MSEG 667
Nanophotonics: Materials and Devices
8: Nonlinear Optics
Prof. Juejun (JJ) Hu hujuejun@udel.edu
“The study of non-linear physics is like the study of nonelephant biology.”
-- Stanislaw Ulam, a mathematician
References

Fundamentals of Photonics


Nonlinear Optics


Ch. 18 and 19
R. W. Boyd
Photonics: Optical Electronics in Modern Communications


A. Yariv and P. Yeh
Ch. 8, Ch. 9 and Ch. 14
Atomic origin of optical nonlinearity
Induced dipole moment P  ne  e  r
Charge density

Electron
charge
D  0E  P
Displacement
Simple harmonic oscillator model (linear)
d2r
dr
m  2  m 
 b 2 r  eE
dt
dt
Linear polarization:
parabolic potential
order nonlinearity:
Pockels media
Restoring
force
Lorentz-Drude
Damping Restoring Driving
2nd
Atomic
nucleus
3rd
order nonlinearity:
Kerr media
Electronic
charge
2nd order nonlinearity
is absent in crystals
with centro-symmetry!
F = kx
F = kx + k2x2
F = kx + k3x3
Nonlinear polarization

Linear medium: low field intensity
D  0E  P
D    E   r 0 E
P  0  E
r  1 
Linear polarization

Nonlinear medium: high field intensity
D  0E  P
P   0   E   ( 2 )  E 2   ( 3)  E 3  ...  PL  PNL
Nonlinear polarization
Pi   0  ij  E j  2dijk  E j Ek  4  ijkl E j Ek El  ...  PL  PNL
Linear
2nd order
susceptibility nonlinear
tensor
susceptibility
tensor
3rd order
nonlinear
susceptibility
tensor
i, j, k = x, y, z
Summation over
repeated indices
Symmetry and nonlinearity

Symmetry of susceptibility tensors
d ijk  dikj

 jkl 
permutation of the indices jkl
In lossless media, the nonlinear coefficients are invariant
under reshuffling of their indices


ijkl  ikjl 
The 2nd order susceptibility tensor of a lossless medium
contains 10 independent nonlinear coefficients
2nd order nonlinearity is absent in crystals with centrosymmetry as well as in glasses/liquids
Pi
 2
 Pi  2  2dijk    E j     Ek 

 2dijk  E j Ek   2
 
P

 i  2dijk    E j     Ek 
Centro-symmetry
 Pi  2  0
Symmetry and nonlinearity: Si vs. GaAs
Inversion center
Silicon: diamond structure
GaAs: zinc blende structure
3rd order nonlinearity only
2nd & 3rd order nonlinearity
Glass poling produces 2nd order nonlinearity
 ( 2 )  3EDC   ( 3 )
Glass bulk
DC (kV)
T < Tg
Thermal poling:
Apply a high voltage (EDC ~
106 V/m) at T < Tg and then
quench the sample to RT w/o
removing the electric field to
“freeze in” the polarization.
where the frozen-in EDC
results from movement of
alkaline ions
Material
d (2)
(pm/V)
Component
GaAs
181
d14
LiNbO3
8.8 / 5.8
d15 / d22
Ge-Sb-S
8.0
d22
Adv. Funct. Mater. 17, 3284-3294 (2007).
Opt. Express 14, 1524-1532 (2006).
Nonlinear optical effects

2nd order optical nonlinear effects





Pockels/electro-optic effect
Second harmonic generation (SHG)
Sum/difference frequency generation (SFG/DFG)
Optical parametric amplification/oscillation (OPA/OPO)
3rd order optical nonlinear effects





Optical Kerr effect/quadratic Pockels effect
Third harmonic generation (THG)
Four wave mixing (FWM)
Two photon absorption (TPA)
Stimulated Raman/Brillion scattering (SRS/SBS)
General methodology for nonlinear optics

Write the expression of electric field in medium

e.g. in an optical waveguide
E  Re[ E0 U ( x, y )  exp(ik  z  i  t )] 

1
E0 U ( x, y )  exp(ik  z  i  t )  c.c.
2
Calculate the linear and nonlinear polarization
P   0   E   ( 2 )  E 2   ( 3)  E 3  ...  PL  PNL

Substitute in to the electromagnetic wave equation
2
2 E
2
 E  0 2 ( 0 E  P )  0 0 2  0 2 PNL
t
t
t
2

Source term
Focus on the terms
with relevant frequencies
And then
go do the
math !
Pockels effect / Electro-optic (EO) effect


2nd order optical nonlinearity
Externally applied electric field
modifies the optical properties of
materials

Refractive index, bifringence, etc.
Total electric field: E  Elight  Eex
1
E0  exp(i  t )  c.c.  Eex
2
1
1
PNL   (2)  E 2   (2) { Re[ E02  exp(i 2  t )]  2 Eex Re[ E0  exp(i  t )]  Eex2  E0 E0*}
2
2
1
~  (2) {2 Eex  Re[ E0 exp(i  t )]  Eex2  E0 E0*} ~ 2  (2) Eex Elight (E0 << Eex)
2
 Re[ E0  exp(i  t )]  Eex 
Dielectric constant change
(2)
Polarization
 r oscillating
 1    2 Static
Eexpolarization
due to 2nd order nonlinearity
at the optical frequency
Electro-optic materials

Electro-optic
coefficient rij
 1 
  2    rij Eex, j
 n i
j
n3
 1 
n      2 
2
 n i
 (2) Eex
n3
    rij E j 
2 j
n
FOM 
n3 r
 static
MIT 6.731 Semiconductor Optoelectronics: Theory and Design
http://www.rle.mit.edu/sclaser/6.731F06/6.731_F_06.htm
Electro-optic modulators


Encodes 1/0 signal streams
onto an optical beam
Electro-optic materials:
LiNbO3, III-V, EO polymers
Mach-Zehnder
interferometer
E-O modulator
Modulator characteristics

Contrast/extinction ratio:
Ron / off 

Pout Voff

Insertion loss:
Loss 

Pout Von 
Pin  Pout Von 
Pin
Modulation bandwidth (speed)

Nonlinear/carrier dynamics

RC delay
3 dB bandwidth
3 dB
Modulation frequency (Hz)
A. Liu et al., Opt. Express 15, 660-668 (2007).
Second harmonic generation (SHG)


2nd order optical nonlinearity
Use light with wavelength l = l0 to
generate light with l = l0 / 2
 Frequency doubling  = 20
Energy is
conserved
Total electric field: E  Elight  Re[ E0  exp( i0  t )]
PNL  
(2)
E ~ 
2
(2)
 Re[ E exp( i 20  t )]
2
0
Radiating dipole with a frequency  = 20
Rigorous solution:
E1  E10  U1 ( x, y )  exp( ik1  z  1  t )
E2  E20  U 2 ( x, y )  exp( ik 2  z  2  t )
2  21
2 E
2
 E  0 0 2  0 2 PNL
t
t
2
D   dijkU 2iU1 jU1k
ijk

d
E1  i1 0  DE 2 E1*  exp[ i(2k1  k 2 ) z ]
dz
1
 1
d
E2  i2 0  DE12  exp[ i(2k1  k 2 ) z ]
dz
1 2
Phase matching condition

Only when 2k1 = k2 will SHG be efficient

n (l 1 ) = n ( l 2 )
~ 100% SHG conversion
efficiency is possible by
optimizing phase matching!
2k1 = k2
2k1 ≠ k2

General rule for parametric processes

SHG, SFG/DFG, THG, FWM

Momentum conservation
Green lightsaber ???
Sum frequency generation (SFG)
Difference frequency generation (DFG)


2nd order optical nonlinearity
Start with two beams ω = ω1
and ω = ω2



SFG: ω3 = ω1 + ω2 , k3 = k1 + k2
DFG: ω3 = ω1 - ω2 , k3 = k1 - k2
SHG
Laser emission
Sum frequency
Pump laser
SFG/DFG for photodetection

Use a 1060 nm laser to convert 10
μm mid-infrared radiation to 960
nm near-infrared radiation that can
be handled by low-cost detectors
"Mid-infrared single-photon counting,"
Opt. Lett. 31, 1094-1096 (2006).
Image courtesy of Institut
für Angewandte Physik
Optical Kerr effect
Third harmonic generation (THG)


3rd order optical nonlinear effects
3rd order optical nonlinearity is present in all materials
Total electric field: E  Elight  Re[ E0  exp( i  t )]
Optical Kerr effect: light-induced refractive index change
PNL   ( 3)  E 3 ~  ( 3) Re[ 3E02 E0*  exp( i  t )]
I 0  E0  E0 E0*
2
Consider the ω term
of the maginary part of non n  n2 I 0  I 0  Change
linear index: two photon absorption
Third harmonic generation (THG): frequency tripling
PNL   ( 3)  E 3 ~  ( 3) Re[ E03  exp( i 3  t )]
Consider the 3ω term
Two photon absorption (TPA)

Bimolecular process



Resonant enhancement of
nonlinear index n2


Absorption depends
quadratically on light intensity
Absorption coefficient TPA  I 0
TPA enhanced near Elight = Eg/2
Superior spatial confinement
of photo-physical and photochemical reactions

3-d patterning using TPAinduced polymerization
Optical Kerr nonlinearity Figure-of-Merit (FOM)
 Time scale of electronic Kerr effect: ~ 50 fs
Material
Nonlinear index
n2 (10-20 m2/W)
TPA
2 (10-12 m/W)
FOM
( n2/2l )
Silica (SiO2)
2.2
‒
‒
c-Si
440
8.4
0.4
a-Si
7400
41
1.4
a-As2S3
290
< 0.01
> 10
a-As2Se3
1200
1.0
2
Data quoted for l = 1550 nm: B. Luther-Davies et al., Opt. Express 15,
9205 (2007); K. Wang and A. Foster, Opt. Lett. 37, 1331-1333 (2012).
Stimulated Raman scattering (SRS)

3rd order optical nonlinearity

Scales with pump light intensity


Interaction of photons with phonons

Photon – phonon = Stokes line

Photon + phonon = anti-Stokes line
Resonant Raman scattering

When the virtual levels align
with a “real” energy level

Significant enhancement
of Raman scattering
Chandrasekhara
Venkata Raman
(1888-1970)
Raman spectra of materials
 Amorphous materials typically have broad Raman peaks
 Phonon energy dispersion
 Heavy atoms: low phonon energy and small Raman shift
Confocal Raman microscopy

Diffraction-limit spatial
resolution



Thin film characterization:
substrate selection
Excitation wavelength
selection

O. Hollricher, Confocal Raman microscopy teams high-resolution capabilities
with powerful materials analysis

Minimum sample thickness
~l
Avoid sample absorption
bands (fluorescence,
heating, signal attenuation)
Integration time
Raman study of single-wall carbon nanotubes


Radial breathing mode (RBM) and tangential mode
Determination of chiral vector (n, m) via Raman spectroscopy
 Nanotube radius: RBM mode peak position: RBM  1 rSWNT

Electronic density of states: resonant Raman spectroscopy
http://academic.pgcc.edu/~ssinex/nanotubes/
Z. Liu et al., Chem. Commun. 45, 6902-6918 (2009).
Si Raman lasers that make the headline

First silicon laser pulses with life

Gain: Raman amplification
Loss: free carrier absorption due to TPA
Solution 1: pulsed operation
 Pulse width << tcarrier << pulse period


O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269 (2004).
Si Raman lasers that make the headline



A continuous-wave Raman silicon laser
Loss: free carrier absorption due to TPA
Solution 2: reverse biased p-i-n diode

Sweep out free carriers generated by TPA
H. Rong et al., “A continuous-wave Raman silicon laser,” Nature 433, 725 (2005).
Si Raman lasers that make the headline



Loss: free carrier absorption due to TPA
What is your solution to the loss issue?
Solution 3: reduce free carrier life-time




Introduce deep level impurities: e.g. Au/Ag
Reduce waveguide size: enhanced surface
recombination
Increase recombination center defect
density
Solution 4?
?