Optical Properties of Materials

C. Optical Properties of Materials

Contents

• light (9.1)

• refractive index, n (9.2)

• dispersion, n( l ) (9.3)

– v

_g

, N

_g

, zero dispersion (9.4)

• principles

– Snell (9.6): direction

– Fresnel (9.7): amplitudes, phases

• losses & origins

– n-jK, e

_r

'-j e

_r

", k'-jk" (9.8) – lattice absorption (9.9)

– VB  CB absorption (9.10) – scattering (9.11)

• case studies

– optical fibre (9.12)

– luminescence, phosphor (9.13)

Objectives

• What happens when light shines on a material?

• colors of materials

• Why are some materials transparent and others not?

• Optical applications:

– optical fibers

– optical amplification – luminescence, phosphors – nanostructures

v.2019.AUG

Reference: S.O. Kasap (3^rd,4^thEd.) Chap. 9; RJD Tilley (Understanding solids, 2^ndEd) Chap. 14.

• 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)

9.1 Light

^(L0)

(L1)

“Light -”

y

x

B

E 



_o



o

x

E t kz

E  cos    

k is propagation constant or wavenumber

monochromatic plane wave:

phase

Which (E, B) is more important?

- Electric and magnetic fields always co-exist by virtue of Faraday’s Law:

time-varying ⁽B field  Efield) - Most materials (atoms in the periodic

table) only respond to E, thus B is usually ignored

- Optical field refers to E field an electromagnetic wave

in 1D:

space time

E

t

l

z

T

time

space

  





 













l







/ 2

/ 2

/ 1

k

T T f

light-related vocabularies:

- monochromatic: single or very narrow range of l (red laser yes, the sun no) - coherent: photons constituting light

beam are in phase (laser yes, LED no) - polarization: direction of E field,

light sources can be (*) unpolarized, (a) plane polarized, (b) elliptically polarized, (c) circularly polarized,

*

(L2)

“planes”

for any given plane, the phase is a constant which moves dz at every dt  phase velocity:

 







 



















l







 l

/ 2

/ 2

/ 1

k

T T f

k f dt

v dz kz

o

t 



   

- propagation

monochromatic plane wave propagates in space at velocity v

 

^{,t E}_o



^t _o



E r  cos  kr 1D(far from source)

3D(close to source) plane wave

spherical wave

(L3)

H

₂

GAS SOLID

semiconductors

direct indirect

InP Si

- generation

(L4)

- generation by hot bodies (incandescence)

H₂, He W C

• 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

 (L5)

- 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)

- 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)

R A

T

o

I I I

I   

Light-Matter Interaction RAT (I0)

(I1)

Transmission Absorption Reflection

Cloaking

R

A

(smooth)

T

(rough) surface condition

Light-Matter Interaction

“interaction” overview of applications

Absorption

• Metals have fine succession of energy states.

• Near-surface electrons absorb visible light.

• Metals appear reflective (shiny) because

• Reflectivity = IR/Io is ~ 0.90-0.95.

(reflected light same frequency as incident)

Energy of electron

filled states unfilled states

E

IR

“conducting” electron

re-emitted photon from material surface

Pd Au

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

Light interaction with metals

Reflection

Colors

Color of metals dictated by interfacial property (metal/insulator interface) called

“surface plasmon resonance (SPR)”

Transmission (none)

RAT

^(I2)

Absorption (conditional): if hn > E_gap

• If E

G

< 1.8 eV, full absorption; color is black (Si@1.1, GaAs@1.4 eV)

• If E

_G

> 3.1 eV, no absorption; colorless (diamond@5.5, glass@9 eV), transparent (!), see (I5)

• If E

_G

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

incident photon energy hn

Energy range of visible light:

Light interaction with non-metals (insulators, semiconductors)

Luminescence

(re-emitted light)

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)

RAT

^(I3)

• Colour determined by sum of frequencies of - transmitted light, scattered light

- re-emitted light from electron transitions

• Ex. 1 (Semiconductor): Cadmium Sulfide (CdS) - E

_gap

= 2.4 eV,

- absorbs higher energy visible light (blue, violet), - Red/yellow/orange is transmitted, gives color.

CdS

Origin of color:

Boron(0-8 ppm depends on position, av. 0.36 ppm)

• Ex. 2 (Insulator)

The Hope Diamond = C (+ % B) - E

_gap

= 5.6 eV

- luminesce after UV exposure ( l <220nm) - phosphorescence

(re-emission occurs with delay time > 1 s)

LDR

Diamond

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

(I4)

Light-Matter Interaction: Results

(2.48 eV)

(1.88 eV)

Ruby (mineral)

Ruby (laser)

• Ex. 3 (Insulator)

Ruby = Sapphire (Al

₂

O

₃

) + Cr

₂

O

₃

- E

_gap

> 3.1eV

- pure sapphire is colorless - adding Cr

₂

O

₃

:

(0.5-2) at. %

• alters the band gap

• blue light is absorbed

• yellow is absorbed

• red is transmitted

• Result: Ruby is deep red.







  

Transparency of insulators dictated by microstructure:

single-crystal

poly-crystal (dense)

poly-crystal (porous)







Intrinsically transparentdielectrics can be made translucent/opaqueby “interior”

reflection/refraction/scattering.

Opacity/Transparency

(I5)

Light-Matter Interaction: Results

(E_G>3.1eV)

• 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

Material Lead glass Silica glass Soda-lime glass Quartz

Plexiglas

Polypropylene Diamond

n

~ 2 1.46 1.51 1.55 1.49 1.49 2.41 - Adding large, heavy ions (e.g. lead, Pb)

can decrease the speed of light.

• Result 2: Intensity of transmitted light decreases with distance traveled (thick pieces less transparent) -- see later in Losses (9.8)

• Result 3: Light can be “bent”

or “refracted”, see later Snell (9.6)

Refraction (q_i ≠ 0): a change in propagation direction of a wave when it changes a medium

Si O Pb Si O

Si O Na Ca Si O

C O H C O H C

(I6)

Light-Matter Interaction: Results



 







• 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 e

_r

 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 crystals (birefringent) which show optical anisotropy (9.14) (R2,R3)

• n is not a constant, but can change with incident wavelength, thus n( l ), see example (R4), for details next section 9.3 Dispersion

r

r r

r o

r o r

v n c

c v c

e

e m

e m

m e e









 1





9.2 Refractive Index (n)

^(R0)

optical materials are non-magnetic (m_r≅1)

10² 10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10¹⁶ ƒ Orientational,

Dipolar Interfacial and

space charge

Ionic

Electronic e_r'

e_r''

e_r' = 1

1 10^-2

Radio Infrared Ultraviolet light

10² 10⁴ 10⁶ 10⁸ 10¹⁰ 10¹² 10¹⁴ 10¹⁶ ƒ Orientational,

Dipolar Interfacial and

space charge

Ionic

Electronic e_r'

e_r''

e_r' = 1

1 10^-2

Radio Infrared Ultraviolet light

Dielectric properties of materials

- complex relative permittivity as a function of frequency - contribution of various polarization mechanisms

) ( )

( f

n n

r r

e l

e





(R1)

- Estimate nin Film 1, Film 2 - Film 1 is lossy, 2 lossless vacuum medium

f f

c = l_of v = lf l_o l  l_o/n k_o k = nk_o

o o

o o

k k f

f v

n  c    

l

 l

 l

l l

l

/ 2

/ 2

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

Fig. 9.25: A line viewed through a cubic sodium chloride (halite) crystal (optically isotropic) and a calcite crystal (optically

anisotropic).

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

Optical anisotropy (9.14)

- 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 transmission axes, along the long edges, at right angles to each other. The ordinary ray, undeflected, goes through the left polarizer whereas the extraordinary wave, deflected, goes through the right polarizer. The two waves therefore have orthogonal polarizations.

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

Applications: polarising beam splitter, phase contrast microscopy...

(R2)

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

Optically isotropic n = no

Glass (crown) 1.510

Diamond 2.417

Fluorite (CaF₂) 1.434

Uniaxial - Positive n_o n_e

Ice 1.309 1.3105

Quartz 1.5442 1.5533

Rutile (TiO₂) 2.616 2.903

Uniaxial - Negative n_o n_e

Calcite (CaCO₃) 1.658 1.486

Tourmaline 1.669 1.638

Lithium niobate (LiNBO₃) 2.29 2.20

Biaxial n1 n2 n3

Mica (muscovite) 1.5601 1.5936 1.5977

 non-crystalline

 crystalline (cubic) In noncrystalline materials, n is isotropic.

In crystalline materials, n is anisotropic.

o

e

n

n 

e

o

n

n 

wiki/Birefringence

n is frequency dependent n maybe direction dependent

   

o r

r

N n

e



  e

e







1

ISOTROPICANISOTROPIC

(R3)

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)

How are these measured in practice?

LF: kHz; e.g. by C-V measurement HF: (Optical) 10¹⁵Hz; Snell’s

 e_r 

Dielectric resonance electronic

dipolar

Physical origin of dispersion relation:

• Examples

case 1 (=): same polarization mechanism ( )

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

(R4)

) ( )

( f

n n

r r

e l

e





?

=

9.3 Dispersion n( l )

- 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)

(D0)

 

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

dispersion in optical fibercommunication

≠

   

₂ ² ₂

2

1

o

n l l

l l

 

 

Brief Physics of dispersion relation

 

^













 













2 2 0 2 induced

induced 2

/ 1











e e 

e

e e

o e r

r

m x Ze Ze p

p N n

(D1)

- 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:

- Example n(l) for GaAs (0.89-4.1 mm) at 300K (obtained empirically):

2767 0

78 10 3

7

₂

2 2

. . .

n   

l

l

[l in mm]

3 2

2 3 2

2 2 2 1

2 2 2 1

)

( C

B C

B C

A B

n  

 

 

 l

l l

l l

l l

B_1,2,3, C_1,2,3 : Sellmeier coefficients

Sources: Batop.com, Wiki

dispersion relation in a material with a singlepolarization mechanism with a singleresonant wavelength l_o

The cause of dispersion: engineering & physics

Dispersionoccurs 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

Colordepends on material (E_G)

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 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 FWHM

(D2)

E_G

E

The effect of dispersion: 9.4 Group velocity and group index

Information:

phase travels @ phase velocity (v) E

_max

travels @ group velocity (v

_g

)

Note that eyes/detectors measure intensity (E²)

v = c/n and  = vk v

_g

= d / d k

v = c and  = ck

 v

_g

= d / d k = c = phase velocity for all l ’s

 in a medium: n > 1 ( l dependent)

 

g g

g

N v c

d n dn

c v k

n v c















l l d

d

l

N_g group index

Dispersionoccurs 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)





^x

x E

E²

 in vacuum: n = 1 ( l independent)

(D3)

The remedy: eliminate / minimise?

l l d n  dn

group index



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 N_g= 1.462  v_g= 2.052 10⁸ m/s

Zero-dispersion









l l

l l

d dN c Ld

dt

d dt L c d

dN

L ct v N c

g g

g g

g g

g

 1









(D4)

9.6 Snell’s law and TIR

Trigonometry:

i & t

1 2

1 2

sin sin

sin sin

n n

t v B

A B

B

t v B

A A

A

t i

i t





 

 

 

 

q q

q q

n

₁

> n

₂

Requirement for wave propagation = Constructive interference: wavefronts for reflected and refracted (transmitted) waves

must be in phase with incidence wave

=

i & r

r i

r

v t

B A A

A

q q

q





 

  sin

₁

ALL other angles interfere destructively

=

Snell’s law





/



=

(1621)

   90

^



when q

_i

 q

_c

 q

_t



TIR leads to wave propagation in a dielectric medium

   

2 1 2

2

2 1 2 0

2 1

/

i y

, t

n sin n n

kz t j exp e

t , z , y E











  

 



 



 ^



l q

 





penetration depth = 1/

from Snell’s law:



 



 ^

1 1 2

sin n n qc

Fiber axis

n₂

n₁ n₂ q q

Cladding TIR

Light ray

Core

Emitter

Input Output

Optical Fiber

Photodetector Digital signal

Information Information

t

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.

9.7 Fresnel’s equations

- 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)

- the results (mainly reflection coefficients) are plotted in two example cases:

- internal reflection, when n

₁

> n

₂

, see (F2) - external reflection, when n

₁

< n

₂

, see (F3)

- human eyes (and optical detectors) sense intensity (E

²

), 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 ( q

_i

= 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)

(F0)

1788---1827

n₂

z y

xinto paper q_i n

1>n q_r 2

Incident wave

q_t q_t= 90^o

Transmitted wave

E_t,

Evanescent wave

Reflected wave

Reflected wave Incident

wave

q_i q_r E_i,

(a) q_i <q_cthen some of the wave is transmitted into the less dense medium.

Some of the wave is reflected.

(b)q_i >q_cthen the incident wave suffers total internal reflection. There is a decaying evanescent wave into medium 2

E_i,

y

k_i

k_t

k_r E_t,

E_i, E_i,

E_r,

E_r,

k_r E_t,

E_r,

E_r,

Governing principles:

A. Snell’s law:

B. electromagnetism:

C. Boundary conditions:

E_tangential (1) = E_tangential (2)

B_tangential (1) = B_tangential (2); for non magnetic media

2 2

1

1sinq n sinq

n 



 



 







 



 E

c B n

c E

B n _// and _//

→q_i= qrand

1 2

n n  n

i i

i i

t

i i

i i

i r

E n t E

n n E

r E

q q

q

q q

q q

2 , 2

,

2 2

2 2

, ,

sin cos

cos 2

sin cos

sin cos



 









 















i i

i i

t

i i

i i

i r

n n

n E

t E

n n

n n

E r E

q q

q

q q

q q

cos sin

cos 2

cos sin

cos sin

2 2

2 //

, //

, //

2 2

2

2 2

2

//

, //

, //



 









 



1

;

1 _//  _// 



r_ t_ r nt

/ 

/ 

 / 

 / 

(F1)

Example:

n

₁

= 1.44 and n

₂

= 1 

normal incidence :

2 1

2 1

//

n n

n r n

r 

 



_

(n

₁

> n

₂

)

Internal Reflection

set r_//= 0



 



 



 tan^  34.8

1 1 2

n n qp







 



 

 , sin^ 44

44 . 1

1

1 1 2

n

n q_c n

(r_, r_//complex)

(+, real)

(+, real)

(–, real)

polarization angle Brewster’s angle

 = 0 @ q_i = 0

set q_i= 0

(F2)

Transmitted light in both internal and external reflection does not experience phase shift (t is always +ve, q

_i

< q

_c

)

(main concern is reflected light for propagation)

External Reflection

Example:

n

₁

= 1 and n

₂

= 1.44   no critical angle

c

n n

n n

q

q q q

no

) 1 (sin

) (

sin sin

2 2

1

2 1

2 1











 

^{ }

 tan^1 1.44 55.2 qp

 =  @ q_i = 0

(n

₁

< n

₂

)

0 sin

1 , 44 .

1 ₂  ₂ 

 n _i

n q

(F3)

Reflectance (R) and Transmittance (T)

R+ T= 1

2 //

1 2 2 //

, 1

2 //

, 2 //

2 1 2 2 , 1

2 , 2

2 2 //

//

, 2 //

, //

2 2

, 2 ,

and

and

n t n E

n E T n

n t n E

n E T n

r E

R E r

E R E

i t

i t

i r

i r

































At normal incidence ( q

_i

= 0):



₁ ¹ ₂²



²

//

2

2 1

2 1 //

4 n n

n T n

T T

n n

n R n

R R

 







 







 









(practical angle in laser diodes, or cavity structures)

Ex:

Reflectance (%)

(F4)

Ex. 9.8

Ex. 9.9

n = 1.44 n = 1.46

l = 1300 nm

i) Find minimum angle for TIR

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

air glass 1 n 1.5

Find reflection coefficients and reflectance in each case

Internal Reflection

Internal and External Reflection

ii) use: ² ₂ ^1/2

2 1

2 sin 1

2











  

 



  i

n n

n q

l

  q_i () 1/ (mm)

87 0.906

90 0.859

thicker than 5 * penetration depth (F5)

Ex. 9.10: n

₁

= 1, n

₃

= 3.5, calculate R without ARC layer.

Insert ARC layer to minimise reflection. A and B must be:

of similar amplitude n₂= (n₁n₃)^1/2 *** Proof

out of phase d = ?

700 nm Requirement: maximum light transmitted to

PV for electricity generation n₁ < n₂ < n₃

Max Absorption (PV)

air/ARC = ARC/semic.

(F6)

 

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.

n₁< n₂

Show that waves A, B and C interfere constructively.

note: l₁ = l₀/n₁, l₂ = l₀/n₂

Max Reflection (LD,VCSEL)

(F7)

9.8 Loss and complex refractive index

- for transparent insulators (E

_G

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

- for metals (no E

_G

) and semiconductors (E

_G

 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

(L0)

(linear) Absorption Coefficient 

 

 

 

 k z  exp j  t k z 

exp E

z k j k t

j exp E

E

kz t

j exp E

E

o o o

 

 



 

 











 Plane wave

Lossless:

Lossy:

Intensity:

 

 

K k k

z I

I

z k E

I

o o

2 2

where

exp

or

2

2

exp

 







 









EXP.



determine K( l )

from absorption /transmission measurements at normal incidence as a function of freq.

Since e

_r

is complex  n is also complex  written as N In fact, propagation constant k is complex: k  k' – jk''

k' describes propagation:

phase velocity v =  /k' k" describes absorption:

attenuation along z

k

₀

is propagation constant in a vacuum (no loss, real #)

o

o

k

k j k

k n – jK k

N   







n refractive index K absorption index

ko

k



ko

k

 note:

o o

o o

k k f

f v

n  c    

l

 l

 l

l l

l

/ 2

/ 2

(extinction coefficient)

Beer’s law*

* Bouguer, Beer, Lambert

R A T

R,T A

(L1)

a-Si

E

_G

photon energy

lossless:

lossy:

'' '

2 2

'' '

2 and

, 0

r r

r r

r

nK K

n

j jK

n

n k

e e

e e

e















 

EXP.



determine R( l )

from reflectance measurement at normal incidence as a function of freq.

 

 

^{2 2}

2 2 2

1 1 1

1

K n

n – K N

R N





 



 

Hence, complex refractive index:

(n,K) collectively called optical constants but they vary with wavelength, hence not constants as name suggest



&



get n – jK (l)

from 9.7 (Fresnel)

Optical constants (nand 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)

K

r n

r ' , e ''  , e

dielectric / optical

Kramers-Konig relation

(L2)

I_o

I_t I_r

broadband lightsource

monochromator

material under test

photodetector

l , photon energy



l

l , photon energy

l , photon energy

l , photon energy

I

_t

/ I

_o

I

_r

/ I

_o

, R

(k", K) n

beamsplitter (50:50)

R (n,K)

z

 

Setup for (n,K) measurements

(L3)

R

K n

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.

(L4)

z

Propagation direction

k

Ions at equilibrium positions in the crystal

Forced oscillations by the EM wave

E_x

9.9 Lattice absorption

vibrate at frequency of E

- 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)

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.

9.10 Band-to-band absorption

Indirect materials:

 h E

hv 

_G



phonon energy - 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:

- 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:

E

G

hv 

) (

24 . ) 1

( m E eV

E hc

G off

cut

G  

l

_

m



l

(B0)



Most photon (63% or 1-1/e) absorbed over penetration depth d  1

) exp(

)

( x I x

I 

_o

 

Semiconductor E_g (eV) lg (mm) Type

InP 1.35 0.91 D

GaAs_0.88Sb_0.12 1.15 1.08 D

Si 1.12 1.11 I

In_0.7Ga_0.3As_0.64P_0.36 0.89 1.4 D In_0.53Ga_0.47As 0.75 1.65 D

Ge 0.66 1.87 I

InAs 0.35 3.5 D

InSb 0.18 7 D

Table 9.2Band gap energy E_gat 300 K, cut-off wavelengthl_gand type of bandgap (D = Direct and I = Indirect) for some photodetector materials.

Si photodiodes cannot be used for optical communication @ 1.3, 1.55 mm. Ge OK.

Ex 9.17: A GaAs LED emits at 860nm. Si

photodetector is to be used. What should be the thickness of Si that absorbs most of the radiation?

Given (Si) @ 860nm is 610⁴ m^-1.

 (= 2k

^

= 2k

_o

K)

NIR

MIR (B1)

surface recombination (device limit)

bandgap limit (material limit) Photodetectors

- materials: energy gap (E_G) 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)

(B2)

- 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 i) eye sensitivity (peaks around green, 555 nm)

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)

9.11 Light scattering in materials

https://bilimfili.com/hiperfizik/hbase/atmos/blusky.html#c1









 GMT

GMT + 7 i ii

iii

9.12 Attenuation in optical fibers

- 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)

water byproducts

Resonance @ 2.7 mm (first harmonic @ 1.4 mm)

(second @ ~1 mm, negligible in high-quality fibre)

Stretching of Si-O bond (resonance @ 9 mm)

(for Ge-O: 11 mm, graded index)

- 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)

(9.9) (9.11)

(9.10)

Glass optical fiber (GOF)

/ 4

1 l

 I

(A0)

wavelength (mm) )

0 (

) log ( 10

P x P

 x



Plastic optical fiber (POF)

Material

PMMA: Poly (methyl methacrylate) (C₅O₂H₈)_n

Other names:

Acrylic glass, Plexiglas

(A1)

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=3x10⁸m/s - light in glass: 0.67c

- electrons in Cu: 0.01c - electrons in Si: 0.001c

(v_sat= 10⁷cm/s at 300 K)





(A2)

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)