The design and modeling of microstructured optical fiber Steven G. Johnson

The design and modeling of
microstructured optical fiber
Steven G. Johnson
MIT / Harvard University
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)
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)
Optical Fibers Today
(not to scale)
losses ~ 0.2 dB/km
more complex profiles
to tune dispersion
“high” index
doped-silica core
n ~ 1.46
silica cladding
n ~ 1.45
at l=1.55µm
(amplifiers every
50–100km)
“LP01”
confined mode
field diameter ~ 8µm
protective
polymer
sheath
[ R. Ramaswami & K. N. Sivarajan, Optical Networks: A Practical Perspective ]
but this is
~ as good as
it gets…
The Glass Ceiling: Limits of Silica
Loss: amplifiers every 50–100km
…limited by Rayleigh scattering (molecular entropy)
…cannot use “exotic” wavelengths like 10.6µm
Nonlinearities: after ~100km, cause dispersion, crosstalk, power limits
(limited by mode area ~ single-mode, bending loss)
also cannot be made (very) large for compact nonlinear devices
Radical modifications to dispersion, polarization effects?
…tunability is limited by low index contrast
Long Distances
High Bit-Rates
Compact Devices
Dense Wavelength Multiplexing (DWDM)
Breaking the Glass Ceiling:
Hollow-core Bandgap Fibers
Bragg fiber
1000x better
loss/nonlinear limits
[ Yeh et al., 1978 ]
(from density)
1d
crystal
+ omnidirectional
= OmniGuides
2d
crystal
Photonic Crystal
(You can also
put stuff in here …)
PCF
[ Knight et al., 1998 ]
Breaking the Glass Ceiling:
Hollow-core Bandgap Fibers
Bragg fiber
[ Yeh et al., 1978 ]
[ figs courtesy
Y. Fink et al., MIT ]
white/grey
= chalco/polymer
+ omnidirectional
= OmniGuides
silica
[ R. F. Cregan
et al.,
Science 285,
1537 (1999) ]
5µm
PCF
[ Knight et al., 1998 ]
Breaking the Glass Ceiling:
Hollow-core Bandgap Fibers
Guiding @ 10.6µm
[ figs courtesy
Y. Fink et al., MIT ]
(high-power CO2 lasers)
loss < 1 dB/m
white/grey
= chalco/polymer
(material loss ~ 104 dB/m)
[ Temelkuran et al.,
Nature 420, 650 (2002) ]
silica
[ R. F. Cregan
et al.,
Science 285,
1537 (1999) ]
5µm
Guiding @ 1.55µm
loss ~ 13dB/km
[ Smith, et al.,
Nature 424, 657 (2003) ]
OFC 2004: 1.7dB/km
BlazePhotonics
Breaking the Glass Ceiling II:
Solid-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) ]
Breaking the Glass Ceiling II:
Solid-core Holey Fibers
endlessly
single-mode
[ T. A. Birks et al.,
Opt. Lett. 22,
961 (1997) ]
polarization
-maintaining
[ K. Suzuki,
Opt. Express 9,
676 (2001) ]
nonlinear fibers
[ Wadsworth et al.,
JOSA B 19,
2148 (2002) ]
low-contrast
linear fiber
(large area)
[ J. C. Knight et al.,
Elec. Lett. 34,
1347 (1998) ]
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)
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)
Sequence of Computation
1
Plot all solutions of infinite cladding as w vs. b
w
“light cone”
b
empty spaces (gaps): guiding possibilities
2
Core introduces new states in empty spaces
— plot w(b) dispersion relation
3
Compute other stuff…
Conventional Fiber: Uniform Cladding
c
2
2
w
b  kt
n
cb

n
uniform cladding, index n
b
kt
(transverse wavevector)
w
light cone
light line:
w=cb/n
b
Conventional Fiber: Uniform Cladding
c
2
2
w
b  kt
n
cb

n
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
PCF: Periodic Cladding
periodic cladding e(x,y)
Bloch’s Theorem for periodic systems:
fields can be written:
b
a
E(x,y) ei(bz+kt xt – wt), H(x,y) ei(bz+kt xt – wt)
periodic functions
on primitive cell
transverse (xy)
Bloch wavevector kt
primitive cell
1
w2
satisfies
 k t ,b   k t ,b  H  2 H
e
c
eigenproblem
(Hermitian
constraint: 
k t ,b  H  0
if lossless)
where:
k t ,b    ikt  ibzˆ
PCF: Cladding Eigensolution
Finite cell  discrete eigenvalues wn
Want to solve for wn(kt, b),
& plot vs. b for “all” n, kt
1
 k t ,b   k t ,b  H n 
e
constraint:
where:
w
w n2
c
2
k t ,b  H  0
k t ,b    ikt  ibzˆ
H(x,y) ei(bz+kt xt – wt)
b
1
Limit range of kt: irreducible Brillouin zone
2
Limit degrees of freedom: expand H in finite basis
3
Efficiently solve eigenproblem: iterative methods
Hn
PCF: Cladding Eigensolution
1
Limit range of kt: irreducible Brillouin zone
—Bloch’s theorem: solutions are periodic in kt
K
first Brillouin zone
= minimum |kt| “primitive cell”
M
G
4
a 3
k
y
kx
irreducible Brillouin zone: reduced by symmetry
2
Limit degrees of freedom: expand H in finite basis
3
Efficiently solve eigenproblem: iterative methods
PCF: Cladding Eigensolution
1
Limit range of kt: irreducible Brillouin zone
2
Limit degrees of freedom: expand H in finite basis
— must satisfy constraint: k ,b  H  0
t
Planewave (FFT) basis
H(x t )   HG e
Finite-element basis
constraint, boundary conditions:
iGxt
Nédélec elements
G
constraint:
[ Nédélec, Numerische Math.
35, 315 (1980) ]
HG  G  k  bzˆ   0
uniform “grid,” periodic boundaries,
simple code, O(N log N)
3
[ figure: Peyrilloux et al.,
J. Lightwave Tech.
21, 536 (2003) ]
nonuniform mesh,
more arbitrary boundaries,
complex code & mesh, O(N)
Efficiently solve eigenproblem: iterative methods
PCF: Cladding Eigensolution
1
Limit range of kt: irreducible Brillouin zone
2
Limit degrees of freedom: expand H in finite basis (N)
N
H  H(xt )   hm bm (x t )
2
ˆ
solve: A H  w H
m1
finite matrix problem:
f g   f* g
3
Ah  w Bh
Am  bm Aˆ b
2
Bm  bm b
Efficiently solve eigenproblem: iterative methods
PCF: Cladding Eigensolution
1
Limit range of kt: irreducible Brillouin zone
2
Limit degrees of freedom: expand H in finite basis
3
Efficiently solve eigenproblem: iterative methods
Ah  w Bh
2
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)
PCF: Cladding Eigensolution
1
Limit range of kt: irreducible Brillouin zone
2
Limit degrees of freedom: expand H in finite basis
3
Efficiently solve eigenproblem: iterative methods
Ah  w Bh
2
Many iterative methods:
— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …,
Rayleigh-quotient minimization
PCF: Cladding Eigensolution
1
Limit range of kt: irreducible Brillouin zone
2
Limit degrees of freedom: expand H in finite basis
3
Efficiently solve eigenproblem: iterative methods
Ah  w Bh
2
Many iterative methods:
— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …,
Rayleigh-quotient minimization
for Hermitian matrices, smallest eigenvalue w0 minimizes:
“variational
theorem”
h' Ah
w  min
h h' Bh
2
0
minimize by conjugate-gradient,
(or multigrid, etc.)
PCF: Holey Silica Cladding
r = 0.1a
2r
n=1.46
a
w (2πc/a)
light cone
dimensionless units:
Maxwell’s equations
are scale-invariant
b (2π/a)
PCF: Holey Silica Cladding
r = 0.17717a
w (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.22973a
w (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.30912a
w (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.34197a
w (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.37193a
w (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.4a
w (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
r = 0.42557a
w (2πc/a)
light cone
b (2π/a)
2r
n=1.46
a
PCF: Holey Silica Cladding
n=1.46
2r
a
r = 0.45a
light cone
w (2πc/a)
gap-guided modes
go here
index-guided modes
go here
b (2π/a)
PCF: Holey Silica Cladding
r = 0.45a
light cone
2r
n=1.46
a
above air line:
w (2πc/a)
guiding in air core
is possible
b (2π/a)
below air line: surface states of air core
Bragg Fiber Cladding
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
at large radius,
becomes ~ planar
nhi = 4.6
Bragg fiber gaps (1d eigenproblem)
w
nlo = 1.6
b
radial kr
(Bloch wavevector)
kf
0 by conservation
of angular momentum
wavenumber b
b = 0: normal incidence
Omnidirectional Cladding
are
G
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
omnidirectional
(planar) reflection
e.g. light from
fluorescent sources
is trapped
Bragg fiber gaps (1d eigenproblem)
w
for nhi / nlo
big enough
and nlo > 1
[ J. N. Winn et al,
Opt. Lett. 23, 1573 (1998) ]
b
wavenumber b
b = 0: normal incidence
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)
Sequence of Computation
1
Plot all solutions of infinite cladding as w vs. b
w
“light cone”
b
empty spaces (gaps): guiding possibilities
2
Core introduces new states in empty spaces
— plot w(b) dispersion relation
3
Compute other stuff…
Computing Guided (Core) Modes
1
b  b  H n 
e
constraint:
where:
w n2
c
2
Hn
b  H  0
b    ibzˆ
Same differential equation
as before,
…except no kt
— can solve the same way
magnetic field = H(x,y) ei(bz– wt)
New considerations:
1
Boundary conditions
2
Leakage (finite-size) radiation loss
3
Interior eigenvalues
Computing Guided (Core) Modes
1
computational cell
Boundary conditions
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
2
Leakage (finite-size) radiation loss
3
Interior eigenvalues
Guided Mode in a Solid Core
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
small computation: only lowest-w band!
(~ one minute, planewave)
0.12
1.46 – bc/w = 1.46 – neff
holey PCF light cone
flux density
0.1
0.08
0.06
fundamental
mode
0.04
(two polarizations)
2r
0.02
0
0.3
endlessly single mode: Dneff decreases with l
0.4
0.5
0.6
0.7
0.8
l/a
0.9
1
1.1
1.2
n=1.46
a
r = 0.3a
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 Newton’s method:
w w
b  b 
dw db
(Factor of 3–4 in time.)
group velocity = power / (energy density)
(a.k.a. Hellman-Feynman theorem,
a.k.a. first-order perturbation theory,
a.k.a. “k-dot-p” theory)
Computing Guided (Core) Modes
1
computational cell
Boundary conditions
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
…except when we want
(small) finite-size losses…
2
Leakage (finite-size) radiation loss
3
Interior eigenvalues
Computing Guided (Core) Modes
1
Boundary conditions
2
Leakage (finite-size) radiation loss
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-Davidson, …
Or imaginary-distance BPM:
[ Saitoh, IEEE J. Quantum Elec. 38, 927 (2002) ]
in imaginary z, largest b (fundamental) mode grows exponentially
3
Interior eigenvalues
Computing Guided (Core) Modes
d
n=1.45
1
Boundary conditions
2
Leakage (finite-size) radiation loss
imaginary-distance BPM
L
[ Saitoh, IEEE J. Quantum Elec. 38, 927 (2002) ]
2 rings
3
3 rings
Interior eigenvalues
Computing Guided (Core) Modes
1
Boundary conditions
2
Leakage (finite-size) radiation loss
3
Interior eigenvalues
[ J. Broeng et al., Opt. Lett. 25, 96 (2000) ]
2.4
fundamental & 2nd order
(~ N states for N-hole cell)
…but most methods
compute smallest w
(or largest b)
guided modes
2.0
w (2πc/a)
Gap-guided modes
lie above continuum
1.6
fundamental
air-guided
mode
bulk
crystal
continuum
1.2
0.8
1.11
1.27
1.43
1.59
1.75
1.91
b (2π/a)
2.07
2.23
2.39
Computing Guided (Core) Modes
1
Boundary conditions
2
Leakage (finite-size) radiation loss
3
Interior (of the spectrum) eigenvalues
i
Gap-guided modes
lie above continuum
(~ N states for N-hole cell)
…but most methods
compute smallest w
(or largest b)
Compute N lowest states first: deflation
(orthogonalize to get higher states)
[ see previous slide ]
ii
Use interior eigensolver method—
…closest eigenvalues to w0 (mid-gap)
Jacobi-Davidson,
Arnoldi with shift-and-invert,
smallest eigenvalues of (A–w02)2
… convergence often slower
iii Other methods: FDTD, etc…
Interior Eigenvalues by FDTD
finite-difference time-domain
Simulate Maxwell’s equations on a discrete grid,
+ PML boundaries + eibz z-dependence
• Excite with broad-spectrum dipole ( ) source
Dw
signal processing
complex wn
[ Mandelshtam,
J. Chem. Phys. 107, 6756 (1997) ]
Response is many
sharp peaks,
one peak per mode
decay rate in time gives loss: Im[b] = – Im[w] / dw/db
Interior Eigenvalues by FDTD
finite-difference time-domain
Simulate Maxwell’s 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
mode field profile
narrow-spectrum source
An Easier Problem:
Bragg-fiber Modes
In each concentric region,
solutions are Bessel functions:
c Jm (kr) + d Ym(kr)
 eimf
w  2
k    e  b2
 c
“angular momentum”
At circular interfaces
match boundary conditions
with 4  4 transfer matrix
…search for complex b that satisfies: finite at r=0, outgoing at r=
[ Johnson, Opt. Express 9, 748 (2001) ]
Hollow Metal Waveguides, Reborn
G
are
Qraphics
uickTim
needed
e™
decom
toand
see
pressor
athis picture.
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
metal waveguide modes
OmniGuide fiber modes
frequency w
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
1970’s microwave tubes
@ Bell Labs
wavenumber b
wavenumber b
modes are directly analogous to those in hollow metal waveguide
An Old Friend: the TE01 mode
lowest-loss mode,
just as in metal
(near) node at interface
= strong confinement
= low losses
non-degenerate mode
— cannot be split
= no birefringence or PMD
E
Bushels of Bessels
—A General Multipole Method
[ White, Opt. Express 9, 721 (2001) ]
Each cylinder has its own Bessel expansion:
only cylinders allowed
M
field ~  cm Jm  dmYm
m
(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 4–5 periods is difficult
future: “Fast Multipole Method”
should reduce to O(N log N)?
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)
All Imperfections are Small
(or the fiber wouldn’t work)
• Material absorption: small imaginary De
• Nonlinearity: small De ~ |E|2
• 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
Perturbation Theory
and 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
Perturbation Theory
for Hermitian eigenproblems
Oˆ u  u u
Du & D u for small DOˆ
given eigenvectors/values:
…find change
Solution:
DOˆ
(1)
(2 )
Du  0  Du  Du  
expand as power series in
Du
(1)
u DOˆ u

uu
(1)
& D u  0  D u 
(first order is usually enough)
Perturbation Theory
for electromagnetism
Dw
c H DAˆ H

2w H H
2
(1)
w  De E

2 e E2
Db
(1)
 Dw / vg
(1)
2
…e.g. absorption
gives
imaginary Dw
= decay!
dw
vg 
db
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
Suppressing Cladding Losses
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
Material absorption: small imaginary De
-2
1x1
0
Mode Losses
÷
Bulk Cladding Losses
EH11
-3
1x1
0
Large differential loss
TE01 strongly suppresses
cladding absorption
-4
1x1
0
TE01
(like ohmic loss, for metal)
1x1
0
-5
1.2
1.6
2
l (mm)
2.4
2.8
High-Power Transmission
at 10.6µm (no previous dielectric waveguide)
Polymer losses @10.6µm ~ 50,000dB/m…
…waveguide losses ~ 1dB/m
Log. of Trans. (arb. u.)
-3.0
Transmission (arb. u.)
8
6
4
-3.5
[ B. Temelkuran et al.,
Nature 420, 650 (2002) ]
-4.0
slope = -0.95 dB/m
R2 = 0.99
-4.5
2.5
3.0
3.5
Length (meters)
4.0
2
0
5
5
6
6
7
7
8
8
9
9 10 10
10 11 11
11 12 12
12
Wavelength(mm)
(mm)
Wavelength
[ figs courtesy Y. Fink et al., MIT ]
Quantifying Nonlinearity
Kerr nonlinearity: small De ~ |E|2
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)
Suppressing Cladding Nonlinearity
G
are
Qraphics
uickTim
needed
decom
e™
toand
see
pressor
athis picture.
-6
1x1
0
Mode Nonlinearity*
÷
Cladding Nonlinearity
-7
1x1
0
TE01
Will be dominated by
nonlinearity of air
-8
1x1
0
~10,000 times weaker
than in silica fiber
(including factor of 10 in area)
1x1
0
* “nonlinearity” = Db(1) / P = g
-9
1.2
1.6
2
l (mm)
2.4
2.8
Acircularity & Perturbation Theory
(or any shifting-boundary problem)
De = e1 – e2
e2
e1
De = e2 – e1
… just plug De’s 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) ]
Acircularity & Perturbation Theory
(or any shifting-boundary problem)
De = e1 – e2
e2
De = e2 – e1
e1
(continuous field components)
Dh
Dw
(1)


w surf.
2

1
2
2
Dh De E||  D D 


e
eE
2
[ S. G. Johnson et al.,
PRE 65, 066611 (2002) ]
Loss from Roughness/Disorder
imperfection acts like a volume current
J ~ De E0
volume-current method
or Green’s functions with first Born approximation
Loss from Roughness/Disorder
imperfection acts like a volume current
J ~ De E0
For surface roughness,
including field discontinuities:
J ~ De E||  e De
1
D
Loss from Roughness/Disorder
uncorrelated disorder adds incoherently
So, compute power P radiated by one localized source J,
and loss rate ~ P * (mean disorder strength)
Effect of an Incomplete Gap
on uncorrelated surface roughness loss
some radiation blocked
reflection
same reflection
radiation
Conventional waveguide
(matching modal area)
…with Si/SiO2 Bragg mirrors (1D gap)
50% lower losses (in dB)
same reflection
Considerations for Roughness Loss
• Band gap can suppress some radiation
— typically by at most ~ 1/2, depending on crystal
• Loss ~ De2 ~ 1000 times larger than for silica
• Loss ~ fraction of |E|2 in solid material
— factor of ~ 1/5 for 7-hole PCF
— ~ 10-5 for large-core Bragg-fiber design
• Hardest part is to get reliable statistics for disorder.
Using perturbations to design
big effects
Perturbation Theory and Dispersion
when two distinct modes cross & interact,
unusual dispersion is produced
w
mode 1
mode 2
no interaction/coupling
b
Perturbation Theory and Dispersion
when two distinct modes cross & interact,
unusual dispersion is produced
w
mode 1
mode 2
coupling: anti-crossing
b
Two Localized Modes
= Very Strong Dispersion
w
core mode
localized
cladding–defect mode
weak coupling
= rapid slope change
= high dispersion
(> 500,000 ps/nm-km
+ dispersion-slope matching)
[ T. Engeness et al., Opt. Express 11, 1175 (2003) ]
b
Slow-light Modes
= Anomalous Dispersion
(Different-Symmetry)
w
slow-light
band edges
at b=0
TM
TE
b0
b=0 point has additional symmetry:
modes can be purely TE/TM polarized
— force different symmetry modes together
b
[ M. Ibanescu et al., Phys. Rev. Lett. 92, 063903 (2004) ]
Slow-light Modes
= Anomalous Dispersion
(Different-Symmetry)
ultra-flat (w4)
backward wave slow light
non-zero
velocity
[ M. Ibanescu et al., Phys. Rev. Lett. 92, 063903 (2004) ]
Slow-light Modes
= Anomalous Dispersion
(Different-Symmetry)
Uses gap at b=0:
perfect metal [1960]
or Bragg fiber
or high-index PCF
(n > 2.5)
[ M. Ibanescu et al., Phys. Rev. Lett. 92, 063903 (2004) ]
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/tutorial