1. Introduction

Porous nanomaterials, such as nanoporous silicon (Si) and nanoporous titanium dioxide (TiO₂), are shown to possess considerable potential for photonic applications [1,2]. Quantum size effects in such media lead to pronounced optical nonlinearities. Improvement in optical properties of porous silicon is observed on embedding conjugated polymers [3]. Various porous III-V compounds, fabricated through the electrochemical etching process, are shown to exhibit significant second- and third-order optical nonlinearities [4]. By creating porous nanostructures, thus, it is possible to alter or enhance the usual optical property exhibited by a bulk material. For instance, porous nanosized LiNbO₃ polycrystal has shown remarkable third-order nonlinear optical property as against the customary second-order nonlinearity of LiNbO₃ crystal [5]. Apart from the optical utility, porous materials like nanoporous metal organic frameworks (MOFs) are becoming ideal candidates for recognizing analytes in chemical sensing applications [6]. In recent years, dielectric nanoporous structures such as anodized aluminum oxide (AAO) have emerged as preferred templates to grow semiconductor quantum dots for optoelectronic applications [7], carbon nanotubes for electrochemical double layer capacitors (EDLCs) [8], and metal nanowires for nonlinear optical applications [9]. However, we note that optical nonlinearities of the porous AAO templates alone, unlike that in the case of porous semiconductors, have not yet systematically been investigated.

In this work, we show that a dielectric AAO nanoporous film can possess large optical nonlinearity which varies with pore number density, pore size and pore shape. The long-range-ordered, hexagonally close-packed nanoporous AAO thin films were fabricated using the well-known two-step anodization process. The number density of pores and the pore size could be controlled by the anodization voltage, while the film thickness was changed by varying the anodization time and considering the dependence of growth rate on the applied voltage and the electrolyte [7]. Their third-order optical nonlinearity and its dependence on film thicknesses, number of pores and pore sizes were investigated using the time-resolved differential optical Kerr gate (DOKG) technique at 800 nm by employing a Ti:sapphire laser. No thickness dependence for optical nonlinearity was observed, however, the nonlinearity was influenced by the number density of pores, the pore size and the shape. The origin of large and ultrafast optical nonlinearity is discussed by considering the porous AAO as a composite of AAO and air, and utilizing the extended Maxwell Garnet (MG) effective medium theory. The local electric field confinement due to varying pore size, pore shape and surface states has been considered in addition to the MG theory.

2. Experiment

2.1. Sample preparation

AAO samples were prepared by a typical two-step anodization process using a grain-free aluminum foil (99.99%, 100 μm in thickness, Tokai) [7]. The aluminum electrode was anodized by applying voltage, in oxalic acid or sulfuric acid solution. The hexagonally close-packed nanoporous AAO thin films were obtained, in which the pore size and the interpore distance were controlled by the anodization voltage, while the film thickness was varied by changing the anodization time. After growing the through-hole nanochannel AAO samples, the remaining substrate layers at the bottom of the samples were uniformly removed and etched out in a saturated HgCl₂ solution and an aqueous 5 wt % phosphoric acid.

AAO films with different thicknesses, number densities of pores and pore sizes were fabricated through the above process in order to analyze the mechanism of origin of third-order optical nonlinearity of the AAO nanostructures. Four AAO samples, APL1–4, were fabricated with different thicknesses but maintaining the same pore number density and pore size, to check their homogeneity. The applied voltage was set to 40 V to maintain the same pore number density (1.19×10¹⁰ cm⁻²) and the anodization was carried out for 10, 15, 20 and 30 min to obtain films with thickness of 615 (APL1), 900 (APL2), 1107 (APL3) and 1607 nm (APL4), respectively. Three AAO films with different pore sizes, 25 (APS1), 34 (APS2) and 62 nm (APS3), were prepared by varying the hole-widening time as 10, 30 and 50 min, respectively, while setting the applied voltage and the anodization time to 40 V and 10 min in order to maintain a fixed pore number density and a film thickness of 615 nm. Finally, three more AAO samples with different pore number densities, 1.19×10¹⁰ (APD1), 2.12×10¹⁰ (APD2) and 2.96×10¹⁰ cm⁻² (APD3), were prepared by varying the anodization voltage as 40, 30 and 25 V and fixing the anodization time to 10 min, and by maintaining the film thickness at 615 nm. The morphologies of these AAO samples, examined through a field emission scanning electron microscope (FE-SEM), are shown in Figs. 1 , 2 and 3 . Figure 1 reveals the cross sectional view of the 615-nm-thick AAO film (APL1). FE-SEM images of APD1, APD2 and APD3 are shown in Fig. 2, while those of APS1, APS2 and APS3 are given in Fig. 3. The thickness, the average pore number density and the average pore size (pore diameter, determined by approximating each hexagonal pore with a spherical cylinder) were estimated from the SEM images. The linear refractive indices and the linear absorption coefficients were calculated using the linear transmission, the linear absorption and the reflection curves recorded by a spectrophotometer (VARIAN, Cary-5000). The UV-Vis-NIR absorption spectra for AAO films with varying pore number are shown in Fig. 4 .

 

Fig. 1 SEM images of the 615 nm-thick AAO film: (a) top- (b) side- and (c) cross-view.

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Fig. 2 Morphologies of the AAO films with different number densities of pores: (a) 1.19×10¹⁰ (APD1), (b) 2.12×10¹⁰ (APD2) and (c) 2.96×10¹⁰ cm⁻² (APD3), and pore size in each film is (a) 34, (b) 31.2 and (c) 19 nm, respectively. The film thickness of all the three films is 615 nm.

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Fig. 3 Morphologies of the AAO films with different pore sizes: (a) 25 (APS1), (b) 34 (APS2) and (c) 62 nm (APS3) (diameter is estimated by approximating each hexagonal pore with a spherical cylinder). The film thickness of all the three films is 615 nm.

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Fig. 4 Linear absorption coefficients of AAO samples APD1–3 with varying pore number density.

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2.2. Differential optical Kerr gate

The time-resolved DOKG experiment [10] with a Ti:sapphire laser delivering 90 femtosecond (fs) pulses at a repetition rate of 92 MHz at 800 nm was employed to investigate the third-order optical nonlinearity of the AAO films. The laser beam was divided into pump and probe beams with 20:1 intensity ratio by a beam splitter. The polarization of the probe beam was set to 45° with respect to that of the pump beam by a half wave plate. The two beams were focused on the sample by a convex lens of focal length of 7 cm. The time delay of the probe with respect to the pump was controlled by a PC-driven linear translator. At zero time delay, the two beams overlap spatially and temporally, and the polarization of the probe beam rotates due to the birefringence induced in the sample by the pump beam. The pump beam, after passing through the sample, was blocked and the probe beam was passed through a quarter wave plate. The circularly polarized probe beam was then split into two beams by a polarizing cube beam splitter and the two beams were detected by a photodetector pair connected to a lock-in amplifier. CS₂ was used to check the reliability of the setup as well as to estimate the${\chi }^{(3)}$of the samples.

3. Results and discussion

Results of DOKG experiment performed at 800 nm are displayed in Figs. 5 , 6 and 7 . In all cases, the optical Kerr effect (OKE) signal is symmetric about the zero time delay indicating that the response time is either comparable to or faster than the width (90 fs) of the laser pulse. The reliability of our DOKG set up was tested by performing experiment on CS₂ and ensuring that the known form of OKE signal for CS₂ was reproduced; the nonlinear response of CS₂ was a bi-exponential decay curve with decay times of 0.14 and 1.58 ps that matched with the previous reports [11,12]. The ${\chi }^{(3)}$of samples was estimated using the following relation [10]

 

Fig. 5 OKE signals for AAO films APL1–4 with different thicknesses. The nonlinear response is not dependent on the film thickness, revealing the homogeneity of the samples (see Table 1).

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Fig. 6 OKE signals for AAO films APD1–3 with different pore number densities. The nonlinear response slightly decreased as the pore number density was increased.

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Fig. 7 OKE signals for AAO films APS1–3 with different pore sizes. The nonlinear response doubled when the pore size was increased from 25 nm (APS1) to 62 nm (APS3).

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Magnitudes of${\chi }^{(3)}$ estimated for AAO samples with different morphologies, along with the estimated structural details such as pore number density, pore size, film thickness and volume fill fraction of pores, are tabulated in Tables 1 , 2 and 3 . As shown in Table 1, ${\chi }^{(3)}$did not depend on the thickness of the films.

Table 1. Structural details and magnitudes of${\chi }^{(3)}$ for samples APL1–4 with different film thicknesses.

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Table 2. Structural details and magnitudes of${\chi }^{(3)}$ for AAO films with different pore number densities.

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Table 3. Structural details and magnitudes of${\chi }^{(3)}$ for AAO films with different pore sizes.

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Note that the magnitude of ${\chi }^{(3)}$for the bulk aluminum oxide (Al₂O₃), which is a dielectric, has been reported to be of the order of 10⁻¹³ esu [10]. However, in this case, we believe that creating nanopores in alumina film certainly has brought in remarkable enhancement to the ${\chi }^{(3)}$of the material. Enhanced surface to volume ratio in the porous AAO could be a reason for the enhancement in nonlinear response. In solids with certain periodic structures, such as photonic crystals, the existence of surface states in the interface between the solid and vacuum or another medium is observed [13,14]. The distinct spatial field profile of such localized surface states leads to the strong ${\chi }^{(3)}$ sensitivity, especially when the dielectric function of the photonic crystal includes Kerr nonlinearity [14]. Such surface states in the periodic porous AAO could be one considerable source of enhanced nonlinear polarization. Additionally, despite a thorough cleaning of nanoporous AAO films, there can be residual charged surface species, such as OH^–, H⁺ and O^2– on the surface and in pores that contribute to the field enhancement and nonlinear polarization of the sample. There are reports on the enhanced nonlinear optical properties of nanoporous semiconductors, where the enhancement came as an additional gain along with the notable intrinsic optical nonlinearity of the bulk semiconductors. But the nonlinear response of bulk dielectric AO is significantly lower than those of bulk semiconductors such as Si and ZnSe [10]. The enhancement in nonlinearity through nanopores in semiconductors, such as TiO₂ and Si, is interpreted in terms of the resonant excitation of oxygen vacancies and related surface defect states [2] and the quantum size effects [1].

Further, varying the pore number density and the pore diameter was also found to have profound influence on the optical nonlinearities of the AAO samples. In order to identify the origin of variation of the optical nonlinearity in the nanoporous AAO, we treat it as nanocomposite of air and AAO and assign it with a Maxwell Garnet-type topology. The Maxwell Garnet theory, which predicts the nonlinear optical property of a composite consisting of a metal and a dielectric, is extended by Sipe and Boyd [15] to the case of a dielectric inclusive in a dielectric host. Their intention was to test if the enhancement of optical nonlinearity was possible when both the inclusive and the host were dielectrics. They found that, since no surface plasmon resonance was possible, the entire enhancement through the local field correction factor (f) could come only through the volume fill fraction (p) of the inclusions and the ratio between the dielectric constants of the constituents. In addition to this, they found that the nonlinear response varies with the type of host and inclusion, i.e., whether the host and the inclusion are optically linear or nonlinear in nature. We utilize their important findings, based on certain conditions related to the dielectric constants of the individual constituents of the composite, and proceed to analyze the results obtained for AAO.

It was found for the case of an optically nonlinear host and a linear inclusion (which is the case with AAO), that the enhancement in nonlinear response depends on the relation between the dielectric constant of the inclusion (${\epsilon }^i$) and the dielectric constant of the host (${\epsilon }^h$). For the case ${\epsilon }^i>{\epsilon }^h,$a strong enhancement in nonlinear response is observed as p increases from 0 to 1. This effect arises because the presence of linear dielectric inclusion in the composite enhances the electric field distribution in the host. When${\epsilon }^i<{\epsilon }^h,$the enhancement in nonlinearity becomes weak and decreases to zero as p increases from 0 to 1. When${\epsilon }^i={\epsilon }^h,$the enhancement in nonlinear response decreases linearly to zero as p goes from 0 to 1 and this happens since the amount of nonlinear material (host) in the composite decreases as the fill fraction of the inclusion increases. Since the nanoporous AAO is regarded as an MG-type composite, we treat air as the inclusion, which is optically linear, and the AAO as the optically nonlinear host. As the refractive index of air is less than that of AAO (n _AAO ≈1.8-2.0), above discussion of the case where${\epsilon }^i<{\epsilon }^h$ should hold well for these AAO samples. Further, from Table 1, it is clear that the nonlinear response of APL1–4 is independent on the film thickness, i.e. the pore length. Therefore, apparently only the number density of pores and the pore diameter have played a key role in controlling the nonlinearity of the AAO films (Tables 2 and 3). The magnitude of ${\chi }^{(3)}$decreased in APD1–3 when the fill fraction of the inclusion increased from 0.08 to 0.16, and this effect is in good agreement with the MG theory; In APD3, note that, the pore size (19 nm) also has an additional influence on decreasing nonlinearity (Table 2). On the contrary, in samples with different pore sizes (APS1–3), with the pore number density remaining almost constant, the magnitude of ${\chi }^{(3)}$increased as the pore size increased even though the estimated p increased from 0.06 to 0.41. This is just in contradiction with the prediction by the extended MG theory. At this point, therefore, one has to consider the additional influence of pore size and shape on the nonlinear response of the AAO films. The contributions from pore size and pore shape to the local field confinement in pores are described below. We consider also the possible contribution from surface states and any residual charged states on the surface of pores and AAO film that result during the preparation of the films.

From the DOKG results (Fig. 7), it is clear that increasing pore size has brought improvements to the optical nonlinearity of AAO. In dielectrics, as it is well-known, only the bound electrons can take part in polarization of the material, in contrast to free electrons in metals and free charge carriers in semiconductors, when exposed to electromagnetic fields. Further, since the length of pores (film thickness) is seen to have least effect on the enhancement in nonlinear polarization, only the number of atoms around the pores might be having profound influence on the nonlinear response of the AAO film. Therefore, neglecting the surface states in the AAO pores, we can only think of the contribution of bound electrons in those atoms that are close to the surface of each pore. The field incident on the film is the applied electric field (E), while the field polarizing the atoms around the pore can be thought of as the local field (E _loc). In order to calculate this local field, we make use of the classical ‘hypothetical spherical cavity’ approach, and estimate the electric field due to the polarized material outside the pore [16].

Assuming the material to be homogeneous as well as linear and isotropic, one can calculate the total electric field at the location of an imagined molecule (air molecules here) in the cavity except for that arising from the molecule itself. We equate each pore in AAO to the hypothetical spherical cavity. This is equivalent to drawing a sphere of some appropriate radius R about the point (molecule) in question, and hence dividing space into two regions–outside the sphere (cavity), where the distribution of dipoles is continuous and can be described in terms of the uniform dipole moment per unit volume P, while inside the sphere the dipoles are individuals. Considering the contributions from bound surface charges around each pore and the dipole moments of any dipoles in the porous AAO, we find that the local electric field is ${\text{E}}_{loc}=\text{E}+{\text{E}}_o+{\text{E}}_I,$ where E is the macroscopic average field, E _o is from the dipoles outside the cavity and E _I due to any individual dipoles inside the pore. The surface charge density of bound surface charges on the sphere arising from the discontinuity of P there can be obtained through${\sigma }_b=\text{P}\cdot \text{n}$. Further it can be shown that the field produced by the dipoles around the sphere is${\text{E}}_o=\text{P}/3{\epsilon }_0,$ with ${\epsilon }_0$being the dielectric constant of free space. Hence, ${\text{E}}_{loc}=\text{E}+(\text{P}/3{\epsilon }_0)+{\text{E}}_I$. In most general cases, E _I can be different from zero, however, it would be zero if the positions of the dipoles were completely random as would be the case in a liquid or gas (in the present case it is air). Thus, we get${\text{E}}_{loc}=\text{E}+(\text{P}/3{\epsilon }_0)$. By following a simple analogy, we obtain an expression for P as$\text{P=}N{\alpha }_e\left(\text{E+}\raisebox{1ex}{$\text{P}$}\!\left/ \!\raisebox{-1ex}{$3{\epsilon }_0$}\right.\right),$ where N is the number density of atoms around the sphere (pore) and ${\alpha }_e$the electronic polarizability. The polarization P includes linear and nonlinear polarizations. Thus, increased pore size will essentially increase the number density of atoms around the pores and hence the local field. The contribution of bound electrons to the nonlinear polarization could be justified since the nonlinear response time is found to be ≤ 90 fs; the electronic nonlinear response time is of the order of 10⁻¹⁵ s. Note, however, that the origin of polarization includes also contributions from the surface states and the residual charged species on the pore surface, which are speculated to be the major reasons for the enhanced nonlinear response in the porous AAO, and therefore the nonlinear response deviates from what is expected by the modified MG theory.

One more feature that is evident from the SEM images of APS1–3, as pore widens, is the well-defined hexagonal pore shape. It is generally known that electric fields near sharp edges are stronger than those near spherical surfaces. Therefore, corner edges in the hexagonal shape might also be playing a considerable role in enhancing the local field across the pore. In a similar experimental report on certain nanoporous semiconducting materials, it was suggested that the non-cylindrical pores could result in larger third-order nonlinearity compared with the cylindrical pores for the same reason as stated above [4]. Thus, in the case of APS1–3, shape can also be considered in addition to the pore size effect.

In the present work we treat the thickness of the film as the pore length in order to understand the enhanced nonlinear response of the porous AAO films and to find negligible pore-length dependence for nonlinearity. However, Apel et al. [17] have previously shown that the nonlinear absorption in bulk Al₂O₃ films, prepared by identical coating procedures, near 193 nm varies with thickness of the film. A nonlinear correlation was observed between two-photon absorptance and Al₂O₃ film thickness and it was explained based on a simple model using excited states. On the contrary, no significant thickness-dependent optical nonlinearity was observed for nanoporous AAO films at an off-resonant spectral region. The porous AAO has a honeycomb structure with well-ordered domains of a few nanometers to microns in size [18]. Therefore, comparison of nonlinear response of porous AAO with that of Al₂O₃ films would not be appropriate.

We make a note that our study has not set the upper bound for maximum pore diameter that can enhance the nonlinear response of the porous AAO and at or above which the enhancement in nonlinearity saturates. The analysis also has a limitation from increasing the pore size further, by maintaining a constant pore number density in a given area in the AAO film. Nevertheless, the study shows that the nonlinearity of porous AAO can also play a notable role while investigating the nonlinearity of nanocomposites comprising AAO and other materials like semiconductors and metals, depending on the pore number density and the pore size. However, when filling other materials with larger dielectric constants into these pores, one has to consider the appropriate case in the MG theory to understand the resulting nonlinear optical response. Shapes of pores in these AAO films will also have an additional influence on the nonlinear response in such composites.

4. Conclusions

A systematic investigation of the third-order optical nonlinearity of the nanoporous anodized aluminum oxide films has been presented. The nonlinearity is considerably influenced by the number of pores, the pore size and the pore shape. A very large value of ${\chi }^{(3)}$ of the order of 10⁻¹⁰ esu has been determined for the nanoporous AAO. Maxwell Garnett theory is utilized to evaluate the results and it is shown that the MG theory is not predicting the enhanced nonlinear optical response when the pore size is increased in AAO. The local electric field enhancement due to pore size, pore shape and surface state effects has been considered to explain this enhancement in nonlinear response.