Intensity Doubling Feature in Loop-Soups & Photonics

• Intensity Doubling Feature is a phenomenon where dual mechanisms yield intensity-doubled outputs, both as a macroscopic cycle decomposition in loop-soups and frequency conversion in SHG.
• In high-dimensional loop-soups, scaling limits and switching properties split clusters into genuine large loops and emergent ghost cycles, each contributing equally to a doubled Poisson intensity.
• In nonlinear photonics, coherent quadratic polarization in nonlinear media drives efficient second harmonic generation, converting input photons into output at twice the fundamental frequency.

The term intensity doubling feature refers to rigorous phenomena in two distinct scientific domains: (1) high-dimensional statistical mechanics, specifically the scaling limits of critical Brownian loop-soups on cable-graphs, and (2) nonlinear photonics, where it describes the generation of light at twice the fundamental frequency via second harmonic generation (SHG). In the former, intensity doubling emerges as a universality property of macroscopic cycles in loop soups and Gaussian free fields (GFF). In the latter, it reflects coherent wave-mixing processes that convert input photon flux into output at doubled energy. Both instances rely on deep properties of the underlying physical or mathematical systems, such as Poissonian loop ensembles, self-avoiding cycle formation, constructive interference, and phase matching.

1. Intensity Doubling in Critical Brownian Loop-Soups

A critical Brownian loop-soup on the cable-graph of $\mathbb{Z}^d$ (for $d \geq 7$), as defined in (Lupu et al., 26 Nov 2025), is a Poissonian ensemble of unrooted Brownian loops with intensity parameter $\lambda_c$, the unique critical value where percolation (appearance of infinite clusters) occurs. Each cluster corresponds to a connected component of the union of loop traces. "Macroscopic loops" are defined as those of diameter comparable to the system size $N$; "proper macroscopic cycles" are simple closed paths (self-avoiding cycles) of diameter at least $\epsilon N$ that are not reducible to tree-like structures at large scales.

The key intensity doubling feature uncovered is as follows: In the scaling limit, the collection of all clusters containing proper macroscopic cycles in a large box $[-N, N]^d$ decomposes into two asymptotically equally probable and law-identical families:

• (a) Clusters containing exactly one macroscopic Brownian loop;
• (b) Clusters with no loop above a mesoscopic cutoff but supporting macroscopic cycles assembled from chains of small loops (the "ghost loop-soup").

The random collection of macroscopic cycles, rescaled, converges to a continuum Brownian loop-soup of intensity $2\lambda_c$; one "copy" arises from actual large loops, the other from these emergent "ghost" cycles (Lupu et al., 26 Nov 2025).

2. Main Theorem and Structural Decomposition

Let $\mathcal{C}(\epsilon, N)$ denote the set of clusters in $\Lambda_N = [-N, N]^d$ containing a proper macroscopic cycle, and let $\mathcal{B}(\epsilon,N)$ be those Brownian loops whose trace carries such a cycle. For suitable thresholds $\beta > 4/(d-2)$ and $\gamma > 2/(d-4)$, and as $N \to \infty$, the probability that a cluster in $\mathcal{C}(\epsilon, N)$ falls into each of the two types (a) or (b) approaches $1/2$. Furthermore, the law of the collective rescaled cycles tends to that of a Poisson random collection of continuum Brownian loops in $[-1,1]^d$ with intensity $2\lambda_c$—half from genuine large loops and half, independently, from the "ghost loop-soup" generated by chains of small loops (Lupu et al., 26 Nov 2025).

The mechanism relies crucially on the switching property: Conditioned on the presence of a nontrivial topological winding (around a codimension-2 subspace), the parity of the total winding number of constituent loops is uniform. As a result, the cluster "chooses" with equal probability to host either a large loop or, alternatively, only an emergent macroscopic cycle assembled from many small loops.

3. Mathematical Framework and Key Formulas

At the critical value, cable-graph loop-soup clusters exhibit two-point connectivity

$$P[x \leftrightarrow y] = \text{const} \cdot g(x,y) \approx \text{const} \cdot |x-y|^{2-d},$$

paralleling high-dimensional critical percolation. The regime is "mean-field," facilitating precise probabilistic estimates.

Critical scaling parameters:

• Loops above diameter $N^\beta$, with $\beta > 4/(d-2)$, are sufficiently rare to ensure clusters are dominated by a single macroscopic loop or none at all.
• Proper macroscopic cycles can be formed in the absence of such loops by concatenating roughly $N/N^\gamma$ small loops ($\gamma > 2/(d-4)$).

The "ghost" loop-soup, denoted $\mathcal{L}_{\rm ghost}$, is realized as the limit of macroscopic cycles of type (b) and is in law a Poisson ensemble of intensity $\lambda_c$. Thus, the sum of both mechanisms results in a total intensity $2\lambda_c$ for macroscopic cycles in the scaling limit (Lupu et al., 26 Nov 2025).

4. Gaussian Free Field Connection and Scaling Limit

By prior results (Le Jan; Lupu), at $\lambda_c$, the collection of loop-soup clusters is equidistributed with the excursion (sign-cluster) structure of the Gaussian free field (GFF) on the cable-graph. The intensity doubling feature therefore translates to the following: In high dimensions ($d\geq7$), boundaries of GFF sign-clusters in the continuum limit form a Brownian loop-soup with double the critical Conformal Loop Ensemble (CLE) intensity. That is, macroscopic interfaces in the GFF carry the statistical law of a Poisson process of Brownian loops at intensity $2\lambda_c$ (Lupu et al., 26 Nov 2025).

This proves a prediction from Lupu (2022) (Lupu, 2022), establishing rigorously the presence of doubled intensity in the scaling limit for these models.

5. Proof Techniques and Structural Insights

The demonstration of intensity doubling necessitates:

• A loop-soup switching lemma on the cable-graph, which ensures clusters with nontrivial windings have equally probable decompositions into the two macroscopic cycle types.
• First moment and BK-type bounds that exclude multiple macroscopic loops per cluster and guarantee macroscopic loops are unpinched (their only non-trivial large cycles are essentially themselves).
• Universal-cover arguments to estimate the probability and tightness of macroscopic cycles assembled from small-loop chains.
• Multi-scale analysis distinguishing macroscopic ($N$), mesoscopic ($N^\beta$), and small-loop ($N^\gamma$) regimes, essential for controlling unwanted cluster events.

The methods rely on mean-field behavior and are robust for $d \gg 6$, where the two-point function decay is $|x-y|^{2-d}$.

6. Broader Context, Universality, and Outlook

This intensity doubling—where the scaling limit produces a law corresponding to twice the critical Poisson intensity—reflects a deep universality in high dimensions. The phenomenon extends, with model-dependent variations, to other percolation-type systems displaying similar two-point decay, as shown in work by Carpenter–Werner (2025). For $d=3,4,5,6$, the situation remains open, with expectations of non-mean-field geometry and possible breakdown of the doubling feature.

A key implication is that sign-cluster boundaries for the GFF (in sufficiently high $d$) do not merely correspond to the critical CLE law, but rather its intensity-doubled version. The thresholds on $\beta$ and $\gamma$ are sharp, and attempts to modify them yield failures in the decomposition: for instance, mesoscopic loops above $N^{\beta'}$ appear as soon as $\beta' < 4/(d-2)$.

In summary, the intensity doubling feature for high-dimensional Brownian loop-soups and associated GFF models reflects an exact symmetry between clusters containing a genuine large loop and those where a macroscopic cycle is built from small loops, yielding a scaling limit in which macroscopic cycles are described by a Poisson loop-soup of intensity $2\lambda_c$ (Lupu et al., 26 Nov 2025).

7. Intensity Doubling in Nonlinear Photonics

In a distinct context, intensity doubling also references frequency-doubling (SHG) in nonlinear optics. Here, the output intensity at $2\omega$ results from coherent quadratic polarization in a nonlinear medium:

$$P^{(2)}(2\omega) = \varepsilon_0 \chi^{(2)} E(\omega)^2$$

In continuous-wave, high-power regimes, frequency-doubling is often realized via intracavity SHG, as in ring-cavity setups with lithium triborate (LBO) crystals. For example, as demonstrated in (Kwon et al., 2019), 18.8 W of 1540 nm input is frequency-doubled to yield 14.0 W at 770 nm (conversion efficiency $\eta = 74\%$).

The intensities of the input and output fields obey a nonlinear relation dictated by the mode structure, cavity enhancement factor, phase matching, and effective nonlinear coefficient $d_{\rm eff}$. In such systems, the physical process results in an optical output at exactly twice the frequency (half the wavelength) of the input.

The system performance depends critically on impedance matching, mode-matching efficiency, thermal management, and robust cavity locking; see the detailed parameter table below.

System Component	Value or Description	Reference
Input Power	18.8 W at 1540 nm	(Kwon et al., 2019)
Output Power	14.0 W at 770 nm (SHG)	(Kwon et al., 2019)
Conversion $\eta$	74%	(Kwon et al., 2019)
Nonlinear Crystal	LBO, 30 mm, Type-I NCPM	(Kwon et al., 2019)
Cavity Build-up	$\sim$19× (Finesse $F$ up to 102)	(Kwon et al., 2019)

This approach extends efficiently to other wavelengths and higher output powers, especially when using appropriate nonlinear crystals and cavity designs.

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References:

• (Lupu et al., 26 Nov 2025): Lupu–Werner, "Intensity doubling for Brownian loop-soups in high dimensions"
• (Kwon et al., 2019): Naik et al., "Generation of 14.0W of single frequency light at 770 nm by intracavity frequency doubling"