"Design and Fabrication of Photonic Crystal Nano

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 1, JANUARY 1, 2014
91
Design and Fabrication of Photonic Crystal
Nano-Beam Resonator: Transmission Line Model
M. Miri, M. Sodagar, K. Mehrany, A. A. Eftekhar, A. Adibi, and B. Rashidian
Abstract—We present a new method for modeling and design of
photonic crystal nano-beam resonators (PCNBRs) based on cascaded transmission lines. The proposed model provides an accurate estimate of the PCNBRs properties such as resonance wavelength and quality factor (Q) with much smaller computation cost
as compared to the brute-force numerical methods. Furthermore,
we have developed a straightforward technique for the design of
high-Q PCNBRs based on resonance modes with Gaussian electromagnetic field profiles. The results obtained by using the proposed
transmission line model are compared against numerical and experimental results and the accuracy of the model is verified. The
proposed model provides an insight to silicon cavity design and
significantly reduces computational burden.
Index Terms—Nano-beam resonators, photonic crystals (PCs),
transmission line (TL) model.
I. INTRODUCTION
HOTONIC crystal (PC) nano-beam resonators
(PCNBR)—thanks to their high quality factor and low
mode volume—are among the promising candidates for
different applications ranging from linear and nonlinear optical
signal processing [1], [2] to cavity quantum electrodynamics [3], [4]. It is no wonder that they have been the subject
of many investigations in the past decade [1]–[10]. In particular, usage of PCNBR for nano-laser applications has been
notable [11]–[13]. Quality factors as high as 7 × 105 for air
bridge nano-beams [6], and mode volumes as low as 0.2(λ/2n)3
for slotted nano-beams [7] are reported in the literature. Since
PCNBR is formed by introducing either local or distributed
defect in an otherwise periodic waveguide structure, a fast and
efficient design strategy is highly desirable to find the proper
defect that can fully exploit the potentials of PCNBRs.
There are only two prominent design strategies hitherto reported. One is based on the Fabry–Perot model [14], and the
other on mode gap modulation [6], [8], [10], and [15]. The
former strategy tries to increase the quality factor by increasing
P
Manuscript received May 17, 2013; revised October 25, 2013; accepted
November 6, 2013. Date of publication November 13, 2013; date of current
version November 29, 2013.
M. Miri, K. Mehrany, and B. Rashidian are with the Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran (e-mail:
m101miri@gmail.com; mehrany@sharif.ir; rashidia@sharif.edu).
M. Sodagar, A. A. Eftekhar, and A. Adibi are with the Department of
Electrical Engineering, Georgia Institute of Technology, Atlanta, GA 30332
USA (e-mail: majidsodagar@gmail.com; eftekhar@ece.gatech.edu; ali.adibi@
ece.gatech.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JLT.2013.2290947
the reflection at the boundary of the resonant region (as if the PCNBR is a simple Fabry–Perot resonator). Unfortunately, the radiation loss, which is inevitably incurred because of the unwanted
sharp transitions of the electromagnetic field at the boundary
of the resonant region, is neglected in this approach. Therefore,
the achievable quality factor when the cavity is simply formed
by introducing a local defect in a 1-D periodic waveguide is
not usually ultrahigh. It can be substantially improved by fine
tuning of the optical properties of the reflecting regions (e.g., by
changing the radii and locations of the holes when the reflecting regions are made of PC waveguide formed by perforating
a silicon waveguide) [16], [17]. The fine tuning of the reflecting regions is however a time consuming numerical procedure.
The latter strategy, on the other hand, tries to blur the unwanted
sharp field transitions by introducing the appropriate distributed
defect that can reduce the radiation loss and increase the quality
factor substantially. It is shown that the appropriate distributed
defect should have a Gaussian-like field profile. Realization of
the desired Gaussian-like field profile is possible via modulation of the band edge of PC waveguide modes in the distributed
defect region and is not as expensive as fine tuning of the optical properties of the reflecting regions in the former strategy.
Nevertheless, the extraction of the appropriate distributed defect profile requires running brute-force numerical simulations,
e.g., 3-D finite difference time domain (FDTD) method, which
happens to be very time-consuming task that complicates the
adoption of this strategy.
The latter strategy is adopted in this paper. Conventionally,
the desired Gaussian-like field profile is realized by linearly
increasing the mirror strength along the reflecting region [15].
This can be achieved by decreasing the radii of the air holes
when the reflecting regions are made of PC waveguide formed
by perforating a silicon waveguide. Extraction of mirror strength
in the reflecting region is however a painstaking step and needs
3-D band diagram simulations. Since pursuing such an approach
takes about tens of hours on an ordinary desktop computer, a
more efficient alternative is in want. Here, the standard transmission line (TL) theory is invoked to extract the appropriate
distributed defect profile that can realize the desired Gaussianlike field profile. TL models have been proposed as a successful
approximate tool for analysis of 2-D periodic structures (2D
PCs) that are uniform along the third dimension [18], [19].
These structures form a 2-D computation problem. To the best
of our knowledge, no such model is reported for analysis of
the similar PC structures that are not uniform along the third
dimension, i.e., when these structures form a 3-D computation
problem. The existing TL models of 2-D PC structures cannot be straightforwardly extended to include the out-of-plane
0733-8724 © 2013 IEEE
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Fig. 1.
JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 1, JANUARY 1, 2014
Schematic view of a typical silicon PCNBR.
inhomogeneity and finite thickness of 3-D structures. This is
carried out in this paper and the PCBNR is modeled by a TL
circuit. The here-proposed TL model can considerably reduce
the runtime of the process.
The structure of this paper is as follows. First, the details of
modeling PCNBRs by TL are given in Section II. The PCNBR
is divided to cascaded PC waveguide sections. Each section is
modeled by a TL segment of the same length. The propagation
constant and characteristic impedance of the TL segment are
obtained by solving a set of equations that relates the numerically extracted reflection coefficient of the waveguide section to
the sought-after parameters of the TL segment. Once the parameters of the TL segment modeling each PC waveguide section
are obtained, the PCNBR is automatically modeled by cascaded
TL segments. The accuracy of the proposed TL model in extraction of the resonance wavelength and its corresponding quality
factor is verified by analysis of a typical Fabry–Perot PCBNR.
The zero frequency of the imaginary part of the network’s total
impedance and its corresponding frequency bandwidth give the
resonance wavelength and its corresponding quality factor, respectively. It is shown that the results obtained by the proposed
model are as accurate as those obtained by brute force numerical method. In Section III, the proposed model is employed to
design resonators of very high quality factor. To this end, the
field profile of the PCNBR mode is first related to the voltage
distribution within the cascaded TL model. The parameters of
the TL segments are then obtained in such a manner that the
voltage distribution becomes Gaussian. In Section IV, the designed structure is fabricated and characterized. The obtained
results demonstrate the accuracy and efficiency of the proposed
TL model for analysis and design of PCBNRs. Finally, the conclusions are made in section V.
II. TRANSMISSION LINE MODEL FOR PCNBR
Without loss of generality, we focus on PCNBR that is formed
by perforating equidistant circular air holes in a silicon rib waveguide. The schematic of such a structure is shown in Fig. 1. The
resonant region is formed by changing the radii of 2n holes
sandwiched between two PC waveguide mirrors, whose periodicity and radii of holes are denoted by a and rm , respectively.
In accordance with Fig. 1, there is a plane of symmetry around
which the radii of holes within the resonant region are shown
by ri , i = 1, 2, . . . , n.
Given that the distance between the center of two adjacent
holes is a, the PCNBR in Fig. 1 can be regarded as a cascaded
sequence PC waveguide sections with length a; each with its
Fig. 2. FDTD simulation structure for the calculation of reflection coefficient.
Waveguide section is illuminated by a rib waveguide mode while perfectly
matched layer (PML) boundary conditions are used at both ends of the waveguides and reflection coefficient is calculated at indicated plane.
specific PC hole radius (see Fig. 1). As mentioned earlier if a
TL model is developed for each PC waveguide section, then
the PCNBR can be modeled by cascaded TL segments. In the
next section, we use cascaded TL model to calculate the PC
waveguide transmission/reflection.
A. Transmission Line Model for PC Waveguide Sections
Consider a typical waveguide section whose length is a, and
has a circular air hole of radius r. Two steps are to be taken to
replace such a waveguide section by a TL segment. The propagation constant of the TL segment γ and its impedance zc are to
be determined. In the first step, the reflection of the waveguide
section when it is created by perforating an air hole of radius
r within the silicon rib waveguide is numerically extracted by
using 3-D FDTD. Fig. 2 shows the details of FDTD reflection
calculation. The waveguide section is illuminated by a rib waveguide mode while perfectly matched layer boundary conditions
are used at both ends of the waveguides and reflection coefficient is calculated at the plane that is indicated in Fig. 2. The rib
waveguide mode in the FDTD simulation is modulated by a temporal Gaussian envelope and therefore reflections is calculated
at a range of frequencies.
For each PC waveguide section two simulations should be
performed to extract two unknown parameters γ and zc . These
two simulation steps are shown in Fig. 3. In following calculations all impedances are normalized in respect to characteristic
impedance of the rib waveguide. In accordance with Fig. 3(a),
if the waveguide section is to be replaced by a TL segment of
length a then the reflection coefficient Γ1 can be written in terms
of the propagation constant γ and the normalized characteristic
impedance zc :
Γ1 =
zin(1) − 1
zin(1) + 1
(1)
where zin(1) is the normalized input impedance of the TL
segment that models the waveguide section
zin(1) = zc
1 + zc tanh γa
.
zc + tanh γa
(2)
In the second step, the reflection coefficient of the waveguide
section when it is cascaded by the same waveguide section is
numerically extracted. In accordance with Fig. 3(b), the cascaded waveguide sections are to be replaced by a TL segment
of length 2a. This time, the reflection coefficient of the structure
MIRI et al.: DESIGN AND FABRICATION OF PHOTONIC CRYSTAL NANO-BEAM RESONATOR: TRANSMISSION LINE MODEL
93
expected when r = 0, the normalized characteristic impedance
of the TL model should be zc = 1. Moreover, since the electrical length of the TL model for r = 0 should not be different
from the electrical length of the silicon rib waveguide, the propagation constant of the TL model γ should be equal to that of
the silicon rib waveguide β. Calculated propagation constant
and characteristic impedance for PC waveguide sections considered in following examples are complex number. This is in
agreement with the fact that since desired frequency range is in
the photonic mode gap of PC waveguides the equivalent TLs
should have complex characteristics impedances and propagation constants. In the next section the proposed TL model will
be used for designing PCNBRs based on algorithm discussed in
Section II. But before considering the design problem, accuracy of the proposed model is tested by modeling a Fabry–Perot
based PCNBR in the following section.
B. Modeling of the PCNBR Using Transmission
Line Approach
Fig. 3. Extraction of the TL model for a typical waveguide section of the
PCNBR shown in Fig. 1. (a) The first step in extraction of the model parameters.
(b) The second step in extraction of the model parameters.
Γ2 can be written as
Γ2 =
zin(2) − 1
zin(2) + 1
(3)
where zin(2) is the normalized input impedance of the TL segment that models the cascaded waveguide sections
zin(2) = zc
1 + zc tanh 2γa
.
zc + tanh 2γa
(4)
Given that the reflection coefficients Γ1 and Γ2 are numerically extracted, the propagation constant γ and the normalized
characteristic impedance zc of the TL model of the waveguide
section can be straightforwardly obtained by solving the following set of equations:
zc
1 + Γ2
1 + tanh2 γa + 2zc tanh γa
=
2
1 − Γ2
zc + zc tanh γa + 2 tanh γa
zc
1 + Γ1
1 + zc tanh γa
=
.
zc + tanh γa
1 − Γ1
(5.a)
(5.b)
This nonlinear equation system can be solved by using numerical methods. Here we have used MATLAB software to
solve these equations. It is worth noting that for r = 0, when
no hole is perforated in the silicon rib waveguide, the abovementioned set of equations fail to render the normalized characteristic impedance and the propagation constant. To resolve
this issue, other facts are to be used. Since no reflection is
To demonstrate how the TL model of PC waveguide sections
can be employed to model PCNBR and to verify the accuracy
of the proposed model, a typical Fabry–Perot PCNBR is considered. Here the TL model for PC waveguide sections will be
used to calculate resonance wavelength and quality factor the
Fabry–Perot PCNBR. These values are then compared against
results of 3-D FDTD simulations. The Fabry–Perot PCNBR is
made by two PC waveguide mirrors that sandwich a resonant
region with no holes. In accordance with the previous section,
the waveguide sections of the Fabry–Perot PCNBR—a silicon
rib waveguide together with PC waveguide mirrors- are substituted by their corresponding TL models. This is shown in
Fig. 4(a). It is assumed that the silicon rib waveguide in which
the Fabry–Perot PCNBR is to be fabricated has thickness of h =
240 nm, width of w = 500 nm, and pedestal layer of thickness
t = 50 nm. The PC waveguide mirrors have five equidistant
air holes of radius r = 80 nm and a = 350 nm. The length of
the resonant region is L = 1.1 μm. To extract the resonance
wavelength and the quality factor of such a typical PCNBR by
using the proposed TL model, the normalized impedance of PC
waveguide zc and the propagation constants of PC waveguide
and the silicon rib waveguide γ and β are to be calculated first.
Then, the resonance condition of the TL model is to be satisfied.
At the resonance wavelength, the imaginary part of the total
impedance of the network should be zero [20]
Im {ztot } = 0; ztot = zL + zR
(6)
where zR and zL are the normalized impedances seen at the
right- and left-hand sides of an arbitrary interface in the circuit, respectively. Here, the chosen interface corresponds to the
boundary between the resonant region and the PC waveguide
mirror
zL =
zload + j tan βL
1 + jzload tan βL
zR = zload = zc
1 + zc tanh 5γa
.
zc + tanh 5γa
(7.a)
(7.b)
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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 1, JANUARY 1, 2014
transition at the boundary between the resonant region and the
PC waveguide mirror is noticeable.
III. PCNBR DESIGN
Fig. 4. (a) Schematic view of a typical Fabry–Perot PCNBR together with its
equivalent TL model. (b) The x component of the electric field profile (E x ) of
the resonance mode extracted from FDTD and absolute value of E x in x = 0
along the resonator.
Therefore by knowing propagation constant and characteristic impedance of the PC waveguide sections we can calculate zR , zL , and ztot from above equations. However since the
proposed TL model is a frequency domain model we need to
perform a frequency sweep and calculate ztot for a range of frequencies to find the resonance frequency at which equation (6)
is satisfied. Our calculations show that the resonance condition
is satisfied when the wavelength of light at free space is λ0 =
1548.60 nm. At this wavelength, the normalized impedance of
PC waveguide is zc = 0.0132 − 0.0100 i, and the propagation
constants of PC waveguide and the silicon rib waveguide are γ
= 3.0971 × 104 +1.5914 × 104 i and β = 1.0164 × 107 , respectively. The corresponding quality factor can then be easily
obtained by finding the full-width at half maximum of the total
impedance ztot [20]. From bandwidth of total impedance at the
abovementioned resonance wavelength the quality factor of the
PCNBR is found to be Q = 134.47.
Using the brute-force 3-D FDTD, the resonance wavelength
and the quality factor of the same PCNBR are λ0 = 1553.12 nm,
and Q = 133.15, respectively. These results are obtained by
using the commercially available Lumerical software. It can be
seen that the error of using the proposed model in calculation
of the resonance wavelength and quality factor is below 0.3%,
and 1%, respectively.
As it mentioned earlier one reason to have a rather low value
for the quality factor in the Fabry–Perot PCNBR is the sharp
transition of the electric field at the boundaries of the resonant
region. This is shown in Fig. 4(b), where a snap shot of the x
component of the resonant electric field profile; Ex , is plotted
together with its absolute value at x = 0 and z = 0. The sharp
Given that the proposed TL model provides the resonance
wavelength and quality factor of the PCNBR in a simple, quick
and efficient manner, standard optimization procedures can be
followed to design the desired PCNBR. Still, finding the optimum PCNBR having the maximum possible quality factor at a
specified resonance wavelength requires a rather complex optimization procedure and is not a straightforward task. Here,
a more heuristic approach based on physical principles is followed instead of pursuing a standard optimization technique in
a blind manner. The most appropriate values for the geometrical
parameters of the silicon rib waveguide and of the PC waveguide mirror are obtained to ensure that the radiation loss of the
structure reaches the minimum possible level. Then, the resonant region is designed in two steps. First, air hole radius of
the innermost PC waveguide section is chosen to support the
desired resonance wavelength. Since the resonance wavelength
of the PCNBR principle mode is close to the photonic band edge
of the innermost PC waveguide section [15], the air hole radius
is found by bringing the spectral position of the photonic band
edge near the desired resonance wavelength. Second, air hole
radii of the outer PC waveguide sections within the resonant region are found by trying to suppress the field profile mismatch
between PC waveguide mirrors and the resonant region. In this
way, the unwanted radiation loss is reduced. Since the least
mismatch is observed when the electromagnetic field energy is
optimally confined around a specific wave vector point in the
momentum space [21], [22], the unknown radii of the air holes
within the resonant region should support a Gaussian electromagnetic field profile [15], [21]. This step can be considerably
expedited, when the proposed TL model is employed.
The width of the silicon rib waveguide in the to-be-designed
PCNBR should be large enough to ensure that the waveguide
mode is well below the light line (to reduce the radiation loss)
while it should be small enough to ensure that the waveguide
remains single mode. Since the PCNBR is to be fabricated on
a SOI wafer whose silicon thickness is h = 240 nm and since
the pedestal layer has a t = 50 nm thickness, the width of the
waveguide can be w = 500 nm to support a single propagating
mode around the communication wavelength 1550 nm lying
well below the light line.
The periodicity of the PC waveguide mirrors a should be small
enough to ensure that the edge of the Brillouin zone π/a is farther
than the edge of the radiation zone 2π/λ0 [21], [22]. However, it
cannot be too small otherwise the fabrication process becomes
complicated. The periodicity of the to-be-designed PCNBR is
here set to a = 340 nm. The air hole radius of the PC waveguide
mirror should be rm = 80 nm to make sure that the desired
communication wavelength 1550 nm is at the center of the
photonic gap. This is shown in Fig. 5, where the band structure of
the PC waveguide mirrors and the desired resonance wavelength
λ0 = 1550 nm are depicted.
MIRI et al.: DESIGN AND FABRICATION OF PHOTONIC CRYSTAL NANO-BEAM RESONATOR: TRANSMISSION LINE MODEL
Fig. 5. Band structure of the photonic crystal waveguide mirrors with rm =
80 nm (black solid lines), and the innermost photonic crystal waveguide section
with r1 = 110 nm (blue dashed lines).These values are chosen to make sure
that the desired communication wavelength 1550 nm will be at the center of the
photonic gap of PC waveguide mirror and at the band edge of innermost PC
waveguide sections.
95
of the central waveguide coincides with the desired resonance
wavelength λ0 = 1550 nm. This is also shown in Fig. 5, where
the band structure of the central waveguide region is depicted
by dashed line.
The outer PC waveguide sections form a transitional waveguide region connecting the central waveguide region to PC
waveguide mirrors. Determination of the air hole radii of the
transitional waveguide region is the most difficult step. This
step is carried out by using the proposed TL model of the structure shown in Fig. 6(b). To support a Gaussian electromagnetic
field profile within the resonant region, the voltage distribution
should follow a Gaussian pattern. The appropriate Gaussian
voltage distribution for the PCNBR of Fig. 6(a) is depicted in
Fig. 6(c). It can be easily shown that the following ratio is to be
held:
2 2
2 2
Vn +1 n a
= exp − (n + 1) a
exp − 2
Vn σ2
σ
(2n + 1)a2
= exp −
σ2
n = 2, . . . , N
(8)
where σ is the standard deviation of the Gaussian profile and
N is the number of the PC waveguide sections within each
transitional region. It is worth noting that even though large
σ is desirable because it results in large N and thus increases
the quality factor by decreasing the radiation loss, we cannot
increase σ beyond a certain value because by doing so we increase the mode volume of the resonant mode. A typical value
at the resonance wavelength λ0 = 1550 nm is σ = 2 μm.
Given that the ratio of the forward and backward waves
within nth TL sections can be written as Vn+ /Vn− =
(zc,n −zc,n −1 )/(zc,n +zc,n −1 ), equation (8) can be further simplified and be written in terms of the characteristic impedance
and propagation constant of the (n−1)th and nth TL sections
2
zc,n cosh γc,n a + zc,n −1 sinh γc,n a = exp − (2n + 1) a
zc,n
σ2
n = 2, . . . , N.
Fig. 6. (a) Schematic view of the PCNBR. (b) TL model of the PCNBR.
(c) The desired Gaussian voltage distribution.
Now that the silicon rib waveguide and the PC waveguide
mirrors are designed, the details of the resonant region characteristics are to be worked out. These details are schematically
depicted in Fig. 6(a). In accordance with the abovementioned
strategy, the radius of the two innermost waveguide sections
(forming the central waveguide region in Fig. 6(a)) should be r1
= 110 nm. In this manner, the lower edge of the photonic gap
(9)
Fortunately, the characteristic impedance and the propagation
constant of the TL model representing the central waveguide region is already known, and thus the characteristic impedance
and the propagation constant of the remaining TL models can
be easily determined in a recursive manner. Each of iterations
has a search step therefore a database of propagation constants
and characteristic impedances for a number of TL segments
should be prepared. Starting from n = 2 in equation (9) which
corresponds to closest PC waveguide section to the central PC
waveguide section, we search the database for the TL segment
(PC waveguide section) whose characteristic impedance and
propagation constant satisfies equation (9). Then the same procedure should be repeated for the next segment (n = 3) and so
forth. The recursive algorithm is terminated when the characteristic impedance and the propagation constant of one of the
outer waveguide sections is close enough to the characteristic
impedance and the propagation constant of the PC waveguide
96
TABLE I
RADII OF AIR HOLES IN THE TRANSITIONAL REGION OF THE DESIGNED PCNBR
Fig. 7. (a) A snapshot of the x component of the electric field profile (E x )
of the resonance mode extracted from 3-D FDTD for a PCNBR with photonic
crystal waveguide mirrors that are composed of 15 unit cells. (b) The absolute
value of E x in x = 0 along the resonator (blue solid line) and a Gaussian
distribution with the standard deviation of σ = 2 μm (dashed-dotted black line).
mirror. The difference is expected to be below 1%. In following
example we have used a database of 30 different PC waveguide
sections with air hole radiuses ranging from 110 to 80 nm.
Numerical calculations shows that for w = 500 nm, h =
240 nm, t = 50 nm, σ = 2 μm, rm = 80 nm, and r1 = 110 nm,
the transitional region should be made of 12 waveguide sections
whose radii of air holes are tabulated in Table I.
To demonstrate that the designed PCNBR has a Gaussian filed
profile, the FDTD simulation is employed to analyze the structure when the PC waveguide mirrors are composed of 15 unit
cells. The resonance wavelength and the quality factor of the
structure are 1566.48 nm, and 1.46 × 106 , respectively (difference between resonance wavelength and quality factor calculated by using the proposed TL model and results of FDTD
simulations is less that 0.1% and 5%, respectively). In Fig. 7(a)
snap shot of the x component of the resonant electric field profile Ex is plotted together with its absolute value at x = 0 and
z = 0. For the sake of comparison, a Gaussian envelop with the
standard deviation of σ = 2 μm is also plotted in Fig. 7(b). The
similarity between the desired Gaussian profile and the actual
shape of the electric field is striking.
It is worth noting that the high level of the quality factor of
the designed PCNBR can be partly attributed to the rather large
number of unit cells in the PC waveguide mirror section. Unfortunately, having large number of unit cells in the mirror section
decreases the coupling efficiency and increases the device foot
JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 1, JANUARY 1, 2014
print. The transmission efficiency of the designed PCNBR in the
direct coupling arrangement is below 4% (calculated by using
proposed TL model). This is too low to be of practical importance. To improve the transmission efficiency, the same PCNBR
with only six unit cells in the PC waveguide mirror section is
considered. From TL model the resonance wavelength, the quality factor, and the transmission efficiency of this structure in the
direct coupling arrangement are 1566.82 nm, 33900, and 33%,
respectively. Although the quality factor of the structure is lowered, it is still high enough for miscellaneous applications. The
transmission efficiency of the structure in the direct coupling
arrangement; on the other hand, is considerably increased. The
transmission efficiency can be further improved without deteriorating the quality factor of the structure. To this end, the radii of
the air holes in the mirror section should be gradually decreased.
But the gradual decrement of the air hole radii should be done in
such a manner that no sharp transition be imposed on the electromagnetic field profile. Therefore, the same recursive formula
should be used to guarantee the Gaussian field distribution. In
this way, the PC waveguide mirror section in its original form
is removed while the transitional region is extended. Since the
lowest air hole radius that our fabrication process can realize
with acceptable quality is about 50 nm, the air hole radii of the
six unit cell sections in the extended transitional region are rm 1
= 80.5 nm, rm 2 = 75.5 nm, rm 3 = 70.5 nm, rm 4 = 65.0 nm,
rm 5 = 59.0 nm, and rm 6 = 53.0 nm. These values are obtained
by using equation (9). TL model based calculations show transmission efficiency of 96% for this PCBNR. It worth noting that
while calculating transmission efficiency with 3-D FDTD simulation is a time consuming task (nearly impossible for ultra-high
quality factor PCNBRs [10]) TL model can be used to directly
calculate the transmission efficiency of PCNBRs. From the theory of microwave networks we know that each TL section with
characteristic impedance zc , propagation constant γ, and length
l, can be represented by a transmission matrix (T -matrix). The
T -matrix of a network made of cascaded TL sections Ttot can
be calculated by multiplying the T -matrices of subsequent TL
sections. Once the overall T -matrix of the PCNBR structure is
obtained by multiplication of the T -matrix of PC waveguide
sections, the transmission efficiency of the PCNBR can be calculated straightforwardly.
In previous studies ultra-high quality factor PCNBRs are designed based on air bridge dielectric waveguide or dielectric
waveguides without pedestal [6], [15]. In this paper we have
focused on designing high quality factor PCBNRs based on
silicon rib waveguides. Presence of a 50 nm pedestal layer in
the rib waveguide increases the radiation loss of the structure
and therefore decreases the quality factor of designed PCNBRs.
This radiation loss is why the qualify factor of the designed PCNBRs is lower than the quality factor of the PCNBRs reported
in [6], [15]. It is by the way worth noting that extraction of the
quality factor and the resonance wavelength by using the proposed approach remains valid as long as the quality factor of the
PCNBR is in between 106 and 103 . Besides, PCNBRs with low
quality factor suffer from radiation loss, which is not properly
modeled by the TL model. The quality factor cannot be too high
because otherwise the FDTD simulations needed for extraction
MIRI et al.: DESIGN AND FABRICATION OF PHOTONIC CRYSTAL NANO-BEAM RESONATOR: TRANSMISSION LINE MODEL
97
TABLE II
CHARACTERIZATION, FDTD, AND TL MODEL RESULTS FOR DIRECT COUPLED
NANO-BEAMS
intersection of the input/output waveguides and the PCNBR)
have been appropriately tapered and hence extinction of about
25 dB was achievable for the first mode. Fig. 8 shows the
schematic diagram of device, the transmission spectrum, and
the micro-graphs of the fabricated PCNBR. Two peaks have
been observed in the transmission response centered at λ0 =
1564.2 nm and at λ0 = 1589.2 nm with quality factors of 25000
and 4000, respectively. The measured quality factors of both
modes are in good agreement with the prediction of the proposed model and also FDTD simulation (see Table II).
V. CONCLUSION
Fig. 8. Direct coupling scheme, characterization curve, and SEM image of
the direct coupled nano-beam.
of the TL model parameters becomes very time consuming.
One other important point is that the TL model considers one
fundamental mode only. Therefore, the overall behavior of the
structure should be dictated by the fundamental mode of the
structure. In other words, higher order modes supported by the
PC waveguide sections should remain below cut-off. Obviously,
the PC waveguide cannot be multi-mode.
We proposed a TL model for photonic crystal waveguide
section and successfully used this model to design high quality
factor PCNBR. This model provides an insight to silicon cavity
design which significantly expedites simulation runtime. One
particular PCNBR was designed to have maximum quality factor
with a specific mode volume and resonance wavelength. The
design procedure was based on realizing Gaussian field profile
within the resonant region. Measurements show the excellent
accuracy of the model in the estimation of the PCNBR resonance
wavelength and quality factor.
REFERENCES
IV. FABRICATION AND CHARACTERIZATION RESULTS
PCNBR designed in the last section is formed by perforating
13 pairs of air holes (with radii r1 −r13 given in Table I) as the
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the previous section) as each mirror section in a silicon rib waveguide with a 50 nm pedestal layer (see Fig. 8). The PCNBR has
been fabricated on a standard SOI wafer with 3 micron BOX and
240 nm top silicon layer. Patterns were defined by electron beam
lithography (EBL) using 6% HSQ resist, spun at 5000 r/min for
60 s followed by prebaked at 90◦ for 3 min. After developing the
EBL resist in 25% TMAH at an elevated temperature for 30 s,
top silicon layer was dry etched by inductively coupled plasma
tool using chlorine chemistry. The etching step is carefully controlled to etch 190 nm of the top Si layer and keep a pedestal
layer with a thickness of 50 nm.
For characterization, the transmission spectrum over the
wavelength range from 1510 to 1640 nm was measured using a wideband tunable laser and using a set of grating couplers
to couple light to the PCNBR input/output waveguides. To improve the coupling efficiency from the input/output ridge waveguide to the PCNBR the air holes in the mirror section (at the
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Authors’ biographies not available at the time of publication.