Efficient generation of a perfect optical vortex by using a phase optical element

Abstract

We consider generation of a perfect optical vortex by three elements: (i) amplitude-phase element with its transmission being proportional to the Bessel function, (ii) optimal element with the transmission proportional to the sign of the Bessel function, and (3) helical axicon. Maximal intensity of light on the ring is shown to be achieved by using the optimal element. Thickness of the light ring generated by the axicon is approximately two times higher than that for other elements. For perfect optical vortices with the radius of several wavelengths we detect the range of topological charges, within which the ring radius is almost independent on them.

Keywords

• optical vortex
• perfect optical vortex
• topological charge
• axicon
• Bessel function
• Fourier optics
• Fraunhofer diffraction

1. Introduction

"Perfect" optical vortex [1] has the following complex amplitude:

Computer Optics and Nanophotonics / V.V. Kotlyar, A.A. Kovalev, A.P. Porfirev In the focus of the lens, the field from the complex amplitude ( 3 ) reads as where r0 = 0, rm  n,m  , m  1, 2,..., N , rN  R . Putting  = f/k in Eq. ( 4 ), the argument of the Bessel function becomes independent on the parameters f and k and equals r. It can be shown that at 0 = n, Nf/(kR) all terms in the sum are positive and their contribution into the light field on the ring is maximal. For the radius of the POV ring to be independent of the topological charge, close roots of the Bessel function need to be chosen: n, N = m, M.

4. Generation of a "perfect" optical vortex by using a axicon

The POV is often generated by using a conical axicon and a SPP [2]. When light passes through such setup, it is equivalent to passing an element with the following transmittance function:

 r  F3 r,   circ   exp ir  in .

 R 

For the first time, optical element ( 5 ) was considered in [6] for generation of light pipes. It was also studied in [7-9]. Instead of scaling factor of the Bessel function, here the parameter  determines an axicon parameter, related with the vertex-angle of the generated conical wave. It is supposed in [2] that in the focus of a spherical lens the light field ( 5 ) generates a POV with its amplitude distribution described by the function E3 ,  exp    0 2 2  exp in . It is a simplification and the complex amplitude of the POV is defined by a Fourier transform of the function ( 5 ), taking a much more complex form [6]. The intensity on the ring is lower than that for the optimal element ( 3 ), but higher than for the amplitude element. The intensity on the ring depends on the topological charge, while the ring thickness is approximately two times larger than in previous cases.

5. Simulation results

In this section, we describe the simulation results of generating the POV by using the considered above optical elements. The simulation parameters are as follows: wavelength of light λ = 532 nm, circular aperture radius is R = 20 mm, and the focal length of an ideal spherical lens f = 100 mm, while the Bessel function’s scale factor  is chosen so that R = γ1,20 = 63,6114, where γ1,20 is 20th zero of the first-order Bessel function ( = 20, n = 1). The POV was simulated for two topological charges: n = 1 and n = 14, while the other parameters were kept unchanged. Note that for the Bessel function of order n = 14 we chose the 14th root (ν = 14) because γ14;14 ≈ γ1;20 = 63.6114. Figure 1 shows the absolute values of two Bessel functions, |J1(γ1;20x/R)| and |J14(γ14;14x/R)|. It is seen in Fig. 1 that both functions are seen to take a zero value at x = R. ( 4 ) ( 5 ) a)

b)

Now we consider generation of the POV by using the optimal phase element ( 3 ). Fig. 3 shows the intensity distributions for the initial light field ( 3 ). Table 2 shows the computed parameters of the POV from Fig. 3. From Table 2, the POV radius became slightly smaller than that in Fig. 2 (less by just 0.3%). The radius remained almost unchanged when the topological charge was increased by a factor of 14. The intensity at the ring is almost 100 times greater than the intensity for the POV in Fig. 2. We note that with an increase of the topological charge by a factor of 14, the intensity on the ring decreased by only 2%. The ring thickness became smaller by approximately 14% compared to the thickness of the ring in Fig. 2. The thickness of the ring is preserved when the topological charge of the optical vortex changes. Fig. 3 shows that side lobes have increased.

Next, we consider generation of the POV by using a helical axicon ( 5 ). Fig. 4 shows intensity distributions for the initial light field ( 5 ), while the Table 3 shows the computed parameters of this POV. From Fig. 4 and Table 3, the POV ring thickness is approximately 2.5 times larger than the thickness of the ring in Fig. 2.

In addition, with increasing topological charge of the vortex, the ring thickness increases by a factor of 1.3.

Computer Optics and Nanophotonics / V.V. Kotlyar, A.A. Kovalev, A.P. Porfirev

An increase of the ring thickness (Fig. 4) with increasing n leads to the decreasing intensity on this ring. From Table 3, it is seen that the maximal intensity on the ROV ring (Fig. 3) decreases 1.7 times as the number n increases by a factor of 14. And even the radius of the maximal intensity ring increases by 8%. In this case, the thickness of the ring and the maximum intensity almost do not change with increasing topological charge of the optical vortex from 1 to 14. The only drawback of the POV generated by the element ( 3 ) is an increased level of side lobes, which constitute about 20% of the maximum intensity.

So, the simulation has shown that among the three optical elements for generating the POV, the optimal phase element in Eq. ( 3 ) is the best one, since the narrowest ring (FWHM = 1,96 = 0.39λf /R) is generated in this case with the maximum intensity being 1.6 times higher than that from the spiral axicon in Eq. ( 5 ).

Above, the topological charge n and the scaling factor of the axicon  were chosen so that the product R remained approximately the same and was equal to the root of the Bessel function. Now we consider the case when this is not so. Let this product be arbitrary and let the optimal element or axicon with a diameter of 2R = 40 to generate a POV by using a lens with a focal length f = 100. Fig. 5 shows the dependence of the light ring thickness (at half-maximum of the intensity) on the radius of the ring.

For the POV radius r0 = 2, both elements work identically. Both of them generate a light ring with a radius of about r0 and with about same thickness. At the same time, for the optimal element the maximal energy is approximately 30% higher. This is explained by the fact that, despite the same width at the level of half-maximum, the thickness at a lower intensity level is larger for the axicon. When r0 = 3, both elements form a ring of wrong radius of about 2. However, the maximum energy for the optimal element is 86% higher. When r0 = 4, the optimal element generates a ring of radius 4.1, while the axicon generates two light rings, one of which has a radius 1.9, and the second - 4.7. When r0 = 5, both elements generate two light rings with radii 5 and 1.5. Further, for r0> 5, both elements generate a light ring of radius r0. With increasing r0 up to r0 = 70, the ring thickness remains practically unchanged, but in all cases the ring generated by the axicon is about 2 to 2.5 times wider.

Now we consider the dependence of the radius of the generated light ring on the topological charge. Let the optimal element ( 3 ) or the axicon ( 5 ) of the diameter 2R to generate a POV by using a lens with a focal length f = 100. Fig. 6 shows the dependence of the radius of the generated light ring on the topological charge for elements of radius R = 10 (Fig. 6a), R = 20 (Fig. 6b), and R = 30 (Fig. 6c).

a) Fig. 6. Radius of the generated light ring vs. the topological charge for different element radii: R = 10 (a), R = 20 (b), and R = 30 (c). Upper curve – axicon ( 5 ), lower curve – optimal element ( 3 ).

Computer Optics and Nanophotonics / V.V. Kotlyar, A.A. Kovalev, A.P. Porfirev

It is seen in Fig. 6 that with using the axicon ( 5 ) the light ring radius rises with the topological charge nearly linearly. When using the optimal element ( 3 ), the radius is almost constant for the topological charges up to R/. With larger topological charges, the radius begins to increase linearly at about the same rate as for the axicon.

6. Experiment

For experimental study of the optical elements for generating the POVs we used an optical setup shown in Fig. 7. The fundamental Gaussian beam was a light source generated by a solid-state laser L (λ=532 nm). The laser beam was expanded and collimated using a system composed of a 40-μm pinhole PH and a lens L1 (f1 = 250 mm). Then the beam illuminated the display of a spatial light modulator SLM (PLUTO VIS, 1920 × 1080 resolution, with 8 μm pixels). The diaphragm D1 was used to separate the central bright spot from the surrounding dark and bright rings caused by diffraction by the pinhole. Further, using the lenses L2 (f2 = 350 mm) and L3 (f3 = 150 mm) and the diaphragm D2, spatial filtering of the phase-modulated laser beam reflected at the SLM display was performed. Using a lens L4 (f4 = 500 mm), the laser beam was focused on the CCD array of a video camera LOMO TC 1000 (pixel size 1.67 × 1.67 μm). To generate the POVs, we used the phase masks shown in Fig. 8, which were output to the SLM display. To separate the non-modulated beam reflected at the display and the phase-modulated beam, a linear phase mask was superimposed on the initial phase mask.

Fig. 7 shows the intensity distributions in the focus of lens L4 generated by using phase masks for the optimal phase elements with topological charges n = 1 and n = 14. The values of the parameters of the resulting POV are given in Table 4.

Figure 8 depicts the intensity distributions in the focus of lens L4 generated using phase masks corresponding to the spiral axicons with n = 1 and n = 14. The values of the parameters of the resulting POV are given in Table 5. Analyzing the experimentally measured parameters of the POV, we can conclude that their relative values are in good agreement with the simulation results. 7. Conclusion

In this work, generation of a perfect optical vortex by three different optical elements is considered: amplitude-phase element with a transmission proportional to the Bessel mode, an optimal phase element and a vortex axicon. It is shown that using any of these three optical elements leads to generation of light rings with the same radius, which depends little on the topological charge of the optical vortex. However, if the POV radius is equal to several wavelengths, then it depends on the topological charge. For the axicon, this dependence is almost linear, while for the optimal element this dependence is almost absent for topological charges that are smaller than the ratio of the element's radius to the wavelength. The intensity of light on the ring is greater (with other conditions being equal) for the optimal phase element. The intensity of all three rings depends little on the topological charge. The thickness of the light ring generated by the vortex axicon is approximately twice larger compared to that of the other two rings. Thus, the optimal filter ( 3 ), studied for the first time in [3], is the best candidate for generation of a perfect optical vortex.

Acknowledgements References

The work was funded by the Russian Science Foundation (RSF) grant # 17-19-01186 (the results pertaining to the experimental generation of perfect optical vortices presented in Section 6), as well as by the Russian Foundation for Basic Research (RFBR) grant # 17-47-630420 (the results pertaining to the numerical simulation of the perfect optical vortices generated by different elements presented in Sections 1-5).

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