1. Introduction

In 1959, Wolf and Richard derived the integral propagation formulas of electric and magnetic fields for linearly polarized light under the tight focusing condition based on the Debye approximation. They pointed out that the beams’ polarization state and intensity will be significantly changed [1]. Since then, more and more work has shown that by modulating the polarization, phase, amplitude, or coherence of tightly focused beams, complex and versatile structured electric field distributions can be achieved [2–7], which have critical applications in optical data storage, microscopy, particle beam trapping, and material processing [8–12]. For example, it is well known that when a radially polarized beam is focused by a high numerical aperture (NA), it creates a strong longitudinal, non-propagating electric field component near the focus and can break the diffraction limit [13,14]. The strong longitudinal electric field has essential applications in microscopy, particle trapping, and material processing. For instance, in 2022, Li et al. achieved materials processing resolution down to 10 nm (${\lambda }/{80}\;$) at an infrared laser wavelength of around 800 nm in the far field, in air, by creating a high purity longitudinal electric field at focus, well beyond the optical diffraction limit [15]. Moreover, it is also well known that when an azimuthally polarized light beam is focused by a high numerical aperture (NA), it creates a hollow pure transverse electric field and a strong solid longitudinal non-propagating magnetic field component near the focus [14]. Furthermore, the strong solid longitudinal magnetic field component can enhance the magnetic dipole absorption rate in semiconductor quantum dots [16,17]. Since the intensity of magnetic field is very small than the electric field, few people pay attention to the properties of the magnetic field. However, a recent research which directly detects photoinduced magnetic force in the nanoscale size indicated that the magnetic field can play a significant role in the case of azimuthal polarization [18].

The mutation is a primary natural phenomenon, and there has been a tremendous development of interest in the area now known as catastrophe theory in mathematics since the first studies in the mid-1960s of Rene Thom’s Stabilite Structurelle et Morphogenesis, which finally appeared in 1974 [19].An essential application of catastrophe theory in optics is that all possible forms of the caustics of light can be found with the help of catastrophe theory [20]. With the help of catastrophe theory, the caustics of the propagation-invariant Airy beam [21], Bessel beam [22,23], and Mathieu beam [24] are studied, and by extending the "fold catastrophe" in the Airy beam a general method for designing beams with arbitrary acceleration trajectories was proposed [25], by extending the integer vortex topological charge in the Gaussian Bessel beam to fraction, fractional order vortex TE and TM vector beams with far-field diffraction symmetry are proposed [26], by using Bessel beams as a pencil a family of propagation-invariant caustic optical beams with arbitrary shape is proposed [27]. Moreover, it was well known that when disturbed by obstacles, some well-designed caustic optical beams showed good self-healing property [28,29]. However, the most direct application of catastrophe theory in optics is constructing special structured beams (catastrophe beams). For example, the complex amplitude function of the Airy beam, known for its non-diffraction, self-healing, and auto-focusing properties, is the most simple function of the four univariate catastrophe functions, and related research has shown that the catastrophe beams constructed by the remaining three univariate catastrophe functions also exhibit unique properties, which are well known as the Pearcey beams, Swallowtail beams, and Butterfly beams [30–35]. While four univariate catastrophe functions have been extensively studied in optics, research on three bivariate higher-order catastrophe functions is still scarce. In 2022, Zhang et al. investigated the hyperbolic umbilic catastrophe beams and demonstrated its characteristics of multi-focus autofocusing. This finding suggests that higher-order catastrophe functions with dual integral variables are worthy of careful study in optics [36].

Since the studies on bivariate higher-order catastrophe functions are still scarce, and more attention needs to be paid to the magnetic field characteristics when an azimuthally polarized beam passes through a high NA objective, in this article, we demonstrate in detail the propagation dynamics of the electric and magnetic fields of the azimuthally polarized vortex circular hyperbolic umbilic beam (APVCHUB) passing through a high NA objective. These unique characteristics of the electric and magnetic field suggest that the APVCHUBs can be used in magnetic particle trapping, semiconductor quantum dot excitation, super-resolution microscopy imaging, weak magnetic measurement, etc.

2. Theory

The electric field of an azimuthally polarized circular hyperbolic umbilic beams with $m$-order concentric vortex can be expressed as [37,38]:

Moreover, from the Maxwell equations, the magnetic field can be expressed as [6]:

3. Simulation

When the APVCHUB is passing through a high NA objective, we are interested in the focusing characteristics of the electric and magnetic fields, the intensity distribution, the phase distribution (vortex topological charge) and the polarization state distribution of the vector field at the foci, so next, we will focus our research on APVCHUB in these aspects. In our simulation, we modify the value of $A$ to adjust the optical power to $1\text {mW}$ while keeping the scaling factor ${{w}_{0}}=0.1\text {mm}$, the wavelength $\lambda =512\text {nm}$, ${{r}_{m}}={r}_{lens}=4\text {mm}$, $n=1$ constant, and if there is no special declaration, the other parameter settings are as follows:$\xi =12$, ${{\text { }\!\!\varphi \!\!\text { }}_{\text {intp}}}=0.002\pi$, ${{\text {r}}_{\text {1}}}=112{{w}_{0}}$, $\text {NA}=0.95$.

Define ${\text {I}\scriptsize {{\text {MZ}}}_{\text {E}}}=\max \left ( {{\left | E\left ( z \right ) \right |}^{2}} \right )$ and ${\text {I}\scriptsize {{\text {MZ}}}_{\text {H}}}=\max \left ( {{\left | H\left ( z \right ) \right |}^{2}} \right )$ as the maximum total electric field intensity and maximum total magnetic field intensity at the propagation distance-$z$, respectively. The effect of NA on the focusing property of the APVCHUB was first studied. Figs. 1(a)-(b) show the variation of ${\text {I}\scriptsize {{\text {MZ}}}_{\text {E}}}$ and ${\text {I}\scriptsize {{\text {MZ}}}_{\text {H}}}$ of APVCHUB with different $\text {NA}$ when $m=0$. From Figs. 1(a)-(b), it can be seen that both the electric and magnetic fields of APVCHUB focus multiple times before the focus of the objective. As $\text {NA}$ decreases, the distance between the foci of the APVCHUB and the focus of the objective increases; the relative intensity between different foci of the electric and magnetic fields of APVCHUB also changes (the intensities of foci farther away from the focus of the objective significantly increased); unlike the electric field, the intensities of the foci of the magnetic field near the focus of the objective show a significantly decrease with the decrease of $\text {NA}$. The distance between the foci of the APVCHUB and the focus of the objective increases with the decrease of NA can be explained by the principle of the combined lens. Because the beam has the auto-focusing property, it can be regarded as a lens with a focal length of ${{f}_{light}}$. So the beam’s focus after passing through the objective can be regarded as the focus of two convex lenses combined, and the combined focal length is equal to ${{f}_{c}}={1}/{\left ( {1}/{{{f}_{ligth}}}\;+{1}/{{{f}_{lens}}}\; \right )}$. Then, the distance between the combined focal length and the focal length of the objective lens is equal to ${{f}_{lens}}-{{f}_{c}}={{{f}_{lens}}^{2}}/{\left ( {{f}_{ligth}}+{{f}_{lens}} \right )}$. It can be seen that when the focal length of the objective lens increases, the distance between the two focal lengths increases, so when the $\text {NA}$ of the objective decrease, the foci of APVCHUB are far away from the focus of the objective. The magnetic field focusing intensity significantly decreases with the decrease of NA can be explained as follows: the magnetic field contains both transverse and longitudinal components, and the longitudinal component is much larger than the transverse component (which will be shown in Figs. 3), while the intensity of the longitudinal component will significantly decrease with the decrease of NA [13–15], so the focusing intensity of the total magnetic field decreases significantly with reducing NA. Subsequently, we investigated the vortex phase’s unique effect on the APVCHUB’s multiple focusing characteristics. Figs. 1(c)-(d) show the variation of ${\text {I}\scriptsize {{\text {MZ}}}_{\text {E}}}$ and ${\text {I}\scriptsize {{\text {MZ}}}_{\text {H}}}$ of APVCHUB with different $m$ when $\text {NA}=0.95$. From Figure 1(c)-(d) it can be seen that with the increasing of $m$, the focusing intensity of electric of APVCHUB first increases then decreases; however, the focusing intensity of the total magnetic field decreases monotonously.

 

Fig. 1. Maximum intensity of the APCHPVBs with different $\text {NA}$ and topological charge $m$ as functions of propagation distance $z$( unit:1$\lambda$). (a) Maximum intensity of total electric field with $m=0$ for different $\text {NA}$; (b) Maximum intensity of total magnetic field with $m=0$ for different $\text {NA}$. (c) Maximum intensity of total electric field with $\text {NA}=0.95$ for different $m$; (d) Maximum intensity of total magnetic field with $\text {NA}=0.95$ for different $m$. The focal length of objective lens with different NA: $\text {NA}=0.95\left ( {{f}_{lens}}=4.2\text {mm} \right ),\text {NA}=0.85\left ( {{f}_{lens}}=4.7\text {mm} \right ),\text {NA}=0.75\left ( {{f}_{lens}}=5.3\text {mm} \right )$. The unit of y-axis in (a) and (c) is ${{10}^{15}}{{\left | {V}/{m}\; \right |}^{2}}$ and the unit of y-axis in (b) and (d) is ${{10}^{10}}{{\left | {A}/{m}\; \right |}^{2}}$

Download Full Size | PDF

 

Fig. 2. Propagation of electric field of a APVCHUB when $m=0$. (a) side view of the total electric field intensity in xz plane; (b) autofocusing capability as a function of propagation distance $z$($\lambda$)(the unit of y-axis is 1); (c1)-(c4),(d1)-(d4) cross-sectional intensity distributions of x and y components of the electric field in the four focal planes with maximum intensity marked 1-4 in (b); The unit of colorbar in each picture is ${{\left | {V}/{m}\; \right |}^{2}}$.

Download Full Size | PDF

 

Fig. 3. Propagation of magnetic field of a APVCHUB when $m=0$. (a) side view of the total magnetic field intensity in xz plane; (b) autofocusing capability as a function of propagation distance $z(\lambda )$(the unit of y-axis is 1); (c1)-(c4),(d1)-(d4) cross-sectional intensity distributions of x and y components of the magnetic field in the four focal planes with maximum intensity marked 1-4 in (b); The unit of colorbar in each picture is ${{\left | {A}/{m}\; \right |}^{2}}$.

Download Full Size | PDF

The reason of electric field’s focusing ability is strongest when $m=1$ is explained as follows: first, the azimuthally polarized electric field only has the transverse component; second, when $m=1$, the central polarization singularity and the vortex phase singularity of the transverse electric field will annihilate, making a solid bright spot in the center. So the focusing intensity of the electric field with $m=1$ is strongest. These explanations can be proved by Figs. 4 and Figs. 5. The total intensity of the magnetic field decreases monotonously with the increase of $m$ because of the following reason: the magnetic field is composed of the transverse magnetic field and the longitudinal magnetic field, and the longitudinal magnetic field only has the vortex phase singularity, the transverse magnetic field has both phase singularity and polarization singularity; the intensity of the longitudinal magnetic field is much larger than the transverse magnetic field, so when $m=1$, the intensity of the transverse magnetic field increases due to the disappearance of the central singularity is lower than that of the longitudinal magnetic field decreasing due to the increase in the number of vortex topological charges. So the intensity of the total magnetic field decreases monotonously with increasing the vortex topological charge. These explanations can also be proved by Figs. 4 and Figs. 5. So, the modulation of the focusing abilities of the electric field and magnetic fields by the mutual induction of the vortex phase and azimuthal polarization is different.

 

Fig. 4. Electric field and magnetic field’s intensity distributions of APVCHUBs at the fourth focus plane marked 4 in Fig. 2(b) with different topological charge $m$. The unit of colorbar in (a1-a4) is ${{\left | {V}/{m}\; \right |}^{2}}$ and the unit of colorbar in (b1-b4,c1-c4) is ${{\left | {A}/{m}\; \right |}^{2}}$.

Download Full Size | PDF

 

Fig. 5. Phase patterns of electric field and magnetic field of APVCHUBs at the fourth focus plane marked 4 in Fig. 2(b) for different topological charge m. The unit of the x-axis and y-axis in each picture is 1 $\lambda$.

Download Full Size | PDF

The variation of the electric field of APVCHUB with $m=0$ versus propagation $z$ is shown in Fig. 2. Figure 2(a) depicts the side-view intensity of the total electric field in the $x-o-z$ plane $(y = 0)$. Figure 2(b) plots the normalized maxmum intensity of the total electric field ${{\eta }_{E}}\left ( z \right )={\max \left ( {{\left | E\left ( \text {z} \right ) \right |}^{2}} \right )}/{\max \left ( {{\left | E \right |}^{2}} \right )}\;$. Figure 2(a) and Fig. 2(b) show that after passing through the high NA objective, multiple hollow foci are generated in front of the objective’s focus. Since the beam is azimuthally polarized, the electric field only has the $x$-vibration component $Ex$, $y$-vibration component $Ey$, and the longitudinal component $Ez$ is 0. Then, the $x$-vibration and $y$-vibration components of the focused electric field of APVCHUB in the four focal planes marked in Fig. 2(b) are shown in Figs. 2(c1-c4) and Figs. 2(d1-d4) respectively. Figures 2(c1-c4) and Figs. 2(d1-d4) show that the electric field is formed symmetrically by two strong inner lobes and several weak outer lobes. The axis of symmetry of foci is the vibration direction because the electric field’s vibration direction is still the azimuth angle direction, which will be shown in Figs. 6.

 

Fig. 6. Vibration state of transverse electric field and magnetic field of APVCHUBs at the fourth focus marked 4 in Fig. 2(b) for different topological charge $m$. The intensity profile underneath each vibration pattern. The red line denotes linear vibration. The blue (or yellow) ellipsoid denotes left- (or right-) handed elliptical vibration. The unit of the x-axis and y-axis in each picture is 1 $\lambda$. The unit of colorbar in (a1-a4) is ${{\left | {V}/{m}\; \right |}^{2}}$ and the unit of colorbar in (b1-b4) is ${{\left | {A}/{m}\; \right |}^{2}}$.

Download Full Size | PDF

Next, the propagation dynamics of the magnetic field of APVCHUB are displayed. Figure 3(a) depicts the $x-o-z$ plane cross-section diagram of the total magnetic field intensity. Figure 3(b) depicts the normalized maximum intensity diagram of the total electric and magnetic fields (${{\eta }_{H}}\left ( z \right )=\max \left ( {{\left | H\left ( \text {z} \right ) \right |}^{2}} \right )/\max \left ( {{\left | H \right |}^{2}} \right )$). From Fig. 3(a) and Fig. 3(b), it can be seen that the APVCHUB’s magnetic field also focuses multiple times, and the focal lengths are consistent with the electric field. However, unlike the electric field, the magnetic field appears as a solid bright spot. Similarly, the transverse and longitudinal magnetic field components of the four focal spots marked in Fig. 3(b) were plotted in Figs. 3(c1-c4), Figs. 3(d1-d4), and Figs. 3(e1-e4), respectively. From Figs. 3(c1-c4) and Figs. 3(d1-d4), it can be seen that similar to the transverse electric field, the transverse magnetic field $Hx$ and $Hy$ are also composed of symmetrical lobes, and the central intensity is 0. However, the symmetry axis of the transverse magnetic field is perpendicular to the vibration direction of the magnetic field because the transverse magnetic field is radially vibrated, which will be shown in Figs. 6. From Figs. 3(e1-e4), it can be seen that the longitudinal magnetic field is a solid bright spot. Comparing Fig. 3(e4) and Fig. 3(d4), it can be seen that the longitudinal magnetic field intensity is about 13 times that of the transverse magnetic field intensity. If we define the ratio of the longitudinal magnetic field’s energy to the total magnetic field’s energy as purity(${{p}_{q}}=\frac {\int {{{\left | {{H}_{z}} \right |}^{2}}dxdy}}{\int {{{\left | {{H}_{x}} \right |}^{2}}+{{\left | {{H}_{y}} \right |}^{2}}+{{\left | {{H}_{z}} \right |}^{2}}dxdy}}$), then through caculation, we find the purity of the longitudinal magnetic field at that focus is $80{\% }$ and it will not change with the vortex topological charge. Since the center of the electric field intensity is 0 when m=0 and the primary energy of the magnetic field is concentrated in a small area in the center($70{\% }-0.5{{\lambda }^{2}}$), the gradient force of the magnetic field at the foci is very large. Hence the APVCHUB may have particular applications in magnetic particle capture [40,41], weak magnetic measurement [18], and quantum dot excitation [16,17].

Due to the transverse electric field and magnetic field of APVCHUB possessing good symmetry, the $x$-component transverse field is rotated by 90 degrees to be the $y$-component transverse field, and the electric and magnetic fields at different focal points are similar in shape. So next, when we study the modulation of the vortex topological charge $m$ on the intensity distribution of the APVCHUB’s foci, only the $Ex$, $Hx$, and $Hz$ component of the fourth focus marked in Fig. 3(b) are plotted. Figures 4 show the intensity distribution of the electric and magnetic fields of APVCHUB with different $m$. From Fig. 4(a2), it can be seen that when $m$=1, the center of the beam becomes a solid bright spot, and around the central solid bright spot, symmetrically distributed two lobes, which indicates that the phase singularity and polarization singularity in the center of the beam are annihilated, and it also explains the reason why the focusing intensity of the beam is enhanced. From Fig. 4(a3), it can be seen that when $m=2$, the intensity of the center of the electric field is zero again, showing a dark notch, and the electric field consists of two main lobes perpendicular to the polarization direction and multiple secondary lobes parallel to the polarization direction. From Fig. 4(a4), it can be seen that when the vortex topological charge increases to 3, the shape of the electric field is similar to that of $m=2$, but the dark notch in the center will be larger. Comparing Figs. 4(a1-a4) and Figs. 4(b1-b4), it can be seen that the shape of the transverse magnetic field is similar to that of the electric field under different vortex topological charges, except that the direction is rotated by 90 degrees, and the energy on the secondary lobes is more. From Figs. 4(c1-c4), it can be seen that the longitudinal magnetic field appears as a hollow ring when the vortex $m>0$, there is a dark notch in the center, and the size of the dark core increases with the vortex topological charge $m$ increasing. As demonstrated above, the combined induction of vortex topological charge, azimuthal polarization and tight focusing conditions achieved unique modulation on the distribution of transverse and longitudinal electric and magnetic fields of the APVCHUBs.

In order to explore the changes of the vortex topological charge of foci, we draw the phase diagram of the electric field and magnetic field of the fourth focal spot corresponding to Figs. 4. From Figs. 5(a1-a4) and Figs. 5(b1-b4), it can be seen that when the vortex topology $m=0$, there is a polarization singularity in the center of the transverse electric field and the transverse magnetic field; when $m=1$, the central phase singularity disappears; when $m\ge 2$, the central phase singularity appears, and the modulus of total topological charge of the transverse electric field and the magnetic field is $\left | m-1 \right |$. From Figs. 5(c1-c4), it can be seen that in the center of the longitudinal magnetic field, the vortex topological charge of the phase singularity is equal to $m$; this also explains the different adjustment characteristics of the vortex topological charges on the focusing ability of the electric and magnetic fields shown in Fig. 1.

The mutual induction of the vortex phase and azimuthal polarization will change the polarization state of the foci of APVCHUB. Figure 6 show the vibration elliptic state of the transverse electric and transverse magnetic fields. From Figs. 6(a1-a4) and Figs. 6(b1-b4), it can be seen that when $m=0$, the electric field maintains azimuthal polarization, and the magnetic field maintains radial vibration; when $m>0$, the polarization state of the transverse field varies with azimuth, and this explains the unique shape of the transverse focus field, i.e., the shape of the main lobes and the secondary lobes. In addition, it should be pointed out that the orientation angle about the vibration ellipse of the transverse magnetic field is perpendicular to the orientation angle about the polarization ellipse of the transverse electric field, which explains that the transverse magnetic pattern shown in Fig. 4 is similar to the transverse electric field strength but rotated by 90 degrees.

Finally, we investigate the effect of parameters on the beam’s ability to focus multiple times. It can be seen from Fig. 7(a) that with the increase of $\xi$, the maximum focus intensity remains the same, but the number of beam’s focus increases. It can be seen from Fig. 7(b) that with the increase of ${{r}_{1}}$, the maximum focus intensity first increases and then decreases, and the number of focus increases slowly and tends to be stable. From Fig. 7(c), it can be seen that with the change of ${{\text { }\!\!\varphi \!\!\text { }}_{\text {intp}}}$, the beam can be focused from the back to the front of the len’s focus, from multiple foci to a single strong focus, and then to multiple foci. The multiple focusing properties of the beam can be flexibly adjusted by adjusting the parameters.

 

Fig. 7. Maximum intensity of magnetic field of the APVCHUBs normalizated by the maxium intensity in the source plane(the maximum magentic intensity of the input APVCHUB in object space) with different combinations of parameters as functions of propagation distance $z$ when $m=0$. (a) with different $\xi$ when ${{\varphi }_{\text {intp}}}=0.002\pi$, ${{r}_{1}}=112{{w}_{0}}$; (b) with different ${{r}_{1}}$ when ${{\varphi }_{\text {intp}}}=0.002\pi$, $\xi =12$; (c) with different ${{\varphi }_{\text {intp}}}$ when $\xi =12$, ${{r}_{1}}=112{{w}_{0}}$. The unit of the z-axis in each picture is ${10}^{9}$.

Download Full Size | PDF

Finally, to intuitively feel the influence of the parameters on the focus of the beam, two cross-section diagrams of the total magnetic field of APVCHUB with different parameters are drawn. From Fig. 8, it can be seen that by changing the $\xi$, the beam’s magnetic field can be changed from a continuous long magnetic needle to a segmented long magnetic needle.

 

Fig. 8. Side view of magnetic field intensity in x-z plane; (a1) $\xi =3$, ${{\varphi }_{\text {intp}}}=0.002\pi$, ${{r}_{1}}=112{{w}_{0}}$; (a2) $\xi =6$, ${{\varphi }_{\text {intp}}}=0.002\pi$, ${{r}_{1}}=112{{w}_{0}}$. The unit of the z-axis in each picture is ${{10}^{9}}{{\left | {A}/{m}\; \right |}^{2}}$.

Download Full Size | PDF

4. Conclusion

Using the Richards-Wolf theory and Maxwell equations, we study the propagation dynamics of the APVCHBs through a high NA objective. First, APVCHB exhibits the multi-focusing characteristic. Second, the vortex phase shows a unique modulation characteristic of the multi-focusing ability of APVCHB; the transverse magnetic and electric fields increase first and then decrease with the increase of vortex topological charge, but the longitudinal magnetic field decreases monotonically as the vortex topological charge increases. Third, the longitudinal magnetic field of APVCHUB is much larger than the transverse magnetic field; under appropriate parameters, the purity of the longitudinal magnetic field of the beam can reach $80{\% }$, and the purity will not change with the change of the vortex topological charge. Fourth, the vortex topological charge will induce the elliptic polarization state of the transverse magnetic field and electric field to change, as well as their shapes; moreover, the polarization states of the transverse electric field and the transverse magnetic field are orthogonal. Fifth, the modulus of the total topological charge of the transverse magnetic field and the transverse electric field is equal to $\left | m-1 \right |$, the longitudinal magnetic field’s topological charge is $m$. Finally, the multi-focusing characteristics of the APVCHUBs can be flexibly adjusted by modulating the parameters. Considering these properties, the APVCHUB may have promising applications in magnetic particle manipulation, extremely weak magnetic detection, data storage, semiconductor quantum dot excitation, etc.