experiments, we have
generated pulses of 80 kW peak power having 70 fs du-
ration. Parabolic amplifiers thus allow access to a conve-
nient fiber-based method of generating and transmitting
high-power optical pulses, rivaling soliton propagation,
stretched-pulse Gaussian pulse propagation [9], as well as
existing chirped pulse amplification systems.
6010 0031-90070084(26)6010(4)$15.00 © 2000 The American Physical Society

VOLUME
84, NUMBER 26 PHYSICAL REVIEW LETTERS 26J
UNE
2000
Our theoretical analysis considers the evolution of
pulses in an optical amplifier in the absence of gain
saturation and for pulses with spectral bandwidths less
than the amplifier bandwidth. In this case, propagation
can be described by the NLSE with gain [10]:
i
A
z
1
2
b
2
2
A
T
2
2gjAj
2
A 1 i
g
2
A . (1)
Here, Az, T is the slowly varying pulse envelope in a co-
moving frame, b
2
is the group velocity dispersion (GVD)
parameter, g is the nonlinearity parameter, and g is the dis-
tributed gain coefficient. In the absence of gain g 0,it
is possible to solve the NLSE exactly using the inverse
scattering method to obtain the well-known soliton so-
lutions [7], but, in the presence of gain, solutions usu-
ally require numerical simulations. However, the NLSE
with gain can also be analyzed using symmetry reduction,
with the solutions obtained in this way representing exact
self-similar solutions which appear in the asymptotic limit
z ! ` [14].
For the NLSE with gain in Eq. (1), this technique yields
an asymptotic self-similar solution in the limit z ! `, pro-
vided that g 0 and that gb
2
. 0. The solution is
Az, T A
0
z兲兵1 2 T T
0
z兲兴
2
1
2
expiwz, T 兲兴,
jTj # T
0
z , (2)
with Az, T 0 for jT j . T
0
z. This corresponds to a
compactly supported pulse with a parabolic intensity pro-
file, and a quadratic phase given by
wz, T w
0
1 3g2g
21
A
2
0
z 2 g6b
2
21
T
2
, (3)
where w
0
is an arbitrary constant. The corresponding
constant linear chirp is given by dvT 2≠wz, T 兲兾
T g3b
2
21
T. In the asymptotic regime, this pulse
propagates self-similarly, maintaining its parabolic shape
subject to the exponential scaling of its amplitude A
0
z
jAz,0j and effective width parameter T
0
z according to
A
0
z 0.5gE
IN
13
gb
2
2
216
expgz3 , (4)
T
0
z 3g
223
gb
2
2
13
E
13
IN
expgz3 , (5)
where E
IN
is the energy of the input pulse to the ampli-
fier. Significantly, these results imply that it is only the
energy of the initial pulse (and not its specific shape) which
determines the amplitude and width of the asymptotic para-
bolic pulse.
These theoretical predictions have been confirmed by
numerical simulation of the NLSE using the standard split-
step Fourier method [8]. Gaussian input pulses having a
range of pulse durations (FWHM) from 100 fs5 ps, but
fixed energy E
IN
12 pJ, were propagated in a 6 m long
fiber amplifier with realistic parameters corresponding to
Yb-doped fiber: g 5.8 3 10
23
W
21
m
21
, b
2
25 3
10
23
ps
2
m
21
, g 1.9 m
21
[11]. Figure 1(a) compares
the evolution of the amplitude of the propagating pulse
FIG. 1. (a) Simulation results showing the evolution of pulse
amplitude as a function of propagation distance for Gaussian
pulses of duration 100 fs5 ps, compared with calculated
asymptotic result (see legend). (b) Simulated output intensity
(circles, left axis) and chirp (circles, right axis) corresponding to
the 200 fs input pulse, compared with the asymptotic parabolic
pulse results (dotted lines).
obtained from simulations with the analytic prediction for
A
0
z given by Eq. (4). The evolution of the pulse in
the amplifier approaches the asymptotic limit in all cases.
Indeed, Fig. 1(b) shows the output pulse characteristics for
the input 200 fs pulse, illustrating the excellent agreement
(over 10 orders of magnitude) between the intensity and
chirp of the simulation output (circles) and the expected
asymptotic pulse profile from Eq. (2) (dashed line).
Additional simulations have been carried out to investi-
gate the dependence on fiber parameters and pulse initial
conditions in more detail. As the fiber gain is increased
for a given input pulse, the exponential growth of the pulse
amplitude and width is correspondingly increased in agree-
ment with Eqs. (4) and (5), and the parabolic asymptotic
limit is reached in a shorter propagation distance. Simu-
lations also show that for a fiber of fixed gain, while the
effect of intensity or phase modulation on an input pulse
modifies the length scale over which the evolution to the
asymptotic limit occurs, the asymptotic parabolic pulse so-
lution is nonetheless reached in all cases after sufficient
propagation distance. In this context, we also note that,
although at jTj T
0
z the solution in Eq. (2) has infinite
slope, this is the case only in the asymptotic limit. At in-
termediate propagation distances, simulations and analysis
predict low amplitude wings on the parabolic pulse which
decay exponentially as a function of T , and which vanish
in the limit z ! `. Indeed, these wings can be seen in
the simulation results in Fig. 1(b) at instantaneous power
levels less than 10
25
W.
To experimentally verify that parabolic pulses are indeed
generated in fiber amplifiers, we injected femtosecond
6011

VOLUME
84, NUMBER 26 PHYSICAL REVIEW LETTERS 26J
UNE
2000
pulses into a high gain Yb-doped fiber amplifier, and
carried out FROG characterization of the amplified pulses.
Figure 2 shows the experimental setup. Here, a fiber-
based pulsed seed source was used to generate Gaussian
input pulses of 200 fs FWHM at a wavelength of 1.06 mm
and at a repetition rate of 63 MHz [11]. These pulses were
then injected into a 3.6 m length of Yb-doped fiber co-
directionally pumped at 976 nm, with a gain of 30 dB in
this geometry. The input pulse energy in the fiber was
estimated at 12 pJ. Complete pulse characterization of
the output pulses was carried out using FROG based on
second-harmonic generation (SHG) in a KDP (potassium
dihydrogen phosphate) crystal, with the experimental
configuration used being similar to that described in
[7]. FROG measurements were carried out on the pulses
directly after the Yb-doped fiber amplifier, as well as
after subsequent propagation in 2 m of standard undoped
single mode fiber (SMF). Intensity and chirp retrieval
from the measured FROG traces was carried out using the
standard FROG retrieval algorithm, with the root-mean-
squared error between the measured FROG trace and that
associated with the retrieved pulse being acceptably low
G , 0.007 in all cases [7].
We first discuss the characterization of the pulses di-
rectly from the amplifier. The solid lines in Fig. 3 show
the measured intensity and chirp for an amplifier gain of
30 dB, corresponding to a distributed gain coefficient of
g 1.9 m
21
. In this case, the output pulse energy was
12 nJ, the temporal FWHM was Dt 2.6 ps, the spec-
tral FWHM was Dl 32 nm, and the corresponding du-
ration bandwidth product was DtDn 22. Note that,
in the figure, the arbitrary intensity profile obtained from
the FROG retrieval algorithm has been scaled to show the
instantaneous power in kilowatts. The experimental in-
tensity and chirp are compared with the results of NLSE
simulations (circles) and the predicted asymptotic para-
bolic pulse characteristics (short dashes) for this length
of fiber. Both the measured intensity and chirp are in
good agreement with the results of NLSE simulations. The
experimentally observed weak oscillations in the wings
are attributed to higher order dispersion and resonant ef-
fects not included in Eq. (1). More significantly, however,
the measured intensity profile is also in agreement (over
FIG. 2. Experimental setup used for parabolic pulse genera-
tion and measurement. Pulse characterization via FROG was
carried out for the pulses directly from the 3.6 m Yb-doped fiber
amplifier as well as after propagation in 2 m of undoped fiber
(enclosed by dashed lines).
2 orders of magnitude) with the asymptotic parabolic pulse
predicted by Eq. (2), using the experimental fiber param-
eters given above. To emphasize the parabolic nature of
these pulses, the figure also includes a sech
2
fit to the mea-
sured intensity profile (long dashes). These parabolic pulse
characteristics are consistent with the results in Fig. 1 for
a 200 fs input pulse, where asymptotic behavior would be
expected after 3.6 m of propagation. We therefore interpret
these results as the first direct experimental characteriza-
tion of a parabolic pulse from an optical fiber amplifier,
and as a confirmation of our theory of asymptotic pulse
evolution described above.
As previously predicted in Ref. [5], an attractive feature
of high power parabolic pulses is that they propagate self-
similarly in normally dispersive fiber, allowing for highly
nonlinear propagation over substantial fiber lengths with-
out optical wave breaking. We have been able to verify
this prediction experimentally by launching the amplified
pulses shown in Fig. 3(a) into a 2 m length of undoped
fiber (SMF) and using FROG to characterize the output
pulses. The output pulses after propagation had broad-
ened both temporally and spectrally with Dt 4.4 ps,
Dl 50.5 nm, and DtDn 60. Figure 3(b) shows the
measured intensity and chirp (solid lines), together with
parabolic (short dashes) and sech
2
(long dashes) fits. The
pulse intensity profile was found to remain parabolic, con-
firming the self-similar nature of pulse propagation, al-
though we note that the dynamic range of the parabolic
profile is reduced due to the presence of a low energy
background having its origin in the weak oscillations in
FIG. 3. (a): Intensity (left axis) and chirp (right axis) for pulses
directly from Yb-doped amplifier for a gain of 30 dB. The
solid lines are the experimental results, compared with NLSE
simulation (circles), asymptotic parabolic pulse profile (short
dashes), and sech
2
fit (long dashes). (b) The solid lines show
measured intensity and chirp after propagation through 2 m of
SMF, compared with parabolic (short dashes) and sech
2
(long
dashes) fits.
6012

VOLUME
84, NUMBER 26 PHYSICAL REVIEW LETTERS 26J
UNE
2000
the wings of the amplified pulses. Importantly, despite the
significant temporal and spectral broadening in this nonlin-
ear regime, the chirp remains linear, a characteristic feature
of parabolic pulse propagation [5]. To demonstrate the po-
tential of high power parabolic pulses in ultrafast optics,
we used a simple dispersive grating pair to compress these
parabolic pulses, obtaining a minimum pulse duration of
Dt 68 fs with a corresponding peak power of 80 kW.
The pulses do not compress to the expected transform lim-
ited pulse duration of around 30 fs because of third order
dispersion in the bulk grating compressor, but we note that
this can easily be eliminated with an improved compressor
design [12].
In conclusion, we have developed a theoretical treatment
of the amplification of pulses in high gain fiber ampli-
fiers which predicts the formation of high power parabolic
pulses from any input pulse. We have also demonstrated
experimentally that a Yb-doped fiber amplifier with 30 dB
gain does indeed yield parabolic pulses, whose intensity
and chirp characteristics are in quantitative agreement with
our theoretical predictions. This is the first experimental
demonstration of the existence of parabolic pulses. In view
of their self-similar propagation and the ease with which
they can be compressed, we expect that these parabolic
pulses will find wide application. Indeed, we anticipate
that parabolic pulse propagation in optical fibers may well
become as important and as widely studied as the propa-
gation of optical fiber solitons.
Note added in proof.After acceptance of this manu-
script, we learned of recent results studying the genera-
tion of self-similarity and Cantor set fractals in nonlinear
soliton-supporting systems [13,14]. We also note that self-
similarity techniques have been used to analyze the evolu-
tion of self-written waveguides in photosensitive materials
[15]. It is likely that the study of self-similarity phenomena
in nonlinear optics will become an increasingly important
field of research in the near future.
[1] G. I. Barenblatt, Scaling, Self-Similarity, and Intermedi-
ate Asymptotics (Cambridge University Press, Cambridge,
England, 1996).
[2] P.J. Olver, Applications of Lie Groups to Differential Equa-
tions (Springer, New York, 1986).
[3] A. A. Afanasev et al., J. Mod. Opt. 38
, 1189 (1991).
[4] C. R. Menyuk, D. Levi, and P. Winternitz, Phys. Rev. Lett.
69
, 3048 (1992); D. Levi, C. R. Menyuk, and P. Winternitz,
Phys. Rev. A 49
, 2844 (1994).
[5] D. Anderson et al., J. Opt. Soc. Am. B 10
, 1185 (1993).
[6] K. Tamura and M. Nakazawa, Opt. Lett. 21
, 68 (1996).
[7] R. Trebino et al., Rev. Sci. Instrum. 68
, 3277 (1997).
[8] G. P. Agrawal, Nonlinear Fiber Optics (Academic Press,
San Francisco, 1995).
[9] D. J. Jones et al., IEICE Trans. Electron. E81-C
, 180
(1998).
[10] E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles
and Applications (Wiley, New York, 1994).
[11] M. E. Fermann et al., Opt. Lett. 24
, 1428 (1999).
[12] S. Kane and J. Squier, IEEE J. Quantum Electron. 31
, 2052
(1995).
[13] M. Soljacic, M. Segev, and C. R. Menyuk, Phys. Rev. E
61
, R1048 (2000).
[14] S. Sears, M. Soljacic, M. Segev, D. Krylov, and K.
Bergman, Phys. Rev. Lett. 84
, 1902 (2000).
[15] T.M. Monro, P. D. Millar, L. Poladian, and C. M. de Sterke,
Opt. Lett. 23
, 268 (1998).
6013
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Cites background from "Self-similar propagation and amplif..."

  • ...The unambiguous experimental observation of parabolic-pulse generation was first reported by Fermann et al.(33) in 2000 in an ytterbium-doped fibre amplifier with normal dispersion at 1....

    [...]

  • ...As well as representing a major experimental advance, the results in Fermann et al. also applied theoretical analysis on the basis of symmetry reduction to the nonlinear Schrödinger equation (NLSE) with gain, formally demonstrating the self-similar nature of the generated parabolic pulses....

    [...]

  • ...The unambiguous experimental observation of parabolic-pulse generation was first reported by Fermann et al.33 in 2000 in an ytterbium-doped fibre amplifier with normal dispersion at 1.06 µm....

    [...]

References
More filters
Book
Govind P. Agrawal1
01 Jan 1989
TL;DR: The field of nonlinear fiber optics has advanced enough that a whole book was devoted to it as discussed by the authors, which has been translated into Chinese, Japanese, and Russian languages, attesting to the worldwide activity in the field.
Abstract: Nonlinear fiber optics concerns with the nonlinear optical phenomena occurring inside optical fibers. Although the field ofnonlinear optics traces its beginning to 1961, when a ruby laser was first used to generate the second-harmonic radiation inside a crystal [1], the use ofoptical fibers as a nonlinear medium became feasible only after 1970 when fiber losses were reduced to below 20 dB/km [2]. Stimulated Raman and Brillouin scatterings in single-mode fibers were studied as early as 1972 [3] and were soon followed by the study of other nonlinear effects such as self- and crossphase modulation and four-wave mixing [4]. By 1989, the field ofnonlinear fiber optics has advanced enough that a whole book was devoted to it [5]. This book or its second edition has been translated into Chinese, Japanese, and Russian languages, attesting to the worldwide activity in the field of nonlinear fiber optics.

15,770 citations

Book
01 Jan 1986
TL;DR: In this paper, the Cauchy-Kovalevskaya Theorem has been used to define a set of invariant solutions for differential functions in a Lie Group.
Abstract: 1 Introduction to Lie Groups- 11 Manifolds- Change of Coordinates- Maps Between Manifolds- The Maximal Rank Condition- Submanifolds- Regular Submanifolds- Implicit Submanifolds- Curves and Connectedness- 12 Lie Groups- Lie Subgroups- Local Lie Groups- Local Transformation Groups- Orbits- 13 Vector Fields- Flows- Action on Functions- Differentials- Lie Brackets- Tangent Spaces and Vectors Fields on Submanifolds- Frobenius' Theorem- 14 Lie Algebras- One-Parameter Subgroups- Subalgebras- The Exponential Map- Lie Algebras of Local Lie Groups- Structure Constants- Commutator Tables- Infinitesimal Group Actions- 15 Differential Forms- Pull-Back and Change of Coordinates- Interior Products- The Differential- The de Rham Complex- Lie Derivatives- Homotopy Operators- Integration and Stokes' Theorem- Notes- Exercises- 2 Symmetry Groups of Differential Equations- 21 Symmetries of Algebraic Equations- Invariant Subsets- Invariant Functions- Infinitesimal Invariance- Local Invariance- Invariants and Functional Dependence- Methods for Constructing Invariants- 22 Groups and Differential Equations- 23 Prolongation- Systems of Differential Equations- Prolongation of Group Actions- Invariance of Differential Equations- Prolongation of Vector Fields- Infinitesimal Invariance- The Prolongation Formula- Total Derivatives- The General Prolongation Formula- Properties of Prolonged Vector Fields- Characteristics of Symmetries- 24 Calculation of Symmetry Groups- 25 Integration of Ordinary Differential Equations- First Order Equations- Higher Order Equations- Differential Invariants- Multi-parameter Symmetry Groups- Solvable Groups- Systems of Ordinary Differential Equations- 26 Nondegeneracy Conditions for Differential Equations- Local Solvability- In variance Criteria- The Cauchy-Kovalevskaya Theorem- Characteristics- Normal Systems- Prolongation of Differential Equations- Notes- Exercises- 3 Group-Invariant Solutions- 31 Construction of Group-Invariant Solutions- 32 Examples of Group-Invariant Solutions- 33 Classification of Group-Invariant Solutions- The Adjoint Representation- Classification of Subgroups and Subalgebras- Classification of Group-Invariant Solutions- 34 Quotient Manifolds- Dimensional Analysis- 35 Group-Invariant Prolongations and Reduction- Extended Jet Bundles- Differential Equations- Group Actions- The Invariant Jet Space- Connection with the Quotient Manifold- The Reduced Equation- Local Coordinates- Notes- Exercises- 4 Symmetry Groups and Conservation Laws- 41 The Calculus of Variations- The Variational Derivative- Null Lagrangians and Divergences- Invariance of the Euler Operator- 42 Variational Symmetries- Infinitesimal Criterion of Invariance- Symmetries of the Euler-Lagrange Equations- Reduction of Order- 43 Conservation Laws- Trivial Conservation Laws- Characteristics of Conservation Laws- 44 Noether's Theorem- Divergence Symmetries- Notes- Exercises- 5 Generalized Symmetries- 51 Generalized Symmetries of Differential Equations- Differential Functions- Generalized Vector Fields- Evolutionary Vector Fields- Equivalence and Trivial Symmetries- Computation of Generalized Symmetries- Group Transformations- Symmetries and Prolongations- The Lie Bracket- Evolution Equations- 52 Recursion Operators, Master Symmetries and Formal Symmetries- Frechet Derivatives- Lie Derivatives of Differential Operators- Criteria for Recursion Operators- The Korteweg-de Vries Equation- Master Symmetries- Pseudo-differential Operators- Formal Symmetries- 53 Generalized Symmetries and Conservation Laws- Adjoints of Differential Operators- Characteristics of Conservation Laws- Variational Symmetries- Group Transformations- Noether's Theorem- Self-adjoint Linear Systems- Action of Symmetries on Conservation Laws- Abnormal Systems and Noether's Second Theorem- Formal Symmetries and Conservation Laws- 54 The Variational Complex- The D-Complex- Vertical Forms- Total Derivatives of Vertical Forms- Functionals and Functional Forms- The Variational Differential- Higher Euler Operators- The Total Homotopy Operator- Notes- Exercises- 6 Finite-Dimensional Hamiltonian Systems- 61 Poisson Brackets- Hamiltonian Vector Fields- The Structure Functions- The Lie-Poisson Structure- 62 Symplectic Structures and Foliations- The Correspondence Between One-Forms and Vector Fields- Rank of a Poisson Structure- Symplectic Manifolds- Maps Between Poisson Manifolds- Poisson Submanifolds- Darboux' Theorem- The Co-adjoint Representation- 63 Symmetries, First Integrals and Reduction of Order- First Integrals- Hamiltonian Symmetry Groups- Reduction of Order in Hamiltonian Systems- Reduction Using Multi-parameter Groups- Hamiltonian Transformation Groups- The Momentum Map- Notes- Exercises- 7 Hamiltonian Methods for Evolution Equations- 71 Poisson Brackets- The Jacobi Identity- Functional Multi-vectors- 72 Symmetries and Conservation Laws- Distinguished Functionals- Lie Brackets- Conservation Laws- 73 Bi-Hamiltonian Systems- Recursion Operators- Notes- Exercises- References- Symbol Index- Author Index

8,118 citations


Additional excerpts

  • ...In particular, the presence of self-similarity can be exploited to obtain asymptotic solutions to partial differential equations describing a physical system by using the mathematical technique of symmetry reduction to reduce the number of degrees of freedom [2]....

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Book ChapterDOI
01 Dec 1996
TL;DR: The application of dimensional analysis to the construction of intermediate asymptotic solutions to problems of mathematical physics can be found in this article, where the authors describe the application of similarity analysis to scaling in deformation and fracture in solids.
Abstract: Preface Introduction 1. Dimensions, dimensional analysis and similarity 2. The application of dimensional analysis to the construction of intermediate asymptotic solutions to problems of mathematical physics. Self-similar solutions 3. Self-similarities of the second kind: first examples 4. Self-similarities of the second kind: further examples 5. Classification of similarity rules and self-similarity solutions. Recipe for application of similarity analysis 6. Scaling and transformation groups. Renormalization groups. 7. Self-similar solutions and travelling waves 8. Invariant solutions: special problems of the theory 9. Scaling in deformation and fracture in solids 10. Scaling in turbulence 11. Scaling in geophysical fluid dynamics 12. Scaling: miscellaneous special problems.

1,844 citations

Journal ArticleDOI
TL;DR: In this article, the authors present a comprehensive overview of the fundamental principles and applications of erbium-doped fiber amplifiers (EDFAs) in optical fiber communications.
Abstract: It is now widely recognized that erbium-doped fiber amplifiers have revolutionized optical fiber communications. EDFAs not only made single-channel, multigigabit-rate, long-distance optical communications possible, but they also opened up a wide variety of additional possibilities Such as soliton generation and transmission and multichannel wavelength-division multiplexing communications. While at AT&T Bell Labs (he is now at Alcatel-Alsthom Recherche in a suburb of Paris, France) Emmanuel Desurvire became heavily involved in and contributed enormously to the theoretical and experimental investigation of EDFA characteristics and system applications. His pioneering work has been internationally recognized. In my view, Desurvire is one of those best qualified to cover the subject of EDFAs, in Erbium-Doped Fiber Amplifiers: Principles and Applications, he has accepted the challenge. According to the author, the purpose of the book is "to provide the basic materials of a comprehensive introduction to the principles and applications of EDFAs." The book is divided into three major parts, which to some extent can be considered independently. Nonetheless, it keeps its cohesion throughout. It provides a thorough understanding of the fundamentals in optical amplification while considering the practical issues related to the device and system performance of EDFAs. The first part of the book explores all the fundamental issues related to EDFAs. It introduces the main concepts necessary for the modelling of the erbium atomic transition. The analysis is detailed and covers such parameters as field distributions and overlap integrals under different operating conditions. This section and the numerous relevant appendices contain a number of useful generalizations of existing models that are published for the first time. The author also considers the fundamental quantum properties of noise generation and accumulation in single- and multiple-stage amplification of classical light. The analysis discusses in great depth the nature, origin and inevitability of noise associated with optical amplification - it also provides useful engineering formulas for the measurement of the noise introduced by amplification. I found the treatment of noise and photon statistics particularly detailed and original. Researchers working on this subject can benefit enormously from the analysis. The second part is primarily experimental and focuses on EDFA device characteristics. However, when specific characteristics of the erbium transition are discussed, the necessary theoretical modifications and additions, supplementary to the general formulations given in the first part, are provided. I found that on the important issue of pulse amplification requirements, the book considers briefly only the special (but very exciting) case of solitons and misses the many problems associated with general pulse amplification. The third and final section of the book, on applications, is primarily concerned with some of the up-to-date linear and nonlinear communication systems and local area networks and the enormous impact that EDFAs have had on their successful implementation. The significance of the EDFAs in optically preamplified receivers is stressed. The most significant digital (linear) and soliton (nonlinear) system experiments performed to date are also reviewed. Initations imposed on linear systems by fiber nonlinearities and dispersion are briefly mentioned. The ground that Desurvire sought to cover in the third part is quite diverse and could well have been the subject of several separate volumes. Therefore, its inclusion in this book is inevitably of a review type. However, the book clearly points out how and to what degree these applications are benefited or enhanced by EDFAs. Overall the book gives one of the most comprehensive and detailed accounts of the physics and fundamental principles of erbium-doped fiber amplifiers published so far. I have not the slightest doubt that the book will be of great help to all scientists and engineers working in the field who are struggling to understand EDFAs. The unified and in-depth presentation of the subject will benefit in particular researchers and graduate students who are dealing with problems involving optical amplification. The book imparts the fundamental concepts quite skilfully and can be used as collateral reading The sections dealing with modelling and the entire second part could well be used in undergraduate courses. I do not hesitate to recommend the book enthusiastically to anybody having an interest in EDFAs and their applications.

1,128 citations


"Self-similar propagation and amplif..." refers background in this paper

  • ...In this case, propagation can be described by the NLSE with gain [10]:...

    [...]

Frequently Asked Questions (16)
Q1. What contributions have the authors mentioned in the paper "Self-similar propagation and amplification of parabolic pulses in optical fibers" ?

Ultrashort pulse propagation in high gain optical fiber amplifiers with normal dispersion is studied by self-similarity analysis of the nonlinear Schrödinger equation with gain. 

At intermediate propagation distances, simulations and analysis predict low amplitude wings on the parabolic pulse which decay exponentially as a function of T , and which vanish in the limit z ! 

despite the significant temporal and spectral broadening in this nonlinear regime, the chirp remains linear, a characteristic feature of parabolic pulse propagation [5]. 

The experimentally observed weak oscillations in the wings are attributed to higher order dispersion and resonant effects not included in Eq. (1). 

It is likely that the study of self-similarity phenomena in nonlinear optics will become an increasingly important field of research in the near future. 

As the fiber gain is increased for a given input pulse, the exponential growth of the pulse amplitude and width is correspondingly increased in agreement with Eqs. (4) and (5), and the parabolic asymptotic limit is reached in a shorter propagation distance. 

As previously predicted in Ref. [5], an attractive feature of high power parabolic pulses is that they propagate selfsimilarly in normally dispersive fiber, allowing for highly nonlinear propagation over substantial fiber lengths without optical wave breaking. 

—After acceptance of this manuscript, the authors learned of recent results studying the generation of self-similarity and Cantor set fractals in nonlinear soliton-supporting systems [13,14]. 

To demonstrate the potential of high power parabolic pulses in ultrafast optics, the authors used a simple dispersive grating pair to compress these parabolic pulses, obtaining a minimum pulse duration of Dt 68 fs with a corresponding peak power of 80 kW. 

This corresponds to a compactly supported pulse with a parabolic intensity profile, and a quadratic phase given byw z, T w0 1 3g 2g 21A20 z 2 g 6b2 21T2, (3)where w0 is an arbitrary constant. 

In the asymptotic regime, this pulse propagates self-similarly, maintaining its parabolic shape subject to the exponential scaling of its amplitude A0 z jA z, 0 j and effective width parameter T0 z according toA0 z 0.5 gEIN 1 3 gb2 2 21 6 exp gz 3 , (4)T0 z 3g22 3 gb2 2 1 3E 1 3 IN exp gz 3 , (5)where EIN is the energy of the input pulse to the amplifier. 

In this case, the output pulse energy was 12 nJ, the temporal FWHM was Dt 2.6 ps, the spectral FWHM was Dl 32 nm, and the corresponding duration bandwidth product was DtDn 22. 

The authors have also demonstrated experimentally that a Yb-doped fiber amplifier with 30 dB gain does indeed yield parabolic pulses, whose intensity and chirp characteristics are in quantitative agreement with their theoretical predictions. 

In this context, the authors also note that, although at jT j T0 z the solution in Eq. (2) has infinite slope, this is the case only in the asymptotic limit. 

Complete pulse characterization of the output pulses was carried out using FROG based on second-harmonic generation (SHG) in a KDP (potassium dihydrogen phosphate) crystal, with the experimental configuration used being similar to that described in [7]. 

The solid lines in Fig. 3 show the measured intensity and chirp for an amplifier gain of 30 dB, corresponding to a distributed gain coefficient of g 1.9 m21.