1. 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)

2. 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

3. Contents
Photonic Crystals
Negative Refraction
Sonic Crystals
Bloch Water Wave
Conclusion

4. Famous People
Eli Yablonovitch
Sajeev John

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

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

7. Photonic Crystals

8. 3D Photonic Crystal?

9. Photonic Band Structure

10. Photonic Band Structure

11. Photonic Band Structure

13. 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”).

14. 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.

15. 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.

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

17. 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”.

18. Elastic Waves Pressure field & Shear Force
Longitudinal & Transverse waves

19. 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.

20. AC wave and SH wave
Leads to
Define
Define
then

21. Universal Wave Equation

22. Triangular Lattice
Square Lattice
Reduced frequency
Bloch Theorem

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

24. Long Wavelength Limit

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

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

27. Surface wave + Photonic waveguide
吳明昌

31. Coupled-Resonator Waveguide

32. Snell’s Law

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

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

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

36. Phononic Band Structures

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

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

39. 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, (2002)

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

42. 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:

43. 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)

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

45. 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)

46. 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, (2000)
PRL 90, (2003)

47. 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

48. Thank You for Your Attention !