Superposition. Fourier Series Constructive Interference of a pulse.
Complex representation of the electric field Pulse description --- a propagating pulse A Bandwidth...
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Transcript of Complex representation of the electric field Pulse description --- a propagating pulse A Bandwidth...
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
-6 -4 -2 0 2 4 6
-1
0
1
-20 -10 0 10 20
Delay (fs)
Chirped pulse
z
t
z = ctz = vgt
A propagating pulse
t
A Bandwidth limited pulse
Actually, we may need the Fourier transforms (review)
0
Properties of Fourier transforms
Shift
Derivative
Linear superposition
Specific functions: Square pulse Gaussian Single sided exponential
Real E(E*(-
Linear phase
Product Convolution
Derivative
Construct the Fourier transform of
Pulse Energy, Parceval theorem
Poynting theorem
Pulse energy
Parceval theorem
Intensity?
Spectral intensity
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
Frequency and phase - CEP
Instantaneous frequency
In general one chooses:
And we are left with
0 2-2 44
Time (in optical periods)
-1
1
0
-1
Field (Field)7
0 2-2 44
Time (in optical periods)
1
0
-1
Field(Field)7
Slowly Varying Envelope Approximation
Meaning in Fourier space??????
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)
Pulse duration, Spectral width
Two-D representation of the field: Wigner function
-2 -1 0 1 2
-2
-1
0
1
2
-2 -1 0 1 2
-2
-1
0
1
2
Time TimeF
requ
ency
Fre
quen
cy
-2 -1 0 1 2
-2
-1
0
1
2
-2 -1 0 1 2
-2
-1
0
1
2
Time TimeF
requ
ency
Fre
quen
cyGaussian Chirped Gaussian
Wigner Distribution
Wigner function: What is the point?
Uncertainty relation:
Equality only holds for a Gaussian pulse (beam) shape free of anyphase modulation, which implies that the Wigner distribution for aGaussian shape occupies the smallest area in the time/frequencyplane.
Only holds for the pulse widths defined as the mean square deviation