(I)

Bloch ,, , , Pi-Gang Luan ()Wave Engineering Lab ()Institute of Optical Sciences National Central University()

: () , (Energy Bandgaps). , (Frequency Bandgaps)?1987, Eli Yablonovitch Sajeev John (Phys. Rev.Lett. 20, 2059 (1987) Phys. Rev.Lett. 23, 2486 (1987 )). (Photonic Crystals).Eli Yablonnovitch Bandgaps , . Sajeev John Bandgaps, , Anderson (Anderson Localization).

Famous People

, . , , ., , ., , , , , , , , . , .

(Photonic Crystals)a b c

Photonic Crystals

() , (2D System)Photonic band structure, Photonic band gap

() : (Defect Mode) (Guiding Mode)

(Photonic Crystal Fiber)

, . : 1. , , , . 2. , .

Photonic crystals as optical components

P. Halevi et.al.Appl. Phys. Lett.75, 2725 (1999)

See alsoPhys. Rev. Lett. 82, 719 (1999)(I) (Long-wavelength Limit): (Anisotropic Lens)

:

(II) (Beyond the Long-wavelength Limit) (Negative Refraction Lens and Subwavelength Imaging)

, , : Fourier (Reciprocal Lattice)Bloch (Blochs Theorem) (Plane Wave Expansion Method) (Photonic energy flow), (Energy velocity) (Group velocity) (Constant frequency curves)

Periodic function, Fourier Series and Reciprocal Lattice

() Lattice Bases vs. Reciprocal Lattice Bases (2D)Square Lattice Triangular Lattice

Binary System and Structure Factor (2D)

Example 1: Square Lattice, Circular Rods/Holes

Example 2: Triangular Lattice, Circular Rods/Holes

Example 3: Simple Cubic Lattice, Spheres

Blochs Theorem (Electron Systems):Blochs Theorem and Brillouin zoneFirst Brillouin Zone

Proving Blochs Theorem

Band Structure (Electron System)

Blochs Theorem and Photonic Band Structure

Two-Dimensional Inhomogeneous Wave SystemsClassical WavesUnified Treatment

Band Structure Calculation (2D, Scalar Wave)

Reduced Brillouin Zone and Dimensionless Frequency

(Bloch Modes)

(Photonic Band Gaps) (Transmission)Left: Photonic Band , Right: Transmission (20layers)

(Photon Energy Flow) (Group Velocity)

Frequency ContoursSquare Lattice

(Multiple Scattering Theory)1. An array of dielectric cylinders in a uniform medium.2. In response to the incident wave from the wave source and the scattered waves from other scatterers, each scatterer will scatter waves repeatedly. Scattered waves can thus be expressed in terms of a modal series of partial waves. 3. Regarding these scattered waves as the incident wave to other scatterers, a set of coupled equations can be formulated and computed rigorously. 4. The total wave at any spatial point is the sum of the direct wave from the source and the scattered waves from all scatterers. 5. The details about MST can be found in J. Appl. Phys. 94, 2173 (2003) (B. Gupta and Z. Ye ()). 6. For brevity, we only consider the E-polarized wave.

(wave source) (scatterers)

:

(Incident Waves), (Scattered Waves)

Addition Theorem and Coordinate Transformation

We also have the relations

(Solving the Total Waves)

(Plane Wave source)

()