#### Abstract

We present a theory of open- and closed-aperture Gaussian beam Z-scan for nonlinear optical materials with saturable absorption and high-order nonlinear refraction. We show that an approximate expression for a transmitted intensity through the nonlinear optical material is possible by means of the Adomian’s decomposition method and the thin film approximation. The theory is applied to semiconductor CdSe quantum dot-polymer nanocomposite films. It is shown that the theory well explains measured results of open- and closed-aperture transmittances in the Z-scan setup. It is also shown that the nanocomposite film possesses simultaneous third- and fifth-order nonlinear refraction as well as saturable absorption of a homogeneously broadened type.

#### 1. Introduction

A Z-scan technique originally developed by Sheik-Bahae et al*.* [1] has been widely used for characterizing the nonlinear optical properties of various nonlinear optical materials whose nonlinear optical responses include nonlinear absorption and nonlinear refraction. Nonlinear absorption may be classified into four types two-photon absorption (TPA), multiphoton absorption, saturable absorption (SA), and reverse SA. Nonlinear refraction includes the third- and higher-order nonlinear refraction as well as cascaded second-order nonlinear refraction. The original Z-scan theory treats nonlinear optical materials having either the third-order nonlinear refraction with TPA or the fifth-order nonlinear refraction without TPA. Later, the theory was extended to the open-aperture Z-scan theories for nonlinear optical materials having multiphoton absorption [2], simultaneous TPA and three-photon absorption [3], and SA [4]. In addition, Z-scan analyses for nonlinear optical materials having simultaneous third- and fifth-order nonlinear refraction with or without TPA [5–9] were reported previously. These extensions are important for materials characterization and device design of, for example, optical limiters and all-optical photonic devices using third- and high-order nonlinear optical effects. However, no Z-scan theory has been reported so far for nonlinear optical materials having both SA and simultaneous third and fifth-order nonlinear refraction. Such a new nonlinear optical phenomenon has been indeed observed in our recently developed inorganic-organic nanocomposites, semiconductor CdSe quantum-dot- (QD-) polymer nanocomposite films [10, 11]. This new class of inorganic-organic nanocomposite material is capable of optically constructing multidimensional and functional (e.g., nonlinear) photonic lattice structures by the so-called holographic assembly of nanoparticles [12]. Indeed, we recently demonstrated holographic assembly of semiconductor CdSe QDs in photopolymer for constructing volume Bragg grating structures with diffraction efficiency near 100% [10]. This composite material has potential nonlinear optical applications such as optical switching/limiting, signal/image processing, and nonlinear photonic crystals [13, 14].

The electronic states of QDs are strongly influenced by the quantum confinement effect when the radius of QDs is smaller than approximately three times of the exciton Bohr radius [15]. The band gap () of QDs increases with decreasing their size and the quantum confinement effect can strongly enhance the third-order optical nonlinearity [16, 17]. The II-VI bulk semiconductor CdSe has the direct band gap eV at 300 K and has the exciton Bohr radius of 5.6 nm [17]. Therefore, the strong quantum confinement effect of CdSe QDs plays an important role in a large enhancement of the optical nonlinearity as compared with that of the bulk CdSe. Most of past studies with semiconductor QDs reported the third-order optical nonlinearity with and without TPA, although high-order optical nonlinearities in CdS_{x}Se_{1-x} QDs doped glasses [18–20] and colloidal CdTe QDs [21] were also reported. Saturation of the cubic Kerr nonlinearity due to high-order nonlinearities prominently at high optical intensities provides interesting nonlinear phenomena such as the formation of stable multidimensional optical solitons, but it sometimes causes a detrimental effect [22]. Therefore, it is necessary to characterize the third- and, if any, high-order optical nonlinearities in nanocomposite materials.

In this paper, we present a theory of the open and closed-aperture Gaussian beam Z-scan for nonlinear optical materials with both SA and simultaneous third- and fifth-order nonlinear refraction. We show that an approximate expression for a transmitted intensity through a nonlinear optical material possessing SA can be found by means of the Adomian’s decomposition method [23]. The theory is applied to semiconductor CdSe QD-polymer nanocomposite films. It is shown that the theory well explains measured results of open- and closed-aperture transmittances in the Z-scan setup. It is also shown that the nanocomposite film possesses simultaneous third- and fifth-order nonlinear refraction as well as SA of a homogeneously broadened type.

#### 2. Theory

Consider a single transverse mode Gaussian beam propagating along the direction in a thin nonlinear medium of thickness with SA of a homogeneously broadened type. In this case the intensity-dependent absorption coefficient is given by where is the linear absorption coefficient and is the saturable intensity. Note that the intensity is a function of the radial distance from the beam axis, time , the propagation distance from the Gaussian beam spot, and the local distance measured from the entrance surface of the medium can be decomposed by means of the Adomian’s decomposition method [23] and is given in a polynomial form as [4] where in which the incident intensity is given by with being the on-axis input intensity in the medium at the beam focus, which contains the temporal intensity envelope peaked at , and . Here, is the beam waist radius and is the diffraction length given by with being the wavenumber of the incident beam. We therefore note that is a function of , and . The first eight terms of () in (2), which are numerically confirmed to approximately represent a solution given by (2) with a high accuracy, are given by When the third- and fifth-order nonlinear refraction is taken into account (these contributions are sufficient for our experiment with semiconductor CdSe QD-polymer nanocomposite films as is shown later), the nonlinear phase shift under the thin film and slowly varying envelope approximations is found from the following differential equation [6, 9]: where and are the third- and fifth-order nonlinear refraction coefficients, respectively. Integration of (5) yields to the nonlinear phase shift given by where the third- and fifth-order nonlinear phase shifts, and are expressed as in which

The electric field at the exit surface of the medium that experiences the nonlinear phase shift is given by The expansion of the exponential term on the right-hand side of (9) in a Taylor series yields to where In the second equation on the right-hand side of (10), the approximation is made such that (8a) and (8b) are evaluated at . Since in practice, the quadratic- and higher-order terms in for the expansion of (10) may be neglected. Then, using the treatment by Tsigaridas et al. [6] with the Gaussian decomposition method [1], the electric field of the transmitted Gaussian beam at the distance from the medium exit plane is expressed as where the electric field of each individual Gaussian beam propagated at the distance is given by In (13), where in which

Note that the physical meanings of the parameters shown above are described in [6]. The normalized Z-scan transmittance [1] is defined as where the transmitted power through the aperture is given by the input power is expressed as , and the linear transmittance of the aperture is given by with and being the aperture radius and the Gaussian beam radius at the aperture, respectively. Note that although we have described the theory for SA of a homogeneously broadened type given by (1), it is also possible to analyze this problem for SA of an inhomogeneously broadened type.

In what follows, we show numerical calculations for of the closed-aperture Z-scan for a nonlinear medium possessing both SA of a homogeneously broadened type and the simultaneous third- and fifth-order nonlinear refraction. Figure 1 shows with and having the same signs for negative (positive) value for as shown in Figure 1(a) (Figure 1(b)), where the third- and fifth-order nonlinear phase shifts at the temporally and spatially peak input intensity are defined as and , respectively. It can be seen from Figures 1(a) and 1(b) that, unlike the pure third-order case, the peak-to-valley (or the valley-to-peak) configuration becomes asymmetric and its peak-to-valley difference in increases with an increase in because and have the same signs. It can also be seen that the fifth-order effect separates the valley more than the peak. The single peak and valley configuration is always maintained although the symmetry of the configuration is opposite each other for and . The deviation of the valley fromunity is larger than that of the peak with an increase in . The peak and valley tend to approach to the beam waist position () with an increase in as seen in Figures 1(a) and 1(b).

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Figure 2 presents with and having the opposite signs for negative (positive) value for as shown in Figure 2(a) (Figure 2(b)). It can be seen from Figures 2(a) and 2(b) that the peak-to-valley (or the valley-to-peak) configuration becomes asymmetric and its peak-to-valley difference in decreases with an increase in because and have the opposite signs. It can also be seen that the fifth-order effect separates the valley more than the peak as similar to the case of the same signs of and . The single peak and valley configuration is always maintained although the symmetry of the configuration is opposite each other. This trend is also the same as the case of the same signs of and for and . The peak and valley tend to separate from the beam waist position () with an increase in as seen in Figures 2(a) and 2(b). This trend is opposite to the case of the same signs of and shown in Figures 1(a) and 1(b).

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Peak-to-valley differences in , , are plotted as a function of in Figure 3 for different values of when maintains the single peak and valley configuration. It can be seen that with cm^{2}/GW and cm^{4}/GW^{2} increases faster than the case of the pure third-order effect without the influence of the fifth-order effect with an increase in . On the other hand, the fifth-order effect results in a slow increase of as increases with cm^{2}/GW and cm^{4}/GW^{2}.

A dependence of the normalized transmittance on is shown in Figure 4. The calculation was made with and = 0.1 and at (0)/ = 0.33. It can be seen that the peaks and valleys of decrease as increases. It means that the closed-aperture Z-scan is highly sensitive when is small as reported in [1].

#### 3. Application to Semiconductor Quantum Dot-Polymer Nanocomposites

##### 3.1. Sample Preparation

CdSe QDs (the average core size of 3 nm) were synthesized by the aqueous synthetic method and then extracted into an ionic liquid monomer. Detailed descriptions of preparation for the CdSe QD-monomer syrup were given in [10]. The syrup was cast on a glass plate loaded with a 10-*μ*m spacer and was covered with another glass plate. For our Z-scan measurement, this uncured film sample was exposed to a green LED to prepare a uniformly cured polymer nanocomposite film sample having the quantum-confined electron-hole transition at ~480 nm.

##### 3.2. Experimental Setup

A conventional Z-scan technique [1] was employed to evaluate nonlinear absorption and nonlinear refraction of our uniformly cured polymer nanocomposite film samples. We used a frequency-doubled, passively and actively mode-locked Nd:YAG laser (Continuum, YG601) with the FWHM pulse width of 35 ps at a wavelength of 532 nm and at a repetition rate of 10 Hz. The incident laser beam was divided into two beams. One laser beam was taken as a reference, while the other one was focused onto the film sample by a 200 mm convex lens to a spot with its beam waist radius ~32 *μ*m. A film sample was moved along the beam propagation direction (-axis) by a computer-controlled moving stage. An aperture and a detector were placed at a distance far away from a film sample. Linear transmittance of the aperture was set to be 0.06 in the closed-aperture Z-scan setup. Individual transmitted pulses at a given pulse energy within ±10% were selected and averaged to obtain one data point.

##### 3.3. Experimental Results

Typical open-aperture Z-scan results for a film sample with 0.91 vol.% CdSe QDs at , 1.2, and 1.8 GW/cm^{2} are shown in Figures 5(a), 5(b), and 5(c). It can be seen that is peaked at the beam focus. Since no nonlinear absorption was observed in a neat polymer film without CdSe QDs, the observed data was caused by the optical nonlinearity of CdSe QDs. The open Z-scan model with SA of a homogeneously broadened type [4] can clearly fit the data better than with TPA [1] having a negative TPA coefficient . We note that the model with SA of an inhomogeneously broadened type could not fit well with the closed-aperture Z-scan data (not shown here). Figure 5(d) shows transmittance change at as a function of . It can be seen that the SA model gives the best fit value for to be 0.54 GW/cm^{2}, which is consistent with that ( GW/cm^{2}) estimated from Figures 5(a)–5(c). It follows that the saturable absorption observed in our film sample is well described by SA of a homogeneously broadened type. This result implies excellent size uniformity of CdSe QDs in our sample as confirmed by a transmission electron microscope image of CdSe QDs [11]. We consider that the physical origin of the light-induced transparency is caused by transient bleaching of the CdSe QDs [24].

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Figure 6 shows the closed-aperture Z-scan result for the same film sample as that shown in Figure 5 at = 1.8 GW/cm^{2}, which is fitted by our closed-aperture Z-scan model without and with the fifth-order nonlinear refraction in the absence of nonlinear absorption (Figure 6(a)) and in the presence of saturable absorption (Figure 6(b)). It can be clearly seen from Figures 6(a) and 6(b) that the film sample exhibits the simultaneous third- and fifth-order nonlinear refraction as well as saturable absorption.

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Figures 7(a), 7(b), and 7(c) show the closed-aperture Z-scan results at several levels of intensities for the same film sample as that shown in Figure 5. It can be seen that the Z-scan data exhibit the peak-and-valley configuration, indicating the negative nonlinear refraction. It is well known that is proportional to for a Kerr material [1]. However, we found that for our film sample approximately had a linear dependence of (see Figure 7(d)), indicating that the third-order effect was not the sole contribution to the observed optical nonlinearity. The fifth-order contribution to as well as SA should be taken into account in the analysis of the closed-aperture Z-scan data.

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Based on our model given in Section 2, we performed the curve fitting of the present theoretical model with , together with and as fitting parameters, to the data. The best-fit values for and were found to be (−) × 10^{−3} cm^{2}/GW and () × 10^{−3} cm^{4}/GW^{2}, respectively. For comparison, a theoretical fitting curve in which only third- and fifth-order nonlinear refraction is taken into account (without nonlinear absorption) is also shown in Figure 7. Poor fit with the data is obvious, showing the need for the inclusion of nonlinear absorption (i.e., SA in our case).

The magnitude of the obtained for the film sample with 0.91 vol.% (3.6 wt%) CdSe QDs is approximately two orders of magnitude larger than that ( cm^{2}/GW) of a bulk CdSe at 1064 nm [25] and is approximately one order of magnitude larger than that ( cm^{2}/GW) at 794 nm for a polymer CR39 composite film with 1.5 wt% CdSe QDs [26]. Generally, electronic and thermal effects mainly contribute to the optical nonlinearities. However, the thermal contribution to the observed optical nonlinearities of the film sample was negligible in our experiment using a picosecond laser at a low repetition rate and therefore the dominating nonlinear response is attributed to the electronic origin as was also confirmed by degenerating multiwave mixing (DMWM) [11]. We also found from a dependence of on the concentration of CdSe QDs that the observed fifth-order nonlinearity had contributions mainly from the intrinsic fifth-order optical nonlinearity and partly from macroscopic and microscopic cascaded fifth-order optical nonlinearities [11, 27]. The occurrence of the intrinsic fifth-order optical nonlinearity is attributed to TPA-induced free carriers in bulk semiconductors [25]. In our case, however, it may originate from the state filling effect in CdSe QDs.

#### 4. Conclusion

We have developed a closed-aperture Z-scan theory for nonlinear optical materials with SA of a homogeneously broadened type and with simultaneous third- and fifth-order nonlinear refraction. The shape and symmetry of the closed-aperture Z-scan transmittance are strongly influenced by the relative sign and magnitude of the fifth-order optical nonlinearity with respect to the third-order one. Because the light-induced transparency takes place due to SA, the peak (the valley) increases (decreases) in the closed-aperture Z-scan transmittance. The theoretical model has been applied to characterize the nonlinear optical properties of our newly developed uniformly cured CdSe QD-polymer nanocomposites. We have shown that the films exhibit SA as well as the negative third-order and the positive fifth-order nonlinear refraction. Such coexistence of the third- and fifth-order optical nonlinearities has also been confirmed by our recent DMWM measurements [11]. Finally, it should be stressed that the Z-scan theory presented in this paper is proved to be useful for characterizing the nonlinear optical properties of novel QD-polymer nanocomposite materials capable of constructing nonlinear photonic crystal structures by holographic patterning. In addition, the observed large nonlinear refraction and induced transparency as well as the capability of holographic nanoparticle assembling make QD-polymer nanocomposite materials promising for nonlinear photonics applications (e.g., optical switching, limiting, and signal processing) by use of holographic Bragg grating structures that provide the electromagnetic nonlinear feedback mechanism.

#### Acknowledgment

This work was supported by the Ministry of Education, Culture, Sports, Science, and Technology of Japan under Grant no. 20360030.