waveguide loss - national tsing hua university photonic devic… · to quantitatively describe the...

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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)


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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Scattering Loss

• Volume Scattering

• Surface Scattering (Dominant)

voidCrystalline Defect contamination


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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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


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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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


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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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.


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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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


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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Absorption Loss

• Interband absorption electron and hole pairs (photodetector)

• Intraband absorption free carrier scattering (metal)

Valence Band

Conduction Band

gh Eν >

gh Eν <


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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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


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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Mobility of Semiconductor

Mob

ility

(cm

2 /V-s

ec)



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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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


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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Temperature-Dependent Free Carrier Absorption


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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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?)


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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Measure dielectric function of gold

Combine dipole dispersion and free-carrier dispersion


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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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



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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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 =



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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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

= − + +


Excitation of Surface Plasmonic Wave

In this case, SPP can not be directly coupled from air

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Radiation Loss

0

)(βωθ

=+dtdXR r

zdtdR

βωθ

=

RX zr

0

0

βββ −

=

The angular phase velocity should be the same.

RadiationrX


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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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∆



))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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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



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Other Losses --- Intersection


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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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


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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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= +


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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Scattering Loss Measurement

Prism Coupler


Scattering loss Measurement by Image Analysis

McNab S.J.

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