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Orbital and spin current density backflow in unidirectional monochromatic electromagnetic fields in vacuum

This paper investigates energy backflow in the Poynting vector and its orbital and spin current density components within 2-dimensional causal unidirectional monochromatic electromagnetic waves in vacuum, revealing unexpectedly strong spin current backflow and analyzing its correlation with local wavenumbers, intensity zeros, and vortices across various polarization states.

Original authors: Peeter Saari, Ioannis Besieris

Published 2026-04-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Peeter Saari, Ioannis Besieris

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 "River That Flows Backward"

Imagine you are standing on the bank of a massive river. You know for a fact that the water is flowing downstream (let's say, North). You throw a leaf in, and it drifts North. This is how light usually works: it travels in one direction, from the source to the destination.

However, this paper discovers something mind-bending: in certain very specific, tightly focused beams of light, tiny patches of the river actually flow backward (South), even though the entire river is supposed to be going North.

This phenomenon is called Energy Backflow. It's like seeing a wave in the ocean crash toward the shore, but then, for a split second, a tiny ripple inside that wave suddenly surges out to sea.

The Cast of Characters

To understand the paper, we need to meet the "characters" inside the light beam:

The Light Beam (The River): The researchers created a perfect, mathematical model of a light beam that only moves forward. It's a "unidirectional" wave.
The Poynting Vector (The Current): This is the measure of how much energy is moving and in which direction. Usually, it points forward. In this study, they found spots where it points backward.
Orbital Current (The Spin of the Water): Imagine the water in the river swirling around like a whirlpool. This is the "orbital" part of the light. It's like the light is spinning as it moves.
Spin Current (The Tug-of-War): This is a bit trickier. Light has an intrinsic "spin" (like a spinning top). The paper looks at how this spin contributes to the flow of energy.

The Big Discovery: The "Surprise" Backflow

The researchers looked at three types of light beams:

TE (Transverse Electric): Like a river where the water only moves side-to-side.
TM (Transverse Magnetic): Like a river where the magnetic forces move side-to-side.
Circular Polarization: Like a river where the water spirals as it moves.

The Shocking Result:
In most previous studies, backward flow was a tiny, weak glitch—maybe 1% of the forward flow. But in this specific setup, the researchers found that the Spin Backflow was incredibly strong.

The Analogy:
Imagine a giant conveyor belt moving boxes forward at 100 mph.

Old View: Occasionally, a single box might slide backward 1 mph.
This Paper's View: In the "Spin" version of the conveyor belt, the boxes are sliding backward at 100 mph right alongside the forward-moving boxes! It's a massive, powerful reverse current happening right next to the forward current.

The "Democracy" of Light

The paper introduces a concept called "Electric-Magnetic Democracy."

Usually, scientists might look at just the electric part of light or just the magnetic part to understand how it moves. It's like trying to understand a car by only looking at the engine, or only looking at the wheels.

The authors say, "No, we must look at both equally." They treated the electric and magnetic fields as equal partners. When they did this, they found that the electric and magnetic parts often cancel each other out or push against each other.

In Linear Polarization (TE/TM), the "Spin" part was a monster, pushing backward as hard as the light pushed forward.
In Circular Polarization, the electric and magnetic parts fought each other so much that they smoothed out the backward flow, making it much weaker.

The "Vortex" and the "Whirlpool"

The paper also maps out where these backward flows happen. They found that backward flow doesn't happen randomly; it happens at specific "traffic jams" in the light.

The Vortices: These are points where the light intensity drops to zero (dark spots). Think of these as the eye of a storm.
The Connection: The backward flow always happens right next to these dark spots. It's as if the light gets so confused trying to navigate around a "hole" in the beam that it spins backward for a moment.
Superoscillations: In these tiny dark spots, the light waves wiggle so fast and so tightly that they create "super-wiggles." It's like a guitar string vibrating so fast it creates a note higher than the string is physically capable of producing.

Why Does This Matter?

You might ask, "Who cares if a tiny bit of light flows backward?"

Particle Manipulation: If you are trying to use light to push tiny particles (like DNA or bacteria) around, knowing where the light flows backward is crucial. You could use these "backward currents" to trap particles in place or pull them toward you, acting like a microscopic vacuum cleaner.
Super-Resolution: This helps us see things smaller than the wavelength of light. By understanding these backward flows and "super-wiggles," we might be able to build microscopes that see details we thought were impossible to see.
Understanding Nature: It challenges our intuition. We think light is simple: it goes from A to B. This paper shows that light is actually a complex, swirling, sometimes contradictory dance of energy.

Summary in One Sentence

This paper reveals that in very specific, tightly focused beams of light, the "spin" of the light can create a powerful backward flow of energy right next to the forward flow, acting like a hidden reverse current in a river that only flows one way, which could be used to trap tiny particles or see the invisible.

1. Problem Statement

The paper investigates the phenomenon of energy backflow (reverse flow), where the Poynting vector points opposite to the predominant direction of wave propagation, specifically within unidirectional monochromatic electromagnetic waves in a vacuum.

Context: While energy backflow is known in tightly focused beams and superpositions of plane waves, it is counterintuitive in unidirectional waves where all constituent plane waves travel forward.
Gap in Literature: Previous studies (e.g., by Kotlyar et al.) focused on Transverse Electric (TE) fields but often neglected the magnetic contribution to spin flow or failed to apply "electric-magnetic democracy" (symmetric treatment of electric and magnetic fields). Furthermore, previous models often relied on numerical integrations or were limited to specific axes rather than the full 2D space.
Objective: To provide a comprehensive, analytical study of both orbital and spin current density backflows, including their separate electric and magnetic contributions, using a "democratic" decomposition of the Poynting vector.

2. Methodology

The authors employ an exact analytical approach based on a specific 2D unidirectional wave function derived in their previous work [17].

Wave Function: The study utilizes a closed-form solution $W(x, z)$ $W (x, z)$ expressed via fractional-order Bessel functions ( $J_{\pm 1/4}$ $J_{\pm 1/4}$ ). This function describes a highly non-paraxial, unidirectional wave propagating along the $z$ $z$ -axis.
- Unlike previous works, this expression is valid over the entire 2D space, allowing for exact computation without numerical integration.
Field Construction:
- TE Fields: Constructed using a magnetic Hertz vector $\vec{\Pi}_m = (0,0,1)W(x,z)$ .
- TM Fields: Constructed using an electric Hertz vector $\vec{\Pi}_e = (0,0,1)W(x,z)$ .
- Circular Polarization: Constructed using $\vec{\Pi}_m = (1, i, 0)W(x,z)$ .
Decomposition of the Poynting Vector:
- The time-averaged Poynting vector $\vec{P}$ is decomposed into orbital ( $\vec{P}_o$ ) and spin ( $\vec{P}_s$ ) components.
- Crucially, the authors apply "electric-magnetic democracy," expressing both orbital and spin components as the average of their electric-only and magnetic-only parts:
  $\vec{P}_o = \frac{1}{2}(\vec{P}_{oe} + \vec{P}_{om}), \quad \vec{P}_s = \frac{1}{2}(\vec{P}_{se} + \vec{P}_{sm})$
- This ensures a unique and physically consistent splitting of the energy flux.
Analysis: The authors analyze the longitudinal projection ( $z$ -component) of these vectors, mapping backflow regions ( $P_z < 0$ ), vortices, local wavenumbers ( $k_{z,loc}$ ), and intensity zeros.

3. Key Contributions

Exact Analytical Framework: The use of a closed-form wave function allows for the precise calculation of Poynting vector components and their derivatives across the entire 2D plane, avoiding the approximations and numerical errors of previous studies.
Democratic Decomposition: The study rigorously applies the "electric-magnetic democracy" principle, revealing that neglecting magnetic contributions (as done in some TE-only studies) leads to an incomplete picture of spin flow.
Discovery of Strong Spin Backflow: The paper identifies that spin current backflow can be as strong as the forward flow, a finding that contrasts sharply with the typical behavior where backflow is usually an order of magnitude weaker.
Correlation with Singular Optics: The authors systematically correlate the locations of backflow peaks with intensity zeros, phase singularities (vortices), and "superoscillations" (gigantic local wavenumbers).

4. Key Results

A. Scalar Wave Analysis

In the scalar version of the wave, backflow peaks are located near intensity zeros and are accompanied by vortices.
Backflow is approximately 25 times weaker than the forward flow.
Vortices occur at points where intensity is zero and the local wavenumber diverges (superoscillations).

B. Linear Polarization (TE and TM)

Spin Backflow: In TE fields, the electric part of the spin flow vanishes, but the magnetic part ( $\vec{P}_{sm}$ ) is non-zero. Remarkably, the spin backflow is as strong as the forward flow near the origin.
Orbital Backflow: The orbital backflow is much weaker (an order of magnitude) than the forward flow. The electric and magnetic parts of the orbital flow exhibit mutually exclusive maxima and minima.
Total Poynting Vector: The total backflow is suppressed by the strong forward orbital flow but remains present.
Spatial Distribution: Unlike scalar backflow (island-like), spin backflow in TE fields occurs in river-like areas.

C. Circular Polarization

Spin Flow: Both electric and magnetic parts of the spin flow are present. The total spin backflow is roughly half the strength of the forward flow (weaker than in the TE case).
Origin Behavior: Unlike the TE case, spin backflow is absent exactly at the origin; instead, the spin flow circulates within the origin region.
Total Backflow: The total energy backflow is significantly suppressed (by a factor of ~10) by the strong forward orbital flow. The backflow is restricted to a small region near $z=0$ .
Vortices vs. Saddle Points: The study identifies that while vortices coincide with intensity zeros and flow sign flips, saddle points (hyperbolic points) also appear where intensity vanishes and local wavenumbers diverge, alternating with vortices.

D. Correlation with Local Wavenumbers

Backflow peaks are consistently found near intensity zeros where the phase derivative (local wavenumber) exhibits singularities.
In scalar and TE/TM cases, the sign of the local wavenumber $k_{z,loc}$ matches the sign of the energy flow $P_z$ .
In circular polarization, this simple sign relation breaks down due to the vector nature of the fields, but the spatial coincidence of singularities, zeros, and flow boundaries remains.

5. Significance

Fundamental Physics: The paper challenges the intuition that unidirectional waves cannot exhibit significant backflow. It demonstrates that spin angular momentum can drive energy flow in the reverse direction with magnitude comparable to forward propagation.
Singular Optics: It provides a detailed map of the relationship between energy backflow, optical vortices, and superoscillations, reinforcing the concept that these phenomena are intrinsically linked at intensity zeros.
Applications: The findings are relevant for:
- Particle Manipulation: Understanding reverse forces in optical tweezers.
- Superresolution: Utilizing superoscillatory regions for imaging beyond the diffraction limit.
- Beam Design: Informing the design of non-paraxial beams where controlling spin-orbit interactions is critical.

In conclusion, the authors demonstrate that while total energy backflow in these specific unidirectional fields is often suppressed by strong forward orbital flow, the spin current density backflow is unexpectedly robust, reaching the magnitude of forward flow in linearly polarized cases. This highlights the critical importance of including magnetic contributions and spin dynamics in the analysis of optical energy transport.