1. Electric field amplitude time A Bandwidth limited pulse
Many frequencies in phase construct a pulse
Electric field
amplitude
time

2. A Bandwidth limited pulse
Time and frequency considerations: stating the obvious
TIME E
FREQUENCY

3. A Bandwidth limited pulse
The spectral resolution of the cw wave is lost
TIME E
FREQUENCY

4. A Bandwidth limited pulse
Some (experimental) displays of electric field versus time
Delay (fs)
-20
-10 10 20 1 -1 -6 -4 -2 2 4 6

5. A Bandwidth limited pulse
Some (experimental) displays of electric field versus time
Delay (fs)
-20
-10 10 20 1 -1 -6 -4 -2 2 4 6

6. A Bandwidth limited pulse
Some (experimental) displays of electric field versus time
Delay (fs)
-20
-10 10 20

7. Chirped pulse
c t

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

9. A Bandwidth limited pulse of 50 fs – what is its? bandwidth

10. Mathematical approach: Fourier transforms

11. Description of an optical pulse
Real electric field:
Fourier transform:
Positive and negative frequencies: redundant information
Eliminate
Relation with the real physical measurable field:
we summarize the essential notations and definitions.
Inside the laser typically, only one pulse circulate. A complex representation
of the field amplitude is particularly convenient in dealing with
propagation problems of electromagnetic pulses.
Instantaneous frequency

12. Wave equations One dimensional wave equation
Complex numbers and trigonometric functions
Three dimensional wave equation
Spherical wave
Electromagnetic wave E & B ( S Poynting vector)
Maxwell’s wave equation, second order

13. Maxwell’s second order wave equation
forward
backward
SOLUTIONS?
Principle of superposition: all solutions can be written as a linear combination of
(complex numbers)
Substitute in wave equation to find the wave velocity

14. Maxwell’s equation: ONLY IF
Be aware of some approximations made in the derivation of Maxwell’s propagation equation.
implies
ONLY IF
Maxwell’s equation:
Consequence: the wave may not be transverse anymore!
Why is the wave transverse?
Gadi Fibich and Boaz Ilan
PHYSICAL REVIEW E 67, (2003)

15. Complex number Im
Z = a + i b
a = Re (z)
b = Im (z)
Z* = a - i b b z q Re a
Add and subtract exponentials and trigonometric functions
Plane waves, spherical waves

16. Plane waves Solution of: Substitute:
Surfaces of constant phase perpendicular to the vector k
Relation between k and l
At t = cst, the field repeats itself
after l

17. We have defined the surface of constant phase, perpendicular to the vector k
These surfaces move along the direction of the k vector.
At which velocity?

18. Spherical wave Wave equation:
Write in spherical coordinates, assuming spherical symmetry:
Conservation of energy

19. Electromagnetic waves