Representation-induced superposition breakdown in linear physics

This paper demonstrates that the superposition principle can fail in multilayered media when using conventional evanescent wave bases due to non-normalizable components, and proposes a power flux mode basis that ensures unitary scattering and restores convergence without regularization.

The Gist

The Big Idea: When Math Tricks Fail Reality

Imagine you are trying to predict how a crowd of people moves through a series of hallways. In physics, we have a golden rule called the Superposition Principle. It says: "If you know how one person moves, and you know how another moves, you can just add their paths together to see how the whole crowd moves."

This rule works perfectly for most things. It's the foundation of how we design everything from fiber optic cables to earthquake-resistant buildings.

However, the authors of this paper discovered a weird glitch. When light (or sound, or vibrations) hits a stack of three or more layers of material, and some of that light gets "stuck" (becoming what physicists call an evanescent wave), the math breaks. If you try to add up all the little bounces and reflections using the standard math tools, the numbers don't settle down—they explode to infinity.

It's like trying to add up a list of numbers that keeps getting bigger and bigger, never stopping. The paper asks: Why does the math fail when the physics is still happening?

The Problem: The "Ghost" Waves

To understand the glitch, we need to look at the "ingredients" the mathematicians are using.

1. The Standard Recipe: Usually, physicists describe waves as "traveling waves" (like a surfer riding a wave) and "ghost waves" (evanescent waves).
2. The Ghost Waves: These are waves that don't actually travel far; they just wiggle a little bit and then die out quickly near a surface. They are like the echo in a room that fades away instantly.
3. The Glitch: When you have a complex stack of layers, these "ghost waves" bounce back and forth. In the standard math recipe, these ghosts are treated as if they carry infinite energy or have no clear size. When you try to add up an infinite number of these "ghostly" bounces, the sum goes haywire.

The Analogy: Imagine you are counting money.

• Traveling waves are like $10 bills. You can count them easily.
• Ghost waves are like IOUs written on napkins that say "I owe you infinity."
• The standard math tries to add up the $10 bills and the infinite IOUs. The total becomes "Infinity," which makes no sense in the real world. The system isn't actually exploding; the way we are counting is broken.

The Solution: A New Way to Count

The authors realized the problem wasn't with the light or the layers; it was with the ruler they were using to measure the waves.

They invented a new "ruler" called Power Flux Modes.

Instead of measuring the waves by how big their electric field is (which is messy for ghost waves), they measure them by how much energy they actually carry.

The Analogy:

• Old Method: Measuring a crowd by counting how many people are wearing red hats. If the hats are invisible (ghosts), you get confused and the count goes crazy.
• New Method: Measuring the crowd by how much footprint they leave on the floor. Even if a person is a ghost, if they are standing there, they take up space. If they aren't moving, they don't take up "energy space."

By switching to this "Energy Ruler," the authors found that:

1. Every wave they count has a clear, finite amount of energy.
2. The "ghost waves" are redefined so they don't carry infinite energy.
3. When you add them up now, the numbers stop growing and settle down to a real, sensible answer.

Why This Matters

This isn't just a math puzzle; it fixes real-world problems.

• Nanotechnology: As we build smaller and smaller devices (like chips for computers), light often gets "trapped" in tiny layers. The old math fails here, making it hard to design perfect lenses or sensors.
• Earthquakes: Sound waves behave similarly in layers of rock. If we want to predict how an earthquake wave bounces through different soil layers, we need math that doesn't explode.
• Quantum Computing: In the quantum world, particles act like waves. If the math describing how they scatter is broken, our quantum computers might not work correctly.

The Takeaway

The paper teaches us a profound lesson: Just because an equation is linear (simple and additive) doesn't mean the way we choose to write it down is correct.

Sometimes, the universe is fine, but our "language" for describing it is flawed. By changing the language to one that respects energy conservation (making sure we only count what actually carries energy), the math suddenly works again, and the "superposition breakdown" is fixed.

In short: They found a broken calculator, realized it was using the wrong units, switched to the right units, and now the calculator works perfectly again—even for the weirdest, most "ghostly" waves in the universe.

Technical Summary

1. Problem Statement

The superposition principle is a cornerstone of linear wave theory, asserting that solutions to linear wave equations can be represented as linear combinations of basis functions. However, the authors identify a fundamental failure of this principle in multilayered media when fields are expanded using conventional basis sets (e.g., standard plane waves or modal amplitudes) in the presence of evanescent or inhomogeneous waves.

• The Phenomenon: In systems with three or more interfaces (e.g., a two-layer stack), the infinite series of partial waves used in Airy-type formulations (summing multiple reflections and transmissions) can diverge.
• The Cause: This divergence is not a numerical artifact or a failure of linearity. Instead, it arises because evanescent waves decay spatially and do not carry net power flux across layers. Consequently, they cannot be normalized with respect to conserved energy flux in the conventional basis. When these non-normalizable components are included in infinite series expansions, the spectral radius of the event scattering operator exceeds unity, leading to mathematical divergence.
• Limitations of Existing Solutions: Traditional methods to handle this, such as analytic continuation, Abel summation, or introducing artificial absorption (regularization), are viewed by the authors as mathematical devices that lack physical causality or require ad hoc assumptions.

2. Methodology

The authors propose a rigorous reformulation of the wave representation to restore convergence without external regularization.

• Power-Flux Eigenmodes: Instead of using standard field amplitudes, the authors construct a new basis set: power-flux eigenmodes. These are defined as eigenfunctions of the power flux operator (specifically, the normal component of the Poynting vector for optics, or the Umov-Poynting vector for elasticity).
• Orthonormality: The new basis is orthonormal with respect to the power flux measure. This ensures that every mode in the expansion contributes a finite, physically meaningful amount of energy.
• Scattering Matrix Formalism:
  • The interface scattering matrices are reformulated in this new basis. Crucially, these matrices become unitary, preserving energy conservation at interfaces regardless of whether the waves are propagating or evanescent.
  • The propagation matrices within layers are also reformulated. In this basis, the eigenvalues of the propagation matrix are bounded by unity ($\leq 1$) for all wave types (propagating, evanescent, and inhomogeneous) in lossless, passive media.
• Global Event Operator: The total scattering process is described by a global event matrix ($E = P \cdot S$). The authors prove that in the power-flux basis, the spectral radius of this operator is $\leq 1$, guaranteeing the convergence of the Neumann series (the infinite sum of scattering events).

3. Key Contributions

1. Identification of Representation-Induced Breakdown: The paper demonstrates that the breakdown of superposition in multilayer systems is a consequence of the choice of basis representation, specifically the inability to normalize evanescent modes with respect to conserved energy flux.
2. Construction of Flux-Orthonormal Basis: The authors derive explicit mathematical forms for power-flux eigenmodes for scalar, electromagnetic, and elastic waves. These modes are constructed to be orthonormal with respect to the Poynting vector, ensuring finite energy contribution.
3. Proof of Unitarity and Boundedness: It is mathematically proven that in the flux-orthonormal basis:
   • Interface scattering is unitary.
   • Propagation eigenvalues are bounded ($\leq 1$).
   • The resulting series expansion for reflection and transmission coefficients is guaranteed to converge.
4. Generalization Across Wave Physics: The framework is shown to be applicable not just to optics (scalar and electromagnetic), but also to elastic waves (acoustics/phononics) and quantum systems (where probability current plays the role of power flux).

4. Results

The authors validate their theory through analytical derivations and numerical simulations:

• Analytical Verification: For a three-interface system ($n_2|n_1|n_2|n_1$), the authors calculate the eigenvalues of the event scattering matrix. In the standard basis, eigenvalues exceed unity in specific parameter regions (near critical angles), causing divergence. In the power-flux basis, all eigenvalues remain $\leq 1$.
• Numerical Simulation (Optics):
  • A simulation of an ultra-short optical pulse reflecting off a two-layer structure (TiO$_2$ and SiO$_2$) was performed.
  • Standard Basis: The reconstructed reflected field diverged as the number of scattering events increased, particularly when the transverse wavevector corresponded to evanescent modes in one layer.
  • Power-Flux Basis: The reconstruction remained stable and converged to a physically meaningful result, accurately capturing the pulse shape and energy conservation.
• Elastic Wave Extension: Similar results were obtained for elastic waves in a titanium layer. The standard mode expansion showed divergence and power imbalance (non-conservation of energy flux) in evanescent regimes. The power-mode expansion restored unitarity, ensured energy conservation, and eliminated divergence.
• Power Imbalance Resolution: The study highlights that in the standard basis, incident and reflected powers are not orthogonal in the presence of evanescent waves, leading to non-local dependencies. The power-flux basis restores orthogonality and local energy conservation.

5. Significance

This work provides a robust theoretical framework for modeling complex wave systems where evanescent fields are dominant, such as:

• Nanophotonics and Plasmonics: Where near-field coupling and evanescent modes dictate device performance.
• Quantum Information: Where unitarity is a fundamental constraint for scattering and transport.
• Seismology and Non-Destructive Testing: Where elastic wave scattering in layered media is critical.

The paper fundamentally shifts the perspective on wave superposition: it argues that for linear systems to be physically consistent and mathematically convergent, the basis set must be chosen to respect the system's conserved quantities (energy/power flux). By doing so, it eliminates the need for artificial regularization techniques and provides a unified, stable method for analyzing multilayer scattering across diverse physical domains.