4. Waveform synthesis and its measurement

As the spectral bandwidth of this coherent laser source exceeds two octaves or 32,200 cm�<sup>1</sup> , conventional methods for ultrafast waveform synthesis is not adequate. We used the shaperassisted linear correlation method [19] for such a task. This method is particularly suited for diagnostics of multiwave synthesized waveforms.

The basic concept is the use of an effective delta function waveform to retrieve the waveform. To begin with, the output electric field of a coherent multiwave synthesized optical waveform, for example, a mode-locked laser can be expressed as:

$$E\_{\mathfrak{a}}(t) = \sum\_{n=1}^{N} a\_n \cos \left( n\omega t + \phi\_{an\omega} + \phi\_{a\ominus EP} \right) \tag{4}$$

where an and ϕan<sup>ω</sup> are the amplitude and phase of each component at the frequency nω, n is a positive integer. ϕaCEP is the carrier envelope phase. Considering two such waveforms, one is the reference with field Eað Þt above and the target waveform with field Ebð Þt , given by

$$E\_b(t) = \sum\_{n=1}^{N} b\_n \cos\left(n\omega t + \phi\_{bm\omega} + \phi\_{b\angle EP}\right). \tag{5}$$

The interference of the two with a relative temporal delay τcan be described as follows:

$$Er(t, \tau) = \frac{1}{2} \sum A\_n e^{j\left(n\omega t + \phi\_m\right)} + c.c.\tag{6}$$

$$\text{(where) } A\_n = \sqrt{a\_{n+}^2 b\_n^2 + 2a\_n b\_n \cos\left(n\omega \tau + \left(\phi\_{bm\nu} - \phi\_{am\nu}\right) + \phi\_{b\subset EP} - \phi\_{a\subset EP}\right)}$$

$$\phi\_n = \cos^{-1}\left[\left(\left(a\_n \cos\left(\phi\_{am\nu} + \phi\_{a\subset EP}\right) + b\_n \cos\left(n\omega \tau + \phi\_{bm\nu} + \phi\_{b\subset EP}\right)\right)/A\_n\right]$$

An and ϕ<sup>n</sup><sup>ω</sup> are the amplitude and phase of the nth Fourier component of the interference signal. The linear cross-correlation function of the reference and target signals with a relative time delay of τ. The time-averaged intensity of ET is then given by

$$\begin{split} \mathcal{I}(\mathbf{r}) &= \frac{1}{T} \Big[ E\_T(t, \mathbf{r}) E\_T^\*(t, \mathbf{r}) dt = \frac{1}{4} \sum\_n A\_n^2 \\ &= \sum \left( a\_{n+}^2 b\_n^2 + 2a\_n b\_n \cos \left( n\omega \tau + n \left( \phi\_{bm\omega} - \phi\_{am\omega} \right) + \phi\_{b\subset EP} - \phi\_{a\subset EP} \right) \right) \end{split} \tag{7}$$

If the reference waveform is a transform-limited cosine pulse function of finite duration or a delta function of unity amplitude, that is, an ¼ a0, phase ϕan<sup>ω</sup> ¼ 0,ϕaCEP ¼ 0

signal is maintained at certain level, for example, the half, the maximum, and the minimum of the magnitude of the interference signal. For example, we fixed Δϕ<sup>532</sup> = π∕2. When the interference signal at 355 nm is at half of the maximum intensity, the phase difference Δϕ<sup>355</sup> is 0.5π. According to Eq. (3), the relative phase relationship is ϕ<sup>355</sup> ¼ 3ϕ<sup>1064</sup> which is the phasematching condition. The carrier envelope phase of the synthesized wave or CEP is zero. Similarly, if we set the phase difference Δϕ<sup>355</sup> to be 0. ϕ<sup>355</sup> ¼ 3ϕ<sup>1064</sup> þ π∕2. Therefore, the CEP

Figure 9. The relative phase between fundamental and the third harmonic is determined as shown on the right of the

figure. Left of the signal shows the experimentally measured interference signal. CEP: carrier envelope phase.

Figure 8. The flow chart for measuring the relative phase through the interference signal.

146 High Power Laser Systems

of the synthesized waveform is π/2.

$$\mathcal{I}(\pi) = \sum \left( a\_{0+}^2 b\_n^2 \right) + 2a\_0 b\_n \cos \left( n\omega \tau + \phi\_{bm\omega} + \phi\_{b\subset\text{EP}} \right) \tag{8}$$

That is, the time-varying part of I(τ) is directly proportional to the target field, Eb(t) (see Eq. (6)). If the reference pulse and target one are delta and square pulse, Eq. (5) can be written as

$$\begin{split} E(t,\tau) &= E\_{\delta}(t) + E\_{\text{square}}(t,\tau) \\ &= A\_{\delta} \sum\_{n} e^{i(\omega\_{n}t - k\_{a}d)} + B\_{\text{squ}} \sum\_{n=1,3,5,\dots} \frac{2}{n\tau} e^{i\left(\omega\_{n}(t-\tau) - k\_{a}d - \frac{n}{2}\right)} \\ &= \sum\_{n} \mathbf{e}^{i\omega\_{n}t} A'\_{n}(\tau) e^{i\boldsymbol{\wp}'\_{n}(\tau)} \end{split} \tag{9}$$

where

$$A\_{n}^{'} = \sqrt{A\_{\delta}^{2} + \left(\frac{B\_{s\eta}}{n}\right)^{2} + 2A\_{\delta}\frac{B\_{sq\eta}}{n}\cos\left(\omega\_{n}\tau - \frac{\pi}{2}\right)},$$

$$\varphi\_{n}^{'} = \tan^{-1}\left[\frac{-A\_{\delta}\sin(k\_{n}d) + \frac{B\_{sq\eta}}{n}\sin\left(\omega\_{n}\tau - \frac{\pi}{2} - k\_{n}d\right)}{A\_{\delta}\cos(k\_{n}d) + \frac{B\_{sq\eta}}{n}\cos\left(\omega\_{n}\tau - \frac{\pi}{2} - k\_{n}d\right)}\right] \qquad \text{for } \mathbf{n} = 1, 3, 5\ldots \tag{11}$$

$$A\_{\mathbf{n}}^{'} = A\_{\delta\nu} \quad \varphi\_{\mathbf{n}}^{'} = -k\_{\delta}d \qquad \text{for } \mathbf{n} = 2, 4, 6\ldots \tag{12}$$

Our laser system can generate fundamental through the fifth harmonics with pulse energies of 380, 178, 70, 41, and 22 mJ. If these can be fully utilized, the synthesized transform-limited pulse will exhibit a temporal FWHM of 480 attoseconds. The intensity envelope will be just 700 attoseconds. The intensity of each attosecond pulse will exceed 1014 W/cm2 when it is focused to a spot size of 20 μm. Such high-power pulses would induce interesting nonlinear effect in materials. Opportunities in novel laser processing should arise. These will be discussed in the

Figure 11. (a) Synthesis of a square waveform with the fundamental, second and third harmonics of the Nd:YAG laser. (b) Synthesis of a sawtooth waveform with the fundamental, second, third and fourth harmonics of the Nd:YAG laser. The solid squares are experimental data. The blue curves are theoretical curves. (reproduced with permission from [11]).

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

http://dx.doi.org/10.5772/intechopen.78269

149

We proposed and demonstrated a new high-power attosecond light source by frequency synthesis. The laser system consists of a narrow-band transform-limited high-power Qswitched Nd:YAG laser and its second (λ = 532 nm) through fifth harmonics, (λ = 213 nm). The laser system was designed such that the cascaded harmonics spatially overlap and copropagate to the far fields. The spectral bandwidth of this coherent laser source thus exceeds

frequency components can be independently controlled. Sub-single-cycle ( 0.37 cycle) subfemtosecond (360 attosecond) pulses with carrier-envelope phase (CEP) control can be generated in this manner. The peak intensity of each pulse exceeds 1014 W/cm2 with a focused spot size of 20 μm. It is also possible to synthesize arbitrary optical waveforms, for example, a square wave. The synthesized waveform is stable at least for thousands of nanosecond.

This work was supported by grants sponsored by the National Science Council of Taiwan (NSC 98-2112-M-009-015-MY3) and Phase II of the Academic Top University Program of the

.The amplitude and phase of the comb consisting of the five

part II of this work.

two octaves or 32,200 cm<sup>1</sup>

Acknowledgements

Ministry of Education, Taiwan.

5. Summary

The linear cross-correlation measurement can be performed using any interferometric arrangement, for example, a Michelson interferometer. Equivalently, it can be conducted by adjusting the amplitudes and phases of the frequency components of the waveform. The experimental setup is shown in Figure 10. A thermal pile power meter, which can detect light from the fundamental (λ = 1064 nm) to the fifth harmonic (λ = 213 nm) of the laser system.

We have shown previously that it is possible to synthesize attosecond pulse train and arbitrary waveforms using this approach [11]. For example, Figure 11(a) shows the synthesized square waveform by the fundamental, second and third harmonics of the Nd:YAG laser. The normalized amplitudes of the harmonics are respectively, 1, 0 and 1/3. Figure 11(b) shows the synthesized sawtooth waveform by the fundamental through the fourth harmonics of the Nd: YAG laser. The normalized amplitudes of the fundamental and harmonics are respectively, 1, 1/2, 1/3 and 1/4. The measured waveforms are in good agreement with the theoretical estimates (solid curves in Figure 11). Although we just used three of four waves in this experiment, the synthesized waveforms already reproduce these familiar mathematical functions.

Figure 10. The experimental arrangement for linear cross-correlation measurement of the synthesized waveform. Second, third, fourth and fifth indicate the nonlinear crystals that generated the cascaded harmonics.

Frequency-Synthesized Approach to High-Power Attosecond Pulse Generation and Applications: Generation… http://dx.doi.org/10.5772/intechopen.78269 149

Figure 11. (a) Synthesis of a square waveform with the fundamental, second and third harmonics of the Nd:YAG laser. (b) Synthesis of a sawtooth waveform with the fundamental, second, third and fourth harmonics of the Nd:YAG laser. The solid squares are experimental data. The blue curves are theoretical curves. (reproduced with permission from [11]).

Our laser system can generate fundamental through the fifth harmonics with pulse energies of 380, 178, 70, 41, and 22 mJ. If these can be fully utilized, the synthesized transform-limited pulse will exhibit a temporal FWHM of 480 attoseconds. The intensity envelope will be just 700 attoseconds. The intensity of each attosecond pulse will exceed 1014 W/cm2 when it is focused to a spot size of 20 μm. Such high-power pulses would induce interesting nonlinear effect in materials. Opportunities in novel laser processing should arise. These will be discussed in the part II of this work.

### 5. Summary

E tð Þ¼ ; τ Eδð Þþ t Esquareð Þ t; τ

e<sup>i</sup>ωnt A0 <sup>n</sup>ð Þτ e iφ0 <sup>n</sup>ð Þ τ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>r</sup> � �

Bsqu n

> Bsqu n

Bsqu n

0

cos <sup>ω</sup>n<sup>τ</sup> � <sup>π</sup>

sin <sup>ω</sup>n<sup>τ</sup> � <sup>π</sup>

cos <sup>ω</sup>n<sup>τ</sup> � <sup>π</sup>

The linear cross-correlation measurement can be performed using any interferometric arrangement, for example, a Michelson interferometer. Equivalently, it can be conducted by adjusting the amplitudes and phases of the frequency components of the waveform. The experimental setup is shown in Figure 10. A thermal pile power meter, which can detect light from the fundamental (λ = 1064 nm) to the fifth harmonic (λ = 213 nm) of the

We have shown previously that it is possible to synthesize attosecond pulse train and arbitrary waveforms using this approach [11]. For example, Figure 11(a) shows the synthesized square waveform by the fundamental, second and third harmonics of the Nd:YAG laser. The normalized amplitudes of the harmonics are respectively, 1, 0 and 1/3. Figure 11(b) shows the synthesized sawtooth waveform by the fundamental through the fourth harmonics of the Nd: YAG laser. The normalized amplitudes of the fundamental and harmonics are respectively, 1, 1/2, 1/3 and 1/4. The measured waveforms are in good agreement with the theoretical estimates (solid curves in Figure 11). Although we just used three of four waves in this experiment, the synthesized waveforms already reproduce these familiar mathematical functions.

Figure 10. The experimental arrangement for linear cross-correlation measurement of the synthesized waveform. Second,

third, fourth and fifth indicate the nonlinear crystals that generated the cascaded harmonics.

2

,

<sup>2</sup> � knd � �

3 7

¼ �knd for n ¼ 2, 4, 6:… (11)

<sup>5</sup> for n <sup>¼</sup> <sup>1</sup>, <sup>3</sup>, <sup>5</sup>::

<sup>2</sup> � knd � �

þ 2A<sup>δ</sup>

�Aδsinð Þþ knd

Aδcosð Þþ knd

¼ Aδ, φ<sup>n</sup>

<sup>i</sup>ð Þ <sup>ω</sup>nt�knd <sup>þ</sup> Bsqu

X <sup>n</sup>¼<sup>1</sup>, <sup>3</sup>, <sup>5</sup>, ::

2 nπ e

<sup>i</sup> <sup>ω</sup>nð Þ� <sup>t</sup>�<sup>τ</sup> knd�<sup>π</sup> ð Þ<sup>2</sup>

(9)

(10)

¼ A<sup>δ</sup> X n e

<sup>¼</sup> <sup>X</sup> n

where

148 High Power Laser Systems

An 0 ¼

> φn 0

laser system.

Aδ

<sup>¼</sup> tan�<sup>1</sup>

<sup>2</sup> <sup>þ</sup> Bsqu n � �<sup>2</sup>

> An 0

2 6 4

> We proposed and demonstrated a new high-power attosecond light source by frequency synthesis. The laser system consists of a narrow-band transform-limited high-power Qswitched Nd:YAG laser and its second (λ = 532 nm) through fifth harmonics, (λ = 213 nm). The laser system was designed such that the cascaded harmonics spatially overlap and copropagate to the far fields. The spectral bandwidth of this coherent laser source thus exceeds two octaves or 32,200 cm<sup>1</sup> .The amplitude and phase of the comb consisting of the five frequency components can be independently controlled. Sub-single-cycle ( 0.37 cycle) subfemtosecond (360 attosecond) pulses with carrier-envelope phase (CEP) control can be generated in this manner. The peak intensity of each pulse exceeds 1014 W/cm2 with a focused spot size of 20 μm. It is also possible to synthesize arbitrary optical waveforms, for example, a square wave. The synthesized waveform is stable at least for thousands of nanosecond.

## Acknowledgements

This work was supported by grants sponsored by the National Science Council of Taiwan (NSC 98-2112-M-009-015-MY3) and Phase II of the Academic Top University Program of the Ministry of Education, Taiwan.
