2. ChapterTwo wave nature light 0
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
1
Plane Electromagnetic Wave
An electromagnetic wave traveling along z has time-varying electric and magnetic
fields that are perpendicular to each other.
Ex = E0 cos(ωt − kz + φ0)
k = 2π / λ
ω = 2πν
φ = ωt − kz + φ0
Ex(z, t) = Re[ E0 exp(jφ0)expj(ωt − kz)]
propagation constant, or wave number
angular frequency
phase
The optical field generally refers to the electric field.
PHY4320 Chapter Two
2
Plane Electromagnetic Wave
A surface over which the phase of a wave is constant is referred to as a
wavefront.
PHY4320 Chapter Two
3
Phase Velocity
φ = ωt − kz + φ0 = constant
During a time interval δt, the wavefronts of contant phases move a
distance δz. The phase velocity v is therefore
δz ω
v = = = νλ
δt k
PHY4320 Chapter Two
4
Plane Electromagnetic Wave
At a given time, the phase
difference between two points
separated by Δr is simply k⋅Δr. If
this phase difference is 0 or
multiples of 2π, then the two points
are in phase.
E(r, t) = E0 cos(ωt − k⋅r + φ0)
k⋅r = kxx + kyy + kzz
k is called the wave vector.
PHY4320 Chapter Two
5
Plane Waves
Ex = E0 cos(ωt − kz + φ0)
The plane wave has no angular separation (no
divergence) in its wavevectors.
E0 does not depend on the distance from a
reference point, and it is the same at all points on
a given plane perpendicular to k.
The plane wave is an idealization that is useful in
analyzing many wave phenomena.
Real light beams would have finite cross sections.
The plane wave obeys Maxwell’s EM wave
equation.
PHY4320 Chapter Two
6
Maxwell’s EM Wave Equation
∂2E
∂2E ∂2E ∂2E
+ 2 + 2 = ε 0ε r μ 0 2
2
∂t
∂z
∂y
∂x
For a perfect plane wave: Ex = E0 cos(ωt − kz + φ0)
Left side:
Right side:
∂2E
2
E0 cos(ωt − kz + φ0 )
=
−
ω
2
∂t
∂ E
=0
2
∂x
2
∂2E
=0
2
∂y
∂2E
2
(
)
cos(ωt − kz + φ0 )
=
−
−
E
k
0
2
∂z
= −k 2 E0 cos(ωt − kz + φ0 )
From EM:
k 2 = ε 0ε r μ 0ω 2
1
v = λν =
PHY4320 Chapter Two
ε 0ε r μ 0
7
Diverging Waves
∂2E ∂2E ∂2E
∂2E
+ 2 + 2 = ε 0ε r μ 0 2
2
∂x
∂y
∂z
∂t
E = (A/r) cos(ωt − kr)
•
•
•
A spherical wave is described by a traveling wave that emerges from a point EM
source and whose amplitude decays with distance r from the source.
Optical divergence refers to the angular separation of wavevectors on a given
wavefront.
Plane and spherical waves represent two extremes of wave propagation behavior
from perfectly parallel to fully diverging wavevectors. They are produced by two
extreme sizes of EM wave sources: an infinitely large source for the plane wave
and a point source for the spherical wave. A real EM source would have a finite
size and finite power.
PHY4320 Chapter Two
8
Gaussian Beams
Many laser beams can be described by Gaussian beams.
Gaussian beam has an exp[j(ωt − kz)] dependence to describe propagation
characteristics but the amplitude varies spatially away from the beam axis.
The beam diameter 2w at any point z is defined as the diameter at which the
beam intensity has fallen to 1/e2 (13.5%) of its maximum.
PHY4320 Chapter Two
9
Gaussian Beams
Diffraction causes light waves to spread transversely as they propagate, and it is
therefore impossible to have a perfectly collimated beam. The spreading of a laser
beam is in precise accord with the predictions of pure diffraction theory. Even if a
Gaussian laser beam wavefront were made perfectly flat at some plane, it would
quickly acquire curvature and begin spreading out. The finite width 2w0 where the
wavefronts are parallel is called the waist of the beam. w0 is the waist radius and 2w0
is the spot size. The increase in beam diameter 2w with z makes an angle of 2θ,
which is called the beam divergence. Spot size and beam divergence are two
important parameters when choosing lasers.
PHY4320 Chapter Two
10
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
11
Refractive Index
When an EM wave is traveling in a dielectric medium, the oscillating electric
field polarizes the molecules of the medium at the frequency of the wave.
The field and the induced molecular dipoles become coupled. The net
effect is that the polarization mechanism delays the propagation of the EM
wave. The stronger the interaction between the field and the dipoles, the
slower the propagation of the wave. The relative permittivity εr measures
the ease with which the medium becomes polarized and indicates the
extent of interaction between the field and the induced dipoles.
The phase velocity in a medium is:
In vacuum:
c=
ε 0 μ0
v=
1
ε 0ε r μ 0
1
The refractive index of the medium is:
c
n = = εr
v
PHY4320 Chapter Two
12
Refractive Index
kmedium = nk
λmedium = λ/n
The frequency ν remains the same in different mediums.
In non-crystalline materials (glasses and liquids), the material structure is
the same in all directions and n does not depend on the direction. The
refractive index is isotropic.
Crystals, in general, have anisotropic properties. The refractive index
seen by a propagating EM wave will depend on the direction of the electric
field relative to crystal structures, which is further determined by both the
propagation direction and the electric field oscillation direction
(polarization).
PHY4320 Chapter Two
13
Relative Permittivity and Refractive Index
The relative permittivity for many materials can be vastly different at high
and low frequencies because different polarization mechanisms operate at
these frequencies. Al low frequencies, all polarization mechanisms (dipolar,
ionic, electronic) contribute to εr, whereas at high (optical) frequencies only
the electronic polarization can respond to the oscillating field.
Low frequency (LF) relative permittivity εr(LF) and refractive index n
Material
εr(LF)
Si
11.9
ε r (LF ) n (optical)
Comment
3.44
3.45 (at 2.15 μm) Electronic polarization up to
optical frequencies
Diamond 5.7
2.39
2.41 (at 590 nm)
GaAs
13.1
3.62
SiO2
3.84 2.00
1.46 (at 600 nm) Ionic polarization contributes to εr(LF)
Water
80
8.9
3.30 (at 5 μm)
Electronic polarization up to UV light
Ionic polarization contributes to εr(LF)
1.33 (at 600 nm) Dipolar polarization contributes to
εr(LF), which is large
PHY4320 Chapter Two
14
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
15
Group Velocity
Since there are no perfect monochromatic waves in practice, we
have to consider the way in which a group of waves differing slightly
in wavelength travel along the same direction. For two harmonic
waves of frequencies ω − δω and ω + δω and wavevectors k − δk and
k + δk interfering with each other and traveling along the z direction:
E x ,1 ( z , t ) = E0 cos[(ω − δω )t − (k − δk )z ]
E x , 2 ( z , t ) = E0 cos[(ω + δω )t − (k + δk )z ]
Using
⎤
⎡1
⎤
⎡1
cos A + cos B = 2 cos ⎢ ( A − B )⎥ cos ⎢ ( A + B )⎥
⎦
⎣2
⎦
⎣2
PHY4320 Chapter Two
16
Group Velocity
E x = E x ,1 + E x , 2 = 2 E0 cos[(δω )t − (δk )z ]cos(ωt − kz )
We obtain:
Generate a wave packet containing an oscillating field at the mean
frequency ω with the amplitude modulated by a slowly varying field of
frequency δω.
PHY4320 Chapter Two
17
Group Velocity
E x = 2 E0 cos[(δω )t − (δk )z ]cos(ωt − kz )
or
The maximum amplitude
moves with a wavevector
δk and a frequency δω.
Phase velocity v =
dω
Group velocity v g =
dk
The group velocity defines the speed with
which energy or information is propagated
since it defines the speed of the envelope of
the amplitude variation.
PHY4320 Chapter Two
ω
c
=
k n
In vacuum:
ω = ck
dω
vg =
dk
=c
group velocity
phase velocity
18
In a medium with n as a function of λ:
ω = ck / n
k = nω / c
cdk d (nω )
dn
=
= n +ω
dω
dω
dω
c /ν 2πc
=
λ=
n
ωn
dn
dn dλ
dn 2πc
dn
ω
=ω
= −ω
= −λ
2
dω
dλ dω
dλ ω n
dλ
dω
c
c
vg =
=
=
dk n − λ dn N g
dλ
dn
Ng = n − λ
dλ
Ng is the group index of the medium.
PHY4320 Chapter Two
19
Group Index of the Medium
Pure SiO2
For silica, Ng is minimum around 1300
nm, which means that light waves
with wavelengths close to 1300 nm
travel with the same group velocity
and do not experience dispersion.
Both n and Ng are functions of the free space wavelength. Such a medium is
called a dispersive medium. n and Ng of SiO2 are important parameters for
optical fiber design in optical communications.
PHY4320 Chapter Two
20
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
21
Snell’s Law
Willebrord van Roijen Snell
(1580-1626)
Incident light waves
Ai and Bi are in
phase. Reflected
waves Ar and Br and
refracted waves At
and Bt should also be
in phase, respectively.
PHY4320 Chapter Two
22
Reflection
v1t
v1t
AB' =
=
sin θ i sin θ r
θi = θ r
Refraction
v1t
v2 t
AB' =
=
sin θ i sin θ t
sin θ i v1 n2
= =
sin θ t v2 n1
PHY4320 Chapter Two
23
sin θ i n2
=
sin θ t n1
Refraction
Total Internal Reflection
When n1 > n2 and θt = 90°, the incidence angle is
called the critical angle θc:
n2
sin θ c =
n1
PHY4320 Chapter Two
24
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
25
Fresnel’s Equations
The incidence plane is
the plane containing the
incident and reflected
light rays.
Transverse electric field (TE) waves: Ei,⊥, Er,⊥, and Et,⊥.
Transverse magnetic field (TM) waves: Ei,//, Er,//, and Et,//.
Ei = Ei 0 exp j (ωt − k i • r )
Er = Er 0 exp j (ωt − k r • r )
Et = Et 0 exp j (ωt − k t • r )
Augustin Fresnel
(1788-1827)
Our goal is to find Er0 and Et0 relative
to Ei0, including any phase changes.
PHY4320 Chapter Two
26
θi = θ r
⎫
⎬
n1 sin θ i = n2 sin θ t ⎭
Snell’s law
B ⊥ = (n / c) E// ⎫
⎬ Maxwell’s equations
B// = (n / c) E⊥ ⎭
Etangential (1) = Etangential (2)⎫⎪
⎬ Boundary conditions
Btangential (1) = Btangential (2) ⎪⎭
PHY4320 Chapter Two
27
Incidence plane
− Ei ,⊥ − Er ,⊥ = − Et ,⊥
− Bi ,⊥ + Br ,⊥ = − Bt ,⊥ ⇒ −(n1 / c )Ei , // + (n1 / c )Er , // = −(n2 / c )Et , //
− Ei , // cos θ i − Er , // cos θ r = − Et , // cos θ t
Bi , // cos θ i − Br , // cos θ r = Bt , // cos θ t ⇒
(n1 / c )Ei ,⊥ cos θ i − (n1 / c )Er ,⊥ cos θ r = (n2 / c )Et ,⊥ cos θ t
PHY4320 Chapter Two
28
de − bf
⎧
x=
⎪
⎧ax + by = e
⎪
ad − bc
⇒
⎨
⎨
⎩cx + dy = f
⎪ y = af − ce
⎪⎩
ad − bc
⎧− Ei ,⊥ − Er ,⊥ = − Et ,⊥
⎨
⎩n1 Ei ,⊥ cos θ i − n1 Er ,⊥ cos θ r = n2 Et ,⊥ cos θ t
⎧ Er ,⊥ − Et ,⊥ = − Ei ,⊥
⇒⎨
⎩n1 Er ,⊥ cos θ r + n2 Et ,⊥ cos θ t = n1 Ei ,⊥ cos θ i
− n2 cos θ t + n1 cos θ i
⎧
⎪ Er ,⊥ = n cos θ + n cos θ Ei ,⊥
⎪
2
1
t
r
⇒⎨
⎪ E = n1 cos θ i + n1 cos θ r E
⎪⎩ t ,⊥ n2 cos θ t + n1 cos θ r i ,⊥
PHY4320 Chapter Two
29
de − bf
⎧
x=
⎪
⎧ax + by = e
⎪
ad − bc
⇒
⎨
⎨
⎩cx + dy = f
⎪ y = af − ce
⎪⎩
ad − bc
⎧− n1 Ei , // + n1 Er , // = −n2 Et , //
⎨
⎩− Ei , // cos θ i − Er , // cos θ r = − Et , // cos θ t
⎧n1 Er , // + n2 Et , // = n1 Ei , //
⇒⎨
⎩ Er , // cos θ r − Et , // cos θ t = − Ei , // cos θ i
− n1 cos θ t + n2 cos θ i
⎧
⎪ Er , // = − n cos θ − n cos θ Ei , //
⎪
1
2
t
r
⇒⎨
⎪ E = − n1 cos θ i − n1 cos θ r E
⎪⎩ t , // − n1 cos θ t − n2 cos θ r i , //
PHY4320 Chapter Two
30
⎧n = n2 / n1
⎪
⇒ sin θ i = n sin θ t
⎨θ i = θ r
⎪n sin θ =n sin θ
2
i
t
⎩ 1
[
⇒ n cos θ t = n − n sin θ t
2
2
2
]
1/ 2
[
= n − sin θ i
2
[
[
2
]
1/ 2
]
]
1/ 2
2
2
⎧
− n2 cos θ t + n1 cos θ i
cos θ i − n − sin θ i
Ei ,⊥ =
Ei ,⊥
⎪ Er , ⊥ =
1/ 2
2
2
n2 cos θ t + n1 cos θ r
cos θ i + n − sin θ i
⎪
⎨
2 cos θ i
⎪ E = n1 cos θ i + n1 cos θ r E =
Ei ,⊥
⊥
,
⊥
i
,
t
1
/
2
2
2
⎪
n
n
+
cos
cos
θ
θ
cos
n
sin
+
−
θ
θi
2
1
t
r
i
⎩
[
PHY4320 Chapter Two
]
31
⎧n = n2 / n1
⎪
⇒ sin θ i = n sin θ t
⎨θ i = θ r
⎪n sin θ =n sin θ
i
t
2
⎩ 1
[
⇒ n cos θ t = n − n sin θ t
2
2
2
]
1/ 2
[
[
[
= n − sin θ i
2
2
]
]
]
1/ 2
1/ 2
2
2
⎧
n − sin θ i − n 2 cos θ i
− n1 cos θ t + n2 cos θ i
Ei , // =
Ei , //
⎪ Er , // =
1/ 2
2
2
2
− n1 cos θ t − n2 cos θ r
n − sin θ i + n cos θ i
⎪
⎨
2n cos θ i
⎪ E = − n1 cos θ i − n1 cos θ r E =
E
⎪ t , // − n cos θ − n cos θ i , // n 2 − sin 2 θ 1/ 2 + n 2 cos θ i , //
t
r
1
2
i
i
⎩
[
PHY4320 Chapter Two
]
32
Ei = Ei 0 exp j (ωt − k i • r )
Er = Er 0 exp j (ωt − k r • r )
r⊥ and r// are reflection coefficients. t⊥
and t// are transmission coefficients.
Et = Et 0 exp j (ωt − k t • r )
r⊥ =
t⊥ =
r// =
t // =
Er 0 , ⊥
Ei 0,⊥
Et 0,⊥
Ei 0,⊥
Er 0, //
Ei 0, //
Et 0, //
Ei 0, //
[
=
cos θ + [n
]
θ]
cos θ i − n − sin θ i
=
i
[n
r// + nt // = 1
r⊥ + 1 = t ⊥
2
− sin
2 cos θ i
[
2
2
2
2
2
]
θ]
− sin θ i
− sin
2
2
1/ 2
1/ 2
i
− sin θ i
2
]
1/ 2
t ⊥ = t ⊥ exp(− jφt ,⊥ )
t // = t // exp(− jφt , // )
]
1/ 2
− n cos θ i
2
+ n 2 cos θ i
2n cos θ i
i
r// = r// exp(− jφr , // )
1/ 2
1/ 2
cos θ i + n 2 − sin 2 θ i
[
n
=
[n
=
2
r⊥ = r⊥ exp(− jφr ,⊥ )
+ n 2 cos θ i
Complex coefficients can only be
obtained when n2 − sin2θi < 0.
n1 > n2 ,θi > θc
Phase changes other than 0 or 180°
occur only when there is TIR.
PHY4320 Chapter Two
33
r⊥
[
]
=
cos θ + [n − sin θ ]
[
n − sin θ ] − n cos θ
=
[n − sin θ ] + n cosθ
cos θ i − n − sin θ i
2
2
1/ 2
2
2
1/ 2
i
1/ 2
2
2
i
2
1/ 2
2
]
i
2
i
[n
1 − n n1 − n2
r⊥ =
=
1 + n n1 + n2
n − n 2 n1 − n2
=
r// =
2
n+n
n1 + n2
i
2
r//
When θi = 0 (normal incidence):
i
When r// = 0 (reflected light is polarized):
− sin θ p = n 2 cos θ p ⇒ n 2 − sin 2 θ p = n 4 cos 2 θ p
2
2
sin
θp
n
4
2
2
2
4
n
n
n
1
tan
tan
⇒
−
=
⇒
+
−
=
θ
θ
p
p
cos 2 θ p cos 2 θ p
n2
θp is called the polarization angle or
⇒ tan θ p =
n1
Brewster’s angle.
2
2
1/ 2
n2
sin θ c =
n1
(
)
PHY4320 Chapter Two
34
Brewster’s Angle (Polarization Angle)
n1 = 1.44, n2 = 1.00
David Brewster
(1781-1868)
θ p = 34.78°
θ c = 43.98°
When θi = θp, reflected light is linearly polarized and is perpendicular to the
incidence plane.
When θp < θi < θc, there is a phase shift of 180° in r//.
|r⊥| and |r//| increases with θi when θi > θp.
There are phase shifts (other than 0 or 180°) in r⊥ and r// when θi > θc.
PHY4320 Chapter Two
35
Example: calculate r⊥ and r// when n1 = 1.44, n2 = 1.00, and θi = 40°.
Solution:
n2 1.00
=
= 0.694
n=
n1 1.44
[
]
r =
cos θ + [n − sin θ ]
cos 40° − [0.694 − sin 40°]
=
cos 40° + [0.694 − sin 40°]
[
n − sin θ ] − n cos θ
r =
[n − sin θ ] + n cosθ
[
0.694 − sin 40°] − 0.694
=
[0.694 − sin 40°] + 0.694
⊥
cos θ i − n − sin θ i
2
2
1/ 2
2
2
1/ 2
i
2
i
2
2
1/ 2
2
2
1/ 2
1/ 2
2
2
i
//
2
i
1/ 2
2
= 0.490
2
i
i
2
2
1/ 2
2
2
2
1/ 2
2
cos 40°
cos 40°
PHY4320 Chapter Two
= −0.170
36
Example: calculate r⊥ and r// when n1 = 1.00, n2 = 1.44, and θi = 40°.
Solution:
n2 1.44
n=
=
= 1.44
n1 1.00
[
]
r =
cos θ + [n − sin θ ]
cos 40° − [1.44 − sin 40°]
=
= −0.255
cos 40° + [1.44 − sin 40°]
[
n − sin θ ] − n cos θ
r =
[n − sin θ ] + n cosθ
[
1.44 − sin 40°] − 1.44 cos 40°
=
= −0.104
[1.44 − sin 40°] + 1.44 cos 40°
⊥
cos θ i − n − sin θ i
2
2
1/ 2
2
2
1/ 2
i
2
i
2
2
2
1/ 2
2
2
1/ 2
1/ 2
2
i
//
2
i
1/ 2
2
2
i
i
2
2
1/ 2
2
2
2
1/ 2
2
PHY4320 Chapter Two
37
n2
tan θ p =
n1
n1 = 1.00, n2 = 1.44
θ p = 55.22°
PHY4320 Chapter Two
38
What are the phase changes at TIR (n1 > n2 and θi > θc)?
r⊥
[
]
=
cos θ + [n − sin θ ]
[
n − sin θ ] − n cos θ
=
[n − sin θ ] + n cosθ
cos θ i − n − sin θ i
2
2
1/ 2
2
2
1/ 2
i
r//
1/ 2
2
2
1/ 2
2
r⊥
2
2 1/ 2
2
2 1/ 2
i
2
2 1/ 2
2
i
2
i
2 1/ 2
2
i
z = x +y
i
PHY4320 Chapter Two
2
)
2 1/ 2
⎛ y⎞
φ = tan ⎜ ⎟
⎝x⎠
i
]
θ −n ]
− n cos θ + j [sin θ − n ]
=
n cos θ + j [sin θ − n ]
cos θ i − j sin θ i − n
i
r//
i
2
i
[
=
cos θ + j [sin
(
r// = 1
2
i
z = z exp(− jφ )
r⊥ = 1
i
2
z = x − jy
−1
z1 exp(− jφ1 )
z1
z= =
z 2 z 2 exp(− jφ2 )
=
z1
z2
exp[− j (φ1 − φ2 )]
39
r⊥
[
=
cos θ + j [sin
2
i
[
]
θ −n ]
cos θ i − j sin 2 θ i − n
2 1/ 2
i
[
]
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟
φ1 = tan
⎟
⎜
θ
cos
i
⎠
⎝
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟
φ2 = − tan
⎟
⎜
cos
θ
i
⎠
⎝
2 1/ 2
[
]
]
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟
φ⊥ = φ1 − φ2 = 2 tan
⎟
⎜
θ
cos
i
⎠
⎝
[
⎛ 1 ⎞ sin θ i − n
tan⎜ φ⊥ ⎟ =
cos θ i
⎝2 ⎠
2
]
2 1/ 2
PHY4320 Chapter Two
40
r// =
[
− n 2 cos θ i + j sin 2 θ i − n
[
n cos θ i + j sin θ i − n
2
2
]
[
]
2 1/ 2
2 1/ 2
[
]
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟ −π
φ1 = tan
2
⎜ n cos θ i ⎟
⎝
⎠
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟
φ2 = − tan
⎜ n 2 cos θ i ⎟
⎝
⎠
]
[
]
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟ −π
φ// = φ1 − φ2 = 2 tan
⎜ n 2 cos θ i ⎟
⎝
⎠
[
]
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟
φ// + π = 2 tan
⎜ n 2 cos θ i ⎟
⎝
⎠
[
1 ⎞ sin θ i − n
⎛1
tan⎜ φ// + π ⎟ =
2
n
2
2
cos θ i
⎝
⎠
2
PHY4320 Chapter Two
]
2 1/ 2
41
Consider now:
ktz = kt sin θ t =
kty = kt cos θ t =
Et = Et 0 exp j (ωt − k t ⋅ r )
= Et 0 exp j (ωt − k ty y − ktz z )
2π
λ
2π
λ
n2 sin θ t =
[
2π
λ
n2 1 − sin θ t
2
]
n1 sin θ i = ki sin θ i = kiz
1/ 2
⎤
⎡ ⎛n ⎞
2
1
n2 ⎢1 − ⎜⎜ ⎟⎟ sin θ i ⎥
=
λ ⎢ ⎝ n2 ⎠
⎥⎦
⎣
2π
PHY4320 Chapter Two
2
1/ 2
42
Evanescent Wave
⎛ n1 ⎞
⎛ n1 ⎞
2
θ i > θ c ⇒ ⎜⎜ ⎟⎟ sin θ i > ⎜⎜ ⎟⎟ sin 2 θ c = 1
⎝ n2 ⎠
⎝ n2 ⎠
2
2
At TIR:
⎡⎛ n ⎞
⎤
2
1
kty = − j
n2 ⎢⎜⎜ ⎟⎟ sin θ i − 1⎥
λ ⎢⎝ n2 ⎠
⎥⎦
⎣
2π
2
1/ 2
= − jα 2
PHY4320 Chapter Two
Why do we put a
minus here?
43
Evanescent Wave
⎤
2πn2 ⎡⎛ n1 ⎞
2
⎢⎜⎜ ⎟⎟ sin θ i − 1⎥
α2 =
λ ⎢⎝ n2 ⎠
⎥⎦
⎣
2
1/ 2
Et ( y, z , t ) = Et 0 exp j (ωt − k ty y − ktz z )
= Et 0 exp j (− kty y )exp j (ωt − kiz z )
= Et 0 exp(− α 2 y ) exp j (ωt − kiz z )
λ 2 2
2 −1 / 2
[
=
n1 sin θ i − n2 ]
δ=
α 2 2π
1
λ = 514.5 nm (green light)
n1 = 1.44 (glass)
n2 = 1.00 (air)
θi = 50° (>θc = 43.98°)
δ=
[
the penetration depth
( )
514.5
1.44 2 sin 2 50o − 1.00 2
2π
PHY4320 Chapter Two
]
−1 / 2
= 176nm
44
Total Internal Reflection Fluorescence (TIRF) Microscopy
TIR illumination can greatly improve signal-to-noise ratios in
applications requiring imaging of minute structures or single
molecules in specimens having large numbers of fluorophores
located outside of the optical plane of interest, such as
molecules in solution in Brownian motion, vesicles undergoing
endocytosis or exocytosis, or single protein trafficking in cells.
Carl Zeiss
PHY4320 Chapter Two
45
PHY4320 Chapter Two
46
θi
48.7°
49.7°
51.1°
53.9°
57.8°
66.5°
I z = I 0e
dp
300 nm
145 nm
100 nm
70 nm
55 nm
40 nm
−z/dp
λ 2 2
2 −1 / 2
[
dp =
n1 sin θ i − n2 ]
4π
PHY4320 Chapter Two
47
Fluorescence intensity
Distance
Evanescent nanometry
Achieves a resolution of ~ 1 nm.
Requires highly photostable fluorescent materials.
PHY4320 Chapter Two
48
Intensity, Reflectance, and Transmittance
1
I = vε r ε o E02 ∝ nE02
2
Light intensity
Reflectance
R⊥ =
Er 0 , ⊥
2
Ei 0,⊥
2
Transmittance
T⊥ =
n2 Et 0,⊥
2
n1 Ei 0,⊥
2
= r⊥
R// =
2
⎛ n2 ⎞
= ⎜⎜ ⎟⎟ t ⊥
⎝ n1 ⎠
2
T// =
PHY4320 Chapter Two
Er 0, //
Ei 0, //
2
2
= r//
n2 Et 0, //
n1 Ei 0, //
2
2
2
⎛ n2 ⎞
= ⎜⎜ ⎟⎟ t //
⎝ n1 ⎠
49
2
r⊥ =
r// =
t⊥ =
t // =
Er 0 , ⊥
Ei 0,⊥
Er 0, //
Ei 0, //
Et 0,⊥
Ei 0,⊥
Et 0, //
Ei 0, //
cos θ − [n − sin θ ]
=
cos θ + [n − sin θ ]
[
n − sin θ ] − n cos θ
=
[n − sin θ ] + n cosθ
2
i
2
1/ 2
2
i
i
2
1/ 2
2
2
=
=
1/ 2
2
[n
i
2
− sin
i
− sin
2
2
2n cos θ i
2
θ]
1/ 2
i
i
2
2 cos θ i
cos θ + [n
n − n 2 n1 − n2
=
=
2
n+n
n1 + n2
2
i
θ]
i
=
1/ 2
i
⎛ n1 − n2 ⎞
⎟⎟
R = R⊥ = R// = ⎜⎜
⎝ n1 + n2 ⎠
2
2n1
=
1 + n n1 + n2
=
+ n 2 cos θ i
2
1 − n n1 −n 2
=
1 + n n1 + n 2
=
1/ 2
2
i
When θi = 0 (normal incidence):
2n
2n1
=
n + n 2 n1 + n2
4n1n2
T = T⊥ = T// =
2
(n1 + n2 )
PHY4320 Chapter Two
50
Example: A ray of light traveling in a glass medium of refractive index n1 =
1.450 becomes incident on a less dense glass medium of refractive index n2 =
1.430. Suppose the free space wavelength (λ) of the light ray is 1 μm. (a)
What is the critical angle for TIR? (b) What is the phase change in the
reflected wave when θi = 85° and 90°? (c) What is the penetration depth of the
evanescent wave into medium 2 when θi = 85° and 90°?
sin θ c = n2 / n1 = 1.430 / 1.450 ⇒ θ c = 80.47°
Solution:
2
⎡ 2
⎛ 1.430 ⎞ ⎤
⎟ ⎥
⎢sin 85 ° − ⎜
θi = 90°
2
2 1/ 2
1
.
450
⎝
⎠
⎢
⎛ 1 ⎞ sin θ i − n
⎦⎥ = 1.614 ⇒ φ = 116 .45 °
tan ⎜ φ ⊥ ⎟ =
=⎣
⊥
cos θ i
cos 85 °
⎝2 ⎠
2 1/ 2
φ⊥ = 180°
⎡ 2
⎛ 1.430 ⎞ ⎤
⎟ ⎥
⎢sin 85 ° − ⎜
2
2 1/ 2
1
.
450
⎠ ⎥⎦
⎝
1 ⎞ sin θ i − n
⎢⎣
⎛1
=
tan ⎜ φ // + π ⎟ =
= 1.660 ⇒ φ // = −62 .13 °
2
2
n cos θ i
2 ⎠
⎝2
⎛ 1.430 ⎞
⎜
⎟ cos 85 °
φ// = 0°
⎝ 1.450 ⎠
[
]
[
δ=
1/ 2
]
−1 / 2
−1 / 2
1μm
λ 2 2
[
[
n1 sin θ i − n22 ] =
1.450 2 sin 2 85° − 1.430 2 ] = 0.780 μm
2π
2π
PHY4320 Chapter Two
δ = 0.663μm
51
Example: Consider a light ray at normal incidence on a boundary between a
glass medium of refractive index 1.5 and air of refractive index 1.0. (a) If light
is traveling from air to glass, what are the reflectance and transmittance? (b) If
light is traveling from glass to air, what are the reflectance and transmittance?
(c) What is the Brewster’s angle when light is traveling from air to glass?
2
2
⎧
⎛ n1 − n2 ⎞ ⎛ 1.0 − 1.5 ⎞
⎟⎟ = ⎜
⎪ R = ⎜⎜
⎟ = 0.04
⎧n1 = 1.0
n1 + n2 ⎠ ⎝ 1.0 + 1.5 ⎠
⎪
⎝
⇒⎨
⎨
4n1n2
4 ×1.0 ×1.5
⎩n2 = 1.5 ⎪
⎪T = (n + n )2 = (1.0 + 1.5)2 = 0.96
1
2
⎩
Solution:
2
2
⎧
⎛ n1 − n2 ⎞ ⎛ 1.5 − 1.0 ⎞
⎟⎟ = ⎜
⎪ R = ⎜⎜
⎟ = 0.04
1
.
5
n
=
⎧ 1
⎪
⎝ n1 + n2 ⎠ ⎝ 1.5 + 1.0 ⎠
⇒
⎨
⎨
4n1n2
4 ×1.5 ×1.0
⎩n2 = 1.0 ⎪
⎪T = (n + n )2 = (1.5 + 1.0 )2 = 0.96
1
2
⎩
n 1.5
tan θ p = 2 =
⇒ θ p = 56.31°
n1 1.0
PHY4320 Chapter Two
Light will lose 4% of
its intensity whenever
it travels through an
interface between air
and glass.
52
Antireflection Coatings
Current commercial solar cells are made of silicon (n ≈ 3.5 at wavelengths of
700–800 nm).
2
2
(
(
n1 − n2 )
1.0 − 3.5)
R=
=
2
(n1 + n2 ) (1.0 + 3.5)2
Δ cφ = kc 2d =
2πn2
⎛ λ ⎞
⎟⎟
⇒ d = m⎜⎜
⎝ 4n2 ⎠
λ
= 0.31
Air Coating (Si3N4) Si (solar cell)
1.0
1.9
3.5
(2d ) = mπ
r12 = r23 in order to obtain efficient
destructive interference.
n1 − n2 n2 − n3
=
⇒ n2 = n1n3
n1 + n2 n2 + n3
r12,⊥ = r12,// = −0.31
r23,⊥ = r23,// = −0.30
n2 = n1n3 = 1.0 × 3.5 = 1.87
d=
λ
4n2
=
700nm
= 92nm
4 ×1.9
PHY4320 Chapter Two
53
Dielectric Mirrors
n1 < n2
We can find that A, B, and C
n1 − n2
2n1
< 0 t12 =
r12 =
> 0 waves are in phase with each
n1 + n2
n1 + n2
other. They interfere with each
n2 − n1
2n2
other constructively.
> 0 t 21 =
r21 =
>0
The total reflectance will be
n2 + n1
n2 + n1
close to unity.
PHY4320 Chapter Two
54
Photonic Crystals
1-D
2-D
periodic in
one direction
periodic in
two directions
3-D
periodic in
three directions
Photonic crystals are ordered structures in which two media with different
dielectric constants or refractive indices are arranged in a periodic form, with
lattice constants comparable to the wavelength of light. The light propagation is
completely forbidden within photonic crystals if they are properly designed
(refractive index contrast, crystal structure, and lattice constant).
PHY4320 Chapter Two
55
Defects in Photonic Crystals
cavity
waveguide
PHY4320 Chapter Two
56
Low-Pumping-Current Lasers
SEM image
H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju,
J.-K. Yang, J.-H. Baek, S.-B. Kim, Y.-H. Lee,
Science 2004, 305, 1444.
PHY4320 Chapter Two
57
Low-Pumping-Current Lasers
radii of air holes:
0.28a (I)
0.35a (II)
0.385a (III)
0.4a (IV)
0.41a (V)
Designed according
to FDTD (finite
difference time
domain) calculations.
electric field intensity profile
PHY4320 Chapter Two
58
Low-Pumping-Current Lasers
These low threshold current, highly efficient lasers might function as
single photon sources, which are useful for cavity quantum
electrodynamics and quantum information processing.
PHY4320 Chapter Two
59
Wavelength Channel Drop Devices
B.-S. Song, S. Noda, T. Asano, Science 2003, 300, 1537.
PHY4320 Chapter Two
60
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
61
Goos-Hänchen Shift
Careful experiments show that the reflected wave at
TIR appears to be laterally shifted from the incidence
point at the interface. This lateral shift is known as the
Goos-Hänchen shift.
The reflected wave appears to be reflected from a
virtual plane inside the optically less dense medium.
The spacing between the interface and the virtual
plane is approximately the penetration depth of the
evanescent wave δ when n1 is very close to n2.
Δz = 2δ tan θ i
PHY4320 Chapter Two
λ = 1 μm
n1 = 1.45
n2 = 1.43
θi = 85°
δ = 0.78 μm
Δz ≈ 18 μm
62
Optical Tunneling
At TIR, we shrink the thickness d of the medium B. When B is sufficiently
thin, an attenuated beam emerges on the other side of B in C. This
phenomenon, which is forbidden by simple geometrical optics, is called
optical tunneling or frustrated total internal reflection. Optical tunneling
is due to the evanescent wave, the field of which penetrates into medium B
and reaches the interface BC.
PHY4320 Chapter Two
63
Beam Splitters Based on Frustrated TIR
The extent of light intensity division between
the two beams depends on the thickness of the
thin layer B and its refractive index.
PHY4320 Chapter Two
64
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
65
Diffraction
Diffraction occurs when light waves encounter small objects or apertures.
For example, the beam passed through a circular aperture is divergent and
exhibits an intensity pattern (called diffraction pattern) that has bright and
dark rings (called Airy rings).
Two types of diffraction: (1) In Fraunhofer diffraction, the incident light is a
plane wave. The observation or detection screen is far away from the
aperture so that the received waves also look like plane waves. (2) In Fresnel
diffraction, the incident and received light waves have significant wavefront
curvatures. Fraunhofer diffraction is by far the most important.
PHY4320 Chapter Two
66
Diffraction
Diffraction can be understood using the Huygens-Fresnel principle. Every
unobstructed point of a wavefront, at a given instant in time, serves as a source of
spherical secondary waves (with the same frequency as that of the primary wave).
The amplitude of the optical field at any point beyond is the superposition of all these
wavelets (considering their amplitudes and relative phases).
PHY4320 Chapter Two
67
Diffraction Pattern through a 1D Slit
The one-dimensional (1D) slit has a width of a. Divide the width a into N (a large
number) coherent sources δy (= a/N). Because the slit is uniformly illuminated, the
amplitude of each source will be proportional to δy. In the forward direction (θ = 0°),
they will be in phase and constitute a forward wave along the z direction. They can
also be in phase at some other angles and therefore give rise to a diffracted wave
along such directions.
The intensity of the received wave at a
point on the screen will be the sum of all
waves arriving from all the sources in the
slit. The wave Y is out of phase with
respect to A by kysinθ.
δE ∝ (δy ) exp(− jky sin θ )
E (θ ) = C ∫ δy exp(− jky sin θ )
a
0
PHY4320 Chapter Two
68
Let x = y − a/2, we obtain
E (θ ) = C ∫ δy exp(− jky sin θ )
a
= C∫
0
a/2
exp[− jk ( x + a / 2 )sin θ ]dx
−a / 2
1
− j ka sin θ
2
= Ce
1
− jkx sin θ
dx =
∫−a / 2e
− jk sin θ
a/2
∫
a/2
e − jkx sin θ dx
−a / 2
a
jk sin θ ⎞
⎛ − jk a2 sin θ
2
⎜e
⎟
−
e
⎜
⎟
⎝
⎠
⎛1
⎛1
⎞
⎞
− 2 j sin ⎜ ka sin θ ⎟ a sin ⎜ ka sin θ ⎟
2
2
⎝
⎝
⎠
⎠
=
=
1
− jk sin θ
ka sin θ
2
PHY4320 Chapter Two
69
E (θ ) =
Ce
1
− j ka sin θ
2
I (θ ) = E (θ )
⎛1
⎞
a sin ⎜ ka sin θ ⎟
⎝2
⎠
1
ka sin θ
2
2
⎡
⎛1
⎞⎤
sin
sin
θ
Ca
ka
⎜
⎟⎥
⎢
⎝2
⎠⎥
=⎢
1
⎢
⎥
ka sin θ
⎢⎣
⎥⎦
2
2
The zero intensity occurs when:
1
π
ka sin θ = a sin θ = mπ
λ
2
m = ±1,±2,L
mλ
sin θ =
λ = 1.3 μm
a
a = 100 μm
divergence 2θ = 1.5°
PHY4320 Chapter Two
Bright and dark regions
The central bright region is
larger than the slit width (the
transmitted beam is
diverging)
70
The diffraction patterns from two-dimensional (2D) apertures such as
rectangular and circular apertures are more complicated to calculate but they
use the same principle based on the interference of waves emitted from all
point sources in the aperture.
The diffraction pattern from a circular aperture of diameter D has a central
bright region. The divergence of this bright region is determined by
sin θ = 1.22
λ
D
Diffraction from rectangular apertures
Why is the central bright region rotated relative to the rectangular aperture?
PHY4320 Chapter Two
71
Diffraction-Limited Resolution
Consider two neighboring coherent sources with an angular separation of Δθ
examined through an imaging system with an aperture of diameter D. The aperture
produces a diffraction pattern of the sources. As the sources get closer, their
diffraction patterns overlap more.
According to Rayleigh criterion, the two spots are just resolvable when the principle
maximum of one diffraction pattern coincides with the minimum of the other, which is
given by the condition:
sin (Δθ min ) = 1.22
λ
D
PHY4320 Chapter Two
72
Diffraction-Limited Resolution
Objectives are key components in an optical microscope.
PHY4320 Chapter Two
73
Diffraction-Limited Resolution
NA = n ⋅ sin (α )
The numerical aperture (NA) of a microscope objective is a measure
of its ability to gather light and resolve fine specimen detail at a fixed
object distance.
PHY4320 Chapter Two
74
Diffraction-Limited Resolution
0.61λ
resolution r =
NA
The resolution of an optical microscope is defined as the shortest distance between two
points on a specimen that can still be distinguished by the observer or camera system as
separate entities. It is the most important feature of the optical system and influences the
ability to distinguish between fine details of a particular specimen.
PHY4320 Chapter Two
75
Overcome Diffraction-Limited Resolution
If a sufficient number of photons are collected from a single fluorescent
molecule, the center of this molecule can be determined up to 1.5-nm
resolution by curve-fitting the fluorescence image to a Gaussian function.
A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E.
Goldman, P. R. Selvin, Science 2003, 300, 2061.
PHY4320 Chapter Two
76
Overcome Diffraction-Limited Resolution
Discernable 30-nm and 7-nm steps are readily observed upon moving the
cover slip, either at a constant rate or a Poisson-distributed rate.
PHY4320 Chapter Two
77
Overcome Diffraction-Limited Resolution
Myosin (肌球蛋白，肌凝蛋白) V is a cargo-carrying motor that takes 37-nm center of
mass steps along actin (肌动蛋白) filaments. It has two heads held together by a
coiled-coil stalk. Hand-over-hand and inchworm models have been proposed for its
movement along actin filaments. For a single fluorescent molecule attached to the
light chain domain of myosin V, the inchworm model predicts a uniform step size of 37
nm, whereas the hand-over-hand model predicts alternating steps of 37 – 2x, 37 + 2x,
where x is the in-plane distance of the dye from the midpoint of the myosin.
PHY4320 Chapter Two
78
Overcome Diffraction-Limited Resolution
The dye is 7 nm from the center of mass along the direction of motion.
This result indicates that myosin V moves in a hand-over-hand fashion along
actin filaments.
PHY4320 Chapter Two
79
Diffraction Grating
A diffraction grating in its simplest form has a periodic series of slits. An
incident beam of light is diffracted in certain well-defined directions that
depend on the wavelength and the grating periodicities.
Zero diffraction intensity
condition for a single slit:
Bragg diffraction condition
(maximum intensity):
sin θ = m
λ
d
m = ±1,±2, L
sin θ =
PHY4320 Chapter Two
mλ
a
m = ±1,±2,L
80
Let’s try to derive the actual diffraction intensity from the grating by assuming
a plane incident wave and that there are N slits.
For the diffraction from a single slit:
Eslit (θ ) =
=
Ce
1
− j ka sin θ
2
⎛1
⎞
a sin ⎜ ka sin θ ⎟
⎝2
⎠
1
ka sin θ
2
C1e
− jα
α
sin α
⎞
⎛ C1 = Ca
⎟
⎜
1
⎜ α = ka sin θ ⎟
2
⎠
⎝
Egrating (θ ) = Eslit (θ ) + Eslit (θ )e − jkd sin θ + Eslit (θ )e − j 2 kd sin θ + L + Eslit (θ )e − j ( N −1)kd sin θ
For the diffraction from the grating:
[
= Eslit (θ ) 1 + e − jkd sin θ + e − j 2 kd sin θ + L + e − j ( N −1)kd sin θ
]
A geometrical progression
PHY4320 Chapter Two
81
1 − e − jNkd sin θ
Egrating (θ ) = Eslit (θ )
1 − e − jkd sin θ
⎛ sin α ⎞ 1 − e
I grating (θ ) = Egrating (θ ) = C12 ⎜
⎟
− jkd sin θ
α
1
e
−
⎠
⎝
2
1− e
1 − e − jkd sin θ
− jNkd sin θ 2
=
− jNkd sin θ 2
2
1 − cos( Nkd sin θ ) + j sin ( Nkd sin θ )
1 − cos(kd sin θ ) + j sin (kd sin θ )
2
2
=
2 − 2 cos( Nkd sin θ )
2 − 2 cos(kd sin θ )
⎞
⎛1
2 sin 2 ⎜ Nkd sin θ ⎟
2
⎛ sin Nβ ⎞
1 − cos( Nkd sin θ )
2
⎠
⎝
⎟⎟
=
= ⎜⎜
=
1 − cos(kd sin θ )
sin β ⎠
⎞
⎛1
2 sin 2 ⎜ kd sin θ ⎟ ⎝
⎠
⎝2
1
⎞
⎛
⎜ β = kd sin θ ⎟
2
2
⎠
⎝
2
⎛ sin α ⎞ ⎛ sin Nβ ⎞
⎟⎟
I grating (θ ) = C12 ⎜
⎟ ⎜⎜
⎝ α ⎠ ⎝ sin β ⎠
PHY4320 Chapter Two
82
When
⎧sin Nβ = 0
⎨sin β = 0
⎩
⎛ sin Nβ ⎞
⎜⎜
⎟⎟ = N 2
⎝ sin β ⎠
1
⎛
⎞
⎜ β = kd sin θ ⎟
2
⎝
⎠
2
β = mπ
sin θ = m
When
λ
d
⎧sin Nβ = 0
⎨sin β ≠ 0
⎩
k ⎞λ
⎛
sin θ = ⎜ m + ⎟
N⎠d
⎝
is maximum.
m = ±1,±2, L
⎛ sin Nβ ⎞
⎜⎜
⎟⎟ = 0
⎝ sin β ⎠
⎧m = ±1,±2, L
⎨k = 1,2, L, N − 1
⎩
2
N−1 zero points, N−2 maxima
PHY4320 Chapter Two
83
⎛ sin α ⎞
I grating (θ ) = C12 ⎜
⎟
⎝ α ⎠
2
⎛ sin Nβ
⎜⎜
⎝ sin β
1
⎧
⎪α = 2 ka sin θ
⎨
1
⎪β = kd sin θ
2
⎩
⎞
⎟⎟
⎠
The amplitude of the diffracted
beam is modulated by the
diffraction amplitude of a single
slit since the latter is spread
substantially than the former.
PHY4320 Chapter Two
84
2
The diffraction grating provides a means of
deflecting an incoming light by an amount that
depends on its wavelength. This is of great
use in spectroscopy.
In reality, any periodic variations in the
refractive index would serve as a
diffraction grating.
sin θ m − sin θ i = m
λ
d
m = ±1,±2, L
PHY4320 Chapter Two
sin θ m + sin θ i = m
λ
d
85
Reading Materials
S. O. Kasap, “Optoelectronics and Photonics: Principles and
Practices”, Prentice Hall, Upper Saddle River, NJ 07458, 2001,
Chapter 1, “Wave Nature of Light”.
PHY4320 Chapter Two
86
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
1
Plane Electromagnetic Wave
An electromagnetic wave traveling along z has time-varying electric and magnetic
fields that are perpendicular to each other.
Ex = E0 cos(ωt − kz + φ0)
k = 2π / λ
ω = 2πν
φ = ωt − kz + φ0
Ex(z, t) = Re[ E0 exp(jφ0)expj(ωt − kz)]
propagation constant, or wave number
angular frequency
phase
The optical field generally refers to the electric field.
PHY4320 Chapter Two
2
Plane Electromagnetic Wave
A surface over which the phase of a wave is constant is referred to as a
wavefront.
PHY4320 Chapter Two
3
Phase Velocity
φ = ωt − kz + φ0 = constant
During a time interval δt, the wavefronts of contant phases move a
distance δz. The phase velocity v is therefore
δz ω
v = = = νλ
δt k
PHY4320 Chapter Two
4
Plane Electromagnetic Wave
At a given time, the phase
difference between two points
separated by Δr is simply k⋅Δr. If
this phase difference is 0 or
multiples of 2π, then the two points
are in phase.
E(r, t) = E0 cos(ωt − k⋅r + φ0)
k⋅r = kxx + kyy + kzz
k is called the wave vector.
PHY4320 Chapter Two
5
Plane Waves
Ex = E0 cos(ωt − kz + φ0)
The plane wave has no angular separation (no
divergence) in its wavevectors.
E0 does not depend on the distance from a
reference point, and it is the same at all points on
a given plane perpendicular to k.
The plane wave is an idealization that is useful in
analyzing many wave phenomena.
Real light beams would have finite cross sections.
The plane wave obeys Maxwell’s EM wave
equation.
PHY4320 Chapter Two
6
Maxwell’s EM Wave Equation
∂2E
∂2E ∂2E ∂2E
+ 2 + 2 = ε 0ε r μ 0 2
2
∂t
∂z
∂y
∂x
For a perfect plane wave: Ex = E0 cos(ωt − kz + φ0)
Left side:
Right side:
∂2E
2
E0 cos(ωt − kz + φ0 )
=
−
ω
2
∂t
∂ E
=0
2
∂x
2
∂2E
=0
2
∂y
∂2E
2
(
)
cos(ωt − kz + φ0 )
=
−
−
E
k
0
2
∂z
= −k 2 E0 cos(ωt − kz + φ0 )
From EM:
k 2 = ε 0ε r μ 0ω 2
1
v = λν =
PHY4320 Chapter Two
ε 0ε r μ 0
7
Diverging Waves
∂2E ∂2E ∂2E
∂2E
+ 2 + 2 = ε 0ε r μ 0 2
2
∂x
∂y
∂z
∂t
E = (A/r) cos(ωt − kr)
•
•
•
A spherical wave is described by a traveling wave that emerges from a point EM
source and whose amplitude decays with distance r from the source.
Optical divergence refers to the angular separation of wavevectors on a given
wavefront.
Plane and spherical waves represent two extremes of wave propagation behavior
from perfectly parallel to fully diverging wavevectors. They are produced by two
extreme sizes of EM wave sources: an infinitely large source for the plane wave
and a point source for the spherical wave. A real EM source would have a finite
size and finite power.
PHY4320 Chapter Two
8
Gaussian Beams
Many laser beams can be described by Gaussian beams.
Gaussian beam has an exp[j(ωt − kz)] dependence to describe propagation
characteristics but the amplitude varies spatially away from the beam axis.
The beam diameter 2w at any point z is defined as the diameter at which the
beam intensity has fallen to 1/e2 (13.5%) of its maximum.
PHY4320 Chapter Two
9
Gaussian Beams
Diffraction causes light waves to spread transversely as they propagate, and it is
therefore impossible to have a perfectly collimated beam. The spreading of a laser
beam is in precise accord with the predictions of pure diffraction theory. Even if a
Gaussian laser beam wavefront were made perfectly flat at some plane, it would
quickly acquire curvature and begin spreading out. The finite width 2w0 where the
wavefronts are parallel is called the waist of the beam. w0 is the waist radius and 2w0
is the spot size. The increase in beam diameter 2w with z makes an angle of 2θ,
which is called the beam divergence. Spot size and beam divergence are two
important parameters when choosing lasers.
PHY4320 Chapter Two
10
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
11
Refractive Index
When an EM wave is traveling in a dielectric medium, the oscillating electric
field polarizes the molecules of the medium at the frequency of the wave.
The field and the induced molecular dipoles become coupled. The net
effect is that the polarization mechanism delays the propagation of the EM
wave. The stronger the interaction between the field and the dipoles, the
slower the propagation of the wave. The relative permittivity εr measures
the ease with which the medium becomes polarized and indicates the
extent of interaction between the field and the induced dipoles.
The phase velocity in a medium is:
In vacuum:
c=
ε 0 μ0
v=
1
ε 0ε r μ 0
1
The refractive index of the medium is:
c
n = = εr
v
PHY4320 Chapter Two
12
Refractive Index
kmedium = nk
λmedium = λ/n
The frequency ν remains the same in different mediums.
In non-crystalline materials (glasses and liquids), the material structure is
the same in all directions and n does not depend on the direction. The
refractive index is isotropic.
Crystals, in general, have anisotropic properties. The refractive index
seen by a propagating EM wave will depend on the direction of the electric
field relative to crystal structures, which is further determined by both the
propagation direction and the electric field oscillation direction
(polarization).
PHY4320 Chapter Two
13
Relative Permittivity and Refractive Index
The relative permittivity for many materials can be vastly different at high
and low frequencies because different polarization mechanisms operate at
these frequencies. Al low frequencies, all polarization mechanisms (dipolar,
ionic, electronic) contribute to εr, whereas at high (optical) frequencies only
the electronic polarization can respond to the oscillating field.
Low frequency (LF) relative permittivity εr(LF) and refractive index n
Material
εr(LF)
Si
11.9
ε r (LF ) n (optical)
Comment
3.44
3.45 (at 2.15 μm) Electronic polarization up to
optical frequencies
Diamond 5.7
2.39
2.41 (at 590 nm)
GaAs
13.1
3.62
SiO2
3.84 2.00
1.46 (at 600 nm) Ionic polarization contributes to εr(LF)
Water
80
8.9
3.30 (at 5 μm)
Electronic polarization up to UV light
Ionic polarization contributes to εr(LF)
1.33 (at 600 nm) Dipolar polarization contributes to
εr(LF), which is large
PHY4320 Chapter Two
14
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
15
Group Velocity
Since there are no perfect monochromatic waves in practice, we
have to consider the way in which a group of waves differing slightly
in wavelength travel along the same direction. For two harmonic
waves of frequencies ω − δω and ω + δω and wavevectors k − δk and
k + δk interfering with each other and traveling along the z direction:
E x ,1 ( z , t ) = E0 cos[(ω − δω )t − (k − δk )z ]
E x , 2 ( z , t ) = E0 cos[(ω + δω )t − (k + δk )z ]
Using
⎤
⎡1
⎤
⎡1
cos A + cos B = 2 cos ⎢ ( A − B )⎥ cos ⎢ ( A + B )⎥
⎦
⎣2
⎦
⎣2
PHY4320 Chapter Two
16
Group Velocity
E x = E x ,1 + E x , 2 = 2 E0 cos[(δω )t − (δk )z ]cos(ωt − kz )
We obtain:
Generate a wave packet containing an oscillating field at the mean
frequency ω with the amplitude modulated by a slowly varying field of
frequency δω.
PHY4320 Chapter Two
17
Group Velocity
E x = 2 E0 cos[(δω )t − (δk )z ]cos(ωt − kz )
or
The maximum amplitude
moves with a wavevector
δk and a frequency δω.
Phase velocity v =
dω
Group velocity v g =
dk
The group velocity defines the speed with
which energy or information is propagated
since it defines the speed of the envelope of
the amplitude variation.
PHY4320 Chapter Two
ω
c
=
k n
In vacuum:
ω = ck
dω
vg =
dk
=c
group velocity
phase velocity
18
In a medium with n as a function of λ:
ω = ck / n
k = nω / c
cdk d (nω )
dn
=
= n +ω
dω
dω
dω
c /ν 2πc
=
λ=
n
ωn
dn
dn dλ
dn 2πc
dn
ω
=ω
= −ω
= −λ
2
dω
dλ dω
dλ ω n
dλ
dω
c
c
vg =
=
=
dk n − λ dn N g
dλ
dn
Ng = n − λ
dλ
Ng is the group index of the medium.
PHY4320 Chapter Two
19
Group Index of the Medium
Pure SiO2
For silica, Ng is minimum around 1300
nm, which means that light waves
with wavelengths close to 1300 nm
travel with the same group velocity
and do not experience dispersion.
Both n and Ng are functions of the free space wavelength. Such a medium is
called a dispersive medium. n and Ng of SiO2 are important parameters for
optical fiber design in optical communications.
PHY4320 Chapter Two
20
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
21
Snell’s Law
Willebrord van Roijen Snell
(1580-1626)
Incident light waves
Ai and Bi are in
phase. Reflected
waves Ar and Br and
refracted waves At
and Bt should also be
in phase, respectively.
PHY4320 Chapter Two
22
Reflection
v1t
v1t
AB' =
=
sin θ i sin θ r
θi = θ r
Refraction
v1t
v2 t
AB' =
=
sin θ i sin θ t
sin θ i v1 n2
= =
sin θ t v2 n1
PHY4320 Chapter Two
23
sin θ i n2
=
sin θ t n1
Refraction
Total Internal Reflection
When n1 > n2 and θt = 90°, the incidence angle is
called the critical angle θc:
n2
sin θ c =
n1
PHY4320 Chapter Two
24
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
25
Fresnel’s Equations
The incidence plane is
the plane containing the
incident and reflected
light rays.
Transverse electric field (TE) waves: Ei,⊥, Er,⊥, and Et,⊥.
Transverse magnetic field (TM) waves: Ei,//, Er,//, and Et,//.
Ei = Ei 0 exp j (ωt − k i • r )
Er = Er 0 exp j (ωt − k r • r )
Et = Et 0 exp j (ωt − k t • r )
Augustin Fresnel
(1788-1827)
Our goal is to find Er0 and Et0 relative
to Ei0, including any phase changes.
PHY4320 Chapter Two
26
θi = θ r
⎫
⎬
n1 sin θ i = n2 sin θ t ⎭
Snell’s law
B ⊥ = (n / c) E// ⎫
⎬ Maxwell’s equations
B// = (n / c) E⊥ ⎭
Etangential (1) = Etangential (2)⎫⎪
⎬ Boundary conditions
Btangential (1) = Btangential (2) ⎪⎭
PHY4320 Chapter Two
27
Incidence plane
− Ei ,⊥ − Er ,⊥ = − Et ,⊥
− Bi ,⊥ + Br ,⊥ = − Bt ,⊥ ⇒ −(n1 / c )Ei , // + (n1 / c )Er , // = −(n2 / c )Et , //
− Ei , // cos θ i − Er , // cos θ r = − Et , // cos θ t
Bi , // cos θ i − Br , // cos θ r = Bt , // cos θ t ⇒
(n1 / c )Ei ,⊥ cos θ i − (n1 / c )Er ,⊥ cos θ r = (n2 / c )Et ,⊥ cos θ t
PHY4320 Chapter Two
28
de − bf
⎧
x=
⎪
⎧ax + by = e
⎪
ad − bc
⇒
⎨
⎨
⎩cx + dy = f
⎪ y = af − ce
⎪⎩
ad − bc
⎧− Ei ,⊥ − Er ,⊥ = − Et ,⊥
⎨
⎩n1 Ei ,⊥ cos θ i − n1 Er ,⊥ cos θ r = n2 Et ,⊥ cos θ t
⎧ Er ,⊥ − Et ,⊥ = − Ei ,⊥
⇒⎨
⎩n1 Er ,⊥ cos θ r + n2 Et ,⊥ cos θ t = n1 Ei ,⊥ cos θ i
− n2 cos θ t + n1 cos θ i
⎧
⎪ Er ,⊥ = n cos θ + n cos θ Ei ,⊥
⎪
2
1
t
r
⇒⎨
⎪ E = n1 cos θ i + n1 cos θ r E
⎪⎩ t ,⊥ n2 cos θ t + n1 cos θ r i ,⊥
PHY4320 Chapter Two
29
de − bf
⎧
x=
⎪
⎧ax + by = e
⎪
ad − bc
⇒
⎨
⎨
⎩cx + dy = f
⎪ y = af − ce
⎪⎩
ad − bc
⎧− n1 Ei , // + n1 Er , // = −n2 Et , //
⎨
⎩− Ei , // cos θ i − Er , // cos θ r = − Et , // cos θ t
⎧n1 Er , // + n2 Et , // = n1 Ei , //
⇒⎨
⎩ Er , // cos θ r − Et , // cos θ t = − Ei , // cos θ i
− n1 cos θ t + n2 cos θ i
⎧
⎪ Er , // = − n cos θ − n cos θ Ei , //
⎪
1
2
t
r
⇒⎨
⎪ E = − n1 cos θ i − n1 cos θ r E
⎪⎩ t , // − n1 cos θ t − n2 cos θ r i , //
PHY4320 Chapter Two
30
⎧n = n2 / n1
⎪
⇒ sin θ i = n sin θ t
⎨θ i = θ r
⎪n sin θ =n sin θ
2
i
t
⎩ 1
[
⇒ n cos θ t = n − n sin θ t
2
2
2
]
1/ 2
[
= n − sin θ i
2
[
[
2
]
1/ 2
]
]
1/ 2
2
2
⎧
− n2 cos θ t + n1 cos θ i
cos θ i − n − sin θ i
Ei ,⊥ =
Ei ,⊥
⎪ Er , ⊥ =
1/ 2
2
2
n2 cos θ t + n1 cos θ r
cos θ i + n − sin θ i
⎪
⎨
2 cos θ i
⎪ E = n1 cos θ i + n1 cos θ r E =
Ei ,⊥
⊥
,
⊥
i
,
t
1
/
2
2
2
⎪
n
n
+
cos
cos
θ
θ
cos
n
sin
+
−
θ
θi
2
1
t
r
i
⎩
[
PHY4320 Chapter Two
]
31
⎧n = n2 / n1
⎪
⇒ sin θ i = n sin θ t
⎨θ i = θ r
⎪n sin θ =n sin θ
i
t
2
⎩ 1
[
⇒ n cos θ t = n − n sin θ t
2
2
2
]
1/ 2
[
[
[
= n − sin θ i
2
2
]
]
]
1/ 2
1/ 2
2
2
⎧
n − sin θ i − n 2 cos θ i
− n1 cos θ t + n2 cos θ i
Ei , // =
Ei , //
⎪ Er , // =
1/ 2
2
2
2
− n1 cos θ t − n2 cos θ r
n − sin θ i + n cos θ i
⎪
⎨
2n cos θ i
⎪ E = − n1 cos θ i − n1 cos θ r E =
E
⎪ t , // − n cos θ − n cos θ i , // n 2 − sin 2 θ 1/ 2 + n 2 cos θ i , //
t
r
1
2
i
i
⎩
[
PHY4320 Chapter Two
]
32
Ei = Ei 0 exp j (ωt − k i • r )
Er = Er 0 exp j (ωt − k r • r )
r⊥ and r// are reflection coefficients. t⊥
and t// are transmission coefficients.
Et = Et 0 exp j (ωt − k t • r )
r⊥ =
t⊥ =
r// =
t // =
Er 0 , ⊥
Ei 0,⊥
Et 0,⊥
Ei 0,⊥
Er 0, //
Ei 0, //
Et 0, //
Ei 0, //
[
=
cos θ + [n
]
θ]
cos θ i − n − sin θ i
=
i
[n
r// + nt // = 1
r⊥ + 1 = t ⊥
2
− sin
2 cos θ i
[
2
2
2
2
2
]
θ]
− sin θ i
− sin
2
2
1/ 2
1/ 2
i
− sin θ i
2
]
1/ 2
t ⊥ = t ⊥ exp(− jφt ,⊥ )
t // = t // exp(− jφt , // )
]
1/ 2
− n cos θ i
2
+ n 2 cos θ i
2n cos θ i
i
r// = r// exp(− jφr , // )
1/ 2
1/ 2
cos θ i + n 2 − sin 2 θ i
[
n
=
[n
=
2
r⊥ = r⊥ exp(− jφr ,⊥ )
+ n 2 cos θ i
Complex coefficients can only be
obtained when n2 − sin2θi < 0.
n1 > n2 ,θi > θc
Phase changes other than 0 or 180°
occur only when there is TIR.
PHY4320 Chapter Two
33
r⊥
[
]
=
cos θ + [n − sin θ ]
[
n − sin θ ] − n cos θ
=
[n − sin θ ] + n cosθ
cos θ i − n − sin θ i
2
2
1/ 2
2
2
1/ 2
i
1/ 2
2
2
i
2
1/ 2
2
]
i
2
i
[n
1 − n n1 − n2
r⊥ =
=
1 + n n1 + n2
n − n 2 n1 − n2
=
r// =
2
n+n
n1 + n2
i
2
r//
When θi = 0 (normal incidence):
i
When r// = 0 (reflected light is polarized):
− sin θ p = n 2 cos θ p ⇒ n 2 − sin 2 θ p = n 4 cos 2 θ p
2
2
sin
θp
n
4
2
2
2
4
n
n
n
1
tan
tan
⇒
−
=
⇒
+
−
=
θ
θ
p
p
cos 2 θ p cos 2 θ p
n2
θp is called the polarization angle or
⇒ tan θ p =
n1
Brewster’s angle.
2
2
1/ 2
n2
sin θ c =
n1
(
)
PHY4320 Chapter Two
34
Brewster’s Angle (Polarization Angle)
n1 = 1.44, n2 = 1.00
David Brewster
(1781-1868)
θ p = 34.78°
θ c = 43.98°
When θi = θp, reflected light is linearly polarized and is perpendicular to the
incidence plane.
When θp < θi < θc, there is a phase shift of 180° in r//.
|r⊥| and |r//| increases with θi when θi > θp.
There are phase shifts (other than 0 or 180°) in r⊥ and r// when θi > θc.
PHY4320 Chapter Two
35
Example: calculate r⊥ and r// when n1 = 1.44, n2 = 1.00, and θi = 40°.
Solution:
n2 1.00
=
= 0.694
n=
n1 1.44
[
]
r =
cos θ + [n − sin θ ]
cos 40° − [0.694 − sin 40°]
=
cos 40° + [0.694 − sin 40°]
[
n − sin θ ] − n cos θ
r =
[n − sin θ ] + n cosθ
[
0.694 − sin 40°] − 0.694
=
[0.694 − sin 40°] + 0.694
⊥
cos θ i − n − sin θ i
2
2
1/ 2
2
2
1/ 2
i
2
i
2
2
1/ 2
2
2
1/ 2
1/ 2
2
2
i
//
2
i
1/ 2
2
= 0.490
2
i
i
2
2
1/ 2
2
2
2
1/ 2
2
cos 40°
cos 40°
PHY4320 Chapter Two
= −0.170
36
Example: calculate r⊥ and r// when n1 = 1.00, n2 = 1.44, and θi = 40°.
Solution:
n2 1.44
n=
=
= 1.44
n1 1.00
[
]
r =
cos θ + [n − sin θ ]
cos 40° − [1.44 − sin 40°]
=
= −0.255
cos 40° + [1.44 − sin 40°]
[
n − sin θ ] − n cos θ
r =
[n − sin θ ] + n cosθ
[
1.44 − sin 40°] − 1.44 cos 40°
=
= −0.104
[1.44 − sin 40°] + 1.44 cos 40°
⊥
cos θ i − n − sin θ i
2
2
1/ 2
2
2
1/ 2
i
2
i
2
2
2
1/ 2
2
2
1/ 2
1/ 2
2
i
//
2
i
1/ 2
2
2
i
i
2
2
1/ 2
2
2
2
1/ 2
2
PHY4320 Chapter Two
37
n2
tan θ p =
n1
n1 = 1.00, n2 = 1.44
θ p = 55.22°
PHY4320 Chapter Two
38
What are the phase changes at TIR (n1 > n2 and θi > θc)?
r⊥
[
]
=
cos θ + [n − sin θ ]
[
n − sin θ ] − n cos θ
=
[n − sin θ ] + n cosθ
cos θ i − n − sin θ i
2
2
1/ 2
2
2
1/ 2
i
r//
1/ 2
2
2
1/ 2
2
r⊥
2
2 1/ 2
2
2 1/ 2
i
2
2 1/ 2
2
i
2
i
2 1/ 2
2
i
z = x +y
i
PHY4320 Chapter Two
2
)
2 1/ 2
⎛ y⎞
φ = tan ⎜ ⎟
⎝x⎠
i
]
θ −n ]
− n cos θ + j [sin θ − n ]
=
n cos θ + j [sin θ − n ]
cos θ i − j sin θ i − n
i
r//
i
2
i
[
=
cos θ + j [sin
(
r// = 1
2
i
z = z exp(− jφ )
r⊥ = 1
i
2
z = x − jy
−1
z1 exp(− jφ1 )
z1
z= =
z 2 z 2 exp(− jφ2 )
=
z1
z2
exp[− j (φ1 − φ2 )]
39
r⊥
[
=
cos θ + j [sin
2
i
[
]
θ −n ]
cos θ i − j sin 2 θ i − n
2 1/ 2
i
[
]
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟
φ1 = tan
⎟
⎜
θ
cos
i
⎠
⎝
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟
φ2 = − tan
⎟
⎜
cos
θ
i
⎠
⎝
2 1/ 2
[
]
]
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟
φ⊥ = φ1 − φ2 = 2 tan
⎟
⎜
θ
cos
i
⎠
⎝
[
⎛ 1 ⎞ sin θ i − n
tan⎜ φ⊥ ⎟ =
cos θ i
⎝2 ⎠
2
]
2 1/ 2
PHY4320 Chapter Two
40
r// =
[
− n 2 cos θ i + j sin 2 θ i − n
[
n cos θ i + j sin θ i − n
2
2
]
[
]
2 1/ 2
2 1/ 2
[
]
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟ −π
φ1 = tan
2
⎜ n cos θ i ⎟
⎝
⎠
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟
φ2 = − tan
⎜ n 2 cos θ i ⎟
⎝
⎠
]
[
]
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟ −π
φ// = φ1 − φ2 = 2 tan
⎜ n 2 cos θ i ⎟
⎝
⎠
[
]
2
2 1/ 2 ⎞
⎛
−1 ⎜ sin θ i − n
⎟
φ// + π = 2 tan
⎜ n 2 cos θ i ⎟
⎝
⎠
[
1 ⎞ sin θ i − n
⎛1
tan⎜ φ// + π ⎟ =
2
n
2
2
cos θ i
⎝
⎠
2
PHY4320 Chapter Two
]
2 1/ 2
41
Consider now:
ktz = kt sin θ t =
kty = kt cos θ t =
Et = Et 0 exp j (ωt − k t ⋅ r )
= Et 0 exp j (ωt − k ty y − ktz z )
2π
λ
2π
λ
n2 sin θ t =
[
2π
λ
n2 1 − sin θ t
2
]
n1 sin θ i = ki sin θ i = kiz
1/ 2
⎤
⎡ ⎛n ⎞
2
1
n2 ⎢1 − ⎜⎜ ⎟⎟ sin θ i ⎥
=
λ ⎢ ⎝ n2 ⎠
⎥⎦
⎣
2π
PHY4320 Chapter Two
2
1/ 2
42
Evanescent Wave
⎛ n1 ⎞
⎛ n1 ⎞
2
θ i > θ c ⇒ ⎜⎜ ⎟⎟ sin θ i > ⎜⎜ ⎟⎟ sin 2 θ c = 1
⎝ n2 ⎠
⎝ n2 ⎠
2
2
At TIR:
⎡⎛ n ⎞
⎤
2
1
kty = − j
n2 ⎢⎜⎜ ⎟⎟ sin θ i − 1⎥
λ ⎢⎝ n2 ⎠
⎥⎦
⎣
2π
2
1/ 2
= − jα 2
PHY4320 Chapter Two
Why do we put a
minus here?
43
Evanescent Wave
⎤
2πn2 ⎡⎛ n1 ⎞
2
⎢⎜⎜ ⎟⎟ sin θ i − 1⎥
α2 =
λ ⎢⎝ n2 ⎠
⎥⎦
⎣
2
1/ 2
Et ( y, z , t ) = Et 0 exp j (ωt − k ty y − ktz z )
= Et 0 exp j (− kty y )exp j (ωt − kiz z )
= Et 0 exp(− α 2 y ) exp j (ωt − kiz z )
λ 2 2
2 −1 / 2
[
=
n1 sin θ i − n2 ]
δ=
α 2 2π
1
λ = 514.5 nm (green light)
n1 = 1.44 (glass)
n2 = 1.00 (air)
θi = 50° (>θc = 43.98°)
δ=
[
the penetration depth
( )
514.5
1.44 2 sin 2 50o − 1.00 2
2π
PHY4320 Chapter Two
]
−1 / 2
= 176nm
44
Total Internal Reflection Fluorescence (TIRF) Microscopy
TIR illumination can greatly improve signal-to-noise ratios in
applications requiring imaging of minute structures or single
molecules in specimens having large numbers of fluorophores
located outside of the optical plane of interest, such as
molecules in solution in Brownian motion, vesicles undergoing
endocytosis or exocytosis, or single protein trafficking in cells.
Carl Zeiss
PHY4320 Chapter Two
45
PHY4320 Chapter Two
46
θi
48.7°
49.7°
51.1°
53.9°
57.8°
66.5°
I z = I 0e
dp
300 nm
145 nm
100 nm
70 nm
55 nm
40 nm
−z/dp
λ 2 2
2 −1 / 2
[
dp =
n1 sin θ i − n2 ]
4π
PHY4320 Chapter Two
47
Fluorescence intensity
Distance
Evanescent nanometry
Achieves a resolution of ~ 1 nm.
Requires highly photostable fluorescent materials.
PHY4320 Chapter Two
48
Intensity, Reflectance, and Transmittance
1
I = vε r ε o E02 ∝ nE02
2
Light intensity
Reflectance
R⊥ =
Er 0 , ⊥
2
Ei 0,⊥
2
Transmittance
T⊥ =
n2 Et 0,⊥
2
n1 Ei 0,⊥
2
= r⊥
R// =
2
⎛ n2 ⎞
= ⎜⎜ ⎟⎟ t ⊥
⎝ n1 ⎠
2
T// =
PHY4320 Chapter Two
Er 0, //
Ei 0, //
2
2
= r//
n2 Et 0, //
n1 Ei 0, //
2
2
2
⎛ n2 ⎞
= ⎜⎜ ⎟⎟ t //
⎝ n1 ⎠
49
2
r⊥ =
r// =
t⊥ =
t // =
Er 0 , ⊥
Ei 0,⊥
Er 0, //
Ei 0, //
Et 0,⊥
Ei 0,⊥
Et 0, //
Ei 0, //
cos θ − [n − sin θ ]
=
cos θ + [n − sin θ ]
[
n − sin θ ] − n cos θ
=
[n − sin θ ] + n cosθ
2
i
2
1/ 2
2
i
i
2
1/ 2
2
2
=
=
1/ 2
2
[n
i
2
− sin
i
− sin
2
2
2n cos θ i
2
θ]
1/ 2
i
i
2
2 cos θ i
cos θ + [n
n − n 2 n1 − n2
=
=
2
n+n
n1 + n2
2
i
θ]
i
=
1/ 2
i
⎛ n1 − n2 ⎞
⎟⎟
R = R⊥ = R// = ⎜⎜
⎝ n1 + n2 ⎠
2
2n1
=
1 + n n1 + n2
=
+ n 2 cos θ i
2
1 − n n1 −n 2
=
1 + n n1 + n 2
=
1/ 2
2
i
When θi = 0 (normal incidence):
2n
2n1
=
n + n 2 n1 + n2
4n1n2
T = T⊥ = T// =
2
(n1 + n2 )
PHY4320 Chapter Two
50
Example: A ray of light traveling in a glass medium of refractive index n1 =
1.450 becomes incident on a less dense glass medium of refractive index n2 =
1.430. Suppose the free space wavelength (λ) of the light ray is 1 μm. (a)
What is the critical angle for TIR? (b) What is the phase change in the
reflected wave when θi = 85° and 90°? (c) What is the penetration depth of the
evanescent wave into medium 2 when θi = 85° and 90°?
sin θ c = n2 / n1 = 1.430 / 1.450 ⇒ θ c = 80.47°
Solution:
2
⎡ 2
⎛ 1.430 ⎞ ⎤
⎟ ⎥
⎢sin 85 ° − ⎜
θi = 90°
2
2 1/ 2
1
.
450
⎝
⎠
⎢
⎛ 1 ⎞ sin θ i − n
⎦⎥ = 1.614 ⇒ φ = 116 .45 °
tan ⎜ φ ⊥ ⎟ =
=⎣
⊥
cos θ i
cos 85 °
⎝2 ⎠
2 1/ 2
φ⊥ = 180°
⎡ 2
⎛ 1.430 ⎞ ⎤
⎟ ⎥
⎢sin 85 ° − ⎜
2
2 1/ 2
1
.
450
⎠ ⎥⎦
⎝
1 ⎞ sin θ i − n
⎢⎣
⎛1
=
tan ⎜ φ // + π ⎟ =
= 1.660 ⇒ φ // = −62 .13 °
2
2
n cos θ i
2 ⎠
⎝2
⎛ 1.430 ⎞
⎜
⎟ cos 85 °
φ// = 0°
⎝ 1.450 ⎠
[
]
[
δ=
1/ 2
]
−1 / 2
−1 / 2
1μm
λ 2 2
[
[
n1 sin θ i − n22 ] =
1.450 2 sin 2 85° − 1.430 2 ] = 0.780 μm
2π
2π
PHY4320 Chapter Two
δ = 0.663μm
51
Example: Consider a light ray at normal incidence on a boundary between a
glass medium of refractive index 1.5 and air of refractive index 1.0. (a) If light
is traveling from air to glass, what are the reflectance and transmittance? (b) If
light is traveling from glass to air, what are the reflectance and transmittance?
(c) What is the Brewster’s angle when light is traveling from air to glass?
2
2
⎧
⎛ n1 − n2 ⎞ ⎛ 1.0 − 1.5 ⎞
⎟⎟ = ⎜
⎪ R = ⎜⎜
⎟ = 0.04
⎧n1 = 1.0
n1 + n2 ⎠ ⎝ 1.0 + 1.5 ⎠
⎪
⎝
⇒⎨
⎨
4n1n2
4 ×1.0 ×1.5
⎩n2 = 1.5 ⎪
⎪T = (n + n )2 = (1.0 + 1.5)2 = 0.96
1
2
⎩
Solution:
2
2
⎧
⎛ n1 − n2 ⎞ ⎛ 1.5 − 1.0 ⎞
⎟⎟ = ⎜
⎪ R = ⎜⎜
⎟ = 0.04
1
.
5
n
=
⎧ 1
⎪
⎝ n1 + n2 ⎠ ⎝ 1.5 + 1.0 ⎠
⇒
⎨
⎨
4n1n2
4 ×1.5 ×1.0
⎩n2 = 1.0 ⎪
⎪T = (n + n )2 = (1.5 + 1.0 )2 = 0.96
1
2
⎩
n 1.5
tan θ p = 2 =
⇒ θ p = 56.31°
n1 1.0
PHY4320 Chapter Two
Light will lose 4% of
its intensity whenever
it travels through an
interface between air
and glass.
52
Antireflection Coatings
Current commercial solar cells are made of silicon (n ≈ 3.5 at wavelengths of
700–800 nm).
2
2
(
(
n1 − n2 )
1.0 − 3.5)
R=
=
2
(n1 + n2 ) (1.0 + 3.5)2
Δ cφ = kc 2d =
2πn2
⎛ λ ⎞
⎟⎟
⇒ d = m⎜⎜
⎝ 4n2 ⎠
λ
= 0.31
Air Coating (Si3N4) Si (solar cell)
1.0
1.9
3.5
(2d ) = mπ
r12 = r23 in order to obtain efficient
destructive interference.
n1 − n2 n2 − n3
=
⇒ n2 = n1n3
n1 + n2 n2 + n3
r12,⊥ = r12,// = −0.31
r23,⊥ = r23,// = −0.30
n2 = n1n3 = 1.0 × 3.5 = 1.87
d=
λ
4n2
=
700nm
= 92nm
4 ×1.9
PHY4320 Chapter Two
53
Dielectric Mirrors
n1 < n2
We can find that A, B, and C
n1 − n2
2n1
< 0 t12 =
r12 =
> 0 waves are in phase with each
n1 + n2
n1 + n2
other. They interfere with each
n2 − n1
2n2
other constructively.
> 0 t 21 =
r21 =
>0
The total reflectance will be
n2 + n1
n2 + n1
close to unity.
PHY4320 Chapter Two
54
Photonic Crystals
1-D
2-D
periodic in
one direction
periodic in
two directions
3-D
periodic in
three directions
Photonic crystals are ordered structures in which two media with different
dielectric constants or refractive indices are arranged in a periodic form, with
lattice constants comparable to the wavelength of light. The light propagation is
completely forbidden within photonic crystals if they are properly designed
(refractive index contrast, crystal structure, and lattice constant).
PHY4320 Chapter Two
55
Defects in Photonic Crystals
cavity
waveguide
PHY4320 Chapter Two
56
Low-Pumping-Current Lasers
SEM image
H.-G. Park, S.-H. Kim, S.-H. Kwon, Y.-G. Ju,
J.-K. Yang, J.-H. Baek, S.-B. Kim, Y.-H. Lee,
Science 2004, 305, 1444.
PHY4320 Chapter Two
57
Low-Pumping-Current Lasers
radii of air holes:
0.28a (I)
0.35a (II)
0.385a (III)
0.4a (IV)
0.41a (V)
Designed according
to FDTD (finite
difference time
domain) calculations.
electric field intensity profile
PHY4320 Chapter Two
58
Low-Pumping-Current Lasers
These low threshold current, highly efficient lasers might function as
single photon sources, which are useful for cavity quantum
electrodynamics and quantum information processing.
PHY4320 Chapter Two
59
Wavelength Channel Drop Devices
B.-S. Song, S. Noda, T. Asano, Science 2003, 300, 1537.
PHY4320 Chapter Two
60
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
61
Goos-Hänchen Shift
Careful experiments show that the reflected wave at
TIR appears to be laterally shifted from the incidence
point at the interface. This lateral shift is known as the
Goos-Hänchen shift.
The reflected wave appears to be reflected from a
virtual plane inside the optically less dense medium.
The spacing between the interface and the virtual
plane is approximately the penetration depth of the
evanescent wave δ when n1 is very close to n2.
Δz = 2δ tan θ i
PHY4320 Chapter Two
λ = 1 μm
n1 = 1.45
n2 = 1.43
θi = 85°
δ = 0.78 μm
Δz ≈ 18 μm
62
Optical Tunneling
At TIR, we shrink the thickness d of the medium B. When B is sufficiently
thin, an attenuated beam emerges on the other side of B in C. This
phenomenon, which is forbidden by simple geometrical optics, is called
optical tunneling or frustrated total internal reflection. Optical tunneling
is due to the evanescent wave, the field of which penetrates into medium B
and reaches the interface BC.
PHY4320 Chapter Two
63
Beam Splitters Based on Frustrated TIR
The extent of light intensity division between
the two beams depends on the thickness of the
thin layer B and its refractive index.
PHY4320 Chapter Two
64
Chapter Two: Wave Nature of Light
Light waves in a homogeneous medium
Plane electromagnetic waves
Maxwell’s wave equation and diverging waves
Refractive index
Group velocity and group index
Snell’s law and total internal reflection (TIR)
Fresnel’s equations
Goos-Hänchen shift and optical tunneling
Diffraction
Fraunhofer diffraction
Diffraction grating
PHY4320 Chapter Two
65
Diffraction
Diffraction occurs when light waves encounter small objects or apertures.
For example, the beam passed through a circular aperture is divergent and
exhibits an intensity pattern (called diffraction pattern) that has bright and
dark rings (called Airy rings).
Two types of diffraction: (1) In Fraunhofer diffraction, the incident light is a
plane wave. The observation or detection screen is far away from the
aperture so that the received waves also look like plane waves. (2) In Fresnel
diffraction, the incident and received light waves have significant wavefront
curvatures. Fraunhofer diffraction is by far the most important.
PHY4320 Chapter Two
66
Diffraction
Diffraction can be understood using the Huygens-Fresnel principle. Every
unobstructed point of a wavefront, at a given instant in time, serves as a source of
spherical secondary waves (with the same frequency as that of the primary wave).
The amplitude of the optical field at any point beyond is the superposition of all these
wavelets (considering their amplitudes and relative phases).
PHY4320 Chapter Two
67
Diffraction Pattern through a 1D Slit
The one-dimensional (1D) slit has a width of a. Divide the width a into N (a large
number) coherent sources δy (= a/N). Because the slit is uniformly illuminated, the
amplitude of each source will be proportional to δy. In the forward direction (θ = 0°),
they will be in phase and constitute a forward wave along the z direction. They can
also be in phase at some other angles and therefore give rise to a diffracted wave
along such directions.
The intensity of the received wave at a
point on the screen will be the sum of all
waves arriving from all the sources in the
slit. The wave Y is out of phase with
respect to A by kysinθ.
δE ∝ (δy ) exp(− jky sin θ )
E (θ ) = C ∫ δy exp(− jky sin θ )
a
0
PHY4320 Chapter Two
68
Let x = y − a/2, we obtain
E (θ ) = C ∫ δy exp(− jky sin θ )
a
= C∫
0
a/2
exp[− jk ( x + a / 2 )sin θ ]dx
−a / 2
1
− j ka sin θ
2
= Ce
1
− jkx sin θ
dx =
∫−a / 2e
− jk sin θ
a/2
∫
a/2
e − jkx sin θ dx
−a / 2
a
jk sin θ ⎞
⎛ − jk a2 sin θ
2
⎜e
⎟
−
e
⎜
⎟
⎝
⎠
⎛1
⎛1
⎞
⎞
− 2 j sin ⎜ ka sin θ ⎟ a sin ⎜ ka sin θ ⎟
2
2
⎝
⎝
⎠
⎠
=
=
1
− jk sin θ
ka sin θ
2
PHY4320 Chapter Two
69
E (θ ) =
Ce
1
− j ka sin θ
2
I (θ ) = E (θ )
⎛1
⎞
a sin ⎜ ka sin θ ⎟
⎝2
⎠
1
ka sin θ
2
2
⎡
⎛1
⎞⎤
sin
sin
θ
Ca
ka
⎜
⎟⎥
⎢
⎝2
⎠⎥
=⎢
1
⎢
⎥
ka sin θ
⎢⎣
⎥⎦
2
2
The zero intensity occurs when:
1
π
ka sin θ = a sin θ = mπ
λ
2
m = ±1,±2,L
mλ
sin θ =
λ = 1.3 μm
a
a = 100 μm
divergence 2θ = 1.5°
PHY4320 Chapter Two
Bright and dark regions
The central bright region is
larger than the slit width (the
transmitted beam is
diverging)
70
The diffraction patterns from two-dimensional (2D) apertures such as
rectangular and circular apertures are more complicated to calculate but they
use the same principle based on the interference of waves emitted from all
point sources in the aperture.
The diffraction pattern from a circular aperture of diameter D has a central
bright region. The divergence of this bright region is determined by
sin θ = 1.22
λ
D
Diffraction from rectangular apertures
Why is the central bright region rotated relative to the rectangular aperture?
PHY4320 Chapter Two
71
Diffraction-Limited Resolution
Consider two neighboring coherent sources with an angular separation of Δθ
examined through an imaging system with an aperture of diameter D. The aperture
produces a diffraction pattern of the sources. As the sources get closer, their
diffraction patterns overlap more.
According to Rayleigh criterion, the two spots are just resolvable when the principle
maximum of one diffraction pattern coincides with the minimum of the other, which is
given by the condition:
sin (Δθ min ) = 1.22
λ
D
PHY4320 Chapter Two
72
Diffraction-Limited Resolution
Objectives are key components in an optical microscope.
PHY4320 Chapter Two
73
Diffraction-Limited Resolution
NA = n ⋅ sin (α )
The numerical aperture (NA) of a microscope objective is a measure
of its ability to gather light and resolve fine specimen detail at a fixed
object distance.
PHY4320 Chapter Two
74
Diffraction-Limited Resolution
0.61λ
resolution r =
NA
The resolution of an optical microscope is defined as the shortest distance between two
points on a specimen that can still be distinguished by the observer or camera system as
separate entities. It is the most important feature of the optical system and influences the
ability to distinguish between fine details of a particular specimen.
PHY4320 Chapter Two
75
Overcome Diffraction-Limited Resolution
If a sufficient number of photons are collected from a single fluorescent
molecule, the center of this molecule can be determined up to 1.5-nm
resolution by curve-fitting the fluorescence image to a Gaussian function.
A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E.
Goldman, P. R. Selvin, Science 2003, 300, 2061.
PHY4320 Chapter Two
76
Overcome Diffraction-Limited Resolution
Discernable 30-nm and 7-nm steps are readily observed upon moving the
cover slip, either at a constant rate or a Poisson-distributed rate.
PHY4320 Chapter Two
77
Overcome Diffraction-Limited Resolution
Myosin (肌球蛋白，肌凝蛋白) V is a cargo-carrying motor that takes 37-nm center of
mass steps along actin (肌动蛋白) filaments. It has two heads held together by a
coiled-coil stalk. Hand-over-hand and inchworm models have been proposed for its
movement along actin filaments. For a single fluorescent molecule attached to the
light chain domain of myosin V, the inchworm model predicts a uniform step size of 37
nm, whereas the hand-over-hand model predicts alternating steps of 37 – 2x, 37 + 2x,
where x is the in-plane distance of the dye from the midpoint of the myosin.
PHY4320 Chapter Two
78
Overcome Diffraction-Limited Resolution
The dye is 7 nm from the center of mass along the direction of motion.
This result indicates that myosin V moves in a hand-over-hand fashion along
actin filaments.
PHY4320 Chapter Two
79
Diffraction Grating
A diffraction grating in its simplest form has a periodic series of slits. An
incident beam of light is diffracted in certain well-defined directions that
depend on the wavelength and the grating periodicities.
Zero diffraction intensity
condition for a single slit:
Bragg diffraction condition
(maximum intensity):
sin θ = m
λ
d
m = ±1,±2, L
sin θ =
PHY4320 Chapter Two
mλ
a
m = ±1,±2,L
80
Let’s try to derive the actual diffraction intensity from the grating by assuming
a plane incident wave and that there are N slits.
For the diffraction from a single slit:
Eslit (θ ) =
=
Ce
1
− j ka sin θ
2
⎛1
⎞
a sin ⎜ ka sin θ ⎟
⎝2
⎠
1
ka sin θ
2
C1e
− jα
α
sin α
⎞
⎛ C1 = Ca
⎟
⎜
1
⎜ α = ka sin θ ⎟
2
⎠
⎝
Egrating (θ ) = Eslit (θ ) + Eslit (θ )e − jkd sin θ + Eslit (θ )e − j 2 kd sin θ + L + Eslit (θ )e − j ( N −1)kd sin θ
For the diffraction from the grating:
[
= Eslit (θ ) 1 + e − jkd sin θ + e − j 2 kd sin θ + L + e − j ( N −1)kd sin θ
]
A geometrical progression
PHY4320 Chapter Two
81
1 − e − jNkd sin θ
Egrating (θ ) = Eslit (θ )
1 − e − jkd sin θ
⎛ sin α ⎞ 1 − e
I grating (θ ) = Egrating (θ ) = C12 ⎜
⎟
− jkd sin θ
α
1
e
−
⎠
⎝
2
1− e
1 − e − jkd sin θ
− jNkd sin θ 2
=
− jNkd sin θ 2
2
1 − cos( Nkd sin θ ) + j sin ( Nkd sin θ )
1 − cos(kd sin θ ) + j sin (kd sin θ )
2
2
=
2 − 2 cos( Nkd sin θ )
2 − 2 cos(kd sin θ )
⎞
⎛1
2 sin 2 ⎜ Nkd sin θ ⎟
2
⎛ sin Nβ ⎞
1 − cos( Nkd sin θ )
2
⎠
⎝
⎟⎟
=
= ⎜⎜
=
1 − cos(kd sin θ )
sin β ⎠
⎞
⎛1
2 sin 2 ⎜ kd sin θ ⎟ ⎝
⎠
⎝2
1
⎞
⎛
⎜ β = kd sin θ ⎟
2
2
⎠
⎝
2
⎛ sin α ⎞ ⎛ sin Nβ ⎞
⎟⎟
I grating (θ ) = C12 ⎜
⎟ ⎜⎜
⎝ α ⎠ ⎝ sin β ⎠
PHY4320 Chapter Two
82
When
⎧sin Nβ = 0
⎨sin β = 0
⎩
⎛ sin Nβ ⎞
⎜⎜
⎟⎟ = N 2
⎝ sin β ⎠
1
⎛
⎞
⎜ β = kd sin θ ⎟
2
⎝
⎠
2
β = mπ
sin θ = m
When
λ
d
⎧sin Nβ = 0
⎨sin β ≠ 0
⎩
k ⎞λ
⎛
sin θ = ⎜ m + ⎟
N⎠d
⎝
is maximum.
m = ±1,±2, L
⎛ sin Nβ ⎞
⎜⎜
⎟⎟ = 0
⎝ sin β ⎠
⎧m = ±1,±2, L
⎨k = 1,2, L, N − 1
⎩
2
N−1 zero points, N−2 maxima
PHY4320 Chapter Two
83
⎛ sin α ⎞
I grating (θ ) = C12 ⎜
⎟
⎝ α ⎠
2
⎛ sin Nβ
⎜⎜
⎝ sin β
1
⎧
⎪α = 2 ka sin θ
⎨
1
⎪β = kd sin θ
2
⎩
⎞
⎟⎟
⎠
The amplitude of the diffracted
beam is modulated by the
diffraction amplitude of a single
slit since the latter is spread
substantially than the former.
PHY4320 Chapter Two
84
2
The diffraction grating provides a means of
deflecting an incoming light by an amount that
depends on its wavelength. This is of great
use in spectroscopy.
In reality, any periodic variations in the
refractive index would serve as a
diffraction grating.
sin θ m − sin θ i = m
λ
d
m = ±1,±2, L
PHY4320 Chapter Two
sin θ m + sin θ i = m
λ
d
85
Reading Materials
S. O. Kasap, “Optoelectronics and Photonics: Principles and
Practices”, Prentice Hall, Upper Saddle River, NJ 07458, 2001,
Chapter 1, “Wave Nature of Light”.
PHY4320 Chapter Two
86