2010. Spring: Electro-Optics (Prof. Sin-Doo Lee, Rm. 301

Week Chapter Week Chapter

Mar. 03 * Basics of Crystal Structures Apr. 26, 29 Periodic Medium II

Mar. 08, 10 Chap. 1: EM Fields

Chap. 3: Polarization of LightMay 03 Chap. 7: Electro-Optics I

Mar. 15, 17 Chap. 4: EM Propagation I May 10, 12 Electro-Optics II

Mar. 22, 24 EM Propagation II May 17, 19 Chap. 8: EO Devices

Mar. 29, 31 Chap. 5: Jones Calculation I May 24, 26 Take-Home Exam (Midterm II)

Apr. 05, 07 Midterm Exam I May 31, Jun. 2 Chap. 8: EO Devices

Apr. 12, 14 Jones Calculation II Jun. 07, 09 Chap. 11: Guided Optics

Apr. 19, 21 Chap. 6: Periodic Medium I Jun. 14 Term Paper (Final)

2010. Spring: Electro-Optics (Prof. Sin-Doo Lee, Rm. 301-1109, http://mipd.snu.ac.kr) Optical Waves in Crystals A. Yariv and P. Yeh (John Wiley, New Jersey, 2003)

Interatomic separation, r

+

Attra

ctio

nR

epul

sion

0

FA = Attractive force

FR = Repulsive force

FN = Net force

ro

r = ∞Molecule

Separated atoms

ro

(a) Force vs r

r

+

Rep

ulsi

onAt

tract

ion

0

EA = Attractive PE

ER = Repulsive PE

E = Net PE

Eoro

(b) Potential energy vs r

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Fig. 1.3: (a) Force vs interatomic separation and (b) Potential energy vsinteratomic separation.

Forc

e

Pote

ntia

l En

ergy

, E

(r)

Chap. 0 Bonding and Types of Solids

0.1 Molecules and General Bonding Principles

- Net force = attractive and repulsive

H

109.5°

C

HH

H(c)

From Principles of Electronic Materials and Devices, Second Edition, S .O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

H

H H

H

L shell

K shell

Covalent bond

C

(a)

Fig. 1.5: (a) Covalent bonding in methane, CH4, involves fourhydrogen atoms sharing electrons with one carbon atom. Eachcovalent bond has two shared electrons. The four bonds areidentical and repel each other. (b) Schematic sketch of CH4 onpaper. (c) In three dimensions, due to symmetry, the bonds aredirected towards the corners of a tetrahedron.

C

H

H H

H

covalentbonds

(b)

0.2 Covalently Bonded Solids: H2, CH4, diamond


Free valenceelectrons forming an

electron gas

Positive metalion cores

Fig. 1.7: In metallic bonding the valence electrons from the metalatoms form a "cloud of electrons" which fills the space between themetal ions and "glues" the ions together through the coulombicattraction between the electron gas and positive metal ions.

0.3 Metallic Bonding: electron gas or cloud (collective sharing of electrons)

Cl

3p

Closed K and L shells

Na

Closed K and L shells(a)

Cl-

3pNa+

FA

r

FA

(b)

Na+

ro

(c)

Cl-


Fig. 1.8: The formation of an ionic bond between Na and Cl atomsin NaCl. The attraction is due to coulombic forces.

3s3s

3s

0.4 Ionically Bonded Solids: salt (cation-anion)

Cl- Na+

ro = 0.28 nm

6

-6

0

-6.3

0.28 nm

Pote

ntia

l ene

rgy

E(r)

, eV

/(ion

-pai

r)

Separation, r1.5 eV

r = ∞

C l Nar = ∞

Na+Cl-

Fig. 1.10: Sketch of the potential energy per ion-pair in solid NaCl.Zero energy corresponds to neutral Na and Cl atoms infinitelyseparated. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Coh

esiv

e en

ergy

- Potential energy per ion pair in solid NaCl

Fig. 1.12: The origin of van der Waals bonding betweenwater molecules. (a) The H2O molecule is polar and has anet permanent dipole moment. (b) Attractions between thevarious dipole moments in water gives rise to van der Waalsbonding.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

(a) (b)

H

O

H

0.5 Secondary Bonding : hydrogen bonds (polar), van der Waals bonds (induced dipolar)

Time averaged electron (negative charge)distribution

Closed L Shell

Ionic core(Nucleus + K-shell)

Ne


A B

Synchronized fluctuationsof the electrons

van der Waals force

Instantaneous electron (negative charge)distribution fluctuates about the nucleus.

Fig. 1.13: Induced dipole-induced dipole interaction and the resultingvan der Waals force.

Lattice

Basis

Crystal

Unit cell Unit cell

a

a

(a) (b) (c)

90°

x

y

Basis placement in unit cell(0,0)

(1/2,1/2)

(d)

Fig. 1.70: (a) A simple square lattice. The unit cell is a square with aside a. (b) Basis has two atoms. (c) Crystal = Lattice + Basis. Theunit cell is a simple square with two atoms. (d) Placement of basisatoms in the crystal unit cell.

0.6 The Crystalline State

0.6.1 Type of Crystals : periodic array of points in space - lattice

Face centered cubic

Simple cubic Body centered cubic

Simplemonoclinic

Simpletetragonal

Body centeredtetragonal

Simpleorthorhombic

Body centeredorthorhombic

Base centeredorthorhombic

Face centered orthorhombic

RhombohedralHexagonal

Base centeredmonoclinic

Triclinic

UNIT CELL GEOMETRY

Fig. 1.71: The seven crystal systems (unit cell geometries) and fourteen Bravaislattices.

CUBIC SYSTEMa = b = c α = β = γ = 90°

Many metals, Al, Cu, Fe, Pb. Many ceramics andsemiconductors, NaCl, CsCl, LiF, Si, GaAs

TETRAGONAL SYSTEMa = b - c α = β = γ = 90°

In, Sn, Barium Titanate, TiO2

ORTHORHOMBIC SYSTEMa - b - c α = β = γ = 90°

S, U, Pl, Ga (<30°C), Iodine, Cementite(Fe3C), Sodium Sulfate

HEXAGONAL SYSTEMa = b - c α = β = 90° ; γ = 120°

Cadmium, Magnesium, Zinc,Graphite

RHOMBOHEDRAL SYSTEMa = b = c α = β = γ - 90°

Arsenic, Boron, Bismuth, Antimony, Mercury(<-39°C)

TRICLINIC SYSTEMa - b - c α - β - γ - 90°

Potassium dicromate

MONOCLINIC SYSTEMa - b - c α = β = 90° ; γ - 90°

α−Selenium, PhosphorusLithium SulfateTin Fluoride

a

a

a

(c)

2R

(b)

a

FCC Unit Cell(a)

Fig. 1.30: (a) The crystal structure of copper is Face Centered Cubic(FCC). The atoms are positioned at well defined sites arranged periodicallyand there is a long range order in the crystal. (b) An FCC unit cell withclosed packed spheres. (c) Reduced sphere representation of the FCC unitcell. Examples: Ag, Al, Au, Ca, Cu, γ-Fe (>912°C), Ni, Pd, Pt, Rh

- Face-Centered Cubic (FCC) Structure:


a

Fig. 1.31: Body centered cubic (BCC) crystal structure. (a) A BCCunit cell with closely packed hard spheres representing the Featoms. (b) A reduced-sphere unit cell.

Examples: Alkali metals (Li, Na, K, Rb), Cr,Mo, W, Mn, α-Fe (< 912°C), β-Ti (> 882°C).

a b

- Body-Centered Cubic (BCC) Structure:

Layer ALayer BLayer A

(a) (b)

Layer A

Layer B

Layer A

c

a

(c) (d) Examples: Be, Mg, α-Ti ( < 882°C ), Cr, Co, Zn, Zr, Cd

Fig. 1.32: The Hexagonal Close Packed (HCP) Crystal Structure. (a) TheHexagonal Close Packed (HCP) Structure. A collection of many Zn atoms.Color difference distinguishes layers (stacks). (b) The stacking sequence ofclosely packed layers is ABAB (c) A unit cell with reduced spheres (d) Thesmallest unit cell with reduced spheres.

- Hexagonal Closed Packed (HCP) Structure:

a

C

a

a

Fig. 1.33: The diamond unit cell is cubic. The cell has eightatoms. Grey Sn (α-Sn) and the elemental semiconductorsGe and Si have this crystal structure.


S

Zn

a

a

a

Fig. 1.34: The Zinc blende (ZnS) cubic crystal structure. Manyimportant compound crystals have the zinc blende structure.Examples: AlAs, GaAs, GaP, GaSb, InAs, InP, InSb, ZnS,ZnTe.

- Diamond & Zinc Blende Cubic Structure:

Table 1.3 Properties of some important crystal structures CrystalStructure

a and R(R is the radius

of the atom).

Coordination Number (CN)

Number of atoms per unit cell

Atomic Packing Factor

Examples

Simple cubic

a = 2R 6 1 0.52 None

BCC a = 4R/√3 8 2 0.68 Many metals: α-Fe, Cr, Mo, W

FCC a = 4R/√2 12 4 0.74 Many metals Ag, Au, Cu, Pt

HCP a = 2Rc = 1.633a

12 2 0.74 Many metals: Co, Mg, Ti, Zn

Diamond a = 8R/√3 4 8 0.34 Covalent solids:Diamond, Ge, Si, α-Sn.

Zinc blende 4 8 0.34 Many covalent and ionic solids. Many compund semiconductors. ZnS, GaAs, GaSb, InAs, InSb

NaCl 6 4 cations

4 anions

0.67

(NaCl)

Ionic solids such as NaCl, AgCl, LiF MgO, CaO

Ionic packing factor depends on relative sizes of ions.

CsCl 8 1 cation1 anion

Ionic solids such as CsCl, CsBr, CsI

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)

http://Materials.Usask.Ca



of the atom).




Examples

Simple cubic

a = 2R 6 1 0.52 None



HCP a = 2Rc = 1.633a




NaCl 6



of the atom).




Examples

Simple cubic

a = 2R 6 1 0.52 None



HCP a = 2Rc = 1.633a




NaCl 6 4 cations

4 anions

0.67

(NaCl)

Ionic solids such as NaCl, AgCl, LiF MgO, CaO

Ionic packing factor depends on relative sizes of ions.

CsCl 8 1 cation1 anion

Ionic solids such as CsCl, CsBr, CsI

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)

http://Materials.Usask.Ca

Covalently bondednetwork of atoms

Cubic crystal

(a) Diamond unit cell

Covalently bonded layer

Layers bonded by van der Waalbonding

Hexagonal unit cell

Covalently bondedlayer

(b) Graphite

Buckminsterfullerene (C60) molecule (the"buckyball" molecule)

The FCC unit cell of theBuckminsterfullerene crystal. Eachlattice point has a C60 molecule

(c) BuckminsterfullereneFig. 1.42: The three allotropes of carbon.

0.6.2 Three Phases of Carbon

Chap. 1. Electromagnetic Fields

1.1. Maxwell's Equations and Wave Equations

- E and H: the electric and magnetic field vectors (E, H) <-> (D, H)

In MKS units, ∇×

∇×

∇⋅ ∇⋅

with constitutive equations

- Poynting's Theorem and Conservation Law: ∇⋅ ⋅

⋅ ⋅ ×

U = the energy density of the EM fields (joules/m3) S = the Poynting vector, energy flow (joules/m2.sec) ∇⋅ = the net EM power flowing out of a unit volume.

[Homework] Conservation of a linear momentum of the EM fields: Problem 1.4 (p. 18)

1.4. Wave Equations

- From Maxwell's equations, ∇×∇×

∇×

Differentiating the above eqn. wrt to time and eliminating H,

∇×∇×

Use the vector identities ∇×∇×

∇×∇× ∇

× ∇×∇×∇× ∇∇⋅∇

→∇

∇× ∇×∇∇⋅

Since ∇⋅ ∇⋅⋅∇,

∇

∇× ∇×∇⋅∇

Similarly, ∇

∇× ∇×∇⋅∇

- Inside a homogeneous and isotropic medium, ∇ →

∇ ∇

→ ⋅

→

×

the refractive index of the medium

1.5. Propagation of Laser Pulse; Group Velocity

- Denoting A(k) = the amplitude of the plane-wave component with k,

∞

∞

with the Fourier spectrum

- For a laser pulse,

(see the next page)

then, ≈

∞

∞

with the envelope function

- The laser pulse travels along undistorted in shape with a velocity

; the group velocity

- In optics, , the phase velocity

Chap. 3. Polarization of Light Waves

3.2. Polarization of Monochromatic Plane Waves

- In a complex-function representation,

→

where

In the x-y plane,

where

- Let x' and y' be a new set of axes along the principal axes of the ellipse:

′

′

, with the rotation angle

where

and

3.2.1 Linear and Circular Polarization

- Linearly polarized: straight lines, and

- Circularly polarized: right-handed and left-handed ; - Ellipticity of polarization: ±

, (+) when the rotation of E is right-handed.

3.3 Complex-Number Presentation

- The amplitudes and the phase angles of the x and y components of E:

- The inclination angle and the ellipticity angle ≡ of the polarization ellipse:

and

3.4 Jones Vector Presentation

- 1941, R. C. Jones,

→

with normalization ⋅

- Mutually orthogonal polarization:

⇔

, ⇔

⇔

; ⋅

- For a general elliptical polarization,

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2010. Spring: Electro-Optics (Prof. Sin-Doo Lee, Rm. 301