1. SPACE
TIME
Fourier transform in time
Fourier transform in space

2. Fourier transform of Maxwell’s equation in TIME
Plane wave
F. T.
solution
Shift
Centered at w
k is the first order of the expansion (in frequency domain)
Shift property:

3. First derivative of k(W): group velocity.
Instead of
we have
In the Fourier domain: next order in the development of k(W)

4. SPACE
TIME
Fourier transform in time
Fourier transform in space

5. Pulse broadening, dispersion
A positive GVD corresponds to
Linearly chirped Gaussian pulse:
Multiply field spectrum
by propagator
To obtain
where
and
Back in time domain:

6. To make a long story short…
Compare with:

8. Combination of both: can be pulse broadening, compression,
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

9. e(t,0) eik(t)d e(t,0) 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)

10. e(DW,0) e(DW,0)e-ik(DW)z 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:

11. PHASE MODULATION
DISPERSION

12. Application to a Gaussian pulse

16. 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

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

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

19. In a linear medium:

20. the E field is no longer transverse
Maxwell’s equations, nonlinear propagation
Maxwell’s equation:
Since
the E field is no longer transverse
Gadi Fibich and Boaz Ilan
PHYSICAL REVIEW E 67, (2003)
Is it important?
Only if

21. Study of propagation from second to first order
apply this first, to give (– 2 i k E)

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