Determine the refractive index profile.
To find out the horizontal refractive index distribution, we repeated the experiment in Fig.
3 of the main text with a square cuvette whose effective side length is 16mm. The He-Ne
probe beam (632.8 nm) is weakly focused in to solution using a 250 mm lens to get a finer
probe beam (diameter<0.5mm). Thus the measured horizontal deflection can be more
accurate than that in Fig. 3(b) and 3(c) of the main manuscript. Then, we changed the
distance between the focused probe beam and the center of the pump beam by moving the
pump beam and the cuvette step by step synchronously while the probe beam remained
unchanged, and the deflection of the probe beam was recorded. Based on Fermat’s
principle, the deflection of the probe beam is caused by the refractive index difference in
the plane perpendicular to the probe beam, therefore, the deflection is directly related to
the index gradient of the material. Fig. 1 is the schematic diagram of the propagating
probe beam, where
dx is the probe beam size, Lc is the length of the cuvette, L is
the distance between the cuvette and the screen ( L  Lc ), and
deflection of the probe beam. Obviously,
x
x
represents the
is proportional to the refractive index
gradient, since deflection angle θ is a small quantity.
Fig. 1. Schematic diagram of the propagation of the probe beam.
In our experiment, the data shows that
x
is always positive (deflecting into the
pump area), which illustrates that the horizontal refractive index decreases from the center
of the pump beam to the edge of the cuvette. Fig. 2 shows the deflection angle θ vs. x, the
distance of the probe beam from the center of the pump beam. We can see that θ increases
from the center of the pump beam to the boundary of it, while decreases as the probe
beam further moving away from the boundary of the pump beam to the edge of the
cuvette, which means the maximum index gradient is just at the boundary of the pump
beam.
Fig. 2. Deflection angle θ under different distance of the probe beam from the center of
the pump beam. The radius of the pump beam (532 nm) is 4 mm, and the probe beam
(632.8 nm) diameter is focused to <0.5 mm.
In order to calculate the refractive index profile, we use infinitesimal method to get
the variation of the refractive index
n ; the formulas are as below:
dn  Lc
x
 tan  
,
n0 dx
L
n 
n0 xxr
,
LLc
where xr is the step length we moved every time, and n0 =1.361 is used as the center
refractive index, so the refractive indexes of the outer measurement points are:
n j  n0  n1  n2 
 n j .
The refractive index profile along x direction is demonstrated in Fig. 3.
Fig. 3. Refractive index of different distance from the center of the pump beam.
It should be pointed out that the refractive indexes we get above are the average
ones through the whole length of the cuvette (i.e., averaged over the y direction). To get
the true result inside the pumped area, further calculations are made using the average
indexes of both inside and outside the pumped area. First, we get the fitting curve
equation
n( x ) of the data outside the pumped area, that is from r=4 mm to 8 mm. Then
the average refractive index inside the pump beam region nin is calculated. The process
is shown below.
Since the thermal convection is isotropy, the index profile must be isotropy, therefore
n( r )  n( x ) . In the center, the optical path out of the pumped area (see inset of Fig. 4)
is:
8
nout (0)  2   n(r )dr .
4
So the average refractive index inside the pumped area is
nin (0) 
n0  16  nout (0)
.
8
Except center, the optical path out of the pumped area of the other x is
nout ( x)  2  
8
42  y 2
n( y 2  x 2 )dy ,
where y represents the vertical distance to the center of the pump beam, and the average
refractive index inside the pump beam is
nin ( x) 
n j 16  nout ( x)
2  42  x 2
.
The final result is shown in Fig. 4.
Fig. 4. Horizontal refractive index profile of the solution layer 1.5 mm below the surface.
Inset shows the relationship of different refractive index we calculate.
As we mentioned above, the refractive index profile is isotropy. After rotating the
curve around r=0mm and one can get the 2D index profile shown in the main text (Fig.
3(f)).