We demonstrate a new concept to vary the carrier-envelope phase of a mode-locked laser by a composite plate while keeping all other pulse parameters practically unaltered. The effect is verified externally in an interferometric autocorrelator, as well as inside the cavity of an octave-spanning femtosecond oscillator. The carrier-envelope frequency can be shifted by half the repetition rate with negligible impact on pulse spectrum and energy.
©2006 Optical Society of America
Only recently it has become possible to completely control the temporal evolution of the electric field of a train of mode-locked laser pulses [1,2]. Mastering the manipulation of the phase and magnitude of the electric field has been made possible by technological advances in femtosecond laser technology and nonlinear optics [1,2], together with ground-breaking ideas in the field of precision spectroscopy with pulsed laser sources [3,4]. This unprecedented high level of control enables a wide range of new applications in science and technology. Time domain applications focus on studies of physical phenomena that depend directly on the electric field rather than the pulse envelope only. Examples are carrier-wave Rabi-flopping , quantum interference of photocurrents , photoemission from metal surfaces , or electron emission from ionized atoms . Furthermore, attosecond physics has been made accessible by using carrier-envelope (CE) phase ϕCE controlled femtosecond pulses to generate coherent light in the XUV spectral regions in a well controlled manner [9,10,11]. Analogously, the high degree of control of the electric field is also beneficial for applications in the frequency domain, where the laser spectrum, composed of discrete longitudinal modes at frequencies f=f CE+m f rep is used in optical frequency metrology [12,13]. Here, frep is the repetition rate of the laser and f CE=f rep ϕ CE/2π is the carrier-envelope frequency, which is the rate of change of the carrier-envelope phase.
In this paper we report a novel composite plate that allows for an arbitrary choice of the CE phase while keeping dispersion in transmission essentially constant, therefore not altering other pulse parameters. Using this plate inside the laser cavity allows one to set f CE to any desired value between zero and the repetition frequency f rep while keeping pulse energy and spectrum unaltered. This is valuable in many applications, such as phase-sensitive nonlinear experiments with few-cycle laser pulses where the CE phase needs to be changed while keeping the pulse width constant . In frequency metrology systems, the CE frequency needs to be controlled to be able to shift the optical comb frequencies to a given reference wavelength without impacting the pulse width or spectrum and therefore the strength of the f CE beat signal . From a technical point of view, many experiments require particular values for f CE due to frequency selective detection schemes (lock-in techniques), pulse-picking requirements, or constraints in control electronics.
The paper first reviews the concept and implications of the carrier-envelope phase in femtosecond oscillators. After presenting the idea of a composite plate for carrier- envelope phase control, we discuss an extra cavity demonstration inside an interferometric autocorrelator. Finally, we implement the plate as part of the dispersion management of a 200 MHz octave-spanning laser to prove the usefulness of the concept for arbitrary manipulation of the CE frequency.
2. Carrier-envelope phase and frequency
The real electric field E(z,t) of a laser pulse may be decomposed into an envelope and a carrier-wave
with A(z,t) as the pulse envelope and the exponential describing the oscillation at the carrier frequency ω 0 whereas the time dependent phase term ϕ(t) describes the chirp of the pulse and ϕ CE the phase between the carrier-wave and the maximum of the envelope, the so-called carrier-envelope phase. During propagation of the wave-packet the carrier-wave propagates with the phase velocity v p and the envelope with the group velocity v g. Since all media including air exhibit a wavelength dependent index of refraction, phase and group velocity are generally different. As a consequence, ϕ CE changes continuously over time. The CE phase shift ϕ CE caused by passing through a dispersive medium of length z can be expressed as
with n the index of refraction of the medium described by the corresponding Sellmeier equations. The above phase shift ϕ CE is solely due to linear propagation of the wave-packet.
Besides this linear effect, nonlinear effects also give rise to a relative phase shift between carrier and envelope. The most prominent effect in femtosecond lasers is the third order Kerr nonlinearity responsible for self-phase modulation (SPM) leading to spectral broadening, whereas the spatial Kerr effect is exploited in Kerr-lens mode-locking (KLM). It can be shown that the Kerr effect leads to a self-phase shift of the carrier, similar to the soliton self-phase shift in fiber optics . Furthermore, the Kerr effect induces a distortion of the envelope called self-steepening causing a group delay of the envelope with respect to the underlying carrier .
During the periodic propagation of the laser pulses inside a laser cavity, the CE phase is different for each emitted laser pulse since the total phase shift ϕ CE accumulated per round trip is generally not an integer multiple of 2π. In other words, the envelope repeats itself after each roundtrip while the carrier-wave is different for successive pulses and repeats itself with the frequency f CE - the carrier-envelope frequency
where f rep is the fundamental pulse repetition frequency. Changing the linear or nonlinear contributions to the round-trip phase shift ϕ CE inside the laser cavity changes the CE frequency.
3. Composite plate for arbitrary carrier-envelope phase control
There are two well established ways to influence the value of the carrier-envelope frequency f CE . Changing the pump power of the laser changes the pulse energy and hence the nonlinear contribution to the phase shift Δϕ nonl. As a typical value for the coefficient describing the carrier-envelope frequency change due to a pump power variation, we experimentally determined 15–20 MHz per Watt of pump power variation for an octave-spanning 200 MHz laser. Varying f CE over the whole repetition rate of 200 MHz is therefore impracticable since one cannot vary the pump power by 100% or 10 W in the given case. For lower and higher repetition rates, the conversion factor scales accordingly and never allows a variation of f CE over the full repetition frequency.
However, modulating the pump power via a fast acousto-optic modulator to lock f CE to a reference frequency has proven very successful, since only small f CE modulations are necessary for locking. Operation within a closed control loop only necessitates pump power modulation on the order of a few percent.
A second way to change f CE is via material dispersion according to Eq. (2). This is, for example, done by moving a wedged BaF2 plate, thereby changing the material insertion. Varying the CE phase by 2π (and hence f CE between zero and f rep) necessitates introducing (or removing) roughly 80 µm of BaF2. An alternative is to rotate a glass plate, such as fused silica, that is operated close to Brewster’s angle. In this case a material thickness variation of approximately 60 µm is necessary.
The main problem associated with the above approach using material insertion or removal is the fact that one is not only changing the ratio between group and phase velocity but also the second order dispersion in the beam path, which impacts the pulse shaping. Fig. 1 shows the calculated change in the group delay as a function of wavelength if material is removed to achieve a CE phase variation of 2π. The curves are shown over the spectral range supporting a short few-cycle pulse on the order of ~5 fs . Due to material insertion (or removal) of 80 µm BaF2 or 60 µm fused silica, the pulse experiences a group delay variation of around 3.5 fs and 2.5 fs respectively, which is not negligible considering the pulse duration of only ~5 fs. In terms of second order dispersion, this corresponds to a change of 3 fs2 and 2.2 fs2 respectively in the group delay dispersion (GDD) at 800 nm. The detrimental effect on carrier-envelope phase sensitive experiments of such a control method has been observed in the experiments by Apolonski et al., who examined the CE phase dependence of electron photoemission by external variation of the CE phase. A significant decrease in signal strength was observed when varying material insertion . In the context of intracavity applications, octave-spanning Titanium-Sapphire (Ti:sapphire) lasers are sensitive to even such minute amounts of dispersion changes, since their dynamics is completely determined by operating within a few fs2 around the point of zero average GDD . It is therefore obvious that a change of 2–3 fs2, as introduced by a few tens of microns of BaF2 or fused silica, can have an impact on the pulse forming mechanism.
Due to the varying dispersion, the pulse spectrum and energy may be modified and the laser may finally become unstable, with the signal-to-noise ratio of the f CE beat signal decreasing.
The dependence of the GDD on the refractive index n(λ) (GDD∝λ 3·d 2 n/dλ 2·z) is quite different from the dependence of the CE phase on n(λ)(ϕ CE∝dn/dλ·z). It is therefore, in principle, conceivable that the CE phase can be changed while keeping second order dispersion constant over the relevant wavelength range. This may, for example, be done by exploiting the dispersive characteristics of two different materials, where their absolute contributions may be adapted independently to keep the GDD constant while varying the CE phase. To first approximation, we implemented such an approach by a composite plate made out of two oppositely wedged materials, where their ratio may be changed continuously by moving the plate relative to the laser beam. Our approach is therefore not to remove or insert material but replace one kind of material by a different kind of material. By moving the plate we vary the CE phase but keep the overall thickness constant, minimizing the impact on pulse energy, spectrum, and f CE beat signal strength.
Figure 2 presents a schematic of the novel composite plate. In this case, it is composed of a thick BaF2 wedge combined with a corresponding thinner fused silica part wedged in opposite direction. Both are glued together to form one common plate with a height of 10 mm, a length of 25 mm and a total thickness of around 1.5 mm.
Using a white-light interferometer, we measured the dispersion of the composite wedge. An average of four measurements is shown in Fig. 2(b) in red which represents the dispersion in the center of the plate. A fit gives information about the composition of the plate shown in blue. BaF2 is the preferred material for the octave-spanning laser to achieve optimum intracavity dispersion management .
Moving the composite plate along its longer axis continuously replaces BaF2 by fused silica or vice versa. Eq. (2) predicts a CE phase shift of 2π, when 200 µm of fused silica is replaced by the same amount of BaF2, while the impact on second order dispersion is negligible (± 0.4 fs2 at 800 nm). The red curve in Fig. 1 confirms that the alteration in group delay is significantly smaller, about one order of magnitude, in comparison to the direct use of pure material insertion or removal.
4. Adjusting the carrier-envelope phase in an interferometric autocorrelator
We first prove the functionality of the composite plate by using it in an interferometric autocorrelator suitable for sub-10 fs pulse characterization . Femtosecond autocorrelators (ACs) are usually built such that dispersion is balanced in both arms to avoid distortion of the measurement results. In interferometric autocorrelators in particular, identical optical paths in both arms of the Michelson interferometer are also required for a symmetrical interferometric autocorrelation (IAC) trace, as predicted by theory. Introducing our composite plate into one arm of the AC varies the CE phase with respect to the second arm, leading to asymmetric and/or double-peaked IACs. The layout of the AC used in the experiments is sketched in Fig. 3(a). It is a standard balanced AC with an additional input port for a calibration laser to measure the time axis. A thin type I KDP crystal is used to generate the second harmonic detected by a photo-multiplier tube.
The composite plate is introduced in one arm, whereas a homogeneous BaF2 plate of the same thickness is placed into the second arm. By doing so, the additional chirp introduced by the composite plate and the BaF2 plate in each arm can be precisely compensated by one bounce on a dispersion compensating mirror. The laser used is a mirror-only ultrabroadband Ti:sapphire laser with a FWHM optical bandwidth of more than 300 nm (VENTEON UB, NanoLayers Optical Coatings GmbH, Germany). The graph to the right in Fig. 3(b) shows the measured IAC without the plates. A phase retrieval algorithm leads to an estimated pulse duration of 6 fs. After placing the glass plates into the AC, the CE phase is tuned to an integer multiple of the CE phase in the second arm, producing a symmetric single-peak trace as expected from theory. It is interesting to notice how this IAC trace in Fig. 4(c) resembles the measurement without the plates Fig. 4(a) which means the new composite plate does not do any harm in terms of dispersion.
A series of measurements are then taken, where the position of the composite plate is varied in steps of 4 mm corresponding to shifts of the CE phase of π/2. The graphs in Fig. 4 (b)–(d) illustrate a continuous shift of the CE phase by 2π. During this measurement, we noticed that the glass plate exhibits structural inhomogeneities that change the alignment of the AC. After moving the plate a few mm, realignment of the AC (by adjusting one of the silver mirrors in the arm with the composite glass plate) became necessary. After doing so, the original power level is recovered within ±10%. The observed structural inhomogeneities are due to difficulties in manufacturing of this composite plate resulting in wavefront distortions. This was confirmed by putting the plate into a Michelson interferometer using a green solid-state laser to image the wave-front distortions. When comparing the spatial interference pattern with a regular BaF2 plate from the same manufacturer we could clearly observe a deviation from a homogeneous plate although it is difficult to quantify.
To obtain a carrier-envelope phase shift of 2π, we had to move the plate in the AC by 16 mm. Eq. (2) predicts that this must correspond to removal of 200 µm of BaF2 (and an addition of 200 µm of fused silica) we deduce a wedge angle of 0.72°.
5. Varying fCE in a 200 MHz octave-spanning femtosecond laser
After the successful test of the composite plate in the AC, another plate from the same manufacturing run was introduced into a 200 MHz octave-spanning Ti:sapphire laser (see Ref.  for details). The plate is part of the dispersion management of the laser cavity and is placed into the short arm of the z-folded asymmetric, standing wave resonator. In the other arm, two wedged BaF2 plates are located to optimize intracavity dispersion for generation of octave-spanning spectra maximizing the SNR of the f CE beat signal in a 1f-to-2f interferometer.
with the composite glass plate. Fig. 5 illustrates how f CE is varied over half the repetition rate. Initially, f CE and its mixing product with the repetition frequency (f rep - f CE) are close to half the repetition rate. By moving the composite plate stepwise by 0.2 mm, we registered a series of measurements until f CE ends up to be close to zero (the used RF-analyzer did not display signals below ~12 MHz).
The initial SNR is about 35dB and stays practically unaltered until we get close to f CE=0. During the measurement sequence, the laser stayed in mode-locked operation and did not need any kind of realignment. In Fig. 6(a) we plot f CE against the wedge position, where the right vertical axis is normalized to the CE phase shift per roundtrip. A linear fit to the data results in an inverse slope of 4.93 mm/2π. Taking into account a factor of two for the double pass inside the laser, and another factor of 1.2 due to Brewster’s angle, we end up with a value of about 12 mm per 2π phase shift. In comparison to the result of 16 mm/2π in the IAC investigation, we obtain a slightly larger gradient. There are several explanations for this observation. First, the two plates used for the two experiments were not the same and deviations may be due to fabrication tolerances. Second, due to the wave front distortions mentioned above, the intracavity pulse energy and laser beam properties vary slightly when moving the glass plate. These changes affect the carrier-envelope frequency f CE via the Kerr effect as discussed before, and manifest themselves in a variation of the local slope of the data set in Fig. 6(a). Figure 6(b) shows the corresponding spectra for each of the data points depicted to the left. Only the last two data points close to f CE=0, corresponding to the black and light blue curve, deviate from the original spectrum at the beginning of the tuning range. Average power and, equivalently, the pulse energy varied within ±10% over the whole tuning range except for the last two data points, which we attribute to the wavefront distortions caused by the plate.
In many applications, utilizing CE phase controlled oscillators with or without successive amplification, it is important to fully control the carrier-envelope phase or its temporal evolution f CE without alternating the pulse energy, pulse spectrum or pulse duration which is not possible by pure material insertion or removal.
By implementation of a novel composite glass plate arbitrary shifts of the carrier-envelope phase can be accomplished while keeping dispersion in transmission practically constant. We demonstrated the functionality of this device by varying the carrier-envelope phase differentially in the arms of an interferometric autocorrelator measuring a series of ultrashort (~6 fs) pulses. Complete carrier-envelope frequency control was demonstrated by tuning the f CE beat note of a 200 MHz, octave-spanning Ti:sapphire laser over half the repetition rate without losing mode-locking or significant changes in pulse energy and spectrum.
Besides the demonstrated applications, we anticipate that such a composite glass plate will be very helpful in compensating for long term drifts of the CE phase, as well as in many experiments where a precise and “neutral” control of the CE phase is desirable, such as in high-harmonic generation and other carrier-envelope sensitive experiments.
This work is supported by ONR N00014-02-1-0717 and DARPA HR0011-05-C-0155.
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