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Josef Matz
science forum Guru Wannabe

Joined: 08 May 2005
Posts: 255

Posted: Mon Jul 17, 2006 6:35 am    Post subject: The geometric representation of spin for elliptic polarized light

A very general form of electromagnetic waves is the so called elliptic
polarized light.
The electrical field of the propagating beam or plane wave rotates around
the propagation direction
on an ellipse. This is called the real valued interpretation of light
according to Maxwell theory.

In order to visualize the spin of an elliptic polarized wave we draw an
ellipse onto an paper in a
x - y coordinate system. The center of the ellipse we locate at the origin.
The big semiaxis of the the
ellipse we call a1 and rotate the ellipse so that a1 is oriented in y
direction.. The small semiaxis we call
a2 and it looks in x direction.
The field of this light beam rotates with contsant angular velocity around
this ellipse ether clockwise or
counterclockwise. The electric field vector at any time is a vector from the
origin to a point on the ellipse.

The energyflux of this plane wase is given by the formula

S = a abs(E)**2 and is pointing in propagation direction.. In the following
we set a = 1.

Then

S = abs(E)**2 = a1**2 + a2**2

According to pythagoreas a1'**2 + a2**2 = a3**2 where a3 is the hypothenuse
of the rectangle.

Thus we have S = a3**2 is the energy flux.

What is spin now ? As i said any elliptic polarized wave carries spin. In
our simple picture
the spin simply is the area of rectangle a1 * a2.

The correct formula is:

L = sgn * a1 * a2 / w

w = 2 Pi v and v the frequency of rotation of the field.
a1 * a2 is the area of the rectangle between big and small semiaxis of the
ellipse.

L shows in direction of propagation or the opposite. For clockwise rotation
of the electric field
L shows in direction of propagation. Hence here the sign sgn = +1.
For counterclockwise rotation of the electric field the Spin L shows
oppostite to the direction of
propagation. Hence here the sidn sgn = - 1.

If you set the rotation andular frequency w = 1 and only view clockwise
rotating light beam, then
you have simply

L = a1*a2

The spin simply is the area of the rectangle of big and small semiaxis of
the ellipse. Thats the
geometric representation of spin of an elliptic electromagnetic wave.

In the special case of linear polarized light a2 = 0. Therefore linear
polarized light carries no spin.

Best regards

Josef Matz

josefmatz@arcor.de

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