Enhancement of 3rd-order nonlinearities in nanoplasmonic metamaterials: figures of merit Jacob B...

Enhancement of 3rd-order nonlinearities in nanoplasmonic metamaterials: figures of merit

Jacob B Khurgin Johns Hopkins University, Baltimore

Greg Sun University of Massachusetts, Boston

Scope

• Rationale• Can one engineer nonlinearity in metal

nanostructures?• Coupled mode theory of enhancement• Assessment of nonlinearity enhancement• Conclusions

Rationale:Nonlinear optical interactions are quite interesting and important, yet are also very weak – how can one improve it?

Ag

It is well known that if one used pulsed (mode-locked) laser and concentrate the same average power into the high peak power with low duty cycle (d.c) efficiency of nonlinear processes will increase

t

P

2( )2( )

1~ . . ~

( . .)

n n

n nout peak n

PP d c P

d c

Can we do the same in the space domain and concentrate the same power into higher local power density to increase the efficiency ?

2( )2( )

1~ ( . .) ~

( . .)

n n

n nout n

PP d c I

d c

Plasmonics as a ”silver bullet” for nonlinear optics

“Mode-locking in space?”

+-

++

+

-

--

+-

++

+

-

--

+-

++

+

-

--

+- +-

Plasmonic concentrators

4

22 2 3~ ~ 10 10localE

QE

24 4 5~ ~ 10 10localE

QE

M. Stockman, P. Nordlander

+-

++

+

-

--

2

2( ) 1 p

j

But:In space there is an additional factor of modal overlap – the field of pump(s) must overlap with field of signal (conceptually similar to the phase-matching)

Plasmonic concentration always brings loss

~ ~ ~ 10 20r

i

Q

Recent work

Recent work

F. B. P. Niesler et al , OPTICS LETTERS 34, 1997 (2009)

Palomba et al J. Opt. A: Pure Appl. Opt. 11 (2009) 114030

Yu Zhang et al, Nano Lett., 2011, 11 (12), pp 5519–5523

6

“Prior to the prior” works

7

H. J. Simon et al, Optical Second-Harmonic Generation with Surface Plasmons in Silver Films,PRL, 1974

Hache, Flytzanis et al, Optical nonlinearities of small metal resonance and quantum size effects, JOSA B 1986

P. N. Butcher and T. P. MacLean, Proc. Phys. Soc. 81, 219 (1963).

S. H. Jha, Theory of Optical Harmonic Generation at a Metal Surfaces Phys Rev 140, 1965

Scope



Can one engineer nonlinearity in metal?In QW’s or QD’s….anharmonic potential-giant dipole of this “artifical atom” or “molecule”

In QW Electron moves up to a few nm

How about electrons in SPP giant “artificial atoms” or “molecules”

+++++

+

++++

How far do the carriers move? 2 2 2 2mv m x ω

NV = NV = hω2 2

1/ 2

2x ~ A

(NV)

22 -3N = 6 × 10 cm

In 30 nm sphere…NV~106 electrons ; Electrons move less than 0.001A!!!!

Conduction electrons do not move, see no anharmonicity, and possess practically no nonlinearity except for the very few ones at the surface One must either use interband transitions (no different from saturable absorber except for much higher loss) or better revert to nonlinear dielectrics

9

SPP modes analogy with giantatoms and molecules is quite superficial

Say we have 1 SPP per modePower dissipation is P γhω ~ 1eV / 10fs ~ 10μW

Power density - very high! 12 3P 10 W / cm

Scope



Four wave interactions

(3)

3

1

2

11j tE e

22

j tE e

33j tE e

1 2 34 ( )(3) *4 1 2 3~ j tj tP e E E E e

4 1 2 3

44,

j toutE e

4 1 2 3

FWM (Four Wave Mixing)

(3)

2 22(3)

2, 1 2~j t j tnlP e E E e

XPM (Cross Phase Modulation)

21

2

11j tE e

11j tE e

32,

j tinE e 1 2 4 3;

22,

j toutE e

2 2(3) *4

1 23

~FWM

EL E E

E c

Efficiency

2(3)2 1( / )

2, 2,~ j t c E Lout inE E e

(3) *4, 1 2 3~outE j L E E E

c

Nonlinear phase shift 2(3)1 2 1~nl L E Ln I

c c

Nonlinear index

(3) 22 0 /n n

2

2~ pumpLn Ic

Practical figure of merit

Switching

pumpI

signalI

For nonlinear switching using XPM or SPM

2 max~ 2 ~nl

LLn I n

c

For wavelength conversion2 2

2 max

2~ ~ ~ 1pumpLn I nc

Maximum interaction length is determined by absorption hence the ultimate figure of merit is what is the a maximum phase shift achievable :

max max~ 2Ln

And how close it is to 1…

Mechanism for the enhancement of nonlinearity

Ag Ag

Ag

Ag Ag

Ag

Ag

Ag

(3)pumpI

sigI

pumpE

sigE

sigEpumpE Average values of fields

Stage 0


pumpp Nanopartcles get polarized at both pump and signal frequencies

Stage 1

sigp

(3)pumpI

sigI

pumpE

sigE

+

-

pumpp sigp

+

-

+

-

+

-

+

-

+

-+

-

+

-

0~ 3sig sigV Qp E

0~ 3pump pumpV Qp E~ / ~ 10 20r imQ


Stage 2

+

-

(3)

pumpI

sigI

pumpE

sigE

pumpp sigp

locE

+

-

+

-

+

-

+

-

+

-

+

-

+

-

Locally enhanced field at both pump and signal frequencies, , ( ); ( )loc pump loc sigE r E r

, ( ) 2loc sig sigQE r ~ E

, ( ) 2loc pump pumpQE r ~ E


Stage 3

Local nonlinear polarization is established, ( )loc nlP r

+

-

(3)

pumpI

sigI

pumpE

sigE,loc nlP

+

-

+

-

+

-

+

-

+

-

+

-

+

-

2 2(3) 3 (3), , 0( ) ~ ( ) ( ) ~ 8loc nl loc pump loc pump sigQ P r E r E r E E


Stage 4

4

2 2,max

( )~ ~ 0.1

( )

loc

loc loc

E r dV

E E r dV

21 4 (3), 0 ,( ) ~ ~ 8loc nl loc nl pump sigQ Q E r P E E

Local nonlinear field is established, ( )loc nlE r

(3)

pumpI

sigI

pumpE

sigE,loc nlP+

-

,loc nlE

+

-

+

-

+

-

+

-

+

-

+

-

+

-

Third order nonlinear polarization does not exactly match the mode


Stage 5

4

2 2,max

( )~ ~ 0.1

( )

loc

loc loc

E r dV

E E r dV

23 4 (3), 0 , 02

~ ~ 12sig nl loc nl pump sigV V Q p E E EAccordingly, each nanoparticle acquires nonlinear dipole moment (at signal frequency)

Third order nonlinear polarization does not exactly match the mode

(3)

pumpI

sigI

pumpE

sigE

+

-

,loc nlE

+

-

+

-

+

-

+

-

+

-

+

-

+

-

,sig nlp


Stage 6

24 (3), , 0~ 12sig nl sig nl pump sigN f Q P p E E

The whole medium then acquires average nonlinear polarization at the signal frequencyf – filling factor

(3)

pumpI

sigI

pumpE

sigE

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

,sig nlp

,sig nlP

Introduce effective nonlinear susceptibility

2(3), 0sig nl eff pump sig P E E

(3) 4 (3)~ 12eff f Q

Scope



Assessing nonlinearity enhancement (3)

4 3(3)

~ 12 ~ 10eff f Q

This sounds mighty good…..

What about absorption?2

3deff

nfQ

Maximum phase shift

2, 4

2

~ 12effnf Q

n

3,max 2, 2

2~ 4nl eff

eff

n I Q n I

Enhanced as much as few hundreds times This sounds really good…..except

still, assuming 13 22 10 /n cm W (chalcogenide glass)

10,max 10nl I

indicating that the input pump pump density must be in excess of 10GW/cm2 in order to attain switching or efficient frequency conversion, meaning that while the length of the device can get reduced manyfold, the switching power cannot and remains huge….

Local “intensity” is now in excess of 1000 GW/cm2 –way past break down!

So, what is the real limit?

and the things only go further downhill from here on once it is realized that all of the enhancement is achieved because the pump field is really concentrated by a factor of Q2 >100!

A better figure of merit24 (3)

, 0~ 12sig nl pump sigf Q P E E

22 (3)0 ,3 loc pump sigf Q E E

20 2 ,6 loc pump sigf Q n I E

206 local sigf Q n E

Assuming that maximum index change is limited by material properties to max 0.01localn n

2,max max max

2~ 3 0.01nl

eff

f Q n Q n

the maximum phase shift is…

There is no way to achieve either all-optical switching or efficient frequency conversion!

Factor of Q2 makes perfect sense –because SPP mode is a harmonic oscillator with a given Q –changing local index shifts resonant frequency and causes change in polarizability proportional to Q2

0

,Re( )sig nlp

0

What if we use dimers or “nano-lenses”?

(3)pumpI

sigI

Field enhancement occurs in two steps –first the larger dipole mode gets excited then the gap mode near smaller nanoparticle

2

, ( ) 2loc sig sigQE r ~ E

2

, ( ) 2loc pump pumpQE r ~ E

But the relation between the average nonlinear polarization and maximum index change is still almost the same, therefore ,maxmax ~0.20.01 nlQn

(3)6 5

(3)~ 5 ~ 10eff f Q

P=1.6mW

Length (m)10-1

100

101

102

103

104

10 -8

10 -7

10 -6

10 -5

10 -4

10 -3

10 -2

10 -1

10 0

10

Non

linea

r Pha

se S

hift

(rad

)

1

P=1.6mWP=1.6mW

P=0.8W

P=0.8W

P=8W

1m2 13 2

2 10 /n cm W

P

1m2 13 2

2 10 /n cm W

P

1m2 13 2

2 10 /n cm W

P

What does it mean?

At low powers and plasmonic enhancement allows one to achieve still small nonlinear phase shift at very short distance, but this shift always saturates well below .

Scope



Two ways to define figure of merit

26

Scientific approach

Engineering approach What would be the overall maximum attainable result at ~one absorption length?

For the nonlinear index type process – what is the maximum phase shift attainable at 10dB loss?

What is the maximum attainable enhancement of nonlinear susceptibility?

For 2) enhancement is fQ3 ~102-103

For (3) enhancement is fQ6 ~105-106

max~Qnmax~10-2<<

Not enough for all-optical switch(or frequency conversion)

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--

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--+- +-

Why such a conflicting result ?

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Scientific approach: what matters is the relative improvement

Engineering approach: what matters is the end result

Take very weak process with efficiency approaching 0….then if the end result is <<1

a very large powerResult= 10

0

a very large powerResult = 0 × 10 << 1

Using metal nanoparticles for enhancement of second order nonlinear processes may not be a “silver bullet” we are looking for.

Ag

Plasmonic enhancement is an excellent technique for study of nonlinear optical properties (the higher order the better) and sensing using it, but not for any type efficient switching, conversion, gating etc.

Enhancement of 3rd-order nonlinearities in nanoplasmonic metamaterials: figures of merit Jacob B...

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