Design, manufacturing, and performance analysis
of mid-infrared achromatic half-wave plates
with diamond subwavelength gratings
Christian Delacroix,
1,
* Pontus Forsberg,
2
Mikael Karlsson,
2
Dimitri Mawet,
3
Olivier Absil,
4
Charles Hanot,
4
Jean Surdej,
4
and Serge Habraken
1
1
Hololab, Université de Liège, Allée du 6 Août 17 B5a, B-4000 Liège, Belgium
2
Ångström Laboratory, Uppsala University, Lägerhyddsvägen 1, SE-751 21 Uppsala, Sweden
3
European Southern Observatory (ESO), Alonso de Córdova 3107, Vitacura, 763 0355 Santiago, Chile
4
IAGL, Université de Liège, Allée du 6 Août 17 B5c, B-4000 Liège, Belgium
*Corresponding author: delacroix@astro.ulg.ac.be
Received 3 May 2012; revised 10 July 2012; accepted 10 July 2012;
posted 20 July 2012 (Doc. ID 167832); published 16 August 2012
In this paper, we present a solution for creating robust monolithic achromatic half-wave plates (HWPs)
for the infrared, based on the form birefringence of subwavelength gratings (SWGs) made out of diamond.
We use the rigorous coupled wave analysis to design the gratings. Our analysis shows that diamond,
besides its outstanding physical and mechanical properties, is a suitable substrate to manufacture
mid-infrared HWPs, thanks to its high refractive index, which allows etching SWGs with lower aspect
ratio. Based on our optimized design, we manufactured a diamond HWP for the 11–13.2 μm region, with
an estimated mean retardance ∼3.143 0.061 rad (180.08 3.51°). In addition, an antireflective grating
was etched on the backside of the wave plate, allowing a total transmittance between 89% and 95% over
the band. © 2012 Optical Society of America
OCIS codes: 050.1950, 050.5080, 050.6624.
1. Introduction
Wave plates are essential tools for modulating the
polarization of an incoming light beam. They are
commonly used in many applications as retarders
or phase shifters. In astrophysics, when it comes to
observing a very high-contrast scene, one must be
able to cancel the bright source, or at least drastically
reduce its intensity. This attenuation can be achieved
by combining two portions of the incident light beam,
one of which having to undergo a half wavelength
(or π) phase shift. Therefore, a half-wave plate (HWP)
is by definition the ideal tool. The retardance of a
zero-order wave plate varies hyperbolically with the
wavelength, which limits the operation of single
wave plates to monochromatic light. However, a
large spectral bandwidth is needed in many cases,
especially in astrophysics, both to increase the sig-
nal-to-noise ratio and to allow spectrophotometry.
Different varieties of achromatic wave plates exist.
Achromatic prism retarders [
1], for instance, operate
in total internal reflection. They are voluminous
and not suitable for applications operating in trans-
mission. Another way of achieving broadband per-
formance is stacking several crystal wave plates
together [
2] and orienting their birefringent axes
using a Pancharatnam method [
3]. A thick combina-
tion of multiorder wave plates is needed, because of
the weak natural birefringence of crystals, which re-
sults in an increased absorption of IR radiation.
Achromatic retarders can also be produced by using
1559-128X/12/245897-06$15.00/0
© 2012 Optical Society of America
20 August 2012 / Vol. 51, No. 24 / APPLIED OPTICS 5897
liquid crystals (LCs) [4], liquid crystal polymers
(LCPs) [
5], or photonic crystals (PhCs) [6,7] as the
birefringent materials. A weakness of these liquid
or photonic crystal retarders is their bandwidth: they
do not transmit at wavelengths beyond the near-
infrared (H-band centered at ∼1.65 μm, K-band
∼2.2 μm) [
8]. Meanwhile, the demand for instru-
ments in the mid-infrared (L-band ∼3.8 μm, M-band
∼4.8 μm, N-band ∼10.5 μm) is increasing in many
domains of astrophysics. Therefore, we are pursuing
a different technological route to synthesize the π
phase shift. We use the dispersion of form birefrin-
gence of subwavelength gratings (SWGs) [
9,10], which
are particularly adapted to longer wavelengths.
In this paper, we present the results of our work on
achromatic wave plates. After a brief introduction to
SWGs (Section
2), we demonstrate in Section 3 that
diamond is a good choice of material for mid-IR
SWGs. In Section
4, we attempt to optimize a design
for a diamond HWP with respect to manufacturabil-
ity and performance. In Section
5, we briefly describe
the fabrication of a diamond achromatic HWP
dedicated to the mid-IR (11–13.2 μm). We calculate
its theoretical efficiency with computer simulations
based on the rigorous coupled wave analysis (RCWA).
Finally, we conclude with the perspectives for present
and future instruments.
2. Achromatic Half-wave Plates (HWPs)
A. Subwavelength gratings (SWGs)
Many new optical devices have been made with
SWGs, such as high-efficiency diffraction gratings,
polarization-selective gratings, wave plates, and
monolithic antireflective structures [
11]. SWGs are
micro-optical structures with a period Λ smaller
than λ ∕ n, λ being the observed wavelength of the in-
cident light and n the refractive index of the grating
substrate. Such structures do not diffract light as a
classical spectroscopic grating does. Instead, only
zeroth transmitted and reflected orders are allowed
to propagate outside the grating, and the incident
wavefront is not affected by further aberrations.
The condition for having an SWG is defined by the
grating equation, which determines whether a dif-
fraction order propagates or not through the grating,
Λ
λ
≤
1
n
I
sin θ maxn
I
;n
II
; (1)
where θ is the incidence angle and n
I
and n
II
are the
refractive indices of the incident (superstrate) and
transmitting (substrate) media, respectively.
One can employ these SWGs to synthesize artifi-
cial birefringent achromatic wave plates [
9]. A bire-
fringent medium, such as a grating (see Fig. 1), has
two different refractive indices, n
TE
and n
TM
, with
regard to the polarization states TE (transverse elec-
tric, parallel to the grating grooves) and TM (trans-
verse magnetic, orthogonal to the grating grooves).
The phase retardation ΔΦ introduced by a birefrin-
gent SWG between the two polarization components
is dependent on the wavelength, and is given by
ΔΦ
TE–TM
λ
2π
λ
Δn
form
λ; (2)
where
Δn
form
λn
TE
λ − n
TM
λ: (3)
h is the optical path through the birefringent med-
ium. In order to produce an achromatic wave plate,
the product of the two factors in λ on the right-hand
side of Eq. (
2) needs to be a constant over a wave-
length range as large as possible. By varying the
grating parameters (geometry, material, incidence),
the wavelength dependence in Δn
form
should be
tuned to be closely proportional to the wavelength
across a wide spectral band.
B. Rigorous Coupled Wave Analysis (RCWA)
Common scalar theories of diffraction in gratings do
not work in the subwavelength domain. In order to
simulate the grating response and to calculate its
form birefringence, we must consider the vectorial
nature of light. For this purpose, we have performed
numerical simulations using the RCWA. RCWA is an
analysis method that is applicable to any multilayer
grating profile [
12,13]. The algorithm converts the
grating-diffraction problem into a matrix problem,
and solves the Maxwell equations from layer to layer.
The solution corresponds to the reflection and trans-
mission matrices in the two modes (TE/TM), de-
scribing the entire diffractive characteristics of the
simulated structure: the diffraction efficiencies
(η
m
TE
) and (η
m
TM
), and the phase shift (ΔΦ
TE–TM
).
3. Choosing an Appropriate Material: Diamond
For an HWP to keep its efficiency over a wide spec-
tral band, the component must be optimized in order
to minimize the phase shift error with respect to π:
ελΔΦ
TE–TM
λ − π: (4)
Since the error is a function of the wavelength, we
minimize the root mean squared (RMS) error ε
rms
Fig. 1. Schematic diagram of a SWG. The incident light beam
vector k
inc
is perpendicular to the grating lines. The filling factor
F is such that FΛ corresponds to the width of the grating walls.
5898 APPLIED OPTICS / Vol. 51, No. 24 / 20 August 2012
over the entire spectral band. We have performed
RCWA simulations to optimize the grating para-
meters for a refractive index ranging from 1.5 to
2.8, and for several specific mid-IR spectral bands,
corresponding to the transmission windows of the
Earth atmosphere: L (3.5–4.1 μm), M (4.5–5.1 μm),
and N (8–13 μm). Other solutions beyond 3 μm are
cumbersome and involve exotic materials. Consider-
ing an ideal rectangular profile, we optimized the
parameters F, h, and Λ (Fig. 1) using a multidimen-
sional nonlinear minimization method (Nelder-Mead
algorithm, also called downhill simplex method) from
the MATLAB software. The period was constrained
to be smaller than the SWG limit λ ∕ n. With these
optimized parameters, the mean phase shift error
is very small (ε
rms
≃ 10
−4
rad) in the considered spec-
tral bands. We also introduced another parameter,
the aspect ratio ρ,
ρ
h
minFΛ; 1 − FΛ
; (5)
which is the height-to-width ratio of either the walls
or the grooves. This parameter must be made as
small as possible since features with high aspect ra-
tio are difficult to manufacture. In our simulations,
the N-band was separated into two parts of compar-
able bandwidth (∼20%), thereby avoiding the strong
ozone absorption band around 10 μm. In Fig.
2, the
optimal aspect ratio is plotted against refractive
index. Minima in these plots indicate a suitable sub-
strate refractive index for the grating. For the longest
wavelengths (N-band mostly), the aspect ratio ρ is
much lower for high refractive indices (>2.25), while
for the shorter wavelengths, the filling factor F varies
strongly, which results in a very fluctuating aspect
ratio due to its definition as a minimum of either the
wall or void thickness. We conclude that using sub-
strates with high refractive indices (e.g., diamond,
ZnSe, Ge) can make the microfabrication of SWGs
for achromatic HWPs much easier. Some exotic
materials with lower refractive index exist, but they
are not compatible with our applications because of
many disadvantages (brittle, deform easily, etc.).
Materials with very high refractive index such as si-
licon (∼3.4) are not compatible either, because they
necessarily require complex antireflective coatings
on top of the SWG.
Among the materials commonly used in the N-
band, one good candidate is diamond, with a refrac-
tive index ∼2.38 for the region of interest (from 3 to
13 μm). The use of diamond substrates leads to many
advantages. First and foremost, it has a wide trans-
mission window [
14]. In addition, its mechanical prop-
erties are outstanding (low density 3.52 kg ∕ m
3
;
very high hardness 10 on the Mohs scale; very high
elasticity). Also, its thermal (excellent conductor, iner-
tia) and chemical (resistant to usual chemicals, acids,
and most of the alkalies) properties make it an
excellent candidate to be space qualified.
4. Design of a Diamond Wave Plate
The larger periods and lower aspect ratios of the
N-band HWP should make this grating easier to fab-
ricate compared to bands of shorter wavelengths, as
shown in Section
3. Here we will focus on a HWP for
the long-wavelength part of the N-band (11–13.2 μm)
for a forthcoming astronomical application (VLT/
VISIR) [
15]. The design of the grating was conducted
in synergy with the manufacturing [
16]. In particu-
lar, the slope of the sidewalls (see Fig. 3) must be
taken into account and the aspect ratios must
be kept within the range of what can be etched in
the material.
The etch process used induces a slope α ≃ 2.7°
(Fig.
3). During the fabrication process, small errors
in line width, slope, and grating depth occur. In par-
ticular, the depth h and the slope α are difficult to
measure precisely. A design that perfor ms well even
under small changes in these parameters was there-
fore sought. We have computed two-dimensional (2D)
maps of the RMS error (ε
rms
) as a function of the fill-
ing factor (F) and of the depth (h), for several values
of α ranging from 2.6 to 2.8°. We also calculated the
mean and standard deviation of all these maps
(see Fig.
4).
To obtain a good compromise between the mean
value and the standard deviation, the optimum va-
lues are h 13.7 μm and F 0.4, which correspond
to a line width (on the top) FΛ 1.84 μm. In the first
Fig. 2. (Color online) RCWA simulation: the aspect ratio of the
gratings for a minimized mean phase shift error as a function
of the refractive index.
Fig. 3. Schematic diagram of a trapezoidal grating. The grating
walls have a slope α and an average width F
equiv
Λ.
20 August 2012 / Vol. 51, No. 24 / APPLIED OPTICS 5899
approximation, one can tell that the relation between
the optimum line width and depth is quasi linear,
with a regression coefficient approximately equal
to 10. This is very useful for the manufacturing
process, to compensate line width errors with the
etching depth. For insta nce, if the line width is smal-
ler than the optimum (e.g., −50 nm), one can etch a
shallower grating ( −500 nm).
5. Manufacturing and Performance Analysis
A. Manufacturing
The pattern was first written in photoresist on a si-
licon wafer by direct laser writing. This pattern was
then transferred to the diamond substrate using a
nanoimprint lithography process and finally dry
etched in the diamond via several masking layers
in an inductively coupled plasma reactor. The details
of this process will be published elsewhere. The dense
oxygen/argon plasma used is very stable, but the etch
rate varies with depth and width of the grooves.
As mentioned before, precise measurements of the
depth and profile of the subwavelength structure are
difficult. The grooves are too narrow for using atomic
force microscopy or white light interferometry. The
best measurements were obtained by cracking the
sample and observing the cross section with a scan-
ning electron microscope (SEM). As can be seen in
Fig.
5 the sidewalls have a slight angle of 2.6–2.8°
from the vertical. Ions deflected in glancing impacts
with the sidewalls cause trenching at the bottom of
the wall. In a narrow groove such as this, trenching
gives rise to the triangular ridge along the center of
the groove.
B. Expected Performances
Cracking the sample is naturally not possible for the
final HWPs. For these an estimate of the depth was
acquired by comparing several SEM images taken at
different angles. The distance between two recogniz-
able features, one at the top and one at the bottom of
a groove, was measured in micrographs recorded at
tilt angles between 5 and 26°. From the variation in
this distance with angle, the depth could be calcu-
lated. The depth measured by this method on
cracked samples was in good agreement with cross
section images (within 2%). Our prototypes have
met the specifications: line width ≃1.8 μm (sidewall
angle ≃2.7°), period 4.6 μm, and depth ≃13.7 μm. As
shown in Fig.
6, the retardance of the manufactured
HWP is nearly ideal in the center of the spectral
band. The mean and standard deviation of the phase
shift over the whole upper N-band (11–13.2 μm)
equal ∼3.143 0.061 rad (180 .08 3.51 °).
C. Antireflective Grating (ARG)
Incoherent reflections with different phase shift may
interfere with the main beam and degrade perfor-
mance of the component. The natural reflection of
the diamond at N-band is ∼17% for one interface.
Although the achromatic HWP was not designed
with reducing reflections in mind, it still serves this
purpose to some degree. As shown in Fig.
7, the the-
oretical transmittance is quite good between 11.5
and 13 μm because the diffraction grating actually
acts as an antireflective layer at this wavelength.
Fig. 4. (Color online) RCWA multiparametric simulation: mean (left) and standard deviation (right) of the RMS phase shift error
(logarithmic scale) over the upper N-band, with α ranging from 2.6 to 2.8°. The period is set to Λ 4.6 μm (SWG limit).
Fig. 5. SEM-micrographs of a diamond achromatic HWP. Left:
cross sectional view of the grooves. Right: antireflective structure
on the backside.
5900 APPLIED OPTICS / Vol. 51, No. 24 / 20 August 2012
In addition, an ARG was designed to reduce sur-
face reflections using a diffraction grating analysis
program (GSOLVER, version 4.20c., Grating Solver
Development Co., USA). The program uses algo-
rithms based on RCWA to calculate values of the
zero-order transmission. A 2D SWG formed by bin-
ary square (2.6 × 2.6 × 2 μm) shaped structures
with a 4 μm period was etched on the backside of
the HWP to reduce the reflection from 17% to less
than 0.5% in the wavelength region of interest (see
Fig.
8). Etching the ARG was much less demanding
than the HWP structure, bu t there is still trenching
and the sidewalls are not perfectly vertical [
11]
(Fig.
5). Since there was some deviation from the
calculated structure, the etch time was optimized
by testing the performance. The transmittance of
diamond substrates with a single sided ARG was
measured in a Perkin Elmer 983 infrared spectro-
photometer. Afte r removing the effects of interfer-
ence within the sample and comparing with an
unetched sample, the transmittance of a single AR
interface has been determined (Fig.
8). As can be
seen, the bandwidth of the manufactured structure
was slightly larger than the calculated one. This
may be due to the somewhat tilted sidewalls. The
total transmittance of the finished HWP components
was between 89% and 95% over the band.
6. Conclusion and Directions for Future Research
Diamond is a good material for manufacturing achro-
matic HWPs for mid-IR wavelengths. By optimiz-
ing the gratings with manufacturing limitations in
mind, components dedicated to the upper N-band
(∼12 μm) could be achieved including an antireflec-
tive solution on their backside. Diamond HWPs are
likely to be used for many applications, particularly
for optical vortices in astrophysics. The diamond
HWP shown in this paper has been developed to en-
able the manufacturing of an N-band annular groove
phase mask (AGPM), which will be installed in 2012
on the VISIR instrument at the VLT, and is also an
excellent candidate for METIS at the future Eur-
opean Extremely Large Telescope. This specific ap-
plication will be the subject of a forthcoming paper.
Moreover, a new process using e-beam lithography
is now being explored to reach smaller grating peri-
ods, and thereby shorter operating wavelengths.
These AGPMs are being evaluated for implementa-
tion on high-contrast imaging instruments such as
NACO (L-ban d ∼3.8 μm) and SPHERE (K-band
∼2.2 μm) at the Very Large Telescope in Chile.
The first author is grateful to the financial sup-
port of the Belgian Fonds de la Recherche Scientifi-
que (FRIA) and Fonds de solidarité ULg. We also
gratefully acknowledge financial support from the
Swedish Diamond Center (financed by Uppsala
University), and the Communauté française de
Fig. 7. Transmission spectrum of a SWG etched on diamond
with HWP optimal specifications, with sidewall angle (2.6–2.8°).
The dotted line shows the natural transmission of the diamond
at N-band, without the SWG (∼83%).
Fig. 8. Transmission spectrum of one diamond interface with
ARG measured with a spectrophotometer. The calculated trans-
mission values for three different depths are also shown.
Fig. 6. (Color online) Retardance for a diamond HWP with opti-
mal specifications at α 2.6–2.8° (slope of the walls). Bandwidth:
11–13.2 μm.
20 August 2012 / Vol. 51, No. 24 / APPLIED OPTICS 5901