1. Introduction

The epsilon-near-zero (ENZ) material is used for enhancing nonlinear optical effects and has drawn a wide attention in scientific researches and significant applications due to its tunable ENZ frequency and phase-matching-free characteristics. For example, as for transparent conducting oxides (TCOs), by modulating their dopant concentrations [1–5], the ENZ frequency can extend from ultraviolet to mid-infrared [6,7]. An ENZ material provides a non-resonant enhancement of the normal electric field component across the vacuum-ENZ dielectric interface [8–11], in which the formation of a narrow ENZ plasmon channel results in field enhancement [12]. It has been successfully used for the demonstration of soliton excitation [13], transmissivity directed hysteresis [14], and enhancement of second and third order harmonic generation [15–17], especially the occurrence of ninth-order harmonics under a much lower laser intensity (∼10¹⁰ W/cm²) than expected [18].

Now if the exciting laser is replaced by a vortex one and oblique incidence by normal incidence, what would be expected? Whether does the nonlinear enhancement still occur or not and if is there some new findings? So far, people's pursuit of communication capacity and quality is constantly increasing to find faster and more convenient means of communication [19]. The emergence of a vortex laser with tunable orbital angular momentum has knocked a new door for the development of electromagnetic communication [19–23] due to the possibility to increase the communication capacity [24–26]. Especially, the high-order vortex harmonics with carrying the topological charge number information have also attracted much attention [23–26]. According to the criteria [19,22], the topological charge number l_q is proportional to its harmonic order ql (i. e., l_q = ql, l being the fundamental topological charge number of the incident beam). However, in the previous investigations, the nonlinear optical media are adopted by traditional media such as atoms and molecules [19,27,28]. Here the interaction between a vortex laser and an ENZ material is numerically investigated by solving the coupled Maxwell-paradigmatic-Kerr equations. The results show that the occurrence of pronounced seventh order harmonics for a vortex laser pulse of long duration, when the laser frequency equals to the ENZ frequency (the real part of the dielectric constant ε is exactly zero). When the laser frequency is tuned away from this ENZ frequency, the harmonic radiation is weakened obviously. Interestingly, when the pulse duration is reduced, a spectral redshift (frequency downwards) occurs for the vortex harmonic radiation. The narrower the pulse and the larger the harmonic order, the more obvious the redshift. It should be indicted that, due to the chirp effects [29] or propagation effects [30], the redshift can be found in other media such as atoms or semiconductor systems. However, to the best of our knowledge, the phenomenon of redshift due to pulse shortening disclosed in this paper has only been found in ENZ materials. According to the law that the topological number of each harmonic is equal to its harmonic order (if the topological charger number is one for the incident LG vortex beam), the harmonic peaks, although with an obvious redshift shown in the following, still possess the exact odd orders, indicated by the transverse electric field distribution of each harmonic radiation from which one can easily figure out the topological charger numbers. These results may be an important guide for modern optical communication, the understanding of ENZ material properties, and strong field control.

2. Theory

In the following numerical demonstration, a vortex laser field polarized along the x direction propagates through the ENZ material along the z direction. First, the incident field is assumed to have a sech-shaped envelope and is expressed as [25,27]

Z_R=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, L^l_p is the associated Laguerre polynomial with p denoting the transverse radial node number (p = 0). 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. E₀ and ω₀ denote the field amplitude and central frequency, respectively. In our subsequent numerical demonstration, the Rayleigh range is Z_R ∼1260 µm with a₀ = 35 µm and λ = 3.055 µm. Thus the condition of a₀/Z_R << 1 is absolutely satisfied and other polarization components beyond E_x can be safely neglected [31–33]. Due to the unavoidable loss at the ENZ frequency, the medium thickness must be not two large [34,35]. In our numerical simulation, as for the propagation distance (thickness ∼4 µm), only 1 µm is exemplified, which is smaller than incident laser wavelength in order to reduce the loss. As a result, because the condition of Z_R >> z is satisfied, one can say a paraxial and quasi-plane-wave description of the electric field is in fact enough for our purposes.

Second, the interaction between the vortex laser and the ENZ material can be described by the coupled three-dimensional Maxwell equations and the paradigmatic-Kerr equation, which are expressed as [9,36,37]

P_s, ω_e, δ_e and ε_s are the saturation polarization, resonant frequency, loss coefficient and static dielectric permittivity, respectively. For large $|\overrightarrow P |$, it accounts for physically important high order nonlinear terms. In the linear regime $|\overrightarrow P |$<< P_s, the dielectric permittivity experienced by a time-frequency field can be directly obtained as

To choose its real part and set it is 0, one can get the ENZ frequency (corresponding to the zero-crossing-point as shown in Fig. 1(a)),

 

Fig. 1. Spectral distributions of (a) dielectric permittivity and (b) enhancement factor from the front surface of ENZ medium (internal electric field E_int and incident electric field E_inc).

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Third, the three-dimensional Maxwell equations are solved by employing Yee's finite-difference time-domain (FDTD) discretization scheme and the paradigmatic-Kerr equation by the Ruge-Kutta algorithm [9,36–39], and finally the transmitted electric field E(t, z) is recorded and its spectral characteristics are investigated via the fast Fourier transformation.

3. Results and discussion

As for the numerical simulation, the ENZ material is 60 µm away from z₀ = 90 µm along the propagation direction to ensure the pulse penetrates negligibly into the ENZ material initially. With suitable parameter values of δ_e = 0.01, ε_s = 1.2 and ω_e = 5.64 × 10¹⁴ s⁻¹ [9], the ENZ frequency is obtained ω₀ = 1.095ω_e from Eq. (6), at which the real part of the dielectric constant ε is exactly zero as shown in Fig. 1(a), but with a non-negligible absorption loss indicated by the non-zero imaginary part of ε. To highlight the nonlinear enhancement effects of this medium at ENZ frequency ω₀ = 1.095ω_e, another frequency point ω₀ = 1.2ω_e is chosen for comparison in nonlinear performance. In addition, the laser amplitude is E₀ = 1 × 10⁸ V/m whose laser intensity is I₀ = 1.33 × 10⁹ W/cm² and then the corresponding third order nonlinear susceptibility is χ⁽³⁾∼1.2 × 10⁻²⁰ m²/V² [9].

First, a laser pulse with long duration (τ₀ = 50 fs) interacts with the ENZ materials with the laser central frequency ω₀ = 1.2ω_e. As shown in Fig. 2(a) (see dashed line), just after a 1 µm distance of propagation (z = 151 µm), the sharp third- and fifth-order harmonic peaks occur and their intensities change of harmonic orders follows an exponential decay pattern based on the low-order perturbation theory. However, if we adjust the incident laser frequency equal to the ENZ frequency ω₀ = 1.095ω_e, the situation changes obviously (see solid line in Fig. 2(a)). one can see that the ENZ wavelength (ω₀ = 1.095ω_e, where the real part of permittivity ε is zero in Fig. 1(a)), the internal field (just transmitted after the front surface of the ENZ medium) is enhanced about 1.7 times (see Fig. 1(b)), indicating that the occurrence of the ENZ field enhancement effects. With the decrease of laser wavelength (the increase of ω₀), the field enhancement factor decreases obviously. Since the excitation field within the material is enhanced, the nonlinear harmonic radiation is naturally enhanced. Beyond this, another enhancement mechanism is attributed to the ENZ enhancement of nonlinearity coefficients [2,9,18], where the nonlinear refractive index $\Delta n = {{\Delta \varepsilon } / {\left( {2\sqrt \varepsilon } \right)}}$ becomes larger when the permittivity ε is small.

 

Fig. 2. (a) The spectral distribution at different laser frequencies ω₀ for laser pulse durations (a) τ₀ = 50 fs and (b) τ₀ = 20 fs.

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Second, when the incident pulse duration is reduced (for example τ₀ = 20 fs) as shown in Fig. 2(b), except the harmonic peaks are broadened as expected, both high order vortex harmonic signals for two cases of ω₀ = 1.2ω_e and ω₀ = 1.095ω_e are redshifted. The third order harmonic generation is near the position of 2.7ω₀ (0.3ω₀ redshift). The fifth order harmonic is shifted near the position of 4.5ω₀ (0.5ω₀ redshift), which indicates the more obvious redshift with the larger harmonic order. If the pulse duration is further reduced, the redshift of harmonic signals are checked to be more prominent (not shown here). Moreover, following the same method in the Ref. [35], the energy conversion efficiency of each order harmonic is numerically estimated using the ratio between the peak intensities of each harmonic and the incident laser. For the results in Fig. 2(b) in the case of 20 fs incident laser, the energy conversion efficiencies of the third-, fifth- and seventh-order harmonics are ∼3.3 × 10⁻⁵, ∼2.9 × 10⁻⁸, and ∼1.2 × 10⁻¹⁰, comparable with the theoretical and experimental values of ENZ AZO (∼2.8 × 10⁻⁴, ∼2.8 × 10⁻⁷, and ∼2.4 × 10⁻⁸) [35] and ICO (∼10⁻⁵, ∼10⁻⁸, and ∼10⁻¹⁰) [18].

In order to disclose the mechanism for harmonic redshift, the electric field waveforms are shown in Fig. 3. The shorter laser pulse leads to a high dω/dt over a short period of time which results in a sharp optical permittivity change [40,41]. The laser pulse after interaction with the ENZ materials has a strongly asymmetric time domain profile and slight reduction of the peak amplitude. In contrast, a longer laser pulse leads to a smaller dω/dt with pulse shape changed little in time domain [40,41]. So the rise and falling parts of the electric field waveforms have different degrees of steepness for different pulse durations. The steep rise part of the laser waveform introduces a spectral frequency redshift due to the self-phase modulation effects based on the third order nonlinearity, while the steep falling part of the electric field waveform contributes to the spectral blueshift [42]. As shown in Figs. 3(a) and 3(b) of the long laser duration (τ₀ = 50 fs), the rising and falling parts of the laser waveform are almost symmetric. In contrast, if the pulse duration is reduced (τ₀ = 20 fs), the symmetry of the laser waveform between the rising and falling parts breaks down. The rise part of waveform is steeper than the falling part, which introduces a more obvious redshift. So the electric field waveform can be modulated by changing laser duration and then the spectral shift can be controlled. Moreover, because the peak of the electric field amplitude modulated by shorter pump pulse is higher than that by longer pump pulse [40], the intensity of third- and fifth- order harmonics for laser pulse τ₀ = 20 fs are less than that for laser pulse τ₀ = 50 fs. However, because the absorption coefficient of fifth-order harmonic (5ω₀) is smaller than that of third-order harmonic (3ω₀), the intensity of third-order harmonics decreases more obvious than that of the fifth order harmonics. Another reason for the redshift maybe attribute to the non-constant field enhancement factor around the ENZ frequency (Fig. 1(b)), where the enhancement factor for the laser field at exactly ENZ frequency is not the maximal due to the existence of dephasing, and the maximum enhancement occurs at a little smaller frequency from the ENZ frequency. As for a Gaussian-like pulse with central frequency ω₀ is at ENZ frequency ω_new, after the front surface of ENZ material, the fundament laser field would peaked at a frequency which is larger than ω_new but smaller than ω₀, and correspondingly the harmonics also present redshifts.

 

Fig. 3. Instantaneous electric waveforms corresponding to spectra in Fig. 2.

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It is worth noting that the transverse electric field distributions of the high order vortex harmonics should be investigated (see Fig. 4) for ω₀ = 1.095ω_e and pulse duration τ₀ = 20 fs. According to the law that the topological number of each harmonic is equal to its harmonic order [19,22], the harmonic peaks with an obvious redshift are still the exact odd orders, which can be further indicated by the transverse electric field distribution of each harmonic radiation. As a result, near the positions of 2.7ω₀, 4.5ω₀, 6.2ω₀ and 8ω₀, the corresponding harmonic generations of third-, fifth-, seventh- and ninth-order can be found like the “windmill”, which may further support the explanations for the redshift. For the laser frequency ω₀ = 1.2ω_e, the similar phenomenon can also be obtained as shown in Fig. 5. However, the corresponding spectral intensities will decrease significantly compared that with ω₀ = 1.095ω_e.

 

Fig. 4. Transverse field distributions of (a) third-, (b) fifth-, (c) seventh- and (d) ninth-order harmonics in the transverse x-y plane with ω₀ = 1.095ω_e and τ₀ = 20 fs.

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Fig. 5. Same as in Fig. 4, but for ω₀ = 1.2ω_e.

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Finally, such phenomena in ENZ materials are not limited to the 3.055 µm as the wavelength case. One reason why we still use this value is that the start of our ENZ topic’s research follows the Ref. [9], thus we initially want to adopt the same parameters including those for laser and for ENZ materials just for comparing the results between us and them. The 3.055 µm wavelength is the ENZ wavelength, which can be tuned by doping concentration or annealing parameters [43]. In fact, the research conclusion drawn in this work is quite general irrespective of ENZ wavelength, therefore the predicted behavior must be realized in other wavelengths.

4. Conclusion

In conclusion, we have studied the high order vortex harmonic generations by the interaction between a vortex laser field and an ENZ medium. When the vortex laser of long duration interacts with the ENZ medium, the high-order vortex harmonics are enhanced due to the nonlinear enhancement effect near ENZ frequency. Interestingly, for the vortex laser of short duration, in addition to the harmonics enhancement phenomena, the obvious frequency redshifts occur, which becomes more obvious with the harmonic order increasing. Moreover, the harmonics of redshift are still the corresponding odd harmonics by the transverse electric field distribution of each order harmonic, which may further proved the explanations about the redshifts. These interesting results are helpful to extending the application of ENZ materials in ultrafast laser photon-electronics.