1. WHY ???? Ultrashort laser pulses

2. (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  high accuracy

3. F=ma  0 ~ 31 Å 10 15 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.

4. 0 1 To do this you need to control a single cycle

5. Resolve very fast events - “Testing” Quantum mechanics Probing chemical reactions Pump probe experiments All applications require propagation/manipulation of pulses

6. 0 1 MANIPULATION OF THIS PULSE

7. Chirped pulse LEADS TO THIS ONE: Propagation through a medium with time dependent index of refraction Pulse compression: propagation through wavelength dependent index

8. Train of pulses in time and frequency -- CEO Brief review of pulse propagation Pulse trains and frequency combs Evolution of a single pulse in an ``ideal'' cavity How unequally spaced modes lead to a perfect frequency comb The real thing (the laser, the “real comb”) Measuring the CEO The frequency comb as seen by the experimentalist

9. Propagation in the time domain PHASE MODULATION n(t) or k(t) E(t) =  (t) e i  t-kz  (t,0) e ik(t)d  (t,0)

10. DISPERSION n(  ) or k(  )  (  )  (  ) e -ik  z Propagation in the frequency domain Retarded frame and taking the inverse FT:

11. PHASE MODULATION DISPERSION

12. Train of pulses in time and frequency CEP and CEO Do the definitions extend to a train of  -functions?

13. FREQUENCY Stating the obvious TIME E CW radiation, short pulses, pulse trains

14. FREQUENCY The spectral resolution of the cw wave is lost TIME E Splicing a CW wave:

15. FREQUENCY What if… we apply a periodic modulation to the cw wave? TIME E E (CEP) (CEP) = CARRIER TO ENVELOPE PHASE CEP/ = CEO = CARRIER TO ENVELOPE OFFSET f 0

16. Electric field Frequency     Coherence  RT  b  av f0f0  RT  p Why is the  (CEP) related to the first tooth of the mode comb?  p Phase between successive pulses:

17. Concluding: Yes, you can combine high spectral resolution with femtosecond temporal resolution 1) 2) The Carrier to Envelope Phase (CEP) applies to a single pulse –  Mode-locked laser 0 4)The engine for the pulse train is the mode-locked laser --- Tooth spacing: What controls the pulse train is the CEO The mode comb does not start at zero but at  = 2  f 0 { 3) The Carrier to Envelope Phase Offset (CEO) applies to a pulse train the description of a modulated carrier applies. Train of pulses in time and frequency -- CEO

18. Electric field Time ee  RT a train of  functions? Fourier transform f r =  RT f0f0 pp  RT pp = ee (i + 1) - ee (i ) f0f0 -f 0 f r - f 0 Frequency Do the definitions extend to

19. Train of pulses in time and frequency -- CEO Brief review of pulse propagation Pulse trains and frequency combs Evolution of a single pulse in an ``ideal'' cavity How unequally spaced modes lead to a perfect frequency comb The real thing (the laser, the “real comb”) Measuring the CEO The frequency comb as seen by the experimentalist

20. Locking and measurement of CEO mixer Laser f 0 control Reference Frequency ref Control of f 0 to ref AOM Control of f 0 to  All pulses have the same CEP! Big shortcoming: method applies only to ultrashort pulses The CEO exists for a train of any pulse duration! D. J. Jones et al. Science 288, 635-639 (2000) S. Rausch et al., Ultrafast Phenomena Conf., Snowmass, MB5, 2010.

21. Other method: interferometry Generate two pulse trains, one with zero CEO the other with the CEO f 0 to be determined, Make the two pulse trains interfere: beat frequency = fspulses D Beat note time A B Coherence time = 1 bandwidth of beat note both pulse trains at the same repetition rate. f 0 Measurement of CEO

22. Train of pulses in time and frequency -- CEO Brief review of pulse propagation Pulse trains and frequency combs Evolution of a single pulse in an ``ideal'' cavity How unequally spaced modes lead to a perfect frequency comb The real thing (the laser, the “real comb”) Measuring the CEO The frequency comb as seen by the experimentalist

23. AMPLIFIER (a) LASER AMPLIFIER (b) (c) The real thing (nearly): the laser

24. time 0 Electric field amplitude Equally spaced modes in phase, make pulses periodic in time

25. The real thing: the laser Tuning the wavelength, the mode and the CEO L. Arissian and J.-C. Diels, “Carrier to envelope and dispersion control in a cavity with prism pairs”, Physical Review A, 75:013824 (2007).

26. Round -trip frequency wavelength MODE-LOCKED LASER FREQUENCY TUNABLE LASER Frequency counter 700800900 100 200 Rep. Rate - 101 884 000 Hz Wavelength [nm] FREQUENCY COUNTER ORTHODONTIST SPECTROMETER D Tuned cw laser Mode locked laser The laser as an orthodontist

27. The laser cavity having dispersion, How can the modes of the frequency comb be exactly equally spaced? Answer in L Arissian and J.-C. Diels Investigation of carrier to envelope phase and repetition rate: fingerprints of mode-locked laser cavities J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 183001 (25pp) Procedure: 1) write the quadratic spectral chirp coefficient induced by Kerr effect: Multiply the chirped frequency spectrum by the dispersion: at each round-trip, to find the condition: The laser as an orthodontist

28. Two related questions: As a pulse circulates in the cavity, does it evolve towards a steady state? Which mechanism makes the unequally spaced cavity modes equidistant? Evolution of a single pulse in an ``ideal'' cavity How unequally spaced modes lead to a perfect frequency comb SAME CONDITION Balance phase modulation by Kerr effect and phase modulation by dispersion

29. Evolution of a single pulse in an ``ideal'' cavity Dispersion Kerr effect Kerr-induced chirp

30. How unequally spaced modes lead to a perfect frequency comb Phase delay Group delay Cavity modes: not equally spaced because n av = n av (  ) Unequally spaced modes, is contradictory to the fact that comb teeth are equally spaced. where dispersion A cavity with ONLY Kerr modulation generates the pulse train: F.T.

31. Train of pulses in time and frequency -- CEO Brief review of pulse propagation Pulse trains and frequency combs Evolution of a single pulse in an ``ideal'' cavity How unequally spaced modes lead to a perfect frequency comb The real thing (the laser, the “real comb”) Measuring the CEO The frequency comb as seen by the experimentalist

32. THE PULSE TRAIN Both fundamental and second harmonic: a straight line. Electronic Spectrum analyzer The frequency comb as seen by the experimentalist

33. THE PULSE TRAIN What we should not see: Modulation of the train on a  s scale (Shows as a sideband on spectrum analyzer on a 100 KHz scale) Q-switched-mode-locked train The frequency comb as seen by the experimentalist

34. The CEO is not a CEP! Carrier to Envelope Phase CEP property of a single ultrashort pulse Carrier to Envelope Offset CEO property of a pulse train Dimensionless! Dimension: frequency A CEO can be accurately measured, even when a CEP cannot be resolved Because the pulse train is a modulated carrier, one can perform Concluding remarks Intracavity Phase Interferometry with unprecedented phase sensitivity.