# Lecture 4

## Pełen tekst

(1)

### Lecturer:

Royal Institute of Technology (KTH) Sweden, bj@kth.se

### Introduction to nonlinear optics

Lectures co-financed by the European Union in scope of the European Social Fund

(2)

Saleh Ch.21

(3)

(4)

### Where from the nonlinear effects come?

In contrast to electrons photons can only interact through polarization of the medium

(5)

Gatech

(6)

### Origin of nonlinear effects

Gatech

Green light obtained by illuminating the crystal with infrared light

(7)

### Invention of the first lasers

(1960) Ali Javan The first He-Ne Laser

(1960) Theodore Maiman Invention of the first Ruby Laser

(8)

### Birth of nonlinear optics

The advent of the laser as an intense, coherent light source gave birth to nonlinear optics

Optical Second-Harmonic Generation – the first nonlinear effect observed with coherent input generating coherent output

Peter Franken

(9)

### First theoretical description of nonlinear phenomena

Rem Khokhlov Sergey Akhmanov

Nicolas Bloembergen

(10)

### Induced polarization – medium response

Polarization is a driving term for the wave equation:

2 2 2

2 2 2 0 2

1

E E P

x c t

dt

  

 

 

0

0

(11)

) 3 ( )

3 ( 0 )

2 (

0

### P

NL - nth order susceptibility tensor

### P

L

Note alternative definition in some books (e.g. Saleh) :

### Generally

((1)E >> (2)E E >>(3)E E E )

(2)

(3)

)

( n

(1) (2) (3)

0

### P    

In centrosymmetric media:

(12)

### Wave mixing by 2nd order nonlinearity

Let us look at polarization frequecies generated by a sum of two waves:

Can light at those frequencies be efficiently generated?

Answer: only if the conservation laws are fullfiled – next slide

## 

2 2 2

1

) (

)

* ( 2 1

) (

) (

2 1

2 2 2

2 2

2 2 1 ) 2 (

2 1 2

1

2 1 2

1

2 2

1 1

x k k i t i

x k k i t i

x k i t i x

k i t i

### 

2nd-harmonic generation

Sum-frequency generation

Difference-frequency generation

dc rectification

### 

Eei te ik x E ei te ik x E e i teik x E e i teik x

### 

x k t E

x k t E

E 1 1 1 2 2 2 1 1 1 2 2 2 1* 1 1 2* 2 2

2 ) 1 cos(

)

cos(

 

 

(2)

(13)

*

1 2 3 4 5

1 2 3 4 5 sig

1 2 3 4 5 sig

### k  k  k  k  k  k N-wave-mixing (or N-photon) mixing

N - number of photons involved (including the emitted one)

(2) can mix 3 waves, (3) - 4 waves, etc

- contribute to complex conjugate of their field Absorbed photons

Emitted photons

### (

ħ’s are cancelled

### )

Typically, ksig DOES NOT correspond to a light wave at frequency ωsig !

Satisfying these two relations simultaneously is called "phase-matching"

Energy must be conserved:

Momentum must also be conserved:

Automatic !

(14)

Frequency

Refractive index

sig

2

2

sig

### 

”Tricks” to inforce phase matching will be discussed later

(15)

### 2nd order NL tensor – contracted notation

Due to intrinsic permutation symmetries one can reduce the NL polarization tensor to its contracted form. For the 2nd order polarization it is:

K

iK jK

K j k

j ijk k

j ijk

i

(2)

(16)

is the

### Units of d (F/m or m/V)

http://phys.strath.ac.uk/12-370/sld018.htm

(17)

### Three-Wave Mixing - mathematical description

Maxwell equations:

Wave equations:

Consider interaction of three harmonic fields via (2) 2dijk with 3 = 1  2

Express each of the fields as:

Unit polarization vectors

Crystal axes

Mix the total field:

to obtain nonlinear polarization: PiNL=

Group terms at different frequencies:

(18)

### Coupled equations for Three Wave Mixing (1)

Substituting nonlinear polarizations to the wave equation, and applying Slowly Varying Amplitude Approximation:

Energy is conserved:

(19)

### Coupled equations for Three Wave Mixing (2)

Introducing variables: n-index of refraction, one obtains:

Momentum mismatch

Coupling coefficient:

∞ photon flux

s i

p c

###  1 

Interaction most efficient for: Δk=0 (phase matching)

Collinear propagation - large field overlaps and large interaction lengths Constructive interaction only possible over coherence length Lc:

c

(20)

### 

Photon flux density

(21)

1

3

2

1

1

3

2

1

3

2

1

3

2

3

1

2

3

1 )

2

3

###  

1

Strong pump laser at

### 

3 amplifies a weak, phase matched, signal at

1

(22)

nd

*

0 0

2 2 2 *2

0 0 0

Since

(23)

2 3 1 2

1

kz

i

2

3

i k L

kL i

2 2 ( /2)

3

( )

2 2 / 3

0 0 3

2 2 2 )

( ) 2 (

SHG

### 

Conversion efficiency for SHG

Note interaction length and intensity dependence

(24)

C

B

Beat length

Coherence length

i k L

2

) 2 / ( 2

3

(25)

3 2 1

3 2 1 0

0

1 1 1

2 1 )

( ) 2 (

3

2 1

(26)

coh

2

2

coh

###  

Phase matched – undepleted pump approximation

Phase matched –pump depletion considered

Phase mis-matched

L

2

1 0

(27)

Closer to

phase-matching

### :

SHG crystal Far from

phase-matching: SHG crystal

B

a a

b b

b b

b a

a a

b a

b a

0

(28)

2

2 max

2

m ax 2

2

C

### 

Mismatch at which the crystal length = coherence length

B

### 

Mismatch at which the crystal length = beat length

### for fixed crystal length

(29)

Quantum Electronics, Warsaw 2010

### SHG – birefringence phase-matching

One can utilize this for

satisfying phase-matching condition

In positive uniaxial crystals ne > no chose the extraordinary polarization for  and the ordinary for 2:

ne depends on propagation angle, so we can tune for a given .

o e

Frequency

Refractive index

e

o

(30)

2 2 2

2 2

e o

e

e

o

crit

e2ω

crit

oω

crit

(31)

p

s

i

g

g

c

###  

In ferroelectrics (e.g., LiNbO3 or KTP), reversed permanent electric dipole moment is obtained by applying high voltage of electric field – so called

”periodic poling”

Output Power

c

### +-+-+

Phasematched

Non-phasematched QPM

Propagation Distance Period = 2Lc

(32)

### Third order nonlinear effects

ENERGY CONSERVATION:

MOMENTUM CONSERVATION:

### Requires phasematching

Phase-matched effects build up

(33)

it

(3) 2

) 3

(

(1) (3) 2

TOT

eff

(1)

2

(3)

2

###  1  4 

eff

with:

Intensity dependent refractive index

0

2

Define: 3

2 2 0 2

0

2

(34)

###  (

The laser beam has Gaussian intensity profile. It can induce a Gaussian refractive index profile inside the NLO sample – a “Kerr lens” !

### Optical Kerr effect:

Self-focusing of Gaussian beam in Kerr medium Utilized e.g. for passive mode-locking

(35)

(36)

###  

The phase is modulated according to the pulse time

envelope

### I(t),

and increases with the propagation distance

0 2

sig sig sig

### Ez t  Eti nk z  Eti n  n I t k z

Instaneneous frequency shift:

0

2

red

blue

m ax

2

###  n

Spectral broadening Generation of new frequencies

### I (ω)

In the absence of GVD only spectrum broadens - temporal shape is preserved !

(37)

2 2 2 2

2

### C  d     

In analogy to bird sounds the pulse is called ”chirped”

Pulse frequency varies in time - the leading edge is shifted to lower frequencies (red shift), the trailing edge to the higher ones (blue shift)

red

blue

Frequency chirp:

(38)

### SPM and GVD induced frequency chirp

Frequency chirp generated by SPM in optical fiber

With normal GVD -

Frequency chirp generated by a grating or prism pair Transform limited

laser pulse

(39)

### Self Phase modulation

Compressor

S. Ghimire With normal GVD – enhanced pulse broadening

With anomalous GVD – pulse compression!

Note: at a given spectral width the shortest possible pulse is unchirped (transform limited)

Chirped pulses have a

”compression potential”

(40)

### Negative (anomalous)dispersion needed to form solitons

-4 -2 0 2 4

0 0.2 0.4 0.6 0.8 1

2

### Soliton

Transform-limited (no chirp) In lossless media preserves its shape !

(41)

### SPM

In anomalous dispersion regime pulse forms a bounding potential

Pulse is trapped in time (shape does not change)

(42)

0

as

i

i

s

0

0

0

i

0

0

s

as

(43)

### Stimulated Raman scattering

When the Stokes power becomes large enough it can act as a pump to the next order Stokes

New phenomenon observed in 1962 by Gisela Eckhardt et al:

for intense pump the Stokes wave rapidly grows

Spontaneous Raman Scattering provides a weak signal that is

amplified by the pump

Stimulated Raman Scattering

### SRS

SRS is utilized for single pass or cascaded amplifiers and lasers

(44)

13.2 THZ

40 THZ

(45)

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