Introduction – Nonlinear optics
 A few examples of nonlinear devices
 This lecture: Nonlinear guided wave optics in dielectrics
 Produces a variety of exotic effects
 Modification of the « color » of a light beam
-Frequency doubling, frequency conversion
-Supercontinuum generation
 Amplification of optical signals/ OPO
 Temporal and Spatial control of light
-Generation of ultra-short pulses
-Self-focusing, solitons
 Switching in Telecommunication systems
 Electro-optic effet
 All-optical information processing
…
P=235 W, τ=35 ps, Lfiber=100 m
Introduction – Nonlinear optics
 Origin of nonlinear effects: Response of dielectric media becomes nonlinear
when subjected to intense EM fields. So nonlinearity is a material property
 One signature of this nonlinearity is the emission of new optical frequencies
 First nonlinear experimental demonstration: Second Harmonic
Generation (SHG) by Franken et al. in 1961
P.A. Franken, A.E. Hill, C.W.
Peters and G. Weireich, Phys.
Rev. Lett. (1961)
λ [10nm]
Introduction – Nonlinear optics
Nonlinear optics: various novel phenomena arising from the interaction
btw. coherent & intense light beams + matter  Variety of applications
Introduction – Nonlinear optics
Linear medium
Linear & Nonlinear optics : a property of the medium
• Optical properties (e.g. refraction & absorption) independent of the optical
intensity
• Superposition principle
• υ unchanged : light frequency not modified by propagation in the medium
(only velocity changes)
• Two light beams do not see each other  No light-light interaction
Nonlinear medium
• The speed of light in the medium (i.e. refractive index) changes with intensity:
n=f(I)
• Superposition principle no longer valid
• Modification of υ : frequency (color) can change when light propagates through
a nonlinear optical medium
• Light can control light: photons interact (via the medium)
The material optical properties are modified by the light beam
Learning objectives of Lecture 11 & 13
 Understand the origin of nonlinear effects; know the distinction between 2nd
and 3rd order nonlinearities as well as the kind of materials in which they occur
 Be able to describe the main phenomena associated with 2nd order
nonlinearities: SHG, EO effect, three-wave mixing etc…
 Be able to describe the main mechanisms associated with 3rd order
nonlinearities: SPM, all-optical control of light, four-wave mixing, THG etc…
 Be able to explain how nonlinear effects can be enhanced in nanophotonic
devices (light confinement,…) and guided-wave optics (γ parameter)
 Be able to give the two conditions upon which nonlinear effects become
significant and derive them for specific phenomena : namely the photon
energy conservation and phase matching condition…
 Understand the interpretation of nonlinear effects via photon interactions
Outline of Lecture 13 &15 on Nonlinear optics
I.
Nonlinear optical media
I.1 Material response : harmonic oscillator – Linear/ Nonlinear regime
I.2 Nonlinear polarization
I.3 Wave equation in a nonlinear medium
I.4 Generation of new frequencies in nonlinear media
II.
2nd order Nonlinearities
II.1 Second-harmonic generation
II.2 Electro-optic effect
II.3 Three wave mixing
II.4 Phase matching
III.
3rd order Nonlinearities
III.1 Third-Harmonic generation
III.2 Optical Kerr Effect
III.3 Four-wave mixing
III.4 Application to switching
IV.
Solitons: dispersion + NLO
spatial and temporal solitons
V.
Coupled mode theory (BE for ECL 3rd year students)
I. Nonlinear optical Media
Reminders from Lecture 1: Lorenz model (Harmonic oscillator)
 Incident electrical field displaces the electronic cloud around fixed nuclei
 Charge separation  Dipolar momentum (-ex)
 Polarization: Σ all dipolar momentum per volume unit
Classical Model
Bound electrons = Dampened harmonic oscillators
x
e-
r
r
uur
d x
dx
2r
me 2 + meγ e
+ meωe x = −eE0 u x
dt
dt
2
 Dipolar momentum:
r
E0
ur
r
E = E0u x
-
+
-
+
 Polarization (density)
-
+
vector (collective effect)
with Ne electron density
-+
-+
I. Nonlinear optical media
-+
ur
r
p = −ex
r
r
P = − Nee x
I.1 Material nonlinear response
NL behaviour of the material under the influence of an intense EM field
Dipole oscillation under the action of an EM field
 Small oscillations:
dipole follows the oscillations of E
r
r
v
p = −ex = αE
(spring constant ∝ elongation)
Polarization P ∝ Incident electric field E
r
r
P = ε 0 χE
Linear Medium
ε0 – vacuum permittivity, 8.85x10-12 As/Vm
χ- Complex susceptibility of the medium
- Real part : medium refractive index
- Imaginary part : absorption of the medium
Linear dielectric medium
 Large oscillations: dipole cannot follow the oscillations of E
 P not ∝ E: Nonlinear relationship btw P and E
Medium becomes nonlinear
Nonlinear dielectric medium
8
I. Nonlinear optical media
I.2 Nonlinear polarization
In general: External electric field (E) << Interatomic Fields (105 to 108 V/m)
Thus nonlinearities are weak
r
r
 Expansion of P = ε 0 χE in a Taylor series around E=0
1
1
2
P = a1 E + a2 E + a3 E 3 + ...
2
6
a1, a2 and a3 : 1st, 2nd and 3rd derivatives of P(E) at E=0
Nonlinear dielectric medium
Usual expression of the polarization density in the nonlinear regime
P = ε 0 χE + 2dE 2 + 4 χ (3 ) E 3 + ...
χ (=a1/ε0)
ε
= 1+ χ
Linear susceptibility with n =
ε0
d (=a2/2)
2nd order nonlinearity
χ(3) (=a3/24)
3rd order nonlinearity
2
I. Nonlinear optical media
I.2 Nonlinear polarization
Nonlinear Polarization density response
P = ε 0 χE + 2dE 2 + 4 χ (3 ) E 3 + ...
or
r
r
r
r
(3) 3
(1)
(2 ) 2
P = ε0 [ χ E + χ E + χ E + L] = PL + PNL
χ(2) ∼ 10-13 - 10-10 m/V
d∼
10-24
-
10-21
A.s/V2
liquid crystals, SC, organic materials
χ(3) ∼ 10-23 -10-18 SI
•
Subsequent Terms lower and lower and require more and more intense EM fields
•
Estimation of d and χ(3) : for electric fields ~ 108 V/m (interatomic)
 1st (χE) and 2nd order terms (dE2) comparable
•
In centrosymmetric materials, 2nd
order nonlinearity (d) disappears
and χ(3) dominates
I. Nonlinear optical media
I.3 Wave equation in a nonlinear medium
Reminders: Maxwell’s Equations and wave equation
Maxwell’s Equations in ra linear dielectric medium
r
r r
r
r
∂B
∂D
∇∧E =−
∇∧H =
∂t
∂t
r r
r r
∇. B = 0
∇. D = 0
D = ε0 E + P
B = µ0 H + µ 0 M
Consequence: Wave equation – propagation at a speed reduced by n
r
v
2
2
ur 2 r 1 ∂ E
∂ P
∇ E − 2 2 = µ0 2
c0 ∂t
∂t
c0 =
r
r
P = ε 0 χE

P proportional to E
r
ur 2 r 1 ∂ E
∇ E− 2 2 =0
c ∂t
2
(isotropic homogeneous linear media)
1
ε0µ0
c = c0 / n
P-E relationship characteristic of the medium
E(r,t)
( )
∇ ∧ (∇ ∧ a) = ∇ ∇.a − ∇ 2 a
I. Nonlinear optical media
Medium
P(r,t) = f (E)
I.3 Wave equation in a nonlinear medium
Propagation of the EM field in a nonlinear material
Nonlinear dielectric Material : Nonlinear wave equation
r
r
2
ur 2 r 1 ∂ E
∂ P
∇ E − 2 2 = µ0 2
c0 ∂t
∂t
r
r r
P = ε 0 χE + PNL
2
linear
nonlinear
r
r2
r
(3 ) 3
PNL = 2dE + 4 χ E + ...
Inhomogeneous Wave equation
r
ur 2 r 1 ∂ E
r
∇ E − 2 2 = −S
c ∂t
2
Interpretation
r
r
∂ PNL
S = − µ0
∂t 2
2
(Isotropic homogeneous media)
Radiation source (accelerated charges) in a material with
index n  PNL produces radiations at new frequencies
Two approaches for (approximatively) solving this differential nonlinear equation for E
 Born Approximation (this lecture → Qualitative estimation of the new fields created)
 Coupled wave theory (see BE for 3A ECL → Estimation of conversion efficiency)
I. Nonlinear optical media
I.4 Generation of new frequencies in NL media
Born approximation
S
radiation
E
S(E)
Iterative process : E 0
→
S (E 0 )
→
E1
S (E1 )
→
→L
1st Born approximation
1st Born approx valid if E-field intensity is not too strong (weak NL)
Propagation in a NL material = scattering of the incident EM field E0 by the medium
• From E0 → evaluation of PNL → Radiation source S(E0)
• E1 calculated from the radiation source by adding to the incident field the contribution
from the field scattered by the different point sources in the medium
E1
E0
Incident
Field at ω
Radiated field
at ω and ωi
S(E0)
Radiation Source (NL function  new frequencies ωi)
I. Nonlinear optical media
Assumption for
the following:
Isotropic
medium
r
E=E
I.4 Generation of new frequencies in NL media
Nonlinear polarization = a radiation source

Nonlinear response of the medium  each atom develops a dipolar momentum
oscillating at new frequencies

Isolated atom radiates at this new frequency with a radiation diagram

In a medium: N atomic dipoles, each oscillating with a specific phase as driven
by the incident field

If the dipoles oscillate with a correct relative phase shift – the fields radiated by
each dipole adds up constructively in the direction of propagation
Resultant Electric field α N
Resultant Intensity α N2
Light interact with light via the material
I. Nonlinear optical media
Outline of Lecture 14 &16 on Nonlinear optics
I.
Nonlinear optical media
I.1 Material response : harmonic oscillator – Linear/ Nonlinear regime
I.2 Nonlinear polarization
I.3 Wave equation in a nonlinear medium
I.4 Generation of new frequencies in nonlinear media
II.
2nd order Nonlinearities
II.1 Second-harmonic generation
II.2 Electro-optic effect
II.3 Three wave mixing
II.4 Phase matching
III.
3rd order Nonlinearities
III.1 Third-Harmonic generation
III.2 Optical Kerr Effect
III.3 Four-wave mixing
III.4 Application to switching
IV.
Solitons: dispersion + NLO
spatial and temporal solitons
V.
Coupled mode theory (BE for ECL 3rd year students)
II. 2nd order optical nonlinearities
• Optical properties of a 2nd order nonlinear material
(Negligible higher order nonlinearities)
PNL = ε 0 χ (2 )E 2 = 2dE 2
PNL = 2dE
PNL frequency components
 Components from the source S
 Components from the radiated field
• Associated phenomena
–
–
–
–
Second Harmonic Generation (SHG)
Optical Rectification
Electro-optic Effect
Three-wave mixing
II. 2nd order optical nonlinearities
2
(non centrosymmetric
material)
E-field contains one or
several frequency
components
II.1 Second harmonic generation
Incident EM Field: monochromatic E at ω and complex amplitude E(ω)
~
E (t ) = E0 cos(ωt + ϕ )
(E (ω ) = E0e jϕ )
1
~
Using complex expression
E (t ) = E (ω )e jωt + c.c. = Re E (ω )e jωt
c.c. complex conjugate
2
~
~2
PNL = 2dE (t )
1
~
1

jω t
*
− jω t
PNL = 2d  E (ω )e + E (ω )e
* E (ω )e jωt + E * (ω )e − jωt 
2
2

2 − 2 jωt
~
1 2

2 jω t
*
PNL = 2d  E (ω )e
+ E (ω ) e
+ 2 E (ω )E * (ω ) 
4

 E 2 (ω )e 2 jωt + c.c.

~
*
+ E (ω )E (ω )
PNL = d 
2


~
PNL (2ω ) = dE (ω )E (ω )
PNL = Re{PNL (2ω )e 2 jωt }+ PNL (0 ) with
[
]
{
[
[
}
] [
(
)
]
]
PNL (0 ) = dE (ω )E * (ω )
II. 2nd order optical nonlinearities
II.1 Second harmonic generation
 Incident EM field:
E (ω )eiωt
{
~
~2
PNL = 2dE (t ) = PNL (0 ) + Re PNL (2ω )e 2 jωt
DC component
E (ω )eiωt
}
2ω component
PNL (0 ) = dE (ω )E * (ω )
PNL (2ω ) = dE (ω )E (ω )
Field at ω (linear)
Field at 2ω
Electrostratic field
2nd order NL
nonlinear
 Ouput: Continuous electric field (DC) + EM Field at 2 × incident frequency (SH)
P
PNL(t)
PNL0(t)
PNL2ω(t)
=
E
t
Optical Rectification
t
II. 2nd order optical nonlinearities
+
t
t
II.1 Second harmonic generation

∂ 2 PNL
e 2 iωt
2
(
)
S(t ) = − µ 0
=
4
µ
ω
P
2
ω
+
c
.
c
.
0
 NL
 = S (2ω ) cos(2ωt )
2
∂t 2


Radiation source S (2ω ) = 4 µ 0ω 2 dE 2 (ω )
I (2ω ) ∝ S (2ω )
2
 Intensity of the field scattered at 2ω ∝ d2I2 with I= incident EM field intensity (∝ E2)
 Conversion efficiency from ω to 2ω ∝ I = P/A [W/m2]
 Conversion to
another range of
frequencies
 Efficient in the
right conditions
P = incident power at ω
A = Cross-section area
2ω
ω
Rubis 694 nm
347 nm
KDP crystal
2ω
ω
YAG 1,06 µm
Optical fiber (Ge or P doped)
ω
2ω
Laser AlGaAs
II. 2nd order optical nonlinearities
780 nm
390 nm
530 nm
II.1 Second harmonic generation
2ω
ω
YAG 1,06 µm
II. 2nd order optical nonlinearities
Ge or P doped optical fiber
530 nm
II.2 Electro-optic effect
Incident field = EM Field @ optical frequency ω
+ Electrostatic Field EDC E~ (t ) = E DC + Re E (ω )e jωt
{
{
}
Static field
{
Optical field
~
~2
PNL = 2dE (t ) = PNL (0 ) + Re PNL (ω )e jωt + Re PNL (2ω )e 2 jωt
(
2
PNL (0 ) = d 2 E DC
+ E (ω )
PNL (ω ) = 4dE DC E (ω )
PNL (2ω ) = dE (ω ) 2
2
)
}
Static component (continuous)
Component ω
Component 2ω
Bias voltage DC
E (ω )eiωt
2nd order NL
II. 2nd order optical nonlinearities
Field at ω (linear)
Field at ω
Field at 2ω
Electrostatic Field
nonlinear
}
II.2 Electro-optic effect
If E (ω ) << E DC
2
2
PNL(2ω) << PNL(ω) << PNL(0)

4dE DC 
Total polarization (L + NL) at ω : P (ω ) = ε 0 χE (ω ) + 4dE DC E (ω ) = ε 0 E (ω )  χ +

ε
0


P (ω ) = ε ∆χE (ω ) (≡ linearized P versus E)
NL
NL
0
Associated index variation
n = 1+ χ
 2n∆n = ∆χ
2
∆χ 2dE DC
∆n =
=
2n
nε 0
χeff= χ+∆χ
Refractive index
varies with EDC
 Calculation without dispersion
PNL=2dE2
Effective linear medium whose index
is linearly controled by electric field
PNL(0)
t
+
E
EDC
Ε(ω)
t
II. 2nd order optical nonlinearities
EDC
-
Ε’(ω)
II.2 Electro-optic effect
Application of the electro-optic effect : Light intensity modulator
∆n =
L
E In + EIn e jϕ
EOut =
2
*
POut ∝ EOut EOut
POut
POut
Return to zero
data format
PIn
[cos(ϕ ) + 1]
=
2
PIn
[cos(π V Vπ ) + 1]
=
2
II. 2nd order optical nonlinearities
2dE DC
nε 0
ϕ = k0 ∆nL
II.3 Three-wave mixing
Incident field = 2 monochromatic waves with optical frequencies ω1 & ω2
~
E (t ) = Re E (ω1 )e jω1t + E (ω 2 )e jω2t
~
~
PNL = 2dE 2 (t ) = PNL (0 ) + Re{PNL (2ω1 )e j 2ω t + PNL (2ω 2 )e j 2ω t }...
{
}
1
(
{
2
+ Re PNL (ω1 + ω 2 )e j (ω1 +ω2 )t + PNL (ω1 − ω 2 )e j (ω1 −ω2 )t
PNL (0 ) = d E (ω1 ) + E (ω 2 )
2
PNL (2ω1 ) = dE (ω1 )E (ω1 )
PNL (2ω 2 ) = dE (ω 2 )E (ω 2 )
2
)
}
Static component
Component at 2ω1
Component at 2ω2
PNL (ω1 + ω 2 ) = 2dE (ω1 )E (ω 2 ) Component ω1 + ω2 (up conversion)
PNL ( ω1 − ω 2 ) = 2dE (ω1 )E * (ω 2 ) Component at ω1 - ω2 (down conversion)
E (ω1 )e
jω1t
E (ω2 )e
jω 2 t
2nd order NL
II. 2nd order optical nonlinearities
ω1
ω2
2ω1
2ω2
ω1−ω2
ω1+ω2
Electrostatic field
linear
nonlinear
II.3 Three-wave mixing
 In practice : even if NL polarisation term comprises 5 frequencies (0,2ω1,2ω2,
ω1+ω2, ω1-ω2), the corresponding waves are actually generated only if dipoles
interfere constructively
 “Phase matching” condition (see later)
 Typically, generation of only another frequency component (for which
phase matching condition met)
EXAMPLE – Frequency conversion : up-conversion (ω1 + ω2 )
YAG 1.06 µm
ω1
CO2 10.6 µm
ω2
II. 2nd order optical nonlinearities
Proustite
crystal
ω3 = ω1 + ω2 (up-conversion)
II.3 Three-wave mixing
ω3
+ NL medium  ω2 = ω3 - ω1
ω1
3 interaction processes
ω1
+ NL medium  ω3 = ω1 + ω2
ω2
ω3
+ NL medium  ω1 = ω3 - ω2
ω2
Phase matching for the 3 processes
APPLICATIONS
Sum Frequency
Conversion (SFG)
signal ω1
ω3 = ω1 + ω2
pump ω2
NL Crystal
pump ω3
Converted Signal
filter
ω3
Parametric amplifier
signal ω1
ω1, ω2
NL Crystal
ω1 Amplified signal
ω2
filter
Parametric oscillator
pump ω3
ω2
ω1
II. 2nd order optical nonlinearities
ω1
NL Crystal
Feedback
II.3 Three-wave mixing
OPO (Optical Parametric Oscillators)
pump ω3
(laser)
M3
ω2
crystal
OPO
M2
M1
Wide (new) range of λ (UV-IR) and temporal characteristics
(cw and fs)
OPO versus diodes laser
II. 2nd order optical nonlinearities
Wavelength [µm]
II.4 Phase matching condition
Example for Three-wave mixing
Wave description
 Electric field of each dipole defined both temporally and spatially:
(
r r
r
Ei (t ) = Ei 0 cos ωi t − ki • r
)
for plane waves with amplitude Ei0, frequency ωi and wavevector ki
 The field radiated by each dipole must be in phase with the polarization
(source function) for them to add up constructively
ω3 = ω1 + ω 2 Frequency matching
r r r
k3 = k1 + k 2 Phase matching
For SHG
II.
2nd
order optical nonlinearities
II.4 Phase matching condition
Example for Three-wave mixing
E (ω1 ) = A1e
r r
− j k1 ⋅ r
PNL (ω3 ) = 2 dE (ω1 )E (ω2 ) = 2 dA1 A2 e
r r
− jk 2 ⋅ r
E (ω 2 ) = A2 e
Medium acts as a source
generating a wave at frequency
ω3 and wavevector k3 such that
Wave description
r r r
k3 = k1 + k 2 Phase matching (spatial)
ω3 = ω1 + ω2 Frequency matching (temporal)
k2
k1
r r r
− i ( k 1 + k 2 )⋅ r
Interaction over long
distance and for a long time
k2
k3
k1
k3
Ex: For waves in the same direction, phase matching : n
ω3
c
=n
ω1
c
+n
ω2
c
⇔ ω3 = ω1 + ω2
Frequency matching ensures phase matching?!
n3ω3 = n1ω1 + n2ω 2 Phase matching is an independent condition on the n dispersion!
II. 2nd order optical nonlinearities
II.4 Phase matching condition
Example for Second harmonic generation
ω SHG = 2ω  *Phase matching
Phase matching through birefringence
ne
Refractive index
no
r
r
k SHG = 2kω
requires
ω ), θ< )n=
o (2
nneo ((ω
nωo ()2ω )
ω
2ω
E1
Frequency ω
no2
Optical axis
r
E1
θa
Birefringent crystal
II. 2nd order optical nonlinearities
no1
r
E2
ne1(θ)
θa
ne2(θ)
II.4 Phase matching condition
Example for Second harmonic generation
Phase mismatch
∆k = k 2ω − 2kω = 2
ω
c
Phase matching
condition: ∆k = 0
(n2ω − nω )
The coherence length is defined as the length across
which the phase shift between P2ω and E2ω is π.
∆k .Lc = π
 Lc =
λ
4(n2ω − nω )
Phase mismatch   Distance L beyond which conversion efficiency
starts decreasing becomes shorter
I 2ω
ω 2 d 2 L2
2
2
=
I
sin
c
( ∆kL )
ω
3
2
2 ( ε 0 c ) nω n2ω
1
Perfect phase
matching
2π
L
0.5
(Coupled wave theory)
0
0
II. 2nd order optical nonlinearities
2π
L
∆k
II.4 Phase matching condition
Example for Second harmonic generation
SHG crystal
Output
Input
SHG crystal
I 2ω
ω 2 d 2 L2
=
Iω2 sin c 2 ( ∆k .L )
3
2
2 ( ε 0 c ) nω n2ω
II. 2nd order optical nonlinearities
II.5 Quantum description of nonlinearities
Nonlinearities : description of photon interaction
hω
Ex: Second harmonic generation
ω
2hω
ω
Nonlinear crystal
hω
2ω
hω
hω3 = hω1 + hω2
 Photon energy conservation
r
r
r
 Photon momentum conservation hk = hk + hk
3
1
2
Three wave mixing
hω2
hω1
hω3
hω2
II. 2nd order optical nonlinearities
hω3
hω1
Virtual state
Virtual state
2hω
Ground state