## Abstract

A nonlinear imaging technique with phase object, which can deduce nonlinear absorption and refraction coefficients by single laser-shot exposure, is expanded to a time-resolved pump-probe system by introducing a pump beam with a variable temporal delay. This new system, in which both degenerate and nondegenerate pump and probe beams in any polarization states can be used, can simultaneously measure dynamic nonlinear absorption and refraction conveniently. In addition, the sensitivity of this new pump-probe system is more than twice that of the Z-scan-based system. The semiconductor ZnSe is used to demonstrate this system.

© 2008 Optical Society of America

## 1. Introduction

The nonlinear-imaging technique with phase object (NIT-PO) is a newly developed technique for measurement of the nonlinearity of materials [1]. Because the NIT-PO is a single-shot technique and the sample does not need to move during the measurement, it easily can be extended to a time-resolved pump-probe system by the introduction of a pump beam. Since the amplitude and sign of the nonlinear absorption and refraction coefficients can be extracted from a single laser shot in the NIT-PO, the time-resolved pump-probe system based on the NIT-PO can measure the dynamic nonlinear absorption and refraction simultaneously. In our time-resolved pump-probe system, the pump and probe beams are crossed at a small angle, so their separation is very easy after they have passed through the sample. This means that the system can be applied to both degenerate and nondegenerate beams in any polarization states. J. Wang *et al.* have measured dynamic nonlinear absorption and refraction by use of a time-resolved pump-probe system that is based on a Z-scan system [2,3]. Numerical simulation shows that the sensitivity of our time-resolved pump-probe system is more than twice that of the Z-scan-based ssystem. In this paper, we demonstrate the time-resolved pump-probe system based on the NIT-PO for a standard sample of ZnSe at the wavelength of 532 nm. The bound electronic nonlinear refraction and two-photon absorption (TPA) as well as the two-photon-generated free-carrier refraction and absorption of ZnSe are measured.

## 2. Experiment setup

The experimental arrangement of a time-resolved pump-probe system based on the NIT-PO is shown schematically in Fig. 1. The extracted 22-ps FWHM double-frequency pulse (*λ*=532 nm) from a Q-switched and mode-locked Nd:YAG laser is separated into two beams: an intense pump beam and a much weaker probe beam. In our experiments, the polarization of the pump beam is adjusted perpendicular to that of the probe beam by a half-wave plate. A variable time delay is introduced into the pump path. The probe branch of the arrangement is a NIT-PO system. The probe beam is first expanded from 8 to 32 mm in diameter by the convex lenses *L*
_{1} and *L*
_{2} with focal lengths *f*
_{1}=10 cm and *f*
_{2}=40 cm, respectively, and then passes through the 4f system, which consists of convex lenses *L*
_{3} and *L*
_{4} with equal focal length *f*
_{3}=*f*
_{4}=40 cm. As shown in Fig. 2(a), an aperture with a radius of *R _{a}*=1.7mm, PO radius of

*L*=0.5 mm, and phase retardation of

_{p}*ϕ*=0.4

_{L}*π*is placed at the front focal plane of

*L*

_{3}. It allows only a small portion of the expanded probe beam to pass through at the central part. Since the size of the aperture is very small compared with the expanded probe beam, the part of the beam illuminated inside the aperture can be seen as a top-hat beam. The nonlinear sample is placed at the Fourier plane of the 4f system. A charge-coupled device (CCD) camera is used to collect the probe beam at the rear focal plane of

*L*

_{4}. The CCD camera (Imager QE of Lavision Company in Germany) has 1040×1376 pixels and a 4095 gray level. The size of each pixel is 6.4×6.4

*µ*m

^{2}. The PO can modulate the nonlinear phase shift in the nonlinear sample into the amplitude change of the electric field at the CCD plane. Figure 2(b) is the profile of a classical nonlinear image. We define the difference between the mean value of the intensity inside the PO and the one outside as Δ

*T*. The amplitude of Δ

*T*increases with the nonlinear phase shift inside the nonlinear sample ΔΦ

_{0}. When ΔΦ

_{0}is positive, the image will have an increased intensity inside the PO,

*i.e.*, Δ

*T*>0. Inversely, Δ

*T*<

_{0}when ΔΦ

_{0}is negative. In the measurement, the value of ΔΦ

_{0}can be deduced by fitting the numerically simulated Δ

*T*well with the experimental one. So the nonlinear refraction index

*n*

_{2}can be obtained from ΔΦ

_{0}=

*kn*

_{2}

*I*

_{0}

*L*, where the wave-vector

*k*, the peak intensity

*I*

_{0}, and the sample thickness

*L*are know quantities.

In our system, the relative sizes of the pump and probe beams inside the nonlinear sample should be considered, because they are main factors with which to decide the sensitivity of the measurement. We do the numerical simulation using the parameters in the above paragraph. Figure 3 shows that |Δ*T*| varies with the ratio of the pump beam radius *ω*
_{e0} and the probe beam radius *ω*
_{p0}. The solid curve in Fig. 3 is plotted with the reported parameters of ZnSe *β*=5.8 cm/GW (TPA coefficient) and *n*
_{2}=-6.8×10^{-14} cm^{2}/*W* at zero time delay, and the dashed curve is obtained with the parameters of CS_{2}, *β*=0 cm/GW, and *n*
_{2}=3.2×10^{-14} cm^{2}/*W*. We can see that both curves reach the highest sensitivity at *ω*
_{e0}/*ω*
_{p0}≈1.25. It means that whether or not the nonlinear sample has nonlinear absorption, the highest sensitivity of the nonlinear refraction measurement can reach around *ω*
_{e0}/*ω*
_{p0}≈1.25. Figure 4 shows that the sensitivity of nonlinear absorption increases with the value of *ω*
_{e0}/*ω*
_{p0} in the TPA measurement in which *T _{ν}* is the normalized transmittance at zero time delay. There are two reasons that make us consider that

*ω*

_{e0}at two to three times greater than

*ω*

_{p0}is the best choice for the measurement. One reason is that though the sensitivity of nonlinear refraction decreases to about 1.5 times less than the highest sensitivity, the sensitivity of nonlinear absorption increases to about 1.3 times greater. So, both the nonlinear absorption and the refraction can reach relatively high sensitivity. Another more important reason is that when a larger pump beam is used, the probe beam can detect a relatively homogeneous area. Thus, the error caused by misalignment of the pump and probe beams will be smaller than when they have approximately the same radii.

In the experiment, the Airy radius of the probe beam at the focal plane of *L*
_{3} is *ω*
_{p0}=1.22*λf*/(2*R _{a}*)≈76

*µ*m. The pump beam with a spatial Gaussian profile is focused to a spot the size of

*ω*

_{e0}=180

*µ*m (HW1/

*e*

^{2}) onto the sample by lens

*L*

_{5}. Considerable care was taken to ensure accurate spatial overlap of the pump and the probe beams within the sample with the aid of a pinhole. The small angle between the pump beam and the probe beam is 4.5°. The peak intensity of the probe beam is approximately 1.5% the intensity of the pump beam.

In the NIT-PO system, three images are needed to deduce the nonlinear absorption and refraction coefficients of the sample. The first image is a linear image, which is obtained when a neutral filter is placed before the nonlinear sample to attenuate the laser intensity that is too weak to induce the nonlinearity. The second one is a nonlinear image, which is obtained by placing the same neutral filter, which was used before, after the nonlinear material. The last one is a no-sample image, which is acquired by taking away the nonlinear material while leaving the neutral filter in the optical setup. We integrate all of the pixels of the linear image to get *En _{l}*, which is in proportion to the transmitted energy of the linear image. Similarly, we can get

*En*and

_{nl}*En*, which are in proportion to the nonlinear transmitted energy and without the sample transmitted energy, respectively, by integrating all of the pixels of the nonlinear image and the no-sample image. The linear transmittance of the sample is

_{ns}*T*=

_{l}*En*/

_{l}*En*. Note that the energy loss because of the reflection of the front and rear surfaces of the sample cell in the linear image should be considered. The nonlinear transmittance of the sample is

_{ns}*T*=

_{nl}*En*/

_{nl}*En*. The nonlinear absorption coefficient

_{l}*β*can be deduced by fitting the numerically calculated nonlinear transmittance to the experimentally measured

*T*by varying the value of

_{nl}*β*. During the calculation of

*β*, the value of

*n*

_{2}is unknown. But since we know that the nonlinear refraction does not affect the nonlinear transmittance, the value of

*n*

_{2}can be set arbitrarily. After the value of

*β*has been obtained, the nonlinear refractive index

*n*

_{2}, the only unknown parameter, can be deduced by fitting the numerically calculated Δ

*T*to the experimentally calculated one. More details about the measurement can be found in Ref. [1].

In our time-resolved pump-probe system based on the NIT-PO, a neutral filter is used in the probe beam to attenuate the intensity of the probe beam at the Fourier plane weakly enough so as to avoid nonlinearity. First, a linear image and a no-sample image are obtained when the pump beam is blocked. The linear transmittance of the material is *T _{l}*=

*En*/

_{l}*En*. Then the pump beam is unblocked, and a series of nonlinear images are taken while the temporal delay is scanning. For each nonlinear image at a different temporal delay,

_{ns}*En*and Δ

_{nl}*T*are extracted. By plotting

*En*as a function of time delay

_{nl}*t*, we obtain a curve that reveals nonlinear absorption alone. On the other hand, the curve Δ

_{d}*T*versus

*t*exhibits nonlinear refraction as well as nonlinear absorption, if present. It is very difficult to extract a signal that is produced by pure nonlinear refraction alone. So, first we have to analyze the normalized curve of

_{d}*En*versus

_{nl}*t*to obtain the photophysical parameters of nonlinear absorption. With the parameters related to nonlinear absorption already known, the parameters of nonlinear refraction can be obtained by fitting the curve of Δ

_{d}*T*versus

*t*.

_{d}In addition to the time-resolved pump-probe system based on the NIT-PO, a time-resolved pump-probe system can also be realized based on a Z-scan. Because the pump beams of the two kinds of systems are both tightly focused Gaussian beams, the sensitivities of the pump-probe system are determined by the probe beams. By comparing the sensitivity of the NIT-PO and the Z-scan, we conclude that the time-resolved pump-probe system based on the NIT-PO has higher sensitivity than the system based on a Z-scan. For small nonlinear phase shift |ΔΦ_{0}|≤*π* with ΔΦ_{0} denoting the on-axis nonlinear phase change at beam waist and small aperture Δ*T _{p-ν}*≈0.406|ΔΦ

_{0}| in the Z-scan, Δ

*T*is the difference between the peak and valley transmittances [3]. On the other hand, the sensitivity of the NIT-PO is determined by

_{p-ν}*φ*(the phase shift of the PO) and

_{L}*ρ*(the ratio of the radii of the PO and the aperture). The sensitivity increases with the decrease of

*ρ*. Considering the conveniently achievable

*ρ*of 0.345 and

*φ*=0.39, we get Δ

_{L}*T*=0.889ΔΦ

_{0}(in Ref. [4]), where Δ

*T*is the difference between the mean intensity within the PO radius on the CCD camera and that outside of the PO radius. So the sensitivity of NIT-PO is more than twice that of the Z-scan (0.889/0.406).

## 3. Measurement and discussion

As described in Ref. [2], the optical nonlinearities contained in the semiconductor ZnSe have two mechanisms: an ultrafast bound electronic nonlinearity that can be regarded as instantaneous, and a much slower TPA-induced free-carrier nonlinearity that has a long recovery time determined by the free-carrier lifetime. A time-resolved study of these processes can identify and characterize the various contributions.

The change of the absorption coefficient and refractive index induced by the bound electronic effect are:

where *I _{e}* is the irradiance of the pump beam,

*β*is the TPA coefficient, and

*n*

_{2}is the nonlinear refractive index. The factor 2 comes from weak-wave retardation [5]. The free-carrier absorption and refraction naturally depend on the density of the photo-generated carriers (Δ

*N*) produced by TPA

where *η* denotes the change of the refractive index per unit carrier density and *σ* is known as the free-carrier absorption cross-section.

In the pump-probe experiment, the probe beam is very weak compared with the excitation beam, so the TPA of the pump beam can be seen as the only source of the carrier generation. The carrier-generation rate is given by

where *τ _{r}* is the carrier lifetime.

By invoking slowly varying envelope approximation and thin-sample approximation [2], the propagation of the pump and probe beams in the sample can be described as

where *I _{p}* and

*ϕ*are the intensity and phase of the probe beam and

_{p}*α*is the linear absorption coefficient of ZnSe.

The linear refraction index of ZnSe is 2.7 at 532 nm, the sample thickness is 2 mm, and the linear transmittance is 0.55 (this includes the surface loss of the sample). During the measurement, the energy of a single pump pulse is 1.48 *µ*J, and it produces a peak intensity of 0.10 GW/cm^{2}. The CCD camera is very sensitive to background light, so the experiments are done in a darkroom. Before the experiments, the background light is eliminated by the software. The energy fluctuation of the laser is ±3%. Five images are taken at each temporal delay. Figures 5(a) and 5(b) are the linear image and nonlinear image at the zero time delay, respectively. It can be found that both the PO and the aperture are in nearly circular symmetry, whether for the linear image or the nonlinear image, so we use polar coordinates in the numerical simulation to simplify the calculation. *En _{nl}* and Δ

*T*are extracted from each nonlinear image. During the extraction, the pixels below 20 counts are set to 0 to reduce the background noise. The mean intensity of the background light is 4.2 counts, and the mean intensity of the laser spot is above 800 counts. The normalized curve of

*En*versus

_{nl}*t*is shown in Fig. 6. The division on of Δ

_{d}*T*by the mean intensity of the linear image is shown in Fig. 7. Each data in the two curves is an average of five images.

Next we deduce the photophysical parameters of ZnSe by numerically simulating the experimental curves. Eqs. (1)–(8) as well as the beam propagation described in Ref. [1] are used in the simulation, including a Fourier transform from the input plane of the 4f system to the sample and an inverse Fourier transform from the sample to the CCD plane. The curve in Fig. 6 only relates to nonlinear absorption. The sharp valley at the zero temporal delay is the result of TPA, and the slow recovery after the pump beam has passed through is the absorption of the free carrier. The depth of the valley at the zero time delay is mainly dependent on the TPA, and the free-carrier absorption has very little contribution. So the TPA coefficient *β*=5.4 cm/GW can be obtained by fitting the valley at zero temporal delay. The free-carrier absorption in Fig. 6 is very small (normalized transmittance reduces to about 0.99), so it is difficult to deduce the lifetime of the free carrier accurately. Fortunately, the recovery in the curve Δ*T* versus *t _{d}* (Fig. 7) is clear to see. The lifetime of the free carrier can be obtained from

*τ*=2.5 ns. Numerical simulate of the curve in Fig. 6 once again by substituting

_{r}*β*and

*τ*, and the free-carrier absorption cross-section is obtained from

_{r}*σ*=6.6×10

^{-17}cm

^{2}. Then we use a similar method of determining

*β*and

*σ*, and it is easy to obtain the nonlinear refractive index

_{α}*n*

_{2}=-6.4×10

^{-14}cm

^{2}/W and

*η*=-9.5×10

^{-21}cm

^{3}by numerical simulating Fig. 7. The photophysical parameters measured in the use of our pump-probe system are in agreement with the ones reported in earlier literature (listed in Table 1). The differences between the values of

*η*in our paper and the ones in Refs. 6 and 7 may be caused by the overestimation of the lifetime of the free carrier

*τ*through use of the indirect Z-scan measurement technique.

_{r}## 4. Conclusion

In this paper, we introduce a time-resolved pump-probe system based on the NIT-PO. Dynamic nonlinear absorption and refraction can be measured simultaneously in this system. This system is suitable for nonlinearity measurement of both degenerate and nondegenerate beams in any polarization state. In addition, the sensitivity of our time-resolved pump-probe system based on the NIT-PO is more than twice that of the Z-scan-based system.

## Acknowledgments

We gratefully acknowledge support by the National Natural Science Fund of China grant 10774109, the Program for New Century Excellent Talents in University grant NCET-04-0333, and the Excellent Youth Fund of Heilongjiang Province grant JC-04-04.

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