Compact bends for multi-mode
photonic crystal waveguides with
high transmission and suppressed
modal crosstalk
Victor Liu∗ and Shanhui Fan
Department of Electrical Engineering, Stanford University, Stanford, California 94305 USA
∗ [email protected]
Abstract: We demonstrate an extremely compact bend for a photonic
crystal waveguide supporting three spatial modes. The bend exhibits nearly
100% transmission over a relative bandwidth of 1% with less than 1%
crosstalk. We show that our design is robust with respect to fabrication
errors. Our design method is applied to create a structure consisting of
dielectric rods, as well as a structure consisting of air holes in a dielectric
background.
© 2013 Optical Society of America
OCIS codes: (350.4238) Nanophotonics and photonic crystals; (130.3120) Integrated optics
devices; (130.5296) Photonic crystal waveguides.
References and links
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1.
Introduction
The use of multiple spatial modes is gaining interest for increasing the throughput of optical
information processing systems [1]. A basic component for integrated systems is a waveguide
bend that can simultaneously redirect multiple spatial modes with near total transmission for
each mode and low crosstalk between the modes. It is possible to use the self-collimation effect
to achieve bending that preserves modal content [2, 3]. However, integrated optics is typically
based on waveguide structures. Recent work on transformation optics approaches has achieved
a waveguide bend structure with high transmission and low crosstalk, but the device has a
spatial extent of many wavelengths [4].
Photonic crystal structures have been widely used for creating very compact integrated optical components. In integrated 2D photonic crystal systems, the design of bends for single-mode
waveguides has been extensively studied both theoretically [5–9] and experimentally [10–15].
In this paper we present the design of a bend for a photonic crystal waveguide supporting three
modes with nearly total transmission and almost no crosstalk. The physical size of the bend is
less than 3 wavelengths long on a side.
2.
The optimized bend structure
Figure 1 shows the optimized device structure, consisting of an input waveguide on the left
and an output waveguide on the bottom, highlighted in blue. The structure is embedded within
a background photonic crystal of silicon rods (refractive index n = 3.4) of radius 0.2a in air
(n = 1) on a square lattice of lattice constant a. The waveguides are formed by removing three
rows of rods and support three modes. The dispersion relations of the three modes are shown
in Fig. 2. Also shown in Fig. 2 are the out-of-plane electric field profiles for the three modes at
the operating frequency.
The bend region is highlighted in orange in Fig. 1 and consists of rods centered on lattice
sites but with varying radii listed in the table on the right of Fig. 1. Although the presented
design is an idealized 2D structure, there have been recent efforts to implement such designs
using effective index models in 3D for planar fabrication [16].
The structure in Fig. 1 has a mirror symmetry plane along the diagonal dashed line. In general, in our design we considered only structures with such a mirror symmetry. We enforced
such a constraint since single-mode photonic crystal waveguide bends with 100% transmission
coefficients all have such a symmetry [5]. In [5], it was argued that a single-mode photonic
crystal waveguide bend can be described by a resonator model, with a resonance located at
the bend region coupled to the two waveguides. Such a resonator model supports 100% trans-
#185003 - $15.00 USD Received 13 Feb 2013; revised 18 Mar 2013; accepted 22 Mar 2013; published 27 Mar 2013
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Rod Radius (a)
1 4
3
7 8 9
9
6
8
5
6 7
2
3
4
1
1
2
3
4
5
6
7
8
9
0.148054
0.177218
0.151771
0.171812
0.209037
0.194607
0.202831
0.200353
0.199508
Normalized frequency ω (2πc/a)
Fig. 1. Schematic of the optimized bend structure. The blue highlighted regions show the
waveguides, and the orange highlighted region shows the bend region where search and
optimization were performed. Solid gray circles indicate dielectric rods (refractive index
3.4) surrounded by air. Unlabeled rods have radius 0.2a where a is the lattice constant. The
table on the right lists the radii of the labeled rods. The dashed green diagonal line shows
the plane of mirror symmetry.
0.45
Mode 1
0.4
Mode 2
0.35
0.3
Mode 3
0.25
0
0.5
Propagation constant k (2π/a)
Fig. 2. Band structure of the waveguide modes within the bandgap of the background photonic crystal. The out-of-plane electric field profiles of the three modes at the operating
frequency are shown on the right. The profiles are taken along the edge of the waveguide
unit cell and are on the same horizontal scale as the cell shown at the bottom right.
#185003 - $15.00 USD Received 13 Feb 2013; revised 18 Mar 2013; accepted 22 Mar 2013; published 27 Mar 2013
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mission only when the resonance couples to the two waveguides with the same strength. The
most straightforward way to implement such a symmetric resonator model is to have a structure
with mirror symmetry. For multi-mode waveguide bends, there is no strict theoretical basis for
enforcing such a symmetry; we simply carried over the intuition that was developed from the
case of a single-mode waveguide bend. The success in obtaining a functional structure in such
a restricted structural space provides some support of this intuition.
We further restricted the design space by considering only structures with rods centered on
the 7 × 7 possible lattice sites within the bend region. It was assumed that all rods have the same
refractive index as the rods in the background photonic crystal, but the radius of the rods in the
bend region was allowed to vary. Combined with the mirror symmetry constraint, only 28 of
the 49 rods in the bend region were free parameters for design.
3.
The design process
Within the design space, we searched for structures with the desired performance by using an
aperiodic design methodology involving combinatorial search followed by continuous parameter adjustment. The process was discussed in more detail in Refs. [17–19]. We will briefly
review the process here, focusing primarily on those aspects that are specific to our current
design.
All bend structures were evaluated using the following error metric:
J=
3
∑ (1 − pm )2
(1)
m=1
where pm is the power transmission coefficient for the m-th waveguide mode. For the device
shown, we selected an operating frequency of ω = 0.4 × 2π c/a.
The design process proceeded in two phases. In the first phase, a combinatorial search was
performed where every possible combination of rods (presence or absence) on the lattice sites in
the bend region was considered, requiring evaluation of 228 possible structures. In this phase,
the rods considered were identical to those in the background photonic crystal (with radius
0.2a). Any structures encountered in the combinatorial search that showed promising behavior
(J < 0.05) were considered initial candidates for further optimization.
In the second phase, the initial candidates were fine tuned by adjusting the rod radii in the
bend region using a simple gradient descent method to minimize the error metric J in Eq. (1).
The gradient of J with respect to rod radius is straightforward to calculate by applying the
adjoint variable method throughout the computational process [20,21]. For the structure shown,
the second phase reduced the initial value of J = 0.0155 to a final value of J = 0.000798.
4.
Numerical methods
In order to efficiently simulate all 228 possible rod configurations, we used an efficient simulation method tailored to the particular class of structures being analyzed [19]. The Dirichlet-toNeumann (DtN) map was used to model each unique type of unit cell [22]. Because the cells
in the structures we considered contain only circular inclusions, the DtN maps were computed
using cylindrical wave expansions, which provided sufficient accuracy with only 5 field samples per unit cell edge. By enforcing continuity constraints on the fields of adjacent cells, a
relatively small linear system Ax = b is formed. Here x represents the fields on the boundaries
of the unit cells, b represents the incident wave source, and A is the system matrix formed from
the DtN maps. Once a single structure is solved by computing A−1 , subsequent modifications
such as addition or removal of a single rod or changing of a rod radius, can be accounted for
straightforwardly by computing the modified A−1 using a low rank update [18]. We performed
#185003 - $15.00 USD Received 13 Feb 2013; revised 18 Mar 2013; accepted 22 Mar 2013; published 27 Mar 2013
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the entire combinatorial search over 228 structures within 12 hours using 1024 cores on the
SDSC Trestles cluster.
5.
Discussion
Transmission coefficient
1
0.8
0.6
0.4
1
0.98
0.96
0.2
0.395
0
0.39
0.4
0.405
0.4
Normalized frequency (2πc/a)
Mode 1
Mode 2
0.41
Mode 3
Fig. 3. Transmission spectra for each of the three waveguide modes shown in Fig. 2 for the
structure shown in Fig. 1. The inset shows details of the transmission peaks.
The transmission spectra of the bend structure shown in Fig. 1 for each of the three modes
is shown in Fig. 3. At the design frequency of 0.4 × 2π c/a, the transmission coefficient for all
three modes exceeds 98%. The transmission peaks have a sizable relative bandwidth of 1% at
a 95% transmission threshold. For operation at λ = 1.55 μ m, the bend region has dimensions
of 4.34 μ m on a side, and a bandwidth of 15 nm. Because the bend structure is lossless, the
high transmission coefficients necessarily imply that very little power is reflected or converted
into other waveguide modes. For the device shown, the cross-talk parameters (transmission
coefficients into other modes) are all less than 1%.
Representative field profiles of the device in operation at the center frequency are shown in
Fig. 4. The incident mode enters from the left and exits from the bottom. We indeed see that
the bend achieves very high transmission while preserving the modal content. Due to the mirror
symmetry of the device, the transmission behavior in the reverse direction is identical.
Fig. 4. Representative field patterns (out-of-plane electric field) for each waveguide mode
of the optimized structure shown in Fig. 1 at the operating frequency of ω = 0.4 × 2π c/a.
In our first combinatorial search phase where we evaluated 228 structures, with all rods in the
#185003 - $15.00 USD Received 13 Feb 2013; revised 18 Mar 2013; accepted 22 Mar 2013; published 27 Mar 2013
(C) 2013 OSA
8 April 2013 / Vol. 21, No. 7 / OPTICS EXPRESS 8073
Transmission coefficient
7 × 7 bend region having the same radius, only 10 structures with J < 0.05 were found. Using
a smaller 6 × 6 bend region did not produce any promising candidates. It is clear that for the
relatively small search space shown in Fig. 1, a randomized search procedure would probably
not encounter any working device, since the probability of encountering such a device is very
low (∼ 4 × 10−8 for the 7 × 7 bend region). The use of heuristic search procedures such as simulated annealing is unlikely to significantly improve the likelihood of encountering a working
device due to the very low correlation in transmission properties between devices differing by
adding or removing even a single rod. Thus, an exhaustive search is actually practical for device
design for this class of structures.
1
Mode 2
0.98
Mode 3
0.96
0.395
Mode 1
0.4
Normalized frequency (2πc/a)
3 digits
0.405
2 digits
Fig. 5. Transmission spectra as in Fig. 3, except for the structure in Fig. 1 with rod radii
rounded to three (dashed lines) and two (solid lines) significant digits.
The six digits of accuracy presented in Fig. 1 is not actually required for a functioning device.
Practical fabrication tolerances limit the precision of the rod radii to two or three significant
digits. In Fig. 5 we show the transmission spectra for devices with hole radii rounded to three
(dashed lines) and two (solid lines) significant digits. In both cases, the transmission coefficient
at the target frequency remains above 98%, suggesting that the design is robust to perturbations
in radius.
The design method is not limited to structures of dielectric rods in air. We have also applied
the same method to design an analogous structure in a system with air holes surrounded by
dielectric. We consider the TE polarization with an out-of-plane magnetic field. In Fig. 6(a), an
optimized structure is shown. The structure is embedded in a background photonic crystal of
air holes (n = 1, r = 0.4a) in a dielectric (n = 3.481) background. The holes in the bend region
after optimization have the radii listed in the table. The corresponding transmission coefficient
for each of the three modes is shown in Fig. 6(b). All three modes achieve greater than 98%
transmission at the design frequency of 0.268 × 2π c/a.
Finally, we comment briefly on the size of the device with respect to fundamental limits. The
resonator model of [5] may be used to estimate a lower limit on the size of the bend region
required. The ability to pass three spatial modes with total transmission at a single frequency
implies the presence of at least three distinct resonances in the bend region. Assuming a typical
estimate of one square wavelength per resonant mode, the bend region would have to be at
least 3λ 2 in area. The bend region we have presented is approximately 8λ 2 in area, making
it near the theoretical optimum. Note that simply having three resonances is not sufficient to
guarantee total transmission of three modes since the field patterns of the resonances must also
be compatible with the waveguide spatial modes, likely increasing the minimum device area
required.
#185003 - $15.00 USD Received 13 Feb 2013; revised 18 Mar 2013; accepted 22 Mar 2013; published 27 Mar 2013
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6 8 11 13 15
3 5
10 14 15
12 13
4 7 9
10 11
8
7
2
4
5 6
1 2
3
(a)
Radius (a)
0.399329
0.394799
0.398081
0.406390
0.414316
0.404690
0.400388
0.398726
0.396619
0.395911
0.395497
0.407046
0.396383
0.390133
0.394339
1
Transmission coefficient
Rod
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.8
0.6
0.4
0.2
0
0.267
0.268
Normalized frequency (2πc/a)
Mode 1
Mode 2
0.269
Mode 3
(b)
Fig. 6. (a) Schematic of an alternate optimized bend structure. Solid white circles indicate
air holes (refractive index 1) surrounded by dielectric (refractive index (3.481). Unlabeled
rods have radius 0.4a where a is the lattice constant. The table on the right lists the radii of
the labeled rods. (b) Transmission spectra of the bend for each of the three modes.
6.
Conclusion
We have described the design of a multi-mode photonic crystal waveguide bend structure exhibiting over 98% transmission, and less than 1% reflection and crosstalk. The structure was
shown to be robust to fabrication error with respect to deviations in the rod radii. In addition to
a structure of dielectric rods in air, we also demonstrated a second device implemented with air
holes in a dielectric background, which may be more experimentally relevant. Finally, we note
that the size of these structures is very near the fundamental size limit derived from heuristic
arguments.
Acknowledgments
This research was supported in part by the National Science Foundation through XSEDE resources provided by the XSEDE Science Gateways program. This work was also supported in
part by the United States Air Force Office of Scientific Research (USAFOSR) grant FA955009-1-0704, and the National Science Foundation (NSF) grant DMS-0968809.
#185003 - $15.00 USD Received 13 Feb 2013; revised 18 Mar 2013; accepted 22 Mar 2013; published 27 Mar 2013
(C) 2013 OSA
8 April 2013 / Vol. 21, No. 7 / OPTICS EXPRESS 8075