2009. 05. Hanjo Lim School of Electrical & Computer Engineering hanjolim @ajou.ac.kr

1. 2009. 05. Hanjo Lim School of Electrical & Computer Engineering hanjolim@ajou.ac.kr Lecture 4. Electrons and lights in 1D periodic structures

2. Electrons in a 1D potential : Nearly free and Kronig-Penny model. • Free electrons ; electrons at BZ boundary meets total reflection • corresponding to interference effect • represented by standing waves of • the form • and • Charge density peaks at • Charge density is zero there and peaks between the atoms. • Electrons in state see more of the attractive potential than those in free • electron state which have Note that • Electrons in see less of the attractive potential than free electrons. • ∴ Electrons in state lie lower (higher) in energy than the free • electron value at the BZ boundary.

3. At BZ boundary with the crystal potential satisfying • with an integer • Let (Note Fourier theorem) • and normalization of the wave length of the box. • multiply • Then, kinetic term, let • equals not zero only when • Since the potential is real and symmetric the integral becomes

4. Likewise, we can prove that normalized wave function • satisfying the Schroedinger eq. • gives the eigenenergy which is higher than • We could also extend our calculation to evaluate and at the 2nd, 3rd, • etc, BZ boundaries, i.e. at with the bandgaps given • by the Fourier components of the crystal potential. • Kronig-Penny Model • with • => Sol.: with and

5. => Show the existence of bandgap and the dependence of on • When electron energy the bandgap appears at the BZ boundaries. • As electrons are confined more around the atoms, becomes larger. • Note the correlation between Fourier components of the and value. • What if the electrons are far from the BZ boundary? Free electrons.

6. Multilayer Film: Physical origins of the PBGs • 1D PhC ; alternating layers of materials with and and a period commonlyusedfor dielectric mirror, optical filters, and resonators • 1) Traditional approach; propagation of plane • wave and multiple reflections at the interfaces. • 2) PhC approach ; symmetry approach with index • of the modes; and band number • Let the modes have as a Bloch form with the • translational invariance for • The CTS in the xy-plane can take any value. • The DTS in the z-direction representation of in the 1st BZ • => photonic band(PB) diagram.

7. consider • i) If uniform dielectric medium, • with an assigned artificial period • bands are continuous. • ii) If nearly uniform dielectric medium light line and a small PBG between the upper and lower branches of the PB structure. • PBG : frequency range in which no mode can exist regardless of value. • iii) If periodic medium with high dielectric constant a PB diagram showing a large PBG. • Note) Most of the promising applications of 2D or 3D PhCs rely on the location and width of PBG. a wave propagating in the z-direction for the 3 cases of periodic dielectric films in the z-direction.

8. Physical origin of the PBG formation ; understandable considering the field mode profiles for the states immediately above and below the gap • Occurrence of the gap between bands at the BZ edge means that the PBG appears at • Note) Standing wave formation at is the origin of the band gap (nearly free electron model in solid state theory). • ∴ PBG is formed by the multiple reflections forming the standing waves.

9. The way of standing wave formation; from the EM variational theorem. • i) If nearly uniform dielectric medium • standing waves at • Note) Any other distribution with same frequency violates the symmetry. • Origin of frequency difference ; due to field concentration to a high- • and low- dielectrics (not fully sinusoidal). => dielectirc band, air band. • ii) If periodic medium with a higher dielectric • contrast the field energy for both • band is primarily concentrated in the high- • layers but the 1st being more concentrated in • the high- material. high- material: lowest energy distribution, low- material: field distribution normal to ground state.

10. Note that, in 1D PhCs, i) PBG always appears for any dielectirc contrast • the smaller the contrast, the smaller the gaps. ii) Occurs between • every set of bands at BZ’s edge or its center. Why? • Evanescent modes in PBGs: defect or surface state. • EM wave propagating in the 1D PhC; Bloch wave • Meaning of no states in PBG; no extended states given by Bloch form. • What happens if an EM wave whose frequency falls in the PBGs is sent • to the surface? No EM modes are allowed in the PhC: No purely real • exists for any mode at that frequency. Then is it reflected just • from the surface or exists in the PhC as an evanescent modes localized • at the surface? What determinates the field distribution in the reflection phenomena? If evanescent modes from the surface, how behaves?

11. Decaying field, i.e., evanescent wave from the surface should have a complex wave vector as giving the skin depth as • If normal incidence, • Consider near the band minimum at • for band minimum • with ◉ likewise Reflecting metal Non ideal conductor 1D photonic crystal

12. For (i.e. in the 2nd band), real Bloch states. • For (i.e. within the gap), purely imaginary decay of the wave with attenuation coefficient). • As ∴ band gap must be wide enough for a good reflection. • Note ; Evanescent modes • There is no way to excite them in a perfect crystal of infinite extent. But • a defect or edge in the PhCs might sustain such a mode. => defect states, • defect modes, surface states, surface modes. • One or more evanescent modes localized at the defect (defect states) may • be compatible depending on the symmetry of a given defect. • The states near the middle of the gap are localized much more tightly • than the states near the gap’s edge. are solutions of the eigenvalue problem, do not satisfy the translational symmetry.

13. Localized states near the surface: surface states • Similarity of localized states between the PhCs and semiconductors; • shallow donors and acceptors, extrinsic or intrinsic defects. • Off-axis propagation in the 1D PhCs (ex: let ) • 1) Because of non-existence of periodic dielectric arrangements in the • off-axis direction, there are no band gaps for off-axis propagation when • all possible are considered. • 2) For on-axis propagation (normal incidence), field in the x-y plane; • degenerate, i.e., x- or y-polarization differ only bya rotationalsymmetry. • ∴ We may take field (polarization) as x- or y-direction as convenient. • * For a mode propagating insome off-axis -direction, broken symmetry • → lifted degeneracy • must be wide enough for a good reflection. • exist a perfect mode • Off-axis propagation • ex) • 1) Nonexistence of band gaps for off-sxis propagation when all possible • are considered. Because of no periodic dielectric arrangement.

14. ex) A wave propagating in y-direction (reflect. invariance on yz-plane) • Possible polarizations; x-direction or in the yz-plane. • Absence of rotational symmetry between the x-polarized wave and yz- • polarized wave → different relations for x- and yz-polarized waves. • ∴ Degenerate bands for the waves propagating in the z-direction split into two distinct polarizations. • ① Different slopes for different polarization • means different velocity, i.e., from • with the band and polarization index smaller • slope of the photonic band => smaller velocity • due to different field confinement. • ②Approximately linear relations for any band in the long-wavelength • limit => homogeneous dielectric medium.

15. The variation of in the photonic crystal is smoothed out in the scale of • the long wavelength EM wave: homogenization phenomena. => effective • dielectric constants depending on and polarization direction. • ex) x-polarized modes have a lower frequency than the modes polarized in the yz-plane for the wave with on 1D PhC of • The field distributions at a long-wavelength limit show the reason. • The field lies in the high- regions for the x-polarized wave and crosses • the low- & high- regions for the wave polarized in the yz-plane. • Asymptotic behavior of the modes for large • (short ) region: Bandwidth • for large value, especially below the line • because of the exponential decay of the modes.

16. Defect modes: modes localized at a defect. • Defects: a structure that destroys a perfectly periodic lattice (ex: a layer • having different width or than the rest in 1D PhCs). • Consider the on-axis propagation of a mode with the frequency in the • PBG via a defect in 1D PhCs. • Introducing a defect will not change the fact that there are no extended • modes with freq. inside the periodic lattice, since the destruction of • periodicity prevents describing the modes • of the system with wave vector • Then a resonant mode of the defect ↔ • extended states inside the rest of PhC? (Yes)

17. Defect state: can be interpreted as localized at defect and exponentially • decay inside the rest, i.e. a wave surrounded by two dielectric mirrors. • If the thickness of a defect becomes of the order of quantized modes • → Fabry-Perot resonator/filter (band pass filter) • If a defect is the high- material, as increases (why?) with the • increase of decay rate as • Density of states : # of allowed states per unit increase in frequency • Interaction (or interference) between two different localized states. • Interaction of modes at the interface between two different PhCs: • possible if the two PBG overlap. Existence of a mode having • Surface states: localized modes at the surface of a PhC. • Surface: there is a PBG only in the PhC, and no PBG in the air.

18. Therefore, we should consider four possibilities depending on whether • the EM wave is decaying or extended in the air or PhC for all possible • If an EM mode is decaying in the PhC • (a mode whose lies in the PBG) and • also in the air ( below the light line) • → EM mode is localized at the surface • → Surface states. • Note: All four cases are possible in the case of • the structure described at the legend of left • figure. • It can be shown that every layered material (1D PhC) has surface modes • for some termination. Band structure of 1D PhC with =13( =0.2a) and =1( =0.8a) with the termination of high dielectic layer with 0.1a thickness.