PHYSICAL REVIEW B 98, 054303 (2018)

Excitation of multiple phonon modes in copper metaborate CuB2 O4
via nonresonant impulsive stimulated Raman scattering
Kotaro Imasaka,1 Roman V. Pisarev,2 Leonard N. Bezmaternykh,3 Tsutomu Shimura,1Alexandra M. Kalashnikova,1,2,* and Takuya Satoh1,41Institute of Industrial Science, The University of Tokyo, 153-8505 Tokyo, Japan
2
Ioffe Institute, 194021 St. Petersburg, Russia
3
L. V. Kirensky Institute of Physics, SB RAS, 660036 Krasnoyarsk, Russia
4
Department of Physics, Kyushu University, 819-0395 Fukuoka, Japan(Received 14 February 2017; revised manuscript received 11 July 2018; published 7 August 2018)
The excitation of four coherent phonon modes of different symmetries is realized in copper metaborate CuB2 O4
via impulsive stimulated Raman scattering (ISRS). The phonons are detected by monitoring changes in the linear
optical birefringence using the polarimetric detection (PD) technique. We compare the results of the ISRS-PD
experiment to the polarized spontaneous Raman scattering spectra. We show that agreement between the two
sets of data obtained by these allied techniques in a wide phonon frequency range of 4–14 THz can be achievedby taking into account the symmetry of the phonon modes and corresponding excitation and detection selection
rules. It is also important to account for the difference between incoherent and coherent phonons in terms of theircontributions to the Raman scattering process. This comparative analysis highlights the importance of the ratio
between the frequency of a particular mode, and the pump and probe spectral widths. We analytically demonstratethat the pump and probe pulse durations of 90 and 50 fs, respectively, used in our experiments limit the highest
frequency of the excited and detected coherent phonon modes to 12 THz, and define their relative amplitudes.
DOI: 10.1103/PhysRevB.98.054303
I. INTRODUCTION

Impulsive stimulated Raman scattering (ISRS) [1] is a
powerful technique, allowing the generation of collective
excitations in a medium via inelastic scattering of a single(sub-)picosecond laser pulse. It has become a widely used
tool for exciting coherent phonons [2–5], magnons [6,7],
phonon-polaritons [8], and coherent charge fluctuations [9].
ISRS is essentially the stimulated inelastic scattering of aphoton of frequency ωi into a photon of frequency ωj =ωi −  accompanied by the creation of a quasiparticle offrequency . This process is, on the one hand, governedby selection rules, i.e., by medium properties described bya Raman tensor, and, on the other hand, highly responsive
to the spectral and temporal characteristic of the exciting
laser pulse. As a result, ISRS enables selective excitationof particular coherent quasiparticles achieved by choosing
proper polarization of the laser pulse. Control of the coherent
quasiparticle amplitude can be realized by pulse shaping[10–15]. Importantly, ISRS can be realized in both opaque andtransparent media. While in the former case ISRS competes
with other excitation mechanisms based on impulsive lightabsorption [16], in the latter case, nonresonant ISRS is the solemechanism driving coherent excitations of electrons, lattice, orspins [17].
ISRS and spontaneous Raman scattering (RS) are closely
related processes. The selection rules for RS define the polarization of exciting femtosecond pulses to be used for triggering*kalashnikova@mail.ioffe.ru

2469-9950/2018/98(5)/054303(13)impulsively specific excitation in a medium. However, sincethe (sub-)picosecond pulse should possess a sufficiently broad
spectrum to excite a coherent mode of a given frequency, thepulse duration defines the efficiency of the ISRS. In contrast
to RS experiments, ISRS employs probe pulses to monitor
excited coherent quasiparticles in the time domain. Therefore
the polarization and duration of these pulses, although often
disregarded, are of no lesser importance for the outcomeof the conventional ISRS experiments. The ISRS and RS
processes differ in the character of the quasiparticles addressed;
coherent versus incoherent, respectively. Understanding how
the interplay of all these factors affects the results of the
ISRS experiment, and establishing vivid links between Raman tensor components and values measured in the ISRS
experiments are crucial in light of recent developments of
ISRS-based techniques. Thus a novel approach was recently
suggested to obtain information about Raman tensors from
ISRS-based coherent lattice fluctuation spectroscopy [18]. It
has been shown that a comparison of RS and pump-probe
data can be used to identify the processes underlying coherent
phonon-plasmon mode generation in doped GaN [19]. The
comparison between RS data and the outcome of pumpprobe experiments was also recently made for the case ofdisplacive excitation of coherent phonons (DECP) in opaquebismuth and antimony to obtain insights into ultrafast processes
triggered by femtosecond laser pulses [20]. It was recently
suggested in Ref. [21] and later disputed in Ref. [22] thatthe changes in the coherent phonon amplitude with pumppulse duration may shed light on the excitation mechanismand help distinguish resonant ISRS from DECP mechanisms.
Finally, the excitation and detection of a plethora of coherent

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PHYSICAL REVIEW B 98, 054303 (2018)quasiparticles in a single experiment, as well as access toother types of high-frequency collective excitations [23], hasrecently become possible owing to the availability of laser
pulses of ever-shorter durations [24]. Therefore tuning the polarization [25], spectral [17], temporal [15], and phase [26,27]
characteristics of laser pulses is actively exploited nowadays for realizing selective excitation of particular collectivemodes.
In this paper, we demonstrate how an intrinsic interconnection between the values measured by RS and ISRS techniques
can be achieved by designing a single ISRS experiment. Thisreveals the roles of the pump and probe polarizations anddurations, as well as the incoherent and coherent natures of
involved quasiparticles. All prerequisites for such an experiment are met by impulsively exciting in a dielectric copper
metaborate CuB2 O4 multiple coherent optical phonon modesof different symmetries, and probing them via a polarizationsensitive optical effect. The choice of CuB2 O4 is motivatedby its crystallographic structure [28] yielding an exceptionallyrich phonon spectrum, unique optical [29], magnetic [30], andmagneto-optical properties [31,32]. We demonstrate that laser
pulses of 90-fs duration can effectively excite, in CuB2 O4 , atleast four optical phonon modes of A1 and B1 symmetries
with frequencies between 4 and 12 THz. These are detectedin the ISRS experiment with the polarimetric detection (PD)technique, in which the polarization modulation of the 50-fs
probe pulses is monitored. By comparing the results of ourexperiment to the RS spectra [33,34], we establish the link
between the values measured by these two complimentary
techniques, and show that an analysis of the efficiencies of
excitation of multiple modes via ISRS has to be performedby taking into account the symmetry of each mode, the corresponding excitation and detection selection rules, and the ratio
between the frequency of the particular mode and the pumpand probe spectral widths. We note that our approach of
revealing the role of the pump and probe pulse durations in
ISRS is an alternative to the conventional one, when one tunes
the duration of the pump or probe pulses and monitors thecorresponding changes in a particular excited coherent phononmode in a medium [22,35]. In the latter studies, special care
had to be taken to account for the positive or negative chirpof either pump or probe pulses, having different effects on theamplitudes of excited [26,36,37] and detected [38] coherent
quasiparticles.
This paper is organized as follows. In Sec. II, we briefly
discuss the copper metaborate properties. In Sec. III, we
describe the sample of CuB2 O4 and the details of the ISRS-PD
experiment. In Sec. IV, we present the experimental data ofthe excitation and detection of multiple coherent phononsin CuB2 O4 by femtosecond laser pulses. In Sec. V A, we
introduce the formalism for describing the ISRS excitation andpolarimetric detection of coherent phonons. In Sec. V B, we
compare the outcomes of the ISRS-PD experiments with the
spontaneous RS spectra and analyze the excitation mechanismand specific detection features of the techniques employed.
This is followed by an analysis and discussion in Sec. V C
regarding the effect of the pump and probe pulse durations on
the excitation and detection of coherent phonons. In Sec. VI,
we summarize our findings and discuss their eventual impact
on further studies of ultrafast laser-induced processes.

II. CRYSTAL STRUCTURE AND LATTICE EXCITATIONSIN A COPPER METABORATE CuB2 O4

CuB2 O4 crystallizes in the tetragonal noncentrosymmetricspace group I 4̄2d, and its primitive unit cell contains 42
atoms [28]. This results in 126 zone-center phonon modes,
including three acoustical and 123 optical. Jahn-Teller Cu2+
(3d 9 ) ions occupy two nonequivalent crystallographic positions, 8d and 4b, in a strongly elongated [Cu2+ O2−
6 ] octahedronand planar [Cu2+ O2−]
complex,respectively.
This unique4
structure yields nontrivial optical [29], phonon [33,34,39], andmagnon [34,40,41] spectra of this compound.
The fundamental optical band gap of copper metaborate is∼4 eV [29]. The polarized optical absorption spectra below thefundamental band gap are characterized by an exceptionallypronounced set of zero-phonon lines arising from 3d-3d localized electronic transitions in Cu2+ ions in two positions, andaccompanied by multiple phonon-assisted sidebands [29]. Thisobservation has naturally triggered an interest in experimental
and theoretical analyses of the phonon modes in CuB2 O4
by means of infrared and Raman spectroscopy in a wide
temperature range of 4–300 K [33,34,39]. All theoretically
predicted optical phonon modes in the center of the Brillouinzone were observed in the frequency range above 4 THz andassigned to particular atomic motions [33]. Below ∼15 THz,phonon spectra are dominated by vibrations in Cu-O complexes, while the dynamics of B-O complexes contributes to thehigher-frequency phonon modes. Some of these modes involve
vibrations within [Cu2+ O2−
4 ] exclusively, while no such puremodes exists for the other complex.
It is worth noting that an intricate magnetic structure of
CuB2 O4 is described by two magnetic subsystems comprised
by Cu2+ ions in the 8d and 4b positions. Strong exchangeinteractions were found only within the subsystem formed
by Cu2+ (4b) magnetic moments below the Néel temperature, TN = 21 K. Intersublattice interactions yield a partial
ordering of the second subsystem below ≈10 K [40,41].
As a result, CuB2 O4 possesses a very rich magnetic phase
diagram, the details of which were clarified experimentally
only recently and are not yet fully understood [30,32,42–45].
Driving coherent phonon modes, in particular pure [Cu2+ O2−
4 ]modes, can be seen as a prospective approach for exploiting phonon-magnon interactions in such a complex
system. In the last few years, there have been a number of intriguing and controversial reports of the magnetooptical properties of copper metaborate [31,32,46–51],
including the very recent demonstration of laser-induced nonreciprocal light absorption and magnetoelectricity [52].

III. EXPERIMENTAL

The sample was a plane-parallel single crystal plate cut
perpendicular to the [010] axis from a large boule grown by
the Kyropoulos method from the melt of the oxides B2 O3 ,CuO, Li2 O, and MoO3 [53]. We used a coordinate systemwith the x, y, and z axes directed along the [100], [010], and[001] crystallographic axes, respectively [Fig. 1(a)]. We notethat the [100] and [010] axes cannot be distinguished in theparamagnetic phase, and the assignment of x and y to these054303-2

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PHYSICAL REVIEW B 98, 054303 (2018)

FIG. 1. (a) Relative orientation of the crystallographic axes andlaboratory frame xyz. The pump pulses are incident along the y axis.
The angle of incidence of the probe beam is 7◦ . The pump (Ep ) andprobe (Epr ) pulses are linearly polarized with the azimuthal angles
θ and φ, respectively. (b) Elliptically polarized probe pulse after
transmission through the sample. (c) Ellipticity, η, of the probe
pulses converted to a rotation φ of the polarization plane after
transmission through the quarter-wave plate.

FIG. 2. Probe ellipticity change η as a function of the pumpprobe time delay, t, measured at three different polarizations, θ , ofthe pump pulses. The probe polarization is φ = 45◦ .

axes was performed for the sake of convenience. The thicknessof the sample was d0 = 67 μm.
The experimental studies of the excitation of the coherent
phonons by femtosecond laser pulses were performed using
an optical pump-probe technique. The 50-fs laser pulses witha central photon energy of 1.55 eV at a repetition rate of
1 kHz were generated by a regenerative amplifier. Here andbelow, we define the pulse duration as the full width at thehalf maximum (FWHM) of its intensity profile. Part of theoutput beam was steered to the optical parametric amplifier,producing τp = 90 fs pump pulses of h̄ωp = 1.08 eV central
photon energy. CuB2 O4 is transparent in this spectral range,and so ISRS was the dominant mechanism for coherent phonon
excitation. Linearly polarized pump pulses at the azimuthal
angle θ with respect to the x axis [see Fig. 1(a)] propagatedalong the sample normal. The pump spot size at the sample was
70 μm (FWHM). A second part of the regenerative amplifier
output beam was used as the probe pulses (h̄ωpr = 1.55 eV,
τpr = 50 fs) and was delayed with respect to the pump pulses
by the variable time, t. The incident probe pulses were linearlypolarized with the azimuthal angle φ with respect to the xaxis [Fig. 1(a)]. The probe spot size at the sample was 40 μm(FWHM).
Coherent phonons excited by the pump pulses modulate thedielectric tensor components of CuB2 O4 , which can be seenin the experiment as a modulation either of the probe pulse
ellipticity [Fig. 1(b)] or the probe pulse polarization azimuthal
angle due to the pump-induced changes in the crystallographic
linear birefringence or dichroism, respectively. We note that theabsorption coefficient of CuB2 O4 at the probe photon energy
was measured to be of ∼60 cm−1 , which suggests that thedichroism experienced by the probe pulses is relatively weak.
In the experiments we measured the pump-induced probe
ellipticity changes η by employing the PD technique. A
quarter-wave plate (QWP) placed in the probe beam behind thesample was used to convert the ellipticity η to a polarization
rotation φ [Fig. 1(c)]. The probe beam transmitted throughthe sample and QWP was split by a Wollaston prism into

vertically and horizontally polarized beams and their intensities were detected by two Si-photodiodes. In this way, thechange of the polarization of the probe pulses, measured as thedifference between the signals at the two diodes, was monitored
as a function of the pump-probe time-delay t. The pump-probe
traces were recorded in steps of 0.02 ps up to 10 ps, including
a −2 ps negative pump-probe delay. All measurements were
performed at T = 293 K.
We would like to emphasize that the detection of coherent phonons, excited via ISRS, is usually realized via themonitoring of changes in the reflectivity or transmitivity.
Instead, we employed the PD scheme to reveal transientpolarization changes, which is more common for experimentson coherent magnons excited via ISRS [7]. To emphasize
the differences with the conventional ISRS experiments withcoherent phonons, further on we refer to our experimental
layout as to ISRS-PD experiment. An advantage of such ascheme for coherent phonon detection is that it allows theanalysis of the symmetry of particular phonon modes and candiscriminate between them if necessary by choosing a proper
probe polarization, as we discuss in detail below. We note thatsuch a scheme is a powerful alternative to the reflective electrooptical sampling technique [54,55]. The polarization sensitive
detection of coherent phonons was reported in, e.g., Ref. [56],where the polarization dependent reflectivity was analyzed.
Here we employ measurements of the transient birefringence,
which is an advantageous technique for transparent media.
IV. EXCITATION AND DETECTION OF COHERENTPHONONS IN CuB2 O4

Figure 2 shows the time-delay dependence of the probe
polarization excited by the pump pulses of three different
polarizations, θ = 0◦ (Ep x), θ = 90◦ (Ep z), and θ = 45◦ .
The probe was polarized at φ = 45◦ . In all three excitationgeometries, a strong coherent artifact was observed at thepump-probe overlap (t = 0), followed by a pronounced oscillatory signal, consisting of several superimposed harmonic

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FIG. 3. (a)–(c) Spontaneous RS spectra measured in the geometries (a) z(xx )z̄, (b) y(xz)ȳ, and (c) x(zz)x̄ (adapted from Ref. [33]). (d)–(f)FFT spectra of the ISRS-PD experimental data (Fig. 2) measured in the geometries (d) Ep x (θ = 0), (e) θ = 45◦ , and (f) Ep z (θ = 90◦ ).
In the data sets (a) and (c), the spectra were normalized by the intensity of the 10.00-THz line, while in the data set (b)—by the intensityof the strong 11.92-THz line. In the data sets (d)–(f), the spectra were normalized by the amplitude of the 10.00-THz line. The symbols
represent the experimental data, and the lines show their fit using Lorentzian functions [Eq. (9)]. The numbers indicate the phonon frequencies
k /2π . The amplitudes Ck and FWHMs σk of the phonon lines are given in Tables III and IV. Different colors indicate the phonon modes of
the A1 (black), B1 (red), and E(y ) (blue) symmetries. The green color denotes the lines to which the phonons of both A1 and B1 symmetries
contribute. The inset shows the pump fluence dependence on the amplitudes of the 7.52- (squares), 10.00- (circles), and 12.03-THz (triangles)
lines in the FFT spectra shown in (f). The lines are linear fits.

components. The fast Fourier transforms (FFT) amplitude
spectra of the traces at positive delays t > 1 ps are plotted in
Figs. 3(d)–3(f). In the FFT spectra, four lines with frequencies
of 4.38, 7.52, 10.00, and 12.03 THz are clearly distinguished.
To reveal the nature of the observed oscillations of theprobe polarization and to examine their relation to the coherent
phonons excited by the pump pulses, we compared the FFTspectra [Figs. 3(d)–3(f)] to the spontaneous RS spectra at
293 K reported in Ref. [33] [Figs. 3(a)–3(c)]. The frequenciesof the lines appearing in the FFT spectra in our experiments
at θ = 0, 90◦ [Figs. 3(d) and 3(f)] are in excellent agreementwith the lowest phonon lines in the RS spectra [Figs. 3(a)and 3(c)]. In particular, the phonons appearing in the x(zz)x̄
RS spectrum [Fig. 3(c)] are excited by the pump pulses
polarized along the z axis and propagating along the y axis
[Fig. 3(f)]. We note that such a comparison is justified because
the [100] (x) and [010] (y) crystallographic axes in CuB2 O4 areequivalent. Analogously, there is a correspondence between
the spontaneous y(xx)ȳ RS spectra [33,34] and our dataobtained for the pump pulses polarized along the x axis. Sucha good agreement allows us to assign the observed oscillationsof the probe polarization to the modulation of the dielectric
permittivity by coherent phonons excited by the pump pulses.
From the comparison of our experimental data with theresults of the spontaneous RS experiments, we can determine
the particular coherent phonon modes that were excited ineach geometry. Pump pulses polarized along the z axis excite
nonpolar A1 modes with frequencies of 7.52, 10.00, and12.03 THz, while the pump pulses polarized along the x axis
excite the 4.38- and 10.00-THz nonpolar B1 modes in addition
to the A1 modes (7.52 and 10.00 THz).
For the geometry with the pump pulses polarized at an angle
θ = 45◦ , the excitation of the modes with E(y) symmetry is
expected, which appears in the RS spectrum [33] measured in
the y(xz)ȳ configuration [Fig. 3(b)]. However, this is not thecase, as can be seen in Fig. 3(e).
In Fig. 4(a), we show the pump-probe traces obtained
for the pump polarization θ = 90◦ for three distinct probe
polarizations; φ = 45◦ , 90◦ , and 135◦ . In Fig. 4(b), we plot
the amplitude of the oscillatory signals η0 versus the incident probe polarization azimuthal angle φ at a frequencyof 0 /2π = 10.00 THz (the most pronounced oscillatory

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PHYSICAL REVIEW B 98, 054303 (2018)

ergy, which suppresses the alternative DECP mechanism [54].
Furthermore, DECP is expected to only drive symmetric A1
modes [16]. Indeed, when the DECP mechanism is involved,
the ions are driven from the equilibrium positions at the groundstate to the nonequilibrium positions at the excited state. As aresult, the symmetry of the crystal remains unchanged and only
the symmetric vibrational A1 mode can be excited.
The ISRS-driven excitation of a coherent phonon withfrequency 0 and normal coordinate Q(t ) can be phenomenologically described by the equation of motion [3,25]
d 2Q
1+ 20 Q =R0 (ωp )Re[Ei (t )Ej (t )∗ ]2dt
16π ij
1 0=
R (ωp )Ip0 Re[ei ej∗ ]α(τp , 0 ), (1)4nc ijFIG. 4. (a) Probing ellipticity, η, versus the pump-probe
time delay, t, measured for three initial polarizations, φ, of theprobe pulses, with a pump polarization of θ = 90◦ . (b) Amplitudeof the pump-induced oscillations at a frequency 10.00 THz of theprobe polarization η0 as a function of probe azimuthal angle φ.
(c) Calculated amplitude of the pump-induced oscillations of theprobe polarization as a function of φ (see the text for details).

contribution to the signal [Fig. 3(f)]). The amplitudes, η0 ,were extracted from the fit of the experimental data to the sinefunction. We note that here and elsewhere in the text η0 isdefined as the signed amplitude of the measured signal. The
oscillation amplitude is strongly enhanced when the incoming
polarization of the probe pulse makes an angle of φ = 45◦
with the x axis, thus demonstrating the crucial role of the probe
pulse polarization in the ISRS-PD experiment. In particular, theresults in Figs. 4(a) and 4(b) suggest that the probe polarized
at a nonzero angle with respect to the pump polarization planefavors detection of the coherent phonon mode excited by thelatter [25]. This provides a hint for explaining the absence ofthe coherent phonon modes in the E(y) symmetry for the signal[Figs. 2 and 3(e)] measured with θ = 45◦ and φ = 45◦ .
The most evident difference between the results of the
ISRS-PD experiments and spontaneous RS data is that therelative values of the amplitudes of the coherent phonons
excited via ISRS cannot be directly related to the amplitudesof the corresponding lines in the spontaneous RS spectra. Furthermore, no coherent phonons with frequencies above 12.03
THz could be reliably observed in the ISRS-PD experiments,
despite the fact that some of these yielded very strong lines
in the RS spectra (Fig. 3). In Secs. V A–V C, we consider
the excitation and detection of coherent phonons in detail toaccount for these observations.

0
where Rij (ωp ) = ∂εij (ωp )/∂Q is the Raman tensor andεij (ωp ) is the dielectric permittivity tensor. Note that Eq. (1)does not include energy dissipation. It is also assumed thatthe dielectric permittivity dispersion is negligible within the
spectral width of the pump pulse. Ei (t ) is the time-dependent
electric field envelope defined as Ei (t ) ≡ Re[Ei (t )eiωp t ] withamplitude E0 and FWHM τ . The pump intensity Ip0 is introduced as Ip0 = nc|E0 |2 /(4π ), where n is the refractive index at
the pump wavelength. e = (ex , ey , ez ) is the polarization unit
vector. α(τp , 0 ) includes the relation between the duration of
the pump pulse, or its spectral width σp , and the frequency ofthe coherent phonon mode. The coefficient α(τp , 0 ) in Eq. (1)for the phonon mode 0 can be obtained by recalling that the
product of two functions in the time domain can be expressed
via the convolution of their Fourier-transforms:∞
Ep (ω)Ep∗ (ω − 0 )dω∞.
(2)α(τp , 0 ) = −∞∗−∞ Ep (ω)Ep (ω)dωThe nominator in the expression for α(τp , 0 ) reflects thephysical description of the ISRS process, in which pairs ofphotons with frequencies that differ by 0 contribute to thegeneration of the corresponding coherent phonon mode. Boththe nominator and denominator reduce to 4π Ip0 /nc for theδ-pulse, and α(τp → 0, 0 ) = 1. The solution of Eq. (1) takes
the simple form of a sine function0∗0Q(t ) ∼ α(τp , 0 )Rij (ωp )Ip Re[ei ej ] sin (0 t ).

In the ISRS-PD experiments reported here, the measured value
is the change of the ellipticity η of the probe polarization
occurring due to modulation of the dielectric permittivity by
excited coherent phonons, and converted to the polarization
rotation by QWP [Figs. 1(b) and 1(c)]. The temporal evolutionof η can be then expressed as (see Appendix for details)
0
Ipr (t )π d0 Rij (ωpr )Q(t ) ∗ 0 ,η(t ) =
√
2λpr
εij
Ipr

V. THEORETICAL BACKGROUND AND DISCUSSION
OF ISRS-PD IN CuB2 O4
A. ISRS as the excitation mechanism of coherent
phonons in CuB2 O4

In general, there are two mechanisms, ISRS and DECP,
which can mediate the excitation of coherent phonons by the
femtosecond laser pulse. We argue that, in our experiments,
ISRS is the mechanism responsible for the excitation. First,
the sample is transparent for the pump central photon en-

(3)

(4)

where Ipr (t ) is the temporal profile of the probe pulse, theexact form of which is introduced below, and ∗ denotes the
convolution operation. The convolution with the probe pulse
temporal profile Ipr (t ) is required to account for the particular
probe duration τpr . λpr is the probe wavelength. Here, thedielectric permittivity dispersion is assumed negligible within
the spectral width of the probe pulse.

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TABLE I. Raman tensor components Rij and the corresponding driving forces in Eq. (1) for the phonon modes of a particular symmetry
under the assumption of infinitesimally short pump pulses. Also listed are the dielectric tensor components δεij (t ) modulated by coherent
phonons of the particular symmetry. For convenience, we have omitted the factor Ip0 /4nc in the expressions for the driving forces.
Rij

ISRS driving force

δεij (t ) = Rij Q(t )Rxx = Ryy , Rzz
Rxx = −Ryy
Rxy = Ryx
Ryz = Rzy
Rzx = Rxz

Rxx Re(ex ex∗ + ey ey∗ ) + Rzz Re(ez ez∗ )Rxx Re(ex ex∗ − ey ey∗ )Rxy Re(ex ey∗ + ey ex∗ )Ryz Re(ey ez∗ + ez ey∗ )Rxz Re(ez ex∗ + ex ez∗ )δεxx (t ) = δεyy (t ), δεzz (t )δεxx (t ) = −δεyy (t )δεxy (t ) = δεyx (t )δεyz (t ) = δεzy (t )δεzx (t ) = δεxz (t )Phonon symmetry
A1
B1
B2
E(x )
E(y )

The amplitude of the coherent phonons excited and detected in the ISRS-PD process [Eq. (4)] has four main con0
stituents. (i) The specific values of Raman tensors R
ij (ωp )0
and R
ij (ωpr ) are determined by the material properties at
the frequencies ωp and ωpr of the pump and probe pulses,
respectively. Generally speaking, the absolute values of the0R
ij components at these two optical frequencies can differ
due to a dispersion in the corresponding spectral range. (ii) The
0∗0
product R
ij (ωp )Ip Re[ei ej ] describes the dependence of thedriving force on the intensity Ip0 and polarization e of thepump pulses. (iii) The parameter α(τp , 0 ) describes the roleof the limited spectral width of the pump pulse in the excitation
process. (iv) The convolution with Ipr (t ) allows us to accountfor the polarization changes of the probe pulse due to themodulation of the dielectric permittivity within the range τprnear the time delay t.
B. Role of the pump and probe polarizations in
the ISRS-PD experimentFirst, we consider the excitation and detection of coherent
phonons in CuB2 O4 , neglecting the duration of the pump andprobe pulses, i.e., setting α(τp , 0 ) = 1 and Ipr (t ) = Ipr0 δ(t ).
In this case, Eq. (4) is simplified and takes a form η(t ) =η0 sin(0 t ), where
0

η0 =

0

π d0 Rij (ωp )Rij (ωpr ) 0
Ip Re[ei ej∗ ].
√
2λpr
εij

(5)

The Raman tensor components for a mode of particularsymmetry in CuB2 O4 belonging to the point group 4̄2m arelisted in Table I [33]. The modes of the symmetry A2 are silent.
Considering the expressions for the Raman tensor components
Rij , we can write the right-hand side (r.h.s.) of Eq. (1) in theform given in Table I. From this symmetry analysis, one can see
directly that the B2 and E(x) modes can be excited under theconditions ex ey∗ + ey ex∗ = 0 and ey ez∗ + ez ey∗ = 0, respectively.
This is not the case in our experiments since ey = 0 when thepump pulses propagate along the y axis.
The driving force for the nonpolar A1 modes can benonzero for any polarization. In the considered geometry, thepump pulse of any linear polarization excites these modes,
provided the Raman tensor components Rxx and Rzz are
nonzero for that particular mode. Indeed, this is observed inour experiment, where the three lowest A1 modes at 7.52,
10.00, and 12.03 THz are all excited by the pump pulse
polarized along the z axis [Fig. 3(f)], because all relevant
Raman tensor components Rzz are nonzero [Fig. 3(c)]. In

contrast, only the 7.52- and 10.00-THz A1 modes are excited by
the pump pulses polarized along the x axis [Fig. 3(d)], which
agrees well with the observation that the line corresponding
to the 12.03-THz A1 (xx) phonon is also very weak in the
spontaneous RS spectra [Fig. 3(a)]. Nonpolar B1 modes can be
excited when |ex |2 = |ey |2 (see Table I). In our experiment, thiscorresponds to the pump pulse polarization making a nonzero
angle with the z axis. Indeed, the B1 mode is excited by thepump pulses with azimuthal angles θ = 0 and 45◦ [Figs. 3(d)and 3(e)].
Polar E(y) coherent phonon modes are expected to be
excited under the condition ez ex∗ + ex ez∗ = 0, which is met
at θ = 45◦ . However, as follows from a comparison of thespectra in Figs. 3(b) and 3(e), no line associated with the E(y)phonon modes appears in the FFT spectrum of the ISRS-PD
data measured in this geometry. This happens because in the
ISRS-PD experiment the detection process is as important as
the excitation. In our experimental geometry, optically excitedcoherent phonons are detected via the modulation of the probe
polarization [Eq. (4)] originating from transient changes of thelinear birefringence. The components of the dielectric tensor
modulated by the coherent phonons of a particular symmetry
are listed in Table I. Clearly, the detection of the A1 and B1
coherent phonons requires the probe polarization to make anangle φ = 0 and 90◦ (see Appendix for the details). In contrast,
the modulation of the probe polarization by E(y) coherent
phonons vanishes at φ = 45◦ .
The relevant amplitudes η0 [Eq. (5)] of the ellipticity
modulation at the phonon frequency 0 for the probe pulses
initially polarized at φ = 45◦ for each pump pulse polarization
θ are

054303-6θ =0:Rzz (ωpr ) Rxx (ωpr )π d0;
A1 : η0 =
Rxx (ωp ) √
− √
εxx
2λpr
εzz
B1 : η0 =π d0 Rxx (ωp )Rxx (ωpr )
;
√
εxx
2λpr

(6)

θ = 90◦ :Rzz (ωpr ) Rxx (ωpr )π d0;
A1 : η0 =
Rzz (ωp ) √
− √
2λpr
εzz
εxx
(7)
θ = 45◦ :√2π d0(Rxx (ωp ) + Rzz (ωp ))A1 : η0 =4λpr

EXCITATION OF MULTIPLE PHONON MODES IN COPPER …

PHYSICAL REVIEW B 98, 054303 (2018)

TABLE II. Initial phases of the probe ellipticity oscillations η(t )extracted from the fit of the data in Fig. 2 to two or three damped
sine-functions.

TABLE III. Amplitudes Ck and FWHMs σk of the A1 + B1
phonon lines in the RS (z(xx )z̄) and ISRS (Ep x) spectra [Figs. 3(a)and 3(d)]. The Raman tensor Rxx values were extracted from thespontaneous RS data taking into account the thermal occupation
numbers N (k ) + 1 at T = 293 K. The amplitudes of the oscillations
η0calc were calculated from the ISRS model, accounting for the pumpand probe pulse durations (see text). All amplitudes were normalized
to the corresponding amplitude of the 10.00-THz line.

θ4.38 THz

7.52 THz

10.00 THz0◦ (160.6 ± 4.9)◦ (142.9 ± 4.6)◦ (326.7 ± 0.7)◦–(35.0 ± 0.6)◦45◦ (186.3 ± 6.6)◦◦◦–(358.4 ± 2.1) (13.2 ± 0.9)◦
9012.03 THz––
(199.1 ± 3.2)◦

ISRS (Ep x)

RS (z(xx )z̄)Rzz (ωpr ) Rxx (ωpr )
;
− √
√
εzz
εxx
√
2π d0 Rxx (ωp )Rxx (ωp )B1 : η0 =
.
√
4λpr
εxx
E(y) : η0 = 0.

calc
k /2πCk
σkRzz
Ck ∼ η0;kσkη0;k(THz) (rel.unit) (GHz) (rel.unit) (rel.unit) (GHz) (rel.unit)(8)Note that all εij components should be taken at the opticalfrequency of the probe pulse here.
It is worth noting that the distinct pump and probe polarizations provide the most efficient conditions for the excitationand detection of coherent phonons or magnons [25], as clearlyseen from the probe polarization dependence of the ISRS-PD
signal shown in Figs. 4(a) and 4(b). We numerically analyzed
the dependence of the amplitude of the oscillations of the probe
ellipticity measured in ISRS-PD scheme on the probe polarization azimuthal angle, φ, when a coherent phonon of a symmetry
A1 modulates the real part of the dielectric permittivity tensor,
thus changing the linear crystallographic birefringence. The
calculations were performed using the Jones matrix method,
taking the experimentally obtained value of 3.1 × 10−3 of thestatic birefringence in the xz plane of the studied sample atthe probe photon energy. Calculations show [Fig. 4(c)] thatthe oscillatory signals are indeed caused by modulation of thelinear birefringence of CuB2 O4 due to coherent phonons, anddemonstrate the importance of the correct choice of the probe
polarization for detection of a laser-driven coherent phononmode of a particular symmetry.
To conclude the discussion of the excitation mechanismof the coherent phonons, we note that ISRS is the solemechanism of excitation of the B1 coherent phonon mode,
while the A1 mode, in general, can be excited via the DECP
mechanism as well. The modulation of the probe ellipticity
by the coherent phonons excited via ISRS and DECP should
possess sine- [Eq. (4)] and cosine-like temporal behaviors,
respectively [57,58]. We have therefore fitted the data in Fig. 2
to the sum of two or three damped oscillations [59]. The
resulting initial phases are shown in Table II. All phononmodes show nearly sinelike behaviors, i.e., the initial phasesare ∼0 or ∼180◦ rather than ∼90◦ or ∼270◦ . We note alsothat the amplitude of the lines in the FFT spectra show alinear dependence on the pump fluence Ip0 (see the inset in
Fig. 3), as expected for the coherent phonons excited via ISRS
[Eqs. (1)–(4)].
C. Role of the pump and probe durations in the excitation ofcoherent phonons via ISRS

Now we analyze the significant differences between theamplitudes of the phonon lines in the FFT spectra of the

4.38
7.52
10.00
10.71
12.03
14.13

0.04
0.02
1
0.08
0.04
1.31

68
77
77
88
108
124

0.16
0.13
1
0.29
0.21
1.21

0.1
0.09
1
–
–
–

40
48
79
–
–
–

0.48
0.07
1
0.04
0.01
0.04

ISRS-PD data and those of the spontaneous RS spectra (Fig. 3).
While both RS and ISRS processes are described by thesame Raman tensor components Rij , there is an essential
difference between them [4]. This is due to the character of thelight used in the RS and ISRS experiments; continuous wave
versus ultrashort pulse, and due to the different registration
techniques. These factors are accounted for in Eqs. (1)–(4)by the constituents (iii) and (iv) that are dependent on thetemporal/spectral profiles of the pump and probe pulses.
Furthermore, in the RS experiment, light scattering fromincoherent phonons takes place in thermal equilibrium, and theBose-Einstein thermal occupation factor enters the expression
for the Raman line intensity [60]. This is not the case when thecoherent phonons are driven by the ISRS process [61].
We consider four A1 and three B1 lowest phonon modes, observed in the spontaneous RS spectra below 15 THz [Figs. 3(a)and 3(c)] [33]. The lines assigned to these modes are fitted by
the sets of Lorentzian functions
 2Ck
σk
,
(9)
IRS () = c +1
π π 2 ( − k )2 + σk2kwhere k and σk are the frequency and FWHM of the k-th
phonon line, respectively. These are listed in Tables III and IV.
The amplitudes Ck of the Stokes lines
 k
2
(10)
Ck ∼ (N (k ) + 1) R
ij (ω)
can be used as a measure of the magnitude of the Raman tensor
kcomponent R
ij (ω) describing the scattering of light at thefrequency ω by the corresponding phonon. Here N (k ) =(exp( h̄k /kB T ) − 1)−1 is the Bose-Einstein thermal occupation factor. Note that at room temperature, the coefficient
N (k ) + 1 deviates from unity and affects the intensity ratio
between different Raman lines when the broad frequency
range is considered. As an example, we mention that theratio R4.38 /R14.13 extracted from the RS data appears to beoverestimated by ∼25% when the thermal occupation factor
is not taken into account.
The peak intensity of the particular phonon line is related0
to the amplitude and FWHM as IRS(k ) = 2Ck (π σk )−1 . We

054303-7

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PHYSICAL REVIEW B 98, 054303 (2018)

TABLE IV. Amplitudes Ck and FWHMs σk of the A1 phonon lines
in the RS (x(zz)x̄) and ISRS (Ep z) spectra [Figs. 3(c) and 3(f)]. The
Raman tensor Rzz values were extracted from the spontaneous RS
data taking into account the thermal occupation numbers N (k ) +1 at T = 293 K. The amplitudes of the oscillations η0calc were
calculated from the ISRS model, accounting for the pump and probe
pulse durations (see text). All amplitudes were normalized to thecorresponding amplitude of the 10.00-THz line.
ISRS (Ep z)

RS (x(zz)x̄)calc
k /2πCk
σkRzz
Ck ∼ η0;kσkη0;k(THz) (rel.unit) (GHz) (rel.unit) (rel.unit) (GHz) (rel.unit)

7.52
10.00
12.03
14.13

0.09
1
1.2
3.12

54
57
75
115

0.25
1
0.88
1.65

0.39
1
0.24
–

56
70
77
–

0.35
1
0.23
0.08

note that a less intense but broader Raman line may havea larger contribution to the Raman tensor. We also notethat the line at k /2π = 10.00 THz in the z(xx)z̄ spectrum[Fig. 3(a)] contains contributions from both the A1 and B1
2
2
10.00
phonons, i.e., C10.00 ∼ (R10.00
xx(A1 ) (ω)) + (Rxx(B1 ) (ω)) . As wasfound experimentally in Ref. [33], the A1 phonon contribution
dominates.
The RS data reported in Ref. [33] were obtained at thephoton energy of h̄ω = 2.41 eV. However, as demonstrated
in Ref. [34], the change of the excitation energy in the RS
experiments from 2.71 to 1.96 eV did not yield any significant
redistribution of the intensity of the phonon lines in the
spontaneous RS spectra. This observation indicates that the
RS process at the photon energies as high as 2.71 eV is ofa nonresonant nature. Therefore we can consider the relative
intensities of the Raman lines reported in Ref. [33] as a reliablemeasure of the Raman tensor components for the case of
nonresonant scattering at the pump (h̄ωp = 1.08 eV) and probe(h̄ωp = 1.55 eV) photon energies used in our experiments:kkRij (ωp ) = Rij (ωpr ).
The normalized amplitudes of the phonon lines in the FFTspectra obtained for the pump-probe traces [Figs. 3(d)–3(f)]using the fit function [Eq. (9)] are summarized in Tables III
and IV. The relation between the Lorentzian line amplitudes
Ci ∼ η0 and the Raman tensor components can be obtained
from Eqs. (6)–(8) and is, in general, less straightforward thanin the case of RS because of the chosen detection technique.
For the sake of simplicity, we assume the ratio Rzz /Rxx ≈ 3
for the A1 mode of frequency /2π = 7.52 THz, as estimatedfrom Eqs. (6) and (7) and from the data in Figs. 3(d) and 3(f).
Then Eqs. (6) and (7) are reduced to

FIG. 5. (a) Amplitudes Ck ∼ η0 (blue bars) of the coherent
phonon modes, as extracted from the FFT of the ISRS-PD experimental data [Figs. 3(d)–3(f)]. Also shown (hatched bars) are thevalues of the corresponding squared Raman tensor components R2ij
(see Tables III and IV). [(b) and (c)] Amplitudes of the probe ellipticity
η0 calculated using Eq. (4) for the cases (b) τpr = 50 fs, τp = 90 fs,(c) τpr = 50 fs, τp → 0 (open bars), and τpr → 0, τp = 90 fs (graybars). All amplitudes are normalized by the amplitude of the 10.00THz mode in the relevant spectra. Dashed lines in (c) show thefrequency dependence of the normalized coefficient α(90 fs, ).



1
1
.
A1 : Ci ∼ η0 ∼ R2zz √ − √
εzz
3 εxx

From Eqs. (11) and (12), it follows that the amplitudes
Ci obtained for the ISRS-PD data should be compared withthose from the RS spectra. In Fig. 5(a), the amplitudes Ci arenormalized with respect to the 10.00-THz line in the relevant
spectra. There are two general trends. First, the 10.00- and12.03-THz A1 phonon lines in the x(zz)x̄ RS spectrum have
similar amplitudes, while in the ISRS spectrum the amplitudeof the 12.03-THz coherent phonon line excited by the pumppulses with Ep z appears to be ∼4 times smaller than that of
the 10.00-THz line. Furthermore, no coherent phonons withfrequencies above 12.03-THz could be detected in the ISRSPD experiment, while some of the phonon lines in this range
gain high intensities in the RS spectra, e.g., the 14.13-THz line.
7.52 2
Second, from the RS data, it follows that (R10.00
zz /Rzz ) ∼ 10
for the A1 modes. In the ISRS spectrum, the corresponding
ratio appears to be only η010.00 /η07.52 ∼ 3.
To analyze the observed difference in the RS and ISRS
spectra, we consider how the durations of both the pump andprobe pulses affect the results. We assume that the pump andprobe pulses have Gaussian temporal profiles,
E (t ) = E0 e−2 ln 2(t /τ ) .
2

θ =0:
A1 : Ci ∼ η0 ∼ R2xx




3
1
;
√ −√
εxx
εzz

R2
B1 : Ci ∼ η0 ∼ √ xx .
εxx
θ = 90◦ :

(11)

(12)

2

(13)

In the frequency domain, both these pulses are assumed to be
Fourier-limited, and their spectral profiles are obtained as a
Fourier transform of Eq. (13),√22 ln 2 −2 ln 2 ω2 /σp(pr),(14)
E (ωp(pr) ) = E0 √eπ σp(pr)054303-8

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PHYSICAL REVIEW B 98, 054303 (2018)−1
where σp(pr) = 4 ln 2 τp(pr)is the FWHM of the pump and probe
pulse intensity profiles in the spectral domain.
The amplitude η0 of the measured probe polarization
oscillations in the ISRS-PD experiment increased as the probe
pulse duration τpr decreased, as compared with the period of aparticular coherent phonon. To illustrate this effect, we calculated the expected change η(t ) in the ISRS-PD experimentusing Eqs. (4), (11), and (12), as well as assuming infinitesimally short pump pulses, i.e., α(τp → 0) = 1. In Fig. 5(c),we plot the normalized FFT intensities of A1 (Ep z), A1 , andB1 (Ep x) coherent phonons calculated in this way. The probe
duration was taken to be τpr = 50 fs (σpr /2π = 9 THz), i.e.,equal to that used in our experiments. The resulting 12.03-THzline (τpr i ≈ 0.7) is noticeably suppressed as compared withthe 10.00-THz line (τpr i ≈ 0.5). This is despite the factthat the intensities of these two phonon modes in the x(zz)x̄
RS spectra [Fig. 5(a)] and, consequently, the corresponding
Raman tensor components, are nearly equal. The effect ofthe probe duration is even more pronounced when the ratios
between the intensities of the 10.00- and 14.13-THz lines in the
RS spectrum are compared with those in the calculated ISRS
spectrum. Thus, considering only the probe pulse duration,
one can account for, to a certain extent, the differences in theintensities of the phonon lines in the RS and ISRS spectra in
the high-frequency part of the spectrum.
While the duration of the probe pulse affects the detection part of the pump-probe experiment, the duration of thepump pulse determines the efficiency of the coherent phonon
excitation. This is accounted for by the coefficient αk (τp , 0 )[Eq. (2)], which has the following form for the Gaussian pulses:

mode, in which the amplitude appears to be overestimated.
Importantly, our calculations show that a reasonable agreement
between the calculated ISRS-PD spectra and those obtained
from the pump-probe experiment is obtained only by including
the durations of both the pump and probe pulses.
To complete the comparative analysis of the spontaneous
RS and ISRS-PD data, we note that the phonon lines in the RS
spectra have very narrow widths (see Tables III and IV) [33].
Thus, in the x(zz)x̄ RS spectra, the line at 10.00 THz ischaracterized by a FWHM of ≈60 GHz. In the FFT spectraof the ISRS-PD data measured with Ep z, this line has a
FWHM of 70 GHz, i.e., broader by only ∼15%. We note that,as discussed in Ref. [5], the shorter decay time (broader line
width) of the coherent phonons excited via ISRS, as comparedto that of incoherent phonons contributing to the RS spectra,
is caused by different energy dissipation channels. However,some of the lines appear to be even narrower in the ISRS-PD
experiments compared with those of the RS spectra. This ismost probably related to the experimental limitations in thetemporal and spectral resolutions in these two cases.

αk (τp , 0 ) = e−2 ln 2 20 /σp2

.

(15)

The normalized coefficient α(90 fs, ) in the range below
15 THz is shown in Fig. 5(c) using the dashed lines. Thisgraph clearly demonstrates that the efficiency of the excitationof, e.g., the 10.00-THz phonon mode would be ∼3.5 times
greater than that of the 12.03-THz mode, given equal Raman
tensor components associated with these modes. We illustrate
the effect of the pump pulse duration on the outcome of thepump-probe experiments by calculating the ISRS-PD spectrausing Eqs. (4), (11), (12), and (15), as well as assuming
an infinitesimally short probe pulse Ipr (t ) = Ipr0 δ(t ) and apump pulse of duration τp = 90 fs (σp /2π ≈ 5 THz). We notethat the amplitudes η0 obtained at τpr → 0 are proportional
to the squared amplitudes of the normal coordinates Q0 , i.e., tothe corresponding atomic displacements excited by the pumppulse.
Finally, we calculate the ISRS-PD amplitudes of thephonons by considering the durations of both the pump (τp =90 fs) and probe (τpr = 50 fs) pulses. The results are shown in
Fig. 5(b) and listed in Tables III and IV. For the excitation of theA1 mode (left side of Fig. 5(b)), our model adequately describes
the main trends observed in the experiments. As shown, for thethree A1 modes with frequencies lying in the range 7–15 THz,
this model accounts for either the partial or total suppressionof the phonon lines with frequencies above 10 THz in the
ISRS-PD spectrum, in all considered geometries. In the lowerfrequency range, the discrepancy between the experimental
results and model occurs only for the 4.38-THz B1 phonon

VI. CONCLUSIONS

We have performed a detailed study of laser-induced excitation and detection of multiple coherent phonon modes in
the ISRS-PD experiment in the dielectric copper metaborate
CuB2 O4 , characterized by a large primitive unit cell containing
42 atoms. In total, three nonpolar A1 and one nonpolar B1
modes were distinguished in the frequency range of 4–13 THz.
We have shown that 90-fs linearly polarized laser pulses with acentral photon energy in the optical transparency range (h̄ωp =1.08 eV) excite the coherent phonons via ISRS. By comparing
the results of the ISRS-PD experiment to spontaneous RS
spectra in this material, we demonstrated that the relationshipbetween the amplitudes and intensities of the phonon lines inthese two types of experiments, is determined by both the excitation and detection conditions, respectively. Namely, theamplitude of the excited coherent phonon is determined by thepolarization, intensity, and duration of the pump pulses. Probepulse polarization and duration are as important as those of
the pump pulses. They determine how the transient changesof the dielectric permittivity due to excited coherent phonons
lead to the modulation of the probe polarization detected in the
ISRS-PD experiment.
A comparison between the spontaneous RS spectra and
ISRS-PD data also allowed us to analyze in detail the limitations imposed by the durations of the pump and probe pulses
on the excitation and detection of the coherent phonons. Accounting for both durations is required to adequately calculate
the modulation of the probe polarization in the pump-probe
experiment using the spontaneous RS data. It is importantto understand the role played by the pump and probe pulse
durations in ISRS, because ever-shorter laser pulses are currently employed in pump-probe experiments, providing accessto high-energy collective excitations in solids. To the best of ourknowledge, no such detailed analysis for the case of multiple
coherent phonon modes excitation has been reported to date.
Finally, we would like to note that the reported details of
the ultrafast coherent lattice dynamics in copper metaborate
CuB2 O4 are of importance in light of recent attention given

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KOTARO IMASAKA et al.

PHYSICAL REVIEW B 98, 054303 (2018)

to the high-frequency phonon-magnon interaction and theirrole in the ultrafast dynamics driven by femtosecond laser
pulses [62–64]. We demonstrated the excitation of coherent
vibrations of Cu2+ ions belonging to different magnetic sublattices. In particular, the B1 4.38-THz coherent phonon isthe vibration in a Cu(4b)O4 complexes solely with periodic
displacement of Cu2+ (4b) ions along the z axis. Magnetic
ions in this complex provide the strongest contribution to themagnetic ordering in CuB2 O4 .

we obtain the ellipticity η(t ) [Fig. 1(b)] that is related to theacquired phase shift between two eigenwaves as
tan 2η(t ) =APPENDIX: PROBE POLARIZATION CHANGES IN THE
ISRS-PD EXPERIMENT

Here we derive the expression that relates the changes ofthe ellipticity of the probe pulses measured in the ISRS-PD
experiment to the Raman tensor components, and the pumpand probe parameters.
1. A case of A1 and B1 coherent phonons

Let the coherent phonon only contribute to the modulationof the diagonal components of the dielectric permittivity tensor,
as in the case of the A1 or B1 phonons. Then, the expression
for the dielectric tensor is εij + δεij (t ), where δεij (t ) = 0 if
i = j . δεij is related to the coherent phonon normal coordinate
as δεij (t ) = Rij Q(t ).
We consider the light propagating along the j axis. The
eigenwaves in this case are two orthogonal linearly polarizedwaves Ei and Ek , where i and k are the unit vectors in the
directions of the i and k axes, respectively. The corresponding
complex refractive indices for these eigenwaves are ni(k) −iκi(k) = εii(kk) + δεii(kk) . After traveling the distance d0 thesewaves acquire additional phases of 2π ni(k) d0 /λ and theiramplitudes are decreased by e−2πκi(k) d0 /λ .
First, for the sake of clarity, we consider a nondissipative
medium (κi(k) = 0). Therefore the diagonal components ofthe dielectric permittivity tensor and, consequently, refractive
indices, are real values. We assume that the light is initiallylinearly polarized at an angle φ to the i axis [Fig. 1(a)]. Then,
upon traveling the distance d0 , the light becomes ellipticallypolarized. Considering that the relationship between the complex amplitudes of the orthogonal components of the lightpolarization is given by [65]Ek /Ei =tan φ + i tan η,
1 − i tan φ tan η(A1)

For small η(t ), this yields
η + η(t ) =ACKNOWLEDGMENTS

We thank V. Yu. Davydov for help with the spontaneous
Raman scattering experiments. A.M.K. acknowledges support
from the Japanese Society for Promotion of Sciences (JSPS)
via the Short-Term Fellowship Program for European andNorth-American young researchers during her stay at the
University of Tokyo. T.S. was supported by JSPS KAKENHI(No. JP15H05454 and No. JP26103004) and JSPS Core-toCore Program (A. Advanced Research Networks). R.V.P. acknowledges the support from the Russian Science Foundation(Grant No. 16-12-10456).

2π (ni (t ) − nk (t ))d0
sin 2φ.
λπ d0(ni (t ) − nk (t )) sin 2φλπ d0 √√≈( εii − εkk ) sin 2φλπ d0 δεii (t ) δεkk (t )sin 2φ, (A2)
+
− √
√
2λ
εii
εkk

where η is the static contribution to the ellipticity due tocrystallographic birefringence. This expression was obtainedby taking into account that the modulation of the dielectric
permittivity tensor induced by coherent phonons is significantly weaker than the value of the corresponding unperturbed
component, i.e., δεij
εij .
As can be seen from Eq. (A2), there are two contributions to
the ellipticity of light passing though the medium. The first one
is related to the birefringence of the unperturbed medium anddoes not contribute to the measured ISRS-PD signal. Thus thechanges in the ellipticity are related to the normal coordinateof the corresponding coherent phonon by
π d0 Rii
Rkk
Q(t ) sin 2φ,(A3)η(t ) ≈
√ −√
εkk
2λ
εii
where we took into account the relationship between the
dielectric permittivity tensor components and phonon normal
coordinate.
As can be seen from Eq. (A3), in order to detect the A1
and B1 coherent phonons by measuring the changes of theprobe pulses ellipticity η(t ), the geometry must be chosenwith φ = 0, 90o , and φ = 45o ensuring the best sensitivity. It
also follows from Eq. (A3), and Table I, that the A1 coherent
phonons would not manifest themselves in the experimentswith pump or probe light propagating along the z axis of thecrystal, since δεxx = δεyy .
In our experiments, the QWP was placed after the sample
with its axis parallel to the incoming probe beam polarization.
It is convenient to consider this experimental geometry in thecoordinate frame, with two axes directed along the light propagation direction and incoming light polarization. In this frame
the Jones vector for the light passing through the medium, withan acquired ellipticity η(t ), has the form [E0 ; E0 η(t )eiπ/2 ].
After passing through the QWP, which introduces a π/2 phaseshift between the components of the Jones vector, the light becomes linearly polarized with the Jones vector [E0 ; E0 η(t )].
Thus the azimuthal angle of the probe pulses appear to berotated by the angle φ(t ) ≈ tan(φ(t )) = η(t ).
2. A case of B2 , E(x), and E( y) coherent phonons

For the case of a coherent phonon mode that modulates theoff-diagonal components of the dielectric permittivity tensor
δεik , e.g., E(y), the eigenwaves are two linearly polarizedwaves with azimuthal angles arctan (δεik /(εkk − εii )) with

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EXCITATION OF MULTIPLE PHONON MODES IN COPPER …

the i and k axes. Here we assumed that the changes of thedielectric permittivity tensor components are smaller than the
difference between the diagonal tensor components of theunperturbed medium. The corresponding refractive indices
are n1(2) = εii(kk) ± (δεik (t ))2 /(εkk − εii ) (κi(k) = 0 for a
nondissipative medium).
The detailed expression for the changes of the light polarization are rather complex in this case, and we simplify it bynoting the following. In contrast to the previously consideredcase, here the excited coherent phonons perturb the basis
formed by the eigenwaves, while the refractive indices for√the eigenwaves can be treated as unchanged n1(2) ≈ εii(kk) .
Therefore it is convenient to consider this scenario as if thebasis remains unchanged and coincides with the i and kaxes, but the light polarization azimuthal angle is modulatedby the angle arctan (δεik (t )/(εkk − εii )) ≈ δεik (t )/(εkk − εii ).
By replacing φ by φ + δεik (t )/(εkk − εii ) in Eq. (A1), we
obtain the expression for the oscillatory part of the probe
ellipticity:η(t ) ≈

2Rik
π d0
√
√ Q(t ) cos 2φλεkk + εii
2π d0 Rik Q(t ) 2 √√( εkk + εii ) sin 2φ.
+λεkk − εii
(A4)

PHYSICAL REVIEW B 98, 054303 (2018)

FIG. 6. Ellipticity of the probe polarization η ± δη0 occurring due
to static linear birefringence (LB) (black solid line), and modulationof the dielectric tensor by nonpolar (red lines) and polar (blue lines)
coherent phonons. The dashed and dash-dotted lines capture theellipticity at maximal positive and negative changes of the dielectric
permittivity by coherent phonons, respectively. The red and blue solidlines show the expected amplitudes of the ellipticity modulation by
nonpolar and polar coherent phonons at the frequency . The inset
displays a magnification of the range close to φ = 45o to show that
only a weak modulation of the ellipticity at the frequency 2 isexpected for this polarization for the case of a polar coherent phonon.

Thus, in contrast to the A1 or B1 coherent phonons, detectionof the, e.g., E(y), coherent phonons can be realized with probe
pulses polarized at φ = 0, while at φ = 45o , their effect on theprobe polarization is quadratic on small perturbations.
To summarize, we plot in Fig. 6 the ellipticity of theprobe pulses as a function of incoming polarization φ fortwo considered cases. If the nonpolar phonons modulate thediagonal elements of the dielectric permittivity tensor, thismodulation manifests itself in a periodic change of the amplitude of the η versus φ dependence, while the knots and maxima
of the dependence remain at their positions. The amplitude of
the modulation η0 increases as the incoming polarization
reaches 45o , and is zero if φ = 0, 90o . In contrast, the polar
coherent phonon periodically shifts the knots and maxima ofthe η versus φ dependence, leaving its amplitude unchanged.
As a result, the amplitude of the ellipticity modulation η0 at
the phonon frequency  reaches its maximum at φ = 0, 90o .
At φ = 45o , only a weak modulation at doubled phonon
frequency 2 is expected (see inset in Fig. 6).

Thus the coherent phonon mode can be detected by directly
measuring the rotation of the probe polarization passing
through the crystal. We note here that the rotation [Eq. (A5)]induced by coherent phonons via linear dichroism is sensitive
to the incoming probe polarization in the same way as theellipticity occurring due to linear birefringence.

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3. A case of opaque medium

When the absorption in a medium cannot be neglected, thediagonal dielectric tensor components are complex (κi(k) > 0).
We consider here the limiting case when the imaginary partof the dielectric permittivity dominates, and the A1 coherent
phonon of a symmetry is detected. Then the light propagatinga distance d0 acquires the modulated change of the azimuthal
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√ −√
2λ
εkk
εii

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