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Complex representation of the electric field

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

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Complex representation of the electric field

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  1. Complex representation of the electric field Pulse description --- a propagating pulse A Bandwidth limited pulse No 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

  2. Delay (fs) -20 -10 0 10 20 1 0 -1 -6 -4 -2 0 2 4 6

  3. Chirped pulse

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

  5. A Bandwidth limited pulse t

  6. 0 Actually, we may need the Fourier transforms (review)

  7. 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

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

  9. 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:

  10. 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 In general one chooses:

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

  12. 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)

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

  14. Wigner Distribution Chirped Gaussian Gaussian

  15. 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

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