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The optical force on a dielectric slab

This paper computes the optical force on a lossless dielectric slab by reconciling Lorentz force and external momentum flux methods, revealing that their general difference arises from the time-dependent electromagnetic momentum stored within the slab, which generates a velocity-dependent force in cavities and resolves the Abraham-Minkowski momentum debate in vacuum environments.

Original authors: Andrew Laeuger, Kevin Kuns, Yanbei Chen, Nancy Aggarwal

Published 2026-10-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrew Laeuger, Kevin Kuns, Yanbei Chen, Nancy Aggarwal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Light does more than just illuminate; it pushes. When a beam of light strikes an object, it transfers momentum, exerting a gentle pressure known as radiation pressure. This phenomenon is the engine behind solar sails and a critical factor in the most sensitive instruments on Earth, such as the detectors used to listen for ripples in spacetime. For decades, physicists have relied on a straightforward method to calculate this push: they measure the momentum of the light entering an object and the momentum of the light leaving it. The difference between the two, they reasoned, must be the force applied to the object. This approach works perfectly for thin mirrors or tiny particles, but it begins to falter when applied to a solid block of transparent material, like a glass slab, that is thick enough for light to spend a measurable amount of time traveling through it.

The core of the problem lies in what happens to the light while it is inside the material. As light passes through a transparent medium, it interacts with the atoms, creating a temporary storage of electromagnetic momentum within the material itself. For a very thin object, this storage is negligible, and the simple "in minus out" calculation holds true. However, for a slab of finite thickness, the light takes time to cross the gap. During this transit, the momentum stored inside the slab changes, especially if the light intensity is fluctuating or if the slab itself is moving. If one ignores this internal storage, the calculation of the force becomes incomplete. The question of how to correctly account for this hidden momentum has been a subject of long-standing debate in physics, with different schools of thought proposing different ways to divide the total momentum between the light and the matter it travels through.

In a recent study, researchers set out to resolve this ambiguity for a specific, practical scenario: a flat, transparent slab of arbitrary thickness submerged in a fluid. They approached the problem using two distinct methods to see if they would arrive at the same answer. The first method involved calculating the force directly on the atoms inside the slab, treating the light as a collection of electric and magnetic fields that push on the bound charges within the material. The second method looked at the slab from the outside, calculating the net flow of momentum entering and leaving the slab's surface. When the researchers applied these methods to a static situation where the light intensity was constant, both approaches agreed perfectly. However, when the light varied over time, or when the slab moved, the two methods initially produced different results.

The discrepancy was not an error in calculation but a missing piece of the physical puzzle. The researchers found that the difference between the two methods was exactly equal to the rate at which the momentum stored inside the slab was changing. When the light intensity fluctuates, the amount of momentum held within the glass changes. This change acts as a force on the slab itself. By adding this "storage term" to the external momentum balance, the two methods reconciled perfectly. The study confirmed that the force on the slab is not just determined by the light hitting its faces, but also by how the light's momentum evolves while it is trapped inside the material.

This finding has a specific and measurable consequence for moving objects. When a slab moves through a cavity where the light intensity depends on its position, the changing storage of momentum creates a force that depends on the slab's velocity. This velocity-dependent force acts as a form of damping, slowing the motion of the slab. The researchers calculated that this effect is proportional to the thickness of the slab and the rate at which the power flowing through it changes. While this damping might be weak compared to other forces in typical setups, it becomes significant in high-precision environments, such as the optical cavities used in gravitational wave detectors, where even tiny forces can alter the behavior of the system.

The study also provided a clear resolution to the historical debate regarding the momentum of light in matter. For years, physicists have argued over whether the momentum of light inside a material is best described by one formula or another. The researchers showed that both formulas are correct, but they describe different partitions of the total momentum. One formulation includes the momentum associated with the material's polarization as part of the light's momentum, while the other treats it separately. The key insight is that the force on the slab depends on which part of the momentum is actually able to cross the boundary of the material. The momentum associated with the material's internal polarization cannot leave the slab, so it does not contribute to the net push on the object's center of mass. Once this distinction is made, the different formulas yield the same physical force.

Ultimately, the work demonstrates that to accurately predict how light moves a transparent object, one must account for the time it takes light to travel through the object. Ignoring the time-dependent storage of momentum inside the material leads to an incomplete picture of the forces at play. By carefully tracking how momentum is stored and released within the slab, the researchers have provided a unified framework that works for both static and dynamic situations. This clarity is essential for the next generation of experiments involving levitated objects and high-precision optical instruments, ensuring that the subtle push of light is understood with the same rigor as the pull of gravity.

Technical Summary: The Optical Force on a Dielectric Slab

Problem Statement
Accurately modeling optical forces on dielectric elements is critical for optomechanical systems, ranging from tethered membranes to the transmissive optics in gravitational-wave detectors. While radiation-pressure forces are typically calculated using the momentum flux of external fields, this approach often neglects the time dependence of electromagnetic momentum stored within a dielectric of finite thickness. Previous treatments have largely focused on subwavelength particles (point-dipole regime) or wavelength-scale particles under monochromatic illumination. This work addresses the specific case of planar dielectric slabs with lateral extents exceeding the beam waist, where the element acts as a cavity component rather than a perturbation. In such systems, the propagation delay across the slab relative to field fluctuation timescales introduces a new physical scale that standard momentum-flux methods may fail to capture consistently.

Methodology
The authors derive the optical force on a lossless dielectric slab of arbitrary thickness ( $L_s$ ) and refractive index ( $n_1$ ), submerged in a dielectric fluid of index ( $n_2$ ). The analysis employs a two-photon formalism, decomposing the electric field into carrier (DC) and sideband (AC) components, and further separating them into amplitude and phase quadratures. This approach allows for the treatment of arbitrary time-dependent cavity fields relevant to linearized optomechanics.

The force is computed via two distinct methods:

Lorentz Force Approach: Calculating the force density on the bound charges and currents within the slab and integrating over the volume.
Momentum Flux Approach: Calculating the net electromagnetic momentum flux through a surface enclosing the slab, utilizing the vacuum-form Maxwell stress tensor evaluated just outside the dielectric.

The study also examines the problem through the lens of continuum mechanics, applying both the Abraham and Minkowski formulations of electromagnetic momentum in matter to reconcile the differences between the two primary calculation methods.

Key Contributions and Results

Reconciliation of Force Methods: The paper demonstrates that the Lorentz force and the external momentum flux methods yield different results for time-varying (AC) fields. The discrepancy is exactly accounted for by the rate of change of the electromagnetic momentum stored within the slab ( $d\vec{p}_{vac}/dt$ ). For static (DC) fields, the storage term vanishes, and both methods agree.
Velocity-Dependent Force: In the AC limit, the internal momentum storage term introduces a velocity-dependent force. For mechanical frequencies well below the cavity linewidth, this manifests as an additional mechanical damping rate:
$\Delta\Gamma = \frac{L_s}{mc^2} \frac{\partial P_{net}}{\partial x}$
where $P_{net}$ is the net power flowing through the slab. This term scales as $(L_s/c^2) dP/dt$ for low frequencies.
Abraham-Minkowski Reconciliation: The authors resolve the apparent conflict between Abraham and Minkowski momentum formulations in the context of a finite slab. They show that the difference arises from how momentum is partitioned between electromagnetic and material degrees of freedom. Specifically, the Minkowski formulation includes the polarization momentum of the slab material, which cannot cross the slab boundary. By recognizing that only the vacuum electromagnetic component contributes to the net mechanical recoil of the slab's center of mass, the authors show that the Abraham formulation (which treats the material contribution separately) and the Lorentz force approach yield consistent results for the slab's acceleration.
Environmental Effects: When the slab is immersed in a dielectric fluid ( $n_2$ ), the Lorentz force calculation includes a correction term proportional to the fluid's susceptibility ( $\chi_{e,2}$ ). This term accounts for momentum transferred to the fluid particles rather than the slab. In the high-reflectivity limit, this recovers the known $n_2$ -fold enhancement of radiation pressure. However, for general slab configurations, the exact force depends on the fluid's mechanical response (pressure and electrostriction), which is not modeled in this work.

Significance and Claims
The paper claims to provide a self-consistent derivation of optical forces for finite-thickness dielectric slabs, essential for accurate optomechanical modeling. The primary significance lies in identifying that neglecting the time-dependence of stored field momentum leads to an incomplete description of the force, particularly in the AC regime where it generates a velocity-dependent damping term.

The authors emphasize that the choice between Abraham and Minkowski formulations depends on the physical partition of the system: if the bound charges are part of the moving object (the slab), the Abraham prescription combined with pure momentum flux balance correctly describes the force. If the charges are external, the Minkowski prescription is more appropriate. This distinction clarifies the "Abraham-Minkowski controversy" for finite dielectrics by focusing on which momentum components can physically cross the boundary to influence the object's center-of-mass motion.

The work notes that while this velocity-dependent damping is likely weak compared to cavity-induced damping in typical setups, it may become relevant in high-finesse cavities or when the phase accumulation within the slab is a substantial fraction of the cavity length. The authors state that these results will be implemented in the gravitational wave detector simulation software Finesse. They explicitly limit their scope by not modeling the accompanying mechanical response (pressure and electrostriction) of the surrounding fluid, noting that a full agreement between formulations in a fluid environment requires such modeling.

Technical Summary: The Optical Force on a Dielectric Slab

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