Introduction

The nonlinear optical properties of semiconductors, used for a variety of applications, are the first studied by Braunstein R et al. [1] and extensively investigated [2]. Some of the largest nonlinearities have ever been reported in semiconductors [3] and involve near-gap excitation. However, these resonant nonlinearities, by their nature, involve significant linear absorption, which is undesirable in many applications. The nonlinear optical behavior in the transparency region of solids due to the anharmonic response of bound valence electrons has been studied extensively in the past [4]. Nonlinear refraction associated with this process is known as the bound-electronic Kerr effect. It is described by a change of refractive index

△n = n2I, where I is the light irradiance (W/cm²) and n2(cm²/W) is the optical Kerr coefficient of the solid. This type of nonlinearity results from virtual intermediate transitions [5] as opposed to real intermediate transitions that occur in resonant (electron-hole plasma)

nonlinearities.

In the language of quantum mechanics, a virtual carrier lifetime can be defined from the uncertainty principle as 1/ω-ωg. Here, ω is the optical frequency, and ωg = Eg/h, where Eg

is the band-gap energy of the solid, and h is Planck’s constant. This equality means that in the transparency region where ω << ωg, the response time is very fast (<< 10^-14 s) and can be regarded as essentially instantaneous. This ultrafast response time has been exploited in applications such as soliton propagation in glass fibers [6] and in the generation of femtosecond pulses in solid-state lasers (Kerr lens mode locking) [7]. Another significant application is the development of ultrafast all-optical-switching (AOS) devices [8].

Although much progress has been made in this area, development of a practical switch has been hindered by the relatively small magnitude of bound-electronic nonlinearities.

1-1 Background

1-1-1 Ultrafast all-optical switching

One of the applications of ultrafast nonlinear refraction is in the role of all-optical switching. In order to obtain switching, the required nonlinear phase shift must be achieved before losses reduce the irradiance. For the transparent spectral region beneath the one-photon band gap in semiconductor and for the high irradiances required for ultrafast nonlinear refraction, the dominant loss mechanism is usually two-photon absorption. In order to obtain all-optical switching, the figure of merit (FOM) n2/βλ (where β is the two-photon absorption coefficient) must exceed some critical value; this value depends on the device geometry but is always of the order of unity [9]. The two horizontal lines in Fig. 1-1 represent the minimum acceptable FOM for nonlinear directional couplers (NLDC) and Fabry-Perot (FP) interferometers. Although it demands a larger FOM, the NLDC scheme is the preferred practical geometry. From Fig. 1-1 we see that the FOM requirement is satisfied either just below the 2PA edge or very near resonance ( hω ˜ Eg ). Since n2 ∝ Eg -4

, a low switching threshold at a given wavelength demands a material with the smallest possible band-gap energy. The theory then suggests that the ideal operating region is just below the band gap. predicted from the two-parabolic-band model. NLDC stands for nonlinear directional coupler, and FP stands for Fabry-Perot etlon.

1-1-2 Optical limiting

A passive optical limiting uses a material’s nonlinear response to block the transmittance of high-irradiance light while allowing low-irradiance light to be transmitted (an operation similar to that of photochromic sunglasses). The primary application of optical limiting is to protect sensitive optical components from being damaged by the high-intensity input light.

The ideal optical limiter has a high linear transmission for low inputs (e.g., energy), a variable limiting input energy, and a large dynamic range defined as the ratio of the linear transmittance to the minimum transmittance obtained for high input (prior to irreversible damage). Since a primary application of optical limiting is to protect sensors, and fluence (energy per unit area) almost always determines damage to detectors; this is the quantity of interest for the output of a limiter.

1-1-3 Effective

by cascaded second-order effect

There has recently been a growing interest in a phenomenon known as cascading of second-order nonlinearities to obtain large, intensity-dependent phase distortions in optical beams. This phenomenon can be briefly described as follows: when a beam propagates in a medium with a χ⁽²⁾ nonlinearity out of phase matching, part of the generated second harmonic is converted back to the fundamental frequency, in phase quadrature with respect to the original wave. There is therefore a phase shift on the fundamental frequency beam proportional to its intensity, as in a medium with a χ⁽³⁾ nonlinearity (Kerr medium).

Depending on the phase mismatch, the equivalent χ⁽³⁾ can be positive or negative, corresponding to a self-focusing or a self-defocusing medium. Although the effect has been known for a very long time [10], its potentiality of generating large third-order nonlinearities of controllable sign has been exploited just recently [11]. The applications of cascaded second-order nonlinearities have all-optical switching in waveguides [12], the generation of spatial solitons in quadratic media [13] and mode-locking of lasers [14].

1-2 Review of χ⁽³⁾ measurement

Several experimental techniques are available for measuring the bound-electronic nonlinear response of semiconductors, ie., β and n₂. These measurements include: degenerate four-wave mixing (DFWM) and Z-scan [15] along with its derivatives [16][17].

1-2-1 Four-wave mixing

Four-wave mixing, where three beams are input to a material and a fourth wave (beam) is generated, can be used for determining the magnitude of material’s nonlinear response and its response time. If the response is known to be third-order and ultrafast, can be determined along with some of its symmetry properties by varying the relative polarizations of the input beams (as well as by monitoring the polarization of the fourth wave). In addition, the frequencies of the input beams can be changed independently to determine the frequency dependence of the nonlinear response, but this can result in the need for a complex geometry to satisfy phase-matching requirements. Equal frequencies are often used, resulting in a much simpler geometry for phase matching, and this is referred to as degenerate four-wave mixing (DFWM). Fig. 1-2 shows one simple geometry for DFWM where two of the input beams (the forward and backward pumps) are oppositely directed. If these beams are nearly plane waves (i.e., well collimated), this geometry ensures phase matching for any third input beam (the signal). Introducing delay arms into each of the beams also allows the temporal dynamics of the nonlinearities to be measured for short optical input pulses. A particularly useful measurement is to monitor the energy of the fourth beam (so-called phase-conjugate beam) as a function of the time delay of the perpendicularly polarized backward pump (signal and forward pump have the same linear polarization) [18].

One of the difficulties in the interpretation of DFWM data for third-order nonlinearities is

that the signal is proportional to χ⁽³⁾² = Re{χ⁽³⁾}²+ Im{χ⁽³⁾}², and so 2PA and n₂ both contribute. Separating the effects is difficult without performing additional experiments.

Higher-order nonlinearities also can contribute, making separation of absorptive and refractive effects difficult.

Fig. 1-2 DFWM geometry to allow temporal dynamics measurements. Detector D2 monitors the conjugate beam energy.

1-2-2 Z-scan

Z-scan was developed for measuring nonlinear refraction (NLR) and determining its sign by Sheik-Bahae et al. [19]. It was soon realized that it also was useful for measuring nonlinear absorption (NLA) and separating the effects of NLR from NLA. A single beam Z scan setup is depicted in Fig. 1-3. The transmittance of a sample in the far field is measured through an aperture (Z-scan) or around an obscuration disk (EZ-scan). The transmittance is determined as a function of the sample position Z measured with respect to the focal plane.

Using a Gaussian spatial profile beam simplifies the analysis. The following example qualitatively describes how such data (Z-scan or EZ-scan) are related to the NLR of the sample.

Fig. 1-3 Z-scan geometry with reference detector to minimize background and maximize the signal-to-noise ratio.

Assume, for example, a material with a positive nonlinear refractive index. Starting the Z-scan (i.e., aperture) from a distance far away from the focus (negative Z), the beam irradiance is low, and negligible NLR occurs; hence the transmittance remains relatively constant. The transmittance here is normalized to unity, as shown in Fig. 1-4. As the sample is brought closer to focus, the beam irradiance increases, leading to self-focusing in the sample. This positive NLR moves the focal point closer to the lens, leading to a larger divergence in the far field. Thus the aperture transmittance is reduced. Moving the sample to behind the focus, the self-focusing helps to collimate the beam, increasing the transmittance of the aperture. Scanning the sample further toward the detector returns the normalized transmittance to unity. Thus the valley followed by peak signal is indicative of positive NLR, whereas a peak followed by a valley shows self-defocusing. Figure 1-4 shows the expected result for both negative and positive self-lensing. The EZ-scan reverses the peak and valley because, in the far field, the largest fractional changes in irradiance occur in the wings of a Gaussian beam.

-5.0 -2.5 0.0 2.5 5.0 0.97

0.98 0.99 1.00 1.01 1.02 1.03

Normalized T

Z/Z0

Fig. 1-4 Predicted Z-scan signal for positive (solid line) and negative (dashed line) nonlinear phase shifts.

In the preceding picture we assumed a purely refractive nonlinearity with no absorptive nonlinearities such as 2PA that will suppress the peak and enhance the valley. If NLA and NLR are present simultaneously, a numerical fit to the data can in principle extract both the nonlinear refractive and absorptive coefficients. However, a second Z-scan with the aperture removed and care taken to collect all the transmitted light can determine the NLA independently. For 2PA alone and a Gaussian input beam, the loss nearly follows the symmetric Lorentzian shape as a function of the sample position Z. The magnitude of the loss determines the NLA. This so-called open aperture Z-scan is only sensitive to NLA. A further division of the apertured Z-scan data (referred to as closed-aperture Z-scan) by the open-aperture Z-scan data gives a curve that for small nonlinearities is purely refractive in nature. In this way we have separate measurements of the absorptive and refractive nonlinearities without the need for computer fits of the Z-scans.

1-2-5 Excite-probe Z-scan

Excite-probe techniques in nonlinear optics have been employed to deduce information

that is not accessible with a single-beam geometry. By using two collinear beams in a Z-scan geometry, we can measure nondegenerate nonlinearities, we can temporally resolve these nonlinearities, and we can separate the absorptive and refractive contributions. There have been several investigations that have used Z-scan in an excite-probe scheme. Z-scan can be modified to give nondegenerate nonlinearities by focusing two collinear beams of different frequencies into the material and monitoring only one of the frequencies (different polarizations can be used for degenerate frequencies). The general geometry is shown in Fig.

1-5. After propagation through the sample, the probe beam is then separated and analyzed through the far-field aperture. Due to collinear propagation of the excitation and probe beams, we are able to separate them only if they differ in wavelength or polarization. The former scheme, known as a two-color Z-scan, has been used to measure the nondegenerate n2

and β in semiconductors.

Fig. 1-5 Optical geometry for a two-color Z-scan.

The most significant application of excite-probe techniques in the past concerned the ultrafast dynamics of nonlinear optical phenomena. The two-color Z-scan can separately monitor the temporal dynamics of NLR and NLA by introducing a temporal delay in the path of one of the input beams. These time-resolved studies can be performed in two fashions.

In one scheme, Z-scans are performed at various fixed delays between excitation and probe pulses. In the second scheme, the sample position is fixed (e.g., at the peak or the valley

position), while the transmittance of the probe is measured as the delay between the two pulses is varied. The analysis of two-color Z-scan is naturally more involved than that of a single-beam Z-scan. The measured signal, in addition to being dependent on the parameters discussed for the single-beam geometry, also will depend on parameters such as the excite-probe beam waist ratio, pulse-width ratio, and the possible focal separation due to chromatic aberration of the lens.

1-3 Merits of ZnO

ZnO is a kind of metal-oxide material with its melting point of around 2250 ⁰K and self-activated crystal of hexagonal wurtzite structure with lattice constant of a=0.3249 nm and c=0.5207 nm in the space group C⁴_6v . The deposited thin film usually belongs to c-axis-oriented textures and was drawn much attentions because of its ultraviolet emission.

The notable properties of ZnO are due to its wide band gap at room temperature and a high exciton binding energy (~60 meV) that is much higher than that of ZnSe and GaN.

Furthermore, the high exciton binding energy permits excitonic recombination even at room temperature. Due to these properties, ZnO can be used as UV or blue emitting materials [20].

In addition to optical transparency throughout the visible region of the spectrum and the observed large piezo-optic and piezoelectric effects in films with c-axis oriented, trivalent cation-doped ZnO exhibits marked electrical conductivity. The combination of these characteristics makes ZnO a system of choice for thin film photo-electronic device applications [21]. As a candidate for ultraviolet optoelectronic device applications, the nonlinear properties of ZnO are attractive.

1-4 Aim of this research

In this thesis we concentrate on the nonlinear response in the transparency range of semiconductors, i.e., for photon energies far enough below the band-gap energy E_g that bound-electronic nonlinearities either dominate the nonlinear response or are responsible for initiating free-carrier nonlinearities (e.g., two-photon absorption-created carrier nonlinearities).

The bound-electronic nonlinearities of two-photon absorption (2PA) and the optical Kerr effect are the primary nonlinearities of interest. In ZnO thin film, a systematic study of second-harmonic generation as a function of the film thickness was reported [22].

Surprisingly, for very think films χ⁽²⁾ as large as 18 pm/V was obtained this was larger than the value 14 pm/V for bulk single-crystal ZnO. And in polycrystalline ZnO with c-axis orientation a larger 2PA coefficient β (8.6 cm/GW) than that (4.2 cm/GW) of bulk single-crystal ZnO was measured with wavelength of incident laser pulse at 532 nm [23].

Besides, ZnO has large exciton bind energy and this electron-hole Coulomb interaction (effect of excitons) could lead to an enhancement of the 2PA coefficient near the two-photon resonance. Thus, a highly-qualify c-axis orientation ZnO thin film is interested to study its nonlinearities near two photon resonance with enhancement of excitons. A large two photon absorption coefficient and nonlinear refraction is observed near two photon resonance in our highly-qualify c-axis orientation ZnO thin film. A detail investigation of this ZnO thin film is worth us to study to put it in use for the field of nonlinear optical application.

In Chapter 2 we briefly describe the simple two-parabolic-band second-order perturbation theory which is used to predict the dispersion of bound-electronic nonlinearity and the two-photon absorption coefficient. In the later of Chapter 2, we describe the determination of two photon absorption coefficient and bound-electronic nonlinearity from open-aperture Z-scan and closed-aperture one respectively. Experiments used to measure this ZnO thin film are given in Chapter 3. In Chapter 4 our measured values of this ZnO thin film are compared with various theoretical models. We also compare our measured ß and n2 values with theoretical values. At last conclusions and perspective are both given in Chapter 5.