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Soliton Self-Frequency Shift
Cancellation in Photonic
Crystal Fibers
D. V. Skryabin,* F. Luan, J. C. Knight, P. St. J. Russell
We report the cancellation of the soliton self-frequency shift in a silica-core
photonic crystal fiber with a negative dispersion slope. Numerical and experimental results show that stabilization of the soliton wavelength is accompanied by exponential amplification of the red-shifted Cherenkov radiation emitted by the soliton. The spectral recoil from the radiation acts on the soliton to
compensate for the Raman frequency shift. This phenomenon may find applications in the development of a family of optical parametric amplifiers.
Solitary waves, or solitons, are self-localized
regions of energy or matter that occur in a
large variety of fundamental processes in
many natural and artificially created nonlinear systems, such as the sea surface,
nerve fibers, superconducting transmission
lines, optical fibers, elementary particles,
and outer space (1). Optical solitons are
self-localized pulses or beams of light with
temporal or spatial dispersion suppressed
by the action of a nonlinear medium in
which they propagate (2, 3). Soon after the
experimental observations of temporal solitons in dispersive optical fibers (4 ), it was
found that fiber solitons can exhibit a
strong self-frequency shift (5, 6 ). That is, a
short (⬍1 ps) soliton propagating in a
Raman-active medium like silica is continuously red-shifted because the lowfrequency end of the soliton spectrum experiences Raman gain at the expense of the
high-frequency end. Although this has been
considered an immutable feature of subpicosecond soliton propagation in silica optical fibers, we show theoretically and experimentally that the soliton self-frequency
shift can be canceled in a silica-core fiber
with a negative dispersion slope. Cancellation of the frequency shift arises because of
the exponential amplification and subsequent saturation of the new radiation band
red-shifted with respect to the soliton and
emitted by the soliton itself through the
Cherenkov mechanism.
Cherenkov radiation (7–9) by a charged
particle propagating in a dispersive medium
occurs when the particle travels at speeds
faster than the phase velocity of light (10).
Equivalently, one can say that the Cherenkov
effect takes place if the wavenumber of the
wave created by the particle becomes smaller
than the wavenumber of the dispersive wave
Department of Physics, University of Bath, Bath BA2
7AY, UK.
*To whom correspondence should be addressed. Email: [email protected]
for the same frequency. Then, the wavenumber matching condition is satisfied and radiation is emitted under some angle to the
direction of the particle motion (9).
Because propagation in fibers is restricted
to one spatial direction, time is the only other
relevant coordinate. The angle in the spacetime plane, which any dispersive wave makes
with the soliton trajectory, characterizes the
frequency of the wave relative to the soliton
frequency. One of the most important properties of idealized optical solitons is their
robustness against perturbations (11). The
soliton locally modifies the dispersion characteristics so that there are no dispersive
waves with real frequencies having wavenumbers close to the soliton ones. This prohibits energy leakage from the soliton to the
waves and back. However, when the group
velocity dispersion (GVD) of the fiber changes significantly over the spectral bandwidth
of the pulse, the soliton loses its immunity to
dispersive waves, because a critical frequency or set of frequencies can be found, such
that the wavenumbers of the waves are
matched with the solitonic wavenumbers
(12). This leads to Cherenkov radiation, and
the soliton is no longer “ideal,” but becomes a quasi-soliton coupled to dispersive
waves (13). This effect gets more pronounced for shorter pulses and steeper
GVD slopes. In what follows GVD will be
denoted as ␤2(␻), defined as the second
derivative of wavenumber k with respect to
angular frequency ␻.
Recently developed photonic-crystal fibers (PCFs) (14–16) give rise to previously
unattainable optical characteristics. One of
the most prominent features of PCFs is the
range of unusual dispersion curves that have
been demonstrated (17, 18), which have a
profound effect on the nonlinear optics within
the fiber (18–23). To reach the balance between dispersion and nonlinearity required
for the solitonic regime, one needs to operate
at a frequency where the GVD is anomalous,
i.e., ␤2(␻) ⬍ 0. We used a PCF with anom-
alous GVD over a finite band of frequencies,
rather than a semi-infinite band as in conventional fibers (Fig. 1A). The dispersion in the
standard telecom fiber (SMF28) is anomalous
at wavelengths longer than 1.3 ␮m, whereas
the PCF dispersion is anomalous between 0.6
and 1.3 ␮m. We have calculated the dependence of the Cherenkov resonances on the
soliton central frequency from the wavenumber matching conditions (12, 21) (Fig. 1B).
There are two branches of Cherenkov radiation. One branch is blue-shifted and the other
one is red-shifted with respect to the soliton
frequency, whereas for the conventional fiber
there is only a blue-shifted branch. The blueshifted radiation arises because there is a
region of ␻ where the slope of ␤2(␻) is
positive and ␤2(␻) changes its sign from
negative to positive. Similarly, the existence
of the frequency range where ␤2(␻) changes
from positive to negative, i.e., its slope is
negative, is a crucial condition for observation of the red-shifted radiation. Such a frequency range is absent in conventional fibers
and in PCFs with larger cores, and its presence in our PCF is a direct result of the very
small core diameter of 1.2 ␮m.
It has been shown theoretically (12) that
solitons emitting Cherenkov radiation lose
energy slowly (i.e., nonexponentially) with
increasing propagation distance z, transferring it to the resonant dispersive wave. To
conserve the overall energy of the photons,
the carrier soliton frequency gets shifted in
the spectral direction opposite to that of the
radiation. This is the so-called spectral recoil
effect (2, 12). It is important to note for the
following that the analysis of (12) disregards
the soliton self-frequency shift due to the
Raman effect, which is a process in which
photons cascade their energy to photons at a
larger wavelength through scattering from
optical phonons. However, the Raman response of the silica accounts for about 20% of
the overall nonlinear response (2) and is a
strong effect in the solitonic propagation regime, so it cannot be neglected. The frequency of the Cherenkov radiation approaches the
central soliton frequency close to the zero
GVD wavelengths (Fig. 1B). Energy exchange between the soliton and the resonant
dispersive wave is expected to reach its maximum in these regions, where energy is fed
into the wave from the most intense central
part of the soliton spectrum. This is because
the amplitude of the emitted radiation is primarily determined by the spectral amplitude
of the soliton at the radiation frequency ␻r.
The spectral amplitude at the frequency ␻r
for an ideal soliton with the central frequency
␻s is proportional to 1/{exp[⫺(␻r ⫺ ␻s)␲␶/
2)] ⫹ exp[(␻r ⫺ ␻s)␲␶/2]}, where ␶ is the
soliton duration (24). Thus, for values of ␻r
approaching ␻s, the intensity of the emitted
wave increases exponentially.
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The Raman scattering generates a red shift
in the soliton carrier frequency, which is directly proportional to the propagation distance z (2).
Therefore, as the soliton approaches the redshifted zero GVD point with a negative GVD
slope, the intensity of the red-shifted branch of
the Cherenkov radiation is expected to increase
exponentially in z. The exponential growth of
the radiation should saturate, however, as the
soliton recoils against the radiation toward the
blue side of the spectrum. Thus, there exists the
possibility of a balance between the red Raman
self-frequency shift and the blue recoil on the
soliton from the red-shifted Cherenkov radiation. This balance will then produce frequencylocked solitary pulses with a growing tail of
Cherenkov radiation.
To verify the above considerations we carried out a series of numerical experiments modeling propagation of optical solitons in the PCF.
We used the generalized nonlinear Schrödinger
(NLS) equation with the linear dispersion fitted
to the experimentally measured one (Fig. 1A),
and a nonlinear response function that included
both the instantaneous electronic and noninstantaneous Raman responses (18, 22, 23). We
initialized this model with a single-soliton solution (2) of the ideal NLS equation having
peak power ⬵ 215 W, duration 53 fs, and
carrier frequency 2␲ ⫻ 250 THz. For this frequency, ␤2 ⬵ ⫺70 ps2/km and the derivative of
␤2 is ⫺0.25 ps3/km, and the soliton is expected
to emit red-shifted radiation with an intensity
much stronger than the intensity of the blue-
shifted radiation, which has a larger detuning
from the soliton frequency. Thus, the overall
recoil effect is expected to push the soliton
toward the blue side of the spectrum.
The results of our modeling (Fig. 2, A and
B) show the evolution of the absolute value
of the electric field in (t ⫺ z/v, z) plane and
the logarithm of the absolute value of the
Fourier transform of the same field in the (␻,
z) plane. Here t, z, and v represent time,
coordinate along the fiber, and soliton group
velocity, respectively. Up to ⬵2.5 m the
pulse evolves as expected of Raman-shifted
solitons (2). However, for z ⬎ 2.5 m, the
pulse dynamics in the spectral and temporal
domains undergo a dramatic change. The
self-frequency shift of the soliton is suddenly
Fig. 1. (A) GVD plots for the telecommunication fiber (SMF 28) and PCF
used in our experiments. (Inset) Scanning electron micrograph of the PCF
transverse section. The core diameter of the PCF is 1.2 ␮m. (B) Dependence of the frequency of the Cherenkov resonances from the soliton
frequencies for the telecom fiber and PCF. Two vertical dashed lines mark
the zero GVD points in the PCF. The diagonal line marks the boundary
where the radiation and soliton frequencies coincide. The radiation
branches above/below this line are, respectively, blue-/red-shifted relative to the soliton carrier frequency. The top axes in (A) and (B) are
marked in wavelength units.
Fig. 2. Results of the
numerical modeling
showing spatiotemporal and spectral
dynamics of the femtosecond
soliton
propagating
down
the PCF. (A) Spatiotemporal plot of the
absolute value of the
field. The gray shaded region appearing
for z ⬎ 2.5 m is the
Cherenkov radiation.
The gray-scale mapping chosen for this
plot exaggerates the
strength of the radiation tail in order to
show it clearly. For
quantitative comparison of the soliton and radiation intensities, see fig. S1. (B) Plot of the
logarithm of the absolute value of the Fourier transform of the field in the
(␻, z) plane. White dashed line marks the zero GVD frequency with the
negative slope. The radiation and solitonic parts of the field are located,
respectively, to the left and the right of the white line. The top axis is
marked in wavelength units.
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canceled almost completely. This happens at
the expense of a bright new spectral line (Fig.
2B), which appears on the red side of the
spectrum in the normal GVD (␤2 ⬎ 0) region.
Simultaneously, the soliton acquires a pronounced tail of Cherenkov radiation (Fig.
2A). The radiation field needs less time than
the soliton to reach a given distance z in the
fiber, so the radiation propagates ahead of the
soliton in the laboratory reference frame. It is
clear that we can directly associate the radiation tail with the new red-shifted spectral
line. From Fig. 2B it is seen that the radiation
exists also for z ⬍ 2.5 m, but its amplitude is
relatively small compared to the strong radiation line rapidly emerging and continuing to
exist for z ⬎ 2.5 m. Taking the soliton frequency measured from our modeling, the radiation frequency can be computed directly
from the wavenumber matching conditions;
the corresponding points are marked by white
diamonds (Fig. 2B). The excellent agreement
between results obtained with these two
methods strongly suggests that the new spectral band is indeed Cherenkov radiation emitted by the soliton.
Thus, we have established that, after some
propagation distance, the rate of energy transfer
from the soliton to the red-shifted Cherenkov
radiation builds up to a point when it balances,
through the spectral recoil mechanism, the
Fig. 3. (A) Experimentally measured spectral evolution of the
femtosecond optical pulse propagating down the PCF. (B) Corresponding numerical modeling.
The top axes are marked in
wavelength units. Cherenkov radiation band appears to the left
of the vertical dashed line marking the zero GVD frequency.
Raman-induced soliton self-frequency shift.
After this point, no further spectral shift of the
soliton occurs. After the balance has been
reached, the radiation contains ⬃ 50% of the
initial soliton energy. For z ⬎ 2.5 m, the length
of the radiation tail increases (Fig. 2A), leading
to a nonexponential decay of the solitonic part
of the field. The latter, however, remains
strongly localized during this process (fig. S1).
To verify the above numerical results, we
carried out a series of experiments in which
we launched optical pulses into the PCF. The
laser source used in the experiments was a
mode-locked Ti:sapphire system emitting
200-fs pulses at a wavelength of 0.86 ␮m.
The fiber length was progressively reduced
from 4 to 0.5 m, and the output spectrum was
recorded for each length with an optical spectrum analyzer. Figure 3A shows measurement results for a peak pump power of 230
W. These initial conditions correspond to the
fourth-order solitonic solution of the ideal
NLS equation (2). For the chosen pump
wavelength, the dispersion slope is still positive, but both red and blue radiation fields
are very weak, because of the large detuning
from the soliton frequency. Therefore, the
initial soliton dynamics is dominated by the
Raman effect. Initially, the pulse splits into
two Raman-shifting solitons, and some residual radiation, which retains the pump fre-
quency. The more intense soliton acquires the
stronger Raman shift. It passes the minimum
point of ␤2(␻) and enters the spectral region
with negative dispersion slope, where the
red-shifted Cherenkov frequency quickly approaches the central part of the soliton spectrum. This leads to the exponential amplification of the red-shifted Cherenkov radiation,
which stabilizes the soliton frequency at
⬵2␲ ⫻ 236 THz, corresponding to the wavelength of ⬵1.27 ␮m, through the recoil
mechanism. Figure 3B shows the corresponding modeling results: They are in excellent
agreement with experiment. Note that the
blobs in the low middle part of the experimental plot (Fig. 3A) are an artifact of the
numerical interpolation of the data.
Thus, our numerical and experimental results show that we have identified and observed an efficient mechanism to amplify the
Cherenkov radiation emitted by optical solitons in PCFs with negative dispersion slopes.
Saturation of this amplification, in its turn,
leads to the cancellation of the soliton selffrequency shift. The detuning between the
solitonic pump and the Cherenkov radiation
in the frequency-locking regime is sensitive
to the value of the GVD slope, which is
determined by the PCF geometry. This suggests the possibility of tailoring the latter to
achieve a desired output frequency, and thus,
of developing a family of optical parametric
amplifiers based on the solitonic Cherenkov
radiation. Methods to control the frequency
of the Cherenkov radiation emitted by a
charged particle inside a photonic crystal by
varying the crystal geometry have also been
recently suggested (25).
The effect of the soliton self-frequency
shift cancellation described above is very robust and has been observed for a broad range
of pump frequencies and powers. The most
likely reason why it has not been previously
identified is that in telecommunication fibers,
for commonly used frequencies, the radiation
is always blue-shifted, so that the recoil and
Raman effects act in the same spectral direction. The same is true for the recent experiments with PCFs with larger core diameters
(18, 21). In these experiments, the blueshifted radiation was a dominant feature of
generated spectra, when pulses were launched
in the proximity of the zero GVD point with
positive slope. However, Raman and recoil effects quickly pull the emergent solitons away
from the zero GVD point and into the region of
the less steep positive or practically flat GVD
slopes, which leads to the decay rather than
amplification of the radiation intensity with
propagation distance.
Results of this work show how novel
and unexpected effects can be discovered in
photonic crystal fibers due to the unique
combination of their dispersive and nonlinear properties.
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References and Notes
1. A. Scott, Nonlinear Science: Emergence and Dynamics of
Coherent Structures (Oxford Univ. Press, Oxford, 1999).
2. A. Hasegawa, M. Matsumoto, Optical Solitons in Fibers (Springer, Berlin, 2003).
3. G. I. Stegeman, M. Segev, Science 286, 1518 (1999).
4. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev.
Lett. 45, 1095 (1980).
5. E. M. Dianov et al., JETP Lett. 41, 294 (1985).
6. F. M. Mitschke, L. F. Mollenauer, Opt. Lett. 11, 659
(1986).
7. P. A. Cherenkov, Dokl. Akad. Nauk SSSR 2, 451 (1934).
8. S. Vavilov, Dokl. Akad. Nauk SSSR 2, 457 (1934).
9. I. Frank, I. Tamm, Dokl. Akad. Nauk SSSR 14, 109
(1937).
10. Cherenkov radiation at speeds below the light threshold has also been recently reported for a spatially
extended system of electric dipoles created by a
femtosecond optical pulse (26).
11. V. E. Zakharov, A.B. Shabat, Sov. Phys. JETP 34, 62 (1971).
12. N. Akhmediev, M. Karlsson, Phys. Rev. A 51, 2602 (1995).
13. For numerous reasons such as, e.g., Cherenkov radiation
or dissipation, most if not all solitons observed in nature
are not the “ideal” ones. To stress the importance of the
nonideal features of the solitons, the term “quasi-solitons” has been introduced and widely used over the last
decade [see, e.g., (27)].
14. J. C. Knight, T. A. Birks, P. St. J. Russell, D. M. Atkin,
Opt. Lett. 21, 1547 (1996).
15. J. C. Knight, J. Broeng, T. A. Birks, P. St. J. Russell,
Science 282, 1476 (1998).
16. R. F. Cregan et al., Science 285, 1537 (1999).
17. J. C. Knight et al., IEEE Photon. Technol. Lett. 12, 807
(2000).
18. W. H. Reeves et al., Nature 424, 511 (2003).
19. J. K. Ranka, R. S. Windeler, A. J. Stenz, Opt. Lett. 25,
25 (2000).
20. W. J. Wadsworth et al., Electron. Lett. 36, 53 (2000).
21. J. Herrmann et al., Phys. Rev. Lett. 88, 173901 (2002).
22. A. L. Gaeta, Opt. Lett. 11, 924 (2002).
23. J. M. Dudley et al., J. Opt. Soc. Am. B 19, 765
(2002).
The Anatomy of the World’s
Largest Extinct Rodent
Marcelo R. Sánchez-Villagra,1* Orangel Aguilera,2 Inés Horovitz3
Phoberomys is reported to be the largest rodent that ever existed, although it
has been known only from isolated teeth and fragmentary postcranial bones.
An exceptionally complete skeleton of Phoberomys pattersoni was discovered
in a rich locality of fossil vertebrates in the Upper Miocene of Venezuela.
Reliable body mass estimates yield ⬃700 kilograms, more than 10 times the
mass of the largest living rodent, the capybara. With Phoberomys, Rodentia
becomes one of the mammalian orders with the largest size range, second only
to diprotodontian marsupials. Several postcranial features support an evolutionary relationship of Phoberomys with pakaranas from the South American
rodent radiation. The associated fossil fauna is diverse and suggests that
Phoberomys lived in marginal lagoons and wetlands.
Phoberomys belongs to the Caviomorpha, a diverse and endemic group of South American
rodents that includes arboreal, cursorial, and
fossorial forms and that ranges today in size
between ⬃200 g and ⬃50 kg (1). The evolution
of caviomorphs is recorded in a rich but geographically biased fossil record. The southern
portion of South America contains most of the
record (2); hence, discoveries in the northern
tropics are of special significance. The Urumaco
Formation in northwestern Venezuela contains
one of the few examples of a diverse fauna of
Upper Miocene vertebrates in the continent. Recent explorations resulted in the discovery of
additional vertebrates in the upper and middle
members of this formation, including the rodent
reported here (table S1). Old and new discoveries make Urumaco one of the best-documented
1
Universität Tübingen, Spezielle Zoologie, Auf der
Morgenstelle 28, D-72076 Tübingen, Germany. 2Universidad Nacional Experimental Francisco de Miranda,
CICBA, Complejo Docente Los Perozos, Carretera
Variante Sur, Coro, 4101, Estado Falcón, Venezuela.
3
Department of Organismic Biology, Ecology, and
Evolution, 621 Young Drive South, University of California, Los Angeles, CA 90095–1606, USA.
*To whom correspondence should be addressed. Email: [email protected]
1708
tropical Miocene fossil fauna of vertebrates in
the world after La Venta in Colombia (3).
The Urumaco Formation is characterized
by diverse faunal associations in continental
(savannas), freshwater (swamps and rivers),
estuarine (brackish), and marine (coastal lagoon, salt marsh, and sandy littoral) environments (table S1). Each assemblage can be
correlated with a distinctive sedimentary environment. The following facies are apparent:
shallow-water marine sediments rich in mollusks and fishes; brackish water rich in marine catfish; and swampy paleoenvironments
rich in crocodilians and gavialids, in freshwater and marine turtles, and in freshwater
catfish. These general sequences repeat several times in the outcrop (4). The skeleton
reported here was found in brown shales
interbedded with thin layers of coal.
Two specimens of Phoberomys pattersoni
Mones 1980 (5) provide the basis for this report. One consists of an almost complete associated skeleton (Fig. 1A). The skull is poorly
preserved and consists of a deformed palate
with the upper molariform series and most of
the dentaries, with molariform teeth and fragments of the incisors. An additional shattered
partial skull, preserving most of the occipital
24. Temporal profile of the amplitude of an ideal fiber
soliton is given by secant hyperbolic: sech(t/␶) ⫽
2/(et/␶ ⫹ e⫺t/␶). It is, however, not commonly
known that the Fourier transform of the sechfunction is again a sech-function. This can be
checked, e.g., using Mathematica 4.0 or in (28).
25. C. Luo, M. Ibanescu, S. G. Johnson, J. D. Joannopoulos,
Science 299, 368 (2003).
26. T. E. Stevens, J. K. Wahlstrand, J. Kuhl, R. Merlin,
Science 291, 627 (2001).
27. V. E. Zakharov, E. A. Kuznetsov, JETP 86, 1035
(1998).
28. W. Feller, An Introduction to Probability Theory and
Its Applications (Wiley, New York, 1966), vol. 2, p.
503.
Supporting Online Material
www.sciencemag.org/cgi/content/full/301/5640/1705/
DC1
Fig. S1
27 June 2003; accepted 13 August 2003
and portions of the basicranial region, was also
collected (Fig. 1B). Based on the degree of
tooth wear and sutural fusion, we estimate that
the specimens were adults at the time of death.
The proximal epiphysis of the tibia and the
distal epiphysis of the ulna are not fused to the
diaphysis. However, it is possible that the animal was an adult, because no sutures can be
recognized in the occipital region. In the pakarana Dinomys, probably the closest extant
relative of Phoberomys, the epiphyses of long
bones fuse late in ontogeny, some during adulthood (6). A description of the anatomy of the
postcranial skeleton of P. pattersoni is presented in the supporting online material.
Allocation of the specimens to P. pattersoni
is secured based on two diagnostic features of
the last upper molar (5): the narrowing of the
posteriormost portion at the level of the last
three prisms, and the size (mesiodistal length: 41
mm; width: 20.7 mm) and relative proportions
of this tooth. Based solely on tooth dimensions,
P. pattersoni is slightly smaller than P. insolita
and P. lozanoi, which have a M3 with a mesiodistal length of 47 and 48 mm, respectively.
Phoberomys, together with the genera Neoepiblema and Eusigmomys, belongs to the fossil
Family Neoepiblemidae, distributed in Argentina, Chile, Brazil, and Venezuela (7). Of all other
species of Neoepiblemidae, cranial remains of
only Neoepiblema ambrosettianus have been
reported to date (7). This animal had a prominent sagittal crest, absent in P. pattersoni. Based
on fragmentary dental remains, Phoberomys
(and therefore the neoepiblemids) has been classified either with the chinchillas and viscachas
(8), with the pakarana (9), or as the sister group
to both (10). We plotted a set of 13 postcranial
characters on a preexisting phylogenetic tree
based on molecular data and found that several
postcranial features support the association of
Phoberomys with the pakarana (Fig. 2). This
position for Phoberomys was the one that required the least number of steps.
P. pattersoni is reported to have been the
size of a rhinoceros (1, 11, 12). This rough
estimate, based on isolated teeth, can be
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