Unraveling the Angular Symmetry of Optical Force in a Solid Dielectric

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 number of both 2 and 0, which cannot be explained solely by the Lorentz or the Einstein-Laub formulation. Instead, we found a modified Helmholtz theory by combining the Lorentz force with additional electrostrictive force could 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.

The force exerted by electromagnetic fields is of fundamental importance in broad sciences and applications [1][2][3], but its exact formulation inside media is still controversial and unclear [4][5][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][9][10]. More recently, the LO law was also 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, there still lack wide agreements on this issue because the hidden momentum is experimentally unobservable with current technique. At the same time, another formulation originally proposed by Einstein and Laub (EL) has also been widely used and was suggested as an alternative of the electromagnetic force formulation [4,[11][12][13][14][15][16][17][18][19][20][21], as it complies with both the special relativity and universal conservation laws without needing the hidden momentum [11,22,23]. The EL formulation is also consistent with the Maxwell's equations, and 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][5][6] failing to distinguish these two formulations. To date, the debates on the LO and EL formulations are still going on because rigorous experimental investigations on distinguishing them are still absent.
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 F LO = ( ∇⋅P)E + ∂P/∂t × B, while the EL force has the form F EL = (P⋅∇)E + ∂P/∂t × B, with E the electric field, B the magnetic induction, and = ε 0 (ε r 1)E the polarization (Supplementary Sec. 2). ε 0 and ε 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 the same total force on an object, they actually produce different force distributions inside a dielectric medium [19,21]. This feature can be harnessed in experiments to 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 may explain the experimental results. Our experiment in a solid dielectric represent a fundamental leap forward in experimental exploration of the 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 an important role in determining the correct formulation of electromagnetic force, but also provide a scheme to solve some other issues in classical electrodynamics, such as the Abraham-Minkowski controversy.
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]. Such force density distribution in the LO formulation has a form in the cylindrical coordinates (r: r, θ, and the force density in the EL formulation is 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 linearly polarized optical fields propagating in the core of the fiber. The intensity (E 0 2 ) of the optical field was sinusoidally modulated (with frequency Ω, modulation depth A, and RF modulation phase 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 r Due to the resonant enhancement effect, the mechanical modes could have amplified mechanical motion in response to the force oscillating at the mechanical eigenfrequencies. The intensities of the actuated mechanical modes were obtained with ultrahigh sensitivity from optomechanical transduction by using an ultrahigh-Q optical whispering-gallery mode traveling in the circumference of the transverse plane, which was supported by the slightly fused cladding of the optical fiber ( Fig. 1a) [25].
According to Eqs. (1) and (2), a critical property of optical force is that the LO force with C = 2 is dependent on the optical polarization angle  while the EL force with C = 0 is independent of  . Therefore, the mechanical modes actuated by the optical force would have different predicted response to optical polarization angle  for these two theories. Solving the elastic equation, it is found the actuated amplitude xamp of a mechanical mode is proportional to the spatial overlap integral of the force density distribution and mechanical modal profile (Supplementary Sec. 3.1): where u(r) is the displacement modal profile of the mechanical mode. Here, we focus on the response of the mechanical wine-glass mode and breathing mode of the fiber (Fig. 1d). The mechanical displacement modal profile of these mechanical eigenmodes can be expressed as [26] ( )  , while that by a force with C = 0 is polarization-independent.
(II) For dual pump beams with polarization angles 1  and 2  and the same modulation phase φ RF , the intensity of mechanical mode actuated by the two forces with C = 2 is proportional to , while that by forces with C = 0 is polarizationindependent.
(III) For dual orthogonally polarized pump beams with a RF modulation phase difference   for EL force with C = 0. Therefore, the intensity of mechanical mode actuated by such two forces with C = 2 would be proportional to 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  with a dependence of 2 cos (2 )  , with >20 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 1  and 2  . It was found that the mechanical intensity follows To further investigate the angular mode number of optical force, we also measured the actuation results of the breathing mode (n = 0) with 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 RF   . Figure 5

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 that by Einstein and Laub. Actually, the Hakim-Higham experiment only showed the directionless 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 can also be compatible 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 done in liquids [4,5,29,30]. The fluidic nature of liquids makes them challenging to measure the angularly antisymmetric force component with C = 2, and also make them unable to provide detailed microscopic information about the force distribution. 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 device enable the first unraveling of the detailed microscopic properties of optical force inside a medium. We expect that these results will not only generate long-term impact on understanding of the light-matter interactions, but also update the fundamental working principle for many applications in science and engineering branches involving optical forces.
Although the experiments were planned based on 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][5][6] besides the LO and EL formulations. The force density distribution of a Gaussian beam in an optical fiber predicted by these existing theories can also have an angular mode number C = 2 or C = 0. Exhaustive scrutiny of all the force formulations, however, is beyond the scope of this work. Here, we found that a modified Helmholtz theory by combining the Lorentz formulation with the electrostrictive force [31,32] could account for the coexistence of force components with C = 2 and C = 0, which possibly explains our experimental results (Supplementary Sec. 8). On the other hand, since the EL formulation has already included the electrostrictive interaction [4,24], it may require other types of modification to explain the experimental results. We believe that the angular symmetry of the 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.