Download Demonstration: quarter-wave plate and half-wave plate

APPLIED COMPUTATIONAL OPTICS GROUP
released on 23.03.2015
Demonstration:
quarter-wave plate and half-wave plate
Site Zhang* and Frank Wyrowski
Abstract
A waveplate, made out of optically anisotropic materials, is an optical
device that manipulates the polarization of light propagating through
it. For example, a quarter-wave plate converts linearly polarized
light into circularly polarized light and vice versa; a half-wave plate
changes the polarization direction of linearly polarized light. We
demonstrate simulations of light propagation through the two types
of waveplates in the software VirtualLabTM .
* Correspondence:
[email protected]
Keywords: Quarter-wave plate, half-wave plate, polarization
Introduction
∆φ =
2π∆nL
,
λ0
uniaxialcrystal
y
op
tic
al
ax
is
x
E
e-wave
Waveplates are usually made out of uniaxial crystals cut into the
shape of a plate with properly chosen optical axis orientation and
thickness. The optical axis of the crystal should be parallel to the
plate surface, as in Fig. 1 we set the x-axis parallel to the optical axis.
In such a way, two polarization directions can be defined for a normal
incident plane wave: the ordinary wave polarized along x-axis with
the refractive index no , and the extraordinary wave polarized along
y-axis with the refractive index ne . They travel at different speeds
inside the crystal and that leads to a phase difference between them
when they leave the crystal. The phase difference ∆φ depends on the
thickness of the crystal and it can be expressed as
o-
z
wa
ve
(1)
where ∆n = |ne − no | is the difference between ordinary and extraordinary refractive indices, L is the thickness of the crystal plate and
λ0 is the vacuum wavelength of the light.
By letting ∆φ = (2m + 1) π2 or ∆φ = (2m + 1)π in Eq. (1), a quarterwave plate or a half-wave plate is obtained respectively. If one chooses
m = 0, a zero-order waveplate is obtained, which is very thin and the
fabrication becomes difficult. For non-zero m, one gets a multipleorder waveplate with larger thickness, which brings convenience in
the mounting but higher (undesired) sensitivity to wavelength and
temperature [1].
L
Figure 1: Sketch of a typical
waveplate made out of uniaxial crystal.
VirtualLabTM Demostrations
Based on the techniques described in [2,3], fully vectorial simulations
of light propagation through waveplates have been carried out in the
1
software VirtualLabTM .
Quarter-wave plate
We design a quarter-wave plate for the wavelength λ0 =633 nm, with
quartz crystal with no = 1.543 and ne = 1.552 [4]. We choose, as an
example, ∆φ = 8.5π in Eq. (1), and obtain a multiple-order quarter
waveplate with m = 8. Then it is trivial to find the corresponding
thickness L =299 µm. Using these parameters we set up a quarterwave plate and use a linearly polarized Gaussian wave 2 mm in front
of it as the incidence. Fig. 2 shows the effect that a quarter-wave plate
converts linear polarization into circular.
Quarter-wave plate:
• vacuum wavelength:
λ0 =633 nm
• refractive indices:
no = 1.543 and
ne = 1.552
• crystal thickness:
L =299 µm
• Gaussian wave waist
radius: 100 µm
• incident polarization:
linear 45◦
Figure 2: Effect of a quarter-wave plate: the incident Gaussian wave
(left) has a 45◦ linear polarization, while the transmitted wave (right)
has a circular polarization.
Half-wave plate
Keeping all parameters the same as for the quarter-wave plate, but
doubling the thickness i.e. letting L =598 µm, a half-wave plate is
obtained, and its effect is shown in Fig. 3.
Half-wave plate:
• vacuum wavelength:
λ0 =633 nm
• refractive indices:
no = 1.543 and
ne = 1.552
• crystal thickness:
L =598 µm
• Gaussian wave waist
radius: 100 µm
• incident polarization:
linear 45◦
Figure 3: Effect of a half-wave plate: the incident Gaussian wave (left)
has a 45◦ linear polarization, while the transmitted wave (right) has
a −45◦ linear polarization i.e. 90◦ rotation of the incident one.
As expected, Fig. 3 sees a rotation of polarization by 90◦ .
2
Bibliography
[1] Mounted multi-order wave plates. from http://www.thorlabs.de (online).
[2] Site Zhang, Daniel Asoubar, and Frank Wyrowski. Rigorous modeling of laser light propagation
through uniaxial and biaxial crystals. Proc. SPIE, 9346:93460N–93460N–13, 2015.
[3] S. Zhang and F. Wyrowski. Introduction to field tracing in homogeneous anisotropic media.
www.applied-computational-optics.org (online), March 2015.
[4] Quartz crystal - refractive index. from http://www.crystran.co.uk (online).