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Equation (1.4) also contains third order polarization components in ˜P(3)(t). While being more complicated, interesting effects like self-phase modulation (SPM), cross-phase modulation (XPM), third harmonic generation (THG), two photon absorption (TPA) and self-focusing (Kerr-lensing; a spatially dependentn2(x,y) withx,ytransversal to the propagation direction) are caused by the

third order susceptibility χ(3)_{i jkl}. As there are four waves participating (that may be degenerate), these effects can all be considered so-called four-wave mixing (FWM) effects. While for sec- ond order processes, media need to show some kind of anisotropy, third order processes can be observed in virtually all media, including dielectrics and gases.

In the absence of a second field ( ˜E(t)= Ee−iωt +_c._c._{), the degenerate processes of ˜P}(3)_{(t) can be}

separated by their frequency into ˜P(3)(t)=P

nP(ωn)e−iωnt. The interesting components here are

P(ω)=0χ(3)(3EE∗)E SPM (1.22)

P(3ω)=0χ(3)E3 THG (1.23)

With introduction of a second field (that could have the same or different polarization, this only changes the indexes inχ(3)_{i jkl}) ˜E(t) = E1e−iω1t+E2e−iω2t+c.c. new terms become relevant, giving

rise to XPM

P(ω1)=0χ(3) 3E2E∗2

E1 XPM (1.24)

When we plug in eq. (1.22) (similar for eq. (1.24)) into the source term of eq. (1.13), we see that the refractive index has experienced an intensity dependent shift∆n:

∇2E˜− 1 c2 ∂ ∂t2 n(ω)2+3χ(3)|E˜(ω)|2E˜ =0 (1.25)

20 1. Theoretical foundations of OPCPA

If we define the nonlinear refractive indexn2 as a correction to the linear refractive index ∆n =

2n2|E|2, it follows for SPM (fields only along one polarization)

n2=

3 4nχ

(3)

xxxx (1.26)

Likewise, for XPM we find a factor of 6 in eq. (1.25) for all fields being polarized along the same axis in an isotropic medium.

As usually we are more interested in the response of the medium to the intensity, there is a second definition forn2(mostly used in the literature, also dubbedγ):

∆n= 2n2|E|2≡ n I

2I (1.27)

that can be converted byγ≡ nI 2=

n2

0n0c.

B-integral

Often the accumulation of pulse distortions induced by SPM (or XPM) are simply characterized by the B-integral [76] B= 2π λ Z l nI₂I(z)dz (1.28) which adds up the nonlinear phase shift of eq. (1.27) over the path length l in the transmitted medium. As n2 is usually positive (unless close to two photon absorption) in most materials,

the B-integral is monotonically growing. In space it promotes the small-scale self-focusing of spatial distortions and can lead to optical damage [76]. Also it can distort the temporal profile of the pulse, like shown in fig. 3.5, especially for ultrashort pump lasers that are typically used in OPCPA. ThereforeBshould be kept low.

Spectral broadening and self-steepening

For a pulse with ˜E(z,t) = A˜(z,t)ei(ωt−kz) +_c._c. _{a shift in the (time-dependent) refractive index}

∆n(t) over the absolute value ofk0 = n_cω leads to a shift ink =k0+ ∆n(t)ω_c. Since this leads to a

(time-dependent) phase-shift for the whole pulse, we need to shift the overall phase, to find the relative modulation. We do this using the phase velocity in the substitution:t0 =t−zn_c. This gives the modulated pulse ˜E(z,t0) = A˜(z,t0)ei(ωt0−∆nωcz)+c.c.and therefore a time-dependent nonlinear

phase-shift of φNL(t)= −∆n(t) ω cz=− ω cn I 2I(t)z (1.29)

For the instantaneous frequency ω(t) of the pulse, this means a shift of δω(t) = _dtdφNL(t) and

results in ω(t)=ω0−nI2 ω cz dI(t) dt (1.30)

1.2 Nonlinear optics 21

As the refractive index n = n0 + n2I(t) changes with the time-dependent intensity I(t) and n2

is usually positive, the instantaneous frequency in eq. (1.30) will create red-shifted frequency- components at the leading edge of the pulse and blue-shifted at the trailing. This mimics a linear chirp in the center of the pulse, but increases the bandwidth elsewhere at the same time. The modulation strength is obviously given by the temporal gradient of the pulse. Note that the phase of the pulse is modulated according to the temporal pulse shape. For non-ideal pulses, this can often not be compensated. As a consequence, the spectral broadening then naturally leads to satellite pulses or pedestals and therefore comes at the cost of contrast [129]. In fact, the temporal shape of the pulse determines how well the broadened spectrum can be compressed [137].

The most broadband spectra generated from broadening are in general not symmetric around the center wavelength. This is caused by higher order dispersion and the effect of self-steepening. As the peak of the pulse experiences a larger group velocity, it is delayed with respect to the center, leading to an even steeper temporal gradient of the trailing edge and (from eq. (1.30)) consequently more spectral extension into the blue from this so-called shock wave. The deriva- tion is beyond the scope of this work and can be found in [11], chapter 13. Self-steepening is significant when the input spectral bandwidth of a pulse is already significant compared to the output spectrum, for example in super-continuum generation (like performed in section 2.2 or section 7.4).

Kerr lensing

An intensity dependent refractive index can also lead to modulations in the spatial domain. This includes effects like nonlinear holography using several beams (transient grating), which can be used for ultrafast pulse characterization. The beam can even focus itself due to a spatial gradient in its beam profile, which is called Kerr-lens self-focusing. For pulses exceeding the critical power

Pcr =

π(0.61)2λ2

8n0n2

(1.31)

this ultimately leads to self-trapping of the beam (filamentation), where one filament carries exactlyPcrand the diffraction (enhanced by plasma generation) is compensated by self-focusing.

As it increases the interaction length of SPM at high intensity, filamentation is useful for spectral broadening. To enhance the self-focusing tendency (that often only leads to very long focal lengths), usually one focuses weakly into a medium. However diffraction and the dispersion of the medium limit the length of the filament.

Two-photon absorption

Four-wave mixing can modulate not only the phase, but also the amplitude of a pulse. The imaginary part ofχ(3)_{leads to two-photon absorption (TPA) according to [133]} dI

dz = −αI −βI 2

22 1. Theoretical foundations of OPCPA and β= 3π 0n2cλ Imχ(3)xxxx(−ω;ω, ω,−ω) (1.32)

The effect is only considerable when the medium contains a transition which is resonant near twice the photon energy. This is the case for the silicon plates used for pulse compression in section 5.4 and demands low intensities (large spot sizes) as significant fractions of the pulse energy can be absorbed. Besides, it further motivates long wavelength few-cycle pulses for probing ultrafast processes in moderate band-gap dielectrics at high fields (see chapter 6) without breaking the sample under investigation.

Photo-refraction

The properties of lithium niobate are very favorable in general, as it does not show any absorption in the near infrared (see table B.2) and is not hygroscopic. However, especially for shorter wavelengths it suffers from photo-refraction, a spatial laser-induced distribution of carriers. The local fields of these charges then lead to a modulated refractive index. While this is a very strong (and therefore rather slow) nonlinear effect, it cannot be described by means of higher- order susceptibilities. To prevent it to some extent, MgO-doping and heating help to elevate the damage threshold to levels that can be used for OPCPA. However, the damage threshold is still much lower than for crystals like BBO, while n2 is significantly higher (table B.1), leading to

large B-integrals in the OPA stages of chapter 5.