Origin of unusual even-order harmonic generation by a vortex laser

Chaojin Zhang; Chengpu Liu

doi:10.1364/OE.27.037034

1. Introduction

With the development of few-cycle laser technology, novel investigations have been focused on the so-called extreme nonlinear optics [1,2], in which carrier-wave Rabi flopping [3,4], third-harmonic generation in disguise of second-harmonic generation [5] and on the other novel phenomena are disclosed. Recently, if the above few-cycle excitation laser is simultaneously spatially-inhomogeneity, even-order harmonics occur besides the general odd-order ones due to the broken inversion symmetry [6–9]. Moreover, the existing results show that the inhomogeneity of the excitation field can obviously extend the cutoff energy in the high-order harmonic generation process [6,7] and introduce an appreciable modification to the energy-resolved photoelectron spectra [8].

Beyond this, if the excitation laser possesses a phase singularity, that is a vortex laser (for example the simplest Laguerre-Gaussian (LG) beam), this phase singularity can transfer to high-order vortex harmonics [10–16], THz beams [17,18] or nonlinear precursors [19] via vortex transformation. Due the phase wind imprints an orbital angular momentum (OAM) to the beam [11], the high-order optical vortex beam has more potential applications than low-order ones such as in quantum cryptography [13] and photo-excitation [14]. In addition, based on the criterion that the topological charge number l_q of a certain q-order harmonic is directly proportional to its harmonic order q, l_q=ql (l being the topological charge number of the fundamental incident LG beam) [11–20], the vortex laser excitation finds the application in the origin clarification of unusual spectral components [9,16]. In the following, we consider the excitation laser both with the spatial inhomogeneity and with the phase singularity interacting with quantum wells, a nice target with convenient energy-gap tunability [9]. It is found that there exist unusual ‘even-order’ vortex harmonics, not real even-order ones. The origin of unusual ‘even-order’ harmonics is that the spectral width of odd-order harmonics is so broad that it tails and enhances at the spectral positions of the so-called even-order harmonics, whose descriptions about this origin is different with that in previous works [6–10].

2. Theory

This incident spatially inhomogeneous LG pulse polarized along x direction and propagated through the square quantum well along z direction is written as [6–9,19],

(1)$$\vec{E}(t = 0,\;r,\;\phi ,\;z) = {E_{lp}}\,{\mathop{\textrm {sech}}\nolimits} \left[ {\frac{{1.76({z - {z_0}} )}}{{c{\tau_0}}}} \right]\cos \left[ {\frac{{{\omega_0}({z - {z_0}} )}}{c}} \right][{1 + \varepsilon \cdot h(x )} ]{\vec{e}_x}.$$

Here ω₀ and τ₀ are a central frequency and pulse duration of full width at half maximum (FWHM) of intensity, respectively. h(x) represents the functional form of the inhomogeneous field, which usually approximated as a power series $h(x )= \sum\nolimits_{i = 1}^N {{b_i}{x^i}} ,$ with the coefficients b_i obtained by fitting the results from a finite-element simulation. Generally speaking, sometimes only the linear order term of h(x) retained, i.e., h(x) =x is applicable [6–10]. ɛ is called the inhomogeneous factor characterizing the spatial-inhomogeneity degree of the incident field. The field amplitude E_lp is defined as,

(2)$$\begin{aligned} {E_{lp}}(t = 0,\;r,\;\phi ,\;z) & = \frac{{{E_0}}}{{{{({1 + {{\widetilde z}^2}/z_R^2} )}^{1/2}}}}{\left( {\frac{r}{{a({\widetilde z} )}}} \right)^{|l |}}L_p^{|l |}\left( {\frac{{2{r^2}}}{{{a^2}({\widetilde z} )}}} \right)\exp \left( {\frac{{ - {r^2}}}{{{a^2}({\widetilde z} )}}} \right)\\ & \times \exp \left( {\frac{{ - ik{r^2}\widetilde z}}{{2({{{\widetilde z}^2} + z_R^2} )}}} \right)\exp ({ - il\phi } )\exp \left( { - i({2p + |l |+ 1} ){{\tan }^{ - 1}}\frac{{\widetilde z}}{{{z_R}}}} \right), \end{aligned}$$

with ${\widetilde z}$=z-z₀, Z_R the Rayleigh range, $a({\widetilde z} )$ the beam radius and a₀=a(0) the beam waist at z₀, $L_p^{|l |}$ the associated Laguerre polynomial. Here the most important factor charactering the laser vortex property is exp(-ilϕ) with l (l = 0, ±1, ±2,…) being the topological charge (TC) number, ϕ being the azimuthal angle, and lϕ indicating the helical phase of vortex laser. In addition, the p denotes the transverse radial node number [13–16]. E₀ and k=ω₀/c are the amplitude of the electric field and carrier wave number, respectively.

The Maxwell-Bloch (M-B) model describing the whole laser-matter interaction takes the form [10,16,19]

(3)$$\frac{{\partial \vec{H}}}{{\partial t}} ={-} \frac{1}{{{\mu _0}}}\nabla \times \vec{E},\quad \frac{{\partial \vec{E}}}{{\partial t}} = \frac{1}{{{\varsigma _0}}}\nabla \times \vec{H} - \frac{1}{{{\varsigma _0}}}\frac{{\partial \vec{P}}}{{\partial t}},$$

(4)$$\frac{{\partial {\rho _{12}}}}{{\partial t}} ={-} i\left( {{\omega_{12}}{\rho_{12}} + \frac{{d{E_x}}}{\hbar }n} \right) - \frac{1}{{{\tau _1}}}{\rho _{12}},\quad \frac{{\partial n}}{{\partial t}} = i\frac{{2d}}{\hbar }{E_x}({\rho _{12}} - \rho _{12}^\ast ) - \frac{1}{{{\tau _2}}}n.$$

Equation (3) is Maxwell equations used for describing laser propagation in media with $\vec{E}$, $\vec{H}$ and $\vec{P}$ being are the electric, magnetic vectors and macroscopic polarization, respectively. ς₀ and μ₀ are vacuum permittivity and permeability, respectively. In contrast, Eq. (4) is Bloch equations used for describing the medium response to the induced laser, with ρ₁₂ being the complex microscopic polarization and n=ρ₂₂-ρ₁₁ indicating the particle population difference at two intersubband energy levels with an energy difference of ħω₁₂ [21,22], which is related to the quantum well width, and the corresponding dipole transition matrix moment d [9]. The transverse dephasing time τ₁ and the longitudinal excited-state lifetime τ₂ are generally in the picosecond time scale [4,19] and can be safely ignored in the simulation due to the short-time laser excitation. The choice of z₀ ensures the excitation laser e penetrate negligibly into the medium at t = 0 [21]. The M-B equations are solved by employing Yee's finite-difference time-domain (FDTD) discretization [23] scheme combined with the predictor-corrector or Runge-Kutta algorithm [24–27]. In the following, the laser-matter interaction parameters with those are adopted [9,10,22]: ω₀=6.0×10¹⁴ s⁻¹ (∼3.14 µm), τ₀=18 fs (∼1.72 laser period), τ₁=1.0 ps, τ₂=0.5 ps, z₀=20 µm, a₀=12 µm, N = 6.0×10¹⁶ cm⁻³ and E₀=3.57×10⁸ V/m.

3. Results and discussion

The interaction between a few-cycle LG₁₀ (i.e., l = 1 and p = 0) beam and symmetric quantum wells is numerically simulated, in which the well-width is 62 Å (ω₁₂=1.11ω₀) [9,10]. The laser spectra under different spatial-inhomogeneity factors ɛ are first investigated. Without inhomogeneity ɛ=0 (solid line in Fig. 1(a)), there exist only the first and the third order harmonics due to the weak and short pulse duration. These two adjacent harmonics with an exponentially decaying intensity change and little increasing peak width, clearly indicating the characteristics of the low-order perturbative theory. When the spatial inhomogeneity of the excitation field is introduced (ɛ=0.006 for example), the additional spectral components appear between the adjacent odd-order harmonics (dashed line in Fig. 1(a)). When the inhomogeneity is further increased (ɛ=0.012), these spectral components grows into clear peaks positioning exactly at even multiple of the fundamental frequency (dash-dotted line in Fig. 1(a)). They can thus call as even-order harmonics. However, if we turn to check their corresponding transverse field distributions, the TC number for each harmonics is not consistent with the prediction from the vortex transformation criterion [10–16] that the topological charge number of each harmonic is directly proportional to its harmonic order, as shown in Fig. 2.

Fig. 1. Laser spectra under different (a) inhomogeneity degree, (b) propagation distances, and (c) pulse durations with τ₀=18 fs, ɛ=0.012, z = 50 µm and well width 62 Å (ω₁₂=1.11ω₀). (d) Same with (a), but for well width of 60 Å (ω₁₂=1.18ω₀).

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Fig. 2. Transverse field distributions of the (a) first, (b) “second”, (c) third, (d) “fourth”, (e) fifth, and (f) “sixth” order harmonics in the x-y plane, which are obtained by using a spectral filter with a width of 0.2ω₀. ɛ=0.012 and the other parameters are same with those in Fig. 1(a).

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From Fig. 2, one can find that for the first, third, and fifth -order harmonics, the transverse field distributions in the x-y plane perpendicular to the propagation direction indicate the TC number is the value as expected. For example, Fig. 2(a) clearly shows one positive and one negative beads, whose total shape looks like a Chinese Tai Chi diagram, indicating its TC number is one, which is consistent with its harmonic order 1. As for that for the third-order harmonic (see Fig. 2(c)), there exist three positive and three negative beads, indicating the TC number is 3, same with its corresponding harmonic order 3. The same point can be as well as for that in Fig. 2(e), where TC number is 5, equal to the harmonic order 5. Thus the TC number of any odd-order harmonic is directly proportional to its harmonic order, i.e., l_q=ql (here l = 1). However, if turning to the even-order ones (see Figs. 2(b), 2(d), and 2(f)), the TC numbers indicated by them are not equal to the corresponding harmonic order values. That shown in Fig. 2(b) exhibits 1 positive and 1 negative peaks and the TC number is 1, which is same with the first-order harmonic, but the harmonic number is 2. It seems that the first-order harmonic shows some contribution to the so-called second-order harmonic. The similar things happened to the other even-order harmonics (Figs. 2(d) and (2f)). This phenomenon seems propose a controversy, and why it occurs and how to solve it?

Natural focus will be on the spatial-inhomogeneity degree of the excitation field and propagation effects and maybe one wonder whether they make the above phenomenon occur. The transverse field distributions (see Fig. 3) for a smaller inhomogeneity factor show the similar pattern with those in Fig. 2, except weaker intensity, as well as those after a new propagation distance (see Fig. 4). Both inhomogeneity degree and propagation show certain influence on the spectral characteristics, but the above controversy is still retained. In addition, when the well-width changes from 62 Å to 60 Å (ω₁₂=1.11ω₀ to 1.18ω₀), the above phenomenon of transverse field distributions for a certain order harmonic can also be found as shown in Fig. 1(d) and Fig. 5.

Fig. 3. Same as in Fig. 2, but for ɛ=0.006.

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Fig. 4. Same as in Fig. 2, but for z = 32 µm.

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Fig. 5. Same as in Fig. 2, but for well-width 60 Å.

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However, thinking that the first-order harmonic shows the same TC number with the second-order harmonic (see Figs. 2(a) and 2(b)), the former seems to give some contribution to the latter. If we turn to changing the pulse duration of the excitation field, for example doubling from 18 fs from 38 fs, as seen from the spectrum in Fig. 1(c), the above unusual even-order harmonics significantly weaken, indicating the contribution from the large spectral width of a short excitation laser. For an even longer laser pulse, the corresponding calculation (not shown here) confirms that the spatial-inhomogeneity introduced even-order harmonics are normal with the TC numbers directly proportional to its harmonic order.

4. Conclusion

In summary, the spatially inhomogeneous few-cycle vortex beam propagating in quantum wells has been investigated with the introduce of the spatial-inhomogeneity of the excitation field, the even-order harmonics should occur beyond the conventional odd-order ones. However, the transverse field distributions of these even-order vortex harmonics presents that the corresponding TC numbers are not those predicted from the vortex transformation criterion that the topological charge number of each harmonic is directly proportional to its harmonic order. This controversy has been solved step by step via the investigations of the contributions from inhomogeneity and pulse bandwidth. If the contribution from large pulse bandwidth is dominant, the controversy is emerging. In contrast, if that from spatial-inhomogeneity is dominant, the transverse field distribution for each even-order vortex harmonics would show normal TC number. In addition, propagation effects and spatial-inhomogeneity both show certain influence on these unusual harmonics.

Funding

National Natural Science Foundation of China (Grants Nos. 61775087,11674312).

Acknowledgments

C.J.Z. gratefully acknowledges the support of open fund of the state key laboratory of high field laser physics of SIOM and the Key Program in the Youth Elite Support Plan in Universities of Anhui Province (gxyqZD2018039).

Disclosures

The authors declare no conflicts of interest.

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Origin of unusual even-order harmonic generation by a vortex laser

Abstract

1. Introduction

2. Theory

3. Results and discussion

4. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (5)

Equations (4)

Optics Express