Introduction to Introduction to

Photonic/Sonic CrystalsPhotonic/Sonic Crystals

Pi-Gang LuanPi-Gang Luan(( 欒丕綱欒丕綱 ))

Institute of Optical Sciences Institute of Optical Sciences

National Central UniversityNational Central University

(( 中央大學光電科學研究所中央大學光電科學研究所 ))

Collaborators:Collaborators: Wave Phenomena:Wave Phenomena:• Chii-Chang ChenChii-Chang Chen ( ( 陳啟昌陳啟昌 , NCU), Since 2002 , NCU), Since 2002

• Zhen YeZhen Ye ( ( 葉真葉真 , NCU), Since 1999, NCU), Since 1999

• Tzong-Jer YangTzong-Jer Yang ( ( 楊宗哲楊宗哲 , NCTU) , Since 2001 , NCTU) , Since 2001

Quantum and Statistical Mechanics:Quantum and Statistical Mechanics:• Yee-Mou KaoYee-Mou Kao ( ( 柯宜謀柯宜謀 , NCTU), Since 2002, NCTU), Since 2002

• Chi-Shung TangChi-Shung Tang ( ( 唐志雄唐志雄 , NCTS), Since 2002, NCTS), Since 2002

• De-Hone LinDe-Hone Lin ( ( 林德鴻林德鴻 , NCTU), Since 2000, NCTU), Since 2000

ContentsContents

• Photonic CrystalsPhotonic Crystals• Negative RefractionNegative Refraction• Sonic CrystalsSonic Crystals• Bloch Water WaveBloch Water Wave• ConclusionConclusion

Famous PeopleFamous People

Eli Yablonovitch Sajeev John

Famous PeopleFamous People

J. D. Joannopoulos 沈平（ Ping Sheng ）

Photonic/Sonic CrystalsPhotonic/Sonic Crystals

1D Crystal

2D Crystal

3D Crystal

Photonic Crystals

3D Photonic Crystal?

Photonic Band Structure

Photonic Band Structure

Photonic Band Structure

ArtificialArtificial Structures and their PropertiesStructures and their Properties

• 1. Photonic Crystals:1. Photonic Crystals: • Man-made Man-made dielectric periodic structuresdielectric periodic structures.. According to According to Bloch’s Bloch’s

theoremtheorem, any , any eigenmode eigenmode of the wave equation propagating in this of the wave equation propagating in this kind of medium must satisfy:kind of medium must satisfy:

• Usually the frequency spectrum of a photonic crystal has the “Usually the frequency spectrum of a photonic crystal has the “band band structurestructure”, that is, there are “”, that is, there are “pass bandspass bands” (which correspond to the ” (which correspond to the situation that the eigenmode equation has the “situation that the eigenmode equation has the “real k solutionreal k solution”) and ”) and ““stop bandsstop bands” (also called ” (also called forbidden bandsforbidden bands or or band gapsband gaps, in which the , in which the eigenmode equation eigenmode equation has nohas no “real k solution”). “real k solution”).

( , ) exp[ ( )] ( ), ( ) ( ) is a period .ic functiont i t U U U k k kr k r r r R r

• The complex-k mode is a kind of The complex-k mode is a kind of evanescent waveevanescent wave (or the so called (or the so called ““near fieldnear field”), which ”), which cannotcannot survivesurvive in an in an infinitely extended photonic infinitely extended photonic crystalcrystal regionregion (ruled out by the boundary conditions at + (ruled out by the boundary conditions at +∞ and -∞∞ and -∞). ).

• However, near a However, near a surfacesurface (interface) or a (interface) or a defect defect (for example, a (for example, a cylinder or a sphere with a different dielectric constant or radius), the cylinder or a sphere with a different dielectric constant or radius), the evanescent wave evanescent wave can existcan exist (the (the surface modesurface mode or the or the defect modedefect mode). ).

• The EM waves do not propagate along the direction that the wave The EM waves do not propagate along the direction that the wave amplitude decays. Using this property one can control the propagation amplitude decays. Using this property one can control the propagation of the light. Examples: of the light. Examples: photonic insulatorsphotonic insulators (omni-directional (omni-directional reflectors, filters), reflectors, filters), waveguideswaveguides, , resonance cavityresonance cavity, , fibersfibers, , spontaneous emission inhibitionspontaneous emission inhibition, etc., etc.

• Even a pass band is useful, since it provides a Even a pass band is useful, since it provides a different dispersion different dispersion relationrelation (the w-k relation) . We can design some “ (the w-k relation) . We can design some “effective mediaeffective media”, ”, usually they are usually they are anisotropic mediaanisotropic media. We can even use them to design . We can even use them to design novel novel lenseslenses and and wave plateswave plates..

Electromagnetic WavesElectromagnetic Waves

• Assuming that J = Assuming that J = ρρ = 0 (charge free and current free) in the system, t = 0 (charge free and current free) in the system, then hen Faraday’s Law + Ampere’s LawFaraday’s Law + Ampere’s Law lead to the lead to the wave equationswave equations for t for the E-field and H-field.he E-field and H-field.

• In a two-dimensional system, the In a two-dimensional system, the permittivitypermittivity (the dielectric constant (the dielectric constant εε) and the ) and the permeability permeability ((μμ) become ) become z-independent functionsz-independent functions. .

• If k_z = 0 , then we have E-polarized wave (nonzero E_z, H_x, H_y) 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 and the H-polarized wave (nonzero H_z, E_x, E_y). These two kinds of waves are of waves are decoupleddecoupled..

• For For monochromatic monochromatic EM waves with a EM waves with a time factortime factor exp(-iwt), we have exp(-iwt), we have D proportional to (curl H), and B proportional to (curl E), thus the twD proportional to (curl H), and B proportional to (curl E), thus the two divergence equations o divergence equations div D=0 and div B=0 are redundantdiv D=0 and div B=0 are redundant. .

E- and H-polarized EM WavesE- and H-polarized EM Waves

ˆ and 0E Ezz

E-polarized wave

H-polarized wave

• 2. Phononic/Sonic (or Acoustic) Crystals:2. Phononic/Sonic (or Acoustic) Crystals:

• Man-made Man-made elastic periodic structureselastic periodic structures. In them both the . In them both the mass mass densitydensity and the and the elastic constantselastic constants ( (Lam’e coefficientsLam’e coefficients) are ) are periodic functions of position. periodic functions of position.

• All the effects (except the quantum effects) discussed before (i.e., All the effects (except the quantum effects) discussed before (i.e., the band structures, the band gaps, the evanescent waves, the the band structures, the band gaps, the evanescent waves, the different dispersion relations) can happen here. In addition, different dispersion relations) can happen here. In addition, there there areare moremore material parametersmaterial parameters (both the mass density and the (both the mass density and the elastic constants can be varied).elastic constants can be varied).

• The main research interests include the “The main research interests include the “sound barrierssound barriers” , “” , “noise noise filtersfilters”, and “”, and “vibration attenuatorsvibration attenuators”. ”.

There are also some researches on “There are also some researches on “acoustic lensacoustic lens” and “” and “negativenegative refractionrefraction”.”.

Elastic WavesElastic Waves

• Pressure field & Shear ForcePressure field & Shear Force • Longitudinal & Transverse wavesLongitudinal & Transverse waves

22

2 2or , , 1, 2,3jii

j

Tui j

t t x

u

T

( ) ( ), , : Lam'e Constantsij e ij e i j j i e eT u u u

Helmholtz Theorem : u ψ

22

2 2

22

2 2

210,

10,

e el

l

et

t

cc t

cc t

ψ

ψ

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 waveAC wave and SH wave

Leads to

Define

Define then

Universal Wave Equation

22

massdensity

1, e t

t

cc

Universal wave equation

Triangular Lattice

Square Lattice

Reduced frequency

Bloch Theorem

Photonic crystals as optical components

P. Halevi et.al.Appl. Phys. Lett.75, 2725 (1999)

See alsoSee alsoPhys. Rev. Lett. Phys. Rev. Lett. 8282, 7, 719 (1999)19 (1999)

Long Wavelength LimitLong 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

吳明昌 2004.06

Surface wave + Photonic waveguide

Coupled-Resonator Waveguide

Snell’s LawSnell’s Law

1 1 2 2' or sin siny yk k n nc c

Constant Frequency CurveConstant 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 InsulatorSonic Insulator

Sculpture Rod Array

Phys. Rev. Lett. 80, 5325 (1998)

Phononic Band StructuresPhononic 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

2 2

221 | | 1 | |

, ( ), ( )2 2cell cell

dA r c c rc

dA

From the universal wave equation, we can derive:

Or E (type I) = E (type II)

Varyingα(r) and c (r), we obtain:

2 22

2 2

2

1 1| | | |

1

| | | |2cell cell

cell cell

dA dAc

dA dAc

SeeZ. Q. ZhangPRB 61,1892(2000)APL 79,3224(2001)R. D. MeadeJ. Opt. Soc. Am. B 10, 328 (1993)

Acoustic Band Gap formationAcoustic Band Gap formation

• Soft material (small Soft material (small ρρc^2c^2) ) Soft spring Soft spring Elastic po Elastic potential energytential energy

• Heavy material (largeHeavy material (largeρρ) ) Lead sphere Lead sphere Kinetic en Kinetic energyergy

• Soft-light material (region I)—Hard-heavy material (rSoft-light material (region I)—Hard-heavy material (region II) system egion II) system Phonon (2 atoms per primitive bas Phonon (2 atoms per primitive basis) is) A gap appears between the 1st and the 2nd band A gap appears between the 1st and the 2nd bands, just like the gap between the “phonon branch” and “s, just like the gap between the “phonon branch” and “optical branch”optical branch”

• Separation of these two kinds of energy Separation of these two kinds of energy Large gap Large gap• Region I should be disconnected (hard to move), and rRegion I should be disconnected (hard to move), and r

egion II should be connected (easy to move)egion II should be connected (easy to move)

Water Background-Air Cylinders Sonic Crystal

C_w = 1490m/s,C_w = 1490m/s,C_a = 340m/s,C_a = 340m/s,ρρ_a/_a/ρρ_w = 0.00129_w = 0.00129

Filling fraction=1/1000Filling 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 po

int 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 See C. F. Chien and R. V. WaterhouseJ. Acoust. Soc. Am. 101,705 (1996),705 (1996)

Bloch Water WaveBloch 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 Wave Propagation in Periodic StructuresStructures — — Electric Filters Electric Filters and Crystal Latticesand Crystal Lattices

““Waves always behave in a similar Waves always behave in a similar way, whether they are longitudinal way, whether they are longitudinal or transverse, elastic or electric. or transverse, elastic or electric.

Scientists of the last (19th) century Scientists of the last (19th) century always kept this idea in mind.”always kept this idea in mind.”

--- --- L. BrillouinL. Brillouin

Thank You for Your Attention !Thank You for Your Attention !