Poincaré Sphere
The Poincare sphere, conceived by Henri Poincare in about 1892 provides a convenient way of representing polarized light and predicting how any given retarder will change the polarization form. The upper and lower poles represent left and right-circularly polarized light. Points on the equator indicate linear polarization. Other points on the sphere represent elliptical polarization. An arbitrarily chosen point H on the equator designates horizontal polarization, and the diametrically opposite point V designates vertical polarization. A general point P on the surface of the (unit radius) Poincare sphere is specific in terms of the longitude and the latitude.
The ellipse of polarisation (figure 1), i.e. the
path traced by the end of the E-vector, as seen by an observer looking
towards
the source of the light, describes the state of polarisation, and so if
the
ellipse of the polarisation can be drawn then one can determine certain
quantities related to the state of polarisation.
The quantities a and b (as shown in figure 1) can be obtained easily by
measuring the intensity transmitted through a linear polariser aligned
along
both the x-axis and then the y-axis. The angle PHI can be obtained by
passing
the light through a linear polariser and rotating the axis of the
polariser
until a maximum intensity is measured, in which case the polariser axis
will be
aligned along PHI. Once PHI and a and b are known the two ellipse axes, A
and B,
can be calculated. Clearly if either quantity A or B tends to zero then
the
light is linearly polarised, and if A and B are equal the light is
circularly
polarised. If we then define a quantity CHI=arctan(=/- B/A) then we have
two
angular quantities (known as the ellipsometry angles) PHI and CHI which
uniquely
describe the ellipse of polarisation, and hence the state of polarisation.
Thus
if we project two angular quantities onto a sphere (for ease of
interpretation
the convention is as shown in figure 2.), we have a workable
representation of
the state of polarisation, known as the Poincare representation.
Every point on the surface of the Poincari sphere represents a unique
state
of polarisation, and general patterns can be found . For example, all
points
along the equatorthe northern(and vice-versa in the southern hemisphere).
The north polerepresents a purely right-hand (left-hand) circular state of
polarisation. The
change of state of polarisation during any optical interaction can be
therefore
described by a curve of the surface of the sphere connectiong the state of
polarisation before the interaction with that after.
Q, U and V are the Stokes parameters, used in the Mueller matrix
calculus.