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We present an extended theoretical background of so-called fluence scan (f-scan or F-scan) method, which is frequently being used for offline characterization of focused short-wavelength (EUV, soft X-ray, and hard X-ray) laser beams [J. Chalupský et al., Opt. Express 18, 27836 (2010)]. The method exploits ablative imprints in various solids to visualize iso-fluence beam contours at different fluence and/or clip levels. An f-scan curve (clip level as a function of the corresponding iso-fluence contour area) can be generated for a general non-Gaussian beam. As shown in this paper, fluence scan encompasses important information about energy distribution within the beam profile, which may play an essential role in laser-matter interaction research employing intense non-ideal beams. Here we for the first time discuss fundamental properties of the f-scan function and its inverse counterpart (if-scan). Furthermore, we extensively elucidate how it is related to the effective beam area, energy distribution, and to the so called Liu’s dependence [J. M. Liu, Opt. Lett. 7, 196 (1982)]. A new method of the effective area evaluation based on weighted inverse f-scan fit is introduced and applied to real data obtained at the SCSS (SPring-8 Compact SASE Source) facility.
© 2013 Optical Society of America
1. Introduction
Rigorous spot-size and fluence profile characterization of focused short-wavelength (EUV, soft and hard X-ray) laser beams belongs to important prerequisites for experiments employing such sources. Contrary to conventional UV/vis/NIR lasers, which are nowadays capable of producing almost perfect Gaussian beams [1], the homogeneity of short-wavelength laser beams still requires some improvements. Due to fundamental (largely single-pass) mechanisms of X-ray lasing (e.g. ASE – Amplified Spontaneous Emission or SASE – Self-amplified Spontaneous Emission) and absence of resonator cavity, the short-wavelength laser beams often exhibit non-uniformities in their intensity profiles and wave-front. Furthermore, a specific behavior of short-wavelength and highly coherent radiation at “imperfect” reflective or refractive optical surfaces may introduce additional distortions to the laser beam. Therefore, X-ray laser beams need to be treated as non-Gaussian [2] rather than Gaussian.
At present days a few techniques already exist enabling rigorous out-of-focus X-ray laser beam characterization, e.g. use of luminescence crystals [3–5], Hartmann sensor [6–8], X-ray CCDs [9], and phosphor-coated EUV cameras [8]. Nevertheless, entering the focal region is inevitably accompanied with several difficulties originating in high radiation intensity and reduced spot-size. Hence immense demands are being placed on dynamic range and spatial resolution of diagnostic methods and devices capable of in-focus beam characterization. One of possible ways to characterize intense (sub)micron-sized X-ray laser beams is represented by methods exploiting ablative imprints in various solids [10–12]. Formerly it was shown [11] that under certain circumstances the shape of an ablative imprint in poly(methyl methacrylate) created by a single 10-fs laser pulse is related to the transverse profile of the incident beam. From accurate AFM (atomic force microscope) measurements of the ablative imprint’s topography, the transverse beam profile can be reconstructed within an interval ranging from the limiting threshold fluence Fth up to the peak fluence F0. Nevertheless, the minor low-intensity part of the beam below the threshold fluence remains not visualized.
An existence of a sharp and invariable ablation threshold is the most important prerequisite for all methods utilizing ablation imprints. Since applications of PMMA targets are spectrally restricted to EUV and soft X-ray radiation below the carbon K-edge, a novel fluence scan method [10] was developed. It employs the fact that the ablation threshold contour imprinted on a flat target surface visualizes an iso-fluence contour of the beam at fixed threshold fluence Fth, i.e. beam cross-section at clip level f = Fth/F0 of maximum. By varying the laser pulse energy Epulse (and thus the peak fluence), we record iso-fluence contours at different threshold-to-peak fluence ratios f (different fractional levels of maximum) and scan the beam in terms of fluence (from here the designation “fluence scan” follows). Values of the threshold-to-peak fluence ratio f always fall into a left-open and right-closed interval ranging from zero to unity and have a meaning of the so called normalized fluence. For the purposes of normalization, the threshold pulse energy Eth must be known (see [10] for more details) since f = Eth/Epulse. By plotting the normalized fluence f as a function of the corresponding ablation contour area S, we obtain normalized fluence scan (f-scan) curve f(S), which has several important properties thoroughly discussed in this paper. Contrary to the former method of the beam profile reconstruction exploiting ablative imprints in PMMA, the fluence scan method greatly facilitates issues connected with in-focus X-ray laser beam characterization. Time consuming AFM measurements are not needed as a standard optical microscopy can be used for ablation (threshold) contours inspection. Furthermore, ablation contours occur at relatively low and fixed threshold fluence which efficiently prevents thermal and hydrodynamic processes from a harmful influence on results. This essentially extends accessible dynamic range. Finally, spectral range of utilization can be flexibly varied through a proper choice of the target material.
It should be noted that the first ablation imprints method was reported by Liu [13] in 1982. In this study a silicon wafer was used for focused Gaussian beam characterization in visible and ultraviolet spectral domain. It was shown that the slope of the so called Liu’s plot (designation for the relation between ablation contour area and natural pulse energy logarithm) represents the beam spot area at 1/e of maximum. Nevertheless, Liu’s approach was primarily developed for Gaussian beams as their Liu’s plots are always linear, whereas non-Gaussian Liu’s plots are generally nonlinear. In this work we focus on fundamental properties of a general non-Gaussian fluence scan, its relation to the respective Liu’s plot, and energy distribution within the beam profile. Furthermore, a new method of weighted inverse f-scan fitting is developed and applied to real data obtained at the SCSS (SPring-8 Compact SASE Source) facility [14]. The effective area of the focused SCSS beam at wavelength of 60 nm is evaluated in this paper.
2. Theory
Let us first define the spatial fluence (time-integrated intensity) distribution of a general non-Gaussian beam propagating along the z-axis in Cartesian coordinates as F(x,y,z) = F0(z)f(x,y,z). Here F0(z) is the peak fluence and f(x,y,z) is the normalized fluence profile. As we do not study phenomena connected with beam propagation, we can focus on the transverse beam profile f(x,y) at a given z-position; therefore, we do not use the z-coordinate in the following equations. By definition, the normalized fluence profile f(x,y) satisfies a condition 0 ≤f(x,y) ≤ 1 for all points in the transverse xy-plane. By integrating the fluence profile F(x,y) and normalized fluence profile f(x,y) over the entire transverse plane, we obtain pulse energy and the effective beam area [10,15], respectively. Hence we can write a relation:
which is applicable to a general beam, whether it be Gaussian or non-Gaussian.For the purposes of illustration, normalized fluence profiles of an ideal Gaussian beam and exemplary simple non-Gaussian beam are depicted in Fig. 1. Effective areas of these two beam profiles are identical (Aeff = 155 μm2), albeit full widths at half maxima (FWHM) and overall beam shapes may significantly differ. This is a consequence of the non-Gaussian nature of the beam in Fig. 1(b), which has been modeled as an incoherent sum of a narrow Gaussian-like central peak and broad surrounding Gaussian-like background.
Fig. 1 (a) Normalized fluence profile of an ideal Gaussian beam defined as f(r) = exp(−|r|2/ρ2), where ρ = 7.02 μm is the radius at 1/e of maximum and r = (x,y) is the transverse coordinate vector. (b) Normalized fluence profile of a simple non-Gaussian beam defined as f(r) = f1exp(−|r|2/ρ12) + f2exp(−|r|2/ρ22), where f1 = 0.7, ρ1 = 3.99 μm, f2 = 0.3, and ρ2 = 11.28 μm.
In the following we use these two representative model beam profiles as illustrative examples. Firstly, we define the normalized and non-normalized fluence scan (f-scan and F-scan) and the corresponding inverse counterparts (if-scan and iF-scan). Secondly, we consider statistical properties of the fluence scan, which will immediately lead us to a description of the effective area and energy distribution within the beam profile. Next, we make a note on the connection between the fluence scan and the Liu’s plot. Eventually, we introduce a new method of the so called weighted inverse f-scan fitting, mathematically prove its correctness, and apply this method to real data.
2.1 Fluence and inverse fluence scan
As stated in previous paragraphs, the normalized fluence scan (f-scan) curve represents a relation between normalized fluence level f and area S being encircled by the corresponding iso-fluence contour. In Fig. 1(a) and Fig. 1(b) a few iso-fluence contours are highlighted as dark solid lines. Manifestly, as both model beams in Fig. 1 are rotationally symmetric, the iso-fluence contour area can be expressed as S = π|r|2, whence it follows that the corresponding fluence scans obtain analytical forms in these particular cases:
Here AG = πρ2 is the beam cross-section area at 1/e of maximum and equals to the effective beam area (Aeff = AG) since the beam is Gaussian. Parameters f1 and f2 specify amplitudes of the two mutually incoherent Gaussian modes constituting the simple non-Gaussian beam with A1 = πρ12 and A2 = πρ22 being the respective mode cross-section areas at 1/e of maximum. Sum of the amplitudes must always equal unity, i.e. f1 + f2 = 1, in order to meet the normalization condition. In accordance with Eq. (1) or (3a), the effective area of the simple non-Gaussian beam is given by Aeff = f1A1 + f2A2. It should be noted that the function (2b) represents a useful model exploitable for a wide class of non-Gaussian laser beams evincing broadened background.In Fig. 2 normalized fluence scans of the two model beam profiles are depicted. Normalized fluence scan f(S) is generally a monotonically decreasing function of S (within the definition domain 0 ≤ S < ∞) with a global maximum f(0) = 1. Consequently, this implies an existence of the so called inverse normalized fluence scan S(f), i.e. if-scan, which is defined in an interval 0 < f ≤ 1 (the iso-fluence contour area diverges as f tends to zero). Up to now we have been concerned with a normalized fluence scan. Analogously, a non-normalized fluence scan (F-scan or just fluence scan) can be defined for single pulses with known pulse energy Epulse through a relation F(S) = F0 f(S), where F0 = Epulse/Aeff is the peak fluence. Except for the normalization factor F0, the non-normalized fluence scan has the same properties as f-scan; therefore, inverse (non-normalized) fluence scan S(F), i.e. iF-scan, can be defined in an interval 0 < F ≤F0. The most important property of the above introduced functions is that the area below their curves corresponds to the effective area, in case of normalized scans (f-scan and if-scan), and pulse energy, in case of non-normalized scans (F-scan and iF-scan). Hence we can write:
Fig. 2 Normalized fluence scan (f-scan) and corresponding cumulative f-scan of the ideal Gaussian beam (black solid curve and black dashed curve) and simple non-Gaussian beam (red solid curve and red dashed curve).
2.2 Effective area and energy distribution within the beam profile
Normalized and non-normalized fluence scan function is closely related to a distribution of the effective area and energy within the beam profile. Rewriting Eqs. (3a) and (3b) into differential forms, we get:
Translated into a statistical language, this means that values f(S)dS and F(S)dS represent a contribution of an interval (S, S + dS) to the total effective area and pulse energy, respectively. Similarly, latter parts of Eq. (4a) and (4b) stand for contributions of bins (f, f + df) and (F, F + dF) to the effective area and pulse energy. Evidently, derivatives dSAeff = f(S) and df Aeff = f |df S| have a meaning of the effective area density and derivatives dSEpulse = F(S) and dFEpulse = F|dFS| represent the energy density as a function of iso-fluence contour area and (normalized) fluence, respectively. In Fig. 3 the effective area density as a function of normalized fluence is shown for the ideal Gaussian and simple non-Gaussian beam. Patently, in case of a Gaussian beam, the effective area (as well as pulse energy) is uniformly distributed within the entire fluence interval, whereas in the latter case, substantial part of the effective area (and thus pulse energy) is concentrated in the low-fluence (intensity) part of the beam profile. This is an immediate consequence of the broadened background containing a large fraction of the total pulse energy.Fig. 3 Effective area density and cumulative effective area density as a function of normalized fluence for the ideal Gaussian beam (black solid curve and black dashed curve) and simple non-Gaussian beam (red solid curve and red dashed curve).
Such non-Gaussian inhomogeneities need to be considered in various laser-matter experiments attempting to study nonlinear intensity-dependent phenomena with non-ideal beams. Enumerating the cumulative effective area (energy) density (analogous to cumulative distribution function) as a function of normalized fluence f (fluence F) we figure out (see dashed curves in Fig. 3) that a Gaussian profile contains 50% of the effective area (pulse energy) in the “low-intensity” interval 0 < f ≤ 1/2 (0 < F ≤F0/2), whereas 75% of the effective area (pulse energy) is accommodated in the same interval in case of our model simple non-Gaussian profile. Such an imbalance may greatly distort results of various (e.g. spectroscopic) experiments if incorrect, typically Gaussian, assumptions of the beam profile are a priori accepted. Unlike Gaussian beam, the non-Gaussian disproportionality may lead to a prevailing contribution of low-intensity-induced, e.g. fluorescence, signal (within the interval 0 < f ≤ 1/2) over the complementary high-intensity-induced signal acquired during the experiment. Precise spectroscopic measurements have demonstrated that considerably better agreement between numerical simulations and measurements can be achieved if real beam profiles are taken into account [16]. In this particular experiment X-ray emission from warm and dense aluminum plasma created by focused LCLS (Linac Coherent Light Source) beam was studied and compared to numerical simulations. Real energy density distributions, measured by means of ablative imprints in lead tungstate (PbWO4), were used as input values for the simulation code.
Dashed curves in Fig. 2 represent cumulative f-scans (or cumulative effective area densities) as functions of iso-fluence contour area. Cumulative f-scan (F-scan) can be considered as a generalization of the so called “power in the bucket” function formerly addressed by Siegman [2] as it quantifies a fractional amount of the effective area (energy) being contained within a closed iso-fluence contour. Evidently, cumulative f-scan (F-scan) converges to the effective beam area (pulse energy) as the iso-fluence contour area tends to infinity. The same applies to cumulative effective area (energy) density in Fig. 3 as the normalized fluence tends to unity (fluence tends to the peak fluence F0).
2.3 Relation between the fluence scan and Liu’s plot
By plotting the ablation contour area S as a function of natural pulse energy logarithm ln(Epulse), we obtain the so called Liu’s plot [13]. Ablation threshold energy Eth can be obtained through an appropriate, in most cases linear, extrapolation of S(ln(Epulse)) to zero ablation contour area (no ablative damage), i.e. S = 0 μm2. Without loss of generality we can express the Liu’s dependence as a function of the peak-to-threshold fluence ratio p = F0/Fth = Epulse/Eth, i.e. as S(ln(p)). Using a substitution u = ln(p), we can express the slope of the Liu’s plot by means of the first derivative as A(u) = duS(u). Provided that the ablation contour corresponds to the iso-fluence beam contour at normalized fluence level f = 1/p = exp(−u), the slope A(u) stands for the beam spot area at 1/e of maximum. This statement usually (but not exclusively) holds true for Gaussian beams as the slope is independent of pulse energy, i.e. A(u) = πρ2 = Aeff. Non-Gaussian Liu’s plots are generally nonlinear, as shown for example in Fig. 4, and so the corresponding derivatives are not constant, i.e. assigned a single value attributable to the beam spot area. However, with regard to the second integral in Eq. (3a), we can write:
whence it follows that the effective area can be considered as a mean value of ablation (iso-fluence) contour S(u) or first derivative of the Liu’s dependence A(u) in the exponential distribution ~exp(−u). Here the absolute term emerging from the per partes (p.p.) integration is zero.Fig. 4 Liu’s plots and corresponding first derivatives derived from the ideal Gaussian beam (black solid curve and black dashed curve) and simple non-Gaussian beam (red solid curve and red dashed curve).
Liu’s plots and respective derivatives derived for the Gaussian and simple non-Gaussian model beams are depicted in Fig. 4. Noticeably, the non-Gaussian Liu’s plot derivative varies in an interval approximately bounded by effective areas of the two constitutive Gaussian modes (A1 < A(u) < A2), whereas the Gaussian Liu’s plot derivative is constant and equal to the effective beam area (A(u) = AG = Aeff).
2.4 Weighted inverse f-scan fit
Ideally, if the normalized fluence scan is known in its continuous form, the effective area can be evaluated by means of Eq. (3a). However, real f-scan measurements are usually represented by a discrete, very often noisy and incomplete, sequence {fi, Si}i = 1..N of N data points. Each data point corresponds to a single ablative imprint created by an individual pulse of variable energy. Albeit not measured, the data point {1, 0} can be artificially but legally added to the data set as the fluence scan is normalized. Discretizing the integral relation (3a), the effective area can be calculated by means of numerical integration, e.g. by the trapezoidal rule of integration [10].
It would be worthy to determine the effective area and the measurement error as a fitting parameter. Though not generally, the model function (2b) can be used in some particular cases. Usually, but not exclusively, the model function (2a) is to be applied to Gaussian beams; however, it can be proven that the corresponding inverse f-scan SG(f) = −AGln(f) can be fitted to a general (non-Gaussian) inverse f-scan S(f) through the least squares method employing a suitably chosen weight w(f) = −1/ln(f). First, we express the weighted sum of squares χw2(AG) in a continuous form as:
The weighted sum of squares can be minimized through differentiation with respect to the fitting parameter AG. The first derivative must equal zero in a local extreme; hence, interchanging the order of the integral and derivative and using Eq. (3a), we can write:The second derivative of χw2(AG) with respect to AG is positive which indicates that minimum has been found. An equivalence between the fitting parameter AG and effective area Aeff is hereby proven, provided that the fitting is done in terms of the weighted least squares method.In order to apply the method of weighted inverse f-scan fitting to a discrete sequence of measured data points, the integral expression (6a) must be properly discretized and weight factors must be suitably defined. Let the sequence {fi, Si}i = 1..N be sorted in an ascending order with respect to fi in an interval 0 < fi ≤ 1; values falling outside this interval must be excluded from the data set. The first data point {f1, S1} corresponds to an ablative imprint obtained at the highest pulse energy, whereas the last one {fN, SN} = {1, 0} represents the artificially added “zero ablation contour” at the threshold pulse energy. By means of the rectangle rule of integration, the weighted sum of squares in Eq. (6a) is to be expressed as:
where the corresponding weight factors are of the following form:Without loss of generality the last weight factor wN can equal to zero as it generally holds true that [SN −SG(fN)]2 = 0. More accurate expressions can be obtained if trapezoidal or higher-order integration rules are used. The fitting parameter AG, which has been proven to be equal to the effective area, is then to be determined by minimizing the weighted sum of squares in Eq. (7a). If applied to our model beams in Fig. 1, the method results in Aeff = (155 ± 0) μm2 and Aeff = (154.3 ± 1.5) μm2 for the ideal Gaussian and simple non-Gaussian beam, respectively.3. Results and discussion
The method of weighted inverse f-scan fit has been applied to experimental data obtained at the SCSS (SPring-8 Compact SASE Source, RIKEN/Japan) facility [14], which has been recently operated as a test facility for the new SACLA (SPring-8 Angstrom Compact free-electron LAser, RIKEN/Japan) laser [17]. SCSS was successfully put into operation in 2006 when self-amplified spontaneous emission (SASE) at wavelength of 49 nm, i.e. in the extreme ultraviolet (EUV) spectral domain, was observed [18]. The major purpose of the SCSS facility was to develop and test novel technologies for its hard X-ray successor SACLA, which started operation in 2011 and currently runs in user mode. Newly developed components, namely in-vacuum short-period undulator with variable gap, high-gradient C-band accelerator cavities, and low-emittance thermionic electron gun [17,18], made it possible to construct a compact X-ray free-electron laser, which routinely crosses the frontiers of the hard X-ray spectral domain. Nevertheless, both the SCSS and SACLA still belong to the family of linac-based free-electron lasers, sharing very similar conceptual design. In the following paragraphs we briefly describe fundamental principles of the SCSS facility [14,18] and performance of its focusing optics [19].
The laser chain starts with an injector section which employs a CeB6 thermionic cathode as a source of electrons, being pre-accelerated by pulsed high-voltage. This leads to formation of individual electron bunches being further compressed in the longitudinal direction and accelerated up to 45 MeV with use of S-band accelerator modules. Pre-accelerated electron bunches are further propagated to a C-band linear accelerator (linac) section where 5712-MHz (C-band) radio-frequency electromagnetic field is synchronously applied. The effective electrical field can be as high as 35 MV/m; hence the electrons are forced to accelerate up to 250 MeV during one passage and at a relatively short distance. Linear accelerator also accommodates bunch compressors (magnetic chicanes) which compress the passing electron bunches in the longitudinal direction and thus raise the peak current. This is an important prerequisite for the subsequent SASE (self-amplified spontaneous emission) process occurring in undulators, placed just behind the linac section. The SCSS facility employs two in-vacuum short-period undulators with the undulator period of 15 mm. Static but alternating magnetic undulator field forces the electrons to oscillate in the transverse direction and to radiate spontaneously. Simultaneously, the generated electromagnetic field exerts additional transverse forces on the electron bunch causing significant modifications of the charge distribution. The electron bunch splits into micro-bunches which within a single micro-bunch radiate coherently. As the electron bunch propagates through the undulator, the radiated intensity exponentially increases until saturation is reached.
After leaving the undulator, the electron bunch is dumped in an absorber and the optical pulse further travels through an optical beamline towards photon diagnostics and focusing optics. Pulse energy can be flexibly varied with use of a gas-filled attenuator while being continuously monitored with a calibrated ion chamber [19]. For the purposes of additional attenuation, thin metallic foils can be utilized. The focusing element is located approx. 24 m downstream the undulator and consists of two separate focusing mirrors coated with a thin layer of silicon carbide (SiC). Both mirrors are operated in grazing incidence geometry in order to maximize the reflectivity. In Table 1 specifications of both focusing mirrors are listed (see ref [19]. for more details).
Table 1. Focusing mirrors at the SCSS facility1
Due to high incidence angles, grazing incidence optics is usually very sensitive to adjustment as even a small misalignment can imply aberrations, especially astigmatism and coma. Occurrence of aberrations usually leads to increased focal spot size and reduced peak fluence; however, focusing performance can be significantly improved by adjusting the mirrors appropriately. SCSS focusing optics was iteratively optimized in order to suppress aberrations to a minimum. Furthermore, the optimized focus size was measured with use of 10-μm scanning pinhole at wavelength of 60 nm. Horizontal and vertical full widths at half maximum (FWHM) were determined to be 26 μm and 22 μm, respectively [19].
During the experiment the SCSS was tuned at 60 nm within a bandwidth of 1%, average pulse energy was ~10 μJ, and pulse duration was ~100 fs. The source is able to work at maximum repetition rate of 60 Hz. The primary aim of the experiment was to investigate response of various solid-state targets to weakly penetrating EUV radiation, which mostly deposits in the near-surface layer. Initially, a 500-nm thin layer of poly(methyl methacrylate) spin-coated on a silicon wafer (Silson, UK) was used to characterize the incident focused beam by means of ablative imprints method. Among all the other tested materials, a Cu/Nb multilayer, consisting of 25 copper-niobium bilayers magnetron-sputtered on a silicon wafer (each individual layer is 30 nm thick; manufactured by T. Polcar, Czech Technical University in Prague), was irradiated under the same beam conditions. This allows not only to study the target response under well-defined beam conditions, but also to test the ablative imprints methods with use of two distinctly different materials.
The experiment was carried out in a movable vacuum chamber allowing longitudinal translation in range of several tens of centimeters across the focal region. Samples were mounted on a motorized triaxial stage enabling independent and micro-precise target movements in both the transverse and longitudinal direction. Samples were irradiated in the tight focus by individual laser shots (single-shot mode) and under normal incidence conditions.
Ablative imprints in both the PMMA and Cu/Nb multilayer are displayed in Fig. 5 as an ascending sequence sorted with respect to increasing pulse energy. Evidently, at comparable pulse energies the ablative imprints in PMMA appear larger than imprints in Cu/Nb multilayer. This is a consequence of a significant difference in ablation threshold fluences. Ablation threshold fluence for PMMA was found to be as low as (6 ± 1) mJ/cm2, whereas ablation threshold fluence for Cu/Nb multilayer is (116 ± 7) mJ/cm2 at wavelength of 60 nm. Presuming a Gaussian laser pulse with FWHM of 100 fs, the ablation threshold can be quantified in terms of intensity as (56 ± 9) GW/cm2 and (1.09 ± 0.07) TW/cm2 for PMMA and Cu/Nb multilayer, respectively. Contrary to Cu/Nb multilayer, PMMA imprints are capable to visualize much larger iso-fluence contours, although equivalent laser pulses, carrying the same amount of energy, were used to damage the targets. Nonetheless, iso-fluence contours corresponding to equal normalized fluence levels are almost identical for both materials. Potential discrepancies are usually caused by shot-to-shot fluctuations of the beam and its profile (e.g. caused by positional and pointing jitter) originating in the source itself and being even more pronounced by vibrations of optical components. If apertures are placed into a jittering beam, the Airy diffraction pattern (see the PMMA imprint in Fig. 5(a) at 13.6 μJ) may undergo noticeable shot-to-shot changes.
Fig. 5 Ablative imprints in PMMA (a) and Cu/Nb multilayer (b) in dependence on increasing pulse energy. The images were obtained by means of Nomarski (DIC – differential interference contrast) microscopy and are in scale.
It turns out that Cu/Nb multilayer performs very well at this particular wavelength and thus can be used for beam profile characterization. This can be supported by the corresponding inverse f-scan curve plotted in Fig. 6 which is almost identical to the if-scan obtained from PMMA ablation imprints. The effective areas Aeff, PMMA = (345 ± 18) μm2 and Aeff, Cu/Nb = (365 ± 22) μm2, measured by means of the weighted if-scan fitting, are in a very good agreement, albeit Cu/Nb data are evidently incomplete (low fluence data are missing). This is due to the fact that the weight factors in Eq. (7b) decline as the normalized fluence approaches zero; hence the beam tail is only lightly weighted and its contribution to the weighted sum of squares is reduced. On the contrary, the top part of the beam is heavily weighted since the weight factors increase as the normalized fluence tends to unity. Nevertheless, the best results can be obtained, provided that the domain of the if-scan function (0 < f ≤ 1) is uniformly covered with measured data points.
Fig. 6 Inverse f-scans obtained from PMMA (black circles) and Cu/Nb (red circles) ablation imprints. The logarithmic curves (black and red solid curve) were fitted to experimental data by means of the weighted least squares method in order to measure the effective area of the beam.
Once the (inverse) normalized fluence scan curve is known, the beam cross-section area at a half of maximum (a two-dimensional analogue of FWHM), i.e. the iso-fluence contour area S(1/2), can be evaluated and compared to the spot size measurements done with use of the scanning pinhole. From the horizontal and vertical full widths at half maximum reported in [19], we obtain the area of the corresponding elliptical contour SFWHM ≈449 μm2, whereas the f-scan measurements, done for the same wavelength, result in S(1/2) ≈235 μm2. The results seem to be markedly different; however, the discrepancy does not mean that one of the measurements was incorrect. The explanation resides in the non-Gaussian nature of the beam, largely caused by diffraction effects in this particular case. Ablative imprints in Fig. 5 clearly evidence an asymmetric beam shape which cannot be modeled with use of elliptical contours. Therefore, the value of SFWHM overestimates the real beam cross-section area at f = 1/2, albeit the horizontal and vertical full widths at half maximum could have been measured correctly (here we have to assume a negligible influence of the pointing and positional jitter on the scanning pinhole measurements). In case of a general non-Gaussian beam, FWHM provides only a partial and rather a qualitative measure of the real spot size. Rigorous spot size characterization would require FWHM measurements in all possible directions. Therefore, considering the value of SFWHM as an upper estimate, the area S(1/2) fulfills our expectations of the beam cross-section area at f = 1/2. The same applies to the measured effective area.
In order to derive the effective area (energy) density from a discontinuous f-scan (F-scan) measurement, an appropriate beam model must be chosen and fitted to f-scan (F-scan) data. Starting with the double-exponential model (2b), we immediately come to a conclusion that the measured f-scan tends rather to a single exponential dependence in this particular case, albeit the transverse beam profile is evidently non-Gaussian (see Fig. 5). By fitting the exponential model (2a) to all (Cu/Nb and PMMA) data merged together, we obtain the effective area Aeff = (339.0 ± 6.9) μm2. The effective area and energy density is then to be calculated according to Eqs. (2a), (4a), and (4b), whence it follows that the effective area as well as energy is almost uniformly distributed within the entire fluence interval 0 < f ≤ 1 and 0 < F ≤F0, respectively. It is a surprising finding that even a strongly distorted non-Gaussian beam profile can (in some particular cases) have similar properties as an ideal Gaussian beam.
4. Conclusions
It has been mathematically demonstrated that the fluence scan method provides not only a good measure of the effective beam area, but also it characterizes energy distribution within the beam profile. Great emphasis should be placed on non-Gaussian energy distributions which may significantly influence results of intensity-dependent laser-matter experiments. This is especially important on the field of high energy density physics, where nonlinear phenomena play a key role. Furthermore, a link between fluence scan curve and Liu’s plot has been clarified. A new method of weighted inverse f-scan fitting has been introduced and successfully applied to real experimental data obtained at the SCSS facility. In comparison with former measurements done with use of a scanning pinhole, it was shown that the obtained effective area fulfills the expectations of the spot size. It has also been found that in some particular cases real non-Gaussian beam profiles can be modeled with use of a single exponential f-scan, which is normally ascribed to a Gaussian beam.
Acknowledgments
This work was partially funded by FP7 project RADINTERFACES (NMP3-SL-2011-263273). The Czech co-authors appreciate funding by grants LG13029, CZ.1.07/2.3.00/30.0057, 13-28721S, P108/11/1312, P205/11/0571, P208/10/2302, and M100101221. J.Ch. thanks to Academy of Sciences of the Czech Republic for postdoctoral financial support. T.B. thanks to Grant Agency of the Charles University in Prague (GAUK - 1374213). Authors are grateful to Professor Tetsuya Ishikawa and the SCSS team for providing the beamtime and making this work possible.
References and links
1. A. E. Siegman, Lasers (University Science Books, Mill Valley, California 1986).
2. A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues, M. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics (Optical Society of America, 1998), pp. 184–199, paper MQ1. http://www.opticsinfobase.org/abstract.cfm?uri=DLAI-1998-MQ1
3. M. Kirm, A. Andrejczuk, J. Krzywinski, and R. Sobierajski, “Influence of excitation density on luminescence decay in Y3Al5O12:Ce and BaF2 crystals excited by free electron laser radiation in VUV,” Phys. Status Solidi C 2(1), 649–652 (2005). [CrossRef]
4. Y. Dong, G. Zhou, J. Xu, G. Zhao, F. Su, L. Su, G. Zhang, D. Zhang, H. Li, and J. Si, “Luminescence studies of Ce:YAG using vacuum ultraviolet synchrotron radiation,” Mater. Res. Bull. 41(10), 1959–1963 (2006). [CrossRef]
5. M. Nikl, “Wide band gap scintillation materials: Progress in the technology and material understanding,” Phys. Status Solidi A 178(2), 595–620 (2000). [CrossRef]
6. J. Hartmann, “Bemerkungen über den Bau und die Justirung von Spektrographen,” Z. Instrum. 20, 47–58 (1900).
7. S. Le Pape, Ph. Zeitoun, M. Idir, P. Dhez, J. J. Rocca, and M. François, “Electromagnetic-field distribution measurements in the soft X-ray range: Full characterization of a soft X-ray laser beam,” Phys. Rev. Lett. 88(18), 183901 (2002). [CrossRef] [PubMed]
8. B. Flöter, P. Juranić, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedtke, “EUV Hartmann sensor for wavefront measurements at the Free-electron LASer in Hamburg,” New J. Phys. 12(8), 083015 (2010). [CrossRef]
10. J. Chalupský, J. Krzywinski, L. Juha, V. Hájková, J. Cihelka, T. Burian, L. Vysín, J. Gaudin, A. Gleeson, M. Jurek, A. R. Khorsand, D. Klinger, H. Wabnitz, R. Sobierajski, M. Störmer, K. Tiedtke, and S. Toleikis, “Spot size characterization of focused non-Gaussian X-ray laser beams,” Opt. Express 18(26), 27836–27845 (2010). [CrossRef] [PubMed]
11. J. Chalupský, L. Juha, J. Kuba, J. Cihelka, V. Hájková, S. Koptyaev, J. Krása, A. Velyhan, M. Bergh, C. Caleman, J. Hajdu, R. M. Bionta, H. Chapman, S. P. Hau-Riege, R. A. London, M. Jurek, J. Krzywinski, R. Nietubyc, J. B. Pelka, R. Sobierajski, J. Meyer-Ter-Vehn, A. Tronnier, K. Sokolowski-Tinten, N. Stojanovic, K. Tiedtke, S. Toleikis, T. Tschentscher, H. Wabnitz, and U. Zastrau, “Characteristics of focused soft X-ray free-electron laser beam determined by ablation of organic molecular solids,” Opt. Express 15(10), 6036–6043 (2007). [CrossRef] [PubMed]
12. J. Chalupsky, P. Bohacek, V. Hajkova, S. P. Hau-Riege, P. A. Heimann, L. Juha, J. Krzywinski, M. Messerschmidt, S. P. Moeller, B. Nagler, M. Rowen, W. F. Schlotter, M. L. Swiggers, and J. J. Turner, “Comparing different approaches to characterization of focused X-ray laser beams,” Nucl. Instrum. Methods Phys. Res. A 631(1), 130–133 (2011). [CrossRef]
13. J. M. Liu, “Simple technique for measurements of pulsed Gaussian-beam spot sizes,” Opt. Lett. 7(5), 196–198 (1982). [CrossRef] [PubMed]
14. T. Shintake, H. Tanaka, T. Hara, T. Tanaka, K. Togawa, M. Yabashi, Y. Otake, Y. Asano, T. Bizen, T. Fukui, S. Goto, A. Higashiya, T. Hirono, N. Hosoda, T. Inagaki, S. Inoue, M. Ishii, Y. Kim, H. Kimura, M. Kitamura, T. Kobayashi, H. Maesaka, T. Masuda, S. Matsui, T. Matsushita, X. Marechal, M. Nagasono, H. Ohashi, T. Ohata, T. Ohshima, K. Onoe, K. Shirasawa, T. Takagi, S. Takahashi, M. Takeuchi, K. Tamasaku, R. Tanaka, Y. Tanaka, T. Tanikawa, T. Togashi, S. Wu, A. Yamashita, K. Yanagida, C. Zhang, H. Kitamura, and T. Ishikawa, “A compact free-electron laser for generating coherent radiation in the extreme ultraviolet region,” Nat. Photonics 2(9), 555–559 (2008). [CrossRef]
15. ISO 11254–1:2000, “Laser and laser-related equipment - Determination of laser-induced damage threshold of optical surfaces - Part 1: 1-on-1 test
16. S. M. Vinko, O. Ciricosta, B. I. Cho, K. Engelhorn, H. K. Chung, C. R. D. Brown, T. Burian, J. Chalupský, R. W. Falcone, C. Graves, V. Hájková, A. Higginbotham, L. Juha, J. Krzywinski, H. J. Lee, M. Messerschmidt, C. D. Murphy, Y. Ping, A. Scherz, W. Schlotter, S. Toleikis, J. J. Turner, L. Vysin, T. Wang, B. Wu, U. Zastrau, D. Zhu, R. W. Lee, P. A. Heimann, B. Nagler, and J. S. Wark, “Creation and diagnosis of a solid-density plasma with an X-ray free-electron laser,” Nature 482(7383), 59–62 (2012). [CrossRef] [PubMed]
17. T. Ishikawa, H. Aoyagi, T. Asaka, Y. Asano, N. Azumi, T. Bizen, H. Ego, K. Fukami, T. Fukui, Y. Furukawa, S. Goto, H. Hanaki, T. Hara, T. Hasegawa, T. Hatsui, A. Higashiya, T. Hirono, N. Hosoda, M. Ishii, T. Inagaki, Y. Inubushi, T. Itoga, Y. Joti, M. Kago, T. Kameshima, H. Kimura, Y. Kirihara, A. Kiyomichi, T. Kobayashi, C. Kondo, T. Kudo, H. Maesaka, X. M. Marechal, T. Masuda, S. Matsubara, T. Matsumoto, T. Matsushita, S. Matsui, M. Nagasono, N. Nariyama, H. Ohashi, T. Ohata, T. Ohshima, S. Ono, Y. Otake, C. Saji, T. Sakurai, T. Sato, K. Sawada, T. Seike, K. Shirasawa, T. Sugimoto, S. Suzuki, S. Takahashi, H. Takebe, K. Takeshita, K. Tamasaku, H. Tanaka, R. Tanaka, T. Tanaka, T. Togashi, K. Togawa, A. Tokuhisa, H. Tomizawa, K. Tono, S. Wu, M. Yabashi, M. Yamaga, A. Yamashita, K. Yanagida, C. Zhang, T. Shintake, H. Kitamura, and N. Kumagai, “A compact X-ray free-electron laser emitting in the sub-ångström region,” Nat. Photonics 6(8), 540–544 (2012). [CrossRef]
18. T. Shintake and SCSS Team, “Status of Japanese XFEL Project and SCSS test accelerator,” Proc. FEL 2006, BESSY, Berlin, Germany, 33–36 (2006).
19. H. Ohashi, Y. Senba, M. Nagasono, M. Yabashi, K. Tono, T. Togashi, T. Kudo, H. Kishimoto, T. Miura, H. Kimura, and T. Ishikawa, “Performance of focusing mirror device in EUV beamline of SPring-8 Compact SASE Source (SCSS),” Nucl. Instrum. Methods Phys. Res. A 649(1), 163–165 (2011). [CrossRef]
