Reflection and Refraction

A short overview

Plane waveA plane wave can be written

as follows:Here A represent the E or B

fields, q=i,r,t and j=x,y,zSo this is a representation of

the waves that is valid i all three cases, i.e. the incoming, the reflected and the transmitted wave

)(Re wtiqj

qj

q

eaA rk

Boundary conditions

For a wave moving from one medium to another medium we have:

(i)

(ii)

2211 EE

112

111 EE

Boundary conditions (iii)

(iv)

21 BB

112

2

111

1

11BB

Form of E and B fields

Electric field and Magnetic fields are of the form;

wtrkiqoj

qj

q

eEE .Re

wtrkiqoj

qj

q

eBB .Re

Boundary conditions

Direction of the wave vectors

Optical laws All the three waves have the same

frequency

Combined fields in medium (1) should be joined to the fields in medium (2)

Boundary conditions should hold at all times and at all points so exponential factors are equal.

tri kn

nkk

2

1 …………………………. 1

Optical laws

Spatial terms give

when z = 0. This holds if components are

separately equal.If incident vector is in x-z plane, wave vector in y is zero.

rkrkrk tri ...

Optical laws The first law is

xtx

rx

i kkk

ttrrii kkk sinsinsin 0.1

Optical laws Apply equation (1) to this equation (1.0) we get two

results:

the optical laws apply to all waves Reflection and Snell's law can in general apply to non-

planar waves incident upon non-planar interface. This is shown below

ri ti nn sinsin 21

Generalisation of the laws

REFLECTION AND REFRACTION

From boundary condition 2 above

From boundary condition(4)

)1.1....(........................................coscoscos tot

ror

ioi EEE

)2.1....(......................................................................11

220

11otroi E

vEE

v

Fresnel Equations

solve the two equations

ti

ti

ti

ti

oi

r

n

nn

n

E

ER

tan

tan

coscos

coscos

1

2

1

2

011

ti

i

oi

t

n

nE

ET

coscos

cos2

1

2

011

Continutation

When

We get

ti 2

n

iB

n

n2tan

011 R

Reflection and refraction At angle of incidence E vector has no component in plane of incidence. This makes it possible to get lineally polarized light from

an unplarized beam.

This fact is used in polarized sun glasses, the filter is oriented in such away that only light that is polarized vertically is transmitted, hence avoiding glare or annoying reflections from horizontal surfaces.

iB

Total internal reflection

If we have 1 and at some point if

From Snell’s law we have 1

(transmitted ray glazes the surface.)

nn

n

1

2

ici

it n sinsin 1

2

t

Total internal reflection

we have total internal reflection (no refracted ray at all). This phenomena is used in light pipes, fibre optics, and studying micro waves. In this case we have an evanescent wave which is rapidly attenuated and transports

1

1sin

nic

ici

END

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