# An efficient low-density grating setup for monochromatization of XUV
ultrafast light sources

Qinda Guo  1 Maciej Dendzik  1 Magnus H. Berntsen  1 Antonija Grubišić-Čabo
1,2 Cong Li  1 Wanyu Chen  1 Yang Wang  1 and Oscar Tjernberg1  1Department of
Applied Physics, KTH Royal Institute of Technology, Hannes Alfvéns väg 12, 114
19 Stockholm, Sweden
2Zernike Institute for Advanced Materials, University of Groningen, 9747 AG
Groningen, The Netherlands
<EMAIL_ADDRESS>
###### Abstract

Ultrafast light sources have become an indispensable tool to access and
understand transient phenomenon in material science. However, a simple and
easy-to-implement method for harmonic selection, with high transmission
efficiency and pulse duration conservation, is still a challenge. Here we
showcase and compare two approaches for selecting the desired harmonic from a
high harmonic generation source while achieving the above goals. The first
approach is the combination of extreme ultraviolet spherical mirrors with
transmission filters and the second approach uses a normal-incidence spherical
grating. Both solutions target time- and angle-resolved photoemission
spectroscopy with photon energies in the 10-20 eV range but are relevant for
other experimental techniques as well. The two approaches for harmonic
selection are characterized in terms of focusing quality, efficiency, and
temporal broadening. It is demonstrated that a focusing grating is able to
provide much higher transmission as compared to the mirror+filter approach
(3.3 times higher for 10.8 eV and 12.9 times higher for 18.1 eV), with only a
slight temporal broadening (6.8% increase) and a somewhat larger spot size
($\sim$30% increase). Overall, our study establishes an experimental
perspective on the trade-off between a single grating normal incidence
monochromator design and the use of filters. As such, it provides a basis for
selecting the most appropriate approach in various fields where an easy-to-
implement harmonic selection from high harmonic generation is needed.

††journal: oe††articletype: Research Article

##  1 Introduction

Extreme ultraviolet (XUV) radiation with a narrow bandwidth is one of the key
ingredients for many high-resolution spectroscopic techniques used in research
today. Beyond this, XUV light also has considerable application value for the
industry, for example in lithographic patterning adopted in semiconductor
manufacturing processes. With the advent of femtosecond high-power lasers and
the progress of high harmonic generation (HHG) techniques, table-top XUV light
sources with pulse duration in the femtosecond range have become a reality.
These sources offer a more compact and affordable alternative compared to the
synchrotron radiation facilities, or free-electron lasers, while
simultaneously supporting spectroscopic, imaging or scattering techniques with
high temporal resolution. Presently, HHG has become a viable approach to
generate bright and ultrafast XUV-light sources, but monochromatization of the
generated radiation still remains a challenge – in particular from the point
of view of maintaining a high photon transmission through the monochromator,
while simultaneously minimizing time broadening of the light pulses, as well
as keeping the optical design robust and compact.

HHG is a well-developed technique used to convert long-wavelength laser light
into radiation with shorter wavelengths, typically in the XUV or soft X-ray
regime[1, 2, 3]. The HHG process can be understood conceptually by the three
step model[4, 5]: firstly the electrons experience the ionization tunneling,
followed by a propagation in the electric field from the driving light, and
finally recombination with the ionic nuclei, yielding high energy and coherent
photons during the process. The HHG spectrum consists of a series of discrete
harmonics spaced by twice the driving frequency, 2$\omega$, up to a maximum
photon energy determined by the single atom cut-off relation,
$h\nu_{\mathrm{cut-off}}=I_{\mathrm{p}}+3.17U_{\mathrm{p}}$, where the $h$ is
Planck’s constant, $I_{\mathrm{p}}$ is the ionization potential and
$U_{\mathrm{p}}$ is the ponderomotive energy of the electron. This relation
implies that the energy cut-off can be increased by using a longer driving
wavelength, as it scales with the relation $U_{\mathrm{p}}\propto
I_{\mathrm{L}}\lambda_{\mathrm{L}}^{2}$, where $I_{\mathrm{L}}$ is the
intensity of the driving laser and $\lambda_{\mathrm{L}}$ is the wavelength.
On the other hand, the HHG efficiency scales proportional to
$\lambda_{\mathrm{L}}^{-(5-6)}$[6]. This inherent trade-off means that, in
practice, one needs to choose suitable driving conditions based on the light
source requirement. In our case, to maximize the photon flux at photon
energies up to 25 eV, a cascaded scheme[7], using the third-harmonic of the
driving laser, is chosen for HHG in a gas target.

The monochromatization requires a selective response to different frequency
components, and gratings are extensively adopted for this purpose. A typical
monochromator layout is the X-ray Czerny-Turner (XCT) design[8], which is a
single-grating based grazing incidence monochromator (GIM), that provides the
most necessary elements needed for the collimation, diffraction and re-
focusing. The line density is usually chosen based on the trade-off between
efficiency and spectral resolution. With high line-density, very high
resolving power and energy resolution have been achieved [9, 10]. Another
common GIM layout is a spherical grating based configuration, which serves as
a compact layout[9, 11]. As the counterpart of GIM, the normal incidence
monochromator (NIM) has also been employed in synchrotrons to access the low
photon energy range and provide high energy resolution[10]. Compared to GIM,
NIM is quite advantageous in terms of compact layout, ease of alignment, and
good XUV imaging properties.

Stepping into the ultrafast regime in the XUV range, monochromators based on
diffractive optics exhibit intrinsic drawbacks, as the induced pulse-front
tilt broadens the pulse length. Nevertheless, there have been designs
targeting HHG [12, 13, 14, 15, 16, 17, 18] and free-electron-laser (FEL) light
sources [19, 20, 21]. One approach is to use a single-grating, but lowering
the line density to reach the design requirements [17, 22, 18], concerning the
pulse length trade-off for a certain spectral linewidth. The double-grating
design is an alternative approach that permits complete pulse stretching
compensation and ideally provides no extra temporal broadening [12, 14, 16,
21]. Pulses as short as 8 fs have been reported [13] using this design.
However, such a configuration reduces the transmission and makes the design
and alignment somewhat complicated. The pulse stretching induced by a single
grating can, however, be reduced by using the off-plane-mount. In this case
the diffraction occurs in the direction perpendicular to the direction of
propagation thus minimizing the number of illuminated groves and thus
decreasing the temporal broadening [15, 23, 22].

In this work, we compare the performance of two types of NIM configurations,
as illustrated in Fig. 1. The mirror+filter approach and the spherical grating
monochromator are compared in terms of their focusing properties, efficiencies
and the introduction of temporal broadening of the HHG pulses. The ultrafast
light source is part of an experimental setup for time- and angle-resolved
photoemission spectroscopy (tr-ARPES) [24]. For the comparison, we designed a
spherical grating NIM specially for 10.8 eV and 18.1 eV photon energies. In
this particular case, it is found that the mirror+filter design results in a
slightly better focus, 100 µm spot size, while the focusing grating focus is
about 30% larger ($\sim$33 µm). The larger spot size could be due to
fabrication imperfections of the grating profiles or overcoating. On the other
hand, in terms of transmission efficiency, the focusing grating is remarkably
effective for both photon energies, resulting in 3.3 times higher efficiency
for 10.8 eV and 12.9 times higher efficiency for 18.1 eV as compared to the
mirror+filter solution. Consequently, this enabled us to achieve a photon flux
on the level of 1011 photons/s. As for the temporal response, the
mirror+filter combination, which is used as a reference, is assumed to provide
negligible temporal broadening. Pump-probe measurements on graphene show an
overall temporal resolution of 205.3$\pm$7.6 fs for mirror+filter and
219.5$\pm$1.9 fs for the spherical grating design. By excluding the temporal
contribution from the pump, this corresponds to a temporal broadening of
72.4$\pm$27.4 fs. In Table 1, the various performance parameters are listed,
to compare the applicability of the grating and filter designs for a specific
experimental design.

##  2 Monochromatization of an ultrafast light souce

For monochromatization of ultrafast light sources, the mirror+filter
configuration is preferred in terms of time resolution as it can eliminate the
temporal broadening issue, providing a significant advantage compared to
monochromators based on gratings. In our setup, the mirror+filter
configuration is comprised of a concave normal incidence mirror, and a thin
film filter that is used to further suppress unwanted harmonics. The mirror
coating method is critical for the harmonic selection, but it is difficult to
completely eliminate unwanted harmonics with the coating alone. For this
reason, the thin film filter with an appropriate transmission window is
necessary. The drawback of this approach is that the performance and working
wavelength range are primarily determined by the available thin film filters,
and they do not always fit the experimental demands. Since thin film filters
need to be as thin as hundred nanometers to maintain reasonable transmission,
they are delicate and can easily be damaged mechanically or by the beam.
Oxidation and contamination are other issues often experienced in the use of
thin film filters.

Here we propose a novel and efficient grating-based NIM that has been
designed, fabricated and experimentally compared with the mirror+filter
solution. It is in principle a spherical focusing grating, which serves as a
single optical element-based NIM, with a low line density of 40 l/mm. The main
design criteria is to have as low line density as possible while still being
able to separate the harmonics geometrically at the sample position. In the
present context, the low resolving power caused by the low line density is not
an issue, as the spectral resolution is intrinsically determined by the HHG
conditions. The purpose of using the lowest possible line density is to
mitigate the pulse-front tilt to a negligible level. Note that the value of 40
l/mm adopted here, is dependent on the particular geometry of the setup and
the trade-off between pulse stretching and focal properties. In the present
comparison, the two compared solutions have similar geometry and focusing
properties so that the performance can be compared in a relatively
straightforward manner. Depending on the availability of filters and coatings
the two solutions are also more or less difficult to implement for various
wavelengths, in the present case we target photon energies ($\sim$10-20 eV).
For the spherical grating efficiency, the simulation with a groove depth of
about $\sim$ 17 nm, together with the 80 nm SiC coating, shows a 15% overall
efficiency centered at 18.1 eV photon energy, and overall, above 10%
efficiency in the range from 10 to 20 eV, as presented in Fig. 2(c).

Figure 1: Schematic drawing of the two types of monochromatization for
ultrafast applications. a) The focusing mirror plus the thin film filter. b)
The focusing grating. ROC stands for the radius of curvature, m is the
diffraction order, $\sigma$ is the line density and $\beta$ is the beam
incidence angle. HH stands for high harmonic. Table 1:  Criteria for two
configurations of ultrafast XUV pulse monochromatization.

| Mirror+filter | Focusing grating
---|---|---
Pulse-front tilt | Negligible | Below 100 fs for $h\nu>10$ eV†
Spectral broadening | Negligible | Negligible
Efficiency |  | Moderate. Limited by reflectivity
---
of the mirror and transmission
through the thin filter
| High. Limited by reflectivity of
---
grating, which is set by the groove
depth, groove-to-land ratio
and coating material
Focusing |  | Determined by source size
---
and spherical aberration
from focusing mirror
| Broadened compared to spherical
---
mirror. Mainly limited by
the fabrication errors
Costs |  | High. Requires multiple
---
band pass mirrors as
well as thin film filters
| Moderate. Single optical element
---
low-line-density grating produced
by standard nanofabrication
processes

† This is for the experimental design presented in this article. The temporal
broadening can be further decreased by lowering the number of illuminated
grooves through lower line density, or smaller beam spot size.

##  3 Mirror+filter configuration

For the direct comparison of the filter and grating-based monochromator
solutions, we focus on two photon energies, 10.8 eV and 18.1 eV, generated by
the HHG setup. For the mirror+filter configuration, and for a photon energy of
10.8 eV, a concave MgF2-coated Al mirror with a radius of curvature (ROC) of
1000 mm is used to focus the diverging high harmonic radiation. The filter in
this case is a thin (500 µm) LiF (Eksma Optics) window that blocks photon
energies higher than 11 eV. For 18.1 eV photons, the previous mirror and
filter were replaced with a concave silicon mirror substrate (ROC=1000 mm)
coated with SiC and a 150 nm thick Sn filter (Lebow), respectively. The Sn
filter provides a band-pass window for photon energies between 17 eV and 24
eV. For more information on the band-pass windows for different thin film
filters, see Supplementary Material Fig. S4.

##  4 Focusing grating configuration

With only a few changes, the experimental setup of the mirror+filter
monochromator described above can be turned into a grating-based NIM. The key
is to replace the spherical focusing mirror with a spherical grating with the
same ROC and instead of thin film filters use a slit to select the desired
diffraction order from the grating. In the following section, we elaborate in
detail on the focusing grating part, in particular regarding the design
principles, and the fabrication recipes.

###  4.1 Design principles

To design a NIM using a focusing grating, and to match the specific
requirements regarding high efficiency and temporal resolution, several
constraints need to be considered. Compared to the mirror+filter solution, one
major challenge for designing the focusing grating is to limit temporal
broadening of laser pulses. The pulse-front tilt experienced on the grating is
proportional to the number of grooves that are illuminated, resulting in a
correlation between the line density, $\sigma$, and the illuminated area, $A$,
on the grating. The spot size $A$ is given by the HHG divergence and the
focusing geometry of the setup and, therefore, in this work, it is regarded as
a fixed parameter. The illuminated area on the grating can be modulated by an
aperture at the cost of reduced photon flux. Keeping the line density $\sigma$
low can minimize time broadening of the pulses, but a sufficiently large
$\sigma$ is required to spatially separate the different diffraction orders of
the harmonics at the position of the focus. The groove depth, $d$, and the
ratio of the groove and peak widths, $r$, determine the efficiency and the
band-pass center of the grating. The choice of a coating, and its thickness,
are also factors affecting the final efficiency.

Figure 1b) depicts the concept of the spherical grating-based monochromator
that reflects and refocuses the diverging HHG radiation generated in the gas
target, while simultaneously separating the harmonics into different
diffraction spots. The desired harmonic can then be selected with a slit or an
aperture. For a grating with $N$ grooves illuminated by the incident beam, the
resolving power, $R$, is expressed as:

$R=\frac{\lambda}{\Delta\lambda}=|m|N=\frac{\Delta t}{\lambda}$ (1)

where $m$ is the diffraction order, $\Delta t$ is the total pulse-front tilt
and $\lambda$ is the radiation wavelength. The expression implies a
proportional relation between the resolving power and the induced time
broadening for a grating-based monochromator. In our case the spectral
linewidth is determined by the HHG conditions, so as a design principle, one
should pursue a low resolving power for the minimum temporal broadening, which
directly links to the line density $\sigma$. Another constraint for $\sigma$
is the wavelength spacing between the different harmonics, as presented in
Figure 2b), since there needs to be sufficient spatial separation for an
aperture to select the harmonics.

Figure 2:  The simulation results for the spherical grating NIM based on the
following principle parameters: 40 l/mm line density, 17 nm groove depth, 1:1
land to groove ratio, and 80 nm SiC coating. a) The distance of the first
order higher harmonic spots with respect to the position of the zero
diffraction order. b) The pulse-front tilt, and c) the overall efficiency,
plotted as a function of the photon energy in range from 5 to 25 eV. The focal
length is set to 3 m.

Following the aforementioned design principles, a focusing grating is
explicitly designed for a photon energy of 18.1 eV. The chosen line density
$\sigma$ is 40 l/mm. With the distance from the grating to the sample in the
photoemission chamber being 3 m at the sample position, the first order
diffraction spots for 18.1 eV and 10.8 eV are positioned 8 mm and 14 mm away
from the zero-order diffraction maximum, respectively. This separation is
sufficient so that an aperture can be used to select the desired harmonics and
block the remaining undesired diffraction spots. The 17 nm groove depth $d$
and the 1:1 ratio value of $r$, provide optimal efficiency for the 18.1 eV
photon energy. The grating pattern is etched into a silicon mirror substrate
with a ROC of 1000 mm, and finally, a SiC coating layer with a thickness of
minimum 30 nm was deposited. This minimum coating layer thickness yields the
best performance according to calculations (see Supplement Material Section 1
and Figure S1.). Figure 2 presents calculations of key performance parameters
for the focusing grating for the photon energy range between 5 eV and 25 eV,
using the parameters presented above. Figure 2a) presents the calculated
space-separation of the diffracted harmonics. The distances between the zeroth
order and the first diffraction order for specific photon energies, are
plotted. The focal length is set to 3 m. The inset shows the results of
separation for the HHG conditions and setup geometry in our case. Figure 2b)
shows the pulse-front tilt calculated based on Eq. 1, as a function of photon
energy. The total efficiency of the grating is illustrated in Fig. 2c).

###  4.2 Fabrication recipe

The fabrication of the spherical grating was conducted in a clean-room using
standard semiconductor processing steps such as lithography, chemical etching,
and sputter deposition of the coating material. The different steps of the
manufacturing process are schematically shown in Fig. 3a). The plano-concave 1
inch silicon mirror substrate (EKSMA Optics), is first spin-coated with a
photoresist (Microposit, S1813) on the concave surface. A photomask made of
chromium is used in the exposure step with a high dose exposure (30 mW/cm2 at
365 nm) from a helium lamp (Karl Suss MJB3). The exposure time was 120
seconds. The substrate was then transferred into a reactive-ion-etching (RIE)
system (Oxford instruments) where argon plasma milling for $\sim$ 6 min was
used to etch the grating pattern and reach the desired groove depth ($\sim$17
nm). The etching was followed by a soft ashing procedure with oxygen in order
to remove the residual photoresist. As the final step, the substrate was
coated with SiC in a thin-film deposition system (AJA Orion-8) to enhance the
reflectivity in the XUV range. The fabricated piece, and the overall quality
of the etched pattern, was then examined with an optical microscope and a
scanning electron microscope (SEM; FEI Nova 200). The results are shown in
Fig. 3c) and d), respectively. The depth and the ratio of the grooves were
measured with a stylus profilometer (KLA Tencor) and are shown in Fig. 3e).

Figure 3: Fabrication of the spherical grating. a) Schematic drawing of the
fabrication processes. b) The final piece after the fabrication. c) and d)
present the optical microscope and SEM images taken on the grating shown in
b), respectively. e) Profilometer characterization results showing an average
groove depth of 18.4$\pm$2.5 nm.

One major issue limiting the fabrication quality for such a grating is the
nature of the curved surface, as this causes non-uniformity in terms of the
groove depth $d$ and the ratio $r$. With a ROC of 1000 mm for the concave
surface of the mirror substrate, there is a height difference of $\sim 80$ µm
between the rim and the center of the mirror. This makes it challenging to
achieve a homogeneous thickness of the photoresist layer, as well to have an
even exposure of the photoresist since the curvature leads to the UV light
being defocused towards the center compared to the rim of the mirror.
Simulations (see Supplementary Material Section 2 and Figures S2 and S3) show
that the value of the groove depth dominantly affects the band-pass center,
but the efficiency is almost unchanged for a $\pm$2.5 nm tolerance of the
depth. This requirement can be fulfilled in the fabrication used here. As Fig.
3e) shows, the groove depth is about 18.4$\pm$2.5 nm which is within the range
of optimal performance.

##  5 Experimental setup

Figure 4: The experimental setup used for characterizing the monochromator. a)
The schematic drawing of the beam path. b) HHG spectral flux on sample as a
function of photon energy. c) Model of the HHG and refocusing chamber showing
detailed mounting and placement of the optical elements for the example
configuration. Mono-components are marked by golden color.

The performance of the two types of monochromatization is characterized using
the HHG-based light source and time- and angle- resolved photoemission
spectroscopy (tr-ARPES) setup described in Ref. 24. This system yields
ultrafast, few-hundred-femtosecond long XUV pulses. Figure 4 shows the
schematic diagram of the setup, illustrating the specific optical layout.
Briefly, the HHG light source consists of two Ytterbium-Doped Fiber Amplifiers
(YDFA) lasers, one (Tangor, Amplitude Systems) for driving the HHG of the
probe source, and another (Tangerine, Amplitude Systems) for driving an
optical parametric amplifier (OPA) which provides the pump beam. Both lasers
deliver pulses with a wavelength centered at 1030 nm, and are physically
synchronized to share one common oscillator. For the pump line, the infrared
(IR) beam has a duration of about 280 fs and is compressed from the OPA to
$\sim$100 fs at an output wavelength of 1.2 µm, which is used for the temporal
response measurement shown in the present work. For the probe line, the
fundamental light source has maximum pulse energy of $\sim$300 µJ at a 250 kHz
repetition rate. The pulse duration is $\sim$461 fs. Prior to the HHG, the IR
output is firstly tripled in frequency through a two stage up-conversion using
non-linear optical crystals, to 343 nm, with an efficiency of about 30%. The
UV light is focused by an off-axis parabolic mirror into a gas jet for HHG
process, which generates output in the XUV range from 10.8 eV to 32.5 eV. The
HHG is conducted in the tight focusing geometry with argon gas as the medium.
In this work, we focus on the 3rd ($\sim$10.8 eV) and 5th ($\sim$18.1 eV)
harmonic. Following the HHG, a mirror wheel carrying the refocusing optics is
placed, with the incidence angle of $\sim$1∘.

##  6 Performance

Three fundamental properties are characterized to evaluate the performance of
the two types of monochromatization investigated in this work. These are:
optical quality (focusing), efficiency, and temporal broadening. The
experimental results for the mirror+filter based and the grating-based optical
setups are presented, and compared, below.

###  6.1 Optical quality and focusing

The focusing properties of both the simple spherical mirror and the grating
are characterized by observing the light spots on a fluorescence screen, and a
YAG ($Y_{3}Al_{5}O_{12}$) crystal. The fluorescence screen, which is mounted
as a powder-coated vacuum viewport on the vacuum setup, is used to view the
diffraction patterns for the corresponding photon energies. Figure 5a)
displays, from top to bottom, the results obtained from the SiC/Si mirror in
combination with an Al filter, the grating, and the grating in combination
with the Al filter, respectively. For these measurements, the zeroth order
diffraction spot from the grating is aligned to the same position on the
fluorescence window as the spot from the mirror. It is clearly seen that the
grating generates resolvable diffraction orders with spatial separation. For
the bottom panel of Fig.5a), the spots on the far left and far right are
blocked by the Al filter. This verifies that the remaining spots are indeed
the fifth harmonic ($h\nu$ = 18.1 eV) spots from the HHG, as the Al filter
provides a band-pass filter for photon energies above $\sim$15 eV, thus
blocking the third harmonic light ($\sim$10.8 eV). The spatial separation
between the first order spots for 10.8 eV and 18.1 eV, obtained from the
image, is about 7 mm. The experimental result is consistent with the
simulation presented in the inset of Fig.2a). The 7 mm distance is
sufficiently large so that the two harmonics can be selectively filtered out
using a small aperture, in case one would like to prevent the unwanted higher
orders from reaching the sample. The latter is not always necessary since the
distance puts the unwanted orders outside the view of the electron analyzer of
the photoemission setup.. In the middle panel of Fig.5a), the very bright zero
order spot, located in the center of the image, is blocked by a metallic foil
to avoid damaging the fluorescence window and to allow the weaker first order
spots to be imaged with the camera.

A YAG crystal is used for accurate determination of the light spot size
produced by the two NIM setups. The YAG crystal is placed at the sample
position in the photoemission vacuum chamber, and its exact position can be
controlled by a motorized manipulator (SPECS) with a calibrated millimeter
scale. Figure 5b) and c) show the results of the fifth harmonic ($h\nu$ = 18.1
eV), as imaged by the spherical SiC/Si mirror and the grating, respectively.
The spherical mirror provides a spot size of about 104$\times$95 µm2
(H$\times$V), while the grating gives 123$\times$144 µm2. From ray-tracing
simulations, presuming a beam waist of the HHG radiation of 10 µm and a
divergence of 6 mrad, we obtain a focus of 47.7$\times$48.1 µm2 (H$\times$V),
$\sim$2.1 and $\sim$2.7 times smaller than the measured values, respectively.
The deviation between simulated and measured spot sizes for the mirror+filter
configuration, is possibly due to underestimation of the HHG plasma size, or
the experimental spherical aberration being larger than the ones used in the
simulation. The 10 µm HHG source size, determined by the focus of the off-axis
parabolic mirror ($f$ = 101.6 mm), is in practice difficult to realize due to
several reasons, including misalignment, UV beam quality, HHG phase-matching,
as well as the roughness of a mirror coating. These uncertainties are present
for both types of NIMs, but the additional broadening of the spot produced by
the spherical grating mainly comes from the fabrication imperfections, as
discussed previously.

Figure 5: The analysis of the diffracted beam spots from the fluorescence
screen. a) The observed spot pattern based on different combinations:
mirror+filter (top), grating (middle) and grating/filter (bottom). The ability
to separate the harmonics is clearly demonstrated. b) and c) depict the spot
size measured with a YAG crystal for the mirror+filter and the focusing
grating setups, respectively. Horizontal and vertical spot size is marked in
the figures.

###  6.2 Efficiency

Figure 6 shows the photon flux measurement for the two types of monochromators
at 10.8 eV and 18.1 eV photon energies, reflecting the efficiency. The photon
flux is determined by measuring the drain current from a piece of annealed
tantalum foil placed in the beam path. The yield efficiency is estimated based
on the Ref. 25. The laser that drives the HHG operates at a repetition rate of
250 kHz, and the pulse energy is tunable up to approximately 80 µJ. The flux
measurement using the focusing grating is based on the first order diffraction
spots from the corresponding harmonics. Overall, the focusing grating provides
a considerable enhancement of the monochromator efficiency compared to the
mirror+filter solution. In the case of our design that is specifically
optimized for the 18.1 eV photon energy, there is an order of magnitude
improvement in the photon flux compared the mirror+filter based solution.

Figure 6: Flux measurement of the mirror+filter and focusing grating NIM for
a) 10.8 eV and b)18.1 eV photon energy. The results are plotted as a function
of the pulse energy driving the HHG. The specific configuration of the
monochromator, as well as the efficiency enhancement are specified in the
figure.

###  6.3 Temporal broadening

Figure 7 compares pump-probe measurements of the electronic excitation in
graphene measured with the tr-ARPES setup described in Ref. 24, and using the
two different types of monochromatization. Since the fifth harmonic with 18.1
eV photon energy has the shortest pulse duration of the two wavelengths
considered here[24], the temporal properties were investigated using this
photon energy. A p-type graphene sample was selected as a test specimen since
it has sufficiently fast intrinsic electron dynamics to precisely reflect the
system-limited temporal resolution[26, 27]. The sample used is a quasi-
freestanding monolayer graphene on a 6H-SiC (0001) substrate[28]. The
measurement was conducted at room temperature, and the data acquisition time
for each delay point is 5 min.

Figure 7a) shows the excitation spectra with integrated momentum as a function
of delay time, taken with the focusing grating NIM. The pump wavelength is 1.2
µm, and the fluence is approximately 35 µJ/cm2. In Fig. 7b) the energy-
integrated intensity within the purple box drawn in Fig. 7a) is plotted as dot
markers. The fit (red line) to the data points is a convolution of a
2-$\tau$-parameter exponential decay curve and a Gaussian function, using the
two $\tau$’s, the decay curve amplitude, and the Gaussian standard deviation
$\sigma$ as fitting parameters. The full width at half maximum (FWHM) of the
fitted Gaussian distribution (gray line) represents the overall temporal
resolution of the system, while the $\tau$-parameters describe the rate of
decay after the excitation. Similar to Fig. 7b), the data in Fig. 7c) is
acquired by using the mirror+filter solution. The same energy window for
integration is selected as for the grating case.

For the focusing grating NIM, an overall temporal resolution of about
219.5$\pm$1.9 fs is found, while the mirror+filter results in a 205.3$\pm$7.6
fs temporal resolution. These results suggest that the focusing grating NIM
introduces a temporal broadening of the probe pulse, as expected. By
deconvoluting the broadening from the pump beam, the pulse duration of the
probe beam is found to be $\sim$195 fs and $\sim$179 fs, for the focusing
grating and mirror+filter, respectively. In comparison, the contribution from
the grating monochromator is about 72.4$\pm$27.4 fs, which agrees well with
the calculation presented in Fig. 2b). As the additional broadening from the
grating based NIM is only 14 fs (increase by 6.8%) compared to the
mirror+filter, this shows that the influence of the grating on the temporal
resolution is negligible in many circumstances. The two sets of data were
collected with the same pulse energy used to drive the HHG ($\sim$60µJ), and
the same acquisition time. The statistics presented here, as indicated by the
noise level and error bars, reflects the increased photon flux at 18.1 eV
photon energy when using the focusing grating (error bars for grating data is
smaller than the measurement points).

Figure 7: Time- and angle-resolved photoemission spectroscopy measurements on
a monolayer p-doped graphene, performed with 18.1 eV (third harmonic). a) The
momentum-integrated spectrum as a function of time delay, taken by using
focusing grating NIM. The pump beam wavelength is 1.2 µm. Purple box indicates
the energy integration window used to obtain data shown in b) and c). b) The
excitation intensity as a function of delay (black dots) and fit to the data
(red line). The gray line shows the fitted Gaussian peak representing the
temporal resolution. c) Same as in b), but for the mirror+filter NIM. FWHM
stands for full width at half maximum. All data taken at room temperature.

##  7 Discussion and Conclusion

Figure 6 shows that using a low-density focusing grating for
monochromatization of 10.8 eV and 18.1 eV photon energies from a HHG source
provides higher efficiency compared to a solution based on a focusing mirror
in combination with thin film filters. With the grating, a temporal broadening
of $<$10% is found for laser pulses with a duration on the order of 200 fs.
Since the particular HHG setup used in this work is designed to achieve a high
energy resolution, and thus have relatively long laser pulses, the pulse
duration of the driving laser is 460 fs [24], and the pulse stretching caused
by the grating has little influence on the overall temporal resolution of the
system. For HHG sources with shorter pulses, however, the pulse-front tilt
induced by the focusing grating can be minimized by reducing the line density.
However, this has to be weighted against the reduction in resolving power of
the grating, as well as the spatial separation of the diffraction orders at
the sample position. In the present work, a line density of 40 lines/mm was
selected since this gave the best trade-off between the temporal broadening
and the separation of the harmonics. The pulse-front tilt can also be reduced
by decreasing the beam size on the grating by an iris or an aperture. Doing
so, however, comes at the cost of reduced photon flux. Since the grating based
solution provides considerably higher transmission this is however a viable
trade-off strategy that provides increased flexibility.

It is worth noting that very few thin film filters work well in the low photon
energy range below 20 eV due to high absorption at these wavelengths. In
practice, band-pass thin film filters normally require ultra-thin thickness to
have decent transmission. Some materials, such as Sn or Si filters ($\sim$ 100
nm), are very delicate and can easily be destroyed by handling or by
absorption of residual light from the driving laser. Furthermore, frequent
replacement is time-consuming and costly. As a filter-less solution, the
focusing grating circumvents these challenges. Applying the focusing grating
in the higher photon energy range above 20 eV could be achieved by optimizing
the groove depth, in combination with a multi-layer coating specifically
tailored for the photon energy of interest. In terms of the fabrication
quality of the grating, the most pressing issue is the curvature of the
surface, as that can cause a non-uniformity of the etched pattern. This can be
addressed by further optimizing the specific procedures, in particular the
lithography process, or by using an error-compensated photo-mask.

In summary, we have compared a monochromator design based on a normal-
incidence spherical grating and compared it to a solution based on a spherical
mirror and filter. The experimental comparison targets the characteristics
important for applications in ultrafast electron spectroscopy. We have
designed and fabricated a spherical grating as a compact, single-element
monochromator, specifically optimized for a photon energy of 18.1 eV. The
performance of the focusing grating monochromator is compared to a solution
based on a spherical mirror in combination with a thin film filter – the
latter is a commonly used monochromatization scheme for HHG-based ultrafast
light sources. Overall, the mirror+filter solution can provide a smaller spot
size at the focus with no temporal broadening. The focusing grating can, on
the other hand, provide an order of magnitude higher efficiency in terms of
photon flux with only a small amount of focus size increase and temporal
broadening. The single-element design of the grating enables a very compact
optical layout with minimal need for maintenance. The focusing grating,
furthermore, requires only a low-cost, commercially available Si-mirror
substrate, and can be produced using standard nano-fabrication processing. The
final product is both mechanically robust, and can handle a high power loads.
In short, the mirror+filter solution is more appropriate in cases where there
is no tolerance in temporal broadening, while the focusing grating is more
advantageous in efficiency. The presented grating solution also presents
significant advantages in terms of low-cost, compact size and high efficiency,
and could be relevant for applications outside of photoelectron spectroscopy
as well.

Funding This work was financially supported by the Knut and Alice Wallenberg
foundation (No.2018-0104) and the Swedish research council VR (No.2019-00701).
Q.G acknowledges the fellowship from China scholarship council
(No.201907930007). M.D. acknowledges financial support from the Göran
Gustafsson foundation.

Acknowledgments We acknowledge the Albanova nanolab for the device
fabrication. We thank Dr. Bharti Matta and professor Ulrich Starke for
providing the high-quality graphene sample.

Disclosures The authors declare no conflicts of interest.

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

Supplemental document See Supplement 1 for supporting content.

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# Notes for Neighborly Partitions

Kathleen O’Hara and Dennis Stanton  317 N. Broad St. Apartment 820,
Philadelphia, PA 19107<EMAIL_ADDRESS>School of Mathematics,
University of Minnesota, Minneapolis, Minnesota 55455, USA
<EMAIL_ADDRESS>
(Date: June 2, 2022)

###### Abstract.

A proof of the first Rogers-Ramanujan identity is given using admissible
neighborly partitions. This completes a program initiated by Mohsen and
Mourtada. The admissible neighborly partitions involve an unusual mod 3
condition on the parts.

##  1\. Introduction

Using commutative algebra, Mohsen and Mourtada [4] gave combinatorial
interpretations of the numerator infinite products of the Rogers-Ramanujan
identities [1, p. 104]

$\sum_{k=0}^{\infty}\frac{q^{k^{2}}}{(q;q)_{k}}=\frac{(q^{2},q^{3},q^{5};q^{5})_{\infty}}{(q;q)_{\infty}},\quad\sum_{k=0}^{\infty}\frac{q^{k^{2}+k}}{(q;q)_{k}}=\frac{(q^{1},q^{4},q^{5};q^{5})_{\infty}}{(q;q)_{\infty}}.$

To do so they defined a set of integer partitions $\lambda$, called
neighborly, a related set of graphs $H_{\lambda},$ and a signature for each
graph $G\in H_{\lambda}.$

######  Theorem 1.1.

[4] Assuming the first Rogers-Ramanujan identity, the numerator infinite
product is

$\sum_{\lambda\in Neighborly}q^{|\lambda|}\sum_{G\in
H_{\lambda}}signature(G)=(q^{2},q^{3},q^{5};q^{5})_{\infty}=1+\sum_{k=1}^{\infty}(-1)^{k}q^{5k^{2}-k/2}(1+q^{k}).$

They ask [4, p. 3] for a proof of Theorem 1.1 without assuming the Rogers-
Ramanujan identities. The purpose of this note is twofold:

  1. (1)

to provide such as proof (see Theorem 4.3),

  2. (2)

to simplify the double sum in Theorem 1.1 to a single sum of signed admissible
neighborly partitions (see Proposition 3.4).

Along the way we give a combinatorial interpretation for the classical
generalization Theorem 4.3 of Theorem 1.1.

We use the standard notation for $q$-series found in [1] and [3], and write
the parts of an integer partition in increasing order.

##  2\. Neighborly partitions

######  Definition 2.1.

A neighborly partition $\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{s})$
has all multiplicities at most 2, and for any part $\lambda_{i}$, there is a
part $\lambda_{j}$, $j\neq i,$ such that $|\lambda_{i}-\lambda_{j}|\leq 1.$

A neighborly partition $\lambda$ can be considered as an ordered pair of
partitions: $\lambda=(\mu_{1},\mu_{2})$, a distinct partition $\mu_{1}$ and
another distinct partition $\mu_{2}$ whose parts are a subset of the parts of
$\mu_{1}$.

######  Example 2.2.

If the neighborly partition is $\lambda=(1,2,3,3,6,6,8,8,9,9,14,14),$ then

$\lambda=((1,2,3,6,8,9,14),(3,6,8,9,14))=(\mu_{1},\mu_{2}).$

The partition $\mu_{1}$ consists of some runs, with singletons possible. In
the example

$\mu_{1}=(1,2,3,6,8,9,14),$

the runs are $1\leftrightarrow 2\leftrightarrow 3,6,8\leftrightarrow 9,{\text{
and }}14.$ Note that if $x$ is a singleton in $\mu_{1}$, then $x$ must appear
in $\mu_{2}.$

Mohsen and Mourtada defined a signature on a graph $G_{\lambda}$ defined by a
neighborly partition $\lambda$.

######  Definition 2.3.

The graph $G_{\lambda}$ of a neighborly partition $\lambda$ has vertices which
are the parts of $\lambda$, and edges from the consecutive parts in runs of
$\mu_{1}$, called the backbone, along with edges between equal parts, called
hanging edges.

######  Example 2.4.

If $\lambda=((1,2,3,6,8,9,14),(3,6,8,9,14))$ the backbone of $G_{\lambda}$ is

$1\leftrightarrow 2\leftrightarrow 3\quad 6\quad 8\leftrightarrow 9\quad 14$

with hanging edges

$3\leftrightarrow 3\quad 6\leftrightarrow 6\quad 8\leftrightarrow 8\quad
9\leftrightarrow 9\quad 14\leftrightarrow 14,{\text{ or}}$
$G_{\lambda}=\begin{matrix}1\leftrightarrow&2\leftrightarrow&3&&6&\quad
8\leftrightarrow&9&&14\\\
&&\updownarrow&&\updownarrow&\updownarrow&\updownarrow&&\updownarrow\\\
&&3&&6&8&9&&14\end{matrix}$

######  Definition 2.5.

The signature of the graph $G_{\lambda}$ is the signed sum over all vertex
spanning forests $H$ of $G_{\lambda}$

$signature(G_{\lambda})=\sum_{H\in VS(G_{\lambda})}(-1)^{\\#edges\ in\ H}.$

######  Example 2.6.

If $\lambda=(1,2,3,4)$, $G_{\lambda}=1\leftrightarrow 2\leftrightarrow
3\leftrightarrow 4$, there are two vertex spanning forests: the entire graph,
or the graph with the edge $2\leftrightarrow 3$ deleted. So
$signature(G_{\lambda})=(-1)^{3}+(-1)^{2}=0.$

First we compute the signature on a backbone in the shape of a chain, i.e.
without hanging edges.

######  Definition 2.7.

Let $\lambda_{n}=(1,2,3,...,n+1)$ so that $G_{\lambda_{n}}$ is a chain with
$n$ edges.

######  Proposition 2.8.

The signature of a chain with $n$ edges is

$B_{n}=signature(G_{\lambda_{n}})=\begin{cases}-1{\text{\ if $n\equiv 1\mod
3$}},\\\ 1{\text{\quad if $n\equiv 2\mod 3$}},\\\ 0{\text{\quad if $n\equiv
3\mod 3$}}.\\\ \end{cases}$

######  Example 2.9.

If $n=5$, let

$e_{1}=1\leftrightarrow 2,\ e_{2}=2\leftrightarrow 3,\ e_{3}=3\leftrightarrow
4,\ e_{4}=4\leftrightarrow 5,\ e_{5}=5\leftrightarrow 6.$

The vertex spanning subgraphs are

$\\{e_{1},e_{2},e_{3},e_{4},e_{5}\\},\\{e_{1},e_{3},e_{4},e_{5}\\},\\{e_{1},e_{2},e_{4},e_{5}\\},\\{e_{1},e_{2},e_{3},e_{5}\\},\\{e_{1},e_{3},e_{5}\\},$

so $B_{5}=1.$

###### Proof.

Let $B_{n}(x)$ be the generating function for vertex spanning forests $H$ of
$G_{\lambda_{n}}$ according to the number of edges,

$B_{n}(x)=\sum_{H\in VS(G_{\lambda_{n}})}x^{\\#{{\text{ edges in H}}}},$

By counting the number of edges in connected components from left to right,
the coefficient of $x^{n-k}$ in $B_{n}(x)$ is the number of compositions of
$n-k$ into $k+1$ parts. So

$B_{n}(x)=\sum_{k=0}^{\lfloor(n-1)/2\rfloor}\binom{n-k-1}{k}x^{n-k}.$

The generating function of $B_{n}(x)$ is

(2.1) $\sum_{n=1}^{\infty}B_{n}(x)t^{n}=xt/(1-xt-xt^{2}),$

so

$\sum_{n=1}^{\infty}B_{n}(-1)t^{n}=-t/(1+t+t^{2})=-t\frac{1-t}{1-t^{3}},$

which proves the mod 3 behavior of $B_{n}$. ∎

######  Remark 2.10.

One can also see the generating function as compositions of 1’s and 2’s
(Fibonacci numbers), by counting the number of new vertices each successive
edge gives. So one would see (2.1) almost immediately.

We now consider the case when $G_{\lambda}$ of a neighborly partition
$\lambda$ has one connected component.

######  Proposition 2.11.

Suppose $\lambda=(\mu_{1},\mu_{2})$ where $\mu_{1}=(1,2,...,n),$ and
$\mu_{2}=(a_{1},a_{2},...,a_{s}),$ $s\geq 1.$ Then

$signature(G_{\lambda})=(-1)^{s}B_{a_{1}}B_{n-a_{s}+1}\prod_{k=2}^{s}B_{a_{k}-a_{k-1}+2}.$

Thus the signature of any connected component of any $G_{\lambda}$ is +1, -1
or 0.

###### Proof.

The hanging edges must be in any vertex spanning forest $H$. Thus we need
spanning forests for the chains

$\displaystyle 1\leftrightarrow 2\leftrightarrow\cdots\leftrightarrow
a_{1}\leftrightarrow a_{1},$ $\displaystyle a_{1}\leftrightarrow
a_{1}\leftrightarrow a_{1}+1\leftrightarrow\cdots\leftrightarrow
a_{2}\leftrightarrow a_{2},\cdots$ $\displaystyle a_{s}\leftrightarrow
a_{s}\leftrightarrow a_{s}+1\cdots\leftrightarrow n,$

which have respectively

$a_{1},a_{2}-a_{1}+2,a_{3}-a_{2}+2,\cdots,n-a_{s}+1{\text{\quad edges.}}$

The choices for spanning forests in these smaller chains may be done
independently. Each hanging edge has been used twice, so the factor $(-1)^{s}$
compensates. ∎

######  Example 2.12.

If $\lambda=((1,2,3,4,5,6,7),(3,6,7)),$
$signature(G_{\lambda})=(-1)^{3}B_{3}B_{5}B_{3}B_{1}=0.$

Finally we need to keep track of the signatures of the connected components of
$G_{\lambda}$.

######  Definition 2.13.

Let $\lambda$ be a neighborly partition. The signature multiset $SIG(c)$ of a
connected component

$c=((k,k+1,k+2,\cdots,n),(a_{1},a_{2},...,a_{s})),\quad s\geq 1$

of $G_{\lambda}$ is the multiset

$SIG(c)=\\{a_{1}-k+1,a_{2}-a_{1}+2,a_{3}-a_{2}+2,\cdots,a_{s}-a_{s-1}+2,n-a_{s}+1\\}.$

If $s=0$ then

$SIG(c)=\\{n-k\\}.$

The signature multiset $SIG(G_{\lambda})$ for a general neighborly partition
$\lambda$ is the multiset union over all connected components of the
individual signature multisets.

######  Example 2.14.

If $\lambda=((2,4,5,6,7,10,12,13,14),(2,4,6,10,14)),$

$G_{\lambda}=\begin{matrix}&\ \ \ 2\quad&\ \ \
4\leftrightarrow&5\leftrightarrow&6\leftrightarrow&7&\quad
10\quad&12\leftrightarrow&13\leftrightarrow&14\\\
&\updownarrow&\updownarrow&&\updownarrow&&\updownarrow&&&\updownarrow\\\
&2&4&&6&&10&&&14\end{matrix}$

the connected components are

$2\leftrightarrow 2,\quad 4\leftrightarrow 4\leftrightarrow 5\leftrightarrow
6\leftrightarrow 6\leftrightarrow 7,\quad 10\leftrightarrow 10,\quad
12\leftrightarrow 13\leftrightarrow 14\leftrightarrow 14.$

Because the signature is independent of labels, Proposition 2.11 can be
applied to each connected component.

$SIG(G_{\lambda})=\\{1,1,1,4,2,1,1,3,1\\}=\\{1,1\\}\cup\\{1,4,2\\}\cup\\{1,1\\}\cup\\{3,1\\}.$

######  Remark 2.15.

One may find the signature multiset by counting the edges in the chains that
the parts of $\mu_{2}$ cut in the runs of $\mu_{1}$.

The signature of any neighborly partition is always $0,1,$ or $-1.$

######  Theorem 2.16.

Let $\lambda=(\mu_{1},\mu_{2})$ be a neighborly partition. Then
$signature(G_{\lambda})=0$ exactly when $SIG(G_{\lambda})$ contains an element
$x\equiv 0\mod 3.$ Otherwise,

$signature(G_{\lambda})=(-1)^{t+s}$

where $t$ is the number of elements $x\in SIG(G_{\lambda})$ such that $x\equiv
1\mod 3$, and $s$ is the number of parts of $\mu_{2}$.

######  Remark 2.17.

If $\lambda=(\mu_{1},\mu_{2})$ is neighborly and $signature(G_{\lambda})\neq
0$, then $\mu_{2}$ does not contain consecutive parts, and thus $\mu_{2}$ is a
difference 2 partition.

##  3\. Admissible neighborly partitions

Theorem 2.16 shows that signature$(B_{\lambda})$ is $\pm 1$ or $0$ for any
neighborly partition. Thus we can eliminate the inner sum in Theorem 1.1, and
replace the the set of neighborly partitions by the smaller set of partitions
when signature$(B_{\lambda})\neq 0.$ These are admissible neighborly
partitions.

######  Definition 3.1.

A neighborly partition $\lambda=(\mu_{1},\mu_{2})$ is admissible if
$SIG(B_{\lambda})$ contains no elements which are congruent to 0 modulo 3.

######  Example 3.2.

The neighborly partition $\lambda$ in Example 2.14 is not admissible. A chain
with $n$ edges is admissible if $3$ does not divide $n$.

Since admissible neighborly partitions have signature $\pm 1,$ we may rename
the signature by the sign.

######  Definition 3.3.

The sign of an admissible neighborly partition $\lambda=(\mu_{1},\mu_{2})$ is

$sign(\lambda)=(-1)^{t+s}$

where $t$ is the number of elements $x\in SIG(G_{\lambda})$ such that $x\equiv
1\mod 3$, and $s$ is the number of parts of $\mu_{2}$.

Then Theorem 1.1 is equivalent to the following propositions.

######  Proposition 3.4.

The generating function for all signed admissible neighborly partitions
$\lambda$ is

$\sum_{\lambda\in
AdmNeighborly}sign(\lambda)q^{|\lambda|}=\prod_{k=0}^{\infty}(1-q^{5k+2})(1-q^{5k+3})(1-q^{5k+5})=\sum_{k=-\infty}^{\infty}(-1)^{k}q^{k(5k+1)/2}$

######  Example 3.5.

There are 4 admissible partitions of $n=8$, two positive and two negative, so
the coefficient of $q^{8}$ in Proposition 3.4 is 0.

$\displaystyle positive:$ $\displaystyle\lambda=((2,3),(3)),\quad
SIG(\lambda)=\\{2,1\\},\quad\lambda=((1,3),(1,3)),\quad
SIG(\lambda)=\\{1,1,1,1\\},$ $\displaystyle negative:$
$\displaystyle\lambda=((4),(4)),\quad
SIG(\lambda)=\\{1,1\\},\quad\lambda=((1,2,3),(2)),\quad
SIG(\lambda)=\\{2,2\\}.$

##  4\. Generating functions

In this section we use generating functions to prove the Main Theorem 4.3. It
accomplishes goal (1) of the Introduction by choosing $x=1.$.

######  Definition 4.1.

Let $GF_{n}(q)$ denote the generating function for all signed admissible
neighborly partitions with exactly $n$ parts,

$GF_{n}(q)=\sum_{\lambda\in{\text{Admissible Neighborly with }}n{\text{
parts}}}sign(\lambda)q^{|\lambda|}.$

We shall later prove the following recurrence.

######  Proposition 4.2.

The generating function $GF_{n}(q)$ satisfies the recurrence

$(1-q^{n})GF_{n}(q)=-(q^{2n-2}+q^{3n-3})GF_{n-2}(q)+(q^{2n-2}+q^{3n-4}+q^{3n-3})GF_{n-3}(q)-q^{3n-4}GF_{n-4}(q).$

Our main result is the generating function for admissible neighborly
partitions according to number of parts and the sum of the parts.

######  Theorem 4.3.

The generating function for all signed admissible neighborly partitions is

$GF(x,q)=\sum_{n=0}^{\infty}GF_{n}(q)x^{n}=1+\sum_{k=1}^{\infty}\frac{(-1)^{k}x^{2k}}{(q;q)_{k}}q^{(5k^{2}-k)/2}(xq;q)_{k-1}(1-xq^{2k}).$

###### Proof.

Let $H(x)$ be the right side in Theorem 4.3. Then $H(x)$ has a well-known
functional equation [5],

$\frac{H(x)}{(xq;q)_{\infty}}-\frac{H(xq)}{(xq^{2};q)_{\infty}}=qx\frac{H(xq^{2})}{(xq^{3};q)_{\infty}}.$

This implies that $H_{n}$, the coefficient of $x^{n}$ in $H(x),$ satisfies

(4.1)
$(1-q^{n})H_{n}=-q^{n}(1-q^{n-1})H_{n-1}-(q^{2n-2}+q^{2n-1})H_{n-2}+q^{2n-2}H_{n-3}.$

Iterating (4.1) on $(1-q^{n-1})H_{n-1}$ gives the same recurrence as in
Proposition 4.2, so $H_{n}=GF_{n}(q).$ ∎

##  5\. Another realization of $sign(\lambda)$

In order to prove Proposition 4.2, we need to simplify the graphs
$G_{\lambda}$, keeping the same vertices and labels, but defining a new sign.
This will be done by deleting edges in the chains cut out by the hanging edges
to obtain a new graph $G^{\prime}_{\lambda},$ so that $sign(\lambda)$ is now
just $(-1)^{\\#edges(G^{\prime}_{\lambda})}.$

Via Theorem 2.16, an admissible neighborly partition $\lambda$ has
$SIG(G_{\lambda})$ with no elements that are multiples of 3. The elements of
$SIG(G_{\lambda})$ are the lengths of chains cut out by the hanging edges. We
will delete edges in $G_{\lambda}$ on each subchain by the following rule,
always preserving the hanging edges.

If a chain has $n$ edges $\\{e_{1},e_{2},\cdots,e_{n}\\}$ from left to right,

  1. (1)

delete edges $\\{e_{3},e_{6},\cdots,e_{n-2}\\}$ if $n\equiv 2\mod 3$

  2. (2)

delete edges
$\\{e_{3},e_{6},\cdots,e_{3m}\\}\cup\\{e_{3m+2},\cdots,e_{6m-1}\\}$ if
$n=6m+1$,

  3. (3)

delete edges
$\\{e_{3},e_{6},\cdots,e_{3m}\\}\cup\\{e_{3m+2},\cdots,e_{6m+2}\\}$ if
$n=6m+4$,

######  Definition 5.1.

Let $G^{\prime}_{\lambda}$ denote the graph $G_{\lambda}$ with these edges
deleted.

######  Example 5.2.

If $\lambda=((1,2,3,4,5,6,7),(1,3,6)),$

$G_{\lambda}=\begin{matrix}\ \ \ \
1\leftrightarrow&2\leftrightarrow&3&\leftrightarrow&4&\leftrightarrow&5\leftrightarrow&\
\ \ 6\leftrightarrow&7\\\ \updownarrow&&\updownarrow&&&&&\updownarrow&\\\
1&&3&&&&&6&\end{matrix}$

In the chain $1\leftrightarrow 1\leftrightarrow 2\leftrightarrow
3\leftrightarrow 3$ we delete the third edge $2\leftrightarrow 3$ to obtain
$1\leftrightarrow 1\leftrightarrow 2\quad 3\leftrightarrow 3.$ In the chain
$3\leftrightarrow 3\leftrightarrow 4\leftrightarrow 5\leftrightarrow
6\leftrightarrow 6$ we delete the third edge $4\leftrightarrow 5$ to obtain
$3\leftrightarrow 3\leftrightarrow 4\quad 5\leftrightarrow 6\leftrightarrow
6,$ so

$G_{\lambda}^{\prime}=\begin{matrix}\ \ \ \
1\leftrightarrow&2\quad&3&\leftrightarrow&4\quad&5\leftrightarrow&\ \ \
6\leftrightarrow&7\\\ \updownarrow&&\updownarrow&&&&\updownarrow&\\\
1&&3&&&&6&\end{matrix}$

Note that the third edge is deleted, along with every next third edge, except
for the middle, and the initial and final edges are preserved. Thus all
hanging edges and vertex labels are preserved. We need to see how the sign can
be preserved.

######  Proposition 5.3.

For any admissible neighborly partition $\lambda$

$sign(\lambda)=(-1)^{\\#edges(G^{\prime}_{\lambda})}.$

###### Proof.

Let’s first check that any chain in $G_{\lambda}$ with $3m+1$ edges has an odd
number of edges in $G_{\lambda}^{\prime}$, while chains in $G_{\lambda}$ with
$3m+2$ edges have an even number of edges in $G_{\lambda}^{\prime}.$

In the second case, if $n=3m+2$, the sign is $+1$ and the number of edges is
$n-(n-2)/3=2m+2$ which is even. For the first case, if $n=6m+1$, the sign is
$-1$ and the number of edges is $n-2m=4m+1$ which is odd. For the first case,
if $n=6m+4$, the sign is $-1$ and the number of edges is $n-2m-1=4m+3$ which
is odd. Finally, $sign(\lambda)$ in Theorem 2.16 also includes a factor of
$(-1)^{s}$, where $s$ is the number of hanging edges. Each hanging edge occurs
in 2 chains, so this factor compensates for double counting these edges. ∎

Since we are deleting every third edge from $G_{\lambda}$ to obtain
$G^{\prime}_{\lambda}$ , the connected components of $G^{\prime}_{\lambda}$
are small and limited.

######  Proposition 5.4.

For any admissible neighborly partition $\lambda$, the connected components of
$G^{\prime}_{\lambda}$ are one of six types

$\displaystyle a\leftrightarrow a,$ $\displaystyle\quad a\leftrightarrow
a+1,\quad a\leftrightarrow a\leftrightarrow a+1,\quad a\leftrightarrow
a+1\leftrightarrow a+1,$ $\displaystyle a\leftrightarrow a+1\leftrightarrow
a+2,\quad a\leftrightarrow a+1\leftrightarrow a+1\leftrightarrow a+2.$

Finally we use these six possible connected components to prove Proposition
4.2.

###### Proof of Proposition 4.2.

Since $q^{n}GF_{n}(q)$ is the signed generating function with $n$ parts and no
$1$, $(1-q^{n})GF_{n}(q)$ is the generating function for signed admissible
neighborly partitions with $n$ parts that include a part of size 1. The first
connected component in any $G^{\prime}_{\lambda}$ must contain a $1$ and be
one of the six graphs in Proposition 5.4.

  1. (1)

If the first component is $1\leftrightarrow 1$, the remaining $n-2$ vertices
have labels at least 3, and the signed generating function is
$-q^{2}q^{2(n-2)}GF_{n-2}(q).$

  2. (2)

If the first component is $1\leftrightarrow 2$, the remaining $n-2$ vertices
have labels at least 4, and the signed generating function is
$-q^{3}q^{3(n-2)}GF_{n-2}(q).$

  3. (3)

If the first component is $1\leftrightarrow 1\leftrightarrow 2$, the remaining
$n-3$ vertices have labels at least 3, and the signed generating function is
$q^{4}q^{2(n-3)}GF_{n-3}(q).$ This is because deleting $1\leftrightarrow
1\leftrightarrow 2$ removes 3 vertices, possibly from the first chain, so its
mod 3 value is unchanged, and remains admissible

  4. (4)

If the first component is $1\leftrightarrow 2\leftrightarrow 2$, the remaining
$n-3$ vertices have labels at least 4, and the signed generating function is
$q^{5}q^{3(n-3)}GF_{n-3}(q).$

  5. (5)

If the first component is $1\leftrightarrow 2\leftrightarrow 3$, the remaining
$n-3$ vertices have labels at least 4, and the signed generating function is
$q^{6}q^{3(n-3)}GF_{n-3}(q).$ As before we are deleting 3 vertices, so
admissibility is preserved.

  6. (6)

If the first component is $1\leftrightarrow 2\leftrightarrow 2\leftrightarrow
3$, the remaining $n-4$ vertices have labels at least 4, and the signed
generating function is $-q^{8}q^{3(n-4)}GF_{n-4}(q).$

These are the six terms in Proposition 4.2. ∎

##  6\. Remarks

A topological explanation of Proposition 2.8 via an Euler characteristic is
given in [2, Cor. 6.3].

The second Rogers-Ramanujan identity has a similar interpretation.

######  Proposition 6.1.

The signed generating function for all admissible neighborly partitions
$\gamma$ without a part of size 1 is

$\displaystyle\sum_{\gamma}sign(\gamma)q^{|\gamma|}=$
$\displaystyle\prod_{k=0}^{\infty}(1-q^{5k+4})(1-q^{5k+5})(1-q^{5k+6})$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}(-1)^{k}q^{k(5k+3)/2}(1+q+\cdots+q^{2k}).$

One may use a version of Proposition 4.2 which counts edges in
$G^{\prime}_{\lambda}$ to prove the next proposition.

######  Proposition 6.2.

The generating function for signed admissible partitions $\lambda$ such that
$G^{\prime}_{\lambda}$

  1. (1)

has $2n$ vertices and $n+j$ edges is

$(-1)^{n+j}\frac{(-q;q^{2})_{n-j}(q^{2n-2j-1};q^{-2})_{j}}{(q^{2};q^{2})_{2j}(q^{2};q^{2})_{n-2j}}q^{2(n-j)^{2}+4j^{2}+2j},$

  2. (2)

has $2n+1$ vertices and $n+j+1$ edges is

$(-1)^{n+j}\frac{(-q;q^{2})_{n-j}(q^{2n-2j-1};q^{-2})_{j}}{(q^{2};q^{2})_{2j+1}(q^{2};q^{2})_{n-2j-1}}q^{2(n-j)^{2}+4j^{2}+6j+2}.$

We do not know a proof of Theorem 4.3 using Proposition 6.2.

It is classically known [5] that

(6.1)
$GF(x,q)=(xq;q)_{\infty}\sum_{k=0}^{\infty}\frac{q^{k^{2}}}{(q;q)_{k}}x^{k}=1+\sum_{k=1}^{\infty}\frac{(-1)^{k}x^{2k}}{(q;q)_{k}}q^{(5k^{2}-k)/2}(xq;q)_{k-1}(1-xq^{2k})$

also satisfies Proposition 4.2.

## References

  * [1] G. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Massachusetts, 1976.
  * [2] R. Ehrenborg and G. Hetyei,  The topology of the independence complex,  Eur. J. Comb. (27).  2006, p. 906-923.
  * [3] G. Gasper and M. Rahman, Basic Hypergeometric Series, second edition Cambridge University Press, Cambridge, 2004.
  * [4] Z. Mohsen and H. Mourtada,  Neighborly partitions and the numerators of Rogers-Ramanujan identities,  arXiv:2201.02481, January 2022.
  * [5] S. Ramanujan,  Proof of certain identities in combinatory analysis,  Proc. Camb. Phil. Soc. (19) 1919, p. 214-216.