# Heteroepitaxial growth of high optical quality, wafer-scale van der Waals
heterostrucutres

Katarzyna Ludwiczak  [<EMAIL_ADDRESS>Aleksandra Krystyna Dąbrowska
[  Johannes Binder  [  Mateusz Tokarczyk  [  Jakub Iwański  [  Bogusława
Kurowska  [  Jakub Turczyński  [  Grzegorz Kowalski  [  Rafał Bożek  [  Roman
Stępniewski  [  Wojciech Pacuski  [  Andrzej Wysmołek  [

###### Abstract

Transition metal dichalcogenides (TMDs) are materials that can exhibit
intriguing optical properties like a change of the bandgap from indirect to
direct when being thinned down to a monolayer. Well-resolved narrow excitonic
resonances can be observed for such monolayers, however only for materials of
sufficient crystalline quality, so far mostly available in the form of
micrometer-sized flakes. A further significant improvement of optical and
electrical properties can be achieved by transferring the TMD on hexagonal
boron nitride (hBN). To exploit the full potential of TMDs in future
applications, epitaxial techniques have to be developed that not only allow to
grow large-scale, high-quality TMD monolayers, but allow to perform the growth
directly on large-scale epitaxial hBN. In this work we address this problem
and demonstrate that MoSe2 of high optical quality can be directly grown on
epitaxial hBN on an entire two-inch wafer. We developed a combined growth
theme for which hBN is first synthesized at high temperature by Metal Organic
Vapor Phase Epitaxy (MOVPE) and as a second step MoSe2 is deposited on top by
Molecular Beam Epitaxy (MBE) at much lower temperatures. We show that this
structure exhibits excellent optical properties, manifested by narrow
excitonic lines in the photoluminescence spectra. Moreover, the material is
homogeneous on the area of the whole two-inch wafer, with only $\pm 0.14$ meV
deviation of excitonic energy. Our mixed growth technique may guide the way
for future large-scale production of high quality TMD/hBN heterostructures.

###### keywords:

layered materials, transition metal dichalcogenides, epitaxy, Metalorganic
Vapour Phase Epitaxy, Molecular Beam Epitaxy, Raman spectroscopy

University of Warsaw] Faculty of Physics, University of Warsaw, ul. Pasteura
5, 02-093 Warsaw University of Warsaw] Faculty of Physics, University of
Warsaw, ul. Pasteura 5, 02-093 Warsaw University of Warsaw] Faculty of
Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw University of
Warsaw] Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093
Warsaw University of Warsaw] Faculty of Physics, University of Warsaw, ul.
Pasteura 5, 02-093 Warsaw Polish Academy of Sciences] Institute of Physics
Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw Polish Academy
of Sciences] Institute of Physics Polish Academy of Sciences, Al. Lotników
32/46, 02-668 Warsaw University of Warsaw] Faculty of Physics, University of
Warsaw, ul. Pasteura 5, 02-093 Warsaw University of Warsaw] Faculty of
Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw University of
Warsaw] Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093
Warsaw University of Warsaw] Faculty of Physics, University of Warsaw, ul.
Pasteura 5, 02-093 Warsaw University of Warsaw] Faculty of Physics, University
of Warsaw, ul. Pasteura 5, 02-093 Warsaw TMD, MOVPE, MBE

##  1 Introduction

Transition metal dichalcogenides (TMDs) - representatives of 2D layered
materials, are intensively studied as promising candidates for future
realizations of optoelectronic devices 1, 2, photodetectors 3, 4, 5, sensors
6, 7, 8, energy and memory storages 9, 10, 11 or transistors 12, 13, 14. Such
realizations are commonly demonstrated on micrometer-sized flakes obtained by
mechanical exfoliation from bulk crystals and consecutively stacked to a
heterostructure by time-consuming deterministic transfer processes 15, 1, 16,
17, 18. This approach allows to produce high quality samples in terms of
electronic properties such as carrier mobility or conductivity 19.

Two-dimensional materials are, however, extremely sensitive to interlayer
interactions and prone to environmental factors. Thus, it is important to
deposit these materials on flat substrates with homogeneous dielectric
properties and without dangling bonds. Hexagonal boron nitride (hBN) appears
to be the perfect candidate for this purpose. The encapsulation in thin layers
of hBN can significantly improve optical properties of the flakes and prolong
their lifetime 20, 21, 22, 23, 24.
The next necessary step towards the practical utilization of the properties of
TMDs requires however, large-area sample synthesis. Significant efforts have
been made to demonstrate wafer-scale growth 25, 26. Approaches based on CVD
(chemical vapour deposition) and PVD (physical vapour deposition) techniques
result in promising large area monolayers 27, 28, 29, 30, 31, 32, but
excellent optical quality could be only achieved after additional post-growth
processing - by a sequence of transfer and encapsulation 33, 34.
Epitaxial method like metalorganic vapour phase epitaxy (MOVPE) was also used
to address the TMD scalability problem 35 and it has been shown that it is
possible to produce wafer-scale materials, however so far not with the same
optical properties. Molecular beam epitaxy (MBE) appears to be a promising
candidate for the development of large-area samples, as it allows to produce
high purity materials with atomic precision 36, 37, 38, 39, 40, 41, 42.
However, the optical quality of the samples is still far behind the best
results obtained for mechanically exfoliated flakes. A way to overcome this
obstacle is to combine TMD layers with hBN. Successful realizations of this
approach such as MoSe2 growth by MBE or CVD on initially exfoliated hBN flakes
43, 44 or an entirely grown hBN/MoS2 heterostructure by CVD 45 have been
already presented.
Here, we present a method that combines two epitaxial techniques: MBE and
MOVPE to provide large-scale crystals of high optical quality. We first use
MOVPE to grow hBN layers of a few nanometres thickness on a two-inch sapphire
wafer 46. Subsequently, we use this sapphire/hBN wafer as a substrate to grow
a monolayer of MoSe2 by MBE. A schematic illustration of the growth sequence
is presented in Figure 1.

Figure 1: Schematic illustration of the combined growth approach. A two-inch
sapphire wafer is used as a substrate for MOVPE hBN and MBE MoSe2 growth.

The quality of the TMDs is revealed by performing optical measurements such as
photoluminescence and Raman spectroscopy. In particular, photoluminescence is
a very sensitive probe of layer thickness, because materials like MoS2, MoSe2,
WS2, WSe2 and MoTe2 exhibit an indirect–direct band gap transition, which
results in a bright and intense photoluminescence from monolayers, in contrast
to photoluminescence from thicker samples 47, 48, 49, 50.

Low-temperature photoluminescence spectra are a suitable indicator of the
optical quality of MoSe2, as excitonic lines can be resolved into two
components, corresponding to the neutral A exciton and trion, only for high-
quality samples 51. In our work, we show that the grown material is of
excellent, homogeneous optical quality, confirmed by Raman spectroscopy and
photoluminescence mapping of the two-inch wafer. We demonstrate the unique
result of well-resolved excitonic lines at low temperatures on the whole
wafer. The proposed approach provides a reliable template for the growth of
high optical performance TMD layers by utilizing large area epitaxial hBN as a
substrate. Our method of combined MBE–MOVPE growth may pave the way for future
applications of van der Waals heterostructures on the wafer scale.

##  2 Results and discussion

###  2.1 Growth and characterization of hBN/MoSe2 heterostructures

Hexagonal boron nitride (hBN) layers were grown on 2-inch sapphire wafers by
MOVPE. The properties of the obtained hBN layers depend on a set of growth
parameters including growth time, temperature, III-V ratio or the employed
growth mode 52. One of the key parameters in designing van der Waals
heterostructures is the thickness of the hBN spacers. In this work we study
the impact of hBN layer thickness on the properties of the subsequently grown
MoSe2 monolayer. To be able to obtain hBN layers of different thicknesses we
employed two different growth modes: Continuous Flow Growth (CFG) and a two
stage epitaxy to obtain three different hBN samples. CFG growth mode yields
ultrathin high quality hBN layers, but shows a self-limiting behaviour 53.
Therefore, this mode cannot be used to obtain samples thicker than a few
nanometres. The grown material shows good optical and structural properties
with thicknesses ranging from about $1$ nm to $6$ nm.

To overcome the obstacle of limited thicknesses we employ the recently
introduced two stage epitaxial growth 46, which is a two-step mode for which
first a high-quality CFG layer is grown followed by a pulsed growth step that
requires an alternate switching of ammonia and TEB flows. With this mode it
becomes possible to grow thicker hBN layers of exceptionally good structural
quality (samples hBN1, hBN2) 46. Samples grown using two stage epitaxy can
reach thicknesses from several to tens of nanometers with a lattice constant
close to the theoretical value of 3.33 Å 46. The growth parameters of the
studied samples are summarized in Table 1.

Table 1: MOVPE and MBE growth parametres of the samples presented in this
study. In each MBE process about 1 ML MoSe2 is depostied, but the growth rate
is 3 times slower in case of the first process (A). MOVPE
---
sample | thickness | growth mode
hBN1 | 13.6 nm | two stage epitaxy
hBN2 | 3.5 nm | two stage epitaxy
hBN3 | 1.5 nm | CFG
MBE
sample | time of growth
MoSe2A | 15 h
MoSe2B | 5 h
MoSe2C | 5x1 h

Figure 2 presents SEM and AFM images showing the morphology of the thickest
(hBN1) and thinnest (hBN3) samples. The MOVPE growth of hBN at high
temperature is unavoidably connected to the appearance of characteristic
wrinkles on the material. Their presence can be attributed to the post-growth
cooling process and different thermal expansion coefficients of sapphire and
hBN. This wrinkle pattern shows a larger mesh size in the case of thicker
samples. Other typical objects on the surface of the material are three-
dimensional precipitates. Their appearance is most probably the result of the
formation of out-of-plane nucleation centres. There is a strong dependence of
the precipitates size. The thinner the sample, the smaller the three-
dimensional precipitates.

To measure the layer thickness and to further characterize the hBN, we
performed infrared reflectance spectra measurements using a FTIR microscope
and TEM imaging (see supporting information Figure S1). By analysing the
spectra within the Dynamic Dielectric Function approximation we were able to
determine the thicknesses of the hBN layers (Figure 2(e)) 46. The
characteristic peak in the FTIR spectra corresponds to the E1u mode of hBN
around $1367$ cm-1 confirming the formation of a sp2-BN layer 54.

Figure 2: Characterization of hBN substrates obtained in different growth
modes. SEM images of (a) hBN1, (b) hBN3; AFM images of (c) hBN1, (d) hBN3 show
the topography of the samples – the thinner the hBN layer, the smaller the
precipitations and wrinkles. (e) FTIR spectra for hBN samples: E1u mode with
corresponding energy and thickness. (f) Results of a $2\theta$/$\omega$ scan
of the samples with extracted interplanar spacing. (g) Low-temperature
photoluminescence spectra obtained for various hBN substrates and MBE growth
processes.

X-ray diffraction (XRD) was used to investigate the structural quality of the
samples. The XRD results are presented in Figure 2(f). The peak position of
around $26.3^{\circ}$ for sample hBN1 corresponds to hBN with some
turbostratic addition. The peak observed for the thinner sample is less
intense and broader, and its angular position of $24.7^{\circ}$ suggest mostly
turbostratic stacking 55.

The hBN samples with various thickness were used as substrates for the
consecutive growth of MoSe2 by MBE. Various growth processes were performed to
optimize monolayer formation. We studied the effect of different growth times
(growth rates) as well as using one or several annealing steps.

Figure 2(g) depicts low temperature photoluminescence spectra for MoSe2 layers
grown with various MBE processes (MoSe2A, MoSe2B, MoSe2C) and on various
epitaxial hBN substrates (hBN1, hBN2, hBN3). At low temperatures, the
photoluminescence spectrum of MoSe2 unveils additional information about the
optical quality of the material. For high-quality samples, the excitonic line
separates into two well-resolved peaks corresponding to the A exciton and
charged exciton (trion).

Such spectra were mostly observed for mechanically exfoliated material onto
selected substrates, 20, 56, 57, 58 but recently high optical quality samples
were also obtained for MBE grown MoSe2 on exfoliated hBN 44.

The analysis of low-temperature PL spectra of different samples (Figure 2(g))
allows us to optimize the process of heteroepitaxial growth. The growth
parameters for the MoSe2A sample were adopted from previous experiments on
exfoliated hBN 44. This starting point of parameters resulted in a broad
spectrum, which can indicate the formation of more than a single atomic layer.
The reduction of the growth time for sample MoSe2B resulted in a notable
improvement of the optical quality of the material and a visible separation of
excitonic lines into two peaks (A exciton and trion). A further introduction
of several annealing steps into the process resulted in an even clearer
separation of the exciton and trion peaks for sample MoSe2C. This growth
process was performed on three different hBN substrates (hBN1, hBN2, hBN3)
during the same growth run. It can be seen that the thinner the hBN layer, the
higher the optical quality of the material.

The inset in Figure 3(a) shows a typical spectrum obtained for MoSe2C
processes grown on hBN3 MOVPE samples in a larger spectral range. The
additional background stems from the photoluminescence of defects in the hBN
layer. The peaks around $1.7-1.8$ eV correspond to the presence of chromium in
the sapphire substrates 59. Figure 3(a) depicts two distinct peaks typical for
MoSe2 corresponding to the neutral A exciton and trion with the subtracted
background. Peaks are visible at energies of $1655$ meV and $1627$ meV,
respectively. These results correspond well with the excitonic complexes
reported in previous works 56, 60, 20, 61. The peak width for the A exciton
and trion is $22$ meV, which is significantly smaller than the peak width
measured for the MoSe2 grown on a SiO2 substrate 3(b) and only a factor of two
larger than for MoSe2 grown on flakes exfoliated from high-quality bulk hBN
44.

Figure 3: Low temperature (5 K) photoluminescence spectra of MoSe2 grown by
MBE on a MOVPE grown hBN compared to material grown by the same MBE process
directly on a silicone dioxide substrate. (a) Photoluminescence spectrum of
MoSe2 grown using optimized MBE and MOVPE processes. The lines of the neutral
A exciton (X${}_{A}^{0}$) and charged exciton (XC) can be resolved. Blue dots
show the data, while red and green curves show results of Lorentzian fits.
Inset – photoluminescence spectrum measured in a larger spectral range showing
an additional background which emerges from the defects in hBN. (b)
photoluminescence spectrum obtained for MBE MoSe2 grown on a SiO2 substrate.
Excitonic lines are not resolved and the line width of the neutral exciton
around $1654$ meV is significantly larger (FWHM $52$ meV) as compared to the
material grown on MOVPE hBN.

The temperature dependence of the excitonic resonances observed in our samples
is typical for monolayer MoSe2. Figure 4(a) depicts the evolution of the
photoluminescence of the neutral exciton A and trion as a function of
temperature (laser excitation wavelength $\lambda=532$). At low temperatures
we observe a binding energy of about $30$ meV. The trion signal abruptly
decays at around $100$ K. Both peaks shift towards lower energies as the
temperature rises, due to the reduction of the band gap, shown in Figure
4(b)56, 62. The blue and green lines are not fits to our data, but show the
typical band-gap energy dependence with parameters taken from 56 indicating
that our material indeed behaves like typical MoSe2 without significant
influence of stress due to the interaction with hBN layer.

Figure 4: (a) Temperature dependence of the photoluminescence of a MoSe2/hBN
sample. The spectra were shifted vertically for clarity. At T=$10$ K the
spectra consists of both, the charged and neutral excitonic lines. The charged
exciton contribution disappears at around $100$ K. (b) Temperature dependence
of the energy of the charged and neutral exciton. The blue and green curves
present a typical temperature dependence of the excitonic lines of MoSe2. The
parameters of the curves were taken from 56.

###  2.2 Wafer-scale growth of a hBN/MoSe2 heterostrucutre

To further prove that our method can be scaled up, we chose the best MOVPE and
MBE processes and grew the layers on an entire two-inch sapphire substrate.
Figure 5(a) presents a picture of the 2-inch wafer after hBN and MoSe2 growth.
The material can be observed with the naked eye. The MoSe2 layer covers the
sample in a circular, visibly darker, brownish area. The transparent,
uncovered outer ring is the result of applying a frame that holds the
substrate in the MBE chamber. It also allows to examine the properties of hBN
after the MBE growth.

The sample was further studied using Raman spectroscopy. Figure 5(b) presents
the outcome of room temperature Raman spectroscopy mapping of the whole wafer.
The out-of-plane A1g mode of MoSe2 at $241$ cm-1 was chosen for the map
analysis, as it is more intense than the in-plane E2g peak (Figure 5 (c)). The
presented map confirms that the material indeed uniformly covers the whole
area of the sample. Raman spectroscopy is also an excellent tool to assess the
number of MoSe2 layers 49, 63. As compared to the bulk material the out-of-
plane mode softens (redshifts) while the in-plane mode stiffens (blueshifts)
with decreasing thickness. A typical Raman spectrum for this sample is
presented in Figure 5(c). The highlighted Raman peak positions indicate the
presence of a single layer of MoSe2. While the peak energy can also vary due
to different substrates and exfoliation/growth techniques 64, 65, the mean
value of the A1g peak energy of $E_{A_{1g}}=241.06$ cm-1 for the whole area of
the sample with a standard deviation as small as $\sigma_{E}=0.85$ cm-1 indeed
indicates the formation of a monolayer material. The distribution of the peak
energy quantitatively depicted in Figure 5(d) additionally shows the
homogeneity of the material.

Figure 5: (a) Picture of an epitaxially grown two-inch hBN/MoSe2 van der Waals
heterostructure. The visibly darker reddish color indicates MoSe2 coverage.
(b) Peak energy of the A1g mode obrained by Raman spectroscopy mapping at room
temperature. (c) Typical Raman spectrum of the two characteristic MoSe2 peaks.
(d) Distribution of the peak energy with a mean value of $E_{A_{1g}}=241.06$
cm-1 and standard deviation $\sigma_{E}=0.85$ cm-1. The results from the Raman
map analysis show that the whole area of the sample is uniformly covered by a
single layer of the MoSe2.

Further evidence for the presence of monolayer MoSe2 can be provided by
photoluminescence mapping. The bandgap becomes direct in the monolayer limit
as there are no more vdW interlayer interactions 66 rendering
photoluminescence a very sensitive tool with regard to thickness fluctuations.
At room temperature, such a spectrum consists of one distinct peak
corresponding to the neutral A exciton with an energy of $1.57$ eV for
monolayer and $1.54$ eV for bilayer MoSe2 49. Figure 6(a)-(d) depicts the
results of PL mapping of the whole wafer at room temperature. The extracted
peak width (Figure 6 (a)) is uniform across the sample with a mean value of
$68\pm 6$ meV, which is very low for spectra obtained at room temperature 56.
The energy of the excitonic line does not vary much between different
positions on the sample. Its mean value of $1569.1\pm 2.2$ meV corresponds
perfectly to previously reported results for monolayer MoSe2 49. Another
important factor in determining the thickness of the material is the
photoluminescence intensity. It decreases usually by one order of magnitude
with every additional layer. In our case, the intensity is homogeneous over
the whole area of the sample (Figure 6 (c)). Transmission Electron Microscope
(TEM) characterization of the final sample is in agreement with the optical
studies indicating the formation of a MoSe2 monolayer (see supporting
information Figure S1). Further characterization of the final sample
morphology is provided by AFM (see supporting information Figure S2).

Figure 6: Photoluminescence mapping at room temperature ($500$ $\mu$m step,
$633$ nm laser). Panels (a),(b),(c) show results of Lorentzian fits to the
obtained spectra. A peak width of $67.9\pm 6.1$ meV and an energy of
$1569.1\pm 2.1$ meV for the excitonic line were obtained. The low standard
deviation indicates sample homogeneity across the whole two-inch wafer. (d)
Histograms of the excitonic line energy (top) and width (bottom).

Figure 7(a)-(d) depicts the distributions of the peak energy of the A exciton
and trion as well as their widths acquired from mapping the sample at liquid
helium temperature. Due to the limited size of the cryostat window, we
measured $100$ spectra uniformly along a straight line ($1$ cm). A mean energy
of $1655.43\pm 0.14$ meV and $1626.49\pm 0.17$ meV was extracted for the peaks
corresponding to the A exciton and trion, which confirms the large homogeneity
of the sample. This aspect is especially important for any future industrial
application of TMDs requiring reproducible results, such as a well-defined
band gap energy.

Figure 7: Photoluminescence peaks statistics at T= $5$ K. The spectra were
measured along a line of 1 cm. The histograms show the result of Lorentzian
fits to the neutral (X${}^{0}_{A}$) and charged exciton (XC). (a) distribution
of the energy of X${}^{0}_{A}$. (b) Distribution of peak width of
X${}^{0}_{A}$. (c) Distribution of the energy of XC. (d) Distribution of peak
width of XC.

##  3 Conclusions

We present a method to produce wafer-scale monolayer TMD samples of high
optical quality. Our method combines two epitaxial growth methods: MOVPE,
which enables us to grow hBN on a sapphire substrate and MBE which was used to
cover the so-grown hBN layer with a monolayer of MoSe2. A thorough analysis of
the optical properties examined by photoluminescence and Raman spectroscpy
mapping reveals a large homogeneity of the epitaxially grown van der Waals
heterostructure on the whole two-inch wafer. We also demonstrate that well-
resolved and narrow excitonic peaks at low temperature can be achieved even
for such large heteroepitaxial samples. The uniform energy distributions with
small standard deviations indicate an excellent homogeneity and repeatability,
which is very difficult to achieve with other techniques like mechanical
exfoliation. Our results constitute a large step towards a possible industrial
implementation of van der Waals heterostructures by enabling wafer-scale
production with precisely defined properties.

##  4 Experimental section

###  4.1 hBN growth by MOVPE

An Aixtron Close Coupled Showerhead 3×2” system with ARGUS thermal mapping
device was used to grow hBN samples. 2” sapphire wafers purchased from
ProCrystal were used as a substrate. Triethylboron (TEB) and ammonia (NH3)
were used as boron and nitrogen precursors, with H2 as carrier gas. Two growth
modes were applied to produce the samples discussed in this work:
\- Continuous Flow Growth (sample hBN3): precursors are simultaneously
injected to the reactor at the same time at high temperatures around
$1300^{\circ}$C with supersaturation of ammonia 52, 46.Such a mode exhibit a
self-limiting behavior. The obtained samples show excellent optical quality,
but cannot be grown thicker than $\sim$6 nm.
\- Two Step Epitaxy (samples hBN1, hBN2) 46: The growth of a CFG buffer is
followed by a pulsed growth step requiring the switching of ammonia and TEB
flows. The obtained samples exhibit excellent structural quality and this way
thicker layers than $\sim$6 nm can be grown.

###  4.2 MoSe2 growth by MBE

A MBE machine fabricated by SVT Associates was used to grow MoSe2 layers. To
optimize the process, three hBN samples (hBN1, hBN2, hBN3) were used as
substrates for subsequent MBE growth.
\- Sample MoSe2A was grown in 3 growth steps – each of which lasted for 5
hours and was followed by $20$ min of heating up to $700^{\circ}$C and $10$
min annealing in Se flux.
\- Sample MoSe2B was grown just for 5 hours (with 3 times higher growth rate)
with only one annealing step at the very end of the process. This time the
temperature of the sample was slowly increased up to $700^{\circ}$C and the
annealing in Se flux lasted for two hours.
\- Sample MoSe2C was grown in 5 consecutive cycles. Each step contained one
hour of growth at $300^{\circ}$C, a slow heating up to $760^{\circ}$C and $10$
min annealing in Se flux. The last annealing was however significantly longer
and lasted for 2 hours.
The selection of growth parameters resulted from previous experiences with
MoSe2 growth on exfoliated hBN as well as optimization to achieve the best
optical quality of the samples.

###  4.3 Characterization of the samples

For low temperature ($5$ K) measurements, samples were placed in liquid helium
continuous flow cryostat (MicrostatHires Oxford Instruments). Luminescence
spectra at $\sim$5 K were measured with a HORIBA T64000 spectrometer with
Nd:YAG $532$ nm continuous wave laser.
Raman and photoluminescence mapping of the sample at room temperature was
performed with Renishaw inVia spectrometers. The material was excited by a
$\lambda$=$532$ nm laser with a power of $320$ $\mu$W and a 20x objective (NA
0.4) for Raman mapping and by $\lambda$=$633$ nm laser, $550$ $\mu$W laser
power and a 20x objective (NA 0.4) for photoluminescence mapping.
XRD measurements were performed using a Panalytical X’pert diffractometer
equipped with Cu X-ray tube.
FTIR reflectance spectra were obtained by Thermo Fischer Scientific iS50,
Nicolet Continuum and analyzed within the Dynamic Dielectric Functin
approximation.
SEM images were acquired using FEI Helios NanoLab 600 and AFM images were
acquired by Digital Instruments MMAFM-2 atomic force microscopy.
The morphology and structure of MoSe2/BN on sapphire were investigated using a
FEI Titan 80-300 transmission electron microscope operating at 300 kV,
equipped with an image corrector. The electron transparent cross-sections of
the heterostructure were cut using a focused gallium ion beam with a Helios
Nanolab 600 focused ion beam (FIB) system. Prior to the ion cutting, a gold
cap was deposited by magnetron sputtering, which was necessary to eliminate
sample charging during TEM lamella preparation.

This work was supported by the National Science Centre, Poland under decisions
2019/33/B/ST5/02766, 2017/27/B/ST5/02284 and 2020/39/D/ST7/02811.

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# Pair filamentation and laser scattering in beam-driven QED cascades

Kenan Qu<EMAIL_ADDRESS>Department of Astrophysical Sciences, Princeton
University, Princeton, New Jersey 08544, USA  Alec Griffith  Nathaniel J.
Fisch  Department of Astrophysical Sciences, Princeton University, Princeton,
New Jersey 08544, USA

###### Abstract

We report the observation of longitudinal filamentation of an electron-
positron pair plasma in a beam-driven QED cascade. The filaments are created
in the“pair-reflection” regime, where the generated pairs are partially
stopped and reflected in the strong laser field. The density filaments form
near the center of the laser pulse and have diameters similar to the laser
wavelength. The simulations do not show sizable magnetostatic fields, ruling
out the onset of two-stream instability or Weibel instability. Instead, we
attribute the origin of pair filamentation to laser ponderomotive forces. The
small plasma filaments induce strong scattering of laser energy to large
angles, serving as a signature of collective QED plasma dynamics.

##  I Introduction

The global race to build ultrastrong lasers worldwide [1, 2, 3] has brought us
closer to laboratory testing of strong-field QED effects [4, 5, 6]. Although
directly producing the Schwinger field, $E_{\mathrm{cr}}\sim
10^{18}\,\mathrm{Vm^{-1}}$, remains obscure, different schemes [7, 8, 9, 10,
11, 12, 13, 14] to magnify the impact of existing technologies have been
proposed. The most promising technique [7, 8] involves the collision of an
ultrastrong laser pulse with a highly energetic electron beam, boosting the
field by a large Lorentz factor in the rest frame of the electron beam to
reach the Schwinger limit. This method was adopted in the seminal QED
experiment at SLAC [15, 16] to yield measurable electron-positron pairs.
Similar experiments have been enabled by the development of laser-wakefield
accelerator, which recently reported exploring the quantum radiation reaction
using petawatt (PW) lasers and GeV electron beams generated by the laser [17,
18]. If the charge in the electron beam can be increased to the nC level
using, e.g., a conventional electron accelerator, its collision with multi-PW
laser is predicted to create a QED plasma [19, 20, 21, 22, 23, 24].

A series of studies [25, 23, 24, 26, 27] have been conducted to investigate
how a QED plasma can be created and, perhaps more importantly, observed using
the least stringent experiment parameters. It was found [23, 24] that the
beam-driven QED cascade can not only produce pairs that are significantly
slowed down but also induce an upshift of the input laser frequency, depending
on the pair density. The laser frequency upshift [28, 29, 30, 31] arises from
the change in the plasma dispersion relation. The pair plasma suddenly induces
a transverse current, which couples to the laser field, emitting at a higher
frequency. However, because of the minuscule plasma dimension, the upshift
signal is localized within a finite region with the size of the laser
wavelength. Although the plasma could expand in the laser ponderomotive
potential, the expansion reduces plasma density, suppressing its contribution
to the frequency upshift.

Despite its small volume, the pair plasmas created in the beam-driven QED
cascade exhibit rich dynamics. In this paper, we report the observation of
pair plasma filamentation and large-angle laser scattering through numerical
simulations. The pair density filamentation develops when the laser intensity
reaches the “pair reflection” threshold [23]. Under this condition, the
created pairs are temporarily stopped, exhibiting the lowest Lorentz factor
and thus the maximum plasma frequency. The highly inhomogeneous plasma with a
scale of the laser wavelength causes Mie scattering [32]. The scattered light
is intensity modulated at different scattering angles, expelling the pairs to
low intensity regions. On the other hand, the laser is refracted towards
regions of low plasma density. This coupled process induces the ponderomotive
filamentation instability [33, 34, 35], which is one of the fastest growing
modes [36, 37] in relativistic plasma streams. The reflection of pairs reduces
their slippage rate with the laser pulse, which enhances the growth of
instability. Because the pair filaments are automatically aligned with the
laser peak, they continuously scatter the laser towards higher angles, causing
an exponential decrease in peak laser intensity. The coherent scattered light,
as a result of laser-plasma interaction, serves as another signature of QED
plasma effects. The flexibility of detecting the scattering at large off-axis
angles reduces the experimental complexity.

This paper explains the pair filamentation instability through the analysis of
results from a 3D QED PIC simulation. The details of the simulation parameters
are presented in Sec. II. The section also describes the evolution of pair
filaments including the pair density, distribution function, and the
corresponding magnetic fields. The dynamics of laser pulse is presented in
Sec. III, in which we provide analytical estimation of the laser intensity
decrease. Finally, we present in Sec. IV our conclusions and discuss the
implication of QED plasma experiments.

Figure 1: Pair filamentation (green) and laser scattering (red) as a result of
$e^{-}$-beam driven QED cascade. The laser is polarized in the $y$-direction
and propagates to $+x$-direction. The $e^{-}$ beam (not shown) propagates to
the $-x$-direction. The snapshot is taken at $t=0.22\,\mathrm{ps}$.

##  II Formation of pair filamentation

The pair filaments are observed in 3D QED PIC simulations that we described in
Refs. [23, 24]. A snapshot of the laser beam and created pairs are plotted in
Fig. 1 to show the simulation schematics. The simulations consider collision
of a $300\,\mathrm{GeV}$ electron beam and a $24\,\mathrm{PW}$ laser pulse.
The electron beam has $1\,\mathrm{nC}$ charge distributed in a Gaussian sphere
with rms radius of $1\,\mathrm{\mu m}$. Its peak density is $4\times
10^{20}\,\mathrm{cm^{-3}}$. The Gaussian laser pulse has
$\lambda=0.8\,\mathrm{\mu m}$ wavelength, $6\times 10^{22}\,\mathrm{Wcm^{-2}}$
peak intensity (correspondingly $a_{0}\approx 170$), $50\,\mathrm{fs}$ rms
duration, and $5\,\mathrm{\mu m}$ waist. The laser is linearly polarized in
the $y$-direction and propagates to the $-x$-direction. The electron beam
propagates to the $+x$-direction. The numerical parameters are similar to
those in Ref. [23], but we increase the transverse simulation window size to
$(40\,\mathrm{\mu m})^{2}$ to capture the pair expansion. The transverse grid
size is correspondingly increased to $(\lambda/6)^{2}$.

The electrons have a maximum quantum parameter $\tilde{\chi}_{e}\approx 220$
at the Gaussian waist in the focal plane, and $\tilde{\chi}_{e}\approx 600$ at
the laser focus. The high quantum parameter $\tilde{\chi}_{e}$ enables a beam-
driven QED cascade which creates electron positron pairs through the Bret-
Wheeler process. The stochastic nature leads to a broad distribution function
of the pair momentum. As the pairs continue to lose energy through radiation
reaction, those in the low energy spectrum begin to become reflected by the
ponderomotive pressure assuming that the laser intensity exceeds the “pair
reflection” threshold [23, 26]. Thus, we can divide the pair evolution into
three stages, including forward moving, being stopped, and being reflected.
All the pairs at different stages can coexist but the pairs in each stage
contains both electrons and positrons moving in the same longitudinal
direction, making the plasma quasineutral and anisotropic.

Figure 2: Pair density $(\mathrm{cm^{-3}})$ at $t=0.18\,\mathrm{ps}$ (a,b),
$0.2\,\mathrm{ps}$ (c,d), and $0.22\,\mathrm{ps}$ (e,f), respectively. The
left column shows the $y=0$ cross section and the right column show the
$x=-2\,\mathrm{\mu m}$ cross section indicated by the vertical line on the
left.

The pair plasma, illustrated in green color in Fig. 1, exhibits filamentation
in the region where it overlaps with the laser pulse. To show more details of
the filaments, we plot in Fig. 2 the evolution of pair density profile in the
$x$-$z$ plane at $y=0$ (left column) and $y$-$z$ plane at $x=-2\,\mathrm{\mu
m}$ (right column). Although the simulation is conducted in a
$100\,\mathrm{\mu m}$-long box, we only show the center part where pair
density is finite. The pairs are initially created with a spherical profile
similar to the injected electron beam. The sphere then expands under the laser
radiation pressure. When the “pair reflection” condition is reached, a portion
of the pairs begin to reverse their moving direction. This is shown in Fig.
2(a) as cavitation of the pair sphere at $t=0.18\,\mathrm{ps}$. The pair
density first exhibits fringes with a wavevector in the $z$-direction, shown
in Fig. 2(b). At $t=0.2\,\mathrm{ps}$, the fringes condense into filaments and
gain a wavevector in the $y$-direction, which can be seen in Fig. 2(d). As the
same time, the cavitation continues to expand. The pair plasma develops three
structures, including a dense core in the front, an expanding shell following
the core, and five observable filaments inside the shell. All the structures
remain through the rest of interaction.

Filamentation of plasmas could arise from two-stream instability, Weibel
instability, or ponderomotive filamentation instability [37, 36]. Although the
formation of pair filamentation is correlated to counterpropagating pair
streams, we cannot directly attribute the filamentation to the plasma
streaming instability. The streaming instability exists in conventional
electron-ion plasmas when counterpropagating electron streams carry a
longitudinal current and a “return current” which produce strong transverse
magnetic field to focus the streams [38, 39, 40]. However, the electron
positron pairs respond symmetrically to the longitudinal laser pressure,
including both the radiation reaction and the laser ponderomotive force. The
longitudinally flowing pair streams are charge neutral and current neutral, at
least in the center, and cannot produce a current or a “return current”. Thus,
the pair plasma streams are not likely produce streaming instabilities.

Figure 3: Pair momentum distributions in the transverse direction (left
column) and longitudinal direction (right column) at $t=0.16\,\mathrm{ps}$
(a,b), $0.18\,\mathrm{ps}$ (c,d), and $0.2\,\mathrm{ps}$ (e,f), respectively.

More details of the filamentation process are shown in the pair momentum space
in Fig. 3. The transverse momentum distribution $p_{y}$, plotted in the left
column, reveals the pair oscillation in the laser field. Interestingly, the
distribution in $p_{y}$ shows narrow width compared to the maximum momentum.
This can be explained from the factor that the creation and deceleration of
pairs are synchronized with the peak laser amplitude and their initial
transverse momentum is negligible. The conservation of canonical momentum
$p_{y}+a_{0}m_{e}c$ leads to synchronized oscillation of $p_{y}$.

The right column of Fig. 3 show the longitudinal momentum distributions
between $0.16\,\mathrm{ps}-0.2\,\mathrm{ps}$. The pairs start with
unidirectional propagation in the $+x$-direction at $t=0.16\,\mathrm{ps}$,
illustrated in Fig. 3(a). They begin to show negative longitudinal momenta
before $t=0.18\,\mathrm{ps}$ in Fig. 3(d), indicating partial pair reflection
in the region near $x=0$. It corresponds to the cavitation of pair sphere in
Fig. 2(a,b). The distribution, however, does not show bump-on-tail
distribution, which eliminates the possibility of two-stream instability.

Figure 4: Magnetic field $B_{x}$ (a,b), $B_{y}$ (c,d), and $B_{z}$ (e,f) in
the $y=0$ cross section at $t=0.18\,\mathrm{ps}$ (left column), and
$0.2\,\mathrm{ps}$ (right column), respectively.

To rule out onset of Weibel instability, we next present the magnetic field
profile in the $y=0$ cross section in Fig. 4. Indeed, the transverse
components $B_{y}$ does not show filament structures. However, filament
structures are observed in the longitudinal field $B_{x}$. They also manifest
in the decrease of laser field amplitude $B_{z}$ in the regions of pair
filaments. Note that the background fields $B_{x}$ and $B_{z}$ extending
between $z=-10\,\mathrm{\mu m}$ and $z=10\,\mathrm{\mu m}$ are the intrinsic
laser field [41] and more details can be found in the Appendix.

The decrease of $B_{z}$ field in the region of pair filaments is an indication
of ponderomotive filamentation [33], which is supported by two main features
of the pair filamentation process illustrated in Fig. 2. First, the formation
of filaments is strongly associated with its interaction with the laser field
rather than a counterpropagating plasma. As seen in Fig. 2(c,d), the forward
moving plasma shell and backward moving fringes do not overlap when the
fringes condensate to filaments.

Second, the filaments exhibit higher contrast in the $z$ direction, which is
perpendicular to the laser polarization direction. The anisotropic
filamentation is associated with Thomson scattering of the linearly polarized
laser when the pair plasma dimension is smaller than the laser wavelength. The
laser drives fundamental pair oscillation in the polarized direction $\hat{y}$
and second harmonic oscillation in the longitudinal direction $\hat{x}$. But
only the fundamental mode beats with the input laser and it emits most
strongly in the orthogonal transverse direction $\hat{z}$. Thus, fringes in
the polarization direction $\hat{y}$ are suppressed. Laser scattering in the
$\hat{y}$ direction only becomes strong when the plasma volume reaches near
the laser wavelength and the scattering becomes Mie scattering. Mie scattering
has a weak dependence on polarization but still has a strong dependence on the
scattering angle. The anisotropic laser scattering serves as a seed for the
quickly growing pair filamentation. The laser ponderomotive force pushes the
plasma to regions of lower laser intensity. The plasma density inhomogeneity
refracts the laser to regions with lower density and hence leads to
filamentation instability.

Figure 5: Poynting vector (in unit $\mathrm{Wm^{-2}}$) $S_{y}$ in the $z=0$
cross section (a,b) and $S_{z}$ in the $y=0$ cross section (c,d) at
$t=0.18\,\mathrm{ps}$ (a,c), and $0.26\,\mathrm{ps}$ (b,d), respectively.

Mie scattering of the laser can be visualized using the Poynting vectors
plotted in Fig. 5. As soon as pair plasma is formed at $t=0.18\,\mathrm{ps}$,
the Poynting vectors in Fig. 5(a,c) show divergences in both transverse
directions $\hat{y}$ and $\hat{z}$. As the filamented pairs copropagate with
the laser, they continuously differact the wave which, shown in Fig. 5(b,d),
deviates from the laser pulse. Each panel of Fig. 5 also shows a low amplitude
$S$ beyond the region of the filaments. They illustrate the transverse energy
outflow of the Gaussian laser pulse $S_{r}\propto|E|^{2}r/R(x)$ where $R(x)$
is the radius of curvature of the beam at $x$. Compared to the background
Poynting vector for the input laser (more details can be found in the
Appendix), the diffraction modes are an order of magnitude stronger. The
diffraction modes exhibit filament structure mainly in the $x$-$z$ plane,
signifying the role of polarization of Thomson scattering.

##  III Laser energy scattering

The pair filamentation has important implications to the QED cascade. It
focuses the pairs in a few filaments of diameters similar to the laser
wavelength, which otherwise expand continuously due to transverse laser
ponderomotive force. The small filaments cause scattering of laser to large
angles. Because the filaments are formed in the region of maximum laser
intensity, they induce scattering of the laser energy which can be observed in
the tail of laser in Fig. 1 and in Fig. 5(b,d). More clearly, The peak laser
intensity, illustrated as blue solid curve in Fig. 6, shows a $40\%$ decrease
during the pair reflection between $t=0.18\,\mathrm{ps}$ to
$0.2\,\mathrm{ps}$. The figure also presents the total laser energy
$\mathcal{E}=(\epsilon_{0}E^{2}+B^{2}/\mu_{0})/2$ as red solid curve, which
shows only a $0.3\%$ decrease due to reflecting the pairs and loss at
simulation window boundaries. Here $\epsilon_{0}$ and $\mu_{0}$ are the vacuum
permittivity and permeability, respectively. Thus, the significant drop in
peak laser intensity combined with approximately constant pulse energy
comports with the theory of laser energy scattering.

Figure 6: Evolution of the peak laser intensity (blue solid) and total energy
(red dashed).

The process of laser scattering is closely related to the laser frequency
upshift when the mediating plasmas change density or Lorentz factor. In the
problem of laser frequency upshift, the plasma is assumed homogeneous in the
transverse direction. The transverse current thus strictly emits in the laser
propagation direction or anti-parallel to it. The emission couples to the
original laser field to cause laser frequency upshift. However, pair
filamentation breaks the symmetry and the oscillating pairs emit to the whole
space due to finite current dimension.

We can analytically estimate the rate of laser intensity decrease. The
transverse current carried by the created pairs is

$\displaystyle\bm{J}$ $\displaystyle=\int_{0}^{t}dt^{\prime}\frac{2\partial
n_{p}}{\partial
t^{\prime}}\int_{t^{\prime}}^{t}\frac{e^{2}}{\gamma(t^{\prime\prime})m_{e}}\bm{E}(t^{\prime\prime})dt^{\prime\prime}$
$\displaystyle=\frac{2e^{2}}{m_{e}}\int_{0}^{t}\frac{n_{p}\bm{E}}{\gamma}dt^{\prime}.$
(1)

The pair density $n_{p}$ is dependent on both time and space. In the beginning
of QED cascade, $n_{p}$ can be modeled as a $\delta$ function in space.
Because the pairs oscillate synchronously, they behave as a relativistic
dipole to induce Thomson scattering111The scattering is inelastic because we
focus on photon emissions with the same frequency which can only beat with the
input laser to create the intensity fringes.. The angle dependence of the
scattered light changes from $\sin^{2}\theta_{1}$ at $\gamma\sim 0$ to
$[(1-\cos\theta_{2})^{2}-\sin^{2}\theta_{2}\cos^{2}\phi/\gamma^{2}]/(1-\cos\theta_{2})^{5}$
at $\gamma\gg 1$, where $\theta_{1}$ and $\theta_{2}$ are the angles between
the scattered light and the laser polarization direction $\hat{y}$ and the
laser propagation direction $\hat{x}$, and $\phi$ is angle between $\hat{y}$
and the plane of the scattering direction and $\hat{x}$. When the pair volume
grows to the scale of $\lambda^{3}$, the scattering needs to be described by
Mie theory. It is strongly dependent on the scattering angle and requires
numerical treatment. But because both Thomson scattering and Mie scattering
are anisotropic, the scattered light beats with the input laser to cause
filamentation of the pair plasmas.

According to the Poynting theorem, the decrease of laser energy density $U$ is
described with $dU/dt={-\nabla\cdot\bm{S}}-{\bm{J}\cdot\bm{E}}$, where
$U\equiv\epsilon_{0}|E|^{2}$ in dispersive media. Note, however, that because
Poynting flux $\bm{S}$ of the scattering arises from the emission of the
transverse current $\bm{J}$, the decrease of energy density $U$ can be found
by analyzing the term ${\bm{J}\cdot\bm{E}}$. We describe laser field as
$\bm{E}=\bm{E}_{0}\cos\varphi$ and the pair density as
$n_{p}=n_{p}\Theta(\varphi-\varphi_{0})$ where $\varphi=\omega t-\bm{k}x$ and
$\varphi_{0}=0$ is the instant of pair stopping at the laser peak. The
transverse current can be expressed as $\bm{J}\cong 2\epsilon_{0}\omega
n_{p}\bm{E}_{0}\sin\varphi\Theta(\varphi-\varphi_{0})/(n_{c}\gamma)$, where
$n_{c}$ is the critical density of frequency $\omega$. Thus the laser energy
density decreases as

$\frac{dU}{d\varphi}=-\frac{2\epsilon_{0}n_{p}}{\gamma
n_{c}}E_{0}^{2}\sin\varphi\cos\varphi\Theta(\varphi-\varphi_{0}).$ (2)

The differential equation shows that energy is transferred from the laser to
the transverse current when $0<\varphi<\pi$ and is transferred back to the
wave when $\pi<\varphi<2\pi$. But the latter process radiates to the whole
space and hence its contribution to the input laser pulse can be neglected.
Therefore, the decrease of peak laser energy density near the pair stopping
point can be found by taking average of $dU/d\varphi$ in half a cycle

$\left\langle\frac{dU}{d\varphi}\right\rangle=-\frac{n_{p}}{\gamma
n_{c}}\langle U\rangle.$ (3)

The result shows that the diffracted laser energy increases exponentially with
the effective plasma density $n_{p}/\gamma$ and interaction time $\omega t$.
For the presented simulation, the pair plasma has a length of $\sim 12\lambda$
with $n_{p}/(\gamma n_{c})\sim 6.7\%$. It results a laser energy density
decrease of $\sim 50\%$, which agrees well with simulation results. Note that
frequency upshift process in a strict 1D scenario also causes a decrease of
laser energy density, but the decrease scales with $1/\omega$ which is an
order of magnitude lower than $40\%$ in the simulation.

##  IV Conclusions and discussions

In conclusion, we investigate the formation of pair filamentation in an
electron-beam-driven QED cascade through a 3D QED PIC simulation. The pair
filamentation is observed in the “pair reflection” regime, in which an above-
threshold laser decelerates the pair plasmas and reverts their propagation
direction. The simulation shows the formation of pair filaments in the
longitudinal direction when the laser traverses through the pair plasma. With
a linearly polarized laser, the pair density first exhibits periodic
modulation in the plane perpendicular to the polarization direction. It then
condenses into filaments with a diameter similar to the laser wavelength.
Because the results do not show the creation of strong transverse magnetic
field perpendicular to the laser, we rule out the possibility of two-stream
instability or Weibel instability.

We attribute the cause of pair filamentation to the laser ponderomotive force.
The initially small pair plasma causes highly anisotropic scattering of the
laser, which interfere with the input laser to create a pressure gradient for
the pair plasma. Pairs are expelled from regions of high laser intensity, and
pair cavitation causes locally high refractive index to focus the laser pulse.
The inhomogeneous plasma density also leads to refraction of laser energy.
Thus, filamentation is formed from the region where dense pair plasma is
stopped. Because the reflected pairs and laser pulse copropagate, their
slippage is reduced, allowing the pair filaments to grow quickly.

Laser polarization critically influences the formation of pair plasma
filamentation. In the initial stage of interaction, the small plasma
oscillates synchronously to cause stronger scattering in the direction
perpendicular to the polarization direction. Because the scattering is
negligible in the polarization direction, a linearly polarized laser only
creates 2D gratings in the initial stage when the pair plasma is smaller than
the laser wavelength. However, a circularly polarized laser is scattered in
both of the transverse directions and thus creates filaments immediately. This
is verified through a separate 3D PIC simulation using circularly polarized
laser. As the plasma volume grows, the scattering becomes less polarization-
dependent, and filaments grow with both linearly and circularly polarized
lasers.

The pair filamentation has important implications to the joint problem of
creating and observing QED plasmas. First, filamentation focuses pairs and
creates high pair density regions inside the laser field. Compared to
homogeneously distributed pairs, filamentation reduces the total pair number
needed to reach high pair density. Second, despite being highly localized, the
dense pair plasma interacts with the most intense part of the laser. The
strong scattering of laser energy due to small filament dimensions causes an
exponential decrease in laser intensity within the filament length. All the
scattered laser energy is redistributed and can be detected at large angles.
Because the scattering arises from the interaction of collective pair motion
and the laser fields, it serves as one more signature of QED plasma effect.

###### Acknowledgements.

This work was supported by NSF PHY-2206691.

Figure 1*: The laser magnetic field $B_{x}$ (a,b), $B_{y}$ (c,d), and $B_{z}$
(e,f) in the $y=0$ cross section without interacting with electron beam at
$t=0.18\,\mathrm{ps}$ (left column), and $0.2\,\mathrm{ps}$ (right column),
respectively.

## Appendix

Figure 2*: Poynting vector (in unit $\mathrm{Wm^{-2}}$) $S_{y}$ in the $z=0$
cross section (a,b) and $S_{z}$ in the $y=0$ cross section (c,d) of the laser
without interacting with electron beam at $t=0.18\,\mathrm{ps}$ (a,c), and
$0.26\,\mathrm{ps}$ (b,d), respectively.

The linearly polarized Gaussian laser pulse contains fields in multiple
directions due to its finite waist size, $w_{0}=5\,\mathrm{\mu m}$. With its
electric field polarized in the $y$-direction, the electric and magnetic
fields in vacuum can be written to the second order of $(w_{0}/z_{R})\approx
0.05$ as [41]

$\displaystyle E_{x}$
$\displaystyle=\tilde{E}\frac{y}{z_{R}}\frac{w_{0}}{w(x)}\cos(\varphi+2\varphi_{G}-\varphi_{R}).$
(1*) $\displaystyle E_{y}$
$\displaystyle=\tilde{E}\Big{\\{}\sin(\varphi+\varphi_{G})+\frac{w_{0}^{2}}{z_{R}^{2}}\Big{[}\frac{y^{2}}{w^{2}(x)}\sin(\varphi+3\varphi_{G})$
$\displaystyle\qquad\qquad-\frac{(y^{2}+z^{2})^{2}}{4w_{0}w^{3}(x)}\sin(\varphi+4\varphi_{G})\Big{]}\Big{\\}},$
(2*) $\displaystyle E_{z}$
$\displaystyle=\tilde{E}\frac{yz}{z_{R}^{2}}\frac{w_{0}^{2}}{w^{2}(x)}\sin(\varphi+3\varphi_{G}),$
(3*) $\displaystyle B_{x}$
$\displaystyle=\frac{\tilde{E}}{c}\frac{z}{z_{R}}\frac{w_{0}}{w(x)}\cos(\varphi+2\varphi_{G}),$
(4*) $\displaystyle B_{y}$ $\displaystyle=0,$ (5*) $\displaystyle B_{z}$
$\displaystyle=\frac{\tilde{E}}{c}\Big{\\{}\sin(\varphi+\varphi_{G})+\frac{w_{0}^{2}}{z_{R}^{2}}\Big{[}\frac{y^{2}+z^{2}}{2w^{2}(x)}\sin(\varphi+3\varphi_{G})$
$\displaystyle\qquad\qquad-\frac{(y^{2}+z^{2})^{2}}{4w_{0}w^{3}(x)}\sin(\varphi+4\varphi_{G})\Big{]}\Big{\\}},$
(6*)

where $k=-2\pi/\lambda$, $w(x)=w_{0}\sqrt{1+(x/x_{R})^{2}}$ is the spot size
at distance $x$, $\varphi=\omega t-kx-k(y^{2}+z^{2})/(2R)$, $R=x+x_{R}^{2}/x$,
$\varphi_{G}=-\arctan(x/x_{R})$ is the Gouy phase, and
$\tilde{E}=E_{0}[w_{0}/w(x)]\exp[-(y^{2}+z^{2})/w^{2}(x)]\exp[-(t-x/c)^{2}/\tau^{2}]$
is the envelope of the Gaussian beam, with $\tau=50\,\mathrm{fs}$.

Figure 1* shows the magnetic fields of laser in vacuum for direct comparison
with Fig. 4.

Figure 2* shows the Poynting vectors of laser in vacuum for direct comparison
with Fig. 5. The decrease of energy at $t=0.26\,\mathrm{ps}$ is caused by
laser exiting the simulation window.

Figure 3* shows the evolution of the peak laser intensity and total energy in
vacuum for direct comparison with Fig. 6.

Figure 3*: Evolution of the peak laser intensity (blue solid) and total energy
(red dashed).

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