Tuning the Ellipticity of High-Order Harmonics from Helium in Orthogonal Two-Color Laser Fields

Abstract

High-order harmonic generation in atomic systems driven by laser fields with tailored symmetries provides a powerful approach for producing structured ultrafast light sources. In this work, we theoretically investigate the ellipticity control of high-order harmonics emitted from helium atoms exposed to orthogonally polarized two-color laser pulses with a 1:3 frequency ratio. The polarization properties of the harmonics are governed by the interplay between the spatial symmetry of the driving field and the atomic potential. By numerically solving the time-dependent Schrödinger equation, we show that fine-tuning the relative phase and amplitude ratio between the fundamental and third-harmonic components enables selective symmetry breaking, resulting in the emission of elliptically and circularly polarized harmonics. Remarkably, we achieve near-perfect circular polarization (ellipticity ≈ 0.995) for the 5th harmonic, as well as highly circularly polarized 17th (0.945), 21st (0.96), and 23rd (0.935) harmonics, demonstrating a level of polarization control and efficiency that exceeds previous schemes. Our results highlight the advantage of using a 1:3 frequency ratio orthogonally polarized two-color laser field over the conventional 1:2 configuration, offering a promising route toward tunable attosecond light sources with tailored polarization characteristics.

1. Introduction

High-order harmonic generation (HHG) arises when atoms or molecules are exposed to a strong laser field, leading to the emission of high-frequency radiation [1]. These harmonics can serve as coherent light sources in the extreme ultraviolet (XUV) and even soft X-ray regimes [2,3,4,5]. Due to their broad spectral bandwidth, HHG is currently one of the most effective methods for producing attosecond (${10}^{-18}$ s) pulses [6,7,8]. The harmonic spectra carry information about the ionic structure of atoms and molecules, and have thus been widely applied in probing molecular structures and ultrafast dynamics [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. HHG is a highly nonlinear process that can be understood using the three-step model [24]: first, the electron escapes from the atom or molecule through quantum tunneling, triggered by the strong external laser field; second, the electron undergoes oscillatory motion in the laser field; and finally, it may return and recombine with the parent ion, emitting a high-energy photon.

Light inherently possesses the property of polarization. Compared with linearly polarized sources, circularly polarized XUV and soft X-ray sources provide an additional degree of freedom, which significantly broadens the application scope of attosecond light sources. By employing various experimental techniques, circularly polarized harmonics can be used to investigate material chirality, magnetism, and structural properties. For instance, they are employed in chiral-sensitive light–matter interactions such as photoelectron circular dichroism to identify chiral [25,26,27,28,29], angle-resolved photoemission driven by circularly polarized pulses [30,31], and ultrafast chiral dynamics in molecules molecules [32]. Additionally, magnetic circular dichroism spectroscopy in the X-ray regime allows for the study of magnetic materials and time-resolved magnetic structure imaging [33]. Circularly polarized attosecond pulses also play an important role in generating attosecond electron wavepacket currents that induce strong, time-dependent magnetic fields inside matter [34,35,36].

In the presence of a linearly polarized laser field, the liberated electron predominantly travels along a straight-line trajectory in one dimension. During recollision, the electron’s velocity is aligned with the polarization direction, resulting in linearly polarized harmonic emission. In contrast, under circularly polarized driving fields, the ionized electron moves in a two-dimensional plane with a continuously changing velocity direction, making recollision with the parent ion unlikely and thus suppressing efficient HHG [37,38,39]. However, by using bicircular fields composed of two circularly polarized pulses with different frequencies [40,41], the electron’s motion can still span a two-dimensional plane while allowing for recollision from various angles [42,43]. This enables the generation of circularly polarized high-order harmonics [44]. Due to their broad applications, generating circularly polarized harmonics has become a hot topic in the field of HHG.

In 1994, S. Watanabe et al. [45] experimentally demonstrated two-color phase control of the total ion yield in the tunneling regime, employing a 100 fs Ti:sapphire laser and its third harmonic. Their results showed that the inclusion of the third harmonic significantly enhanced the harmonic yield. More recently, in 2019, Milošević et al. [46] investigated the effect of the relative phase between the two frequency components of an orthogonally polarized two-color (OTC) laser field with a frequency ratio of 1:3 on the shape of the high-order harmonic spectrum, using the strong-field approximation (SFA). They reported that the typical plateau structure observed in single-color linearly polarized fields could be significantly altered depending on the phase delay, and that the harmonics generated under the 1:3 field could exhibit much higher ellipticity and energy than those from bicircular fields.

In previous studies, it has been challenging to generate circularly polarized high-order harmonics when using linearly or elliptically polarized driving laser fields. Although counter-rotating circularly polarized two-color fields can efficiently produce circularly polarized harmonics, the experimental setups involved are relatively complex. When using orthogonally polarized two-color fields, most studies have employed a 1:2 frequency ratio for the driving field. In contrast, our study adopts a less commonly explored 1:3 frequency ratio OTC field and successfully achieves the efficient generation of nearly circularly polarized harmonics.

In this work, we investigate orthogonally polarized two-color laser pulses in which the fundamental and harmonic components have a frequency ratio of 1:3. Notably, unlike the 1:2 configuration that predominantly generates linearly polarized high-order harmonics, the 1:3 configuration allows the generation of elliptically polarized harmonics due to the asymmetry introduced by the third harmonic component. We focus on the ellipticity control of high-order harmonics generated from helium atoms driven by an OTC laser field with a frequency ratio of 1:3. To investigate this process, we solve the time-dependent Schrödinger equation (TDSE) numerically. Our results demonstrate that by adjusting the relative phase and field strength ratio of the two laser components, specific harmonic orders can achieve circular or near-circular polarization.

The paper is organized as follows: Section 2 introduces the theoretical framework employed in this work. Section 3 presents the simulation results along with detailed analysis and discussion. Finally, Section 4 provides an overview of the primary outcomes of this study.

2. Materials and Methods

Under the electric dipole approximation and in the length gauge, the time-dependent wave function $\mathrm{\Psi }(x,y,t)$ of a helium (He) atom driven by an orthogonally polarized two-color laser field satisfies the TDSE (throughout this work, we adopt atomic units for all calculations):

$$i{\displaystyle \frac{\partial }{\partial t}}\mathrm{\Psi }(x,y,t)=\widehat{H}(x,y,t)\mathrm{\Psi }(x,y,t)$$

(1)

The corresponding Hamiltonian is:

$$\widehat{H}(x,y,t)=-{\displaystyle \frac{(p_x^2+p_y^2)}{2}}+V(x,y)+x·E(x,t)+y·E(y,t)$$

(2)

In our simulation, the helium atom is modeled using a simplified two-dimensional single-active-electron (SAE) approach, in which the electron motion is confined to a plane and governed by a soft-core Coulomb potential:

$$V(x,y)=-{\displaystyle \frac{q}{\sqrt{x^2+y^2+a^2}}}$$

(3)

where we set $q=1.5$ and $a^2=0.6$. These parameter values are chosen to model a helium atom. This model has been extensively applied in studies of strong-field phenomena and HHG [5,47,48], as it captures the essential physical mechanisms while significantly reducing computational cost. Note that in this work, we employ a two-dimensional single-active-electron model, where both the electron dynamics and the driving laser field are confined to the x-y plane. Accordingly, the discussion of harmonic polarization refers to the ellipticity or circularity of the electric field vector within this plane, rather than to the full three-dimensional polarization state with a defined propagation direction. In this context, the circularly polarized (laser) wave is assumed to have polarization components along the x and y directions. The propagation direction (z axis) is perpendicular to the $x-y$ plane in which the quantum dynamics of the wave packets are being calculated. The time-dependent dipole moments of the wave packet in the $x-y$ plane are then assumed to generate the emitted radiation. This approach is widely used in reduced-dimensionality strong-field simulations and provides a practical and insightful means to analyze polarization-resolved harmonic emission [49,50,51,52]. The calculated ionization potential of the helium atom in the ground state is 0.9 a.u. The corresponding ground-state wave function is shown in Figure 1.

The orthogonally polarized two-color laser electric field vector E(t) rotates in the x-y plane, and its components along the x and y axes are given by:

$$\begin{array}{cc}\hfill E_x\left(t\right)& =E_{0}f\left(t\right){\displaystyle \frac{1}{\sqrt{1+{\eta }^2}}}cos(\omega t+\phi ),\hfill \\ \hfill E_y\left(t\right)& =E_{0}f\left(t\right){\displaystyle \frac{\eta }{\sqrt{1+{\eta }^2}}}cos\left(3\omega t\right),\hfill \end{array}$$

(4)

where $f\left(t\right)$ is the trapezoidal envelope function of the laser pulse with one optical cycle of linear turn-on and turn-off and a total duration of eight optical cycles. $E_0$ is the peak amplitude of the fundamental field, $\omega$ is the angular frequency of the fundamental field, $\eta$ is the ratio of the field strengths between the third harmonic and the fundamental, and $\phi$ is the relative phase between the two components. The intensity ratio of the fields is 1:$\eta$.

We solve Equation (1) using the split-operator method to obtain the time-dependent wave function at each moment. To avoid unphysical reflection from the boundaries of the numerical grid, a masking function ${cos}^{1/8}$ is applied to the wave function at each time step.

The key computational settings and laser parameters used in our simulations are summarized in

Table 1. Computational and laser parameters.

Parameter	Value/Description
Spatial grid size	409.6 a.u. (radius) with 0.2 a.u. spacing
Absorbing boundary	51.2 a.u. thick cosine mask starting at 358.4 a.u.
Pulse duration	8 optical cycles (≈21.1 fs at 800 nm), trapezoidal envelope
Time step	0.1 a.u. (≈2.4 attoseconds)
Laser wavelength	Fundamental: 800 nm; third harmonic: 266 nm
Intensity	$2.5\times {10}^{14}$ W/cm2 for both fields ($E_0$ = 0.085 a.u.)

.

The acceleration dipole matrix elements in the x and y directions are calculated from the wave function as follows:

$$\begin{array}{cc}\hfill a_x\left(t\right)& ={\displaystyle \frac{d^2}{d{t}^2}}〈\psi (x,y,t)\left|x\right|\psi (x,y,t)〉\hfill \\ \hfill & =〈\psi (x,y,t)|{\displaystyle \frac{\partial V(x,y)}{\partial x}}-E_x\left(t\right)|\psi (x,y,t)〉,\hfill \\ \\ \hfill a_y\left(t\right)& ={\displaystyle \frac{d^2}{d{t}^2}}〈\psi (x,y,t)\left|y\right|\psi (x,y,t)〉\hfill \\ \hfill & =〈\psi (x,y,t)\left|{\displaystyle \frac{\partial V(x,y)}{\partial y}}-E_y\left(t\right)\right|\psi (x,y,t)〉\hfill \end{array}$$

(5)

The corresponding high-order harmonic spectra are obtained via Fourier transform:

$$S_{x,y}\left(\omega \right)={\left|\int a_{x,y}\left(t\right)e^{-i\omega t}dt\right|}^2$$

(6)

The ellipticity $\epsilon$ of the emitted harmonics is given by:

$$\epsilon ={\displaystyle \frac{\left|a_{+}\right|-\left|a_{-}\right|}{\left|a_{+}\right|+\left|a_{-}\right|}}$$

(7)

where $a_{\pm }={\displaystyle \frac{1}{\sqrt{2}}}(a_x\pm i{a}_y)$, $a_x$ and $a_y$ represent the frequency-domain components of the dipole acceleration in the x and y directions, respectively [53,54].

3. Results and Discussion

The numerical simulations presented in this article were implemented using FORTRAN (90 code), with a focus on optimizing harmonic generation. Figure 2 shows the harmonic spectra produced by irradiating a helium atom with an OTC (800–266 nm) laser field and a monochromatic linearly polarized 266 nm laser pulse, both with a peak amplitude of $E_0=0.085$. As seen in the figure, the harmonics generated by the orthogonal two-color field are odd harmonics. Moreover, beyond the 25th harmonic order, the y-component of the harmonic signal (red solid line) is significantly weaker than the x-component (black solid line). This is because the 25th harmonic is close to the cutoff of the monochromatic 266 nm field, as indicated by the blue solid line in Figure 2.

To generate purely circularly polarized harmonics, two conditions must be satisfied: (1) the phase difference between the x and y components of the harmonics must be $\pm 0.5\pi$, and (2) the magnitudes of the x and y components must be nearly equal. According to Figure 2, beyond the 25th order, the magnitude of the harmonic components in the x and y directions differ significantly. Therefore, in the following discussion of harmonic ellipticity modulation, we focus on harmonic orders below the 25th.

Figure 3a shows the variation of harmonic ellipticity with different relative phases between the two colors, under an orthogonal two-color field (800–266 nm) with $E_0=0.085$. The color map indicates the ellipticity: $+1$ denotes right-handed circular polarization, $-1$ denotes left-handed circular polarization, and 0 indicates linear polarization. It is evident that the harmonic ellipticity is strongly influenced by the relative phase of the laser fields. As the phase varies, the generated harmonics transition between left-circular, linear, and right-circular polarization.

To better illustrate the phase-dependent behavior, we show in Figure 3b the ellipticity of the fifth harmonic as a function of the relative phase. The ellipticity reaches 0.83 at a relative phase of $0.8\pi$, and $-0.83$ at $1.8\pi$. At $0.95\pi$ and $1.95\pi$, the ellipticity drops to zero.

To further investigate the origin of this modulation, Figure 3c presents the phase difference and amplitude ratio of the x and y components of the fifth harmonic as functions of the relative phase. The vertical axis denotes the phase difference, while the color indicates the amplitude ratio, with 1 representing equal magnitudes. At a relative phase of $0.8\pi$, the phase difference is $-0.5\pi$ and the amplitude ratio is close to 1, resulting in an ellipticity of 0.83. At $1.8\pi$, the phase difference is $+0.5\pi$ and the amplitude ratio is again near 1, yielding an ellipticity of $-0.83$. However, at $0.95\pi$, although the amplitude ratio is still close to 1, the phase difference is 1, and thus the ellipticity is 0.

This demonstrates that the relative phase modulates the harmonic ellipticity by altering both the phase difference and the amplitude ratio of the x and y components of the harmonics.

It is known that to obtain purely circularly polarized harmonics, two conditions must be satisfied simultaneously: the phase difference between the harmonic components should be $\pm 0.5\pi$, and the amplitude ratio of the x and y components must be 1. In Figure 3, we discussed that the ellipticity of the fifth harmonic is 0.83 when the relative phase is $0.8\pi$. At this point, the phase difference of the fifth harmonic is $-0.5\pi$, but the amplitude ratio is not exactly 1. To achieve purely circularly polarized harmonics, we adjust the laser field amplitude ratio to make the amplitude ratio of the harmonics equal to 1.

In Figure 4a, we show the ellipticity of the fifth harmonic at a relative phase of $0.8\pi$ for different amplitude ratios. It can be seen that the ellipticity of the fifth harmonic reaches 1 when the amplitude ratios are 1.08 and 1.1, indicating purely circular polarization. Figure 4b presents the variation of the amplitude of the fifth harmonic in the x direction (black diamonds) and y direction (red squares), as well as the phase difference (blue triangles), with changing laser field amplitude ratios. It is clear that at amplitude ratios of 1.08 and 1.1, the magnitudes of the harmonic components in the x and y directions are equal, i.e., the amplitude ratio is 1, and the absolute value of the phase difference is close to $-0.5\pi$, i.e., the phase difference is $\pm 0.5\pi$. Therefore, at this point, the ellipticity of the harmonic is 1.

Based on the results and analysis of Figure 3 and Figure 4, we can draw the following conclusions: for the fifth-order harmonic, adjusting the relative phase of the incident laser can change the phase difference between the harmonic components in the x and y directions, as well as alter the intensity variation of the harmonics in both directions. Moreover, the ellipticity of the harmonic exhibits a periodic dependence on the relative phase. By selecting a relative phase with a large ellipticity, such as $\phi =0.8\pi$, the ellipticity of the harmonic can be effectively optimized by tuning the amplitude ratio of the incident laser fields, thereby achieving circularly polarized high-order harmonics.

Above, we achieved circularly polarized fifth harmonic by adjusting the relative phase and amplitude ratio of the laser field. What about the other harmonics? How do they change with the relative phase and amplitude ratio of the laser field? In Figure 5, we show the ellipticity of the 5th, 17th, 21st, and 23rd harmonics as a function of the relative phase and amplitude ratio of the laser field. The horizontal axis represents the amplitude ratio of the laser field, the vertical axis represents the relative phase of the laser field, and the color represents the ellipticity of the harmonic. As shown in Figure 5a, for lower-order harmonics in the falling region (e.g., the fifth order), although the ellipticity $\epsilon$ exhibits a clear periodic dependence on the relative phase $\phi$, it remains stable with respect to variations in the amplitude ratio $\eta$ near certain values of $\phi$. This indicates that there exists a tolerance region with good robustness, which is favorable for experimental implementation. In contrast, for medium-to-high-order harmonics in the plateau region (e.g., the 17th, 21st, and 23rd orders in Figure 5b–d), the periodic response of $\epsilon$ to $\phi$ and $\eta$ becomes significantly weaker, and the overall trend is smoother. This weakening of periodicity implies that the ellipticity varies less across a broader parameter space, i.e., the system is less sensitive to parameter fluctuations and exhibits greater stability. As demonstrated in Figure 5, proper adjustment of the laser field parameters, specifically its relative phase and amplitude ratio, allows the ellipticity to reach 0.995 for the 5th harmonic, along with 0.945, 0.96, and 0.935 for the 17th, 21st, and 23rd harmonics, respectively.

Figure 5 shows that the possibility of achieving certain ellipticities in high-order harmonics arises from the interplay between the amplitude ratio and relative phase of the two-color orthogonally polarized driving laser fields. Specifically, these parameters control the vectorial composition of the driving electric field, which in turn determines the relative amplitudes and phases of the harmonic components along the x and y directions. By tuning the relative phase $\phi$ and amplitude ratio $\eta$, one can manipulate both the phase difference and intensity ratio of the harmonic emissions, enabling constructive or destructive interference that shapes the polarization state. When these parameters are optimized within specific ranges, near-perfect circular polarization (high ellipticity) is achievable. This tunability is rooted in the coherent nature of harmonic generation and the symmetry-breaking introduced by the two-color field configuration, allowing selective enhancement or suppression of harmonic components to produce the desired ellipticity.

The above investigation is based on the incident laser field with an amplitude of 0.085 a.u. This field strength is chosen to ensure both a relatively high efficiency of high-order harmonic generation and a sufficiently broad harmonic plateau. If the laser field is too weak, such as 0.04 a.u. shown in Figure 6a,b, the ionization is insufficient, making it difficult to generate strong harmonics, and the resulting plateau is narrow. On the other hand, if the laser field is too strong, for example, 0.1 a.u. as shown in Figure 6d, excessive ionization depletes the ground-state population, thereby reducing the harmonic yield. Therefore, considering the ionization potential of the helium atom, we select a field strength of 0.085 a.u. as the driving laser amplitude. At this field strength, as well as at nearby values such as 0.07 a.u. shown in Figure 6c, the harmonic plateau becomes broader while maintaining a sufficient ground-state population. In addition, under all these field strengths, harmonics with ellipticity close to circular polarization can be observed, as indicated by the blue dotted lines in Figure 6. By tuning the relative phase and amplitude ratio of the incident laser fields, harmonics with high ellipticity, approaching circular polarization, can be achieved.

4. Conclusions

In summary, our theoretical investigation of helium atoms driven by orthogonally polarized two-color (1:3 frequency ratio) laser pulses reveals three significant advances in harmonic ellipticity control. First, we identify optimal parameter ranges for generating near-perfect circular polarization ($\epsilon >0.99$) in the fifth harmonic. Second, we demonstrate that this control extends to higher harmonics (17th–23rd) while maintaining $\epsilon >0.93$, thereby overcoming previous limitations in harmonic order scalability. This effect arises because the relative phase and amplitude ratio of the driving field influence both the amplitude and phase difference of the harmonic components. These findings provide concrete experimental guidelines through our parameter optimization maps (Figure 5 and Figure 6), and suggest a new approach for generating femtosecond circularly polarized XUV pulses without requiring helical undulators.