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Unformatted text preview: rn ej (ωt+k1x x)
Ey,II (x, t) = Ae−αx ejωt + B eαx ej ωt
Ey,III (x, t) = tn ej (ωt−k1x x)
Assume that µ = µ0 in all three regions.
(a) Using the boundary condition on the tangential electric ﬁeld at x = 0 and x = d, ﬁnd two equations
relating the unknowns rn , A, B , and tn .
(b) Find the tangential magnetic ﬁelds using Faraday’s Law:
∂ Bz
∂ Ey
=−
∂x
∂t 3 6.007 Spring 2011 Problem Set 8: Electromagnetic Waves at Boundaries Then, using the boundary condition on the tangential magnetic ﬁeld at x = 0 and x = d, ﬁnd two
equations relating the unknowns rn , A, B , and tn .
x
(c) Now that you have four equations and four unknowns, set up a matrix equation in the form of Mx = C ,
where:
⎛
⎞
rn
⎜A⎟
⎟
x =⎜
⎝B⎠
tn
With this matrix equation, you could solve for the complex amplitudes rn , A, B , and tn . We’ll see
this again when we cover tunneling in quantum mechanics.
2 (d) Using MATLAB, ﬁnd the transmitted intensity, tn  , if the incident angle in region I (glass) is θ = 45◦ ,
the index of glass is n1 = 1.5, the index of air is n2 = 1, the free space wavelength is λ0 = 640 nm, and
the air gap is d = 100 nm.
You’ll need to numerically solve the system of equations represented by your matrix in (c) using
MATLAB. To ﬁnd the inverse of M, use the MATLAB function inv() (e.g., x=inv(M)*C). Problem 8.3 – Thin Film Interference
In class we have seen the Fresnel equations for reﬂected and transmitted wave amplitudes. These equations
assume that the materials are semiinﬁnite (that they continue for ever). If we look at reﬂections from
sections of materials with ﬁnite thickness we have to take into account interference phenomenon in addition
to the transmission/reﬂection amplitudes for semiinﬁnite material boundaries. In this problem we will look
at the reﬂection of light from a ﬁlm of oil on top of water. See the diagram below. Geometry for Problem 8.4, not drawn to scale.
There can be a peak in reﬂected intensity only if the diﬀerence in phase gained between path AB and path
ACD is equal to some i...
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This document was uploaded on 03/17/2014 for the course ELECTRICAL 6.007 at MIT.
 Spring '11
 VladimirBulovic
 Electromagnet

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