Conservation of orbital angular momentum for high harmonic generation of fractional vortex beams

Shasha Li; Baifei Shen; Xiaomei Zhang; Zhigang Bu; Weifeng Gong

doi:10.1364/OE.26.023460

1. Introduction

The orbital angular momentum (OAM) is related to the spatial structure of wave fronts. In 1992, it was shown by Allen [1] that beams with an azimuthal phase term $\exp (i l φ)$ had OAM that was linked to the azimuthal component of the Poynting vector [2], where l is the topological charge (TC) and $φ$ is the azimuthal angle. The OAM, a fundamental property of light, has added a new independent degree of freedom to light, giving rise to a rich variety of applications such as the optical trapping of atoms [3], optical tweezers [4,5], optical communications [6,7], quantum information and computation [8], super-resolution microscopy [9], and even astrophysics [10].

The majority of current studies have investigated integral vortex beams, but only a handful of studies have focused on fractional vortex beams [11–17]. It has been verified by many researchers [18–27] that OAM is conserved during harmonic generation of integral vortex beams, and the TC scales with the harmonic order. What’s more, the conservation of fractional angular momentum has already been studied theoretically in high harmonics generation in gases [11,28]. In [11], extreme-ultraviolet beams carrying fractional OAM is obtained driven by infrared conical refraction (CR) beams interacting with gas target, where CR beams is composed of two components with an average OAM of ± 1/2. It is demonstrated that the OAM of the q_th harmonic is ± q/2 out of the OAM conservation. In [28], HHG in gases is driven by a double-TC beam with fractional OAM. The OAM content of the harmonic vortex is broader in comparison to that of the driving field. Besides, the conservation of SAM is also studied theoretically and experimentally [29,30]. Different from these studies, the average OAM of the fractional vortex beam can take any value. Besides, since the target is an overdense plasma target, the harmonics generation is far more different with gases.

It is well known that integral vortex beam has a definite OAM. However, when it comes to the fractional vortex beam, composed of photons with different OAM values, the average OAM is available and has an important physical significance. In this paper, we demonstrate that the average OAM follows momentum conservation in plasmas using theoretical analysis and three dimensional (3D) particle-in-cell (PIC) simulation. In a 3D PIC simulation, a linearly polarized relativistic fractional vortex beam impinges on a solid foil. The harmonics are generated in the reflected beams after the beam interacts with the foil. The average orbital momentum of the fractional vortex beam is calculated, which shows that the average orbital momentum of the n_th harmonic is n times that of the fundamental frequency beam. This proves that the average orbital angular momentum is conserved during harmonic generation for fractional vortex beams. The conservation law has been proven analytically. Based on the vortex oscillating mirror (VOM) model [18], the front electron layer not only oscillates in the longitudinal direction, but also twirls in the azimuthal direction, and the azimuthal motion of electrons transfers the average OAM of the driving beam to high harmonics. In this work, we study high harmonic generation of the fractional vortex beam and extend the conservation law for the average OAM for an integral vortex beam to a fractional vortex beam. In the end, we give a further explanation of the source of the mirror’s vortex phase in the VOM model.

2. 3D PIC simulations

We begin with 3D PIC simulations based on the EPOCH [30] code. A relativistic fractional vortex beam is of normal incidence on the foil from the left at t = 0 [Fig. 1(a)] and is reflected [Fig. 1(b)] after interacting with the plasma target. At t = 0, the mode of the driving fractional vortex beam is $α = 1.7$ , which can be expanded in integral Laguerre-Gaussian (LG) modes. Here, the LG modes are described as

a ({LG}_{p l}) = a_{0} {(\frac{\sqrt{2} r}{r_{0}})}^{| l |} \exp (- \frac{r^{2}}{r_{0}^{2}}) \exp (i l φ) {(- 1)}^{p} L_{p}^{l} (\frac{2 r^{2}}{r_{0}^{2}}) \sin^{2} (\frac{π t}{2 t_{0}}),

where l is the azimuthal index and p is the number of radial nodes in the intensity distribution. In this simulation, p = 0, the spot size

r_{0} = 4 μ m

, and the pulse duration t₀ = 10T, where T is the driving laser period and

φ

is the azimuthal coordinate. The laser wavelength is

λ_{0} = 800 n m

, and

a_{0} = e E_{y} / m_{e} ω_{0} c = 5

is the normalized dimensionless laser electric field, where m_e is the electron mass, and e is the electron charge,

ω_{0}

is the fundamental frequency, c is the speed of light in vacuum. The foil occupies the region

10 λ < x < 12 λ

,

- 10 λ < y < 10 λ

, and

- 10 λ < z < 10 λ

. The foil density is 10n_c, where

n_{c} = m_{e} ω_{0}^{2} / 4 π e^{2}

is the critical density. The simulation box is

12 λ (x) \times 20 λ (y) \times 20 λ (z)

, and there are two macroparticles per cell. The simulation mesh size is

d x = λ / 60

in the x direction and

d y = d z = λ / 30

in the y and z directions.

Fig. 1 (a) Electric field E_y of the incident pulse in the x − y plane at z = 0 and at t = 10T. (b) Electric field E_y of the reflected pulse in the x − y plane at z = 0 at t = 20T after the incident beam is completely reflected from the solid target. The black dotted lines in (a) and (b) show the position $x = 5 λ$ , and the gray box shows the target position. (c) The OAM spectrum of the driving fractional vortex beam. The incident and reflected electric field E_y in the z − y plane at $x = 5 λ$ is shown in (d) and (e), which occurs at the same time as (a) and (b), respectively. (f) Transverse intensity distribution of the incident beam integrated for entire pulse at the same time used in (a) and (d).

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Figure 1 shows the incident beam details [see Fig. 1(a), 1(c), 1(d), and 1(f)] and reflected electric field distributions [see Fig. 1(b) and 1(e)]. At t = 10T, the incident electric field in the x – y plane at z = 0 and in the z – y plane at $x = 5 λ$ are shown in Fig. 1(a) and Fig. 1(d), respectively. At t = 20T, when the incident beam is completely reflected after interacting with the solid target, the reflected electric fields in the x – y plane at z = 0 and in the z – y plane at $x = 5 λ$ are shown in Fig. 1(b) and Fig. 1(e), respectively. The transverse distribution of the reflected electric field is not the same as the incident electric field because it is the superposition of different harmonics that are generated during the pulse interacts with the target.

The fractional vortex beam can be expanded into the superposition of eigen-modes with integer OAM [12], and the probabilities are peaked around the integer nearest to $α$ . As can be seen in Fig. 1(c), the probability is peaked at TC = 2 for the fractional vortex beam with $α = 1.7$ . Therefore, the transverse electric field [see Fig. 1(d)] is more similar to the distribution of the eigen-mode LG₀₂. The transverse intensity distribution of the incident beam integrated for entire pulse at t = 10T is shown in Fig. 1(f). On the one hand, the transverse intensity distribution of fractional beam has a gap with “C” shape, which is different from the circularly symmetric intensity of integral vortex beam. The characteristics of the fractional vortex beams have unique advantages in many applications. For example, the gap could be used to guide and transport particles [31] and improve the ability of optical sorting [32]. Because the fractional vortex beam can gradually change the OAM, it has a particular application in the study of high-dimensional entanglement states [33–36]. On the other hand, the fractional vortex beams have two intensity enhancement points where the pondermotive force is very strong, which may offer more advantages in particle manipulation and other potential applications.

The frequency spectrum of the laser field at t = 20T after interaction between the input pulse and target is shown in Fig. 2. The spectrum shows that the main harmonics are of odd order, which agrees well with the VOM model results [18].

Fig. 2 Frequency spectrum of the laser field after interaction between the input pulse and target. The field signal corresponds to $y = 5 λ$ and z = 0 at t = 20T.

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Figure 3 shows the transverse electric field distributions (the first row) and intensity distributions (the second row) of the first, third, fifth, and seventh harmonics of the reflected field at $x = 5 λ$ at t = 20T. Different from the integral vortex beam, the electric fields of harmonics are asymmetric, and the corresponding modes are expected to be fractional. There is an exceptional case when the incident pulse is of half-integer modes, which will be discussed later. The transverse intensity distributions of the harmonics still show a gap on the bright ring with the familiar “C” shape with the incident beam, as well as two intensity enhancement points.

Fig. 3 Reflected electric field distribution of the (a1) first, (b1) third, (c1) fifth, and (d1) seventh harmonics in the z − y plane at $x = 5 λ$ and t = 20T. Corresponding transverse intensity distribution of harmonics integrated for entire pulse are shown in (a2) - (d2).

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For the integral vortex beam, it has a definite OAM because the integral vortex beam is an eigen-mode of the angular momentum operator. Its average OAM is just equals to its TC (for convenience, let $ℏ = 1$ in our paper). The TC can be roughly obtained by counting the number of intertwined helices, and then the OAM can be deduced. But, this method is not suitable to fractional vortex modes since they are not the eigen-modes. However, we can calculate the average value of OAM of fraction harmonics. In the simulation we calculate the average OAM of fractional vortex harmonics by considering the entire pulse. Firstly, we calculate the total OAM of the entire pulse by $L_{x} = {\int (\vec{r} \times \vec{p})}_{x} d \vec{r} = {\int (\vec{r} \times (ε_{0} \vec{E} \times \vec{B}))}_{x} d \vec{r}$ . Secondly, the photons number N_p is obtained by dividing the total energy E_n by the single photon energy $ℏ ω_{n}$ . Finally, the average OAM of harmonics can be obtained by dividing the total angular momentum by the number of photons. These can be described by the following formula,

\bar{L_{x}} = \frac{L_{x}}{N_{p}} = \frac{L_{x}}{E_{n} / ℏ ω_{n}} = \frac{ω_{n} L_{x}}{E_{n}} ℏ .

Here

\bar{L_{x}}

and

L_{x}

are the average OAM and total OAM of the entire pulse in the propagation direction, respectively. N_p is the total number of photons. E_n is the total electromagnetic energy. And

ω_{n}

is the frequency of n_th harmonic.

In addition to the $α = 1.7$ mode, the 0.5 and 2.3 mode are also studied for comparison. The results of first, third, fifth, and seventh harmonics are shown in Table 1, where all the average OAM results for different harmonics are calculated at the same time (t = 20T). The $\bar{L_{x}} (ω_{0})$ and $\bar{L_{x}} (ω_{n})$ are the average OAM in the x direction for the fundamental frequency beam and n_th harmonic. RE is the relative error between the calculated results of the n_th harmonic and the fundamental frequency results multiplied by n.

Table 1. Average OAM of n_th Harmonics for Different Fractional Vortex Modes.

View Table

Figure 4(a) shows the results in Table 1, which clearly illustrates the relationship between $\bar{L_{x}} (ω_{n})$ , $n \bar{L_{x}} (ω_{0})$ , and the harmonic orders for different fractional vortex modes. The relative error can be ignored based on our calculated results. From these results, one can concluded that $\bar{L_{x}} (ω_{n}) = n \bar{L_{x}} (ω_{0})$ , i.e., the average OAM of the fractional vortex beam is conserved during harmonic generation.

Fig. 4 Relationship between $\bar{L_{x}} (ω_{n})$ , $n \bar{L_{x}} (ω_{0})$ , and harmonic orders for different fractional modes (a) and integral eigen-modes (b).

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The integral vortex eigen-modes (l₀ = 1,2,3) are also checked using the same method. The relationships between $\bar{L_{x}} (ω_{n})$ , $n \bar{L_{x}} (ω_{0})$ and the harmonic orders are demonstrated in Fig. 4(b) which confirmed that average OAM is conserved during harmonic generation.

3. Theoretical analysis

When an intense relativistic laser beam imprints on an overdense (n_e>>n_c) plasma surface, harmonic generation can be explained well by oscillating mirror (OM) model [37–48]. In the OM model, ions are treated as a fixed background, and the electron density is regarded as a rigid step function oscillating harmonically. The electron surface can reflect the beam like a mirror. Zhang et al. [18] proposed the so-called “vortex oscillating mirror” (VOM) model when a relativistic intense integral vortex beam incidence on an overdense plasma surface resulting in harmonic generation. Here, we extend this model to a fractional vortex beam. In the VOM model, the electron surface still oscillates harmonically though a fixed ion background. However, its oscillation phase $\exp (i 2 ω_{0} t + i \bar{L_{x}} (ω_{0}) ϕ)$ consists of two parts. The first part $\exp (i 2 ω_{0} t)$ arises from the pondermotive force, which leads to the mirror oscillation in the longitudinal direction. The second part $\exp (i \bar{L_{x}} (ω_{0}) ϕ)$ is derived from the transversally nonuniform pondermotive force due to the vortex beam, where the $\bar{L_{x}} (ω_{0})$ is the average OAM of the incident vortex beam, which results in the helical of the electron density in the azimuthal direction. As shown in Fig. 5(a) – 5(d), the driving vortex beam is the $α = 1.7$ mode, the transverse electron density distribution at $x = 10 λ$ when t = 14.25 T, 14.5 T, 14.75 T, and 15 T during one laser period rotates as the phase of the incident light changes.

Fig. 5 Transverse electron density distribution at $x = 10 λ$ when (a) t = 14.25 T, (b) 14.5 T, (c) 14.75 T, and (d) 15 T. The incident vortex beam is the $α = 1.7$ mode.

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For the fractional vortex beam, the incident beam can be expanded in integral eigen-modes, which can be described as

\begin{array}{l} E (ω_{0}) ~ a_{0} \exp (i ω_{0} t + i k x) \exp (i α ϕ) \\ ~ \frac{a_{0}}{π} \exp (i ω_{0} t + i k x) \exp (i π α) \sin (π α) \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m}, \end{array}

where

ω_{0}

is the frequency of the fundamental beam, and m is the integral vortex order. The average OAM of the incident laser can be calculated using

\bar{L_{x}} (ω_{0}) = \frac{〈 E (ω_{0}) | {\hat{L}}_{x} | E (ω_{0}) 〉}{〈 E (ω_{0}) | E (ω_{0}) 〉} = \frac{〈 \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m} | {\hat{L}}_{x} | \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m} 〉}{〈 \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m} | \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m} 〉} .

During the interaction of incident fraction vortex beam and the solid foil in the VOM model, we believe that the average rotation speed of the mirror is equal to the average rotation speed of the incident light, and therefore the vortex phase of the mirror is

\exp (i \bar{L_{x}} (ω_{0}) ϕ)

. Consequently, the reflected field can be expressed by

E_{r} ~ \frac{a_{0}}{π} \exp (i k x) \exp [i (ω_{0} t + κ \exp (i 2 (ω_{0} t + \bar{L_{x}} (ω_{0}) ϕ)))] \exp (i π α) \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m},

where κ is a parameter associated with the oscillating amplitude. After Fourier expansion of Eq. (5), we obtain

\begin{array}{l} E_{r} ~ \frac{a_{0}}{π} \exp (i k x) \exp (- i \bar{L_{x}} (ω_{0}) ϕ) \sum_{n = 0}^{\infty} [J_{q} (κ) \exp [i (2 n + 1) (ω_{0} t + \bar{L_{x}} (ω_{0}) ϕ)]] \\ \times \exp (i π α) \sin (π α) \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m} \end{array}

where J_q is the Bessel function of the first kind. From Eq. (6), it can be found that the reflected field indeed includes odd order harmonics, which agrees with results from previous studies.

Therefore, the electric field of the n_th odd harmonic is

\begin{array}{l} E (ω_{n}) ~ \frac{a_{0}}{π} \exp (i k x) \exp (i (n - 1) \bar{L_{x}} (ω_{0}) ϕ) J_{\frac{n - 1}{2}} (κ) \exp (i n ω_{0} t), \\ \times \exp (i π α) \sin (π α) \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m} \end{array}

where

ω_{n}

is the frequency of n_th harmonic, and n is the harmonic order. Its average OAM can be calculated using

\begin{array}{l} \bar{L_{x}} (ω_{n}) = \frac{〈 E (ω_{n}) | {\hat{L}}_{x} | E (ω_{n}) 〉}{〈 E (ω_{n}) | E (ω_{n}) 〉} \\ = \frac{〈 \exp (i (n - 1) \bar{L_{x}} (ω_{0}) ϕ) \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m} | {\hat{L}}_{x} | \exp (i (n - 1) \bar{L_{x}} (ω_{0}) ϕ) \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m} 〉}{〈 \exp (i (n - 1) \bar{L_{x}} (ω_{0}) ϕ) \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m} | \exp (i (n - 1) \bar{L_{x}} (ω_{0}) ϕ) \sum_{m = - \infty}^{\infty} \frac{\exp (i m ϕ)}{α - m} 〉} . \\ = n \bar{L_{x}} (ω_{0}) \end{array}

Equation (8) shows that the average OAM of the fractional vortex beam is conserved during harmonic generation and proves that the VOM model is suitable to integral and fractional vortex beams.

4. Discussion

The average OAM of a fractional vortex beam is $\bar{L_{x}} = α - \sin (2 α π) / 2 π$ [16]. When the TC is integer or half-integer, the average OAM is equal to its TC. The half-integer case is special because its even-order harmonics are integral eigen-modes while its odd-order harmonics are fractional modes.

Based on the conservation law for the average OAM, the $α_{n}$ value for the nth harmonic obeys

n (α - \sin (2 α π) / 2 π) = α_{n} - \sin (2 α_{n} π) / 2 π .

The blue solid line in Fig. 6 shows the relationship between the

α_{n}

value for the n_th harmonic and the harmonic order for the fundamental frequency beam with

α_{0} = 1.7

, while the gray dotted line shows the relationship between

n α_{0}

and the harmonic order n. Which indicates it does not follow a linear relationship.

Fig. 6 The relationship between $α_{n}$ of n_th harmonic and its harmonic order n for the fundamental frequency beam with $α_{0} = 1.7$ . The gray dotted line is the relationship between $n α_{0}$ and the harmonic order n.

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The OAM spectrums of harmonics in Fig. 7 give us an insight of the microscopic mechanism of the fractional vortex harmonic generation. The OAM spectrums of harmonics in Fig. 7 give us an insight of the microscopic mechanism of the fractional vortex harmonic generation. From the OAM spectrum of fundamental beam in Fig. 1(c) in our manuscript, it is shown that it is mainly composed by photons with l₁ = 1 and l₂ = 2, but also photons with other charges, including l = 0, higher chargers and even negative chargers. Therefore, there are many channels to produce a specific harmonic. That’s the reason why there are many eigen modes for the high order harmonics. The complex problem of the harmonics generation needs a further study.

Fig. 7 The OAM spectrum of reflected 3rd (a), 5th (b), 7th (c) order harmonics with fundamental beam of the mode α = 1.7. The OAM spectrum is obtained by performing the Fourier transformation of the laser field along the azimuth coordinate and integrating along the radius.

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Limited by the short simulation distance, the focusing effect hasn’t been shown. The fractional vortex beam can be expressed by the linear superposition of eigen-modes with integer OAM. Although the transmission process of each mode is linear and different eigen-modes can be focused on the same position in principle, the intensity at the focus may be affected since the relative phase of different eigen-modes may change during the propagation.

5. Conclusion

In conclusion, high harmonics are generated when a relativistic linearly polarized fractional vortex beam irradiates on a solid foil. Both simulation and theoretical analysis show that the average OAM of the n_th harmonic is n times that of the fundamental frequency beam, which demonstrates that the conservation law for the average OAM is suitable for not only integral but also fractional vortex beam during harmonic generation. This progress can be explained well by the VOM model theoretically.

For the integral vortex beam, based on the conservation law, the TC of harmonic scales with its harmonic order because the eigenvalue of the OAM is just its TC. For the fractional vortex beam, the average value of the OAM for the superposition state can be obtained according to the conservation law, and subsequently the modes of harmonics can be determined from the relationship between the mode order and the average OAM.

This work demonstrates momentum conservation for the average OAM of the fractional vortex beam during harmonic generation. The results open up the prospect of further studies of the harmonic generation of fractional vortex beams, as well as possible applications in quantum information and multiple microparticle trapping and manipulation.

Funding

National Natural Science Foundation of China (11674339 and 11335013); Strategic Priority Research Program of the Chinese Academy of Sciences (XDB16); Ministry of Science and Technology of the People’s Republic of China (2016YFA0401102 and 2018YFA0404803).

Acknowledgments

Thanks for the support of Innovation Program of Shanghai Municipal Education Commission and Shanghai Supercomputer Center.

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$α$	$ω_{0}$	$3 ω_{0}$			$5 ω_{0}$			$7 ω_{0}$
$α$	$\bar{L_{x}} (ω_{0})$	$\bar{L_{x}} (ω_{3})$	$3 \bar{L_{x}} (ω_{0})$	RE	$\bar{L_{x}} (ω_{5})$	$5 \bar{L_{x}} (ω_{0})$	RE	$\bar{L_{x}} (ω_{7})$	$7 \bar{L_{x}} (ω_{0})$	RE
0.5	0.4892	1.5233	1.4675	3.75%	2.5332	2.4458	3.57%	3.4582	3.4242	0.99%
1.7	1.8623	5.4057	5.5870	3.24%	9.1874	9.3116	1.33%	12.782	13.036	1.95%
2.3	2.1492	6.4744	6.4476	0.42%	10.778	10.746	0.30%	14.961	15.044	0.55%

Conservation of orbital angular momentum for high harmonic generation of fractional vortex beams

Abstract

1. Introduction

2. 3D PIC simulations

3. Theoretical analysis

4. Discussion

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (9)

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