1. Complex representation of the electric field Pulse description --- a propagating pulse A Bandwidth limited pulseNo Fourier Transform involved Actually, we may need the Fourier transforms (review) Construct the Fourier transform of Pulse Energy, Parceval theorem Frequency and phase - CEP Slowly Varying Envelope Approximation Pulse duration, Spectral width

4. Chirped pulse

5. z t z = ct z = v g t A propagating pulse

6. t A Bandwidth limited pulse

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

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

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

10. 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

11. Frequency and phase - CEP Instantaneous frequency In general one chooses: And we are left with 02 -2 4 4 Time (in optical periods) 1 0 Field (Field) 7 02 -2 4 4 Time (in optical periods) 1 0 Field (Field) 7

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

13. Robin K Bullough Mathematical Physicist Robin K. Bullough (21 November 1929-30 August 2008) was a British Mathematical Physicist famous for his role in the development of the theory of the optical soliton. J.C.Eilbeck J.D.Gibbon, P.J.Caudrey and R.~K.~Bullough, « Solitons in nonlinear optics I: A more accurate description of the 2 pi pulse in self-induced transparency », Journal of Physics A: Mathematical, Nuclear and General, 6: 1337--1345, (1973)

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

15. Gaussian Chirped Gaussian Wigner Distribution

16. 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