wave nature of lightocw.nctu.edu.tw/course/physics/optoelectronics_lecture...ex. consider a hene...

OPTOELECTRONICS Prof. Wei-I Lee 1

Wave Nature of Light


Fight About Light – I

Ref : NTU 黃鼎偉教授

Pre-Newton Era ( 1600 – 1700 )


Fight About Light – II

Ref. : NTU 黃鼎偉教授

Post-Newton Era ( 1700 – 1800 )


得罪了不該得罪的人?


Proof of Light as Waves


Post-Young Era ( 1800 – 1900 )


Wave Nature of Light



Proof of Light as Particles


Post-Einstein Era ( 1900 – ? )


The Understanding of Light

Laser

Source : NTU 黃鼎偉教授


Particle-Wave Duality of Light

光有時以波的型態出現，有時以粒子的型態出現。當以波的型態出現時，

即不具粒子性。當以粒子的型態出現時，即不具波動性。

The photon nature of light will be important when discussing

semiconductor optoelectronic devices. For many other optical devices, the

wave nature of light will be more important and will be discussed first.

Dual Nature of Light


a monochromatic EM plane wave :

a traveling wave along z-dir.

Eo : amplitude , w : angular frequency , k : wave number, 2p/l

fo : phase constant , ( wt – kz + fo ) : phase , f

The interaction of a light wave with a non-conducting matter usually

described through the electric field optical field refers to electric field

Traveling Wave in Z-Direction

Plane Wave in Homogenous Medium


Wavefront

wavefront : a surface over which the phase of a wave is constant

(wavefront of a plane wave) (the direction of wave propagation)



More General Traveling Wave Expression

exponential notation of traveling waves

direction of wave propagation can be indicated by the wave vector, k ( k )



Phase Velocity

relationship between time and space for a given phase f

during a time interval dt this const. phase wavefront moves dz

phase velocity of the wave = dz/dt



Maxwell’s Equations

In an isotropic and linear dielectric medium

er ( relative permitivity ) : 1. the same in all directions

2. indep. of electric field

E must obey the following Maxwell’s EM wave equation

( assumes the conductivity of the medium is zero, s = 0 )

There are many possible waves that can satisfy the above Maxwell’s eq.

e.g. A plane wave :

but a perfect plane wave does not exist in reality



Optical Divergence

optical divergence : angular separation of wave vectors on a given

wavefront

for a perfect plane wave optical divergence = 0o

for a spherical wave, from a point EM source optical divergence = 360o



Gaussian Beam

A more practical example of a light beam :

over a small spatial region at far distance

~ plane wave

- Many light beams, e.g. output of a laser, Gaussian beam

beam diameter : 2w

(pw2 contains 85% power)

waist, spot size : 2wo

waist radius : wo

beam divergence : 2q

waist beam divergence



Gaussian Beam


this equ. also defines the min. spot size to which a Gaussian

beam can be focused

Ex. Consider a HeNe laser beam at 633 nm with a spot size of 10mm.

Assuming a Gaussian beam, what is the divergence of the beam?

<Sol.> use

we can find


EM Waves in Dielectric Mediums

dielectric material : a poor conductor of electricity, but an efficient

supporter of electrostatic fields

EM wave travels in a dielectric medium polarize the molecules

EM field and molecular dipoles coupled

slow downs the EM wave w.r.t to its speed in a vacuum, i.e. c

stronger EM field/dipoles interaction slower EM wave

relative permittivity (dielectric const.) er : measure the ease of polarization

the phase velocity of EM wave in a nonmagnetic dielectric medium :

( in vacuum, er = 1,

)

er due to electronic polarization for u in optical range

(electronic + ionic) polarization for u in IR range slower v

er is a function of u, er(u)

Refractive Index


Refractive Index

refractive index of a medium, n :

(speed of light in vacuum) / (speed of light in the medium)

n light propagates slower, material probably denser

u remains the same as in vacuum l changes in different mediums

lmedium = l/n

kmedium = nk

In noncrystalline materials, e.g. glasses er and n are usually isotropic

in crystals ( excluding cubic crystals such as diamonds )

er and n are usually anisotropic

n depends on the value of er along the direction of the oscillating

electric field

Refractive Index


Relative Permittivity and Refractive Index in Material

er is a function of u n is a function of u

Refractive Index


Group Velocity

Group Velocity and Group Index

In reality, there is no perfect monochromatic light

have to consider a group of waves differing slightly in w and k

2 perfectly harmonic waves with w+dw, k+dk interfere with each other

w-dw, k-dk

wave packet

wave packet (max. amplitude)

travels at group velocity

vg = dw/dk = dw/dk

vg defines the speed of

energy/information

propagation


Group Velocity

2 perfectly harmonic waves with w+dw, k+dk interfere with each other

w-dw, k-dk


vg = dw / dk


Group Refractive Index

in vacuum v = c for all u and l, and w = vk

In many other materials, in general, n = n(l) and then

,

Ng : group refractive index of a medium

if n is not a function of l Ng = n



Dispersive Medium

in many materials, n is a function of l

v (phase velocity) and vg (group velocity) depend on l

dispersive mediums

Silica (pure SiO2)

near 1300 nm

Ng is minimum

and relatively l independent

light waves with l ~ 1300 nm

travel with the same vg

( no dispersion )



v and vg in Dispersive Medium

Ex. l = 1mm, refractive index (l=1mm) = 1.45 v and vg = ?

<Sol.>

near 1000 nm

Ng = 1.463

vg is ~0.9% slower than v



Energy Density in EM Wave

from EM theory :

in an isotropic dielectric medium with a refractive index n

v = (eoermo)-1/2 , n = er

energy density (energy

per unit volume) in Ex

= energy density in By

total energy density in the wave = eoer Ex2

Magnetic Field, Irradiance and Poynting Vector


S = energy flow per unit time per unit area ( instantaneous irradiance,

intensity ) =

Poynting vector :

energy flow per unti time

per unit area in the direction

of energy propagation

average irradiance :

( v = c/n , er = n2 )

all practical measurements yield the average irradiance

Irradiance and Poynting Vector

Magnetic Field, Irradiance and Poynting Vector


Snell’s Law and Total Internal Reflection (TIR)

Snell’s Law and Total Internal Reflection

kr = ki , qr = qi

Snell’s Law :

when n1 > n2

qt > qi

when qi reaches critical angle qc

sin qc = n2/n1

qt = 90o no transmitted wave total internal reflection (TIR)


Transverse Electric and Transverse Magnetic Fields

Fresnel’s Equations

plane of incidence : plane containing the incident and the reflected rays

Ei,, Er,, Et, ( normal to the plane of incidence and z-direction ) :

transverse electric field (TE) waves

Ei,//, Er,//, Et,// ( parallel to the plane of incidence ) their companion

magnetic field components are perpendicular to the plane of incidence

and z-direction transverse magnetic field (TM) waves


Solving Boundary Conditions


( phase change of Er and Et w.r.t Ei

included in the complex amplitudes

Ero and Eto )

Apply two B.C. :

1. Etangential(1) = Etangential(2)

2. Btangential(1) = Btangential(2) [ Note : B// = (n/c) E , B = (n/c) E// ]

Ero and Eto w.r.t Eio can be obtained

1. qr = qi

2. n1sinq1 = n2sinq2

3. Fresnel’s equations

n2

n1 > n2




Fresnel’s equations :

amplitude of the Er and Et w.r.t. Ei

in terms of n1, n2, and qi

r : reflection coefficients

t : transmission coefficients

n = n2/n1

r// + nt// = 1 , r + 1 = t

- amplitudes and phases of the reflected and transmitted waves can be

determined from the above reflection and transmission coefficients

n2

n1 > n2


Internal Reflection


n1 > n2 internal reflection ; n2 > n1 external reflection

Ex. n1 = 1.44, n2 = 1.00

from reflection coefficients

magnitude and phase change in the reflected waves

w.r.t. the incident wave

e.g. r = rexp(-jF)

r: % of the incident wave amplitude

F : phase change w.r.t. the incident wave

n2

n1 > n2


Internal Reflection


when qi < qc ( n2 – sin2qi ) > 0 ( note : sin qc = n2/n1 )

r// r real numbers r > 0 : f =0o , r < 0 : f = -180o

when qi > qc ( n2 – sin2qi ) < 0

phase change, other than 0o and 180o, in reflected

waves w.r.t. the incident wave

phase change other than 0o and 180o occur only when

there is total internal reflection

n2

n1 > n2

( n = n2/n1 )


Polarization and Brewster’s Angle


qi ~ 0o (normal incidence) f =0o , no phse change

> 0, for n1>n2 (internal reflection)

qi r , r//

when qi = qp , tan qp = n2/n1

r//= 0, (field in reflected wave) (plane of incidence)

at qi = qp : polarization angle or Brewster’s angle

reflected wave is linearly polarized

n2

n1 > n2

( n = n2/n1 )


Total Internal Reflection ( TIR )


qi > qc ( TIR ) amplitude of the reflected wave from

TIR equals to the amplitude of the incident wave

reflected wave phase shifts determined by the

following equ. :

n2

n1 > n2

( n = n2/n1 )


Evanescent Wave


What happens to the transmitted

wave when qi > qc? (TIR)

according to the B.C.

there must still be an electric

field in medium 2

evanescent wave :

: attenuation coefficient

l : free space wavelength

d = 1/a2 : penetration depth

- evanescent wave propagates along the boundary ( along z ) with the same

speed as the z-component velocity of the incident and reflected waves


External Reflection


external reflection : light reflection

when light approaches from the lower

index side to the higher index side

Ex. n1 = 1.00, n2 = 1.44 =====

in external reflection at normal

incidence 180o phase shift

r// = 0 at Brewster angle

reflected wave polarized in

the E component only

transmitted light does not experience phase shift, similar to internal

reflection when qi < qc


Reflectance


Power flow per unit area, intensity or irradiance, of a traveling light wave :

v : light speed in the medium

er : medium’s relative permittivity

nEo2 ( since v = c/n and er = n2 )

Reflectance, R : (reflected light intensity) / (incident light intensity)

( note : reflectance is always a real number, reflection coefficients can be complex numbers )

r2 = (r) (r)* , r//2 = (r//) (r//)*

for normal incidence :

Ex. n of glass : ~ 1.5 4% of the incident light will be reflected back

into the glass when light enters perpendicularly into air from glass

( ½ ereoEo2 : energy per unit volume )


Transmittance


Transmittance, R : (transmitted light intensity) / (incident light intensity)

( note : transmitted light is in a different medium and generally in a different direction )

for normal incidence :

( I nEo2 )

R + T = 1


Antireflection Coatings on Solar Cells


semiconductor

solar cell

sun light

electrical

energy

P-t

yp

e

N-t

yp

e

reflected

light

n1(air) = 1 , n2(Si @ 700 – 800 nm) = 3.5

antireflection coating : n2 , n1(air) < n2 < n3(Si )

phase difference between wave A and B :

[ 180o (@ n1/n2 interface) + 180o (@ n2/n3 interface) ]

+ kc2d = (2p/lc) 2d = (2pn2/l) 2d

to reduce the reflected light,

A and B should interfere destructively

d = multiples of quarter wavelength

when n2 = n1n3 ( ~ 1.87 ) r12 (@ n1/n2) = r23(@ n2/n2)

Aand Bcomparable good degree of destructive interference


Dielectric Mirrors


n1 < n2 , l1 = lo/n1 , l2 = lo/n2

lo = free space wavelength

phase difference between wave A and B :

180o (@ n1/n2 interface) + k2 2 (l2/4)

= p + (2p/l2) 2 (l2 /4) = 2p

A , B in phase and interfere constructively

similarly, B and C interfere constructively

all reflected waves from the consecutive

boundaries interfere constructively

after several layers (depending on n1 and n2),

transmitted intensity negligible and

reflected intensity close to unity

widely used in vertical cavity surface emitting lasers (VCSEL)


Fabry-Perot Optical Resonator

Multiple Interference and Optical Resonators

(metal coated) M1 and M2 in perfect parallel with free space in between

stationary/standing EM waves in the cavity

each allowed lm : a cavity mode

with the resonant freq. um

uf : fundamental mode freq. , freq. separation between

two neighboring modes (Dum), free spectral range

if no loss from the cavity and the mirrors

are perfectly reflecting

intensity peaks at um will be sharp lines

serves to (1) “store” radiation energy or

(2) filter light at certain freq.


Optical Resonator with Non-perfect Reflectors - I


M1 and M2 : identical with r = r

(phase of B) – (phase of A) = k (2L)

(magnitude of B) = r2 (magnitude of A)

A + B = A + A r2 exp(-j 2k L)

after infinite round-trip reflections

with Icavity = Ecavity2 and R = r2

Io = A2 : original intensity

um = m (c/2L)


Optical Resonator with Non-perfect Reflectors - II


Dum

R (mirror reflectance) radiation loss from the cavity

broader mode peaks and smaller difference between max. and min.

intensities

spectral width, dum : full width at half maximum (FWHM) of a mode

intensity

when R > 0.6

F, finesse : uf / dum = Dum / dum

= (mode separation) / (spectral width)

cavity losses ( R ) F

sharper mode peaks


Optical Resonator as Optical Filters


Fabry-Perot cavities widely used in laser, inference filter, and

spectroscopic applications

adjust L “tuning

capability” to scan

different wavelengths

Icavity = (1 – R )Iincidnet

Itransmitted = (1 – R)Icavity

for a cavity filled with a medium with a refractive index n

use nk for k in the above eq.


Goos-Haenchen Shift

Goos-Haenchen Shift and Optical Tunneling

when qi > qc ( TIR )

reflected wave appears

to be laterally shifted at

the interface

appears to be reflected

from a virtual plane

Goos-Haenchen shift

lateral shift ( Dz ) effect caused by :

(1) phase change f at the interface of total internal reflection

(2) electric field extends into n2 by a penetration depth d = 1/a2

Dz = 2d tanqi : depends on qi and penetration depth

Ex. l = 1 mm, qi = 85o, n1 = 1.45, n2 = 1.43 (glass/glass interface),

d = 0.78 mm Dz ~ 18 mm


Optical Tunneling


shrink d to sufficiently small

attenuated beam emerges in C

transmitted beam in C carriers

some of the light intensity

intensity of the reflected beam reduced

frustrated total internal reflection (FTIR)

FTIR utilized in beam splitters

extent of energy division

between two beams depends

on : (1) thickness of layer B

(2) refractive index of B


Beam Splitter Cubes



Temporal Coherence

Temporal and Spatial Coherence

consider a traveling EM wave represented by a pure sinusoidal wave :

perfectly coherent

perfect coherence : one can predict the phase of any portion of the wave

from any other portion of the wave

temporal coherence : the extent to which two points, such as P and Q,

separated in time at a given location in space can be correlated

A more practical since wave exist only over a time duration Dt

this wave-train has coherence time = Dt

coherence length l = cDt

( observed at

a fixed point )


Spectrum of F(t)


Any arbitrary time dependent function f(t) can be represented by a sum of

pure sinusoidal waves with varying frequencies, amplitudes, and phases.

( Fourier transform )

spectrum of f(t) : the amplitudes of various sinusoidal oscillations that

constitute the function f(t)

Du : spectral width

Du = 1/Dt


Coherence Length and Spectral Width


Du = 1/Dt coherence and spectral width are intimately linked

Ex. The orange radiation at 589 nm emitted from a sodium lamp has

spectral width Du ~ 5 x 1011 Hz coherence time Dt ~ 2 ps and

coherence length ~ 0.6 mm

Ex. Red lasing emission from a He-Ne laser operating in multimode has

spectral width Du

~ 1.5 x 109 Hz

coherence

length ~ 200 mm

Ex. A continuous

wave laser operating

in a single mode will have very narrow spectral width coherence length

of several hundred meters light waves from laser devices have

substantial coherence lengths and are therefore widely used in wave-

interference studies and applications


White Light / White Noise


ideal white light : consists all frequencies of sinusoidal waves or lights

for white light / white noise which contains a wide range of frequencies

knowing P can not predict the phase or the signal at any other point Q,

unless Q is very close to P

no coherence

light in the real

world lies between

( a ) and ( c )


Coherence Between Waves


coherence between two waves : extent of correlation between two waves

Ex. Two identical wave trains of coherence length l travel different optical

paths when arrive at the same destination, they can interfere only over

a time period Dt the 2 waves have mutual temporal coherence over the

time interval Dt

spatial coherence :

the extent of coherence

between waves radiated

from different locations

on a light source

spatially coherent source :

emits waves that are in

phase over its entire

emission surface


Double Slit Interference of Light

Diffraction Principles

each slit acts a point source

R >> d s1p // s2p

Constructive interference

when : d sinq = nl , n = 0, 1, 2, ….

Destructive interference

when : d sinq = (n + ½ ) l , n = 0, 1, 2, …..

R


When Slit Width Can Not Be Ignored


Haygen’s Principle :

每一個波前 ( wavefront ) 的點形同另一個波點源 ( point source )

當狹縫寬度( a ) 波長 ( l )

when a >> l

what will happen when a > l ?~


Diffraction Phenomena


Diffraction :

Fraunhofer diffraction

Fresnel diffraction

Fraunhofer diffraction :

1. incident light beam

a plane wave

( collimated light beam )

2. observation or detection done far away from the aperture

also look like plane waves

Fresnel diffraction :

both incident light beam and received light waves are curvature waves

( not plane waves )

e.g. light source and detection screen are both close to the aperture

Fraunhofer diffraction much more important than Fresnel diffraction


Fraunhofer Diffraction


zero intensity occurs when

sinq = ml/a , m = 1, 2, 3

center bright region > a

( beam divergence )


Fraunhofer Diffraction


• 當滿足以下條件時 , 屏幕上會出現暗帶 :

CP – EP ( path difference ) = l/2

AP – EP = l a sinq = l

• 假設 a << R

• 當滿足以下條件時 , 屏幕上會出現另一個暗帶 :

DP – EP ( path difference ) = l/2 AP – EP = 2l a sinq = 2l

• 繞射弱點發生在 : a sinq = ml , m = 1, 2, 3, …..

sin q1 = l/a , sin q3 = 2l / a

A

B

C

D

E

a

a

to point P screen

R


Airy Rings


diffraction pattern from a

circular aperture Airy rings

( described by a Bessel func. )

angular radius of airy disk :

sinq = 1.22 • l/D

- divergence angle from aperture

center to Airy disk circumference

= 2q


Resolving Power of Imaging Systems


Rayleigh criterion : the two spots are just resolvable when the principle

maximum of one diffraction pattern coincides with the minimum of the

other

sin( Dqmin ) = 1.22 • l/D ( D : aperture diameter )

wave nature of lightocw.nctu.edu.tw/course/physics/optoelectronics_lecture...ex. consider a hene...

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