Battling Loss in Plasmonics and Metamaterials Jacob B Khurgin Johns Hopkins University 1RBNI-14...

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Battling Loss in Plasmonics and Metamaterials

Jacob B KhurginJohns Hopkins University

Towards Low Loss in Plasmonics and MetamaterialsAway from High Loss in Plasmonics and Metamaterials

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1. Denial2. Anger3. Bargaining4. Despair5. Acceptance

5 stages of dealing with the loss

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“hoc

key s

tick c

urve

”1990 1995 2000 2005 2010 20150

1000

2000

3000

4000

5000

6000

7000

8000

9000

year

Publ

icati

ons

in P

lasm

onic

s an

d M

etam

ater

ials

(W

eb o

f Sci

ence

)

10,000 BC(discovery of Ag)

Elec

tron

sca

tter

ing

time

in A

g (f

s)

10

20

30

40

50

60

70

80

?

Motivation

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Scope•Why are the metals necessary for sub-wavelength confinement?•What are the surface plasmons and polaritons?•Why subwavelength confinement in optical range always means high loss?•Why reducing loss is so important? 3 Case studies

1. Who needs negative index?2. How does loss impact plasmonic enhancement of the emission?3. Plasmons and nonlinear optics –a winning combination?

•Why and how do the metals absorb and reflect?•Can a metal be made lossless?•Does it have to be a metal?•Can metal loss be compensated by gain?•Can one make a true sub-wavelength laser /spaser?

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Scope

•Why are the metals necessary for sub-wavelength confinement?•What are the surface plasmons and polaritons?•Why subwavelength confinement in optical range always means high loss?•Why reducing loss is so important? 3 Case studies



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Why metal?Metal’s claim to fame is its negative real part of the dielectric constant –Drude Formula

2

2( ) 1 p

j

22

0p

Ne

m

Plasma frequency is defined as

Metal Plasma Frequencies and Scattering Rates

gold Au 2081 THz 8.5 eV 12.31013 s-1

silver Ag 2182 THz 9.0 eV 3.2 1013 s-1 aluminum Al 3231 THz 13.2 eV 27.31013 s-1

Plasma frequency Scattering rate

Plasma frequency

Scattering rate

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Permittivity of Gold

K. Busch et al., Phys. Rep. 444, 101 (2007)

experimental

Drude fit

Interband absorption

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Material Q-factor

Re( )

Im( )MQ

Realistically…QM~10-20 because of surface scattering

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What does negative mean?

( ) ( ) ( ) ( )n n j

The electrons move to screen the electro-magnetic field – hence the field gets expelled from the metal and the field inside becomes evanescent

MetalFree Electronsx

------------

++++++++++++

Surface Charge

ES Screening field

Total Field=0

EExternal field Refractive index –complex, mostly

imaginary

Evanescent Field

( ) ( )

0 0~ ~j n z zc cE E e E e

z

E

Bulk Charge Oscillations2

2( ) 1 p

j

( ) 0p

In bulk metal when the frequency exceeds plasma frequency the electrons can no longer follow the electric field and screen it – metal becomes transparent

At plasma frequency the bulk metal supports longitudinal charge oscillations

------------

++++++++++++

E

------------

++++++++++++

E

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Why do we need free carriers?

min ~ / 2L nComes from the uncertainty principle

/ 2p x

/ 2k x

min

21/ 2

nL

min ~ / 4L n

We want to concentrate optical field on sub-wavelength scale…but are prevented by the diffraction limit

To see how we can beat diffraction limit …let us re-derive it from the energy conservation considerations

Easy to understand for real n…but what about imaginary part?

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t

E

H =0

nEH E

k

2

2e

EU

Electric “Potential”Energy

2 2

0 2 2M

H EU

Magnetic“Kinetic”Energy

sin( )sin( )kz tE

Ewt=0

al

cos( )cos( )kz tH

Hwt=p/2

al

UMUE

0

2 n nk

c

Energy balance in a mode

Energy oscillates between potential (electric) and kinetic (magnetic)

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UM

UE

sin sin( )z ta

E

Ewt=0

a l

cos cos( )z ta

H

Hwt=0

a l

t

E

H =0 0 0

2 2a a na nEH E c E

0 2

2

ca

n n

2

2e

EU


2 22 2

00 0

2 2

2 2M E

H na E naU U


Lack of energy balance in a sub-l mode

If a<<l0/2n there is almost no magnetic field (quasi-static limit) UM<<UE –energy is not conserved

The energy will radiate because it cannot all fit into magnetic energy –this is diffraction limit!

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0 0

2na nEH

0 2

2

ca

n n

2

2e

EU


2

0

2M E

naU U


Free carriers restore balance in a sub-l mode

sin sin( )z ta

E

Ewt=0

a l

++

+ cos cos( )

cos( )

z ta

t

H

J

Hwt=0

a l

v

J

2 2

2

2 2

~

KK

K E

Nmv L JU

L CU

TrueKinetic”Energy of electrons LK=Kinetic Inductance

(inertia of electrons)

UE UM UK

At some resonant frequency w0 the balance is achieved

If a<<l0/2n there is almost no magnetic field (quasi-static limit) UM<<UE – hence energy oscillates between electric energy and kinetic energy of free carriers

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Scope•Why are the metals necessary for sub-wavelength confinement?

•What are the surface plasmons and polaritons?•Why subwavelength confinement in optical range always means high loss?•Why reducing loss is so important? 3 Case studies



Surface charge oscillations

1p

sp

D

d>0

m<0---

+++

+++

+++

---

---

---

+++

+++

+++

---

---

---

+++

+++

+++

---

---

---

+++

+++

+++

---

---

Surface plasmon (SP)

( ) ( ) 0m sp d sp

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Interface surface plasmon polaritons (SPP)

d mspp

d mc

>0

<0

weff

0 5 10 15 20 252.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

Wave number (relative units)

En

erg

y (e

V)

Ligh

t lin

e

g

cv

n

Unfortunately…in a real metal

Surface plasmons couple with electromagnetic waves

SPP is a TM wave propagating along the interface and evanescent in both mediums, with dispersion

Near the resonance interface SPP is characterized by a very small (sub-wavelength) effective width weff, large effective index c/ and small group velocity

The losses are important!!!!!

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Localized SPP of a sub-wavelength nanoparticle

max, 1(cos )

1

l

l l l l

rr aaa

P Er al a

r

+

2 0 For sub-wavelength dimensions one can use electro-static approximation and solve Laplace equation in stead of wave equation

2max,

cos 2

l l

rr a

aaE

ar a

r

Surface charge density oscillations coupled with electric field

0 max,

2 1(cos )

1l l l

lE P

l

Most Important is the Dipole mode l=1

0M

0D

1 1( ) 2 0 2 1

pm D

D

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Dipole SPP of a sub-wavelength nanoparticle

Electric Field

31 0 max,12p a E Dipole Moment

Larger particle-larger antenna

Emax

+

3

, 2

8

( 1)eff lD

aV

l

Larger particle – less field concentration

For dipole mode3

3,1

2~eff

D

aV a

Radiative Decay:

32 2

3radD D

a

The field is confined in a small effective volume !

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Scope•Why are the metals necessary for sub-wavelength confinement?•What are the surface plasmons and polaritons?

•Why does subwavelength confinement in optical range always means high loss?•Why reducing loss is so important? 3 Case studies



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The heavy price of having free carriers

wt=p/2J In the sub-wavelength metallic structures (in all three

dimensions !) half of the time almost all the energy is stored in kinetic motion of electrons –where it is being lost with the decay rate 2 of the order of 10 fs-1. Therefore the rate of energy loss in truly sub-wavelength structure ,geff, is always of the order of .

Case of propagating SPPHere sub-wavelength simply means very large wave-vector

0 5 10 15 20 252.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

Wave number (relative units)

Ene

rgy

(eV

)

g

cv

n

Long Range SPP

High Loss in “true plasmon” region

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Kinetic and Magnetic Inductances

2 2 2

20

22

2 2 2K Kp

v a I IU Nm A a L

A

20

2; K

p

aL

A

Magnetic Inductance 0

8lnM

aL a

r

The ratio of energies goes to zero as the physical size is decreasedM M

K K

U L

U L

As particle gets progressively smaller more and more energy gets stored in the form of kinetic energy of electrons

Kinetic inductance is caused by the inertia of electrons

2r

a

J

E

H

Split Ring ResonatorA-current area with skin effect

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Finding Effective Q of plasmonic mode

M Keff

eff K

L LQ

L

2r

a

J

E

H

Split Ring Resonator -example

10-1

100

101

102

10310

0

101

102

103

2a=/4

/8

/16/32

Wavelength (m)

Q-f

acto

rp b

Qm0

a=4r

Au

Border wavelength w~ g(where optics – displacement current and electronics - conductivity current meet

opticselectronics

0rj H E + E2 2

2 2 20

0

~M

E r

U aa

U

When conductivity current dominates there is no wavelength dependence between E and H fields, hence H field can be large enough to store the energy

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Estimate of effective loss

Q of Drude metal structure actually gets reduced with wavelength and only starts getting better when < - THz region which is not really optics –and even then LC circuit is not a very high Q resonator!!!!!

That is why high Q resonators in electronics include:low loss inductances with high (energy stored in spin magnetization and not kinetic energy of electrons)Quartz crystalsSurface acoustic waves/2 cavities ….

The true sub-wavelength region where kinetic inductance dominates (plasmonics) is inherently lossy and various geometric tricks will not mitigate this loss significantly!

optics

electronics

100

101

10210-4

10-3

10-2

10-1

100

101

102

2a=/4

/8/16/32

Wavelength (m)b

eff

()/

(b)

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Plasmonic and loss are inseparable

Frequency (n/l)

Confinement (1/a)

/a n

/a n /a n

1/100nm1/50mm

p

/a n

/a n

Conductivity currentFree carriersELECTRONICS

Conductivity currentBound carriersAbsorber

Reactivecurrent, free carriersPLASMONICS

Reactive (Displacement) current, bound carriersOPTICS

Reactive current of free carriers can be confined on sub wavelength scale but then it cannot engender magnetic field strong enough to store the energy without excessive loss

RBNI-14 26

Scope•Why are the metals necessary for sub-wavelength confinement?•What are the surface plasmons and polaritons?•Why subwavelength confinement in optical range always means high loss?

•Why reducing loss is so important? 3 Case studies1. Who needs negative index?2. How does loss impact plasmonic enhancement of the emission?3. Plasmons and nonlinear optics –a winning combination?


RBNI-14 27

What are plasmonic metal nanostructures good for?

1. High field concentration near the metal-dielectric interface enhance linear and nonlinear optical properties , such as electro-luminescence, photoluminescence, Raman effect and so on.

3. New “meta-materials” with unusual properties including negative refractive index

0 0 0 =377 0n

Bending with no reflection – perfect lensing

2. High effective index and strong confinement of interface SPP can be used to develop sub-wavelength passive and active photonic devices (nano-laser and “spaser”)

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Why do we need negative index material?

Presumably to get to the super-resolution……-Pendry’s superlens.

nn

object image

-n2

21 p

Perfect near field focus

But we know that the moment we introduce finite loss in the metal g, the resolving power of this near field lens deteriorates drastically

2

21 p

j

0

50

100

150

-150

-100

-50

0 10 20 30

X(nm))

Z(nm

)

Case study 1: Who needs negative index?

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What if we reduce (or eliminate) loss?Create “artificial dielectric” with metal nanoparticles playing the role of atoms with large polarizability

20

2 2 2 20 0(1 / )eff d d

Qf f

j Q j

2 20 /p K Resonant frequency (K=3 for spheres)

f-filling fraction /Q

Near the resonance we can get fairly large values of

Then we can just make a conventional lens with high resolution l/neff that would give us a magnified image in the far field

1/2 1/2~ ~eff effn Q

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What does it mean?

-1Loss(s )

1

2

3

4

x 1013

6 3 1 0.2

superlens

metal - dielectric composite lens

Silver

nn

object image

-n

With the current high metal loss the Superlens does not really perform

If the loss is reduced one can get higher resolution and magnified image in far field without negative n!

Reduced loss would change everything!

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Meta-Catch 22

As long as metal loss stays what it is, negative index materials in optical range are highly unfeasible

If one could reduce metal loss by an order of magnitude or more the negative index materials and devices may become highly unnecessary

Probably makes sense to see what we can do about loss

A catch-22 is a paradoxical situation in which an individual cannot avoid a problem because of contradictory constraints or rules. Often these situations are such that solving one part of a problem only creates another problem, which ultimately leads back to the original problem.

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Case study 2: how loss impacts plasmonic enhancement of emission, absorption, Raman….

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Plasmonic Enhancement of Radiative Efficiency-why loss is critical

Surface-plasmon-enhanced light emitters based on InGaN quantum wellsKOICHI OKAMOTO, ISAMU NIKI1, ALEXANDER SHVARTSER, YUKIO NARUKAWA,TAKASHI MUKAI2 AND AXEL SCHERERNature Materials 3, 601 (2004)

>10 –fold enhancement(some form the enhanced absorption)

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Origin of plasmonic enhancement

Radiative decay is proportional to the density of states

3 2

3 2 3 3

4D

n

c

About 1/THz.m3 @400nm

Radiative time even for the allowed transition~100ps

21 3 3 2

3 30 0

5 10 /2rad D D

e ff m ps

m

Oscillator strength

Nonradiative time can be much shorter…because density of final states is orders of magnitude higher

1 33

4~ ~D

nrad phononph ph

radnrad

nrad

nradrad

radrad

11

1

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“The bottleneck”

Density of states

Nonradiativeprocesses

radiation

radnrad

nrad

nradrad

radrad

11

1

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Removing bottleneck

3D photon statesNonradiative states1

nrad 1rad

High density “photon” states

1 11r rad

How to increase photon density?

1. Reduce the volume V<<λ3 or2. Reduce the frequencies range Q=/>>1

3 3

4D

1

1~

V

Purcell Factor:1

1 3

~ ~radP

r D

F

Original 1946 Paper –28 lines, 5 equations

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“Dense photons”Density of states can be changed only by coupling to other media(polariton effect) or by coupling between counter-propagating photons (microcavity, photonic crystals)

Propagating “dense photons” – polaritons, slow light, SPP

0

k

c

Photon states

0 resonance can be atomic, Bragg…

gap

vg=d/dk

Polariton states

Dense States are Slow States –Do not couple well into true photons

1

3

~ ~PP g

D

F cv

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“Dense photons II”Density of states can be changed only by coupling to other media(polariton effect) or by coupling between counter-propagating photons (microcavity, photonic crystals)

Localized “dense photons” – microcavity, localized SP

Dense States are ConfinedDo not couple well into true photons

Free photon statesn-cavity resonance

Veff

3

3

~ ~cavP

D eff

FV

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Re-emergence of the bottleneck


nrad 1rad

“dense photons”

1 11r P radF

3 3

4D

1

1~

V

radTransfer from “dense” (high impedance) to normal (low impedance) photons

More nonradiative states

nradNonradiative decay of “dense photons”

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The bottleneck is alive and well…just shifted down the line

Density of states


radiation

Reservoir of dense photons


radiation

rad

nrad

Nonradiative decay

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So…what is the enhancement?


nrad 1rad

“dense photons” rad

nrad

radnrad

nrad

nradrad

radrad

11

1Original radiative efficiency

1 11r P radF

Purcell enhancement

radout

nrad rad

Out-coupling efficiency

New radiative efficiency1

,1 1 1P rad

rad outP rad nrad

F

F

Overall enhancement

,11 1(1 )

rad out out

rad rad rad P rad

FF

Good emitter cannot be improved!!!!

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Trade offsUse effective index ' /p p Dk

as a parameter

More of a PlasmonMore of a PhotonThe tight confinedand slow SPP offer large enhancementof radiation rate, butthey are lossy and difficult to couple outside

0 5 10 15 20 252.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

Wave number (relative units)E

nerg

y (e

V)

g

cv

n

1 2 3 4 510-2

100

102

104

Effective Index ’

Purcell Factor

Group Index=c/vg

Effective width

Loss

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Main result for Propagating Interface SPPF 1

Substantial improvement can be achieved only for very inefficient emitters with rad<1%

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Enhancement with localized SPP

+

31 0 max,12p a E Dipole Moment

Larger particle-better antenna

Effective Volume3

effD

aV

Smaller particle – better resonator

It is easy to see that the same particle cannot be a good antenna and a good resonator at once!

Only the emitters whose efficiency is originally quite poor can be significantly enhanced by SPP

Similarly, when it comes to absorption only very weak absorbers can be enhanced by SPP

Maximum possible enhancement is on the order of

rad nrad

Q

RBNI-14 45

Case study 3: Plasmons and nonlinear optics

RBNI-14 46

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

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

Plasmonics as a ”silver bullet” for nonlinear optics

“Mode-locking in space?”

RBNI-14

+-

++

+

-

--

+-

++

+

-

--

+-

++

+

-

--

+- +-

Plasmonic concentrators

47

22 2 3~ ~ 10 10localE

QE

24 4 5~ ~ 10 10localE

QE

M. Stockman, P. Nordlander

+-

++

+

-

--

2

2( ) 1 p

j

But:

Plasmonic concentration always brings loss

~ ~ ~ 10 20r

i

Q

RBNI-14 48

Practical figure of merit

Df

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 n

c

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

max max~ 2L

n

And how close it is to 1…

RBNI-14 49

Enhancing nonlinear index

Ag Ag

Ag

Ag Ag

Ag

Ag

Ag

c(3)pumpI

sigI

pumpE

sigE

f

f – volume filling factor k– mode overlap

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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

3max 2, 2

2~ 4eff

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!

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!

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Saturation of nonlinearityEven if one disregards optical damage nonlinear index change saturates

Borchers et al, “Saturation of the all-optical Kerr effect in solids”, Opt Lett 37, 1541 (2012)So, what is the real limit?

RBNI-14 52

A slightly better figure of merit24 (3)

, 0~ 12sig nl pump sigf Q P E 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…

The path to achieve either all-optical switching or efficient frequency conversion is less than obvious

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

RBNI-14 53

10-5

10-4

10-3

10-2

10-1

10010

-6

10-5

10-4

10-3

10-2

10-1

100

Phas

e Sh

ift (r

ad) f=10-3

f=10-4

f=10-5

f=10-6

Length (cm)

1mm2 13 2

2 10 /n cm W

P=1W1mm2

13 22 10 /n cm W

P=1W

Phase shift vs. distance

RBNI-14 54

Two ways to define figure of meritScientific 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 c(2) enhancement is kfQ3 ~102-103

For c(3) enhancement is kfQ6 ~105-106

DFmax~kQDnmax~10-2<<p

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

+- +++-

--+

- +++-

--

+- +++-

--+- +-

RBNI-14 55

Why such a conflicting result ?

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 it perhaps less stellar for any type efficient switching, conversion, gating etc.

RBNI-14 56




RBNI-14 57

How do the metals absorb?

EF

Valence Band

1

Conduction Band 1

Conduction Band 2

k

E

Ec2

3

3

Intraband transitions –assisted by scattering

2

22 4

2( ) (E E )initial final scat final initial

all cstates

eR f f M

m

E k

At even higher frequencies also Interband transitions –band edge

There have to be Real transitions into the Real states

Involves the states within from the Fermi level

23

( ) 2Im( ) ~ ( ) ~ scat finalM

At small frequencies only the states near Fermi level are involved , but at optical frequencies –all bets are off and everything depends on density of final states

RBNI-14 58

Why are the metals lossier than Semiconductors?

EF

Valence Band

k

E

Metal

EF

Valence Band

Semiconductor

k

E

~ finalR r

E

,final met

,final sem

Density of final states is much higher in the metalsOf course plasma frequency is higher in metals, hence trade–off is inevitable, but semiconductor is a viable alternative in the THz regime

RBNI-14 59

Why at optical frequencies loss does not decrease dramatically with temperature?

Valence Band

k

ETo maintain energy conservation phonons must be involved

To maintain momentum conservationPhonons must have wave vector commensurate with the Fermi wave vector

kF

For low frequencies only absorption of phonons is possible – this process is proportional to the number of phonons and thus highly temperature dependent

EF

( )

~ 1 5

ph Fk

THz

ph

( )ph Fk ph

For high frequencies both absorption and emission of phonons is possible – the later process has a spontaneous component that does not depend on phonon number and thus is temperature independent

RBNI-14 60

Temperature independent absorption with e-e- scattering

2( ) ~ /ee FE

k

E

EF

k1

k2

k3

Hee h

(a) (b)

k

E

EF

k1

k2

k3

k4Hee

h

k

E

EF

k1

k2

k3

k4

Hee

h

(c)

k

E

EF

k1

k2

k3

k4

Hee

h

(d)

Two electrons are excited by a single photon

Only Umklapp scattering is contributing

3 1 4 2 k k k k g

The process becomes very important at high frequencies

RBNI-14 61

How does the metal reflects-refracts?

EF

Valence Band

Conduction Band 1

Conduction Band 2

k

E

Ec

Intraband transitions-onto itself

2

2

2Im( ) ~ ~ ( )v initial initial

initialstates

eR f f

k

k

E

Looks very Drude-like

Interband transitions

Virtual transitions to real states

Negative detuning

Positive detuning

2/2

~ 0E E

if

ibinitial final initialstates

P Ef

Involves only the states near the Fermi level

Involves all the states below the Fermi level

2

2 3

( )( ) ( ) p

ib j

2 2 20/ 3p f fe v Depends only on Fermi-level properties

22( ) ~ scat finalM

Depends only on density of states on all final states

RBNI-14 62



•Why and how do the metals absorb and reflect?

•Can a metal be made lossless?•Does it have to be a metal?•Can metal loss be compensated by gain?•Can one make a true sub-wavelength laser /spaser?

RBNI-14 63

Decoupling absorption and reflection

22( ) ~ 0scat finalM

Valence Band

Conduction Band 2

EF

E

Ec

E12

Ecv

Conduction Band 1

There is no final state for the Absorption to take place

2 2 20/ 3p f fe v

2

2 3

( )( ) ( ) p

ib j

So intraband absorption is 0, but what about interband?

Still 0 because there are no real states in the gap

RBNI-14 64

Lossless, yet metal?

Valence Band

Conduction Band 2

EF

E

Ec

E12

Ecv

Conduction Band 1

There is no final state for the absorption to take place

12,c cvE E E

2

2( ) ( ) p

ib

2 2 2 2

0/ 3p f fe v

But we still want negative e

Thus we want narrow bands with wide gaps

We need large Fermi velocity (small effective mass)

As always in Nature we see two contradictory demands and thus should see if some type of trade-off and compromise is possible

RBNI-14 65

Consider b.c.c. lattice (Na)

a

Vss

Coupling strength Vss

Brillouin Zone –f.c.c.

EFEc

Tight Binding Model

RBNI-14 66

Lossless metal condition

2

0

1/20.6 /16 ssss ibe V aV

a

Vss~exp(-a/a0)

a0/a

Solution is possible but it take place at large inter-atomic distances that are bigger than in typical metals (and also prone to Mott localization)

Lossy metal

p

cE

0

Lossless metal

Dielectric

Lossy dielectric12E

RBNI-14 67

Example:Na

2 4 6 8 10 12 14-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Ener

gy (e

V)

Lattice Constant (A)

3s

3p

Ec

E12Ef

Need Lattice constant of 8 Angstrom instead of 4.3 Angstrom

That is why metals absorb….

RBNI-14 68

Hypothetical 8 A lattice spacing Na

1500 2000 2500 3000-3

-2

-1

0

1

2

3

Wavelength (nm)

FreeCarrierAbsorption

r

InterbandAbsorption

i

p c12

0.2

0.4

0.6

RBNI-14 6969

Big question: what do we do?

• We need to use stoichiometric arrangement of metal atoms separated by the non-metal atoms

• ITO –like materials are not solution – need stoichiometry!

• Example: AlO –metal • Maybe 2D structures?

AlZnO2

Al AlZn

O

AlZnO2

Al AlZn

O

RBNI-14 7070

Prospective materials

Na7MgF8

Mg

Na

F

Mg

Mg

Mg

NaF

Na

F

Mg

9.7eV

RBNI-14 71



•Why and how do the metals absorb and reflect?•Can a metal be made lossless?

•Does it have to be a metal?•Can metal loss be compensated by gain?•Can one make a true sub-wavelength laser /spaser?

RBNI-14 72

Potential and Kinetic energy of the oscillator

2202

d r dr er E

dt dt m

Equation of motion0

0 2 20

/( )

e mr

j

Amplitude2

02 20

/( ) 1

Ne m

j

Dielectric constant

22 20 0 2 2 0

0 0 22 2 2 20

1 /

4 4P

Nm r Ne mU E

Potential Energy of electrons Kinetic Energy of electrons

22 20 2 2 0

0 22 2 2 20

1 /

4 4K

Nm r Ne mU E

Far from resonance 0 2

2 00 2

0

1 /

4P

Ne mU E

0KU 2

020

/( ) 1

Ne m

Electric field Energy

2 2 20 02 2

0 0 22 2 2 20

/1 11

4 4E V P K

Ne mU E E U U U

20

1

4VU E

“Pure” Electric field Energy

22 0

0 20

1 /1

4E V P

Ne mU E U U

All of the electric energy is potential

UV

UMUP

00

RBNI-14 73

In the dispersive region

Electric field Energy

2 2 2 22 2 0 0 0

0 0 2 22 2 2 2 2 2 2 20 0

1 1 / /1

4 4E

Ne m Ne mU E E

2 2 2 2

2 2 20 0 00 0 02 222 2 2 2 2 2 2 22 2 2 2

0 00

1 / 1 / /

4 4P

Ne m Ne m Ne mU E E

Potential Energy of electrons

Kinetic Energy of electrons

22 2 0

0 22 2 2 20

1 /

4K

Ne mU E

20

1

4E “Non-dispersive (static)contribution”

“Equal dispersive (dynamic)contributions”

UV

UMUP,S

With dispersion – additional energy storage in dipole oscillations –less need for magnetic field

UP,DUK

UP,S UP,D

00

RBNI-14 74

In the Reststrahlen region

2,

10

4P S VU U E

, ,V P S P D M KU U U U U

Balance

0

0

0

,P D KU USince 0?MU If we add the second dielectric with e>0

e<0

e2>0

, , ,2P V P S P D P M KU U U U U U U

UP UM UKNo different from metal:

In metal w0=0 hence there is no potential energy

RBNI-14 75

Improvement?What if we use dielectric, such as SiC in Reststrahlen region (10-12 microns)

The width of SPP resonance geff~g gets reduced relative to the metal as it depends on QM1

2 2

2 21 LO TO

TO j

1MQ

2

Re( )2

Im( )TO

MQ

>0

<0

SiC

SiO2TO

LO

Interface phonon polariton decay constant g is a few ps!!!! (not 10fs as in metal)

It means that phonon polariton decays in time slower than plasmon polariton

e<0

e>0

But the propagation length depends on QM2< QM1 hence the propagation length is not much longer(slow light effect) 1/prop effL d d Similarly, field enhancement is proportional to QM2< QM1 hence it is not as high as in metals

2~ /U E Purcell effect is weak

RBNI-14 76

Another way?

>0

<0

Ag

SiO2

e<0

e>0 SiO2

Ag

Introduce highly dispersive dielectric component: Rb atoms, Quantum dots into the metal-dielectric plasmonic structure…..

00

Now significant amount of energy is in the form o fkinetic energy of bound electrons – less energy goes into conduction electrons oscillations-less scattering

UV

UP,S UP,DUK,M

UV- “pure E-field energy”

UP,S- “non-dispersive potential energy of bound electronsUP,D- “highly-dispersive potential energy of bound electrons

UM- Magnetic energy

UK,M-Kinetic Energy of electrons in metal

UK,D-Kinetic Energy of bound electrons

UM

UK,D

UM

RBNI-14 77

“Slow-light loaded plasmon poariton”

Most Energy is contained in the oscillations of bound electrons in dielectric – less in the oscillations of conduction electrons in the metal

The width of SPP resonance geff gets reduced relative to the metal

It means that SL loaded SPP decays in time slower than SPP with non-dispersive dielectric

But the propagation length does not change at all in SL loaded SPP (slow light effect)

1/prop effL d d

Similarly, field enhancement is SL-loaded SPP is no better than in normal SPP

2~ /U E

RBNI-14 78

Can metal loss be compensated by gain?

RBNI-14 79



•Why and how do the metals absorb and reflect?•Can a metal be made lossless?•Does it have to be a metal?

•Can metal loss be compensated by gain?•Can one make a true sub-wavelength laser /spaser?

RBNI-14 80

(1)Can one compensate the SP loss with gain?

The rate of loss in the metal is on the scale of 1/(10 fs) – it can only be compensated by the gain medium which has high density of active atoms and allowed transitions - semiconductor

>0

<0

Semiconductor

loss

gain

Metal loss

Semiconductor gain

“spaser”Can one make a small “Nanolaser” or Spaser?

Two relevant questions

RBNI-14 81

Purcell FactorSpontaneous decay is proportional to the density of photon states

w

∂∂g

ωv =

β

Metal

Localized SPP mode

effV1

3 3

8~spon D

=In 3D space

For propagating SPP 1 ~ ~SPP SPP

effd

Purcell Enhancement3 3

2 4,

3

~ ~ ~ ~ 10 10spon LSPP LSP

LSP D eff eff

F QV V

For localized SP 1 1

~ ~LSP LSPeffV

Purcell Enhancement3

1 3,

3

~ ~ ~ 10 10spon LSPP LSP

SPP D eff g

Fd v

>0

<0

Metal

Dielectric

Propagating SPP mode

effd

But is it good or bad?

RBNI-14 82

Can one compensate the plasmon loss with gain?

>0

<0Ag

Semiconductor

loss

gain

Rate equations

( )

el elp

eff rad

el tr

dn nIF

dt eV

g a n n

gain

Density of electron current Purcell’s Factor

differential gain Transparency density

19 3

3

10

1/3 1 2

~ 10

/ 20

~ 100; ~ 10

~ ~ /

tr

eff

p rad

tr eff tr P rad

n cm

V

F s

J eV n F MA cm

Purcell Enhancement does play the role of a spoiler – not surprising because in lasers we always want to reduce spontaneous emission and not enhance it

It is the current density and not the carrier density that matters.

RBNI-14 83

Compensation of loss in propagating SPP E

Rsp

Egap

eM

eS

N2/qS

z

xJ

J kx

lx=2p/kx

<0

>0

n-doped

+

p-doped

-Loss g

Gain g

Weff

Wa

2zE

zE

Weff-90% energy effective width normalized to l/n

/

/x

effx

k nn

n c

Effective modal index

Lx

Weff

RBNI-14 84

“Rea

l Pla

smon

”How does it look?

Metal: Silver. To fit the SPP to the bandgap for wide range of wavelengths we need to use “hypothetical” InxGaxNyAs1-y semiconductor

700 900 1100 1300 1500 1700 1900500 l(nm)Weff

Weff

Weff

Weff

Weff

n eff

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71

1.4

1.8

2.2

Effective modal index

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71018

1019 Transparency carrier density

Ntr(c

m-3

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71013

1014

Modal loss

geff

(s-1

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7104

105

106

107

J tr(A

/cm

2 )

Transparency current density

F p

0 0.1 0.2 0.3 0.4 0.5 0.6 0.71

10

100

Purcell’s factor

As expected, loss goes up as(1-1/neff

2)

Transparency density is reasonable

Purcell’s factor goes up as(1+neff

5)

Transparency current goes up as(1+neff

5) (1-1/neff

2)

Long

Ran

ge S

P

RBNI-14 85



•Why and how do the metals absorb and reflect?•Can a metal be made lossless?•Does it have to be a metal?•Can metal loss be compensated by gain?

•Can one make a true sub-wavelength laser /spaser?

RBNI-14 86

Enter The Spaser

LASERLightAmplification byStimulatedEmission ofRadiation

SPASERSurfacePlasmonAmplification byStimulatedEmission ofRadiation

Stockman, 2003

RBNI-14 87

Is Spaser Unique?

Very small amount of radiation is coming out for small SPASER

+++

+++

Energy is mostly contained in the matter, not field

d

Fraction of energy contained in free electron oscillations

2

1 dM

d gd

naf

Fraction of energy contained in bound electron oscillations in dielectric (polariton)

1gdD

d gd

f

dgd

Total fraction of energy in the electron vibrations:

2

1 1d gd

ed gd

na

f

For 40 nm GaAs Ag Spaser operating at 550nm 0.8ef

Ag

barrier

Intrinsic GaAs

BarrierAlAs

BarrierAlAs

N-doped AlGaAs

P-doped AlGaAs10nm

RBNI-14 88

Is Spaser Unique?

Consider a small semiconductor laser with high reflectivity mirrors Very small amount of radiation is coming out of it

Energy is mostly contained in the matter, not field

Fraction of energy contained in free electron oscillations 0Mf Fraction of energy contained in

bound electron oscillations in dielectric (polariton)

1gsS

s gs

f

2 2 ~ 22s s

gs s s s g

nn nn

Total fraction of energy in the electron vibrations: 1gse

s gs

f

For GaAs laser operating at 880 nm –strong dispersion!

0.60ef

+

+

+

+ +R>99%

In any laser operating in a dispersive material not a photon but a polariton is emitted and most of the energy is contained in electronic vibrations! The only difference is that in the SPASER it is free electrons. So what’ s so special? Big loss!

RBNI-14 89

Spaser and VCSEL

VCSEL

Ag

barrier

Intrinsic GaAs

BarrierAlAs

BarrierAlAs

N-doped AlGaAs

P-doped AlGaAs10nm

SPASER

Electric Field Energy10% 16%

Magnetic Field Energy30%3%

Free Electrons Energyin metal

45%

Bound Electrons Energyin semiconductor

60%

36%

FIELD

MATTER

RBNI-14 90

A very simple way to look at the spaser….

Rate equation for SP – bosonic particle

( 1)SPeff SP loss SP

dNg N N

dt

Metal loss

Semiconductor gain

g

g

loss metal rad

Spontaneous radiation

Spontaneous SPP emission –energy transfer from semiconductor to metal is at the rate of

/ 2eff lossg

Electrons flow

Holes flow

Current is

~ ~ 20lossI e A

Current density 2 6 2J ~ I / πr ~ 10 A / cm

This best case result does not depend on shape, size( as long as it is truly sub-wavelength, i.e. single mode) or on the gain material

The threshold is defined as 1SPN Equal probability of stimulated and spontaneous emission (linewidth is decreased by a factor of 2)

loss

RBNI-14 91

The current scales down with the volume only until the dimensions become sub-wavelength – after that – the same current has to go into progressively ever smaller volume –not a very good idea!

What happens?

1~tr eff tr P radI eV n F

We want to reduce the current by reducing the volume

Unfortunately…Purcell factor is inversely proportional to the volume

The current is proportional to the number of modes (confined and free space) into which the gain medium emits. The less is the volume the less is the number of modes. Until you have just one mode left. Then the volume stops being important.

RBNI-14 92

80 nm Single mode Au-InGaAs Spaser “emitting” at 1320 nm

P-contact N-contactAu

Active layer

10-7

10-6

10-5

10-4

1012

1013

1014

1015

Current (A)

SPP

emis

sion

rate

(1/s

) an

d Li

new

idth

(1/s

)

loss

effγ1 2 loss

tI

SPR

d

cb

a

-effective linewidth

“output”

0 100 200 3000

2

4

6

8

10

Current (A)

Carr

ier d

ensi

ty (1

018cm

-3)

tI

d

c

b

a

Input output characteristics shows no threshold

Linewidth narrowing indicates “spasing threshold” of 28 mA

Carrier density (gain) is clamped near threshold –sign of “spasing”

0 2 4 6 8 100

2

4

6

8

10

Carrier density (1018cm-3)

Num

ber o

f SPP

’s in

the

mod

e

,c thN

thre

shol

d

Near threshold half of SP’s are coherent and half have random phases – so it more like“SPED” rather than “SPASER”

RBNI-14 93

Lineshape evolution

0.7 0.8 0.9 1-6

-5

-4

-3

-2

-1

0

1

Energy (eV)

Mat

eria

l G

ain

(1014

s-1)

Mod

al g

ain

spec

tral

den

sity

(a.u

)(a)

Low pump

0.7 0.8 0.9 1-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Energy (eV)

Mat

eria

l G

ain

(1014

s-1)

Mod

al g

ain

spec

tral

den

sity

(a.u

)

(b)

Below threshold

0.7 0.8 0.9 1-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Energy (eV)

Mat

eria

l G

ain

(1014

s-1)

Mod

al g

ain

spec

tral

den

sity

(a.u

)

(c)

At threshold

0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

Energy (eV)

Mat

eria

l Gai

n (1

014

s-1)

Mod

al ga

in sp

ectr

al d

ensit

y (

a.u)

(d)

Way above threshold

RBNI-14 94

The threshold current depends on

The threshold current does not depend on

Scattering Rate in Metal

•Shape of the Mode•Size of the mode as long as it is sub-wavelength in all 3 Dimensions•Confinement factor of the mode•Gain material! (as long as it works in a “normal laser”)•Temperature (weakly)•Wavelength (red, orange, IR, polka-dot) as long it is less than ~20mm•Longitude, Latitude, Altitude•Attitude of the scientist•Amount of money spent

thr eff spI = eγ n ~ 20μA

RBNI-14 95

Do we need a nanolaser?

• True sub-diffraction structure always require metal and thus inherently very lossy

• Therefore, threshold of the sub-diffraction laser will always be very high (20 mA in a tiny volume)

• There will be only a few photons (plasmons) in a mode – no coherence (linewidth of THz)

• But the device will be very fast (THz)

What if we just use spontaneous emission –SPED (Surface Plasmon Emitting Diode)?

RBNI-14 96

Characteristics of SPED

• Small volume –high density of integration• Low power consumption • Easier to cool• High speed due to Purcell Effect

RBNI-14 97

VCSEL, SPASER and SPED

Pump Current density (A/cm2)

Intr

aca

vity

Po

we

r (W

)L

inew

idth

(Hz)

102 103 104 105 106 107 10810-810-710-610-510-410-310-2

102 103 104 105 106 107 108106

108

1010

1012

1014

102 103 104 105 106 107 108108

109

1010

1011

1012

1013

Ma

xim

um

freq

ue

ncy

(H

z)

(a)

(b)

(c)



LED

VCSEL

SPED

SPASER

LEDVCSEL

SPED SPASER

LED

VCSEL

SPED

SPASER

RBNI-14 98

Conclusions:•Loss affects and limits everything that is purportedly good about plasmonics and metamaterials Reducing loss would change the paradigm.•To get sub-wavelength concentration of light in all three dimensions one does need material with e<0 (usually a metal)•In optical and near IR region it always leads to losses commensurate with rate of decay in metal -1/10 fs, but far IR and THz the situation is less dire –perhaps mid-IR I swhere the action should be•Primarily because of loss only very weak optical processes can be enhanced by plasmonic means (SERS) – any optical device that requires high efficiency cannot be improved by plasmonic means. Hence sensing is the most promising niche.•Loss in metals in optical range is fundamentally different from loss (resistance) at low frequencies.•Loss and negative dielectric constant can be decoupled (in the future)•Using phonon polaritons and other resonant schemes reduces loss but has drawbacks of its own•Compensating loss with gain is tricky and probably unrealistic•Incoherent sub-wavelength sources are just as good as lsub-waveelngth lasers and might have a future.

RBNI-14 99

Additional readingJ. B. Khurgin, G. Sun “In search of elusive lossless metal, Appl. Phys. Lett, 96, 181102 (2010) J. B. Khurgin, G. Sun, Enhancement of optical properties of nanoscaled objects by metal nanoparticles”, J. Opt. Soc. B, 26, 83 (2009) J. B. Khurgin, G. Sun, Impact of high-order surface plasmon modes of metal nanoparticles on enhancement of optical emission, Appl. Phys. Lett., 95, 171103 (2009)

J. B. Khurgin, G. Sun, “Enhancement of light absorption in quantum well by surface plasmon polariton”, Appl. Phys. Lett, 94, 191106 (2009) J. B. Khurgin, G. Sun, R. A. Soref, “Plasmonic enhancement of photoluminescence by metal nanoparticles”, Appl. Phys. Lett. 94, 101103 (2009); J. B. Khurgin, G. Sun, R. A. Soref, “Practical limits of absorption enhancement near metal nanoparticles”, Appl. Phys. Lett. 94, 071103 (2009) G. Sun, J.B. Khurgin, R. A. Soref, “Plasmonic light-emission enhancement with isolated metal nanoparticles and their coupled arrays “, J. Opt. Soc. Am. B 25, 1748 (2008) J.B. Khurgin, G. Sun, R. A. Soref “Electroluminescence efficiency enhancement using metal nanoparticles”, Appl. Phys. Lett., 93 021120 (2008) Khurgin JB, Sun G, Soref RA ”Enhancement of luminescence efficiency using surface plasmon polaritons: figures of merit” J. Opt. Soc. Am. B 24: 1968-1980 (2007) Khurgin JB ”Surface plasmon-assisted laser cooling of solids”, Phys. Rev. Lett 98, Art. No. 177401 (2007) Sun G, Khurgin JB, Soref RA ”Practicable enhancement of spontaneous emission using surface plasmons” Appl. Phys. Lett., 90 Art. No. 111107 (2007)

RBNI-14 100

J. B. Khurgin, A. Boltasseva, “Reflecting upon the losses in plasmonics and metamaterials”, MRS Bulletin , 37, 768-779 (2012)J. B. Khurgin , G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium”, Appl. Phys. Lett, 100, 011105 (2012)

J.B. Khurgin and G. Sun: “How small can “ Nano ” be in a “ Nanolaser ”?”, Nanophotonics, 1, 3-8 (2012) J. B. Khurgin, G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence”, Optics Express 20 15309-15325 (2012) G. Sun, J. B. Khurgin, “Origin of giant difference between fluorescence, resonance, and nonresonance Raman scattering enhancement by surface plasmons”, Phys. Rev. A 85, 063410 (2012 JG. Sun, J. B. Khurgin, and D. P. Tsai, “Comparative analysis of photoluminescence and Raman enhancement by metal nanoparticles”, Opt. Letters, 37, 1583-1585 (2012) G. Sun, J. B. Khurgin, A. Bratkovsky “Coupled-mode theory of field enhancement in complex metal nanostructures “, Phys. Rev, B 84, 045415 (2011)J. B. Khurgin, G. Sun, Scaling of losses with size and wavelength in Nanoplasmonics” Appl. Phys. Lett, 99, 211106 (2011) B. Zhang, J. B. Khurgin, “Eigen mode approach to the sub-wavelength imaging with surface plasmon polaritons” Appl. Phys. Lett, 98, 263102 (2011) G. Sun, J. B. Khurgin, Optimization of the nanolens consisting of coupled metal nanoparticles: An analytical approach”, Appl. Phys. Lett, 98, 153115 (2011) G. Sun, J. B. Khurgin, “Plasmon Enhancement of Luminescence by Metal Nanoparticles : IEEE J. of Selected Topics In Quantum Electronics , 17 ,110 (2011) J. B. Khurgin, G. Sun , “Theory of optical emission enhancement by coupled metal nanoparticles: An analytical approach”, Appl. Phys. Lett 98, 113116 (2011) G. Sun, J. B. Khurgin, Comparative study of field enhancement between isolated and coupled metal nanoparticles: An analytical approach”, Appl. Phys. Lett. 97, 263110 (2010)

RBNI-14 101

Extra

RBNI-14 102

Higher order modes

l=2 l=5 l=10

High order modes tend to cling to the surface they are non-radiative

1

1

1 max,1 2

~ (cos )

l

j tl l

rr a

aE e P E

ar a

r

E

r

a

3

, 2

8

( 1)eff lD

aV

l

Small volume– high field concentration

RBNI-14 103

Impedance mismatchIn electronics in order to go from the high impedance circuit element (output stage of power amplifier-10K ) to the low impedance medium (coaxial cable -75, strip line) one usesImpedance transformer e.g. source follower

VDD

-VSS

IREF

vin

vout

Now we want “dense photons”to serve as a buffer between the emitting medium and the propagating photons

Can this buffer be lossless?

RBNI-14 104

Impedance mismatchIn electronics in order to go from the high impedance circuit element (output stage of power amplifier-10K ) to the low impedance medium (coaxial cable -75, strip line) one usesImpedance transformer e.g. source follower

VDD

-VSS

IREF

vin

vout

Now we want “dense photons”to serve as a buffer between the emitting medium and the propagating photons

Can this buffer be lossless?

RBNI-14 105

Threshold of a laser with just one mode

We propose that the threshold of a laser is more appropriately described by the pump power (or current) needed to bring the mean cavity photon number to unity, rather than the conventional“ definition" that it is the pump power at which the optical gain equals the cavity loss. In general the two definitions agree to within a factor of 2, but in a class of microcavity lasers with high spontaneous emission coupling efficiency and high absorption loss, the de6nitions may differ by several orders of magnitude.

Bjork, G., Karlsson, A. & Yamamoto, Y. Definition of a laser threshold.Phys. Rev. A 50, 16751680 (1994).

..it corresponds to the number of SPs at threshold roughly Nsp,t~1 i.e., on average, just about one SP in the mode, which is the threshold definition according to [36].”

J. B. Khurgin, G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence”, Optics Express 20 15309-15325 (2012)

RBNI-14 106

Single mode spaser as a “threshold-less” laser

1

SPeff loss SP eff spon

cex eff SP eff sp spon A

dNg N g n

dtdN

e I g N g n r rdt

Rate equations

SP

Carriers

Spontaneous emission into other modes

Auger recombination(proportional to volume)

Excess noise factor ~1

Excitation currentIf we neglect emission in other modes and nonradiative recombination… /SP ex lossN I e

Rate of SP generation is / /SP SP loss exR N I e No threshold! Since due to small volume SP emission into the single mode is the dominant recombination mechanism, all the energy goes into SP’s, but are they coherent?

Effective linewidth eff loss effg

We choose the critical effective linewidth narrowing , , / 2eff c loss eff c lossg It corresponds to , / 2eff c lossg and , 1SP c spN n one SP in the mode

Critical (threshold?) current ~ 20cr eff spI e n A

Battling Loss in Plasmonics and Metamaterials Jacob B Khurgin Johns Hopkins University 1RBNI-14...

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