### Near-field distribution

Evaluation of Eq. 4 shows that depending on each *N*, certain terms of the series are equal zero. Therefore, only a finite number of azimuthal harmonics \(\tilde {n}\) have to be taken into account generating the resulting field distribution in the near-field region. Figure 2 gives an overview on the corresponding intensity distribution of each contributing azimuthal harmonic \(\tilde {n}\) along the radial component *r* in the transversal observation plane.

The analysis shows that in each case the contributing azimuthal harmonics can be expressed by \( \tilde {n}=1, qN\pm 1\) with \( q \in \mathbb {N} \backslash \{0\} \). Each contribution results in a weighted Bessel beam with a TC of \(\mid l \mid =\tilde {n}\). Here, these Bessel modes are referred to as fundamental vortex modes (FVMs). Successive FVMs have alternating algebraic sign. The first sign results from the orientation of the object as for an increasing slope of the structure (Fig. 1b) it is positive, for an negative slope negative, respectively. Thus, the result can also be seen as axial superposition of different isotropic fundamental vortex modes [32]. Note that not all high harmonics \(\tilde {n}\) have to be taken into account as their contribution is either comparatively low or near zero as they cause high orders of the Bessel function (see Eq. 4). With respect of a high number of blades (*N*), the field distribution of the discretized spiral axicon resembles the not discretized spiral axicon, as the second contribution does not influence the contribution of the first harmonic \(\tilde {n}=1\). Therefore, for high *N*, the resulting field distribution near the optical axis solely consists of a Bessel beam with the TC *l*=1.

The resulting field distributions with corresponding symmetry axis for different *N* of the object structure are illustrated in Fig. 3. All intensity distributions exhibit zero intensity in the center. Comparing the zero contours of real- and imaginary part (see right column in Fig. 3) and the phase distributions (see middle column in Fig. 3), all distributions contain an on-axis screw dislocation (SD) in the center.

Further examination of the real and imaginary zero contour identifies additional off-axis SD at their intersections. The SDs are arranged symmetrically around the center leading to an optical vortex lattice. The optical vortices with positive and negative charge are indicated by blue and red circles in the right column of Fig. 3. Each phase profile around these singularities exhibits a decreasing/increasing phase with a phase step of *Δ**ϕ*=±2*π* corresponding to an optical vortex. Here, the ring integral was used for the determination of the SDs. The result leads in all cases to an single charged optical vortex with the corresponding TC of

$$\begin{array}{@{}rcl@{}} l=\frac{1}{2\pi} \oint \nabla \phi \,d \vec s =1 \quad \text{or} \quad l=-1 \end{array} $$

(9)

depending on the orientation. Both, the positions of the PS, and the closed path integral were determined numerically based on the analytical calculations. Note the geometrical position of the SD depending on the symmetry given by *N* and the corresponding anticorrelated (alternating) charge so that OVs of the same charge are no direct neighbours. For *N*=3, Fig. 3 does not indicate all OVs as their spatial extend is small, as they are to close to neighbouring OVs or they are connected to a low intensity level.

The angles of the intersection of the zero contour lines of the real- and imaginary part reveal further characteristics of the vortices [36, 37]. While for a conventional isotropic on-axis vortex, the contour lines for ℜ(*u*)=0 and *I*(*u*)=0 intersect under an angle of 90°, this is not true for the off-axis vortices shown in Fig. 3. Instead, here one observes anisotropic characteristics. Such optical vortices are referred to as non-canonical vortices, which here also correlates with a non circular symmetry of the intensity distribution of off-axis vortices. This behaviour is also depicted in Fig. 4a– c, which outlines the phase distribution along one circumference around the SDs with the radius *r*_{ps}. With smaller division ratio in azimuthal direction of the grating further off-axis vortices occur as the amplitudes of the higher azimuthal harmonics rise [30]. More over, the first order off-axis vortices shown in Fig. 3 develop a more isotropic behaviour whereas the second order off-axis vortices have comparable anisotropic behaviour like presented in Fig. 4.

The on-axis vortex of the modulated spiral axicon with *N*=3 exhibits rising anisotropic character with increasing *r*_{ps}. The strong anisotropic character in Fig. 4c, on the one hand, results from the low distance of the circumference to the adjacent negative charged off-axis SD and on the other hand can also be derived from the strong curvature of the real and imaginary contour. Equivalent but weaker behaviour with rising *r*_{ps} can be observed for *N*=4 and *N*=5. The phase distribution’s inclination becomes more constant with increasing *N* at higher *r*_{ps}. Note that the higher the number of blades *N* of the structure gets, the more the field distribution corresponds to the unmodulated spiral axicon.

In the following, the structure with *N*=6 will be considered in detail. In contrast to the on-axis vortex, the off-axis vortices are unlikely to be isotropic. Figure 4d shows the phase distribution around one particular off-axis singularity (*x*_{ps}=*y*_{ps}=26.38 μm for different values of *r*_{ps} at *z*=12 mm for *N*=6 blades.

For better comparison, Fig. 5a and b depict the azimuthal phase gradient along one circumference with different radii *r*_{ps} around the SD of the on-axis and off-axis vortex, respectively.

The azimuthal gradient of the on-axis vortex exhibits sixfold rotational symmetry, which correlates to the six corners of the on-axis intensity maximum with donut shape. Moreover, the OV can be seen as isotropic within a certain range, as the oscillations are comparatively small for *r*_{ps}≤9.25 *μ**m*, where the intensity of the on-axis vortex reaches its maximum. The off-axis vortex shown in Fig. 5b has in general strong anisotropic behaviour which increases for higher *r*_{ps} of the circumference, as in this case, the circumferences also approach adjacent off-axis SDs.

Further investigations reveal that neither the position of the SDs changes along *z*, nor the azimuthal phase gradient illustrated in Fig. 5 changes along the propagation direction. The calculated gradient in *z*-direction *∂**ϕ*(*r*,*θ*,*z*)/*∂**z*, resulting from Eq. 4, has no azimuthal dependencies:

$$\begin{array}{@{}rcl@{}} \frac{\partial \phi(r,\theta,z)}{\partial z} = k-\frac{1}{2k} \left(\frac{2\pi}{r} \right)^{2}- \frac{kr^{2}}{2z^{2}} \end{array} $$

(10)

The first and the second terms are the derivative of the phase connected to the traveling plane wave and the longitudinal phase variation, respectively. The third term is the derivative of a quadratic phase front which remains from the approximation by the method of stationary phase [35]. Both, the second and the third term are negligible compared to the first term. Therefore, Eq. 10 shows that the transversal vortex lattice structure persists along propagation in *z*-direction in the above defined near-field region.

The resulting 2D - phase gradients in Cartesian coordinates are depicted in Fig. 6 for the on- (top row) and off-axis vortex (bottom row). The value of the gradient is limited to ±5×10^{5} for a better overview.

In case of an isotropic vortex, the azimuthal phase gradient, given by \(\frac {1}{r} \frac {\partial \phi }{ \partial \theta }\), is proportional to the inverse of the radial coordinate, irrespective of the angle *θ*. In general, the phase gradient in azimuthal coordinates is associated with the Cartesian coordinates by:

$$\begin{array}{@{}rcl@{}} \frac{\partial \phi (x,y)}{\partial x} &=& \frac{\partial \phi (r,\theta)}{\partial r} \cos \left(\theta \right) - \frac {1}{r} \frac{\partial \phi (r,\theta)}{\partial \theta} \sin \left(\theta \right) \end{array} $$

(11)

$$\begin{array}{@{}rcl@{}} \frac{\partial\phi (x,y)}{\partial y} &=& \frac{\partial \phi (r,\theta)}{\partial r} \sin \left(\theta \right) + \frac {1}{r} \frac{\partial \phi (r,\theta)}{\partial \theta} \cos \left(\theta \right). \end{array} $$

(12)

For an isotropic vortex the gradient distributions on the coordinates axes are linked by:

$$\begin{array}{@{}rcl@{}} \frac{\partial \phi(x,y)}{\partial x} &=& - \frac{1}{r}\frac{\partial \phi (r,\theta)}{\partial \theta} \sin \left(\theta \right) \quad \text{for} \quad x=0 \end{array} $$

(13)

$$\begin{array}{@{}rcl@{}} \frac{\partial \phi(x,y)}{\partial y} &=& \frac{1}{r}\frac{\partial \phi (r,\theta)}{\partial \theta} \cos \left(\theta \right) \quad \text{for} \quad y=0 \end{array} $$

(14)

Therefore, in both cases, the gradient distributions in Cartesian coordinates exhibit lobes with different algebraic sign above and underneath the *x*-axis, and right and left of the *y*-axis, respectively. This behaviour can be seen for the *x*- and *y*- components of the gradient for the on-axis vortex (see Fig. 6a, b). The black curve in Fig. 6 indicates the zero-value on the *x*- and *y*-axis, respectively. Furthermore, the dashed lines indicate the axis, where the absolute value of the *x*- and *y*-component equals the azimuthal phase gradient without respect to the sign. The orientation of the TC (positive/negative), is given by the direction of the azimuthal phase gradient. This can in turn be calculated from the gradient distribution in Cartesian coordinates, where characteristic lobes are recognizable. Near the middle point, the solid (which is straight near the middle point) and the dashed lines are perpendicular to each other. In the outer region of the distribution shown in Fig. 6a, b, the distribution loses its mirror symmetry correlating to the results from Fig. 5.

Figure 6 (bottom row) illustrates the distribution of the *x*- and *y*- component of the gradient for the off-axis vortex in Cartesian coordinates. Here, this distribution again exhibits lobes similar to Fig. 6 (top row). In the off-axis case, the dashed lines in the *x*- and *y* - gradient distribution are curved and not perpendicular to the zero lines. The gradient distribution exhibits no mirror symmetry. This represents the anisotropy and correlates with the results indicated in Fig. 5. The amount of the *x*-component of the gradient in Cartesian coordinates does not equal the azimuthal phase gradient on the dashed line in this case, neither does the *y*-component on the other dashed line. This is owed to the fact of an off-axis vortex which requires a coordinate transformation for further investigation. Here, we concentrate on an qualitative considerations.

### Experimental results

Figure 8 depicts the experimental results obtained with an helium-neon laser at *λ*=632.8 nm. The experimental setup is presented in Fig. 7. The structured spiral axicon with *N*=6 blades has an radial period *r*_{p}=32 μm and a diameter *D*_{0}=960 μm. The measured intensity distribution at *z*=18 mm can be seen in Fig. 8a showing one intensity maximum in donut shape in the center (on-axis) and six maxima in donut shape arranged symmetrically around the on-axis maximum. The transversal intensity distribution was 20× magnified. The pixel size of the CCD camera was 4.75 μm×4.75 μm. The corresponding wavefront measurement, obtained with a Shack-Hartmann-sensor, can be seen in Fig. 8b and c for the on-axis and off-axis vortex, respectively.

The Shack-Hartmann sensor used in the measurement has a pitch of 130 μm. In order to resolve the wavefront of the vortices, a sufficient magnification had to be used. In our experiments, this magnification was 50× to yield a donut diameter on the sensor camera about 1.5 mm.

The vector fields shown in Fig. 8b and c demonstrate the isotropic character of the on-axis vortex and the anisotropy of the off-axis vortex, respectively. Both are in agreement with Fig. 6. We would like to explain the asymmetric behaviour of the off-axis vortices in more detail with a qualitative and geometrical evaluation of the experimental data. The resulting phase gradient can easily be calculated from the measured spot displacement shown in Fig. 9.

The axes shown here (continuous and dashed lines) are used to demonstrate the same behavior of the phase gradient for the on- and off-axis situation. By comparison with the calculated results shown in Fig. 6, on can observe that the experimental distribution in Fig. 9 match the predictions.