Anomalous Refraction and Photonic Crystal Lenses
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Transcript of Anomalous Refraction and Photonic Crystal Lenses
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Anomalous Refraction and
Photonic Crystal Lenses
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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
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J. B. Pendry
Famous PeopleFamous People
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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)
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Focusing of electromagnetic waves by periodic arrays of dielectric cylinders
Bikash C. Gupta and Zhen Ye,Phys. Rev. B 67,153109 (2003)
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Long Wavelength LimitLong Wavelength Limit
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Negative Refraction
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Permittivity, Permeability Permittivity, Permeability Reflection, and Refraction Reflection, and Refraction
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Principle of the Negative RefractionPrinciple of the Negative Refraction
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Left-Handed Materials
D. R. Smith et. al., Physics Today, 17, May (2000).
Phys. Rev. Lett. 84, 4184 (2000) ; Science, 292, 77 (2001)
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The Building Blocks of LHM
2
2( ) 1 p
2
2 20
( ) 1F
Electric Dipoles Magnetic Dipoles+
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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
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J. B. Pendry’s “Perfect Lens”J. B. Pendry’s “Perfect Lens”
0, 0, 0n J. B. Pendry, Phys. Rev. Lett. 80, 3966 (2000)
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Surface-Plasmon-Polaritons (SPP)
SPP exists whenε<0 or μ<0 in the blue region
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Subwavelength Focusing EffectSubwavelength Focusing Effect Surface-Plasmon-Polariton (SPP) Surface-Plasmon-Polariton (SPP)
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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)
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Negative RefractionNegative Refractionof Modulated of Modulated EM WavesEM WavesAPL 81, 2713 (2002)APL 81, 2713 (2002)
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Simple ExplanationSimple Explanation
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Gaussian BeamGaussian Beam
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Refraction of a Wave PacketRefraction of a Wave Packet
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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)
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Perfect Lens ?Perfect Lens ?
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(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
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Calculation of the Electric FieldCalculation of the Electric Field
Total E field:
Green’s function:
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Fourier TransformFourier Transform
Boundary conditions:
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Solution of Green’s FunctionSolution of Green’s Function
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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
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Realizable vs. Unrealizable situationsRealizable vs. Unrealizable situations
0| |d z0| |d z
Virtual images
SourceNo solution can exist in this blank region
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Absorptive Lens (I)Absorptive Lens (I)1.0 0.0001i 11.7d d
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Absorptive Lens (II)Absorptive Lens (II)
11.1d d 11.3d d12.3d d
1.0 0.0001i
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Subwavelength FocusingSubwavelength Focusing
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Field Strength --- Type IField Strength --- Type I
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Field Strength --- Type IIField Strength --- Type II
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Field Strength --- Type IIIField Strength --- Type III
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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
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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.
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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)
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Snell’s Law—The Generalized FormSnell’s Law—The Generalized Form
1 1 2 2' or sin siny yk k n nc c
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Negative Refraction by Calcite ( Yau Negative Refraction by Calcite ( Yau et.alet.al. ). )
http://arxiv.org/abs/cond-mat/0312125
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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)
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Negative Refraction --- ExperimentNegative Refraction --- Experiment
Costas M. Soukoulis et. al., Nature 423, 604, 5 June 2003
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Subwavelength Imaging
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Subwavelength Focusing by PC
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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)
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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)
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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)
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Negative refraction or partial band gap effect? Square lattice, rotated by 45˚ (II)
Phys. Rev. B 70, 113101 (2004)
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Negative Negative Refraction?Refraction?
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Negative refraction ? (very large incidence angle ) Square lattice, rotated by 45˚
73˚ incidence
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Constant Frequency Curve—Triangular latticeConstant Frequency Curve—Triangular lattice
Phys. Rev. B 67, 235107 (2003)
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Negative refraction and left-handed behavior in two-dimensional photonic crystalsS. Foteinopoulou and C. M. SoukoulisPhys. Rev. B 67 235107
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Constant Frequency CurveSquare Lattice v.s. Triangular Lattice
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Negative Refraction—Triangular Lattice
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Negative refraction Triangular lattice, strong reflection
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Negative refraction Reducing reflection by proper termination of the surfaces
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Negative Refraction Beam propagation, proper termination
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PC Slab Lens – Triangular Lattice (with proper termination of the slab surfaces)
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Superluminal Phenomenon?
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Anomalous Reflection
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Left-Handed Materials Does it really work at the long-wavelength regime?
λ/a = 5~7
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APL, 85, 341 (2004)
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APL, 85, 1072(2004)
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Beyond the Long-wavelength LimitBeyond the Long-wavelength Limit
a/λ= 0.49 a/λ= 0.58
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Convex Photonic Crystal Lens (Triangular Lattice)
a/λ= 0.49 a/λ= 0.58
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Concave Photonic Crystal Lens (Triangular Lattice)
a/λ= 0.49 a/λ= 0.58
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Terraced V shaped PC Lens operating at an NR frequency
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Calculating the Spot Size and Focal Length
Source field
Distribution Width
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NR-PC Lens as Wave Coupler
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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
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Thanks for Your Attention !