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Abstract
A theoretical analysis based on mirror symmetry is proposed to analyze and predict the symmetry in intensity, phase and polarization distributions of the tightly focused vector optical field (VOF). We extend the analysis to more cases including more complicated polarization states and weak focusing cases. We further show the symmetric tightly focused fields of the eccentric cylindrical VOF and the redesigned VOF with a radially variant polarization state, which are achieved by redesigning the polarization state of the incident VOF based on the symmetry analysis. We also take the laser fabrication as an example to further show how to apply this symmetry analysis in a specific application area. Such a theoretical analysis can improve the calculation efficiency, provide new insights into the tight focusing process and offer a convenient way to engineer the field distributions in the focal plane, which may have potential applications in areas needing flexibly controllable tightly focused fields, such as laser fabrication, optical trapping, and optical storage.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Vector optical fields (VOFs) with spatially variant polarization states have attracted great attention, due to the flexible properties and various applications [1]. The focused VOFs have been proved to be flexibly controllable [2–8] and have many applications such as the far-field focal spots beyond the diffraction limit [9–11], the light needle [12–15], the optical cage [16–18], the optical chain [19,20], the subwavelengh sharp line [21–23], the information transmission [24–26], the laser processing [27–29] and the forming of reverse energy flow [30]. Most of the past researches dealt with cylindrical VOFs with cylindrical symmetric states of polarization (SoPs) [1–7,9–14,16,17,19,20,24–26,28,30], such as the well-known radially polarized VOF (RP-VOF) and azimuthally polarized VOF (AP-VOF) [2]. Besides the RP-VOF and AP-VOF, the tightly focused fields of the other symmetric VOFs have also attracted attention [15,18,21–23,27,29,31–33]. As a matter of fact, all other VOFs, including the higher-order cylindrical VOF [7], have no cylindrical symmetric SoPs. As a result, the symmetry of the tightly focused VOFs becomes a complicated and important topic.
In this paper, we propose a new theory to analyze the mirror symmetry of the tightly focused VOFs, and further extend the analysis to more cases including more complicated SoPs and weak focusing case. We prove that symmetric tightly focused fields can be generated by redesigning the SoP in the input plane, and we also take the laser fabrication as an example to show how to apply this symmetry analysis in a specific application area. Such theoretical analysis based on mirror symmetry can help us to have deep insight into the tight focusing process, offer a new way to analyze the properties of the focused field, and assist in designing and manipulating the tightly focused fields, which can be applied in many areas needing flexibly controlled tightly focused fields.
2. Mirror symmetry of tightly focused vector optical fields with space-variant linear polarizations
2.1 Principle of symmetry analysis
Richards and Wolf first introduced the contribution of polarization for calculating the focal field with a high numerical-aperture (NA) lens [34], which is widely used to theoretically study the property of the tightly focused optical fields in past two decades [1–7,9–13,15–23,25,29–32]. The schematic of the tight focusing process is shown in Fig. 1(a). For the incident VOF with expression of $\mathbf {E}_{i}=E_{x_{i}} \hat {\mathrm {\textbf {e}}}_{x}+E_{y_{i}} \hat {\mathrm {\textbf {e}}}_{y}$ ($\hat {\mathrm {\textbf {e}}}_{x}$ and $\hat {\mathrm {\textbf {e}}}_{y}$ are the unit vectors in the $x$- and $y$-directions, and $E_{x_{i}}$ and $E_{y_{i}}$ are the complex amplitudes of $x$- and $y$-components of the incident field, respectively), its tightly focused field can be expressed by six components based on the Richards-Wolf vector diffraction theory [1,2,34], as follows
Fig. 1. Tight focusing process. (a) Schematic of the tight focusing process. (b) Four symmetric area elements we choose in the input plane as ${A}(\rho , \varphi )$, ${B}(\rho , \pi -\varphi )$, ${C}(\rho , \pi +\varphi )$ and ${D}(\rho , 2 \pi -\varphi )$. The symmetric linear polarizations are also drawn on them. (c) Arbitrarily-chosen point $A'(r, \phi )$ in the focal plane.
The subscripts of the focused field in Eqs. (1)–(6) represent the components of the incident field and the focused field, respectively. For instance, $E_{y x}$ means the $x$-component of the focused field contributed by the $y$-component of the incident field. Thus, the tightly focused field can be written as
Based on the above equations from the Richards-Wolf diffraction theory, we propose a new method to analyze the mirror symmetry of the tightly focused field. We should point out that unless otherwise stated, all the symmetries we mention in this paper are mirror symmetry. The key idea of our method is to analyze the symmetry of the whole tightly focused field by using the contribution of the symmetric area elements of the incident VOF in the input plane to the tightly focused field. We first choose two/four symmetric area elements as a group in the input plane and then calculate their contributions to the tightly focused field at arbitrary point in the focal plane. We should state that there are two reasons why our method is valid by choosing the symmetric area elements in the input plane to analyze the symmetry of the tightly focused field. First, the integral calculation is essentially summation and the group of symmetric area elements in the input plane is chosen arbitrarily. Second, the symmetric points in the focal plane are also chosen arbitrarily.
As is well known, the VOFs with space-variant linear polarization distribution have been widely studied and used [1–7,9–14,16–31]. Hence, we first discuss the tightly focused field of the incident VOF with space-variant linear polarizations and uniform amplitude and phase distributions, and then discuss the more complicated cases in Section 3.1.
In this paper, we consider two basic situations that the SoP of the incident VOF has one symmetric axis and two orthogonal symmetric axes, as shown in Fig. 1(b). We can choose four symmetric area elements $A$, $B$, $C$ and $D$, with their coordinates of ${A}(\rho , \varphi )$, ${B}(\rho , \pi -\varphi )$, ${C}(\rho , \pi +\varphi )$ and ${D}(\rho , 2 \pi -\varphi )$. The incident fields at these area elements satisfy $E_{x_{i}} (A)=-E_{x_{i}} (B)=-E_{x_{i}} (C)=E_{x_{i}} (D)$ and $E_{y_{i}} (A)=E_{y_{i}} (B)=-E_{y_{i}} (C)=-E_{y_{i}} (D)$ with the symmetric linear polarizations. For the case of one symmetric axis, we can choose two area elements $A$ and $B$ without loss of generality, and the contribution of the incident fields in the two area elements $A$ and $B$ to the tightly focused field at any point $A'(r, \phi )$ of the focal plane should be
For the case of two orthogonal symmetric axes of the SoP in the input plane, we should choose four symmetric area elements $A$, $B$, $C$ and $D$. In this case, the contribution of the incident fields in the four area elements $A$, $B$, $C$ and $D$ to the tightly focused field at any point $A'(r, \phi )$ of the focal plane should be
With Eqs. (8)–(19), we can calculate the focal field contributed by the incident VOFs in a group of symmetric area elements. The total tightly focused field at any point in the focal plane is equal to the sum of the focal fields contributed by all the area elements in the input plane. In this way, we can analyze the symmetry of the tightly focused field when there is one symmteric axis or two orthogonal symmetric axes of the SoP in the input plane.
2.2 Symmetry of the tightly focused fields
When analyzing the symmetry of the tightly focused field, we focus on the symmetry of the intensity, phase and polarization distributions in the focal plane. We first discuss the incident VOF with only one symmetric axis. In this case, we need to choose other two symmetric points of $A'(r, \phi )$ in the focal plane as $B'(r, \pi -\phi )$ and $D'(r, 2\pi -\phi )$. Furthermore, we can use Eqs. (8)–(13) to calculate the tightly focused fields at points $A'$, $B'$ and $D'$ contributed by the incident fields in the group of area elements $A$ and $B$. After summing the focal fields contributed by all the groups of area elements to calculate the total tightly focused field, we find that the relationship of the tightly focused fields at points $A'$, $B'$ and $D'$ is the same before and after summation. As a result, we can deduce the relationship of the total tightly focal fields at points $A'$, $B'$ and $D'$ as
We readily calculate the symmetry relationship of the intensity of the tightly focused field asTherefore, we can conclude that if there is one symmetric axis in the SoP of the incident VOF with space-variant linear polarization distribution, the total intensity of the tightly focused field has two symmetric axes: parallel and perpendicular to the original symmetric axis in the input plane, respectively. In addition, the intensity distributions of the three components of the tightly focused field are also symmetric about the two orthogonal symmetric axes. Obviously, the amount of the symmetric axes of the intensity of the tightly focused field is twice that of the symmetric axes in the input plane. Only one exception is that when there are orthogonal symmetric axes of the SoP in the input plane, the amount of the symmetric axes of the intensity of the focused field may keep unchanged. This special case will be discussed below.
Except for the intensity symmetry, we can also come to the important conclusion about the phase distribution of the tightly focused field from Eqs. (20)–(22): when there is one symmetric axis of the SoP of the incident VOF (for instance, $y$ axis), the $x$-component of the tightly focused field at the two symmetric points about the $y$ axis are out of phase, while the phases of their $y$- and $z$-components exhibit the symmetric distribution about the $y$ axis. In contrast, the phases of the three components have no obvious symmetry or regularity about the $x$ axis.
Now we will further investigate the symmetry of the phase distribution of the tightly focused field when there are two orthogonal symmetric axes of the SoP of the incident VOF with space-variant linear polarization distribution. Here we assume that the two symmetric axes of the SoP are $x$ and $y$ axes. With Eqs. (14)–(19), we can get the relationship of the total tightly focused fields at points $A'$, $B'$ and $D'$ as
Clearly, the intensity exhibits two orthogonal symmetric axes in the focal plane when there are two orthogonal symmetric axes of the SoP in the input plane, which is in agreement with the conclusion when there is one symmetric axis of the SoP in the input plane. Furthermore, we readily find the symmetry relationship of the phase of the tightly focused field as
This means that when the SoP of the incident VOF has two symmetric axes (for instance, $x$ and $y$ axes), the $x$-component is in phase about the $x$ axis but out of phase about the $y$ axis; in contrast, the $y$-component is out of phase about the $x$ axis but in phase about the $y$ axis; the $z$-component of the tightly focused field is in phase about both $x$ and $y$ axes.
With the aid of the above analysis about the intensity and phase symmetries of the tightly focused field, we can analyze the polarization symmetry of the tightly focused field. Figure 2 shows the polarization distribution on symmetric points in the focal plane when there is one or two symmetric axes of the SoP of the incident VOF with space-variant linear polarization distribution. We find from Fig. 2(a) that when there is one symmetric axis of the SoP of the incident VOF, the SoP of the tightly focused field is also symmetric about the original symmetric axis. When there are two symmetric axes of the SoP of the incident VOF, the SoP of the tightly focused field is also symmetric about the original two symmetric axes, as shown Fig. 2(b). These conclusions can also be directly obtained by Eqs. (20)–(22) and Eqs. (26)–(28).
Fig. 2. The SoPs of the tightly focused field. (a) The SoP of the incident VOF is symmetric about $y$ axis. (b) The SoP of the incident VOF is symmetric about both $x$ and $y$ axes. The red and green arrows represent the $x$- and $y$-components, and the black spots represent the direction of the $z$-component of the tightly focused field.
3. Analysis for more cases based on mirror symmetry
3.1 More cases of the SoPs of the incident VOFs
All the above discussions about the symmetry of the tightly focused field is limited to the case of symmetric linear polarizations in the input plane, which can be represented by the case shown in Fig. 3(a). In fact, we can also obtain certain symmetry of the tightly focused field in some other cases of symmetric polarizations in the input plane. Here we list three possible cases of linear polarizations at two symmetric positions about the $y$ axis: the opposite polarizations in Fig. 3(b); the opposite $y$-polarized component in Fig. 3(c); the same polarizations in Fig. 3(d). For the four cases in Fig. 3, the $x$- and $y$-components of the incident VOFs have the following relationships: (a) $E_{x_{i}} \! = \! -\overline {E}_{x_{i}}$, $E_{y_{i}} \! = \! \overline {E}_{y_{i}}$; (b) $E_{x_{i}} \! = \! -\overline {E}_{x_{i}}$, $E_{y_{i}} \! = \! -\overline {E}_{y_{i}}$; (c) $E_{x_{i}} \! = \! \overline {E}_{x_{i}}$, $E_{y_{i}} \! = \! -\overline {E}_{y_{i}}$; (d) $E_{x_{i}} \! = \! \overline {E}_{x_{i}}$, $E_{y_{i}} \! = \! \overline {E}_{y_{i}}$.
Fig. 3. Four cases of linear polarizations in symmetric area elements about the $y$ axis in the input plane. (a) The mirror-symmetric polarizations, (b) the opposite polarizations, (c) the opposite $y$-polarized component, (d) the same polarizations.
Now we will first consider the $x$-component of the tightly focused field $E_{x} \! = \! E_{x x}+E_{y x}$. We can get from Section 2 that $E_{x_{i}} \! = \! \overline {E}_{x_{i}}$ $\Rightarrow$ $E_{x x}(A') \! = \! E_{x x}(B')$; $E_{x_{i}} \! = \! -\overline {E}_{x_{i}}$ $\Rightarrow$ $E_{x x}(A') \! = \! -{E}_{x x}(B')$; $E_{y_{i}} \! = \! \overline {E}_{y_{i}}$ $\Rightarrow$ $E_{y x}(A') \! = \! -E_{y x}(B')$; $E_{y_{i}} \! = \! -\overline {E}_{y_{i}}$ $\Rightarrow$ $E_{y x}(A') \! = \! E_{y x}(B')$. As a result, we can conclude: for the case in Fig. 3(a), $E_{x}(A') \! = \! -E_{x}(B')$; for the case in Fig. 3(b), $\left |E_{x}(A')\right | \! \neq \! \left |E_{x}(B')\right |$; for the case in Fig. 3(c), $E_{x}(A') \! = \! E_{x}(B')$; for the case in Fig. 3(d), $\left |E_{x}(A')\right | \! \neq \! \left |E_{x}(B')\right |$.
Obviously, the intensity of the $x$-component of the tightly focused field is symmetric about the $y$ axis for cases in Figs. 3(a) and (c), and similar conclusion can be deduced for the $y$- and $z$-components of the tightly focused field: for the case in Fig. 3(a), ${E}_{y}(A')={E}_{y}(B')$ and $E_{z}(A')=E_{z}(B')$; for the case in Fig. 3(b), $\left |E_{y}(A')\right | \neq \left |E_{y}(B')\right |$ and $\left |E_{z}(A')\right | \neq \left |E_{z}(B')\right |$; for the case in Fig. 3(c), $E_{y}(A')=-E_{y}(B')$ and $E_{z}(A')=-E_{z}(B')$; for the case in Fig. 3(d), $\left |E_{y}(A')\right | \neq \left |E_{y}(B')\right |$ and $\left |E_{z}(A')\right | \neq \left |E_{z}(B')\right |$. The total intensity of the tightly focused field is symmetric about the $y$-axis for the two cases in Figs. 3(a) and (c). Similarly, we can conclude that the total intensity of the tightly focused field is also symmetric about the $x$-axis for these two cases. Hence, the linear polarizations in Figs. 3(a) and (c) can be considered as the symmetric polarizations when studying the symmetry of the intensity of the tightly focused field. However, the linear polarizations in Figs. 3(b) and (d) cannot lead to the mirror symmetry of the tightly focused field. As is well known, any polarization can be decomposed into two orthogonal components: one parallel and one perpendicular to the symmetric axis. Compared with the original linear polarization, we can define the symmetric polarization as the polarization with one component same and the other component opposite with the original polarization. Such a definition is interesting and useful for understanding the reasons why the tightly focused field of the AP-VOF is a cylindrical symmetric ring [2] and the tightly focused field of the incident AP-VOF with the symmetric fan-like obstacle has also the mirror-symmetric intensity distribution [35].
Besides the symmetry of the intensity, we should also discuss the symmetry of the phase for the case in Fig. 3(c). Based on the above conclusions, it is obvious that when there is one symmetric axis of SoP of the incident VOF (for instance, $y$ axis) in Fig. 3(c), the $x$-components of the tightly focused field are in phase about the $y$ axis, while the $y$- and $z$-components are out of phase about the $y$ axis.
Up to now, we have primarily considered the symmetry of the tightly focused field of the VOF with space-variant linear polarization distributions, which is the most commonly used VOFs. However, the cylindrical VOFs with hybrid SoPs [36–38], full Poincaré beam [32,39], and other new kinds of VOFs with SoPs including linear, elliptical, and circular polarizations [40–43] have also attracted attention. This raises a significant question: what happens to the symmetry of the tightly focused field, when the incident field has a more complicated SoP? To answer this question, we must first analyze the symmetry of elliptical polarizations. The analysis is similar to the case of linear polarizations. When the elliptical polarization in the area element $B$ in Fig. 4 is expressed as $\mathbf {E}_{i}=\mathbf {E}_{1}+j \mathbf {E}_{2}$, there are four basic cases of the elliptical polarizations in the symmetric area element $A$: $\mathbf {E}_{i}=\mathbf {\overline {E}}_{1} +j \mathbf {\overline {E}}_{2}$ in Fig. 4(a), $\mathbf {E}_{i}=\mathbf {\overline {E}}_{1} -j \mathbf {\overline {E}}_{2}$ in Fig. 4(b), $\mathbf {E}_{i}=-\mathbf {\overline {E}}_{1} +j \mathbf {\overline {E}}_{2}$ in Fig. 4(c) and $\mathbf {E}_{i}=-\mathbf {\overline {E}}_{1} -j \mathbf {\overline {E}}_{2}$ in Fig. 4(d). We should point out that all kinds of elliptical polarizations can be decomposed into two orthogonal linearly polarized components $\mathbf {E}_{1}$ and $\mathbf {E}_{2}$ ($\mathbf {\overline {E}}_{1}$ and $\mathbf {\overline {E}}_{2}$) with a $\pi /2$ phase difference.
Fig. 4. Four cases of symmetric elliptical polarizations in area element $A$ in the input plane: (a) $\mathbf {E}_{i}=\mathbf {\overline {E}}_{1} +j \mathbf {\overline {E}}_{2}$, (b) $\mathbf {E}_{i}=\mathbf {\overline {E}}_{1} -j \mathbf {\overline {E}}_{2}$, (c) $\mathbf {E}_{i}=-\mathbf {\overline {E}}_{1} +j \mathbf {\overline {E}}_{2}$ and (d) $\mathbf {E}_{i}=-\mathbf {\overline {E}}_{1} -j \mathbf {\overline {E}}_{2}$, where $\mathbf {\overline {E}}_{1}$ and $\mathbf {\overline {E}}_{2}$ represent two components on the long and short axes of the polarization ellipse, respectively. The elliptical polarization in area element $B$ is the same for the four cases as $\mathbf {E}_{i}=\mathbf {E}_{1}+j \mathbf {E}_{2}$.
Now we will explore the symmetry of the tightly focused field for the case of the elliptical polarizations in the input plane. The incident field in the area element $B$ is the same in four cases of Fig. 4, which can also be written as
For the case in Fig. 4(a), the incident field in the area element $A$ in the input plane can be written as
Using the same method in Section 2, we can give the $x$-components of the tightly focused field at the point $A'$ contributed by the $x$- and $y$-components of the incident fields in the area elements $A$ and $B$ as
After summing the contributions by all the area elements in the input plane, we can calculate the tightly focused fields at the points $A'$, $B'$ and $D'$, and then give the relationships of the tightly focused fields as
For the intensity, we can readily findSimilarly, for the cases in Figs. 4(b) and (c), $I_{x}(A')=I_{x}(B') \neq I_{x}(D')$; for the case in Fig. 4(d), $I_{x}(A')=I_{x}(D') \neq I_{x}(B')$. By following the same approach as done for the $x$-component of the tightly focused field, the intensity distributions of $y$- and $z$-components of the tightly focused field can also be calculated, and the conclusions are similar.
Based on the above analysis, we can find that if the long and short axes of the two elliptical polarizations with the same ellipticity are symmetric about certain axis ($y$ axis), the intensity of the tightly focused field is symmetric about the original symmetric axis ($y$ axis) for the cases in Figs. 4(b) and (c), and the intensity of the tightly focused field is symmetric about the axis ($x$ axis) perpendicular to the original symmetric axis for the cases in Figs. 4(a) and (d). Moreover, the intensity distributions of the three components follow the same rule.
For the special case of circular polarizations in the SoP of the incident VOF, which we will not discuss in detail here, is also similar to the case of linear polarizations. When considering the circular polarizations in the symmetric area elements in the input plane, we can come to a conclusion that if there is one symmetric axis of the SoP of the incident field, there are two symmetric axes of the intensity of the tightly focused field: the parallel axis and the orthogonal axis of the original symmetric axis of the incident VOF.
For the incident VOF with the hybrid SoPs including linear, elliptical and circular polarizations, the symmetry of the intensity distribution of the tightly focused field is determined by the symmetry of the elliptical polarization in the input plane, because the amount of the symmetric axes in the focal plane is the least for the case of elliptical polarizations in our analysis. The analysis of phase and polarization of the tightly focused field is similar to the analysis in Section 2, and the detailed derivation will not be discussed here.
In the above analysis, we focus on the effect of the SoP symmetry of the incident VOF on the symmetry of the tightly focused field. Now we briefly discuss the influence of the symmetry of the amplitude and phase of the incident VOF on the tightly focused field. For instance, if the amplitude of the incident VOF is not symmetric about $y$ axis in Fig. 1(b), we cannot say the polarization is symmetric about $y$ axis and all the above conclusions are invalid. That is to say, the symmetric polarizations in the input plane we introduced above should also possess the symmetric intensity. For example, when a fanlike obstacle is attached to the RP-VOF and AP-VOF with cylindrical symmetric SoPs, the symmetric axis of the polarization is determined by the obstacle, and symmetry of the focused field satisfies the above conclusions [35,44,45]. As for the phase, $\pi$ phase can bring opposite directions of the polarizations, which means the phase can also affect the symmetric SoPs of the incident field, and this is similar to the discussion of the intensity. For more complicated VOF with space-variant amplitude, phase and polarization distributions, the symmetry of the tightly focused field may need to be analyzed following the basic theory introduced in Section 2.
3.2 Analysis based on mirror symmetry for the weakly focused fields
Although the tightly focused VOF is of great importance in various realms [1–7,9–23,25,29–32], the weakly focused field with lens of low numerical aperture is more commonly used. The calculation theory for weakly focused field is the scalar diffraction theory [46] instead of the vectorial diffraction theory proposed by Richards and Wolf above. This raises a significant question: is our above analysis for the tightly focused field still valid for the weakly focused field? Here we will briefly compare the tightly and weakly focusing cases. The weakly focused field has two distinct characteristics different from the tightly focused field: (i) there has no longitudinal component, i.e., no $E_{x z}$ or $E_{y z}$ are involved; (ii) the $x$ ($y$)-component of the incident optical field only contributes to the $x$ ($y$)-component of the focused field, i.e., no $E_{x y}$ or $E_{y x}$ are involved. Thus, only $E_{x x}$ and $E_{y y}$ components need to be calculated. The method of symmetry analysis is the same using the two diffraction theories. For simplicity, we consider the incident VOF with uniform intensity and phase distributions, and space-variant local linear polarization distribution here. Therefore, the weakly focused field is expressed by the Fresnel diffraction formula
To apply this Fresnel diffraction formula to symmetry analysis and demonstrate the relationship of the symmetry of weakly and tightly focused fields, we consider the case of local linearly polarized VOF with one symmetric axis ($y$ axis) of polarization in the incident plane as an example. We still choose two symmetric area elements $A (x_{i}, y_{i})$ and $B (-x_{i}, y_{i})$ in the input plane, and then analyze the focal fields at three points in the focal plane: $A'(x_{f}, y_{f})$, $B'(-x_{f}, y_{f})$ and $D'(x_{f}, -y_{f})$, respectively. The $x$-components of the weakly focused field at points $A'$, $B'$ and $D'$ contributed by the $x$-components of the incident fields in the area elements $A$ and $B$ can be written as
After summing the contributions by all the area elements in the input plane, we can calculate the weakly focused fields at the points $A'$, $B'$ and $D'$, and then give the relationships as
Considering the intensity, we can get This means that the intensity of the $x$-component is symmetric about both $x$ and $y$ axes in the focal plane. The intensity distribution of the $y$-component of the weakly focused field can be calculated in the same way as the above, which is also symmetric about two symmetric axes. As a result, we demonstrate that when the SoP of the incident VOF with space-variant linear polarization distribution is symmetric about one axis, the weakly focused field has two symmetric axes: the parallel and perpendicular axes of the symmetric axis of the incident VOF. Moreover, all other conclusions of the weakly focused field are the same as the tightly focused field, which will not be discussed in detail here.4. Applications of the theoretical analysis based on mirror symmetry
Based on the above the theoretical analysis for focused VOFs, we proceed to explore the applications of the symmetry analyais in this section. Here we introduce how to redesign the symmetric SoP in the inupt plane in order to generate symmetric tightly focused fields, and further take laser fabrication as an example to illustrate how to apply the symmetry analysis in a specific area.
4.1 Achieve symmetric tightly focused field by redesigning the symmetric SoP of the incident VOF
For the cylindrical VOFs introduced above, the SoPs of the VOFs change along the azimuthal direction, and there always exists a singularity in the center of the incident field. If we change the location of the central singularity of the incident cylindrical VOF, the original symmetry of the SoP of the cylindrical VOF will be broken, as shown in the first column of Fig. 5. This kind of VOF is called the eccentric cylindrical VOF, which can be written as
Fig. 5. Focal intensity patterns of the eccentric cylindrical VOFs. The first column shows the SoPs of eccentric cylindrical VOFs, other four columns are the total intensity patterns and the intensity patterns of $x$-, $y$-, and $z$-components of the tightly focused field, respectively. Three rows correspond to the cases when $m$ = 1, 2 and 3, respectively. The singularity in the input plane is located at ($-0.5\rho _{m}$, 0), where $\rho _{m}$ is the maximum radius of the incident VOF. The numerical aperture is NA = 0.9 and any image of the focused field has a dimension of $4 \lambda \times 4 \lambda$.
From the discussion of the eccentric cylindrical VOF with one symmetric axis of the SoP, we can find that the tightly focused field with new symmetry property can be generated by simply redesigning the SoP of the traditional VOF. Obviously, the symmetry of the SoP decreases from cylindrical VOF to eccentric cylindrical VOF. Now we will intorduce another case of redesigning the SoP of the incident VOF to obtain more symmetric axes in the focal plane. We mainly consider the VOFs with the same polarization distribution at the symmetric points about the symmetric axis ($y$ axis), as shown in Fig. 6(a). In order to redesign the symmetric SoP of the incident VOF, we need to decompose the polarization into two orthogonal components: one ($y$-component) parallel to the symmetric axis and the other one ($x$-component) perpendicular to the symmetric axis. We further change the direction of the component perpendicular to symmetric axis ($x$-component) to the opposite direction, and keep the other component unchanged, as shown in Fig. 6(b). As a result, the new polarization distribution is symmetric about the $y$ axis which is pictured in Fig. 6(c).
Fig. 6. The method of redesigning the SoP distribution when the polarizations are the same at the symmetric points about the symmetric axis ($y$ axis). (a) Original polarization distribution. (b) The diagram of redesigning process. (c) The redesigned polarization distribution.
Besides the VOFs with azimuthally variant SoPs introduced above, the VOFs with the radially variant SoPs also appear in the family of vector optical fields and have been attracted attention [18,47,48]. However, there is no mirror symmetric axis in the SoPs of this kind of VOFs, which may limit some potential applications of the tightly focused fields. The cylindrical VOFs with the radially variant SoPs can be expressed as
Fig. 7. The intensity and phase distributions of the tightly focused field of the VOFs with radially variant SoPs when $n = 0.5$. The SoPs of the incident VOFs are shown in the blue circles. (a) The original VOF with radially variant SoPs, (b) and (c) the redesigned VOFs with one and two symmetric axes of the SoPs, respectively. The numerical aperture is NA = 0.9 and any image has a dimension of $4 \lambda \times 4 \lambda$.
According to the method introduced in Fig. 6, we only need to change the $x$-component of the VOF in one side of $y$ axis to achieve symmetric SoPs with one symmetric axis. In this case, the expression of the VOF can be rewritten as $\mathbf {E}_{i}= \operatorname {sgn}(x_{i}) \cos (2 \pi n \rho / \rho _{m}) \hat {\mathbf {e}}_{x}+\sin (2 \pi n \rho / \rho _{m}) \hat {\mathbf {e}}_{y}$. As shown in Fig. 7(b), the SoPs of this kind of VOF are symmetric about the $y$ axis, and the intensity of the tightly focused field is symmetric about both the $x$ and $y$ axes. The total intensity pattern exhibits a rhombic strong spot surrounded by four stripe-shaped sub-strong spots in the focal plane. The $x$-components (the $y$-/$z$-components) at the symmetric points about the $y$ axis are out of phase (in phase).
In order to obtain more kinds of intensity and phase distributions in the focal plane, we further change the VOF as $\mathbf {E}_{i}= \operatorname {sgn}(x_{i}) \cos (2 \pi n \rho / \rho _{m}) \hat {\mathbf {e}}_{x}+\operatorname {sgn}(y_{i}) \sin (2 \pi n \rho / \rho _{m}) \hat {\mathbf {e}}_{y}$. In this case, the SoP of the incident VOF is both symmetric about the $x$ and $y$ axes, and the intensity patterns of the tightly focused fields are also symmetric about $x$ and $y$ axes. As shown in Fig. 7(c), the total intensity pattern of the tightly focused field appears a hollow intensity pattern with two strong spots in the vertical direction and two sub-strong spots in the horizontal direction, respectively. The $x$-components are in phase at the symmetric points about the $x$ axis while they are out of phase at the symmetric points about the $y$ axis, respectively. The $y$-components are in phase at the symmetric points about the $y$ axis while they are out of phase at the symmetric points about the $x$ axis, respectively. The $z$-components are in phase at the symmetric points about both $x$ and $y$ axes. All these results are in agreement with the conclusions analyzed above in Section 2. We should also point out that several experimental methods can be used to achieve the redesigning method we introduce, including the experimental method of generating VOFs with arbitrary SoPs [47,49–53], and using additional anisotropic sector plates [54] and binary phase plates [55].
Obviously, new kinds of VOFs with symmetric SoPs can be created by the redesigning methods, and symmetric tightly focused fields can be generated, which can be directly applied in the areas needing various symmetric tightly focused fields. However, for more cases, the tightly focused fields must satisfy the specific requirement in many applicaitons. This raises a significant question: how can our symmetry analysis work in the application areas with specific requirements? We will take the area of laser fabrication as an example to address this question below.
4.2 How to apply the symmetry analysis: take laser fabrication as an example
Now we take an example to illustrate how to apply the symmetry analysis in a specific area, taking laser fabrication (process multi-holes with $N$ symmetric axes) as an example without losing generality, as shown in Fig. 8. With the aim of processing multi-holes with $N$ symmetric axes, we need to analyze the symmetry of the input VOF (usually with new scheme to design and modulate optical fields). If the input field is with $N/2$ pairs of orthogonal symmetric axes or $N/2$ none-orthogonal symmetric axes, we calculate the tightly focused field with vectorial diffraction theory until achieving the desired focal pattern, and further apply the result to laser fabrication. To make the flowchart easier to understand, we show a specific example in the insets of Fig. 8. When we want to fabricate 6 holes with 6 symmetric axes shown in Fig. 8(a) [27,29], the tightly focused field should exhibit six focal spots with 6 symmetric axes. As a result, the input field should own 3 pairs of orthogonal symmetric axes as shown in Fig. 8(b), or 3 non-orthogonal symmetric axes as shown in Fig. 8(c). In this way, we can optimize the simulation of tightly focused fields by reducing the possible choices of the input VOFs when we aim at specific applications needing certain symmetry in the focal plane.
Fig. 8. Flowchart of applying the symmetry analysis to laser fabrication as an example. The insets: (a) the microstructures with 6 holes in laser fabrication and the red dotted lines are symmetric axes; (b) and (c) the schematic of SoPs of the VOFs with 3 pairs of orthogonal symmetric axes and 3 non-orthogonal symmetric axes in the input plane, respectively.
It should be pointed out that with the theoretical analysis based on mirror symmetry, we can verify the correctness of the simulated focal fields fast, as the symmetry property is fundamental to avoid mistakes when calculating the focal fields. If the calculated results do not satisfy the symmetry property, the simulation must be wrong, and we need to further check the symmetry of the input field and the calculation process. We should point out that this symmetry analysis cannot guarantee the correctness of the calculation, but it is still a useful tool to avoid many mistakes in designing the input VOF and simulating the focused field. As a result, the symmetry analysis can improve the efficiency of the calculation process when applying the focused fields in various areas.
5. Conclusion
We propose a new method for analyzing the tightly focused fields, based on the mirror symmetry, which can be used to discuss the relationship of the symmetry between the tightly focused field and the incident vector optical field. Furthermore, we provide the discussion for more cases of SoPs of the incident VOFs, and analyze the symmetry property of the weakly focused fields. We also present the tightly focused eccentric cylindrical VOF and the redesigned VOF with radially variant SoP, in order to demonstrate the way to achieve symmetric tightly focused field by redesigning the symmetric SoP of the incident VOF. Moreover, we take laser fabrication as an example to introduce how to apply the symmetry analysis in a specific application area. In general, there are three specific ways to apply the symmetry analysis in focal engineering: (i) the symmetry analysis can assist in redesigning the SoP of the incident VOF to achieve symmetric tightly focused field. (ii) The symmetry analysis can provide a way to reduce the possible choices of the input field, which can be applied to application area with specific requirements. (iii) The symmetry analysis can provide a way to improve the calculation efficiency, as it can be a useful tool to avoid many mistakes in designing input VOF and simulating the focused field. In this paper, we introduce a kind of symmetry transfer from the incident plane to the focal plane, which can provide new insight in studying focusing problem, as we can predict, analyze and apply the tightly focused field in the view of mirror symmetry. In this way, the theory can be considered as a reference to guide the researchers to analyze the symmetry properties and achieve certain symmetry in the focal plane. We hope this theoretical analysis based on mirror symmetry can bring new applications in the areas needing symmetric tightly focused fields such as laser fabrication, optical trapping, optical storage and so on.
Funding
National Natural Science Foundation of China (11534006, 11674184, 11774183, 11804187, 11904199); Natural Science Foundation of Shandong Province (ZR2019BF006); A Project of Shandong Province Higher Educational Science and Technology Program (J18KA229).
Acknowledgments
We acknowledge the support by Collaborative Innovation Center of Extreme Optics.
Disclosures
The authors declare no conflicts of interest.
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