A self referencing attosecond interferometer with zeptosecond precision
AA self referencing attosecond interferometerwith zeptosecond precision J AN T ROSS , , G EORGIOS K OLLIOPOULOS , AND C ARLOS
A.T
RALLERO -H ERRERO J.R. Macdonald Laboratory, Department of Physics, Kansas State University, Manhattan, Kansas 66506,USA Extreme Light Infrastructure - Nuclear Physics, Romania Department of Physics, University of Connecticut, Storrs, CT 06268, USA * [email protected] Abstract:
In this work we demonstrate the generation of two intense, ultrafast laser pulses thatallow a controlled interferometric measurement of higher harmonic generation pulses with 12.8attoseconds in resolution (half the atomic unit of time) and a precision as low as 680 zeptoseconds(10 − seconds). We create two replicas of a driving femtosecond pulse which share the sameoptical path except at the focus where they converge to two foci. An attosecond pulse trainemerges from each focus through the process of HHG. The two attosecond pulse trains fromeach focus interfere in the far field producing a clear interference pattern in the XUV region.By controlling the relative optical phase between the two driving laser pulses we are able toactively influence the delay between the pulses and are able to perform very stable and precisepump-probe experiments. Because of the phase operation occurs across the entire spatial profilewe effectively create two indistinguishable intense laser pulses or a common path interferometerfor attosecond pulses. Commonality across the two beams means that they are extremely stable toenvironmental and mechanical fluctuations up to a Rayleigh range from the focus. In our opinionthis represents an ideal source for homodyne and heterodyne spectroscopic measurements withsub-attosecond precision. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It takes photons, carrying information about the electromagnetic interaction between electronsand atoms ≈ . a r X i v : . [ phy s i c s . a t o m - ph ] M a y ig. 1. (a) Experimental setup scheme. Light is reflected from an SLM and is focused into agas jet using a lens (f=50 cm) to produce harmonics from two foci. The harmonics propagateand interfere in the far field (spectrometer detector). (b) Working schematics of the appliedphase masks. The two masks are applied to the entire beam in a check pattern. The phasesare wrapped in multiples of 2 π . (c) Profiles of the 11th and 19th harmonic showing theinterference pattern. The number of peaks changes with the wavelength (harmonic order)and with the distance between foci. (d) Evolution of the interference pattern of harmonic 11as a function of the relative offset phase between the two masks ∆Φ SLM . quantum state of the target or more precisely the phase of the induced dipole and electron wavepacket.
2. Experimental setup
In this work we generate harmonics from two optical foci ( f and f ) produced by a system of atwo-dimensional spatial light modulator (SLM) and a lens (Fig. 1 a). The two foci are generatedusing two intertwined phase masks (Fig. 1 b) with opposite wave front tilt. We start with anintense 30 fs pulse incident on the SLM to produce peak intensities at the foci of ≈ W/cm .At these intensities, when interacting with a dense gas target, harmonics are generated at eachfocus. As the harmonics propagate towards the detector they interfere in the far field. An exampleof the measured interference pattern of harmonics 11th (71.4 nm, 17.4 eV) and 19th (41.3 nm, 30eV) is shown in Fig. 1 c). Similar interferometric schemes have been used in the past [20, 24, 26]with HHG, but have to compromise between stability or flexibility. With our method it is possibleto control the generation of the one XUV beam (spatially and temporally) independently from theother. This is the novelty of this scheme, which consists an advantage in comparison to previouslyproposed methods [20, 23, 27]. By applying an offset to one mask relative to the other we cancontrol the delay or phase (within one cycle) of one pulse relative to the other. This is simply ig. 2. Spatial profiles of the laser at the focus. In panel a), the intensity distribution isgiven for two laser foci separated by 100 µ m and with no additional phase offset, as seenin panel b), where we display the phase distribution of the light. In panel c), we show theintensity distribution of a phase mask that imprints an phase offset of π / π / π / σ away fromthe center of focus (as indicated by the circles) understood in terms of Fourier optics [28]. Since the SLM is limited in the amount of overallphase (2 × π at 785 nm), we wrap the phase similar to Fresnel lenses [28]. Because the mask is aphase change across the entire beam profile, there is only one beam with two diverging wavefronts. With this, we are able to generate two beams that share almost exactly the same beampath. They only become distinguishable at approximately one Rayleigh range before the focus.As the beam is focused in vacuum, the difference in optical path is only dictated by fluctuationsin the background pressure over a distance of a few millimeters. The method of pulse shapinghas been actively used as a means of generating zeptosecond precision interferometers [29]in a frequency domain setup, which does not allow the generation of two spatially separated ig. 3. Phase of harmonics 9 to 19 (left axis) measured from the interferograms as shownin Fig. 1 c) as a function of the relative SLM phase ∆Φ SLM (bottom axis). The right axisis the phase in units of 2 π showing that the phase evolution of each harmonic 2 q + ∆ φ ( q + ) = ( q + ) × ∆Φ SLM . Top axis is the corresponding delay time for a 785 nmwavelength pulse. The SLM has access to 205 phase values thus capable of a resolution of12.8 attoseconds. pulses. It is important to remark that the phases are applied to the entire beam and not splitin distinguishable halves, like in previous works [20, 27]. The phase and tilt of each pulse isidentified by using a checkerboard phase pattern where f is controlled by “white" squares and f is controlled by “black" squares. The main advantage of using an intertwined pattern is that thetwo beams are completely indistinguishable from each other up to a few millimeters from thefocus, which was also the case in previous methods [20, 27]. Nevertheless, with the proposedscheme, the sampling on the laser mode is much better, the alignment process much easierand the whole optics setup more resistant to slight beam pointing instabilities. In addition, thedistance in between the two foci (the two secondary sources) can be tuned dynamically, withoutany modification on the optical setup and without any realignment, only by applying on the SLMa new set of sub-patterns with a different tilt with each other. This possibility can go so faras to isolate one of the two sources. One has just to increase the tilt of the sub-pattern whichcorresponds to the second source so much as to transpose this last one far away from the target(for an application which makes use of this, see section 4). Consequently, it is possible to checkthe emission from each one of the two sources separately before letting them interfere in the farfield. To our knowledge this is a completely new possibility in the field of HHG spectroscopywhich can be utilized in order to in situ check if the two secondary sources are indeed identical.To preserve phase matching in HHG the Rayleigh range is chosen to be much larger than theinteracting region (500 µ m in our case). Also, the distance between the two foci is smaller thanthe inner diameter of the glass capillary used to generate the gas jet. Practically, this means thatdifferences in the optical path will be dictated by changes in pressures of 10 − to 10 − Torr overone Rayleigh range. . Stability and precision measurement
The overall phase of the driving field has a great impact on the generation of harmonics [30]. InFigure 2, we show that our tilted Fresnel masks produce Gaussian shaped foci with transversalphase distributions which are almost flat. In the figure and with a separation of 100 µ m , thetwo laser fields still overlap slightly in the focus and the spatial phase of the two sources isaltered (through interference effects). The intensity distribution is shown in the left panel. There,intensity differences of less than 1% can be observed between phase masks of altered delay. Inthe middle and right panel, we show the spatial phase distribution of two separate masks. There,the phase profile across the foci is flat, except at the wings (we define wings as intensities of lessthan 10% of the peak intensity) for a few values of the relative phase between the two foci(e.g. pi / π / ∆Φ SLM . The evolutionof the interference fringes in harmonic 11 as a function of ∆Φ SLM is shown in Fig. 1 d). In ourSLM we can control ∆Φ SLM with a step size of ≈ . ∆ φ q , asa function of the relative delay of the two foci we need to calculate the phase evolution foreach harmonic. We can do this calculation in two ways. One option is to measure the relativedisplacement of the maxima and minima in the fringe pattern keeping in mind that a 2 π shiftcorresponds to the distance between two peaks [25]. Our method makes use of Fourier analysisof the fringe pattern. In this approach ∆ φ q is the phase of the fringe frequency of the Fourierspectrum, which corresponds to the inverse spacing between peaks in the detector. Becausethe fringe separation in the detector is proportional to λ ( q + ) / D with D the spacing betweenthe two sources, the fringe spacing reduces with harmonic order. For this study, we kept theseparation between the two foci at 100 µ m. The relative phase evolution for harmonics 9 to 19(left axis) using the Frequency Fourier analysis is shown in Fig. 3. In addition, we show on theright axis the phase in units of 2 π for each harmonic. From this plot the scaling of the phase ∆ φ ( q + ) , for each harmonic of order 2 q +
1, as a function of ∆Φ SLM is clear. This scaling isanother demonstration of the self-balancing geometry of our interferometer. For each oscillationperiod of the fundamental there are exactly ( q + ) oscillations in the electromagnetic field forharmonic ( q + ) . Again, we are able to observe this thanks to the self-referencing nature of theinterferometer. To be able to observe oscillations in the harmonics we of course need attosecondresolution in our time steps. The top axis in Fig. 3 shows the corresponding delay time for a785 nm center wavelength pulse. As mentioned before the SLM has access to 205 effectivephase values thus capable of a resolution of 12.8 attoseconds. This resolution was confirmed bymeasuring the amount the fringes moved for each harmonic. However, the main limitation in mostinterferometers is not in the resolution but rather in the precision. Temperature, humidity andmechanical instabilities are often enough to cause errors in the orders of hundreds of attoseconds.To further demonstrate the power of our method, we calculate the jitter for harmonics 9 to 19 ig. 4. Histogram of the jitter for all delays as a function of harmonic. The jitter is calculatedas the standard error, for each delay value taking into account the 17 measurements done ateach delay. Each of the 17 measurements are made up 600 laser shots. using 17 independent experiments at each delay. The jitter is calculated from the standard errorof the phase using the 17 measurements. Each one of the 17 measurements is comprised of 600lasers shots, the number of laser shots chosen to form a harmonic picture. Figure 4 shows thehistogram of the jitter values for the 205 delay values for harmonics 11 to 19. The results areplotted as a histogram of the jitter values. Harmonic 9 is not included in the figure but is includedin the summary below. HO SE [as] deviation [as]9 3.10 2.811 0.80 0.7213 0.68 0.6015 0.71 0.6317 0.80 0.7119 1.10 0.94 Table 1. Column “SE" are the standard error estimates calculated from 17 images takenfor each harmonic with each image consisting of 600 laser shots. Since we can assumeuncorrelated errors for the temporal jitter, we can also use the 205 delay “experiments" asmeasurements for the error.
The results from Fig. 4 are summarized in Table 1. The SE column reports the standard error
Harmonic 13 Harmonic 16 Harmonic 19 (a)(b) (c) (d)
Fig. 5. (a) Experimental setup scheme. The 2nd harmonic of the 785 nm light is generatedin a BBO crystal. 785 nm and 392 nm beams are separated by a dichroic mirror. The IRpart is reflected from an SLM and its 2nd harmonic is reflected from a silver mirror. In theSLM masks is now added a Fresnel lens pattern (f = 11 m). After the recombination, 785nm and 393 nm beams are focused together into a gas jet at a single focus. (b), (c) and (d)Evolution of the harmonics 13, 16 and 19 as a function of the offset phase applied on theSLM sub-pattern. This corresponds to a tunable phase-delay between the 785 nm and the393 nm electric fields. of the jitter from the histogram and the deviation column is the standard deviation from the samedistribution. From both, the histogram and the table, we can clearly observe that we have aprecision better than one attosecond. Harmonic 13 has the best precision of 680 zeptosecondsfollowed by harmonic 15 with a precision of 710 zeptoseconds. We should mention that harmonic9 and 19 have the worst resolution of 3.1 and 1.1 attoseconds. These larger uncertainties are dueto poor signal levels in harmonic 9 and due to a lack of resolution in the fringes in harmonic19. The low signal of harmonic 9 is due to the response of the holographic grating used. Asmentioned, the resolution of the fringes can be improved by changing the distance between the foci.
4. A two-color Michelson-type interferometer
We have already discussed on the possibility to use the above scheme as a self-referencinginterferometer. Here we will show that the phase delay induced by our method can be consideredas well in respect to an external field. In other words, we will show that, with our scheme, theSLM can successfully substitute the movable arm of a Michelson-type interferometer, that theSLM is able to perfectly act as a delay stage of high precision and stability.Here we retrieve the delay dependence on HHG, when driven by a two color laser field inargon gas. The experimental setup is shown in in Figure 5 panel(a), where an incoming 785 nmlaser field produces its second harmonic in a 200 µ m thick beta barium borate (BBO) crystal. Aichroic mirror separates the 393 nm radiation from the 785 nm one. The former is reflected on aflat silver mirror and the last is reflected on the SLM surface. They recombine and they are bothfocused on the target. On the SLM is applied a series of masks which induce in the gas target areaa single focal spot identical to the one of the pair of spots used before. This is feasible simply byincreasing by one order of magnitude the tilt angle of one of the subpatterns. As a result, thesecond focal spot is formed far away from the gas-target. The two components (785 nm and 393nm) are focused using a lens with a focal length equal to 50 cm for a wavelength equal to 800 nm.The focal length of the lens is not the same for the 393 nm centered light. We are able to bring thetwo beams to focus together by simply adding a fresnel lens pattern (with focal length equal to 11m) on the top of each one of the masks that we apply on the SLM. With this small modification,in the series of masks, and not in the experimental setup, the IR is brought to focus together withits 2nd harmonic on the gas-target. This proves the high versatility of the proposed method. Thepolarizations of the IR and its 2nd harmonic are perpendicular to each other and the amplituderatio of the 393 nm field in respect to the 785 nm field is 0.22. As demonstrated in numerousprevious works [31, 32], the second harmonic can act as a gate on the process of high harmonicgeneration. Depending on the delay between 785 and 392 nm, high-Harmonic generation of anyspecific order is either suppressed or enhanced. This appears in the characteristic form of yieldoscillations (see Figure 5 panels(b)-(d)), as it has been confirmed before.
5. Summary
To conclude, we have presented in this work a novel interferometer, for trains of attosecond pulsesgenerated through HHG, with tunable arms and high stability. An impressive delay resolutionwith high precision has been proven by the fact that the phase difference of each harmonic order ( q + ) , φ ( q + ) , scales with the phase of the fundamental as φ ( q + ) = ( q + ) × φ f undamental .This interferometer resembles a common path homodyne detector for the fundamental as twoidentical copies of the driving pulse are created at the focus by using an SLM and a lens. Sincethe two beams are indistinguishable until ≈ one Rayleigh range from the focus, their optical pathare identical and almost jitter free. Because of this, the phase between the two foci is perfectlylocked. For the first time, the relative phases of the driving field in each one of the two foci canbe manipulated. The trains of attosecond pulses, which emanate from these secondary sources,can be delayed with respect to each other with a resolution of 12.8 attoseconds (half of the atomicunit of time). This delay can be controlled with a precision of 680 zeptoseconds for harmonic 13. Funding
This work was supported by the Chemical Sciences, Geosciences, and Biosciences Division,Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy (DOE) underGrant No. DE-FG02-86ER13491.
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