Discussed above can be generalized to other cases of wave equations with periodic potentials Periodicity Fourier transformation- Reciprocal lattice Dispersion relation, Bandgaps, Separated bands Brillouin zones(Bz) -Wigner-Seitz cells of the reciprocal lattice Born-von Karman cyclic boundary conditions Home Page Title Page Contents Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 10 of 41 Go Back Full Screen Close Quit Discussed above can be generalized to other cases of wave equations with periodic potentials. Periodicity → Fourier transformation→ Reciprocal lattice Dispersion relation, Bandgaps, Separated bands Brillouin zones (BZ)–Wigner-Seitz cells of the reciprocal lattice Born-von Karman cyclic boundary conditions
5.1.3. Revival of the study on Classical Waves Classical waves-an old scientific problem Elastic waves and Electromagnetic waves Waves in water--Solitons, Nonlinear physics X-ray diffraction in crystals demonstrated wave property of the rays and periodic structures of crystals Home Page Bragg equation, kinematical theory Title Page 2 dsin e=n入 (5.1.12) Contents wavelength X and lattice parameter d Can be used to X rays, electrons, and neutrons diffraction in crystals age II of 4 Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 11 of 41 Go Back Full Screen Close Quit 5.1.3. Revival of the Study on Classical Waves Classical waves—an old scientific problem: Elastic waves and Electromagnetic waves Waves in water—Solitons, Nonlinear physics X-ray diffraction in crystals demonstrated: wave property of the rays and periodic structures of crystals Bragg equation, kinematical theory 2d sin θ = nλ (5.1.12) wavelength λ and lattice parameter d Can be used to X rays, electrons, and neutrons diffraction in crystals
Bandgaps for X rays in crystals? Frequency-wavevector relation of electromagnetic waves stationary equation for electric displacement vector D ⅴ2D-××(r)D= D (5.1.13) Home Page (5.1.14) Title Page Photonic bandgaps Contents a window of frequencies, electromagnetic wave cannot propagate Crucial step to find the propagation gaps Theoretically, John. Yablonovitch. in 1987 Experimentally, Yablonovitch and gmitter, in 1989, microwaves, three- age 12 of 4. dimensional dielectric structures, called photonic crystals, pushed in Go Back the late 1990s to light waves, application in photonics Full Screen Close
Home Page Title Page Contents JJ II J I Page 12 of 41 Go Back Full Screen Close Quit Bandgaps for X rays in crystals ? Frequency-wavevector relation of electromagnetic waves stationary equation for electric displacement vector D −∇2D − ∇ × ∇ × [χ(r)D] = ω 2 c 2 D (5.1.13) χ(r) = 1 − 1 (r) (5.1.14) Photonic bandgaps: a window of frequencies, electromagnetic wave cannot propagate Crucial step to find the propagation gaps Theoretically, John, Yablonovitch, in 1987; Experimentally, Yablonovitch and Gmitter, in 1989, microwaves, threedimensional dielectric structures, called photonic crystals, pushed in the late 1990s to light waves, application in photonics
Renewed interests Classical waves in both Periodic and Aperiodic structures Electronic wave: Band structure(Bloch, 1928 )- localization(Ander- son, 1958; Edwards, 1958) Classical wave: localization John, 1984; Anderson, 1985)- band structure(Yabolonovitch. 1987: John, 1987 Home Page Electron and photon both have characteristics of waves Titie Contents age 13 of 4. Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 13 of 41 Go Back Full Screen Close Quit Renewed interests: Classical waves in both Periodic and Aperiodic structures Electronic wave: Band structure (Bloch, 1928) → localization (Anderson, 1958; Edwards, 1958); Classical wave: localization (John, 1984; Anderson, 1985) → band structure (Yabolonovitch, 1987; John, 1987) Electron and photon both have characteristics of waves
Renewed interests Classical waves in both Periodic and Aperiodic structures Electronic wave: Band structure(Bloch, 1928 )- localization(Ander- son, 1958; Edwards, 1958) Classical wave: localization John, 1984; Anderson, 1985)- band structure(Yabolonovitch. 1987: John, 1987 Home Page Electron and photon both have characteristics of waves Titie There are some basic differences Dispersion relation: electrons is parabolic, photons is linear Contents Angular momentum: electrons is 1/2, scalar-wave, photons have spin 1. vector-wave Band theory: electrons, approximated, Coulomb interactions, photons exact, no interactions age 13 of 4. Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 13 of 41 Go Back Full Screen Close Quit Renewed interests: Classical waves in both Periodic and Aperiodic structures Electronic wave: Band structure (Bloch, 1928) → localization (Anderson, 1958; Edwards, 1958); Classical wave: localization (John, 1984; Anderson, 1985) → band structure (Yabolonovitch, 1987; John, 1987) Electron and photon both have characteristics of waves There are some basic differences: Dispersion relation: electrons is parabolic, photons is linear; Angular momentum: electrons is 1/2, scalar-wave, photons have spin 1, vector-wave; Band theory: electrons, approximated, Coulomb interactions, photons, exact, no interactions