The <Taylor expansion> of the square root is
$$
\sqrt{1+\alpha}=1+\frac{\alpha}{2}
-\frac{\alpha^2}{8}+O(\alpha^3).
$$
Substitution into the exact <reflection coefficient> gives
$$
R=\frac{1-\sqrt{1+\alpha}}
{1+\sqrt{1+\alpha}}
=-\frac{\alpha}{4}+\frac{\alpha^2}{8}
+O(\alpha^3).
$$
Consequently
$$
\boxed{
\psi_r=-\frac{\alpha}{4}e^{i(px+q_0z)}
+O(\alpha^2)}.
$$
This is exactly the reflected field furnished by both the <Born approximation> and the term linear in $α$ in the <Rytov approximation>. Thus the exact, Born, and Rytov fields agree through first order. If the Rytov exponential is retained without re-expansion, its higher powers generate spatial harmonics $e^{i(px+(2m-1)q_0z)}$; those terms are part of the approximation and should not be confused with the exact interface's single reflected wave.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 335 / 1 / iii / Solution

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The reflection coefficient is the complex ratio between reflected and incident wave amplitudes. Its modulus determines amplitude attenuation and its argument determines the phase shift on reflection.


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 Incoming links: Reflection coefficient