Compact Terahertz Wave Collimator Design with Photonic Crystal Slabs

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JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 9, NO. 3, SEPTEMBER 2011
227
Compact Terahertz Wave Collimator Design with
Photonic Crystal Slabs

Abstract⎯A compact terahertz (THz) wave
collimator is proposed, which works under the
frequency from 2.4

THz to 2.7

THz with a photonic
crystal (PC) slab based on the self-collimation effect.
The plane wave expansion (PWE) method is used to
calculate the dispersion surfaces and the
equal-frequency contours (EFCs) and optimize the
structure. The propagation of the THz waves in the
structure is simulated and the normalized transmission
is calculated by using the finite-difference time-domain
(FDTD) method with perfectly matched layer (PML)
absorbing boundary conditions. Numerical simulations
show that the designed collimator has a good collimation
property and a high transmittance.

Index Terms⎯Photonic crystals, self-collimation,
Terahertz.
1. Introduction
Terahertz (THz) waves, which lie between far-infrared
and millimeter waves (frequency spanning from 0.1

THz to
10

THz), have attracted increasing interests and studies over
the past several years. Various THz apparatus are
developed to satisfy the growing needs of applications in
many fields, such as spectroscopy
[ ]
1
,

medical applications
[ ]
2
,
security screening
[ ]
3
, space communications
[ ]
4
, atmospheric
studies
[ ]
5
, and sensing
[ ],[ ]
6
7
. Controlling the THz waves in a
compact integrated circuit is required for these applications.
Photonic crystals (PCs), which were first introduced by
Yablonovitch and John in 1987
[ ],[ ]
8
9
, have the natural
advantages of controlling the propagation of light waves.
Therefore, numerous THz devices based on the PCs have
been studied
[ ],[ ]
10
11
. But until now, most of these devices are


Manuscript received January 28, 2011; revised June 8, 2011. This work
was supported by the National Natural Science Foundation of China under
Grant No.60588502.
Z.-H. Wu, K. Xie, and P. Jiang are with the School of Optoelectronic
Information, University of Electronic Science and Technology of China,
Chengdu 610054, China (e-mail: wu.zhenhai@hotmail.com, kangxie@
uestc.edu.cn, and jiangp@uestc.edu.cn).
J.-C Wan is with the School of Information Engineering, Huanghe
Science and Technology College of China, Zhengzhou 450005, China
(e-mail: xyb@hhstu.edu.cn).
Color versions of one or more of the figures in this paper are available
online at http://www.intl-jest.com.
Digital Object Identifier: 10.3969/j.issn.1674-862X.2011.03.006
based on the photonic bandgap (PBG) properties of PCs. In
recent years, the unusual dispersion properties of PCs, such
as negative refraction
[ ]
12
, self-collimation
[ ],[ ]
13
14
, and
superprism
[ ]
15
, have attracted more and more attentions.
Some devices based on the dispersion properties are also
exploited.
In this paper, a compact 2.6

THz wave collimator with
optimized structure based on the self-collimation effect of
PCs is proposed. The collimator will be useful for the
integrated circuit of THz waves.
2. Structure Design and Analysis
2.1 Theory of Self-Collimation
The propagation of electromagnetic wave in a PC
structure is governed by its dispersion surfaces. Incident
electromagnetic waves propagate in directions normal to
the dispersion surface. The divergence and convergence of
the wave beam depend on the curvature of the dispersion
surface. This concept is different from the nonlinear
propagation. The nonlinear propagation (such as soliton
propagation) requires high-intensity incident light, and the
balance between diffraction broadening and focusing easily
collapses due to the absorption
[ ]
16
. However, with the
self-collimation phenomena, the propagation is independent
of the light intensity. A cross section of the dispersion
surface at a constant frequency is referred to as an
equal-frequency contour (EFC)
[ ]
13
, as shown in Fig. 1. The
circular solid line describes the EFC in a homogeneous
material, while the round square shaped with the dashed
line describes the EFC in a PC slab. The EFCs can be
obtained by using the PWE method.
The group velocity v
g
is perpendicular to the EFCs,
which is given by v
g
=

k
ω(k)
[ ]
14
, where ω is the frequency at
the wavevector k. When the EFC corresponding to a
frequency is a straight line, it leads to the self-collimation
phenomenon, as indicated in Fig. 1.







Fig. 1. Schematic illustration of the self-collimation phenomenon.
Zhen-Hai Wu, Kang Xie, Ping Jiang, and Jian-Chen Wan

k
ω
(k)

k
ω
(k)
k
p
k
p
k
0
k
0
JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 9, NO. 3, SEPTEMBER 2011

228
Fig. 2 shows the numerical simulation results of the
electromagnetic wave propagation of a point source in the
center of a homogeneous silicon slab and a rod-based PC
silicon slab with the finite-difference time-domain (FDTD)
method, respectively. From the simulation, we can see
clearly that the propagation in the silicon slab is isotropic as
shown in Fig. 2 (a), while the propagation in the designed
PC slab is confined in the self-collimation directions as
shown in Fig. 2 (b).
2.2 Structure Design and Analysis
Based on the theory of self-collimation in the PCs, a 2.6

THz (λ=11.534

μm) wave collimator with a silicon PC slab
is designed as shown in Fig. 3.
Most materials have a strong absorption in the THz
band and are not appropriate for THz devices. However,
there are sill some materials having lower absorbance, such
as high-density polyethylene (HDPE), teflon, and silicon. In
this paper, silicon is chosen as the material for its lower
absorbance and the mature processing technology. The
refractive index of silicon at 2.6

THz is n =3.42.
The structure consists of square array circular air holes
in a silicon slab and the square side is 45
°
tilted to the slab
side. The ΓΜ direction in the reciprocal lattice is along the
side direction of the slab as shown in Fig. 3. The
parameters of the slab are optimized by using the plane
wave expansion (PWE) method for self-collimation
properties. The side length of the square lattice is a=21.00

μm. The diameter of the holes is d=0.70a=14.70

μm (radius
r=0.35a=7.35

μm). The slab has a very compact size of
30.70a×22.63a×0.60a (644.70

μm×475.23

μm×12.6

μm).
By computing the eigenfrequencies for wavevctors at
all k-points in the irreducible Brillouin zone (IBZ) with the
PWE method, we can obtain the dispersion surfaces, the
EFCs, and the band diagrams of odd mode for the PC slab
as shown in Fig. 4.





(a) (b)
Fig. 2. FDTD simulation of the electromagnetic wave propagation
of a point source in different slabs: (a) a homogeneous silicon slab
and (b) a silicon PC slab.





Fig. 3. Structure of the PC slab, with parameters n=3.42, a=21.00
μm, r=0.35a=7.35

μm, and h=0.60a=12.60

μm.














(a)
Normalized frequency
(
ω
a/2
π
c)
0.8
0.6
0.4
0.2
0
0.5
0.5















(b)















(c)
Fig. 4. Dispersion surface and band diagrams: (a) the first
dispersion surface and the light cone, (b) the EFCs, and (c) the
band diagrams of odd mode for the PC slab.
Fig. 4 (a) shows the dispersion surface of the first band
and the light cone for the slab. It is observed that there is a
planar part of the dispersion surface out of the light cone
and this part is related to the self-collimation frequency.
Fig. 4 (b) describes the EFCs of the first band of the
slab. We can see that EFCs of the normalized frequency
from 0.170 to 0.190 (2.429

THz to 2.714

THz) are
perpendicular to the ΓΜ direction. This frequency range
corresponds to the planar part in the dispersion surface in
Fig. 4 (a). The electromagnetic waves in this range will be
converged in a line in the ΓΜ direction. The frequency out
of this range will diverge in the PC slab.
Fig. 4 (c) illustrates the band diagrams of the slab. It
0
0
0.5
−0.5

k
x
a/2
π

k
y
a/2
π
0.8
0.4
0.3
0.2
0
0
0.1

0.
2

0.
3

0.
4

kya/2
π


0.
1


0.
3

0.
1
0.1
0.3
0.5

0.
5
k
x
a/2
π


0.
5
X
M
0
0.1
0.7

0.5
0.6
0.3
0.2
0.4
Normalized frequency (
ω
a/2
π
c)
Γ
Γ
M

X
WU et al.: Compact Terahertz Wave Collimator Design with Photonic Crystal Slabs

229
can be seen that in the ΓΜ direction, the normalized
frequencies from 0.22 to 0.24 are in a band gap and can not
propagate in this direction. The dash-dotted line is the light
line, which is relative to the light cone in the dispersion
surface. The frequencies above the light line can not be
coupled into the PC slab
[ ]
17
.
3. Numerical Simulation
To verify the structure designed above, the THz wave
propagation in the PC slab is simulated and the
transmission property is calculated by using the FDTD
method with the perfectly matched layer (PML) absorbing
boundary conditions.
To simulate the propagation, an odd polarized Gaussian
beam source is placed before the slab with the distance
about 4a. The width and height of the source are 5a and
0.6a. The propagation in a homogeneous silicon slab is also
simulated as a comparison.
Fig. 5 shows the FDTD simulation results of the
propagation of a Gaussian beam in a homogeneous silicon
slab and the designed PC slab, respectively. It can be seen
clearly that in the silicon slab the beam diverges rapidly;
while in the designed PC slab, the beam converges in a
straight line. The collimated beam is confined within a
about constant 3-to-4-lattice width tightly and shows a high
collimating property. The wave propagation in a
self-collimating PCs looks like the propagation in a PC
waveguide. But different from waveguide, it needs no
defect and special incident point, which makes it easier to
fabricate.
To calculate the transmission spectrum, we accumulate
the field Fourier transforms at the incident plane as an
incident flux and that at the output plane as a transmitted
flux with and without the slab structure, respectively. The
normalized transmission is the transmitted flux divided by
the incident flux. Fig. 6 shows the normalized transmission
for the designed PC slab. The normalized frequencies from
0.17 to 0.19 in the self-collimation frequency range show a
high transmittance of 99.45%. It demonstrates that the
designed PC slab has a good transmittance in this band. On
the other hand, the frequencies from 0.21 to 0.24 have an
ultra-low transmittance of nearly 0%. These frequencies are
in a band gap in the ΓΜ direction as shown in Fig. 4 (c).
The transmission spectrum coincides with the EFCs and
band diagrams analyzed in Section 2 very well.
To illustrate the application of the designed THz wave
collimator, a THz wave bend and a THz wave splitter
experiences are proposed as two instances by introducing
an air gap line defect into the PC slab. The defect widths
for the bend and the splitter are w=2.0a and w=0.2a,
respectively. It can be seen clearly that the collimated wave
beam is bent and split perfectly as shown in Fig. 7 (a) and
Fig. 7 (b), respectively.








(a) (b)
Fig. 5. FDTD simulations of the propagation of an odd polarized
Gaussian beam in different slabs: (a) a homogeneous silicon slab
and (b) the designed PC slab.

0.1

0.5
0.6
0.3
0.2
0.4
Normalized frequency (
ω
a
/
2
π
c)
Normalized transmission
1.0
0.8
0.9
0.7
0.10
0
0.30
0.14
0.18
0.22
0.26
















Fig. 6. Normalized transmission of the designed PC slab
calculated by FDTD method.

(a) (b)
Fig. 7. FDTD simulations of the propagation of an odd polarized
Gaussian beam in the PC slab with an air gap line defect. The
defect widths are (a) w=2.0a and (b) w=0.2a, respectively.

4. Conclusions
By using the PWE method, a 2.6

THz wave collimator
with a silicon PC slab has been designed. The designed PC
slab has a compact size of 644.70

μm×475.23

μm×12.6

μm,
which is much smaller than the traditional THz devices. By
using the FDTD method, the transmittance of the PC slab is
calculated and the propagation is simulated. The simulation
results show that the designed PC slab has a good
collimation property and a high transmittance. Moreover, a
THz wave bend and a THz wave splitter experience are
proposed to illustrate the application of the collimator. This
work will be useful for the design of compact integrated
THz circuits and devices, such as THz wave power splitters,
polarization splitters, and switches.
JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY, VOL. 9, NO. 3, SEPTEMBER 2011

230
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Zhen-Hai Wu was born in Sichuan
Province, China, in 1977. He received the
B.S. degree from Sichuan Normal
University (SCNU), Chengdu, in 2000. He
is currently pursuing the Ph.D. degree with
the School of Optoelectronic Information,
University of Electronic Science and
Technology of

China (UESTC). His
research interests include photonic crystals, nonlinear fiber optics,
and free space optical communication.

Kang Xie was born in Sichuan Province,
China, in 1965. He received the B.S and
M.S. degrees from Chongqing University,
Chongqing, in 1986 and Xi’an Jiaotong
University, Xi’an, in 1988, respectively.
He received the Ph.D. degree from Salford
University, Manchester, UK. Now, he is a
professor

with

the

School of
Optoelectronic Information, UESTC. His recent research interests
include nonlinear optics, optical communication, microwave
engineering, etc.

Ping Jiang was born in Sichuan Province,
China, in 1980. She received the B.S.
degree from SCNU, in 2003 and the M.S.
degree from UESTC, in 2007. She is
currently pursuing the Ph.D. degree with
the School of Optoelectronic Information,
UESTC. Her recent research interests
include electromagnetic bandgap structures
and microstrip antenna designs.

Jian-Chen Wan was born in Henan
Province, China, in 1973. He received the
M.S. degree from the Information
Engineering University of the People’s
Liberation Army, Zhengzhou, in 2009.
Now, he is a teacher with the School of
Information Engineering, Huanghe Science
and Technology College of China. His
recent research interests include electrical
engineering, optical communication, etc.