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Abstract
When a two-color Laguerre-Gaussian laser beam propagates through an indium tin oxide (ITO) material, the spatial distributions of odd- and even-order vortex harmonics carrying orbital angular momentum (OAM) are studied. The origin of vortex harmonics can be directly clarified by investigating their dependence on the incident laser field amplitude and frequency. In addition, it is shown that the spectral intensities of vortex harmonics are sensitive to the epsilon-near-zero nonlinear enhancing effects and the thickness of ITO materials. Thus the vortex harmonics can be conveniently tunable, which provides a wider potential application in optical communications based on high-order OAM coherent vortex beams.
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1. Introduction
Due to the potential applications of high-order harmonic generation in molecular tomography and production of ultrafast coherent EUV light sources, it has attracted the researchers’ wide attention [1–4]. The high-order harmonic generation relies on extreme nonlinear optics where the still very weak nonlinear effects requiring a longer laser-matter interaction length and/or a higher excitation laser intensity, which is usually impossible for a laser beam directly from oscillator without multi-pass amplification [1–4]. The occurrence of epsilon-near-zero (ENZ) materials provides an alternative scheme to make the above route become possible [5,6], because the internal field within the front surface and the third-order nonlinearity coefficients of this kind of material is significantly enhanced [7,8]. For example, a ninth-order harmonic is generated when cadmium oxide (CdO), a kind of indium tin oxide (ITO) ENZ material, driven by an around 10 GW/cm2 laser [5]. Moreover, an obvious harmonic redshift phenomenon is disclosed in ENZ materials under a few-cycle laser excitation [6]. It is pointed out that the three-step model of high-order harmonic generations in atom, the electrons should be three process: ionization, acceleration and recombination [3,9,10]. In contrast, the origin of high-order vortex harmonics in the work is results from the macroscopic coherent polarization Px(t) acting as a source of the re-emitted field [11,12].
Though the odd-order harmonics can be obtained in the above works [5,6], the investigation is limited to the single-color laser excitation case. Here we investigate the harmonic generation in the ENZ materials but under two-color laser excitation. The two color field is composed of the fundamental laser field and its double-frequency field. Via the two-color excitation, even-order harmonics are also generated in any materials without the necessity of breaking the centro-symmetry of materials based on polar molecules [11], surface effects [12,13] or nonlocal bulk effects [14,15]. The synthesized two-color field via a fundamental laser and its double-frequency field for high-order harmonic generation has been widely adopted and many numerical simulation and experimental investigation confirm its feasibility [16–28]. Up to now, there exist three kinds of resources for the occurrence of even-order harmonics: a) breaking of symmetry in the field of the fundamental and its second harmonic [16–20], b) modification in field coherence [21–24], and c) contribution from many different nonlinear processes such as four-wave mixing and sum-frequency/difference frequency [25–28]. It is meaningful to clearly explore the real generation origin of even-order harmonics driven by a two-color field.
Inspired by the above question, one can use a two-color Laguerre-Gaussian (LG) vortex beam to interact with ITO materials under the case of very weak incident laser intensity (∼1010 W/cm2). The topological charge numbers of odd- and even-order vortex harmonics with the spatial distributions can be directly distinguished by their transverse field distributions [29–33] and with the help of time-frequency analysis. At the same time, the information carried by OAM in high-order vortex harmonics may be potential application in optoelectronics and optical information processing. Importantly, the spectral intensities of even-order harmonics depend sensitively on the incident field amplitude and frequency. In addition, an optimum thickness of ITO materials is determined to support the strongest harmonic signal. For example, the fourth-order vortex harmonic, the energy conversion efficiency is up to 2.3 × 10−4.
2. Theoretical model
A synthesized field via a two-color LG vortex laser, polarized along the x direction, first freely propagates in the vacuum and then is incident on the ITO material with thickness of d = 900 nm along the z direction. When the laser arrives at the front surface (zs = 60 µm), it partially penetrates the ITO material and continue to propagate until again into the vacuum. The widely adopted formula for a LG vortex laser with amplitude Elp is written as [6,29–31,34]
ZR=πa02/λ is the Rayleigh length, a(z’) is the radius of beam waist at position z'=z-z0 and a(0) =a0 at z0. E0 is the electric field amplitude. z0 is the initial laser peak position and a suitable choice of z0 should ensure that the LG pulse penetrates negligibly into the media at t = 0 [11,13,29], and here z0 is set as 30 µm. Llp is the associated Laguerre polynomial with p denoting the transverse radial node number (here p = 0 just for convenient simulation). $r = \sqrt {{{({x - {x_0}} )}^2} + {{({y - {y_0}} )}^2}} $ (x0 and y0 are the axial center coordinates on the transverse x-y plane). exp(-ilϕ) describes the helical phase profile of an optical vortex, with l (0, ± 1, ± 2,…) the topological charge number and ϕ the azimuthal angle.
Because in our following numerical simulation, the Rayleigh length is ZR =πa02/λ∼552 µm with λ=1.28 µm and a0 = 15 µm, which is much larger than the propagation distance investigate here, i.e., ZR >> zs is satisfied, a paraxial plane-wave description of the above electric field (1) is in fact enough for our purpose. In our numerical simulation, the incident field is synthesized via a fundamental laser of LG10 (l = 1) beam and its double-frequency laser of LG20 (l = 2) beam. This incident field is assumed to have a sech-shaped envelope [6,29] and simplified as,
Here ω2 = 2ω0, and τ0 denotes the full width at half maximum (FWHM) of pulse intensity. ω0 denote the central frequency. A is the field amplitude ratio of the two components of the incident two-color field. c is the light speed at vacuum.
The interaction between this vortex laser and the ITO material can be described by combining three-dimensional Maxwell equations [6,7,35] with the paradigmatic-Kerr equation [6,35],
Ps, ωe, δe, ɛs, and ɛ∞ are the saturation polarization, resonant frequency, loss coefficient, static dielectric permittivity, and high frequency permittivity, respectively. Here ωe = 1.327 × 1015 s-1, δe = 0.01371, ɛs = 3.846698, and ɛ∞=3.1178 are adopted [8,35–40]. In addition, in the ENZ material investigation, the important factor is the so-called ENZ frequency. It can be obtained by setting the real part of dielectric permittivity as zero, which is corresponding to the zero-crossing-point in the dispersion relation curve [7,35],
Here based on the above values of ωe, δe, ɛs, and ɛ∞, the ENZ frequency ω0 = 1.1108ωe, corresponding to a wavelength of λ=1.28 µm [41]. The above Maxwell equations (3) are first solved by employing Yee's finite-difference time-domain (FDTD) discretization scheme [42–44] and the paradigmatic-Kerr Eq. (4) by the Runge-Kutta algorithm [45,46], respectively. Then, the transmitted electric field E(t, z=∞) after the ITO material is recorded. Finally, a Fourier transformation is conducted to the transmitted field for investigating the spectral characteristics of harmonics. The laser amplitude is E0 = 5 × 108 V/m (corresponding to laser intensity I0 = 3.33 × 1010 W/cm2), and a0 = 15 µm, A = 0.1, τ0 = 20 fs are used for the following numerical simulation.
3. Results and discussion
First, the temporal and spatial evolution of a single-color LG pulse with frequency equal to ω0 propagating through the ITO material is numerically investigated. The field waveform is recorded at a position of z = 65 µm, as shown in Fig. 1(a). the laser pulse after interaction with the ITO materials has a significantly asymmetric time-spatial domain profile, which is consistent with previous work [5,6]. It is pointed out that the laser duration (τ0 = 20 fs) is much larger than the optical cycle of incident laser pulse (∼ 4.27 fs), the redshift in high-order harmonic generation should disappear [6]. Due to the waveform change and inversion symmetry of the system, only the odd-order harmonics are found as shown in Fig. 2(a) (solid line). When another weaker single-color laser field (2ω0) whose intensity is only one percent of the former ω0 field is incident, the change in the time-spatial domain profile is trivial after interacting with the ITO material, and the corresponding harmonics cannot be found as shown in Fig. 2(a) (dashed line). But when these two fields are superposed to form a two-color beam and then as a whole interacting with the ITO material, as shown in Fig. 1(b), the rise and falling parts of the temporal waveform changes significantly if compared with the case in Fig. 1(a), where there exists obvious peak splitting in the electric field waveform. The results show that not only the odd-order harmonic generation is found, but also the even-order harmonic generation can be obtained (see dash-dotted line in Fig. 2(a)). It should be pointed out that the intensities of odd-order harmonics are nearly unchanged due to the adding of the weaker 2ω0 beam. This phenomenon can be further proved by the subsequent time frequency analyses to these vortex harmonics, as shown in Fig. 3. As shown in Fig. 3(a), the only odd-order harmonics can be found when the only single-color field (ω0) is present. In contrast, the odd- and even-order harmonics simultaneously occur under the two-color excitation as shown in Fig. 3(b).
Fig. 1. Temporal and spatial evolution of a LG pulse propagating through the ITO material under (a) single-color ω0 or (b) two-color ω0 + 2ω0 excitation with fundamental laser frequency equal to ω0.
Fig. 2. Spectral distributions of vortex harmonics (a) at different incident LG pulses and (b) at ω0 + 0.3ω0. The 4ω0 harmonic signal intensity versus the incident laser peak amplitude of (c) ω0 laser for fixing 2ω0 field intensity and (d) 2ω0 field or fixing ω0 field intensity.
Fig. 3. Time-frequency analysis to high-order vortex harmonics in Fig. 2(a) under (a) single-color ω0 and (b) two-color ω0 + 2ω0 excitation.
Second, the origin of even-order harmonics maybe comes from the four-wave mixing effects [25–28]. Here we take the fourth-order harmonic as an example to present a further confirmation in the case of harmonic generation from a two-color field interacting with the ITO material. First, we artificially change the original double-frequency ω2 = 2ω0 to a non-integer lower frequency ω2 = 0.3ω0 and make the same simulation procedure to investigate the harmonic distribution, as shown in Fig. 2(b). Now, beside the other peaks in high-order vortex harmonics, the fourth-order harmonic peaks at 2.3ω0 instead of the previous 4ω0. Such phenomenon keeps if we try other different frequency lasers beyond 0.3ω0 (not shown here). Second, corroborative numerical simulation and fitting is shown in Figs. 2(c) and (d), which is based on adjusting the incident laser peak amplitude of ω0 field or 2ω0 field, respectively. As shown in Fig. 2(c), fixing the 2ω0 field peak amplitude and just tuning the ω0 field peak amplitude, when the ω0 field intensity is not two high, it conforms to the law of absorbing two ω0 photons: the fourth-order harmonic signal is proportional to the square of the ω0 field intensity indicating a fourth-order harmonic photon comes from the contribution two ω0 photons because E4th∽E12E2. Similarly, if we fixing the ω0 field intensity and tuning the 2ω0 field peak intensity, as shown in Fig. 2(d), we find that the harmonic signal is proportional to the 2ω0 laser intensity indicating that the fourth-order harmonic photon forms by absorbing a 2ω0 photon except for two ω0 photons. Above all, the four-wave mixing mechanism for fourth-order harmonic is confirmed. Importantly, as shown in Fig. 2(c), when the incident ω0 field intensity is two high, the dependence of harmonic signal on the laser intensity deviates from the above quadratic pattern. As the laser intensity increases the signal increases to a maximum and then decreases. The reason is the nonlinear saturation absorption effect and the non-negligible absorption loss existing in the ENZ materials.
Third, in order to further assist the investigation of high-order vortex harmonics, we turn to the transverse field distributions. By utilizing the time-frequency analysis based on the wavelet-transformation technique exemplified by the widely adopted Gabor transformation [6,13,29], the whole vortex harmonic spectrum in Fig. 2(a) is decomposed by filtering each harmonic with a spectral window of 0.2ω0 width around its peak position to conveniently investigate the transverse field distributions of each harmonic. According to the criteria [29–33] of vortex harmonics: the topological charge number lq of the qth-order harmonic is directly proportional to its harmonic order, i.e., lq = ql (l = 1 for a fundamental incident ω0 field of LG10), one can directly distinguish certain harmonic order from the transverse field distribution where the topological charger number is easily determined. The transverse field spatial distributions for the different order vortex harmonics are shown in Fig. 4. One can clearly distinguish the fourth-order and sixth-order harmonics from Figs. 4(c) and 4(e). It maybe in turn helpful to proving the above explanation about the generation origin of even-order harmonics.
Fig. 4. Transverse field distributions of (a) second, (b) third, (c) fourth, (d) fifth, (e) sixth and (f) seventh order harmonics under two-color excitation with fundamental laser frequency equal to ω0.
Finally, to highlight the nonlinear enhancement effects of ITO materials at ENZ frequency ω0 = 1.1108ωe, we try other laser frequencies of 1.3ωe and 1.5ωe for a comparison investigation. As shown in Fig. 5(a), the intensities of vortex harmonics are obviously reduced when the incident ω0 laser frequency in the two-color field deviates a little more from the ENZ frequency of 1.1108ωe. For example, the peak of the fourth-order harmonic signal for laser frequency at 1.1108ωe is about 5 times that at 1.5ωe. In contrast, the fifth-order harmonic signal is even reduced by 30 times. Correspondingly, the intensities of the transverse field spatial distributions about the fourth-order and fifth-order harmonics will decrease simultaneously (not shown here). The mechanism for high-order vortex harmonic enhancement is attributed to the ENZ enhancement of nonlinearity coefficients [7,8]. Thus, the spectral intensities and spatial distributions of high-order vortex harmonics can be controlled effectively by the incident laser frequency.
Fig. 5. (a) Spectral distributions of high-order harmonic signals at different laser frequencies ω0. Intensities of (b) fourth-order harmonic and (c) sixth-order harmonic at different material thickness d.
In addition, the ITO material thickness d has an important influence on the intensities of vortex harmonics. Let's take the fourth-order harmonic as an example. As shown in Fig. 5(b), with the thickness increase from 600 nm to 900 nm, the harmonic signal is first enhanced as the consequence of the phase-matching-free coherent enhancement effect. But with the further increase of material thickness d to 1200 nm, the intensity of harmonic signal decreases. This is due to the accumulated absorption loss becoming stronger and stronger for a long propagation distance, predicted from the Beer–Lambert law. Similarly, as for the sixth-order harmonic, with the increase of propagation distance from 300 nm to 500 nm, the signal is also enhanced and further increase the harmonic signal in turn decreases as shown in Fig. 5(c).
From the perspective of harmonic application in optical communications based on high-order OAM coherent vortex beams, the energy conversion efficiency of a high-order harmonic signal from the excitation laser is essential. The conversion efficiency can be estimated by using the ratio between the peak intensity of each harmonic and the incident laser intensity [40,41,47]. At this optimal ITO material thickness d∼900 nm (ω0 = 1.1108ωe) as shown in Fig. 5(c), the energy conversion efficiency can be numerically estimated as ∼2.3 × 10−4. It is worth noting that, such an enhancement phenomenon of high-order vortex harmonics is not limited to the ITO materials shown here. In fact, the research results obtained in this work are quite general irrespective of ENZ media, therefore the predicted harmonic behaviors must be realized in other ENZ materials.
4. Conclusion
In conclusion, we have studied the spatial distributions of high-order vortex harmonics driven by a two-color LG vortex beam in ITO materials. The results show that the generation origin of odd- and even-order harmonics can be clarified by transverse field distributions and time-frequency analyses. Interestingly, the origin of vortex harmonics can be directly clarified by investigating their dependence on the incident laser field amplitude and frequency. In addition, it is shown that the spectral intensities of vortex harmonics are sensitive to the epsilon-near-zero nonlinear enhancing effects and the thickness of ITO materials. Thus the vortex harmonics can be conveniently tunable, which provides a wider potential application in optical communications based on high-order OAM coherent vortex beams.
Funding
National Natural Science Foundation of China (12074398, 12174161).
Acknowledgments
The authors are grateful to Dr. Yi Wu for inspired discussions.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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