# Chiral-sensitive frequency mixing in valley-excited two-dimensional semiconductors

Prachi Venkat, Leon Schlemmer, Amar Bharti, David Gauthier, Willem Boutu, Gopal Dixit

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Prachi Venkat, Leon Schlemmer, Amar Bharti, David Gauthier, Willem Boutu, et al., Chiral-sensitive frequency mixing in valley-excited two-dimensional semiconductors. Applied Physics Letters, 2026, 128 (2), pp.021102. 10.1063/5.0308260. cea-05464529

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# Chiral-sensitive frequency mixing in valley-excited two-dimensional semiconductors EP

Prachi Venkat ID; Leon Schlemmer ID; Amar Bharti ID; David Gauthier; Willem Boutu ID; Gopal Dixit ID

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Appl. Phys. Lett. 128, 021102 (2026)

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# Chiral-sensitive frequency mixing in valley-excited two-dimensional semiconductors EP

Cite as: Appl. Phys. Lett. **128**, 021102 (2026); doi: 10.1063/5.0308260

Submitted: 21 October 2025 · Accepted: 24 December 2025 ·

Published Online: 15 January 2026

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Prachi Venkat,<fnref n="1" /> ID Leon Schlemmer,<fnref n="2" />,<fnref n="3" /> ID Amar Bharti,<fnref n="1" /> ID David Gauthier,<fnref n="2" />,<fnref n="3" /> Willem Boutu,<fnref n="2" />,<fnref n="3" /> ID and Gopal Dixit<fnref n="1" />,<fnref n="4" />,⁵

## AFFILIATIONS

## ABSTRACT

Two-dimensional (2D) semiconductors endowed with valley degree of freedom offer potential applications in next-generation petahertz val- leytronics and photonics for emerging quantum technologies. We experimentally and theoretically uncover striking signatures of nonpertur- bative nonlinear optical response in valley-selective photodoped monolayer molybdenum disulfide, highlighting the interplay between strong-field dynamics and valley polarization. The interplay of valley-assisted excitation and circularly polarized probe leads to chiral- sensitive sum and difference frequency mixing of the pump and probe photons, appearing as sidebands in the high-harmonic spectra. Interestingly, the sideband's strength depends on the photodoped valley's chirality and the weaker pump's helicity. Additionally, a circular dichroic signal results from valley-selective photodoping when the pumps's helicity is changed. Sensitivity of the sideband's strength on the pump-probe delay bears an imprint of the temporal aspect of the electron-hole coherence in 2D semiconductors.

© 2026 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0308260

Motivated by successful synthesis of two-dimensional (2D) gra- phene, researchers proliferated various 2D materials with diverse nota- ble properties and applications.<fnref n="1" /> Special focus has been given to 2D transition-metal dichalcogenide semiconductors as these materials exhibit direct bandgap in the visible regime, which makes them ideal candidates to fabricate optoelectronics and photonics devices.<fnref n="2" />⁻<fnref n="4" /> In addition, these 2D semiconductors are potential candidates for quan- tum technologies as they possess valley degrees of freedom from their two degenerate valleys at the Brillouin zone's edge: K and K'.⁵,⁶ Valley- based devices are poised to revolutionize quantum devices and peta- hertz electronics as the valleys are capitalized to read, write, and process information at ambient conditions.⁷⁻⁹ Owing to the selection rules stemming from the Berry curvature, left- and right-handed circu- larly polarized light can selectively excite one of the two valleys.¹⁰,¹¹ Furthermore, selective valley-excitations can effectuate circular dichro- ism, Hall effect as well as magnetization without an external magnetic field in 2D molybdenum disulfide (MoS₂).¹²⁻¹⁴ In addition, 2D MoS₂ exhibits intriguing features in nonlinear optical processes in perturba- tive and nonperturbative regimes.¹⁵,¹⁶ The present work delves in an

interplay of the perturbative and nonperturbative nonlinear processes in valley-excited 2D MoS₂.

The availability of ultrashort intense laser pulses in mid-infrared regimes has allowed us to observe strong-field-driven higher-harmonics generation (HHG) in solids. Following the first observation of HHG in zinc oxide crystal,¹⁷ HHG shortly became the go-to method to probe electronic motion in solids on its intrinsic timescale.¹⁸ High-harmonic spectroscopy has been employed to gain new perspectives on nonequi- librium phenomena in solids,¹⁹⁻³⁷ including 2D materials.³⁸⁻⁶⁶ Recent years have witnessed incessant significant research activities in nonper- turbative HHG from 2D MoS₂.⁶⁷⁻⁷⁹ Reis and co-workers demonstrated that monolayer MoS₂ exhibits significantly higher efficiency of HHG compared to its bulk counterpart, and further revealed the emergence of anomalous even-order harmonics arising from the material's nonzero Berry curvature.⁶⁷ The reason behind the higher efficiency of HHG is attributed to the reduced scattering effects and direct bandgap in mono- layer MoS₂. It has been experimentally shown that the valley pseudospin in monolayer MoS₂ affects HHG significantly.⁶⁹ In addition, the role of intraband and interband dynamics as well as polarization-resolved

<footnote n="1">¹Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India</footnote>

<footnote n="2">²Université Paris-Saclay, CEA, LIDYL, 91191 Gif-sur-Yvette, France</footnote>

<footnote n="3">³CY Cergy Paris Université, CEA, LIDYL, 91191 Gif-sur-Yvette, France</footnote>

<footnote n="4">⁴Max-Born Institut, Max-Born Straße 2A, 12489 Berlin, Germany</footnote>

<footnote>a) Author to whom correspondence should be addressed: gdixit@phy.iitb.ac.in</footnote>
harmonics in MoS₂ have been discussed in Refs. 78, 80, and 81. These works have provided ways to tailor properties of the emitted harmonics and its underlying mechanism. Recently, a resonant linearly polarized pump was used to photodope MoS₂, which was driven by a nonresonant linear probe to generate harmonics.⁸²,⁸³ A delay of several hundreds of femtoseconds between pump and probe pulses leads to a suppression of harmonics, which is attributed to enhanced electron-hole decoherence time.⁸² Additionally, the incoherent photocarriers also boost the lower-order harmonics by enhancing the intraband current, which is otherwise suppressed by momentum relaxation.⁸⁴

This Letter endeavors to address nonperturbative HHG in MoS₂ in the valley-selective photodoped 2D MoS₂. A resonant circularly polarized pump produces a valley-selective excitation in MoS₂, which is subsequently interrogated by a nonresonant circularly polarized probe pulse as illustrated in Fig. 1. We unequivocally demonstrate an interplay of the perturbative and nonperturbative nonlinear processes, i.e., generation of sidebands of different orders from frequency mixing of the pump and probe pulses along with the main harmonics. In the present case, the sidebands' strength associated with frequency mixing is sensitive to pump-probe delay, which can be used as a marker to extract temporal aspect of the electron-hole coherence. Additionally, the sidebands' strength critically depends on the polarization of pump and probe pulses, and exhibits circular dichroism as one exchanges the helicity of the pump and probe pulses. Moreover, our work differs from the previous study of Nagai et al.⁷⁹ in regard to the resonant aspect of the pump pulse that generates significant excited populations. Furthermore, the pump and probe wavelengths used in our study are incommensurate.

We begin by experimentally studying the response of MoS₂ to an overlapping resonant pump and nonresonant probe. The sample is composed of monolayer flakes of MoS₂ grown by chemical vapor deposition (CVD) on sapphire and redeposited on a fused silica substrate Fig. 2(a). In our pump-probe setup, the probe pulse corresponds to the main laser beam centered around 1860 nm with a pulse duration of 30 fs FWHM. Intensities up to 0.5 TW/cm² are used to drive non-perturbative dynamics that result in HHG. The pump beam is generated via third harmonic generation of the 1800 nm laser in a BBO crystal. The resulting broad spectrum is spectrally filtered using 10 nm bandwidth optical filters. Three central wavelengths have been used: 635 nm (1.95 eV), 680 nm (1.8 eV), and 800 nm (1.55 eV). The first two wavelengths correspond to the B and A exciton resonances in 2D MoS₂,²,⁸⁵ whereas the 800 nm excitation lies well below the resonances. The temporal duration of the pump, measured by cross correlation with the probe, is approximately of 200 fs FWHM, varying slightly depending on the chosen central wavelength. The pump pulse is left- or right-handed circularly polarized (referred to, respectively, as

LCP and RCP) with a measured helicity higher than 84%. The pump intensity is in the range of a few GW/cm². At 635 and 680 nm, when the pump precedes the probe by hundreds of femtoseconds, we observe a significant reduction in the harmonic yield. In contrast, no change in harmonic yield is observed when pumping at 800 nm, compared to the case without the pump. This behavior is consistent with previous studies⁸²⁻⁸⁴ and is attributed to a significant number of carriers excited to the conduction band by the resonant pump. Although we have not directly verified whether the excitation is selective to a single valley, extensive literature¹⁰,¹¹ has established that resonant excitation ensures valley-selective excitation. When the pump and probe pulses overlap in time, we observe the generation of sidebands next to the HHG spectrum, as shown in Fig. 2(b). These experimental spectra are the result of subtraction of the spectra measured when the pump and the probe are focused on a flake and the spectra measured when the pump and the probe are focused next to a flake (bare substrate). When the pump and the probe pulses temporally overlap, the intensity of the harmonics is not strongly affected; however, the intensities of the observed sidebands depend on the pump and the probe's polarization. In our experimental configuration, resonant excitation with left or right circularly polarized light is expected to selectively populate the K or K' valleys, respectively. Selective photodoping of 2D MoS₂ creates coherent carrier populations, which are subsequently driven by the intense probe field to generate sidebands. These sidebands are only visible while the pump and probe pulses overlap temporally, within the temporal resolution allowed by our pump-probe setup. Sidebands are clearly visible when the pump is at resonance with either 635 nm or 680 nm. However, within our experimental conditions, no sidebands are visible with the pump centered at 800 nm.

The experimental results in Fig. 2 highlight the signature of valley selectivity in the form of sidebands in the emitted spectra. To further understand the intricacies of this interaction, the equation of motion within the density matrix framework is simulated to describe the interaction of the pump and probe pulses with 2D MoS₂.³⁸,⁸⁶ The electronic structure of monolayer MoS₂ is modeled using a tight-binding Hamiltonian, characterized by a bandgap of 1.8 eV and a lattice constant of 2.46 Å.⁸⁶ The laser-driven electronic motion in MoS₂ is numerically simulated by the following equation:

$$i\partial_t \rho_{nn'}^{\mathbf{k}}(t) = \mathbf{E}(t) \cdot \sum_{n'} \left[ \rho_{nn'}^{\mathbf{k}}(t) \mathbf{d}_{n'n}^{\mathbf{k}_i} - \mathbf{d}_{nn'}^{\mathbf{k}_i} \rho_{n'n}^{\mathbf{k}}(t) \right] \\ + \left( \varepsilon_{nn'}^{\mathbf{k}_i} + \frac{(1 - \delta_{nn'})}{T_2} \right) \rho_{nn'}^{\mathbf{k}}(t). \quad (1)$$

Here, $\rho_{nn'}$ is the density matrix element between energy bands n and n'. $\mathbf{d}_{nn'}^{\mathbf{k}_i}$ and $\varepsilon_{nn'}$ denote the dipole matrix element and the energy gap

<figure><img src="image_13.png" /><figcaption>FIG. 1. Schematic of frequency mixing from 2D MoS₂. A resonant circularly polarized pump (Ω) selectively excites electronic population in one valley, which is probed by a nonresonant circularly polarized (ω) pulse to generate higher-order harmonics along with side-peaks associated with sum and difference frequencies of the pump and probe photons.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**FIG. 2.** (a) Optical microscope image of the MoS₂ sample. The sample consists of many MoS₂ flakes fixed on a fused silica substrate. The pump and probe beams are focused such that they spatially overlap and fit inside one flake. (b) Experimentally measured spectrum captured in three different pump-probe conditions. Gray: only the right circularly polarized (RCP) probe. Red: temporal overlap between the pump and the probe, both are RCP. Blue: same as red but with the pump in left circular polarization (LCP). For this data set, the pump is centered at 680 nm. Sidebands resulting from sum-frequency generation are labeled SI and SII, respectively. In order to isolate the response of MoS₂, we have subtracted the contribution of the background from the total spectrum.</figcaption></figure>

between energy bands *n* and *n'*, respectively. The time-dependent crystal momentum is defined as $\mathbf{k}_t = \mathbf{k} + A(t)$, where $A(t)$ and $E(t)$ represent the vector potential and electric field of the laser pulse. To account for dissipative mechanisms, a phenomenological dephasing time of $T_2 = 4$ fs is introduced. The coupled differential equations are numerically solved using a fourth-order Runge-Kutta algorithm with a time step of 0.02 fs, along with a 300 × 300 k-point grid to discretize the Brillouin zone. The population in *n*th band is given by $n(t) = \int \rho_{nn}(\mathbf{k}, t)d\mathbf{k}$. The total time-dependent current at any k-point is estimated as $\mathbf{J}_k(t) = \sum_{nn'} \mathbf{P}_{nn'}(\mathbf{k}_t; t) \rho_{nn'}^k(t)$, where $\mathbf{P}_{nn'}^k = \langle n\mathbf{k} | \nabla_k \mathcal{H}(\mathbf{k}; t) | n'\mathbf{k} \rangle$ is momentum matrix element and $\mathcal{H}(\mathbf{k})$ is the tight-binding Hamiltonian.

In our simulations, both pump and probe pulses are right-handed circularly polarized throughout this paper unless stated otherwise. A right-handed circularly polarized pump pulse with wavelength of 685 nm, resonant with the minimum direct bandgap, and intensity $5 \times 10^9$ W/cm$^2$ yields 96% electronic excitation in the K valley, which is consistent with earlier works.$^{12,13}$ The valley-excited MoS$_2$ is probed by an intense nonresonant circularly polarized pulse having 1800 nm wavelength and intensity $3.8 \times 10^{11}$ W/cm$^2$. Those parameters are overall close to the experimental conditions. The nonresonant probe pulse leads competing pathways to excite both valleys, which results in K and K' valleys occupation up to ~70% and 30%, respectively. As the individual pump and probe pulses result in different valley populations, it is imperative to analyze the current generated by both the pulses.

**Figure 3(a)** presents the component-resolved current generated by individual pump and probe pulses. The current generated by the probe pulse dominates over the one from the pump pulse. The temporal profile of both components of the current are nearly identical in the case of the combined pump-probe pulse configuration and of the sole probe pulse, as visible from **Fig. 3(b)**. Thus, identical current profiles indicate that the subcycle electron dynamics and the resultant high-harmonic spectra would be mainly governed by the probe pulse in a combined pump-probe configuration. This conclusion also suggests that HHG is insensitive to any valley-selective excitation by the

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>**FIG. 3.** Component-resolved simulated current originating from pump, probe and pump-probe interactions. (a) x and y components of current due to the individual pump and probe pulses. (b) Comparison of the current's components stemming from an individual probe and combined pump-probe pulses. Here, both pump and probe pulses are ~100 fs long with right-handed circular polarization.</figcaption></figure>
resonant pump pulse. To address whether this assumption is correct or not, let us analyze the high-harmonic spectra in detail.

High-harmonic spectra of the valley-excited MoS₂ with the harmonics generated by the individual pump and probe pulses as a background are presented in Fig. 4. A circularly polarized probe pulse generates (3n±1) harmonic orders with n being a positive integer.⁸⁷ The absence of 3n harmonic orders is a consequence of the interaction of circularly polarized light with threefold symmetric MoS₂.⁸⁸,⁸⁹ Notably, the circular pump pulse is weak, which produces a fundamental and its feeble second harmonic, at wavelengths shifted with respect to the main harmonics from the probe field.

Valley-excited MoS₂ results in the generation of sidebands along with the main (3n±1) harmonics as evident from Fig. 4. A careful inspection reveals that the sidebands are positioned at the sum and difference of the frequencies of the pump (Ω) and probe (ω) fields. The spectra display three sidebands corresponding to sum-frequency generation (SFG), i.e., Ω + ω (SI), Ω + 2ω (SII), and Ω + 3ω (SIII). In addition, two sidebands associated with the difference-frequency generation (DFG), i.e., Ω − ω (DI) and Ω − 2ω (DII), are also present. The frequency mixing is linear with the pump field as the intensity of

<figure><img src="image_17.png" /><figcaption>**FIG. 4.** Simulated high-harmonic spectrum of 2D MoS₂ driven by a resonant pump and nonresonant probe pulses at zero and 20 fs pump-probe delay times. (a) Both pump and probe pulses are right-handed circularly polarized (RCP). (b) The pump pulse is LCP, whereas the probe pulse is right-handed circularly polarized (RCP). Harmonics originating from individual pump (green) and probe (gray) pulses are in the background. Sidebands resulting from sum-frequency generation are labeled SI, SII, and SIII, whereas DI and DII stand for the sidebands associated with difference-frequency generation.</figcaption></figure>

the resonant pump is relatively weak. However, the sidebands' inten-
sity varies nonlinearly with the probe intensity. This conclusively indi-
cates the non-perturbative nature of the observed frequency mixing
with respect to the probe pulse. The appearance of sidebands despite
the seemingly identical currents, generated by the sole probe and com-
bined pump-probe in Fig. 3, establishes that HHG is indeed sensitive
to valley-selective photodoping in 2D semiconductors. Sidebands
above the bandgap are dominated by interband dynamics, whereas
sidebands below the bandgap primarily originate from intraband
motion.

It should be noted that all the main characteristic features from
the experimental result in Fig. 2(b) are reproduced in the simulated
spectra from Fig. 4. The simulation adequately mirrors the decreased
generation efficiency of the fifth harmonic, as compared to the fourth
harmonic for a circularly polarized probe. Moreover, consistent with
experimental results, SII is more efficiently generated in the LCP-RCP
configuration than SI in the RCP-RCP configuration.

The sidebands' strength distinctively depends on the pump-
probe delay. We observe that a delay of roughly three probe optical
cycles does not affect the sidebands' strength significantly as visible
from Fig. 4. However, the strengths of the sidebands as well as har-
monics reduce drastically for a delay beyond eight optical cycles of
probe and completely disappear for a larger temporal delay. The sensi-
tivity of the sideband strength to longer temporal delays reflects elec-
tron-hole decoherence as a dephasing time of 4 fs is used. Note that
the reduction in the harmonic's yield for larger delay in the present
case is consistent with recent experiments.⁸²,⁸³ However, the sidebands
are sensitive to delays of the order of tens of femtoseconds. This behav-
ior of sidebands can be crucial in understanding electron-hole
coherence.

So far, we have focused on how the pump intensity and pump-
probe delay affect SFG and DFG in the photodoped MoS₂. It is also
important to explore how the helicity of the laser pulse impacts fre-
quency mixing as it is a key factor in preferential valley excitation.
Figure 4(a) shows two prominent SI and SIII sidebands in the case of
the right-handed pump and probe pulses, which changes as the helicity
of the pump changes. For left-handed circular pump, the SI and SIII
sidebands are significantly weaker, while the SII sideband is enhanced
as shown in Fig. 4(b). In addition, DI and DII sidebands for the
left-handed pump are evident, but DI is completely suppressed as the
helicity of the pump changes from left to right. The enhancement or
suppression of sidebands at different energies critically depends on the
helicity of the pump pulse, which is a signature of circular dichroism
in the photodoped MoS₂. Thus, left- and right-handed pump pulses
result in distinct DFG and SFG sidebands with different intensities. It
is known that both **K** and **K'** valleys in 2D semiconductors are con-
nected by time-reversal symmetry. This means that the pump and
probe pulses, both with left-hand polarization, yield identical side-
bands to the right-handed polarized pump and probe, which is con-
firmed by our simulation.

Before concluding this article, let us analyze the polarization
properties of the emitted harmonics and sidebands. By virtue of the
selection rules for threefold symmetric 2D materials, emitted (3n + 1)
and (3n − 1) harmonics should have the same and opposite helicities
as the probe pulse, respectively.⁸⁷,⁸⁸ Thus, first (second), fourth (fifth),
and seventh (eighth) harmonics exhibit the same (opposite) helicity as
the probe pulse, as evident from Fig. 5. First SI sideband is left-handed
<figure><img src="image_18.png" /><figcaption>**FIG. 5.** Helicity-resolved high-harmonic spectra from the simulations for (a) right-handed circularly polarized (RCP) pump and probe pulses. (b) The pump pulse is LCP, whereas the probe pulse is right-handed circularly polarized (RCP).</figcaption></figure>

polarized as it originates from addition of the two right-handed photons from the pump and probe pulses as shown in Fig. 5(a). Similarly, the addition of one pump and three probe photons results in SIII sideband with right-handed polarization.⁹⁰,⁹¹ On the other hand, the difference between one pump and two probe right-handed photons yields left-handed DII sideband [see Fig. 5(a)]. Additionally, all the DFG and SFG sidebands, DI, DII, and SII, are right-handed polarized as the helicity of the pump pulse reverses from right to left, which dominates the nature of sidebands [see Fig. 5(b)]. Thus, the helicity of the pump pulse not only suppresses or enhances the sidebands' strength but also controls their polarization attributes. Interestingly, previously observed sidebands in Floquet systems⁷⁹ also follow the same selection rules as in the present case where the excitation is resonant.

It is worth noting that frequency mixing is also observed when both the pump and probe pulses are linearly polarized. In this case, however, the linear pump excites both valleys identically, and therefore the properties of the generated sidebands cannot be tailored. Furthermore, no circular dichroism is detected when a linearly polarized pump and a circularly polarized probe are used. These findings demonstrate that valley-selective pre-excitation is a prerequisite for observing and controlling the sideband properties in photodoped MoS₂.

In summary, we have investigated the impact of valley-selective excitation on HHG driven by circular pulses. A resonant pump pulse

photodopes one of the valleys in MoS₂, which is subsequently probed using nonresonant intense circular pulses. It has been observed that the temporal profiles of the currents induced by a sole probe and combined pump-probe pulses are alike. Despite being similar currents, harmonic spectra corresponding to the valley-excited and pristine MoS₂ are drastically different. Valley-selective photodoping in MoS₂ generates higher-order harmonics of the circular probe pulse along with the sidebands corresponding to SFG and DFG of the pump and probe photons. The intensity of the pump and probe and the delay between them determine the sidebands' strength. Moreover, the helicity of the pump pulse critically influences the sidebands' polarization. Thus, the chirality of the valley-excited MoS₂ can be probed by circular dichroism, induced by circular pulses with opposite helicities. The polarization and intensity of both pump and probe can be tuned to obtain the desired properties of the sidebands associated with SFG and DFG in valley-excited MoS₂. Studying the generation and nature of frequency mixed sidebands can be useful in fine-tuning the experimental setups. The strength of the sidebands can indicate complete overlap of the pump and probe pulses. Additionally, the sidebands' strength as a function of the pump-probe delay can potentially be a novel method to determine the electron-hole coherence time in solids. The present work opens new avenues for light-based ultrafast valleytronics, optoelectronics, and quantum technologies capitalizing on the valley attributes of 2D semiconductors.

We acknowledge receiving support from the IRCC, IIT Bombay, through SCPP Project No. RD/0523-IOE00I0-112. P.V. acknowledges receiving financial support from the Anusandhan National Research Foundation (ANRF), India (NPDF Project No. PDF/2023/001102). L.S., D.G., and W.B. acknowledge receiving financial support from the French Agence Nationale de la Recherche through the ATTOCOM project (Grant No. ANR-21-CE30-0036) and from the European Commission through the OPTOLOGIC project (H2020-FET OPEN, Grant No. 899794).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Prachi Venkat:** Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). **Leon Schlemmer:** Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). **Amar Bharti:** Investigation (equal); Validation (equal). **David Gauthier:** Project administration (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). **Willem Boutu:** Conceptualization (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). **Gopal Dixit:** Conceptualization (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.
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