Third-order nonlinear process (
In part I of this work, we present the design, construction and diagnostics of a new scheme of generating high-power attosecond pulses and arbitrary waveforms by multicolor synthesis. In this chapter, we demonstrate selected applications of this novel source, such as coherently controlled harmonic generation as well as phase-sensitive two-color ablation of copper and stainless steel by this multicolor laser system.
- arbitrary waveform synthesis
- coherent control
- perturbative nonlinear optics
- optical harmonic generation
- four-wave mixing
- laser ionization and plasma
- laser material processing
- laser ablation
In part I of this work , we reported a new high-power laser system for generating attosecond light pulses and arbitrary waveforms by frequency synthesis. The laser system can generate up to five amplitude and phase-controlled collinear beams with wavelengths from the fundamental output of the Nd:YAG laser (λ = 1064 nm) and its second (λ = 532 nm) through the fifth harmonic (λ = 213 nm). 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. Stable square and saw-tooth waveforms were also demonstrated .
The coherent control of nonlinear optical processes such as harmonic generation by waveform-controlled 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 . 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 of the above two processes.
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 case for fourth-harmonic generation (FHG) in Section 2.
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 show that hole drilling is more efficient for optimized waveforms.
2. Nonlinear frequency conversion by coherently controlled three-color excitation of inert gases
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 = 2ω1 and third harmonic ω3 = 3ω1) excitation, the fourth-harmonic signal can be generated by three nonlinear optical processes (ω1 + ω1 + ω2 = ω4, −ω1 + ω2 + ω3 = ω4, and −ω2 + ω3 + ω3 = 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, ω2 and ω3 vary, combinations of three sinusoidal modulation due to interference in the output intensity of the fourth harmonic at frequency is predicted. We will also show that the relative amplitude of the fundamental, second-harmonic and third-harmonic driving laser field influences the fourth-harmonic signal.
We assume plane-waves propagating in the +z direction. The three-color field can be represented as:
where , and 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
With the phase or wave-vector mismatch given by , and . In this section, the symbol “I”,“II”, “III” represent the three possible four-wave mixing (FWM) processes with corresponding nonlinear susceptibilities:
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 in Figure 2. Clearly, the modulation is more complex than the two-color case.
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 (ω3 = ω1 + ω1 + ω1) and four-wave mixing (FWM, ω3 = ω2 + ω2 – ω1) processes and a cross term of the two. As the relative phase between ω1 and ω2 varies, a sinusoidal modulation in output intensity at frequency ω3 is expected and was demonstrated in our previous work . 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. . More than two-order-of-magnitude increase of the efficiency of third-harmonic generation occurs due to the plasma-enhanced third-order susceptibility . 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
The experimental setup for studying the effect of plasma formation on generation of third-harmonic 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−1. 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 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 ω1 and second harmonic ω2) of the Nd:YAG laser, the third-harmonic signal can be generated by two optical processes, i.e., ω1 + ω1 + ω1 = ω3 and −ω1 + ω2 + ω2 = 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
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,
In Eq. (9),
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 ϕ1 = 0. Therefore, Δϕ = ϕ2 − 2ϕ1 = ϕ2 is the relative phase between ω1 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, , 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.
|Excitation Source||The modulation of phase or contrast (Normalized)||The fluctuation in TH power without phase delay (normalized)||Enhancement ratio (two-colour/one color)|
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
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 . That is,
The ionization probability of the atoms by the laser pulse can be calculated by solving the rate equation
This allows us to calculate the plasma density in terms of the density of the neutral gas.
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.
In the above two equations, the ratios γI,p and γII,p are values determined by the experiment. Here, we assume γI,p = γII,p = 410−49 and
The four-wave mixing process is dominant in the third-harmonic signal. For our experimental conditions, the THG component is approximately 10−4 that of the FWM process. In the low plasma density limit, the FWM term can be written as
Thus, in the high plasma density limit, the four-wave mixing term becomes
4. Laser-material processing with multi-color synthesized light field
Lately, high-energy laser beams have been increasingly been used for processing and fabrication of material and devices. These include the fabrication of micro electro mechanical systems, optoelectronic components, biomedical micro fluid chips and silicon chip processing, electronic packages and drilling of circuit boards, to name just a few.
There are two kinds of mechanisms occurring during laser processing of materials: a photo-thermal one and a photo-chemical one. In the photo-thermal mechanism, laser beams with 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 be advantageous for laser processing.
Ablation of materials with multiple lasers, for example, lasers with dual colors were reported recently [14, 15, 16, 17, 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 . 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  demonstrated that superimposed fs pulses at the fundamental (
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, 20, 21]. Schumacher and Bucksbaum  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.  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 . The observed phenomenon was attributed to the field-dependence of the ionization cross section. In this work, we focus on the ablation study of metals under phase-controlled dual color ns pulses with the relative delay between the two color being less than one oscillation period.
Results of preliminary experiments on drilling of copper and stainless steel with the multi-color laser system used in this work (see Figure 7). The nonlinear optical crystals for harmonic generation are arranged in a cascaded layout. The crystals are KD*P type II for the second harmonic,
We studied two-color laser ablation of cooper and steel to demonstrate the feasibility of the approach. In the plane-wave approximation, the synthesized dual-color laser field can be written as,
During the experiment, we applied the dual color ns pulses with the same total energy (∼100 mJ) to 150 μm thick copper and stainless steel foil. We adjusted the phase modulators only, so, varied the relative phase between the harmonics. Then we measured the time required to make a pass through hole in a foil and estimated the ablation rate. In Figure 9(a), we have plotted diameters of holes drilled in copper sheetsas a function of relative phases between the fundamental (
Figure 10(a)–(d) shows simulated results the peak strengths of the synthesized laser field with various relative phases (Δ
In another experiment, we fixed the exposure time at 10 s, varied the relative phase of the two-color beams and examined the ablated holes afterwards. In Figure 11(a), we have plotted diameters of holes drilled in copper sheets as a function of relative phases between the fundamental (
As an application of the high-power laser system based on synthesized waveforms, we studied harmonic generation by three-color waveform synthesis in inert gas systems. In third-order nonlinear optics, the interaction between three-color beam and inert gases can be used to generate fourth to ninth harmonics of the laser fundamental output. For fourth-harmonic generation, there are three kinds of four-wave mixing processes: ω4 = ωi + ωj + ωk, ω4 = ωi + ωj − ωk, where i, j, k = 1, 2, 3. For fifth-harmonic generation, there are three possible processes: ω5 = ω1 + ω2 + ω2, ω5 = ω1 + ω1 + ω3, ω5 = ω3 + ω3 − ω1. For the sixth and seventh harmonic, there are two kinds of four-wave mixing processes, and so on. To illustrate, we present in detail the simulation results for fourth-harmonic generation using three-order nonlinear processes. It is shown that the fourth-harmonic signal varies with the phase of the fundamental beam.
Previously, we studied the influence of relative phases and intensities of the two-color pump on the third-order nonlinear frequency conversion process. It is shown that the third-harmonic (TH) signal oscillates periodically with the relative phases of the two-color driving laser fields due to the interference of TH signals from a direct third-harmonic-generation (THG) channel and a four-wave mixing (FWM) channel. In intense laser field, however, plasma can be generated through the ionization process. In the multiphoton ionization region, the plasma density was estimated by the Perelomov, Popov, and Terent’ev (PPT) model where the instantaneous laser field and frequency of laser are taken into account. Under the assumption that susceptibility and wave-vector mismatch depend on the plasma density, we show that plasma plays a significant role in the generated third-harmonic signal. The simulation results are in good agreement with the experiments.
Finally, we showed preliminary data indicating that the synthesized two-color laser fields are powerful in enhancing the conversion efficiency of HHG and VUV spectra. We also demonstrated phase-sensitive two-color ablation of copper and stainless steel. Our results show that hole drilling is more efficient with the use of optimized waveforms.
This work was supported by grants sponsored by the National Science Council (now Ministry of Science and Technology or MoST) of Taiwan (NSC 98-2112-M-009-015-MY3) and Phase II of the Academic Top University Program of the Ministry of Education, Taiwan.
Pan C-L, Yang C-S, Zatazev A, Chen W-J, Lee C-K. Frequency-synthesized approach to high-power Attosecond pulse generation and applications (I). In: Harooni M, editor. High Power Laser Systems. Rijeka, Croatia: InTech; 2018. ISBN: 978-953-51-5267-5
Chen W-J, Wang H-Z, Lin R-Y, Lee C-K, Pan C-L. Attosecond pulse synthesis and arbitrary waveform generation with cascaded harmonics of an injection-seeded high-power Q-switched Nd:YAG laser. Laser Physics Letters. 2012; 9(3):212
Suntsov S, Abdollahpour D, Papazoglou DG, Tzortzakis S. Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air. Optics Express. 2009; 17(5):3190-3195
Fedotov A, Gladkov S, Koroteev N, Zheltikov A. Highly efficient frequency tripling of laser radiation in a low-temperature laser-produced gaseous plasma. JOSA B. 1991; 8(2):363-366
Feng L-B, Lu X, Xi T-T, Liu X-L, Li Y-T, Chen L-M, Ma J-L, Dong Q-L, Wang W-M, Sheng Z-M. Numerical studies of third-harmonic generation in laser filament in air perturbed by plasma spot. Physics of Plasmas. 2012; 19(7):072305
Suntsov S, Abdollahpour D, Papazoglou D, Tzortzakis S. Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility. Physical Review A. 2010; 81(3):033817
Rodríguez C, Sun Z,Wang Z, RudolphW. Characterization of laser-induced air plasmas by third harmonic generation. Optics Express. 2011; 19(17):16115-16125
Mahon R, McIlrath TJ, Myerscough VP, Koopman DW. Third-Harmonic Generation in Argon, Krypton, and Xenon: Bandwidth Limitations in the Vicinity of Lyman-(alpha). Quantum Electronics, IEEE Journal. 1979; 5(6):444-451
Marr G, West J. Absolute Photoionization Cross-Section Tables For Helium, Neon, Argon, and Krypton In The VUV Spectral Regions. Atomic Data and Nuclear Data Tables. 1976; 18(5):497-508
Chang Z. Fundamentals of Attosecond Optics. 1st ed. CRC Press; February 16, 2011. ISBN-10: 1420089374. ISBN-13: 978-1420089370
Sapaev U, Husakou A, Herrmann J. Combined action of the bound-electron nonlinearity and the tunnel-ionization current in low-order harmonic generation in noble gases. Optics Express. 2013; 21(21):25582-25591
Cain SR. A photothermal model for polymer ablation: Chemical modification. Journal of Physical Chemistry. 1993; 97:051902-1-5
Pan C-L, Lin C-H, Yang C-S, Zaytsev A. Laser ablation of polymethylmethacrylate (PMMA) by phase-controlled femtosecond two-color synthesized waveforms. In: Yang D, editor. Applications of Laser Ablation—Thin Film Deposition, Nanomaterial Synthesis and Surface Modification. London, UK: InTech; 2016. ISBN: 978-953-51-2812-0, Print ISBN: 978-953-51-2811-3
Théberge F, Chin SL. Enhanced ablation of silica by the superposition of femtosecond and nanosecond laser pulses. Applied Physics A: Materials Science & Processing. 2005; 80:1505
Okoshi M, Inoue N. Laser ablation of polymers using 395 nm and 790 nm femtosecond lasers. Applied Physics A: Materials Science & Processing. 2004; 79:841
Chowdhury IH, Xu X, Weiner AM. Ultrafast two-color ablation of fused silica. Applied Physics A: Materials Science & Processing. 2006; 83:49
Zoppel S, Merz R, Zehetner J, Reider GA. Enhancement of laser ablation yield by two color excitation. Applied Physics A: Materials Science & Processing. 2005; 81:847
Zoppel S, Zehetner J, Reider GA. Two color laser ablation: Enhanced yield, improved machining. Applied Surface Science. 2007; 253:7692
Schumacher DW, Bucksbaum PH. Phase dependence of intense-field ionization. Physical Review A. 1996; 54(5):4271
Gao L, Li X, Fu P, Guo D-S. Phase-difference effect in two-color above-threshold ionization. Physical Review A. 1998; 58(5):3807
Schwarz E, Reider GA. Laser-induced plasma by two-color excitation. Applied Physics B: Lasers and Optics. 2012; 107(1):23