2.1.1 Mathematical Description
The description of all classical optic phenomena is founded on the set of partial differential equations that were introduced by James Clerk Maxwell in 1861 [20].
The macroscopic version of these equations, which describe the interplay of electric and magnetic fields with matter, are given in SI-units as [21]
∇ ·D = ρ, (2.1.1)
∇ ·B = 0, (2.1.2)
∇ ×E = −∂B
∂t, (2.1.3)
∇ ×H = J+∂D
∂t, (2.1.4)
where ρ is the charge density, J the current density, D =0E+P =E the dis-placement field in a medium with the polarizationP, and H=B/µ0−M=B/µ is the magnetic field with magnetizationM. While 0 and µ0 are the electric per-mittivity and magnetic permeability of the vacuum, respectively, they are replaced in a medium by the absolute quantitiesandµ, which account for the response of the material. In general, and µ are tensors, however, in the case of an isotropic material they become scalar quantities.
In the absence of free charges and free currents (ρ = 0,J= 0), Maxwell’s equa-tions give rise to the wave equaequa-tions of the electric and magnetic fields,
∇2E− 1
Possible solutions of the wave equations are monofrequent, continuous electro-magnetic waves that are infinitely extended and propagate at the phase velocity c= 1/√
µ. Moreover, all superpositions of such plane waves represent solutions as well. In fact, ultrashort laser pulses which are subject of this thesis can be under-stood as a superposition of plane waves of different frequencies that have a fixed phase relation. Mathematically, the electric field of an optical pulse as a function of space and timeE(t, z) is formally given by the Fourier transform of the complex spectral amplitude ˜E(ω, z) [22]
E(t, z) =FhE(ω, z)˜ i(ω) = 1
√2π Z ∞
−∞
E(ω, z)e˜ iωtdω. (2.1.7) Note that the electric field is now described as a scalar quantity, which is justified for linearly polarized light. In case of elliptically polarized light, one has to employ the full vectorized description if nonlinear effects come into play; otherwise the orthogonal components can be treated independently. As the electric field E(t, z) is a real physical quantity its spectral amplitude has to be self-adjoint: ˜E(ω, z) = E˜∗(−ω, z). In many situations it is convenient to describe the electric field by the complex field amplitude E(t, z) that is obtained by the Fourier transformation of the single-sided spectrum ˜Ess(ω, z)
E(t, z) =FhE˜ss(ω, z)i(ω) = 1
√2π Z ∞
0
E˜(ω, z)e−iωtdω, (2.1.8) where the connection of the complex and the real electric field is then given by
E(t, z) =E(t, z) +c.c.= 2Re[E(t, z)]. (2.1.9) For a given optical pulse one can define the spectral bandwidth ∆ω as well as the pulse duration τ, which typically refer to the full width at half maximum (FWHM). It is a general outcome of Eq. (2.1.7) that a broader spectrum enables shorter minimum pulse durations. However, for a given spectral width, the shortest possible pulse duration is only obtained if the spectral phase is flat over the entire spectrum. In this case, the pulse duration is called bandwidth-limited or Fourier-limited. The product of this pulse duration and the spectral width, also referred to as the time-bandwidth product, is a constant parameter and is characteristic for a certain temporal pulse shape: τ∆ω = const.
2.1 Ultrashort Laser Pulses
In most cases, the spectrum of an optical pulse is centered around a certain mean frequency or carrier frequency, which is defined by
ωc= If the spectral amplitude contains only frequencies in a small interval around this carrier frequency (∆ω ωc), the substitution ω → ωc+ ∆ω separates the field amplitude in a complex envelopeA(t, z) and a term oscillating at ωc
E(t, z) = FhA˜ss(∆ω, z)i(∆ω)·e−iωct (2.1.11)
= A(t, z)eiωct (2.1.12)
= A(t, z)eiϕ(t,z)e−iωct. (2.1.13) Here ˜Ass(∆ω, z) is the complex spectral amplitude shifted to the baseband, while A(t, z) andϕ(t, z) represent the envelope and phase of the pulse. This description of optical pulses is only correct if the above mentioned criterion of a narrow spectrum is fulfilled, which is usually referred to as the slowly-varying envelope approximation (SVEA) [4]. As the term implies the approximation only holds if the temporal variation of the envelope is small within one oscillation of the optical field
where k(ωc) is the wavenumber at the carrier frequency. The generation of ever shorter optical pulses with modern ultra-broadband laser systems down to the few-cycle regime renders the validity of the SVEA more and more questionable. In the near infrared pulse durations of 10 fs, which correspond to only 4 optical cycles, represent the threshold down to which the SVEA and therefore the separation into an envelope and a carrier oscillation still holds. For even shorter laser pulses (∆ω≈ ωc) the carrier frequency and the phase of an optical pulse are no longer independent quantities and further use of the SVEA would violate the energy conservation and lead to unphysical DC electric fields. To this end, more sophisticated theories for the description of ultrashort laser pulses have been developed that are applicable even down to the single-cycle regime [23–25].
2.1.2 Propagation in Dispersive Media
Based on the approximations made in the previous section, the propagation of optical pulses is now investigated in dispersive media. In case of linear propagation in a transparent medium, the wave equations (2.1.6) can be transferred into a much simpler differential equation [4]
∂
∂z
A(∆ω, z) +˜ ikA(∆ω, z) = 0,˜ (2.1.15)
which is readily solved by the ansatz
A(ω, z) = ˜˜ A(ω,0)eikz. (2.1.16) Here k=k(ω) =ωn(ω)/c0 is the dispersion relation of the material, which can be expanded in a Taylor series around the carrier frequency
k(ω) = being the propagation parameters. All of these propagation parameters represent different orders of material dispersion and describe the impact on different pulse parameters. Specifically, β0 and β1 are related to the phase velocity vp = ωc/β0 and the group velocity vg = 1/β1, respectively. While the former defines the traveling speed of the carrier wave, the latter indicates the speed of the envelope of the pulse. Furthermore, β2 and β3 are referred to as group-velocity dispersion (GVD) and third-order dispersion (TOD). Higher order dispersion beyond β3 will not be considered here. Due to the difference of the phase and group velocity, propagation of a pulse in a dispersive medium causes a phase slip of the carrier wave with respect to the envelope. This so called group-phase offset (GPO) changes a pulse parameter that is typically referred to as the carrier-envelope phase (CEP) or sometimes absolute phase [12, 26]. More precisely, the CEP is defined as the phase difference of the maximum of the pulse envelope and the closest field maximum.
Propagation in a medium with the refractive index n(ω) will induce a GPO
∆ϕGPO(z) = ωc z
where ng(ωc) is the group index at the carrier frequency, which is defined as ng(ωc) =n(ωc) +ωc
While the propagation parametersβ0 andβ1 solely act on the CEP and the timing of a pulse, all higher orders will also influence the temporal pulse shape. The group velocity dispersion β2, for instance, induces a linear chirp to an initially
2.1 Ultrashort Laser Pulses
Fourier-limited Gaussian pulse and increases the pulse duration according to [27]
τ(z) =τ0
whereτ0is the Fourier-limited duration (FWHM) of a Gaussian pulse. Accordingly, third order or even higher order dispersion causes a nonlinear chirp. It should be noted that dispersion can also be used for the inverse process, i.e., compressing an initially chirped pulse. For this purpose, the dispersion and chirp parameters must have opposite signs.
The previous treatment of linear propagation has fully neglected material absorp-tion. However, as real dispersive materials with a refractive indexn(ω) inherently possess also an absorption function α(ω), the ansatz in Eq. (2.1.16) has to be ex-panded by an additional absorption term. Generally, dispersion and absorption are related to the real and imaginary part of the dielectric function according to
(ω) = 1 +χ(1)(ω) =
Hereχ(1)(ω) is the linear susceptibility that describes the materials response in the frequency domain, which is connected to the polarization by ˜P(ω) =0χ(1)(ω) ˜E(ω).
The fact that absorption and dispersion are two intrinsically related phenomena is quantified by the Kramers-Kronig relation [28, 29]
n(ω)−1 = c
which is a general consequence of causality and is observed also in electronic or acoustic wave systems.
Nonlinear Propagation
If the optical intensityI =cn0|E|2/2 of an ultrashort laser pulses becomes as high as some TW/cm2 the response of optical media starts to deviate from the linear behavior that has been assumed so far. Under these conditions the polarization is expanded in a Taylor series for the electric field
P(t) =PL(t) +PNL(t) =0
χ(1)E(t) +χ(2)E2(t) +χ(3)E3(t) +. . ., (2.1.26) withχ(n)being the nth order susceptibility, which is a tensor of rankn+ 1. While even-order susceptibilities vanish in all centro-symmetric materials [30], such as gases, liquids and most crystals, they are observed solely in materials without inversion symmetry. Propagation under the influence of such nonlinearities can lead to the generation of new frequency components. A prerequisite for efficient
frequency conversion, however, is that the radiation generated at different positions within the material interferes constructively with each other, which is referred to as phase matching. The nonlinear processes that are most relevant for the experiments presented in this thesis are second harmonic generation (SHG), difference frequency generation (DFG) and four-wave mixing (FWM) and will be treated later in more detail.