Momentum Change of Electromagnetic Waves in Reflection and Refraction

In classical electromagnetic theory, electromagnetic waves are not only carriers of energy but also transport momentum. Understanding this fundamental property is essential for investigating phenomena such as radiation pressure, optical tweezers, and the complex interactions between electromagnetic fields and matter.

The energy flux density of an electromagnetic wave is defined by the Poynting vector $\mathbf{S}$:
$$\mathbf{S} = \mathbf{E} \times \mathbf{H}$$
where $\mathbf{E}$ is the electric field intensity and $\mathbf{H}$ is the magnetic field intensity. In a vacuum, the momentum density $\mathbf{g}$ stored in the electromagnetic field per unit volume is related to the Poynting vector by:
$$\mathbf{g} = \frac{1}{c^2} \mathbf{S}$$
in which $c$ is the speed of light in a vacuum. Consequently, energy flux and momentum flux in electromagnetic waves are intimately coupled. When these waves encounter a material interface and undergo reflection or refraction, their propagation direction, phase velocity, and local medium parameters change. This spatial and directional alteration results in a net vector change of electromagnetic momentum. In accordance with Newton's third law and the law of momentum conservation, any variation in the momentum of the electromagnetic field must be accompanied by mechanical forces exerted upon the material boundary.
When an electromagnetic wave impinges upon an opaque reflective interface—such as an ideal metallic mirror—its propagation direction is altered.

1. Calculating the Momentum Increment

Consider a monochromatic plane electromagnetic wave striking an ideal reflective surface at normal incidence. Let the momentum vector of the incident wave be $\mathbf{p}{inc}$ and that of the reflected wave be $\mathbf{p}{ref}$. Assuming an ideal reflection where the wave amplitude remains unchanged and only the direction is reversed, we can evaluate the momentum transfer.

Defining the incident direction along the $+z$-axis, the reflected wave travels along the $-z$-axis. The momentum change $\Delta \mathbf{p}$ experienced by the field is:
$$\Delta \mathbf{p} = \mathbf{p}{inc} - \mathbf{p}{ref} = p_{inc} \hat{z} - (-p_{inc} \hat{z}) = 2p_{inc} \hat{z}$$

This indicates that during the reflection process, the electromagnetic wave delivers twice its initial momentum to the boundary.

2. Radiation Pressure

Because the time rate of change of momentum equates to a force, this continuous redistribution of momentum macroscopically manifests as radiation pressure acting on the interface. For a continuous plane wave of intensity $I$, the energy delivered per unit area per unit time is $I$. In the case of normal incidence on a perfectly reflecting surface, the resulting pressure $P$ is:
$$P = \frac{2I}{c}$$
Conversely, if the boundary is purely absorptive—meaning the electromagnetic wave is completely absorbed without reflection—the reflected momentum vanishes, halving the momentum change and reducing the pressure to:
$$P_{abs} = \frac{I}{c}$$

Application Example: Solar sail technology harnesses this subtle yet continuous radiation pressure from sunlight to propel spacecraft through deep space, eliminating the need for conventional chemical propellants.

Momentum Variation and Medium Effects in Refraction

Refraction is inherently more intricate than reflection because the electromagnetic wave transitions from one medium (with refractive index $n_1$) into another (with refractive index $n_2$). This transition alters not only the wave vector's direction but also its phase velocity, sparking the long-standing debate regarding the Abraham-Minkowski momentum dilemma.

1. Snell's Law and Momentum Direction

According to Snell's Law of refraction:
$$n_1 \sin \theta_1 = n_2 \sin \theta_2$$
As the wave refracts, the direction of its momentum vector shifts alongside the wave vector $\mathbf{k}$. Because the momentum aligns with the direction of wave propagation, this directional shift can induce a tangential force on the refracting surface.

2. The Abraham-Minkowski Controversy

Inside a dielectric medium, the exact definition of electromagnetic momentum splits into two primary theoretical frameworks, depending on how one partitions total momentum between the electromagnetic field and the surrounding matter:

Minkowski Momentum: Assumes the medium's momentum is given by $p_M = n \frac{E}{c}$. This formulation proves highly intuitive for wave equations and kinematics of photons traversing media, neatly conforming to standard refraction geometry.
Abraham Momentum: Assumes the momentum is $p_A = \frac{E}{nc}$. This viewpoint preserves a direct proportionality between energy flux and momentum flux akin to vacuum conditions, emphasizing a strict link to the Poynting vector.

Physical Essence: This ongoing debate fundamentally highlights that within a material medium, electromagnetic fields and matter (specifically bound polarization charges and magnetization currents) are strongly coupled. The Minkowski momentum is generally interpreted as the total momentum of the "field-plus-matter" composite system, whereas the Abraham momentum isolates the purely electromagnetic field component.

During refraction, the vector change $\Delta \mathbf{p}$ generates both normal and tangential forces on the boundary. For instance, when light travels from air into glass, the decrease in the refraction angle alters the normal momentum component, thereby exerting a net mechanical pressure on the surface.

Momentum Conservation and the Maxwell Stress Tensor

To rigorously quantify the mechanical forces exerted by electromagnetic waves during reflection and refraction, physicists utilize the Maxwell Stress Tensor $\mathbf{T}$.

The principle of momentum conservation within an electromagnetic framework is expressed as:
$$\mathbf{f} = \nabla \cdot \mathbf{T} - \frac{\partial \mathbf{g}}{\partial t}$$
where $\mathbf{f}$ represents the electromagnetic force density.

When analyzing boundary value problems involving reflection and refraction, integrating the Maxwell stress tensor across either side of the interface yields the total force:

Reflection Scenarios: Evaluating the tensor discontinuity across the boundary accurately retrieves the surface radiation pressure.
Refraction Scenarios: Calculating the jump in the stress tensor across the two distinct media accounts for both normal pressures and tangential shear forces, explaining why obliquely incident light tends to exert a lateral drag on surfaces.

Conclusion

The momentum dynamics of electromagnetic waves during reflection and refraction represent core mechanisms governing field-matter interactions.

Reflection generates significant radiation pressure via directional reversal, with momentum transfer directly tied to surface reflectivity.
Refraction introduces medium-dependent refractive indices, leading to diverse theoretical descriptions that illustrate the profound complexity of field-matter coupling.
Mechanical Forces are comprehensively unified through the Maxwell stress tensor, establishing a robust theoretical framework for modern optical micromanipulation and optomechanics.