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Persson, A I., Enquist, H., Jurgilaitis, A., Andreasson, B P., Larsson, J. (2015)Real-time observation of coherent acoustic phonons generated by an acousticallymismatched optoacoustic transducer using x-ray diffractionJournal of Applied Physics, 118(18): 185308https://doi.org/10.1063/1.4935269

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© 2015 Author(s).

Real-time observation of coherent acousticphonons generated by an acousticallymismatched optoacoustic transducer usingx-ray diffractionCite as: J. Appl. Phys. 118, 185308 (2015); https://doi.org/10.1063/1.4935269Submitted: 09 September 2015 . Accepted: 24 October 2015 . Published Online: 13 November 2015

A. I. H. Persson, H. Enquist, A. Jurgilaitis, B. P. Andreasson , and J. Larsson

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Real-time observation of coherent acoustic phonons generatedby an acoustically mismatched optoacoustic transducer using x-raydiffraction

A. I. H. Persson,1 H. Enquist,2 A. Jurgilaitis,2 B. P. Andreasson,1,a) and J. Larsson1,2,b)

1Department of Physics, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden2MAX IV Laboratory, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden

(Received 9 September 2015; accepted 24 October 2015; published online 13 November 2015)

The spectrum of laser-generated acoustic phonons in indium antimonide coated with a thin nickel

film has been studied using time-resolved x-ray diffraction. Strain pulses that can be considered to

be built up from coherent phonons were generated in the nickel film by absorption of short laser

pulses. Acoustic reflections at the Ni–InSb interface leads to interference that strongly modifies the

resulting phonon spectrum. The study was performed with high momentum transfer resolution to-

gether with high time resolution. This was achieved by using a third-generation synchrotron radia-

tion source that provided a high-brightness beam and an ultrafast x-ray streak camera to obtain a

temporal resolution of 10 ps. We also carried out simulations, using commercial finite element soft-

ware packages and on-line dynamic diffraction tools. Using these tools, it is possible to calculate

the time-resolved x-ray reflectivity from these complicated strain shapes. The acoustic pulses have

a peak strain amplitude close to 1%, and we investigated the possibility to use this device as an x-

ray switch. At a bright source optimized for hard x-ray generation, the low reflectivity may be an

acceptable trade-off to obtain a pulse duration that is more than an order of magnitude shorter.VC 2015 Author(s). All article content, except where otherwise noted, is licensed under a CreativeCommons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4935269]

I. INTRODUCTION

When a short laser pulse impinges on the surface of a

material, the resulting heating and thermal expansion gener-

ate an acoustic strain pulse.1,2 This strain pulse is built up by

a range of acoustic phonon modes that can be studied indi-

vidually using time-resolved x-ray diffraction (TRXD).3,4

Different strategies have been devised to modify and control

the phonon spectrum. Coherent control by multiple-laser-

pulse excitation has been demonstrated by Lindenberg et al.5

and Synnergren et al.6 Multilayered structures can modify

the phonon spectrum,7 giving rise to phonon folding,8 which

can be observed using TXRD. There is a need to control the

phonon spectrum for two main reasons: first to be able to

observe and diagnose objects “buried” in materials9 and sec-

ond to create a simple device that can act as a fast x-ray

switch.10,11 Such a switch could be used to create short x-ray

pulses at a storage ring without having to perturb the elec-

trons in the storage ring.

The phonon spectrum can be modified by using an opto-

acoustic transducer.1,12–15 Such a transducer can be imple-

mented as a thin film with acoustic and optical properties

that differ from the underlying material to be investi-

gated.12,15,16 This causes strain to be generated in the thin

film, and the strain pulse is partially reflected at the interface

with the underlying material. The interface between the film

and vacuum acts as a reflector for the back-propagating

pulse, and thus a train of pulses propagates into the underly-

ing material. Strain pulses from optoacoustic transducers

consisting of a thin layer of gold on germanium have been

investigated by x-ray diffraction with high temporal resolu-

tion12 and high momentum transfer resolution.13 Loether

et al.13 also suggested that a transducer consisting of a thin

metal layer on a semiconductor crystal could be used as an

x-ray switch. They investigated this with low temporal reso-

lution and modelled the behavior. A different approach has

been used by, e.g. Shayduk et al.17 They used acoustically

matched multilayer structures to control the phonon

spectrum.

In the present study, we investigated strain pulses from

an optoacoustic transducer consisting of a nickel-coated in-

dium antimonide wafer, using high momentum transfer reso-

lution at the MAX II electron storage ring, achieving high

temporal resolution with a streak camera. The high temporal

and momentum resolution allowed the direct investigation of

how the phonon spectrum is influenced by the optoacoustic

transducer. Simulations were carried out to ensure that we

could model the acoustic propagation in the optoacoustic

transducer and the scatting from the substrate. We also

investigated the possibilities and limitations of implementing

the x-ray switch proposed by Loether et al.13

II. STRAIN GENERATION AND DETECTION

Before discussing the probing of the strain, we describe

the strain generation. First we discuss how a strain wave is

generated at the vacuum-sample interface and then we illus-

trate how the strain pulse entering into the material to be

a)Current address: School of Information Technology, Halmstad University,

P.O. Box 823, SE-301 18 Halmstad, Sweden.b)Author to whom correspondence should be addressed. Electronic mail:

Jorgen.Larsson@fysik.lth.se

0021-8979/2015/118(18)/185308/6 VC Author(s) 2015118, 185308-1

JOURNAL OF APPLIED PHYSICS 118, 185308 (2015)

probed is modified by the acoustic properties of the two

materials.

The energy of a laser pulse is deposited in the sample by

excitation of carriers, which equilibrate with the lattice in a

few ps. Thus the laser heats a thin surface layer of the sam-

ple, leading to expansion. This creates a static strain and two

strain waves that propagate in opposite directions.1 The

shape of the static strain and the two counter-propagating

strain pulses is a replica of the effective absorption profile,

which is typically an exponential decay. The shape and am-

plitude of the strain pulse is determined by the optical

absorption depth and the diffusion of the electrons before

they thermalize with the lattice. The diffusion depth of elec-

trons was not previously known for an amorphous Ni film

and was determined in this study. Each of the propagating

strain pulses has the opposite sign to the static strain profile,

and half the amplitude, so that the initial condition of no

strain is fulfilled. The pulse propagating toward the vacuum

surface is reflected with an associated change of sign, which

gives rise to the strain pulse shape illustrated in Fig. 1(a). It

should be noted that for an uncoated sample, the electronic

strain associated with an abrupt laser-induced bandgap

change gives a larger contribution than the thermal strain.18

On the short time scale during which this process was

observed, only longitudinal expansion takes place. To find

the strain in indium antimonide (InSb), we start by calculat-

ing the reflection and transmission coefficients for the stress

at the boundary. The reflection coefficient for the stress (Rs)

and strain (Rg) is given by19,20

Rs ¼ Rg ¼qInSbvInSb � qNivNi

qNivNi þ qInSbvInSb; (1)

where q and v denote the density and the speed of sound in

the material in question. The transmission coefficient for the

stress, Ts, is defined as 1þ Rs19,20 and can be written as

Ts ¼2qInSbvInSb

qNivNi þ qInSbvInSb: (2)

The strain, g, is given by the stress, s, divided by the bulk

modulus, B. This together with Eqs. (1) and (2) gives the

transmission coefficient for strain, Tg, when the strain pulse

propagates from Ni into InSb

Tg ¼gt

gi

¼ BNi

BInSb

2qInSb�InSb

qNi�Ni þ qInSb�InSb; (3)

B is the bulk modulus, which is given by:19 B ¼ v2q. The rele-

vant material parameters for Ni and InSb are given in Table I.

Using the values for density and speed of sound, it can be seen

that the transmitted strain is a factor of 2.2 higher in InSb than

the initial strain in Ni. Using the values in Table I, it was found

that 41% of the strain pulse is reflected at the Ni–InSb inter-

face and propagates back, with the opposite sign, into the Ni

film toward the vacuum, where 100% of the strain is reflected,

also with a change of sign. The process is repeated, leading to

a series of diminishing strain pulses in the InSb, as illustrated

in Fig. 1. The period of the pulse train depends on the thick-

ness of the film and the speed of sound in it. The ratio of the

amplitudes of the strain pulses in the two materials depends on

the acoustic properties of the materials. The relative extension

of the pulses in Ni and InSb is determined by the relative speed

of sound in the two materials.

In this study, we investigated the phonon spectrum in

the strain pulse train generated by the optoacoustic trans-

ducer in a series of experiments on single-phonon modes.

FIG. 1. Strain propagation in the sample. (a) Shortly after laser excitation,

(b) after the first reflection of the strain wave at the Ni–InSb interface, (c) af-

ter the strain wave inside Ni has been reflected at the vacuum–sample inter-

face, and (d) after the second reflection at the Ni–InSb interface.

TABLE I. Optical and acoustic properties of indium antimonide and nickel (both as bulk and a thin �150 nm film).

Optical absorption depth at 800 nm Density (kg/m3) Speed of sound (m/s) (longitudinal, into sample) Acoustic impedance (kg/s m2)a

InSb 91 nmb 5774.7c 3880d 2.240 � 107

Ni (bulk) 14 nme 8900c 6040f 5.376 � 107

Ni (thin film) 17.6 nmg 8900h 6040g 5.376 � 107

aAcoustic impedance, Z, calculated as v � q(speed of sound � density).bAspnes and Studna.31

cPhysical constants of inorganic compounds, Section IV, Handbook of Chemistry and Physics.32

dSlutsky and Garland.33

eLynch et al.34

fSpeed of Sound in Various Media, Section 14, Handbook of Chemistry and Physics.35

gSaito et al.36

hKoenig and Carron.

185308-2 Persson et al. J. Appl. Phys. 118, 185308 (2015)

This was achieved by tuning the x-ray energy away from the

Bragg condition, so that the sum of the phonon wave vector

and the reciprocal lattice vector was equal to the momentum

difference between the incident and scattered x-ray photons.

This scattering condition has been explained in detail by

Lindenberg et al.3 and Larsson et al.4 for two different geo-

metries, and is expressed in the following equations:

G6K ¼ ks � k0; (4)

jKj ¼ 2pdhkl

DE

E; (5)

T ¼ 2pvjKj ; (6)

where G is the reciprocal lattice vector, K is the phonon

wave vector, k0 is the momentum vector of the incoming

photon, and ks is the momentum vector of the scattered pho-

ton. DE is the deviation in photon energy from the resonant

scattering condition, E is the photon energy needed to fulfill

the Bragg condition without absorbing or emitting phonons,

and dhkl is the lattice spacing (hkl are the Miller indices of

the crystal lattice). The frequency of the phonon is mani-

fested in the time domain and can be measured as a periodic

oscillation using a time-resolved detector such as a streak

camera. The phonon period, T, can be calculated from the

phonon wave vector and the speed of sound, v, using Eq. (6).

III. EXPERIMENTAL SETUP AND PROCEDURE

The measurements were performed at beamline D611 at

the MAX II electron storage ring at the MAX IV Laboratory

in Lund, Sweden. The experimental setup is illustrated in

Fig. 2. The monochromator used was an InSb monochroma-

tor with a bandwidth of DE/E¼ 4� 10�4, and the size of the

focal spot of the x-rays at the sample position was

400� 200 lm2. A more detailed description of the beamline

has been given by Harbst et al.21 The repetition rate of the

x-ray pulses was 100 MHz. The laser was a titanium-doped

sapphire oscillator and amplifier system with 800 nm wave-

length, a pulse duration of 50 fs, and a repetition rate of

4.25 kHz. The laser oscillator and the x-rays were

synchronized at 100 MHz to an accuracy of 50 ps. A me-

chanical chopper synchronized to the laser amplifier was

used to reduce the x-ray repetition rate at the beamline to

4.25 kHz. The chopper opening time was about 6 ls, and ad-

jacent x-ray pulses were suppressed with electrical gating

using pulsed microchannel plates (MCPs) in the detector.

The detector was an in-house developed streak camera;22 the

time window used was 300 ps long and the time resolution

was 9 ps. In the experiment, the x-ray pulse is a backlighter,

and the temporal resolution comes from the streak camera

including the jitter between the exciting laser pulse and the

pulse triggering the photo-conductive switch.23

The sample investigated was an InSb (111) bulk sample

with a nickel film deposited on one surface using thermal

evaporation. The thickness of the film was measured (moni-

tored) with an oscillating crystal and found to be

150 6 20 nm. An uncoated InSb (111) reference sample was

also used. The angle of the sample was fixed, and the x-ray

energy could be scanned around the Bragg condition for the

InSb 111 reflection, which was fulfilled at 5.58 keV. In this

geometry, the attenuation length of the x-rays in Ni was

2.6 lm and in InSb 0.83 lm.24 The Gaussian laser beam was

focused by two cylindrical lenses to an elliptical spot. The

size of the laser beam on the sample was 4.7� 0.7 mm2

FWHM. The x-ray footprint at a Bragg angle of 17.3� was

1.3� 0.2 mm2 (FWHM). The x-ray beam and the laser beam

were spatially overlapped on the sample. This resulted in a

laser fluence of 11.7 mJ/cm2 incident on the Ni-coated sam-

ple and a fluence of 3 mJ/cm2 incident on the uncoated sam-

ple. The uncertainty in the absolute fluence is about 50%,

partly related to a non-Gaussian beam shape and experimen-

tal difficulties in maintaining laser x-ray overlap over the

extensive time needed to carry out a set of measurements.

IV. MEASUREMENTS AND RESULTS

To find the energy at which the Bragg condition is was

fulfilled and to obtain the absolute number of photons, an

energy scan was carried out with a silicon photodiode. This

scan is shown in Fig. 3.

Time-resolved data were acquired with a streak camera,

with and without laser excitation. A streak camera provides

FIG. 2. Experimental setup. A laser pulse impinges on the sample and is

absorbed in the nickel film. The coherent acoustic phonons building up the

generated strain pulse propagates into the sample and are probed in the InSb

crystal using x-ray diffraction. The long pulse x-rays from the MAX II stor-

age ring serves as a backlighter, and high temporal resolution is obtained

using an x-ray streak camera which records the transient. Many such record-

ings provides the time-evolution of the x-ray reflectivity with sufficient sig-

nal to noise ratio.

FIG. 3. Energy scan. The graph represents the time-integrated x-ray reflec-

tivity of the unexcited sample as the x-ray energy is scanned while the sam-

ple is stationary. The black dots are measured values and the black line is a

fitted Voigt profile.

185308-3 Persson et al. J. Appl. Phys. 118, 185308 (2015)

images with time along one axis and the position along an

entrance slit on the other. In the geometry used, the position

along the entrance slit images the diffracted x-ray beam ver-

tically. Each image consisted of data accumulated over 100 s

at 4.25 kHz. The images without laser excitation were used

as a background that was subtracted to produce differential

images, which were analyzed. Lineouts of the x-ray exposed

region of the differential images yielded a representation of

the intensity as function of time. A set of measurements

were taken for different energy offsets around the peak of

the energy scan. To visualize the full data set, we plotted the

intensity as function of both time and energy offset in a

false-color image. We refer to these images as time-

dependent energy scans, which are analogous to the time-

resolved rocking curves shown in previous x-ray studies of

acoustic phonons.25 Absolute calibration in terms of pho-

tons/ps at a given x-ray energy was obtained by comparing

the streak camera signal prior to the laser exposure of the

sample to the current measured in the static energy scan.

The intensity variations with energy offset and time can

be seen in Fig. 4. Several qualitative differences were

observed between the time-dependent energy scans of the

coated and uncoated samples. The most striking feature of

the coated sample is the interference, which manifests as a

modulation of the intensity as a function of energy after a 50

ps delay, as shown in Fig. 4(a). As an example, the modula-

tion at 615 eV offset is very low compared to the high, oscil-

lating modulation at 621 eV. This is demonstrated by

lineouts for the offsets of þ15eV and �21 eV in Fig. 5. The

initial strain observed in the InSb substrate in the Ni-coated

sample is due to the propagating strain wave generated in the

Ni film, which results in initial compression of the InSb.

This can be seen as an initial increase in x-ray scattering in-

tensity for the positive energy offsets in Fig. 4(a). In the

uncoated sample, on the other hand, the initial strain is pre-

dominantly due to the static strain at the surface, thus

yielding a decrease in x-ray scattering intensity for positive

energy offsets, as shown in Fig. 4(b).

V. SIMULATIONS AND INTERPRETATION

A series of simulations was carried out for different film

thicknesses and absorption depths to verify the acoustic

model and to find the best match for the unknown effective

photon absorption depth in Ni, including electron diffusion.

Another aim of the simulations was to confirm our ability to

predict time-resolved reflectivities for different experimental

conditions. The strain simulations, including heat conduc-

tion, were carried out using ComsolTM

, using the “solid

mechanics” and “heat transfer in solids” modules. Due to the

difference in length scale of the longitudinal and transverse

FIG. 4. Experimental data for Ni-

coated (a) and uncoated (b) InSb sam-

ples are shown above the correspond-

ing (c), (d) simulated data using an

acoustic model and x-ray scattering

calculations. To see the rich structure

in the wings of the energy scan on a

linear scale, the range between �4eV

and 8 eV has been omitted. To have

the same scale for the coated and

uncoated sample, the intensity is satu-

rated close to the peak in (b) and (d).

The left side of the color scale shows

reflectivity difference and the right

side shows photons/ps.

FIG. 5. Phonon interference illustrated by lineouts with energy offsets of

þ15 eV (a) and �21 eV (b) for the Ni-coated sample, showing the experi-

mental data (solid, black curves) and the simulations (dashed, blue curves).

The reflectivity is defined as the number of photons before and after the sam-

ple. The reflectivity without laser excitation of phonons is 1.5 � 10�3 for a

þ15 eV offset (a) and 7 � 10�4 for a �21 eV offset (b).

185308-4 Persson et al. J. Appl. Phys. 118, 185308 (2015)

directions, 1D simulations were used to describe the genera-

tion and propagation of acoustic waves. The temporal step

size in the simulations was 0.1 ps and the spatial step size

was 1 nm. A simple MatlabTM

program was used to generate

a reduced set of strain profiles with one profile per ps. The x-

ray reflectivity for offsets around the Bragg peak was calcu-

lated for each strain profile using the tool GID_sl on Sergey

Stepanov’s x-ray server.26 The data from the simulations

were merged to a time-dependent energy scan and renormal-

ized for direct comparison to the experimental data. The

acoustic simulation, including heat transport, took 15 min on

a 2nd generation Core i5 computer. The execution of the

MatlabTM

code, which sends the strain profiles to the

Stepanov server, executes in about 45 min, most of the time

being taken by calculations on the server. The simulations

were subsequently convoluted in both energy and time to

account for the experimental resolution in time and energy.

The results of the simulations are given in Fig. 4(c) for

Ni-coated InSb and in Fig. 4(d) for uncoated InSb. As can be

seen, there is excellent agreement between the simulations

and the experiments. This means that both the complex pho-

non spectrum and the x-ray scattering intensities can be

described by the models described in the preceding text. We

have carried out simulations for different film thicknesses.

The best match is the simulation shown in Fig. 4(c) for a

film thickness of 155 nm. Simulations were performed for

different absorption depths in Ni to account for electron dif-

fusion. The width of the first feature in Fig. 5 is sensitive to

the effective absorption depth. An absorption depth of 30 nm

gave the best agreement with the experimental data. With

this absorption depth the expected strain in Ni is 0.45%. To

obtain the same modulation depth in the simulation as in the

experiments an initial peak strain of 0.23% was used.

After being able to reproduce the experimental data with

our simulations, we make the following observations. The

constructive and destructive interference in the time-

dependent energy scans for the coated sample are related to

the acoustic round-trip time, which is 51.5 6 1 ps in the

155 nm Ni film. In Fig. 6, we show the interference effect.

Fig. 6(a) shows the strain profile as function of time at a

depth of 300 nm in the substrate. In Fig. 6(b), we show the

modified phonon spectrum, which is obtained by the Fourier

transform of Fig. 6(a). In the experiment, we select one pho-

non wave vector by choosing the energy offset (Eq. (5)).

This results in a periodic oscillation with the period time

given by Eq. (6). By band-pass filtering Fig. 6(b) and taking

the inverse Fourier transform, we observe the temporal shape

of this particular “phonon mode.” At offsets of 621 eV, the

phonon period, as given by Eq. (6), is �25 ps, and the inter-

ference is constructive as the consecutive acoustic pulse

arrives in phase with the previous oscillation, see Fig. 6(c).

At offsets of 615 eV, the phonon period is �35 ps, and the

interference is destructive as the consecutive acoustic pulse

arrives out of phase with the previous oscillation. This is

seen in Fig. 6(d).

VI. DISCUSSION

By varying the thickness and the material of the film

acoustic pulse trains with different pulse spacing (in time)

and different amplitudes of the pulses can be created. In the

coated sample, the heat from the laser pulse is deposited

only in the nickel layer (due to the short optical absorption

depth and the thickness of the film). This leads to no direct

heating of the InSb, compared to the case in the uncoated

sample, where heating from the laser shifts the peak in the

energy scans toward lower energy, due to expansion of the

sample.

We investigated the suggestion by Loether et al.13 to use

this device as an x-ray switch. There are severe limitations

with this concept. The best example of destructive interfer-

ence was shown by offset þ15 eV as seen in the lineout

shown in Fig. 5(a). For comparison, we show the construc-

tive interference that occurs at an offset of �21 eV in

FIG. 6. (a) The strain profile 300 nm

into the sample. (b) The phonon spec-

trum at a depth of 300 nm; the dotted

and dashed vertical lines indicate pho-

non frequencies of 29 and 39 GHz,

respectively. (c) and (d) The observed

phonon oscillations for the frequencies

indicated in (b).

185308-5 Persson et al. J. Appl. Phys. 118, 185308 (2015)

Fig. 5(b). The pulse shape in Fig. 5(a) has an overall duration

of 38 ps and a long tail exceeding 200 ps. Furthermore, the

maximum number of photons is 0.2% of the number incident

on the sample. For typical bunch durations in storage rings

of 200 ps, this means that the total number of reflected pho-

tons is 0.04%.

In conclusion, we find that it is possible to control the

generated phonon spectra with an optoacoustic transducer

and observe constructive and destructive interference of the

x-rays emitted from the sample. These interferences can be

used to implement an x-ray switch, but the efficiency of such

a device, as for other similar schemes27–29 will be low. This

trade-off may be worthwhile at high-flux storage rings such

as the Advanced Photon Source, USA or European

Synchrotron Radiation Facility (ESRF), France. At beamline

ID09 at ESRF, one could anticipate >104 photons per pulse

in this mode of operation. Alternative methods of shortening

pulses at 3rd generation synchrotron radiation sources may

involve modifying machine parameters that influence all

users. One such example is low-alpha mode where a low-

charge bunch with short-pulse duration can be produced.

Low-alpha mode for storage rings is discussed by Martin

et al.30 and references therein. However, a lowering of the

pulse duration to 5 ps typically involves lowering the bunch

current and thereby the x-ray flux by one to two orders of

magnitude. Future improvements of an optoacoustic trans-

ducer x-ray switch could yield higher speed by utilizing

higher wavevector phonons as well as higher reflectivity by

increasing the strain.

ACKNOWLEDGMENTS

The authors would like to thank the Swedish Research

Council (VR), Knut and Alice Wallenberg’s Foundation, the

Crafoord Foundation, Stiftelsen Olle Engkvist byggm€astareand Marie Skłodowska-Curie ITN within Horizon 2020 for

the financial support. We also thank Aaron M. Lindenberg,

Jesper Nygaard, and Maher Harb for valuable discussions.

1C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, Phys. Rev. B 34(6),

4129 (1986).2C. Rose-Petruck, R. Jimenez, T. Guo, A. Cavalleri, C. W. Siders, F. Rksi,

J. A. Squier, B. C. Walker, K. R. Wilson, and C. P. J. Barty, Nature

398(6725), 310 (1999).3A. M. Lindenberg, I. Kang, S. L. Johnson, T. Missalla, P. A. Heimann, Z.

Chang, J. Larsson, P. H. Bucksbaum, H. C. Kapteyn, H. A. Padmore, R.

W. Lee, J. S. Wark, and R. W. Falcone, Phys. Rev. Lett. 84(1), 111

(2000).4J. Larsson, A. Allen, P. H. Bucksbaum, R. W. Falcone, A. Lindenberg, G.

Naylor, T. Missalla, D. A. Reis, K. Scheidt, A. Sj€ogren, P. Sondhauss, M.

Wulff, and J. S. Wark, Appl. Phys. A 75(4), 467 (2002).5A. M. Lindenberg, I. Kang, S. L. Johnson, R. W. Falcone, P. A. Heimann,

Z. Chang, R. W. Lee, and J. S. Wark, Opt. Lett. 27(10), 869 (2002).6O. Synnergren, T. N. Hansen, S. Canton, H. Enquist, P. Sondhauss, A.

Srivastava, and J. Larsson, Appl. Phys. Lett. 90(17), 171929 (2007).7M. Bargheer, N. Zhavoronkov, Y. Gritsai, J. C. Woo, D. S. Kim, M.

Woerner, and T. Elsaesser, Science 306(5702), 1771 (2004).

8P. Sondhauss, J. Larsson, M. Harbst, G. A. Naylor, A. Plech, K. Scheidt,

O. Synnergren, M. Wulff, and J. S. Wark, Phys. Rev. Lett. 94(12), 125509

(2005).9M. Trigo, Y. M. Sheu, D. A. Arms, J. Chen, S. Ghimire, R. S. Goldman, E.

Landahl, R. Merlin, E. Peterson, M. Reason, and D. A. Reis, Phys. Rev.

Lett. 101(2), 025505 (2008).10S. Fahy and R. Merlin, Phys. Rev. Lett. 73(8), 1122 (1994).11P. H. Bucksbaum and R. Merlin, Solid State Commun. 111(10), 535

(1999).12Y. Gao and M. F. DeCamp, Appl. Phys. Lett. 100(19), 191903 (2012).13A. Loether, Y. Gao, Z. Chen, M. F. DeCamp, E. M. Dufresne, D. A.

Walko, and H. Wen, Struct. Dyn. 1(2), 024301 (2014).14M. Nicoul, U. Shymanovich, A. Tarasevitch, D. von der Linde, and K.

Sokolowski-Tinten, Appl. Phys. Lett. 98(19), 191902 (2011).15B. C. Daly, N. C. R. Holme, T. Buma, C. Branciard, T. B. Norris, D. M.

Tennant, J. A. Taylor, J. E. Bower, and S. Pau, Appl. Phys. Lett. 84(25),

5180 (2004).16H. T. Grahn, H. J. Maris, and J. Tauc, IEEE J. Quantum Electron. 25(12),

2562 (1989).17R. Shayduk, M. Herzog, A. Bojahr, D. Schick, P. Gaal, W. Leitenberger,

H. Navirian, M. Sander, J. Goldshteyn, I. Vrejoiu, and M. Bargheer, Phys.

Rev. B 87(18), 184301 (2013).18P. Sondhauss, O. Synnergren, T. N. Hansen, S. E. Canton, H. Enquist, A.

Srivastava, and J. Larsson, Phys. Rev. B 78(11), 115202 (2008).19S. N. Rschevkin, A Course of Lectures on the Theory of Sound (Pergamon

Press Ltd., London, 1963), p. 464.20H. Enquist, H. Navirian, T. N. Hansen, A. M. Lindenberg, P. Sondhauss,

O. Synnergren, J. S. Wark, and J. Larsson, Phys. Rev. Lett. 98(22),

225502 (2007).21M. Harbst, T. N. Hansen, C. Caleman, W. K. Fullagar, P. J€onsson, P.

Sondhauss, O. Synnergren, and J. Larsson, Appl. Phys. A 81(5), 893

(2005).22H. Enquist, H. Navirian, R. N€uske, C. von Korff Schmising, A. Jurgilaitis,

M. Herzog, M. Bargheer, P. Sondhauss, and J. Larsson, Opt. Lett. 35(19),

3219 (2010).23J. Larsson, Z. Chang, E. Judd, P. J. Schuck, R. W. Falcone, P. A.

Heimann, H. A. Padmore, H. C. Kapteyn, P. H. Bucksbaum, M. M.

Murnane, R. W. Lee, A. Machacek, J. S. Wark, X. Liu, and B. Shan, Opt.

Lett. 22(13), 1012 (1997).24B. L. Henke, E. M. Gullikson, and J. C. Davis, At. Data Nucl. Data Tables

54(2), 181 (1993).25D. A. Reis, M. F. DeCamp, P. H. Bucksbaum, R. Clarke, E. Dufresne, M.

Hertlein, R. Merlin, R. Falcone, H. Kapteyn, M. M. Murnane, J. Larsson,

Th. Missalla, and J. S. Wark, Phys. Rev. Lett. 86(14), 3072 (2001).26S. A. Stepanov, Proc. SPIE 5536, 16 (2004).27J. M. H. Sheppard, P. Sondhauss, R. Merlin, P. Bucksbaum, R. W. Lee,

and J. S. Wark, Solid State Commun. 136(3), 181 (2005).28M. Herzog, W. Leitenberger, R. Shayduk, R. M. van der Veen, C. J.

Milne, S. L. Johnson, I. Vrejoiu, M. Alexe, D. Hesse, and M. Bargheer,

Appl. Phys. Lett. 96(16), 161906 (2010).29P. Gaal, D. Schick, M. Herzog, A. Bojahr, R. Shayduk, J. Goldshteyn, W.

Leitenberger, I. Vrejoiu, D. Khakhulin, M. Wulff, and M. Bargheer,

J. Synchrotron Radiat. 21(2), 380 (2014).30I. P. S. Martin, G. Rehm, C. Thomas, and R. Bartolini, Phys. Rev. Spec.

Top.–Accel. Beams 14(4), 040705 (2011).31D. E. Aspnes and A. A. Studna, Phys. Rev. B 27(2), 985 (1983).32“Physical constants of inorganic compounds,” in CRC Handbook of

Chemistry and Physics 95th Edition (Internet Version 2015), edited by W.

M. Haynes (CRC Press/Taylor and Francis, Boca Raton, FL, 2015).33L. J. Slutsky and C. W. Garland, Phys. Rev. 113(1), 167 (1959).34D. W. Lynch, R. Rosei, and J. H. Weaver, Solid State Commun. 9(24),

2195 (1971).35“Speed of sound in various media,” in CRC Handbook of Chemistry and

Physics, 95th Edition (Internet Version 2015), edited by W. M. Haynes

(RCR Press/Taylor and Francis, Boca Raton, FL, 2015).36T. Saito, O. Matsuda, and O. B. Wright, Phys. Rev. B 67(20), 205421

(2003).

185308-6 Persson et al. J. Appl. Phys. 118, 185308 (2015)