1. WHY ???? Ultrashort laser pulses

2. (Very) High field physics Highest peak power, requires highest concentration of energy E L I Create … shorter pulses (attosecond) Create x-rays (point source) Imaging High fields  high nonlinearities  high accuracy

3. F=ma  0 ~ 31 Å 10 15 W/cm 2, 800 nm 2020 Electrons ejected by tunnel ionization can be re-captured by the next half optical cycle of opposite sign. The interaction of the returning electron with the atom/molecule leads to high harmonic generation and generation of single attosecond pulses.

4. 0 1 To do this you need to control a single cycle

5. Resolve very fast events - “Testing” Quantum mechanics Probing chemical reactions Pump probe experiments All applicatons require propagation/manipulation of pulses

6. 0 1 MANIPULATION OF THIS PULSE

7. Chirped pulse LEADS TO THIS ONE: Propagation through a medium with time dependent index of refraction Pulse compression: propagation through wavelength dependent index

8. Why do we need the Fourier transforms? Construct the Fourier transform of “Linear” propagation in frequency domain “Non-Linear” propagation in time domain

9. Actually, we may need the Fourier transforms (review) 0

10. Properties of Fourier transforms Shift Derivative Linear superposition Specific functions: Square pulse Gaussian Single sided exponential Real E(  E*(-  Linear phase Product Convolution Derivative

11. Description of an optical pulse Real electric field: Fourier transform: Positive and negative frequencies: redundant information Eliminate Relation with the real physical measurable field: Instantaneous frequency

12.  0z t t What is important about the Carrier to Envelope Phase?

13. Slowly Varying Envelope Approximation Meaning in Fourier space??????

14. Forward – Backward Propagation Maxwell Equation s = t – n/c Z r = t + n/c z No scattering No coupling between E F & E B No linear assumption Slowly varying envelope Study of linear propagation (Maxwell second order)

15. Solution of 2 nd order equation Propagation through medium No change in frequency spectrum To make F.T easier shift in frequency Expand k value around central freq  l z  Z=0 Dispersion included k real Study of linear propagation

16. Expansion orders in k(  Material property II) Second Study of linear propagation

17. Propagation in the time domain PHASE MODULATION n(t) or k(t) E(t) =  (t) e i  t-kz  (t,0) e ik(t)d  (t,0)

18. DISPERSION n(  ) or k(  )  (  )  (  ) e -ik  z Propagation in the frequency domain Retarded frame and taking the inverse FT:

19. PHASE MODULATION DISPERSION

20. Application to a Gaussian pulse

21. Evolution of a single pulse in an ``ideal'' cavity Dispersion Kerr effect Kerr-induced chirp

23. Study of propagation from second to first order

24. From Second order to first order (the tedious way) (Polarization envelope)

25. Pulse duration, Spectral width Two-D representation of the field: Wigner function

26. Gaussian Chirped Gaussian Wigner Distribution

27. Wigner function: What is the point? Uncertainty relation: Equality only holds for a Gaussian pulse (beam) shape free of any phase modulation, which implies that the Wigner distribution for a Gaussian shape occupies the smallest area in the time/frequency plane. Only holds for the pulse widths defined as the mean square deviation