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Feb 16, 2011

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Feb 16, 2011

Finish Polarization

The Radiation Spectrum

HW & Questions

LCD Displays

Bees can see polarized light

polarization of blue sky enables them to navigate

Humans:

Haidinger’s Brush

Vikings:

Iceland Spar?

Many invertebrates can see polarization, e.g. the Octopus

Not to navigate (they don’t go far)

Perhaps they can see transparent jellyfish better?

unpolarized polarized

Polarization and Stress Tests

In a transparent object, each wavelength of light is polarized by

a different angle. Passing unpolarized light through a polarizer,

then the object, then another polarizer results in a colorful pattern

which changes as one of the polarizers is turned.

CD cover seen in polarized light from monitor

3D movies

Polarization is also used in the entertainment industry to produce and show 3-D movies. Three-dimensional movies are actually two movies being shown at the same time through two projectors. The two movies are filmed from two slightly different camera locations. Each individual movie is then projected from different sides of the audience onto a metal screen. The movies are projected through a polarizing filter. The polarizing filter used for the projector on the left may have its polarization axis aligned horizontally while the polarizing filter used for the projector on the right would have its polarization axis aligned vertically. Consequently, there are two slightly different movies being projected onto a screen. Each movie is cast by light which is polarized with an orientation perpendicular to the other movie. The audience then wears glasses which have two Polaroid filters. Each filter has a different polarization axis - one is horizontal and the other is vertical. The result of this arrangement of projectors and filters is that the left eye sees the movie which is projected from the right projector while the right eye sees the movie which is projected from the left projector. This gives the viewer a perception of depth.

Stokes Parameters:

Linear Polarized:

Linear Polarized:

Linear Polarized (45 deg):

Linear Polarized

(- 45 degrees):

Left-hand

Circular:

Right-hand

Circular:

Unpolarized:

The Radiation Spectrum

Rybicki & Lightman, Section 2.3

Consider

Transverse E-field

What is the spectrum?

Energy / time as a function of frequency

Note:

Radians / sec

Define the Fourier Transform of

Eqn. 1

The inverse is

Eqn. 2

Take complex conjugate of Eqn. (1):

So….

and

But since

We have

So

eqn. (b)

eqn. (a)

Also

Now

From Eqn. (a), (b)

So…

Parseval’s Theorem for Fourier Transforms:

Proof:

Poynting Theorem: Energy /time/area

Integrate over pulse:

So…

Electromagnetic Potentials

Rybicki & Lightman, Chapter 3

Instead of worrying about E and B, we can use the scalar and vector

potentials

Simpler:

1 scalar and 1 vector quantity instead of 2 vector quantities.

Relativistic treatment is simpler.

(1)

So

Therefore, there exists a φ such that

(2)

or

Equations (1) & (2) already satisfy 2 of Maxwell’s Equations –

what about the others?

becomes

For reasons which will become clear in a minute, we re-write this

last equation as

(3)

The 4th Maxwell equation

becomes

Since

we get

(4)

GUAGE TRANSFORMATIONS

(1)-(4) do not determine A and φ uniquely:

one can add the gradient of an arbitrary scalar ψ to A

and leave B unchanged

Likewise E will be unchanged if you add

These are called GUAGE TRANSFORMATIONS

For the LORENTZ GUAGE we take

so that (3) and (4) simplify to

(5)

(6)

RETARDED POTENTIALS

It turns out that the solutions to (5) and (6) can be expressed

as integrals over sources of charge, provided you properly

take into account the fact that changes in the E and B fields

can move no faster than the speed of light.

RETARDED POTENTIALS

At point r =(x,y,z), integrate over charges at positions r’

(7)

(8)

where [ρ] evaluate ρ at retarded time:

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