光子晶體原理與計算 (II)

光的負折射及異常傳播 (Negative refraction and anomalous propagation of light)

光子晶體透鏡 (Photonic crystal lenses)

次波長聚焦 (Subwavelength focusing)

Pi-Gang Luan ( 欒丕綱 )Wave Engineering Lab ( 波動工程實驗室 )

Institute of Optical Sciences

National Central University

( 中央大學光電科學研究所 )

負折射 (Negative Refraction)

西元 1968 年由蘇聯科學家V. G. Veselago 提出 “左手介質” (Left-handed materials) 理論

西元 2000 年 , 美國物理學家D. R. Smith 做出 “人工的” (artificial) 左手介質

同年 , 英國物理學家 John. B. Pendry 在 Phys. Rev. Lett. 提出 “完美透鏡” 構想 , 而日本 科學家 M. Notomi 仔細分析了光子晶體在 “非長波極限” 下的光傳播行為

近年 , “ 光子晶體負折射” 與 “次波長成像” 成為熱門的研究領域

Left-Handed MaterialsLeft-Handed Materials

D. R. Smith et. al., Physics Today, 17, May (2000).

Phys. Rev. Lett. 84, 4184 (2000) ; Science, 292, 77 (2001)

The The Building BlocksBuilding Blocks of LHM of LHM

2

2( ) 1 p

2

2 20

( ) 1F

Electric DipolesElectric Dipoles Magnetic DipolesMagnetic Dipoles+

左手介質的負折射機制 : TM 波

0, 0

, , , t t n nE H B D 在邊界連續2

* | |Re

8 8

c c

kS E H

兩種負折射的比較

左手介質 光子晶體 ( 利用特殊的色散特性 )ε < 0, μ < 0利用

Negative Refraction by LHM: Beam Propagation

Negative Refraction of PC: Beam propagation

完美透鏡 ( Perfect Lens)

J. B. Pendry, Phys. Rev. Lett. 80, 3966 (2000)

0, 0, 0n

“All this was pointed out by Veselago some time ago. The new message in this Letter is that, remarkably, the medium can also cancel the decay of evanescent waves. The challenge here is that such waves decay in amplitude, not in phase, as they propagate away from the object plane. Therefore to focus them we need to amplify them rather than to correct their phase. We shall show that evanescent waves emerge from the far side of the medium enhanced in amplitude by the transmission process.”

vector (phase velocity)k

Poynting vector (energy flow)First proposed by V. G. Veselago (1968)

sf1 f2

表面電漿子 (Surface-Plasmon-Polaritons (SPP))

SPP exists whenε<0 or μ<0 in the blue region

Subwavelength Focusing EffectSubwavelength Focusing Effect Surface-Plasmon-Polariton (SPP) Surface-Plasmon-Polariton (SPP)

對 “負折射” 與 “完美透鏡” 的質疑對 “負折射” 與 “完美透鏡” 的質疑

• Negative Refraction Makes a Perfect Lens

J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).

• Left-Handed Materials Do Not Make a Perfect Lens

N. Garcia et al., Phys. Rev. Lett. 88, 207403 (2002)

• Perfect lenses made with left-handed materials: Alice’s mirror?

Daniel Maystre and Stefan Enoch, J. Opt. Soc. Am. A, 21, 122 (2004)

Is it Possible?Is it Possible?

• ““Left-Handed Materials Do Not Make a Perfect Lens”, ”, N. Garcia and M. Nieto-Vesperinas, PRL 88, 207403 (2002)

• “Wave Refraction in Negative-Index Media: Always Positive and Very Inhomogeneous”, P.M. Valanju, R. M. Walser, and A. P. Valanju, PRL 88, 187401 (2002)

Negative RefractionNegative Refractionof Modulated of Modulated EM WavesEM WavesAPL 81, 2713 (2002)APL 81, 2713 (2002)

Simple ExplanationSimple Explanation

斯乃爾定律 (Snell’s Law)

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

一般晶體 ( 方解石 Calcite) 所導致的負折射

http://arxiv.org/abs/cond-mat/0312125 ( 游漢輝教授等 )

能量速度 (Energy velocity) 與 群速度 (Group velocity)

spacetimee

spacetimeU

S

vEnergy velocity :

( )g kv kGroup velocity :

e gv vIt can be shown that

Wave energy flows along the normal direction of the constant frequency curve (surface)

等頻率曲線 (Constant Frequency Curve)Square Lattice v.s. Triangular Lattice

Subwavelength Focusing by PCSubwavelength Focusing by PC

All-angle negative refraction without negative effective index Chiyan Luo, Steven G. Johnson, and J. D. Joannopoulos, J. B. Pendry, Phys. Rev. B 65, 201104 (2002)

See also:

Phys. Rev. Lett. 90, 107402 (2003)Phys. Rev. B. 67 235107 (2003)Phys. Rev. B. 68 045115 (2003)

“Negative refraction and left-handed behavior in two-dimensional photonic crystals” S. Foteinopoulou and C. M. Soukoulis

次波長成像真的需要負折射嗎 ?

L. S Chen, C. H. Kuo, and Z. Ye, Phys. Rev. E 69, 066612 (2004)

Z. Y. Li and L. L. Lin, Phys. Rev. B 68, 245110 (2003)

S. He, Z. Ruan, L. Chen, and J. Shen, Phys. Rev. B 70, 115113 (2004)

Structure and Parameters

0

Dielectric cylinders in air, square lattice

(lattice constant )

Parameters: 14, 1, 0.3

The square lattice has been rotated by 45 .

Each layer contains 50 cylinders.

There are 8 layers in

a

r a

this structure.

Interlayer distance:

2 / 2 0.707

Source-surface distance 0.5

Working frequency (dimensionless) :

/(2 ) / 0.192

ˆE-polarized waves ( )

d a a

a

a c a

E

E z

Increasing the interlayer distance

1.5d 2d 2.5d 3d

d=0.707a

Decreasing the interlayer distance

d-2D d-5D d-8D d-10D

d = 0.707a

r = 0.3a

D = (d-r)/10

Negative Refraction?

See also Phys. Rev. E 70, 056608 (2004)  

Phys. Rev. B 70, 113101 (2004)

大角度入射時 , 正方晶格光子晶體中的負折射現象Square lattice, rotated by 45˚

73˚ incidence

負折射現象—三角晶格

光子晶體透鏡 –三角晶格

相位波的超光速傳播現像 (Superluminal propagation of phase wave)

異常反射 (Anomalous Reflection)

Beyond the Long-wavelength LimitBeyond the Long-wavelength Limit

a/λ= 0.49 a/λ= 0.58

Convex Photonic Crystal Lens (Triangular Lattice)

a/λ= 0.49 a/λ= 0.58

Concave Photonic Crystal Lens (Triangular Lattice)

a/λ= 0.49 a/λ= 0.58

Negative Refraction by PC

• Subwavelength imaging does not imply negative refraction

• Anomalous refraction, anomalous reflection and strong anisotropy are common features for wave propagation in artificial media beyond the long-wavelength limit

• Mesoscopic phenomena can happen in both nanoscale world and macroscopic world, only the relative size between the wavelength and the wave-environment interaction range is important

Wave-Environment Interaction in Mesoscopic World

General Features

• Wave coherence is important• Complex boundaries or many scatterers • Wavelength ~ Mean scattering distance (Mean free path)• Scattering strength (coupling constant) cannot be too small• Multiple scattering (the bare waves are repeatedly scattered)• The renormalized wave can be very different from the bare

waves• The actual size is irrelevant, the relative size is the key

parameter. So “Mesoscopic” does not imply “Nanoscale”• Similar phenomena can happen in quantum and classical

(electromagnetic and acoustic) systems• Wave equations + Boundary conditions = Physics

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