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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