Tight focusing of azimuthally polarized Laguerre–Gaussian vortex beams by diffractive axicons

This study presents a comprehensive theoretical investigation into the focusing properties of azimuthally polarized Laguerre–Gaussian vortex (APLGV) beams when interacting with different optical elements, including a linear axicon, binary axicon, and lens based on the Debye approximation. The research findings highlight the intriguing combination of polarization and vortex singularities within the APLGV beam, which result in distinctive focal shapes when interacting with these optical elements. The focal shapes achieved include multiple tightly focused spots and optical needles, which can be controlled by adjusting optical system parameters and beam characteristics such as the numerical aperture (NA), truncation parameter, beam order, and annular obstruction. These parameters can be carefully selected to achieve specific focal shapes with applications in multi-optical manipulation, particle acceleration, and trapping. By harnessing the unique properties of APLGV beams and optimizing the optical setup, researchers can explore new possibilities for advanced optical manipulation and control.

While refractive axicons are well-known for their ability to achieve tightly focused beams with high energy efficiency in the near-field region, diffractive axicons present unique advantages.Diffractive axicons enable precise focusing with high energy efficiency and a substantially higher NA, excel in the generation of longitudinally polarized beams, the attainment of uniform axial intensity, and the production of light segments that extend beyond the near-field zone without experiencing intensity reduction due to total internal reflection [1,4,10].Refractive axicons exhibit low chromatic dispersion but are limited by a NA value of approximately 0.75 which can be approach unity by fabricating refractive axicon from optically dense materials that are not transparent to visible wavelengths bands such as silicon [1,4,20].Furthermore, the fabrication of a refractive axicon of the desired quality poses other challenges attributed to the intricacy of the technology and the absence of effective methods for ensuring and certifying conical surface precision [21], so fabricating refractive axicon are more complicated than the diffractive axicon.On the other hand, diffractive axicons enable precise focusing with high energy efficiency and a substantially higher NA [1].Spatial light modulators can be employed in experimental setups to implement diffractive axicons under specific conditions [16,22], and they can be fabricated on curved surfaces using a direct laser writer [23].Consequently, diffractive axicons have practical applications in the precise focusing of polarized vortex beams and in polarization conversion.
On the other hand, the focusing of cylindrically polarized vortex beams has attracted significant attention due to their diverse focal shapes and unique properties, which have found extensive applications in particle manipulation, microscopy, nonlinear optics, and lithography [4,16].Radially polarized vortex beams have been effectively utilized to generate a range of focal shapes, including uniform long light tunnels [24], dark spots [25], and longitudinally polarized annular fields [26].Similarly, the focusing of azimuthally polarized vortex beams has led to the creation of super-long transversely polarized optical needles [27,28], tightly confined focal spots [2], and various other focal shapes [29].Notably, Khonina and Degtyarive made a significant discovery in their research, highlighting that diffractive axicons are the optimal optical elements for generating longitudinally polarized optical needles when circularly or radially polarized Gaussian vortex beams with high numerical aperture (NA) are employed as incident light sources [4].
Laguerre-Gaussian vortex beams with different radial modes (p > 0), known as multi-ring-shaped beams, have garnered growing attention due to their distinctive characteristics, making it a promising model in the field of beam shaping.The destructive interference between the rings in high-order Laguerre-Gaussian beams is effectively reduced the focal spot [30].For instance, without phase singularity, polarized multi-ring-shaped beams have been used to generate different focal shapes such as multiple focal segments [31,32], dark channels [33], and sharp focal spots [30].Furthermore, in our recent study, double focal spots and flat-topped focal segments are generated by focusing a radially polarized double-ring-shaped beam using a linear axicon in the presence of the phase singularity [3].Therefore, introducing a phase singularity in a tightly focused azimuthally polarized Laguerre-Gaussian vortex beam can produce novel and intriguing focal shapes with distinct properties.
In this paper, tight focusing of azimuthally Laguerre-Gaussian vortex beams by diffractive axicons were numerically investigated based on vector Debye theory.Vector diffraction theory is described in section 2. In addition to the axicon parameters, the effects of the beam parameters on focused beam shapes are presented and discussed in section 3. Finally, we present a conclusion and summarize our findings in section 4.

Theoretical model
Based on the Debye approximation, the vector electric field in the focal region of an aplanatic optical system in cylindrical coordinate system can be described by [34]: sin sin sin cos 0 exp cos sin cos . 1

*
Here, E(θ, j) represents the transmission function of the pupil, which describes the distribution of the incident beam at the aperture.T(θ) represents the transmission function of the axicon lens, which accounts for the specific properties of the axicon.The coordinates (r, j, z) correspond to the cylindrical coordinates in the focal region, while (θ, f) represent the angular spherical coordinates in the pupil.Besides, the parameter f denotes the focal length of the optical system, λ represents the wavelength of the incident beam, NA arcsin( ) a = corresponds to the convergence angle associated with the radius of the incident optical aperture, and NA represents the numerical aperture of the optical system.The parameter σ refers to the inner focusing angle or the annular obstruction, depending on the specific case, and k = 2π/λ represents the wave number.By integrating over the angular and axial dimensions, the equation allows for the calculation of the electric field distribution in the focal region.The terms involving the transmission functions and the exponential factors account for the effects of the pupil and axicon lens, as well as the phase and amplitude variations introduced by the optical system.
In cases where a vortex phase with radial symmetry is present, the transfer function can be expressed as follows: where m denotes the topological charge of the incident vortex beam.The transmittance function of the diffractive linear axicon or classical axicon is given by [10,16]: where α 0 refers to the axicon parameter that associated with the numerical aperture of the optical element and the signs '+' and '−' correspond to the defocusing and focusing axicons, respectively.It has been found that the maximum value of the intensity distribution of the focused beam located either to the right of focus, herein the focus is at z = 0, in the case of a defocusing axicon or to the left of the focus for the focusing axicon [16,29].When a binary axicon is implemented for focusing APLGV beams the complex transmission function can be written as [16] The binary axicon has been employed to create binary focal segments before and after the focal plane, such as Bessel beam with a double ring in the focal plane [16].This can be explained by approximating the complex transmission function as a superposition of two complex conjugated defocusing and focusing linear axicons.Consequently, the complex transmission function can be rewritten as [10,16]: By substituting equation (2) in (1) and using the following relation [35] where J is the Bessel function of the first kind of order m.The expression in equation (1) can be reduced to a single integral as follows:

*
From the final equation, it can be deduced that the longitudinal component is zero for all values of m, and the radial component is equal to zero only when m equals 0, while preserving azimuthal polarization and maintaining zero intensity on the optical axis.In addition, for |m| = 1 the first-order vortex phase singularity, there is a maximum intensity on the optical axis; otherwise, there is zero intensity on the optical axis i.e. hollow profile is formed along the optical axis [30].
In this work, Laguerre-Gaussian beam with radial mode number p and azimuthal mode number m is chosen as a vortex model which is given as: where E 0 represents an amplitude constant, and β is the truncation parameter, which denotes the ratio of the pupil radius to the beam waist.It is important to note that β should be greater than 1 to ensure proper beam truncation.The term L m p represents the generalized Laguerre polynomial of radial mode p and order m.For the special case where p = m = 0, equation (8) reduces to a Gaussian beam.When p is greater than 0, equation (8) generates a multi-ring-shaped beam with p + 1 concentric rings.For example, setting p = 1 yields a doublering-shaped beam, which has been utilized in various applications to generate unique beam shapes in the focal region of axicons or spherical lenses.These applications include multiple optical cages [31], double focal spots [3], and flat-topped beams [36].By carefully selecting the beam and optical system parameters, these configurations can be achieved.

Numerical calculation and discussions
In this section, numerical calculations are accomplished using Mathematica 12.0 based on equation (7) with equations (3), ( 5) and ( 8) for E 0 = 1, λ = 532 nm, and f = 150λ with different values of the annular obstruction σ, numerical aperture NA, axicon parameter α 0 , and truncation parameter β.In addition to these parameters, the beam order p and topological charge m can be changed to form different focal shapes in the focal region of lenses for the APLGV focused beam.
Figure 1 illustrates the intensity distribution of the APLGV beam in the focal region of spherical lens, linear axicon, and binary axicon with β = 1.1, p = 1, m = 1, and α 0 = 0.01 for different values of annular obstruction σ.It is obvious that a focal spot, in the case of the spherical lens first row, is generated in the focal region with a longitudinal full-width at half-maximum (LFWHM) of 3.67λ when σ = 0 i.e. without an obstacle, as shown in figure 1(a-1).This focal spot extends as σ increases, and the LFWHM becomes 4.03λ when σ = 0.25 figure 1(a-2), for instance.Then it is divided to form double spots for σ = 0.75 with LFWHM of approximately 3.48λ for each of them figure 1(a-4).As shown in the second row of figure 1, a tight focal spot is formed between the linear axicon and focal plane z = 0.This focal spot is extended, thereby, with increasing the σ value and forming separated twin spots.Their properties are controlled by carefully choosing values of parameters, namely NA, β, and σ.In the case of the binary axicon, double spots are generated on both sides of the focal plane z = 0 forming an optical cage between them figure 1(c-1) and (2) then they join together by increasing the obstruction value σ until forming an optical needle when σ = 0.45 with LFWHM of 8λ.Subsequently, a focal spot is formed when σ = 0.50 figure 1(c-3), and then extends to form a longer optical needle.
In figure 2, the intensity distribution of the APLGV focused beam in the focal region of a linear axicon is shown.The parameters used are m = 1, p = 1, NA = 0.95, α 0 = 0.01, σ = 0.672, and β = 1.1.The figure clearly demonstrates the generation of separated identical twin double spots in the focal region.These spots have a maximum intensity separation of 3.78λ, a longitudinal full-width at half-maximum (LFWHM) of 2λ, and a full width at half maximum (FWHM) of 0.43λ.Figures 2(a), (b), and (c) depict the cross-sectional intensity distribution of the APLGV beam at planes z = − 2.67λ, z = 0, and z = λ, respectively.The intensity distribution is symmetric around the optical axis, and both spots have the same FWHM.It is worth noting that the numerical aperture (NA) plays a significant role in controlling the focusing characteristics of the APLGV beam.As shown in figure 2(e), decreasing the numerical aperture causes the focal spots to expand, and the distance between them decreases until they merge together.However, reducing the numerical aperture can increase the LFWHM of the spots.These results highlight the influence of the numerical aperture on the focal characteristics of the APLGV beam.By adjusting the NA, it is possible to control the size, separation, and merging behavior of the focal spots.This provides flexibility in tailoring the focal region for specific applications and desired spot configurations.
To investigate the effect of the axicon parameter on the APLGV focused beam, figure 3 showcases the focal region shapes of the APLGV beam for different axicon parameters in both the linear and binary axicon configurations.The parameters used are m = 1, p = 1, NA = 0.85, and β = 1.1, with annular obstruction values of σ = 0.50 and 0.75.The figure reveals that the intensity shape and the maximum on-axis position of the APLGV focused beam undergo significant changes as the axicon parameter varies, as well as when the annular obstruction σ is altered.For the linear axicon with σ = 0.50, the focal region moves away from the axicon as the axicon parameter decreases.Comparing figure 3(a-7) with figure 1(a-3), it can be observed that the focal region shapes are almost similar, indicating that the linear axicon can produce a focal region similar to that of a spherical lens when a small axicon parameter α 0 is used.In the case of the binary axicon, the two focal regions move toward the focal point, forming a focal spot and subsequently a focal region similar to that of the lens and linear axicon.Figure 3(b) shows that as the annular obstruction increases, the focal region becomes divided, forming multiple focal spots.The linear axicon generates double focal spots, which move away from the axicon as the axicon parameter is decreased, and ultimately, these spots align at the focal point.Similarly, the binary axicon produces tightly quadrupled spots in the focal region when appropriate parameter values are used.It is also observed that these spots approach each other as the axicon parameter increases, ultimately forming double focal spots.To further explore the focal region shapes between the states of σ = 0.50 and σ = 0.75, figure 4 displays the intensity distribution in the focal region of both linear and binary axicons for α 0 = 0.055.Different annular obstruction values of σ = 0.55, 0.60, 0.65, and 0.70 are considered.The intensity gradually redistributes until identical double focal spots are formed by the linear axicon, and nearly identical quadruple spots are formed by the binary axicon.These spots have a mean FWHM of 0.46λ and a mean LFWHM of 3.44λ when σ = 0.742 and 0.745, respectively.These results demonstrate the ability to control and manipulate the focal region shapes by varying the parameters of the axicon and annular obstruction.By adjusting the axicon parameter and annular obstruction, various focal configurations such as double spots and quadruple spots can be achieved.The specific focal region shapes depend on the chosen parameters, providing flexibility in tailoring the focal characteristics for different applications.
The truncation parameter β plays a crucial role in determining the properties of the incident beam, along with the radial beam order p and topological charge m.In figure 5, we present the intensity distribution of the APLGV focused beam in the focal region of both linear and binary axicons.The parameters used are m = 1, p = 1, NA=0.85, α 0 = 0.055, and σ = 0.75, while varying the truncation parameter as β = 1.12, 1.14, 1.16, 1.18, and β = 1.2.The shape of the focal region undergoes significant changes with the variation in the truncation parameter.In the first row of figure 5, corresponding to the linear axicon, it can be observed that the maximum intensity of the left spot in figure 3(b-3) decreases as the value of β increases (figures 5 (a-1:3)).Further increasing β leads to a tightly focused spot with an FWHM of 0.45λ and an LFWHM of 6.61λ when β = 1.2 (figure 5 (a-5)), for example.In the second row, the intensity distribution in the focal region of the binary axicon experiences significant changes as the value of β increases.The distribution transforms from quadruple spots in figure 3(b-4) to extended double spots (figure 5(b-3) and (b-4)), formed on both sides of the focal plane, which become longitudinally squeezed with higher values of β.These results demonstrate the impact of the truncation parameter β on the focal region shape of the APLGV focused beam.By adjusting β, it is possible to control the size, intensity, and longitudinal characteristics of the focal spots.The choice of β allows for tailoring the focal region to achieve desired spot configurations and beam properties, offering versatility in various applications.with the linear axicon.The needle shape remains identical behind and after the spherical lens focus at z = 0, but with the linear axicon, the on-axis intensity and the width of the needle (lateral full width at half maximum) decrease along the focal line.In the case of the binary axicon, the beam shape is symmetric on both sides of the focus at z = 0.As the NA decreases, the  beam changes from a focal spot in figure 1 (c-3) with NA = 0.85 to separated focal spots in figure 6 (c-1) with NA = 0.75.As the NA continues to decrease, the separated focal spots merge to form unseparated triple spots in figure 6 (c-4) with NA = 0.65, and eventually transform into an optical needle in figure 6 (c-4) with NA = 0.45.On the other hand, a high value of NA is beneficial for generating a tightly focused beam, as demonstrated in figure 2 with NA = 0.95 for the linear axicon.The focal shape of an APLGV focused beam can be altered not only by the annular obstruction (parameter σ) but also by the numerical aperture.By adjusting the NA, it is possible to manipulate the focal characteristics and achieve different focal shapes.This flexibility in shaping the focal region opens up possibilities for various applications.
Figure 7 presents the intensity distribution of the APLGV beam in the focal region of linear and binary axicons with different radial beam orders p = 2, 3, and 4, considering m = 1, α 0 = 0.055, and β = 1.16.It is evident that a high-order APLGV beam can generate a tightly focused spot using a linear axicon, with altered characteristics depending on the beam and axicon parameters.In figure 7-(a-1), the LFWHM is 2.92λ, and this value increases as p and/or σ are increased.For instance, the LFWHM becomes 7.19λ in figure 7-(b-6).In the case of the binary axicon, double focal spots are formed on both sides of the focal plane at z = 0.The distance between these spots and their properties undergo significant changes by varying p and/or σ.The results presented in figure 7 demonstrate that the focal characteristics of the APLGV beam can be tailored by using highorder beams and axicons.The focal spot size and properties can be controlled by adjusting the radial beam order and the truncation parameter.This provides versatility in shaping the focal region and enables applications that require specific focal spot configurations.

Conclusion
In conclusion, based on vector diffraction theory, tight focusing of azimuthally polarized Laguerre-Gaussian vortex (APLGV) beam by a lens, linear axicon, and binary axicon was theoretically investigated.Careful selection of optimal parameters for both the beam and optical system enable the generation of multiple focal spots and an optical needle, with the ability to tailor their key characteristics such as the full width at half maximum and longitudinal full width at half maximum through parameter manipulation.Simulation result showed that focal spot is generated in the focal region of lens and linear axicon that is expanding by introducing the obstacle then divided to double spots whose longitudinal full width at half maximum decreases as the numerical aperture increases.Furthermore, by focusing the APLGV beam with appropriate parameters values using a binary axicon, tightly quadruplicated spots appear within the focal region.As the truncation parameter β escalates or the axicon parameter decreases, these spots converge into double focal spots and the double focal spots generated in the focal region of the linear axicon converge to one spot.Furthermore, a narrow focal spot can be achieved using a linear axicon, and twin spots on both sides of the focus z = 0 can be obtained with a binary axicon by increasing the beam order p.It is worth noting that this study utilized a single beam form in combination with various lenses to create those distinct focal geometries easily.We propose that these versatile focal congurations hold promise for applications in laser-based manipulation of multiple particles, material processing, and laser machining.The ability to precisely control and shape the focal region of APLGV beams opens up new possibilities for advanced optical manipulation and fabrication techniques.

Figure 6
Figure6illustrates the intensity distribution in the r-z plane of the APLGV focused beam with parameters m = 1, p = 1, α 0 = 0.01, σ = 0.50, and β = 1.1, for different numerical aperture (NA) values using a spherical lens, linear axicon, and binary axicon.The figure clearly demonstrates that the intensity shape undergoes significant changes with varying NA.Generally, reducing the NA value leads to an expansion of the focal region[3].In the case of the spherical lens and linear axicon, transversely polarized optical needles are formed in the focal region of the APLGV beam, as shown in figure6(a-1:4) and (b-1:4), respectively.The focal depth increases as the numerical aperture decreases.For example, when NA is 0.55, the focal depth or LFWHM is approximately 15λ in figure6 (a-3) with the spherical lens and 15.3λ in figure6 (b-3) with the linear axicon.The needle shape remains identical behind and after the spherical lens focus at z = 0, but with the linear axicon, the on-axis intensity and the width of the needle (lateral full width at half maximum) decrease along the focal line.In the case of the binary axicon, the beam shape is symmetric on both sides of the focus at z = 0.As the NA decreases, the