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With carrier-envelope phase (CEP) stabilized mid-infrared (MIR) laser pulse, the CEP-controlled supercontinuum generation can be distinctly observed in a very small distance range when the focus of the laser pulse closes to the exit surface of the fused silica (FS). This CEP effect will be gradually weakened and finally disappears if the laser focus moves out of this range. With numerical simulation, we find that although the CEP effect starts from the tunneling ionization of the electron, it can be observed only when the supercontinuum mainly comes from the self-phase modulation (SPM) and self-steepening of the laser pulse and too much electrons will make it ambiguous.
© 2014 Optical Society of America
1. Introduction
Many novel and important phenomena can be observed when intense laser field are focused into atomic [1,2] or molecular gases [3], transparent liquids [4] or solids [5] because of the nonlinear interaction between the material and the strong laser field. The nonlinear interaction between intense femtosecond laser fields and gases has been widely investigated in decades, e.g. high order harmonics generation [6–10], high-order above-threshold ionization in various gases [11–13] and single-cycle infrared pulses generation [14]. The study on the nonlinear interaction between femtosecond laser fields and solid material [15–20] is also very popular during recent years. But so far, the CEP effect of tunneling ionization during the laser filamentation is rarely reported. D. E. Laban et al used CEP-stabilized few-cycle light pulses to investigate the CEP effect of the self-focusing by observing a filament in air [21]. C. Gong et al observed CEP effect during the interaction between the few-cycle MIR laser pulses and the bulk solid transparent materials [20].
For solid material, ionization is understood as an electron transition from the valence band into the conduction band, which can be analogous to the electron ionization to the continuum state in atoms and molecules [22]. According to the Keldysh parameter for solid materials [23] (here ω and E are angular frequency and electric field strength of the laser pulse, respectively. Ip is the ionization potential of the material, m is the reduced mass of the electron and e is charge of the electron.), multiphoton ionization is the main part when γ >> 1. Conversely, γ << 1 implies that tunneling ionization is dominating. For tunneling ionization, the electron tunnels out of the potential barrier mainly around the crest of the electric field of the laser pulse, which dependents on the CEP of the laser pulse greatly. But for multiphoton ionization, this dependence is very weak. As we known, during the filamentation process, the yield of the electron is important to balance the self-focusing of the laser pulse. So, the CEP effect on tunneling ionization should be observed on the supercontinuum generation during the filamentation process.
For fused silica, Ip and m are equal to 9eV and 0.64me, respectively. The Keldysh parameter for solid materials is inversely proportional to the wavelength and electric field strength of the laser pulse. Using 800nm wavelength and the maximum intensity in the filament [24], the value of γ is still greater than 1, which means that the multiphoton ionization would always be dominant for 800nm laser pulse. If replacing 800nm with 1750 nm, γ will equal unity as long as the laser intensity reaches 7.3 × 1012 W/cm2. With the increasing of the laser intensity, the tunneling ionization plays more and more important role during ionization process.
The observed CEP effect in the work of C. Gong et al [20] is very weak, which should be retrieved with special technique. As we known, the CEP of 1750 nm laser pulse will change 2π for the pulse propagating about 75 μm in the fused silica. So, if the length of the filament is long, the CEP effect of the supercontinuum generation may be weakened greatly by the CEP shift of the 1750 nm laser pulse during propagation. In this work, we use the tightly focusing scheme to shorten the length of the filament to investigate the CEP effect of supercontinuum generation. We focus 2 cycles (12fs), CEP stabilized laser pulses at 1.75 μm central wavelength into fused silica to generate supercontinuum in the experiment. We find that the spectral range and intensity of the supercontinuum varies periodically with the CEP of the laser pulse only when the focus of the laser pulse closes to the exit surface of the fused silica, otherwise this phenomenon will disappear. A model including Keldysh’s formulation to calculate the ionization rate has been employed to simulate the nonlinear propagation for ultrashort laser pulses in solid material in order to explain this phenomenon.
2. Experimental setup and result
The schematic of the experimental setup is shown in Fig. 1(a). CEP stabilized 1.6mJ, 40fs laser pulses at 1.8μm central wavelength are obtained from a home-made three-stage mid-IR optical parametric amplifier (OPA) pumped by a commercial Ti:sapphire laser amplifier (Coherent LEGEND-HE-Cryo) which provides 40fs pulses with pulse energy up to 8mJ at 1kHz repetition rate. By focusing 1.8um femtosecond laser pulses into an argon-filled hollow fiber, the spectrum is broadened due to the self-phase modulation (SPM) effect, ranging from 1200nm to 2100nm. A pair of fused silica wedges is used to compensate the chirp of the pulse due to its negative group delay dispersion (GDD) in the MIR region [25]. Finally, 1.0mJ, 12fs (about 2 cycles) laser pulses at 1.75μm central wavelength are obtained. The CEP of the few-cycle pulses is passive-stabilized with fluctuations of ~547mrad (root-mean-squared, rms) [25].
Fig. 1 (a) The schematic of the sample geometry in the experiment. The laser pulses are focused by an off-axis parabolic mirror (PM) into the fused silica (FS). (b) The measured spectra as a function of moving distance of the fused silica. The exit surface of the fused silica is placed near the focus in the beginning, corresponding to zexp = 0. (c) Measured spectra at moving distance of 525μm (solid blue curve) and 1625μm (solid red curve).
In this experiment, only about 450nJ pulse energy is used. The CEP-stabilized few-cycle laser pulses are focused into the fused silica to generate supercontinuum by an off-axis parabolic mirror with a focal length of 50mm. The focus spot size is about 18μm and the thickness of the fused silica is 2mm. Spectroscopic measurements for supercontinuum are carried out by using the spectrometers of Ocean Optics, USB 2000 + in the range of 200–1100nm and Ocean Optics NIR256-2.5 in the range of 900–2500nm, respectively. The signal integral time in the measurement is 100ms, in which 100 shots are accumulated to obtain a spectrum. The CEP of the laser pulse is changed by adjusting the thickness of the wedge in our experiment.
Firstly, we place the exit surface of the fused silica around the laser focus in the air and define this position as zexp = 0, where supercontinuum almost cannot be generated. Then we can move the fused silica forward along the direction of the laser propagation and define the moving distance as positive zexp [as shown in the Fig. 1(a)]. With the increase of zexp, the laser focus will be moved into the fused silica gradually. The generated supercontinuum of each moving distance zexp can be measured as shown in Fig. 1(b), in which the observed spectrum range shifts from 500nm-650nm to 650nm-800nm when zexp increases from 350μm to 1800μm. Figure 1(c) shows the measured entire spectra of two different positions (525μm and 1625μm), demonstrating that supercontinuum generation depends on the relative position between the fused silica and laser focus. In this experiment, the laser wavelength is in the anomalous dispersion regime of the fused silica, the effect of which has been studied in some earlier works [26–30]. The blue supercontinuum generation of this regime may be explained as the interference of supercontinuum undergoing anomalous group velocity dispersion in fused silica [called an isolated anti-Stokes wing by E. O. Smetanina et al [26] or the three-wave mixing model and phase-matching arguments [27].
Figures 2(a)-2(c) show spectral range and intensity of the supercontinuum varying with the CEP of the laser pulse when the fused silica moves 375μm, 425μm and 500μm, respectively. The CEP changes from 0 to 3π with a step of π/16 radian. The supercontinuum changes with a period of π at first when the MIR pulse is focused near the exit surface of the fused silica at normal incidence and this variation becomes ambiguous gradually and disappears finally when the fused silica moves forward. Both the spectral range and intensity change with the CEP of the laser pulse in Figs. 2(a) and 2(b). Three different wavelengths (523nm, 548nm and 576nm) in Fig. 2(b) are selected to show clearly the spectral intensity varying with CEP, where the measured intensity are normalized to its maximal intensity for each wavelength as
and the results are shown as the blue stars in Fig. 2(d). The max{.} in Eq. (1) denotes choosing the maximum value. The sine function is used to fit the blue stars and the result is shown as the red solid curve in Fig. 2(d), in which the CEP of the peak shifts to small value with the increase of the wavelength, indicated by the black curve. We also insert a 100μm thick fused silica into the optical path (before the off-axis parabolic mirror), with which a horizontal shift of about 0.62π can be observed for the black curve, closing to the extra added CEP introduced by the 100μm fused silica (0.66π), which is a further evidence of the CEP dependence. When zexp is 500μm in Fig. 2(c), the CEP of the laser pulse has almost no effect on the supercontinuum generation in the fused silica. To keep moving forward the fused silica until the supercontinuum vanishes, the periodic variation of the supercontinuum never re-appears by changing the CEP of the laser pulse. So, this phenomenon caused by CEPs can be observed only when the laser focus closes to the exit surface of the fused silica. In our experiment, the fused silica moves about 1.4mm from the appearance to the disappearance of the supercontinuum, while the supercontinuum varies periodically with CEPs of the laser pulse only in a very short range of about 0.1mm. If replacing the 2mm thick fused silica with a 0.5mm thick one, supercontinuum can be generated in the distance ranges of 0.275mm and 0.325mm when the pulse energy is 0.40μJ and 0.465μJ, respectively, while supercontinuum varying periodically with the CEP of the laser pulse still can only be observed in a short distance range of about 0.1mm.Fig. 2 Generated supercontinuum varies with the CEP when the fused silica moves forward of 375μm (a), 425μm (b), and 500μm (c), respectively. (d) Normalized intensity varies with the CEP for three wavelengths (523nm, 548nm and 576nm) in Fig. 2(b). The red solid curve is the sine function fit to the experimental data (blue stars). The black solid curve marks the shift of intensity peak.
3. Numerical simulation and discussion
To understand the physical process above in more details, the numerical simulation is performed. The equation for nonlinear propagation of an axially symmetric laser pulse in optical media can be expressed as [20]
The four terms on the right side of Eq. (2) are diffraction, all dispersion (index curve is used in the simulation directly), SPM (no CEP effect included) and plasma behavior (CEP effect included), respectively. Here ω is the angular frequency, n(ω) is the refractive index of different frequency calculated by Sellmeier equation, k(ω) = n(ω) ω/c, c is the light speed in the vacuum. The ε0 is the dielectric constant in the vacuum. The m and e are the effective mass and charge of the electron, respectively. I(r, z, t) is the intensity of the laser pulse. represents the Fourier transformation and is Fourier transformation of electric field . The effective bandgap of the fused silica Ip is 9eV. The nb and n2 are linear refractive index and nonlinear coefficient, respectively. The nonlinear index n2 at 1.75μm cannot be found in the previous work, so n2 = 3.54 × 10−16 cm2/W for 800 nm laser pulse is used in the simulation [31]. The time evolution of the ionized electron density can be given by the equationThe two terms in the right side of Eq. (3) denote the ionization and the electron recombination, respectively. The ionization rate WMPI can be calculated with the Eq. (7) in Ref [23]. The characteristic time of electron recombination τr is 150fs in fused silica [32]. The atomic density nat is 2.1 × 1022 cm−3. Differential Eq. (2) is numerically solved in axially symmetric coordinate by 4th order Runge-Kutta method in the frequency domain.In the simulation, the initial beam profile of incident laser pulse is a spatiotemporal Gaussian profile without chirp, which can be written as
Here φCEP describes the effect of the CEP. The laser parameters are close to those used in the experiment. The diameter of the beam waist, 2w0, is chosen as 20μm and the peak intensity is 1 × 1013W/cm2.The central wavelength λ and the duration of laser pulse tp are 1750nm and 12fs, respectively. The distance from the incident surface of the fused silica to the laser focus is symbolized by zsim in the simulation, which is equal to zero when the beam waist is located on the incident surface of the fused silica and positive if the fused silica moves forward along the direction of the laser propagation. The detail geometry of the simulation is shown in Fig. 3(a).Fig. 3 (a) The simulation schematic of the beam focus in the vacuum, the incident surface of sample and the spatial-temporal distribution of the pulse in the focus. (b) Simulated on-axis temporal field strength as a function of the propagation distance when zsim and CEP are −650μm and zero, respectively. A, B and C label propagation distance of 500μm, 600μm and 705μm, respectively. (c) Simulated supercontinuum corresponds to Fig. 3(b), but only the spectrum from 400 nm to 1000 nm is shown here. The spatial distribution of field strength of the pulse at A(d), B(e) and C(f) position. The insets show on-axis full electric field.
When zsim and CEP of the laser pulse equal −650μm and zero, respectively, on-axis pulse profile and supercontinuum from 400nm to 1000nm varying with the propagating distance are shown in Figs. 3(b) and 3(c), respectively. Figures 3(d)-3(f) show temporal-spatial profile of the pulses of three different propagating distances (500μm, 600μm, 705μm) which are indicated as A, B and C in the Fig. 3(b), respectively. When the propagation distance is shorter than 680μm, the supercontinuum is mainly from the self-phase modulation and the self-steepening of the pulse, while the main contribution will come from the plasma after that [Fig. 3(f)].
To analyze the CEP effect during the propagation, we define a parameter α (modulation depth) to describe the intensity change of the supercontinuum for different CEPs, which is written as
where and represent the maximum value and minimum value, respectively. The number j marks the initial pulse with different CEP. Figure 4(a) shows α varying with the propagating distance in the fused silica for 500nm wavelength component. When zsim equals −625μm [the black solid curve in Fig. 4(a)], the modulation depth α oscillates very fast for propagating distance larger than 680μm because the supercontinuum generation is sensitive to the laser parameter. If we look at the intensity of 500nm radiation as a function of CEP, the intensity peak will shift greatly for the propagating distance larger than 680μm even when the pulse duration changes very small, and the sum of the intensity of different pulse durations will greatly decrease the CEP effect [as shown in right figure of Fig. 4(b)]. But if the propagating distance is less than 680μm, there is no peak shift for 500nm radiation when slightly changing the pulse duration, e.g. from 11.5fs to 12.5fs, and integral of the intensity of different pulse durations can keep the modulation depth because the intensity oscillations as a function of CEP for different pulse durations are accordant [as shown in left figure of Fig. 4(b)]. As we known, the laser parameters cannot be very stable in the experiment. The pulse energy and duration will always be changing. So if the generation of the supercontinuum is very sensitive to the laser parameters, we cannot measure the CEP effect even if we can control the CEP of the laser pulse. That is why we cannot observe the CEP effect in Fig. 2(c).Fig. 4 (a) The modulation depth α of 500 nm radiation with propagating distance for four different zsim. D and E label propagation distance of 566μm and 700μm, respectively. (b) Intensity of 500nm as a function of the CEP for three different pulse durations (11.5fs, 12fs, 12.5fs) and the average value at D (left figure) and E (right figure) position when zsim equals −625μm. (c) Generated supercontinuum spectrum at the exit surface of the fused silica [dotted black line in the Fig. 4(a)] as a function of CEP when zsim equals −710μm. (d) Normalized intensity varies with the CEPs in the exit surface of the fused silica [dotted black line in the Fig. 4(a)] for three wavelengths (475 nm, 500 nm and 528 nm) when zsim equals −710μm (upper figure) or −625μm (bottom figure). The intensity peak moves to small CEP with the increase of the wavelength as represented by an arrow in the upper figure.
By comparing Fig. 4(a) to Fig. 3(a), a good CEP effect can only be observed for zsim = −650μm when the supercontinuum is mainly generated from the self-phase modulation and self-steepening of the laser pulse, while the CEP effect will be very weak if the contribution of the plasma becomes significant [after the propagation distance of about 700μm in Fig. 4(a)]. Besides, when the propagation distance is shorter than 440μm, the intensity of generated supercontinuum is too low to be measured in the experiment. For the propagation distance from 440μm to 680μm, the modulation depth α first increases, then decreases. A large modulation depth can be found between 520μm and 625μm, which can explain why the CEP effect can only be observed in a short distance range in our experiment. Similarly, this propagating distance is from 575μm to 675μm when zsim equals −710μm.
The CEP effect originates from the free electron, not from the SPM and SS directly, but too much electrons will destroy the measured CEP effect because the supercontinuum generation will become sensitive to the laser parameter. A dotted black line is plotted in Fig. 4(a) as the exit surface of the fused silica, we can see the change of modulation depth when zsim is adjusted. Figure 4(c) show the effect of CEP offset on the generated supercontinuum spectrum at the exit surface of the fused silica when zsim equals −710μm. Figure 4(d) shows the normalized intensity as a function of CEP for three wavelengths (475nm, 500nm and 528nm) when zsim equals −710μm (larger α) or −625μm (small α). When α is large [zsim = −710μm, the upper figure of Fig. 4(d)], the CEP of the peak shifts as indicated by the arrow while no shift can be observed and the variation of the intensity decreases when α is small [zsim = −625μm, the lower figure of Fig. 4(d)].
4. Conclusion
In summary, spectral range and intensity of supercontinuum change with focal position of the laser pulse in the transparent material, while the CEP-controllable supercontinuum can only be observed in a very short distance range. With numerical simulation, we find that distinct CEP effect can only be observed in a short distance range where the supercontinuum is mainly generated from the self-phase modulation and self-steepening. But too much electron will destroy the measured CEP effect because the supercontinuum generation will become very sensitive to the laser parameter. In our experimental scheme, this short distance range is about 100μm in the fused silica.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 11127901, 11134010, 11227902, 11222439, 11274325, and 61221064), National 973 Project (Grant No. 2011CB808103), and Shanghai Commission of Science and Technology (Grant No. 12QA1403700).
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