Thesis Optical and acoustical properties of band gap materials.pdf

8/14/2019 Thesis Optical and acoustical properties of band gap materials.pdf

1/144

______________________________________________________________

97 06 20


2/144


3/144


4/144


5/144

i

(Lamb

wave)


6/144

ii

Optical and acoustical properties of band gap materials.

ABSTRACT

This thesis deals with the propagation properties of the electromagnetic and the

elastic waves in artificial crystals which are called photonic and phononic crystals,

respectively.

In the first part, the three dimensional photonic crystals formed by microspheres

with Ag nano particles coating are studied. The light can be reflected by crystals due

to the partial band gap of the photonic crystals arranged in face-centered cubic

structure. The results reveal that the reflectance is wavelength dependant and can be

enhanced by suitable Ag coating on the microspheres.

In the second part, the bulk elastic waves propagating in two-dimensional and

three-dimensional phononic crystals are studied experimentally and theoretically. The

two-dimensional phononic crystals are constructed using steel cylinders. The

cylinders are arranged periodically in water. The three-dimensional phononic crystal

consists of close-packed periodic arrays of spherical beads of steel embedded in an

epoxy matrix. The forbidden band gap can be observed in the experimentalmeasurement of the transmission spectra. The results agree with the theoretical

calculations. The defect modes in two-dimensional phononic crystals are also studied.

The results show that the bulk elastic waves could be well controlled and confined by

the phononic crystal structures.

The propagation of acoustic waves in a square-lattice phononic crystal slab

consisting of a single layer of spherical steel beads in a solid epoxy matrix is studied

experimentally and theoretically. The transmission and the field image of acoustic

wave are investigated. The transmission attenuation caused by absorption and bandgap effects is measured as a function of frequency and propagation distance. We also

demonstrate experimentally that the acoustic waves are well confined and propagate

inside a line-defect waveguide.


7/144

iii

Mr. BAHRAM

DJAFARI-ROUHANI

Abdelkrim Choujaa

Abdelkrim Khelif Vincent Laude Choujaa

Khelif

Laude


8/144

iv

Hichem Karim


9/144

v

Contents

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

1 Introduction to photonic and phononic crystals 1

2 Introduction to photonic and phononic crystals 4

2.1 History and development of photonic crystals . . . . . . . . . . . . . . . 4

2.1.1 Photonic crystals in Nature . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Photonic crystal fiber . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.4 Colloidal microsphere base photonic crystal . . . . . . . . . . . . 15

2.2 History and development of phononic crystals . . . . . . . . . . . . . . . 17

2.3 Elastic wave propagation in materials . . . . . . . . . . . . . . . . . . . 26

2.4 Theoretical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.1 Finite-Difference Time-Domain method . . . . . . . . . . . . . . 29

2.4.2 Finite element method . . . . . . . . . . . . . . . . . . . . . . . 32

3 Optical properties of metallodielectric opals 34

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Plane Wave Expansion (PWE) . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Fabrication of opals and experimental samples . . . . . . . . . . . . . . . 37


10/144

3.3.1 Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Bulk waves in two dimensional water-steel and three dimentional epoxy-steel

phononic crystals 49

4.1 Introduction to this chapter . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Two-dimentional water-steel phononic crystals . . . . . . . . . . . . . . 50

4.2.1 The geometric and elastic properties of the structure . . . . . . . 504.2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.3 Triangular lattice structure . . . . . . . . . . . . . . . . . . . . . 55

4.2.4 Honeycomb lattice structure . . . . . . . . . . . . . . . . . . . . 63

4.2.5 Discussion of complete band gap in triangular and honeycomb

lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 Line defect waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4 Three-dimensional epoxy-steel phononic crystals . . . . . . . . . . . . . 75

4.4.1 Structure and experimental setup . . . . . . . . . . . . . . . . . . 75

4.4.2 Complete band gaps . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Square lattice steel-epoxy phononic crystal slab 83

5.1 Introduction to this chapter . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Introduction to the experiment method . . . . . . . . . . . . . . . . . . . 845.2.1 Fabrication and properties of the structures . . . . . . . . . . . . 84

5.2.2 Optical characterization setup . . . . . . . . . . . . . . . . . . . 87

5.2.3 Principle of laser interferometer . . . . . . . . . . . . . . . . . . 89

5.3 Complete band gap characterization . . . . . . . . . . . . . . . . . . . . 90

5.4 Attenuation behavior versus propagation distance . . . . . . . . . . . . . 93

vi


11/144

5.5 Line defect waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6 Unusual refraction effect . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6 General Conclusion 106

Bibliography 109

vii


12/144

List of Figures

2.1 From left to right diagram, PCs are in one, two, and three dimensions. The

red and yellow colors indicate the different refraction index. (http://ab-

initio.mit.edu/photons/) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Somewhere in a beam of sunlight in the Amazon basin, the three-inch

wings of a morpho butterfly glow with deep blue iridescence. Then the

morpho shifts its wings and, as if a switch flipped, shows its other self,

dull brown. [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 High-resolution SEM images showing the fine structure of the scales with

rounded ends: (a) blue (BL) male, blue region; (b) blue (BL) male, blue-

violet region; (c) brown (BR) male.[2] . . . . . . . . . . . . . . . . . . . 6

2.4 (a)Sea mouse, (b) its iridescent threads (c) SEM cross-section of a spine.[2]

7

2.5 The opal is the example of photonic band structures in mineral worlds.[2] 8

2.6 Cross-sectional micrograph of the 2D PC. The holes were formed in a

1.85m thick Si core layer. TheSiO2layer was2m thick. [3] . . . . . 9

2.7 (a) Scanning electron micrograph top view of 38 period triple-line-defect

linear waveguide. The defect hole diameter is 0.8 times of lattice constant

(352nm). The rectangular boxes indicate the interfaces between the ridgeand the photonic band gap waveguide. (b) Scanning electron micrograph

image of the Y-splitter sample, which consists of a 120o Y- splitter and

two60o bends (indicated by the red circles). (c) Infrared camera image of

the two Y-splitter outputs at = 1650nm. They are both Gaussian-like

and equally bright, indicating a near 50/50 splitting ratio.[4] . . . . . . . 11

viii


13/144

2.8 Five photonic crystal cavities coupled together lithographically by ar-

rangement in the same slab.[5] . . . . . . . . . . . . . . . . . . . . . . . 12

2.9 Scanning electron microscope image of the end of a photonic crystal fiber,

showing the central core where a hole has been omitted. [6] . . . . . . . 13

2.10 Scanning electron micrograph image of the inner cladding and core of the

airsilica microstructure fiber. [7] . . . . . . . . . . . . . . . . . . . . . 142.11 Optical spectrum of the continuum generated in a 75-cm section of mi-

crostructure fiber. The dashed curve shows the spectrum of the initial

100-fs pulse. [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.12 SEM image of a6 10m2 region of the surface of a sample made from415 nm spheres. [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.13 SEM images of the photonic crystals after different heating durations.

(A) More than 15 layers of microspheres were orderly packed. (B) The

thickness of the opals is about 17m. (C) The thickness of the opals is

about78m. (D) The thickness of the opals would be more than200m.

(E) This shows that the surface of the photonic crystals in (D) is well

packed in the FCC structure. [9] . . . . . . . . . . . . . . . . . . . . . . 17

2.14 Band structure for a period array of aluminum cylinders in nickel back-

ground. The inset shows the unit cell [10]. . . . . . . . . . . . . . . . . 20

2.15 Dark and white represent positive and negative vibrations, respectively.

(a) Experimentally observed patterns of liquid surface waves. Black areas

in the middle stands for the slab of copper cylinders. (b) Simulated results.Parameters used in simulations are the same as in experiment. Red dots

denote copper cylinders. [11] . . . . . . . . . . . . . . . . . . . . . . . . 25

2.16 Radiation amplitude field distribution over the phononic crystal: (a)by

experiment and (b) by simulation. [12] . . . . . . . . . . . . . . . . . . . 27

2.17 The diagram of the computational space . . . . . . . . . . . . . . . . . . 31

ix


14/144

3.1 SEM picture of pure PMMA microspheres. . . . . . . . . . . . . . . . . 38

3.2 SEM picture of PMMA microspheres with10wt%Ag nanoparticles. . . . 39

3.3 SEM picture of PMMA microspheres with20wt%Ag nanoparticles. . . . 39

3.4 The diagram of the first Brillouin zoom of FCC structure. . . . . . . . . . 40

3.5 Photonic band structure of the PMMA microspheres in fcc lattice. . . . . 41

3.6 The setup of supercontinuum source. . . . . . . . . . . . . . . . . . . . . 41

3.7 The spectrum of light is indicated by red line. Blue line indicates the dark

noise. The unit of intensity axis is arbitrarily unit. . . . . . . . . . . . . . 42

3.8 The experiment setup of (a)10o to45o (b) normal incident opal reflective

spectra measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.9 The reflection spectrum of sample with incident angle0o. . . . . . . . . 44





3.14 (a)Reflection peak wavelength and (b) the coefficient of the microspheres

for different incident angles. . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1 Two-dimensional phononic crystal used in the experiments. . . . . . . . . 51

4.2 The definition of the lattice directions. K and M are the two highest

symmetric directions in (a) the triangular and (b) the honeycomb structure. 52

4.3 Gap maps for (a) the triangular lattice and (b) the honeycomb lattice as a

function of the cylinder radius to lattice constant ratio. . . . . . . . . . . 53

4.4 The experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 Signal intensity in the time domain with (blue dash line) and without (red

solid line) a phononic crystal structure. . . . . . . . . . . . . . . . . . . . 54

4.6 Reference spectra of two tranducers couples with central frequency is

450kHz (blue line) and 900kHz (red line). . . . . . . . . . . . . . . . . . 55

x


15/144

4.7 Band structure of the infinite triangular-lattice phononic crystal composed

of steel cylinders in water, plotted along the -K-M-path of the first

irreducible Brillouin zone. . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.8 (a) Real part of the pressure field in water for the eigen mode labeled

A in Fig.4.7. (b) and (c) are the real part of x and y displacement field

respectively in the steel cylinders for the eigen mode labeled B in Fig.4.7. 57

4.9 TheM direction transmssion spectrum of triangular lattice with 5, 6 and

7 layers. The insert at the doen left is the definition of layer. . . . . . . . 58

4.10 (a) Experimental (thick red solid line) and theoretical (thin blue solid line)

transmission through a triangular-lattice phononic crystal of steel cylin-

ders in water, along the M direction. (b) Band structure of the infinite

phononic crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.11 Real part of the pressure field in water for the eigenmode labeled C in

the band structure. The arrow shows the direction of incidence of waves

launched by the transducer. . . . . . . . . . . . . . . . . . . . . . . . . 60

4.12 TheK direction transmssion spectrum of triangular lattice with 5, 6 and

7 layers. The insert at the doen left is the definition of layer. . . . . . . . 61


transmission through a triangular-lattice phononic crystal of steel cylin-

ders in water, along the K direction. (b) Band structure of the infinite


4.14 (c) Real part of the pressure field in water for the eigenmode labeled D inthe band structure. The arrow shows the direction of incidence of waves

launched by the transducer. This mode belongs to a deaf band and is not

excited by a plane-wave transducer. . . . . . . . . . . . . . . . . . . . . 62

4.15 K direction transmission spectrum for the honeycomb lattice with 5, 6

and 7 layers. The inset at the down left is the definition of the layer. . . . 64

xi


16/144


transmission through a honeycomb-lattice phononic crystal of steel cylin-

ders in water, along the K direction. (b) Band structure of the infinite


4.17 Real part of the pressure field in water for the eigenmode labeled E in




4.18 The M direction transmssion spectrum of honeycomb lattice with 5, 6

and 7 layers. The insert at the doen left is the definition of layer. . . . . . 67


transmission through a honeycomb-lattice phononic crystal of steel cylin-

ders in water, along the M direction. (b) Band structure of the infinite


4.20 Real part of the pressure field in water for the eigenmode labeled F in




4.21 The (a) 1 line, (b) 2 lines and (c) 3 lines defect waveguides in triangular

lattice. The red dash circles indicate the position of steel cylinders that

was removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.22 The experimental transmission spectra (solid line) of 1 line defect waveg-uide with length 15mm, 25mm and 35mm. The dash line shown the trans-

mission spectrum for the triangular lattice K direction for the perfect

structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

xii


17/144

4.23 The experimental transmission spectra (solid line) of 2 line defect waveg-

uide with length 15mm, 25mm and 35mm. The dash line shown the trans-


structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.24 The experimental transmission spectra (solid line) of 3 line defect waveg-

uide with length 15mm, 25mm and 35mm. The dash line shown the trans-


structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.25 The experimental (red solid line) and theoretical (blue dash line) trans-

mission spectra of (a) 1 line, (b) 2 lines and (c) 3 lines defect waveguide. . 74

4.26 Experiment setup of acoustic wave field inside waveguide. . . . . . . . . 75

4.27 The acoustic wave real part images of (a) 2 lines and (b) 3 lines defect

waveguides. The frequency of the wave is 950kHz. . . . . . . . . . . . . 75

4.28 Structure orientation with [100] and [111] direction. . . . . . . . . . . . . 76

4.29 The picture of the 3-D phononic crystal with top surface perpendicular to

(a) [100] and (b) [111] directions. . . . . . . . . . . . . . . . . . . . . . 77

4.30 Experimental set-up used to measure the transmission spectra of phononic

crystals using acoustic transducers in acoustical contact with the phononic

crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.31 The transmission of [100] direction. . . . . . . . . . . . . . . . . . . . . 79

4.32 Transmission in the [111] direction. . . . . . . . . . . . . . . . . . . . . 80

4.33 Transmission in the [110] direction. . . . . . . . . . . . . . . . . . . . . 80

5.1 Pictures of the structures which edge is parallel to (a)X and (b) M

direction. (c) Definition of the lattice direction. . . . . . . . . . . . . . . 85

5.2 The fabrication process of the phononic crystal slab structure. . . . . . . . 86

xiii


18/144

5.3 Sketch of the experiment. Waves are excited in the phononic crystal slabs

using an ultrasonic transducer through a prism with an incidence angle of

30 degrees. The phononic crystal slab is formed by a one layer thick array

of spherical steel beads with a diameter of 4 mm arranged according to a

square lattice in an epoxy matrix. . . . . . . . . . . . . . . . . . . . . . . 88

5.4 Principle of the laser interferometer. . . . . . . . . . . . . . . . . . . . . 89

5.5 (a) Scanning region position. (b) Measured transmission spectrum (left

hand side) and computed band structure (right hand side) in theX direction 91

5.6 (a) Scanning region position. (b) Measured transmission spectrum (left

hand side) and computed band structure (right hand side) in the M direction 92

5.7 Dependence of the displacement amplitude with frequency and propaga-

tion distance measured for (a) theX and (b) theM oriented phononic

crystal slabs. The white lines in the gray maps on the left mark the com-

plete band gap frequency range, while the black cycles indicate the po-

sition of steel beads. The illustrations in the left side of these two color

maps indicate the scanning region. . . . . . . . . . . . . . . . . . . . . . 93

5.8 The graphs (a) and (b) are extracted from the data in Fig. 5.7(a) and

5.7(b), respectively, and show line scans at three particular frequencies.

The chosen frequencies lie before the complete band gap, inside it, and

above it. The blue cycles indicate the position of steel beads. . . . . . . . 95

5.9 Dependence of the displacement amplitude with frequency and propaga-

tion distance measured for the pure epoxy slabs with 4mm thickness. . . . 965.10 The picture of the line defect waveguide . . . . . . . . . . . . . . . . . . 97

xiv


19/144

5.11 Experimental transmission through a line defect waveguide managed along

theX direction of the phononic crystal slab and the corresponding theo-

retical band diagram. On the left, the transmission spectrum of waveguide

is shown with a red line, while the transmission spectrum of the perfect

phononic crystal slab in X direction is shown in blue line. The right

hand side is theoretical band diagram of the waveguide. The complete

band gap frequency range is located between the two gray areas. The

illustrations in the left side indicates the scanning region. . . . . . . . . . 99

5.12 Spatial scanning region of displacement field of acoustic waves in the line

defect waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.13 Real part of the displacement field of acoustic waves frequency within

complete band gap in the defect line waveguide. The wave frequencies at

275 kHz, 300 kHz and 325 kHz, respectively. The white circles indicate

the positions of the steel beads. . . . . . . . . . . . . . . . . . . . . . . . 100

5.14 Real part of the displacement field of acoustic waves in the defect line

waveguide at a frequency of 135 kHz. The white circles indicate the po-

sitions of the steel beads. . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.15 (a)Experimental set up for negative refraction. (b) Illustration of the neg-

ative refraction effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.16 Real part of the displacement field with frequenies (a) 133 kHz and (b)

380 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.17 Real part of the displacement field with frequencies (a) 280 kHz and (b)300 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.18 Real part of the displacement field with frequencies (a) 333 kHz and (b)

340 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

xv


20/144

List of Tables

2.1 Band-structure-related properties of three periodic cyctem. [10] . . . . . 19

4.1 Material constants of the phononic crystal constituents. cSand cLare the

velocities for shear and longitudinal waves, respectively, is the mass

density, and ZL is the acoustic impedance for longitudinal waves. . . . . . 524.2 Material constants of the phononic crystal constituents. cSand cLare the


density, and ZL is the acoustic impedance for longitudinal waves. . . . . . 78

5.1 Material constants of the phononic crystal constituents. cSand cLare the


density, and ZL is the acoustic impedance for longitudinal waves. . . . . . 87

xvi


21/144

Chapter1 General Introduction

Since the first experiment of X-ray refraction experiment is observed in 1962, the the-

ories of solid state were developed rapidly. In the solid state theories, the electrons are

considered as rather wave than particle. In the crystalline solid, the periodic potential

applied to the electron wave and causes some energy gaps. The electron waves with fre-

quencies within those gaps can not existence inside the solid. The same methods can beapplied to traditional waves as electromagnetic waves or acoustic waves. The first con-

cept was presented in 1987 for electromagnetic waves and in 1993 for acoustic waves.

The analogy energy gaps in traditional wave are called forbidden band gaps or complete

band gaps. The structures, which can generate complete band gaps for electromagnetic

and acoustic waves, are called photonic and phononic crystals, respectively. The elec-

tromagnetic or acoustic wave with frequencies with in the complete band gaps can not

propagate inside the crystals. As the crystalline solid with periodic potential, the photonicand phononic crystals are periodic structure geometrically. The optical and elastic param-

eters are periodically varying inside the photonic and phononic crystals, respectively.

This thesis presented the studies in photonic and phononic crystals. The works are

cooperated by Department of Optics and Photonics, National Central University, Taiwan

and Institude FEMTO-ST, Universit de Franche-Comt, France. Three major subjects

are presented in this thesis. First subject focuses on the optical properties of three-

dimensional photonic crystal. Second subject includes the studies of the acoustic bulk

waves in two and three-dimensional phononic crystals. Finally, the third subject includes

the studies of Lamb wave mode in phononic crystal slab which is consisted of plate and

plannar periodic structure is studied.

We presented in the studies of three-dimensional photonic crystals a method to en-

hance the reflectance of the artificial opal. The experiment is demonstrated by coating

1


22/144

Ag nanoparticles on the polymethylmethacrylate microspheres to form a metallodielec-

tric artificial opal. The diameter of microspheres is 320nm. The angle-resolved reflection

spectra of the metallodielectric opals are studied. The experimental results are verified

by the three dimensional theoretical band structures which can predict the band gap fre-

quency range corresponding to different incident directions. By a suitable coating of

Ag nanoparticles on the microspheres, the reflectance of the opals is enhanced without

changing the reflection wavelength. This metallodielectric material can be used as the

high reflectance mirror. The device works in visible light wavelength range.

In the chapter of acoustic bulk waves, two-dimensional phononic crystals with tri-

angular and honeycomb lattices are investigated experimentally and theoretically. The

structures used in our experiment are composed of arrays of steel cylinders immersed in

water. The geometrical parameters of these two structures are the same. The measured

transmission spectra revealed the existence of complete band gaps but also of deaf bands.

Band gaps and deaf bands are identified by comparing band structure computations, ob-

tained by a periodic-boundary finite element method, with transmission simulations, ob-

tained using the finite difference time domain method. The appearance of flat bands and

the polarization of the associated eigenmodes is also discussed. Triangular and honey-

comb phononic crystals with equal cylinder diameter and smallest spacing are compared.

As previously obtained with air-solid phononic crystals, it is found that the first com-

plete band gap opens for the honeycomb lattice but not for the triangular lattice, thanks to

symmetry reduction.

Three-dimensional epoxy-steel photonic crystals are studied experimentally also inthis chapter. The structures used in our experiment are composed of face-center cubic

package spherical steel beads embedded in an epoxy matrix. The complete band gaps

are identified by measuring the transmission in different orientations of structure. A large

complete band gaps for longitudinal waves are observed. The60% relative width of the

band gap is larger than expected based on the contrast in material properties. The cou-

2


23/144

pling between shear and longitudinal polarizations is proposed to be the mechanism for

enlarging the band gap by comparing with the steel beads in water matrix case.

Finally, the propagation of acoustic waves in a square-lattice phononic crystal slab

consisting of a single layer of spherical steel beads in a solid epoxy matrix is studied ex-

perimentally. The acoustic wave, which is guided between two surfaces of slab, is plate

mode, also called Lamb wave mode. The three dimensional confinement and control of

acoustic waves is achieved by the single layer periodic structure and the two face of slab,

so this kind of structure is also called quasi-three dimensional phononic crystal structure.

Acoustic waves are excited by an ultrasonic transducer and fully characterized on the slab

surface by laser interferometry. The transmission and field image of acoustic wave are

investigated. A complete band gap is found to extend around 300 kHz, in good agreement

with theoretical predictions. The transmission attenuation caused by absorption and band

gap effects is obtained as a function of frequency and propagation distance. Well confined

acoustic wave propagation inside a line-defect waveguide is further observed experimen-

tally. An unusual refraction effect is observed in the wave frequencies near the higher

band edge of complete band gaps.

3


24/144

Chapter2 Introduction to photonic and phononic

crystals

2.1 History and development of photonic crystals

The wave propagation inside periodic structure can be traced to X-ray refraction ex-

periment. The first experiment of X-ray diffraction by crystal is observed. The experiment

of X-ray introduced the concept of forbidden band. After, the theories of solid-state and

semi-conductor were developed widely. The concept was applied to the case of electro-

magnetic waves and elastic waves.

The concepts of photonic crystals and photonic band gap were first presented by Eli

Yablonovitch and Sajeev Jhon in Physical Review Letters [13, 14] in 1987. They purposed

to suppress spontaneous emission and to achieve localization of light. Recently, photonic

crystals attracted much attention of researchers in not only optics but also chemistry,

physics, and microelectronics beacuse of the potential of light control. Photonic crystals

are potent to control the flows of photons. The purpose could be achieved by designing

the band gap frequency range properly.

Photonic crystals (PCs) are dielectric materials in which the refraction index is period-

ically modulated. The validity of Bloch-Floquet theorem for Maxwell equations implies

the existence of photonic bands, in analogy to electrons in crystalline solids. Photonic

crystal structures can produce complete band gaps of electromagnetic wave, in analogy

to energy gaps of electrons. The electromagnetic wave with frequencies within band gaps

range can not propagate inside photonic crystal structure. By introducing some defects or

deformations inside structures, the photonic crystals can be used to control the propaga-

tion direction and properties of light in many approaches.

4


25/144

Figure 2.1: From left to right diagram, PCs are in one, two, and three dimen-

sions. The red and yellow colors indicate the different refraction index. (http://ab-

initio.mit.edu/photons/)

The photonic crystals types could be divided into one, two and three dimensional.

Fig. 2.1 indicates the three type of structure. In one dimensional structure, the refraction

index is periodically modulated along one dimension, as multilayer structure. Two di-

mensional structure could be achieved by arranging dielectric cylinders in air [15, 16, 17]

periodically or inverse [17, 18]. Three dimensional structure could be fabricated by pack-

aging dielectric cylinders or spheres. There are various techniques for the fabricationof three dimensional PCs [19]. For instance, silicon micromachining [20], wafer fusion

bonding [21], holographic lithography [22], self-assembly [23, 24], angled etching [25],

micromanipulation [26], glancing angle deposition [27] and autocloning [28].

2.1.1 Photonic crystals in Nature

Photonic crystals structure is not only artificial. It existed in nature also, in many

plants, animals and mineral. These photonic crystals periodic structures in nature causethe surface of materials to gleam colors when the light incident on them. The colors do

not result from the existence of pigment.

Here are some examples of the photonic crystals in nature. Fig. 2.2 shows the iri-

descent colors of butterfly wings caused by the periodic structures. The SEM picture of

butterfly wings are shown in Fig. 2.3. [1, 2]

5


26/144

Figure 2.2: Somewhere in a beam of sunlight in the Amazon basin, the three-inch wings

of a morpho butterfly glow with deep blue iridescence. Then the morpho shifts its wings

and, as if a switch flipped, shows its other self, dull brown. [1]

Figure 2.3: High-resolution SEM images showing the fine structure of the scales with

rounded ends: (a) blue (BL) male, blue region; (b) blue (BL) male, blue-violet region; (c)brown (BR) male.[2]

Fig. 2.4 (a), (b) and (c) shows the colorful spine pf sea mouse and the SEM picture

of cross-section of a spine, respectively.[1, 29] Sea mouse is a marine worm and living

in moderately deep water. [1, 29, 30] Scientists discovered that the spine of sea mouse

consisted of an array of regularly arranged hollow cylinders (Fig. 2.4 (c)), and this simple

6


27/144

structure gives rise to a spectacular iridescence. [30]

Figure 2.4: (a)Sea mouse, (b) its iridescent threads (c) SEM cross-section of a spine.[2]

Fig. 2.5 shows the picture of opal. This is the oldest and best-known example of

photonic crystals structure in mineral word. The colorful surface of opal is caused by thephotonic crystals structure also.

2.1.2 Defects

Electromagnetic waves with frequencies within the photonic crystal band gaps are

forbidden to propagate in the photonic crystal. However, suitable defects introduced into

the photonic crystal can act as waveguides or cavities for those electromagnetic waves.

Electromagnetic waves can be controlled (guided or confined) by those defects. Thecombination of those defects can act as the novel planar light wave circuit with higher

efficiency than traditional methods.

Some briefly introductions of the waveguide and cavity defects in photonic crystal are

presented in follow. Meadeet al[31] presented the waveguides and cavities in two dimen-

sional photonic crystal structure theoretically. Compare the results to conventional struc-

7


28/144

Figure 2.5: The opal is the example of photonic band structures in mineral worlds.[2]

tures, the results of photonic crystal defects shown lower bend losses and higher Q values

than conventional bending waveguides and cavities, respectively. Mekiset al[32] demon-

strated highly efficient transmission of light around sharp corners in photonic bandgapwaveguides theoretically. The transmittance is higher than 95 %. High transmission is

observed even for90o bends with zero radius of curvature, with a maximum transmission

of98%as opposed to30%for analogous conventional dielectric waveguides. Tokushima

et al[3] demonstrated1.55m wavelength lightwave propagation through a 120o sharply

bent waveguide formed in a triangular-lattice two-dimensional photonic crystal which

was fabricated in a silicon-on-insulator (SOI) wafer with the top silicon layer of the wafer

used as a core layer. Fig. 2.6 shows the cross sectional micrograph of the structure.C. Martijn de Sterke et al [33] studied the the modes of coupled photonic crystal

waveguides. Liet al[34] employed a highly efficient transfer-matrix method to inves-

tigate the propagation loss in a photonic crystal waveguide without complete wave con-

finement because of limited cladding wall thickness. An anomalous loss phenomenon is

found where the loss for guided modes near the upper band edge can be several orders of

8


29/144

Figure 2.6: Cross-sectional micrograph of the 2D PC. The holes were formed in a1.85m

thick Si core layer. TheSiO2layer was2m thick. [3]

magnitude smaller than that for modes in the middle of the band gap, and it is attributed to

a different localization degree of the guided mode at different frequency domains. Gersen

et al[35] presented a real-space observation of fast and slow pulses propagating inside

a photonic crystal waveguide by time-resolved near-field scanning optical microscopy.For a specific optical frequency, a localized pattern associated with a flat band in the dis-

persion diagram. During at least 3 ps, movement of this field is hardly discernible: its

group velocity would be at mostc/1000. They also measured the eigenfield distribution

and the band structure of a photonic crystal waveguide with a phase-sensitive near-field

scanning optical microscope [36]. Bloch modes are visualized in the waveguide. In the

band structure, multiple Brillouin zones due to zone folding are observed, in which pos-

itive and negative dispersion is seen. The negative slopes are shown to correspond toa negative phase velocity but a positive group velocity. Chienet al[37] introduced the

point-defect coupling under the tight-binding approximation to describe the behavior of

dispersion relations of the guided modes in a single photonic crystal waveguide and two

coupled identical photonic crystal waveguides.

Some optical devices consisted of photonic crystal waveguides were presented also.

9


30/144

Linet al[4] reported a successful experimental realization of a photonic-crystal Y split-

ter operating at 1.6mwith a large splitting angle of120o and a miniature sizeof3mX3m. The device is based on a triple-line-defect in 2D photonic crystal slab

structure. The material of light-guiding layer is GaAs. Fig. 2.7 shows the SEM picture

of the devices. Sharkawy et al[38] presented an electro-optical switch implemented in

coupled photonic crystal waveguides theoretically. The device is designed in a square

lattice of silicon posts in air as well as in a hexagonal lattice of air holes in a silicon slab.

The switching mechanism is a change in the conductance in the coupling region between

the waveguides and hence modulating the coupling coefficient and eventually switching

is achieved. Wavelength-selective operation of an optical filter (add/drop) based on a

contra-directional photonic crystal waveguide coupler is demonstrated by Qiuet al[39]

experimentally and theoretically. The waveguides are defined as line defects in a two-

dimensional triangular photonic crystal fabricated in an InP/GaInAsP heterostructure.

Chien et al [40] demonstrated a dual-wavelength demultiplexer with a coupling length

of only two wavelengths and output power ratio as high as 15 dB. The fundamental mode

of the two coupled photonic crystal waveguides can be odd parity in a triangular photonic

crystal and their dispersion curves do intersect. Thus, the photonic crystal waveguides are

decoupled at the crossing point.

The optical cavity with high Q value could be used to confine light and is useful to fab-

ricate laser or other optical devices. Tayebet al[41] analyzed diffraction by a finite set of

parallel cylinders to study the influence of defects in a photonic crystal. They describedl

the near-field map, scattered field, and energy flow of the electromagnetic field and alsolocalized resonant modes. Schereret al[5] presented some examples of high-Q value op-

tical nanocavities in InGaAsP experimentally. The structures can be used to fabricate sur-

face plasmon enhanced light-emitting diode (LEDs) and vertical-cavity surface-emitting

lasers. The photonic crystal cavity are shown in Fig. 2.8

Robinsonet al[42] theoretically demonstrate a mechanism for reduction of mode vol-

10


31/144

Figure 2.7: (a) Scanning electron micrograph top view of 38 period triple-line-defect

linear waveguide. The defect hole diameter is 0.8 times of lattice constant (352nm).

The rectangular boxes indicate the interfaces between the ridge and the photonic band

gap waveguide. (b) Scanning electron micrograph image of the Y-splitter sample, which

consists of a120o Y- splitter and two60o bends (indicated by the red circles). (c) Infrared

camera image of the two Y-splitter outputs at = 1650nm. They are both Gaussian-like

and equally bright, indicating a near 50/50 splitting ratio.[4]

ume in high index contrast optical microcavities to below a cubic half wavelength. The

principle can be applied to nearly every existing microcavity resonator to enhance not only

11


32/144

Figure 2.8: Five photonic crystal cavities coupled together lithographically by arrange-ment in the same slab.[5]

light emission but also nonlinear effects. Englundet al[43] observed a large spontaneous

emission rate modification of individual InAs quantum dots in a 2D photonic crystal with

a modified, high-Q single-defect cavity. The emission rate increase of up to 8 times than

quantum dots in a bulk semiconductor. Chaloupka et al [44] shown how to construct

circular photonic crystals and variable-size circular photonic crystals cavities, which can

support well-localized modes with high-quality factors by local density of states Istrate et

al[45] presented a method which can obtain the equivalent of the Fresnel coefficients for

photonic crystals by the band structure and Bloch modes. They used these coefficients to

derive the reflection of light from a photonic crystal of finite size and the resonant modes

of photonic crystal cavities. Xuet al[46] presented theoretically a nanowire-array-based

optical microcavity high Q values and small Veffcan find applications in fields such as

microlasers, compact nonlinear frequency conversion, alloptical signal processing, andquantum information processing. The material is consist of III-V and II-VI semicon-

ductor. OBrien et al[47] presented experimentally tunable optical delay in a coupled

resonator structure consisting of a chain of three heterostructure nanocavities. The speed

of light in the device is varied over a range from c/75 to c/120 using effective pump powers

below100W.

12


33/144

2.1.3 Photonic crystal fiber

Optical fiber is a traditional and useful optical device. It is applied to optical commu-

nication, optical measurement, laser and many other categories. Many works presented

that by adding the photonic crystal structure in the core of optical fiber, the novel applica-

tion and phenomena could be observed.

In 1996, Knightet al[48] presented the photonic crystal fiber consists of a pure silica

core surrounded by a silica-air photonic crystal material with a hexagonal symmetry. The

fiber supports a single robust low-loss guided mode over a very broad spectral range ofat least from 458 to 1550 nm. Birks et al [6] presented an all-silica optical fiber by

embedding a central core in a two-dimensional photonic crystal with a micrometer-spaced

hexagonal array of air holes. An effective-index model confirms that such a fiber can be

single mode for any wavelength. The SEM picture of fiber is shown in Fig. 2.9.

Figure 2.9: Scanning electron microscope image of the end of a photonic crystal fiber,

showing the central core where a hole has been omitted. [6]

Ranka et al [7] demonstrated experimentally air-silica microstructure optical fibers

can exhibit anomalous dispersion at visible wavelengths. The output spectra of fiber

13


34/144

extending from the violet to the infrared, by propagating pulses of 100-fs duration and

kilowatt peak powers through a microstructure fiber near the zero-dispersion wavelength.

Fig. 2.10 shows the Electron micrograph picture of fiber Fig. 2.11 shows the optical spec-

Figure 2.10: Scanning electron micrograph image of the inner cladding and core of the

airsilica microstructure fiber. [7]

trum of fiber. Also shown is the spectrum of initial pulse. Jhonson et al[49] presented

Figure 2.11: Optical spectrum of the continuum generated in a 75-cm section of mi-

crostructure fiber. The dashed curve shows the spectrum of the initial 100-fs pulse. [7]

present the light-propagation characteristics of OmniGuide fibers, which guide light by

14


35/144

concentric multi-layer dielectric mirrors having the property of omnidirectional reflection.

The multi-layer dielectric acts one-dimensional photonic crystal structure. Shephardet al

[50] discussed the damage threshold and practical limitations of current hollow-core fibers

for the delivery of short optical pulses. Coen et al[51] presented the generation of a spa-

tially single-mode white-light supercontinuum has been observed in a photonic crystal

fiber pumped with 60-ps pulses of subkilowatt peak power experimentally.

2.1.4 Colloidal microsphere base photonic crystal

Recently, three-dimensional photonic crystal fabricated by microspheres attracted much

attenuation. The structure is also called artificial opal. The advantages of photonic crystal

based in colloidal microspheres are fabricated easily and with small size which can be

used to high frequency range.

In 1996, Tarhan et al[52] studied the photonic band structure of a polystyrene col-

loidal fcc crystal by transmission measurements. Mguezet al[8] investigated the optical

properties of packed monodisperse silica submicron spheres by means of optical transmis-sion measurements. The lattice parameters of these structures can be easily tuned through

the sphere size form 200 to 700 nm thus covering the whole visible and near infrared

spectrum. Fig. 2.12 shows the SEM picture of the structure. In almost the same time,

Bogomolovet al[53] reported on the photonic band gap phenomenon in the visible range

in a three-dimensional dielectric lattice formed by close-packed spherical silica clusters

experimentally. Manifestations of the photonic pseudogap have been established for both

transmission and emission spectra.Liet al[54] investegated the full band gap of inverse-opal (air-spheres in host) theo-

retically. They also found that, in the presence of disorder such as variations in the radii of

air spheres and their positions, the band gap reduces significantly. This imposes a severe

requirement on the uniformity of the crystal lattice. Lpez Tejeira et al[55] analyzed the

symmetry properties of eigenstates along the high-symmetry directions of close-packed

15


36/144

Figure 2.12: SEM image of a610m2 region of the surface of a sample made from415 nm spheres. [8]

bare opals according to the group theory. The results shows that some bands cannot be

coupled with an external plane wave along several directions, because of symmetry rea-

sons, and is not expected to provide any observable features in the transmittance Wang

et al[56] studied the effect of different stacking sequences on the optical properties of

inverted opal photonic crystals composed of close-packed air spheres embedded in high

dielectric matrix of GaAs by electromagnetic wave multiple scattering technique. Chenet al [57] presented theoretically a method to reduce the propagation loss of Si-based

photonic crystal slab waveguides. The transmission are enhanced to twice by stacking

silica and polystryrene microspheres on the top of waveguides. Next year, they [9] pro-

posed a technique to fabricate a free-standing three-dimensional colloidal crystal by self-

assembling the colloidal microspheres with controllable thickness from the air and liquid

interface. The structures can be used for nano-photonic circuits, white-light LEDs or as a

photocatalyst. Fig. 2.13 shows the opal with different thicknesses.

Pavariniet al[58] presented a theoretical approach for the interpretation of reflectance

spectra of opal photonic crystals with fcc structure and (111) surface orientation. It is

based on the calculation of photonic bands and density of states corresponding to a spec-

ified angle of incidence in air and yield a clear distinction between diffraction in the

direction of light propagation by (111) family planes and diffraction in other directions

16


37/144

Figure 2.13: SEM images of the photonic crystals after different heating durations. (A)

More than 15 layers of microspheres were orderly packed. (B) The thickness of the opals

is about17m. (C) The thickness of the opals is about 78m. (D) The thickness of the

opals would be more than200m. (E) This shows that the surface of the photonic crystals

in (D) is well packed in the FCC structure. [9]

by higher-order planes. The theoretical results were verified by reflectance measurements

on artificial opals made of self-assembled polystyrene spheres

2.2 History and development of phononic crystals

A phononic crystal is an artificial crystal which is composed of a finite-size periodic

array of sonic scatterers embedded in a homogeneous host material. The host material

may be solid [59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69] or fluid [61, 70, 71, 72, 73, 74,75, 76]. Phononic crystals can produce full band-gaps, or complete band-gaps, where any

sound wave is not allowed to propagate into the crystal. The theory of phononic crystals is

similar to that of photonic crystals. The analogy between behaivors of the electromagnetic

wave in photonic crystals and acoustic wave in phononic crystals were well discussed by

Miyashita [77].

17


38/144

The concept of the phononic crystal is first presented by Kushwaha et al. [10] and

Economou and Sigalas [78] in 1993. They discuss classical wave propagation in periodic

structures. Classical waves include electromagnetic waves and elastic (acoustic) waves.

Kushwaha et al. list a comparison of ordinary crystals, dielectric composites and elastic

composites (Tab. 2.1). Tab. 2.1 clearly shows the analogy and difference between quan-

tum, electromagnetic and acoustic waves.

Kushwaha also studied theoretically the acoustic band gap of straight, infinite cylin-

ders embedded in an elastic background. First, the cylinder is made by Ni and the back-

ground is made by Al. In the second structure, the materials of cylinders and background

are exchanged. The band structure of the second structure is displayed in Fig. 2.14. This

is the first "band gap" results for acoustic waves.

Kushwaha and Halevi [79] then presented the acoustic band gap in periodic, binary

composites of long, elastic cylinders that form a hexagonal lattice. They concluded sev-

eral general rules for the occurrence of acoustic band gaps. First, the wave velocity in

the inclusion should be lower than that in the matrix material. Second, The diamond and

FCC structure are preferable, especially the FCC structure. Third, the difference between

acoustic wave velocity in inclusion and matrix should be large. Fourth, the difference

between the velocity of transverse and longitudinal wave should be as small as possible.

Lastly, The material density is also a key point to decide band gap effects. Further works

have shown that these rules are not absolute.

The geometric parameters of phononic crystals are directly dependent on the acoustic

wave length. For long wave length (low frequency), the size of unit cell (for example: theinset of Fig. 2.14) could be very large. The first experimental result of the acoustic band

gap measure is achieved by Martinez-Sala et al[80]. The experiment sample is a sculpture

in Juan March Fundation at Madrid. The sculpture is consisted of square array steel

cylinders with diameter 2.9cm. The acoustic wave frequency is 2 kHz. Afterwards, many

different composite materials were studied, for instance, carbon/epoxy and water/mercury

18


39/144

Table 2.1: Band-structure-related properties of three periodic cyctem. [10]

Property "Electronic" Crystal "Photonic" Crystal "Phononic" Crystal

Material Crytalline Constructed of two di-

electric materials

Constructed of two

elastic materials

Parameters Universal constants,

atomic numbers

Dielectric constants of

constituents

Mass densities, sound

speeds of constituents

Lattice Con-

stant

1-5 0.1m - 1cm Mesoscopic or macro-

scopicWaves de Broglie (electrons)

Electromagnetic or

light (photons) E,B

Vibrational or sound

(phonons) u

Polarization Spin , Transverse: D =0( E= 0)

Coupled trans.-longit.

(u= 0, D= 0)Differential

equation

h2m

2+ V(r) =ih

t

2E ( E) =(r)c2

2Et2

2uit2

= 1{ xi

(ulxl

) +

xl

[(uixl

+ ulxi

)]}

Free particlelimit

W = h2

k2

2m = ck =cl,tk

Band gap Increases with crystal

potential; no electron

states

Increase with |ab|; no photons, no light

Increase with| ab|; no phonons, novibration; no sound

Spectral re-

gion

Radio wave, mi-

crowave, optical, x

ray

Microwave, optical 1GHz

[81, 82]. The first experiment result of a bulk acoustic wave complete band gap was

presented by Sanchez-Perezet al[73]. The results presented the complete band of two-

dimensional periodic array of rigid cylinders in air with square and triangular geometrical

configurations. The authors also observed the deaf band effect. Deaf bands are acoustic

19


40/144

Figure 2.14: Band structure for a period array of aluminum cylinders in nickel back-ground. The inset shows the unit cell [10].

branches which exist in the band structure. But the modes of these branches cannot be

excited depending on the symmetry of the mode with respect to the source, hence they

do not transport acoustic energy through the crystal. At almost the same time, Montero

de Espinosa et al[83] presented the complete band gap effect of mercury cylinders in

aluminum matrix without deaf band.

Different kinds of material were also presented. For example, Wenet al[84] presented

the band structure of flexural waves in a periodic binary straight beam with different cross

sections. It is worth to mention that Liuet al[85] presented the locally resonant material

for which the size is 12 times smaller than the wave length. These results can be extended

to the large wave length range. The size of the unit cell can also be reduced.

20


41/144

So far, the results were presented here are all in the case of acoustic bluk wave. The

acoustic bulk wave is an elastic wave propagating in an infinite solid material. Generally,

the studies of three-dimensional phononic crystals or two-dimensional phononic crystals

with the periodic structure in x-y plane and infinite in z-axis fall in this category.

Phononic crystals for surface acoustic waves and Lamb waves

In contrast to the bulk wave, there are some kinds of acoustic waves that are confined

at the surface or the interface of a material. Tanaka et al[86] present the theoretical eluci-dation of surface acoustic waves characteristics in 2D phononic crystals. They studied the

wave propagating on the surface perpendicular to the axes of elastic cylinders (z-direction)

forming a 2D lattice. The system occupying z>0 is periodic within the flat surface (z=0),

which is taken to be the x-y plane and homogeneous in the z direction. They also compare

the acoustic stop bands of surface and bulk modes in two-dimensional phononic crystal

[87]. Some examples of the acoustic surface wave researchs include different materials

and measurement techniques will be introduced below. Mesegueret al[88] studied thesurface-elastic waves attenuation by a periodic array of cylindrical holes. The experi-

ments were performed in a marble quarry by drilling cylindrical holes. The attenuation

spectra of the surface waves show the existence of absolute band gaps for elastic waves in

these semi-infinite two-dimensional crystals. Vienset al[89] presented the frequency gap

of surface acoustic wave in phononic structures multilayers which consisted of slots cut in

an aluminum substrate filled with a polymer. The complete band gap of surface acoustic

wave in water is presented by Jeong et al[90]. Dhar et al[91] studied the frequencies

dispersion of surface acoustic modes in aluminum-coated glass plates with square-wave

surface relief. The relief is fabricated by laser-based transient grating method. Profunser

et al[92] persented the imaging of ripples in one-dimensional phononic crystal consists

of of a grating of alternate copper and silicon oxide lines of thickness both deposited on

a silicon substrate by ultrafast optical pump and probe technique. The phononic crystal

21


42/144

devices using surface acoustic wave are received much attenuation also [93, 94, 95]. The

functions included the applications in high frequency wave, filter devices and acoustic

mirrors or transducers.

Lamb waves have also received much attenuation recently, They can be generated in

the elastic plate with finite thickness in the direction perpendicular to the propagation sur-

face. The vibration exists on the two faces of the plate and the acoustic energy distributes

inside all the plate, not just near the surface. Like the surface acoustic wave, the propa-

gation direction of the Lamb wave is already fixed in the direction parallel to the material

surface. The wave can be further controled by adding a structure varying periodically

along the surface of material with phononic crystal effect. It means that the influence

of two-dimensional phononic crystals for surface acoustic waves or Lamb waves is like

the influence of three-dimensional phononic crystals for the bulk waves. Below are some

examples of phononic crystals for Lamb waves.

Hsuet al[96] studied theoretically the propagation for the lower bands of Lamb waves

in two-dimensional phononic-crystal plates consisting of square array of crystalline gold

cylinders in the epoxy matrix. The theory is based on Mindlins plate theory and on the

plane wave expansion method. Zhanget al[97] studied experimentally a new type of

phononic crystal produced by patterning holes on thin metal plates with optical charac-

terization. Khelifet al[98] examined theoretically the propagation of elastic waves in a

phononic crystal slab consisting of piezoelectric inclusions placed periodically in a host

material by finite element method. The system is composed of a square lattice of quartz

cylinders embedded in an epoxy matrix. Gaoet al[99] presented numerically the prop-agation of Lamb waves in one-dimensional quasiperiodic composite thin plates made of

tungsten and silicon resin arranged according to a Fibonacci sequence. The propagation

of Lamb waves in silicon plates coated by a very thin two-dimensional phononic film was

presented by Bonelloet al[100] by using a laser ultrasonics technique.

All the studies above considered a one dimensional structure or a two dimenstional

22


43/144

structure with cylindrical inclusions. A phononic crystal slab with two dimensional spher-

ical inclusions will be presented later in chapter 3.

Guide and Cavity

To achieve acoustic wave control, the complete band gap effect can be extended to guiding

or confinement the acoustic energy by adding specific defects. Further, various applica-

tions can be developed by studying the interaction between these defect states.

The first study of a defect state of a phononic crystal was presented theoretically bySigalas[101] and Torreset al[102]. There are many theoretical and experimental studies

of different defects in numerous kinds of phononic crystals. Generally, the theoretical

methods used in these studies are based on the plane wave expansion or the finite differ-

ence time domain methods [103, 104]. The defect band in the band structure could be

calculted by plane wave expansion. The transmission can be predicted by finite differ-

ence time domain methods. Some examples are introduced here. Chandraet al[105] pre-

sented the acoustic wave propagation along waveguides in three-dimensional phononiccrystals by finite difference time domain methods. The phononic crystal is constituted

of lead spherical inclusions on a face-centered cubic lattice embedded in an epoxy ma-

trix. The cavity mode and sharp bending waveguide transmission were studied by khelif

et al[106, 107] in a two dimensional phononic crystal constituted by a square array of

circular parallel steel cylinders in water. Sharp bending and the coupling phenomenon

of joined parallel phononic crystal waveguides in two-dimensional steel/epoxy phononic

crystal was presented by Sun et al [108, 109]. Pennec et al[110] studied the coupling

effect between two continuum waveguides and two cavities inserted in a phononic crystal

composed of steel cylinders in water.

Based on these researches, functional and tunable phononic devices can be developed.

Waveguides with stubs which induce zeros of transmission were studied by Khelifet al

[62]. The device have potential applications in filtering and wavelength demultiplexing

23


44/144

phenomena. A waveguide with tunable narrow pass bands is studied by khelifet al[63]

and Pennec et al [111]. Kafesaki et al [112] showed that the transmittivity (T) in the

waveguide as a function of frequency has an oscillating behavior with regions ofT 1and regions ofT 0. Wanget al[113] studied a quasi-one-dimensional waveguide withperiodical double stubs with tunable complete spectral gaps in the band structure.

Negative refraction in phononic crystal

Negative refraction received great attention in photonic crystal in recent years. Thesame effect is studied in phononic crystal also. The negative refraction effect can be used

to realize perfect imaging and superlenses. There are many theoretical studies based on

the multiple-scattering method. Liet al[114] presented the negative refraction effect in

two dimensional square lattice, rubber-coated tungsten cylinders placed in water. Zhang et

al[115] presented the All-Angle-Negative-Refraction (AANR) in phononic crystals with

steel cylinders in air background and water cylinders in mercury background. Far-field

images of two-dimensional, hexagonal arrays of steel cylinders in air, phononic crystalsuperlens are presented by Qiuet al[116].

Experimental observation of negative refraction of acoustic wave was realized by Feng

et al[117] in two-dimensional triangular sonic crystal constituted by drilling triangular

array of holes on aluminum plate. Hu et al [11] observated directly the point source

imaging in liquid surface waves. The phononic crystal is constituted by square lattice

copper cylinders. Fig. 2.15 shown the simulated and experimental results. Fig. 2.15 (a)

shown the patterns of liquid surface waves with different frequencies. The black areas

inside these figures marked the position of phononic crystal. The acoustic point source is

placed at the left side and near to the phononic crystal. Fig. 2.15 (b) shown the simulated

results with the same parameters as in experiment.

Ke et al [118] observed negative refraction behavior and imaging effect in a two-

dimensional phononic crystal which consists of a triangular array of steel rods immersed

24


45/144

Figure 2.15: Dark and white represent positive and negative vibrations, respectively. (a)

Experimentally observed patterns of liquid surface waves. Black areas in the middle

stands for the slab of copper cylinders. (b) Simulated results. Parameters used in simula-

tions are the same as in experiment. Red dots denote copper cylinders. [11]

in water. The same effect was observed in three dimensional phononic crystals consist

of tungsten carbide beads surrounded by water, with the beads close packed in a face

centered cubic crystal structure, by Yanget al[119].

Other applications and studies

There are many other kinds of studies in phononic crystal. For instance, the tunneling

effect, high frequency phononic crystal and highly directional acoustic sources based on

phononic crystals.

The tunneling effect is one of the most striking phenomena in quantum mechanics.

Since the theory of photonic and phononic crystals is deduced from the "electric" crystal,the existence of tunneling effect with classical waves is possible. Yanget al[120] studied

experimentally and theoretically the tunneling effect in 3D phononic crystals, consisting

of fcc arrays of close-packed tungsten carbide beads in water. Qiuet al[121] found that

the resonant tunneling of longitudinal waves can be distinguished from those of transverse

waves in two-dimensional double phononic potential barriers.

25


46/144

High frequency phononic crystals consisting of micro structures are achieved by mi-

cro fabrication processes or nano structures. Gorishnyyet al[122] demonstrated a new

method for fabrication and characterization of hypersonic phononic crystals by combi-

nation of interference lithography and Brillouin light scattering. The frequency range

extends to the GHz range. Lin et al[123] studied the characteristics of a nanoacoustic

mirror and a nanophononic cavity by femtosecond optical pulse. The first phononic band

gap frequency is centered at 280 GHz.

Recently, there have been some researches for highly directional acoustic wave sources

which can be applied as acoustic transducers. Qiuet al[124] demonstrated a highly di-

rectional acoustic source with a large radiation enhancement, operating at the band-edge

frequency of the phononic crystal by placing a line acoustic source inside a phononic

crystal with a square lattice. Wuet al[125] presented an optimal amplitude magnification

which is more than 86.5 times in comparison with the amplitude of the original source

freely radiating based on the planar resonant cavity of two-dimensional phononic crystals.

Fig. 2.16 displayed the experiment results which were presented by Ke et al [12]. The

structure they used is steel rods in water arranged in a square lattice. The results shown a

highly directional radiation with a half-power angular width of6o.

2.3 Elastic wave propagation in materials

This section is a brief introduction to acoustic wave propagation inside elastic materi-

als [126].

The propagation of acoustic waves affects the deformation of materials. The dis-

placement vector at each point of a material is presented by the vector u. The strain-

displacement relation can be expressed by:

Sij =1

2

uixj

+ujxi

(2.1)

The displacements of a material are accompanied with restoring forces which are

26


47/144

Figure 2.16: Radiation amplitude field distribution over the phononic crystal: (a)by ex-

periment and (b) by simulation. [12]

called the stress field T. The inertial and elastic restoring forces in a free vibrating

medium are thus related through the translational equation of motion:

T= 2u

t2 (2.2)

Hookes Law states the strain is linearly proportional to the stress, or conversely, that

the stress is linearly proportional to the strain. In general, the stress can be expressed by :

27


48/144

Tij =cijklSkl (2.3)

Wherei, j, k,l = x, y,zand with implied summation over the repeated subscriptsk

and l. The "microscopic spring constants" cijklare called elastic stiffness constants. These

stiffness constants are not all independent since

cijkl =cjikl =cijlk =cjilk (2.4)

and

cijkl =cklij (2.5)

Therefore, the stress field can be writen as:

Tij =cijklukxl

(2.6)

In isotropic material, The stress reduces to:

Tij =ullij+ 2uij (2.7)

where(r)and(r)are so-called Lam coefficients.

The relations above describe elastic wave propagation in media. The theoretical meth-

ods for calculating wave propagation inside inhomogeneous media are based on these

relations.

2.4 Theoretical methods

This section introduces some theoretical calculation methods for phononic crystas.

The methods include the Finite-Difference Time-Domain (FDTD) method and Finite El-

ement Methods (FEM). In our study, The FDTD method is used to compute the transmis-

28


49/144

sion of phononic crystals; FEM is used to compute the band structure. FEM also is used

to calculate the eigen-modes of acoustic branches in the band structure.

2.4.1 Finite-Difference Time-Domain method

The Finite-Difference Time-Domain method used in phononic crystals was first de-

veloped by Sigalas and Garca [103, 104]. The FDTD method can handle inhomogeneous

mixed system cases such as fluids in solids or the converse. It is a code for the temporal

integration of incident wave packets for the full elastic equation that converges well in all

cases investigated. Compared to other methods, the FDTD method can calculate the wave

varying in the structure with time. The simulation is suited to finite dimension structures

and can be made most similar to the real experiment environment.

The propagation of elastic waves in isotropic inhomogeneous media is described by:

2ui

t

2 =

1

Tij

xj

(2.8)

Tij = ullij+ 2uij (2.9)

Sij = 1

2

uixj

+ ujxi

(2.10)

whereuiis theith component of the displacement vectoru(r),Tijis the stress tensor,

(r)and(r)are the so-called Lame coefficients, and(r)is the density.

The FDTD method uses discretization of the elastic wave equations in both the spa-

tial and time domains, sets appropriate boundary conditions, and explicitly calculates the

evolution of the displacement vectoru in the time domain. More specifically, real space

is discretized into a cubic grid. The separation distance between grids isx,yor z.

Propagation time is also discretized into many temporal steps with interval t. For exam-

ple, a function(x)is discretized using the method above. The differential of(x)with

respect toxcan be expressed as:

29


50/144

(x)x

= (x + x,y,z) (x,y,z)x

(2.11)

For elastic wave equations, the displacement vectorucan be expressed by uxx+uyy+

uzz. The componentsux,uy anduzare spatially interlaced by half a grid cell; they can

be approximated by center differences in both space and time. The equation for the ux

displacement becomes:

ux(l ,m,n,k+ 1) = 2ux(l ,m,n,k) ux(l ,m,n,k 1) + t2

(l ,m,n)x

[Txx(l+12

, m , n, k) Txx(l 12

, m , n, k)]

+ t2

(l ,m,n)y[Txy(l, m +

1

2, n , k) Txy(l, m 1

2, n , k)]

+ t2

(l ,m,n)z

[Txz(l ,m,n +12

, k) Txz(l ,m,n 12

, k)] (2.12)

where theTxxcomponent of the stress tensor is given by

Txx(l+1

2, m , n, k) =

1

x[(l+

1

2, m , n, k) + 2(l+

1

2, m , n, k)]

[ux(l+ 1, m , n, k) ux(l ,m,n,k)]+

(l+ 12

, m , n, k)

y [uy(l+

1

2, m +

1

2, n , k) uy(l+1

2, m 1

2, n , k)]

+

(l+ 12

, m , n, k)

z

[uz(l+12

, m , n +1

2, k) uz(l+1

2, m , n 1

2, k)] (2.13)

where k is the index for the time step. In a similar way one can get the other components

of the stress tensor. The y component of the displacement vector is calculated at the

(l + 12

, m + 12

, n)grid point and the z component at the (l + 12

, m , n + 12

)grid point. That

way, we can get second order accuracy in our finite difference scheme.

30


51/144

Figure 2.17: The diagram of the computational space

The computational space is sketched in Fig. 2.17. The computational cell contains

the phononic crystal structure in its central part. The structure consists of inclusions

embedded in a matrix of a different material. The inclusions form a lattice which we want

to study. On the left and right sides of the composite, we put a homogeneous material,

indentical to the matrix material of the composite. The acoustic wave is launched in the

homogeneous (excitation) region in the left of composite and propagates along the z axis.

The components of the displacement vector as a function of time are collected in the otherhomogeneous (detection) region on the other side of the composite. Periodic boundary

conditions have been used at the edges of cell along x and y directions. In order to close

the space along the z axis, Murs first order absorbing boundary conditions are applied at

two edges in z direction which are given by the following expression:

31


52/144

ux(l,m, 1, k+ 1) = ux(i,m, 2, k) +c0t xc0t + x

[ux(l,m, 2, k+ 1) ux(i,m, 1, k)] (2.14)

forn = 1;c0is the longitudinal sound velocity of the matrix material forux,uyand

uz.

The time series results collected at the detection region are converted into the fre-

quency domain using a fast Fourier transform. By normalizing these results relative to the

incident wave, one can find the transmission coefficient.

2.4.2 Finite element method

The finite element method has more accuracy than the other methods in phononic

crystals with sharp spatial changes of impedance, e.x. air/solid. In our studies, it is the

major tool to calculate the band structures of bulk waves and Lamb waves in phononic

crystals.

In the finite element method, the phononic crystal is assumed to be infinite and ar-

ranged periodically along the surface. The whole domain is split into successive unit cells

containing one hole surrounded with the matrix material. Each unit cell is indexed by m

,p. The unit cell is meshed and divided into finite elements connected by nodes. Accord-

ing to the Bloch-Floquet theorem, all fields obey a periodicity law, yielding for instance

the following relation between the mechanical displacementsuifor nodes lying on the

boundary of the unit cell:

ui(x + ma1, y+pa2) = ui(x,y,z)exp[j(kxma1)] exp[j(kxpa2)] (2.15)

where kx and ky are the components of the Bloch wave vectors in the x and y di-

rections, respectively. a1anda2are the pitches of the array. Considering the periodical

32


53/144

boundary conditions above allows us to reduce the model to a single unit cell which can

be meshed using finite elements. A mechanical displacement and electrical potential fi-

nite element scheme is used. Considering a monochromatic variation of mechanical and

electrical fields with a time dependence in exp(jt), whereis the angular frequency,

the general piezoelectric problem with no external applied force can be written:

Kuu 2Muu Ku

Ku K

u

=

0

0

(2.16)

whereKuuandMuuare the stiffness and mass matrices of the purely elastic part of

the problem,Kuand Kuare piezoelectric-coupling matrices, andKaccounts for the

purely dielectric problem.u and represent, respectively, all displacements and electrical

potential at the nodes of the mesh, gathered together in vector form. As the angular

frequencyis a periodical function of the wave vector, the problem can be reduced to the

first Brillouin zone. The dispersion curves are eventually built by varying the wave vector

on the first Brillouin zone for a given propagation direction. The full band structure is

then deduced using symmetries.

For our case, there are no piezoelectric-coupling and dielectric problems. The problem

can be reduced to:

Kuu 2Muu

(u) = 0 (2.17)

The band diagram is obtained by solving an eigenvalue problem inside the first irre-

ducible Brillouin zone.

33


54/144


55/144

most important, because it is the most stable against disorder.

In this chapter, the synthesis and the angle-resolved reflectance spectra of metallodi-

electric opals by coating the Ag nanoparticles onto the dielectric micro-spheres are pre-

sented. The opals are made by polymethylmethacrylate(PMMA). With the suitable con-

centration of the Ag nanoparticles, the reflectance of the opals can be enhanced without

changing the photonic band gap of the PC structures. The results are presented experi-

mentally and theoretically. The theoretical band structure of microspheres is calculated by

plane-wave expansion method. It shows the relation between band gap frequency range

and light incident angle. The experimental one is measured the angle-resolved reflectance

spectra directally. The experiment setup will be presented in a later section.

3.2 Plane Wave Expansion (PWE)

Plane wave expansion [132] is widely adapted to calculate to band structure of pho-

tonic crystals. In order to verify to our experimental results, the band structure of our opal

are calculated by PWE also. Band structure is the dispersion relation of the periodicallyarranged dielectric structure. The band gap frequency range, phase and group velocity

can be extracted from band structure. A brief introduction of PWE is presented in follow

paragraph.

E(r, t) =1c

B(r, t)

t =(r)

c

H(r, t)

t (3.1)

H(r, t) =1c

D(r, t)t

= 0(r)

cE(r, t)

t (3.2)

where Eis the electric field, His the magnetic field, Dis the electric displacement

and Bthe magnetic induction.

35


56/144

The time-indepent form could be get by set E, H, Dand Bas follow:

E(r, t) = E(r)eit (3.3)

H(r, t) = H(r)eit (3.4)

D(r, t) = D(r)eit (3.5)

B(r, t) = B(r)eit (3.6)

The magnetic permeability of the photonic crystal is equal to that in free space, i.e.,

(r) =0. The electric constant is assumed real, isotropic, and spatial periodically, i.e.,(r) =(r+ ai), i= 1, 2, 3, where ais the lattice vector of the photonic crystal.

According the description above, the equations 3.1 and 3.2 can be rewrite as:

1

(r) ( E(r)) =

2

c2E(r) (3.7)

( H(r, t)

(r) ) =

2

c2H(r) (3.8)

This is the Helmholtz equation of electromagnetic wave.

The Blochs theorem is applied to the equation above. Blochs theorem indicated that

The eigenfunctions of the wave equation for a periodic potential can be expressed as:

(r) = uk(r)eikr, whereuk(r) = uk(r+ a)and ais the lattice vector of the lattice of

periodic potential.

Because of(r) is periodical. Thus the Blochs theorem can be applied to the Helmholtz

equation of electromagnetic wave. The electric field and magnetic field are thus charac-

terized by a wave vector kin first Brillouin zone and band index as:

E(r) = Ekn(r) = ukn(r)eikr (3.9)

H(r) = Ekn(r) = vkn(r)eikr (3.10)

whereukn(r)andvkn(r)are periodic vectorial function.

36


57/144

ukn(r+ ai) = ukn(r) (3.11)

vkn(r+ ai) = vkn(r) (3.12)

fori = 1, 2, 3.

Because of the spatial peridodicity

Thesis Optical and acoustical properties of band gap materials.pdf

Documents

Transcript of Thesis Optical and acoustical properties of band gap materials.pdf