= Solution
Translation invariance in $y$, together with a $y$-independent incident field, permits a $y$-independent solution. In homogeneous space Maxwell's equations then separate into two scalar sectors.
For a <TE wave>, set $\mathbf E=\widehat{\mathbf y}E_y(x,z)$. Its component satisfies $(\partial_x^2+\partial_z^2+k^2)E_y=0$. The $y$ direction is tangent to the conducting surface, so vanishing tangential <electric field> imposes $E_y(x,h(x))=0$. Therefore \b[TE scattering is exactly the two-dimensional Dirichlet scalar problem], with $E_y$ replacing the acoustic scalar. Its first-order boundary expansion and angular-spectrum formula are those above.
For a <TM wave>, set $\mathbf H=\widehat{\mathbf y}H_y(x,z)$. Maxwell's equation $\nabla\times\mathbf H=-i\omega\epsilon\mathbf E$ gives
$$
\mathbf E=\frac{i}{\omega\epsilon}(-\partial_zH_y,0,\partial_xH_y).
$$
A surface tangent is $(1,0,h')$, so vanishing tangential <electric field> gives
$$
\boxed{(\partial_z-h'\partial_x)H_y(x,h(x))=0,\quad\text{equivalently }\partial_nH_y=0.}
$$
Thus \b[TM also reduces to two dimensions, but to a Neumann problem, not the <Dirichlet problem> in part (a)]. This is <scalar polarization reduction at a conducting corrugation>.
For example the flat TM reflection has positive sign, $H^{[0]}=A_i e^{ip_ix}(e^{-iq_iz}+e^{iq_iz})$. Its linearized rough forcing is
$$
\partial_zH_s^{[1]}(x,0)=2A_i e^{ip_ix}\bigl(q_i^2h+ip_i h'\bigr).
$$
In the transform domain this becomes $2A_i(k^2-p_i\xi)\widehat h(\xi-p_i)$, and propagation divides by $i\beta(\xi)$ rather than prescribing the electric Dirichlet trace. The explicit slope term and different reflection sign show why simply reusing the TE boundary formula for TM would be incorrect.
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