A p-polarised plane wave of unit amplitude travels in a non-magnetic medium of real…

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Wave propagation and light in media · Wave propagation and light in media

What an Instrument Can See, Ch The boundary as a two-port

Problem

A p-polarised plane wave of unit amplitude travels in a non-magnetic medium of real index

n_1

and meets, at a plane interface, a second non-magnetic medium of real index

n_2

. The angle of incidence is

\theta_i

and the angle of refraction

\theta_t

obeys Snell's law

n_1\sin\theta_i = n_2\sin\theta_t

. Writing

E_i

E_r

E_t

for the electric-field amplitudes of the incident, reflected and transmitted waves, continuity of the tangential components of

\mathbf{E}

and

\mathbf{H}

across the interface, with the positive directions chosen so that

r_p=r_s

at normal incidence, gives the pair of equations

(E_i+E_r)\cos\theta_i = E_t\cos\theta_t, \qquad n_1(E_i-E_r) = n_2 E_t.

((a)) Solve this pair of equations for

r_p = E_r/E_i

and

t_p = E_t/E_i

, expressing both purely in terms of

n_1

n_2

\theta_i

and

\theta_t

. ((b)) A microscope slide of index

n_1

is immersed in an oil of exactly the same index,

n_2 = n_1

, so that a real interface exists (the oil and glass differ chemically) but the scalar refractive index does not change across it. Evaluate your expressions for

r_p

and

t_p

in this limit for a general angle of incidence

\theta_i

, and state what you find.

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