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For constant Electron density, common path length $L$, and a solenoidal isotropic spectral tensor, the rotation-measure correlation contains the window $|{\int }_0^L{e}^{i{k}_{z}z}dz{|}^2$. As $L$ becomes large compared with the correlation length, this window acts as $2\pi L\delta (k_z)$ within the integral. The transverse projector then has $zz$ component one, yielding the displayed two-dimensional Fourier transform. For finite $L$, the windowed integral retains longitudinal modes and endpoint corrections. The method of using such rotation measure correlations to infer magnetic spectra is developed by Enßlin and Vogt.

The shell-integrated magnetic energy spectrum distributes magnetic energy over wavenumbers. With a solenoidal isotropic spectral tensor normalized to ${\hat{C}}_{ii}=2H(k)$ and the inverse Fourier transform measure $d^{3}k/(2\pi {)}^3$, one has $\mathbb{E}|\mathbf{B}{|}^2={\pi }^{-2}\int k^{2}H(k)dk$. In Gaussian units, $E_B(k)=k^{2}H(k)/(8{\pi }^3)$ integrates to the energy density $\mathbb{E}|\mathbf{B}{|}^2/(8\pi )$. A spectrum per unit mass or normalized to $B^2/2$ has different constant factors. Scale-local fluctuation estimates give $\delta B_l^2$ proportional to $k{E}_B(k)$ at $k\sim l^{-1}$ when the spectrum is normalized consistently.

Statistical isotropy and statistical reflection symmetry allow the spectral tensor to contain only ${\delta }_{ij}$ and $k_i{k}_j$:The solenoidal magnetic-field constraint is essential here. Taking its Fourier transform gives $k_i{\hat{B}}_i=0$, hence $k_i{\hat{C}}_{ij}=0$ and $A+D{k}^2=0$. Thus the spectral tensor is a scalar multiple of the transverse projector of a vector field. Its trace is $2A$, so the prescribed normalization givesThis is the solenoidal isotropic spectral tensor. Without the solenoidal condition, isotropy and reflection symmetry would leave two scalar functions. Without reflection symmetry, an antisymmetric term proportional to $i{ϵ}_{ijm}{k}_m$ could additionally encode magnetic helicity.

Write the transverse displacement as $\mathbf{s}$ and take the common line-of-sight interval to be $[0,L]$. The rotation measure is a line integral of the magnetic field, so its correlation isSubstitute the Fourier inversion of the solenoidal isotropic spectral tensor. Keeping the finite observation length gives the exact windowed expressionThe allowed extension of the longitudinal displacement integral to the whole real line is the long-path projection of a magnetic correlation. It applies when $L$ is much larger than the correlation length: equivalently, the squared window becomes $2\pi L\delta (k_z)$ inside the integral. The Dirac delta function sets $k_z=0$, leavingStatistical isotropy makes this a function of $s=|\mathbf{s}|$. The displayed two-dimensional relation is the intended long-path approximation; the preceding windowed formula accounts for finite-length edge effects.