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Ming-Chang Lee, Integrated Photonic Devices
Waveguide Loss
Class: Integrated Photonic DevicesTime: Fri. 8:00am ~ 11:00am.
Classroom: 資電206Lecturer: Prof. 李明昌(Ming-Chang Lee)
Ming-Chang Lee, Integrated Photonic Devices
Optical Loss in Waveguides
• Scattering Loss– Due to surface roughness
• Absorption Loss– Due to photons annihilated in materials
• Radiation Loss – Due to waveguide bending
Three major losses in waveguide
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Ming-Chang Lee, Integrated Photonic Devices
Scattering Loss
• Volume Scattering
• Surface Scattering (Dominant)
voidCrystalline Defect contamination
Ming-Chang Lee, Integrated Photonic Devices
Surface Scattering Loss (Tien’s Model)
• Each reflection induce scattering light
interface
'mθ
' 24exp[ ( cos ) ]r i mP P πσ θλ
= −
iP rP
Rayleigh Criterion
σ
σ : variance of surface roughness
'
0 1
sin mm k n
βθ =
1n
m: mode number
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Ming-Chang Lee, Integrated Photonic Devices
Surface Scattering Loss
To quantitatively describe the optical loss, the exponential attenuation coefficient is generally used. In this case, the intensity (power per unit length) decays along the waveguide.
)exp()( 0 zIzI α−= 0I is the initial intensity at z = 0
12σ
13σ
'mθ
3γ
2γ
n1
n2
n3
))/1()/1(
1)(sin
cos21(
32'
'32
γγθθα
++=
gm
ms tA 2 2 1/ 2
13 124 ( )A π σ σλ
= +
efft/1Average roughness
Ming-Chang Lee, Integrated Photonic Devices
Scattering Loss Analysis by Tien’s Model
Consider a planar waveguide with TE polarization
'cos mθ
'mθ
1 cm
Substrate, n2
Waveguide, n1
The power carried by the incident beam hit on the unit length (1 cm)
yE
2 '1 cos
8 y mc n E θπ
Ey is the field amplitude
According to the Rayleigh criterion, the reflected beam from the upper film surface
22 ' '
14cos exp cos
8 y m mc n E πσθ θπ λ
⋅ −
σ: variation of surfaceroughness
Rayleigh criterion
Cover cladding, n3
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Ming-Chang Lee, Integrated Photonic Devices
Scattering Loss Analysis by Tien’s Model
Consider the two film surface
4π σλ
( )1/ 213 12
4π σ σλ
+
The power lost by surface scattering per unit length is
( ){ }22 ' '1
2 2 3 '1
cos 1 exp cos8
cos8
y m m
y m
c n E A
c n E A
θ θπ
θπ
⋅ − −
≈
The planar waveguide mode power flow
2 '1
2 3
1 1sin4 y m gc n E tθπ γ γ
⋅ + +
power flow tg
3
1γ
2
1γ
Ming-Chang Lee, Integrated Photonic Devices
Scattering
The power attenuation per unit length
3 '2
'2 3
cos1 12 sin (1/ ) (1/ )
m
m g
At
θαθ γ γ
= + +
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Ming-Chang Lee, Integrated Photonic Devices
Surface Scattering Loss
• High order modes have more reflections from the surface
Low-order mode
High-order mode
mgR t
LNθcot2
= Where m is mode no.
Ming-Chang Lee, Integrated Photonic Devices
Mode Effect
The loss for the m=3 waveguide mode is as much as 14 times that of the m = 0 waveguide mode.
Ta2O5
λ= 632.8 nm
Surface Scattering
Volume absorption
1: ( / ) 4.3 dB cm cmα −=Tien, 1971
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Ming-Chang Lee, Integrated Photonic Devices
Sidewall Scattering Loss
• Sidewall roughness is created during etching process• The propagation loss is highly related to the roughness for a
small-dimension waveguide
Rough Sidewall Smooth Sidewall Measured Result
Tsuchizawa T. JSTQE 2005
Ming-Chang Lee, Integrated Photonic Devices
Optical Loss due to Surface Roughness
• Single mode waveguide
• Surface roughness required to achieve low loss
w× t < 0.18 µm2
w
t
Waveguide Width (um)
5 nm
3 nm
1 nm
Wav
egui
de L
oss
(dB
/cm
) t = 0.25um
SurfaceRoughness
TM-like Mode
Si
δrms < 1 nm
Wavelength : 1550nm
Optical Field
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Ming-Chang Lee, Integrated Photonic Devices
Absorption Loss
• Interband absorption electron and hole pairs (photodetector)
• Intraband absorption free carrier scattering (metal)
Valence Band
Conduction Band
gh Eν >
gh Eν <
Ming-Chang Lee, Integrated Photonic Devices
Free Carrier Absorption (Drude Model)
E
electrons
)exp()exp( 002
2
tjeEtjqEdtdxmg
dtxdm ωω −==+
Not harmonic oscillator!
g : damping coefficient due to scattering
m : mass of carrier
)exp(/)(2
0 tjgjmeEx ω
ωω −=
EχχPPED )1( 100100 ++=++= εεRecall
Free Carrier EffectDielectric polarization(Dipole Moment) (electron or hole)
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Ming-Chang Lee, Integrated Photonic Devices
Free Carrier Absorption
E
electrons
)exp(/)(02
2
1 tjEgjmNeNex ω
ωω −−
=−=P
''1
'12
02
1)/()( χχχ j
gjmNe
+=−
−=
ωωε
20
1 2 2
20
1 2 2
( ) /( )χ '
( ) /( )χ ''
Ne mg
Ne g mg
εω
ωεω
= − +
= +
( )' ' ''0 0 1 0 0 1 1 0 1 1[(1 ) ]
2''0D E P P χ χ χ E Ej n n jnε ε ε= + + = + + + = + +
'''' 1
1 '0 12( )
nn n
χ=
+'' '
1 11χ χ χ<< + +where
Ming-Chang Lee, Integrated Photonic Devices
What is g?
At steady state, electron move as a constant speed.
eEdtdxmg =
The definition of mobility µ
Edtdx µ=
(1)
(2)
µmeg =
That is, d2x/dt2 = 0
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Ming-Chang Lee, Integrated Photonic Devices
Mobility of Semiconductor
Mob
ility
(cm
2 /V-s
ec)
Ming-Chang Lee, Integrated Photonic Devices
Free Carrier Absorption
2 2 ' '' ' ''0 0 0
2 2''0 0 0
exp[ ( ) ] exp[ ( ) ]
exp[ 2 ] exp[ ]
I E E jk n jn z jk n jn z
E k n z E zα
∝ = + ⋅ − −
= − = −
'' 3''
0 0 2 20
2fcNek n k
n m n cχα
ε ω µ= ≈ =
• Free carrier absorption is proportional to the carrier density
• The refractive index is also affected by the free carriers
( )gω >> For optical wavelength
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Ming-Chang Lee, Integrated Photonic Devices
Free Carrier Absorption
17 34.6 10 cm−×
18 31.4 10 cm−×
18 32.5 10 cm−×
19 31.68 10 cm−×
p-Si, 300k
Hole Concentration
16 3(1) 1.4 10 cm−× 16 3(2) 8 10 cm−× 17 3(3) 1.7 10 cm−×17 3(4) 3.2 10 cm−× 18 3(5) 6.1 10 cm−× 19 3(6) 1 10 cm−×
n-Si, 300k
Ming-Chang Lee, Integrated Photonic Devices
Resistivity vs. Impurity Concentration
Sze and Irvin
Impurity Concentration (cm-3)
Res
istiv
ity(O
hm-c
m)
0.6
2
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Ming-Chang Lee, Integrated Photonic Devices
Temperature-Dependent Free Carrier Absorption
Ming-Chang Lee, Integrated Photonic Devices
Free Carrier Absorption on Proton Bombardment Waveguide
Proton
GaAs or GaP
GaAs or GaP (Heavily Doped)
* 2 20
2 1 2 2( )4 g
m cN Nt e
ε π− ≥
0 1.3 mλ µ=
0 10.6 mλ µ=
The major loss comes from the evanescent wave penetrating in substrate
tg
d=tg
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Ming-Chang Lee, Integrated Photonic Devices
Surface Plasmons
• The interaction of metals with electromagnetic radiation is largely dictated by the free electrons in the metal.
• Most metals possess a negative dielectric constant at optical frequency
• Only the surface can support optical wave propagation (why?)
Ming-Chang Lee, Integrated Photonic Devices
Optical Properties of Nobel Metals
2' 0
2 2
2" 0
2 2
( ) /( )χ ( )
( ) /( )χ ( )
free carrier
free carrier
Ne mg
Ne g mg
εωω
ωεωω
−
−
= − +
= +
2 2 2' 0
2 2 2 2 20 0
2''
2 2 2 2 20 0
( / )( )χ ( )[( ) ]
( / )( )χ ( )[( ) ]
dipole
dipole
Ne m
Ne m
ω ωωε ω ω ζ ω
ζωωε ω ω ζ ω
−= − +
= − +
Free-Carrier DispersionDipole Dispersion
goldgold
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Ming-Chang Lee, Integrated Photonic Devices
Measure dielectric function of gold
Combine dipole dispersion and free-carrier dispersion
Ming-Chang Lee, Integrated Photonic Devices
Surface Plasmons at plane interface
• EM wave and surface charge are oscillating.• The fields in the perpendicular direction decay
exponentially.• The momentum of SP is larger than the free space
photon.
W.L. Banes, nature review article
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Ming-Chang Lee, Integrated Photonic Devices
Surface Plasmons at plane interface
Metal
Dielectric
1 1,ε µ
2 2,ε µ
2
2E( , ) ( , )E( , ) 0r r rcωω ε ω ω∇×∇× − =
Mathematically, the solution has to satisfy the wave equation
where 1( , ) ( )rε ω ε ω= z < 0 2( , ) ( )rε ω ε ω= z > 0
Ming-Chang Lee, Integrated Photonic Devices
Surface Plasmons at plane interface
TE wave or s-wave can not be a solution of surface plasmonic wave (H can not satisfy the boundary condition)
Consider a TM wave or p-wave,
,
,
0 exp( )exp( )i x
i x z
i z
EE jk x j t jk z
Eω
= −
1, 2i =where
Since the x-direction wave vector is conserved or Snell’s law
2 2 2,x i z ik k kε+ = k
cω
=where
Since both spaces are source-free; that is, D = 0∇ ⋅
, , , 0x i x i z i zk E k E+ = (a)
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Ming-Chang Lee, Integrated Photonic Devices
Surface Plasmons at plane interface
Consider the boundary condition,
1, 2,
1 1, 2 2,
00
x x
z z
E EE Eε ε
− =
− =(b)
Combine (a) and (b), since the electric fields are not trivial solutions, the determinant of respective matrix has to be zero; then
1 2, 2 1, 0z zk kε ε− =
Therefore,
22 21 2 1 2
21 2 1 2
xk kc
ε ε ε ε ωε ε ε ε
= =+ +
22
,1 2
ii zk kε
ε ε=
+and
2 2 2,x i z ik k kε+ =recall
1, 2i =
Ming-Chang Lee, Integrated Photonic Devices
Surface Plasmons at plane interface
Since the surface plasmonic mode are evanescent on the two sides of interface
xk should be real ,i zk should be imaginary and
Therefore
1 2
1 2
( ) ( ) 0( ) ( ) 0
ε ω ε ωε ω ε ω
⋅ <+ <
The dielectric functions must be negative with an absolute value exceeding that of the other.
Nobel metals such as gold and silver, have a large negative real part of the dielectric constant along with small imaginary part
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Ming-Chang Lee, Integrated Photonic Devices
Properties of surface plasmonic waves
Consider the metal dielectric
' "1 1 1jε ε ε= +
Suppose the imaginary part is much smaller than the real part and ε2 is positive real, the wave number of SP mode
' "x x xk k jk= +
'' 1 2
'1 2
' "" 1 2 1 2
' ' '1 2 1 1 22 ( )
x
x
kc
kc
ε ε ωε ε
ε ε ε ε ωε ε ε ε ε
≈+
≈+ +
where
and'2 "
1 11, ' '
1 2 1
12zk j
cε εω
ε ε ε
= + +
2 "2 1
2, ' '1 2 1 2
12( )zk j
cε εω
ε ε ε ε
= − + +
Ming-Chang Lee, Integrated Photonic Devices
Excitation of Surface Plasmonic Wave
In this case, SPP can not be directly coupled from air
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Ming-Chang Lee, Integrated Photonic Devices
Radiation Loss
0
)(βωθ
=+dtdXR r
zdtdR
βωθ
=
RX zr
0
0
βββ −
=
The angular phase velocity should be the same.
RadiationrX
Ming-Chang Lee, Integrated Photonic Devices
What is Attenuation Coefficient (α)?
zzP
zP
zPzPdzzdP
zP
∆∆
−≈
−=−=
)()(
1
))exp()(()()(
1
0
0 αα Because
Dissipated Power
Propagation length
Z∆
0P
P∆
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Ming-Chang Lee, Integrated Photonic Devices
What is α due to Radiation Loss?
radP
rX
Total Power
Suppose the field
≥
−−=
≤≤−=
2)2/(
exp[)2
cos()(
22)cos()(
0
0
axaxhaCxE
axahxCxE
γ
∫∞
∞−
++== )]2
(cos)sin(21
2[)( 2
02 haha
haCdxxEPtotal γ
Radiated Power
∫∞
−−==rX
rradaXhaCdxxEP ))2
(2exp()2
cos(2
)( 02
γγ
0( )P z
( )P z∆
Ming-Chang Lee, Integrated Photonic Devices
What is Attenuation Coefficient (α)?
))2
sin((2
2
aaaZc
λφλφ
=== Because a: waveguide widthλ: wavelength
The propagation length of unguided mode (analogy to a truncated waveguide)
22
10
02
)]2
(cos)sin(21
2[
)exp(2)2exp()2
(cos2
ahahah
a
aRha z
γ
γλ
βββ
γγ
α++
−−
=
The attenuated coefficient:
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Ming-Chang Lee, Integrated Photonic Devices
What is α due to Radiation Loss?
• The attenuation coefficient decreases with the bending radius
• The attenuation coefficient decrease with the index contrast
)exp()]
2(cos)sin(
21
2[
)exp(2)2exp()2
(cos2
2122
10
02
RCCahaha
ha
aRha z
−=++
−−
=γ
γλ
βββ
γγ
α
0
02
2β
ββγ
−= zC
Ming-Chang Lee, Integrated Photonic Devices
What is Attenuation Coefficient (α)?
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Ming-Chang Lee, Integrated Photonic Devices
Other Losses --- Intersection
Ming-Chang Lee, Integrated Photonic Devices
Waveguide Loss Measurement
• How to distinguish the loss?1. Waveguide loss or coupling loss?2. Waveguide loss of fundamental mode or
high-order modes?3. Scattering loss, absorption loss, or
radiation loss?
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Ming-Chang Lee, Integrated Photonic Devices
End-Fire Coupling Loss Measurement
1 2
2 1
ln( / )P PZ Z
α =−
for 2 1Z Z>
• Advantage– Simple and direct
• Disadvantage– Alignment sensitive– End face condition should be
consistent– Can’t distinguish the loss
associated with different mode number
Ming-Chang Lee, Integrated Photonic Devices
Prism-Coupled Loss Measurement
m = 0
m = 1
m = 0
m = 1
• Advantage– Can measure the loss from different modes– Alignment insensitive– End face quality is not required
• Disadvantage– Less accurate (It is difficult to reproduce the coupling loss)
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Ming-Chang Lee, Integrated Photonic Devices
Fabry-Perot
1U
1 exp( )exp( )2iU U t j Lαϕ= ⋅ − −
2 2*
0 2 2
(1 ) exp( )(1 ) 4 sino o
LI U UR Rγ α
ϕ− −
= ⋅ =− +
22 1 exp( 2 )exp( )U U j Lγ ϕ α= ⋅ − −
2UiU oU
2 21t γ= −
, tγ , tγ
: reflectionγ
:t transmission
2 exp( )R Lγ α= −where
0 nn
U U= ∑
max min2
max min
11 1ln1
I IL I I
αγ
−= − +
max :I nϕ π=
min1: (2 1)2
I nϕ π= +
Ming-Chang Lee, Integrated Photonic Devices
Fabry-Perot Loss Measurement
max min2
max min
11 1ln1
I IL I I
αγ
−= − +
2γ : Reflectivity
• Advantage– Alignment insensitive
• Disadvantage– End face condition should
be consistent– Only for single mode
waveguide– Light source should be
single frequency
L
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Ming-Chang Lee, Integrated Photonic Devices
Scattering Loss Measurement
Prism Coupler
Ming-Chang Lee, Integrated Photonic Devices
Scattering loss Measurement by Image Analysis
McNab S.J.