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Abstract
The angular momentum (AM) of light has important implications for many fields of optics research, such as optics communication, quantum information and laser-scan microscopy. Thus, in this paper, we intend to investigate the average AM of the vortex ultrashort pulsed beam (VUPB), which is generated and widely applied in the field of high-intensity and ultrafast regimes. According to the vector potential, we firstly derive the electromagnetic field of the VUPB, then analyze the spatial intensity, the phase profile and the polarization state of the beam. We explore that the total AM per photon of VUPB is not $n\hbar$ value when the beam waist w0 is of wavelength or sub-wavelength order, since the polarization of VUPB is changed from linear to circular polarization with the decrease of beam waist, which introduces the conversion of the orbital AM to the spin AM. Moreover, based on the conservation of the total AM, the minimum waist of VUPB can be obtained, which is dependent on the pulse duration time α and topological charge n. Finally, the average AM of the fractional VUPB expressed by the superposition of the integer VUPB with different weights, is analyzed, which is not equal to the AM of fractional CW beam μ − sin (2μπ)/2π (μ is the fractional topological charge), but is associated with the beam waist w0 and α. Therefore, we think that the AM of VUPB can be controlled by adjusting the α and w0.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is well known that the optical vortex beam can carry spin and orbital angular momentum (OAM) [1,2]. The spin angular momentum (SAM) is purely intrinsic and associated with the polarization of the optical vortex beam, while OAM can be separated into intrinsic and extrinsic components, the former one is characterized by azimuthal phase and the latter is related to the beam propagation trajectory [3]. In 1992, Allen group [1] proved that a Laguerre-Gaussian mode possess OAM of $n\hbar$ where $n$ is the integer topological charge. Nowadays, researchers are devoted to control the average AM of the beam [4–12], In 2016, Wang et al. reported the fractional OAM carried by the azimuthal varying polarized vector fields [4]. Partially coherent vortex beams are utilized to control the OAM by adjustment of the beam parameters [6], increase the amount of OAM by contributions of vortex and twist phase [7] and explore the existence of the correlation-induced OAM changes [5]. Off-axis LG beams can also be employed to control the total AM of the beam by adjusting the length of the displacement vector [8,9]. The AM of the off-axis beam can even be fractional because of the extrinsic OAM resulting from a net linear transverse momentum [10]. Furthermore, as reported, the average AM of the fractional vortex beam is equal to $\mu -\sin (2\mu \pi )/2\pi$ ($\mu$ is the fractional topological charge) [11], which has been verified in experiment [12,13], thus, the average total AM can also be fractional [12].
The ultrashort vortex pulsed beam (VUPB), which introduces a vortex into an ultrashort pulse, has attracted great interest for research. The integer VUPB can be generated by many conventional methods such as cylindrical lenses [14,15], spiral phase plates [16,17], and computer-generated holograms [18]. Recently, for the VUPB, it has been proved that the topological charge, OAM and pulse duration time are strong correlated due to the spatio-temporal coupling [19,20]. The amount of OAM carried by the VUPB can affect its temporal properties, namely, it varies its time duration and its carrier frequency [21]. The pulse duration time somehow will influence the total vortex strength, which represents the signed sum of all the vortices [22]. Moreover, the topological charge $n$ does not completely indicate $n\hbar$ value of the angular momentum of the VUPB in some cases [23].
Recently, tight focusing is widely studied for its potential applications in microscopy, trapping and material processing. The beam is always focused into an extremely small waist: wavelength order or even sub-wavelength order. Numerous works have been carried out to explore the smallest focusing spot size. In 2003, Quabis et al. reported that a radially polarized field can be focused into a spot size significantly smaller than for linear polarization [24]. However, by adding a $\pi$-phase shift to one-half of a linearly polarized beam, the roles of the transversal and longitudinal field components of the focused beam are interchanged, resulting in better focusing [25]. Moreover, the smallest achievable feature size in petal beam can reach subwavelength dimension, and it depends on the OAM carried by the superposed light beams [26]. Generally, the beam size of the ultrashort pulse is smaller for achieving high power density. In our paper, we mainly focus on exploring the total angular momentum of the integer and fractional VUPB, and how the pulse duration time and topological charge $n$ influence the total AM and the minimum beam waist.
The paper proceeds as follow, firstly, the electric and magnetic components of the VUPB are derived through a Lorentz-gauged vector potential, which is the exact solution of the Maxwell equations beyond the paraxial approximation. The spatial transverse intensity and phase profiles are present and the polarization states are discussed. Then, the analytical formula of the average AM of the integer VUPB is obtained and utilized to analyze the minimum beam waist $w_{0,\min }$ in detail. Moreover, the numerical results of the average AM of the fractional VUPB are acquired. Finally, the spatial transverse intensity distributions are utilized to discuss the average OAM of the fractional VUPB depending on the pulse duration time and the beam waist.
2. AM of integer VUPB
In this section, firstly, we discuss the derivation mathematics for the three dimensional electric-magnetic field components of VUPB in free space. For simplifying the calculation, we set a vector potential that only distributes along the $x$ direction [1,20,27]:
Under the Lorentz condition $\nabla \cdot {{\mathbf {A}}} + \partial \varphi /\partial \left ( {{c^2}t} \right ) = 0$, combining with Maxwell equations ${{{\mathbf {E}}}_{\textrm {n,up}}} = - \nabla \varphi - \partial {{\mathbf {A}}}/\partial t$ and ${{{\mathbf {B}}}_{\textrm {n,up}}} = \nabla \times {{\mathbf {A}}}$, we can finally derive the expressions of electric-magnetic field components of the integer VUPB. The normalized temporal distribution of the integer VUPB is shown in Fig. 1 combining with indicated values of $\alpha$ and $n$. Here we should emphasis that there is a fundamental restriction to the topological charge of the vortex, and hence to the OAM, carried by a pulse beam, which is different compared to the continuous light, its topological charge can be infinite. According to the theory of M. A. Porras [28], the chosen parameters for the following calculation satisfy the relationship of $\left | n \right |<2\alpha$.
Figures. 1(a) and 1(c) reveal that the topological charge $n$ influences the pulse duration time and the shape even with the same $\alpha$ in the ultrashort pulsed beam region. The bigger $n$ broadens the pulse duration time. In other words, as the amount of OAM carried by the pulse grows, the pulse itself adapts to it by increasing the number of optical cycles so that it is able to perform under its envelope [21]. Nevertheless, by comparing with the curves in Fig. 1(a), such OAM-shape coupling effect vanishes as the increase of $\alpha$. So, we think that it is interesting to explore how the pulse duration time $\alpha$ and the average AM of the VUPB influence each other.
Fig. 1. The normalized temporal distribution of pulse shape Re[E] (solid) and its modulus |E| (dashed) with $\lambda _0=800nm$ and the indicated values of $\alpha$ and topological charge $n$. Re[ ] represents the real part of the VUPB.
To discuss the transverse vortex structure of the VUPB, we calculate the normalized intensity and phase distributions of the VUPB at $z=0$ plane, as shown in Fig. 2 and Fig. 3. From Fig. 2, we can see that when the beam waist $w_0=1mm$, the $x$-component of the VUPB dominates the intensity profiles, consistent with the vector potential. Then, the spatial intensity distribution is a ring-like pattern. We calculate the polarization of this VUPB (Fig. 4(a)-(b)) and find that it is a linear polarized beam, which resulting in zero longitudinal SAM. As $w_0$ decreases, the three components of VUPB are almost of the same order, as shown in Fig. 3 with the beam waist $w_0=0.4\lambda$. Especially, the $y$-component of the VUPB cannot be ignored compared with the $x$-component demonstrated in Fig. 3(a) and Fig. 3(b). We discuss the polarization distribution of the VUPB (Fig. 4(c)-(d)) and find that it is nearly a circular polarized beam, which results in the appearance of SAM [29]. Furthermore, contrasting with the first row and the second row of Fig. 3, the pulse duration time slightly alters the spatial profiles of the VUPB and weights of the $x$ and $y$ components. Thus, we conceive of the average AM varying with $\alpha$.
Fig. 2. Normalized intensity and phase(inserts) profiles with $z=0,w_0=1mm,n=3$, (a-c) $\alpha =3$ and (d-f) $\alpha =300$.
Fig. 3. Normalized intensity and phase(inserts) with $z=0,w_0=0.4\lambda ,n=3$, (a-c) $\alpha =3$ and (d-f) $\alpha =300$.
Fig. 4. Polarization states on the transverse plane of the VUPB with $n=5$ (a) $\alpha =3,w_0=1mm$, (b) $\alpha =300,w_0=1mm$, (c) $\alpha =3,w_0=0.4\lambda$, (d) $\alpha =300,w_0=0.4\lambda$.
The beam possesses linear momentum ${{\mathbf {p}}} = {{{\varepsilon _0}} \mathord {\left / {\vphantom {{{\varepsilon _0}} 2}} \right.} 2}\left ( {{\mathop {\textrm {Re}}\nolimits }\left [ {{\mathbf {E}}} \right ] \times {\mathop {\textrm {Re}}\nolimits } \left [ {{\mathbf {B}}} \right ]} \right )$ and angular momentum density ${{\mathbf {j}}} = {{\mathbf {r}}} \times {{\mathbf {p}}}$ [30], where $\mathbf {r}$ is the position vector. The angular momentum density of the VUPB with respect to a reference point ${{{\mathbf {r}}}_{\textrm {ref}}}$ is given by $\left ( {{{\mathbf {r}}} - {{{\mathbf {r}}}_{\textrm {ref}}}} \right ) \times {{\mathbf {p}}}$ . The average total AM of the pulsed beam per photon with respect to the origin (${{{{\mathbf {r}}}_{\textrm {ref}}} = 0}$) is thus found to be [31]:
For simplifying the calculation of the total AM, we set $z=0$ and $n>0$. Subscribing the electromagnetic component of the VUPB into Eq. (2), the $z$ component of the average total AM per photon $\left \langle {{J_z}} \right \rangle$ can be obtained. We should emphasize that the calculated AM $\left \langle {{J_z}} \right \rangle$ is the mixture of both SAM and OAM, and the OAM of the integer VUPB is compeletely intrinsic because of the perfect cylindrical symmetry of the spot distribution [10], as shown in Fig. 2 and Fig. 3. Under the conditon of $n\ge 2$, Eq. (3) where we set $\hbar =1$ shows the total AM per photon as a function of the beam waist $w_0$:
From Eq. (3), we can clearly note that the total AM is not directly proportional to $n$, but associated with the pulse parameters $w_0$ and $\alpha$. For the continuous wave (CW) condition, where we set $\alpha \to + \infty$, Eq. (3) can be transformed into:
Based on Eq. (4), we find that even for the CW beam, the average AM also depends on the beam waist, which is different from the Allen’s work [1], where the OAM of CW LG beam is equal to $n$ ($\hbar =1$). In our paper, only if $w_0$ is much bigger than wavelength, OAM is equal to $n$. When the beam waist of the VUPB decreases into the wavelength or sub-wavelength, the propagation of the VUPB no longer follows the paraxial approximation, causing the appearance of the three electromagnetic components. Then, the polarization state of the VUPB changes from linear polarizatin to elliptical polarization or even circular polarization, both the elliptical and the circular VUPB possess SAM, as shown in Fig. 4. Thus, we believe that the orbital-to-spin conversion leads to the appearance of the SAM. With the increase of the beam waist of the VUPB, the propagation of the VUPB satisfies the paraxial approximation and the spin-orbital interaction vanishes. As a result, the polarization state of the VUPB keeps linear, which is consistent with the potential vector, and the longitudinal SAM is zero. Thus, the average AM of the VUPB is equal to $n$. Analogous with [32], the proportionality of the intrinsic OAM to the topological charge $n$ holds true only in the paraxial approximation.
Undoubtedly, the total AM needs to be conserved in the spin-orbital interaction process. Thus, we assume that the SAM can be -1 when the beam waist is small enough. That is to say, the minimum AM of the VUPB after the orbital-to-spin conversion is $n-1$. Under this condition, we can discuss the minimum beam waist based on Eq. (3), which is a function of the pulse duration time $\alpha$ and the topological charge $n$.
Figure 5(a) explores that the topological charge and pulse duration time affect the minimum beam waist $w_{0,\min }$ of the VUPB. The minimum beam waist increases with the increase of $n$, and is almost proportional to $n$, as demonstrated in Fig. 5(a). Also, the pulse duration time $\alpha$ changes the amount of the minimum beam waist. Figure 5(b) shows the $w_{0,\min }$ with $n=4$ (insert) and $n=15$ as a function of $\alpha$. The value of the $w_{0,\min }$ decreases with the increase of $\alpha$ and tends to a constant value. In conclusion, both the value of topological charge $n$ and the pulse duration time $\alpha$ affect the minimum beam waist of the VUPB, and it can reach sub-wavelength order, which is consistent with the results in [20,24–26,33].
The average AM of the VUPB therefore varies with $w_0$, which is due to the orbital-to-spin conversion and the angular momentum conservation, as shown in Fig. 6. In the region of smaller $w_0$, the value of the average AM varies with $w_0$. For instance, when $w_0=0.4\lambda$, the polarization state of the VUPB is almost of right-hand circular polarization, thus the extreme value of the AM can be set as $n-1$. Then, as $w_0=0.8\lambda$, the polarization state of the VUPB is of slightly left-handed elliptical polarization, the value of the AM is a few larger than $n$. Finally, if $w_0$ is much bigger than the wavelength, the polarization state of the VUPB is linear polarization, then the value of the AM is fixed at $n$. Moreover, as shown in Fig. 3, the pulse duration parameter $\alpha$ changes the weight of the three components of the VUPB, i.e. orbital-to-spin conversion efficiency, causing the polarization state to vary with $\alpha$. As a result, the average SAM is associated with the pulse duration time $\alpha$. Qualitatively, this phenomenon is the manifestation of the spatio-temporal coupling effect. While analying the ultrashort pulsed beam, the temporal envelope also affects the orbital-to-spin conversion efficiency, especially for the smaller beam waist. Thus, as demonstrated in Fig. 6, with the increase of $n$, the maximum difference of average AM between the different duration time $\alpha$ enlarges, which is caused by the orbital-to-spin conversion efficiency depending on $\alpha$ and $n$.
Fig. 5. Minimum beam waist $w_{0,\min }$ as a function of (a) $n$ with $\alpha =6,15$ and $300$, (b) $\alpha$ with $n=4$ and $15$.
Fig. 6. The average AM of the VUPB as a function of $w_0$ with different pulse duration time $\alpha$ and topological charge $n$.
3. AM of fractional VUPB
In section 2, we have discussed the total AM of the integer VUPB, including the SAM and OAM. The intrinsic OAM is characterized by azimuthal angle dependence and phase singularity. The extrinsic OAM is related to the calculation axis whose underlying reason is the presence of a net linear momentum in the transverse plane [10]. For the integer VUPB, its extrinsic OAM is zero, but the fractional vortex beam whose transverse linear momentum is not equal to zero, has the complicated extrinsic OAM.
Following Berry’s theory [34], the fractional VUPB is expressible as the superposition of the integer VUPB with different weights, which is given by:
For analyzing the fractional VUPB, we give the numerical results of the transverse intensity distributions as shown in Fig. 7. Analogous with the integer VUPB, the beam waist highly influences the weight of the three components of the fractional VUPB. Additionally, the beam waist also alters the transverse spatial intensity pattern. For larger beam waist with $w_0=1mm$, the intensity distributions are not spatial symmetric any longer, the beam spot is a C-like pattern. And for smaller beam waist with $w_0=0.6\lambda$, the transverse intensity distributions are also spatial asymmetric as shown in Fig. 7(a) and Fig. 7(c). Thus, fractional vortex modes cause net linear transverse momentum resulting in extrinsic OAM. Compared with the first row in Fig. 7, the pulse duration time $\alpha$ also influences the spatial intensity distribution, i.e. transverse momentum of the VUPB. Therefore, the extrinsic OAM is related to $\alpha$. Moreover, when the beam waist is small, $\alpha$ also affects the orbital-to-spin conversion efficiency. Thus, the average AM of the fractional VUPB is the function of both $\alpha$ and $w_0$, as shown in Fig. 8.
Fig. 7. The normalizd intensity distributions of the three components of the fractional VUPB under the condition of $z=0$, $\mu =6.5$ and (a) $\alpha =6$, $w_0=0.6\lambda$, (b) $\alpha =6$, $w_0=1mm$, (a) $\alpha =300$, $w_0=0.6\lambda$, (d) $\alpha =300$, $w_0=1mm$ .
Fig. 8. The relationship between the fractional topological charge $\mu$ and the average AM of the fractional CW and VUPB with different $\alpha$, (a) $w_0=1mm$, and (b) $w_0=0.6\lambda$ .
Figure 8 shows the relationship between fractional topological charge and the total AM of the fractional VUPB. Here, we should mention that according to the theory of M. A. Porras [28], the fractional topological charge $\mu$ should be smaller than $\alpha$, so the various range of $\mu$ is set from 1 to 5 in this simulation. As for fractional CW vortex beam, the value of the angular momentum per photon is given by $\mu -\sin (2 \mu \pi )/2 \pi$ [11]. Siemens et al. have reported that the fractional vortex modes can be separated into coherent sums of intrinsic and extrinsic components, which in the case of fractional OAM means separating into light and dark modes [35]. The solid red line in Fig. 8(a) represents the average AM of the CW which is given by $\mu -\sin (2\mu \pi )/2\pi$. First of all, we compare the average AM of the fractional CW and fractional VUPB with $\alpha =6$ and $\alpha =300$ under the condition of a larger $w_0$, e.g. 1mm in Fig. 8(a). Then, we can explore that the difference of average AM between the CW and fractional VUPB is very weak. When the beam waist becomes smaller which is set as $0.6\lambda$ in Fig. 8(b), the results are more complicated. According to Fig. 6, the values of the average AM of the integer VUPB are associated with the pulse duration time parameter $\alpha$ when $w_0$ is smaller. Thus, in the calculation of the average AM of the fractional VUPB, although $w_0$ is fixed, the relationship of the value of the average AM between different $\alpha$ still varies with $\mu$. Furthermore, in Fig. 8(b), with the increasing of $\mu$, the difference of average AM between the different pulse duration time $\alpha$ enlarges, which is the same as the integer topological charge shown in Fig. 6.
4. Conclusion
In summary, we have derived the electromagnetic field of the integer and fractional VUPB beyond the paraxial approximation. When the beam waist is of wavelength or subwavelength order, the $y-$ and $z-$ components of the VUPB appear, causing the appearance of the average SAM. Thus, the average AM can be controlled by adjusting the beam waist of the VUPB. Moreover, we have obtained the analytical expression of the average AM of both the UVPB and the CW. Then, the minimum focusing beam waist can be derived by minimum average AM hypothesis ${\left \langle {{J_z}} \right \rangle _{\min }} = n - 1$, which is induced by the orbital-to-spin conversion. The total average AM of the fractional VUPB also have been analyzed in section 3. Analogous with the integer VUPB, the beam waist affects the average AM of the fractional VUPB. Furthermore, the average AM of the fractional VUPB contains both intrinsic and extrinsic AM. The pulse duration time and the beam waist alters the transverse spatial profiles of the fractional VUPB, resulting in changing extrinsic AM. These results may introduce some possible applications in the manipulation of particles by the VUPB.
Funding
National Key Research and Development Program of China (2017YFC0601602, 2018YFA0307200).
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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