Introduction to Photonic/Sonic Crystals Pi-Gang Luan (欒丕綱) Institute of Optical Sciences National Central University (中央大學光電科學研究所)

Collaborators: Wave Phenomena: Quantum and Statistical Mechanics: Chii-Chang Chen (陳啟昌, NCU), Since 2002 Zhen Ye (葉真, NCU), Since 1999 Tzong-Jer Yang (楊宗哲, NCTU) , Since 2001 Quantum and Statistical Mechanics: Yee-Mou Kao (柯宜謀, NCTU), Since 2002 Chi-Shung Tang (唐志雄, NCTS), Since 2002 De-Hone Lin (林德鴻, NCTU), Since 2000

Contents Photonic Crystals Negative Refraction Sonic Crystals Bloch Water Wave Conclusion

Famous People Eli Yablonovitch Sajeev John

Famous People J. D. Joannopoulos 沈平（Ping Sheng）

Photonic/Sonic Crystals 1D Crystal 3D Crystal 2D Crystal

Photonic Crystals

3D Photonic Crystal?

Photonic Band Structure

Photonic Band Structure

Photonic Band Structure

Artificial Structures and their Properties 1. Photonic Crystals: Man-made dielectric periodic structures. According to Bloch’s theorem, any eigenmode of the wave equation propagating in this kind of medium must satisfy: Usually the frequency spectrum of a photonic crystal has the “band structure”, that is, there are “pass bands” (which correspond to the situation that the eigenmode equation has the “real k solution”) and “stop bands” (also called forbidden bands or band gaps, in which the eigenmode equation has no “real k solution”).

The complex-k mode is a kind of evanescent wave (or the so called “near field”), which cannot survive in an infinitely extended photonic crystal region (ruled out by the boundary conditions at +∞ and -∞). However, near a surface (interface) or a defect (for example, a cylinder or a sphere with a different dielectric constant or radius), the evanescent wave can exist (the surface mode or the defect mode). The EM waves do not propagate along the direction that the wave amplitude decays. Using this property one can control the propagation of the light. Examples: photonic insulators (omni-directional reflectors, filters), waveguides, resonance cavity, fibers, spontaneous emission inhibition, etc. Even a pass band is useful, since it provides a different dispersion relation (the w-k relation) . We can design some “effective media”, usually they are anisotropic media. We can even use them to design novel lenses and wave plates.

Electromagnetic Waves Assuming that J = ρ = 0 (charge free and current free) in the system, then Faraday’s Law + Ampere’s Law lead to the wave equations for the E-field and H-field. In a two-dimensional system, the permittivity (the dielectric constant ε) and the permeability (μ) become z-independent functions. If k_z = 0 , then we have E-polarized wave (nonzero E_z, H_x, H_y) and the H-polarized wave (nonzero H_z, E_x, E_y). These two kinds of waves are decoupled. For monochromatic EM waves with a time factor exp(-iwt), we have D proportional to (curl H), and B proportional to (curl E), thus the two divergence equations div D=0 and div B=0 are redundant.

E- and H-polarized EM Waves E-polarized wave H-polarized wave

2. Phononic/Sonic (or Acoustic) Crystals: Man-made elastic periodic structures. In them both the mass density and the elastic constants (Lam’e coefficients) are periodic functions of position. All the effects (except the quantum effects) discussed before (i.e., the band structures, the band gaps, the evanescent waves, the different dispersion relations) can happen here. In addition, there are more material parameters (both the mass density and the elastic constants can be varied). The main research interests include the “sound barriers” , “noise filters”, and “vibration attenuators”. There are also some researches on “acoustic lens” and “negative refraction”.

Elastic Waves Pressure field & Shear Force Longitudinal & Transverse waves

Acoustic Wave and SH (shear) Wave Two-Dimensional Wave Crystal In an ideal (composite) fluid, shear force = 0, thus only the longitudinal wave (i.e., the pressure wave) can propagate inside. In a 2D system , the mass density and Lam’e constants are z-independent functions. If the wave propagation direction k has zero component along the z axis (i.e., k_z=0), then u_xy (i.e., the component lying on the xy plane ) and u_z (the component that parallel to the z axis) are decoupled.

AC wave and SH wave Leads to Define Define then

Universal Wave Equation

Triangular Lattice Square Lattice Reduced frequency Bloch Theorem

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

Long Wavelength Limit

Focusing of electromagnetic waves by periodic arrays of dielectric cylinders Bikash C. Gupta and Zhen Ye, Phys. Rev. B 67, 153109 (2003)

Light at the End of the Tunnel 19 March 2004 Phys. Rev. B 69, 121402 Phys. Rev. Lett. 92, 113903

Surface wave + Photonic waveguide 吳明昌 2004.06

Coupled-Resonator Waveguide

Snell’s Law

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

“Negative refraction and left-handed behavior in two-dimensional photonic crystals” S. Foteinopoulou and C. M. Soukoulis

Sonic Insulator Sculpture Rod Array Phys. Rev. Lett. 80, 5325 (1998)

Phononic Band Structures

Acoustic Band Gaps J. O. Vasseur et. al., PRL 86, 3012 (2001)

“Giant acoustic stop bands in two-dimensional periodic arrays of liquid cylinders” M. S. Kushwaha and P. Halevi Appl. Phys. Lett. 69, 31 (1996)

Acoustic Lens Using the pass band (Propagating Modes) A Lens-like structure can focus sound Refractive Acoustic Devices for Airborne Sound Phys. Rev. Lett. 88, 023902 (2002)

Locally Resonant Sonic Material Ping Sheng et. al., Science 289, 1734 (2000)

Application (I): Band Gap Engineering From the universal wave equation, we can derive: See Z. Q. Zhang PRB 61,1892 (2000) APL 79,3224 (2001) R. D. Meade J. Opt. Soc. Am. B 10, 328 (1993) Or E (type I) = E (type II) Varyingα(r) and c (r), we obtain:

Acoustic Band Gap formation Soft material (small ρc^2)  Soft spring  Elastic potential energy Heavy material (largeρ)  Lead sphere  Kinetic energy Soft-light material (region I)—Hard-heavy material (region II) system  Phonon (2 atoms per primitive basis)  A gap appears between the 1st and the 2nd bands, just like the gap between the “phonon branch” and “optical branch” Separation of these two kinds of energy  Large gap Region I should be disconnected (hard to move), and region II should be connected (easy to move)

Water Background-Air Cylinders Sonic Crystal C_w = 1490m/s, C_a = 340m/s, ρ_a/ρ_w = 0.00129 Filling fraction=1/1000

Application (II) Energy Flow Vortices in Wave Crystals A singular point is a vortex if and only if it is an isolated zero of Φ. The vorticity is nonzero. A singular point is a saddle point if it is an isolated zero of Q , an isolated point at which the phases of Q and Φ differ by odd multiples of π/2 or a combination of the previous two situations. The vorticity is zero. See C. F. Chien and R. V. Waterhouse J. Acoust. Soc. Am. 101,705 (1996)

Bloch Water Wave “Visualization of Bloch waves and domain walls” by M. Torres, et. al. Nature, 398, 114, 11 Mar. 1998 See also: PRE 63, 011204 (2000) PRL 90, 114501 (2003)

Scientists of the last (19th) century always kept this idea in mind.” Wave Propagation in Periodic Structures — Electric Filters and Crystal Lattices “Waves always behave in a similar way, whether they are longitudinal or transverse, elastic or electric. Scientists of the last (19th) century always kept this idea in mind.” --- L. Brillouin

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