Single photon nano-antennasLearning objectives:- scatterering- emission control

Resonant NanophotonicsAnnemarie BerkhoutKevin CogneeHugo DoelemanBeniamino FerrandoTomas KaandorpRadoslaw KolkowskiRuslan RohrichIsabelle PalstraTom Wolterink

CollaborationsPhilippe LalanneEwold VerhagenAlbert PolmanStefan Witte

Femius Koenderink – f.koenderink@amolf.nl

Center for Nanophotonics, AMOLF, Amsterdam

Motivation

Optical microscopybelow l/2 limit

Spectroscopy

Liu & AlivisatosBates & Zhuang [PALM, STORM]

Single photon sourcesQuantum information

Lozano, Verschuuren, Rivas

LED lighting & phosphors

Nanophotonics to control emission, absorption, lasing…- Directionality, polarization and phasefront of emission- Efficiency, brightness and Purcell factor- Spatial and temporal coherence

2

Making a light source

Space• Whereto does the photon go ?• With what polarization ?

Time• How long does it take for the photon to appear ?

Matter• Selection rules – what color comes out? Engineering wave functions

Engineering light

3

Basic scenarios

Microcavity - barely leaks light- high Q [ stores light ~ 105 optical cycles]- volume encloses a standing wave (l3)

Antenna - excellent radiator - low Q [ looses light in 10 optical cycles]- `antenna gain & directivity’

4

Antenna achievement chart

Brighter per fluorophore

1000x brighter

[total counts /sec on detector]

Directional sources

From 4p sr to 15o beams

[counts /sr / sec on detector]

1000x faster and brighter

[fluorescence decay rate ]

Directional & faster

Orrit, Moerner, Wenger van Hulst, Wenger, AMOLF Greffet, AMOLF Mikkelsen, Baumberg

ACS Photonics 4, 710–722, (2017) 5

Antenna achievement chart

Orrit, Moerner, Wenger van Hulst, Wenger, AMOLF Greffet Mikkelsen, Baumberg

Why are metal particles extremely strong scatterers?What is an `extremely strong’ scatterer anyway?

How do you generate directivity from a point source?How does emission get accelerated? What determines brightness?What is a good measurement for performance?

6

Scattering & observables

Extinction cross section [m2]

Power removed from beamIncident intensity

Extinction = scattering + absorption

removed from the beam

Re-radiated intoall angles

Lost as heat in the scatterer

7

Scattering figures of meritC

ross

sec

tio

n /

ph

ysic

al a

rea

Physical area

Rayleigh regime: (r/l)4

Resonances create s > pr2

Geometric resonances require size

Diagram: astronomy.stackexchange.com - actual theory: G. Mie, H. C. van de HulstThe diagram is not quite correct – in the geometric limit, you actually find twice the physical cross section (Bohren & Huffman) 8

Diagram: astronomy.stackexchange.com - actual theory: G. Mie, H. C. van de HulstThe diagram is not quite correct – in the geometric limit, you actually find twice the physical cross section (Bohren & Huffman)

Scattering figures of meritC

ross

sec

tio

n /

ph

ysic

al a

rea

Plasmonics

- Deep sub-l size, no geometry-only resonances

- Material resonance ensures largecross section

9

Scattering resonance

Murray & Barnes, Adv Mat 2007

Cross section

>

geometric area

aV Polarizability

> Physical volume

Figures of Merit

2

13

a V

Circa 103-104 free electrons

Incident field separates e- from ionic backbone

Linear restoring force implies a resonance

Resonant dipole scatterers l ~300-1000 nm, Q ~ 5-30

Induced dipoles

E0 m

z

r

q

a

Qualitatively:

Incident field induces local polarization density in the medium

If the object is so small that Ein hardly varies over it ( a << l ), the local induced response adds up to an induced dipole

11

Electrostatic sphere

20203

02

0001

42

2

3

2

r

prE

rEarE

rErErE

mm

m

m

m

m

m

p

qq

q

q

q

q

q

coscos

coscos

coscoscos

E0 m

z

r

q

a

Electrostatic treatment (frequency = 0)Sphere in a homogeneous electric field

Inside sphere: homogeneous field along incident field (Ez)Outside sphere: background field plus field of a dipole

In the ball:

Outside:

Incidentplane wave Ez Scattered, dipole

Check: (1) this potential is itself continuous, yet (2) its derivative jumps by , since Div D=0

12

3

0 0 with 42

mSI SI m

m

p E a

a a p

http://people.ee.duke.edu/~drsmith/plasmonics/enhancement.htm

|Ez| |Ex|

x

z

Electrostatic solution – scattered field

13

Drude model

200 400 600 800 1000 1200 1400 1600 1800-150

-100

-50

0

50

Measured data:

'

"

Drude model:

'

"

Modified Drude model:

'

"

Wavelength (nm)

'

Silver /optical constants

14

2 23 0

2 2

0

1 means ( )

p

easy ai i

a

Electrostatics: free electrons shield all fields

= - ∞ at = 0

Natural response “plasma” frequency in UV

flips sign, will go through - 2m

Deep UV: electrons can not keep up

=1 (transparency) for > p

Metal sphere

Drude model for a metal: Lorentzian `plasmon resonance’

2 23 0

2 2

0

1 means ( )

p

easy ai i

a

Units of volume [m3]

15Note: Now I chose the medium to be vacuum m=1. Otherwise you still get a resonance, but not quite Lorentzian

• Color tunable in visible +2m , shape effects

• Cross section ~ 10x pr2

• Strong dipolar near field

500 550 600 650 7000.00

0.01

0.02

0.03

Extin

ction c

rosssect. (

m

2)

Wavelength (nm)

|E|2/|Ein|2

30 nm Ag particle in glass

General properties

16

Scattering

2 2| | | |P p a

Radiated power P

Extinction

in Imdp

W Edt

a

Work done to drive p

≥

Optical theorem

Extinction = scattering + absorption

Proportional to volume V Proportional to V2

17In these units: extinction: sext = 4pk Ima and sscat = 8p/3k4 |a|2 - see for units Phys. Rev. B 83, 245102, (2011).

Optical theorem

Dimensional analysis

• Rayleighs’ law: scattering cross section [m2] scales as V2 ~ r6 sscat ~ r2 (r/l)4

• Absorption or scattering? extinction [m2] scales as V ~ r3 sext ~ r2(r/l)

small particles only absorb

• Since extinction > scattering, polarizability is limited in magnitude

More polarizability means more radiative loss - limiting polarizability

3 22Im | |

3ka a k=2p/l

18Note: Since clearly |Im a | < |a|, you immediately find | a| is bound to be no greater than 3/2 k-3

Example: simple spheres

Calculated exact cross sectionof Au spheres r=10 -50 nm

(1) Long wavelength - Rayleigh

(2) Increase in s with volume

(3) Until s approaches

(4) Radiation damping lowers antenna Q

(5) Absorption bounds Q, radiation limits s

Exti

nct

ion

cro

ss s

ecti

on

(

m2)

450 650 850

Wavelength (nm)

0

.04

.08

.12

19

Example: simple spheres

Exti

nct

ion

cro

ss s

ecti

on

(

m2)

450 650 850

Wavelength (nm)

0

.04

.08

.12

Addition of “radiation damping”

20

Beyond point polarizability

Polarizability intuition extends to:Magnetic scatterersMagneto-electric, chiral scatterersMagnetic modes in Mie scatterers Isabelle Staude

Similar cross sections but far larger local |E|2

Gap-plasmonsNanocube patch antennasNanosphere on mirror antennas

Javier AizpuruaMikkelsen, Baumberg

Oligomers & arraysHybrid, higher Q multipolar modesPhased arrays, metasurfaces

Curto, GiessenEngheta, AMOLF

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What is a source ?

S0

S1

Antenna – driven dipoleFixed current source

Spontaneous emitter ħ at a time

22

Three distinct effects

• Source brightness [counts/sec] is enhanced because

pump field is concentrated at the emitter

• Given that the source has been excited and emits a photon,

antenna effects redirect the light into a narrow beam

• Fluorescence decay rates are enhanced - Purcell effects on rate and IQE

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Radiation provides resistance

Radiation resistance – environment sets power to current ratio

The work you need to do keep current j going depends on environment

Conversely, the same current can generate more power (or less)

Dipole above a mirror in classical terms

- +

-+

-

+

-

+

Interference of the dipole field with that of its mirror image The same current radiates a different far field power

25

Backaction view point

-+

In order to maintain constant currentone does work against ones own field

Suppose we call the dipole field

Resistance is due to field Im pT.G(r’,r’).p that comes back to the source“Imaginary part of the Green function”

26

Single quantum emitter

• After one excitation, emits just one quantum of light

• Probabilistic timing of when emission occurs

• Fermi’s Golden rule: exponential decay, characteristic rate [inverse time]

Laser pulses

Hits ondetector

Hits onAPD 2

Time

S0

S1

Time (ns)

Lounis & Orrit, Single photon sources, Rep. Prog. Phys (2005)

Drexhage experiment on single quantum emitter

Note how: the power is not the variable of choice: one photon out per photon in (QE=1)the decay rate varies with mirror-geometry

K.H. Drexhage and many since - ensembles of emitters (1966)Single emitters: Buchler PRL (2005), Frimmer (2013), Huck (2016)

0 40 80t (ns)

10

100

1000

Even

ts

slope

Single NV center in nanodiamond

28

Understanding Fermi’s Golden Rule

2

2all finalstates

2( )f i f i

f

V E Ep

Energy conservationMatrix elements:Transition strengthSelection rules

Spontaneous emission of a two-level atom:

Initial state: excited atom + 0 photons.

Final state: ground state atom + 1 photon in some photon state

Question: how many states are there for the photon ???(constraint: photon energy = atomic energy level difference)

29

How many photon states are there in a box of vacuum ?

( , ) sin( ) with ( , , )i tE x t Ae l m nL

p k r kStates in an LxLxL box:

l,m,n positive integers

Number of states with |k|between k and k+dk:

3

24( ) 2

8

LN k dk k dkp

p

l,m,n > 0fill one octant

fudge 2 for polarization

As a function of frequency (ck):

2 23 3

2 2 2 3( )

dkN d L d L d

c d c

p p

Picture fromhttp://britneyspears.ac

k

dk

30

Fluorescence decay rates

Fermi’s Rule: Fluorescence rate number of photon states

0 2 4 60

50000

100000

150000

Photo

n s

tate

s p

er

m3, per

Hz

Frequency (1015

s-1)

Visible light: ~105 photon states per Hz, per m3 of vacuum

Loudon, The Quantum Theory of Light

Cavity

Fluorescence in a cavity

0 2 4 60

50000

100000

150000

Photo

n s

tate

s p

er

m3, per

Hz

Frequency (1015

s-1)

Fermi’s Rule: Fluorescence rate number of photon states

Microcavity: Exactly one extra state per D=/Q in a volume V

Gérard & Gayral, J. Lightw. Technol. (1999)

Cavity

Fluorescence in a cavity

0 2 4 60

50000

100000

150000

Photo

n s

tate

s p

er

m3, per

Hz

Frequency (1015

s-1)

Fermi’s Rule: Fluorescence rate number of photon states

Microcavity: Exactly one extra state per D=/Q in a volume V

Purcell factor

3

2

3

4

QF

Vl

p

Gérard & Gayral, J. Lightw. Technol. (1999)

From LDOS to actual performance

S0

S1

A unit efficiency emitter, emits exactly a single ħ per pump photonHigh LDOS will make the photon come out faster, but the source is not more bright

A low-efficiency sources gains efficiency by outcompeting nonradiative loss

LDOS itself may contain new loss channels - quenching

nonradiative

34

Antenna achievement chart

Brighter per fluorophore

1000x brighter

Intrinsically poor emitters- 10x boost in QE- 100x boost in pump

Intrinsically good emitters- 500 - 1000x faster- Brighter by pump boost

Directional & faster

Orrit, Moerner, Wenger van Hulst, Wenger, AMOLF Greffet Mikkelsen, Baumberg

Brighter by beaming

Factor ~ 5

35

Doing a good measurement is not easy

• Make sure you have a single emitter (at a time)

• Of known quantum efficiency

• At the right location

• Calibrate collection efficiency, directivity, rate, and efficiency

• All that in a limited photon budget [ < 107 total detected photons usually]

Averaging over many emitters is highly misleading 1. Average over a diffraction limit is useless - very far from peak numbers2. Average is biased – fluorescence signals select on unquenched emitters3. Product of averages and average of product is very different

36

Actual calculation – Au sphere in a pump focus

ACS Photonics 4, 710–722, (2017) 37

Record reported antenna

Hoang, Akselrod & Mikkelsen

Ag nanopatch on template-stripped Au, CdSe/CdS quantum dots- Lifetime suggests Fp > 540 – quantum efficiency presumed 20%- 1900x overal brightness increase in cw excitation

Factorized as 225 [pump ] x 2 [ Q.E. ] x 4 [collection]

Nature Photonics 8, 835–840, Nano Letters 16, 270–275 38

Next steps

Orrit, Moerner, Wenger van Hulst, Wenger, AMOLF Greffet Mikkelsen, Baumberg

Better emission controlQuenching & efficiency controlIntegration with optical circuitsQ-factor control & indistinguishable photonsCooperative emission & lasing

Beyond emission controlMolecular optomechanics & SERSQuantum optics with single moleculesBreaking selection rules Extreme-gap nanophotonics

Nature Materials (2019), DOI: 10.1038/s41563-019-0290-y

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