Diffraction of relativistic vortex harmonics with fractional average orbital angular momentum Download: 1241次
Light beams carry both spin angular momentum (SAM) and orbital angular momentum (OAM) in their direction of propagation. The average value of the SAM is associated with the polarization state, zero for linear polarization and
However, the majority of previous studies have investigated integer vortex beams, and focused optical vortices have also been studied[13,14]. Very few studies have examined the relativistic mechanism of fractional vortex beams[15]. In contrast to the circularly symmetric intensity of integer vortex beams, fractional vortex beams have a gap on the bright ring, which means that fractional vortex beams may provide more controllable parameters. Increasing attention has been given to fractional vortex beams[16
In this study, the diffraction of fractional vortex harmonics is investigated for a relativistic fractional vortex beam irradiating a plasma target, which is successfully explained by a two-step model. In the first step, the fractional vortex beam is incident on the target and produces a spot-shaped hole during the hole-boring (HB) stage, and harmonics are generated simultaneously. In the second step, harmonics are diffracted by the hole and propagate to the far field, which is calculated using the Fraunhofer diffraction theory. Besides, the diffraction of harmonics driven by the fractional vortex beams is compared with cases of integer vortex beams, which show that the diffraction effects on the generation of harmonics become apparent with an increase in the harmonic order for both fractional and integer vortex beams. This work facilitates not only a basic recognition of fractional vortex beams but also suggests some possible detection procedures for the vortex beams with fractional average OAM in the relativistic region according to the far-field diffraction patterns of harmonics, which is important for potential applications in quantum information and multiple microparticle trapping and manipulation.
In the first step, three dimensional (3D) particle-in-cell (PIC) simulation based on EPOCH[28] code is performed to elucidate the mechanism of the fractional vortex beam irradiating an over-dense plasma target. At
In this simulation,
Fig. 1. (a) Electric field of the incident pulse in the x –y plane at at . (b) Transverse intensity distribution of the incident beam averaged for the entire pulse at [the black dotted line in (a)] at . (c) Electron density distribution at when the pulse arrives at the target surface. The black lines in (b) and (c) are the outlines of the 1/e maximum intensity of the incident beam and the hole of the target. (d) Frequency spectrum of the laser field after interaction between the input pulse and target. The field signal corresponds to and at .
At
When the relativistic fractional vortex beam interacts with the target, both a spot-hole and harmonics are generated simultaneously. The harmonics are reflected from the target and propagate to the far field. As shown in the frequency spectrum in Fig.
The electric fields of the harmonics for the fractional vortex beam are asymmetric, and the corresponding average OAM is expected to be fractional, which is different from the case of the integer vortex beam. The transverse electric fields and intensity distributions of harmonics in the near-field are shown in Fig.
Fig. 2. Reflected transverse electric field distributions of the (a1) first, (b1) third, (c1) fifth, and (d1) seventh harmonics in the z –y plane at and . Corresponding transverse intensity distributions of harmonics averaged on entire pulse are shown in (a2)–(d2).
The 3D PIC simulation illustrates the first step well, where a fractional vortex beam is incident on a target and produces a spot-shaped hole during the HB stage, and harmonics are generated simultaneously. However, what can be measured by a charge-coupled device (CCD) in actual experiments are transverse patterns in the far field. Because the accuracy and simulation distance in the PIC simulation are limited, only harmonics in the near-field can be obtained. Therefore, harmonics propagating to the far field should be considered, which can be realized by diffraction theory in the second step.
Before analyzing the diffraction in the second step, it is necessary to determine the specific expressions of the fundamental and harmonics of the fractional vortex beam. The expressions can be obtained by the following steps. Firstly, the mode
Firstly, for the vortex beam with fractional average OAM, its mode is related to its average OAM, which is expressed as[20]
It has been verified that the average OAM is conservative during the harmonic generation of fractional vortex beams[15,31,32], where
Figure
Fig. 3. Relation between the mode of the th harmonic and its harmonic order for the fundamental frequency beam with . The gray dotted line is the scaling relation between and the harmonic order .
Secondly, the fractional vortex beam can be expanded by the superposition of integer vortex beams, and the superposition is determined by the Fourier series
Thus, the probabilities of each integer mode are
Fig. 4. Superposition of (a) first, (b) third, (c) fifth, and (d) seventh harmonics for different integer modes.
Finally, it has been demonstrated that the paraxial fractional vortex field propagating in the
Thus, the specific expressions of the fundamental and harmonics of the fractional vortex beam have been acquired.
The far-field diffraction patterns of the harmonics can be obtained by performing a Fourier transform of the complex amplitude distribution on the hole plane in space, given by
According to the aforementioned theory, the diffraction patterns of the harmonics for the fractional vortex beam in the far field can be obtained. As shown in Figs.
Fig. 5. Fraunhofer diffraction patterns of first, third, fifth, and seventh order harmonics of fractional (first row) and integer (second row) vortex beams under different conditions. The illustrations in the bottom left of the first column are corresponding holes. (a1)–(d1) Fundamental fractional vortex beam with diffracted by the hole shape of . (a2)–(d2) Fundamental integer vortex beam diffracted by the hole shape of .
In the case of the integer vortex beam, considering the LG beam for example, an annular hole[33] can be produced in the first step, as shown in the bottom left of Fig.
Comparing the diffraction patterns of harmonics for fractional and integer vortex beams, it can be found that different harmonics have their own unique characteristics. Consequently, the diffraction patterns may provide some potential detection methods for the vortex beam with fractional average OAM during relativistic interaction according to the far-field diffraction patterns of the harmonics.
From the calculations and previous discussions, it can be concluded that the Fraunhofer diffraction pattern is mainly determined by the hole shape, while the hole shape is determined by the shape of the incident laser beam. Therefore, the far-field diffraction patterns of the harmonics are mainly related to the incident laser, as demonstrated by our two-step model.
It has been shown that harmonics can be generated and propagate to the far field when a relativistic fractional vortex beam irradiates a solid target. The details of this process were successfully calculated using a two-step model. In the first step, a fundamental spot-shaped hole is produced by an incident fractional vortex beam, and harmonics are generated simultaneously. In the second step, harmonics are diffracted by this hole and propagate to the far field. This work not only is of fundamental importance for the understanding of the relativistic fractional vortex beam interaction with plasmas, but also suggests some potential detection procedures for the vortex beam with fractional average OAM during relativistic interaction based on the far-field diffraction patterns of harmonics, which is important for potential applications in quantum information and multiple microparticle trapping and manipulation.
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Shasha Li, Baifei Shen, Wenpeng Wang, Zhigang Bu, Hao Zhang, Hui Zhang, Shuhua Zhai. Diffraction of relativistic vortex harmonics with fractional average orbital angular momentum[J]. Chinese Optics Letters, 2019, 17(5): 050501.