Impact of Laterally-Coupled Grating Microstructure on Effective

Coupling Coefficients

R. Millett, K. Hinzer, A. Benhsaien, T. J. Hall and H. Schriemer

(hschriem@uottawa.ca)

School of Information Technology and EngineeringCentre for Research in PhotonicsUniversity of Ottawa, Ottawa, ON

Canada K1N 6N5

OutlineOutline

•

Introduction: Laterally-coupled distributed feedback laser

•

Extended coupled-mode theory

•

Impact of grating shape on coupling strength

•

Some experimental results

•

Conclusions

Introduction: LC-DFB laserIntroduction: LC-DFB laser• Epitaxial layer structure…

• Ridge waveguide…

• Lateral

grating – in the upper ridge sidewalls…sets desired frequency.

• Active region: multiple InGaAsP quantum wells.

• Electrodes: provide injected carriers for the material gain

LC-DFB vs DFB lasersLC-DFB vs DFB lasers

Standard DFB growthLC-DFB growth

Lateral coupling has advantages:

Optimal grating conformationOptimal grating conformation• Optimizing integration density:

– functionality / footprint requires both

performance and

fabrication-tolerance.

• Optimizing performance:– Expanded design space for

control over grating shape;– Can exploit

fabrication trends

for enhanced robustness.

• Volume manufacturing focus.

Sinusoidal

Triangular

Trapezoidal

Higher-order gratings & fab toleranceHigher-order gratings & fab tolerance• CPFC* 5x i-line stepper

lithography has a resolution of 365 nm: – Higher order gratings relaxes

fabrication tolerances;– High order grating periods are

longer than first-order:

• Process limitations:– Corner rounding & shape

modification…

CPFC: Canadian Photonics Fabrication Centre, Ottawa ON

2D extended CMT – eigenvalue problem2D extended CMT – eigenvalue problem

2 2 20( , , ) ( , , ) ( , , ) 0,x xE x y z k n x y z E x y z∇ + =

( ) ( ) ( ) ( )2 20

0

, , , , exp 2qqq

n x y z n x y A x y j qzπ∞

=−∞≠

= + Λ∑

( )( )( , , ) ( , , ) expmx x m

mE x y z E x y z j zβ

∞

=−∞

= ∑

•

Begins with Helmholtz wave equation:

•

Periodic medium → Fourier series…

•

…Floquet-Bloch:

Partial waves

Fourier coefficients

•

For p

= –N, obtain forward and backward propagating modes:

•

Solve quasi-TE fundamental mode

using FEA

( ) ( ) ( ) ( )(0) ( )0 0, ,p

x xE A z x y E B z x yε ε= =

( )0 ,x yε

•

Mode

is evanescently coupled within grating region

•

The more the mode extends into the grating region, the stronger the coupling.

•

Partial waves have solutions of the form

•

Solve:

( ) ( ) ( ) ( )( ) (0) ( ), , 0,m px m mE A z x y B z x y m pε ε= + ≠

2 ( ) 2 ( )2 2 2 ( )0 02 2

20 0

( , ) ( , )( , ) ( , )

( , ) ( , ), , 0,

i iim m

m m

m i

x y x yk n x y x y

x yk A x y x y m i i p

ε εβ ε

ε−

∂ ∂ ⎡ ⎤+ + −⎣ ⎦∂ ∂

=− ≠ =

E.g.

radiating mode for a 3rd

order grating

Modified coupled mode equationsModified coupled mode equations• With 2D extended Streifer correction terms ζ1,…,4

•

Solve for

A, B

= longitudinal mode fields

κp

= Coupling coefficient

α

= modal gain

δ

= Bragg frequency detuning

( ) ( )

( ) ( )

*1 2

3 4

p

p

dA j j A j BdzdB j j B j Adz

α δ ζ κ ζ

α δ ζ κ ζ

+ − − − = +

− + − − − = +

( ) ( )

( ) ( )

*1 2

3 4

p

p

dA j j A j BdzdB j j B j Adz

α δ ζ κ ζ

α δ ζ κ ζ

+ − − − = +

− + − − − = +

Coupling coefficient κp

Coupling coefficient κp

κp

measures amount of coupling between forward- and backward-propagating

fundamental modes.

Streifer correction terms ζ1 = ζ3 Streifer correction terms ζ1 = ζ3

( ) ( )

( ) ( )

*1 2

3 4

p

p

dA j j A j BdzdB j j B j Adz

α δ ζ κ ζ

α δ ζ κ ζ

+ − − − = +

− + − − − = +

ζ1

term –

coupling of partial waves generated by forward-

propagating mode to the forward- propagating mode

Streifer correction terms ζ2 = ζ4 Streifer correction terms ζ2 = ζ4

( ) ( )

( ) ( )

*1 2

3 4

p

p

dA j j A j BdzdB j j B j Adz

α δ ζ κ ζ

α δ ζ κ ζ

+ − − − = +

− + − − − = +

ζ2

term –

coupling of partial waves generated by forward-

propagating mode to the backward-propagating mode

Effective coupling coefficient κeff

Effective coupling coefficient κeff

•

A measure of grating strength:

•

Contributions from:

and

where:

( )( ) ( )*2 4 expeff p p eff effjκ κ ζ κ ζ κ φ κ⎡ ⎤= + + = ⎣ ⎦

( ) ( )2

200

0

, ,2p p

G

k A x y x y dxdyP

κ εβ

= ∫∫

( )2 ,

0,

pq q

qq p

ζ η∞

−=−∞≠ −

= ∑

( ) ( )2

( ) ( )0, 0

0

, ( , ) ,2

i ir s r s

G

k A x y x y x y dxdyP

η ε εβ

= ∫∫

(0)4 ,

0,

q p qqq p

ζ η∞

−=−∞≠

= ∑

Grating strengthGrating strength•

Effect of correction terms…E.g.,

3rd order: WN

/ WW

= 1.5 / 3 (μm), 0.7 duty cycle:

Shape functions and Fourier coefficientsShape functions and Fourier coefficients• Grating:

• Average refractive index:

• Fourier coefficients:

( ) ( ){( )}

2 1 20 1 2 1

22 1 2

( ) ( )

( ) ( )

n x n w x w x

n w x w x

−= Λ −

+ Λ + −

( ) ( ) ( )

( ) ( ){ ( )}

22

2

2 22 1

2 1

1 , , exp 2

exp 2 ( ) exp 2 ( )2

qA x n x y z j qz dz

n nj qw x j qw x

j q

π

π ππ

Λ

−Λ

= − ΛΛ

−= − Λ − − Λ

∫

1( )w x2 ( )w x

x

z1n

2n

Results – 1st order gratings…Results – 1st order gratings…• The wide ridge width is fixed at 4 µm. The narrow ridge width

is varied from 1 to 4 µm.• The grating fades away when WN

approaches WW

and the coupling thus vanishes

…Higher order gratings…Higher order gratings• The relative behaviour

of |κeff

| markedly changes at higher grating orders…

• Rectangular gratings no longer provide the strongest coupling:

– Dramatic improvements for higher order.

Why?

Field and Fourier coefficientField and Fourier coefficient• Effect of dominates κeff

, but…

3rd

order gratings with WN

/ WW

= 1.5 / 4 (μm)

Rect Sin TriΓ < Γ < Γ

( )( ) ,is x yε

(0.43%, 0.72%, 0.98%)

TE mode ~ exponential decay

Contribution of Fourier coefficient nearest the ridge dominates.

Trapezoidal gratingsTrapezoidal gratingsσ

= 0: rectangular

σ

= 0.5: triangularσ

= 0.42: optimal

Rounded gratingsRounded gratings

•

3rd

order: WN

/ WW

= 1.5 / 4 (μm) h1

= h3

, 0.5 duty cycle

•

Elliptical rounding

•

Rectangular grating: h1

& h3

→ 0

•

1 2h h

Rect Round SinΓ < Γ < Γ

L-I

measurementsL-I

measurements• Threshold currents similar to FP on same wafer

– minimal optical loss penalty for using higher order gratings• Threshold currents larger than expected due to additional loss

from proximity of optical mode to metal contacts

SMSRSMSR• 3rd order:

WN

/ WW

= 1.5 / 4.5 (μm) 200 mA

bias, room temp.

Extract κeff

Extract κeff

• Fit the below-threshold spontaneous emission spectrum…– A 10-dimensional constrained nonlinear optimization

• Predict:• Fit: 112.5 1.38 1.4 0.05 cmeffκ −= ∠ ± ∠

112.2 1.41 cmeffκ −= ∠

ConclusionsConclusions

• Extending CMT to incorporate 2D treatment of partial waves is

critical to the design process

• Grating conformation plays a key role

– Shape optimization presently underway

• “Alpha” experimental results extremely promising

• Effective coupling coefficient…

– Theory and experiment agree

AcknowledgmentsAcknowledgmentsSupport has been provided by:• CMC Microsystems• Ontario Centres

of Excellence (OCE)

• Canadian Foundation for Innovation (CFI)• Natural Sciences and Engineering Research Council (NSERC)• OneChip

Photonics

• Canada Research Chairs Program• Canadian Photonics Fabrication Centre• Ontario Graduate Scholarship Program• Communications Research Centre Canada