Presentation of third- and fifth-order optical nonlinearities measurement using the D4Sigma-Z-scan Method. I present a resolution of propagation equation in general case (with third- and fifth-order nonlinearities) and a numerical inversion. This presentation is conclude with experimental results.

1. 1
V. Besse, C. Cassagne, H. Leblond, G. Boudebs
Laboratoire de Photonique d’Angers EA 4464, Université d’Angers,
2 Bd Lavoisier, 49000 Angers, France
valentin.besse@univ-angers.fr
Third- and Fifth-order Optical
Nonlinearities Characterization Using the
D4σ-Z-scan Method

2. 2
OUTLINE
• Introduction : Propagation equation, Newton’s method, D4σ-Z-
scan technique …
• Theory : analytic solution of the equation, numerical inversion
• Nonlinear coefficients measurement using D4σ-Z-scan method
• Conclusion

3. 3
PROPAGATION OF A BEAM
NLM
Linear absorption
α (m-1)
Two photon absorption
β (m/W)
Three photon absorption
γ (m3
/W2
)
Third-order refractive index
n2 (m2
/W)
Fifth-order refractive index
n4 (m4
/W2
)
Linear refractive index
n0
?
χ(3)
χ(5)
χ(1)

4. 4
D4σ-Z-SCAN INSIDE A 4f SYSTEM
C.C.D.
f1 f1 f2
L1 L2
L3M1 M2
NLM
BS1 BS2
f2
O(x,y)
L2: far field diffraction
x
z
y
[M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, E. W. Stryland, "Sensitive measurement of optical nonlinearities using a single
beam", IEEE J. Quantum Electron. 26, 4, 760-769, (1990)]
[K. Fedus, G. Boudebs, "Experimental techniques using 4f coherent imaging system for measuring nonlinear refraction", Opt. Comm.
292, 140–148 (2013)]
[G. Boudebs, V. Besse, C. Cassagne, H. Leblond and C.B. de Araújo, “Nonlinear characterization of materials using the D4σ method
inside a Z-scan 4f-system”, Opt. Lett. 38, 13, 2206-2208 (2013)]

5. 5
CCD
PRINCIPLE OF THE D4σ METHOD
The detector is a CCD to allow distance measurements
y
NL regime :
Linear regime :
x
NL L
L
ω − ω
ω
NLyω
Lyω
Beam Waist Relative Variation
(BWRV)
z(mm)
NL L
L
ω − ω
ω
Δωpv
NLphase
shift
BWRV

6. 6
WHY WE CHOOSE THE D4σ METHOD
• The sensitivity is independent from the experimental setup and CCD
pixel size.
• The relation between Δω and the effective phase shift at the focus
remains valid in the presence of relatively high nonlinear absorption
• Does not need to divide two different profile to obtain the nonlinear
refractive response
( ) ( )
( )
2
im
x
im
I x, y x x dxdy
2
I x, y dxdy
+∞ +∞
−∞ −∞
+∞ +∞
−∞ −∞
−
ω =
∫ ∫
∫ ∫
( )
( )
im
im
I x, y xdxdy
x
I x, y dxdy
+∞ +∞
−∞ −∞
+∞ +∞
−∞ −∞
=
∫ ∫
∫ ∫
where

7. 7
OUTLINE
• Introduction : Beam propagating equation, Newton’s method,
D4σ-Z-scan technique …
• Theory : analytic solution of the equation, numerical inversion
• Nonlinear coefficients measurement using D4σ-Z-scan method
• Conclusion

8. 8
PROPAGATION EQUATION
• under the slow varying envelope
approximation
• thin sample approximation
Modulus of the wave vector : k = 2π/λ Wavelength : λ
(1)
2 3d
d
= − − −
I
αI βI γI
z
Optical intensity I (W/m2
)
( )2d
d
= +
ϕ
2 4k n I n I
z
(2)
Phase ϕ
2 2
2
2 2 2 2
0
1 1∂ ∂
∇ − =
∂ ∂εc t c t
% % %E E P
[R. W. Boyd, Nonlinear Optics, third edition (Academic Press, New York 2007)]
• : electric field
• : nonlinear polarization
%E
%P

9. 9
SOLUTION OF (1)
( ) ( )
ln ln lnL L L
0 0 0
I I X I X
I I X I X
L
X X X X X X X X
− +
− +
+ − − − + + − +
      − −
  ÷  ÷  ÷− −      = − + −
 − −
 
 
γ γ γ
• Exactly solve after partial fraction decomposition
• Each coefficient is nonzero
2
4>β αγ( ) ( )2
4 2± = ± − −β αγ β γX assuming
[V. Besse, G. Boudebs and H. Leblond, “Determination of the third-and fifth-order optical nonlinearities: the general case”, Appl.
Phys. B, (2014)]
• Sample located between : z = 0 and z = L, with boundary conditions I (z = 0) = I0 and
I (z = L) = IL

10. 10
SOLUTION OF (1)
Problem !
We have : z ( I ) We need : I ( z )
?

11. 11
NEWTON’S METHOD
• Find approximations to the root
• Iterative method
• Using first-order Taylor expansion
( ) ( ) ( )( )'
n 1 ,n ,n ,n 1 ,nL, L L L Lz I z I z I I I+ += + −
( )( ) ( )'
,n 1 ,n 1 ,n ,nL+ += − −L L L LI I z I z I
• Easy to implement
• Need an adequate choice of the
starting values
( )1
n
lim L,nz I L+
→∞
=
[T. R. Oliveira, L. de S. Menezes, C. B. de Araújo and A. A. Lipovskii, “Nonlinear absorption of transparent glass ceramics containing sodium
niobate nanocrystals”, Phys. Rev. B 76, 134207 (2007)]

12. 12
SOLUTION OF (2)
( )
( )
( )
2 4
2 4
ln
ln
+
+
+ − +
−
−
−
  −
∆ = − +  ÷
− − 
 −
+ +  ÷
− 
ϕ
γ
L
0
L
0
I Xk
n n X
X X I X
I X
n n X
I X
( )2d
d
= +
ϕ
2 4k n I n I
z
(2)
Phase ϕ
2 3
d
d =
− − −
I
z
αI βI γI
by using (1) :
( )2
2 3
d
d
+
=
ϕ 2 4k n I n I
IαI+βI +γI
(2)
Phase ϕ
2
4>β αγ( ) ( )2
4 2± = ± − −β αγ β γX assuming
andwhere ∆ϕ = ϕ − ϕL 0

13. 13
OUTLINE
• Introduction : Beam propagating equation, Newton’s method,
D4σ-Z-scan technique …
• Theory : analytic solution of the equation, numerical inversion
• Nonlinear coefficients measurement using D4σ-Z-scan method
• Conclusion

14. 14
CARBON DISULFIDE (CS2)
• Considered as a reference material for nonlinear characteristics measurement.
• at λ = 532 nm, α = 0 and β = 0.
• Three photon absorption phenomenon becomes predominant at high intensity
• Fifth-order (n4 and γ) coefficients values previously published.
[S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas and X. Michaut, “An experimental investigation of the nonlinear
refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques”, Chem. Phys. Lett., 369, 3, 318-
324 (2003)]

15. 15
NONLINEAR ABSORPTION COEFFICIENTS
Pure Two-Photon Absorption (2PA)
0 L
L 0
I I
L
βI I
−
=
Pure Three-Photon Absorption (3PA)
2
2 2
0 L
2 2
L 0
I I
L
γI I
−
=

16. 16
NONLINEAR ABSORPTION COEFFICIENTS
λ = 532 nm and I0 = 25 GW/cm2
Pure 3PA: γ = (9.3 ± 1.9)×10-26
m3
/W2
(red solid line)
Pure 2PA: β = (8.5 ± 0.9)×10-12
m/W
(blue dotted line)
λ = 1,064 nm and I0 = 65 GW/cm2
Pure 3PA: γ = (4.6 ± 0.9)×10-26
m3
/W2
(red
solid line)
Pure 2PA: β = (1.7 ± 0.2)×10-12
m/W (blue
dotted line)
[G. Boudebs, V. Besse, C. Cassagne, H. Leblond and C.B. de Araújo, “Nonlinear characterization of materials using the
D4σ method inside a Z-scan 4f-system” Opt. Lett., 38, 13, 2206-2208 (2013)]
[G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, F. Sanchez, “Experimental and theoretical study of
higher-order nonlinearities in chalcogenide glasses” Opt. Comm., 219, 1, 427-433 (2003)]

17. 17
NONLINEAR REFRACTIVE INDEX
Pure third-order susceptibility contribution (only χ(3)
)
Mixed third- and fifth-order susceptibilities contribution (χ(3)
and χ(5)
)
ln
  −
∆ =   ÷
  
ϕ L
2
0
Ik
n
β I
ln
  −−
∆ = +  ÷
  
ϕ L 0 L
2 4
L 0 0
I I Ik
n n
γ I I I

18. 18
NONLINEAR REFRACTIVE INDEX
Measured at low intensity
λ = 532 nm and I0 = 1.6 GW/cm2
n2 = (1.5 ± 0.3)×10-18
m2
/W
β < 0.2 cm/GW
λ = 1,064 nm and I0 = 4.5 GW/cm2
n2 = (4.5 ± 1.3)×10-19
m2
/W
β < 0.05 cm/GW
Fifth-order contribution is INSIGNIFICANT

19. 19
NONLINEAR REFRACTIVE INDEX
λ = 532 nm and I0 = 25 GW/cm2
χ(3)
and χ(5)
: γ = (9.3 ± 1.9)×10-26
m3
/W2,
n2
= (1.5 ± 0.3)×10-18
m2
/W and n4 =
(1.2 ± 0.3) ×10-32
m4
/W2
(red solid line)
χ(3)
: β = (8.5 ± 0.9)×10-12
m/W and
n2 = (2.7 ± 0.3)×10-18
m2
/W (blue dotted
line)
λ = 1,064 nm and I0 = 65 GW/cm2
χ(3)
and χ(5)
: γ = (4.6 ± 1.9)×10-27
m3
/W2,
n2
= (4.5 ± 1.3)×10-19
m2
/W and n4 =
(2.2 ± 0.4) ×10-33
m4
/W2
(red solid line)
χ(3)
: β = (1.7 ± 0.2)×10-12
m/W and
n2 = (1.4 ± 0.2)×10-18
m2
/W (blue dotted
line)

20. 20
OUTLINE
• Introduction : Beam propagating equation, Newton’s method,
D4σ-Z-scan technique …
• Theory : analytic solution of the equation, numerical inversion
• Nonlinear coefficients measurement using D4σ-Z-scan method
• Conclusion

21. 21
CONCLUSION
• Analytical solution of the z (I) and ϕ (I) with third- (n2 and β) and fifth-order (n4 and
γ) nonlinearities.
• Allows to measure the fifth-order nonlinearities.
•The fitting method is easy to implement.
• Confusion could appear when considering ONLY data at high intensity leading to a
overstimation of the n2 values of CS2 at 532 nm and 1064 nm.

22. 22
THANK YOU
FOR YOUR
ATTENTION