## Abstract

Diffraction patterns formed by axicons with different tip (vertex) angles are analytically and numerically investigated. Results show that the axicon (or tapered dielectric probe) can form an extended axial light beam, a compact evanescent field, a hollow beam, and a collimated beam, depending on the vertex angle. Two-dimensional and three-dimensional models of a tapered dielectric probe show that, with small changes to the vertex angle, light transmitted by the probe is scattered rather than focused, and vice versa. Angle meanings corresponded to boundary transitions have a quantum character and densify as the angle approaches zero. These features should be taken into consideration when manufacturing microaxicons intended for various applications.

© 2017 Optical Society of America

## 1. Introduction

More than half of a century has passed since a dielectric cone that forms an extended axial beam was named an “axicon” [1]. Though axicons were used and studied long before they got their official name [2], this optical element has provoked discussion, and aroused heightened interest, related to the non-diffraction properties of Bessel beams formed using this component, particularly in the second half of the last century [3].

The ability to produce microaxicons [4,5] including tapered fiber probes [6,7] has expanded the range of applications in which these optical components can be employed, giving them a prominent position in micro- and nano-optics [8–14].

The simplest tapered dielectric probe is a piece of optical fiber conically elongated on one side [15,16]. Note that the vertex angle of such tapered cones is much less than the total internal reflection angle [17], and in contrast to classical axicons [1–3], they are not designed to form non-diffraction light beams with extended focal depth [18].

Tapered probes with small vertex angles are widely used in near-field optics [15,19–24]. These near-field probes are usually coated with a thin layer of metal, and can have an aperture window, depending on the application. The resolution of a near-field scanning microscope depends on the size of the aperture, or the probe ending, which can be a nanoscale structure. Although, such nanostructures can be used for sharp focusing [20,22,23], the energetic efficiency of the light beam when this is done is very low [25].

For such applications as subwavelength surface micropatterning, it is desirable to increase the transmitted energy, without deteriorating the resolution. In this case, uncoated dielectric tapers with a very small vertex angle, are used [26,27]. Similar devices are used to capture and move microparticles [18,28].

In [29], using a ray optics approach it is shown that axicons with small angles can form both converging and diverging beams. The generation of real and imaginary focuses varies periodically as the vertex angle decreases. Moreover, for smaller angles, this change occurs faster. Changes to the angle by fractions of degrees, result in significant changes to the diffraction pattern. In this research, a two-dimensional simulation that confirms these theoretical results has also been executed, using the finite element method.

Though the two-dimensional model is sometimes used for estimating light propagation in tapered waveguides [30], it would be more reasonable to perform a three-dimensional simulation, particularly when polarization effects have to be considered [31,32].

In this paper, using the finite element method implemented in the COMSOL Multiphysics software package, and the finite-difference time-domain (FDTD) method implemented in Meep, a flexible free-software package, a three-dimensional calculation of Gaussian beam diffraction, caused by tapered microaxicons with different vertex angles, has been performed. Results show that depending on the vertex angle, the axicon can form an extended axial light beam, a compact evanescent field, a hollow beam, and a collimated beam. These features attract theoretical interest and should be taken into consideration in manufacturing microaxicons for various applications.

## 2. Theoretical analysis based on the ray-optics approach

We consider a refractive axicon, as depicted in Fig. 1. α_{0} is the half angle at the vertex where radiation is emitted, in the direction of the flat surface.

If α_{0} is sufficiently large, internal reflection does not occur at the surface of the axicon and the transmitted rays form a bright light needle in the optical axis (Fig. 3, Line 1).

The numerical aperture (*NA*) of the axicon is defined by, β, the angle at which the beams leave the axicon [Eq. (1)]:

In [33], based on ray optics analysis, it was shown that if light illuminates the refractive axicon, the numerical aperture is limited by total internal reflection:

In Eq. (2), ${n}_{ax}$is the refractive index of the axicon, and ${\alpha}_{tir}$is the total internal reflection angle. In particular, when ${n}_{ax}=1.5$, ${\alpha}_{tir}=48.19\xb0$.

According to Eq. (2), the numerical aperture of the axicon cannot exceed the following value:

For ${n}_{ax}=1.5$, *NA*_{lim} = 0.75. It follows from Eq. (3) that it is possible to increase the limit of the numerical aperture by increasing the refractive index of axicon. This can be achieved by changing the material used for fabrication the axicon.

If the axicon angle is less than the total internal reflection angle i.e.,${\alpha}_{0}<{\alpha}_{tir}$, the geometric-optical model states that light cannot be transmitted from the axicon. This axiom explains internal reflection arising from tapered surfaces of the axicon. When ${\alpha}_{0}=45\xb0$ (reflective symmetry), the beam is reflected; there is no light transmission. In wave theory, this situation is exclusive to the generation of evanescent waves (Fig. 3, Lines 2 and 3).

This situation remains unchanged until the beam is reflected from only one tapered side of the axicon. The transition to this regime, ${\alpha}_{halo}$, is given below:

In this regime, beams are transmitted alongside the axicon, creating a light halo surrounding the component. When *n* = 1.5, ${\alpha}_{halo}=43.94\xb0$, as dictated by Eq. (4).

As the axicon angle is further reduced, horizontal projections of the outgoing beam will decrease, and at a certain angle, ${\alpha}_{0}={\alpha}_{ver}$, will equal zero: the outgoing beam is transmitted vertically (maximum halo— see Fig. 3, Line 4). This angle is determined from Eq. (5):

From Eq. (5), when *n* = 1.5, ${\alpha}_{ver}\approx 38\xb0$.

As the axicon angle is further reduced, horizontal projections of the outgoing beam, travel along the direction of propagation of the incident beam, i.e. the radiation will go forward. The beams do not intersect with the optical axis. In other words, a hollow beam will be formed (Fig. 3, Line 5).

The hollow beam remains unchanged until the outgoing beam moves horizontally (parallel to the optical axis), corresponding to the behavior of a collimated beam (Fig. 4, Line 1). A reduction in the axicon angle will result in the generation of first a convergent beam (Fig. 4, Line 2), then a divergent beam (Fig. 4, Line 3).

In [29] it is shown that this situation is regularly repeated due to the availability of several total internal reflection angles. We can determine the angles that correspond to the transmission of collimated beams using Eq. (6), for *p*-fold internal reflection:

When *p* = 1 or *p* = 2, Eq. (6) can be solved analytically as follows:

Thus, when *n* = 1.5,${\alpha}_{col}^{1}\approx 16.78\xb0$, ${\alpha}_{col}^{2}\approx 9.785\xb0$.

Values for $p=\overline{1,4}$ are shown in Fig. 2.

The transition between divergent and convergent beams (i.e., angular positions of new total reflection) may be obtained from the following equation:

Using Eq. (8), we can write down the following:

In particular,${\alpha}_{tir}^{2}={16.06}^{o}$, ${\alpha}_{tir}^{3}={9.638}^{o}$.

Note that when *p* = 1, Eq. (9) is also correct, and coincides with Eq. (2). From the above equation, we can define regions for the formation of convergent and divergent beams. Focusing is observed for ranges of axicon angles,${\alpha}_{0}$, determined by the following inequalities, for *p*-fold internal reflection:

Divergent (hollow) beams are formed for ranges of axicon angles determined by the following inequalities, for *p*-fold internal reflection:

Thus, as the vertex angle of the axicon is reduced, there will be infinite alternation of configurations with imaginary and real focuses. The limits of these ranges are the new positions of total reflection, and positions of the outgoing horizontal beam.

Figure 2 shows the range of axicon angles (marked in red) corresponding to the formation of an extended axial light beam (Bessel beam). The evanescent-field domain is marked in grey. For a halo region (marked in violet), there is a glow around the element. After the halo region, the recurring alternating states begin: formation of a hollow beam (green), and a focused beam (blue). Note that as this alternation occurs, the width of domains is gradually reduced. A collimated beam (orange lines) is obtained at well-defined angles, corresponding to the regions to the right of the focused beam domain. The positions of these orange lines are visually reminiscent of the positions of atom energy lines—they densify as the angle approaches zero.

The proper presentation of these effects is not obtained by changing the axicon angle due to the dispersion of the refractive index of the axicon. In this case, we need to change the wavelength of the axicon-illuminating radiation.

Note that a focused beam is observed for narrower ranges of axicon angles, rather than a divergent beam. Because of the narrow range of angles at which the true focus occurs, a resemblance to resonance effects can be observed with this phenomenon. As a result of this narrow range, some problems may occur when using a polychromatic illuminating beam. Due to dispersion, transmission intervals for different wavelengths do not necessarily have a common overlap area. This dispersion results in the generation of an imaginary focus rather than the real focus, for a part of the spectral range. Thus, spectral distribution of a focused beam becomes distorted.

## 3. Wave calculation in the framework of the 2D axicon model

This section presents the results of numerical modeling of Gaussian beam diffraction by the axicon. These results were obtained by solving the Helmholtz equation using the finite element method in a two-dimensional model, implemented in COMSOL. For convenience of calculations, the numerical aperture of the axicons was calculated in this model using the following equation [33]:

*D*is the diameter of the base of the axicon, and

*h*is the height of the axicon. For simulation we use

*h*= 8um,

*D*= 2

*h*tan α.

Figures 3 and 4 show distributions of electric field amplitudes for a range of axicon angles, ${\alpha}_{0}$, from 8°–60°. The radiation wavelength, λ, is 633 nm, and the refractive index of the axicon is 1.5.

From the above calculations, it is clear that if the axicon angle is gradually changed, a periodically recurring situation occurs: first, rays emitted by the axicon converge, and then disperse. In addition, the angular range of such regions is narrowed as the axicon angle is reduced. It can be observed that axicons with similar angles differ greatly in transmission characteristics. For example, when ${\alpha}_{0}=10\xb0$, the axicon is convergent, whereas when ${\alpha}_{0}=8.5\xb0$, the axicon is dispersive.

For better comparison of the results of geometric analysis, the graph in Fig. 5 shows the change in the amplitude, with respect to the angle, at a point on the optical axis that is seven wavelengths from the apex of the axicon.

It is clearly seen that the maximum amplitudes are formed on the optical axis when ${\alpha}_{0}\approx 9.75\xb0$, and ${\alpha}_{0}\approx 16.45\xb0$. The minimum value is achieved when${\alpha}_{0}\approx 13.85\xb0$. Note that between the two maxima, there is also an area with a slightly varying amplitude (between 11 and 13 degrees). The difficulty in predicting this kind of behavior using the geometrical approximation explains why a significant number of numerical calculations are performed using wave theory.

It is clear that when using microaxicons, effects, which cannot be predicted by geometric optics, arise. However, we note that the qualitative correlation of predictions made using the geometric approach is satisfactory.

Note also that at smaller axicon angles, the value of the maximum amplitude is smaller. From a geometric optics point of view, this reduced amplitude can be explained by the shallower angle of emergence of the rays emitted from the axicon. This shallow angle tends to result in elongation of the focal beam, and consequently, in a reduction in intensity at each point of the beam. From the wave theory perspective, the shallow angle is matched by a decreasing transmission coefficient. In addition, the multiple internal reflections on the sidewalls of the axicon results in a loss of energy, as this is transmitted to decaying waves.

Thus, the results of modeling are consistent with geometric constructions, and confirm the analytical outcomes obtained in the previous section.

## 4. Numerical calculation using a 3D-model of an axicon

The modeling parameters for Meep are as follows: the wavelength is λ = 0.532 μm, the computational domain, x, y, and z ∈ [–5.83λ; 5.83λ], is a network, 6.2 × 6.2 × 6.2 μm in size (~11.65λ), encircled by a 0.6 μm (1.13λ) perfectly matched layer (PML). The spatial discretization interval is λ/15, and the temporal discretization interval is λ/(30*c*), where *c* is the speed of light in a vacuum. We considered the Gaussian and the first-order Laguerre-Gaussian mode as being the input laser radiation (circular “+” polarization). The refractive index is 1.5.

The modeling parameters for COMSOL are as follows: the wavelength is λ = 0.532 μm. The computational domain is cylindrically shaped: the height of the cylinder is 10λ, and the radius is 3.233λ. The computational domain contains the considered axicon, which is made of glass with a refractive index of 1.5. The axis of the axicon coincides with the axis of the cylindrically shaped computational domain. The base of the axicon is combined with the base of the cylindrical domain. The height of the axicon is fixed and is equal to 3λ. The radius of the base of the axicon depends on the vertex angle of the axicon. A tetrahedral network is laid on the computational domain. The maximum size of a tetrahedral element is λ/5. The base of the axicon was illuminated by a Gaussian beam. The radius of the beam waist is 3λ.

Figure 6 shows the results of numerical modeling in Meep and COMSOL when ${\alpha}_{0}={60}^{o}\ge {\alpha}_{tir}$(Focused), and ${\alpha}_{halo}<{\alpha}_{0}={47}^{o}<{\alpha}_{tir}$(Evanescent). We note the similarity of the diffraction patterns obtained when calculating in different software packages. We will use COMSOL in further studies of Gaussian beam diffraction on a 3D structure. The result of numerical modeling is shown in Fig. 7.

Figure 8 shows changes in the amplitude at a point on the optical axis, for focusing and scattering beams.

Comparison of the results for 2D and 3D models shows that for the vertex angle more than 16 degrees, the distributions formed by axicons are in qualitative agreement with each other and the geometrical optics theory. For some angles, similar results were obtained in [4, 5, 25]. However, these studies did not investigate the entire range of angles, and also did not emphasize the periodic character of the properties of the axicon at small angles.

The main differences between 2D and 3D models appear at vertex angles of less than 15 degrees, when periodic properties appear. These differences in 2D and 3D simulation results are inevitable due to the difference in the structure of the optical elements. Particularly noticeable effect of the configuration of the element will be at micro-dimensions, namely, in the case under investigation (in the simulation, the decrease in the angle was made by decreasing the radius of the axicon, and not by increasing its length in order to maintain the micro-dimensions in all dimensions). We believe that for large dimensions (with element’s radius of the order of a millimeter) the difference between 2D and 3D structures will be significantly smaller, as well as the difference from the ray-tracing theory.

There are experimental results in papers [4] and [5], which are measured for laser beams focusing by microaxicons with angle ranges 52.5-75.0 degrees and 42.5-85.0 degrees respectively. In this ranges an axicon works like a conventional refractive optical element in accordance with theoretical and numerical results of our work. Experimentally achieved [4] transverse sizes of formed Bessel beams are in a close agreement with the sizes which are numerically received in our work both in 2d and 3d models. Particularly, for the angle of 60 degrees experimentally measured [4] transverse light spot size is 1.42λ and we calculated 1.37λ and 1.41λ in 2d and 3d models respectively.

Notice that the transverse size of the beam is determined with a numerical aperture, which is defined with an angle of an axicon. Focusing axicons (α_{0} > α_{tir}) generate Bessel beams which length depends not only on axicon’s numerical aperture but also it is linearly proportional to the radius of illuminated part of an axicon.

Numerical results for small axicon’s angles have been received with another software package in [26]. The authors primarily investigate sizes of created near-field light spots and field propagation are considered only for angles of 4, 15, 25, and 35 degrees (for these angles results of [26] are in close agreement with our results). Note, the periodical character of focusing and defocusing property changes has not been revealed in [26]. Experimental measurements have been made only in the near field, so it disables direct comparison with our results.

Notice, that the difference between focused and defocused field is negligible in the immediate vicinity of axicon’s vertex. We focus on investigation of longitudinal light field distributions and transverse distributions (graphs in Figs. 5 and 8) are observed at distances which are greater than the wavelength where evanescent waves are negligible. Consequently, a novel result of our work is that: even for narrow axicons a significant light energy may be exist beyond the near-field region.

## 5. Conclusion

This paper investigates both analytically and numerically the diffraction patterns of axicons with a small opening angle, which approximate fiber probes of near-field microscopes. For numerical solution of Maxwell's equations, we used the finite element method, implemented in COMSOL Multiphysics, and the finite-difference time-domain (FDTD) method, implemented in Meep.

Calculations performed using 2D models agree with geometrical optic approximation much better than those performed with 3D models. In the 2D models, an axicon with a vertex angle of 20° (a standard angle for the NT-MDT tapered fiber probe) is scattering, and at 16 degrees the axicon is focusing. It is found in 3D modeling that an axicon with a vertex angle of 24.5° is scattering, whereas an axicon with an angle of 14° is focusing. Thus, the observed difference between 2D and 3D models occurs only for angles of less than 15 degrees, which is due to the difference in the shape of the elements, as well as their small size.

When axicon’s angle α_{0} > α_{tir}, an axicon works like a conventional refractive optical element and generate Bessel beams. In this case, our numerically received results both in 2d and 3d models are in a close agreement with experimentally achieved results [4].

Calculations have shown that, depending on the vertex angle, the same tapered feature can form an extended axial light beam (Bessel beam), a compact optical-axis evanescent field, a hollow beam, and even a collimated beam. Moreover, after a critical value, further reduction of the axicon angle leads to recurrent changes to the types of the fields being formed: divergent, collimated, and focused. In addition, the beam changing period is inversely proportional to the number of reflections inside the axicon.

A similar variety of fields output by the axicon can be obtained by changing the wavelength of the illuminating beam, due to the dispersion of the axicon material. These features should be taken into consideration when producing microaxicons intended for various applications.

## Funding

Russian Foundation for Basic Research (grants 16-07-00825, 16-29-11698, 16-37-00241 mol_a) and by the Ministry of Education and Science of Russian Federation.

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