= Solution
Choose the time dependence $e^{-i\omega t}$ and measure distances in background-wavenumber units. Thus $L_0=\Delta+1$, the <incident field> satisfies $L_0\psi_i=0$, and the <scattering potential> convention is $V=n^2-1$, giving $L_0\psi=-V\psi$. To match the minus sign in the paper's integral, take the outgoing <Green function> to satisfy $L_0G=\delta$. This is the negative of the frequently used outgoing <Helmholtz equation> fundamental solution satisfying $L_0G=-\delta$. In physical coordinates one restores $k_0^2$ in the <scattering potential>, or absorbs it in the integral kernel; the sign convention must remain consistent.
On a region where the incident and total fields are nonzero, introduce the <logarithmic wave perturbation>
$$
\psi=\psi_i e^\chi,\qquad \chi=\log(\psi/\psi_i).
$$
Choose a continuous logarithm branch connected to the unperturbed field. Substitution into the <Helmholtz equation> gives
$$
\mathcal L_i\chi+\nabla\chi\cdot\nabla\chi=-V,\qquad \mathcal L_i=\Delta+2\frac{\nabla\psi_i}{\psi_i}\cdot\nabla.
$$
The quadratic term uses the complex bilinear dot product, not the squared modulus of the gradient. Write $V=\varepsilon V_1$ and $\chi=\varepsilon\chi^{[1]}+\cdots$. The first-order <Rytov approximation> discards that quadratic term. Since $L_0(\psi_i\chi)=\psi_i\mathcal L_i\chi$, the outgoing first correction is
$$
\chi_1(\mathbf r)=-\frac1{\psi_i(\mathbf r)}\int G(\mathbf r,\mathbf r')V(\mathbf r')\psi_i(\mathbf r')\,d\mathbf r',
$$
and hence
$$
\boxed{\psi_1^{(R)}(\mathbf r)=\psi_i(\mathbf r)\exp\left[-\frac1{\psi_i(\mathbf r)}\int G(\mathbf r,\mathbf r')V(\mathbf r')\psi_i(\mathbf r')\,d\mathbf r'\right].}
$$
The <validity of the first Rytov approximation> concerns the omitted logarithmic correction. Its next contribution satisfies $\mathcal L_i\chi_2=-\nabla\chi_1\cdot\nabla\chi_1$, so a useful explicit criterion is that the outgoing solution $\chi_2$ be small in the region of interest. Weak <refractive index> contrast and small <wave phase> gradients on a wavelength scale provide the usual perturbative regime, with weak <amplitude> fluctuations and no strong focusing or zeros that destroy the logarithm. A sufficient local source comparison is $|\nabla\chi_1\cdot\nabla\chi_1|\ll|V|$ where the <scattering potential> is nonzero, together with control of propagation of that error. This is not a universal pointwise test at zeros of $V$.
\b[Small accumulated <wave phase> is not required in the same way as in a linear field approximation]: the exponential retains that <wave phase> accumulation. Large gradients, strong multiple-scattering <amplitude> effects or field zeros can still invalidate the <Rytov approximation>. The integrals also require a finite scattering region or appropriate convergence conditions.
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