From Linear
description
Transcript of From Linear
IAPT Workshop 2nd August ISP, CUSAT
V P N [email protected]
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