2.3. Role of the fundamental frequency

In frequency-domain description of the pulse train, mode spacing or frequency difference of adjacent modes dictates pulse spacing in time or pulse repetition rate in frequency [13]. Therefore, lower-frequency components of the pulse decide the main structure of the pulse waveform while higher-frequency components provide the fine structure or details of the pulse. To illustrate, Figure 3 shows a square wave synthesized by 10 modes with frequencies from the fundamental to 10th harmonic of the pump laser. Severe distortion is observed if the amplitude of the fundamental is attenuated merely by 10% (see Figure 3(b)). This reveals the significance of the component at the fundamental frequency. Significantly, high-quality square Frequency-Synthesized Approach to High-Power Attosecond Pulse Generation and Applications: Generation… http://dx.doi.org/10.5772/intechopen.78269 141

ϕ<sup>0</sup> = ϕ<sup>3</sup> 3ϕ<sup>m</sup> = 4ϕ<sup>3</sup> 3ϕ<sup>4</sup> are random in time as well. Although CEP of generated pulse trains with 1 ns pulse every 33 ms is fixed. The CEP of attosecond pulses in different ns pulse envelopes varies randomly as well. This severely limits the application of this type of attosecond light source. Cross-correlation by four-wave-mixing interaction among attosecond pulses within the same ns pulse, which are commensurate. Therefore, correlation behavior

Alternatively, CEP will be fixed if all phase-controlled frequency components of the pulse train are optical harmonics from the same laser, rather than through Raman sideband generation. It is clear that the relative phase among generated higher-order harmonics and the lower ones are fixed. For example, relative phase among ϕ5, ϕ<sup>2</sup> and ϕ<sup>3</sup> of the fifth, second and third harmonic of the same laser will not be changing, if light of frequency ω<sup>5</sup> is generated from

At this junction, it is instructive to note that Hansch proposed that sub-femtosecond pulse could be synthesized by nonlinear phase locking of lasers nearly a decade ago [16]. Later, his group further demonstrated the feasibility of this approach with three cw phase-locked semiconductor lasers [17]. This approach, however, was not pursued since primarily because of the

In the following, we summarize the potential advantages and unique features of attosecond pulse generation through pulse synthesis of harmonics of the same laser in contrast to the

This is expected since all the wavelength conversion processes for generation of the harmonics up to the near UV are from second order nonlinearity, instead of third order nonlinearity in the

Only one pump laser is required in the present scheme (see Figure 2(b)) rather than two in the

In frequency-domain description of the pulse train, mode spacing or frequency difference of adjacent modes dictates pulse spacing in time or pulse repetition rate in frequency [13]. Therefore, lower-frequency components of the pulse decide the main structure of the pulse waveform while higher-frequency components provide the fine structure or details of the pulse. To illustrate, Figure 3 shows a square wave synthesized by 10 modes with frequencies from the fundamental to 10th harmonic of the pump laser. Severe distortion is observed if the amplitude of the fundamental is attenuated merely by 10% (see Figure 3(b)). This reveals the significance of the component at the fundamental frequency. Significantly, high-quality square

could still be observed.

140 High Power Laser Systems

SFG of ω<sup>2</sup> and ω3.

low power generated.

2.1. Higher efficiency

Raman sideband approach.

Raman sideband approach.

2.2. Simplicity and compactness

2.3. Role of the fundamental frequency

earlier Raman sideband approach:

Figure 2. (a) Schematic of the Raman sideband generation approach (b): Schematic of the cascaded harmonics approach A Q-switched Nd:YAG laser and its harmonics up to the fifth order is used as the laser source.

Figure 3. (a) Synthesis of a square wave with modes at frequency from the fundamental to the 10th harmonic of the laser output. (b) Same as (a) except that the amplitude of the mode at the fundamental frequency is attenuated by 10%.

waveform can already be synthesized with five frequency components from the fundamental frequency to fifth harmonic frequency.

#### 2.4. Bandwidth

Compared to the Raman sideband approach, shortest pulse duration generated by the present cascaded harmonics synthesis method is inevitably limited since fewer numbers of channels are available in practice. However, this does not severely limit the application of the latter in generating attosecond pulses. A pulse 0.6 fs in duration could be obtained by synthesis of the fundamental wavelength of 1064 nm and its second, third, fourth and fifth harmonics. This can be understood by realizing that bandwidth is independent of the number of channels physically. These five components already span sufficient bandwidths.

### 2.5. Pulse quality

Arbitrary waveform synthesis is of importance for attosecond science. As an example, we show the synthesis of a Gaussian pulse with various numbers of frequency components or channels. The image-quality-index, which is widely used in image pattern recognition, is used to gauge the quality of the shaped pulse [18]. The quality increases step-wise only when number of channels is equal to 2n (n is an integer), that is, 2, 4, 8, 16…). This implies that we do not have to put too much effort into generating the sixth and seventh harmonic unless the 8th harmonic (133 nm) can be generated efficiently as well (Figure 4).

In our lab, we have constructed a system for the demonstration of attosecond pulse generation by synthesis of cascaded harmonics. This is shown in Figure 6. The fundamental frequency component at ω<sup>1</sup> is from a custom-made injection-seeded Quanta Ray PRO- 290 Q-switched Nd:YAG laser (λ = 1064 nm) operating at 10 Hz. The pulse duration is about 10 ns. The laser

Figure 5. (a) Synthesis of a sawtooth waveform with various numbers of channels (b) the perfect sawtooth wave, those synthesized with 4 channels (from the fundamental to the fourth harmonics) and 32 channels (from the fundamental to

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

2nd through the 5th harmonics of the laser fundamental beam are arranged in a cascaded layout. The crystals are KD\*P type II for the second harmonic ω2, KD\*P type I for the third harmonic ω3, BaB2O4 (BBO) type I for the fourth harmonic ω4, and BBO type I for the fifth harmonic ω5. Thus the five-color output of the laser system covers optical spectra from the near infrared (NIR) or 1064 nm to the ultraviolet (UV), that is, 213 nm. The cascade setup was adopted to ensure that the second-order nonlinear optical process all occurred collinearly. As a result, the fundamental and harmonics overlapped spatially. The pulse energy of each harmonic was 380, 178, 70, 41, and 22 mJ, respectively. The polarizations of the five colors were elliptic, horizontal, vertical, vertical, and horizontal for the fundamental through the fifth harmonic in that order. Eventually, all five

Precision control of the amplitude and the phase of each frequency components are essential. To this end, we first spatially dispersed the five colors by a fused silica prism. The dispersed beams were then recollimated but spatially separated by using another, larger fused silica prism. In the parallel co-propagating region of the five colors, we inserted an amplitude modulator and a phase modulator each. Therefore, it is possible to adjust the amplitude and relative phase of these harmonics separately. Each amplitude modulator was the assembly of a half waveplate and a polarizer. The polarization directions of the harmonic frequencies were all horizontal after passing through the polarizers. We can adjust the pulse energy of each harmonic by rotating the orientation of the half waveplate. To deal with the elliptical polarization of the fundamental frequency of the laser, we used a quarter waveplate to rotate the

. The nonlinear optical crystals for generating

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143

frequency bandwidth is narrower than 0.003 cm<sup>1</sup>

the 32th harmonics).

colors will be converted into horizontally polarized light (see below).

elliptical polarization back to the linear polarization.

As a further example, we show the synthesis of a sawtooth waveform with various numbers of channels (see Figure 5(a)). Figure 5(a) also indicates that a quality factor of 92% could be achieved already with 4 channels. The perfect sawtooth wave, those synthesized with 4 channels (from the fundamental to the fourth harmonics) and 32 channels (from the fundamental to the 32nd harmonics) are illustrated in Figure 5(b). The sawtooth waveform synthesized with 4 channels is already recognizable, while that generated with 32 channels is indistinguishable from the mathematical function.

Figure 4. Image quality index of a Gaussian pulse generated with different number of channels.

Frequency-Synthesized Approach to High-Power Attosecond Pulse Generation and Applications: Generation… http://dx.doi.org/10.5772/intechopen.78269 143

waveform can already be synthesized with five frequency components from the fundamental

Compared to the Raman sideband approach, shortest pulse duration generated by the present cascaded harmonics synthesis method is inevitably limited since fewer numbers of channels are available in practice. However, this does not severely limit the application of the latter in generating attosecond pulses. A pulse 0.6 fs in duration could be obtained by synthesis of the fundamental wavelength of 1064 nm and its second, third, fourth and fifth harmonics. This can be understood by realizing that bandwidth is independent of the number of channels physi-

Arbitrary waveform synthesis is of importance for attosecond science. As an example, we show the synthesis of a Gaussian pulse with various numbers of frequency components or channels. The image-quality-index, which is widely used in image pattern recognition, is used to gauge the quality of the shaped pulse [18]. The quality increases step-wise only when number of channels is equal to 2n (n is an integer), that is, 2, 4, 8, 16…). This implies that we do not have to put too much effort into generating the sixth and seventh harmonic unless the

As a further example, we show the synthesis of a sawtooth waveform with various numbers of channels (see Figure 5(a)). Figure 5(a) also indicates that a quality factor of 92% could be achieved already with 4 channels. The perfect sawtooth wave, those synthesized with 4 channels (from the fundamental to the fourth harmonics) and 32 channels (from the fundamental to the 32nd harmonics) are illustrated in Figure 5(b). The sawtooth waveform synthesized with 4 channels is already recognizable, while that generated with 32 channels is

cally. These five components already span sufficient bandwidths.

8th harmonic (133 nm) can be generated efficiently as well (Figure 4).

Figure 4. Image quality index of a Gaussian pulse generated with different number of channels.

indistinguishable from the mathematical function.

frequency to fifth harmonic frequency.

2.4. Bandwidth

142 High Power Laser Systems

2.5. Pulse quality

Figure 5. (a) Synthesis of a sawtooth waveform with various numbers of channels (b) the perfect sawtooth wave, those synthesized with 4 channels (from the fundamental to the fourth harmonics) and 32 channels (from the fundamental to the 32th harmonics).

In our lab, we have constructed a system for the demonstration of attosecond pulse generation by synthesis of cascaded harmonics. This is shown in Figure 6. The fundamental frequency component at ω<sup>1</sup> is from a custom-made injection-seeded Quanta Ray PRO- 290 Q-switched Nd:YAG laser (λ = 1064 nm) operating at 10 Hz. The pulse duration is about 10 ns. The laser frequency bandwidth is narrower than 0.003 cm<sup>1</sup> . The nonlinear optical crystals for generating 2nd through the 5th harmonics of the laser fundamental beam are arranged in a cascaded layout. The crystals are KD\*P type II for the second harmonic ω2, KD\*P type I for the third harmonic ω3, BaB2O4 (BBO) type I for the fourth harmonic ω4, and BBO type I for the fifth harmonic ω5. Thus the five-color output of the laser system covers optical spectra from the near infrared (NIR) or 1064 nm to the ultraviolet (UV), that is, 213 nm. The cascade setup was adopted to ensure that the second-order nonlinear optical process all occurred collinearly. As a result, the fundamental and harmonics overlapped spatially. The pulse energy of each harmonic was 380, 178, 70, 41, and 22 mJ, respectively. The polarizations of the five colors were elliptic, horizontal, vertical, vertical, and horizontal for the fundamental through the fifth harmonic in that order. Eventually, all five colors will be converted into horizontally polarized light (see below).

Precision control of the amplitude and the phase of each frequency components are essential. To this end, we first spatially dispersed the five colors by a fused silica prism. The dispersed beams were then recollimated but spatially separated by using another, larger fused silica prism. In the parallel co-propagating region of the five colors, we inserted an amplitude modulator and a phase modulator each. Therefore, it is possible to adjust the amplitude and relative phase of these harmonics separately. Each amplitude modulator was the assembly of a half waveplate and a polarizer. The polarization directions of the harmonic frequencies were all horizontal after passing through the polarizers. We can adjust the pulse energy of each harmonic by rotating the orientation of the half waveplate. To deal with the elliptical polarization of the fundamental frequency of the laser, we used a quarter waveplate to rotate the elliptical polarization back to the linear polarization.

set of photodiodes. Every harmonic was heterodyned with a signal at the same frequency derived by optically summing two lower harmonics in a particular BBO crystal. Then the resulting interference signal can be used to calibrate the phase modulator and to align the phases of the harmonic frequencies. Since the polarization of the summed output is orthogonal to that of the harmonics, a polarizer is used to project the polarization of the two states onto a common axis in order to maximize the heterodyning signal. With five frequency components, four measurements are needed for determining of their relative phases. The setup of the

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

The flow chart of determining the relative phase of the second harmonic (532 nm) with respect to the fundamental by measuring the interference signal is shown in Figure 8. First, light from

(SFG) in a BBO crystal. By tuning the phase modulator inserted in the beam path of the light at ω<sup>3</sup> from the laser signal, we can introduce a phase difference Δϕ<sup>3</sup> between the harmonic from the laser system and light of the same frequency from the sum-frequency generation process.

<sup>355</sup> � ϕ<sup>355</sup> ¼ ϕ<sup>1064</sup> þ ϕ<sup>532</sup> þ

As the phase modulator is tuned, the interference signal shows the expected sinusoidal behavior (see Figure 9). After the relative phase changes over a few cycles, the scan is stopped (middle of Figure 9). This part of the interference record reflects the phase stability of the

The phase stability of the third-harmonic beam at 355 nm is 0.0407 π, while that of the second harmonic is 0.1103 π. It is possible to control the phase modulator such that the interference

Figure 7. The experimental setup for relative phase measurement. Four BBO crystals were used. Second, third, fourth and

π <sup>2</sup> � <sup>ϕ</sup><sup>355</sup>

ϕ<sup>355</sup> ¼ 3ϕ<sup>1064</sup> þ π � Δϕ<sup>532</sup> � Δϕ<sup>355</sup> (3)

<sup>0</sup> through the sum-frequency generation

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145

relative phase between each harmonic is shown as Figure 7.

the laser system at ω<sup>1</sup> and ω<sup>2</sup> generates the signal at ω<sup>3</sup>

Δϕ<sup>355</sup> ¼ ϕ<sup>0</sup>

system, as the phase and power of the harmonics do vary in practice.

fifth indicate the nonlinear crystals in the cascaded generation process.

For the case of 355 nm light (see Figure 9),

Or

Figure 6. First-generation NTHU Attosecond source based on frequency synthesis of cascaded in-line harmonics of a single-frequency Q-switched Nd:YAG laser. Amplitude and phase modulation of each of the harmonics are provided. Insets (a) and (b) show predicted and experimentally generated square waveforms.

Each phase modulator consisted of a pair of right-angle triangle prisms. Adjusting the relative positions of each prism in the pair independently along the direction of their hypotenuse will change the effective path length traveled by each harmonic. The phase of each harmonic wave will be altered by Δϕ<sup>i</sup> = 2π (nprism � nair)l/λi, where l is the relative displacement of the two prisms nprism, and nair are refractive indices of the prism and air, respectively. This scheme allows variation of the phase Δϕ<sup>i</sup> of the ith harmonic, i = 1–5, but will not affect the beam alignment. Finally, these five beams of fundamental output and cascaded harmonics of the Nd: YAG laser were recombined and collimated by another prism set at a symmetry position to the first prism set. The whole setup is similar to a 4-f imaging system.

## 3. Relative phase measurement

For waveform control and pulse synthesis, we need to determine the relative phase among the harmonics. This was accomplished as follows: First, the fifth harmonic was used as the reference. We then proceed to adjust the relative phase of all other four harmonic frequencies to the reference. Four type I BBO crystals were employed. These were cut at (a) θ = 22.9� and ϕ = 0� for 1064 nm + 1064 nm ! 532 nm, (b) θ = 31.3� and ϕ = 0� for 1064 nm + 532 nm ! 355 nm, (c) θ = 47.7� and ϕ = 0� for 532 nm + 532 nm ! 266 nm, and (d) θ = 51.2� and ϕ = 0� for 1064 nm + 266 nm ! 213 nm, respectively. The Nd:YAG laser harmonic frequencies and summed frequencies generated from the BBO crystal were then dispersed and detected by a set of photodiodes. Every harmonic was heterodyned with a signal at the same frequency derived by optically summing two lower harmonics in a particular BBO crystal. Then the resulting interference signal can be used to calibrate the phase modulator and to align the phases of the harmonic frequencies. Since the polarization of the summed output is orthogonal to that of the harmonics, a polarizer is used to project the polarization of the two states onto a common axis in order to maximize the heterodyning signal. With five frequency components, four measurements are needed for determining of their relative phases. The setup of the relative phase between each harmonic is shown as Figure 7.

The flow chart of determining the relative phase of the second harmonic (532 nm) with respect to the fundamental by measuring the interference signal is shown in Figure 8. First, light from the laser system at ω<sup>1</sup> and ω<sup>2</sup> generates the signal at ω<sup>3</sup> <sup>0</sup> through the sum-frequency generation (SFG) in a BBO crystal. By tuning the phase modulator inserted in the beam path of the light at ω<sup>3</sup> from the laser signal, we can introduce a phase difference Δϕ<sup>3</sup> between the harmonic from the laser system and light of the same frequency from the sum-frequency generation process. For the case of 355 nm light (see Figure 9),

$$
\Delta \phi\_{355} = \phi'\_{355} - \phi\_{355} = \phi\_{1064} + \phi\_{532} + \frac{\pi}{2} - \phi\_{355}
$$

Or

Each phase modulator consisted of a pair of right-angle triangle prisms. Adjusting the relative positions of each prism in the pair independently along the direction of their hypotenuse will change the effective path length traveled by each harmonic. The phase of each harmonic wave will be altered by Δϕ<sup>i</sup> = 2π (nprism � nair)l/λi, where l is the relative displacement of the two prisms nprism, and nair are refractive indices of the prism and air, respectively. This scheme allows variation of the phase Δϕ<sup>i</sup> of the ith harmonic, i = 1–5, but will not affect the beam alignment. Finally, these five beams of fundamental output and cascaded harmonics of the Nd: YAG laser were recombined and collimated by another prism set at a symmetry position to the

Figure 6. First-generation NTHU Attosecond source based on frequency synthesis of cascaded in-line harmonics of a single-frequency Q-switched Nd:YAG laser. Amplitude and phase modulation of each of the harmonics are provided.

For waveform control and pulse synthesis, we need to determine the relative phase among the harmonics. This was accomplished as follows: First, the fifth harmonic was used as the reference. We then proceed to adjust the relative phase of all other four harmonic frequencies to the reference. Four type I BBO crystals were employed. These were cut at (a) θ = 22.9� and ϕ = 0� for 1064 nm + 1064 nm ! 532 nm, (b) θ = 31.3� and ϕ = 0� for 1064 nm + 532 nm ! 355 nm, (c) θ = 47.7� and ϕ = 0� for 532 nm + 532 nm ! 266 nm, and (d) θ = 51.2� and ϕ = 0� for 1064 nm + 266 nm ! 213 nm, respectively. The Nd:YAG laser harmonic frequencies and summed frequencies generated from the BBO crystal were then dispersed and detected by a

first prism set. The whole setup is similar to a 4-f imaging system.

Insets (a) and (b) show predicted and experimentally generated square waveforms.

3. Relative phase measurement

144 High Power Laser Systems

$$
\phi\_{355} = 3\phi\_{1064} + \pi - \Delta\phi\_{532} - \Delta\phi\_{355} \tag{3}
$$

As the phase modulator is tuned, the interference signal shows the expected sinusoidal behavior (see Figure 9). After the relative phase changes over a few cycles, the scan is stopped (middle of Figure 9). This part of the interference record reflects the phase stability of the system, as the phase and power of the harmonics do vary in practice.

The phase stability of the third-harmonic beam at 355 nm is 0.0407 π, while that of the second harmonic is 0.1103 π. It is possible to control the phase modulator such that the interference

Figure 7. The experimental setup for relative phase measurement. Four BBO crystals were used. Second, third, fourth and fifth indicate the nonlinear crystals in the cascaded generation process.

4. Waveform synthesis and its measurement

diagnostics of multiwave synthesized waveforms.

for example, a mode-locked laser can be expressed as:

EaðÞ¼ <sup>t</sup> <sup>X</sup> N

EbðÞ¼ <sup>t</sup> <sup>X</sup> N

n¼1

ETð Þ¼ t; τ

time delay of τ. The time-averaged intensity of ET is then given by

<sup>T</sup>ð Þ <sup>t</sup>; <sup>τ</sup> dt <sup>¼</sup> <sup>1</sup>

delta function of unity amplitude, that is, an ¼ a0, phase ϕan<sup>ω</sup> ¼ 0,ϕaCEP ¼ 0

<sup>0</sup>þb<sup>2</sup> n 4 X n A2 n

<sup>n</sup> þ 2anbncos nωτ þ n ϕbn<sup>ω</sup> � ϕan<sup>ω</sup>

If the reference waveform is a transform-limited cosine pulse function of finite duration or a

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

a2 <sup>n</sup>þb<sup>2</sup>

ETð Þ <sup>t</sup>; <sup>τ</sup> <sup>E</sup><sup>∗</sup>

<sup>I</sup>ð Þ¼ <sup>τ</sup> <sup>X</sup> <sup>a</sup><sup>2</sup>

<sup>n</sup>þb<sup>2</sup>

ð Þ where An ¼

<sup>I</sup>ð Þ¼ <sup>τ</sup> <sup>1</sup> T ð

<sup>¼</sup> <sup>P</sup> <sup>a</sup><sup>2</sup>

n¼1

As the spectral bandwidth of this coherent laser source exceeds two octaves or 32,200 cm�<sup>1</sup>

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

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

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,

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

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

<sup>X</sup>Ane

<sup>n</sup> þ 2anbncos nωτ þ ϕbn<sup>ω</sup> � ϕan<sup>ω</sup>

<sup>ϕ</sup><sup>n</sup> <sup>¼</sup> cos �<sup>1</sup> ancos <sup>ϕ</sup>an<sup>ω</sup> <sup>þ</sup> <sup>ϕ</sup>aCEP � � <sup>þ</sup> bncos <sup>n</sup>ωτ <sup>þ</sup> <sup>ϕ</sup>bn<sup>ω</sup> <sup>þ</sup> <sup>ϕ</sup>bCEP � � � � <sup>=</sup>An

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� � <sup>þ</sup> <sup>ϕ</sup>bCEP � <sup>ϕ</sup>aCEP <sup>q</sup> � �

�� �

� � <sup>þ</sup> <sup>ϕ</sup>bCEP � <sup>ϕ</sup>aCEP � � � �

� � <sup>þ</sup> <sup>2</sup>a0bncos <sup>n</sup>ωτ <sup>þ</sup> <sup>ϕ</sup>bn<sup>ω</sup> <sup>þ</sup> <sup>ϕ</sup>bCEP � � (8)

1 2

ancos <sup>n</sup>ω<sup>t</sup> <sup>þ</sup> <sup>ϕ</sup>an<sup>ω</sup> <sup>þ</sup> <sup>ϕ</sup>aCEP � � (4)

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

bn cos nω<sup>t</sup> <sup>þ</sup> <sup>ϕ</sup>bn<sup>ω</sup> <sup>þ</sup> <sup>ϕ</sup>bCEP � �: (5)

i nð Þ <sup>ω</sup>tþϕn<sup>ω</sup> <sup>þ</sup> <sup>c</sup>:c: (6)

,

147

(7)

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

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.

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 of the synthesized waveform is π/2.
