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A Unified Perspective on Causality and One-Sided System Responses in Time and Space Across Physical and Fourier Domains

This paper reviews the connection between causality and one-sided system responses in time and frequency domains, then investigates the feasibility of achieving one-sided spatial nonlocality in nonlocal flat optics, where fundamental obstacles are uncovered that raise the open question of whether such responses are merely platform-incompatible or fundamentally forbidden by nature.

Original authors: Mohammadreza Salehi, Francesco Monticone

Published 2026-07-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohammadreza Salehi, Francesco Monticone

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

The Big Picture: The "One-Way Street" of Physics

Imagine the universe has a set of traffic rules. The most famous rule is Causality: you can't get an effect before you have a cause. If you throw a ball (the cause), the window breaks (the effect) after the throw, not before. In physics, this is usually about time.

This paper explores a fascinating idea: What if we could create "one-way streets" not just in time, but in frequency (color of light), space (direction of waves), and even non-locality (how things talk to each other across a distance)?

The authors, Salehi and Monticone, act like detectives mapping out a four-quadrant grid. They look at how systems respond to inputs in different domains (Time vs. Space, and Real World vs. Frequency World). They find that three of these four quadrants are possible to engineer, but the fourth one hits a wall.

Quadrant 1: The Classic Rule (Time)

The Concept: This is the standard rule of the universe.
The Analogy: Think of a domino chain. You push the first domino (cause), and the rest fall in a line (effect). The dominoes never fall before you push them.
The Science: In materials, this means if you shine light on something, the material's reaction happens after the light hits it. This creates a mathematical link (called Kramers-Kronig relations) between how much a material absorbs light and how it bends it. You can't change one without changing the other.

Quadrant 2: The "Color Shifter" (Frequency)

The Concept: One-sided frequency conversion.
The Analogy: Imagine a magic DJ booth. Usually, when music plays, it can go up or down in pitch. But this is a DJ that only plays songs faster than the original. If you play a slow song, it instantly speeds it up. It never slows it down.
The Science: The authors explain that if you rapidly change a material's properties over time (like a time-varying lens), you can force light to change its color (frequency) in only one direction (e.g., always turning red light into blue light, never blue into red).
The Result: This allows for "reflectionless absorbers." Imagine a wall that swallows a sound wave completely because it keeps shifting the sound's pitch so fast that the sound can never bounce back to you. It's like a black hole for specific frequencies.

Quadrant 3: The "One-Way Mirror" (Space/Momentum)

The Concept: One-sided wavevector conversion.
The Analogy: Imagine a river with a strong current. If you throw a leaf in, it flows downstream. If you try to throw a leaf upstream against the current, the river pushes it back. But in this engineered material, the "river" is the material itself. It allows waves to pass through in one direction (absorbing them) but acts like a solid wall if they try to come from the other side.
The Science: By carefully designing how a material changes as you move across it (spatially), they can create a slab that absorbs all waves coming from the left, no matter the angle, but reflects waves coming from the right. It's a "one-way" absorber.

Quadrant 4: The Mystery (Spatial Nonlocality)

The Concept: One-sided spatial nonlocality.
The Analogy: Imagine a telepathic conversation. Usually, if Person A talks to Person B, the message goes both ways. "Nonlocality" means Person A can talk to Person B even if they are far apart, without a direct wire.
"One-sided nonlocality" would be a scenario where Person A can hear Person B, but Person B cannot hear Person A, even though they are in the same room. Or, imagine a camera that only takes pictures of things to its left, but is completely blind to things on its right, even though the lens is wide open.
The Science: The authors tried to build this using multilayer flat optics (stacks of thin glass layers). They wanted to see if they could make a system where the output at one point depends only on inputs from one side (e.g., only from the left), ignoring everything on the right.
The Finding: They hit a fundamental roadblock. When they tried to calculate the math for these stacks of layers, they found that the "traffic rules" of physics (specifically Titchmarsh's theorem) break down. The math shows that poles and "branch cuts" (mathematical singularities) appear in places they shouldn't.
The Conclusion: It seems nature might fundamentally forbid this specific type of "one-sided telepathy" in these flat optical structures. The paper asks: Is this just a failure of our current building blocks (the glass layers), or is it a law of nature that says "You cannot have a one-sided nonlocal response"?

Summary

The paper is a tour of what is possible in physics:

Time: We know this works (cause before effect).
Frequency: We can engineer materials to shift colors in only one direction.
Space: We can engineer materials to absorb waves from one side but reflect them from the other.
Nonlocal Space: We tried to make materials that "listen" only to one side of space, but the math suggests this might be impossible, at least with the technology we currently use.

The authors leave us with a big question: Is this impossibility just a limitation of our current tools, or is it a fundamental rule of the universe that we haven't discovered yet?

Technical Summary: A Unified Perspective on Causality and One-Sided System Responses

Problem Statement
The principle of causality is a fundamental constraint in physics, mathematically formalized by Titchmarsh's theorem and the Kramers–Kronig relations. In the temporal domain, this principle dictates that a system's response cannot precede its excitation, linking the real and imaginary parts of frequency responses. While this temporal causality is a universal law, recent research has demonstrated that analogous "one-sided" responses can be engineered in other domains:

Spectral Causality: One-sided frequency conversion in time-varying structures.
Spatial Momentum Conversion: One-sided wavevector conversion in spatially inhomogeneous structures.

However, a critical gap remains in the taxonomy of one-sided responses: one-sided spatial nonlocality. This refers to a scenario where a system's spatial response depends on inputs from an extended region in only one direction (e.g., the output at a point depends only on inputs to its left, but not its right). While one-sided responses in time, frequency, and wavevector have been successfully realized, the feasibility of one-sided spatial nonlocality, particularly within the framework of nonlocal flat optics (multilayer structures), remains an open and unexplored problem.

Methodology
The authors employ a unified mathematical framework based on Titchmarsh's theorem to analyze one-sided responses across four quadrants: time, space, and their respective Fourier domains.

Review of Established Analogues:
- Spectral Causality (Time-Varying): The paper reviews how time-modulated materials satisfying a temporal analog of Kramers–Kronig relations can achieve one-sided frequency conversion (e.g., exclusively up-conversion). This is achieved by ensuring the susceptibility kernel $\tilde{\chi}(\omega, \omega')$ vanishes for $\omega < \omega'$ , effectively suppressing backward scattering and enabling applications like reflectionless absorption and temporal cloaking.
- One-Sided Wavevector Conversion (Spatially Varying): The authors examine spatially inhomogeneous structures where the susceptibility $\chi(z)$ satisfies spatial Kramers–Kronig relations. This allows for the engineering of materials that suppress back-scattering for waves propagating in a specific direction, realizing angle-independent reflectionless absorbers.
Investigation of One-Sided Spatial Nonlocality:
- The core investigation focuses on realizing a one-sided spatial response in nonlocal flat optics, specifically using multilayer dielectric structures.
- The system is modeled as a convolution in real space ( $E_{out} = F * E_{in}$ ), where the nonlocal kernel $F(x)$ is determined by the structure's transfer function $\tilde{F}(k_x)$ in the Fourier (wavevector) domain.
- To achieve one-sided nonlocality, $\tilde{F}(k_x)$ must satisfy a Kramers–Kronig relation that enforces analyticity in a specific half-plane, analogous to the conditions for temporal or spectral causality.
- The authors utilize a scattering-matrix formulation to calculate the transfer function of arbitrary multilayer stacks composed of anisotropic (and potentially nonreciprocal) media. This method ensures numerical stability by avoiding the exponential growth/decay issues associated with standard ABCD matrix approaches.
- The analysis considers both lossless and lossy cases, examining the location of poles and branch cuts of the transfer function in the complex $k_x$ plane.

Key Results

Validation of Known Domains: The paper confirms that one-sided responses in frequency (via time modulation) and wavevector (via spatial grading) are achievable through careful engineering of Kramers–Kronig relations in the respective domains.
Obstacles in Spatial Nonlocality: When applying this framework to multilayer structures for spatial nonlocality, the authors find fundamental obstacles.
- In the lossless case, the transfer function $\tilde{F}(k_x)$ exhibits poles and branch points on the real axis, preventing the necessary analyticity in the complex half-plane required for one-sidedness.
- In the lossy case, introducing infinitesimal loss moves these singularities (poles and branch cuts) into both the upper and lower halves of the complex $k_x$ plane.
- Consequently, the conditions of Titchmarsh's theorem required to enforce a one-sided spatial response are violated in both lossless and lossy scenarios for standard multilayer flat optics.

Significance and Claims
The paper does not claim to have successfully realized one-sided spatial nonlocality. Instead, its primary contribution is the identification of a fundamental barrier to its realization within the widely used platform of nonlocal flat optics.

Unification: The work provides a cohesive theoretical perspective linking causality, spectral causality, and spatial momentum conversion, organizing them into a unified taxonomy of one-sided system responses.
Negative Result as Insight: By demonstrating that standard multilayer structures cannot support one-sided spatial nonlocality due to the unavoidable presence of singularities in the complex wavevector plane, the authors raise a profound open question: Is one-sided spatial nonlocality merely incompatible with current flat-optics platforms, or is it fundamentally forbidden by nature itself?
Future Direction: The paper suggests that if such a response is possible, it likely requires a platform beyond conventional multilayer dielectrics, potentially pointing toward the rich field of metamaterials and artificial spatial nonlocality as the necessary ground for further exploration.

In summary, the paper serves as a rigorous review of existing one-sided phenomena and a critical investigation into a new frontier, concluding that while the mathematical framework exists, the physical implementation in multilayer optics faces insurmountable analyticity constraints, leaving the fundamental possibility of one-sided spatial nonlocality as an open scientific inquiry.