Noise sources and upconversion

As the main interest in this work is to provide a frequency comb suitable for high resolution spectroscopy in a wavelength range previously inaccessible to ultranarrowband lasers, one main question that arises is, weather the frequency comb will survive the nonlinear conversion or not. As was already stated earlier, if the nonlinear conversion is elastic that is if the effective photonic

equation of motion9_{is time-independent, energy conservation requires that the generated photons}

are sums of integer numbers of photons from the driving field. If that driving field is an ideal frequency comb with frequencies (2.8) the output of thenth_{harmonic has to contain frequencies}

ω(2_j n+1) =jωr+ (2n+ 1)ωCE (3.20)

and no other components.

However, a real frequency comb suffers from noise both in phase and amplitude due to fun- damental limits (like shot noise) and technical noise sources (like acoustic vibrations modulating the repetition rate of the laser, air currents, thermal fluctuations and the like). Any of these will add other spectral components that may be broadband (like shot noise, which has a white fre- quency spectrum). Also the time independence of the conversion dynamics has to be checked critically. The theoretical derivation presented in appendix A completely neglects other channels than recombination back to the ground state. One major other channel is ionization, which does not contribute to the generated radiation but changes ground state population and the dispersion relation for propagation of the electric field inside the medium. This section tries to discuss the impact of all these noise sources on the expected frequency structure of the generated radiation.

A simple, purely classical analysis of the process of nonlinear upconversion of a harmonic signal shows that the phase noise power in the signal will be amplified in the upconverted signal with respect to its carrier by a factor n2 _{with n being the harmonic order of the upconverted} signal. In brief, the argument is as follows. Let’s assume an input signal

V(t) = V0exp(i(ω0t+φ(t)) +c.c., (3.21) whereφ(t)is the phase noise term. If the phase noise is sufficiently smallhφ2_{(t)− hφ}2_{ii 1} the power spectrum of this signal is given by Rutman (1978)

|Vˆ(ω)|2 = V 2 0 4 (δ(ω−ω0) +| ˆ φ(ω−ω0)|2). (3.22)

Under the assumption that the nonlinear conversion coefficient is determined by the carrier only, thenthharmonic of that signal will take the form

Vn(t) =Vn(V0, ω0) exp(i(nω0t+nφ(t))) (3.23) and the power spectrum of the upconverted signal becomes

|Vˆn(ω)|2 = V2 n(V0, ω0) 4 (δ(ω−nω0) +n 2_{|φˆ(ω−ω} 0)|2). (3.24)

If the harmonic order becomes too high, the noise background will eventually become larger than the carrier signal atω0(within some bandwidth) so that the carrier can no more be identified. This phenomenon is customarily referred to as carrier collapse in harmonic upconversion. Based on this argument, Telle (1996) have predicted that a phase cohenrent radio frequency to optical link would not be possible with frequency combs.

Amplitude noise in the driving signal will produce amplitude changes in the upconverted signal that may be very large as the conversion efficiency depends highly nonlinearly on the driving field strength. The driving amplitude is also coupled to the phase of the generated field via (A.24). For example, for a driving field with an intensity high enough that the ponderomotive energy is 10 times the photon energy of the fundamental field, an amplitude change of 10% could already cause a nonlinear phase change of2πfor the long trajectory.

Another effect that affects the phase of the generated light (as well as the fundamental radi- ation field) is the linear refractive index of the generating medium. As for high order harmonic generation the driving field is so strong that many atoms are ionized, the dispersion inside the medium is dominated by the plasma refractive index which in turn is dominated by the contribu- tion from the electrons. The electron plasma dispersion relation is given by

k(ω) = ω c r 1− ω 2 el ω2, (3.25)

whereωelis the plasma frequency

ωel=

r Nele

0me

, (3.26)

andNel is the electron density. Fluctuations in the plasma density lead to phase fluctuations in

both the driving wave and the generated field. Such fluctuations may come from density fluctua- tions of the generating medium but they are also induced by fluctuations in the driving intensity. This is an additional mechanisms that couples amplitude and phase of the generated XUV radia- tion. The plasma dispersion relation approaches the vacuum dispersion relation quadratically as the frequencyωincreases. Therefore the plasma density has the largest effect on the fundamental wave propagation. For the 11thharmonic the effect is a factor of 121 weaker.

Other sources of noise in the HHG radiation include spontaneous emission from the plasma recombination and decay from excited states. These spontaneous emission processes are however typically nondirectional, so that the power gets distributed across the entire solid angle 4π and only a small amount is emitted into the well-collimated Gaussian beam of HHG. There may also be other bound states than the ground state, to which the atom can recombine, resulting in an emission of photons with a different energy. However such processes are rather improbable and additionally, as the atom can carry away momentum, this results in an emission that will go into a large solid angle, so that it will have only small power inside the well collimated beam of HHG radiation.

The experiments by Zerne et al. (1997), Bellini et al. (1998) and citetLynga1999 clearly show that the noise contributions from fluctuations of the gas density and other external factors to the phase and amplitude of the generated radiation are not large. However, these experiments only test such external noise sources as two identical replica of one laser pulse are used to generate XUV radiation in two different media and observing the interference between the two XUV beams. By doing that, any fluctuations in the driving field are rejected as they affect both XUV beams in the same way. Therefore it remains to be proven that the temporal coherence of the driving frequency comb actually survives the high harmonic process.

Gain medium λ[nm] Pav[W] fr[MHz] τ [fs] Pp [MW] Reference

Ti:Sapphire 790 2 11 30 6 Fernandez et al. (2004)

Yb:Fibre 1060 76 75 400 2.5 Limpert et al. (2003)

Yb:Fibre 1060 10 75 80 1.7 Limpert et al. (2002)

Yb:YAG Disk 1030 60 34.3 810 2.1 Innerhofer et al. (2003)

Yb:KY(WO4)2 1028 22 24.6 240 3.7 Brunner et al. (2002)

Yb:YAG + fiber 1030 18 34 33 12 Innerhofer et al. (2004)

Ti:Sa RegAmp 800 0.7 0.1 35 200 Lindner et al. (2003)

Table 3.1: Currently available ultrafast high power laser oscillators and their parameters. λ: Wavelength,Pav: average power,fr: repetition frequency,τ: pulse duration,Pp: peak power