The design and modeling of microstructured optical fiber

1
The design and modeling ofmicrostructured
optical fiber

• Steven G. Johnson
• MIT / Harvard University

2
Outline
What are these fibers (and why should I care)?
The guiding mechanisms index-guiding and band
gaps
Finding the guided modes
Small corrections (with big impacts)
3
Outline
What are these fibers (and why should I care)?
The guiding mechanisms index-guiding and band
gaps
Finding the guided modes
Small corrections (with big impacts)
4
Optical Fibers Today(not to scale)
silica cladding n 1.45
R. Ramaswami K. N. Sivarajan, Optical
Networks A Practical Perspective
5
The Glass Ceiling Limits of Silica
Radical modifications to dispersion, polarization
effects? tunability is limited by low index
contrast
High Bit-Rates
Compact Devices
Long Distances
Dense Wavelength Multiplexing (DWDM)
6
Breaking the Glass CeilingHollow-core Bandgap
Fibers
Photonic Crystal
7
Breaking the Glass CeilingHollow-core Bandgap
Fibers
Bragg fiber
Yeh et al., 1978
figs courtesy Y. Fink et al., MIT
omnidirectional OmniGuides
white/grey chalco/polymer
silica
R. F. Cregan et al., Science 285, 1537
(1999)
5µm
PCF
Knight et al., 1998
8
Breaking the Glass CeilingHollow-core Bandgap
Fibers
Guiding _at_ 10.6µm (high-power CO2 lasers) loss lt 1
dB/m (material loss 104 dB/m)
figs courtesy Y. Fink et al., MIT
white/grey chalco/polymer
Temelkuran et al., Nature 420, 650 (2002)
silica
Guiding _at_ 1.55µm loss 13dB/km
R. F. Cregan et al., Science 285, 1537
(1999)
5µm
Smith, et al., Nature 424, 657 (2003)
OFC 2004 1.7dB/km BlazePhotonics
9
Breaking the Glass Ceiling IISolid-core Holey
Fibers
solid core
holey cladding forms effective low-index material
Can have much higher contrast than doped silica
strong confinement enhanced nonlinearities,
birefringence,
J. C. Knight et al., Opt. Lett. 21, 1547 (1996)

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Breaking the Glass Ceiling IISolid-core Holey
Fibers
nonlinear fibers
endlessly single-mode
Wadsworth et al., JOSA B 19, 2148 (2002)
T. A. Birks et al., Opt. Lett. 22, 961 (1997)
polarization -maintaining
low-contrast linear fiber (large area)
K. Suzuki, Opt. Express 9, 676 (2001)
J. C. Knight et al., Elec. Lett. 34, 1347
(1998)
11
Outline
What are these fibers (and why should I care)?
The guiding mechanisms index-guiding and band
gaps
Finding the guided modes
Small corrections (with big impacts)
12
Universal Truths Conservation Laws
an arbitrary-shaped fiber
(1) Linear, time-invariant system (nonlinearities
are small correction)
z
frequency w is conserved
cladding
(2) z-invariant system (bends
etc. are small correction)
wavenumber b is conserved
core
electric (E) and magnetic (H) fields can be
chosen
E(x,y) ei(bz wt),
H(x,y) ei(bz wt)
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Sequence of Computation
Plot all solutions of infinite cladding as w vs. b
1
w
light cone
b
empty spaces (gaps) guiding possibilities
Core introduces new states in empty spaces
plot w(b) dispersion relation
2
Compute other stuff
3
14
Conventional Fiber Uniform Cladding
uniform cladding, index n
kt
b
(transverse wavevector)
w
light cone
light line w c b / n
b
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Conventional Fiber Uniform Cladding
uniform cladding, index n
b
w
light cone
higher-order
core with higher index n pulls down index-guided
mode(s)
fundamental
w c b / n'
b
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PCF Periodic Cladding
periodic cladding e(x,y)
Blochs Theorem for periodic systems
fields can be written
b
a
E(x,y) ei(bzkt xt wt),
H(x,y) ei(bzkt xt wt)
transverse (xy) Bloch wavevector kt
periodic functions on primitive cell
primitive cell
where
satisfies eigenproblem (Hermitian if lossless)
constraint
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PCF Cladding Eigensolution
Finite cell ? discrete eigenvalues wn
Want to solve for wn(kt, b), plot vs. b for
all n, kt
constraint
where
H(x,y) ei(bzkt xt wt)
Limit range of kt irreducible Brillouin zone
1
Limit degrees of freedom expand H in finite basis
2
Efficiently solve eigenproblem iterative methods
3
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PCF Cladding Eigensolution
Limit range of kt irreducible Brillouin zone
1
Blochs theorem solutions are periodic in kt
M
K
first Brillouin zone minimum kt primitive
cell
G
irreducible Brillouin zone reduced by symmetry
Limit degrees of freedom expand H in finite basis
2
Efficiently solve eigenproblem iterative methods
3
19
PCF Cladding Eigensolution
Limit range of kt irreducible Brillouin zone
1
Limit degrees of freedom expand H in finite basis
2
must satisfy constraint
Planewave (FFT) basis
Finite-element basis
constraint, boundary conditions
Nédélec elements
Nédélec, Numerische Math. 35, 315 (1980)
constraint
nonuniform mesh, more arbitrary
boundaries, complex code mesh, O(N)
uniform grid, periodic boundaries, simple code,
O(N log N)
figure Peyrilloux et al., J. Lightwave
Tech. 21, 536 (2003)
Efficiently solve eigenproblem iterative methods
3
20
PCF Cladding Eigensolution
Limit range of kt irreducible Brillouin zone
1
Limit degrees of freedom expand H in finite
basis (N)
2
solve
finite matrix problem
Efficiently solve eigenproblem iterative methods
3
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PCF Cladding Eigensolution
Limit range of kt irreducible Brillouin zone
1
Limit degrees of freedom expand H in finite basis
2
Efficiently solve eigenproblem iterative methods
3
Slow way compute A B, ask LAPACK for
eigenvalues requires O(N2) storage, O(N3) time
Faster way start with initial guess
eigenvector h0 iteratively improve O(Np)
storage, O(Np2) time for p eigenvectors
(p smallest eigenvalues)
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PCF Cladding Eigensolution
Limit range of kt irreducible Brillouin zone
1
Limit degrees of freedom expand H in finite basis
2
Efficiently solve eigenproblem iterative methods
3
Many iterative methods Arnoldi, Lanczos,
Davidson, Jacobi-Davidson, ,
Rayleigh-quotient minimization
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PCF Cladding Eigensolution
Limit range of kt irreducible Brillouin zone
1
Limit degrees of freedom expand H in finite basis
2
Efficiently solve eigenproblem iterative methods
3
Many iterative methods Arnoldi, Lanczos,
Davidson, Jacobi-Davidson, ,
Rayleigh-quotient minimization
for Hermitian matrices, smallest eigenvalue w0
minimizes
minimize by conjugate-gradient, (or multigrid,
etc.)
variational theorem
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PCF Holey Silica Cladding
n1.46
2r
a
r 0.1a
light cone
w (2pc/a)
w bc
b (2p/a)
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PCF Holey Silica Cladding
n1.46
2r
a
r 0.17717a
light cone
w (2pc/a)
w bc
b (2p/a)
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PCF Holey Silica Cladding
n1.46
2r
a
r 0.22973a
light cone
w (2pc/a)
w bc
b (2p/a)
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PCF Holey Silica Cladding
n1.46
2r
a
r 0.30912a
light cone
w (2pc/a)
w bc
b (2p/a)
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PCF Holey Silica Cladding
n1.46
2r
a
r 0.34197a
light cone
w (2pc/a)
w bc
b (2p/a)
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PCF Holey Silica Cladding
n1.46
2r
a
r 0.37193a
light cone
w (2pc/a)
w bc
b (2p/a)
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PCF Holey Silica Cladding
n1.46
2r
a
r 0.4a
light cone
w (2pc/a)
w bc
b (2p/a)
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PCF Holey Silica Cladding
n1.46
2r
a
r 0.42557a
light cone
w (2pc/a)
w bc
b (2p/a)
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PCF Holey Silica Cladding
n1.46
2r
a
r 0.45a
light cone
w (2pc/a)
w bc
b (2p/a)
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PCF Holey Silica Cladding
n1.46
2r
a
r 0.45a
light cone
w (2pc/a)
air light line w bc
b (2p/a)
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Bragg Fiber Cladding
Bragg fiber gaps (1d eigenproblem)
at large radius, becomes planar
w
nhi 4.6
nlo 1.6
b
radial kr (Bloch wavevector)
0 by conservation of angular momentum
kf
wavenumber b
b 0 normal incidence
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Omnidirectional Cladding
Bragg fiber gaps (1d eigenproblem)
w
omnidirectional (planar) reflection
e.g. light from fluorescent sources is trapped
for nhi / nlo big enough and nlo gt 1
J. N. Winn et al, Opt. Lett. 23, 1573 (1998)
wavenumber b
b
b 0 normal incidence
36
Outline
What are these fibers (and why should I care)?
The guiding mechanisms index-guiding and band
gaps
Finding the guided modes
Small corrections (with big impacts)
37
Sequence of Computation
Plot all solutions of infinite cladding as w vs. b
1
w
light cone
b
empty spaces (gaps) guiding possibilities
Core introduces new states in empty spaces
plot w(b) dispersion relation
2
Compute other stuff
3
38
Computing Guided (Core) Modes
Same differential equation as before, except no
kt
constraint
can solve the same way
where
magnetic field H(x,y) ei(bz wt)
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Computing Guided (Core) Modes
Boundary conditions
1
computational cell
Only care about guided modes exponentially
decaying outside core
Effect of boundary cond. decays exponentially
mostly, boundaries are irrelevant!
periodic (planewave), conducting, absorbing all
okay
Leakage (finite-size) radiation loss
2
Interior eigenvalues
3
40
Guided Mode in a Solid Core
small computation only lowest-w band!
( one minute, planewave)
holey PCF light cone
flux density
1.46 bc/w 1.46 neff
fundamental mode (two polarizations)
n1.46
2r
endlessly single mode Dneff decreases with l
a
r 0.3a
l / a
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Fixed-frequency Modes?
Here, we are computing w(b'), but we often want
b(w') l is specified
No problem! Just find root of w(b') w', using
Newtons method
(Factor of 34 in time.)
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Computing Guided (Core) Modes
Boundary conditions
1
computational cell
Only care about guided modes exponentially
decaying outside core
Effect of boundary cond. decays exponentially
mostly, boundaries are irrelevant!
periodic (planewave), conducting, absorbing all
okay
Leakage (finite-size) radiation loss
2
Interior eigenvalues
3
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Computing Guided (Core) Modes
Boundary conditions
1
Leakage (finite-size) radiation loss
2
Use PML absorbing boundary layer
perfectly matched layer
Berenger, J. Comp. Phys. 114, 185 (1994)
with iterative method that works for
non-Hermitian (dissipative) systems Jacobi-David
son,
Interior eigenvalues
3
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Computing Guided (Core) Modes
Boundary conditions
1
n1.45
Leakage (finite-size) radiation loss
2
d
imaginary-distance BPM
L
Saitoh, IEEE J. Quantum Elec. 38, 927 (2002)
2 rings
3 rings
Interior eigenvalues
3
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Computing Guided (Core) Modes
Boundary conditions
1
Leakage (finite-size) radiation loss
2
Interior eigenvalues
3
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Computing Guided (Core) Modes
Boundary conditions
1
Leakage (finite-size) radiation loss
2
Interior (of the spectrum) eigenvalues
3
Compute N lowest states first deflation (orthogon
alize to get higher states)
i
see previous slide
Gap-guided modes lie above continuum ( N states
for N-hole cell)
but most methods compute smallest w (or largest
b)
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Interior Eigenvalues by FDTD
finite-difference time-domain
Simulate Maxwells equations on a discrete
grid, PML boundaries eibz z-dependence
Excite with broad-spectrum dipole ( ) source
Dw
Response is many sharp peaks, one peak per mode
signal processing
complex wn
Mandelshtam, J. Chem. Phys. 107, 6756 (1997)
decay rate in time gives loss Imb Imw /
dw/db
48
Interior Eigenvalues by FDTD
finite-difference time-domain
Simulate Maxwells equations on a discrete
grid, PML boundaries eibz z-dependence
Excite with broad-spectrum dipole ( ) source
Dw
Response is many sharp peaks, one peak per mode
narrow-spectrum source
mode field profile
49
An Easier Problem Bragg-fiber Modes
In each concentric region, solutions are Bessel
functions c Jm (kr) d Ym(kr) ? eimf
angular momentum
At circular interfaces match boundary
conditions with 4 ? 4 transfer matrix
search for complex b that satisfies finite at
r0, outgoing at r?
Johnson, Opt. Express 9, 748 (2001)
50
Hollow Metal Waveguides, Reborn
OmniGuide fiber modes
wavenumber b
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An Old Friend the TE01 mode
lowest-loss mode, just as in metal
non-degenerate mode cannot be split no
birefringence or PMD
52
Bushels of Bessels
A General Multipole Method
White, Opt. Express 9, 721 (2001)
Each cylinder has its own Bessel expansion
only cylinders allowed
(m is not conserved)
With N cylinders, get 2NM ? 2NM matrix of
boundary conditions
Solution gives full complex b, but takes O(N3)
time
more than 45 periods is difficult
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Outline
What are these fibers (and why should I care)?
The guiding mechanisms index-guiding and band
gaps
Finding the guided modes
Small corrections (with big impacts)
54
All Imperfections are Small
(or the fiber wouldnt work)
Material absorption small imaginary De
Nonlinearity small De E2
Acircularity (birefringence) small e boundary
shift
Bends small De Dx / Rbend
Roughness small De or boundary shift
Weak effects, long distances hard to compute
directly use perturbation theory
55
Perturbation Theoryand Related
Methods(Coupled-Mode Theory, Volume-Current
Method, etc.)
Given solution for ideal system compute
approximate effect of small changes
solves hard problems starting with easy
problems provides (semi) analytical insight
56
Perturbation Theoryfor Hermitian eigenproblems
given eigenvectors/values
find change for small
57
Perturbation Theoryfor electromagnetism
e.g. absorption gives imaginary Dw decay!
58
A Quantitative Example
but what about the cladding?
Gas can have low loss nonlinearity
some field penetrates!
may need to use very bad material to get high
index contrast
59
Suppressing Cladding Losses
Material absorption small imaginary De
Mode Losses Bulk Cladding Losses
EH11
Large differential loss
TE01 strongly suppresses cladding
absorption (like ohmic loss, for metal)
TE01
l (mm)
60
High-Power Transmissionat 10.6µm (no previous
dielectric waveguide)
Polymer losses _at_10.6µm 50,000dB/m
waveguide losses 1dB/m
B. Temelkuran et al., Nature 420, 650 (2002)
figs courtesy Y. Fink et al., MIT
61
Quantifying Nonlinearity
Kerr nonlinearity small De E2
Db power P 1 / lengthscale for nonlinear
effects
g Db / P nonlinear-strength parameter
determining self-phase modulation (SPM),
four-wave mixing (FWM),
(unlike effective area, tells where the field
is, not just how big)
62
Suppressing Cladding Nonlinearity
Mode Nonlinearity Cladding Nonlinearity
TE01
Will be dominated by nonlinearity of air 10,000
times weaker than in silica fiber (including
factor of 10 in area)
l (mm)
nonlinearity Db(1) / P g
63
Acircularity Perturbation Theory
(or any shifting-boundary problem)
e2
e1
just plug Des into perturbation formulas?
FAILS for high index contrast!
beware field discontinuity fortunately, a simple
correction exists
S. G. Johnson et al., PRE 65, 066611 (2002)
64
Acircularity Perturbation Theory
(or any shifting-boundary problem)
e2
e1
(continuous field components)
Dh
S. G. Johnson et al., PRE 65, 066611 (2002)
65
Loss from Roughness/Disorder
volume-current method or Greens functions with
first Born approximation
66
Loss from Roughness/Disorder
For surface roughness, including field
discontinuities
67
Loss from Roughness/Disorder
So, compute power P radiated by one localized
source J, and loss rate P (mean disorder
strength)
68
Losses from Point Scatterers
strip
69
theoremBand Gaps Suppress Radiationwithout
increasing reflection
M. Povinelli et al., Appl. Phys. Lett., in
press (2004)
70
Effect of an Incomplete Gap
on uncorrelated surface roughness loss
some radiation blocked
radiation
same reflection
reflection
Conventional waveguide
with Si/SiO2 Bragg mirrors (1D gap) 50 lower
losses (in dB) same reflection
(matching modal area)
71
Considerations for Roughness Loss
Band gap can suppress some radiation
typically by at most 1/2, depending on crystal
72
Using perturbations to designbig effects
73
Perturbation Theory and Dispersion
when two distinct modes cross interact, unusual
dispersion is produced
w
mode 1
mode 2
no interaction/coupling
b
74
Perturbation Theory and Dispersion
when two distinct modes cross interact, unusual
dispersion is produced
w
mode 1
mode 2
coupling anti-crossing
b
75
Two Localized Modes Very Strong Dispersion
w
core mode
localized claddingdefect mode
b
T. Engeness et al., Opt. Express 11, 1175
(2003)
76
(Different-Symmetry) Slow-light Modes Anomalous
Dispersion
w
slow-light band edges at b0
b
b0
M. Ibanescu et al., Phys. Rev. Lett. 92, 063903
(2004)
77
(Different-Symmetry) Slow-light Modes Anomalous
Dispersion
slow light
non-zero velocity
backward wave
ultra-flat (w4)
M. Ibanescu et al., Phys. Rev. Lett. 92, 063903
(2004)
78
(Different-Symmetry) Slow-light Modes Anomalous
Dispersion
Uses gap at b0 perfect metal 1960 or Bragg
fiber or high-index PCF (n gt 2.5)
M. Ibanescu et al., Phys. Rev. Lett. 92, 063903
(2004)
79
Further Reading
Reviews J. D. Joannopoulos, R. D. Meade, and
J. N. Winn, Photonic Crystals Molding the Flow
of Light (Princeton Univ. Press, 1995). P.
Russell, Photonic-crystal fibers, Science 299,
358 (2003).
This Presentation, Free Software, Other
Material http//ab-initio.mit.edu/photons/tutori
al