Polarization and Polarization Controllers - NTNU

where + denotes right-handed circular polarization.
2.3 Inner Product and Orthogonality
Two normalized polarization states J 1 and J 2 are called orthogonal if their inner product defined as
* *
J1 xJ
2x
+ J1y
J 2 y
J =
(14)
1 J 2
is equal to zero. * denotes complex conjugation. Two useful observations are:
2 J1
J1
J 2
*
J = (15)
and
J = 1 . (16)
If two vectors are normalized and orthogonal, they form an orthonormal set.
1 J 1
2.4 Orthogonal Expansion Basis
An arbitrary Jones vector J can be written as a weighted superposition of two orthonormal Jones
vectors J 1 and J 2 (the expansion basis) [1], often using linearly or circularly polarized waves as basis.
J J + , (17)
= 1J
1 + J 2J
2 = J 1 J J 1 J 2 J J 2
where we have indicated that the expansion weights J 1 and J 2 are the inner products
J 2
=
J 2
J
J = J and
. If we want to express linearly polarized light in the basis of circular polarized light, we
can use (11) and (13) with (14) to obtain J J J J =1/ 2 and
+ x = − x
1
J 1
J x
Similarly, J J − J J = −i / 2 and
+ y = − y
= 1
2
J
( J + + −
). (18)
1
J y = J
2
( J + − −
). (19)
2.5 Changing Expansion Basis
Assuming that we have a set of orthogonal Jones vectors J 1 , J 2 , and a second set of Jones vectors J 1 ’
J 2 ’, we can write an arbitrary Jones vector as
' ' ' '
1J1 + J 2J
2 = J1J1
J 2J
2
J = J +
. (20)
The relation between the two sets of expansion basis is now found by taking the inner product with J 1 ’
and J 2 ’, producing
4