Beam tube boundary effects in stray light modeling of long Fabry-Perot arm cavities for third-generation gravitational-wave detectors

This paper introduces a waveguide-like mode description to model beam tube boundary effects in third-generation gravitational-wave detectors, demonstrating that such effects are subdominant in densely baffled cavities and thereby validating the continued use of standard FFT-based tools for their design.

The Gist

The Big Picture: The "Long Hallway" Problem

Imagine you are trying to send a laser beam down a hallway that is 40 kilometers long (about 25 miles). This isn't a normal hallway; it's a vacuum tube inside a future gravitational wave detector (like the "Cosmic Explorer").

The goal is to keep the laser beam perfectly straight and focused, bouncing back and forth between two mirrors at the ends. However, the hallway has walls (the beam tube), and along the way, there are "baffles" (like trash cans or shields) meant to catch any stray light that bounces off the walls.

The Problem:
Scientists use computer programs to design these hallways. The most popular program (called SIS) treats the hallway as if it were open air. It assumes the laser beam can expand infinitely without hitting anything.

• The Risk: In reality, the laser does hit the walls of the tube. If the computer ignores the walls, it might miss how the light scatters, which could create "noise" that hides the gravitational waves we are trying to detect.

The Solution:
The authors of this paper built a new computer model that treats the hallway like a giant, hollow pipe (a waveguide). They forced the light to obey the rule: "You cannot go through the walls."

They then compared their "Pipe Model" against the popular "Open Air Model" to see if the difference mattered.

---

The Key Analogy: The Pipe vs. The Open Field

Think of the laser beam like a giant, invisible water hose spraying water.

1. The "Open Air" Model (Old Way):
   Imagine spraying water in a massive, empty field. The water spreads out in a perfect circle. If you put a small bucket (a baffle) in the way, the water splashes off it, creating a messy spray pattern. The computer calculates this splash perfectly, assuming the water can go on forever.

2. The "Pipe" Model (New Way):
   Now, imagine that same hose inside a giant, narrow metal pipe. When the water hits the pipe walls, it bounces back. The shape of the water spray changes completely because it's confined.

The Question: Does it matter if we are in the field or the pipe?

• The Answer: For the main stream of water (the part hitting the mirrors), it doesn't matter much. Both models look the same.
• The Catch: For the "splashes" and "mist" near the edges (the stray light), the pipe model is very different. The walls contain the mess.

---

What They Actually Did

The authors did three main things:

1. They invented a new "Language" for light.

Instead of describing the light as a simple smooth wave (like a Gaussian beam), they described it as a stack of specific "vibrations" inside a pipe.

• Analogy: Think of a guitar string. It can vibrate in simple ways (one bump) or complex ways (many bumps). The authors listed all the possible ways light can vibrate inside a cylindrical tube. This allowed them to calculate exactly how the light behaves when it hits the walls.

2. They tested the "Baffles" (The Trash Cans).

They simulated a hallway filled with 200 circular shields (baffles) spaced out along the 40km tube.

• The Discovery: These baffles act like a sieve. As the light travels down the tube, the baffles catch the "messy" outer edges of the light beam.
• The Result: By the time the light reaches the end, the "messy" part that would have hit the walls has been filtered out. The light inside the main area looks almost exactly the same in both the "Pipe Model" and the "Open Air Model."

3. They checked for "Wobbles" and "Dents."

What if a baffle is slightly off-center (wobbly) or the tube has a small dent?

• The Finding: If the baffles are dense (close together), they act as a shield. They stop the light from hitting the walls, so a wobble or a dent doesn't cause much noise.
• The Warning: If the baffles are sparse (far apart), the light has more room to hit the walls. In this case, a small wobble creates a big noise problem.

---

The Bottom Line: Why Should We Care?

The authors wanted to know: "Can we keep using the old, simpler computer code (SIS) for designing these massive new detectors?"

The Verdict: Yes, but with conditions.

• Good News: As long as the detectors are designed with many baffles (dense shielding) and the components are stable (not wobbling wildly), the "Open Air" model is accurate enough. The walls of the tube don't ruin the experiment.
• The Caveat: If you design a detector with very few baffles, the walls do matter, and you need the fancy new "Pipe Model" to get it right.

Summary in a Sentence

The paper proves that for future giant gravitational wave detectors, as long as you pack the vacuum tubes with enough "trash cans" (baffles) to catch stray light, you can safely ignore the fact that the light is bouncing off the walls, and the simpler computer models will still work perfectly.

Technical Summary

1. Problem Statement

Third-generation (3G) gravitational-wave (GW) detectors, such as the Einstein Telescope (ET) and Cosmic Explorer (CE), will utilize extremely long Fabry–Perot arm cavities (10–40 km) housed within vacuum beam tubes.

• Stray Light Noise: Scattered light (stray light) from optical components is a critical noise source. It can recombine with the resonant laser field, introducing phase and amplitude noise that couples to mechanical motions of non-optical components, degrading detector sensitivity.
• Modeling Limitations: Current design tools rely heavily on Fast Fourier Transform (FFT)-based paraxial codes (e.g., the Stationary Interferometer Simulation, SIS). These tools propagate fields in free space and do not explicitly enforce the boundary conditions of the vacuum beam tube.
• The Core Question: For 3G detectors with long baselines and dense baffle arrays, is the interaction between the optical field and the beam tube wall negligible? If wall interactions are significant, free-space FFT models may yield inaccurate predictions for stray light noise and baffle layouts.

2. Methodology

The authors introduce a waveguide-like modal description of the optical field to explicitly incorporate the beam tube boundary condition.

• Waveguide Basis: Instead of using standard free-space Gaussian modes (Hermite-Gauss or Laguerre-Gauss), the authors solve the scalar Helmholtz equation in cylindrical coordinates with Dirichlet boundary conditions ($\psi=0$ at the tube radius $R$).
  • The eigenmodes are defined as $\psi_{mn}(r, \phi, z) = J_m(\frac{\alpha_{mn}r}{R}) \cos(m\phi) e^{-i\beta_{mn}z}$, where $\alpha_{mn}$ are roots of Bessel functions.
  • This basis naturally accounts for the confinement of the field within the tube.
• Modal Mixing Matrices:
  • Propagation: Modeled as a diagonal matrix modifying the phase of each mode based on propagation distance.
  • Optical Elements (Mirrors/Baffles): Modeled as multiplicative transverse operators. The authors derive analytical closed-form series for the mode-mixing matrices of axisymmetric circular apertures (mirrors and baffles).
  • Miscentering: They derive expressions for baffles displaced from the tube axis, which break azimuthal symmetry and induce mode coupling between different angular indices ($m$).
• Numerical Implementation:
  • The steady-state intracavity field is computed by solving the self-consistency equation $(I - A)\mathbf{c}_I = \mathbf{T}_{ITM}\mathbf{c}_{in}$, where $A$ is the round-trip operator.
  • To handle numerical instabilities with high-order modes, the authors implement an efficient numerical integration strategy using Gauss-Legendre quadrature, which is significantly faster and more stable than direct numerical integration or high-order analytical series summation.

3. Key Contributions

1. Independent Benchmarking Tool: The waveguide model serves as an independent benchmark for existing FFT-based tools (like SIS), allowing for a systematic assessment of when beam tube boundary effects can be safely neglected.
2. Analytical Clipping Matrices: Derivation of closed-form series for the coupling matrices of circular apertures within a waveguide basis, validated against numerical integration.
3. Noise Coupling Metrics: Development of a method to quantify strain-equivalent noise couplings resulting from:
   • Transverse miscentering of baffles.
   • Localized defects in the beam tube wall (e.g., ovalization or protrusions).

4. Key Results

• Field Reconstruction: The waveguide basis accurately reconstructs Gaussian beams and clipped fields. While the fundamental mode dominates inside the mirror aperture, the waveguide model correctly captures the field behavior near the tube wall, which differs from free-space diffraction patterns (e.g., the absence of free-space Airy rings outside the aperture).
• Baffle Filtering Effect: In cavities with dense baffle arrays (e.g., 200 baffles over 40 km):
  • The baffles act as spatial filters, progressively suppressing the diffractive "halo" (large-radius field components).
  • Consequently, the field incident on the beam tube wall is significantly reduced.
  • Agreement with SIS: For the resonant field within the mirror aperture and for small baffle motions, the waveguide results agree closely with SIS predictions. The beam tube boundary condition has a subdominant impact in the regime of densely baffled cavities.
• Noise Sensitivity:
  • Miscentering: The phase noise coupling from a displaced baffle is suppressed as baffle density increases. Sparse baffles allow the diffracted halo to propagate further, making the system more sensitive to miscentering and tube defects.
  • Tube Defects: Localized defects in the tube wall (modeled as off-center apertures) induce measurable phase noise only when the baffle density is low. In dense configurations, the baffles shield the tube, mitigating the defect's impact.
• Quantitative Comparison: In the relevant regime for 3G designs (densely baffled, small perturbations), the waveguide and SIS models show consistent qualitative and quantitative behavior, validating the continued use of SIS for design guidance.

5. Significance

• Validation of Current Design Tools: The study confirms that for the specific design parameters of 3G detectors (long arms with dense baffle arrays), the assumption of free-space propagation used in current FFT codes is valid. The beam tube boundary effects do not significantly alter the resonant field or the primary noise couplings of interest.
• Design Guidance: The results support the continued use of SIS for guiding baffle layout studies. However, the authors emphasize that for unbaffled cavities or configurations with sparse baffles, the beam tube boundary becomes critical, and waveguide modeling is necessary.
• Noise Mitigation Strategy: The paper quantifies the importance of baffle density. It demonstrates that increasing baffle density not only blocks stray light directly but also acts as a filter that suppresses the large-radius field components responsible for coupling with tube imperfections and misaligned components.

In conclusion, this work provides a rigorous theoretical framework and numerical validation that reassures the GW community that current modeling tools are sufficient for 3G detector design, provided that the cavities are sufficiently baffled to suppress diffractive halo interactions with the vacuum tube.