Excitation of multiple phonon modes in copper metaborate CuB 2 O 4 via non-resonant impulsive stimulated Raman scattering
Kotaro Imasaka, Roman V. Pisarev, Leonard N. Bezmaternykh, Tsutomu Shimura, Alexandra M. Kalashnikova, Takuya Satoh
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J u l Excitation of multiple phonon modes in copper metaborate CuB O via non-resonant impulsive stimulated Raman scattering Kotaro Imasaka, Roman V. Pisarev, Leonard N. Bezmaternykh, Tsutomu Shimura, Alexandra M. Kalashnikova, , and Takuya Satoh , Institute of Industrial Science, The University of Tokyo, 153-8505 Tokyo, Japan Ioffe Institute, 194021 St. Petersburg, Russia L. V. Kirensky Institute of Physics, SB RAS, 660036 Krasnoyarsk, Russia Department of Physics, Kyushu University, 819-0395 Fukuoka, Japan (Dated: July 13, 2018)The excitation of four coherent phonon modes of different symmetries are realized in coppermetaborate CuB O via impulsive stimulated Raman scattering (ISRS). The phonons are detectedby 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 Ramanscattering spectra. We show that agreement between the two sets of data obtained by these alliedtechniques in a wide phonon frequency range of 4–14 THz can be achieved by taking into account thesymmetry of the phonon modes and corresponding excitation and detection selection rules. It is alsoimportant to account for the difference between incoherent and coherent phonons in terms of theircontributions to the Raman scattering process. This comparative analysis highlights the importanceof the ratio between the frequency of a particular mode, and the pump and probe spectral widths.We analytically demonstrate that 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 phononmodes to 12 THz, and define their relative amplitudes. PACS numbers: 78.47.D-, 78.30.-j, 74.72.Cj, 78.20.Fm
I. INTRODUCTION
Impulsive stimulated Raman scattering (ISRS) is apowerful technique, allowing the generation of collectiveexcitations in a medium via inelastic scattering of a sin-gle (sub-)picosecond laser pulse. It has become a widelyused tool for exciting coherent phonons, magnons, phonon-polaritons, and coherent charge fluctuations. ISRS is essentially the stimulated inelastic scattering ofa photon of frequency ω i into a photon of frequency ω j = ω i − Ω accompanied by the creation of a quasipar-ticle of frequency Ω. This process is, on the one hand,governed by selection rules, i.e., by medium propertiesdescribed by a Raman tensor, and, on the other hand,highly responsive to the spectral and temporal charac-teristic of the exciting laser pulse. As a result, ISRSenables selective excitation of particular coherent quasi-particles achieved by choosing proper polarization of thelaser pulse. Control of the coherent quasiparticle ampli-tude can be realized by pulse shaping.
Importantly,ISRS can be realized in both opaque and transparentmedia. While in the former case ISRS competes withother excitation mechanisms based on impulsive lightabsorption, in the latter case, non-resonant ISRS is thesole mechanism driving coherent excitations of electrons,lattice, or spins. ISRS and spontaneous Raman scattering (RS) areclosely related processes. The selection rules for RS de-fine the polarization of exciting femtosecond pulses tobe used for triggering impulsively specific excitation ina medium. However, since the (sub-)picosecond pulseshould possess a sufficiently broad spectrum to excite a coherent mode of a given frequency, the pulse dura-tion defines the efficiency of the ISRS. In contrast toRS experiments, ISRS employs probe pulses to moni-tor excited coherent quasiparticles in the time domain.Therefore, the polarization and duration of these pulses,although often disregarded, are of no lesser importancefor the outcome of the conventional ISRS experiments.The ISRS and RS processes differ in the character of thequasiparticles addressed; coherent versus incoherent, re-spectively. Understanding how the interplay of all thesefactors affects the results of the ISRS experiment, andestablishing vivid links between Raman tensor compo-nents and values measured in the ISRS experiments arecrucial in light of recent developments of ISRS-basedtechniques. Thus, a novel approach was recently sug-gested to obtain information about Raman tensors fromISRS-based coherent lattice fluctuation spectroscopy. It has been shown that a comparison of RS and pump-probe data can be used to identify the processes underly-ing coherent phonon-plasmon mode generation in dopedGaN. The comparison between RS data and the out-come of pump-probe experiments was also recently madefor the case of displacive excitation of coherent phonons(DECP) in opaque bismuth and antimony to obtain in-sights into ultrafast processes triggered by femtosecondlaser pulses. It was recently suggested in Ref. 21 andlater disputed in Ref. 22 that the changes in the coher-ent phonon amplitude with pump pulse duration mayshed light on the excitation mechanism and help distin-guish resonant ISRS from DECP mechanisms. Finally,the excitation and detection of a plethora of coherentquasi-particles in a single experiment, as well as accessto other types of high-frequency collective excitations, has recently become possible owing to the availability oflaser pulses of ever-shorter durations. Therefore, tun-ing polarization, spectral, temporal, and phase characteristics of laser pulses are actively exploited nowa-days for realizing selective excitation of particular collec-tive modes.In this Article we demonstrate how an intrinsic inter-connection between the values measured by RS and ISRStechniques can be achieved by designing a single ISRS ex-periment. This reveals the roles of the pump and probepolarizations and durations, as well as the incoherent andcoherent natures of involved quasiparticles. All prerequi-sites for such an experiment are met by impulsively excit-ing in a dielectric copper metaborate CuB O multiplecoherent optical phonon modes of different symmetries,and probing them via a polarization-sensitive optical ef-fect. The choice of CuB O is motivated by its crystallo-graphic structure yielding an exceptionally rich phononspectrum, unique optical, magnetic, and magneto-optical properties. We demonstrate that laser pulsesof 90-fs duration can effectively excite, in CuB O , atleast four optical phonon modes of A and B symmetrieswith frequencies between 4 and 12 THz. These are de-tected in the ISRS experiment with the polarimetric de-tection (PD) technique, in which the polarization modu-lation of the 50-fs probe pulses is monitored. By compar-ing the results of our experiment to the RS spectra, we establish the link between the values measured bythese two complimentary techniques, and show that ananalysis of the efficiencies of excitation of multiple modesvia ISRS has to be performed by taking into accountthe symmetry of each mode, the corresponding excita-tion and detection selection rules, and the ratio betweenthe frequency of the particular mode and the pump andprobe spectral widths. We note that our approach of re-vealing the role of the pump and probe pulse durations inISRS is an alternative to the conventional one, when onetunes the duration of the pump or probe pulses and mon-itors the corresponding changes in a particular excitedcoherent phonon mode in a medium. In latter stud-ies special care had to be taken to account for the positiveor negative chirp of either pump or probe pulses, havingdifferent effects on the amplitudes of excited anddetected coherent quasiparticles.This Article is organized as follows. In Sec. II we brieflydiscuss the copper metaborate properties. In Sec. IIIwe describe the sample of copper metaborate CuB O and the details of the ISRS-PD experiment. In Sec. IVwe present the experimental data of the excitation anddetection of multiple coherent phonons in CuB O byfemtosecond laser pulses. In Sec. V A we introduce theformalism for describing the ISRS excitation and polari-metric detection of coherent phonons. In Sec. V B wecompare the outcomes of the ISRS-PD experiments withthe spontaneous RS spectra and analyze the excitationmechanism and specific detection features of the tech-niques employed. This is followed by an analysis and dis- cussion in Sec. V C regarding the effect of the pump andprobe pulses durations on the excitation and detection ofcoherent phonons. In Sec. VI we summarize our findingsand discuss their eventual impact on further studies ofultrafast laser-induced processes. II. CRYSTAL STRUCTURE AND LATTICEEXCITATIONS IN A COPPER METABORATECuB O CuB O crystallizes in the tetragonal non-centrosymmetric space group I ¯42 d , and its primitiveunit cell contains 42 atoms. This results in 126zone-center phonon modes, including three acousticaland 123 optical. Jahn-Teller Cu (3 d ) ions occupytwo non-equivalent crystallographic positions, 8 d and4 b , in a strongly elongated [Cu O − ] octahedron andplanar [Cu O − ] complex, respectively. This uniquestructure yields nontrivial optical, phonon, andmagnon, spectra of this compound.The fundamental optical band gap of copper metabo-rate is ∼ The polarized optical absorption spectrabelow the fundamental band gap are characterized by anexceptionally pronounced set of zero-phonon lines arisingfrom 3 d − d localized electronic transitions in Cu ionsin two positions, and accompanied by multiple phonon-assisted sidebands. This observation has naturally trig-gered an interest in experimental and theoretical analysesof the phonon modes in CuB O by means of infrared andRaman spectroscopy in a wide temperature range of 4–300 K. All theoretically predicted optical phononmodes in the center of the Brillouin zone were observed inthe frequency range above 4 THz and assigned to partic-ular atomic motions. Below ∼
15 THz phonon spectraare dominated by vibrations in Cu-O complexes, whilethe dynamics of B-O complexes contributes to the higher-frequency phonon modes. Some of these modes involvevibrations within [Cu O − ] exclusively, while no suchpure modes exists for the other complex.It is worth noting that an intricate magnetic struc-ture of CuB O is described by two magnetic sublat-tices comprised by Cu ions in the 8 d and 4 b positions.Strong exchange interactions were found only within thesublattice formed by Cu (4 b ) magnetic moments belowthe N´eel temperature, T N =21 K. Intersublattice inter-actions yield an ordering of the second sublattice be-low ≈
10 K.
As a result, CuB O possesses a veryrich magnetic phase diagram, the details of whichwere clarified experimentally only recently and are notyet fully understood. Driving coherent phonon modes,in particular pure [Cu O − ] modes, can be seen as aprospective approach for exploiting phonon-magnon in-teractions in such a complex system.In the last few years, there have been a numberof intriguing and controversial reports of the magneto-optical properties of copper metaborate, includ-ing the very recent demonstration of laser-induced non-reciprocal light absorption. III. EXPERIMENTAL
The sample was a plane-parallel single crystal plate cutperpendicular to the [010] axis from a large boule grownby the Kyropulos method from the melt of the oxidesB O , CuO, Li O, and MoO . We used a coordinatesystem with the x -, y -, and z -axes directed along the[100], [010], and [001] crystallographic axes, respectively[Fig. 1(a)]. We note that the [100] and [010] axes can-not be distinguished in the paramagnetic phase, and theassignment of x and y to these axes was performed forthe sake of convenience. The thickness of the sample was d =67 µ m.The experimental studies of the excitation of the coher-ent phonons by femtosecond laser pulses were performedusing an optical pump-probe technique. The 50-fs laserpulses with a central photon energy of 1 .
55 eV at a repe-tition rate of 1 kHz were generated by a regenerative am-plifier. Here and below we define the pulse duration asthe full width at the half maximum (FWHM) of its inten-sity profile. Part of the output beam was steered to theoptical parametric amplifier, producing τ p = 90-fs pumppulses of ¯ hω p = 1 . O is transparent in this spectral range, and so ISRS wasthe 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)] propagated alongthe sample normal. The pump spot size at the samplewas 70 µ m (FWHM). A second part of the regenera-tive amplifier output beam was used as the probe pulses(¯ hω pr = 1 .
55 eV, τ pr = 50 fs) and was delayed with therespect to the pump pulses by the variable time, t . Theincident probe pulses were linearly polarized with theazimuthal angle φ with respect to the x -axis [Fig. 1(a)].The probe spot size at the sample was 40 µ m (FWHM).Coherent phonons excited by the pump pulses modu-late the dielectric tensor components of CuB O , whichcan be seen in the experiment as a modulation either ofthe probe pulse ellipticity [Fig. 1(b)] or the probe pulsepolarization azimuthal angle due to the pump-inducedchanges in the crystallographic linear birefringence ordichroism, respectively. We note that the absorption co-efficient of CuB O at the probe photon energy is ∼ − , which suggests that the dichroism experienced bythe probe pulses is relatively weak. In the experiments wemeasured the pump-induced probe ellipticity changes ∆ η by employing the PD technique. A quarter-wave plate(QWP) placed in the probe beam behind the sample wasused to convert the ellipticity ∆ η to a polarization rota-tion ∆ φ [Fig. 1(c)]. The probe beam transmitted throughthe sample and QWP was split by a Wollaston prism intovertically and horizontally polarized beams and their in-tensities were detected by two Si-photodiodes. In thisway, the change of the polarization of the probe pulses,measured as the difference between the signals at the two z ||[001] E p E pr (a) y x ||[100] θ x z xz ab Δφ = Δ η φ tan( Δ η )= b/a probe behind the sample:probe behind the QWP:incident pump and probe: (b)(c) FIG. 1: (Color online) (a) Relative orientation of the crystal-lographic axes and laboratory frame xyz . The pump pulsesare incident along the y -axis. The angle of incidence of theprobe beam is 7 ◦ . The pump ( E p ) and probe ( E pr ) pulsesare linearly polarized with the azimuthal angles θ and φ , re-spectively. (b) Elliptically polarized probe pulse after trans-mission through the sample. (c) Ellipticity, ∆ η , of the probepulses converted to a rotation ∆ φ of the polarization planeafter transmission through the quarter-wave plate. diodes, was monitored as a function of the pump-probetime-delay t . The pump-probe traces were recorded insteps of 0.02 ps up to 10 ps, including a − T = 293 K.We would like to emphasize that the detection of co-herent phonons, excited via ISRS, is usually realized viathe monitoring of changes in the reflectivity or transmi-tivity. Instead, we employed the PD scheme to revealtransient polarization changes, which is more commonfor experiments on coherent magnons excited via ISRS. To emphasize the differences with the conventional ISRSexperiments with coherent phonons, further on we referto our experimental layout as to ISRS-PD experiment.An advantage of such a scheme for coherent phonon de-tection is that it allows the analysis of the symmetry ofparticular phonon modes and can discriminate betweenthem if necessary by choosing a proper probe polariza-tion, as we discuss in detail below. We note that such ascheme is a powerful alternative to the reflective electro-optical sampling technique.
The polarization sensi-tive detection of coherent phonons was reported in e.g.,Ref. 56, where the polarization dependent reflectivity wasanalyzed. Here we employ measurements of the tran-sient birefringence, which is an advantageous techniquefor transparent media.
IV. EXCITATION AND DETECTION OFCOHERENT PHONONS IN CuB O Figure 2 shows the time-delay dependence of the probepolarization excited by the pump pulses of three differ-ent polarizations, θ = 0 ◦ ( E p k x ), θ = 90 ◦ ( E p k z ), and θ = 45 ◦ . The probe was polarized at φ = 45 ◦ . In all three -1 0 1.5 2.0 4 6 8-0.50.00.51.01.5 E p ||z ( =90 o ) =45 o P r obe e lli p t i c i t y c hange ( m r ad ) Time delay t (ps) E p ||x ( =0) FIG. 2: (Color online) Probe ellipticity change ∆ η as a func-tion of the pump-probe time delay, t , measured at three dif-ferent polarizations, θ , of the pump pulses. The probe polar-ization is φ = 45 ◦ . excitation geometries, a strong coherent artifact was ob-served at the pump-probe overlap ( t = 0), followed by apronounced oscillatory signal, consisting of several super-imposed harmonic components. The fast Fourier trans-forms (FFT) amplitude spectra of the traces at positivedelays t > θ = 0 , ◦ [Figs. 3(d,f)] arein excellent agreement with the lowest phonon lines inthe RS spectra [Figs. 3(a,c)]. In particular, the phononsappearing in the x ( zz )¯ x RS spectrum [Fig. 3(c)] are ex-cited by the pump pulses polarized along the z -axis andpropagating along the y -axis [Figs. 3(f)]. We note thatsuch a comparison is justified because the [100] ( x ) and[010] ( y ) crystallographic axes in CuB O are equivalent.Analogously, there is a correspondence between the spon-taneous y ( xx )¯ y RS spectra and our data obtained forthe pump pulses polarized along the x -axis. Such a goodagreement allows us to assign the observed oscillationsof the probe polarization to the modulation of the di-electric permittivity by coherent phonons excited by thepump pulses.From the comparison of our experimental data with theresults of the spontaneous RS experiments, we can de-termine the particular coherent phonon modes that wereexcited in each geometry. Pump pulses polarized alongthe z -axis excite non-polar A modes with frequencies of7.52, 10.00, and 12.03 THz, while the pump pulses po-larized along the x -axis excite the 4.38- and 10.00-THz non-polar B modes in addition to the A modes (7.52and 10.00 THz).For the geometry with the pump pulses polarizedat an angle θ = 45 ◦ , the excitation of the modeswith E ( y ) symmetry are expected, which appear in theRS spectrum measured in the y ( xz )¯ y configuration[Fig. 3(b)]. However, this is not the case, as can be seenin Fig. 3(e).In Fig. 4(a) we show the pump-probe traces obtainedfor the pump polarization θ = 90 ◦ for three distinct probepolarizations; φ = 45 ◦ , 90 ◦ , and 135 ◦ . In Fig. 4(b) weplot the amplitude of the oscillatory signals ∆ η versusthe incident probe polarization azimuthal angle φ at afrequency of Ω / π = 10 .
00 THz (the most pronouncedoscillatory contribution to the signal [Fig. 3(f)]). Theamplitudes, ∆ η , were extracted from the fit of the ex-perimental data to the sine-function. We note that hereand elsewhere in the text ∆ η is defined as the signedamplitude of the measured signal. The oscillation am-plitude is strongly enhanced when the incoming polar-ization of the probe pulse makes an angle of φ = 45 ◦ with the x -axis, thus demonstrating the crucial role ofthe probe pulse polarization in the ISRS-PD experiment.In particular, the results in Figs. 4(a,b) suggest that theprobe polarized at a nonzero angle with respect to thepump polarization plane favors detection of the coher-ent phonon mode excited by the latter. This providesa hint for explaining the absence of the coherent phononmodes in the E ( y ) symmetry for the signal [Figs. 2 and3(e)] measured with θ = 45 ◦ and φ = 45 ◦ .The most evident difference between the results ofthe ISRS-PD experiments and spontaneous RS data isthat the relative values of the amplitudes of the coherentphonons excited via ISRS cannot be directly related tothe amplitudes of the corresponding lines in the spon-taneous RS spectra. Furthermore, no coherent phononswith frequencies above 12.03 THz could be reliably ob-served in the ISRS-PD experiments, despite the fact thatsome of these yielded very strong lines in the RS spec-tra (Fig. 3). In Sec. V A–V C we consider the excitationand detection of coherent phonons in detail to accountfor these observations. V. THEORETICAL BACKGROUND ANDDISCUSSION OF ISRS-PD IN CuB O A. ISRS as the excitation mechanism of coherentphonons in CuB O In general, there are two mechanisms, ISRS and DECP,which can mediate the excitation of coherent phononsby the femtosecond laser pulse. We argue that, in ourexperiments, ISRS is the mechanism responsible for theexcitation. First, the sample is transparent for the pumpcentral photon energy, which suppresses the alternativeDECP mechanism. Furthermore, DECP is expectedto only drive symmetric A modes. Indeed, when the
Frequency /2 (THz) y(xz)y (e) =45 o (c) x(zz)x (a) N o r m a li z ed I S R S - P D a m p li t ude N o r m a li z ed R S i n t en s i t y (b) experiment Lorentzian fit z(xx)z ( =90 o )E p ||z (f) modes symmetry:A ; B ;A +B ; E(y) ( =0)E p ||x (d) Frequency /2 (THz) E p ||z N o r m a li z ed I S R S - P D a m p li t ude I p (mJ/cm )7.52 FIG. 3: (Color online) (a–c) Spontaneous RS spectra measured in the geometries (a) z ( xx )¯ z , (b) y ( xz )¯ y , and (c) x ( zz )¯ x (adapted from Ref. 33). (d–f) FFT spectra of the ISRS-PD experimental data (Fig. 2) measured in the geometries (d) E p k x ( θ = 0), (e) θ = 45 ◦ , and (f) E p k z ( θ = 90 ◦ ). In the data sets (a,c) the spectra were normalized by the intensity of the10.00-THz line, while in the data set (b) – by the intensity of the strong 11.92-THz line. In the data sets (d–f) the spectra werenormalized by the amplitude of the 10.00-THz line. The symbols represent the experimental data, and the lines show their fitusing Lorentzian functions [Eq. (9)]. The numbers indicate the phonon frequencies Ω k / π . The amplitudes C k and FWHMs σ k of the phonon lines are given in Tables III and IV. Different colors indicate the phonon modes of the A (black), B (red), and E ( y ) (blue) symmetries. The green color denotes the lines to which the phonons of both A and B symmetries contribute. Theinset 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 panel (f). The lines are linear fits. DECP mechanism is involved, the ions are driven fromthe equilibrium positions at the ground state to the non-equilibrium positions at the excited state. As a result,the symmetry of the crystal remains unchanged and onlythe symmetric vibrational A mode can be excited.The ISRS-driven excitation of a coherent phonon withfrequency Ω and normal coordinate Q ( t ) can be phe-nomenologically described by the equation of motion: d Qdt + Ω Q =116 π R Ω ij ( ω p )Re[ E i ( t ) E j ( t ) ∗ ] =14 nc R Ω ij ( ω p ) I Re[ e i e ∗ j ] α ( τ p , Ω ) , (1)where R Ω ij ( ω p ) = ∂ε ij ( ω p ) /∂Q is the Raman tensorand ε ij ( ω p ) is the dielectric permittivity tensor. Notethat Eq. (1) does not include energy dissipation. Itis also assumed that the dielectric permittivity disper-sion is negligible within the spectral width of the pumppulse. E i ( t ) is the time-dependent electric field enve- lope defined as E i ( t ) ≡ Re[ E i ( t )e iω p t ] with amplitude E and FWHM τ . The pump intensity I is introduced as I = nc |E | / (4 π ), where n is the refractive index at thepump wavelength. e = ( e x , e y , e z ) is the polarizationunit vector. α ( τ p , Ω ) includes the relation between theduration of the pump pulse, or its spectral width σ p , andthe frequency of the coherent phonon mode. The coeffi-cient α ( τ p , Ω ) in Eq. (1) for the phonon mode Ω can beobtained by recalling that the product of two functionsin the time domain can be expressed via the convolutionof their Fourier-transforms: α ( τ p , Ω ) = R ∞−∞ E p ( ω ) E p ( ω − Ω ) dω R ∞−∞ E p ( ω ) E p ( ω ) dω . (2)The nominator in the expression for α ( τ p , Ω ) reflects thephysical description of the ISRS process, in which pairsof photons with frequencies that differ by Ω contributeto the generation of the corresponding coherent phononmode. Both the nominator and denominator reduce to4 πI p /nc for the δ − pulse, and α ( τ p → , Ω ) = 1. The A m p li t ude ( m r ad ) Probe polarization (deg) experiment (b)E p ||z Time delay t (ps) 135 o o (E pr ||z) P r obe e lli p t i c i t y c hange ( m r ad ) o (a) (c) A m p li t ude ( a r b . un i t ) Probe polarization (deg)
FIG. 4: (Color online) (a) Probing ellipticity, ∆ η , versus thepump-probe time delay, t , measured for three initial polar-izations, φ , of the probe pulses, with a pump polarization of θ = 90 ◦ . (b) Amplitude of the pump-induced oscillations at afrequency 10.00 THz of the probe polarization ∆ η as a func-tion of probe azimuthal angle φ . (c) Calculated amplitude ofthe pump-induced oscillations of the probe polarization as afunction of φ (see the text for details). solution of Eq. (1) takes the simple form of a sine-function Q ( t ) ∼ α ( τ p , Ω ) R Ω ij ( ω p ) I Re[ e i e ∗ j ] sin (Ω t ) . (3)In the ISRS-PD experiments reported here, the measuredvalue is the change of the ellipticity ∆ η of the probepolarization occurring due to modulation of the dielectricpermittivity by excited coherent phonons, and convertedto the polarization rotation by QWP [Figs. 1(b,c)]. Thetemporal evolution of ∆ η can be then expressed as (seeApp. A for details)∆ η ( t ) = πd λ pr R Ω ij ( ω pr ) √ ε ij Q ( t ) ∗ I pr ( t ) I , (4)where I pr ( t ) is the temporal profile of the probe pulse,the exact form of which is introduced below, and ∗ de-notes the convolution operation. The convolution withthe probe pulse temporal profile I pr ( t ) is required to ac-count for the particular probe duration τ pr . λ pr is theprobe wavelength. Here, the dielectric permittivity dis-persion is assumed negligible within the spectral widthof the probe pulse. The amplitude of the coherent phonons excited anddetected in the ISRS-PD process [Eq. (4)] has four mainconstituents:(i) Specific values of Raman tensors R Ω ij ( ω p ) and R Ω ij ( ω pr ) are determined by the material proper-ties at the frequencies ω p and ω pr of the pump andprobe pulses, respectively. Generally speaking, theabsolute values of the R Ω ij components at these twooptical frequencies can differ due to a dispersion inthe corresponding spectral range.(ii) The product R Ω ij ( ω p ) I Re[ e i e ∗ j ] describes the de-pendence of the driving force on the intensity I and polarization e of the pump pulses.(iii) The parameter α ( τ p , Ω ) describes the role of thelimited spectral width of the pump pulse in theexcitation process.(iv) The convolution with I pr ( t ) allows us to account forthe polarization changes of the probe pulse due tothe modulation of the dielectric permittivity withinthe range τ pr near the time delay, t . B. Role of the pump and probe polarizations inthe ISRS-PD experiment
First, we consider the excitation and detection of co-herent phonons in CuB O , neglecting the duration ofthe pump and probe pulses, i.e., setting α ( τ p , Ω ) = 1and I pr ( t ) = I δ ( t ). In this case Eq. (4) is simplified andtakes a form ∆ η ( t ) = ∆ η sin(Ω t ), where:∆ η = πd λ pr R Ω ij ( ω p ) R Ω ij ( ω pr ) √ ε ij I Re[ e i e ∗ j ] . (5)The Raman tensor components for a mode of particularsymmetry in CuB O belonging to the point group ¯42 m are listed in Table I. The modes of the symmetry A aresilent. Considering the expressions for the Raman tensorcomponents R ij , we can write the r.h.s. of Eq. (1) inthe form given in Table I. From this symmetry analysisone can see directly that the B and E ( x ) modes canbe excited under the conditions e x e ∗ y + e y e ∗ x = 0 and e y e ∗ z + e z e ∗ y = 0, respectively. This is not the case inour experiments, since e y = 0 when the pump pulsespropagate along the y -axis.The driving force for the non-polar A modes can benonzero for any polarization. In the considered geometrythe pump pulse of any linear polarization excites thesemodes, provided the Raman tensor components R xx and R zz are nonzero for that particular mode. Indeed, this is observed in our experiment, where the three lowest A modes at 7.52-, 10.00-, and 12.03 THz are all excitedby the pump pulse polarized along the z -axis [Fig. 3(f)],because all relevant Raman tensor components R zz arenonzero [Fig. 3(c)]. In contrast, only the 7.52- and 10.00- TABLE I: Raman tensor components R ij and the corresponding driving forces in Eq. (1) for the phonon modes of a particularsymmetry under the assumption of infinitesimally short pump pulses. Also listed are the dielectric tensor components δε ij ( t )modulated by coherent phonons of the particular symmetry. For convenience we have omitted the factor I / nc in theexpressions for the driving forces.Phononsymmetry R ij ISRS driving force δε ij ( t ) = R ij Q ( t ) A R xx = R yy , R zz R xx Re (cid:0) e x e ∗ x + e y e ∗ y (cid:1) + R zz Re ( e z e ∗ z ) δε xx ( t ) = δε yy ( t ) , δε zz ( t ) B R xx = −R yy R xx Re (cid:0) e x e ∗ x − e y e ∗ y (cid:1) δε xx ( t ) = − δε yy ( t ) B R xy = R yx R xy Re (cid:0) e x e ∗ y + e y e ∗ x (cid:1) δε xy ( t ) = δε yx ( t ) E ( x ) R yz = R zy R yz Re (cid:0) e y e ∗ z + e z e ∗ y (cid:1) δε yz ( t ) = δε zy ( t ) E ( y ) R zx = R xz R xz Re ( e z e ∗ x + e x e ∗ z ) δε zx ( t ) = δε xz ( t ) THz A modes are excited by the pump pulses polar-ized along the x -axis [Fig. 3(d)], which agrees well withthe observation that the line corresponding to the 12.03-THz A ( xx ) phonon is also very weak in the spontaneousRS spectra [Fig. 3(a)]. Non-polar B modes can be ex-cited when | e x | = | e y | (see Table I). In our experimentthis corresponds to the pump pulse polarization makinga nonzero angle with the z -axis. Indeed, the B mode isexcited by the pump pulses with azimuthal angles θ = 0and 45 ◦ [Figs. 3(d,e)].Polar E ( y ) coherent phonon modes are expected to beexcited under the condition e z e ∗ x + e x e ∗ z = 0, which ismet at θ = 45 ◦ . However, as follows from a compari-son of the spectra in Figs. 3(b,e), no line associated withthe E ( y ) phonon modes appear in the FFT spectrum ofthe ISRS-PD data measured in this geometry. This hap-pens because in the ISRS-PD experiment the detectionprocess is as important as the excitation. In our experi-mental geometry, optically excited coherent phonons aredetected via the modulation of the probe polarization[Eq. (4)] originating from transient changes of the linearbirefringence. The components of the dielectric tensormodulated by the coherent phonons of a particular sym-metry are listed in Table I. Clearly, the detection of the A and B coherent phonons requires the probe polariza-tion to make an angle φ = 0 and 90 ◦ (see App. A for thedetails). In contrast, the modulation of the probe polar-ization by E ( y ) coherent phonons vanishes at φ = 45 ◦ .The relevant amplitudes ∆ η [Eq. 5] of the ellipticitymodulation at the phonon frequency Ω for the probepulses initially polarized at φ = 45 ◦ for each pump pulse polarization θ are θ = 0 : (6) A : ∆ η = πd λ pr R xx ( ω p ) (cid:20) R zz ( ω pr ) √ ε zz − R xx ( ω pr ) √ ε xx (cid:21) ; B : ∆ η = πd λ pr R xx ( ω p ) R xx ( ω pr ) √ ε xx ; θ = 90 ◦ : (7) A : ∆ η = πd λ pr R zz ( ω p ) (cid:20) R zz ( ω pr ) √ ε zz − R xx ( ω pr ) √ ε xx (cid:21) ; θ = 45 ◦ : (8) A : ∆ η = √ πd λ pr ( R xx ( ω p ) + R zz ( ω p )) (cid:20) R zz ( ω pr ) √ ε zz − R xx ( ω pr ) √ ε xx (cid:21) ; B : ∆ η = √ πd λ pr R xx ( ω p ) R xx ( ω p ) √ ε xx .E ( y ) : ∆ η = 0 . Note that all ε ij components should be taken at the op-tical frequency of the probe pulse here.It is worth noting that the distinct pump and probepolarizations provide the most efficient conditions forthe excitation and detection of coherent phonons ormagnons, as clearly seen from the probe polarizationdependence of the ISRS-PD signal shown in Figs. 4(a,b).We numerically analyzed the dependence of the ampli-tude of the oscillations of the probe ellipticity measuredin ISRS-PD scheme on the probe polarization azimuthalangle, φ , when a coherent phonon of a symmetry A mod-ulates the real part of the dielectric permittivity tensor,thus changing the linear crystallographic birefringence.The calculations were performed using the Jones ma-trix method, taking the experimentally obtained valueof 3.1 × − of the static birefringence in the xz planeof the studied sample at the probe photon energy. Cal-culations show [Fig. 4(c)] that the oscillatory signals areindeed caused by modulation of the linear birefringenceof CuB O due to coherent phonons, and demonstratethe importance of the correct choice of the probe polar- TABLE II: Initial phases of the probe ellipticity oscillations∆ η ( t ) extracted from the fit of the data in Fig. 2 to two orthree damped sine-functions. θ ◦ (160.6 ± ◦ (142.9 ± ◦ (326.7 ± ◦ -45 ◦ (186.3 ± ◦ - (35.0 ± ◦ -90 ◦ - (358.4 ± ◦ (13.2 ± ◦ (199.1 ± ◦ ization for detection of a laser-driven coherent phononmode of a particular symmetry.To conclude the discussion of the excitation mecha-nism of the coherent phonons, we note that ISRS is thesole mechanism of excitation of the B coherent phononmode, while the A mode, in general, can be excitedvia the DECP mechanism as well. The modulation ofthe probe ellipticity by the coherent phonons excited viaISRS and DECP should possess sine- [Eq. (4)] and cosine-like temporal behaviors, respectively. We have there-fore fitted the data in Fig. 2 to the sum of two or threedamped oscillations. The resulting initial phases areshown in Table II. All phonon modes show nearly sine-like behaviors, i.e., the initial phases are ∼ ∼ ◦ rather than ∼ ◦ or ∼ ◦ . We note also that theamplitude of the lines in the FFT spectra show a lin-ear dependence on the pump fluence I (see the inset inFig. 3), as expected for the coherent phonons excited viaISRS [Eqs. (1–4)]. C. Role of the pump and probe durations in theexcitation of coherent phonons via ISRS
Now we analyze the significant differences between theamplitudes of the phonon lines in the FFT spectra of theISRS-PD data and those of the spontaneous RS spec-tra (Fig. 3). While both RS and ISRS processes are de-scribed by the same Raman tensor components R ij , thereis an essential difference between them. This is due tothe character of the light used in the RS and ISRS ex-periments; continuous wave versus ultrashort pulse, anddue to the different registration techniques. These fac-tors are accounted for in Eqs. (1–4) by the constituents(iii–iv) that are dependent on the temporal/spectral pro-files of the pump and probe pulses. Furthermore, in theRS experiment light scattering from incoherent phononstakes place in thermal equilibrium, and the Bose-Einsteinthermal occupation factor enters the expression for theRaman line intensity. This is not the case when thecoherent phonons are driven by the ISRS process. We consider four A and three B lowest phononmodes, observed in the spontaneous RS spectra below15 THz [Fig. 3(a,c)]. The lines assigned to these modes
TABLE III: Amplitudes C k and FWHMs σ k of the A + B phonon lines in the RS ( z ( xx )¯ z ) and ISRS ( E p k x ) spectra[Figs. 3(a,d)]. The Raman tensor R xx values were extractedfrom the spontaneous RS data taking into account the ther-mal occupation numbers N (Ω k ) + 1 at T = 293 K. The am-plitudes of the oscillations ∆ η calc0 were calculated from theISRS model, accounting for the pump and probe pulse du-rations (see text). All amplitudes were normalized to thecorresponding amplitude of the 10.00-THz line.Ω k / π (THz) RS ( z ( xx )¯ z ) ISRS ( E p k x ) C k σ k R zz C k ∼ ∆ η k σ k ∆ η calc0; k (rel.unit) (GHz) (rel.unit) (rel.unit) (GHz) (rel.unit)4.38 0.04 68 0.16 0.1 40 0.487.52 0.02 77 0.13 0.09 48 0.0710.00 1 77 1 1 79 110.71 0.08 88 0.29 - - 0.0412.03 0.04 108 0.21 - - 0.0114.13 1.31 124 1.21 - - 0.04 are fitted by the sets of Lorentzian functions I RS (Ω) = c + X k C k π σ k π (Ω − Ω k ) + σ k , (9)where Ω k and σ k are the frequency and FWHM of the k -th phonon line, respectively. These are listed in Tables IIIand IV. The amplitudes C k of the Stokes lines C k ∼ ( N (Ω k ) + 1) h R Ω k ij ( ω ) i (10)can be used as a measure of the magnitude of the Ra-man tensor component R Ω k ij ( ω ) describing the scatter-ing of light at the frequency ω by the correspondingphonon. Here N (Ω k ) = (exp(¯ h Ω k /k B T ) − − is theBose-Einstein thermal occupation factor. Note that atroom temperature the coefficient N (Ω k )+1 deviates fromunity and affects the intensity ratio between different Ra-man lines when the broad frequency range is considered.As an example, we mention that the ratio R . / R . extracted from the RS data appears to be overestimatedby ∼
25 % when the thermal occupation factor is nottaken into account.The peak intensity of the particular phonon line isrelated to the amplitude and FWHM as I (Ω k ) =2 C k ( πσ k ) − . We note that a less intense but broader Ra-man line may have a larger contribution to the Ramantensor. We also note that the line at Ω k / π =10.00 THzin the z ( xx )¯ z spectrum [Fig. 3(a)] contains contribu-tions from both the A and B phonons, i.e., C . ∼ (cid:16) R . xx ( A ) ( ω ) (cid:17) + (cid:16) R . xx ( B ) ( ω ) (cid:17) . As was found experi-mentally in Ref. 33, the A phonon contribution domi-nates.The RS data reported in Ref. 33 were obtained at thephoton energy of ¯ hω = 2 .
41 eV. However, as demon-strated in Ref. 34, the change of the excitation energyin the RS experiments from 2.71 to 1.96 eV did not
TABLE IV: Amplitudes C k and FWHMs σ k of the A phonon lines in the RS ( x ( zz )¯ x ) and ISRS ( E p k z ) spectra[Figs. 3(c,f)]. The Raman tensor R zz values were extractedfrom the spontaneous RS data taking into account the ther-mal occupation numbers N (Ω k ) + 1 at T = 293 K. The am-plitudes of the oscillations ∆ η calc0 were calculated from theISRS model, accounting for the pump and probe pulse du-rations (see text). All amplitudes were normalized to thecorresponding amplitude of the 10.00-THz line.Ω k / π (THz) RS ( x ( zz )¯ x ) ISRS ( E p k z ) C k σ k R zz C k ∼ ∆ η k σ k ∆ η calc0; k (rel.unit) (GHz) (rel.unit) (rel.unit) (GHz) (rel.unit)7.52 0.09 54 0.25 0.39 56 0.3510.00 1 57 1 1 70 112.03 1.2 75 0.88 0.24 77 0.2314.13 3.12 115 1.65 - - 0.08 yield any significant redistribution of the intensity of thephonon lines in the spontaneous RS spectra. This ob-servation indicates that the RS process at the photonenergies as high as 2.71 eV is of a non-resonant nature.Therefore, we can consider the relative intensities of theRaman lines reported in Ref. 33 as a reliable measureof the Raman tensor components for the case of non-resonant scattering at the pump (¯ hω p = 1 .
08 eV) andprobe (¯ hω p = 1 .
55 eV) photon energies used in our ex-periments: R Ω k ij ( ω p ) = R Ω k ij ( ω pr ).The normalized amplitudes of the phonon lines inthe FFT spectra obtained for the pump-probe traces[Figs. 3(d–f)] using the fit function [Eq. (9)] are sum-marized in Tables III and IV. The relation between theLorentzian line amplitudes C i ∼ ∆ η and the Ramantensor components can be obtained from Eqs. (6–8) andis, in general, less straightforward than in the case ofRS because of the chosen detection technique. For thesake of simplicity, we assume the ratio R zz / R xx ≈ A mode of frequency Ω / π =7.52 THz, as estimatedfrom Eqs. (6),(7) and from the data in Figs. 3(d,f). ThenEqs. (6),(7) are reduced to θ = 0 : (11) A : C i ∼ ∆ η ∼ R xx (cid:20) √ ε zz − √ ε xx (cid:21) ; B : C i ∼ ∆ η ∼ R xx √ ε xx .θ = 90 ◦ : (12) A : C i ∼ ∆ η ∼ R zz (cid:20) √ ε zz − √ ε xx (cid:21) . From Eqs. (11),(12) it follows that the amplitudes C i obtained for the ISRS-PD data should be compared withthose from the RS spectra. In Fig. 5(a) the amplitudes C i that are plotted are normalized with respect to the 10.00-THz line in the relevant spectra. There are two gen-eral trends. First, the 10.00- and 12.03-THz A phonon ISRS-PD R (b) (a) E p ||z E p ||xcalculationsexperiment N o r m a li z ed a m p li t ude (r e l . un i t ) p =90fs pr =50fs (c) Frequency /2 (THz) p ->0; pr =50 fs p =90 fs; pr ->0 FIG. 5: (Color online) (a) Amplitudes C k ∼ ∆ η (blue bars)of the coherent phonon modes, as extracted from a FFT ofthe ISRS-PD experimental data [Figs. 3(d–f)]. Also shown(hatched bars) are the values of the corresponding squaredRaman tensor components R ij (see Tables III and IV). (b–c) Amplitudes of the probe ellipticity ∆ η calculated usingEq. (4) for the cases (b) τ pr =50 fs, τ p =90 fs, (c) τ pr =50 fs, τ p → τ pr → τ p =90 fs (gray bars). Allamplitudes are normalized by the amplitude of the 10.00-THzmode in the relevant spectra. Dashed lines in the panel (c)show the frequency dependence of the normalized coefficient α (90 fs , Ω). lines in the x ( zz )¯ x RS spectrum have similar amplitudes,while in the ISRS spectrum the amplitude of the 12.03-THz coherent phonon line excited by the pump pulseswith E p k z appears to be ∼ (cid:0) R . zz / R . zz (cid:1) ∼
10 for the A modes. Inthe ISRS spectrum, the corresponding ratio appears tobe only ∆ η . / ∆ η . ∼ E ( t ) = E e − ( t /τ ) . (13)In the frequency domain, both these pulses are assumedto be Fourier-limited, and their spectral profiles are ob-tained as a Fourier transform of Eq. (13) E ( ω p(pr) ) = E √ √ πσ p(pr) e − ( ω /σ ) , (14)where σ p(pr) = 4 ln 2 τ − is the FWHM of the pumpand probe pulse intensity profiles in the spectral domain.The amplitude ∆ η of the measured probe polariza-tion oscillations in the ISRS-PD experiment increased as0the probe pulse duration τ pr decreased, as compared withthe period of a particular coherent phonon. To illustratethis effect we calculated the expected change ∆ η ( t ) inthe ISRS-PD experiment using Eqs. (4),(11), and (12),as well as assuming infinitesimally short pump pulses,i.e., α ( τ p →
0) = 1. In Fig. 5(c) we plot the normalizedFFT intensities of A ( E p k z ), A and B ( E p k x ) coher-ent phonons calculated in this way. The probe durationwas taken to be τ pr =50 fs ( σ pr / π =9 THz), i.e., equal tothat used in our experiments. The resulting 12.03-THzline ( τ pr Ω i ≈ .
7) is noticeably suppressed as comparedwith the 10.00-THz line ( τ pr Ω i ≈ . x ( zz )¯ x RS spectra (Fig. 5(a)) and, consequently,the corresponding Raman tensor components, are nearlyequal. The effect of the probe duration is even morepronounced when the ratios between the intensities ofthe 10.00- and 14.13-THz lines in the RS spectrum arecompared with those in the calculated ISRS spectrum.Thus, considering only the probe pulse duration, one canaccount for, to a certain extent, the differences in the in-tensities of the phonon lines in the RS and ISRS spectrain the high-frequency part of the spectrum.While the duration of the probe pulse affects the detec-tion part of the pump-probe experiment, the duration ofthe pump pulse determines the efficiency of the coherentphonon excitation. This is accounted for by the coeffi-cient α k ( τ p , Ω ) (2), which has the following form for theGaussian pulses: α k ( τ p , Ω ) = e − ( Ω /σ ) . (15)The normalized coefficient α (90 fs , Ω) in the range be-low 15 THz is shown in Fig. 5(c) using the dashedlines. This graph clearly demonstrates that the efficiencyof the excitation of, e.g., the 10.00-THz phonon modewould be ∼ I pr ( t ) = I δ ( t ) and apump pulse of duration τ p =90 fs ( σ p / π ≈ η obtained at τ pr → Q , i.e. , to the corresponding atomic dis-placements excited by the pump pulse.Finally, we calculate the ISRS-PD amplitudes of thephonons by considering the durations of both the pump( τ p =90 fs) and probe ( τ pr =50 fs) pulses. The results areshown in Fig. 5(b) and listed in Tables III and IV. Forthe excitation of the A mode (left side of Fig. 5(b)), ourmodel adequately describes the main trends observed inthe experiments. As shown, for the three A modes withfrequencies lying in the range 7–15 THz, this model ac-counts for either the partial or total suppression of thephonon lines with frequencies above 10 THz in the ISRS-PD spectrum, in all considered geometries. In the lower frequency range the discrepancy between the experimen-tal results and model occurs only for the 4.38-THz B phonon mode, in which the amplitude appears to beoverestimated. Importantly, our calculations show thata reasonable agreement between the calculated ISRS-PDspectra and those obtained from the pump-probe experi-ment is obtained only by including the durations of boththe pump and probe pulses.To complete the comparative analysis of the sponta-neous RS and ISRS-PD data, we note that the phononlines in the RS spectra have very narrow widths (seeTables III and IV). Thus, in the x ( zz )¯ x RS spectra,the line at 10.00 THz is characterized by a FWHM of ≈
60 GHz. In the FFT spectra of the ISRS-PD data mea-sured with E p k z , this line has a FWHM of 70 GHz, i.e.,broader by only ∼ VI. CONCLUSIONS
We have performed a detailed study of laser-inducedexcitation and detection of multiple coherent phononmodes in the ISRS-PD experiment in the dielectric cop-per metaborate CuB O , characterized by a large prim-itive unit cell containing 42 atoms. In total, three non-polar A and one non-polar B modes were distinguishedin the frequency range of 4–13 THz. We have shown that90-fs linearly polarized laser pulses with a central photonenergy in the optical transparency range (¯ hω p =1.08 eV)excite the coherent phonons via ISRS. By comparing theresults of the ISRS-PD experiment to spontaneous RSspectra in this material, we demonstrated that the re-lationship between the amplitudes and intensities of thephonon lines in these two types of experiments, is deter-mined by both the excitation and detection conditions,respectively. Namely, the amplitude of the excited coher-ent phonon is determined by the polarization, intensity,and duration of the pump pulses. Probe pulse polariza-tion and duration are as important as those of the pumppulses. They determine how the transient changes of thedielectric permittivity due to excited coherent phononslead to the modulation of the probe polarization detectedin the ISRS-PD experiment.A comparison between the spontaneous RS spectra andISRS-PD data also allowed us to analyze in detail the lim-itations imposed by the durations of the pump and probepulses on the excitation and detection of the coherentphonons. Accounting for both durations is required toadequately calculate the modulation of the probe polar-1ization in the pump-probe experiment using the spon-taneous RS data. It is important to understand therole played by the pump and probe pulse durations inISRS, because ever-shorter laser pulses are currently em-ployed in pump-probe experiments, providing access tohigh-energy collective excitations in solids. To the bestof our knowledge, no such detailed analysis for the caseof multiple coherent phonon modes excitation has beenreported to date.Finally, we would like to note that the reported de-tails of the ultrafast coherent lattice dynamics in coppermetaborate CuB O are of importance in light of recentattention given to the high-frequency phonon-magnon in-teraction and their role in the ultrafast dynamics drivenby femtosecond laser pulses. We demonstrated theexcitation of coherent vibrations of Cu ions belongingto different magnetic sublattices. In particular, the B b )O complexes solely with periodic displacement of Cu (4 b )ions along the z -axis. Magnetic ions in this complex pro-vide the strongest contribution to the magnetic orderingin CuB O . VII. ACKNOWLEDGEMENTS
We thank V. Yu. Davydov for help with the sponta-neous Raman scattering experiments. AMK acknowl-edges support from the Japanese Society for Promo-tion of Sciences (JSPS) via the Short-Term FellowshipProgram for European and North-American young re-searchers during her stay at the University of Tokyo. TSwas supported by JSPS KAKENHI (No. JP15H05454and JP26103004) and JSPS Core-to-Core Program (A.Advanced Research Networks). RVP acknowledges thesupport from the Russian Science Foundation (grant No.16-12-10456).
Appendix A: Probe polarization changes in theISRS-PD experiment
Here we derive the expression that relates the changesof the ellipticity of the probe pulses measured in theISRS-PD experiment to the Raman tensor components,and the pump and probe parameters.
1. A case of A and B coherent phonons Let the coherent phonon only contribute to the mod-ulation of the diagonal components of the dielectricpermittivity tensor, as in the case of the A or B phonons. Then, the expression for the dielectric ten-sor is ε ij + δε ij ( t ), where δε ij ( t ) = 0 if i = j . δε ij is related to the coherent phonon normal coordinate as δε ij ( t ) = R ij Q ( t ). We consider the light propagating along the j -axis.The eigenwaves in this case are two orthogonal linearly-polarized waves E i and E k , where i and k are the unitvectors in the directions of the i - and k -axes, respectively.The corresponding complex refraction indices for theseeigenwaves are n i ( k ) − i κ i ( k ) = p ε ii ( kk ) + δε ii ( kk ) . Aftertravelling the distance d these waves acquire additionalphases of 2 πn i ( k ) d /λ and their amplitudes are decreasedby e − πκ i ( k ) d /λ .First, for the sake of clarity we consider a nondissipa-tive medium ( κ i ( k ) = 0 ) . Therefore, the diagonal com-ponents of the dielectric permittivity tensor and, conse-quently, refractive indices, are real values. We assumethat the light is initially linearly polarized at an angle φ to the i -axis [Fig. 1(a)]. Then, upon traveling the dis-tance d , the light becomes elliptically polarized. Con-sidering that the relationship between the complex am-plitudes of the orthogonal components of the light polar-ization is given by E k /E i = tan φ + i tan η − i tan φ tan η , we obtain the ellipticity η ( t ) [Fig. 1(b)] that relates theacquired phase shift between two eigenwaves bytan 2 η ( t ) = 2 π ( n i ( t ) − n k ( t )) d λ sin 2 φ. (A1)For small η ( t ) this yields η + ∆ η ( t ) = πd λ ( n i ( t ) − n k ( t )) sin 2 φ ≈ (A2) πd λ ( √ ε ii − √ ε kk ) sin 2 φ + πd λ (cid:18) δε ii ( t ) √ ε ii − δε kk ( t ) √ ε kk (cid:19) sin 2 φ, where η is the static contribution to the ellipticity dueto crystallographic birefringence. This expression wasobtained by taking into account that the modulationof the dielectric permittivity tensor induced by coher-ent phonons is significantly weaker than the value of thecorresponding unperturbed component, i.e. δε ij ≪ ε ij .As can be seen from Eq. (A2), there are two con-tributions to the ellipticity of light passing though themedium. The first one is related to the birefringenceof the unperturbed medium and does not contribute tothe measured ISRS-PD signal. Thus, the changes in theellipticity are related to the normal coordinate of the cor-responding coherent phonon by∆ η ( t ) ≈ πd λ (cid:18) R ii √ ε ii − R kk √ ε kk (cid:19) Q ( t ) sin 2 φ, (A3)where we took into account the relationship between thedielectric permittivity tensor components and phononnormal coordinate.As can be seen from Eq. (A3), in order to detect the A and B coherent phonons by measuring the changes2of the probe pulses ellipticity ∆ η ( t ) the geometry mustbe chosen with φ = 0 , o , and φ = 45 o ensuring thebest sensitivity. It also follows from Eq. (A3), and theTable I, that the A coherent phonons would not manifestthemselves in the experiments with pump or probe lightpropagating along the z -axis of the crystal, since δε xx = δε yy .In our experiments, the QWP was placed after thesample with its axis parallel to the incoming probe beampolarization. It is convenient to consider this experimen-tal geometry in the coordinate frame, with two axes di-rected along the light propagation direction and incom-ing light polarization. In this frame the Jones vectorfor the light passing through the medium, with an ac-quired ellipticity ∆ η ( t ), has the form [ E ; E ∆ η ( t ) e iπ/ ].After passing through the QWP, which introduces a π/ E ; E ∆ η ( t )]. Thus, the azimuthal angleof the probe pulses appear to be rotated by the angle∆ φ ( t ) ≈ tan(∆ φ ( t )) = ∆ η ( t ).
2. A case of B , E ( x ) , and E ( y ) coherent phonons For the case of a coherent phonon mode, whichmodulates the off-diagonal components of the dielec-tric permittivity tensor δε ik , e.g., E ( y ), the eigenwavesare two linearly polarized waves with azimuthal anglesarctan ( δε ik / ( ε kk − ε ii )) with the i - and k -axes. Herewe assumed that the changes of the dielectric permit-tivity tensor components are smaller than the differencebetween the diagonal tensor components of the unper-turbed medium. The corresponding refractive indices are n = p ε ii ( kk ) ± ( δε ik ( t )) / ( ε kk − ε ii ) ( κ i ( k ) = 0 for anondissipative medium).The detailed expression for the changes of the light po-larization are rather complex in this case, and we simplifyit by noting the following. In contrast to the previouslyconsidered case, here the excited coherent phonons per-turb the basis formed by the eigenwaves, while the re-fractive indices for the eigenwaves can be treated as un-changed n ≈ √ ε ii ( kk ) . Therefore, it is convenient toconsider this scenario as if the basis remains unchangedand coincides with the i - and k -axes, but the light po-larization azimuthal angle is modulated by the anglearctan ( δε ik ( t ) / ( ε kk − ε ii )) ≈ δε ik ( t ) / ( ε kk − ε ii ). By re-placing φ by φ + δε ik ( t ) / ( ε kk − ε ii ) in Eq. (A1) we obtainthe expression for the oscillatory part of the probe ellip-ticity:∆ η ( t ) ≈ πd λ R ik √ ε kk + √ ε ii Q ( t ) cos 2 φ + (A4)2 πd λ (cid:18) R ik Q ( t ) ε kk − ε ii (cid:19) ( √ ε kk + √ ε ii ) sin 2 φ. Thus, in contrast to the A or B coherent phonons, de-tection of the, e.g., E ( y ), coherent phonons can be re-alized with probe pulses polarized at φ = 0, while at P r obe e lli p t i c i t y m odu l a t i on a m p li t ude ( a r b . un i t ) O u t c o m i ng p r obe e lli p t i c i t y + ( a r b . un i t ) Incoming probe polarization (deg) static LBLB modulated by a nonpolar (A ) phonon: ii + ii ii - ii (at ) LB modulated by a polar (E(y)) phonon: ii + ik ii - ik (at ) A m p li t ude at 2 (deg) at FIG. 6: (Color online) Ellipticity of the probe polarization η ± δη occurring due to static linear birefringence (LB) (blacksolid line), and modulation of the dielectric tensor by non-polar (red lines) and polar (blue lines) coherent phonons. Thedashed and dash-dotted lines capture the ellipticity at maxi-mal positive and negative changes of the dielectric permittiv-ity by coherent phonons, respectively. The red and blue solidlines show the expected amplitudes of the ellipticity modu-lation by non-polar and polar coherent phonons at the fre-quency Ω. The inset displays a magnification of the rangeclose to φ = 45 o to show that only a weak modulation of theellipticity at the frequency 2Ω is expected for this polarizationfor the case of a polar coherent phonon. φ = 45 o their effect on the probe polarization is quadraticon 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 non-polar phonons modulatethe diagonal elements of the dielectric permittivity ten-sor, this modulation manifests itself in a periodic changeof the amplitude of the η versus φ dependence, while theknots and maxima of the dependence remain at their po-sitions. The amplitude of the modulation ∆ η increasesas the incoming polarization reaches 45 o , and is zero if φ = 0 , o . In contrast, the polar coherent phonon peri-odically shifts the knots and maxima of the η versus φ de-pendence, leaving its amplitude unchanged. As a result,the amplitude of the ellipticity modulation ∆ η at thephonon frequency Ω reaches its maximum at φ = 0 , o .At φ = 45 o only a weak modulation at doubled phononfrequency 2Ω is expected (see inset in Fig. 6).
3. A case of opaque medium
When the absorption in a medium cannot be neglected,the diagonal dielectric tensor components are complex( κ i ( k ) > A coherent phonon of a symmetry is detected.3Then the light propagating a distance d acquires themodulated change of the azimuthal angle of:∆ φ ( t ) ≈ πd λ (cid:18) R ii √ ε ii − R kk √ ε kk (cid:19) Q ( t ) sin 2 φ, (A5)Thus, the coherent phonon mode can be detected by di-rectly measuring the rotation of the probe polarization passing through the crystal. We note here that the rota-tion (Eq. (A5)) induced by coherent phonons via lineardichroism is sensitive to the incoming probe polarizationin the same way as the ellipticity occurring due to linearbirefringence. S. De Silvestri, J. G. Fujimoto, E.P. Ippen, E. B. GambleJr., L. R. Williams, and K. A. Nelson, Chem. Phys. Lett. , 146 (1985). L. Dhar, J. A. Rogers, and K. A. Nelson, Chem. Rev. ,157 (1994). R. Merlin, Solid State Commun. , 207 (1997). Y.-X. Yan and K. A. Nelson, J. Chem. Phys. , 6257(1987). O. V. 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