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Broadband Photo-Excited Coherent Acoustic Frequency Combs and Mini-Brillouin-Zone Modes in a MQW-SESAM Structure

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A multiple quantum-well semiconductor saturable absorber mirror (MQW-SESAM) structure has been investigated by femtosecond pump-probe laser spectroscopy at a central wavelength of around 1050 nm. Coherent acoustic phonons are generated and detected over a wide frequency range from ~15 GHz to ~800 GHz. In the optical absorption region, i.e., in the multiple quantum wells (In0.27Ga0.73As), acoustic frequency combs centered at ~365 GHz, with a comb spacing of ~33 GHz, are generated. Most importantly, in the transparent region, i.e., in the distributed Bragg reflector, which is formed by a non-doped long-period semiconductor GaAs/Al0.95Ga0.05As superlattice, the mini-Brillouin-zone center, as well as zone-edge acoustic modes, are observed. The mini-zone-center modes with a fundamental frequency of 32 GHz can be attributed to the spatial modulation of the pump optical interference field with a period very close to that of the distributed Bragg reflector, in combination with the periodic spatial modulation of the electrostriction coefficient in the distributed Bragg reflector. The excitation of mini-zone-edge modes is attributed to the stimulated subharmonic decay of the fundamental center modes. Their subsequent back-folding to the mini-Brillouin-zone center makes them Raman active for the probe light.
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applied
sciences
Article
Frequency Combs and Mini-Brillouin-Zone Modes
in a MQW-SESAM Structure
Changxiu Li 1, *, Vitalyi Gusev 2, Emmanouil Dimakis 3, Thomas Dekorsy 1,4
and Mike Hettich 1,5
1Department of Physics and Center of Applied Photonics, University of Konstanz, D-78457 Konstanz,
Germany; thomas.dekorsy@uni-konstanz.de (T.D.); mike.hettich@recendt.at (M.H.)
2LAUM, UMR-CNRS 6613, Le Mans Université, Avenue O. Messiaen, 72085 Le Mans, France;
vitali.goussev@univ-lemans.fr
3Helmholtz-Zentrum Dresden-Rossendorf, D-01328 Dresden, Germany; e.dimakis@hzdr.de
4Institute for Technical Physics, German Aerospace Center (DLR), D-70569 Stuttgart, Germany
5Research Center for Non-Destructive Testing GmbH, 4040 Linz, Austria
*Correspondence: changxiu.li@uni-konstanz.de
Received: 4 December 2018; Accepted: 8 January 2019; Published: 15 January 2019


Featured Application: Semiconductor saturable absorber mirrors can be characterized
non-destructively by their coherent acoustic response in the sub-THz range. Optical below-bandgap
generation of coherent acoustic phonons opens a new way to investigate light-matter interaction
dynamics in a long-period semiconductor superlattice on picosecond time scales.
Abstract:
A multiple quantum-well semiconductor saturable absorber mirror (MQW-SESAM)
structure has been investigated by femtosecond pump-probe laser spectroscopy at a central
wavelength of around 1050 nm. Coherent acoustic phonons are generated and detected over a wide
frequency range from ~15 GHz to ~800 GHz. In the optical absorption region, i.e., in the multiple
quantum wells (In
0.27
Ga
0.73
As), acoustic frequency combs centered at ~365 GHz, with a comb spacing
of ~33 GHz, are generated. Most importantly, in the transparent region, i.e., in the distributed
Bragg reﬂector, which is formed by a non-doped long-period semiconductor GaAs/Al
0.95
Ga
0.05
As
superlattice, the mini-Brillouin-zone center, as well as zone-edge acoustic modes, are observed.
The mini-zone-center modes with a fundamental frequency of 32 GHz can be attributed to the
spatial modulation of the pump optical interference ﬁeld with a period very close to that of the
distributed Bragg reﬂector, in combination with the periodic spatial modulation of the electrostriction
coefﬁcient in the distributed Bragg reﬂector. The excitation of mini-zone-edge modes is attributed to
the stimulated subharmonic decay of the fundamental center modes. Their subsequent back-folding
to the mini-Brillouin-zone center makes them Raman active for the probe light.
Keywords:
coherent acoustic phonons; pump-probe spectroscopy; quantum well; semiconductor
superlattice; mini-Brillouin-zone; high-speed asynchronous optical sampling
1. Introduction
Periodic structures such as multiple quantum wells (MQWs) and superlattices (SLs) have
been used as attractive tools for the observation of folded longitudinal coherent acoustic phonons
(LCAPs) [
1
,
2
]. A broad range of possible applications based on periodic structures have been
proposed and investigated ranging from tailored heat transport for thermoelectric applications to
new optomechanical systems and the development of acoustic phonon ampliﬁcation, i.e., sasers [
3
5
].
Appl. Sci. 2019,9, 289; doi:10.3390/app9020289 www.mdpi.com/journal/applsci
Appl. Sci. 2019,9, 289 2 of 22
Despite the intense research in this ﬁeld, even periodic planar ﬁlm systems still exhibit new and
surprising features regarding their acoustic behavior, in particular in pump-probe experiments.
All-optical femtosecond pump-probe spectroscopy allows us to gain insightful knowledge of the
acoustic phonon behavior in both the light absorbing and transparent regions of a semiconductor SL
structure by the selection of optimal pump and probe wavelengths [
6
]. If light absorption is involved
in semiconductors, then coherent acoustic phonons are excited mainly due to the deformation potential
interaction or the inverse piezoelectric effect related to electron-hole excitation via inter-band light
absorption [
7
,
8
]. If the deformation potential mechanism is suppressed by the process of ultrafast
electron-hole recombination as in amorphous semiconductors, the thermo-elastic effect can also play
a role in coherent acoustic phonon generation through energy transfer from excited hot electrons
to phonons in semiconductors [
7
]. Although direct semiconductors do not exhibit photoexcitation
of carriers for optical pump below-bandgap excitation, the electric ﬁeld of the incident light itself
can induce stress via electrostriction [
7
]. So far, a great number of observations have been obtained
regarding folded LCAPs in light absorbing periodic materials [
9
,
10
], and coherent acoustic phonons
have also been studied near the absorption region of spacer materials sandwiched by two distributed
Bragg reﬂectors (DBRs) in cavity optomechanical semiconductor devices [
11
,
12
]. When the laser
energy was around 10 meV below the bandgap of the spacer material, electrostriction was ruled out
as the driving mechanism for mechanical cavity modes, while optoelectronic deformation potential
interactions turned out to play an important role in the near-bandgap acoustic phonon excitation.
There are also some observations of acoustic phonon excitation through electrostriction in ferroelectric
materials and liquids [
13
,
14
]. However, it is not common to ﬁnd experimental demonstrations of the
electrostriction excitation mechanism in transparent semiconductor heterostructures and SLs where
the LCAPs spectrum exhibits mini-Brillouin-zone (MBZ) center and edge modes. Because the artiﬁcial
periodic structure introduces a much larger period than the lattice constant of bulk materials (e.g.,
GaAs, AlAs), the original Brillouin zone is required to be folded into a MBZ with a new boundary
of k=
π
/d
SL
, where kdenotes the wavevector and d
SL
denotes the SL period. Therefore, multiple
high frequency modes can appear at and near the MBZ-zone center k= 0. The folded modes with
wavevectors 0
k
2q
probe
, where q
probe
denotes the wavevector of the probe light in the SL, can fulﬁll
the wavevector conservation selection rule, and thus become Raman active in optical spectroscopy [
15
].
Once the wavevector 2q
probe
exceeds the boundary of the MBZ, Umklapp processes have to be taken
into account, making it possible to detect LCAP modes with k= |2q
probe
2m
π
/d
SL
|, where mis an
integer [
16
]. Nonetheless, mini-zone-edge modes are usually not observable in optical spectroscopy
because the wavevector conservation rule cannot be satisﬁed at k=kmax =π/dSL.
However, there are indeed a few experiments showing that MBZ-edge modes can be observed
in some structures [
17
20
]. In quantum cascade laser structures consisting of SL formed by a
doped and non-doped GaAs and AlGaAs layers with a thickness range from several nm to around
35 nm, edge modes were observed above a temperature of 150 K and at room temperature, which
is attributed to a symmetry breakdown and the effects of the doping of some of the constituting
layers [
17
]. Beardsley et al. [
18
] observed edge modes in a n
+
-GaAs/AlAs SL at low temperature and
proposed that the free-carriers in the SL enable momentum conservation through photon-phonon
scattering. The electrons conﬁned in quantum wells (QWs) can absorb a momentum that is less than
kmax π/LQW, where LQW means the thickness of the QW. Thus, the excess momentum k2qprobe in
the backward scattering of probe light can be absorbed if |
π
/d
SL
2q
probe
|
π
/L
QW.
In stationary or
frequency-domain Raman experiments on thermal phonons, the observations of mini-zone-edge
phonons are attributed to disorder-induced q
z
-non-conserving scattering, which can be caused,
for example, by interface roughness [
19
] and interface defects [
20
] in GaAs/AlAs superlattices.
Finally, Brillouin zone edge phonons in KTaO
3
and TiO
2
crystals, coherently and resonantly induced
by crossed two near-infrared laser beams, are observed in coherent anti-Stokes Raman scattering
(CARS) spectroscopy [
21
24
]. The observation of the overtones of Brillouin edge modes in those
Appl. Sci. 2019,9, 289 3 of 22
crystals is supported by the folding of X-point phonons onto the
Γ
-point due to grating formation by
coherent phonons.
In this paper, we report on the generation and detection of CLAPs in a MQW- semiconductor
saturable absorber mirror (SESAM) structure performed by femtosecond time-domain spectroscopy
(high-speed asynchronous optical sampling) in a reﬂection geometry at a wavelength of around
1050 nm for both pump and probe laser sources. The MQW-SESAM structure exhibits multiple QWs
as embedded saturable absorber layers and a distributed Bragg mirror (DBR) which are spatially
separated by GaAs and AlGaAs layers. Both of the pump and probe photon energies are above the
bandgap of the saturable absorber layers but below that of constituting materials of the DBR, whose
reﬂectance is close to 100% for our laser wavelengths. Due to the high detection sensitivity of our
pump-probe system, we are able to observe not only very pronounced LCAPs excited in the MQW
region, but also much weaker acoustic phonon contributions which can be attributed to folded LCAPs
excited in the transparent DBR region.
Our ﬁndings demonstrate the ﬁrst simultaneous observation of MBZ-center and -edge modes in
a transparent non-doped long-period semiconductor SL. The detailed discussion of the MBZ mode
excitation and detection in our experiment will be given in Section 3.2. In addition, the broadband
coherent acoustic frequency combs ranging from ~30 GHz to ~800 GHz monitored in the light
absorbing MQW region enables us to non-destructively characterize our SESAM structure (Section 3.1).
Finally, the structure-related complexity of our experimental phonon spectrum will be discussed in
Appendix A.
2. Experimental Procedure
The MQW-SESAM sample used in our experiment is schematically shown in Figure 1.
The structure of the sample from top to bottom (light incident direction) consists of a GaAs cap layer,
nine In
0.27
Ga
0.73
As QWs separated by GaAs barriers and a DBR composed of 23 GaAs/Al
0.95
Ga
0.05
As
bilayers, which is grown on a GaAs substrate oriented along the (100) direction. The neighboring
triple-QW stacks are separated by 112-nm thick GaAs layers. Each triple-QW stack consists of an
In
0.27
Ga
0.73
As/GaAs/In
0.27
Ga
0.73
As/GaAs/In
0.27
Ga
0.73
As structure with 7-nm thickness for each QW
and 6-nm thickness for each barrier. The MQW structure is separated from the sample surface by a
GaAs cap layer with the thickness equal to approximately half of the separation distance between
triple-QW stacks. Each bilayer in the DBR consists of 73.7-nm GaAs and 86.7-nm Al
0.95
Ga
0.05
As.
The DBR and the last QW are spaced by a 9.6-nm GaAs layer and a 68.7-nm Al
0.95
Ga
0.05
As layer.
Two GHz-repetition-rate Yb:KYW lasers with 5 kHz repetition rate difference emitting both at a central
wavelength of ~1050 nm are employed to conduct femtosecond pump-probe spectroscopy on this
sample in a reﬂection conﬁguration. One Yb:KYW laser is used as pump laser and another is used as
probe laser. Both pump and probe beams are incident on the air/cap layer surface. The reﬂected probe
beam is received by a 125-MHz photodetector, which is connected to a 100-MHz A/D converter for
data acquisition. Thus, the photo-induced reﬂectivity change (
R/R) is monitored as a function of time
delay between the pump pulse and probe pulse which is automatically introduced by their repetition
rate difference. Unless a speciﬁc note is given, the incident pump power and probe power are always
set to 100 mW and 4.5 mW, respectively. All experiments are performed at room temperature. A more
detailed description of the experimental set-up can be found in Ref. [
25
]. Tables 1and 2list relevant
properties of the lasers and the sample, respectively. From both tables, it is clear that photon energies
(1.184 eV, 1.183 eV) of pump and probe lasers are above the bandgap of In
0.27
Ga
0.73
As (1.052 eV (bulk),
1.177 eV (conﬁnement effects considered)); meanwhile, the photon energies are signiﬁcantly below the
bandgap of GaAs (1.424 eV (bulk at room temperature)) and the bandgap of Al
0.95
Ga
0.05
As (2.148 eV
(bulk)). Therefore, the optical absorption region refers only to MQWs (In
0.27
Ga
0.73
As) represented by
violet areas in Figure 1, while the bottom DBR materials are transparent to both pump and probe light.
Both these features will be of relevance for our following discussion regarding CLAPs photo-induced
generation and detection.
Appl. Sci. 2019,9, 289 4 of 22
Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 22
Figure 1. Schematic diagram of the sample structure. The beige-colored areas denote GaAs layers.
The violet areas denote In
0.27
Ga
0.73
As layers while the green areas denote Al
0.95
Ga
0.05
As layers. Optical
pulses are incident from the left.
Table 1. Parameters of lasers.
Laser
Central
Wavelength
λ
0
(nm)
Bandwidth
Δλ (nm)/ΔE
(meV)
Pulse
Width
Δτ (fs)
Output
Power P
(mW)
Repetition Rate
f
rep
Photon
Energy
E
p
(eV)
Pump laser 1047.5 5.1/6 210 260 1 GHz + 5 kHz 1.184
Probe laser 1048.3 5.9/7 280 310 1 GHz 1.183
Table 2. Properties of MQW-SESAM sample at a wavelength of λ = 1048.3 nm.
Material Refractive
Index n Extinction Coefficient K Energy Gap E
g
(eV) Acoustic
Velocity v (m/s)
Density ρ
(g/cm
3
)
GaAs 3.48 [26] 0 (α = 7.4 × 10
3
cm
1
) [26] 1.424 [27] 4730 [27] 5.317 [28]
In
0.27
Ga
0.73
As 3.60
[29,30]
0.028 (α = 3.4 × 10
3
cm
1
)
[31]
1.052 [32]
(1.177 quantization) 4487 [33] 5.410 [33]
Al
0.95
Ga
0.05
As 2.98 [34] 0 [35] 2.148 [26] 5592 [36] 3.838 [36]
DBR
(GaAs/Al
0.95
Ga
0.05
As) 3.22 [37]
5160 [37]
QW-heterostructures
(GaAs/In
0.27
Ga
0.73
As) 3.54 [37]
4596 [37]
3. Results and Discussions
The obtained experimental signal is shown in the inset of Figure 2a. After a background
subtraction conducted via a smoothing average, rich acoustic features become visible in the main
panel of Figure 2a, where several distinct wave-packet-like structures are present in the signal.
Regarding the numerical signal processing, we note that the extraction procedure is first optimized
for the higher frequency components, and thus, some lower frequency components are suppressed
in Figure 2a. In order to address the lower frequency component contributions more quantitatively,
the extraction procedure is adjusted in the later investigations in Section 3.2 It is important to
mention that the applied extraction procedure is only used to increase the visibility of the small
acoustic contributions to the overall signal (detected relative reflectivity change induced by
superimposed acoustic signal R/R~10
7
while the reflectivity change from the electronic
background R/R~10
4
, i.e., 3 orders of magnitude larger), without introducing artifacts in the
acoustic spectrum. The two different frequency domains will be discussed in the following
paragraphs, starting with the high frequency contributions.
3.1. High Frequency Coherent Acoustic Phonon Combs
3.1.1. Generation and Detection Mechanisms
Figure 1.
Schematic diagram of the sample structure. The beige-colored areas denote GaAs layers.
The violet areas denote In
0.27
Ga
0.73
As layers while the green areas denote Al
0.95
Ga
0.05
As layers. Optical
pulses are incident from the left.
Table 1. Parameters of lasers.
Laser Central Wavelength
λ0(nm)
Bandwidth λ
(nm)/E(meV)
Pulse Width
τ(fs)
Output Power
P(mW)
Repetition
Rate frep
Photon Energy
Ep(eV)
Pump laser 1047.5 5.1/6 210 260
1 GHz + 5 kHz
1.184
Probe laser 1048.3 5.9/7 280 310 1 GHz 1.183
Table 2. Properties of MQW-SESAM sample at a wavelength of λ= 1048.3 nm.
Material Refractive
Index n
Extinction
Coefﬁcient K
Energy Gap Eg
(eV)
Acoustic
Velocity v(m/s)
Density ρ
(g/cm3)
GaAs 3.48 [26]0 (α= 7.4 ×103cm1)
[26]1.424 [27] 4730 [27] 5.317 [28]
In0.27Ga0.73 As 3.60 [29,30]0.028 (α= 3.4 ×103
cm1) [31]
1.052 [32]
(1.177 quantization)
4487 [33] 5.410 [33]
Al0.95Ga0.05 As 2.98 [34] 0 [35] 2.148 [26] 5592 [36] 3.838 [36]
DBR
(GaAs/Al
0.95
Ga
0.05
As)
3.22 [37] 5160 [37]
QW-heterostructures
(GaAs/In
0.27
Ga
0.73
As)
3.54 [37] 4596 [37]
3. Results and Discussions
The obtained experimental signal is shown in the inset of Figure 2a. After a background
subtraction conducted via a smoothing average, rich acoustic features become visible in the main panel
of Figure 2a, where several distinct wave-packet-like structures are present in the signal. Regarding
the numerical signal processing, we note that the extraction procedure is ﬁrst optimized for the higher
frequency components, and thus, some lower frequency components are suppressed in Figure 2a.
In order to address the lower frequency component contributions more quantitatively, the extraction
procedure is adjusted in the later investigations in Section 3.2 It is important to mention that the
applied extraction procedure is only used to increase the visibility of the small acoustic contributions
to the overall signal (detected relative reﬂectivity change induced by superimposed acoustic signal
R/R~10
7
while the reﬂectivity change from the electronic background
R/R~10
4
, i.e., 3 orders of
magnitude larger), without introducing artifacts in the acoustic spectrum. The two different frequency
domains will be discussed in the following paragraphs, starting with the high frequency contributions.
3.1. High Frequency Coherent Acoustic Phonon Combs
3.1.1. Generation and Detection Mechanisms
Since the pump light energy is absorbed in the QWs in our sample, stresses could potentially be
photo-excited by three mechanisms including the inverse piezoelectric effect, the thermo-elastic effect
and the deformation potential,
σ(z,t)=σPE(z,t)+σT E(z,t)+σDP (z,t). (1)
Appl. Sci. 2019,9, 289 5 of 22
However, the inverse piezoelectric effect only dominates when the semiconductor heterostructure
is grown along a high index piezo-electrically-active direction where large built-in electric ﬁelds can be
formed, especially in the [111] direction. The [100] direction is non-piezo-electric (our sample) [
38
].
Additionally, although the laser light can heat up the sample and induce a thermal expansion, unlike
in metals, in semiconductors such a thermo-elastic effect is usually considerably weaker than the
deformation potential effect, especially when the energy of the pump optical quantum is close to the
semiconductor band gap [
39
]. Therefore, it makes sense to ignore the contributions from
σPE
and
σTE
.
In other words, stresses related to light absorption are considered to take place in the nine QW layers
in our sample mainly through deformation potential interaction,
σDP(z,t)=deh N(z,t), (2)
where d
eh
denotes the electron-hole phonon deformation potential parameter and N(z,t) denotes the
photo-excited carrier concentration [
4
]. Equation (2) indicates that the generation of coherent acoustic
phonons largely depends on the concentration of photo-excited electrons and holes in the bottom of
the conduction band and in the top of the valence band when the photon energy is very close to the
material bandgap, which holds true for our QWs with a bandgap only ~7 meV below the pump photon
energy (1.184-eV photon energy and 1.177-eV bandgap of QW). In terms of detection, the presence
of acoustic phonons is able to induce a displacement of the surface and interfaces of the medium
(interferometric effect) and perturb the optical refractive index of the medium (photo-elastic effect),
which are monitored by the reﬂected probe light via the optical reﬂectivity change. Because the probe
light exhibits approximately the same wavelength as the pump light, the generated forward and
backward propagating stress waves are also detected in the QWs layers through the photo-elastic
effect. The interferometric effect can in principle contribute to the detection of the acoustic waves.
The MQW-SESAM structure is designed to maximize the optical electric ﬁeld at the position of the
QWs, and thus, efﬁcient light absorption, generation, and detection of elastic waves can be expected.
3.1.2. Overview of Experimental Results
Due to the periodic structure of the MQWs, a train of acoustic wave-packets over the whole
time window is detected, as depicted in Figure 2a. The detected acoustic signal is proportional to the
integral of f
g
(z,t) * f
d
(z), i.e., the convolution of two functions, along the acoustic propagation axis z,
where f
g
(z,t) represents the generated stress in each triple-QW stack and f
d
(z) represents the detection
sensitivity function in each triple-QW stack. The main feature of the detected acoustic signal can be
explained quite well by the analysis conducted in [
25
], where we used the same sample for an initial
test of the pump-probe system performance.
We will quickly reiterate some of the previous ﬁndings which are important to establish a
deeper and more quantitative understanding of the rich acoustics in the sample. Firstly, as shown
in Figure 2b the central frequency of the acoustic spectrum (the peak frequency of the main lobe
of the wave-packet FFT depicted by the magenta curve) is 366.5 GHz, which is determined by the
travelling time of the stress wave between neighboring QWs. Due to the resemblance of our results
with optical frequency combs, we also apply the same terminology here, where the central frequency
(or wavelength) refers to the frequency (or wavelength) of the carrier (wave-packet) and spectral lines
are equally spaced by the inverse of the pulse repetition rate, i.e., the inverse of time delay between
neighboring wave-packets. In theory, the central acoustic frequency induced by the periodic QWs can
be calculated as f
0
= (d
QW
/v
eff
)
1
= 353.5 GHz, where v
eff
denotes the effective longitudinal acoustic
velocities in GaAs/In
0.27
Ga
0.73
As heterostructure which can be taken from Table 2, and d
QW
denotes
the distance of adjacent QWs (13 nm) within the triple-QW stack. Secondly, the acoustic frequency
comb shows a spacing of around 33 GHz determined by the travelling time of the stress wave between
neighboring triple-QW stacks, which is also equal to the round trip time between the ﬁrst triple-QW
stack and the air/GaAs interface. The comb spacing induced by the spatial periodicity of the triple-QW
Appl. Sci. 2019,9, 289 6 of 22
stacks can be approximated by
f
0
= (d
ss
/v
GaAs
)
1
32.6 GHz, where d
ss
represents the spatial period
of the triple-QW stacks (145 nm), and v
GaAs
represents the longitudinal sound velocity in GaAs, whose
contribution to the SESAM structure width largely dominates over that of In
0.27
Ga
0.73
As. In a word,
the experimental results ﬁt well with the above estimations. In addition, the ﬁve peaks in the detected
individual wave-packets can be explained by the convolution of the stress bursts, consisting of three
pulses incident on the triple-QW structures, and the detection sensitivity function, containing three
peaks in each of the triple-QW structures (see Figure 3).
Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 22
acoustic frequency comb shows a spacing of around 33 GHz determined by the travelling time of the
stress wave between neighboring triple-QW stacks, which is also equal to the round trip time
between the first triple-QW stack and the air/GaAs interface. The comb spacing induced by the
spatial periodicity of the triple-QW stacks can be approximated by f
0
= (d
ss
/v
GaAs
)
1
32.6 GHz, where
d
ss
represents the spatial period of the triple-QW stacks (145 nm), and v
GaAs
represents the
longitudinal sound velocity in GaAs, whose contribution to the SESAM structure width largely
dominates over that of In
0.27
Ga
0.73
As. In a word, the experimental results fit well with the above
estimations. In addition, the five peaks in the detected individual wave-packets can be explained by
the convolution of the stress bursts, consisting of three pulses incident on the triple-QW structures,
and the detection sensitivity function, containing three peaks in each of the triple-QW structures (see
Figure 3).
Figure 2. (a) Coherent acoustic phonon wave-packets after background removal. Top inset: the
original time trace. Bottom inset: zoom in of the wave-packet at around 30 ps. (b) Acoustic spectrum
and dispersion relation in GaAs/In
0.27
Ga
0.73
As SL. The brown line represents the acoustic spectrum.
The magenta curve depicts the FFT of the wave-packet at 30 ps. The z-like blue line represents the
dispersion relation and the horizontal blue line indicates the wavevector position k = 2q
probe
GaAs/InGaAs
.
The vertical gray lines are spaced by the comb spacing of 33.15 GHz.
3.1.3. Generation, Propagation and Detection of Acoustic Waves in MQW Region
In III-V semiconductor multilayer structures, the acoustic reflection from the internal interface
is usually insignificant, due to the small acoustic impedance mismatch between constituent
materials. If the densities ρ
GaAs
and ρ
InGaAs
, as well as longitudinal acoustic velocities v
GaAs
and v
InGaAs
,
are taken from Table 2, the reflection coefficient of the GaAs/In
0.27
Ga
0.73
As interface can be calculated
as Γ =|(ρ
GaAs
v
GaAs
ρ
InGaAs
v
InGaAs
)/(ρ
GaAs
v
GaAs
+ ρ
InGaAs
v
InGaAs
)| = 1.8 × 10
2
. As illustrated in Figure 3, if the
small acoustic impedance mismatch between GaAs and In
0.27
Ga
0.73
As is neglected, in total six discrete
wave-packets should be detected after the sample is irradiated by the pump light. The first
wave-packet, which is not presented in Figure 2a because of the difficulties in removing the
background signal at times close to the pump/probe overlap, stems from the detection of the
forward and backward propagating stress bursts (S
ib
and S
ir
, i = 1, 2, 3) generated in each of the
triple-QW stacks when they are escaping from the corresponding local stacks (Figure 3, line 1). The
second detected wave-packet stems from the detection of five stress bursts in the neighboring/local
triple-QW stacks at the same time when the photo-generated stresses after the propagation between
the triple-QW stacks start to move across the stacks. These five stress bursts include two pairs of
bursts excited in the first and the second triple-QW stacks (S
ib
and S
ir
, i = 1, 2) and the backward
propagating burst excited in the third triple-QW stack, i.e., S
3r
(Figure 3, line 2). The third
wave-packet stems from the detection of four stress bursts (Figure 3, line 3), and so on. In
Figure 2.
(
a
) Coherent acoustic phonon wave-packets after background removal. Top inset: the original
time trace. Bottom inset: zoom in of the wave-packet at around 30 ps. (
b
) Acoustic spectrum and
dispersion relation in GaAs/In
0.27
Ga
0.73
As SL. The brown line represents the acoustic spectrum. The
magenta curve depicts the FFT of the wave-packet at 30 ps. The z-like blue line represents the dispersion
relation and the horizontal blue line indicates the wavevector position k= 2
qGaAs/InGaAs
probe
. The vertical
gray lines are spaced by the comb spacing of 33.15 GHz.
3.1.3. Generation, Propagation and Detection of Acoustic Waves in MQW Region
In III-V semiconductor multilayer structures, the acoustic reﬂection from the internal interface is
usually insigniﬁcant, due to the small acoustic impedance mismatch between constituent materials.
If the densities
ρGaAs
and
ρInGaAs
, as well as longitudinal acoustic velocities v
GaAs
and v
InGaAs
, are
taken from Table 2, the reﬂection coefﬁcient of the GaAs/In
0.27
Ga
0.73
As interface can be calculated
as
Γ
=|(
ρGaAs
v
GaAs ρInGaAs
v
InGaAs
)/(
ρGaAs
v
GaAs
+
ρInGaAs
v
InGaAs
)| = 1.8
×
10
2
. As illustrated in
Figure 3, if the small acoustic impedance mismatch between GaAs and In
0.27
Ga
0.73
As is neglected, in
total six discrete wave-packets should be detected after the sample is irradiated by the pump light.
The ﬁrst wave-packet, which is not presented in Figure 2a because of the difﬁculties in removing the
background signal at times close to the pump/probe overlap, stems from the detection of the forward
and backward propagating stress bursts (S
ib
and S
ir
, i = 1, 2, 3) generated in each of the triple-QW
stacks when they are escaping from the corresponding local stacks (Figure 3, line 1). The second
detected wave-packet stems from the detection of ﬁve stress bursts in the neighboring/local triple-QW
stacks at the same time when the photo-generated stresses after the propagation between the triple-QW
stacks start to move across the stacks. These ﬁve stress bursts include two pairs of bursts excited in
the ﬁrst and the second triple-QW stacks (S
ib
and S
ir
, i = 1, 2) and the backward propagating burst
excited in the third triple-QW stack, i.e., S
3r
(Figure 3, line 2). The third wave-packet stems from the
detection of four stress bursts (Figure 3, line 3), and so on. In consequence, a smaller and smaller
number of stress bursts contribute to the detection as stress waves propagate along the z-axis, which
can also partly explain the damping of the wave-packet train (see Figure 2a). Thereby, in the end, there
Appl. Sci. 2019,9, 289 7 of 22
is only the detection of S
3r
in the third triple-QW stack, which corresponds to the sixth wave-packet
signal. After S
3r
leaves the last triple-QW stack, acoustic wave-packets excited in the QWs are not
expected to be further detected in the MQW region, under the condition of their negligible reﬂections
in the DBR region. However, our experimental results, presented in Figure 2a, clearly show more than
ﬁve wave-packets (the 2nd~6th wave-packets) in the time domain (the ﬁrst wave-packet at 0 ps is
not shown there). In order to explain the existence of wave-packets beyond the expected number of
occurrences given by the sample geometry, the small acoustic impedance mismatch between GaAs
and In
0.27
Ga
0.73
As should be taken into account. In this way, the reﬂection of acoustic phonons from
the interfaces between these materials can contribute to wave-packets bouncing between neighboring
triple-QW stacks and between the ﬁrst triple-QW stack and the air/GaAs interface. These multiple
reﬂections produce “ringing” of the triple-QW stacks/air region, which lasts longer than the time
necessary for the acoustic wave just to traverse it twice. A detailed theoretical model and comparison
with experimental results will be presented in [40].
Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 22
consequence, a smaller and smaller number of stress bursts contribute to the detection as stress
waves propagate along the z-axis, which can also partly explain the damping of the wave-packet
train (see Figure 2a). Thereby, in the end, there is only the detection of S
3r
in the third triple-QW
stack, which corresponds to the sixth wave-packet signal. After S
3r
leaves the last triple-QW stack,
acoustic wave-packets excited in the QWs are not expected to be further detected in the MQW
region, under the condition of their negligible reflections in the DBR region. However, our
experimental results, presented in Figure 2a, clearly show more than five wave-packets (the 2nd~6th
wave-packets) in the time domain (the first wave-packet at 0 ps is not shown there). In order to
explain the existence of wave-packets beyond the expected number of occurrences given by the
sample geometry, the small acoustic impedance mismatch between GaAs and In
0.27
Ga
0.73
As should
be taken into account. In this way, the reflection of acoustic phonons from the interfaces between
these materials can contribute to wave-packets bouncing between neighboring triple-QW stacks and
between the first triple-QW stack and the air/GaAs interface. These multiple reflections produce
“ringing” of the triple-QW stacks/air region, which lasts longer than the time necessary for the
acoustic wave just to traverse it twice. A detailed theoretical model and comparison with
experimental results will be presented in [40].
Figure 3. The illustration of the generation, propagation, and detection of acoustic stress bursts in the
sample. Beige areas represent GaAs, violet areas represent QWs. Green area represents Al
0.95
Ga
0.05
As.
S
ir
and S
ib
(i = 1, 2, 3) denote the backward and forward emitted from the i-th triple-QW stack stress
bursts presented in red and blue colors, respectively.
In addition to the temporal evolution of the acoustic signal, its frequency spectrum is also
worthy of scrutiny, given that our previous result [25] revealed only a part of the features visible in
the spectrum detected in the current experiments. After 25 times magnification in the frequency
range from 0 to 232 GHz and 15 times magnification in the frequency range from 497 GHz to 1000
GHz, the acoustic spectrum revealed a wide sub-THz comb-like structure with a regular comb
spacing of around 33 GHz, which covered the range from ~30 GHz to ~800 GHz, as illustrated in
Figure 2b. The comb spacing ~33 GHz (marked by gray vertical lines) corresponds to the reciprocal
of time delay of neighboring wave-packets (30.25 ps), which agrees very well with the experimental
result in the vicinity of 365 GHz, while there are small offsets in other spectral ranges. The visible
non-constant comb spacing can be attributed to several factors. Firstly, one has to notice that the
spectral amplitudes in the range 0–232 GHz and in the range 497–1000 GHz are at least one order of
magnitude smaller than those around 365 GHz, which could possibly introduce a few GHz offset
Figure 3.
The illustration of the generation, propagation, and detection of acoustic stress bursts in the
sample. Beige areas represent GaAs, violet areas represent QWs. Green area represents Al
0.95
Ga
0.05
As.
S
ir
and S
ib
(i = 1, 2, 3) denote the backward and forward emitted from the i-th triple-QW stack stress
bursts presented in red and blue colors, respectively.
In addition to the temporal evolution of the acoustic signal, its frequency spectrum is also worthy
of scrutiny, given that our previous result [
25
] revealed only a part of the features visible in the
spectrum detected in the current experiments. After 25 times magniﬁcation in the frequency range
from 0 to 232 GHz and 15 times magniﬁcation in the frequency range from 497 GHz to 1000 GHz,
the acoustic spectrum revealed a wide sub-THz comb-like structure with a regular comb spacing of
around 33 GHz, which covered the range from ~30 GHz to ~800 GHz, as illustrated in Figure 2b.
The comb spacing ~33 GHz (marked by gray vertical lines) corresponds to the reciprocal of time delay
of neighboring wave-packets (30.25 ps), which agrees very well with the experimental result in the
vicinity of 365 GHz, while there are small offsets in other spectral ranges. The visible non-constant
comb spacing can be attributed to several factors. Firstly, one has to notice that the spectral amplitudes
in the range 0–232 GHz and in the range 497–1000 GHz are at least one order of magnitude smaller than
those around 365 GHz, which could possibly introduce a few GHz offset from the marked positions.
Secondly, the low frequency modes could be nearly degenerate with the mini-Brillouin-zone modes
Appl. Sci. 2019,9, 289 8 of 22
which will be discussed in Section 3.2. Thirdly, there are uncertainties in the determination of the
longitudinal acoustic velocity in the sample and in the quantum well stack-to-stack distance. This will
lead to slight deviations from the expected constant comb spacing. The mode amplitudes of the
frequency comb are modulated by the envelope derived from the ﬁrst wave-packet FFT (magenta
line in Figure 2b), which produces a main lobe with a bandwidth of
B= 109.6 GHz (full width at
half maximum) and a few side lobes with at least an order of smaller amplitudes than that of the
main lobe. Therefore, the ﬁve frequency components in the vicinity of 365 GHz dominate the acoustic
temporal features, while the lower frequency components (below 100 GHz) and higher frequency
components (around 700 GHz) have a much smaller impact on the overall acoustic oscillations. It is
worth noting that some of the predicted comb components are missing, which can be explained by
the amplitude modulation of their envelope. Because the observed wave-packet is a result of the
convolution between the generated stress burst and the detection sensitivity function of the QW-stack,
the FFT envelope of the frequency comb is associated with the generation and detection processes.
If the generation and detection functions are assumed to be of the rectangular spatial shape with a
33-nm width equal to the QW-stack thickness, due to the spatial restriction, the corresponding FFT
envelope is thus sampling-function-like with a main lobe bandwidth
B= 124 GHz, which means the
amplitude modulation leads to the suppression of comb components close to the dips of the envelope
such as those around 260 GHz and 460 GHz. In total, up to 24 periodic frequency components spaced
by 33 GHz are resolved in our experiment. Table 3lists the main high frequency components from
experiment and calculation. In comparison with calculations based on m
f(mis an integer), the
experimental frequency components show a small shift, i.e., of around 6 GHz, for each component in
the vicinity of 365 GHz, which can be attributed to the uncertainties of longitudinal sound velocities
in the sample and stack-to-stack distances. In order to understand the generated coherent acoustic
phonons from the perspective of a SL rather than individual QWs, a folded CLAP dispersion relation
calculated by the Rytov model in the SL formed by the GaAs (6 nm)/In
0.27
Ga
0.73
As (7 nm) units is
plotted in Figure 2b. The dispersion relation of folded acoustic modes in a SL consisting of alternative
layer A and layer B takes the following form in the slightly modiﬁed Rytov model [37]
cos(kd)=cosωdA
vA
+dB
vBε2
2sinωdA
vAsinωdB
vB, (3)
where εis expressed as
ε=ρBvBρAvA
(ρBvBρAvA)1/2 . (4)
In Equations (3) and (4), k,d,
ω
,d
A
,v
A
,d
B
,v
B
,
ρA
and
ρB
denote the wavevector, the SL period,
the angular frequency, the thickness of layer A, the longitudinal sound velocity in layer A, thickness of
layer B, longitudinal sound velocity in layer B, density of layer A and density of layer B, respectively.
In pump-probe experiments in reﬂection geometry, typical triplet-like mode structures usually arise
due to the k= 0 modes at the Brillouin zone center and the detection of two additional modes of the
nearby dispersion branches [
37
], as marked by solid dots in Figure 2b. The calculation yields the
expected triplet mode frequencies at 323.7, 354.7, and 386.2 GHz. The 354.7 GHz mode belongs to
Raman active MBZ-center mode. 323.7 GHz and 386.2 GHz modes belong to lower branch p=
1 and
upper branch p= +1 at k= 2
qGaAs/InGaAs
probe
= 0.176
π
/d
QW
, respectively, where prepresents the acoustic
phonon order, and krepresents the wavevector. The calculated triplet (323.7, 354.7, 386.2) is nearly
coincident with three frequency comb center components (332.1, 364.6, 397.1) if the deviation between
experimental and calculated comb spacings is considered. In addition, the unfolded mode of 31.2 GHz
at k= 0.176
π
/d
QW
is nearly coincident with the ﬁrst comb component at around 33 GHz. However,
due to the overlapping frequency region with the acoustic frequency comb generated in the QWs, it is
difﬁcult to say how much those folded/unfolded CLAP modes contribute to the detected frequency
components. In addition, the frequency amplitudes also exhibit uncertainty arising from difﬁculties
of acquiring the exact proﬁle of the detection spectral sensitivity function. Further investigations are
Appl. Sci. 2019,9, 289 9 of 22
required in order to distinguish coherent acoustic phonons generated in the GaAs/In
0.27
Ga
0.73
As SL
from those generated in the QWs. For example, the same pump-probe experiment can be applied to
a new sample where the triple-QW stacks distance d
ss
is carefully adjusted to result in an acoustic
frequency comb with a spacing distinctly differing from 33 GHz, while the rest of the structure
remains equal.
Table 3. Acoustic phonon spectrum in the high frequency region
Experiment Calculation MQWs Calculation InGaAs/GaAs
SL [37,41,42]
f0(GHz) Comb (GHz) f0(GHz) Comb (GHz) fc,fe(GHz) f2q (k= 2qprobe)
(0.176π/dQW)
366.5 299.9 633.0 353.5 293.4 619.4
k=π/dQW
L1175.5
U1179.4
Lowest
31.2
B(GHz) 332.1 666.1 B(GHz) 326.0 652.0
k= 0
L 354.7
U 355.3
p=1 323.7
p= +1 386.2
109.6 364.6 700.1 86.9 (G 2)
124 (R 2)358.6 684.6
k=π/dQW
L 530.5
U 534.3
f(GHz) 397.1 732.0 f(GHz) 391.2 717.2
k= 0
L 709.3
U 710.6
p=2 678.7
p= +2 741.2
32.6, 33.3 430.3 766.0 32.6 423.8 749.8
k=π/dQW
L 885.7
U 889.3
1
L denotes the lower frequency of phonon bandgap, while U denotes the upper frequency of phonon bandgap.
2
G
stands for Gaussian proﬁle and R stands for rectangular proﬁle of triple-QW in the calculation.
In summary, a comb-like acoustic spectrum is generated and detected in the QWs in the
MQW-SESAM sample where optical absorption takes place. Consisting of up to 24 detectable frequency
components spaced by ~33 GHz, it covers the frequency domain from ~30 GHz to ~800 GHz, but
is most pronounced in the vicinity of 365 GHz. The detectability of a large number of frequency
components can be attributed to the fact that the spectral sensitivity functions are nearly identical to
the spectral generation functions in our structure by applying the two almost degenerate ~1050-nm
lasers for pump and probe. The observed spectral features can be readily understood in terms of single
QWs or a SL description. In this case, we prefer the former, because the spatial excitation and detection
functions are directly available and immediately result in the observed comb spectra. Still, we think
the discussion of the latter is instructive in this scenario.
3.2. Low Frequency Mini-Brillouin-Zone Mode
3.2.1. Overview of Experimental Results
If the extraction of the acoustic phonon contributions from the original signal trace is numerically
optimized for the lower frequencies in the signal via a larger smoothing scale for electronic and thermal
background subtraction, a damped long-period oscillation (around 15 ps) over a long time delay is
revealed, as shown in Figure 4a. The corresponding acoustic spectrum in Figure 4b exhibits multiple
low frequency modes, which are the origin of the weak beatings in the time domain signal. Since the
DBR in the sample has a long period d
DBR
= 160.4 nm which could, in theory, pave the way for the
generation of the folded CLAP starting from 32 GHz, we propose that those low frequency modes
referring to harmonics and subharmonics of 32 GHz are photo-excited in the DBR for some reason.
Thus, the acoustic dispersion relation in the DBR is plotted in Figure 4b, represented by the blue
line, and the calculated potential modes are listed in Table 4. As a result, the experimental modes
at 32.0 GHz, 63.1 GHz, and 95.1 GHz (marked by blue dots) agree well with the MBZ-center modes
Appl. Sci. 2019,9, 289 10 of 22
32.17/32.18 GHz, 64.33/64.35 GHz and 96.50/96.53 GHz at k= 0. Due to the small forbidden gap
(0.01 GHz, 0.02 GHz and 0.03 GHz for the 1st, 2nd and 3rd dispersion branches, respectively) at k= 0,
it is hard to resolve if the center modes belong to lower or upper branches. In addition, the long period
of the DBR requires Umklapp processes for the wavevector k= 2
qDBR
probe
, and thus, after the Umklapp
processes the wavevector k= |2
qDBR
probe
2
π
/d
DBR
| = 0.028
π
/d
DBR
is very close to k= 0 as the blue
horizontal line indicates. Thus, this is a possible explanation why seemingly only the center modes
are present in the spectrum in the reﬂection geometry. Meanwhile, the experimental modes 15.1 GHz,
78.3 GHz, and 110.1 GHz (marked by green dots) show a remarkable agreement with the MBZ-edge
modes 15.28 GHz, 79.62 GHz and 110.8 GHz at k=
π
/d
DBR
. Normally, the generation of folded CLAPs
is not expected in a DBR undergoing a below bandgap light irradiation due to the absence of light
absorption. MBZ-edge modes are not detectable due to non-fulﬁllment of the wavevector conservation
rule. Hence, in the following section, we will elaborate on the interpretation of our experimental
ﬁndings and propose potential mechanisms that facilitate MBZ-center and -edge mode excitation and
detection in the DBR of the MQW-SESAM structure.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 22
represented by the blue line, and the calculated potential modes are listed in Table 4. As a result, the
experimental modes at 32.0 GHz, 63.1 GHz, and 95.1 GHz (marked by blue dots) agree well with the
MBZ-center modes 32.17/32.18 GHz, 64.33/64.35 GHz and 96.50/96.53 GHz at k = 0. Due to the small
forbidden gap (0.01 GHz, 0.02 GHz and 0.03 GHz for the 1st, 2nd and 3rd dispersion branches,
respectively) at k = 0, it is hard to resolve if the center modes belong to lower or upper branches. In
addition, the long period of the DBR requires Umklapp processes for the wavevector k = 2qprobe
DBR , and
thus, after the Umklapp processes the wavevector k = |2qprobe
DBR 2π/d
DBR
| = 0.028π/d
DBR
is very close to
k = 0 as the blue horizontal line indicates. Thus, this is a possible explanation why seemingly only the
center modes are present in the spectrum in the reflection geometry. Meanwhile, the experimental
modes 15.1 GHz, 78.3 GHz, and 110.1 GHz (marked by green dots) show a remarkable agreement
with the MBZ-edge modes 15.28 GHz, 79.62 GHz and 110.8 GHz at k = π/d
DBR
. Normally, the
generation of folded CLAPs is not expected in a DBR undergoing a below bandgap light irradiation
due to the absence of light absorption. MBZ-edge modes are not detectable due to non-fulfillment of
the wavevector conservation rule. Hence, in the following section, we will elaborate on the
interpretation of our experimental findings and propose potential mechanisms that facilitate
MBZ-center and -edge mode excitation and detection in the DBR of the MQW-SESAM structure.
Figure 4. (a) Coherent acoustic phonon oscillation from 70 ps to 800 ps when background subtraction
is optimized for low frequencies extraction. (b) FFT of (a) and dispersion relations in SL formed by
DBR with period d
DBR
= 160.4 nm as well as SL formed by MQWs and GaAs spacers with period
d
MQW-SL
= 145 nm. Brown line represents FFT, where MBZ-center modes at 32.0 GHz, 63.1 GHz and
95.1 GHz and MBZ-edge modes at 15.1 GHz, 78.3 GHz and 110.1 GHz are marked. Blue line
represents dispersion relation in the DBR (horizontal blue line represents k = |2q
probe
DBR
2π/d
DBR
| =
0.028π/d
DBR
in the DBR). Magenta line represents dispersion relation in MQWs and GaAs spacers
formed SL (horizontal magenta line represents k = |2q
probe
MQW-SL
2π/d
MQW-SL
|
= 0.066π/d
MQW-SL
in the SL
formed by MQWs and GaAs spacers). (c) Calculated electric intensity field (red line) in the
MQW-SESAM structure given the incident pump wavelength of 1047.5 nm. The interface of air/cap
layer is at position 0. Beige areas denote GaAs. Violet areas denote In
0.27
Ga
0.73
As. Green areas denote
Al
0.95
Ga
0.05
As.
Figure 4.
(
a
) Coherent acoustic phonon oscillation from 70 ps to 800 ps when background subtraction is
optimized for low frequencies extraction. (
b
) FFT of (
a
) and dispersion relations in SL formed by DBR
with period d
DBR
= 160.4 nm as well as SL formed by MQWs and GaAs spacers with period d
MQW-SL
=
145 nm. Brown line represents FFT, where MBZ-center modes at 32.0 GHz, 63.1 GHz and 95.1 GHz and
MBZ-edge modes at 15.1 GHz, 78.3 GHz and 110.1 GHz are marked. Blue line represents dispersion
relation in the DBR (horizontal blue line represents k= |2
qDBR
probe
2
π
/d
DBR
| = 0.028
π
/d
DBR
in the
DBR). Magenta line represents dispersion relation in MQWs and GaAs spacers formed SL (horizontal
magenta line represents k= |2
qMQWSL
probe
2
π
/d
MQW-SL
| = 0.066
π
/d
MQW-SL
in the SL formed by MQWs
and GaAs spacers). (
c
) Calculated electric intensity ﬁeld (red line) in the MQW-SESAM structure given
the incident pump wavelength of 1047.5 nm. The interface of air/cap layer is at position 0. Beige areas
denote GaAs. Violet areas denote In0.27Ga0.73As. Green areas denote Al0.95Ga0.05 As.
3.2.2. MBZ-Center Mode Excitation
One potential mechanism for the photo-excitation of MBZ-center modes is electrostriction.
This mechanism can manifest itself in transparent media when, for example, two oblique pump
Appl. Sci. 2019,9, 289 11 of 22
laser beams are incident on the transparent medium to create an optical interference pattern inside the
medium for excitation of counter-propagating acoustic waves [
13
]. When the acoustic phonons are
excited through electrostriction, the energy is directly coupled from electro-magnetic ﬁelds of the laser
to the acoustic ﬁeld. The excited phonon momentum then matches the interference electric ﬁeld via the
momentum conservation law
kg=±
k1
k2
, where the subscript gdenotes generated acoustic
phonon and subscripts 1 and 2 denote the two incident photons. In the case of two laser beams with
the same frequency, the generated phonon wavelength
Λ
is matched to the light ﬁeld wavenumber by
kg=2qpumpsin(θ/2)
, where
kg
denotes the excitation (pump) light wavenumber in the medium and
θ
denotes the angle between the two pump beams. Therefore, the generated phonon wavevector is
θ
in such a conﬁguration. When two incident pump beams propagate in opposite
directions, the acoustic phonon exhibits a wavelength of λpump/2, corresponding to kg= 2qpump .
In our experiment, the incident pump beam is reﬂected by the DBR, whose reﬂectance is up to
99.7% for the pump wavelength of ~1050 nm. Multiple scattering of the incident light in the DBR
creates two evanescent counter-propagating light ﬁelds with comparable amplitudes, opening the way
for the generation of high frequency CLAPs in the DBR via the electrostriction mechanism. As shown
in Figure 4c, the pump light ﬁeld intensity distribution in the DBR contains oscillating and evanescent
average components covering nearly the whole DBR thickness. The sinusoidal component of the light
ﬁeld generates the phonons with
k=kg=2qDBR
pump
, where
qDBR
pump
denotes the wavevector of the pump
light in the DBR, which is modelled as an inﬁnite SL. However, the magnitude of the evaluated k
g
appears to be outside the ﬁrst Brillouin zone of the DBR, which can also be treated as a phononic crystal.
This means that the generation of the phononic crystal eigenmode by the sinusoidally modulated
part of the interference pattern created by our pump light should include Umklapp processes and the
wavenumber of the generated phonon is
k=k0g=
2qDBR
pump 2π/dDBR
=0.026π
dDBR
<< π
dDBR
, where d
DBR
denotes the DBR period. As a result, because in our DBR the wavenumber of the pump light is very
close to the MBZ boundary, in other words, half of the pump light wavelength is close to the DBR
period, the photo-generated phonon is very close to MBZ center. However, this is not the only channel
for the generation of the GHz CLAPs in the DBR. In the DBR, not only the periodically oscillating
part, but also the averaged part of the intensity distribution of the pump light interference pattern
can generate high frequency CLAPs due to the spatial periodic modulation of the electrostriction
parameter in the DBR. This mechanism is absent in spatially-homogeneous, non-structured media.
In principle, in this second channel, all k= 0 modes of the phononic DBR are photo-generated. From the
above analysis, it follows that, because in our DBR
λDBR
pump/
2 and d
DBR
are very close, the constructive
generation of the lowest energy
k
=
0 mode in the MBZ of the DBR via two described channels could
be possible, in spite of transparency of the DBR for the pump light. The local electrostriction force
f
can be presented as [43]
fes =1
2(ρde
dρE2), (5)
where
ρ
denotes the total mass per unit volume,
e
denotes the permittivity and
E
denotes the electric
ﬁeld. The right side of Equation (5) can be expanded into two terms:
fes =1
2ρde
dρE2+1
2E2(ρde
dρ). (6)
The ﬁrst term describes the force
fes1
stemming from the spatial modulation of the electric intensity
E2
;
the second term describes the force
fes2
caused by the spatial modulation of the electrostriction
coefﬁcient ρde
dρ. The combined effects described by Equation (6) cause the resonant excitation of MBZ
mode at
k
=
0 (fundamental ~32 GHz mode and its higher-order harmonics) in the DBR, given the
condition
λDBR
pump/2
d
DBR
in our sample. Thus, the peaks at 32.0 GHz, 63.1 GHz and 95.1 GHz in
Figure 4b can be attributed to MBZ-zone center modes, which are resonantly coherently excited by
Appl. Sci. 2019,9, 289 12 of 22
the light ﬁeld in the DBR via simultaneous spatial modulation of the electric ﬁeld intensity and of the
electrostriction coefﬁcient.
One may notice that amplitudes of the MBZ center modes do not show a monotonic decrease
with increasing order. On the contrary, the second order mode 63.1 GHz signiﬁcantly dominates in
the spectrum. The amplitude ratios among the three modes are approximately 0.2:1:0.05. In order to
explain this unusual behavior, we need to take two factors into account.
Firstly, as demonstrated in Section 3.1, the acoustic frequency comb excited in the In
0.27
Ga
0.73
As
QWs has multiple components equidistantly spaced by
f
33 GHz over a wide frequency range,
which means the ﬁrst three components are at the locations
f, 2
fand 3
f, respectively. Therefore,
the ﬁrst three comb components are approximately coincident with the MBZ center modes of the ﬁrst
three orders in the DBR. In consequence, the experimental frequency at around 63.1 GHz could acquire
more contribution from the photo-excited acoustic frequency comb in the QWs than the experimental
frequencies at around 32.0 GHz and 95.1 GHz do. This is due to the fact that in the lowest frequency
region, the side lobe of the envelope determined by the individual wave-packet FFT shows a peak
close to the comb component 2
f, while it exhibits dips close to the comb components
fand 3
f(see
Figure 2b). The combined frequency at around 63.1 GHz is thus enhanced greatly compared to the
pure second order MBZ-center mode in the DBR, which could partwise explain why the second order
MBZ-center mode dominates rather than the ﬁrst order MBZ-center mode (we cannot ﬁlter out the
pure contribution of MBZ center modes from the spectrum).
Secondly, the amplitudes of the spectral components of the signal depend not only on the
generation but also on the detection processes. While all the above considered three frequencies could
potentially be detected both in the MQWs spatial region and in the DBR spatial region, the lowest of
these frequencies could be additionally efﬁciently generated and detected in the parts of the samples
which are not spatially structured. In other words, the generation and detection of the phonons
around 32–33 GHz frequency could take place in spatially-homogeneous materials constituting our
samples, for example in spacers between the MQWs. The counter-propagating light ﬁelds of pump
laser in the MQW-SESAM structure could efﬁciently excite phonons at these frequencies by stimulated
Brillouin scattering (via the electrostriction mechanism due to intensity modulation in their interference
pattern, Figure 4c). Probe light could be efﬁciently scattered by the propagating acoustic waves at
these frequencies in our sample via the acousto-optic effect because the probe light and acoustic wave
vectors are matched via momentum conservation law in the backward Brillouin scattering process. We
found that in all materials of our sample, the so-called Brillouin frequencies of phonons, satisfying the
momentum conservation law which is essentially the same for the efﬁcient generation and the efﬁcient
detection, are very close to the MBZ center mode 32 GHz at
k
=
0 (31.4 GHz in GaAs, 30.8 GHz in
In
0.27
Ga
0.73
As and 31.8 GHz in Al
0.95
Ga
0.05
As). The listed Brillouin frequencies here and in Table 4
have been calculated using
fi=2nivi/λprobe
[
44
], where
ni
denotes the refractive index,
vi
denotes
the longitudinal sound velocity in sample constituent materials and
λprobe
denotes the probe light
wavelength in vacuum. Thus, the Brillouin frequencies could probably suppress the MBZ center mode
at 32 GHz due to destructive interference between them when the phase difference is close to
π
, which
could potentially explain why the ﬁrst-order mode has a weaker amplitude than the second order one.
3.2.3. MBZ-Edge Mode Excitation and Detection
We will now move on to the discussion of the unexpected occurrence of MBZ-edge modes
at 15.1 GHz, 78.3 GHz and 110.1 GHz (see Figure 4b marked by green solid dots). Concerning
our experimental observation of MBZ-edge modes, ﬁrstly, we discuss a scenario that momentum
conservation for edge mode could be fulﬁlled due to the absorption of the excess momentum by free
carriers in the doped SL sample, when |
π
/d
DBR
2q
probe
|
π
/L
QW
(L
QW
denotes the thickness of
QW layer in SL) [
18
]. In our structure, the DBR period d
DBR
= 160.4 nm is more than one order of
magnitude larger than the GaAs/AlAs SL period in Ref. [
18
], which causes the wavenumbers of the
probe photon and edge phonon being nearly equal q
probe
=π
/d
DBR
. If the refractive index of the
Appl. Sci. 2019,9, 289 13 of 22
DBR n
DBR
3.22 (from Table 2) is used for q
probe
, then it appears that q
probe
<
π
/d
DBR
and |
π
/d
DBR
2q
probe
|
=π
/d
DBR
. Even if the refractive index of GaAs n
GaAs
3.48 (from Table 2) is used for
estimating q
probe
, then it appears that q
probe
>
π
/d
DBR
, requiring Umklapp processes in light scattering.
So, the photons with wavenumber q’
probe
=q
probe
2
π
/d
DBR
could be scattered by the MBZ-edge
phonons with wavenumber k’ =
π
/d
DBR
, leading to the same momentum excess |k’
2q’
probe
|
=
π
/d
DBR
. Therefore, the momentum conservation law in our experiments could be potentially restituted.
However, our DBR is non-doped, and already possesses considerable thicknesses for the constituent
layers, which makes it unlikely to induce electron conﬁnement effects and signiﬁcant interactions
between the electrons in the neighboring GaAs layers required for absorbing excess momentum.
Consequently, the zone-edge phonons in our experiments should not be detected in the frame of the
hypothesis proposed in [
18
]. Secondly, we discuss the scenario that the observation of the zone-edge
modes could be attributed to disorder-induced partial breakdown of momentum conservation in
the semiconductor SL [
19
]. The roughness-induced layer thickness ﬂuctuations cause a distribution
of electronic conﬁnement energies; thus, Raman scattering could be enabled in individual wells by
conﬁned electrons. In this situation, the wavevector conservation along the growth direction is no
longer required. However, the precision in the fabrication of SLs has been considerably improved
during the last 25 years and our DBR constituent layers (73.7 nm and 86.7 nm) are too thick to consider
the inﬂuence of the atomic-scale thickness ﬂuctuations. Lastly, the detectability of zone-edge mode
could be possible in the case of Brillouin zone (BZ) folding of the BZ edge to the
Γ
-point in crystals
according to Refs. [
21
24
]. For example, in Ref. [
24
] multi-order phase-matching-free signals due to
BZ edge phonons with comparable intensities are observed in a TiO
2
crystal by CARS spectroscopy.
By the introduction of the dynamic BZ folding, originally Raman-inactive phonons at the opposing
X-points become observable, and can be well understood. We consider this scenario to be the most
plausible explanation for MBZ-edge modes observation in our experiments.
In the framework of the hypothesis proposed by Hanamura et al. [
24
], ﬁrstly, the lowest energy
k
=
0 Raman active modes are resonantly and coherently excited by two pump laser ﬁelds at different
frequencies, when the frequency difference
ω1ω2
is tuned to the phonon mode frequency at
ω
(0).
Compared to the above proposed hypothesis, the fundamental difference in our system lies in that
the resonant Raman active scattering takes place in
k
-space rather than
ω
-space. In our pump-probe
spectroscopy set-up, differing from two crossed incident pump light pulses with different frequencies
ω1
and
ω2
applied in CARS approaches, our incident pump light is perpendicular to the sample
surface, and it is nearly 100% reﬂected back by the DBR. Thus, these are counter-propagating light
ﬁelds that can excite zone-center phonon modes
ω
(0). In other words, the Raman-active
ω
(0) phonon
excitation is realized not by tuning pump pulses frequency difference to
ω
=
ω1ω2
=
ω
(0), but
by spatial modulation of the light ﬁeld intensity and electrostriction coefﬁcient in the DBR with
approximately equal periods of the SL and the light ﬁeld intensity distribution, d
DBR λDBR
pump/2
, as
previously demonstrated. In our experiments, the folded zone-center mode
ω
(0)
32 GHz and two
higher-order zone-center modes, 63.1 GHz and 95.1 GHz, are excited and observed due to the ﬁnite
spectral width of the ultrafast lasers pulse providing the required frequencies from the processes of
difference frequency generation within the spectral width of our pump light (see Figure 4b), which
paves the way for further MBZ-edge phonons excitation. Secondly, a pair of phonons with frequencies
ω
(0)/2
±ω
(
ωω(0)/2
) are excited due to subharmonics decay of
ω
(0) (
ω
(0)
(
ω
(0)/2 +
ω
)+(
ω
(0)/2
ω
), k= 0
(+
kmax
)+(
kmax
). In the DBR, the wavevector at MBZ edge is
given by k=k
max
=
π
/d
DBR
. Furthermore, owing to the high amplitude of the resonantly excited
coherent k= 0 mode, the process of decay can become stimulated, leading to the excitation of coherent
zone-edge modes with k
=±
kmax
,
ω
=ω(0)/2
=ω
k
max
. We illustrate this process in Figure 5.
As can be seen, a pair of phonons (
ω
(0)/2, +k
max
) and (
ω
(0)/2,
k
max
) at the opposite MBZ edges are
coherently excited by the MBZ-center mode
ω
(0) via decay processes. The analysis of the temporal
dynamics of spectral modes will later conﬁrm the plausibility of the subharmonic decay process
ω
(0)
Appl. Sci. 2019,9, 289 14 of 22
ω(0)/2 +ω(0)/2
. Thirdly, because the edge mode excited via stimulated subharmonic decay
process is coherent, it introduces a new periodic phonon grating whose period is twice as large as
the lattice period. The doubling of the phonon grating period is attributed to the above described
excitation of phonon pairs at opposite MBZ edges, which causes lattice vibrations with
π
phase shift
0.95
Ga
0.05
units along z-axis. As a consequence, a new zone folding is required
by the dynamically doubled grating. Thus, with respect to the middle of the mini Brillouin zone,
±
k
max
/2, the original zone-edge phonons are folded to near-center at
k
=
0,
ω
=ω(0)/2
where, being
Raman-active, they are observable by the probe light. Fourthly, the probe light scattering by dynamic
folded phonons with
k
=0, ω
=ω(0)/2
is a Raman-Nath-like process which enables the detection of
multiple higher-order harmonics of zone-edge modes with comparable amplitudes [23,24].
Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 22
ω(0) ω(0)/2 + ω(0)/2. Thirdly, because the edge mode excited via stimulated subharmonic decay
process is coherent, it introduces a new periodic phonon grating whose period is twice as large as
the lattice period. The doubling of the phonon grating period is attributed to the above described
excitation of phonon pairs at opposite MBZ edges, which causes lattice vibrations with π phase shift
0.95
Ga
0.05
units along z-axis. As a consequence, a new zone folding is required
by the dynamically doubled grating. Thus, with respect to the middle of the mini Brillouin zone,
k
max
/2, the original zone-edge phonons are folded to near-center at k 0, ω ω(0)/2 where, being
Raman-active, they are observable by the probe light. Fourthly, the probe light scattering by
dynamic folded phonons with k 0, ω ω(0)/2 is a Raman-Nath-like process which enables the
detection of multiple higher-order harmonics of zone-edge modes with comparable amplitudes
[23,24].
Figure 5. Illustration of zone folding induced by a pair of phonons on the opposite MBZ edges
forming phonon grating. A pair of phonons is excited by the subharmonic decay of mode ω(0). In the
DBR, at the MBZ edge wavevector is expressed as k = k
max
= π/d
DBR
. The mirror symmetry operation
with respect to k
max
/2 boundaries enables the MBZ-edge modes (ω(0)/2 + ω, +k
max
) and (ω(0)/2
ω, k
max
) to be folded into the MBZ-center k 0. In our situation, due to ω ω(0)/2, the difference
between ω(0)/2 + ω,ω(0)/2 and ω(0)/2 + ω are not given in the plot. We assume that all those modes
are ω(0)/2.
As a result of the above hypothesis, the MBZ-edge mode around 15 GHz at k = π/d
DBR
,
and its
higher-order replicas in the signal up to those at 78.3 GHz and 110.1 GHz with the same-order
amplitudes could be possibly detectable due to the subharmonic decay of the ω(0) 32 GHz mode,
subsequent dynamic zone folding, and a Raman-Nath-like probe light scattering process in our
experiments (see Figure 4b). So far, we have only discussed the core principle of detectability of
MBZ-edge modes; however, the actual processes are much more complicated.
As illustrated in Figure 6, basically, three steps are involved to create a quite complex MBZ
acoustic spectrum in the DBR. At first, a series of MBZ-center modes ω
i
(0) (i = 1, 2, 3, …,
approximately at 32, 64, 96 GHz, …) are excited due to the electrostriction mechanism, and those
modes can become detectable by stimulated Brillouin scattering or Raman scattering (see line 1 in
Figure 6). Subsequently, the stimulated subharmonic decay of MBZ-center modes ω
i
(0) (i = 1, 2, 3, …)
excites the edge modes ω
i
(0)/2 (i = 1, 2, 3, …, approximately at 16, 32, 48, 64, 80, 96, 112 GHz, …), and
then the edge modes become detectable at the zone center due to dynamical zone folding (see lines 2
and 3 in Figure 6). At last, the detection of dynamically-folded ω
i
(0)/2 (i = 1, 2, 3, …) mode at the zone
center is a Raman-Nath-like probe light scattering process, which means several series of
Raman-Nath-like phonon signals mω
i
(0)/2 (m and i = 1, 2, 3, …) can be present at the same time, for
example, mω
1
(0)/2 = 16m (16, 32, 48, 64, 80, 96, 112 GHz, …, see line 4 in Figure 6), mω
2
(0)/2 = 32m (32,
64, 96, …, see line 5 in Figure 5), mω
2
(0)/2 = 48m (48, 96, 144 GHz, … see line 6 in Figure 6), and so on.
As can be seen, in particular, the subharmonic decay of the first-order MBZ center mode ω
1
(0) = 32
GHz provides an explanation to the otherwise unexplainable ~15 GHz frequency. In terms of other
Figure 5.
Illustration of zone folding induced by a pair of phonons on the opposite MBZ edges forming
phonon grating. A pair of phonons is excited by the subharmonic decay of mode ω(0). In the DBR, at
the MBZ edge wavevector is expressed as k=k
max
=
π
/d
DBR
. The mirror symmetry operation with
respect to
±
k
max
/2 boundaries enables the MBZ-edge modes (
ω
(0)/2 +
ω
, +k
max
) and (
ω
(0)/2
ω
,
k
max
) to be folded into the MBZ-center k
=
0. In our situation, due to
ωω(0)/2
, the difference
between
ω
(0)/2 +
ω
,
ω
(0)/2 and
ω
(0)/2 +
ω
are not given in the plot. We assume that all those
modes are
=ω(0)/2.
As a result of the above hypothesis, the MBZ-edge mode around 15 GHz at k=
π
/d
DBR
, and
its higher-order replicas in the signal up to those at 78.3 GHz and 110.1 GHz with the same-order
amplitudes could be possibly detectable due to the subharmonic decay of the
ω
(0)
32 GHz mode,
subsequent dynamic zone folding, and a Raman-Nath-like probe light scattering process in our
experiments (see Figure 4b). So far, we have only discussed the core principle of detectability of
MBZ-edge modes; however, the actual processes are much more complicated.
As illustrated in Figure 6, basically, three steps are involved to create a quite complex MBZ acoustic
spectrum in the DBR. At ﬁrst, a series of MBZ-center modes
ωi
(0) (i= 1, 2, 3,
. . .
, approximately
at 32, 64, 96 GHz,
. . .
) are excited due to the electrostriction mechanism, and those modes can
become detectable by stimulated Brillouin scattering or Raman scattering (see line 1 in Figure 6).
Subsequently, the stimulated subharmonic decay of MBZ-center modes
ωi
(0) (i= 1, 2, 3,
. . .
) excites
the edge modes
ωi
(0)/2 (i= 1, 2, 3,
. . .
, approximately at 16, 32, 48, 64, 80, 96, 112 GHz,
. . .
), and
then the edge modes become detectable at the zone center due to dynamical zone folding (see lines
2 and 3 in Figure 6). At last, the detection of dynamically-folded
ωi
(0)/2 (i= 1, 2, 3,
. . .
) mode at
the zone center is a Raman-Nath-like probe light scattering process, which means several series of
Raman-Nath-like phonon signals m
ωi
(0)/2 (mand i= 1, 2, 3,
. . .
) can be present at the same time, for
example, m
ω1
(0)/2 = 16m(16, 32, 48, 64, 80, 96, 112 GHz,
. . .
, see line 4 in Figure 6), m
ω2
(0)/2 = 32m
(32, 64, 96,
. . .
, see line 5 in Figure 5), m
ω2
(0)/2 = 48m(48, 96, 144 GHz,
. . .
see line 6 in Figure 6),
and so on. As can be seen, in particular, the subharmonic decay of the ﬁrst-order MBZ center mode
Appl. Sci. 2019,9, 289 15 of 22
ω1
(0) = 32 GHz provides an explanation to the otherwise unexplainable ~15 GHz frequency. In terms
of other frequency components at
ωi
(0) (i= 1, 2, 3,
. . .
) and
ωi
(0)/2 (i= 2, 3, 4,
. . .
) detected in
our experiment, several sources can contribute compared to that of ~15 GHz mode. Based on the
demonstration in Figure 6, the frequency components at 32, 64 and 96 GHz consist of the MBZ-center
modes
ωi
(0) (i= 1 for 32 GHz, i= 2 for 64 GHz, i= 3 for 96 GHz) generated by electrostriction
and the modes generated via the subharmonic decay process at
ωi
(0)/2 (i= 2 for 32 GHz, i= 4 for
64 GHz, and i= 6 for 96 GHz), Subsequently, those modes become detectable through dynamic zone
folding and Raman-Nath processes. Those frequencies at 32, 64 and 96 GHz can also be high-order
replicas at m
ωi
(0)/2 provided by probe light Raman-Nath-like scattering process ((m,i) = (2,1) for
32 GHz, (m,i) = (4,1), (2,2) for 64 GHz, and (m,i) = (6,1), (3,2), (2,3) for 96 GHz). Due to the involvement
of MBZ-center modes
ωi
(0), the frequencies at approximately 32, 64 and 96 GHz can be observed
without the hypothesis of a subharmonic decay, dynamic zone folding and Raman-Nath-like detection.
In contrast, the frequencies at approximately 80, 112 GHz can only be observable by assumption of
the above-discussed hypothesis. As can be seen from Figure 6, the frequency components 80 GHz
and 160 GHz consist of the modes generated via the subharmonic decay process at
ωi
(0)/2 (i= 5 for
80 GHz and i= 7 for 160 GHz), and detected via dynamic zone folding and Raman-Nath processes.
Those frequencies at 32, 64 and 96 GHz can be present as high-order replicas at m
ωi
(0)/2 ((m,i) =
(5,1) for 80 GHz and (m,i) = (7,1) for 160 GHz) in the Raman-Nath-like probe light scattering process.
The previous discussion shows that the observed frequency spectrum is most probably formed by a
non-trivial interplay between different excitation and detection mechanisms in the DBR.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 15 of 22
frequency components at ω
i
(0) (i = 1, 2, 3, …) and ω
i
(0)/2 (i = 2, 3, 4, …) detected in our experiment,
several sources can contribute compared to that of ~15 GHz mode. Based on the demonstration in
Figure 6, the frequency components at 32, 64 and 96 GHz consist of the MBZ-center modes ω
i
(0) (i = 1
for 32 GHz, i = 2 for 64 GHz, i = 3 for 96 GHz) generated by electrostriction and the modes generated
via the subharmonic decay process at ω
i
(0)/2 (i = 2 for 32 GHz, i = 4 for 64 GHz, and i = 6 for 96 GHz),
Subsequently, those modes become detectable through dynamic zone folding and Raman-Nath
processes. Those frequencies at 32, 64 and 96 GHz can also be high-order replicas at mω
i
(0)/2
provided by probe light Raman-Nath-like scattering process ((m,i) = (2,1) for 32 GHz, (m,i) = (4,1),
(2,2) for 64 GHz, and (m,i) = (6,1), (3,2), (2,3) for 96 GHz). Due to the involvement of MBZ-center
modes ω
i
(0), the frequencies at approximately 32, 64 and 96 GHz can be observed without the
hypothesis of a subharmonic decay, dynamic zone folding and Raman-Nath-like detection. In
contrast, the frequencies at approximately 80, 112 GHz can only be observable by assumption of the
above-discussed hypothesis. As can be seen from Figure 6, the frequency components 80 GHz and
160 GHz consist of the modes generated via the subharmonic decay process at ω
i
(0)/2 (i = 5 for 80
GHz and i = 7 for 160 GHz), and detected via dynamic zone folding and Raman-Nath processes.
Those frequencies at 32, 64 and 96 GHz can be present as high-order replicas at mω
i
(0)/2 ((m,i) = (5,1)
for 80 GHz and (m,i) = (7,1) for 160 GHz) in the Raman-Nath-like probe light scattering process. The
previous discussion shows that the observed frequency spectrum is most probably formed by a
non-trivial interplay between different excitation and detection mechanisms in the DBR.
Figure 6. Illustration of MBZ coherent acoustic phonon distribution in the DBR. The plot
demonstrates MBZ-center modes ω
i
(0) distribution, the subharmonic decay of ω
i
(0)/2 and
subsequent new zone folding (MBZ edge is folded onto MBZ center), and the detectable
Raman-Nath-like (RN-like) phonon sequences mω
i
(0)/2, where m (m = 1, 2, 3, …) denotes the replicas
order of each RN-like sequence and i (i = 1, 2, 3, …) denotes the order of MBZ center mode. (Here, the
phonon bandgaps on MBZ center and edge are ignored for simplicity and values of frequency are
approximately regarded as n-times 16 GHz based on calculation, where n = 1, 2, 3, …).
In order to strengthen our main assumptions, starting from the next paragraph, some
arguments will be presented to provide strong indications that the 15 GHz mode is excited via a
stimulated subharmonic decay. Additionally, although we only discussed MBZ-center and -edge
modes in the DBR, the process could also potentially take place beyond the DBR in our sample due
to structure-related feature. In order to obtain a full view of the low frequency MBZ modes, Table 4
lists all the low frequencies that could be potentially detected in this sample including MBZ modes
in the DBR and MBZ modes in the SL formed by triple-QW stacks and the 112-nm spacers, and
Brillouin frequencies in all materials. The presence of a significant number of loosely-separated
frequencies enhances the complexity of our interpretation, and will be discussed in Appendix A.
Figure 6.
Illustration of MBZ coherent acoustic phonon distribution in the DBR. The plot demonstrates
MBZ-center modes
ωi
(0) distribution, the subharmonic decay of
ωi
(0)/2 and subsequent new zone
folding (MBZ edge is folded onto MBZ center), and the detectable Raman-Nath-like (RN-like) phonon
sequences m
ωi
(0)/2, where m(m= 1, 2, 3,
. . .
) denotes the replicas order of each RN-like sequence and
i(i= 1, 2, 3,
. . .
) denotes the order of MBZ center mode. (Here, the phonon bandgaps on MBZ center
and edge are ignored for simplicity and values of frequency are approximately regarded as n-times
16 GHz based on calculation, where n= 1, 2, 3, . . . ).
In order to strengthen our main assumptions, starting from the next paragraph, some arguments
will be presented to provide strong indications that the 15 GHz mode is excited via a stimulated
subharmonic decay. Additionally, although we only discussed MBZ-center and -edge modes in
the DBR, the process could also potentially take place beyond the DBR in our sample due to
structure-related feature. In order to obtain a full view of the low frequency MBZ modes, Table 4lists
all the low frequencies that could be potentially detected in this sample including MBZ modes in the
DBR and MBZ modes in the SL formed by triple-QW stacks and the 112-nm spacers, and Brillouin
frequencies in all materials. The presence of a signiﬁcant number of loosely-separated frequencies
enhances the complexity of our interpretation, and will be discussed in Appendix A.
Appl. Sci. 2019,9, 289 16 of 22
Table 4. Acoustic phonon spectrum in the low frequency region.
Experiment Calculation DBR [41,42]
(dDBR = 160.4 nm)
Calculation SL Formed by QW
Stacks and Spacers
(dMQW-SL = 145 nm)
Calculation
Brillouin
Frequency [44]
fc,fe
(GHz) B(GHz) fc,fe(GHz) f2k (GHz) (k= 2qprobe)
(0.028π/dDBR)fc,fe(GHz) f2k (GHz) k= 2qprobe
(0.06π/dMQW_SL)fBin GaAs (GHz)
15.1 2.1
k=π/dDBR
L 15.28
U 16.90
Lowest
0.65
k=π/dDBR
16.20
Lowest
1.02 31.4
32.0 2.2
k= 0
L 32.17
U 32.18
p=1 31.56
p= +1 32.53
q= 0
L 32.40
U 32.41
p=1 31.20
p= +1 33.54
fBin
In0.27Ga0.73 As
(GHz)
- -
k=π/dDBR
L 47.45
U 49.07
k=π/dDBR
48.61 30.8
63.1 2.2
k= 0
L 64.33
U 64.35
p=2 63.82
p= +2 64.68
k=0
L 64.808
U 64.814
p=2 63.67
p= +2 65.86
f
B
inAl
0.95
Ga
0.05
As
(GHz)
78.3 3.4
k=π/dDBR
L 79.62
U 81.24
k=π/dDBR
L 81.01
U 81.02
31.8
95.1 2.2
k= 0
L 96.50
U 96.53
p=3 95.97
p= +3 96.94 k= 097.22 p=3 96.18
p= +3 98.23
110.1 1.8
k=π/dSL
L 111.80
U 113.40
k=π/dSL
L 113.41
U 113.43
3.2.4. Indications of Stimulated Subharmonic Decay of MBZ-Center Mode
Our ﬁrst approach of providing an indication of the stimulated subharmonic decay of the mode
ω
(0)
32 GHz into the mode
ω
(0)/2
15 GHz consists in the evaluation of the dependencies of
the observed mode amplitudes on pump laser power. The pump power is varied from 43 mW to
67 mW in steps of 4 mW. In the considered interval of pump powers, the amplitudes of the evaluated
individual spectral lines are well above the noise level. A small time interval ranging from 500 ps
to 800 ps is chosen to capture the decaying center mode at 32 GHz. The acoustic signals and their
spectra are presented in Figure 7a,b. The dependencies of the 15 GHz, 32 GHz, 63 GHz, 80 GHz
and 95 GHz modes amplitudes on the pump power are presented in Figure 7c. Those modes are
normalized by their amplitudes at 43 mW power. This plot demonstrates the following tendency: the
decrease of the 32-GHz mode amplitude is accompanied by an increase of the amplitudes of several
other frequency components, which could be related to the detection of the zone-edge mode around
15 GHz frequency through a Raman-Nath-like probe light scattering process which accounts for the
presence of frequencies at m
ω
(0)/2 (m= 1, 2, 3,
. . .
). The experimental dependences in Figure 7c
are importantly nonlinear and multiple nonlinear processes could contribute to the details of these
dependencies, including the pump absorption saturation in the QWs. However, it is clear that the
decay in the amplitude of the 32 GHz mode is overall consistent with the hypothesis that its energy
loss via a nonlinear process of the stimulated subharmonic decay could provide a contribution to
the total dynamics of the ~15 GHz mode. In Figure 7d, the amplitude ratio between the 32 GHz and
15 GHz modes decreases from 1.5 to 0.5, which is clearly a sign that the mode ~32 GHz could play a
role in the excitation of the otherwise unexplainable mode at ~15 GHz.
Appl. Sci. 2019,9, 289 17 of 22
Appl. Sci. 2019, 9, x FOR PEER REVIEW 17 of 22
Figure 7. (a) Coherent acoustic phonon and (b) spectrum in the time interval from 500 ps to 800 ps at
the pump power from 43 mW (bottom) to 67 mW (top) in steps of 4 mW. (c) Normalized frequency
amplitude to the amplitude at lowest pump power for each frequency. (d) Frequency amplitude ratio
between 32 GHz center mode and 15 GHz edge mode.
Our second approach to find additional support for the hypothesis of the stimulated
subharmonic decay process is to monitor the time dependence of the ~15 GHz mode and compare it
to the other modes of interest, in particular, the mode at ~32 GHz. We analyze the low-frequency
part of the oscillations starting from the delay time of 400 ps, where the signal coming from the
detection of high-amplitude high-frequency combs in MQWs stops dominating (see Figure 2a). The
idea is to extract amplitudes of the individual spectral lines as a function of the time delay from a
short-time FFT result, shown in the right-hand-side of Figure 8a. The outcome, i.e., a rising
amplitude of the 15 GHz oscillation and simultaneously decaying amplitude of 32 GHz oscillation
over time delay, as plotted in Figure 8b, supports the argument that the excitation of the 15 GHz
edge mode could stem from the stimulated decay of the ~32 GHz center mode. The analysis of the
decay of the ~32 GHz mode amplitude presented in Figure 8b demonstrates that the attenuation of
the mode in the interval from 400 ps to about 450 ps is amplitude-dependent and diminishing with
time. Only in the time-delay interval after around 450 ps can the curve be fitted by an exponent with
a constant decay time. This is an additional indication of the nonlinearity of the processes leading to
the decay of ~32 GHz mode and the plausibility of the stimulated subharmonic emission of the ~15
GHz edge mode.
All the above-presented demonstrations (Figures 7 and 8) and arguments indicate that the
source of the MBZ-edge mode excitation is a near k = 0 eigenmode of the DBR, which, in turn,
provides evidence for monitoring folded CLAPs and edge modes in a long-period non-doped
transparent semiconductor SL.
In summary, up to three of the MBZ-center at k 0 and -edge modes at k = π/d
DBR
starting from
15 GHz to 110.1 GHz are excited and observed in the optically transparent DBR region of the
MQW-SESAM sample. A possible explanation for this unusual behavior is presented which assumes
the generation mechanism of the zone-center modes is electrostriction in the DBR. Stimulated
subharmonic decay of these zone-center modes then leads to the excitation of the zone edge modes
in the DBR, which can be detected because of dynamic zone folding and Raman-Nath-type light
scattering following the hypothesis. The nonlinearities revealed in the experimentally-observed
acoustic dynamics of the MQW-SESAM structure provide strong a indication of the dominant role of
the stimulated subharmonic decay processes in the detection of the zone-edge mode frequencies.
Figure 7.
(
a
) Coherent acoustic phonon and (
b
) spectrum in the time interval from 500 ps to 800 ps at
the pump power from 43 mW (bottom) to 67 mW (top) in steps of 4 mW. (
c
) Normalized frequency
amplitude to the amplitude at lowest pump power for each frequency. (
d
) Frequency amplitude ratio
between 32 GHz center mode and 15 GHz edge mode.
Our second approach to ﬁnd additional support for the hypothesis of the stimulated subharmonic
decay process is to monitor the time dependence of the ~15 GHz mode and compare it to the other
modes of interest, in particular, the mode at ~32 GHz. We analyze the low-frequency part of the
oscillations starting from the delay time of 400 ps, where the signal coming from the detection of
high-amplitude high-frequency combs in MQWs stops dominating (see Figure 2a). The idea is to
extract amplitudes of the individual spectral lines as a function of the time delay from a short-time FFT
result, shown in the right-hand-side of Figure 8a. The outcome, i.e., a rising amplitude of the 15 GHz
oscillation and simultaneously decaying amplitude of 32 GHz oscillation over time delay, as plotted
in Figure 8b, supports the argument that the excitation of the 15 GHz edge mode could stem from
the stimulated decay of the ~32 GHz center mode. The analysis of the decay of the ~32 GHz mode
amplitude presented in Figure 8b demonstrates that the attenuation of the mode in the interval from
400 ps to about 450 ps is amplitude-dependent and diminishing with time. Only in the time-delay
interval after around 450 ps can the curve be ﬁtted by an exponent with a constant decay time. This is
an additional indication of the nonlinearity of the processes leading to the decay of ~32 GHz mode
and the plausibility of the stimulated subharmonic emission of the ~15 GHz edge mode.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 18 of 22
Figure 8. (a) Left: FFT amplitude of the acoustic signal in the time interval from 400 ps to 800 ps for
the pump power of 67 mW. Right: short time FFT analysis by using a long moving window length.
(b) Dependences on time of 15 GHz, 32 GHz and 63 GHz mode amplitudes, which are obtained from
short time FFT image (see right-hand figure of (a)).
We have to bear in mind that the structure we investigated is complex, which potentially leads
to the overlap of multiple coherent acoustic phonon sources in the frequency domain and a possible
subsequent interplay between them such as enhancement or cancellation. For the time being, it is
hard to identify the contributions from different mechanisms of the CLAPs excitation and detection
in the individual spectral components of the signal. We discuss some of these peculiarities in
Appendix A, where one can not only gain a deeper understanding of potential acoustic mechanisms,
but can also find an explanation for important missing modes and additional modes in Figure 4b, as
well as the reason why only the edge mode of ~15 GHz excited via stimulated harmonic decay is
revealed through the nonlinearities in the experiment.
4. Conclusions
In summary, we have investigated a MQW-SESAM structure by femtosecond time-resolved
spectroscopy at a wavelength of ~1050 nm, and found a wide variety of coherent acoustic features. In
particular, we observed two distinct frequency regimes: one with its main frequency components
appearing around 365 GHz, and the other with frequencies ranging from ~15 GHz to ~100 GHz.
The high frequency components can be attributed to the coherent acoustic phonons generated
and detected in the In
0.27
Ga
0.73
As QWs of the sample, which also give rise to an acoustic frequency
comb with a spacing of ~33 GHz. The lower frequency components show the signature of SL
eigenmodes and a comparison with the calculated acoustic dispersion of the DBR superlattice yields
good agreement. This is surprising, because the DBR materials are transparent at the used pump
wavelength, and thus, no acoustic response is expected. In addition, several unexpected modes are
present in the acoustic spectrum, in particular, MBZ edge-modes which are usually not observable
due to optical detection selection rules.
We propose an explanation based on the excitation of CLAPs in the DBR by spatial
electrostriction coefficient modulation and the spatially modulated pump electric field. A
subsequent stimulated subharmonic decay of the MBZ-center modes is then able to excite the
MBZ-edge modes including the 15 GHz mode. The dynamical folding of the zone-edge mode onto
the zone center, together with Raman-Nath-type probe light scattering, can account for the
appearance of the MBZ-edge modes in the measured acoustic spectra. The observation of
simultaneous CLAPs in MQW region and the weak MBZ modes in the DBR is enabled by the
high-detection-sensitivity in our system (R/R~10
7
).
Our results not only provide a way to non-destructively characterize SESAM structures which
are extensively employed in the realization of mode-locked lasers, but also provide new insights into
the light-matter interaction in non-doped long-period superlattices formed by optically
non-absorbing materials.
Figure 8.
(
a
) Left: FFT amplitude of the acoustic signal in the time interval from 400 ps to 800 ps for
the pump power of 67 mW. Right: short time FFT analysis by using a long moving window length.
(
b
) Dependences on time of 15 GHz, 32 GHz and 63 GHz mode amplitudes, which are obtained from
short time FFT image (see right-hand ﬁgure of (a)).
Appl. Sci. 2019,9, 289 18 of 22
All the above-presented demonstrations (Figures 7and 8) and arguments indicate that the source
of the MBZ-edge mode excitation is a near k= 0 eigenmode of the DBR, which, in turn, provides
evidence for monitoring folded CLAPs and edge modes in a long-period non-doped transparent
semiconductor SL.
In summary, up to three of the MBZ-center at k0 and -edge modes at k=
π
/d
DBR
starting
from 15 GHz to 110.1 GHz are excited and observed in the optically transparent DBR region of
the MQW-SESAM sample. A possible explanation for this unusual behavior is presented which
assumes the generation mechanism of the zone-center modes is electrostriction in the DBR. Stimulated
subharmonic decay of these zone-center modes then leads to the excitation of the zone edge modes
in the DBR, which can be detected because of dynamic zone folding and Raman-Nath-type light
scattering following the hypothesis. The nonlinearities revealed in the experimentally-observed
acoustic dynamics of the MQW-SESAM structure provide strong a indication of the dominant role of
the stimulated subharmonic decay processes in the detection of the zone-edge mode frequencies.
We have to bear in mind that the structure we investigated is complex, which potentially leads
to the overlap of multiple coherent acoustic phonon sources in the frequency domain and a possible
subsequent interplay between them such as enhancement or cancellation. For the time being, it is hard
to identify the contributions from different mechanisms of the CLAPs excitation and detection in the
individual spectral components of the signal. We discuss some of these peculiarities in Appendix A,
where one can not only gain a deeper understanding of potential acoustic mechanisms, but can also
ﬁnd an explanation for important missing modes and additional modes in Figure 4b, as well as the
reason why only the edge mode of ~15 GHz excited via stimulated harmonic decay is revealed through
the nonlinearities in the experiment.
4. Conclusions
In summary, we have investigated a MQW-SESAM structure by femtosecond time-resolved
spectroscopy at a wavelength of ~1050 nm, and found a wide variety of coherent acoustic features.
In particular, we observed two distinct frequency regimes: one with its main frequency components
appearing around 365 GHz, and the other with frequencies ranging from ~15 GHz to ~100 GHz.
The high frequency components can be attributed to the coherent acoustic phonons generated and
detected in the In
0.27
Ga
0.73
As QWs of the sample, which also give rise to an acoustic frequency comb
with a spacing of ~33 GHz. The lower frequency components show the signature of SL eigenmodes and
a comparison with the calculated acoustic dispersion of the DBR superlattice yields good agreement.
This is surprising, because the DBR materials are transparent at the used pump wavelength, and thus,
no acoustic response is expected. In addition, several unexpected modes are present in the acoustic
spectrum, in particular, MBZ edge-modes which are usually not observable due to optical detection
selection rules.
We propose an explanation based on the excitation of CLAPs in the DBR by spatial electrostriction
coefﬁcient modulation and the spatially modulated pump electric ﬁeld. A subsequent stimulated
subharmonic decay of the MBZ-center modes is then able to excite the MBZ-edge modes including
the 15 GHz mode. The dynamical folding of the zone-edge mode onto the zone center, together with
Raman-Nath-type probe light scattering, can account for the appearance of the MBZ-edge modes in
the measured acoustic spectra. The observation of simultaneous CLAPs in MQW region and the weak
MBZ modes in the DBR is enabled by the high-detection-sensitivity in our system (R/R~107).
Our results not only provide a way to non-destructively characterize SESAM structures
which are extensively employed in the realization of mode-locked lasers, but also provide new
insights into the light-matter interaction in non-doped long-period superlattices formed by optically
non-absorbing materials.
Author Contributions:
Investigation, C.L.; Methodology, V.G.; Resources, E.D.; Software, C.L. and M.H.;
Supervision, T.D.; Visualization, C.L.; Writing original draft, C.L. and V.G.; Writing review & editing, V.G.,
T.D. and M.H.
Appl. Sci. 2019,9, 289 19 of 22
Funding:
This research is ﬁnancially supported by the Center for Applied Photonics at the University of Konstanz
(CAP-02). It is partially supported by the Deutsche Forschungsgemeinschaft (DFG) through SFB 767. It is also
supported by Chinese Scholarship Council (CSC).
Acknowledgments:
Changxiu Li is grateful to versatile experimental supports from Nico Krauß and Gerhard
Schäfer, and helpful discussions with Lukas Ebner and Oliver Kliebisch.
Conﬂicts of Interest: The authors declare no conﬂict of interest.
Appendix A. Complexity of Acoustic Spectrum
(1) The triple-QW stacks and the GaAs spacers between triple-QW stacks together are forming
a periodic structure with a period of d
MQW-SL
= 145 nm, which is rather close to the DBR period
d
DBR
= 160.4 nm. Therefore, resembling the MBZ-center modes excitation in the DBR, in principle the
observable acoustic frequencies MBZ-center mode
ω
(0) (32, 63, 95 GHz,
. . .
) could also be excited
in this long-period SL due to the electrostriction mechanism induced by the counter-propagating
light ﬁeld, where the triple-QW stacks and their spacers form a SL whose period also satisﬁes the
relation d
MQW-SL λMQWSL
pump /
2. However, due to the light absorption in QWs (In
0.27
Ga
0.73
As) of
this SL with a period of 145 nm, the zone center modes could also be potentially excited through
deformation potential. Meanwhile, resembling MBZ-edge modes excitation and detectability in the
DBR, the observable MBZ-edge modes (15, 78, 110 GHz,
. . .
) could also be excited through the
stimulated subharmonic decay of zone center mode generated in this 145-nm SL and be detectable
through dynamic zone folding (the presence of modes higher than 15 GHz could also be due to
higher-order replicas via Raman-Nath probe light scattering process). Interestingly, as illustrated by
the acoustic dispersion relation depicted by the magenta line for the 145-nm SL and the blue line for the
DBR in Figure 4b, if we take a close look at the MBZ-edge modes, the experimental peaks marked by
green solid dots are always closer to the lower frequencies at the forbidden gap edge in the DBR than
to the edge frequencies in the 145-nm SL (here forbidden gaps are very small), so the MBZ edge modes
(15, 78, 110 GHz,
. . .
) are more likely excited in the DBR than in the 145-nm SL. Furthermore, there are
two pronounced closely adjacent peaks with comparable amplitudes spaced by approximately 2 GHz
at ~78 GHz and also at ~110 GHz, which are consistent to the frequency difference between the lower
frequency at the forbidden gap edge in the DBR and the edge mode of 145-nm SL at the corresponding
order (as calculated in Table 4and illustrated in Figure 4b), which means the upper peaks ~80 GHz and
~112 GHz could be possibly assigned to edge modes excited in the 145-nm SL. However, MBZ-zone
center modes excited in DBR and those modes excited in 145-nm SL at the corresponding order are too
close to be distinguished in the experiment.
(2) Multiple contributions to each MBZ-center and edge frequency component in the DBR or in the
145-nm SL are involved by means of the subharmonic decay of center modes, dynamic zone folding and
Raman-Nath type detection processes. In terms of zone center frequency, there are potential three kinds
of contributions to the detected signals including zone center modes generated via electrostriction,
the subharmonic decay of zone center mode, and the higher replicas of the dynamically folded
lower frequencies through the Raman-Nath-like probe light scattering process. In terms of zone edge
frequency, their occurrences in the acoustic spectra requires the hypothesis of a subharmonic decay
of the zone center mode, dynamic zone folding and Raman-Nath-like probe light scattering process.
Except the case of the 15 GHz mode, all other frequencies equal to edge mode frequencies contain
simultaneous contributions from the subharmonic decay of the zone center mode and contributions
from higher-order replicas of dynamically folded lower frequencies through the Raman-Nath-like
probe light scattering process, which is why only the 15 GHz mode rather than the 80 GHz and
the 110 GHz modes excitation via stimulated subharmonic decay is demonstrated through a few
approaches in Section 3.2. Despite the theoretically predicted occurrence of a mode with a frequency
of ~48 GHz as demonstrated in Figure 6, this phonon mode is absent in our experiment. The absence
of ~48 GHz mode can be potentially attributed to the cancellation of contributions in the detection
Appl. Sci. 2019,9, 289 20 of 22
of the modes
ω3
(0)/2 via stimulated subharmonic decay and the presence of the third-order replicas
3ω1(0)/2 of 15 GHz mode in the probe.
(3) The resonances of individual layers in our structure can also be potentially observed
through the nonstationary process detection in time-domain spectroscopy, which are induced by the
propagating phonons generated in QWs (In
0.27
Ga
0.73
As) or in the DBR. In the experiment, the detected
~24 GHz (see Figure 4b) between MBZ-edge mode 15 GHz and -center mode 32 GHz can be attributed
to the half-wavelength resonant oscillations in 112-nm GaAs layers conﬁned between two QWs, which
are estimated to take place at ~ 21 GHz. Besides, the ~38 GHz mode (the closest peak on the right
side of 32 GHz in Figure 4b) could be a result of the resonance in Al
0.95
Ga
0.05
As layer with a thickness
of 68.7 nm (in front of the DBR) where the oscillation at ~40 GHz can take place. Additionally, the
resonances that could potentially exist in the individual layer of GaAs (73.7 nm) and Al
0.95
Ga
0.05
As
(86.7 nm) in the DBR are both close to 32 GHz, which are coincident with the ﬁrst-order MBZ-center
mode in the DBR. The above mentioned modes can be excited from time to time when the propagating
phonons are incident on the corresponding layers.
(4) Considering the frequencies of phonons that could be excited via stimulated Brillouin
backscattering processes in all constituent materials of the structure are close to 32 GHz mode in
our experiment. As a result, the lowest in energy of k
=
0 Raman-active modes in the stationary folded
MBZ could be additionally resonantly pumped both in the DBR and the 145-nm SL by the coherent
acoustic waves at the Brillouin frequency ω=ωBincident on them.
(5) The frequency comb spacing of the coherent acoustic phonons from the MQWs are very close
to the lowest MBZ-center mode, i.e.,
f
=ω
(0)
32 GHz, which hampers the determination of the
decay time and amplitude for the phonon contribution from the DBR.
(6) All DBR eigenmodes could in theory be excited also by the CLAPs generated in QWs through
light absorption when those CLAPs are incident on the DBR.
Despite the above discussed multitude of possible processes for excitation and detection of CLAPs
in our sample, the occurrence of MBZ-edge mode in our experiments can be well explained by the
hypothesis proposed in [2124] and is corroborated by the nonlinearities in the process of stimulated
subharmonic decay of the MBZ-center mode in our experiments.
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... Recently, our work also covered hybrid structures that consist of QW regions and an underlying superlattice [106,107]. To demonstrate that a free-running dual-comb laser is well-suited for such studies, we prepare an experimental pump-probe setup as shown in Fig. 4(a). ...
... Previous results of SESAM samples have revealed the complex acoustic wave excitation, propagation, and detection mechanisms in these structures [107,109]. Here, we will focus on the extraction of the structural and material properties to demonstrate the capabilities of the new measurement system. ...
... We will first turn to the discussion of the acoustic pulses. These originate from the QW region of the sample [107,110], which is corroborated by the time separation of subsequent pulses at 32.0 ± 0.2 ps. This is in very good agreement with the calculated propagation time from the middle of one QW to the middle of another nearby QW being 31.3 ...
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... Recently, our work also covered hybrid structures that consist of QW regions and an underlying superlattice [106,107]. To demonstrate that a free-running dual-comb laser is wellsuited for such studies, we prepare an experimental pump-probe setup as shown in Fig. 4(a). ...
... Previous results of SESAM samples have revealed the complex acoustic wave excitation, propagation, and detection mechanisms in these structures [107,109,110]. Here, we will focus on the extraction of the structural and material properties to demonstrate the capabilities of the new measurement system. ...
... We will first turn to the discussion of the acoustic pulses. These originate from the QW region of the sample [107,109], which is corroborated by the time separation of subsequent pulses at 32.0 ± 0.2 ps. This is in very good agreement with the calculated propagation time from the middle of one QW to the middle of another nearby QW being 31.3 ...
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We present a free-running 80-MHz dual-comb polarization-multiplexed solid-state laser which delivers 1.8 W of average power with 110-fs pulse duration per comb. With a high-sensitivity pump-probe setup, we apply this free-running dual-comb laser to picosecond ultrasonic measurements. The ultrasonic signatures in a semiconductor multi-quantum-well structure originating from the quantum wells and superlattice regions are revealed and discussed. We further demonstrate ultrasonic measurements on a thin-film metalized sample and compare these measurements to ones obtained with a pair of locked femtosecond lasers. Our data show that a free-running dual-comb laser is well-suited for picosecond ultrasonic measurements and thus it offers a significant reduction in complexity and cost for this widely adopted non-destructive testing technique.
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... As we will show in the following this does not hinder an already in depth understanding of the undergoing acoustic dynamics. Based on the structure of the triple-QW stack where the thickness of InGaAs is very close to that of the GaAs barrier d InGaAs ≈d barrier , and the wave-packet-like acoustic oscillations displayed in our earlier measurements [39,40], it is reasonable to assume that the stress wave generated in each triple-QW stack is a cosine wave enveloped by the profile of the stack. For simplicity, we will initially assume that the stack generation profile exhibits a Gaussian shape. ...
... where a is a constant, indicating the stress wave amplitude, v denotes the longitudinal acoustic velocity in the sample, d QW denotes the QW-to-QW distance. Because the sound velocities are nearly equal v InGaAs ≈v GaAs [40], for simplicity in the modelling, we assume that acoustic waves are travelling everywhere in the MQW structure at the speed of v = v GaAs . The standard deviation σ in the Gaussian formula can be given as ...
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... The bilayer h-BN/MoS 2 geometry used here provides a point of comparison with a similar transduction approach based on spatially-inhomogeneous optical absorption that has been used to generate a broadband frequency comb in GaAs/AlGaAs/InGaAs systems 36,37 . A distinct advantage offered by 2D materials is the higher level of spatial confinement, which is illustrated by comparing our 53 nm thick h-BN/MoS 2 stack versus the combined 407 nm thick multiple quantum well structures (plus 3.76 μm for the DBR reflector) in ref. 37 . ...
... The bilayer h-BN/MoS 2 geometry used here provides a point of comparison with a similar transduction approach based on spatially-inhomogeneous optical absorption that has been used to generate a broadband frequency comb in GaAs/AlGaAs/InGaAs systems 36,37 . A distinct advantage offered by 2D materials is the higher level of spatial confinement, which is illustrated by comparing our 53 nm thick h-BN/MoS 2 stack versus the combined 407 nm thick multiple quantum well structures (plus 3.76 μm for the DBR reflector) in ref. 37 . More importantly, given the wide frequency range of excited overtones, our measurement provides experimental evaluation for the frequency dependence of the longitudinal phonon lifetimes in h-BN (Fig. 3b inset), as the energy dissipation in our bilayer structure can be separated into individual contributions from h-BN and MoS 2 (see for example ref. 38 and Supplementary Note 4 with the elastic energy distribution for different vibrational modes in the bilayer shown in Supplementary Fig. 8, and energy partitioning coefficients listed in Supplementary Table 1). ...
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