New Era of Discreteness and Periodicity in Optics George Stegeman , KFUPM Chair Professor

1. New Era of Discreteness and Periodicity in Optics George Stegeman, KFUPM Chair Professor Professor Emeritus College of Optics and Photonics/CREOL University of Central Florida, USA

2. 1887 What Is Meant By Discreteness and Periodicity? 1990s 1-D Periodic in 1D “Bragg grating” Periodic in 1D “Waveguide array” Periodic in 2D “Photonic crystal fiber” Collection of similar, discrete optical structures, materials, devices etc. which as an ensemble create new phenomena, functionalities or applications. Frequently periodicities are involved in discreteness, i.e. the structure arrangement is (quasi-) periodic in space.

3. n1 n1 n2 White Light Incidence Classic example of periodicity involving interference – dielectric mirror (frequency filter) Multi-layer structure - different refractive indices (optical impedances) in each layer, layer thicknesses /2 Single large (many optical wavelengths  in size) block of material (e.g. glass) n1n2  Essentially nothing happens • - Periodicities of order a few wavelengths • Analogy to solid state physics, but now with complete control over band structure • Multiple bands for propagation, i.e. Floquet-Bloch analysis of modes and dispersion • Guiding of radiation requires introducing defects into regular structures • Fabrication tolerances can be very demanding but technology available For an excellent discussion, see http://ab-initio.mit.edu/photons/

4. - Where Did Awareness of New Optics From Discreteness Start? Pioneering paper: E. Yablanovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics”, Phys. Rev. Lett., 58, 2059 (1987) Atom in infinite medium radiates in all directions at atwhen electron in an excited state (lifetime gm) drops to ground state via spontaneous emission (fundamental process) Inside a cavity, only radiation in cavity modes cav=md/2is allowed n=1, 2, 3…, i.e. standing waves |m> |g> d

5. - - Discreteness  New Science at cav=md/2 atcav |m> |m> atcav |g> |g> Lifetime of excited state altered  Fundamental process inhibited!

6. Length Scales of Periodicity and Consequences 1. Optical Wavelength - periodically modulated (in space) dispersion relations - prime examples are photonic crystals and waveguide arrays - many new wavevectors available for wavevector-conserving interactions - control of anomalous diffraction (space) and dispersion (time) possible - many new solitons - basic concepts closely related to solid state physics 2. Sub-optical Wavelength - modified optical properties when averaged over a wavelength - prime example is “meta-materials”  unique optical properties - negative refractive index - novel dispersion relations and propagation properties - etc. - effective medium theories important Photonic Crystal Fibers Waveguide Arrays Negative Index Photonic Crystals

7. White Light Incidence n1 Frequency Gap n2 band gap irreducible Brillouin zone 1 D “Photonic Crystal”  d d=/2 w Constructive interference on reflection at gap! k 0 –π/a 

8. 1887 1987 1-D 2-D 3-D Periodic in 1D “Bragg grating” Periodic in three dimensions Periodic in two dimensions

9. 3D Photonic Crystal: MIT 0 . 8 I. II. 0 . 6 2 1 % g a p z L' 0 . 4 U' G X K' W U'' U L W' 0 . 2 K 0 G U Õ L X W K • “Super-prism” Effect • Lasers • “Negative Refraction”

10. Prisms: Bulk Optics Snells Law: n1sin(1)=n2sin(2) Refractive Index Dispersion in Visible 2 1 n1 n2 Angular Dispersion

11. “Super” Prism: Photonic Crystal Bulk gap Photonic Crystal z L' U' G X K' G  X W U'' U L W' K Angular Dispersion

12. Super Prism Effect: Control Both Magnitude and Sign of Angular Dispersion SOI Planar Photonic Lattice Input Guides Output Guides PC Near -K direction 0.4o/nm Near -M direction 1.3o/nm (100x normal glass prism) A. Lupu, E. Cassan, S. Laval, et. al., Opt. Expr. 12, 5690 (2004)

13. y x z “2D Optical Circuits in Quasi-3D” Photonic Crystals Guided Wave Planar Device Concept “Light Channels” introduced by eliminating one row and/or column Note: – This 2D circuit must be imbedded in a 3D photonic crystal to avoid radiation loss along the z-axis!

14. kz “2D Optical Circuits in Quasi-3D” Photonic Crystals If height zis finite, we must couple to out-of-plane wavevectors… Make it as tall as possible!! z

15. Un-optimized Optimized Optimized Reducing BendingLosses: Technical University of Denmark Two Bend Loss/Loss of Straight Guide L.H. Frandsen, A. Harpøth, P.I. Borel, M. Kristensen, J.S. Jensen and O. Sigmund, Opt. Expr., 12, 5916 (2004)

16. Photonic Crystal Fibres: Cylindrical 1D Photonic Crystals • geometry, shape and filling material can be varied • fabrication improved to loss of 0.58dB/km at 1550nm • exquisite control of dispersion in effective index • - zero group velocity dispersion (GVD) wavelength • - multiple zero GVD wavelengths • - phase-matching of nonlinear interactions • photonic band gaps fibers • wavelength size modal areas  enhanced NLO Defect Necessary for Guiding Courtesy of Phillip Russell, Bath University

17. silica webs reduceGVD in PCF 300 silica strand (computed) 200 ) m k . 100 m n / s 0 p ( 100 - - 200 - 300 0.5 0.6 0.7 0.8 0.9 1.0 bulk silica m wavelength ( m) Group Velocity Dispersion (GVD) Control: Bath University Temporal Pulse Fiber Transmission Line anomalous GVD (ps/nm-km) normal Wavelength (m)

18. d = 0.58 mΛ = 2.59 m d = 0.57 mΛ = 2.47 m PCF With Ultra-Flat and Ultra-Small Dispersion (GVD): Bath d = hole size = hole separation 11 periods Control of dimensions to better than 1% required 2 anomalous 0 normal 2 dispersion (ps/nm.km) 4 6 8 10 1 1.2 1.4 1.6 wavelength (m)

19. Supercontinuum Generation: Opt. Expr. May 2006, Bath Fiber A nonlinear optics feast of effects!! Self- and cross-phase modulation Multi-wave mixing Stimulated Raman, Anti-stokes Raman Raman Self-Frequency Shift Third Harmonic Generation Etc. pump=1550nm 100fs pulses From <350nm to >3000 nm!

20. Waveguide Arrays: Coupling Between Waveguides an • anis field at n-th channel center • βis propagation constant of single isolated channel • E(x) is the transverse channel waveguide field. • c is coupling constant due to field overlap an+1 E(x) n n+1 nn+1

21. 20 Waveguide Number -20 Distance Arrays of Weakly Coupled Waveguides: “Discrete” Diffraction • Light spreads (diffracts) through array by “discrete diffraction”, via coupling c Discrete diffraction • 1D or 2D lattices of waveguides • feature dimensions of order of the wavelength of light • periodicity  multiple (Floquet-Bloch) bands for propagation • - negative refraction - normal, zero or anomalous diffraction • - discrete Talbot effect - photonic Bloch oscillations • Many novel “discrete” spatial solitons • - solitons with fields in-phase or out-of-phase in adjacent channels • - “interface “ solitons at edges, corners and between dissimilar arrays

22. Normal diffraction 1-3 degrees 1D Diffraction in Waveguide Arrays kx – Bloch wavevector (momentum) D < 0 “normal” diffraction D > 0 “anomalous” diffraction Homogeneous Medium Zero diffraction kz Normal diffraction  “Anomalous” diffraction  - kxd

23. 20 Waveguide Number 0 -20 Distance Finite Beam Excitation W z Diffraction in Bulk Media

24. Length Scales of Periodicity and Consequences • 2. Sub-optical Wavelength • - modified optical properties when averaged over a wavelength • - prime example is “meta-materials” •  unique optical properties • - negative refractive index • - novel dispersion relations and propagation properties • - etc. • - effective medium theories important • - basic concepts closely related to solid state physics

25. Negative Index Materials: Problem in Materials Science Pioneer: V. G. Veselago, Soviet Physics USPEKI 10, 509 (1968). • Negative Magnetic Permiability • Not found in nature • Negative Dielectric Constant • Found in nature (metals) • due to electron plasma resonances • Composite Materials with Metals • For metallic (sub-wavelength) inclusions • Plasmon (collective electron) resonances with resonant frequencies depending • on shape and size •  both electric and magnetic properties changed

26. Negative Index Materials in the Near Infrared =2000nm Au Al2O3 Shuang Zhang, Wenjun Fan, N. C. Panoiu,K. J. Malloy, R. M. Osgood and S. R. J. Brueck,Phys. Rev. Lett., 95, 137404 (2005)

27. Examples of Repercussions and Possible Applications: Contra-directional Energy and Phase Velocity Maxwell’s Equation predict: Wave vectors Wave vectors Poynting vectors (energy flow) Poynting vectors (energy flow) Courtesy of Allan Boardman, Salford University

28. “Cloaking” J. B. Pendry, D. Schurig, D. R. Smith, Science, 312, 1780 (2006) Quote: “it is now conceivable that a material can be constructed whose permittivity and permeability values may be designed to vary independently and arbitrarily throughout a material, taking positive or negative values as desired.” “Each of the rays intersecting the large sphere is required to follow a curved, and therefore longer, trajectory than it would have done in free space, and yet we are requiring the ray to arrive on the far side of the sphere with the same phase.”  Works over a narrow spectral bandwidth

29. Summary • Discreteness with periodicity introduces new paradigms into optics • Fundamental wave properties, namely refractive index, dispersion, • scattering and interference, and diffraction can be controlled • and/or eliminated • 3. New physical phenomena are introduced and well-known effects • are changed. • New ultra-compact optical devices possible • And much, much more…..

31. Negative Index Materials: Martin Wegeners Group  = 1500nm Optics Letters, 31, 1800, (2006)

32. 1 0.8 0.6 frequencyw (2πc/d) = d / l 0.4 TM Photonic Band Gap 0.2 TM bands Ez 0 G G X M irreducible Brillouin zone M Orthogonal Field Distributions E E X – + TM TE G H H 2-D Photonic Crystals - Array of Pillars: MIT d Ez (+ 90° rotated version) n2/n1=3.5

33. Electrically Pumped, Semiconductor Photonic Crystal Laser: Park et. al. Science Sept 3, 2004 2007 – 100A and 0.9V threshold

34. hydrogenfilled 0.3 0.020 P pump(532 nm) 0.2 S Stokes(683 nm) relative coupled power 0.010 anti-Stokes(435 nm) 0.1 AS 0.0 0.000 10 20 30 40 50 60 70 80 fibre length (cm) Highly Efficient Raman Shifter: Bath Un. Science 2002 coupled energy 5.6 Jpulse duration 6 nsec

35. Band 1: Band 2: kz (1/m) Band 3: Band 4: =kxd (units of ) Floquet-Bloch Bands

36. Bulk Crystal Band Diagram 0.6 0.4 Photonic Band Gap frequency (c/a) 0.2 0 G G X M M (k not conserved) X G 2D Photonic Crystal Cavity Modes: MIT A point defect can push up a single mode from the band edge High Q cavities

37. endlessly single-mode “Endlessly” Single Mode Fibres Normal fibers: single mode for • The smaller the , the smaller the influence • of the air holes the larger the effective ncl •  the smaller • with proper design V <2.405 • - Measured single mode for 0.35m<<1.55m 2a T. A. Birks, J. C. Knight, and P. St. J. Russell, Opt. Expr., 22, 961 (1997)

38. Example of Shape Dependence 3 4 2 FOM Effective Index 0 1 -4 0 Zahyun Ku and S. R. J. Brueck, Opt. Expr., 15, 4515 (2007)