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V. Besse, C. Cassagne, H. Leblond, G. Boudebs
Laboratoire de Photonique d’Angers EA 4464, Université d’Angers,2 Bd Lavoisier, 49000 Angers, France
Third- and Fifth-order Optical Nonlinearities Characterization Using the
D4σ-Z-scan Method
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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
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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)
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D4σ-Z-SCAN INSIDE A 4f SYSTEMC.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)]
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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 − ωω
NLyω
Lyω
Beam Waist Relative Variation(BWRV)
z(mm)
NL L
L
ω − ωω
Δωpv
NL
pha
se
shif
t
BW
RV
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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
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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
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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 γIz
Optical intensity I (W/m2)
( )2d
d= +ϕ
2 4k n I n Iz
(2)
Phase ϕ
2 22
2 2 2 20
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
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SOLUTION OF (1)
( ) ( )
ln ln lnL L L
0 0 0
I I X I XI I X I X
LX 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
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SOLUTION OF (1)
Problem !We have : z ( I ) We need : I ( z )
?
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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
( )1nlim 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)]
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SOLUTION OF (2)
( ) ( )
( )
2 4
2 4
ln
ln
++
+ − +
−−
−
−∆ = − + ÷− − −+ + ÷−
ϕγ
L
0
L
0
I Xkn n X
X X I X
I Xn n X
I X
( )2d
d= +ϕ
2 4k n I n Iz
(2)
Phase ϕ
2 3
dd =
− − −I
zαI 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
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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
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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)]
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NONLINEAR ABSORPTION COEFFICIENTS
Pure Two-Photon Absorption (2PA)
0 L
L 0
I IL
βI I
−=
Pure Three-Photon Absorption (3PA)
2
2 20 L
2 2L 0
I IL
γI I
−=
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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)]
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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
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Ikn
β I
ln −−∆ = + ÷
ϕ L 0 L
2 4L 0 0
I I Ikn n
γ I I I
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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
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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)
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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
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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.
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ATTENTION