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?
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
2
P
Pout ~  d .c. 
(n) 2
n
Ppeak
~
 (n) P n
(d .c.)n 1
t
Can we do the same in the space domain and concentrate the same power into higher
local power density to increase the efficiency ?
Pout ~ ( d .c.) 
(n) 2
In ~

(n) 2
Pn
( d .c.) n 1
Ag
Plasmonics as a ”silver bullet” for
nonlinear optics
“Mode-locking in space?”
Plasmonic concentrators
-
+
-
-
+
+
+
-
+
+
+
+
+
+
 ( )  1 
+
+
M. Stockman, P. Nordlander
-
Q~
+
-
+
+
+
- +
- +
 2  j
r 
~ ~ 10  20
i 
2
 Elocal 
4
4
5
~
Q
~
10

10


 E 
2
 Elocal 
2
2
3

 ~ Q ~ 10  10
 E 
-
 p2
But:
In space there is an additional factor of modal overlap k – the field of pump(s) must
overlap with field of signal (conceptually similar to the phase-matching)
Plasmonic concentration always brings loss
4
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
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
7
Scope
• Rationale
• Can one engineer nonlinearity in metal
nanostructures?
• Coupled mode theory of enhancement
• Assessment of nonlinearity enhancement
• Conclusions
Can one engineer nonlinearity in metal?
In QW’s or QD’s….anharmonic
potential-giant dipole of this
“artifical atom” or “molecule”
How about electrons in SPP giant
“artificial atoms” or “molecules”
+
+ N = 6 × 10 cm +++
+
+
+
+
+
Say we have 1 SPP per mode
22
-3
Power dissipation is P  γhω ~ 1eV / 10fs ~ 10μW
12
3
Power density P  10 W / cm - very high!
How far do the carriers move?
2
mv 2
m 2 x 2 ω2
x
~
A
1/ 2
NV
= NV
= hω
(NV)
2
2
In 30 nm sphere…NV~106 electrons ;
Electrons move less than 0.001A!!!!
In QW Electron moves up to a few nm
SPP modes analogy with giantatoms and molecules is quite superficial
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 9
saturable absorber except for much higher loss) or better revert to nonlinear dielectrics
Scope
• Rationale
• Can one engineer nonlinearity in metal
nanostructures?
• Coupled mode theory of enhancement
• Assessment of nonlinearity enhancement
• Conclusions
Four wave interactions
FWM (Four Wave Mixing)
E1e j1t
j2t
E2e
E3e
j3t
Efficiency
j1t
E1e
j1t
E1e
E2,ine
j3t
(3)
1
2
E4,out e j4t
P4e j4t ~  (3) E1E2*E3e j (1 2 3 )t
 (3)
*
4  1  2  3
E
~
j
L

E
E
E
1 2 3
3 24,out
c
2
2
4  1  2  3

E4
 (3)
*
FWM 
~ L  E1E2 ~ Ln2 I pump
E3
1
2
2
Nonlinear phase shift
c
c
XPM (Cross Phase Modulation)
E2,out e j2t
(3)
P2,nl e j2t ~  (3) E1 E2e j2t
2
1  2 ; 4  3
j ( 2t   ( 3) / c E1 L )
2
E2,out ~ E2,in e
nl ~

c
L  (3) E1 
2

c
Nonlinear index
Ln2 I1
n2   (3)0 / n2
Practical figure of merit
Switching
I pump

For nonlinear switching using XPM or SPM
nl 
I signal

c
Ln2 I ~ 2
L

nmax ~ 
For wavelength conversion
~

c
2
Ln2 I pump ~
2

2
nmax ~ 1
Maximum interaction length is determined by absorption hence the ultimate figure of merit is
what is the a maximum phase shift achievable :
max ~ 2
And how close it is to 1…
L

nmax
Mechanism for the enhancement of
nonlinearity
I pump
Ag
(3)
E pump
Ag
Ag
Stage 0
I sig
Ag
Ag
E pump Esig
Esig
Ag
Ag
Average values of fields
Ag
Mechanism for the enhancement of
nonlinearity
I pump
I sig
+
E pump
+
-
p pump psig
(3)
+
-
-
+
+
-
-
Stage 1
+
+
+
-
-
-
p pump psig
Esig
Nanopartcles get polarized at both pump and signal frequencies
psig ~ 30VEsigQ
ppump ~ 3 0VE pumpQ
Q ~  r / im ~ 10  20
Mechanism for the enhancement of
nonlinearity
I pump
I sig
E pump
+
+
-
-
p pump psig
+
-
(3)
-
+
+
-
-
Eloc
Esig
+
Stage 2
+
+
-
-
enhanced field at both pump and signal
Eloc, pump (r ); Eloc,sig (r ) Locally
frequencies
Eloc,sig (r ) ~ 2QEsig
Eloc, pump (r) ~ 2QEpump
Mechanism for the enhancement of
nonlinearity
I pump
+
E pump
+
-
-
I sig
+
-
+
-
-
Ploc,nl
Esig
Ploc ,nl ( r ) ~ 
+
-
+
Ploc,nl (r )
(3)
(3)
Stage 3
+
+
-
-
Local nonlinear polarization is established
2
Eloc , pump ( r ) Eloc ( r ) ~ 8 0Q 
3
(3)
2
E pump Esig
Mechanism for the enhancement of
nonlinearity
+
I pump
I sig
E pump
+
-
-
+
+
-
-
Ploc,nl
-
+
-
Eloc ,nl
+
(3)
Esig
Stage 4
+
+
-
-
Eloc,nl (r ) Local nonlinear field is established
2
1
4 (3)
Eloc ,nl ( r ) ~ k Q 0 Ploc,nl ~ k 8Q  E pump Esig
k~
4
E
 loc (r)dV
2
loc ,max
E
E
2
loc
( r )dV
~ 0.1Third order nonlinear polarization does not exactly
match the mode
Mechanism for the enhancement of
nonlinearity
I pump
I sig
+
E pump
+
-
Eloc ,nl
psig ,nl
(3)
+
-
-
+
+
-
-
Stage 5
+
+
+
-
-
-
Esig
Accordingly, each nanoparticle acquires nonlinear dipole moment (at signal frequency)
2
3
4 (3)
sig ,nl
loc ,nl
0
pump
sig
2 0
p
k~
~  VE
4
E
 loc (r)dV
2
2
Eloc
E
,max  loc ( r )dV
~ 12 V k Q 
~ 0.1
E
E
Third order nonlinear polarization does not exactly
match the mode
Mechanism for the enhancement of
nonlinearity
I pump
E pump
+
+
-
I sig
+
-
(3)
+
-
psig ,nl
-
+
+
-
-
+
Esig
-
Stage 6
+
Psig ,nl
-
The whole medium then acquires average nonlinear polarization at the signal frequency
f – filling factor
2
4 (3)
Psig ,nl ~ Npsig ,nl  12 0 f k Q  E pump Esig
2
(3)
Introduce effective nonlinear
Psig ,nl   0  eff E pump Esig
susceptibility
(3)
4 (3)
eff ~ 12 f kQ 
Scope
• Rationale
• Can one engineer nonlinearity in metal
nanostructures?
• Coupled mode theory of enhancement
• Assessment of nonlinearity enhancement
• Conclusions
Assessing nonlinearity enhancement
 eff(3)
n2,eff
4
3
4
~
12
f
k
Q
~
10
~
12
f
k
Q
n2
 (3)
This sounds mighty good…..
 eff 
What about absorption?
Maximum phase shift
nl ,max 
2
eff
Enhanced as much as few hundreds times
still, assuming
2 nd

3 fQ
n2,eff I ~ 4k Q 3n2 I
This sounds really good…..except
n2  1013 cm2 / W (chalcogenide glass)
nl ,max  1010 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….
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! Local “intensity” is now in excess of 1000 GW/cm2 –way past break down!
So, what is the real limit?
A better figure of merit
Re( psig ,nl )
2
Psig ,nl ~ 12 0 f k Q  E pump Esig
2
2
(3)
 3 f k Q  0  Eloc, pump Esig
4
(3)

 6 f kQ  n I
2
0
2 loc , pump
 E
sig
 6 f kQ20  nlocal  Esig
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 0

Assuming that maximum index change is limited by material properties to nlocal  nmax  0.01
the maximum phase shift is…
nl ,max ~
2
eff
3 f k Q 2 nmax
k Qnmax  0.01
There is no way to achieve either all-optical switching or efficient frequency
conversion!
What if we use dimers or “nano-lenses”?
(3)
I pump
I sig
Field enhancement occurs in two steps –first the larger dipole mode gets excited then the gap
mode near smaller nanoparticle
Eloc ,sig ( r ) ~  2QEsig 
2
Eloc , pump ( r ) ~  2QE pump 
2
 eff(3)
6
5
~
5
f
k
Q
~
10
 (3)
But the relation between the average nonlinear polarization and maximum index change is
nl ,max ~ 0.2kQnmax  0.01
still almost the same, therefore 
What does it mean?
1mm2
10
P=8W

10 0
P=0.8W
Nonlinear Phase Shift (rad)
n2  1013 cm2 / W
1
10 -1
P
P=0.8W
P=1.6mW
P=1.6mW
10 -2
10 -3
10 -4
1mm2
13
n2  10 cm / W
2
P
P=1.6mW
10 -5
10 -6
10 -7
10 -8
-1
1mm2
n2  1013 cm2 / W
10
0
10
1
10
2
10
3
10
4
10
Length (mm)
P
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
• Rationale
• Can one engineer nonlinearity in metal
nanostructures?
• Coupled mode theory of enhancement
• Assessment of nonlinearity enhancement
• Conclusions
Two ways to define figure of merit
Scientific approach
---
+ + +
+ --
What is the maximum attainable enhancement of
nonlinear susceptibility?
+
+
+
+
For 2) enhancement is kfQ3 ~102-103
For (3) enhancement is kfQ6 ~105-106
+
+ -- +
+
+ +
Engineering approach
For the nonlinear index type process
– what is the maximum phase shift
attainable at 10dB loss?
What would be the overall maximum attainable
result at ~one absorption length?
Fmax~kQnmax~10-2<<
Not enough for all-optical switch
(or frequency conversion)
26
Why such a conflicting result ?
Scientific approach: what matters is the relative improvement
Take very weak process with efficiency approaching 0….then if the end result is <<1
Result
a very large power
= 10
0
Engineering approach: what matters is the end result
Result = 0 × 10
Ag
a very large power
<< 1
Using metal nanoparticles for enhancement of second order
nonlinear processes may not be a “silver bullet” we are looking for.
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.
27