## Abstract

We report a novel method to freely transform the modes of a perfect optical vortex (POV). By adjusting the scaling factor of the Bessel–Gauss beam at the object plane, the POV mode transformation can be easily controlled from circle to ellipse with a high mode purity. Combined with the modulation of the cone angle of an axicon, the ellipse mode can be freely adjusted along the two orthogonal directions. The properties of the “perfect vortex” are experimentally verified. Moreover, fractional elliptic POVs with versatile modes are presented, where the number and position of the gaps are controllable. These findings are significant for applications that require the complex structured optical field of the POV.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Recently, optical vortex beams that carry an orbital angular momentum (OAM) have been extensively studied owing to their applications in optical communication [1–3], chiral microstructures [4], particles manipulations [5, 6], astronomical observations [7], optical measurements [8, 9], etc [10–12]. However, the central dark hollow of a conventional optical vortex is strongly dependent on its topological charge (TC). This property limits the applications that simultaneously require a small vortex diameter and a large TC, in particular, for coupling multiple OAM beams into a fiber used in communication technologies [13, 14]. In order to overcome this challenge, in 2013, Ostrovsky et al. [15, 16] proposed the concept of a perfect optical vortex (POV), where the diameter is independent of the TCs. After this finding, research has been performed on POV’s generation [17–19], verification [20], modulation [21–23], and applications [16, 24, 25].

However, POV has only a single sample mode in the observation plane, i.e., a bright ring that has a constant diameter, independent of the TC. Therefore, it is challenging to satisfy the requirement to form a complex structured optical field. Versatile modes are significant for optical vortex beams, owing to the potentials for advanced applications [26], such as the optical cage formation [27–29], optical microfluidic sorting [30], and micro-particles regulation and acceleration [31]. Therefore, it is of a key importance to realize a POV with multiple modes. In 2017, Mazilu et al. [32] proposed a fractional POV with multiple gaps, which enriches the modes of POVs. Further, Kovalev et al. [33] reported an elliptic perfect optical vortex (EPOV), and derived the exact analytical expressions for the total OAM and OAM density. However, it is inconvenient to modify the ellipticity of EPOV due to the two dependent parameters. Therefore, they cannot supply EPOV versatile modes.

The EPOV is a type of asymmetric optical vortex field, suitable for manipulation of biological cells [34]. Therefore, it is interesting to study the mode transformation of the POV and its modulation from circle to ellipse. For this purpose, we propose a novel method, called coordinate transformation, in order to freely transform the modes from POV to EPOV and simultaneously to provide versatile spatial modes for the fractional EPOV.

In this study, we deduce the analytic expression for the EPOV, where the eccentricity is determined only by a scaling factor. The EPOV modes can be freely transformed from circular to elliptical. During the mode transformation, the properties of the “perfect vortex” are preserved, and the mode purities maintain high values. Further, the modes of the fractional EPOV (half-integer) are obtained; it is shown that the location of the gap and its number can be freely controlled. This study reveals a method to enrich the modes of the POV, which could facilitate novel applications such as complex optical tweezers.

## 2. Coordinate transformation method to freely modulate POV modes

First, we consider the POV generation process, which can be easily obtained through the Fourier transformation (FT) of a Bessel–Gauss beam using a spatial light modulator (SLM) [17]. The complex amplitude of the Bessel–Gauss beam at the SLM plane (object plane) can be written in polar coordinates (*ρ*, *φ*) as:

*J*is the

_{l}*l*th-order Bessel function of first kind,

*k*is the radial wave number, and

_{ρ}*ω*

_{g}is the beam waist of the Gaussian beam.

Using a convex lens that has a focal length of *f*, in order to conduct the FT of Eq. (1), the POV beam can be produced at the recording plane, which can be written as [17]:

*r*,

*θ*) represents the polar coordinates at the recording plane,

*ω*

_{0}( = 2

*f*/

*kω*

_{g}) is the Gaussian beam waist at the focus,

*l*is the topological charge, and

*R*is a constant that determines the radius of the POV. For a small

*ω*

_{0}, Eq. (2) approximately describes a perfect vortex beam [18].

Intuitively, if we want to change a circle to an ellipse, in Cartesian coordinates, we can fix one axis and stretch out the other axis. One can ask the following question: Can we obtain an EPOV by stretching the Bessel–Gauss beam to an elliptic Bessel–Gauss beam and then apply an FT operation? In order to find the answer, the coordinates can be transformed as *mx* = *ρ*cos(*φ*) and *y* = *ρ*sin(*φ*), where *m* is a positive constant, called a scaling factor, and (*x*, *y*) denotes the Cartesian coordinates at the object plane. In this case, (*ρ*, *φ*) is a type of elliptic coordinate, and the polar coordinates in Eq. (1) are the particular case for *m* = 1. In contrast, the third term in Eq. (1) changes from a Gaussian to an elliptical Gaussian distribution.

Similar to the FT procedure for polar coordinates [35], the elliptic Bessel–Gauss beam in Eq. (1) can be deduced from the FT definition in Cartesian coordinates:

*u*,

*v*) denote the Cartesian coordinates at the Fourier plane (recording plane), and

*k*is the wave number.

In order to achieve a coordinate transformation for Eq. (3) and obtain an analytical expression, at the Fourier plane, the elliptic coordinates are defined as *u* = *r*cos(*θ*) and *mv* = *r*sin(*θ*), rotated by π/2 with respect to those at the object plane.

By substituting the variables of Eq. (3), after simplifications, the equation can be rewritten in elliptic coordinates as:

After a mathematical operation similar to that reported in Ref [17], the complex amplitude of the optical field is:

*ω*

_{0}is replaced with

*ω*

_{m}. For a small value of

*ω*

_{m}, Eq. (5) represents the complex amplitude of an elliptic POV. Compared with the POV, the EPOV generation requires a smaller

*ω*

_{0}, due to the relation

*ω*

_{m}=

*mω*

_{0};

*R*represents the semi-axis of the EPOV along the

*u*-direction.

Next, we determine the location of the EPOV elliptic ring. The quadratic sum of *u* = *r*cos(*θ*) and *mv* = *r*sin(*θ*) can be reduced to:

This expression represents a series of ellipses that have the same eccentricity *e* = sqrt(1-*m*^{2}) for 0<*m*<1, and *e* = sqrt(1-1/*m*^{2}) for *m*>1, respectively. For *r* = *R*, this expression reveals the bright ellipse of the EPOV.

Therefore, the approximate EPOV can be easily generated by a coordinate transformation from Cartesian to elliptic coordinates. The advantage of this method is that the EPOV modes can be freely modulated from circle to ellipse by adjusting the scaling factor *m*.

## 3. Experiment

The experimental setup is illustrated in Fig. 1. A reflective liquid-crystal SLM (HOLOEYE, PLUTO-VIS-016, pixel size: 8μm × 8μm, resolution: 1920 × 1080 pixels) is used to generate an EPOV beam. A laser beam that has a wavelength of 532 nm and a power of 50 mW (Laserwave Co., LWGL532) is expanded and illuminates the SLM plane. Then, the FT is conducted by a lens (*f* = 20cm) and the EPOV beam is recorded by a CCD camera (Basler acA1600–60gc, pixel size of 4.5 μm × 4.5 μm).

However, after passing through the beam expander and the aperture A1, the laser beam becomes an approximate flat-top beam that does not satisfy the elliptic Gaussian term in Eq. (1). In order to overcome this challenge, an elliptic aperture is introduced and multiplied with the phase mask to obtain an approximately elliptic flat-top beam. Therefore, the combined phase mask introduced at the SLM is designed according to the following equation:

*α*and

*n*are the cone angle and refractive index of the axicon, respectively.

*D*is the period of the blazed grating.

The generation process of the phase mask is illustrated in Fig. 2. A spiral phase pattern [Fig. 2(a)] adds the axicon phase [Fig. 2(b)], which is generated in the elliptic coordinates (*ρ*, *φ*). Subsequently, a blazed grating is added [Fig. 2(c)], and then truncated by an elliptic aperture [Fig. 2(d)]. In order to maintain the imaging quality, the area around the formed elliptic phase mask is designed as a “checkerboard” pattern, as illustrated in Fig. 2(f). The phase mask shown in Fig. 1(e) is written at the SLM. It is noted that the elliptic Bessel-Gauss is approximately generated since the Bessel function *J _{l}*(

*k*) is replaced by the axicon exp[

_{ρ}ρ*ik*(

*n*-1)

*αρ*].

## 4. Results and discussions

Using the proposed method, EPOVs can be generated experimentally, as illustrated in Fig. 3. In order to obtain an accurate mode transformation, the optical path should be preliminarily aligned using the criterion *e* = ~10^{−3} for *m* = 1, where the mode pattern is a bright circle ring that corresponds to the POV mode. The EPOV modes can be easily transformed from circle to ellipse by adjusting the scaling factor *m*, when the TC has *l* = 1 and *l* = 10. When 0<*m*<1, the circle is squeezed along the horizontal direction to an ellipse and the eccentricity increases with the decrease of the scaling factor *m*. In contrast, the circle is stretched to an ellipse along the horizontal direction when *m*>1.

After calculating the eccentricities of the patterns shown in Fig. 3, the correlation coefficients of the eccentricity *e* between the experimental and theoretical results are obtained as 0.9993 and 0.9997 for *l* = 1 and *l* = 10, respectively, which indicates that the spatial structures of the experimental patterns are desirable. Researchers can obtain an arbitrary ellipse mode if they are provided with a suitable scaling factor *m*. A dynamic mode transformation is demonstrated in Visualization 1.

For beams, the mode purity *η* is an important parameter that can be estimated using a semi-quantitative method through the correlation coefficients between the observed patterns and fitting calculation modes [36].

We assumed that the fitting formula for the observed mode patterns is *I*(*r*, *θ*) = *A*_{0}exp[-2(*r*-*R*)^{2}/*ω*^{2}] + *B*_{0}. Using the least squares criterion, the optimal parameters including the ring radius *R*, ring width *ω*, beam center position(*O*_{x}, *O*_{y}), background *B*_{0}, and intensity scale factor *A*_{0}, are obtained by fitting the experimental patterns shown in Fig. 3. Then, the mode purity *η* is calculated by fitting each whole pattern and using the above formula, as presented in Fig. 4. For a better visualization, Fig. 4 shows only the cross-sections of the mode patterns of Fig. 3 (dots) and their fitting curves (solid curves). Figure 4 shows that the mode purity *η* is greater than 0.9 for each mode pattern of Fig. 3, which indicates that the mode patterns maintain the highest mode purity during the EPOV mode transformation using the proposed method.

Intuitionally, Fig. 3 shows that the shape and size of the bright ring are independent of the TCs under the same conditions. However, are the EPOVs perfect vortices? This is a key issue that should be addressed. For this purpose, Fig. 5 shows the intensity patterns of the EPOV with different TCs of *l* = 1, 3, 5, 7, and 9. In the patterns, the values in the parenthesis are the semi-major and semi-minor axes, respectively. The calculations show that the relative error of the semi-major and semi-minor axes are both lower than 2.5% for *m* = 0.5 and *m* = 2. These results demonstrate that the EPOV satisfies the property of the perfect vortex; their semi-major and semi-minor axes are independent of the TCs.

Next, we analyze whether the spiral phase structure is preserved during the mode transformation of the EPOV from circle to ellipse. In order to verify the existence of the vortex, a simple and effective in situ method is employed to provide an interference between the EPOV and its conjugate beam using the phase shift method [20]. Figure 6 illustrates the interference patterns between the EPOVs in Fig. 5 and their conjugate beams. As expected, elliptic spiral interference fringes are observed in each pattern, which demonstrate the existence of the spiral phase of EPOV. A careful analysis shows that the fringe number is twice the corresponding TC; the same result has been reported in Ref [20].

At this stage, the experimental EPOV is verified to be a real elliptic “perfect optical vortex”, which needs to be maintained while transforming their modes. As shown in Figs. 3 and 5, the vertical axis is fixed when the modes transform. In order to increase the flexibility of the mode transformation, the cone angle of the axicon is used as an adjusting parameter to change the fixed elliptic axis. The mode transformation for different cone angles *α* is illustrated in Fig. 7. Obviously, the semi-fixing axis increases with the angle, and decreases along the vertical direction. The fitting relationship between the semi-major axis *b* (units: micrometer) and cone angle *α* (units: degree) can be expressed as a linear function: *b* = 7.55 + 1.05 × 10^{4}*α*; the correlation coefficient is 0.99994. By adjusting the parameters *m* and *α*, one can obtain any desired mode from circle to ellipse.

A fractional vortex can provide more manipulation degrees of freedom and higher resolution of the OAM force. Therefore, it is of importance to generate and transform a fractional EPOV with TC of *l'*. Figure 8 shows a fractional EPOV (half-order) for *l'* = 1.5 and *l'* = 10.5, along with the designed phase masks. For comparison, the experimental and theoretical intensity patterns, as well as the corresponding phase patterns, are simultaneously shown. The experimental intensity patterns are in a good agreement with the theoretical results. However, another two smaller gaps appear near the main gap for higher half-order beams (TC = 10.5). This phenomenon emerges due to the split of the high-order vortex and slight mismatch of the vortex beam after the propagation, with respect to its initial state; the location of the gap is slightly displaced with the increase of TC, as shown in Ref [37]. The phase patterns and their subset magnifications ( × 3) show the formation of a half-order vortex (gap in the intensity ring). The indeterminacy of the phase pattern number is equal to that of the gaps.

The left two columns of Fig. 8 are obtained based on the coordinates mentioned above, where the gap is always located on the left vertex, similar to that in a conventional fractional optical vortex [38]. In order to change the position of the gap, the coordinates transform as *mx* = *ρ*sin(*φ*), *y* = *ρ*cos(*φ*), and *u* = *r*sin(*θ*), *mv* = *r*cos(*θ*) at the object and Fourier planes, respectively. The results are shown in the right two columns in Fig. 8. Compared with the previous coordinate transformation, the angular coordinate rotates clockwise by π/2, which causes a shift in the gaps and reverses the TC’s sign, as shown in Figs. 8 (c) and (d). However, the eccentricity and directions of the major and minor axes of the ellipse are unaltered, owing to the invariability of the ellipse equation [Eq. (6)]. Therefore, there are more choices for the selection of a suitable mode for the fractional EPOVs.

The detailed properties of these two types of fractional EPOV are listed in the first and second rows in Table 1. If we stretch the Bessel–Gauss beam along the other direction, the fixed axis of the ellipse will alter to its orthogonal direction and we can obtain another two modes of the fractional EPOV, which are listed in the third and fourth rows in Table 1.

However, the intensity patterns in Table 1 show that the gap is located only at the right or top vertex of the ellipse. It is desirable to be able to freely adjust the gap position on the ellipse. By applying a spiral phase, exp[i*l'* × rem(*φ* + *ψ*,2π)] where rem(·) is the remainder after division, instead of exp(i*l'φ*) when generating the phase mask, the gap can be positioned at a desired position, as demonstrated in the upper panel of Fig. 9, as well as in the animation shown in Visualization 2.

To obtain the fractional EPOV with multi-gaps, a special spiral phase is designed instead of exp(i*l'φ*) when generating the phase mask. The multi-gaps spiral phase of *F _{v}* is designed as,

*N*≥1 is the number of the gaps of the EPOV;

*n'*is an integer; step(·) is the Heaviside step function;

*φ'*is a variable,

The spiral phases calculated by Eq. (8) and the fractional EPOV patterns with multi-gaps, i.e. *N* = 1~5, are demonstrated in the two lower panels of Fig. 9. This method could enrich the modes and expands the potential applications of EPOVs. Simultaneously, the mode purity maintains a high value of ~0.9 for both gap-moved mode patterns and multi-gaps mode patterns.

However, there is a fascinating and important issue: can the TC value be preserved a constant of 1.5 in all cases of Fig. 9? As is well known, a fractional optical vortex can be obtained by the superposition of a series of integer optical vortices weighted by Fourier coefficients [38]. Based on this theory, the fractional EPOV of *F _{v}* is decomposed into a series of integer spiral phases using the similar process of the Ref [39]. Therefore, Eq. (8) can be rewritten by the Fourier series,

*l*is an integer;

*c*is the Fourier coefficients;

_{l}*φ*

_{0}is an initial phase. The Fourier spectrum components of fractional optical vortices can be written as

*N*|

*c*|

_{l}^{2}, which are shown in Fig. 10. For an integer spiral phase

*l*= 1, the component is a single mode as seen in Fig. 10(a). Figure 10(b)~(f) show the situations of the fractional spiral phase used to generate different multi-gaps. The fractional spiral phase

*l'*= 1.5 with a single gap demonstrates that the energy shifts from the

*l*= 1 vortex to the

*l*= 2 vortex, which is shown in Fig. 10(b). As increasing the number of gaps,

*N*, the energy shifts from the

*l*= 1, 2 vortices to the other vortices. However, when we count all of the compositions and obtain a total TC

*L*= ∑

*lN*|

*c*|

_{l}^{2}, which always approaches to 1.5 with the data range of

*l*expanding for different multi-gaps of

*N*. As the data range of

*l*is from −150 to 153, the total TCs are shown in Fig. 10.

Moreover, the Fourier decompositions of the fractional spiral phase with different gap locations (shown as the second panel of Fig. 9) are all as same as Fig. 10(b). Therefore, the TC is a constant during modulating the gap locations. Notice that the integer optical vortices used in Fourier series in Eq. (9) are all of the elliptic integer optical vortices. These results demonstrate that the parameter *N* only influences the energy distribution within different integer optical vortices. Consequently, in all these cases of Fig. 9 the topological charge of the generated EPOV is equal to 1.5.

For considering the influence of the phase distribution on the mode purity, the fractional EPOV can be decomposed into a finite number of eigenmodes with *l* within the range [-10, 13] as shown in Fig. 10. The ratio of the component of each *l* (component mode purity) is calculated in this range. Then the mode purity of the fractional EPOV is also obtained by summing all component mode purities, which is denoted as *η'* and marked in Fig. 10. Obviously, these mode purities of *η'* are bigger than those obtained using the semi-quantitative method as the effect of nonideal behavior of the SLM and the environmental vibration [40] is not considered. One can also find that the mode purity *η'* decreases with the number of the gap increasing because the energy shifts to higher order optical vortices.

Actually, the designing parameters of the phase mask will influence the mode purity of the output beam [41] using the semi-quantitative method for either integer or fractional EPOV. The effect of this issue should be considered in the future work.

In general, using the proposed method, the POV modes can be freely modulated from circle to ellipse. Simultaneously, the property of the perfect vortex is preserved. In a future research, different methods should be developed to generate more complex modes, for example, using the axis-off method [42]. Moreover, the properties of the OAM and gradient force should be further studied during the dynamic process of the mode transformation, in particular, for fractional EPOVs.

## 5. Conclusions

In summary, the POV modes can be freely transformed from circle to ellipse using the coordinate transformation method. By adjusting the scaling parameter *m*, the ellipse intensity ring becomes stretched/squeezed along one direction. By modulating the cone angle of the axicon, the ellipse can be adjusted along the other direction. Moreover, versatile modes of the fractional EPOV (half-order) are obtained; the number and position of the gap can be controlled. During the mode transformation, the observed mode pattern always maintains a higher mode purity. For a special fractional EPOV, it is proved that its TC maintains a constant during the mode transformation. This technique can provide versatile POV modes for various potential applications such as optical tweezers, laser micro-machining, and laser beam shaping.

## Funding

National Natural Science Foundation of China (NSFC) (Grant Numbers: 61775052, 61205086, and 11704098).

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