Nanophotonics modelling: FDTD and eigenmode expansion Peter Bienstman
http://photonics.intec.UGent.be
Acknowledgements Photonics group at Ghent University
http://photonics.intec.ugent.be P. Vandersteegen, D. Taillaert, B. Maes, B. Luyssaert, P. Sanchis, K. Huang (MIT) © intec 2004
http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion • 3D eigenmode expansion
© intec 2004
http://photonics.intec.UGent.be
FDTD • finite differences • time domain • explicit scheme • very general • brute force • calculates fields
© intec 2004
http://photonics.intec.UGent.be
Finite differences Derivatives Æ central differences f(x)
x x0-∆x/2
x0
x0+∆x/2
f (x0 + ∆x / 2) − f (x0 − ∆x / 2) ∂f (x0 ) = + O(∆x 2 ) ∂x ∆x © intec 2004
http://photonics.intec.UGent.be
Maxwell’s curl laws E or J
H
E
∂H ∇ × E = −µ ∂t © intec 2004
H
∂E ∇× H = J +ε ∂t http://photonics.intec.UGent.be
Yee cell
© intec 2004
http://photonics.intec.UGent.be
Leapfrogging E
E
E
E
E
E
t=0 H
t=0.5∆t E
H
E
H
E
H
E
H
E
E
t=∆t
t=1.5∆t
© intec 2004
H
H
H
H
H
http://photonics.intec.UGent.be
Stability Maximum speed of ‘information transmission”:
t=0
x
vmax t=∆t
x
Courant limit: c ≤ vmax
© intec 2004
∆x = ∆t
∆x ⇒ ∆t ≤ c http://photonics.intec.UGent.be
Courant limit e.g. pick ∆t = 2 ∆x / c (i.e. twice too big)
© intec 2004
t=0
x
t=∆t
x
http://photonics.intec.UGent.be
Numerical dispersion • discretised plane wave:
e
jkx
→e
jkˆxˆ
• problem: k ≠ kˆ Normalised phase velocity 1.00 20 points / λ 10 points / λ
0.95
5 points / λ 0
© intec 2004
45
90
angle
http://photonics.intec.UGent.be
Applications
© intec 2004
http://photonics.intec.UGent.be
FDTD conclusions • very general • brute force • well-suited for non-linearities • can take a long time to run z
doubled resolution: 2x2x2 x 2 = 16 x more work
• can suffer from numerical dispersion • can suffer from staircasing • interpretation not always straightforward © intec 2004
http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion z
eigenmode expansion in a nutshell
• 3D eigenmode expansion
© intec 2004
http://photonics.intec.UGent.be
Modelling approaches
spatial discretisation
© intec 2004
eigenmode expansion
http://photonics.intec.UGent.be
Eigenmode expansion in a nutshell • layer-by-layer approach • eigenmode = ‘natural’ field profile • write field as sum of eigenmodi =1x
• zeer compact: [1, 0.5] © intec 2004
1 + 0.5 x 2 E↔A http://photonics.intec.UGent.be
Interface between two layers
[1, 0.5]
a b 1 [?, ?] = ⋅ c d 0.5 T and R-matrices (scattering matrices) © intec 2004
http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion z
eigenmode expansion in a nutshell
z
eigenmode expansion in more detail
• 3D eigenmode expansion
© intec 2004
http://photonics.intec.UGent.be
Eigenmode expansion in more detail Flowchart: • find eigenmodes in each layer • calculate S matrices of interfaces • calculate S matrices of entire stack
© intec 2004
http://photonics.intec.UGent.be
Eigenmodes • consider single frequency • eigenmode: field profile that keeps its transverse shape:
z
E ( x, y, z ) = E ( x, y )e − jβz • β (or kz) : propagation constant
© intec 2004
http://photonics.intec.UGent.be
Finding eigenmodes z β
• β: unknown, but fixed • each material: fw and bw plane waves z
2 * L unknowns
• impose continuity of E and H, and BC • transcendental equation: f(β) = 0 © intec 2004
http://photonics.intec.UGent.be
Modes in open waveguides z E
x
Re(kz) guided modes
E
x
radiation modes
Im(kz) © intec 2004
http://photonics.intec.UGent.be
Modes in closed waveguides
Re(kz) guided modes radiation modes
Im(kz) © intec 2004
http://photonics.intec.UGent.be
Eigenmode expansion in more detail Flowchart: • find eigenmodes in each layer • calculate S matrices of interfaces • calculate S matrices of entire stack
© intec 2004
http://photonics.intec.UGent.be
S matrix of interface incident field
reflected field
A refl = R12 ⋅ A inc A trans = T 12 ⋅ A inc • matrices calculated by imposing
1
(E inc + E refl )tan = (E trans )tan (Hinc + Hrefl )tan = (H trans )tan
2
(mode matching) transmitted field
• expressions involve overlap integrals
∫E ×H i
© intec 2004
j
⋅ u z dS
http://photonics.intec.UGent.be
S matrix of stack incident field
reflected field
A refl = R stack ⋅ A inc A trans = T stack ⋅ A inc • matrices contain contributions from • R and T’s of interfaces • propagation across layers
• contributions can be accounted for by • T-matrix formalism (fast, unstable) • S-matrix formalism (slower, stable) transmitted field
© intec 2004
Issue: mix of
e jβ z , e − j β z http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion z
eigenmode expansion in a nutshell
z
eigenmode expansion in more detail
z
boundary conditions
• 3D eigenmode expansion
© intec 2004
http://photonics.intec.UGent.be
Problem
• parasitic reflections • absorbing boundary conditions needed © intec 2004
http://photonics.intec.UGent.be
Example m etal wall
I
T
R
L parasitic reflec tions m etal wall
© intec 2004
http://photonics.intec.UGent.be
With absorbing BC
© intec 2004
http://photonics.intec.UGent.be
Without absorbing BC
© intec 2004
http://photonics.intec.UGent.be
Absorbers Traditional absorber: material with loss
medium
absorber metal
Reflections now occur at the absorber interface © intec 2004
http://photonics.intec.UGent.be
Perfectly matched layers PML: material with a complex thickness e.g. plane wave
d-ja
e
medium
PML
metal
−j
2π n⋅r λ
complex r provides absorption
No more reflections at the absorber interface © intec 2004
http://photonics.intec.UGent.be
Other PML formulations α α • anisotropic medium ε , µ = ε 0ε r , µ 0 µ r
1 α
• split field formalism (Bérenger, FDTD) z
e.g. Hz = Hzx + Hzy
© intec 2004
http://photonics.intec.UGent.be
Modal spectrum imag(d)=-0.2 imag(d)=-1.6 imag(d)=-0.4 imag(d)=-0.8 imag(d)=0 0
-1
-2
-3
-4
-5
-6
-7
© intec 2004
0
0.5
1
1.5
2
2.5
3
3.5
4
http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion z
eigenmode expansion in a nutshell
z
eigenmode expansion in more detail
z
boundary conditions
z
finite structures
• 3D eigenmode expansion
© intec 2004
http://photonics.intec.UGent.be
Finite structure: PhC splitter
© intec 2004
http://photonics.intec.UGent.be
Comparison with FDTD • eigenmode expansion z
can be faster (more data reuse)
z
calculation time logarithmic in # periods
z
field profiles only calculated when needed
z
very physical interpretation
z
trivial to handle dispersive materials
z
can handle semi-infinite media
z
calculation time: (# modes)^3
z
staircasing
© intec 2004
http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion z
eigenmode expansion in a nutshell
z
eigenmode expansion in more detail
z
boundary conditions
z
finite structures
z
infinite structures: band diagrams
• 3D eigenmode expansion
© intec 2004
http://photonics.intec.UGent.be
Band structure calculation uses eigenvalues of transfer matrix of period T F1
F2
B1
B2
F2 F1 − jk z d F1 B = T ⋅ B = e B 2 1 1 © intec 2004
http://photonics.intec.UGent.be
Square lattice of rods
wall z wall
TE © intec 2004
http://photonics.intec.UGent.be
Polaritonic materials (K.C. Huang)
ε
ωT © intec 2004
ωL http://photonics.intec.UGent.be
Band diagrams
© intec 2004
http://photonics.intec.UGent.be
Flux expulsion
f = 0.3916 © intec 2004
f = 0.4030 http://photonics.intec.UGent.be
Plane wave methods • e.g. MPB (MIT Photonic Bands) • Bloch theorem is invoked • fields are expanded in plane waves:
e
(
j N xGx x + N y G y y + N z Gz z
)
• eps is fourier decomposed • structure is discretised • leads to eigenvalue problem for each kz:
A⋅x =ω x 2
© intec 2004
http://photonics.intec.UGent.be
Comparison with plane-wave method • eigenmode expansion z
fewer basis functions needed
z
independent variable is frequency
z
can do finite structures too
z
easiest along high symmetry direction in Brillouin zone
z
very useful as ‘auxiliary’ calculation
© intec 2004
http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion z
eigenmode expansion in a nutshell
z
eigenmode expansion in more detail
z
boundary conditions
z
finite structures
z
infinite structures: band diagrams
z
semi-infinite structures
• 3D eigenmode expansion © intec 2004
http://photonics.intec.UGent.be
Semi-infinite media Useful e.g. when modelling tapers:
© intec 2004
http://photonics.intec.UGent.be
Reflections • not a perfect taper: Fabry-Perot effect
© intec 2004
http://photonics.intec.UGent.be
Fabry-Perot resonances Power transmission + Fabry-Perot curve 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
10
20
30
40
50
60
70
Periods
© intec 2004
http://photonics.intec.UGent.be
Modelling semi-infinite crystals
… © intec 2004
http://photonics.intec.UGent.be
Semi-infinite media Generalised reflection matrix from field profiles of Bloch modes:
… © intec 2004
R = B⋅F
−1
http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion z
eigenmode expansion in a nutshell
z
eigenmode expansion in more detail
z
boundary conditions
z
finite structures
z
infinite structures: band diagrams
z
semi-infinite structures
z
laser cavities
• 3D eigenmode expansion © intec 2004
http://photonics.intec.UGent.be
Laser modes of cavities Resonance condition:
A res = R top ⋅ R bot ⋅ A res Rbot
find eigenvectors of
Q ≡ R top ⋅ R bot with eigenvalue 1 Rtop more stable: SVD: find singular values of
Q −I with value 0
© intec 2004
http://photonics.intec.UGent.be
Airpost VCSEL
© intec 2004
http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion z
principles
z
boundary conditions
z
finite structures
z
infinite structures: band diagrams
z
semi-infinite structures
z
laser cavities
z
spontaneous emission
• 3D eigenmode expansion © intec 2004
http://photonics.intec.UGent.be
Current source inside cavity
Rtop Aup,0
Aup
…
d=0 …
Rbot
© intec 2004
Ado,0
Ado
http://photonics.intec.UGent.be
Spontaneous emission rate • electric dipole between two metal plates wall
wall
• transform to closed circular geometry using PML
© intec 2004
http://photonics.intec.UGent.be
Spontaneous emission rate
Spontaneous emission rate 3.5
CAMFR
3.0
analytic
2.5 2.0 1.5 1.0 0.5 0.0 0
1
2
3
4
5
Cavity length (wavelenghts)
© intec 2004
http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion z
principles
z
boundary conditions
z
finite structures
z
infinite structures: band diagrams
z
semi-infinite structures
z
laser cavities and spontaneous emission
z
example of device modelling
• 3D eigenmode expansion © intec 2004
http://photonics.intec.UGent.be
Fiber coupler (D. Taillaert) Our fiber coupler z
use a grating to couple light from/to a fiber perpendicular to the PIC
z
use a spot-size convertor in plane
z
1.55µm wavelength
z
wafer scale, no need to cleave/polish the devices
z
good alignment tolerances
z
high vertical index-contrast
© intec 2004
Single mode fiber core
spot size convertor
(artist’s impression)
http://photonics.intec.UGent.be
Experimental results paper submitted to Optics Express
© intec 2004
http://photonics.intec.UGent.be
Experimental results Transmission measurement ⇒ coupling efficiency Polarization maintaining fibers, tilted 10 degrees, for incoupling and outcoupling
Fiber coupler grating for coupling to single-mode fiber
© intec 2004
12µm wide, 1mm long ridge waveguide
http://photonics.intec.UGent.be
Experimental results Transmission measurement ⇒ coupling efficiency Polarization maintaining fibers, tilted 10 degrees, for incoupling and outcoupling
Fiber coupler grating for coupling to single-mode fiber
© intec 2004
12µm wide, 1mm long ridge waveguide
http://photonics.intec.UGent.be
Experimental results fiber coupling efficiency
0.40
0.30 620nm period 630nm period 620nm theory 630nm theory
0.20
0.10
0.00 1520
1560
1600
1640
wavelength (nm)
33% efficiency (4.8dB coupling loss) 35-40nm 1dB bandwidth © intec 2004
http://photonics.intec.UGent.be
Experimental results Experiments agree well with theory, difference is due to z
theoretical model is 2-D
z
also the distance between the fiber end and the grating is larger in experiment than in theory
The efficiency is limited because z
coupling to substrate (up/down ratio=0.9)
z
mode mismatch with gaussian beam
z
with an optimized oxide thickness, the up/down ratio can be improved to 1.4-1.5
© intec 2004
http://photonics.intec.UGent.be
Experimental results z
70% , but 1µm thick wg and very long grating
! ! !
! ! ! !! !
T.W. Ang, G.T. Reed, A. Vonsovici, A.G.R. Evans, P.R. Routley, and M.R. Josey, “Effects of grating heights on highly efficient unibond SOI waveguide grating couplers,” IEEE Photon. Tech. Lett. 12, 59–61 (2000)
z
H T D I
57%, beam diameter 48µm, through substrate, metal mirror on top (27% for beam diameter 28µm)
W D 60-65%, top dielectric stack N A B W 50% claimed O (E=1-R-T) , approx. 30x30µm R R A N
R. Orobtchouk, A. Layadu, H. Gualous, D. Pascal, A. Koster, and S. Laval, “High efficiency light coupling in a submicrometric silicon-on-insulator waveguide,” Appl. Opt. 39, 5773–5777 (2000).
z
A. Narashima and E. Yablonovitch, ``Efficient optical coupling into single mode Silicon-on-Insulator thin films using a planar grating coupler embedded in a high index contrast dielectric stack.'', presented at the conference on Lasers and Electro-Optics (CLEO), Baltimore, USA, June 2003.
z
Optical interconnects on SOI: a front end approach ,Orobtchouk, R.; Schnell, N.; Benyattou, T.; Lardenois, S.; Pascal, D.; Cordat, A.; Laval, S.; Cassan, E.; Koster, A.; Bouchier, D.; Bouzaida, N.; Dal'zotto, B.; Florin, B.; Heitzmann, M.; Lamouchi, Z.; Louis, D.; Mollard, L.; Renaud, D.; Gautier, J.; Interconnect Technology Conference, 2002. Proceedings of the IEEE 2002 International , 3-5 June 2002 , Pages:83 - 85
z 33%, © intec 2004
1dB bandwidth > 35nm http://photonics.intec.UGent.be
Improved designs paper submitted to Optics Letters θ
fiber core
y x
z
n=1.5
1µm
n=3.476 220nm n=1.444
n=3.476 © intec 2004
http://photonics.intec.UGent.be
Improved designs 3D-problem reduced to 2 x 2-D problems,
E x(x,y)
E x(y) 0.22 0
y-axis (µm )
thanks to large width of ridge wg, Ex(x,y)≈Ex (x).Ex (y)
E x(x)
-6
© intec 2004
0 x-axis (µm)
6
http://photonics.intec.UGent.be
Improved designs Uniform grating z
theoretical efficiency limited to 80% because of overlap exponential - gaussian profile
z
P=P0exp(-2αz)
Non-uniform grating z
different etch depth or duty cycle
z
α becomes function of z
z
theory and experiments in literature for weak gratings
© intec 2004
http://photonics.intec.UGent.be
Improved designs Use design procedure for weak gratings and optimize using genetic algorithm (76 → 97% overlap with gaussian profile) 1 0.9 0.8
G 2 (z) G 2 (z)s imu la tio n 2 ∫G (z)dz -1 α (z) (µm )
0.7
2α ( z ) =
2
G ( z) z
1 − ∫ G 2 (t )dt 0
0.6 0.5 0.4 0.3 0.2 0.1 0
© intec 2004
2
4
6 z-a xis (µm)
8
10
12
http://photonics.intec.UGent.be
Optimise groove lengths d1 d2 …
Note: modes can be reused!
© intec 2004
http://photonics.intec.UGent.be
Genetic algorithm generate 100 random couplers (random L) calculate all the couplers generation0 = all 100 couplers sort generation with descending transmission 2 first elements of generation are allowed to mate -> 2 extra couplers calculate extra couplers add extra couplers to generation sort generation with descending transmission generation1 = best 100 couplers © intec 2004
http://photonics.intec.UGent.be
Cross-over and mutation
1
2
Parent1
Parent2
[B1, B2, …, BN,L1, L2, …, LN]
[B1, B2, …, BN,L1, L2, …, LN]
[B1, B2, …, BN,L1, L2, …, LN]
[B1, B2, …, BN,L1, L2, …, LN]
Child 1
Child 2
Afterwards child1 en child2 are slightly mutated (independently) using a Gauss distribution around the cross-over value
© intec 2004
http://photonics.intec.UGent.be
Improved designs Results 1 S OI S OI+DBR
0.9 0.8
← 1dB co upling los s→
e fficie ncy
0.7 0.6 0.5 0.4 0.3 0.2 1 500
© intec 2004
1520
1 54 0 1560 wavele ngth (nm)
158 0
1 600
http://photonics.intec.UGent.be
Improved designs 90% coupling efficiency
smallest groove width is 30nm, but can be increased to 60nm with only 1-2 percent decreased efficiency © intec 2004
http://photonics.intec.UGent.be
Improved designs Sensitivity to fabrication errors 1 90nm 100nm 110nm
0.9 e fficie ncy
etch depth
0.8 0.7 0.6 0.5 1 500
groove width
e fficien cy
0.9 2
1520
1 54 0 1560 wavele ngth (nm)
5nm error dis tribution
158 0
1 600
10nm e rro r dis trib utio n
0.9 1 0.9 0 0.8 9 0.8 8
© intec 2004
http://photonics.intec.UGent.be
Improved designs Is it possible to use a top dielectric stack to improve the efficiency ?
Normal SOI
SOI with buried DBR
SOI with top mirror
Demonstrated by Yablonovitch et al.
© intec 2004
http://photonics.intec.UGent.be
Improved designs Back to basics : source in a (± symmetric) cavity
(2n+1)π λ/4 2mπ
λ /4≈120nm ⇒ Works only for shallow gratings •cavity doesn’t only change directivity but also extraction efficiency •rigorous grating calculations agree with the simple model © intec 2004
http://photonics.intec.UGent.be
Interference Coupler (B. Luyssaert)
Bout
B3 L3 Bin
B = [B1, B2, …, BN] L = [L1, L2, …, LN]
Put a sequence of waveguide sections with different widths and lengths between input and output waveguide © intec 2004
http://photonics.intec.UGent.be
Result
• 2D simulation • width IN = 1 µm, width OUT = 13 µm • length = 33 µm • efficiency = 94% (mode to mode) © intec 2004
http://photonics.intec.UGent.be
PhC coupler (P. Sanchis)
© intec 2004
http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion z
principles
z
boundary conditions
z
finite, infinite and semi-infinite structures
z
laser cavities and spontaneous emission
z
example of device modelling
z
modelling non-linearities
• 3D eigenmode expansion © intec 2004
http://photonics.intec.UGent.be
Non-linear mode expansion (B. Maes) Kerr effect Intensity dependent refractive index:
n ( r ) = n Linear + n Kerr I ( r ) → spatially dependent ⇒ spatial discretisation
© intec 2004
http://photonics.intec.UGent.be
Non-linear mode expansion (2) Iterative procedure
n0 → I 0 → n1 → I1 → n2 L 144424443 (1)
( 2)
(1)
( 2)
One step
(1) Linear eigenmode calculation (e.g. CAMFR) (2) Non-linear index equation:
n1 = n Linear + n Kerr I 0 If n1 − n0 < ε solution reached, else repeat above step © intec 2004
http://photonics.intec.UGent.be
Photonic crystal cavity
Efficiency :
Small non-linear sections
Linear parts need only one calculation
M. Soljacic, M. Ibanescu, S.G. Johnson, Y. Fink, and J.D. Joannopoulos, ‘Optimal bistable switching in nonlinear photonic crystals’, Phys. Rev. E 66, 055601(R) 2002. © intec 2004
http://photonics.intec.UGent.be
Photonic crystal cavity (2) 1 0.8
T
0.6 0.4
0.2 0 0
50000
100000
150000
200000
250000
300000
Input power (a.u.)
Low transmission
© intec 2004
↔
High transmission
http://photonics.intec.UGent.be
Characteristics & comparison Flexibility :
Generic 2D structures
Saturable Kerr, absorption, ...
Rigorous Bidirectional ↔ standard BPM Efficiency
Linear parts need only one calculation
Small non-linear sections
CW-solutions ↔ FDTD: long pulses
© intec 2004
http://photonics.intec.UGent.be
PhC Flip-Flop (1) Szöke’s flip-flop Signal CW Memory signal ON pulse A
B
OFF pulse
A and B : bistable resonators © intec 2004
http://photonics.intec.UGent.be
PhC Flip-Flop PhC version
CW signal
OFF pulse
ON pulse
Memory signal
© intec 2004
http://photonics.intec.UGent.be
PhC Flip-Flop Principle Transmission of one switch 1.2
Pout
1
CW
0.8
OFF
ON
0.6
CW
0.4
OUT
0.2 0 0
0.5
1
1.5
2
2.5
Pin
3
ON-pulse: reflection of CW keeps first switch ON OFF-pulse: eliminates reflection → both switches OFF © intec 2004
http://photonics.intec.UGent.be
PhC Flip-Flop Results
CW
OFF
ON
CW simulations
OUT 1.2
POUT
1
0.8 0.6
ON - signal
0.4
Switching OFF
0.2 0 0
© intec 2004
2
4
6
8
POFF
10
http://photonics.intec.UGent.be
Kerr effect - Bloch modes Efficiency ↑↑ as size of nonlinear sections ↓ → finite structures e.g. resonators → periodic infinite structures ⇒ Only one period needed
Example : PhC bands
a/λ
0.6 0.5 0.4
Linear Nonlinear
0.3
z 0.2 0.1 0 0
© intec 2004
2
4
6
8
kz
10
http://photonics.intec.UGent.be
Self-localized waveguides Frequency within the bandgap • Linear : Exponential decay in perfect PhC, but : Line defect → waveguide
• Nonlinear : Field can ‘carve’ its own channel ⇒ Gap soliton or self-localized waveguide © intec 2004
http://photonics.intec.UGent.be
Self-localized waveguides Start with ‘seed’ from linear mode Rapid parameter analysis
z
© intec 2004
http://photonics.intec.UGent.be
Second Harmonic Generation ω
ω
χ(2) 2ω
2ω
Phase Matching : Efficient SHG if : 2k(ω) - k(2ω) = 0 or n(ω) = n(2ω)
Quasi PM :
|E2ω|2
Periodicity in n or χ(2)
Quasi PM
zc no PM
zc= π/∆k 0
© intec 2004
1
2
3
4
5
6
7
z
http://photonics.intec.UGent.be
8
Modified matrix formalism F1
F2
B1
B2
Linear: F2 B = 1
S
Propagation and interfaces
Becomes: F2 B = 1
F1 B 2
S
F1 ∗ B + ∗ 2
Remember: Ck ( z ) = Ck (0) + term © intec 2004
Generation (and scattering)
http://photonics.intec.UGent.be
Second harmonic generation Longitudinal periodicity: n = 3.15
ω
1.25µm
2ω n = 3.3 Tω
1
Flux2ω
0.9 0.8
160000 140000 120000
0.7 0.6
100000
0.5
80000
0.4
60000
0.3
40000
0.2
20000
0.1 0 1.545
1.547
1.549
1.551
1.553
1.555
λ (µm) © intec 2004
0 1.545
1.547
1.549
1.551
1.553
1.555
2λ (µm) http://photonics.intec.UGent.be
Outline • FDTD • 2D eigenmode expansion z
principles
z
boundary conditions
z
finite, infinite and semi-infinite structures
z
laser cavities and spontaneous emission
z
example of device modelling
z
modelling non-linearities
• 3D eigenmode expansion © intec 2004
http://photonics.intec.UGent.be
Challenges for 3D model • need to find modes of waveguides with arbitrary 2D cross-section • need to handle radiation losses (PML) • need to handle material losses complex propagation constants
© intec 2004
http://photonics.intec.UGent.be
Waveguide with 2D cross section
z © intec 2004
http://photonics.intec.UGent.be
Rotated basis functions y
x
z
kz
© intec 2004
http://photonics.intec.UGent.be
Dispersion relation
Rright
Rleft
Ares = R left ⋅ R right ⋅ Ares © intec 2004
http://photonics.intec.UGent.be
Modes in the complex plane 0.0 0
0.5
1
1.5
-0.5
-1.0
-1.5
-2.0
© intec 2004
http://photonics.intec.UGent.be
Option 1: mode tracking imag(d)=-0.2 imag(d)=-1.6 imag(d)=-0.4 imag(d)=-0.8 imag(d)=0 0
-1
-2
-3
-4
-5
-6
-7
© intec 2004
0
0.5
1
1.5
2
2.5
3
3.5
4
http://photonics.intec.UGent.be
Option 2: contour integration
0.0 0
0.5
1
1.5
-0.5
-1.0
-1.5
-2.0
© intec 2004
http://photonics.intec.UGent.be
Option 3: two-stage method • Stage 1 z
Construct coarse estimate of betas
z
Expand them in “plane waves”
z
Algebraic eigenvalue problem
z
All modes are found at the same time
A⋅x = β x 2
• Stage 2 z
Refine these estimates
z
Expand them in 1D film modes
z
Roots of transcendental complex function
z
Better accuracy
© intec 2004
http://photonics.intec.UGent.be
Stage 1: plane wave methods Several methods are available: • Omar / Schuenemann: uniform modes • Noponen / Turunen: Fourier • Li: Fourier taking care with discontinuities
© intec 2004
http://photonics.intec.UGent.be
Omar-Schueneman
• Expand in eigenmodes of uniform waveguide • Horribly slow convergence © intec 2004
http://photonics.intec.UGent.be
Noponen-Turunen / Li • grating Fourier methods: need adapting to waveguide configuration • Li: Fourier factoring of ‘best’ function to satisfy boundary conditions
© intec 2004
http://photonics.intec.UGent.be
Convergence: square waveguide Effective index 1.17
1.165
Noponen-Turunen
Film modes 1.16
Li 1.155
Omar-Schuenemann 1.15 0
100
200
300
400
500
600
700
number of modes
© intec 2004
http://photonics.intec.UGent.be
Convergence: photonic wire Effective index
Li
2.50
Film modes
2.25
2.00
1.75
Noponen-Turunen
1.50 0
200
400
600
800
1000
number of modes
220x415 nm n=3.47
n=1.0
n=1.44 © intec 2004
http://photonics.intec.UGent.be
Field profiles
E © intec 2004
http://photonics.intec.UGent.be
Field profiles
40 eigenmodes © intec 2004
500 plane waves http://photonics.intec.UGent.be
Photonic Crystal Fibre
Too few airholes Lossy fibre Radiation losses handled through PML Boundary conditions
© intec 2004
http://photonics.intec.UGent.be
Photonic wire n=1.0 n= 3.5 n= 1.45
n=3.5
PML © intec 2004
http://photonics.intec.UGent.be
Photonic wire losses Propagation losses for 400 nm buffer
dB/mm
150 100 50 0 0.3
0.4
0.5
Wire width (micron)
© intec 2004
http://photonics.intec.UGent.be
3D model • sequence of waveguides with 2D crosssection • need to calculate overlap integrals z
Film modes: slow (N4)
z
Plane waves: fast (N)
• hybrid model: z
Use plane wave field representation when npw ≈ nfilm (e.g. radiation modes)
© intec 2004
http://photonics.intec.UGent.be
Flow chart • calculate plane wave estimates • choose which modes to refine with film modes (e.g. all guided modes) • calculate overlap integrals, for each mode either using a plane wave expansion or a film mode expansion Better trade-off between speed and accuracy
© intec 2004
http://photonics.intec.UGent.be
Example: double step Transmission 0.90 0.88 0.86 0.84
Plane wave
0.82 0.80
Hybrid
0.78 0.76
Film modes
0.74 0
20
40
60
80
100
number of modes © intec 2004
http://photonics.intec.UGent.be
3D interference coupler
© intec 2004
http://photonics.intec.UGent.be
Transmission Transmission 1.00 0.80 0.60 0.40 0.20 0.00 1.4
1.5
1.6
1.7
1.8
1.9
2
wavelength
© intec 2004
http://photonics.intec.UGent.be
CAMFR
Our eigenmode expansion simulator freely available from htpp://camfr.sourceforge.net
© intec 2004
http://photonics.intec.UGent.be
