Polynomial Closed-Form Model for Evaluating Nonlinear Interference in Any Island

Polynomial closed-form GN model is proposed by expressing the spatial power profile of each channel along a span as a polynomial. In this paper, we present the generic closed-form expression for all contributions of self-, cross-, and multi-channel interference. The full derivation is provided.

💡 Research Summary

The paper introduces a Polynomial Closed‑Form Model (PCFM) for predicting nonlinear interference (NLI) in fiber‑optic communication systems. Traditional closed‑form GN‑model approximations rely on simplifying the spatial power profile (SPP) of each channel to a constant or exponential decay, often neglecting intra‑span NLI coherence, using infinite series expansions, or assuming rectangular channel spectra and flat NLI PSD. These approximations limit accuracy, especially when Raman amplification, inter‑channel stimulated Raman scattering (ISRS), lumped loss, or short spans are present.

PCFM overcomes these limitations by representing the SPP as a general polynomial:
p(z)=∑{n=0}^{N_p} p_n z^n,
where N_p can be any non‑negative integer. Substituting this expression into the GN‑model reference formula (Eq. 1) transforms the core double integral K{ns,x}(f_CUT) into a sum of terms involving polynomial coefficients and a set of integrals Q_{nm}. By pairing Q_{nm} with its complex conjugate Q_{mn}, the authors define a real quantity I_{nm}=Q_{nm}+Q_{mn}, which eliminates the need for complex arithmetic in the final result.

The frequency integration is carried out analytically using cosine‑integral (Ci) and sine‑integral (Si) functions. After introducing the parameter B=4π²β_{ns}^{2,eff} and defining λ_k combinations of the frequency limits a_k, b_k, c_m, d_m, the authors derive a compact expression for the function F(u) that appears in I_{nm}. This expression reduces to a finite sum of Si terms multiplied by rational coefficients, valid for B u≠0, and to a simple product of interval lengths when B u=0.

The remaining spatial integration depends only on the polynomial powers n and m. By changing variables to u=z₁−z₂ and t=z₂, the double integral collapses to a single integral I_{p,q}(L;λ) of the form ∫₀ᴸ u^{p−1}(L−u)^{q} Si(λu) du. The authors treat two cases: p≥1, which yields a combination of Beta functions and auxiliary integrals S_k(L;λ) that can be expressed with elementary sin‑integrals and known constants; and p=0, which involves the generalized hypergeometric function ₂F₃.

All of these steps culminate in the generic closed‑form expression (Eq. 18) that is valid for any “island” – i.e., any region of the frequency plane contributing to SCI (self‑channel interference), XCI (cross‑channel interference), or MCI (multi‑channel interference). By simply inserting the appropriate frequency limits and polynomial coefficients for a given island, one obtains K_{ns,x} and consequently the NLI PSD G_{ns,NLI,x}(f_CUT).

The paper demonstrates that, for the special case N_p=0 (constant power along the span), the derived SCI expression reduces exactly to the result previously reported in reference