## Abstract

We find a two-parameter family of astigmatic elliptical Gaussian (AEG) optical vortices, which are free space modes up to scale and rotation. We calculate total normalized orbital angular momentum of AEG vortices, which can be an integer, fractional and zero, and which is equal to the algebraic sum of two terms reflecting the contribution of the vortex and astigmatic components of the light field. In any transverse plane, such a beam has an isolated *n*-fold degenerate intensity null on the optical axis (an optical vortex) embedded into an elliptical Gaussian beam. In addition to the quadratic elliptical phase, a beam has the phase of a cylindrical lens rotated by an angle of 45 degrees with respect to the principal axes of the ellipse of the Gaussian beam intensity distribution. The degenerated central intensity null in these beams does not split it into *n* spatially separated intensity nulls, as is usually assumed for elliptical astigmatic beams.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

It is known [1–5] that a cylindrical lens can be used to determine the topological charge of an optical vortex. This property of the cylindrical lens was noticed long ago. For example, a Hermite-Gaussian laser beam of an order (0, *n*) with the zero orbital angular momentum (OAM) was transformed by using a cylindrical lens into a Laguerre-Gaussian laser beam [6] that has *n*-fold degenerate intensity null and possesses the OAM. Using a cylindrical lens, it is possible to generate vortex-free laser beams with the OAM [7,8]. There are no isolated intensity nulls (singular points) in such beams. These astigmatic beams are described by a superposition of an infinite number of optical vortices with only even positive and negative topological charges [8].

It is also known that a linear combination of even and odd Mathieu [9], Ince-Gaussian [10] and Hermite [11,12] beams with a phase shift of π/2 generates elliptical optical vortices with their OAM depending on the degree of ellipticity. Both vortex and astigmatic components contribute to the OAM of such beams and these contributions of both components (vortex and astigmatic) to the OAM can change with the propagation of such elliptical beams [2,13].

The OAM of optical vortices, including the fractional OAM [14,15], can be measured not only by using the cylindrical lens [1–5], but also by many other ways, for example, by using interferograms [8,14] and a triangular aperture [16].

In this paper, we consider new laser beams with combined properties of vortex elliptic Gaussian beams [9–13] and of astigmatic vortex-free laser beams [7,8]. We call such a family of laser beams as astigmatic elliptical Gaussian (AEG) optical vortices. In the initial plane, the AEG-vortex is an *n*-fold degenerate circularly symmetric intensity null embedded into the center of the waist of an elliptic Gaussian beam, whose waist radii along the Cartesian axes are related by a certain relation, and then passed through a cylindrical lens rotated in the initial plane around the optical axis by an angle of 45 degrees with respect to the Cartesian axes. Such a beam propagates in free space preserving its structure up to scale and rotation. Moreover, the far field (Fourier transform) for AEG beams is located from the initial plane by a distance of double focal length of the cylindrical lens. The normalized total OAM of such beams is calculated. It turned out to be equal to the algebraic sum of two terms, one of which is equal to the topological charge of the optical vortex, and the second is equal to the OAM of the astigmatic elliptical Gaussian beam. These two terms can both strengthen and compensate each other up to zero. In contrast to [13], the magnitude of the contributions of the vortex and astigmatic components to the OAM does not vary with the distance.

## 2. Beam amplitude at double focal length from cylindrical lens

When a circularly symmetric optical vortex with an integer topological charge *n*, embedded into the waist of an elliptical Gaussian beam, propagates through a cylindrical lens, rotated in the transverse plane by an angle α, the complex amplitude of such a light field immediately after the cylindrical lens has the form:

We used the following designations in Eq. (1): (*x*, *y*) are Cartesian coordinates in the transverse plane *z* = 0 (*z* is the longitudinal Cartesian coordinate), *w* is the scaling factor for the optical vortex, *w _{x}* and

*w*are the waist radii of the elliptic Gaussian beam along the Cartesian coordinates,

_{y}*f*is the focal length of the cylindrical lens,

*α*is the tilt angle of the lens axis with respect to the vertical axis

*y*,

*k*= 2π/

*λ*is the wavenumber of light with the wavelength

*λ*. At a distance

*z*from the initial plane, the complex amplitude of the beam of Eq. (1) is defined by the Fresnel transform:

If the tilt angle of the cylindrical lens is 45 degrees (*α =* π/4) and the distance after the lens equals the double focal length (*z* = 2*f*), then the Fresnel transform in Eq. (2) becomes the Fourier transform (up to a phase factor before the integrals) of the amplitude describing an optical vortex, embedded into an elliptical Gaussian beam and passed a cylindrical lens:

Our next goal is to calculate the integral in Eq. (3). We rewrite it in the dimensionless variables: *x*/*w* →*x*, *y*/*w* →*y*, *u*/*w* →*u*, *v*/*w* →*v*, *w*/*w _{x}* =

*γ*,

*w*/

*w*=

_{y}*β*,

*z*

_{0}= 2

*f*, where

*z*

_{0}=

*kw*

^{2}/2. Then, Eq. (1) reads as

This integral in Eq. (5) can be evaluated and the complex amplitude is

The symbol *H _{n}*(

*x*) in Eq. (6) is the Hermite polynomial. It is seen in Eq. (6) that the argument of the Hermite polynomial is complex. It was shown in [1] that when a circularly symmetric Gaussian optical vortex with a topological charge

*n*passes through a cylindrical lens, then at a distance of double focal length behind the lens on a certain straight line in the transverse plane the Hermite polynomial has a real argument, and therefore on this line there are

*n*isolated nulls (roots of the polynomial). Using these intensity nulls, the topological charge of the optical vortex can be determined. Equation (6) extends the method of the topological charge determination proposed in [1] to optical vortices (

*x*+

*iy*)

*embedded in elliptic Gaussian beams of the form exp(–*

^{n}*x*

^{2}–

*β*

^{2}

*y*

^{2}). Indeed, it is seen in Eq. (6) that the argument of the Hermite polynomial becomes real for γ = 1 and there are

*n*real roots (nulls) of the polynomial on the axis

*u*.

Now we find the normalized orbital angular momentum (OAM) of the beam of Eq. (6). The OAM can be found by using the known formulas [7] (up to constant multipliers):

*J*is axial component of the OAM vector,

_{z}*W*is the energy (power) density of light, Im is the imaginary part of a complex number, $\overline{E}$ is the complex conjugate to the amplitude from Eq. (1). Substituting Eq. (1) into Eqs. (7) and (8), we obtain a simple expression for the normalized OAM of the light field of Eq. (1):

*ξ*= (

*w*/

_{x}*w*+

_{y}*w*/

_{y}*w*)/2 ≥ 1 and ${P}_{n}^{m}\left(\xi \right)$ denote the associated Legendre polynomials:

_{x}Despite the imaginary unit in Eq. (9), the OAM has a real value since in our case *ξ* ≥1, and the associated Legendre polynomial ${P}_{n+1}^{1}\left(\xi \right)$ is purely imaginary at *ξ* ≥1.

At the conditions *α* = *π*/4, *z* = *z*_{0} = 2*f*, *w* = *γw _{x}* =

*βw*, we obtain instead of Eq. (9):

_{y}*ξ*= (

*β*/

*γ*+

*γ*/

*β*)/2.

It is seen in Eqs. (9) and (11) that the OAM of the beam of Eq. (1) consists of two terms [2,13]. The first term is determined by the vortex component of the beam and is equal to the vortex topological charge, whereas the second term is determined by the astigmatic component of the beam, induced by a cylindrical lens, and in the absence of vortex (*n* = 0) is equal to $\left(k\mathrm{sin}2\alpha \right)\left({w}_{y}^{2}-{w}_{x}^{2}\right){(8f)}^{-1}$ [8]. It is also seen in Eqs. (9) and (11) that choosing parameters of the Gaussian beam ellipticity allows compensation of the vortex part of the OAM. For example, at *n* = 1 the total OAM of Eq. (9) of the beam from Eq. (1) equals zero (*J _{z}/W* = 0) at

## 3. Family of astigmatic elliptic Gaussian vortices

Now we return to Eq. (6). For a certain relation between the ellipticity parameters of a Gaussian beam (*γ*^{2} = *β*^{2} + 2), since ξ* ^{n}H_{n}*(

*a*/ξ) → (2

*a*)

*at ξ → 0, it follows from Eq. (6) that such vortex astigmatic elliptic Gaussian beams almost preserve their structure after the Fourier transform:*

^{n}It is seen in Eq. (13) that the beam size changes in the Fourier plane. In addition, the elliptical Gaussian beam rotates by 90 degrees, whereas the wavefront of the cylindrical lens is inverted. As a particular example, we consider the following parameters of a Gaussian beam: *γ*^{2} = 3 and *β*^{2} = 1. Then, instead of Eq. (13), we get:

Instead of Eqs. (5) and (15), we can write the Fourier transform in the usual form. To this end, we denote $\overline{u}=2u,\text{\hspace{1em}}\overline{v}=2v$ in Eq. (15), and then we obtain:

It is seen in Eqs. (13) and (16) that, first, for certain ellipticity parameters of a Gaussian beam (*γ*^{2} = *β*^{2} + 2), at a double focal length from a cylindrical lens, there are no *n* intensity nulls, which can be used to determine the topological charge of the original vortex. The optical vortex is reconstructed and still has an isolated intensity null in the center, which is *n*-fold degenerate. And, second, we obtained a two-parameter (*n*, *β*) family of astigmatic elliptical Gaussian (AEG) vortices that have the Fourier-invariance property (up to scale change and 90-degree rotation).

The normalized OAM of AEG vortices of Eq. (14) is fractional. It follows from Eq. (11) and reads as

*ξ*= (1 +

*β*

^{2}) (2 +

*β*

^{2})

^{–1/2}/

*β*.

Using from Eq. (14), we can obtain the complex amplitude of the light field at any distance *z* and in dimensional quantities:

It is seen in Eqs. (18) and (19) that the AEG-vortices preserve their structure on propagation not only at *z* = 2*f*, but also at other values of *z*, though the divergence is different along the *x* and *y* coordinates.

Moreover, the waist radii of an elliptical Gaussian beam are not arbitrary and should meet the condition

In dimensionless variables, the condition in Eq. (20) is equivalent to the previously given (*γ*^{2} = *β*^{2} + 2).

It can be shown that the light field described by the amplitude of Eq. (18) is a solution of the paraxial Helmholtz equation:

From Eq. (18), an expression follows for the complex amplitude of the AEG vortex at a distance *z* = 2*f*:

*z*

_{0}

*=*

_{y}*kw*

_{y}^{2}/2 is the Rayleigh range along the axis

*y*. If we impose a condition on the parameters of the Gaussian beam and on the focal length of the cylindrical lens (

*z*

_{0}

*= 2*

_{y}*f*), then Eq. (22) becomes simpler:

Expression (23) coincides with the right side of Eq. (15) in dimensionless variables.

Tending *z* in Eq. (18) to zero, we obtain in the limiting case the initial amplitude that coincides in the dimensionless variables with Eq. (14), where ${\beta}^{2}=2f/{z}_{0y}=4f/\left(k{w}_{0y}^{2}\right)$.

## 4. Simulation results

Figure 1 shows the intensity and phase distributions calculated by Eq. (18) for the AEG vortices for different topological charges and for different propagation distances *z* from the initial plane. Calculation parameters: the initial waist radii of the elliptical Gaussian beam *w _{x}* = 123 μm and

*w*= 3

_{y}*w*= 369 μm, wavelength λ = 532 nm. It is seen in Fig. 1 that the intensity pattern of the AEG vortices (for

_{x}*n*> 0) looks like two quasi-elliptic Gaussian beams, whose major axes lie on the same line and which are separated by a distance

*L*that depends on the square root of the topological charge

*n*:

*y*) has the following form for any

*z*:

It is seen in Eq. (25) that the dependence of the waist radius of AEG-vortices on the distance *z* differs from such dependence for the usual Gaussian beam.

When propagating in free space, the AEG vortices preserve their transverse shape, changing in scale (the distance between the maxima of the two bright spots increases) and rotating around the optical axis. At the same time, on the optical axis an isolated *n*-fold degenerate intensity null remains with the phase around him changing by 2*πn*. But even at *n* = 0, although the AEG vortex does not have a vortex component (there is no intensity null at the center), nevertheless it possesses the OAM (since the astigmatic component remains). From Eq. (17) and from the ratio between the initial waist radii *w _{y}* = 3

*w*(

_{x}*β*

^{2}= 1/4, since

*γ*

^{2}–

*β*

^{2}= 2 and therefore, according to the definition of

*β*and

*γ*, ((

*βw*/

_{y}*w*)

_{x}^{2}–

*β*

^{2}= 8

*β*

^{2}= 2) follows that the OAM of the beams in Fig. 1 is equal to:

In particular, for *n* = 0, 1, 3, 5, 7 the normalized OAM is *J _{z}*/

*W*= 16/9 ≈1.78, 19/3 ≈6.33, 16.42, 26.46, and 36.47, respectively.

It is also seen in Fig. 1 that for any *n* the AEG vortices rotate counter-clockwise on propagation independently of the value *n*. It can be shown that the angle *θ* of rotation of the whole AEG vortex depends on the distance *z* as follows:

It is seen in Eq. (27) that at *z* = 0 the rotation angle is *θ* = 0 (the beam is stretched along the vertical axis, since *w _{y}* = 3

*w*), at

_{x}*z*= 2

*f*the rotation angle is

*θ*= π/2 (the beam is stretched along the horizontal axis), and at

*z*=

*f*the angle is slightly larger than π/4 (since

*z*>> 2

_{y}*f*for the chosen parameters) and is equal to:

## 5. Experimental results

To confirm the theory and the simulation, we observed the propagation of AEG vortices experimentally. Figure 2 shows the optical setup used in the experiment. An elliptical Gaussian laser beam (λ = 532 nm) with the ellipticity parameter of 1: 3 was directed onto the display of a spatial light modulator *SLM* (HOLOEYE, PLUTO-VIS). The light modulator was used to implement the phase mask with the transmission function τ(*r*, ϕ) = exp(*i*[*n*ϕ + α*x*]) ((*r*, ϕ) are the polar coordinates, *r*^{2} = *x*^{2} + *y*^{2}, *α* is the spatial carrier frequency, *n* is the topological charge), which generates a given off-axis vortex beam. After reflecting from the modulator, the phase modulated elliptical beam was directed by the lenses *L*_{1}, *L*_{2} (*f*_{1} = 350 mm, *f*_{2} = 150 mm) and by the mirrors *M*_{1}, *M*_{2} to a cylindrical lens *CL*_{1} (*f*_{3} = 100 mm) tilted by an angle of 45 degrees in a plane orthogonal to the beam propagation axis. System of lenses was used to build the image of the plane, conjugated with the plane of modulator's display, in the plane of the cylindrical lens *z* = 0. Using the CMOS camera on an optical rail, the generated beam intensity distributions were recorded at different distances *z* from the plane of the cylindrical lens. Images obtained at the distances *z* = 0, 100, 150 and 200 mm are shown in Fig. 3. It is seen in Fig. 3 that for *n* = 0 only one elliptical light spot is generated, whereas for *n* = 1, 3, 5, 7 there are two spots, and at *n* = 1 these spots interfere with each other.

A comparison of Fig. 1 with Fig. 3 shows a good agreement between the theory and the experiment.

Figure 4 shows two interferograms obtained for AEG vortices from Fig. 3 at *z* = 2*f* and *n* = 1, 3. It is seen in Fig. 4 that, indeed, the optical vortex with a topological charge *n* = 1 [Fig. 4(a)] and *n* = 3 is remained in the center of the beam [Fig. 4(b)].

Interferograms for *n* = 5, 7 have low contrast of fringes because of almost zero intensity of light in the interval between two local maxima [Fig. 3].

## 6. Conclusion

In conclusion, we have found a new modal solution of the paraxial Helmholtz equation. This solution describes a two-parameter family of laser modes – astigmatic elliptical Gaussian optical vortices. These modes are orthogonal with respect to one parameter (topological charge) and non-orthogonal with respect to the other parameter (ellipticity of the Gaussian beam). In the initial plane, such beams have an isolated *n*-fold degenerate intensity null embedded into the center of the waist of an elliptic Gaussian beam, whose principal axes are directed along the Cartesian axes, and who passed through a cylindrical lens with its axis rotated in the initial plane around the optical axis by 45 degrees with respect to the Cartesian axes. The waist radii of the Gaussian beam are not arbitrary, but are related by a definite relation. A total orbital angular momentum normalized to the beam power is found. It is equal to the sum of two terms, one of which equals the topological charge *n* of the optical vortex, and the second term describes at *n* = 0 the contribution of the vortex-free astigmatic elliptic Gaussian beam [8]. The orbital angular momentum of such beams can be fractional, integer, or zero. The found modal beams preserve their structure upon propagation, varying only in scale and rotating around the optical axis. At a double focal length from the cylindrical lens, the beam rotates by 90 degrees. The intensity distribution of such beams has a characteristic form: two almost elliptical light spots, whose major axes lie on the same line, and the distance between the maxima of these light spots depends as the square root on the topological charge and increases when the beam propagates in free space. We note that because the waist radii of the elliptical Gaussian beam and the tilt angle of the cylindrical lens are matched in such beams, the cylindrical lens does not eliminate the degeneracy of the central intensity null and does not split it into *n* spatially separated intensity nulls, as is usually assumed [1].

## Funding

Ministry of Science and Higher Education (State assignment); Russian Science Foundation (17-19-01186); Russian Foundation for Basic Research (project 18-29-20003).

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