Anomalous Refraction and Photonic Crystal Lenses

Wave-Environment Interaction in Mesoscopic World Important 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

Famous People J. B. Pendry

Photonic crystals as optical components P. Halevi et.al. Appl. Phys. Lett. 75, 2725 (1999) See also Phys. Rev. Lett. 82, 719 (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 Limit

Negative Refraction

Permittivity, Permeability Reflection, and Refraction

Principle 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 + Electric Dipoles Magnetic Dipoles

The Idea of the “Perfect Lens” First proposed by V. G. Veselago (1968) “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.” J. B. Pendry, Phys. Rev. Lett. 80, 3966 (2000)

J. B. Pendry’s “Perfect Lens” Phys. Rev. Lett. 80, 3966 (2000)

Surface-Plasmon-Polaritons (SPP) SPP exists whenε<0 or μ<0 in the blue region

Subwavelength Focusing Effect Surface-Plasmon-Polariton (SPP)

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 Refraction of Modulated EM Waves APL 81, 2713 (2002)

Simple Explanation

Gaussian Beam

Refraction of a Wave Packet

Perfect Lens ? 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)

Perfect Lens ?

System Description Slab thickness: d Permittivity and permeability: Line Source, located at (0, – d/2) Radiation field from the source: The radiation field satisfies the Helmholtz equation:

Calculation of the Electric Field Green’s function: Total E field:

Fourier Transform Boundary conditions:

Solution of Green’s Function

Thickness Limitation on an Ideal LHM Lens Ideal lens: I II III IV V p1 p2 Divergenceless condition: Phase matching problem: p1 and p2

Realizable vs. Unrealizable situations Virtual images Source No solution can exist in this blank region

Absorptive Lens (I)

Absorptive Lens (II)

Subwavelength Focusing

Field Strength --- Type I

Field Strength --- Type II

Field Strength --- Type III

Two Cases of Imaging

Uncertainty Principle vs. Subwavelength Focusing

Energy velocity vs. Group velocity It can be shown that Wave energy flows along the normal direction of the constant frequency curve (surface)

Snell’s Law—The Generalized Form

Negative Refraction by Calcite ( Yau et.al. ) http://arxiv.org/abs/cond-mat/0312125

Negative Refraction by PC in Media with a Negative Refractive Index” S. Foteinopoulou, E. N. Economou, C.M. Soukoulis Phys. Rev. Lett. 90, 107402 (2003)

Negative Refraction --- Experiment Costas M. Soukoulis et. al., Nature 423, 604, 5 June 2003

Subwavelength Imaging

Subwavelength Focusing by PC

negative effective index 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˚ (I) Phys. Rev. E 70, 056608 (2004)  

Negative refraction or partial band gap effect Negative refraction or partial band gap effect? Square lattice, rotated by 45˚ (II) Phys. Rev. B 70, 113101 (2004)

Negative Refraction?

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

Constant Frequency Curve—Triangular lattice Phys. Rev. B 67, 235107 (2003)

Negative refraction and left-handed behavior in two-dimensional photonic crystals S. Foteinopoulou and C. M. Soukoulis Phys. Rev. B 67 235107

Constant Frequency Curve Square 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?

APL, 85, 341 (2004)

APL, 85, 1072 (2004)

Beyond the Long-wavelength Limit

Convex Photonic Crystal Lens (Triangular Lattice)

Concave Photonic Crystal Lens (Triangular Lattice)

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

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