## Abstract

The textbook-accepted formulation of electromagnetic force was proposed by Lorentz in the 19th century, but its validity has been challenged due to incompatibility with the special relativity and momentum conservation. The Einstein–Laub formulation, which can reconcile those conflicts, was suggested as an alternative to the Lorentz formulation. However, intense debates on the exact force are still going on due to lack of experimental evidence. Here, we report the first experimental investigation of angular symmetry of optical force inside a solid dielectric, aiming to distinguish the two formulations. The experiments surprisingly show that the optical force exerted by a Gaussian beam has components with the angular mode numbers of both 2 and 0, which cannot be explained solely by the Lorentz or the Einstein–Laub formulation. Instead, we found that a modified Helmholtz theory by combining the Lorentz force with additional electrostrictive force can explain our experimental results. Our results represent a fundamental leap forward in determining the correct force formulation and will update the working principles of many applications involving electromagnetic forces.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The force exerted by electromagnetic fields is of fundamental importance in broad sciences and applications [1–3], but its exact formulation inside media is still controversial and unclear [4–6]. The Lorentz (LO) law of electromagnetic force is widely adopted and regarded as one of the foundations of classical electrodynamics. However, this century-old physical law has been in crisis [7]. In the 1960s, Shockley pointed out that the LO law contradicts the universal momentum conservation in certain systems involving magnetic media [8–10]. More recently, the LO law was found to be incompatible with the special relativity, as it predicts different results in different reference frames [11]. These problems of the LO law could be avoided by introducing an additional hidden momentum of electromagnetic field in magnetic media [8,11]. However, some people have concerns because the hidden momentum is experimentally unobservable with current techniques. In the meantime, another formulation originally proposed by Einstein and Laub (EL) has been widely used as an alternative of the electromagnetic force formulation [4,11–21], as it complies with both the special relativity and universal conservation laws without the need for a hidden momentum [11,22,23]. The EL formulation is also consistent with the Maxwell’s equations, and it agrees with the existing measurement results of the total force or torque that support the LO formulation [19,24]. Their equivalence on the total force or torque measurements leads to most of the existing experiments [4–6] failing to distinguish these two formulations. To date, the debates on the LO and EL formulations are still going on because of the lack of rigorous experiments to distinguish them.

The underlying difference between the LO and EL formulations lies in their different descriptions of the quantum nature of media and electromagnetic fields: the LO formulation treats the electric and magnetic dipoles inside a medium as distributions of ordinary charges and currents, while the EL formulation treats the electric and magnetic dipoles as two individual constituents that are distinct from ordinary charges and currents [17,19]. Due to the different treatments, the LO force in a nonmagnetic dielectric material has the form ${{\textbf F}_{{\rm{LO}}}} = (- \nabla \cdot {\textbf P}){\textbf E} + \partial {\textbf P}/\partial t \times {\textbf B}$, while the EL force has the form ${{\textbf F}_{{\rm{EL}}}} = ({\textbf P} \cdot \nabla){\textbf E} + \partial {\textbf P}/\partial t \times {\textbf B}$, with ${\textbf{E}}$ the electric field, ${\textbf{B}}$ the magnetic induction, and ${\textbf P =}{\varepsilon _0}({\varepsilon _{\rm{r}}} - 1){\textbf E}$ the polarization (Supplement 1, Section 2). ${\varepsilon _{{0}}}$ and ${\varepsilon _{\rm{r}}}$ are the vacuum permittivity and the relative permittivity of the material, respectively. Note that the hidden momentum problem can be avoided naturally in nonmagnetic dielectric media, inside which the hidden momentum is always zero. It was recently discovered that although these two formulations predict identical total force on an object, the predicted force distributions inside a dielectric medium are different [19,21]. This feature can be harnessed to experimentally distinguish the two formulations. However, the predicted differences are microscopic and exist only inside a medium, which were thought to be too weak to be detected.

Here we investigated for the first time the optical force distribution inside a solid dielectric by employing an optomechanical approach with ultrahigh detection sensitivity. Theoretically, the optical force distribution exerted by a linearly polarized optical Gaussian beam inside a dielectric has angular symmetry with angular mode number $C = {{2}}$ by the LO formulation or $C = {{0}}$ by the EL formulation (Fig. 1). We derived three criteria for determining the angular symmetry of optical force distribution inside a single-mode optical fiber. Surprisingly, multiple experiments based on these three criteria all show that the optical force distribution of a Gaussian beam in an optical fiber has components of both $C = {{2}}$ and $C = {{0}}$. These results cannot be explained solely by the LO or the EL formulation, indicating the necessity of a modification or a new theory. We found that a modified Helmholtz theory by supplementing the LO force with additional electrostrictive force can explain the experimental results. Our experiment in a solid dielectric is important in the experimental studies of optical force distribution, because it can avoid many spurious effects in previous experiments [4,21], can have ultrahigh sensitivity, and can identify different optical force components separately. Our results will not only play a crucial role in determining the correct formulation of electromagnetic force but also provide a scheme for solving some other issues in classical electrodynamics, such as the Abraham–Minkowski controversy.

## 2. ANGULAR MODE NUMBER OF THE OPTICAL FORCE

For a linearly polarized optical beam of Gaussian profile propagating in a dielectric medium, such as a single-mode fiber [Fig. 1(a)], the LO formulation predicts a force density distribution tending to stretch (compress) the medium along (perpendicular to) the light polarization direction [Fig. 1(b)], and the EL formulation predicts a force density distribution tending to compress the medium radially inward [Fig. 1(c)] [19,21]. The force density distribution in the LO formulation has a form in the cylindrical coordinate system (${\textbf{r}}$: $r$, $\theta$, $z$) as

We employed an optical-fiber-based system to identify the angular symmetry of the optical force in a slightly modified single-mode fiber [Fig. 1(a)]. In the system, the optical force was exerted by a linearly polarized optical beam propagating in the core of the fiber. The intensity $E_0^2$ of the optical field was sinusoidally modulated (with frequency $\Omega$, modulation depth $A$, and RF modulation phase ${\varphi _{{\rm{RF}}}}$) to generate oscillating optical force to actuate the mechanical modes [Fig. 1(d)] of the fiber. The oscillating part of the optical force can be described as ${\textbf F}({\textbf r},t) = AE_0^2\cos (\Omega t + {\varphi _{{\rm{RF}}}}) \cdot {{\textbf F}_{{\rm{LO/EL}}}}({\textbf r})$. Due to the resonant enhancement effect, the mechanical modes would have amplified mechanical motion in response to the force oscillating at their eigenfrequencies. The intensities of the actuated mechanical modes were obtained with ultrahigh sensitivity from optomechanical transduction of an ultrahigh-quality optical whispering-gallery mode traveling along the circumference of the cross section, which was supported by the slightly fused cladding of the optical fiber [Fig. 1(a)] [25].

According to Eqs. (1) and (2), the LO force with $C = {{2}}$ depends on the optical polarization angle $\phi$ while the EL force with $C = {{0}}$ is independent on $\phi$. Therefore, the mechanical modes would respond differently to the actuating optical force under these two theories. The actuated amplitude ${x_{\text{amp}}}$ of a mechanical mode is proportional to a spatial overlap integral of the force density distribution and the mechanical modal profile (Supplement 1, Section 3.1):

- (I) For a single pump beam with polarization angle $\phi$, the intensity of the mechanical mode actuated by a force with $C = {{2}}$ is proportional to ${| {\cos (2\phi)} |^2}$, while that by a force with $C = {{0}}$ is polarization independent.
- (II) For dual pump beams with polarization angles ${\phi _1}$ and ${\phi _2}$ and the same modulation phase ${{{\varphi}}_{{\rm{RF}}}}$, the intensity of the mechanical mode actuated by two forces with $C = {{2}}$ is proportional to ${| {\cos (2{\phi _1}) + \cos (2{\phi _2})} |^2} = {| {2\cos ({\phi _1} + {\phi _2})\cos ({\phi _1} - {\phi _2})} |^2}$, while that by two forces with $C = {{0}}$ is polarization independent.
- (III) For dual orthogonally polarized pump beams with a RF modulation phase difference $\Delta {\varphi _{{\rm{RF}}}} = {\varphi _{{\rm{RF2}}}} - {\varphi _{{\rm{RF1}}}}$, the superimposed force is $AE_0^2[\cos (\Omega t + {\varphi _{{\rm{RF2}}}}) - \cos (\Omega t + {\varphi _{{\rm{RF1}}}})] \cdot {{\textbf F}_{{\rm{LO}}}}({\textbf r})$ or $AE_0^2[\cos (\Omega t + {\varphi _{{\rm{RF2}}}}) + \cos (\Omega t + {\varphi _{{\rm{RF1}}}})] \cdot {{\textbf F}_{{\rm{EL}}}}({\textbf r})$ because of the relations ${{\textbf F}{_{{\rm{LO}}}(\phi)}} = - {{\textbf F}{_{{\rm{LO}}}(\phi + \pi /2)}}$ and ${{\textbf F}{_{{\rm{EL}}}(\phi)}} = {{\textbf F}{_{{\rm{EL}}}(\phi + \pi /2)}}$. The amplitude of the superimposed force is proportional to $\sin (\Delta {\varphi _{{\rm{RF}}}}/2)$ for the LO force with $C = {{2}}$ or to $\cos (\Delta {\varphi _{{\rm{RF}}}}/2)$ for the EL force with $C = {{0}}$. Therefore, the intensity of the mechanical mode actuated by such two forces with $C = {{2}}$ is proportional to $|\!\sin (\Delta {\varphi _{{\rm{RF}}}}/2){|^2}$, while that by two forces with $C = {{0}}$ is proportional to $|\!\cos (\Delta {\varphi _{{\rm{RF}}}}/2){|^2}$.

## 3. MEASUREMENT OF THE ANGULAR SYMMETRY OF OPTICAL FORCE

To experimentally examine the angular mode number of the optical force under the three criteria, we fabricated a bottle-like microstructure on a standard single-mode optical fiber (Fig. 2; Supplement 1, Section 1). We slightly fused the cladding of the fiber to create two necks for the bottle-like microstructure [Fig. 2(a)], whose diameters range from 100 to 120 µm for different sample devices tested in the experiment (Supplement 1, Fig. S1). Such bottle-like device configuration forms optical and mechanical energy potentials for supporting high-quality optical probe modes and mechanical modes (Supplement 1, Sections 1.3 and 4.3). Obtaining high quality factors from these optical probe modes and mechanical modes is crucial for achieving high sensitivity in detecting the mechanical motion, because the optomechanical transduction and resonant amplification of the mechanical motion depend respectively on the optical and mechanical quality factors. The pump light beam that exerts an optical force to actuate the mechanical modes propagates in the fiber core [Fig. 2(b)], which preserves a quasi-Gaussian profile when passing through the bottle-like microstructure [Figs. 2(c) and S2]. The pump light is also experimentally confirmed to be quasilinearly polarized (Supplement 1, Section 4.4). Such quasilinearly polarized Gaussian pump light beam well satisfies all the critical experimental requirements for the theoretical analysis on force density distribution and the three criteria to examine the force formulations (Supplement 1, Sections 3 and 6).

The angular mode number of the optical force density was experimentally investigated by measuring the intensity of the wine-glass mode ($n = {{2}}$) according to Criteria I and II. First, we measured the response of mechanical intensity to the polarization angle of a single pump beam. It was found that the mechanical intensity follows the pump beam’s polarization angle $\phi$ with a dependence of ${| {\cos (2\phi)} |^2}$, with ${\gt}{{20}}\;{\rm{dB}}$ extinction ratio [Fig. 3(a)]. Next, we applied two pump beams and measured the response of the same mechanical mode to the two pump beams’ polarization angles ${\phi _1}$ and ${\phi _2}$. It was found that the mechanical intensity follows ${| {\cos ({\phi _1} + {\phi _2})\cos ({\phi _1} - {\phi _2})} |^2}$ [Fig. 3(b)], with ${\gt}$20 dB extinction ratio. When ${\phi _1}$ is fixed at 0°, the measured mechanical intensity follows a dependence of ${| {\cos {\phi _2}} |^4}$ [Fig. 3(c)]. Specifically, for two orthogonally polarized pump beams (${\phi _1} = {0^ \circ}$, ${\phi _2} = {90^ \circ}$), the measured mechanical intensity is much weaker than that actuated by a single pump beam ($\phi = {0^ \circ}$ or ${90^ \circ}$), indicating that the forces of two orthogonally polarized pump beams cancel each other out [Figs. 3(d)–3(f)]. According to Criteria I and II, these results indicate the existence of force component with $C = {{2}}$.

To further investigate the angular mode number of optical force, we also measured the actuation results of the breathing mode ($n = {{0}}$) under the same experimental configuration. With a single pump beam, the mechanical intensity does not vary with the polarization angle [Fig. 4(a)]. In addition, the mechanical intensity also remains constant under actuation by dual pump beams with different polarization angles [Fig. 4(b)]. According to Criteria I and II, these results indicate that the optical force also has a component with $C = {{0}}$.

Next, the angular mode number of optical force was also investigated under the condition in Criterion III, where the wine-glass mode ($n = {{2}}$) and the breathing mode ($n = {{0}}$) each were actuated by two orthogonally polarized pump beams modulated at the same RF frequency but with a constant phase difference $\Delta {\varphi _{{\rm{RF}}}}$. Figure 5(a) shows the measured mechanical intensity of the wine-glass mode ($n = {{2}}$) as a function of $\Delta {\varphi _{{\rm{RF}}}}$, which follows the dependence of $|\!\sin (\Delta {\varphi _{{\rm{RF}}}}/2){|^2}$ and confirms the existence of a force component with $C = {{2}}$. On the other hand, the mechanical intensity of the breathing mode ($n = {{0}}$) follows $\Delta {\varphi _{{\rm{RF}}}}$ with a dependence of $|\!\cos (\Delta {\varphi _{{\rm{RF}}}}/2){|^2}$ [Fig. 5(b)], which confirms the existence of a force component with $C = {{0}}$.

The above experimental results indicate that the optical force by a linearly polarized Gaussian beam in a solid dielectric medium has components with angular mode number of both $C = {{2}}$ and $C = {{0}}$. Such results are highly reproducible and are confirmed to be valid even when the optical and mechanical modes are imperfect due to some moderate distortion of the fabricated device structure (Supplement 1, Sections 5 and 6). This is because the polarization dependence of the optical force density in the LO or EL formulation is not affected by slight geometric perturbation of the device structure. Nonetheless, a small portion of crosstalk exists due to the perturbation terms caused by the geometric imperfection, yielding actuation of the mechanical modes with $n \ne C$. By taking this factor into account, we numerically simulated the actuated mechanical intensities and compared them with the measured results, concluding that the ratio between the force components with $C = {{2}}$ and $C = {{0}}$ is between 1:3 and 1:1 (Supplement 1, Section 7). Therefore, these two force components are comparable in magnitude. Since the LO and EL formulations each predict an optical force with a unique angular mode number ($C = {{2}}$ or $C = {{0}}$), neither of them can explain our experimental results.

## 4. DISCUSSION

Although the unraveled angular symmetry of optical force contradicts the predictions of both the LO and EL formulations, our results are consistent with previous experimental observation by Ashkin and Dziedzic in 1973 [27]—a bulge appeared on water surface at the spot where a focused laser beam entered, which was ever taken as an evidence supporting the EL formulation [20,21]. According to our experimental results, such a bulge can be generated as long as the angularly symmetric compressive force component with $C = {{0}}$ exists. It should also be noted that the Hakim–Higham experiment in 1962 [28] was believed to support the Helmholtz force over the Einstein–Laub force. Actually, the Hakim–Higham experiment only showed the strength of electric pressure along the $y$ axis in their setup. Such a one-dimensional scalar measurement is not enough to determine the distribution and angular mode number of electromagnetic force. Our findings are also consistent with their results.

The coexistence of angular mode number $C = {{2}}$ and $C = {{0}}$ of the optical force density inside a dielectric has not been experimentally identified before, because most relevant experiments are conducted in liquids [4,5,29,30]. The fluidic nature of liquids makes measurement of the angularly antisymmetric force component with $C = {{2}}$ challenging, and also makes acquirement of the microscopic information about the force distribution impossible. Additionally, those conventional experiments based on liquids are mostly phenomenological with some spurious effects [4,21]. By contrast, our experiment based on a lossless solid dielectric avoids most of the ambiguous effects encountered previously, and the mechanical modes of the investigated devices unveil the microscopic properties of optical force inside a medium. We expect that these results will not only generate long-term impact on understanding of light–matter interactions but also update the fundamental working principle for many applications in science and engineering branches involving optical forces.

Note that the electrostrictive effect of the optical field can also contribute to mechanical motion in our experiments. In this regard, we found that a modified Helmholtz theory by combining the Lorentz formulation with the electrostrictive force [31,32] can account for the coexistence of force components with $C = {{2}}$ and $C = {{0}}$, which possibly explains our experimental results (Supplement 1, Section 8). Since the EL formulation has already included the electrostrictive interaction [4,24], it may require other types of modification to explain the experimental results. On the other hand, although our experiments were planned based on distinguishing the force distributions inside a medium predicted by the LO and EL formulations, the unraveled angular symmetry can be used to examine any other related theories [4–6] beyond the LO and EL formulations. The force density distribution of a Gaussian beam in an optical fiber predicted by these existing theories has an angular mode number $C = {{2}}$ or $C = {{0}}$. Exhaustive scrutiny of all the force formulations, however, is beyond the scope of this work. We believe that the angular symmetry of optical force unraveled in this work will serve as a crucial step in the ultimate determination of the correct electromagnetic force formulation inside media in the future.

## Funding

Research Grants Council of Hong Kong (14208717, 24208915); National Key Research and Development Program of China (2016YFA0301303); National Natural Science Foundation of China (11722436); Anhui Initiative in Quantum Information Technologies (AHY130200).

## Acknowledgment

This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication. The authors thank Dr. Zhen Shen and Rui Niu for discussions.

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## Supplemental document

See Supplement 1 for supporting content.

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