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 Incoming links: Scalar polarization reduction at a conducting corrugation

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 70 / 3 / c / Solution Created 2026-10-03 Updated 2026-10-14

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

E = 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 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

H = y H_{y} (x, z)

. Maxwell's equation

\nabla \times H = - iω ϵ E

gives

E = \frac{i}{ω ϵ} (- \partial_{z} H_{y}, 0, \partial_{x} H_{y}) .

A surface tangent is

(1, 0, h^{'})

, so vanishing tangential electric field gives

(\partial_{z} - h^{'} \partial_{x}) H_{y} (x, h (x)) = 0, equivalently \partial_{n} H_{y} = 0.

Thus 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^{i p_{i} x} (e^{- i q_{i} z} + e^{i q_{i} z})

. Its linearized rough forcing is

\partial_{z} H_{s}^{[1]} (x, 0) = 2 A_{i} e^{i p_{i} x} (q_{i}^{2} h + i p_{i} h^{'}) .

In the transform domain this becomes

2 A_{i} (k^{2} - p_{i} ξ) h (ξ - p_{i})

, and propagation divides by

i β (ξ)

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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