2. Nonlinear frequency conversion by coherently controlled three-color excitation of inert gases

We assume plane-waves propagating in the +z direction. The three-color field can be repre-

Table 1. Third-order nonlinear process (ω<sup>n</sup> = ω<sup>i</sup> + ω<sup>j</sup> + ωk, ω<sup>n</sup> = ω<sup>i</sup> + ω<sup>j</sup> � ωk,n=4–9, i, j, k = 1–3) that can contribute to the generation of 4th to 9th harmonics of the laser fundamental output of a three-color field (the fundamental ω1, second

where φ1, φ<sup>2</sup> and φ<sup>3</sup> are the modulated phases of the three colors, respectively. As a source, the nonlinear polarization term in the medium induced by the three-color fields will generate several new frequency components. If we only consider third-order nonlinear optical processes only, assuming no pump depletion, the electric field of the fourth harmonic can then be

<sup>1</sup>E2L sin c

<sup>1</sup>E2E3L sin c

i k ð Þ <sup>2</sup>z�ω2tþφ<sup>2</sup> <sup>þ</sup> <sup>E</sup>1<sup>e</sup>


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

ω1+ω1+ω3!ω<sup>5</sup> -ω1+ω3+ω3!ω<sup>5</sup>

ω1+ω2+ω3!ω<sup>6</sup>

ω1+ω3+ω3!ω<sup>7</sup>

Δk4IL 2 � �<sup>e</sup>

Δk4IIL 2 � �<sup>e</sup>

Δk4IIIL 2 � �<sup>e</sup>

h i (1)

E4ð Þ¼ z E4Ið Þþ z E4IIð Þþ z E4IIIð Þz , (2)

<sup>i</sup>Δk4ILe

<sup>i</sup>Δk4IILe

<sup>i</sup>Δk4IIILe

i k ð Þ <sup>3</sup>z�ω3tþφ<sup>1</sup> <sup>þ</sup> <sup>c</sup>:c:

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

155

<sup>i</sup>ð Þ <sup>2</sup>φ1þφ<sup>2</sup> : (3)

<sup>i</sup>ð Þ �φ1þφ2þφ<sup>3</sup> , (4)

<sup>i</sup>ð Þ �φ2þ2φ<sup>3</sup> , (5)

i k ð Þ <sup>1</sup>z�ω1tþφ<sup>1</sup> <sup>þ</sup> <sup>E</sup>2<sup>e</sup>

sented as:

rewritten as

where

<sup>E</sup>~ð Þ¼ <sup>z</sup>; <sup>t</sup>

harmonic ω<sup>2</sup> = 2ω<sup>1</sup> and third harmonic ω<sup>3</sup> = 3ω1).

1 <sup>2</sup> <sup>E</sup>1<sup>e</sup>

Harmonic generation Processes Fourth ω1+ω1+ω2!ω<sup>4</sup>

Fifth ω1+ω2+ω2!ω<sup>5</sup>

Sixth ω2+ω2+ω2!ω<sup>6</sup>

Seventh ω2+ω2+ω3!ω<sup>7</sup>

Eighth ω2+ω3+ω3!ω<sup>8</sup> Ninth ω3+ω3+ω3!ω<sup>9</sup>

E4Ið Þ¼ z i

E4IIð Þ¼ z i

E4IIIð Þ¼ z i

12π<sup>2</sup> n4λ<sup>1</sup>

> Nχð Þ<sup>3</sup> 4IIE<sup>∗</sup>

> > Nχð Þ<sup>3</sup> 4IIIE<sup>∗</sup> 2E2 <sup>3</sup>L sin c

24π<sup>2</sup> n4λ<sup>1</sup>

> 12π<sup>2</sup> n4λ<sup>1</sup>

Nχð Þ<sup>3</sup> <sup>4</sup><sup>I</sup> <sup>E</sup><sup>2</sup>

In this section, we investigate the use of three-color laser fields as a source to generate harmonic signals in an isotropic media, for example, inert gases. With three-color pump and consider only the lowest order nonlinear processes in isotropic systems, that is, third-order nonlinear process, one can expect to generate 4th to 9th harmonics of the laser fundamental output. A richness of nonlinear effects and complicated quantum interference phenomena is predicted. This summarized in Table 1.

Using perturbative nonlinear formulism, we first derived the general formula of the harmonic electric field as well as the corresponding intensity. The coherent effect manifests itself through the interference of two frequency conversion pathways. In the following, we will use the case of FHG to illustrate the physical phenomena expected.

With three-color field (the fundamental ω1, second harmonic ω2 = 2ω<sup>1</sup> and third harmonic ω<sup>3</sup> = 3ω1) excitation, the fourth-harmonic signal can be generated by three nonlinear optical processes (ω<sup>1</sup> + ω<sup>1</sup> + ω<sup>2</sup> = ω4, ω<sup>1</sup> + ω<sup>2</sup> + ω<sup>3</sup> = ω4, and ω<sup>2</sup> + ω<sup>3</sup> + ω<sup>3</sup> = ω4). The conversion efficiency for the fourth-harmonic signal can be modulated by the interference between each two of three FWM processes. As the relative phase between ω1, ω<sup>2</sup> and ω<sup>3</sup> vary, combinations of three sinusoidal modulation due to interference in the output intensity of the fourth harmonic at frequency ω<sup>4</sup> is predicted. We will also show that the relative amplitude of the fundamental, second-harmonic and third-harmonic driving laser field influences the fourthharmonic signal.

Frequency-Synthesized Approach to High-Power Attosecond Pulse Generation and Applications: Applications http://dx.doi.org/10.5772/intechopen.78270 155


Table 1. Third-order nonlinear process (ω<sup>n</sup> = ω<sup>i</sup> + ω<sup>j</sup> + ωk, ω<sup>n</sup> = ω<sup>i</sup> + ω<sup>j</sup> � ωk,n=4–9, i, j, k = 1–3) that can contribute to the generation of 4th to 9th harmonics of the laser fundamental output of a three-color field (the fundamental ω1, second harmonic ω<sup>2</sup> = 2ω<sup>1</sup> and third harmonic ω<sup>3</sup> = 3ω1).

We assume plane-waves propagating in the +z direction. The three-color field can be represented as:

$$\tilde{\mathbf{E}}(z,t) = \frac{1}{2} \left[ E\_1 e^{i\left(k\_1 z - a\gamma t + \varphi\_1\right)} + E\_2 e^{i\left(k\_2 z - a\gamma t + \varphi\_2\right)} + E\_1 e^{i\left(k\_3 z - a\gamma t + \varphi\_1\right)} + c.c.\right] \tag{1}$$

where φ1, φ<sup>2</sup> and φ<sup>3</sup> are the modulated phases of the three colors, respectively. As a source, the nonlinear polarization term in the medium induced by the three-color fields will generate several new frequency components. If we only consider third-order nonlinear optical processes only, assuming no pump depletion, the electric field of the fourth harmonic can then be rewritten as

$$E\_4(z) = E\_{4I}(z) + E\_{4II}(z) + E\_{4III}(z),\tag{2}$$

where

The coherent control of nonlinear optical processes such as harmonic generation by waveformcontrolled laser field is important for both fundamental science and technological applications. Previously, we have studied the influence of relative phases and intensities of the two-color pump (1064 and 532 nm) electric fields on the third-order nonlinear frequency conversion process in argon [2]. It was shown that the third-harmonic (TH) signal oscillates periodically with the relative phases of the two-color driving laser fields. The data are in good agreement with a perturbative nonlinear optical analysis of the TH signal, which consists of contribution of the direct third-harmonic-generation (THG), four-wave mixing (FWM) and the interference

As an extension of this work, we have studied generation of harmonics by three-color synthesized waveform in inert gas systems. We will illustrate the physics involved by examining the

Anomalous enhancement of the THz signal in the presence of the 532 nm beam was observed, however. In this work, we show that plasma generated through the ionization process during laser-matter interaction plays a significant role in the enhancement of the TH signal. We also demonstrated phase-sensitive two-color ablation of copper and stainless steel. Our results

2. Nonlinear frequency conversion by coherently controlled three-color

In this section, we investigate the use of three-color laser fields as a source to generate harmonic signals in an isotropic media, for example, inert gases. With three-color pump and consider only the lowest order nonlinear processes in isotropic systems, that is, third-order nonlinear process, one can expect to generate 4th to 9th harmonics of the laser fundamental output. A richness of nonlinear effects and complicated quantum interference phenomena is

Using perturbative nonlinear formulism, we first derived the general formula of the harmonic electric field as well as the corresponding intensity. The coherent effect manifests itself through the interference of two frequency conversion pathways. In the following, we will use the case

With three-color field (the fundamental ω1, second harmonic ω2 = 2ω<sup>1</sup> and third harmonic ω<sup>3</sup> = 3ω1) excitation, the fourth-harmonic signal can be generated by three nonlinear optical processes (ω<sup>1</sup> + ω<sup>1</sup> + ω<sup>2</sup> = ω4, ω<sup>1</sup> + ω<sup>2</sup> + ω<sup>3</sup> = ω4, and ω<sup>2</sup> + ω<sup>3</sup> + ω<sup>3</sup> = ω4). The conversion efficiency for the fourth-harmonic signal can be modulated by the interference between each two of three FWM processes. As the relative phase between ω1, ω<sup>2</sup> and ω<sup>3</sup> vary, combinations of three sinusoidal modulation due to interference in the output intensity of the fourth harmonic at frequency ω<sup>4</sup> is predicted. We will also show that the relative amplitude of the fundamental, second-harmonic and third-harmonic driving laser field influences the fourth-

of the above two processes.

154 High Power Laser Systems

excitation of inert gases

harmonic signal.

predicted. This summarized in Table 1.

of FHG to illustrate the physical phenomena expected.

case for fourth-harmonic generation (FHG) in Section 2.

show that hole drilling is more efficient for optimized waveforms.

$$E\_{4l}(z) = i \frac{12\pi^2}{n\_4 \lambda\_1} N \chi\_{4l}^{(3)} E\_1^2 E\_2 L \sin c \left(\frac{\Delta k\_{4l} L}{2}\right) e^{i\Delta k\_{4l} L} e^{i\left(2\varphi\_1 + \varphi\_2\right)}.\tag{3}$$

$$E\_{4\text{II}}(z) = i \frac{24\pi^2}{n\_4\lambda\_1} N \chi\_{4\text{II}}^{(3)} E\_1^\* E\_2 E\_3 L \sin c \left(\frac{\Delta k\_{4\text{II}}L}{2}\right) e^{i\Delta k\_{4\text{II}}L} e^{i\left(-\varphi\_1 + \varphi\_2 + \varphi\_3\right)}\,\tag{4}$$

$$E\_{4\text{III}}(z) = i \frac{12\pi^2}{n\_4\lambda\_1} N \chi\_{4\text{III}}^{(3)} E\_2^\* E\_3^2 L \sin c \left(\frac{\Delta k\_{4\text{III}}L}{2}\right) e^{i\Delta k\_{4\text{II}}L} e^{i\left(-\varphi\_2 + 2\varphi\_3\right)},\tag{5}$$

With the phase or wave-vector mismatch given by Δk4<sup>I</sup> ¼ k<sup>4</sup> � 2k<sup>1</sup> � k2, Δk4II ¼ k4þ k<sup>1</sup> � k<sup>2</sup> � k<sup>3</sup> and Δk4III ¼ k<sup>4</sup> þ k<sup>2</sup> � 2k3. In this section, the symbol "I","II", "III" represent the three possible four-wave mixing (FWM) processes with corresponding nonlinear susceptibilities: <sup>χ</sup>(3)4I <sup>=</sup> <sup>χ</sup>(3)(ω4; <sup>ω</sup>1, <sup>ω</sup>1, <sup>ω</sup>2), <sup>χ</sup>(3)4II <sup>=</sup> <sup>χ</sup>(3)(ω4; �ω1, <sup>ω</sup>2, <sup>ω</sup>3), <sup>χ</sup>(3)4III <sup>=</sup> <sup>χ</sup>(3)(ω4; �ω2, <sup>ω</sup>3, <sup>ω</sup>3), respectively. L stands for the nonlinear medium length. The intensity of the fourth-harmonic signal can then be written as

$$I\_4(z) = \frac{c\eta\_4}{8\pi} \left\langle \left| \tilde{E}\_{4l}(z,t) + \tilde{E}\_{4l\text{I}}(z,t) + \tilde{E}\_{4\text{III}}(z,t) \right|^2 \right\rangle \tag{6}$$

or

I4ð Þ¼ z 9216π<sup>6</sup>N<sup>2</sup> L2 c<sup>2</sup>λ<sup>2</sup> <sup>1</sup>n<sup>4</sup> 1 n2 <sup>1</sup>n<sup>2</sup> χð Þ<sup>3</sup> 4I � �<sup>2</sup> I 2 1I2sinc<sup>2</sup> <sup>Δ</sup>k4IL 2 � � þ 4 n1n2n<sup>3</sup> χð Þ<sup>3</sup> <sup>4</sup>II � �<sup>2</sup> <sup>I</sup>1I2I3sinc<sup>2</sup> <sup>Δ</sup>k4iIL 2 � � þ 1 n2n<sup>2</sup> 3 χð Þ<sup>3</sup> <sup>4</sup>III � �<sup>2</sup> I2I 2 3sinc<sup>2</sup> <sup>Δ</sup>k4IIIL 2 � � þ 2 ffiffiffiffiffiffiffiffiffi n1n<sup>3</sup> <sup>p</sup> <sup>n</sup>1n<sup>3</sup> χð Þ<sup>3</sup> <sup>4</sup><sup>I</sup> <sup>χ</sup>ð Þ<sup>3</sup> 4II ffiffiffiffiffiffiffiffi I1I<sup>3</sup> <sup>p</sup> <sup>I</sup>1I2sinc <sup>Δ</sup>k4IL 2 � � �sinc <sup>Δ</sup>k4IIL 2 � � cos Δk4IL <sup>2</sup> � <sup>Δ</sup>k4IIL <sup>2</sup> <sup>þ</sup> <sup>3</sup>φ<sup>1</sup> � <sup>φ</sup><sup>3</sup> � � þ 2 ffiffiffiffiffiffiffiffiffi n1n<sup>3</sup> <sup>p</sup> <sup>n</sup>2n<sup>3</sup> χð Þ<sup>3</sup> 4IIχð Þ<sup>3</sup> 4III ffiffiffiffiffiffiffiffi I1I<sup>3</sup> <sup>p</sup> <sup>I</sup>2I3sinc <sup>Δ</sup>k4IIL 2 � � �sinc <sup>Δ</sup>k4IIIL 2 � � � cos Δk4IIL <sup>2</sup> � <sup>Δ</sup>k4IIIL <sup>2</sup> � <sup>φ</sup><sup>1</sup> <sup>þ</sup> <sup>2</sup>φ<sup>2</sup> � <sup>φ</sup><sup>3</sup> � � þ 1 n1n2n<sup>3</sup> χð Þ<sup>3</sup> 4IIIχð Þ<sup>3</sup> <sup>4</sup><sup>I</sup> <sup>I</sup>1I2I3sinc <sup>Δ</sup>k4IIIL 2 � � �sinc <sup>Δ</sup>k4IL 2 � � cos Δk4IIIL <sup>2</sup> � <sup>Δ</sup>k4IL <sup>2</sup> � <sup>2</sup>φ<sup>1</sup> � <sup>2</sup>φ<sup>2</sup> <sup>þ</sup> <sup>2</sup>φ<sup>3</sup> � � 8 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>: 9 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>= >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>; (7)

In Eq. (7), the first, second and third terms are the three FWM processes, I, II and III, respectively. The last three terms are cross-terms due the interference of the optical fields generated by FWM processes I and II, II and III and III and I, in that order.

In the simulation, we used three-color laser fields (the fundamental, second harmonic, and third harmonic of the Nd:YAG laser) to generate fourth-harmonic signal in gaseous argon. For simplicity, we further assumed that the phase mismatch for all of FWM processes is equal and negligible. Further, the fundamental and second-harmonic power is the same and their sum is normalized.

In Figure 1 we show the fourth-harmonic signal as function of the power ratio of the fundamental beam and that of the fundamental and second harmonic combined. The third-harmonic beam is held constant. Examining Figure 1, one can see clearly that much higher conversion efficiency of the fourth-harmonic signal would be generated if the normalized power ratio is around 0.8.

The dependence of the fourth-harmonic signal on the phase of the fundamental beam is shown

Figure 1. The fourth-harmonic signal versus the power ratio P1/P1 + P2. Contributions by the three FWM processes and

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

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

157

the cross-terms are shown as different colors.

in Figure 2. Clearly, the modulation is more complex than the two-color case.

Figure 2. The dependence of the fourth-harmonic signal on the phase of the fundamental beam.

Frequency-Synthesized Approach to High-Power Attosecond Pulse Generation and Applications: Applications http://dx.doi.org/10.5772/intechopen.78270 157

With the phase or wave-vector mismatch given by Δk4<sup>I</sup> ¼ k<sup>4</sup> � 2k<sup>1</sup> � k2, Δk4II ¼ k4þ k<sup>1</sup> � k<sup>2</sup> � k<sup>3</sup> and Δk4III ¼ k<sup>4</sup> þ k<sup>2</sup> � 2k3. In this section, the symbol "I","II", "III" represent the three possible four-wave mixing (FWM) processes with corresponding nonlinear susceptibilities: <sup>χ</sup>(3)4I <sup>=</sup> <sup>χ</sup>(3)(ω4; <sup>ω</sup>1, <sup>ω</sup>1, <sup>ω</sup>2), <sup>χ</sup>(3)4II <sup>=</sup> <sup>χ</sup>(3)(ω4; �ω1, <sup>ω</sup>2, <sup>ω</sup>3), <sup>χ</sup>(3)4III <sup>=</sup> <sup>χ</sup>(3)(ω4; �ω2, <sup>ω</sup>3, <sup>ω</sup>3), respectively. L stands for the nonlinear medium length. The intensity of the fourth-harmonic

<sup>E</sup>~4Ið Þþ <sup>z</sup>; <sup>t</sup> <sup>E</sup>~4IIð Þþ <sup>z</sup>; <sup>t</sup> <sup>E</sup>~4IIIð Þ <sup>z</sup>; <sup>t</sup>

1I2sinc<sup>2</sup> <sup>Δ</sup>k4IL

<sup>2</sup> � �

2 � �

<sup>I</sup>1I2I3sinc<sup>2</sup> <sup>Δ</sup>k4iIL

3sinc<sup>2</sup> <sup>Δ</sup>k4IIIL 2 � �

> ffiffiffiffiffiffiffiffi I1I<sup>3</sup>

Δk4IL

ffiffiffiffiffiffiffiffi I1I<sup>3</sup>

<sup>4</sup><sup>I</sup> <sup>I</sup>1I2I3sinc <sup>Δ</sup>k4IIIL

Δk4IIIL

Δk4IIL

2 � �

<sup>p</sup> <sup>I</sup>1I2sinc <sup>Δ</sup>k4IL

<sup>p</sup> <sup>I</sup>2I3sinc <sup>Δ</sup>k4IIL

2 � �

<sup>2</sup> � <sup>Δ</sup>k4IL

<sup>2</sup> � <sup>Δ</sup>k4IIIL

� �

<sup>2</sup> � <sup>Δ</sup>k4IIL

2 � �

<sup>2</sup> <sup>þ</sup> <sup>3</sup>φ<sup>1</sup> � <sup>φ</sup><sup>3</sup>

2 � �

� �

� �

<sup>2</sup> � <sup>φ</sup><sup>1</sup> <sup>þ</sup> <sup>2</sup>φ<sup>2</sup> � <sup>φ</sup><sup>3</sup>

<sup>2</sup> � <sup>2</sup>φ<sup>1</sup> � <sup>2</sup>φ<sup>2</sup> <sup>þ</sup> <sup>2</sup>φ<sup>3</sup>

� � �

(6)

(7)

9

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

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

signal can then be written as

156 High Power Laser Systems

I4ð Þ¼ z

9216π<sup>6</sup>N<sup>2</sup>

c<sup>2</sup>λ<sup>2</sup> <sup>1</sup>n<sup>4</sup> L2

or

I4ð Þ¼ z

8

cn<sup>4</sup> 8π

1 n2 <sup>1</sup>n<sup>2</sup>

þ 4 n1n2n<sup>3</sup>

þ 1 n2n<sup>2</sup> 3

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

þ

þ

þ 1 n1n2n<sup>3</sup>

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

� � �

χð Þ<sup>3</sup> 4I � �<sup>2</sup> I 2

> χð Þ<sup>3</sup> 4II � �<sup>2</sup>

> > I2I 2

χð Þ<sup>3</sup> <sup>4</sup><sup>I</sup> <sup>χ</sup>ð Þ<sup>3</sup> 4II

χð Þ<sup>3</sup> 4IIχð Þ<sup>3</sup> 4III

cos

� cos

cos

In Eq. (7), the first, second and third terms are the three FWM processes, I, II and III, respectively. The last three terms are cross-terms due the interference of the optical fields generated

In the simulation, we used three-color laser fields (the fundamental, second harmonic, and third harmonic of the Nd:YAG laser) to generate fourth-harmonic signal in gaseous argon. For simplicity, we further assumed that the phase mismatch for all of FWM processes is equal and negligible. Further, the fundamental and second-harmonic power is the same and their sum is normalized. In Figure 1 we show the fourth-harmonic signal as function of the power ratio of the fundamental beam and that of the fundamental and second harmonic combined. The third-harmonic beam is held constant. Examining Figure 1, one can see clearly that much higher conversion efficiency of the fourth-harmonic signal would be generated if the normalized power ratio is around 0.8.

χð Þ<sup>3</sup> 4III � �<sup>2</sup>

2 ffiffiffiffiffiffiffiffiffi n1n<sup>3</sup> <sup>p</sup> <sup>n</sup>1n<sup>3</sup>

�sinc <sup>Δ</sup>k4IIL 2 � �

> 2 ffiffiffiffiffiffiffiffiffi n1n<sup>3</sup> <sup>p</sup> <sup>n</sup>2n<sup>3</sup>

�sinc <sup>Δ</sup>k4IIIL 2 � �

�sinc <sup>Δ</sup>k4IL 2 � �

by FWM processes I and II, II and III and III and I, in that order.

χð Þ<sup>3</sup> 4IIIχð Þ<sup>3</sup>

Figure 1. The fourth-harmonic signal versus the power ratio P1/P1 + P2. Contributions by the three FWM processes and the cross-terms are shown as different colors.

Figure 2. The dependence of the fourth-harmonic signal on the phase of the fundamental beam.

The dependence of the fourth-harmonic signal on the phase of the fundamental beam is shown in Figure 2. Clearly, the modulation is more complex than the two-color case.
