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OPTOELECTRONICS Prof. Wei-I Lee 1
Wave Nature of Light
OPTOELECTRONICS Prof. Wei-I Lee 2
Fight About Light – I
Ref : NTU 黃鼎偉 教授
Pre-Newton Era ( 1600 – 1700 )
OPTOELECTRONICS Prof. Wei-I Lee 3
Fight About Light – II
Ref. : NTU 黃鼎偉 教授
Post-Newton Era ( 1700 – 1800 )
OPTOELECTRONICS Prof. Wei-I Lee 4
得罪了不該得罪的人?
OPTOELECTRONICS Prof. Wei-I Lee 5
Proof of Light as Waves
Ref. : NTU 黃鼎偉 教授
Post-Young Era ( 1800 – 1900 )
OPTOELECTRONICS Prof. Wei-I Lee 6
Wave Nature of Light
Ref. : NTU 黃鼎偉 教授
OPTOELECTRONICS Prof. Wei-I Lee 7
Proof of Light as Particles
Ref. : NTU 黃鼎偉 教授
Post-Einstein Era ( 1900 – ? )
OPTOELECTRONICS Prof. Wei-I Lee 8
The Understanding of Light
Laser
Source : NTU 黃鼎偉 教授
OPTOELECTRONICS Prof. Wei-I Lee 9
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
OPTOELECTRONICS Prof. Wei-I Lee 10
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
OPTOELECTRONICS Prof. Wei-I Lee 11
Wavefront
wavefront : a surface over which the phase of a wave is constant
(wavefront of a plane wave) (the direction of wave propagation)
Plane Wave in Homogenous Medium
OPTOELECTRONICS Prof. Wei-I Lee 12
More General Traveling Wave Expression
exponential notation of traveling waves
direction of wave propagation can be indicated by the wave vector, k ( k )
Plane Wave in Homogenous Medium
OPTOELECTRONICS Prof. Wei-I Lee 13
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
Plane Wave in Homogenous Medium
OPTOELECTRONICS Prof. Wei-I Lee 14
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
Plane Wave in Homogenous Medium
OPTOELECTRONICS Prof. Wei-I Lee 15
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
Plane Wave in Homogenous Medium
OPTOELECTRONICS Prof. Wei-I Lee 16
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
Plane Wave in Homogenous Medium
OPTOELECTRONICS Prof. Wei-I Lee 17
Gaussian Beam
Plane Wave in Homogenous Medium
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
OPTOELECTRONICS Prof. Wei-I Lee 18
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
OPTOELECTRONICS Prof. Wei-I Lee 19
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
OPTOELECTRONICS Prof. Wei-I Lee 20
Relative Permittivity and Refractive Index in Material
er is a function of u n is a function of u
Refractive Index
OPTOELECTRONICS Prof. Wei-I Lee 21
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
OPTOELECTRONICS Prof. Wei-I Lee 22
Group Velocity
2 perfectly harmonic waves with w+dw, k+dk interfere with each other
w-dw, k-dk
Group Velocity and Group Index
vg = dw / dk
OPTOELECTRONICS Prof. Wei-I Lee 23
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
Group Velocity and Group Index
OPTOELECTRONICS Prof. Wei-I Lee 24
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 )
Group Velocity and Group Index
OPTOELECTRONICS Prof. Wei-I Lee 25
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
Group Velocity and Group Index
OPTOELECTRONICS Prof. Wei-I Lee 26
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
OPTOELECTRONICS Prof. Wei-I Lee 27
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
OPTOELECTRONICS Prof. Wei-I Lee 28
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)
OPTOELECTRONICS Prof. Wei-I Lee 29
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
OPTOELECTRONICS Prof. Wei-I Lee 30
Solving Boundary Conditions
Fresnel’s Equations
( 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
OPTOELECTRONICS Prof. Wei-I Lee 31
Fresnel’s Equations
Fresnel’s Equations
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
OPTOELECTRONICS Prof. Wei-I Lee 32
Internal Reflection
Fresnel’s Equations
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
OPTOELECTRONICS Prof. Wei-I Lee 33
Internal Reflection
Fresnel’s Equations
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 )
OPTOELECTRONICS Prof. Wei-I Lee 34
Polarization and Brewster’s Angle
Fresnel’s Equations
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 )
OPTOELECTRONICS Prof. Wei-I Lee 35
Total Internal Reflection ( TIR )
Fresnel’s Equations
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 )
OPTOELECTRONICS Prof. Wei-I Lee 36
Evanescent Wave
Fresnel’s Equations
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
OPTOELECTRONICS Prof. Wei-I Lee 37
External Reflection
Fresnel’s Equations
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
OPTOELECTRONICS Prof. Wei-I Lee 38
Reflectance
Fresnel’s Equations
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 )
OPTOELECTRONICS Prof. Wei-I Lee 39
Transmittance
Fresnel’s Equations
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
OPTOELECTRONICS Prof. Wei-I Lee 40
Antireflection Coatings on Solar Cells
Fresnel’s Equations
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
OPTOELECTRONICS Prof. Wei-I Lee 41
Dielectric Mirrors
Fresnel’s Equations
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)
OPTOELECTRONICS Prof. Wei-I Lee 42
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.
OPTOELECTRONICS Prof. Wei-I Lee 43
Optical Resonator with Non-perfect Reflectors - I
Multiple Interference and Optical Resonators
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)
OPTOELECTRONICS Prof. Wei-I Lee 44
Optical Resonator with Non-perfect Reflectors - II
Multiple Interference and Optical Resonators
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
OPTOELECTRONICS Prof. Wei-I Lee 45
Optical Resonator as Optical Filters
Multiple Interference and Optical Resonators
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.
OPTOELECTRONICS Prof. Wei-I Lee 46
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
OPTOELECTRONICS Prof. Wei-I Lee 47
Optical Tunneling
Goos-Haenchen Shift and 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
OPTOELECTRONICS Prof. Wei-I Lee 48
Beam Splitter Cubes
Goos-Haenchen Shift and Optical Tunneling
OPTOELECTRONICS Prof. Wei-I Lee 49
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 )
OPTOELECTRONICS Prof. Wei-I Lee 50
Spectrum of F(t)
Temporal and Spatial Coherence
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
OPTOELECTRONICS Prof. Wei-I Lee 51
Coherence Length and Spectral Width
Temporal and Spatial Coherence
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
OPTOELECTRONICS Prof. Wei-I Lee 52
White Light / White Noise
Temporal and Spatial Coherence
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 )
OPTOELECTRONICS Prof. Wei-I Lee 53
Coherence Between Waves
Temporal and Spatial Coherence
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
OPTOELECTRONICS Prof. Wei-I Lee 54
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
OPTOELECTRONICS Prof. Wei-I Lee 55
When Slit Width Can Not Be Ignored
Diffraction Principles
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 ?~
OPTOELECTRONICS Prof. Wei-I Lee 56
Diffraction Phenomena
Diffraction Principles
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
OPTOELECTRONICS Prof. Wei-I Lee 57
Fraunhofer Diffraction
Diffraction Principles
zero intensity occurs when
sinq = ml/a , m = 1, 2, 3
center bright region > a
( beam divergence )
OPTOELECTRONICS Prof. Wei-I Lee 58
Fraunhofer Diffraction
Diffraction Principles
• 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
OPTOELECTRONICS Prof. Wei-I Lee 59
Airy Rings
Diffraction Principles
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
OPTOELECTRONICS Prof. Wei-I Lee 60
Resolving Power of Imaging Systems
Diffraction Principles
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 )
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