Anomalous Refraction and

Photonic Crystal Lenses

Wave-Environment Interaction in Mesoscopic World

Important Features

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

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

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

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

J. B. Pendry

Famous PeopleFamous People

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)

Focusing of electromagnetic waves by periodic arrays of dielectric cylinders

Bikash C. Gupta and Zhen Ye,Phys. Rev. B 67,153109 (2003)

Long Wavelength LimitLong Wavelength Limit

Negative Refraction

Permittivity, Permeability Permittivity, Permeability Reflection, and Refraction Reflection, and Refraction

Principle of the Negative RefractionPrinciple of the Negative Refraction

Left-Handed Materials

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

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

The Building Blocks of LHM

2

2( ) 1 p

2

2 20

( ) 1F

Electric Dipoles Magnetic Dipoles+

The Idea of the “Perfect Lens”The Idea of the “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

J. B. Pendry’s “Perfect Lens”J. B. Pendry’s “Perfect Lens”

0, 0, 0n J. B. Pendry, Phys. Rev. Lett. 80, 3966 (2000)

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)

Is it Possible?Is it Possible?

• ““Left-Handed Materials Left-Handed Materials Do Not Make a Perfect Do Not Make a Perfect LensLens”, N. Garcia and M. ”, N. Garcia and M. Nieto-Vesperinas, PRL Nieto-Vesperinas, PRL 8888, 207403 (2002), 207403 (2002)

• ““Wave Refraction in Wave Refraction in Negative-Index Media: Negative-Index Media: Always Positive and Always Positive and Very InhomogeneousVery Inhomogeneous”, ”, P.M. Valanju, R. M. P.M. Valanju, R. M. Walser, and A. P. Walser, and A. P. Valanju, PRL Valanju, PRL 8888, , 187401 (2002)187401 (2002)

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

Simple ExplanationSimple Explanation

Gaussian BeamGaussian Beam

Refraction of a Wave PacketRefraction of a Wave Packet

Perfect Lens ?Perfect Lens ?

• Negative Refraction Makes a Perfect LensNegative Refraction Makes a Perfect Lens

J. B. Pendry, Phys. Rev. Lett. J. B. Pendry, Phys. Rev. Lett. 8585, 3966 (2000). , 3966 (2000).

• Left-Handed Materials Do Not Make a Perfect LensLeft-Handed Materials Do Not Make a Perfect Lens

N. Garcia N. Garcia et al.et al., Phys. Rev. Lett. , Phys. Rev. Lett. 8888, 207403 (2002), 207403 (2002)

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

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

Perfect Lens ?Perfect Lens ?

(1)0 0 0

20 0

ˆ( ) ( | |)

/ , / .

i t i trad e A H k e

k c A J c

E r y r r

Radiation field from the source:

System Description

Slab thickness: dPermittivity and permeability:

Line Source, located at (0, – d/2)

(2)0 0ˆ( ) ( )i t i te J e J r y r r

1 , 1i i

The radiation field satisfies the Helmholtz equation:

2 22

4( ) ( ) ( )radk i

c

E r J r

Calculation of the Electric FieldCalculation of the Electric Field

Total E field:

Green’s function:

Fourier TransformFourier Transform

Boundary conditions:

Solution of Green’s FunctionSolution of Green’s Function

Thickness Limitation on an Ideal LHM LensThickness Limitation on an Ideal LHM Lens

Divergenceless condition:

0

No source inside and behind the slab

: Time-averaged Poynting vector

S

S

Ideal lens: 1n

Phase matching problem: p1 and p2

I II III IV V0

p1 p2

I II III IV V0

p1 p2

I II III IV V0 I II III IV V0 I II III IV V0

p1 p2p1 p2

Realizable vs. Unrealizable situationsRealizable vs. Unrealizable situations

0| |d z0| |d z

Virtual images

SourceNo solution can exist in this blank region

Absorptive Lens (I)Absorptive Lens (I)1.0 0.0001i 11.7d d

Absorptive Lens (II)Absorptive Lens (II)

11.1d d 11.3d d12.3d d

1.0 0.0001i

Subwavelength FocusingSubwavelength Focusing

Field Strength --- Type IField Strength --- Type I

Field Strength --- Type IIField Strength --- Type II

Field Strength --- Type IIIField Strength --- Type III

Two Cases of ImagingTwo Cases of Imaging

0

Case 1:

1, 2,

2 / 0.3,

1.0 0.001

x

z d

k

i

0

Case 2:

1, 2,

2 / 2,

1.00 0.000001

x

z d

k

i

Uncertainty Principle vs. Uncertainty Principle vs. Subwavelength FocusingSubwavelength Focusing

This decaying behavior can be easily explained by the

According to this principle, we

must have the relation , here represents

the width

uncertainty principle

of the image, and represe

.

1

ntsx

xx k x

k

2 2 2 2

A subwavelength image is mainly formed by summing

over the Fourier components

the

fluctuation

of .

Since = / , these compone

.

nt

of tho

s must

se | | /

have i

t

maginary

's. This lea

ermsx

x z

z

xk

c

c

k

k k

k

ds to the decaying profile of the field strength.

Energy velocity vs. 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)

Snell’s Law—The Generalized FormSnell’s Law—The Generalized Form

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

Negative Refraction by Calcite ( Yau Negative Refraction by Calcite ( Yau et.alet.al. ). )

http://arxiv.org/abs/cond-mat/0312125

Negative Refraction by PCNegative Refraction by PC

“Refraction in Media with a Negative Refractive Index”

S. Foteinopoulou, E. N. Economou, C.M. Soukoulis

Phys. Rev. Lett. 90, 107402 (2003)

Negative Refraction --- ExperimentNegative Refraction --- Experiment

Costas M. Soukoulis et. al., Nature 423, 604, 5 June 2003

Subwavelength Imaging

Subwavelength 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)

Does subwavelength focusing need negative refraction?

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)

Negative refraction or partial band gap effect ? Negative refraction or partial band gap effect ? Square lattice, rotated by 45 Square lattice, rotated by 45˚̊ (I) (I)

Phys. Rev. E 70, 056608 (2004)  

Negative refraction or partial band gap effect? Square lattice, rotated by 45˚ (II)

Phys. Rev. B 70, 113101 (2004)

Negative Negative Refraction?Refraction?

Negative refraction ? (very large incidence angle ) Square lattice, rotated by 45˚

73˚ incidence

Constant Frequency Curve—Triangular latticeConstant Frequency Curve—Triangular lattice

Phys. Rev. B 67, 235107 (2003)

Negative refraction and left-handed behavior in two-dimensional photonic crystalsS. Foteinopoulou and C. M. SoukoulisPhys. Rev. B 67 235107

Constant Frequency CurveSquare Lattice v.s. Triangular Lattice

Negative Refraction—Triangular Lattice

Negative refraction Triangular lattice, strong reflection

Negative refraction Reducing reflection by proper termination of the surfaces

Negative Refraction Beam propagation, proper termination

PC Slab Lens – Triangular Lattice (with proper termination of the slab surfaces)

Superluminal Phenomenon?

Anomalous Reflection

Left-Handed Materials Does it really work at the long-wavelength regime?

λ/a = 5~7

APL, 85, 341 (2004)

APL, 85, 1072(2004)

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

Terraced V shaped PC Lens operating at an NR frequency

Calculating the Spot Size and Focal Length

Source field

Distribution Width

NR-PC Lens as Wave Coupler

Conclusion• Subwavelength imaging does not imply negative

refraction• Surface termination is important for reducing the

reflection• 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

Thanks for Your Attention !