EE485 Introduction to Photonics Superposition of Waves and Interference

1. Two-beam interference and interferometry

2. Multi-wave interference

3. Fabry-Perot interferometer

4. Group/phase velocity and dispersion

Reading: Pedrotti3, Sec. 5.1-5.2, 5.5-5.6, 7.1-7.2, 7.4, 7.9, 8.1-8.7, 8-9, 10.7-10.8

Reference: J. S. Verdeyen, “Laser Electronics,” 3rd ed., Sec. 6.1-6.3, Prentice Hall

Lih Y. Lin 2

Superposition Principle

+

1 1( , ), t I r 2 2( , ), t I r

1 2( , ) ( , ) ( , )t t t r r r 1 2I I I or or Both? √ X X

Formally speaking,

If 1 and 2 are independently solutions of the wave equation, 2

2

2 2

1

v t

then the linear combination,

1 2

, : constants

a b

a b

is also a solution.

For electromagnetic waves,

1 2 1 2 E E E H H H(Orientation of the fields must be taken into account.)

Lih Y. Lin 3

Two-Beam Interference

1 01 1 1 01 1

2 02 2 2 02 2

1 2

exp[ ( )] exp[ ( )]

exp[ ( )] exp[ ( )]

i t i t

i t i t

E E k r E

E E k r E

E E E

Same frequency, different phase

Recall from Light as Electromagnetic Waves, optical intensity =

21| | Re ( ) ( )* ( )

2I

S E r H r E r

For the combined wave,

* *1 2 1 2( ( ) ( )) ( ( ) ( ))I E r E r E r E r

1 2 1 2 1 2

1 2 2 1

1 2 1 2

1 2

exp[ ( )]

exp[ ( )]

2 cos

I I I I I i

I I i

I I I I I

If E01 and E02 are parallel to each other,

max min

max min

Fringe contrastI I

I I

Lih Y. Lin 4

Young’s Double-slit Experiment

0

20

2 [1 cos( sin )]

sin 4 cos

I I ka

aI

sinka

For points P near the optical axis,

204 cos

ayI I

L

Discussion

If a thin plate of glass is placed over one

of the slits, what will happen to the fringe

pattern?

Lih Y. Lin 5

Michelson Interferometer (I)

λ = 632.8 nm λ = 420 nm

02 1 cos 2I I

: Optical path length difference

Dark fringes: 2 cosd m

| |2 sin

m

d

Lih Y. Lin 6

Michelson Interferometer (II) Sensing small refractive index change

Input

x

n 0

( 1)2( ) 2 1 cos 2

n xI n I

Start with dark fringe in the center,

Example:

5

632.8 nm

0.5 10 3.164 mmx

→n of 10-5 results in

2 phase change.

Resolving small wavelength difference (Spectrometer)

Input Start with coincident bright fringe in the center

for both 1 and 2.

d

Move the mirror by d until the next coincidence

occurs. 2

2 d

Lih Y. Lin

Mach-Zehnder Interferometer

Free-space Fiber-optic

Example: Design a fiber-optic Mach-Zehnder interferometer that can

demultiplex two light signals of free-space wavelength 1 = 1551 nm and

2 = 1550 nm. Through which output port would light of 3 = 1549 nm and

4 = 1548 nm exit?

Lih Y. Lin 8

Interference in Dielectric Films

Additional phase shift occurs when

reflected by higher index material.

In normal incidence (from n1 material to

n2 material), the reflection coefficient

2 1

1

1

/ : Relative index

nr

n

n n n

Anti-reflection (AR) coating (0-th order):

0 , / 4ff sn n n t

High-reflection coating (dielectric mirror)

high/4

low/4

high/4

low/4

high/4

low/4

Lih Y. Lin 9

Multi-Wave Interference ― Equal Amplitude and Equal Phase Difference

1

22

0 2

exp[ ( 1) ], 1,2,...,

1 exp( )

1 exp( )

sin ( / 2)

sin ( / 2)

m o

M

m om

U I i m m M

iMU U I

i

MI U I

Interesting features:

• Mean intensity 2

0

0

1

2I Id MI

• Peak intensity = M2I0

• Intensity drops to zero at =

2/M from peak intensity.

• Sensitivity to increases with M.

• (M - 2) minor peaks between

major peaks.

M = 5

Lih Y. Lin 10

Multi-Wave Interference Example: Bragg Reflection

Intensity of the reflected light maximizes

at

sin2

n

d

One of the applications:

Characterizing the lattice constant of a

crystal.

= 1.16 Å

1.162.48

2sin(27 / 2)d

Å

For (200) direction,

Lih Y. Lin 11

Multi-Wave Interference ― Progressively Smaller Amplitude and Equal Phase Difference

Parallel plate Fabry-Perot Interferometer

r, r’: Reflection coefficient

t, t’: Transmission coefficient

Two high-reflection plates separated by

a distance d.

d is often tunable.

Lih Y. Lin 12

Fabry-Perot Interferometer ― Principle

r1, r2: Reflection coefficients

t1, t2: Transmission coefficients

(These all work on fields.)

0

20 1 2

0

1

2

n i

EE

e

nd

r r

Consider just to the right of

mirror M1,

0 1

2

i

itr n

E t E

E t e E

Furthermore,

Lih Y. Lin 13

Fabry-Perot Interferometer ― Spectra

Define power reflectivity of the mirror 2 2, 1R t R r

Transmittance

2

1 2

2 21 2 1 2

(1 ) (1 )

(1 ) 4 sin

tr

i

E R RT

E R R R R

Reflectance

2

1rnet

i

ER T

E

Quiz: For sharper transmission spectrum,

do we want higher R1,2 or lower R1,2?

Lih Y. Lin 14

Fabry-Perot Interferometer ― Parameters

1 2

1/2 1/41 2

1

2 ( )

R Rc

nd R R

Free spectral range (FSR)

20,

2FSR FSR FSR

c

nd c

FWHM

Cavity Q-factor

1/ 40 1 2

1/ 2 0 1 2

( )2

1

R RndQ

R R

Example

A HeNe laser ( = 632.8 nm) cavity is defined by d ~ 1 m,

R1 = R2 = 0.99

→ 0 = 4.74 x 1014, 1/2 = 480 kHz, Q = 9.88 x 108,

and F = 313.

Finesse

1/41 2

1/2 1 2

( )

1

FSR R R

R R

F

Lih Y. Lin 15

Fabry-Perot Interferometer ― Tunable Filter and Spectrometer

0

2 ndd m

c

Resonant condition T

Resonant wavelength

0

2nd

m

But the Fabry-Perot interferometer has finite passband width …

T

d

1 2

Resolving criterion:

Spacing less than what’s defined by T = 0.5 Tmax

()min

21 20

min 1/41 2

1

2 ( )

R R

nd R R

Exercise:

Design a tunable filter that can separate 1 =

1550.918 nm and 2 = 1552.524 nm (Two standard

wavelengths in optical fiber communication systems). min

Resolving power

R

Lih Y. Lin 16

MEMS Tunable Filters and Modulators

V

Via-hole

silicon

Output fiber

Input fiber

Ground

Bottom mirror

Top curved mirror

100 mm

Filter Tuning plot

-60

-50

-40

-30

-20

-10

0

1500 1520 1540 1560 1580

Wavelength (nm)

Tra

ns

mis

sio

n (

dB

) 8 V12 V15 V18 V21 V

0 < Vdrive < 30V

3/4 < gap < /2

input

/4 SiNx

Silicon

PSG

reflect

transmit

Vdrive

/4

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Re

fle

ctivity

1600 1500 1400 1300

Wavelength (nm)

gap 3 o

o /2 gap

o

100 mm

22 mm

Ford, Walker, Greywall & Goossen, IEEE J. Lightwave Tech. 16, 1998

Lih Y. Lin 17

Two-Wave Interference ― Different f and

1 0 1 1

2 0 2 2

1 2 0

1 2 1 2

1 2 1 2

cos( )

cos( )

2 cos( )cos( )

, 2 2

| | | |,

2 2

p p g g

p p

g g

E E k x t

E E k x t

E E E E k x t k x t

k kk

k kk

cos( )p pk x t : Average

cos( )g gk x t : Modulation, beating

Beat frequency 1 22b g

constructive

interference

destructive

interference

Lih Y. Lin 18

Phase Velocity, Group Velocity, and Dispersion

Phase velocity:

Assume the two waves are close in and k.

1 2

1 2

p

p

p

cv

k k k k n

Group velocity: 1 2

1 2

g

g

g

dv

k k k dk

1g p

dnv v

n d

Dispersion. n = n(). Light with different wavelength

travels with different velocity in a medium.

Normal dispersion: dn/d < 0, vg < vp.

0 2 4 6 8 101

0

11

1

E x( )

Envp1 x( )

Envp2 x( )

100 x

0 2 4 6 8 101

0

11

1

E x( )

Envp1 x( )

Envp2 x( )

100 x

0 2 4 6 8 101

0

11

1

E x( )

Envp1 x( )

Envp2 x( )

100 x

t0

t1

t2

vg determines the speed with which energy is

transmitted. It is the directly measurable speed of

the wave.

Lih Y. Lin 19

Dispersion in an Optical Fiber

Dispersion coefficient:

→ A measure of time delay per wavelength

(nm) after certain distance.

Group index: g

cv

N

dnN n

d

2

2 (s/m-nm)

d nD

c d

Bandwidth limited by dispersion:

max

0.5

( / )L

L

If the original pulse width cannot be

neglected compared to the broadening,

2 2 2

0 ( )f

max 0.5 / f

Lih Y. Lin 20

Pulse Broadening in an Optical Fiber

or,

√ X

√ X

Lih Y. Lin 21

Multi-Wave Interference ― Different Frequencies

|Uf|

f f0 f

1, 2, …, M At a given position r,

( 1) / 2

0 0( 1) / 2

22

0 2

( ) exp[ 2 ( ) ]

sin ( / )( ) | ( ) |

sin ( / )

1

M

q M

F

F

F

U t I i f q f t

M t TI t U t I

t T

Tf

1

M f

Total frequency bandwidth

E.g., 1 ps pulse can be generated by combining 1000 waves separated by 1 GHz from each other. (Note: The exact relation between pulse width and bandwidth depends on the pulse shape. This is a subject of

Fourier Optics.)

(Compare this with multi-wave interference

of same frequencies but different phases.)

M = 5