Light in air ( ) strikes the flat glass window of a fish tank filled with water ( )…

PHYSICS · Subjective · Fermi University Physics · Optics & Photonics · Wave propagation and light in media · Wave propagation and light in media

Optics and Photonics, Ch Reflection and transmission at interfaces

Problem

Light in air (

n_i = 1.00

) strikes the flat glass window of a fish tank filled with water (

n_t = 1.33

) at an angle of incidence

\theta_i = 40.0^\circ

(treat this as a single air-water interface; ignore the glass). The wave is p-polarised. Follow the same reasoning the chapter used for s-polarisation, but with the roles of

\mathbf{E}

and

\mathbf{H}

exchanged, to derive the Fresnel p-coefficients and check energy conservation. (a) For p-polarisation

\mathbf{H}

lies entirely along the surface (no

\cos\theta

factor) while

\mathbf{E}

has a component along the surface that carries a

\cos\theta

factor. Write the tangential-

H

continuity condition, using

H_0 = nE_0/(\mu_0 c)

, as

n_i(E_i - E_r) = n_tE_t

, and the tangential-

E

continuity condition as

(E_i + E_r)\cos\theta_i = E_t\cos\theta_t

. (These signs are the ones convention CV5 requires; you are not asked to justify the signs, only to use them.) (b) Eliminate

E_t

between the two equations to obtain

r_p = E_r/E_i

in terms of

n_i, n_t, \theta_i, \theta_t

. (c) Substitute back to obtain

t_p = E_t/E_i

. (d) Using Snell's law, evaluate

\theta_t

, then

r_p

and

t_p

numerically to three significant figures for this interface. (e) Compute

\mathcal{R}_p = r_p^2

and

\mathcal{T}_p = \dfrac{n_t\cos\theta_t}{n_i\cos\theta_i}\,t_p^2

, and verify that

\mathcal{R}_p + \mathcal{T}_p = 1

to within rounding.

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