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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

valentin.besse@univ-angers.fr

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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