VUV Spectroscopy of Atoms, Molecules and Surfaces

1.2 Frequency mixing 5
the frequency domain with the susceptibilities being equal and χ (n) providing
the connection between the frequency generated in the nth order non-linear
proces and that of the driving field [12]. In principle, the third- or higher order
harmonic radiation could just as well result from sum-frequency mixing of the
second harmonic with the fundamental and in general, the (n-1)th harmonic
generation proces is considered a special case of n-wave mixing of n different
driving fields.
From the above expansion and Maxwell’s equations the field amplitudes
and intensities as a function of propagation length of the generated harmonic
radiation can be calculated. This assumes a knowledge of the frequency,
intensity and focusing conditions for the driving field (or fields) and of the
dispersion properties of the medium. The intensity IVUV of the generated
field depends on the density of the medium N, the non-linear susceptibility
χ (n) , the focusing conditions and the intensities of the driving fields in a
manner indicated by stating it for the case of four-wave sum- and difference
frequency mixing, ωVUV =2ω1±ω2, with driving fields of frequencies ω1 and
ω2 and intensities I1 and I2, respectively [13]:
IVUV ∝ N 2 [χ (3) ] 2 I1 2 I2F (b∆k). (1.2)
Here, b is the confocal parameter, assuming Gaussian beams, and ∆k =
kVUV − (2k1±k2) is the wavevector mismatch with kVUV, k1 and k2 denoting
the wavevectors of the generated- and driving fields, respectively. F (b∆k)
is the so-called phase-matching factor which is maximum for b∆k = −2 and
∆k = 0 for sum- and difference frequency mixing, respectively [14]. The
difference between the two phase-matching relations originates from the difference
in driving polarizations and the Gouy phaseshift φ(z) experienced
along the direction of propagation z of a Gaussian beam traversing a focal
point [15]. Since the polarizations responsible for sum- and difference frequency
mixing are proportional to E1 2 E2 and E1 2 E2 ∗ (∗ denoting complex
conjugate), respectively, the phase differences between the polarizations and
the generated (Gaussian) fields will be 3φ − φ =2φ and φ − φ = 0, respectively,
assuming Gaussian beams of the form Ej(z) ∝ exp(iφ(z)), j =1,2
[10]. The Gouy phaseshift thus introduced in sum-frequency mixing must be
compensated by a negative dispersion as expressed by the phase-matching
conditions given above [16].
Frequency mixing can be performed in non-linear crystals down to ∼190
nm, below which most materials become absorbing [17]. The simplest of
the mixing processes, second-harmonic generation, works down to ∼200 nm,
with the cut-off being dictated by the phase-matching conditions [18]. This
limit may be pushed closer to the absorption edge by the application of
sum-frequency mixing [19]. The current short-wavelength limit of ∼170 nm