Supplementary figure 1. Propagation of an intense beam at high voltage, corresponding to

0

90

. A reduced self-focusing (i.e. vanishing nonlinear coefficient T in eq. (1)) allows beam diffraction.
Supplementary figure 2. Measured SS trajectories for various applied voltages.

Supplementary figure 3. Measured and calculated apparent walk-off for various powers.
Error bars relate to the pixel size of the CCD camera.

Supplementary figure 4. Calculated coefficients in equations (1) and (2) Vs dimensionless figure B

A /

2

0
 
LF
( V / L )
2

normalizes the non locality A , with

0
. The
 
LF the low-frequency dielectric anisotropy and L the cell thickness (see supplementary equations). The parameters are those adopted in the text.

Derivation of the leading equations for the optical envelope
By adopting a “multiple scale expansion”,
28
one can derive a single equation for the nonlinear dynamics of the e-wave. The ordinary wave is orthogonally polarized to the wave-vector, hence -at the lowest order of approximation- self-action is limited to the e-wave (polarized in the plane
 
, see figure 2). In fact, as long as higher order effects can be neglected (e.g, the optically induced
Freedericksz transition
7-9
), the ordinary wave propagates linearly. This is in perfect agreement with the experimental evidence (figure 2b).
In this section we briefly review the theoretical approach used to derive eqs. (1) and (2) in the text.
The reported derivation takes into account only the extraordinary wave, and is developed at the lowest order in the optically-induced perturbation. Full details will be reported elsewhere.
Considering the wave equation in an anisotropic medium, with a relative permittivity tensor ε and k
0

2

/

:
   
E
 k
0
2 ε 
E the plane wave solution propagating along z can be found in the form
E

A exp( ik
0 n ) and, adopting a dyadic notation,
L ( n )

A

 n
2
( zz

I )
 ε


A

0
Considering the dielectric susceptibility of the liquid crystal
 ij
 

 ij
   ˆ i j with i , j

 x , y , z

and n x , y , z the components of the director field, oriented as in figure 1. The following refractive indices for the ordinary and extraordinary waves can be readily obtained,

n o
2  
 n e
2
(

0
)

2



2




(






 
) cos( 2

0
) being

0
the angle between z and n

is given by tan

 

 sin

   
 
0 cos

0

0
 
2 with the ordinary-wave polarized orthogonally to the plane ( z along
, n
ˆ t
 
0
and propagating in the walk-off direction s
ˆ
 
0
, see figure 2.
As common practice in nonlinear optics, in the presence of an optically induced reorientation we adopt a perturbative approach, with
ε  ε  
2 δε
, being

a “smallness” parameter to set equal to 1 at the end of the derivation (see standard perturbative methods and multiple scale approaches to nonlinear systems
28
). Following the experimental evidence, we look for a solution in terms of a spatial wave-packet propagating in the walk-off direction, and modulating the e-wave solution:
E

 t
ˆ
E
 
F
 
2
G

...
 exp
 ik
0 n e z
 with E , F and G depending on the “slow spatial scales” defined by r n
  n r t n
  n t s n
  n s n

1 , 2 ,...
where we introduced the reference system ( r, t, s ), with t s
ˆ as in figure 1. Note that the reference system rotates as the voltage is varied, hence r is the coordinate ortogonal to the plane
( z , n

0
. From the wave equation at order O
 
: k
0
2
L ( n e
)

F
 ik
0 n e t
ˆ



2 cos
 

E
 s
1


 ik
0 n e s
ˆ

 sin

E
 s
1
 cos
 

E
 t
1


 ik
0 n e

 sin

E
 r
1


The solvability condition (

E /
 s
1

0 ) yields the first-order vectorial corrections to the e-wave:

F r
F t


0 in k
0
 e r sin
F s
 k in
0
 e s

E
 r
1 cos
  
E
 t
1 with
 r , s
  r , s
 
0
the non vanishing eigenvalues of L ( n e
) .
At the second order: k
0
2
L ( n e
)

G
 
1
 
1
 ik
0 n e

( E t
ˆ
)

 k
0
2
 
1
δε

 ˆ t E
F
 
1
 
F

 ik
0 n e

 
2

( E
ˆ t )
 
2
 
( E
ˆ t )

Finally, taking into account the results at the first order and going back to the original variables, the solvability condition yields the equations appearing in the text (expressed in terms of

instead of
0

, see below):
0
2 ik
0 n e
(

0
) cos

  
0
 
E
 s

D t
(

0
)
 2 E
 t
2

D r
(

0
)
 2 E
 r
2
 k
0
2
(
ˆ t
 δε  t
ˆ
) E

0
The relevant coefficients are
D t
D r



 
 
2
2




(
   

)

 
2
 

2






 


2

2


2

2
2






2


   cos
 
0

2
 

 
(
 
  cos
(
 


2


 
2

0


)
) cos( 2
 cos( 2

0
0
)
)
2

At the lowest order in the optically-induced director tilt

, in the bulk, we find
( t
ˆ  δε  t
ˆ
)
  
T (

0
)
 with
T (

0
)


   


2
2

 

2


 




 



 sin(



2
2


0

)
2

 cos( 2

0
)
While the previous expressions are in terms of

0
, to directly compare them with experimental results it is more convenient to use

0
, i.e. the director elevation angle with respect to the bottom

plane of the cell (see figure 1). In our geometry they are simply related by: cos(

)

0 cos(

4
) cos(

0
) .
The resulting trend is shown in supplementary figure 4.
Derivation of the leading equation for the director field
The overall free energy of the nematic liquid crystal is written as the sum of the elastic ( F
K
), lowfrequency ( F ), and optical (
LF
F opt
) contributions:
F

F
K

F
LF

F opt
The expressions of F
K and F can be found in reference books;
LF
8,9
in the presence of the extraordinary wave E , the optical portion is
F opt
 

0


4
| E |
2 

0
 
4
(
ˆ  t
ˆ
)
2
| E |
2
Once the low-frequency problem has been solved, the leading equation in the optically-induced perturbation

is
K
 2  
A (

0
)
 

0
 
4 sin

2

0

2

(

0
)

| E |
2 
0

0
can be expressed in terms of

0
, thus obtaining the equation used in the text. In general, A is derived from the actual profile of the director tilt
  
as induced by the bias and the cell geometry, numerically calculated as outlined below. However, a simple expression can be derived by assuming that the low-frequency electric-field is constant across the cell. In this case, the voltage distribution resembles that of an ideal capacitor and, by generalizing the treatment in Ref. 14 and
Ref. 25, we get:
A

2

0
 
LF

V L

2

 sin
2
( 2

0

0
)
 cos
 
0

 with L the cell thickness across x and
 
LF the low-frequency anisotropy. The validity of this expression was confirmed by experiments on MI.
25
Supplementary figure 2 graphs the normalized coefficient

B

2

0
 
LF
A

V L

2


 sin ( 2
2


0
0
)
 cos
 
0


As outlined in the text, A measures the degree of nonlocality, i.e. when A is small (as for small
 and V ) the index perturbation is much wider than the optical intensity profile, the opposite being
0 true at high bias (

0

90

).
Equations of the director profile due to the applied voltage
This requires the solution of coupled non-homogeneous and nonlinear equations, i.e. the minimization of the NLC elastic energy in the presence of a quasi-static electric field.
3 Note that, in order to avoid disclinations in the NLC bulk, a small pre-tilt was introduced at the cell interfaces,
(i.e.

close to 2° at zero bias) and included in our treatment. The equations were numerically
0 integrated and the results are shown in figure 3. The calculated apparent walk-off Vs applied voltage is compared with experimental data. The parameters adopted are those for E7 at
 
1064 nm : K
1

12 N , K
3

19 .
5 N (the elastic constants for splay and twist were individually accounted for in the low-frequency model), n
0
2 
1 .
50 , n
2 e
(

0

90

)

1 .
70 ,
 
LF

14 .
5 and



5 .
1 .
29
Considering a cell with planar alignment as in figure 1, in the presence of a low-frequency bias the displacement vector components are
D x
D y
D z
 
0


LF

 
0
 
LF
  
LF sin(

)
2

E x
LF sin(

) cos(

) E x
LF

0 with E x
LF  
V ( x ) /
 x and V ( x ) the applied voltage, and
 
LF and

LF

defining the permittivity tensor at low frequency. The coupled system given by the divergence-free condition for D and the
Euler-Lagrange equation 8,9 for the director reads
 


K
1
LF

  
LF
 sin
2
 
 
LF
 cos
2 cos
2 
K
3 sin
2
 d
2
V dx
2
  
LF sin
 d 2
 dx 2

K
3

K
1 sin
2 d
 dx d
 dV

0
2
 dx
 
LF
2 dx dV dx

2 sin( 2

)

0

with K and
1
K the pertinent elastic coefficients, and
3

( 0 )
 
( L )

2

/ 180 ; V ( 0 )

0 ; V ( L )

V the boundary conditions (including a 2° pre-tilt). The system can be solved numerically and the results compared with the experimental data, as described in the text.