Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers.
ABSTRACT We show theoretically that the photoionization process in a hollow-core photonic crystal fiber filled with a Raman-inactive noble gas leads to a constant acceleration of solitons in the time domain with a continuous shift to higher frequencies, limited only by ionization loss. This phenomenon is opposite to the well-known Raman self-frequency redshift of solitons in solid-core glass fibers. We also predict the existence of unconventional long-range nonlocal soliton interactions leading to spectral and temporal soliton clustering. Furthermore, if the core is filled with a Raman-active molecular gas, spectral transformations between redshifted, blueshifted, and stabilized solitons can take place in the same fiber.
- SourceAvailable from: John Colin Travers[show abstract] [hide abstract]
ABSTRACT: We review the use of hollow-core photonic crystal fibers (PCFs) in the field of ultrafast gas-based nonlinear optics, including recent experiments, numerical modeling, and a discussion of future prospects. Concentrating on broadband guiding kagomé-style hollow-core PCF, we describe its potential for moving conventional nonlinear fiber optics both into extreme regimes—such as few-cycle pulse compression and efficient deep ultraviolet wavelength generation—and into regimes hitherto inaccessible, such as single-mode guidance in a photoionized plasma and high-harmonic generation in fiber.Journal of the Optical Society of America B 01/2011; 28(12):A11-A26. · 2.21 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: Solitary waves have consistently captured the imagination of scientists, ranging from fundamental breakthroughs in spectroscopy and metrology enabled by supercontinuum light, to gap solitons for dispersionless slow-light, and discrete spatial solitons in lattices, amongst others. Recent progress in strong-field atomic physics include impressive demonstrations of attosecond pulses and high-harmonic generation via photoionization of free-electrons in gases at extreme intensities of 10(14) W/cm(2). Here we report the first phase-resolved observations of femtosecond optical solitons in a semiconductor microchip, with multiphoton ionization at picojoule energies and 10(10) W/cm(2) intensities. The dramatic nonlinearity leads to picojoule observations of free-electron-induced blue-shift at 10(16) cm(-3) carrier densities and self-chirped femtosecond soliton acceleration. Furthermore, we evidence the time-gated dynamics of soliton splitting on-chip, and the suppression of soliton recurrence due to fast free-electron dynamics. These observations in the highly dispersive slow-light media reveal a rich set of physics governing ultralow-power nonlinear photon-plasma dynamics.Scientific Reports 01/2013; 3:1100. · 2.93 Impact Factor
Theory of Photoionization-Induced Blueshift of Ultrashort Solitons in Gas-Filled Hollow-Core
Photonic Crystal Fibers
Mohammed F. Saleh,1Wonkeun Chang,1Philipp Ho ¨lzer,1Alexander Nazarkin,1John C. Travers,1Nicolas Y. Joly,1,2
Philip St. J. Russell,1,2and Fabio Biancalana1
1Max Planck Institute for the Science of Light, Gu ¨nther-Scharowsky Strasse 1, 91058 Erlangen, Germany
2Department of Physics, University of Erlangen-Nuremberg, 91054 Erlangen, Germany
(Received 27 June 2011; published 7 November 2011)
We show theoretically that the photoionization process in a hollow-core photonic crystal fiber filled
with a Raman-inactive noble gas leads to a constant acceleration of solitons in the time domain with a
continuous shift to higher frequencies, limited only by ionization loss. This phenomenon is opposite to the
well-known Raman self-frequency redshift of solitons in solid-core glass fibers. We also predict the
existence of unconventional long-range nonlocal soliton interactions leading to spectral and temporal
soliton clustering. Furthermore, if the core is filled with a Raman-active molecular gas, spectral trans-
formations between redshifted, blueshifted, and stabilized solitons can take place in the same fiber.
DOI: 10.1103/PhysRevLett.107.203902 PACS numbers: 42.65.Tg, 05.45.Yv, 32.80.Fb
Introduction.—Hollow-core (HC) photonic crystal fibers
(PCFs)  based on a kagome-lattice cladding have re-
cently been shown to be very interesting for the investiga-
tionof broadbandlight-matter interactions betweenintense
optical pulses and gaseous media. The fibers typically
show transmission bands covering the visible and near-IR
parts of the spectrum with relatively low loss and low
group velocity dispersion (GVD), absence of surface
modes, and high confinement of light in the core. Filled
with a noble gas, they have recently been used in high-
harmonic and efficient deep UV generation from femto-
second pump pulses at 800 nm [2–4]. It has been previ-
ously shown that the Raman threshold can be drastically
reduced in a HC PCF filled with a Raman-active gas (such
as H2) . The system can be used for detailed experimen-
tal studies of, e.g.,self-similar solutionsof the sine-Gordon
equation , backward stimulated Raman scattering [5,7].
The concept of soliton self-frequency blue-shift has
been introduced and predicted in Ref. . Soliton blueshift
has been recently observed in tapered solid-core photonic
crystal fibers  as a result of the variation of the zero-
dispersion wavelength along the fiber. In conventional
band-gap-guiding gas-filled HC PCFs, which have narrow
bands of transmission, a limited ionization-induced blue-
shift of guided ultrashort pulses has been reported [10,11].
Very recently, ultrafast nonlinear dynamics in the ioniza-
tion regime has been studied experimentally in Ar-filled
kagome-style HC PCF  (a detailed account of these
experiments is available in a parallel submission ).
The reasons for the success of kagome HC PCF in
these applications are (i) a GVD that is remarkably small
(j?2j < 10 fs2=cm ? 1 ps2=km from 400 to 1000 nm) in
comparison to solid-core fibers [Fig. 1(a)] [3,13] and
(ii) the gas and waveguide contributions to the GVD can
be balanced by varying the pressure, unlike in large-bore
capillary-based systems where the normal dispersion of the
gas dominates over the waveguide dispersion .
Photoionization in gases is traditionally modeled by
using the full electric field of the pulse . In this
Letter, we first develop a new model to study pulse propa-
gation in gas-filled HC PCFs in terms of the complex
envelope of the pulse. Using this model, we show analyti-
cally for the first time that intrapulse photoionization leads
to (i) a soliton self-frequency blueshift; (ii) long-range
‘‘nonlocal’’ soliton correlations and clustering; and
(iii) spectral transformations of redshifted, blueshifted,
FIG. 1 (color online).
pressures between 1 and 9 bar (calculated from Ref. ). All
subsequent calculations in this Letter assume 5 bar pressure.
Inset: Cross section of a broadband-guiding HC PCF with a
kagome-lattice cladding and a core diameter 30 ?m. Typical
experimental transmission losses for the fundamental mode are
1 dB=m at 800 nm. (b) Comparison of the dependence of the Ar
ionization rate on the pulse intensity using the full model of
Eq. (1) and the linearized model.
(a) GVD of an Ar-filled HC PCF for gas
PRL 107, 203902 (2011)
PHYSICAL REVIEW LETTERS
11 NOVEMBER 2011
? 2011 American Physical Society
and stabilized solitons in Raman-active gas-filled HC
Governing equations.—Photoionization can take place
by either tunneling or multiphoton processes. These re-
gimes are characterized by the Keldysh parameter pK
[14,15]. In the tunneling regime (pK? 1) the time-
averaged ionization rate WðIÞ is given by [16,17]
WðIÞ ¼ dðIH=IÞ1=4exp½?bðIH=IÞ1=2?;
where d ? 4?0½3=??1=2½UI=UH?7=4, b ? 2=3½UI=UH?3=2,
?0¼ 4:1 ? 1016Hz is the characteristic atomic frequency,
UIis the ionization energy of the gas (? 15:76 eV for
argon), UH? 13:6 eV is the ionization energy of hydro-
gen, IH¼ 3:6 ? 1016W=cm2, and I is the laser pulse
intensity. For values of I in the range of 100 TW=cm2,
the Keldysh parameter is pK& 1 for noble gases.
However, experiments show that tunneling models provide
excellent agreement with the experimental measurements
even for pK? 1 [18,19]. As shown in Fig. 1(b), Eq. (1)
predicts an ionization rate that is exponential-like for pulse
intensities above a threshold value. Loss due to absorption
of photons in the plasma is proportional to the ionization
rate. Hence, any pulse with I ? Ithwill have its intensity
strongly driven back to near the threshold value, resulting
in drastically reduced the ionization loss. This allows us to
use the first-order Taylor series to linearize the tunneling
model just above I ¼ Ith, where the optical pulses can
survive for a relatively long time without appreciable
attenuation. Expanding Eq. (1) in its linear regime around
an arbitrary point a ¼ Ia=IHresults in W ? ~ ??I?ð?IÞ,
where ?I ? I ? Ith, ~ ? ¼ de?xð2x ? 1Þ=½4a5=4IH?, Ith¼
aIHð2x ? 5Þ=ð2x ? 1Þ is the threshold intensity, x ¼
, and a is chosen to reproduce the physically
observed threshold intensity in the fiber of Fig. 1(a),
a ffi 2 ? 10?3. The purpose of the Heaviside function ?
is to set the ionization rate to zero below the threshold
intensity. As shown in Fig. 1(b), the linearized model
underestimates the ionization rate then the ionization
loss, in comparison to the full model. This yields to a
similar qualitative behavior between the two models even
for I > Ith, since the ionization rate and the ionization loss
are the key factors in the photoionization process.
One can prove from first principles that propagation
of light in a HC PCF filled with an ionized Raman-active
gas can be then described by the following coupled
@tne¼ ½~ ?=Aeff?½nT? ne??j?j2?ð?j?j2Þ;
i@zþ^Dði@tÞ þ ?KRðtÞ ? j?ðtÞj2?
? ¼ 0;
where ?ðz;tÞ is the electric field envelope, z is the longi-
tudinal coordinate along the fiber, t is the time in a refer-
ence frame moving with the pulse group velocity,
lated at an arbitrary reference frequency !0, ?Kis the Kerr
nonlinear coefficient of the gas, RðtÞ ¼ ð1 ? ?Þ?ðtÞ þ
?hðtÞ is the normalized Kerr and Raman response function
of the gas, ?ðtÞ is the Dirac delta function, ? is the relative
strength of the noninstantaneous Raman nonlinearity, hðtÞ
is the causal Raman response function of the gas [10,20],
the symbol ? denotes the time convolution, c is the speed
of light, k0¼ !0=c, !0is the pulse central frequency,
!p¼ ½e2ne=ð?0meÞ?1=2is the plasma frequency associated
with an electron density neðtÞ, e and meare the electron
charge and mass, respectively, and ?0is the vacuum per-
mittivity, ? ¼ ?1þ ?2is the total loss coefficient, ?1is
the fiber loss, ?2¼AeffUI
loss term, Aeffis the effective mode area, ?j?j2¼ j?j2?
number density of ionizable atoms in the fiber, associated
with the maximum plasma frequency !T? ½e2nT=
ð?0meÞ?1=2. In these coupled equations, the recombination
process is neglected since the pulse duration (of the order
of tens of femtoseconds) is always shorter than the recom-
bination time . If j?j2is measured in watts, ~ ?=cAeff?
?Ihas the dimensions of W?1m?1. This is the nonlinearity
associated with the plasma formation in the fiber.
According to recent experimental measurements , ?K
shows a linear dependence on the gas pressure. These
coupled equations (2) are the first contribution of this
Letter. The validity of Eqs. (2) has been verified by using
a more complete ionization model based on the unidirec-
tional wave equation .
Perturbation theory for floating pulses.—In order to
extract useful analytical information from Eqs. (2), further
simplifications are necessary. For pulses with maximum
intensities just above the ionization threshold (which we
dub floating pulses, a new concept introduced in this Letter
for the first time), the ionization loss is not large and can be
neglected as a first approximation. For such pulses, only a
small portion of energy above the threshold intensity con-
tributes to the creation of free electrons. Furthermore, for
floating pulses one can remove the ? function from the
equations, provided that the cross section ~ ? is replaced by
a properly reduced ~ ?0that takes into account the over-
estimation of the ionization rate . Introducing the
following rescalings and redefinitions: ??z=z0, ??t=t0,
½Aeff?Kz0?, where z0? t2
dispersion length at the reference frequency !0and t0is
the input pulse duration . Hence, the two coupled
equations for floating pulses can be replaced by
is thefull dispersion
operator, ?mis the mth order dispersion coefficient calcu-
2j?j2@tneis the ionization-induced
th, j?j2¼ IAeff, j?j2
th¼ IthAeff, and nTis the total
c ? ?=?0,
2k0z0½!T=!0?2, and ? ? ~ ?0t0=
0=j?2ð!0Þj is the second-order
rð?Þ ? RðtÞt0,
½i@?þ^Dði@?Þ þ rð?Þ ? jcð?Þj2? ??c ¼ 0;
@?? ¼ ?ð?T? ?Þjcj2:
PRL 107, 203902 (2011)
11 NOVEMBER 2011
The total number of photons is conserved in this set of
coupled equations—in contrast to Eqs. (2)—since losses
are neglected for floating pulses.
The effect of the Raman and ionization perturbations on
the soliton dynamics in HC PCFs can be studied by using
Eqs. (3). The second equation can be solved analytically,
?ð?Þ ¼ ?Tf1 ? e??R?
tion ?ð?1Þ ¼ 0, corresponding to the absence of any
plasma before the pulse arrives. For a small ionization
cross section, ?ð?Þ ’ ?R?
jcð?Þj2? ?0@?jcð?Þj2. This allows the two coupled
equations to be reduced to a single partial integro-
i@?c þ^Dði@?Þc þ jcj2c ? ?Rc@?jcj2
that the effect of ionization is exactly opposite to that of
the Raman effect: The fourth term in Eq. (4) involves a
derivative of the field intensity, while the fifth term in-
volves an integral on the same quantity. One can then
conjecture that the last term will lead to a soliton self-
frequency blueshift due to ionization, instead of a redshift
due to Raman self-scattering [25–27]. To prove this state-
ment, we use the perturbation theory described in
Ref. . First, the soliton functional shape is assumed
to be unchanged during the action of the perturbations
induced by the Raman effect and the photoionization
process (this must be verified a posteriori): cSð?;?Þ ¼
A0sechfA0½? ? ?pð?Þ?ge?i?ð?Þ?, with ?pð?Þ is the temporal
location of the soliton peak and ?ð?Þ is the self-frequency
shift. When this ansatz is inserted into Eq. (4), simple
ordinary differential equations can be obtained for both
?ð?Þ and ?pð?Þ, results in ?ð?Þ ¼ ?Ramanð?Þ þ ?ionð?Þ ¼
?g?, ?pð?Þ ¼ g?2=2, and g ¼ gredþ gblue, where gred¼
positive, negative, or even zero, depending on the value of
?, ?R, and A0. By using the exact solution for ?ð?Þ given
previously, one obtains the more precise rate g0
tends to gbluefor small values of ? but starts to differ
considerably from it for A0> ??1. The above solution
clearly shows that, in the range of validity of perturbation
theory (i.e., for floating solitons), photoionization leads to
a soliton self-frequency blueshift. This blueshift is accom-
panied by a constant acceleration of the pulse in the time
domain—opposite to the Raman effect, which produces
pulse deceleration . This blueshift (distinct from the
effects discussed in Refs. [8,9]) is limited only by ioniza-
tion loss, which slowly decreases the pulse intensity until it
falls below the threshold value.
In the presence of ionization-induced losses above the
threshold intensity, Eqs. (2) must be numerically solved
?1jcð?0Þj2d?0g, with the initial condi-
?1jcð?0Þj2d?0, where ? ?
??T. Moreover, in the long-pulse limit jcð? ? ?0Þj2’
0?0rð?0Þd?0. This equation shows clearly
0and gblue¼ ?ð2=3Þ?A2
0. Note that g can be
0?T½ð1 ? ?A0Þ ? ð1 þ ?A0Þexpð?2?A0Þ?, which
to study the full dynamics of floating pulses. Figures 2(a)
and 2(b) show the temporal and spectral evolution of a
high-order input soliton, closely following the results
reported in the companion experimental paper .
When the intensity of the energetic pulse exceeds the
threshold value as a result of self-compression, a funda-
mental soliton is ejected from the main pulse and continues
to blueshift until ionization loss reduces its amplitude
below the threshold value. At longer distances, another
compression occurs and a second soliton is generated.
The use of a kagome-style HC PCF is essential to observe
the soliton blueshift, since conventional photonic-band-
gap fibers have much stronger dispersion variations, which
would quickly destabilize any possible solitary wave as in
Long-range nonlocal soliton forces and clustering.—An
interesting and unexpected interaction occurs between two
solitons when their temporal separation is shorter than the
recombination time, due to the nonvanishing electron den-
sity tail. Using the exact formula for the ionization field
?ðtÞ, one can see that a leading soliton with amplitude A0
can slow down the acceleration of a trailing soliton by an
exponential factor expð?2?A0Þ. The reason is that the
ionization field ?ðtÞ, created by the first soliton, decays
at a relatively slow rate. This establishes a unique nonlocal
interaction between this soliton and other temporally dis-
Figures 3(a) and 3(b) show the output temporal and
spectral dependence of a pulse Nsech? on the soliton order
N. In the presence of ionization loss, when the intensity
of the leading soliton decreases to the threshold value,
FIG. 2 (color online).
of an energetic pulse propagating in an Ar-filled HC PCF. The
temporal profile of the input pulse is Nsech?, with N ¼ 8. The
panels show the ejection of two fundamental solitons that con-
tinuously blueshift until ionization loss reduces their intensities
below the threshold value. Contour plots in this Letter are given
in a logarithmic scale.
Temporal (a) and spectral (b) evolution
PRL 107, 203902 (2011)
11 NOVEMBER 2011
i.e., the blueshifting process ceases, the trailing soliton will
recover its expected blueshift. The reason is the
disappearance of the exponential decaying factor at that
particular point. Also, there is a maximum frequency at-
tained by each soliton that depends on the initial soliton
intensity. These interactions may lead, at some ‘‘magic’’
input energy, to clustering two or more distinct solitons in
both temporal and spectral domains, as shown in Figs. 3(a)
Soliton spectral transformations.—Interestingly, this
perturbation theory for floating solitons predicts the for-
mation of spectrally stabilized solitons in Raman-active
gases due to the different signs and A0dependence of the
Raman and photoionization shifts, gred/ A4
known result ) and gblue/ ?A2
time in this Letter), respectively. If one launches a suffi-
ciently energetic pulse into the fiber, soliton fission takes
place, independent of the particular perturbation applied
[28,29]. This generates a train offundamental solitons with
progressively decreasing peak amplitudes.
The temporal and spectral evolution of such a pulse
when it propagates in a mixture of argon and air
(Raman-active) is depicted in Figs. 4(a) and 4(b). After
the fission process, solitons with intensities less than the
threshold intensity are redshifted by the undisturbed
Raman process. However, solitons possessing intensities
above the threshold value are influenced simultaneously by
Depending on their initial intensities, these solitons can
be initially blueshifted, redshifted, or stabilized. The
Raman self-frequency redshift will be more pronounced
than the ionization self-frequency blueshift for floating
0(reported for the first
solitons possessing initially larger amplitudes (Aj> Acr),
where Acris a critical amplitude. However, for less intense
floating solitons (Aj< Acr) it may happen that exactly the
opposite phenomenon occurs; i.e., the blueshift will domi-
nate. The critical intensity can be estimated from the
equation gredþ gblue¼ 0, giving A2
the ionization loss arrests the photoionization process, the
initially blueshifted solitons start to reverse their self-
frequency shift towards the red. At a certain point, these
solitons can become frequency-stabilized over a short dis-
tance. This may result in multiple collisions between float-
ing solitons if their temporal trajectories intersect. When
the instantaneous intensity exceeds the threshold value
upon collision, a second blueshift event may occur.
Conclusions.—A direct photoionization process can act
on solitons by constantly blueshifting their central frequen-
cies representing the exact counterpart of the Raman self-
frequency redshift when the intensity of solitons is slightly
above the photoionization threshold. This spectral trans-
formation is limited by the ionization loss that restricts the
pulse intensity to the threshold value, hence arresting the
soliton blueshift. The new theoretical model, presented by
Eqs. (2), is suitable for analytical manipulations and has
led us to predict a number of new phenomena such as long-
range nonlocal correlation forces and spectral transforma-
tion between red- and blueshift in Raman-active gases. The
results reveal new physics and offer novel opportunities for
themanipulationand controlofthe solitondynamicsinside
these versatile optical waveguides.
cr¼ 5?=ð4?RÞ. When
FIG. 3 (color online).
an energetic pulse Nsech? after propagating inside an Ar-filled
HC PCF with length ? ¼ 1=4 versus the soliton order N.
Temporal and spectral clustering occur at N ¼ 9:2 due to the
long-range ‘‘nonlocal’’ soliton interactions described in the text.
Temporal (a) and spectral (b) outputs of
FIG. 4 (color online).
of an energetic pulse propagates in a HC PCF filled with argon
and air. The temporal profile of the input pulse is Nsech?, with
N ¼ 4. Soliton temporal and spectral trajectories show an initial
acceleration and blueshift due to plasma formation, an inter-
mediate stabilization below-threshold intensity, and finally a
deceleration and redshift due to the Raman effect. Near the fiber
end, a second blueshift event takes place due to a soliton
collision, generating a second surge of ionized plasma.
Temporal (a) and spectral (b) evolution
PRL 107, 203902 (2011)
11 NOVEMBER 2011
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