Download Guiding and Confining Light in Void Nanostructure

Guiding and Confining Light in Void Nanostructure
Vilson R. Almeida, Qianfan Xu, Carlos A. Barrios, and Michal Lipson
School of Electrical and Computer Engineering, Cornell University, 411 Phillips Hall, Ithaca, New York
14853
Email: [email protected]
Abstract:
We present a novel waveguide geometry for enhancing and confining light in a nanometer-wide low-index
material. Light enhancement and confinement is caused by large discontinuity of the electric field at
high-index-contrast interfaces. We show that by using such a structure the field can be confined in a
50-nm-wide low index region with a normalized intensity of 20 µm-2. This intensity is approximately 20
times higher than that can be achieved in SiO2 with conventional rectangular waveguides.
Recent results in integrated optics have shown the ability to guide, bend, split and filter [1-5] light on
chips using optical devices based on high-index-contrast waveguides. In all of these devices, the guiding
mechanism is based on total internal reflection (TIR) in a high-index-material (core) surrounded by a
low-index material (cladding). TIR mechanism can confine light very strongly in the high-index material.
In recent years, a number of structures have been proposed to guide or enhance light in low-index materials
[6-11]. All of them rely on the external reflections provided by interference effects. Unlike TIR, the
external reflection cannot be perfectly unity; therefore, the modes in these structures are inherently leaky
modes. In addition, since interferences are involved, these structures are strongly wavelength dependent.
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Here we show that the field can be enhanced and confined in the low-index material even when light is
guided by TIR. Considering a high-index-contrast interface, Maxwell’s equations state that in order to
satisfy the continuity of the normal component of electric flux density D, the correspondent electric field
(E-field) has to undergo a large discontinuity with much higher amplitude in the low-index side. We show
that this discontinuity can be used to strongly enhance and confine light in a nanometer-wide region of
low-index material. Since the guiding mode is indeed an eigenmode, the proposed structure is
fundamentally lossless and has very low wavelength sensitivity. Furthermore, we show that this structure is
compatible with highly integrated photonics technology.
The principle of operation of the novel structure can be illustrated by analyzing the slab-based structure
shown in Fig. 1(a), where a low-index slot is embedded between two high-index slabs (the shadowed
regions). The novel structure is hereafter referred as slot-waveguide. The slot-waveguide eigenmode can
be seen as formed by the interaction between the fundamental eigenmodes of the individual
slab-waveguides. Rigorously, the analytical solution for the transverse E-field profile Ex of the
fundamental transverse-magnetic (TM) eigenmode of the slab-based slot-waveguide is

 1 cosh(γ S x), x < a
 nS 2
 1
γ
E x ( x) = A ⋅  2 cosh(γ S a ) cos[κ H ( x − a )] + 2 S sinh(γ S a ) sin [κ H ( x − a )], a < x < b
nS κ H
 nH
2
 1 

n γ
 2 cosh(γ S a ) cos[κ H (b − a )] + H2 S sinh(γ S a ) sin [κ H (b − a )] ⋅ exp[− γ C ⋅ ( x − b)],
nS κ H
 nC 

, (1)
x >b
where κH is the transverse wavenumber in the high-index slabs, γC is the field decay coefficient in the
cladding, γS is the field decay coefficient in the slot, and the constant A is given by
2
A = A0
µ0
ε0
k 0 nH − κ H
k0
2
2
2
where A0 is an arbitrary constant, k0 = 2π/λ0 is the vacuum wavenumber, and ε0 and µ0 are the electric
permittivity and magnetic permeability in vacuum, respectively. The transverse parameters κH, γS, and γC
simultaneously obey the following relations
β 2 = k 0 2 nH 2 − κ H 2 = k 0 2 nC 2 + γ C 2 = k 0 2 nS 2 + γ S 2 ,
(2)
where β is the eigenmode propagation constant, which can be calculated by solving the transcendental
characteristic equation

 γ C nH 2  γ S nH 2
 =
tan κ H (b − a ) − arctan
tanh(γ S a ) .
2 
2
κ
n
 H C  κ H nS

(3)
One can see from equation (1) that at the interface between the slot and the slabs ( x = a ) the E-field
2
immediately inside the slot ( x = a − ) is n H / n S
2
high-index slabs ( x = a + ). The ratio n H / n S
2
2
times higher than that immediately inside the
equals 6 for a Si/SiO2 interface, and equals 12 for a Si/air
interface. When the width of the slot is much smaller than the characteristic decay length inside the slot
( a << 1 / γ S ), the field remains high all across the slot. By assuming that nH = 3.48, nS = nC = 1.44, a = 25
nm, and b = 205nm, we calculated the TM Ex distribution at λ0 = 1.55 µm. Fig. 2 shows the Ex distribution,
which evidences the large discontinuity and the high E-field confinement in the slot; in this case,
a = 25 nm << 1 / γ S = 140 nm . We also show in Fig. 2 (dotted and dashed lines), the fundamental
eigenmodes of the individual slab-waveguides.
Hereafter we shall consider a 3D slot-waveguide structure with finite height [see Fig. 1(b)]. In this case,
the quasi-Transverse-Electric (quasi-TE) eigenmode is defined as the one that presents the major E-field
component along the x-direction. Therefore, the quasi-TE eigenmode in the 3D slot-waveguide structure
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presents optical properties similar to those of the TM eigenmode in the studied slab-based slot-waveguide
structure. We numerically simulated the quasi-TE eigenmode of the slot-waveguide structure shown in Fig.
1(b) using a full-vectorial finite difference mode solver [12]. Non-uniform grid mesh is used in the
simulations, which simultaneously enables fine grids near the slot for maximizing the accuracy, and large
domains of simulation for eliminating the influence of boundary conditions. In the simulations, the
smallest grid size is one-twentieth of the slot width, and the total domain is 10 µm × 10 µm. For
concreteness, we assume that the slot-waveguide is built on the widely used Silicon-on-Insulator (SOI)
platform, with the silicon being the high-index material, and SiO2 composing the low-index surrounding
cladding. Therefore nH = 3.48, nC = 1.44, and λ0 = 1.55 µm are assumed, unless otherwise specified. We
also assume that the rectangular cross-section slot region, with height h and width wS, is filled with SiO2
(nS = 1.44). In practice, the slot can be filled with any low-index material of interest, and the platform and
the working wavelength can be chosen according to the application; in any of these variants, the principle
of the E-field enhancement in the slot remains the same.
The transverse E-field distribution of the quasi-TE mode is shown in Fig. 3. In all simulations we have
assumed a slot-waveguide with default cross-section parameters wH = 180 nm, wS = 50 nm, and h = 300 nm,
unless otherwise specified. Figure 3(a) shows the contours of the E-field amplitude and the E-field lines.
The center bright region shows a strong E-field inside the slot. The directions of the E-field lines in the slot
confirm that the major component of the E-field is normal to the high-index-contrast interface, which
induces the strong discontinuity of the E-field. The amplitude profile of the E-field is shown in the 3D
surface plot shown in Fig. 3(b). One can see that the amplitude profile of the E-field along the x-direction
at y = 0 strongly resembles the one in Fig. 2 for the analytical expression of the TM eigenmode of the
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slab-based structure. The vertical confinement of the E-field in the slot region is dictated by that in the
high-index regions. Note that since the slot-waveguide structure has a well-defined eigenmode, it is
theoretically lossless.
As a result of the E-field enhancement in the slot, the optical intensity (i.e., optical power density) in
the slot is much higher than that in the high-index region of the structure. Figure 4 shows the optical power
Pslot and the average optical intensity Islot = Pslot/(h⋅wS) inside the slot as a function of its width wS and the
width of the silicon region wH. Both Pslot and Islot are normalized with respect to the total optical power in
the waveguide. For comparison, the normalized average intensity in the silicon region ISi is plotted as well.
One can see from Fig. 4 that Islot decreases monotonically with the increase of the slot width wS, but it
remains very high when the slot is narrow (wS < 40 nm); under this condition, Islot stays about 10 times
higher than ISi. Figure 4 shows that Pslot remains nearly constant around 30% for wS ≥ 50 nm. For wS = 50
nm, Islot is as high as 20 µm-2, which is 6 times higher than ISi. One can also see from Fig. 4 that wH does not
significantly affect the slot-waveguide performance.
Light propagating in the slot-waveguide, not only is significantly confined to a low-index material, but
also has an intensity much higher than that achievable using conventional rectangular waveguides. With
the same finite difference mode solver, we simulated the eigenmodes of the conventional channel
(rectangular cross-section with step-index profile) waveguides [1], and calculated the average intensities in
the cores of the waveguides. When the cross-section of the waveguide is large, the intensity is low because
of the large mode size. When the cross-section is too small, the intensity is also low because the mode
becomes delocalized [13]. Therefore, there is an optimal waveguide dimension for achieving the highest
intensity. The highest normalized average intensity in a conventional channel SOI waveguide is less than 9
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µm-2, which is obtained for a cross-section of 360 nm × 200 nm. If the light is to be confined in a low-index
material such as SiO2 with a conventional channel waveguide structure, the highest possible index contrast
that can be achieved is through the SiO2/air platform. The highest normalized average intensity that can be
obtained in such a structure is less than 1.1 µm-2, which is obtained for a cross-section of 900 nm × 500 nm;
this is about 20 times lower than that we have calculated for the slot-waveguide with default cross-section
parameters. For the leaky-mode waveguides based on external reflections, such as antiresonant reflecting
optical waveguide (ARROW) and photonic crystal waveguide, the size of the low-index core is limited to
be larger than half of the wavelength in the low-index material. Therefore, the normalized intensity can
hardly exceed 1 µm-2 at λ0 = 1.55 µm wavelength.
In order to investigate the wavelength dependence, we simulated both the Pslot and Islot as a function of
wavelength over a broad spectral range, which is shown in Fig. 5. The simulations show that the structure
has very low wavelength sensitivity. This is because there is no interference effect involved in the guiding
and confinement mechanism. The material dispersion [14] has been taken into account in the simulations.
One can see that Pslot and Islot show very low wavelength sensitivity, varying only less than 10% over a
wavelength span of more than 400 nm. Therefore, the same slot-waveguide can be used to guide and
confine light in a low-index material over a broad wavelength range, what widens its application scope. As
a comparison, Fig. 5 shows the optical intensity levels for the alternative approaches discussed in the
previous paragraph.
With the purpose of verifying the slot-waveguide compatibility with highly integrated photonics
technology, we calculated the bending losses for a slot-waveguide with default cross-section parameters
and bending radius of only R = 5 µm. We found out that the transmission in a 360º turn is of 99.2 %, which
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corresponds to propagation bending losses of only 11 dB/cm. This parameter is particularly relevant for the
fabrication of high-quality ring-resonators, where such bending losses can potentially translate into a
quality-factor of about Q = 20,000.
In summary, we present in this paper a novel waveguide geometry for guiding and confining light in
low index region. The waveguide discussed has two unique properties. First, it produces high E-field
amplitude, optical power and optical intensity, in low-index materials on levels that cannot be achievable
with conventional waveguides. This property enables highly efficient interaction between fields and active
materials, which may lead to all-optical switching and parametric amplification on integrated photonics.
Second, a strong E-field confinement is localized in a nanometer-sized low-index region, which could be
filled with low-index materials such as liquids or gases. Therefore, the slot-waveguide can be used to
greatly increase the sensibility of compact optical sensing devices or to enhance the efficiency of near-field
optical probes.
Acknowledgement
The authors would like to thank Christina Manolatou for her guidance and assistance in the simulations
and Roberto R. Panepucci for fruitful discussions. This work was supported by the Air Force Office of
Scientific Research under grant number AFOSR F49620-03-1-0424. V. R. Almeida acknowledges
sponsorship support provided by the Brazilian Defense Ministry.
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nC
nS
nH
nH
y
z~
nC
nC
x
-b
-a
0 a
h
nH
b
(a)
(b)
wH
nS
wS
nH
wH
y
-z
x
Fig. 1. Schematic of the slot-waveguide structure. (a) With infinite height. (b) With finite height.
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1.0
Normalized Ex
0.8
0.6
0.4
0.2
0.0
-b
-400
-a a
-200
0
b
200
400
x (nm)
Fig. 2. Normalized transverse E-field (Ex) distribution of the fundamental TM eigenmode (solid line) for
the slab-based slot-waveguide at λ0 = 1.55 µm, with nH = 3.48, nS = nC = 1.44, a = 25 nm, and b = 205 nm.
Also shown are the individual slab-waveguide TM eigenmodes (dotted and dash-dot lines).
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Fig. 3. Transverse E-field profile of the quasi-TE mode in a SOI-based slot-waveguide. The origin of the
coordinates system is located at the center of the waveguide, with a horizontal x-axis and a vertical y-axis.
(a) Contour of the E-field amplitude and the E-field lines. (b) 3D surface plot of the E-field amplitude.
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wH=160 nm
wH=180 nm
wH=200 nm
Normalized Power (%)
-2
Pslot
80
60
30
20
40
Islot
20
10
ISi
0
0
20
40
Normalized Intensity (µm )
40
60
80
100
120
0
140
Slot Width wS (nm)
Fig. 4. The normalized optical power in the slot Pslot, the normalized average optical intensity in the slot Islot,
and the normalized average optical intensity in silicon ISi, for the fundamental quasi-TE eigenmode of the
slot-waveguide. All three quantities are normalized with respect to the total waveguide optical power.
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30
-2
Pslot
I slot
20
20
10
0
1.2
SOI Waveguide
ARROW and PBG Waveguide
SiO2 Waveguide
1.3
Normalized Intensity (µm )
Normalized Power (%)
30
1.4
1.5
Wavelength (µm)
1.6
10
0
Fig. 5. Wavelength dependence of the normalized optical power Pslot and the normalized average
optical intensity Islot in the slot, for wH = 180 nm, wS = 50 nm, and h = 300 nm. Optimal values for the
normalized optical intensity of alternative waveguide approaches are indicated for comparison purposes.
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