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Third- and Fifth-Order Optical Nonlinearities Characterization Using the D4σ-Z-Scan Method

Valentin Besse (*), Christophe Cassagne, Hervé Leblond, and Georges Boudebs LUNAM Université, Université d’Angers, Laboratoire de Photonique d’Angers,

EA 4464, 2 Boulevard Lavoisier, 49045 Angers CEDEX 01, France (*) Tel: (0033) (0)2 41 73 50 44, e-mail: valentin.besse@univ-angers.fr

ABSTRACT We solved analytically the equations governing the evolution of the intensity and phase shift inside a nonlinear medium taking into account third- and fifth-order nonlinear susceptibilities. We give explicit expression of the propagation distance and the phase shift versus the output intensity. This solution is inverted numerically by means of Newton's method. The result does not apply to some special cases due to the vanishing of coefficients, then analytical solutions are given, so that all the cases are covered. Combined with D4σ-Z-scan method, the third- and fifth-order nonlinear coefficients are measured at 532 nm and 1064 nm in the picosecond regime. Keywords: nonlinear absorption, nonlinear refractive index, Newton’s method, D4σ-Z-scan technique.

1. INTRODUCTION

Large nonlinearities are essential for many applications in optics. For example highly nonlinear (NL) absorbing materials can find applications in the field of optical limiting [1], and large nonlinear refractive index is used for ultrafast all-optical switching [2, 3]. Another application is the generation and the propagation of stable two- or three- dimensional dissipative solitons [4], theoretically predicted and experimentally achieved [5] with specific third-order and fifth-order optical nonlinearities. Many methods allow to measure third-order nonlinear refraction and two-photon absorption (2PA) [6, 7]. In reference [7], we have shown using D4σ method that the numerical calculations allow to obtain simple relations that can be used for the measurements simplifying the procedure especially for NL absorbing material. We will show hereafter that using this method, one can determine high-order susceptibilities in all cases.

Previously, we have solved analytically, in references [8] and [9] with adequate approximation, the differential equation which governs the evolution of the intensity in the presence of nonlinear third-order refractive index (n2) and both 2PA and three photon one (3PA). Using NL fitting combined to Runge-Kutta numerical integration method we solved numerically the beam propagation equation. In [10] the case of third- and fifth-order (n4) NL refractive indices with 2PA only was considered. Here we consider the general case with materials exhibiting all the NL coefficients up to the fifth-order including 2PA, 3PA, n2 and n4.

2. THEORY

We consider a beam propagating along the z-axis in a sample exhibiting linear absorption [coefficient α (m−1)], 2PA [coefficient β (m/W)], 3PA [coefficient γ (m3/W2)], Kerr effect [third-order NL refractive index n2 (m

2/W)], and fifth-order NL refractive index [n4 (m4/W2)]. Under the slowly varying envelope and thin sample approximations, the optical intensity I and the phase ϕ satisfy the equations

2 3d

d

IαI βI γI

z= − − − (1)

( )2d

d 2 4k n I n Iz

= +ϕ (2)

Where k = 2π/λ is the modulus of the wave vector and λ is the wavelength. We consider here the general case where each coefficient is nonzero. The differential equation (1) governs the evolution of the intensity as a function of the propagation distance z in the medium. The sample is located between the plane z = 0 and the plane z = L. The boundary conditions are I (z = 0) = I0 and I (z = L) = IL. It is possible to exactly solve this equation by separating the variables. After partial fraction decomposition and integration from 0 to L, (1) leads to

( ) ( )

,

ln ln lnL L L +

0 0 0 +

+ + + +

I I X I X

I I X I Xz

γX X γX X X γX X X

−

−

− − − −

− − − − = − + − − −

(3)

where ( ) ( )2 4 2X β αγ β γ± = ± − − (assuming that 2 4β αγ> )

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Expression (3) can be inverted numerically using Newton’s method. Newton's method is iterative and based on the approximation of the function z ( I ) using a first order Taylor expansion. Solving the equation z(I) = L with z(I) given by Newton’s method yields the recurrence relation IL,n+1 = IL,n – (z(IL,n) – L)/z′(IL,n) which is used in the iterative process (the prime holds for the derivative with respect to I). Then the NL phase shift Δϕ is obtained by solving equation (2) and using equation (1). After decomposing this solution in partial fractions, it is integrated from I0 to IL to yield

( ) ( ) ( )2 4 2 4ln lnL L

0 0

I X I Xkn n X n n X

X X I X I Xϕ

γ+ −

+ −+ − + −

− −Δ = − + + + − − −

. (4)

Here Δϕ = ϕL – ϕ0 is the NL phase shift due to the propagation through the NL medium. We replaced IL by the value IL,n that we found previously. By fitting Z-scan or D4σ traces we are able to determine experimentally the value of each coefficient: α , β , γ (with Eq. (3)), 2n and 4n (with Eq. (4)).

The various cases and corresponding solutions are reported in tables 1 and 2 (see appendix) in reference [11].

3. EXPERIMENTAL RESULTS

The experimental data were obtained using D4σ method inside a Z-scan 4f-system (see Fig. 1). The procedure is extensively described in references [7] and [11].

Figure 1. The 4f imaging system. The nonlinear material (NLM) is moved around the focal plane ( 0Z = ). The labels refer to: lenses (L1, L2 and L3), beam splitters (BS1 and BS2), and mirrors (M1 and M2) .The object at the

entry is a circular aperture. The image is acquired by a CCD camera.

We used a 1 mm thick cell filled with carbon disulfide (CS2) excited by a Nd:YAG laser delivering linearly polarized picosecond pulses at λ = 1.064 μm (17 ps) and 532 nm (12 ps) with a repetition rate less than 1 Hz. In the image plane, we use a cooled (-30°C) 1000×1018 pixels CCD camera with a fixed linear gain. Each pixel 12×12 µm2 shows 4095 gray levels. It is important to note that we perform absolute measurements [12] [13] which avoids an intensity calibration depending on a specific material.

3.1 Nonlinear absorption coefficients

The CS2 does not exhibit linear absorption (α = 0) or 2PA (β = 0) at 532 nm. However, with increasing incident intensity the phenomenon of 3PA becomes predominant at 800 nm [14]. We measures the NL absorption using a central peak intensity at the focus inside the cell I0 = 25 GW/cm2. The empty circles in Fig. 2 correspond to the experimental data. The numerical fitting is realised by using Newton’s method with two different analytical expressions related to lines six and seven in appendix A in reference [11]. As we can see in Fig. 2, considering only 2PA we obtain β = (8.5±0.9)×10-12 m/W (blue dotted line) but the best fit is found for 3PA with coefficient γ = (9.3±1.9)×10-26 m3/W2 (solid red line). In reference [14] the authors found a value of γ = 1.37×10-27 m3/W2 using 120 picoseconds pulses at 800 nm which is in good agreement with our value considering the difference in the experimental parameters. Moreover, one of the CS2 absorption bands is approximately centered at 200 nm (see Fig. 1a in reference [5]), that is approximately λ/3 for λ = 532 nm. The proximity with this resonance band leads to increase |χ(5)| ∝ γ which is supported by the fitting of the 3PA (solid red line in Fig. 2), more close to the experimental data than the 2PA one.

CC

D

f1 f1 f2

L1 L2

L3 M1

NLM BS1

BS2

f2

Z

x y

M2

Image plane

Object plane

+Z 0

-Z

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Figure 2. (Color online) Open aperture Z-scan normalized transmittance (empty circles) of a 1 mm thick cell filled with CS2 measured at λ = 532 nm with I0 = 25 GW/cm2. The solid line (red) shows the numerical fitting

considering only 3PA where γ = (9.3±1.9)×10-26 m3/W2, the dotted line (blue) shows the fitting considering only 2PA: β = (8.5±0.9)×10-12 m/W.

The same measurement was made at λ = 1.064 µm leading to β = (1.7±0.2)×10-12 m/W with pure 2PA effect: and to γ = (4.6±0.9)×10-27 m3/W2 with pure 3PA effect. Again it appears that the fitting considering only γ is better than the one with β. These measurements confirm that CS2 does not exhibit significant 2PA characteristic but mainly 3PA at both considered wavelengths.

3.2 Nonlinear refractive index

Following the D4σ method described in details in [7] we measured the beam waist relative variation (BWRV) using the first and the second order moments of the image profile versus Z, the position of the sample in the focus. In both cases, at λ = 532 nm and at λ = 1.064 µm, we measured n2 and β at low intensity where the fifth-order contribution is insignificant. The obtained results are n2 = (1.5±0.3)×10-18 m2/W, β < 0.2 cm2/GW at I0 = 1.6 GW/cm2 for λ = 532 nm, and n2 = (4.5±1.3)×10-19 m2/W, β < 0.05 cm2/GW at 4.5 GW/cm2 for λ = 1064 nm. Then these measured values were fixed for the second set of acquisitions performed at higher intensity to determine the fifth-order coefficients.

At λ = 532 nm, the images are acquired using I0 = 25 GW/cm2, which corresponds to the previous open aperture Z-scan transmittance shown in Fig. 2. We measured the BWRV and the results are shown on Fig. 3 (empty circles).

Figure 3: (Color online) Beam waist relative variation (empty circles) versus Z of a 1 mm thick cell filled with

CS2 measured at λ = 532 nm with I0 = 25 GW/cm2. The blue dotted line is obtained considering only χ(3) contribution: β = (8.5±0.9)×10-12 m/W and n2 = (2.7±0.3)×10-18 m2/W. The solid line (red) is found

with 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.

In order to take into account the third- and fifth-order contributions at this relatively high intensity the analytical formula line six in appendix B in reference [11] is used to fit the data where all parameters, except n4, are already known. Using the values measured at low intensity and considering the value obtained above for the

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3PA coefficient, the solid (red) line shown in Fig. 3 is obtained for n4 = (1.2±0.3)×10-32 m4/W2. Considering only third order contribution, we compute the BWRV assuming that the NL refraction is limited to third-order susceptibility only. The blue dotted line in Fig. 3, is obtained with n2 = (2.7±0.3)×10-18 m2/W using the 2PA coefficient found above. The comparison between the blue and the red lines shows that the NL refraction contribution of the fifth-order can be attributed to the third one.

The same experiment was performed at λ = 1.064 µm by considering third-order susceptibility only: β = (1.7±0.2)×10-12 m/W and n2 = (1.4±0.2)×10-18 m2/W; and considering together third- and fifth-order nonlinearities, we obtain n4 = (2.2±0.4)×10-33 m4/W2 using the above result for 3PA and n2 = (4.5±1.3)×10-19 m2/W obtained at low intensity 4.6 GW/cm2.

In both cases (λ = 532 nm or λ = 1.064 µm) the agreement between the fits and the experimental data is very good, showing again that confusion could appear when considering data at high intensity. If we consider that the CS2 response at high intensity is only due to n2 contribution, one can overestimate the n2 values for both considered wavelengths. In conclusion we point out a relatively high dependence response of the CS2 following the intensity.

4. CONCLUSIONS

In summary, we have found the analytical solution of the propagation distance and the phase shift inside the nonlinear medium versus the output intensity in the case of third- and fifth-order NL susceptibilities. We introduced a numerical inversion by means of Newton's method. The solutions are given in all the cases whatever is the indeterminacy due to vanishing coefficients. Combined with D4σ-Z-scan method in the case of CS2, this analysis allowed us to measure the third- and fifth-order nonlinear optical susceptibilities.

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