IAPT Workshop 2nd August ISP, CUSAT

V P N Nampoorinampoori@gmail.com

Simple pendulum

Oscillate

Rotate

Simple pendulum

Oscillate

Rotate

Double pendulum

22

2

0

0 2

2

1

2

d

dtg

Sin tL

LT

g

Taking the amplitude small

θ

1 2

1 2

2

sup

nd order linear DE

solutions and

a b

erposition principle

The hallmark of linear equations.

We can predict state of the simple pendulum at any future time

Linearlty helps us to predict future.

x

Alternate representation of damped motion of simple pendulumPhase space plot

3

22 3

2

1sin

3

1( ) 0

3

d

dt

Second order nonlinear differential equationSuperposition principle is not valid Prediction is not possible

Linear dynamics – prediction possible

Traffic jam –Nonlinear dynamics - prediction is impossible

One-dimensional maps

One-dimensional maps, definition: - a set V (e.g. real numbers between 0 and 1) - a map of the kind f:VV

Linear maps:

- a and b are constants

- linear maps are invertible with no ambiguity

Non-linear maps: The logistic map

One-dimensional maps

Non-linear maps: The logistic map

with

Discretization of the logistic equation for the dynamics of a biological population x

Motivation:

b: birth rate (assumed constant)

cx: death rate depends on population (competition for food, …)

How do we explore the logistic map?

Geometric representation

x

f(x)

0 1

1

0.5

Evolution of a map: 1) Choose initial conditions2) Proceed vertically until you hit f(x) 3) Proceed horizontally until you hit y=x4) Repeat 2)5) Repeat 3) . :

Evolution of the logistic map

fixed point ?

Phenomenology of the logistic map

y=x

f(x)

0 1

1

0.5

y=x

f(x)

0 1

1

0.5

0 10.5

1

0 10.5

1

fixed point

fixed point

2-cycle? chaos?

a) b)

c)d)

What’s going on? Analyze first a) b) b) c) , …

Geometrical representation

x

f(x)

0 1

1

0.5

x

f(x)

0 1

1

0.5

fixed pointEvolution of the logistic map

How do we analyze the existence/stability of a fixed point?

Fixed points

- Condition for existence:

- Logistic map:

- Notice: since the second fixed point exists only for

Stability

- Define the distance of from the fixed point

- Consider a neighborhood of

- The requirement implies

Logistic map?

Taylor expansion

Stability and the Logistic Map

- Stability condition:

- First fixed point: stable (attractor) for

- Second fixed point: stable (attractor) for

x

f(x)

0 1

1

0.5

x

f(x)

0 1

1

0.5

- No coexistence of 2 stable fixed points for these parameters (transcritical biforcation)

What about ?

Period doubling

x

f(x)

0 1

1

0.5

Evolution of the logistic map

0 10.5

1) The map oscillatesbetween two values of x

2) Period doubling:

Observations:

What is it happening?

Period doubling

0 10.5 and thus:

- At the fixed point becomes unstable, since

-Observation: an attracting 2-cycle starts (flip)-bifurcation The points are found solving the equations

These points form a 2-cycle forHowever, the relation suggeststhey are fixed points for the iterated map

Stability analysis for :

and thus:

For , loss of stability and bifurcation to a 4-cycle

Now, graphically..

>

Why do these points appear?

Bifurcation diagram Plot of fixed points vs

International Relations and Logistic Map

Let A and B are two neighbouring countries

Both countries look each other with enmity.

Country A has x1 fraction of the budget for the Defence for year 1Country B has same fraction in its budget as soon as A’s budget session Is overNext year A has increased budget allocation x2Budget allocation goes on increasing.If complete budget is for Defence , it is not possible since no funds for Other areas

1 ( ) (1 )n n n nx f x x x Fund allocation for subsequent years for the country A

x

f(x)

0 1

1

0.5

As time progresses, budget allocation for defense decreases.Peace time . A and B are friends.

1 ( ) (1 )n n n nx f x x x

Parameter μ is called enmity parameter. Now let a third country C intervenesIn the region to modulate the enmity parameter and μ = μ(t)

Phenomenology of the logistic map

y=x

f(x)

0 1

1

0.5

y=x

f(x)

0 1

1

0.5

0 10.5

1

0 10.5

1

fixed point

fixed point

2-cycle? chaos?

a) b)

c)d)

What’s going on? Analyze first a) b) b) c) , …

x

f(x)

0 1

1

0.5

x

f(x)

0 1

1

0.5

Budget allocation stabilises To a fixed value.Caution time. Yellow

Budget allocation decreases and goes to zeroFull peace time Green

Period doubling

x

f(x)

0 1

1

0.5

Evolution of the logistic map

0 10.5

1) The map oscillatesbetween two values of x

2) Period doubling:

Observations:

What is it happening?

Bifurcation diagram Plot of fixed points vs

Peace time

WAR!!!

Tension builds up

Evolution of International Relationships between three countriesTwo countries are at enmity and the third is the controlling country

From Peace time to War time

Interaction leads to modification of dynamics.A, B and C are three components of a system with two states YES (1) or NO ( 0)

Case 1 A, B and C are non interactingFollowing are the 8 equal probable states of the system1=( 000), 2=( 001), 3=(010), 4=( 100),5=(110),6=(101),7=(011), 8=(111)

Probability of occurrence is 1/8 for all the states.

States evolve randomly.

Case II Let A obeys AND logic gate while B and C obey OR logic gate

T1 T2 T3

(000) (000) (000) (000)

(010) (001) (010) (001)

(001) (001)(010)(010)

(100) (011) (111) (111)

(110) (011) (111) (111)

(101) (011) (111) (111)

(011) (111) (111) (111)

(111) (111) (111) (111)

I

II

III

bistable

1/8

2/8

5/8

Evolution to Fixed stateBlissful state!!!

Bistable picture

Bistable picture

Rabbit and duck – bistable state

Photograph of melting ice landscape – Face of Jesus Christ - evolution leading to fixed state

Evolution – Esher’s painting

Nonlinearity in Optics

Linear OpticsMaxwell’s Equations : Light -- Matter Interaction

Maxwell’s equations for charge free, nonmagnetic medium.D = 0 .B = 0 XE = -B/t XH = D/t D = 0E + P and B = 0H

In vacuum, P = 0 and on combining above eqns2E - 00 2 E/t2 = 0 or 2E - 1/c2

2 E/t2 = 0 In a medium, 2E - 00 2 E/t2 - 02P/t2 = 0 writing P = 0 E, 2E - 1/v2

2 E/t2 = 0 where, v = (0 )-1/2

Defining c/v = n, 2E - n2/c2 2 E/t2 = 0

we get a second order linear diffl eqn describing what is called, Linear Optics

P=(1) E + (2) E2 + (3)E3 +…..

(n)/ (n+1) << 1

For isotropic medium, (n) will be scalar.

(n) represents nth order non linear optical coefficient

Polarisation of a medium P = PL +PNL

where,PL = (1) E and PNL = (2) E2 + (3)E3 +…..

On substituting P in the Maxwell’s eqns, we get2E - n2/c2

2 E/t2 = 02PNL/t2

This is nonlinear differential equation and describes various types of Nonlinear optical phenomena. Type of NL effects exhibited by the medium depend the order of nonlinear optical coefficient.

Maxwell’s eqnsE Polarisation

feedback

2

Optical second harmonic generation

1

2

3

Sum (difference) frequency generation

OPC by DFWM

Consequence of 3rd order optical nonlinearity intensity dependent complex refractive index

Inn)I(n

I)(2/1

)I2/11(

)I1(ngiveswhich

)E(indexrefractive

E

E

E)E(

EEP

21

32/1102/11

0

1

32/11

0

2/11

32/11

0

2/1230

10

230

10

230

10

330

10

One of the consequences of 3rd order NL

PNL = (1)E + (3)E3 =( (1) + (3)E2)E = (n1 + n2 I)E

We have n(I) = (n1 + n2 I)

n

I

n2> 0

n2< 0

Intensity dependent ref.index has applications inself induced transparency, self focussing, optical limitingand in optical computing.

n1

Saturable absorbers– Materials which become transparent above threshold

intense light pulses

I

α

T

Absorption decreases with intensity

Materials become transparent at high intensity

Is

It

I

Optical Limiters : Materials which are opaque above a threshold laserintensity

Materials become opaque at higher intensity

Is

Materials become opaque at higher pump intensity – optical limiter

Materials become transparent at higher pump intensity- saturable absorber

Protection of eyes and sensitive devices from intense light pulsesLaser mode lockingOptical pulse shapingOptical signal processing and computing

.

Optical limiters are used in……

Intensity dependent refractive index

Laser beam :Gaussian beam

I(r ) = I0exp(-2r2/w2)

r

I(r )

Beam cross section

I( r) = I0exp(-2r2/w2) T ( r) =T0exp(-2r2/w2) n(r )=n0exp(-2r2/w2)

n = n(I)

n= n1-n2I

Fabry - Perot etalon as NLO device

n=n1+n2I

L

Condition for transmission peak L = m / 2n = m / 2(n1+n2I) m = 2nL/m is the resonant condition

It

It

i/p#1

i/p#2 o/p

Ii

It

Saturable abspn

Materials become transparent at higher pump intensity- saturable absorber

Ii=I1+I2 It

i/p#1

i/p#2 o/p

Ii

It

Optical limiting( extra abspn at high intensity)

Ii=I1+I2 It

Materials become opaque at higher pump intensity – optical limiter

+I/p 1

0 or 1

I/p 20 or 1

0,1 or 2 Binary o/p

Threshold logic

1

0

nonlinearity

1 2

O/p

I/p 1 2

1 2 1 2

1

NOT/NORAND

OR XOR

A method to vary the intensity of pump beam incident on a sample:The sample is moved along a focused gaussian beam. Laser Intensity can be varied continuously with maximum at the focal point.

Sample

z

Z-scan Experiment

Open aperture z-scan : Sensitive to nonlinear absorption of sample

Closed aperture z-scan : Sensitive to both nonlinear absorption & refraction

B

Detector

Beam splitter

Aperture

M S Bahae et.al; “High-sensitivity, single-beam n2 measurements”, Opt Lett, 14, 955 (1989)

(a) Open aperture z scan

The propagation through the sample is given by

IIdz

dI

1. Reverse Saturable Absorption RSA- transmittance valley

Application: Optical limiting

2. Saturable Absorption SA- transmittance peak

Application: Optical pulse compression

Self assembled nano films :

Drop casting & solvent evaporation

Reaction solution dropped onto a

substrate @ 120oC

Solvent evaporation helps particles

to spontaneously assemble into

periodic structures

Temperature and solvent – significantly

influence the process of self-assembly

self assembled films of ZnO

RSA in ColloidsThe observed nonlinear absorption is attributed to TPA followed by FCA

SA in Self assembled films

Open aperture

Size dependence

Intensity-220 MW/cm2 and Irradiation wavelength-532 nm

self assembled films of ZnO

Colloids Self assembled thin films

0z

Normalized Transmittance Normalized

Transmittance

0z

Negative nonlinearity : peak-valley Positive nonlinearity: valley-peak

For a positive (self-focusing) nonlinearity, The positive lensing in the sample placed before the focus moves the focal

position closer to the sample resulting in a greater far field divergence and a reduced aperture transmittance corresponds to valley

When the sample is after focus, the same positive lensing reduces the far field divergence allowing for a larger aperture transmittance corresponds to peak

The opposite occurs for a self-defocusing nonlinearity

Closed Aperture Z scan

To Conclude

Nature is NonlinearLinearity is an approximation for our convenience