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EE16.468/16.568 Lecture 1
Class overview:
1. Brief review of physical optics, wave propagation, interference, diffraction, and polarization
2. Introduction to Integrated optics and integrated electrical circuits
3. Guide-wave optics: 2D and 3D optical waveguide, optical fiber, modedispersion, group velocity and group velocity dispersion.
4. Mode-coupling theory, Mach-zehnder interferometer, Directional coupler, taps and WDM coupler.
5. Electro-optics, index tensor, electro-optic effect in crystal, electro-optic coefficient
6. Electro-optical modulators
7. Passive and active optical waveguide devices, Fiber Optical amplifiers and semiconductor optical amplifiers, Photonic switches and all optical switches
8. Opto-electronic integrated circuits (OEIC)
EE16.468/16.568 Lecture 1
Complex numbers
ibac iCeor
a
b
C: amplitude : angle
C
Real numbers do not have phases
Complex numbers have phases
)()( 212121 bbiaacc
iCe is the phase of the complex number
EE16.468/16.568 Lecture 1
)(212121
2121 iii eCCeCeCcc
Physical meaning of multiplication of two complex numbers
11
ieC 22
ieC21 CC Number:
Phase:21
2C times
Phase delay 2
Why ?1ii
21iei
)()22
(11
iieeii
1
i
90
0180
270
-i
-1
ii 3 14 i
i 1 1;14 i
EE16.468/16.568 Lecture 1
)( kxiAe
?16
1
90
0180
270
-1
5...1,0,1)
6
2(
6 neni
11...1,0,22)
12
2(
1212 neni
Optical wave, frequency domain
xk,
Vertical polarization,
Horizontal polarizationInitial phase
Phase delay by position
Amplitude, I=|A|2, Optical intensity
k vector, to x-direction
k polarization, transwave
EE16.468/16.568 Lecture 1
11
ieC 2ie
Optical wave propagation in air (free-space)
k, )( kxiAe )( delaykxiAe
xxkdelay
2Phase delay
Time and frequency domain of optical signal
)( kxiAe
EE16.468/16.568 Lecture 1
Phase distortion in atmosphere
Optical wave propagation in air (free-space)
xxk
2
Phase is the same on the plane --- plane wave
)( kxiAek
∆x
)( kxiAek
∆x
Wave front
Additional phase delay, non ideal plane wave
Wave front distortion
EE16.468/16.568 Lecture 1
Optical wave to different directions
)( 1 kxiAe
)( 2 kxiAe
1x
2x
)(2
21xxxk
delay
Phase delay
Refractive index n
2
k
In vacuum,
In media, like glass,
fT
cV c: speed of light
nn
k
2
/
2 nf
T
nncV /
//
EE16.468/16.568 Lecture 1
Optical wave to different directions
)( 1 kxiAe
)( 2 kxiAe
1x
2x
)(2
21xxxk
delay
Phase delay
In media with refractive index n
)( 1 knxiAe
)( 2 knxiAe
1nx
2nx
)(2
21xxnxkn
delay
Phase delay
n
nx1, nx2, optical path
EE16.468/16.568 Lecture 1
Interference of two optical waves
a1a2
a3
b1 b2
50%
)( 321
2
1 aaaiknAe
)( 21
2
1 bbiknAe
A Aout
Aout =)( 321
2
1 aaaiknAe + )( 21
2
1 bbiknAe
Upper arm :
Lower arm :
Phase delay between the two arms:
)(32121aaabbkn
If phase delay is 2m, then:)( 321
2
1 aaaiknAe = )( 21
2
1 bbiknAe
If phase delay is 2m +, then: )( 321
2
1 aaaiknAe = - )( 21
2
1 bbiknAe
EE16.468/16.568 Lecture 1March-Zehnder interferometer
a1a2
a3
b1b2
50%
)( 321
2
1 aaaiknAe
)( 21
2
1 bbiknAe
A Aout
Upper arm :
Lower arm :
321321aaabbb
If phase delay is 2m, then:)( 321
2
1 aaaiknAe = )( 21
2
1 bbiknAe
If phase delay is 2m +, then: )( 321
2
1 aaaiknAe = - )( 21
2
1 bbiknAe
b3n2
n
Phase delay between the two arms:
)(222
nabnk
Electro-optic effect, n2 changes with E -field
EE16.468/16.568 Lecture 1
Electro-optic modulator based on March-Zehnder interferometer
If phase delay is 2m, then:)( 321
2
1 aaaiknAe
=
)( 21
2
1 bbiknAe
If phase delay is 2m +, then: )( 321
2
1 aaaiknAe = -
)( 21
2
1 bbiknAe
Phase delay between the two arms:
)(222
nabnk
Electro-optic effect, n2 changes with E -fielda1a2
a3
b1b2
50%A Aout
b3n2
n
EE16.468/16.568 Lecture 1
Reflection, refraction of light
n1
n2
1 1
2
2211sinsin nn
reflection
Refraction
n2 > n1 2 < 1
n1
n2
1 1
2
2211sinsin nn
reflection
Refraction
n2 < n1 2 > 1
EE16.468/16.568 Lecture 1
Total internal reflection (TIR)
n1
n2
1 1
2
2211sinsin nn
reflection
Refraction
n2 < n1 2 > 1
,|sinsin209022211nnn
When 2 = 900, 1 is critical angle
121/sin nn
EE16.468/16.568 Lecture 1
Application – optical fiber
n1
n2
1 1
2 = 900
Total reflection
,|sinsin209022211nnn
When 2 = 900, 1 is critical angle
121/sin nnNo refraction
Total internal reflection (TIR)
n2
n1
n1
Low loss, TIR
Flexible
Communication system
Laser EO modulator
Electrical signal
Optical fiber
Detector Electrical signal
EE16.468/16.568 Lecture 1
n1
n2
Intensity reflection = [(n1-n2)/(n1+n2)]2
Reflection percentage
Example 1:
Refractive index of glass n = 1.5;
Refractive index of air n = 1;
The reflection from glass surface is 4%;
Example 2: detector is usually made from Gallium Arsenide (GaAs), n = 3.5;
n1
N2 = 3.5 Detector, GaAs
Intensity reflection = [(n1-n2)/(n1+n2)]2
The reflection from GaAs surface is 31%;
EE16.468/16.568 Lecture 1
Interference of two optical beams, applications
1, Mach-Zehnder interferometer
Electro-optic effect, n2 changes with E -fielda1a2
a3
b1b2
50%A Aout
b3n2
n
EE16.468/16.568 Lecture 1
2, Michelson interferometer
d1 d2Beam splitter
Detector
)22(1122dndnk
n1 n2
Phase difference:M1
Mirror 2Measuring small moving or displacement
If the detected light changes from bright to dark,Then the distance moved is half wavelength/2
Michelson interferometer measuring speed of light in ether, speed of light not depends on direction
d1d2
Beam
splitter
Detector
n2n1
M1
Mirror 2
)22(1221dnbnk
Phase difference:
EE16.468/16.568 Lecture 1
3, Measuring surface flatness
n2
)(2dnk )'(
2dnk
n2
4, Measuring refractive index of liquid or gas
Accuracy: 0.1 wavelength = 0.1*500nm = 50nm
Gas in
EE16.468/16.568 Lecture 1Wave optics
0
Dielectric materials
Maxwell equations:
t
BE
D
t
DJH
0 B
ED
0J HB
t
BE
0 D
t
DH
0 B
Maxwell equations in dielectric materials:
BjE
0 D
DjH
0 B
phasor
)()( BjE
EEE
22)(
EE16.468/16.568 Lecture 1Wave optics approach
022 EE
Helmholtz Equation:
022 EkE
22 kFree-space solutions
ikzeEyE 0ˆ
ikzeExE 0ˆ
xxk
2
Phase is the same on the plane --- plane wave
)( kxiAek
∆xWave front
0220
2 EnkE
EE16.468/16.568 Lecture 1Wave optics
2-D Optical waveguide
n2 < n1
Core, refractive index n1
Cladding, n2
Cladding, n2
y
x
z
d
)(ˆ xEyE
TE mode:
022 EE
Helmholtz Equation:
022 EkE
22 k
)(ˆ xHyH TM mode:
0220
2 EnkE
EE16.468/16.568 Lecture 1Normal reflection at material interface
Helmholtz Equation:
0220
2 EnkE
x
z
n1
n2
Ein
Et
Er
trin EEE
trin EEE
inE tr 1
t
BE
HjE
jE
H
00
ˆˆˆ
xEzyx
zyx
E
)0(ˆˆ)0(ˆ
)(ˆˆ)0(ˆ
zz
Eyx
y
Ez
z
Eyx
x
xx
znikr
znikin eExeExE 1010 ˆˆ1
znikteExE 20ˆ2
H
EE16.468/16.568 Lecture 1Normal reflection at material interface
Helmholtz Equation:
0220
2 EnkE
x
z
n1
n2
Ein
Et
Er
trin EEE
tr 1
z
Ex
znikr
znikin eExeExE 1010 ˆˆ1
znikteExE 20ˆ2
H
Continuous @ z=0 Continuous @ z=0
Why?znik
rznik
in eExeExE 1010 ˆˆ1
znikteExE 20ˆ2
)(ˆ 10101010
1 znikr
znikin eEnikeEnikx
z
E
znikteEnikx
z
E20
202 ˆ
EE16.468/16.568 Lecture 1Normal reflection at material interface
Helmholtz Equation:
0220
2 EnkE
x n1
n2
EinEr
trin EEE
tr 1
znikr
znikin eExeExE 1010 ˆˆ1
znikteExE 20ˆ2
)(ˆ 10101010
1 znikr
znikin eEnikeEnikx
z
E
znikteEnikx
z
E20
202 ˆ
Z = 0
Z = 0
)(ˆ 1010 rin EnikEnikx
tEnikx 20ˆ
EE16.468/16.568 Lecture 1Normal reflection at material interface
Helmholtz Equation:
0220
2 EnkE
x n1
n2
EinEr
trin EEE
tr 1
znikr
znikin eExeExE 1010 ˆˆ1
znikteExE 20ˆ2
trin EnikxEnikEnikx 201010 ˆ)(ˆ
tnrnn 211
tr 1
tnrnn 111 21
12
nn
nt
21
21
nn
nnr
EE16.468/16.568 Lecture 1
Reflection, refraction of light
n1
n2
1 1
2
x
z
y
Ein
Er
Et
znkxnkiin
rnkiinin eEyeEyE 11011010 cossinˆ ˆˆ
znkxnkirr eEyE 110110 cossinˆ
znkxnkitt eEyE 220220 cossinˆ
0| ztrin EEE
0
cossincossincossin |220220110110110110
zznkxnki
tznkxnki
rznkxnki
in eEeEeE
xnkit
xnkir
xnkiin eEeEeE 220110110 sinsinsin
2211sinsin nn
trin EEE
EE16.468/16.568 Lecture 1
Reflection, refraction of light
n1
n2
1 1
2
x
z
y
Ein
Er
Et
t
BE
HjE
jE
H
00
ˆˆˆ
yEzyx
zyx
E
)0(ˆ)0(ˆˆ
)(ˆ)0(ˆˆ
zyz
Ex
x
Ezy
z
Ex
y
yy
z
Ey
H
Continuous @ z=0 Continuous @ z=0
Why?
EE16.468/16.568 Lecture 1
Reflection, refraction of light
n1
n2
1 1
2
x
z
y
Ein
Er
Et
znkxnkiin
rnkiinin eEyeEyE 11011010 cossinˆ ˆˆ
znkxnkirr eEyE 110110 cossinˆ
znkxnkitt eEyE 220220 cossinˆ
z
Ey
Continuous @ z=0
rrin EnkEnkEnk 220110110 coscoscos
trin EnkEnkEnk 110110110 coscoscos
trin EEE
tin EnknkEnk 220110110 coscoscos2
int Enknk
nkE
220110
110
coscos
cos2
EE16.468/16.568 Lecture 1
Reflection, refraction of light
n1
n2
1 1
2
x
z
y
Ein
Er
Et
znkxnkiin
rnkiinin eEyeEyE 11011010 cossinˆ ˆˆ
znkxnkirr eEyE 110110 cossinˆ
znkxnkitt eEyE 220220 cossinˆ
z
Ey
Continuous @ z=0
rrin EnkEnkEnk 220110110 coscoscos
0coscoscoscos 110220110220 rin EnknkEnknk
trin EEE
trin EnkEnkEnk 220220220 coscoscos
inr Enknk
nknkE
220110
220110
coscos
coscos
EE16.468/16.568 Lecture 1
Reflection, refraction of light
n1
n2
1 1
2
x
z
y
Ein
Er
Et
2211
2211
coscos
coscos
nn
nnr
2211sinsin nn
2211
11
coscos
cos2
nn
nt
rt
11
EE16.468/16.568 Lecture 1
n1
n2
1
1
2
x
z
y
Ein
Er
Et
znkxnkiininin eEzExE 110110 cossin
11 sinˆcosˆ
2211sinsin nn
211 coscoscos trin EEE
1
2
1 znkxnkirrr eEzExE 110110 cossin
11 sinˆcosˆ
znkxnkittt eEzExE 220220 cossin
22 sinˆcosˆ
xnkit
xnkir
xnkiin eEeEeE 220110110 sin
2sin
1sin
1 coscoscos
EE16.468/16.568 Lecture 1
n1
n2
1
1
2
x
z
y
Ein
Er
Et
znkxnkiininin eEzExE 110110 cossin
11 sinˆcosˆ
1
2
1 znkxnkirrr eEzExE 110110 cossin
11 sinˆcosˆ
znkxnkittt eEzExE 220220 cossin
22 sinˆcosˆ
t
BE
zx EEzyx
zyx
E
0
ˆˆˆ
)0(ˆ)(ˆ0ˆ
)(ˆ)(ˆˆ
zx
E
z
Eyx
y
Ez
x
E
z
Ey
y
Ex
zx
xzxz
x
E
z
E zx
continuous
EE16.468/16.568 Lecture 1
n1
n2
1
1
2
x
z
y
Ein
Er
Et
znkxnkiininin eEzExE 110110 cossin
11 sinˆcosˆ
1
2
1 znkxnkirrr eEzExE 110110 cossin
11 sinˆcosˆ
znkxnkittt eEzExE 220220 cossin
22 sinˆcosˆ
x
E
z
E zx
continuous
xnkir
xnkiin
xnkir
xnkiin
enkEenkE
enkEenkE110110
110110
sin1101
sin1101
sin1101
sin1101
sinsinsinsin
coscoscoscos
xnkit
xnkit enkEenkE 220220 sin
2202sin
2202 sinsincoscos =
201010 nkEnkEnkE rrin
EE16.468/16.568 Lecture 1
n1
n2
1
1
2
x
z
y
Ein
Er
Et
znkxnkiininin eEzExE 110110 cossin
11 sinˆcosˆ
1
2
1 znkxnkirrr eEzExE 110110 cossin
11 sinˆcosˆ
znkxnkittt eEzExE 220220 cossin
22 sinˆcosˆ
201010 nkEnkEnkE trin
211 coscoscos trin EEE
121111 coscoscos nEnEnE trin
211111 coscoscos trin EnEnnE
1221
11
coscos
cos2
nn
nt
EE16.468/16.568 Lecture 1
n1
n2
1
1
2
x
z
y
Ein
Er
Et
znkxnkiininin eEzExE 110110 cossin
11 sinˆcosˆ
1
2
1 znkxnkirrr eEzExE 110110 cossin
11 sinˆcosˆ
znkxnkittt eEzExE 220220 cossin
22 sinˆcosˆ
222121 coscoscos nEnEnE trin
211 coscoscos trin EEE
201010 nkEnkEnkE trin
221212 coscoscos nEnEnE trin
0coscoscoscos 21122112 nnEnnE rin
2112
1221
coscos
coscos
nn
nnr
1221
11
coscos
cos2
nn
nt
EE16.468/16.568 Lecture 1
n1
n2
1
1
2
x
z
y
Ein
Er
Et
1
2
1
1221
11
coscos
cos2
nn
nt
2112
1221
coscos
coscos
nn
nnr
)tan(
)tan(
coscos
coscos
12
12
2112
1221
nn
nnr
Brewster’s angle n1
n2
1
1
2
Ein
Er
Et
1
http://buphy.bu.edu/~duffy/semester2/c27_brewster.html
EE16.468/16.568 Lecture 1
n1
n2
1
1
2
x
z
y
Ein
Er
Et
1
2
1 2112
1221
coscos
coscos
nn
nnr
1221
11
coscos
cos2
nn
nt
EE16.468/16.568 Lecture 1Normal reflection at material interface
Helmholtz Equation:
0220
2 EnkE
x n1
n2
EinEr
znikr
znikin eExeExE 1010 ˆˆ1
znikteExE 20ˆ2
21
12
nn
nt
21
21
nn
nnr
x n1
EinEr
n2
n2n1
rr
t
t
tr
t
t
t
r2t
r4t
rt
r3t
r5t
EE16.468/16.568 Lecture 1
Equal difference series
......, 21 naaa
baa 12
bnaan )1(1
baa 23
baa nn 1+
=
nn aaaS ...21
11 ...aaaS nnn +
= )(2 1aanS nn
nnn aaabaaa 11112
nnn aaabaaa 12123 2
1111 )1(2
)(2
)(2
nabnn
naaan
aan
S nnn
EE16.468/16.568 Lecture 1
Power series......, 21 naaa
12 baa
23 baa
nn aaaS ...21
13221 ...... nnn aaababababS-
=11)1( nn aaSb
b
aba
b
aaS
nn
n
111111
1 nn baa…
11aba n
n
x
EE16.468/16.568 Lecture 1
Power series......, 21 naaa
b
a
b
ba
aaaSn
n
n
11
)1(lim
......
11
21
When |b|<1
b can be anything, number, variable, complex number or function
Optical cavity, multiple reflections
n2n1
rr
t
t
tr
t
t
t
r2t
r4t
rt
r3t
r5t
21
21
1
......
r
t
tttS nt
21
21
1
......
r
rt
ttrtS nr
EE16.468/16.568 Lecture 1
Optical cavity, wavelength dependence, resonant
n2n1
rr
t
r
t
t
Lkni
Likn
t er
etAAA
2
2
22
2
21 1...
Likne 2
L
A0=1
tLikne 2
tte Likn2
tetr Lkni 222
tetr Lkni 244
Lknierb 222
2
2
22
22
22
22
2
Re11 LkniLkni
Likn
tt
t
er
etSI
R, intensity reflection
EE16.468/16.568 Lecture 1
Optical cavity, wavelength dependence, resonant2
2
22
22
22
22
2
Re11 LkniLkni
Likn
tt
t
er
etSI
R, intensity reflectionExample: n2 = 1.5, R=0.04, L = 0.05mm, =0.55µm
0.55 0.552 0.554 0.556 0.558 0.56 0.562 0.5640.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
0.55 0.552 0.554 0.556 0.558 0.56 0.562 0.5640
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength (um)Wavelength (um)
Inte
nsity
EE16.468/16.568 Lecture 1
Optical cavity, wavelength dependence, resonant
2
2
22
22
22
22
2
Re11 LkniLkni
Likn
tt
t
er
etSI
Resonant condition
mLkn 222
Constructively Interference
)12(22
mLkn Destructively Interference
mLn 22 Resonant condition, m = 1, 2, 3, …
mLn 2
2
m=1, half- cavity
m=2, cavity
EE16.468/16.568 Lecture 1
Optical cavity, Free-spectral range (FSR)
mLn 22 Resonant condition, m = 1, 2, 3, …
mLn
m
22
12
1
2
mLn
m-
122 2
1
2 mm
LnLn
1)(2
1
12
mm
mmLn
12
22
LnLn2
2
2
Example: n2 = 1.5, R=0.04, L = 1mm, =0.55µm
LnFSR
2
2
2
EE16.468/16.568 Lecture 1
Optical cavity, Free-spectral range (FSR)
Example: n2 = 1.5, R=0.04, L = 0.05mm, =0.55µm
LnFSR
2
2
2
mLn
FSR 02.0
2 2
2
L increase, FSR decrease, FSR not dependent on R
0.55 0.552 0.554 0.556 0.558 0.56 0.562 0.5640
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FSR FSR
L = 0.5mm
mLn
FSR 002.0
2 2
2
EE16.468/16.568 Lecture 1
Loss, and photon lifetime of an resonant cavity
21RR
21/ RRe t
c
Lnt 22
r1
r1r2
Intensity left after two mirror reflection:
)ln(/ 21RRt
)ln(
/2
21
2
RR
cLn
21/ 1 RR
te t
21
2
1
/2
RR
cLn
EE16.468/16.568 Lecture 1
Quality factor Q of an resonant cavity
c
LnRRIdtdI 2
21
2/)1(/
r1
r1r2
Intensity left after two mirror reflection:
dtdI
IQ
/
)1(
/2
21
2
RR
cLnQ
21
2
1
/2
RR
cLn
Q
/0
0 ttj eeII
Time domain
Frequency domain
Q
])/1()[(
)/1(
]/1)([
1
)(
220
00
0)/1(
0
/0
0
jI
jIdteI
dteeeII
tj
tjtjt
/1
EE16.468/16.568 Lecture 1Transmission matrix analysis of resonant cavity
n2
n1 A1
Likne 2
B1
A2
B2
L
At this interface
2211122 BrAtA
1122211 ArBtB
1122212 AtBrA 2211122 BrAtA
212
212
121
1B
t
rA
tA
1122211 ArAtB
2
12
212
12122211
1B
t
rA
trBtB
212
211221122
12
121 B
t
rrttA
t
rB
EE16.468/16.568 Lecture 1Transmission matrix analysis of resonant cavity
n2
n1 A1
Likne 2
B1
A2
B2
L
In Matrix form:
2
2
2112211212
21
121
1 11
B
A
rrttr
r
tB
A
212
212
121
1B
t
rA
tA
212
211221122
12
121 B
t
rrttA
t
rB
1
))((42
21
12212
21
2121122112
nn
nnnn
nn
nnrrtt
2
21
1
1
B
AM
B
A
2112211212
21
121
11
rrttr
r
tM
EE16.468/16.568 Lecture 1Transmission matrix analysis of resonant cavity
n2
n1 A1 Lnike 20
B1
A2
B2
L
LnikeAA 2023 A3
B3 LnikeBB 2032
In Matrix form:
3
3
2
2
20
20
0
0
B
A
e
e
B
ALnik
Lnik
3
32
2
2
B
AM
B
A
Lnik
Lnik
e
eM
20
20
0
02
EE16.468/16.568 Lecture 1Transmission matrix analysis of resonant cavity
n2
n1 A1 Lnike 20
B1
A2
B2
L
A3
B3
In Matrix form:
4
43
3
3
B
AM
B
A
A4
B4
4123214 BrAtA
3214123 ArBtB
4
4
2112211221
12
213
3 11
B
A
rrttr
r
tB
A
2112211221
12
213
11
rrttr
r
tM
EE16.468/16.568 Lecture 1Transmission matrix analysis of resonant cavity
n2
n1 A1 Lnike 20
B1
A2
B2
L
A3
B3
In Matrix form:
4
4
1
1
B
AM
B
A
A4
B4
4123214 BrAtA
3214123 ArBtB
4
4321
3
321
2
21
1
1
B
AMMM
B
AMM
B
AM
B
A
321 MMMM
EE16.468/16.568 Lecture 1Transmission matrix analysis of resonant cavity
n2
n1 A1 Lnike 20
B1
A2
B2
L
A3
B3
In Matrix form:
4
4
2221
1211
1
1
B
A
MM
MM
B
A
A4
B4
4123214 BrAtA
3214123 ArBtB
4111 AMA
04
4
2221
1211
1
1
B
A
MM
MM
B
A
111
4
1A
MA
111
214211 A
M
MAMB
2
11
1
MT
2
11
21
M
MR
EE16.468/16.568 Lecture 1E-field profile inside the resonant cavity
n2
n1 A1 Lnike 20
B1
A2
B2
L
A3
B3
A4
B4
04
4
2221
1211
1
1
B
A
MM
MM
B
A
111
4
1A
MA
04
2221
1211
1
1 A
MM
MM
B
A
2
21
1
1
B
AM
B
A
2
2
1
111 B
A
B
AM
EE16.468/16.568 Lecture 1E-field profile inside the resonant cavity
n2
n1 A1 Lnike 20
B1
A2
B2
L
A3
B3
A4
B4
znikznik eBeAE 1010111
znikznik eBeAE 2020222
znikeAE 1044
EE16.468/16.568 Lecture 1E-field profile inside the resonant cavity
n2
n1 A1 Lnike 20
B1
A2
B2
L
A3
B3
A4
B4
znikznik eBeAE 1010111
znikznik eBeAE 2020222
znikeAE 1044
n2
n1 A5 Lnike 20
B5
A6
B6
A7
B7
A8
B8
Ld
EE16.468/16.568 Lecture 1
Ring cavity, Ring resonator
LnFSR
2
2
2
95%, r
5%, t
R0
L = 2R0
2
2
22
22
22
22
2
Re11 LkniLkni
Likn
tt
t
er
etSI
mLn 22
mLn 2
2
EE16.468/16.568 Lecture 1
Lens and optical path
lens
f ffocus
Focal length
Focal plane
Same optical path
EE16.468/16.568 Lecture 1
Lens and optical path
lens
f ffocus
Focal length
Focal plane
Same optical path
lens
f ffocus
Same optical path
f
EE16.468/16.568 Lecture 1
Interference of multiple Waves, gratings
x1
x2
101iknxeAA
202iknxeAA
210021iknxiknx
totoal eAeAAAA
)}(sin)]cos(1{[
)sin()cos(1
)1()1(
222
0
22
0
2)(2
0
2)(0
2
00
2
12121
21
xknxknA
xknixknA
eAeeA
eAeAAI
xxiknxxikniknx
iknxiknxtotaltotoal
Screen
Near field pattern
EE16.468/16.568 Lecture 1
Diffraction of Waves
x1
x2
Screen
Far field pattern
f
Focal plane
∆x
d
∆x = d*sin()
∆ = k*∆x =(2/) d*sin()
When ∆ = 2m, bright
When ∆ = 2m+ , dark
∆L
∆L = f*sin()
2m = (2/) d*sin()
m = d*sin()
m = d* ∆L /f
∆L= m *f/d, bright spots
EE16.468/16.568 Lecture 1
Multiple slots, grating
Screen
Far field pattern
f
Focal plane
∆xm
d
∆xm = md*sin()
∆m = k*∆xm =(2/) md*sin()
When ∆ = 2m, bright
When ∆ = 2m+ , dark
∆L
∆L = f*sin()
2m = (2/) d*sin()
m = d*sin()
m = d* ∆L /f
∆L= m *f/d, bright spots
d
∆x1
∆x2
tan() = ∆L /f
~ tan() ~ sin(), when is small
Bright spot still at small position
EE16.468/16.568 Lecture 1
Multiple slots, grating
Screen
f
Focal plane
∆xm
d
∆xm = md*sin()
∆m = k*∆xm =(2/) md*sin()
∆L
∆L = f*sin()
d
∆x1
∆x2
sin
sin)1(
0
sinsin2sin0
0
0000
210
1
1
)...1(
)...1(
...
...
0
0
210
210
ikd
dmikikx
ikmddikikdikx
xikxikxikikx
ikxikxikxikx
mtotoal
e
eeA
eeeeA
eeeeA
eAeAeAeA
AAAAA
m
m
)sin(sin)sincos(1
)sin(sin)sincos(1
1
1
22
222
0
2
sin
sin)1(
0
20
kdkd
kmdkmdA
e
eeAAI
ikd
dmikikx
totaltotoal
EE16.468/16.568 Lecture 1
Optical wave, photon
Photon energy: ,hvE is the frequency of the photon, , vfc
,2
/ pckcchcE
momentum of the photon,
,2
pcmccmcE
,kp
For photon:,hvpcE
,kp
,2
k
EE16.468/16.568 Lecture 1
More on grating, Grating period, Phase matching condition
∆xm
d
∆xm = md*sin()
∆m = k*∆xm =(2/) md*sin()
d
∆x1
∆x2
d is called grating period, often use symbol: ,
k vector, k = (2/), along the beam propagation direction
Grating vector = 2 /,
k out = 2
/
kgrating = 2/
kout *sin() = kgrating = 2/
d*sin() =
Phase-matching condition
EE16.468/16.568 Lecture 1
Distributed feedback grating (DFB grating)
kin
kout k
kout = kin – k
Phase-matching condition
2nout / out = 2nin / in – 2/
kin
kout k
kout = kin – k = - kin
Phase-matching condition
2/ = 2*2/
Effective reflection
= /2
EE16.468/16.568 Lecture 1
Single slot diffraction:
)sin(
)1(
...
...
)sin(0
0
)sin(0
0000
210
210
ikd
eA
d
dzeA
eAeAeAeA
AAAAA
ikddikz
ikxikxikxikx
mtotoal
m
,)(sin
2/sin
)2/sinsin(
)sin(
)2/sinsin(2
)sin(
)(
)sin(
)1(
2
2
0
2
0
2
0
22/sin2/sin
0
2)sin(02
2/sin
I
kd
kdI
kd
kdiI
kd
eeeI
kd
eAAI
ikdikdikd
ikd
totoaltotoal
sin2/sin dkd
EE16.468/16.568 Lecture 1
Single slot diffraction:
,)(sin
2
2
0
II totoal
sin2/sin dkd
When 2
sin
md bright
When md sin dark
EE16.468/16.568 Lecture 1
Single slot diffraction:
,)(sin
2
2
0
II totoal
sin2/sin dkd
When 2
sin
md bright
When md sin dark
EE16.468/16.568 Lecture 1
Angular width :
,)(sin
2
2
0
II totoal
sin2/sin dkd
md sinfirst dark:
d
sind
Half angular width
EE16.468/16.568 Lecture 1
Resolution of optical instrument
d Focus sizedff
*
d
d
22.1'
eye
∆’d
22.1' d = 16mm.
EE16.468/16.568 Lecture 1
Polarization of light: linear polarization
Why decompose into two primary polarization directions?
x y
z
no
x y
z
ne
In crystal, the refractive index is different along different polarizations
no
ne
Refractive index ellipsoid
Any linear polarization can be decomposed into two primary polarization with the same phase
)cos(0 tA
)cos(0 tB
EE16.468/16.568 Lecture 1
Polarization of light: circular polarization
)cos(0 tA
)cos(0 tB
Lag /2
)cos(0 tA
)2/cos(0 tA
)cos(0 tA
)2/cos(0 tA
0t
)cos(0 tA
)2/cos(0 tA2/ t
)cos(0 tA
)2/cos(0 tA t
]/[tan
)]cos(/)cos([tan
001
001
AB
tAtB
EE16.468/16.568 Lecture 1
Polarization of light: circular polarization
)cos(0 tA
)2/cos(0 tA2/3 t
)cos(0 tA
)2/cos(0 tA 2t
)cos(0 tA
)sin(0 tA
)cos(0 tA
)2/cos(0 tA
)cos(0 tA
t
Clockwise
EE16.468/16.568 Lecture 1
Polarization of light: circular polarization
)sin(0 tA
)cos(0 tA
)2/cos(0 tA
)cos(0 tA
t
Count-clockwise
)sin(0 tB
)cos(0 tA
)2/cos(0 tB
)cos(0 tA
t
elliptical polarization
EE16.468/16.568 Lecture 1
Delays along different directions:
x y
z
no
x y
z
ne
no
ne
Refractive index ellipsoid
d
no
d
ne
Phase difference: dnnk e )( 0
lagno
ne
2/)( 0 dnnk ewhen
4/)( 0 dnn eQuarter-wavelength /4 plate
Right (clockwise) circular polarization
EE16.468/16.568 Lecture 1
Right (clockwise) circular polarization
no
d
ne
Phase difference: dnnk e )( 0
lagno
ne
dnnk e )( 0when Change the direction of the polarization
2/)( 0 dnn ehalf-wavelength /2 plate
/4 plate
)cos(0 tA )2/cos(0 tA
)cos(0 tA
)cos(0 tA
45º
EE16.468/16.568 Lecture 1
/2 plate
)cos(0 tA
)cos(0 tA
)cos(0 tA
)cos(0 tA
Change the direction of the polarization by 2
EO modulator
polarizer
ne
analyzer
dark
no
Brightno
EE16.468/16.568 Lecture 1
)cos(0 tA EO modulator
analyzer
dark
polarizer
)cos(0 tA Bright
neno
dnnk e )( 0
t
VEn 0 V
EE16.468/16.568 Lecture 1
Circular polarizations and linear polarizations
)2/cos(0 tA
)cos(0 tA
)cos(0 tA
)2/cos(0 tA
)2/cos(0 tA
)cos(0 tA
EE16.468/16.568 Lecture 1
Circular polarizations and linear polarizations
)2/cos(0 tA
)cos(0 tA
)2/cos(0 tA
)cos(0 tA
+
)2/cos()cos( 00 tAtA
)2/cos()cos( 00 tAtA
Linear polarization
EE16.468/16.568 Lecture 1
Circular polarizations and linear polarizations
)2/cos(0 tA
)cos(0 tA
)2/cos(0 tA
)cos(0 tA
+
)4/2/cos()4/2/cos(2
)2/cos()cos(
0
00
tA
tAtA
)2/4/cos()2/4/cos(2
)2/cos()cos(
0
00
tA
tAtA
Linear polarization
)2
cos()2
cos(2)cos()cos(
)2/4/cos(
)2/4/cos()'tan(
EE16.468/16.568 Lecture 1
Faraday rotation
)2/cos(0 tA
)cos(0 tA
)2/cos(0 tA
)cos(0 tA
Apply B field generate delay for left or right polarization
EE16.468/16.568 Lecture 1Jones Matrix
i1
2
1
)cos(0 tE
)2
cos(0
tE
)cos(0 tE
)2
cos(0
tE
i
1
2
1
)cos( ta
)2
cos( tb
ib
a
EE16.468/16.568 Lecture 1Jones Matrix
i1
2
1
)cos(0 tE
)2
cos(0
tE
)cos(0 tE
)2
cos(0
tE
i
1
2
1
)cos( ta
)2
cos( tb
ib
a
+
0
1
Linear
EE16.468/16.568 Lecture 1Jones Matrix
0
1
0E
)cos(0 E
4/
1
1
2
1
/2 phase delay
/4 plate
i0
01
0
1
0
1
0
01
i
/2 phase delay
/4 plate
i0
01
ii
1
2
1
1
1
0
01
2
1
)cos(0 tE
)2
cos(0
tE
i
1
2
1
EE16.468/16.568 Lecture 1Jones Matrix
x
x’yy’
)sin('cos' yxx
cos'sin' yxy
'
'
cossin
sincos
y
x
y
x
'
'
cossin
sincos
y
x
y
x
y
x
y
x
cossin
sincos
'
'
y
xR
y
x)(
'
'
cossin
sincos)(R
EE16.468/16.568 Lecture 1Jones Matrix
090x
x’yy’
01
10)90( 0R
0
1
0E
1
0
0
1
01
10
0
1)90( 0R
cossin
sincos)(R
EE16.468/16.568 Lecture 1Jones Matrix
cossin
sincos)(R
)cos( ta
)2
cos( tb
ib
a
)cos( ta
)2
cos( tb
cossin
sincos
cossin
sincos)(
iba
iba
ib
a
ib
aR
EE16.468/16.568 Lecture 1Jones Matrix
)cos(0 tE
)2
cos(0
tE
i
1
2
1
cossin
sincos)(R
cossin
sincos1
cossin
sincos1)(
i
i
iiR
EE16.468/16.568 Lecture 1Jones Matrix
i1
2
1
)cos(0 tE
)2
cos(0
tE
+
Linear
)cos(0 tE
)2
cos(0
tE
cossin
sincos
i
i
ii
i
cossin
sincos1
)2/(cos2)2/cos()2/sin(2
)2/cos()2/sin(2)2/(cos22
2
i
i
)2/sin()2/cos()2/sin(2
)2/sin()2/cos()2/cos(2
i
i
0
1
)2/cos()2/sin(
)2/sin()2/cos()2/sin()2/cos(
i