EE16.468/16.568Lecture 1 Class overview: 1.Brief review of physical optics, wave propagation, interference, diffraction, and polarization 2.Introduction

$Page 1: EE16.468/16.568Lecture 1 Class overview: 1.Brief review of physical optics, wave propagation, interference, diffraction, and polarization 2.Introduction$
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


2

1 aaaiknAe = )( 21

2

1 bbiknAe


2

1 aaaiknAe = - )( 21

2

1 bbiknAe

b3n2

n


)(222

nabnk

Electro-optic effect, n2 changes with E -field

EE16.468/16.568 Lecture 1

Electro-optic modulator based on March-Zehnder interferometer


2

1 aaaiknAe

=

)( 21

2

1 bbiknAe


2

1 aaaiknAe = -

)( 21

2

1 bbiknAe


)(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


Helmholtz Equation:

0220

2 EnkE

x

z

n1

n2

Ein

Et

Er

trin EEE

tr 1

z

Ex

znikr


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


Helmholtz Equation:

0220

2 EnkE

x n1

n2

EinEr

trin EEE

tr 1

znikr


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


Helmholtz Equation:

0220

2 EnkE

x n1

n2

EinEr

trin EEE

tr 1

znikr


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


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


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


n1

n2

1 1

2

x

z

y

Ein

Er

Et

znkxnkiin




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


n1

n2

1 1

2

x

z

y

Ein

Er

Et

znkxnkiin




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


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


11 sinˆcosˆ

1

2


11 sinˆcosˆ


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


11 sinˆcosˆ

1

2


11 sinˆcosˆ


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


11 sinˆcosˆ

1

2


11 sinˆcosˆ


22 sinˆcosˆ

201010 nkEnkEnkE trin


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


11 sinˆcosˆ

1

2


11 sinˆcosˆ


22 sinˆcosˆ



201010 nkEnkEnkE 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


Helmholtz Equation:

0220

2 EnkE

x n1

n2

EinEr

znikr


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


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


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


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


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


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


n2

n1 A1 Lnike 20

B1

A2

B2

L

A3

B3

A4

B4

znikznik eBeAE 1010111


znikeAE 1044


n2

n1 A1 Lnike 20

B1

A2

B2

L

A3

B3

A4

B4



znikeAE 1044

n2

n1 A5 Lnike 20

B5

A6

B6

A7

B7

A8

B8

Ld

EE16.468/16.568 Lecture 1Bragg gratings

n1 n2

d1 d2

/4

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

Ring cavity, Ring resonator filter

LnFSR

2

2

2

L = 2R0

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

Lens and optical path Focal length

f

Focal plane

Same optical path

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

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


2nout / out = 2nin / in – 2/

kin

kout k

kout = kin – k = - kin


2/ = 2*2/

Effective reflection

= /2

EE16.468/16.568 Lecture 1

Diffraction: Multiple beam interference

EE16.468/16.568 Lecture 1

Diffraction:

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


,)(sin

2

2

0

II totoal

sin2/sin dkd

When 2

sin

md bright

When md sin dark

EE16.468/16.568 Lecture 1


,)(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

Diffraction of a lens

d

d

Focus size

Focus size

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

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


)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


)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


)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


)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

0

1

0E

)cos(0 E

)sin(0 E

0E

1

0

sin

cos


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


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


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


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


090x

x’yy’

01

10)90( 0R

0

1

0E

1

0

0

1

01

10

0

1)90( 0R

cossin

sincos)(R


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


)cos(0 tE

)2

cos(0

tE

i

1

2

1

cossin

sincos)(R

cossin

sincos1

cossin

sincos1)(

i

i

iiR


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

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

Documents

EE16.468/16.568Lecture 1 Class overview: 1.Brief review of physical optics, wave propagation, interference, diffraction, and polarization 2.Introduction