Free electron laser measurement of liquid carbon reflectivity in the extreme ultraviolet

: Ultrafast time-resolved extreme ultraviolet (EUV) reﬂectivity measurements of optically pumped amorphous carbon (a-C) have been performed with the FERMI free electron laser (FEL). This work extends the energy range used in previous reﬂectivity studies and adds polarization dependence. The EUV probe is known to be sensitive to lattice dynamics, since in this range the reﬂectivity is essentially una ﬀ ected by the photo-excited surface plasma. The exploitation of both s- and p-polarized EUV radiation permits variation of the penetration depth of the probe; a signiﬁcant increase in the characteristic time is observed upon increasing the probing depth (1 vs. 5 ps) due to hydrodynamic expansion and consequent destruction of the excited region, implying that there is only a short window during which the probed region is in the isochoric regime. A weak wavelength dependence of the reﬂectivity is found, consistent with previous measurements and implying a maximum electronic temperature of 0.8 eV ± 0.4.


Introduction
Ultrafast laser irradiation has become a standard way to induce phase transitions in condensed matter [1,2]. Depending on the material studied and the power of the laser, ultrafast melting, sublimation, or production of extreme states such as warm dense matter or plasmas can be effected [3][4][5][6]. Laser-driven excited states typically exist for only a few picoseconds before the excited region of the sample starts to equilibrate with the bulk, leading to extensive thermal expansion, which destroys the sample in cases such as the present. Thus, it is necessary to probe many identical samples, as is the case in this study, employing ultrafast probing and sample rastering in order to study the system before its destruction by thermal expansion [7].
Here, we detail ultrafast free electron laser (FEL) probing of laser-excited amorphous carbon(a-C) in the extreme ultraviolet (EUV) wavelength range of 21-42 nm over delay times of up to 15 ps and using both s-and p-polarized light. The goal is to elucidate the nature of highly excited carbon, as it has a particularly interesting predicted phase diagram, with unresolved questions as to whether a liquid state actually exists, if there are multiple liquid states, or if it is just a highly-driven plasma state [8][9][10][11][12].
Carbon under extreme thermodynamic conditions was first studied by Bundy, employing flash heating to induce ultrafast melting in graphite [13]. Others have performed ultrafast optical electrons can strongly affect the time evolution of the reflectivity and transmission, making data interpretation much less straightforward [15,16]. Here, an extreme ultraviolet probe in the range 25-60 eV is used so that the photons are much less affected by the photo-induced plasma. Moreover, we employ polarization as an additional probe of the system, which changes the probe depth [17,18].

Results
The measurements were carried out at EIS-TIMEX beamline of the FERMI seeded FEL [19]. The pump-probe timing has a jitter of less than 10 fs [20]. The probe wavelength and polarization were varied using the harmonic and Apple-II-type undulator at FERMI [21]. The samples comprised 100 nm thick coatings of a-C on silicon substrates. The optical (251 nm) pump power is sufficient to induce sample damage, while the EUV probe power is not. This has been verified previously, based on power densities, visual inspection, and reflectivity measurements [22].
The full dataset comprises seven FEL wavelengths with s-polarization and five with ppolarization. Representative time curves are plotted in Figure 1, showing two different polarizations and two different probe energies. The probe pulse reflectivity decreases sharply following the excitation pulse (Δt t 0), but remains roughly constant for ambient samples (data at t t 0). The long  term value measured here approaches that found at 100 ms following the pump pulse, evidencing that all of the sample dynamics occur in the time range studied here.
While s-polarization and p-polarization measurements both yield time constants that generally increase with increasing wavelength, a larger effect is found from switching from s-polarization to ppolarization. As shown in Figure 2, s-polarization has a time constant of ~1 ps, while p-polarization has a time constant of ~5 ps. Previous work estimated the plasma frequency to correspond to an energy t 5 eV, which is substantially below that of the probe energy [22]. However, at the longer probe wavelengths used here, the reflectivity should be higher which may lead to our observed change in time constant. The expected reflectivity changes are due to dynamics of the lattice, with spolarization and p-polarization providing different probe depths [23]. The p-polarized data at 28 nm appears to be aberrant compared to the rest of the data, and we believe that it is due to random fluctuations.  While s-polarization and p-polarization measurements both yield time constants that generally increase with increasing wavelength, a larger effect is found from switching from s-polarization to p-polarization. As shown in Figure 2, s-polarization has a time constant of~1 ps, while p-polarization has a time constant of~5 ps. Previous work estimated the plasma frequency to correspond to an energy < 5 eV, which is substantially below that of the probe energy [22]. However, at the longer probe wavelengths used here, the reflectivity should be higher which may lead to our observed change in time constant. The expected reflectivity changes are due to dynamics of the lattice, with s-polarization and p-polarization providing different probe depths [23]. The p-polarized data at 28 nm appears to be aberrant compared to the rest of the data, and we believe that it is due to random fluctuations.

Discussion
Transient EUV reflectivity exhibits markedly different trends then what is observed in the optical regime, which has a significantly faster reflectivity increase followed by a slow decay [14]. The optical results have been attributed to a large increase in free electron density, leading to a metallic-like surface, initially increasing the reflectivity, followed by the relaxation of the electrons and lattice expansion [24]. In contrast, EUV reflectivity is sensitive primarily to lattice dynamics, simplifying the interpretation.
Previous work has shown that the decay lifetime as a function of probe wavelength can be approximated assuming a constant surface expansion velocity at short times as where is a parameter depending on the probe polarization and the exact form of the density gradient, and ̅ is the velocity of the expanding surface [14,22,25]. This result implies that the time constant should increase linearly with the wavelength of the probe. This result can be used in conjunction with the simplification of the expression for a surface experiencing a shockwave (Equation (2)) to calculate the electronic temperature.
In this case, Z is the effective charge, is the electron temperature, M is mass of the carbon atom, and is the Boltzmann constant. Given the large scatter in the experimental data, when including all of the points and using the s-polarization value of 0.6, an expansion velocity of 9.5 ± 1 nm/ps is obtained, substantially lower than both the previously determined 17 ± 4 nm/ps and the speed of sound in graphite 18 nm/ps [14,22,26]. However, due to the limited number of points and the large error bars, this value is sensitive to the removal of outlier points, and a value within error of the speed of sound can also be obtained. Using wavelengths between 22 and 32 nm, the expansion velocity is calculated to be 13 ± 0.5 nm/ps, within the error bars of the previous measurement. The ppolarized data are even more difficult to interpret, given the large scatter and error bars, but a value consistent with expansion at the speed of sound is reasonable. The calculated surface expansion velocity can then be used to calculate the electronic temperature, as detailed in Ref. [22]. While the surface temperature calculated from a fit of all data is closer to 0.4 eV than to the 0.8 eV value reported

Discussion
Transient EUV reflectivity exhibits markedly different trends then what is observed in the optical regime, which has a significantly faster reflectivity increase followed by a slow decay [14]. The optical results have been attributed to a large increase in free electron density, leading to a metallic-like surface, initially increasing the reflectivity, followed by the relaxation of the electrons and lattice expansion [24]. In contrast, EUV reflectivity is sensitive primarily to lattice dynamics, simplifying the interpretation.
Previous work has shown that the decay lifetime τ as a function of probe wavelength can be approximated assuming a constant surface expansion velocity at short times as where f is a parameter depending on the probe polarization and the exact form of the density gradient, and υ is the velocity of the expanding surface [14,22,25]. This result implies that the time constant should increase linearly with the wavelength of the probe. This result can be used in conjunction with the simplification of the expression for a surface experiencing a shockwave (Equation (2)) to calculate the electronic temperature.
In this case, Z is the effective charge, T E is the electron temperature, M is mass of the carbon atom, and K b is the Boltzmann constant. Given the large scatter in the experimental data, when including all of the points and using the s-polarization f value of 0.6, an expansion velocity of 9.5 ± 1 nm/ps is obtained, substantially lower than both the previously determined 17 ± 4 nm/ps and the speed of sound in graphite 18 nm/ps [14,22,26]. However, due to the limited number of points and the large error bars, this value is sensitive to the removal of outlier points, and a value within error of the speed of sound can also be obtained. Using wavelengths between 22 and 32 nm, the expansion velocity is calculated to be 13 ± 0.5 nm/ps, within the error bars of the previous measurement. The p-polarized data are even more difficult to interpret, given the large scatter and error bars, but a value consistent with expansion at the speed of sound is reasonable. The calculated surface expansion velocity can then be used to calculate the electronic temperature, as detailed in Ref. [22]. While the surface temperature Photonics 2020, 7, 35 4 of 8 calculated from a fit of all data is closer to 0.4 eV than to the 0.8 eV value reported previously [22], the large scatter in the data indicates that a definitive assignment is not possible at present.
The large difference between s-polarization and p-polarization results in terms of decay time constant can be understood in terms of differences in what is being probed, as the p-polarization has a larger penetration depth [17,18], implying that it is actually sensitive to the underlying silicon substrate and related energy diffusion. However, expanding the films would make the experiment significantly harder to interpret as it would lead to a significantly less homogeneous phase of carbon.
The real and imaginary optical constants corresponding to refractive index n and extinction coefficient k from an amorphous film are shown in Figure 3 [27]. Over the 20-45 nm range, the extinction coefficient (k) increases monotonically and the refractive index (n) decreases. While there is a noticeable increase in extinction as the probe length increases, it alone does not explain the observed data. The p-polarization data at 28 nm appear to be due to random fluctuations in the experiment.
Photonics 2020, 7, 35 4 of 8 previously [22], the large scatter in the data indicates that a definitive assignment is not possible at present. The large difference between s-polarization and p-polarization results in terms of decay time constant can be understood in terms of differences in what is being probed, as the p-polarization has a larger penetration depth [17,18], implying that it is actually sensitive to the underlying silicon substrate and related energy diffusion. However, expanding the films would make the experiment significantly harder to interpret as it would lead to a significantly less homogeneous phase of carbon.
The real and imaginary optical constants corresponding to refractive index n and extinction coefficient k from an amorphous film are shown in Figure 3 [27]. Over the 20-45 nm range, the extinction coefficient (k) increases monotonically and the refractive index (n) decreases. While there is a noticeable increase in extinction as the probe length increases, it alone does not explain the observed data. The p-polarization data at 28 nm appear to be due to random fluctuations in the experiment. In principle, it is possible to invert the reflectivity measurements to obtain n and k (the real and imaginary components of the complex index of refraction) from the Kramers-Kronig relation. However, given the quantity of data here, it was not possible to solve due to discontinuities related to solution hopping [28]. A direct solution of the real and imaginary parts of the optical constants would allow for the determination of reflectivity and transmission as a function of time, and given additional related experiments, temperature. This would effect a more detailed understanding of the physical properties of liquid carbon. While this can, in principle, be addressed by making large approximations, it is difficult to justify such approximations here.

Materials and Methods
The experimental setup has been detailed previously [22,29]. Briefly, at the EIS-TIMEX endstation, an optical pump pulse (251 nm, 130 fs, pulse energy ~35 µJ) focused to approximately 70 × 100 µm 2 FWHM and an FEL pulse (very roughly 29-59 eV, pulse length ~70 fs, ~1 µJ) focused to 30 × 15 µm impinged upon a 100 nm amorphous carbon film of 2.1 g/cm 3 density with roughness less than 1 nm, and oriented at approximately 40° relative to the pump pulse [19]. The fluence of the optical laser had a large variation and may be related to the noise in the observed signal. The FEL and optical pulses were divergent by approximately 10°, which was exploited to separate the beams and measure the reflected FEL intensity on an AXUV photo-diode (Opto-Diode) with a response time of In principle, it is possible to invert the reflectivity measurements to obtain n and k (the real and imaginary components of the complex index of refraction) from the Kramers-Kronig relation. However, given the quantity of data here, it was not possible to solve due to discontinuities related to solution hopping [28]. A direct solution of the real and imaginary parts of the optical constants would allow for the determination of reflectivity and transmission as a function of time, and given additional related experiments, temperature. This would effect a more detailed understanding of the physical properties of liquid carbon. While this can, in principle, be addressed by making large approximations, it is difficult to justify such approximations here.

Materials and Methods
The experimental setup has been detailed previously [22,29]. Briefly, at the EIS-TIMEX end-station, an optical pump pulse (251 nm, 130 fs, pulse energy~35 µJ) focused to approximately 70 × 100 µm 2 FWHM and an FEL pulse (very roughly 29-59 eV, pulse length~70 fs,~1 µJ) focused to 30 × 15 µm impinged upon a 100 nm amorphous carbon film of 2.1 g/cm 3 density with roughness less than 1 nm, and oriented at approximately 40 • relative to the pump pulse [19]. The fluence of the optical laser had a large variation and may be related to the noise in the observed signal. The FEL and optical pulses were divergent by approximately 10 • , which was exploited to separate the beams and measure the reflected FEL intensity on an AXUV photo-diode (Opto-Diode) with a response time of significantly less than Photonics 2020, 7, 35 5 of 8 0.1 s. The photodiode was shielded from the pump pulse and other stray radiation by a 100 nm thick Al foil. Polarization of the FEL has been characterized previously [30].
As the pump pulse damaged the sample, a series of FEL probe pulses were used to optimize the measurement: (i) Three FEL pulses well before the arrival (unpumped reflectivity); (ii) One value followed very shortly after the pump pulse (pump-probe); (iii) One value 100 ms after the pump pulse (post damage reflectivity).
The sample was then rastered to a new spot and the sequence repeated. We observed no evidence of sample damage from step (i). Error bars are one standard-deviation. Shown in Figure 4 is the different reflectivity curves for the entire series of pump pulses with horizontal polarization at 41.83 nm.
Photonics 2020, 7, 35 5 of 8 significantly less than 0.1 s. The photodiode was shielded from the pump pulse and other stray radiation by a 100 nm thick Al foil. Polarization of the FEL has been characterized previously [30]. As the pump pulse damaged the sample, a series of FEL probe pulses were used to optimize the measurement: (i) Three FEL pulses well before the arrival (unpumped reflectivity); (ii) One value followed very shortly after the pump pulse (pump-probe); (iii) One value 100 ms after the pump pulse (post damage reflectivity).
The sample was then rastered to a new spot and the sequence repeated. We observed no evidence of sample damage from step (i). Error bars are one standard-deviation. Shown in Figure 4 is the different reflectivity curves for the entire series of pump pulses with horizontal polarization at 41.83 nm.  The pre-shots (black triangles), pump-probe (red circles) and post-shots (blue triangles) corresponding to steps (i) (ii) and (iii), respectively. Only the pump-probe shows a large change in reflectivity on the timescale studied here. The reflectivity observed as a function of delay was fit using the following empirical model: The dynamics studied are significantly slower than the pulse durations used in the experiment, so such a simple model is well justified.

Conclusions
Irradiation by intense ultrafast optical laser pulses causes a large decrease in the EUV reflectivity of amorphous carbon films. The process is significantly simpler than what is found with reflectivity in the optical regime, and is consistent with previous measurements, revealing a maximum electron temperature of 0.8 eV, while adding to the energy ranges and polarization dependence explored. The s-polarized FEL probe decays significantly faster (~1 ps) then the p-polarized (~5 ps) probe. The p-polarized FEL probe has a larger penetration depth and is thus sensitive to different dynamics than the s-polarized probe is. Ultimately, given additional data, it should be possible to completely invert the Kramers-Kronig relationship and solve for n and k.