Decreasing ultrafast X-ray pulse durations with saturable absorption and resonant transitions Sebastian Cardoch1Fabian Trost2Howard A. Scott3Henry

2025-05-06 0 0 799.05KB 19 页 10玖币

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Decreasing ultrafast X-ray pulse durations with saturable absorption and resonant

transitions

Sebastian Cardoch,1, ∗Fabian Trost,2Howard A. Scott,3Henry

N. Chapman,2, 4, 5 Carl Caleman,1, 2 and Nicusor Timneanu1, †

1Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden

2Center for Free-Electron Laser Science, Deutsches-Elektronen Synchrotron (DESY), Hamburg, Germany

3Lawrence Livermore National Laboratory, L-18, P.O. Box 808, 94550, Livermore, CA, USA

4The Hamburg Center for Ultrafast Imaging, Universit¨at Hamburg,

Luruper Chaussee 149, 22761 Hamburg, Germany

5Department of Physics, Universit¨at Hamburg, Luruper Chaussee 149, 22761, Hamburg, Germany

(Dated: October 12, 2022)

Saturable absorption is a nonlinear eﬀect where a material’s ability to absorb light is frustrated due

to a high inﬂux of photons and the creation of electron vacancies. Experimentally induced saturable

absorption in copper revealed a reduction in the temporal duration of transmitted X-ray laser pulses,

but a complete understanding of this process is still missing. In this computational work, we employ

non-local thermodynamic equilibrium plasma simulations to study the interaction of femtosecond

X-rays and copper. Following the onset of frustrated absorption, we ﬁnd that a K–Mresonant

transition occurring at highly charged states turns copper opaque again. The changes in absorption

generate a transient transparent window responsible for the shortened transmission signal. We also

propose using ﬂuorescence induced by the incident beam as an alternative source to achieve shorter

X-ray pulses. Intense femtosecond X-ray pulses are valuable to probe the structure and dynamics

of biological samples or to reach extreme states of matter. Shortened pulses could be relevant for

emerging imaging techniques.

Keywords: X-ray, copper, saturable absorption, frustrated absorption, temporal shape, femtosecond pulse,

NLTE theory, K-shell ﬂuorescence, incoherent diﬀractive imaging, warm dense matter, free-electron lasers

I. INTRODUCTION

X-ray free-electron lasers (XFELs) can generate pulses

with unprecedented characteristics suitable to study the

structure and dynamics of biological samples [1], ultrafast

phase transitions [2], or exotic states of matter [3]. A cur-

rent goal is to produce high-intensity (1017–1019 W/cm2)

extremely short pulses of tens of femtosecond that can

image matter at ˚

Angstr¨om resolution before the onset of

radiation damage or atomic motion [4, 5]. Recent sug-

gestions for a new technique, incoherent diﬀractive imag-

ing [6], require the development of X-ray pulses shorter

than the coherence time of ﬂuorescence emission [7]. The

intense pulses from XFELs can alter the structure and

optical properties of materials, resulting in nonlinear ef-

fects. Taking advantage of this material response, Inoue

et al. [8] experimentally demonstrated temporal shorten-

ing of X-rays by inducing saturable absorption in a solid

copper target, thus uncovering a potential approach to

satisfy the pulse constraints for incoherent imaging.

Saturable absorption, which describes ﬂuence-induced

transparency, has been investigated in the soft and hard

X-ray regimes on transitions metals such as aluminum [9]

and iron [10]. The initially opaque target attenuates the

incoming radiation until depletion of electrons in the K-

shell weakens Coulomb interactions with the core, caus-

∗sebastian.cardoch@physics.uu.se

†nicusor.timneanu@physics.uu.se

ing broadening and shifting of the K-edge to higher en-

ergies [8]. The sample achieves this transparent state

if the photoionization rate is comparable to the Auger-

Meitner and ﬂuorescence decay rates [11]. Inoue et al. [8]

indirectly measured the transmission of X-rays through

the material and found a detectable temporal decrease

compared to the incident beam at a few selected ﬂu-

ences. The study opened interesting questions about the

dynamic processes inside the material. With a greater

photon ﬂux, we expect an increased formation of single

core-hole states, a faster shift in copper’s K-edge, and

the material will reach transparency sooner. If the time

it takes to go from cold absorption to saturation exclu-

sively dictates the transmission of X-rays, the resulting

pulse duration should increase at higher ﬂuence, contra-

dicting experimental evidence. We identify a more com-

plete description of the electronic damage that governs

transmission is needed.

In this paper, we computationally investigate why

XFEL beams transmitted through copper have shorter

temporal durations. We also explore Cu ﬂuorescence,

induced by absorption of the incident beam, as an alter-

native source of X-rays that might exhibit similar tem-

poral characteristics. We chose a copper target to com-

pare our calculations with the results of the experiment

performed by Inoue et al. [8]. Copper has a ﬂuorescence

yield comparable to Auger-Meitner electron yield with its

Kαemission found above iron’s, cobalt’s, and nickel’s K-

edge. Transmission or ﬂuorescence originating from the

copper target can generate core vacancies on these lower

Z elements, found in crystals or biomolecules, whose ﬂuo-

arXiv:2210.04938v1 [physics.plasm-ph] 10 Oct 2022

FIG. 1. Schematic representation of a 1D simulation that fol-

lows the eﬀects of an incident Gaussian pulse I0(t) in the ma-

terial and monitors transmission IT(t) and ﬂuorescence IF(t)

intensities. Electron/ion temperatures, radiation landscape,

and electronic state are sampled at 9 diﬀerent nodes. The

transmission and ﬂuorescence spectra were taken from the

back node along the forward direction over a 2 πsolid angle.

rescence could be applied for structure determination [6].

High-intensity X-rays with wavelengths just above cop-

per’s K-edge experience signiﬁcant absorption in the ma-

terial (absorption coeﬃcient 103cm−1). Large quantities

of energy are deposited mainly from 1selectron ioniza-

tion leading to further damage to the electronic struc-

ture, and the sample becomes a plasma within femtosec-

onds after exposure [12]. Photon-matter collisions create

a cascade of secondary processes and a dynamic radia-

tion energy landscape that results in notable tempera-

ture diﬀerences between ions and electrons and between

the front (facing the beam) and back of the sample [13].

Thermalization and cooling through expansion occur on

much longer timescales (1–10 ps), so the material ex-

ists in a transient warm-dense-matter state that can be

studied by non-local thermodynamic equilibrium (NLTE)

theory [14–18].

We carried out NLTE simulations with a collisional-

radiative model to study a 10 µm-thick copper sample

that is illuminated by X-rays. We chose a range of ﬂu-

ences (5×103–7×107J/cm2) that are relevant in exper-

imental settings of present-day XFELs. The incident

beam’s time proﬁle I0(t) was deﬁned as a Gaussian func-

tion with 7 fs full width at half maximum (FWHM),

centered at 30 fs, with a 9 keV photon energy, and

∆E/E = 1×10−3bandwidth [19]. Using a screened hy-

drogenic model, the material was described by a set of en-

ergy levels and transition rates for radiative, collisional,

and autoionization/electron capture events. Based on

the setup shown in ﬁgure 1, we computed the transmis-

sion time proﬁle IT(t), ﬂuorescence time proﬁle IF(t),

absorption, and occupations resulting from the photo-

induced electronic ﬂuctuations. In an experiment, we

expect a delay in the radiation path along the thickness

of the material that follows the speed of light (approx.

30 fs for 10 µm). In the simulations, radiation is ap-

plied instantaneously at each simulation time step with

a magnitude that reﬂects the material’s current optical

properties along the radiation path.

7500

8000

8500

9000

0 10 20 30 40 50 60

(a) (b)

(c)

7500

8000

8500

9000

0 10 20 30 40 50 60

01234567

×105

Energy [eV]

Time [fs] Time [fs]

1013

1014

1015

1016

1017

1018

1019

Intensity [W/cm2]

Pulse duration FWHM [fs]

Incident ﬂuence [J/cm2]

Incident

Transmitted

Fluorescence

FIG. 2. Calculated intensity at the (a) front and (b) back of

the copper slab irradiated with a 3.5×105J/cm2pulse. We did

not consider a speciﬁc detector distance and neglected inten-

sity decays following the inverse square law. Kα1= 8012 eV,

Kα2= 7992 eV and Kβ1= 8868 eV. (c) Incident, transmitted

(9000 eV), and ﬂuorescent (8006 eV) pulse durations with in-

creasing ﬂuence. Error bars represent a 95% conﬁdence bound

of the best ﬁt’s width.

II. RESULTS AND DISCUSSION

A. X-ray transmission and ﬂuorescence

We initially calculated the duration of the angle-

averaged intensity proﬁles IT(t) and IF(t) as a function

of incident ﬂuence. The resulting pulse durations are

shown in ﬁgure 2(c). The intensity at any node (sampling

planes) consists of radiation from two origins: transmis-

sion of the X-ray beam and emission from the mate-

rial. These two contributions along the forward direc-

tion made up the detected intensity spectra that, for a

single ﬂuence, are shown in ﬁgure 2. Panels (a) and (b)

correspond to the front and back nodes of the sample,

respectively. We deﬁned the ﬂuorescence as the signal

that yielded the shortest FWHM and highest peak in-

tensity over the photon energy range between 7–9 keV.

We divided the spectra in bins of 9 eV (identical to the

I0(t) bandwidth), computed the aspect ratio as peak in-

tensity/duration, and found 8006 eV to be the strongest.

See the Supporting Information for results. We employed

a single Gaussian best ﬁt to determine the FWHM.

The simulated transmission proﬁle FWHM followed

experimental results from [8], where we found pulse du-

rations of roughly 4–5 fs at ﬂuences of 2–3×105J/cm2.

We observed some discrepancies at low ﬂuences, where

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

01234567

(a) (b)

×105

01234567

×105

Max. fractional K-shell pop.

Fluence [J/cm2]

1s2

1s1

1s0

Avg. ionization (t > 40 [fs])

Fluence [J/cm2]

leading node

trailing node

FIG. 3. (a) Average ion charge at the end of the pulse and

(b) maximum K-shell population created in the material. The

incident photon energy is not large enough to create a second

core hole state. We attribute the non-zero 1s0state popula-

tion to electron-ion collisional ionization. If the rate of this

process is faster than the relaxation time of photoionization,

a single core hole state can become doubly ionized.

experiments showed pulse times longer or similar to the

incident beam. Our simulations instead returned shorter

pulse times compared to the incident beam. At higher

ﬂuences between 6–7×105J/cm2, the simulations pre-

dicted longer durations than the incident X-rays. The

transmission in these cases featured a double peak that

was not well captured by a single Gaussian best ﬁt, re-

sulting in large uncertainty in the FWHM. In the low ﬂu-

ence limit, the ﬁnal average charge in the copper atoms

was below +8, and the generation of core holes was less

than 10%, as shown in ﬁgure 3. The screened hydro-

genic model reliably describes a system with signiﬁcant

ionization but loses accuracy for closed-shell and neutral

atoms [20]. These artifacts can be corrected by scal-

ing energies to match more detailed calculations [20],

but we expect a less accurate system representation in

the low ionization regime. The simulations also revealed

marginally shorter ﬂuorescence proﬁle FWHM at ﬂuences

below 2.0 J/cm2.

B. Temporal suppression mechanism

To understand the calculated IT(t) and IF(t) inten-

sity proﬁle durations with a 9 keV incident beam, we ex-

plored the dynamics of the transmission and ﬂuorescence

relative to the initial pulse. Figure 4(a) shows normal-

ized proﬁles for a single ﬂuence of 3.5×105J/cm2. We

found transmission peaked and died out earlier than the

incident signal, while the ﬂuorescence persisted over the

entire duration of the incident signal. Figures 4(b) and

varying ﬂuences. For transmission, transparency and ter-

mination tended to happen at earlier times as ﬂuence in-

creased. Fluorescence FWHM increased with increasing

ﬂuence and peak times shifted earlier in time at ﬂuences

below 3.5×105J/cm2and shifted to later times at higher

ﬂuences. Peak times for all three proﬁles are summarized

in ﬁgure 4(d).

20 25 30 35 40

(a) (b)

01234567

×105

0.5

1.5

20 25 30 35 40

×1019

20 25 30 35 40

×1017

Norm. intensity [arb. units]

Time [fs]

Incident

Transmitted

Fluorescence

Peak time [fs]

Fluence [J/cm2]

Incident

Transmitted

Fluorescence

Intensity [W/cm2]

Time [fs]

5.00×104J/cm2

1.00×105

1.25×105

1.50×105

1.75×105

2.00×105

2.50×105

3.00×105

3.50×105

4.00×105

Intensity [W/cm2]

Time [fs]

5.00×104J/cm2

1.00×105

1.25×105

1.50×105

1.75×105

2.00×105

2.50×105

3.00×105

3.50×105

4.00×105

FIG. 4. (a) Radiation dynamics for a 3.5×105J/cm2incident

beam revealed transmission occurred early in the radiation

exposure and extinguished before the peak of the incident

pulse at 30 fs. (b) Transmission proﬁles shifted earlier in time

with increasing ﬂuence and (c) emission at 8006 eV became

wider with increasing ﬂuence. (d) Summary of peak intensity.

a. Transmission proﬁle We found saturable absorp-

tion oﬀered an incomplete description of transmission

proﬁles. When absorption saturates, the transmitted X-

rays should match the incident pulse. Instead, our calcu-

lations revealed transmission terminated well before the

incident beam. Figures 5(a)-(c) display the absorption

coeﬃcient of the material near the copper K-edge as a

function of time for ﬂuences of 1.5, 3.5, and 7×105J/cm2,

respectively. In all cases, we observed shifts in the edge

plus an opaque feature at photon energies below the edge

corresponding to a K–Mtransition. We found the most

dominant contribution at 9 keV came from a 1s–3ptran-

sition, where the Cu atoms reached ionization levels be-

tween +9 to +17. For low ﬂuences, the shortening of

the transmission proﬁle duration was uniquely a conse-

quence of frustrated absorption. The initial section of

the beam was absorbed until the K-edge moved to larger

energies. For suﬃciently high ﬂuences, the opaque transi-

tion shifted into the photon energy range of the incoming

X-rays, eﬀectively forming a transient transparent win-

dow in the material. The outcome was an even shorter

transmission. At more extreme ﬂuences, the resonant

transition shifted into the photon energy range of the in-

coming X-rays but was promptly suppressed by the sheer

number of incident photons resulting in a double peak

proﬁle with a large FWHM.

We believe the reason for the resonant K–Mstate’s

proliferation and motion along the path of the beam

is similar to that of photon energy shifts in emission

spectra for high-temperature plasma discussed in liter-

ature [11, 21, 22]. The main mechanism for absorption is

101

102

103

104

8000 8500 9000 9500 10000

(a)

(b)

(c)

(d)

(e)

(f)

101

102

103

104

8000 8500 9000 9500 10000

101

102

103

104

8000 8500 9000 9500 10000

8800 8900 9000 9100 9200

Absorption [1/cm]

total

bound-bound

bound-free

10 fs

Absorption [1/cm]

27 fs

Absorption [1/cm]

Energy [eV]

33 fs

Time [fs]

500

1000

1500

2000

2500

3000

1.5×105J/cm2

Time [fs]

500

1000

1500

2000

2500

3000

3.5×105J/cm2

Time [fs]

Energy [eV]

500

1000

1500

2000

2500

3000

7×105J/cm2

FIG. 5. (a)-(c) Opacity averaged over all zones showing displacement of the K-edge and resonant K–Mtransition for three

incident ﬂuences. The vertical dashed line indicates the XFEL pulse photon energy. Horizontal lines in (b) are cuts through

the absorption shown in the three panels to the right. (d)-(f) Opacities at 3.5×105J/cm2averaged over all zones at three

instances during the simulation. The 10 fs snapshot shows opacities for a cold sample, the 27 fs shows a dip in the opacity at

9 keV, and the 33 fs snapshot shows an increased opacity at 9 keV.

K-shell ionization resulting in a single core hole. K-shell

ﬂuorescence and electron ejection are competing pro-

cesses with comparable probabilities. In copper, ﬂuores-

cence accounts for 44.5% of the total recombination while

the remaining holes are ﬁlled mainly via KLL Auger-

Meitner decay [23]. Electron impact is another source of

ionization. Hot electrons ejected by collision with the X-

ray beam or through Auger-Meitner decay generate fur-

ther vacancies in the material, triggering an ionization

cascade. Primary and secondary ejected electrons equili-

brate through collisions with the cold electron reservoir

(conduction band). Cold electrons also gain kinetic en-

ergy and begin to ionize outer valance states in the ma-

terial. As more bound electrons exit the atoms, screen-

ing of higher levels is reduced, and deep states move

closer to the nucleus. For high enough charged states

the 1s–3ptransition (most dominant around 9 keV) in-

creases and shifts into the range of the incident beam.

Photo-electrons are no longer ejected to the continuum

and are instead resonantly pumped to the M-shell [11].

The ﬁnal transmitted proﬁle duration is determined by

the time delay between the K-edge and 1s–3pphoton

energy changes along the thickness of the sample.

b. Fluorescence proﬁle For a given ﬂuence we ﬁt-

ted a linear combination of the normalized ﬂuorescence

proﬁle given by the simulations using ˆ

IF(t) = a1ˆ

I0(t) +

a2ˆ

IT(t), where a1and a2are coeﬃcients that changed

based on the incident ﬂuence. We found ˆ

IF(t) at low ﬂu-

ence was mainly described by the transmission proﬁle and

at high ﬂuence by the incident proﬁle. More information

is found in the Supporting Information. These changes in

the coeﬃcients suggested absorption caused by the K–M

resonance extended the temporal duration of the result-

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DecreasingultrafastX-raypulsedurationswithsaturableabsorptionandresonanttransitionsSebastianCardoch,1,FabianTrost,2HowardA.Scott,3HenryN.Chapman,2,4,5CarlCaleman,1,2andNicusorTimneanu1,y1DepartmentofPhysicsandAstronomy,UppsalaUniversity,Box516,SE-75120,Uppsala,Sweden2CenterforFree-ElectronLaserScie...

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Decreasing ultrafast X-ray pulse durations with saturable absorption and resonant transitions Sebastian Cardoch1Fabian Trost2Howard A. Scott3Henry

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