Enhanced stimulated Raman scattering in temperature controlled
liquid water
Yuval Ganot, 1,a Shmuel Shrenkel, 2 Boris D. Barmashenko,3 and Ilana Bar3
1Department
of Engineering, Sapir Academic College, D. N. Hof Ashkelon 79165, Israel
of ElectroOptics Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
3Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
aCorresponding author: yuvalga@sapir.ac.il
2Unit
Supplementary Information
A. Side view photograph of the forward stimulated
Raman scattering
Figure S1(a) presents a side view of the generated forward
stimulated Raman scattering (FSRS) beam, for pump energy of
100 mJ/pulse and water temperature of 30 oC. The photograph
was taken with a commercial digital single-lens reflex camera
fitted with a high optical density red filter. The exposure time
was 10 s, averaging 100 laser pulses. The photograph was
analyzed using a code written in Matlab for obtaining the
approximated beam width, w, as a function of z, as shown in
panel (b). As seen from Fig. S1(a), the FSRS beam at z > 20 cm
becomes brighter, implying that the FSRS generation at this
point increases substantially, resulting in a strong heat release
and leading to a gradient in the refractive index. Thus, the
photograph and its analysis show that at a temperature of 30 oC
the beam undergoes defocusing and that there is correlation
between the onset of FSRS and its defocusing.
Fig. S1. Forward stimulated Raman scattering (FSRS) characteristics as
a function of z, the distance from the entrance window: (a) Side view of
the beam intensity in the water cell, photographed at pump energy of
100 mJ/pulse and at a water temperature of 30 oC, and (b) the
numerically retrieved beam width. The spatial resolution in the
photograph is ~ 0.2 mm.
B. Detailed description of the model used to simulate the
forward stimulated Raman scattering
As mentioned in the article, the computer code written to
simulate the propagation of the FSRS beam for a single-pass in
water accounted for the following physical processes: i)
conversion of a part of the pump energy into FSRS and
backward stimulated Brillouin scattering (BSBS) beams; ii)
diffraction of the beams; iii) attenuation of the beams by
absorption and scattering in water; iv) formation of a nonuniform distribution of the refractive index of water across the
beam propagation direction by heat release due to the FSRS
process and v) defocusing of the pump, FSRS and BSBS beams
by the resulting non-uniform refractive index distribution. Yet,
the following processes were not considered in the model: i)
backward SRS, the experimental conditions did not support its
gain, or it was too small to be observed, ii) water
photoionization, since special care was taken to choose focusing
conditions minimizing it and iii) nonlinear self-focusing and
cross-phase modulation in water are assumed to be negligible
for the current experimental setup, based on the following
reasons.
Based on our measurements and simulations it is roughly
estimated that the heat deposition during SRS generation
corresponds to an effective attenuation coefficient for the pump
beam, which is about two orders of magnitude larger than the
absorption coefficient in water (see Table S1). Following Ref. 1,
it is found that for this attenuation coefficient, self-focusing is
totally suppressed by thermal defocusing at the point where
SRS starts to build up, ~10 cm prior to the expected beam waist
position. It is therefore reasonable to neglect it for practicality of
the numerical simulations. The SRS beam intensity side view
photograph, Fig. S1, supports this assumption. It shows that as
the SRS starts to build up, defocusing of the SRS takes over,
preventing the beam from reaching the pump focal spot size. As
can be seen, the beam waist is located at about z = 22 cm with a
radius, w0, of 0.45 mm. Thus, for a100 mJ/pulse pump energy
and
for
this
w,
the
pump
intensity
Ip is
estimated to be 3 x 109 W/cm2. Furthermore, the change in the
index of refraction due to the optical Kerr effect, Δ𝑛𝐾𝑒𝑟𝑟, which is
responsible for the nonlinear focusing, can be estimated by
Δ𝑛𝐾𝑒𝑟𝑟 = 𝑛2𝐼𝑝 = 5x10−7, where 𝑛2=1.7 𝑥 10−16 cm2/W is the
nonlinear index of refraction.2 Since the change of the index of
refraction due to temperature increase of 1 oC (the typical
temperature difference across the beam), is 10-4 (see Refs. 3 and
4), i.e., more than two orders of magnitude larger than Δ𝑛𝐾𝑒𝑟𝑟, its
effect on the FSRS beam can be assumed to be negligible in our
simulations. However, as the pump beam continues to focus,
this process may play a role in its propagation. Under different
experimental conditions, self-focusing in FSRS beam
propagation
might
be
significant5
and
additional
investigation is required.
The output of the code was the FSRS pulse energy at the
exit window from which the ratio of the FSRS pulse energy to
the pump pulse energy was calculated to yield the simulated
FSRS conversion efficiency,  R , defined as:
and n( , T ) is the refractive index of the water at frequency ω
E
 R  FSRS ,
E pum p
 B, abs are the absorption coefficients of the pump, FSRS and
(1)
We set the z coordinate to coincide with the main axis of the
cylindrical cell, containing the water so that the pump and
FSRS beams propagate in the +z direction, whereas the BSBS
wave travels in the –z direction. Therefore, the total electric
field, E, can be represented as a superposition of the pump,
FSRS and BSBS fields: 6
E  Ap e
i ( k p z  p t )
 AR ei ( k R z  R t ) 
(2)
AB ei (  k B z  B t )  c.c.
where,
Ap , AR and AB are the amplitudes of the interacting
pump, FSRS and BSBS waves, respectively, slowly varying in
the z direction and in time and 𝑘𝑝,𝑅,𝐵 and 𝜔𝑝,𝑅,𝐵 are the wave
numbers and the corresponding angular frequencies. The
optical field distribution of the pump beam at the cell entrance
is assumed to be axially symmetric. Therefore, the amplitudes
Ap , AR and AB depend only on the geometrical factors z and r,
where the latter is the distance from the optical axis, and are
described by the paraxial wave equation,6,7 When all the
aforementioned physical processes are considered, the following
equations are obtained:
Ap
z

i 1  
r
Ap  ik p n p Ap 
2k p r r r
gR  p
2 0cn(R , T )  AR 2 Ap 
2 R
(3)


gB
2 0n(B , T )  AB 2 Ap  p, abs p, scat Ap ,
2
2
AR
i 1  

r AR  ik R nR AR 
z
2k R r r r
 R,abs   R, scat
2
gR
2 0 cn( p , T ) Ap AR 
AR ,
2
2


(4)
AB
i 1  

r AB  ik B nB AB 
z
2k B r r r
 B,abs   B,scat
2
gB
2 0 cn( p , T ) Ap AB 
AB ,
2
2

where:
n p, R, B 

n( p, R, B , T )  n( p, R, B , T0 )
n( p, R, B , T0 )
,
(5)
(6)
and temperature T , T0 is the set temperature of the water
before the interaction with the laser beam, g R and g B are the
FSRS and BSBS gain factors, respectively,  p,abs ,  R,abs and
BSBS beams, respectively, and  p,scat ,  R,scat and  B, scat are
the scattering coefficients of these beams.
and g B
gR 
gB 
The values of g R
are given by:6,8
8cN
 R2 n 2 ( R , T ) R
d
,
d
(7)
 e2 B2
n( B , T )uc 3 ( B   p )
,
(8)
where N is the number density of the water molecules, d / d
is the spontaneous Raman cross-section,  R (cm-1) is the
spontaneous Raman linewidth,
 e    /  T
is the
electrostrictive constant, ε is the water dielectric constant,  is
the water density, u is the sound velocity in water and  B and
 p are the BSBS and pump laser linewidths, respectively. The
 B value depends on the water viscosity,  , as:6,9
 B 
 B2 n 2 ( B , T )
,
c 2
(9)
In fact, the first and second terms in the right hand side of
these equations describe the diffraction and defocusing of the
pump, FSRS and BSBS beams; the third and fourth terms in
Eq. (3) and the third terms in Eqs. (4) and (5) describe the FSRS
and BSBS processes and the last terms in these equations
describe the absorption and scattering losses. As described in
the article, radiation at ~ 649 nm was observed only in the
forward direction. Therefore, the existence of the FSRS of the
Brillouin wave or the BSBS of the FSRS wave,10 both
propagating in the –z direction, can be excluded and were not
accounted for in the simulations.
It is worth noting that the transit time of the beam through
the collimated beam range near the waist of length, 𝑏 = 2𝑘𝑝 𝑤02
(where the SRS conversion process takes place), is bn/c, and is
assumed to be shorter than the pump pulse length, τ. As seen
from the photograph, see Fig. S1 of the supplementary material,
Sect. A, showing the dependence of the beam width on z, the
value of b is ~10 cm, implying transit time (~ 0.5 ns), which is an
order of magnitude shorter than the pulse length (5 ns) and
hence allowing to use the steady state approximation,
The assumed boundary conditions at the cell entrance (z = 0)
were that the pump beam has a Gaussian spatial and temporal
profile, whereas the intensity of the Raman Stokes signal was
equal to the spontaneous Raman scattering intensity emitted
into a diffraction solid angle:7
z  0,
lasts from t = -2to t = 2 The pulse duration, 4τ, was divided
 E p ln 2 / 
Ap   2
 w n( , T )c
p 0
0
 i
1/ 2





2t 2 ln 2
2
e
1/ 2


hc 2  R k p

AR  
 4wi2 n 2 ( p , T0 )c 0 



k p wi2
Ri
r2
2 wi2

e

(10)
into N (>>1) very short time intervals of length t4NAt
each time interval i, the value of Ap (z = 0) was assumed to be
(11)
N = 700 was used, but we checked that the increase of N does
not affect the calculated results. The spatial step in the z
ik p r 2
2 Ri
,
2
r
2 wi2
,
where E p is the pump pulse energy, wi and Ri are the radii of
the pump beam and the wave front curvature, respectively, at z
= 0, which are determined using the given f.l. of the lens and the
cell window thickness. The boundary condition for the AB
amplitude at the cell exit, z = L, was set by assuming that the
spontaneous Brillouin scattering intensity is 10-12 of the pump
intensity:2
AB  10  6 A p
,
(12)
The temperature distribution T (r , z, t )
in water was
calculated using the adiabatic approximation and assuming
that the heat release per second and per unit volume, J h , in the
FSRS and in the absorption processes is equal to the change of
the local thermal energy:
t
T (r , z, t )  T0 
 J
h
/ c w dt ,
direction was chosen by the “pdpe” program so that the relative
error at each step was smaller than 10-6. The adiabatic
approximation, Eq. (13), is valid since the pulse temporal step
size, t = 7 ps, is longer than the phonon relaxation time ~1 ps10
and much shorter than the heat transfer time > 1 ms.1
An iterative method of trial and error was used to find the
AB (z = L) value for which the boundary condition (Eq. 12)
holds and several iterations were needed for it. For low water
temperatures, i.e., 4 and 10 oC, the experimentally observed
BSBS intensity was negligibly small (see Fig. 4 of the article)
and AB was assumed to be zero. For these temperatures only
Eqs. (3) and (4) were solved.
The output of the program contained the spatial-temporal
distributions of Ap, R, B (r, z, t ) and T (r , z, t ) , from which the
FSRS pulse energy was attained by integrating the Stokes
intensity at the exit window plane, z = L, over r and t. The ratio
of the FSRS pulse energy to the pump pulse energy is the
calculated value of the measured FSRS conversion efficiency,  R
, defined in Eq. (1) and is given by:

(13)

where c w is the water heat capacity per unit volume and the
heat release J h is given by:
 p

J h  
 1 g R I p I R 
.
 R

 p ,abs I p   R ,abs I R   B ,abs I B
(14)

 
dt I R (r , z  L, t )2rdr
R

0
Ep
.
(16)
The most important parameter values used in the calculations
are summarized in Table S1. The refractive index of water,
n( , T ) is given in Refs. 2,3. Note that the assumed value of
 R  19,000 m-1 (resulting in g R  1.6 x 10-12 m/W) is in good
The pump, I p , FSRS, I R , and BSBS, I B , intensities are
given by:
I p, R, B  2 0cn( p, R, B , T ) Ap, R, B
independent on t and equal to A p z = 0, t  τ  i - 1 Δt  . Usually
2
.
(15)
It is important to note that since the Raman shift in water is
relatively high, corresponding to a 𝜔𝑃 ⁄𝜔𝑅 ≅ 1.2 ratio, it is
implied that a large fraction of the electromagnetic energy of the
beam is converted into thermal energy. Therefore, even in the
absence of the losses for the non-resonant absorption and
scattering, the electromagnetic power is not conserved.
The systems of partial differential equations, Eqs. (3) – (5),
with the boundary conditions, Eqs. (10) – (12), and temperature,
T, defined by Eqs. (13) – (14), were solved numerically using the
“pdepe” Matlab computer program. For the aforementioned
steady state approximation the boundary conditions should be
independent of t. To get rid of the temporal dependence in the
right hand side of Eq. (10), we assumed that the pump pulse
agreement with the width measured in previous works.10,11 The
values of the scattering coefficients  p , R , B ,scat have been chosen
so that the calculated values of the total radiation loss in the cell
are consistent with the experimental values. The pump in our
experiment was a multi-longitudinal mode laser, implying that
 p >  B and therefore resulting in a g B value, smaller by
an order of magnitude than that measured by Shi et.al.9 for a
single mode pump laser.
The beam expansion following the onset of SRS is mainly
caused by defocusing rather than by diffraction. For example, in
the region next to the waist the second term of Eqs. (3) - (5),
related to the defocusing is larger than the first term, which is
responsible for diffraction by a factor R  k p n p w0 >>1. At a
2
2
temperature of 30 oC the maximum value of R is ~1000,
though the peak temperature difference is only about 1
oC.
Table S1. Parameters used in the model
Parameter
Value
Ref.
d/dm2/molecule/sr
5.48 x10-34
19,000
13
This paper
4.4 x 10-2
14
 R , m-1
 p,abs ,  B,abs , m-1
 R,abs m-1
3.4 x 10-1
14
 p , m-1
20
This paper
 B , m-1
g B , m/W
5
This paper
2.4x10-12
This paper
5
0.55
0.7
This paper
This paper
This paper
τ, ns
L, m
f.l., m
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