Pulse description --- a propagating pulse

1. OPTICS OF SHORT PULSES with minimum of equations, maximum of analogies and hand waving. A Bandwidth limited pulse No Fourier Transform involved Pulse description --- a propagating pulse Fourier transforms review Slowly Varying Envelope Approximation

2. A Bandwidth limited pulse Many frequencies in phase construct a pulse Electric field amplitude time 0

3. A Bandwidth limited pulse E TIME FREQUENCY Time and frequency considerations: stating the obvious

4. A Bandwidth limited pulse E TIME The spectral resolution of the cw wave is lost FREQUENCY

5. A propagating pulse t z z = ct z = vgt

6. A Bandwidth limited pulse t

7. 0 We may need the Fourier transforms (review)

8. Properties of Fourier transforms Linear superposition Shift Linear phase Real E(W) = E*(-W) Convolution Product Derivative Derivative Specific functions: Square pulse Gaussian Single sided exponential

9. Construct the Fourier transform of w W 0 -w

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

11. How to correctly propagate an ultrashort pulse without phase and group velocity We have to return to Maxwell's propagation equation: In frequency It is only if That the pulse propagates unchanged at velocity n/c Group velocity is a concept that is clearly related to the SVEA

13. Propagation of the complex field Maxwell’s equations, linear propagation Pulse broadening, dispersion Maxwell’s equations, nonlinear propagation

14. Maxwell’s equations, linear propagation Dielectrics, no charge, no current: can be a tensor  birefringence Medium equation:

15. In a linear medium:

16. the E field is no longer transverse Since Gadi Fibich and Boaz Ilan PHYSICAL REVIEW E 67, 036622 (2003) Only if Maxwell’s equations, nonlinear propagation Maxwell’s equation: Is it important?

17. Study of propagation from second to first order

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

19. Pulse broadening, dispersion

20. Study of linear propagation Propagation through medium No change in frequency spectrum W Z=0 z Solution of 2nd order equation To make F.T easier shift in frequency Expand k value around central freq wl Expand k to first order, leads to a group delay:

21. Study of linear propagation Expansion orders in k(W)--- Material property

22. Propagation in dispersive media: the pulse is chirped and broadening Propagation in nonlinear media: the pulse is chirped Combination of both: can be pulse broadening, compression, Soliton generation

23. Propagation in the time domain PHASE MODULATION E(t) = e(t)eiwt-kz n(t) or k(t) e(t,0) eik(t)d e(t,0)

24. Propagation in the frequency domain DISPERSION n(W) or k(W) e(DW,0) e(DW,0)e-ik(DW)z Retarded frame and taking the inverse FT:

25. PHASE MODULATION DISPERSION

27. Normalization: and Soliton equation in space Eigenvalue equation (normalized variables. Solution of type: Townes’ soliton 2D nonlinear Schroedinger equation

28. In space: In time

29. Back to linear propagation: Gaussian pulse

30. Delay (fs) -20 -10 0 10 20 1 0 -1 -6 -4 -2 0 2 4 6 Pulse propagation through 2 mm of BK7 glass

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

32. Wigner Distribution Chirped Gaussian Gaussian

33. Uncertainty relation: Wigner function: What is the point? 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

34. A Bandwidth limited pulse Some (experimental) displays of electric field versus time Delay (fs) -20 -10 0 10 20 1 0 How was this measured? -1 -6 -4 -2 0 2 4 6

35. A Bandwidth limited pulse Some (experimental) displays of electric field versus time Delay (fs) -20 -10 0 10 20

36. Chirped pulse

37. Construct the Fourier transform of Pulse Energy, Parceval theorem Poynting theorem Pulse energy Parceval theorem ? Intensity Spectral intensity