⇦Chinese Physics Letters, 2020, Vol. 37, No. 12, Article code 124201 Generation and Tunable Focal Shift of the Hybridly Polarized Vector Optical Fields with Parabolic Symmetry Xu-Zhen Gao (高旭珍)1, Meng-Shuai Wang (王梦帅)1, Jia-Hao Zhao (赵嘉豪)1, Peng-Cheng Zhao (赵鹏程)1, Xia Zhang (张霞)1, Yue Pan (潘岳)1*, Yongnan Li (李勇男)2, Chenghou Tu (涂成厚)2, and Hui-Tian Wang (王慧田)3,4* Affiliations 1School of Physics and Physical Engineering, Shandong Provincial Key Laboratory of Laser Polarization and Information Technology, Qufu Normal University, Qufu 273165, China 2School of Physics and Key Laboratory of Weak-Light Nonlinear Photonics, Nankai University, Tianjin 300071, China 3National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China 4Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China Received 31 August 2020; accepted 21 October 2020; published online 8 December 2020 Supported by the National Natural Science Foundation of China (Grant Nos. 11534006, 11674184, 11774183, 11804187 and 11904199), the Natural Science Foundation of Shandong Province (Grant No. ZR2019BF006), and the Collaborative Innovation Center of Extreme Optics.
*Corresponding authors. Email: panyue.89@163.com or panyue@qfnu.edu.cn; htwang@nju.edu.cn Citation Text: Gao X Z, Wang M S, Zhao J H, Zhao P C, and Zhang X et al. 2020 Chin. Phys. Lett. 37 124201    Abstract Based on a parabolic coordinate system, we theoretically design and experimentally generate hybridly polarized vector optical fields with parabolic symmetry of the first and second kinds, which can further enrich the family of vector optical fields. The wavefront of this new-kind vector optical field contains circular, elliptic and linear polarizations, and the polarizations can keep the same or change along the parabolic curves. Then we present the realization of tunable focal shift with the hybridly polarized vector optical field, and show a specific law of the focal shift of the focused hybridly polarized vector optical field with the parabolic symmetry. We hope these results can provide a new way to flexibly modulate focal fields, which can be applied in realms such as optical machining, optical trapping and information transmission. DOI:10.1088/0256-307X/37/12/124201 PACS:42.25.Fx, 42.25.Ja, 42.30.Kq © 2020 Chinese Physics Society Article Text In the past few years, vector optical fields (VOFs) with spatially variant polarization states[1] have attracted great interest due to many distinctive features away from traditional scalar optical fields and their novel applications, including optical trapping and manipulation,[2–4] optical microfabrication,[5] single molecule imaging,[6] nonlinear optics,[7,8] quantum optics,[9] near-field optics,[10] optical microscopy,[11] enhancement of longitudinally polarized fields,[12] and so on. Meanwhile, with increasing interest in VOFs, hybridly polarized vector optical fields (HP-VOFs) with linear, elliptic and circular polarizations in wavefronts of the fields simultaneously have also drawn attention.[3,8,13–18] In this Letter, we theoretically design and experimentally generate HP-VOF with the parabolic symmetry in the parabolic coordinate system. This new kind of HP-VOFs exhibits the parabolic symmetry with circular, elliptic and linear polarizations on the wavefront, respectively. Moreover, properties of the tunable focal shift of the HP-VOFs with the parabolic symmetry are numerically investigated. It is shown that the focused field exhibits two focal spots at different symmetric locations, and the locations of the two focal spots are controllable, depending on the indices of the HP-VOFs with the parabolic symmetry. We hope these results can provide a new way to flexibly modulate the focal fields, which may be applied in realms such as optical machining, optical trapping and information transmission. The parabolic coordinate system is one of the basic two-dimensional coordinate systems, and its most representative characteristic is that the constant curves of the two coordinates are parabolic curves. In the parabolic coordinate system, researchers have designed different kinds of useful optical fields with space-invariant polarization distribution or space-variant linear polarizations.[19–21] Here, we design the HP-VOFs with the parabolic symmetry in parabolic coordinates. The HP-VOFs with the parabolic symmetry of the first kind can be expressed as $$ \boldsymbol{E}(u, v)=\cos(m \pi u + n \pi v) \hat{\boldsymbol{e}}_{x}+i \sin(m \pi u + n \pi v) \hat{\boldsymbol{e}}_{y},~~ \tag {1} $$ where $m $ and $n $ are the indices of the HP-VOFs, and $\{\hat{\boldsymbol{e}}_{x} , \hat{\boldsymbol{e}}_{y} \}$ are the unit vectors in Cartesian coordinates; $(u, v)$ are the parabolic coordinates, which have the following relationship with the Cartesian coordinates $(x, y)$: $$ u^{2}=\sqrt{x^{2}+y^{2}}-y, ~~~ v^{2}=\sqrt{x^{2}+y^{2}}+y.~~ \tag {2} $$ We can see clearly from Eq. (2) that the parabolic coordinates $(u, v)$ can be either positive or negative, and the sign of the coordinates $(u, v)$ can be included in the parameters $(m, n)$ without loss of generality, because $(m, n)$ can also be either positive or negative. In this way, the parabolic coordinates $(u, v)$ in Eq. (1) can be expressed as $$ u=\sqrt{\sqrt{x^{2}+y^{2}}-y}, ~~~ v=\sqrt{\sqrt{x^{2}+y^{2}}+y}.~~ \tag {3} $$ Using Eqs. (1) and (3), we can easily design the HP-VOFs with the parabolic symmetry of the first kind, whose polarization distributions along the constant curves of $m \pi u + n \pi v $ on the wavefront keep the same. When $n = 0 $ or $m = 0 $, the polarization distributions of the HP-VOFs along the parabolic curves of $\sqrt{x^{2}+y^{2}}-y=k$ or $\sqrt{x^{2}+y^{2}}+y=k$ keep the same, where $k$ is a constant.
cpl-37-12-124201-fig1.png
Fig. 1. Theoretically simulated and experimentally measured intensity patterns and normalized Stokes parameters of the HP-VOFs with the parabolic symmetry of the first kind when $(m, n) = (1, 0) $ and $(1, 1)$, respectively. The corresponding polarization distributions are superimposed with simulated intensity patterns in the first column. The colorbar values of 1 and $-1$ correspond to horizontal and vertical polarizations for $S_{1}$, $\pm 45^{\circ}$ polarizations for $S_{2}$, and right- and left-handed circular polarizations for $S_{3}$. The red ellipse (circle): right-handed elliptic (circular) polarization; yellow ellipse (circle): left-handed elliptic (circular) polarization; white line: linear polarization. The size of the optical field is $2\,\mathrm{mm} \times 2\,\mathrm{mm}$.
To experimentally generate the HP-VOFs with the parabolic symmetry, a common path interferometer implemented with a spatial light modulator (SLM) and a $4f$ system are employed.[3,14,22] Figure 1 shows the experimentally measured intensity patterns and the normalized Stokes parameters of the HP-VOFs with the parabolic symmetry when $(m, n) = (1, 0)$ and $(1, 1)$, respectively. For comparison, the corresponding theoretical results are displayed in the first and third rows of Fig. 1. Meanwhile, the corresponding polarization distribution of the HP-VOF with the parabolic symmetry is schematically superimposed with the theoretical intensity patterns in the first column. The total intensity patterns of the two cases are both uniform without singularities, which is quite different from the case of traditional cylindrical VOFs.[1] When $(m, n) = (1, 0)$, the intensity patterns of the $x$- and $y$-components both exhibit parabolic curves, which means that the polarizations along the parabolic curves keep the same. The Stokes parameter $S_{2}$ of the HP-VOF is close to zero, suggesting that the polarizations on the wavefront only include two orientations of long axes ($x$ and $y$ directions). The Stokes parameters $S_{1}$ and $S_{3}$ also exhibit parabolic curves which agree with the above discussion. We should point out that when $m$ increases, the changing period of the polarization increases, and the polarizations along a set of open-upward parabolas keep the same when $m \neq 0 $ and $n = 0 $. For the other case of $(m, n) = (1, 1)$ in Fig. 1, the polarizations along the parabolic curves do not keep the same, and the intensity patterns of the $x$- and $y$-components as well as the Stokes parameters $S_{1}$ and $S_{3}$ exhibit an eyelid-like shape, which are symmetric about both the $x$ and $y$ axes. The experimental results are in good agreement with the theoretical simulations. Apart from the HP-VOFs discussed above, we also design the HP-VOFs with the parabolic symmetry of the second kind in order to bring a new kind of VOFs as well as the more flexible focal shift property. The expression of this new HP-VOF is $$\begin{align} \boldsymbol{E}(x, y)={}&\cos (p \pi \sqrt{x^{2}+y^{2}}-q \pi y) \hat{\boldsymbol{e}}_{x}\\ &+i\sin (p \pi \sqrt{x^{2}+y^{2}}-q \pi y) \hat{\boldsymbol{e}}_{y},~~ \tag {4} \end{align} $$ where $p $ and $q $ are the indices of the HP-VOFs with the parabolic symmetry of the second kind. In this case, the polarizations along the constant curves of $p \pi \sqrt{x^{2}+y^{2}}-q \pi y=t$ keep the same, where $t $ is a constant.
cpl-37-12-124201-fig2.png
Fig. 2. Measured HP-VOFs with the parabolic symmetry of the second kind when $(p, q) = (0, 4)$, $(2, 4)$, $(4, 4)$, $(4, 2)$ and $(4, 0)$, respectively. The first row shows the total intensity patterns, and the second and third rows show the intensity patterns behind the horizontal and $\pi /4 $ polarizers, respectively. The first column shows the directions of the polarizers. The size of the optical field is $2\,\mathrm{mm} \times 2\,\mathrm{mm}$.
Figure 2 shows the VOFs with the parabolic symmetry of the second kind when $(p, q) = (0, 4)$, $(2, 4)$, $(4, 4)$, $(4, 2)$ and $(4, 0)$, respectively. It is clear that the total intensity patterns are always uniform without singularities, ignoring the interference fringes. The intensity patterns behind the $\pi /4$ polarizer exhibit no obvious extinction and the intensity is half of the total intensity, proving that this is a kind of HP-VOFs. When $p = 0$, the polarization distribution is a function of coordinate $y$ only, so the intensity of the $x$-component is with straight fringes. When $p$ increases and $q = 4$, as shown from the 2–4 columns in Fig. 2, the intensity patterns of $x$-components gradually change into a set of open-upward parabolas. When $p = 4 $ and $q$ decreases to 0, as shown from the 4–6 columns in Fig. 2, the intensity patterns of $x$-components change from a set of parabolas into a concentric annulus structure. All the experimentally measured results are in good agreement with the above discussions. Based on the observation of Fig. 2, we can summarize three interesting special cases of the HP-VOFs as follows: (a) When $p = q $, the polarizations along the constant curves of $y=\frac{\pi q^{2}}{2\,t} x^{2}-\frac{t}{2 \pi q}$ keep the same, which are obvious parabolic curves. (b) When $p = 0 $, the polarizations along the constant curves of $y=-\frac{t}{\pi q}$ keep the same, which are straight lines. (c) When $q = 0 $, the polarizations along the constant curves of $r=\sqrt{x^{2}+y^{2}}=\frac{t}{\pi p}$ keep the same, which are circular rings. Obviously, the design of the HP-VOFs with the parabolic symmetry of the second kind can provide more degrees of freedom in modulating polarization states on the wavefront of the VOFs. It should be pointed out that the experimentally generated HP-VOF introduced above is transversely confined with square boundary due to the effective area of the SLM. In the experiment, the coordinates $(x, y)$ can be normalized as $(0.8 x / r_{m}, 0.8 y / r_{m})$, where $2 r_{m} = 2$ mm is the size of the square boundary of the optical field.
cpl-37-12-124201-fig3.png
Fig. 3. The intensity patterns of the focused HP-VOFs with the parabolic symmetry of the first kind when $n = 0 $, $m = 1,\, 2,\, 3,\, 4$ and 5 in five rows, respectively. The numerical aperture is ${\rm NA} = 0.1$, and the coordinates $(x, y)$ in the incident plane are normalized to $(x / r_{m}, y / r_{m}) $, where $r_{m} = f \cdot \mathrm{NA} /n_{0} $ is the radius of the incident optical field. The full widths of half maximum (FWHMs) of each focal spot in $y $ and $z $ directions are (0.52$\lambda /\mathrm{NA}$, 1.79$\lambda/\mathrm{NA}^2$), (0.53$\lambda/\mathrm{NA} $, 1.842$\lambda/\mathrm{NA}^2 $), (0.62$\lambda/\mathrm{NA} $, 1.978$\lambda/\mathrm{NA}^2 $), (0.86$\lambda/\mathrm{NA} $, 2.078$\lambda/\mathrm{NA}^2 $), (0.72$\lambda/\mathrm{NA} $, 2.105$\lambda/\mathrm{NA}^2 $) when $m = 1,\, 2,\, 3,\, 4$ and 5, respectively. Any picture has a size of $600 \lambda \times 50 \lambda$ with $\lambda $ being the wavelength.
As is well known, one of the most representative applications of vector optical fields is the flexibly controllable focal engineering, as the space-variant polarization distribution can always provide additional degree of freedom in controlling focal distribution. In the investigation of the focal properties of optical beams, focal shift has attracted interest for several decades.[23–30] Focal shift is defined as the phenomenon that the point of absolute maximum focal intensity does not coincide with the geometrical focus but shifts along optical axis,[28] and the more generalized definition can be considered as the focal spots move away from the geometrical focus. For the HP-VOFs with the parabolic symmetry we propose, there is obvious rule of the tunable focal shift of the focal fields, and the value of the focal shift can be flexibly controlled. Thus, the properties of the focal shift of the newly proposed HP-VOFs with the parabolic symmetry will be investigated. To calculate the focal shift, we use the Fresnel scalar diffraction theory in the following simulation.[31] The input HP-VOFs with the parabolic symmetry is with circular boundary due to the focal lens, and the maximum radius of the aperture is $r_{m} = f \cdot \mathrm{NA} /n_{0} $, where $f$ is the focal length of the objective, NA is the numerical aperture of the focal lens, and $n_{0} = 1 $ is the refractive index in air. It should also be pointed out that the following calculation results are also verified by the Richards–Wolf vectorial diffraction theory,[32,33] and the longitudinal component of the focused field is sufficiently small so that it can be neglected. For the HP-VOFs with the parabolic symmetry of the first kind, the focal field splits into two focal spots at different locations. Figure 3 shows the focal fields in $z$–$y $ plane when $n = 0$. In this case, we can see that when $m $ increases from 1 to 5, the two focal spots gradually move away from the original focus towards opposite directions in the $z$–$y $ plane, which means that the focal shift can be controlled flexibly with different values of $m $. The focal shifts of one focal spot away from the original center in $z $ and $y $ directions are $(148\lambda , 3.4\lambda)$, $(191 \lambda , 6.8 \lambda)$, $(226.7 \lambda , 9.8 \lambda)$, $(247.3 \lambda , 11.6 \lambda)$, $(279.9 \lambda , 13.2 \lambda)$ when $m = 1,\, 2,\, 3,\, 4$ and 5, respectively. The case of $m = 0 $ is similar to the case of $n = 0 $, because both $m $ and $n $ in front of the terms $u $ and $v $ have similar roles, which can be easily derived from Eqs. (1)-(3). To explain this focal shift property, we perform a simple theoretical analysis as follows. The HP-VOFs with the parabolic symmetry of the first kind can also be expressed as $$ \boldsymbol{E}(u, v)=e^{i(m \pi u+n \pi v)} \hat{\boldsymbol{e}}_{\pi / 4}+e^{-i(m \pi u+n \pi v)} \hat{\boldsymbol{e}}_{-\pi / 4},~~ \tag {5} $$ where $\{\hat{\boldsymbol{e}}_{\pi / 4}, \hat{\boldsymbol{e}}_{- \pi / 4}\}$ are the unit vectors in $\pm \pi / 4$ directions in Cartesian coordinates, in the relationship with $\{\hat{\boldsymbol{e}}_{x}, \hat{\boldsymbol{e}}_{y}\}$ as $\hat{\boldsymbol{e}}_{\pi / 4}=\frac{\sqrt{2}}{2}\left(\hat{\boldsymbol{e}}_{x}+\hat{\boldsymbol{e}}_{y}\right)$ and $\hat{\boldsymbol{e}}_{-\pi / 4}=\frac{\sqrt{2}}{2}\left(\hat{\boldsymbol{e}}_{x}-\hat{\boldsymbol{e}}_{y}\right)$, respectively. When $n = 0 $, $\boldsymbol{E}(u, v)=\exp (i m \pi \sqrt{r-y}) \hat{\boldsymbol{e}}_{\pi / 4}+\exp (-i m \pi \sqrt{r-y}) \hat{\boldsymbol{e}}_{-\pi / 4}$, which means that the two orders carry opposite phases that are functions of the coordinates $r = \sqrt{x^{2}+y^{2}}$ and $y $. It is obvious that the two opposite phases are the reason why the two focal spots move towards opposite directions. Moreover, the radial coordinate $r $ induces the focal shift in propagation direction ($z $ direction), while the $y $ coordinate induces the focal shift in $y $ direction. The above analysis demonstrates that two focal spots of the HP-VOF with the parabolic symmetry of the first kind exhibit the focal shifts in $z $ and $y $ directions when $m = 0 $ or $n = 0 $.
cpl-37-12-124201-fig4.png
Fig. 4. The curves of the focal shift $T_{z}$ in $z $ direction and $T_{y}$ in $y $ direction of the focused HP-VOFs with the parabolic symmetry of the first kind. The index $m $ changes from 0 to 10 and $n = 0 $, and the length scales of $T_{z}$ and $T_{y}$ are in units of wavelength.
In order to get insights into the influence of the indices $m $ or $n $ on the focal shift of the HP-VOFs with the parabolic symmetry of the first kind, we numerically simulate the focal shifts of the two focal spots with different indices $m $ (the case of different values of $n $ is totally the same). Figure 4 plots the curves of the focal shift $T_{z}$ in the $z $ direction and the focal shift $T_{y}$ in the $y $ direction as $m $ varies when $n = 0 $. It can be found that with the increasing value of $m $, both focal shifts $T_{z}$ and $T_{y}$ increase gradually and the variation tendency is similar. However, the curves of focal shift exist one exception of no focal shift when $m = 0.5 $. Actually, there are still two focal spots in this exception, but the moving distance of these two focal spots is so small that the two focal spots still overlap with each other. As a result, the maximum intensity of the superposed focal field occurs at the position of the original geometrical focus. In addition, the focal shift $T_{y}$ almost keeps the same when $m $ increases from 7 to 8. In fact, the focal shift also increases very slightly when $m $ increases from 7 to 8, which means that the focal shift $T_{y}$ keeps increasing when $m $ increases and $m > 0.5 $. For the HP-VOFs with the parabolic symmetry of the second kind, the modulation of the focal shift of the two focal spots is more flexible, as the focal shift in $y $ and $z $ directions can be controlled separately. Figure 5(a) shows the curve of the focal shift $T_{z}$ in $z $ direction as the index $p $ varies from 0 to 10 when $q = 5 $. Figure 5(b) shows the curve of the focal shift $T_{y}$ in $y $ direction as the index $q $ varies from 0 to 10 when $p = 5 $. It can be seen that the focal shifts in $z $ and $y $ directions possess approximately linear relation, providing a feasible and quantitative method for modulating focal shifts. In addition, the focal shift $T_{z}$ in the $z $ direction with varying $p $ can reach approximately $700 \lambda$, which is obviously larger than the focal shift $T_{y}$ in the $y $ direction with varying $q $. We have discussed the focal shift of the HP-VOFs with the parabolic symmetry of the second kind when $q = 5 $ or $p = 5 $ in Fig. 5, respectively. What we now concern is whether the value of $q $ (or $p $) affects the variation tendency of the focal shift with varying $p $ (or $q $). Figures 6(a) and 6(b) show the focal shift of the focused HP-VOFs with the parabolic symmetry of the second kind in $z $ and $y $ directions as both the indices $p $ and $q $ vary. All the values of the focal shift are normalized by the maximum value of the focal shift. It can be seen from Fig. 6(a) that the focal shift in $z $ direction is determined by the value of $p $ and is approximately irrelevant with the value of $q $. Meanwhile, the focal shift in $y $ direction is dependent on the value of $q $, which is approximately uncorrelated with the value of $p $, as shown in Fig. 6(b). Several values at special positions in Fig. 6 do not satisfy the above law, which is mainly due to the deviation in the simulation.
cpl-37-12-124201-fig5.png
Fig. 5. The curves of the focal shift of the focused HP-VOFs with the parabolic symmetry of the second kind with (a) the index $p $ of 0 to 10 when $q = 5 $ and (b) the index $q $ of 0 to 10 when $p = 5 $. The length scales of $T_{z}$ and $T_{y}$ are in units of wavelength.
cpl-37-12-124201-fig6.png
Fig. 6. The focal shift of the focused HP-VOFs with the parabolic symmetry of the second kind with different values of $p $ and $q $ changing from 0 to 10 in steps of 0.5: (a) the focal shift $T_{z}$ in $z $ direction; (b) the focal shift $T_{y}$ in $y $ direction. All the values of the focal shift are normalized by the maximum value of the focal shift, and the length scales of $T_{z}$ and $T_{y}$ are in units of wavelength.
One conclusion drawn from Fig. 6 is that the curves in Fig. 5 can represent the focal shift modulated separately by the indices $p $ and $q $ for the HP-VOFs with the parabolic symmetry of the second kind. As the focal shift can be modulated flexibly by the values of the indices $(p , q)$, the focal field of the HP-VOF with the parabolic symmetry of the second kind can be widely used in the realms needing flexible focal engineering, including the optical manipulation, optical machining and information transmission. It should be pointed out that, although the focal shift of the HP-VOF with the parabolic symmetry we introduce is in $z$ and $y$ directions, the actual focal shift can be modulated in longitudinal direction ($z$ direction) and arbitrary directions in the transverse plane, including $x$ and $y$ directions as special cases. This modulation can be realized by rewriting Eq. (4) into $$\begin{alignat}{1} \boldsymbol{E}_{2} (x, y)={}& \cos{ [ p \pi r-q \pi (x \cos{\varphi_0} + y \cos{\varphi_0})} ] \hat{\boldsymbol{e}}_{x} \\ &+ i \sin{ [ p \pi r-q \pi (x \cos{\varphi_0} + y \cos{\varphi_0})} ] \hat{\boldsymbol{e}}_{y},~~ \tag {6} \end{alignat} $$ where $r $ is the radial coordinate in the incident plane, $\varphi_0 $ is the azimuthal angle of focal spots located in the transverse plane, and the two cases of $\varphi_0 = 0 $ and $\varphi_0 = \pi/2 $ correspond to the cases in which the focal shifts are in $x$ and $y$ directions, respectively. In this way, we can achieve the arbitrary tunable modulation of the focal shift with this new kind of HP-VOFs, as the two focal spots can be set at any location. In conclusion, we have theoretically and experimentally presented hybridly polarized vector optical fields with parabolic symmetry of the first and second kinds, and further discussed the tunable focal shift of the focused HP-VOFs. For the HP-VOFs with the parabolic symmetry of the first kind, the focal field splits into two focal spots, and the two focal spots move toward opposite directions in $z$–$y$ plane. For the HP-VOFs with the parabolic symmetry of the second kind, the focal shift of the two focal spots can be flexibly controlled and modulated separately by the indices $p $ and $q $. We hope the controllable focal shift of the newly proposed HP-VOFs with the parabolic symmetry can bring variable applications in realms needing focal engineering, such as optical manipulation, optical machining and information transmission. References Cylindrical vector beams: from mathematical concepts to applicationsTrapping metallic Rayleigh particles with radial polarizationOptical orbital angular momentum from the curl of polarizationControlled negative energy flow in the focus of a radial polarized optical beamRevealing Local Field Structure of Focused Ultrashort PulsesLongitudinal Field Modes Probed by Single MoleculesNear-Field Second-Harmonic Generation Induced by Local Field EnhancementTaming the Collapse of Optical FieldsRemote Preparation of Single-Photon “Hybrid” Entangled and Vector-Polarization StatesAzimuthally Polarized Spatial Dark Solitons: Exact Solutions of Maxwell's Equations in a Kerr MediumHarnessing the Point-Spread Function for High-Resolution Far-Field Optical MicroscopyTightly autofocusing beams: an effective enhancement of longitudinally polarized fieldsGeneration and tight focusing of hybridly polarized vector beamsA new type of vector fields with hybrid states of polarizationFull Poincaré beamsHigher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of LightPolarization evolution characteristics of focused hybridly polarized vector fieldsSpatial-Variant Geometric Phase of Hybrid-Polarized Vector Optical FieldsParabolic nondiffracting optical wave fieldsNonparaxial Mathieu and Weber Accelerating BeamsParabolic-symmetry vector optical fields and their tightly focusing propertiesGeneration of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangementFocal shift in Laguerre–Gaussian beamsFocal shift in vector beamsFocal shifts in focused nonuniformly polarized beamsFocal shift in vector Mathieu-Gauss beamsFocal Shift of Paraxial Gaussian Beams in a Left-Handed Material Slab LensFocal shift in radially polarized hollow Gaussian beamFocal shift in tightly focused Laguerre–Gaussian beamsOptical twists and transverse focal shift in a strongly focused, circularly polarized vortex fieldElectromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic systemFocusing of high numerical aperture cylindrical-vector beams
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