3. Third-harmonic generation by coherently controlled two-color excitation of inert gases: plasma effect

With two-color excitation, the third-harmonic signal is contributed by the direct THG (ω<sup>3</sup> = ω<sup>1</sup> + ω<sup>1</sup> + ω1) and four-wave mixing (FWM, ω<sup>3</sup> = ω<sup>2</sup> + ω<sup>2</sup> – ω1) processes and a cross term of the two. As the relative phase between ω<sup>1</sup> and ω<sup>2</sup> varies, a sinusoidal modulation in output intensity at frequency ω<sup>3</sup> is expected and was demonstrated in our previous work [2]. In intense laser field, plasma can be generated through the ionization of gases. Optical harmonic generation in plasmas has been studied for a long time. Recently, significant enhancement of the third-harmonic emission in plasma has been reported by Suntsov et al. [3]. More than twoorder-of-magnitude increase of the efficiency of third-harmonic generation occurs due to the plasma-enhanced third-order susceptibility [5]. More specifically, the presence of charged species (free electrons and ions) can effectively increase the third-order nonlinear optical susceptibility [4, 5]. This indicates that the susceptibility can be expressed as a function of the plasma density Ne induced by laser field. Additionally, the refractive index of the target, for example, gases or solids, is also changed in the presence of the plasma. The wave-vector mismatch Δk, in plasma, between the fundamental and the third-harmonic signal can be derived by using the Drude model. Enhanced third-harmonic signal that eventually saturates at higher plasma density was predicted [6, 7]. In this chapter, we observed more than ten orders of magnitude enhancement of third-harmonic generation in argon plasma by employing the fundamental (1064 nm) and second-harmonic (532 nm) fields of an injectionseeded Q-switched Nd:YAG laser. Under the assumption that susceptibility and wave-vector mismatch depend on the plasma density, we show that plasma plays a significant role in the third-harmonic signal by an analysis based on the formulism of perturbative nonlinear optics. Significant enhancement of the TH signal is caused by the plasma-enhanced susceptibility of the dominant four-wave mixing process. When the plasma density is high enough, the TH signal becomes saturated and drops primarily due to the detrimental effect of the wave-vector mismatch.

prisms. With the desired amplitude ratio and relative phase, the two-color laser fields are recombined with an identical pair of prisms and then focused into a vacuum chamber filled with argon (10 Torr) by a 10-cm-focal lens to generate the third-harmonic (355 nm) signal. To overlap two foci of the fundamental and second-harmonic beam, the dispersion of the lens is compensated by a telescope in the fundamental arm. The third-harmonic generation is filtered by a monochromator (VM-502, Acton Research) and detected by a photomultiplier tubes

Figure 3. The experimental setup for studying the effect of plasma formation on generation of third-harmonic signal by

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159

With excitation by the two-color field (the fundamental ω<sup>1</sup> and second harmonic ω2) of the Nd: YAG laser, the third-harmonic signal can be generated by two optical processes, i.e., ω<sup>1</sup> + ω<sup>1</sup> + ω<sup>1</sup> = ω<sup>3</sup> and �ω<sup>1</sup> + ω<sup>2</sup> + ω<sup>2</sup> = ω3. We assume plane-waves propagating in the +z direction. The theoretical formulism is similar to the three-color case in Section 1. In the slow-varying enve-

lope approximation and assume no pump depletion, the TH field can be written as

þi 9π<sup>2</sup> n3λ<sup>1</sup>

8 >>><

>>>:

i 3π<sup>2</sup> n3λ<sup>1</sup>

χð Þ<sup>3</sup> <sup>I</sup> <sup>E</sup><sup>3</sup>

χð Þ<sup>3</sup> II <sup>E</sup><sup>∗</sup> 1E2

where the subscripts "I" and "II" denote the two nonlinear processes, namely the direct THG and FWM; χ(3) is the third-order nonlinear susceptibility, L is the length of the nonlinear material; φ<sup>1</sup> and φ<sup>2</sup> are the phases of the fundamental and second-harmonic beams, respectively. Δk<sup>13</sup> = 3k<sup>1</sup> � k<sup>3</sup> and Δk<sup>213</sup> = 2 k<sup>2</sup> � k<sup>1</sup> � k<sup>3</sup> are wave-vector mismatch due to dispersion in the gaseous media. The refractive index of the gas can be calculated by using Sellmeier

<sup>1</sup>Lsinc <sup>Δ</sup>k13<sup>L</sup> 2 � �<sup>e</sup>

<sup>2</sup>Lsinc <sup>Δ</sup>k213<sup>L</sup> 2 � �<sup>e</sup>

i Δk13L <sup>2</sup> e i3φ<sup>1</sup>

i Δk213L <sup>2</sup> e

<sup>i</sup>ð Þ <sup>2</sup>φ2�φ<sup>1</sup>

9 >>>=

>>>;

(8)

(R11568, Hamamatsu).

phase-controlled two-color excitation.

equation,

Eb <sup>3</sup>ð Þ¼ z Eb3Ið Þþ z Eb3IIð Þ¼ z

The experimental setup for studying the effect of plasma formation on generation of thirdharmonic signal by phase-controlled two-color excitation is shown in Figure 3. It is a simplified version of the multicolor laser system described in part I of this work and our previous papers [2, 3]. To reiterate, we employed a Q-switched Nd:YAG laser system (Spectra Physics GCR Pro-290) that generates intense 1064 nm pulses with a pulse duration of 10 ns (FWHM) and a line width of <0.003 cm<sup>1</sup> . The laser pulse repetition rate is 10 Hz, and the maximum pulse energy is 1.9 J/pulse. The second-harmonic (532 nm) beam was generated by using the nonlinear optical crystal KD\*P (type I phase matching). The maximum pulse energy of the second-harmonic signal is around 1 J/pulse. The fundamental and second-harmonic pulses propagate collinearly with a fixed relative phase. This two-color laser beams are separated by a prism pair into two arms. A power tunable two-color system can be generated with two amplitude modulators for each arm. The relative phase and amplitudes of these two-color laser fields can be timed independently by amplitude and phase modulators. The fundamental and second-harmonic beams are first angularly separated and then made parallel by a pair of Frequency-Synthesized Approach to High-Power Attosecond Pulse Generation and Applications: Applications http://dx.doi.org/10.5772/intechopen.78270 159

3. Third-harmonic generation by coherently controlled two-color excitation

With two-color excitation, the third-harmonic signal is contributed by the direct THG (ω<sup>3</sup> = ω<sup>1</sup> + ω<sup>1</sup> + ω1) and four-wave mixing (FWM, ω<sup>3</sup> = ω<sup>2</sup> + ω<sup>2</sup> – ω1) processes and a cross term of the two. As the relative phase between ω<sup>1</sup> and ω<sup>2</sup> varies, a sinusoidal modulation in output intensity at frequency ω<sup>3</sup> is expected and was demonstrated in our previous work [2]. In intense laser field, plasma can be generated through the ionization of gases. Optical harmonic generation in plasmas has been studied for a long time. Recently, significant enhancement of the third-harmonic emission in plasma has been reported by Suntsov et al. [3]. More than twoorder-of-magnitude increase of the efficiency of third-harmonic generation occurs due to the plasma-enhanced third-order susceptibility [5]. More specifically, the presence of charged species (free electrons and ions) can effectively increase the third-order nonlinear optical susceptibility [4, 5]. This indicates that the susceptibility can be expressed as a function of the plasma density Ne induced by laser field. Additionally, the refractive index of the target, for example, gases or solids, is also changed in the presence of the plasma. The wave-vector mismatch Δk, in plasma, between the fundamental and the third-harmonic signal can be derived by using the Drude model. Enhanced third-harmonic signal that eventually saturates at higher plasma density was predicted [6, 7]. In this chapter, we observed more than ten orders of magnitude enhancement of third-harmonic generation in argon plasma by employing the fundamental (1064 nm) and second-harmonic (532 nm) fields of an injectionseeded Q-switched Nd:YAG laser. Under the assumption that susceptibility and wave-vector mismatch depend on the plasma density, we show that plasma plays a significant role in the third-harmonic signal by an analysis based on the formulism of perturbative nonlinear optics. Significant enhancement of the TH signal is caused by the plasma-enhanced susceptibility of the dominant four-wave mixing process. When the plasma density is high enough, the TH signal becomes saturated and drops primarily due to the detrimental effect of the wave-vector

The experimental setup for studying the effect of plasma formation on generation of thirdharmonic signal by phase-controlled two-color excitation is shown in Figure 3. It is a simplified version of the multicolor laser system described in part I of this work and our previous papers [2, 3]. To reiterate, we employed a Q-switched Nd:YAG laser system (Spectra Physics GCR Pro-290) that generates intense 1064 nm pulses with a pulse duration of 10 ns (FWHM)

pulse energy is 1.9 J/pulse. The second-harmonic (532 nm) beam was generated by using the nonlinear optical crystal KD\*P (type I phase matching). The maximum pulse energy of the second-harmonic signal is around 1 J/pulse. The fundamental and second-harmonic pulses propagate collinearly with a fixed relative phase. This two-color laser beams are separated by a prism pair into two arms. A power tunable two-color system can be generated with two amplitude modulators for each arm. The relative phase and amplitudes of these two-color laser fields can be timed independently by amplitude and phase modulators. The fundamental and second-harmonic beams are first angularly separated and then made parallel by a pair of

. The laser pulse repetition rate is 10 Hz, and the maximum

of inert gases: plasma effect

158 High Power Laser Systems

mismatch.

and a line width of <0.003 cm<sup>1</sup>

Figure 3. The experimental setup for studying the effect of plasma formation on generation of third-harmonic signal by phase-controlled two-color excitation.

prisms. With the desired amplitude ratio and relative phase, the two-color laser fields are recombined with an identical pair of prisms and then focused into a vacuum chamber filled with argon (10 Torr) by a 10-cm-focal lens to generate the third-harmonic (355 nm) signal. To overlap two foci of the fundamental and second-harmonic beam, the dispersion of the lens is compensated by a telescope in the fundamental arm. The third-harmonic generation is filtered by a monochromator (VM-502, Acton Research) and detected by a photomultiplier tubes (R11568, Hamamatsu).

With excitation by the two-color field (the fundamental ω<sup>1</sup> and second harmonic ω2) of the Nd: YAG laser, the third-harmonic signal can be generated by two optical processes, i.e., ω<sup>1</sup> + ω<sup>1</sup> + ω<sup>1</sup> = ω<sup>3</sup> and �ω<sup>1</sup> + ω<sup>2</sup> + ω<sup>2</sup> = ω3. We assume plane-waves propagating in the +z direction. The theoretical formulism is similar to the three-color case in Section 1. In the slow-varying envelope approximation and assume no pump depletion, the TH field can be written as

$$\hat{E}\_{3}(\mathbf{z}) = \hat{E}\_{3l}(\mathbf{z}) + \hat{E}\_{3l\text{I}}(\mathbf{z}) = \left\{ \begin{aligned} i\frac{3\pi^{2}}{n\_{3}\lambda\_{1}}\chi\_{l}^{(3)}E\_{1}^{3}L\text{sinc}\left(\frac{\Delta k\_{13}L}{2}\right)e^{\frac{\Delta k\_{13}l}{2}}e^{i3\varphi\_{l}} \\ + i\frac{9\pi^{2}}{n\_{3}\lambda\_{1}}\chi\_{l}^{(3)}E\_{1}^{\*}E\_{2}^{2}L\text{sinc}\left(\frac{\Delta k\_{213}L}{2}\right)e^{\frac{\Delta k\_{13}l}{2}}e^{i(2\varphi\_{l}-\varphi\_{l})} \end{aligned} \right\} \tag{8}$$

where the subscripts "I" and "II" denote the two nonlinear processes, namely the direct THG and FWM; χ(3) is the third-order nonlinear susceptibility, L is the length of the nonlinear material; φ<sup>1</sup> and φ<sup>2</sup> are the phases of the fundamental and second-harmonic beams, respectively. Δk<sup>13</sup> = 3k<sup>1</sup> � k<sup>3</sup> and Δk<sup>213</sup> = 2 k<sup>2</sup> � k<sup>1</sup> � k<sup>3</sup> are wave-vector mismatch due to dispersion in the gaseous media. The refractive index of the gas can be calculated by using Sellmeier equation,

$$\begin{split} n(\lambda) - 1 &= (n - 1)\_{\text{lines}} + (n - 1)\_{\text{cont}} \\ &= \frac{N\_{\text{g}} r\_{\text{c}}}{2\pi} \sum\_{i} \frac{f\_{i}}{\lambda\_{i}^{-2} - \lambda^{-2}} + \frac{N\_{\text{g}}}{2\pi^{2}} \left\{ \frac{\sigma d \overline{v}\_{i}}{\overline{v}\_{i}^{2} - \overline{v}^{2}} \right. \end{split} \tag{9}$$

In Eq. (9), N<sup>g</sup> = P/kBT is the gas density related to the pressure of the gas by ideal gas law N<sup>g</sup> in which k<sup>B</sup> is Planck constant; r<sup>e</sup> is classical electron radius. The first term or (n � 1)lines refers to the contribution by discrete energy levels of the atom while the second term or (n � 1)cont is that by the continuum states. For the oscillator strengths f of argon, we used those listed in Ref. [8]. In addition, we take the photoionization cross-section from Ref. [9]. The intensity of TH signal, therefore, can be written as

$$I\_{3}(\boldsymbol{q}) = \frac{c\eta\_{3}}{8\pi} \left\langle \left| \hat{\boldsymbol{E}}\_{3\rm I}(\boldsymbol{z},t) + \hat{\boldsymbol{E}}\_{3\rm II}(\boldsymbol{z},t) \right|^{2} \right\rangle = I\_{3\rm I} + I\_{3\rm II} + I\_{3\rm III}$$

$$= \frac{576\pi^{6}L^{2}I\_{1}}{c^{2}n\_{3}n\_{1}\boldsymbol{\lambda}\_{1}^{2}} \left\{ \begin{aligned} &\frac{1}{n\_{1}^{2}} \left(\boldsymbol{\chi}\_{1}^{(3)}\right)^{2} I\_{1}^{2} \text{sinc}^{2} \left(\frac{\Delta k\_{13}L}{2}\right) \\\\ &+ \frac{9}{n\_{2}^{2}} \left(\boldsymbol{\chi}\_{1}^{(3)}\right)^{2} I\_{2}^{2} \text{sinc}^{2} \left(\frac{\Delta k\_{23}L}{2}\right) \\\\ &+ \frac{6}{n\_{1}n\_{2}} \boldsymbol{\chi}\_{1}^{(3)}\boldsymbol{\chi}\_{1}^{(3)}I\_{1}I\_{2} \text{sinc} \left(\frac{\Delta k\_{13}L}{2}\right) \text{sinc} \left(\frac{\Delta k\_{23}L}{2}\right) \\\\ &\times \cos\left(\frac{\Delta k\_{13}L}{2} - \frac{\Delta k\_{23}L}{2} + 4\rho\_{1} - 2\rho\_{2}\right) \end{aligned} \tag{10}$$

In Eq. (10), the first, second and third term corresponds to THG. FWM and a cross-term due to the interference of the former two processes. For the sake of simplicity, we can set ϕ<sup>1</sup> = 0. Therefore, Δϕ = ϕ<sup>2</sup> � 2ϕ<sup>1</sup> = ϕ<sup>2</sup> is the relative phase between ω<sup>1</sup> and ω2. In media with normal dispersion, for example, the non-resonant excitation of room-temperature argon gas, the relative magnitude of the wave vectors is k1 < k2 < k3. Accordingly, the phase mismatch, j j Δk<sup>13</sup> > j j Δk<sup>123</sup> ≈ 0, is negligible. A sinusoidal dependence of the TH signal on the relative phase is thus expected. An example is shown in Figure 4. The pulse energy of the 1064 nm and 532 nm beams were 70 and 1 mJ, respectively. The pressure of the argon gas was 100 Torr. As the beams are slightly elliptical, we measured the TH signal in two transverse directions. The percentile errors in the X- and Y-directions are shown. Note that the TH signal is very weak if only the fundamental beam is used for excitation.

It was found that the TH signal can be enhanced by more than one order of magnitude with two-color excitation. In Table 2, we summarize the phase modulation and enhancement of the TH signal with two-color excitation for several ratios of fundamental and second-harmonic pulse energies. The fluctuations of the TH signal when the relative phase of the fundamental and second-harmonic beams is a constant is also shown.

We observed the enhancement of the TH signal is substantial for two-color excitation. Plasma emission was found to be visible to the naked eye in such cases. It is reasonable to assume that laser-induced ionization in the inert gases, for example, argon. With our experimental conditions, the ionization process is in the multiphoton ionization regime, which occurs when an atoms simultaneously absorbing several photons. The multiphoton ionization rate w(ω,F) can

Table 2. The phase modulation and enhancement of the TH signal with two-color excitation for several ratios of

Figure 4. Typical trace of the TH signal plotted as a function of the relative phase between the fundamental and secondharmonic beams. The system noise level corresponding to situation in which the slit of the monochrometer was closed is

Frequency-Synthesized Approach to High-Power Attosecond Pulse Generation and Applications: Applications

The fluctuation in TH power without

Enhancement ratio (twocolour/one color)

http://dx.doi.org/10.5772/intechopen.78270

161

phase delay (normalized)

0.45 0.065 9.02~16.38

0.6258 0.166 2.28~5.52

0.3446 0.1457 13.41~20.58

also shown.

Excitation Source

(1) 1064 (70mJ/ pulse) +532 (1mJ/ pulse)

(2) 1064 (110mJ/ pulse) +532 (1mJ/ pulse)

(3) 1064 (70mJ/ pulse) +532 (20mJ/ pulse)

The modulation of phase or contrast (Normalized)

fundamental and second-harmonic pulse energies.

Frequency-Synthesized Approach to High-Power Attosecond Pulse Generation and Applications: Applications http://dx.doi.org/10.5772/intechopen.78270 161

nð Þ� λ 1 ¼ ð Þ n � 1 lines þ ð Þ n � 1 cont

X i

f i λ�<sup>2</sup>

In Eq. (9), N<sup>g</sup> = P/kBT is the gas density related to the pressure of the gas by ideal gas law N<sup>g</sup> in which k<sup>B</sup> is Planck constant; r<sup>e</sup> is classical electron radius. The first term or (n � 1)lines refers to the contribution by discrete energy levels of the atom while the second term or (n � 1)cont is that by the continuum states. For the oscillator strengths f of argon, we used those listed in Ref. [8]. In addition, we take the photoionization cross-section from Ref. [9]. The intensity of TH

> � � �

> > χð Þ<sup>3</sup> I � �<sup>2</sup> I 2

χð Þ<sup>3</sup> II � �<sup>2</sup> I 2

Δk13L

In Eq. (10), the first, second and third term corresponds to THG. FWM and a cross-term due to the interference of the former two processes. For the sake of simplicity, we can set ϕ<sup>1</sup> = 0. Therefore, Δϕ = ϕ<sup>2</sup> � 2ϕ<sup>1</sup> = ϕ<sup>2</sup> is the relative phase between ω<sup>1</sup> and ω2. In media with normal dispersion, for example, the non-resonant excitation of room-temperature argon gas, the relative magnitude of the wave vectors is k1 < k2 < k3. Accordingly, the phase mismatch, j j Δk<sup>13</sup> > j j Δk<sup>123</sup> ≈ 0, is negligible. A sinusoidal dependence of the TH signal on the relative phase is thus expected. An example is shown in Figure 4. The pulse energy of the 1064 nm and 532 nm beams were 70 and 1 mJ, respectively. The pressure of the argon gas was 100 Torr. As the beams are slightly elliptical, we measured the TH signal in two transverse directions. The percentile errors in the X- and Y-directions are shown. Note that the TH signal is very

It was found that the TH signal can be enhanced by more than one order of magnitude with two-color excitation. In Table 2, we summarize the phase modulation and enhancement of the TH signal with two-color excitation for several ratios of fundamental and second-harmonic pulse energies. The fluctuations of the TH signal when the relative phase of the fundamental

We observed the enhancement of the TH signal is substantial for two-color excitation. Plasma emission was found to be visible to the naked eye in such cases. It is reasonable to assume that

II <sup>I</sup>1I2sinc <sup>Δ</sup>k13<sup>L</sup>

<sup>2</sup> � <sup>Δ</sup>k213<sup>L</sup>

1 n2 1

þ 9 n2 2

� cos

<sup>i</sup> � <sup>λ</sup>�<sup>2</sup> <sup>þ</sup>

Ng 2π<sup>2</sup>

¼ I3<sup>I</sup> þ I3II þ I3III

1sinc<sup>2</sup> <sup>Δ</sup>k13<sup>L</sup> 2 � �

2sinc<sup>2</sup> <sup>Δ</sup>k213<sup>L</sup> 2 � �

> sinc <sup>Δ</sup>k213<sup>L</sup> 2 � �

2 � �

� �

<sup>2</sup> <sup>þ</sup> <sup>4</sup>φ<sup>1</sup> � <sup>2</sup>φ<sup>2</sup>

ð σdvi v2 <sup>i</sup> � <sup>v</sup><sup>2</sup> : (9)

9

>>>>>>>>>>>>>>>>=

(10)

>>>>>>>>>>>>>>>>;

<sup>¼</sup> Ngre 2π

Eb3Ið Þþ z; t Eb3IIðz; tÞ

<sup>2</sup> � �

signal, therefore, can be written as

160 High Power Laser Systems

I3ð Þ¼ φ

cn<sup>3</sup> 8π

<sup>¼</sup> <sup>576</sup>π<sup>6</sup>L<sup>2</sup>

c<sup>2</sup>n3n1λ<sup>2</sup> 1

I1

weak if only the fundamental beam is used for excitation.

and second-harmonic beams is a constant is also shown.

8

>>>>>>>>>>>>>>>><

þ 6 n1n<sup>2</sup> χð Þ<sup>3</sup> <sup>I</sup> <sup>χ</sup>ð Þ<sup>3</sup>

>>>>>>>>>>>>>>>>:

� � �

Figure 4. Typical trace of the TH signal plotted as a function of the relative phase between the fundamental and secondharmonic beams. The system noise level corresponding to situation in which the slit of the monochrometer was closed is also shown.


Table 2. The phase modulation and enhancement of the TH signal with two-color excitation for several ratios of fundamental and second-harmonic pulse energies.

laser-induced ionization in the inert gases, for example, argon. With our experimental conditions, the ionization process is in the multiphoton ionization regime, which occurs when an atoms simultaneously absorbing several photons. The multiphoton ionization rate w(ω,F) can be calculated by the Perelomov-Popov-Terent'ev (PPT) model, where F is the laser fluence. The rate is a function of the laser oscillation frequency and laser field strength. For the two-color case, we assume an effective frequency which is calculated from the power distribution of laser frequency to describe the influence of the two-color electric field on the ionization rate.

$$
\omega\_{\rm eff} = \frac{\int\_0^\infty \omega |E(\omega)|^2 d\omega}{\int\_0^\infty |E(\omega)|^2 d\omega} \tag{11}
$$

Besides, for our nanosecond pulse, there are several million cycles inside the pulse envelope for fundamental beam in the near infrared. The cycle-averaged ionization rate is thus used in this work [10]. That is,

$$
\overline{w}\_{PPT}(F\_a) = \frac{1}{T\_0} \int\_0^{T\_0} w\_{PPT}(t)dt\tag{12}
$$

or

$$
\overline{w}\_{\rm PPT}(F\_a) = \sqrt{\frac{2}{\pi}} \sqrt{\frac{3F\_a}{2F\_0}} w\_{\rm PPT}(F\_a) \tag{13}
$$

The ionization probability of the atoms by the laser pulse can be calculated by solving the rate equation

$$p = 1 - e^{-\int\_{-\infty}^{\omega} w\_{PPT}(t)dt}.\tag{14}$$

Additionally, we note that the refractive index of the media would also be changed when the plasma is generated. This can be calculated, in the first approximation, by using the Drude

Figure 5. Ionization probability calculated by PPT model for argon at 100 Torr, excited by two-color field with funda-

Frequency-Synthesized Approach to High-Power Attosecond Pulse Generation and Applications: Applications

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<sup>n</sup>ω,p <sup>¼</sup> <sup>n</sup>ω,<sup>g</sup> � Ne

equal. The subscripts p and g represent plasma and neutral gas, respectively. For argon, nω, <sup>g</sup> = nω,Ar. The wave-vector mismatch becomes Δk13, <sup>p</sup> = 3k1, <sup>p</sup> – k3,<sup>p</sup> and Δk213,<sup>p</sup> = 2k2, <sup>p</sup> – k1,<sup>p</sup> – k3,p. The intensity of third-harmonic signal can then be calculated using Eq. (10). This is plotted as a function of the energy of the second-harmonic pulse for two values of the pulse energies for the fundamental beam in Figure 6. The experimental data are in good agreements with the

The four-wave mixing process is dominant in the third-harmonic signal. For our experimental conditions, the THG component is approximately 10�<sup>4</sup> that of the FWM process. In the

that the enhancement of the third-harmonic signal is due to the plasma-enhanced susceptibility for the FWM process. On the other hand, when the plasma density is high enough, the wave-vector mismatch Δk becomes significant due to the plasma-induced refractive index

Thus, in the high plasma density limit, the four-wave mixing term becomes I3II ∝ sinc<sup>2</sup> (Δk213,pL/2). This is one of the reasons why the third-harmonic signal saturates at high plasma density. As a result, the TH signal is higher when the pulse energy of the

2Nc

<sup>2</sup> is the critical plasma density when the laser and plasma frequencies are

II,p) <sup>2</sup> ∝ N<sup>e</sup> 2 (18)

163

. This indicates

model.

where Nc = ε0meω<sup>2</sup>

/e

mental pulse energies at 150 mJ and 200 mJ.

simulated values using the above theoretical formulism.

low plasma density limit, the FWM term can be written as I3II ∝ (χ(3)

change, which is linearly proportional to the plasma density.

This allows us to calculate the plasma density in terms of the density of the neutral gas.

$$N\_{\epsilon} = p \times N\_{\S} \tag{15}$$

The step-like behavior for the ionization probability as shown in Figure 5 is caused by the increase of the effective frequency when the number of the second-harmonic photons increases. That is, there are new absorption processes occurring when the effective photon energy of the pulse reaches the threshold of the ionization process.

Now, we consider influence of the plasma on the third-harmonic signal. We assume that the third-order optical susceptibility is a sum of the susceptibilities for the neutral and ionized gas atoms.

$$
\chi\_{l,p}^{(3)} = \chi\_{l,\mathbf{g}}^{(3)} + \mathcal{\mathcal{V}}\_{l,p} \mathcal{N}\_e \tag{16}
$$

$$
\chi\_{\rm II,p}^{(3)} = \chi\_{\rm II,g}^{(3)} + \chi\_{\rm II,p} N\_e \tag{17}
$$

In the above two equations, the ratios γI,p and γII,p are values determined by the experiment. Here, we assume <sup>γ</sup>I,p <sup>=</sup> <sup>γ</sup>II,p = 4�10�<sup>49</sup> and <sup>χ</sup>(3) I,g = χ(3) II,g <sup>=</sup> <sup>χ</sup>(3)II,Ar = 3.8 �10�<sup>26</sup> m2 /V<sup>2</sup> [11].

Frequency-Synthesized Approach to High-Power Attosecond Pulse Generation and Applications: Applications http://dx.doi.org/10.5772/intechopen.78270 163

be calculated by the Perelomov-Popov-Terent'ev (PPT) model, where F is the laser fluence. The rate is a function of the laser oscillation frequency and laser field strength. For the two-color case, we assume an effective frequency which is calculated from the power distribution of laser

frequency to describe the influence of the two-color electric field on the ionization rate.

Ð ∞ <sup>0</sup> ωj j Eð Þ ω

Ð ∞ <sup>0</sup> j j Eð Þ ω 2 dω

Besides, for our nanosecond pulse, there are several million cycles inside the pulse envelope for fundamental beam in the near infrared. The cycle-averaged ionization rate is thus used in this

> 1 T0 ð<sup>T</sup><sup>0</sup> 0

ffiffiffi 2 π r ffiffiffiffiffiffiffi 3Fa 2F<sup>0</sup>

The ionization probability of the atoms by the laser pulse can be calculated by solving the rate

� Ð ∞

The step-like behavior for the ionization probability as shown in Figure 5 is caused by the increase of the effective frequency when the number of the second-harmonic photons increases. That is, there are new absorption processes occurring when the effective photon

Now, we consider influence of the plasma on the third-harmonic signal. We assume that the third-order optical susceptibility is a sum of the susceptibilities for the neutral and ionized gas

In the above two equations, the ratios γI,p and γII,p are values determined by the experiment.

I,g = χ(3)

s

2 dω

wPPTð Þt dt (12)

wPPTð Þ Fa (13)

�<sup>∞</sup> wPPT ð Þ<sup>t</sup> dt: (14)

Ne ¼ p � Ng (15)

I,<sup>g</sup> þ γI,pNe (16)

II,<sup>g</sup> þ γII,pNe (17)

II,g <sup>=</sup> <sup>χ</sup>(3)II,Ar = 3.8 �10�<sup>26</sup> m2

/V<sup>2</sup> [11].

(11)

ωeff ¼

wPPTð Þ¼ Fa

wPPTð Þ¼ Fa

p ¼ 1 � e

energy of the pulse reaches the threshold of the ionization process.

Here, we assume <sup>γ</sup>I,p <sup>=</sup> <sup>γ</sup>II,p = 4�10�<sup>49</sup> and <sup>χ</sup>(3)

χð Þ<sup>3</sup> I, <sup>p</sup> <sup>¼</sup> <sup>χ</sup>ð Þ<sup>3</sup>

χð Þ<sup>3</sup> II, <sup>p</sup> <sup>¼</sup> <sup>χ</sup>ð Þ<sup>3</sup>

This allows us to calculate the plasma density in terms of the density of the neutral gas.

work [10]. That is,

162 High Power Laser Systems

or

equation

atoms.

Figure 5. Ionization probability calculated by PPT model for argon at 100 Torr, excited by two-color field with fundamental pulse energies at 150 mJ and 200 mJ.

Additionally, we note that the refractive index of the media would also be changed when the plasma is generated. This can be calculated, in the first approximation, by using the Drude model.

$$n\_{\omega,p} = n\_{\omega,\text{g}} - \frac{N\_{\text{c}}}{2N\_{\text{c}}} \tag{18}$$

where Nc = ε0meω<sup>2</sup> /e <sup>2</sup> is the critical plasma density when the laser and plasma frequencies are equal. The subscripts p and g represent plasma and neutral gas, respectively. For argon, nω, <sup>g</sup> = nω,Ar. The wave-vector mismatch becomes Δk13, <sup>p</sup> = 3k1, <sup>p</sup> – k3,<sup>p</sup> and Δk213,<sup>p</sup> = 2k2, <sup>p</sup> – k1,<sup>p</sup> – k3,p. The intensity of third-harmonic signal can then be calculated using Eq. (10). This is plotted as a function of the energy of the second-harmonic pulse for two values of the pulse energies for the fundamental beam in Figure 6. The experimental data are in good agreements with the simulated values using the above theoretical formulism.

The four-wave mixing process is dominant in the third-harmonic signal. For our experimental conditions, the THG component is approximately 10�<sup>4</sup> that of the FWM process. In the low plasma density limit, the FWM term can be written as I3II ∝ (χ(3) II,p) <sup>2</sup> ∝ N<sup>e</sup> 2 . This indicates that the enhancement of the third-harmonic signal is due to the plasma-enhanced susceptibility for the FWM process. On the other hand, when the plasma density is high enough, the wave-vector mismatch Δk becomes significant due to the plasma-induced refractive index change, which is linearly proportional to the plasma density.

Thus, in the high plasma density limit, the four-wave mixing term becomes I3II ∝ sinc<sup>2</sup> (Δk213,pL/2). This is one of the reasons why the third-harmonic signal saturates at high plasma density. As a result, the TH signal is higher when the pulse energy of the

high-power density are used as a thermal source which is focused on an object for a period of time. The energy absorbed on the surface of the object is transferred into the bulk of the object via thermal conduction. Thereafter, a part of the object is melted or vaporized by the deposited thermal energy. The laser spot is moved to another part of the work piece ready for further processing. In the photo-chemical mechanism, the bonding of molecules in the material to be processed is broken after absorption of one or more photons, which make electrons hop between energy levels and molecular bonds in the material can be broken as a result [12, 13]. In laser processing, the laser is chosen according to characteristics such as energy absorption, thermal diffusion and melting point of the material. For example, ablation is performed on various materials using lasers with appropriate wavelength. It is interesting, therefore, to investigate whether synthesized waveforms proposed and demonstrated in our work could

Frequency-Synthesized Approach to High-Power Attosecond Pulse Generation and Applications: Applications

http://dx.doi.org/10.5772/intechopen.78270

165

Ablation of materials with multiple lasers, for example, lasers with dual colors were reported recently [14–18]. Incoherent or coherent summation of multi-color beams can be implemented. With incoherent summation of two femtosecond and nanosecond class pulsed lasers, an enhancement of volume of the vaporized material was observed by Théberge and Chin [14]. In this work, the free electrons and defect states induced by intense fs pulses were exploited by the ns pulses. In another work, Okoshi and Inoue [15] demonstrated that superimposed fs pulses at the fundamental (ω) and small fraction of the second-harmonics (2ω) output of the Ti: sapphire laser with the relative fluence ratio 1/39 was able to etch polyethylene (PE) much deeper and faster. They attributed the observe phenomena by the higher photon energy of 2ω pulses which can cut the chemical bonds of PE to form a modified layer of PE on the ablated surface. However, this article did not discuss about the temporal dynamics of the laser ablation process. On the other hand, the enhancement of absorption/reflection was observed in fused silica with coherent summation of dual-color pulses at zero delay [16]. This is because of defect states formation or free electron plasma generated in the material this way. For silicon, the ablation process was reported in the case of nanosecond and picosecond laser pulses where a small portion of the (2ω) beam can excite electrons into the conduction band [17]. For femtosecond pulses, this effect became insignificant because a sufficient population in the conduction band is created by multiphoton absorption in silicon. However, on the scale of carrier lifetime, all of the above-mentioned works consider relatively long time delays between the

We note that tunable relative-phase control between the two dual-color exciting laser was applied in order to study the physical mechanism of intense-field photoionization in the gas phase [19–21]. Schumacher and Bucksbaum [19] reported that number of photoelectrons created in a regime that both multiphoton and tunneling ionization mechanisms are present is indeed dependent on the relative phase of the dual-colors. Later, Gao et al. [20] showed that the observed phase-dependence represents a quantum interference (QI) between the different channels corresponding to different number of photons involved. Recently, in comparison with monochromatic excitation, the threshold of plasma creation in the material to be ablated has been identified to be significantly reduced with the use of a ns infrared laser pulses and its second-harmonic one [21]. The observed phenomenon was attributed to the field-dependence

be advantageous for laser processing.

beams of two colors (picoseconds).

Figure 6. The comparison of the experimental and simulation results for two-color excited third-harmonic signal in argon as a function of the second-harmonic pulse energy.

fundamental output was lower. This is in good agreement with the experimental results for fundamental pulse energies of 150 and 200 mJ. However, the theoretically predicted threshold for plasma enhancement does not match that of the experimental data. The may be explained by the dependence of the threshold on the step-like enhanced ionization probability. The step-like behavior caused by new absorption processes becomes dominant when the effective photon energy (effective frequency) reaches the threshold of this process. However, in reality, there actually exist many quantum processes involving the absorption of several photons at frequencies of ω and 2ω. The different quantum processes have different ionization rate. When the power ratio of the two-color field is changed, the ionization probability of the different quantum process is also changed. It could be argued that the variation of ionization rate with the second-harmonic pulse energy is continuous rather than step-like when the plasma density increases. This in turn should shift the threshold pulse energy.
