SHORT PULSES

1. SHORT PULSES AS INTRODUCTION TO FOURIER TRANSFORMS SPACE TIME ANALOGY: What applies to pulses in time can be transposed to beams in space In time: dispersion In space: diffraction

2. Complex representation of the electric field A Bandwidth limited pulse No Fourier Transform involved Pulse description --- a propagating pulse Actually, we may need the Fourier transforms (review) Construct the Fourier transform of Pulse Energy, Parceval theorem Slowly Varying Envelope Approximation

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

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

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

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

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

8. Chirped pulse

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

10. A Bandwidth limited pulse t

11. 0 We may need the Fourier transforms (review)

12. 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

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

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

15. 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:

16. 1 And we are left with 1 0 (Field)7 Field (Field)7 Field 0 Instantaneous frequency -1 -1 -1 4 0 2 4 0 2 4 4 -2 -2 Time (in optical periods) Time (in optical periods) Frequency and phase – CEP – is it “femtonitpicking”? In general one chooses:

17. Frequency and phase – CEP – is it “femtonitpicking”? W 0 -w w

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

19. Traditional CEP measurement through high order nonlinear interaction High order effects depend on the CEP Two pulses of 2.5 optical cycle. The blue line is the electric field. The green dotted line is the seventh power. w = 2p/T

20. The CEP – how to “measure” it? G.G. Paulus et al, Phys. Rev. Lett. 91, 253004 (2003)

21. Complex representation of the electric field A Bandwidth limited pulse No Fourier Transform involved Pulse description --- a propagating pulse Actually, we may need the Fourier transforms (review) Construct the Fourier transform of Pulse Energy, Parceval theorem Frequency and phase – CEP – is it “femtonitpicking”? Slowly Varying Envelope Approximation

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

23. Maxwell’s equations, linear propagation Dielectrics, no charge, no current: Medium equation:

24. In a linear medium:

25. 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?

26. Study of propagation from second to first order

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

28. Pulse broadening, dispersion

29. Pulse broadening, dispersion Spectral phase Electric field amplitude w1 w5 w3 W w5 w4 w3 w2 0 time w1 Spectral phase W z = v1t (fast) z = v2t (slow) z = ct

30. Pulse broadening, dispersion E(t) Broadening and chirping time Electric field amplitude time z = v1t (fast) z = v2t (slow) z = ct

31. 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:

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

33. 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

34. 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)

35. 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:

36. PHASE MODULATION DISPERSION

37. Application to a Gaussian pulse