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Abstract
We present a numerical study of the intensity and polarization structure of vector helical Ince-Gaussian (VHIG) modes, which present a distinct subclass of vector Ince-Gaussian modes with defined parameter settings. The intensity profile of VHIG beams has an elliptic hollow structure, while the polarization distribution shows multiple single-charge polarization vortices arranged along a line. By selecting the mode order, phase factor and ellipticity of the VHIG beams, we can control the number of elliptic rings, the number of polarization vortices, and the topology of the vector singularity. Furthermore, we simulate the focusing properties of VHIG beams based on vector diffraction theory. Our results indicate that the ellipticity parameter of VHIG beams could be a valuable degree of freedom to generate attractive transverse profiles and longitudinal distributions under focusing, which may have implications for lithography, material processing, optical communication, and even optical trapping and manipulation.
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1. Introduction
Structured light containing phase and polarization singularities has been an area of increasing interest for the past decades [1,2], with implications for fundamental research as well as scientific applications. The most representative examples of light modes defined in their spatial degrees of freedom are Hermite-Gaussian (HG) [3], Laguerre-Gaussian (LG) [4], and Ince-Gaussian (IG) [5] modes. They present the exact analytical solutions of the free-space paraxial wave equation (PWE) in Cartesian, circular and elliptic cylindrical coordinates, respectively. HG and LG modes can be understood as subsets of the IG modes: Changing the ellipticity parameter of the Ince polynomials, which define the spatial distribution, constitutes a continuous transitional mode change between HG and LG modes. IG beams have been generated in a stable resonator [6], and their propagation properties have been studied in free space, turbulent atmosphere and fibers to test the information transmission capacity of these beams [7–9]. Furthermore, IG beams are widely applied in optical trapping and manipulating due to their flexible spatial structure [10]. In the quantum regime, entanglement properties of photons in IG modes have been studied [11,12], as well as possible applications in quantum memories [13] and nonlinear frequency conversion [14,15].
With the appropriate weighting and phase factors, a superposition of IG beams can generate light fields with tailored structures. The superposition of two scalar IG beams with the same polarization can generate more complex phase structures, e.g. rectangular [16] and flower-shaped [17] phase vortex arrays. Experimentally, such mode superpositions can be efficiently and conveniently realized directly from a diode-pumped laser [18,19], and most studies have been conducted on phase vortices generated by superimposed scalar IG beams. This concept can easily be generalized to vector IG modes by combining two IG beams with different polarizations. In 2018, the first vector IG modes were constructed [20], demonstrating singularity networks that significantly enhance the number of degrees of freedom for information technologies. Since then, the vector IG mode have been explored more widely, including explorations of their generation [21], tightly focusing characteristics [22,23], classical entanglement characters [24] and application to nonlinear frequency conversion [25].
In this paper, we concentrate on a subclass of vector IG modes, namely, the vector helical Ince-Gaussian modes (VHIG). Their scalar equivalent, the helical Ince-Gaussian (HIG) mode, has been shown to display elliptical intensity rings and to contain several in-line phase vortices (phase singularity associated with orbital angular momentum) [26]. Inspired by these scalar studies, here we introduce the VHIG beams and study their topologies and focusing properties. Similar to previous work on vector IG modes [20], VHIG beams are generated from the superposition of even and odd IG modes with orthogonal linear polarization and identical parameters. The mode orders and phase factors of the constituent IG modes are chosen such that the resulting intensity distribution of the VHIG modes display elliptic rings with controllable numbers and ellipticity. The associated polarization distribution shows multiple in-line polarization vortices. Increasing the ellipticity parameter of our VHIG beams allows us to split a central high-order polarization singularity into multiple unitary ones arranged along a line, mirroring the behavior of scalar HIG beams. Interestingly, the different IG mode combinations and phase factors can generate four types of vector point singularities (V-points) [27]. Compared with commonly used vector modes, these tightly focused VHIG beams exhibit unique longitudinal and transverse electric field components controllable via the ellipticity parameter, generating succinct trapping potentials for applications in tweezing and optical maniupulation.
2. Properties of VHIG beams
In this section, we discuss details about the properties of VHIG beams. The even and odd scalar IG modes are defined as [5]
Thus, the VHIG beams can be constructed by combining even and odd IG beams with orthogonal linear polarization states as
Stokes parameters $(S_{0}, S_{1}, S_{2}, S_{3})$ are widely used to describe the state of polarization for any optical field. Following Eq. (2) they can be expressed as
The corresponding polarization ellipses of the light field can be obtained from the Stokes parameters: The ellipticity is given by $\chi =\frac {1}{2}\textrm{arctan}[S_{3}/\sqrt {S^{2}_{1}+S^{2}_{2}}]$, and the alignment of the major axis by $\Psi =\frac {1}{2}\textrm{arctan}(S_{2}/S_{1})$. Note that $\phi _{12}=2\Psi$ is also referred to as the Stokes phase [27]. For our VHIG beams, the Stokes parameters, and hence the polarization ellipses, vary across the beam profile. In the figures throughout this paper we show the intensity distribution as linear colormap, overlayed by black lines which indicate the local polarization directions. The associated Stokes phases, where shown, are indicated in separate plots in hue-colours.
In Fig. 1 we analyze the polarization and intensity structure of VHIG beams for varying ellipticity parameters $\epsilon$ and specific charges $p$ and $m$. We find that the number of V-points and their charge indices vary with the ellipticity parameter $\epsilon$. A V-point is characterized by its winding number, or Poincaré-Hopf index, quantified by the net rotation angle of polarization vectors surrounding the singularity [29]. An inhomogeneous distribution in the orientation of linear polarization can lead to the generation of a V-point [30]. For $\epsilon \rightarrow 0$, the obtained vector modes are the familiar cylindrical vector beams (CVBs), featuring a single V-point of topological charge $m$ as discussed in [20]. For $\epsilon \rightarrow \infty$, the spatial beam structure this V-point splits into $m$ independent singly charged V-points situated along a line. Intermediate ellipticity values $\epsilon$ illustrate the splitting process: The separation between the single-charged in-line V-points along the major axis of the ellipse increases with $\epsilon$, for finite values of $\epsilon$ positioned within the hollow core of the intensity ellipse. The number of elliptic rings is characterized by $N=[1+(p-m)/2]$, as already observed for scalar HIG beams [26], whereas for our VHIG we observe in addition the formation of $m$ in-line polarization singularites.
Fig. 1. Intensity and polarization profiles for VHIG beams as a function of the ellipticity parameter $\epsilon$, charge $p$ and degree $m$. Insets show the corresponding Stokes phase. In all cases, $\alpha = \pi$, $j = e$ and $j' = o$.
For scalar IG beams, the ellipticity parameter $\epsilon$ controls the intensity profile of beams. In contrast, for VHIG beams with a charge index of $p=m=1$ neither the intensity nor the polarization distributions change with $\epsilon$, as shown in the first column of Fig. 1. This is because for $p=m=1$ ordered VHIG beams the constituent (even and odd components) of the modal superposition also do not change with $\epsilon$. It is worth noting that the resulting light fields have similarities to first-order CVBs, following the same superposition principle of LG or HG beams with a phase shift. Instead, for VHIG beams with larger charge indices, increasing the ellipticity parameter $\epsilon$ modifies the intensity profile and leads to the generation of noncanonical in-line singularities. The latter is a manifestation of the general equivalence between interference patterns arising from superpositions of homogeneously polarized light modes, and polarization modulations arising from superpositions of orthogonally polarized light modes.
Thus far, we have considered only the intensity and polarization distributions of VHIG beams. In the following we will highlight their topological structure and nature of the polarization singularities. Specifically, we will discuss the role of the phase factor $\exp (i\alpha )$ and the choice of even and odd IG modes in Eq. (2) in determining the singularity type. We illustrate this for VHIG beams generated as superpositions of $\textrm{IG}_{222}^o$ and $\textrm{IG}_{222}^e$. For $p=m=2$ and an ellipticity $\epsilon >0$, each superposition generates a pair of single-charged V-points, but their configuration depends on the different polarization assignment and phase factors as shown in Fig. 2. We realize four different polarization distributions, classified as type-I, type-II, type-III and type-IV (for Figs. 2(a)–2(d) respectively). These four polarization distributions have a different angular dependence of the local polarization direction. For a unitary V-point, types I and III correspond to the well-known azimuthal and radial polarization distributions, whereas types II and IV correspond to conjugated modes of the previous two types [27]. The associated Stokes phases are shown in the bottom row, witnessing the existence of the four distinct topological structures. Here, we use the Stokes index (winding number of Stokes vortices) to characterize the singularity, as shown in Fig. 2 on each phase profile. The positive sign of the Stokes index corresponds to an increase of the Stokes phase in the counterclockwise direction as realized in Figs. 2(a) and 2(c), and the negative sign of the Stokes index corresponds to the clockwise direction as realized in Figs. 2(b) and 2(d). Again we note the analogy to properties of singularities of CVBs: Changing the phase ($\alpha$ in our work) or the basis on the Higher-Order Poincaré Sphere, also changes the polarization profile of the corresponding superposition. Phase values other than $\alpha = n \pi$ will lead to polarization patterns that contain elliptical polarizations, here and in the remainder of this paper we have chosen instead to concentrate on VHIG beams with purely linear polarization states.
Fig. 2. V-points and Stokes phases for selected superpositions of $\textrm{IG}_{222}^o$ and $\textrm{IG}_{222}^e$, showing (a) $\textrm{IG}_{222}^o \hat {x} + \textrm{IG}_{222}^e \hat {y}$, (b) $\textrm{IG}_{222}^o \hat {x} - \textrm{IG}_{222}^e \hat {y}$, (c) $\textrm{IG}_{222}^e \hat {x} + \textrm{IG}_{222}^o \hat {y}$ and (d) $\textrm{IG}_{222}^e \hat {x} - \textrm{IG}_{222}^o \hat {y}$. Each superposition corresponds to a pair of type-I, type-II, type-III and type-IV V-points, respectively, with intensity patterns and polarization distributions shown in the top row, and Stokes phases $\phi _{12}$ and Stokes indices in the bottom row.
Experimentally, the VHIG modes discussed in Fig. 1 and Fig. 2 could be generated by an interferometer combined with spatial light modulators [31] or digital micromirror devices [32], as they only contain the superposition of two different spatial modes, or alternatively by using dynamic modulation technique for automatically aligned vector modes with superior propagation behavior [33]. We note, however, that V-points are unstable under minor perturbation to the coherent superposition, which would render the theoretically predicted pure linearly polarized V-points to C-points (ellipse-field singularity).
3. Tightly focusing of VHIG beams
It is known from previous work that tight focusing of vector modes can exhibit unique properties due to the structured transverse component which generates non-negligible and spatially varying longitudinal components [34–39]. Here we simulate the tight focusing properties of VHIG beams, creating focus structures that change with the ellipticity parameter $\epsilon$. According to the Richards and Wolf vector diffraction theory in a vacuum environment [40], the field in the focal plane can be expressed as
Fig. 3. Focusing properties of VHIG beams $\textrm{IG}^{o}_{2,2,\epsilon } \hat {x}- \textrm{IG}^{e}_{2,2,\epsilon } \hat {y}$ with varying ellipticity parameter $\epsilon$. (a) represents the normalized intensity distribution of $|\vec {E}|^{2}$, $|{E}_{x+y}|^{2}$, $|{E}_{x}|^{2}$, $|{E}_{y}|^{2}$ and $|{E}_{z}|^{2}$ in focal field, respectively. The number in each image indicates the peak ratio. (b) and (c) show the cross section of the normalized intensity distribution for $|{E}_{x+y}|^{2}$ indicating by green dashed lines.
Fig. 4. Focusing properties of VHIG beams $\textrm{IG}^{o}_{6,6,\epsilon } \hat {x}- \textrm{IG}^{e}_{6,6,\epsilon } \hat {y}$ with varying ellipticity parameter $\epsilon$. Each column represents the normalized intensity distribution of $|\vec {E}|^{2}$, $|{E}_{x+y}|^{2}$, $|{E}_{x}|^{2}$, $|{E}_{y}|^{2}$ and $|{E}_{z}|^{2}$ in focal field, respectively. The number in each image indicates the peak ratio. The last column shows the 3D perspective of the normalized intensity distribution for $|{E}_{z}|^{2}$.
Figures 3 and 4 illustrate that tightly focused vectorial structures can exhibit spatially varying longitudinal and transverse electric field components. Here, the ellipticity parameter can provide a new degree of freedom to generate a desired intensity distribution at the focal plane. The spatial profile of the transverse component $|{E}_{x+y}|^{2}$ of VHIG beams under tight focusing resembles the intensity profile of the input light (see Fig. 1). It is instructive to observe the individual different behaviors of the $x$ and $y$ polarized components, the former combines centrally with increasing $\epsilon$, whereas the latter is unaffected. Moreover focusing partially converts radial polarization components into longitudinal ones, while azimuthally polarized light remains azimuthal. For the case of Fig. 3 the longitudinal light intensity arises predominantly from the contribution of $|{E}_{x}|^{2}$ along the $x$-axis, leading to a two-lobed pattern. An increase in ellipticity increases this contribution, and consequently increases the amount of light converted into longitudinal light intensity, as visible in the increasing peak ratio of $|{E}_{z}|^{2}$, accompanied by a decrease of the $|{E}_{x}|^{2}$ (and to a much lesser extend the $|{E}_{y}|^{2}$) peak ratio. This is of course a very simplified qualitative discussion, and more generally it is not only the local polarization direction, but also its phase distribution over the beam profile that determines the interference between the light distribution upon focusing [44–46].
The dependence on $\epsilon$ of the transverse light intensity $|{E}_{x+y}|^{2}$ is shown in detail in the intensity cross section along the x-axis in Fig. 3(b), and along the y-axis in Fig. 3(c). The former shows that with increasing ellipticity, the left and right lobes will be slightly compressed and move outward, and an additional central intensity peak appears. This is also reflected in the profile along the y-axis, with a double peak combining into a single peak with increasing ellipticity. Compared with the most commonly used CVBs [47], we can observe more structured intensity distributions at the focal plane, benefitting from the ellipticity parameter.
The possibility of generating structured focused fields becomes more striking for higher order superpositions, as demonstrated in Fig. 4 for a high order VHIG beams ($p=m=6$) with varying ellipticity parameter $\epsilon$. Here, changing the ellipticity parameter does not simply compress the intensity distribution, but provides different geometries of both the transverse and longitudinal intensity components on focus. The transverse component $|{E}_{x+y}|^{2}$ transforms from a bright ring around a star-shaped dark core to spatially separated intensity regions arranged horizontally, when the ellipticity parameter is increased. The lobes of the $|{E}_{x}|^{2}$ component above and below the horizontal line will merge together, whereas the $|{E}_{y}|^{2}$ component will be stretched in the horizontal direction without the concatenation. Due to the larger variation in local polarization, both $|{E}_{x}|^{2}$ and $|{E}_{y}|^{2}$ contain significant radial polarization contributions, and lose energy to the longitudinal light component, as can be observed in the varying peak intensity ratios. The last column shows the 3D view of the $|{E}_{z}|^{2}$ components, illustrating that the ellipticity $\epsilon$ provides an efficient transformation of transverse to longitudinal field components, generating geometries that encompass rotational as well as linear symmetries. Thus the distribution of optical force and torque will have an $\epsilon$ dependent structure, which can be applied to the specific manipulation and trapping of particles [48–50]. We also note that tailoring the longitudinal component of the light field could become useful for improving depth resolution in microscopy applications [51,52], testing the vectorial response of optically active materials [53,54] and other related fields [55].
4. Conclusion
In summary, we theoretically analyse the singularity profiles and focusing properties of VHIG beams, a subclass of vector IG modes with controllable elliptic hollow structures and unitary polarization vortices. In contrast to previous studies of elliptic symmetry vector modes [56–58], our VHIG modes are exact analytical solutions of the paraxial wave equation in elliptic cylindrical coordinates. We have illustrated that VHIG beams allow us to tailor the number of elliptic rings and V-points by appropriately choosing their mode indices. Moreover, an additional phase factor and the assignment of orthogonal polarization components can change the topology of V-point singularities. VHIG beams expand the range of the elliptic symmetry vector mode and establish the transformation of a multiple-order V-point into several in-line unitary V-points by adjusting the ellipticity parameter. Finally, we have shown that tightly focused VHIG beams could generate unique intensity distributions of the transverse and longitudinal light fields, which should be of significant interest for applications in material processing, optical communications and optical trapping and manipulation.
Funding
National Natural Science Foundation of China (11534008, 12033007, 12104358, 61875205, 92050103); H2020 Marie Skłodowska-Curie Actions (721465); Newton Fund (NIF/R1/192384); Engineering and Physical Sciences Research Council (EP/M01326X/1); China Scholarship Council (201906280228).
Acknowledgments
S. F.-A. acknowledges financial support from the European Training Network ColOpt, funded by the European Union Horizon 2020 program under the Marie Sklodowska-Curie Action. C. M. C. acknowledges support by the Royal Society Newton International Fellowship. M.A.-K. is supported through a QuantIC scholarship from EPSRC. J. W. acknowledges support by the China Scholarship Council.
Disclosures
The authors declare no conflicts 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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