Photonic Crystal Defects as Basic Building Blocks For Metamaterials Design Development of novel electromagnetic-moment elements as basic building blocks for the design of metamaterials with desired permittivities and permeabilities using defects in photonic crystals.

Negative Refraction Phenomena in Photonic Crystals

Exploration of negative refraction and superlensing phenomena in photonic crystal systems.

Extension of Left-Handed Material Behavior to Optical Frequencies

J. D. Joannopoulos Massachusetts Institute of Technology DoD MURI on Metamaterials Photonic Crystal Defects as Basic Building Blocks For Metamaterials Design

Point-defects in photonic crystals consisting of dielectrics and/or metals, can be designed to have single mode resonance properties that can be either primarily electric field or magnetic field in nature.

In order to optimize a defect state’s electric field or magnetic field nature, it is preferable to work in a photonic crystal environment that can distinguish, as much as possible, between TM-like modes (where the electric field will be of primary interest) and TE-like modes (where the magnetic field is of primary importance). A natural way to distinguish between TM-like modes and TE-like modes is to employ 2D photonic crystal slab systems, two examples of which are illustrated below.

Dielectric rods in air are useful for isolating TM-like modes, while air holes in dielectric are useful for isolating TE-like modes. Unfortunately, point-defects in such 2D slab configurations are intrinsically lossy because of their coupling to the radiation manifold.

DoD MURI on Metamaterials Recently, we designed a 3D photonic crystal that possesses a large omnidirectional photonic band gap (up to 27% using Si at 1.5 microns) and consists of alternating stacks of the two basic 2D slab configurations already discussed. The key advantage of this structure is that point-defects can now be introduced systematically in the layers consisting of dielectric-rods-in-air (to create electric moments) or in the layers consisting of air-holes-in-dielectric (to create magnetic moments), without incurring intrinsic radiation losses due to the lack of a complete photonic band gap.

DoD MURI on Metamaterials Thus metamaterials designed to consist of arrays of these defects (or moments) could lead to novel materials with interesting new electric field or magnetic field responses, and provide a means for tailoring permittivity or permeability to desired specifications. DoD MURI on Metamaterials cutting the crystal to have cylindrical border with defect at center

fields at t=6000

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DoD MURI on Metamaterials “Magnetic Dipole” Radiation

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PBG Defect Radiation DoD MURI on Metamaterials defect: larger computational cell

fields at t=6000

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0 1000 2000 3000 4000 5000 6000 7000 DoD MURI on Metamaterials Triangular lattice of air holes in dielectric with triangular lattice of defects

r: radius def-r: defect radius

a: center-to- center hole separation

center-to- center defect separation is 3a

DoD MURI on Metamaterials Band Structure: TE modes (H perpendicular to plane) ε=9, r=0.3a, def-r=0.65a

0.34

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„ n<-1 ν = 0.260-0.268 c/a DoD MURI on Metamaterials Negative Refraction Phenomena in Photonic Crystals

Negative refraction of electromagnetic waves in Left-Handed materials is the foundation for a variety of novel phenomena, including super-lensing as shown in the figure below.

super-lens conventional lens Veselago (1968)

normal refraction medium

object image object image

negative refraction medium

Recent work (Notomi et el.) indicates that negative refraction phenomena in photonic crystals are possible in regimes of negative group velocity and negative effect index above the first band near the Brillouin zone center. However, lower frequencies in the band structure may be more desirable in high-resolution superlensing. Consequently, the ability to obtain negative refraction in the lowest band is of particular interest. DoD MURI on Metamaterials In a collaborative effort with John Pendry we have recently discovered that single-beam negative refraction in photonic crystals is indeed possible for all incoming angles for the lowest photonic band if one operates at a region near a Brillouin zone corner.

Interestingly, this occurs in a regime of positive effective index of refraction! The band has a positive group velocity and a positive refractive index, but a negative photonic “effective mass”. We have identified a frequency range so that for all incident angles one obtains only a single, negative-refracted beam. Such all-angle negative refraction (AANR) is essential for superlens applications. DoD MURI on Metamaterials To illustrate this phenomenon, we have designed and numerically simulated photonic crystal micro-superlenses. We are working with our experimental colleagues Gang Chen, Shelly Schultz, and Dave Smith in order to realize this for the TM modes at microwave length scales.

DoD MURI on Metamaterials Extension of Left-Handed Material Behavior to Optical Frequencies

The goal is to design a 3D photonic crystal with a large frequency range in which the effective negative- index concept is still valid. In particular, we study again the possibility of All-Angle Negative Refraction (AANR), i.e. negative refraction for beams of all incident angles from air. To realize AANR, suffcient criteria are that the frequency range be near a negative “photonic-mass” region in the bandstructure and below the diffraction threshhold, and the photonic-crystal constant-frequency contour be all-convex and larger than that of air. Clearly, this is only possible in the first few bands. In addition, above the first two bands care must be taken to ensure that the symmetry of the photonic modes allows good coupling from external planewaves. The geometric lattice of the required 3D photonic crystal can be determined from the following intuitive argument. In the periodic zone scheme, the constant-frequency contours for the first few bands of the photonic crystal can be constructed by joining all the spherical contours of an effective uniform medium which are centered on the reciprocal lattice sites and rounding the sharp parts of the joint surface across Brillouin zone boundaries. For a given Brillouin zone corner C, we expect that the more neighboring reciprocal-lattice sites C has, the stronger the resulting rounding effect and the easier it is for the constant-frequency contours to become all-convex around C. Thus, a rough rule to choose the geometric lattice for AANR is just to maximize the number N of C’s nearest-neighbor reciprocal-lattice sites. If AANR is to be realized in the fundamental (i.e. the first two) bands, then C is a corner of the first Brillouin zone. In this case, a simple-cubic (SC) reciprocal lattice with N = 8 should be used, resulting in a SC photonic crystal with (111) surface termination. If AANR is to be realized in the bands after folding once, then C is a corner of the second Brillouin zone, which in most lattices is just Γ after translation of a reciprocal-lattice vector. This is the usual effective negative-index situation, and the Face-Centered Cubic (FCC) reciprocal lattice which has N = 12 should be chosen, giving a Body-Centered Cubic (BCC) structure in real space. DoD MURI on Metamaterials Band structure of a BCC lattice of air cubes in dielectric ε = 18.The cubes have sides 0.75a and are oriented with sides parallel to those of the conventional BCC cell.In the shaded AANR frequency range, the photonic crystal exhibits negative refraction for incoming radiation of all angles.The dashed lines are light lines along ΓH and ΓN.

DoD MURI on Metamaterials A systematic method to modify the photonic crystal for layer-by-layer fabrication. (a) is a side cross-section of the original design.(b) is an approximation of (a) by replacing each cubic void by two block voids.(c) has three block voids to form the stairs.(d) uses four block voids to approximate (a).The numbers in each figure shows the AANR size achievable in each of the approximate designs.The shaded region indicates high index materials.

DoD MURI on Metamaterials