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Abstract
Magneto-optical effects, which have been known for over a century, are among the most fundamental phenomena in physics and describe changes in the polarization state of light when it interacts with magnetic materials. When a polarized plane wave propagates in or through a homogeneous and isotropic transparent medium, it is generally accepted that its transverse polarization structure remains unchanged. However, we show that a strong radial polarization component can be generated when an azimuthally polarized sine-Gaussian plane wave is tightly focused by a high numerical aperture lens, resulting in a magneto-optical-like effect that does not require external magnetic field or magnetic medium. Calculations show that the intensity structure and polarization distribution of the highly confined electric field strongly depend on the parameters m and φ0 in the sinusoidal term, where m can be used to control the number of the multifocal spots and φ0 can be used to control the position of each focal spot. Finally, we show that this peculiar electric field distribution can be used to realize multiple particles trapping with controllable numbers and locations.
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1. Introduction
Magneto-optical effects, which refer to changes in the polarization state of light when it interacts with magnetic matter, are among the most fundamental phenomena in physics and have been known for more than a century. In 1846, Faraday discovered the first magneto-optical phenomenon in which the plane of linearly polarized light is rotated as is passes through a piece of glass exposed to an external magnetic field [1]. It was some thirty years later, in 1877, that Kerr discovered a similar effect in reflected light [2]. The interaction between band exchange splitting and spin-orbit coupling is thought to be the microscopic origin of the magneto-optical effect. As an important consequence of the Zeeman effect, band exchange splitting is induced by an external magnetic field or by spontaneous magnetization of the magnetic material; spin-orbit coupling further splits the bands and couples the orbital motion of the spin-polarized electrons to the incident polarized light; the simultaneous presence of band exchange splitting and spin-orbit coupling leads to different responses of the magnetic medium to left- and right-handed polarized light, manifested in the magneto-optical Faraday effect and the Kerr effect [3,4,5,6,7,8]. After much research, they have evolved into useful and attractive spectroscopic tools for measuring ferromagnetism in two-dimensional (2D) systems, identifying and modifying magnetic sequences, and visualizing magnetic domains [9–11].
Polarization, as an intrinsic optical degree of freedom, is one of the most remarkable properties of light. In two dimensions, paraxial light is vectorial, with an inhomogeneous polarization structure throughout the transverse plane. The polarization state of a non-paraxial light beam is also vectorial in three dimensions, giving rise to novel states of structured light such as optical needles [12–14], Möbius strips [15–20], polarization knots [21–23], optical skyrmions [24–26], and transverse spin density [27–35]. Due to the importance of these structured light fields, their propagation in various media has attracted much interest [36–43]. There is strong evidence for the stability of the vectorial structure as well as strong data contradicting it. Generalizations about the robustness of vectorial light in arbitrarily complex media are not possible due to the particular nature of each study. However, it is generally believed that the transverse polarization structure of a polarized plane wave remains unchanged when it propagates paraxially in or through a homogeneous and isotropic transparent medium [36]. Under non-paraxial conditions, such as tight focusing through a high numerical aperture (NA) lens, a longitudinally polarized component may appear, but the transverse polarization structure is still preserved. For example, a radially polarized beam can be focused to produce a strong longitudinal and non-propagating electric field, and the transverse polarization will be completely radially polarized [44–47]. An azimuthally polarized beam, on the other hand, can be focused into a hollow dark spot with full azimuthal polarization [44–47]. To the best of our knowledge, there is an absence of the magneto-optical-like effect for plane waves propagating in a homogeneous and isotropic transparent medium.
In this Letter, we show that when an azimuthally polarized sine-Gaussian (APSG) plane wave is tightly focused by a high NA lens, a strong radial polarization component can indeed be generated in a homogeneous and isotropic transparent medium, leading to a magneto-optical-like effect that does not require external magnetic field or magnetic medium. A new type of structured optical field, the APSG beam, is proposed. We then present the explicit expressions for calculating the strength vector and energy flux of the electric and magnetic fields in image space using the vectorial diffraction methods of Richards and Wolf. Calculations show that the intensity structure and polarization distribution of the highly confined electric field are found to depend strongly on the parameters m and φ0 in the sinusoidal term, where m can be used to control the number of the multifocal spots and φ0 can be used to control the position of each focal spot. To gain a better understanding of the magneto-optical-like effect here, the energy fluxes of such highly confined fields are also studied in detail. Lastly, we demonstrate how multiple particles trapping with adjustable numbers and positions may be achieved using this unusual electric field distribution.
2. Theoretical model
The famous cylindrical vector beam has been well studied and found to be useful in various applications [48–51]. Mathematically, its electric field can be described as [52]
A highly confined electromagnetic field, such as that produced by a high NA objective lens focusing system, is extremely desirable in a wide range of applications, from microimaging to optical tweezers and high-density storage. When the input field is represented by Eq. (2), the electric and magnetic fields at an arbitrary observation point P(ρ, ϕ, z) in the focal volume of a high NA objective lens can be calculated using Richards and Wolf’s vector diffraction theory [53]
Here, $\textrm{B} = \sqrt {\varepsilon /\mu } $, where ε and µ are, respectively, the electric permittivity and magnetic permeability. er is a unit vector along the radial direction. Clearly, the radial polarization component in the electric field and the azimuthal polarization component in the magnetic field emerge, as can be seen in Eq. (4), which is very different from the traditional azimuthally polarized input field [44,54]. In terms of the 3D electric and magnetic fields expressed above, the energy flux can be obtained from a determination of the time-averaged Poynting vector [53–58]
where the asterisk represents complex conjugation. Based on the above equations, we can now study the focal behavior of the new type of structured optical field. All length measurements are in unit of wavelength, therefor, $\lambda = 1$; $\textrm{NA} = 1.26$, $n = 1.33$, $\beta = 1$, and $\textrm{A} = 1$ are used in the following calculations.3. Results and discussions
Now, we calculate the electric field intensity distributions in the focal plane of the proposed APSG beams under the above focusing conditions. Figure 1 shows the radial, azimuthal, and total electric field intensity distributions with $m = 1$, 3, 5, and 7 when ${\varphi _0} = \pi /2$. For $m = 1$ and ${\varphi _0} = \pi /2$, the input field is the traditional azimuthally polarized beam, so we can see that no radial polarization component can be found [Fig. 1(a)], the total field [Fig. 1(c)] is completely contributed by the azimuthal polarization component [Fig. 1(b)]. For other values of m, however, the situation is reversed [Figs. 1(d)-1(l)]. The radial polarization component becomes much stronger compared to the azimuthal polarization component and plays the dominant role in the total fields, very different from the traditional azimuthally polarized beam. Thus, a strong polarization rotation occurs between the field in the input plane and the focal plane, similar to the magneto-optical effect observed in the transmission or reflection of light when it interacts with magnetic matter. But no magnetic matter is needed here. Furthermore, with increasing m, the azimuthal polarization component becomes weaker and weaker and can even be neglected at $m = 7$ [Fig. 1(k)], so that the magneto-optical-like effect becomes more pronounced for large values of m. It should be emphasized, however, that the change in m also leads to a change in the intensity structure of the light field. For $m = 1$, they are ring-shaped patterns for the azimuthal and total fields. For other values of m, however, they are multifocal spot distributions for all the radial, azimuthal and total fields. The number of the multifocal spots is controllable and depends on m, which is equal to $2|{m - 1} |$. The input beam with $m = 5$ and ${\varphi _0} = \pi /2$ is taken as an example to study the propagation properties of the APSG beam, shown in Fig. 2. Clearly, the radial component is always stronger and dominates the total field at different propagation locations, so that the magneto-optical effect found here can be maintained during propagation. The results found here are quite different from those reported previously [54], because the polarization of the incident beam together with its amplitude distribution determines the intensity distribution in the focal volume.
Fig. 1. Normalized electric field intensity distributions in the focal plane of four types of APSG beams with $m = 1$ (first row), 3 (second row), 5 (third row), and 7 (fourth row) when ${\varphi _0} = \pi /2$. The columns from left to right correspond to the radial component, azimuthal component and their total.
Fig. 2. Normalized electric field intensity distributions in the x-y planes at $z ={-} 0.9\lambda $ (first column), −0.6λ (second column), -0.3λ (third column), 0 (fourth column), 0.3λ (fifth column), 0.6λ (sixth column) and 0.9λ (seventh column) of a type of APSG beam with $m = 5$ and ${\varphi _0} = \pi /2$. The rows from top to bottom correspond to the radial component, the azimuthal component and their total. All the intensity distributions are normalized to the maximum of the total intensity distributions in the x-y plane at $z = 0$.
The polarization distributions of the four types of APSG beams shown in Fig. 1 are calculated as shown in Fig. 3. It is clear that this is an on-axis polarization vortex for $m = 1$, so no radial polarization component can be found. For other values of m, they exhibit off-axis multiple polarization vortex distributions, the number of which coincides with the number of multiple foci. These distinct polarization distributions are expected to have novel and distinctive applications in light-matter interactions. Figure 4 shows the normalized longitudinal Poynting vector distributions in the focal plane for the beams shown in Figs. 1 and 3. Calculations show that there is no transverse energy flow, the Poynting vectors are all along the longitudinal direction for the four input fields, resulting from the fact that the local polarization is linearly polarized and the wave front is a plane wave. Overall, the distribution of the longitudinal energy flow for all four input fields is similar to the distribution of their total electric field intensity.
Fig. 3. Polarization distributions in the focal plane of four types of APSG input fields identical to those shown in Fig. 1.
Fig. 4. Longitudinal Poynting vectors distributions in the focal plane of four types of APSG input fields identical to those shown in Fig. 1.
All of the above studies focus on the condition that ${\varphi _0} = \pi /2$. We now study the effect of the parameter ${\varphi _0}$ on the intensity, polarization and energy flow distributions. When m is equal to 1, the input field is always the traditional azimuthally polarized beam. The parameter ${\varphi _0}$ therefore only affects the magnitude of the focused electric field intensity distribution and does not change its appearance and shape. In addition, the polarization and longitudinal energy flow distributions do not change either. For other values of m, calculations show that the parameter ${\varphi _0}$ can be used to control the location of the multiple foci. As an example, Fig. 5 show the normalized total electric field intensity, polarization, and longitudinal Poynting vector distributions in the focal plane of four types of APSG beams with ${\varphi _0} = 0$, $\mathrm{\pi }/8$, $2\mathrm{\pi }/8$, and $3\mathrm{\pi }/8$ when $m = 7$. Obviously, they are always multiple foci distributions and their number is unchanged by the change of ${\varphi _0}$, so the number of multiple foci depends only on m. However, as ${\varphi _0}$ increases, the intensity distribution exhibits a global clockwise rotation, so the parameter ${\varphi _0}$ can be used to control the location of each hotspot. And the polarization and longitudinal energy flow rotate accordingly. The angle of the ith radial direction (where $i = 1$,…, $2({m - 1} )$) with respect to the x direction anticlockwise can be given by ${\phi _i} = ({i - 1} )\pi /({m - 1} )$ for ${\varphi _0} = 0$. The above directions become the center of the ith polarization vortex as ${\varphi _0}$ grows from 0 to $\mathrm{\pi }/2$, and the ith radial direction can now be expressed as ${\phi _i} = ({2i - 1} )\pi /({2m - 2} )$, where $i = 1$,…, $2({m - 1} )$.
Fig. 5. Normalized total electric field intensity (left column), polarization (middle column), and longitudinal Poynting vector distributions (right column) in the focal plane of four types of APSG beams with ${\varphi _0} = 0$ (first row), $\mathrm{\pi }/8$ (second row), $2\mathrm{\pi }/8$ (third row), and $3\mathrm{\pi }/8$ (fourth row) when $m = 7$.
We now investigate the generation of optical force when the tightly focused APSG beam interacts with particles. Here, we assume that the particles are much smaller than the trapping wavelength, so the Rayleigh approximation can be used. And the optical response of a nanostructure can be represented as a dipole or a collection of dipoles. The dipolar polarizability controls the strength of the interaction with an optical field. The equation for a spherical Rayleigh particle with radial a1 and permittivity ε1 in a focused APSG field propagating in a medium with permittivity ε and permeability µ is as follows [59]:
where α0 is the point-like particle polarizability given by [58]An optical force is generated when linear momentum is transferred from a light beam to a particle, based on the conservation of momentum. This optical force is expressed as the sum of three terms [60–62]
The force resulting from the electric intensity gradient, which allows three-dimensional confinement in optical tweezers and accounts for the majority of the total optical forces, is represented by the first term in Eq. (8). A force in the direction of propagation corresponds to the second term, which is responsible for the radiation pressure. The curl force associated with the non-uniform distribution of the spin angular momentum is the third term.
We can now investigate the mechanical effects of the interaction of the focused field with the particles using the above equations. Consider a Rayleigh particle with complex refractive index ${n_1} = 1.59 + 0.005i$ and radius ${a_1} = 30nm$. Optical tweezers often use polystyrene particles as samples, and the value of these particles determines the real part of the complex refractive index, n1. To describe the absorption of light by the particle, we now give the refractive index n1 a small imaginary part. To investigate the ability of the focused APSG beams to trap in three dimensions, we first calculate the longitudinal force distribution in the x-z plane ($y = 0$) for four different input fields, as shown in Figs. 6(a), 6(b), 6(c), and 6(d), where $({m,\textrm{}{\varphi_0}} )= ({1,\; \pi /2} ),$ (3, 0), (5, 0), and (7, 0). The direction of the longitudinal forces is indicated by their positive and negative values, pointing towards the + z and − z axes, respectively. The equilibrium location where where ${F_z} = 0$ for the four input fields listed above is calculated to be approximately at $z = 0.113\mathrm{\lambda }$, 0.109λ, 0.105λ, and 0.101λ, respectively. The longitudinal forces point in opposite directions to the equilibrium position on either side of the equilibrium position. Consequently, a stable longitudinal trap for the Rayleigh particle is possible for each of the four input fields. The corresponding transverse force distributions in the x-y plane at the aforementioned longitudinal equilibrium points are shown in Figs. 6(e)-6(h), where the arrow indicates the direction of the transverse force. From the arrows in Fig. 6(e) it can be seen that the transverse force produces the force equilibrium on a circle, which means that particles can be trapped there. On the contrary, for higher-order APSG beams, multiple equilibrium positions appear that are symmetrical with respect to the optical axis. As a result, a fixed number of particle traps can be achieved depending on the polarization order of m. The angle of the ith trapping direction (where $i = 1$,…, $2({m - 1} )$) with respect to the x direction anticlockwise can be given by ${\phi _i} = ({i - 1} )\pi /({m - 1} )$.
Fig. 6. Longitudinal force distributions in the x-z planes (left column, all at $y = 0$) and transverse force distributions in the x-y planes (right column, from top to bottom at $z = 0.113\mathrm{\lambda }$, 0.109λ, 0.105λ, and 0.101λ, respectively) on the Rayleigh particle with ${n_1} = 1.59 + 0.005i$ generated by the tightly focused APSG beams with $({m,\textrm{}{\varphi_0}} )= ({1,\; \pi /2} )$ (first row), (3, 0) (second row), (5, 0) (third row), and (7, 0) (fourth row), respectively. The blue arrows represent the direction of the force.
An optical force is generated when light’s linear momentum is transferred to a particle, while an optical torque is produced when its angular momentum is transferred. For the harmonically varying external electric field, the time-averaged spin torque is [63–65]
This allows us to investigate the optical torque distributions using Eqs. (9) and (10). Calculations show that transverse spin torque cannot be found for all the APSG beams due to the absence of longitudinal electric field distributions. Furthermore, there are no longitudinal spin torque distributions since all of the local polarizations are linearly polarized throughout. For the four input fields same as that in Fig. 6, the corresponding longitudinal orbital torque distributions in the x-y plane at the aforementioned longitudinal equilibrium points are shown in Figs. 7(a)-7(d), where the yellow arrows indicate the direction of particle motion. In Fig. 7(a), it is the traditional azimuthally polarized beam and no longitudinal orbital torque distribution can be found. However, the longitudinal orbital moment distributions of the remaining three beams are all very strong and have a multi-petal shape. As a result, the particles will rotate either clockwise or anticlockwise, depending on their position. Finally, all of the particles will move to equilibrium positions driven by the longitudinal orbital torque, where the longitudinal orbital torque is zero, demonstrating once again the multiple particle trapping capability of higher order APSG beams.
Fig. 7. The longitudinal orbital torque distributions on the Rayleigh particle with ${n_1} = 1.59 + 0.005i$ in the x-y plane at (a) $z = 0.113\mathrm{\lambda }$, (b) 0.109λ, (c) 0.105λ, and (d) 0.101λ for the four input fields same as that in Fig. 5. The yellow arrows indicate the direction of particle motion due to optical torque.
It is possible to control both the number and position of the trapped particles. Figure 8 shows the transverse force and longitudinal orbital torque distributions in the x-y planes at the longitudinal equilibrium positions $z = 0.109\mathrm{\lambda }$, 0.105λ, and 0.101λ for another three APSG input fields with $({m,\textrm{}{\varphi_0}} )= ({3,\; \pi /2} )$, (5, $\pi /2$), and (7, $\pi /2$). Changes in ${\varphi _0}$ compared to Fig. 6 resulted in changes in the trapping positions for multiple particles. The ith trapping direction with respect to the x direction counterclockwise can now be expressed as ${\phi _i} = ({2i - 1} )\pi /({2m - 2} )$, where $i = 1$,…, $2({m - 1} )$. The longitudinal orbital torque distributions vary in a similar way. Specifically, the positions of the positive and negative values are reversed compared to that in Fig. 7, resulting in the change of the equilibrium positions.
Fig. 8. The transverse force (left column) and longitudinal orbital torque (right column) distributions in the x-y planes (from top to bottom at $z = 0.109\mathrm{\lambda }$, 0.105λ, and 0.101λ, respectively) on the Rayleigh particle with ${n_1} = 1.59 + 0.005i$ generated by the tightly focused APSG beams with $({m,\textrm{}{\varphi_0}} )= ({3,\; \pi /2} )$ (first row), (5, $\pi /2$) (second row), and (7, $\pi /2$) (third row), respectively. The blue arrows indicate the direction of the local transverse force, while the yellow arrows show the direction of particle motion caused by optical torque.
4. Conclusions
To summarize, we proposed and demonstrated a magneto-optical-like effect in the tight focusing of an APSG beam. From the vector diffraction theory of Richards and Wolf, we obtained explicit expressions for all components of the electric and magnetic field strength vectors in the focal volume of a high NA aplanatic focusing system. The calculations showed that, in a homogeneous and isotropic transparent medium, a strong radial polarization component can indeed be generated for the higher order APSG input fields, leading to a magneto-optical-like effect that does not require magnetic materials. The intensity structure and polarization distribution of the highly confined electric field were found to be strongly dependent on the parameters m and φ0 in the sinusoidal term, where m can be used to control the number of the multifocal spots and φ0 can be used to control the position of each focal spot. The corresponding energy flux distributions were also studied in detail. Finally, we showed that this peculiar electric field distribution can be used to realize multiple particles trapping with controllable numbers and locations.
Funding
National Natural Science Foundation of China (12074224, 61975128); Natural Science Foundation of Shandong Province (ZR2020MA087, ZR2020QA066, ZR2021YQ02).
Disclosures
The authors declare no conflicts of interest.
Data availability
All the initial data of figures that support the findings of this study are available from the corresponding author upon reasonable request.
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