Tratamiento y selección de las variables independientes

tendency toward THG in air and other isotropic media will be investigated more in chapter 7.

5.5

CEP characterization

The CEP was characterized using a collinear f-2f setup, since interferometers using two arms were observed to drift too much over extended measuring periods. The pulses were first broad- ened by focusing into a (2 mm thick, uncoated) YAG crystal and then doubled using a (1 mm thick, uncoated) BBO crystal. The fringe contrast was optimized by a cube polarizer, since it provides an indication of fast CEP jitter (see section 4.2.1). The modulation depth shown in the bottom graph of fig. 5.9 is quite good given the integration time of 100 ms. This is especially important to exclude large hidden high-frequency CEP jitter. The dead time of the spectrometer is about 2%, so the interferograms are recorded at a rate of almost 10 Hz. Analyzing the green fringes, we evaluate a CEP jitter of 155 mrad, shown in the middle of fig. 5.9. It is obvious that almost no drift occurs in the measured period of 2895 s. In fact the fitted CEP drift is only 100mrad/_{h, and for most experiments a slow loop is therefore not necessary. However, using a}

rolling average filter, we estimate that a slow loop with a bandwidth of 0.5 Hz, would reduce the CEP jitter to 91 mrad. The shot-to-shot jitter accounts for 51 mrad. This could be reached by controlling the CEP directly using an upgraded version of the AOPDF.

Disabling the delay stabilization of section 4.3.5, the CEP stability decreased considerably. The interferogram in fig. 5.10 is fluctuating and the retrieved CEP jitter is 402 mrad. This allows us to calculate the timing jitter to CEP jitter coupling (discussed in section 4.3.2). Assuming an unstabilized jitter of 127 fs (from section 4.3.5), together with all remaining sources of fast CEP fluctuations that amount to the 51 mrad shot-to-shot jitter mentioned above, this would lead to a coupling constant of 320 mrad_{100 fs} . It compares well with [48], if the different seed chirp is considered. Again, using a 0.5 Hz slow loop, the CEP fluctuations could be decreased to 106 mrad. Also the shot-to-shot jitter is increased to 59 mrad, confirming the high regulation bandwidth of the delay stabilizer described in section 4.3.5. Over longer periods, the overlap between seed and pump pulses was even lost slowly.

As discussed in [4], the filament used for broadening couples intensity and CEP jitter in the typ- ical f-2f interferometer. To avoid this, for the spectrum given, a 3f-2f interferometer would be necessary, that overlaps the third harmonic of 2.4µm and the second harmonic of 1.6µm without any filament. Doing both measurements synchronized (maybe even using the very same spec- trometer, as in section 4.3.6), together with a simple photo diode measurement of the intensity would yield the coupling constant between intensity and CEP. At the time of the measurement, this equipment was not available to us, but should be considered in the future.

In comparison to other (actively and passively) CEP stabilized OPCPAs, our system (without slow-loop) is very comparable. While we experience more jitter than [55] (78.5 mrad), which has the advantage of being very compact and pumped by a fiber laser (as beam-pointing couples to CEP as well, see [4]), the cited system also suffers from considerable CEP drift. This may

82 5. Building an infrared few-cycle OPCPA

be explained by the idler being generated in a high average power stage. Another system [90] centered at 2.2µm and using DFG in a PPLN, following power amplification in PPLN/PPLT reports 150 mrad over 10 s. And an OPCPA system around 2.1µm [125] employing BiB3O6

pumped with Ti:Sa pulses showed 260 mrad over 35 s and 410 mrad over 10 min, illustrating how important long-term assessment of the CEP stability is.

After the publication [119] of a part of chapter 4, a NIR OPCPA system [48] (pumped by fiber lasers again) reported explicit timing stabilization in their first OPCPA stage as a means of re- ducing the CEP jitter from 100 mrad to 86 mrad over a period of 40 min.

While actively stabilized amplifier systems also report CEP jitter on the level of 100–200 mrad, they do exhibit faster oscillations from the fast loop usually hidden by the integrating f-2f mea- surement, but can be correctly characterized with a single-shot method like in [77]. Since passive stabilization removes this source of fast jitter, spectral interference methods that integrate over several shots are more appropriate here for characterizing the CEP stability.

In summary, the delay stabilization of chapter 4 stabilizes the CEP of the OPCPA to a large extent. While the good CEP stability could still be improved to excellent performance (especially given the large dimensions of the setup), the drift is little enough to omit another feedback loop altogether for the following experiment of chapter 6.

5.5 CEP characterization 83 10000 1010 1020 1030 1040 1050 1060 0.2 0.4 0.6 0.8 1 wavelength (nm) intensity (norm)

Figure 5.9: f-2f measurement of CEP stability with OPCPA delay stabilization enabled, from top to bottom: f-2f interferogram, retrieved CEP and interferometer spectrum (with analyzed part in green)

84 5. Building an infrared few-cycle OPCPA time (s) wavelength (nm) 0 100 200 300 400 500 600 700 800 900 1000 1026 1028 1030 1032 1034 1036 500 1000 1500 2000 2500 50 100 150 200 250 300 350 400 450 500 550 600 −π/2 −π/4 0 π/4 π/2 phase (rad) time (s)

Figure 5.10: Degradation of CEP stability with OPCPA delay stabilization disabled, from top to bottom: f-2f interferogram and retrieved CEP

Chapter 6

First prototype experiment: controlling

currents in a dielectric using IR few-cycle

pulses

This chapter will add laser field control of current in dielectric media as an example experi- ment to prove that the OPCPA can be used for the experiments it was built for according to the introduction.

Currents induced by optical fields in dielectrics [114] were discovered very recently in our group and the almost instantaneous carrier injection (within≈1 fs) promises applications in high-speed metrology, like a simple light-phase detector [98] or a petahertz oscilloscope [71]. The latter could serve in super-octave synthesis for measurement and control of a sub-cycle NIR pulse, seeded from the source described in chapter 7. The present chapter simply aims to reproduce the first measurement of Schiffrin et al. [114], previously performed with a sub-4 fs actively stabilized and HCF-broadened Ti:Sapphire amplifier. The OPCPA of chapter 5 is suited for this experiment since it amplifies phase-stable infrared pulses with a duration below 2 cycles and reaches similar intensities in the dielectric. Also we are able to switch the CEP by use of our AOPDF, which allows lock-in amplification of the pA currents, that would otherwise be hidden in the noise background.

6.1

Current control theory

In order to explain the injection of carriers into the dielectric qualitatively, there are two very recent approaches both from the original publication (Schiffrin et al. [114]) and a theoretical investigation of Kruchinin et al. [80]. The latter approach is based on perturbation theory and includes the interference of multiple quantum channels (by multi-photon ionization) that lead to an asymmetry of the electron wave-packet in the conduction band in k-space, which means a

86

6. First prototype experiment: controlling currents in a dielectric using IR few-cycle pulses

non-vanishing electron velocity~~k. While it fits well to the previously published data of [114] for

low fields, for high fields the approximations did break down. This is where the theory of [114] steps in, using a quasi-static electric field on the modified band structures (that show attosecond response time). These energy structures do correspond to Wannier-stark localization and explain the increased polarizability at highest fields by a combination of Zener tunneling and the process of adiabatic transfer. For IR fields and consequently smaller Keldysh parameters γ = q IP

2Up

(with the ponderomotive potentialUpgiven in eq. (1)), naturally the probability of multi-photon

ionization decreases, while the probability of tunneling increases (as the field oscillation period is extended). Therefore the theory given in [114] (which is appropriate for very large fields over 1V/_{Å), is considered more relevant here.}

In order to measure a charge after an optical pulse, the injection process needs to be a nonlinear one (that means, it must happen at some instant during the pulse), since in a linear transport over the whole pulse there would be no net charges transported. Similar to the situation in attosecond streaking [69], the vector potential in the coulomb gaugeAL(t)=

R∞

t F(t 0

)dt0 would just average out to zero fort→ −∞, leading to no acquired momentum∆p= eAL(t) of the electrons.

When a field F is applied to a periodic crystal, this lifts the translation invariance inside the bulk dielectric, leading to wave functions that localize at different crystal sites l, where N is the number of lattice periods andl = −N₂,· · · , N₂. The energies of the bands of these so-called Wannier-stark states (WS), are equally spaced in energy on the so-called Wannier-stark ladder. Now for both valence and conduction band such a WS ladder does exist, with energy states

Enl = En− |eF|al (6.1)

whereais the lattice constant andz = lathe coordinate of such an electron wave-packet. With increasing field strengths, the energy states of valence (n = v) and conduction (n= c) band will eventually cross. For small fields the WS wave-functions do not overlap in space, as they are from very different crystal sites in

Ec−Ev

|eF|a =|lc−lv| (6.2)

As the fields are low, the crossing therefore must occur at large distances∆z = ∆la, leading to a very small dipole coupling. Consequently, this also means at larger fields the distances are close and the wave-functions do overlap more. This leads to a anti-crossing splitting∆ =~γ

∆ = |eF|a 2π exp      − ma∆2g 4~2|eF|       (6.3)

with the bandgap∆gbetween the bands. This splitting and therefore the tunneling rateγis expo-

nentially small for low fields and the level crossing is diabatic (with little inter-band population transfer).

For increasing fields, the splitting∆increases, but the transition rates are still low as the wave- functions barely overlap. For yet larger fields, the rate can become significant and the level