# Ultrafast Four-Wave Optical Parametric Amplification in Transparent Condensed Bulk Media

Authors: Audrius Dubietis; Gintaras Tamošauskas; Gintaras Valiulis; Algis Piskarskas
Journal: Laser Chemistry
 (2008)

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## Abstract

We present a short overview of recent advances in ultrashort pulse four-wave optical parametric
amplification in transparent condensed bulk media with Kerr nonlinearity. Highly efficient (10% to 15% pump-to-signal energy conversion) four-wave optical parametric amplification in water and fused silica is experimentally demonstrated. The amplification process highly benefits from 1-dimensional spatial soliton propagation regime, which sets in under proper combination of cylindrical beam focusing and noncollinear phase matching geometry with millijoule pumping. Under these operating conditions, strong four-wave coupling quenches catastrophic beam break-up and filamentation, and the setup operates reasonably below the damage threshold of the nonlinear medium. The proposed methodology offers a number of advantages as compared to guided-wave configurations in gaseous media.


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## Body

## 1. Introduction

Four-wave mixing in transparent isotropic media with
instantaneous nonlinear response (Kerr nonlinearity) is widely used in many
fields of modern photonics: wide band telecommunications, imaging, metrology,
and nonlinear optics [1]. In the ultrashort pulse regime, four-wave mixing
manifests itself through generation of a broadband radiation, often termed
white-light continuum, which can be produced in any media regardless of its
physical state (e.g., solids, liquids, and gases) [2]. Generation of the
white-light continuum is associated with four-wave mixing, and, more precisely,
with four-wave parametric amplification of the quantum noise, where multicolor
radiation emerges at certain directions predefined by the phase-matching
condition in bulk dispersive media. Far from frequency resonances, the
nonlinear third-order susceptibility is rather low and therefore substantially
high laser field intensity is required to induce an appreciable nonlinear
response of the medium. On the other hand, high intensity gives rise to a
manifold of competing nonlinear optical effects: stimulated Raman scattering,
self- and cross-phase-modulation, and self-focusing, simultaneously occurring
almost at the same intensity threshold [3]. In extended, wide-aperture bulk media, four-wave
mixing is univocally linked to femtosecond filamentation phenomena, which
represent an ultimate medium response to high optical intensity, and which
recently experience a boost of interest associated with femtosecond pulse
nonlinear optics [4, 5].The discovery of the four-wave-mixing-driven optical
parametric amplification dates back to the dawn of nonlinear optics [6]. However, real progress in
the field by employing bulk transparent amplifying media has made a sensible
breakthrough only recently. In 1997, Durfee et al. proposed a scheme for the
four-wave parametric amplification in the ultrashort pulse regime, which
exploits self-guided propagation of intense pump beam in noble gas-filled
capillary waveguide [7]. The proposed technique successfully combined all the
essential prerequisites for efficient energy exchange between the interacting
waves through cubic response of the nonlinear medium: almost collinear
phase-matching, high pump intensity, and long interaction length. Moreover, use
of low dispersion gaseous media allowed to achieve broad amplification
bandwidth [8, 9]. More recently, four-wave
optical parametric amplification was demonstrated in the filamentation regime,
where the signal was amplified along the intense femtosecond filament in noble
gas or air [10, 11] and has led to amplification of optical pulses with
duration down to few optical-cycles [12].In condensed bulk media, the conditions for efficient
four-wave optical parametric amplification are much less favored [13]. Due to high material
dispersion, nondegenerate four-wave parametric interactions could be phase
matched only noncollinearly. This severely reduces useful beam overlap length
in tight focusing geometry and implies using of short samples of the nonlinear
media. On the other hand, applicable pump intensity is much limited by the
occurrence of competing nonlinear effects: self-focusing, self- and cross-phase
modulation, stimulated Raman scattering, and eventually, optical damage, which
come into play almost at the same intensity threshold. So far, typical energy
of the pulses amplified via parametric four-wave processes in bulk solid state
media was in the order of ∼1 μJ in the visible spectral
range [14] and few hundreds of nJ in the midinfrared
spectral
range [15]. A new
route in practical implementation of efficient four-wave optical parametric amplification
in water by means of cylindrical focusing and accessing 1-dimensional spatial
soliton propagation regime, which benefits from extended beam overlap distance
in the noncollinear phase-matching geometry, has been proposed [16].In this paper, with an example of water as the
nonlinear medium, we briefly overview the underlying principles that lead to
efficient four-wave optical parametric amplification in bulk condensed media
with Kerr nonlinearity. We demonstrate that the proposed methodology might be
successfully applied to isotropic solid-state media (fused silica, in
particular) to achieve the amplification of high-energy and broadband optical
pulses. We also discuss further directions in development of the four-wave
optical parametric amplifiers (FWOPAs) operating in the ultrashort pulse
regime.


## 2. Phase-Matching Geometry and Experimental Setup

In isotropic medium, the cubic nonlinearity couples
together four frequency components of the optical field through the induced
nonlinear polarization. In the most general case, the nonlinear polarization
might oscillate at positive 22 different frequencies and the observed process
is termed four-wave mixing [17]. Confining ourselves to the particular case of
interaction: four-wave parametric amplification with ωi=ωp+ωp−ωs (here subscripts p,s,i stand for pump, signal, and idler waves), we
will obtain a set of four equations describing the complex amplitude behavior
versus propagation distance z for the signal, idler, and two pump waves. For
simplicity, if we assume that the two pump waves are identical, the interaction
is fully described by the set of three coupled equations:(1)∂As∂z+Ls(As)=−iσsAp2Ai*e−i(Δkz)−iσs|As|2As−iσs(2|Ai|2+2|Ap|2)As,∂Ai∂z+Li(Ai)=−iσiAp2As*e−i(Δkz)−iσi|Ai|2Ai−iσi(2|As|2+2|Ap|2)Ai,∂Ap∂z+Lp(Ap)=−iσpAsAiAp*ei(Δkz)−iσp|Ap|2Ap−iσp(2|Ai|2+2|As|2)Ap,where Aj is the complex amplitude with j=p,s,i referring to pump, signal, and idler waves. σj is the nonlinear coupling coefficient
expressed through third-order nonlinear material susceptibility χ(3) :(2)σj=38ωjcnjχ(3), and Lj is the linear propagation operator, which
describes pulse propagation with the group velocity uj=∂ωj/kj and accounts for the group velocity
dispersion, lateral walk-off, and diffraction:(3)Lj=1uj∂∂t−i2gj∂2∂t2+βj∂∂x+i2kj∇⊥,where uj is the pulse group velocity, gj is the group velocity dispersion coefficient, βj is the propagation angle with respect to the
pump beam axis (in small-angle approximation it takes values of θ for the signal, ϕ for the idler, and 0 for the pump waves), kj=njωj/c is the wave number, and ∇⊥ is the Laplace operator. The first term on the
right-hand side of (1) describes the four-wave parametric amplification, while
the second and third terms describe the effects of self-focusing and
cross-phase modulation, respectively. For the pump beam, cross-phase modulation
term is weak and can be neglected, whereas it plays a relevant role in spatial
reshaping of the signal and idler waves. By contrast, self-focusing term is
essential for the pump beam due to its high intensity, and may not be accounted
in the equations describing propagation of the signal and idler waves. For what
concerns the parametric amplification (first term on the right-hand side), the
strength of induced nonlinear polarization, oscillating at frequency ωi,
crucially depends on the phase mismatch of the interacting waves, which is
expressed as Δk=2kp−ks−ki.
Assuming normal group velocity dispersion, in collinear configuration kp+kp is always smaller than ks+ki, therefore the condition of perfect phase matching Δk=0 is achievable in noncollinear geometry only.
The wave-vector matching diagram for the four-wave parametric amplification is
depicted in Figure 1. It considers an intense pump beam, represented by the sum
of two collinear wave-vectors kp,
signal (ks), and idler (ki) beams propagating at some angles θ and ϕ with respect to the pump beam propagation
axis. The phase-matching condition is described by a vector equation: 2kp=ks+ki,
which, projected in the transverse and longitudinal directions with respect to
the pump beam axis, reads as
(4)2kp=kscosθ+kicosϕ,kssinθ=kisinϕ.Figure 1
Wave-vector diagram describing the
noncollinear four-wave parametric amplification. Color notation is for
illustration of frequencies:ωs>ωp>ωi; note that the pump wave is composed of two photons of ωp.After simple algebra, phase-matching angleθ between the pump and signal beams is expressed
as(5)cosθ=2kp2+ks2−ki24kskp,and which may be found with
account for particular wavelengths and dispersion relation of the nonlinear
medium used.In the experiments, we used a commercial Nd:glass
laser system (Twinkle, Light Conversion Ltd., Vilnius,
Lithuania) which delivered 1-picosecond pulses at 1054 nm with
energy up to 6 mJ at 10 Hz repetition rate. A fraction of either fundamental or
second laser harmonics served as the pump beam for the FWOPA. The seed signal
was delivered by a broadly (320–2600 nm) tunable, second-harmonic pumped optical
parametric generator/amplifier system (Topas, Light Conversion Ltd.). Energy of
the pump beam was adjusted by means of a half-wave plate and a thin-film
polarizer, whereas energy of the seed signal was varied using a variable
neutral density filter. Both pump and signal beams, parallel to
each other, had linear polarization. The beam size of both pump
and seed signal was aligned using suitable telescopes, and the temporal delay
was matched using motorized mechanical delay lines. The beams were loosely
focused onto the input face of a thick nonlinear medium by means of a
cylindrical lens. The complete geometrical arrangement is illustrated in Figure2, providing top and side views of the interaction. The focusing was performed
in the vertical (y) plane, while the beams were crossed in the
horizontal (x) plane, thus ensuring a good spatial overlap
along entire sample length. Note that the input face of the nonlinear medium is
placed exactly at the geometrical focus of the lens, and further 1-dimensional
beam contraction along y-axis inside the medium is due to
self-focusing effect. The beam crossing angle was calculated according to (5),
with accounts for dispersive properties of the particular nonlinear medium and
wavelengths used.Geometrical beam arrangement in the FWOPA
using cylindrical beam focusing.θext is the external phase matching angle, θext≈nθ. CL1 and CL2 are cylindrical lenses for beam focusing and restoring,
respectively.
(a)(b)


## 3. Four-Wave Parametric Amplification in Water: Stable 1-Dimensional Spatial Solitons versus Multiple Filamentations

The first set of experiments was performed using water
as the nonlinear amplifying medium, which was contained within a
variable-length cuvette with 1-mm-thick fused silica windows. The construction
of the cuvette allowed to vary its length and therefore to precisely monitor
beam propagation dynamics inside the nonlinear medium. Second harmonic of the
laser served as the pump beam (λp=527 nm) and cylindrical focusing with fy=+500 mm, fx=∞ lens, yielded almost identical pump and seed
signal beam dimensions of 1.2 mm × 130 μm at the input face of the water cuvette, as
depicted in Figure 3(a). Energy of the pump beam was set to Ep=75 μJ, that is ∼65Pcr in terms of critical power for self-focusing,
where Pcr=3.77λp2/(8πn0n2)=1.15 MW, with n0=1.33 and n2=2.7×10−16 cm/W being linear and nonlinear refractive indexes of water,
respectively. In the absence of the seed signal, highly elliptic pump beam
undergoes gradual contraction in the y-plane, and after passing the nonlinear focus
located at z=25 mm, it breaks up into an array of almost
equally spaced multiple filaments, as shown in Figure 3(b). This break-up
pattern is universal, and has been observed under different operating
conditions and in a variety of nonlinear media [18–21]. Interestingly, further propagation of periodic
multifilamentary structures in water gives rise to phase-matched stokes and
antistokes emission at large angles mediated by the stimulated Raman scattering
[22].Near-field intensity profiles of the pump
beam: (a) at the input and at the output of 40-mm-long water cell; (b) in the
absence; (c) in the presence of the seed signal launched at the phase-matching
angle; (d) measured FWHM diameter of the pump beam (circles and solid curve)
versus propagation distancez within the nonlinear medium and calculated
diffraction spreading of the Gaussian beam of the same diameter (dashed
curve).
(a)(b)(c)(d)Injection of a weak signal beam (λs=490 nm) at the phase-matching angle (θext=1.0 deg) immediately quenches the beam break-up
and filament formation. In this case, pump beam still contracts along its short
(y-axis) direction and transforms into a uniform
light stripe with 24 μm FWHM diameter (Figure 3(c)). This structure
propagates over a considerable distance without apparent diffraction (Figure
3(d)) inside the nonlinear medium, as measured by imaging the output face of
the variable water cuvette onto a 10-bit dynamic range CCD camera (COHU-6612
linked to Spiricon LBA-400PC frame grabber). The diffraction-free propagation
regime of the pump beam extends over a distance of 15 mm, that is almost 4.5
times larger than the diffraction length for a Gaussian beam of the same
diameter (zR≈3.3 mm). As a consequence, the pump beam preserves
its high intensity and remains optimally overlapped with the noncollinear seed,
thus providing an ideal condition for efficient four-wave parametric
amplification to take place. The amplified signal (λs=490 nm) and idler (λi=570 nm) beams exhibit high spatial quality and
diffraction-free propagation as well, with their propagation being driven by
spatial cross-phase modulation imposed by the intense pump beam. The overall
observation suggests formation of stable 1-dimensional
multicolor parametric solitons due to strong
four-wave coupling. Here, off-axis propagating signal and idler waves act as a
damping mechanism that efficiently suppresses any local spike that forms in the
pump beam as a result of modulational instability. In our experiment, the
soliton regime is sustained within a wide range of input parameters, with pump
energy as high as 90μJ and seed signal energy as low as 10 nJ.Figure4 illustrates some of the autocorrelation
measurements of the pump and signal pulses performed by a scanning
autocorrelator, which also employs beam imaging. It is worth mentioning that in
the absence of the seed signal, the pump breaks up not only in the spatial
domain, but also in the temporal domain, resulting in pulse splitting within
each individual filament (the result is presented elsewhere [23]). The split pulses further
undergo complex propagation, similar to that observed in a single filament
propagation dynamics [24]. However, the spatial as well as the
temporal break-ups of the pump are quenched after injection of
the seed pulse. The only observed slight deterioration of the pump pulse
temporal profile (Figure 4(b)) occurs due to significant pump energy depletion.
Under these settings, the amplified signal retains a clean Gaussian intensity
profile as shown in Figure 4(d) and exhibits some slight pulsewidth narrowing
from 0.7 picosecond to 0.5 picosecond as a result of parametric gain.Normalized autocorrelation functions of (a)
input pump pulse, (b) depleted pump pulse at the FWOPA output, (c) input seed
signal, and (d) output amplified signal. Red curves represent a corresponding
Gaussian fit. Output pulses were measured after propagation in 40-mm-long water
cuvette, withEp=60μJ and pump-to-signal conversion of 15%.
(a)(b)(c)(d)Summarizing the above observations, we conclude that
noncollinear four-wave coupling is a highly efficient mechanism
of the removal of space-time instabilities of
1-dimensional spatial solitons in bulk nonlinear medium in the ultrashort pulse
regime. It is worth mentioning that the stability problem of 1-dimensional
spatial solitons in bulk media attracts a great attention from the fundamental
and applied research (see [25] and references therein for an overview on the related
topics) and stable 1-dimensional spatial solitons in the ultrashort pulse
regime were never observed before.


## 4. Energy Scaling

Water FWOPA provides extremely high energy conversion
efficiency from pump to parametric waves with a figure as high as 25% in the
gain saturation regime withEp=60μJ. The obtained result suggests a signal gain
factor of ∼100,
with almost 15% pump-to-signal energy conversion. A distinct property of water
FWOPA, operating at signal and idler wavelengths close to
those of the pump, concerns a strong cascading
effect, which manifests itself as an emergence of a series of frequency
upshifted and downshifted components at the output, as shown in Figure 5.Figure 5
FWOPA output imaged on the screen after the
wavefront restoring cylindrical lens. The intense green spot in the center is
the pump, the closest cyan and green spots are the signal and idler waves,
respectively, while the remaining colorful spots are additional frequency
components generated by the cascading.The cascaded frequency components are generated
through the multistep four-wave mixing processesks1=2ks−kp, ks2=2ks1−ks,
and so forth [14].
Here, genuine signal and idler waves and their cascaded counterparts, indicated by subscript numbering, serve as a new pump in the particular four-wave
mixing process. The unequal number and uneven energy distribution between the
generated frequency components can be explained by a particular phase-mismatch,
which is established under predefined propagation
angles and nontrivial impact of increasing linear absorption of water towards
lower frequencies. The impact of these processes is investigated in more detail
in [16, 23]. Here we just mention that
under present operating conditions, the cascaded components comprise only up to
20% of the total output energy and contribute with
less than 5% of the overall energy conversion. The number as well as energy of
the cascaded components can be suppressed by tuning wavelength of the seed
signal farther from that of the pump. The amplified signal energy is 10μJ, that compares to the signal energy obtained
via four-wave mixing in noble gases and air under
millijoule-pulse pumping [9–11]. Being
different from guided-wave schemes, the pump beam size in
1-dimensional spatial soliton regime is not limited along one of the transverse
dimensions (along x-axis), so present setup offers an excellent
possibility of easy beam-size and energy scaling.In order to demonstrate energy scaling capabilities of
the FWOPA, we performed a second experiment on the four-wave optical parametric
amplification in 12-mm-long fused silica (UV-grade, type KU-1) sample with
pumping at fundamental laser frequencyλp=1054 nm and seeding with λp=730 nm. General settings in the interaction scheme
are very similar as those depicted in Figure 2. The initial beam sizes of the
pump and seed signal were expanded by suitable telescopes, focused with fy=+750 mm, fx=∞ lens, yielding beam FWHM dimensions at the
input face of the fused silica sample of 3.7 mm × 195 μm and 2.8 mm × 135 μm, respectively, and crossed at θext=1.59° so as to ensure perfect phase matching.The operational characteristics of the fused silica
FWOPA at high energy pumping (Ep=1.8 mJ) are depicted in Figure 6. By varying
energy of the seed pulse from 0.3 μJ to 16 μJ, we access the complete coverage of the
amplification regimes: from the small signal gain up to the gain saturation.
The highest small signal gain of ∼80 is measured within seed energy interval of
0.3–1.5 μJ. The gain factor gradually drops with
increasing seed signal energy and saturates at ∼10 with 15 μJ seed. In this condition, the largest
amplified signal energy of 180 μJ is achieved, pointing to an appreciable
pump-to-signal energy conversion efficiency of 9.2%. As compared to previous
results of water-based FWOPA, the present configuration is able to support
high-energy amplification regime in condensed medium, and what is to the best
of our knowledge the first demonstration of four-wave parametric amplification
in transparent bulk medium with millijoule pumping.
We also underline that such an impressive amplified signal energy extracted
from the present scheme outreaches those produced so far in gaseous media by
almost 20 times [7–12].Figure 6
Amplified signal energy versus seed pulse
energy measured atEp=1.8 mJ in 12-mm-long fused silica sample.Figure7 illustrates the central portion of the intensity
profiles of input (seed) and output (amplified)
signals, demonstrating how it reshapes into a
smooth light stripe having ∼17 μm FWHM diameter imposed by the propagation
dynamics of the intense pump beam, which experiences 1-dimensional
self-focusing. The pump beam, as measured at the output face of the fused
silica sample, contracts along the y-axis to a dimension of ∼25 μm, whereas its wider (x-axis) dimension remains almost the same. The
details of its spatial dynamics inside the sample were not captured due to
fixed sample length, however, the observation of the spatial profile at the
sample output suggests that no break-up into multiple filaments and no optical
damage neither inside the sample nor on its output face there occurs at
energies as high as 1.9 mJ. We have evaluated the pump beam intensity of 210 GW/cm2 and 1.7 TW/cm2 assuming Gaussian spatial and temporal profiles at the input and
output faces of the fused silica sample, respectively. The latter converts to a
maximum fluence of 1.8 J/cm2 without an account of possible multiphoton absorption and energy conversion
into the parametric waves. We note that the damage fluence for fused silica is
2.5 J/cm2 for 1-picosecond pulses [26], so our operating conditions are safely below the
optical damage threshold.Near-field intensity profiles of (a) seed
signal and (b) amplified signal at the fused silica FWOPA input and output,
respectively.
(a)(b)


## 5. Broadband Four-Wave Optical Parametric Amplification

One of the most important characteristics of any
optical parametric amplifier is the gain bandwidth, which essentially
determines its ability to amplify ultrashort light pulses. In the framework of the
ultrashort pulse propagation, the amplification bandwidth is defined by the
group-velocity (vg=∂ω/∂k) differences between the signal and idler
waves that convert to a wave-vector mismatch in the
first-order approximation [27]:(6)Δk=(1vgi−1vgs)Δω.In conventional three-wave optical parametric
amplifiers, the broadband phase-matching is achieved by setting an appropriate
angle between the pump and signal beams so as to match their wave-vector
projections along the signal propagation direction:vgs=vgicosΘ [28], assuming that vgs<vgi (that is the case of normal group-velocity
dispersion) and where Θ denotes the angle between the signal and
corresponding idler frequency components. If the propagation direction of the
signal is fixed, the angle Θ depends solely on its wavelength. The same
considerations hold also for the four-wave optical parametric amplification.
Figure 8 shows the phase matching curve in fused silica pumped by the
fundamental harmonic of the Nd:glass laser (λp=1054 nm). The character of the curve prompts that
for signal wavelengths around 730 nm, there exists an optimum beam crossing
angle of θext=1.59°,
which keeps nearly constant over considerable signal wavelength range
(indicated by a bold red curve). The corresponding idler waves (with wavelength
from 1675 nm to 2240 nm) fan out within a wide range of angles (3.5°–5°,
bold blue curve) when tuning the signal from 770 nm to 690 nm.Figure 8
Phase-matching curve for the four-wave
optical parametric amplification in fused silica. Wave-vectors of the pump withλp=1054 nm lie on the propagation axis and are
indicated by an arrow. Expected broadband amplification range for the signal is
highlighted by the red curve, whereas the blue curve marks the corresponding
idler wavelength range.Experimentally, in the first approach, we have scanned
the amplification bandwidth of the FWOPA by launching a tunable narrow-band (3 nm FWHM) seed signal from the parametric generator/amplifier system Topas,
maintaining a constant delay between the pump and seed signal pulses and fixed
crossing angle. Wavelength of the seed signal was varied in 5 nm step from 680 nm to 820 nm. The result is plotted in Figure9, where the amplification
bandwidth was measured in the small-signal (Es=1 μJ) amplification regime and in the gain
saturation regime with Es=8 μJ. Both scanned spectral gain profiles are
quite similar, indicating FWHM amplification bandwidth of ∼75 nm.Figure 9
Scanning of the FWOPA amplification bandwidth
with a narrow-band seed signal in the small signal amplification regime (red
curve) and in the gain saturation regime (blue curve).And finally, we have seeded the 12-mm-long fused
silica FWOPA with a broadband white-light continuum signal with central
wavelength at 740 nm, which was generated by self-focusing of a pulsed Bessel
beam. The broadband seed signal was produced as follows. The Gaussian output
pulse of 1-picosecond duration atλ=740 nm from the commercial optical parametric
generator/amplifier system was directed to a BK7-glass axicon with apex angle
of 175.4°,
which produced a Bessel beam of the same pulsewidth, with diameter of the
central core of ∼10 μm. The ultrashort-pulsed Bessel with energy of ∼50 μJ was made to propagate in 20-mm-long fused
silica sample, where it generated a narrow axial beam of high spatial quality
with white-light spectral extent [29]. Here we note that the broadband axial radiation is
easily separable from the intense spectral component of the pump Bessel beam by
placing an iris aperture outside the Bessel zone. The spectral envelope of the
white-light continuum radiation averaged over 100
laser shots is depicted by a dashed curve in Figure 10, and has a width of 170 nm estimated at 1/e2 level. Spatial dimensions of the white-light
continuum beam were matched to those of the pump in a similar manner as
described in the previous experiments. The energy of the broadband seed signal
at the FWOPA input was 0.75 μJ.Figure 10
Amplified spectrum of the white-light
continuum signal. Dashed curve indicates the input spectrum of broadband
radiation.The experimental conditions (focusing geometry and the
input beam sizes) were identical to the previous experiment, and the beam
crossing angle was kept atθext=1.59° so as to ensure broadband phase matching.
Under these settings, the seed signal was amplified up to 11.5 μJ energy with 1.8 mJ pump. We note here that
the amplified signal energy could be considerably increased optimizing the seed
pulse energy as demonstrated with a narrow-band seed experiment; see the
results depicted in Figure 6. The spectrum of the amplified signal is shown in
Figure 10. Its estimated FWHM bandwidth is 73 nm, that is almost identical to
that obtained with narrow-band signal scanning.The pulse compression experiments with precise
measurements of the pulse duration are in progress; we note, however, that the
amplified spectrum width corresponds to a transform-limited pulse of∼10 fs duration.


## 6. Conclusions and Outlook

To summarize the results presented above, we have
demonstrated an operating condition and methodology for highly efficient
four-wave optical parametric amplification in isotropic transparent condensed
media with Kerr nonlinearity. Our approach is based on cylindrical focusing
geometry and utilization of 1-dimensional spatial soliton propagation regime,
which comprise a complex solution for the problems generally encountered in the
four-wave optical parametric amplification in bulk media so far
as follows.(i)
Cylindrical focusing enables exact fulfillment
of the noncollinear phase matching condition without reduction of the
interaction length and allows high conversion efficiency to be achieved.(ii)
Stable 1-dimensional spatial soliton
propagation regime ensures high beam quality and suppresses self-action effects
in the space-time domain that lead to beam break-up, filamentation, and pulse
splitting. In what follows, the setup operates reasonably below the damage
threshold.(iii)
Being different
from guided-wave configurations (hollow fibers and light filaments),
1-dimensional spatial solitons are not limited in aperture and therefore can be
made to carry a large amount of energy, which can be easily scaled by
increasing one (wider) transverse dimension without altering the propagation
dynamics.(iv)
There is a great flexibility in achieving
broad gain bandwidth in the four-wave parametric amplification process. It can
be set either by making use of natural flattening of the noncollinear
phase-matching curve or suitable signal pulse-front tilting (so-called
achromatic phase matching) or eventually combining both possibilities
altogether.(v)
Methodology of the four-wave optical
parametric amplification is highly flexible in terms
of wavelength range and therefore laser sources
are to be used, and can be easily adapted to broad
wavelength range from the UV up to mid-IR.(vi)
Condensed isotropic media have several
advantages over gaseous media concerning their smaller dimensions and higher
third-order nonlinearity. This property is essential for designing compact and
high power four-wave optical parametric amplifiers.The proposed four-wave optical parametric
amplification methodology could be adapted to virtually all transparent media:
solids, liquids, and gases and throughout a variety of wavelengths, especially
those where conventional phase-matching techniques based on material birefringence
are of limited use or not applicable at all. Using second-harmonic pump pulses
from Nd:glass or Ti:sapphire laser systems, four-wave optical parametric
amplification in transparent bulk isotropic media may be readily shifted into
the UV spectral region, where conventional phase matching techniques based on
material birefringence suffer from many serious constraints. Employing
achromatic phase matching technique [30] (that is imposing angular dispersion on the seed signal
such that individual spectral components that comprise the broadband pulse are
made to propagate at different angles to fulfill the phase-matching condition),
it is possible to further extend the gain bandwidth and/or achieve tunability
of the amplified signal. Taking an advantage of energy scaling capabilities,
the proposed methodology might foresee the amplification of few-optical cycle
pulses to millijoule energy. And finally, our
finding of quenching the spontaneous nature of beam break-up and filamentation
by means of launching a weak seed signal opens a new route towards controllable
propagation of intense laser beams.



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*Source: 534951-2008-09-21.xml*