Four-wave mixing and secondary radiations generated by nonharmonic two-color filaments in air: Influence of the Kerr and plasma nonlinearities

This study investigates the generation of tunable mid-infrared radiation and secondary satellites in air via two-color femtosecond filaments, revealing through experiments and simulations that while plasma nonlinearities broaden frequencies, the Kerr nonlinearity plays the dominant role in amplifying four-wave mixing signals prior to the emergence of weaker secondary radiations.

The Gist

Imagine you have two different musical instruments playing in a room: one is a deep, rumbling bass drum (the fundamental wave, around 800 nanometers), and the other is a higher-pitched flute (the seed wave, around 1.3 micrometers). When you blast these two sounds together at incredibly high volumes through a specific type of air, something magical happens. They don't just play side-by-side; they crash into each other and create entirely new notes that neither instrument could play on its own.

This paper is about discovering exactly how these new notes are made, specifically focusing on two types of "new music" that appear in the air: a deep, invisible bass note called mid-infrared radiation (around 3.3 micrometers) and some faint, ghostly echoes called secondary radiations.

Here is the breakdown of what the scientists found, using simple analogies:

1. The Two "Conductors" of the Air

The researchers discovered that the air itself acts like a conductor, but it has two different "modes" of reacting to the loud laser lights.

• The "Kerr" Conductor (The Instant Reaction): Think of this as the air's immediate, reflexive reaction. When the light hits the air molecules, they squish together instantly and bounce back. This is a fast, clean interaction. The paper shows that for the main "new notes" (the mid-infrared and visible light), this instant reaction is the primary engine. It's like the drumbeat that starts the song. Without this initial squish-and-bounce, the new notes wouldn't get loud enough to hear.
• The "Plasma" Conductor (The Electric Storm): If the light gets really intense, it strips electrons off the air molecules, turning the air into a tiny, glowing electric storm (plasma). This is a slower, messier reaction. The paper found that while this storm isn't the main reason the new notes are created, it acts like a soundboard. It takes the notes created by the "Kerr" conductor and stretches them out, making them broader and wider. It also helps create the faint "ghostly echoes" (secondary radiations).

2. The Main Discovery: Tuning the "Mid-IR" Radio

The team successfully built a setup where they could tune the "bass drum" (the seed wave) to different pitches. By doing this, they could tune the resulting mid-infrared "new note" to appear anywhere between 3 and 8 micrometers.

• The Analogy: Imagine you have a radio that only picks up one station. By tweaking the two original laser beams, the scientists showed they could tune this "air radio" to pick up a whole range of new stations in the mid-infrared spectrum. This is useful because it creates a powerful, tunable source of light that is hard to make with standard lasers.

3. The "Ghostly Echoes" (Secondary Radiations)

This is the most surprising part of the paper. Besides the main new notes, they found much fainter signals appearing at frequencies that are simple sums and differences of the original two beams (like $Frequency A + Frequency B$).

• The Catch: The paper claims these "ghostly echoes" cannot happen unless the main "visible" note (created by the Kerr effect first) is already loud and "broad" (spread out in frequency).
• The Analogy: Think of the visible light as a wide, loud shout. The "secondary radiations" are like the faint whispers that only appear if that shout is loud enough to shake the walls. If the shout is too quiet or too narrow (too focused), the whispers never appear. The "electric storm" (plasma) is needed to turn that loud shout into the whispers, but the shout itself must be created first by the "Kerr" reaction.

4. The Two Experiments

The scientists used two different setups to figure this out:

• Setup A (The "Long Focus"): They used lenses to focus the beams at slightly different spots. This let them see how the "electric storm" (plasma) stretched out the mid-infrared light. They saw that as the air got more ionized, the light got broader.
• Setup B (The "Short Focus"): They focused both beams to the exact same spot with lower energy. This allowed them to see the faint "ghostly echoes" clearly. They confirmed that these echoes only appeared when the visible light was broad enough and the air was slightly ionized.

The Bottom Line

The paper concludes with a clear rule for how this light mixing works:

1. First, the air's instant reaction (Kerr) must create and amplify the main new colors (visible and mid-infrared).
2. Second, if the light is strong enough to create an electric storm (plasma), that storm will stretch those colors out.
3. Third, the faint "ghostly echoes" (secondary radiations) only appear if the first step created a wide, broad visible light and the second step (plasma) happened to mix it further.

In short: The "Kerr" effect builds the house, and the "Plasma" effect adds the windows and the attic. You can't have the attic (the secondary radiations) without the house being built first.

Technical Summary

Technical Summary: Four-wave mixing and secondary radiations generated by nonharmonic two-color filaments in air

Problem Statement
The generation of coherent mid-infrared (mid-IR) radiation, particularly beyond 3 µm, is critical for applications in terahertz science, atto-science, and ultrafast spectroscopy. While solid-state emitters and fiber sources exist, they often suffer from limited output power and spectral tunability. Gaseous media, specifically air, offer broad transparency and nonlinear conversion capabilities via mechanisms like four-wave mixing (FWM). However, the dominant conversion mechanism for generating mid-IR radiation in air filaments—whether driven by the third-order Kerr nonlinearity or plasma nonlinearities (photoionization)—remains a subject of debate. Furthermore, while primary FWM modes (Stokes and anti-Stokes) are well-documented, the generation and origin of weaker "secondary radiations" (cascaded satellites at frequencies such as $\omega_1 \pm \omega_2$) in photocurrent-driven regimes have not been clearly identified or characterized.

Methodology
The study employs a dual approach combining two distinct experimental setups with comprehensive numerical modeling:

1. Experimental Setup 1 (ELI Beamlines, Czech Republic): A two-lens configuration focuses a fundamental wave (FW, $\sim$800 nm) and a seed wave (SW, tunable $\sim$1.3 µm) with different focal lengths ($f_1=96$ cm, $f_2=75$ cm). This setup allows for the optimization of mid-IR generation and the investigation of ionization-induced spectral broadening at higher pump energies (up to 3.1 mJ FW).
2. Experimental Setup 2 (LRC, Vilnius, Lithuania): A setup focusing both color components at the same propagation distance ($f=20$ cm or $f=100$ cm) but at lower pump energies (FW $\le$ 0.45 mJ). This configuration is designed to isolate and measure weaker secondary radiations emerging at frequencies $\omega_1 \pm \omega_2$ and their multiples, distinct from the primary FWM modes.
3. Numerical Modeling:
   • Local Current (LC) Model: Used to compute radiated spectra based on free electron density driven by photoionization (PPT rate) and bound electron nonlinearities (instantaneous Kerr and Raman-delayed responses).
   • Unidirectional Pulse Propagation Equation (UPPE): A 3D axis-symmetric solver used to simulate the full propagation dynamics, including linear dispersion, self-focusing, plasma defocusing, and the interplay between Kerr and plasma effects along the filament.

Key Contributions and Results

• Tunable Mid-IR Generation: The authors demonstrate the production of tunable mid-IR radiation centered around 3.3 µm (generated via $2\omega_{SW} - \omega_{FW}$) using two-color femtosecond filaments in air. By tuning the SW wavelength between 1.25 and 1.45 µm, they achieved mid-IR wavelengths ranging from 2.86 to 7.73 µm. The conversion efficiency was measured at approximately $3 \times 10^{-5}$, yielding mid-IR energies of 80–90 nJ.
• Role of Kerr vs. Plasma Nonlinearities:
  • Kerr Dominance: For pump energies limited to a few mJ (specifically up to $\sim$1–2 mJ), the generation and amplification of primary FWM modes (both visible and mid-IR) are driven primarily by the instantaneous Kerr nonlinearity. The energy scaling follows Kerr-type behavior ($\sim E_{SW}^2 E_{FW}$).
  • Plasma Broadening: While the Kerr effect seeds and amplifies the FWM modes, the plasma response (triggered by photoionization at higher intensities) does not necessarily increase the amplitude of these modes but significantly broadens their spectra.
  • Saturation: At higher pump energies (>5 mJ), saturation effects and destructive interferences between Kerr and plasma contributions limit the conversion efficiency.
• Characterization of Secondary Radiations: The study identifies and characterizes weaker "secondary radiations" located at frequency combinations $\omega_1 \pm \omega_2$ (e.g., $\sim$2 µm, $\sim$1 µm, and $\sim$492 nm) and their harmonics.
  • These emissions are observed only when a broadened visible FWM radiation is present.
  • Numerical simulations reveal that the early Kerr stage is essential to seed and amplify the primary FWM visible mode. Without this broadened visible component, the secondary radiations do not develop, even in the presence of plasma.
  • The secondary radiations act as "plasma spectral markers," emerging specifically when the plasma regime is active and coupled with the broadened visible spectrum.
• Mechanism of Secondary Radiation: Analytical calculations and simulations indicate that generating these secondary frequencies requires the visible FWM component to be spectrally detuned (broadened). A strict resonance condition without detuning fails to produce these frequencies; the spectral broadening induced by the Kerr and plasma stages allows the necessary frequency mixing to occur.

Significance and Claims
The paper claims to resolve the debate regarding the dominant mechanisms in two-color filamentation by disentangling the specific roles of Kerr and plasma nonlinearities. It establishes that:

1. The Kerr nonlinearity is the primary driver for the initial generation and amplification of FWM modes in the low-to-moderate energy regime.
2. Plasma nonlinearities are responsible for spectral broadening and the generation of secondary radiations, but only if the primary FWM modes have been sufficiently amplified and broadened by the Kerr stage.
3. The detection of these secondary radiations serves as a diagnostic tool ("plasma markers") to distinguish emissions triggered by optical nonlinearities from those created directly by the plasma.

The work provides a proof-of-principle for generating tunable mid-IR beams in air and offers a theoretical framework explaining the emergence of complex spectral features (secondary radiations) in filamentation, emphasizing the necessity of a broad visible spectrum to trigger these weaker plasma-driven emissions.