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Abstract
Vector vortex beams are widely used because of their anisotropic vortex polarization state and spiral phase. Constructing mixed mode vector vortex beams in free space still requires complex designs and calculations. We propose a method for generating mixed mode vector Elliptical perfect optical vortex (EPOV) arrays in free space by mode extraction and optical pen. It is demonstrated that the long axis and short axis of EPOVs are not limited by the topological charge (TC). Flexible modulation of parameters in the array is achieved, including number, position, ellipticity, ring size, TC, and polarization mode. This approach is simple and effective, it will provide a powerful optical tool for optical tweezers, particle manipulation, and optical communication.
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1. Introduction
An optical vortex (OV) beam has a helical phase profile described by exp (ilφ), where l is the TC and φ is the azimuthal angle [1]. OVs have been widely used in optical communication [2,3], optical manipulation [4,5] and microscopic imaging [6,7] owing to their unique properties. It is well known that the radius of OV is closely related to TC, which greatly limits its application. To solve this problem, Ostrosky et al. proposed the concept of the perfect optical vortex (POV) [8]. POV exhibits a constant intensity distribution and the radius isn't controlled by the TC [9,10], This breakthrough has also opened up more opportunities [11–14]. However, conventional circular POVs are not versatile in complex structural optics. Designing different shapes of POV can meet specific application scenarios [15,16]. So, EPOVs offer additional possibilities due to their unique light field structure.
EPOVs carry both spin angular momentum (SAM) and orbital angular momentum (OAM) [17]. Compared with traditional circular POVs, the OAM of EPOVs is affected by parameters such as ellipticity and angular direction [18,19]. Kovalev et al. gave exact expressions for the OAM density and the total OAM of the EPOV, and also proved that the OAM density is maximal on the smaller side of the EPOV and is minimal on its larger side [20]. This special property allows more flexible manipulation of particles, as well as the generation of OAM-entangled photons. As an extension of the concept of POV, EPOV can be generated by a variety of methods. For example, EPOV can be generated by the elliptic Bessel beam or the spiral axicon [20], the coordinate transformation [21], and the phase-only diffraction optical element [22]. Unfortunately, the above approaches can only generate one independent mode of EPOV, and these studies mainly involve scalar optical fields. The conversion from scalar to vector can provide higher controllable degrees of freedom for high-volume information encoding [23], particle capture [24], and super-resolution imaging [25]. Delin Li et al. proposed a method to generate vector EPOV by modulating the dynamics and geometric phase [26]. But the vector EPOV generated in this way also has only one independent mode. After that, they used vector optical field generator and phase type computer-generated hologram to shape the vector EPOV and obtained higher purity vector EPOV arrays [27]. However, this approach requires the algorithm to iterate repeatedly, adds a tedious computational process. Therefore, it has great practical significance to study mixed mode vector POV arrays.
In this paper, we propose a method for generating mixed-mode coexisting vector EPOV arrays. In our project, the circular POV is converted to EPOV by coordinate transformation. Then, the mode extraction [28] is used to extract the arbitrary order vector beam in the 30-th order vortex polarizer (VP), and the optical pen [29] is used to modulate multiple degrees of freedom of the vector EPOV array. We have experimentally verified the feasibility of this approach. In this way, the numbers, position, ellipticity, and polarization order of vector EPOVs can be flexibly adjusted to generate vector EPOV arrays with controllable mixing modes. It will provide methodological guidance in the fields of particle manipulation, optical tweezers, high-capacity optical communication, and quantum optics, and also provides a new direction for higher degree of freedom in optical field modulation.
2. Theory
In the focusing system, the electric field distribution of the beam near the focus according to the Debye vector diffraction theory can be expressed as [30]
A vector beam with inhomogeneous polarization can be described as a combination of two orthogonal polarization modes. The two polarization modes can be simplified into x and y linear polarization modes. Therefore, the electric field of the m-order vector beam can be written as [31,32]
where $|x \rangle$ and $|y \rangle$ denote the x and y linear polarization modes, respectively. Thus, the propagation unit vector of the incident beam immediately after having passed through the lens can be expressed as where Vx and Vy are the electric vectors of $|x \rangle$ and $|y \rangle$, respectively. They can be written as [30]Next, we discuss the construction of EPOV. It is well known that the Gaussian beam can be converted into a Bessel-Gaussian beam by an axicon, and the POVs can be obtained at the back focus plane after the Fourier transform. Its transmittance function can be expressed as
where r is the polar diameter, d is the period of the axicon and in this article d = 0.1 mm. To obtain the EPOV, we stretch a circular Bessel beam into an elliptical Bessel beam by a coordinate transformation.Figure 1(a) shows the process of coordinate transformation, where a = 1, b = 0.6, and both phase and focus fields are stretched along the x-axis direction. Therefore, POV also can be regarded as a special kind of EPOV when a = b.
Fig. 1. Schematic diagram of EPOV generation. (a) The results of coordinate transformation of phase and focus fields. (b) Generation process of vector EPOV phase mask. (c) Focusing of the vector EPOV.
Vector EPOVs have complex spatial polarization patterns. To identify the polarization modes of the EPOVs, we use the mode extraction approach. By adding a modulation term Td to modulate the m-order vector beam, the modulation symmetry of the polarization is broken and the target modes are separated from other modes. The modulation term Td is expressed as [28]
where m is the order of the vortex polarizer, and m = 30 in this paper, ne is the polarization order of the generated vector beam, β is the initial polarization direction, and β = 0 in this paper.To implement the vector EPOV array with mixed modes in space, we combine the previously studied optical pen with it [28,33]. The total transmittance function of the vector EPOV array is
Figure 1(b) depicts the generation process of the vector EPOV phase mask. From left to right, it shows the axicon phase, the vortex phase, the mode extraction phase, and the total phase, respectively. Eventually, the focused light intensity of the vector EPOV array can be obtained using I=|E|2. Figure 1(c) shows the focusing process of the vector EPOV. The incident beam is a 30-th order vector beam, P0 is the modulated phase, and finally the focused field of the vector EPOV is obtained through the lens L0.
3. Experiment and discussion
In order to verify the above principle, we design an experiment shown in Fig. 2. A laser with the wavelength of 780 nm is used as the light source. The generated Gaussian beam is converted to linear polarization by the polarizer P. The beam is expanded by lenses L1 (f = 75 mm) and L2 (f = 150 mm), then it passes through a reflector M to the beam splitter BS. After the beam splitter, one of the beams is directed to the phase-only spatial light modulator (SLM) (HOLOEYE, model: PLUTO-2-NIR-011, pixel: 1920 × 1080, image element: 8.0 µm). The combined beam passes through a 4f system consisting of L3 (f = 100 mm) and L4 (f = 100 mm). The 30-order VP is conjugated to the SLM through the 4f system, and the VP is a vortex polarizer realized by the Q-plate technique. The modulated vector EPOV output is focused by the lens L5 with NA = 0.15 in the focusing system and the light intensity of the EPOV is recorded using a CCD (DMK 33GJ003e, pixel: 3856 × 2764, image element: 1.67 µm).
Fig. 2. Schematic diagram of the experimental setup for generating the vector EPOV.P: polarizer; L1 - L5: lens, L1 (f = 75 mm), L2 (f = 150 mm), L3 - L5 (f = 100 mm); pin1 - pin2: pinhole; M: mirror; SLM: Spatial light modulator; BS: beam splitter; VP: vortex polarizer. The calculated phase hologram is loaded into the SLM and the EPOV image is acquired by CCD.
Using the experimental setup shown in Fig. 2, we verify the perfect characteristics of EPOV firstly. Experimental plots of EPOVs with different TCs and their calculated sizes when a = 1, b = 0.7 are given in Fig. 3. Figures 3(a1) - (a5) are the phase diagrams and the TCs are l = -6, -3, 0, 3, 6 from left to right, respectively. (b1) - (b5) are the corresponding experimental results. (c) shows the defined long axis and short axis of EPOV, and (d) shows the calculated long axis and short axis corresponding to the different TCs. In the experiments, we unify the ellipticity and verify the long axis and short axis of EPOV for five different TCs. Figure 3(d) clearly illustrates the calculation results. The maximum error of the long axis is only 0.0034 mm and the maximum error of the short axis is 0.0201 mm. It can be determined that, as the circular POV, the EPOV also conforms to the “perfect” characteristics and its long axis and short axis are independent of the TCs.
Fig. 3. EPOV with different TCs. (a1) - (a5) are phase diagrams, (b1) - (b5) are experimental diagrams, (c) shows the defined long axis and short axis of EPOV, and (d) is the calculated long axis and short axis corresponding to the different TCs.
To investigate the effect of ellipticity on the light field of EPOV, we generate the vector EPOV by mode extraction. In our experimental setup VP is 30-th order (m = 30) and ne represents the order of the generated vector EPOV. Figure 4 shows the experimental results of vector EPOV with different ellipticity, where ne = 2 and the shape of the ellipse can be controlled by varying the parameters a and b. The first column in Fig. 4 shows the phase diagram, the column (a2) shows the experimental results without the analyzer. Columns 3 to 5 are experimental results when the analyzer is 0°, 45° and 90°, respectively. Figure 4 (c2) illustrates the 2nd order vector POV with a = 1, b = 1, which has a uniform light intensity distribution of the ring. With the increasing ellipticity, the side where the long axis of the ellipse located always has higher energy intensity, and the short axis is opposite. Therefore, we can conclude that the ellipticity modulation factor can change the shape of the optical field, as well as change the light intensity distribution and OAM density.
Fig. 4. Experimental diagram of vector EPOV for different ellipticity with ne = 2. (a1), (b1), (c1), (d1), and (e1) are the phase diagrams for different ellipticity, respectively. Column (a2) shows the experimental results without the analyzer, a and b represent the ellipticity parameters. Columns (a3)-(a5) are the experimental results when the analyzer is 0°, 45°, and 90°, respectively. The light intensities are normalized to a unit value, the phase scale is 0 - 2π, and the direction of the arrow in the circle indicates the direction of the analyzer.
In order to illustrate the effect of ellipticity on the light field energy clearly, we plot the vector EPOV energy density distribution curves with different ellipticity as shown in Fig. 5. Figures 5(a1), (b1), (c1) and (d1) represent the light field energy distribution when (a = 1, b = 0.6), (a = 1, b = 0.8), (a = 0.6, b = 1), (a = 0.8, b = 1), respectively. As shown in Fig. 5(a1), the center point of EPOV is defined as the coordinate origin, and a circle of the ellipse (0 - 2π) is divided equally into N (N = 90) parts. Then, accumulate the intensities in each part and count the number of pixel points of light intensity. Finally, a mean value is taken to obtain the average energy density distribution curve on the circumference of the EPOV. Figures 5(a2), (b2), (c2), and (d2) are the calculated results of energy density distributions for different ellipticity, respectively. According to the calculation results detailed in Supplement 1, it can be seen that the light intensity and OAM are uniformly distributed on the circumference when the focus is a POV. When a > b, the long axis of the ellipse is in the x-axis direction. The highest energy density of EPOV is found at φ = 0, π. As a < b, the long axis of the ellipse falls in the y-axis direction, EPOV has the highest energy density while φ = π/2, 3π/2. The larger the ellipticity, the higher the energy density at the peak. It is also proved that the OAM density is maximum on the smaller side of the EPOV and minimum on its larger side.
Fig. 5. The diagram of the vector EPOV energy density distribution for different ellipticity. (a1), (b1), (c1), (d1) are the energy distributions for (a = 1, b = 0.6), (a = 1, b = 0.8), (a = 0.6, b = 1), (a = 0.8, b = 1), respectively. (a2), (b2), (c2), and (d2) are the energy density distribution curves with different ellipticity, respectively.
Arbitrary polarization mode vector beams can be generated by mode extraction, and this method is efficient and flexible. To verify its feasibility, we generate vector EPOVs of different orders as shown in Fig. 6. Figures 6(a1), (b1), (c1) and (d1) are phase diagrams. Figures 6(a2), (b2), (c2) and (d2) are the vector EPOV without analyzer, ne indicates the polarization order of the generated vector EPOV. Columns 3 to 5 represent the experimental results when the analyzer is 0°, 45° and 90°, respectively, and the direction of the arrow in the circle indicates the direction of the analyzer. Figure 6 gives the 1st to 4th order vector EPOV generated using mode extraction when a = 1, b = 0.7, ne is the polarization order of the generated vector EPOV. The vector EPOV will be divided into multiple flaps through the analyzer. The above experiments also verify that it is feasible to generate arbitrary orders of polarization vector EPOV using mode extraction. Because of the flexibility of this approach, it will be more beneficial to generate vector EPOV arrays with mixed modes.
Fig. 6. Experimental diagram of the vector EPOV with different polarization modes. (a1), (b1), (c1), (d1) are phase diagrams. (a2), (b2), (c2), (d2) are the vector EPOV without analyzer, ne represents the polarization order of the generated vector EPOV. Columns (a3) - (a5) represent the experimental plots at 0°, 45° and 90°, respectively. The light intensities are normalized to a unit value, the phase scale is 0 - 2π, and the direction of the arrow in the circle indicates the direction of the analyzer.
In some complex applications, such as multichannel optical communication and multidimensional particle manipulation, single-mode EPOV can't reach the expected results. In this case, researchers prefer optical field arrays with controllable modes. Next, we combine the optical pen technique to modulate vector EPOV arrays with controlled mixing modes. First of all, we have to determine the number and position of the vector EPOVs. From Eq. (10), N represents the number of EPOVs, and xj, yj and zj represent the positions in the focal region. Then, we superimpose the phase functions of EPOV and mode extraction at each optical pen position to modulate the vector EPOV array in space.
Finally, we construct vector EPOV arrays with mixed modes coexisting experimentally. We can adjust one parameter of the vector EPOV array and also adjust multiple parameters of the array independently. The modulation of independent parameters in the array is given in Supplement 1. Figure 7 illustrates the vector EPOV array with controllable mixed modes. Figures 7(a1), (b1) are phase diagrams. (a2), (b2) are vector EPOV arrays without analyzer. Columns 3 to 5 shows the experimental results when the polarizer is 0°, 45° and 90°, respectively. As shown in Fig. 7 (a2) and (b2), it is possible to modulate several parameters of EPOV simultaneously, such as ellipticity, ring size, TCs, numbers and polarization order. The generated mixed-mode EPOV array has flexible modulation capability and can be user-defined according to the requirements. The specific parameters of Fig. 7 are given in Supplement 1.
Fig. 7. Vector EPOV array with controlled mixed mode. (a1), (b1) are phase diagrams. (a2), (b2) are vector EPOV arrays without analyzer. (a3), (b3) are the experimental results when the polarizer is 0°. (a4), (b4) are the experimental results when the polarizer is 45°. (a5), (b5) are the experimental results when the polarizer is 90°. The light intensities are normalized to a unit value, the phase scale is 0-2π, and the direction of the arrow in the circle indicates the direction of the analyzer.
The mode extraction can extract vector beams of arbitrary order less than the VP order, while the optical pen provides an efficient method for multi-degree-of-freedom optical field modulation. The combination of the two methods provides extremely high flexibility and controllability. Controlled mixed mode EPOV arrays generated in this way offer great advantages as it does not require expensive optical processing devices and complex optical systems. The vector EPOV array provides a new approach to some complex application scenarios, it has important guiding significance in optical field modulation technology and optical applications.
4. Conclusion
In summary, we propose a simple method to construct vector EPOV arrays with mixed modes in free space. Specifically, arbitrary modes of vector EPOVs are obtained by mode extraction. Then, combined with an optical pen, we can independently and precisely modulate multiple degrees of freedom of the vector EPOV arrays in free space. This work will provide a novel optical tool for several fields such as multichannel optical communication, multidimensional particle manipulation, and optical tweezers. Due to the flexible controllability and versatility of this approach, it will also offer potential applications in quantum optics, singularity optics, and other fields.
Funding
National Key Research and Development Program of China (2018YFC1313803).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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