ULTRAFAST PHENOMENA – LINEAR AND NONLINEAR To present nonlinear optics as successive approximations of the semi-classical interaction between light and matter To give the experimentalist view of the physics, with minimum of equations, maximum of analogies and hand waving. “Blessed the feeble minded, because the kingdom of heaven belongs to them” OBJECTIVES

WHY ???? Ultrashort laser pulses

F=ma  0 ~ 31 Å W/cm 2, 800 nm 2020 Electrons ejected by tunnel ionization can be re-captured by the next half optical cycle of opposite sign. The interaction of the returning electron with the atom/molecule leads to high harmonic generation and generation of single attosecond pulses.

(Very) High field physics Highest peak power, requires highest concentration of energy E L I Create … shorter pulses (attosecond) Create x-rays (point source) Imaging High fields  high nonlinearities Relativistic electrons in on optical cycle: W/cm 2 Generate electron-position pairs from light: W/cm 2 Vacuum nonlinear optics 

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 – is it “femtonitpicking”? Slowly Varying Envelope Approximation More femtonitpicking about the CEP Complex representation of the electric field

time 0 Electric field amplitude Many frequencies in phase construct a pulse A Bandwidth limited pulse

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

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

Some (experimental) displays of electric field versus time Delay (fs)

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

Chirped pulse c t

z t z = ct z = v g t A propagating pulse

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

We may need the Fourier transforms (review) 0

Shift Derivative Linear superposition Specific functions: Square pulse Gaussian Single sided exponential Real E(  E*(-  Linear phase Product Convolution Derivative Properties of Fourier transforms

Construct the Fourier transform of 0   

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

In general one chooses: And we are left with Time (in optical periods) 1 0 Field (Field) Time (in optical periods) 1 0 Field (Field) 7 Frequency and phase – CEP – is it “femtonitpicking”? for bandwidth limited.

0    Frequency and phase – CEP – is it “femtonitpicking”? Does cw radiation propagate?

Slowly Varying Envelope Approximation Meaning in Fourier space??????

Robin K Bullough Mathematical Physicist Robin K. Bullough (21 November 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: , (1973) This guy says yes Does cw radiation propagate?

Short pulse: what is the carrier?Long pulse: what is the envelope center? CEP: a tricky definition Electric field Time Electric field Frequency Electric field Time Electric field Frequency ? ? Witty and inexperienced Experienced and slower… C E P The areful xperimental hysicist The arrier to nvelope hase + c.c. Traditional CEP definition does not work for chirped pulses

We may not know what CEP is, but we (think to) know how to control it. Propagation through glass of thickness d, the group delay is: For  CEP) = 2  (group minus phase) delay = T =   /c: The change in CEP is proportional to the thickness of glass: CEP: a tricky definition The CEP changes with propagation!

Example for an ultrashort pulse In time: In frequency:  E      s -1 (800 nm)] 3) The CEP --- traditional definition and control Red: real field E(t) Blue:envelope

Definition of CEP: difference between phase    (t) TIME t and phase    (t) at peak of real field at peak of amplitude, TIME t Electric field is just an oscillation in time CEP independent of C and E In time: FREQUENCY  In frequency:

We have to return to Maxwell's propagation equation: In frequency How to correctly propagate an ultrashort pulse without phase and group velocity

Single pulse Propagation maxwell Eq. Definition of CEP: difference between phase   (t) at peak of amplitude and phase   (t) at peak of real field CEP = CEP = CEP = CEP = Red: real field E(t) Blue:envelope

Two pulses of 2.5 optical cycle.The blue line is the electric field. The green dotted line is the seventh power.  T Traditional CEP measurement through high order nonlinear interaction High order effects depend on the CEP

The CEP – how to “measure” it? G.G. Paulus et al, Phys. Rev. Lett. 91, (2003)

What the right (black) and left (red) MCP measure, for = CEP / G.G. Paulus et al, Phys. Rev. Lett. 91, (2003) Electron energy (eV) counts CEP Total yield High energy

Measuring CEP changes optically, through interferometry? Traditional CEP measurement rely on high order nonlinear interaction Ionization rate of sufficient high order to sense a difference between and. + = + = 0 Repeat every time interval “Beat note” Intracavity Phase Interferometry Converting a phase shift  (CEP) into a frequency 