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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, mode
dispersion, 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, electrooptic 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
c  a  ib or Cei
C: amplitude
: angle
b
C

a
c1  c2  (a1  a2 )  i(b1  b2 )
Real numbers do not have phases
Complex numbers have phases
Cei
 is the phase of the complex number
EE16.468/16.568
Lecture 1
Physical meaning of multiplication of two complex numbers
c1  c2  C1ei1  C2 ei2  C1  C2 ei (1 2 )
C1e
i1
Why
C2e
i2
Number:
C1  C2
Phase:
1  2
C2
times
Phase delay
i i  1?
90
i

i  1 e 2
i
i  i  11 e
i  i
3
 
i(  )
2 2
 ei (  )
i 1
4
180
0
-1
1
-i
270
1  i
4
1  i;1
2
EE16.468/16.568
Lecture 1
90
1?
6
6
1e
i(
2
n)
6
, n  0,1...5
180
12
2  12 2e
2
i(
n)
12
0
-1
1
, n  0,1...11
270
Optical wave, frequency domain
Aei ( kx  )
Initial phase
Vertical polarization,
k  polarization, transwave
k, 
x
k vector, to x-direction
Horizontal polarization
Phase delay by position
Amplitude, I=|A|2, Optical intensity
EE16.468/16.568
Lecture 1
Optical wave propagation in air (free-space)
Ae
k, 
i ( kx  )
C1ei1
i ( kx  delay)
Ae
e i2
Time and frequency domain of optical signal
Ae
i ( kx  )
Phase delay
d ela y  k  x 
2

x
EE16.468/16.568
Lecture 1
Optical wave propagation in air (free-space)
Ae
k
∆x
i ( kx  )
Wave front
Phase is the same on the plane --- plane wave
  kx
2

x
Phase distortion in atmosphere
k
∆x
Aei ( kx  )
Wave front distortion
Additional phase delay,  non ideal plane wave
EE16.468/16.568
Lecture 1
Optical wave to different directions
i ( kx1  )
x1
Ae
Phase delay
delay  kx 
x2
Ae
i ( kx 2  )
Refractive index n
In vacuum,
k 
2

V c

T
 f
c: speed of light
In media, like glass,
2
2
k 

n
 /n

V  c/n 
 /n
T
 f / n
2

( x1  x 2 )
EE16.468/16.568
Lecture 1
Optical wave to different directions
i ( kx1  )
x1
Ae
Phase delay
delay  kx 
x2
Ae
2

( x1  x 2 )
i ( kx 2  )
In media with refractive index n
n
nx1
Ae
i ( knx1  )
Phase delay
d ela y  knx 
nx 2
nx1, nx2, optical path
Ae
i ( knx2  )
2

n ( x1  x 2 )
EE16.468/16.568
Lecture 1
Interference of two optical waves
a2
a1
A
Upper arm :
a3
50%
b1
Aout
b2
Lower arm :
Aout =
1 ikn( a1 a2 a3 )
Ae
2
1 ikn( a1 a2 a3 )
Ae
2
+
1 ikn( b1 b2 )
Ae
2
1 ikn( b1 b2 )
Ae
2
Phase delay between the two arms:
kn ( b1  b2  a1  a 2  a 3 )
1 ikn( a1 a2 a3 )
1 ikn( b1 b2 )
=
Ae
Ae
If phase delay is 2m, then:
2
2
1 ikn( b1 b2 )
1 ikn( a1 a2 a3 )
Ae
If phase delay is 2m +, then:
=Ae
2
2
EE16.468/16.568
March-Zehnder interferometer
n
a2
a1
b1
n2
b2
Upper arm :
a3
50%
A
Lecture 1
1 ikn( a1 a2 a3 )
Ae
2
Aout
b3
Lower arm :
b1  b2  b3  a1  a 2  a 3
1 ikn( b1 b2 )
Ae
2
Phase delay between the two arms:
k ( n2 b2  na2 )
1 ikn( a1 a2 a3 )
1 ikn( b1 b2 )
=
Ae
Ae
If phase delay is 2m, then:
2
2
1 ikn( b1 b2 )
1 ikn( a1 a2 a3 )
Ae
If phase delay is 2m +, then:
=Ae
2
2
Electro-optic effect, n2 changes with E -field
EE16.468/16.568
Lecture 1
Electro-optic modulator based on March-Zehnder interferometer
n
a2
a1
a3
50%
A
b1
n2
b2
Aout
Electro-optic effect, n2 changes with E -field
b3
Phase delay between the two arms:
k ( n2 b2  na2 )
1 ikn( a1 a2 a3 )
1 ikn( b1 b2 )
Ae
Ae
If phase delay is 2m, then:
=
2
2
1 ikn( a1 a2 a3 )
1 ikn( b1 b2 )
If phase delay is 2m +, then:
Ae
Ae
=2
2
EE16.468/16.568
Lecture 1
Reflection, refraction of light
reflection
n1 sin 1  n2 sin  2
1 1
n1
n2 > n1
n2
2 < 1
2
Refraction
reflection
1 1
n1
n2
n1 sin 1  n2 sin  2
n2 < n1
2
Refraction
2 > 1
EE16.468/16.568
Lecture 1
Total internal reflection (TIR)
reflection
n1 sin 1  n2 sin  2
1 1
n1
n2
n2 < n1
2
Refraction
When 2 = 900, 1 is critical angle
n1 sin 1  n 2 sin  2 |
 2  900
sin 1  n2 / n1
 n2 ,
2 > 1
EE16.468/16.568
Lecture 1
Total internal reflection (TIR)
1 1
Total reflection When 2 = 900, 1 is critical angle
n1
n1 sin 1  n 2 sin  2 |
 2  900
2 = 900
n2
 n2 ,
sin 1  n2 / n1
No refraction
Application – optical fiber
n1
Low loss, TIR
n2
Flexible
n1
Communication system
Electrical signal
Laser
EO modulator
Detector
Optical fiber
Electrical signal
EE16.468/16.568
Lecture 1
Reflection percentage
Intensity reflection = [(n1-n2)/(n1+n2)]2
Example 1:
n1
Refractive index of glass n = 1.5;
n2
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;
Intensity reflection = [(n1-n2)/(n1+n2)]2
n1
N2 = 3.5
The reflection from GaAs surface is 31%;
Detector, GaAs
EE16.468/16.568
Lecture 1
Interference of two optical beams, applications
1, Mach-Zehnder interferometer
n
a2
a1
a3
50%
A
b1
n2
b2
Aout
b3
Electro-optic effect, n2 changes with E -field
EE16.468/16.568
Lecture 1
2, Michelson interferometer
M1
Beam splitter
n1 d1
n2 d2
Phase difference:
k ( n2 2 d 2  n1 2 d1 )
Measuring small moving or displacement
Mirror 2
Detector
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
M1
n2 d1
Mirror
Beam splitter
n1 d2
Detector
Phase difference:
k ( n1 2b2  n2 2 d1 )
EE16.468/16.568
Lecture 1
3, Measuring surface flatness
k ( n2 d )
k ( n 2 d ' )
Accuracy: 0.1 wavelength = 0.1*500nm = 50nm
n2
n2
4, Measuring refractive index of liquid or gas
Gas in
EE16.468/16.568
Wave optics
Lecture 1
Maxwell equations:


B
 E  
t

D  


D
H  J 
t

 B  0
Dielectric materials


 0
D  E

J 0


B  H
Maxwell equations in dielectric materials:


B
 E  
t

 D  0

 D
H 
t

 B  0


  (  E )   j (  B )
phasor


  E   jB

 D  0


  H  jD

 B  0



2
2
(  E )   E   E
EE16.468/16.568
Wave optics approach
Helmholtz Equation:


2
 E k E 0


 E   2 E  0
2
2
Free-space solutions

E  yˆ E0eikz
 2 2
 E  k0 n E  0
2
k 2   2 

E  xˆE0eikz
k
∆x
Aei ( kx  )
Wave front
Phase is the same on the plane --- plane wave
  kx
2

x
Lecture 1
EE16.468/16.568
Wave optics
Lecture 1
Helmholtz Equation:


2
 E k E 0


2 E   2 E  0
k 2   2 
2


2 E  k02 n2 E  0
2-D Optical waveguide
x
z
y
n2 < n1
Cladding, n2
d
Core, refractive index n1
Cladding, n2
TE mode:

E  yˆ E ( x )
TM mode:

H  yˆ H ( x )
EE16.468/16.568
Normal reflection at material interface
Helmholtz Equation:


2
2 2
 E  k0 n E  0
Er

E1  xˆEineik0n1z  xˆEr eik0n1z

E2  xˆEt eik0n2 z
Ein
n1
x



Ein  Er  Et



Ein  Er  Et

Ein
n2
z
Et


B
 E  
t
xˆ


 E 
x
Ex


  E   jH


 E
H 
j
Lecture 1
yˆ

y
0
E x
E x
 zˆ ( 
)
z
y
E x
 xˆ (0)  yˆ
 zˆ (0)
z
 xˆ (0)  yˆ

H
zˆ

z
0
1 r  t
EE16.468/16.568
Normal reflection at material interface
Helmholtz Equation:


2
2 2
 E  k0 n E  0
Er

E1  xˆEineik0n1z  xˆEr eik0n1z

E2  xˆEt eik0n2 z
Ein
n1
x
Lecture 1



Ein  Er  Et
1 r  t
n2
z
Et

H Continuous @ z=0
Why?

E2  xˆEt eik0n2 z
E x
z
Continuous @ z=0

E1  xˆEineik0n1z  xˆEr eik0n1z

E1
 xˆ (ik 0 n1 Eineik0 n1 z  ik 0 n1 Er e ik0 n1z )
z

E2
 xˆik 0 n2 Et e ik0 n2 z
z
EE16.468/16.568
Normal reflection at material interface
Helmholtz Equation:


2
2 2
 E  k0 n E  0
Er
x
Lecture 1

E1  xˆEineik0n1z  xˆEr eik0n1z

E2  xˆEt eik0n2 z
Ein
n1



Ein  Er  Et
1 r  t
n2

E1
 xˆ (ik 0 n1 Eineik0 n1 z  ik 0 n1 Er e ik0 n1z )
z
Z=0
 xˆ (ik 0 n1 Ein  ik 0 n1 Er )

E2
 xˆik 0 n2 Et e ik0 n2 z
z
 xˆik 0 n2 Et
Z=0
EE16.468/16.568
Normal reflection at material interface
Helmholtz Equation:


2
2 2
 E  k0 n E  0
Er
x

E1  xˆEineik0n1z  xˆEr eik0n1z

E2  xˆEt eik0n2 z
Ein
n1



Ein  Er  Et
1 r  t
n2
xˆ (ik 0 n1 Ein  ik 0 n1 Er )  xˆik 0 n2 Et
n1  n1r  n2t
1 r  t
n1  n1r  n1t
t
2n1
n1  n2
r
n1  n2
n1  n2
Lecture 1
EE16.468/16.568
Lecture 1
Reflection, refraction of light




ik0 n1 rˆ
i k0 n1 sin1 x  k0 n1 cos1 z 
Ein  yˆEine
 yˆEine
Ein
1 1
y
x
n1
Er



i k0 n2 sin 2 x  k0 n2 cos 2 z 
Et  yˆEt e
n2
z
2
Et
Eine 


i k0 n1 sin1 x  k0 n1 cos1 z
Eine

ik 0 n1 sin1 x
 Er e



i k0 n1 sin1 x  k0 n1 cos1 z 
Er  yˆEr e


Ein  Er  Et | z 0
  E ei k0 n1 sin1x k0 n1 cos1z   E ei k0 n2 sin 2 x  k0 n2 cos 2 z  |
r
t
z 0

ik 0 n1 sin1 x
n1 sin 1  n 2 sin  2

 Et e


ik 0 n2 sin 2 x
Ein  Er  Et


EE16.468/16.568
Lecture 1
Reflection, refraction of light
Ein
1 1
y
x
z
n1
Er
n2
2


B
 E  
t
xˆ


 E 
x
0
Et
  xˆ
  xˆ

H Continuous @ z=0
Why?
E y
z
E y
z
E y
z
yˆ

y
Ey


  E   jH
zˆ

z
0
 yˆ (0)  zˆ (
E y
x
)
 yˆ (0)  zˆ (0)
Continuous @ z=0


 E
H 
j
EE16.468/16.568
Lecture 1
Reflection, refraction of light




ik0 n1 rˆ
i k0 n1 sin1 x  k0 n1 cos1 z 
Ein  yˆEine
 yˆEine
Ein
1 1
y
x
z
n1
Er
n2
2
Et



i k0 n1 sin1 x  k0 n1 cos1 z 
Er  yˆEr e



i k0 n2 sin 2 x  k0 n2 cos 2 z 
Et  yˆEt e
E y
z
Continuous @ z=0
k0 n1 cos 1Ein  k0 n1 cos 1 Er  k0 n2 cos  2 Er
k0 n1 cos 1Ein  k0 n1 cos 1 Er  k0 n1 cos 1 Et
2k0 n1 cos 1Ein  k0 n1 cos 1  k0 n2 cos  2 Et
Et 
2k0 n1 cos 1
Ein
k0 n1 cos 1  k0 n2 cos  2 
Ein  Er  Et
EE16.468/16.568
Lecture 1
Reflection, refraction of light




ik0 n1 rˆ
i k0 n1 sin1 x  k0 n1 cos1 z 
Ein  yˆEine
 yˆEine
Ein
1 1
y
x
z
n1
Er
n2
2
Et



i k0 n1 sin1 x  k0 n1 cos1 z 
Er  yˆEr e



i k0 n2 sin 2 x  k0 n2 cos 2 z 
Et  yˆEt e
E y
z
Continuous @ z=0
k0 n1 cos 1Ein  k0 n1 cos 1 Er  k0 n2 cos  2 Er
Ein  Er  Et
k0 n2 cos  2 Ein  k0 n2 cos  2 Er  k0 n2 cos  2 Et
k0 n2 cos 2  k0 n1 cos1 Ein  k0 n2 cos 2  k0 n1 cos1 Er
Er 
k0 n1 cos 1  k0 n2 cos  2
Ein
k0 n1 cos 1  k0 n2 cos  2 
0
EE16.468/16.568
Lecture 1
Reflection, refraction of light
n1 sin 1  n 2 sin  2
Ein
1 1
y
x
z
n1
Er
r
n1 cos 1  n2 cos  2
n1 cos 1  n2 cos  2 
t
2n1 cos 1
n1 cos 1  n2 cos  2 
n2
2
Et
r
t
1
1
EE16.468/16.568
Lecture 1
Ein
y
x
z
n1
1 
1
1
1
Er
2
n2
2

Ein  xˆEin cos1  zˆEin sin 1 ei k0n1 sin1x k0n1 cos

Er   xˆEr cos 1  zˆEr sin 1 e i k0 n1 sin1 x  k0 n1 cos1 z

Et  xˆEt cos 2  zˆEt sin  2 ei k0n2 sin2 xk0n2 cos2 z
Et
Ein cos 1e i k0 n1 sin1x   Er cos 1e i k0 n1 sin1x   Et cos  2 ei k0 n2 sin 2 x 
n1 sin 1  n 2 sin  2
Ein cos 1  Er cos 1  Et cos  2
EE16.468/16.568
Lecture 1
Ein
y
x
z
n1
1 
1
1
1
Er
2
n2
2


B
 E  
t

Ein  xˆEin cos1  zˆEin sin 1 ei k0n1 sin1x k0n1 cos

Er   xˆEr cos 1  zˆEr sin 1 e i k0 n1 sin1 x  k0 n1 cos1 z

Et  xˆEt cos 2  zˆEt sin  2 ei k0n2 sin2 xk0n2 cos2 z
Et
xˆ


 E 
x
Ex
yˆ

y
0
zˆ

z
Ez
E
E x
E z
E z
 yˆ ( x 
)  zˆ (
)
y
z
x
y
E
E z
  xˆ 0  yˆ ( x 
)  zˆ (0)
z
x
  xˆ
E x E z

z
x
continuous
EE16.468/16.568
Lecture 1
Ein
y
x
n1
1 
1
1
2
n2
z
1
Er
2
Et

Ein  xˆEin cos1  zˆEin sin 1 ei k0n1 sin1x k0n1 cos

Er   xˆEr cos 1  zˆEr sin 1 e i k0 n1 sin1 x  k0 n1 cos1 z

Et  xˆEt cos 2  zˆEt sin  2 ei k0n2 sin2 xk0n2 cos2 z
E x E z

z
x
continuous
Ein cos 1k0 n1 cos 1ei k0 n1 sin1x   Er cos 1k0 n1 cos 1ei k0 n1 sin1x 

  Ein sin 1k0 n1 sin 1ei k0 n1 sin1x   Er sin 1k0 n1 sin 1ei k0 n1 sin1x 
=
Et cos  2 k0 n2 cos  2 ei k0 n2 sin 2 x   Et sin  2 k0 n2 sin  2 ei k0 n2 sin 2 x 
Eink0 n1  Er k0 n1  Er k0 n2

EE16.468/16.568
Lecture 1
Ein
y
x
z
n1
1 
1
1
1
Er
2
n2
2
Et

Ein  xˆEin cos1  zˆEin sin 1 ei k0n1 sin1x k0n1 cos

Er   xˆEr cos 1  zˆEr sin 1 e i k0 n1 sin1 x  k0 n1 cos1 z

Et  xˆEt cos 2  zˆEt sin  2 ei k0n2 sin2 xk0n2 cos2 z
Ein k0 n1  Er k0 n1  Et k0 n2
Ein cos 1  Er cos 1  Et cos  2
Einn1 cos 1  Er n1 cos 1  Et n2 cos 1
Einn1 cos 1  n1Er cos 1  n1Et cos  2
2n1 cos 1
t
n1 cos  2  n2 cos 1
EE16.468/16.568
Lecture 1
Ein
y
x
z
n1
1 
1
1
1
Er
2
n2
2
Et

Ein  xˆEin cos1  zˆEin sin 1 ei k0n1 sin1x k0n1 cos

Er   xˆEr cos 1  zˆEr sin 1 e i k0 n1 sin1 x  k0 n1 cos1 z

Et  xˆEt cos 2  zˆEt sin  2 ei k0n2 sin2 xk0n2 cos2 z
Ein k0 n1  Er k0 n1  Et k0 n2
Ein cos 1  Er cos 1  Et cos  2
Ein n1 cos  2  Er n1 cos  2  Et n2 cos  2
Ein n2 cos 1  Er n2 cos 1  Et n2 cos  2
Ein n2 cos 1  n1 cos  2   Er n2 cos 1  n1 cos  2   0
n1 cos  2  n2 cos 1
r
n2 cos 1  n1 cos  2
2n1 cos 1
t
n1 cos  2  n2 cos 1
EE16.468/16.568
Lecture 1
Ein
y
x
z
n1
1 
1
1
1
Er
2
n2
2
Et
n1 cos  2  n2 cos 1
r
n2 cos 1  n1 cos  2
2n1 cos 1
t
n1 cos  2  n2 cos 1
n1 cos  2  n2 cos 1
tan(  2  1 )
r

n2 cos 1  n1 cos  2
tan(  2  1 )
Ein
Brewster’s angle
n1
1 
1
1
Er
n2
2 Et
http://buphy.bu.edu/~duffy/semester2/c27_brewster.html
EE16.468/16.568
Lecture 1
Ein
y
x
z
n1
1 
1
1
1
Er
2
n2
2
Et
n1 cos  2  n2 cos 1
r
n2 cos 1  n1 cos  2
2n1 cos 1
t
n1 cos  2  n2 cos 1
EE16.468/16.568
Normal reflection at material interface
Helmholtz Equation:


2
2 2
 E  k0 n E  0
Er

E1  xˆEineik0n1z  xˆEr eik0n1z

E2  xˆEt eik0n2 z
Ein
n1
x
n2
Er
Lecture 1
2n1
t
n1  n2
n1
Ein
n1
x
n2
r
rt
r3t
r5t
n1  n2
n1  n2
n2
t
t
t
t
r
r
t
r
t
r2 t
r4 t
EE16.468/16.568
Lecture 1
Equal difference series
a1 , a2 ...an ...
a2  a1  b
a3  a2  b
+
=
an  an1  b
an  a1  (n  1)b
Sn  a1  a2  ...an
a2  an1  a1  b  an1  a1  an
+
Sn  an  an 1  ...a1
a3  an2  a1  2b  an2  a1  an
=
2Sn  n(an  a1 )
Sn 
n
n
n
(an  a1 )  (an  a1 )  na1  (n  1)b  na1
2
2
2
EE16.468/16.568
Power series
a1 , a2 ...an ...
a2  ba1
a3  ba2
…
x
an  ban 1
an  b n 1a1
-
=
Sn  a1  a2  ...an
bSn  ba1  ba2  ...ban  a2  a3  ...an1
(1  b) Sn  a1  an1
a1  an 1 a1  bn a1
Sn 

1 b
1 b
Lecture 1
EE16.468/16.568
Lecture 1
Power series
a1 , a2 ...an ...
S  a1  a2  ...an  ...
a1 (1  bn )
a1
 lim

n
1 b
1 b
When |b|<1
b can be anything, number, variable, complex number or function
Optical cavity, multiple reflections
n1
Sr  t1r  t2  ...tn  ...

t1r
1  r2
rt
r3t
r5t
n2
t
t
t
r
r
t
r
t
t
St  t1  t2  ...tn  ...
r2 t

r4 t
t1
1  r2
EE16.468/16.568
Lecture 1
Optical cavity, wavelength dependence, resonant
n1
A0=1
t
n2
r
eikn2 L
eikn2 L
t
teikn2 Lt
r
r
t
tr2ei 2 kn2 Lt
t
tr4ei 4 kn2 Lt
L
b  r 2 ei 2 kn2 L
t 2eikn2 L
At  A1  A2  ... 
1  r 2ei 2 kn2 L
I t  St
2
2 ikn2 L
2
2
te
t


2 i 2 kn2 L
1 r e
1  Re i 2 kn2 L
2
R, intensity reflection
EE16.468/16.568
Lecture 1
Optical cavity, wavelength dependence, resonant
I t  St 
2
2 ikn2 L
2
2
te
t

1  r 2ei 2 kn2 L
1  Re i 2 kn2 L
2
Example: n2 = 1.5, R=0.04, L = 0.05mm, =0.55µm
R, intensity reflection
1
1
0.9
0.99
0.8
Intensity
0.98
0.7
0.97
0.6
0.96
0.5
0.4
0.95
0.3
0.94
0.2
0.93
0.92
0.55
0.1
0.552
0.554
0.556
0.558
0.56
Wavelength (um)
0.562
0.564
0
0.55
0.552
0.554
0.556
0.558
Wavelength (um)
0.56
0.562
0.564
EE16.468/16.568
Lecture 1
Optical cavity, wavelength dependence, resonant
I t  St 
2
2 ikn2 L
2
2
te
t

1  r 2ei 2 kn2 L
1  Re i 2 kn2 L
2
Resonant condition
2 kn 2 L  ( 2 m  1) Destructively Interference
2 kn 2 L  2 m
2

n 2 L  m
m=1, half- cavity
m=2,  cavity
Constructively Interference
2n2 L  m
Resonant condition, m = 1, 2, 3, …
EE16.468/16.568
Lecture 1
Optical cavity, Free-spectral range (FSR)
2n2 L  m
2 n2 L
m
2 n2 L
 m 1
m
-
Resonant condition, m = 1, 2, 3, …
m 1
2n2 L

m1
2n2 L
m
2n2 L

2
1
1
2n2 L(m  m 1 )
m1m
 
2
2n2 L
1
FSR 
2
2n2 L
Example: n2 = 1.5, R=0.04, L = 1mm, =0.55µm
EE16.468/16.568
Lecture 1
Optical cavity, Free-spectral range (FSR)
FSR 
2
2n2 L
L increase, FSR decrease, FSR not dependent on R
Example: n2 = 1.5, R=0.04, L = 0.05mm, =0.55µm
1
FSR 

2
2n2 L
0.9
 0.02 m
0.8
0.7
0.6
0.5
L = 0.5mm
FSR 
2
2n2 L
0.4
 0.002m
0.3
0.2
0.1
0
0.55
0.552
0.554
FSR FSR
0.556
0.558
0.56
0.562
0.564
EE16.468/16.568
Lecture 1
Loss, and photon lifetime of an resonant cavity
Intensity left after two mirror reflection:
r1
R1R2
r1r2
e
t /
 R1R2
 t /   ln( R1R2 )
e t /   1 
2n2 L / c

 ln( R1 R2 )
t

2n2 L / c

1  R1 R2
2 n2 L
t
c
 R1 R2
EE16.468/16.568
Lecture 1
Quality factor Q of an resonant cavity
Intensity left after two mirror reflection:
r1
r1r2
Q
I
 dI / dt
 dI / dt  I (1  R1 R2 ) /
Time domain
I  I 0 e  j 0 t e  t / 
Q
Frequency domain
 2n2 L / c
(1  R1 R2 )
Q  
I ( )  I 0  e t / e  j0t e jt dt
 I 0  e ( j 1/ )t dt  I 0
  1/ 
1
[ j (  0 )  1 /  ]
 I0

2n2 L
c
2n2 L / c
1  R1 R2

Q

(  j  1 /  )
[(  0 ) 2  (1 /  ) 2 ]
EE16.468/16.568
Transmission matrix analysis of resonant cavity
Lecture 1
L
n1
A1
A2  t12 A1  r21B2
A2
n2
B1
eikn2 L
B1  t21B2  r12 A1
B2
At this interface
A2  t12 A1  r21B2
B1  t21 A2  r12 A1
1
r21 

B1  t21B2  r12  A2  B2 
t12 
 t12
A2  r21B2  t12 A1
1
r21
A1 
A2  B2
t12
t12
r12
t12t 21 r 12r21
B1 
A2 
B2
t12
t12
EE16.468/16.568
Transmission matrix analysis of resonant cavity
Lecture 1
L
n1
A1
A2
n2
B1
eikn2 L
B2
1
r21
A1 
A2  B2
t12
t12
r12
t12t 21 r 12r21
B1 
A2 
B2
t12
t12
In Matrix form:
 r21
 A1  1  1
 A2 
   
 
 B1  t12  r12 t12t21  r12r21  B2 
4n1n2
(n1  n2 )(n2  n1 )
t12t21  r12r21 

1
2
2
n1  n2 
n1  n2 
 A1 
 A2 
   M1  
 B1 
 B2 
 r21

11


M1  
t12  r12 t12t21  r12r21 
EE16.468/16.568
Transmission matrix analysis of resonant cavity
L
n1
A1
B1
A2
B2
ik0 n2 L
A3
A3  A2 eik 0n2 L
n2
B3
B2  B3eik 0n2 L
e
In Matrix form:
 A2   e ik 0n2 L
   
 B2   0
0  A3 
 
ik 0n2 L 
e
 B3 
 A3 
 A2 
   M 2  
 B2 
 B3 
 e ik 0n2 L
M 2  
 0
0 

ik 0n2 L 
e

Lecture 1
EE16.468/16.568
Transmission matrix analysis of resonant cavity
L
n1
A1
A2
eik0 n2 L
n2
B1
B2
A3
A4
A4  t21 A3  r12 B4
B3
B4
B3  t12 B4  r21 A3
In Matrix form:
 r12
 A3  1  1
 A4 
   
 
 B3  t21  r21 t12t21  r12r21  B4 
 A3 
 A4 
   M 3  
 B4 
 B3 
 r12

11


M3  
t21  r21 t12t21  r12r21 
Lecture 1
EE16.468/16.568
Transmission matrix analysis of resonant cavity
L
n1
A1
A2
eik0 n2 L
n2
B1
B2
A3
A4
A4  t21 A3  r12 B4
B3
B4
B3  t12 B4  r21 A3
In Matrix form:
 A3 
 A1 
 A2 
 A4 
   M 1    M 1M 2    M1M 2 M 3  
 B1 
 B2 
 B4 
 B3 
 A1 
 A4 
   M  
 B1 
 B4 
M  M 1M 2 M 3
Lecture 1
EE16.468/16.568
Transmission matrix analysis of resonant cavity
Lecture 1
L
n1
A1
A2
eik0 n2 L
n2
B1
A3
A4
A4  t21 A3  r12 B4
B3
B4
B3  t12 B4  r21 A3
B2
In Matrix form:
 A1   M11 M12  A4 
   
 
 B1   M 21 M 22  B4 
1
A4 
A1
A1  M11 A4
M 11
M 21
B1  M 21 A4 
A1
M 11
 A1   M11 M12  A4 
   


 B1   M 21 M 22  B4  0 
1
T
M 11
2
M 21
R
M 11
2
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E-field profile inside the resonant cavity
Lecture 1
L
n1
A1
A2
eik0 n2 L
n2
B1
A3
A4
B3
B4
B2
 A1   M11 M12  A4 
   


 B1   M 21 M 22  B4  0 
1
A4 
A1
M 11
 A1   M11 M12  A4 
   
 
 B1   M 21 M 22  0 
 A1 
 A2 
   M1  
 B1 
 B2 
 A1   A2 
M     
 B1   B2 
1
1
EE16.468/16.568
E-field profile inside the resonant cavity
L
n1
A1
A2
eik0 n2 L
n2
B1
A3
A4
B3
B4
B2
E1  A1eik0n1z  B1eik0n1z
E2  A2eik0n2 z  B2eik0n2 z
E4  A4eik0n1z
Lecture 1
EE16.468/16.568
E-field profile inside the resonant cavity
L
n1
A1
B1
A2
d
ik0 n2 L
A3
A4
n2
B3
B4
e
B2
E1  A1eik0n1z  B1eik0n1z
E2  A2eik0n2 z  B2eik0n2 z
E4  A4eik0n1z
Lecture 1
n1
L
A5
A6
eik0 n2 L
n2
B5
B6
A7
A8
B7
B8
EE16.468/16.568
Bragg gratings
/4
n1
n2
d1
d2
Lecture 1
EE16.468/16.568
Lecture 1
Ring cavity, Ring resonator
5%, t
L = 2R0
95%, r
I t  St 
2
R0
2

n 2 L  m
FSR 
2
2n2 L
2n2 L  m
2 ikn2 L
2
2
te
t

1  r 2ei 2 kn2 L
1  Re i 2 kn2 L
2
EE16.468/16.568
Lecture 1
Ring cavity, Ring resonator filter
L = 2R0
2

n 2 L  m
FSR 
2
2n2 L
2n2 L  m
EE16.468/16.568
Lecture 1
Lens and optical path
Focal length
Focal plane
Same optical path
f
f
focus
lens
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Lecture 1
Lens and optical path
Focal length
f
Focal plane
Same optical path
f
f
focus
lens
Same optical path
f
f
focus
lens
EE16.468/16.568
Lens and optical path
Lecture 1
Focal length
f
Focal plane
Same optical path
EE16.468/16.568
Lecture 1
Interference of multiple Waves, gratings
Near field pattern
x1
A1  A0eiknx1
A2  A0eiknx2
Atotoal  A1  A2  A0eiknx1  A0eiknx2
x2
I totoal  Atotal  A0e
2
 A0e
iknx1
(1  e
iknx1
ikn ( x2  x1 )
2
 A0e
iknx2 2
)  A0 (1  e
2
 A0 1  cos( knx )  i sin( knx )
2
ikn ( x2  x1 )
2
 A0 {[1  cos( knx )]2  sin 2 (knx )}
2
Screen
)
2
EE16.468/16.568
Lecture 1
Diffraction of Waves
Far field pattern
When ∆ = 2m, bright
When ∆ = 2m+ , dark
2m = (2/) d*sin()
x1

d
m  = d*sin()


m  = d* ∆L /f
∆L
∆L= m *f/d, bright spots
∆L = f*sin()
x2
f
∆x
Focal plane
∆x = d*sin()
∆ = k*∆x =(2/) d*sin()
Screen
EE16.468/16.568
Lecture 1
Multiple slots, grating
Far field pattern

d
d
When ∆ = 2m, bright
Focal plane

When ∆ = 2m+ , dark
2m = (2/) d*sin()
∆x1
m  = d*sin()
∆x2
∆L

tan() = ∆L /f
 ~ tan() ~ sin(), when  is small
∆L = f*sin()
f
m  = d* ∆L /f
∆L= m *f/d, bright spots
Bright spot still at small position
∆xm
∆xm = md*sin()
∆m = k*∆xm =(2/) md*sin()
Screen
EE16.468/16.568
Lecture 1
Multiple slots, grating
Atotoal  A0   A1  A2  ...  Am

d
d
Focal plane

 A0eikx0  A0eikx1  A0eikx2  ...  A0eikxm
 A0eikx0 (1  eikx1  eikx2  ...  eikxm )
∆x1
 A0eikx0 (1  eikd sin  eik 2 d sin  ...  eikmd sin )
∆x2
∆
L
∆L = f*sin()

f
∆xm
∆xm = md*sin()
∆m = k*∆xm =(2/) md*sin()
Screen
 A0e
ikx0
1  eik ( m1) d sin
1  eikd sin
I totoal  Atotal
 A0
2
2
ik ( m 1) d sin
ikx0 1  e
 A0e
1  eikd sin
2
1  cos( kmd sin  )2  sin 2 (kmd sin  )
1  cos( kd sin  )2  sin 2 (kd sin  )
EE16.468/16.568
Lecture 1
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Lecture 1
Optical wave, photon
Photon energy:
E  hv,
 is the frequency of the photon,
E  hc /  
p  k ,
E  mc
For photon:
2
2

momentum of the photon,
 mcc  pc,
E  pc  hv,
p  k ,
k 
2

c  kc  pc,
,
c  f  v ,
EE16.468/16.568
Lecture 1
More on grating, Grating period, Phase matching condition
k vector, k = (2/), along the beam propagation direction

d
d

d is called grating period, often use symbol: ,
∆x1
Grating vector = 2 /,
Phase-matching condition
∆x2
kgrating = 2/

∆xm
∆xm = md*sin()
∆m = k*∆xm =(2/) md*sin()
kout *sin() = kgrating = 2/
d*sin() = 
EE16.468/16.568
Lecture 1
Distributed feedback grating (DFB grating)
Effective reflection


kin
kin
k
k
kout
kout = kin – k
Phase-matching condition
2nout / out = 2nin / in – 2/
kout
kout = kin – k = - kin
Phase-matching condition
2/ = 2*2/
 = /2
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Diffraction: Multiple beam interference
Lecture 1
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Diffraction:
Lecture 1
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Single slot diffraction:
Lecture 1
Atotoal  A0   A1  A2  ...  Am
 A0eikx0  A0eikx1  A0eikx2  ...  A0eikxm
d
  A0e
ikz sin( )
0


dz A0 (1  eikd sin( ) )

d
ikd sin(  )
I totoal  Atotoal 
2
 I0
e
ikd sin / 2
ikd sin( )
A0 (1  e
)
kd sin(  )
ikd sin / 2
(e
e
kd sin(  )
ikd sin / 2
2
2
2
)
2i sin( kd sin  / 2)
sin( kd sin  / 2)
 I0
 I0
kd sin(  )
kd sin  / 2
 I0
sin 2 ( )

2
,
  kd sin  / 2 

d sin 

2
EE16.468/16.568
Lecture 1
Single slot diffraction:
I totoal  I 0
sin 2 ( )

2
  kd sin  / 2 


,

d sin 



When   d sin   m 

2
When


d sin   m

bright
dark
EE16.468/16.568
Lecture 1
Single slot diffraction:
I totoal  I 0
sin 2 ( )

2
  kd sin  / 2 


,

d sin 



When   d sin   m 

2
When


d sin   m

bright
dark
EE16.468/16.568
Angular width :


Lecture 1
I totoal  I 0
sin 2 ( )

2
,

  kd sin  / 2  d sin 


first dark:   d sin   m



sin  

d
Half angular width
d
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Lecture 1
Diffraction of a lens
d
Focus size
d
Focus size
EE16.468/16.568
Lecture 1
Resolution of optical instrument


d
Focus size f *   f
d
 '  1.22
 '  1.22
∆’
eye

d

d
d = 16mm.

d
EE16.468/16.568
Polarization of light: Linear polarization
Lecture 1
EE16.468/16.568
Lecture 1
Polarization of light: linear polarization
Any linear polarization can be
decomposed into two primary
polarization with the same phase
B0 cos(t )
A0 cos(t )
Why decompose into two primary polarization directions?
In crystal, the refractive index is different along different polarizations
no
z
x
z
y
no
x
ne
Refractive index ellipsoid
y
ne
EE16.468/16.568
Lecture 1
Polarization of light: circular polarization
B0 cos(t )
  tan 1[ B0 cos(t ) / A0 cos(t )]
 tan 1[ B0 / A0 ]
t  0
A0 cos(t )
A0 cos(t   / 2)
A0 cos(t   / 2)
Lag /2
A0 cos(t )
A0 cos(t   / 2)
A0 cos(t )
t   / 2
t  
A0 cos(t   / 2)
A0 cos(t )
A0 cos(t )
EE16.468/16.568
Lecture 1
Polarization of light: circular polarization
A0 cos(t   / 2)
Clockwise
t  3 / 2
A0 cos(t   / 2)
A0 cos(t )
t  2
A0 cos(t )
 A0 sin( t )
A0 cos(t   / 2)

  t
A0 cos(t )
A0 cos(t )
A0 cos(t )
EE16.468/16.568
Lecture 1
Polarization of light: circular polarization
Count-clockwise
A0 sin( t )
A0 cos(t   / 2)

  t
A0 cos(t )
A0 cos(t )
elliptical polarization
B0 sin( t )
B0 cos(t   / 2)

A0 cos(t )
  t
A0 cos(t )
EE16.468/16.568
Lecture 1
Delays along different directions:
z
d
y
x
no
d
no
no
z
x
y
ne
ne
Refractive index ellipsoid
no lag
Phase difference: k (n0  ne )d
ne
ne
when k (n0  ne )d   / 2
(n0  ne )d   / 4
Right (clockwise) circular polarization
Quarter-wavelength /4 plate
EE16.468/16.568
Lecture 1
Right (clockwise) circular polarization
A0 cos(t   / 2)
A0 cos(t )
/4 plate
45º

A0 cos(t )
A0 cos(t )
d
no
no lag
Phase difference: k (n0  ne )d
ne
ne
when
k (n0  ne )d  
(n0  ne )d   / 2
Change the direction of the polarization
half-wavelength /2 plate
EE16.468/16.568
Lecture 1
Change the direction of the polarization by 2
A0 cos(t )
/2 plate

A0 cos(t   )

A0 cos(t )
A0 cos(t )
EO modulator
polarizer
analyzer
dark
no
no
Bright
ne
EE16.468/16.568
Lecture 1
EO modulator
A0 cos(t )
dark
analyzer
polarizer
A0 cos(t )
no
k (n0  ne )d  
n0  E 
V
t
V

ne

Bright
EE16.468/16.568
Lecture 1
Circular polarizations and linear polarizations
A0 cos(t   / 2)
A0 cos(t   / 2)


A0 cos(t )
A0 cos(t )
A0 cos(t )

A0 cos(t   / 2)
EE16.468/16.568
Lecture 1
Circular polarizations and linear polarizations
A0 cos(t )
A0 cos(t   / 2)


A0 cos(t   / 2)
A0 cos(t )
+
A0 cos(t )  A0 cos(t   / 2)
Linear polarization
A0 cos(t )  A0 cos(t   / 2)
EE16.468/16.568
Lecture 1
Circular polarizations and linear polarizations
A0 cos(t   )
A0 cos(t   / 2)


A0 cos(t   / 2   )
A0 cos(t )
+
cos( )  cos(  )  2 cos(
A0 cos(t   )  A0 cos(t   / 2)
 2 A0 cos(t   / 2   / 4) cos( / 2   / 4)
tan(  ' ) 
cos( / 4   / 2)
cos( / 4   / 2)


2
) cos(
 
2
Linear polarization
A0 cos(t )  A0 cos(t   / 2   )
 2 A0 cos(t   / 4   / 2) cos( / 4   / 2)
)
EE16.468/16.568
Lecture 1
Faraday rotation
Apply B field generate delay
for left or right polarization
A0 cos(t   )

A0 cos(t   / 2)

A0 cos(t   / 2   )
A0 cos(t )
EE16.468/16.568
Lecture 1
Jones Matrix
1
 
0
E0
E0
0
 
1
E0 sin(  )

E0 cos( )
 cos  


 sin  
EE16.468/16.568
Lecture 1
Jones Matrix

E0 cos(t  )
2

1 1
 
2 i
E0 cos(t )

E0 cos(t  )
2

E0 cos(t )
1 1
 
2 i 

b cos(t  )
2

a cos(t )
a
 
 ib 
EE16.468/16.568
Lecture 1
Jones Matrix

E0 cos(t  )
2
1 1
 
2 i

Linear
E0 cos(t )

E0 cos(t  )
2
1
 
0
+

E0 cos(t )
1 1
 
2 i 

b cos(t  )
2

a cos(t )
a
 
 ib 
EE16.468/16.568
Lecture 1
Jones Matrix
1
 
0
1 0


0 i
/2 phase delay 
/4 plate
 1 0  1   1 

    
 0 i  0   0 
E0
1 1
 
2 1
/4 plate
 /4
1 0


0 i 
/2 phase delay
E0 cos( )

E0 cos(t  )
2

E0 cos(t )
1 1
 
2 i 
1  1 0 1

  
2  0 i 1
EE16.468/16.568
y’
Lecture 1
Jones Matrix
y
x  x' cos   y ' ( sin  )
x’

y  x' sin   y ' cos 
x
 x   cos 
   
 y   sin 
 cos 

  sin 
 x'   cos 
   
 y '    sin 
sin   x 
 
cos   y 
 cos 
R( )  
 sin 
 sin  

cos  
 x' 
 x
   R( ) 
 y' 
 y
 sin   x' 
 
cos   y ' 
sin   x   x' 
    
cos   y   y ' 
EE16.468/16.568
y’
y
Lecture 1
Jones Matrix
x’
  90 0
1
 
0
 cos 
R( )  
 sin 
x
 0  1

R(  90 )  
1 0 
0
 1   0  1 1   0 
    
R(  90 )   
 0   1 0  0   1 
0
E0
 sin  

cos  
EE16.468/16.568
Lecture 1
Jones Matrix

b cos(t  )
2

a
 
 ib 
a cos(t )
 cos 
R( )  
 sin 

b cos(t  )
2

 sin  

cos  

a cos(t )
 a   cos 
R( )   
 ib   sin 
 sin   a   a cos   ib sin  
   

cos   ib   a sin   ib cos  
EE16.468/16.568

E0 cos(t  )
2


E0 cos(t )
1 1
 
2 i 
Lecture 1
Jones Matrix
1  cos 
R( )   
 i   sin 
 cos 
R( )  
 sin 
 sin  

cos  
 sin  1  cos   i sin  
   

cos   i   sin   i cos  
EE16.468/16.568
Lecture 1
Jones Matrix

E0 cos(t  )
2
1 1
 
2 i

Linear
E0 cos(t )

E0 cos(t  )
2
+


E0 cos(t )
1  cos

 sin  
 cos   i sin  


 sin   i cos  
 2 cos( / 2)cos( / 2)  i sin(  / 2) 


 2 sin(  / 2)cos( / 2)  i sin(  / 2)  
 cos( / 2)  sin(  / 2)  1 
 
 cos( / 2)  i sin(  / 2) 
 sin(  / 2) cos( / 2)  0 
 2 cos 2 ( / 2)  i 2 sin( 

 2 sin(  / 2) cos( / 2) 

EE16.468/16.568
Lecture 1
Faraday rotation
Apply B field generate delay
for left or right polarization
A0 cos(t   )

A0 cos(t   / 2)

A0 cos(t   / 2   )
A0 cos(t )