Abstract
Precise knowledge of the behaviour of the phase of light in a focused beam is fundamental to understanding and controlling laserdriven processes. More than a hundred years ago, an axial phase anomaly for focused monochromatic light beams was discovered and is now commonly known as the Gouy phase^{1,2,3,4}. Recent theoretical work has brought into question the validity of applying this monochromatic phase formulation to the broadband pulses becoming ubiquitous today^{5,6}. Based on electron backscattering at sharp nanometrescale metal tips, a method is available to measure light fields with subwavelength spatial resolution and subopticalcycle time resolution^{7,8,9}. Here we report such a direct, threedimensional measurement of the spatial dependence of the optical phase of a focused, 4fs, nearinfrared pulsed laser beam. The observed optical phase deviates substantially from the monochromatic Gouy phase—exhibiting a much more complex spatial dependence, both along the propagation axis and in the radial direction. In our measurements, these significant deviations are the rule and not the exception for focused, broadband laser pulses. Therefore, we expect wide ramifications for all broadband laser–matter interactions, such as in highharmonic and attosecond pulse generation, femtochemistry^{10}, ophthalmological optical coherence tomography^{11,12} and lightwave electronics^{13}.
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Main
In ultrafast light–matter interactions, the phase of the optical carrier field with respect to the pulse envelope’s maximum—the carrierenvelope phase (CEP, see Supplementary Information)^{14}—is one of the fundamental controls that allows one to steer chemical reactions^{15}, the generation of attosecond pulses via highharmonic generation^{16}, and electron emission and acceleration from solid surfaces and nanostructures^{7,9,17}, among others. Hence, determining and controlling the CEP is mandatory in many fields using lasers, but taking into account the broadband and often intense and ultrashort nature of these pulses is challenging and an active area of research^{18,19}. Further, nonlinear light–matter interactions usually take place in the focus of a beam where the CEP shows a strong spatial dependence. Thus, for a detailed understanding of and field control over these processes, it is essential to take the focal phase evolution, target position, and target extent into account^{13,20}. This is as important as controlling the CEP of the input beam itself.
For a focused monochromatic beam, the onaxis phase shift due to diffraction is described by the familiar Gouy phase, which follows a simple arctangent curve^{1,2}. However, many of today’s coherent light sources, even some as common place as those used in ophthalmological diagnostics, are far from being monochromatic. Rather, they can span close to, and many even exceed, an octave of spectral bandwidth^{12,21,22}. Moreover, recent theoretical studies, based on diffraction theory for pulsed Gaussian beams, yielded spatially dependent phases that significantly deviate from the simple Gouy phase and show a much more complex behaviour that is dictated by the wavelengthdependent input beam geometry^{5,6}. The need for further investigation is underscored by experimental studies, which strongly suggest deviations from the monochromatic Gouy phase^{23,24,25,26}.
In recent years, significant advancements have been made in CEP detection and control^{14,16,18,27}, which have facilitated and driven the discovery of more and more processes that are dependent on and can be controlled via the CEP. One such phenomenon is the strongfield, fewcyclelaserdriven backscattering of photoelectrons; namely, those electrons freed by the laser field that can then be driven back to and scattered off their parent matter when the field flips sign within an optical cycle. The large kinetic energy obtained by these electrons strongly depends on the CEP, which we utilize as an experimental signature (see Supplementary Information). This extremely CEPsensitive effect was observed in noble gases and recently also at solidstate nanotips^{7,8,9}. For the latter, strongfieldinduced photoemission happens almost exclusively in the enhanced optical nearfield region at the apex of the sharp tip, with a radius of ∼10 nm (ref. 28). Thus, electrons from such a highly localized source are particularly well suited to be used as a sensor to probe focused light fields with resolution better than their natural length and timescales. Namely here, we obtain a spatial resolution of ∼10 nm (≪λ/2 ≈ 350 nm, the typical length scale of a focus of light with wavelength of λ ≈ 700 nm), and a CEP resolution of ∼80 mrad, corresponding to ∼60 as (≪2.3 fs, the optical period of light with λ ≈ 700 nm).
In this work we present a quantitative, direct, threedimensional mapping of the CEP evolution of a focused broadband laser beam, spanning a range of seven times the Rayleigh range along the propagation axis and one and a half times the local beam radius perpendicular to the optical axis. To achieve this, we have combined a nanotipbased focus characterization setup with a xenongasbased, stereographic, abovethreshold ionization CEP meter^{27,29}. Every laser pulse from a hollowcore fibre compressor is split between the two separate, but synchronized and parallel measurements: one for individual characterization of the random CEP of each shot and the other for recording electron spectra from the laser–nanotip interaction (see Methods and Supplementary Information). In this way, every detected electron from the nanotip can be associated with the CEP value for the specific laser pulse producing that electron. The beam towards the nanotip focus measurement was spatially filtered to remove higher transverse modes and provide a good ground state (TEM_{00}) approximation as required for the following model. Measurements were done to probe the relative CEP shift both along the laser’s propagation axis and radially outward, mapping the cylindrically symmetric threedimensional space. Both tungsten and gold tips were used. In comparison with previous experimental work^{23,24,25}, in this measurement we probe the focus of a typical intense fewcycle pulse laser beam with a significantly better spatial resolution, scan a larger range of the focal volume and attain a welldefined focus with minimized aberrations by use of an offaxis parabolic mirror and a spatial filter element upfront. In combination with considerable efforts to flatten the spectral phase, these factors allow for a clear and thorough experimental and theoretical investigation of deviations from the monochromatic Gouy phase and additionally reveal the transverse phase structure.
The longitudinal phase difference for a focused, continuous and monochromatic Gaussian beam as compared with a plane wave is described along the propagation axis by
and is known as the Gouy phase, where z is the laserpropagation direction and z_{R} is the Rayleigh length^{1,6,23}. Laser pulses on the other hand are composed of a coherent superposition of a broad range of wavelengths, leading to a much more complex, spatially dependent phase profile. Moreover, the transverse shape and the divergence of the Gaussian beam in front of the focusing element are, in general, wavelengthdependent. Thus, it is necessary to take into account the wavelengthdependent input beam geometry to determine the spatial dependence of the CEP after the focusing element. A more generalized treatment of the strong focusing of chirpfree pulsed Gaussian beams in terms of their properties prior to focusing combines the concept of enveloped carrier oscillations with fundamental diffraction theory^{2}, which introduces isocarrierphase fronts and pulsepeak fronts^{5,6}. Their difference results in a relative CEP shift in the focal area of
where Δφ is defined to be 0 in the focal reference plane (defined by z = 0) for all radii r; here, z_{R} is the Rayleigh length at the centre frequency; w(z) is the zdependent beam radius, and r/w(z) is the normalized radial coordinate (see Supplementary Information). The last term of equation (2) describes the difference to the axial Gouy phase, equation (1), and accounts for the wavelengthdependent geometry of the input beam and the difference in curvature of the carrierphase fronts as compared with that of the pulse fronts. This term scales with g_{0}—a dimensionless geometry factor of the input beam evaluated at the central angular frequency of the laser spectrum ω_{0}, which we call the Porras factor^{5,6}.
The Porras factor is given by
and represents the normalized first derivative of the input beam’s Rayleigh length, Z_{R}(ω) (capital letters are used for input beam parameters), with respect to the laser’s spectral angular frequencies, ω, evaluated at ω_{0}. The Rayleigh length before the focusing element is linked to the frequencydependent input beam waist, W(ω), by Z_{R}(ω) = ω × W(ω)^{2}/(2c), where c is the speed of light. Three characteristic cases can be highlighted: g_{0} = −1 for an isodiverging input beam, that is, with a constant divergence angle for all ω; g_{0} = 0 for an isodiffracting beam, that is, with a constant Rayleigh length; and g_{0} = +1 for a beam with a frequencyindependent waist radius^{6} (see Fig. 1 for example phase profiles and Fig. 3 for the case g_{0} = 0). Note that at g_{0} = 0, all frequencies are diffracted identically in the firstorder approximation and hence the phase shift consists merely of the Gouy component. However, one ought not simply assume or expect g_{0} to be 0 as this is just one special and specific case, which is within a continuous range of possible g_{0} values and rarely found in practice.
Figure 3 shows the results of two onaxis measurements together with the monochromatic Gouy phase curve. The measured CEP shows extrema at z ≈ 1.7 times the Rayleigh length both before and after the focus, resulting in a much steeper slope in the focal plane as compared with the arctangent curve of the Gouy phase. The data can be fitted well by equation (2) at r = 0, resulting in g_{0} = −2.1 ± 0.2 for the first (I) and −1.8 ± 0.3 for the second (II) data set. These g_{0} values are corroborated by independent measurements of the spectral input beam properties outside the vacuum chamber, namely the spectrally resolved Rayleigh length calculated from the beam diameter. This provides an attractive and more easily attainable alternative way to estimate g_{0} via equation (3) and, therefore, obtain an approximate idea of the focal CEP evolution in the interaction region (see Supplementary Information and Fig. 1). Note that further work is warranted and needs to determine whether this technique will fulfil its promise. In addition, the similarity of the experimental curves in Fig. 3 shows the reproducibility of the measurement.
We were also able to produce a threedimensional map of the CEP by scanning both z and r with the tip (Fig. 4). We observe that there is a strong radial dependence of the CEP, again differing from the monochromatic Gouy phase. Fitting equation (2) to the data points yields g_{0} = −1.2 ± 0.3, where g_{0} and the offsets in z, r and Δφ are free parameters and the Rayleigh length z_{R} was determined independently by a knifeedge measurement to z_{R} = 350 ± 50 μm (see Supplementary Information). For negative g_{0} values, as observed in all measurements here, characteristic features of the CEP around the focus can be recognized (see Figs 4 and 1). First, the radial dependence of the CEP surface is concave before, for example, at z = −800 μm, and convex after the focus, for example, at z = 800 μm. Second, radially further out from the axis, that is, r ≳ w(z), the CEP changes sign, implying that the vector potential at the pulse peak changes sign as well (see Supplementary Equation 9). In other words, the CEP evolves with opposite slopes on and off the optical axis. Third, in the focal plane at the waist radius, that is, at (z, r) = (0, w_{0}), the phase exhibits a saddle point. In this measurement, the value for g_{0} differs from that obtained previously in the onaxis case as it was done with different laser tuning. This underscores the need to properly characterize the light source in use and determine the influence of laser tuning parameters, such as hollowcore fibre pressure, on the spectral geometry of the output beam and g_{0} to facilitate advanced phase control in the future.
In the present experiment, the nanotip samples such a small volume of the laser focus in comparison with the Rayleigh range and the focal waist size that it can be assumed that only a single intensity and phase are effective. However, this is not the case in most ultrafast, laserinduced processes. To account for the fact that common targets often experience a large range of laser intensities, one must properly weight the results with the intensity and targetdensity profiles, that is, the intensityvolume effect needs to be taken into account^{30}. Further, here we see that this problem is significantly exacerbated for broadband lasers as there is an analogous and coupled phasevolume effect due to the twodimensional phase profile. In some situations, these complications could be reduced by physically limiting the extent of the target or effectively limiting it by selecting a highorder process. However, they are by no means eliminated. For example, even for a target, very thin in the laserpropagation direction, Δz ≪ z_{R}, the phase is often strongly radially dependent even within the beam width. Thus, the common assumption of an arctangent dependence of the CEP is often unwarranted and insufficient.
Although these effects can complicate interpretation, they also have the potential to be used to enhance desired effects and perhaps even be utilized in novel ways, for instance, to improve phase matching in highorder harmonic and attosecond pulse generation. Further, one could optimize these effects by tailoring the interaction region via the input beam geometry, expressed by g_{0}. In addition, we expect that recent developments such as particle trapping and acceleration of atoms with femtosecond laser pulses may benefit from the CEP control demonstrated here through an interesting welllike structure formed by the CEP gradients^{31,32,33}. Hence, new ways of controlling atoms with large forces through ultrashort, strong laser fields may result. The detailed knowledge of the phase evolution found here impacts many fields where knowledge and control of the optical phase is vital and fascinating developments can be foreseen.
Methods
We employ a phasetagging method^{29} for recording the CEPdependent electron backscattering at tungsten and gold nanotips, placed at different points near the focus (see Supplementary Figs 1 and 2). This method utilizes simultaneous measurements of electron timeofflight spectra from the metal nanotip in event mode on the one hand, and of the CEP of every single shot of the fewcycle pulse train on the other, the latter being determined by a phasemeter^{27}. The events in both measurements are synchronized by triggering on the same laser pulse. Both measurements rely on the strong CEP dependence of backscattered, laserdriven photoelectrons; namely, those electrons that return to their parent matter and scatter elastically back from it after acceleration in the optical field.
In the CEP meter the two electron spectra that build up from xenon in the left and right direction of the horizontal laser polarization are recorded and compared by calculating the asymmetry parameter A = (N_{L} − N_{R})/(N_{L} + N_{R}), that is, the contrast, where N is the electron yield left and right, respectively. From this parameter, the CEP of the laser shot can be evaluated ^{27}. Exploiting the high spatial resolution and pronounced CEP sensitivity of the electron backscattering at the metal nanotip, we recorded phasetagged kineticenergy spectra at positions on the optical axis and offaxis on hyperbolic curves in a plane along the focus (see Fig. 1 and Supplementary Information 2).
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
We gratefully acknowledge M. Förster for supplying nanotips, and P. Dombi and S. Thomas for support in the measurement campaign. This work has been supported by the DFG Grants PA 730/5 and SPP1840 QUTIF, LaserlabEurope EUH2020 654148, ERC Grant NearFieldAtto, and the DFG Cluster of Excellence Munich Center for Advanced Photonics. D.H. acknowledges the Helmholtz Association for financial support. M.K. acknowledges the Minerva Foundation and the Koshland Foundation for financial support.
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Hoff, D., Krüger, M., Maisenbacher, L. et al. Tracing the phase of focused broadband laser pulses. Nature Phys 13, 947–951 (2017). https://doi.org/10.1038/nphys4185
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DOI: https://doi.org/10.1038/nphys4185
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