PSFC/JA-01-14
Global Photonic Band Gaps in
Two-Dimensional Metallic Lattices
E. I. Smirnova and C. Chen
July 2001
Plasma Science and Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
This work was supported by the Air Force Office of Scientific Research, Grants No. F4962099-0197 and No. F49620-00-1-0007, and by the Department of Energy, Office of High
Energy and Nuclear Physics, Grant No. DE-FG02-95ER-40919. Reproduction, translation,
publication, use and disposal, in whole or part, by or for the United States government is
permitted.
Submitted for publication in Physical Review Letters.
Global Photonic Band Gaps in Two-Dimensional Metallic Lattices
E. I. Smirnova and C. Chen
Plasma Science and Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
ABSTRACT
The fundamental and higher frequency global photonic band gaps are obtained for
transverse electric (TE) and transverse magnetic (TM) modes in photonic band gap (PBG)
structures consisting of two-dimensional (2D) square and triangular lattices of perfectly
conducting cylinders. The global photonic band gaps in metallic lattice, which are found to
differ qualitatively as well as quantitatively from those in dielectric lattices, not only explain
the observations of single-mode confinement and high mode selectivity in recent microwave
PBG experiments, and but also provide a useful guide for the design of metallic PBG-based
devices in the future.
PACS: 42.70.Qs, 41.20.Jb
2
Metallic photonic band gap (PBG) structures have important applications in high-power
microwave devices such as microwave/millimeter wave sources and rf accelerators. In
addition to their abilities to support high field strength, their band gaps are strikingly
different, in many aspects, from those in dielectric PBG structures. In this Letter, we present
new results on the fundamental and high-frequency global photonic band gaps in 2D square
and triangular lattices of perfectly conducting cylinders. These global photonic band gaps,
which are band gaps for wave propagation with zero longitudinal wave vector but an
arbitrary transverse wave vector, are especially useful properties in the design of metallic
PBG devices such as metallic PBG cavities with high mode selectivity.
Since the pioneering work of Yablonovitch in the late 1980s [1,2], photonic band gap
(PBG) structures have emerged as a growth area for research and development [3-13]. The
existence of local and global photonic band gaps in periodic dielectric and/or metallic
structures has enabled physicists and engineers to find novel areas of applications. Previously
proposed or demonstrated applications of dielectric PBG structures range from the
controlling of spontaneous emission in optical devices [1], to the applications of photonic
crystals in semiconductor lasers and photovoltaic cells [8], to the omniguide formed with
alternating dielectric layers [9-11]. While initial studies of PBG structures were primarily
focused on dielectric PBG structures [1-8], metallic PBG structures [14-16], as well as
dielectric-metallic hybrids, have received considerable attention recently, because of their
applications in rf accelerators [17,18] and high-power microwave vacuum electron devices
[19] and in the transmission and filtering of microwaves [12]. In particular, single-mode PBG
rf accelerating cells [17,18] have been demonstrated by creating a single defect in PBG
lattices, and a PBG resonator gyrotron [19] has been demonstrated experimentally with high
mode selectivity.
There are two important aspects that need to be studied in order to facilitate the design of
metallic PBG devices. One involves the wave propagation in the bulk of the metallic PBG
structure, and the other concerns the wave interaction with the interface between the metallic
PBG structure and vacuum, e.g., mode confinement in a metallic PBG cavity. For analyses of
metallic PBG cavities formed by single or multiple defects in the PBG structure, finiteelement codes such as SUPERFISH [20] and HFSS [21] are ideally suited. For studies of
3
wave propagation in the bulk of metallic PBG structures, on the other hand, the generalized
Rayleigh expansion method [14], plane-wave expansion method [15], and finite-difference
time-domain scheme [16] have been used.
One of the most important and computationally challenging problems is the calculation
of the global photonic band gaps in metallic lattices. While a number of papers have dealt
with the global photonic band gaps in dielectric lattices [1,6,7], results for a metallic lattice
have only been obtained for TM modes in the lowest gap of a square lattice [14]. The present
results represent a major extension to triangular as well as square lattices and to a very wide
range of frequencies.
We consider the square and triangular lattices shown in Fig. 1 with a being the radius of
the conducting cylinder, and b , the lattice spacing. The conductivity profile in the lattice
satisfies the periodic condition
σ (x ⊥ + Tmn ) = σ (x ⊥ ) ,
(3)
where x ⊥ = xe$ x + ye$ y is the transverse displacement, m and n are integers, and Tmn is the
periodicity vector defined as
Tmn
m beˆ x + n beˆ y


= 
3
n
ˆ
+
+
m
b
e
n beˆ y


x

2
2

(square lattice),
(triangular lattice).
(4)
In the present analysis, we set σ = ∞ inside the perfectly conducting cylinders.
It is readily shown from Maxwell’s equations that the wave field in the two-dimensional
PBG structures can be decomposed into two independent classes of modes: the transverse
electric (TE) mode and the transverse magnetic (TM) mode. All the field components in the
TM (TE) modes can be expressed through the axial component of the electric (magnetic)
field, which we will further denote by ψ . Since the system is homogeneous along the z-axis,
we take the Fourier transform of ψ in axial coordinate z and time t and consider
ψ (x ⊥ , k z , ω ) = ∫∫ψ (x ⊥ , z , t )e i (k z −ωt ) dzdt ,
z
which we denote hereafter simply by ψ (x ⊥ ) assuming that the frequency ω and the
longitudinal wave number k z are fixed. The Helmholtz equation for ψ follows from the
Maxwell equations
4

ω2 
∇ 2⊥ψ (x ⊥ ) =  k z2 − 2 ψ (x ⊥ ) .
c 

(5)
The boundary conditions on the surfaces S of the conducting poles are
ψ S =0
(TM mode),
(6)
∂ψ
∂n
(TE mode),
(7)
=0
S
where n is the normal vector to the pole surface.
The discrete translational symmetry of the conductivity profile allows us to write the
fundamental solution of the Helmholtz equation in Bloch form
ψ (x ⊥ + T ) = ψ (x ⊥ )e ik
⊥ ⋅T
,
(8)
where T is any vector of Tmn , k ⊥ = k x eˆ x + k y eˆ y an arbitrary transverse wave number. Thus
we need to solve (5) only inside the fundamental unit cell. Equation (5) together with the
boundary condition (6) or (7) defines the eigenvalue problem of finding λ2 = ω 2 / c 2 − k z2 as
a function of k ⊥ , where c is the speed of light.
The eigenvalue problems defined in Eqs. (5), (6) and (8) and Eqs. (5), (7) and (8) are
solved numerically using a newly developed Photonic Band Gap Structure Simulator
(PBGSS) code, which employs the standard coordinate-space finite-difference method. In the
code, the fundamental unit cell of the square (triangular) lattice is covered with a
(2 N + 1) × (2 N + 1)
square (triangular) mesh. The Helmholtz equation (5) is approximated by
a set of linear relations among the values ψ ij of the function ψ (x ⊥ ) at grid points (i , j ) on
the mesh. The coordinate-space finite-difference method is chosen because of its high
accuracy and high reliability when fine mesh is used in the simulations. The code has been
validated in the limit a << λ using some quasi-static estimates. A detailed description of the
PBGSS code is available elsewhere [22].
To determine the global band gaps, we perform extensive numerical analyses using the
PBGSS code. For each value of a / b , it is sufficient to search through all k ⊥ on the
boundary of the Brillouin zone and find the minimum and maximum of each dispersion
curve. Then we check if a gap exists between any two adjacent propagating modes, i.e., if the
5
minimum of the higher order mode is above the maximum of the lower-order propagating
mode. It is important to perform simulations with small grid steps to assure the accuracy of
the simulation results. For the results presented below, we use a grid with 2 N + 1 = 41 to
cover one unit cell, because the results are nearly identical as we further increase the number
of grid points. Results of the global band gap calculations are summarized in Figs. 2 and 3
for the TM and TE modes, respectively.
Shown in Fig. 2(a) are five lowest-order global TM band gaps for the square lattice. The
zeroth-order global TM band gap exists below the first propagating mode, that is, there is a
cutoff frequency for the TM mode. The cutoff frequency exists even for very small
conducting cylinders and approaches to zero as 1 / ln(b / a ) for a / b → 0 , which is
highlighted with the dashed curve in Fig. 2. The first-order global TM band gap occurs
between the first and second-lowest propagating modes. There is a threshold for first-order
global TM band gap opening at a / b ≅ 0.1 . For the first-order global TM band gap in the
square lattice, our results agree with the previously reported results [14]. Higher-order global
TM band gaps occur between the third and fourth, fourth and fifth, and fifth and sixth
propagating modes. There is no global band gap between the second and third propagating
modes in the square lattice. In Fig. 2(b), three lowest-order global TM band gaps are plotted
for the triangular lattice. The zeroth-order global TM band gap exists below the first
propagating mode, which is similar to the case of the square lattice. The threshold for
occurrence of the first-order global TM band gap, which is between the second and third
propagating modes, is at a / b ≅ 0.2 . The second-order global TM band gap occurs between
the sixth and seventh propagating modes. We find that the width of each global TM band gap
increases as the ratio a / b increases.
We have also calculated the global TE band gaps in both types of lattices. The results are
shown in Fig. 3 up to the seventh propagating mode. For the square lattice [Fig. 3(a)], we
find that the first global TE band gap occurs when a / b > 0.3 . This is the band gap between
the first and second propagating modes, which are tangent at k ⊥ =
π
b
(eˆ
x
+ eˆ y ) for lower
ratios of a / b . Unlike the first global TM band gap, the lower boundary of this band gap
decreases in frequency with increasing a / b . The higher order band gap opens and then
6
closes for even lower ratio of a / b , and this gap is between the sixth and seventh propagating
modes.
For the triangular lattice, we find three global TE band gaps as shown in Fig. 3(b). All of
these gaps tend to close with increasing a / b except for the lowest one, which occurs
between the second and third propagating modes for a / b > 0.35 . The second global TE band
gap, between the third and fourth propagating modes, appears for lower ratios of a / b than
those for the lowest global TE band gap. The third global TE band gap occurs between the
sixth and seventh propagating modes.
While the global TE band gaps in the metallic lattice resemble qualitatively the
previously reported global TE and TM band gaps in dielectric lattices [7], which typically
close with increasing a / b , there are two striking differences between the metallic global
TM band gaps obtained here and the dielectric global band gaps calculated previously [7].
First, there is a zeroth-order global TM band gap in metallic lattices, which is a cutoff
analogous to that in a conventional waveguide and exists for all values of a / b , whereas
there is no such cutoff in dielectric lattices for either TE or TM mode. Second, the width of
the global TM band gap in the metallic lattice increases with increasing a / b , whereas the
global TE and TM band gaps in dielectric lattices typically close as the ratio a / b increases.
The results of the global photonic band gap calculations are in good agreement with two
microwave PBG experiments performed recently at MIT [18,19]. The PBG cavities in both
experiments were designed using SUPERFISH [20] and/or HFSS [21] in a trial-and-error
manner. The first experiment [18] was a 17 GHz PBG accelerating cell formed by a single
defect in a triangular copper lattice with conductor radius a = 0.079 cm and lattice spacing
b = 0.64 cm. The operating point with a / b = 1.23 and ω b / c = 2.28 , marked by the solid
dot in Fig. 2(b), falls into the zeroth-order global TM band gap, and there is no global band
gap at higher frequencies. The observation of a single TM010 mode confinement in the PBG
accelerating cell [18] is consistent with the results shown in Fig. 2(b). The second experiment
[19] was a 140 GHz gyrotron with a PBG cavity. The cavity was formed by removing several
rods in a triangular copper lattice with rod radius a = 0.795 mm and lattice spacing b = 2.03
mm. The operating point with a / b = 0.39 and ω b / c = 5.95 , marked by the solid dot in Fig.
3(b), is in the middle of the lowest-order global TE band gap. A mode resembling the TE041
7
mode in a conventional cavity was excited in the experiment [19]. The observations of such
mode confinement and frequency selectivity are consistent with our analysis.
Evidently, the maps of global photonic band gaps in Figs. 2 and 3 will be very useful in
the design of future metallic PBG experiments. As a first example, the results in Fig. 2(a)
suggest that increasing the conductor size up to a / b = 0.19 in a PBG accelerating cell with a
triangular lattice will not only preserve the confinement of a single mode, but also make
lattice cooling easier, which is important in high-power rf accelerator operation. As a second,
perhaps more interesting example, the results in Fig. 3(a) predict that operating a PBG
resonator gyrotron with a square lattice of a / b = 0.29 and a normalized operating frequency
around ωb / c = 9.3 , which is in the second global TE band gap, will suppress mode
competition from lower frequencies, because there is no global TE band gap below the
operating frequency. Similarly, operating a PBG resonator gyrotron in the second and third
global TE band gaps with the triangular lattice may prove to be effective in suppression of
mode competition.
To summarize, the fundamental and higher frequency global photonic band gaps were
determined for both TE and TM modes in square and triangular lattices of perfectly
conducting cylinders. Striking differences were found between metallic and dielectric global
photonic band gaps, especially for TM modes. Good agreement was found between the
global photonic band gap calculations and two microwave PBG experiments performed
recently. It is anticipated that the maps of global TE and TM band gaps will provide a useful
guide for the design of metallic PBG-based devices for a wide variety of applications in the
future.
8
ACKNOLEDGMENTS
This work was supported AFOSR and DOE. The authors wish to thank V. Khemani for
his assistance in the development of PBGSS code, and W. Brown, M. Hess, M. A. Shapiro, J.
R. Sirigiri and R. J. Temkin for helpful discussions.
9
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10
FIGURE CAPTIONS
Fig. 1 Scheme of PBG structures representing (a) square lattice and (b) triangular lattice of
perfectly conducting cylinders with radius a and spacing b .
Fig. 2 Plots of global frequency band gaps for TM mode as a function of a / b obtained
from PBGSS calculations for (a) square lattice and (b) triangular lattice. Here, the
solid dot represents the operating point of the MIT 17 GHz PBG accelerating cell
experiment.
Fig. 3 Plots of global frequency band gaps for TE mode as a function of a / b obtained
from PBGSS calculations for (a) square lattice and (b) triangular lattice. Here, the
solid dot represents the operating point of the MIT 140 GHz PBG resonator gyrotron
experiment.
11
a)
y
a
x
b
b
b)
y
a
b
x
b
Fig. 1
12
b
20.0
ωb/c
15.0
10.0
5.0
(a)
0.0
0.0
0.1
0.2
0.3
0.4
0.5
a/b
25.0
ωb/c
20.0
15.0
10.0
5.0
(b)
•
0.0
0.0
0.1
0.2
0.3
a/b
Fig. 2
13
0.4
0.5
10.0
ωb/c
8.0
6.0
4.0
2.0
0.0
0.0
(a)
0.1
0.2
a/b
0.3
0.4
0.5
15.0
ωb/c
10.0
•
5.0
(b)
0.0
0.0
0.1
0.2
0.3
a/b
Fig. 3
14
0.4
0.5