Asymmetric Fano resonance and bistability in a two-ring resonator switch

by:Landobasa Y.M. Tobing

M. K. Chin

Photonic Research CentreNanyang Technological University

Singapore

Illustration about bistability•As IIN ↑, the resonant wavelength λ0 moves.

•The movement depends on the sign of Kerr nonlinearity n2.

•For n2<0, it is pulled towards the operating wavelength λi.

•As λ0 moves, the build-up factor increases accordingly.

•The build-up factor B starts to decrease when λ0 < λi.

λi λ0

n2 < 0 n2 > 0

Cav

ity

build

-up f

acto

r B

(1) λ0( IIN)

(2) At n2 < 0, the situation in (1) becomes unstable at the increasing B. It would suddenly jump to a stable situation in (2).

Introduction

Parameters in nonlinear switching

Extinction ratio ER = ON/OFF

Modulation depth MD = ON-OFF

n2IIN

T

ON

OFF

IONIOFF

Nonlinear

T

λLinear

Introduction

Bistability in various configurations

……

WG1

WG2

(1) (2) (3)

(4)

(5)

(6)

1. F. Sanchez, Opt. Commun. 142, 211 (1997). 2. B. Maes, et. al.. J. Opt. Soc. Am. B 22(8), 1778-1784 (2005).3. Y. Dumeige, et. al.. Opt. Commun. 250 (2005) 376-383. 4. Y. Dumeige, et. al.. Phys. Rev. B, 72 066609 (2005).5. J. Danckaert, et. al., Phys. Rev. B, 44, 15, 8214 (1991).6. M. Soljačić, et. al., Phys. Rev. B, 66, 055601 (2002).

Outline

Bistability in symmetric resonance

IIN

IR

IOUTIIN ID

ITlimitation

Bistability in asymmetric resonance

Comparison of switching performance

One ring coupled to one bus

2 2

' '

1

b r jt ab jt r a

r t

⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠+ =

IIN

IR

IOUT

Critical couplingr = a

100.8 100.9 101 101.1 101.20

0.2

0.4

0.6

0.8

1

δ/2π

r = a

r ≠ a

TH

RO

UG

H

a’ b’

a b

Assumptions:

Phase matched and lossless coupling.

The effective index is constant within narrow range of frequency.

( )/

2eff C

C

c n L

L R

δ ω

π

=

=

Symmetric resonance

Parametric formulation

1 exp( )LLαη

α− −

=

One ring coupled to one waveguide bus

2R

2 2IN

11 2 cos

I rBI ra r aδ

−= =

− +

IIN

IR

IOUT

( )eff 2

( ) ( )2

R L NL R

R

I IL n n I

δ δ δπ ηλ

= + Δ

= +

IR

IR/IIN

•The line slope becomes flatter at increasing input intensity IIN.

•Multistability starts to occur when the there are more than one intersections.

•The corresponding IIN and IOUT can be found through IR.

Only Kerr nonlinearity n2 is assumed to present.

Nonlinear Spectrum

One ring coupled to one waveguide bus

10-4 10-3

0.2

0.4

0.6

0.8

n2IIN

T

10-40

0.2

0.4

0.6

0.8

1

n2IIN

λ = 1567nm

(a) (b)T

0.950.920.89

1567.4nm1567.6nm1567.8nm

•Different curves in (a) correspond to different initial detuning. Bistability occurs only when detuning is larger than the critical detuning.

•Extinction ratio (ER) increases when detuning is closer to critical detuning.

•Figure (b) shows ER increases when the loss is increased, which broadens the linewidth. This has the same effect as if the detuning is closer to critical detuning. But broader linewidth also increases the input threshold.

The nonlinear sensitivity is at best in the critical coupling condition.

0| | 3 2ω ω ω− ≥ Δ

Initial detuning condition

Limitation of symmetric resonance

Generally, high ER can be obtained if we place the operating wavelength very close to critical detuning. This makes, in critical coupling condition, the initial amplitude spectrum to be low.High ER consequently yields low ‘modulation depth’.

Difficult to obtain both high ER and large MD

The Two Ring Configuration

IIN ID

IT

•The upper ring provides optical feedback to the bottom ring.

•This induces significant phase perturbation at the vicinity of the upper ring resonance.

•This ‘loading effect’induces a large phase swing, where additional resonance is possible.

•The resonance is mainly due to the interference between the denoted pathways Path 1

Path 2

2 1/R Rγ =

Asymmetric resonance

Linear spectrum

Two Ring Configuration

102.6 102.7 102.8 102.9 103 103.1 103.2 103.3 103.4

-206

-207

-208

-205

-209

102.5 103 103.50

0.5

1

δ/π

T

δ1/2π

γ = 1.05 γ = 1.0

1 12 1 2 2 11

2 2 1 2 2 1

sin( ) sin( )tan tan

cos( ) 1 cos( )a a r

r a a rγδ γδ

δ δγδ γδ

− −⎧ ⎫ ⎧ ⎫= − −⎨ ⎬ ⎨ ⎬

− −⎩ ⎭ ⎩ ⎭

The modified round trip phase

1.55 1.552 1.5540

0.5

1

λ (μm)

(2) (1)

(1) (2)

γ = 0.95

γ = 1.0

γ = 1.05

(3) (4)

T (1) (2)

(3) (4)

Resonance Splitting

The tuning of Asymmetricity

Two Ring Configuration Linear Spectrum

•At γ = 1, the resonances are split symmetrically. At γ ≠ 1, the resonances are split asymmetrically, giving rise to asymmetric Fano-like resonance.

•The Fano resonance becomes narrower as it detunes farther from the bottom ring resonance.

•The asymmetricity can be tailored upon changing the ‘envelope’ from the bottom ring spectrum as denoted by changing r1.

1.55 1.552 1.554 1.5560

0.5

1

DR

OP

1.55 1.552 1.554 1.556

1.55 1.552 1.554 1.5561.55 1.552 1.554 1.556λ (μm)

γ = 1.0 γ = 1.03

γ = 1.05 γ = 1.07

1.553 1.5535 1.554 1.5545 1.555 1.5555 1.556 1.5565 1.5570

0.2

0.4

0.6

0.8

1

λ (μm)

DR

OP

(D) r1 = 0.75

r1 = 0.85

r1 = 0.95

r2 = 0.99

Linewidth ‘shrinking’

1.535 1.54 1.545 1.55 1.5550

0.2

0.4

0.6

0.8

1

λ (μm )

D

γ = 2 .0

1.5452 1.5454 1.54560

0.5

1

λ (μm )

2 22 1

2 2 4 2 21 1 2

21

FWHM 21 2

(1 )| | ,

(1 ) (1 )

2(1 )with

(1 )

rD d

r r B

rr B

γ θ

θγ

−= ≅

− + +

−=

+

•A Vernier effect at integer γ makes resonance splitting around the bottom ring resonance.

• At the upper ring resonance (just right in the middle), the symmetric Fano-like resonance arises, with the linewidth ‘shrunk’by the build-up factor B2 in the upper ring, relative to the bottom ring resonance linewidth.

Two Ring Configuration Linear Spectrum

Nonlinear Spectrum (parametric formulation)

Two Ring Configuration Nonlinear Spectrum

E8 E9

E1E6

E2 E3E4

E5

ET

EIN ED

Ring 2

Ring 1

1/ 2 14 21 2 1 1 1 1

16 21 2 1 1 1 1 1

2 16 21 1 2 1 1 1 1 1

exp( | | ) exp( / 2)

[1 exp( )exp( | | )]( / )

[1 exp( )exp( | | )]( / )

D NL

T NL

IN NL

E it T a i E i E

E a T i i E r it E

E a r T i i E r it E

γ δ

δ γ

δ γ

= −

= − −

= − −98 2

3 2 2 2 9 9 2

98 22 2 2 2 9 9 2

[ exp( ) exp( | | )] /

[1 exp( )exp( | | )] /NL

NL

E r a i i E E it

E a r i i E E it

δ γ

δ γ

= − −

= − −2 1/ 2 2

2 1 1 2 3 2 1,2 1,2 1,2| | | | , / , exp( )E a E T E E a Lα− = = = −

0 eff 2 [1 exp( )] /(2 )ijNL o ijk n n c Lγ ε α α= − −

•For convenience, the E9 is chosen as the parametric field.

•The loading effect is denoted by the T2 in the set of equations.

•The n2 < 0, R ~ 15 μm (*)

Set of equations

(*) We followed the n2 and dimensions used in Y. Dumeige, et. al., Opt. Commun. 250 (2005) 376-383

Nonlinear Through (T) Output for different asymmetrical resonances

Two Ring Configuration Nonlinear Spectrum

•In the critical coupling condition, the resonant absorption totally quenches the interference between the two pathways, thus degrading the nonlinear response.

•At non-critical coupling, the intensity build-up is faster as it increases but slower as it decreases.

•Therefore, the ER is significantly improved in a more asymmetric case.

10 -6 10-5 10-4 10-3 10-20

0.2

0.4

0.6

0.8

1

n2IIN

T λ = 1555 nm

r1,2 = 0.95

r1 = 0.85 r2 = 0.95

r1 = 0.85 r2 = 0.95 a1,2 = r2

r1 = 0.85, r2 = 0.95

r2=a1,2=0.95r1 = 0.85

r1,2 = 0.95

Comparison between the two configurations (with the same optical bandwidth)

symmetric

asymmetric

•Both configurations exhibit ~0.1nm (10GHz) optical bandwidth, with ER > 10dB.

•Observably, the asymmetric resonance has ‘wider critical detuning’ as compared to the symmetric one.

•The asymmetric case has more modulation depth.

•If the relatively high MD is maintained, the input threshold for asymmetric case is one order lower.

10-10 10-5 1000

0.2

0.4

0.6

0.8

1

n2IIN

T

ConclusionsIntroducing asymmetricity increases the extinction ratio and modulation depth at the same time.To achieve a desired optical switching bandwidth, a compromise with the input threshold is necessary. Critical coupling degrades the switching performance by quenching the inter-pathways interference.The two-ring configuration can be used to generate ultra-narrow resonances.

THANK YOU

Q&A

Verification with coupled mode theory from other works

1.55 1.552 1.5540

0.5

1

λ (μm)

(2) (1)

(1) (2)

γ = 0.95

γ = 1.0

γ = 1.05

(3) (4)

T (1) (2)

(3) (4)

From Yanik et. al. (CMT) Transfer matrix

1.55 1.552 1.554 1.556 1.558 1.5610 -2

10 -1

10 0

10 1

10 2

λ (mum)

B 1 B2

B21

( )( )

9 221

2 2 2 2

1 1 11 2

1 1 2 1

1/ 49 22 1 1 21 1

2

1 exp( )/

1 exp( )

exp( / 4)

IN

IN

E itFE

E a r iE it r

FEE a r T i

E EFE a i FE FE

E E

δ

δ

δ

= =− −

= =− −

⎛ ⎞⎛ ⎞= = −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

Build-up factor in the respective rings

E8 E9

E1E6

E2 E3E4

E5

ET

EIN ED

Ring 2

Ring 1

0

0.2

0.4

0.6

0.8

1

1.554 1.5545 1.555 1.5555 1.556-1000

-500

0

500

1000

λ (μm)

dB21(λ)/dδ1

dB2(λ)/dδ1

T

Build-up slope in the upper rings