Solitons on vortex filaments generated by ns laser pulse on metal surface
1
S. Lugomer and A. Maksirnovic
Rudjer Boskovic Institute, Bijenicka c. 54. 10000 Zagreb. Croatia
(Received
6 March
1997; accepted
for publication
Laser-induced
vaporization in the nonstationary
(oscillatory) regime, on the ns time scale, that
generates the nonlinear dynamics of the molten surface layer of tantalum. was studied. Microscale
vortex filaments with the cascade of splittings raise up the bushlike structure. The cascade of vortex
filament angle-locking
splittings was shown to represent the period doubling bifurcation to chaos.
Torsion of the filaments generates two-dimensional
solitons in the side zones of the bush and
three-dimensional
solitons in the central zone. Two-dimensional
solitons are obtained from a
modified Korteweg-de
Vries equation, while three-dimensional
ones are obtained from nonlinear
Schrodinger equation. The surface dynamics associated with metal vaporization appears to be more
complex than usually assumed.
© 1997 American Institute of Physics. [S0021-8979(97)06515-8]
I. INTRODUCTION
Laser-matter
interaction
(LMI) on a short time scale
causes abrupt vaporization of the target, which depending on
the pulse power density, wavelength, and pulse duration, as
well as on the target material, may be in a stationary or in a
nonstationary
regime.i " Approaching
the transition from
planar to volume boiling, the vaporization process enters a
nonstarionary
(oscillatory) regime. In this regime the homogeneous surface vaporization turns into the inhomogeneous
one in space time. This process, characteristic for the time
scale of IOns and the laser power density Q below the
threshold of the volume boiling (Q < Qv.s), gives rise of a
spectrum of dynamically
formed structures. The structures
start to appear local in diameter, inside the area of planar
vaporization.
By the increase of Q, the size of the zones
changes as well as the type of the structures inside them. At
Q = QVR. = 108 W/cm2, the surface is subjected to volume
boiling; the dynamical structures are lost and turned into
microbubbles,
and craters formed by their explosion. Finally,
this process ends with formation of one, single crater the size
of which is comparable to the beam size-well
known as the
. 'ablation crater.' [A number of phenomena associated with
laser ablation (or laser sputtering) have been studied by Miotello ('I (11.5-7 They include boiling and superhearing'
primary and secondary mechanism in sputtering," gas-dynamic
effects with recondensation.'
etc].
By the decrease of Q to Q = /06-107
W/cm2 various
types of dynamically
formed structures are generated for
which the shear layer in the liquid metal surface associated
with nonstationary
vaporization, is responsible. The surface
shear layer is subjected to fast nonlinear dynamics that generate a reach spectrum of morphological structures characteristic for unbounded gaseous and fluid medium. Some of the
structures generated on the surface of refractory metals by ns
laser pulses have been discussed by Lugomer8-10
and Lugomer and Maksimovic.11
To generate new complex structures we have started a
series of LMI experiments by changing the laser parameters
and the target materials. Among the interesting structures are
the vortex filaments-the
objects "embedded
in an infinite
domain,"
and their dynamics. The micron-scale vortex fila1374
2 May 1997)
J. Appl. Phys. 82 (3). 1 August 1997
ments were generated in the shear layer of molten tantalum
surface. After termination of IOns laser pulse, these structures stay frozen-in permanently because of ultrafast coolins
[dTldt = 3000 Kil0 lO-slO 109_1OIO(Kis); dt = hulse d::
ration, for rough estimation],
thus enabling a posteriori
analysis by the scanning electron 'microscope (SEM).
It was realized that the precise variation of laser power
density (by the variation of the spot diameter - 3 nun for
2R <S J 0%, at constant laser power), leads to the appearance
of different kinds of surface organization of the vortex filaments, in different regions orto their disappearance from the
spot. Although it is not precisely established, it may be said
that a window of laser parameters exists in which every type
of structure appears spontaneously.
The vortex filament
structures
of the bush type were generated
by the
Q-switched Nd:YAP (P == Perovskite) laser on the Ta surface. These structures, which have not been reported yet, are
the subject of this article.
The basic condition for their generation is the existence
of the liquid shear layer on molten metal surface which enables the formation of the vortex filaments; the shock waves
cause their splitting. while torsion generates the solitons on
filaments. Topological analysis of these structures gives their
basic characteristics
which have to be put in the context of
dynamics and the phenomena occurring during the laser
pulse. In principle, such experiments open up the possibility
of studying the change of self-organization
of the surface
flow of molten metal as a function of laser parameters: pulse
energy, pulse duration, and laser wavelength.
=
A. Generation of the shear layer
A short and powerful laser pulse of a Gaussian profile
generates a molten surface layer with a characteristic
gradients of pressure V P and temperature V T. For the discussion
of surface dynamics it is convenient to divide both gradients
on the components, vertical and parallel, with respect to the
surface: 10
0021-8979/97/p.?n\/1
VP==VPII+VP.l
(I)
The vertical thermal gradient V T.l ' raises the upper sur~7<1/11'1/<;:1I) 1'11'1
p=
~,I)
face layer to (almost) the boiling temperature. T8 (T = T8).
while the lower surface layer is at T 8> T ~ T M, melting tem-
,~.
perature. The surface layer behaves as a bilayer characterized
by TI.PI.KI.·
..• VI , ... and T2.P2'K2 •...• V2 •... with TI
,:. > T2; PI < P2; KI = K2 •.. ·VI > V20r. .. U', ~ V2, wherep
:.:': ::::density, K = thermal conductivity, and U = fluid velocity.
'\; Evidently. the upper layer (index 1) is a high momentum m
"
layer with m 1 = P I
while the lower one is the low mo~~~" mentum layer with m2 = P2V~, Thus a shear layer is formed
':.~
~.: ...t.".,:·.\:·.;.:·'
-
. ~
vi.
'~.;~!._~,:
..,.:.:.: that establishes mixing of both layers and generates the vor~.
.
ticity at their interface. 10
The dynamics of the molten surface layer are driven by
V PII• For the Gaussian beams V PII is directed from the periphery toward the center of the spot. while the driving forces
f C( - V P are oriented in counter gradient direction,
The shear flow can be described by the dimensionless
parameters. namely the velocity and density ratios r is.
respectively. defined aslO
>""
U2
r= -
and
VI
P2
s= -.
PI
(2)
The third parameter. called momentum
defined according to10,12
In=
nl2 =sr2=P2
Inl
(U2)2
PI
ratio m, can be
(3)
VI
.....~
It should be noted that the top stream is always the faster
one, so th~t r < 1. while sand m are allowed to assume ~ny
'; ~t§. non-negative value, The shear layer may also be defined In a
different way. for example, through the characteristic thickness. The most commonly used are the vorticity thickness &u
..' .-, and the momentum thickness 8: 13
.~Ii;,
i~::·
~1i.~.
(4)
and
6=
HI2
J
U 1 - f.L(y)
-H12
V1-
U2
.
f.L(y.) - V 2
V1-
V2
d y,
(5)
where H is the thickness of the fluid layer. Various assumptions have been made for f.L(y) as well as for Ow and 6 by
various authors in their numerical simulations,
taking into
;~:;' account low and medium Reynolds numbers, The 33 number
;~~ is defined 14
irs:
.72=--
,
~
(6)
lI'
where V' = (V 2 - U 1 )12. and 1I is the kinematic viscosity, In
many cases 33 is in the interval of 102 and 103.
In laser melting of surfaces, one has to take into account
that the shear layer characteristics
are different for the
Gaussian
profile
transmission
electron
microscope
(TEMoo), and for the "top hat" profile (realized in multimode beams).
For the Gaussian profile, (V P)II is maximum in the center of the spot (i.e. P~=o > p;=ri, while for the top hat profile the pressure component 6. PII is constant for any position
in the spot, i.e .. P~=o = p~=ri), Therefore, the momentum
ratio (varies within the spot in the first case
the second case), Consequently,
the flow
different and will depend on position in the
to the center, i.e., for the Gaussian profile
change from center to the periphery.
and is constant in
patterns will be
spot with respect
the patterns will
II. EXPERIMENT
A series of experiments was performed in order to generate a micron-scale vortex filaments, to study their spatial
organization, interaction with the shock waves, and their dynamics, A Q-switched Nd:YAP laser of E = 0.3 J of pulse
radiation of 10 ns was used to irradiate the target. The pulse
power density Q was varied below the volume boiling
threshold of 108 W/cm2 by the variation of the spot size,
until the filament structures start to appear in the surface
morphology. For the target material various refractory metals
have been used, but the best results were obtained with tantalum plates of 1 X 1 X 0.05 ern in size, The plates were mechanically polished and washed in alcohol.
Characteristics
of the surface morphology
have been
studied by the SEM, To make topological details visible, the
micrographs have been numerically filtered, and the objects
compared with those obtained by numerical simulation, Image analysis of the micrograph
was done by using the
Phorc -Styler program,
III. RESUL IS AND DISCUSSiON
A. Vortex filament bush
A micrograph obtained by the SEM on the surface after
irradiation by 10 ns laser pulse, shows a radially oriented
vortex filament with cascade of angle-locking
bifurcations
[Fig, 1(a)]. The filament thickness is - 150-250 f.Lm, while
the angle between the splitted filaments is ¢ = 5r - 60° in all
bifurcation stages, The filament segments
are of variable
length with rather irregular position of the splitting points,
The cascade of vortex filaments splittings generates a three
dimensional (3D) dimensional bush, the branches of which
show many overcrossings
and undercrossings.
Filaments
generated in the last few cascades that are immersed in the
background fluid can be made more visible by the numerical
filtration of the micrograph what reveals many details of the
splitting process relating to the core size, 0' etc.
Reconstruction of the bush given in Fig, l(b) reveals the
existence of solitons on filaments, which are 3D in the central zone of the bush, and 2D on it's sides, This indicates that
filaments are subjected to torsion, which generates solitons
and puts them into motion, (The solitons on the right side of
the bush are scarcely visible, because they are fully immersed in the background fluid).
How is [his unusual structure generated?
In the regime of nonstationary
laser vaporization,
the
process of material removal from the surface is associated
with regular or irregular oscillations which generate a shock
wave with the series of spikes in the surface layer.2,3 The
surface shock waves travel in radial direction and interact
with vortex filarnent.P which is also radially oriented. Thus
an axial compression on the filament is generated, This interaction may lead to filament bending, or breaking or to
i •.
~.
J. Appl. Phys., Vol. 82, No.3,
~.
I{."
L·
1 August 1997
S. Lugomer and A. Maksirnovic
1375
..~
,
.'
111 ~
(b)
-_~_u_/_
FIRST CHAOTIC
REGION
e-,
"'
,.;
---~'---~::l
FILAMENT
POPULATION
FIG. 1. (A) Vortex filament bush generated by the single pulse of
Q-switched Nd: Y AP laser on Ta surface M - 200 X . (B) Reconstruction of
the above micrograph. Illustration on the right side reveals the cascade bifurcation into chaos (period doubling bifurcations) in the parameter space.
splitting, as recently described by Erlebacher et at. IS Oscillating shock, generated in the nonstationary vaporization
driven by the single laser pulse, will cause the cascade of the
vortex splittings.
In the following we briefly touch on three aspects of the
vortex filament splitting and generation of bush: hydrodynarnical, topological, and the fractal aspect.
B. Vortex filament splitting
According to Frost and Biue." in the ideal case (no viscosity), when the. total momentum is conserved, the axial
compression of the vortex filament (caused by the shock
1376
J. Appl. Phys., Vol. 82, No.3,
1 August 1997
r ,
II
FIG. 2. (a) Numerically filtered micrograph (M '" 400 X ) showing the filament splitting region. (b) Phase portrait of the fluid motion before bifurcation, at bifurcation and after bifurcation of vortex filament. (i) The lowest
illustration: T pq is a sinchronism of the system i.e it is periodic solution of
the equation, for which the following relation is fulfilled: ¢(I + 2r.p)
= ¢(I) + 2r.q where p and q are integers (p = 1,2, ... ; q = 0, :.!: I,:.!: 2 .... ).
Period of motion is 2 71p. (ii) Cross-section of the vortex filament by the
plane containing the shock wave which is normal to the filament axis. The
shock front may cause instability and change the periodic solution of the
equilibrium state Op·q into a new state or=:'? with periodic solution
rP-I.q+2.
The equilibrium
state Op·q transforms into a new state
OP-2.q+2
with the separation of the periodic solution p.q+l. (iii) Phase
portrait of trajectories in the plane before the bifurcation point. The three
types of trajectories exist: Periodic solution around two fixed centers (type
I' P.q)' The "eight type" trajectory common for both centers and the outer
trajectory which is a common envelope of all other trajectories. (iv) Phase
portrait of trajectories in the plane of the bifurcation point. The set of solutions in the "eight formation"
with one fixed point, containing periodic
solutions in the left and the right loop. No common solution for trajectories
around two centers exist beyond this point. (v) Beyond the bifurcation point.
the two new synchronising rp.q (corresponding to the same core size 0")
appear (based on Ref. 18).
wave) leads to a decrease of velocity u. This occurs in abrupt
jumps in which decrease of vorticity w occurs too.
Assuming a Gaussian distribution of vorticity in the
filament: 17
(7)
where Wo and 0" are constants, it follows that bifurcation of
the filament is associated with the increase of the vortex core
size 0". Really, the core size of the parent filament increases
about twice just before splitting (Figs. 1 and 2) thus playing
a role of a precursor of splitting [Figs. 2(a) and 2(b)]. The
core size of a daughter filament is equal to the parent core
S. Lugomer
and
A. Maksirnovic
_::
X·
!/ize .. (f. before splitting. The process repeats with daughter
t}:~laments; their core size jumps to 2(f. and the new splitting
into four vortices occurs.
: .
For the ideal case. the fluctuation in vorticity w can be
~~onnected with fluctuation in a vortex deformation through
the tensor of oscillating deformation velocity S'j: 16
!
:
dto,
-dt = wJS'J
(8)
•
where
Sij==~
2
(9)
au
where oU; and
j are the components
of the velocity flue_ tuations. It is quite obvious that decrease of U (and w) in our
(nonideal) case caused by the shock may be connected with
Sij in Eq. (8) which leads to splitting.
It is obvious that for the oscillating shock. the cascade of
filament splittings will take place at the critical values of
vorticity wcl.wc2.wc) •...• or at the critical values of the Rey-
i
I..
I
~c
I
.~c
2
•....
I
This indicates
that gradient
of
Reynolds numbers exists on the radially oriented filament in
the laser spot. Maybe more correct expression would be that
filament in the interaction with a series of axial shocks suffers a series of jumps in it's vorticity. and suffers a cascade
of splittings at corresponding .9Bc .~c •....
.'
I
I
C. Local phase portrait of the filament splitting
Synchronism
of tlie nonlinear dynamical system r pq
near the bifurcation point is shown in Fig. 2(b)(i). where r
I,
!. denotes the periodic solution of motion while parameters p
"
and q have the following meaning: q =F 0 denotes the rotational type of dynamics. and p denotes the number of attractors in the phase portrait. 18 Therefore. Fig. 2(b)(i) relates to
the circulation r of the fluid around the point attractor in the
center of the filament cross-section.
The phase portrait of disturbed periodical motion of the
fluid induced by the shock is shown in Fig. 2(b)(ii). Periodic
orbits are still stable but their circular shape is deformed
(extended) under axial compression."
Evolution of the system is shown in Fig. 2(b)(iii) which
shows the appearance of two attractors and three types of
orbits: closed orbits around one attractor, closed orbits
around second attractor and common orbits around both attractors. Separation of orbits leads to the appearance of the
"eight" type of separatrices with one fixed point. O. (saddle
point) Fig. 2(b)(iv).
In the last stage. a total separation of orbits gives rise of
two systems of synchronism r p q » with periodic. circular orbits Fig. 2(b)(v). For details of the phase portrait evolution
see Neurnark and Landa 18 or Tufillaro et al. 19
i
D. Vortex filament bush as an illustration of period
doubling bifurcation into chaos
Reconstruction of the bush shown in Fig. 1(b) suggests a
similarity with the branching process characteristic for period doubling bifurcation into chaos. Bifurcation points are
J. Appl. Phys., Vol. 82, No.3,
if
xn+l==rxn(l-x,,)~rxn
(a(OllJ + a(OUj»).
aXj
ax;
nolds number:
not regularly distributed and the following procedure
is
needed to make this similarity real. Connecting the points of
the same cascade level by irregular curves. a beautiful regularity of the branching process is obtained in the parameter
space. Figure 1(b) (right) shows the number of filaments
(generated in cascades) as a function of r, where r is the
bifurcation parameter. Thus, every curve connecting the bifurcation points of the same level has the same value of the
bifurcation parameter r. This brings us to the general, well
known problem formulation by the difference equationr"
1 August 1997
Xn
is in [0.1].
(10)
In the discussion of solution of various values of r, it
should be pointed out that for r > 3, a "time dependent"
solution of period 2 occurs. That is. the equation Xn+2 = Xn
develops two real roots in addition to the steady states. Solution is stable and attracting for r ranging from r = r2
= 3.0 until r == r3 = 3.45. At r3 a period 4 solution develops
and the period 2 solution looses its stability.i" Between r)
and r cc = 3.56998. a stable solution of period 2k is successively supplanted by the stable solution of period 2k+ I. The
lower period solutions become unstable. The behavior of iterates of Eq. (10) is reminiscent of Landau's infinite sequence of transitions in that the bifurcations to higher and
higher periodicities. and thus to more complex behaviors.
occurs with even smaller increases in r. At r; = 3.57 there
are no stable periodic solutions.r'' The oscillations all settle
down to what appears to be period 2. However, the period
number changes. depending on how many digits in rare
taken into account. This indicates that at r; the system is no
more sensitive on initial state. Thus, the system enters in the
chaotic behavior.j" Since the illustration in the phase space
of the bush [Fig. l(b) right] is typical for period doubling
bifurcation to chaos. we attribute the above values to the
bifurcation parameters as given in this diagram.
E. Fractal description of vortex filament bush
Vortex filament bush formed by the cascade of anglelocking bifurcations (Fig. 1) may be assumed a self-similar
structure which infinitely repeats at smaller and smaller
scales. The repetition of the bifurcation, as shown in Fig. I,
may be followed up to stage 3 or 4. Beyond that bifurcated
segments are immersed in the background fluid. and even the
numerical filtration does not give clear details. The most appropriate system to describe the vortex filament branching as
the fractal object is Lindenmayer-systern
bush or the
L-system bush." The L system are a particular type of symbolic dynamical system with the added feature of a geometrical interpretation of the evolution of the system. The L system is defined by the alphabet set V. by the axiom w, and by
the productions." p.
The order of the system is defined by the number of
times to execute all the transformation rules (or productions).
The commands + and - denote the rotation of the mapping
in the counterclockwise.
or the clockwise direction. respectively. Axiom, w = + + + + FX denotes the initial string on
which the productions are applied.
The L system of the order 3 can be writterr"
S.
Lugomer
and A. Maksirnovic
1377
I
i
/ I
/
liquid bilayer (3-7 times). and consequently these solitons
are two dimensional. They do not travel along the filaments
in a helical way. but simply by sliding or "slipping" in a
surface plane. Figure 1(a) indicates that different filaments
fl.hJ3.···.!;,·
.. of the bush have a different torsion
71(fl) ={ 72(f2) ={ 7;(/;). Torsion 7 is the largest (7 = 7
)
max
on the filaments of the central zone. and decreases toward the
side zones of the bush; it falls to the zero volume (7 = 0) in
the side zones. and consequently these solitons are two dimensional (see below). We shall assume a filament to be a
curve in space. and the soliton motion treat as a motion of a
curve.
as described
by differential
geometry
and
topo1ogy?2.23
The spatial motion of the twisted curve may be described by specifying the curvature K. and torsion 7. at each
point on the curve as a function of time t. The spatial variations of a twisted curve are described by the Frenet-Serret
(F-S) equations:24-26 .
an
at
-=Kn·
as
FIG. 3. L-system
bush of the order 3.
x[
as = -
(I2)
Tn,
ar
( aaar) . do:'
g(a.t)=
w=++++++X.
:X--+@.6--F+
'
ab
where t, n, and b are the unit tangent, normal, and binormal
vectors, respectively. Defining a metric g on the curve parameterized by a in 3 space:24-26
V={F.X.A. -. +},
PI
-=-Kt+7b
as
'
(13)
the arclength, s, along the curve is given by
+[ - -F+FF+X]
Joa ~g(a'.t)da'.
s(a.t)=
+ +F-[FA]-X].
(11)
and can be seen in Fig. 3. Part of the production P I (before
parentheses) generates a "stap" at 60° and then resets the
angle. The first square parenthesis represents the right
branch. while the second one represents the left branch. The
symbol A in the parentheses of P I is put to assure that the
left branch continues to grow toward the inner side of the
bush.
In spite of the fact that the L-system bush in Fig. 3 is not
exactly the bush in Fig. 1. their similarity is significant. In
the spirit of this similarity one can say that vortex filament
cascade branching is well described by the fractal of the type
L-system bush.
!
(I4)
Then the above relations are defined by the metric notation:
ar
ar
as
aa
t= _=g-II2_.
at
n= as'
(IS)
and b = t x n, where r(a,t) is a position vector of a point on
a curve. It follows from differential geometry and topology"
that motion of a point on a curve can be specified in the
form: 22,24-26
arl
r=at
(16)
=Un+Vb+Wt,
a
which describes the local motion if {U. V, W} depend only
on local values of {K.-r} and their derivatives.
IV. MICRO-SOLITONS ON VORTEX FILAMENTS IN 2D
AND 3D
A. Soliton on vortex filament in 2D (side zones of the
bush)
Vortex filaments in the bush (Fig. 1) are twisted. showing many small loops in the central zone and a few larger
loops on their side zones. Such loops represent solitons or
the solitary waves on the filamentts). The solitons in the
central zone appear in helical topology (helical twisting)
which move along the filament as 3D multi-solitons. since
their loop size is smaller or comparable to the thickness of
the molten surface layer. On the other hand. the loop size of
the solitons in the side zones of the bush are the single solitons with the loop size much larger than the thickness of the
Planar motion of the curve occurs if V = 0 and 7 = 0 in
Eq. (16). In addition the binormal vector b (b = t x n) is also
O. Then, the evolution in time, as 2D motion is described by
(Nakayama et aI.26):
1378
J. Appl. Phys .• Vol. 82, No.3,
1 August 1997
. (au
t= ~+KW
) n,
(17)
K=
a2 U 2
aK)
( --2
+ K U+ -a
\ as
s
W,
S. Lugomer and A. MaksimoviC
".
.1
..~
.~
,.
~~After some mathematics.
and using the arguments of Gold·stein and Petrich+' and Nakayama
et al.26 the last equation
(the evolution
aK
-=
at
equation)
2
(a--2
U
as
3
2.5
2.5
may be written?6
+K 2)U -- aK
1.5
JS xUds",
as
1.5
(I8)
0
0.5
BY using U
= -
= -
(aKI as)
K' , the equation
of motion (17)
0.5
0
·3
becomes
0
·1
·2
·3
3
3
2.5
2.5
1.5
1.5
We look
0.5
0.5
in the form of trav-
0
elingwaveK(s,t)
= K(t), where j = s - ct(cistheconstant
. velocity along the filament),26.28 assuming the boundary con'dition K-+O as s-+oo.28 Integrating
twice the mKdV equa-
3
2.5
~ tion and using
1.5
1.5
0.5
0.5
...:.
%!.'.:;
~;;
3 , K 2-0
K. + K "'+ 'iK
- ,
known as modified
I:fand
~Jl~.
(19)
Korteveg
de Vrie equation
where
«, = K
«, = K'
are derivations
afat and alas, respectively.
The solution of Eq. (19) describes the propagation
·~.loop without
changing
it's shape along the filament.
of the mKdV
."for the solution
the above
equation
boundary
condition
·1
of a
one obtains
-4
·3
·2
·1
0
1
2
3
4
0
·3
·2
·1
·2
·1
3
2.5
~ Y..B~:..·
~~~'
K'=:t~
(20)
~4C-K2.
0
-a
-4
r~,flntegrating
this equation
solution for the curvature,
once more, one obtains a general
K, which generates
the loop on the
'.f1~
·2
·1
0
1
2
3
4
o.
·3
(a)
(b)
~~,-.curve:
K=2~
sechf z ~~+Cl)'
(21)
'i~' where C I is the integration
t~t
constant. Without loss of generality, we can put C I = 0 and from now on, to look only at a
positive argument (because the sech is an even function).
:'~';
mined,
~:;';i{
i·;~B}
jlll
Now,
when
the
which
K
generates
we turn to the F-S system
the loop
is deter-
of Eq. (12) for t and n,
which may be written
fl
:(;r~'
(22)
FIG. 4. Two-dimensional soliton on vortex filament. (a)
creases if the parameter c, increases from c = 0.45 to c =
2D soliton on filament in time (T = 0.7), (c) Numericaly
nified micrograph of2D soliton on filament (M = 300 X)
The loop size de0.9. (b) Motion of
filtered and mag.
.'"'.:'
This equation
can be solved
for the unit tangent
vector
(23)
which
has the following
value for curvature
t" + a
form
(if we insert
and use the substitution
Ot' + 4a2 sech(ag)t=
tanh(a
~
t>=sin{
the calculated
The coordinates
=
tion:
a):
(24)
O.
4 arctan[ tanh(
simplify
this equation
with
From the known
+4
solution
curve by computing
for
K
of the mKdV,
t and coordinates
d ()f ds = K, which gives29
struct the unit tangent
()= 4 arctan[ tanh(
Setting29 t = (cos
sech( TJ) t= O.
vector
i) ].
=-
cos { 4 arctan[ tanh(
J. Appl. Phys., Vol. 82. No.3,
i)]}
1 August 1997
integra-
(28)
a
Graphical
ment
presentation
in 2D for variable
of a single soliton
velocity,
c, is shown
on vortex
fila-
in Fig. 4(a).
Motion of 2D soliton in time, for torsion T = 0.7 is given in
Fig. 4(b). These results of numerical simulation can be compared with the real 2D single soliton in Fig. 4(c) obtained
from micrograph
[Fig. I (a)] by numerical filtration and magnification.
duration
tr
by further
2 sech( 71)
a
we can recon-
(26)
{),sin ()), one has
2
y= --
(25)
of the plane
of the curve are obtained
a
the substitution
a~= TJ:
t" + tanh( 71) t'
(27)
2 tanh( 71)
x=sWe can further
i)]}.
Supposing
that first 3-4 ns of 10 ns laser pulse
is spent on the formation
the soliton,
The soliton
of the vortex
filament
and
the rest of 6- 7 ns is spent on the soliton motion.
is frozen in the position reached at the end of the
S. Lugomer and A. Maksirnovic
1379
/
0.4
.
0.1
o
-0.1
1
-0.2
(a)
-0.3
-0.4
-10
1
-8
-6
-4
-2
0
2
4
6
8
10
(a)
0.8
0.6
0.4
0.2
o
-0.2
-0.4
-0.6
(b)
-0.8
-1
-10
2
-8
-6
-4
-2
0
2
4
6
8
10
(b)
1.5
1
0.5
o
-0.5
-1
(c)
-1.5
-2
-10
-8
-6
-4
-2
0
2
4
6
FIG. 5. Projections of 3D soliton on the plane: (a) 'T = 2.0, (b)
'T = 0.5. (x.z) plane, -_. (x,y) plane and ... envelope.
8
'T
(c)
10
= 1.0, (c)
pulse at - 750 f.Lm from the beginning of the filament, what
gives it's velocity v - 750(f.Lm)/(6-7)·1O-9(s)-IOOf.Lm/ns
-105m/s.
(d)
B. Soliton on vortex filament in 3D (central zone of
the bush)
Solitons observed in the central zone of the bush (Fig. 1)
are 3D multi-solitons with a loop diameter comparable to the
thickness of the liquid bilayer. Spatial motion of the soliton
in 3D is described by the unit vectors t, n, b which have
three arguments. By generalization of Eq. (17) and following
Nakayama= one finds
.
t=
(au
-a.;-TV+KW ) n+ (av
a;+rU ) b,
FIG. 6. 3D soliton on filament in (
= 1.0, (d) 'T = 2.0.
=
0: (a)
'T
= 0.3, (b)
'T
=
0.5, (c)
'T
(29)
where
(->:al .
a
av
n=- ( --TV+KW
as
) 1+ [1-- a (av
-+TU )
K as
as
+;(~~
-TV+KW)
.
b=-
(av
-+rU
as
-TV+KW)
1380
r(s,t)
and
one finds the evolution
equations
l-
) t- [1-- a (av
-+rU
K as
as
+;(~~
For K(S,t)
)
ln,
J. Appl. Phys .. Vol. 82. No.3. 1 August 1997
a-r = !...
at
as
+r
[~ !... (av
K
I
S'
as as
1
+ TU) + ~ ( au
as
K
_ TV)
av
KUds +KTU+Ka;'
S. Lugomer
(30)
and A. Makslmovic
",
z
0.2
0.1
-0.1
·.,;
'i
FIG. 8. Helical motion of 3D soliton on filament with time step tlr
= D.3( 'T = 2). Larger 'T and larger time step cause that soliton vanish fast
from the observed space.
.~:~r
:..«~,.,:
~:.'
'}1~'
FIG. 7. Helical motion of 3D soliton on filament with time step tlr = D.l
(T= 0.5).
x
By using the substitution of the new complex variable
'¥, for K and T (Lamb,2-l.25Nakayama." and Hasimoto "),
'¥=
,:
»,
":~'
K(s,t)ei!'r(s'.r'Jds',
(NLS):28,30
(32)
By a similar procedure as before (for the 2D case), we
get the solution for x.y.z components of the vector
r:24,25.28.3 I
X.
J. Appl. Phys .. Vol. 82, No.3,
l
1 August 1997
2:)
tanh 7J
0
Sin
-
V
(31)
and putting U = 0, V = K, the evolution equations (30) are
transferred into the complex nonlinear Schrodinger equation
(
2J.L.
y= -
..
.¥,
.
=s-
2J.L
z= -
v
sinh y cos 0,
(33)
which represent a helical motion of a loop along the x axis.
For details of calculation see Refs. 24, 25, 28, 30. We shall
not discuss all possible consequences of the solution which
are numerous (for constant c, K, for variable K and T, as well
as for various values of V and lV in the equation of motion
etc.). Instead, we shall pay attention to the aspect of solution
S. Lugomer and A. Maksirnovlc
1381
/
-4
0.5
Z
o
y
0.5
Z
-0.5
1.S
(b)
x
Y
FIG. 10. Two solitons generated on two parallel filaments extending from
00 to + ce, seen from different spatial
angles: (a) view point .....•{4.0,
- 2.4,2.0}. (b) view point .....•( 1.3, - 2.4,2.0}.
(b)
FIG. 9. (A) Numerically filtered micrograph of a 3D solitons on filaments
taken under various conditions of filtration parameters in the central zone of
the bush. which remind on the so-called "pigtail" formations. (B) Illustration of the numerically generated soliton seen from various spatial angles.
relevant for our experimental case. We have varied the parameter 7 in simulations and observed the soliton evolution
in time. Projections of 3D soliton into x, y plane, for 7
= 2.0, 1.0, and 0.5, are given in Figs. 5(a), 5(b), and 5(c)
respectively. The soliton size (amplitude) decreases with increasing 7, as clearly seen in Fig. 6, for t = O.
The soliton motion on the filament for 7 = 0.5 in the time
sequences of 6. t = 0.1 is shown in Fig. 7. In contrast to the
2D soliton, the 3D soliton travels along the x axis by helical
motions, which Fig. 7 clearly demonstrates.
Motion of 3D soliton with much larger torsion (7 = 2) in
the time sequence 6. t = 0.3 is shown in Fig. 8. This soliton
moves fast and after four sequences practically comes out
and vanishes from the observed spatial interval. Namely, 7
:x: c, and larger T means larger soliton velocity.
The above figures enable a comparison with the real
soliton observed in the central zone of the bush in Fig. 1. By
1382
J. Appl. Phys., Vol. 82, No.3, 1 August 1997
using the Photo-Styler program, the micrograph of the central zone is numerically filtered and magnified, and shown in
Fig. 9(a). The filaments containing multi-solitons remind one
of the so-called "pigtail" tormations.V Figure 9(b) shows a
single soliton obtained by numerical simulation in different
spatial views. Figure 10 shows two parallel filaments extending from - co to + (X) with 3D solitons, observed in different
spatial views.
V. CONCLUSION
This article puts more light on the process of laser vaporization on ns time scale, at the transition from planar to
volume vaporization, in particular-s-on the nonlinear dynamics of the molten shear layer. In the nonstationary laser vaporization a shear layer of molten metal is generated and
vortex filaments on the micron-scale are observed. The oscillatory shock wave generated in the nonstationary process
of material removal causes the cascade of angle-locking
splittings and formation of the vortex filament bush. Torsion
of vortex filaments generates the solitons and causes their
motion along the filaments. The solitons in the central zone
of the bush are three-dimensional multi-solitons that travel in
a helical way, while the solitons in the side zones of the bush
are two-dimensional single solitons that travel by simple
"slipping" along the filaments. This indicates that torsion T,
is not constant in all the branches of the bush but has the
maximum in the central zone and vanishes (7 = 0) in the side
zones. Two-dimensional solitons are described by the modified Korteweg-de Vries equation in parameter K, while the
three-dimensional ones are described by the nonlinear scnrcdinger equation in complex parameter '1'. The solitons on
S. Lugomer and A. Maksimovic
': filaments generated in laser experiments are fast and moving
.without friction along the filament, and reach the velocity of
. .: 105 m/s. These experiments clearly demonstrate that very
, complex structure may appear as a consequence of nonlinear
.: dynamics in the liquid molten layer of metal in the process of
nonstationary vaporization caused by the single laser pulse.
~ They offer the possibility of simply generating complex unusual structures and studying them. Finally, they also show
that the process of laser vaporization on short time scale,
l
~t~~;
of target, is more com-
.!~~:~.
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I