Taane Stevenson has not received any gifts yet
PHYSICAL REVIEW X 6, 031037 (2016)
21603308=16=6(3)=031037(13) 0310371 Published by the American Physical Society
in a frame moving with the pulse group velocity vg, and wξ
is the axial pulse length. This field has a moving null at
(x ¼ 0, ξ ¼ 0) along the y axis, perpendicular to the
direction of propagation, and the circulation Γ on closed
spatiotemporal contours in (x, ξ) around the null is
quantized. We refer to this apparently previously unexplored
object as a spatiotemporal optical vortex (STOV).
We contrast the STOV construction with other recent
work, such as by Eilenberger et al. [9] or Mihalache et al.
[10], where beams with nontrivial spatiotemporal coupling
are created with embedded vorticity, but the
vorticity is fundamentally similar to the LG construction
given above, where phase and energy circulation from the
vortex occur only in the transverse spatial dimensions.
We refer to optical vortices with phase circulation solely
in spatial dimensions as spatial optical vortices, and note
that, as far as we know, this definition encompasses all
optical vortices studied to date. Outside of optics there
has been theoretical and experimental work on systems
with analogs of the STOV construction, such as work by
Nicholls and Nye in acoustics [11] and Kartashov et al. in
quantum fluids [12].
In this paper, we demonstrate that STOVs are a
fundamental and universal feature of optical pulse collapse
and arrest in selffocusing media. Their existence
in nonlinear ultrafast pulse propagation appears to be
ubiquitous, and their creation, motion, and destruction is
strongly linked to the complex spatiotemporal evolution
of the pulse.
II. SPATIOTEMPORAL OPTICAL VORTICES
Optical collapse is a fundamental phenomenon in nonlinear
optics [13]. It occurs when the laserpulseinduced
change in the medium’s refractive index generates a selflens
whose focusing strength increases with intensity. Above a
critical power level (Pcr), selflensing exceeds diffraction,
and the pulse experiences runaway selffocusing. In the
absence of “arrest” mechanisms terminating selffocusing,
the pulsewould collapse to a singularity. In reality, additional
physical effects intervene. For example, in the case of
femtosecond filamentation in ionizing media [14], plasma
generation acts to defocus the pulse when the peak selffocused
beam intensity reaches the ionization threshold.
Other collapse arrest mechanisms include dispersioninduced
pulse lengthening [15], cascaded thirdorder nonlinearities
[16], vectorial effects from beam nonparaxiality
[17], and, in the case of relativistic selffocusing, electron
cavitation [18].
In both calculations and simulations [19,20], it has been
found that the following modified paraxial equation for
wave evolution, in the form of a timedependent nonlinear
Schrödinger equation, is well suited to describe optical
pulse collapse and collapse arrest for beam propagation
along z:
2
∂
∂z
ik −
∂
∂ξ
ψ þ ∇⊥
2ψ − β2
∂2ψ
∂ξ2 þ k2Vfψg ¼ 0. ð1Þ
Here, E ¼ ψðr⊥; z; ξÞeiðkz−ωtÞ is the dimensionless
electric field component, ψ is the pulse envelope,
β2 ¼ c2k0ð∂2k=∂ω2Þ0 is the dimensionless group velocity
dispersion at ω ¼ ω0, r⊥ and ξ are as before, and z is the lab
frame axial position of the pulse. The physics of selffocusing
and arrest is contained in the functional Vfψg. To demonstrate
toroidal STOV generation in the arrest of selffocusing
collapse, we perform propagation simulations using Eq. (1),
imposing azimuthal symmetry, and include electronic, rotational,
and ionization nonlinearities in Vfψg [19,21]. The
Gaussian input pulse is 3mJ, 45 fs (Gaussian FWHM) with an
input waist w0 ¼ 1 mm, and the propagation medium is
atmospheric pressure air.As our experiments (see Sec. III) are
intentionally operated in the single filamentation regime,
where multifilamentary modulational instabilities are precluded,
azimuthal symmetry is a good approximation [14].
Figure 1(a) is a postcollapse plot of the pulse phase at
z ¼ 120 cm, which shows the emergence of two oppositely
wound and oppositely propagating STOVs [v1 (þ1 charge,
forward) and v2 (−1 charge, backward)] entrained between
the higher intensity core of the beam and the beam periphery.
The delayed rotational nonlinearity from the N2 and O2 air
constituents [22] forms an additional vortex pair ∼100 fs
behind the main pulse, v3 (þ1, forward) and v4 (−1,
backward), where v3 is shown bisected in Fig. 1(a) and v4
has exited the simulation window. Further evolution of the
simulation shows that v2 and v3 collide and annihilate, while
v1 continues propagating with the most intense part of the
pulse (which we discuss later). We note that the delayed
generation of v3 and v4, where the pulse intensity is many
orders of magnitude weaker than at its peak, shows that
STOVs can also be generated linearly by an imposed
spatiotemporal index transient. STOVs are not merely mathematical
freeriders on intense propagating pulses: as we
show, a real energy flux j circulates either as a saddle (for
β2 > 0) or as a spiral (for β2 < 0) in the (r⊥, ξ) plane
surrounding the STOV core.
To understand the generation of these spatiotemporal
toroidal structures, recall that phase vortices in fields are
closely associated with localized field nulls [6]. In arrested
selffocusing, field nulls occur as a natural part of the
dynamics and spawn toroidal vortices of opposite charge.
This can be illustrated by the toy model of Fig. 2, which
shows the effect on a beam of abruptly spatially varying
selfinduced change in refractive index, which we model
here as a sharp transverse step. We consider lefttoright
propagation of the “half plane wave” pulse Eðx; z; ξÞ ¼
E0½δ þ hðxÞe−ðξ=ξ0Þ2ei½zðk−ω=vgÞ−ωξ=vgeiϕNLðx;z;ξÞ, neglecting
dispersion and diffraction, where ξ and z are as before,
δ ≪ 1, hðxÞ ¼ 1 for x < 0 and 0 otherwise, ϕNLðx; z; ξÞ ¼ kn2jEðx; z; ξÞj2z is the sharply stepped nonlinear Kerr
phase, and n2 is the nonlinear index of refraction. At
N. JHAJJ et al. PHYS. REV. X 6, 031037 (2016)
0310372
z ¼ 0, the phase fronts are aligned for all x. As the pulse
propagates, the Gaussian intensity distribution causes the
front of the pulse to redshift and the back of the pulse to
blueshift, with spatiotemporal phase front shear developing
between the pulse “core” at x < 0 and periphery at x > 0.
When the propagation reaches z ¼ zv ¼ πðkn2E20
Þ−1, the
peak of the pulse in the core is π out of phase with the
periphery, and the electric field magnitude nulls out at
the single point (ξ ¼ 0, x ¼ 0, z ¼ zv), marked by a circle,
forming an “edge dislocation” [6]. Upon further propagation,
continued phase shearing spawns two null points of opposing
(1) phasewinding, marked by circles (z ¼ 2zv). These two
STOVs, whose axes are along y (perpendicular to page),
immediately begin moving apart: one vortex advances in time
towards the front of the pulsewhile the other vortex moves to
the back. Similar dynamics are theorized to exist in monochromatic
breather solitons, where ringshaped vortices are
formed quasiperiodically throughout propagation [23].
These general features occur in simulations of
selffocusing collapse arrest. In Fig. 3, we show 2D
profiles of the pulse’s phase ϕðr; z; ξ ¼ ξvÞ and intensity
cjEðr; z; ξ ¼ ξvÞj2=8π for a simulation using Eq. (1),
tracking the moving plane ξ ¼ ξv, where the phase singularity
and field null first appear. Log lineouts of the
intensity (normalized to 1013 W=cm2) are overlaid on
the 2D profiles. As the initial (z ¼ 0) Gaussian input beam
selffocuses, a strongly peaked highintensity central core
(similar to the Townes profile [24]) develops, surrounded
by a lowerintensity periphery, with a sharp transition knee
between them (z ¼ 156 cm). The associated phase plot
shows the central core having accumulated a much larger
nonlinear phase shift than the periphery. During collapse,
the knee moves radially inward, preventing phase shear
from accumulating substantially at any particular radial
location. However, as the core peak intensity rises to the
point where ionization begins (z ¼ 160 cm, not shown),
the location of the intensity knee stabilizes and highly
localized phase shear builds up, greatly steepening the
transition between core and periphery, with the field
beginning to dip towards a null (z ¼ 166 cm). Finally,
(a)
(b)
FIG. 1. (a) Phase and intensity projections of simulated pulse, in local coordinates, showing STOV generation. Propagation medium:
atmospheric pressure air. Black dashed lines encircle vortices v1, v2, and v3, and arrows point in the direction of increasing phase. Our
phase winding convention considers a ðξ − ξ0Þ þ iðr − r0Þ winding about a null at (ξ0, r0) to denote a þ1 STOV, giving the v1 STOVa
þ1 charge. The white dots in the intensity projection are centered on the locations of vortices v1, v2, and v3. (b) Simulation of an air
filament crossing the airhelium boundary for the conditions of (a), showing the beam fluence and plasma density. Nonlinear
propagation terminates as the beam transitions from air to helium, whereupon the beam and a reference pulse (not shown; see
Appendix A) are directed to an interferometer.
SPATIOTEMPORAL OPTICAL VORTICES PHYS. REV. X 6, 031037 (2016)
0310373
only a short propagation distance later (z ¼ 167 cm),
sufficient shear develops that the core and periphery are
π out of phase at the slice ξ ¼ ξv, with the dip in the
simulation becoming orders of magnitude deeper, forming
a ringshaped null surrounding a core of relatively flat
intensity and phase. Note that the coreperiphery phase
difference Δϕcp jumps by 2π between z ¼ 166 and 167 cm.
[As seen in the computation of Fig 1(a), the null then
spawns two oppositely charged (1) toroidal vortices that
propagate forward and backward.] The plane ξ ¼ ξv is still
shown at z ¼ 178 cm, but the vortices have now migrated
out of that plane. Later in the paper, we derive and discuss
equations of motion describing vortex dynamics within the
moving reference frame.
III. EXPERIMENT
A. Experimental concept
In order to experimentally confirm the existence of
STOVs, we image the spatiospectral phase and intensity
profiles of femtosecond laser pulses midflight during their
precollapse and postcollapse evolution in air. Until now,
we have been discussing STOVs as a spatiotemporal
phenomenon, but they are also vortices in their spatiospectral
representation, which has enabled us to unambiguously
observe them. Why should a beam with a STOV have a
vortex in the spatiospectral domain? For a small temporal
chirp of the electric field where the vortex is embedded,
there is a simple linear mapping between time and
z = 0
core
periphery
ξ 0
0
x
z = zv = π (kn2E0
2)1
ξ 0
1 +1
+12π
+14π
z = 2zv
ξ 0
FIG. 2. Toy model showing birth of a vortex pair via spatiotemporal phase front shear. The white curve and arrow depicts the axial
(temporal) intensity profile and propagation direction, while the “core” and “periphery” labels denote the spatial intensity step. The
z ¼ 0 panel shows the initial condition where phases are aligned, the z ¼ zv panel shows the birth of the null (vortices overlap), and the
z ¼ 2zv panel shows continued shear carrying the vortices apart, with the þ1 vortex moving to the temporal front and the −1 vortex
moving backward. Our phase winding convention considers a ðξ − ξ0Þ þ iðx − x0Þ winding about a null at (ξ0, x0) to denote a þ1 STOV.
In the third panel, red and black arrows indicate the direction of increasing phase. The two arrows connect the same lines of constant
phase; the arrows show that the phase difference between head and tail is ill defined due to the vorticity of the −1 STOV.
input
z = 0 cm
250 m
0 
1 
2 
3 
4 
10(I/I0)
collapsing
z = 156 cm
0 
1 
2 
3 
4 
before vortex
z = 166 cm
0 
1 
2 
3 
4 
vortex creation
z = 167 cm
0 
1 
2 
3 
4 
vortex moved
z = 178 cm
rad
1
0
1
2
3
4
5
0 
1 
2 
3 
4 
TW/cm2
0
10
20
30
40
FIG. 3. Simulations of beam phase (top) and intensity (bottom) at the axial or temporal slice ξv, where the STOV pair first appears.
From left to right, the pulse is advancing along z, with the input shown at z ¼ 0, a collapsing beam at z ¼ 156 cm, ionization onset at
160 cm (not shown), just before the vortices spawn at 166 cm, just after the vortices spawn at 167 cm, and an image where the vortex
pair has moved out of the ξ ¼ ξv plane. The linear intensity images are overlaid with centered lineouts of log10ðI=I0Þ, where
I0 ¼ 1013 W=cm2. Experimental parameters are used as code inputs: w0 ¼ 1.3 mm, pulse energy 2.8 mJ (P=Pcr ¼ 6.4), pulse
FWHM 45 fs.
N. JHAJJ et al. PHYS. REV. X 6, 031037 (2016)
0310374
frequency. This leads to the vortex appearing in the
spatiospectral as well as the spatiotemporal domain. In
Appendix A1, we discuss a Gaussian beam with a
temporally centered toroidal STOV and show that the
spatiospectral representation of the beam possesses a vortex
with the same spatial radius. We note that the relationship
between vorticity in the spatiotemporal and spatiospectral
domains is, in general, complex, with the full field
distribution (including vortices) in one domain contributing
to an individual vortex in the conjugate domain. Because
our experimental images identify STOVs after optical
collapse arrest has occurred, our results avoid this complexity,
as simulations show that by this point there is only one
(1) STOV propagating with the filamenting pulse.
Direct measurement of the spatial phase and intensity
profile of a filament in midflight is not amenable to standard
techniques; the use of relay imaging or beam splitters is
subject to the severe distorting effects of nonlinear propagation
andmaterial damage.However, by interrupting nonlinear
beam propagation by an airhelium interface, the inflight
beam intensity and phase profile can be linearly imaged
through helium, taking advantage of the very large difference
in instantaneous nonlinear response between helium and
air (n2;He=n2;air ≈ 0.04 [22,25]). This helium cell technique
was first employed by Ting et al. [26] to measure the inflight
intensity profile of a femtosecond filament. Here, we extend
the technique to also enable measurement of the pulse
transverse phase profile. In Appendix A, we discuss the
details of how we produce the airhelium interface and how
the <4mmthick airhelium transition layer is sufficiently
thin to enable distortionfree imaging of the air filament cross
section. The interface is movable along the laser propagation
axis to allow midflight filament intensity and phase imaging
over the full propagation path.
A conceptual view of the experiment is presented in
Fig. 1(b), where we show the postcollapse pulse from the
simulation of Fig. 1(a) encountering the airhelium interface
and terminating nonlinear propagation, after which it is
relay imaged through a folded wave front interferometer
and combined there with a weak femtosecond reference
beam with flat spatial and spectral phase. The resulting
interferogram encodes the 2D spatial phase and intensity
of the inflight filamenting (signal) beam at the airhelium
interface, averaged over the ∼20 nm bandwidth of the
λ ¼ 800 nm reference arm. In effect, the reference arm
spectrally gates the STOV, resulting in a spatiospectral
interferogram centered at the reference pulse bandwidth.
Spectral gating is crucial to our measurements. If the
reference arm had the same wide bandwidth as nonlinearly
generated in the signal (filamenting) arm, the signature
of the STOV, which is present over a smaller spectral
window, would be washed out. A detailed discussion of the
experimental setup, the spatiospectral representation of
STOVs, and the interferometric analysis is found in
Appendix A.
B. Experimental results
The experiment is performed by scanning the helium cell
axial position over a range covering both precollapse and
postcollapse propagation for all energies used. The data
consist of a densely spaced collection of beam intensity and
phase images at various cell positions ðzhÞ and input pulse
energies ðεiÞ. For all measurements, collimated beams are
launched with w0 ¼ 1.3 mm and FWHM intensity pulse
width τ ¼ 45 fs.
Figure 4 shows the beam onaxis phase shift Δϕ with
respect to the interferometric reference pulse as a function
of P=Pcr at a fixed position of zh ¼ 150 cm after launch,
where Pcr ¼ 3.77λ2=8πn0n2 for our Gaussian input beam
profile and P ¼ ϵi=τ is the input power. The red points are
averages over 2600 shots (blue points) in 150 energy bins.
It is important to note that the scatter in Δϕ of roughly
1 rad at any given laser power is constant across all
powers measured, including P=Pcr ≪ 1, where we could
not detect any nonlinear phase. Therefore, the scatter is due
to the shottoshot interferometric instability of the measurement
and is not intrinsic to the filamentation process.
The most striking aspect of the plot is the abrupt jump in
beam central phase of approximately ∼2π at P=Pcr ∼ 5.
The phase goes from positive and rapidly increasing
(increasing selffocusing) to abruptly negative (defocusing),
providing a clear signature of the transition from the
precollapse to the postcollapse beam. For nominally constant
laser power right at the jump, the phase fluctuates in
the range ∼ π, showing the extreme sensitivity of the
phase flip.
A more revealing way to display what is happening at the
collapse is shown in Fig. 5. Here, for given zh, the phase
images (zh, εi) are searched for εi or P=Pcr, where the
P/Pcr
phase (rad)
4
2
0
2
4

0 5 10
FIG. 4. Beam onaxis phase shift (with respect to flatphase
reference arm) as a function of pulse power at zh ¼ 150 cm. The
phase jumps abruptly by ∼2π at P=Pcr ∼ 5, providing a clear
signature of the transition from the precollapse to the postcollapse
beam. The red points are averages over 2600 shots (blue points)
in 150 energy bins.
SPATIOTEMPORAL OPTICAL VORTICES PHYS. REV. X 6, 031037 (2016)
0310375
central phase appears to randomly flip sign from shot to
shot. These are the powers at which pulse collapse is
observed for each position, just as P=Pcr ∼ 5 is for zh ¼ 150 cm in Fig. 4. The top row of Fig. 5 shows beam phase
and intensity images for input power P=Pcr ¼ 4.4 (at
zh ¼ 165 cm). Because the onset of collapse arrest is
extremely sensitive to fluctuations in the beam energy
(as seen in Fig. 4), these images span the possibilities of
prearrest through postarrest of the collapse, and typically
three types of images appear. Figure 5(i) shows strongly
peaked intensity and positive phase; the beam is collapsing,
but arrest has not yet begun. Figure 5(ii) shows radically
different images, as does Fig. 5(iii): the intensity images
show narrow ringshaped nulls embedded in a relatively
smooth background, and the phase images show a sharp yet
smoothly transitioning jump close to π or −π across the
rings, with the phase jumps flipped between Figs. 5(ii)
and 5(iii). We note that the smooth phase transition from
the periphery to the core rules out 2π phase ambiguities in
interferometric phase extraction.
The bottom row of Fig. 5 plots, for a range of P=Pcr, the
phase difference jΔϕcpj between the core and periphery of
the beam. To do this, for each phase image we compute the
difference between the maximum and minimum values of
the phase within a 60μm box centered about the largest
spatial phase gradient, the radial location of which defines
core and periphery. For each nominal value of P=Pcr, it is
clear that as the phase gradient becomes large, the phase
difference saturates at π. Near saturation, roughly 50% of
the shots have the core phase advanced from the periphery
while the others show the reverse.
The evidence from Figs. 4 and 5, and comparison to the
simulation of Fig. 3, strongly suggests that we are imaging
spatiotemporal vortex rings: the abrupt appearance of ringshaped
nulls in the field magnitude accompanied by 2π
phase jumps in jΔϕcpj across the null—these are exactly the
signatures of a vortex. Because the circulation around a
general singly charged vortex is 2π, examining our vortex
in the spatiospectral domain ðr⊥;ωξÞ, one would expect,
depending on the sign of vortex winding, that the coreperiphery
phase difference jΔϕcpj jumps by 2π from ωξ
slices just before the vortex (Δϕcp ¼ π) to ωξ slices just
after (Δϕcp ¼∓ π). For example, before a vortex of charge
þ1, the core is phase advanced with respect to the
periphery; after the vortex, it is phase retarded. This is
exactly what we observe experimentally and what is
predicted in the simulation of Fig. 3 and its spatiospectral
counterpart, where even the ∼400μm diameter of the
vortex ring is accurately determined. Of the four STOVs
that simulations show are generated at collapse arrest in air
[see discussion of Fig. 1(a)], only the temporally foremost
þ1 STOV does not annihilate or separate from the bulk of
the pulse. Using Fig. 1(a) as a guide, we interpret our
results as a spectral “flyby” of a þ1 STOV from the blue to
the red side of our reference pulse spectrum centered
at 800 nm. We note that a similar flyby of a −1 STOV
from red to blue would present itself in an identical manner.
How are STOVs born in real collapsing pulses? In our
(r, z, ξ) simulations, vortices are immediately born as
tori around the beam owing to azimuthally symmetric
(φindependent) phase shear. In real beam collapse, where
there is φ variation in the laser field, topological considerations
lead us to expect that shear in higher Efield
locations will first lead to a point null, followed by an
expanding and distorted torus on one side of the beam
that progressively wraps to the other side of the beam and
then, meeting itself, splits into two toroidal STOVs of
opposite phase winding. As might be expected from an
electromagnetic field with spiral phase, STOVs possess
angular momentum density, which we will discuss in a
phase
phase (rad) 4.4 Pcr
2 250 m
0
2
4
0 0.2 0.4
/3
2 /3
11.1 Pcr
 cp (rad)
(i) intensity
0 0.2 0.4
10.4 Pcr
phase
0 0.2 0.4
9.4 Pcr
 max (rad/ )
(ii) intensity
0 0.2 0.4
8 Pcr
phase
0 0.2 0.4
6.4 Pcr
(iii) intensity
intensity (a.u.)
0
0.5
1
0 0.2 0.4
/3
2 /3
4.4 Pcr
FIG. 5. Top row: Retrieved spatial phase (radians) and intensity (arbitrary units) images at z ¼ 165 cm, P=Pcr ¼ 4.4 for (i) a
precollapse beam and (ii) and (iii) beams where a vortex ring is on either side of the reference central wavelength of 800 nm. The bottom
row shows that as the maximal phase gradient in the images increases, the maximal phase shift saturates at π for all cases of P=Pcr
leading to beam collapse.
N. JHAJJ et al. PHYS. REV. X 6, 031037 (2016)
0310376
future publication. The onset of these STOVs, aligned with
planes of constant ξ, has a beamregularizing influence, as
seen in the images of Fig. 5, which show remarkably flat
phase and intensity profiles inside the ring. This could be
the reason for the notably highquality spatial modes and
supercontinuum beams (socalled spatial cleaning [27])
observed in filamentation. We are performing 3D propagation
simulations to verify this scenario. We also note that
the ring null forms a natural and welldefined boundary
between what had been qualitatively labeled the “core” and
“reservoir” regions [14] in femtosecond filaments.
IV. STOV DYNAMICS AND ENERGY FLOW
Once STOVs are generated, it is important to understand
how they propagate. Following the method of Ref. [28]
as applied to spatial vortices, we approximate the local
form of the STOV as a spacetime “Rvortex,” ψvortex≡
ðξ − ξ0Þ iðr − r0Þ, of charge 1 with a linear phase
winding about (ξ0, r0), embedded in a background field
envelope ψbg, such that ψ ¼ ψbgψvortex. If we take
ψbg ¼ ρeiχ , where ρ and χ are the real amplitude and
phase of the background field, then as the pulse propagates
along z, the nonlinear Schrödinger equation in (r, z, ξ)
moves the vortex location rvortex ¼ ðr0; ξ0Þ according to
k
∂rvortex
∂z ¼
1
ρ
ˆr
∂ρ
∂r
− ˆξβ2
∂ρ
∂ξ
× φˆþ
ˆr
∂χ
∂r
− ˆ ξβ2
∂χ
∂ξ
ˆξ
1
2r
ðr;ξ;zÞ¼ðr0;ξ0;z0Þ
; ð2Þ
where the derivatives are evaluated at the present vortex
core location (r0, ξ0, z0) (see Appendix B). Equation (2)
demonstrates interesting analogies with fluid vortices.
The term ˆξ=2r propels the vortex forward or backward
depending on its charge and radius (curvature), and
strongly resembles the speed ∼Γ=4πr of a toroidal
fluid vortex (such as a smoke ring) [29]. Identifying
ˆr ∂χ=∂r ¼ ∇⊥χ ¼ k j⊥=ρ2 as the local effective velocity
associated with the background electromagnetic flux
(see Appendix C), we interpret it as a chargeindependent
draglike term, expanding (contracting) the STOV for power
outflow (inflow) for ∂χ=∂r > 0 (<0). The term ˆr ×
φˆ ρ−1∂ρ=∂r is a Magnuslike motion [1], here propelling
the STOValong ˆ ξ, perpendicular to the vortex circulation
vector φˆ and the ring expansion or contraction direction rˆ.
The terms −ˆξ β2
∂χ=∂ξ and ∓ˆξ × φˆ β2ρ−1∂ρ=∂ξ are their
spatiotemporal analogues. In gases, small β2 (∼10−5)
makes these terms negligible; they contribute much more
significantly in solid media.
We note that ψbg is not a fixed field independent of vortex
motion. Equation (2) should be understood as a stepwise
predictor of vortex motion based on an updated ψbg.
To understand energy flow near STOVs, it is useful to
consider the electromagnetic energy flux associated with
the full field envelope ψ ¼ ueiΦ in the moving frame of the
pulse (see Appendix C),
j ¼
1
k
u2
∇⊥Φ − β2
∂Φ
∂ξ
ˆξ
; ð3Þ
where one can see that the sign of β2 determines whether
the energy flow near the STOV is spiral (β2 < 0) or saddle
(β2 > 0). What are the relative contributions of longitudinal
and transverse energy flow about a STOV? For filamentation
in air, β2 ∼ 10−5, the characteristic axial pulse length
and width are ∼c × 10 fs and ∼100 μm (filament core), and
one finds jξ=jr ∼ 10−4 ≪ 1. Here, the distinction between
saddle and spiral is not important, as we are in the
degenerate case. For solids, however, where β2 ∼ 10−2,
we expect the distinction between saddle and spiral energy
flow to be very important. Figure 6 provides intuitive
visualization of the energy flow pattern for the saddle,
degenerate, and spiral cases.
Equations (2) and (3) are useful for an intuitive picture of
the governing dynamics of STOVs, especially when viewed
2>0, j /jr = 1
r
j /jr 1 2<0, j /jr = 1
rad

0
FIG. 6. Demonstration of energy flow about a þ1 “Rvortex” STOV. White arrows correspond to size and direction of j in Eq. (3).
There are three distinct regimes: the lefthand panel shows the saddle regime, which exists in regularly dispersive media (β2 > 0), the
middle panel shows the degenerate regime, which can exist in regular or anomalously dispersive media and has a dominant axis for
energy flow, and the righthand panel shows the spiral regime, which exists in anomalously dispersive media (β2 < 0).
SPATIOTEMPORAL OPTICAL VORTICES PHYS. REV. X 6, 031037 (2016)
0310377
together with propagation simulations. For example, for the
four STOVs seen in the simulations of Fig. 1(a), their
dominant early movement is governed by ˆξρ−1∂ρ=∂r in
Eq. (2), which propels the þ1 (−1) STOV temporally
forward (backward), with the forward motion initially
being superluminal. (We will explore the detailed implications
of superluminal STOV motion in a future publication.)
A consequence of the opposing directions for
the charges is collision and annihilation of v2 and v3 of
Fig. 1(a), as discussed earlier. Remarkably, the v3 STOV
superluminally climbs from a region of negligible intensity
through many orders of magnitude of increasing intensity
to reach and annihilate v2, whereupon a local depression is
left in the field that more gradually dissipates. The v1
STOV eventually settles in the temporal middle of the
highest intensity portion of the pulse, propagating at nearly
vg. Evidently, ρ selfconsistently evolves to balance the ˆξ
terms in Eq. (2) and χ flattens along the radial dimension
(as indicated in the experiment), preventing expansion or
contraction of the STOV. Our simulations show that a
surviving þ1 STOV is always coupled to the filamenting
pulse. This is no coincidence, as the energy flow for a þ1
STOV is toward (away from) the pulse axis temporally in
front of (behind) the STOV. This is exactly as expected,
where the front of the pulse draws energy in by Kerr selffocusing,
and energy at the back of the pulse is directed
outward by plasma refraction. A link and descriptions of
STOV movies are in Ref. [30]. There appears to be a deep
connection between the STOV picture of filamentation—
spontaneous generation of STOVs followed by STOVgoverned
energy flow in the beam—and the spontaneous
formation of conical nonlinear waves (X and O waves),
which have been used to explain propagation dynamics of a
filamenting beam [31–33].
In real beams without φ symmetry, we expect collisions
of oppositely charged STOVs to be much more complex,
although the beam regularization observed in experiments
may conspire to promote collisions. Auxiliary 3D þ 1 linear
propagation simulations, in which we impose STOVs as
initial conditions on Gaussian beams, show repulsion of
likecharged STOVs, which pass around each other, and
splitting of higher charge STOVs into multiple STOVs of
single charge. We note that our measured airbased STOVs
are not solitons, as diffraction does not balance selffocusing
for a dark object. STOV solitons could exist, however, in an
anomalously dispersive, selfdefocusing medium.
V. CONCLUSIONS
We have introduced the concept of the spatiotemporal
optical vortex to ultrafast optics and demonstrated its
existence. A STOV is an optical vortex with phase
circulation in a spatiotemporal plane. STOVs form naturally
as a consequence of arrested selffocusing collapse
and their dynamics influences subsequent pulse propagation.
STOVs can also be imposed linearly via prescribed
spatiotemporal or spatiospectral phase shifts, making possible
their engineering for applications. While evidence for
STOV generation was demonstrated in experiments and
simulations of short pulse filamentation in air, we expect
that STOVs, whose dynamics are subject to topological
constraints, are a fundamental and ubiquitous element of
nonlinear propagation of intense pulses. STOVSTOV
interactions should prove to be a fundamental mediator
of intrabeam and interbeam dynamics.
ACKNOWLEDGEMENTS
The authors thank John Palastro (NRL) for the use of
his propagation simulation (and permission to modify the
source code), and Tony Ting (NRL), Dan Lathrop (UMd),
and Peter Megson (UMd) for useful discussions. This
work is supported by the Defense Advanced Research
Projects Agency (Grant No. W911NF1410372), the
Air Force Office of Scientific Research (Grant
No. FA95501310044), the National Science Foundation
(Grant No. PHY1301948), and the Army Research Office
(Grant No. W911NF1410372).
APPENDIX A: EXPERIMENTAL
CONFIGURATION AND ANALYSIS
1. Experimental setup
The experimental apparatus makes possible the
reconstruction of the transverse spatial phase and intensity
profiles of a femtosecond optical air filament in midflight.
To do this, we use an abrupt airhelium transition to halt
nonlinear propagation, as ionization yield and selffocusing
are both negligible in helium. The beam is then relayed
linearly from the transition zone through the helium and
interfered with a reference pulse in an interferometer,
enabling extraction of the transverse amplitude and phase
profiles of the filament.
The experimental setup is shown in Fig. 7. Our filamentation
source is a chirped pulse amplification Ti:
sapphire amplifier (λ ¼ 800 nm, 45 fs, 0–5 mJ). The beam
from the laser is spatially filtered using a pinhole to produce
a Gaussian mode with flat phase fronts—this is important
for the reference arm in the experiment, which requires
a flat spatial phase. After spatial filtering, the pumpfilamenting
(signal) arm and reference arm are generated
using uncoated wedges in a MachZehnder (MZ) configuration
to create a large difference in power between the
two beams (∼104∶1). Here, the lowpower reference arm
reflects off the front faces of the wedges, while the highpower
signal arm transmits, creating a dispersion imbalance
that is corrected further downstream. The beams are then
compressed, with the signal arm now at 45fs FWHM
intensity, rotated 90 deg using a periscope (converting P
polarization to S), and down collimated to a waist of
1.3 mm using a reflective offaxis telescope.
N. JHAJJ et al. PHYS. REV. X 6, 031037 (2016)
0310378
After down collimation, the beams are launched a variable
distance spanning 50–225 cm beyond the last optic of the
telescope before nonlinear propagation terminates inside the
nozzle of the translatable helium cell. Past the airhelium
transition, both beams propagate linearly in helium 50 cm,
with the intense filament core of the highpower beam
expanding transversely in size. Both beams are then wedge
attenuated before being directed out of the cell through a
200μmthickBK7window. Outside the cell, the highpower
beam is attenuated to match the power of the reference arm
via reflections from a second set of wedges in MZ configuration.
In order to maintain polarization purity, polarization
rotation from the upstream periscope is necessary, as wedge
reflections preferentially select for s polarization. Wedge
transmissions by the reference arm fix the dispersion
mismatch created in the precompressor MZ interferometer.
The beams are recombined at the output of the interferometer
and sent through a lens that images an upstream plane, just
before the nozzle’s gas transition region, to a CCD camera.
The airhelium interface is formed by the nonturbulent
flow of helium, at slightly positive pressure, into the
ambient air through a 1=4”diameter nozzle on a translatable
railmounted cell. The filament propagates from air
into the nozzle and nonlinear propagation terminates over
the sharp 4mm transition from air to helium. The heliumair
transition is measured, as in Ting et al. [26], by
monitoring the strength of the 33D–23P, λ ¼ 587.6 nm
helium line as the helium cell nozzle is moved through a
tightly focused 800nm, 45fs ionizing beam. As we show
in Fig. 8(a), the rapid dropoff of the helium line indicates
that there is negligible helium beyond a 4mm 10%to90%
transition layer at the nozzle.
To confirm the fidelity of imaging and phase
reconstruction through the helium cell, we use a time
domain propagation code [19] in 2 þ 1 dimensions (r, z, ξ)
to model the propagation of a filamenting beam through the
4mm airhelium transition into the far field in the bulk
helium at atmospheric pressure. The accuracy of phase and
intensity reconstruction is verified by reverse propagating
the beam via phase conjugation through vacuum back to
the air region just before the transition. The results are
displayed in Figs. 8(b) and 8(c), which show that the
reconstructed spatial intensity and phase at 800 nm (red)
closely track that of the input electric field just before the
transition region (black). In addition, we verify that small
deviations from the correct imaging plane (1 cm) do not
affect the results.
2. STOV in spatiospectral space
Although the STOV is a vortex in the spatiotemporal
domain, our experiment measures the spatial phase of the
filamenting pulse by interference with a reference pulse that
is centered at λ ¼ 800 nm (see Appendix A3). What is the
signature of a STOV in spatiospectral space?
Consider an “R” vortex ring centered at (r0, ξ0)
embedded in a Gaussian background field with a temporal
chirp,
Eðr; ξÞ ¼ E0e−r2=w2r
−ξ2=w2ξ
e−iaξ2 ½ξ − ξ0 iðr − r0Þ; ðA1Þ
where wr and wξ are transverse and longitudinal widths,
respectively, and a is a chirp parameter. The Fourier
transform along the ξ axis is, up to a complex coefficient,
MachZehnder
interferometer
pinhole spatial
filter
CCD
single filament
and reference
downcollimating telescope
Helium cell
rail translation
compressor
MachZehnder
interferometer
signal (pump)
and reference
arms
FIG. 7. Experimental setup for measuring the inflight intensity
and spatial phase profiles of collapsing and filamenting femtosecond
pulses over a ∼2 m range.
axial position (mm)
5 0 5 10 15
line strength (arb. units)
0
0.5
1
(a)
0 0.5 1
0.2
0.4
0.6
0.8
1 (b)
transverse position (mm)
intensity (arb. units)
original
reconstructed
0 0.5 1
4
2
0 (c)
phase (rad)
FIG. 8. (a) Line emission (in arbitrary units) of the neutral
helium 33D–23P, λ ¼ 587.6 nm transition induced by a tightly
focused 800 nm pulse as a function of helium cell position.
(b) Simulated filament intensity and (c) spatial phase at 800 nm
(black) just before the ∼4mm airhelium transition, as well as
reconstructed intensity and phase (red). The reconstruction is
performed by propagating the solution 4 mm through the
transition region followed by 50 cm of helium. The simulated
pulse is then backpropagated through 50.4 cm of vacuum to
simulate reconstruction from imaging optics.
SPATIOTEMPORAL OPTICAL VORTICES PHYS. REV. X 6, 031037 (2016)
0310379
E~ ðr;ωÞ ∝ e−r2=w2r
−ðω−ωcÞ2=½v2g
ðia−w−2
ξ Þ
×
ð2 þ 2iaw2ξ
Þξ0
−
iw2ξ
ðω − ωcÞ
vg
2ðr − r0Þðaw2ξ
− iÞ
; ðA2Þ
where ω ≡ kξvg, the variable conjugate to ξ is kξ, and ωc is
the central frequency of the pulse. In spatiospectral space
the vortex is centered at
r00 ¼ r0
1 ∓ ξ0
r0
1
aw2ξ
;
ω0 ¼ 2ξ0vg
1
aw4ξ
þ a
þ ωc: ðA3Þ
It is seen that the STOV is manifested at a definite spectral
location ω0 depending on its axial position ξ0 and pulse
chirp a. For a STOV perfectly centered in the pulse
(ξ0 ¼ 0), its spatiospectral representation is located at
the central frequency (ω ¼ ωc) and at the same spatial
radius (r00 ¼ r0). For an air filament, the STOV is, indeed,
positioned temporally about the peak intensity of the
filamenting pulse, as this is where the Kerr effect‐driven
inward flow of optical energy yields to the plasma
refraction‐induced outflow.
Figure 9 displays the spatiospectral representation of a
STOVusing simulation output.We use the same simulation
parameters presented in Fig. 3: w0 ¼ 1.3 mm, pulse
energy ¼ 2.8 mJ (P=Pcr ¼ 6.4), and pulse FWHM 45 fs.
The top row shows the spatiospectral phase, while the
bottom row displays the spatiospectral intensity. The leftmost
column shows the complete spatiospectral phase and
intensity, and makes clear that there is phase vorticity in the
spatiospectral domain. The middle (rightmost) column
shows the phase and intensity spectrally fore (aft) of the
vortex, where we present the data in a manner similar to the
experimental images of Fig. 5. Here, “fore” and “aft” refer
to left and right of the dashed black line in the upper lefthand
panel. The diameter of the core is well reproduced, as
are the abrupt phase jumps of ∼ π from core to periphery
on both sides of the vortex. The spectral intensity images
show the dark ring of the vortex core. The location of the
vortex core changes as the beam propagates. The phase flip
seen in Fig. 9 by sampling spectrally to either side of the
core would also be seen by observing a fixed frequency
while scanning the input power or by scanning along the
propagation axis of the filament, which is how we observe
the phase flip experimentally in Figs. 4 and 5.
phase (rad)
2.1 2.2 2.3 2.4 2.5
radius (μm)
0
100
200
300
400
3
2
1
0
1
2
3
angular frequency (PHz)
2.1 2.2 2.3 2.4 2.5
radius (μm)
0
100
200
300
400
phase (rad)
250 μm
2
0
2
4
6
intensity (arb. units)
0
0.2
0.4
0.6
0.8
1
FIG. 9. Spectral phase (radians, top) and spectral intensity (arbitrary units, bottom) taken from simulation at z ¼ 180 cm as the vortex
core in the spatiospectral domain crosses 2.35 PHz (λ ¼ 800 nm). Simulation parameters are the same as Fig. 3 of the main text:
w0 ¼ 1.3 mm, pulse energy 2.8 mJ (P=Pcr ¼ 6.4), pulse FWHM 45 fs. From left to right, the top row (bottom row) shows the full
spatialspectral phase (intensity), as well as spatial phase (intensity) slices spectrally fore and aft of 800 nm. The dashed black line in the
top lefthand panel intersects the vortex core.
N. JHAJJ et al. PHYS. REV. X 6, 031037 (2016)
03103710
3. Interferometric reconstruction
Since the collimated lowpower reference arm has a flat
spatial phase, the spatial phase difference, extracted from
the interferogram using standard techniques [34,35], is just
the spatial phase accumulated by the nonlinearly propagating
beam. Such a pulse can develop a complicated
time (and, therefore, frequency) dependence [14,36,37].
However, our interferograms are spectrally gated by the
narrower reference pulse spectrum. What is actually measured,
therefore, is a weighted average of the spatial phase
as a function of frequency, where the weighting factor is the
product of the (narrower) spectral amplitude of the lowpower
reference arm and the (broader) spectral amplitude
of the filamenting arm. The oscillatory portion of the signal
on the CCD is given by
intðx; yÞ ¼ 2Re
e−ikx sin θ
Z
∞
−∞
dωArefðx; y;ωÞAsigðx; y;ωÞ
;
ðA4Þ
where x and y are transverse coordinates in the beam, Aref
and Asig are Fourier transforms of the field envelopes of the
reference and signal pulses, k is the central wave number
of the reference pulse, and θ is the crossing angle of the
two beams. As the spatially filtered reference pulse
propagates linearly, Aref is fully known, and is well
approximated by a Gaussian with flat spectral phase,
Arefðx; y;ωÞ ¼ A0e−ðx2þy2Þ=w2−ω2=ω2
0 , and the weighted spatiospectral
phase of the signal beam is then extracted.
What is the difference between the weighted spatial phase
given in Eq. (A4) and the spatial phase at a single frequency
(such as at the spectral center λ ¼ 800 nm), as discussed in
Appendix A2? Figure 10 uses simulation output to directly
compare the spatiospectral phase at λ ¼ 800 nm and the
weighted spatiospectral phase. We simulate using the same
input parameters as Fig. 3: w0 ¼ 1.3 mm, pulse energy
2.8 mJ (P=Pcr ¼ 6.4), pulseFWHM45 fs. The figure tracks
the evolution of the two different spatial phase quantities as
the STOV in spatiospectral space crosses λ ¼ 800 nm. It is
apparent from Fig. 10 that the two quantities follow each
other closely, and, critically, they both exhibit the flip in
phasewe use to identify the vortex. The twomain differences
in the phase quantities are that the 800nm spatial phase
flips first at z ¼ 179 cm, followed by the weighted phase at
z ¼ 185 cm (likely due to asymmetric dispersion about
800 nm), and that the weighted phase gives a core region
∼20% larger than the exact spatial phase at 800 nm.
Figure 10 establishes that the weighted spatial phase is a
good proxy for the spatial phase at λ ¼ 800 nm and can be
used to detect the STOV in spatiospectral space as outlined
in Appendix A2.
APPENDIX B: SPATIOTEMPORAL OPTICAL
VORTEX EQUATIONS OF MOTION
We assume azimuthal symmetry and consider the evolution
of the complex envelope ψ associated with the scalar
electric field E,
Eðr; ξ; zÞ ¼ ψðr; ξ; zÞeiðkz−ωtÞ; ðB1Þ
where k ¼ 2π=λ, ξ ¼ vgt − z, and vg is the group velocity.
In the paraxial and slowly varying envelope approximation,
the propagation equation for ψ is
2ik
∂ψ
∂z þ ∇2
⊥ψ − β2
∂2ψ
∂ξ2 þ k2VðψÞψ ¼ 0; ðB2Þ
where ∇2
⊥¼∂2=∂r2þð1=rÞð∂=∂rÞ, β2 ¼ c2k0ð∂2k=∂ω2Þ0
is the dimensionless group velocity dispersion, and VðψÞ is
a nonlinear term. For the case β2 ¼ 0, Eq. (B2) drives
transverse displacement of purely spatial optical vortices
through the term ∇2
⊥ψ, as shown in Refs. [28,38]. Here, the
term containing β2 additionally drives vortex motion in the
local space (ξ) direction. Therefore, as the beam propagates,
the vortex moves temporally as well.
Suppose that at z0 the position of the vortex ring is
rvortex ¼ ðξ0; r0Þ, and at z0 þ dz the position of the vortex
ring is rvortex þ drvortex ¼ ðξ0 þ dξ; r0 þ drÞ. Then, ψðr0þ dr; ξ0 þ dξ; z0 þ dzÞ ≈ ψðr0; ξ0; z0Þ þ ð∂ψ=∂rÞdr þ ð∂ψ=
∂ξÞdξ þ ð∂ψ=∂zÞdz. But since ψ ¼ 0 at the vortex, this
leads to
∇STψ · drvortex ¼ −
∂ψ
∂z
dz; ðB3Þ
radius (μm)
0 500 1000 1500
phase (rad)
2
0
2
4
6
z = 174 cm
z = 178 cm
z = 179 cm
z = 184 cm
z = 185 cm
z = 190 cm
800 nm
weighted
FIG. 10. Comparison of the spatialspectral phase of the beam
at 800 nm to the weighted spatial phase measured about 800 nm
using the same simulation parameters considered in Fig. 3:
w0 ¼ 1.3 mm, pulse energy 2.8 mJ (P=Pcr ¼ 6.4), pulse FWHM
45 fs. Flipping of the spatial phase occurs at z ¼ 179 cm at
800 nm and z ¼ 185 cm for the weighted spatial phase.
SPATIOTEMPORAL OPTICAL VORTICES PHYS. REV. X 6, 031037 (2016)
03103711
where the spacetime gradient is defined as ∇ST ¼
ð∂=∂rÞˆr þ ð∂=∂ξÞˆξ.
Following the method of Ref. [28] as applied to spatial
vortices, we approximate the local form of the STOV
as a spacetime Rvortex, ψvortex ≡ ðξ − ξ0Þ iðr − r0Þ, of
charge 1 with a linear phase winding about (ξ0, r0),
embedded in a background field envelope ψbg such that
ψ ¼ ψbgψvortex. If we take ψbg ¼ ρeiχ , where ρ and χ are
the real amplitude and phase of the background field, then
substitution of ψ ¼ ψbgψvortex into Eq. (B2) yields
k
∂rvortex
∂z ¼
1
ρ
ˆr
∂ρ
∂r
− ˆξβ2
∂ρ
∂ξ
× ˆϕ
ˆr
∂χ
∂r
− ˆξβ2
∂χ
∂ξ
ˆξ
1
2r
: ðB4Þ
This is Eq. (2) in the main text. As the pulse propagates
along z, Eq. (B4) gives the next move of the STOV based
on the current selfconsistent background field ψbg.
APPENDIX C: ENERGY CURRENT DENSITY
Without loss of generality, we take ψ ¼ ueiΦ, where u
and Φ are the real amplitude and phase of the field. Based
on conservation of energy, we can derive an equation of the
form ð∂=∂zÞu2 ¼ −∇ · j, where j is the energy current
density in the laser pulse frame and u2 ¼ jψj2 is the
normalized energy density. Following an argument similar
to Ref. [39], in order to derive j we start with ð∂=∂zÞu2 ¼
ð∂=∂zÞðψψÞ ¼ ψð∂=∂zÞψ þ c:c: and replace ð∂=∂zÞψ
using Eq. (B2). Near the vortex, the amplitude of the
field is close to zero, so loss mechanisms like ionization
and molecular rotations can be neglected, resulting in
ψð1=2ikÞVψ þ c:c: ¼ 0 (Vfψg is a real quantity).
Furthermore, for other terms, after canceling out the purely
imaginary terms with their complex conjugates, we obtain
ð−1=2ikÞψ∇2
⊥ψ þ c:c ¼ −ð1=kÞðu2∇2
⊥Φ þ ∇⊥u2 · ∇⊥ΦÞ and ðβ2=2ikÞψð∂2=∂ξ2Þψ þc:c: ¼ ðβ2=kÞ½u2ð∂2=∂ξ2ÞΦþ
ð∂=∂ξÞu2ð∂=∂ξÞΦ, resulting in an energy current
density of
j ¼
1
k
u2
∇⊥Φ − β2
∂Φ
∂ξ
ˆξ
: ðC1Þ
[1] K. Batchelor, An Introduction to Fluid Dynamics
(Cambridge University Press, Cambridge, England, 1977).
[2] M. J. H. Ku, W. Ji, B. Mukherjee, E. GuardadoSanchez,
L.W. Cheuk, T. Yefsah, and M.W. Zwierlein, Motion of a
Solitonic Vortex in the BECBCS Crossover, Phys. Rev.
Lett. 113, 065301 (2014).
[3] S. Donadello, S. Serafini, M. Tylutki, L. P. Pitaevskii, F.
Dalfovo, G. Lamporesi, and G. Ferrari, Observation of
Solitonic Vortices in BoseEinstein Condensates, Phys. Rev.
Lett. 113, 065302 (2014).
[4] G. P. Bewley, D. P. Lathrop, and K. R. Sreenivasan, Superfluid
Helium: Visualization of Quantized Vortices, Nature
(London) 441, 588 (2006).
[5] M. R. Dennis, Y. S. Kivshar, M. S. Soskin, and G. A.
Swartzlander, Jr., Singular Optics: More Ado About
Nothing, J. Opt. A 11, 090201 (2009).
[6] J. F. Nye and M. V. Berry, Dislocations in Wave Trains,
Proc. R. Soc. A 336, 165 (1974).
[7] L. Allen, M.W. Beijersbergen, R. J. C. Spreeuw, and J. P.
Woerdman, Orbital Angular Momentum of Light and the
Transformation of LaguerreGaussian Laser Modes, Phys.
Rev. A 45, 8185 (1992).
[8] M.W. Beijersbergen, R. P. C. Coerwinkel,M.Kristensen, and
J. P. Woerdman, HelicalWavefront Laser Beams Produced
with a Spiral Phaseplate, Opt. Commun. 112, 321 (1994).
[9] F. Eilenberger, K. Prater, S. Minardi, R. Geiss, U. Röpke, J.
Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann,
and T. Pertsch, Observation of Discrete, Vortex Light
Bullets, Phys. Rev. X 3, 041031 (2013).
[10] D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.C.
Crasovan, L. Torner, and B. A. Malomed, Stable Vortex Tori
in the ThreeDimensional CubicQuintic GinzburgLandau
Equation, Phys. Rev. Lett. 97, 073904 (2006).
[11] K.W. Nicholls and J. F. Nye, The Paths of Dislocations in
Wave Pulses: An Experimental Test, J. Phys. A 19, 375
(1986).
[12] Y. V. Kartashov, B. A. Malomed, Y. Shnir, and L. Torner,
Twisted Toroidal Vortex Solitons in Inhomogeneous Media
with Repulsive Nonlinearity, Phys. Rev. Lett. 113, 264101
(2014).
[13] G. Fibich, The Nonlinear Schrödinger Equation: Singular
Solutions and Optical Collapse (Springer, New York, 2015).
[14] A. Couairon and A. Mysyrowicz, Femtosecond Filamentation
in Transparent Media, Phys. Rep. 441, 47 (2007).
[15] D. Strickland and P. B. Corkum, Resistance of Short Pulses
to SelfFocusing, J. Opt. Soc. Am. B 11, 492 (1994).
[16] P. Panagiotopoulos, P. Whalen, M. Kolesik, and J. V.
Moloney, Super High Power MidInfrared Femtosecond
Light Bullet, Nat. Photonics 9, 543 (2015).
[17] G. Fibich and B. Ilan, Vectorial and Random Effects in
SelfFocusing and in Multiple Filamentation, Physica
(Amsterdam) 157D, 112 (2001).
[18] A. J. Goers, G. A. Hine, L. Feder, B. Miao, F. Salehi, J. K.
Wahlstrand, and H. M. Milchberg, MultiMeV Electron
Acceleration by Subterawatt Laser Pulses, Phys. Rev. Lett.
115, 194802 (2015).
[19] J. P. Palastro, T. M. Antonsen, Jr., and H. M. Milchberg,
Compression, Spectral Broadening, and Collimation in
Multiple, Femtosecond Pulse Filamentation in Atmosphere,
Phys. Rev. A 86, 033834 (2012).
[20] M. Kolesik and J. V. Moloney, Nonlinear Optical Pulse
Propagation Simulation: From Maxwell’s to Unidirectional
Equations, Phys. Rev. E 70, 036604 (2004); J. Andreasen
and M. Kolesik, Nonlinear Propagation of Light in
Structured Media: Generalized Unidirectional Pulse
Propagation Equations, Phys. Rev. E 86, 036706 (2012).
N. JHAJJ et al. PHYS. REV. X 6, 031037 (2016)
03103712
[21] E.W. Rosenthal, J. P. Palastro, N. Jhajj, S. Zahedpour,
J. K. Wahlstrand, and H. M. Milchberg, Sensitivity of
Propagation and Energy Deposition in Femtosecond Filamentation
to the Nonlinear Refractive Index, J. Phys. B 48,
094011 (2015).
[22] J. K. Wahlstrand, Y.H. Cheng, and H. M. Milchberg,
Absolute Measurement of the Transient Optical Nonlinearity
in N2, O2, N2O, and Ar, Phys. Rev. A 85, 043820 (2012).
[23] A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S.
Kivshar, Spontaneous Knotting of SelfTrapped Waves, Sci.
Rep. 2, 771 (2012).
[24] K. D. Moll, A. L. Gaeta, and G. Fibich, SelfSimilar Optical
Wave Collapse: Observation of the Townes Profile, Phys.
Rev. Lett. 90, 203902 (2003).
[25] J. K. Wahlstrand, Y.H. Cheng, and H. M. Milchberg, High
Field Optical Nonlinearity and the KramersKronig Relations,
Phys. Rev. Lett. 109, 113904 (2012).
[26] A. Ting, D. F. Gordon, E. Briscoe, J. R. Penano, and P.
Sprangle, Direct Characterization of SelfGuided Femtosecond
Laser Filaments in Air, Appl. Opt. 44, 1474 (2005).
[27] B. Prade, M. Franco, A. Mysyrowicz, A. Couairon, H.
Buersing, B. Eberle, M. Krenz, D. Seiffer, and O. Vasseur,
Spatial Mode Cleaning by Femtosecond Filamentation in
Air, Opt. Lett. 31, 2601 (2006).
[28] D. Rozas, C. T. Law, and G. A. Swartzlander, Jr., Propagation
Dynamics of Optical Vortices, J. Opt. Soc. Am. B 14,
3054 (1997).
[29] I. S. Sullivan, J. J. Niemela, R. E. Hershberger, D. Bolster,
and R. J. Donnelly, Dynamics of Thin Vortex Rings, J. Fluid
Mech. 609, 319 (2008).
[30] See STOV movies at http://lasermatter.umd.edu. Movie
captioned “STOV birth and collision”: Initial pulse is
collimated with Gaussian beam waist w0 ¼ 1.3 mm,
45 fs FWHM intensity beam, and energy 2.8 mJ. The beam
is propagating in air. From z ¼ 130 cm to z ¼ 160 cm, the
beam undergoes optical collapse, and then generates 4
STOVs from z ¼ 163 cm to z ¼ 167 cm. There is a STOV
annihilation event at z ¼ 204 cm. Movie captioned “STOV
settles with main filamenting pulse”: Same simulation input
parameters as “STOV birth and collision,” but with window
zoomed in on the temporally foremost filamenting pulse.
We see the þ1 STOV, which forms with the pulse at optical
collapse (z ¼ 167 cm), settle around the foremost pulse.
There is an additional vortex pair that forms at z ¼ 219 cm.
The þ1 STOV subsequently annihilates on axis at
z ¼ 232 cm, while the −1 STOV is annihilated by another
vortex that forms on axis at z ¼ 243 cm and is then almost
immediately annihilated by another collision at z ¼ 244 cm.
[31] M. Kolesik, E.W. Wright, and J. V. Moloney, Dynamic
Nonlinear X Waves for Femtosecond Pulse Propagation in
Water, Phys. Rev. Lett. 92, 253901 (2004).
[32] D. Faccio, A. Matijosius, A. Dubietis, R. Piskarskas, A.
Varanavicius, E. Gaizauskas, A. Piskarskas, A. Couairon,
and P. Di Trapani, Near and FarField Evolution of Laser
Pulse Filaments in Kerr Media, Phys. Rev. E 72, 037601
(2005).
[33] D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon,
D. Papazoglou, and S. Tzortzakis, Ultrashort Laser Pulse
Filamentation from Spontaneous X Wave Formation in Air,
Opt. Express 16, 1565 (2008).
[34] M. Takeda, H. Ina, and S. Kobayashi, Fourier Transform
Method of FringePattern Analysis for ComputerBased
Topography and Interferometry, J. Opt. Soc. Am. 72, 156
(1982).
[35] T. R. Clark and H. M. Milchberg, Time and SpaceResolved
Density Evolution of the Plasma Waveguide, Phys. Rev.
Lett. 78, 2373 (1997).
[36] A. Couairon, J. Biegert, C. P. Hauri, W. Kornelis, F.W.
Helbing, U. Keller, and A. Mysyrowicz, SelfCompression
of UltraShort Laser Pulses Down to One Optical Cycle by
Filamentation, J. Mod. Opt. 53, 75 (2006).
[37] S. Skupin, G. Stibenz, L. Bergé, F. Lederer, T. Sokollik,
M. Schnürer, N. Zhavoronkov, and G. Steinmeyer, Self
Compression by Femtosecond Pulse Filamentation: Experiments
versus Numerical Simulations, Phys. Rev. E 74,
056604 (2006).
[38] G. Indebetouw, Optical Vortices and Their Propagation, J.
Mod. Opt. 40, 73 (1993).
[39] D. Faccio, A. Lotti, A. Matijosius, F. Bragheri, V.
Degiorgio, A. Couairon, and P. Di Trapani, Experimental
EnergyDensity Flux Characterization of Ultrashort Laser
Pulse Filaments, Opt. Express 17, 8193 (2009).
SPATIOTEMPORAL OPTICAL VORTICES PHYS. REV. X 6, 031037 (2016)
03103713
© 2018 Created by Anton Novikov. Powered by
Comment Wall
You need to be a member of Vortex Based Math to add comments!
Join Vortex Based Math