**2. Z-scan technique**

There are several different methods for determining the nonlinear optical response of material. The most commonly used technology is Z-scan invented by Sheik-Bahae et al. [11, 12]. Z-scan is also one of the simplest experimental methods to be employed. Because Z-scan signal can provide information not only on the magnitude of optical nonlinearity but also on its sign, the use of the Z-scan technique for nonlinear optical measurement is increasing.

The typical Z-scan setup is shown in **Figure 1**, where the lens L is used to focus the laser beam, and the smallest section of the beam crosses at the 0 point of the Z axis. O stands for the test sample placed near the 0 of the Z axis. BS is a splitter, which splits the laser into two beams.

**Figure 1.** Setup of the Z-scan technique.

and large nonlinear properties have been used in various photonic and optoelectronic applications such as optical communication, optical information processing, optical data storage,

Among various nonlinear effects, nonlinear absorption and nonlinear refraction attract more attention. Ultrafast nonlinear absorption properties are of importance since the nonlinearities considerably change the propagation of intense light through the medium, which can induce novel applications in optoelectronics, optical switchers, and limiters, as well as in optical computing, optical memories, and nonlinear spectroscopy. In fact, there is also much intrinsic interest in nonlinear refractive phenomena, particularly self-focusing and self-defocusing.

The search for new materials is one of the defining characteristics of modern science and technology. Of course, in nonlinear optics field, it is the case. Seeking and investigating new nonlinear optical materials with large nonlinear optical properties and fast nonlinear response is still one of the important works concerning nonlinear optics studies. Generally, the nonlinear optical materials should exhibit high transmission at normal light, so as not to degrade normal vision while exhibiting low or high transmission at intense light to serve as optical limiting materials for protecting human eyes and sensors or as saturable absorber for mode locking. In addition, nonlinear optical materials must exhibit fast response over a broad wavelength

The rapid development of nanoscience and nanotechnology has provided a number of new opportunities for nonlinear optics. A growing number of nanomaterials have been shown to possess remarkable nonlinear optical properties; this promotes the design and fabrication of nanoscale optoelectronic and photonic devices. Specially, metal nanoparticles (MNPs) have attracted considerable attention as potential nonlinear optical materials. Among them, gold and silver nanoparticles (NPs) have been paid more attention because they both exhibit a broad surface plasmon resonance (SPR) absorption band in the visible region of the electro-

In the following sections, the technologies used to measure the amplitude of nonlinear absorption and nonlinear refraction of nonlinear optical materials will be introduced. And some investigations concerning nonlinear optical properties of metal nanoparticles will be discussed.

There are several different methods for determining the nonlinear optical response of material. The most commonly used technology is Z-scan invented by Sheik-Bahae et al. [11, 12]. Z-scan is also one of the simplest experimental methods to be employed. Because Z-scan signal can provide information not only on the magnitude of optical nonlinearity but also on its sign, the use of the Z-scan technique for nonlinear optical measurement is increasing. The typical Z-scan setup is shown in **Figure 1**, where the lens L is used to focus the laser beam, and the smallest section of the beam crosses at the 0 point of the Z axis. O stands for the test sample placed near the 0 of the Z axis. BS is a splitter, which splits the laser into two beams.

pulsed laser deposition, and optical limiters.

44 Laser Technology and its Applications

range and a high damage threshold.

magnetic spectrum [1–10].

**2. Z-scan technique**

S is a small aperture used for the measurement of nonlinear refraction. D1 and D2 are photodetectors. During the test, the object to be tested moves along the Z axis, and the relationship between the light intensity and Z value is recorded.

When there is no aperture before photodetector D1 , Z-scan measurement is called open aperture Z-scan, which can provide the information about the nonlinear absorption. When there is an aperture before photodetector D2 , Z-scan measurement is called closed aperture Z-scan, which can provide the information about the nonlinear refraction of materials.

#### **2.1. Open aperture Z-scan technique**

When studying the materials' nonlinear absorption such as saturable absorption (SA) and reverse saturable absorption (RAS), we need to use open aperture Z-scan technique. Normalized open aperture Z-scan data is insensitive to beam distortion and is only a function of nonlinear absorption.

In the case of SA, the nonlinear absorption coefficient may be written as:

$$\alpha(\mathbf{I}) = \frac{a\_0}{1 + \left(\mathbf{I}/I\_\circ\right)}\tag{1}$$

where *α*<sup>0</sup> is the linear absorption coefficient, *I* is the excitation intensity, and *I S* is the saturation intensity. It is assumed that two-photon absorption (TPA) does not take place simultaneously with SA. The transmitted intensity is obtained from the equation:

$$\frac{d\mathbf{l}}{dz} = -a\mathbf{I} \tag{2}$$

Here, *z* corresponds to the sample thickness. As shown in **Figure 2**, saturation intensity *I S* can be obtained by fitting the experimental curve according to Eqs. (1) and (2). When materials show only RSA or TPA, according to open aperture Z-scan theory, the normalized transmission can be expressed as [12]:

$$T(\mathbf{z}) \ = \sum\_{m=0}^{n} \frac{[-q\_o(\mathbf{z})]^m}{(m+1)^{\vee}} \approx 1 - \frac{\beta \, \, l\_o \, L\_{\, eff}}{2 \sqrt{2} (1 + z^2/z\_o^2)} \tag{3}$$

**Figure 2.** Theoretical and experimental result of the open aperture Z-scan for SA.

where *β* is the nonlinear absorption coefficient, *I* 0 is the on-axis peak intensity at the focus, *Leff* = (1 − *e*<sup>−</sup>*α*<sup>0</sup> *lL*)/*α*<sup>0</sup> , *Leff* is the effective interaction length, *α*<sup>0</sup> is the linear absorption coefficient, *z* is the longitudinal displacement of the sample from the focus (z = 0), *L* is the sample length, and *z*<sup>0</sup> is Rayleigh diffraction length. From Eq. (3), nonlinear absorption coefficient *β* can be obtained to be:

$$\beta = 2\sqrt{2}\left[1 - T(z=0)\right]/I\_oL\_{\eta\ell} \tag{4}$$

where *α*<sup>0</sup>

*α*0

*T* = 1 −

be obtained by fitting experimental data.

**2.2. Closed aperture Z-scan technique**

*T*(z) = 1 +

the focus (z = 0), *L* is the sample length, and *z*<sup>0</sup>

*<sup>n</sup>*<sup>2</sup> <sup>=</sup> *<sup>α</sup>*<sup>0</sup>

can be obtained.

**Figure 4.** Normalized transmission as a function of position for open aperture Z-scan.

where *Δϕ*<sup>0</sup> = *kn*<sup>0</sup> *Leff*, *k* = 2*π*/*λ* is wave vector, *Δn*<sup>0</sup> = *n*<sup>2</sup> *I*

the on-axis peak intensity at the focus, *Leff* = (1 − *e* <sup>−</sup>*α*<sup>0</sup>

data can be expressed as [11]:

values can be obtained to be:

tion index of samples *n*<sup>2</sup>

is the linear absorption coefficient, *I* is the laser intensity, *I*

(

So normalized transmission can be obtained to be *TN* <sup>=</sup> *<sup>T</sup>*/*T*<sup>0</sup>

and *β* is the nonlinear absorption coefficient. The transmission of material can be deduced to be:

*<sup>α</sup>* \_\_\_\_\_\_\_\_\_ <sup>0</sup> <sup>1</sup> <sup>+</sup> *<sup>I</sup>* \_\_\_\_\_\_\_\_ <sup>0</sup> (1 + *z* <sup>2</sup> /*z*<sup>0</sup> 2 ) *I s*

When the nonlinear refraction of materials needs to be obtained, closed aperture Z-scan experiments should be conducted. The normalized transmission of closed aperture Z-scan

4V *<sup>ϕ</sup>*<sup>0</sup> *<sup>x</sup>* \_\_\_\_\_\_\_\_\_

0 , *n*<sup>2</sup>

is Rayleigh diffraction length.

*<sup>L</sup>*)/*α*<sup>0</sup> , *Leff*

is the linear absorption coefficient, *z* is the longitudinal displacement of the sample from

The difference between normalized peak and valley in closed aperture Z-scan curve is *Δ TP*−*<sup>V</sup>* = 0.406 (1 − *S*)0.25|*Δϕ*0|, and *S* is aperture transmittance. The nonlinear refractive index

As shown in **Figure 5**, by fitting the experimental data using Eqs. (7) and (8), nonlinear refrac-

*ΔTp*−*<sup>v</sup>* \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 0.406 *kI*<sup>0</sup> (1 <sup>−</sup> *<sup>S</sup>*)0.25(1 <sup>−</sup> *<sup>e</sup>* <sup>−</sup>*a*<sup>0</sup>

<sup>+</sup> *<sup>β</sup> <sup>I</sup>* \_\_\_\_\_\_ <sup>0</sup> 1 + *z* <sup>2</sup> /*z*<sup>0</sup> 2 ) *S*

Nonlinear Optical Response of Noble Metal Nanoparticles

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. As shown in **Figure 4**, *I*

(*x*<sup>2</sup> <sup>+</sup> 1)(*x*<sup>2</sup> <sup>+</sup> 9) (7)

is nonlinear refractive index, *I*

is the effective interaction length,

*<sup>L</sup>*) (8)

is the saturation intensity,

*S*

and *β* can

47

0 is

*L* (6)

Selectively, as shown in **Figure 3**, *β* can also be obtained by fitting the experimental curve.

But when materials show transformation from SA to RSA, a nonlinear absorption coefficient including SA coefficient and TPA coefficient should be defined as [13]:

 *<sup>α</sup>*(*I*) <sup>=</sup> *<sup>α</sup>* \_\_\_\_\_\_ <sup>0</sup> 1 + (*I*/*I s* ) <sup>+</sup> *<sup>I</sup>* (5)

**Figure 3.** Theoretical and experimental result of the open aperture Z-scan for RSA.

where *α*<sup>0</sup> is the linear absorption coefficient, *I* is the laser intensity, *I S* is the saturation intensity, and *β* is the nonlinear absorption coefficient. The transmission of material can be deduced to be:

$$T = 1 - \left(\frac{a\_0}{1 + \frac{l\_0}{\left(1 + z^2/z\_0^2\right)I\_\star}} + \frac{\beta \, l\_0}{1 + z^2/z\_0^2}\right)L\tag{6}$$

So normalized transmission can be obtained to be *TN* <sup>=</sup> *<sup>T</sup>*/*T*<sup>0</sup> . As shown in **Figure 4**, *I S* and *β* can be obtained by fitting experimental data.

#### **2.2. Closed aperture Z-scan technique**

When the nonlinear refraction of materials needs to be obtained, closed aperture Z-scan experiments should be conducted. The normalized transmission of closed aperture Z-scan data can be expressed as [11]:

$$T(\mathbf{z}) = 1 + \frac{4\mathbf{V}\,\phi\_0 \mathbf{x}}{(\mathbf{x}^2 + 1)(\mathbf{x}^2 + 9)}\tag{7}$$

where *Δϕ*<sup>0</sup> = *kn*<sup>0</sup> *Leff*, *k* = 2*π*/*λ* is wave vector, *Δn*<sup>0</sup> = *n*<sup>2</sup> *I* 0 , *n*<sup>2</sup> is nonlinear refractive index, *I* 0 is the on-axis peak intensity at the focus, *Leff* = (1 − *e* <sup>−</sup>*α*<sup>0</sup> *<sup>L</sup>*)/*α*<sup>0</sup> , *Leff* is the effective interaction length, *α*0 is the linear absorption coefficient, *z* is the longitudinal displacement of the sample from the focus (z = 0), *L* is the sample length, and *z*<sup>0</sup> is Rayleigh diffraction length.

The difference between normalized peak and valley in closed aperture Z-scan curve is *Δ TP*−*<sup>V</sup>* = 0.406 (1 − *S*)0.25|*Δϕ*0|, and *S* is aperture transmittance. The nonlinear refractive index values can be obtained to be:

 $n\_p
$${}^{1}$$
 --  $n\_p$ {}^{1}$  --  $n\_p
$${}^{1}$$
 --  $n\_p$ {}^{1}$  --  $n$ - $n$ - $n$ - $n$ - $n$ - $n$  values can be obtained to be: 
$$a\_2 = \frac{a\_o \Lambda T\_{pv}}{0.406 \, k l\_0 (1 - S)^{0.2} (1 - e^{-uL})}\tag{8}$$

As shown in **Figure 5**, by fitting the experimental data using Eqs. (7) and (8), nonlinear refraction index of samples *n*<sup>2</sup> can be obtained.

**Figure 4.** Normalized transmission as a function of position for open aperture Z-scan.

**Figure 3.** Theoretical and experimental result of the open aperture Z-scan for RSA.

where *β* is the nonlinear absorption coefficient, *I*

*β* = 2 √

*<sup>α</sup>*(*I*) <sup>=</sup> *<sup>α</sup>* \_\_\_\_\_\_ <sup>0</sup>

, *Leff* is the effective interaction length, *α*<sup>0</sup>

**Figure 2.** Theoretical and experimental result of the open aperture Z-scan for SA.

(1 − *e*<sup>−</sup>*α*<sup>0</sup>

be:

*lL*)/*α*<sup>0</sup>

46 Laser Technology and its Applications

0

longitudinal displacement of the sample from the focus (z = 0), *L* is the sample length, and *z*<sup>0</sup>

Selectively, as shown in **Figure 3**, *β* can also be obtained by fitting the experimental curve.

But when materials show transformation from SA to RSA, a nonlinear absorption coefficient

1 + (*I*/*I s*

\_\_

including SA coefficient and TPA coefficient should be defined as [13]:

Rayleigh diffraction length. From Eq. (3), nonlinear absorption coefficient *β* can be obtained to

2[1 − *T*(*z* = 0)]/*I*

is the on-axis peak intensity at the focus, *Leff* =

is the linear absorption coefficient, *z* is the

<sup>0</sup> *Leff* (4)

) <sup>+</sup> *<sup>I</sup>* (5)

is

**3.1. Nonlinear optical studies of Au nanoparticles**

nonlinear optical properties in Au nanoparticles [43].

positive nonlinear refraction.

The first experimental results on the nonlinear optical effects of Au nanoparticle were obtained by Ricard et al. in 1985 [32]. They prepared the Au nanoparticle with an average diameter of 10 nm and measured the third-order nonlinear susceptibility using phase conjugation to be 1.5 × 10<sup>−</sup><sup>9</sup> esu at 530 nm. They traced the enhancement to the nonlinearities of the electrons in Au particles.

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49

Among all composite materials, those made out of gold NPs embedded in a dielectric matrix are more important, because of their strong SPR absorption band in the visible region [33]. The coexistence of unique linear and nonlinear (especially third-order) optical properties makes the material be well suited for the potential applications ranging from optical limiter [20, 34], quantum information processing [35, 36], cancer treatment [37–39], on to all-optical switching [33, 40, 41]. In this direction, Au NPs embedded in dielectric media have been widely put more attention for their SPR, which depends strongly on the NPs environment and geometry [42].

Many investigations were performed in Au nanoparticles to study the nonlinear refractive index and nonlinear absorption [2, 16–22]. Moreover, the optical limiting of Au nanoparticles has also been studied widely for protection of human eyes and optical devices from laser damage. The contents studied mainly include the effects of sizes, matrices, and shapes on the

Sánchezdena O et al. studied the size dependence of nonlinear optical response in Au metallic nanoparticles with diameters of 5.1, 13.4, and 14.2 nm synthesized and embedded in sapphire by using ion implantation [43]. Under 532-nm, 26-ps pulses, they found that the Au NPs exhibited a negative nonlinear absorption, which increases with size and size-independent

For larger Au nanoparticles than those above, a systematic study of the size-related nonlinear optical properties of triangular Au particles was performed by S.H. Yoon et al. who fabricated the triangular Au nanoparticle arrays with four larger sizes of 37, 70, 140, and 190 nm on SiO2 substrates using nanosphere lithography [44]. **Figure 6** shows the absorption spectra of the Au nanoparticles of different sizes. It can be seen that the SPR absorption peaks lie at 552, 566, 580, and 606 nm for the 37, 70, 140, and 190 nm Au nanoparticles, respectively. With the

**Figure 7** shows the typical OA Z-scan experiment results of the four samples. The curve of the 37-nm sample showed a TPA with an additional SA component. For the samples with size of 70 and 140 nm, TPA component turned weaker and SA became dominant. The curve of the 190-nm sample showed only the SA component. These differences occur because the absorption in the excitation region is much weaker than that at 400 nm for the Au nanoparticles sized 37 nm, and herein, the interband transition to the TPA process plays a key role. However, the absorption at 800 nm is larger than that at 400 nm for the 190-nm Au nanoparticles; this is because the SA process becomes dominant. The curves of the samples of 70 and 140 nm

showed a transition in this variation of the two nonlinear mechanism contributions.

**Figure 8** shows the CA Z-scan data for four Au nanoparticles of 37, 70, 140, and 190 nm. For the 37 and 70 nm Au nanoparticles, a self-defocusing occurs, and the nonlinear refraction index

increasing particle size, the absorption peak shifts to longer wavelength.

**Figure 5.** Normalized transmission curves of Z-scan data with an aperture divided by those without an aperture.
