Fabrication and tolerances of moth-eye structures for perfect ...

Fabrication and tolerances of moth-eye structures for perfect
antireflection in the mid-infrared wavelength region
Hiroaki Imada a , Takashi Miyata b , Shigeyuki Sako b , Takafumi Kamizuka b , Tomohiko
Nakamura c , Kentaro Asano b , Mizuho Uchiyama b , Kazushi Okada b , Takehiko Wada d , Takao
Nakagawa d , Takashi Onaka c and Itsuki Sakon c
a Department of Physics, University of Tsukuba, 1-1-1 Ten-nodai, Tsukuba, Japan;
b Institute of Astronomy, University of Tokyo, Mitaka, Japan;
c Department of Astronomy, University of Tokyo, Bunkyo, Japan;
d Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1
Yoshinodai, Chuo-ku, Sagamihara, Japan
ABSTRACT
Mid-infrared, 25 - 45 microns, is a very important wavelength region to investigate the physics of lower temperature
environments in the universe. There are few transparent materials in the range of mid-infrared except
silicon. However, the reflection on a silicon surface reaches 30 % because of its high refractive index (∼3.4).
To apply silicon to mid-infrared astronomical instruments, we need a way of antireflection and have adopted
a moth-eye structure. This structure keeps durable under cryogenic environments, which is advantageous to
mid-infrared instruments. We have fabricated three samples of the moth-eye structure on plane silicon surfaces
by electron-beam lithograph and reactive ion etching. The structures consist of many cones standing on silicon
surfaces. We have substantiated the transmittance of 96 % or higher in the wide range of 20 - 50 microns and
higher than 98 % at the maximum. The transmittance of moth-eye surfaces, however, is theoretically expected
as 100 %. We have examined the discrepancy between the transmittance of the theory and fabrications with
electromagnetic simulations. It has been revealed that shapes of the cones and gaps at the bottom of the cones
seriously affect the transmittance. We have estimated a few tolerances for manufacturing the moth-eye structures
achieving sufficient transmittance of nearly 100 %.
Keywords: Subwavelength structures, Moth-eye structure, Optical fabrication, Diffractive optics
1. INTRODUCTION
Mid-infrared is one of the most important wavelength ranges for astronomy because the physics of lower temperature
environments in the universe can be investigated with mid-infrared. Many ambitious projects for the
mid-infrared astronomy, such as SPICA 1 and TAO, 2 are in progress. Optical systems of telescopes for midinfrared
consist of many mirrors so far due to lack of transparent materials transmitting faint light from stars
without loss in the mid-infrared range, especially 25 - 45 µm, although optical systems with mirrors are generally
larger than those with refractive elements like lenses. Refractive elements are necessary in order to make optical
systems smaller than ever. Smaller systems are desirable to space telescopes in particular. Silicon, exceptionally,
is almost completely transparent at mid-infrared wavelengths longer than 25 µm. Silicon does not have deliquescence
and toxicity like CsI and KRS5, which means that silicon is very stable and easy to process. Silicon
therefore is the only and most promising material for mid-infrared optics.
One of the most serious disadvantages of silicon, however, is reflecting loss. For 25 µm or longer, the reflectance
R is
Further author information: (Send correspondence to H. I.)
H. I.: E-mail: s1120251@u.tsukuba.ac.jp, Telephone: +81 29 853 5600 (ext. 8299)
R = (n Si − 1) 2
∼ 0.30, (1)
(n Si + 1)
2

Figure 1. (a) A schematic illustration of moth-eye structure. The transmittance depends on D and H. (b) The SEM
image of the sample No. 1 taken right above the protuberances. (c) The side image of the sample No. 1. The flared
shape is observed. The sample No.2 has a similar shape to that of No. 1. (d) The side image of the sample No. 3. The
shape of the protuberances is straight.
where n Si is the refractive index of silicon, and n Si ∼ 3.42. 3 This high reflectance arises from its extremely high
refractive index. Antireflection is strongly needed for astronomical applications. Although there are many ways
for antireflection, most of them cannot be applied to mid-infrared instruments. Muti-layer coating, for example,
is one of the most popular ways to achieve antireflection but needs various kinds of materials, each having a
different refractive index. In addition, multi-layer structure can be broken due to bimetal effect under cryogenic
environment which is needed in order to suppress the thermal radiation from instruments. We therefore have
difficulty in applying multi-layer coating. Another example is a sub-wavelength structure which consists of bulk
and porous layers piled by turns. 4 It is suitable for antireflection but not easy to fabricate because it is necessary
to pile many layers in order to achieve broadband antireflection. A moth-eye structure is also one of effective
structures for antireflection. This structure is a surface-relief structure which needs only one material. We think
this structure is the best because it does not have problems mentioned above. We have designed moth-eye
structures on silicon for 25-40 µm.
The moth-eye structure has many protuberances placed on a plane surface next to each other like Fig. 1(a).
The protuberances have nearly a conical shape. A similar structure can be found on corneas of nocturnal insects,
such as moths, after which it is called moth-eye structure. Bernhard 5 showed that the structures on the corneas
of moths worked to suppress reflection by experiments using a model scaled up properly for microwave. In case
of visible light, they have been applied to a surface on crystalline silicon for solar cells (e.g. Forberich et al. 6 ).
Moth-eye structures on a GaAs substrate have been analyzed in detail in the near-infrared up to 17 µm. 7
In this paper, 25-40 µm is the band in which we would like to attain transmittance of more than 99 %
for normal incidence. The next section includes how we have determined the design of the moth-eye structure
and have fabricated them. In section 3, it is shown how we have made measurements and have corrected the
measurements for getting precise transmittance. In section 4, the results of measurements after analysis are

shown. Simulations with Rigorous Coupled Wave Analysis (RCWA) have been carried out to investigate what
makes the spectra of the measurements in section 5. Finally, we summarize a few tolerances for fabrication of
moth-eye structures.
2. DESIGN AND FABRICATION OF MOTH-EYE STRUCTURE
The principle of antireflection by moth-eye structure is understood qualitatively well. The essence of reflection is
that a refractive index is discontinuous at an interface between two media. If the wavelength λ is longer than the
period D between the protuberances (Fig. 1 (a)), the refractive index appears to be an effective value between
those of silicon n Si and vacuum. Moreover, if the effective refractive index varies so gradually at the interface
that the light cannot recognize the discontinuity or variation of it, reflection should not be generated. Rayleigh 8
investigated the relation between the wavelength λ and the effective thickness H of the layer of graded refractive
index. H corresponds to the height of the moth-eye protuberances. Clapham et al. 9 calculated the reflectance of
the graded interface and showed the results (see fig. 3 in Clapham et al.). When H/λ ≪ 1, the refractive index
appears to be discontinuous and the reflectance is not zero. As H/λ increases, the reflectance falls monotonically
and reaches 0% at H/λ ∼ 0.4. When H/λ is larger than 0.3, that is, λ < ∼ 3H, the reflectance is less than 1 %.
Diffraction is another concern to design the shape of the protuberances. When the wavelength λ is equal
to or shorter than n Si D, the moth-eye structure acts as a grating and the higher orders of the transmitted
diffraction are generated, which makes the 0th order diffraction decreased. Detailed and strict discussions about
the effective index of refraction and diffraction can be found in Brückner et al. 10 In the consideration above,
approximately n Si D < λ < 3H is the range in which the transmittance of the 0th order diffraction is almost 100
% theoretically.
We have designed moth-eye structures to attain transmittance of more than 99% in the wavelengths from 25
µm to 40 µm. For normal incidence, one of the limits of the transmitted wavelength is given by n Si D ≤ 25 µm,
which yields D ≤ 7.3 µm. The other is 40µm ≤ 3H, that is, H ≥ 13.3 µm. We have adopted D = 5 µm and
H =15 - 20 µm as a goal in fabricating.
The shape of protuberances we have intended to fabricate is a cone, which is easier to process. The substrates
are high resistivity silicon, the diameter is 4 inches and the thickness is 600 µm. One side of the substrates was
processed and the other side was not. Electron-beam lithograph and Reactive Ion Etching (RIE) was used. The
first process was to make metal masks by electron-beam lithograph on the silicon surface where the protuberances
should stand. Then, a silicon surface was etched by optimizing ion energies of reactive ions. As a result, the
area without the metal masks was carved, then protuberances like cones appeared on the surface of the silicon.
Finally, the metal masks were removed.
We have acquired three moth-eye samples. The diameter of the processed area of No.1 and No.2 is 25mm, and
that of No. 3 is 40mm. Fig. 1 (b) is a scanning electron microscope (SEM) image taken right above protuberances.
The period D and the height H of all the samples were measured as 5 µm and 16 µm, respectively. Fig. 1 (c)
and Fig. 1 (d) of SEM images show the side views of the protuberances. There is a difference in the shape
of the cones among the samples. The samples No.1 and No.2 have flared shape: they have spindly tops and
become fat as getting close to the bottom (Fig. 1 (c)). By contrast, No. 3 is straight shapes from the tops to
the bottoms (Fig. 1 (d)). Note that the tops of the protuberances remain flat, not pointed, due to the metal
masks in processing. Fig. 2 was a photo taken in visible light. The area with moth-eye structure seems to have
no reflection in visible light range and the other area looks like a mirror.
3. MEASUREMENT AND ANALYSIS
Transmittance spectra of the samples were obtained by Fourier Transform Infrared spectroscopy (FT-IR). The
incident angle was 0 degrees. The 0th order transmitted diffraction was measured in a vacuum. The mylar beam
splitter was used. The samples were at the room temperature around 300 K. The resolution was 2 cm −1 . For
getting better S/N, we have averaged 6 successive measurement points relative to wavenumbers, therefore the
effective resolution was 12 cm −1 .

Figure 2. The appearance of No.3. The moth-eye structure is in the central area and there seems to be no reflection, whereas
the area without moth-eye structure reflects visible light like a mirror. The yellow crosses represent the measurement
points. The intervals between two points were 10 mm for No. 3.
The measurements must be corrected for getting precise transmittance spectra because one side of the substrates
remains bare and the absorption of the silicon is not completely negligible. For this purpose, we have
measured spectra on several parts of the silicon with and without the moth-eye structure (Fig. 2).
First, we have estimate absorption of a sample. The measurable transmittance of a bare silicon board T Si is
given by
T Si = (1 − R) 2 exp(−κd)
∞∑
[R exp(−κd)] 2m , (2)
where R = (n Si − 1) 2 /(n Si + 1) 2 is the reflectance on a bare silicon surface in a vacuum, κ is the absorption
coefficient of the sample and d is the thickness of it. n Si is the refractive index of silicon. We have used Edwards’
data 3 for the refractive index n Si and the data is reproduced by
m=0
n Si (λ) = 3.41861 + 0.02661 + 2.6939 × 10 −5 λ − 2.6405 × 10 −7 λ 2
λ
− 1.6729 × 10 −9 λ 3 + 1.2612 × 10 −11 λ 4 , (3)
where the unit of λ is µm. In the range of 10µm - 70µm, equation (3) replicates Edwards’ data well (Fig. 3 (a)).
We can solve equation (2) for the unknown variable κ.
Second, we have calculated the transmittance on a surface with the moth-eye structure using the absorption
coefficient κ solved in advance. If there is no absorption on the surface with moth-eye, the measurable
transmittance T is
T = (1 − R ME )(1 − R) exp(−κd)
∞∑
[R ME R exp(−2κd)] m , (4)
where R ME is the reflectance on the surface with moth-eye. By solving equation (4) for R ME , the transmittance
m=0

Figure 3. (a) The refractive index of Edwards’ data and its approximation. In the range of 10 - 70 µm, equation (3)
replicates Edwards’ data well. (b) The transmittance of the samples. It is extremely improved.
at the surface with moth-eye T ME is obtained,
Equation (5) was used in the analysis of all measurements.
T ME = 1 − R ME
T [1 − R exp(−2κd)]
=
(1 − R) exp(−κd) − T R exp(−2κd) . (5)
4. RESULTS
Fig. 3 (b) displays the results of the transmittance spectra of the moth-eye structure samples after the correction
with equation (5). The spectra of any sample do not depend on the measurement points, so that we show the
spectra at the center of every sample. Fig. 3 (b) clearly shows that the moth-eye structure extremely improved
the transmittance of all samples. The best sample is No.3 which has the straight shape. No. 3 achieves more
than 98 % at the maximum and more than 96 % in the wide range of 20-50 µm. For No.1 and No.2 which have
the flared shape, their transmittance is also more than 98 % at the maximum but their profiles are different from
that of No.3. Both the transmittance of No.1 and No.2 are decreasing rapidly as wavelength gets longer. Their
spectra seem similar but are not completely identical. Note that several peaks around 14, 15, 16, 23 and 26 µm
are artificial. These peaks were due to the absorption in FT-IR.
5. SIMULATIONS
The measured transmittance of more than 98 % is very high, but it falls short of the theoretical expectation
of more than 99 %. The shape of the protuberances could not make the transmittance reach to 99 %. To
investigate qualitatively how far the shape affects the transmittance, we carried out a numerical analysis with
RCWA. Models of the protuberances used in this calculation were based on the fabricated samples; the height
and period of models was 16 µm and 5 µm, respectively.
5.1 Flared model
An ideal moth-eye structure is illustrated in Fig. 1 (a). The protuberances are perfect cones and next to each
other. A flared model corresponding to the samples No.1 and No.2 is shown in Fig. 4 (a). The simulated spectra
of the ideal model and flared model are Fig. 4 (b); the solid line (red) represents the ideal model, the longest
dashed line (green) is the model for No.1 and No.2, the second longest (blue) is the measurement of No. 2, and
the shortest (magenta) is the measurement of No.3. We can see that the models approximately reproduce the

Figure 4. (a) A flared model for No.1 and No.2. (b) The results of simulation and measurements. The solid line (red)
is for the ideal cone model and the longest dashed line (green) for the flared cone model. The second longest dashed
line (blue) represents the measurement of No.2 and the shortest (mazenta) represents the measurement of No. 3. The
simulation reproduces the measurements well.
Figure 5. (a) A flat top model. (b) The result of simulation as a function of the filling factor. The solid line (red)
corresponds to 30 µm and the dashed line (green) 40 µm.

Figure 6. (a) A model of straight shape cones having gap space between protuberances at the bottom. (b) The result of
simulation as a function of the porosity. The solid line (red) corresponds to 30 µm and the dashed line (green) corresponds
to 40 µm.
transmittance of No.2 and No.3, respectively. We deduced that the rapid decrease of the transmittance of No.1
and No. 2 at longer wavelengths is caused by the flared shape. This is qualitatively explained as follows: the
protuberances are so spindly that the effective index of refraction is almost 1 near the tops, whereas the index
rapidly increases near the foot because of the flared shape. The effective height of the protuberances therefore
becomes lower than the actual height. This causes the decrease of the transmittance at longer wavelengths.
5.2 Flat top model
We have taken notice of effects of the flat tops shown in Fig. 5 (a). The size of the flat area is a parameter of
this model. To assess it, a filling factor at the top is introduced as a ratio of the flat area to the area of one
section surrounded by the black lines in Fig. 1 (b). Fig. 5 (b) shows the relation between the filling factor and
the transmittance at 30 µm (solid line, red) and 40 µm (dashed line, green). The filling factor of 3 %, which
is estimated from the SEM image (Fig. 1 (b)), corresponds to the transmittance of more than 99.5 % at both
30 and 40 µm. The transmittance does not become 98 % unless the filling factor exceeds ∼20 %, not observed
in the SEM images. The flat tops can hardly affect the transmittance in our measurements, that is, the filling
factor of a few percent allows the transmittance to reach more than 99 %.
5.3 Gap space model
We have tried models with gap space at the bottom between the cone. The model we have made is shown in
Fig. 6 (a) and the result is in Fig. 6 (b). To assess the size of gap space, a porosity at the bottom is defined as
a ratio of the area of gap space to that of the section. The porosity corresponding to the transmittance of 98 %
is about 23 %, which is realized when the diameter of the protuberances at the bottom is 4.6 µm. It is probable
that there are gaps of 0.4 µm between the cones at the bottom as far as we have observed the SEM images.
Thus we conclude that the shortage of the transmittance is caused by the gaps at the bottom rather than by the
flat tops. To achieve 99 %, it is important that the porosity is less than 15 %.
The differences between the spectra No.1 and No.2 can also be attributed to this gap effect. We have calculated
the flared model with and without gaps at the bottom. The result is shown in Fig. 7. The transmittance of the
gap model also decreases more rapidly than the no-gap model.
6. CONCLUSION
We have developed antireflection for mid-infrared astronomical instruments with moth-eye structures. Electronbeam
lithograph and RIE was used to fabricate the moth-eye structures on silicon. We have obtained three
samples which have different shapes of the protuberances. The profiles of the transmittance are also different.

transmittance [%]
110
100
90
80
no gap space
gap space
No. 1
No. 2
70
65
10 20 30 40 50 60 70
wavelength [µm]
Figure 7. The result of simulation for flared models with and without gap space. The solid line (red) corresponds to the
case without gap and the longest dashed line (green) with gap. The second longest (blue) represents the measurement of
No. 1 and the shortest (magenta) represents the measurement of No.2.
The best sample has transmitted more than 98 % at the maximum and more than 96 % in the wide range of
20-50 µm. To investigate what made the spectra different, and why the transmittance did not reach more than
99 % in the wide range, we have carried out a numerical analysis with RCWA. Consequently, we have acquired
a few tolerances for fabricating moth-eye structures:
1. The shape of the protuberances must be straight from the tops to the bottoms.
2. The tops may be flat if the filling factor is a few percent.
3. Gap space at the bottom must be as little as possible, that is, the porosity is less than 15 %.
If the moth-eye structure can be made within the allowed tolerances, the transmittance of more than 99 % will
be achieved.
In the near future, moth-eye structures will be applied to both sides of silicon substrates and finally to silicon
lens for mid-infrared instruments.
ACKNOWLEDGMENTS
We are grateful to NTT Advanced Technology Corporation for fabricating the moth-eye structure samples which
meet our requirements. This work was supported by Grant-in-Aid for Young Scientists (A) 21740131 and Grant
for development of on-board instruments from JAXA.
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