Nonlinear characterization of materials using the D4σ method inside a Z-scan 4f-system

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Nonlinear characterization of materials using the D4 σ method inside a Z-scan 4 f -system

Georges Boudebs,^1,* Valentin Besse,¹Christophe Cassagne,¹Hervé Leblond,¹and Cid B. de Araújo²

1LUNAM Université, Université d’Angers, LPhiA, Laboratoire de Photoniques d’Angers, EA 4464, 49045 Angers Cedex 01, France

2Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil

*Corresponding author: georges.boudebs@univ‑angers.fr Received April 16, 2013; revised May 23, 2013; accepted May 24, 2013;

posted May 24, 2013 (Doc. ID 188925); published June 19, 2013

We show that direct measurement of the beam radius in Z-scan experiments using a CCD camera at the output of a 4f-imaging system allows higher sensitivity and better accuracy than Baryscan. One of the advantages is to be insensitive to pointing instability of pulsed lasers because no hard (physical) aperture is employed as in the usual Z-scan.

In addition, the numerical calculations involved here and the measurement of the beam radius are simplified since we do not measure the transmittance through an aperture and it is not subject to mathematical artifacts related to a normalization process, especially when the diffracted light intensity is very low. © 2013 Optical Society of America OCIS codes: (160.4330) Nonlinear optical materials; (190.4420) Nonlinear optics, transverse effects in; (260.5950) Self-focusing; (050.5080) Phase shift.

http://dx.doi.org/10.1364/OL.38.002206

Inspection of linear and nonlinear (NL) optical parameters, such as the refractive index and the absorption coefficient is essential for most applications in optics (optical limiting, lasers, optical amplifiers, all-optical switching, etc.). Measurements of NL refraction and NL absorption using far-field beam diffraction as in the Z-scan method [1] are widely performed today. Recently, a variant of this method called Baryscan [2], reporting an increased sensitivity, has been published. The authors measured the displacement of the beam centroid caused by the focusing (or defocusing) effect due to the nonlinearity and achieved by a razorblade positioned close to a position sensitive detector (PSD). The razorblade truncates half of the beam, indicating an increase (or de- crease) of the beam size. Then the PSD shows a change in the position of the beam centroid. The authors of [2]

claim what they call sensitivity up toλ∕50;000using sta- ble CW solid-state laser input. According to [1] and more generally to measurement characteristic definitions, this quantity should be called phase distortion resolution (PDR) and should not be considered as sensitivity, which is defined as the ratio of the output of a measurement system with respect to an input. More recently, Ferdinan- dus et al. reported a dual-arm Z-scan technique [3] im- proving the PDR up to λ∕1000 using pulsed lasers. The dual-arm scheme allowed determination of NL refractive signals considerably smaller than in the single-arm Z-scan method (λ∕300). These recent studies show that improvement of the PDR is still relevant, especially considering that normalization of signals in the Z-scan method plays an important role, as pointed out first in [4]. Measuring the diffraction efficiency allowed us to compare the sensitivity of different techniques inside a Z-scan4f system [5] and had the precious advantage of avoiding a“division by zero” with nonexisting energy as in EZ-scan [6]. Ex- ploiting the diffraction efficiency allowed us to increase both the sensitivity and the signal-to-noise (S/N) ratio by properly matching the object with the field stop. It was shown that the Z-scan is the most sensitive technique with the highest S/N ratio allowing measurements of NL phase distortions as small asλ∕1000. But there still

remain open questions to fully understand the physical phenomenon contributing to sensitivity improvement and what should be the PDR of the Baryscan method (BM) using pulsed lasers.

We demonstrate here that the sensitivity of Baryscan is two times lower than that of Z-scan, and we show that the use of hard (physical) apertures (such as a razorblade) with pulsed lasers presenting pointing fluctuations could be a severe limitation leading to both lower sensitivity and small S/N ratio when compared to direct measurement of the output beam waist variation using a CCD sensor. Instead of measuring the beam radius as a function of the peak intensity as in [7], the D4σ method con- sists in determining the first- and second-order moments of the beam profile, providing higher precision in the measurements. The NL image formation inside the shown in Fig.1is described using a model based on Fou- rier optics (see, for example, [8]).

The general scheme of beam propagation inside the 4f system is described in detail in [5]. Although the method presented here can be applied to arbitrary beam profiles, to make comparison with [1,2], we assume that the electric field at the object plane is Gaussian, Ex; y E₀ exp−x²y²∕ω²_e, where xand yare the spatial coordinates,E0denotes the on-axis amplitude, and ωeis the beam waist at the entry of the setup. LetSu; v be the spatial spectrum of E: Su; v F~Ex; y R_∞

−∞

R_∞

−∞ Ex; yexp−j2πuxvydxdy, where F~

Image plane M

L1 L2

0 +z -z

Object plane Gaussian

beam

f f f f

z' y x

Z-scan

Baryscan

CCD

Fig. 1. Schematic of the4f system. The sample (M) is moved along the focal region. The labels refer to lenses (L1−L2).

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denotes the Fourier transform operation, anduandvare the normalized spatial frequencies. S is propagated over distancez⁰, by taking into account the transfer function of the wave propagation phenomenon Hu; v

expj2πz⁰

1−λu²−λv²

p ∕λ, where λ is the wave- length. Then the field amplitude at z⁰ is obtained by computing the inverse Fourier transform: Ex; y; z⁰ F~⁻¹Su; vHu; v. To calculate the output beam after passing through a lens with focal lengthf, we apply the phase transformation related to the thickness variation:

tLx; y exp−jπx²y²∕λf. The first propagation is performed on a distance z⁰f from the object plane to the first lens. Then we propagate the beam up to the sample located at z using z⁰fz in H, the optical transfer function. Next the NL response of the material is taken into account before continuing the propagation using the same method up to the image plane. We assume cubic nonlinearity and a thin NL medium of thickness L exhibiting (i) linear absorption defined by αm⁻¹, (ii) two-photon absorption defined byβm∕W, and (iii) NL refraction defined byn2m²∕W. In these conditions, the transmittance of the sample is described as

Tu; v; z fe^αL1qu; v; zg^−1∕2expjΔϕêffNLu; v; z; (1) where qu; v; z βL_{ef f}Iu; v; z with Leff 1−e^−αL∕α andIu; v; zdenotes the intensity of the laser beam within the sample. The latter quantity is related to the input electric field by Iu; v; z∝jSu; v; zj². ΔϕêffNLu; v; z 2πn₂LeffIeffu; v; z∕λ is the NL phase shift where we define the effective intensity Ieffu; v; z Iu; v; z log1qu; v; z∕qu; v; z. The on-axis NL absorption and NL phase shift at the focus are q₀βLeffI₀ and φêffNL0 ΔϕêffNL0;0;0, respectively, and I0 is the focal on-axis intensity. We consider hereafter the low excitation regime (q0<1;φêffNL0<1).

The beam waist measurement is performed using the ISO standard definition [9]. Based on the second moment ofIx; y, the D4σmethod gives four times the standard deviation of the intensity distribution. For example, the beam radius in thex direction is

ωx2

R_∞

−∞

R_∞

−∞ Ix; yx−x¯²dxdy R_∞

−∞

R_∞

−∞ Ix; ydxdy s

; (2)

where x¯R_∞

−∞ R_∞

−∞ Ix; yxdxdy∕R_∞

−∞ R_∞

−∞ Ix; ydxdy is the centroid of the beam profile in thex direction.

For Gaussian beams, the D4σ method gives the same result as the1∕e²method, whereas for other beam shapes there can be significant deviations. The1∕e² width measurements are noisier than D4σ width measurements depending on the integral of all the pixels contained in each frame. Also for multimodal marginal distributions (beam profile with multiple peaks), the D4σwidth would be a better choice.

To calculate the energy inside the aperture in the Z-scan configuration we must integrate over a radius defined by S, the closed aperture linear transmittance related to ωi, the beam waist in the image plane in the linear regime. The latter is physically equal to the beam

waist at the entry (ω_iω_e) because the magnification of the 4f system is equal to 1.

The simulation results of ωNL−ωL∕ωL, the beam waist relative variation (BWRV) versus z, the position of the sample in the focus, are shown in Fig.2(a), where ωNL andωL denote, respectively, the mean values (measured alongxandy) of the beam waist defined in Eq. (2) associated to the NL and linear profiles at the output of the4f system.

The difference between the peak and the valleyΔω_pv, shown in Fig. 2(b), is a function of the effective phase shift at the focus for a Gaussian input beam showing a linear relation (Δω_pv0.34×φ^effNL0). Note that this for- mula allows us to measuren2when the other parameters inφ^effNL0are known. Moreover the linearity remains valid in the presence of relatively high NL absorption (in the present case we haveq0 0.58).

A detailed description of our experimental setup is given in [5]. Excitation is provided by a Nd:YAG laser delivering linearly polarized 17 ps single pulses at λ 1064μm with 10 Hz repetition rate. In the image plane, we use a cooled (−30°C)1000×1018pixel CCD camera.

We perform Z-scan, BM, and D4σ profiles out from the same acquired images at eachzposition by numerically changing the soft aperture. Two sets of acquisitions are performed in the linear and the NL regimes to correct for the diffraction effects due to sample inhomogeneities and imperfections by subtracting the data related to the scans in each regime. When using the D4σ method, one has to calculate the centroid of the beam before processing the beam waist using Eq. (2). This processing inside the frames is equivalent to follow the pointing instability of the pulsed laser and thus reduces the noise that could appear with hard apertures such as the ones used with Z-scan and BM.

Figure3(a)shows the experimental results for glassy As₂Se₃, which present high NL response due to bound and free electrons. The D4σ results (filled squares) can be compared with those of BM (circles) that is buried inside the noise due to laser pointing instability, while the D4σmethod gives a cleaner signal with a higher S/N ratio.

The same behavior would be found for Z-scan using hard aperture. The calculated profiles (solid line for D4σ and dashed line for BM) are shown in the same figure. The agreement with the D4σ results is very good, while the BM produces a poor S/N ratio. Moreover, the D4σmethod is unambiguously two times more sensitive than the BM.

Physically we consider the totality of the pixels in order

Fig. 2. (a) BWRV versusz. The other parameters areφ^effNL0 0.8 and q00.58. (b) Calculated Δωpv, versus the effective phase shift at the focus.

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to measure the beam waist. So the quantity of informa- tion is twice that obtained with the razorblade in BM.

This is illustrated in Fig.3(a)by the difference between the peaks and the valleys inside the two scans processed for the same acquisitions showing approximately a factor of 2 in the sensitivity. Moreover the PDR in BM using a pulsed laser with hard aperture in our experiment is approximately λ∕180, which is a factor of 5 lower than the PDR obtained with the D4σ method.

To obtain a good comparison between Z-scan and D4σ and to improve the S/N ratio in the Z-scan, the centroid calculation is used for both profiles in Fig.3(b)to mon- itor the beam transversally for pointing laser fluctuations.

The NL parameters obtained wereq00.6[see the inset in Fig. 3(b)] and φ^effNL00.27. Considering α0m⁻¹, L0.64mm, and I00.34GW∕cm², we obtain n2 234×10⁻¹⁸ m²∕W andβ223cm∕GW. Note that the calibration of I0 is done according to the n2 value

obtained for CS₂ in [1] in the same experimental conditions. The calculations and processing in Fig.3(b)have been made using the same acquired images showing unambiguously the same sensitivity. The radius of the cir- cular closed aperture in the Z-scan is calculated for S0.73 to maximize the optical diffraction efficiency [5]. The advantage in the D4σ method when compared to Z-scan is that there is no need to divide two different profiles to obtain the NL refractive response.

For an arbitrary (not Gaussian) beam, it is recom- mended to use the D4σor second-moment beam width, which follows the ISO standard definition [9], after calibration using a reference NL material, because the sensitivity depends on the spatial profile of the incident beam. Note that the BWRV is a ratio for which the pixel size (12μm ×12 μm) does not alter the results in Fig.3(b)when the sampling theorem is valid for a given beam at the output, while the accuracy in determining Δωpv is related to the high dynamic range of the CCD (4095 gray levels in our case).

In summary, the D4σmethod is insensitive to pointing instability of the pulsed laser because no hard aperture is employed as in Z-scan or BM. Numerical calculations allow us to obtain simple relations that can be used for measurements simplifying the procedure, especially for NL absorbing material.

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Fig. 3. (a) Comparison between D4σmeasurements (squares) with those of Baryscan (circles). The solid and dashed lines show the numerical results. (b) Comparison of the BWRV (squares; vertically shifted to 1) and the usual Z-scan transmittance (stars) for a highly NL absorbing material. The inset shows the open aperture Z-scan transmittance.

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