Insert something into the laser cavity that favors high intensities.

1. fspulses – Passive mode-locking: The saturable absorber Saturable absorber Insert something into the laser cavity that favors high intensities.  strong maxima will grow stronger at the expense of weaker ones  eventually, all of the energy is concentrated in one packet loss gain gain > loss Time another mechanism – based on non-linear optical phenomenon

2. x x A lens and a lens f(x) = n k L(x) A lens is a lens because the phase delay seen by a beam varies quadratically with x: L(x) Typical laser beam with a Gaussian intensity profile: I0 exp(-x2/w2) f(x) = n(x) k L In both cases, a quadratic variation of the phase with x yields a lens. Now what if L is constant, but n varies quadratically with x: n(x)

3. Passive mode-locking via Kerr lensing: fs pulses • Just as absorption varies with intensity (a≈a0 – a2I ), a medium’s other optical characteristic, the refractive index, will also change with intensity. Kerr effect The nonlinear refractive index is the basis of most ultrafast lasers today.

4. High power Kerr-lens mode-locking Solid Low power • - Titanium:Sapphire not only lases, but it has a large n2 ! If the pulse is more intense in the center, it undergoes self-lensing. Placing an aperture at the focus favors a short pulse: Losses (due to the aperture) are too high for a low-intensity cw mode to lase, but not for high-intensity fs pulse. Large n2 - Kerr-lensing is the passive mode-locking mechanism of the Ti:Sapphire laser.

5. Modeling Kerr-lens mode-locking

6. Mirror Additional focusing optics can arrange for perfect overlap of the high-intensity beam back in the Ti:Sapphire crystal. But not the low-intensity beam! Kerr-lensing is a type of saturable absorber. If a pulse experiences additional focusing due to high intensity and the nonlinear refractive index, and we align the laser for this extra focusing, then a high-intensity beam will have better overlap with the gain medium. High-intensity pulse Ti:Sapph Low-intensity pulse This is a type of saturable absorption.

7. Absorption and emission spectra of Ti:Sapphire (nm) Titanium Sapphire (Ti3+:Al2O3 Titanium doped Sapphire crystal) It can be pumped with a (continuous) Argon laser (~450-515 nm) or a doubled-Nd laser (~532 nm). Upper level lifetime: 3.2 msec Ti:Sapphire lases from ~700 nm to ~1000 nm. It is currently the main laser of the ultrafast community - pulses as short as a few fs and average power in excess of a Watt.

8. Slit for tuning Ti:Sapphire gain medium cw pump beam Prism dispersion compensator The Ti:Sapphire laser (including dispersion compensation) Adding two prisms compensates for dispersion in the Ti:Sapphire crystal and mirrors. This is currently the workhorse laser of the ultrafast optics community.

9. GVD - the main mechanism that limit pulse shortening • Group-velocity dispersion: • GVD in crystal and mirrors. • GVD spreads the pulse in time. • All fs lasers incorporate dispersion-compensating components.

10. Dispersionis critical in ultrafast optics. Dispersion, the dependence of the refractive index on wavelength, has two effects on a pulse, one in space and the other in time. Angular dispersion disperses a beam in space (angle): Group-velocity dispersion (GVD) disperses a pulse in time (“chirp”): vg(blue)<vg(red) Both effects play major roles in ultrafast optics. Longer wavelengths almost always travel faster than shorter ones, i.e., the pulse gets up-chirped (the instantaneous freq. increases).

11. w = pulse frequency Dwp = spectral width In the visible and near-IR, GVD is positive in all materials. Pulses get longer and longer (and become more chirped) as they pass through more and more glass or other material. The Group-delay dispersion (GDD) = Distance through glass x GVD. Its units are fs2.

12. Pulse Compressor Positively-chirped input pulse compressed output pulse Angular dispersion yields negative GDD. This device has negative group-delay dispersion and hence can compensate for propagation through materials (i.e., for positive chirp). The additional prisms are required to put the pulse back together again.

13. A simpler, two-prism pulse compressor Uncompressed input pulse Reflecting the pulse back through the first two prisms also works and is easier to set up. Compressed output pulse Mirror This design is particularly convenient inside a laser.

14. Adjusting the GVD Any prism in the compressor can be translated perpendicular to the beam path to add glass and reduce the magnitude of negative GVD. Remarkably, moving a prism varies the GVD, but does not misalign the beam. The output path is independent of prism position. Input beam Output beam

15. Slit for tuning Ti:Sapphire gain medium cw pump beam Prism dispersion compensator The Ti:Sapphire laser (including dispersion compensation) Adding two prisms compensates for dispersion in the Ti:Sapphire crystal and mirrors. This is currently the workhorse laser of the ultrafast optics community.

16. Mode Locking 1+cos(wMt) modulator transmission Time Active Mode Locking Insert something into the laser cavity that sinusoidally modulates the amplitude of the pulse.  mode competition couples each mode to modulation sidebands  eventually, all the modes are coupled and phase-locked Passive Mode Locking Saturable absorber Insert something into the laser cavity that favors high intensities.  strong maxima will grow stronger at the expense of weaker ones  eventually, all of the energy is concentrated in one packet loss gain gain > loss Time

17. Active Mode Locking Insert something into the laser cavity that sinusoidally modulates the amplitude of the pulse.  couples each mode to modulation sidebands  eventually, all the modes are coupled and phase-locked modulator transmission 1+cos(wMt) Time

18. Active mode-locking: the Electro-optic modulator Applying a voltage to a crystal changes its refractive indices and introduces birefringence. A few kV can turn a crystal into a half- or quarter-wave plate. Polarizer If V = 0, the pulse polarization doesn’t change. “Pockels cell” (voltage may be transverse or longitudinal) If V = Vp, the pulse polarization switches to its orthogonal state. V Applying a sinusoidal voltage yields sinusoidal modulation to the beam. An electro-optic modulator can also be used without a polarizer to simply introduce a phase modulation, which works by sinusoidally shifting the modes into and out of the actual cavity modes.

19. Active mode-locking: the Acousto-optic modulator An acoustic wave induces sinusoidal density, and hence sinusoidal refractive-index, variations in a medium. This will diffract away some of a light wave’s energy. Acoustic transducer Pressure, density, and refractive-index variations due to acoustic wave w Output Beam Input beam w Quartz Diffracted Beam (Loss) Sinusoidally modulating the acoustic wave amplitude yields sinusoidal modulation of the transmitted beam.

20. Active Mode-Locking 1+cos(wMt) modulator transmission Time wn+wM wn-wM w0 pc/L cavity modes Frequency In the frequency domain, a modulator introduces side-bands of every mode. For mode-locking: choose wM = mode spacing. This means that: wM = 2p/cavity round-trip time = 2p/(2L/c) = pc/L (period=round-trip time) • Side-bands coincide with adjacent cavity modes !!! • Each mode competes for gain with adjacent modes. • - Most efficient operation is for phases to lock -> global phase locking. • - Result is global phase locking. • - n coupled equations: En En+1, En-1

21. wM w w0 Modeling the laser modes and gain Gain profile and resulting laser modes (mode spacing) • Let the zeroth mode be at the center of the gain, w0. • The nth mode frequency is then: where n = …, -1, 0, 1, … Let an be the amplitude of the nth mode and assume a wide Lorentzian gain profile, G(n): an+ After the gain medium: Before the gain medium

22. Modeling an amplitude modulator An amplitude modulator causes losses at the laser round-trip frequency, wM. Multiplies the laser light (i.e., each mode) by M[1-cos(wMt)] This actually spreads the energy from the nth to the (n+1)th and (n-1)th modes. Including the loss, , we can write: where the superscript k indicates the kth round trip.

23. where, in this continuous limit, where: Solve for the steady-state solution In steady state, Also, the finite difference becomes a second derivative when the modes are many and closely spaced: This differential equation has the solution: with the constraints: In practice, the lowest-order solution occurs (n =0) : A Gaussian pulse !

24. Fourier-transforming to the time domain Recalling that multiplication by -w2 in the frequency domain is just a second derivative in the time domain (and vice versa). So, becomes: which has a steady-state solution of: (This makes sense because Hermite-Gaussians are their own Fourier transforms.)