### Phys 402: Nonlinear Spectroscopy: SHG and Raman Scattering

```Requirements: Polarization of Electromagnetic Waves
 General consideration of polarization
 How Polarizers work
Phys 402:
Nonlinear Spectroscopy:
SHG and Raman Scattering
 Representation of Polarization: Jones Formalism
Polarization of light and materials properties are important to
understand nonlinear effects !
Spring 2008
Andrei Sirenko, NJIT Lecture 2
1
2
Linear vs. Nonlinear Spectroscopy
G
|P|
G
G
P = ε0χ E
 Linear spectroscopy:
G
G
G
G
G
D = ε 0ε E = ε 0 (1 + χ ) E = ε 0 E + ε 0 P
⎡ε xx ε xy ε xz ⎤ ⎡ Ex ⎤
⎡ Dx ⎤
⎢
⎥ ⎢ ⎥
⎢ ⎥
⎢ Dy ⎥ = ε 0 ⋅ ⎢ε yx ε yy ε yz ⎥ ⋅ ⎢ E y ⎥
⎢ε ε ε ⎥ ⎢ E ⎥
⎢D ⎥
⎣ z⎦
⎣ zx zy zz ⎦ ⎣ z ⎦
 Nonlinear effects:
We will consider in details only SHG and Raman Scattering
3
G
|E|
G
G
P = ε0χ E
G
G G
P = ε 0 χ E + PNL
PNL = χ (2) E 2
G
( PNL )i = 2dijk E j Ek
G
|P|
G
G G
P = ε 0 χ E + PNL
G
|E|
Induced polarization vs. electric field in linear dielectric and in a
crystal without center of inversion, where electrons move in 4
asymmetric potential.
Microscopic understanding of nonlinearity
Electric field and Polarization
GG
G
G
E (r , t ) = E0 exp[i (kr − ωt )]
In vacuum
In a materials media
Electric field
G
|P|
 Electronic contribution to susceptibility (linear response)
G
G
P = ε0χ E
Displaced electronic cloud feels a restoring
force, which is linear (for small displacements)
Polarization
G
|E|
time
Electric field
For simplicity consider one-dimensional case (∆r parallel to x)
G
|P|
∆r
e
G
E
+
p
G
G G
P = ε 0 χ E + PNL
Polarization
time
G
|E|
5
6
Microscopic understanding of nonlinearity
Now, have electromagnetic wave with field E(t)=E0e-iωt
 Electronic contribution to susceptibility (nonlinear response)
Force F(t)=eE0e-iωt
Equation of motion becomes
(forced oscillator)
eEo e −iωt − mω02 x = m
d 2x
dt 2
and get
Look for a solution x(t)=x0e-iωt
em
em
x(t ) = 2
E e −iωt = 2
E (t )
2 o
ω0 − ω
ω0 − ω 2
Expect strong response (large x), ⇒ large susceptibility χ ⇒
large refractive index n at ω ≈ ω0
Dipole moment p = qx, so polarization P = eNZx
(N atoms per unit volume, Z electrons per atom) ⇒
P=
e 2 ZN m
ω02 − ω 2
E
Electron is moving in an asymmetric potential with damping
NZe 2
1
Recall P = ε0χE and get χ (ω ) =
2
ε 0 m (ω07 − ω 2 )
Linear response:
∂ 2 x(t )
∂x(t )
e
+γ
+ ω02 x(t ) + Dx 2 (t ) =
Eo e − iωt
2
∂t
∂t
2m
x – deviation from potential minimum
mDx 2 (t ) Anharmonic restoring force
Linear response:
∂x(t )
Damping
γ
∂t
x(ω , t ) = q1e − iωt + q2 e − i 2ωt
Solution:
(
q1 =
q2 =
eE0
1
⋅
m ω02 − ω 2 + iγ ⋅ ω
− De E
2
2
2
0
2m ⎡⎣ω − ω + iγ ⋅ ω ⎤⎦ ⎡⎣ω02 − (2ω ) 2 + iγ ⋅ 2ω ⎤⎦
2
2
0
2
)
NonLinear
response:
(second harmonic
8
generation)
Why nonlinear effects are usually weaker than linear ones?
For NL polarization at the second harmonic frequency:
(2)
P 2ω = Neq2 e− i 2ωt = χ NL
⋅ E02 e − i 2ωt
V ( x)
For correct power consideration we need to take the complex
conjugate part of the electromagnetic wave
P 2ω =
r0
x
1
1
Neq2 (e− i 2ωt + e+ i 2ωt ) = d NL ⋅ E02 (e − i 2ωt + e+ i 2ωt )
2
2
x << r0
For nonlinear susceptibility we have:
mD ( χ L (ω ) ) ⋅ χ L (2ω ) ⋅ ε 03
=
2 N 2 | e3 |
2
χ
(2)
NL
−3e2
; r0 ≈ 0.5 nm
D≈
ε 0 mr04
m 2 m 3
e 2 −5.83
x2
x3
V ( x) = x + Dx =
(
+ 24.1 2 − 13.3 3 + ...)
2
3
4πε 0
r0
r0
r0
Why nonlinear effects are weaker than linear effects?
9
Symmetry of nonlinear susceptibility tensor
G
G G
| PNL |= χˆ (2) ⋅ | E1 ⋅ E2 |
G
( PNL )i = 2dijk E j Ek
Two-wave mixing
General case of two-wave mixing:
d13 d14 d15
d 23 d 24 d 25
d33 d34 d35
⎡E
⎤
⎢ 2
⎥
⎢ Ey
⎥
d16 ⎤ ⎢ 2
⎥
Ez
⎥
⎢
⎥
d 26 ⎥ ⋅
⎢ 2 Ez E y ⎥
d36 ⎥⎦ ⎢
⎥
⎢ 2 Ez Ex ⎥
⎢
⎥
⎣ 2 Ex E y ⎦
Wave propagation along z;
i, j, k index are permutations of x and y coordinates
dE1i
σ µ
µ
=− 1
E1i − iω1
d ' E E * e − i ( k3 − k2 − k1 ) z
2 ε1
ε1 ijk 3 j 2 k
dz
For cubic, tetragonal, and orthorhombic crystals:
⎡0 0
⎢
⎢0 0
⎢⎣0 0
0 ⎤ ⎡0 0
0 d 25 0 ⎥⎥ = ⎢⎢0 0
0
0 d36 ⎥⎦ ⎢⎣0 0
0 ⎤
0 d ' 0 ⎥⎥
0
0 d "⎥⎦
0 d14 0
0 d' 0
0
0
0
0
ω1 + ω2 = ω3
(ω3 = ω1 + ω2 )
G G G
k1 + k2 = k3
G
G
G
∂2E
∂E
2
∇ E = εµ 2 + σµ
Wave equation for linear process
∂t
∂t
G
Using Laplace operator:
G
G
2
G
∂ 2 PNL
∂ E
∂E
2
Wave equation for nonlinear ∇ E = εµ 2 + σµ
+µ
t
t
∂
∂
∂t 2
process:
2
x
⎡ Px ⎤ ⎡ d11 d12
⎢ ⎥ ⎢
⎢ Py ⎥ = ⎢ d 21 d 22
⎢ P ⎥ ⎢⎣ d31 d32
⎣ z⎦
10
dE *2 k
σ
=− 2
2
dz
dE3 j
11
dz
=−
σ3
2
µ *
µ
E − iω2
d ' E E * e−i ( − k + k
ε 2 2k
ε 2 ijk 1i 3 j
3
µ
µ
E − iω3
d ' E E e−i (− k +k
ε3 3 j
ε 3 ijk 1i 2 k
3
2 + k1 ) z
2 + k1 ) z
12
Second-harmonic generation
SHG: ω + ω = 2ω (ω3 = 2ω1 )
G
G
2kω = k2ω
Second-harmonic generation
dE1i
Small loss of power in the primary beam
≈0
dz
dE3 j
µ
= −i 2ω
d ' E E e − i∆kz
dz
ε ijk 1i 1k
e − i∆kz − 1
µ
d 'ijk E1i E1k
i ∆k
ε
E3 j ( z ) = −i 2ω
Power (2ω ) = E3 j ( L ) E *3 j ( L) = 4
Coherence length:
l=
Phase matching requirement:
⎡ε xx ε xy ε xz ⎤ ⎡ n 2 e 0 0 ⎤
⎥
⎢
⎥ ⎢
ε = ⎢ε yx ε yy ε yz ⎥ = ⎢0 n 2O 0 ⎥
⎢ε ε ε ⎥ ⎢0 0 n 2 ⎥
O⎦
⎣ zx zy zz ⎦ ⎣
G
G
2kω = k2ω
⎞
ε
NZe 2 ⎛
1
= Re 1 +
⎜
⎟
ε0
ε 0 m ⎝ ω02 − ω 2 + iωγ ⎠
Strong absorption
nO (ω )
ne (2ω )
G
ω
2ω
2⋅ | kω |= 2n (ω ) = n (ω )
c
c
G
2ω
| k2ω |= n (2ω )
c
n(ω ) ≠ n (2ω ) !
⇒
G
G
2kω = k2ω
ω
2ω
ω0
ω
Blue waves propagate with the same velocity in the crystal
13
Experimental setup for Second-harmonic generation
G
G
2kω = k2ω
ne (2ω ) = nO (ω )
n (ω )
⇒
2
If in birefringent crystal ne (2ω ) = nO (ω )
ω + ω = 2ω
nR (ω ) = Re
Transparent crystal
µ 2
sin ( ∆kL / 2)
ω ( d 'ijk ) 2 E 21i E12k
ε3
(∆kL / 2) 2
2π
∆k
If in birefringent crystal ne (2ω ) = nO (ω )
KH 2 PO4
KDP crystal
14
Applications of Second-harmonic generation
 Lasers (Nd:YAG, second harmonic)
 Coherent anti-Stokes Raman scattering
 Bio-imaging
 Materials Physics
Fine tuning of refractive index for
Phase matching in uniaxial
Nonlinear crystals:
 Solar Physics
 Quantum cryptography (two-wave mixing)
1
cos θ sin θ
=
+ 2
2
n(θ )
nO 2
ne
⎡n2e 0 0 ⎤
⎥
⎢
ε = ⎢0 n 2O 0 ⎥
⎢0 0 n 2 ⎥
O⎦
⎣
15
16
Brillouin and Raman spectroscopy
Raman scattering in crystalline solids
Inelastic light scattering mediated by the electronic polarizability of the medium
• a material or a molecule scatters irradiant light from a source
• Most of the scattered light is at the same wavelength as the laser source
(elastic, or Raileigh scattering)
• but a small amount of light is scattered at different wavelengths
(inelastic,
or Raman scattering)
α
β
ћωi
β
α
ћΩ
ћωs
Stokes
ћωi
0
0
Raileigh
ћΩ
I
=ωi = =ω s ± =Ω;
λi
λi ~ 5000 Å, a0 ~ 4-5 Å ⇒ λphonon >> a0
ω
Stokes
Raman
Scattering
ωi- Ω(q)
1. Energy conservation:
2. Momentum conservation:
4πn
ki = k s ± q ⇒ 0 ≤ q ≤ 2 k ⇒ 0 ≤ q ≤
Elastic
(Raileigh)
Scattering
ћωs
AntiStokes
Not every crystal lattice vibration can be probed by Raman
scattering. There are certain Selection rules:
ωi Anti-Stokes
ks
q≈0
ki
ks
q ≈ 2k
ki
ks
q
ki
⇒ only small wavevector (cloze to BZ center) phonons are seen in
the 1st order (single phonon) Raman spectra of bulk crystals
Raman
Scattering
ωi+ Ω(q)
Analysis of scattered light energy, polarization, relative intensity
provides information on lattice vibrations or other excitations17
3. Selection rules determined by crystal symmetry
18
Example of Raman scattering in crystalline solids
Raman scattering in crystalline solids
Phonon Energy
Raman
scattering
G
G G
G
q = ±∆k = ± | ki − ks |
3S = 15 modes
3 acoustic modes
12 optical modes; 3 × 4
Mandelstam-Brillouin
scattering
Phonon wavevector
19
2 × TO1 + LO1
2 × TO2 + LO2
2 × TO3 + LO3
2 × TO4 + LO4
20
```