Download Playing Tricks with Waves

Introduction to
Photonic/Sonic Crystals
Pi-Gang Luan
(欒丕綱)
Institute of Optical Sciences
National Central University
(中央大學光電科學研究所)
Collaborators:
Wave Phenomena:
• Chii-Chang Chen (陳啟昌, NCU), Since 2002
• Zhen Ye (葉真, NCU), Since 1999
• Tzong-Jer Yang (楊宗哲, NCTU) , Since 2001
Quantum and Statistical Mechanics:
• Yee-Mou Kao (柯宜謀, NCTU), Since 2002
• Chi-Shung Tang (唐志雄, NCTS), Since 2002
• De-Hone Lin (林德鴻, NCTU), Since 2000
Contents
• Photonic Crystals
• Negative Refraction
• Sonic Crystals
• Bloch Water Wave
• Conclusion
Famous People
Eli Yablonovitch
Sajeev John
Famous People
J. D. Joannopoulos
沈平（Ping Sheng）
Photonic/Sonic Crystals
1D Crystal
3D Crystal
2D Crystal
Photonic Crystals
3D Photonic Crystal?
Photonic Band Structure
Photonic Band Structure
Photonic Band Structure
Artificial Structures and their Properties
• 1. Photonic Crystals:
• Man-made dielectric periodic structures. According to Bloch’s
theorem, any eigenmode of the wave equation propagating in this
kind of medium must satisfy:
 (r, t )  exp[i(k  r   t )]Uk (r), Uk (r  R)  Uk (r) is a periodic function.
• Usually the frequency spectrum of a photonic crystal has the “band
structure”, that is, there are “pass bands” (which correspond to the
situation that the eigenmode equation has the “real k solution”) and
“stop bands” (also called forbidden bands or band gaps, in which the
eigenmode equation has no “real k solution”).
• The complex-k mode is a kind of evanescent wave (or the so called
•
•
•
“near field”), which cannot survive in an infinitely extended photonic
crystal region (ruled out by the boundary conditions at +∞ and -∞).
However, near a surface (interface) or a defect (for example, a
cylinder or a sphere with a different dielectric constant or radius), the
evanescent wave can exist (the surface mode or the defect mode).
The EM waves do not propagate along the direction that the wave
amplitude decays. Using this property one can control the propagation
of the light. Examples: photonic insulators (omni-directional
reflectors, filters), waveguides, resonance cavity, fibers,
spontaneous emission inhibition, etc.
Even a pass band is useful, since it provides a different dispersion
relation (the w-k relation) . We can design some “effective media”,
usually they are anisotropic media. We can even use them to design
novel lenses and wave plates.
Electromagnetic Waves
• Assuming that J = ρ = 0 (charge free and current free) in the system,
then Faraday’s Law + Ampere’s Law lead to the wave equations for
the E-field and H-field.
• In a two-dimensional system, the permittivity (the dielectric constant ε)
•
and the permeability (μ) become z-independent functions.
If k_z = 0 , then we have E-polarized wave (nonzero E_z, H_x, H_y)
and the H-polarized wave (nonzero H_z, E_x, E_y). These two kinds
of waves are decoupled.
• For monochromatic EM waves with a time factor exp(-iwt), we have
D proportional to (curl H), and B proportional to (curl E), thus the
two divergence equations div D=0 and div B=0 are redundant.
E- and H-polarized EM Waves

E  Ezˆ and

0 
z
E-polarized wave
H-polarized wave
• 2. Phononic/Sonic (or Acoustic) Crystals:
• Man-made elastic periodic structures. In them both the mass
density and the elastic constants (Lam’e coefficients) are periodic
functions of position.
• All the effects (except the quantum effects) discussed before (i.e.,
the band structures, the band gaps, the evanescent waves, the
different dispersion relations) can happen here. In addition, there
are more material parameters (both the mass density and the
elastic constants can be varied).
• The main research interests include the “sound barriers” , “noise
filters”, and “vibration attenuators”.
There are also some researches on “acoustic lens” and “negative
refraction”.
Elastic Waves
 2ui T ji
 2u
 2    T or  2 
, i, j  1, 2,3
t
t
x j
Tij  e (  u) ij  e ( i u j   j ui ), e , e : Lam'e Constants
• Pressure field & Shear Force
• Longitudinal & Transverse waves
Helmholtz Theorem : u      ψ
2
1


2
  2
 0,
2
cl t
2
1

ψ
2
 ψ 2
 0,
2
ct t
cl 
ct 
e  2 e

e

Acoustic Wave and SH (shear) Wave
• In an ideal (composite) fluid, shear
force = 0, thus only the longitudinal
wave (i.e., the pressure wave) can
propagate inside.
• In a 2D system , the mass density and
Lam’e constants are z-independent
functions. If the wave propagation
direction k has zero component along
the z axis (i.e., k_z=0), then u_xy
(i.e., the component lying on the xy
plane ) and u_z (the component that
parallel to the z axis) are decoupled.
Two-Dimensional
Wave Crystal
AC wave and SH wave
Define
Leads to
Define
then
Universal Wave Equation
  mass density
1

,
2
 ct

e   ct2
Universal wave equation
Bloch
Theorem
Reduced frequency
Square Lattice
Triangular Lattice
Photonic
crystals
as optical
components
P. Halevi et.al.
Appl. Phys. Lett.
75, 2725 (1999)
See also
Phys. Rev. Lett. 82,
719 (1999)
Long Wavelength Limit
Focusing of electromagnetic waves by
periodic arrays of dielectric cylinders
Bikash C. Gupta
and Zhen Ye,
Phys. Rev. B 67,
153109 (2003)
Light at the End of the Tunnel
19 March 2004
Phys. Rev. B 69,
121402
Phys. Rev. Lett. 92,
113903
Surface wave + Photonic waveguide
吳明昌
2004.06
Coupled-Resonator Waveguide
Snell’s Law
k y  k 'y


or
n1 sin 1  n2 sin  2
c
c
Constant Frequency Curve
Phys. Rev. B 67, 235107 (2003)
“Negative refraction and left-handed
behavior in two-dimensional photonic
crystals”
S. Foteinopoulou and C. M. Soukoulis
Sonic Insulator
Sculpture
Rod Array
Phys. Rev. Lett. 80, 5325 (1998)
Phononic Band Structures
Acoustic Band Gaps
J. O. Vasseur et. al., PRL 86, 3012 (2001)
“Giant acoustic
stop bands in
two-dimensional
periodic arrays of
liquid cylinders”
M. S. Kushwaha
and P. Halevi
Appl. Phys. Lett.
69, 31 (1996)
Acoustic Lens
Using the pass band
(Propagating Modes)
A Lens-like structure
can focus sound
Refractive Acoustic
Devices for Airborne
Sound
Phys. Rev. Lett. 88,
023902 (2002)
Locally Resonant Sonic Material
Ping Sheng et. al., Science 289, 1734 (2000)
Application (I): Band Gap Engineering
From the universal wave equation, we can
derive:
1 |  |2
1 2 |  |2
dA    2 dA,    (r ), c  c(r )

2 cell 
2 cell  c
Or E (type I) = E (type II)
Varyingα(r) and c (r), we obtain:


1
 1 
2
2

|

|
dA

|

|
dA
   

cell   c 2 
 1  cell
 



2
2

|  |
||
 2
dA
dA


2




c
cell
cell


See
Z. Q. Zhang
PRB 61,1892
(2000)
APL 79,3224
(2001)
R. D. Meade
J. Opt. Soc. Am. B
10, 328 (1993)
Acoustic Band Gap formation
• Soft material (small ρc^2)  Soft spring  Elastic
•
•
•
•
potential energy
Heavy material (largeρ)  Lead sphere  Kinetic
energy
Soft-light material (region I)—Hard-heavy material
(region II) system  Phonon (2 atoms per primitive
basis)  A gap appears between the 1st and the 2nd
bands, just like the gap between the “phonon branch”
and “optical branch”
Separation of these two kinds of energy  Large gap
Region I should be disconnected (hard to move), and
region II should be connected (easy to move)
Water Background-Air Cylinders Sonic Crystal
C_w = 1490m/s,
C_a = 340m/s,
ρ_a/ρ_w = 0.00129
Filling fraction=1/1000
Application (II)
Energy Flow Vortices in Wave Crystals
A singular point is a vortex if and only
if it is an isolated zero of Φ. The
vorticity is nonzero.
A singular point is a saddle point if it
is an isolated zero of Q , an isolated
point at which the phases of Q and Φ
differ by odd multiples of π/2 or a
combination of the previous two
situations. The vorticity is zero.
See C. F. Chien and R. V. Waterhouse
J. Acoust. Soc. Am. 101,705 (1996)
Bloch Water Wave
“Visualization of
Bloch waves and
domain walls” by
M. Torres, et. al.
Nature, 398, 114,
11 Mar. 1998
See also:
PRE 63, 011204
(2000)
PRL 90, 114501
(2003)
Wave Propagation in Periodic
Structures — Electric Filters and
Crystal Lattices
“Waves always behave in a similar
way, whether they are longitudinal
or transverse, elastic or electric.
Scientists of the last (19th) century
always kept this idea in mind.”
--- L. Brillouin
Thank You for Your Attention !