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

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

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

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

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

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

Chirped pulse c t

A propagating pulse t z z = ct z = vgt

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

Mathematical approach: Fourier transforms

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

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

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

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, 036622 (2003)

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

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

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?

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

Electromagnetic waves