Fractional vortex ultrashort pulsed beams with modulating vortex strength

Mengdi Luo; Zhaoying Wang

doi:10.1364/OE.27.036259

1. Introduction

The optical vortex beams with spiral phase, phase singularity and hollow intensity distribution have been boosted by the interest owing to their unique properties. The spiral phase plates (SPPs) are common devices for creating vortex beams. The SPPs were first manufactured and investigated in 1992 [1] and then designed to generate optical vortices in 1994 [2]. Also, the diffractive SPPs were manufactured as the “fork” amplitude grating [3] and as the spiral amplitude grating [4] to flexibly generate controllable optical vortex. In 1992, Allen group have demonstrated that the Laguerre-Gaussian beam has a well-defined orbital angular momentum (OAM) equal to $n\hbar$ [5]. The TCs characterize the transverse phase dependence, $exp ({in\varphi } )$. Thus, a complete round trip around the vortex implies a $2\pi n$ phase difference. Usually, the topological charge is an integer. Recently, there has been significant interest in optical vortices with fractional TC $\alpha$, called fractional vortex continuous beams (FVCBs), which can be generated through a non-integer spiral phase plate [6–9]. In a pioneering paper [8], the propagation properties of the FVCBs has only been theoretically studied and then numerous researches have been carried out both in theories and experiments, such as the birth of a vortex [6,7], topologically structured darkness [10], atom manipulation [11], Hilbert’s Hotel [9] and so on. More importantly, FVCBs also play an crucial role in the study of quantum information [12–14] and optical communication [15].

The theoretical work presented by Berry [8] points out that no fractional-strength vortices can propagate for fractional TCs. Thus, it is meaningful and important to study the vortex strength. Analogy with the Aharonov–Bohm effect, the total vortex strength ${S_\alpha }$ is proposed to evaluate the signed sum of all the vortices threading a large loop including the propagation axis [8]. The total vortex strength jumps by unit, which means that a vortex is born. The birth of a vortex for plane wave beams always happens when the fractional TCs are slightly larger than half-integer. Some experiments have carried out and confirmed this result [16–18]. Also, some studies on the birth of a vortex for the beams with finite width at Fraunhofer zone have been carried out and they found that the rule of a half-integer value to the birth of a vortex is not valid for Fraunhofer zone [6,7]. The vortices for a finite size beam (e.g. FVUPBs) is different from that generated by plane-illumination [8]. On the other hand, it is well known that a vortex beam generally carries a fixed OAM during propagation. However, more recently, increasing interest has been paid to control the topological charge of a vortex beam during propagation [19–22]. So, it is still meaningful to further study the TVS, the birth of a vortex and OAM.

When the width of pulsed beam is as short as femtosecond, time-space coupling effect is observed, combining the singular optics with particular features of ultrashort pulses like polychromaticity, high temporal resolution, and extreme intensities [23–26]. Many tools are employed to generate ultrashort vortex pulsed beams with integer TC, such as optical parametric amplification [27,28], cylindrical lenses [29,30], spiral phase plates [31,32], computer-generated holograms [33,34], or spiral multi-pinhole plates [35]. In 2019, Miguel A. Porras studied the upper bound to the orbital angular momentum carried by an ultrashort pulse [36] and the effects of the coupling between the orbital angular momentum and the temporal degree of freedom in pulsed Laguerre-Gauss beams [37]. In the newest research, the extremely-ultraviolet beams are demonstrated that they manifest as a temporal OAM variation along a pulse [38]. However, few studies have been carried out to explore the propagation properties of FVUPBs. Hernández García group theoretically investigated the extreme-ultraviolet attosecond pulse beams carrying fractional orbital momentum from high-order harmonic generation [39].

In most papers about the FVCBs, the relationship between the TVS ${S_\alpha }$ and the fraction TCs are only analyzed at Fraunhofer diffraction region. On the other hand, the total OAM of the entire optical field in each propagation plane remains unvaried but the local OAM is tunable [19–22], which is a very interesting phenomenon. Moreover, we do not find the researches on the study of the propagation properties of the FVUPBs and its TVS, which will be demonstrated in this paper. We would like to explore how the TVS of the FVUPBs varies during propagation and what kind of factor it depends on. The fields of the FVUPBs are obtained by the superposition of the ultrashort pulsed beams with integer TCs n [6–9] at arbitrary propagation plane. An analysis of the propagation of intensity and phase distributions of the FVUPBs for different durations and peak wavelengths has been presented in free space. We also calculate the vortex structure of FVUPBs and discover that the FVUPBs carry decreasing TVS during propagation. This special vortex structure depends on the propagation distance, peak wavelengths and the pulse durations in free space, which is totally different from the characteristics of FVCBs. According to our knowledge, this result about this special vortex structure of FVUPBs is first explored.

2. Theory

Firstly, let us briefly review the method of getting a pulsed beam proposed in [40]. The singly-ringed LG modes with integer TC n can be expressed as [15,40,41]:

(1)$$\begin{aligned}{E_{n,CW}}({\rho ,\varphi ,z} ) = &\sqrt {\frac{2}{{\pi |n |!}}} \frac{1}{w}{\xi ^{|n |}}{L^n}({{\xi^2}} ) exp \left( {ik\frac{{{\rho^2}}}{{2R}} - \frac{{{\rho^2}}}{{{w^2}}}} \right)\\ &{exp \left[ {in\varphi - i({|n |+ 1} )arctan \left( {\frac{z}{{{z_R}}}} \right)} \right]} \end{aligned},$$

where $k = \omega /c$ is the wave number, ${z_R}$ is the Rayleigh distance, $w = {w_0}\sqrt {1 + {z^2}/z_R^2}$ is the beam size and $R = z + z_R^2/z$ is the beam curvature. The function ${L^n}({\sim} )$ is the Laguerre polynomial with a parameter which is a square of the normalized transverse radius $\xi = \sqrt 2 \rho /w$.

To form the isodiffracting pulsed beam solution (${z_R} = {z_{R,{\omega _p}}}$), the broadband spectrum used here is

(2)$$f(\omega )= {({\omega {\tau_0}} )^s} exp ({ - \omega {\tau_0}} ),$$

where $s \ge 1$ and ${\tau _0} > 0$ are real parameters related to the bandwidth and the peak frequency (${\omega _p} = s/{\tau _0}$) of the pulse.

Combined Eq. (1) and Eq. (2), by taking the analytic inverse Fourier transform, we can get the vortex pulsed beam with integer TC [40]:

(3)$$\begin{aligned}{E_{n,UP}}({\rho ,\varphi ,z,t} ) &= \sqrt {\frac{2}{{\pi |n |!}}} \frac{{\xi _T^{|n |}{\cal L}_0^n({s,n} )}}{{\sqrt {1 + {z^2}/z_R^2} }}\frac{1}{{{{({1 + iT} )}^{s + 1}}{\beta ^{s + 1}}}}\\ &{exp \left[ {in\varphi - i({|n |+ 1} )arctan \left( {\frac{z}{{{z_R}}}} \right)} \right]} \end{aligned}.$$

where ${\cal{L}}_0^n({s,n} )= \Gamma ({s + |n |/2 + 1} )/\Gamma ({s + 1} )$, $\Gamma $ is the Euler gamma function, and the parameter ${\xi _T}$ is a time-coupled normalized transverse radius:

{\xi _T} = \sqrt {\frac{2}{{\beta ({1 + iT} )}}} \frac{\rho }{a}.

The parameter T is the function of time t and is defined as following:

(4)$$T = \frac{1}{{{\tau _0}\beta }}\left[ {t - \frac{1}{c}\left( {z + \frac{{{\rho^2}}}{{2R}}} \right)} \right].$$

The quantity $\beta = 1 + {\rho ^2}/{a^2}$ is a transverse scaling parameter, where $a = \sqrt {2c{\tau _0}{z_R}({1 + {z^2}/z_R^2} )}$ is the transverse radius of the pulse. From Eq. (3), we can find that the amplitude and phase of the vortex beam are both space-time coupled. Such a coupling effect between the space and time introduces some unique features which cannot be predicted from any space-time separable solution or from monochromatic waves [37,40].

Secondly, this paper mainly focuses on the study of FVUPBs. Following Berry [8], the fractional phase can be expanded in Fourier series:

(5)$$exp ({i\alpha \varphi } )= \frac{{exp ({i\pi \alpha } )sin ({\pi \alpha } )}}{\pi }\sum\limits_{n ={-} \infty }^{ + \infty } {\frac{{exp ({in\varphi } )}}{{\alpha - n}}} .$$

Then, the fractional vortex electric field is expressible as the superposition of the electric field with integer TCs n. Thus, the electric field for FVUPBs is given by:

(6)$${E_{\alpha ,UP}}({\rho ,\varphi ,z,t} )= \frac{{exp ({i\pi \alpha } )sin ({\pi \alpha } )}}{\pi }\sum\limits_{n ={-} \infty }^{ + \infty } {\frac{{{E_{n,UP}}}}{{\alpha - n}}} .$$

We would like to mention that in Eqs. (5) and (6), the expansion in Fourier series has an infinite number of terms with certain coefficients, which is the case for the FVCBs generated by a non-integer refractive SPP [6–9]. However, using coded phase diffractive optical elements (DOE), it is possible to form the FVCBs representing a finite sum of vortex beams with different weights [42–44]. Moreover, the diffraction pattern of FVCBs is well theoretically described by the superposition of two vortex beams with adjacent integer numbers [45,46]. Analogically, the analysis we proposed in our paper also demonstrates that the weighs of different integer TCs n decrease fast with the increasing of |n|. Thus, in our calculation, the FVUPBs can be obtained by the superposition of finite number of ultrashort vortex pulsed beams with integer TCs n.

Due to the interesting singularity structure of fractional vortex, a quantity to evaluate the signed sum of all the vortices called total vortex strength (TVS) is presented by Berry [8]. The TVS can be expressed as [6–9]:

(7)$$\begin{aligned}{S_\alpha} &= \mathop {lim }\limits_{\rho \to \infty } \left[ {\frac{1}{{2\pi }}\int\limits_0^{2\pi } {d\varphi \frac{\partial }{{\partial \varphi }}arg {E_{\alpha ,UP}}} } \right]\\ &= {\mathop {lim }\limits_{\rho \to \infty } \left\{ {\frac{1}{{2\pi }}\int\limits_0^{2\pi } {d\varphi {\mathop{Re}\nolimits} \left[ {({ - i} )\frac{{\partial {E_{\alpha ,UP}}/\partial \varphi }}{{{E_{\alpha ,UP}}}}} \right]} } \right\}} \end{aligned}.$$

The numerical integration to calculate the TVS ${S_\alpha }$ based on Eq. (7) is very sensitive to the value of radius $\rho$. The radius tends to infinity in Eq. (7), which cannot be realized in numerical calculation. In our paper, the value of TVS will approach to constant when the radius become bigger since the transverse energy is localized. So in our calculation of the TVS based on Eq. (7), we set the value of radius to $\rho = 1000{w_0}$ for any propagation plane and any α, which is large enough to involve all the vortices, instead of infinity. Thus, the TVS is independent with the radius of the loop. For fractional vortex, no fraction-strength vortices can propagate as mentioned by Berry [8]. Therefore, the jump in ${S_\alpha }$ means the birth of a vortex.

3. Simulations and discussions

To discuss the transverse vortex structure of the FVUPBs, we calculate the intensity and phase distributions of the FVUPBs based on Eq. (6) at different propagation planes. All the intensities are normalized by the maximum intensity of the FVUPBs with $s = 1$ at $z = 0.5{z_R}$ plane.

Figures 1(a)–1(d) show the theoretical results for the distributions of the intensity and phase of FVUPBs with $\alpha = 4.4$ at different propagation distances. Figures 1(a) and 1(b) are the distributions at the time when the intensity is maximum (${t_m}$). Figures 1(c) and 1(d) represent the distributions at the time of ${t_m} + 4{\tau _0}$. From Figs. 1(a)–1(d), we can find that the intensity profiles are all not perfectly symmetric in the x-y plane. It is very intriguing that there are five integer vortices on the phase pattern at the propagation distances of $z = 0.5{z_R}$ and $z = {z_R}$, which means a birth of a vortex. But there are only four integer vortices at the plane of $2{z_R}$ at the different propagation time. In the last picture in Fig. 1(d), the positive unit vortex is signed by “+” and the negative unit vortex is signed by “-”. Thus, there are still four vortices because of the generation of a pair of + 1 and −1 charge vortices. Moreover, compared Figs. 1(b) and 1(d), the number of vortex keeps the same for different time at a fixed propagation plane. So we would like to think about the relationship between the vortices number and the propagation distance.

Fig. 1. The normalized intensity and phase distributions of FVUPBs at the propagation distances of $0.5{z_R}$, ${z_R}$ and $2{z_R}$. (a) and (b) are the intensity and phase at the time when the intensity is maximum (${t_m}$). (c) and (d) represent the intensity and phase at the time of ${t_m} + 4{\tau _0}$.

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Figures 2(a) and 2(b) are the intensity and phase distributions of FVUPBs at the propagation plane of $5{z_R}$, $8{z_R}$ and $9{z_R}$ with $t = {t_m}$. We find that the number of vortex decreases from 3 to 1 with the increasing of the propagation distance from $5{z_R}$ to $9{z_R}$. Furthermore, the vortex will disappear in extremely far field. In one word, the propagation distance influences the number of the vortex for the FVUPBs, which is quite different from the FVCBs.

Fig. 2. The intensity (a) and phase (b) distributions of FVUPBs at the propagation distance of $5{z_R}$, $8{z_R}$ and $9{z_R}$ with $t = {t_m}$.

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Figures 3(a)–3(d) present the propagation properties of the FVUPBs for different durations and peak wavelengths. We can find that the pulse peaks appear at the time determined by $t - z/c - {\rho ^2}/({2Rc} )= 0$, which is consistent with Eq. (4), then defines a spherical pulse front of radius R at each distance z. Since $R = z + z_R^2/z$ depends on the wavelength, the pulse peaks located at different time for different peak wavelengths. In Figs. 3(a)–3(d), we also explore that the envelope of the FVUPBs slightly broadens as the peak wavelength increases. Because there are no big differences in the distributions of the FVUPBs for different peak wavelengths, in our following calculation, we set the center wavelength as 800 nm unless otherwise stated.

Fig. 3. The propagation properties of the FVUPBs for different durations and peak wavelengths. The solid curves represent the normalized pulse shape ${\mathop{\rm Re}\nolimits} (E )$, while the dotted curves are its modulus $|E |$. (a) s = 1, $z = 0.5{z_R} \, (\lambda = 800nm)$, (b) s = 10, $z = 0.5{z_R} \, (\lambda = 800nm)$, (c) s = 1, $z = 2.5{z_R} \, (\lambda = 800nm)$, (d) s = 10, $z = 2.5{z_R} \, (\lambda = 800nm)$.

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In [6], Jesus-Silva et al. have observed the jumps in the TVS of FVCBs at $\alpha = m + \varepsilon$, where m is the integer part of $\alpha$ and $\varepsilon$ is a small fraction. They mentioned that the beam waist increases with $\alpha$, and the value of $\varepsilon$ changes with $\alpha$ as well. In [7], Wen et al. argue the results from [6] that the loop radius $\rho$ for the calculation of ${S_\alpha }$ is too small. If the loop radius in the calculation is big enough (for example as 10w₀), the results in [6] will become those in [7]. In [7], they obverses the jumps in ${S_\alpha }$ of FVCBs only when $\alpha$ is around any even number (i.e. α=0, ±2, ±4, ±6, $\ldots $), where $\varepsilon$ is almost zero. In [6] and [7], they both explore the vortex birth at the Fraunhofer diffraction region (at focal plane) because the vortex of FVCBs is not stable in near field. Thus, carefully analysis is present to show the propagation properties of ${S_\alpha }$ in this letter. Also, we study the birth of vortex as a function of α at different propagation planes. And all the calculations of S_α as a function of α are dealt by the step of 0.01 and with $t = {t_m}$.

Firstly, we exhibit S_α for different fractional TCs α at the various propagation planes, as shown in Fig. 4. We can explore that whatever the value of α is, the number of the vortex decreases step by step to zero, which is the same phenomenon as we mentioned in Fig. 2. Meanwhile, when the value of α is smaller, the vortex structure can be stable over a longer-distance propagation. For example, for α=1.5, the vortex structure can remain stable in the region between z = 0 to about $z = 46{z_R}$ where S_α is one or two. While, for α=2.5, the vortex structure can remain stable in the region between z = 0 to about $z = 22{z_R}$, where S_α decreases from three to one.

Fig. 4. The TVS varies as a function of the propagation distance z for the FVUPBs with different fraction TCs α.

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Secondly, in our paper, the FVUPBs have broadband spectrum, it is necessary to analyze the TVS of the FVUPBs varying with different pulse durations and peak wavelengths. We compare the jumps in S_α of FVUPBs with s = 1, 5, 10 at $z = 3{z_R}$ plane in Fig. 5(a). For the FVUPBs, we have observed the jumps in the TVS as α=1.06, 2.11, 3.16, 4.22 and 5.27 for s = 10, α=1.20, 2.29, 3.37, 4.45 and 5.52 for s = 5, and α=1.51, 2.61, 3.70, 4.77 and 5.82 for s = 1. It is concluded that the birth of a vortex happens at $\alpha=m + \varepsilon$, and $\varepsilon$ varies with $\varepsilon$ and s. The results for the birth of vortex with s = 10 are completely different from s = 1 and s = 5. Therefore, the pulse duration has a strong impact on the birth of a vortex. Figure 5(b) shows the TVS as a function of $\alpha $ for the FVUPBs with different peak wavelengths at the plane of $z = 2.5{z_R}$ ($\lambda = 800nm$). We explore that the different peak wavelengths slightly influence the jumps in the TVS of FVUPBs. For λ=632.8 nm, the jumps in S_α happens at α=1.40, 2.49, 3.57, 4.64 and 5.69. For λ=800 nm, S_α jumps at α=1.46, 2.57, 3.65, 4.72 and 5.77. For λ=1064 nm, S_α jumps at α=1.54, 2.65, 3.73, 4.80 and 5.85.

Fig. 5. The TVS varies as a function of α for the FVUPBs (a) with different s, (b) with different peak wavelengths, (c) at different propagation distances.

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Finally, The TVS of FVUPBs as a function of $\alpha $ at different propagation distances z with s = 1 is shown in Fig. 5(c). For various z, the jumps in S_α happen at different ε. When $z = 0.5{z_R}$, there are jumps in S_α at α=1.07, 2.09, 3.12, 4.14 and 5.16. While $z = 3{z_R}$, jumps are at α=1.51, 2.61, 3.70, 4.77 and 5.82. Obviously, the jumps points are totally different with various z.

The characteristics of TVS are determined mainly by the phase of the FVUPBs. The variations of the wavelength of the laser light can be described as a change in the generated TC order. Due to the band spectrum of the ultrashort pulse, the different wavelength components have different features of TVS, so that the TVS of the FVUPBs depends on the pulse durations and peak wavelengths. Meanwhile, the S_α also depends on the propagation distance z. In our opinion, it is because that both the pulse duration and propagation distances influence the intensity and phase distribution of the beams. When z = 0, R→∞ and ${a^2} = sw_0^2$. If we consider the time of t = z/c, then T = 0. If the transverse parameter $\rho \gg a$, then $\beta \approx {\rho ^2}/{a^2}$, the ultrashort pulse in Eq. (3) can be simplified as follows:

(8)$${E_{n,UP}}({\rho ,\varphi } )\approx \sqrt {\frac{2}{{\pi |n |!}}} {2^{\frac{{|n |}}{2}}}{\cal L}_0^n({s,n} ){\left( {\frac{{{\rho^2}}}{{{a^2}}}} \right)^{ - ({s + 1} )}}exp ({in\varphi } ).$$

While in far field $z \gg {z_R}$, $R \approx z$ and ${a^2} \approx 2s{z^2}/({k{z_R}} )$, at the time of t = z/c, if $\rho \gg a$, then $\beta \approx {\rho ^2}/{a^2}$ and $T \approx{-} z/{z_R}$, the ultrashort pulse in Eq. (3) can be written as follows:

(9)$$\begin{aligned}{E_{n,UP}}({\rho ,\varphi } ) &\approx \sqrt {\frac{2}{{\pi |n |!}}} {2^{\frac{{|n |}}{2}}}{\cal L}_0^n({s,n} ){{({z/{z_R}} )}^{ - ({2 + s} )}}{{({z/{z_R}} )}^{ - |n |/2}}\\ &{{\left( {\frac{{{\rho^2}}}{{{a^2}}}} \right)}^{ - ({s + 1} )}}exp \left[ {in\varphi - i\left( {\frac{{|n |}}{2} - s} \right)arctan \left( {\frac{z}{{{z_R}}}} \right)} \right]. \end{aligned}$$

Compared with Eqs. (8) and (9), we can find a different term ${({z/{z_R}} )^{ - |n |/2}}$ in the amplitude. This term is the main reason for the dependence of the birth of vortex on the propagation distance. According to Eq. (6), the electric field of the FVUPBs is the superposition of the electric field with integer TCs n. Thus, we set the field ${E_{\alpha ,UP}} \propto \sum\nolimits_{n ={-} \infty }^{ + \infty } {{A_n}exp (i{\theta _n})}$, where the phase θ_n is strongly associated with the transverse phase $n\varphi$, and the term ${({z/{z_R}} )^{ - |n |/2}}$ manipulates the coefficient A_n for different integer TCs n at the corresponding propagation plane. When $z \gg {z_R}$, A_n is bigger for the field with smaller integer n, which makes the vortex disappear. For example, if α=2.5, $z = 30{z_R}$ and ρ=100w₀, then A₅=-1.79×10⁻⁷, A₄=-1.33×10⁻⁶, A₃=-1.72×10⁻⁵, A₂=6.92×10⁻⁵, A₁=8.41×10⁻⁵, A₀=1.47×10⁻⁴, A-₁=3.60×10⁻⁵, A-₂=7.70×10⁻⁶, A-₃=1.56×10⁻⁶, A-₄=3.07×10⁻⁷ and A-₅=5.93×10⁻⁸. Thus, when $n \le - 2$ or $n \ge 4$, the coefficient A_n is too small that we can almost ignore its intensity and phase contribution to the FVUPBs. Moreover, the coefficient A₀ is the biggest, so that its phase distribution is the key for the value of S_α, which leads to the disappearance of the vortex. But in near propagation field, A_n is not closely related to the propagation distance, so that for α=2.5, the vortex can stably propagates between z = 0 to about $z = 22{z_R}$, where the number of vortex decreases from three to one. Furthermore, for the same reason, ${\cal L}_0^n({s,n} )$ in Eq. (3) also effects the coefficients A_n for different s, which results in the dependence of S_α on the pulse duration. Here, the reason of the relationship between the total vortex strength ${S_\alpha }$ and the propagation distance is similar to [19], where the local topological charge of an anomalous Bessel vortex beam is inversely proportional to the propagation distance. In [19], they think the beam is a mixture of vortex beams with different OAMs, only one of which is focused in the center in a specific propagation plane, and others are out of focus and are in the background, so that the distribution of OAMs in different propagation planes changes.

Although, as all the discussion presented above, the vortex structure of the FVUPBs is complicated, which depends on plenty of parameters, we should not confuse this with the nature of its orbital angular momentum [16]. As these vortices occur at regions of zero intensity they carry no linear or angular momentum in themselves. In free space, the orbital momentum integrated over the whole beam is invariant under propagation. Based on Eq. (3), we integrate the square of the electric field amplitude over the whole transverse plane and time dimension, and confirm that it is conserved during the propagation. As we noted that, the fractional vortex electric field is a mixture of the electric field with different TCs n in Eq. (3). Therefore, the total orbital angular momentum of FVUPBs in any propagation plane remains constant.

4. Summary

In summary, we have theoretically studied the propagation characteristics and the total vortex strength of the FVUPBs obtained by the superposition of the ultrashort pulsed beams with integer TCs n. We present the transverse distributions of the intensity and phase of FVUPBs. The results show that the number of the vortex is propagation-distance-dependent and time-independent. To evaluate this property of the vortex structure of FVUPBs and study the birth of a vortex, we numerically calculate the TVS S_α. We demonstrate that the jumps and disappearance in S_α depend on the pulse durations, peak wavelengths, propagation distance and α for the FVUPBs, which is new features compared with FVCBs. Finally, we figure out that this special vortex structure of FVUPBs appears due to the weight of the pulsed beam with different integer TCs n. However, the total orbital angular momentum is invariant during propagation. According to the analysis, we can control the OAM during the propagation by the different durations and peak wavelengths of the FVUPBs. Our findings will bring distinct perspectives to research the OAM associated with the propagation distance which may find applications in quantum entanglement, optical communication and optical micro-manipulation.

Funding

National Key Research and Development Program in China (2017YFC0601602); Fundamental Research Funds for the Central Universities, China (2018FZA3005).

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Fractional vortex ultrashort pulsed beams with modulating vortex strength

Abstract

1. Introduction

2. Theory

3. Simulations and discussions

4. Summary

Funding

References

Cited By

Figures (5)

Equations (10)

Optics Express