1

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

2

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

3

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

4

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.

5

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.

6

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

7

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

8

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

9

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

10

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

11

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.

12

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].

13

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].

14

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.

15

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.

16

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

17

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

18

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

19

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.

20

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

21

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.

22

(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

23

(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

24

(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.

25

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.

26

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.

27

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.

28

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.

31

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.

32

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,

34

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

References

[1] H. Mosallaei and Y. Rahmat-Samii, ° Peri odi c ban-gap and effective dielectric

materials in electromagnetics: Characterization and applications in nanocavities and

waveguides, ±IEEE Trans. Antennas Propagat., vol. 51, no. 3, pp. 549-563, Mar.

2003.

[2] E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. Crozier, and F. Capasso, ° Pl as moni c

laser antennas and related devices, ±IEEE Journal of Selected Topics in Quantum

Electronics, vol. 14, no. 6, Nov. 2008.

[3] F. Jerome, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A.Y. Cho,

Quantum Cascade Laser ± Science 264 (5158): 553 556.

[4] A.W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno,

High-power and high-temperature THz quantum-cascade lasers based on

lens-coupled metal metal waveguides ±,Optics Letters, Vol. 32, No. 19 /

October 1, 2007

[5] L. N hle, J. Semmel, W. Kaiser, S. H fling, and A. Forchel,

±Tapered quant u m

cascade lasers ±,Applied Physics Letter, Vol. 91, 181122, 2007

[6] D. Hofstetter, J. Faist, M. Beck, and U. Oesterle, ° Surfac-emittiing 10.1 ¦ Ì m

quantum-cascade distributed feedback lasers, ±Appl.Phys.Lett., 5, 3769-3771

(1999).

[7] A. W M. Lee, et al., ° Hi g-power and high-temperature THz quantum-cascade

lasers based on lens-coupled metal-metal waveguides, ±Opt. Lett., vol. 32, pp.

2840-2842, 2007.

53

[8] M. I. Amanti, M. Fischer, C. Walther, G. Scalari, and J. Faist, ° Hor n ant ennas f or

terahertz quantum cascade lasers, ±Electron. Lett., vol. 43, pp. 573-574, 2007.

[9] M. Troccoli, C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho, ° Mi -infrared

quantum cascade laser amplifier for high power single-mode emission and

improved beam quality, ±Appl. Phys. Lett., vol. 80, pp. 4103-4105, 2002.

[10] L. Nahle, J. Semmel, W. Kaiser, S. Hofling, and A. Forchel, ° Tapered quant u m

cascade lasers, ±Appl. Phys. Lett., vol. 91, p. 181122, 2007.

[11] N. Yu, J. Fan, Q. Wang, C. Pflugl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan,

and F. Capasso, ° S mal-divergence semiconductor lasers by plasmonic collimation, ±

Nature Photonics, vol. 2, Sept. 2008.

[12] N. Yu, R. Blanchard, J. Fan, Q. J. Wang, C. P fl¨ ¹ l, L. Diehl, T. Edamura, M.

Yamanishi, H. Kan, and F. Capasso1, ° Quant u m cascade l asers wit h i nt egrat ed

plasmonic antenna-array collimators, ±OPTICS EXPRESS, vol. 16, no. 24, Nov.

2008.

[13] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolf, ° Extraor di nar y

optical transmission through sub-wavelength hole arrays, ±Nature, vol. 391, Feb.

1998.

[14] H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J.

Garcia-Vidal, and T. W. Ebbesen, ° Bea mi ng li ght fr o m a sub wavel engt h aperure, ±

Science, vol. 297, Aug. 2002.

54

[15] S. S. Akarca-Biyikli, I. Bulu, and E. Ozbay, ° Enhanced trans missi on of mi cr o wave

radiation in one-dimensional metallic gratings with sub wavelength aperture, ±

Applied Physics Letters, vol. 85, no. 7, Aug. 2004.

[16] Z. Guo-Ting, L. Juan, H. Chuan-Fei, S. Fang, S. Xiao-Xing, ° Bea mi ng of li ght

from metallic surface plasmon polarities nanostructure, ±Proc. of SPIE , vol. 6624.

[17] S. Enoch, B. S. Gralak, and G. Tayeb, ° Radi ati ng di pol es i n woodpil e and si mpl e

cubic structure, ±Photonic bandgap Materials and Device, vol.4655, 2002.

[18] M. Thevenot, M. S. Denis, A. Reineix, B. Jecko, ° Desi gn of a ne w phot oni c cover

to increase antenna directivity, ±Microwave and Optical Technology Lett., vol. 22,

no. 2, July 1999.

[19] Y. J. Lee, J. Yeo, R. Mittra, and W. S. Park, ° Appli cati on of el ectr o magneti c

Bandgap (EBG) superstrutes with controllable defects for a class of patch antenna

as spatial angular filters, ±IEEE Transaction of Antenna and Propagation, vol. 53,

no. 1. Jan. 2005.

[20] A. R. Weily, L. Horvath, K. P. Esselle, B. C. Sanders, and T. S. Bird, ° A pl anar

resonator antenna based on a woodpile EBG material, ±IEEE Transaction of

Antenna and Propagation, vol. 53, no 1. Jan. 2005.

[21] H. Caglayan, I. Bulu, and E. Ozbay, Highly directional enhanced radiation from

sources embedded inside two and three-dimensional photonic crystals, ±Proc. Of

SPIE, vol. 5733.

55

[22] l. C. Ma, J. Qu, R. Mittra, and N. Farahat, ° Underst andi ng physi cal mechani s m of

performance enhancement of antenna via the use of EBG superstrate, ±IEEE AP-S

International Syposiume, San Diego, CA, July 5-12, 2008.

[23] S. Yarga, K. Sertel, J. L.Volakis, ° Degenerat e band edge cr yst als f or directi ve

antennas, ±IEEE Transactions on Antenna and Propagation, vol. 56, no. 1, Jan.

2008.

[24] A. Taflove and S. C. Hagness, Computational Electrodynamics: The

Finite-Difference Time-Domain Method, Artech House, MA, 3 edition, 2005.

[25] H. Mosallaei, ° F DT-PLRC technique for modeling of anisotropic-dispersive

media and metamaterial devices, ±IEEE Trans. Electromagn. Compat., vol. 49, no.

3, pp. 649-660, Aug. 2007.

[26] H. Mosallaei and Y. Rahmat-Samii, ° Br oadband charact eri zati on of co plex

periodic EBG structures: An FDTD/Prony technique based on the split-field

approach, ± Electromag. J., 23(2):135-151, 2003.

[27] Yee.K.S, ° Nu meri cal sol uti on of i niti al boundar y val ue pr obl e ms i nvol vi ng

Maxwell ¯s equati ons i n i sotr opi c medi c

IEEE Trans.Antenna Propagat.,Vol

14,1966,pp. 302-307.

[28] Roden. J.A, and S.D.Gedney, ° Convol uti onal P ML : An effi ci ent F DTD

implementation of the CFS-PML for arbitrary media ±,Microwave Optical Tech.

Lett., Vol 27,2000,pp.334-339

[29] H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, Inc., 1984.

56

[30] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals, Princeton,

NJ: Princeton Univ. Press, 1995.

[31] E. Yablonovitch, °I nhi bit ed spont aneous e missi on i n soli -state physics and

electronics, ±Phys. Rev. Lett., vol. 58, no. 20, pp. 2059-2062, May 1987.

[32] F. Yang, J. Chen, R. Qiang, Elsherbeni, A. FDTD Analysis of Periodic Structures

at Arbitrary Incidence Angles: A Simple and Efficient Implementation of the

Periodic Boundary Conditions , APS, 2006

[33] J.-Q. Xi, J. K. Kim, E. F. Schubert, D. Ye, T.-M. Lu, and S. Lin, ° Ver y

low-refractive-index optical thin films consisting of an array of SiO2 nanorods ±,

OPTICS LETTERS, vol. 31, no. 5, Mar. 2006.

[34] E. F. Schubert, J. K. Kim, and J.-Q. Xi, ° Lo -refractive-index materials: A new

class of optical thin-film materials, ±Phys. Stat. Sol. B, 244, no. 8, pp. 3002 3008,

2007.