L
L
a
y
3 mm
y
x
(Left) The optical axis orientation in a diffractive waveplate (DW) rotates in the plane of
the waveplate, the (x,y) plane. Elongated ellipses show orientation of anisotropic molecules coinciding with the local orientation of the optical axis. The period of modulation
of the optical axis orientation L is half that of molecular orientation due to equivalency
of the optical states with rotation angle w = p and 2p. (Right) Under a polarizing microscope, the structure of a “cycloidal” waveplate is revealed as a grating with lines shifting
when rotating the polarizer. The modulation period is 3 mm for the sample in the photo.
xL
(a)
(c)
(b)
A cycloidal DW diffracts a circular polarized beam into +1st or –1st order, depending on its
handedness (a) and (b), whereas both orders are present for a linearly polarized beam (c).
Solid arrows indicate propagation direction of the beams; polarization states are shown
as dashed lines. The blue dashed lines in (a) and (b) indicate the polarization of the
incident light. In (c), the LC polymer DW can be seen illuminated by an incident beam
of 199 diameter.
(a)
(b)
(c)
(a) The diffraction on a DW can be cancelled; the angle can
be doubled; or it can be redirected in two dimensions with
a counter-DW, depending on whether or not its cycloidal
orientation pattern is parallel, anti-parallel, or makes an angle with respect to the 1st DW.
The photos (b-d) show this for a linearly polarized laser beam with holographically
generated complex profile. The DWs are identical and parallel in (b), fulfilling full-wave
waveplate condition and thus eliminating diffraction; they are anti-parallel in (c), doubling
the diffraction angle. They make 90° with respect to each other in (d).
(d)
Diffractive waveplates
Consider a linearly polarized light beam
incident normally, along the z-axis, on a
birefringent film in the x,y plane. If the
thickness of the film L and its optical
anisotropy, Dn, are chosen such that
LDn = l/2, and its optical axis is oriented
at 45° with respect to the polarization
direction of the input beam, the polarization of the output beam is rotated by
90°. This is how half-wave waveplates
function. The polarization rotation angle
at the output of such a waveplate, b = 2a,
depends on the orientation of the optical axis d = (dx, dy) = (cosa, sina). LC
materials, both low molecular weight as
well as polymeric, allow continuous rotation of d in the plane of the waveplate at
high spatial frequencies, a = qx, where
the spatial modulation period L = 2p/q
can be comparable to the wavelength of
visible light. Polarization of light at the
output of such a waveplate is consequently
modulated in space, which is revealed
under a polarizing microscope.
The electric field in the rotating
polarization pattern at the output of
this waveplate is averaged out, < E > = 0,
and there is no light transmitted in the
direction of the incident beam. That
pattern, however, corresponds to the
overlap of two circularly polarized beams
propagating at the angles ±l/L. Only
one of the orders is present in the case
of a circularly polarized input beam, the
+1st or –1st, depending on whether the
beam is right- or left-handed. Since a
half-wave waveplate reverses the sign of
circular polarization, the emerging beam
is polarized circularly orthogonal to the
input beam.
Thus, as opposed to CLCs, both
polarization components are diffracted.
The diffraction efficiency of a diffractive
waveplate is determined by the conversion factor of light into the orthogonal
component of polarization at the output
of the waveplate, sin2pLDn /l. This
expression determines the total diffraction efficiency: that of the single diffraction order in the case of a circularly
polarized incident beam, and that of
both diffracted beams for an unpolarized
or linearly polarized input. The validity
42 | OPN Optics & Photonics News
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