Quantitative determination of linear and SHG optical
effective responses of achiral or chiral materials in
planar structures: Theory and Materials
SUPPLEMENTARY INFORMATION
Vincent Rodriguez
Institut des Sciences Moléculaires - UMR 5255 CNRS,
Université Bordeaux I, 351 cours de la Libération
33405 Talence Cedex, France.
v.rodriguez@ism.u-bordeaux1.fr
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Worked out example to illustrate the theory
We consider the simple case of an ideally achiral isotropic planar dielectric
homogeneous film, with thickness d and class symmetry Cv (achiral, NLO active), deposited
on a glass substrate (achiral). The optical matrix of these two layers, Mopt defined by Eq. [4],
reads:
M opt

nil2


   0

0


0

  '  0


0

0 0
0 0 0  


nil2 0    0 0 0 
0 0 0 
0 nil2 

0 0
1 0 0  
0 0
  0 1 0 

0 0 1 
0 0
(S1)
where nil is the index of refraction, at frequency i (i=1 or 2),of the film (n1f, n2f) or the glass
substrate (n1s, n2s). Let us denote (X, Y, Z) the reference frame where the Z axis is
perpendicular to the stratified media as shown in figure A, where nif >nis. Assume a
fundamental laser beam, obliquely propagating in the (X, Z) incidence plane at an angle  0 ,
in the air with a wave vector K 0  c sin  0 , 0,  c cos  0  . The two planar layers are
considered magnetically and optically inactive
X
nif
nis
if
0
air
Film
(NLO)
Z
is
Substrate
(glass)
air
Fig. A: Schematic representation of the two-layer planar system at i (i=1, 2).
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Following the formalism given in Eq. (10), for any anisotropic and/or chiral layer, the
4x4 normalized field matrices at both interfaces are different since there is a phase delay
between the field-eigenvectors, and, for example at interface z-, -Qi may be expressed as
~
  E1T, x,i

~ T ,i
  H
1, y

Qi   ~
  E T , i
1, y

~ T ,i
 H
1, x

~
 E2T, ,xi
~
 H 2T,,yi
~
 E2T, ,yi
~
 H 2T,,xi
~
 E3R, x,i
~
 H 3R, ,yi
~
 E3R, y,i
~
 H 3R, x,i
~
 E4R, x,i 

 ~ R ,i 
 H 4, y

~
 E4R, ,yi 

~
 H 4R, ,xi 
(S2)
The two first column field-eigenvectors correspond to the z-positive solutions (transmission)
and the two last column field-eigenvectors correspond to the z-negative solutions (reflection).
The expression of the 4x4 normalized field matrix, -Qi of any isotropic achiral layer (e.g. the
substrate here) at interface z- is straightforward (Eq. (11) from ref. [22])
 Ci

n
i

Qi  
0

0

0
 Ci
0
ni
1
0
ni C i
0


0 

1 

 ni Ci 
0
(S3)
where C i  cos  i . Since the layer is isotropic and achiral, there is a common phase delay (say
) for the four field-eigenvectors at depth z+. For the special case of the input and output air
media, we have the two following 42 normalized field matrice at z = 0 (z-negative fields)
and z = d (z-positive fields)
  C0

 1

Q0 ( z  0)  
 0

 0

0 
 C0



 1
0
 ; Q ( z  d )  
0
 0
1 



 0
 C0 

0

0

1

C 0 
(S4)
Now, to formulate the boundary conditions at i (i=1 or 2) with 2 layers, following
Eq. (11), let us first detail the Mi band matrix with dimension 4(2+1)4(2+1)=1212 :
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Mi 
  C 0

 
  1
 
 
 
 




















1 

 C 0 
~
   E1T, x,i

~ T ,i
  H
1, y
 ~
   E T ,i
1, y

~
   H T ,i
1, x

 ~ T ,i
  E1, x

~ T ,i
  H
1, y
 ~
   E T ,i
1, y

~
   H T ,i
1, x

~
  E 2T, ,xi
~
  H 2T,,yi
~
  E 2T, ,yi
~
  H 2T,,xi
~
  E 2T, ,xi
~
  H 2T,,yi
~
  E 2T, ,yi
~
  H 2T,,xi
~
  E3R, x,i
~
  H 3R, ,yi
~
  E3R, y,i
~
  H 3R, x,i
~
  E3R, x,i
~
  H 3R, ,yi
~
  E3R, y,i
~
  H 3R, x,i
~
  E 4R, x,i 

~
  H 4R, ,yi 

~
  E 4R, ,yi 

 ~ R ,i 
 H 4, x 
~
  E 4R, x,i 

~
  H 4R, ,yi 

 ~ R ,i 
 E 4, y

 ~ R ,i 
 H 4, x 
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C is

 nis




C is

 nis




 C is
nis
1
n1s C1s
 C is
nis
1
n1s C1s




1 

 nis C is 



 exp( i )
1 

 nis C is 
C 0

1

























1 


C 0   (S5)
For the fundamental wave, the source term column vector , with dimension 121, is
(Eq. (20) from ref. [22])
  E X  C0 E P  
 

  H Y  EP  
  E  E

Y
S
 

  H X  C 0 E S  


0 


0 


 
Ωω  

0 


 


0
 


0 


0 


 


0 


 


0
 


(S6)
where EP and ES are the fundamental input electric field amplitudes (from the air medium).
For the harmonic wave, the source terms are generated from the nonlinear
polarization, P, and the nonlinear magnetisation, M. In this example, we only consider
dipolar
electric
contributions
(eee
terms)
and
assume
Kleinman
conditions
eee
eee
eee
(  xyz
  yzx
  zxyx
 0 ). The non vanishing eee components are zzz, xxz=xzx, zxx and the
nonlinear polarization reads:
Px2   Px2   Px2 
Py2   Py2   Py2 
Pz2   Pz2   Pz2 
  E E
2
i , j 1
eee
zxx
i
x
j
x
  xxzeee E xi E zj  E zi E xj  
  E
i
x
E zj  E zi E xj

   E
i
y
E zj  E zi E yj

2
i , j 1
2
  E
i , j 1

eee
xxz

i
y
E zj  E zi E yj 
4
eee
xxz
i , j 3
4
eee
xxz
i , j 3
   E E
eee
 E yi E yj   zzz
E zi E zj 
4
i , j 3
eee
zxx
i
x
j
x

[S7]

eee
 E yi E yj   zzz
E zi E zj

where +P and –P are the two sets of NLO polarization that come from transmitted fields (i,
j=1, 2) and reflected fields (i, j=3, 4) respectively.
The nonlinear polarization (and also magnetization) radiates the bound-wave fields
which are determined by solving the inhomogeneous wave equation (Eq. (15)). Additionally,
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there are two bound-wave fields per interface, one coming from z-positive fundamental fields
(EB,T, HB,T) and the other one coming from z-negative fundamental fields (EB,R, HB,R). This
particular point has already been commented and is experimentally illustrated in section 3.4
(case of a quartz plate). The column vector , with dimension 121, may contain both the
forward and backward bound-wave fields at both interfaces (z+ and z-) of the film. If we
consider SHG fields that are reflected it reads
Ω 2ω
    E XB ,T     E XB , R  
 
 

   H YB ,T    H YB , R  
    B ,T     B , R  
   EY    EY  
   H XB ,T    H XB , R  

 B ,T
 B,R 
  EX   EX  
   H B ,T    H B , R  
     YB ,T     YB , R  
   EY    EY  
   H B ,T    H B , R  
X 
X 

 

0




0 


 


0 


 


0 


(S8)
Finally, for both frequencies,  then 2, the amplitude field column vector Z, with
dimension 121, can be calculated, first by inverting the M band matrix and second by
multiplying the inverse matrix M-1 with the corresponding column vector  (Eq. (17) from
ref. [22]). As a consequence of this formalism, no obvious analytical solutions can be
provided but the amplitude fields (AT, AR) are determined in each medium, respectively
corresponding to the z-positive (transmitted) fields and to the z-negative (reflected) fields:
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-1
Z iω  M iω
Ωiω
  AXair,i,R  
 
 
  air, R  
  AY ,i  
,T
  A1film
 
, i
 
 
  A film,T  
  2 , i  
  A film, R  
  3 , i  
  A film, R  
4 , i
   substrate,T 
  A1,i



,T 
  A2substrate

, i


  A substrate, R  
  3 , i

  A substrate, R  

  4 , i
air ,T
  A X , i  
 
 
  AYair,i,T  
 
 
(S9)
Therefore, it is straightforward to calculate the amplitude of the four field-eigenvectors in any
medium, either at  or at 2:
 E j , x z  


 H z  
j, y

 j ,i  z   A j ,i 

 E j , y z  


  H j , x  z 
(S10)
The output intensities are calculated from the two last (first) amplitude terms, AXair,i,T and AYair,i,T
( AXair,i,R and AYair,i, R ), to obtain the transmitted (reflected) power at  or 2
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