Abstract
We present a new scheme of generating high-power attosecond pulses and arbitrary waveform synthesis by multicolor synthesis. The full bandwidth of the multicolor laser system extends more than two-octaves and reaches 37,600 cm−1 which can be used to generate sub-single-cycle (∼0.37 cycle) sub-femtosecond (360 attosecond) pulses with carrier-envelope phase (CEP) control. The results show a promising approach for generation of relatively high-power attosecond pulses in the optical region. In this chapter, the design and diagnostics of the laser system are described. In part 2 of this work (the following chapter), we demonstrate selected applications of this novel source, such as coherently controlled harmonic generation as well as phase-sensitive 2-color ablation of copper and stainless steel by this multi-color laser system.
Keywords
- arbitrary waveform synthesis
- harmonic generation
- cascaded harmonics
- laser ablation
- laser sources
- pulse generation
- multicolored
1. Introduction
Human history has taught us that the invention of novel light sources and related technologies would lead to breakthroughs in science and impact the society and civilization tremendously. X-rays and lasers are good examples of such technologies. High-power laser systems are a class of coherent light sources that play a major role in the advancement of science and technology, ranging from inertial nuclear fusion, laboratory astrophysics to laser weapons and 3D printing. Such lasers emit continuous wave (CW), nanosecond (ns), picosecond (ps) and femtosecond (fs) pulsed output. Single or near-single cycle electromagnetic radiation can now be generated by laser-based techniques from the terahertz (1 THz = 1012 Hz) to the soft x-ray regions of the spectrum. The spectra of the latter yield attosecond (1 as = 10–18 s) pulses. Such novel sources are expected to have a wide range of potential applications. Attosecond sources [1, 2] are perhaps among the most exciting new laser sources currently under development. In the near future, controlled light wave can steer electrons inside and around atoms. This emerging technology has been dubbed as “lightwave electronics” [3]. Nonetheless, study of condensed matter with attosecond time-resolution remains a challenge [4]. While the potential of microfabrication and nanostructuring of materials by ultrafast lasers were recognized and demonstrated more than a decade ago [5], there have not been reports of real-world applications of attosecond pulses to date. Primarily, this is limited by the lack of powerful attosecond sources.
Among the approaches that allow generation of attosecond pulses, the high-order harmonic generation (HHG) [6] seems to be the most promising one. HHG can serve as a source of intense attosecond pulses that extending from the Vacuum Ultraviolet (VUV) or extreme ultraviolet (EUV) to the soft X-ray region [7]. Alternatively, Chen et al. [8] and Hsieh et al. [9] showed that carrier-envelope-phase (CEP) controlled sub-cycle pulse train can be generated by high-order stimulated Raman scattering (HSRS) process. Recently, we demonstrated the generation of attosecond pulses through pulse synthesis of harmonics of the same laser up to the fifth order. These harmonics were generated through second-order nonlinear optical processes, that is, second harmonic generation (SHG) or sum frequency generation (SFG). This novel source is able to generate sub-single-cycle (∼0.37 cycle) pulses with peak intensity of a single pulse as high as 1014W/cm2, pulse width as short as 400 attosecond with carrier-envelope-phase (CEP) control [10]. Waveform (purple trace) and intensity (red traces) of such ultrashort pulses are shown in Figure 1.
We further show that the relative phase among the optical fields of the harmonics can be maintained a constant at least for thousands of nanosecond pulses. The worst-case relative phase fluctuation is 0.04 π rad. It is shown that sub-femtosecond (360 attosecond) pulses with carrier-envelope phase (CEP) control can be generated in this manner. Synthesis of arbitrary waveforms, for example, square and sawtooth waveforms are possible [11].
Compared to the mainstream method of generating attosecond pulses by higher-harmonic generation (HHG) of few-cycle femtosecond pulses, this novel light source has advantages of compactness and simplicity. Further, arbitrary optical waveform can be synthesized while the attosecond pulse generated in this way is sub-single-cycle with full CEP control. In Section 2 of the chapter, we describe the basic principle for generation of attosecond pulses by synthesis of cascaded harmonics. The prototype system is described. The diagnostics of such broadband sources is nontrivial. Experimental methods for relative phase control among the harmonics are presented in Section 3. This is followed by a review of the synthesis of arbitrary waveforms and their diagnostics by the linear cross-correlation method. Finally, we summarize in Section 5 of the chapter. Applications of this novel high-power laser system can be found in part 2 of this work (the following chapter), in which we discuss coherently controlled harmonic generation [12] as well as phase-sensitive 2-color ablation of copper and stainless steel by this multi-colour laser system.
2. Generation of attosecond pulses by synthesis of cascaded harmonics
Fundamentally, an optical pulse train with a repetition rate of ωm can be viewed as the sum of a set of frequency components that form an arithmetic series [13]. The electric field of each component can be written in the following form:
where
where
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
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 low power generated.
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 earlier Raman sideband approach:
2.1. Higher efficiency
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 Raman sideband approach.
2.2. Simplicity and compactness
Only one pump laser is required in the present scheme (see Figure 2(b)) rather than two in the Raman sideband approach.
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 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).
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.
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
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.
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 Δ
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)
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 ω1 and ω2 generates the signal at ω3′ through the sum-frequency generation (SFG) in a BBO crystal. By tuning the phase modulator inserted in the beam path of the light at ω3 from the laser signal, we can introduce a phase difference Δ
Or
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 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
4. Waveform synthesis and its measurement
As the spectral bandwidth of this coherent laser source exceeds two octaves or 32,200 cm−1, conventional methods for ultrafast waveform synthesis is not adequate. We used the shaper-assisted 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:
where
The interference of the two with a relative temporal delay
If the reference waveform is a transform-limited cosine pulse function of finite duration or a delta function of unity amplitude, that is,
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
where
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.
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
5. Summary
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 Q-switched 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 co-propagate to the far fields. The spectral bandwidth of this coherent laser source thus exceeds two octaves or 32,200 cm−1.The amplitude and phase of the comb consisting of the five frequency components can be independently controlled. Sub-single-cycle (∼ 0.37 cycle) sub-femtosecond (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.
Acknowledgments
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.
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