Optical dielectric nanoantenna for quantum cascade laser ...1273/fulltext.pdf · injected current...
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Optical Dielectric Nanoantenna for Quantum Cascade Laser Device
Directive Emission
A Thesis Presented
by
Jing Wu
to
The Department of
Electrical and Computer Engineering
for the degree of
Master of Science
in
Electrical Engineering
Northeastern University
Boston, Massachusetts
Summer 2009
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Contents
Contents ---------------------------------------------------------------------------------------- 2
Acknowledgments ---------------------------------------------------------------------------- 4
Abstract ----------------------------------------------------------------------------------------- 5
List of Figures --------------------------------------------------------------------------------- 6
1. Introduction and Background ----------------------------------------------------------- 8
1.1 Motivation--------------------------------------------------------------------------- 8
1.2 Concept of Quantum Cascade Laser--------------------------------------------- 9
1.3 QCL Collimation and its Challenge----------------------------------------------12
1.4 Objective-----------------------------------------------------------------------------14
1.5 Outline of Thesis--------------------------------------------------------------------15
2. QCL Modeling with FDTD Technique -------------------------------------------------16
2.1 FDTD Technique--------------------------------------------------------------------16
2.1.1 Maxwell s Equation and the Yee Algorithm-----------------------------16
2.1.2 Perfect Matched Layer----------------------------------------------------19
2.2 QCL Modeling with FDTD technique-----------------------------------------20
2.1.1 Geometry-------------------------------------------------------------------20
2.1.2 Near Field Performance of QCL---------------------------------------21
2.1.3 Far Field Performance of QCL------------------------------------------25
2.2 Conclusion------------------------------------------------------------------------26
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3 Physical and Modeling of Dielectric Nanoantenna-----------------------------------27
3.1 Photonic Crystals------------------------------------------------------------------27
3.2 PCs Dielectric Pattern and its Dispersion Diagram---------------------------27
3.3 Transmission Characteristics---------------------------------------------------- 30
3.4 Cavity Defect Modes--------------------------------------------------------------31
3.5 Dielectric Pattern Nanoantenna with Waveguide Source--------------------33
4 Nanoantenna Engineering QCL Radiation-------------------------------------------- 37
4.1 Dielectric Nanoantenna Integrated with QCL Device------------------------ 37
4.2 Near Field and Far Field Performance of Dielectric Nanoantenna--------- 39
4.3 Larger Size Facet Aperture------------------------------------------------------ 42
4.4 Optimization on Operating Frequency----------------------------------------- 45
4.5 Alignment-------------------------------------------------------------------------- 47
5 Thesis Conclusion------------------------------------------------------------------------ 50
References---------------------------------------------------------------------------------52
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Acknowledgments
I would like to thank my adviser, Professor Hossein Mosallaei for his constant
help and support during the time spent working on this project. His novel approach to
the problems I encountered was invaluable to my experience.
I would also like to acknowledge and thank the other members of Applied EM &
Optics Devices Laboratory. Without the friendly help from the lab personnel and the
scholarly and constructive work environment within the lab, this process would have
taken much longer.
Finally a special thanks goes out to my family, who have always been behind me
whatever my goal at the time happened to be. Their love and support has helped me to
achieve my goals, and I will forever appreciate it.
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Abstract
This thesis presents the concept and modeling of a dielectric-patterned nanoantenna
integrated with a quantum cascade laser (QCL) device to tailor the radiation beam and
provide directive emission characteristic. A periodic dielectric configuration with
optimized and different periodicities in transverse and propagating directions is
realized to engineer a band-edge dispersion diagram and manipulate the performance of
the source radiation. The structure is integrated with the QCL device. Directive
emission of
with vertical and horizontal narrow beamwidthes of 14o and 12o,
respectively, are demonstrated. This will be about a 2.5-times improvement in the
vertical and 4.5 times-improvement in the horizontal planes of the QCL source
beamwidth, offering enhanced directivity. High-efficiency power output is also
obtained. Full wave analysis based on finite difference time domain (FDTD) technique
is applied to characterize the device comprehensively and exploit novel physical
parameters. The concept of a periodic-pattern dielectric nanoantenna is very general
and can be applied in THz, IR, and visible spectrums, scaling the geometry accordingly.
The obtained nanoantenna can offer optical devices directive emission, enabling
long-range energy communication.
Index Terms- Dispersion diagram, metamaterial, nanoantenna, photonic crystal, quantum cascade laser.
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List of Figures [Fig.]
1. Interband and intersubband transitions
2. Yee ¯s Latti c
3. The geometry of QCL device
4. Normalized E and H field profiles inside the active region of the QCL source along
vertical direction
5. Normalized E and H field profiles inside the active region of the QCL source along
horizontal direction.
6. Near-field distributions of the QCL source
7. Radiation performance of the QCL source
8. 3D PC metamaterial pattern with different periodicities in transverse and vertical
directions
9. Dispersion Diagram for 3D PCs Dielectric pattern
10. 2D kx (or kz)-ky vector plane dispersion diagram
11. Transmission coefficients for normal incident planewaves propagating along y and
z directions inside the PC.
12. A cavity structure constructed from a low dielectric material sandwiched between
two PC layers
13. Normal incident transmission performance of a cavity structure constructed from a
low dielectric material sandwiched between two PC layers.
14. Dielectric pattern nanoantenna and its aperture near-field
15. The radiation characteristic of dielectric pattern nanoantenna
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16. The radiation patterns of dielectric nanoantenna at different frequency
17. QCL device integrated with dielectric pattern nanoantenna.
18. Near-field distributions of the QCL integrated with dielectric nanoantenna
19. Radiation performance of QCL nanoantenna
20. Near-field distributions of the QCL integrated with larger dielectric nanoantenna
21. Radiation performance of larger QCL nanoantenna
22. Radiation performance of QCL nanoantenna at different operating frequency
23. Alignment of dielectric pattern versus QCL device
24. Radiation performance of QCL nanoantenna with different alignment
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Chapter 1 Introduction and Background
1.1 Motivation
Optical energy transmission over a long-range and in a short-period of time attracts
significant interest for nanoscale on-chip and between-systems communication []. A
possible solution to this demand may be realized utilizing the concept of near-field
guiding along the chain of plasmonic nanoparticles. However, the loss of plasmonic
particles and the fact that the near-field propagation depends on
and higher-order
terms reduce the opportunity for long-range guiding. Defect guiding in photonic
crystals [1] can be another option, but obviously one cannot use this approach for
inter-chip connection.
An antenna is a key enabling component in the microwave spectrum for long range
space data communication. It can radiate far field patterns with
dependence
having power density related to . It will be of extreme importance to translate the
concepts of antennas from RF to THz, creating optical nanoantennas and enabling
nanoscale wireless communication and interconnect. There have been some works on
dipole and bowtie THz nanoantennas where one obtains strong near-fields around their
terminals at the resonance lengths (if they are illuminated by a plane-wave) [2]. The
goal of this thesis is to provide a systematic study on nanoantennas. It is demonstrated
that one can use an array of periodic dielectric pattern to manipulate a spherical wave
and transform it into a plane-wave with far-field directive emission. The concept is
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demonstrated well by integrating the antenna with a quantum cascade laser (QCL)
device scanning its radiation beam performance successfully.
1.2 Concept of quantum cascade laser
Quantum cascade lasers (QCLs) are semiconductor lasers that emit in the mid- to
far-infrared portion of the electromagnetic spectrums ( ) and were
first demonstrated by Jerome Faist, Federico Capasso, Deborah Sivco, Carlo Sirtori,
Albert Hutchinson, and Alfred Cho at Bell Laboratories in 1994[3]. Unlike typical
inter-band semi- conductor lasers that emit electromagnetic radiation through the
recombination of electron hole pairs across the material band gap, QCLs are
uni-polar and laser emission is achieved through the use of intersubband transitions in
a repeated stack of semiconductor superlattices.[4,5]
Within a bulk semiconductor crystal, electrons may occupy states in one of two
continuous energy bands - the valence band, which is heavily populated with low
energy electrons and the conduction band, which is sparsely populated with high
energy electrons. The two energy bands are separated by an energy band gap in which
there are no permitted states available for electrons to occupy. Conventional
semiconductor laser diodes generate light by a single photon being emitted when a
high energy electron in the conduction band recombines with a hole in the valence
band. The energy of the photon and hence the emission wavelength of the laser diodes
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is therefore determined by the band gap of the material system used, as shown in
Fig. 1.
(a) (b)
Fig. 1 (a) Interband transitions in conventional semiconductor lasers emit a single photon, and (b) In quantum cascade structures, electrons undergo intersubband
transitions and photons are emitted. The electrons tunnel to the next period of the structure and the process repeats.
A QCL however does not use bulk semiconductor materials in its optically active
region. Instead it comprises a periodic series of thin layers of varying material
composition forming a super lattice. The super lattice introduces a varying electric
potential across the length of the device, meaning that there is a varying probability of
electrons occupying different positions over the length of the device. This is referred to
as one-dimensional multiple quantum well confinement and leads to the splitting of the
band of permitted energies into a number of discrete electronic subbands. By suitable
design of the layer thicknesses it is possible to engineer a population inversion between
two subbands in the system which is required in order to achieve laser emission. Since
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the position of the energy levels in the system is primarily determined by the layer
thicknesses and not the material, it is possible to tune the emission wavelength of QCLs
over a wide range in the same material system.
Additionally, in a unipolar QCL, once an electron has undergone an intersubband
transition and emitted a photon in one period of the superlattice, the electron can tunnel
into the next period of the structure where another photon can be emitted. This process
of a single electron causing the emission of multiple photons as it traverses through the
QCL structure gives rise to the name cascade and makes a quantum efficiency of
greater than unity possible which leads to higher output powers than semiconductor
laser diodes.
The first step in processing quantum cascade gain material to make a useful
light-emitting device is to confine the gain medium in an optical waveguide. This
makes it possible to direct the emitted light into a collimated beam, and allows a laser
resonator to be built such that light can be coupled back into the gain medium. A
ridge waveguide is created by etching parallel trenches in the quantum cascade gain
material to create an isolated stripe of QC material, typically ~10 um wide, and
several mm long. A dielectric material is typically deposited in the trenches to guide
injected current into the ridge, then the entire ridge is typically coated with gold to
provide electrical contact and to help remove heat from the ridge when it is producing
light. Light is emitted from the cleaved ends of the waveguide, with an active area
that is typically only a few micrometers in dimension.
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1.3 QCL collimation and its challenge
Quantum cascade lasers (QCLs) are light sources operating based on intersubband
transitions in multiple quantum well structures. Their emissions wavelengths extend
from mid- to far infrared, which make them very capable devices for applications, such
as gas sensing, free space optical communication, and imaging. Most of these fields of
applications require light to be concentrated in a small solid angle in the far field. QCL,
as an edge-emitting semiconductor, however, usually have a large beam divergence,
due to diffraction at their small light-emitting apertures. Their full-width at a
half-maximum (FWHM) divergence angles are in the range of several tens of degree,
i.e., 300 to 600 [6]. Hence, the main engineering question is if one can reduce its
emission divergent angle and offer a narrow-beam radiation performance.
Conventionally, this divergent beam is collimated with lenses or curved mirrors,
which usually requires meticulous optical alignment. Besides this, there are a limited
number of other methods, which include incorporating a micro-machined lens or horn
antenna onto the laser facet [7, 8], and using tapered laser waveguides with
laterally-expanded ends [9, 10]. However, it is not practical to suppress the vertical
divergence by simply growing thick laser active cores; such devices would require
unrealistically high voltages for operation and would have heat dissipation problems.
Recently, Capasso s group has demonstrated an interesting solution to this problem
by patterning the QCL facet with 1D and 2D plasmonic grating antennas [11, 12].
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Conceptually, the source propagation is transformed into a surface wave guiding along
the facet, where the plasmonic grating with optimized periodicity enhances the far-field
propagation and energy concentration along a direction of interest [13-16]. However,
since the QCL surface wave propagates mostly along one direction, controlling the
beam radiation is mainly achieved in one plane (vertical plane). To spread the surface
wave over the entire facet and control the beam in both planes, one can make the facet
aperture size smaller, but this will reduce the throughout power efficiency of the QCL.
There will be then a trade-off between the required divergent angles in vertical and
horizontal planes, and the QCL power output efficiency. It must also be mentioned that
the proposed design methodology is realized for mid-IR operation where the plasmonic
material acts mostly like a metal and if one is interested to achieve an optical
nanoantenna in for example a visible band, the concept cannot be generalized simply by
scaling the size, as the plasmonic material will have different properties in this
spectrum.
A dielectric-patterned nanoantenna will be an alternative for the plasmonic
configuration. Anisotropic dielectric media and photonic crystals operating in their
band-edge have proven unique properties for directive emission [17-23].
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1.4 Objective
The objective of this paper is to present a systematic study of dielectrics rod
arrangements controlling the amplitude and phase of a source distribution and tailoring
the radiation performance of a QCL device. The basic idea is to transform all the
k-vectors into a specific direction. This can be determined by optimizing a 3D periodic
configuration of dielectric elements having different periodicities to engineer an
anisotropic dispersion diagram supporting high-intensity defect mode [17, 18]. The
structure is integrated with a semiconductor QCL source to examine the concept and
offer a directive emission nanoantenna laser device. The advantages are that the
radiation beams can be controlled in both planes and further one can implement the
concept at any optical frequency of interest by scaling the geometry accordingly.
A finite difference time domain (FDTD) technique [24-26] is applied to fully
characterize the performance of QCL, dielectric-patterned nanoantenna, and the
complex structure of QCL integrated with nanoantenna. The physics and fundamental
concepts are comprehensively investigated to achieve a high-performance QCL
nanoantenna component. Near- and far-fields behaviors are manipulated to optimize
the desired design parameters.
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1.5 Outline of thesis
In the second chapter of this thesis paper, the FDTD technique will be discussed
and applied to provide a 3D full wave numerical analysis. A QCL device model will
be built up based on the FDTD technique. The near-field and far-field performance of
the QCL will be obtained.
Then, physics and modeling of the dielectric nanoantenna will be introduced. The
FDTD full wave analysis with periodic boundary conditions will be applied to
characterize the proposed structure comprehensively and determine the dispersion
diagram of this dielectric nanoantenna. Other characteristics like the transmission
coefficient of dielectric pattern under plane wave excitation will also be obtained. The
objective in chapter 3 is to study the characteristics of the dielectric nanoantenna and
optimize it for QCL device.
In the next chapter, the proposed dielectric pattern nanoantenna will be intergrated
with QCL to collimate the QCL radiation. Near field and far field performance of
nanoantenna with QCL source will be analyzed by FDTD technique. High
directive-emission performance will be demonstrated.
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Chapter2 QCL Modeling with FDTD Technique
2.1 FDTD Technique
Finite-difference time-domain (FDTD) is a popular computational
electrodynamics modeling technique. It is considered easy to understand and easy to
implement in software. Since it is a time-domain method, solutions can cover a wide
frequency range with a single simulation run.
The FDTD method belongs in the general class of grid-based differential
time-domain numerical modeling methods. The time-dependent Maxwell's equations
(in partial differential form) are discretized using central-difference approximations to
the space and time partial derivatives. The resulting finite-difference equations are
solved in either software or hardware in a leapfrog manner: the electric field vector
components in a volume of space are solved at a given instant in time; then the
magnetic field vector components in the same spatial volume are solved at the next
instant in time; and the process is repeated over and over again until the desired
transient or steady-state electromagnetic field behavior is fully evolved.
2.1.1 Maxwell s Equation and the Yee Algorithm
When Maxwell's differential equations [1, 2] are examined, it can be seen that the
change in the E-field in time (the time derivative) is dependent on the change in the
H-field across space (the curl). This results in the basic FDTD time-stepping relation
that, at any point in space, the updated value of the E-field in time is dependent on the
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stored value of the E-field and the numerical curl of the local distribution of the
H-field in space [27]. The H-field is time-stepped in a similar manner. At any point in
space, the updated value of the H-field in time is dependent on the stored value of the
H-field and the numerical curl of the local distribution of the E-field in space.
Iterating the E-field and H-field updates results in a marching-in-time process wherein
sampled-data analogs of the continuous electromagnetic waves under consideration
propagate in a numerical grid stored in the computer memory
[1]
[2]
1
1
1
yx zx
y x zy
y xzz
HE HE
t y z
E H HE
t z x
H HEE
t x y
*
*
*
1
1
1
yx zx
y x zy
y xzz
EH EH
t y z
H E EH
t z x
E EHH
t x y
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Fig. 2 Yee ¯ Lattice
This algorithm solves for both electric and magnetic fields in time and space
using the coupled Maxwell ¯s equati on rather than solving for the electric field alone
(or magnetic field alone) with a wave equation . Moreover, the Yee algorithm centers
its E and H components in three dimensional space so that every E component is
surrounded by four circulating H components, and every H component is surrounded
by four circulating E components. Fig. 2 above shows the E and H components
displacement over what ¯s kno wn as Ye
Lattice. One of 6 expressions for Maxwell
Equation is shown. [3]
[3]
1, 1/ 2, 1/ 2 , 1/ 2, 1/ 2 , 1, 1/ 2 , , 1/ 2 , 1/ 2, 1 , 1/ 2, 2
, ,
1/ 2 1/ 2
, 1/ 2, 1/ 2, 1/ 2, 1/ 2
1i j k i j k i j k i j k i j k i j k
i j k
n n n n n nX X Z Z Y Y n
i j k Xi j k
E E H H H HE
t y z
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2.1.2 Perfect Matched Layer
One of the greatest challenges of the FDTD method has been the efficient and
accurate solution of electromagnetic wave interaction problems in unbound regions.
For such problems, an absorbing boundary condition (ABC) must be introduced at the
outer lattice boundary to simulate the extension of the lattice to infinity. Thus, the
boundary condition must suppress spurious reflections of the outgoing numerical
waves. The PML is a layer used to surround the simulations computational domain in
order to simulate infinite wave propagation. It represents an anechoic chamber
providing reflectionless propagation for all impinging waves (any incident angle) over
their full frequency spectrum. Thus, plane waves of arbitrary incidence, polarization,
and frequency are matched at the boundary.
It should be noted that simulations done for this research use the Convolution
Perfectly Matched Layer (CPML) [28]. The application of the CPML is completely
independent of the host medium. Subsequently, when treating more generalized media
such as lossy, inhomogeneous, dispersive, anisotropic, or nonlinear media, the CPML
formulation is unchanged. Furthermore, for such generalized media, the CPML
formulation can require the same, memory than required by previous formulations, yet
with the added capability of effectively absorbing evanescent waves.
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2.2 QCL Modeling with FDTD technique
2.2.1 Geometry
Fig. 3 illustrates the geometry of a QCL device constructed to operate around the
center wavelength . The light is trapped inside an active region and is
guided towards the end, where it can radiate. The active region has a material of
refractive index , effective length of around , and width
of . It is surrounded by a cladding medium made of indium phosphide (InP) with
. This construction, which can be considered as an optical
waveguide, enables total internal reflection inside the active region guiding the wave
through the medium [29]. Further, since the refractive index of the active region is very
close to that of cladding, an almost
TEM mode profile is secured. The
structure has a substrate of material index .
Fig. 3. The geometry of QCL device. Active region is surrounded by cladding medium and generates an almost TEM wave.
Y
14 ¦Ì
22 ¦Ì
38 ¦Ì
56 ¦Ì
35 ¦Ì
15 ¦Ì
2.7 ¦Ì
2.7¦Ì
cladding
active region
7¦Ì
4¦Ì
substrate
X
Z
30.7¦Ì
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2.2.2 Near Field Performance of QCL
To provide a 3D full wave numerical analysis for this configuration, we excite the
system with an infinitesimal dipole located at the center of the active region, far from
the QCL facet, and apply the FDTD to comprehensively characterize the optical
performance of the structure. The side-walls of the cladding region are tapered with
lossy materials to ensure the generation of dominant mode (as this is the case in
practical realization). The results for the E and H modes profiles inside the active region
are depicted in Fig. 4 and 5, where we see strong
and
components, while
other components are relatively small (less than 0.2 of
, respectively). This
illustrates almost a single TEM mode generation. The power transmission is mostly
inside the active region toward the laser facet.
The near-fields on facet and in vertical and horizontal planes are determined in
Fig. 6. As observed, on the laser facet (YZ plane), the fields focus on the aperture of
the active core as a spot. On ZY and YZ plane, the fields diverge away from the facet,
which results in large beam divergence.
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(a)
(b)
Fig. 4 Normalized E and H field profiles inside the active region of the QCL source
along vertical direction (z-axis). The fields are normalized to the maximum of Ez and
Hx components: (a) E, and (b) H
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(a)
(b)
Fig. 5: Normalized E and H field profiles inside the active region of the QCL source
along horizontal direction (x-axis). (a) E, and (b) H
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(a)
(b)
(c)
Fig. 6: Near-field distributions (Total electric power:
) of the QCL
source on (a) the facet (x-z plane), (b) vertical plane (y-z plane), and (c) horizontal plane (x-y plane). Notice to the beam divergence.
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2.2.3 Far Field Performance of QCL
To better understand the radiation performance of the QCL device, the field
patterns in vertical and horizontal planes are obtained in Fig. 7. Large divergence of
about
and
in those planes are evaluated. The directivity is around
. As mentioned, for many optical applications one requires to concentrate the
beam in a small spot and the main engineering question is how a less-divergent QCL
beam can be manipulated.
Fig. 7 Radiation performance of the QCL source illustrating wide beamwidth and poor
directivity.
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2.3 Conclusion
A QCL device operating around the center wavelength
is modeled
numerically using FDTD technique. This device is properly tapered with lossy
materials to ensure the generation of dominant mode (TEM mode). The near field
distribution indicates that, inside the QCL device, the power will transmit along the
active region, acting like an optical waveguide, while diverge outside the facet, which
results in a divergent beam.
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Chapter 3 Physics and Modeling of Dielectric Nanoantenna
3.1 Photonic Crystals
Photonic crystals (PCs) are periodic arrangements of dielectric elements. They can
stop the electromagnetic waves propagation in some specific frequency spectrum,
providing bandgap phenomenon [30, 31]. There has been significant amount of
research for integrating bandgap PCs into the optical devices. The dielectrics material
has the advantage of low loss and less dependence on frequency, compared to
plasmonic materials in THz, due to its pure dielectric property.
3.2 PCs Dielectric Pattern and its Dispersion Diagram
This thesis presents a unique application of PCs for making directive QCL
emission. Conceptually, to obtain a directive emission, one needs to translate all the
k-vectors of the source spherical wave into a desired direction achieving near-field
plane-wave profile with Fourier-transform far-field narrow-beam radiation.
Transformation optics is required to translate a high-intensity local field into an array of
radiators distributed along an aperture. To achieve this, one can think of realizing a
cavity defect mode made of coupled impedance surfaces where a plane wave can be
tunneled through the structure through the supported surface waves.
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Fig. 8 3D PC metamaterial pattern with different periodicities in transverse and
vertical directions
To illustrate this concept, let us consider a periodic dielectric pattern with different
periodicities in transverse and propagating directions, as depicted in Fig. 8. The
structure is made of dielectric rods having material index
with
square cross-sections of size . The periodicities along x and z
directions are the same of values , where along the propagating
direction, y, a different periodicity of
is designed. Utilizing
different-periodic configuration allows one to realize an anisotropic performance with
engineered dispersion diagram supporting proper band-edge defect mode with
k-vectors along the surface of PC and stop-band for other directions (they will be
decaying modes).
4 ¦Ì
4.27 ¦Ì
4.27 ¦Ì
n=3.26
X
Y
Z
1.07 ¦Ì
29
Fig. 9 shows the irreducible brillouin zone of the crystal lattice. The normalized
values of the vertices in (
format are:
(0, 0, 0), S (0, 1, 0), T (0, 1 ,1), X
(0, 0, 1), M (1, 0, 1), R (1, 1, 1). They are normalized to the vector .
The FDTD full wave analysis with periodic boundary conditions is applied to
characterize the proposed structure and determine the dispersion diagram, as presented
in Fig. 9. A band gap behavior from
to
is obtained. At the band-edge (
only
mode is supported. The 2D dispersion diagram in the
plane can provide better information in this regard, as is plotted in Fig. 10. As obtained,
only the modes with
or
can exist, while propagation along the y direction is
forbidden.
Fig. 9 Dispersion Diagram for 3D PCs Dielectric pattern
S T X M R T0
10
20
30
40
Freq
uenc
y (T
Hz)
Band-Edge
ky
kz
kx P Q
T X
S
R
M
30
Fig. 10 2D kx (or kz)-ky vector plane dispersion diagram. Upper band edge is around .
3.3 Transmission Characteristics
To better highlight the physics, we also investigate the transmission coefficient for
a plane wave propagating along the y and z directions through the crystal. The result is
shown in Fig. 11. A pass-band through the gap region for the wave propagating along
the z (transverse to crystal) at
is obtained. This is the defect mode
determined through the dispersion diagram analysis. There is an about 4% error that
can be attributed to the finite size structure which is considered for transmission
coefficient analysis along the wave propagation.
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Fig. 11 Transmission coefficients for normal incident planewaves propagating along y and z directions inside the PC.
3.4 Cavity Defect Modes
Fig. 12 demonstrates a cavity structure made by low-dielectric material slab (i.e.
MgF2, n=1.18) [33, 34], sandwiched in the middle of two periodic patterns surfaces
(slices of our designed configuration) operating at their band-edge. The cavity can
tunnel the defect mode to the other side through the coupling between band-edge
surface modes supported by the periodic layers. This can happen only for a normal
incident wave, while for a slightly tilted incident wave the k-vector will positioned
inside the bandgap and no transmission occurs.
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Fig. 12 A cavity structure constructed from a low dielectric material sandwiched between two PC layers
In order to optimize the coupling between band-edge surface and the cavity,
transmission coefficient for a plane wave illuminating the structure with different
thickness of MgF2 space are analyzed, as shown in Fig 13. Bandgap performance are
due to the photonic crystal cover at both side, while passband inside are determined
by the thickness of MgF2 space.
For thickness , , , and , the wave can tunnel
through the cavity at 32.2THz, 31.2THz, 30.2 THz, and 28.8THz . The
optimized transmission is around frequency
where the slab thickness is
at
(dielectric wavelength).
8 ¦Ì
35.2 ¦Ì
35.2 ¦Ì PC operating at band-edge
33
Fig. 13: Normal incident transmission performance of a cavity structure constructed from a low dielectric material sandwiched between two PC layers. The defect mode is
around the resonance of the cavity and the band-edge of the PC patterns.
3.4 Dielectric Pattern Nanoantenna with Waveguide Source
Now, let us go back to our original question whether one can translate a point
source radiation into a uniform near-field distribution along a large aperture, enabling
far-field directive emission, by engineering a novel dielectric pattern nanoantenna
(with flat surface).
To address this, we consider a small waveguide source aperture with less than
size opened inside a finite size PEC ( , which is coated by the low-dielectric
material of n=1.18 and thickness of 4
and is covered by one layer of our designed
periodic configuration (about 8 periods in transverse directions). The configuration is
depicted in Fig. 14(a). Equivalent to this is the cavity model presented in Fig. 12. Hence,
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one can expect the point source can be transformed into a relatively uniform field along
the
aperture size.
(a) (b)
Fig. 14 (a) Dielectric pattern nanoantenna and (b) its aperture near-field,
The near-field distributions are shown in Fig. 14(b), validating our expectations.
The field is trapped inside the cavity and can radiate only in the normal direction. The
near field distribution on the facet excites the dielectric pattern and forms an array
antenna, which will result in high-directive emission performance.
The radiation pattern has a directivity of
with vertical and horizontal planes
beamwidth of about
and , respectively, Fig. 15. Note that,
compared to the cavity design where the PC is considered to be periodic in x-z plane,
the operating frequency in this case is slightly shifted and optimized to be at
4 ¦Ì
Backed PEC
35
, in order to compensate for the diffractions from the edges of finite size
structure and consequently reduce the sidelobes.
Fig 15 The radiation characteristic. The beamwidth of about
and
are achieved. Uniform aperture field provides directive radiation.
The radiation patterns at different frequencies are shown in Fig 16. The gain
decreased significantly when operated around 29.6THz, since it locates inside the
band gap of the PCs cover. The gain of side lobes increases and the beamwidth gets
narrower as the operating frequency increases from 30THz to 31THz. The optimized
operating frequency is , which is a trade-off between side-lobe and
beamwidth.
Following a similar optimization process, the designed dielectric pattern nanoantenna
will be integrated in next section to demonstrate a QCL directive emission antenna.
36
(a)
(b)
Fig. 16 The radiation patterns at different frequency: (a) Vertical plane, and (b)
Horizontal plane
37
Chapter 4 Nanoantenna Engineering QCL Radiation
4.1 Dielectric Nanoantenna Integrated with QCL Device
As mentioned earlier, a quantum cascade laser (QCL) device source is a key
component in optics with many potential applications in emerging areas of photonics.
However, one of the major desires (and challenges) is to successfully concentrate its
radiation beam in small spots. This can enable nanoscale imaging and engineered
molecular interactions applications, just to name a few.
The proposed dielectric pattern nanoantenna is an excellent candidate to collimate
the QCL radiation. To accomplish this, the facet of the QCL is coated with a plasmonic
layer (mostly acts like a conductor in this spectrum) and then an aperture with the same
dimensions as the effective region of active layer ( ) is opened through it.
This will ensure maximum efficiency throughout power for the system. The plasmonic
layer is coated by 4
thick slab of low-dielectric index n=1.18 medium (a spacer),
and then the whole structure is covered by our dielectric pattern nanoantenna. Fig. 17
depicts the configuration. The dielectric pattern photonic crystal supports the
band-edge modes with k-vectors propagating along the surface. The small-aperture
source excitation (with almost spherical wave radiation) operates at the defect-mode of
the cavity and hence all the k-vectors are trapped inside the medium with normal
radiation towards the outside. One can envision the engineered dielectric pattern and its
back plasmonic layer, as a cavity constructed from a double-thickness slab medium
sandwiched between two PC layers, where an input plane-wave can be transformed to
38
output port through the cavity defect mode and coupled surface-waves supported by the
PC layers. Hence, a localized spherical wave around the middle of the cavity can
provide a near plane-wave phase-front on the structure aperture (as explained earlier),
enabling far-field directive emission.
Fig. 17: QCL device integrated with dielectric pattern nanoantenna.
39
2 Near Field and Far Field Performance of Dielectric Nanoantenna
Figs. 18 and 19 illustrate the FDTD analysis of the QCL aperture near-field and
radiation pattern characteristics. As observed, the near-field is distributed along the
large size antenna aperture with tapered behavior around the edges, providing an
array-type configuration with directive emission of
gain, and vertical and
horizontal narrow beamwidths of about
and , respectively. This is
an about 2.5-times improvement in the vertical and 4.5-times improvement in the
horizontal planes of the beamwidth performance of the QCL itself. The operating
frequency is optimized at
to provide the best radiation performance.
The surface wave decays while propagating along the vertical direction toward the
bottom edge of facet, which results in a maximum directivity at the direction
, roughly 5 degrees down from the y-axis.
The near-field performances in vertical and horizontal planes illustrate almost
successful beam collimation of the QCL emission. Note that one needs to control the
locations of the dielectric rods around the source aperture to ensure the best array
elements arrangement in this region. The radiation efficacy of the QCL nanoantenna is
around 80% compared to un-patterned case. The power efficiency is obtained by
dividing the total radiated power of nanoantenna with that of unpatterned QCL.
40
(a)
(b)
(c) Fig. 18: Near-field distributions of the QCL integrated with dielectric nanoantenna on
(a) the facet (x-z plane), (b) vertical plane (y-z plane), and (c) horizontal plane (x-y plane). As observed, compared to Figs. 3, the fields are more concentrated
(less-divergent) away from the device.
41
Fig. 19: Radiation performance of QCL nanoantenna demonstrating directive emission ( ) with narrow beamwidth of
and . The beam is tilted down by
in vertical plane due to the geometry asymmetry in this direction. Radiation pattern has been plotted in pattern coordinate system.
The concept of the proposed dielectric nanoantenna is very general and can be
applied at any optical frequency of interest, scaling the geometry accordingly
(obviously the used dielectric materials should have the similar properties). This can
enable advanced point-to-point photonic communications. Although here a about 20
directivity is achieved, one can expect to enhance further the performance in both
vertical and horizontal planes by sophisticated tailoring the dielectric pattern
nanoantenna arrangement (especially around the source), extending the antenna
aperture size, and cascading more PC patterns in front of the device. Plasmonic coating
can also be tailored around the antenna boundary to reduce the edge diffractions.
42
The good news about QCL device is the existence of a large size facet aperture,
where one can pattern it properly to enable very high directive-emission performance.
This is clearly observed from Fig. 11(a), where the near-field is distributed and tapered
along the facet very successfully.
4.3 Larger size facet aperture
The dimension of a practical QCL device can be tens of wavelengths in both
horizontal and vertical direction. If near-field can be distributed along a larger size
antenna aperture with tapered behavior around the edges, we can obtain a higher
directive-emission performance.
Four more periods of PCs are added to the side and bottom edge of the facet, (2 at
each side). As we expected, the near field can be distributed all along the facet, as
shown in Fig. 20, providing an array-type configuration with directive emission of
gain, and vertical and horizontal narrow beamwidthes of about
and
, respectively, shown in Fig 21. The beamwidth
is reduced significantly,
compared to the previous design, because the near field can distributed futher when
the size of the QCL increase in vertical direction. However, the beawidth
keeps
the same level, due to the limitation of size in horizontal direction.
Also, the maximum directivity is at the direction , roughly 10
degrees down from y-axis (compared to 5 degrees shift from the previous design ),
43
which indicate that field distribution pattern on the facet plays a more important role
on the radiation.
(a)
(b)
44
(c)
Fig. 20: Near-field distributions of the QCL integrated with larger dielectric nanoantenna on (a) the facet (x-z plane), (b) vertical plane (y-z plane), and (c)
horizontal plane (x-y plane). As observed, compared to Figs. 3, the fields are more concentrated (less-divergent) away from the device.
Fig. 21: Radiation performance of larger QCL nanoantenna demonstrating directive emission ( ) with narrow beamwidth of
and . The beam is tilted down by
in vertical plane due to the geometry asymmetry in this direction. Radiation pattern has been plotted in pattern coordinate system.
45
4.4 Optimization on Operating Frequency
The key point of this design is to make the nanoantenna operate exactly at the
band edge the dielectric pattern. However, the dispersion diagram and transmission
characteristics of the crystal may change due to edge effect from finite size,
alignments, and the fact that this cover is consist of one period of the crystal. These
factors can result in a frequency shift on band edge and the radiation performance of
QCL at given frequency (e.g.
). In this section, QCL devices excited
by sources with different frequencies are analyzed. The radiation patterns in the
vertical and horizontal planes are shown in Fig. 22 (a) and (b).
As observed, the best operating frequency is , providing directive
emission of
gain, and vertical and horizontal narrow beamwidths of about
and , respectively.
At , which is inside the bandgap of dielectric pattern, the radiation
is blocked. The gain is only 10dB, with wide beamwidth
and ,
respectively.
At , QCL is operation around the band edge. The gain is around
19.4dB,
and
. Hence, operating at the frequency band from
to
can be a good choice for high directive emission
performance.
46
At , the operating frequency has moved out of
the band edge area. Many other modes other than the cavity defect mode are allowed
in the PCs cover. Hence, larger side lobes can be observed .The nanoantenna can not
radiate properly as expected.
(a)
(b) Fig.22 Radiation performance of QCL nanoantenna at different operating frequency:
(a) Vertical plane pattern, and (b) Horizontal plane pattern
47
4.5 Alignment
Conventionally, the divergent beam can be collimated with lenses or curved
mirrors, which usually requires meticulous optical alignment. So, here comes the
interesting question: How will the alignment affect the performance of QCL
integrated with dielectric pattern?
Ideally, the cross of horizontal and vertical rods (black point) should overlap with
the center of the active region, as shown in Fig.23 (a). However, in fabrication
processes, the locations of rods may shift by mistake or deviation. Fig.23 (b)~(d)
show a series of shifting rods with the active region of QCL: Horizontal shift with
Lh=1 ¦Ì, vertical shift with Lv=1 ¦Ì, and on both with Lh=1 ¦Ì and Lv=1 ¦Ì.
Fig. 24 shows the radiation performance of the QCL nanoantenna with different
alignment. As observed, the 1 ¦Ì shift on horizontal or vertical direction can hardly
changes the highly directive emission and narrow beamwidth characteristics.
Compared to lenses or curved mirrors, the far radiation of this nanoantenna is more
dependent on the secondary radiation from the dielectric pattern, rather than the
radiation of the QCL source. This is a great advantage, which provides more
flexibility on the design and fabrication process of dielectric pattern .
48
Lv=1 ¦Ì
Lh=1 ¦Ì
Lv=1 ¦Ì
Lh=1 ¦Ì
(a)
(b)
(c)
(d) Fig. 23 Alignment of dielectric pattern versus QCL device: (a) Original design, the center of active region overlap with the cross of horizontal and vertical rods (black
point), (b) Horizontal shift with Lh=1 ¦Ì, (c) Vertical shift with Lv=1 ¦Ì , and (d) Shift on both sides with Lh=1 ¦Ì, Lv=1 ¦Ì.
49
(a)
(b) Fig.24 Radiation performance of QCL nanoantenna with different alignment: (a)
Vertical plane pattern, and (b) Horizontal plane pattern
50
Chapter 5 Thesis Conclusion
In this paper, a new approach for QCL nanoantenna directive emission with the use of
dielectric patterns is addressed. A periodic dielectric configuration with optimized and
different periodicities in transverse and propagating directions is realized to engineer
the dispersion diagram and manipulate the performance of source radiation. The
nanoantenna dielectric pattern operates at the band-edge and can transform a point
source radiation into a distributed array of radiators along a large size aperture,
enabling directive emission characteristic. The finite difference time domain (FDTD)
technique with periodic boundary conditions is applied to characterize the performance
of complex periodic configurations, successfully obtaining the concept and physical
parameters, and demonstrating novel designs.
The nanoantenna pattern is integrated with the QCL source device. It is
demonstrated that the source radiation is transformed into the tapered array elements
distribution along the antenna aperture realizing efficient far-field radiation with
directivity of
and vertical and horizontal narrow beamwidthes of about 14o and
12o, respectively. This is about 2.5 and 4.5 times improvements in the vertical and
horizontal planes of the radiation characteristics of the source itself, collimating the
beam very successfully.
The obtained QCL nanoantenna with narrow beam radiation can enable
long-range photonics communication. Further, its ability to concentrate the beam in
51
small and nanoscale spots can feature other potential applications in nanophotonics,
such as, nanoimaging and engineered molecular-quantum interactions, among many
others.
52
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