Optical Metamaterials Negative index and perfect lensing air n =1 = - 1 q Propagating waves n=-1 image plane Evanescent waves
Fish in a Pond n=1.3 n=-1.3
Negative permittivity and permeability Maxwell’s Equations for plane monochromatic waves Right-handed triplet Left-handed triplet Veselago, 1968 Why negative n? Energy Does not depend on the sign! (Right-hand) Energy and k are in opposite directions Energy always moves forward k moves backward
Refraction between right and left handness x n>0? z Is not conserved
Refraction between right and left handness x n<0 z conserved
Negative refraction Taken from youtube
Properties of NIM slab Flat lens d ABCD matrix Interface between air n =1 = - 1 q ABCD matrix Interface between Propagation of distance d 2d Imaging with NIM slab
Properties of NIM slab Impedance matching for n=-1 air n =1 = - 1 q Impedance matching for n=-1 2d No reflection for any K-vector
Negative refraction makes a perfect lens Sir John Pendry Propagating waves Object plane n=-1 Evanescent waves air n =1 = - 1 q image Negative-index Metamaterials Super-resolution Imaging Optical Illusion (e.g., cloaking)
Negative refraction makes a perfect lens Pendry assumed lossless NIM with n=-1 Calculated the transmitted field as a function of k (OTF) Fresnel coefficients for magnetic field of TM wave From air to NIM (0=air 1=NIM): , , air n =1 = - 1 q (Impedance matching)
Negative refraction makes a perfect lens Pendry assumed lossless NIM with n=-1 Calculated the transmitted field as a function of k (OTF) Fresnel coefficients for magnetic field of TM wave Transmission through the slab , , air n =1 = - 1 q Symmetric Transmission function
Negative refraction makes a perfect lens Now, what about evanescent waves? evanescent waves amplification?
Negative refraction makes a perfect lens Before going into discussion… what do we have? Reflection coefficient goes to infinity ? Convergence of the geometric series Sign of evanescent kz Enhancement for every kx 1. Reflection coefficient Yes. Reflection coefficient can be infinite (pole in the transfer matrix)! This corresponds to guided mode (like surface plasmon) and can occur even when epsilon and mu different than -1 Example – plasmon dispersion
Negative refraction makes a perfect lens Reflection coefficient goes to infinity ? Convergence of the geometric series Sign of evanescent kz Enhancement for every kx 2. Convergence of the geometric series Works only if q<1 Or q is not real positive (negative or complex)
Negative refraction makes a perfect lens Reflection coefficient goes to infinity ? Convergence of the geometric series Sign of evanescent kz Enhancement for every kx 3. Sign of kz Not sure? Let’s pick the “-” sign
Negative refraction makes a perfect lens How come? Changing the sign of kz1 corresponds to r1/r The sign of kz1 does not matter!
Negative refraction makes a perfect lens Reflection coefficient goes to infinity ? Convergence of the geometric series Sign of evanescent kz Enhancement for every kx 3. Works for every kx This happens in the limit Why? Because this “enhancement” is related to the existence of surface waves For every k?
Negative refraction makes a perfect lens Reflection coefficient goes to infinity ? Convergence of the geometric series Sign of evanescent kz Enhancement for every kx 3. What about the limit for
Evanescent amplification condition Condition for amplification:
Evanescent amplification condition Condition for amplification: Deviation from the resonance condition limits resolution
Evanescent amplification condition
Practical issues Solution – use metallic slab (“poor man’s lens”) No such material exists in nature Loss seems inevitable Dispersion, causality, etc… Solution – use metallic slab (“poor man’s lens”) Only negative permittivity Does not support propagating waves No impedance matching Only works for TM through surface plasmons erxcitation
Slab of lossless “metal” Assume lossless metal with Object plane e=-1 image Take the limit Seems we get a similar result, but…
Slab of lossless “metal” For What happens for
Evanescent amplification in metal?
Metal vs. NIM Single polarization (TM only) Single plasmon wave-vector at high kx Impedance mismatch - reflection Supports evanescent waves only Metal NIM
Metal with loss NIM vs. Lossless metal is a nice theoretical discussion Neither lossless NIM or metal was obtained so far No NIM for super-resolution Real metal Loss can actually help! Permittivity mismatch too! D=35nm PMMA/silver@365nm
Super-resolution imaging with silver slab Zhang (2005) silver superlens (‘poor man’s lens’) 35 nm silver slab 365nm illumination 60/90nm resolution
Superlens at mid-infrared Silicon Carbide – e goes negative between 10.3mm and 12.5mm Phonon polariton Near-field microscopy l/20 resolution
Conclusions So far, only metallic super-lens for near-field imaging Resolution is limited by the loss NIMs have been demonstrated, without super-resolution Their unit cell is typically not small enough for high-k Next lessons: Realization of NIMs by artificial magnetism Extraction of their properties Other metamaterials for super-resolution applications