Quantum Dots in Photonic Structures

1. Lecture 3: Solution of Maxwell equations in a periodicdielectric Quantum Dots in PhotonicStructures Jan Suffczyński Wednesdays, 17.00, SDT Projekt Fizyka Plus nr POKL.04.01.02-00-034/11 współfinansowany przez Unię Europejską ze środków Europejskiego Funduszu Społecznego w ramach Programu Operacyjnego Kapitał Ludzki

2. Zasady zaliczenia przedmiotu • Egzamin ustny na koniec semestru: dwa pytania ze znanego uprzednio zestawu pytań

3. Literatura • Photoniccrystals: molding the flow of light, John D. Joannopoulos, wyd. II, PrincetonUniversity Press, 2008 • Controlling spontaneousemission dynamics in semiconductor microcavities: anexperimentalapproach, Bruno Gayral, Ann. Phys. Fr. 26 No 2, EDP Sciences, 2001 • Cavityquantum electrodynamics, S. Haroche, D. Kleppner, Phys. Today 42, 24 (1989). • Cavityquantum electrodynamics, S. Haroche, J-M. Raimond, ScientificAmarican (April 1993). • Microcavities, A. V. Kavokin, J.J. Baumberg, G. Malpuech,F. P. Laussy, Oxford Press, 2007

4. Reminder • Qualityfactor: a measure of the rateatwhichopticalenergydecays from the cavity (absorption, scattering, leakagedue to imperfectmirrors) Lifetime of the photon within the cavity: τ= 1/Γ= Q/ωc

5. Reminder: Fermi’sGoldenRule • Spontaneousemissionrateis not aninherentproperty of the emitter • Sponteanousemissionrateproportional to: Dipol moment of the emitter Density of photonstates atemitterwavelength Electric field intensity atemitterposition

6. Reminder: Fermi’sGoldenRule • Spontaneousemissionrateis not aninherentproperty of the emitter • Sponteanousemissionrateproportional to: Dipol moment of the emitter Density of photonstates atemitterwavelength Electric field intensity atemitterposition mirror mirror Spontaneousemissioninhibited Spontaneousemissionenhanced

7. Weak vs strongcoupling Out of the cavity

8. Weak vs strongcoupling Out of the cavity

9. Plan for today 2. One-dimensionalphotoniccrystal (Bragg Mirror) Refractiveindex of the matter 3. Two-dimensionalphotoniccrystal

10. EM waveinteracting with dielectric medium In linear, homogeneous, and isotropicmediapolarizationP linearlyproportional to E:  - a scalar constant called the “electric susceptibility” Definerelativedielectricconstant as: Note: In anisotropic media Pand E are not necessarily parallel: In nonlinearmedia:

11. Refractiveindex Allthe materials propertiesresult from P!

12. Refractiveindex - dispersion Waveequation: = The lightslows down in the medium! (phasespeed < c) Refractiveindex:

13. Refractiveindex Material with an index of refractionn The light slows down inside the material, therefore its wavelength becomes shorter and its phase gets shifted In Out • After: AndrásSzilágyi

14. As light travels from one medium to another: • Both the wave speed and the wavelength do change • The wavefronts do not pile up, nor are created or destroyed at the boundary, so, frequency does not change Refractiveindex

15. Refractiveindex - dispersion Waveequation: = The lightslows down in the medium! (phasespeed < c) Refractiveindex: Thedependence of n on λ is called dispersion • n usually decreases with increasing wavelength • violetlight refracts more than red light when passing from air into a material

16. Refractiveindex In the case of absorbing medium: Complexdielectricfunction allowssimultaneousdescription of refraction and absorption Propagation with phasespeed c/n Exponentialdecay

17. Photoniccrystals - introductoryexample 1. Braggscattering Regardless of how small the reflectivity r is from an individual scatterer, thetotalreflection R from a semiinfinitestructure: Complete reflectionwhen: • Propagation of the light in crystalinhibitedwhenBraggconditionsatisfied • Origin of the photonicbang gap

18. Fabry–Pérotinterferometer I0

19. Fabry–Pérotinterferometer I0 Fresnel:

20. Fabry–Pérotinterferometer The reflectance of the etalon Retalon: The transmissionof the etalon Tetalon: Maxima of the transmission for multiples of π

21. Photoniccrystals Photoniccrystal: Periodic arrangement of dielectric (ormetallic…) objects ( periodicrefractiveindexcontrast!) • The periodcomparable to the wavelength of light in the material.

22. 1D photonic crystal: a Bragg mirror

23. 1887 1987 Photoniccrystals Media with periodic refractiveindeces givingrise to photonic band gaps: “optical insulators”

24. can trap light in cavities and act as waveguides Photoniccrystals Media with periodic refractiveindeces Media with periodic refractiveindeces givingrise to photonic band gaps: “optical insulators”

25. 1D photoniccrystal: a Bragg mirror d1n1=d2n2=lBr/4 Quaterwave stack condition d1 n1 d2 n2 d1kn1=d2kn2= lBr/4*(2π/lBr)=π/2  destructiveinterference of the reflectedwave with the incidentwave! Example GaaS/AlAsBragg mirror:

26. 1D photoniccrystal: a Bragg mirror Distributed Bragg Reflector in Transfer Matrix Method formalism Blackboardcalculation (for a referencesee for exampleC. B. Fu, C. S. Yang, M. C. Kuo,Y. J. Lai,J. Lee, J. L. Shen, W. C. Chou, and S. Jeng, High ReflectanceZnTe/ZnSe Distributed Bragg Reector at 570 nm, CHINESE JOURNAL OF PHYSICS 41, 535 (2003). (mind the error in Equation 14: i = 1  i = 2)

27. Summary • Complexrefractiveindex - describesrefraction and absorption • Photoniccrystalsareartificialmedia with a periodic indexcontrast(periodcomparable to the wavelength of light in the material) • Bragg mirror calculatedwithin Transfer Matrixformalism