1 Nanoparticles as Nanoscale Resonators I

1 Nanoparticles as Nanoscale Resonators I

Consider a quantum emitter in proximity to a gold metal nanoparticle of radius 40 nm with dielectric constant _{N P} =−5.9311 +i2.0970 surrounded by air (₁ = 1). The wavelength of the emitted photons is 550 nm.

• Calculate the modification of the spontaneous emission rate for the quantum emitter as a function of distance from the nanoparticle and of the dipole orientation in the quasistatic approximation.

Parameters: a= 40 nm,_m =_{N P},_b = 1,λ= 550 nm The Purcell factor is given by the equation:

Γ

Γ₀ =F (1)

Γ

Γ₀ =η_a 9

16π⁴ ×(1−L)²

V_ph² (2)

Where, L = 1/3 (for a sphere), andVph = ^4π₃ ^a_λ³3 is the volume of the sphere in units of the cubic wavelength. The antenna efficiencyηa is given by

ηa= 1 1 +_ω^γ

p

3 4π²

√bL Vph

, (3)

where,_b= 1 and m can be written as

_m= 1− ω_p²

ω(ω+iγ). (4)

Then

m−1 =− ω²_p

ω(ω+iγ) =−ω_p²(ω−iγ)

ω(ω²+γ²), (5) Re{_m−1}=−ω_p²

ω

ω

(ω²+γ²). (6)

Hence we can write that

Im{_m−1}=− γ ω_p²

ω(ω²+γ²). (7)

Thus we get

Im{_m−1}

Re{_m−1} =−γ

ω. (8)

Consider thatω is close to the resonance of the nano particle. Thus we have ω' ωp

√1 + 2b

= ωp

√3, (9)

Im{_m−1}

Re{_m−1} =−√ 3 γ

ωp

. (10)

Thus

γ

ω '=− 1

√3

Im{_m−1}

Re{_m−1} = 2.1

√3×6.93 '0.175. (11) Thus the Purcell factor can be written as

Γ Γ0

= 1

1 + 0.175(^λ_a)³× ^√1

3

× 9 16π⁴

4 9

9 16π²

λ a

6

, (12)

Γ Γ0

' 1

1 + 0.175(^λ_a)³ 9 64π⁶

λ a

6

, (13)

where

1

1 + 0.175(^λ_a)³ =ηa, (14) and ^λ_a = 550/40 '13.8. Thus we can write ^λ_a6

'6.9×10⁶. Hence we can write the Purcell factor for a dipole in the field orientation:

Γ

Γ₀ = 1

2.6×10² 1.4×10⁻⁴×6.9×10⁶'3.7 (15) If the dipole is not aligned along the field, there will be no field enhancement. In this case _Γ^Γ

0 <1.

• When a photon is emitted, estimate the probability that the photon is radiated to the far field (external quantum efficiency) as a function of distance from the nanoparticle and of the dipole orientation.

The longitudinal fieldE≈1/(kd)³ Γ

Γ₀ ' |E|², (16)

hence

Γ Γ0

' 1

(kd)⁶. (17)

At distances, where the effect drops by a factor of 10 we can write:

1

(ka)⁶ = 10 1

k⁶(a+d)⁶, (18)

wheredis the distance from the surface of the nano particle.

1

(a)⁶ = 10

(a+d)⁶ =⇒ 1 a =

√6

10

a+d (19)

√6

10 =a+d =⇒ d=a(⁶

√

10−1) =⇒ d'0.47×a= 18nm. (20) The antenna efficiency can be written as

ηa= P_rad

(P_rad+P_abs) = P_rad P_tot = Γr

Γ , (21)

η_a= 1 1 +_ω^γ

p

√9

bLAR² 2(ka)³

. (22)

η_a' 1

1 + 0.1(^λ_a)³ = 1

1 + 1.4 '40%. (23)

If you increase the distance, ηa increases and it reaches 100% at about kd = 1, where d≈ _2π^λ '90 nm.

2 Nanoparticles as Nanoscale Resonators II

Figure 1: Interaction between Nanoparticle and an Emitter

(a) In the figure shown above, we have a nanoparticle located in the vicinity of a quantum emitter. That emitter’s dipole could be oriented horizontally or vertically.

The spontaneous emission rate of the emitter without the nanoparticleΓ_oand using the Purcell factor we can then calculate the modification of the spontaneous emission rate _Γ^Γ

o. As the dipole emits radiation in free space, it is associated with the electric field of the radiated emitted, and the relationship is as follows:

Γo ∼ωIm n

~ p·E~^∗

o , whereE~^∗ is the electric field of the dipole itself.

The field of the dipole in the presence of a nanoparticle would be otherwise if the nanoparticle does not exist. Such that the radiation emitted by the dipole propagates to the nanoparticle, a portion of it will be absorbed by the nanoparticle, but the other portion will be reflected back to the emitter (dipole).

Γ

Γ_o = 1 +6πo

k Im n

~ p·S·~p

o ,

where S represents the field emitted by the dipole, which goes to the nanoparticle and gets emitted back by the nanoparticle to the dipole.

The dipole field in presence of the nanoparticle, it corresponds to the Green’s tensor times the dipole.

G(0,0;ω) =Go(0,0, ω) +Go(0, d, ω)oα(ω)Go(d,0;ω),

whered is the distance separating the emitter apart of the nanoparticle, α(ω) is the polariz- ability of the nanoparticle, andG_o(0,0, ω) is the free-space Green’s tensor.

S=G_ooαG_o G_o(R;ω) =

"

(1−RˆR) +ˆ i

kR(1−3 ˆRR) +ˆ 1

k²R²(1−3 ˆRR)ˆ

# e^ikR 4πRo

k ,

whereR=|r−r⁰ |is the difference between the distance vectors from the origin point to the source and the origin to the point of the field.

1=





1 0 0 0 1 0 0 0 1





and

RˆRˆ =



 Rx

R_y R_z



 R_x R_y R_z .

To calculate this as a function of the dipole orientation, let’s consider the orientation clarified below:

Figure 2: Parallel coupling vs. perpendicular coupling

The dipole is alongz-direction, therefore~p= (0,0,1). Now we want to calculate the distance either in case of parallel coupling or perpendicular coupling :

kcoupling : d= ˆzd=⇒Rˆ = (0,0,1)

⊥coupling: d= ˆxd=⇒Rˆ = (1,0,0) RˆRˆ =



 0 0 1



 0 0 1

=





0 0 0 0 0 0 0 0 1



:k

:

RˆRˆ=



 1 0 0



 1 0 0

=





1 0 0 0 0 0 0 0 0



:⊥

k:









1 0 0 0 1 0 0 0 0



+ i

kR − 1 k²R²





1 0 0 0 1 0 0 0 −2







Go(R)k

⊥:









0 0 0 0 1 0 0 0 1



+ i

kR − 1 k²R²





−2 0 0 0 1 0 0 0 1







Go(R)k

Now, we need to calculate the scalar productGo·~pand also ~p·Go , whereGo = _4πR^eikR

o.

~

p·G_o= 0 0 1









1 0 0 0 1 0 0 0 0



+ i

kR − 1 k²R²





1 0 0 0 1 0 0 0 −2







G_o(R)k fork.

~ p·Go =

(0,0,0) + i

kR − 1 k²R²

(0,0,−2)

Go(R)k:k

~ p·G_o =

(0,0,1) + i

kR − 1 k²R²

(0,0,1)

G_o(R)k:⊥

Go·~p=







 0 0 0



+ i

kR − 1 k²R²

+



 0 0

−2







Go(R)k:kGo·~p=







 0 0 1



+ i

kR − 1 k²R²

+



 0 0 1







Go(R)k:⊥

~

p·S·~p=~p·GooαGo.~p

This is what we want to calculate for parallel and perpendicular, so:

k:

i

kR − 1 k²R²

2

4Go(d,0)·Go(0, d)oα(ω)k²

⊥:

1 +

i

kR − 1 k²R²

2

1G_o(d,0)·G_o(0, d)_oα(ω)k² k:

i

kR − 1 k²R²

2

4 e^2ikd 16π²²d²

oαk²

⊥:

1 + i

kd− 1 k²d²

2

e^2ikd 16π²²d²

oαk² Now we can plug it into expression of decay rate:

Γ_k

Γ_o = 1+6π k k²Im

( i

kR − 1 k²R²

2

e^2ikd 4π²_od²α

) ,Γ⊥

Γ_o = 1+6π k k²Im

( i

kR − 1 k²R²

2

e^2ikd 16π²_od²α

) .

After cancellation of similar terms, we can write the equations as the following:

Γ_k

Γ_o = 1+ 3 2πk³Im

( i

kR − 1 k²R²

2

e^2ikd k²d²α

) Γ⊥

Γ_o = 1+ 3 8πk³Im

( 1 + i

kR − 1 k²R²

2

e^2ikd k²d²α

)

Figure 3: Γ/Γo as a function of distance Forkd1(Far Field):

Γ_k

Γ_o '1 + 3 2πk³

−1 k⁴d⁴

Imn

αe^2ikdo , Γ⊥

Γ_o '1 + 3 8πk³ 1

k²d²Im n

αe^2ikd o

.

Forkd1(Near Field):

Γ_k Γo

' 3 2πk³

1 k⁶d⁶

Im{α}, Γ⊥

Γ_o ' 3 8πk³

1 k²d²

Im{α}. The quantum efficiencyη is defined asη = _Γ ^ΓR

R+ΓN R. We can either calculateΓRorΓN R, but it is easier to calculateΓN R, because it is the power absorbed by the nanoparticle.

Figure 4:Γ/Γ_o vs. kd.

ΓN R

Γo

=Power absorbed by the Nanoparticle P_abs ∼ωImn

~

p_{N P} ·E~^∗o ,

E~ =Go(0, d)·p~ Field radiated by the emitter

~

p_{N P} =_oα ~E,

P_abs =ωIm{_oα} |G_o(0, d)·~p|², k:|G_o(0, d)·~p|²=

i kd− 1

k²d²

2 1

4π²²_od²k²,

⊥:|G_o(0, d)·~p|²=

1 + i kd− 1

k²d²

2 1

16π²²_od²k².

Γ^k_{N R} Γ_o = 3k³

2π 1

k⁴d⁴ + 1 k⁶d⁶

Im{α},Γ^⊥_{N R} Γ_o = 3k³

8π 1

k⁴d⁴ + 1

k⁶d⁶ − 1 k²d²

Im{α}.

=⇒ ΓR= Γ−ΓN R

Forkd1 (Far-field): ΓN R '0 Forkd1 (Near-field): ΓN R' _k6¹_d6.

3 Nanoparticles as Nanoantennas

Modification of the radiation pattern by a metal nanoparticle, as indicated below:

The radiation pattern shown above will create superposition of both radiations at the far field.

Figure 5: Interaction between induced dipole and emitter’s dipole.

E(r,~ 0;ω)∼G_o(~r,0;ω)·~p

~

p_{N P} =_oα ~E(d,~ 0;ω)

So, starting fromE(r,~ 0, ω), we calculate ~p_{N P}, then the induced dipole - of the nanoparticle - radiates energy.

E~_{N P}(~r) =G_o(~r, ~d;ω)·~p_{N P}, E(~~ r) =G_o(~r,0;ω)·~p.

E~_tot =E~_{N P} +E ...this creates new radiation pattern~ E~N P(~r) =Go(~r, ~d;ω)oαGo(d,~ 0;ω)·~p,

E~_tot 'h

G_o(~r, ~d;ω)_oαG_o(d,~ 0;ω) +G_o(~r,0;ω)i

·~p.

whereGo(~r, ~d;ω)oαGo(d,~ 0;ω)is for the nanoparticle, whileGo(~r,0;ω)for the free space with- out the nanoparticle.

In the Far-field: kr1

G_o(~r,0;ω)'(1−RˆR)ˆ e^ikR 4π_oRk, whereRˆ =





cosφsinθ sinφsinθ

cosθ



Emitter

Figure 6: Distance vectors.

Rˆ−dˆ= ˆR−zˆ fork,Rˆ−xˆ for⊥ Rˆ−dˆ=





cosφsinθ cosφsinθ cosθ−1



fork nanoparticle

=





cosφsinθ−1 cosφsinφ

cosθ



for⊥ e^{ik|R−~ d|~}

4πo |R~ −d~|' e^{ik|R−~ d|~} 4πor

~

p is along z:



 0 0 p_o





(1−RˆR)ˆ



 0 0 1



=



 RxRy

R_yR_z 1−R²_z



=





cosφsinθcosθ sinφsinθcosθ

sin²θ





G_o(d,0)·~p= i

kd− 1 k²d²

e^ikd 4π_od



 0 0

−2



p_okk

G_o(d,0)·~p=

1 + i kd− 1

k²d²

e^ikd 4π_od



 0 0 1



p_ok⊥ For the NP:

h1−( ˆR−d)( ˆˆ R−d)ˆi



 0 0 1



or



 0 0

−2





=





(Rx−dx)(Rz−dz) (R_y −d_y)(R_z−d_z) 1−(Rz−dz)(Rz−dz)



(1) or (−2)

=





cosφsinθ(cosθ−1) sinφsinθ(cosθ−1)



(−2)





(cosφsinθ−1) cosθ sinφsinθcosθ





E~_tot =









cosφsinθcosθ sinφsinθcosθ

sin²θ



+





cosφsinθ(cosθ−1) sinφsinθ(cosθ−1)

1−(cosθ−1)²



_oα(−2) i kd− 1

k²d² e^ikd 4π_od



 e^ikr 4πr

E~_tot =









cosφsinθcosθ sinφsinθcosθ

sin²θ



+





(cosφsinθ−1) cosθ sinφsinθcosθ

sin²θ



_oα

1 + i kd− 1

k²d²

e^2ikd 4π²od



 e^ikr 4πr

Radiation Pattern:

P(Ω)r²dΩ =|E_transv |² r²dΩ P(Ω) =r² |Etransv |²

Ex,y,z −→Eθ, Eφ, where

|Etransv |²=|E_θ|² +|E_φ|²