# High Harmonics with Controllable Polarization by a Burst of Linearly-Polarized Driver Pulses

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{6}molecules [28], and with bi-chromatic co-propagating orthogonally polarized pumps driving atomic Ne gas [29]. Circular harmonics were also produced by mixing non-collinear circularly polarized counter-rotating pumps [30].

## 2. High Harmonic Generation Scheme

_{N}is the cycle time of the DS, and $\widehat{R}\left(2\pi /N\right)$ is a two dimensional (2D) rotation operator by an angle of 2π/N. This DS leads to circular harmonics [6], and is similar to the DS present in the bi-circular fields, but for a longer period. A general pulse-burst that upholds Equation (1) can be written as:

_{p}+ 1) optical cycles. E

_{0}is the field amplitude, ω

_{0}is the optical frequency, and A(t) is a real envelope function. The burst in total contains M repetitions of T

_{N}long cycles, and each of the individual linearly-polarized pulses is separated by a time τ from the next, such that the overall period of the DS is given in Equation (5), where T is the cycle of the optical frequency T = 2π/ω

_{0}. Figure 1 shows examples of such a pulse-burst for M = 1, and N = 4 and 5.

## 3. Analytical Model

#### 3.1. Circular Harmonics

_{p}is an even integer, and M is an odd integer. The vectorial components of the harmonic emission then take the form:

_{p}+ 1) fundamental cycles, δ(t) is the Dirac delta function, $\overrightarrow{a}\left(t\right)={a}_{x}\left(t\right)\widehat{x}+{a}_{y}\left(t\right)\widehat{y}$ is the vector polarization, and other parameters are as detailed previously. The assumed delta function form can be relaxed to some other broadened function, like a Gaussian, without quantitatively changing the analysis. In particular, broadening adds a spectral envelope function, which does not change the polarization.

_{N}the comb’s density is increased (the limiting factor of spectral resolution is the free spectral range: 2π/T

_{N}). By changing the delay between the linearly-polarized pulses, τ, one can shift the harmonic frequencies such that they are not necessarily rational fractions of the original optical frequency (according to Equations (5) and (12)). This gives exact continuous control of the emitted frequencies, and can be seen for example in Figure 2a compared to Figure 2b.

_{N}, acts the role of the distance between slits. The duration of each pulse (N

_{p}) and the fundamental optical period (T) together give rise to a spectral envelope function (as the first two terms in Equation (11) suggest), and thus correspond to the width of each slit.

#### 3.2. Polarization Control

#### 3.3. Photonic Conservation Laws

_{0}. Other types of photons in the driver are a result of multiple pulses within the burst, but are scarce due to the sinc envelope. We will return to this point later.

_{1}and N

_{2}are the number of right and left circularly polarized photons annihilated in the process, respectively, and the index “N” was omitted since the following analysis is performed for the general case of bursts consisting of N pulses. This is under the assumption that the orbital-angular momentum of the driven atom/molecule is unchanged in the process, otherwise it should be added to the analysis as well [39]. Equation (18) means there must be one more, or one less, right polarized photon than left polarized photons annihilated. Also, this means N

_{1}+ N

_{2}is an odd integer, which conserves parity (thus in this case SAM conservation is a much stronger constraint compared to parity conservation).

_{1}+ N

_{2}driver photons into a single high-harmonic photon, we must know the energies of each of the annihilated photons. However, since the driver is only quasi-monochromatic, it has an infinite amount of photons with different energies, making this analysis complicated. To overcome this complexity, we first assume N

_{1}> N

_{2}, meaning ${\sigma}^{\left(p\right)}=1$. Furthermore, due to the driver’s spectrum and its comprising photons, each annihilated photon can be characterized by a single integer number, q. Then the energy of the p’th emitted harmonic is:

_{1}= N

_{2}+ 1, which reads:

_{2}> N

_{1}results in a similar conclusion: a left circularly polarized photon can only be emitted at energies of $\frac{2\pi}{{T}_{N}}\left(Np+1\right)$.

_{0}are greatly preferred, seeing as there are simply many more of these types of photons. Additionally, since only an odd number of photons is annihilated due to Equation (18), multiple annihilation of these photons leads to odd integer multiples of ω

_{0}at a high probability. In contrast, in order to get emission near even integer multiples of ω

_{0}, two photons with energies close to ω

_{0}, plus an additional low energy photon need to be annihilated, which has a low probability to occur. Gaining intuition through the conservation-law arguments presented above is somewhat limited, since the number of possible channels contributing to the emission of a given harmonic grows combinatorically with the harmonic order, and as such is large. For instance, combinatorial arithmetics shows that there are a total of 21 different channels through which a photon of energy $\frac{2\pi}{{T}_{N}}\left(Np-1\right)$ with an index p = 15 can be created. Nevertheless, since the abundance of driver photons is limited to only few type of photons, not all harmonic channels contribute, and the general qualitative behavior of the HHG emission can still be grasped using the conservation-law picture.

## 4. Quantum Simulations

#### 4.1. Numerical Model

_{p}= 0.793 a.u.). The atomic potential is taken according to Reference [40]:

^{−4}, α = 3, and r

_{0}= 36 a.u. The laser intensity (I

_{0}) was set in the range of 10

^{14}W/cm

^{2}in all calculations, such that the overall ionization does not exceed 4%. Equation (23) was solved with a 3rd order split operator method [41,42]. The time and spatial grids were discretized on an L × L Cartesian grid for L = 120 a.u., with spacing dx = dy =0.2348 a.u., and dt = 0.02 a.u. Convergence was tested with respect to the grid densities and sizes. The dipole acceleration was calculated using Ehrenfest theorem [43]:

#### 4.2. Numerical Results

_{p}, and τ) the numerical spectra match the analytical model derived in Section 3. Simulations show harmonic emission at identical frequencies to those derived analytically in Equation (12). In terms of the ellipticity of the emitted harmonics, for the case of e = 1 all harmonics are fully circular (as seen in Figure 5a). For the non-circular case (e ≠ 1), the ellipticity closely follows the results of the analytical derivation of Section 3.2, and varies from 1 to 0 continuously (as seen in Figure 5a–e). The simplicity of our analytical model is a consequence of the temporal separation of the groups of “linear” HHG events from each pulse within the burst. Hence, the major features in the scheme allowing efficient harmonic polarization control result from simple interference phenomena. Significantly, the case of N = 4 presented in Figure 5, exhibits a “clean” spectrum that contains only the desired elliptical peaks, as expected from the analytical derivation. For other values of N, breaking the circular symmetry is accompanied by the appearance of new (previously forbidden) harmonic peaks, which radiate weakly. However, for N = 4 a 2-fold rotational symmetry prevents this, meaning that no energy is wasted through new harmonic channels. The numerical variation in peak intensities can be seen in Figure 5f for a sample spectral region, and clearly show a minimal coupling between the intensity and the target ellipticity, all the way from fully circular to fully linear (up to ~10%).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Example driving laser burst of linearly-polarized pulses according to Equation (2). (

**a**) N = 4, M = 1, N

_{p}= 2, τ = 0; (

**b**) N = 5, M = 1, N

_{p}= 2, τ = 0. In (

**a**) and (

**b**), A(t) is taken as a trapezoidal envelope with a rise and drop of one optical cycle long. Front view (Lissajous curve) of the pulse-burst is shown in inset, each arrow represents a linearly-polarized pulse within the burst. The pulses are numbered in red according to their chronological order.

**Figure 2.**Analytically calculated spectral power and relative intensity of left and right circular spectral components according to Equation (11) for several values of parameters. (

**a**) N = 4, M = 3, N

_{p}= 2, τ = 0; (

**b**) N = 4, M = 3, N

_{p}= 2, τ = 5.25; (

**c**) N = 5, M = 1, N

_{p}= 2, τ = 0; (

**d**) N = 5, M = 3, N

_{p}= 2, τ = 0. The spectra are shown for representative harmonic orders 20–22 since within the analytical model the spectrum is repetitive due to the delta function responses.

**Figure 3.**Analytical model for N’th order ellipticity control schemes for N = 3–9. The plot shows the deviation of the scheme progression parameter, e, from the ellipticity ${\epsilon}^{\left(N\right)}$ (calculated according to Equations (14) and (15)), where the respective N value is given above each curve. For large values of N the ellipticity closely follows a linear dependence on the scheme progression parameter, e, with continuous control over the polarization of the emitted spectrum, from linear, through elliptic, to fully circular. The inset above the plot shows schematically how the pulses in the burst are rotated upon the variation of $e$ in a manner that is symmetric about the x-axis. Arrows represent the original polarization axis of the pulses in the symmetric case yielding circular harmonics, and black arrows the direction in which these are rotated.

**Figure 4.**Analytically calculated spectral power of the driver field (Equation (17)), with left and right circular spectral components for: (

**a**) N = 3, M = 3, N

_{p}= 2, τ = 0; and (

**b**) N = 9, M = 3, N

_{p}= 2, τ = 0. Power is displayed in logarithmic scale.

**Figure 5.**Numerical time dependent Schrödinger equation (TDSE) simulations for: N = 4, M = 3, N

_{p}= 2, τ = 0, I

_{0}= 3 × 10

^{14}W/cm

^{2}, and λ = 800 nm, where the relative angle is changed according to Equation (14) to give a target ellipticity of: (

**a**) ε

^{(4)}= 1; (

**b**) ε

^{(4)}= 0.75; (

**c**) ε

^{(4)}= 0.5; (

**d**) ε

^{(4)}= 0.25. In each case the target ellipticity should be identical in all peaks in the spectrum. A Lissajous curve describing the shape of the pulse-burst is shown in inset, where blue arrows represent the linearly-polarized pulses in the burst. The numerically calculated ellipticity is indicated in black on top of each spectral peak. The spectra are presented for a selected region in the plateau 33–35ω

_{0}. (

**e**) Ellipticity calculated numerically for each peak in the same spectral region compared to the analytically predicted ellipticity from Equation (16), linear line indicated in dashed black. (

**f**) The intensity variation for all the peaks in the same spectral region as a function of target ellipticity, showing minimal coupling of ellipticity and yield.

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**MDPI and ACS Style**

Neufeld, O.; Bordo, E.; Fleischer, A.; Cohen, O.
High Harmonics with Controllable Polarization by a Burst of Linearly-Polarized Driver Pulses. *Photonics* **2017**, *4*, 31.
https://doi.org/10.3390/photonics4020031

**AMA Style**

Neufeld O, Bordo E, Fleischer A, Cohen O.
High Harmonics with Controllable Polarization by a Burst of Linearly-Polarized Driver Pulses. *Photonics*. 2017; 4(2):31.
https://doi.org/10.3390/photonics4020031

**Chicago/Turabian Style**

Neufeld, Ofer, Eliyahu Bordo, Avner Fleischer, and Oren Cohen.
2017. "High Harmonics with Controllable Polarization by a Burst of Linearly-Polarized Driver Pulses" *Photonics* 4, no. 2: 31.
https://doi.org/10.3390/photonics4020031