•Losses and nonlinear effects
–Introduction
ECE 4006/5166 Guided Wave Optics
Losses and nonlinearities
Why related?
• Loss gives ratio of output to input power per unit guide
length
• Nonlinearities limit input power.
• The application (e.g. SNR at a receiver) sets minimum
output power.
All three, then, determine maximum link length.
Primary loss mechanisms
• Absorption (various sources)
• Scattering (bulk = Rayleigh, surface)
• Leakage (bending, coupling).
Absorption mechanisms and Rayleigh scattering are covered
in other classes. They cause exponential attenuation of the
guided power just as they do for plane waves in homogenous
media.
I will thus focus on the new physics of surface scatter and
bending loss. We will cover coupling loss later.
Robert R. McLeod, University of Colorado
153
•Losses and nonlinear effects
–Bend loss
ECE 4006/5166 Guided Wave Optics
Origins of bend loss
Bend loss can be predicted from ray considerations (increased incidence
angle on boundary) but not quantified.
Straight waveguide
Bent waveguide, radius R
x
θ
z
y
r
x
z
y
The waves propagating in the curved space accumulate extra phase
proportional to distance from the center of the waveguide.
Ε ( x , y , z ) = E ( x, y ,0)e − jk0 δn ( x ) z
Usual propagator
Ε ( x , y , r θ ) = E ( x, y ,0)e − jk0 δn ( x ) r θ
Equivalent in curved space
Map to prev. coordinates
= E ( x, y ,0)e − jk0 δn ( x ) ( R + x ) ( z / R )
− jk0 [δn ( x ) (1+ Rx ) ] z
= E ( x, y ,0)e
Equivalent index
x
x
0
0
δn ( x )
ncl nco
δn ( x ) ( 1 +
nco
x
R
)
Thus we can simulate (and understand) curved waveguides via a scaled δn.
Robert R. McLeod, University of Colorado
Pollock and Lipson 6.1
154
•Losses and nonlinear effects
–Bend loss
ECE 4006/5166 Guided Wave Optics
Propagating in curved spaces
Conformal transform
Consider a field in a 2D (x,z) plane described by
[∇
2
xz
]
+ k 02 n 2 ( x, z ) E (x, y ) = 0
This formally applies only TE propagation of a field Ez given a physical
structure that is invariant in y. With minor modifications, it can describe
TM propagation or waveguides with confinement in the y direction via the
effective index approximation.
Apply a conformal transformation to a new set of coordinates u (radial) and
v (azimuthal). The proper transform is:
W ≡ u + jv = R2 ln
Which gives the new wave equation:
{∇
2
uv
[
Z
x + jz
= R2 ln
R2
R2
]}
2
+ k 02 e u R2 n ( x (u, v ), z (u, v )) E ( x, y ) = 0
Application:
n (u ) = e u R2 n (ρ )
u


≈ 1 + + O (u 2 )n (ρ )
 R

which is the previous expression
u = R2 ln
ρ
R2
v = θ R2
Robert R. McLeod, University of Colorado
Heilblum & Harris, IEEE J. Q. E., VOL. QE-11, pp. 75-83, 1975
155
•Losses and nonlinear effects
–Bend loss
ECE 4006/5166 Guided Wave Optics
BPM in straight guide
Launch mode from
slab waveguide
solver.
Mode travels >
3mm without
significant chage.
a = 5 [ µm] =
λ0 = 1 [ µm]
nco = 1.5
Neff
δn = .001
N eff = 1.50068
|E| vs. x,z
Robert R. McLeod, University of Colorado
156
•Losses and nonlinear effects
–Bend loss
ECE 4006/5166 Guided Wave Optics
BPM in curved guide
nclad (1 +
xescape = R
xescape
R
)=N
N − nclad
nclad
Launch mode from
slab waveguide
solver.
Mode at exit is
leaky, lower total
power and shifted
towards larger
radius
xescape
Neff
Lossy Tail
Radius of curvature,
R 20 mm
|E| vs. x,z . The
plotting coordinate
system has been
warped to match the
curvature of the
space. The original
propagation,
however, is simply
rectangular BPM
with the index
above.
Robert R. McLeod, University of Colorado
157
•Losses and nonlinear effects
–Bend loss
ECE 4006/5166 Guided Wave Optics
Loss due to curvature
Derivation
P (z ) = P (0)e −α z
− 1 dP (z )
P (z ) dz
− 1 PTail
≈
PTot Z Char
α=
Assume an exponential decay with z
Solve for loss coefficient
Approximate value, where
∞
PTail =
2
E
∫ (x )dx
Power in tail, beyond escape radius
x Escape
∞
PTot =
∫ E (x )dx
2
Total power in guide
−∞
Z Char
2
2a )
(
=
Characteristic diffraction length of mode
2λcl
For simplicity, assume mode is unchanged (which will underestimate).
Results for slabs and fibers:
α = C1e − C R
Slab
2
− 12
α = C3 R e − C R
Fiber
4
where the constants depend only on waveguide geometry. The equations
for the constants can be found in Pollock (Eq. 8.13 & 8.14) for the fiber
case and Hunsperger (Eq. 5.3.12) for the slab case. Note that the
dominate behavior is that loss increases exponentially with bending radius.
Robert R. McLeod, University of Colorado
Hunsperger, Sec 5.3
158
•Losses and nonlinear effects
–Bend loss
ECE 4006/5166 Guided Wave Optics
Typical values
Slab waveguide
Putting the Hunsperger equation for a slab waveguide into normalized
variables yields the following expression:
C2 R
C1
αλ0 =
cos 2 (U ) exp (2W )
2
a
2nclad W   1 + 21U sin (2U ) + W1 cos 2 (U )
 λ0 
[
]

R λ0 N − nclad
exp − 4W
a λ0 nclad




Assume we desire a single mode guide with the minimum bend loss. This
will occur for the largest N which will be at the cutoff for the second mode or
V=
π
2
∴
a
λ0
=
1
4 nco2 − ncl2
Taking nclad = 1.5 and solving for N0 at cutoff of m=1 in every case yields:
Not physically
reasonable.
Violates assumptions?
Robert R. McLeod, University of Colorado
Not physically
reasonable.
Violates assumptions?
Hunsperger
159
•Losses and nonlinear effects
–Substrate coupling
ECE 4006/5166 Guided Wave Optics
Coupling to substrate
wcore
air
Si
dclad
Si
SiO2
SiO2 substrate thickness – coupling loss
Robert R. McLeod, University of Colorado
Wim Bogaerts, “Nanophotonic waveguides and photonic crystals in
siliconon- insulator”, PhD Thesis, Ghent University, April 2004
160
•Losses and nonlinear effects
–Edge scatter
ECE 4006/5166 Guided Wave Optics
Edge scatter
SEM
http://photonics.intec.ugent.be/download/ocs75.pdf
Robert R. McLeod, University of Colorado
F. P. Payne, J. P. R. Lacey, “A theoretical analysis of scattering loss from planar
optical waveguides,” Optical and Quantum Electronics 26, 977-986 (1994)
161
•Losses and nonlinear effects
–Edge scatter
ECE 4006/5166 Guided Wave Optics
Edge scatter
Outline of derivation
The field at the edge of the guide interacts with the rough edge which appears
to be a random diffraction grating:
−K = k −k
inc
The strength of
the field at the
guide edge.
The strength of
the effective
grating.
diff
Bragg matching of the grating
power spectrum to all
possible scattering angles.
By the Wiener-Khintchine the power spectrum is the Fourier transform of the
auto-correlation of the roughness
Various statistics could be chosen for this autocorrelation R(u), experiments
suggest an exponential statistic is best:
where Lc is the correlation length and σ is the standard deviation of the
roughness.
Moral: The loss will depend on the field at the edge, the index contrast of the
edge, the standard deviation of the roughness and its correlation length.
Robert R. McLeod, University of Colorado
Size Influence on the Propagation Loss Induced by Sidewall Roughness in
Ultrasmall SOI Waveguides, F. Grillot IEEE PHOTONICS TECHNOLOGY
LETTERS, VOL. 16, NO. 7, JULY 2004 1661
162
•Losses and nonlinear effects
–Edge scatter
ECE 4006/5166 Guided Wave Optics
Impact of sidewall roughness
Numerical results
Silica clad, H = 260 nm.
Typical of raw
RIE process
Increasing
confinement
Single-mode
Multi-mode
Typical of RIE
process followed
by post-processing
Increasing
confinement
Single-mode
Robert R. McLeod, University of Colorado
Multi-mode
163
•Losses and nonlinear effects
–Intro to nonlinearities
ECE 4006/5166 Guided Wave Optics
Nonlinear optics
Introduction
In the temporal frequency domain, the general dielectric response of a
material can be written as a vector Taylor series expansion of the polarization
in the electric field as:
r
P (ωq )
r
= ε 0 χ (ω q )E (ωq )
1
Linear
r
r
+ ε 0 χ (ω q = ωm + ωn )E (ωm )E (ωn )
r
r
r
3
+ ε 0 χ (ωq = ωm + ω n + ω p )E (ωm )E (ωn )E (ω p )
2
2nd order
3rd order
where sums are implied over all vector components and all frequencies,
positive and negative, that obey the constraint of summing to ωq. χ is the
“susceptibility” and is a tensor of order 2 for the linear case, 3 for second
order etc. The linear susceptibility is usually lumped into the relative
dielectric permittivity as εr = 1 + χ1.
In the time domain, susceptibilities relate real electric fields to real
polarization fields and are obviously real quantities. The Fourier transform
of the susceptibilities can have
real parts associated with nonresonant processes that do not exchange
energy with the medium that are referred to as parametric processes
imaginary parts associated with loss or gain of energy to the medium
through material resonances that are referred to as nonparametric.
The Kramers-Kronig relation and its analogs for the higher-order tensors
requires that (for a local nonlinearity):
χ n (ω q = ω1 + ω 2 + K + ω n ) = χ n (− ωq = −ω1 − ω2 − ... − ωn )
∗
Robert R. McLeod, University of Colorado
Photonic Devices, Jia-Ming Liu, Cambridge Press. Chapter 9.
164
•Losses and nonlinear effects
–Intro to nonlinearities
ECE 4006/5166 Guided Wave Optics
Summary of second- and
third-order processes
Second order:
Third order:
Single primes (‘) indicate real part, double primes (“) imaginary part
Robert R. McLeod, University of Colorado
Photonic Devices, Jia-Ming Liu, Cambridge Press. Chapter 9.
165
•Losses and nonlinear effects
–Intro to nonlinearities
ECE 4006/5166 Guided Wave Optics
Noether’s Theorem
Real Domain
Fourier Domain
Translation invariance in time
Conservation of energy
E sin (ω t )
Time
+ hΩ
δχ sin (Ωt )
− hΩ
hω
h (ω − Ω )
δP = ε 0δχE =
1
2 ε 0δχ {cos[(ω + Ω )t ] + cos[(ω − Ω )t ]}
Translation invariance in space
(
r r
δχ sin K ⋅ r
h (ω + Ω )
Conservation of momentum
)
Space
r
+ hK
δP = 12 ε 0δχ
r r r
cos k + K ⋅ r 


+


r
r
r


cos k − K ⋅ r 
[(
r r
E sin k1 ⋅ r
( )
[(
Robert R. McLeod, University of Colorado
) ]
) ]
r r
h k +K
(
)
r
hk
r
− hK
r r
h k −K
(
)
166
•Losses and nonlinear effects
–Raman and Brillouin scattering
ECE 4006/5166 Guided Wave Optics
Spontaneous Rayleigh, Raman
and Brillouin scattering
Rayleigh scattering: elastic scattering of a photon. No energy exchange
with medium so
hωout = hωin
Raman scattering: inelastic scattering of a photon from an “optical
phonon” = an out-of-phase vibration of oppositely-charged atoms
within a single crystal unit cell, creating an oscillating dipole that can
interact with the EM field. These phonons are non-dispersive (any
value of k is allowed for a particular frequency). If the phonon is
created, energy is lost from the photon and thus it emerges with a lower
frequency or Stokes shift, equal to ~14 THz for SiO2
hω out = hωin − hΩ phonon
Maxwell-Boltzman statistics specify that, in equilibrium, most of the
phonon states are unpopulated, so downward Stokes shift dominates.
Anti-Stokes shift occurs when a phonon is destroyed:
hωout = hωin + hΩ phonon
Raman scattering is automatically phase-matched and the non-dispersive
phonons allow any direction of scattering with the same frequency shift.
Brillouin scattering: inelastic scattering of a photon from an “acoustic
phonon” = an in-of-phase vibration of atoms within a single crystal unit
cell in a long-range wave. These dispersive phonons result in a smaller
frequency shift, typically 10 to 20 GHz for SiO2 that depends on
scattering direction (maximum backwards). The physics is similar to the
AO effect but here the photon creates the vibration. Scattering vanishes
for the forward direction.
Note: Stokes shift is also observed in fluorescence but is fundamentally
different because it is only observed at fixed frequencies. The Stokes
shift from scattering occurs at a phonon frequency offset from a broad
range of input optical frequencies.
Robert R. McLeod, University of Colorado
167
•Losses and nonlinear effects
–Raman and Brillouin scattering
ECE 4006/5166 Guided Wave Optics
Acoustic vs. optical phonons
Acoustic phonons
have typical
dispersion curve
Ω = VA K.
Optical nearly
flat, so ~any K
but always ~same
Ω.
Optical
hω phonon =
Acoustic
This should
probably be a/2.
+ charged atoms
a
- charged atoms
Acoustic
ka=0
Finite frequency
at k=0
Optical
Acoustic
k a = 0.3
Optical
2π/K
Density wave
k a = 0.6
Acoustic
Dipole wave
Optical
Robert R. McLeod, University of Colorado
For animations of acoustic vs. optical phonons, see:
http://fermi.la.asu.edu/ccli/applets/phonon/phonon.html
168
•Losses and nonlinear effects
–Raman and Brillouin scattering
ECE 4006/5166 Guided Wave Optics
Spontaneous Brillouin scattering
not allowed in forward direction
Conservation of energy: A pump photon is destroyed causing a Stokes
photon and an acoustic phonon to be created.
ω pump = ω Stokes + Ω phonon
Dispersion relations: Acoustic (not optical) phonons:
ω pump
r
v
v
= c k pump , ω Stokes = c k Stokes , Ω phonon = V A K phonon
Conservation of momentum:
r
v
v
k pump = k Stokes + K phonon
In the forward direction:
k pump = k Stokes + K phonon
Assume pump is forward (k>0), Stokes and phonon k can be either + or r
r
.
k pump
k pump
r
r
k Stokes K phonon
r
k Stokes
k pump = k Stokes + K phonon
ω pump = ω Stokes +
r
K phonon
k pump = k Stokes − K phonon
c
Ω phonon
VA
Energy conservation not possible.
ω pump = ω Stokes −
c
Ω phonon
VA
Energy conservation not possible.
Optical phonon dispersion relation does permit forward Raman scattering.
Robert R. McLeod, University of Colorado
C.S. Wang ,” Vanishing of the forward first stokes line in stimulated Brillouin
scattering “ Physics Letters A Volume 27, Issue 9, 23 September 1968, Pages 633-634
169
•Losses and nonlinear effects
–Raman and Brillouin scattering
ECE 4006/5166 Guided Wave Optics
Spontaneous Brillouin scattering
in other directions
ky
Assuming kpump ~ kStokes
r
k Stokes
θ
r
r
K phonon
k pump
kx
θ
1
K phonon = k pump sin
2
2
θ
1 Ω phonon ω pump
=
sin
cn
2 VA
2
Ω phonon = 2 n
VA
θ
ω pump sin
c
2
• This looks just like Bragg matching condition for an acoustooptic deflector. It essentially is – the acoustic wave is created
by the scattering of the pump photon.
• The frequency shift between the pump and the Stokes waves is
dependent on the scattering angle, reaching a maximum of
V
Ω phonon = 2n A ω pump
c
in the backwards scattering case. The Raman frequency offset
does not depend on scattering angle because all optical phonon
K vectors (K < 0.3/a) have the same frequency.
• The frequency shift is proportional to the optical frequency. In
contrast, the Raman scattering frequency offset is constant with
optical frequency.
Robert R. McLeod, University of Colorado
170
•Losses and nonlinear effects
–Raman and Brillouin scattering
ECE 4006/5166 Guided Wave Optics
Line shape
of Raman and Brillouin processes
χ (t ) ∝ e −γt sin (Ωt )u (t )
Time domain
Ω
Fourier transform in rotating wave aprox
(ω − Ω ) + jγ
Ω(ω − Ω )
Ωγ
= −χ0
+
j
χ
Re and Im parts
0
(ω − Ω )2 + γ 2
(ω − Ω )2 + γ 2
χ (ω ) ≈ − χ 0
Loss, γ > 0
Gain, γ < 0
2γ
ω −Ω
2γ
ω −Ω
ω −Ω
For Brillouin scattering in silica fiber,
the counter-propagating frequency shift
Ω/2π = 10 to 20 GHz and the
bandwidth 2γ/2π = 50 to 100 MHz.
Raman spectrum of silica is result of
superposition of multiple resonances
For Raman scattering in silica fiber,
the frequency shift Ω/2π = 13.2 THz
and the bandwidth 2γ/2π = ~10 THz.
Robert R. McLeod, University of Colorado
D. Hollenbeck and C. D. Cantrell, “Multiple-vibrational-mode model for fiber-optic
Raman gain spectrum and response function”, J. Opt. Soc. Am. B 19 (12), 2886 (2002)
171