We propose a novel scheme for designing and generating kaleidoscope-structured vector optical fields (KS-VOFs) by analogy with the principle of multiple mirror reflection in a kaleidoscope. For KS-VOFs with symmetric polarization states, we show the symmetry properties of the focal fields with various shapes for different applications. The redistributing symmetric local spin angular momentum (SAM) density indicates that the design method of the KS-VOFs plays a role as a catalyst to the redistribution process of polarization states and local SAM conversion in the tight focusing process. Meanwhile, the controllable transverse energy flow in the focal plane can be used to transport multiple absorptive particles and then to be fixed at certain locations. Our results may find applications in optical machining, trapping, and manipulation.

## I. INTRODUCTION

Polarization, as an intrinsic nature of light, plays an important role in engineering the optical field and controlling the interaction of light with matter. There have been a lot of research studies on vector optical fields (VOFs) with space-variant polarization,^{1–3} which have been applied in the areas such as quantum optics and information,^{4–6} nonlinear optics,^{7–9} single molecule imaging,^{10} near-field optics,^{11} spatial filters (SFs),^{12,13} singular optics,^{14,15} and optical trapping and manipulation of particles.^{16–18} Besides these applications, the most attractive property of the VOFs is the focusing property, which leads to various kinds of focal fields such as the far-field focal spot beyond the diffraction limit,^{19,20} the light needle of a longitudinally polarized field,^{21} the optical cage,^{22,23} the optical chain,^{24,25} and tightly focused field of a flattop sharp line.^{26,27} In these research studies, we can always find certain mirror symmetry in the polarization states of the VOFs and the focal intensity patterns.^{19–27} However, to our knowledge, these mirror symmetries in the VOFs and corresponding focal fields cannot be arbitrarily designed, and then, the corresponding spin angular momentum (SAM) and transverse energy flow in the focal plane are not flexibly controllable.

Here, we propose a new kind of VOFs with arbitrary mirror-symmetric polarization states, which are called the kaleidoscope-structured VOFs (KS-VOFs). The geometric configurations of symmetric polarization states of the KS-VOF provide an additional degree of freedom, assisting in studying the focusing property. We show various weakly and tightly focused fields and deeply explore the redistributing symmetric SAM density and transverse energy flow of the tightly focused KS-VOFs, which are useful in realms such as optical machining, trapping, and manipulation.

## II. DESIGN AND GENERATION OF KS-VOFS

The inspiration of designing KS-VOFs is originated from the kaleidoscope by analogy with the principle of multiple reflection. As is well known, when light is reflected by a mirror, the image is reversed in the mirror, that is to say, the image in the mirror and the original object are mirror-symmetric about the mirror. If another mirror is introduced in the system, there occur more than two images. The smaller the angle between the two mirrors is, the more images there occur. The number of images should be *N* = 2*π*/*θ* − 1, where *θ* is the angle between the mirrors. When *N* is an even number, one important property of the images is that adjacent two of them are mirror-symmetric. A kaleidoscope is usually composed of two mirrors set at *θ* = *π*/3 to each other. Thus, the number of images is *N* = 5, and people can observe six patterns including the original object and five images symmetric about each other. As a result, the mirror-symmetric colorful images can be seen in the kaleidoscope.

Here, we design the polarization states of the KS-VOF by analogy with the principle of multiple reflection in the kaleidoscope. Figure 1 shows the schematic of polarization states of the KS-VOF, in which the optical field is divided into even number of sectors, and the polarization state in the red circle is regarded as the original polarization state with arbitrary orientation, while other sectors are filled with polarization states symmetric to the original polarization state. No matter how complex the polarization state in the original sector is, the polarization states of the KS-VOF always exhibit mirror-symmetry. Figure 1 shows the schematic of designing the KS-VOFs with onefold, twofold, threefold, fourfold, fivefold, and sixfold symmetric states of local linear polarization. Correspondingly, there are one, two, three, four, five, and six mirror-symmetric axes for the six cases. We define a parameter *s* to describe the amount of the mirror-symmetric axes of the KS-VOF. In particular, *s* = 0 indicates the original VOF, which is the special case of the KS-VOF. We should state that the design of mirror-symmetric polarization states is realized by the experimental method introduced below, and the analogy of multiple reflection does not mean that the polarization can really be reflected.

Figure 2 shows the experimental setup for generating the KS-VOFs. This scheme is a common path interferometric configuration with the aid of a 4f system composed of a pair of identical lenses (L1 and L2), based on the wavefront reconstruction and the Poincaré sphere.^{3,16} The transmission function of the spatial light modulator (SLM) is $t(x,y)=0.5+0.5\u2061cos2\pi f0x+\delta $, where *δ* is the space-variant phase called additional phase here. An Ronchi phase grating with transmission function of $t(x,y)=2\pi [0.5+0.5\u2061cos2\pi f0x]$ is placed in the output plane of the 4f system to superpose the two orders diffracted from the grating on SLM. With this scheme, we can generate local linearly polarized VOFs with arbitrary spatial distribution. Actually, we can also generate other kinds of VOFs with similar experimental setup.^{28–31} Here, we only consider the local linearly polarized KS-VOFs, which is enough to illustrate their novel properties. The local linearly polarized KS-VOFs can be expressed as $Acos(\delta )e^x+sin\delta e^y$, which means that there is no phase difference between the *x*- and *y*-components of the KS-VOF.

The polarization state of the KS-VOF depends on both the value of *s* and the polarization state in the original sector (as shown in the red circle in Fig. 1), and the polarization state in the original sector can be designed arbitrarily. As a result, numerous KS-VOFs can be flexibly controlled, providing much potentials to achieve useful applications needing certain mirror-symmetry. As the most representative kind of VOFs is cylindrical VOFs,^{3,16} we use them in the original sector of the KS-VOF for further study. For local linearly polarized cylindrical VOFs, we choose the additional phase *δ*_{1} = *mφ* + *nρ*/*ρ*_{0} + *δ*_{0}, where *m* and *n* are the azimuthal index (i.e., topological charge) and the radial index of the cylindrical VOF in the original sector, *ρ*_{0} is the maximum radius of the original optical field, and *δ*_{0} is a constant which can change the polarization at all positions and we can set *δ*_{0} = 0 without losing generality. With the additional phase *δ*_{1} = *mφ* + *nρ*/*ρ*_{0} + *δ*_{0} in the original sector (as the first sector), we can derive the additional phase *δ* of the *p*th sector based on the schematic in Fig. 1 as *δ*_{p+1} = (2*π*/*s* − *δ*_{p}), where *p* = 1, 2, 3, …, *s* − 1. Thus, the KS-VOFs can be generated by loading the additional phase on the SLM. When the original sector is filled with cylindrical VOF with *m* = *n* = 6, six columns in Fig. 3 show the intensity patterns of the KS-VOFs with *s* = 1, 2, 3, 4, 5, and 6, respectively. In the first column (*s* = 1), there are four short singular lines and one singular spot arranged in the horizontal line. For other cases, there is always the same number of short singular lines in the symmetric axes, and one singular spot can always be found in the center of the field. This phenomenon can be understood by the destructive interference on two sides of the symmetric axis. As for the *x*-component intensity pattern of the KS-VOF, the number of symmetric axes is not always the same as the total intensity. This can be speculated by the original polarization states of the KS-VOF. For instance, for the case of *s* = 4 in the fourth column, the *x* and *y* axes are the symmetric axes of the intensity pattern of the *x*-component. However, the *x*-component intensity patterns are complementary instead of symmetric about the ±45° symmetric axes. The reason is based on the fact that the *x*- and *y*-polarizations are symmetric about the ±45° symmetric axes, leading to the complementary *x*-component intensity patterns. We can see that the experimental results are in good agreement with the simulated ones, as shown in Fig. 3.

In the above, we have explored the KS-VOFs with different amount of symmetric axes; now we will investigate the cases of different polarization distributions in the original sector. Figure 4 shows the KS-VOFs with *s* = 4 and *n* = 10 but for different values of *m* = 0, 2, 4, 6, 8, and 10. For the total intensity patterns, the singular spots at the center become bigger as *m* increases due to the limited distinguishability of the SLM. The total intensity patterns have short singular lines in the four symmetric axes including *x*, *y*, and ±45° axes. When *m* = 0, the polarization states change along the radial direction, and the *x*-component intensity pattern exhibits two groups of symmetric circular arcs. As *m* increases, the amount of the bright and dark stripes also increases. All these *x*-component intensity patterns are symmetric about *x* and *y* axes and are complementary about the ±45° axes. As a result, the *x*-component intensity patterns show a three-dimensional sense as if the patterns in the right-and-left sectors are not in the same plane of up-and-down sectors. This is quite interesting because these three-dimensional-sense intensity patterns can bring us strong visual impact. When *m* increases from 0 to 10, the arc shapes in the *x*-component intensity patterns gradually become candle-flame-like (leaflike) shape. The experimental results are in good agreement with the simulated ones.

## III. FOCAL INTENSITY PATTERNS OF KS-VOFS

After investigating the design and generation of the KS-VOFs, we now explore their focusing properties, due to the great interest. For the focusing process, there are two basic theories: scalar diffraction theory is used to calculate weakly focused fields, while vector diffraction theory is used to calculate tightly focused fields. We choose different theories to simulate the focal fields based on the numerical aperture (NA) of the focusing lens. To calculate the weakly focused fields, the incident VOF is written as

where *A* is the amplitude and *δ* = *mφ* + *nρ*/*ρ*_{0} + *δ*_{0} is the additional phase. When considering the VOF with uniform amplitude, the weakly focused field can be given by the Fresnel diffraction formula as^{32}

where (*x*, *y*) are the coordinates in the incident plane and (*x*_{f}, *y*_{f}) are the coordinates in the focal plane. circ(·) is the well-known circular function describing the boundary of the circular incident field.

Figure 5 shows the focal patterns of the specially designed KS-VOFs. It can be seen that when *s* = 1, 2, 3, 4, 5, and 6, the focal intensity patterns have two, two, six, four, ten, and six symmetric axes, respectively. For the onefold symmetric pattern of the polarization state in the first column (*s* = 1), when (*m*, *n*) = (0, 2) and (1.5, 1.5), their focal patterns exhibit two spots and an Arabic number “8,” respectively. The focal intensity pattern of Arabic number “8” is the complementary pattern of the two spots, and the two focal fields can be used in optical machining to fabricate two holes and embossment of two spots. For the twofold symmetric pattern of the polarization state in the second column (*s* = 2), there are two focal spots when (*m*, *n*) = (0, 0), while the focal field becomes two strip-shaped spots when (*m*, *n*) = (3, 5). For the threefold symmetric pattern of polarization state in third column (*s* = 3), the focal field exhibits a nested structure of two hexagons when (*m*, *n*) = (0, 6), while the focal field is similar to a gear when (*m*, *n*) = (5, 5). Similarly, for fourfold, fivefold, and sixfold symmetric patterns of the polarization state, the focal fields exhibit certain amount of focal spots or complementary focal holes. Clearly, for the weakly focused fields of the KS-VOFs, the amount of the symmetric axes is equal to 2*s* if *s* is an odd number but *s* if *s* is an even number.

In the above, we have explored the weak focusing properties. However, the tightly focused VOFs have been proved to be more novel and useful.^{19–27} According to the vector diffraction theory, the tightly focused fields can be expressed as^{33,34}

with

where *E*_{ρ} and *E*_{ϕ} are the radial and azimuthal components of the incident VOF and (*ρ*, *ϕ*) are the radial and azimuthal coordinates in the incident plane. *r*, *φ*, and *z* are the radial, azimuthal, and longitudinal coordinates in the focal plane. *k* = 2*π*/*λ* is the wave vector of light with a wavelength of *λ* in free space. *f* is the focal length of the objective. *P*(*θ*) is the pupil plane apodization function, which is chosen to be $P(\theta )=cos\u2061\theta $ with *θ* = sin^{−1}(*ρ*/*f*). *θ*_{m} is the maximum ray angle passing through the objective, defined as sin *θ*_{m} = *NA*. The incident VOF is a round field with a radius of *ρ*_{m} = *f* sin *θ*_{m} = *f* · *NA*, and we choose *NA* = 0.9 in the simulation of tight focusing below. We should point out that the vectorial diffraction theory is applicable for calculating both weakly and tightly focused fields, and we still choose the Fresnel diffraction formula to calculate the weakly focused fields because it is more commonly used and convenient to simulate the weakly focused fields.

Figure 6 shows the tightly focused fields of the KS-VOFs with onefold, twofold, threefold, fourfold, fivefold, and sixfold symmetric KS-VOFs. For *s* = 2 and (*m*, *n*) = (0, 0.5), the focal field is a flattop sharp line with a dimension of ∼0.58*λ* × 1.76*λ* (full width at half-maximum, FWHM), which has its figure of merit (defined as the full width at 95% of the maximum divided by FWHM) in the *x* direction to be 0.71 and is obviously superior to the former research studies.^{26,27} This high-performance subwavelength sharp line can be used in lithography and optical storage.^{26,27} When *s* = 2 and (*m*, *n*) = (3, 0), the square focal field with four strong spots is the same as that of the traditional cylindrical VOF. This is because the polarization states of the KS-VOF in this case are the same as that of the cylindrical VOF when (*m*, *n*) = (3, 0) and have two symmetric axes. When *s* = 2 and (*m*, *n*) = (3, 0.5), the focal field has two strongest spots in the vertical direction and secondary strong spots in the horizontal direction. This means that the focal field can be adjusted and designed by changing the parameters of the KS-VOF. For the fivefold symmetric KS-VOFs in the fifth column, all three focal fields have ten strong spots in a circle. When (*m*, *n*) = (0, 2.5), the focal field is composed of ten elliptical spots located at a circle. When (*m*, *n*) = (1.5, 2), ten strong spots in a circle exhibit a rectanglelike shape. When (*m*, *n*) = (2.5, 2), one strong spot at the center are surrounded by ten trianglelike strong spots. In Fig. 6, tightly focused fields exhibit the shape of cross, gear, hexagon, and so on. Such flexibly controlled tightly focused fields with various shapes can be used in optical trapping and optical machining and may bring us more properties and applications.

Now we will discuss the difference in the tightly focused fields between the KS-VOFs and the original VOFs used to design the KS-VOFs, as shown in Fig. 7. As is known, when *n* ≠ 0, the polarization states of the cylindrical VOF lack the mirror symmetry about any axis. As shown in the first column, the tightly focused fields of the cylindrical VOFs exhibit the rotation-symmetry instead of the mirror-symmetry. However, by analogy with the principle of multiple reflection, the KS-VOFs with the mirror-symmetric polarization states can be designed based on the cylindrical VOFs when *n* ≠ 0. As a result, the tightly focused fields exhibit two, two, six, four, ten, and six symmetric axes when *s* = 1, 2, 3, 4, 5, and 6, as shown in the second to seventh columns. This is similar to the weakly focused fields in Fig. 5, that is to say, for the tightly focused fields of the KS-VOFs, the amount of the symmetric axes is also equal to 2*s* if *s* is an odd number but *s* if *s* is an even number. Meanwhile, the results in Fig. 7 also inspire us that the design method of the KS-VOFs can provide a convenient route to add mirror symmetry to the polarization states of the VOFs without mirror symmetry, and the focal fields with corresponding mirror symmetry can be generated and modulated as a result. Another interesting thing is, after designing the KS-VOFs by analogy with the principle of multiple mirror reflection in a kaleidoscope, the kaleidoscopelike tight focusing patterns with various symmetry properties can be observed in Figs. 5–7. These symmetric patterns also reflect the connection of the two fundamental concepts of VOF and kaleidoscope.

## IV. REDISTRIBUTION OF POLARIZATION STATES AND SAM DENSITY IN THE FOCAL PLANE

As an intrinsic nature of light, SAM is associated with circular polarization and has two possible quantized values of ±*ℏ*. Recently, the SAM density, as a three-dimensional vector to describe the local spatial characteristic of SAM corresponding to spinning energy flux, has attracted attention for its redistributing behavior and relative properties.^{35–40} For monochromatic light, the time-averaged SAM density can be expressed as^{35–40}

where *ε* and *μ* are the vacuum permittivity and permeability and *ω* is the angular frequency of the input light beam. Im[·] represents the imaginary parts, and **E**^{*} and **H**^{*} denote the complex conjugates of the electric and magnetic fields in the focal plane, respectively. The focusing process is in free space with nonmagnetic surrounding medium, so only the electric part of the focused field is generally considered.

According to the above theory, we study the SAM density of the tightly focused KS-VOFs by the lens of *NA* = 0.9. Figure 8 shows the distributions of polarization states, the transverse and longitudinal component intensities, and the SAM density for the tightly focused KS-VOFs (*s* = 3) with *m* = *n* = 2, and the case of local linearly polarized VOF with same topological charges is also shown as a comparison. For the original VOF and KS-VOF, the polarization states and intensity patterns of the incident VOFs and the transverse and longitudinal components of focal fields are presented in Figs. 8(a1)–8(a3) and 8(c1)–8(c3), respectively. The local SAM density profiles (*S*_{x}, *S*_{y}, and *S*_{z}) are displayed in Figs. 8(b1)–8(b3) for the original VOF and Figs. 8(d1)–8(d3) for the KS-VOF. For the input original VOF, the polarization state of the focal field still exhibits linear polarization state despite redistribution, as shown in Fig. 8(a2). However, the focal field of the KS-VOFs has rich polarization states including linear polarization, and right-handed and left-handed elliptical (circular) polarizations, as displayed in Fig. 8(c2). This means that the local SAM of the photons, corresponding to the circular polarizations, has appeared and redistributed compared with the local linearly polarized VOFs. The redistribution of the polarization state also leads to the symmetric profiles of the SAM density for tightly focused KS-VOFs, as demonstrated in Figs. 8(d1)–8(d3). Meanwhile, the strong symmetrical longitudinal SAM density appears in the focal plane compared with the case of the original VOF. The appearance of the longitudinal component of the SAM density, leading to the resultant three-dimensional spin states, may provide more degrees of freedom in optical manipulation and spin-dependent directive coupling. It should be emphasized that only the SAM density is redistributed, whereas the total SAM is zero and remains the conservation in a tightly focusing process.

In order to gain some further insights into the impact of the symmetric polarization states (represented by the parameter *s*) of the KS-VOF on the SAM density, we explore the SAM density of the tightly focused KS-VOFs with (*m*, *n*) = (1, 2), *s* = 1, 3, and 5 in the focal plane, as depicted in Fig. 9. We can see that the amount of the mirror-symmetric axes of the polarization state of the focal field and the longitudinal SAM density is 2*s*. As *s* increases, the magnitude of the transverse SAM density increases, while the profile remains invariant. The redistributing symmetric polarization states and SAM density may have potential applications in optical switching, polarization-sensitive photodetectors, optical trapping, and microscopy.

Similar to the SAM density, the orbital angular momentum (OAM) density of the tightly focused KS-VOF is calculated and discussed as a comparison by $O=r\xd7Im\epsilon 0E*(\u2207)E+\mu 0H*(\u2207)H/4\omega $,^{41,42} where **r** is the radius vector. The first and second columns in Fig. 10 show the transverse and longitudinal components of the SAM density when (*m*, *n*) = (1, 2), which is the same with the pictures in Fig. 9. The third and fourth columns show the OAM density as a comparison. From the first column in Fig. 10, we can find that the transverse component of the SAM density is with cylindrical symmetric distribution, while the longitudinal component of the SAM density and the OAM density possess 2*s* symmetric axes. The direction of the transverse SAM density is clockwise pointing to the azimuthal orientation, while the transverse OAM density is anticlockwise pointing to the azimuthal orientation.

The evolution characteristics of the SAM and OAM always attract interest, so we analyze the total SAM and OAM of the KS-VOF before and after being tightly focused. As is well known, the total SAM and OAM can be calculated by the spatial integral of the SAM and OAM density vector **J**_{s} = ∫**S***d***r** and **J**_{o} = ∫**O***d***r**.^{38,41} The numerical results indicate that the total SAM and OAM of the incident KS-VOF with local linear polarization and without any space variant phase distribution are zero. Meanwhile, the total SAM and OAM of the tightly focused KS-VOF are also zero. Considering the initial SAM and OAM, the total SAM and OAM remain conservation after being tightly focused. We deduce an interesting conclusion that there is zero OAM to SAM or SAM to OAM conversion in the tightly focusing process, and the SAM and OAM only redistribute in the focal plane. It means that the design method of the kaleidoscope-structured vector optical fields plays a role as a catalyst to the redistribution process of the SAM and OAM in the tight focusing process.

## V. CONTROLLABLE TRANSVERSE ENERGY FLOW IN THE FOCAL PLANE

The transverse energy flow, also referred to as internal energy flow,^{41–44} has attracted attention for the tightly focused VOFs.^{45–52} Now we are interested in the transverse energy flow of the tightly focused KS-VOFs, which are useful in manipulating the particles. Based on the full time-dependent three-dimensional electric and magnetic field vectors, the energy flow can be defined by the time-averaged Poynting vector,^{41,43,45,47,49–52}

where **E** and **H** are the electric and magnetic fields in the focal plane, and **P**_{s} and **P**_{o} are the spin and orbital components of the Poynting vector **P**. We also use the vectorial diffraction theory pioneered by Richards and Wolf^{33,34} to calculate the tightly focused fields under the numerical aperture of NA = 0.95, to give a comparison with the results reported in Refs. 47 and 50–52.

Now we discuss the energy flow of the tightly focused KS-VOFs, in particular the transverse energy flow for manipulating the particles. The calculation results show that the transverse energy flow only exists in the case when the polarization states exhibit the odd-fold symmetry; otherwise, the transverse energy flow is zero. The intensity and the Poynting vectors of the tightly focused fields of the onefold, threefold, and fivefold symmetric KS-VOFs are shown in the three columns in Fig. 11, respectively. The three rows display the total intensity, the transverse component, and longitudinal component of the Poynting vector in the focal plane. We can find that the transverse energy flow of the tightly focused field of the onefold symmetric KS-VOF is composed of two parts and finally flow into one fixed location, which means the particles can be transported to the fixed location along different routes. For the threefold and fivefold symmetric KS-VOFs, the transverse energy finally flows to three and five fixed locations. Compared with the longitudinal component, the relative intensities of transverse energy flow in the focal plane are 0.753, 0.502, and 0.680, which are obviously superior to Refs. 47 and 50. Thus, the manipulation of particles in the *x*–*y* plane will be easy to achieve, and the distribution of transverse energy flow is more uniform.

Figure 12 shows the transverse energy flow of the tightly focused fields of the threefold symmetric KS-VOFs with (*m*, *n*) = (4.5, 7.5), (4.5, 9), (5.5, 7) and (6.5, 7), respectively. When (*m*, *n*) = (4.5, 7.5), the transverse energy flow exhibits six uniform tentacles, which are very useful and powerful to trap and transport the particles. When (*m*, *n*) = (4.5, 9), the transverse energy flow exhibits a pattern similar to six spanners, which can be used to trap and transport the particles such as spanners. When (*m*, *n*) = (5.5, 7), the transverse energy flow consists of two-tiered energy flows and the energy flow finally flows to three fixed locations. When (*m*, *n*) = (6.5, 7), the transverse energy flow also consists of two-tiered annular energy flow rings, and the boundaries become more clear. The design of the transverse energy flow is obviously more flexible than that reported in Refs. 47 and 50, and the relative values are also larger.

As we have discussed, the energy flow can be defined by the time-averaged Poynting vector **P**, and the momentum current is related to the Poynting vector by **M** = *c*^{2}**P**, which means the momentum current has the same distribution as the energy flow. It is known that the momentum current can be divided into spin current and orbital current. Figure 13 shows the transverse and longitudinal components of the momentum current. From the direction of the transverse momentum current in the first row of Fig. 13, we can see that both spin and orbital current contribute to the characteristic of the transverse energy flow to transport particles to fixed locations. We should also state that we only show the magnitude of the longitudinal component of the Poynting vector in the third column in Fig. 11 because the longitudinal components of the Poynting vector and momentum current are approximately positive, and the sign of the spin and orbital current is discussed in Fig. 13.

Furthermore, we explore the rotation behavior of the KS-VOF during the focusing process. As is well known, the vortex optical field with OAM rotates 90° during the focusing process,^{53} and the cylindrical VOF as the superposition of two opposite vortex fields also holds this rotation characteristic.^{54,55} The VOFs in the original sector of the KS-VOFs are cylindrical VOFs, and the VOFs in other sectors are also cylindrical VOFs with same topological charges and symmetric polarization states. This means that the two orders of the KS-VOF are with opposite phase distribution. As a result, the two orders rotate 90° along the opposite directions due to the opposite azimuthally variant phase, while the opposite radially variant phase distributions lead to the radial separation during propagation. These rotation and separation phenomena can be observed experimentally in propagation by inserting fanlike obstacles.^{54,55} Meanwhile, these rotation and separation properties lead to various focal patterns, angular momentum density, and energy flow distributions in the focal plane.

After introducing the focal patterns, the SAM density, the OAM density, and the energy flow of the tightly focused fields of the KS-VOFs, it is interesting to discuss the evolution of the symmetry of the KS-VOFs during the focusing process. In fact, the symmetry of the tightly focused VOFs is due to the symmetry of the polarization states of the incident VOFs. In particular, for the KS-VOFs with odd-fold mirror-symmetric polarization states (*s* is an odd number), the amount of the symmetric axes for the tightly focused field intensity, SAM density, OAM density, and the Poynting vector is 2*s*. Meanwhile, for the case when *s* is even, the amount of the symmetric axes is also *s*. We should illustrate that the conclusion above is for the total intensity of the focal fields, the SAM density, the OAM density, the transverse and longitudinal energy flow, and spin and orbital momentum current as there may be different amounts of symmetric axes for other certain components such as *x*- and *y*-components.

One should also point out that besides the local linearly polarized KS-VOFs, we can also design and generate the complicated KS-VOFs by choosing the more complicated VOF in the original sector. The method for designing these complicated KS-VOFs is similar to the case of local linearly polarized KS-VOFs as long as the long axes of the elliptical polarizations in the adjacent sectors are mirror-symmetric. As for the circular polarization in the original sector, the symmetric polarization should also be circular polarization. For instance, a typical kind of uniformly elliptically polarized vector optical fields^{29} can also be used in designing and generating KS-VOFs. When the ±1st-orders are with different amplitudes in the experimental setup in Fig. 2, we can generate uniformly elliptically polarized KS-VOF, which has the same ellipticity and sense of local elliptical polarization at any location and also has a mirror-symmetric distribution of orientation of the elliptical polarization. This means ellipticity as an additional degree of freedom can be used in modulating the KS-VOFs, leading to certain symmetry and variety of applications. Moreover, vector optical fields with arbitrary polarization states can be designed in the original sector to generate arbitrary KS-VOFs, using the experimental methods to generate arbitrary VOFs.^{30,31}

## VI. CONCLUSION

In conclusion, we have presented and generated a new kind of KS-VOFs, which is designed by analogy with the principle of multiple reflection in the kaleidoscope. Based on our designing method, the KS-VOFs can greatly enrich the family of VOFs because various VOFs can be chosen in the original sector. The most striking feature of such a new kind of VOFs is the flexibly designed symmetric polarization states, which can be reflected in the corresponding focal properties. The various focal intensity patterns including flattop subwavelength sharp line with high figure of merit can be used in optical machining, trapping, and storage. Moreover, we also study the SAM density and transverse energy flow of the tightly focused KS-VOFs. The appearance and redistribution of the local SAM and the symmetric SAM density may be used in optical switching, polarization-sensitive photodetectors, optical trapping, and microscopy. Meanwhile, the controllable transverse energy flow can trap and transport multiple absorptive particles to certain positions. In addition, the analysis of symmetry deepens our understanding of the focusing process and can help us to design various new kinds of VOFs and find new properties and applications. We hope that more applications can be developed by this new kind of KS-VOFs with various symmetry properties in the broad prospect.

## ACKNOWLEDGMENTS

This work was supported by the National Key R&D Program of China (Grant Nos. 2017YFA0303800 and 2017YFA0303700), the National Natural Science Foundation of China (Grant Nos. 11534006, 11674184, 11774183, and 11804187), the Shandong Provincial Natural Science Foundation (Grant No. ZR2019BF006), the Natural Science Foundation of Tianjin (Grant No. 16JCZDJC31300), A Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J18KA229), and the 111 Project (Grant No. B07013). We acknowledge the support by the Collaborative Innovation Center of Extreme Optics.