Isotropic octupolar SHG response in LaBGeO5 glass-ceramic with
spherulitic precipitation.
Lo Nhat Truong1, Marc Dussauze2*, Evelyne Fargin1,*, Luis Santos3, Hélène
Vigouroux1, Alexandre Fargues1, Frédéric Adamietz2, Vincent Rodriguez2*
1
Univ. Bordeaux, ICMCB, CNRS UPR 9048, F-33600 Pessac, France
2
3
Univ. Bordeaux, ISM, CNRS UMR 5255, F-33405 Talence, France
Centro de Química Estrutural and Departamento de Engenharia Química, Instituto Superior
Técnico, Universidade de Lisboa, Av. Rovisco Pais, nº1, Lisboa,Portugal
Appendix
We express the parallel and perpendicular component of the SHG NLO polarization of an
average spherulite as follow (according to the lab frame depicted in Figure 4):
 
Pi 2    i;2, E
2
 
 i;2, E 
2
 2 i;2, E  E
(A1)
with i  ,  .
The incident electric field components are given by
E   E0 sin   cos t   2 

E  E0 cos   cos t 
where  2 is the angle of the half wave plate, the phase shift  2 is induced by the quarter
wave plate and E0 is the incident electric field amplitude. Combining Eq. A1 and A2, the
coherent intensity of the second harmonic signal is given by:
 
I2i  Pi 2    E0
2
4
i;2, cos 2   cos 2 t   2   i;2, sin 2   cos 2 t 
2 i;2, sin   cos   cos t   2  cos t 
1
2
(A3)
After time averaging and straightforward calculus, we obtain the final expression:
I2i  Pi 2  
2
     cos         sin  
 E  8
2  2          .    sin   cos  



2
i ; , 
4
2
0
2
i; ,
4
2
i ; ,
2
2
i ; , 
2
i; ,
2
4
2
(A4)
2
Experimentally, we observe first I 2i  I 2i , which implies  ;2,   ;2, and   2; ,    2; , ,
and second I2  I2 , which implies by identification  ;2,    2; ,
and   2; ,   ;2, . In
other words all the components are equal in module, however since we observe maxima at
2
2
45° modulo 90° it is necessary that i;,   i; , . As a consequence we finally obtain
 
I2i  Pi 2     E0

2
4
 
8   2 

2
cos 4    sin 4    6sin 2   cos 2  
(A5)
And the corresponding 2nd order susceptibility tensor partially writes
 .

 .
 .

.


.


.


.
.
.
.
.
.





(A6)
Clearly this correspond to a local C3h symmetry but with a unique term since isotopic average
occurs in the plane (yz) or more generally
  .
2