FILTER DESIGN FUNDAMENTALS

Electronic Materials Group, Massachusetts Institute of Technology

How to turn wave interference into something useful ?

Luca Dal Negro

Contents

Electronic Materials Group, Massachusetts Institute of Technology

Motivations

Recap on wave optics and interference

Optical components: physics and device parameters

Interferometers

Multiple waves interference

Fabry-Perot resonators (physics and device parameters)

m - Ring resonators

Bragg filters

Diffraction gratings

Filters everywhere: system application from the real world

Fiber grating, Chirp grating, Add/Drop filters, filter-based WDM

Starting Motivations

Electronic Materials Group, Massachusetts Institute of Technology

Filters applications in optical systems = everywhere

• Amplifier noise suppression• Pump laser stabilization• gain compensation• dispersion compensation• Add/Drop devices• multiplex/demultiplexfor WDM functions

• laser devices• light- matter interaction enhancement

Wave optics

Electronic Materials Group, Massachusetts Institute of Technology

[ ]),(Re),( trUtru =

Complex wavefunction

ftierUtrU π2)(),( = )()()( rierarU ϕ=

Complex amplitude

Helmholtz equation(linear as the general wave equation)0)()( 22 =+∇ rUk

Superposition principles ⇒ Interference

Wave interference

Electronic Materials Group, Massachusetts Institute of Technology

)cos(2 2121 ϕIIIII ++=

λπϕ /2 dkd ==

Phase of the wave

2)(rUI =

Light Intensity

Fabry-Perot Etalon

Electronic Materials Group, Massachusetts Institute of Technology

20

max

2/1

)1(

)1(

2

rII

rrF

dc

F

−=

−=

=

π

ν

[ ]22max

)2/sin()/2(1)(

ϕπϕ

FII

+=

Losses in a Fabry-Perot

Electronic Materials Group, Massachusetts Institute of Technology

m-ring filters

Electronic Materials Group, Massachusetts Institute of Technology

Resonant condition:

go nmmr /2 λλπ ==

waveguidebus

ring resonator

Si

St

Aκ

κ

Number of energy circulations in a ring:

rQvQ

rv

rv

rL

n gggeff

πωωππτ

π====

2222

busg

ring Pr

QvP

πω2

=Optical power in a ring:

Wavelength response of a ring

Electronic Materials Group, Massachusetts Institute of Technology

From B. E. Little et al., J. of Light Wave Tech., 15, 6, 998, 1997

Coupling identical rings

Electronic Materials Group, Massachusetts Institute of Technology

Flattening the resonance peak shape

From B. E. Little et al., J. of Light Wave Tech., 15, 6, 998, 1997

Fundamentals of Gratings

Electronic Materials Group, Massachusetts Institute of Technology

[ ][ ]2

2

0 )2/sin()2/sin(

ϕϕMII =

θϕ sin2kd=

Bragg phase change:

λθ nd =)sin(2Bragg condition:

Electronic Materials Group, Massachusetts Institute of Technology

Electronic Materials Group, Massachusetts Institute of Technology

Bragg waveguides

Electronic Materials Group, Massachusetts Institute of Technology

Λ≈=βπλ 2

22 βδ ∆−= g

Bragg scattering condition:

2

)sinh()cosh( LiLT

δβδδδ∆+

=

2g

Electronic Materials Group, Massachusetts Institute of Technology

Electronic Materials Group, Massachusetts Institute of Technology

Electronic Materials Group, Massachusetts Institute of Technology

Suggested Readings

Electronic Materials Group, Massachusetts Institute of Technology

Optics• Born and Wolf, Principles of Optics (especially for T-matrix approach)• E.Rosencher, B.Vinter, Optoelectronics, Cambridge University Press.• Saleh andTeich, Photonics, JW• A. Yariv, Quantum electronics

Optical Systems

• P.C. Becker, N.A. Olsson, J.R. Simpson, Erbium doped Fiber amplifiers, AP• C.K. Madsen, J.H.Zhao, Optical Filter Design and Analysis, JW