PHY4320 Chapter Two 1
Light waves in a homogeneous mediumPlane electromagnetic wavesMaxwell’s wave equation and diverging waves
Refractive indexGroup velocity and group indexSnell’s law and total internal reflection (TIR)Fresnel’s equationsGoos-Hänchen shift and optical tunnelingDiffraction
Fraunhofer diffractionDiffraction grating
Chapter Two: Wave Nature of LightChapter Two: Wave Nature of Light
PHY4320 Chapter Two 2
Plane Electromagnetic WavePlane 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)
propagation constant, or wave numberangular frequencyphase
Ex(z, t) = Re[ E0 exp(jφ0)expj(ωt − kz)]
The optical field generally refers to the electric field.
λπ /2=kπνω 2=
0φωφ +−= kzt
PHY4320 Chapter Two 3
Plane Electromagnetic WavePlane Electromagnetic Wave
A surface over which the phase of a wave is constant is referred to as a wavefront.
PHY4320 Chapter Two 4
Phase VelocityPhase Velocity
During a time interval δt, the wavefronts of contant phases move a distance δz. The phase velocity v is therefore
φ = ωt − kz + φ0 = constant
νλωδδ
===kt
zv
PHY4320 Chapter Two 5
Plane Electromagnetic WavePlane Electromagnetic Wave
E(r, t) = E0 cos(ωt − k⋅r + φ0)k⋅r = kxx + kyy + kzz
k is called the wave vector.
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.
PHY4320 Chapter Two 6
Plane WavesPlane Waves
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.
Ex = E0 cos(ωt − kz + φ0)
PHY4320 Chapter Two 7
MaxwellMaxwell’’s EM Wave Equations EM Wave Equation
2
2
002
2
2
2
2
2
tE
zE
yE
xE
r ∂∂
=∂∂
+∂∂
+∂∂ μεε
Ex = E0 cos(ωt − kz + φ0)For a perfect plane wave:
02
2
=∂∂
xE
02
2
=∂∂
yE
( ) ( )02
02
2
cos φω +−−−=∂∂ kztkE
zE
( )002 cos φω +−−= kztEk
Left side:
( )002
2
2
cos φωω +−−=∂∂ kztE
tE
Right side:
200
2 ωμεε rk =
00
1μεε
λνr
v ==
From EM:
PHY4320 Chapter Two 8
Diverging WavesDiverging Waves
E = (A/r) cos(ωt − kr)
2
2
002
2
2
2
2
2
tE
zE
yE
xE
r ∂∂
=∂∂
+∂∂
+∂∂ μεε
• 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 9
Gaussian BeamsGaussian 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.
Many laser beams can be described by Gaussian beams.
PHY4320 Chapter Two 10
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 2w0is 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.
Gaussian BeamsGaussian Beams
PHY4320 Chapter Two 11
Light waves in a homogeneous mediumPlane electromagnetic wavesMaxwell’s wave equation and diverging waves
Refractive indexGroup velocity and group indexSnell’s law and total internal reflection (TIR)Fresnel’s equationsGoos-Hänchen shift and optical tunnelingDiffraction
Fraunhofer diffractionDiffraction grating
Chapter Two: Wave Nature of LightChapter Two: Wave Nature of Light
PHY4320 Chapter Two 12
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.
Refractive IndexRefractive Index
00
1με
=c
In vacuum:
00
1μεε r
v =
The phase velocity in a medium is:
rvcn ε==
The refractive index of the medium is:
PHY4320 Chapter Two 13
Refractive IndexRefractive Index
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).
kmedium = nk
λmedium = λ/n
PHY4320 Chapter Two 14
Relative Permittivity and Refractive IndexRelative Permittivity and Refractive Index
Low frequency (LF) relative permittivity εr(LF) and refractive index n
Si 11.9 3.44 3.45 (at 2.15 μm) Electronic polarization up tooptical frequencies
Diamond 5.7 2.39 2.41 (at 590 nm) Electronic polarization up to UV light
GaAs 13.1 3.62 3.30 (at 5 μm) Ionic polarization contributes to εr(LF)
SiO2 3.84 2.00 1.46 (at 600 nm) Ionic polarization contributes to εr(LF)
Water 80 8.9 1.33 (at 600 nm) Dipolar polarization contributes toεr(LF), which is large
Material εr(LF) )(LFrε n (optical) Comment
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.
PHY4320 Chapter Two 15
Light waves in a homogeneous mediumPlane electromagnetic wavesMaxwell’s wave equation and diverging waves
Refractive indexGroup velocity and group indexSnell’s law and total internal reflection (TIR)Fresnel’s equationsGoos-Hänchen shift and optical tunnelingDiffraction
Fraunhofer diffractionDiffraction grating
Chapter Two: Wave Nature of LightChapter Two: Wave Nature of Light
PHY4320 Chapter Two 16
Group VelocityGroup Velocity
( ) ( ) ( )[ ]zkktEtzEx δδωω −−−= cos, 01,
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:
( ) ( ) ( )[ ]zkktEtzEx δδωω +−+= cos, 02,
( ) ( )⎥⎦⎤
⎢⎣⎡ +⎥⎦
⎤⎢⎣⎡ −=+ BABABA
21cos
21cos2coscos
Using
PHY4320 Chapter Two 17
We obtain:
( ) ( )[ ] ( )kztzktEEEE xxx −−=+= ωδδω coscos2 02,1,
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.
Group VelocityGroup Velocity
PHY4320 Chapter Two 18
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.
( ) ( )[ ] ( )kztzktEEx −−= ωδδω coscos2 0
nc
kv ==
ωPhase velocityor
dkdvgω
=
The maximum amplitude moves with a wavevector δk and a frequency δω.
Group velocityIn vacuum:
ck=ω
cdkdvg ==ω
group velocity
phase velocity
Group VelocityGroup Velocity
PHY4320 Chapter Two 19
In a medium with n as a function of λ:
Ng is the group index of the medium.g
g Nc
ddnn
cdkdv =
−==
λλ
ωλ
λddnnNg −=
λλ
ωπ
λω
ωλ
λω
ωω
ddn
nc
ddn
dd
ddn
ddn
−=−== 2
2nc
nc
ωπνλ 2/
==
( )ω
ωωω
ω ddnn
dnd
dcdk
+==
nck /=ω
cnk /ω=
PHY4320 Chapter Two 20
Group Index of the MediumGroup Index of the Medium
Pure SiO2
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.
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.
PHY4320 Chapter Two 21
Light waves in a homogeneous mediumPlane electromagnetic wavesMaxwell’s wave equation and diverging waves
Refractive indexGroup velocity and group indexSnell’s law and total internal reflection (TIR)Fresnel’s equationsGoos-Hänchen shift and optical tunnelingDiffraction
Fraunhofer diffractionDiffraction grating
Chapter Two: Wave Nature of LightChapter Two: Wave Nature of Light
PHY4320 Chapter Two 22
SnellSnell’’s Laws Law
Willebrord van Roijen Snell(1580-1626)
Incident light waves Ai and Bi are in phase. Reflected waves Ar and Br and refracted waves Atand Bt should also be in phase, respectively.
PHY4320 Chapter Two 23
ri
ri
tvtvAB
θθθθ
=
==sinsin
' 11
Reflection
Refraction
1
2
2
1
21
sinsin
sinsin'
nn
vv
tvtvAB
t
i
ti
==
==
θθ
θθ
PHY4320 Chapter Two 24
Total Internal ReflectionTotal Internal Reflection
1
2
sinsin
nn
t
i =θθ
RefractionWhen n1 > n2 and θt = 90°, the incidence angle is called the critical angle θc:
1
2sinnn
c =θ
PHY4320 Chapter Two 25
Light waves in a homogeneous mediumPlane electromagnetic wavesMaxwell’s wave equation and diverging waves
Refractive indexGroup velocity and group indexSnell’s law and total internal reflection (TIR)Fresnel’s equationsGoos-Hänchen shift and optical tunnelingDiffraction
Fraunhofer diffractionDiffraction grating
Chapter Two: Wave Nature of LightChapter Two: Wave Nature of Light
PHY4320 Chapter Two 26
FresnelFresnel’’s Equationss Equations
( )( )( )rk
rkrk
t
r
i
•−=•−=•−=
tjEEtjEEtjEE
tt
rr
ii
ωωω
expexpexp
0
0
0
Transverse electric field (TE) waves: Ei,⊥, Er,⊥, and Et,⊥.Transverse magnetic field (TM) waves: Ei,//, Er,//, and Et,//.
Augustin Fresnel(1788-1827)
The incidence plane is the plane containing the incident and reflected light rays.
Our goal is to find Er0 and Et0 relative to Ei0, including any phase changes.
PHY4320 Chapter Two 27
⎪⎭
⎪⎬⎫
=
=⎭⎬⎫
==
⎭⎬⎫
==
⊥
⊥
)2()1(
)2()1(
)/()/(
sinsin
tangentialtangential
tangentialtangential
//
//
21
BB
EE
EcnBEcnB
nn ti
ri
θθθθ Snell’s law
Maxwell’s equations
Boundary conditions
PHY4320 Chapter Two 28
( ) ( ) ( )
( ) ( ) ( ) ttrrii
ttrrii
ttrrii
tritri
tri
EcnEcnEcnBBB
EEEEcnEcnEcnBBB
EEE
θθθθθθ
θθθ
cos/cos/cos/coscoscos
coscoscos///
,2,1,1
//,//,//,
//,//,//,
//,2//,1//,1,,,
,,,
⊥⊥⊥
⊥⊥⊥
⊥⊥⊥
=−
⇒=−
−=−−
−=+−⇒−=+−
−=−−
Incidence plane
PHY4320 Chapter Two 29
⎪⎪⎩
⎪⎪⎨
⎧
++
=
++−
=
⇒
⎩⎨⎧
=+
−=−⇒
⎩⎨⎧
=−
−=−−
⊥⊥
⊥⊥
⊥⊥⊥
⊥⊥⊥
⊥⊥⊥
⊥⊥⊥
,12
11,
,12
12,
,1,2,1
,,,
,2,1,1
,,,
coscoscoscoscoscoscoscos
coscoscos
coscoscos
irt
rit
irt
itr
iittrr
itr
ttrrii
tri
EnnnnE
EnnnnE
EnEnEnEEE
EnEnEnEEE
θθθθθθθθ
θθθ
θθθ
⎪⎪⎩
⎪⎪⎨
⎧
−−
=
−−
=⇒
⎩⎨⎧
=+=+
bcadceafy
bcadbfdex
fdycxebyax
PHY4320 Chapter Two 30
⎪⎪⎩
⎪⎪⎨
⎧
−−−−
=
−−+−
=
⇒
⎩⎨⎧
−=−
=+⇒
⎩⎨⎧
−=−−
−=+−
//,21
11//,
//,21
21//,
//,//,//,
//,1//,2//,1
//,//,//,
//,2//,1//,1
coscoscoscoscoscoscoscos
coscoscos
coscoscos
irt
rit
irt
itr
iittrr
itr
ttrrii
tri
EnnnnE
EnnnnE
EEEEnEnEn
EEEEnEnEn
θθθθθθθθ
θθθ
θθθ
⎪⎪⎩
⎪⎪⎨
⎧
−−
=
−−
=⇒
⎩⎨⎧
=+=+
bcadceafy
bcadbfdex
fdycxebyax
PHY4320 Chapter Two 31
[ ][ ]
[ ]⎪⎪
⎩
⎪⎪
⎨
⎧
−+=
++
=
−+
−−=
++−
=
⊥⊥⊥
⊥⊥⊥
,2/122,12
11,
,2/122
2/122
,12
12,
sincoscos2
coscoscoscos
sincossincos
coscoscoscos
i
ii
ii
rt
rit
i
ii
iii
rt
itr
En
EnnnnE
EnnE
nnnnE
θθθ
θθθθ
θθθθ
θθθθ
[ ] [ ] 2/1222/1222
21
12
sinsincos
sinsinsinsin
/
itt
ti
ti
ri
nnnn
nnn
nnn
θθθ
θθθθ
θθ
−=−=⇒
=⇒⎪⎩
⎪⎨
⎧
===
PHY4320 Chapter Two 32
[ ] [ ] 2/1222/1222
21
12
sinsincos
sinsinsinsin
/
itt
ti
ti
ri
nnnn
nnn
nnn
θθθ
θθθθ
θθ
−=−=⇒
=⇒⎪⎩
⎪⎨
⎧
===
[ ][ ]
[ ]⎪⎪
⎩
⎪⎪
⎨
⎧
+−=
−−−−
=
+−
−−=
−−+−
=
//,22/122//,21
11//,
//,22/122
22/122
//,21
21//,
cossincos2
coscoscoscos
cossincossin
coscoscoscos
i
ii
ii
rt
rit
i
ii
iii
rt
itr
Enn
nEnnnnE
EnnnnE
nnnnE
θθθ
θθθθ
θθθθ
θθθθ
PHY4320 Chapter Two 33
[ ][ ]
[ ] ii
i
i
t
ii
ii
i
r
nnn
EE
t
nnnn
EE
r
θθθ
θθθθ
cossincos2
cossincossin
22/122//,0
//,0//
22/122
22/122
//,0
//,0//
+−==
+−
−−==
[ ][ ]
[ ] 2/122,0
,0
2/122
2/122
,0
,0
sincoscos2
sincossincos
ii
i
i
t
ii
ii
i
r
nEE
t
nn
EE
r
θθθ
θθθθ
−+==
−+
−−==
⊥
⊥⊥
⊥
⊥⊥
( )( )( )rk
rkrk
t
r
i
•−=•−=•−=
tjEEtjEEtjEE
tt
rr
ii
ωωω
expexpexp
0
0
0 r⊥ and r// are reflection coefficients. t⊥and t// are transmission coefficients.
Complex coefficients can only be obtained when n2 − sin2θi < 0.
n1 > n2 ,θi > θcPhase changes other than 0 or 180°occur only when there is TIR.1//// =+ ntr
⊥⊥ =+ tr 1
( )⊥⊥⊥ −= ,exp rjrr φ
( )//,//// exp rjrr φ−=
( )⊥⊥⊥ −= ,exp tjtt φ
( )//,//// exp tjtt φ−=
PHY4320 Chapter Two 341
2sinnn
c =θ
[ ][ ]
[ ][ ] ii
ii
ii
ii
nnnnr
nnr
θθθθ
θθθθ
cossincossin
sincossincos
22/122
22/122
//
2/122
2/122
+−
−−=
−+
−−=⊥
21
212
2
//
21
21
11
nnnn
nnnnr
nnnn
nnr
+−
=+−
=
+−
=+−
=⊥
When θi = 0 (normal incidence):
[ ]( )
1
2
422242
2
2
2
242222/122
tan
tantan1cossin
cos
cossincossin
nn
nnnnnnnn
p
ppp
p
p
pppp
=⇒
=−+⇒=−⇒
=−⇒=−
θ
θθθθ
θ
θθθθ
θp is called the polarization angle or Brewster’s angle.
When r// = 0 (reflected light is polarized):
PHY4320 Chapter Two 35
BrewsterBrewster’’s Angle (Polarization Angle)s Angle (Polarization Angle)n1 = 1.44, n2 = 1.00
David Brewster(1781-1868)
°=
°=
98.43
78.34
c
p
θ
θ
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 36
Example: calculate r⊥ and r// when n1 = 1.44, n2 = 1.00, and θi = 40°.
[ ][ ][ ][ ]
[ ][ ]
[ ][ ] 170.0
40cos694.040sin694.040cos694.040sin694.0
cossincossin
490.040sin694.040cos40sin694.040cos
sincossincos
694.044.100.1
22/122
22/122
22/122
22/122
//
2/122
2/122
2/122
2/122
1
2
−=°+°−
°−°−=
+−
−−=
=°−+°
°−−°=
−+
−−=
===
⊥
ii
ii
ii
ii
nnnnr
nnr
nnn
θθθθ
θθθθ
Solution:
PHY4320 Chapter Two 37
Example: calculate r⊥ and r// when n1 = 1.00, n2 = 1.44, and θi = 40°.
[ ][ ][ ][ ]
[ ][ ]
[ ][ ] 104.0
40cos44.140sin44.140cos44.140sin44.1
cossincossin
255.040sin44.140cos40sin44.140cos
sincossincos
44.100.144.1
22/122
22/122
22/122
22/122
//
2/122
2/122
2/122
2/122
1
2
−=°+°−
°−°−=
+−
−−=
−=°−+°
°−−°=
−+
−−=
===
⊥
ii
ii
ii
ii
nnnnr
nnr
nnn
θθθθ
θθθθ
Solution:
PHY4320 Chapter Two 38
n1 = 1.00, n2 = 1.44
°=
=
22.55
tan1
2
p
p nn
θ
θ
PHY4320 Chapter Two 39
[ ][ ]
[ ][ ] 2/1222
2/1222
//
2/122
2/122
sincossincos
sincossincos
njnnjnr
njnjr
ii
ii
ii
ii
−+
−+−=
−+
−−=⊥
θθθθ
θθθθ
( )( )
⎟⎠⎞
⎜⎝⎛=
+=
−=
−=
−
xy
yxz
jzzjyxz
1
2/122
tan
exp
φ
φ
( )( )
( )[ ]212
1
22
11
2
1
exp
expexp
φφ
φφ
−−=
−−
==
jzz
jzjz
zzz
What are the phase changes at TIR (n1 > n2 and θi > θc)?
[ ][ ]
[ ][ ] ii
ii
ii
ii
nnnnr
nnr
θθθθ
θθθθ
cossincossin
sincossincos
22/122
22/122
//
2/122
2/122
+−
−−=
−+
−−=⊥
1
1
// =
=⊥
r
r
PHY4320 Chapter Two 40
[ ][ ] 2/122
2/122
sincossincos
njnjr
ii
ii
−+
−−=⊥
θθθθ [ ]
[ ]⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
−
−
i
i
i
i
n
n
θθφ
θθφ
cossintan
cossintan
2/1221
2
2/1221
1
[ ]
[ ]i
i
i
i
n
n
θθφ
θθφφφ
cossin
21tan
cossintan2
2/122
2/1221
21
−=⎟
⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=−=
⊥
−⊥
PHY4320 Chapter Two 41
[ ][ ] 2/1222
2/1222
//sincossincos
njnnjnr
ii
ii
−+
−+−=
θθθθ
[ ]
[ ]⎟⎟⎠
⎞⎜⎜⎝
⎛ −−=
−⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
−
−
i
i
i
i
nn
nn
θθφ
πθ
θφ
cossintan
cossintan
2
2/1221
2
2
2/1221
1
[ ]
[ ]
[ ]i
i
i
i
i
i
nn
nn
nn
θθπφ
θθπφ
πθ
θφφφ
cossin
21
21tan
cossintan2
cossintan2
2
2/122
//
2
2/1221
//
2
2/1221
21//
−=⎟
⎠⎞
⎜⎝⎛ +
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=+
−⎟⎟⎠
⎞⎜⎜⎝
⎛ −=−=
−
−
PHY4320 Chapter Two 42
Consider now: ( )rkt ⋅−= tjEE tt ωexp0
[ ]2/1
22
2
12
2/122
12
sin12sin12cos
sinsin2sin2sin
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−=−==
=====
itttty
iziiittttz
nnnnkk
kknnkk
θλπθ
λπθ
θθλπθ
λπθ
( )zkyktjE tzt −−= ty0 exp ω
PHY4320 Chapter Two 43
Evanescent WaveEvanescent Wave
1sinsin 22
2
122
2
1 =⎟⎟⎠
⎞⎜⎜⎝
⎛>⎟⎟
⎠
⎞⎜⎜⎝
⎛⇒> cici n
nnn θθθθ
2
2/1
22
2
12 1sin2 αθ
λπ j
nnnjk ity −=
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
At TIR:
Why do we put a minus here?
PHY4320 Chapter Two 44
Evanescent Wave2/1
22
2
122 1sin2
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛= in
nn θλπα
the penetration depth[ ] 2/122
221
2
sin2
1 −−== nn iθπ
λα
δ
( ) ( )zkyktjEtzyE tztt −−= ty0 exp,, ω
( ) ( )zktjykjE iztyt −−= ωexpexp0
( ) ( )zktjyE izt −−= ωα expexp 20
( )[ ] nm17600.150sin44.12
5.514 2/1222 =−=−o
πδ
λ = 514.5 nm (green light)n1 = 1.44 (glass)n2 = 1.00 (air)θi = 50° (>θc = 43.98°)
PHY4320 Chapter Two 45
Total Internal Reflection Fluorescence (TIRF) MicroscopyTotal Internal Reflection Fluorescence (TIRF) Microscopy
Carl Zeiss
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 fluorophoreslocated 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.
PHY4320 Chapter Two 46
PHY4320 Chapter Two 47
θi dp48.7° 300 nm49.7° 145 nm51.1° 100 nm53.9° 70 nm57.8° 55 nm66.5° 40 nm
[ ] 2/122
221
/0
sin4
−
−
−=
=
nnd
eII
ip
dzz
p
θπλ
PHY4320 Chapter Two 48
Evanescent nanometry
Achieves a resolution of ~ 1 nm.Requires highly photostable fluorescent materials.
Fluorescence intensity
Distance
PHY4320 Chapter Two 49
Intensity, Reflectance, and TransmittanceIntensity, Reflectance, and Transmittance
20
202
1 nEEvI or ∝= εεLight intensity
22
,0
2,0
⊥
⊥
⊥⊥ == r
E
ER
i
r 2//2
//,0
2//,0
// rE
ER
i
r ==
Reflectance
2//
1
22
//,01
2//,02
// tnn
En
EnT
i
t⎟⎟⎠
⎞⎜⎜⎝
⎛==
2
1
22
,01
2,02
⊥
⊥
⊥⊥ ⎟⎟
⎠
⎞⎜⎜⎝
⎛== t
nn
En
EnT
i
t
Transmittance
PHY4320 Chapter Two 50
[ ][ ] 2/122
2/122
,0
,0
sincossincos
ii
ii
i
r
nn
EE
rθθθθ
−+
−−==
⊥
⊥⊥
[ ][ ] ii
ii
i
r
nnnn
EE
rθθθθ
cossincossin
22/122
22/122
//,0
//,0//
+−
−−==
[ ] 2/122,0
,0
sincoscos2
ii
i
i
t
nEE
tθθ
θ−+
==⊥
⊥⊥
[ ] ii
i
i
t
nnn
EE
tθθ
θcossin
cos222/122
//,0
//,0//
+−==
2
21
21// ⎟⎟
⎠
⎞⎜⎜⎝
⎛+−
=== ⊥ nnnnRRR
When θi = 0 (normal incidence):
( )221
21//
4nnnnTTT
+=== ⊥
21
21
11
nnnn
nn
+−
=+−
=
21
212
2
nnnn
nnnn
+−
=+−
=
21
121
2nn
nn +=
+=
21
12
22nn
nnnn
+=
+=
PHY4320 Chapter Two 51
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°?
°=⇒== 47.80450.1/430.1/sin 12 cc nn θθSolution:
[ ]°=⇒=
°⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−°
=−
=⎟⎠⎞
⎜⎝⎛
⊥⊥ 45.116614.185cos
450.1430.185sin
cossin
21tan
2/122
2/122
φθ
θφi
i n
[ ]°−=⇒=
°⎟⎠⎞
⎜⎝⎛
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−°
=−
=⎟⎠⎞
⎜⎝⎛ + 13.62660.1
85cos450.1430.1
450.1430.185sin
cossin
21
21tan //2
2/122
2
2/122
// φθ
θπφi
i
nn
[ ] [ ] mmnn i μπμθ
πλδ 780.0430.185sin450.1
21sin
22/12222/12
222
1 =−°=−=−−
θi = 90°
φ⊥ = 180°
φ// = 0°
δ = 0.663μm
PHY4320 Chapter Two 52
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?
( ) ( )⎪⎪
⎩
⎪⎪
⎨
⎧
=+××
=+
=
=⎟⎠⎞
⎜⎝⎛
+−
=⎟⎟⎠
⎞⎜⎜⎝
⎛+−
=⇒
⎩⎨⎧
==
96.05.10.1
5.10.144
04.05.10.15.10.1
5.10.1
2221
21
22
21
21
2
1
nnnnT
nnnnR
nn
Solution:
°=⇒== 31.560.15.1tan
1
2pp n
n θθ
( ) ( )⎪⎪
⎩
⎪⎪
⎨
⎧
=+××
=+
=
=⎟⎠⎞
⎜⎝⎛
+−
=⎟⎟⎠
⎞⎜⎜⎝
⎛+−
=⇒
⎩⎨⎧
==
96.00.15.1
0.15.144
04.00.15.10.15.1
0.15.1
2221
21
22
21
21
2
1
nnnnT
nnnnR
nn
Light will lose 4% of its intensity whenever it travels through an interface between air and glass.
PHY4320 Chapter Two 53
Antireflection CoatingsAntireflection CoatingsCurrent commercial solar cells are made of silicon (n ≈ 3.5 at wavelengths of 700–800 nm).
( )( )
( )( )
31.05.30.15.30.1
2
2
221
221 =
+−
=+−
=nnnnR Air Coating (Si3N4) Si (solar cell)
1.0 1.9 3.5
( )
⎟⎟⎠
⎞⎜⎜⎝
⎛=⇒
===Δ
2
2
4
222
nmd
mdndkcc
λ
πλπφ
r12,⊥ = r12,// = −0.31r23,⊥ = r23,// = −0.30
r12 = r23 in order to obtain efficient destructive interference.
31232
32
21
21 nnnnnnn
nnnn
=⇒+−
=+−
nmnmn
d 929.14
7004 2
=×
==λ
87.15.30.1312 =×== nnn
PHY4320 Chapter Two 54
Dielectric MirrorsDielectric Mirrors
0
0
12
1221
21
2112
21
>+−
=
<+−
=
<
nnnnr
nnnnr
nn
02
02
12
221
21
112
>+
=
>+
=
nnnt
nnnt
We can find that A, B, and C waves are in phase with each other. They interfere with each other constructively.The total reflectance will be close to unity.
PHY4320 Chapter Two 55
Photonic CrystalsPhotonic Crystals1-D
periodic in one direction
2-D
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 56
Defects in Photonic CrystalsDefects in Photonic Crystals
cavity waveguide
PHY4320 Chapter Two 57
LowLow--PumpingPumping--Current LasersCurrent Lasers
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.
SEM image
PHY4320 Chapter Two 58
LowLow--PumpingPumping--Current LasersCurrent Lasers
radii of air holes:0.28a (I)0.35a (II)
0.385a (III)0.4a (IV)0.41a (V)
electric field intensity profileDesigned according to FDTD (finite difference time domain) calculations.
PHY4320 Chapter Two 59
LowLow--PumpingPumping--Current LasersCurrent 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 60
Wavelength Channel Drop DevicesWavelength Channel Drop Devices
B.-S. Song, S. Noda, T. Asano, Science 2003, 300, 1537.
PHY4320 Chapter Two 61
Light waves in a homogeneous mediumPlane electromagnetic wavesMaxwell’s wave equation and diverging waves
Refractive indexGroup velocity and group indexSnell’s law and total internal reflection (TIR)Fresnel’s equationsGoos-Hänchen shift and optical tunnelingDiffraction
Fraunhofer diffractionDiffraction grating
Chapter Two: Wave Nature of LightChapter Two: Wave Nature of Light
PHY4320 Chapter Two 62
GoosGoos--HHäänchennchen ShiftShift
λ = 1 μmn1 = 1.45n2 = 1.43θi = 85°δ = 0.78 μmΔz ≈ 18 μm
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.
iz θδ tan2=Δ
PHY4320 Chapter Two 63
Optical TunnelingOptical 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 64
Beam Splitters Based on Frustrated TIRBeam 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 65
Light waves in a homogeneous mediumPlane electromagnetic wavesMaxwell’s wave equation and diverging waves
Refractive indexGroup velocity and group indexSnell’s law and total internal reflection (TIR)Fresnel’s equationsGoos-Hänchen shift and optical tunnelingDiffraction
Fraunhofer diffractionDiffraction grating
Chapter Two: Wave Nature of LightChapter Two: Wave Nature of Light
PHY4320 Chapter Two 66
DiffractionDiffractionDiffraction 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 67
DiffractionDiffractionDiffraction 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 68
Diffraction Pattern through a 1D SlitDiffraction Pattern through a 1D SlitThe 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.
( ) ( )θδδ sinexp jkyyE −∝
( ) ( )∫ −=a
jkyyCE0
sinexp θδθ
PHY4320 Chapter Two 69
Let x = y − a/2, we obtain
( ) ( )∫ −=a
jkyyCE0
sinexp θδθ
( )[ ]
∫∫
−
−−−
=
+−=2/
2/
sinsin21
2/
2/sin2/exp
a
a
jkxkaj
a
a
dxeCe
dxaxjkC
θθ
θ
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
−
−
−∫θθθ
θsin
2sin
22/
2/
sin
sin1 ajkajka
a
jkx eejk
dxe
θ
θ
θ
θ
sin21
sin21sin
sin
sin21sin2
ka
kaa
jk
kaj ⎟⎠⎞
⎜⎝⎛
=−
⎟⎠⎞
⎜⎝⎛−
=
PHY4320 Chapter Two 70
( ) ( )
2
2
sin21
sin21sin
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛
==θ
θθθ
ka
kaCaEI
( )θ
θθ
θ
sin21
sin21sin
sin21
ka
kaaCeE
kaj⎟⎠⎞
⎜⎝⎛
=
−
Bright and dark regionsThe central bright region is larger than the slit width (the transmitted beam is diverging)
The zero intensity occurs when:
πθλπθ maka == sinsin
21
amλθ =sin
L,2,1 ±±=m
λ = 1.3 μma = 100 μmdivergence 2θ = 1.5°
PHY4320 Chapter Two 71
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
Dλθ 22.1sin =
Why is the central bright region rotated relative to the rectangular aperture?
Diffraction from rectangular apertures
PHY4320 Chapter Two 72
DiffractionDiffraction--Limited ResolutionLimited ResolutionConsider two neighboring coherent sources with an angular separation of Δθexamined 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.
( )Dλθ 22.1sin min =Δ
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:
PHY4320 Chapter Two 73
DiffractionDiffraction--Limited ResolutionLimited Resolution
Objectives are key components in an optical microscope.
PHY4320 Chapter Two 74
DiffractionDiffraction--Limited ResolutionLimited Resolution
( )αsinNA ⋅= nThe 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 75
DiffractionDiffraction--Limited ResolutionLimited Resolution
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.
NA61.0r λ
=resolution
PHY4320 Chapter Two 76
Overcome DiffractionOvercome Diffraction--Limited ResolutionLimited ResolutionIf 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 77
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.
Overcome DiffractionOvercome Diffraction--Limited ResolutionLimited Resolution
PHY4320 Chapter Two 78
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.
Overcome DiffractionOvercome Diffraction--Limited ResolutionLimited Resolution
PHY4320 Chapter Two 79
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.
Overcome DiffractionOvercome Diffraction--Limited ResolutionLimited Resolution
PHY4320 Chapter Two 80
Diffraction GratingDiffraction GratingA 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.
dm λθ =sin
Bragg diffraction condition(maximum intensity):
amλθ =sin L,2,1 ±±=m
Zero diffraction intensity condition for a single slit:
L,2,1 ±±=m
PHY4320 Chapter Two 81
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:
( )θ
θθ
θ
sin21
sin21sin
sin21
slit
ka
kaaCeE
kaj⎟⎠⎞
⎜⎝⎛
=
−
ααα sin1
jeC −
= ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
=
θα sin21
1
ka
CaC
For the diffraction from the grating:
( ) ( ) ( ) ( ) ( ) ( ) θθθ θθθθθ sin1slit
sin2slit
sinslitslitgrating
kdNjkdjjkd eEeEeEEE −−−− ++++= L
( ) ( )[ ]θθθθ sin1sin2sinslit 1 kdNjkdjjkd eeeE −−−− ++++= L
A geometrical progression
PHY4320 Chapter Two 82
( ) ( ) θ
θ
θθ sin
sin
slitgrating 11
jkd
jNkd
eeEE −
−
−−
=
( ) ( )2
sin
sin22
1
2
gratinggrating 11sin
θ
θ
ααθθ jkd
jNkd
eeCEI −
−
−−
⎟⎠⎞
⎜⎝⎛==
( ) ( )( ) ( )
( )( )θ
θθθ
θθθ
θ
sincos22sincos22
sinsinsincos1
sinsinsincos111
2
22
sin
sin
kdNkd
kdjkd
NkdjNkdee
jkd
jNkd
−−
=+−
+−=
−−
−
−
( )( )
2
2
2
sinsin
sin21sin2
sin21sin2
sincos1sincos1
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
=−−
=ββ
θ
θ
θθ N
kd
Nkd
kdNkd
⎟⎠⎞
⎜⎝⎛ = θβ sin
21 kd
( )22
21grating sin
sinsin⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=
ββ
ααθ NCI
PHY4320 Chapter Two 83
⎟⎠⎞
⎜⎝⎛ = θβ sin
21 kd
⎩⎨⎧
==0sin
0sinββNWhen
is maximum.22
sinsin NN
=⎟⎟⎠
⎞⎜⎜⎝
⎛ββ
dm
mλθ
πβ
=
=
sinL,2,1 ±±=m
⎩⎨⎧
≠=0sin
0sinββNWhen
0sin
sin2
=⎟⎟⎠
⎞⎜⎜⎝
⎛ββN
dNkm λθ ⎟⎠⎞
⎜⎝⎛ +=sin
⎩⎨⎧
−=±±=
1,,2,1,2,1Nk
mL
L
N−1 zero points, N−2 maxima
PHY4320 Chapter Two 84
( )22
21grating sin
sinsin⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=
ββ
ααθ NCI
⎪⎩
⎪⎨
⎧
=
=
θβ
θα
sin21
sin21
kd
ka
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 85
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.
dmimλθθ =− sinsin
dmimλθθ =+ sinsinL,2,1 ±±=m
PHY4320 Chapter Two 86
Reading MaterialsReading Materials
S. O. Kasap, “Optoelectronics and Photonics: Principles and Practices”, Prentice Hall, Upper Saddle River, NJ 07458, 2001, Chapter 1, “Wave Nature of Light”.