v.2019.AUG C. Optical Properties of Materials

Contents Objectives

• light (9.1) • What happens when light shines • refractive index, n (9.2) on a material? • dispersion, n(l) (9.3) • colors of materials – vg, Ng, zero dispersion (9.4) • principles • Why are some materials – Snell (9.6): direction transparent and others not? – Fresnel (9.7): amplitudes, phases • Optical applications: • losses & origins – optical fibers – n-jK, er'-jer", k'-jk" (9.8) – lattice absorption (9.9) – optical amplification – VB  CB absorption (9.10) – luminescence, phosphors – scattering (9.11) – nanostructures • case studies – optical fibre (9.12) – luminescence, phosphor (9.13)

Reference: S.O. Kasap (3rd,4th Ed.) Chap. 9; RJD Tilley (Understanding solids, 2nd Ed) Chap. 14.

Optcal Properties 2102308 1 9.1 Light (L0) • optical properties of matter describe the interaction of light with matter • types of interactions: scattering, reflection, absorption, transmission, diffraction… • properties of light: wave-particle duality – for propagation & scattering, light = EM wave – for absorption & emission, photons = particle • visible light has energy ~1.8-3.1 eV, wavelength ~400-700 nm (visible spectrum) vs. • a much wider electromagnetic spectrum (Fig./Table 14.1) (L1) • wavelength l of interest: 10 nm (X-rays) – 100 mm. Selected applications: – germicide/cosmetic: 300-400 nm (UV) – display: 400-700 nm (VIS) – optical communication: 1.3 and 1.55 mm (IR) • light is an oscillating electromagnetic (EM) field where electric component E ⊥ to magnetic component B, both of which ⊥ to propagation direction k (Fig. 9.1) (L2-L3) – simplest description: monochromatic light travelling in space in 1D (Fig. 9.2) – more complete description must be in 3D (Fig. 9.3) • light can be generated by electrons in atoms, molecules, solids (crystals) dropping from high to low energy levels/bands (L4-L5). Other courses explain the principle of light generation by semiconductor devices (2102385, SKJ), and lasers (2102589, SPY)

Optcal Properties 2102308 2 (L1)

Optcal Properties 2102308 3 (L2) “Light -” an electromagnetic wave Ex  By Which (E, B) is more important? - Electric and magnetic fields always co-exist by virtue of Faraday’s Law: time-varying (B field  E field) - Most materials (atoms in the periodic table) only respond to E, thus B is usually ignored - Optical field refers to E field

light-related vocabularies: - monochromatic: single or very narrow monochromatic plane wave: range of l (red laser yes, the sun no) time space phase - coherent: photons constituting light beam are in phase (laser yes, LED no) in 1D: Ex  Eocost  kzo  - polarization: direction of E field, k is propagation constant or wavenumber light sources can be (*) unpolarized, (a) plane polarized, (b) elliptically E polarized, (c) circularly polarized, T  f 1/T  t time     2 /T  z space k  2 / l  l   * Optcal Properties 2102308 4 (L3) - propagation monochromatic plane wave propagates in space at velocity v

1D (far from source) for any given plane, the phase   t  kz plane wave o is a constant which moves dz at “planes” every dt  phase velocity: dz  v    lf dt k  f 1/T      2 /T  k  2 / l 

3D (close to source) spherical wave

Er,t  Eocost k r o 

Optcal Properties 2102308 5 (L4) - generation

GAS SOLID H semiconductors 2 direct indirect InP Si

Optcal Properties 2102308 6 H2, He W C (L5) - generation by hot bodies (incandescence)  • heated elements/molecules: () electrons excited to E2 and relaxed to E1 (Fig. 14.4), releasing photons • output spectrum depends solely on temperature T described by Planck’s law of blackbody radiation

Optcal Properties 2102308 7 (I0) Light-Matter Interaction RAT - incident light is mostly Reflected, Absorbed, Transmitted (refracted, non-normal incidence), and partly scatterd (Fig. 14.13) (I1). Other types of interactions: diffraction, fluorescent… - conservation of energy: (ignoring scattering and secondary effects)

Io  IT  I A  IR - optical materials/systems often designed to maximize or minimize R, A or T (l) – see pix (I1) - interaction of light with - metals (I2): reflection, mostly - non-metals: absorption, luminescence (I3) - results of light-matter interactions: - colours: (I4) CdS, Hope, (I5) ruby - opacity: internal reflection by microstructure determines opacity (I5) - refractive index: light slowed down, bent (refracted) (I6)

Optcal Properties 2102308 8 (I1) Light-Matter Interaction

“interaction” overview of applications

surface condition Reflection Absorption

(rough) A (smooth) T R

Cloaking Transmission

Optcal Properties 2102308 9 Light interaction with metals RAT (I2) Absorption Reflection

re-emitted photon from material surface Energy of electron I unfilled states R “conducting” electron E

filled states

• Metals have fine succession of energy states. • Metals appear reflective (shiny) because • Near-surface electrons absorb visible light. • Reflectivity = IR/Io is ~ 0.90-0.95. (reflected light same frequency as incident)

Colors Pd Au Color of metals dictated by interfacial property (metal/insulator interface) called Transmission (none) “surface plasmon resonance (SPR)”

since IA + IR ~ 1  IT ~ 0; therefore, all metals are opaque (except thin foil < 100 nm, transmit visible light)

Optcal Properties 2102308 10 Light interaction with non-metals (I3) (insulators, semiconductors) RAT

Absorption (conditional): if hn > Egap Reflection: depends on n, ex. - R ≈ 4% for glass (insulator, n ~ 1.4) - R ≈ 30 % for Si (semicond, n ~ 3.5) - for details, see Fresnel (9.7)

Energy range of visible light:

Luminescence incident photon (re-emitted light) energy hn

• If EG < 1.8 eV, full absorption; color is black ([email protected], [email protected] eV)

• If EG > 3.1 eV, no absorption; colorless ([email protected], glass@9 eV), transparent (!), see (I5)

• If EG in between or has impurity level, partial absorption; material has a color.

Optcal Properties 2102308 11 (I4) Light-Matter Interaction: Results LDR

• Colour determined by sum of frequencies of - transmitted light, scattered light - re-emitted light from electron transitions CdS • Ex. 1 (Semiconductor): Cadmium Sulfide (CdS) - Egap = 2.4 eV, - absorbs higher energy visible light (blue, violet), - Red/yellow/orange is transmitted, gives color. (2.48 eV) Diamond • Ex. 2 (Insulator) The Hope Diamond = C (+ % B)

- Egap = 5.6 eV - luminesce after UV exposure (l<220nm) (1.88 eV) - phosphorescence (re-emission occurs with delay time > 1 s)

Origin of color: Boron (0-8 ppm depends on position, av. 0.36 ppm)

http://nhminsci.blogspot.com/2012/05/hope-diamond-blue-by-day-red-by-night_22.html

Optcal Properties 2102308 12 (I5) Light-Matter Interaction: Results

• Ex. 3 (Insulator) Ruby Ruby = Sapphire (Al2O3) + Cr2O3 Ruby (laser) - Egap > 3.1eV - pure sapphire is colorless (mineral)

- adding Cr2O3 : (0.5-2) at. % • alters the band gap  • blue light is absorbed Opacity/Transparency (EG>3.1eV)  • yellow is absorbed Transparency of insulators  • red is transmitted dictated by microstructure: • Result: Ruby is deep red.  single-crystal  poly-crystal (dense)  poly-crystal (porous)

     

Intrinsically transparent dielectrics can be made translucent/opaque by “interior” reflection/refraction/scattering.

Optcal Properties 2102308 13 (I6) Light-Matter Interaction: Results

• Transmitted light distorts electron clouds. no transmitted + light • Result 1: Light is slower in a material vs vacuum.

Index of refraction (n) = speed of light in a vacuum speed of light in a material - Adding large, heavy ions (e.g. lead, Pb) Material n can decrease the speed of light. Lead glass ~ 2 Si O Pb Silica glass 1.46 Si O • Result 2: Intensity of transmitted light decreases Soda-lime glass 1.51 Si O Na Ca with distance traveled (thick pieces less transparent) Quartz 1.55 Si O -- see later in Losses (9.8) Plexiglas 1.49 C O H Polypropylene 1.49 C O H • Result 3: Light can be “bent” Diamond 2.41 C or “refracted”, see later Snell (9.6)

Refraction (qi ≠ 0): a change in propagation direction of a wave when it changes a medium Optcal Properties 2102308 14    





Optcal Properties 2102308 15 9.2 Refractive Index (n) (R0)

• refractive index (n) measures how much light slows down in a material, with respect to vacuum. Speed of light in i) vacuum = c, ii) material = phase velocity v (), ∴ n  • materials slow down light due to light-matter interaction: atoms/molecules/grains in optical/dielectric materials are polarized by the electric field component of light E (Reminder: polarization mechanisms), Fig. A (R1) • propagation of light (E) in a material is effectively delayed (phase lag) wrt. vacuum

• (from ) material permittivity er  stronger dipoles (drag )  delays  • atoms/molecules in medium are polarized at the same frequency as E, hence the frequency of light does not change, but other wave propagation & parameters do change, Table A (R1). Typical wavefronts in films, Fig. B (R1) • n is not a constant, but can change with incident direction, especially in some 1 c c crystals (birefringent) which show  v    e re omr mo e r mr e r optical anisotropy (9.14) (R2,R3) c  n   e • n is not a constant, but can change with v r optical materials are incident wavelength, thus n(l), see non-magnetic (mr ≅ 1) example (R4), for details next section 9.3 Dispersion

Optcal Properties 2102308 16 Dielectric properties of materials (R1) - complex relative permittivity as a function of frequency - contribution of various polarization mechanisms

Interfacial and space charge Fig. A Orientational, e' r Dipolar n  e r Ionic Electronic n(l)  e r ( f ) er''

er' = 1

ƒ 10•2 1 102 104 106 108 1010 1012 1014 1016 Radio Infrared Ultraviolet light

Table A: Wave propagation parameters Fig. B Wave travelling into optical films

vacuum medium f f

c = lof v = lf

lo l  lo/n

ko k = nko

c l f l 2 / l k n   o  o   v lf l 2 / lo ko - Estimate n in Film 1, Film 2 - Film 1 is lossy, 2 lossless

Optcal Properties 2102308 17 Optical anisotropy (9.14) (R2) - most non-crystalline materials (glasses, liquids) and all cubic crystals are optically isotropic (they “look” the same in all directions) - crystals are different: n of crystals depend on direction of electric field in the propagating light beam - Optically anisotropic crystals are called bi-refringent because incident light beam may be doubly refracted (Figs. 9.25, 9.26)

Figure 9.26 Two polaroid analyzers are placed with their Fig. 9.25: A line viewed through a cubic sodium chloride (halite) transmission axes, along the long edges, at right angles to each crystal (optically isotropic) and a calcite crystal (optically other. The ordinary ray, undeflected, goes through the left anisotropic). polarizer whereas the extraordinary wave, deflected, goes through the right polarizer. The two waves therefore have orthogonal From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) polarizations. http://Materials.Usask.Ca From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002) Applications: polarising beam splitter, phase contrast microscopy...http://Materials.Usask.Ca

Optcal Properties 2102308 18 (R3) n is frequency dependent n maybe direction dependent n  e r N In noncrystalline materials, n is isotropic. e r  1 In crystalline materials, n is anisotropic. e o

Optically isotropic n = no Glass (crown) 1.510  non-crystalline

Diamond 2.417  crystalline (cubic) ISOTROPIC Fluorite (CaF2) 1.434

Uniaxial - Positive no ne Ice 1.309 1.3105 Quartz 1.5442 1.5533 ne  no

Rutile (TiO2) 2.616 2.903

Uniaxial - Negative no ne

Calcite (CaCO3) 1.658 1.486

ANISOTROPIC Tourmaline 1.669 1.638 no  ne Lithium niobate (LiNBO3) 2.29 2.20 wiki/Birefringence

Biaxial n1 n2 n3 Mica (muscovite) 1.5601 1.5936 1.5977

Table 9.3 Principal refractive indices of some optically isotropic and anisotropic crystals (near 589 nm, yellow Na-D line).

Optcal Properties 2102308 19 (R4) Physical origin of dispersion relation:

n  e r

n(l)  e r ( f )

• Examples ? = e  r Material er(LF) er(LF)] n (optical) Diamond 5.7 2.39 2.41 (at 590 nm)

Si 11.9 3.44 3.45 (at 2.15 mm)

SiO2 3.84 2.00 1.46 (at 600 nm)

Dielectric resonance How are these measured in practice? electronic LF: kHz; e.g. by C-V measurement dipolar HF: (Optical) 1015 Hz; Snell’s case 1 (=): same polarization mechanism ( )

case 2 (): additional polarization mechanism ( ) at low frequency

Optcal Properties 2102308 20 9.3 Dispersion n(l) (D0) - dispersion relations: interrelations of wave properties (l, f, v, n), most importantly n(l) () - dispersion occurs when pure plane waves of different wavelengths have different propagation velocities, so that a wave packet of mixed wavelengths tends to spread out in space () - brief physics of dispersion (D1) - the cause: imperfect light sources, they all have finite width l, or FWHM (D2) - the effect: group velocity and group index (9.4) (D3) - the results: pulse spread (), poor system performance (bit-error rate, BER ) - the remedy: zero dispersion (D4)

dispersion in optical fiber communication  

≠ dispersion in prism causes different colors to refract at different angles, splitting white light into a rainbow of colors

Optcal Properties 2102308 21 Brief Physics of dispersion relation (D1)

2 n  e r  dispersion relation in a material with a single polarization  N mechanism with a single resonant wavelength lo e 1 e  r  2 e o  2 l  n l 1  2 2 e  pinduced /  l  lo Ze2  pinduced  Ze x  2 2  me 0   

- in real crystals, atoms interact  very complicated. In general, a material can have polarization mechanisms each with a different resonance frequency. This leads to a semi-empirical dispersion relation: 2 2 2 2 B1l B2l B3l n (l)  A  2  2  2 l  C1 l  C2 l  C3

B1,2,3, C1,2,3 : Sellmeier coefficients

Sources: Batop.com, Wiki

- Example n(l) for GaAs (0.89-4.1 mm) at 300K (obtained empirically): 3.78l2 n2  7.10  l2  0.2767 [l in mm]

Optcal Properties 2102308 22 The cause of dispersion: engineering & physics (D2)

Dispersion occurs when pure plane waves of different wavelengths have different propagation velocities, so that a wave packet of mixed wavelengths tends to spread out in space. (Wiki)

Light sources: - broadband: Sun, incandescent – for general lighting - narrowband: light-emitting diodes (LEDs), (LDs) – for communication

E Color depends on material (EG)

EG Color purity (full-width at half-maximum: FWHM) depends on device (structure / mechanism): -LEDs (homojunction [p-n], spontaneous emission): FWHM ~ 30 nm -LDs (heterojunction [p+-n+], stimulated FWHM emission): FWHM ~ few nm

FWHM = 0 (perfect monochrome) cannot be achieved.

no perfect monochromatic waves  not single value Heisenberg: Et   Typical spectra of RGB LEDs. Source: Wiki

Optcal Properties 2102308 23 The effect of dispersion: 9.4 Group velocity and group index (D3)

Dispersion occurs when pure plane waves of different wavelengths have different propagation velocities, so that a wave packet of mixed wavelengths tends to spread out in space. (Wiki)

Information: phase travels @ phase velocity (v)

Emax travels @ group velocity (vg) Note that eyes/detectors measure intensity (E2) v = c/n and  = vk

vg =d /dk

 E  in vacuum: n = 1 (l independent) v = c and  = ck

 vg = d /dk = c = phase velocity for all l’s x   in a medium: n > 1 (l dependent) c d c v   v   E2 g dn nl dk n  l dl c x vg  Ng group index N g

Optcal Properties 2102308 24 The remedy: eliminate / minimise? (D4)

group index dn  n  l  dl   ct  c g N g   vg L Zero-dispersion dN c dt g  g  dl L dl dt 1 dN g  g Ldl c dl

Ex. 9.6: find the phase velocity, group index, and group velocity of light (1300 nm) travelling in a pure silica glass.

n = 1.447  v = 2.07  108 m/s 8 Ng = 1.462  vg = 2.052  10 m/s Optcal Properties 2102308 25 9.6 Snell’s law and TIR

Requirement for wave propagation = Trigonometry: Constructive interference: wavefronts for reflected and refracted (transmitted) waves must be in phase with incidence wave i & t =

AA  ABsin qt  v2t 

BB  ABsin qi  v1t 

sin qi n2 /   Snell’s law (1621) sin qt n1

n > n 1 2 = i & r

AA  ABsinqr  v1t 

= qi  qr

ALL other angles interfere destructively

Optcal Properties 2102308 26 y from Snell’s law: Et0, y,z,t  e exp jt  kz  2 1/ 2 when qi qc  qt  90    2n2  n1  2     sin qi 1 1 n     2  l  n2  qc  sin       n1  penetration depth = 1/

Optical Fiber

Digital signal Emitter Photodetector Information Information t Input Output

TIR n2

q q Fiber axis n Core 1

Light Cladding ray n2

An optical fiber link for transmitting digital information in communications. The fiber core has a higher refractive index so that the light travels along the fiber inside the fiber core by total internal reflection at the core-cladding interface.

TIR leads to wave propagation in a dielectric medium

Optcal Properties 2102308 27 9.7 Fresnel’s equations (F0) - a set of four equations (Fresnel, 1818) used to determine the magnitudes and phases of reflected and transmitted waves at the interface of two transparent materials with different refractive indices, see (F1) 1788------1827 - the results (mainly reflection coefficients) are plotted in two example cases:

- internal reflection, when n1 > n2, see (F2)

- external reflection, when n1 < n2, see (F3) - human eyes (and optical detectors) sense intensity (E2), not the magnitude of the optical field (E), hence appropriate to plot reflectance and transmittance (F4) - the equations provide mathematical guideline to the designs of some important

normal-incidence (qi = 0°) applications: - low-reflection coating: aims to absorb all incident radiation, useful in solar cells and photodetectors, see anti-reflection coating (ARC) (F6) - high-reflection coating: aims to reflect all incident radiation, useful as mirrors, see quarter wave stack (F7) - various optical filters: low-pass, high-pass, bandpass (similar in principle to electronic filters)

Optcal Properties 2102308 28 (F1)

y E t, Transmitted wave y k t Evanescent n 2 wave q = 90o q t t E t, E z t, n > n x into paper k 1 2 Ei, i q Ei, q q  i qr E  i r E r, r, k kr E r Ei, i, E r, Er, Incident Reflected Incident Reflected wave wave wave wave

Governing(a) qi < principles:qc then some of the wave is (b) qi > qc then the incident wave suffers transmitted into the less dense medium. total internal reflection. There is a decaying A. Snell’s law: n1 sinq1  n2 sinq2 Some of the wave is reflected. evanescenntwave into mediumn2 B. electromagnetism: B   E// and B//   E  c   c  C. Boundary conditions: E (1) = E (2) n tangential tangential n  2 B (1) = B (2); for non magnetic media → qi = qr and tangential tangential n1 2 2 E n2  sin 2 q  n2 cosq Er, cosqi  n  sin qi r,// i i r   r//    /   2 2 2 2 2  /  E Ei,// n  sin q  n cosq i, cosqi  n  sin qi i i E 2ncosq Et, 2cosqi t,// i t   t//    /   2 2 2 2 2  /  E Ei,// n  sin q  n cosq i, cosqi  n  sin qi i i

r 1  t ; r//  nt// 1

Optcal Properties 2102308 29 (F2) Internal Reflection

(n1 > n2) Example: 1  n  1 2  n1 = 1.44 and n2 = 1  n  , qc  sin    44 1.44  n1 

(r, r// complex)

(+, real)

(–, real)

 = 0 @ qi = 0 (+, real)

set r// = 0 1 n2  set qi = 0 q  tan    34.8 polarization angle p   Brewster’s angle  n1  n  n normal incidence : 1 2 r//  r  n1  n2

Optcal Properties 2102308 30 (F3) External Reflection

(n1 < n2) Example: 1.44 n = 1 and n = 1.44  n  , n2  sin 2 q  0  no critical angle 1 2 1 i

n1 sin q1  sin q2 n2

(n1  n2 )  (sin q2 1)

no qc

 =  @ qi = 0 1 q p  tan 1.44  55.2

Transmitted light in both internal and external reflection does not experience

phase shift (t is always +ve, qi < qc) (main concern is reflected light for propagation)

Optcal Properties 2102308 31 Reflectance (R) and Transmittance (T) (F4) 2 2 Er, 2 Er,// 2 R  2  r and R//  2  r// Ei, Ei,// 2 2 n2 Et, n 2 n2 Et,// n 2 T   2 t and T   2 t  2 n  // 2 n // n1 Ei, 1 n1 Ei,// 1

At normal incidence (qi = 0): (practical angle in laser diodes, or cavity structures) 2  n  n   1 2  R  R//  R     n1  n2  R + T = 1 4n1n2 T  T//  T  2 n1  n2 

Ex: Reflectance (%) Reflectance

Optcal Properties 2102308 32 (F5) Internal Reflection Ex. 9.8 n = 1.44 thicker than 5 * penetration depth n = 1.46 l = 1300 nm

i) Find minimum angle for TIR

ii) Find penetration depth of evanescent wave in the upper medium when qi = 87 and 90

2 1/ 2 ii) use: 2n  n     2  1  sin 2 q 1 q () 1/ (mm) l  n  i i  2   87 0.906 90 0.859 Internal and External Reflection Ex. 9.9 air glass Find reflection coefficients and reflectance in 1 n 1.5 each case

Optcal Properties 2102308 33 (F6) Max Absorption (PV)

Ex. 9.10: n1 = 1, n3 = 3.5, calculate R without ARC layer.

Requirement: maximum light transmitted to 700 nm PV for electricity generation

n1 < n2 < n3 Insert ARC layer to minimise reflection. A and B must be: 1/2  of similar amplitude  n2 = (n1n3) *** Proof

  out of phase  d = ?

air/ARC = ARC/semic. 

Optcal Properties 2102308 34 n1 < n2 (F7) Max Reflection (LD,VCSEL)

Ex. 9.11: Dielectric Mirrors Q) Why do we need dielectric mirrors? A) Bend light without absorption (cf metallic mirrors)

Design criterion: A,B,C interfere constructively.

Show that waves A, B and C interfere constructively.

note: l1 = l0/n1, l2 = l0/n2

Optcal Properties 2102308 35 9.8 Loss and complex refractive index (L0)

- for transparent insulators (EG > 3.1 eV), the light-matter interaction can be described simply by the (real) refractive index n

- for metals (no EG) and semiconductors (EG  few eV), large number of free electrons cause absorption (loss) of light and the light-matter interaction must be described by a complex refractive index N = n − jK - loss quantified by absorption coeficient , related to absorption index K, the imaginary part of complex refractive index N (L1) - example (n,K) spectra of amorphous Si (a solar cell material) (L2), using a simple optical test setup (L3). Note relationship between optical and dielectric properties in (L2), with calculation example in (L4) - loss of photons (between source and detector) is caused by absorption and scattering - two intrinsic absorption mechanisms (anything “intrinsic” can’t be removed): - lattice absorption (9.9): photons to phonons (heat), IR active - band-to-band absorption (9.10): photons to carriers (EHPs), UV active - one intrinsic scattering mechanism, Rayleigh (9.11), UV-VIS-IR active

Optcal Properties 2102308 36 (L1)

Since er is complex  n is also complex  written as N R A T In fact, propagation constant k is complex: k  k' – jk'' k k  jk note: c l f l 2 / l k n   o  o   N  n – jK   ko ko v lf l 2 / lo ko R,T A

(linear) Absorption Coefficient  n refractive index  k ko K absorption index  k k Lossless: o E  Eo exp jt  kz Plane wave (extinction coefficient) Lossy: E  Eo exp jt  k  jkz

 Eo exp kzexp jt  kz k' describes propagation: 2 phase velocity v = /k' Intensity: I  E  exp 2kz k" describes absorption: or I  Io expz Beer’s law* attenuation along z where   2k  2ko K

EXP.  determine K(l) k0 is propagation constant in from absorption /transmission measurements a vacuum (no loss, real #) at normal incidence as a function of freq.

* Bouguer, Beer, Lambert

Optcal Properties 2102308 37 Hence, complex refractive index: (L2) (n,K) collectively called optical constants but they vary with wavelength, hence not constants as name suggest a-Si dielectric / optical ' '' er ,er  n, K Kramers-Konig relation

lossless: k  0, n  e r EG ' '' lossy: n  jK  e r  je r photon 2 2 ' '' energy n  K  e r and 2nK  e r

2 2 EXP.  determine R(l) 1 N n – 1  K 2 from reflectance measurement at R   2 2 normal incidence as a function of freq. 1 N n  1  K

 &  get n – jK (l) from 9.7 (Fresnel)

Optical constants (n and K) can also be found from ellipsometry (a kind of reflection measurement, see JA Woollam.com), the principle is based on polarization and angle of incidence (Fresnel’s equations)

Optcal Properties 2102308 38 Setup for (n,K) measurements (L3)

Io z photodetector broadband lightsource l beamsplitter (50:50)

It monochromator material under test Ir

Ir / Io , R It / Io

 

l , photon energy l , photon energy R

R (n,K) K n  (k", K) n

l , photon energy l , photon energy

Optcal Properties 2102308 39 (L4) Ex. 9.12 Spectroscopic ellipsometry measurements on a Si crystal at 826.6 nm show that the real and imaginary parts of complex relative permittivity are 13.488 and 0.038. Find complex refractive index, absorption coefficient  at this wavelength, and phase velocity.

Optcal Properties 2102308 40 9.9 Lattice absorption - the electric component of the lightwave (E) interacts with ions in the lattice - the interaction is strongest when the frequency of the lightwave (color, energy) match the natural lattice vibration frequency (1012 Hz, in the IR region)

Ions at equilibrium positions in the crystal

vibrate at frequency of E Forced oscillations by the EM wave

Ex k Propagation direction z

Lattice absorption through a crystal. The field in the EM wave oscillates the ions, which consequently generate "mechanical" waves in the crystal; energy is thereby transferred from the wave to lattice vibrations.

Optcal Properties 2102308 41 (B0) 9.10 Band-to-band absorption - at sufficiently high energy, light interacts with valence electrons in the bonds, freeing them ( EHPs) - electrons are transferred from (valence) band-to- (conduction) band, requiring minimum photon energy: hc 1.24 EG   lcutoff (mm)   l EG (eV) - for direct materials, Fig. (a), the interaction only involve photons & electrons, but for (b) indirect materials, Fig. (b), phonons are involved too (conservations of energy & momentum) - (band-to-band) absorption spectra, l, show sudden change around l () of materials (B1) - optical detection systems performance limited mainly by choice of material and device (B2)

Direct materials: Indirect materials: hv  E G hv  EG  h phonon energy

Optcal Properties 2102308 42 (B1) Table 9.2 Band gap energy Eg at 300 K, cut-off  wavelength lg and type of bandgap (D = Direct  (= 2k = 2koK) and I = Indirect) for some photodetector materials.

Semiconductor Eg (eV) lg (mm) Type InP 1.35 0.91 D

GaAs0.88Sb0.12 1.15 1.08 D Si 1.12 1.11 I

In0.7Ga0.3As0.64P0.36 0.89 1.4 D In Ga As 0.75 1.65 D 0.53 0.47 NIR Ge 0.66 1.87 I

InAs 0.35 3.5 D MIR InSb 0.18 7 D

Si photodiodes cannot be used for optical communication @ 1.3, 1.55 mm. Ge OK. I(x)  I exp(x) o Ex 9.17: A GaAs LED emits at 860nm. Si photodetector is to be used. What should be the Most photon (63% or 1-1/e) absorbed thickness of Si that absorbs most of the radiation? over penetration depth d  1 Given (Si) @ 860nm is 6104 m-1.

Optcal Properties 2102308 43 (B2) Photodetectors

- materials: energy gap (EG) dictates detection range - devices: structure (p-n, p-i-n, APD, ...) dictates speed (RC), sensitivity - systems: available as 0D (point), 1D (line), 2D (area) detection - commercial products: embedded in tablet, smartphone, robots, cars...

Silicon (UV-VIS-NIR) III-V, II-VI (NIR, MIR)

surface recombination bandgap limit (device limit) (material limit)

Optcal Properties 2102308 44 9.11 Light scattering in materials - scattering is due to small particles embedded (solid) or suspended (liquid, gas) in transparent media - scattering reduces light intensity in the main direction by radiating part of the incident energy in all directions; the radiation pattern (angle dependent intensity: I(q )) depends on the relative size between the incident wavelength l and the particle size: - particle size < l10  Rayleigh scattering (see )

- particle size  l  Mie scattering - colour of sky: blue (daytime, ), pink/red (sunrise/sunset, ) due to ii i) eye sensitivity (peaks around green, 555 nm) iii i ii) the visible spectrum (bell shape, range 400-700 nm), and iii) 1/l4 — shorter l (blue) scattered more, longer l (red) forwarded more, by atmospheric dusts - attenuation in optical fiber (9.12) is fundamentally limited by iii)

   GMT



 https://bilimfili.com/hiperfizik/hbase/atmos/blusky.html#c1 GMT + 7

Optcal Properties 2102308 45 9.12 Attenuation in optical fibers (A0) Glass optical fiber (GOF) - optical fiber materials are mainly glass (A0) and polymer (A1), optical transmission through fibers always suffer from loss, quantified by attenuation (dB/km), see  - intrinsic loss mechanisms in GOF: lattice absorption (9.9) and Rayleigh scattering (9.11), the latter due to non-crystalline state of glass (small density fluctuation) - extrinsic loss mechanisms: hydroxyl (OH−), metallic (Fe2+) impurity, negligible by 1979 - comparison between electrical (Cu) vs optical (GOF, POF) signal transmission, Table 1 (A2) - when nature gives you a loss, it also gives you a gain (EDFA) (A3) Resonance @ 2.7 mm water byproducts (first harmonic @ 1.4 mm) (second @ ~1 mm, negligible in high-quality fibre)  10 P(x)  log (9.11) (9.9) x P(0) Stretching of Si-O bond 4 (9.10) I 1/ l (resonance @ 9 mm) (for Ge-O: 11 mm, graded index)

wavelength (mm) - Not shown, below 500 nm,   sharply since photons excite electrons from VB  CB (9.10) - loss @ 1.5 mm (dB/km): 0.2 in 1979, now = 0.16, intrinsic limit. (Source: Tilley 2011)

Optcal Properties 2102308 46 Plastic optical fiber (POF) (A1)

Material PMMA: Poly (methyl methacrylate)

(C5O2H8)n

Other names: Acrylic glass, Plexiglas

Optcal Properties 2102308 47 (A2) Benefits of optical fibers in communications systems: 1. transmission distance  2. transmission speed 3. information density 4. no electromagnetic interference (EMI)

Speed: - light in space: c=3x108 m/s - light in glass: 0.67c - electrons in Cu: 0.01c - electrons in Si: 0.001c 7 (vsat = 10 cm/s at 300 K)



Optcal Properties 48 Erbium doped fiber amplifier (EDFA) (A3) - active region of EDFA system consists of 30-m section of fiber doped with Er3+ (Fig. 14.35a) - how it works: weak optical signal enters (1), and after being pumped by laser diodes (LD) (2), leaves as output (3) with increased intensity: input (1) + pump (2)  output (3) - material: Erbium (III) ion (Er3+), not atom. The electronic structure of Er3+ is [Xe] 4f11, the 4f-

shell is well shielded (by the outer 5s, 5p shells) from the environment (SiO2), thus energy levels 4 4 4 (not bands): ground state I15/2 (GS), and next 2 excited states I13/2, I11/2 (Fig. 14.35b) 3+ 4 4 - physics: LD 980nm shone  (4) Er excited from I15/2 to I11/2, but (5) quickly relaxes by non- 4 radiative radiation to I13/2 and stays there for a long time, when an input photon (1) l1480 nm 3+ 4 arrives, it entices the excited Er (at I13/2) to return to GS by stimulated emission (6), think laser. Thus, 1 photon in, 2 photons out. Gain!

- effectively, power transfers from LD (2) (5) to output (3), which is an amplified version of input (1)

(1) (3) (4) (7) (6) (2)

(a) (b)

Optcal Properties 2102308 49 (M0) 9.13. Luminescence

• Incandescence: emission of radiation from thermal source (L5) • Luminescence: emission of radiation by a material due to absorption and conversion of energy (non-thermal source) (Fig. 15.9). Note the emission is always lower in energy than excitation. • Luminescence classified by excitation source (M1) (a) Photoluminescence (PL): luminescence from materials excited by photons. Ex. PL fibers in banknotes (b) Cathodoluminescence (CL): luminescence from materials excited by energetic electrons. Ex. CRT • Luminescence classified by time (emission after excitation) – < 10 ns: fluorescence – > 10 ns: phosphorescence • Luminescent materials (phosphors): – fluorescent lamps, TV screens (M1,2) – white LEDs (M2) – quantum dots (QDs) (M3-M5)

Optcal Properties 2102308 50 (M1) Emitted light Emitted light TV Screen phosphor Phosphor Phosphor

Incident Incident light electrons Heat Heat (a) Photoluminescence (b) Cathodoluminescence Phosphor Activators or luminescent centers Al O : Cr3+ Incident (e.g. Cr3+) 2 3 light

Host matrix (e.g. Al2O3) (c) A typical phosphor = host + activators Photoluminescence, cathodoluminescence and a typical phosphor

halophosphate − 3+ 2+ Ca10F2P6O24 : (Cl ,Sb ,Mn )

Optcal Properties 2102308 51 Mercury (Hg) emission spectrum (for fluorescent lamp) (M2) - white light from fluorescent lamp comes from the phosphor (RGB mixture) excited by sharp lines of Hg ions (Hg atoms previously excited by energetic electrons) - the spectrum of Hg (left) and energy level diagram (right)

Phosphor (YAG): InGaN chip: blue Application: White LEDs yellow emission emission 1.0 Total white emission (a) A typical “white” LED structure. (b) Yellow Yellow The spectral distribution of light emitted phosphor by a white LED. Blue luminescence is Blue 0.5 emission emitted by the GaInN chip and “yellow” Blue phosphorescence or luminescence is chip produced by a phosphor. The combined emission spectrum looks “white”. 0 350 450 550 650 750 White LED Wavelength (nm) (a) (b)

Optcal Properties 2102308 52 Quantum Dots (QDs) (M3) - bulk materials, carriers free to move in three dimensions (3D) - nanomaterials, carriers motion limited in 1 or more dimensions. The three types of quantum nanostructures: quantum wells (2D), quantum wires (1D), quantum dots (0D) (Fig. 3.4) (M4) - electron energy levels: see equations [1D], [3D] (M4). Optical properties (absorption, emission) of nanostructures depend on material (atoms), bonding (lattice constant), and size:

- for quantum wells, size means thickness, emission at energy E > EG (Fig. 14.38) (M5) - for QDs, size depends on shape, but generally a×b×c (length×width×height). See energy diagram (Fig. 14.41) and size-dependent emission spectra (Fig. 14.40) (M5) - quantum dots maybe synthesized by various techniques: wet chemistry, chemical vapor deposition (CVD), molecular beam epitaxy (MBE)

Optcal Properties 2102308 53 (Quantum) Nanostructures (M4) - structures with one or more dimensions in the same scale as de Broglie wavelength of electrons (nm in metals, 10s nm in semiconductors) - electrons confined in such structures can have only certain energies with certain wavelength dictated by size. For confinement in: - 1 dimension (Fig. 2.17), i.e. a well of width a, the discrete energy levels is [1D] - 3 dimensions, in a a×b×c dot, the discrete energy levels is [3D] - Size matters: electron energy is inversely proportional to size (E ∝ 1/a2)

[1D]

2 h 2 [1D]: E  n [2D] [3D] n 8m*a2 h2  n2 n2 n2  [3D]: E(n ,n ,n )   x  y  z  x y z *  2 2 2  8m  a b c 

Optcal Properties 2102308 54 (M5)

Optcal Properties 2102308 55 Optcal Properties 2102308 56