1. Supervised by: Prof. Dr. Hassan Elkamchouchi Dr. Adel Elfahhar Presented by: Eng. Islam Mohammed Salah Kotb Alexandria University Faculty of Engineering Electrical Engineering Department 2. Outlines Introduction What are Photonic Crystals? Maxwell Equations and Bandgap Famous topologies Electro-optics and Nonlinear Optics Nonlinear Photonic Crystals Optical limiter 3. Outlines Introduction 4. Is it possible to have an All-Optical Processor? 5. The answer is Photonic Crystals 6. Outlines What are Photonic Crystals? 7. Photonic crystals are regular arrays of materials with different refractive indices arranged in such a way to inhibit the propagation of light Optical Insulators 8. 3m Photonic Crystals in Nature wing scale: Morpho rhetenor butterfly [ P. Vukosic et al., Proc. Roy. Soc: Bio. Sci. 266, 1403 (1999) ] Peacock feather [J. Zi et al, Proc. Nat. Acad. Sci. USA, 100, 12576 (2003) ] [figs: Blau, Physics Today 57, 18 (2004)] http://www.bugguy012002.com/MORPHIDAE.html [ also: B. Gralak et al., Opt. Express 9, 567 (2001) ] 9. Electronic and Photonic Crystals atoms in diamond structure wavevector electronenergy Periodic Medium Blochwaves: BandDiagram dielectric spheres, diamond lattice wavevector photonfrequency 10. planewave E,H ~ ei(kx-wt) k = w / c = 2p l k scattering It is known that light scatters when collides with atoms. 11. here: scattering off three specks of silicon 12. What about many particles? 13. Bloch Theorem states waves in a periodic medium can propagate without scattering the light seems to form several coherent beams that propagate without scattering and almost without diffraction (supercollimation) 14. for most , beam(s) propagate through crystal without scattering (scattering cancels coherently) ...but for some (~ 2a), no light can propagate: a photonic band gap a planewave E,H ~ ei(kx-wt) k = w / c = 2p l k 15. Outlines Maxwell Equations and Bandgap 16. 0H. E. JE t H H t E Maxwell Equations Maxwell equation in linear, isotropic, non-magnetic and homogeneous media can be written as: 17. In optical materials there is no free charges or currents included so: 0H. 0E. E t H H t E J 18. 00 2 2 2 2 2 2 0 2 2 0 1 where 1 )( 1 1 )( 1 )( c H tc H r E tc E r r E t E Eigen Operator (Hermitian): are real (lossless) eigen-states are orthogonal eigen-states are complete (give all solutions) 19. Bloch Theorem The Bloch-Floquet theorem tells us that, for a Hermitian eigenproblem whose operators are periodic functions of position, the solutions can always be chosen of the form )(),( ).( rHetrH k trki Corollary 1: k is conserved, i.e. no scattering of Bloch wave Corollary 2: given by finite unit cell, so are discrete n(k) Hk rki e 20. k k H c Hki r ki E c Ekiki r 2 2 )( )( 1 )( )()( )( 1 The field is periodic so solve over a finite domain a unit cell (First Brilluin Zone) Solve for all frequencies which are : n(k) (for n=1,2,3,. ). Plot as a function of the wavevector k, to form the band structure of the crystal. Solution can be obtained by iterative numerical methods such as Finite Difference Time Domain (FDTD). 21. Blochs theorem: solutions are periodic in k kx kyfirst Brillouin zone = minimum |k| primitive cell 2p aG M X irreducible Brillouin zone: reduced by symmetry Frequency(2c/a)=a/ G GX M a 2d periodicity, =12:1 22. Outlines Famous topologies 23. 1 D Photonic Crystals 1 2 1 2 1 2 1 2 1 2 1 2 (x) = (x+a)a band gap k 0 /a/a irreducible Brillouin zone 24. Famous Applications Dielectric mirrors Fiber Bragg grating 2 3 4 5 6 7 8 9 10 Number of periods n1/n2 = 1.44 Transmission(dB) Frequency (c/a) 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 -5 -10 -15 -20 -25 Varying number of layers Transmission(dB) Varying index ratio Frequency (c/a) 0 -5 -10 -15 -20 -25 -30 -35 -40 -45 0 0.1 0.2 0.3 0.4 0.5 0.6 1.3 1.5 2 3 4 Index ratio 25. 2 D Photonic Crystals There are 2 famous topologies that are mostly used: Square Lattice Triangular Lattice 26. Square Lattice The irreducible Brillouin Zone of this lattice is an isosceles right angled triangle TM band gaps are found using isolated regions of high dielectric constant material imbedded in a background of lower dielectric constant. 27. TE band gaps, significant bands are found for isolated regions of lower dielectric constant immersed in a higher dielectric background. 28. Triangular Lattice TM band gaps tend to occur in lattices formed by isolated high-permittivity regions, and TE band gaps in connected lattices. the connected lattice tends to combine these two advantages in the case where the diameter of the holes approaches the period of the lattice. Easy to fabricate using semiconductors fabrication techniques. 29. Point defects Introducing a point defect in any of the above mentioned topologies will increase the field localization at this point. 30. Line defects When a line defect is introduced into a photonic crystal waveguides may be created. Such waveguides can guide light at optical wavelengths with minimal propagation losses. 31. Applications Beam Splitters 32. resonantfilters channel-dropfilters high-transmission sharpbends waveguidesplitters 33. 3 D Photonic Crystals Yablonovite: The first experimental observation of a 3D complete photonic bandgap was made by Eli Yablonovitch in 1991 using a variant of the diamond lattice structure, now known as the Yablonovite. This structure has a complete gap when the refractive index is n = 3.6 34. Woodpile 35. Rod-hole (MIT) The structure is an fcc lattice of air (or low index) cylinders in dielectric, oriented along the 111 direction. Such a structure results in a system of two types: triangular lattices of air holes in dielectric and dielectric cylinders (rods) in air. 36. This structure offers a bandgap of 21% and even over 8% for Si:SiO2 contrast ( = 12:2); the PBG persists down to contrasts of 4:1 (2:1 index contrast). 37. Outlines Electro-optics and Nonlinear Optics 38. Electro-optics The electro-optic effect is a change in the refractive index that results from the application of a steady or low- frequency electric field. An electric field applied to an anisotropic optical material modifies its refractive indexes and thereby the effect that it has on polarized light passing through it. The dependence of the refractive index on the applied electric field has one of the two following forms: The refractive index changes in proportion to the applied electric field, which is known as the linear electro-optic effect or Pockets effect. The refractive index changes in proportion to the square of the applied electric field, known as the quadratic electro- optic effect or Kerr effect. 39. 3n 2 a - 3n 1 a 2 23 2 13 2 1 )0()( 2)0()( )2 22 1 1 ( 3 2 2 10 asdefinedis)(lityimpermeabielectricThe 0 2 2 2 a, 0 1 a),0(Where ....,2 2 a 2 1 1 an(E) EnEnnEn EEE EaEa n n dn d n E dE nd EdE dn nn EEn is called Packels Coefficient is called Kerr Coefficient 40. Pockels Effect Kerr Effect 41. Nonlinear Optical Materials The refractive index, and consequently the speed of light in a nonlinear optical medium, depends on light intensity. The principle of superposition is violated in a nonlinear optical medium. The frequency of light is altered as it passes through a nonlinear optical medium Photons interact within the confines of a nonlinear optical medium so that light can indeed be used to control light. The properties of a dielectric medium through which an optical electromagnetic wave propagates are described by the relation between the polarization-density vector: P(r, t) and the electric-field vector E(r, t). The mathematical relation between the vector functions P(r, t) and E(r, t) which is governed by the characteristics of the medium, defines the system. The medium is said to be nonlinear if this relation is nonlinear. 42. where and are coefficients describing the strength of the second and third-order nonlinear effects, Second-Order Nonlinear Optical Materials Second harmonic generation 43. Third-Order Nonlinear Optical Materials (kerr Medium) Third harmonic generation 44. Optical Kerr effect n2=1+ is the optical intensity of the initial wave is the impedance of the medium n=/2n which is called Optical Kerr Coefficient 45. Outlines Nonlinear Photonic Crystals 46. Nonlinear optical Photonic Crystals Photonic Crystal (periodic dielectric) Photonic Bandgap Nonlinear Material (Kerr Medium) n changes with light intensity New Outstanding Applications 47. Nonlinear Photonic crystals can either be made of fully nonlinear materials or by just adding nonlinear elements as defects in a linear Photonic Crystal =0 =0 Maxwell Equations Nonlinear media 48. Maxwell Equations for nonlinear periodic media 49. This problem is solved using perturbation theory (r)=0(r)+ 50. Outlines Optical limiter 51. Optical Limiter 52. 0.00E+00 2.00E-03 4.00E-03 6.00E-03 8.00E-03 1.00E-02 0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 OutputPower Input Power 1.16 1.23 1- Rectangular Lattice with line nonlinear defect Lattice constant a = 0.57m; rod radius r = 0.075 m, RI of rods n = 3.5; the three nonlinear rods rnl = 0.12 m, n2 = 2.7x10-9 m2/W. 53. Rods are all nonlinear with nrod=1.87, nmedium=1.25 and (3)=0.0015. 2- Rectangular Lattice with two nonlinear defects 54. 3- Square Coupled Cavity waveguide (rods in low index dielectric): Dielectric rods with nr = 3.5, nb = 1.5. The periodicity of the lattice is given by a, whereas the radius of the holes is r = a/4. The resonant cavity has been created in this system by increasing the radius of the central hole to rd = 5a/3. The two line defects have been introduced by reducing the radius of the corresponding holes to r/3. 55. 4- Triangular Coupled Cavity waveguide (holes in dielectric substrate): a is the lattice constant of the structure, the radius of the holes =0.3a, ns= 3.4 (GaAs), nNL =2.6, n2 = 2.710-9 m2/W and the defect radius is rd = 0.25a, Defect period is 5a 56. All Optical AND gate 57. Future Work Controllable Threshold Optical Hard Limiter More Devices (XOR, OR,.)