MSEG 667
Nanophotonics: Materials and Devices
3: Guided Wave Optics
Prof. Juejun (JJ) Hu
hujuejun@udel.edu
Modes
“When asked, many well-trained scientists and
engineers will say that they understand what a
mode is, but will be unable to define the idea of
modes and will also be unable to remember
where they learned the idea!”
Quantum Mechanics for Scientists and Engineers
David A. B. Miller
Guided wave optics vs. multi-layers
Light

Longitudinal variation of refractive
indices




Refractive indices vary along the light
propagation direction
Approach: transfer matrix method
Devices: DBRs, ARCs
Transverse variation of refractive
indices



The index distribution is not a function
of z (light propagation direction)
Approach: guided wave optics
Devices: fibers, planar waveguides
H
L
H
L
H
L
High-index
substrate
Why is guided wave optics important?

It is the very basis of numerous photonic devices


Optical fibers, waveguides, traveling wave resonators, surface
plasmon polariton waveguides, waveguide modulators…
Fiber optics
The masters of light
“If we were to unravel all of the glass fibers
that wind around the globe, we would get a
single thread over one billion kilometers long
– which is enough to encircle the globe more
than 25 000 times – and is increasing by
thousands of kilometers every hour.”
-- 2009 Nobel Prize in Physics
Press Release
Waveguide geometries and terminologies
2-d optical confinement
1-d optical confinement
cladding
nlow
core
nhigh
cladding
nlow
Slab waveguide
core
nlow
nhigh
cladding
Channel/photonic wire
waveguide
cladding
core
Step-index fiber
Graded-index (GRIN) fiber
nlow
nhigh
nlow
Rib/ridge waveguide
How does light propagate in a waveguide?
light
Question:
If we send light down a channel
waveguide, what are we going to
see at the waveguide output facet?
?
JJ knows the
answer, but
we don’t !
A
B
C
D
E
What is a waveguide mode?
 A propagation mode of a waveguide at a given wavelength is a
stable shape in which the wave propagates.
 Waves in the form of such a mode of a given waveguide retain
exactly the same cross-sectional shape (complex amplitude) as
they move down the waveguide.
 Waveguide mode profiles are
wavelength dependent
 Waveguide modes at any given
wavelength are completely
determined by the crosssectional geometry and
refractive index profile of the
waveguide
Reading: Definition of Modes
1-d optical confinement: slab waveguide
 2 n2  2 
Wave equation:    2 2  E  0
c0 t 

with spatially non-uniform refractive index
Ey  x, z, t   U  x   ei z  eit
x
y
TE: E-field parallel to substrate
z
Helmholtz equation:
2 / 
z
d2
[ 2  k 2   2 ]U ( x )  0
dx
k = nk0 = nω/c
Propagation constant is related to
the wavelength (spatial periodicity)
of light propagating in the waveguide
effective index
Propagation constant: β = neff k0
Field boundary conditions
Quantum mechanics =
? Guided wave optics
… The similarity between physical equations allows physicists to gain
understandings in fields besides their own area of expertise… -- R. P. Feynman
Quantum
mechanics
"According to the
experiment, grad
students exist in
a state of both
productivity and
unproductivity."
-- Ph.D. Comics
Guided wave optics
Quantum mechanics =
? Guided wave optics
… The similarity between physical equations allows physicists to gain
understandings in fields besides their own area of expertise… -- R. P. Feynman

Quantum mechanics
1-d time-independent
Schrödinger equation

2 2
[
 x  V  E ] ( x )  0
2m





ψ(x) : time-independent wave
function (time x-section)
-V(x) : potential energy landscape
-E : energy (eigenvalue)
Time-dependent wave function
(energy eigenstate)
( x, t )   ( x )  exp( iEt )
t : time evolution
Guided wave optics
Helmholtz equation in a slab
waveguide
[2x  k02n 2   2 ]U ( x )  0





U(x) : x-sectional optical mode
profile (complex amplitude)
k02n(x)2 : x-sectional index profile
β2 : propagation constant
Electric field along z direction
(waveguide mode)
E ( x, z )  U ( x )  exp( iz )
z : wave propagation
1-d optical confinement problem re-examined
Helmholtz equation:
Schrödinger equation:
[2x  k02n 2   2 ]U ( x )  0
x
[
V
nclad
ncore
1 2
 x  V  E ] ( x )  0
2m
?
V0
nclad
nclad ncore
n
• Discretized propagation constant β
values
• Higher order mode with smaller β
have more nodes (U = 0)
• Larger waveguides with higher index
contrast supports more modes
• Guided modes: nclad < neff < ncore
E3
E2
Vwell
E1
x
1-d potential well (particle in a well)
• Discretized energy levels (states)
• Wave functions with higher
energy have more nodes (ψ = 0)
• Deeper and wider potential wells
gives more bounded states
• Bounded states: Vwell < E < V0
Waveguide dispersion
waveguide dispersion
short λ
high ω
long λ
low ω
β = neff k0 = neff ω/c0
nclad
ncore
nclad


ncoreω/c0
ncladω/c0
At long wavelength, effective index is small
(QM analogy: reduced potential well depth)
At short wavelength, effective index is large
Group velocity in waveguides
ncoreω/c0
Low vg
ncladω/c0
vg 
d
d
Group index ng 
Phase velocity vp: traveling speed
of any given phase of the wave
Group velocity vg: velocity of
wave packets (information)
c0
d
 c0
vg
d
neff
c0


 c0
vp

 Effective index: spatial periodicity (phase)
 Waveguide effective index is always smaller than core index
 Group index: information velocity (wave packet)
 In waveguides, group index can be greater than core index!
2-d confinement & effective index method
Channel waveguide
Rib/ridge waveguide
nclad
nclad
ncore
ncore


nclad
Directly solving 2-d
Helmholtz equation for
U(x,y)
Deconvoluting the 2-d
equation into two 1-d
problems

Separation of variables
 Solve for U’(x) & U”(y)
 U(x,y) ~ U’(x) U”(y)
nclad neff,core nclad
y
z
x
neff,clad neff,core neff,clad

Less accurate for highindex-contrast
waveguide systems
EIM mode solver:
http://wwwhome.math.utwente.nl/~hammerm/eims.html
http://wwwhome.math.utwente.nl/~hammer/eimsinout.html
Coupled waveguides and supermode
V
x
Anti-symmetric
WG 1
WG 2
Symmetric
x
Cladding
x
Modal overlap!
supermodes
neff + Δn
neff - Δn
http://wwwhome.math.utwente.nl/~
hammer/Wmm_Manual/cmt.html
Coupled mode theory
Symmetric
≈
(U 1  U 2 )  exp[ ikz  ( neff  n )]
Anti-symmetric
+
U1
≈
(U1  U 2 )  exp[ ikz  ( neff  n )]
U2
+
U1
 U 2  U 2 exp( i )
If equal amplitude of symmetric and antisymmetric modes are launched,
coupled mode: (U1  U 2 )  exp[ ikz  (neff   )]  (U1  U 2 )  exp[ ikz  (neff   )]
z=0
2U1
z = π/2kΔn
z = π/kΔn
2U2 exp(iπ/2Δn) 2U1 exp(iπ/Δn)
1
2
z
π/kΔn Beating length
Waveguide directional coupler
Symmetric
coupler
Asymmetric waveguide
directional coupler
Optical power
Optical power
Beating length π/kβ
Propagation distance
Propagation distance
3dB direction
coupler
WG 1
WG 2
Cladding
10 log 10 (0.5) ~ 3.0(dB)
Optical loss in waveguides

Material attenuation






Roughness scattering



Electronic absorption (band-to-band transition)
Bond vibrational (phonon) absorption
Impurity absorption
Semiconductors: free carrier absorption (FCA)
Glasses: Rayleigh scattering, Urbach band tail states
Planar waveguides: line edge roughness due to imperfect
lithography and pattern transfer
Fibers: frozen-in surface capillary waves
Optical leakage


Bending loss
Substrate leakage
Waveguide confinement factor
Consider the following scenario:
A waveguide consists of an absorptive core
with an absorption coefficient acore and an nonabsorptive cladding. How do the mode profile
evolve when it propagate along the guide?
core
cladding
E
E  x, y   exp  i  r z  it   exp  a wg z 
Modal attenuation
coefficient:
a wg   corea core
x
Propagation
ncore c0 0  E dxdy
2
Confinement
 core 
factor:
core
 Re  E  H *  z  dxdy
E
?

J. Robinson, K. Preston, O. Painter, M. Lipson, "First-principle derivation of
gain in high-index-contrast waveguides," Opt. Express 16, 16659 (2008).
x
Absorption in silica (glass) and silicon (semiconductor)



Short wavelength edge:
Rayleigh scattering (density
fluctuation in glasses)
Long wavelength edge: Si-O
bond phonon absorption
Other mechanisms:
impurities, band tail states



Short wavelength edge:
band-to-band transition
Long wavelength edge: Si-Si
phonon absorption
Other mechanisms: FCA,
oxygen impurities (the
arrows below)
Roughness scattering

Origin of roughness:

Planar waveguides: line edge roughness evolution in processing
T. Barwicz and H. Smith, “Evolution of line-edge roughness during fabrication of highindex-contrast microphotonic devices,” J. Vac. Sci. Technol. B 21, 2892-2896 (2003).

Fibers: frozen-in capillary waves due to energy equi-partition
P. Roberts et al., “Ultimate low loss of hollow-core photonic crystal fibres,” Opt.
Express 13, 236-244 (2005).


QM analogy: time-dependent perturbation
Modeling of scattering loss

High-index-contrast waveguides suffer from high scattering loss
F. Payne and J. Lacey, “A theoretical analysis of scattering loss from planar optical
waveguides,” Opt. Quantum Electron. 26, 977 (1994).
T. Barwicz et al., “Three-dimensional analysis of scattering losses due to sidewall
roughness in microphotonic waveguides,” J. Lightwave Technol. 23, 2719 (2005).
a wg , s   ncore 2  nclad 2    2
where  is the RMS roughness
Optical leakage loss
n
x
Single-crystal
Silicon
Silicon oxide cladding
nSi
nSiO2
Silicon substrate
x
QM analogy
V
x
Tunneling!
Unfortunately quantum tunneling
does not work for cars!
Boundary conditions
Guided wave optics
Quantum mechanics
• Continuity of wave function
• Continuity of the first order
derivative of wave function
Polarization dependent!
Core
y
Cladding
Substrate
x
z
 TE mode: Ez = 0 (slab), Ex >> Ey (channel)
 TM mode: Hz = 0 (slab), Ey >> Ex (channel)
Boundary conditions
Guided wave optics
TE mode profile
Polarization dependent!
Core
y
Cladding
Substrate
x
z
 TE mode: Ez = 0 (slab), Ex >> Ey (channel)
 TM mode: Hz = 0 (slab), Ey >> Ex (channel)
Ex amplitude
of TE mode
y
y
x
z
x
Discontinuity
of field due to
boundary
condition!
Slot waveguides
Field concentration in low index material
y
slot
x
z
Cladding
Substrate
TE mode profile
Use low index material for:
• Light emission
• Light modulation
• Plasmonic waveguiding
V. Almeida et al., “Guiding and
confining light in void nanostructure,”
Opt. Lett. 29, 1209-1211 (2004).