The Quasi-Triangle Array of Rectangular Holes with the Completely
Suppression of High Order Diffractions
Lina Shi, Hailiang Li, Ziwei Liu, Tanchao Pu, Nan Gao and Changqing Xie
Key Laboratory of Microelectronic Device & Integrated Technology,
Institute of Microelectronics of Chinese Academy of Sciences, Beijing 100029, China
Jiangsu National Synergetic Innovation Center for Advanced Materials, Nanjing 210009, China
Keywords: Diffraction Gratings, Binary Optics, Optical Design and Fabrication, High-Resolution Spectroscopy.
Abstract: We propose the quasi-triangle array of rectangular holes with the completely suppression of high order
diffractions. The membrane with holes can be free-standing and scalable from X-rays to far infrared
wavelengths. Both numerical and experimental results demonstrate the completely suppression of high order
diffractions. The desired diffraction pattern only containing the 0
th
and +1
st
/-1
st
order diffractions results
from the constructive interference of lights from different holes according to some statistical law
distribution. The suppression effect depends on the number of holes. Our results should be of great interest
in a wide spectrum unscrambling for any wavelength range.
1 INTRODUCTION
Gratings are the key component of the spectrometers.
Spectrum unscrambling only needs the first order
diffraction of the traditional black-white grating.
However, unwanted higher order diffraction always
overlaps the first diffraction, and thus greatly
degrade precision of analysis (Palmer, 2005). The
sinusoidal transmission gratings only have 0
th
and
+1
st
/-1
st
order diffractions (Born, and Wolf, 1980),
but they are much more difficult to fabricate than the
black-white ones (Jin, Gao, Liu, Li, and Tan, 2010,
Vincent, Haidar, Collin, Guérineau, Primot,
Cambril, and Pelouard, 2008). The high order
diffractions can become evanescent waves with a
grating period in the range of the wavelength
λ
(Clausnitzer, Kämpfe, Kley, Tünnermann,
Tishchenko, and Parriaux, 2008, Zhou, Seki,
Kitamura, Kuramoto, Sukegawa, Ishii, Kanai,
Itatani, Kobayashi, and Watanabe, 2014, Warren,
Smith, Vawter, and Wendt, 1995). Unfortunately,
for short wavelengths less than 100 nm, it’s difficult
to scale the grating period down to the wavelength
size by the current nanofabrication technology
(Pease, Deshpande, Wang, Russe, and Chou, 2007).
Therefore, it has been a goal to design the black-
white structure much larger than the wavelength
with the suppression of high order diffractions.
Several structures with the suppression of high
order diffractions have been developed, and the
points are to modulate the groove position or to
introduce structures with complicated shapes (Gao,
and Xie, 2011, Fan, et al, 2015, Cao, Förster,
Fuhrmann, Wang, Kuang, Liu, and Ding, 2007).
Unfortunately, the reported works can’t obtain
complete suppression of high order diffractions
since it’s difficult to realize the complex shapes or
the very small gaps between the two adjacent
grooves. Moreover, these structures based on the one
dimensional grating can’t be free-standing.
Unfortunately, the supporting membrane will absorb
80% energy of the soft X-ray.
Recently, photon sieves with aperiodic
distributed holes have drawn great attention owing
to their novel properties, super-resolution focusing
and imaging beyond the evanescent region (Kipp,
Skibowski, Johnson, Berndt, Adelung, Harm, and
Seemann, 2001, Huang, Liu, Garcia-Vidal, Hong,
Luk’yanchuk, Teng, and Qiu, 2015, Huang, Kao,
Fedotov, Chen, and Zheludev, 2008, Huang,
Zheludev, Chen, and Abajo, 2007). The numerous
holes can be designed that creates constructive
interference, leading to a subwavelength focus of
prescribed size and shape. Photon sieves with
aperiodic distributed holes can acquire rich degrees
of freedom (spatial position and geometric shape of
holes) to realize complex functionalities, which are
not achievable through periodic features with limited
control in geometry.
54
Shi L., Li H., Liu Z., Pu T., Gao N. and Xie C.
The Quasi-Triangle Array of Rectangular Holes with the Completely Suppression of High Order Diffractions.
DOI: 10.5220/0006100700540058
In Proceedings of the 5th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2017), pages 54-58
ISBN: 978-989-758-223-3
Copyright
c
2017 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
Triggered by photon sieves, we propose the
quasi-triangle array of rectangular holes with the
completely suppression of high order diffractions.
The membrane with holes can be free-standing and
scalable from X-rays to far infrared wavelengths.
The location distribution of holes according to some
statistical law results in a desired diffraction pattern.
We numerically and experimentally demonstrate the
diffraction pattern of the quasi-periodic hole array
with only 0
th
and +1
st
/-1
st
orders. Furthermore, we
investigate the effect of the hole number on the
diffraction property.
2 DESIGN AND SIMULATIONS
We start the design from the triangle array of
rectangular holes, and the holes are shifted by
s
along the
ξ
axis (Fig. 1(a)) according to the
probability distribution
() ( / )cos(2 / )sP sP
ξξ
ρπ π
=⋅
,
|| /4sP
ξ
, where
2P
ξ
is the period of the triangle
array along the
ξ
axis. Here, we only consider the
shift
s
along the
ξ
axis since spectral measurement
is usually at one direction. Moreover, we choose the
triangle array rather than the square array since the
spacing between any two adjacent holes of the
triangle array is larger than that of the square array
for the same period and hole size. This will benefit
Figure 1: (a) The quasi-triangle array of rectangular holes:
each hole shifts
s along the
ξ
axis according to the
probability distribution
()s
ρ
. (b) Microphotograph of the
fabricated quasi-triangle array with
44cm cm×
area.
the fabrication, and lead to more stable free standing
structure. In addition, rectangular hole is selected
since its diffraction intensity pattern is beneficial to
the suppression of high order diffractions.
For a membrane that contains a large number of
identical and similarly oriented holes, the light
distribution in the Fraunhofer diffraction pattern is
given by [3]
()
(' ')
(,) ' '.
nn
ik p q
ik p q
n
Upq C e e d d
ξη
ξη
ξη
−+
−+
Α
=

(1)
Here
0
/( )CPz
λ
=
P
is the power density
incident on the hole array,
λ
is the incident light
wavelength,
0
z
is the distance between the hole array
plane and the diffraction plane. The coordinates of
the hole center are
11
(, )
ξη
,
22
(, )
ξη
, …
(, )
NN
ξη
, and
k
is the wavevector.
0
/pxz=
,
0
/qyz=
. The integration
extends over the hole area and the integral expresses
the effect of a single hole. The sum represents the
superposition of the coherent diffraction patterns.
For the quasi-triangle array of
2NN
ξη
rectangular
holes of sides
2/2aP
ξ
=
and
2/2bP
η
=
as shown in
Fig. 1, the diffraction intensity pattern is
*
2
22
22
0
2
22
0
22
22
(,) (,) (,)
(4 ) sin sin
()()
| ( ) exp( ( ) ) |
cos ( /4) cos ( 2 /4 /4)
(1 / 2 / ) (1 / 2 / )
sin 2
sin sin
()()(
sin
nn
n
S
Ipq Upq U pq
Pab kpa kqb
z kpa kqb
s ikp s ikq ds
kpP kp P kqP
I
kpP kpP
Nkp P
kpa kqb
kpa kqb N
ξξη
ξξ
ξξ
ξ
λ
ρξη
ππ
=∗
=⋅
⋅−+
⋅+
=⋅
−+
⋅⋅
22
sin
)( ).
2sin
NkqP
kp P N kqP
ηη
ξη η
(2)
Here
22
0
(2 4 )IC NNab
ξη
=⋅
is the peak irradiance
of the diffraction pattern. For simplicity, in this letter
we set
0
1I =
.
2P
ξ
and
P
η
are respectively the periods
along the
ξ
and
η
axes.
Figure 2 presents the diffraction intensity pattern
according to Eq. (2). As expected, the 0
th
and 1
st
order diffractions are kept along x axis, and the
high-order diffractions disappear. The logarithm of
diffraction intensity along x axis in Fig. 2(b)
presents clearly the complete suppression of the high
order diffractions. Insets in Fig. 2 shows clearly
intensity distributions of the 0th and 1st order
diffractions. Figure 2 shows that the diffraction
pattern of the quasi-triangle array of rectangular
holes along x axis is the same as the ideal sinusoidal
transmission grating.
The Quasi-Triangle Array of Rectangular Holes with the Completely Suppression of High Order Diffractions
55
Figure 2: (a) The far-field diffraction intensity pattern of
the quasi-triangle array of rectangular holes. (b) The
diffraction intensity along the x axis. Insets: the 0
th
and 1
st
order diffractions.
Numerial simulation based on Eq. (2) is carried
out to evaluate the diffraction property of the quasi-
triangle array of 100000 rectangular holes. The
logarithm of diffraction intensity along x axis is
shown in Fig. 3, and high-order diffraction is much
less than the noise of 10-5 between 0th and 1st
diffraction (red dash line in Fig. 3), which agrees
well with the theoretical prediction of Eq. (2) and
Fig. 2.
Figure 3: The diffraction intensity along x axis of the
quasi-triangle array with 100000 rectangular holes.
3 EXPERIMENTAL RESULTS
AND DISCUSSIONS
A proof-of-principle experiment is performed to
confirm our theoretical and numerical results. The
experimental setup is shown in Fig. 4. A plane wave
is produced by expanding a 632 nm laser beam from
Sprout (Lighthouse Photonics) and then illuminating
the quasi-triangle array of holes. The far-field
diffraction pattern from the array is focused by a
lens and recorded by a CCD (ANDOR DU920P-
BU2). The quasi-triangle array of rectangular holes
with
44cm cm×
area is fabricated on a glass substrate
by DESIGN WRITE LAZER 2000 from Heidelberg
Instruments Mikrotechnik GmbH. The microphoto-
graph of fabricated structure is illustrated in Fig.
1(b). Periods
2P
ξ
and
P
η
of the quasi-triangle array
along the
ξ
and
η
axes are respectively
20 m
μ
and
10 m
μ
. The side of square holes is
5 m
μ
.
Figure 4: Experimental setup for the optical measurement.
Figure 5 presents the recorded diffraction pattern
and it is obvious that high order diffractions of the
quasi-triangle array of holes are effectively
suppressed. The diffraction intensity along x axis in
Fig. 5(b) is almost the same as the ideal sinusoidal
transmission gratings. The ratio of 1st order
diffraction intensity to the 0th order diffraction
intensity is 73.56% and much larger than the
theoretical predition and numerical value 25%. This
is because the CCD is saturated by the 0th order
diffraction intenstiy. In addition, the red vertical
lines in Fig. 5(a) are crosstalk along y direction due
to our one-dimension CCD.
Figure 5: (a) The far-field diffraction intensity pattern of
the quasi-triangle array of rectangular holes. (b) The
diffraction intensity along the
ξ
axis.
Now we focus on the complete suppression of
high order diffractions along the x axis. As
N
ξ
is
large enough, the intensity according to Eq. (2)
along the x axis is given by:
PHOTOPTICS 2017 - 5th International Conference on Photonics, Optics and Laser Technology
56
2
0
222
0
0
sinc ( 8 / )
()
(1 2 / ) (1 2 / ) cos ( 4 )
,0
1
,
42
INkpa
Ip
kp a kp a kp a
Ip
Ip
ka
ξ
π
ππ
π
=
−+
=
=
±
=
(3)
Equation (3) demonstrates that the quasi-triangle
array of infinite rectangular holes can generate the
same diffraction pattern as sinusoidal transmission
gratings along the x axis. Only three diffraction
peaks (the 0
th
order and +1
st
/-1
st
orders) appear on
the
xy
plane. The above theoretical results are
scalable from X-ray to far infrared wavelengths.
To obtain physical insight into the diffraction
property of the quasi-triangle array of rectangular
holes, the average transmission function along
ξ
axis
is calculated by integrating the probability
distribution over
η
axis:
/4
|| /4
12
() () (1 cos( )).
2
P
P
Tsds
P
ξ
ξ
ξ
ξ
π
ξρ ξ
==+
(4)
Equation (4) shows that the quasi-triangle array
of infinite rectangular holes has the same
transmission function along the
ξ
axis as sinusoidal
transmission gratings. It is the average diffractive
effect similar to sinusoidal grating that eliminates
high-order diffractions.
We can also understand the suppression of high-
order diffractions by the interference weakening or
strengthening. It’s known that diffraction peaks is
from the constructive interference of lights from the
different holes. The interference of lights from
different rectangular holes is controlled by the hole
position. The desired diffraction pattern only
containing the 0
th
order and +1
st
/-1
st
order diffra-
ctions can be tailored by the location distribution of
holes according to some statistical law.
In the above discussions, we assume the size of
the quasi-triangle array is infinite. In practice, the
number of holes is finite and thus the diffraction
pattern will not be perfect as that described by Eq.
(2) and (3). Numerical simulation based Eq. (2) is
carried out and we obtain the noise intensity at 3rd
order diffraction location for different number of
holes in Fig. 6. It reveals that the noise intensity is
approximately inversely proportional to the number
of holes. This can be attributed to the approaching of
the average transmittance function of quasi-triangle
array to the ideal sinusoidal function with the hole
number increasing. The noise intensity around the
location of the 3rd order diffraction becomes less
than 10-5 as the hole number reaches to 30000.
Figure 6: The maximum noise intensity around the
location of the 3
rd
order diffraction versus the number of
holes.
4 CONCLUSIONS
In conclusion, the binary quasi-triangle array of
rectangular holes has been proposed to suppress the
high-order diffractions which may lead to
wavelength overlapping in spectral measurement.
The new design of the membrane with holes can be
free-standing and scalable from X-rays to far
infrared wavelengths. Both numerical and
experimental results have demonstrated the high-
order diffractions are efficiently suppressed. The
total number of holes illuminated by the light affects
the suppression of the high-order diffractions and the
noise intensity will be less than 10
-5
as the hole
number larger than 30000. The binary quasi-periodic
hole array offers an opportunity for high-accuracy
spectral measurement and will possess broad
potential applications in optical science and
engineering fields.
ACKNOWLEDGEMENTS
This work is supported by National Natural Science
Foundation of China (NSFC) (61107032, 61275170)
and the Opening Project of Key Laboratory of
Microelectronics Devices and Integrated Techno-
logy, Institute of Microelectronics of Chinese
Academy of Sciences.
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