## Fluence scan: an unexplored property of a laser beam |

Optics Express, Vol. 21, Issue 22, pp. 26363-26375 (2013)

http://dx.doi.org/10.1364/OE.21.026363

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### Abstract

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

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

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]

*F*

_{th}, i.e. beam cross-section at clip level

*f*=

*F*

_{th}/

*F*

_{0}of maximum. By varying the laser pulse energy

*E*

_{pulse}(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

*E*

_{th}must be known (see [10

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]

*f*=

*E*

_{th}/

*E*

_{pulse}. 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.

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]

## 2. Theory

*F*(

*x*,

*y*,

*z*) =

*F*

_{0}(

*z*)

*f*(

*x*,

*y*,

*z*). Here

*F*

_{0}(

*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

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]

*A*

_{eff}= 155 μm

^{2}), 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.

### 2.1 Fluence and inverse fluence scan

*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

*A*

_{G}=

*πρ*

^{2}is the beam cross-section area at 1/e of maximum and equals to the effective beam area (

*A*

_{eff}=

*A*

_{G}) since the beam is Gaussian. Parameters

*f*

_{1}and

*f*

_{2}specify amplitudes of the two mutually incoherent Gaussian modes constituting the simple non-Gaussian beam with

*A*

_{1}=

*πρ*

_{1}

^{2}and

*A*

_{2}=

*πρ*

_{2}

^{2}being the respective mode cross-section areas at 1/e of maximum. Sum of the amplitudes must always equal unity, i.e.

*f*

_{1}+

*f*

_{2}= 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

*A*

_{eff}=

*f*

_{1}

*A*

_{1}+

*f*

_{2}

*A*

_{2}. 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.

*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

*E*

_{pulse}through a relation

*F*(

*S*) =

*F*

_{0}

*f*(

*S*), where

*F*

_{0}=

*E*

_{pulse}/

*A*

_{eff}is the peak fluence. Except for the normalization factor

*F*

_{0}, 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*≤

*F*

_{0}. 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:

### 2.2 Effective area and energy distribution within the beam profile

*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*≤

*F*

_{0}/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

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]

_{4}), were used as input values for the simulation code.

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

*F*

_{0}).

### 2.3 Relation between the fluence scan and Liu’s plot

*S*as a function of natural pulse energy logarithm ln(

*E*

_{pulse}), we obtain the so called Liu’s plot [13

13. J. M. Liu, “Simple technique for measurements of pulsed Gaussian-beam spot sizes,” Opt. Lett. **7**(5), 196–198 (1982). [CrossRef] [PubMed]

*E*

_{th}can be obtained through an appropriate, in most cases linear, extrapolation of

*S*(ln(

*E*

_{pulse})) to zero ablation contour area (no ablative damage), i.e.

*S*= 0 μm

^{2}. Without loss of generality we can express the Liu’s dependence as a function of the peak-to-threshold fluence ratio

*p*=

*F*

_{0}/

*F*

_{th}=

*E*

_{pulse}/

*E*

_{th}, 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*) =

*d*(

_{u}S*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*) =

*πρ*

^{2}=

*A*

_{eff}. 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.

*A*

_{1}<

*A*(

*u*) <

*A*

_{2}), whereas the Gaussian Liu’s plot derivative is constant and equal to the effective beam area (

*A*(

*u*) =

*A*

_{G}=

*A*

_{eff}).

### 2.4 Weighted inverse f-scan fit

*f*,

_{i}*S*}

_{i}

_{i}_{= 1..}

*of*

_{N}*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

**18**(26), 27836–27845 (2010). [CrossRef] [PubMed]

*S*

_{G}(

*f*) = −

*A*

_{G}ln(

*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

*χ*

_{w}^{2}(

*A*

_{G}) in a continuous form as:The weighted sum of squares can be minimized through differentiation with respect to the fitting parameter

*A*

_{G}. 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

*χ*

_{w}^{2}(

*A*

_{G}) with respect to

*A*

_{G}is positive which indicates that minimum has been found. An equivalence between the fitting parameter

*A*

_{G}and effective area

*A*

_{eff}is hereby proven, provided that the fitting is done in terms of the weighted least squares method.

*f*,

_{i}*S*}

_{i}

_{i}_{= 1..}

*be sorted in an ascending order with respect to*

_{N}*f*in an interval 0 <

_{i}*f*≤ 1; values falling outside this interval must be excluded from the data set. The first data point {

_{i}*f*

_{1},

*S*

_{1}} corresponds to an ablative imprint obtained at the highest pulse energy, whereas the last one {

*f*,

_{N}*S*} = {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

_{N}*w*can equal to zero as it generally holds true that [

_{N}*S*−

_{N}*S*

_{G}(

*f*)]

_{N}^{2}= 0. More accurate expressions can be obtained if trapezoidal or higher-order integration rules are used. The fitting parameter

*A*

_{G}, 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

*A*

_{eff}= (155 ± 0) μm

^{2}and

*A*

_{eff}= (154.3 ± 1.5) μm

^{2}for the ideal Gaussian and simple non-Gaussian beam, respectively.

## 3. Results and discussion

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]

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]

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]

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]

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]

_{6}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.

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]

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]

**649**(1), 163–165 (2011). [CrossRef]

^{2}, whereas ablation threshold fluence for Cu/Nb multilayer is (116 ± 7) mJ/cm

^{2}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/cm

^{2}and (1.09 ± 0.07) TW/cm

^{2}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.

*A*

_{eff, PMMA}= (345 ± 18) μm

^{2}and

*A*

_{eff, Cu/Nb}= (365 ± 22) μm

^{2}, 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.

*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

**649**(1), 163–165 (2011). [CrossRef]

*S*

_{FWHM}≈449 μm

^{2}, whereas the f-scan measurements, done for the same wavelength, result in

*S*(1/2) ≈235 μm

^{2}. 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

*S*

_{FWHM}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

*S*

_{FWHM}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.

*A*

_{eff}= (339.0 ± 6.9) μm

^{2}. 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*≤

*F*

_{0}, 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

## Acknowledgments

## References and links

1. | A. E. Siegman, |

2. | A. E. Siegman, “How to (maybe) measure laser beam quality,” in |

3. | M. Kirm, A. Andrejczuk, J. Krzywinski, and R. Sobierajski, “Influence of excitation density on luminescence decay in Y |

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. |

5. | M. Nikl, “Wide band gap scintillation materials: Progress in the technology and material understanding,” Phys. Status Solidi A |

6. | J. Hartmann, “Bemerkungen über den Bau und die Justirung von Spektrographen,” Z. Instrum. |

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. |

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. |

9. | |

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 |

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 |

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 |

13. | J. M. Liu, “Simple technique for measurements of pulsed Gaussian-beam spot sizes,” Opt. Lett. |

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 |

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 |

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 |

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 |

**OCIS Codes**

(140.2600) Lasers and laser optics : Free-electron lasers (FELs)

(140.7240) Lasers and laser optics : UV, EUV, and X-ray lasers

(340.7480) X-ray optics : X-rays, soft x-rays, extreme ultraviolet (EUV)

(140.3295) Lasers and laser optics : Laser beam characterization

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: July 22, 2013

Revised Manuscript: September 12, 2013

Manuscript Accepted: October 8, 2013

Published: October 25, 2013

**Citation**

Jaromír Chalupský, Tomáš Burian, Věra Hájková, Libor Juha, Tomáš Polcar, Jérôme Gaudin, Mitsuru Nagasono, Ryszard Sobierajski, Makina Yabashi, and Jacek Krzywinski, "Fluence scan: an unexplored property of a laser beam," Opt. Express **21**, 26363-26375 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-26363

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### References

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- 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]
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- 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,” Nature482(7383), 59–62 (2012). [CrossRef] [PubMed]
- 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. Photonics6(8), 540–544 (2012). [CrossRef]
- T. Shintake and SCSS Team, “Status of Japanese XFEL Project and SCSS test accelerator,” Proc. FEL 2006, BESSY, Berlin, Germany, 33–36 (2006).
- 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. A649(1), 163–165 (2011). [CrossRef]

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