1. Introduction

In recent years, vector optical fields with inhomogeneous polarization in the transverse plane have gained increasing interest due to their unique polarization state distribution and a wide range of applications [1], such as sharper focus beyond diffraction limit [2–4], light needle [5,6], quantum optics and information [7–9], optical trapping and manipulation of particles [10–12], and laser processing [13,14]. As an important kind of novel vector opical fields, the full Poincaré sphere vector optical field (PS-VOF) has been proposed and constantly attracted interest recently [15]. The polarization state of the full PS-VOF maps onto the entire surface of the Poincaré sphere. In other word, any polarization states on the Poincaré sphere can always be found on the wave front of the full PS-VOF. It has been constructed firstly from a coaxial superposition of a fundamental Gaussian mode and a spiral-phase Laguerre-Gauss with orthogonal polarizations [15]. Meanwhile, several methods were proposed to generate the full PS-VOF [16–18]. Such optical field can be used for flattop beam shaping [19], optical trapping [20–22], and analyzing the geometric phase intrinsic to their phase and polarization structure [23].

In this paper, we introduce a new method by flattening the Poincaré sphere surface to generate PS-VOF, and the polarization state of the PS-VOF can cover flexibly controlled region of the Poincaré sphere. We design and generate full and half PS-VOFs, and explore the focusing behavior of the PS-VOF. The conversion and annihilation of spin angular momentum (SAM) in the focusing process are also discussed. Moreover, we propose a method to study the polarization coverage of the Poincaré sphere in the focal plane, and full PS-VOF with high quality in the focal plane is presented by adjusting the proportion of the right-handed polarizations in the input plane.

2. Theoretical design of the PS-VOF

The Poincaré sphere provides an important tool of representing polarizations, as shown in Fig. 1(a). Based on the Poincaré sphere, we propose a new kind of PS-VOF by flattening the Poincaré sphere surface, and the schematic of the flattening process is depicted in Fig. 1(b). The inspiration of designing this new PS-VOF is originated from peeling an orange. The surface of the Poincaré sphere is divided into eight sectors on the northern hemisphere and eight sectors on the southern hemisphere along the equator and longitudes, which are marked as $A_{N}$–$H_{N}$ ($A_{S}$–$H_{S}$) on the northern (southern) hemisphere of the Poincaré sphere. Then these sectors from the sphere surface are flattened to the two-dimensional (2D) plane, and gaps appear between two adjacent sectors, as shown in Figs. 1(c) and 1(d). The appearance of the gaps can be understood, as this flattening process is similar to peeling an orange. In order to achieve uniform intensity distribution, we rescale the size of sectors or combine the sectors on the northern and southern hemispheres of the Poincaré sphere, as shown in Figs. 1(e) and 1(f). This rescaling process would be discussed in detail latter when deducing the expression of the PS-VOF. This new PS-VOF can cover part or entire surface of the Poincaré sphere, and it can be referred to as half or full PS-VOF. Figure 1(e) shows the half PS-VOF designed with flattened northern hemisphere of the Poincaré sphere after rescaling. We can also rescale and combine the flattened northern and southern hemispheres of the Poincaré sphere to achieve full PS-VOF, as shown in Fig. 1(f). It should be pointed out that this is only a simplified schematic to directly show the design method of the half and full PS-VOF, and the number of divided sectors and specific arrangement on the wave front can be more flexible. If there is no other illustration, the Poincaré sphere refers to the surface of the Poincaré sphere in the following.

 

Fig. 1. The schematic of designing PS-VOF. (a) The Poincaré sphere. (b) Dividing the Poincaré sphere along the equator and longitudes. (c) Flattened northern hemisphere of the Poincaé sphere. (d) Flattened southern hemisphere of the Poincaré sphere. (e) The schematic of half PS-VOF designed with flattened northern hemisphere after rescaling. (f) The schematic of full PS-VOF designed with the combination of flattened northern and southern hemispheres after rescaling. $A_{N}$–$H_{N}$ ($A_{S}$–$H_{S}$) represent the sectors on northern (southern) hemisphere of the Poincaré sphere.

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We now theoretically explore the mathematical description of the PS-VOF. For simplicity without loss of generality, we take the half and full PS-VOF in Figs. 1(e) and 1(f) as two cases. The algebraic representation of the polarization states on the surface of Poincaré sphere can be written as [24]

We now deduce the relationship of the coordinates (2$\alpha$, 2$\phi$, $R$) on the Poincaré sphere and the coordinates ($r$, $\varphi$) on the sectors flattened from the Poincaré sphere surface to the 2D plane, as shown in Fig. 2(a). Here we assume the lengths of the three arcs of the sector keep unchanged in the flattening process. That is, the lengths of the arcs $\mathrm {T}_{1}$, $\mathrm {T}_{2}$ and $\mathrm {T}_{3}$ are equal to $\mathrm {L}_{1}$, $\mathrm {L}_{2}$ and $\mathrm {L}_{3}$. For this purpose, we first define the point P(2$\alpha$, 2$\phi$, $R$) as an arbitrary point on the Poincaré sphere surface, and the point Q($r$, $\varphi$) is the corresponding point on the wave front of the PS-VOF after flattening the sphere surface. Obviously, if we set $\mathrm {L}_{4}$ = $\mathrm {T}_{4}$ and $r$ = $\mathrm {T}_{5}$, then $\mathrm {L}_{1}$, $\mathrm {L}_{2}$ and $\mathrm {L}_{3}$ would be equal to $\mathrm {T}_{1}$, $\mathrm {T}_{2}$ and $\mathrm {T}_{3}$. This is because that $\mathrm {L}_{1}$ and $\mathrm {T}_{1}$ are the maximum values of $\mathrm {L}_{4}$ and $\mathrm {T}_{4}$. Meanwhile, $\mathrm {L}_{2}$ = $\mathrm {L}_{3}$ and $\mathrm {T}_{2}$ = $\mathrm {T}_{3}$ are the maximum values of $r$ and $\mathrm {T}_{5}$, respectively. The lengths of the arcs on the Poincaré sphere are

 

Fig. 2. The polarizations and relative parameters on (a) The Poincaré sphere surface. (b) Flattened northern hemisphere of the Poincaré sphere. (c) The wave front of the PS-VOF represented by Eq. (5), and the polarizations on the northern hemisphere correspond to the region when $\varphi$ changes from 0 to 4. (d) The wave front of the PS-VOF represented by Eq. (6) when $m_{N} = 2/\pi$. P(2$\alpha$, 2$\phi$, $R$) is an arbitrary point on the Poincaré sphere surface, Q($r$, $\varphi$) is the corresponding point on the wave front of PS-VOF after flattening the sphere surface.

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With the relationship of $\mathrm {L}_{2}$ = $\mathrm {T}_{2}$, $\mathrm {L}_{4}$ = $\mathrm {T}_{4}$, $r$ = $\mathrm {T}_{5}$, we can get

As a result, the relationship of the coordinates (2$\alpha$, 2$\phi$, $R$) on the northern hemisphere of the Poincaré sphere and the coordinates ($r$, $\varphi$) on the wave front of the PS-VOF is

Based on Eqs. (1) and (4), the algebraic representation of the polarization states on the wave front of the PS-VOF can be expressed as

Due to the relationship between the angles $\varphi$ and 2$\phi$, the polarization states from the flattened Poincaré sphere cannot entirely fill the wave front of the PS-VOF. For the northern hemisphere of the Poincaré sphere, range of 2$\phi$ is [0, 2$\pi$) and the corresponding coordinate $\varphi$ changes from 0 to 4. Although we divide the Poincaré sphere into sectors to present the theory of designing the PS-VOF, the derivation of Eq. (5) is based on the transformation of coordinates, which is independent of the number of the sectors. The actual polarization distribution of the PS-VOF in Eq. (5) is presented in Fig. 2(c), and all the polarizations on the northern hemisphere of the Poincaré sphere are shown in the white region, and the range of $\varphi$ is [0, 4). This means that the range of $\varphi$ is smaller than 2$\pi$, which also indicates that gaps appear on the wave front. To make the gaps disappear and achieve point-to-point correspondence of the polarizations on the Poincaré sphere and the wave front of the PS-VOF, we further introduce $m_{N}$ to modulate the azimuthal variation of the polarization state of the PS-VOF in order to make the range of $\varphi$ to be [0, 2$\pi$) on the wave front. Thus, the PS-VOF flattened from the northern hemisphere of the Poincaré sphere can be represented as

In this way, we can find that for one complete period of the flattened northern hemisphere of the Poincaré sphere, the range of $m_{N} \varphi$ should be [0, 4/$m_{N}$]. In order to make the range of the azimuthal coordinate $\varphi$ to be [0, 2$\pi$) which can cover the whole region of the wave front, we can set $m_{N}$ = 2/$\pi$. As a result, the polarizations from the northern hemisphere can exactly cover the full region of the wave front. In this way, the polarization can achieve point-to-point correspondence on the Poincaré sphere and the wave front of the PS-VOF, which can be seen in Fig. 2(d). Furthermore, $m_{N}$ is also a parameter which can add an additional degree of freedom to modulate the polarization distribution of the PS-VOF. When $m_{N}$ increases, the polarization can change along the azimuthal direction rapidly, which is similar to the case of high-order cylindrical vector optical fields [1,25]. Meanwhile, the assumption that $\mathrm {L}_{1}$, $\mathrm {L}_{2}$ and $\mathrm {L}_{3}$ are equal to $\mathrm {T}_{1}$, $\mathrm {T}_{2}$ and $\mathrm {T}_{3}$ would also change when $m_{N}$ is introduced.

Following the same method, we now briefly discuss the case of the flattened southern hemisphere. The relationship of the coordinates (2$\alpha$, 2$\phi$) on the southern hemisphere of the Poincaré sphere and the coordinates ($r$, $\varphi$) on the wave front of the PS-VOF is $\phi =\pi \varphi /4$ and $\alpha =r/2-\pi /4$. Thus, the PS-VOF flattened from the southern hemisphere of the Poincaré sphere can be expressed as

Two kinds of PS-VOFs discussed above are both half PS-VOF because their polarizations can cover half region of the Poincaré sphere surface. If we set $m_{N}$ = $m_{S}$ = 4/$\pi$, the total azimuthal angles $\varphi$ taken by the flattened northern and southern hemispheres are both $\pi$. After arranging them together on the wave front, we can generate the PS-VOF covering all the polarization states on the Poincaré sphere, which can be called full PS-VOF as shown in Fig. 1(f). When the polarizations from the northern and southern hemispheres are combined together, different ways of dividing sectors on the Poincaré sphere can be chosen. For simplicity, we only consider the case when the flattened northern and southern hemispheres of the Poincaré sphere are divided into equal sectors. In this case, we suppose the numbers of sectors containing flattened northern and southern hemispheres of the Poincaré sphere are both $N_{0}$, and the sectors are distributed uniformly on the wave front for the consideration of symmetric distribution. In this way, the full PS-VOF can be defined as

Next, we provide the discussion for the polarizations on the boundaries of the sectors in Figs. 1 and 2. In Figs. 1(c) and 1(d), one polarization on the Poincaré sphere may appear two times on the boundary of two adjacent sectors. After deducing Eq. 5 and Eq. 6, the polarization become continuous and smooth, which can be seen from Figs. 2(c) and 2(d). This is because there is no parameters of sectors in these euqations, and every point on the wave front (white region in Figs. 2(c) and 2(d)) corresponds to one polarization on the Poincaré sphere. For the full PS-VOF shown in Fig. 1(f), the polarization on the boundaries of two adjacent sectors can be different. As a result, there may occur singular line on the boundary, and the polarizations on the boundary of the sectors on the Poincaré sphere may be destructed on the wave front of the PS-VOF. More discussion of the singular lines on the boundary of the sectors will be given in the experimental part.

3. Experimental generation of the PS-VOF

To generate the PS-VOF, we perform the experiment using a 4f system and spatial light modulator (SLM), which is a universal method for generating vector optical fields [25,26]. We should point out that this experimental method is a common method to generate vector optical fields with arbitrary polarization distribution, and the innovation of this paper is that we achieve the above designed PS-VOF in the experiment instead of the experimental method itself. Figure 3 shows the generated half PS-VOFs with $m_{N}$ = 4/$\pi$ and 8/$\pi$ in Eq. (6), which are designed by flattening northern hemisphere of the Poincaré sphere. The first and third rows show the simulated polarization states and normalized Stokes parameters of the half PS-VOFs, and the second and fourth rows present the corresponding experimental results. For the half PS-VOF with $m_{N}$ = 4/$\pi$ in the first two rows of Fig. 3, the polarization state on the wave front of the PS-VOF corresponds to two periods of the polarizations from the northern hemisphere of the Poincaré sphere. That is to say, the polarizations on the northern hemisphere of the Poincaré sphere always appear twice on the wave front of the PS-VOF. In this case, the orientation of the polarization is along radial direction. As a result, $S_{1}$ and $S_{2}$ keep the same along the radial direction and change along the azimuthal direction, due to the fact that the Stokes parameters $S_{1}$ and $S_{2}$ describes the orientation of polarization. Meanwhile, $S_{3}$ stands for the ellipticity and handness of polarization: for $\left |S_{3}\right |$ = 0 indicates the linear polarization, $0<\left |S_{3}\right |<1$ indicates the elliptical polarization and $\left |S_{3}\right |$ = 1 indicates the circular polarization; while the positive and negative $S_{3}$ correspond to the right-handed and left-handed polarizations, respectively. It can be seen from the $S_{3}$ of the PS-VOF that the ellipticity of polarization changes along radial direction and only right-handed polarizations appear on the wave front. For the third and fourth rows, the half PS-VOF with $m_{N}$ = 8/$\pi$ is similar with the above one, and the only difference is that the parameter $m_{N}$ is two times larger than the former case, which means every polarization on the northern hemisphere of the Poincaré sphere appear four times on the wave front of the PS-VOF. We can also find that there is no singular points or singular lines on the wave front of the PS-VOF, which means the polarization distribution is smooth for this case.

 

Fig. 3. Simulated and measured half PS-VOF designed by flattening northern hemisphere of the Poincaré sphere based on Eq. (6). The first and last two rows show the half PS-VOF with $m_{N}$ = 4/$\pi$ and 8/$\pi$, respectively. The first and third rows show the simulated results of the PS-VOFs, and the second and fourth rows show the corresponding experimental results. The first column shows the polarization states and the experimental intensity patterns, the black and red polarizations represent linear and right-handed polarizations, respectively. Stokes parameters $S_{1}$, $S_{2}$ and $S_{3}$ are given in the second to fourth columns.

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Figure 4 shows the generated full PS-VOF with $m_{N}$ = $m_{S}$ = 4/$\pi$, $\Delta \varphi _{N}$ = $\Delta \varphi _{S}$ = $\pi$ /$N_{0}$, $\Delta \varphi _{0}^{N}$ = $\Delta \varphi _{0}^{S}$ = 0 based on Eqs. (8) and (9). The first and third rows show the simulated polarization states and Stokes parameters of the PS-VOF, and the second and fourth rows give the corresponding experimental results. When $m_{N}$ = $m_{S}$ = 4/$\pi$, the polarization states of the full PS-VOF on the wave front contain one period of polarizations from the flattened northern hemisphere and one period of polarizations from the southern hemisphere of the Poincaré sphere, respectively. The first and last two rows show the full PS-VOF with $N_{0}$ = 3 and 4, respectively. It means that the polarizations from the northern and southern hemispheres are divided into three and four regions, respectively. The Stokes parameters $S_{1}$ and $S_{2}$ keep the same along the radial direction and change along the azimuthal direction in Fig. 4. Thus, the orientation of the polarization changes along the azimuthal direction, which is similar to the case in Fig. 3. In contrast to Fig. 3, $S_{3}$ is not only positive in Fig. 4, which means the corresponding polarization states on the wave front of the full PS-VOF contains the right-handed and left-handed polarizations. It can be clearly seen that there are singular lines on the boundary of the sectors on the wave front of the full PS-VOF, which originates from the destructive interference.

 

Fig. 4. Simulated and measured full PS-VOF with $m_{N}$ = 4/$\pi$, $\Delta \varphi _{N}$ = $\Delta \varphi _{S}$ = $\pi$ /$N_{0}$, $\Delta \varphi _{0}^{N}$ = $\Delta \varphi _{0}^{S}$ = 0. The first and last two rows show the full PS-VOF with $N_{0}$ = 3 and 4, respectively. The first and third rows show the simulated results of the PS-VOFs, and the second and fourth rows give the corresponding experimental results. The first column shows the polarization states and the experimental intensity patterns, the black, green and red polarizations represent linear, left-handed, and right-handed polarizations, respectively. Stokes parameters $S_{1}$, $S_{2}$ and $S_{3}$ are given in the second to fourth columns.

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4. Modulation of the PS-VOF in the focal plane

The above discussion shows that the polarization states of the PS-VOF can cover arbitrary surface of the Poincaré sphere. Meanwhile, as one of the most important application of the vector optical field is the focal engineering, it is also important to construct the full PS-VOF in the focal plane. In this section, we will explore the polarization coverage of the Poincaré sphere of the focused PS-VOFs. The focused field can be calculated by the Richards-Wolf vectorial diffraction theory [27,28], which can be written as

Figure 5 describes the connection between polarization states of the focal fields and the Poincaré sphere. Figure 5(a) shows the polarization state of the PS-VOF in the incident plane, and the focal intensity and corresponding polarization state are shown in Fig. 5(b). By calculating, we obtain the 3-dimensinal (3D) distribution of the polarization coverage of the Poincaré sphere in the focal plane, which is shown as the blue area in Fig. 5(c). This calculation process is introduced below. Based on the Richards-Wolf vectorial diffraction theory with Eq. (10), we can get a matrix of the focal field by simulation with coordinates of F$(x, y)$. To simulate the polarization coverage on the Poincaré sphere, we also construct a matrix of the Poincaré sphere surface with coordinates of P$(2\alpha , 2\varphi )$. Then we calculate the orientation, ellipticity and handness of the polarization of the focal field, and based on these information, we find the corresponding points on the Poincaré sphere. At last, the polarizations of the focal field can be marked on the Poincaré sphere. In this way, the polarization distribution of the focal field is presented on the Poincaré sphere. If there are many points with the same polarization in the focal plane, we would add their energy up and mark them as one point on the Poincaré sphere. By this method of point correspondence, we can present the energy distribution of the focal field on the Poincaré sphere. It is noted that the pixel resolutions of the focal field and the Poincaré sphere should be carefully chosen in order to achieve more accurate result. Generally, the larger pixel resolution of matrix can bring high accuracy in simulation, but the case here is a little different. Firstly, the pixel resolution of the focal field should be large. For the pixel resolution of the Poincaré sphere, it can neither be too large nor too small. If the pixel resolution is too small, there will not be enough kinds of polarizations on the Poincaré sphere, and the result would be rough. If the pixel resolution is too large, we cannot guarantee that the number of the polarizations in the focal field is large enough to cover the Poincaré sphere accurately. In this case, we have summarized a method to choose pixel resolutions in simulation. Large pixel resolution of the focal field and low pixel resolution of the Poincaré sphere are chosen at first. Then with gradually increasing pixel resolution of the Poincaré sphere, the coverage pattern will experience the process of “rough with low pixel resolution, almost invariant, and decrease with incontinuity”. The pixel resolution of the Poincaré sphere would be appropriate when we choose the “almost invariant” part of the coverage pattern. After calculating repeatedly, we choose the pixel resolutions of the focal field and the Poincaré sphere to be $4000 \times 4000$ and $400 \times 400$ in the following calculation to get accurate coverage patterns.

 

Fig. 5. The schematic of studying the polarization coverage of the Poincaré sphere in the focal plane. (a) The polarization state of the incident PS-VOF with $m_{N}$ = 4/$\pi$ in Eq. (6). (b) The polarization state and intensity distribution of the focused PS-VOF with NA = 0.01. The red polarizations represent the right-handed polarizations and the size of the picture is $130 \lambda \times 130 \lambda$. (c) The 3D polarization coverage of the Poincaré sphere. (d) The 2D expanded polarization coverage area of the Poincaré sphere based on the Mollweide projection. The cut-off intensity ratio is $\beta _{c}=0.1$.

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In order to show the polarization coverage more clearly, we use the Mollweide projection to achieve 3D to 2D conversion as shown in Fig. 5(d). Among various projections to project sphere surface, we choose the Mollweide projection because it is a kind of equal-area projection [29], which means that a region on the sphere surface retains its area after Mollweide projection. Meanwhile, Mollweide projection also shows relatively low shape distortion [30] and is quite acceptable for readers [31]. We can calculate the projection with the following equations [32]:

We should now summarize the theoretical framework in Fig. 5. First of all, the focal field can be calculated by Richards-Wolf vectorial diffraction theory based on Eq. (10), and we get a matrix to present the focal field in simulation, as shown in Fig. 5(b). The points of the focal field are marked on the Poincaré sphere, and we can get the polarization distribution of the focal field on the Poincaré sphere, as shown in Fig. 5(c). In order to further study the focal properties of the PS-VOF, we can define a parameter of coverage ratio of the polarization on the Poincaré sphere. The expression is $\gamma =S_{f}/ S_{t}$, where $S_{t}$ is the area of the entire surface of the Poincaré sphere and $S_{f}$ represents the area of the polarization coverage of the Poincaré sphere in focal plane. At last, we present the Poincaré sphere surface on the 2D plane with Mollweide projection, which makes the presentation of polarization more clearly. This is the theoretical framework used in Figs. 6–8.

 

Fig. 6. The focal fields of half and full PS-VOFs. The first row shows the half PS-VOF with $m_{N}$ = 2/$\pi$ in Eq. (6), and the second row shows the full PS-VOF with $\Delta \varphi _{N}$ = $\Delta \varphi _{S}$ = $\pi$ /$N_{0}$, $N_{0}$ = 1 $, \Delta \varphi _{0}^{N}$ = $\Delta \varphi _{0}^{S}$ = 0 in Eqs. (8) and (9). The first column shows the intensity and polarization state of the input PS-VOF, and the second column shows the intensity and polarization state of the focal field with the size of $130 \lambda \times 130 \lambda$. The third column shows the energy coverage of the polarizations on the Poincaré sphere in the focal plane. The fourth column shows the polarization coverage area of the Poincaré sphere in the focal plane, and the colors of white and black represent whether the polarization appears in the focal plane or not. The cut-off intensity ratio of the focal field is $\beta _{c}=0.1$. The red ellipse (circle): right-handed elliptic (circular) polarization; yellow ellipse (circle): left-handed elliptic (circular) polarization; black line: linear polarization.

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In simulated calculation, for the sake of reducing the influence of the polarization with very weak intensity, we set the cut-off intensity of the focal field as $I_{c}=\beta _{c} I_{\max }$, where $\beta _{c}$ is the cut-off intensity ratio and $I_{\max }$ is the maximum intensity of the focal field. When we calculate the polarization coverage of the Poincaré sphere, we only consider the region where intensity of the focal field satisfies $I_{f}>I_{c}$. In addition, the value $\beta _{c}$ is also an important evaluation index of the full PS-VOF in the focal plane, as the higher value of $\beta _{c}$ means the better quality of the full PS-VOF. The cut-off intensity ratio is set to be $\beta _{c}=0.1$ in Fig. 5.

Next, we use the method shown in Fig. 5 to further investigate the law of controlling polarization coverage of the PS-VOF in the focal plane, and try to construct the full PS-VOF in the focal plane. Figure 6 shows the polarization coverage of the Poincaré sphere in the focal plane with two kinds of incident PS-VOFs. The first row shows the half PS-VOF presented by Eq. (6), and the second row shows the full PS-VOF presented by Eq. (8). In these two cases, the cut-off intensity ratios $\beta _{c}$ are both 0.1. The polarization states of these two PS-VOFs in the input plane are shown in the first column, and the focal intensity patterns and the corresponding SoPs are shown in the second column. For each PS-VOF, we choose two kinds of 2D patterns to show the polarization coverage of the Poincaré sphere in third and fourth columns. The third column of Fig. 6 shows the energy distribution of polarization states in the focal plane, and it depicts the energy proportion of each polarization state in the focal plane. In the fourth column of Fig. 6, the two colors represent whether the polarization exsits in the focal plane or not. For the half PS-VOF shown in the first row of Fig. 6, the focal field is obviously with all kinds of polarizations, which means we can achieve full PS-VOF in the focal plane. One interesting phenomenon is that the pure right-handed polarizations in the incident plane can turn into polarizations with different chiralities in the focal plane. This means that the conversion of SAM happens in this focusing process. The SAM is related to the circular polarizations with two quantized values of $\pm \hbar$, which can make the particle spin around its own axis [33,34]. Although the SAM conversion happens in the focusing process, the right-handed polarizations still over the left-handed polarizations in the focal plane, which can be seen from the third column of Fig. 6. For the full PS-VOF shown in the second row of Fig. 6, the focal field contains pure linear polarizations, which means the SAM annihilates in the focusing process. This is easy to understand as the right- and left-handed polarizations counteract in the focusing process, leading the SAM to be zero at any locations and the linear polarizations only cover the equator of the Poincaré sphere. This also means that when we design the full PS-VOF with certain proportion and symmetry in the incident plane, SAM annihilation can occur in the focal plane.

We will focus on how the proportion of the polarizations of the input PS-VOF from the two hemispheres affect the focal properties. Figure 7 shows the dependence of the coverage ratio $\gamma$ on the area proportion $d$, which represents the proportion of northern hemisphere of the Poincaré sphere on the wave front. Figure 7(a) shows the case of the incident PS-VOF with $m_{N}=2 / \pi d$, $m_{S}=2 / \pi (1-d)$ , $\Delta \varphi _{N}=2 \pi d$, $\Delta \varphi _{S}=2 \pi (1-d)$, $N_{0}=1$, $\Delta \varphi _{0}^{N}=0$, $\Delta \varphi _{0}^{S}=2 \pi d$ and $\beta _{c}=0.1$ in Eqs. (8) and (9). When $d$ is not 0 or 1, polarizations on the wave front are consisted of one period of polarizations from the northern hemisphere and one period of polarizations from the southern hemisphere, respectively. When the proportion $d$ is zero in Fig. 7(a1), the polarizations are all from the southern hemisphere of the Poincaré sphere. When $d$ = 1 in Fig. 7(a3), the polarizations are all from the northern hemisphere of the Poincaré sphere, which is also the case in the first row of Fig. 6. When $d$ = 0 or 1, the coverage ratios $\gamma$ are both $100 \%$, which means the focal fields are full PS-VOFs. The second row of Fig. 6 corresponds to the case in Fig. 7(a2), and the annihilation of the SAM leads the coverage ratio to zero. We can find that the coverage ratio is lower when the areas of the right- and left-hand polarizations are closer to each other from Fig. 7(a). Figure 7(b) shows the case of the incident PS-VOF with $m_{N}=2 / \pi d$, $m_{S}=4 / \pi (1-d)$ , $\Delta \varphi _{N}=2 \pi d$, $\Delta \varphi _{S}=2 \pi (1-d)$, $N_{0}=1$, $\Delta \varphi _{0}^{N}=0$, $\Delta \varphi _{0}^{S}=2 \pi d$ and $\beta _{c}=0.1$ in Eqs. (8) and (9). In this case, when $d$ is not 0 or 1, there are always two periods of polarizations from the northern hemisphere and one period of polarizations from the southern hemisphere, respectively. It is very interesting that when the proportion $d$ is approximately the Golden ratio (0.618), and the coverage ratio reaches to $100 \%$, which means we can achieve full PS-VOF in the focal plane.

 

Fig. 7. The dependence of the coverage ratio $\gamma$ of the polarizations on the proportion $d$ of the polarizations coming from northern hemisphere of the Poincaré sphere. (a) The incident PS-VOF is with $m_{N}=2 / \pi d$, $m_{S}=2 / \pi (1-d)$ , $\Delta \varphi _{N}=2 \pi d$, $\Delta \varphi _{S}=2 \pi (1-d)$, $N_{0}=1$, $\Delta \varphi _{0}^{N}=0$, $\Delta \varphi _{0}^{S}=2 \pi d$. (b) The incident PS-VOF is with $m_{N}=2 / \pi d$, $m_{S}=4 / \pi (1-d)$ , $\Delta \varphi _{N}=2 \pi d$, $\Delta \varphi _{S}=2 \pi (1-d)$, $N_{0}=1$, $\Delta \varphi _{0}^{N}=0$, $\Delta \varphi _{0}^{S}=2 \pi d$. The cut-off intensity ratio of the focal field is $\beta _{c}=0.1$, and the numerical aperture is NA = 0.01. The red ellipse (circle): right-handed elliptic (circular) polarization; yellow ellipse (circle): left-handed elliptic (circular) polarization; black line: linear polarization.

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Figure 8(a) shows the histogram of the coverage ratio of the polarizations on the Poincaré sphere with different cut-off intensity ratio $\beta _{c}$ for two cases in Figs. 7(a3) and 7(b2), and Figs. 8(b) and 8(c) depict the intensity and polarization state of the focused PS-VOF of these two cases, respectively. The green bars correspond to the coverage ratio of half PS-VOF in Fig. 7(a3), and the case in Fig. 7(a1) is the same. The orange bars correspond to the coverage ratio of full PS-VOF in Fig. 7(b2). The coverage ratios of the two cases are both $100 \%$ when the cut-off intensity ratio $\beta _{c}=0.05$ and 0.1, and the coverage ratio of the focused full PS-VOF still keeps $100 \%$ when $\beta _{c}$ increases to 0.15 and 0.2. However, the coverage ratio of the focused half PS-VOF is below $100 \%$ when the cut-off intensity ratio $\beta _{c}$ is greater than 0.1. The two coverage ratios both decrease with the increase of the cut-off intensity ratio $\beta _{c}$ from 0.2 to 0.4, and it is worth mentioning that the coverage ratio of the focused full PS-VOF is always larger. Obviously, when the proportion $d$ of the right-handed polarizations of the PS-VOF is the Golden ratio (0.618) as shown in Fig. 7(b2), we can achieve the full PS-VOF with high quality in the focal plane.

 

Fig. 8. (a) The histogram of the coverage ratio of the polarizations on the Poincaré sphere with different cut-off intensity ratio $\beta _{c}$. The orange bars correspond to the half PS-VOF in Fig. 7(a3) with $d$ = 0, and the green bars correspond to the full PS-VOF in Fig. 7(b2) is with $d$ = 0.618. (b) and (c) The intensity and polarization state of the focal field of the incident PS-VOF in Fig. 7(a3) and Fig. 7(b2), and the size of each image is $130 \lambda \times 130 \lambda$.

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Here we should give a brief introduction of the Golden ratio. The Golden ratio, also known as golden section or golden number, is an irrational number attracting attention in various subjects. To define the Golden ratio $\Phi$ [35,36], a straight line AB is divided at a point C, and the line AB is divided to a longer line AC and a shorter line CB. Then we assume that the ratio of the lengths of the two lines (AC/CB) is equal to the ratio of the sum of the two lines (AC+CB) to the longer side (AC), and this ratio can be called Golden ratio $\Phi$ = AC/CB = (AC+CB) /AC $\approx$ 1.618. In order to define the range of the area proportion $d$ to be $[0, 1]$, we choose the ratio of the longer part AC to the whole straight line AB, which is |AC|/|AB| = 1/$\Phi$ $\approx$ 0.618, as the Golden ratio in this paper [36]. Meanwhile, the Golden ratio is very common and appears in various areas, including science, art, as well as in nature [35,36]. It is very interesting that we find when the proportion $d$ is 0.618 in Fig. 7(b), full PS-VOF is achieved in the focal plane, and the property is good as shown in Fig. 8. It is also important to introduce the Golden ratio in modulating polarization proportion, specifically modulating proportion of the right-handed polarizations in this paper.

5. Conclusions

In summary, we design and generate the flexibly modulated Poincaré sphere vector optical field (PS-VOF) based on 2D flattened Poincaré sphere surface. In this way, we can achieve half and full PS-VOFs, which means the polarizations on the wave front of the PS-VOF can cover half or whole surface of the Poincaré sphere. We further study the focal properties of this new kind of PS-VOF, and concentrate on studying the polarization coverage of the Poincaré sphere in the focal plane. In addition, we achieve full PS-VOF in the focal plane when the proportion of the right-handed polarizations on the wave front satisfies Golden ratio, and discuss the phenomenon of the SAM conversion and annihilation in the focusing process. The flexibly modulated PS-VOF in both input and focal planes enriches the family of the vector optical fields, which can be potentially applied in the regions with sensitivity to polarizations. Moreover, we hope the method we propose to study the focal field with Poincaré sphere can be a direct and useful tool, which will bring new insights in presenting and studying polarizations in the focal plane.