complex representation of the electric field pulse description --- a propagating pulse a bandwidth...
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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
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-6 -4 -2 0 2 4 6
-1
0
1
-20 -10 0 10 20
Delay (fs)
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Chirped pulse
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z
t
z = ctz = vgt
A propagating pulse
![Page 6: Complex representation of the electric field Pulse description --- a propagating pulse A Bandwidth limited pulseNo Fourier Transform involved Actually,](https://reader036.vdocuments.us/reader036/viewer/2022070409/56649e745503460f94b74d27/html5/thumbnails/6.jpg)
t
A Bandwidth limited pulse
![Page 7: Complex representation of the electric field Pulse description --- a propagating pulse A Bandwidth limited pulseNo Fourier Transform involved Actually,](https://reader036.vdocuments.us/reader036/viewer/2022070409/56649e745503460f94b74d27/html5/thumbnails/7.jpg)
Actually, we may need the Fourier transforms (review)
0
![Page 8: Complex representation of the electric field Pulse description --- a propagating pulse A Bandwidth limited pulseNo Fourier Transform involved Actually,](https://reader036.vdocuments.us/reader036/viewer/2022070409/56649e745503460f94b74d27/html5/thumbnails/8.jpg)
Properties of Fourier transforms
Shift
Derivative
Linear superposition
Specific functions: Square pulse Gaussian Single sided exponential
Real E(E*(-
Linear phase
Product Convolution
Derivative
![Page 9: Complex representation of the electric field Pulse description --- a propagating pulse A Bandwidth limited pulseNo Fourier Transform involved Actually,](https://reader036.vdocuments.us/reader036/viewer/2022070409/56649e745503460f94b74d27/html5/thumbnails/9.jpg)
Construct the Fourier transform of
Pulse Energy, Parceval theorem
Poynting theorem
Pulse energy
Parceval theorem
Intensity?
Spectral intensity
![Page 10: Complex representation of the electric field Pulse description --- a propagating pulse A Bandwidth limited pulseNo Fourier Transform involved Actually,](https://reader036.vdocuments.us/reader036/viewer/2022070409/56649e745503460f94b74d27/html5/thumbnails/10.jpg)
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
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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
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Slowly Varying Envelope Approximation
Meaning in Fourier space??????
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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)
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Pulse duration, Spectral width
Two-D representation of the field: Wigner function
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-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
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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