Theoretical study of light and sound interaction in phoxonic crystal structures

Dr. José María Escalante Fernández

Thesis link

http://riunet.upv.es/handle/10251/33754

CQED in optomechanical cavities -Jaynes-Cumming model of an indirect gap semiconductor cavity -Theoretical study of two-level systems inside on optomechanical cavity where mechanical oscillation are induced

OUTLINE

Summary of my scientist background

Summary of my scientific research

Slow-wave phenomena in phoxonic structures -CRAW -Slow-waveguide in honeycomb phoxonic crystal structure -Effect of losses on group velocity in phoxonic crystals

Optical gain in silicon -Optical gain in silicon optical cavities -Einstein’s relation fro indirect bandgap semiconductor -Optical gain in silicon acuto-optical cavities

PhD in Telecommunications Engineering -Theoretical study of light and sound interaction in phoxonic crystals structures -My supervisors: Alejandro Martínez Abietar and Vincent Laude

Summary of my scientist background

Bachelor-Degree in Physics -Quantum Mechanics -Solid state physics -Materials science -Electrodynamics

Master in Telecommunication Engineering -Optical communications -Microwave theory -Antenna theory -Wireless communications

Classical Physics

Solid State Physics

Quantum Physics

Slow-wave phenomena in phoxonic structures

Optical gain in silicon

CQED in optomechanical cavities

MA

CRO

SCO

PIC

MIC

ROSC

OPI

CSummary of my scientific research

Slow-wave phenomenain phoxonic structures

The slow wave phenomena is phoxonic structure is a effect very interesting to develop devices implementable in silicon photonic technology.

CRAW (Coupled resonator acoustic waveguide)

CROWAmnon Yariv, Yong Xu, Reginald K.

Lee, and Axel Scherer, Optics Letters, Vol. 24, Issue 11, pp. 711-713 (1999)

Tight-binding model

Control of group velocity with the distance between cavities

Linear chain of coupled harmonic oscillators.

∑+∞

−∞=+−=

nmnm

n Udt

Ud γ2

2

Tight-binding model

José M. Escalante, Alejandro Martínez, and Vincent Laude, “Dispersion relation of coupledresonator

acoustic waveguides formed by defect cavities in a phononic crystal“, J. Phys. D: Appl. Phys. 46 475301

(2013)

Slow-waveguide in a honeycomb phoXonic crystal structure

Photonic Dispersion relation Phononic Dispersion relation

Ladder Waveguide∆=a∆=1.1a ∆=1.2a ∆=a∆=1.1a ∆=1.2

a

Photo-Elastic effect

PHONONIC SLAB MODES

GROUP VELOCITY OF PHOXONIC SS-WB

Optomechanical effect

Ali Adibi et al., Phys Rev. B Vol .64, 033308 (2001)

Slot-Ladder WaveguideH=200nmH=100nm∆=1.2a H=300nm,β=0.8

Optomechanical effect

So, we have a wave guide which supports slow modes with low losses

GROUP VELOCITY OF PHOXONIC SS-WB

José M. Escalante, Alejandro Martínez, and Vincent Laude, “Slow phoXonic waveguides in honeycom lattice", (ACCEPTED Journal of Applied Physics)

Effect of losses on group velocity in phoxonic crystals

The master equation of photonic crystals.

Hc

Hr

2

)(

1

=

∧∇∧∇ ω

ε ( ) ururC

)(:)(ˆ 2ρω−=∇⋅∇

The master equation of phononic crystals.

BxxkA 2)( ω=

In order to obtain a numerical solution and following a Galerkin procedure, we get an eigenvalue problem of the form

By left-multiplying with the left-eigenvector X.

BxxxkAx TT 2)( ω=

0)();,( 2 =−= BxxxkAxkD TT ωµω

Getting the dispersion relation in implicit form

0);,();,( =−−=+ µδδωωδµµω kkDkD''' ; ''' iCCi +−=+ εεδµµ

FLi2

ωδω =

n=2 n=3

n=4

nkkkD )();,( 00 −−−= αωωµω

Frozen Modes

Effect of losses inside bandgap

0

0

2

2 ωωκ −−

= kkB

vg

FL

Bvg

0

2

2κω≈

Classical Hartman effect

( )( ) ( ) 012

);,( 02212=−

−−−

= kk

BkD

κωωωωµω

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and . Sheng, Phys. Rev. Lett. 88 , 104301 (2002)

Vincent Laude, José M. Escalante, and Alejandro Martínez, “Effect of loss on the dispersion relation of photonic and phononic crystals", (ACCEPTED Physical Review B)

Optical gain in silicon

The possibility to develop a silicon laser will allow a low cost of laser implementable with silicon photonic technology.

M. J. Chen, C. S. Tsai, and M. K. Wu, J. J. Appl. Phys., Vol. 45, No. 8B, pp. 6576–6588 (2006).

Optical gain in silicon optical

cavity Purcell factor

Bsp

cavspP B

BF

,

,=

The creation of the cavity affects several parameters by means of

Purcell effect

Bp,cavp, ττ PF=

p,BPp,cav KFK =

BC,cavC,

PF

ττ =

cBspBabBstp

NRRRR

dt

dN

τωωω −−+−= )()()( ,,,

p

pBspBabBst

p NRRR

dt

dN

τωβωω −+−= )()()( ,,,

q

qqBspBabBst

q NNRRR

dt

dN

τωωω 0

,,, )()()(−

−+−=

Threshold condition

2

,

, )(exp)1( N

TK

FnnMF

K

BqqBP

Bp

Bp

∆−Ω+−+= ωτ

∆−Ω++

−⋅=TK

F

n

nR

n

chg

Bq

qcavspcav

)(exp

11)(

)(8)( ,22

23

ωωωπ

ωOptical

gain

Below threshold Above threshold

Carrier density

Photon density

Phonon density

Bc,

pP

RF

Nτ

= thq

q0,

, N

)1n(

≡++

=

q

pBPBp

bp

K

RMF

KN τ

τ

)1(N 20,

2,p pqBpBcB RnM +≈ ττ ( ) , thpBppp RRFN −= τ

0qq NN ≈ pq RN qq0N τ+≈

Pcav Fg ∝)( ω

FCA and Optical gain

T.F. Boggess, J.R. Klaus, M. Bohnert, K. Mansour, S.C. Moss, I.W. Boyd, A.L. Smirl, IEEE J. of Quantum Electronics, Vol. QE. 22, No. 2, February 1986

P

nradrad

bulk

radbulk

F

WW

W

+=η

Increment of quantum efficiency1 2 3 4 5 6 7 8 9 10

x 1019

-2

0

2

4

6

8

10

12

14

16x 10

4

N (carrier density)

Opt

ical

Gai

n ;

FC

A

1 2 3 4 5 6 7 8 9 10

x 1019

-2

0

2

4

6

8

10

12x 10

4

N (carrier density)

Opt

ical

Gai

n ;

FC

A

PCAV Fg ∝

, PcavFCA F∝α

PCAV Fg ∝

José M. Escalante and Alejandro Martínez, “Theoretical study about the gain in indirect bandgap semiconductor cavities", Physica B: Physics of Condensed Matter Vol. 407, 2044-

2049 (2012)

M. J. Chen, C. S. Tsai, and M. K. Wu, J. J. Appl. Phys., Vol. 45, No. 8B, pp. 6576–6588 (2006).

Einstein’s resultsabst BB =

stpsp BDB =

Einstein’s relations for indirect bandgap semiconductors

A. Einstein, "Zur Quantentheorie der Strahlung", Physik Z. 18, 121 (1917)

N.D. Lanzilloti-Kimura, A. Fainstein, B. Perrin, B. Jusserand, A.

Soukiassian, X.X. Xi, D.G. Schlom, Phys. Rev. Lett. 104 (2010) 187402.

A.A.Zadernovskii and L.A. Rivlin, Kvantovaya Elektron (Moscow) 20,

353-362 (April 1993)

A.A.Zadernovskii and L.A. Rivlin, Kvantovaya Elektron (Moscow) 20,

353-362 (April 1993)

Thorsten Trupke and Martin A. Green,

J.Appl. Phys., Vol. 93, No. 11, 1 June 2003

Chin-Yi Tsai, J. Appl. Phys. 99 (2006) 053506.

Both particles are quanta of a classical vibration field

They have no mass

Their interaction with electrons is very similar

They are bosons, so the stimulated emission is proportional to the density

Following the same idea as in the Einstein’s work and basing on the paper of M. J. Chen et al., we get the following rate equations

( )

∆−Ω+−−Ω+= TK

F

gVCqkkqkqBeENNBR

ω

ωπρρ 2

8

( )

∆−Ω+−−Ω+= TK

F

gVCkkkBeENNBR

ω

ωπρ 200 8

( )

∆−Ω+−−Ω+= TK

F

gVCqqqBeENNBR

ω

ωπρ 200 8

( )

∆−Ω+−−Ω+= TK

F

gVCBeENNBR

ω

ωπ 20000 8

( )2

8 gVCqkabab ENNBR −Ω+= ωπρρ

qkkqab RRRRR 0000 +++=The principle of detailed balance requires that the upward transition rate equal the downward transition rate at thermodynamic equilibrium (∆F=0)

kqab BB =

kqqk BDB =0

kqkq BDB =0

kqqkqqkk BDDBDBDB === 0000

Our results

Einstein’s resultsabst BB =

stpsp BDB =

21qkqab BB =

21120 qkqqqk BDB =

2121 0 qkqqkq BDB =

212100 qkqqqk BDDB =

21210 qkqkqq BDB =

22 000 qkkq BDB =

000 11 kqkq BDB =

00000 kk BDB =

What happen with multi-phonon processes?

The relationship between the different coefficients

maintains the same structure.

José M. Escalante and Alejandro Martínez, “Theoretical study about the relations among coefficients of stimulated emission, spontaneous emission and absorption in indirect

bandgap semiconductor", Physica B: Physics of Condensed Matter Vol. 411, 52-55 (2012)

N.D. Lanzilloti-Kimura, A. Fainstein, B. Perrin, B. Jusserand, A. Soukiassian, X.X. Xi, D.G. Schlom, Phys. Rev. Lett. 104 (2010) 187402.

Ω= ,, ppp FFF ω

Bp,,cavp, ττ ωPF=

Bq,,cavq, ττ Ω= PF

, p,BPp,cav KFK ω=

, q,BPq,cav KFK Ω=

,,

BC,cavC,

Ω

=PP FF ω

ττ

Ω= ,,,00

,00 PPBulk

CAV FFB

Bω

ACOUSTIC PURCELL EFFECT

Optical gain in silicon acuto-optical cavity

Photon threshold condition

Phonon threshold condition

( )

∆−Ω+−+=TK

FnnNFM

K

BqqPB

p

p )(exp12 ω

τ

( )

∆−Ω+−+=TK

FnnNFM

K

BppPB

q

q )(exp12 ω

τ

( )1,

,, +

=pBqBP

Bqqth nMF

KN

τ

( )1,

,, +

=qBpBP

Bppth nMF

KN

τ

Below threshold Above threshold

Carrier density

Photon density

Phonon density

Bc,

pP

RF

Nτ

= thq0,

, N )1n(

≡+

=BPBp

bp

MF

KN

τ

0Np ≈ ( ) ,, thpBppp RRFN −= τω

0,,0 qBqPqq nKFNN Ω=≈ 0,,0 qBqPqq nKFNN Ω=≈

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.05

1.1

1.15

nq

∆F (

eV)

Optical gain and population inversion

First Case Spontaneous emission of photon is neglected

00000 kkqqkkq RRRRRR +≈+++

( )

∆−Ω+−+⋅=TK

FnnR

n

chg

Bqq

)(exp1

)(8)( 0022

23

ωωπ

ω

Second Case Co-stimulate emission of photons and phonons

kqqkkq RRRRR ≈+++ 0000

∆−Ω+−⋅=TK

FnR

n

chg

Bq

)(exp1

)(8)( 0022

23

ωωπ

ω

The population inversion is not

necessary!!!

The population inversion is necessary!!!

T. Trupke, et al., Appl. Phys. Lett.

82 (2003) 2996.

FCA and Optical gain

,, ωα PcavFCA F∝

( ) Ω∝ ,, pp FFg ωω

OPF=1000

José M. Escalante and Alejandro Martínez, “Optical gain by simultaneous photon and phonon confinement in indirect bandgap semiconductor

acousto-optical cavities” Opt. Quant. Electron.45, pp. 1045-1056 (2013)

CQED in optomechanical

cavities

CQED offers the prospect of a novel technology, single molecular detection, gravitational wave detection or quantum information processing, opening the new age of quantum devices.

Jaynes-Cumming model of an indirect bandgap semiconductor cavity

How is possible to apply discrete energy level mode to an indirect

bandgap semiconductor?

• Using a non-linear JCM

• Studying the photoluminiscence spectra for a silicon bulk and for a silicon optical cavity

TRANSVERSAL OPTICAL

PHONON (TO)

T=77K

T=112K

T=77K

T=26K

Photoluminesce spectra of silicon bulk

Photoluminiscence spectra of silicon

cavities

The interaction with the resonant modes (M1)

Interband-transition mediated by one transversal optical phonon (TO)

Using a correct design of the cavity it is possible

that both peaks coincide.

The main peak would correspond to the transition by the photons and phonons resonant inside the cavity

With an appropriated cavity

We can “choose” the inter-band transitions

Cleaning the PL spectra of secondary Peaks less likely

So, it is plausible to go from a continuous spectrum of energies to a discrete spectrum of

energies

1000nm=pλ 5nm=qλ

Inter-band transitions are mediated by two particles

The phonon has the correct moment

We consider that the transitions take place between the CB edge and VB edge

There are some processes involving the intermediate states which are highly unlikely

( )abbagbbaaH ++++Ω

++ +−++= σσωωσσω γ ˆˆ0

JCM +

g⊗⊗=Ψ βα)0( )()()( tPtPtW ge −=

Rabi frequency oscillation for indirect bandgap

semiconductor)(

;;;1;122

fiIVB

CBVB EEEE

gmnHemnW −

−−−

=→ δπ

)(,,2

2

fin ni

nnCBVB EE

EE

VBHknknHCBW −

−= ∑→ δπ

C. Y. Tsai, J. Appl. Phys.

99, 053506 (2006)

)',()( kkMkMg qp

=

nmkkMkM qpnm )',()(2

=Ω

Theoretical study of two-level systems inside optomechanical cavity where mechanical

oscillations are induced

( )

( )

++

++∆+

++++

++∆=

=++ nnlnaan

naannnlHH

m

m

ˆˆ2

1

2ˆˆ

ˆˆ)1ˆ(ˆ2

1

2

20

20

βαωδ

δβαω

W. Wang and L. C. Wang, “Dynamics of a coupled atom and optomechanical Cavity”, J. of the Korean Physical S. Vol. 57, No. 4, pp. 704-709 (2010).

Yong-Hong Ma and Ling Zhou, “Enhanced entanglement between a movable mirror and a cavity eld assisted by two-level atoms”, J. Appl. Phys. 111, 103109 (2012).

2

2

2 mmωζα −=

2

2

2 mm

g

ωβ −=

22 mm

g

ωζδ −=

Measures the optomechanical effects

Measures the coupling effects

Measurer of the modification of coupling due to the optomechanical effect

NON-INTERACTION 0, =⇒ δβ

“COMPETITION” 1=⇒δβ

OPTOMECHANICAL 1<⇒δβ

COUPLING 1>⇒δβ

Population Inversion

Field Entropy

Photon Number

NON-INTERACTION REGIME 0, =⇒ δβ

001.0, =δβ

“COMPETITION”REGIIME

1=⇒δβ

1, =δβ

“COMPETITION”REGIME

1=⇒δβ

5.0

01.0

==

δβ

10

01.0

==

δβ

OPTOMECHANICALREGIME

1<⇒δβ

01.0

5.0

==

δβ

COUPLINGREGIME

1>⇒δβ

01.0

10

==

δβ

01.0

100

==

δβ

COUPLINGREGIME

1>⇒δβ

My research idea in NTT

Apply my slow-wave phenomena knowledge to develop buffer devices, electro-optical modulators or dual sensors.

Following the study of optomechanical and photo-elastic effect in phoxonic slab structures, since this structures has a high potential the next generation of computer and optical communications, because of their implementability.

Continue the study of silicon laser from other approaches.

Explore new topics and field of current research, from both a theoretical and experimental point of view.

Thank you for your attention!!!

I welcome your question, suggestions and comments