1. IAPT Workshop 2 nd August ISP, CUSAT V P N Nampoori nampoori@gmail.com

2. Simple pendulum Oscillate Rotate

3. Simple pendulum Oscillate Rotate Double pendulum

6. Taking the amplitude small θ

7. The hallmark of linear equations. We can predict state of the simple pendulum at any future time Linearlty helps us to predict future.

9. Alternate representation of damped motion of simple pendulum Phase space plot

10. Second order nonlinear differential equation Superposition principle is not valid Prediction is not possible

11. Linear dynamics – prediction possible

12. Traffic jam –Nonlinear dynamics - prediction is impossible

13. One-dimensional maps One-dimensional maps, definition: - a set V (e.g. real numbers between 0 and 1) - a map of the kind f:V  V Linear maps: - a and b are constants - linear maps are invertible with no ambiguity Non-linear maps: The logistic map

14. 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?

15. Geometric representation x f(x) 01 1 0.5 Evolution of a map: 1) Choose initial conditions 2) Proceed vertically until you hit f(x) 3) Proceed horizontally until you hit y=x 4) Repeat 2) 5) Repeat 3). : Evolution of the logistic map fixed point ?

16. Phenomenology of the logistic map y=x f(x) 01 1 0.5 y=x f(x) 01 1 0.5 01 1 01 1 fixed point 2-cycle? chaos? a) b) c) d) What’s going on? Analyze first a)  b) b)  c), …

17. Geometrical representation x f(x) 01 1 0.5 x f(x) 01 1 0.5 fixed point Evolution of the logistic map How do we analyze the existence/stability of a fixed point?

19. Fixed points - Condition for existence: - Logistic map: - Notice: sincethe 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

20. Stability and the Logistic Map - Stability condition: - First fixed point: stable (attractor) for - Second fixed point: stable (attractor) for x f(x) 01 1 0.5 x f(x) 01 1 0.5 - No coexistence of 2 stable fixed points for these parameters (transcritical biforcation) What about ?

21. Period doubling x f(x) 01 1 0.5 Evolution of the logistic map 010.5 1) The map oscillates between two values of x 2) Period doubling: Observations: What is it happening?

22. Period doubling 010.5 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 for However, the relation suggests they 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?

23. Bifurcation diagram Plot of fixed points vs

24. 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 1 Country B has same fraction in its budget as soon as A’s budget session Is over Next year A has increased budget allocation x2 Budget allocation goes on increasing. If complete budget is for Defence, it is not possible since no funds for Other areas

25. Fund allocation for subsequent years for the country A x f(x) 0110.5 As time progresses, budget allocation for defense decreases. Peace time. A and B are friends.

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

27. Phenomenology of the logistic map y=x f(x) 01 1 0.5 y=x f(x) 01 1 0.5 01 1 01 1 fixed point 2-cycle? chaos? a) b) c) d) What’s going on? Analyze first a)  b) b)  c), …

28. x f(x) 01 1 0.5 x f(x) 01 1 0.5 Budget allocation stabilises To a fixed value. Caution time. Yellow Budget allocation decreases and goes to zero Full peace time Green

29. Period doubling x f(x) 01 1 0.5 Evolution of the logistic map 010.5 1) The map oscillates between two values of x 2) Period doubling: Observations: What is it happening?

30. Bifurcation diagram Plot of fixed points vs Peace time WAR!!! Tension builds up

31. Evolution of International Relationships between three countries Two countries are at enmity and the third is the controlling country From Peace time to War time

32. 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 interacting Following are the 8 equal probable states of the system 1=( 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

33. T1 T2T3 (000) (010) (001)(010)(001) (010) (100)(011) (111) (110) (011) (111) (101)(011) (111) (011) (111) I II III bistable 1/8 2/8 5/8 Evolution to Fixed state Blissful state!!!

34. Bistable picture

36. Rabbit and duck – bistable state

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

38. Evolution – Esher’s painting

39. Nonlinearity in Optics

40. Linear Optics Maxwell’s Equations : Light -- Matter Interaction Maxwell’s equations for charge free, nonmagnetic medium .D = 0 .B = 0  XE = -  B/  t  XH =  D/  t D =  0 E + P and B =  0 H In vacuum, P = 0 and on combining above eqns  2 E -  0  0  2 E/  t 2 = 0 or  2 E - 1/c 2  2 E/  t 2 = 0 In a medium,  2 E -  0  0  2 E/  t 2 -  0  2 P/  t 2 = 0 writing P =  0  E,  2 E - 1/v 2  2 E/  t 2 = 0 where, v = (  0 ) -1/2 Defining c/v = n,  2 E - n 2 /c 2  2 E/  t 2 = 0 we get a second order linear diffl eqn describing what is called, Linear Optics

41. 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

42. Polarisation of a medium P = P L +P NL where,P L =  (1) E and P NL =  (2) E 2 +  (3) E 3 +….. On substituting P in the Maxwell’s eqns, we get  2 E - n 2 /c 2  2 E/  t 2 =  0  2 P NL /  t 2 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 eqns E Polarisation feedback

43.  22 Optical second harmonic generation 11 22 33 Sum (difference) frequency generation OPC by DFWM

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

45. One of the consequences of 3rd order NL P NL =  (1) E +  (3) E 3 =(  (1) +  (3) E 2 )E = (n 1 + n 2 I)E We have n(I) = (n 1 + n 2 I) n I n 2 > 0 n 2 < 0 Intensity dependent ref.index has applications in self induced transparency, self focussing, optical limiting and in optical computing. n1n1

46. Saturable absorbers– Materials which become transparent above threshold intense light pulses I α T Absorption decreases with intensity Materials become transparent at high intensity IsIs

47. ItIt I Optical Limiters : Materials which are opaque above a threshold laser intensity Materials become opaque at higher intensity IsIs

48. Materials become opaque at higher pump intensity – optical limiter Materials become transparent at higher pump intensity- saturable absorber

49.  Protection of eyes and sensitive devices from intense light pulses  Laser mode locking  Optical pulse shaping  Optical signal processing and computing. Optical limiters are used in……

50. Intensity dependent refractive index Laser beam :Gaussian beam I(r ) = I 0 exp(-2r 2 /w 2 ) r I(r ) Beam cross section

51. I( r) = I 0 exp(-2r 2 /w 2 ) T ( r) =T 0 exp(-2r 2 /w 2 )n(r )=n 0 exp(-2r 2 /w 2 ) n = n(I) n= n 1 -n 2 I

52. Fabry - Perot etalon as NLO device n=n 1 +n 2 I L Condition for transmission peak L = m / 2n = m / 2(n 1 +n 2 I) m = 2nL/m is the resonant condition ItIt ItIt

53. i/p#1 i/p#2 o/p IiIi ItIt Saturable abspn Materials become transparent at higher pump intensity- saturable absorber I i =I 1 +I 2 ItIt

54. i/p#1 i/p#2 o/p IiIi ItIt Optical limiting ( extra abspn at high intensity) I i =I 1 +I 2 ItIt Materials become opaque at higher pump intensity – optical limiter

55. + I/p 1 0 or 1 I/p 20 or 1 0,1 or 2 Binary o/p Threshold logic 1 0 nonlinearity 12 O/p I/p 1 2 1 2 12 1 NOT/NOR AND OR XOR

56. 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)

57. (a) Open aperture z scan The propagation through the sample is given by 1.Reverse Saturable Absorption RSA- transmittance valley Application: Optical limiting 2. Saturable Absorption SA- transmittance peak Application: Optical pulse compression

59. Self assembled nano films : Drop casting & solvent evaporation  Reaction solution dropped onto a substrate @ 120 o C  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

60. RSA in Colloids The observed nonlinear absorption is attributed to TPA followed by FCA SA in Self assembled films Open aperture

61. Size dependence Intensity-220 MW/cm 2 and Irradiation wavelength-532 nm self assembled films of ZnO Colloids Self assembled thin films

62. 0z0z Normalized Transmittance 0z0z 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

63. To Conclude Nature is Nonlinear Linearity is an approximation for our convenience