Optics and Photonics Journal, 2011, 1, 97-100
doi:10.4236/opj.2011.13016 Published Online September 2011 (http://www.SciRP.org/journal/opj)
Copyright © 2011 SciRes. OPJ
Optimization of Duty Ratio of Metallic Grating Arrays for
Dong Liu1, Yongqi Fu1,*, Lechen Yang1, Baoshun Zhang2, Haijun Li2, Kai Fu2, Min Xiong2
1School of Physical Electronics, University of Electronic Science and Technology of China,
2Suzhou Institute of Nano -Technology and Nano-Bionics, CAS, Suzhou , China
Received May 27, 2011; revised June 25, 2011; accepted July 6, 2011
Influence of duty ratio of metallic gratings applied in quantum well infrared photodetector (QWIP) with de-
tection ranging from 3 m to 5 m was studied in this paper. The influence on longer enhanced wavelength
working at infrared waveband was investigated. A relationship between the duty ratio and the enhanced peak
intensity is given. Some results can be applied to optimize the enhanced efficiency of the metallic gratings.
Keywords: Metallic Gratings, Duty Ratio, Infrared Waveband, QWIP
Recently, many research papers report the enhanced and
localized surface plasmon due to the unique feature of
extraordinary transmission [1-5]. As well known, to ex-
cite surface plasmon, approaches of both prism and grat-
ing coupling are applicable. The former needs an appro-
priate angle to form perfect total reflection which de-
pends on the thickness of the metal film [6,7]. However,
period and duty ratio of a metallic grating directly influ-
ence the position of the enhanced resonant peak and cor-
responding values. As a coupling layer of quantum well
infrared photodetector (QWIP), it is necessary to con-
sider the feasibility while manufacture QWIP distributed
as arrays in large-area . Therefore, the grating cou-
pling will be more appropriate for the infrared imaging
devices because of its smaller volume and good control-
lability in comparison to the approach of prim coupling.
To utilize the enhancement of surface plasmon (SP), we
investigated the characteristics and fundamental issues of
the metallic gratings in the view of optimization of the
duty ratio. Investigation of the optimized duty ratio was
carried out for the purpose of improving enhanced reso-
nant peak in the infrared waveband.
2. Theory Background and Computational
For calculation of the metallic grating, there are some
important parameter settings to be considered firstly. The
type of the gratings, the period of the arrays, the ar-
rangement mode and size of the holes are all the crucial
factors [9-12]. Some papers reported the calculation of
the array period by means of the formulas of dispersion
relations as follows [1,7,13,14]:
where, a0 is the period of the grating,
2 is the di-
electric constants of the medium and metal respectively,
and i and j are the integral numbers. Equation (1) is for
the array with square cross section. Equation (2) is the
array with regular triangle cross section which is calcu-
lated by the similar method of phase-matching. The mo-
mentum of SP matches the momentum of regular triangle
crystal lattice the incident photon. However, the radius of
the hole, the thickness of metal film and the hole array
will be the three important factors to be determined. Here,
we choose the metallic gratings consist of Au film with
characteristics of stability and the hollow holes embed-
ded in the sapphire substrate. Though the surface plas-
mon is sensitive to thickness, the point of this paper is
about the duty ratio of the metal gratings. Hence, the Au
film was set as 50 nm, which is an optimized thickness
D. LIU ET AL.
as . Shape of the periodic arrange of air holes is
regular triangle. We chose a regular triangle lattice in-
stead of square lattice because the former has the wide
bandwidth of enhancement as well as large maximal
transmission domain due to its symmetrical geometry,
and thus the Equation (2) will be practical in the nu-
merical calculation . The simulation model in the
software FDTD solutions was established, as shown in
The computational numerical calculation was carried
out for the purpose of deriving optimized duty ratio for
the gratings in infrared waveband and focus on the mid-
infrared as well as far infrared where the photodetector
works. Therefore, we chose the points in the range from
5 m to 9 m and calculated the period for each corre-
sponding enhanced peak firstly. Then we can obtain the
enhanced peak positions and corresponding array periods,
as shown in Table 1 . For each corresponding periods, we
chose a series of radius which match along with the duty
ratio as 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18, 0.20,
0.25, 0.30, 0.35, 0.40, 0.45, and 0.50, respectively. Cal-
culation of the transmission spectrum was performed by
means of finite difference and time domain (FDTD) al-
By calculation, we obtained seven groups of data, as
shown in Figure 2. The x-axis denotes the duty ratio of
the metallic grating, and the y-axis represents the en-
hanced peak value for each duty ratio. With increasing of
the period of the array, the enhanced peak value de-
creases and the optimizing duty ratio increases accord-
ingly. As can be seen in Figure 2, we picked up the data
with the relationship between array period and the en-
hanced peak intensity, and re-plotted, as shown in Fig-
ure 3. It is a quadratic power curve with nearly linear
Figure 1. (a) Vertical view of the model, p is the period of
the array, r is the radius of the holes; (b) the perspective
view of the model and the material from the bottom to the
top is sapphire and Au film respectively.
Table 1. Enhanced peaks and corresponding array periods.
Enhanced Peak (μm) 5 6 7 7.5 8 9 10
Period (μm) 3.4 4.2 5.4 6.1 6.7 8.510.7
0.0 0.1 0.2 0.3 0.4 0.5
Figure 2. Duty ratio vs. peak values for different period
arrays. The duty ratio is ranging from 0.04 to 0.50 and cal-
culated the peak value.
Figure 3. Period of array vs. its corresponding enhanced
relationship. With increasing of the array period, the in-
tensity decreases. This is a qualitative analysis and sug-
gests a rough changing tendency. The intensity is calcu-
lated assuming incident beam intensity with value of 1.
However, the most important point of our work is
searching the relationship between enhanced wavelength
and the duty ratio which can be a reference in the de-
signing of the metallic gratings. As can be seen from
Table 1 and Figure 2, for each period it corresponds to
an enhanced wavelength and an optimizing duty ratio
which indicates the duty ratio when the y-axis value ap-
proaches to be maximum for each curve, as shown in
Figure 2. Hence, it is easy to obtain the relationship of
enhanced wavelength and the optimizing duty ratio.
Figure 4 shows the relationship. And the fitting formula
is shown as follows:
x x (3)
Generally speaking, the enhancement and locality of
the two-dimensional (2D) metallic gratings are always
crucial issues because they play an important role in the
design. Therefore, it requires ensuring its enhanced peak
Copyright © 2011 SciRes. OPJ
D. LIU ET AL.99
0.18 Simulated data
Figure 4. Every enhanced wavelength and its corresponding
optimizing duty ratio.
We can find its optimizing duty ratio from this curve
once the response wavelength is determined. Figure 4
gives us directly the guideline in the design. Corre-
sponding to each response wavelength we can derive the
optimized duty ratio for a strongest enhanced peak value.
3. Results and Discussion
As can be seen, the effect of surface plasmon becomes
less effective while the enhanced peak intensity drops
being weak . However, it can be seen from Figure 3
that certain period value is minimum which will cause
the surface plasmon losing its effectiveness of enhance-
ment obviously when the period increases. The duty ratio
of the most enhanced peak increases in this case. When
the period is large enough, the duty ratio closes to the
case that there is a few holes only on the Au film. The
peak intensity will equal to that of the light source inten-
sity [17-19]. For different sizes of array structures, it
needs different duty ratios so as to obtain the optimized
enhancement which is in the same level of orders of
magnitude of the enhanced wavelength [20,21]. Simi-
larly, the changing rule is a nonlinear curve, when the
curve close to a value which requires duty ratio turns to
be infinite. Then the surface plasmon enhancement loses
its effectiveness [21-24]. This result proves further that
surface plasmon effect dose not fit for jumbo size struc-
ture. To validate whether this changing rules is exact or
not, we collected more results to check it. By choosing
any enhanced wavelength, such as 4.5 m, then calculate
the array period using Equation (2), and the wavelength
is almost 2.918 m. The corresponding optimized duty
ratio for such a responding wavelength will be close to
0.1028. And the radius can be calculated to be 0.4912
m. This is an analysis result from the curve in Figure 4
theoretically, whereas change its radius in the simula-
tions. We calculated the results as shown in Figure 5. It
can be seen that assumes a certain radius which can gen-
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Figure 5. Different radiuses and its corresponding en-
hanced peak value for a metallic grating with the period of
2.9 micron and the enhanced wavele ngth is 4. 5 m.
erate optimized enhancement through surface plasmon. It
is not difficult to derive the optimized radius of 0.5046
m after fitting the data. However, it is apparent that the
two values are close. The difference of several nanome-
ters can be ignored for an array with dimension of sev-
eral microns. Thus, the fitting accuracy is high enough
for the curve of Figure 4. Hence, Equation (3) can be an
empirical formula and provides a rough guide during the
designing of the metallic grating for infrared band at
To investigate basic rules regarding metallic gratings
being used for QWIP, we carried out optimization of
duty ratio of the metallic gratings. For the periodic array
with a certain period, we performed an investigation re-
garding changing trend of enhanced resonant peak and
the duty ratio for different periods in infrared waveband,
and analyzed the relationship between duty ratio and
enhanced peak intensity. In addition, the curves of en-
hanced wavelength and duty ratio were also studied
which can be used as a basic rule of the metallic grating.
To prove validity of the trend and rule, we simulated a
group of models with different radius which mean dif-
ferent duty ratios also. After comparison, the results
demonstrated that the trend and rule are valid, and can be
treated as a guide for further study of metallic grating in
The research work was supported by National Natural
Science Foundation of China (No. 60877021 and
 T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio and P.
Copyright © 2011 SciRes. OPJ
D. LIU ET AL.
Copyright © 2011 SciRes. OPJ
A. Wolf, “Extraordinary Optical Transmission through
Sub-Wavelength Hole Arrays,” Nature, Vol. 391, No.
6668, 1998, pp. 667-669.
 W. L. Barnes, A. Dereux and T. W. Ebbesen, “Surface
Plasmon Subwavelength Optics,” Nature, Vol. 424, No.
14, 2003, pp. 824-829.
 U. Schroter and D. Heitmann, “Surface-Plasmon-Enhan-
Ced Transmission through Metalliclic Gratings,” Physi-
cal Review B, Vol. 58, No. 23, 1998, pp. 419-421.
 E. Popov, M. Nevière, S. Enoch and R. Reinisch, “The-
ory of Light Transmission through Subwavelength Peri-
odic Hole Arrays,” Physical Review B, Vol. 62, No. 23,
2000, pp. 16100-16108. doi:10.1103/PhysRevB.62.16100
 A. Krishnan, T. Thio, T. J. Kim, H. J. Lezec, T. W. Eb-
besen, P. A. Wolff, J. Pendry, L. Martin-Moreno and
Garcia-Vidal, “Evanescently Coupled Resonance in Sur-
face Plasmon Enhanced Transmission,” Optics Commu-
nications, Vol. 200, No. 1-6, 2001, pp. 1-7.
 S. A. Maier, “Plasmonics: Fundamentals and Aplica-
tions,” Springer, Science+Business Media LLC, 2007.
 H. Raether, “Surface Plasmons on Smooth and Rough
Surfaces and on Gratings,” Springer Tracts in Modem
Physics, Springer-Verlag, Berlin, Heidelberg, Vol. 2, p. l,
 H. Schneider and H. C. Liu, “Quantum Well Infrared
Photodetectors,”Springer-Verlag, Berlin Heidelberg, 2007.
 Z.-B. Li, Y.-H. Yang, X.-T. Kong, W.-Y. Zhou and J.-G.
Tian, “Enhanced Transmission through a Subwavelength
Slit Surrounded by Periodic Dielectric Bars above the
Metallic Surface,” Applied Optics, Vol. 10, 2008, Article
 H. P. Paudel, K. Bayat, M. F. Baroughi, S. May and D. W.
Galipeau, “Geometry Dependence of Field Enhancement
in 2D Metalliclic Photonic Crystals,” Optics Express, Vol.
17, No. 24, 2009, pp. 22179-22189.
 J. W. Cleary, G. Medhi, R. E. Peale and W. R. Buchwald,
“Long-Wave Infrared Surface Plasmon Grating Coupler,”
Applied Optics, Vol. 49, No. 16, 2010, pp. 3102-3109.
 W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux and
T.W. Ebbesen, “Surface Plasmon Polaritons and Their
Role in the Enhanced Transmission of Light through Pe-
riodic Arrays of Subwavelength Holes in a Metallic
Film,” Physical Review Letters, Vol. 10, No. 92, 2004,
Article ID: 107401.
 A. Benabbas, V. Halté and J.-Y. Bigot, “Analytical
Model of the Optical Response of Periodically Structured
Metalliclic Films,” Optics Express, Vol. 22, 2005, pp.
 L. Martin-Moreno and F. J. Garcia-Vidal, “Optical
Transmission Through Circular Hole Arrays in Optically
Thick Metallic Films,” Optics Express, Vol. 12, No. 16,
2004, pp. 3619-3628. doi:10.1364/OPEX.12.003619
 Y. Xie, A. R. Zakharian, J. V. Moloney and M. Man-
suripur, “Transmission of Light through Periodic Arrays
of Sub-Wavelength Slits in Metalliclic Hosts,” Optics
Express, Vol. 14, No. 14, 2006, pp. 6400-6413.
 Lumerical Solution Inc., “FDTD Solution, A Commercial
Professional Software,” Lumerical Solution Inc., Van-
couver, Canada. http://www.lumerical.com
 L. Salomon, F. Grillot, A. V. Zayats and F. de Fornel,
“Near-Field Distribution of Optical Transmission of Pe-
riodic Subwavelength Holes in a Metallic Film,” Physical
Review Letters, Vol. 86, 2001, pp. 1110-1113.
 H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen and
H. J. Lezec, “Surface Plasmons Enhance Optical Trans-
mission through Subwavelength Holes,” Physical Review
B, Vol. 58, No. 11, 1998, pp. 6779-6782.
 J. G. Rivas, C. Schotsch, P. H. Bolivar and H. Kurz,
“Enhanced Transmission of Thz Radiation through Sub-
wavelength Hole,” Physical Review B, Vol. 68, No. 20,
2003, Article ID: 201306.
 A. P. Hibbins, J. R. Sambles and C. R. Lawrence, “Grat-
ingless Enhanced Microwave Transmission through a
Subwavelength Aperture in a Thick Metallic Plate,” Ap-
plied Physics Letters, Vol. 84, No. 24, 2002, pp. 4661-
 M. Sarrazin, J.-P. Vigneron and J.-M. Vigoureux, “Role
of Wood Anomalies in Optical Properties of Thin Metal-
liclic Films with a Bidimensional Array of Subwave-
length Holes,” Physical Review B, Vol. 67, 2003, Article
ID: 085415. doi:10.1103/PhysRevB.67.085415
 K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van
Hulst and L. Kuipers, “Strong Influence of Hole Shape on
Extraordinary Transmission through Periodic Arrays of
Subwavelength Holes,” Physical Review Letters, Vol. 92,
2004, Article ID: 183901.
 A. Degiron and T. W. Ebbesen, “The Role of Localized
Surface Plasmon Modes in the Enhanced Transmission of
Periodic Subwavelength Apertures,” Journal of Optics A:
Pure and Applied Optics, Vol. 7, No. 2, 2005, pp. S90-
 L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M.
Pellerin, T. Thio, J. B. Pendry and T. W. Ebbesen, “The-
ory of Extraordinary Optical Transmission through Sub-
wavelength Hole Arrays,” Physical Review Letters, Vol.
86, No. 6, 2001, pp. 1114-1117.